Beam theory

Definition

Let a straight body occupy $\set{(x, y, z) : 0 < x < L, \ (y, z) \in A}$, where the cross-section $A \subset \R^2$ has a diameter much smaller than the length $L$. Let the body be made of a linear elastic material and carry a transverse load $q(x)$ per unit length. A beam theory is a model of this body in which:

1. The unknowns are functions of $x$ alone, such as the transverse deflection $w(x)$ of the axis and the rotation $\varphi(x)$ of the cross-section.
2. The displacement of the body is given in terms of these unknowns by an assumption on the kinematics of the cross-sections.
3. The internal forces are the shear force $V(x)$ and the bending moment $M(x)$. With $q$ and $w$ measured in the same direction they satisfy the equilibrium equations $V' = -q$ and $M' = V$.
4. A constitutive law, derived from the elastic law and the assumption of clause 2, expresses $M$ and $V$ through the unknowns.