Question 18.1: Determine the yield moment, the plastic moment and the shape...

The elastic and plastic neutral axes of a rectangular cross section coincide (Eq. (18.3))

A_{1}=A_{2}={\frac{A}{2}}      (18.3)

and pass through the centroid of area of the section. Thus, from Eq. (18.1)

M_{\mathbf{Y}}={\frac{\sigma_{\mathbf{Y}}I}{y_{1}}}       (18.1)

M_{\mathrm{Y}}=\frac{\sigma_{\mathrm{Y}} b d^{3} / 12}{d / 2}=\sigma_{\mathrm{Y}} \frac{b d^{2}}{6}     (i)

and from Eq. (18.4)

M_{\mathrm{P}}=\sigma_{\mathrm{Y}}{\frac{A}{2}}({\bar{y}}_{1}+{\bar{y}}_{2})      (18.4)

M_{\mathrm{P}}=\sigma_{\mathrm{Y}} \frac{b d}{2}\left(\frac{d}{4}+\frac{d}{4}\right)=\sigma_{\mathrm{Y}} \frac{b d^{2}}{4}       (ii)

Substituting for M_{\mathrm{P}} and M_{\mathrm{Y}} in Eq. (18.7)

f={\frac{M_{\mathrm{P}}}{M_{\mathrm{Y}}}}={\frac{\sigma _{\mathrm{Y}}Z_{\mathrm{P}}}{\sigma_{\mathrm{Y}}Z_{\mathrm{e}}}}={\frac{Z_{\mathrm{P}}}{Z_{\mathrm{e}}}}      (18.7)

we obtain

f=\frac{M_{\mathrm{P}}}{M_{\mathrm{Y}}}=\frac{3}{2}     (iii)

Note that the plastic collapse of a rectangular section beam occurs at a bending moment that is 50 \% greater than the moment at initial yielding of the beam.