Problems $6$: For the beam shown in the figure below, we are interested in determining the location and value of maximum deflection by using the moment$-$area method. To this end, we are going to take the following steps.

$($a$)$ Compute tangential deviation $t_{CA}$ by drawing a line tangent to the elastic curve $($deflection curve$)$ at $A.$

$($b$)$ Compute the slope at $A({}_A)$ using a geometric equation $t_{CA}=L{}_A,$ where $L$ is the length of the beam. You need to add minus sign to your result of ${}_A$ since the rotation is clockwise.

$($c$)$ Assuming that the point of maximum deflection lies between A and $B,$ find its location measured from support $A($the location is denoted by $D$ in figure $6($b$)).$ Note that the slope of the elastic curve equals zero at the point $($that is$,{}_D=0).$ Use the first moment$-$area theorem; ${}_D={}_A+{}_{AD}$

$($d$)$ Compute length $DE$ by a geometric equation $DE=(AD){}_A.$

$($e$)$ Compute maximum deflection of the beam by subtracting $t_{DA}$ from DE$.t_{DA}$ is the tangential deviation of $D$ with respect to the tangential line at $A.$ Use the second moment$-$area theorem.