Question $4$ Fibre Section

Consider the following T$-$section. The material of this section is elasto$-$perfectly$-$plastic

such that its Young's modulus $E$ is ${10}^4$ unit and yield stress ${}_y$ is $10$ unit.

a$)$ Given its normal strain distribution, determine the $2$D section resisting forces

$s_r$ and tangent stiffness $k_s.$ Use the mid$-$point rule with $4$ integration points in

total and comment the accuracy by comparing the numerical solution against the

exact analytical solution.

b$)$ What would be the section forces and stiffness matrix if they were calculated

with respect to the centroid of the section.

c$)$ Increase the curvature by $10\%,$ while keeping the axial force unchanged. De$-$

termine the corresponding bending moment about the centroid, not the original

reference axis. Refer to the transformation process in part $($b$).$

d$)$ Write a script to calculate and plot its moment$-$curvature relationship for this

constant axial force. Report the bending moment about the centroid, not the

original reference axis. Use an appropriate step size for the curvature increment.

Make the curvature large enough to have the bending moment close to the plastic

moment capacity.