Shear Force and Bending Moment

Shear Force and Bending Moment Diagrams for a Simply-Supported Beam Under A Uniform Load

After the support reactions are calculated, the shear force and bending moment diagrams can be drawn.

Shear force is the force in the beam acting perpendicular to its longitudinal (x) axis. For design purposes, the beam's ability to resist shear force is more important than its ability to resist an axial force. Axial force is the force in the beam acting parallel to the longitudinal axis.

The following is a drawing of a simply-supported beam of length L under a uniform load, q:

This beam has the following support reactions:

where Rl and Rr are the reactions at the left and right ends of the beam, respectively.

The shear forces at the ends of the beam are equal to the vertical forces of the support reactions. The shear force F(x) at any other point x on the beam can be found by using the following equation.

where x is the distance from the left end of the beam.

Shear force diagrams are simply plots of the shear force (on the y-axis) versus the position of various points along the beam (on the x-axis). Thus, the following is the generalized shear force diagram for the beam shown above.

The bending moment at any point along the beam is equal to the area under the shear force diagram up to that point. (Note: For a simply-supported beam, the bending moment at the ends will always be equal to zero.)

To calculate the bending moment the beam must be broken up into two sections:

 (a) one from x = 0 to x = L/2 and (b) the other from x = L/2 to x = L.

The bending moment M(x) at any point x along the beam can be found by using the following equations:

Bending moment diagrams are simply plots of the bending moment (on the y-axis) versus the position of various points along the beam (on the x-axis). Thus, the following is the generalized bending moment diagram for the beam shown above.

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