Shear and bending moment diagrams are analytical tools used in structural analyses and structural design to determine the value of shear force and bending moment at different length of an element. Using these diagrams type and size of a member can be determined easily. Another application of shear and moment diagrams is that the deflection can be easily determined using either the moment area method or the conjugate beam method.
Convention
Normal Convention
The normal convention used in most engineering applications is to label a positive shear force one that spins an element clockwise (up on the left, and down on the right). Likewise the normal convention for a positive bending moment is to warp the element in a "u" shape manner (Clockwise on the left, and counterclockwise on the right).
Normal positive shear force convention (left) and normal bending moment convention (right).
Concrete Design Convention
An exception to using the normal convention is used when designing concrete structures. Since concrete is weak in tension the most important part of designing a member with a high bending moment is to show whether the top or bottom of the concrete member is in tension. Because of this the positive moment diagram is alway drawn such that the tension on top is defined to be positive. This is opposite of the normal convention. The shear convention remains the same.
Vertical and Angled Members
For vertical members deciding the convention is to start from the bottom and move up in the same way that horizontal members start from the left and move to the right. In this way a force pushing to the left from the bottom will inspire a positive shear moment which will also be drawn to the left. For angled members if there is a conflict of interest between the normal convention and the vertical convention most often an engineer will follow the normal or horizontal reaction but either can be followed and the engineer should make note of which convention they are following.
For concrete in either vertical or angled members the shear diagrams are drawn as stated above but the the moment diagram should be drawn to show which side the tension of the member will be on.
Procedure
There are three major steps to constructing the shear and moment diagrams. The first is to construct a loading diagram, the second is to calculate the shear force and the bending moment as a function of the position of the beam, and the third is to draw the shear and moment diagrams.
Loading Diagram
A loading diagram shows all loads applied to the beam which includes the service loads as well as the reaction loads. The reaction loads can be determined using several methods including finite element method and static analysis. Once the reaction loads have been determined the loading diagram can be drawn.
Calculating the Shear and Moment
With the loading diagram drawn the next step is to find the value of the shear force and moment at any given point along the element. For a horizontal beam one way to perform this is at any point to "chop off" the right end of the beam and calculate the internal shear force needed to keep the left portion of the beam in static equilibrium. That internal shear force is the value of the shear force needed to plot on the shear diagram. The moment is done in similar method but will generally be more complicated.
Both the shear and moment functions should be written as stepwise functions with respect to position on the beam.
Drawing the Shear and Moment Diagrams
After the value of the shear force and bending moment diagram are known for all positions on the member the diagrams can finally be drawn. Important positions where maximum or minimum values of shear force or bending moment occur should have the distance from one end of the member noted with a dimension.
Example
The example includes a point load, a distributed load, and an applied moment. The supports include both hinged supports and a fixed end support. The first drawing is the situation of the element. The second drawing is the loading diagram with the reaction values given without calculations shown. The third drawing is the shear and final shear force diagram and the fourth drawing is the bending moment diagram. Below is the stepwise functions for the shear force and bending moment with the functions expanded to show the effects of each loading on the shear and bending functions.
Relationships between Load, Shear, and Moment Diagrams
Since this method can become unnecessarily complicated, it can be quite helpful to understand different relations between the loading, shear, and moment diagram. The first of these is the relationship between distributed load and the shear diagram. Since a distributed load varies the shear load according to its magnitude it can be derived that:
Where w is the magnitude of the distributed load. Similarly the slope in the moment diagram at a given distance is equal to the magnitude of the shear diagram at that distance, or:
A direct result of this is that at every point the shear diagram crosses zero the moment diagram will have a local maximum or minimum value. Also if the shear diagram is zero over a length of the member the moment digram will have a constant value over that length.
Applications in Finding Deflections
Using shear and moment diagrams can yield a quick calculation of deflection using either the moment area method or the conjugate beam method.
References
^Beer F: "Mechanics of Materials", page 322-323. McGraw-Hill, 2006
