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

Problem 1: Part 4

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 $C^0$.