Civil engineering topics

Design of beam using Working Stress Method (WSM)

2020-07-28T15:28:00.001+05:30

Working stress method is one of the oldest design philosophies, and now it is applicable at very fewer places. Nowadays, other methods are used instead of working stress method as they rule out the limitations of WSM. The alternative methods that are used are Limit State Method and Ultimate Load Methods which uses loads factors and ultimate strength in the design. In it, we start with the analysis part, which is practised in the Mechanics of Solids. The first task is to find bending moments, shear forces, reaction forces at the supports, axial forces, etc.

Let us look at the bending stresses first as they produce both types of stresses (compressive and tensile). The theory of bending (plane section remains plane before and after bending) is applicable. Still, one thing to note is that this theory is applicable only when the perfect bond between two different materials exists and no-slip occur between the material.

Stress-strain Distribution:

Since it is already assumed there is a perfect bond between the materials,

then the theory is bending is applicable.

The linear variation of strain is also assumed in the materials.

So, the strain (e) will be the same on either side of the neutral axis at the same distance.

So, the stresses in different materials f1 = E1e and f2 = E2e.

Where E1 and E2 are the moduli of elasticity of two materials.

Since the strains were same, from strain compatibility we can say that:

e1 = e2

f1/E1 = f2/E2

f2 = E2*f1/E1 = m*f1 where, m = E2/E1 = modular ratio

Now we have defined a new term here “modular ratio (m)”. Let us know about it first.

Modular Ratio (m):

Since we have to analyse two materials in a section (concrete and steel) so to avoid complications, we have to transform the whole section into a single homogeneous section. This is done through the help of the modular ratio (m).

Take small element “dy” in material 2, which is at a distance y from the neutral axis.

So, the force in that small element (dF2 ) can be written as:

dF2 = f2(b2*dy) but we have defined f2 = E2*f1/E1 = m*f1

dF2 = m*f1(b2*dy) = f1(mb2*dy)

So, material 2 can be transformed into material 1 by just by replacing the original width b2 by mb2.

Modifications in the value of modular ratio:

Since for analysis, we have defined the modular ratio as the ratio of elastic modulus of the two materials. However, this definition of modular ratio does not account for long time analysis like creep, shrinkage and other properties of concrete. However, to account that to some extent IS: 456:2000 suggests a different value of m.

m = 280 / (3 * permissible stress in concrete)

Since another material used in beam other than concrete is steel so we should also discuss

the transformed area and stresses in steel in compression and tension region.

In Tension:

Ast can be converted using the modular ratio.

The equivalent area of concrete can be written as m*Ast.

The equivalent stress in concrete can be written as fst/m.

In Compression:

The modular ratio in compression steel is not the same as in tension steel.

This is because the doubly reinforced beam will result in a higher stress level,

so IS 456:2000 recommends using 1.5*m instead of m.

So, the equivalent concrete area will be 1.5*m*Ast and equivalent stress will be fst/(1.5*m).

The behaviour of Reinforced Concrete in flexure:

When reinforced concrete is loaded, its behaviour varies according to the loading stage. The analysis starts from the uncracked phase thereafter in cracked phase. The crack starts developing when stress in the end fibre reaches the modulus of rupture. Then the cracking moment (Mcr) can be defined as

Mcr = fcr * IT/yT

Where IT is the second moment of area of transformed concrete about

NA and yT is a distance of tension fibre from the neutral axis.

As the crack takes place in concrete, it means that applied moment has exceeded the cracking moment, and the crack will start developing in tension region. The area used for the analysis part will significantly change as the crack has developed. The concrete in the tension region will be neglected, and the resistance will be there only due to reinforcement.

It is quite conservative to assume that concrete does not take any tensile stress and load the

resistance is only due to reinforcement.

This assumption is quite valid because the tensile strength of concrete is very less.

Some assumptions are used in the analysis of beams by working stress method ( at service loads) :

Theory of bending is applicable, i.e. plane section remains plane before and after the bending.
Under working loads, the stress-strain curve of steel and concrete is linear, i.e. Hooke’s law is valid.
Reinforcement takes all the tensile stresses.
The modular ratio is 280/(3*permissible stress in concrete)

Design of Reinforced Concrete Structures:

Basically, all the civil engineering structure design started from hit and trial and some empirical approach. The WSM method is based on linear elastic theory, and it was also assumed deflection would be linear and will obey Hooke’s law. However, it was noticed that the deflection and cracking, i.e. failure occur at a much higher load than the service load.

Here comes the concept of factor of safety (FOS), which is the ratio of failure load to service load.

FOS = Failure load / Service load

Using FOS, the permissible stresses are defined as

Permissible stress = Characteristic Strength / FOS

Design of singly reinforced sections :

The above figure represents the stress-strain distribution of a beam subjected to pure flexure.

