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# Derivation of stress results

##### Table of contents

You can select one or more of the following results:

• Principal residual stress (first principal and second principal stresses).
• Maximum shear stress.
• Mises-Hencky stress.

The mathematical derivation of these results can be illustrated with the aid of the well known Mohr's circle which depicts the stress state at a point.

## First principal and second principal stress

The principal stresses are the extreme values of the normal stresses. Since they characterize the physical state of the stress at a point, they are independent of any coordinates of reference. They are calculated in the following way:

where is the maximum normal stress and is the minimum normal stress. The corresponding directions of the principal stresses are called the first principal and second principal direction respectively. The angle for these is calculated using the following equation:
Note Positive principal stress values represent tension and negative values compression.

## Maximum shear stress

The Maximum shear stress is the extreme value of the shear stress and its value is calculated as follows:

## Mises-Hencky stress

The Mises-Hencky stress is calculated as follows:

Note As can be seen in the above equation, Mises-Hencky stress values will always be positive. Due to the extrapolation from elemental centroids to the edges of elements, it is possible for small negative values to be generated for Mises-Hencky stress. These small negative values should be regarded as equal to zero.

## Interpreting the stress results

In general, when examining stress results, check the distribution of stresses within the part and the maximum stress levels in the part.Stress and Warp outputs results for both the top and bottom of the element (Normalized thickness = 1 and -1 respectively).

These need to be compared against recommended maximum stresses for the material and any relevant design criteria for the part, for example, specified failure criteria.

Non-filled, isotropic materials will in general exhibit either brittle or ductile behavior, as illustrated in the following figure, where (a) represents brittle and (b) ductile stress-strain behavior.

The recommended stress results to consider in each case are:

• For brittle materials, consider the principal stress results.
• For ductile materials, consider the Mises-Hencky result.

For fiber-filled, anisotropic materials, the behavior of the part under load, the mechanics of failure, and the design criteria for failure will be considerably more complex than for an isotropic material. Stress analysis of composite materials, and interpretation of the results obtained, requires special expertise on the part of the user.

Both the fiber orientation and Stress/Warp analyses output results on a per-laminate basis through the thickness of the part.

NotePrincipal stress orientation may not correspond to principal fiber orientation. Stress orientation data is dependent on the stress state and is located in the center of each layer. Fiber orientation data is calculated at the layer interfaces and is a material property.