You are asked to determine two$-$term Ritz solution using variational method. Consider a beam subjected to transverse loading as shown below: $(40$ Pts$.)$

Find appropriate approximate functions ${}_j$ and ${}_0(5$ Points$)$

Compute the generalized stiffness matrix $K(5$ Points$)$

Compute the generalized force vector $\{F\}(5$ Points$)$

Determine a two$-$term Ritz solution for transverse beam deflection. $(10$ Points$)$

Determine a point where the peak deflection occurs. $(5$ Points$)$

Plot your results for the transverse deflection, slope, and internal reactions. $(5$ Points$)$

MUST validate and comment on your results. $(5$ Points$)$

where $L=8($in$),$ EIzz $={10}^6(l{b}^{-}i{n}^2),p_0=10(l\frac{b}{i}n),F=100(lbs)$

Hints:

$\begin{array}{cc}\text{E}\text{B}\text{C}& NBC\\ v(0)& V_y(L)\\ v^{\text{'}}(0)& M_z(L)=E{V}^{\text{'}\text{'}}(L)=0\\ v(\frac{L}{4})=0& ?\end{array}=0=E{V}^{\text{'}\text{'}}(L)=0=0=0=E{V}^{\text{'}\text{'}}(L)=0=0=-EI{V}^{\text{'}\text{'}\text{'}}(L)=-F=0=E{V}^{\text{'}\text{'}}(L)=0=0=-EI{V}^{\text{'}\text{'}\text{'}}(L)=-F=0$