Thank You for Your Consideration

tracker890 Source h · Aug 19, 2020

Beam with a fixed support at A, a hinge at B, a roller support at C, and a uniform distributed load q over the left span of length L/2. Below it is a free-body diagram of the left segment with reactions and a section coordinate x. Then a Mathcad-style worksheet solves for reactions and derives the moment and rotation at A, giving M_A = qL^2/32.

$Solution slide showing a beam free-body diagram with a uniformly distributed load q over length L, followed by a Maple-style worksheet solving four equations for reactions R_Ay, M_A, R_Cy and V_B, with the result sol = {M_A = 3/8 q L^2, R_Ay = q L, R_Cy = 1/2 q L, V_B = 1/2 q L}.$

Thank you for your time and consideration.

tracker890 Source h · Aug 20, 2020

Free-body diagram of a beam segment of length x, with a downward distributed load q on top, labelled axes x and y, and internal force and moment labels in red, with x in [0, L/2].

PhanthomJay · Aug 21, 2020

I'm not sure what energy method you are using, but since the problem is statically determinate, you don't need to know any rotation angles. B is a hinged joint, not a support, so there is no moment in the beam at B. The solution appears correct. Keep it simple.

tracker890 Source h · Aug 22, 2020

Dear PhanthomJay ：
The energy method is Castigliano’s theorem and thank you for your help.

Castigliano’s theorem is ref. as follows：

Table of contents page from Mechanics of Materials, Seventh Edition by James M. Gere and Barry J. Goodno, showing Chapter 9 Deflections of Beams; sections 9.9 Castigliano's Theorem (p. 731) and 9.10 Deflections Produced by Impact (p. 744) are boxed in red.

Thank You for Your Consideration

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