<?xml version='1.0' encoding='UTF-8'?><?xml-stylesheet href="http://www.blogger.com/styles/atom.css" type="text/css"?><feed xmlns='http://www.w3.org/2005/Atom' xmlns:openSearch='http://a9.com/-/spec/opensearchrss/1.0/' xmlns:blogger='http://schemas.google.com/blogger/2008' xmlns:georss='http://www.georss.org/georss' xmlns:gd="http://schemas.google.com/g/2005" xmlns:thr='http://purl.org/syndication/thread/1.0'><id>tag:blogger.com,1999:blog-530621634329509626</id><updated>2024-12-27T02:16:13.499-08:00</updated><category term="Study materials"/><category term="Business ideas"/><category term="Part time jobs"/><title type='text'>shivakrishna</title><subtitle type='html'></subtitle><link rel='http://schemas.google.com/g/2005#feed' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/posts/default'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default?redirect=false'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/'/><link rel='hub' href='http://pubsubhubbub.appspot.com/'/><link rel='next' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default?start-index=26&amp;max-results=25&amp;redirect=false'/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><generator version='7.00' uri='http://www.blogger.com'>Blogger</generator><openSearch:totalResults>41</openSearch:totalResults><openSearch:startIndex>1</openSearch:startIndex><openSearch:itemsPerPage>25</openSearch:itemsPerPage><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-3871469028273458307</id><published>2024-05-01T02:49:00.000-07:00</published><updated>2024-05-01T02:51:49.818-07:00</updated><title type='text'>i knew i  will going to my past and set all the events correct in future </title><content type='html'>

 all these things are came out of my mind asked me to store in online 
hi my name is shivakrishna
i knew tht your alredy seen in my blog page

let me introduce my self 
im shiva krishn young and energitic person in the felid of aeronautic and spae exploration that&#39;s why i chooed my education and career into aerostreem 
after compleeting my masters i have joind as an assistent profeeser in reputed amity universiy in mumbai 

you know if evrything is not going to according to our plan right in 2020 there is corona virus spreed happen 
alredy i have been married at the age of 25 after my masters 
at that situation i have a kid name gunda t@$ v!k  i have resigned my job came to family life 
at that situation  realized my mistakes 
i left my family and went to job for my passion
i havn&#39;t seen my earnings 
i kept my savings in my wife name 
after i realized it had made mark that i&#39;m psycho 

my wife chaeted me twice
i felt first time it was my mistake 
2nd time also keep gioing on silently 
 ichoosed my life and left family like a new bird
 i knew my phd is going finished in november with full security under custody of police and judge, 
i will come from 2050 my kids in the year of 2018 april 29 thats my kid birthday 
or 15 th nov 2022 
 i will dissapier in the next weeks .
 

the very 1st step is meeting alians technology guru bthat year 2028
india will be super power in 2030
but keep safe ther a big virus ahead in this year after that very few colonys are survived  
there is a traditional methods to follow to make technology advanced 
i will give u some books to read at the time i arrived to my kids in 2032 
till this is the update .........................bye bye yonky 
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/3871469028273458307/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2024/05/i-knew-i-will-going-to-my-past-and-set.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3871469028273458307'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3871469028273458307'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2024/05/i-knew-i-will-going-to-my-past-and-set.html' title='i knew i  will going to my past and set all the events correct in future '/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-7727694104931273961</id><published>2024-05-01T02:25:00.000-07:00</published><updated>2024-05-01T02:25:12.688-07:00</updated><title type='text'>Mastering the Art of 3D Printer Extruder: A Comprehensive Guide to Optimize Your Printing Experience</title><content type='html'>

Introduction to 3D printer extruder
When it comes to 3D printing, the extruder plays a crucial role in the printing process. The extruder is responsible for melting the filament and depositing it layer by layer to create a three-dimensional object. Understanding how a 3D printer extruder works and optimizing its settings can greatly enhance your printing experience. In this comprehensive guide, I will take you through everything you need to know about 3D printer extruders, from their types and advantages to troubleshooting tips and recommended brands.

How does a 3D printer extruder work?
A 3D printer extruder consists of several components that work together to ensure smooth and precise filament deposition. First, the filament is fed into the extruder through a tube or a direct drive system. The extruder then uses a motor and a gear assembly to push the filament towards the hot end. At the hot end, the filament passes through a heating element, usually a nozzle, where it is melted. The molten filament is then extruded onto the build plate or the previous layers to create the desired object.

Types of extruders used in 3D printing
There are different types of extruders used in 3D printing, each with its own advantages and limitations. The most common type is the single extruder, which uses a single nozzle to deposit the filament. Single extruders are widely used due to their simplicity and affordability. However, they are limited in terms of color mixing and printing complex objects.

Another type is the dual extruder, which uses two nozzles to deposit different filaments simultaneously. Dual extruders are ideal for multi-color or multi-material printing. They allow you to create objects with intricate designs and vibrant colors. However, they can be more challenging to calibrate and maintain compared to single extruders.

Advantages of a dual extruder 3D printer
Investing in a dual extruder 3D printer can open up a world of possibilities for your 3D printing projects. With a dual extruder, you can easily print objects with multiple colors or materials without needing to manually switch filaments. This is particularly useful for creating detailed models, prototypes, or artistic pieces. Additionally, dual extruders enable you to print support structures using a different material that can be easily removed after printing, resulting in cleaner and more precise prints.

Common issues with 3D printer extruders and troubleshooting tips
Despite their importance, 3D printer extruders can sometimes encounter issues that can disrupt your printing process. One common problem is filament clogging, which can occur if the temperature is too low or if there are impurities in the filament. To troubleshoot this issue, you can try increasing the temperature or cleaning the nozzle with a fine needle. Another common issue is under-extrusion, where the extruder fails to deposit enough filament. This can be caused by a clogged nozzle, a loose extruder gear, or incorrect extruder settings. To address under-extrusion, you can clean the nozzle, tighten the gear, or adjust the extruder&#39;s steps per millimeter settings.

Optimizing your 3D printing experience with the right extruder settings
To achieve the best results with your 3D printer extruder, it is important to optimize its settings. This includes adjusting the temperature, filament flow rate, and retraction settings. The temperature setting determines the melting point of the filament and should be adjusted based on the type of filament you are using. The filament flow rate controls the amount of filament being extruded and can be adjusted to achieve the desired level of detail and strength. Retraction settings, on the other hand, control how the filament is pulled back to prevent oozing and stringing. Finding the right balance of these settings will greatly improve the quality of your prints.

Maintenance and care for your 3D printer extruder
Proper maintenance and care of your 3D printer extruder are essential for its longevity and optimal performance. Regularly cleaning the nozzle and removing any filament debris will prevent clogs and ensure smooth filament flow. Lubricating the gears and bearings will reduce friction and prevent wear. It is also important to regularly check the tension of the extruder gear to ensure proper filament feeding. Additionally, keeping the extruder motor and cooling fan clean and free from dust will prevent overheating and improve overall performance.

Upgrading your single extruder 3D printer to a dual extruder
If you already own a single extruder 3D printer, you may be wondering if it is possible to upgrade it to a dual extruder. The answer is yes, it is possible, but it requires some modifications and additional components. Upgrading to a dual extruder usually involves installing a second hot end and nozzle, as well as adding a second extruder motor and gear assembly. It is important to note that not all 3D printers are compatible with dual extruders, so it is recommended to check with the manufacturer or consult online forums for compatibility information.

Recommended 3D printer extruder brands and models
When it comes to choosing a 3D printer extruder, there are several reputable brands and models to consider. Some popular brands include E3D, Micro Swiss, Bondtech, and Prusa Research. These brands offer a range of extruders with different features and capabilities to cater to various needs and budgets. It is important to research and read reviews to find the best extruder that suits your specific requirements.

Conclusion
Mastering the art of the 3D printer extruder is essential for optimizing your 3D printing experience. Understanding how a 3D printer extruder works, the different types available, and the advantages of a dual extruder can help you make informed decisions when it comes to choosing and using an extruder. Being aware of common issues and troubleshooting tips, as well as optimizing your extruder settings, will ensure high-quality prints. With proper maintenance and care, your 3D printer extruder will continue to deliver exceptional results. Whether you choose to upgrade your single extruder printer or invest in a dual extruder printer, the right extruder brand and model can make all the difference in achieving your desired printing outcomes. So, take your time, do your research, and select the best 3D printer extruder that suits your needs. Happy printing!

CTA: Ready to take your 3D printing to the next level? Explore our range of high-quality 3D printer extruders and accessories to enhance your printing experience. Click here to browse our collection.</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/7727694104931273961/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2024/05/mastering-art-of-3d-printer-extruder.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/7727694104931273961'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/7727694104931273961'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2024/05/mastering-art-of-3d-printer-extruder.html' title='Mastering the Art of 3D Printer Extruder: A Comprehensive Guide to Optimize Your Printing Experience'/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-2238455640626025816</id><published>2017-02-18T01:54:00.001-08:00</published><updated>2017-07-21T02:27:58.302-07:00</updated><category scheme="http://www.blogger.com/atom/ns#" term="Business ideas"/><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;br /&gt;&lt;/div&gt;

MAKE MORE HAPPEN
Need a Business Idea? Here are 55
Need a Business Idea? Here are 55
Image credit: Entrepreneur
 The Staff of Entrepreneur Media, Inc.
This article has been excerpted from 55 Surefire Home Based Businesses You Can Start for Under $5,000 by Entrepreneur Press &amp; Cheryl Kimball (2009).

Today, tens of thousands of people are considering starting a home based business, and for good reasons. On average, people can expect to have two and three careers during their work life. Those leaving one career often think about their second or third career move being to their own home. People who have been part of the traditional nine-to-five work force and are on the verge of retiring from that life are thinking of what to do next. The good news: Starting a homebased business is within the reach of almost anyone who wants to take a risk and work hard.


$1,500 or less to start up

1. ACCOUNTANT
Experience, training or licensing may be needed

Create a flier outlining your services. Before you do that, you need to know what those services will be. Do you want to simply do bookkeeping for a small business? A more involved level of accounting would be do actually work up balance sheets, income statements, and other financial reports on a monthly, quarterly, and/or annual basis, depending on the needs of the business. Other specializations can include tax accounting, a huge area of potential work. Many business owners don&#39;t mind keeping their own day-to-day bookkeeping records but would rather get professional help with their taxes.

2. BICYCLE REPAIR

In many parts of the country, this business tends to be seasonal, but you can find ways around that. Rent a storage unit and offer to store people&#39;s bicycles over the winter after you do a tune-up and any needed repairs on them. If you want to cater to the Lance Armstrong wannabes, you can have business all year round. These road race riders are training through snow, sleet and dark of night. Some of them work on their own bicycles, but many of them don&#39;t, so you can get their business all year. And if you keep Saturday shop hours, you can be sure you will have a group of enthusiasts coming by to talk all things cycling.


3. BOAT CLEANING
Experience, training or licensing may be needed

Boats that are hauled out of the water for the winter or even just for mid-season repairs will need the hull cleaned. And depending on the type of boat, it is a good time to give a major cleaning everything else too--the decks, the sleeping quarters, the head, and the holds. Start by approaching homes that have a boat sitting in the yard. Or you could market your services to the marina to contract you to do the boat cleaning it offers to customers.

4. BUSINESS PLAN SERVICE
Has expansion possibilities

Offer a soup-to-nuts business plan, including market research, the business plan narrative and the financial statements. Plan your fee around the main one that the client will want and offer the others as add-on services. You can give clients an electronic file and allow them to take it from there, or you can keep the business plan on file and offer the service of tweaking it whenever necessary. Have business plan samples to show clients--and make sure to include your own!

Read This: Write Your Business Plan by The Staff at Entrepreneur Media | Amazon | eBooks.com | Barnes &amp; Noble

5. CHIMNEY SWEEP

Learning to be a chimney sweep may mean nothing more than apprenticing with someone already in the business. By becoming a chimney expert, you can combine a chimney sweep business with a chimney inspection service--covering more than just whether or not the chimney needs cleaning but whether the chimney is in good working order or in need of repair.

6. CLEANING SERVICE

There are many directions you can take this business. If you want to work during hours when no one else does, you can focus on office clients. You can focus on retail businesses and keep your customers clumped into one or two blocks. Restaurants are in great need of daily thorough cleaning and can be a great source of steady clients. Perhaps you would be more interested in house cleaning. Many times with cleaning services you don&#39;t have to spend lots of money on advertising or marketing because your customers will come by word of mouth.

Read This: Start Your Own Cleaning Service by Entrepreneur Press and Jacquelyn Lynn | Amazon | eBooks.com | Barnes &amp; Noble

7. COMPUTER REPAIR
Experience, training or licensing may be needed

Study the main types of software that system users will want--word processing, photo manipulation software, mail merge, spreadsheet, design and especially security software. Investigate all the components--monitor types in all their varieties; keyboards, from wired to ergonomic to wireless; mouse types; as well as peripheral components like printers and scanners. Become completely familiar with all the ISPs (internet service providers) available in the market area you plan to cover. Establish yourself as the guru who can meet the needs of the personal computer user, the small business or a larger corporation.

8. CONSULTANT
Has expansion possibilities

To be a consultant, you need to have an expertise in something so you can market yourself as an advisor to others looking to work in that area. Perhaps you managed several large warehouses in your career with a drugstore company, you did all the marketing for many years for a large shoe manufacturer or you set up a chain of beauty supply shops or take-out restaurants. You can use this experience to help others do similar things without making the same mistakes that you made along the way.

Read This: Start Your Own Consulting Business by Entrepreneur Press and Eileen F. Sandlin | Amazon | eBooks.com | Barnes &amp; Noble

9. DOG WALKING
Experience, training, or licensing may be needed

Dog walkers take pooches out for their daily constitutional one or more times a day, either individually or in small groups. In some cities across the United States, like New York, dog walking alone can be a booming business. But it&#39;s actually more common for dog walkers to offer additional services, including playing with and feeding pets, bringing in newspapers and mail, and turning lights on and off.

Read This: Start Your Own Pet Business and More by Entrepreneur Press and Eileen F. Sandlin | Amazon | eBooks.com | Barnes &amp; Noble

10. EBAY ASSISTANT

Do you have items lurking around your household that you could sell on eBay? Figure out your asking price and decide whether to auction it or put it in your eBay store. Then decide if you want a minimum bid and how long you want the auction to last. You will want to establish a PayPal account to use for transactions. The eBay website provides all the information you need to know to get up and running with an eBay business.

Read This: eBay Business All-in-One For Dummies by Marsha Collier | Amazon | eBooks.com | Barnes &amp; Noble 

From Editorial Services to Household Organizer

11. EDITORIAL SERVICES
Has expansion possibilities

Here are some of the editorial services you can provide from the quiet of your own home:

Copyediting. This is where fact checking takes place, and where grammatical, stylistic and typographical errors are caught.
Proofreading. This is the last stop for a &quot;finished&quot; piece. The proofreader makes sure the copyediting changes have been properly made and no new errors are created in the process.
Indexing. There are indexing courses available and you can get indexing software.
Developmental editing. A developmental editor works with a manuscript on big-picture things like organization and content issues.
Book doctoring. This is an editorial service provided for manuscripts written by experts. They create a manuscript as best they can and then a book doctor puts it into publishable shape.
Ghost Writing. As a ghost writer, you actually do the research and write the book and someone else&#39;s name is attached as the author.
Copywriting. Also known as business writing, this is writing that promotes a product or a service.
Book writing. Do you have an expertise in something professional, such as accounting or interior decorating? Or personally, like knitting? Why not write a book about it?
Magazine article writing. Magazines and newspapers are a great way to get your writing published before tackling the daunting task of writing a whole book.
Web page content provider. Providing content for a web site is a good way to make some money writing.
Read This: Start Your Own Freelance Writing Business and More by Entrepreneur Press and George Sheldon | Amazon | eBooks.com | Barnes &amp; Noble

12. ELECTRONICS REPAIR
Has expansion possibilities

This business is similar to the computer repair business, but you will take on all sorts of electronic equipment besides just computers. With smaller electronics, you will need to be prepared to have customers bring their repair projects to you, as you would have difficulty recovering the cost of driving around picking up broken equipment and returning it. You may also want to encourage people to give you their old electronics so you can use them for parts.

13. EVENT PLANNING
Has expansion possibilities

One of the first things you need to do is visit every potential event location with which you plan to work. Work with the marketing manager to tour each site and learn what is available at each location. Start a database that will allow you to sort venues by varying features--the number of people each site holds, if there is AV equipment available on site, will you need to arrange for rental chairs, etc. Then when you are beginning to plan an event with a client, you can find out what the key parameters are for the event and easily pull up the three or four sites that meet the basic criteria. and engagement parties, etc.

Read This: Start Your Own Event Planning Business by The Staff at Entrepreneur Media and Cheryl Kimball | Amazon | eBooks.com | Barnes &amp; Noble

14. EXPERT WITNESS SERVICE
Experience, training or licensing may be needed

One way to make money in this field is by being an expert witness yourself. If you have an expertise that could be useful in legal cases, you can market yourself to attorneys to act as an expert witness. Another way to be active in the expert witness field is to play a sort of matchmaker, matching attorneys up with expert witnesses for their cases--either for the defense or for the prosecution. Expert witnesses for big money cases can be expected to fly anywhere to testify. There&#39;s no reason your database of witnesses can&#39;t be from all parts of the country.

15. FINANCIAL PLANNER
Experience, training or licensing may be needed

To start, you should go through the certification process so that you can label yourself a CFP (Certified Financial Planner). Your certificate shows that you have expertise and credibility, and this differentiation will help people choose you as their financial planner.

16. FLEA MARKET
Has expansion possibilities

People love to spend weekends rummaging through tables full of other people&#39;s unwanted items, looking for treasures. Make sure to change your layout and put new stuff out for sale often. You want people to come back time and again to see what&#39;s new. You don&#39;t even have to have that much new stuff to make things look new. Just moving an item from a table to the top of a bookshelf might get it noticed, even though the item has been in your inventory since you first started having sales.

Read This: Flea Market America: The Complete Guide to Flea Enterprise by Cree McCree | Amazon | Barnes &amp; Noble

17. GOLF COACH
Experience, training or licensing may be needed

Let the local public courses know about your coaching business. Cultivate relationships with the staff and encourage them to recommend you as a coach. Another place to look for customers is the corporate world. Golfing is a game that business people use to develop relationships outside the office. You do need to be a better than average golfer to develop a reputation as a golf coach. You also need to be a good teacher, know how to be motivational and be willing to work with many different types of people.

18. HOME ENERGY AUDITOR
Experience, training or licensing may be needed

All homeowners are always on the lookout for ways to save on their utility bills. You can come to their aid by providing them with an audit of their house and giving them a breakdown of how they could accomplish real savings in heating, cooling and electrical use. You can go one step further and do the implementation and installation of some of your suggestions in their home yourself. Do a complete appliance audit, with efficiency ratings and calculations based on the age of the appliance. And don&#39;t forget the water heater!

Read This: Toward a Zero Energy Home: A Complete Guide to Energy Self-Sufficiency at Home by David Johnston and Scott Gibson | Amazon | Indigo.ca | Barnes &amp; Noble

Amazon | eBooks.com | Barnes &amp; Noble

20. HOUSEHOLD ORGANIZER
Has expansion possibilities

You can choose either to do the organizing work or to come in to a home and consult on the things the homeowner could do to better organize. Have a portfolio of different organizational scenarios in different rooms in the home and talk with the homeowner about the style he or she likes. Create checklists and questionnaires to understand how the family uses the home. Are the kids wildly busy with after-school activities? Or are they usually home after school and want access to their toys? Do they share rooms? All of these things will help you tailor an organizing plan and become the family hero.

From Import/Export to Solar Energy

21. IMPORT/EXPORT SPECIALIST
Experience, training or licensing may be needed

If you don&#39;t already have work experience with importing and/or exporting, you will have a longer learning curve. You can start by learning the basics and hosting educational sessions to teach others what they need to know to get started in import/export. That alone would probably gain you your first couple of clients. If you keep going with educational seminars and expand your reach to outside your immediate region, you could probably develop a sufficient and ongoing customer base very quickly, but be careful not to outpace your learning curve!

Read This: Start Your Own Import/Export Business by Entrepreneur Press and Krista Turner | Amazon | eBooks.com | Barnes &amp; Noble

22. INTERIOR DECORATOR
Experience, training or licensing may be needed

Market your talents to building contractors. People purchasing new homes can often be overwhelmed with the choices and possibilities in home decorating. Design some questionnaires for each major element and each major room in the house. Find out how the homeowner will use the home--are there children? Pets? Does the woman of the house wear high heels? Do the home&#39;s residents neglect to remove shoes? How will each room be used? Where might task lighting and ambient lighting be most appropriate?

23. JEWELRY MAKING
Experience, training or licensing may be needed

There are many different ways of getting into the jewelry business and many different types of materials with which you can work. Working in metal will probably require the most in the way of specific tools. You need to be able to heat the metal to manipulate it, and you need metalworking tools to cut and engrave it. But there are many other materials that you can work with to make jewelry--glass, plastic, beads, feathers, even wood, to name just a few.

Read This: Start Your Own Fashion Accessories Business by Entrepreneur Press and Eileen F. Sandlin | Amazon | eBooks.com | Barnes &amp; Noble

24. MARKETING COPY WRITER
Experience, training or licensing may be needed

If you can write copy that gets people excited about purchasing what your client has to sell, you can make good money in this business. Unless you are highly experienced from working in the copywriting field, take a course. There are online courses or classes at community colleges and universities that can give you a leg up in getting savvy at writing copy for brochures, catalogs, advertising and, of course, marketing copy for the web.

25. NOTARY PUBLIC/JUSTICE OF THE PEACE
Experience, training or licensing may be needed

In most states in the U.S., a notary public is a state officer who is authorized to witness and attest to the legalities of certain documents by signature and stamping a seal. Most states require that you pass an exam and a background check. It costs very little to become a notary and your income from notary work is negligible. A justice of the peace typically performs wedding ceremonies. States have varying rules and procedures for becoming a JP and performing services. Becoming a JP and/or notary public does not cost much money. And it is not a big moneymaking venture! Many states set the fees you can charge for JP services. JPs can add additional fees, and often do, including travel and hourly rates for additional meetings such as rehearsals, other prep time and any special requests.

26. PERSONAL CONCIERGE

This business is for someone who is supremely efficient and has the ability to make things happen. People who hire you will expect things when they want them and you need to be able to come through with not only what they want, but with a personal touch and a smile on your face. The most likely clients for a personal concierge service are top executives who find themselves at the office by 7 a.m. and are there most nights until 9 p.m., leaving them very little time to do all those things that often need to be done during those very hours.

Read This: Start Your Own Personal Concierge Service by Entrepreneur Press and Ciree Linsenman | Amazon | eBooks.com | Barnes &amp; Noble

27. PERSONAL TRAINER
Experience, training or licensing may be needed

Advertise your services in places where everyone goes, like restaurants and grocery stores. Having a website is a good idea--people want some privacy in their decision-making when it comes to getting fit. They can go to your website and determine if your approach to personal training is an approach that would work for them. It is important to emphasize the safety aspect of using a personal trainer. You can help clients get fit and avoid injury.

Read This: Start Your Own Personal Training Business by The Staff of Entrepreneur Media and Cheryl Kimball | Amazon | Indigo.ca | Barnes &amp; Noble

28. PROPERTY MANAGEMENT

Your job, in the case of rental units, will be to make sure the property is running smoothly. For seasonal properties, you will most likely spend your management time making sure the property is ready for seasonal visits and well-maintained when no one is around. If the owners go away for six weeks in the winter, the property manager makes regular checks on the property. You will be the contact number if the security system operator needs to contact someone about a breach in security.

29. SMALL ENGINE REPAIR
Experience, training or licensing may be needed

Most community colleges offer some level of engine-repair courses. Another way to learn would be to take a part-time position at a repair shop or a rental facility where you could learn on the job, although you will want to be open about your plans. You should be prepared to work on push-behind lawn mowers, riding lawn mowers, generators, garden tools such as rototillers and edgers, chainsaws, wood chippers and snowblowers. You need to decide whether you&#39;ll want to take on bigger jobs, such as tractors, snowmobiles and ATVs; space may be your decision-maker.

30. SOLAR ENERGY CONSULTANT
Experience, training or licensing may be needed
Has expansion possibilities

As a solar consultant, you can basically conduct a home inspection and give clients a report on their solar options for their particular home and site. This can range from full-fledged general solar installations that generate electricity to simple solar walkway lighting. You might want to start by working in a solar products company to become knowledgeable in the solar energy field. However, to be a consultant, it is often best not to be affiliated with any one company or product and be able to recommend products and options across the field of solar energy.

Read This: Solar Energy: Technologies and Project Delivery for Buildings by Andy Walker | Amazon | Indigo.ca | Barnes &amp; Noble

From Tax Preparer to Graphic Design

31. TAX PREPARER
Experience, training or licensing may be needed

Most tax preparation franchises offer courses, seminars, and training to get you ready to work for them. You will learn a lot about tax preparation while working for them before going out on your own. There is a lot of educational support out there to learn tax preparation and all its complexities. And there are lots of individuals and businesses willing to spend a few hundred dollars a year to have someone else prepare their taxes and keep watch for tax breaks or tax burdens on their behalf.

32. TAXIDERMIST
Experience, training or licensing may be needed

Today&#39;s world of taxidermy isn&#39;t exclusive to preserving real specimens. Taxidermy also refers to recreating a specimen using completely artificial materials. Taxidermy schools where you can learn the trade are located almost throughout the country, typically as courses over several weeks specializing in certain levels of expertise, from beginner to master&#39;s level. Like any enterprise, there are taxidermy conventions that you can attend and learn about the latest techniques and materials.

Amazon | eBooks.com | Barnes &amp; Noble

34. USED BOOK SALES

Almost everyone has a few boxes of books stashed away in the house somewhere. Why not make a business out of them? In order to gain customers--especially repeat customers--you will need to have some regular shop hours. Make your shop known for something-a specific category (or two) of books, having some first editions for sale, all paperbacks a dollar and all hardcovers two bucks, and/or a swap program. Maps, illustrations, postcards, greeting cards and magazines are good sidelines to include in your shop.

35. WEDDING PLANNER

You will need to be up-to-date on wedding trends and fads, dress styles, color trends--almost everything under the sun! Offer your customers an ala carte menu of services, from helping pick flowers, the wedding gown and bridesmaid dresses to picking the venue and hiring the caterer. Before you open your business, shop at all the wedding shops, and even pretend you are a bride-to-be to see what kinds of services the wedding gown shop provides and how they treat potential customers. You need to know every detail of the business to give the accurate impression that you are the go-to person for anyone planning a wedding.

Read This: Start Your Own Event Planning Business by The Staff of Entrepreneur Media and Cheryl Kimball | Amazon | eBooks.com | Barnes &amp; Noble

$1,500 to 3,000 to start up

36. APPLIANCE REPAIR
Experience, training or licensing may be needed

Every household has a number of appliances, large and small. You can work on your own or on contract with appliance stores to cover their warranty service calls--or, best of all, you can do some of each. Plan to start slow and build your customer base on recommendations and referrals based on work well done. Consider developing relationships with contractors to be the go-to person to install appliances in newly constructed houses.

37. COMPUTER TRAINING
Experience, training or licensing may be needed
Has expansion possibilities

If you are proficient in both Macintosh and PC, you should offer training in both types of computers. You could probably make a living helping seniors learn how to use the internet and e-mail to keep in touch with their loved ones, who are now commonly spread around the country. Err on the side of caution in this business. People do not want to know all the details about what makes a computer work. If you overload them with information from the beginning by explaining bits, bytes, and megapixels, they will stick to their paper and pencil forever.

38. DESKTOP PUBLISHER
Experience, training or licensing may be needed

You can use desktop publishing software to create newsletters, magazines, books or even marketing materials. You can create the content for your desktop publications, or you can pay a writer to create the content for you. Alternatively, you can advertise your desktop publishing services to design and create newsletters and books for others with their content.

39. FENCE INSTALLATIONS

Fences are everywhere. And they don&#39;t last forever, so they need to be repaired and replaced with a certain amount of frequency. The most common fence material is wood. However, vinyl has become a popular fence choice due to its longevity and relative freedom from maintenance. Wrought iron is another common fencing, especially in urban environments. You can have fun shopping for vintage wrought iron fencing at salvage yards.

40. FREELANCE GRAPHIC DESIGNER
Experience, training or licensing may be needed

Despite the proliferation of the internet, print media is here to stay for the foreseeable future! Fliers, newsletters, magazines, information sheets, letters and advertisements are just a few of the types of print media that business hire freelancers to create for them. Websites and online advertising need graphic design services as well. Even if your expertise is only in design, offer the works for potential clients, including the editorial creation and the printing and even mailing of the final piece. You can line up regular freelancers for those parts of the job you can&#39;t do.

Read This: Start Your Own Graphic Design Business by Entrepreneur Press and George Sheldon | Amazon | eBooks.com | Barnes &amp; Noble

From Gift Baskets to Rug Cleaning

41. GIFT BASKET SERVICE
Has expansion possibilities

Finding a niche is the best way to start out in the gift basket business. Are you a dog lover, horse lover, or exercise guru who could put together baskets that hold the things that people with this interest would like? Do you already create a product that a gift basket could be built around? Have you made your own soaps for the past 10 years? A gift basket that included one or two of your soaps, hand lotion, a scrub brush and manicure kit could be a lovely basket to receive.

Read This: Start Your Own Gift Basket Service by Entrepreneur Press and Cheryl Kimball | Amazon | eBooks.com | Barnes &amp; Noble

42. GRAFFITI REMOVAL

Create an arsenal of cleaning products that can clean almost every kind of product (paint, chalk, markers) from every kind of surface (cement, wood, pavement). The best way to conduct a graffiti service is to offer a subscription-like arrangement. Once a month or whatever interval makes sense for your clients, go around to their property and clean off the graffiti. Charge them a monthly or quarterly fee and make it simple for everyone--they don&#39;t have to think about graffiti, and you just do your job.

43. HAIRSTYLIST
Experience, training or licensing may be needed

Hairstyling is a popular business that can be quite lucrative. Generally a home based hairstylist business is likely to be started by someone who has already has a cosmetology career and wants a change. If you already have your cosmetology training and license, and loads of experience under your belt working in a hairstyling salon, you probably have a following that will follow you right home without any hesitation.

Read This: Start Your Own Hair Salon and Day Spa by Entrepreneur Press and Eileen F. Sandlin | Amazon | eBooks.com | Barnes &amp; Noble

4. HERBAL FARM STAND
You need to decide whether you will sell your herbs as live plants, picked or cut in bunches and packed, or dried. If you plan to market to cooks instead of gardeners, you will want to sell your herbs either fresh cut and packed in sealed bags, or dried and sold in baggies. You can also consider a &quot;pick-your-own&quot; arrangement; however, be aware that herbs are more delicate than most P.Y.O. products. You may save your garden a lot of strife and your plants a lot of wear and tear if you do the picking.

45. LANDSCAPER
Experience, training or licensing may be needed

If you have a knack for this type of work, a degree won&#39;t be necessary. Most people want their yards tidied up in the spring, their lawns mowed in the summer, their leaves removed in the fall, and their shrubs and driveways ready for winter snow. You will also want to offer garden work such as spring planting of annuals and perennials; vegetable garden preparation, planting and fall cleanup; pest control and watering. You can offer tree care service. There is plenty to do in the yard that has nothing to do with plants: stone wall restoration, fencing, irrigation system installation.

Read This: Start Your Own Lawn Care or Landscaping Business by The Staff of Entrepreneur Media and Cheryl Kimball | Amazon | eBooks.com | Barnes &amp; Noble

46. MASSAGE THERAPIST
Experience, training or licensing may be needed

You will want to become certified in massage therapy to be able to effectively market your services. Courses that lead to certification include not only information on human anatomy and physiology and the effects that massage has on both, but also on how to make a business out of the field of massage. You could do either a certification program or an associate&#39;s degree and stay within the $5,000 scope of this book.

Read This: Start Your Own Day Spa and More by Entrepreneur Press and Preethi Burkholder | Amazon | eBooks.com | Barnes &amp; Noble

47. MOVING SERVICE

Lots of people who are moving want to hire someone to do the heavy lifting for them. You can leave the large-scale, long-distance moving to the big moving companies. Your work can be the local, moving-across-town or to the town-next-door jobs. These are the ones that people start off thinking perhaps they could do themselves, and it will be your job to convince them otherwise. Your signs around town will tempt them to let you take care of that part of the move, while they are busy taking care of those other 500 items on their list.

48. MUSIC LESSONS
Experience, training or licensing may be needed

You want to stick to the instrument(s) you know, but you may be a skilled enough musician to offer lessons on several different instruments, or those in a particular class, e.g., stringed or woodwind.You can decide to take on individuals or classes, depending on space and availability of instruments. Public schools are continually reducing their commitment to art and music classes for students, so you can try to work with the public school system to supplement their efforts in those areas.

49. PHOTOGRAPHER
Experience, training or licensing may be needed

Making money as a photographer can be done in a number of different ways. You can specialize in one area, the most common being weddings. There are niches you can explore for photography: portraits of people and their pets, families, and homes; photographs of holiday events, birthday parties or Christmas cards; the possibilities are endless.

Read This: Start Your Own Photography Business by Entrepreneur Press and Charlene Davis | Amazon | eBooks.com | Barnes &amp; Noble

50. RUG CLEANING

You will need to learn how to work with all kinds of carpet fabrics, from synthetic to wool carpets. Decide whether you will take on valuable antique carpets and family heirlooms; if so, you will want to get specialized training in how to handle these carpets and the specialized ways of cleaning them. Learn how to get tough stains and odors out of carpets--such as dog and cat odors--and your services will be in great demand.

