Consider the following beam deformation problem with a constrained

transverse displacement at the RHS $($Right Hand Side$).($Note that this problem has two unknown nodal displacements at the RHS of the beam, W and Theta transverse displacement and rotation of the node. There are no applied forces or moments. Define the discrete finite element equilibrium equations in matrix form using the primal functional I$($w$).$ Use the Lagrange multiplier method to apply the displacement constraint. You do not have to solve the equations.

$\{$K$\}\{$D$\}=\{$Gf$\}$

The total potential energy functional I$($w$)$ and the beam displacement

approximation $w(x)$ are shown below:

$I(w)=\frac{1}{2}{\displaystyle \underset{0}{\overset{L}{}}}EI(\frac{d^{2}w}{d{x}^2}{)}^{2}dx-M\frac{dw}{dx}{|}_L$

$w(x)={}_1(x)W+{}_2(x)$

${}_1(x)=3(\frac{x}{h}{)}^2-2(\frac{x}{h}{)}^3$

${}_2(x)=-h[(\frac{x}{h}{)}^2-(\frac{x}{h}{)}^3]$