Where b = width of the beam

d = Effective depth of the beam

fcbc = Actual stress in concrete and less than permissible stress

fst = Actual stress in steel and less than permissible stress

x = kd = depth of neutral axis from top fibre

k = Neutral axis factor

jd = ( 1 - k/3)d = lever arm

Neutral axis:

Calculation of the actual depth of the neutral axis is done by just equating the moment

of the area on both sides of the neutral axis.

The equation can be written based on the figure above:

b*x*x/2 = m*Ast(d-x) , solving x will give neutral axis depth.

Now one should have a question in his/her mind at this point that why are we giving so

much importance to the neutral axis?

The reason behind it is the types of the reinforced section that will result after the design

. If the allowable stress in concrete and allowable stress in steel reaches the same time,

then the depth of the neutral axis achieved is called balanced section neutral axis depth.

However, in construction practices, we always prefer an under-reinforced section which means

the depth of the neutral axis will be less than balanced section neutral axis depth.

The reason behind this approach is that we need a ductile failure for any structure.

Since steel will reach its capacity first, then it will elongate due to its ductile nature,

and thus the owner will be alarmed of the danger.

On the other hand, if the section is over-reinforced, then the failure will be brittle,

and thus there will be a threat to the life of many people.

After calculating the neutral axis depth, the next process is to calculate the moment of resistance

which can be easily calculated by

Mr = Permissible stress in steel * area of steel * lever arm (for under-reinforced section)

The area of steel can be calculated using a balanced section, and then after putting the area,

one should check finally that section is under-reinforced or not.

If not, then change the area of steel or the dimension of the beam.

Many times one will end up with a beam of very high dimensions and to tackle doubly reinforced

sections are designed. This method is used when the beam dimensions are fixed.

The design procedure will be the same for this type of beam, also.

First neutral axis depth will be calculated using m and 1.5m for tension and compression region

respectively. Then the moment of resistance of singly reinforced section will be calculated.

After that, the addition bars in compression region will be provided to resist the extra moment.

Limitations of WSM:

Linear elastic behaviour of concrete does not hold good.
The actual factor of safety is not known.
This may end up with a large steel area, and thus, some of the assumptions made can be violated.

Design Philosophies for design of RC structures

2020-07-09T15:45:00.001+05:30

To have a properly designed structure which fulfils all the requirements like proper durability, functioning, etc. We follow certain design philosophies. Every philosophy has some set of assumptions, and based on the premises, some ideas are suggested. So, let us start one by one.

Working Stress Method (WSM) :

Now it is an outdated method, and it is used occasionally. Today, it is used in a place where the complete analysis of the structure is complex, and no risk is allowed like an overhead water tank, chimney, etc.

This traditional method has the following assumptions:

The material behaves elastically, means stress-strain linear relation will be applicable.
The adequate factor of Safety is applied so that the working load is well below the material strength.

Here, we raised a term factor of Safety ( FOS) which is given by

FOS = Strength of material / Allowable strength of the material

Since in this method, it is assumed that materials will behave elastically so all the forces and stresses can be calculated using techniques and concepts of Strength of Materials. However, using it for a composite section where steel and concrete both are present, one more assumption is taken to have simplicity in the analysis. The bond between steel and concrete is assumed perfect, and the strain in steel and concrete is considered equal at the interface.

By assuming this strain compatibility, we can indirectly relate the stress in concrete and stress in steel using the term modular ratio (m):

m = Stress in steel / Stress in concrete

= (strain in steel * elastic modulus of steel) / (strain in concrete * elastic modulus of concrete)

= elastic modulus of steel / elastic modulus of concrete

Limitations of WSM:

In it, we are not able to understand the relative importance of different loads like wind load, snow load, earthquake load, etc. These loads act with various uncertainties, and thus we end designing over conservative structures.
While designing beams, one can end up with a beam of significant depth which may not be feasible sometimes.

The Ultimate Load Method (ULM) of Design:

This method is a slight improvement in WSM, where non-linear behaviour of steel and concrete are taken into consideration. In this method, the ultimate loads are considered where stresses generated produce failure. Safety of structures is taken into account, and it is ensured by a term called Load Factor.

Load Factor = Ultimate Load / Service Load

This load factor leads us to a different Factor of Safety based upon requirements. Thus, the major drawback of WSM is removed.

Limitations of ULM:

This method uses ultimate loads while designing, and thus, one cannot ensure that both safety and serviceability together will be satisfactory.
This method uses non-linear analysis which is troublesome to analyse.