From Websites to Pet Sitting

51. WEBSITE DEVELOPER
Experience, training or licensing may be needed

Many courses exist (many of which, logically, are offered online) where you can learn the language of website creation and can learn about the details, like how to set up shopping cart systems, security concerns, etc. You will, of course, need to learn about each company you design for. What is the atmosphere of the company that you need to reflect in the website design--is it wild and contemporary, meaning brilliant colors and fun graphics? Or will more classic colors like black, navy blue and maroon be more appropriate?

Offer: Get a 10-day free trial to Lynda.com and learn web development – among many other topics and courses – anytime, anywhere.

$3,000 to $5,000 to start up

52. BED AND BREAKFAST

Do you have a room that has its own bathroom and is private from the rest of the living space? Are you near attractions such as a tourist area, sports stadium or venue for a large annual event? Or is your home in the country with spring peepers, summer crickets and crisp fall nights that could give a city-dweller a weekend of peaceful living? Say you can rent the room for $150 a night for Friday and Saturday nights 48 weeks a year--that&#39;s $14,400 in revenue! Utilize what you have and create a unique experience.

Read This: Start Your Own Bed and Breakfast by Entrepreneur Press and Cheryl Kimball | Amazon | eBooks.com | Barnes &amp; Noble

53. CHRISTMAS TREE SALES

If you want to start a Christmas tree farm, you need to plan ahead. It takes approximately seven years for a Balsam fir--perhaps the most traditional Christmas tree--to grow from a small sapling to a 5- to 6-foot tree. Selling your trees yourself is the best option. Consumers come to the property, pick the one they want, and you harvest it for them. The other option is to buy your trees from a wholesaler and sell them either in your yard or in a vacant lot that you rent from Thanksgiving to Christmas.

54. DAY CARE
Experience, training or licensing may be needed

Perhaps you love children. Perhaps you have children of your own and the idea of taking care of a few more for part of the day appeals to you. Child-care needs continue to soar in the United States. Many people prefer the option of their child being cared for in a home environment while they are at work, opposed to a more institutional-like setting. These things mean that a homebased childcare business can get off and running immediately.

Read This: Start Your Own Child-Care Service by The Staff of Entrepreneur Media and Jacquelyn Lynn | Amazon | eBooks.com | Barnes &amp; Noble

55. PET SITTING
Experience, training or licensing may be needed


Starting a pet sitting service requires almost nothing in start-up costs. You do need some general credentials that will cost little or nothing to acquire. Your list of credentials should probably include personal pet ownership--if not currently, at least in the past--as well as other pet-related experience, including working at a pet food store, an animal hospital or other animal-related business. You will need to spend a little to become &quot;bonded.&quot; This is known as &quot;honesty insurance,&quot; and ensures your clients that you won&#39;t get their house keys and make off with their valuables (or that they&#39;ll get their money back if you do).</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/2238455640626025816/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2017/02/make-more-happenneed-business-idea-here.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2238455640626025816'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2238455640626025816'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2017/02/make-more-happenneed-business-idea-here.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-8630523124140853399</id><published>2017-02-18T01:48:00.002-08:00</published><updated>2017-02-18T01:51:11.279-08:00</updated><title type='text'></title><content type='html'>
101-Best-Side-Business-Ideas-You-Can-Start-While-Working-a-Full 
No matter how rewarding your full-time job may be, finding the right side business idea and eventually becoming fully self-employed is even more meaningful than great pay and solid benefits.

Choosing the path of entrepreneurship working on your business idea is without a doubt riskier than being content with holding a 9-5 job, and requires way more sacrifice. However, once you’re reaping the lifestyle benefits of being your own boss and hustling your way into making significantly more money than you ever could at your day job, the hard work will have all been worth it.

Want to find a profitable niche business idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate today.

Validate Business Idea Online Course CTA Image

The question I&#39;m asked the most (by far) when I meet and speak with fellow entrepreneurs is, &quot;How do I know which business idea is best for me?&quot; This one is always quickly followed by, &quot;How do I start on this side business idea while I&#39;m still working my full-time job

Check out these 101 side business ideas you can start while you still keep a full-time job.
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This makes perfect sense. With how many businesses already exist out in the world, it can be difficult to come up with the right business idea you should be spending your time on. Especially if you&#39;re looking for a home business idea that&#39;ll afford you immense lifestyle flexibility.

The majority of us don&#39;t have the luxury of being able to quit our day jobs to pursue starting a business today, without having to worry about how we&#39;re going to meet our financial obligations moving forward with no immediate income.

That&#39;s why I&#39;m a huge advocate of always starting a business while working full-time, so that you can test your way into your new product or service, get feedback, validate the business idea, and start generating income before you quit your job. If you can master the art of scaling a side business while keeping your day job, you&#39;ll have no trouble succeeding once you&#39;re fully self-employed.

When I set out to start a new business, I always make sure it aligns with both my core competencies and my interests. It starts with learning how to discover your strengths as an entrepreneur, and it&#39;s how I make sure I&#39;ll be engaged, even when the going gets rough. I learned this the hard way when I decided to start a freelance business a few years ago to offer my freelance content marketing services.

From there, you need to learn which tools, resources, and feedback communities you can use to accelerate the pace at which you&#39;re able to launch. Check out these 79 essential tools and resources I use to launch profitable side businesses.

With that in mind, I  put together this massive list of the 101 Best Side Business Ideas You Can Start While You&#39;re Still Working a Full-Time Job, to help give you inspiration with proven business ideas that can be executed on while you still keep your day job—and primary source of income.

The big problem for many of us is that working a full-time job makes it too exhausting to even consider trying to find an alternative outlet. Many of the world&#39;s most successful entrepreneurs struggled with this when they wanted to start a business for the first time while they still had a day job.

Luckily, there are tons of ways to start businesses and make money on the side while still leading a well-rounded, meaningful life. If you play your cards right, you’ll be able to quit your day job and grow your side business into an empowering self-employed career.

Obviously some of these small business ideas have more earning potential than others, but what they all share in common are relatively low barriers to entry and the flexibility to work at them for a limited amount of time per week. I also tried to focus on businesses that have some sort of online component to them, since that&#39;s what I love most.

Want to find a profitable niche business idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate today.

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If I missed any good ones, please share your best side business ideas in the comments below! If you&#39;re ready to start a business while you&#39;re still working full-time, join my course, 30 Days to Validate.

Please note that some of the links below are affiliate links and at no additional cost to you, I will earn a commission. Know that I only recommend tools and learning resources I&#39;ve personally used and believe are genuinely helpful, not because of the small commissions I make if you decide to purchase them. Most of all, I would never advocate for buying something that you can&#39;t afford or that you&#39;re not yet ready to implement.

Here are my picks for the best business ideas you can start right now, while you&#39;re still working full-time.

101 Best Side Business Ideas You Can Start While Working a Full-Time Job
Want to work for yourself? Here&#39;s 101 side business ideas you can start while keeping a day job.
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1. Graphic Design.
While having a formal background in graphic design is absolutely going to be helpful, it’s also relatively easy to learn the foundations of graphic design on your own. An increasingly easy-to-use Adobe Illustrator and websites like Canva and Visme are making it so that just about anyone with two opposable thumbs, a bit of creativity, and motivation can get paid to create or alter images.

The 65 Best Business Ideas You Can Start While You&#39;re Still Working a Full-Time Job with Ryan Robinson Online Course

2. Web Design.
Web designers are incredibly valuable for technology companies—which is why becoming a freelance web designer is a top side business idea today. Web design is all about mastering the art of creating a beautiful, value-driven experience for the people using a website or app. There are always new websites popping up in need of professional web design, and companies like Skillcrush, CareerFoundry, and General Assembly have proven online programs that&#39;ll get up-to-speed quickly with this career path. CreativeLive also has a phenomenal class that regularly plays for free, teaching the foundations of modern web design.

3. Web Development.
As a web developer, you&#39;ll build incredibly valuable skills that are in extremely high demand. You can get up to speed on building websites in as little as a few months with inexpensive or free online education programs like Treehouse, the Web Developer Bootcamp on Udemy and Codecademy. If you don&#39;t find what you&#39;re looking for there, here&#39;s a list of the 45 best places you can learn to code for free. Once you command a knowledge of HTML, Ruby, Python, Javascript, or CSS, you can start a freelance business on the side to build your portfolio while you still hold your full-time job. Over time, you&#39;ll build more relationships, have more experience and eventually take this business idea over to becoming a full-time source of income.

4. Tax Preparation.
It’s not the sort of job that’s covered in glory, but someone needs to make sure all the numbers add up at the end of the year. Every business and most individuals need someone with the domain expertise to help prepare tax returns, especially time or resource-strapped small business owners. The Income Tax School provides an array of training programs that&#39;ll certify you with tax prep in as little as 10 weeks, and once tax season rolls around you&#39;ll be able to charge an average of $229 per return as a freelance tax preparer with this side business idea, reports CNBC.

5. Commission-Only Sales.
If you have a knack for connecting with people and the willingness to take on some risk, a commission-based freelance sales role could be a great fit for you. Many startups seek part-time and commission-only salespeople, especially when they&#39;re just getting started, which means you&#39;ll often be able to make this a home based business idea. Sell for nothing but commission, negotiate yourself a good chunk of equity, and you could profit big time if you&#39;re pitching a solid product and the startup succeeds. Check out Angel List and see if an opportunity aligns with your interests.

6. Online Courses.
If you&#39;re an expert at something, there&#39;s likely an audience of people online who would be willing to pay to become an expert in your field—just like you. If you want to take your skills and turn them into an online course that teaches others how to get the same results you&#39;ve achieved in your life, career, or business, start with How to Create an Awesome Online Course on Udemy, where instructor Miguel Hernandez covers how he makes over $90,000/yr teaching online with more than 8 hours of video instruction. Once you&#39;re ready to teach your own online course, I recommend using Teachable, the easiest and most affordable to use platform for creating, hosting and selling your online courses.

Want to find a profitable online course idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate today.

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Here&#39;s what one of my students, Kevin has to say—just 2 weeks after joining.

Kevin of Inbound for Startups
&quot;I&#39;ve been taking your advice and guest blogging to build my credibility and audience. I was able to get my first pre-sell for my course this week!&quot;



Kevin of Inbound for Startups
Are you ready to get serious about finally building a business? How about getting your first paying customers in the next few weeks? Follow my step-by-step system and launch in a matter of weeks after joining 30 Days to Validate.

7. eBooks.
Packaging your skills and knowledge into a downloadable eBook that delivers value to those seeking to learn a skill, advance in their careers, or start their own businesses, makes for a strong value proposition if you target the right audience. Check out Leslie Samuel&#39;s great guide to selling eBooks online and start building your strategy around this side business idea. This class with Tara Gentile on CreativeLive will also show you how to use your existing body of work to write an eBook within the next week. Put in some serious work with your eBook, build an audience and you&#39;ll have a platform to pitch traditional publishers on landing a book deal—then you can write one of the best business books and really build your personal brand.

8. Instagram Marketing.
Build up a following on your Instagram account and you could quickly be approached by major brands, gear companies, and other relevant businesses that sell products or services related to the type of content you share on Instagram. If you have hundreds of thousands of followers, you can easily charge anywhere between $500 to $5,000 per post or more—which makes for a very profitable side business idea. Check out this fashion Instagrammer on ThePennyHoarder, making a significant income from brand sponsorships. Once you get some traction, to cut down on the amount of time you spend uploading images, you can make your entire workflow more efficient by posting photos from your Mac or PC.

9. Online Coaching.
If you have something you&#39;re skilled at and very passionate about, you can turn that winning combination into offering your services with one-on-one online coaching. Just be sure to implement your own opportunity management system so you don&#39;t get caught up working with clients that you can&#39;t measurably help. Regina Anaejionu will give you a step-by-step plan for putting your skills and experience to work by developing an online coaching business. Check out her content for a jump start on this side business idea.

10. Podcasting.
Podcasting as a Business You Can Start While Working Full-Time Tim Ferriss

If you can create a regular audience for your podcast on a specific topic, this is a great way to get sponsors and fund this side business idea. At CreativeLive, I regularly paid $50-$500 (or more depending on audience size) per episode for a 30 second advertisement on relevant podcasts like The Tim Ferriss Show, the #1 business podcast right now. The podcast has even helped Tim secure his own TV show. Naturally, it helps if you already have an online audience you can tap for listening to your regular podcast, but that hasn&#39;t stopped thousands of people from building successful businesses on the back of podcasting—including Alex Blumberg, founder of Gimlet Media who teaches how to use storytelling and launch a podcast. You can also check out this class about how to make money podcasting, which broadcasts for free every month on CreativeLive.

11. Amazon Reselling.
Anyone can sell goods on Amazon, provided you have products to sell (or buy low, then resell). If you’re the type to hit all the local garage sales each weekend, there’s all sorts of valuable things that can be resold online as a side business idea. If you want to step your Amazon selling game up, check out this detailed guide to Amazon Clearance Arbitrage on SideHustleNation featuring an interview from Travis Scott of StuffParentsLike.com who&#39;s turned this home based business idea into a lucrative money-maker.

12. Local Business Consulting.
If you&#39;ve developed valuable skill sets or certifications within your industry over the years, consider putting your skills to use in your free time by offering your consulting services to local business owners as a potentially lucrative side business idea. Whether you&#39;re an expert marketer, business strategist, or manufacturing aficionado, there&#39;s likely a local business owner who&#39;s willing to pay you to help them solve an issue with their company. Start with this 18-step checklist to becoming a local business consultant from Karyn Greenstreet.

13. Phone Case Business.
There&#39;s a huge growing market for mobile phone accessories, and plenty of handmade sellers are raking in 6 and sometimes 7 figures from their phone case businesses. With turnkey solutions like Case Escape readily available, you can now get ready to go kits that&#39;ll give you everything you need to start your own phone case business. Once you&#39;re up and running, you can sell cases on Etsy, Amazon Handmade, and Fancy. I know one seller who clears around $60,000/yr through Etsy alone.

14. Affiliate Sales and Marketing.
If you already have a website that&#39;s driving in targeted traffic, a great way to make passive income from the content you&#39;re already creating, is through affiliate marketing. ShareASale, Rakuten, Clickbank, and Skimlinks are a few of the best affiliate networks and tools that can help you make money from the content you already produce, thus amplifying your side business idea income. Check out this extensive list by Justine Grey, covering the 59 Best Affiliate Programs for Business Bloggers.

15. Virtual Assistant.
Have a knack for staying organized? If you&#39;re a jack of all trades, you should consider working as a virtual assistant. You can find great gigs on Elance, Indeed, or Upwork. It can be an awesome way to rub shoulders with some very important people, build up your professional network, and you&#39;ll have the added perk of being able to start this as a home based business idea. Becoming a virtual assistant can be a particularly great side business if your partner&#39;s job means you&#39;re constantly on the road—for example if you&#39;re the husband or wife of a coach or athlete, which makes regular moving a way of life.

16. Remote English Teacher/Tutor.
Best-Business-Ideas-to-Start-While-You-Still-Work-a-Full-Time-Job-by-Ryan-Robinson

Teaching and tutoring English as a second language is a great way to make a solid side income, not to mention opening doors for you to travel the world if you&#39;d like. While full ESL (English as a Second Language) accreditation is recommended, as long as you’re a native speaker, there are people in countries such as Hong Kong or the UAE who are willing to pay upwards of $25/hr for you to teach them English via Skype. Indeed, Learn4Good and Remote.co often have remote english tutoring jobs posted, check back frequently. Then once you land that remote gig, you&#39;ll need somewhere more professional than your dining room table to meet with students—check out this post about how to find places to work remotely.

17. SAT Tutor.
If you had a knack for standardized tests and had no trouble acing the SAT, ACT or other college exams, why not start tutoring high schoolers on the side? Parents of all economic backgrounds are more than willing to shell out upwards of $100/hr to the right tutor, if it means their son or daughter will get admitted to the college or university of their choice. See this quick checklist for starting an SAT tutoring business from the Work At Home Mom. Whitney over at Rookiemoms also has a cool story to share about a stay-at-home mom making $40/hr helping kids out with homework and turning it into a profitable side business idea.

18. Social Media Manager.
We&#39;re all guilty of spending too much time on Facebook, Twitter, or Pinterest sometimes, so why not get paid to put your expertise to work? Lots of companies, especially startups or those in retail or travel, have heavy social media presences and are constantly in need of people to help build their brands online. You can find these types of opportunities on sites like Flexjobs and CareerBuilder.

19. Google Paid Ad Specialist.
If you know a thing or two about paid internet marketing and are comfortable with Google, a great way to make some extra income on the side is to manage a company’s Google Ad Campaigns, and gradually start bringing on more clients as your consulting business grows.

20. Start a Blog.
Think blogging is no longer a viable source of income? Think again. Tens of thousands of bloggers, creating content on topics as diverse as scrapbooking, home cooking, travel, film, lifestyle, and more are launching self-employed careers thanks to a combination of blog subscribers, affiliate marketing, sponsorships, and other revenue streams. Your first step is sharpening your writing abilities, understanding what your audience craves and learning how to write a blog post that drives traffic and brings you readers on a regular basis. Even blogging about tech gadgets and products you love can turn into a profitable side business; take AwesomeStuffToBuy for example. If you want to speed up the process of building your blog, I recommend setting up quick, affordable and easy website hosting with a company like HostGator and then hiring a talented freelance designer or developer on a high quality marketplace like Crew.

Want to find a profitable website idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate and let&#39;s make it happen today.

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21. Presentation Design Consultant.
Yes, even the lowly PowerPoint requires outside consulting every now and then. I know I would happily outsource the visual layout of my presentation decks for work meetings, investor pitches and lectures. Adam Noar from Presentation Panda is living proof that you can turn your skills at slideshow presentation design into a legit side business idea.

22. Travel Consultant.
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If you love to travel and find yourself randomly searching for airfare sales or browsing Lonely Planet, why not carve out a niche for yourself as a private travel agent? Take my friend, Mark Jackson&#39;s lead with what he&#39;s doing to build a travel consulting side business idea. Start with word of mouth recommendations from friends who know they can count on you for the cheapest flights, and don&#39;t forget to set up an account with Plansify to start getting paid for your travel expertise in your free time.

23. Landing Page Specialist.
If you have a way with words and know how to make the keyword-friendly, beautifully designed, SEO-optimized landing pages, why not charge other companies for your services and turn it into a money-making side business idea? Even a short landing page is worth a couple hundred bucks in most cases. If you want to get started as a landing page specialist or freelance copywriter, check out Neville Medhora&#39;s awesome post on How to Become a Copywriter with No Experience.

24. Interior Design Consultant.
Someone out there is remodeling their kitchen and needs to know which shade of granite will match best with mahogany flooring. That someone will often be happy to pay you for your advice, especially if you’re the kind of person that subscribes to websites like Contemporist.

25. Housesitter.
It’s not exactly a way to make consistent great money, but housesitting—exactly what it sounds like—is a fantastic way to live in exotic locales around the world without paying a dime in rent. Did I mention it’s a way to travel and live rent free? Here’s a list of four great websites from the legendary Nomadic Matt, to start your housesitting search.

26. Babysitter.
No, babysitting isn’t just for teenagers and college students. Quite on the contrary, if you call yourself an Au Pair instead, you can make some pretty good side money working nights and weekends.

27. Property Manager.
Know someone who rents out property to others? Perhaps they could use a hand managing their property. If so, you can make a decent amount of money on the side with relatively little work on your part. From collecting rental checks, to managing repairs and improvements, or simply being on call for emergencies, a property manager is essential for most real estate investors.

28. Sell on Etsy.
Business Ideas to Start While Working a Full-Time Job 1

Have a talent for crafting or creating other handmade goods? From bracelets to phone cases, rings, furniture, and more, Etsy is one of the world&#39;s largest independent marketplaces that’s perfect for anyone who is creative and willing to sell their handmade creations. As long as you have the space, this can make for one of the best home business ideas that can be started with a very limited investment. Consider these 5 steps to starting an Etsy store, from Handmadeology. Then, after you&#39;re able to build your brand and grow your own audience, you can start an eCommerce site of your own and retain a larger portion of the revenue from your product sales.

29. Ebay Sales.
Similar to selling on Amazon, eBay is a place to make money selling just about anything you can think of, with the added excitement of using the auction selling model to sometimes get way more than you expected for that antique baseball card you picked up at a local street fair.

30. Fiverr Gigs.
Fiverr is a great place for first-time freelancers who might not have tons of experience and want to build up a portfolio of their work. You&#39;ll be able to complete simple tasks ranging from logo design, to creating animations, or even drawing a company logo on your forehead.

31. College Admissions Essay Editor.
If editing and advising college students on how to write compelling 500 word essays on topics such as &quot;You were just invited to speak at the White House. Write your speech,&quot; seems like a compelling way to spend your free time, trust me—tons of parents will pay you to edit admissions essays and offer constructive feedback for their children. Be careful not to blur the ethical line of actually writing their essays, but serving as an editor to help them convey their message can become a great side business idea that has the potential to spread by word-of-mouth referral in your community.

32. Portrait Photographer.
If you own a camera, starting a freelance portrait photography business could be a very natural way to make a side income. Start with doing free shoots for friends and family to build up a strong online portfolio, get familiar with your gear and the editing process, then you&#39;ll be able to get paid for photographing professional head shots and celebratory family moments as your side business idea. Get started by checking out this men&#39;s portrait photography class that regularly broadcasts for free on CreativeLive.

33. Wedding Photographer.
Wedding photographers command premium rates. After all, you are capturing one of a couple&#39;s most important life moments. Many professional wedding photographers charge between $2,500 - $10,000 (or more) to shoot a wedding, a very lucrative side business idea. Check out the Complete Wedding Photography Experience to get up to speed on everything you need to launch a successful wedding photography business.

34. Online Dating Consultant.
Believe it or not, some people have such a difficult time at dating, that they can’t even handle the online component of it. If you’re a smooth talker, why not leverage that skill into a paid match maker? People are paying.

35. Writing Erotic Fiction.
Werewolves, stepbrothers, and tentacles—the demand is high. As odd as it may sound, if you’ve got the imagination (and a clever pseudonym), you can make upwards of $5,000 a month writing erotic fiction in your free time as a side business idea.

36. Writing Greeting Cards.
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Fancy yourself a poet? You can earn $300 for every poem you write that this greeting card company publishes.

37. WordPress Website Consultant.
Countless small businesses start out their web presence using a WordPress hosted website before needing to upgrade to other solutions for various reasons. Many of them will pay several hundreds of dollars for someone to get them set up online. If you have the patience to learn how to do it yourself, it&#39;s an extremely valuable skill.

38. Drive for Uber or Lyft.
Driving for one of the two globally expanding app-centric taxi alternative services, Uber or Lyft can be a very lucrative way to earn money on nights and weekends - working only when you want.

39. Rent Your Car on Turo.
If you own a car but barely use it, you can make some easy cash on the side by renting it out on Turo. Before you freak out, know that there’s a $1 Million insurance policy on all cars, plus drivers are prescreened, so you can have peace of mind.

40. Art Collector.
Not to be confused with hoarding, this one takes a lot of time, patience, and passion. If you have an eye for good art, it’s easy to get in on the ground level by visiting the studio department at your local university—though don&#39;t expect to get rich overnight with this side business idea. Many art students are more than happy to sell their work for a bargain, and in as little as a few years, there&#39;s a chance that piece you bought for a couple hundred bucks may be worth well into the thousands.

41. Catering Business.
EatWith is a great way to test the waters as a chef for your side business idea, and if you have enough rave reviews you might be able to turn your knife skills into a full-time business.

42. Develop an App.
Sometimes it seems like there’s an app for everything. Yet somehow, new ones keep popping up and selling for lots of money, all the time. If you spot a niche that hasn’t been filled to its potential just yet, and you can learn the coding skills (or know someone who already has them), you could be on to something. Today, it&#39;s even possible to build an app without any coding skills whatsoever. Just make sure you validate your app idea before jumping too far in.

Want to find a profitable app idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate today.

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43. Online News Correspondent.
You don’t need to have a degree in journalism to be a reporter these days, and there are many news websites that can always use a bit of help on getting local coverage. Some of them, such as The Examiner or HuffPost, will compensate contributors based on ad revenue generated per article written—a great incentive to provide compelling content for your side business idea.

44. Patent Something.
Kia Silverbrock has been granted over 4,665 US Patents in his lifetime (so far). You don’t have to invent the wheel to make good money from your patented concepts—just make sure that your idea can be produced affordably, or it will never stand up against a lawsuit in the future.

45. Buy and Sell Domain Names.
Domain name trading has been around for the last couple decades, and while most slam-dunk names have long been sold off (Insure.com went for $16 Million in 2009) there’s still plenty of others that you can get your hands on for relatively cheap and broker as your side business idea. But beware: some experts doubt the long-term viability of this business, so you shouldn’t quit your day job just to put all your effort into this one without some successes already in the bag. To get you started, here are some tips from GoDaddy, arguably the world’s largest and most famous repository of domain names. Imagine owning desirable domain names for the next decade&#39;s worth of innovative companies.

46. Start a Popup Shop.
While the barriers to entry for a retail shop can be very high, one way to keep costs down and limit your time investment while you keep your full-time job, is to set up a weekend popup shop. More of whatever it is you’re selling—donuts, clothing, antique baseball cards, or anything else—will stay in your pockets instead of going to pay for operating expenses such as rent or utilities. This guide from Shopify will give you a step-by-step process for getting your popup shop off the ground.

47. Brew Your Own Beer.
Like drinking beer? Why not try making it yourself? With patience and skill you might end up brewing something that others are willing to pay to drink. Pick up an easy-to-use starter kit from Mr. Beer online, invest the time it&#39;ll take to perfect your craft, make a unique brew and start shopping it around to friends &amp; family to see what they think of this side business idea.

48. Freelance Proofreading and Editing.
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As long as there is still the written word, there will always be editors. Freelance editing and proofreading not only pays a decent hourly wage, it also gives you the chance to read about potentially interesting topics too. What&#39;s more, freelance writing &amp; editing can afford you a lifestyle that lets you travel the world as a digital nomad. You can find lots of job postings from companies and individuals in need of writing, proofreading, and editing services on Upwork, which makes this a high-demand side business idea.

49. Buy Used Electronics and Refurbish Them.
Many people give up on their faulty laptops, mobile phones, or cameras without even looking into the cause of their malfunctions. If you have the skills to fix them, consider starting a side business refurbishing and reselling used electronics.

50. Data Analysis.
Got a thing for numbers? Plenty of companies need to hire someone who’s good at data analysis. Upwork and Digiserved are but a two of many websites that are great for freelancers with an analytical prowess, looking for extra work.

51. Acquire Parts from Electronics Stores.
There is a treasure trove of valuable items being thrown away by big box stores around the world. Electronics stores toss out everything from printer cartridges to tablets, and if you’ve got the stomach for rummaging around in dumpsters you can easily make money rummaging than you ever thought possible.

52. Copywriting for Websites.
Every website owner (including myself) hires copywriters to write content for things like about pages, FAQ&#39;s, or blog posts. Hourly wages for novice copywriters are not very high, but with experience and a great portfolio, you can start charging more than you make at your normal job. Pick up my free downloadable Freelance Proposal Template and get started today.

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53. Licensed Product Distributor.
It doesn’t happen overnight, but licensing a foreign product for sale domestically, can lead to a big payoff down the road. Invest intelligently, sell a product you believe in, and you can reap the rewards for years to come.

54. Fill Out Online Surveys.
It’s not very engaging, or mentally stimulating, but online surveys do pay out, believe it or not. Just don&#39;t expect to make bank with this side business idea—I recommend combining it with another source of side income and turning to surveys in your downtime.

55. Airbnb Host.
Not only is Airbnb a great way to make money by renting out your spare bedroom or living room couch, but you also have the benefit of meeting new people and making new friends.

56. Personal Fitness Trainer.
If you’re a fitness buff and have the right combination of charisma and business sense, working as a part-time personal trainer can be both physically and financially rewarding. Once you build up a reputation and client base for yourself, it could easily turn into a full-time endeavor for you. Check out these tips to a successful personal training business by the American Fitness Professionals and this interview with several fitness blog owners who are making a living online, from MonetizePros. Check out this resource if you want to get started with a business plan for your personal fitness trainer business.

57. Yoga or Meditation Instructor.
Yoga is getting ever more popular, which means yoga instructors are more in demand than ever. Pursue emotional and physical balance with others, during your free time while helping your bank balance, too.

58. Start a YouTube Channel.
If you can create value-driven, entertaining video content and grow your subscriber base to a few thousand subscribers, your videos can start generating pretty substantial income from all the ads being displayed on your videos. Many YouTube users make well into the millions each year, so there&#39;s clear potential to take this from side business idea to millionaire with the right combination of content, audience, skills, relationships and timing.

59. Translator.
If your mastery of another language is good enough to have the grammar and spelling down, translating is a great side business to set up for yourself and can be done remotely. Flexjobs has literally hundreds of freelance, remote translator jobs available right now.

60. Tour Guide.
Unique Tour Business You Can Start While Working a Full-Time Job Vantigo SF

Live in a destination where travelers frequently visit? If you love meeting new people from around the world and also love the city you live in, starting your own local tour business will give you both of those perks. Take a unique spin on your local tour business like Erik from Vantigo. He was starting his VW van tour business in San Francisco as just a side business idea while he was still working a full-time job, and grew it to being a sustainable source of full-time income before quitting—now he&#39;s running a multiple van tour business for himself.

61. Music Instructor.
Do you play a musical instrument well enough to teach it to others? How about taking your skills at producing music or sound design and teaching others based on your experience? Private music instructors in all disciplines charge upwards of $20-100/hr, and you don’t necessarily have to do them in person, either.

62. Stock Photographer.
If you have a fondness for taking pictures of smiling families or laughing children blowing bubbles, consider becoming a stock photographer and selling your images to a stock photo company like Unsplash, ShutterStock and iStockPhoto. You’ll get royalties every time someone licenses an image you’ve submitted. To really be successful, build your own photography website to be able to showcase your portfolio and start getting private corporate work.

63. Ghostwriter.
Ghostwriting pays pretty well, and if you&#39;re talented at researching and creating great content within a certain subject domain, you can quickly build a roster of high-paying clientele. Writers like Jeff Haden have created very lucrative careers for themselves by writing for business executives and CEO&#39;s.

64. Online Subcontracting.
Once you’ve established yourself in any number of freelance verticals online, subcontracting your jobs out to other freelancers can help you significantly grow your client list, and revenue generating possibilities.

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65. DJ-ing.
Take the time to develop your skills at either making your own music, or becoming a pro at mixing, and your side business of dj-ing local events could turn into a much bigger business. Though you&#39;ll first need to command a solid knowledge of sound design, mixing and music production before expecting to land your first gig.

66. Clothing Alterations and Tailoring.
Fashion, style, and grooming continue to be a lucrative side business. If you know how to mend clothes, then you have a growing market of budget-conscious and/or personalization-obsessed customers. Entrepreneur says you need around two thousand dollars to start a decent side business idea in this niche but there’s no harm in starting smaller and growing from there, as long as you can make magic with needle and thread.

67. Teach DIYs.
DIY Do It Yourself Handmade Side Business Idea

Nothing beats teaching more novice learners about your passion, hobby, or craft. Explore dozens of DIY portals (such as DIY.org, DIY Network, Instructables and Mahalo) to get ideas on how to earn a healthy side income just by showing others how to do the things you love. You can also sift through the countless ad-supported YouTube channels that teach just about anything from guitar strumming to 3D printing.

68. Baking.
It’s funny how much baking can relate to making money. Start bringing in some side income by kneading flour, mastering the oven, and appeasing everyone’s sweet tooth. While some experts have already turned baking into their sole “bread and butter,” you need not leave your day job just yet. Depending on your experience, you can start by doing something simple on the side like Grandma’s nostalgic cookies before heading on to offer exquisite artisanal fare.

69. Being a Task Rabbit.
If you don’t mind doing other people’s chores, then TaskRabbit may be the right side business for you. Earn money doing odd jobs like walking your neighbor’s dog or mowing Mr. Smith’s lawn. Running errands not your style? Tell that to the top taskers on the portal who reportedly earn as much as $7000 a month.

70. Freelance Content Marketing.
From personal experience as a freelance content marketer, I can vouch for the profitability in this industry - if you have the right skills and experience to back it up. With startups, established brands, and even notable influencers needing to build credibility and grow their brands, professionals who can serve up compelling content enjoy a huge demand for their services. And, since brands will always need high quality content to bring in new customers, your skills will always remain an asset to invest in.

71. Freelance Ebook Writing.
If you’ve always wanted to publish your own book, there isn’t a better time than now to do so. That’s because access to self-publishing tools and marketplaces has never been easier and more affordable. This eBook writer currently earns up to a couple of thousand bucks each month from six ebooks she published; while you can reportedly demand around $1000 per eBook project serving as a ghost writer. My good friend, Caroline Beaton used freelance writing as a side business to eventually go from secretary to self-employed, while focusing on her own personal development and looking inward to discover what she&#39;s truly passionate about.

72. Investing Your Money.
If you keep your money hidden in the attic or earning close to nothing in a savings account, you might want to consider putting it to better use through smart investing. Sure, there are plenty dangers to look out for, but most personal finance experts will advise you to start learning about the stock market if you want to grow your wealth.

73. Investing Others&#39; Money.
Incidentally, helping others grow their wealth is also a good way to grow your own financial worth on the side. If you have stock market and finance credentials, then you can earn freelance commissions or consulting fees advising clients on how to get the best returns for their investments.

74. Accounting and Bookkeeping.
Many small companies and startups now outsource jobs that aren’t part of their core business (such as HR, admin, and accounting). Because companies always need to track their cash flow, freelance accountants and bookkeepers can really benefit from the increasing demand for part-time help.

