Shear Load and Bending Moment Diagrams

Equilibrium of forces: The equilibrium of forces in the vertical direction in the segment shown of the member results in

Taking the limit as gives

Therefore, for continuous shear loads, the change in shear is related to the integral of the distributed load.

Equilibrium of moments: The equilibrium of moments around the centroid C for the section shown yields

Taking the limit as gives

Therefore, for continuous moments, the change in moment is related to the integral of the shear load (the area under the shear diagram is related to the change in moment).

Point loads and point moments: When there is a point load F_o and a point moment M_o applied at a point in the beam, the point load results in a jump in the value of the shear load V and the point moment results in a jump in the value of the bending moment M.

Equilibrium of forces on the element requires that

Taking a limit as Dx goes to zero results in the relation for the jump in the shear load due to an applied downward point load of F_o to be given by

Equilibrium of moments on the element requires that

Taking a limit as Dx goes to zero results in the relation for the jump in the bending moment due to an applied counter-clockwise point moment of M_o to be given by

Therefore, there will be a discontinuity at each applied point load in the shear diagram and a discontinuity at each applied couple in the moment diagram.

Drawing shear force and bending moment diagrams: The shear load and bending moment diagrams are constructed by integrating the distributed load to get the shear diagram (adding jumps at all point loads), and integrating the shear diagram to get the bending moment (adding jumps at all point couples). The following is an example of one shear load and bending moment diagram.

Note:

1. First draw the free-body-diagram of the beam with sufficient room under it for the shear and moment diagrams (if needed, solve for support reactions first).

2. Draw the shear diagram under the free-body-diagram. The distributed load is the slope of the shear diagram and each point load represents a jump in the shear diagram. Label all the loads on the shear diagram

3. Draw the moment diagram below the shear diagram. The shear load is the slope of the moment and point moments result in jumps in the moment diagram. The area under the shear diagram equals the change in moment over the segment considered (up to any jumps due to point moments). Label the value of the moment at all important points on the moment diagram.

ã Mehrdad Negahban and the University of Nebraska, 1996-2000.

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Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526