Problem $1.$

Consider a cable loaded statically by a sinusoidal distribution of transverse load:

$q\frac{?}{b}=ar(q)sin(\frac{x}{L})$

with $\frac{?}{\text{b}\text{a}\text{r}}(q)=70,L=25.$ The prestressing force is

$P=\frac{20}{b}ar(q)L.$

The left$-$hand end is pinned, and there's no force applied at the right$-$hand end. Compute the

approximate solution for the deflection of the wire from the Galerkin formulation. Consider a

one$-$term approximation with the test function ${}_1=x,$ and the basis function $N_1=sin(\frac{x}{2L}).$

a$)$ Obtain the analytical solution of the problem.

b$)$ Compute the multiplier $w_1$ using Galerkin method and plot the analytical and numerical

solutions. Compare analytical and numerical solutions.

c$)$ Compute the force residual $r_F$ at $x=L.$

d$)$ Compute the balance residual $r_B$ for the approximate solution:

$r_B=P\frac{de{l}^2{}_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}}}{del{x}^2}+q(x)$