Design of beam using Working Stress Method (WSM)

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.













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