Problem $4$

The $30cm45cm$ rectangular reinforced concrete cross section is reinfroced by elstoplastic steel bars centered at $5cm$ from the bottom edge. The reinforcing steel is elasto$-$plastic with a yielding stress ${}_y=400$MPa and a modulus of elasticity $E=200$GPa. The concrete crushes at a maximum strain ${}_{cr}=4\mathrm{.}0{10}^{-3}m\frac{m}{m}m$; its peak stress is ${}_0=30$MPa which takes place at a strain ${}_0=2\mathrm{.}0{10}^{-3}m\frac{m}{m}m.$

Assuming that concrete cannot carry any tenile stresses, the compression stress$-$strain relation for concrete is modeled by an equation of the form:

$\frac{}{{}_o}=2\mathrm{.}1(\frac{}{{}_o})-1\mathrm{.}3(\frac{}{{}_o}{)}^2+0\mathrm{.}2(\frac{}{{}_o}{)}^3$

Determine the area of steel $A_s$ required so that the curvature at concrete crushing is $=\frac{{}_{\text{m}\text{a}\text{x}}}{c}=2{10}^{-4}\frac{1}{c}m.$

Determine the maximum moment that the section can carry.

$3$