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^M_x}{\partial x} - y\frac{\partial^2 v^M_y}{\partial x^2} ~. $$

Since second-derivatives of $$v^M_y$$ 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 $$C^0$$.