Shear Force and Bending Moment
To find the shear force and bending moment over the length of a beam, first solve for the external reactions at the boundary conditions. For example, the cantilever beam below has an applied force shown in red, and the reactions are shown in blue at the fixed boundary condition:
After the external reactions have been solved for, take section cuts along the length of the beam and solve for the reactions at each section cut. An example section cut is shown in the figure below:
When the beam is cut at the section, either side of the beam can be considered when solving for the reactions. The side that is selected does not affect the results, so choose whichever side is easiest. In the figure above, the side of the beam to the right of the section cut was selected. The reactions at the section cut are shown with blue arrows.
The signs of the shear and moment are important. The sign is determined after a section cut is taken and the reactions are solved for the portion of the beam to one side of the cut. The shear force at the section cut is considered positive if it causes clockwise rotation of the selected beam section, and it is considered negative if it causes counter-clockwise rotation. The bending moment at the section cut is considered positive if it compresses the top of the beam and elongates the bottom of the beam (i.e. if it makes the beam “smile”).
Based on this sign convention, the shear force at the section cut in the figure above is positive since it causes clockwise rotation of the selected section. The moment is negative since it compresses the bottom of the beam and elongates the top (i.e. it makes the beam “frown”).
Shear and Moment Diagrams
The shear and bending moment throughout a beam are commonly expressed with diagrams. A shear diagram shows the shear along the length of the beam, and a moment diagram shows the bending moment along the length of the beam. These diagrams are typically shown stacked on top of one another, and the combination of these two diagrams is a shear-moment diagram. Shear-moment diagrams for some common end conditions and loading configurations are shown within the beam deflection tables at the end of this page. An example of a shear-moment diagram is shown in the following figure:
General rules for drawing shear-moment diagrams are given in the table below:
|Shear Diagram||Moment Diagram|
|Point loads cause a vertical jump in the shear diagram. The direction of the jump is the same as the sign of the point load.Uniform distributed loads result in a straight, sloped line on the shear diagram. The slope of the line is equal to the value of the distributed load.The shear diagram is horizontal for distances along the beam with no applied load.The shear at any point along the beam is equal to the slope of the moment at that same point:V=dMdxV=dMdx||The moment diagram is a straight, sloped line for distances along the beam with no applied load. The slope of the line is equal to the value of the shear.Uniform distributed loads result in a parabolic curve on the moment diagram.The maximum/minimum values of moment occur where the shear line crosses zero.The moment at any point along the beam is equal to the area under the shear diagram up to that point:M=∫Vdx|