Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 4

Problem 1: Part 4

Why is the Euler-Bernoulli beam theory $C^{1}$? Why is the Timoshenko beam theory $C^{0}$?

In the Euler-Bernoulli theory, the rate of deformation in the $xx$ direction is given by

$D_{xx}={\frac {\partial v_{x}^{M}}{\partial x}}-y{\frac {\partial ^{2}v_{y}^{M}}{\partial x^{2}}}~.$

Since second-derivatives of $v_{y}^{M}$ are involved, any approximate solution needs to be at least twice differentiable (or $C^{1}$) to be acceptable. This is why this theory is called $C^{1}$.

On the other hand, in the Timoshenko theory, only first derivatives of the primary dependent variables appear. Therefore, once differentiable (or $C^{0}$) shape functions are acceptable and the theory is called $C^{0}$.

Nonlinear finite elements/Homework 6/Solutions/Problem 1