Chapter 2, Problem 14E

The boundary-value problem for a clamped-clamped beam can be written as

$\begin{array}{l}\frac{d^{4}w}{d{x}^4}-p\left(x\right)=0,0\le x\le 1\\ w\left(0\right)=w\left(1\right)=\frac{dw}{dx}\left(0\right)=\frac{dw}{dx}\left(1\right)=1\}\text{boundary conditions}\text{.}\end{array}$ a uniformly distributed load is applied, that is, $p\left(x\right)=p_0$, calculate the approximate beam deflection $\tilde{w}\left(x\right)$ using the Galerkin method. Hint: Assume the approximate deflection as $\tilde{w}\left(x\right)=c\varphi \left(x\right)=c{s}^2{\left(1-x\right)}^2$.