A bar assembly is loaded with force $P$ at one end and constrained at the other end, as shown in the figure below. Determine:

a$.$ The displacement at node $2$ and node $3$

b$.$ The stress in the bar assembly

Assume, $P=5000N,A=250m{m}^2,L=200mm,E=2{10}^4\frac{N}{m}{m}^2$

$2.($a$)$

A cylindrical component is subjected to a tensile stress of $100$ MPa in the axial direction $({}_1),$ a compressive stress of $50$ MPa in the radial direction $({}_2),$ and no stress in the hoop direction $({}_3=0).$ Calculate the von Mises stress acting on the component.

$($b$)$

A $7075$ T$6$ aluminum component has a plane stress element shown below, which is found to have the following stresses:

${}_x=10$ksi,${}_y=25$ksi, and ${}_{xy}=15$ksicw.

What is the factor of safety using von Mises theory?

Note: shear in the counter clockwise direction $($ccw$)$ is positive. Shear in the clockwise direction $($cw$)$ is negative.