75. Building a Niche Website.
How to Build an Engaged Audience for Your Website

Find an audience for your passion or hobby and you’re all set to monetize something you love via a niche website. That of course, is easier said than done, something not intended for the faint of heart. Prepare to invest a lot of your time, but if you can hurdle the steep challenges and positively answer a few key questions on whether the website idea you’re thinking about is profitable, then you can begin building it.

76. Car Washing and Detailing.
A lot of car owners simply don’t have time to personally care for their vehicles, much less make them sparkle. With simple equipment such as car shampoo, polishing wax, tire black, sponges, a pail and a sturdy hose, you can start earning a decent side income by making cars look fresh in your free time after work and on weekends.

77. Caregiving.
Providing care for the elderly often requires a broader set of credentials. These may include licenses, training certificates, business permits, and tax forms depending on which state you wish to operate, the level of care you intend to provide, and whether you plan on running a formal business or just hiring yourself out as an independent worker. If you’re already giving unpaid care to an elderly relative or friend, you may want to review Medicaid provisions for some compensation. In all cases, you need genuine affection for the elderly, a lot of common sense, and good knowledge of first aid, CPR, and elderly care basics.

78. Carpet Cleaning.
Carpets make your home warm and cozy. But they also tend to gather dust, grime, stains, allergens and bugs over time. You can tap into this opportunity by launching a carpet cleaning business or serving as an independent contract cleaner in your free time. Either way, you’ll need to invest in proper equipment such as vacuum cleaners, carpet brushes, and cleaning chemicals. Look into the possibility of renting equipment for a full weekend and lining up 5-10 clients to service, to offset your rental costs. If you turn a healthy profit, you&#39;ll eventually be able to invest in owning your own equipment and increase your margins.

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79. House Cleaning.
If you want to broaden your services and earn more, you can opt to make the entire house look neat and clean. House cleaning services cover carpets, floors, lawns, garages, windows, walls, and roofs, requiring a whole range of cleaning tools. You can hire maids and janitors to do the actual cleaning while you handle administrative and customer service tasks. Or, you can start small by actually getting your hands dirty as a contractor, learning on the job before launching your own cleaning operations. Check out Cleaning Zoom&#39;s ultimate guide to starting a cleaning business.

80. Child Care.
Many families with kids — including single-parents — have very limited time resources, creating a demand for part-time child care providers. If you love kids and have some spare time, you can earn plenty of side income by looking after children while their parents are away. Just be sure to identify the particular age group you&#39;re comfortable caring for and you’re all set after any necessary certifications. Establishing a home-based daycare center capable of servicing six or more kids is also a lucrative option.

81. Computer Repair.
Employed techies planning to earn extra money on the side can leverage their software and hardware skills by offering home-based computer repair services. If this rings a bell, you can start with a modest one-man tech team before envisioning a scaled-up operation as massive as Geek Squad. Remember, you can provide home service locally as well as offer remote support through online messaging and video calling services.

82. Modeling.
If you&#39;ve got the attitude, there should be a product or service out there looking for the right model to promote their brand. You can sign up with a large agency, find a freelance agent, or look for your own modelling gigs as an independent talent. Since modeling is a complex field, get all the advice you need before jumping right in and find a mentor who can guide you through some of the early challenges before establishing your brand. Depending on your preference, you can specialize in a single media format such as video or still images, to build a compelling portfolio.

83. Computer Training and Lessons.
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Everyone (even kids and retirees) need to have some level of technical know-how to stay competitive and appreciate the marvels of the digital age. Just observe how learning sites like Codecademy, Treehouse, and Udacity continue to grow and you’ll understand the urgency of getting computer training for our generation. If you’re a techie, you can cash in on this need by offering lessons and tutorials within your neighborhood or across cyberspace through portals like YouTube or Udemy. You can even set up your own tutorial site with an interface for online payments.

84. Contract Customer Service.
Some firms outsource their customer service operations and many of these companies accept home-based contractors. You can start by signing up on freelancing sites such as Upwork. If you’re already a CSR specialist with management skills, then you can form and lead a virtual team online and engage clients as a bona fide customer service company or, provide customer service training to such teams as well as individuals.

85. Dog Walking.
Dogs are amazing. But, owners aren’t always at home to take them for a walk. If you’re a canine fan like I hope you are, this great side business idea can net you a regular profit by just taking some adorable pups for a stroll around the park. With pet-crazed, yet busy Americans having more pooches around, dog walking services have become a thriving industry in some U.S. cities. In fact, there’s already an app called Wag that links up dog walkers and pet owners.

86. Real Estate Sale Consultant.
The real estate market is on the rebound (for today) and people are buying and selling properties in many locations across the U.S. That means a lot of opportunities for professionals who know how to navigate the legal, financial, and commercial aspect of real estate. If you consider yourself an expert in the subject, it won&#39;t be a walk in the park (hehe), but you can earn significant additional income as a home-based consultant.

87. Making Custom Furniture.
Turning your favorite craft or hobby into a revenue-generating business easily counts among the most fulfilling experiences anyone can have. With more and more people demanding a higher degree of personalization in just about anything they own or use, count yourself lucky if your craft happens to be the creation of custom furniture. Not only is the real estate sector regaining its footing, the number of style-conscious customers is also on the rise.

88. Making Handmade Jewelry.
There are at least three teenage entrepreneurs who’ve already earned astonishing amounts of money by launching hand-made jewelry businesses: M3 Girl Designs’ Maddie Bradshaw, Bella Weems of Origami Owl, and the girl behind DesignedByLei. With the industry set to shine in the near term via a brisk annual growth rate of 5-6%, the market is ripe for new entrants who offer sparkling new ideas on how to craft appealing handmade goods.

89. Being a Gigwalker.
Earning money on GigWalk is similar to working on Task Rabbit. You get to choose from an amusingly diverse range of jobs from taking snapshots of restaurant menus to counting the items arrayed on supermarket racks. It won’t make you rich though, with payouts tending to congregate in the lower half of their $3-to-$100 range. But, if you&#39;re looking for some sustainable side income, it can&#39;t hurt doing a strange but fun errand and getting a tip at the end. When you’re bored or a little short on cash, you can access GigWalk anytime, anywhere via their mobile app. Meanwhile, all the jobs specifically vetted for you will be within reach (ideally, just a brisk walk away).

90. Home Appraisal.
Best Business Ideas Home Appraisal on ryrob Ryan Robinson

A real estate appraisal business can be operated from home, on a part-time basis. A perfect option if you want to keep your day job while earning extra on the side. You won’t need a college degree to start one, but you’ll need bankable appraiser credentials (including relevant training and professional licenses), strong knowledge of the sector, and a growing network of industry players including mortgage brokers, real estate agents, banks, and fellow appraisers that&#39;ll want to utilize your services.

91. Human Billboard.
If you don’t mind parading yourself around in public places, and enjoy a little attention, then working as a human billboard can pay a surprising amount of dividends. Renting out your beard, for example can earn you as much as $5 a day while having a brand logo tattooed on your shaven head for five years could pay as much as $7,000. This is definitely not for the faint of heart, and is a little extreme for me, but nonetheless it&#39;s been done. You can start by holding a placard or wearing a sandwich board. If you think this is a good business idea for yourself, you can go all the way and turn it into a full on home business.

92. Purchasing an Existing Website.
Websites aren&#39;t too dissimilar to stocks. Most are junk, but some can generate value for you. That’s why, like stocks, they are bought and sold all the time. You can buy and sell websites in the hope of generating future earnings based on their user traffic, current revenue intake, domain name, or some other factors that might be a hidden cash cow everyone else has overlooked. Interested? Check out marketplaces like Flippa and Flipping Enterprises to learn more.

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93. Mobile Laundry Service.
We all wear clothes (at least some of the time). Which, of course, makes the business of cleaning soiled laundry a hit. In fact, the self-service laundromat industry in the US is worth more than $3 billion, with many related businesses such as a mobile app laundry service, a real rolling mobile laundry service, and home laundry pickup &amp; delivery services springing up each year. Here’s one way you can cash in on the trend.

94. Junk Removal Service.
As the saying goes, one person’s trash is another’s treasure. Without people cleaning up everybody else’s junk, the world would be a lot messier than it already is. The job might not be glamorous, but you get to make the world a cleaner place while earning extra side income. To start, you’ll need a sturdy second-hand truck and standard equipment like shovels, wheelbarrows, sledgehammers, rakes, and garbage cans.

95. Becoming a Notary Public.
Public notaries primarily serve as validators of, and impartial witnesses in public legal documents, especially those related to affidavits, oaths, deeds, estates, special-powers-of-attorney, and business agreements. Depending on the state, a full-time notary public earns an average annual salary of $36,000, and it&#39;s a great business to do on the side as you bring in money through other ways. If you want to get started, here are the requirements and steps you need to consider in the US.

96. Wedding Planning.
Like birthdays, marriages happen all the time. Which means you can treat weddings as a recurring fountain of business opportunities: wedding dresses and coats, jewelers, food caterers, venue providers, photographers and videographers, performers, flower shops, travel agencies, souvenir crafts, and a host of other ventures. Now imagine if you can form a networks of these service providers so you can offer engaged couples a range of hassle-free wedding packages. The process is certainly fun, and so is the pay.

97. Event and Party Planning.
Being a wedding specialist works well for top-notch players, but being a generalist also has its benefits. Event and party planners leverage their network of service providers to cover not only weddings, but also seminars, corporate events, birthdays, baptisms, concerts, conferences, holiday parties, and other events. Here are some tips if you want to run a part-time event planning business at home. And here’s an honest look into the upside and downside of the trade.

98. Becoming a Personal Chef.
Busy schedules can stress people out. It can also disrupt family life to the oint that busy parents and homework-laden kids barely have time to prepare decent dinners and weekend meals. Hence, the surprising demand for part-time family chefs. If cooking healthy and delicious meals is your thing, then this lucrative side business idea can supplement your regular income by helping feed busy homes.

99. Pet Sitting.
Free and Cheap Business Idea Dog Sitting ryrob Ryan Robinson

The prevalence of unattended pets is another side effect of busy life, creating job opportunities for people with time, patience, and affection to spare. Just check out DogVacay and Rover to see what I mean. Would the payout be worth it if you try? One person in New York earns $3,300 a month just pet-sitting for busy animal lovers.

100. Pet Grooming.
If that isn’t enough, you can expand your services to include pet grooming. To do that, you&#39;ll need to have relevant training, an operating license depending on your location, and pet grooming equipment. According to PayScale, pet groomers have an average salary range of $17,600 to $48,500. Not bad as a side gig if you can find a way to get in most of your clients on the weekends.

101. Pool Cleaning.
Spending a leisurely afternoon in a swimming pool can be a lot of fun. But cleaning it can be a pain in the neck. Hence, most pool owners hire people to do the cleaning for them. If you need extra cash and don’t mind getting physical, then pool cleaning is a viable side business for you.

There&#39;s More...

I&#39;ve been getting so much feedback with great business ideas from my community and in the comments, so I frequently add to this post. If you have ideas you&#39;d like to share, please add to them in the comments below!

102. Private Labeling and Selling Products on Amazon.
Counted amongst the most valuable tech giants, Amazon is also a massive global market where virtually anyone can cash in on the rising tide of ecommerce. But, like everything else that involves money, you have to do quite a bit of work to earn it. In this case, you need to do tons of research (looking for generic products such as clocks, key chains and mugs to attach your brand to) as well as a sensible selling strategy to milk profits from your private label. Check out how digital marketer Neil Patel is doing this right now.

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103. Officiating Recreational Sports Games.
Do you love sports? Officiating non-professional games held at community parks and college sport fields can be an exciting way to earn extra income on nights and weekends.

104. Selling on TeeSpring.
T-shirts can be a viable path toward creating an automated side business that makes money for you while you sleep. Even Bloomberg and Forbes feature stories from entrepreneurs who&#39;ve done just that. Want to learn how to make it happen? You can find TeeSpring selling walkthroughs here, here and here.

105. Build Custom Software for Freelance Clients.
Software drives businesses today. This is a fact not lost on individuals and companies attempting to earn a profit, make a name for themselves, or simply get things done. As a result, the job outlook for software developers will be much sunnier than prospects for other occupations. In fact, it won’t take you long to find a good-paying software development project on sites like Upwork, Guru.com, and Freelancer. LinkedIn also recently launched their brand new freelancing platform, ProFinder which has a dedicated section just for hiring talented software developers.

106. Coaching Sports Teams.
Side Business Idea Sports Coach ryrob Ryan Robinson

Nothing beats playing your favorite game. But getting paid for coaching a sports team and staying close to the action is the next best thing. Not only will you learn new skills, you’ll earn money getting your team into shape. The median pay for sports coaches clock in at $30,400 according to the US Bureau of Labor Statistics. Expect to have only a slice of that if you’re signing up for part-time coaching only.

107. T-Shirt Printing Business.
As I’ve already covered with TeeSpring, the T-shirt retailing business is worth millions of dollars. But what if you want to do the manufacturing side of things? T-shirt printing turns out to be quite rewarding. This multi-billion dollar industry counts big corporate factories as well as home-based part-time entrepreneurs.

108. Vehicle Advertising.
Yes, you can turn your car into a money-maker by driving for Uber and Lyft, but there are other ways to convert mileage into money. One way is to do vehicle advertising, a potential side income source of anywhere between $100 to $600 per month. Whether you own a compact sedan or a full-sized bus, your vehicle’s exterior space is prime real estate for ads. Check Wrapify, carvertise, and similar sites for more details.

109. Window Cleaning Services.
From the inside looking out, I want to see the world as clearly as possible. And so do a lot of homeowners and office managers. Now that creates a demand for window cleaners, especially for office buildings. If you want to try it out as a side business, here’s a step-by-step guide on how to start a window-washing business at home.

110. Working on Mechanical Turk.
Mechanical Turk operates much like TaskRabbit and GigWalk: it’s a corner on cyberspace where Amazon gathers tasks to be done, people willing to do them, and people willing to pay for them. On Mechanical Turk, you get to do the oddest jobs you can imagine, made possible by a parallel online universe that runs on hits, visits, surveys, reviews, pins, likes, CPCs, reads, and other metrics. But don’t get too ambitious. The small sums being paid out to human Mechanical Turk users rarely add up to anything substantial, even if you invest most of your spare time into it. It&#39;s a better opportunity for internationally-based people with internet access and lower costs of living than in the US. Here’s one person’s account of his former life as a Mechanical Turk.

111. Yard Work Services.
A large proportion of homeowners and small businesses have lawns to tend to. Many turn to independent providers of yard work services to take care of their lawn care woes. To start and sustain a yard work services side business, you need training, equipment, and a growing network of customers. A full-time landscaping and groundskeeping worker earns a median wage of around $25,000, while those working on the side during evenings and weekends can expect to make substantially less.

112. Public Speaking.
If you have the gift of gab, public speaking is a worthwhile side business to pursue. Many authors, medical practitioners, scientists, artisans, politicians, and other subject matter experts earn extra income by delivering speeches, seminars, and presentations. Professional motivational speakers earn an average annual salary of around $90,000 with just that portion of their business.

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113. College Counseling.
College counsellors come in all shapes and sizes. Their primary duty is to help guide students through an important period in their lives. Some counselors focus on admissions, career advice, financial aid, athletics, extracurricular activities, and other concerns. There are institutions looking out for part-time college counselors, or you can start your own practice and charge anywhere from $25-$100+ per hour for your consultative services if you&#39;re well-versed on the subject.

114. Making Seasonal Decorations.
Seasonal Decorator Side Business Idea ryrob Ryan Robinson

Christmas, Halloween, Easter, Yuletide, Hanukkah, Valentine’s Day, Chinese New Year, 4th of July, Mother’s Day. There are a ton of traditional holidays that count as solid reasons to start a side business crafting and selling seasonal decorations. After all, people and businesses pay good money for them. In fact, total sales of Christmas trees in the U.S. alone amounted to a whopping $1.04 billion in 2014. And you still have holiday lights, nativity scenes, crafted hangings, baskets, wreaths, and other decorations to cover.

115. Snowplowing.
The holidays do bring cheer, but when winter comes, so does the snow. Thick, traffic-hindering piles of it. Snow can be fun for a while but somebody has to get rid of it for our lifestyles to continue onward. Make extra money from home by offering snow and ice removal services.

116. Selling Handmade Clothing and Garments.
With the rise of online craft marketplaces like Etsy, people with decent artisanal skills like sewing and woodworking have an always-open market to sell their products. If you’ve always wanted to design and make clothes by hand, then you can start turning those fashion ideas into real, hand-sewn garments and make some money on the side.

117. Antique Refurbishing.
The antique market is not as lively as it used to be. But there’s still money to be had from the industry if you love rare old stuff and possess the skill of restoring them to their former glory. To start an antique refurbishing business at home, you’ll likely need a few thousand dollars to build out a basic workshop and stock it with all of the right treatments and materials. Start small by borrowing around what you can, and learning the basics of the trade.

118. Dance Instructor.
Most people, even those many won’t admit it, love moving their bodies. It&#39;s also a great workout. If at some point in your life, you had to hang your dancing shoes for a more sensible office job, there&#39;s nothing stopping you from earning money on the side with this passion project. Part-time dance instructors get a fraction of the pie their full-time peers enjoy, but it can add up to hundreds each week if you&#39;re consistent.

119. Part-time Bicycle Delivery.
Believe it or not, this freelance bike messenger makes more money than some people glued to their work desks. You won&#39;t rake in a whopping $50,000 as this guy does by just doing this in your free time, but you’ll surely get a piece of the action. Even Uber is trying out a blueprint for a delivery service, and companies like WunWun and Amazon are stepping up their efforts to expand same-day delivery. If you have a decent bicycle, those pair of pedals might just earn you a healthy side income.

120. Home-Based Makeup Services.
If you have a talent for helping people look their best, and know how to mesmerize with eyeliners and turn heads with lipsticks, consider using your beauty skills to create a profitable side business. For under $2000, you can start your own make-up artist business.

121. Voice-Over Talent.
If you speak like James Earl Jones or Scarlett Johansson, quite a number of digital publishers (including game developers, animated film-makers, and training video producers) pay good money for voice talents. You can find these publishers looking for voice-over artists on sites like PeoplePerHour, Freelancer and Upwork.

122. Buying and Selling Cars.
Car flipping can be a fun way to turn your love for cars into a quick side business idea if you have the capital and a stomach for the inherent risk. Buying and selling cars is similar to other “buy low, sell high” business models, but the profit potential per hour invested is very high. According to Jeremy Fisher at 3HourFlip.com, the trick is to learn how to make car deals come to you, so you can minimize your time invested and maximize your profit on each flip.

123. Small Business Marketing Consultant.
If you have experience with marketing or a knack for getting people excited about the products and services you use on a regular basis, think about refining your skills and putting them to work as a small business marketing consultant in your region. Businesses of all sizes are in constant need of bringing in more customers, which is where you come in. Start by pitching some of these small business marketing ideas by Brian Downard and you&#39;ll be sure to impress the first set of clients you begin working with in your area.

124. Professional Organizing.
If keeping your home organized comes naturally and you enjoy helping other people, consider becoming a professional organizer. As the need for baby boomers to start downsizing grows, you can use your organizing and decluttering skills to start making an industry standard $50-125 an hour with very little startup costs and some free expert advice from Jen Kilbourne. Believe it or not, pro organizers don&#39;t just work with hoarders. More and more people are hiring professional organizers in order stop wasting time and money due to the stress of everyday disorganization.

125. Rent out your space.
People are always searching for one-of-a-kind venues for meetings, parties and weddings. Why not earn some extra money off the space you already have by renting it out for events? If you own a unique venue, like a studio, warehouse or boat, UK-based company Tagvenue will connect you with clients looking for somewhere special to host their event. Not a bad low-effort side business idea.

126. Build a Chrome Extension.
With nearly 100,000 different custom extensions both free and paid available in Google Chrome&#39;s marketplace, there&#39;s a huge opportunity for creating a useful extension for people to use while browsing the web. One of my favorite Chrome extensions, Yet Another Mail Merge, which enables you to send bulk customized emails from within Gmail, gives you 50 free credits to send emails and gives you the chance to upgrade to a paid account to unlock more sending—a great upsell for the people who are getting value from the extension. Even free Chrome extensions like SVRF Tabs by SVRF, which replaces your new tab with stunning VR and 360º images, have the potential to bring in new users and eventually drive revenue for the startup&#39;s core business. Another fun example from a friend of mine, Kathleen Garvin, is Hide images with NOPE, a Chrome extension that hides images on the web pages you&#39;re browsing—to help eliminate distractions and keep you from seeing images (like those of certain politicians) that you may want to hide.

127. Floral Design.
Valentine&#39;s day. Birthdays. Weddings. The list goes on and on—throughout the course of a year, there are dozens of occasions when people need to order flowers. What&#39;s more is that once you&#39;re up to speed in this business, your costs can stay relatively low if you know where to source your flowers, and typical margins are in the neighborhood of 300% (or more) on cut flowers which makes this a particularly profitable side business idea in the right geographic area. Check out this amazing interview with floral designer, Sara Tedford of Ladybird Poppy to hear about how her floral design company started out as a side business idea of doing weddings and events for her friends and family.

128. Recycling.
Though certainly not for everyone, recycling the containers from products left strewn around on the ground can net you a surprisingly decent side income if you put in the time. Attending events like street fairs and outdoor concerts can be a great starting place. Put on a clean shirt, jeans, apply your sunscreen, grab a cheap pick-up tool like this one, sturdy plastic bags, and you can sell them for upwards of $4.00/pound on eBay.

129. What Did I Miss?
Share in the comments if I&#39;m missing any great business ideas that you can start while working full-time! Bonus points if you&#39;ve already gotten started on it, and share the link to your website.

Want to find a profitable business idea and get your first paying customers in the next 30 days? Join me in 30 Days to Validate today.</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/8630523124140853399/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2017/02/working-full-timeryan-robinson442.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8630523124140853399'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8630523124140853399'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2017/02/working-full-timeryan-robinson442.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-1258200515675721118</id><published>2015-11-18T23:55:00.002-08:00</published><updated>2015-11-18T23:55:52.671-08:00</updated><title type='text'>engagement NOVEMBER 15, 2015</title><content type='html'>&lt;iframe id=&quot;vp1kx2dK&quot; 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style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;span style=&quot;color: blue; font-family: Arial; font-size: 12.0pt; font-style: italic; font-weight: bold;&quot;&gt;Flight
Control Systems - Irreversible&lt;/span&gt;
&lt;br /&gt;

&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;An
irreversible flight control system is where there is not a direct
mechanical linkage connection between the control lever in the flight
compartment and the flight control surface. In an irreversible
mechanical system, the control lever in the cockpit moves a spool
valve on a hydraulic power control unit (PCU).  A mechanical linkage
drives the spool.  The mechanical linkage will consist of a
combination of bellcranks, pushrods, cable systems, pogos, summing
linkages, etc.  The design of a mechanical irreversible system is
similar to the design of a reversible flight control system.  The
main difference will be the spool forces are better defined than
surface hinge moments.  Also, an artificial feel system is required
for irreversible flight control systems.  Variation in cable
stiffness characteristics over environmental and operating ranges can
also be critical since spool movements will be small.  As a general
rule, cable pretension values will be higher for an irreversible
system (some commercial aircraft are in the 150 lb pretension range).
   &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Irreversible
flight control systems include both mechanical linkage controlled
hydraulically powered PCUs and fly by wire systems.  Today, most new
irreversible systems are fly by wire.   A description of fly by wire
systems can be found in &lt;span style=&quot;color: blue;&quot;&gt;Flight Control Systems
– Fly By Wire&lt;/span&gt;.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;A
simple example of a mechanical irreversible flight control system is
shown in Figure 1.  The system in Figure 1 shows a rudder system with
bellcranks and pushrods connecting the rudder pedals to the hydraulic
servoactuator (PCU).  The bellcranks and pushrods form a series of
four bar linkages.  The artificial feel system shows a common
approach – spring-loaded cam - for producing force feedback to the
pilot.  Since cables are not used in this example, the system will
have good stiffness.  Of course, system freeplay will need to very
small.  A summing link is used at the connection to the PCU.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;img height=&quot;198&quot; src=&quot;http://www.daerospace.com/FlightControlSystems/FCIrrevFig1.png&quot; style=&quot;border-width: 0px;&quot; width=&quot;470&quot; /&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Figure
1     Irreversible Flight Control System&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Irreversible
flight control systems are used on larger aircraft where the hinge
moment (surface) loads are large and a person cannot supply enough
force.  Keep in mind that the input travel in the cockpit is limited
by the amount of cockpit control travel and this becomes a limit on
the maximum hinge moment available without additional power input.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Pilots
require feedback forces when they fly (to emulate reversible system
stick force per g characteristics and other natural aerodynamic force
feedback).  Therefore, artificial feel systems are installed in all
irreversible flight control systems. Artificial feel systems are done
through springs, pneumatic actuators or, in a few cases, hydraulic
actuators.  Springs loaded in compression are the most reliable and
hence the most common.   Artificial feel systems are always redundant
(sometimes triple redundant) so that artificial feel is not lost
after any single failure in an artificial feel system.  A
spring-loaded cam artificial feel system is shown in Figure 1.&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Irreversible
systems require hydraulic or electromechanical actuators.  However,
electromechanical actuators are only used in fly by wire systems
(i.e., they are not mechanically controlled through linkages as can
hydraulic actuators).  The choice and number of actuators in turn
drive the design of the hydraulic or electric power systems.  The end
result is a combination of increased size, cost and complexity of
these systems.  Hence, irreversible systems are generally more costly
and complex than reversible systems.  Assessment of system hazards
and failure modes are therefore more involved for irreversible
systems. &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Hydraulic
PCUs in powered flight control systems will typically have different
operating modes.  Some examples of different modes and features in
PCUs can be found in &lt;span style=&quot;color: blue;&quot;&gt;Power Control Unit,
Hydraulic - Description&lt;/span&gt;.  Generally, these modes will address
issues such as loss of a hydraulic power source, loss of a partner
PCU (where 2 or more PCUs are connected to a single surface), loss of
electrical control signal, jammed spool or other detected fault
within a PCU.  Typical modes are normal operation, bypass (where
actuator follows surface with minimal resistance and no flutter
damping protection), damped bypass (follows surface movement but with
a hydraulic damping orifice for flutter protection), and locked or
centering (actuator piston is held in failed or neutral position).  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;The
most critical failure conditions within a flight control system are a
surface runaway, jam, disconnection of a mechanical linkage, or loss
of electrical/hydraulic power.  To allow operation after these
failures (perhaps degraded operation), various features are often
included to allow some type of system operation after these failures.
 These include pogos (load limiters), shearouts, jam overrides,
multiple hydraulic actuators with independent hydraulic sources and
disconnect clutches.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;In
irreversible flight control systems, jam protection is provided by
pogos, jam breakout mechanisms, shearouts, and disconnect clutches
when dual flight control runs are used.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Pogos
(or load limiters or bungees) are a very stiff spring installed in
the system.  Under normal operating loads, the pogo acts like a stiff
pushrod in the system.  However, if there is a jam downstream of the
pogo, then with an increase in applied force to the mechanism the
pogo will compress and allow other portions of the system to move. 
For example, if the cockpit control column provide control to dual
elevator mechanisms and a jam occurs in one side of the dual elevator
mechanism, then the pogo will compress in the side with the jam and
the other side can continue to operate (albeit with higher forces
applied at the control column).  More details on pogos can be found
in &lt;span style=&quot;color: blue;&quot;&gt;Mechanism – Pogo&lt;/span&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Jam
breakout devices function similar to a pogo.  A jam breakout device
is shown in Figure 2.  In Figure 2, the cam and link 1 are splined
together and from a rigid part.  The rigid cam/link1 part can rotate
relative to the grey plate but is held in place by the spring force
applied to the cam roller (through lever 1 and lever 2) that pushes
the roller against the cam surface.  In normal operation, the spring
force will be sufficient to maintain the roller in the position shown
with normal operating forces applied to the input pushrod.  Under
normal operation, the entire mechanism shown in Figure 2 - link1,
cam, plate, lever 1, lever 2 and the spring cartridge - form a
bellcrank and rotate together about point 0.  If a jam occurs in
either of the output links, the plate will be held fixed.  When the
plate is fixed, the mechanism upstream of the jam breakout device can
move by applying enough force to push the roller out of the detent
and compressing the spring.  Another means to override a jam is to
have a non-jammable spool in the PCU.  This is usually done with a
spool within a spool design, where the inner spool will move an outer
spool if the inner spool becomes jammed within the outer spool. 
Higher forces will be required to move the outer spool in its sleeve.
 A spool in a spool configuration is a type of jam breakout
mechanism.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;img height=&quot;393&quot; src=&quot;http://www.daerospace.com/FlightControlSystems/FCIrrevFig2.png&quot; style=&quot;border-width: 0px;&quot; width=&quot;370&quot; /&gt;

&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Figure
2     Jam Breakout Device&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;As
the name suggests, a shearout is a structural element, such as a pin
or fastener, which is designed to fail (usually in shear) at a given
load.  Shearouts are normally used to protect against an overload in
a system or part, but may be used to breakaway when a load occurs. 
Thus a shearout is used as means to limit the maximum load that may
be seen in a system and allows a lower limit load design criteria to
be used when structural sizing of parts. &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Means
to protect against disconnection failures include using a dual
(redundant) mechanism, using multiple actuators on a surface or using
lost motion devices.   Using multiple actuators on a surface with
dual control runs provides disconnection protection.  For example, if
one of the dual runs has a disconnection the other control run would
be operational and would provide control to one of the actuators. 
Another feature that may be used for disconnection failures is a lost
motion device.  In a lost motion device, relative motion is allowed
between 2 parts.  Under normal operation, the two parts will not
interact, but if a disconnect failure occurs then after an initial
input motion to take up the slack, the two parts will engage and the
system output will be driven (see &lt;span style=&quot;color: blue;&quot;&gt;Mechanisms –
Lost Motion&lt;/span&gt;). &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;A
disconnect clutch is mechanical clutch that connects two independent
flight control runs.  If a jam occurs in one of the flight control
runs, then the disconnect clutch is pulled (opened) through either a
handle or switch in the flight compartment which allows the
non-jammed side to operate.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Figure
3 shows a generic, single axis flight control system with built-in
redundancy features to address several critical failure conditions. 
The system shown in Figure 3 has a disconnect clutch at the control
column and a pogo connecting the aft quadrants, which protects
against a jam in either the pilot or copilots control runs.  In
addition, there are pogos in the linkage to each hydraulic actuator. 
These pogos would still allow control with a jammed servo in a
hydraulic power control unit.  Protection against the loss of a
single hydraulic source is provided through separate hydraulic
sources for each hydraulic actuator.  Protection against a
disconnection anywhere in the system is accomplished through dual
systems – dual quadrants, dual cables runs and dual actuators.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;img height=&quot;312&quot; src=&quot;http://www.daerospace.com/FlightControlSystems/FCIrrevFig3.png&quot; style=&quot;border-width: 0px;&quot; width=&quot;470&quot; /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;CENTER&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Figure
3     Generic Flight Control System with Redundancy Features&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;br /&gt;
&lt;/div&gt;
&lt;div align=&quot;LEFT&quot; style=&quot;line-height: 100%;&quot;&gt;
&lt;span style=&quot;font-family: Arial, sans-serif;&quot;&gt;&lt;span style=&quot;font-size: x-small;&quot;&gt;Design
considerations for reversible flight control systems can be found in
&lt;span style=&quot;color: blue;&quot;&gt;Flight Control System – Design
Considerations&lt;/span&gt;.  A discussion of design requirements can be
found in &lt;span style=&quot;color: blue;&quot;&gt;Flight Control Systems – Design
Requirements&lt;/span&gt;.  &lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/8438306026804422884/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/08/flightcontrol-systems-irreversible.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8438306026804422884'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8438306026804422884'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/08/flightcontrol-systems-irreversible.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-4089493547643604585</id><published>2013-05-23T00:32:00.001-07:00</published><updated>2017-07-20T22:13:06.409-07:00</updated><category scheme="http://www.blogger.com/atom/ns#" term="Study materials"/><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;h3 class=&quot;post-title entry-title&quot; itemprop=&quot;name&quot;&gt;
Shear centre
&lt;/h3&gt;
&lt;div class=&quot;post-header&quot;&gt;

