Extracting shell element plane stress components relative to a common CSYS

When expressed in terms of the element coordinate system, the stress state of a plane stress shell element is:

\left[ \sigma \right] = \left[ {\begin{array}{{20}{c}} {{\sigma _x}}&{{\tau _{xy}}}&{{\tau _{xz}}}\ {{\tau _{xy}}}&{{\sigma _y}}&{{\tau _{yz}}}\ {{\tau _{xz}}}&{{\tau _{yz}}}&{{\sigma _z}} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} {{\sigma _x}}&{{\tau _{xy}}}&0\ {{\tau _{xy}}}&{{\sigma _y}}&0\ 0&0&0 \end{array}} \right]

where the X- and Y-directions of the element coordinate system lie in the plane of the shell element.

Because the element coordinate system for a shell element depends on the shape and connectivity of the element, the orientation of the X- and Y-axes of the element coordinate systems typically vary from element to element. Thus, the X-axis of one element may differ from the X-axis of a neighboring element, and so on.

When you select the Shell Results in Projected Coordinate System command, the software transforms the stresses for each shell element to a coordinate system whose X- and Y-axes lie in the plane of the shell element, and, for certain cases such as all the shell elements lying in the same plane, whose X-axis is the same for all of the shell elements.

To do so, for each element, the software:

1. Transforms the stress state from the element coordinate system to the coordinate system that you specify using the Coordinate System list options.

2. Derives the new in-plane coordinate axes for the shell element as follows:Projects the +X-axis of the coordinate system that you specify using the Coordinate System list options along the normal to the shell element onto the face of the shell element to define the +X-axis of the new coordinate system.Uses the positive normal for the shell element as the +Z-axis of the new coordinate system.Computes the vector cross product of the unit vector in the +Z-direction of the new coordinate system and the unit vector in the +X-direction of the new coordinate system to obtain the unit vector in the +Y-direction of the new coordinate system.

3. Transforms the stress state from the coordinate system that you specify using the Coordinate System list options to the new coordinate system.

Thus, the plane stress representation of the stress state for the shell element is retained in the new coordinate system is as follows:

\left[ \sigma \right] = \left[ {\begin{array}{{20}{c}} {{\sigma _{x'}}}&{{\tau _{x'y'}}}&{{\tau _{x'z}}}\ {{\tau _{x'y'}}}&{{\sigma _{y'}}}&{{\tau _{y'z}}}\ {{\tau _{x'z}}}&{{\tau _{y'z}}}&{{\sigma _z}} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} {{\sigma _{x'}}}&{{\tau _{x'y'}}}&0\ {{\tau _{x'y'}}}&{{\sigma _{y'}}}&0\ 0&0&0 \end{array}} \right]

where X' and Y' are the in-plane axes of the new coordinate system.

Note:

Because of rounding errors associated with the coordinate transformations, σz, τx'z, and τy'z may not be identically zero.

If the software is unable to project the +X-axis onto the shell element face, it projects the +Y-axis instead and follows a similar procedure to obtain the new coordinate system.

Caution:

To avoid inconsistencies in the algebraic sign of the in-plane shear stress, make sure that all of the element normals point to the same side of the mesh.

Where do I find it?

Application	Pre/Post
Prerequisite	An active margin of safety solution with the Load Extraction dialog box open
Location in dialog box	In the Result group, Shell Results in Projected Coordinate System check box

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Extracting shell element plane stress components relative to a common CSYS, Simcenter 3D 2021.1 Series

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