$"6\mathrm{.}87$ The cantilever beam shown consists of a $Z$ shape of $\frac{1}{4}-$in$.$ thick ness. 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 ares 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$)$

$"6\mathrm{.}88$ For the cantilever beam and loading of Prob. $6\mathrm{.}87,$ determine the distribution of the shearing stresses along line $B^{\text{'}}{D}^{\text{'}}$ in the vertical web of the $Z$ shape.