&lt;/div&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue; font-family: &#39;lucida grande&#39;, tahoma, verdana, arial, helvetica, sans-serif;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue; font-family: &#39;lucida grande&#39;, tahoma, verdana, arial, helvetica, sans-serif;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue; font-family: &#39;lucida grande&#39;, tahoma, verdana, arial, helvetica, sans-serif;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue; font-family: &#39;lucida grande&#39;, tahoma, verdana, arial, helvetica, sans-serif;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;b&gt;Shear
 centre of a section can be defined as a point about which the applied 
force is balanced by the set of shear forces obtained by summing the 
shear stresses over the section.&lt;/b&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue; font-family: &#39;lucida grande&#39;, tahoma, verdana, arial, helvetica, sans-serif;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;b&gt;&lt;div class=&quot;separator&quot; style=&quot;clear: both; text-align: center;&quot;&gt;
&lt;a href=&quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwXLxMpQHXBxUeyyUnOtBgywBSIXxHiPwXwnnY9RlGF-1mTdoxDWqiFSYms2Bl0uT_auup3UsYzzkuvgm0mFJb18Lh13muNpWKZsoNBslvWM__wKxpnsTyqEu2J2z1gYbd6Zz52GMEDNs/s1600/img__ssb1_0014.jpg&quot; style=&quot;margin-left: 1em; margin-right: 1em;&quot;&gt;&lt;img border=&quot;0&quot; height=&quot;230&quot; src=&quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwXLxMpQHXBxUeyyUnOtBgywBSIXxHiPwXwnnY9RlGF-1mTdoxDWqiFSYms2Bl0uT_auup3UsYzzkuvgm0mFJb18Lh13muNpWKZsoNBslvWM__wKxpnsTyqEu2J2z1gYbd6Zz52GMEDNs/s400/img__ssb1_0014.jpg&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: black; font-family: Arial, Helvetica, sans-serif; font-weight: normal;&quot;&gt;&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: purple;&quot;&gt;In
 unsymmetrical sections and in particular angle and channel sections, 
the summation of the Shear Stresses in each leg gives a set of Forces 
which must be in equilibrium with the applied Shearing Force.&lt;/span&gt;&lt;/b&gt;&lt;/span&gt;&lt;/div&gt;
&lt;/b&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;font-family: Arial, Helvetica, sans-serif;&quot;&gt;&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue;&quot;&gt;(a)
 Consider the angle section which is bending about a principle axis and 
with a Shearing Force F at right angle to this axis. The sum of the 
Shear Stresses produces a force in the direction of each leg as shown 
above. It is clear that their resultant passes through the corner of the
 angle and unless F is applied through this point there will be a 
twisting of the angle as well as Bending. This point is known as&amp;nbsp;&lt;/span&gt;&lt;/b&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;font-family: Arial, Helvetica, sans-serif;&quot;&gt;&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: red;&quot;&gt;The Shear Centre or Centre of Twist&lt;/span&gt;&lt;/b&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;Apple-style-span&quot; style=&quot;line-height: 18px;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;font-family: Arial, Helvetica, sans-serif;&quot;&gt;&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: purple;&quot;&gt;(b)
 For a channel section with loading parallel to the Web, the total 
Shearing Force carried by the web must equal F and that in the flanges 
produces two equal and opposite horizontal forces. It can be seen that 
for equilibrium the applied load causing F must lie in a plane outside 
the channel as indicated.&amp;nbsp;&lt;/span&gt;&lt;/b&gt;&lt;/div&gt;
&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: purple;&quot;&gt;&lt;br /&gt;
&lt;/span&gt;&lt;/b&gt;&lt;/div&gt;
&lt;div class=&quot;separator&quot; style=&quot;clear: both; text-align: center;&quot;&gt;
&lt;a href=&quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3dxFICgVwVaaWOm5CfdNWVEjaa8F4x15m7nwNmqbeaG1NAPh9PN6faheUsjba1HnKhwWFwbpadMEkLwhafIwgh6hemtIfRxQ8lKteNgn2dLcwT6K0BZRZatBYvAQQnUUPw976oDZsKuI/s1600/img__ssb1_0005_200_193_thumb.gif&quot; style=&quot;margin-left: 1em; margin-right: 1em;&quot;&gt;&lt;img border=&quot;0&quot; height=&quot;308&quot; src=&quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3dxFICgVwVaaWOm5CfdNWVEjaa8F4x15m7nwNmqbeaG1NAPh9PN6faheUsjba1HnKhwWFwbpadMEkLwhafIwgh6hemtIfRxQ8lKteNgn2dLcwT6K0BZRZatBYvAQQnUUPw976oDZsKuI/s320/img__ssb1_0005_200_193_thumb.gif&quot; width=&quot;320&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;div style=&quot;text-align: justify;&quot;&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;font-size: large;&quot;&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: red;&quot;&gt;Note:&lt;/span&gt;&lt;/span&gt;&lt;/b&gt;&lt;br /&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue;&quot;&gt;&lt;/span&gt;&lt;/b&gt;&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue;&quot;&gt;1. In case of a beam having two axes of symmetry, the shear centre coincides with the centroid.&lt;/span&gt;&lt;/b&gt;&lt;br /&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue;&quot;&gt;2. In case of 
sections having one axis of symmetry, the shear centre does not coinside
 with the centroid but lies on the axis of symmetry.&lt;/span&gt;&lt;/b&gt;&lt;br /&gt;
&lt;b&gt;&lt;span class=&quot;Apple-style-span&quot; style=&quot;color: blue;&quot;&gt;3. When the load passes through the shear centre then there will be only bending in the cross section and no twisting.&lt;/span&gt;&lt;/b&gt;&lt;/div&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/4089493547643604585/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/shear-centre-shear-centre-of-section.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/4089493547643604585'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/4089493547643604585'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/shear-centre-shear-centre-of-section.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwXLxMpQHXBxUeyyUnOtBgywBSIXxHiPwXwnnY9RlGF-1mTdoxDWqiFSYms2Bl0uT_auup3UsYzzkuvgm0mFJb18Lh13muNpWKZsoNBslvWM__wKxpnsTyqEu2J2z1gYbd6Zz52GMEDNs/s72-c/img__ssb1_0014.jpg" height="72" width="72"/><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-2113484486993705254</id><published>2013-05-22T02:02:00.002-07:00</published><updated>2013-05-22T02:02:53.973-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;div class=&quot;post-meta fix&quot;&gt;
 
       
       &lt;div class=&quot;post-header fix post-nothumb&quot;&gt;
                &lt;div class=&quot;post-title-section fix&quot;&gt;

         &lt;div class=&quot;post-title fix&quot;&gt;
          
          &lt;h1 class=&quot;entry-title&quot;&gt;
Jet Engine Detail Design: The Compressor&lt;/h1&gt;
&lt;div class=&quot;metabar&quot;&gt;
&lt;em&gt;&lt;span class=&quot;sword&quot;&gt;&lt;/span&gt;&lt;span class=&quot;categories&quot;&gt;&lt;br /&gt;&lt;/span&gt;  &lt;/em&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
In this post the design of jet engine compressors will be 
discussed leading to the definition of ballpark performance parameters. 
For smaller engines centrifugal (CF) compressors are used since they can
 handle smaller flow rates more effectively and are more compact than 
axial compressors. Axial compressors however have the advantage of a 
smaller frontal area for a given flow rate, can handle higher flow rates
 and generally have higher efficiencies than CF compressors. For larger 
turbines used on civil aircraft the most suitable compressor and turbine
 will be of the axial type. Early axial compressors were able to raise 
the pressure of the incoming area around 5-fold, while modern turbofan 
engines have pressure ratios in excess of 30:1.&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_554&quot; style=&quot;width: 429px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/jet-engine-detail-design-the-compressor/compressorschmetic/&quot; rel=&quot;attachment wp-att-554&quot;&gt;&lt;img alt=&quot;Low pressure axial compressor scheme of the Olympus BOl.1 turbojet. (Photo Credit: Wikipedia)&quot; class=&quot; wp-image-554 &quot; height=&quot;277&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2013/01/compressorschmetic-1024x676.jpg&quot; width=&quot;419&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Low pressure axial compressor scheme of the Olympus BOl.1 turbojet. (Photo Credit: Wikipedia)&lt;/div&gt;
&lt;/div&gt;
Because the pressure rises in the direction of flow through the 
compressor there is an acute risk of the boundary layers separating on 
the compressor blades as they encounter this adverse pressure gradient. 
When this happens the performance of the compressor drops dramatically 
and compressor is said to stall. For this reason the compression is 
spread over a large number of compressor stages such that the smaller 
incremental increases in pressure across each stage allow engineers to 
obtain a large overall pressure ratio without incurring stall. A stage 
consists of a row of rotating blades called the &lt;i&gt;rotor&lt;/i&gt; and a row of stationary blades called the &lt;i&gt;stator&lt;/i&gt;.
 Each of these rows may consist of between 30-100 distinct blades and 
there may be up to 20 stages between the air inlet and compressor 
outlet. The role of the rotor blades is to accelerate the incoming air 
in order to increase the kinetic energy of the fluid. Across the stators
 the fluid is then decelerated and as a consequence the fluid pressure 
is increased. As the pressure and density increase across each stage the
 overall flow velocity is kept relatively constant by reducing the 
height of the blades from stage to stage. Thus the compressor tapers 
down from inlet to outlet.&lt;br /&gt;

In an attempt to reduce the number of compressor stages for a more 
compact engine, a designer’s goal is to maximise the pressure ratio 
across each stage. The stage pressure ratio &lt;i&gt;R&lt;/i&gt; is given by the following expression,&lt;br /&gt;

&lt;img alt=&quot;R_s=\left[1+\eta_s\frac{UC_a}{c_pT_{01}}(\tan b_1 - \tan b_2)\right]^{\frac{\gamma}{\gamma-1}}&quot; class=&quot;latex&quot; src=&quot;http://s0.wp.com/latex.php?latex=R_s%3D%5Cleft%5B1%2B%5Ceta_s%5Cfrac%7BUC_a%7D%7Bc_pT_%7B01%7D%7D%28%5Ctan+b_1+-+%5Ctan+b_2%29%5Cright%5D%5E%7B%5Cfrac%7B%5Cgamma%7D%7B%5Cgamma-1%7D%7D&amp;amp;bg=ffffff&amp;amp;fg=000&amp;amp;s=0&quot; title=&quot;R_s=\left[1+\eta_s\frac{UC_a}{c_pT_{01}}(\tan b_1 - \tan b_2)\right]^{\frac{\gamma}{\gamma-1}}&quot; /&gt;
Where &lt;img alt=&quot;\eta_s&quot; class=&quot;latex&quot; src=&quot;http://s0.wp.com/latex.php?latex=%5Ceta_s&amp;amp;bg=ffffff&amp;amp;fg=000&amp;amp;s=0&quot; title=&quot;\eta_s&quot; /&gt;&amp;nbsp;is the stage isentropic efficiency, &lt;i&gt;T&lt;sub&gt;01&lt;/sub&gt;&lt;/i&gt; is the total (stagnation) temperature, &lt;i&gt;U&lt;/i&gt; the rotary speed of the compressor, &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt;&lt;/i&gt; the axial speed of the fluid, &lt;i&gt;c&lt;sub&gt;p&lt;/sub&gt;&lt;/i&gt; the coefficient of latent fusion at constant pressure, and b&lt;sub&gt;1&lt;/sub&gt; and b&lt;sub&gt;2&lt;/sub&gt; the angle of the rotor blade leading and trailing edge relative to the axial flow direction.&lt;br /&gt;

&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_555&quot; style=&quot;width: 444px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/jet-engine-detail-design-the-compressor/compressorblades/&quot; rel=&quot;attachment wp-att-555&quot;&gt;&lt;img alt=&quot;Diagram of an axial flow compressor. (Photo Credit: Wikipedia)&quot; class=&quot; wp-image-555 &quot; height=&quot;309&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2013/01/CompressorBlades.png&quot; width=&quot;434&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Diagram of an axial flow compressor. (Photo Credit: Wikipedia)&lt;/div&gt;
&lt;/div&gt;
&lt;br /&gt;

The pressure ratio across each stage can be maximised by increasing the rotary speed of the compressor &lt;i&gt;U&lt;/i&gt;, the angle through which the fluid is turned across the rotor blades tan b&lt;sub&gt;1&lt;/sub&gt; –tan b&lt;sub&gt;2&lt;/sub&gt; and the axial speed of the fluid &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt;&lt;/i&gt; through the compressor. However there is a limit on the extent of these three parameters.&lt;br /&gt;

&lt;strong&gt;1.&lt;/strong&gt; The blade tip speed and therefore &lt;i&gt;U&lt;/i&gt; is 
limited by stress considerations at the root. If the fan is assumed to 
be of constant cross-sectional area then the centrifugal stress at the 
root is given by,&lt;br /&gt;

&lt;img alt=&quot;\sigma_r=\int_{r_r}^{r_t}\rho_b\Omega^2 r dr = 0.5\rho_b U_t^2\left(1-\left(\frac{r_r}{r_t}\right)^2\right)&quot; class=&quot;latex&quot; src=&quot;http://s0.wp.com/latex.php?latex=%5Csigma_r%3D%5Cint_%7Br_r%7D%5E%7Br_t%7D%5Crho_b%5COmega%5E2+r+dr+%3D+0.5%5Crho_b+U_t%5E2%5Cleft%281-%5Cleft%28%5Cfrac%7Br_r%7D%7Br_t%7D%5Cright%29%5E2%5Cright%29&amp;amp;bg=ffffff&amp;amp;fg=000&amp;amp;s=0&quot; title=&quot;\sigma_r=\int_{r_r}^{r_t}\rho_b\Omega^2 r dr = 0.5\rho_b U_t^2\left(1-\left(\frac{r_r}{r_t}\right)^2\right)&quot; /&gt;
Where &lt;i&gt;U&lt;sub&gt;t&lt;/sub&gt;&lt;/i&gt; is the tip speed, &lt;img alt=&quot;\rho_b&quot; class=&quot;latex&quot; src=&quot;http://s0.wp.com/latex.php?latex=%5Crho_b&amp;amp;bg=ffffff&amp;amp;fg=000&amp;amp;s=0&quot; title=&quot;\rho_b&quot; /&gt; is the density of the blade, and the ratio r&lt;sub&gt;r&lt;/sub&gt;/r&lt;sub&gt;t&lt;/sub&gt;
 is called the root-to-tip ratio of the blade. To prevent the blades 
from detaching from the hub and destroying the engine this root stress 
is not allowed to exceed a certain proof stress. It can be seen that the
 root stress is proportional to the square of the compressor rotational 
velocity and decreases as the blade length becomes shorter. Since the 
first compressor blades have the highest blade lengths they limit the 
maximum tip speed and therefore the efficiency of the compressor. It is 
therefore common to split the compressor into double or triple spool 
configurations such as a large fan, intermediate-pressure and 
high-pressure compressors that are rotating at three different speeds. 
In this manner the large diameter fan can rotate at lower speeds to 
satisfy the stress restrictions while the shorter blade high-pressure 
compressor may rotate at higher speeds.&lt;br /&gt;

However the rotational speed of the fan is typically constrained by 
more stringent stress considerations. In a turbofan engine the large 
diameter fan at the front of the engine acts as a single-stage 
compressor. In modern turbofan engines the fan divides the flow with 
most of the air going to the bypass duct to a propelling nozzle and only
 a small portion going into the core. The high root stresses caused by 
the long fan blades are often exacerbated by bird strikes. For 
mechanical reasons a lower limit of root-to-tip ratio of 0.35 is often 
employed. The flow impinging onto the fan is also at a very high Mach 
number since the cruising speed of civil aircraft is typically around M =
 0.83. Supersonic flow inevitably terminates in a shock wave with a 
resulting increase in pressure and entropy over the compressor blades. 
Shock waves reduce the efficiency of the compressor blades since they 
disturb the flow over the profile that lead to boundary layer 
separation. Furthermore, these shock waves may cause unwanted vibrations
 of the fan blades that further reduce the efficiency of the compressor 
and increase noise. Therefore for reasons of efficiency, reducing noise 
and limiting the damage of bird strikes the tip speed of the fan is 
restricted, typically a relative Mach number of 1.6 is considered as the
 upper limit.&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;2.&lt;/strong&gt; The axial speed &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt; &lt;/i&gt;has to be 
maximised to optimise the pressure ratio and reduce the frontal area of 
the engine. Similar to the argument given above the axial speed is 
typically limited by compressibility effects of supersonic flow. As the 
pressure, static temperature and therefore the speed of sound increases 
from stage to stage, the compressibility effects are worst in the first 
stages. For the first stage the air enters axially such that by adding 
the orthogonal velocity vectors &lt;i&gt;U&lt;/i&gt; and &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt;&lt;/i&gt; we get &lt;i&gt;V&lt;sup&gt;2&lt;/sup&gt; = U&lt;sup&gt;2&lt;/sup&gt; + C&lt;sub&gt;a&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;/i&gt; where &lt;i&gt;V &lt;/i&gt;is the speed relative to the blade. In modern engines &lt;i&gt;V &lt;/i&gt;may
 be in the transonic region incurring quite large losses. In this 
respect twin-spool engines have the advantage that the lower-pressure 
compressor rotates at a lower speed, which reduces the compressibility 
problem.&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;3.&lt;/strong&gt; The angle through which the fluid is turned across the rotor blades &lt;i&gt;b&lt;/i&gt;
 is limited by the growth of the boundary layers. Compressor blades are 
aerofoils that function in the same manner as aeroplane wings. Therefore
 as the angle of attack or camber of aerofoil is increased to increase 
the rotation of the flow velocity vector, the adverse pressure gradient 
across the suction surface increases, until at some point the boundary 
layer separates. As the boundary layer separates the effective turning 
angle &lt;i&gt;b&lt;/i&gt; is reduced such that the total pressure increase across the stage reduces.&lt;br /&gt;

&lt;br /&gt;

The limits of &lt;i&gt;U&lt;/i&gt;, &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt;&lt;/i&gt; and b&lt;sub&gt;1&lt;/sub&gt; – b&lt;sub&gt;2&lt;/sub&gt; all place limits on the maximum pressure ratio that can be achieved in an axial compressor. Typical examples are &lt;i&gt;U ≈ 350 m/s&lt;/i&gt;, &lt;i&gt;C&lt;sub&gt;a&lt;/sub&gt; = 200 m/s&lt;/i&gt;, b&lt;sub&gt;1&lt;/sub&gt; – b&lt;sub&gt;2&lt;/sub&gt; &amp;lt; 45°.&lt;br /&gt;

Compressor blades are typically quite thin and are constructed from 
lightweight metallic alloys such as aluminium and titanium. The 
compressor blades feature an aerofoil section as shown in the Figure 
below. The centrifugal forces that act on the airflow are balanced by 
high-pressure air towards the tip of the blade. In order to obtain this 
higher pressure towards the tip the blade must be twisted from root to 
tip in order to change the angle of incidence on the flow, and therefore
 control the pressure variation over the blade.&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;Key References&lt;/strong&gt;&lt;br /&gt;

Rolls-Royce (1996). The Jet Engine.&amp;nbsp;Rolls Royce Technical Publications; 5th ed. edition&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/2113484486993705254/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/jet-engine-detail-design-compressor-in.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2113484486993705254'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2113484486993705254'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/jet-engine-detail-design-compressor-in.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-2115685972755759419</id><published>2013-05-22T02:01:00.002-07:00</published><updated>2017-07-20T22:10:44.996-07:00</updated><category scheme="http://www.blogger.com/atom/ns#" term="Study materials"/><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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       &lt;div class=&quot;post-header fix post-nothumb&quot;&gt;
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         &lt;div class=&quot;post-title fix&quot;&gt;
          
          &lt;h1 class=&quot;entry-title&quot;&gt;
A Brief History of Aircraft Structures&lt;/h1&gt;
&lt;div class=&quot;metabar&quot;&gt;
&lt;em&gt;&lt;span class=&quot;sword&quot;&gt;On&lt;/span&gt; &lt;span class=&quot;date time published&quot; title=&quot;2012-08-19T14:40:48+0000&quot;&gt;August 19, 2012&lt;/span&gt;  · &lt;span class=&quot;sword&quot;&gt;In&lt;/span&gt; &lt;span class=&quot;categories&quot;&gt;&lt;a href=&quot;http://aerospaceengineeringblog.com/category/composite-materials/&quot; rel=&quot;category tag&quot; title=&quot;View all posts in Composite Materials&quot;&gt;Composite Materials&lt;/a&gt;, &lt;a href=&quot;http://aerospaceengineeringblog.com/category/history/&quot; rel=&quot;category tag&quot; title=&quot;View all posts in History&quot;&gt;History&lt;/a&gt;, &lt;a href=&quot;http://aerospaceengineeringblog.com/category/structural-optimisation/&quot; rel=&quot;category tag&quot; title=&quot;View all posts in Novel Materials/Tailored Structures&quot;&gt;Novel Materials/Tailored Structures&lt;/a&gt;&lt;/span&gt;  &lt;/em&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
Aircraft have changed enormously over the last century from 
the early Wright Flyer flown at Kittyhawk to the supersonic SR-71 
Blackbird flown today. Of course the developments in aeronautical 
engineering can be broken down into separate divisions that have 
developed at different rates: a) the aerodynamics, b) power plant 
engineering, c) control, radios and navigation aids, d) airframe 
engineering (e.g. hydraulic/electrical systems, interior fittings etc.),
 and finally e) the structural design. For example, power plants have 
developed in two large steps separated by a series of sudden burst of 
ingenuity. In order to facilitate the first successful flight the Wright
 Brothers had to find a light yet powerful engine system. The next 
stride was the &lt;a href=&quot;http://aerospaceengineeringblog.com/the-birth-of-the-jet-the-engine-that-shrunk-the-world/&quot; title=&quot;The Birth of the Jet: The Engine that Shrunk the World&quot;&gt;ingenious invention of the jet engine&lt;/a&gt;
 prior and during WWII by Sir Frank Whittle and Hans von Ohain. In 
between, the power output of piston engines “increased almost 200 times 
from 12 bhp to over 2000 bhp in just 40 years, with only a ten times 
increase in mass (3) “. As will be outlined in this article, the design 
of aerospace structures on the other hand has only made one fundamental 
stride forward, but this change was sufficient to change the complete 
design principle of modern aircraft. Today however, the strict 
environmental legislation and advent of the composite era may induce 
further leaps in structural design.&lt;br /&gt;

&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_429&quot; style=&quot;width: 278px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/flye-lotech.gif&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-429   &quot; height=&quot;234&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/flye-lotech.gif&quot; title=&quot;Wright Flyer&quot; width=&quot;268&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 1. A schematic drawing of the Wright Flyer (1)&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_430&quot; style=&quot;width: 304px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/blcakbird.jpeg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-430 &quot; height=&quot;231&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/blcakbird.jpeg&quot; title=&quot;SR-71 Blackbird&quot; width=&quot;294&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 2. The modern supersonic SR-71 Blackbird (2)&lt;/div&gt;
&lt;/div&gt;
&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;1) Wire Braced Structures&lt;/strong&gt;&lt;br /&gt;

If we look at the early design of aircraft such as the Wright Flyer 
in Figure 1 there can really be no misunderstanding of the construction 
style. The entire aircraft, including most notably the wings, forward 
and rear structures were all constructed from rectangular frames that 
were prevented from shearing (forming a parallelogram) or collapsing by 
diagonally stretched wire. There were two major innovative thoughts 
behind this design philosophy. Firstly, the idea that two parallel wings
 would facilitate a lighter yet stronger structure than a single wing, 
and secondly, that these two wings could be supported with two light 
wires rather than with a single, thicker wooden member. The structural 
advantage of the biplane construction is that the two wings, vertical 
struts and wires form a deep light beam, which is more resistant to 
bending and twisting than a single wing. Much like a composite sandwich 
beam it can be treated as two stiff outer skins for high bending 
rigidity connected by a lightweight “core” to provide resistance to 
shear and torsion.&lt;br /&gt;

&lt;div&gt;
&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_431&quot; style=&quot;width: 458px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Sopwith-Camel.jpg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-431   &quot; height=&quot;371&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Sopwith-Camel-1024x846.jpg&quot; title=&quot;1917 Sopwith Camel&quot; width=&quot;448&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 3. Cutaway drawing of the 1917 Sopwith Camel (3)&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div&gt;
&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_434&quot; style=&quot;width: 517px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Hawker-Hurricane.jpg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-434   &quot; height=&quot;309&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Hawker-Hurricane-1024x624.jpg&quot; title=&quot;1935 Hawker Hurricane&quot; width=&quot;507&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 4. Cutaway drawing of the 1935 Hawker Hurricane (3)&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div&gt;
The biplane construction with wire bracing was the most notable 
feature of aircraft construction for much of the following years and 
paired nicely with lightweight materials such as bamboo and spruce 
(Figure 3). Wood is a composite of cellulose fibres embedded in a matrix
 of lignin and the early aeronautical engineers knew to take advantage 
of its high specific strength and stiffness. Strangely enough, after the
 era of metals we are now returning back to the composite roots of 
aircraft, albeit in a more advanced fashion. The biplane era lasted 
until the 1930s at which point metal was taking over as the prime 
aerospace material. Initially the design philosophy was not adapted to 
take full advantage of thin sheet metal manufacturing techniques such 
that wooden spars and struts were just replaced by thinner metal tubing.
 Consequently there remained a striking similarity in construction 
between a 1917 (Figure 3) and a 1931 (Figure 4) fighter. Even though 
some thin metal sheets were being used these components generally did 
not carry much load such that the main fuselage structure featured 4 
horizontal longerons supported by vertical struts and wire bracing. This
 so called “Warren Girder” design can also be seen in some of earliest 
monoplane wing constructions such as the 1935 Hawker Hurricane. 
Aeronautical engineers were initially “unsure how to combine the new 
metal construction with a traditional fabric covering (3)” used on 
earlier aircraft. The onset of WWII meant that some safe and 
conservative design decisions were made to facilitate monoplane wings 
and the “Warren Girder” principle was directly copied to the internal 
framework of monoplane wings (Figure 5). These early designs were far 
from optimised and perfectly characterise the transition period between 
wire-frame structures and the semi-monocoque structures we use today.&lt;/div&gt;
&lt;div&gt;
&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_435&quot; style=&quot;width: 332px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Hawker-Hurricane-Wing.jpg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-435  &quot; height=&quot;261&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Hawker-Hurricane-Wing.jpg&quot; title=&quot;Hawker Hurricane Wing Cutaway&quot; width=&quot;322&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 5. The Hawker Hurricane wing construction (3).&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;strong&gt;2) Semi-Monocoque Structures&lt;/strong&gt;&lt;br /&gt;

The internal cross-bracing was initially acceptable for the early 
single or double seater aircraft, but would obviously not provide enough
 room for larger passenger aircrafts. To overcome this, inspiration was 
taken from the long tradition and expertise in boat building which had 
already been applied to construct the fuselages of early wooden flying 
boats. The highest standards of yacht construction at the time featured 
“bent wooden frames and double or triple skins…with a clear varnished 
finish…and presented a much more open and usable fuselage interior (3)”.
 The well-established boat building techniques were thus passed on to 
aircraft construction to produce newer aircraft with very smooth, 
aerodynamic profiles.&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_438&quot; style=&quot;width: 273px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Flying-Boat.jpg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot; wp-image-438  &quot; height=&quot;259&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Flying-Boat.jpg&quot; title=&quot;Flying Boat Construction&quot; width=&quot;263&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 6. Semi monocoque fuselage construction of an early wooden flying boat (4)&lt;/div&gt;
&lt;/div&gt;
The major advantage of this type of construction is that the outer 
skin of the fuselage and wing no longer just define the shape and 
aerodynamic profile of the aircraft, but become an active load-carrying 
member of the structure as well. Thus, the structure becomes 
“multifunctional” and more efficient, unlike the braced fuselage which 
would be just as strong without the fabric covering the girders. As a 
consequence the whole structure is generally at a uniform and lower 
stress level, reducing stress concentrations and giving better fatigue 
life. Finally, as the majority of the material is located at the outer 
surface of the structure the second and polar moments of area, and 
therefore the bending and torsional rigidities are much increased. On 
the other hand, the thin-skinned construction means that compression and
 shear buckling become the most likely forms of failure. In order to 
increase the critical buckling loads the skins are stiffened by 
stringers and broken up into smaller sections by spars and ribs.&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_441&quot; style=&quot;width: 397px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Airplane-Wing-Part-Diagram-Terminology.png&quot;&gt;&lt;img alt=&quot;&quot; class=&quot; wp-image-441   &quot; height=&quot;245&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Airplane-Wing-Part-Diagram-Terminology.png&quot; title=&quot;Semi Monocoque Wing&quot; width=&quot;387&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 7. Components of a semi monocoque wing (5)&lt;/div&gt;
&lt;/div&gt;
Because the external skin is now a working part of the structure this type of construction became to be known as&amp;nbsp;&lt;em&gt;stressed skin &lt;/em&gt;or &lt;em&gt;semi-monocoque&lt;/em&gt;,
 where monocoque means &amp;nbsp;”shell in one piece” and “semi” is an english 
addition to describe the discrete discontinuities of internal 
stiffeners. The adoption of the&amp;nbsp;&lt;em&gt;semi-monocoque &lt;/em&gt;construction&amp;nbsp;and
 a change from wood to metal naturally coincided since sheet metal 
production allowed a variety of thin skins to be easily manufactured 
quite cheaply, with better surface finish and superior material 
properties. Furthermore, metal construction was conducive to riveting 
which would overcome the adhesive problems of &lt;a href=&quot;http://aerospaceengineeringblog.com/an-introduction-to-composite-materials/&quot; title=&quot;An Introduction to Composite Materials&quot;&gt;early wooden&amp;nbsp;semi-monocoque aircraft such as the deHavilland Mosquito.&lt;/a&gt;&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_436&quot; style=&quot;width: 505px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/A400m.jpg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot; wp-image-436   &quot; height=&quot;347&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/A400m.jpg&quot; title=&quot;A400m Cutaway Drawing&quot; width=&quot;495&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 8. Cutaway Drawing of the recently released A400M aircraft (7).&lt;/div&gt;
&lt;/div&gt;
Figure 8 shows the typical construction of a modern aircraft. There 
have been numerous different structural arrangements over the past 
number of years but all generally feature some sort of vertical 
stiffener (ribs in the wings and rings in the fuselage) and longitudinal
 stiffener (called stringers). Over the years the main driver has been 
towards a) a reduction in the number of rivets by reverting to bonded 
assembly or ideally manufacturing separate components as a single piece 
and b) understanding &lt;a href=&quot;http://aerospaceengineeringblog.com/the-dehavilland-comet-crash/&quot; title=&quot;The DeHavilland Comet Crash&quot;&gt;the effects and growth of cracks under static and fatigue loading&lt;/a&gt;&amp;nbsp;by
 building structures that can easily be inspected or have multiple 
redundancies (load paths). The design and manufacturing methods of&lt;em&gt; semi-monocoque&lt;/em&gt;
 aircraft are now so automated that the development of a new aluminium, 
medium sized airliner “could be regarded as a routine exercise (1)”. 
However, the continuing legislative pressure to reduce weight and fuel 
consumption provides enough incentive for further development.&lt;br /&gt;

&lt;strong&gt;3) Sandwich Structures and Composite Materials&lt;/strong&gt;&lt;br /&gt;

One of the major disadvantages of thin-skinned structures is their 
lack of rigidity under compressive loading which gives them a tendency 
to buckle. A sheet of paper nicely illustrates this point, since it is 
quite strong in tension but will provide no support under compression. 
One way of improving the rigidity of thin panels is by increasing the 
bending stiffness with the aid of external stiffeners, which at the same
 time break the structure up into smaller sections. The critical 
buckling load is a function of the square of the width of the plate over
 which the load is applied. Therefore skins can be made 4 times stronger
 in buckling by just cutting the width in half. As a wing bends upwards 
the main compressive loads act on the top skin along the length of the 
wing and therefore a large number of stringers are visible across the 
width.&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_443&quot; style=&quot;width: 412px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Untitled.png&quot;&gt;&lt;img alt=&quot;&quot; class=&quot; wp-image-443  &quot; height=&quot;178&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Untitled.png&quot; title=&quot;Buckling of a blade stiffened wing panel&quot; width=&quot;402&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig.
 8. Buckling analysis of a stiffened wing panel. The stiffeners break 
the buckling mode shapes into smaller wavelengths that require higher 
energy to form compared to a single wave (7)&lt;/div&gt;
&lt;/div&gt;
Another technique to provide more rigidity is sandwich construction. 
This generally features a very lightweight core, such as a honeycomb 
lattice or a foam, sandwiched between two thin yet stiff outer panels. 
Here the role of the sandwich core is to carry any shear loads and 
separate the two skins as far as possible. The second moment of area is a
 function of the cube of the depth and therefore the bending rigidity is
 greatly increased with this technique. Ideally, in this manner it would
 be possible to design an entire fuselage without any internal rings or 
stringers and the Beech Starship is an excellent example of a successful
 application. However, there are problems of forming honeycomb cores 
onto doubly curved shells since the material is susceptible to strong 
anticlastic curvature, forming a saddle shape when bent in one 
direction. Furthermore, there are problems with condensation and water 
ingress into the honeycomb cells and the ability to guarantee a good 
bond surface between the core and the outer skins. There is the 
possibility to use foam cores instead, but these tend to be heavier with
 lower mechanical properties. Perhaps the current trend is away from 
sandwich construction (10).&lt;br /&gt;

&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_444&quot; style=&quot;width: 299px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Steinbichler_Shearography_Honeycomb_with_CFRP_Top_Layer_Artificial_failures_that_simulate_layer-core_delaminations_Material.jpeg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-444 &quot; height=&quot;217&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Steinbichler_Shearography_Honeycomb_with_CFRP_Top_Layer_Artificial_failures_that_simulate_layer-core_delaminations_Material-1024x768.jpeg&quot; title=&quot;Carbon Honeycomb Sandwich Panel&quot; width=&quot;289&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 9. A carbon fibre composite/honeycomb sandwich panel (9)&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_440&quot; style=&quot;width: 284px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/NC-6-Over-Foggy-Hills.jpeg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-440    &quot; height=&quot;208&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/NC-6-Over-Foggy-Hills.jpeg&quot; title=&quot;Beech Starship&quot; width=&quot;274&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 10. The Beech Starship whose fuselage was design using sandwich construction with minimal internal bulkheads and ribs (8)&lt;/div&gt;
&lt;/div&gt;
&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

One of the major applications of honeycomb structures has been in combination with &lt;a href=&quot;http://aerospaceengineeringblog.com/an-introduction-to-composite-materials/&quot; title=&quot;An Introduction to Composite Materials&quot;&gt;composite materials&lt;/a&gt;.
 Stiff carbon composite panels are the ideal candidate for the outer 
skins and the whole assembly can be co-cured together in an autoclave 
without having to perform any secondary bonding operations. Furthermore,
 the incredible specific strength and stiffness of carbon composites 
makes this combination an &amp;nbsp;ultra lightweight yet resilient structure for
 aerospace applications. Indeed, we are now at the start of the “black” 
carbon age in commercial aircraft design.&amp;nbsp;Apart from their excellent 
specific strength and stiffness properties composites exhibit the 
ability to tailor optimum mechanical properties by orientating the 
majority of plies in the direction of the load and allowing for less 
material waste during manufacture. &amp;nbsp;As a result, the first generation of
 commercial aircraft that contain large proportions of composite parts, 
such as the Boeing 787 Dreamliner and Airbus A350 XWB, are planned to 
enter service throughout the next years.&lt;br /&gt;

