Timoshenko–Ehrenfest beam theory

Definition

Let a straight beam occupy $0 \le x \le L$, with Young's modulus $E$, shear modulus $G$, density $\rho$, cross-sectional area $A$ and second moment of area $I$. Let $\kappa > 0$ be the shear coefficient of the cross-section. The unknowns are the deflection $w(x, t)$ and the rotation $\varphi(x, t)$ of the cross-section. The load $q(x, t)$ per unit length acts in the direction of $w$. The bending moment and the shear force are $$M = -E I \frac{\partial \varphi}{\partial x}, \qquad V = \kappa A G \left( \frac{\partial w}{\partial x} - \varphi \right).$$ The Timoshenko–Ehrenfest beam equations are $$\rho A \frac{\partial^2 w}{\partial t^2} - \frac{\partial}{\partial x}\left[ \kappa A G \left( \frac{\partial w}{\partial x} - \varphi \right) \right] = q,$$ $$\rho I \frac{\partial^2 \varphi}{\partial t^2} - \frac{\partial}{\partial x}\left( E I \frac{\partial \varphi}{\partial x} \right) - \kappa A G \left( \frac{\partial w}{\partial x} - \varphi \right) = 0.$$