Stress Averaging in 2D elements

The formulation of 2D elements used in GSA means that there will be stress discontinuities between elements. Large discontinuities indicate that the mesh is not as refined as it should be. However, in the cases where the discontinuities are relatively small it is still convenient for viewing purposes to average the stresses at the nodes to give smooth contours between elements.

Stress averaging rules

In deciding which elements to include in the averaging a number of checks are made. The averaging is carried out on an element by element basis. For any 2D element the number of attached elements is established and the averaging is carried out over these elements provided the following conditions are met:

• The element must be a 2D element.
• The element must be of the same material as the reference element.
• The element must be the same thickness as the reference element.
• The element must be the same order as the reference element (both linear or both quadratic).
• The element must lie in the same plane as the reference element (i.e. no averaging over folds).

The stress results for all the elements meeting the above criteria are then averaged to get the mean stress value.

Notes on stress averaging

The stress averaging can be used to get the “best” approximation to the stress at the node. However, stress averaging can also mask poor stress results.

The stress averaging ignores the complexities of the stress field that results from having other elements attached to the reference element, for example beam framing elements, or 2D stiffening elements.

Stress error

It is useful to be able to quantify the error in the different stress values at nodes. This can be done by contouring the stress error based on the von Mises stress in the elements meeting at a node. The stress error is the standard deviation of the element stresses:

$$e = \sqrt{\frac{\sum_n (\sigma^2) - n(\sigma_{av}^2)}{n}}$$

where $n$ is the number of elements meeting at the node and the summation is over the $n$ elements. The error reported at the centre of the element is the average of the nodal errors.