$1\mathrm{.}24$ In Problem $1\mathrm{.}21,$ parts $($b$)$ and $($c$),$ a qualitative plot of the deformations and the rigid$-$body rotations at two locations is sought. Using a spreadsheet program or a mathematics software package, make separate plots of the edges:

$x=0,0ya$

$y=a,0xb$

$x=b,0ya$

in the deformed state. For the problem, let $a=1,b=10,v=0\mathrm{.}3,$ and $p\frac{g}{E}=2{10}^{-8}.$ It may be necessary to exaggerate $($scale$)$ the actual deflections to visualize the difference between the deformed and undeformed shapes. If you solved Problem $1,21$ carlier, do the results agree?

$1\mathrm{.}21$ A thin, uniformly thick plate of thickness $t$ is hanging under its weight as shown. The corresponding displacement field in the $xy$ plane can be approximated by

$(x,y)=\frac{g}{2E}(2bx-x^{2}cdotsv{y}^2),v(x,y)=-v\frac{pg}{E}y(b-x)$

chaptir I $-$ Force, Stress, Strain, and Displacement

where $p$ and $E$ are the mass density and Young's modulus of the plate material, tespectively, and $g$ is the gravity constant.

$($a$)$ Determine the corresponding plane stress field ${}_x(x,y),{}_y(x,y)$ and ${}_{ij}(x,y)$ and comment on the validity of the results.

$($b$)$ Qualitatively, draw the deformed and undeformed shape of the edges on the same drawing. For the sake of clarity, exaggerate the deflections of the edges.

$($c$)$ Determine the rotation of the plate at points A and $B.$ Do the rotations agree with the sketch of part $($b$)?$

Problitum $1\mathrm{.}21$