$*6\mathrm{.}87$ The cantilever beam shown consists of a $Z$ shape of $\frac{1}{4}-$in$.$ thickness. For the given loading, determine the distribution of the shearing stresses along line $A^{\text{'}}{B}^{\text{'}}$ in the upper horizontal leg of the Z shape. The $x^{\text{'}}$ and $y^{\text{'}}$ axes are the principal centroidal axes of the cross section, and the corresponding moments of inertia are $I_{x^{\text{'}}}=166\mathrm{.}3i{n}^4$ and $I_{y^{\text{'}}}=13\mathrm{.}61i{n}^4.$

Fig. P$6\mathrm{.}87$

$($b$)$