A cantilever beam is a structural element that extends horizontally and is supported on only one end. The unsupported end is known as the cantilever, and it extends beyond the support point. Cantilever beams are often used in construction to support balconies, roofs, and other overhangs. They can also be used in bridges and other structures to extend the deck out over a waterway or other obstacle.
Cantilever beams are members that are supported from a single side only – typically with fixed support. In order to ensure the structure is static, the support must be fixed so that it is able to support all forces and moments in all directions. A cantilever beam is usually modeled like so, with the left end being the support and the right end being the cantilevered end:
A good example of a cantilever beam is a balcony. A balcony is supported on one end only, the rest of the beam extends over open space; there is nothing supporting it on the other side. Other examples would be the end of a continuous beam of a high-rise building floor or the cantilevered girders of a bridge segment. Cantilever Beam Equations.
There are a range of equations for how to calculate cantilever beam forces and deflections. These can be simplified into simple cantilever beam formula, based on the following:
Taken from our beam deflection formula and equation page. Cantilever Beam equations can be calculated from the following formula, where:
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So how do we calculate the maximum bending moment force of a cantilever beam? You can do this using the same method as shown in our how to calculate bending moment in a beam article. However, there are short hand equations you can use. For instance, the equation for the bending moment at any point x along a cantilever beam is given by:
\(M_x = -Px\)
\(M_x \) = bending moment at point x
\(P \) = load applied at the end of the cantilever
\(x \) = distance from the fixed end (support point) to point of interest along the length of the beam.
For a distributed load, the equation would change to:
\(M_x = – ∫wx\) over the length (x1 to x2)
where: w = distributed load x1 and x2 are the limits of integration.
This equation is valid for a simple cantilever beam with a point load or a uniformly distributed load applied at the free end of the beam. It should be considered that cantilevers beam can have complex loading and boundary conditions, such as multiple point loads, varying distributed loads, or even inclined loads, in those cases the above equation might not be valid, and a more complex approach might be required, it’s where the FEA comes in handy.
How to calculate stress in a cantilever beam? Cantilever Stress is calculated from the bending force and is dependent on the beam’s cross-section. For instance, if a member is quite small, there is not much cross-sectional area for the force to spread across, so the stress will be quite high. Cantilever beam stress can be calculated from either our tutorial on how to calculate beam stress or using SkyCiv Beam Software – which will show the stresses of your beam.
It’s useful to note that cantilever beams typically result in tension on the upper fibres of the beam. This means that in the case of a concrete cantilever beam, primary tensile reinforcement is typically required along the upper surface. This is in contrast to a conventional concrete beam supported at both ends, where primary tensile reinforcement would typically exist along the bottom surface of the beam.
Cantilevers deflect more than most types of beams since they are only supported from one end. This means there is less support for the load to be transferred. Cantilever beam deflection can be calculated in a few different ways, including using simplified cantilever beam equations or cantilever beam calculators and software (more information on both is below). The equation for the reaction at a fixed support of a cantilever beam is simply given by:
Reaction Force in Y \( = R_y = P\)
Moment Force about Z \( = _ = Px\)
\(F_y \) = reaction force in the Y direction at support A (the fixed support)
\(M_z \) = reaction moment about Z at support A (the fixed support)
\(P \) = the load applied at the end of the cantilever beam
\(x \) = distance of point load from support
This equation applies when the load is a point load on a cantilever. When the load is distributed, it is the summation of all forces in the vertical direction that needs to be zero. The equation becomes:
\(∑F_x = 0\)
Where the reaction force would be the algebraic sum of all the vertical forces acting on the structure. This equation assumes that the support is a fixed support, meaning that it does not have any rotation or translation. If the support has some degrees of freedom, the equation would change and become more complex. It’s important to keep in mind that this equation is just one step in analyzing a structure, in the design process of a real structure, several considerations such as load combinations, safety factors, material properties, etc. will be taken into account before finalizing a design.
