The cantilever is one of the most common beams. We'll explore a complete structural overview of this structure - its model, loads, analysis, and design.
The cantilever beam is a structure most structural engineers will commonly see. This post extensively discusses this structure. We'll discuss it in four parts: (1) model, (2) loads, (3) analysis, and (4) design.
Model
We first start discussing its model. The cantilever beam is a structure of definite length with the following properties:
When referring to a cantilever, we associate it with a beam designed to withstand flexural loads - forces that act perpendicular to the beam's longitudinal axis.
The beam may experience other loads other than the flexural type. We can apply axial or torsional loads on it. It is possible to have a combination of these types.
Analysis
Let's move on to a typical analysis of a cantilever beam.
We will analyze all four aspects: stability, reactions, internal forces, and deflection.
Stability
Let's start with its stability. A structure is stable if it is both externally and internally stable:
The supports are NOT collinear, concurrent, or parallel.
The beam's determinacy is zero.
The member is well arranged. There are also no internal devices that may compromise its stability.
Reactions
Let's now talk about the cantilever beam's reactions.
As we have seen earlier, only one fixed-end support carries the entire member. This plane fixed support has three reaction components: (1) horizontal force component, (2) vertical force component, and (3) moment component.
We can solve these quickly. Since it is a determinate structure, we can solve this by applying the equilibrium concept.
Internal Forces and Stresses
Shear
The cantilever's shear has the following typical properties:
The shear at its free end is usually zero.
The vertical reaction at its fixed-end support is usually the maximum shear.
It is essential to note that these properties are only sometimes accurate. Again, it would depend on the beam's loading conditions.
Moment
The beam's moment will depend on the loads; however, we can observe a fascinating pattern regarding its moment behavior. Using these patterns, we can efficiently compute its area and center of the moment graph (provided it matches the loading condition)—more on this at this link.
Deflection
Lastly, we'll discuss the beam's deflection - a measure of its movement from its original shape.
Assuming we have \(P\) or \(w\) loads acting in one direction, the typical deflection is the following:
No deflection at its fixed end (as per support requirements)
The maximum deflection is usually at its free end.
Again, this is sometimes true. It would entirely depend on the loading condition of the beam.
Rotation and Translation
Like the beam's moment, there is an interesting pattern when observing its rotation and translation, especially at its free end—more at this link.
Design
Finally, let's discuss some key aspects of a cantilever's design.
Let's recall that a structure's design depends on many factors, such as its purpose, design philosophy, costs, and logistics, to name a few.
Despite these factors, there are some general remarks we can take note of when designing a cantilever. Below are some of them:
Usually, the longer it is, the bigger or thicker its size.
The fixed-ended support should be strong enough since it is the only support of the beam.
Summary
Let's summarize:
The cantilever beam is a structure most structural engineers will commonly see.
The cantilever beam is a structure of definite length with the following properties (1) it has only one fixed-end support on one end, and (2) its other end is free.
It is a determinate structure.
In terms of its properties, it can be of any length, shape, or material.
When referring to a cantilever, we associate it with a beam designed to withstand flexural loads.
It is naturally stable because (1) the supports are NOT collinear, concurrent, or parallel, (2) the beam's determinacy is zero, and (3) there are also no internal devices that may compromise its stability.
Only one fixed-end support carries the entire member. A 2D fixed support has three components that can be solved using equilibrium.
The cantilever's shear has the following typical properties: (1) the shear at its free end is usually zero, and (2) the vertical reaction at its fixed-end support is usually the maximum shear.
The cantilever's moment will depend on the loading conditions.
Assuming we have \(P\) or \(w\) in one direction, a typical cantilever beam's deflection is: (1) no translation or rotation at its fixed end (as per support requirements), and (2) the maximum deflection is usually at its free end.
Some critical aspects of designing a cantilever include the following: (1) it must have a thicker section as the beam becomes longer, and (2) the fixed-ended support should be strong enough to support the beam.
