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 δ_B and angle of rotation θ_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 dδ_B and differential angle of rotation dθ_B at the free end are found from the corresponding formulas in Case 5 of Table H-1, Appendix H available online, by replacing P with q dx and a with x; thus,
dδ_B=\frac{(qdx)(x^2)(3L-x)}{6EI} dθ_B=\frac{(q dx)(x^2)}{2EI}
By integrating over the loaded region, we get
δ_B=\int{dδ_B} =\frac{q}{6EI} \int_{{L}/{2}}^{L}{x^{2}(3L-x)dx }=\frac{41qL^{4} }{384EI} (8-54)
θ_B=\int{dθ_B}=\frac{q}{2EI} \int_{{L}/{2}}^{L}{x^{2}dx }=\frac{7qL^{3} }{48EI} (8-55)
Note: These same results can be obtained by using the formulas in Case 3 of Table H-1 and substituting a=b=L/2.