User:Eml5526.s11.team2.stewart/HW1

=Problem 1.1= = Given = $$  \displaystyle \frac{\partial }\left[ {E(x)A(x)\frac} \right] + f(x,t) = m(x)\frac $$ = Find = Prove the given equation using a balance of forces for the finite element shown from lecture slide. = Solution =

=Problem 1.2= = Given = $$  \displaystyle \frac{\partial }\left[ {E(x)A(x)\frac} \right] + f(x,t) = m(x)\frac $$ With a rectangular cross section as shown:



= Find = Prove the given equation using a balance of forces for the finite element shown from lecture slide. = Solution =

$$ \displaystyle \frac + f(x,t) = \underbrace {\rho \left( {x + \frac{2}} \right)A\left( {x + \frac{2}} \right)}_{m\left( {x + \frac{2}} \right)}\frac $$

$$ \displaystyle \mathop {\lim }\limits_{dx \to 0} \frac + f(x,t) = \mathop {\lim }\limits_{dx \to 0} \rho \left( {x + \frac{2}} \right)A\left( {x + \frac{2}} \right)\frac $$

$$ \displaystyle \partial \frac + f(x,t) = \rho (x)h(x)b\frac $$

$$ \displaystyle \sigma (x) = E(x)\frac $$

$$ \displaystyle \frac{\partial }\left[ {E(x)h(x)b\frac} \right] + f(x,t) = \rho (x)h(x)b\frac $$