The Limit State of Design (LSM):

This is the design method which takes care of safety at ultimate loads and serviceability at design load, and thus it is the most used method in today's era. This thing is done by applying different FOS for different kind of building loads, and all of them are decided by studying them with separate approaches.

Limit state refers to the state of " about to collapse" beyond which the structure cannot be used practically means either the serviceability criteria will fail or structure thoroughly will fail.

Limit State of Collapse:

This deals with the strength of the structure, and it is studied using the following parameters:

Flexure
Compression
Shear
Torsion

Limit State of Serviceability:

This limit state deals with the deformation of structure to that extent when the fabric becomes unserviceable. It is studied using the following parameters.

Deflection
Excessive Vibration
Corrosion
Cracking

Important Note - If the structure has attained limit state of serviceability then after removal of loads it can achieve its original position. However, it is not the case with the limit state of collapse.

Load and Resistance Factor Design (LRFD):

LRFD is the simplest method of design to use. In only one condition just needs to be satisfied.

Design Resistance should be greater than or equal to Design Load.

Design resistance is multiplied by a factor (phi) which takes care of all the uncertainties of the property of material characteristics values like elastic modulus and Poisson's ratio. The numerical value of the resistance factor is always less than one.

The design load is multiplied by a factor (gamma) which take cares of the possibility of overloading, and thus this value is always greater than one.

IS 456:2000 Recommendations for LSM of Design:

1. Characteristic strength and characteristic load:

Characteristic strength has been covered in greater detail in my previous blog. Follow this link to read about it:

http://gk1711.blogspot.com/2020/07/characteristic-strength-of-concrete.html

The characteristic load is that load which has 95% probability of not being exceeded during its lifetime.

2. Partial Safety Factor for material:

The design strength is always less than the ultimate strength of the material, and thus FOS is implemented.

For concrete fck is characteristic strength, but in design, the characteristic strength is taken as 0.67*fck.

This reduction is there because while testing, we prepare the cubic specimen, but in reality, there are many different shapes, and sizes and this reduction is to account that.

Now addition FOS of 1.5 is taken for concrete and of 1.15 for steel.

Thus, design stress in concrete = 0.67*fck / 1.5 = 0.446*fck

design stress in steel = fy / 1.15 = 0.87*fy

At serviceability limit state FOS is taken as 1. Because we want to analyse the actual deflection and cracking in the building.

3. Partial Safty Factor for loads:

It can be seen in Table 18 of IS 456:2000:

For different load combinations, it is given as:

1.5( DL + LL )
1.5( DL + EQ )
0.9DL + 1.5EQ
1.2(DL + LL + EQ)

For serviceability criteria it is given as:

1.0(DL + LL)
1.0(DL + EQ)
DL + 0.8LL + 0.8EQ

Behaviour of Concrete under compression and tension

2020-07-05T11:56:00.002+05:30

The behaviour of concrete is different when it comes to compression and tension. Usually, concrete performs very well under compressive stresses, but when it is subjected to tensile stresses, its performance is considered inferior.

Let us look one by one how concrete behaves under different stresses and what are some terms related to it.

The behaviour of concrete under uniaxial compression:

IS 456:2000 has stated that to test the compressive strength of concrete specimen of 150 mm cube should be tested after 28 days of curing. Then the testing on the concrete cube is performed. The loading test of the concrete cube is strain-controlled, and the uniform strain rate applied is 0.001 mm/mm/min.

IS 456:2000 also recommends testing of 3 samples and the average value of maximum stress should be noted as compressive strength.

Now many people may ask why we perform the strain-controlled test and why not stress-controlled test.

The answer to this question when we perform a strain-controlled test, we can see the post-peak effect on the stress-strain curve. In other words, stress will start decreasing after achieving its maximum value, and thus we will be able to note the compressive strength of concrete. However, in the case of the stress-controlled test, we will not be able to see the post-peak effect.

One thing to note also is that in the USA, a standard cylinder of 300 mm length and 150 mm diameter is tested. It is experimentally shown that the compressive strength of the cylindrical specimen comes out to be less than the compressive strength of the cube.

This leads us to discuss on Influence of size and shape of the specimen on the strength of concrete.

While testing a concrete block, you will notice that some confinement/platens are provided at the top and bottom of the specimen. Those platens generate shear stresses in the sample. The effect of the shear stress reduces as the thickness of the specimen increases. The shear stress development increases the compressive strength. So the compressive strength of the cubical specimen is more than the cylindrical sample. As in the cubical example, the height to breadth ratio is 1, and in the cylindrical specimen, that ratio is 2.

The behaviour of concrete under tension:

Concrete is usually not designed for tensile load for that we put reinforcement in concrete. However, there are other ways by which tensile stresses are imposed on concrete so we should have knowledge about the tensile strength of concrete too. How tensile stresses can generate are due to temperature, flexure, shear, shrinkage, etc.

