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Simple beam bending is often analyzed with the Euler–Bernoulli beam equation. The conditions for using simple bending theory are: [4] The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present. The material is isotropic (or orthotropic) and homogeneous.
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
Pure bending occurs only under a constant bending moment (M) since the shear force (V), which is equal to , has to be equal to zero. In reality, a state of pure bending does not practically exist, because such a state needs an absolutely weightless member. The state of pure bending is an approximation made to derive formulas.
The bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location. The beam is composed of an isotropic material. 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:
A statically determinate beam, bending (sagging) under a uniformly distributed load. A beam is a structural element that primarily resists loads applied laterally across the beam's axis (an element designed to carry a load pushing parallel to its axis would be a strut or column).
The attempts to provide precise expressions were made by many scientists, including Stephen Timoshenko, [12] Raymond D. Mindlin, [13] G. R. Cowper, [14] N. G. Stephen, [15] J. R. Hutchinson [16] etc. (see also the derivation of the Timoshenko beam theory as a refined beam theory based on the variational-asymptotic method in the book by Khanh C ...
For a 3-point test of a rectangular beam behaving as an isotropic linear material, where w and h are the width and height of the beam, I is the second moment of area of the beam's cross-section, L is the distance between the two outer supports, and d is the deflection due to the load F applied at the middle of the beam, the flexural modulus: [1]
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