&lt;div class=&quot;wp-caption aligncenter&quot; id=&quot;attachment_445&quot; style=&quot;width: 353px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/delamination.jpeg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;size-full wp-image-445 &quot; height=&quot;220&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/delamination.jpeg&quot; title=&quot;Delamination&quot; width=&quot;343&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 11. Considerable delamination leading to catastrophic failure (11)&lt;/div&gt;
&lt;/div&gt;
Considerable effort has been made to mature composite technology in order to reduce manufacturing costs, &lt;a href=&quot;http://aerospaceengineeringblog.com/composite-manufacturing-autoclave-variability/&quot; title=&quot;Composite Manufacturing – Autoclave Variability&quot;&gt;guarantee reliably high quality laminates&lt;/a&gt;, understand the &lt;a href=&quot;http://www.theengineer.co.uk/sectors/aerospace/in-depth/in-the-line-of-fibre/1012862.article&quot;&gt;highly complex failure criteria&lt;/a&gt;&amp;nbsp;and built &lt;a href=&quot;http://aerospaceengineeringblog.com/carbon-nanotube-hierarchical-composites-for-interlaminar-strengthening/&quot; title=&quot;Carbon Nanotube Hierarchical Composites for Interlaminar Strengthening&quot;&gt;hierarchical&lt;/a&gt;, &lt;a href=&quot;http://aerospaceengineeringblog.com/smart-materials-application-variable-geometry-chevron-for-noise-reduction/&quot; title=&quot;Smart Materials Application: Variable Geometry Chevron for Noise Reduction&quot;&gt;multifunctional&lt;/a&gt; or &lt;a href=&quot;http://www.bristol.ac.uk/composites/news/2012/67.html&quot;&gt;self-healing structures&lt;/a&gt;.
 &amp;nbsp;One of the major shortcomings is that the structural advantages of 
fibre-reinforced plastics must be viewed with respect to applications 
where the primary loads are aligned with the fibre direction. However, 
if a composite plate is subjected to significant out-of-plane stresses 
subsurface delaminations may develop between layers due to the weak 
through-thickness cohesive strength of the composite. These intralaminar
 delaminations are a significant problem as they are difficult to detect
 by visual inspection and may reduce the compressive strength of the 
laminate by up to 60%.&lt;br /&gt;

&lt;strong&gt;4) Novel Designs&lt;/strong&gt;&lt;br /&gt;

With environmental legislation becoming ever so strict it is adamant 
that new concepts for lightweight and fuel efficient aircraft are found 
swiftly. Although the pressure on developing advanced composite 
materials is high it must be remembered that 100 years of innovation 
were required to reach the stage that large metal &lt;em&gt;semi-monocoque&lt;/em&gt;
 structures could be manufactured in the 1940s and another 30 years to 
fully understand all failure criteria. Thus we may still require 
significant research and development before all current issues with 
composite materials are resolved. Apart from carbon fibre and other 
composites other researchers have been looking into completely 
redefining the shape of aircraft. Researchers at MIT have been 
developing the &lt;a href=&quot;http://web.mit.edu/aeroastro/news/magazine/aeroastro-no4/silentaircraft.html&quot;&gt;blended wing concept&lt;/a&gt;&amp;nbsp;and NASA are exploring the technology of &lt;a href=&quot;http://www.engineeringmotion.com/videos/605/nasa-morphing-aircraft&quot;&gt;morphing or shape-changing aircraft&lt;/a&gt;, taking inspiration directly from nature.&lt;br /&gt;

&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_448&quot; style=&quot;width: 348px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/silent-aircraft.jpeg&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-448  &quot; height=&quot;108&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/silent-aircraft.jpeg&quot; title=&quot;MIT Silent Aircraft&quot; width=&quot;338&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 12. Illustration of the MIT Silent Aircraft concept (12).&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;wp-caption alignleft&quot; id=&quot;attachment_450&quot; style=&quot;width: 237px;&quot;&gt;
&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Untitled1.png&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;wp-image-450 &quot; height=&quot;136&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/Untitled1.png&quot; title=&quot;NASA Morphing Wing Aircraft&quot; width=&quot;227&quot; /&gt;&lt;/a&gt;&lt;div class=&quot;wp-caption-text&quot;&gt;
Fig. 13. NASA morphing wing aircraft (13)&lt;/div&gt;
&lt;/div&gt;
&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Whatever the final solution might look like the next 5o years in 
aerospace engineering will be incredibly innovative, ground-breaking and
 an exciting industry to be part of!&lt;br /&gt;

&lt;br /&gt;

&lt;a href=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/f-_0000_AE-logo-c2-e1345059131185.png&quot;&gt;&lt;img alt=&quot;&quot; class=&quot;alignnone size-full wp-image-366&quot; height=&quot;102&quot; src=&quot;http://aerospaceengineeringblog.com/wp-content/uploads/2012/08/f-_0000_AE-logo-c2-e1345059131185.png&quot; title=&quot;Aerospace Engineering Logo 2&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;References&lt;/strong&gt;&lt;br /&gt;

(3) Cutler, John (1992).&amp;nbsp;&lt;em&gt;Understanding Aircraft Structures&lt;/em&gt;. 2nd Edition. Blackwell Scientific Publications, Oxford.&lt;br /&gt;

(10)&amp;nbsp;Potter, Kevin (1996).&amp;nbsp;&lt;em&gt;An Introduction to Composite Products: Design, Development and Manufacture.&amp;nbsp;&lt;/em&gt;Springer, 5th Ed. Chapman &amp;amp; Hall, London.&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;Images&lt;/strong&gt;&lt;br /&gt;

(1)&amp;nbsp;http://www.pbs.org/wgbh/nova/wright/images/flye-lotech.gif&lt;br /&gt;

(2)&amp;nbsp;http://thexodirectory.com/wp-content/uploads/2011/05/Air-to-air-overhead-front-view-of-an-SR-71A-460×361.jpg&lt;br /&gt;

(4)&amp;nbsp;http://imgc.artprintimages.com/images/art-print/j-r-eyerman-workmen-building-flying-boat-that-was-designed-by-millionaire-howard-r-hughes_i-G-37-3793-OAAIF00Z.jpg&lt;br /&gt;

(5)&amp;nbsp;http://www.nomenclaturo.com/wp-content/uploads/Airplane-Wing-Part-Diagram-Terminology.png&lt;br /&gt;

(6) http://pds13.egloos.com/pds/200906/24/60/a0118060_4a4194709ef22.jpg&lt;br /&gt;

(7)&amp;nbsp;http://www.dnv.com/binaries/PULS-buckling_tcm4-284864.JPG&lt;br /&gt;

(8)&amp;nbsp;http://www.bobscherer.com/Images/Pages/Starship/Starship%20page/NC-6%20Over%20Foggy%20Hills.jpg&lt;br /&gt;

(9)http://upload.wikimedia.org/wikipedia/commons/3/3d/Steinbichler_Shearography_Honeycomb_with_CFRP_Top_Layer_Artificial_failures_that_simulate_layer-core_delaminations_Material.jpg&lt;br /&gt;

(11)&amp;nbsp;http://en.wikipedia.org/wiki/File:Delamination-CFRP.jpg&lt;br /&gt;

(12) http://silentaircraft.org/&lt;br /&gt;

(13)&amp;nbsp;http://www.espaciolutacoot.com.mx/images/postcard/large/nave1.jpg&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/2115685972755759419/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/a-brief-history-of-aircraft-structures.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2115685972755759419'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2115685972755759419'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2013/05/a-brief-history-of-aircraft-structures.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-1953669485504702629</id><published>2012-09-24T08:59:00.001-07:00</published><updated>2012-09-24T08:59:54.054-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;br /&gt;
&lt;h1 class=&quot;contentheading&quot; style=&quot;color: #cc0000; font-family: verdana; font-size: 18px; line-height: 21px; margin: 0px 0px 15px; padding: 0px;&quot;&gt;
Shear Lag Effect&lt;/h1&gt;
&lt;div class=&quot;FontN&quot; style=&quot;font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; line-height: 16px; text-align: justify;&quot;&gt;
Plane Beam theory assumes plane section before the bending remains plane after the bending which implies structure should have infinite lateral stiffness or there is no shear force acting at the beam section. Hence bending stress diagram for a beam has to be in a linear pattern.&lt;br /&gt;&lt;br /&gt;In the event of earthquake and extreme wind pressure slender structures (The term Slender has a different meaning for box girder, it means ratio of width to span of a box girder,) often experience tremendous amount of lateral load, because most of the mass being lumped primarily at the first degree of freedom. When slender structures, of which span to width ratio is high, experience larger lateral loads they become prone to non-linear bending and shear stress distribution across the cross-section (Fig. 01). In a box girder, a large shear flow is transmitted from vertical webs to the horizontal flanges, which causes in-plane shear deformation of the flanges and results in unpredicted extra longitudinal displacement at the web-flange junction [1]. Due to which central portion of the flange lag behind that of the web for the response quantities. This phenomenon of lagging is called as Shear Lag. Shear Lag effect is relevant to any slender box element that is loaded laterally such as airplane wing structure and box girder bridges.&lt;br /&gt;&lt;br /&gt;&lt;div align=&quot;center&quot;&gt;
&lt;table border=&quot;0&quot; style=&quot;width: 490px;&quot;&gt;&lt;tbody&gt;
&lt;tr&gt;&lt;td style=&quot;border: 0px; font-family: verdana;&quot;&gt;&lt;div align=&quot;center&quot;&gt;
&lt;img alt=&quot;Shear Lag Effect&quot; src=&quot;http://www.nbmcw.com/images/nbm-media/Articles/Computer-Software/20071-shear-lag-1.jpg&quot; /&gt;&lt;/div&gt;
&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td bgcolor=&quot;#ffcc66&quot; style=&quot;border: 0px; font-family: Verdana; font-size: 10px;&quot;&gt;&lt;div align=&quot;center&quot;&gt;
Figure 1: Schematic Showing Shear-Lag Effect in Box Girder (a) Shear Stress Distribution Across the C/S of Box Girder (b) Non-uniformity of Longitudinal Stress (c) Out-of-plane Warping&lt;/div&gt;
&lt;/td&gt;&lt;/tr&gt;
&lt;/tbody&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;br /&gt;While relying on plane beam theory which underestimates shear lag effect in box girder, gives unre- liable response quantities (Shear, Moment, Displacements etc.).&lt;br /&gt;&lt;br /&gt;&lt;div align=&quot;right&quot; style=&quot;float: right; width: 250px;&quot;&gt;
&lt;table border=&quot;0&quot; style=&quot;width: 240px;&quot;&gt;&lt;tbody&gt;
&lt;tr&gt;&lt;td style=&quot;border: 0px; font-family: verdana;&quot;&gt;&lt;div align=&quot;center&quot;&gt;
&lt;img alt=&quot;Shear Lag Effect&quot; src=&quot;http://www.nbmcw.com/images/nbm-media/Articles/Computer-Software/20071-shear-lag-2.jpg&quot; /&gt;&lt;/div&gt;
&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td bgcolor=&quot;#ffcc66&quot; style=&quot;border: 0px; font-family: Verdana; font-size: 10px;&quot;&gt;&lt;div align=&quot;center&quot;&gt;
Figure 2: Negative Shear Lag&lt;/div&gt;
&lt;/td&gt;&lt;/tr&gt;
&lt;/tbody&gt;&lt;/table&gt;
&lt;/div&gt;
It has also been observed that there is another phenomenon exist which can be named as “Negative Shear Lag” (Fig.02). For a cantilever box girder researchers observed that for the one fourth the cantilever span length stress at the edge of the box girder is less as compared to stresses at the central flange portion [2].&lt;br /&gt;&lt;br /&gt;Significant points about Shear Lag:-&lt;ul class=&quot;FontN&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Conventional Plane Beam Theory underestimates the shear lag effect.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Shear Lag results in non-linear displacement distribution across the cross section.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Due to shear lag longitudinal displacement of the Fig. 02 Negative Shear Lag box girder increases.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;It also leads to stress concentration at web-flange junction and results in out-of-plane warping.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;In cable stayed bridges, forces due to shear lag effect observed maximum at pylon and reduces away from it [3].&lt;/li&gt;
&lt;li&gt;Shear Lag effect increases with the width of the Box Girder i.e. slender section has increased.&lt;/li&gt;
&lt;/ul&gt;
Now a days many bridges are designed as a “Wide Single Cell Box” girder bridge which generally possesses high slenderness ratio and resulting in attracting a higher lateral forces. Recent bridge designs theories find it difficult to maintain slenderness ratio (Width to span ratio) less, as it increases the cost of the project tremendously. To counteract Shear Lag effect following remedial measures can be considered:&lt;ul class=&quot;FontN&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Number of the shells in the box girder would be increased.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Slenderness ratio of the bridge would be reduced.&lt;/li&gt;
&lt;li&gt;Height of the pylon would be reduced. It has been observed that extra-dosed bridges attract lesser lateral forces and hence minimizes shear lag effect.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 class=&quot;FontSH&quot; style=&quot;font-size: 12px; line-height: 12px; margin: 25px 0px 10px;&quot;&gt;
References&lt;/h2&gt;
&lt;ul class=&quot;FontN&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Bazant, P., Vladimir, K., “Shear lag effect and uncertainty in concrete box girder creep”, ASCE&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Leonard, J., “Investigation of shear lag effect in highrise building with dia-grid system”, Thesis report,&lt;/li&gt;
&lt;li&gt;Gao-feng, WU., Hong, XU., “Theoretical and experimental study on shear lag effect of partially cable stayed bridge”, Journal of Zhejiang University Science.&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
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src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-5821701133478500041</id><published>2012-09-24T08:49:00.003-07:00</published><updated>2012-09-24T08:49:55.863-07:00</updated><title type='text'>Electrical and Structure Properties of High-κ Barium Tantalite and Aluminum Oxide Interface with Zinc Oxide for Applications in Transparent Thin Film Transistors : UNT Digital Library</title><content type='html'>&lt;a href=&quot;http://digital.library.unt.edu/ark:/67531/metadc84233/?q=silver+oxide#.UGCBGuSkVK8.blogger&quot;&gt;Electrical and Structure Properties of High-κ Barium Tantalite and Aluminum Oxide Interface with Zinc Oxide for Applications in Transparent Thin Film Transistors : UNT Digital Library&lt;/a&gt;</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/5821701133478500041/comments/default' title='Post Comments'/><link 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src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-4193587478339825113</id><published>2012-09-24T08:49:00.001-07:00</published><updated>2012-09-24T08:49:53.122-07:00</updated><title type='text'>JAC: A Journal of Composition Theory, Volume 16, Number 2, 1996 : UNT Digital Library</title><content type='html'>&lt;a href=&quot;http://digital.library.unt.edu/ark:/67531/metadc28616/?q=aeronautical#.UGCBDNTx9TE.blogger&quot;&gt;JAC: A Journal of Composition Theory, Volume 16, Number 2, 1996 : UNT Digital Library&lt;/a&gt;</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/4193587478339825113/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/jac-journal-of-composition-theory.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' 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application in silver toxicity studies : UNT Digital Library</title><content type='html'>&lt;a href=&quot;http://digital.library.unt.edu/ark:/67531/metadc2613/?q=silver+oxide#.UGCAsbOJPUY.blogger&quot;&gt;The performance of silicon based sensor and its application in silver toxicity studies : UNT Digital Library&lt;/a&gt;</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/9038682721887938796/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/the-performance-of-silicon-based-sensor.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/9038682721887938796'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/9038682721887938796'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/the-performance-of-silicon-based-sensor.html' title='The performance of silicon based sensor and its application in silver toxicity studies : UNT Digital Library'/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-8358192511046023401</id><published>2012-09-24T08:47:00.001-07:00</published><updated>2012-09-24T08:47:51.117-07:00</updated><title type='text'>Electrical and Structure Properties of High-κ Barium Tantalite and Aluminum Oxide Interface with Zinc Oxide for Applications in Transparent Thin Film Transistors : UNT Digital Library</title><content type='html'>&lt;a 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href=&quot;http://digital.library.unt.edu/ark:/67531/metadc54569/m1/6/#.UGCACA67vpw.blogger&quot;&gt;Effect of shear lag on bending vibration of box beams, Sequence: 6 | UNT Digital Library&lt;/a&gt;</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/1233211461798724411/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/effect-of-shear-lag-on-bending.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/1233211461798724411'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/1233211461798724411'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/effect-of-shear-lag-on-bending.html' title='Effect of shear lag on bending vibration of box beams, Sequence: 6 | UNT Digital Library'/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-3833202967306930809</id><published>2012-09-18T11:14:00.002-07:00</published><updated>2012-09-18T11:14:25.043-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;br /&gt;
&lt;h3 class=&quot;post-title entry-title&quot; style=&quot;background-color: white; color: #333333; font-family: Times, serif; font-size: 18px; line-height: 1.1em; margin: 0px; padding: 0px;&quot;&gt;
&lt;a href=&quot;http://free-ebook-download-links.blogspot.in/2008/08/free-books-on-mechanical-engineering.html&quot; style=&quot;color: #333333; text-decoration: none;&quot;&gt;Free Books on Mechanical Engineering&lt;/a&gt;&lt;/h3&gt;
&lt;div class=&quot;post-header-line-1&quot; style=&quot;background-color: white; color: #333333; font-family: Times, serif; font-size: 14px; line-height: 1.3em; margin: 0px 0px 0.75em;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;post-body entry-content&quot; style=&quot;background-color: white; color: #333333; font-family: Times, serif; font-size: 14px; line-height: 1.3em; margin: 0px 0px 0.75em;&quot;&gt;
&lt;ol&gt;
&lt;li&gt;Handbook of thermodynamics, heat transfer and fluid flow-&amp;nbsp;&lt;a href=&quot;http://www.artikel-software.com/file/thermo-v1.pdf&quot; style=&quot;color: #0066cc; font-weight: bold;&quot; wrc_done=&quot;true&quot;&gt;Vol. 1&lt;/a&gt;&amp;nbsp;,&amp;nbsp;&lt;a href=&quot;http://www.artikel-software.com/file/thermo-v2.pdf&quot; style=&quot;color: #0066cc; font-weight: bold;&quot; wrc_done=&quot;true&quot;&gt;Vol. 2&lt;/a&gt;&amp;nbsp;,&amp;nbsp;&lt;a href=&quot;http://www.artikel-software.com/file/thermo-v3.pdf&quot; style=&quot;color: #0066cc; font-weight: bold;&quot; wrc_done=&quot;true&quot;&gt;Vol. 3&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.artikel-software.com/file/Basic_Hydraulic_Systems_and_Components.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Basic Hydraulic Systems and Components&lt;/a&gt;&amp;nbsp;-&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.artikel-software.com/file/ht.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Heat Transfer Textbook&lt;/a&gt;&amp;nbsp;- John H. Lienhard IV and John H. Lienhard V&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.potto.org/FM/fluidMechanics.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Basics of Fluid Mechanics&lt;/a&gt;&amp;nbsp;-Genick Bar&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.potto.org/gasDynamics.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Fundamentals of Compressible Flow Mechanics&lt;/a&gt;&amp;nbsp;-Genick Bar–Meir&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.potto.org/dieCasting.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Fundamentals of Die Casting Design&lt;/a&gt;&amp;nbsp;&amp;nbsp;- Genick Bar–Meir&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.nd.edu/~msen/Teaching/ThConv/THT.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Turbulent Heat Transfer&lt;/a&gt;&amp;nbsp;-Scott Stolpa&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/51/2/cabook.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Fundamentals of Multiphase Flow&lt;/a&gt;&amp;nbsp;-Brennen, Christopher Earls&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/226/&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Feedback Systems: An Introduction for Scientists and Engineers&lt;/a&gt;&amp;nbsp;-Åström, Karl Johan and Murray, Richard M.&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/1/5/BUBBOOK.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Cavitation and Bubble Dynamics&lt;/a&gt;&amp;nbsp;- Brennen, Christopher Earls&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/22/3/pumbook.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Hydrodynamics of Pumps&lt;/a&gt;&amp;nbsp;- Brennen, Christopher Earls&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/53/1/Housner-HudsonDyn80.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Applied Mechanics Dynamics&lt;/a&gt;&amp;nbsp;- Housner, George W. and Hudson, Donald E.&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://caltechbook.library.caltech.edu/54/1/Housner-Vreeland65.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;The Analysis of Stress and Deformation&lt;/a&gt;&amp;nbsp;- Housner, George W. and Vreeland, Thad, Jr.&lt;/li&gt;
&lt;li&gt;Fundamental Handbook of Mechanical Science-&amp;nbsp;&lt;a href=&quot;http://www.hss.energy.gov/NuclearSafety/techstds/standard/hdbk1018/h1018v1.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Vol.1&lt;/a&gt;&amp;nbsp;&amp;nbsp;,&amp;nbsp;&lt;a href=&quot;http://www.hss.energy.gov/NuclearSafety/techstds/standard/hdbk1018/h1018v2.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Vol. 2&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://artikel-software.com/file/Uncertainty%20and%20Dimensional%20Calibration.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Uncertainty and Dimensional Calibration&lt;/a&gt;&amp;nbsp;-Ted Doiron and John Stoup&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.outokumpu.com/applications/upload/pubs_113155252.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Stainless Steels - Their properties and their suitability for welding&lt;/a&gt;&amp;nbsp;&amp;nbsp;-Björn Holmberg&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.supsalv.org/pdf/cut_weld.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Underwater Cutting and Welding Manual&lt;/a&gt;&amp;nbsp;-&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://artikel-software.com/file/noise_and_vibration_control.pdf&quot; style=&quot;color: #0066cc;&quot; wrc_done=&quot;true&quot;&gt;Noise and Vibration Control for Buildings&lt;/a&gt;&amp;nbsp;-&lt;/li&gt;
&lt;/ol&gt;
&lt;/div&gt;
&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/3833202967306930809/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/free-books-on-mechanical-engineering.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3833202967306930809'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3833202967306930809'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/free-books-on-mechanical-engineering.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-4547167390172825404</id><published>2012-09-18T11:12:00.001-07:00</published><updated>2012-09-18T11:12:42.994-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/4547167390172825404/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/what-is-nanotechnology-pdf-nano-books.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/4547167390172825404'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/4547167390172825404'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/what-is-nanotechnology-pdf-nano-books.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-3067846885767414569</id><published>2012-09-18T11:09:00.001-07:00</published><updated>2012-09-18T11:09:18.647-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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&lt;a href=&quot;http://engineeringbooks-ravi.blogspot.in/2009/07/radio-and-electronics-and-cook-book.html&quot; style=&quot;color: #ffa575; text-decoration: none;&quot;&gt;RADIO AND ELECTRONICS AND COOK BOOK&lt;/a&gt;&lt;/h3&gt;
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&lt;a href=&quot;http://engineeringbooks-ravi.blogspot.in/2009/07/dynamics-of-machines.html&quot; style=&quot;color: #ffa575; text-decoration: none;&quot;&gt;DYNAMICS OF MACHINES&lt;/a&gt;&lt;/h3&gt;
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</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/3067846885767414569/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/analytical-chemistry-vogels.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3067846885767414569'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/3067846885767414569'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/analytical-chemistry-vogels.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-zaTY9KlT_gjY8c_kBSGsXkyS19d0YCnAXAkRvpY3n4klFQt6RbmbflCLnpZhWeEI3zTaTmD7XQbYGUFP0BrAezn-4aC65jr3Hsj-EiEkBV8NIwswlREURsr7IHJijEJXIru9sWmUAwc/s72-c/Textbook+of+Quantitative+Chemical+Anlysis+Vogel+5th+edition.JPG" height="72" width="72"/><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-8471311538372081259</id><published>2012-09-18T11:08:00.001-07:00</published><updated>2012-09-18T11:08:18.935-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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&lt;span style=&quot;background-color: transparent;&quot;&gt;Mechanical engineering books | Download for free&lt;/span&gt;&lt;/form&gt;
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The free engineering textbooks in this category are designed to help mechanical engineers to prepare for their exams. Become a master in topics such as computational fluid dynamics, heat transfer, control engineering or automation and robotics engineering with our mechanical engineering books.&lt;/div&gt;
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Get the entire “Mechanical engineering books | Download for free” category&lt;/h3&gt;
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Engineering Mathematics: YouTube Workbook&lt;/h3&gt;
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&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/concepts-in-electric-circuits&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
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Concepts in Electric Circuits&lt;/h3&gt;
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&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/engineering-thermodynamics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
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Engineering Thermodynamics&lt;/h3&gt;
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Wind Turbines&lt;/h3&gt;
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&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/control-engineering-matlab&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Control Engineering&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/control-engineering-matlab.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/manufacturing-processes-and-materials-exercises&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Manufacturing Processes and Materials: Exercises&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/manufacturing-processes-and-materials-exercises.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/heat-transfer&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Heat Transfer&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/heat-transfer.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
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&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/engineering-fluid-mechanics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Engineering Fluid Mechanics&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/engineering-fluid-mechanics.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/introduction-to-digital-signal-and-system-analysis&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Introduction to Digital Signal and System Analysis&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/introduction-to-digital-signal-and-system-analysis.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/key-concepts-in-turbo-machinery&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Basic Concepts in Turbomachinery&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/key-concepts-in-turbo-machinery.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/a-first-course-in-fluid-mechanics-for-engineers&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
A First Course in Fluid Mechanics for Engineers&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/a-first-course-in-fluid-mechanics-for-engineers.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/automation-and-robotics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Automation and Robotics&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/automation-and-robotics.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/cad-cam-rapid-prototyping-application-evaluation&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
CAD-CAM &amp;amp; Rapid prototyping Application Evaluation&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/cad-cam-rapid-prototyping-application-evaluation.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/computational-fluid-dynamics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Computational Fluid Dynamics&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/computational-fluid-dynamics.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/a-first-course-on-aerodynamics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
A First Course on Aerodynamics&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/a-first-course-on-aerodynamics.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/heat-transfer-exercise-book&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Heat Transfer: Exercises&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/heat-transfer-exercise-book.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/hydrocarbons&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Hydrocarbons&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/hydrocarbons.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/chemical-thermodynamics&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Chemical Thermodynamics&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/chemical-thermodynamics.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/introduction-to-polymer-science-and-technology&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Introduction to Polymer Science and Technology&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/introduction-to-polymer-science-and-technology.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;English&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Format:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;PDF&lt;/dd&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Price:&lt;/dt&gt;
&lt;dd style=&quot;border: 0px; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 55px;&quot;&gt;Free&lt;/dd&gt;&lt;/dl&gt;
&lt;strong class=&quot;sec&quot; style=&quot;border: 0px; bottom: 8px; font: inherit; margin: 0px; padding: 0px; position: absolute; right: 8px; vertical-align: baseline;&quot;&gt;Download!&lt;/strong&gt;&lt;/a&gt;&lt;/article&gt;&lt;article class=&quot;book&quot; style=&quot;background-color: #f7f7f7; border-bottom-left-radius: 3px; border-bottom-right-radius: 3px; border-top-left-radius: 3px; border-top-right-radius: 3px; border: 1px solid rgb(247, 154, 75); float: left; font: inherit; height: 126px; margin: 0px 15px 15px 0px; padding: 10px; position: relative; vertical-align: baseline; width: 195px;&quot;&gt;&lt;a href=&quot;http://bookboon.com/en/textbooks/mechanics/introductory-finite-difference-methods-for-pdes&quot; style=&quot;border: 0px; color: #333333; display: block; font-size: 9pt; font: inherit; margin: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;&quot;&gt;&lt;div class=&quot;header&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;&quot;&gt;
&lt;h3 style=&quot;border: 0px; font: inherit; height: 28px; margin: 0px; overflow: hidden; padding: 0px; vertical-align: baseline;&quot;&gt;
Introductory Finite Difference Methods for PDEs&lt;/h3&gt;
&lt;/div&gt;
&lt;img alt=&quot;&quot; src=&quot;http://bookboon.com/content/images/thumb-7c4195a5d3b3e111965c782bcb67bd3e-90/introductory-finite-difference-methods-for-pdes.jpg&quot; style=&quot;border: 0px; font-size: 12px; font: inherit; margin: 10px 0px 0px; padding: 0px; vertical-align: baseline; width: 64px;&quot; /&gt;
&lt;dl style=&quot;border: 0px; font-size: 11px; font: inherit; left: 90px; margin: 6px 0px; padding: 0px; position: absolute; top: 40px; vertical-align: baseline;&quot;&gt;
&lt;dt style=&quot;border: 0px; clear: left; float: left; font-size: 11px; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 60px;&quot;&gt;Language&lt;/dt&gt;
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An Introduction to Nonlinearity in Control Systems&lt;/h3&gt;
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A Wet Look At Climate Change&lt;/h3&gt;
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Introductory Finite Volume Methods for PDEs&lt;/h3&gt;
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Theory of waves in materials&lt;/h3&gt;
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</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/8471311538372081259/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/mechanical-engineering-books-download.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8471311538372081259'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/8471311538372081259'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/mechanical-engineering-books-download.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-2718447895551295416</id><published>2012-09-18T11:05:00.002-07:00</published><updated>2012-09-18T11:05:22.198-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$19.90&lt;/strong&gt;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&amp;nbsp;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;br /&gt;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&amp;nbsp;&lt;/li&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0830639012=&quot;&quot; href=&quot;http://www.amazon.com/Illustrated-Guide-Aerodynamics-Hubert-Smith/dp/0830639012%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0830639012&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51NQcZCYfFL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;125&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$17.21&lt;/strong&gt;&lt;/a&gt;&lt;/li&gt;
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&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_0&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec0&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Introduction&lt;/h2&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;background-color: white; clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
&lt;h3 style=&quot;clear: left; color: #006699; font-size: 15px; margin: 25px 0px 5px;&quot;&gt;
Key Facts&lt;/h3&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;&lt;em style=&quot;color: #006699;&quot;&gt;Definitions&lt;/em&gt;&lt;/strong&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Law of Conservation of Energy&lt;/strong&gt;&amp;nbsp;Energy can neither be created, nor destroyed. It can only be said to change its state from one to another. As such, the total energy in any given system is always a constant.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Bernoullis principle&lt;/strong&gt;&lt;br /&gt;For a perfect incompressible liquid, flowing in a continuous stream, the total energy of a particle remains the same, while the particle moves from one point to another.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Kutta condition&lt;/strong&gt;&lt;br /&gt;A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Lift&lt;/strong&gt;&lt;br /&gt;It is the component of force acting upwards and perpendicular to the line of flight, in a steady, undisturbed stream of air. It is important to note that &#39;upwards&#39; implies to the roof of the aircraft in all attitudes of flight, i.e. Upright, Banked (as during a turn), and Inverted.&lt;/div&gt;
&lt;div id=&quot;content_2&quot; style=&quot;background-color: white;&quot;&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;

