Problem $1$

Your colleague asks you to help them with the design of the beam shown on the right. Specifically, you are asked to perform the following so that the moment connection at A can be designed:

$($a$)$ Use the M$$ller$-$Breslau principle to develop qualitative influence lines $($IL$)$ to determine the live load patterns that will produce maximum $($positive and negative$)$ shear forces and bending moments at $A.$ The live load consists of a distributed load, $w_L=1\mathrm{.}0ki\frac{p}{f}t$ and one concentrated load, $P_L=7\mathrm{.}5$kip.

$($b$)$ Determine the maximum $($positive and negative$)$ factored shear forces, $V_u$ at A and their associated bending moments.

$($c$)$ Determine the maximum $($positive and negative$)$ factored bending moments, $M_u$ at A and their associated shear forces.

For $($b$)$ and $($c$)$: Use the three$-$moment equation for your calculations using the moment at $A$ as the redundant. Assume a dead load, $w_D=125l\frac{b}{f}t.$ Use standard ASCE $7-22$ load factors and combinations.