A short concrete column measuring 12 square inches is reinforced with four #8 bars ( $A_{s} =4\times 0.79 \space in.^2=3.14 \space in.^2$ ) and supports an axial load of 250 k. Steel bearing plates are used top and bottom to ensure equal deformations of steel and concrete. Calculate the stress developed in each material if

$E_{c} =3\times 10^6 \space psi$ and

$E_{s} =29\times 10^6 \space psi$

Question

A short concrete column measuring 12 square inches is reinforced with four #8 bars ( $A_{s} =4\times 0.79 \space in.^2=3.14 \space in.^2$ ) and supports an axial load of 250 k. Steel bearing plates are used top and bottom to ensure equal deformations of steel and concrete. Calculate the stress developed in each material if

$E_{c} =3\times 10^6 \space psi$ and

E_{s} =29\times 10^6 \space psi

[latex]E_{c} =3 imes 10^6 \space psi[/latex] and

[latex]E_{s} =29 imes 10^6 \space psi[/latex]

Accepted Answer

From equilibrium,

[latex]\left[\sum{F_{y}=0 } ight] - 250 \space k + f_{s} A_{s} +f_{c} A_{c} =0[/latex]

[latex] A_{s}=3.14 \space in.^2[/latex]

[latex]A_{c}= (12^{\prime \prime } imes 12^{\prime \prime })-3.14 \space in.^2\cong 141\space in.^2[/latex]

[latex]3.14 \space f_{s}+141 \space f_{c}=250 \space k[/latex]

From the deformation relationship,

[latex] \delta _{s} = \delta _{c} \space ; \space L_{s}=L_{c}[/latex]

[latex] herefore \frac{ \delta _{s}}{L} =\frac{ \delta _{c}}{L} [/latex]

and

[latex]\varepsilon_{s}=\varepsilon_{c}[/latex]

Because

[latex]E=\frac{f}{\varepsilon} [/latex]

and

[latex]\frac{f_{s}}{E_{s}} =\frac{f_{c}}{E_{c}} [/latex]

[latex]f_{s}=f_{c}\frac{E_{s}}{E_{c}}=\frac{29 imes 10^3(f_{c})}{3 imes 10^3} =9.67 \space f_{c}[/latex]

Substituting into the equilibrium equation

[latex]3.14(9.76f_{c})+141f_{c}=250[/latex]

[latex]30.4f_{c}+141f_{c}=250[/latex]

[latex]171.4f_{c}=250[/latex]

[latex]f_{c}=1.46 \space ksi[/latex]

[latex] herefore f_{s}=9.67(1.46) \space ksi [/latex]

[latex] f_{s}=14.1 \space ksi[/latex]

Question 5.21: A short concrete column measuring 12 square inches is reinfo...

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