Lecture $24$ Complete Elasticity with Thermal Expansion

Problem $24\mathrm{.}1$

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, except for applied stresses. 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 the displacement of the $(Lx,Ly,Lz$ corner of the beam would displace in $x,y,$ and $z$ in response to a temperature increase of $10C,$ an axial stress of $4$ MPa in the x $-$direction applied via a force on the $x=Lx$ face, and applied shear stresses to give the beam a uniform shear stress of ${}_{xz}=1$MPa.

b$)$ Use FlexPDE to determine the displacement $(Lx,Ly,Lz$ corner

c$)$ Explain any discrepancy $($or agreement$)$ between part a and b$.$