Prior to Patran Version 9.0, the results processor attempted to recognize whether a stress or strain tensor was 2D (e.g. Plane Stress: iz = 0 and Plane Strain: iz = 0 for i = 1, 2, 3) or 3D and then calculate principal values based either a 2D or 3D formulation respectively. Starting with Patran Version 9.0 the user must choose to use either the 2D or 3D formulation. The mechanism that is provided to allow the user to choose a specific formulation is to either select the derived quantity that includes “2D” as part of its name, which will cause the 2D formulation to be used, or to choose the quantity that does not contain “2D” as part of its name, which will cause the 3D formulation to be used. This change was motivated by requests from our customers who wanted to control which formulation they wanted to apply.

Examples of these 2D or 3D tensor quantities are shown below.

For 2D tensors Patran uses the two in plane principal values as the maximum and the minimum regardless if both of their values are either greater or less than zero. Patran calculates the maximum shear stress to be one half the difference between the maximum and minimum principal values. A consequence of this formulation is that for the cases where both in plane principal values have common signs the maximum shear stress can be under calculated. Similarly, Tresca stress could be under calculated, as shown in the following example.

Example:

Smajor = 30Sminor = 10 where Smajor and Sminor are the maximum and minimum 2D in-plane Principal Stresses respectively.

Using a 2D tensor, ”Tresca 2D” will be 30-10 = 20.

Using a 3D tensor, ”Tresca” will be 30 – 0 = 30 (Sminor = 0)

The following table provides additional definitions for the selected result derivations. These include tensor to vector, tensor to scalar, and vector to scalar resolutions.

Below are the equations and examples of the derivation methods:

von Mises stress is calculated from the following equation:

Example: The elements shown below have the following stress contributions:

The von Mises stress calculated at node 11 when nodal averaging is done first due to the contribution from each element is 78.96. When the von Mises derivation is done first and then averaging at the nodes takes place, the calculated von Mises stress is 79.02. Thus a difference can arise depending on whether the averaging is done first or the derivation. This can be true for all derived results.

 

Octahedral shear stress is calculated from the following equation:

From the von Mises example above the octahedral shear stress is:

Hydrostatic stress is calculated from the following equation:

From the von Mises example above the hydrostatic stress is:

1st, 2nd, and 3rd invariant stresses are calculated from the following equations:

From the von Mises example above the invariant stresses are:

Principal stresses are calculated from either a Mohr’s circle method for 2D tensors or from a 3x3 Jacobian Rotation Eigenvector extraction method for a 3D tensors. The User Interface allows for either a tensor-dependent derivation, or a 2D calculation. The tensor-dependent calculation will choose either a 2D or 3D calculation depending on values of each tensor encountered. A 2D calculation will be used when the ZZ, YZ and ZX are exactly zero (which is the case when the analysis code does not calculate these values), otherwise the full 3D tensor will be considered. Both the magnitudes of the principals and their direction cosines are calculated from these routines.

The magnitudes of the two principal stresses from the 2D Mohr’s circle method are calculated according the following equations:

 

The direction cosines for the 2D Mohr’s circle method are calculated by assembling the following 3x3 transformation matrix:

From the von Mises example above the principal stresses are:

Also the principal stress determinant is the product of the three principals and the major, minor, and intermediate principal deviatoric stresses are calculated from:

Tresca shear stress is calculated from the following equation:

where are calculated as mentioned under Principal stress derivations above.

From the von Mises example above the Tresca shear stress is:

Maximum shear stress is calculated from the following equation

where are calculated as mentioned under Principal stress derivations above.

From the von Mises example above the Tresca shear stress is:

Magnitude (vector length) is calculated from the components with the standard formula: