Question 11.10: Determine the deflection of end A of the cantilever beam AB ...
Determine the deflection of end A of the cantilever beam AB (Fig. 11.34), 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 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
U=\frac{P^{2} L^{3}}{6 E I}\left(1+\frac{3 E h^{2}}{10 G L^{2}}\right)=U_{ s }\left(1+\frac{3 E h^{2}}{10 G L^{2}}\right) (11.24)
\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}.