Input the solutions on Blackboard, and write the derivations and solutions on paper.
PROBLEM 4 (20pts)
Consider a plane stress trapezoidal plate (bottom $a$, top $2a$, height $2a$)
as shown. Suppose the Airy stress function for the plate is given as,
$\Phi(x, y) = \frac{wx^2}{2} \left(1 - \frac{y}{4a}\right)$
(a) Express the stress components, $\sigma_x$, $\sigma_y$ and $\tau_{xy}$
(b) Show they satisfy the equilibrium equations.
(c) Compute the traction vector $t = \sigma n = \begin{pmatrix} t_x \\ t_y \end{pmatrix}^T$ at A ($x = a/2$, $y = 2a$).
(d) Suppose $a = 2 m$, $w = 20 MPa$, compute the von Mises/effective stress $\sigma_e$ at A.