&lt;dl style=&quot;margin: 0px; padding: 0px;&quot;&gt;&lt;dd style=&quot;margin: 0px 15px;&quot;&gt;Bernoulii&#39;s equation, considering the two points &#39;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&#39; and &#39;&lt;img alt=&quot;b&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eac45ceecbfed93aba876de03af882d8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&#39; in the airflow (refer Diagram 2), is given by:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\frac{P_a}{\rho g} + \frac{v_a^2}{2g} + z_a = \frac{P_b}{\rho g} + \frac{v_b^2}{2g} + z_b = K (Constant)&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/8dc831e2050fd60ae4833a67f15e556d.gif&quot; width=&quot;356&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq1&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(1)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
where,&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;P_a&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/ba4be0726df28085026c173d8823b8fb.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;20&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;P_b&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/9aa54edc8bac2dd85ca7346963b7b00c.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;17&quot; /&gt;&amp;nbsp;is the Pressure of the fluid at Point&amp;nbsp;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;b&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eac45ceecbfed93aba876de03af882d8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;respectively&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;v_a&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/cache/eqns/cac11e3218421eb7965cc4d445293138.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;V_b&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/48009f8b34bfef3ffeaac97d41314573.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;is the Velocity of the fluid at Point&amp;nbsp;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;b&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eac45ceecbfed93aba876de03af882d8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;respectively&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;z_a&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/cache/eqns/77898255a44b2c2d8dd79d401182483a.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;z_b&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/cache/eqns/714c27e1110d6ad2b2afbb135044c018.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;14&quot; /&gt;&amp;nbsp;is the vertcal elevation at Point&amp;nbsp;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;b&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eac45ceecbfed93aba876de03af882d8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;respectively&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;\rho&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/57011c20909c8c445250a5f0f67c802d.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;is the Density of the fluid&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;g&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/2823f4c1ae3c299d121df40698450d64.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;is the acceleration due to gravity&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For our calculation, we shall consider the values of&amp;nbsp;&lt;img alt=&quot;\rho&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/57011c20909c8c445250a5f0f67c802d.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;g&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/2823f4c1ae3c299d121df40698450d64.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;to be constant at both points&amp;nbsp;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;b&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eac45ceecbfed93aba876de03af882d8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;. We shall also assume the values of&amp;nbsp;&lt;img alt=&quot;z_a&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/cache/eqns/77898255a44b2c2d8dd79d401182483a.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;z_b&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/cache/eqns/714c27e1110d6ad2b2afbb135044c018.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;14&quot; /&gt;&amp;nbsp;to be negligible in a wind tunnel experiment. The equation can then be written as:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_a + \frac{v_a^2}{2} = P_b + \frac{v_b^2}{2} = K&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/0193ee21728921efdf5d9e382b011972.gif&quot; width=&quot;186&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq2&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(2)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Since&amp;nbsp;&lt;img alt=&quot;a&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/ff854bf18bacca53704e11ed47f413c9.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;is at a point in the airflow that is unaffected by the passage of the aerofoil through the air, the values of pressure and velocity at this point can be considered to be the same as that of ambient air.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV3&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 443px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/img_0498.jpg&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/img_0498.jpg&quot; border=&quot;0&quot; height=&quot;110&quot; src=&quot;http://www.codecogs.com/users/23573/img_0498_439_110_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;439&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/img_0498.jpg&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Diagram 3&lt;/em&gt;&amp;nbsp;Aerofoil section 2 in a Wind Tunnel experiment.&lt;/figcaption&gt;&lt;/figure&gt;Now consider a Point&amp;nbsp;&lt;img alt=&quot;c&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/61e8b6a93696f38116b748b6e5e51c94.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;(Diagram 3 above ) that is located below the aerofoil section.&lt;br /&gt;Using Bernoulli&#39;s equation from (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/aerodynamics/basic_aerodynamics_theory.php#eq1&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;1&lt;/a&gt;), equating the values of pressure and velocity at this point (&lt;img alt=&quot;c&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/61e8b6a93696f38116b748b6e5e51c94.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;) with those of ambient air, we get:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_a + \frac{v_a^2}{2} = P_c + \frac{v_c^2}{2} = K&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/1c7727e191e791eda951452b45a87780.gif&quot; width=&quot;186&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq3&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(3)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
From Eq (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/aerodynamics/basic_aerodynamics_theory.php#eq2&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;2&lt;/a&gt;) and (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/aerodynamics/basic_aerodynamics_theory.php#eq3&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;3&lt;/a&gt;) we have:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_b +\frac{v_b^2}{2}=P_c + \frac{v_c^2}{2} = K&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/8b7839d08927bf4e7df16b6bd462ae39.gif&quot; width=&quot;184&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq4&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(4)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
According to the Kutta condition of circulation, the airflow over the upper surface of the aerofoil is faster than that below the underside of the aerofoil.&lt;br /&gt;i.e.&amp;nbsp;&lt;img alt=&quot;v_b &amp;gt; v_c&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/14975143b903224655444fe88235b0c4.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;51&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore, to satisfy the condition given in Eq(&lt;a href=&quot;http://www.codecogs.com/reference/engineering/aerodynamics/basic_aerodynamics_theory.php#eq4&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;4&lt;/a&gt;),&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;P_b&amp;lt;P_C&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/905b477a3f6c652c56744650a76e799a.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;60&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;The net result is an upward acting force which is termed as &#39;Lift&#39;.&lt;/strong&gt;&lt;/dd&gt;&lt;/dl&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_3&quot; style=&quot;background-color: white;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec3&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Forces Acting In Flight&lt;/h2&gt;
In the following diagram, we see four forces acting on an aircraft at all times.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV4&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 396px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/four_forces_1.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/four_forces_1.png&quot; border=&quot;0&quot; height=&quot;220&quot; src=&quot;http://www.codecogs.com/users/23573/four_forces_1_392_220_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;392&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/four_forces_1.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Diagram 4&lt;/em&gt;&amp;nbsp;Four Forces in Flight&lt;/figcaption&gt;&lt;/figure&gt;When an aircraft is in level flight, all the four fources are balanced and the aircraft can be said to be in a state of Equilibrium.&lt;br /&gt;Lift(&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;) = Weight(&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;), and&lt;br /&gt;Thrust(&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;) = Drag(&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;)&lt;br /&gt;In such state the aircraft will be travelling at a steady height, at a uniform velocity, and fixed heading.&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_4&quot; style=&quot;background-color: white;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec4&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h3 style=&quot;clear: left; color: #006699; font-size: 15px; margin: 25px 0px 5px;&quot;&gt;
Climbing And Descending&lt;/h3&gt;
&lt;dl style=&quot;margin: 0px; padding: 0px;&quot;&gt;&lt;dd style=&quot;margin: 0px 15px;&quot;&gt;The following discussion explains the relationship between these four forces when an aircraft is in the phase of gaining altitude (climbing), and when the aircraft descends.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV5&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 380px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/climbing.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/climbing.png&quot; border=&quot;0&quot; height=&quot;211&quot; src=&quot;http://www.codecogs.com/users/23573/climbing_376_211_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;376&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/climbing.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
Forces during a Climb&lt;/figcaption&gt;&lt;/figure&gt;In the above diagram, consider the angle of climb to be&amp;nbsp;&lt;img alt=&quot;\alpha&quot; height=&quot;10&quot; src=&quot;http://www.codecogs.com/images/eqns/f968db686c0bdc7925c6ed34485a0835.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;12&quot; /&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Resolving the vectors about&amp;nbsp;&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;, we get:&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;component&amp;nbsp;&lt;img alt=&quot;W sin \alpha&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/4a2edfc0c7e582c1068d45a6d23900b5.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;52&quot; /&gt;&amp;nbsp;directly opposing Thrust&amp;nbsp;&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;; and,&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;component&amp;nbsp;&lt;img alt=&quot;W cos \alpha&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/b5650b71ee17a2533d9251827d946f5a.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;51&quot; /&gt;&amp;nbsp;directly opposing Lift&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore,&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;T = D+ W \sin  \alpha&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/fcaaa06d850e4f8fcc92a01e084caf74.gif&quot; width=&quot;133&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq5&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(5)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
Implying that in a climb,&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;T &amp;gt; D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/e724295e0f8e199a0b3932af3e24e2b4.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;48&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;T \propto \alpha&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/95a70f8deeb3573d0f49724ae6ff9287.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;44&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Also,&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;L= W \cos  \alpha&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/ab863a0df97b2fa240fb467dcab24aad.gif&quot; width=&quot;97&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq6&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(6)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
Implying that&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;L &amp;lt; W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/691dff905350a2fed0fce9479df262f0.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;52&quot; /&gt;&amp;nbsp;(It is important to note the definition of Lift in this context)&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;L \propto \frac{1}{\alpha }&quot; height=&quot;21&quot; src=&quot;http://www.codecogs.com/cache/eqns/a8edecaf985bf6cecb98a6644710a7a1.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;44&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV6&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 380px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/descending.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/descending.png&quot; border=&quot;0&quot; height=&quot;211&quot; src=&quot;http://www.codecogs.com/users/23573/descending_376_211_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;376&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/descending.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
Forces during a Glide or Controlled Descent&lt;/figcaption&gt;&lt;/figure&gt;In the above diagram, consider the angle of descent to be&amp;nbsp;&lt;img alt=&quot;\alpha&quot; height=&quot;10&quot; src=&quot;http://www.codecogs.com/images/eqns/f968db686c0bdc7925c6ed34485a0835.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;12&quot; /&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Resolving the vectors we find that the Total Force, acting equal and opposite to the weight (&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;), is a resultant of the Lift&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;and Drag&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;vectors. That is,&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\tan \alpha  = D/L&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/6094ff2fd52960bf0bce0a82ddc09b11.gif&quot; width=&quot;101&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq7&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(7)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
This implies that&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;L/D \propto \frac{1}{\alpha}&quot; height=&quot;21&quot; src=&quot;http://www.codecogs.com/cache/eqns/176119c7341138d956c99a2ccd4bedf1.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;67&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
Or,&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Higher the value of the ratio&amp;nbsp;&lt;img alt=&quot;L/D&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/0cb445543252f2ad151539b393dfdf20.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;33&quot; /&gt;, the flatter the gliding angle - the more appropriate (for man and machinery) the descent from altitude.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;&lt;em style=&quot;color: #006699;&quot;&gt;Note&lt;/em&gt;&lt;/strong&gt;&amp;nbsp;: As we shall see in a later section, the ratio of L and D also has a significant effect on the Range and the Endurance(in other words the Performance) of the aircraft.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Note regarding Numericals&lt;/em&gt;&amp;nbsp;In working out numericals associated with the Forces in Flight, it is important to note that the vectors (&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;,&amp;nbsp;&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;,&amp;nbsp;&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;,&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;) are merely representative, and do not indicate their actual position. This is further explained in the diagram below.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV7&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 380px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/four_forces.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/four_forces.png&quot; border=&quot;0&quot; height=&quot;211&quot; src=&quot;http://www.codecogs.com/users/23573/four_forces_376_211_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;376&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/four_forces.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
The Two Opposing Couples&lt;/figcaption&gt;&lt;/figure&gt;In low wing aircrafts, the&amp;nbsp;&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;&amp;nbsp;vector is often located ahead of the&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;vector; providing a couple with a nose-down pitching moment. Similarly, and directly opposing this (&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;-&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;) couple, the&amp;nbsp;&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;&amp;nbsp;vector is located below the&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;vector; providing a couple with a nose-up pitching moment.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Note: In the event of loss of power (&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;),the nose-down pitching moment takes precedence, allowing the aircraft to enter into a glide. If a nose-up pitching moment was provided in such an event, the aircraft would inevitably stall and descend uncontrollably earthward. Thus the need for Weight and Balance of an aircraft, and the need to ensure that the C of G is within limits. This will be further analysed in future pages.&lt;/em&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
In addition,the forces acting on the Mainplane (Wings) are balanced by similar forces acting on the Tailplane (Stabilizer). All such numericals can be solved withe reference to the law of moments.&lt;/dd&gt;&lt;/dl&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
Aero-dynamics is the study of the dynamics associated with the motions of air over a moving object. When an aircraft accelerates down the runway to take off, it produces a relative motion between the air and the aircrafts wings. It is this rush of airflow over and below the wings, created by the aircrafts forward motion, which produces lift and consequently makes all forms of powered flight possible.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV1&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 306px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/302px-streamlines_around_a_naca_0012_svg.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/302px-streamlines_around_a_naca_0012_svg.png&quot; border=&quot;0&quot; height=&quot;164&quot; src=&quot;http://www.codecogs.com/users/23573/302px-streamlines_around_a_naca_0012_svg.png&quot; style=&quot;border: 0px none;&quot; width=&quot;302&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/302px-streamlines_around_a_naca_0012_svg.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Diagram 1&lt;/em&gt;&amp;nbsp;&lt;strong&gt;An aerofoil in a steady, streamlined, and incompressible flow of air.&lt;/strong&gt;&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;Image Courtesy&lt;/em&gt;&lt;a class=&quot;orange&quot; href=&quot;http://en.wikipedia.org/wiki/Airfoil&quot; rel=&quot;nofollow&quot; style=&quot;color: #ff6600; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;http://en.wikipedia.org/wiki/Airfoil&quot;&gt;http://en.wikipedia.org/wiki/Airfoil&lt;/a&gt;&lt;/figcaption&gt;&lt;/figure&gt;The contours of the upper and lower surfaces of the wing contribute largely to the pressure differences that occur between the two surfaces. The resulting pressure leads to a net force that provides &#39;Lift&#39; to the wing.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
In this section, we use the Bernoulli&#39;s and Euler&#39;s equations to understand how an aeroplane derives lift. We also look at the four principle forces acting on an aircraft in flight, at ALL times.&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_1&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec1&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Airflow Over An Aerofoil&lt;/h2&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_2&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec2&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h3 style=&quot;clear: left; color: #006699; font-size: 15px; margin: 25px 0px 5px;&quot;&gt;
Bernoulli&#39;s Equation&lt;/h3&gt;
&lt;dl style=&quot;margin: 0px; padding: 0px;&quot;&gt;&lt;dd style=&quot;margin: 0px 15px;&quot;&gt;As with all applications of the Bernoulli&#39;s principle, we shall assume that the flow of air around the aerofoil is:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Steady,&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Incompressible, and&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Frictionless.&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV2&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 486px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/img_0497_-_copy.jpg&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/img_0497_-_copy.jpg&quot; border=&quot;0&quot; height=&quot;123&quot; src=&quot;http://www.codecogs.com/users/23573/img_0497_-_copy_482_123_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;482&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/img_0497_-_copy.jpg&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;em style=&quot;color: #006699;&quot;&gt;Diagram 2&lt;/em&gt;&amp;nbsp;Aerofoil section in a Wind Tunnel experiment.&lt;/figcaption&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;h5 style=&quot;background-color: white; clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Level Flight - With No Tail Plane Load&lt;/h5&gt;
&lt;div class=&quot;example&quot; style=&quot;background-color: white; font-size: 13px;&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
A seaplane of mass 20000 Kg. is flying in Straight and Level Flight. The thrust vector is located 1.2m ABOVE the drag vector. If the Lift/Drag ratio is 5:1, calculate the position of the lift vector. Consider the forces on the tail plane to be absent or negligable. Assume&amp;nbsp;&lt;img alt=&quot;g=9.81m/s^2&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/7b443b3535a7a4b03b0f3c6f7ecc1b35.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;97&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
By resolving the vectors it can be seen that in this case the Thrust and drag vectors produce a nose-down pitching moment. To oppose this, the Lift and Weight vectors will produce a nose-up pitching moment, i.e. Lift vector ahead of weight.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV1&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 282px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23573/worked_examples__1.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23573/worked_examples__1.png&quot; border=&quot;0&quot; height=&quot;156&quot; src=&quot;http://www.codecogs.com/users/23573/worked_examples__1_278_156_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;278&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23573/worked_examples__1.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Let&amp;nbsp;&lt;img alt=&quot;X&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/c07e87724f53985bb1cf8898ddb575b1.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;be the moment arm between&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;&amp;nbsp;In st. and level flight we know,&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;=&amp;nbsp;&lt;img alt=&quot;W&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/550c1a02c338472a41bd9878293693f0.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;&amp;nbsp;= 20000g kN&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;L/D&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/0cb445543252f2ad151539b393dfdf20.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;33&quot; /&gt;&amp;nbsp;ratio = 5:1&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore,&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;= 4000g kN =&amp;nbsp;&lt;img alt=&quot;T&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/ee421c15ad9e9557830337fbaf185853.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Resolving the moments we get,&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;4000g  \ast 1.2 = 20000g \ast  X&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/cache/eqns/5ad3cd2f542e2bdb306f434d0ffeba1f.gif&quot; width=&quot;197&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
(1)&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;/dd&gt;&lt;/dl&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/2718447895551295416/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/engineering-aerodynamics-basic.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2718447895551295416'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/2718447895551295416'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/engineering-aerodynamics-basic.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-515690468744661235</id><published>2012-09-18T11:02:00.001-07:00</published><updated>2012-09-18T11:02:17.407-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
&lt;br /&gt;
&lt;div id=&quot;content_1&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
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&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Axial Flow Pumps And Fans&lt;/h2&gt;
The figures below are for an&amp;nbsp;&lt;strong&gt;Axial Flow Propeller&lt;/strong&gt;&amp;nbsp;fan pumping air.&lt;figure id=&quot;imageDIV1&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 404px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/22109/img_axf1.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;22109/img_axf1.png&quot; border=&quot;0&quot; height=&quot;292&quot; src=&quot;http://www.codecogs.com/users/22109/img_axf1_400_292_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/22109/img_axf1.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
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&lt;strong&gt;Leonhard Euler&lt;/strong&gt;&amp;nbsp;(15 April 1707 - 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory.&lt;/div&gt;
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&lt;figure id=&quot;imageDIV2&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 404px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/22109/img_axf2.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;22109/img_axf2.png&quot; border=&quot;0&quot; height=&quot;415&quot; src=&quot;http://www.codecogs.com/users/22109/img_axf2_400_415_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/22109/img_axf2.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Notes:&lt;/strong&gt;&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;The fixed diffuser vanes are used to remove the whirl component of the discharge velocity of the impeller and to convert the energy to Pressure.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;The impeller vanes may be adjustable.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;The machine may be fitted with pre-entry vanes to ensure that there is no pre-rotation and that the flow is purely Axial.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;The bottom diagram is produced by considering a Radius&amp;nbsp;&lt;img alt=&quot;R&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/f364b5e1ae7dafe6f4e7ed2579b40495.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;of the impeller and drawing it out in a flat plane.&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;v_1 = v = \frac{2\pi RN}{60}&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/images/eqns/ff259db6c86521dfc5fe5e5acb2588ac.gif&quot; width=&quot;127&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The flow through the machine&amp;nbsp;&lt;img alt=&quot;q = \pi \left(r_0^2 - r_{boss}^2 \right)\times V_f&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/cache/eqns/6baa8e50935ade67835662467dde9898.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;163&quot; /&gt;&amp;nbsp;(The boss area can be neglected).&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Also,&amp;nbsp;&lt;img alt=&quot;V = V_f = V_f1&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/c42a7839a4a63f36cb835d6d5bc3e961.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;101&quot; /&gt;&amp;nbsp;(Blade area neglected).&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
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&lt;a href=&quot;&quot; name=&quot;sec2&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h3 style=&quot;clear: left; color: #006699; font-size: 15px; margin: 25px 0px 5px;&quot;&gt;
Euler Theory&lt;/h3&gt;
&lt;dl style=&quot;margin: 0px; padding: 0px;&quot;&gt;&lt;dd style=&quot;margin: 0px 15px;&quot;&gt;Work done by the Vanes per lb. of water&amp;nbsp;&lt;img alt=&quot;= \displaystyle\frac{V_{w1}v_1 - V_wv}{g} = \displaystyle\frac{V_{w1}v}{g}\;ft.&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/c9ff6d401c8c8ce755ff02e151183000.gif&quot; style=&quot;margin-bottom: -14px;&quot; width=&quot;197&quot; /&gt;&amp;nbsp;(&lt;img alt=&quot;V_w = 0&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/82784c98c590489ccbc43ac7e9f84ca2.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;52&quot; /&gt;)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Hydraulic or Manometric&amp;nbsp;&lt;img alt=&quot;\eta  = H_m\div \displaystyle\frac{V_{w1}v}{g}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/a0b7b174ec63142a28d743b018705fdb.gif&quot; style=&quot;margin-bottom: -14px;&quot; width=&quot;115&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Where&amp;nbsp;&lt;img alt=&quot;H_m&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/8d7d12cc6c293b10d56deea34b342bf4.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;27&quot; /&gt;&amp;nbsp;= The Manometric head minus the Head developed by the Pump across the Flanges.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Applying&amp;nbsp;&lt;strong&gt;Bernoulli&#39;&lt;/strong&gt;&lt;strong&gt;s equation&lt;/strong&gt;&amp;nbsp;across the Vanes:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\frac{p}{w} + \frac{V^2}{2g}\;+\frac{V_{w1}v}{g} = \frac{p_1}{w} + \frac{v_1^2}{2g}&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/a9b1fd6546be9c51891bc2110b9d1a07.gif&quot; width=&quot;217&quot; /&gt;&lt;/div&gt;
(Neglecting losses in the Vanes)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore the Pressure rise across the Vanes is given by:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\frac{p_1 - p}{w} = \frac{V_{w1}v}{g} - \frac{V_1^2 - V^2}{2g}&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/images/eqns/f2db9b6b5f3d2bd9fe0651913115e025.gif&quot; width=&quot;207&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
But,&amp;nbsp;&lt;img alt=&quot;V_1^2 = V_{w1}^2 + V_{f1}^2 = V_{w1}^2 + V_{f}^2 = V_{w1}^2 + V^2&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/cache/eqns/aea0fe3c4999413aa784fc497966598c.gif&quot; style=&quot;margin-bottom: -8px;&quot; width=&quot;293&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\frac{p_1 - p}{w} = \frac{V_{w1}v}{g} - \frac{V_{w1}^2}{2g}&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/images/eqns/580464ba16846844c49f68cb49ae5256.gif&quot; width=&quot;212&quot; /&gt;&lt;/div&gt;
&lt;/dd&gt;&lt;/dl&gt;
&lt;/div&gt;
&lt;/div&gt;
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&lt;a href=&quot;&quot; name=&quot;sec3&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
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&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Aerofoil Theory Applied To Propeller Pumps&lt;/h2&gt;
The Combined Inlet and Outlet and Inlet Triangles.&lt;figure id=&quot;imageDIV3&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 380px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/22109/img_axf3.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;22109/img_axf3.png&quot; border=&quot;0&quot; height=&quot;280&quot; src=&quot;http://www.codecogs.com/users/22109/img_axf3_376_280_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;376&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/22109/img_axf3.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Blades are considered to be aerofoils in cascade in a fluid stream of Velocity&amp;nbsp;&lt;img alt=&quot;V_r&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/793ad21d076282967d80aa3ded5bf4b3.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;Average in a direction&amp;nbsp;&lt;img alt=&quot;\theta&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/fa3af604b5504f4e45ded5b524b78187.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;to the tangent.&lt;figure id=&quot;imageDIV4&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 454px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/22109/img_axf4.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;22109/img_axf4.png&quot; border=&quot;0&quot; height=&quot;280&quot; src=&quot;http://www.codecogs.com/users/22109/img_axf4_450_280_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;450&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/22109/img_axf4.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Force&amp;nbsp;&lt;img alt=&quot;P&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/5ce933a59b34e187cf233a8056e27182.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;exerted by the Fluid on the Vane element&amp;nbsp;&lt;img alt=&quot;i&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/4392083a6c3bb5ac8196a84839d7d76f.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;7&quot; /&gt;&amp;nbsp;is the vector Sum of the lift&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;and the force exerted on the Vane by the Fluid. The components of&amp;nbsp;&lt;img alt=&quot;P&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/5ce933a59b34e187cf233a8056e27182.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;are&amp;nbsp;&lt;img alt=&quot;P_t&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/2867460b86738495c7c0ccf4f09b75af.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;18&quot; /&gt;&amp;nbsp;in the Tangential Direction and&amp;nbsp;&lt;img alt=&quot;P_a&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/ba4be0726df28085026c173d8823b8fb.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;20&quot; /&gt;&amp;nbsp;in the Axial Direction.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
&lt;strong&gt;Torque&lt;/strong&gt;, also called&amp;nbsp;&lt;strong&gt;moment&lt;/strong&gt;&amp;nbsp;or&amp;nbsp;&lt;strong&gt;moment of force&lt;/strong&gt;, is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Consider an elemental Vane thickness&amp;nbsp;&lt;img alt=&quot;dR&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/a5bddc47d942f333aac9541321d62dab.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;22&quot; /&gt;&amp;nbsp;at Radius&amp;nbsp;&lt;img alt=&quot;R&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/f364b5e1ae7dafe6f4e7ed2579b40495.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
From Aerofoil Theory Lift,&amp;nbsp;&lt;img alt=&quot;L = C_L\;\times \;\displaystyle\frac{1}{2}\rho V_{r\;ave.}^2\;\times C\;dr\;\;lb.&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/3657e0e2da2c7c5be5837259fd76f6dd.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;234&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
And Drag,&amp;nbsp;&lt;img alt=&quot;D = C_D\;\times \frac{1}{2}\rho V_{r\;ave.}^2\;\times C\;{dR}\;\;lb.&quot; height=&quot;21&quot; src=&quot;http://www.codecogs.com/cache/eqns/ce3f8b7cfb78e5d20949aa34e92aebfe.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;237&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;C_L&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/29ca86c483bfa9f0343b802a2d263cae.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;19&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;C_D&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/78db82d00286718d508718c20d9ca618.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;25&quot; /&gt;&amp;nbsp;are lift and drag coefficients depending upon the aerofoil section and the angle of incidence.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Resolving&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;in the direction of motion.&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_t = L\;sin\,\theta _{ave.} + D\;Cos\,\theta _{ave.}\;\;\;\;lb./Vane&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/images/eqns/cb2c4fcdd6c79f28ef643f5041465605.gif&quot; width=&quot;320&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_a = L\;Cos\,\theta _{ave.} - D\;Sin\,\theta _{ave.}\;\;\;\;lb./Vane&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/images/eqns/ae4373bf0774023fb70a9ed3038736ca.gif&quot; width=&quot;326&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;is commonly small compared with&amp;nbsp;&lt;img alt=&quot;L&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0ab35b88dc08040b611034dc12ce90c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;and the second terms are often neglected.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For&amp;nbsp;&lt;img alt=&quot;N&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/df6317d9742a6c975c000155f3b397a7.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;17&quot; /&gt;&amp;nbsp;Vanes the Total Tangential Force =&amp;nbsp;&lt;img alt=&quot;n\;P_t&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/b1f0546f0f0a4b43e47ae6b06fa17efb.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;33&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= n\left(L \sin \theta _{ave.} + D \cos \theta _{ave.} \right)&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/7acd74613f717595d7c6eff8f45c66ac.gif&quot; width=&quot;216&quot; /&gt;&lt;/div&gt;
Therefore, Torque&amp;nbsp;&lt;img alt=&quot;= nP_t\times R&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/4b9b8680a9336c215096f2146ea14cec.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;76&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Horse-Power required to rotate the Vane elements =&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{2\pi NT}{33000}&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/716f0bae1392386acb1f5565d3d6042b.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;46&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Efficiency&amp;nbsp;&lt;img alt=&quot;\eta&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/96850e060c984ac741e0ba426c6965a2.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;= W.H.P. out put / H.P. in put&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Where&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;W&lt;/em&gt;&amp;nbsp;= Weight of flow through the annular ring =&amp;nbsp;&lt;img alt=&quot;w\times 2\,\pi\;RdR \times V_f&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/images/eqns/392a841514889f3ca2efb605dba72c3e.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;132&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Total Axial Force =&amp;nbsp;&lt;img alt=&quot;n Pa&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/02a8699305760c6c66f1c52cb3e07143.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;31&quot; /&gt;&amp;nbsp;If&amp;nbsp;&lt;img alt=&quot;\Delta p&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/images/eqns/afa572f99e1d2f3a6a1b0363732ff0a4.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;24&quot; /&gt;&amp;nbsp;is the pressure drop across the vanes,&amp;nbsp;&lt;img alt=&quot;\Delta p\times 2\pi \,R\,dr = nP_a&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/b2bd0ca12d6de14fe33ff833cd9aa44b.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;146&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
This equation can be solved to find&amp;nbsp;&lt;img alt=&quot;\Delta p&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/images/eqns/afa572f99e1d2f3a6a1b0363732ff0a4.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;24&quot; /&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;example_577&quot;&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
&lt;div class=&quot;example_select&quot; style=&quot;font-weight: bold; margin-bottom: 10px;&quot;&gt;
Example:&amp;nbsp;&lt;button class=&quot;orange&quot; style=&quot;background-color: #ff6600; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(197, 197, 197); color: white; cursor: pointer; font-size: 11px; line-height: 20px; margin: 0px 7px 0px 0px; padding-left: 8px; padding-right: 8px;&quot;&gt;1&lt;/button&gt;&lt;/div&gt;
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&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div class=&quot;blue&quot; style=&quot;color: #006699; float: right;&quot;&gt;
[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Example 1&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
For the Axial Flow Propeller it is required to find the&amp;nbsp;&lt;strong&gt;torque&lt;/strong&gt;&amp;nbsp;/ft. radius using both&amp;nbsp;&lt;strong&gt;Aerofoil&lt;/strong&gt;&amp;nbsp;and&amp;nbsp;&lt;strong&gt;Euler Theories&lt;/strong&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
We know the radius&amp;nbsp;&lt;img alt=&quot;R&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/f364b5e1ae7dafe6f4e7ed2579b40495.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;3.2 ft&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/ab2051a2f6d568b0e78c4fda1a68588c.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;36&quot; /&gt;&amp;nbsp;and the speed&amp;nbsp;&lt;img alt=&quot;N&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/df6317d9742a6c975c000155f3b397a7.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;17&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;450 r.p.m.&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/c70f2a63edbf2015eb4caad6f539e284.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;69&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border-color: rgb(153, 153, 153) rgb(0, 55, 148) rgb(153, 153, 153) rgb(153, 153, 153); border-style: dashed solid dashed dashed; border-width: 1px 2px 1px 1px; margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;img alt=&quot;V = V_f = V_{f1} = 131\;ft./sec&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/1108c5c25992a052e5c627a4d13323de.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;204&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;\rho  = 0.074\;lb/ft^3&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/60d5b1bd6716d4ffcd9bf1879c4afaac.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;116&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Theoretical Head = 114 ft. of air.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;\eta _H = 0.9\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = 12&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/9946f17e22ab0c0020e76f3121eff49b.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;177&quot; /&gt;&amp;nbsp;Vanes&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;br /&gt;&lt;img alt=&quot;C_L = 0.44\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C_d = 0&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/d19d0bb06ffcd783266b4119ed75af7c.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;191&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Chord = 1.1 ft.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
