Analysis and Design of Singly Reinforced Beam By Working Stress Method(WSM)

Here are the steps for analysing and designing singly reinforced section.

A)  ANALYSIS. 

1) Steps for determining Moment of Resistance.

i. First calculate the depth of critical neutral axis `(x_c)`using the following formula.

`sigma_(cbc)/(sigma_(st)/m)=x_c/(d-x_c)`

ii. Find the depth of actual neutral axis `x` using this formula.

`b×x×x/2=mA_(st)×(d-x)`

iii. Compare `x` and `x_c`.

    If `x` < `x_c` ( under reinforced section )

    `MOR = sigma_(st) ×A_(st)×(d-x/3)`

   If `x ` > `x_c` (over reinforced section)

     `MOR = sigma_(cbc)/2×b×x×(d-x/3)`

2) Steps for determining Actual Stresses.

i) First calculate depth of actual neutral axis using following formula.

`b×x×x/2=mA_(st)×(d-x)`

ii) Calculate `sigma_(st)` using following formula.

`MOR = sigma_(st) ×A_(st)×(d-x/3)`

iii) Find ` sigma_(cbc)`  using following formula.

`sigma_(cbc)/(sigma_(st)/m)=x_c/(d-x_c)`

B) DESIGN.

i) Assume balanced section. Assume `d/b=1.5` to ` 2`

i) First calculate position of neutral axis in terms of `d` using this formula.

`sigma_(cbc)/(sigma_(st)/m)=x_c/(d-x_c)`

ii) Find MOR in terms of `b` using following formula.

`MOR = sigma_(cbc)/2×b×x×(d-x/3)`

iii) Equate BM and MOR and find `b` and  `d`.

iv) Find  `A_(st)` using the following formula.

`MOR = sigma_(st) ×A_(st)×(d-x/3)`

v) Find number of bars required assuming a suitable diameter of bar using following formula.

`n= A_(st)/(pi×phi^2/4)`

Where,

 •`sigma_(cbc)` is permissible stress of concrete.

•`sigma_(st)` is permissible stress of steel`

•`A_(sc)` is Area of steel 

•`x_c` is depth of critical neutral axis 

•`x` is depth of neutral axis

•`d` is effective depth of beam

•`b` is width of beam

•`MOR` is Moment of resistance

•`m` is modular ratio `(m=280/(3sigma_(cbc)))`

•`n` number of bars

•`phi` is diameter of bar