Problem #$3-$ Beam Analysis $(20$ points total, $5$ parts$)$

A W$18$x$65$ made of A$36$ structural steel with a yield stress of $36,000$ psi, allowable shear stress limit of $14,400$ psi and Modulus of Elasticity, $E=29,000,000$ psi is subjected to a maximum bending moment of $428k-ft$ and a maximum shear force of $72$kips. Determine the following:

Part A $(4$ points$)$: Calculate the bending stress in the beam. Additionally, calculate the shear stress in the beam using the average web shear approach.

Part B $(4$ points$)$: Find the allowable bending stress by applying a factor of safety of $1\mathrm{.}67$ to the yield stress. The allowable shear stress is given in the problem statement.

Part C $(4$ points$)$: Does the calculated bending stress exceed the allowable bending stress? Does the calculated shear stress exceed the allowable shear stress?

Part D $(4$ points$)$: Calculate the Moment Strength $($MR$)$ and Shear Strength $($VR$)$ of the beam.

Part E $(4$ points$)$: Determine the adequacy of the beam by comparing the moment strength $($MR$)$ vs$.$ maximum bending moment $($Mmax$)$ and the shear strength $($VR$)$ Vs$.$ the maximum shear force $($Vmax$).$ Does the beam fail in bending and$/$or shear?