Problem #$4-$ Beam Analysis $(15$ points total, $4$ parts$)$

A W$24$x$76$ made of A$36$ structural steel with a yield stress of $36$ ksi $,$ allowable bending stress limit of $21\mathrm{.}6$ ksi and shear stress limit of $14\mathrm{.}4$ ksi and Modulus of Elasticity, $E=29,000,000$ is subjected to a maximum bending moment of $174$ k $-$ ft and a maximum shear force of $51$ kips $.$ Show all work and any assumptions clearly. Complete the following:

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

Part B $(3$ points$)$: Does the calculated bending stress exceed the allowable bending stress and does the calculated shear stress exceed the allowable shear stress?

Part C $(5$ points$)$: Calculate the Moment Strength $($MR$)$ and Shear Strength $($VR$)$ of the beam.

Part D $(3$ points$)$: Determine the adequacy of the beam's design 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?