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Three-ways to derive the Euler-Bernoulli-Saint Venant Beam Theory

By Wenbin Yu

Purdue University

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This lecture notes used used to teach graduate structural mechanics course at Utah State University. In the notes, we formulated the complete classical beam model (extension/torsion/bending in two directions), which is also called Euler-Bernoulli-Saint beam theory, in three ways: Newtonian method, variational method, and variational asymptotic method, using 3D elasticity theory as the starting point. Many self-contradictions of the various assumptions used in both Newtonian method and variational method are clearly pointed out. The variational asymptotic method does not rely on any ad hoc assumptions, ending in a self-consistent theory. Many confusion points (at least for myself when I was an undergraduate student) about the beam theory have been clarified. Deriving Timoshenko model was left as a homework problem. It can be considered as a theoretical tutorial for VABS, a tool for blade modeling extensively used in helicopter and wind turbine industries. The VABS code will be hosted on cdmHUB very soon.