It is required to find the&amp;nbsp;&lt;strong&gt;Torque&lt;/strong&gt;&amp;nbsp;per ft radius&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\eta _H = H_M\div \frac{V_{w1}V_1}{g} = 114\div \frac{V_{w1}v_1}{g}&quot; height=&quot;43&quot; src=&quot;http://www.codecogs.com/images/eqns/049488e7c2de52ecfcce4b4b0c165b32.gif&quot; width=&quot;267&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
But,&amp;nbsp;&lt;img alt=&quot;v_1 = v = \displaystyle\frac{2\pi RN}{60} = 151\;ft./sec.&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/b8c9af8c5653489448258ee9d40b6b69.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;226&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;V_{w1} = \frac{114\times 32.2}{0.9\times 151} = 27\;ft./sec.&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/images/eqns/b7cafee7161ce7eb033b455a563beec2.gif&quot; width=&quot;287&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Velocity triangle:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;V_{r\, ave.} = 131^2 + (151 - 13.5)^2 = 190\;ft./sec.&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/images/eqns/f72e9f23a935751c91dd1e133950f4ef.gif&quot; width=&quot;342&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\theta _{ave.} = Tan^{-1}\frac{131}{151 - 13.5} = 43^0\;36&#39;&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/images/eqns/695481b84f4bcee6f063057ec7501bf8.gif&quot; width=&quot;270&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Lift&amp;nbsp;&lt;img alt=&quot;= C_L\times \displaystyle\frac{1}{2}\rho V_{r\,ave.}\times C\times dR&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/64c46417734f2d8b227d405b8bff248e.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;197&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 0.44\times \frac{0.074}{2\times 32.2}\times 190^2\times 1.1\;lb./ft.radius&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/images/eqns/40a1ae3b685211b9949f1297be9530ad.gif&quot; width=&quot;349&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 20.1 lb./ft. radius&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/d28569391f85752a9a52f979b382ca1c.gif&quot; width=&quot;150&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;P_t = 20.1\times Sin\theta _{ave.} = 13.85\;lb./ft. radius&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/6de4d7d07196e1a2ef64bd85d3cfe868.gif&quot; width=&quot;326&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For twelve Vanes the longitudinal Force = 12 Pt&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore, Torque required&amp;nbsp;&lt;img alt=&quot;= 12\;P_t\;R = 12\times 13.85\times 3.2 = 532\lb.ft./ft.\radius&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/d909d2904bba1875ff284fe96433c16b.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;311&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Note:&lt;/strong&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Axial thrust / blade&amp;nbsp;&lt;img alt=&quot;P_a = L\,cos\,\theta _{ave.}&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/images/eqns/e5624b05f610013dcc7a3f5315c00bbb.gif&quot; style=&quot;margin-bottom: -3px;&quot; width=&quot;113&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 20.1\times Cos\,45^0\;36&#39; = 14.55\;lb./ft.\;width&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/images/eqns/eef043f32a3d19c18d16ca4b8d5378e9.gif&quot; width=&quot;330&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Total axial Force for 12 Vanes =&amp;nbsp;&lt;img alt=&quot;14.55 X 12 = 174.6 lb./ft. width&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/d9d7ed0cce9bf13de59f21d8ca9f5b67.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;217&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
If&amp;nbsp;&lt;img alt=&quot;\delta\,p&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/images/eqns/84545cbe7c74474b9f83dedea7edc4d0.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;21&quot; /&gt;&amp;nbsp;is the pressure rise, then&amp;nbsp;&lt;img alt=&quot;\Delta p\times 2\pi \times  R\,dR = 174.6 lb./ft.&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/0ca275ee36d2a7369451ca696545521b.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;224&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\Delta p = \frac{174.6}{2\pi \times 3.2} = 8.68\;lb./ft^2&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/images/eqns/f1c2e124aa690b56d7ae8dcdfb666d34.gif&quot; width=&quot;272&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Or,&amp;nbsp;&lt;img alt=&quot;\;\;\;\;\Delta p = \displaystyle\frac{8.68}{0.074} = 117.3\;ft.\;of\;air&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/3eaeeee18bdb8f7951dd15cf0802f3e7.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;220&quot; /&gt;&lt;h3 style=&quot;clear: left; color: #006699; font-size: 15px; margin: 25px 0px 5px;&quot;&gt;
An Alternative Approach Is:&lt;/h3&gt;
&lt;dl style=&quot;margin: 0px; padding: 0px;&quot;&gt;&lt;dd style=&quot;margin: 0px 15px;&quot;&gt;The weight of Flow through an annular element,&amp;nbsp;&lt;img alt=&quot;W = 2\pi \;R\,dR\times V_f\times w&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/acef84d654905c0ed4fa6e77a123fd49.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;171&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 2\times 3.2\times \pi \times 1\times 131\times 0.074 = 195\;lb./sec.&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/images/eqns/3215225d538b88de15053fa864f26199.gif&quot; width=&quot;364&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Work done per lb per lb.&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \frac{V_{w1} v_1}{g} = \frac{H_M}{\eta }\= \frac{114}{0.9} = 126.7\;\;\;[ft.lb./lb.]&quot; height=&quot;42&quot; src=&quot;http://www.codecogs.com/cache/eqns/8d573ec597c0749e62253e0a090b6a4e.gif&quot; width=&quot;121&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore the work done per second per ft. radius&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= W\times \frac{V_{w1}v_1}{g} = 195\times 126.7\;\;\;[ft.\;lb.\;/sec.]&quot; height=&quot;42&quot; src=&quot;http://www.codecogs.com/cache/eqns/2e5721cd96ed55fec295ff5aef8d0a76.gif&quot; width=&quot;332&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
= Torque * Angular Velocity =&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{T\times 2\pi N}{60}&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/68169c8fb409c5573152bff59e18c8e8.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;67&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;T = \frac{195\times 126.7\times 60}{2\times \pi \times 450} = 524\;lb.ft.&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/images/eqns/ccfed7fa7e26650488cf1c8c42113464.gif&quot; width=&quot;311&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Pressure rise across the Vanes:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\frac{p_1 - p}{w} = \frac{V_{w1}v_1}{g} - \frac{V_{w1}^2}{2g}= 126.7 - \frac{27^2}{64.4} = 115.4 \;ft.\; of\; air&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/5d54a50e2b65ddc37fed7759ccd61d7e.gif&quot; width=&quot;442&quot; /&gt;&lt;/div&gt;
&lt;/dd&gt;&lt;/dl&gt;
&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Solution&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
The&amp;nbsp;&lt;strong&gt;Torque&lt;/strong&gt;&amp;nbsp;required is&amp;nbsp;&lt;img alt=&quot;532\lb.ft./ft.\radius&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/0193354ea662a9eaa4a59475d01a9fb6.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;79&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;br /&gt;&lt;div id=&quot;content_4&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
</content><link rel='replies' type='application/atom+xml' href='http://shivakrishnaaero.blogspot.com/feeds/515690468744661235/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/axial-flow-pumps-and-fans-figures-below.html#comment-form' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/515690468744661235'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/530621634329509626/posts/default/515690468744661235'/><link rel='alternate' type='text/html' href='http://shivakrishnaaero.blogspot.com/2012/09/axial-flow-pumps-and-fans-figures-below.html' title=''/><author><name>dff</name><uri>http://www.blogger.com/profile/11901452372836104498</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-530621634329509626.post-83683269715318980</id><published>2012-09-18T10:54:00.003-07:00</published><updated>2012-09-18T10:54:28.071-07:00</updated><title type='text'></title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
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&lt;div class=&quot;inner&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px; padding-right: 10px; position: relative;&quot;&gt;
&lt;header&gt;&lt;div class=&quot;title text&quot; style=&quot;height: 70px; padding-bottom: 2px; position: relative;&quot;&gt;
&lt;h1 style=&quot;clear: left; color: #006699; float: left; font-size: 50px; font-weight: normal; margin: 0px; padding: 28px 0px 0px;&quot; title=&quot;ref-189&quot;&gt;
Shear Stress in Beams&lt;/h1&gt;
&lt;/div&gt;
&lt;div class=&quot;brief&quot; style=&quot;clear: left; font-size: 15px; font-weight: 600; margin: 0px; padding: 6px 0px 2px;&quot;&gt;
An analysis of Shear Stress in Beams of various cross sections.&lt;/div&gt;
&lt;/header&gt;&lt;div class=&quot;noprint&quot;&gt;
&lt;img alt=&quot;+&quot; class=&quot;arrowdown&quot; id=&quot;versionsIMG&quot; src=&quot;http://www.codecogs.com/images/spacer.gif&quot; style=&quot;background-image: url(http://www.codecogs.com/images/icons.png); background-position: -60px -20px; background-repeat: initial initial; height: 14px; margin-bottom: -2px; width: 13px;&quot; /&gt;&lt;span class=&quot;orange&quot; style=&quot;color: #ff6600;&quot;&gt;&amp;nbsp;View other versions (5)&lt;/span&gt;&lt;/div&gt;
&lt;/div&gt;
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&lt;div class=&quot;groupbox shadow&quot; style=&quot;border: 1px solid rgb(204, 204, 204); margin: 15px 0px; padding: 10px;&quot;&gt;
&lt;strong&gt;Contents&lt;/strong&gt;&lt;ol&gt;
&lt;li&gt;Introduction.&lt;/li&gt;
&lt;li&gt;Variation Of Shear Stress.&lt;/li&gt;
&lt;li&gt;Rectangular Sections.&lt;/li&gt;
&lt;li&gt;I-section&lt;/li&gt;
&lt;li&gt;Principle Stresses In I-beams.&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#sec5&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;Pitch Of Rivets In Built Up Girders.&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#sec6&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;Solid Circular Sections.&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#sec7&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;Thin Circular Tubes.&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#sec8&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;Miscellaneous Sections&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#sec9&quot; style=&quot;color: #006699; text-decoration: none;&quot;&gt;Shear Centre.&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;Page Comments&lt;/li&gt;
&lt;/ol&gt;
&lt;/div&gt;
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Introduction.&lt;/h2&gt;
This page should be read in conjunction to &quot; Engineering Materials, Shear Stress&quot;&lt;/div&gt;
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Variation Of Shear Stress.&lt;/h2&gt;
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The&amp;nbsp;&lt;strong&gt;Shearing Force&lt;/strong&gt;&amp;nbsp;at any cross section of a Beam will set up a&amp;nbsp;&lt;strong&gt;Shear Strain&lt;/strong&gt;&amp;nbsp;on transverse sections which in general will vary across the section.&lt;/div&gt;
In the following analysis it has been assumed that the Stress is uniform cross the width (i.e. parallel to the neutral axis) and that the presence of shear stress does not affect the distribution of Bending Stress.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
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This last point is not strictly true because the presence of Shear Stress will distort the transverse planes which will no longer remain plane. (See &quot;Bending of Beams Part 4). Due to the Shear Stress on transverse planes there will be complementary planes parallel to the neutral axis.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
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In the following diagram two transverse sections are shown at a distance&amp;nbsp;&lt;img alt=&quot;\displaystyle \delta x&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/images/eqns/194a6c738c961c1c0841608ed2276a9a.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;19&quot; /&gt;&amp;nbsp;apart and the Shearing forces will be&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;F + \delta F&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/22e97133b8fc22671e6da956d85dc29d.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;55&quot; /&gt;&amp;nbsp;whilst the Bending Moments are&amp;nbsp;&lt;img alt=&quot;M&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a16d60221263b12e1f668e347e5ba06e.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;19&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;M + \delta M&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/9bbec194edb91cf006a9735c2894a5f2.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;64&quot; /&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
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A&amp;nbsp;&lt;strong&gt;shear stress&lt;/strong&gt;&amp;nbsp;is defined as the component of stress coplanar with a material cross section. Shear stress arises from a force vector perpendicular to the surface normal vector of the cross section.&lt;/div&gt;
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&lt;figure id=&quot;imageDIV1&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 404px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-001.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-001.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-001_400_200_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-001.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;Let&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
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&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;em style=&quot;color: #006699;&quot;&gt;s&lt;/em&gt;&amp;nbsp;be the value of the complementary shear stress and hence the transverse shear stress at a distance&amp;nbsp;&lt;img alt=&quot;y_0&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/5b58570ac4cefa1698b07485ad0fc7e4.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;from the Neutral Axis.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;em style=&quot;color: #006699;&quot;&gt;z&lt;/em&gt;&amp;nbsp;be the width of the cross section at&amp;nbsp;&lt;img alt=&quot;y_0&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/5b58570ac4cefa1698b07485ad0fc7e4.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;16&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;em style=&quot;color: #006699;&quot;&gt;A&lt;/em&gt;&amp;nbsp;be the area of cross section cut off by a line parallel to the Neutral Axis.&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;\bar{y}&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/b133832740bd3f8ff975cb3bfc8a310e.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;8&quot; /&gt;&amp;nbsp;be the distance of the centroid of&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;A&lt;/em&gt;&amp;nbsp;from the Neutral Axis.&lt;/li&gt;
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&lt;figure id=&quot;imageDIV2&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-002.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-002.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-002_300_200_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-002.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;f&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/05e26d3a53df3ae77efb9e717eb7026b.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;f + \delta f&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/a27730639740ae22eb461a72d48c3e69.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;46&quot; /&gt;&amp;nbsp;are the Normal Stresses on an element of Area&amp;nbsp;&lt;img alt=&quot;\delta A&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/72a066522b8e92ab1564ac7e56f4ed53.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;20&quot; /&gt;. There is a difference in Longitudinal Forces equal to&amp;nbsp;&lt;img alt=&quot;\delta f\;.\;\delta A&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/7f42a1270b76d75a1c6c821779848e67.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;52&quot; /&gt;&amp;nbsp;and this summed over the area&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;A&lt;/em&gt;&amp;nbsp;must be in equilibrium with the transverse Shear Stress&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;s&lt;/em&gt;&amp;nbsp;on the longitudinal plane of area&amp;nbsp;&lt;img alt=&quot;z\;\delta x&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/121debd340c3c24fa19d0aa4ccc3c716.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;30&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Thus,&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s\;z\;\delta x = \int \delta f\;\;\delta A&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/3162ecf6c9924ea6a4bc968a607d2a97.gif&quot; width=&quot;142&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq1&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(1)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
But&amp;nbsp;&lt;img alt=&quot;f = \frac{M\;y}{I}&quot; height=&quot;23&quot; src=&quot;http://www.codecogs.com/cache/eqns/81585b3f913f4db5cd219f496d56fc1f.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;58&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
And&amp;nbsp;&lt;img alt=&quot;f + \delta f= \frac{(M + \delta M)\;y}{I}&quot; height=&quot;24&quot; src=&quot;http://www.codecogs.com/cache/eqns/e2d7ffdcc6b68732b4654e33aa94330e.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;136&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;\delta f = \delta M\times \frac{y}{I}&quot; height=&quot;33&quot; src=&quot;http://www.codecogs.com/cache/eqns/4be809af8ee864a64705807eeb16c07e.gif&quot; width=&quot;153&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Substituting in equation (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#eq1&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;1&lt;/a&gt;)&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s\;\;z\;\;\delta x = \left(\frac{\delta M}{I} \right)\int y\;\delta A&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/41423da9445f1202a834b2495862e2fa.gif&quot; width=&quot;199&quot; /&gt;&lt;/div&gt;
or&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \left(\frac{\delta M}{\delta x} \right)\;\frac{A\;\bar{y}}{z\;I} = F\;.\;\frac{A\bar{y}}{z\;I}&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/f7f5df7cfd34b0037721a83a39b4b62c.gif&quot; width=&quot;209&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq2&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(2)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Note:&amp;nbsp;&lt;img alt=&quot;F = \frac{dM}{dx}&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/cache/eqns/6e863a4c43f2e906bfd4f2ff19462a8b.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;58&quot; /&gt;. See &quot;Shearing Force and Bending Moments.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
It should also be noted that&amp;nbsp;&lt;img alt=&quot;Z&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/2afafebc838e36ebb94b41f782c45bf1.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;&amp;nbsp;is the actual width of the section at the position where&amp;nbsp;&lt;img alt=&quot;s&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/6c5fb5d0f40fef15d27bb640dbe54758.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;is being calculated and that&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;is the Total Moment of Inertia about the neutral axis. In some applications it is advantageous to calculate&amp;nbsp;&lt;img alt=&quot;\displaystyle A\;\bar{y}&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/images/eqns/8ebf479b5c65dd6ba0fee8e05a4d3144.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;27&quot; /&gt;&amp;nbsp;as several parts.&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_2&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec2&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Rectangular Sections.&lt;/h2&gt;
For a rectangular section at any distance&amp;nbsp;&lt;img alt=&quot;y&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/4e8ce9e3a6f5c0619114a4b2f94db712.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;from the Neutral Axis :&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A = b\left(\frac{d}{2} - y \right)&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/8225d4402eb3b88475f45f12ec34350a.gif&quot; width=&quot;116&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;bar{y} = \frac{1}{2}\left(\frac{d}{2} + y \right)$ and $z = b&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/f2da28310c89b930312764bd1e7f19c6.gif&quot; width=&quot;227&quot; /&gt;&lt;/div&gt;
&lt;figure id=&quot;imageDIV3&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; float: left; margin: 10px 20px 10px 0px; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0003.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0003.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0003.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0003.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Substituting in equation (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#eq2&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;2&lt;/a&gt;) and putting&amp;nbsp;&lt;img alt=&quot;\displaystyle I = \frac{b\;d^3}{12}&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/0474aaa8a7147676571536d9febe2768.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;61&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \frac{F\timesb(d/2 - y)(d/2 + y)}{b\times (b\;d^3/12)\times 2}&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/f87f4793d8f59de894680c21a5fe8f22.gif&quot; width=&quot;193&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;s = \left(\frac{6\;F}{b\;d^3} \right)\left(\frac{d^2}{4} - y^2 \right)&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/4d9003078939f61582962ea4081d351b.gif&quot; width=&quot;226&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
This shows that there is a parabolic variation of Shear Stress with&amp;nbsp;&lt;img alt=&quot;y&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/4e8ce9e3a6f5c0619114a4b2f94db712.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;10&quot; /&gt;. The maximum Shear Force occurs at the Neutral Axis and is given by:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\hat{s} = \frac{3\;F}{2\;b\;d}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/77139415c8193c5b1d1a054939433d45.gif&quot; width=&quot;70&quot; /&gt;&lt;/div&gt;
If&amp;nbsp;&lt;img alt=&quot;\frac{F}{bd}&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/cache/eqns/c7539acc2353305a60f87bde0be1b34c.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;13&quot; /&gt;&amp;nbsp;is called the&amp;nbsp;&lt;strong&gt;Mean Stress&lt;/strong&gt;&amp;nbsp;then:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\hat{s} = 1.5\times s_{mean}&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/8dfe672c745498c7c42ea4ba9ef63953.gif&quot; width=&quot;117&quot; /&gt;&lt;div class=&quot;eqn_num&quot; style=&quot;margin-top: -20px; padding-right: 10px; text-align: right;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;eq3&quot; style=&quot;color: #006699; font-weight: 600;&quot;&gt;(3)&lt;/a&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_3&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec3&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
I-section&lt;/h2&gt;
&lt;figure id=&quot;imageDIV4&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 223px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-004.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-004.png&quot; border=&quot;0&quot; height=&quot;146&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-004_219_146_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;219&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-004.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The dimensions are shown in the diagram. It is required to find an expression for the&amp;nbsp;&lt;strong&gt;Shear Stress in the Web.&lt;/strong&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;A\bar{y}&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/e2949677aac2c5113394e75d196b4ab3.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;21&quot; /&gt;&amp;nbsp;is made up of two parts as follows:&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;for the flanged area&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px;&quot;&gt;
&lt;img alt=&quot;A\bar{y} = B\;\left(\frac{D - d}{2} \right)\left(\frac{D + d}{4} \right)&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/211f99597a8e748e1a0c353da1f301e1.gif&quot; width=&quot;224&quot; /&gt;&lt;/div&gt;
&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;for the web part&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px;&quot;&gt;
&lt;img alt=&quot;+\;b\;\left(\displaystyle\frac{d}{2} - y \right)\left(\displaystyle\frac{\displaystyle\frac{d}{2}\:+\;y}{2} \right)&quot; height=&quot;79&quot; src=&quot;http://www.codecogs.com/cache/eqns/c5bfeba2c7d948bf3602859a0fd4d5da.gif&quot; width=&quot;193&quot; /&gt;&lt;/div&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Using Equation (&lt;a href=&quot;http://www.codecogs.com/reference/engineering/materials/beams/shear_stress_in_beams.php#eq2&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot;&gt;2&lt;/a&gt;) and noting that&amp;nbsp;&lt;img alt=&quot;z = b&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/f09ace690e7ca53a2b9f8c630ef4fc5e.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;38&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \left(\frac{F}{b\;I} \right)\left[B\;\left(\frac{D^2 - d^2}{8} \right) + \left(\frac{b}{2} \right)\left(\frac{d^2}{4} - y^2 \right) \right]&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/3c2fff9947735269e39b4138a2841cd6.gif&quot; width=&quot;367&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
As with the rectangular section, the maximum transverse Shear Stress is at the neutral axis.&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\hat{s} = \left(\frac{F}{8\;b\;I} \right)\left[B\left(D^2 - d^2 \right) + b\,d^2 \right]&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/5f9d3278e7d4db1e376aebe6526abdfe.gif&quot; width=&quot;258&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
At the top of the web,&amp;nbsp;&lt;img alt=&quot;s = \left(\displaystyle\frac{F}{8\;b\;I} \right)B\left(D^2 - d^2 \right)&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/d74ddff32e03c63019258d0cf0acdfce.gif&quot; style=&quot;margin-bottom: -15px;&quot; width=&quot;180&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Since the Shear Stress has to follow the direction of the boundary, the distribution must be of the form shown becoming horizontal at the flanges.&lt;figure id=&quot;imageDIV5&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 204px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-005.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-005.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-005_200_200_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;200&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-005.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Consequently the complementary Shear Stress in the flanges is on longitudinal planes perpendicular to the neutral axis and the &quot;width&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;z&quot;&lt;/em&gt;&amp;nbsp;is replaced by the flange thickness&amp;nbsp;&lt;img alt=&quot;\frac{D - d}{2}&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/cache/eqns/261cbe244586b35f3c15015e32aa59aa.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;28&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Then,&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \frac{F\times A\;\bar{y}}{\left[\frac{D - d}{2}\right]\;I}&quot; height=&quot;45&quot; src=&quot;http://www.codecogs.com/cache/eqns/36c40f962d9bbc60a904d31bfe9b4ada.gif&quot; width=&quot;98&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Showing that the Shear Stress in the flanges varies from a maximum at the top web to zero at the outer tips.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
In Practice, however it will be found that most of the Shearing Force ( About 95%) is carried by the Web and the Shear Force in the flanges is negligible.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
As the variation over the web is comparatively small ( about 25%) it is convenient for design purposes and also in calculating deflection due to Shear, to assume that all the Shearing Force is carried by the Web and is uniformly distributed. Similarly it is normal practice to assume that, as a first approximation, the Bending Moment is carried wholly by the flanges.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 1430242574=&quot;&quot; href=&quot;http://www.amazon.com/JavaFX-2-0-Introduction-Carl-Dea/dp/1430242574%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D1430242574&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51DPgyF8YeL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;129&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$19.63&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0131580183=&quot;&quot; href=&quot;http://www.amazon.com/Understanding-DB2-Learning-Visually-Examples/dp/0131580183%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0131580183&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51MYlg66IpL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;119&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$136.00&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0596009747=&quot;&quot; href=&quot;http://www.amazon.com/XSLT-Cookbook-Solutions-Examples-Developers/dp/0596009747%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0596009747&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51LhLGmcm7L._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;122&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$17.99&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0137054890=&quot;&quot; href=&quot;http://www.amazon.com/JavaScript-Example-Edition-Ellie-Quigley/dp/0137054890%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0137054890&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51S0vBQ6mPL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;122&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$26.74&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
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&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div class=&quot;blue&quot; style=&quot;color: #006699; float: right;&quot;&gt;
[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Example 2&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;figure id=&quot;imageDIV1&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0007.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0007.png&quot; border=&quot;0&quot; height=&quot;400&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0007.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0007.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;A 12 in. by 5 in. British Standard Beam is subjected to a Shearing Force of 10 tons.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Calculate the&amp;nbsp;&lt;strong&gt;value&lt;/strong&gt;&amp;nbsp;of the transverse Shear Stress at the Neutral Axis and at the top of the Web and&amp;nbsp;&lt;strong&gt;compare&lt;/strong&gt;&amp;nbsp;this with the mean Stress calculated on the assumption that the Stress is uniformly distributed over the Web.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
What&amp;nbsp;&lt;strong&gt;percentage&lt;/strong&gt;&amp;nbsp;of the Shearing Force is carried by the Web?&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;img alt=&quot;I=220\;in^{4}&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/ee44874ef0bd5add8b57bc8f2ce1c00e.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;83&quot; /&gt;&amp;nbsp;Area=&amp;nbsp;&lt;img alt=&quot;9.4\;in^2&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/3f9a82341beb548c0728e98dea9d4ec1.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;48&quot; /&gt;&amp;nbsp;Web Thickness=&lt;img alt=&quot;0.35\;in.&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/681c012ac58eb35ffa3df587c76f014f.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;54&quot; /&gt;&amp;nbsp;and Flange thickness=&lt;img alt=&quot;0.55\;in.&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/deab83313613c3f9dea7aee045065200.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;54&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A\;\bar{y} = (5\times 0.55)\times 5.725 + (5.45 - y)\times 0.35\times \left(\frac{5.45 + y}{2} \right)&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/8516f17039036aef49a9ed8aa0bccd73.gif&quot; width=&quot;458&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 15.75 + \frac{(5.45^2 - y^2)\times 0.35}{2} = 20.95 - 0.175\;y^2&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/d0806d68997c89fb9e8a9d64f8dd1eb3.gif&quot; width=&quot;379&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = F\times \frac{A\;\bar{y}}{z\;I} = \frac{10\;(20.95 - 0.175\;y^2)}{0.35 \times 220}&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/1b349764f094ade502c9cf10f467a855.gif&quot; width=&quot;288&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
At the Neutral Axis&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \frac{10\times 20.95}{0.35\times 220} = 2.71\;tons\;in^{-2}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/cac44f408aa44b2c0839a5ecdad2c0f2.gif&quot; width=&quot;249&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
At the Top of the Web (&lt;img alt=&quot;y=5.45\;in.&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/9459ace13ae6e10380bd0d7e6cc45d84.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;85&quot; /&gt;)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = \frac{10\times 15.75}{0.35\times 220} = 2.01\;tons\;in^{-2}&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/cache/eqns/ea647538922173c9dd1c1e4a9c4526d0.gif&quot; width=&quot;249&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Assuming that all the Shearing Force is carried uniformly by the web.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s_{mean} = \frac{10}{0.35\times 10.9} = 2.62\;tons\;in.^{-2}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/19a0728d55ca4c0add34aebe7f0f5024.gif&quot; width=&quot;292&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Total Shearing Force carried by the web is given by:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\int_{-\frac{d}{2}}^{\frac{d}{2}}{s\,.\,b\;dy} = \int_{-5.45}^{5.45}{\frac{0.35\times10\;(20.95 - 0.175\;y^2)\;dy}{0.35\times 220}}&quot; height=&quot;53&quot; src=&quot;http://www.codecogs.com/cache/eqns/c295647e4825faa36adc830572cc894b.gif&quot; width=&quot;397&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \frac{1}{22}\;\left[20.95 - \frac{0.175\;y^2}{3} \right]_{-5.45}^{5.45}= \frac{228 - 18.9}{22} = 9.5\;tons&quot; height=&quot;49&quot; src=&quot;http://www.codecogs.com/cache/eqns/04c028d63f301992240dfd89db8a8e6c.gif&quot; width=&quot;425&quot; /&gt;&lt;/div&gt;
i.e.&amp;nbsp;&lt;img alt=&quot;\;95\;^0/_0&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/33d3783f01fbff4b9eb4b2f299aad53c.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;42&quot; /&gt;&amp;nbsp;of the Total&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The remaining 5 % of the vertical Shear Stress is presumably accounted for by the component of the Shear Stress at the junction of the flange and the web.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Failure due to Shear in the Web usually takes the form of buckling brought about by the Compressive Stresses on planes at 45 degrees to the transverse section. ( See Compound Stress and Strain). For this reason deep webs are often supported by vertical stiffeners.&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Solution&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;strong&gt;Percentage&lt;/strong&gt;&amp;nbsp;of the Shearing Force is&amp;nbsp;&lt;img alt=&quot;\;95\;^0/_0&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/33d3783f01fbff4b9eb4b2f299aad53c.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;42&quot; /&gt;&amp;nbsp;of the Total.&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
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&lt;a href=&quot;&quot; name=&quot;sec4&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
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Principle Stresses In I-beams.&lt;/h2&gt;
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&lt;strong&gt;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;-&lt;/strong&gt;&lt;strong&gt;beams&lt;/strong&gt;, also known as&amp;nbsp;&lt;strong&gt;&lt;img alt=&quot;H&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/69ecf9478a13979e5d132e22424f3ded.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;17&quot; /&gt;-&lt;/strong&gt;&lt;strong&gt;beams&lt;/strong&gt;are beams with an&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;- or&amp;nbsp;&lt;img alt=&quot;H&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/69ecf9478a13979e5d132e22424f3ded.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;17&quot; /&gt;-shaped cross-section. The horizontal elements of the&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;are flanges, while the vertical element is the web. The web resists shear forces while the flanges resist most of the bending moment experienced by the beam.&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
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When an&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;section beam is subjected to both Bending and Shear Stresses it is normal to find that the Maximum Principle Stress is at the top of the Web. The other possible value is the Maximum Bending Stress which occurs at the outer edge of the Flange.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
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[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Example 3&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
A short vertical column is firmly fixed at the base and is 1 ft high. The column is of&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;section 8 in. by 4 in. The flanges are 0.4 in thick and the web is 0.28 in. thick.&amp;nbsp;&lt;img alt=&quot;\displaystyle I = 55.6;in^4&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/dc27d3fffe4a2da14d7d5404143a8dd0.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;90&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;Area = 5.3;in.^2&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/bf17e18e182aba09bc29fbad73108c38.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;115&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
An inclined load of 8 tons acts on the top of the column in the centre of the section and in the plane which contains the centre-line of the web. The line of action is&amp;nbsp;&lt;img alt=&quot;\displaystyle 30^{0}&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/images/eqns/41733be7301643e4825bda27f71e6aa1.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;24&quot; /&gt;&amp;nbsp;to the vertical.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Determine the&amp;nbsp;&lt;strong&gt;position&lt;/strong&gt;&amp;nbsp;and&amp;nbsp;&lt;strong&gt;magnitude&lt;/strong&gt;&amp;nbsp;of the greatest principle stress on the base of the column.&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
The inclined load will intersect the base cross-section at a distance&amp;nbsp;&lt;img alt=&quot;\displaystyle 12\;\tan\;30^{0} = 6.93\;in.&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/2da16a3204f7e8637248cbe02638376d.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;155&quot; /&gt;&amp;nbsp;from the centroid.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Resolving the load into horizontal and vertical components:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Direct Load =&amp;nbsp;&lt;img alt=&quot;8\;\cos\;30 = 6.92\;tons&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/62f68f4af6babbaf03697f14472b8dbc.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;150&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Shearing Force =&amp;nbsp;&lt;img alt=&quot;8\;\sin\;30 = 4\;tons&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/bbff54a9340689b5ed1ceff0be522366.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;127&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;Bending Moment =&amp;nbsp;&lt;img alt=&quot;6.92\times 6.93 = 47.9\;tons-in.&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/1784a29763b9ab986fa62623312cec84.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;208&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
At the top of the Web:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = F\times \frac{ A\;\bar{y}}{z\;I} = \frac{4\times (4\times 0.4)\times 3.8}{0.28\times 55.6} = 1.565\;tons\;in^{-2}&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/b4438fd064bc2ba230ea49fbb52c63b5.gif&quot; width=&quot;414&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Bending Stress =&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{47.9\times (4 - 0.4)}{55.6} = 3.105\;tons\;in^{-2}&quot; height=&quot;36&quot; src=&quot;http://www.codecogs.com/cache/eqns/e59692938862a19ad2fc20f387cd1f6d.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;248&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Direct Stress =&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{6.92}{5.3} = 1.305\;tons\;in^{-2}&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/07dd13ba3e803ee79dab386af7693678.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;165&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Therefore, Total Normal Stress =&amp;nbsp;&lt;img alt=&quot;3.105 + 1.305 = 4.41\;tons\;in.^{-2}&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/b3108cffe5d4bb64cfa88f0363cdf587.gif&quot; style=&quot;margin-bottom: -2px;&quot; width=&quot;226&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
See the Reference Pages on Compound Stress and Strain (currently in preparation)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Maximum Principle Stress =&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{f}{2} + \displaystyle\frac{1}{2}\sqrt{f^2 + 4s^2}&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/f1110440587f648e52a8dc67507ed308.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;122&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 2.205 + \frac{1}{2}\sqrt{19.4 + 9.8}= 4.91\;tons\; in.^{-2}&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/3b84acc13a88aa6caf181f4463f2ccec.gif&quot; width=&quot;324&quot; /&gt;&lt;/div&gt;
(in Compression)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
This will occur at the top of the Web.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Checking the value of the maximum Bending Stress which is:-&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;47.9\times \frac{4}{55.6} = 3.45\;tons\;in.^{-2}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/4ed0a1a7f5cd34b8a898228d176e18b4.gif&quot; width=&quot;228&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Which together with the direct Stress gives a maximum value of&amp;nbsp;&lt;img alt=&quot;3.45 + 1.305 = 4.705 tons/sq.in.&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/8d56ed1c8ba1647a7ccbaecdc0b81a55.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;229&quot; /&gt;&amp;nbsp;at the outside of the flange and which is less than the value at the top of the Web.&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_5&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec5&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Pitch Of Rivets In Built Up Girders.&lt;/h2&gt;
The load carried by one rivet in a beam section built up as shown in the diagram, is determined by the difference between normal stresses on certain areas of two transverse sections at a distance apart equal to the pitch of the rivets.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV6&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0008.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0008.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0008.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0008.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The are to be used is that part of the cross-section which &quot; comes away&quot; when the particular set of rivets are removed. I.e. in the case of the rivets holding the flange to the angle section, the area is that shaded in (b) and for the rivets holding the angles to the web, the area is shown in (c)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
