Learn : Concept of transformed or equivalent section of a RCC beam,load carried by steel.load carried by concrete,section in terms of concrete

**CONCEPT OF TRANSFORMED OR EQUIVALENT SECTION**

Consider an R.C.C. section shown in Fig. 2.2(a) subjected to a compressive load P.

Let A = Area of cross-section

A_{c} = Area of concrete

A_{st} = Area of steel

m = Modular ratio

s_{s} = Stress in steel

s_{c} = Stress in concrete

e_{s} = Strain in steel

e_{c} = Strain in concrete

P_{s} = Load carried by steel

P_{c} = Load carried by concrete

A_{eqc} = Equivalent area of section in terms of concrete

E_{s} = Yong’s modulus of elasticity of steel

E_{c} = Young’s modulus of elasticity of concrete

P = P_{s} + P_{c}

\[ P = \sigma _{s}A_{st}+\sigma _{c}A_{c} \]

The bond between steel and concrete is assumed to be perfect so the strains in steel and the surrounding concrete will be equal

\[ \varepsilon _{s}=\varepsilon _{c} \]

\[\frac{\sigma _{s}}{E}_{s}=\frac{\sigma _{c}}{E}_{c}\]

\[ \sigma _{s}=\frac{E_{s}}{E_{c}}.\sigma _{c}\]

\[\sigma_{s} =m.\sigma _{c}\]

or \[\sigma_{c}=\frac{\sigma_{s}}{m}\]

* It means that stress in steel is m times the stress in concrete or load carried by steel is m times the load carried by concrete of equal area. Using Eqns. (i) and (ii)*

\[ \therefore P=m.\sigma _{c}.A_{st}+\sigma _{c}.A_{c}=\sigma _{c}\left ( m.A_{st}+A_{c} \right )\]

\[ \sigma _{c}=\frac{P}{\left ( A_{c}+m.A_{st} \right )}=\frac{P}{A_{eqc}}\]

The expression in the denominator \[\left ( A_{c}+m.A_{st} \right )\] is called the equivalent area of the section in terms of concrete. It means that the area of steel Ast, can be replaced by an equivalent area of concrete equal to m.Ast as shown in Fig. 2.2(b)

\[A_{eqc}=A_{c}+m.A_{st}\]

\[=A-A_{st}+m.A_{st} \] \[because A_{c}=A-A_{st}\]

\[=A+\left ( m-1 \right )A_{st}\]

Therefore, the concept of modular ratio makes it possible to transform the composite section into an equivalent homogeneous section, made up of one material.