Question 13.17: A cantilever beam of length L has a rectangular cross sectio...

Using Eq. (13.21) we obtain the form factor β for the cross section of the beam directly.
Thus

\beta =\frac{A}{I_z^2} \int_{y_1}^{y_2}{\frac{(A^\prime\overline{y} )^2}{b_0} } \ dy                                                     (13.21)

\beta =\frac{BD}{(BD^3/12)^2} \int_{-D/2}^{D/2}\frac{1}{B} {\left[B\left(\frac{D}{2} – y \right) \frac{1}{2} \left(\frac{D}{2} + y \right) \right]^2 } \ dy       (see Ex. 10.1)

which simplifies to

\beta =\frac{36}{D^5} \int_{-D/2}^{D/2}{\left(\frac{D^4}{16} – \frac{D^2y^2}{2} + y^4 \right) } \ dy

Integrating we obtain

\beta =\frac{36}{D^5} \left[\frac{D^4y}{16} – \frac{D^2y^3}{6} + \frac{y^5}{5} \right]_{-D/2}^{D/2}

which gives

\beta =\frac{6}{5}

Note that the dimensions of the cross section do not feature in the expression for β. The form factor for any rectangular cross section is therefore 6/5 or 1.2.

Let us suppose that v_s is the vertical deflection of the free end of the cantilever due to shear. Hence, from Eq. (13.22) we have

v_s = \frac{\beta }{G} \int_{L}{\frac{S_y}{A} } \ dx                                                         (13.22)

v_s = \frac{6 }{5G} \int_{0}^{L}{\left(\frac{-W}{BD}\right) } \ dx

so that

v_s = – \frac{6WL}{5GBD}                                              (i)

The vertical deflection due to bending of the free end of a cantilever carrying a concentrated load has previously been determined in Ex. 13.1 and is −WL³/3EI. The total deflection, v_T , produced by bending and shear is then

v_T = – \frac{WL^3}{3EI} – \frac{6WL}{5GBD}                                                (ii)

Rewriting Eq. (ii) we obtain

v_T = – \frac{WL^3}{3EI} \left[1+\frac{3}{10} \frac{E}{G} \left(\frac{D}{L} \right)^2 \right]                                                            (iii)

For many materials (3E/10G) is approximately unity so that the contribution of shear to the total deflection is (D/L)² of the bending deflection. Clearly this term only becomes significant for short, deep beams.