Analysis and Design of Doubly Reinforcement Section by WSM

 

Here are the steps for analysing and designing the RC section by WSM.

A) ANALYSIS.

1) Steps for determining Moment of Resistance.

i) First determine the depth of critical neutral axis using the following formula.

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

ii) Find depth of actual neutral axis using following formula.

`bx^2/2+(1.5m-1)A_(sc)×(x-d')=mA_(st)(d-x)`

iii) Compare `x` & `x_c`.

    If `x` `< x_c` ( Section is under reinforced so limiting value is `sigma_(st)`.)

    So find `sigma'_(cbc)` using `(sigma'_(cbc))/(sigma_(st)/m)=x/(d-x)`

And find `sigma_(sc)` using ` sigma_(sc)=sigma'_(cbc)(x-d')/x`

     Now` MOR = (sigma'_(cbc))/2bx(d-x/3)+(1.5m-1)A_(sc)sigma_(sc)(d-d')`

   If `x` >` x_c` ( Section is over reinforced so limiting value is `sigma_(cbc)`.)

  So find only `sigma_(sc)` using ` sigma_(sc)=sigma_(cbc)(x-d')/x`

   Now `MOR =sigma_(cbc)/2bx(d-x/3)+(1.5m-1)A_(sc)(d-d')`

2) Steps for determining Actual Stresses.

i) Find depth of actual neutral axis. 

`bx^2/2+(1.5m-1)A_(sc)×(x-d')=mA_(st)(d-x)`

ii) Find `sigma_(sc) `in terms of `sigma_(cbc)` using  following formula.

`sigma_(sc)=sigma_(cbc)(x-d')/x`

iii) Also find `sigma_(sc) `in terms of `sigma_(cbc)`using following formula.

`BM=MOR=sigma_(cbc)/2bx(d-x/3)+(1.5m-1)A_(sc)(d-d')`

iv) Solve the equations and find `sigma_(cbc)` & `sigma_(sc)` using calculator.

v) Find `sigma_(st)` using the following formula.

`sigma_(cbc)/(sigma_(st)/m)=x/(d_x)`

vi) Find `sigma_(sc)` using the following formula.

`sigma_(sc)=1.5m×(sigma'_(cbc))`

B) Design.

i) Assume the section is balanced and find depth of neutral  axis using following formula.