# How to Calculate Cantilevers

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Cantilevers are beams that jut out of a structure without a support on the free end, much like a diving board. Cantilevers often carry loads when they are used in buildings--such as for balconies--or bridges or towers. Even the wings of an airplane can be thought of as cantilevered beams. When a load sits on a cantilevered beam, two reactions occur at its support. There is the vertical shear force, which counteracts the object's weight, but the greater force is often the bending moment, which keeps the beam from rotating. You can calculate these loads using a couple equations.

Determine the weight of the beam itself. If this is unknown, you can look up the beam material's density and then multiply that number by the beam's volume.

Calculate the shear force at the beam's support. This is the vertical, upward force that counteracts the weight of the beam and the object. As you might expect, the shear force is simply the sum of the beam's weight and the load it carries.

Calculate the bending moment due to the weight of the beam itself. The bending moment along a cross section equals the distance to a perpendicular force times the magnitude of that force. For example, if a 10 Newton force acts on a beam at 20 m from its cantilevered support, the moment at the support is 200 Newton-meters. Because the center of mass of a beam is at the midpoint of its length, the moment caused by the beam is its weight multiplied by one-half its suspended length.

Calculate the bending moment due to the weight of the load. This equals the load's center of weight times its distance from the beam's support. For example, if 10 kg rectangular flower bed sits on a beam at between 15 and 20 m from the support, its induced bending moment would be:

17.5 m * 10 kg = 175 kg-m.

Add the bending moments induced by the load and the beam itself to obtain the total bending moment.

#### Warnings

• Remember not to directly add shear force and the bending moment. Shear force is a vertical force parallel to the beam's cross section, while the bending moment consists of small, horizontal forces that both push and pull perpendicularly to the beam's cross section.

#### References

• "FE Review Manual: Rapid Preparation for Fundamentals of Engineering Exam"; Michael R. Lindeburg; 2006