A beam can satisfy stress limits and still deform too much to perform its function. Beam deflection describes the transverse displacement of the beam's neutral axis.

For a slender linearly elastic beam undergoing small deflection, the Euler–Bernoulli relation is approximately

$$EI\frac{d^2y}{dx^2}=M(x),$$

where $E$ is Young's modulus, $I$ is the area moment of inertia and $M(x)$ is the bending-moment distribution.

The product $EI$ is the flexural rigidity.

Double integration

If $M(x)$ is known, integrate twice:

$$EI\frac{dy}{dx}=\int M(x),dx+C_1,$$

$$EIy(x)=\int!\int M(x),dx,dx+C_1x+C_2.$$

The constants are determined from support or symmetry boundary conditions.

Slope and displacement

The first derivative $dy/dx$ represents beam slope under the small-angle approximation, while $y(x)$ is the transverse deflection.