Euler Bernoulli Beam Bending

So far we have discussed what a force is and how it causes normal stresses and normal strains on an object. We have also discussed what a bending moment is, but what effects would a bending moment have on a beam? How would it cause the beam to deflect and would it cause stresses on a beam? To answer these questions we will use Euler-Bernoulli Beam Bending Theory

Leonhard Euler was a Swiss polymath who made important discoveries in a wide range of fields, including mathematics, structural mechanics, fluid dynamics and astronomy. His contributions to structural mechanics include contributions to Euler-Bernoulli beam bending and Euler theory of strut buckling. His surname is pronounced “Oiler”, not “Youler”.

Euler-Bernoulli Beam Bending Theory Equation

Without going too much into the derivation, Euler-Bernoulli beam bending theory gives the following equation for how a bending moment effects a beam. A full derivation of the equation can be found here.

$$EI \frac{d^{2}v}{dx^{2}} = M $$

Where \(E\) is the Young’s modulus of the beam, \(I\) is the Second Moment of Area of a beam, \(v\) is the vertical deflection of the beam and \(x\) is the distance along a beam. \(M\) is the moment at the x-distance along the beam that is being examined and so will typically be a different value at each location.

Using Euler-Bernoulli beam bending theory also allows the relationship between shear force, \(V\), and moment to be expressed as:

$$V = \frac{dM}{dx}$$

I.e. the shear force is equal to the rate of change of moment. This is a useful identity for sanity checking bending moment and shear force diagrams:

  • For a linear moment behaviour (typically due to applied point loads) the applied shear force will be constant.
  • For a quadratic moment behaviour (typically due to applied uniformly distributed load) the shear force will change linearly.

If the shapes of the bending moment and shear diagram don't match the above rules of thumb, then further checking of the bending and shear force equations is probably needed.

Euler-Bernoulli Beam Bending Theory Assumptions

It is important to understand the assumptions that are made in Euler-Bernoulli beam bending theory, the main assumptions are:

  • Plane sections remain plane – This is a complicated way of saying that as a beam deflects, sections that started as perpendicular to the neutral axis before bending will remain perpendicular to the neutral axis after bending. This assumes there are no significant shear deformations within the beam, this is generally a valid assumption unless the beam is very deep compared to its span. A beam is typically defined as a “deep beam” if the span to depth ratio is less than 2.
  • Angles of deformations remain small – This assumes that the angle of deformations is sufficiently small to allow the small angle approximation of \(sin \theta \approx \theta \) i.e. \(\theta \approx \frac{dv}{dx}\) to be applied.
  • The beam is formed of a material that is homogeneous (same material properties in all locations) and isotropic (same material properties in all directions). In addition, it is assumed the beam behaves elastically.

Figure 1: Plane sections remaining plane during bending. I.e. section that started perpendicular to the neutral axis, remain perpendicular to the neutral axis during bending.

Euler-Bernoulli beam bending can be seen as simplification of the later Timoshenko beam theory, this more advanced beam theory allows for shear deformations, rotational effects and provides a better model for thick beams.

Elastic Beam Bending Equations

Euler-Bernoulli beam bending theory gives rise to the elastic beam bending equations below, these are incredibly useful equations for structural analysis of beams:

$$ \frac{M}{I} =\frac{E}{R} =\frac{\sigma}{y} $$

Where, M is the applied moment, I is the second moment of area of the beam, E is the Young’s modulus of the beam, R is the radius of curvature of the neutral axis, \(\sigma\) is the resulting stress at a point on the beam cross section, y is the distance from the neutral axis to the location on the cross section where you would like to find your value of stress.

The above equation can be rearranged to link applied moment, the section properties of a beam and the stress resulting from the applied moment:

$$\sigma= \frac{My}{I}$$

Often the section properties of the beam \( \frac{I}{y}\) are grouped into a single section property called the section modulus:

$$Z= \frac{I}{y}$$

Where Z is the section modulus, this is parameter and how it is used is discussed in more detail in a separate article.

Figure 2: Using section modulus to relate applied moment, M, fibre distance, y (the distance from the neutral axis to the location on the cross section where you would like to find your value of stress), to second moment of area, I and stress, σ.

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