The Euler-Bernoulli beam theory is a simple calculation that is used to determine the bending of a beam when a load is applied to it. First introduced in the 18th century, it became a popular theory that was used in the engineering of structures like the Eiffel Tower or the original Ferris Wheel. Since then, it has often been applied in civil and mechanical engineering projects over the last century.

Although there are more advanced methods of calculation that can be used today for modern engineering projects, the Euler-Bernoulli beam theory is often still used because of how simple it is.

### What Is the Euler Bernoulli Beam Theory?

The equation that is used in the Euler Bernoulli beam theory describes the relationship between the deflection of the beam and its applied load. It can be calculated by using the equation below.

**d2/dx2(EI d2w/dx2) = q**

In this equation, w(x) is describing the deflection of the beam in the direction of z, but in the position of x. This means that q is the distributed load. So a force per unit length, or the pressure being applied to a force per area, can be a function of w, x, or other variables that may be in place with equation.

If this process seems familiar, it is because the classical beam theory is a simplification of the linear theory of elasticity. This does mean that the equation is limited to the small deflections of a beam that is subjected to lateral loads only.

### Why Is the Euler Bernoulli Beam Theory Used?

Besides the ability to calculate deflection, the beam equation is able to describe the moments and forces that are interacting with the beam, allowing for the possibility to describe stresses. By filling out the equations within this theory, it becomes possible to know the strength and deflection of the beams that are under a bending moment.

Why does stress form within a beam? Both the bending of a beam and the shear force that is caused by this action will cause stress, with the maximum amount being found at the neutral axis. By knowing where the point of maximum stress will be on a beam, it becomes possible to know where the most strength within the structure will need to be placed.

This applies to all beams except for those that are the largest and stockiest in the design. Shear forces stresses are important, but the bending moment stresses place a greater overall impact on the beam in terms of stress concentration. This means that although a beam is likely to bend underneath the weight of an impact, most of the stress that the beam encounters will be kept along the surface of the beam.

### Strain Within the Euler Bernoulli Beam Theory

When deflection occurs on a neutral surface within the Euler-Bernoulli beam theory, then an expression of strain must be developed. In order to obtain this expression of strain, an assumption must be made regarding the normality of the neutral surface remaining constant during deformation. This would create deflections that are small.

That can be expressed with this equation.

**Ex – dx’-dx/dx = -z/p = -k,z**

For the purpose of this equation, dx is the length of an element of the neutral surface when the beam is in an undeformed state. Small deflections do not change the length after bending, but it will form an arc in the beam. This is expressed by p in the equation.

Once bent, the beam’s length changes from dx to dx’ because of the arc that occurs. This is how the strain in the beam can be calculated, because k becomes the curvature of the beam. You’re able to calculate the initial axial strain as a function of distance from the neutral surface. At this point, you would just need to find the relationship between the radius of the curvature and “w,” which is the beam deflection expression.

### What Happens If There Are Large Deflections?

If we are looking at the original Euler-Bernoulli beam theory, then only small rotations and smaller strains can actually be calculated with accuracy. An extension of the theory is required to be able to determine moderate-to-large rotations, but only under the assumption that the strain remains small.

This is able to be done because the beam theory suggests that plane sections remain plane, normal to the axis of the beam as it leads to the displacement of form.

The Euler Bernoulli beam theory has been instrumental in the development of engineering theory for more than a century. Many bridges and similar structures still employ it today, so take a moment to look at a local bridge and you’ll see this mathematical equation at work.