Write a computer program that employs a finite$-$difference method to approximate the

distribution of stresses inside the dam.

Data: $w_1=50ft,w_2=200ft,h_1=250ft,h_2=50ft$

${}_w=62\mathrm{.}4\frac{lbm}{f{t}^3},{}_c=144\frac{lbm}{f{t}^3},g=32\mathrm{.}2\frac{ft}{se{c}^2}$

Let $x=y=50ft.$ along $AE$ and $ED$ but there is tail$-$water at $CD.$The figure below is a simplified view of the cross section of a wide concrete dam. The

internal stresses ${}_x,{}_y,$ and ${}_{xy}($considered positive as shown in the figure$)$ are given by:

${}_x=+\frac{de{l}^2}{del{y}^2},{}_y=+\frac{de{l}^2}{del{x}^2},{}_{xy}=-\frac{de{l}^2}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}},$ and the potential functions $$ is:

$=-{}_{c}gx$ where ${}_c$ is the density of $c$

The Airy stress function $$ is the solution of the biharmonic equation:

$gra{d}^4=\frac{de{l}^4}{del{x}^4}+2\frac{de{l}^4}{del{x}^{2}del{y}^2}+\frac{de{l}^4}{del{y}^4}=0$

:By introducing the variable $u=gra{d}^2,$ Airy equation is solved by solving Poisson's equation

twice in succession, i$.$e$.gra{d}^{2}u=0,$ with $u=0$ along the edges and,

$gra{d}^2=u,$ with $=0$ along the edges.

Thus,

$\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{v}^2}=u(x,y)$

The boundary conditions are zero normal and tangential stresses along ABC, with ${}_{xy}=0$

and ${}_y=-{}_{w}gx$ along $AE$ and $CD,$ where ${}_w$ is the density of water. Along $DE,$ assume as

an approximation that ${}_{xy}$ is constant and that ${}_x$ at any point is proportional to the height of

concrete above that point.

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