Concept of transformed or equivalent section of a RCC beam

R.C.C. Structure design Working Stress method
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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.

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