$2\mathrm{.}16.$ A $3-$m by $2-$m rectangular thin plate is deformed by the movement of point $B$ to $B^{\text{'}}$ as shown by

the dashed lines in Fig. P$2\mathrm{.}16.$ Assuming a displacement field of the form $u=c_{1}xy$ and $v=c_{2}xy,$

wherein $c_1$ and $c_2$ are constants, determine $($a$)$ expressions for displacements $u$ and u ; $($b$)$ strain

components ${}_x,{}_y,$ and ${}_{xy}$ at point $B$; and $($c$)$ the normal strain ${}_x,$ in the direction of line $QB.$

Verify that the strain field is possible.

Figure P$2\mathrm{.}16.$

Answers

a$.u=500({10}^{-6})xy,v=250({10}^{-6})xy$

b$.({}_x{)}_B=1000,({}_y{)}_B=750,({}_{xy}{)}_B=2000$

c$.{}_{x^{\text{'}}}=875+125cos67\mathrm{.}38+1000sin67\mathrm{.}38=1846$