$5\mathrm{.}10$ Develop a least$-$squares finite element model of the following set of nonlinear equations

that arise in connection with the Timoshenko beam theory $($assume that EA$,$ EI$,$ and

$GA{K}_s$ are constant$)$:

$-\frac{dN}{dx}=f$

$-\frac{d}{dx}[GA{K}_s(+\frac{dw}{dx})]-\frac{d}{dx}(N\frac{dw}{dx})=q$

$-\frac{d}{dx}(EI\frac{d}{dx})+GA{K}_s(+\frac{dw}{dx})=0$