Q $1$ a Show that for a Beam$-$element subjected to uniformly distributed load the Equivalent nodal forces as are as shown in the adjacent diagram

$v(x)=(1-\frac{3x^2}{L^2}+\frac{2x^3}{L^3})v_1+(x-\frac{2x^2}{L}+\frac{x^3}{L^2}),F_{1q}={\displaystyle \underset{0}{\overset{L}{}}}q(x)N_1(x)dx$

$+(\frac{3x^2}{L^2}-\frac{2x^3}{L^3})v_2+(\frac{x^3}{L^2}-\frac{x^2}{L}){}_2,M_{1q}={\displaystyle \underset{0}{\overset{L}{}}}q(x)N_2(x)dx$

$F_{2q}={\displaystyle \underset{0}{\overset{L}{}}}q(x)N_3(x)dx$

$M_{2q}={\displaystyle \underset{0}{\overset{L}{}}}q(x)N_4(x)dx$

Use finite element analysis