Appendix. Equations for nonlinear beam element $($see Homework $-4-$ Solutions.pdf$)$

Shape function:

$\{[u],[v]\}=N*u=\left[\begin{array}{cccccc}{N}_5(x)& 0& 0& N_6(x)& 0& 0\\ 0& N_1(x)& N_2(x)& 0& N_3(x)& N_4(x)\end{array}\right]\{[u_1],[w_1],[{}_1],[u_2],[w_2],[{}_2]\}$

where $N_1=\frac{1}{l^3}(2x^3-3x^{2}l+l^3),N_2=\frac{1}{l^3}(x^{3}l-2x^2{l}^2+x{l}^3),$

$N_3=\frac{1}{l^3}(-2x^3+3x^{2}l),N_4=\frac{1}{l^3}(x^{3}l-x^2{l}^2),$

$N_5=\frac{1-x}{l},N_6=\frac{x}{l}.$

Strain$-$displacement matrix:

$[B^l]=[B_1^l]-z*[B_2^l]$

Where

$[B_1^l]=\left[\begin{array}{cccccc}-\frac{1}{L}& 0& 0& \frac{1}{L}& 0& 0\end{array}\right]$

$[B_2^l]=\frac{1}{L^3}\left[\begin{array}{cccccc}0& 12x-6L& 6xL-4L^2& 0& -12x+6L& 6xL-2L^2\end{array}\right]$

$[G]=\frac{1}{L^3}\left[\begin{array}{cccccc}0& 6x^2-6xL& 3x^{2}L-4x{L}^2+L^3& 0& -6x^2+6xL& 3x^{2}L-2x{L}^2\end{array}\right].$