$(1)$

Consider the total potential energy for a beam described by Timoshenko beam theory:

$={\displaystyle \underset{0}{\overset{L}{}}}[\frac{EI}{2}(\frac{d}{dx}{)}^2+\frac{GA}{2}(\frac{dw}{dx}-{)}^2-qw]dx$

Here $w$ is the vertical displacement, $$ is the rotation angle of a fiber, $q(x)$ is the distributed vertical load, $E,A,G,,$ and I are Young's modulus, the beam cross sectional area, the shear modulus, the shear correction factor, and the beam moment of inertia, respectively.

$($i$)$ Show that the Euler$-$Lagrange equations are:

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

$\frac{d}{dx}[GA(\frac{dw}{dx}-)]+q=0$

$($ii$)$ Show that the boundary conditions at $x=0$ and $x=L$ are:

$\backslash$table$[[$Kinematic BC$,,$Natural BC$],[$ prescribed,or$,EI\frac{d}{dx}=0$