# Euler-Bernoulli versus Timoshenko Beam Theories

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Beam theories provide a means of calculating the load carrying capabilities of and the deflection characteristics of beams. The Euler-Bernoulli and Timoshenko beam theories are described and contrasted in this short essay.
The Euler-Bernoulli beam theory or classical beam theory (pure bending moment) provides for analysis of cases of small deflection of a beam that is relatively long compared to beam depth in the direction of loading. The Euler-Bernoulli equation describes the relationships between beam deflection and the load applied. The beam equation also describes the relationships of applied forces as well as moments. As a result, the equation can be used to describe beam stress under loads.
Strain in the Euler-Bernoulli beam analysis is expressed in terms of the deflection of a “neutral surface”. Under transverse loading, one of the beam surfaces shortens while the other elongates. Therefore, a neutral surface that undergoes no axial strain is established at the centroid (or centroid of an “equivalent” section in the instance that the beam is a composite of different materials) between the surface undergoing axial compression and the surface undergoing axial elongation.
One of the assumptions in the application of the Euler-Bernoulli equation is that the normals to the neutral surface remain normal and that the deflections are small under deformation. Another way of describing these assumptions is that the beam bends in a constant radius arc and that the neutral surface does not change length under deformation (allowing the curvature and beam deflection to be related).
The Euler-Bernoulli beam equation includes a fourth order derivative. A minimum of four boundary conditions are required in order to arrive at a unique so...
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... describing the behavior of short beams (short relative to the beam dimension subjected to transverse loading), composite beams (or beams of other than isotopic materials), and the analysis of beams subjected to dynamic stimulus, particularly when the excitation wavelength approaches the beam thickness.
Like the Euler-Bernoulli beam theory, the Timoshenko beam theory includes a fourth order derivative, but the Timoshenko beam theory includes a second order derivative that takes into account the added effects of shear deformation and rotational inertia on the beam.
As the shear modulus (modulus of rigidity) of the element being analyzed approaches infinity, the Timoshenko beam theory analysis approaches the Euler–Bernoulli analysis. Similarly, if the rotational inertial effects are neglected, the Timoshenko beam theory analysis approaches the Euler–Bernoulli analysis