Problem Description

In the field of structural analysis, accurate stress visualisation is crucial for interpreting material behaviour. However, discrepancies can sometimes occur between theoretical expectations and practical observations in PLAXIS Output, particularly at nodes within models.

Consider a concrete block designed to withstand only compressive forces, with zero tensile strength defined as an input for the material parameters. When analysing the principal stresses, one would expect to see only compressive stresses (Figure 1). However, an inspection of the σ₃ shadings plot suggests the presence of tensile stresses within the block, contradicting its zero tensile strength (Figure 2).

Figure 1: Effective principal stresses and Principal total stresses, σ₃ – coloured principal directions plot (plotted at stress points)

Figure 2: Principal total stress, σ₃ - Shadings plot (plotted at nodes)

A detailed review of the table results (accessible via the “Table” icon in the top toolbar) does not indicate tension in σ₃. However, noticeable tension peaks at the element corners raise questions about the accuracy of stress visualisation and the discrepancy between the values shown and the input of zero tensile strength.

This article aims to explain these stress discrepancies* and the reasons behind this apparent contradiction between the anticipated stress patterns and the observed tensile stresses at specific nodes.

Interpretation

In PLAXIS Output, stress results can be visualised in two ways: at the stress points of the soil element (in the principal direction), as shown in Figure 1, or at the nodes of the soil element (in the shadings plot), as illustrated in Figure 2. The type of points used for stress plotting (stress point or node) is indicated in parentheses next to the maximum and minimum values on the graph (see Figure 3).

Figure 3: Results plotted for stress points and Nodes

In the context of Finite Element Analysis, stresses are computed at the stress points and then, for visualisation purposes,  extrapolated to nodal values to create shaded plots. This extrapolation process may occasionally yield anomalies or spikes in results, influenced primarily by factors such as mesh density and the gradient of stresses (i.e., the rate at which they change). Consequently, the values at the nodes can deviate from those at the stress points, especially in non-uniform stress fields.

When two or more neighbouring finite elements share common nodes, extrapolating stresses from stress points to these nodes can cause stress discontinuities. Each element independently calculates stresses at its stress points. Despite being in the same physical location within the structure, the stresses extrapolated to shared nodes may differ due to variations in integration rules or element shapes. Consequently, stress values at shared nodes might vary between adjacent elements. To derive a nodal stress value, the standard procedure involves computing the weighted average of the extrapolated element stresses at that node (see also “Result Smoothing”).

The example presented in Figures 1 and 2 illustrates a scenario where two adjacent volumes with different material properties (one allowing tension and the other restricting it) extrapolate different stress values to their common nodes. As a result, the nodal values at the highlighted corner in Figure 2, which come from both the concrete block and the neighbouring volume element that allows tension, display tensile stresses. Therefore, the tensile principal total stresses (σ₃) are not merely a visualisation issue but represent weighted average values, as explained above.

Result Smoothing

As mentioned above, the extrapolation of results from stress points to nodes can introduce numerical noise and localised irregularities, particularly in coarse meshes or elements where stresses change rapidly. To mitigate this, PLAXIS applies result smoothing, which computes a weighted average of nodal values from neighbouring elements. The contribution of each adjacent element is influenced by the distance of the stress points from the nodes and accounts for element size. Stress points closest to a node and smaller elements have a greater influence in determining the extrapolated nodal value. 

The Use Result Smoothing option reduces numerical noise from extrapolation, making plots appear more visually consistent. However, when stresses change significantly over a single element, this smoothing process can lead to misleading visual effects. Users should pay particular attention to such cases, as extrapolation can create unexpected stress patterns. The Ignore Transitions option further smooths stress and strain results across structural elements that share nodal positions, even if they differ in orientation. However, this option does not smooth results between elements with different material sets.

The inclusion of nodes in the smoothing process follows specific rules. When "Ignore Transitions" is OFF, smoothing is primarily determined by element connectivity, whereas when it is ON, smoothing is instead based on geometric proximity. Only results belonging to the same material and structure type are smoothed. Notably, results from Plates, Geogrids, Cables, Embedded Beams, and Connections are specifically excluded from smoothing. Additionally, embedded beam nodes located on a soil surface or at the interface between elastic and nonlinear materials, as well as water results on dry clusters, are not smoothed. In 3D structures, smoothing is exclusively performed only on co-planar or aligned elements.
Result smoothing can be controlled using the View > Use Result Smoothing and View > Ignore Transitions options. When these options are deactivated, no smoothing is applied, and raw extrapolated values are displayed.

To reduce the impact of extrapolation and improve result accuracy, the mesh can be refined to shorten the extrapolation distance, leading to more localised and accurate nodal values, or higher-order elements (available in PLAXIS 2D) can be used to improve stress representation.