Problem #$2$: $(30$ points$)$ An overhanging beam ACB of length $3\mathrm{.}7$ m is pinned at point A$,$ and

supported by a roller at point C$,$ as shown in Figure $2.$ The beam is loaded by an $8$ kN downward

force at the free end B and by a uniform downward load of $3$kN$//$m over the overhanging part of

the beam between the roller at C and the free end at B$,$ as shown. In an x$-$y$-$z coordinate system

$($not shown in the figure$)$ with origin at the centroid of the cross section, the positive x$-$direction

measures horizontal distance to the right of point A$,$ and the positive y$-$direction measures vertical

distances above the centroid of the cross section.

a$)$ Find the reactions at pin A and roller C$.$ Is the problem statically determinate or

indeterminate? Explain.

b$)$ Find the equations for the shear force V$=$V$($x$)$ and the bending moment M$=$M$($x$)$ throughout

the beam. For each cut required, draw a clear free body diagram.

c$)$ Neatly sketch the shear force and bending moment diagrams, and provide the important

values and locations of each. In particular, find the magnitude of the maximum shear force

V$\_($max$)$ in the beam, and find the value of x at which it occurs. Find the magnitude of the

maximum bending M$\_($max$)$ in the beam, and find the value of x at which it occurs.

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