$5\mathrm{.}8$ Develop the mixed finite element model of the following set of nonlinear equations

among the variables $(u,w,M)$ of the Euler$-$Bernoulli beam theory with the von K$$rm$$n

strains:

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

$-\frac{d^{2}M}{d{x}^2}-\frac{d}{dx}(N\frac{dw}{dx})=q$

$\frac{M}{EI}+\frac{d^{2}w}{d{x}^2}=0$

where $f(x)$ and $q(x)$ are the distributed axial and transverse forces, respectively, EA

and $EI$ are the axial and bending stiffnesses, and $N$ is known in terms of $u$ and $w$ as

$N=EA[\frac{du}{dx}+\frac{1}{2}(\frac{dw}{dx}{)}^2]$

Also, compute the tangent stiffness coefficients. Present the finite element model in the

standard format $[$see$,$ for example, Eq$.(5\mathrm{.}2\mathrm{.}32)]$ and define all coefficients.