Problem $23\mathrm{.}1$

Suppose a beam with dimensions $(L_x,L_y,L_z)=(2m,1m,1m)$ is fixed on the $x=0$ surface and is otherwise free. The beam has a stiffness of $40$ GPa Poisson's ratio of $0\mathrm{.}3,$ and a linear coefficient of thermal expansion of $25e-\frac{6}{K}.$

a$)$ assuming the internal stress & strain are uniform $($i$.$e$.,$ ignoring anchor effects$)$ calculate how much of a temperature increase would cause it to expand so that its side length in $x$ increases by $0\mathrm{.}5$ mm $($if it was otherwise free$)$

b$)$ apply this increase in FlexPDE with the $x=0$ surface fixed $($but all other surfaces free$)$ and determine how much displacement occurs at the centre of the $x=Lx$ surface.

c$)$ Is the stress in the beam uniform in part b$?$ Explain with the help of at least one painted contour OR elevation plot to demonstrate your point.