$(1)$ Consider a steel beam with a length L of $25$ m and a constant cross$-$section with dimen$-$

sions shown below. The steel has a Young's modulus of $200$GPa and Poisson Ratio of

$0\mathrm{.}30.$ The beam is clamped at its left edge and free at its right edge. In terms of external

loading, it is subject to a downward point force of P$=1$kN halfway its length as well as

an upward constant distributed load Q$\_(0)=5\mathrm{.}0$k$($N$)/($m$).$

$($a$)$ Determine the distribution of the shear stresses in the cross$-$sectional area at x$=8$m$.$

$($b$)$ What percentage of the total resultant shear force is carried by the web in this

cross$-$section?

$($c$)$ Based on your answer for question $1$b$.$ what can you conclude about which features

in a beam's cross$-$section carry the bulk of the shear loading? What do you expect

in that regard of the bending moment, i$.$e$.,$ which features in a beam's cross$-$section

carry the bulk of the loading induced by bending moments?

$(2)$ For a beam with rectangular cross$-$section A$($x$),$ show that the maximum shear stress in

its cross$-$sections is given by:

$\backslash$tau $\_($xy$\_($max$))($x$)=(3)/(2)($V$\_($y$)($x$))/($A$($x$))=(3)/(2)\backslash$tau $\_($xy$)^($avg$)($x$).$

Remark: the above is an often used formula in engineering design and analysis of beams

with rectangular cross$-$sections.