# 2D Element Stress Results

The method of calculating the stresses and forces in 2D elements depends to a certain extent on the solution method. The stresses are based on the strains, which in turn are based on the displacement gradients in the element. The stresses are calculated at the Gaussian integration points and then extrapolated to the nodes. Depending on the solution GSA either calculates the force and moment results from the stresses or by direct integration through the thickness of the element.

2D element results (forces and stresses) are most conveniently viewed as contours or diagrams. Diagrams may be more useful in principal stresses with the direction information are required.

## Sign Convention

- Membrane stresses are positive when tensile (i.e. positive direct strain).
- Positive shear stresses result from positive shear strain.

This is illustrated as follows.

2D Element Stress Sign Convention

## Projected and derived stresses

Stresses are a tensor value so there are six possible stress components that can be reported for any position on the element – three direct stresses and three shear stresses. The x and y stresses are in the plane of the element and z in the through thickness direction. When a user axis is selected the stresses are projected on to the plane of the element so that z remains normal to the element surface.

The derived stresses are useful ways of understanding how the element responds to the loading. The derived stress options are:

**principal stresses**– stresses rotated so that there no shear stresses showing flow of stress in metals, concrete**maximum shear****von Mises**– a measure of the distortional stress used in yield functions to see how close the material is to yield ( $σ_{yield} − σ_{vm} = 0$ )**average**– checking if material is in overall compression/tension

Details of these are given in the GSA Theory.

## Limitations of the stress calculation

The stress calculations show more clearly than the displacement calculation the approximate nature of the finite element method. In particular it should be noted that the stresses are calculated by interpolation involving an extrapolation from the Gauss points to the nodes. This means that for element with high stress gradients the interpolated values will be less accurate than for elements with low stress gradients. It is important to check to see what variation of stress exists over the element and if it is large it may be necessary to refine the mesh locally to overcome this problem.

The stress is calculated on an element by element basis (unlike the displacement) so it is possible, and in fact very likely, that the stresses at a node in one element will not match up with the stress at the same node in an adjacent element. In some cases, where there is a discontinuity in the material properties or the thickness of the elements, it is correct that there is a local stress discontinuity, however in other cases this stress difference will be a measure of the error in the solution.

When contour plots of the stresses are required, or when a stress value is required for further calculation it is better to have a single value. The values from all the elements at a node can be averaged to give a single value at the node. However it should be remembered that the stresses which look smooth and continuous on the plots are in fact calculated as a series of discontinuous stress patches.

## Checking 2D elements stresses

It is useful to be able to carry out some quick checks on the stresses to get an idea of the accuracy of the analysis. There is no simple answer as to how good an analysis is, but a number of qualitative rules can be established.

- check that the
**stress gradients**are low within the elements, particularly in regions of high stress - check that the
**stress discontinuities**between elements are small — look at the stress errors