If&amp;nbsp;&lt;img alt=&quot;p&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/bfcd1111118a91abdb49f5a96fed48d1.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;is the pitch of the rivets and&amp;nbsp;&lt;img alt=&quot;R&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/f364b5e1ae7dafe6f4e7ed2579b40495.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is the force on the rivets over a length&amp;nbsp;&lt;img alt=&quot;p&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/images/eqns/bfcd1111118a91abdb49f5a96fed48d1.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;of beam then proceeding as in the section on the variation of Shear Stress:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;R = \sum{\delta f\times \delta A}&quot; height=&quot;26&quot; src=&quot;http://www.codecogs.com/cache/eqns/b7ffa64cc2e9935d5dc2849019827d32.gif&quot; width=&quot;131&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \sum{\frac{\delta M\times y}{I}}\times \delta A = \left(\frac{\delta M}{I} \right)\times \sum{y\;\delta A}= F\times p\times \frac{A\;\bar{y}}{I}&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/4d27198f4cd4f02ef0cdc65fdc825f47.gif&quot; width=&quot;447&quot; /&gt;&lt;/div&gt;
(Compare with&amp;nbsp;&lt;img alt=&quot;F = \frac{dM}{dx}&quot; height=&quot;22&quot; src=&quot;http://www.codecogs.com/cache/eqns/6e863a4c43f2e906bfd4f2ff19462a8b.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;58&quot; /&gt;&amp;nbsp;and let&amp;nbsp;&lt;img alt=&quot;dx = p\right&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/02176e7277800692b860278304a8037e.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;48&quot; /&gt;)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Note that for the flanges there are two rivets to a pitch length and they are usually staggered so as not to occur together in one cross section. Also note that the web rivets are in double shear.&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;example_769&quot;&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
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&lt;/div&gt;
&lt;div class=&quot;noprint&quot; id=&quot;amazon_768&quot; style=&quot;height: 172px; left: 0px; overflow: hidden; position: relative; visibility: visible; z-index: 2;&quot;&gt;
&lt;ul style=&quot;height: 2236px; list-style: none url(http://www.codecogs.com/images/list-orange.gif); margin: 0px; padding: 0px; position: relative; top: -2064px; z-index: 1;&quot;&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 1454805218=&quot;&quot; href=&quot;http://www.amazon.com/Examples-Explanations-Administrative-Fourth-Edition/dp/1454805218%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D1454805218&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41wUz0o61IL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;111&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$37.39&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0132381826=&quot;&quot; href=&quot;http://www.amazon.com/Perl-Example-Edition-Ellie-Quigley/dp/0132381826%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0132381826&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51yrbezH2eL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;121&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$29.02&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 1936220466=&quot;&quot; href=&quot;http://www.amazon.com/Excel-2011-Mac-intermediate-advanced/dp/1936220466%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D1936220466&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51Jq54JiZBL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;124&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$5.75&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;br /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div class=&quot;blue&quot; style=&quot;color: #006699; float: right;&quot;&gt;
[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Example 4&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
An&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;-section beam is built up of a web plate 10 in. by&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{1}{2}&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/41c29bcfc9b6c5593bf59d45cb43bf08.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;in. with flange plates 6 in. by 1 in. which are secured by rivets through angle sections of 2 in. by 2 in. by&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{1}{4}&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/bf8458745a0dce3f683669758e2822f0.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;in. ( The arrangement is the same as the above diagram)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
If the Bending Stress is limited to&amp;nbsp;&lt;img alt=&quot;t\;tons/sq.in.&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/a766af3f0fa7d6c3fe37b9679f5be86b.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;92&quot; /&gt;&amp;nbsp;estimate the&amp;nbsp;&lt;strong&gt;maximum&lt;/strong&gt;&amp;nbsp;uniformly distributed&amp;nbsp;&lt;strong&gt;load&lt;/strong&gt;&amp;nbsp;which can be carried over a length of 12 ft.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Assuming&amp;nbsp;&lt;img alt=&quot;\displaystyle\frac{1}{2}&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/41c29bcfc9b6c5593bf59d45cb43bf08.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;in. diam. rivets, calculate their&amp;nbsp;&lt;strong&gt;pitch&lt;/strong&gt;&amp;nbsp;if the allowable Shearing Stress is 5 tons/sq.in. and the bearing pressure 10 tons/sq.in.&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
If the loading is&amp;nbsp;&lt;img alt=&quot;w\;ton/ft.&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/26646007293068d98c6d7ee86ceeecab.gif&quot; style=&quot;margin-bottom: -5px;&quot; width=&quot;69&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\hat{M} = \frac{w\;l^2}{8} = \frac{w\times12^2}{8}\;tons-ft.&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/c7e45632c59073d9a99f4c0b8847bbdb.gif&quot; width=&quot;249&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \frac{w\times12^2}{8}\times 12 = 216\;w\;tons-in.&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/61bb22a511e886f7fffdc2fbd94f8665.gif&quot; width=&quot;275&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;I = \displaystyle\frac{1}{2}\times\displaystyle\frac{10^3}{12}&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/bf70d919ee136eaa592ea4b40f34faea.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;89&quot; /&gt;&amp;nbsp;for the web&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;+\;4\left[\displaystyle\frac{2\times0.25^3}{12} + (2\times 0.25)\times4.875^2 + 0.25\times\displaystyle\frac{1.75^3}{12} + (0.25\times 1.75)\times 3.875^2 \right]&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/6ed4af9e027253d400511883bf4a5058.gif&quot; style=&quot;margin-bottom: -15px;&quot; width=&quot;557&quot; /&gt;&amp;nbsp;for the angles&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;= 2\left[6\times \displaystyle\frac{1^3}{12} + (6\times 1)\times 5.5^2 \right]&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/39390cbc4744b2b0c077c6be7a9cdb01.gif&quot; style=&quot;margin-bottom: -15px;&quot; width=&quot;210&quot; /&gt;&amp;nbsp;for the flanges&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 41.7 + 0.01 + 47.5 + 0.45 + 26.2 + 1 = 363 = 480\;in^4&quot; height=&quot;17&quot; src=&quot;http://www.codecogs.com/cache/eqns/f8b70d7451d5ca288df4984d5ef5de03.gif&quot; width=&quot;426&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Now&amp;nbsp;&lt;img alt=&quot;\;\;\;\;\;\;f = 8 = \displaystyle\frac{\hat{M}\hat{y}}{I} = \displaystyle\frac{216\;w\times 6}{480}&quot; height=&quot;40&quot; src=&quot;http://www.codecogs.com/cache/eqns/9fdcfeebff8502460fc528d28bb13781.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;187&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;w = 2.98\;\;say\;\;3.00\;tons/ft.&quot; height=&quot;19&quot; src=&quot;http://www.codecogs.com/cache/eqns/26193f82b6f1cca3279c2defad95cb64.gif&quot; width=&quot;265&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Permissible Load per pitch length.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For one rivet in double shear in the web or two rivets in single shear in the flange&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 2\times\frac{\pi }{4}\times\left( \frac{1}{2} \right)^2\times 5 = 1.965\;tons&quot; height=&quot;47&quot; src=&quot;http://www.codecogs.com/cache/eqns/55cfcf2866b6078a0bd827f11032f2b6.gif&quot; width=&quot;268&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Crushing of the rivets (one in web or two in flange)&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \left(d\;t \right)\times 10 = \left(\frac{1}{2}\times \frac{1}{2} \right)\times 10 = 2.5\;tons&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/b83c633f29a61249be42df032e742e97.gif&quot; width=&quot;318&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For the flange rivets&amp;nbsp;&lt;img alt=&quot;A\;\bar{y} = 6\times 1\times 5.5 = 33\;in.^2&quot; height=&quot;18&quot; src=&quot;http://www.codecogs.com/cache/eqns/d718ffd140e3fcb18ef25470b53970fd.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;197&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The load per pitch length if the smaller of equations&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= F\;p\times \frac{A\;\bar{y}}{I} = \left(3\times\frac{12}{2} \right)\times p\times \frac{33}{480}&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/8c26e2d210a7443518a0dd2b4d906ac4.gif&quot; width=&quot;286&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;p = 1.58\;\;\;say\;\;1\frac{1}{2}\;in.&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/6fa883b9dd7e6c9a6823af40ced261d4.gif&quot; width=&quot;208&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For the Web rivets&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A\;\bar{y} = 33 + 2\times (2\times \frac{1}{4})\times 4.875 + 2\times (1.75\times\frac{1}{4})\times 3.875 = 41.27\;in.^2&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/194dc9d5536b287009d7e9e8aec1e322.gif&quot; width=&quot;539&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;p = \frac{R\;I}{F\;A\;\bar{y}} = \frac{1.965\times 480}{18\times 41.27}&quot; height=&quot;42&quot; src=&quot;http://www.codecogs.com/cache/eqns/34e722c6292eddd9c494c9fcc06d9cbf.gif&quot; width=&quot;242&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 1.26\;in.\;\;\;say\;1\frac{1}{4}in.&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/f239d241a7205cd45adb81db5aac4096.gif&quot; width=&quot;167&quot; /&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Solution&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;1.965\;tons&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/59500449071280c7c08f076867f3d992.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;73&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;The&amp;nbsp;&lt;strong&gt;load per pitch length&lt;/strong&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;p= 1\displaystyle\frac{1}{2}\;in.&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/576c45e9eeec428d1621306e6e0119bd.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;77&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;For the&amp;nbsp;&lt;strong&gt;Web rivets&lt;/strong&gt;,&amp;nbsp;&lt;img alt=&quot;p=1\displaystyle\frac{1}{4}in.&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/9dee25401f9f6fb7c1afc439d54d80e3.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;72&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_6&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec6&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Solid Circular Sections.&lt;/h2&gt;
&lt;figure id=&quot;imageDIV7&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0009.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0009.png&quot; border=&quot;0&quot; height=&quot;300&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0009.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0009.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Let&amp;nbsp;&lt;img alt=&quot;s&quot; height=&quot;9&quot; src=&quot;http://www.codecogs.com/images/eqns/6c5fb5d0f40fef15d27bb640dbe54758.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;9&quot; /&gt;&amp;nbsp;be the Shear Force across a chord parallel to&amp;nbsp;&lt;img alt=&quot;XX&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/7b25b3c5e92f032be439d5a76c7ee08d.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;31&quot; /&gt;&amp;nbsp;defined by the angle&amp;nbsp;&lt;img alt=&quot;\displaystyle \theta&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/89be7211b7cb38952eceb6641782ba6c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;10&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = F\times \frac{A \bar{y}}{z\;I}= \frac{F \int (2x\;dy)\;y}{2R \cos \theta \times \pi R^4/4}= \frac{4\;F}{R^5 \cos \theta }\int_{R\sin\theta }^{R}{\sqrt{(R^2\;-y^2)} y\;dy}&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/cache/eqns/40224a39aea84a8b472324d1984426dd.gif&quot; width=&quot;527&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \frac{4\;F}{\pi R^5 \cos \theta }\left[\frac{1}{3}\left(R^2 - y^2 \right)^{\frac{3}{2}} \right]_R^{R\sin\theta }&quot; height=&quot;49&quot; src=&quot;http://www.codecogs.com/cache/eqns/4d8bad13c84a09b621d6ae7b5f4002e0.gif&quot; width=&quot;253&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= \frac{4\;F}{\pi \;R^5\;\cos\;\theta }\times \frac{1}{3}(1 - \sin^2\theta )^{\frac{3}{2}} = \frac{4\;F\;\cos^2\theta }{3\;\pi \;R^2}&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/f07b102eed0d2112a640774fa572cb62.gif&quot; width=&quot;345&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
At the Neutral Axis the Shear Stress&amp;nbsp;&lt;img alt=&quot;\displaystyle \hat{s}&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/baa63c857ed998c208c0df662f961704.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;is maximum and equals&amp;nbsp;&lt;img alt=&quot;\frac{4\;F}{3\;\pi \;R^2}&quot; height=&quot;23&quot; src=&quot;http://www.codecogs.com/cache/eqns/8d57998eb201fb72f899f1e4be17c26a.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;38&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;\hat{s} = \displaystyle\frac{4}{3}\times&quot; height=&quot;35&quot; src=&quot;http://www.codecogs.com/cache/eqns/8c8a5c765b06b0d1c9acffa388721da3.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;52&quot; /&gt;&amp;nbsp;mean shear stress&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV8&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 154px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0010.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0010.png&quot; border=&quot;0&quot; height=&quot;150&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0010_150_150_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;150&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0010.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The directional distribution of the Shear Stress is as indicated. This does not affect the magnitude of the greatest Shear Stress which is usually the value required.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
This particular case is applicable to&amp;nbsp;&lt;strong&gt;Rivets&lt;/strong&gt;&amp;nbsp;in Shear but the ratio may be assumed to have been incorporated in the allowable stress value which is then taken as uniform over the section.&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_7&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec7&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Thin Circular Tubes.&lt;/h2&gt;
It was shown in the page on Shear Stress that it has to follows the direction of the boundary. In a thin walled circular tube this means that it can be assumed to be tangential.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV9&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); float: right; margin: 10px; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0011.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0011.png&quot; border=&quot;0&quot; height=&quot;300&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0011.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0011.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Assume that&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;XX&lt;/em&gt;&amp;nbsp;is the neutral axis and that&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;P&lt;/em&gt;&amp;nbsp;and&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;Q&lt;/em&gt;&amp;nbsp;are two symmetrically placed positions defined by the angle&amp;nbsp;&lt;img alt=&quot;\theta&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/fa3af604b5504f4e45ded5b524b78187.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;10&quot; /&gt;&amp;nbsp;. Let the Shear Stress be&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;s&lt;/em&gt;.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The complementary Shear Stress is along the longitudinal planes and are balanced by the difference of Normal Stress on the shaded area subtending the angle&amp;nbsp;&lt;img alt=&quot;2\theta&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/images/eqns/fbc00a08dbeab3048eaf5207723386cb.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;18&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For a length of Beam&amp;nbsp;&lt;img alt=&quot;\displaystyle \delta x&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/images/eqns/194a6c738c961c1c0841608ed2276a9a.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;19&quot; /&gt;&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;2\;s\;\;t\;dx = \int_{-\theta }^{\theta}{\delta f\;t\;R\;d\phi }&quot; height=&quot;45&quot; src=&quot;http://www.codecogs.com/cache/eqns/0f51bbaa34618602f54cd7fda51f517d.gif&quot; width=&quot;191&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;s = \frac{1}{2} \int_{-\theta }^{\theta} \left(\frac{\delta f}{\delta x} \right) R\;d\phi&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/cache/eqns/f3bc258b8187a99d09235a28d6c55632.gif&quot; width=&quot;217&quot; /&gt;&lt;/div&gt;
but&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\frac{\delta f}{\delta x} = \left(\frac{\delta M}{\delta x} \right)\times \frac{y}{I_x}= \frac{F\times R\times \cos\phi }{\frac{1}{2}(2\pi R\,t)\,R^2}&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/cache/eqns/3b2fa53826eb04f85366971acfca1239.gif&quot; width=&quot;282&quot; /&gt;&lt;/div&gt;
where&amp;nbsp;&lt;img alt=&quot;I_x = \frac{1}{2}&quot; height=&quot;21&quot; src=&quot;http://www.codecogs.com/cache/eqns/f2858ed709851f6d543be0121a9b4af2.gif&quot; style=&quot;margin-bottom: -6px;&quot; width=&quot;46&quot; /&gt;&amp;nbsp;Polar Moment of inertia&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;I = \frac{1}{2}\;(A)\times(R)^2&quot; height=&quot;37&quot; src=&quot;http://www.codecogs.com/cache/eqns/d7890ba102890bdac666459d5b0a1c9c.gif&quot; width=&quot;135&quot; /&gt;&lt;/div&gt;
where&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;A&lt;/em&gt;&amp;nbsp;represents the area and&amp;nbsp;&lt;em style=&quot;color: #006699;&quot;&gt;R&lt;/em&gt;&amp;nbsp;the mean radius&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;See&lt;/strong&gt;&amp;nbsp;Engineering Materials , Bending Stress, Moments of Inertia.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;\;s = \frac{F\;R^2}{2\pi \;R^3\;t}\int_{-\theta }^{\;\theta }{\cos\;\phi\; d\,\phi }&quot; height=&quot;45&quot; src=&quot;http://www.codecogs.com/cache/eqns/01e19d1ec4be0069be3cfb6d7c9f6760.gif&quot; width=&quot;252&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;\;s = \frac{F\;}{2\pi \;R\;t}\times 2\;\sin\;\theta  = \frac{F\;\sin\;\theta }{\pi \;R\;t}&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/cache/eqns/dc8f65f4b10499fa0323221fa3e05112.gif&quot; width=&quot;305&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Maximum Shear Force will occur at the neutral axis and is given by:&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\hat{s} = \frac{F}{\pi \;R\;t} = 2\times mean\;shear\;stress&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/cache/eqns/5761105a72c00c921f253b657f74f332.gif&quot; width=&quot;278&quot; /&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_8&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec8&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Miscellaneous Sections&lt;/h2&gt;
The Shear Stress at any point in a cross-section can always be calculated from the basic formula&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = F\times \frac{A\;\bar{y}}{z\;I}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/226415fc256630f6e1f72d8db461697e.gif&quot; width=&quot;98&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;example_773&quot;&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 1439207178=&quot;&quot; href=&quot;http://www.amazon.com/Stress-Without-Tears-aircraft-stress-mathematics/dp/1439207178%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D1439207178&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/5102S7V%2BhGL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;126&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$18.49&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;iframe border=&quot;0&quot; frameborder=&quot;0&quot; height=&quot;125&quot; marginwidth=&quot;0&quot; scrolling=&quot;no&quot; src=&quot;http://rcm.amazon.com/e/cm?t=code097-20&amp;amp;o=1&amp;amp;p=21&amp;amp;l=ur1&amp;amp;category=kindle&amp;amp;banner=0RAH4NSD1J4RGX4N6SG2&amp;amp;f=ifr&quot; style=&quot;border-style: none;&quot; width=&quot;125&quot;&gt;&lt;/iframe&gt;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 5511116920=&quot;&quot; href=&quot;http://www.amazon.com/Shear-stress-Jesse-Russell/dp/5511116920%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D5511116920&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41v%2Bxk7RU8L._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;120&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$19.94&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0470048247=&quot;&quot; href=&quot;http://www.amazon.com/Petersons-Stress-Concentration-Factors-Walter/dp/0470048247%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0470048247&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/51bKmM68GTL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;114&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$119.90&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0486485803=&quot;&quot; href=&quot;http://www.amazon.com/Aircraft-Structures-Dover-Aeronautical-Engineering/dp/0486485803%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0486485803&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&amp;nbsp;&amp;nbsp;&lt;/a&gt;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0486610985=&quot;&quot; href=&quot;http://www.amazon.com/Stress-Waves-Solids-H-Kolsky/dp/0486610985%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0486610985&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41WdvYn1zXL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;101&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$6.73&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0080437842=&quot;&quot; href=&quot;http://www.amazon.com/Shear-Deformable-Beams-Plates-Relationships/dp/0080437842%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0080437842&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41r9LVMs%2BQL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;107&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$123.81&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0415258367=&quot;&quot; href=&quot;http://www.amazon.com/Innovative-Shear-Design-Structural-Engineering/dp/0415258367%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0415258367&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41-PFIkF5pL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;107&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$146.99&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 1439207178=&quot;&quot; href=&quot;http://www.amazon.com/Stress-Without-Tears-aircraft-stress-mathematics/dp/1439207178%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D1439207178&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/5102S7V%2BhGL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;126&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$18.49&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;iframe border=&quot;0&quot; frameborder=&quot;0&quot; height=&quot;125&quot; marginwidth=&quot;0&quot; scrolling=&quot;no&quot; src=&quot;http://rcm.amazon.com/e/cm?t=code097-20&amp;amp;o=1&amp;amp;p=21&amp;amp;l=ur1&amp;amp;category=kindle&amp;amp;banner=0RAH4NSD1J4RGX4N6SG2&amp;amp;f=ifr&quot; style=&quot;border-style: none;&quot; width=&quot;125&quot;&gt;&lt;/iframe&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div class=&quot;blue&quot; style=&quot;color: #006699; float: right;&quot;&gt;
[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Shear Stress&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
For the section shown determine the&amp;nbsp;&lt;strong&gt;average Shearing Stress&lt;/strong&gt;&amp;nbsp;at&amp;nbsp;&lt;img alt=&quot;A&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/0aeaea70eee6f9a76340e223202e278c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;,&amp;nbsp;&lt;img alt=&quot;B&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/5e8d03ab6b08ffe0cd49ace81aa6fe29.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;,&amp;nbsp;&lt;img alt=&quot;C&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/3361e92ce74abc85746719d295d27b48.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;, and&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;for a Shearing Force of 20 tons, and find the&amp;nbsp;&lt;strong&gt;ratio&lt;/strong&gt;&amp;nbsp;of the maximum to mean Stress.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV2&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 404px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0012.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0012.png&quot; border=&quot;0&quot; height=&quot;300&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0012.png&quot; style=&quot;border: 0px none;&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0012.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;strong&gt;Draw&lt;/strong&gt;&amp;nbsp;to scale a diagram to show the variation of shearing Stress across the section.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;I = 5\times \frac{6^3}{12} - \pi \times\frac{4^4}{64} = 77.4\;in.^4&quot; height=&quot;41&quot; src=&quot;http://www.codecogs.com/cache/eqns/a8a8606280a87eb1fc1172437bc7fe26.gif&quot; width=&quot;252&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;A&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/0aeaea70eee6f9a76340e223202e278c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;img alt=&quot;A\;\bar{y} = 0&quot; height=&quot;16&quot; src=&quot;http://www.codecogs.com/cache/eqns/edaf02c233653c351b85414ba7b129c2.gif&quot; style=&quot;margin-bottom: -4px;&quot; width=&quot;56&quot; /&gt;&amp;nbsp;and&amp;nbsp;&lt;img alt=&quot;\;\therefore\;\;\;\;\;s = 0&quot; height=&quot;12&quot; src=&quot;http://www.codecogs.com/cache/eqns/b5b81c9c3012639836e3d95fbafed15e.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;77&quot; /&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;B&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/5e8d03ab6b08ffe0cd49ace81aa6fe29.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A\;\bar{y} = (5\times1)\times 2.5 = 12.5\;in^3&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/cache/eqns/9e1c830a611cce3e363bd760bc28491c.gif&quot; width=&quot;235&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;z = 5\;in.&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/66c6ec5ee74370eb2e628d8f40ff9e57.gif&quot; width=&quot;68&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;s = \frac{20\times 12.5}{5\times 77.4} = 0.647\;tons\;in.^{-2}&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/630a6e0e5b2ccd419d46ae4dc5abe730.gif&quot; width=&quot;303&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;figure id=&quot;imageDIV3&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 304px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0013.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0013.png&quot; border=&quot;0&quot; height=&quot;200&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0013.png&quot; style=&quot;border: 0px none;&quot; width=&quot;300&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0013.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;C&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/3361e92ce74abc85746719d295d27b48.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A\;\bar{y} = (5\times 2)2 - \int_{1}^{2}{2x\;dy\;y}= 20 - \int_{1}^{2}{\sqrt{(4 - y^2)}\;\;2y\;dy}&quot; height=&quot;45&quot; src=&quot;http://www.codecogs.com/cache/eqns/fd443b4b96f9c32b0ee44cf0b3065874.gif&quot; width=&quot;448&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 20 + \frac{2}{3}\left[(4 - y^2)^\frac{3}{2} \right]_1^2= 20 - 3.47 = 16.53\.in.^3&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/14aa67816911150cdc20d33f0c9f7514.gif&quot; width=&quot;359&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;s = \frac{20\times 16.53}{\left[5 - 2\sqrt{(4 - 1)} \right]\times 77.4} = 2.77\;tons\;in^{_2}&quot; height=&quot;55&quot; src=&quot;http://www.codecogs.com/cache/eqns/5cc1c8bda07509f8c8f4da8a28dfa8a6.gif&quot; width=&quot;390&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;A\;\bar{y} = (5\times 3)\times1.5 - \int_{0}^{2}{\sqrt{4 - y^2}\;\times 2\;y\;dy }&quot; height=&quot;46&quot; src=&quot;http://www.codecogs.com/cache/eqns/15b74aa3a0fe74fc162f6c8397687232.gif&quot; width=&quot;342&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 22.5 - \frac{2}{3}\times 4^{\frac{3}{2}} = 17.17\;in.^3&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/084261efc54412e25e7ef631724a9352.gif&quot; width=&quot;224&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;\therefore\;\;\;\;\;\;\;s = \frac{20\times 17.17}{1\times 77.4} = 4.44\;tons\;in.^{-2}&quot; height=&quot;39&quot; src=&quot;http://www.codecogs.com/cache/eqns/16f26be0be6e523de52edeca81b48a91.gif&quot; width=&quot;308&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The variations of Shearing Stress are shown on the above graph.&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Solution&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
The&amp;nbsp;&lt;strong&gt;average Shearing Stress&lt;/strong&gt;&amp;nbsp;:&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;A&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/0aeaea70eee6f9a76340e223202e278c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;14&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;0&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/a0e6289118c7cce216123ac768d48c68.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;10&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;B&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/5e8d03ab6b08ffe0cd49ace81aa6fe29.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;0.647\;tons\;in.^{-2}&quot; height=&quot;15&quot; src=&quot;http://www.codecogs.com/cache/eqns/0ca90538899dc956db9b2633e866efb5.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;116&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;C&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/3361e92ce74abc85746719d295d27b48.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;2.77\;tons\;in^{_2}&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/4e3ca6b8f96e8bd903a2fbe4335e11ff.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;92&quot; /&gt;&lt;/li&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;At&amp;nbsp;&lt;img alt=&quot;D&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/9c3855d144e699651baaa2fc6f28eb9c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;16&quot; /&gt;&amp;nbsp;is&amp;nbsp;&lt;img alt=&quot;4.44\;tons\;in.^{-2}&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/feb9da7693a38951a27e3367b74c6389.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;108&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;content_9&quot; style=&quot;background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 18px;&quot;&gt;
&lt;a href=&quot;&quot; name=&quot;sec9&quot; style=&quot;color: #006699;&quot;&gt;&lt;/a&gt;&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;h2 style=&quot;border-bottom-color: rgb(197, 197, 197); border-bottom-style: solid; border-bottom-width: 1px; clear: left; color: #006699; font-size: 18px; margin: 25px 0px 10px;&quot;&gt;
Shear Centre.&lt;/h2&gt;
For unsymmetrical sections and in particular angle and channel sections, the summation of the Shear Stresses in each leg gives a set of Forces which must be in equilibrium with the applied Shearing Force.&lt;figure id=&quot;imageDIV10&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 354px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0014.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0014.png&quot; border=&quot;0&quot; height=&quot;263&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0014_350_263_thumb.gif&quot; style=&quot;border: 0px none;&quot; width=&quot;350&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0014.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Consider the angle section which is bending about a principle axis and with a Shearing Force&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;at right angle to this axis. The sum of the Shear Stresses produces a force in the direction of each leg as shown above.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;yellowbox&quot; style=&quot;background-color: #ffffcc; border: 1px solid rgb(204, 204, 204); clear: left; margin: 5px 5px 2px; padding: 5px;&quot;&gt;
It is clear that their resultant passes through the corner of the angle and unless&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is applied through this point there will be a twisting of the angle as well as Bending. This point is known as&amp;nbsp;&lt;strong&gt;The Shear Centre or Centre of Twist&lt;/strong&gt;.&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
For a channel section with loading parallel to the Web, the total Shearing Force carried by the web must equal&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;and that in the flanges produces two equal and opposite horizontal forces. It can be seen that for equilibrium the applied load causing&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;must lie in a plane outside the channel as indicated. Its position is calculated as in the following example.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div id=&quot;example_772&quot;&gt;
&lt;div class=&quot;rightpanel&quot; style=&quot;clear: right; float: right; font-size: 12px; margin-bottom: 25px; margin-right: 10px; padding-left: 10px; width: 220px;&quot;&gt;
&lt;div class=&quot;example_select&quot; style=&quot;font-weight: bold; margin-bottom: 10px;&quot;&gt;
Example:&amp;nbsp;&lt;button class=&quot;orange&quot; style=&quot;background-color: #ff6600; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(197, 197, 197); color: white; cursor: pointer; font-size: 11px; line-height: 20px; margin: 0px 7px 0px 0px; padding-left: 8px; padding-right: 8px;&quot;&gt;1&lt;/button&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;noprint&quot; id=&quot;amazon_771&quot; style=&quot;height: 172px; left: 0px; overflow: hidden; position: relative; visibility: visible; z-index: 2;&quot;&gt;
&lt;ul style=&quot;height: 2236px; list-style: none url(http://www.codecogs.com/images/list-orange.gif); margin: 0px; padding: 0px; position: relative; top: -2064px; z-index: 1;&quot;&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;a 0735558566=&quot;&quot; href=&quot;http://www.amazon.com/Agency-Partnerships-LLCs-Examples-Explanations/dp/0735558566%3FSubscriptionId%3DAKIAJQSHMIN7TCX3LH2A%26tag%3Dcode027-20%26linkCode%3Dsp1%26camp%3D2025%26creative%3D165953%26creativeASIN%3D0735558566&quot; style=&quot;color: #006699; text-decoration: none;&quot; wrc_done=&quot;true&quot;&gt;&lt;img height=&quot;160&quot; src=&quot;http://ecx.images-amazon.com/images/I/41A1ZAWZOsL._SL160_.jpg&quot; style=&quot;border: 0px none;&quot; width=&quot;105&quot; /&gt;&amp;nbsp;&amp;nbsp;&lt;strong&gt;$23.00&lt;/strong&gt;&lt;/a&gt;&amp;nbsp;&lt;/li&gt;
&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;iframe border=&quot;0&quot; frameborder=&quot;0&quot; height=&quot;125&quot; marginwidth=&quot;0&quot; scrolling=&quot;no&quot; src=&quot;http://rcm.amazon.com/e/cm?t=code097-20&amp;amp;o=1&amp;amp;p=21&amp;amp;l=ur1&amp;amp;category=kindle&amp;amp;banner=1SXH1NWCPDHK3NRJC7R2&amp;amp;f=ifr&quot; style=&quot;border-style: none;&quot; width=&quot;125&quot;&gt;&lt;/iframe&gt;&lt;/li&gt;
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&lt;li style=&quot;float: none; height: 165px; overflow: hidden; padding-bottom: 7px; width: 220px;&quot;&gt;&lt;br /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;cent&quot; style=&quot;margin-right: 240px; position: relative;&quot;&gt;
&lt;div style=&quot;overflow-y: hidden;&quot;&gt;
&lt;div class=&quot;blue&quot; style=&quot;color: #006699; float: right;&quot;&gt;
[imperial]&lt;/div&gt;
&lt;h5 style=&quot;clear: left; color: #006699; font-size: 14px; margin: 12px 0px 2px;&quot;&gt;
Example - Example 6&lt;/h5&gt;
&lt;div class=&quot;example&quot;&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Problem&lt;/div&gt;
&lt;div class=&quot;problem&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;strong&gt;Explain&lt;/strong&gt;&amp;nbsp;why a single section channel with its web vertical, subjected to vertical loading as a beam will be in torsion unless the load is applied through a point outside the section known as the Shear Centre.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Find its&amp;nbsp;&lt;strong&gt;approximate position&lt;/strong&gt;&amp;nbsp;for a channel section&amp;nbsp;&lt;img alt=&quot;\displaystyle 1\frac{1}{2}\;in.&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/8d87d51681ff994ef90d029801da702d.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;44&quot; /&gt;&amp;nbsp;by&amp;nbsp;&lt;img alt=&quot;\displaystyle1\frac{1}{2}\;in.&quot; height=&quot;34&quot; src=&quot;http://www.codecogs.com/cache/eqns/3ec5cef354cab25871fa90c2b30c48fb.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;44&quot; /&gt;&amp;nbsp;outside by&amp;nbsp;&lt;img alt=&quot;\displaystyle \frac{1}{8}\;in.&quot; height=&quot;36&quot; src=&quot;http://www.codecogs.com/images/eqns/6c77dfb26ff154382cd33455fd78431c.gif&quot; style=&quot;margin-bottom: -11px;&quot; width=&quot;37&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Workings&lt;/div&gt;
&lt;div class=&quot;workings&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
If&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is the Shearing Force at the section, then the Total Vertical Force in the Web can be taken to be equal to&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;. It should be mentioned that integrating from the height of the web only will give a value slightly less than&amp;nbsp;&lt;img alt=&quot;F&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/eeed423d85a2258bd8be8eeee2cdc092.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;( Compare with Example 2) but the remaining vertical force is assumed to be carried by the corners of the section.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;I_x = \frac{1}{8}\times \frac{1.5^3}{12} + 2\left[\frac{1.375\times 0.125^3}{12} + 1.375\times0.125\times 0.6875^2 \right]&quot; height=&quot;44&quot; src=&quot;http://www.codecogs.com/cache/eqns/199d8d0bdf8f1276db4cc6209ea9757f.gif&quot; width=&quot;477&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= 0.0352 + 2(0.0002 + 0.0812) = 0.198\;in.^4&quot; height=&quot;20&quot; src=&quot;http://www.codecogs.com/cache/eqns/5f75340a221968a940367673964a3c95.gif&quot; width=&quot;332&quot; /&gt;&lt;/div&gt;
&lt;figure id=&quot;imageDIV4&quot; style=&quot;background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); clear: left; margin: 0px auto; padding: 3px; width: 404px;&quot;&gt;&lt;a class=&quot;group&quot; href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0015.png&quot; rel=&quot;group&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot;&gt;&lt;img alt=&quot;23287/Shear-Stress-0015.png&quot; border=&quot;0&quot; height=&quot;300&quot; src=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0015.png&quot; style=&quot;border: 0px none;&quot; width=&quot;400&quot; /&gt;&lt;/a&gt;&lt;figcaption style=&quot;font-size: 10px; padding: 2px;&quot;&gt;&lt;div class=&quot;magnify noprint&quot; style=&quot;margin-top: -10px; text-align: right;&quot;&gt;
&lt;a href=&quot;http://www.codecogs.com/users/23287/Shear-Stress-0015.png&quot; style=&quot;color: #006699; font-weight: 600; text-decoration: none;&quot; target=&quot;_blank&quot; title=&quot;Enlarge&quot;&gt;&lt;img alt=&quot;+&quot; height=&quot;11&quot; src=&quot;http://www.codecogs.com/images/icons/magnify-clip.png&quot; style=&quot;border: 0px none;&quot; width=&quot;15&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
&lt;/figcaption&gt;&lt;/figure&gt;&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Proceeding as in the section on&amp;nbsp;&lt;img alt=&quot;I&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/740b5ba05033ae2132b5abc5fe826d6b.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;beams, the Shear Stress in the flanges at a distance&amp;nbsp;&lt;img alt=&quot;z&quot; height=&quot;8&quot; src=&quot;http://www.codecogs.com/cache/eqns/5fc666d0a8aedeb6343aec1c3d3dbfee.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;8&quot; /&gt;&amp;nbsp;from the tip is:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;s = F\times \frac{A\;\bar{y}}{t\;I}\;\;\;\;\;\;(t = 0.125\;in.)&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/fe823854eaaf2f37301c1241850174b3.gif&quot; width=&quot;240&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;= F\;(z\;t)\times \frac{0.6875}{t\;l} = 3.47\;F\;z&quot; height=&quot;38&quot; src=&quot;http://www.codecogs.com/cache/eqns/7c67e84c859cbfed4812eb18e36e246d.gif&quot; width=&quot;237&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
The Total force in each Flange&amp;nbsp;&lt;img alt=&quot;R&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/f364b5e1ae7dafe6f4e7ed2579b40495.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;15&quot; /&gt;&amp;nbsp;is given by:&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;R = \int s\;t\;dz = \frac{3.47\timesF}{8}\left[\frac{z^2}{2} \right]_0^{1.4375} = 0.448\;F&quot; height=&quot;50&quot; src=&quot;http://www.codecogs.com/cache/eqns/f7c9a1c09fbfcfc258355ea8c092de05.gif&quot; width=&quot;325&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
If&amp;nbsp;&lt;img alt=&quot;h&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/images/eqns/1e0422619fb37b9133f5d6231921c88c.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;11&quot; /&gt;&amp;nbsp;is the distance of the Shear Centre ( Through which the applied load must act if there is to be no twisting of the section) from the centre line of the Web, then for equilibrium.&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
&lt;div class=&quot;formulaDsp&quot; style=&quot;padding: 6px 0px 6px 30px;&quot;&gt;
&lt;img alt=&quot;F\times h = R\times 1.375&quot; height=&quot;14&quot; src=&quot;http://www.codecogs.com/cache/eqns/f34047fae4e52df6d15bf4b64179b2df.gif&quot; width=&quot;147&quot; /&gt;&lt;/div&gt;
&lt;div style=&quot;margin-bottom: 10px; margin-top: 10px;&quot;&gt;
&lt;/div&gt;
Or&amp;nbsp;&lt;img alt=&quot;\;\;\;\;\;\;\;h = 0.617\;in.&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/f7aed8ee2f0383b6d925e71f007c13c8.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;95&quot; /&gt;&lt;/div&gt;
&lt;div class=&quot;label&quot; style=&quot;color: #999999; font-size: 9px; margin-bottom: -14px; margin-left: 6px;&quot;&gt;
Solution&lt;/div&gt;
&lt;div class=&quot;solution&quot; style=&quot;border: 1px dashed rgb(153, 153, 153); margin-top: 10px; padding: 4px 6px;&quot;&gt;
&lt;ul style=&quot;list-style: url(http://www.codecogs.com/images/list-orange.gif); margin: 2px 4px 2px 30px; padding: 0px;&quot;&gt;
&lt;li style=&quot;padding-bottom: 7px;&quot;&gt;&lt;img alt=&quot;h = 0.617\;in.&quot; height=&quot;13&quot; src=&quot;http://www.codecogs.com/cache/eqns/e39a7847916c00486d80e990d619eb83.gif&quot; style=&quot;margin-bottom: -1px;&quot; width=&quot;94&quot; /&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
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