Question 11.10: Determine the deflection of end A of the cantilever beam AB ...
Determine the deflection of end A of the cantilever beam A B (Fig. 11.31), taking into account the effect of (a) the normal stresses only, (b) both the normal and shearing stresses.
(a) Effect of Normal Stresses. The work of the force \mathbf{P} as it is slowly applied to A is
U=\frac{1}{2} P y_{A}
Substituting for U the expression obtained for the strain energy of the beam in Example 11.03, where only the effect of the normal stresses was considered, we write
\frac{P^{2} L^{3}}{6 E I}=\frac{1}{2} P y_{A}
and, solving for y_{A},
y_{A}=\frac{P L^{3}}{3 E I}
(b) Effect of Normal and Shearing Stresses. We now substitute for U the expression (11.24) obtained in Example 11.05, where the effects of both the normal and shearing stresses were taken into account. We have
\frac{P^{2} L^{3}}{6 E I}\left(1+\frac{3 E h^{2}}{10 G L^{2}}\right)=\frac{1}{2} P y_{A}
and, solving for y_{A},
y_{A}=\frac{P L^{3}}{3 E I}\left(1+\frac{3 E h^{2}}{10 G L^{2}}\right)
We note that the relative error when the effect of shear is neglected is the same that was obtained in Example 11.05, i.e., less than 0.9(h / L)^{2}. As we indicated then, this is less than 0.9 \% for a beam with a ratio h / L less than \frac{1}{10}.