Deflection of cantilever beam

A *cantilever beam* is a beam which is fixed at one end. The other end is able to move freely.
Fixed means that it is unable to translate as well as rotate.
If a load is applied to a cantilever beam, both an internal bending moment and a shear force are developed.
This causes the beam to deform. Suppose the beam is divided into a infinitely large number of infinitely small slices.
For each slice, the deformation can be expressed in a translation and a rotation.
Suppose a force is applied to the end of a cantilever beam.
The force caused by gravity because of the mass of the beam is neglected.
Then at any point the *deflection* can be calculated with equation (1).
The vertical displacement of the beam is indicated with variable *w*, which is positive upwards.
As the vertical displacement is different on every location of the beam, *w* is a function of *x*.
*P* is the magnitude of the force, *L* the length of the beam, *E* the elasticity modulus of the material and *I* the moment of inertia.
The maximum deflection takes place at its end, this is where *x = L*, so the maximum deflection of a cantilever beam is calculated with formula (2).
If the applied force is suddenly taken away, this causes a vibration with a certain frequency. This is called the resonance or natural frequency.
The natural frequency of a cantilever beam can be calculated.

In reality the relationship between force and displacement is not linear. As the force is always applied vertically, the rotation of the beam causes the beam and the force not to be perpendicular. A beam’s resistance is much higher in the longitudinal direction than it is in the lateral direction. The component of the force which is parallel to the beam stretches the beam. This is not captured in the deflection equation. Therefore the formula is only valid for small rotations; in that case the component of the force in the direction of the cantilever beam is virtually equal to zero, causing nearly the whole force to act in the direction of deflection.