Question $3$

Consider a beam supported with a roller on one end and fixed on the other end $($see the Figure $($a$)$ below$).$ The beam is subjected to a linearly increasing distributed load $(w_0).$

The function that represents the elastic curve $($dotted lines$)$ in Figure $($b$)$ is as follows:

$y=\frac{w_0}{120\text{E}\text{I}\text{L}}(-x^5+2L^2{x}^3-L^{4}x)$

where: $L=450cm,E=50000000\frac{N}{c}{m}^2,I=30000c{m}^4$ and $w_0=17500\frac{N}{c}m.$

Apply the bisection method to find the point of maximum deflection, which occurs where $d\frac{y}{d}x=0.$ Once you have identified this value of $x,$ substitute it back into the original equation to calculate the maximum deflection.

Hint: Think carefully about the number of iterations you might need to get to the answer.

$[25$ marks$]$