It is tough to test the direct tensile strength of concrete as it requires perfect alignment, and we have to ensure that secondary stresses are not generating. To encounter that we use some indirect methods like flexure test and cylinder splitting test.

Flexure Test:

In this, a simply supported plain concrete beam is subjected to a three-point loading and the stress distribution here is assumed linear. For linear distribution maximum stress will develop at the extreme fibre of the beam. That stress is also known as modulus of rupture.

Modulus of rupture can be calculated as

fcr = M/Z

Here, M is the moment due to which beam fails.

Z is section modulus.

Note: Since the stress distribution is not linear in the actual scenario IS 456:2000 recommends a different formula.

fcr = 0.7 * ( fck)^0.5

Here, fck is characteristic strength of concrete in MPa.

Splitting Tensile Strength of Concrete:

In this test, the standard size of the cylindrical specimen ( length 30 mm and diameter 15 mm) is loaded along its side on the diametrical plane. This loading ensures uniform tensile stress along the plane of loading.

The splitting tensile strength (fct ) is obtained as :

fct = 2*P / ( 3.14* d * L )

Here, P is the maximum load at which the sample fails.

d is the diameter of the cylinder

and L is the length of the cylinder.

Characteristic Strength of Concrete

2020-07-03T20:23:00.003+05:30

Concrete uses different constituents like coarse aggregate, fine aggregate, water, plasticisers, etc. All of the components come in different variety, and thus the concrete prepared from the same mixture and taken from the same tray can have significantly varying compressive strengths. This variation can be controlled by assuring quality control at the site.

Some ways to have reasonable quality control on the concrete mix can be the following:

Controlling the size of aggregates by proper sieve analysis.
For coarse aggregates, there are tests like flakiness test and elongation test.
For fine aggregates, bulking of sand should be in a reasonable range.
Water content should be proper.
Mixing of components should be uniform.

The graph becomes more and more flatter when quality control is more miserable at the site. If quality control is better, then a sharp peak can be observed. One thing can be concluded that if more specimens are near average compressive strength, your mix is better quality-wise.

The compressive strength of concrete is studied in terms of standard deviation and coefficient of variation.

Coefficient of variation = Standard Deviation / Mean strength

Experimentally it is found that the coefficient of variation of concrete varies between 0.01 to

0.02. Here, we can notice that if the coefficient of variation is near 0.01, then it will infer to reasonable quality control on the concrete mix.

Followings are some essential points regarding testing of concrete block:

150 mm cube specimen is tested under uniaxial compression after 28 days of curing.
As per IS 456: 2000 there should be at least 3 specimens tested, and the final result should be average of three.
The individual strength of blocks should not be more or less than 15% of the average compressive strength. If it happens, the test should be discarded.

Now the question arises why do we need characteristic strength?

From the above statements, we can now understand the uncertainty of the compressive strength values of concrete. So, we need a certain minimum value from a mix. "The characteristic strength is that strength of material below which not more than 5% of tests results are expected to fail".

So, one final conclusion we can make is that the mean strength of concrete is always higher than the characteristic compressive strength of the concrete.

If you want to read about types of concrete, then follow this link:

http://gk1711.blogspot.com/2020/07/types-of-concrete.html

Types of Concrete

2020-07-02T19:20:00.005+05:30

Concrete is the most crucial component of buildings and infrastructure nowadays. It works brilliantly when only compressive forces (stresses) are acting, but its tensile strength is low. Its tensile strength is around 10 - 12% of its compressive strength. Its combination with reinforcement makes its most versatile material. Based on the purpose of use, there are the following types of concrete:

1. Plain Concrete:

It is the most straightforward type of concrete which uses cement (along with adhesive other cementitious materials are used when required), sand, water and coarse aggregate.

It is used where no tensile stress is generated—for example - mass concreting in dams, pedestals, etc.

2. Reinforced Concrete:

This composite material has gained so much popularity because it can sustain tensile stress also. Because of reinforced concrete today, the construction of load-bearing beams and columns are possible to this extent.

Use of reinforcement also provides warning well before the collapse because of its ductile behaviour.

Using less steel can lead to crack propagation, and using more steel will be dangerous too. Because over reinforced member will fail in compression first, and thus the owner will not get the warning of collapse.

To provide the correct amount of reinforcement, there are some provisions in IS 456: 2000 (latest version of code).

3. Prestressed Concrete:

This type of concrete is used when the member is subjected to fatigue loading or in a case where even a hairline crack is prohibited (like pressure vessels, water tanks, etc.).

This concrete uses prestressed steel wires embedded in it so that. While actual loading, there is no tension.