Question 9.7: 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. 9-19a. Obtain formulas for the deflection δ_B and angle of rotation θ_B at the free end (Fig. 9-19c). (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. 9-19b). 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 G-1, Appendix G, by replacing P with q dx and a with x; thus.
dδ_B=\frac{(q\ dx)(x^2)(3L\ -\ x)}{6EI}\quad \quad dθ_B=\frac{(q\ dx)(x^2)}{2EI}
By integrating over the loaded region, we get
δ_B=\int{d\ δ_B}=\frac{q}{6EI}\int_{L/2}^{L}{x^2(3L\ -\ x)dx}=\frac{41qL^4}{38EI} (9-54)
θ_B=\int{d\ θ_B}=\frac{q}{2EI}\int_{L/2}^{L}{x^2\ dx}=\frac{7qL^3}{48EI} (9-55)
Note: These same results can be obtained by using the formulas in Case 3 of Table G-1 and substituting a = b = L/2.