# Potential Energy Functional - Timoshenko Shear Locking

1. Nov 29, 2012

### bugatti79

Folks,

The total potential energy functional for an isolated finite element timoshenko beam is given as

$\displaystyle \Pi_e(w, \Psi)=\int_{x_e}^{x_{e+1}} \left[ \frac{EI}{2} \left (\frac{d \Psi}{dx}\right )^2 + \frac{ G A K_s}{2} \left ( \frac {dw}{dx} + \Psi \right )^2 +...\right]dx +....$

Where the first term in the integral is the bending energy of the element. The author states that a constant state of $\Psi(x)$ is not admissible because the bending energy of the element would be zero leading to the numerical problem of shear locking.

Not sure I understand this concept. It is just a term that will go to 0 on the first derivative but the rest of the integral can still be evaluated. Why is it not admissible?