Example of beam
Example of beam:
The simply supported beam in Fig. 1a is subjected to the three forces F1 = F, F2 = 2F and F3 = −F.
Draw the shear-force and bending-moment diagrams.
Solution As a first step we draw the free-body diagram of the entire beam (Fig. 1b) and compute the support reactions A and B. The equilibrium conditions yield
In the next step, we pass imaginary sections at arbitrary positions x in each span between two concentrated loads. The equilibrium of the forces for the left-hand parts of the beam yields the shear force (the corresponding free-body diagrams are not shown in Fig. 1):
The bending moment is obtained from the equilibrium of the moments:
It should be noted that it would have been simpler to use the right-hand parts instead of the left-hand parts for x > 2a.
The shear-force and bending-moment diagrams are shown in Fig. 1c. The bending-moment diagram has a positive (negative) slope in the regions of a positive (negative) shear force.
The values of the stress resultants at the right-hand end of the beam (x = 4a) may serve as checks:
– the shear force has a jump discontinuity of magnitude B (the diagram should close at x = 4a),
– the bending moment is zero (roller support at the end of the beam).
Since the beam in Fig. 1a is simply supported,