Problem 10 - Concepts and Applications of Finite Element Analysis

Question

Show that initial stresses \(\left\\{\sigma_{0}\right]\) on a \(C^{0}\) element lead 'to nodal loads \(\left\\{\mathbf{r}_{e}\right\\}\) that are self- equilibrating. That is, show that nodal forces in \(\left(\mathbf{r}_{e}\right\\}\) provide a zero resultant.

Step-by-Step Solution

Answer

The nodal forces \({\mathbf{r}_{e}}\) due to initial stress \(\sigma_0\) on a \(C^{0}\) element are self-equilibrating. This is because the total internal force and total internal torque over the element volume are both zero, resulting in a zero resultant of forces and moments due to these nodal loads.

Step 1: Express the Nodal Loads

The nodal loads \({\mathbf{r}_{e}}\) due to the initial stresses \(\sigma_{0}\) can be represented in terms of the stresses. Using the stress-strain-displacement relationship, this can be written as \({\mathbf{r}_{e}} = \int_V \mathbf{B}^T \sigma_0 dV\), where \(V\) is the volume of the element, and \(\mathbf{B}\) is the strain-displacement matrix.

Step 2: Show the Forces Resultant is Zero

The forces resultant due to the nodal loads \({\mathbf{r}_{e}}\) around an arbitrary point can be found by taking the sum of these forces, which is \(\sum {\mathbf{r}_{e}} = \int_V \mathbf{B}^T \sigma_0 dV\). For the forces to be self-equilibrating, their resultant should be zero. This essentially means that the resultant force is a zero vector. This can be shown by demonstrating that the total internal force over the volume of the element is zero, using the equilibrium equation \(\sigma_0 = 0\).

Step 3: Show the Moment Resultant is Zero

The moments resultant due to the nodal loads \({\mathbf{r}_{e}}\) around an arbitrary point can be found by taking the sum of moment vectors, each obtained by taking cross product of position vector from the arbitrary point to each nodal force with corresponding force. The moment resultant should also be zero with zero vector for forces to be self-equilibrating. It could be shown using the equilibrium of moments, by proving that the total internal torque over the volume of the element is zero.