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Determine the deflection of cantilever beams with a single load.

If you read our previous article on what is deflection, you’ll have some idea of deflection and it’s magnitude.  Now we’re going to investigate how we determine deflection.

What is a cantilever beam?

Cantilever beams are constrained at one end and free at their other end. At the fixed end of the cantilever beam, deflection must be zero. Deflection increases as we move away from the support and the maximum deflection will occur at the free end. The image below shows a cantilever beam in a structure:

A cantilever beam in construction

Cantilever beams can either be end-loaded or uniformly loaded, and this video can give you some more information on the slope and deflection of a beam along key points.

End-Loaded Cantilever Beams

In end-loaded cantilever beams, the load is applied at a single point on the beam. A cantilever beam with a deflected shape is shown in the first image, while a cantilever beam carrying a point load at its free end is shown in the second image.

diagram of a deflected beam
diagram of a point loaded beam

Uniformly-Loaded Cantilever Beams

Uniformly loaded cantilever beams have the force acting uniformly throughout the length of the beam. An example of cantilever beam carrying uniformly distributed load (UDL) is mostly found in balconies. A representation of this type of cantilever beam is shown in the image below:

diagram of an end loaded beam

Standard Data Table for Deflection of Cantilever Beams

Using standard tables is the most practical and fastest approach for calculating the deflection and slope of a beam.  For a cantilever beam subjected to a single type of loading, such as point or concentrated load, uniformly distributed load, and couple, deflection and slope can be determined using the formulae in the Table below.

standard data table

Let’s have a look at an example.  Suppose a 3m long cantilever beam is carrying a 25kN point load at its free end.  If the moment of inertia of the beam is 108mm4 and the Young’s Modulus is 2.1 105N/mm2, then how would we find the deflection at the free end?

Length of the beam = 3m

Point Load = 25kN = 25000N

Moment of inertia (i) = 108mm4 = 0.0001m4

Young’s Modulus (E) = 2.1 105 N/mm2 = 2.1 1011N/m2

Deflection at the free end = PI33EI = 25000 333 2.1 1011 0.0001 = 0.01071m

Let’s take another example where the beam is 4m long and subjected to a point load of 5kN at the free end.  Let’s determine the slope and deflection at the free end, taking flexural stiffness EI as 53.3 MNm2.

Length of beam = 4m

Point load on beam = 5 kN = 5000 N

Flexural stiffness = 53.3 MNm2 = 53.3 106 Nm2

From the Table, the deflection at the free end is given by Pl33EI = 5000 433 53.3 106 = 0.002 m

We’re going to continue our series on deflection with more articles on various calculations with various loads on a beam so stay tuned.

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