Concept of transformed or equivalent section of a RCC beam

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.

Letย ย ย ย ย  A ย ย ย ย ย ย ย  = Area of cross-section

Acย ย ย ย ย ย ย  = Area of concrete

Astย ย ย ย ย ย  = Area of steel

mย ย ย ย ย ย ย ย  = Modular ratio

ssย ย ย ย ย ย ย  = Stress in steel

scย ย ย ย ย ย ย  = Stress in concrete

esย ย ย ย ย ย ย ย  = Strain in steel

ecย ย ย ย ย ย ย ย  = Strain in concrete

Psย ย ย ย ย ย ย ย  = Load carried by steel

Pcย ย ย ย ย ย ย ย  = Load carried by concrete

Aeqcย ย ย ย  = Equivalent area of section in terms of concrete

Esย ย ย ย ย ย ย  = Yong’s modulus of elasticity of steel

Ecย ย ย ย ย ย ย  = Young’s modulus of elasticity of concrete

Pย ย ย ย ย ย ย ย ย  = Ps + Pc

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

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

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

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

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

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

orย ย ย ย ย ย ย  \[\sigma_{c}=\frac{\sigma_{s}}{m}\]

ย ย ย ย ย ย ย ย ย ย ย  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)

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

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

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)

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

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

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

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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  1. Best content ๐Ÿ‘๐Ÿ‘ Thank you very much for post such material ๐Ÿ™๐Ÿ™๐Ÿ™

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