Problem $1.(40$ points$)$ For the beam shown below. consider $L=3m.$

$($a$)$ Draw the influence line for

$($i$)$ vertical reaction at $B,$

$($ii$)$ shear at left of $B,$

$($iii$)$ shear at right of $B,$ and

$($iv$)$ moment at B

$($b$)$ The beam is subjected to a distributed dead load of $1k\frac{N}{m}$ and a concentrated live load of $P.$ The shear and moment capacities of beam's cross$-$section are $40kN$ and $80kN.m,$ respectively. These capacities are the same for positive and negative directions of internal forces and moments. Find the maximum downward concentrated force $P$ such that the beam does not fail at $B.$

Note $(1)$: The supports at A and B are a roller, and at D is a roller guide. Also, there is a hinge at C$.$

Note $(2)$: Both the shape of the influence line and the values at key points of the influence lines should be specified.