“Plane-stress” and “Plane-strain” are two terms we commonly come across when using finite element analysis software. These are essentially two variations of 2D analysis, with differences in their assumptions.
Plane-stress, Plane-strain, what are they?
As the names suggest, “plane-stress” means “stress can only exist in-plane”, i.e. no out-of-plane stress will be produced. In plane-stress analysis uses the analogy of a thin shell – the model is basically looking at this thin shell directly from above. The shell can be compressed or stretched within the plane of analysis, and during this process, strain will be generated in the out-of-plane direction, as a result of applying the Poisson’s ratio.
“plane-strain” means “strain can only exist in-plane”, i.e. no out-of-plane strain will be produced. This is because the boundary condition is restrained (meaning movement is prevented) in the out-of-plane direction. Since movement is restrained, there is no strain in the out-of-plane direction. Instead, stress will be generated due to the movement fixity. This is analogous of a cross section of an infinitely long element.
There is also a variation of plane-strain analysis, which is called “axisymmetric” analysis. In this type of analysis, the cross section is assumed to revolve a full circle about the zero axis line. Out-of-plane strain is still non-existent, only the representative geometry is different.
What’s same and what’s different between plane-stress and plane-strain
Things in common:
Both are 2D analysis based on abstraction and simplification of the real geometry. Although being much more abstract and less accurate in comparison, 2D analysis runs multiple times faster than 3D. This offers significant advantages when the accuracy from the 2D analysis is ‘good enough’ in practice.
Both are effectively ‘cross-sectional’ analysis, i.e. based on a ‘slice’ of the real situation. Neither can take into account of discontinuities and changes of cross section.
Neither can simulate out-of-plane bending/shear. Out-of-plane behaviour can only be simulated by 3D analysis.
Differences:
In plane-stress analysis, there can be strain in the thickness of the element. In other words, if the element is stretched, it will get thinner, and if compressed, it will get thicker.
In plane-strain analysis, the deformation in the out-of-plane (thickness) direction is fully fixed so no deformation could take place. Because of this, stress builds up in the out-of-plane direction as the plate takes in-plane stresses.
Range of use:
Plane stress is generally suitable for the analysis of an element with relatively limited depth in the out-of-plane direction, such as a box or stocky cylinder. This analysis is typically offered only in structural or generic FE analysis software, not in geotechnical analysis software.
·Plane strain is generally suitable for cross-sectional analysis of elements of almost infinite depth in the out-of-plane direction. Situations of ‘linear’ structures, such as retaining walls, embankments, bridges and tunnels, which have constant cross-sections, with lengths that can be taken as almost infinite compared to their cross-sectional size, and negligible change in length when subject to loading.