$($a$)3$D$.$ Yield stress $F_y$ of a steel specimen is found by a tensile test depicted in the figure below. Use the von Mises formula to show that shear stress at yield is $\frac{F_y}{\sqrt[\phantom{2}]{3}}.$

$($b$)2$D$.$ In the hypothetical $2$D test below, $f_c$ is the maximum compressive stress exhibited by the concrete in a press. The dotted plane is inclined at angle $$ from the top surface, whose area is A$.$ Show that shear stress along the dotted plane is $q=\frac{1}{2}{f}_c^{\text{'}}$ when $=45.$

Importance. Why $=45?$ Shear stress is maximum at $=45.$ This can be shown by calculus or by graph. In $($a$),$ the $3$D answer is called von Mises stress. In $($b$),$ the $2$D answer is called Tresca stress. Fifty years ago, steel structures were designed with Tresca stress or $q=0\mathrm{.}5F_y.$ Today, they're designed with the more accurate von Mises stress or $q=0\mathrm{.}577F_y.$