Show me the steps to solve GIVEN: A simply$-$supported beam is loaded at its center with a point load $P=40\mathrm{.}0kN.$

The beam is $L=4\mathrm{.}00m$ long. The overall cross$-$section of the beam is $100mm$ wide by $320$

$mm$ deep. Three pieces of wood are used to make the beam: one $100mm200mm$ piece forms

the core, and two $100mm60mm$ pieces are glued to the middle piece $-$ one at the top and one

at the bottom, as shown below.

The wood has Normal $("$yield$")$ Strength $S_y=45$MPa, and Shear Strength ${}_y=25$MPa.

The glue has Shear Strength ${}_y=12$MPa.

REQ'D: Considering the load on the beam, and the given strengths:

$($a$)$ Determine the Factor of Safety for the wood.

$($b$)$ Determine the Factor of Safety for the glue.

For $($a$)$ and $($b$),$ if you calculate that the system will fail, indicate where failure occurs.