Question 11.10: Determine the deflection of end A of the cantilever beam AB ...

(a) Effect of Normal Stresses.     The work of the force P as it is slowly applied to A is

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

and, solving for $y_{A}$,

(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)

and, solving for $y_{A}$,

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 $\frac{1}{10}$.