The Euler-Bernoulli beam equation

The Euler-Bernoulli beam equation defines the behaviour of a beam element (see below). It is based on five assumptions:

(1) continuum mechanics is valid for a bending beam
(2) the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section.
(3) the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location.
(4) the beam is composed of an isotropic material
(5) the applied load is orthogonal to the beam's neutral axis and acts in a unique plane.

A simplified version of Euler-Bernoulli beam equation is:

Here u is the deflection and w(x) is a load per unit length. E is the elastic modulus and I is the second moment of area, the product of these giving the stiffness of the beam.

This equation is very common in engineering practice: it describes the deflection of a uniform, static beam.

Successive derivatives of u have important meaning:

is the deflection.

is the slope of the beam.

is the bending moment in the beam.

is the shear force in the beam.

A bending moment manifests itself as a tension and a compression force, acting as a couple in a beam. The stresses caused by these forces can be represented by:

where σ is the stress, M is the bending moment, y is the distance from the neutral axis of the beam to the point under consideration and I is the second moment of area. Often the equation is simplified to the moment divided by the section modulus (S), which is I/y. This equation allows a structural engineer to assess the stress in a structural element when subjected to a bending moment.