Consider the simply supported beam structure shown in Figure $3.$

Figure $3.$ Simply supported elastic beam.

The beam has constant flexural stiffness EI along its entire length $(2$L$).$ A vertical point load $P$ is applied at

the beam midpoint.

$[3$a$](3$ pts$)$ Write down the bending moment, $M(x),$ as a piecewise linear function.

$[3$b$](7$ pts$)$ Starting from the differential equation,

$\frac{d^{2}y}{d{x}^2}=[\frac{M(x)}{EI}]$

and appropriate boundary conditions, derive a formula for the beam displacement $y(x)$ over the inter$-$

val $0xL.$ What is the location and value of the maximum vertical displacement?

Show all of your working.