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Cantilever: Definition & Terminology

Instructor: Hassan Alsaud

Earned my B.S. in Civil Engineering back in 2011. Have two years of experience in oil and gas fields and two year as a graduate research assistant. Earned my Master degree in Engineering from Tennessee State University in 2016.

In this lesson, we are going to learn about a structural element called the cantilever. We are going to look at some examples of cantilevers and learn how cantilevers are structurally analyzed.

Cantilevers

If you happen to be walking along the sidewalk and look up at a building that has balconies, you will see that these balconies are fixed at one end and unsupported at the other end. While driving down the highway, you might likewise notice that road directional signs are supported at one end and free at the other end. These structures are called cantilevers. Some other examples of cantilevers might include parking shades and swimming pool diving boards.

Cantilevers are rigid structures, such as beams, which are fixed at one end and free at the other end. Some cantilevers can be supported throughout their length by trusses or cables. When a load is applied to the cantilever the cantilever transfers that load to the fixed end by bending.

Structural Analysis of a Cantilever

The fixed end of the cantilever beam has three reaction forces in 2D; there is a force in x direction and a force in y direction, in addition to the bending moments in the z direction. If the other end is free, the cantilever is said to be statically determinate because the number of unknown reactions is equal to the number of equations of equilibrium.

On the other hand, if the other end is supported by a roller or hung from a cable, the cantilever beam is said to be statically indeterminate. This is because the number of unknown reactions exceeds the number of equations of equilibrium. In this case, analyzing the cantilever beam requires more information than statics alone.

Examples

That being said, the reaction force in the x direction and y direction on the fixed end of the cantilever, with the presence or other supports, equal to the net horizontal force and the net vertical force respectively. However, the moment reaction of the fixed end is equal to the net vertical forces multiplied by the distance of the net force to the fixed end.

Example 1:

A 2-meter cantilever beam is fixed at one end and free at the other end. A concentrated vertical downward load of 10 kN is applied at the free end of the beam. What is the value of the reaction loads at the fixed end?

Answer:

There are three equations of equilibrium in two dimensional structures, which are as follows:


sigma f x


sigma f y


sigma m z


Since there are no external forces in the x-direction, Fx, or the fixed reaction, is equal to zero.

Since there is only one reaction force in the y-direction, the value of a reaction force is equal to the net vertical forces, but it is in the opposite direction. In this example, Fy = 10 kN upwards.

Since there is only one vertical force, the moment reaction of the fixed end is equal to the force multiplied by the distance. Here, the moment reaction is equal to 2 m * 10 kN = 20 kN-m.

Example 2:

If the same cantilever beam is subjected to a distributed load of 5 kN/m applied through the whole length of the beam. What are the values of the reaction forces of the fixed end?

Answer:

Since there are no external forces in the x-direction, Fx, or the fixed reaction, is equal to zero.

Fy is equal to the total vertical forces applied to the beam, i.e. the uniformly distributed load times the length at which it is applied.

Fy = 5 kN/m * 2 m = 10 kN

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