Engineers need to be able to predict how beams will deflect under loads so that they can design them appropriately. Excessive deformations could damage parts of the structure connected to the beam, feel unsafe to the user, or prevent the beam from meeting its intended function.
The deflection of a beam can be determined from the deflection differential equation – show below – where $x$ is the distance along the beam, $y(x)$ is the beam displacement, $M(x)$ is the bending moment at $x$ and $EI$ is the flexural rigidity of the beam cross-section.
The video below explains the deflection differential equation in more detail, and takes a look at five different methods that can be used to predict how a beam will deform when loads are applied to it. These are:
- the Double Integration Method
- Macaulay’s Method
- the Principle of Superposition
- the Moment-Area Method
- Castigliano’s Theorem (which is based on strain energy)