Beam ABC has a pin at A and a roller at $C.$ It supports a uniform

load of intensity $q_0$ over one half of it and a concentrated moment

of magnitude $M_0$ at $C.$ The shear and moment expressions are

given. The beam has an "I" shaped cross$-$section shown at right.

Plot the shear and moment along the length of the beam. Find

expressions for the deflection and rotation along the length of the

beam. Find the location of maximum moment and shear in terms

of the dimensionless ratio $=\frac{M_0}{q_0}{L}^2.$ The maximum allowable

normal stress is $50$MPa. Find the dimensions of a suitable cross

section if $M_0=60kN-m,q_0=12k\frac{N}{m},L=6m,$ assuming that $t_w$

$=\frac{d}{40},t_f=\frac{d}{20},$ and $b_f=\frac{d}{2}.$ Calculate the maximum shear

stress for those conditions.

The shear and moment expressions are $($with $x$ measured from the

left end of the beam$)$: