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
Ac = Area of concrete
Ast = Area of steel
m = Modular ratio
ss = Stress in steel
sc = Stress in concrete
es = Strain in steel
ec = Strain in concrete
Ps = Load carried by steel
Pc = Load carried by concrete
Aeqc = Equivalent area of section in terms of concrete
Es = Yong’s modulus of elasticity of steel
Ec = Young’s modulus of elasticity of concrete
P = Ps + Pc
\[ P = \sigma _{s}A_{st}+\sigma _{c}A_{c} \]
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.
