Problem $5\mathrm{.}4$:

Consider the plane stress solution for a semi$-$infinite elastic solid under a concentrated point loading $($see Fig. $5-4($a$)).$

With respect to the axes shown, the Cartesian stress components are given by:

${}_{11}=-\frac{2P{x}_1^2{x}_2}{(x_1^2+x_2^2{)}^2}$

${}_{22}=-\frac{2P{x}_2^3}{(x_1^2+x_2^2{)}^2}$

${}_{12}=-\frac{2P{x}_1{x}_2^2}{(x_1^2+x_2^2{)}^2}$

a$)$ Calculate the maximum shear stress at any point in the body and plot contours of ${}_{\text{m}\text{a}\text{x}}.$

b$)$ Solve the problem with two concentrated forces as shown in Fig. $5-4($b$).$

c$)$ Prove that the stress field far away from the load is equal in both cases.

d$)$ Plot the stresses for each problem at $x_2=10$a and $x_2=100a.$