Question 11.CA.15: Determine the reactions at the supports for the prismatic be......
Determine the reactions at the supports for the prismatic beam and load shown (Fig. 11.41a).
The beam is statically indeterminate to the first degree. The reaction at A is redundant and the beam is released from that support. The reaction R _A is considered to be an unknown load (Fig. 11.41b) and will be determined under the condition that the deflection y_A at A must be zero. By Castigliano’s theorem, y_A=\partial U / \partial R_A , where U is the strain energy of the beam under the distributed load and the redundant reaction. Recalling Eq. (11.57),
x_j=\frac{\partial U}{\partial P_j}=\int_0^L \frac{M}{E I} \frac{\partial M}{\partial P_j} d x (11.57)
y_A=\frac{\partial U}{\partial R_A}=\int_0^L \frac{M}{E I} \frac{\partial M}{\partial R_A} d x (1)
The bending moment M for the load of Fig. 11.41b at a distance x from A is
M=R_A x-\frac{1}{2} w x^2 (2)
and its derivative with respect to R_A is
\frac{\partial M}{\partial R_A}=x (3)
Substituting for M and \partial M / \partial R_A from Eqs. (2) and (3) into Eq. (1), write
y_A=\frac{1}{E I} \int_0^L\left(R_A x^2-\frac{1}{2} w x^3\right) d x=\frac{1}{E I}\left(\frac{R_A L^3}{3}-\frac{w L^4}{8}\right)
Set y_A=0 \text { and solve for } R_A :
R_A=\frac{3}{8} w L \quad R _A=\frac{3}{8} w L \uparrow
From the conditions of equilibrium for the beam, the reaction at B consists of the force and couple:
R _B=\frac{5}{8} w L \uparrow \quad M _B=\frac{1}{8} w L^2 ⤸