Problems $6-8$ Lego man's cousin Duplo man is sitting around when someone puts a $50$ kg weight on his head. Unable to move $($since he doesn't have legs$)$ he just sits there and his body deforms.

Assume that he is made of solid acrylonitrile butadiene styrene, which has a Young's modulus of $350$ MPa and a Poisson's ratio of $0\mathrm{.}4.$ Additionally assume that this head is a cylinder, his upper body is a pyramidal frustum $($four sided pyramid with its top cut off$),$ and this lower body is a rectangular prism, with all dimensions given in the figure below.

Points A$,$ B and C all lie on Duplo man's midline and are located halfway between the top and bottom of their respective body sections.

Write the $3-$D stress tensor at point A after the weight is placed on Duplo man's head

$*7)$ Determine the maximum shearing stress $({}_{x^{\text{'}}{y}^{\text{'}}})$ in a plane perpendicular to the $z-$axis and existing at point $B,$ and the angle of $x^{\text{'}}-y^{\text{'}}$ relative the current $x-y$ coordinate system.

$*8)$ How wide is Duplo Man's lower body section in the $x-$direction after the weight is placed on his head? $($Note: Assume the floor is frictionless$)$