Shear and moment diagram

Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of an element. Using these diagrams the type and size of a member of a given material can be easily determined. 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

Although these conventions are relative and any convention can be used if stated explicitly, practicing engineers have adopted a standard convention used in design practices.

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). Another way to remember this is if the moment is bending the beam into a "smile" then the moment is positive, with compression at the top of the beam and tension on the bottom.

Normal positive shear force convention (left) and normal bending moment convention (right).

This convention was selected to simplify the analysis of beams. Since a horizontal member is usually analyzed from left to right and positive in the vertical direction is normally taken to be up, the positive shear convention was chosen to be up from the left, and to make all drawings consistent down from the right. The positive bending convention was chosen such that a positive shear force would tend to create a positive moment.

A clear understanding of most beams that are analyzed can be found here. This shows most of the conventions, both in forces and supports that we use to understand how beams are loaded.

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 (This can also apply in steel beams, especially rotating beams or shafts). Because of this the positive moment diagram is always drawn such that the tension on top is defined to be positive. This is opposite of the normal convention. The shear convention for reinforced concrete remains the same as the normal convention.

Vertical and angled members

For vertical members 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 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

Loading the diagram starts with analyzing the beam and determining the static loads and finding the reactions to them. There are different types of loads as shown through this link. Some examples of loading in the real world are buildings. After finding the reactions a loading diagram shows all loads applied to the beam which includes the service loads as well as the reaction loads. The service loads are loads put on the building during its use these include dead, live, roof live, snow, wind, earthquake, and other types of loads.^[1] In practice these loads are factored in a way such that they place the maximum reasonable stresses on a structure. From the service loads and the structural configuration the reaction loads can be determined using one several structural analysis 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 by "chopping off" one end and calculating the bending moment at that point but will generally be more complicated.^[2] With these shear and moments calculated we can also determine the stresses on the beam and see if the beam is safe to operate with the loads applied. These stresses have to be less than or equal to the yielding point of the beams material over a factor of safety determined by engineers. The most common factor of safety is 2.

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 defined for all regions of the member the diagrams can finally be drawn. Important positions where maximum or minimum values of shear force or bending moment occur should be dimensioned from one end of the member noted with a dimension. Normally the shear diagram is drawn directly below the loading diagram with the moment diagram drawn directly beneath the shear diagram to show which points on the shear and moment diagrams line up with the different loadings that the member is subjected to. The step functions and any calculations are usually written out below the shear and moment diagrams.^[3]

In a bending moment diagram the point where the bending moment curve "cuts" the zero line is called point of contraflexure.

Example

The example below 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 or what most people call a free body diagram. The second drawing is the loading diagram with the reaction values given without the calculations shown. The third drawing is the shear force diagram and the fourth drawing is the bending moment diagram. For the bending moment diagram the normal sign convention was used. Below the moment diagram 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. The example is illustrated using imperial units.

Relationships between load, shear, and moment diagrams

Since this method can easily become unnecessarily complicated with relatively simple problems, it can be quite helpful to understand different relations between the loading, shear, and moment diagram. The first of these is the relationship between a distributed load on the loading diagram and the shear diagram. Since a distributed load varies the shear load according to its magnitude it can be derived that the slope of the shear diagram is equal to the magnitude of the distributed load. The relationship between distributed load and shear force magnitude is:^[4]

$\frac{dQ}{dx} = -q$

Some direct results of this is that a shear diagram will have a point change in magnitude if a point load is applied to a member, and a linearly varying shear magnitude as a result of a constant distributed load. Similarly it can be shown that the slope of the moment diagram at a given point is equal to the magnitude of the shear diagram at that distance. The relationship between distributed shear force and bending moment is:^[5]

$\frac{dM}{dx} = Q$

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. Also if the shear diagram is zero over a length of the member, the moment diagram will have a constant value over that length. By calculus it can be shown that a point load will lead to a linearly varying moment diagram, and a constant distributed load will lead to a quadratic moment diagram.

Practical considerations

In practical applications the entire stepwise function is rarely written out. The only parts of the stepwise function that would be written out are the moment equations in a nonlinear portion of the moment diagram; this occurs whenever a distributed load is applied to the member. For constant portions the value of the shear and/or moment diagram is written right on the diagram, and for linearly varying portions of a member the beginning value, end value, and slope or the portion of the member are all that are required.^[3]

See also

• Singularity function#Example beam calculation

References

1. ^ American Society of Civil Engineers and Structural Engineering Institute (2005). ASCE/SEI 7-05: Minimum Design Loads for Buildings and Other Structures. American Society of Civil Engineers Publishing. ISBN 0784408092. 
2. ^ Beer, Ferdinand P.; E. Russell Johnston, John T. DeWolf (2004). Mechanics of Materials. McGraw-Hill. pp. 311. ISBN 0072980907. 
3. ^ ^{a b} Hibbeler, R.C (1985). Structural Analysis. Macmillan. pp. 146–148. 
4. ^ Emweb.unl.edu
5. ^ Beer, Ferdinand P.; E. Russell Johnston, John T. DeWolf (2004). Mechanics of Materials. McGraw-Hill. pp. 322–323. ISBN 0072980907. 

Further reading

• Cheng, Fa-Hwa. "Shear Forces and Bending Moments in Beams" Statics and Strength of Materials. New York: Glencoe, McGraw-Hill, 1997. Print.
• Spotts, Merhyle Franklin, Terry E. Shoup, and Lee Emrey. Hornberger. "Shear and Bending Moment Diagrams." Design of Machine Elements. Upper Saddle River, NJ: Pearson/Prentice Hall, 2004. Print.

External links

• To draw the shear and moment diagrams by writing the shear and moment equations.
• To draw the shear and moment diagrams by the relationship between load, shear, and moment.