Problem $25\mathrm{.}1$

A rectangular prism beam made of a material with $E=1$GPa and $v=0\mathrm{.}2$ has one corner at the origin and the opposite one at $(L_x,L_y,L_z)=(3,0\mathrm{.}5,1).$ The beam is fixed on the $x=0$ surface and has an $x-$directed stress distributed over its $x=L_x$ surface of ${}_x=\frac{y-0\mathrm{.}25(m)}{0\mathrm{.}25(m)}50$kPa.

a$)$ Determine the moment applied by this stress distribution

b$)$ Determine this beam's flexural rigidity relevant for bending moments in the z$-$direction.

c$)$ Determine the curvature and radius of curvature the beam will bend into $($ignoring anchor effects$)$

d$)$ Determine the transverse displacement of the end of the beam from this bending.