Question 8.7: A cantilever beam AB with a uniform load of intensity q acti...
A cantilever beam AB with a uniform load of intensity q acting on the right-hand half of the beam is shown in Fig. 8-19a.
Obtain formulas for the deflection \delta_B and angle of rotation \theta_B at the free end (Fig. 8-19c). (Note: The beam has length L and constant flexural rigidity EI.)
In this example we will determine the deflection and angle of rotation by treating an element of the uniform load as a concentrated load and then integrating (see Fig. 8-19b). The element of load has magnitude q dx and is located at distance x from the support. The resulting differential deflection d\delta_B and differential angle of rotation d\theta_B at the free end are found from the corresponding formulas in Case 5 of Table G-1, Appendix G (available online), by replacing P with qdx and a with x; thus,
d\delta _B=\frac{(qdx)(x^2)(3L – x)}{6EI} d\theta _B=\frac{(q dx)(x^2)}{2EI}
By integrating over the loaded region, we get
\delta _B=\int{d\delta _B} = \frac{q}{6EI}\int_{L/2}^{L}{x^2(3L-x)dx} = \frac{41qL^4}{384EI} (8-54)
\theta _B=\int{d\theta _B}= \frac{q}{2EI}\int_{L/2}^{L}{x^2dx} = \frac{7qL^3}{48EI} (8-55)
Note: These same results can be obtained by using the formulas in Case 3 of Table G-1 and substituting a = b = L/2.