Stress linearization formulas for 3D models

For a 3D model, all of the stress components can be non-zero. Thus, the stress state at a point along the SCL is:

\sigma = \left[ {\begin{array}{*{20}{c}} {{\sigma _{xx}}}&{{\sigma _{xy}}}&{{\sigma _{xz}}}\ {{\sigma _{xy}}}&{{\sigma _{yy}}}&{{\sigma _{yz}}}\ {{\sigma _{xz}}}&{{\sigma _{yz}}}&{{\sigma _{zz}}} \end{array}} \right]

where the local x-direction is aligned with the stress classification line (SCL) and the positive sense of both are the same. The local z-direction is obtained by crossing a unit vector in the local x-direction with a vector in the absolute Y-direction. The normalized result is a unit vector in the local z-direction. Then, the local y-direction is found by crossing the unit vector in the local z-direction with the unit vector in the local x-direction.

For the special case where the local x-direction and absolute Y-direction are parallel, the local y- and z-directions are obtained as follows:

• If the positive sense of both the local x-direction and absolute Y-direction are the same, the local y-direction is taken to be the negative absolute X-direction. The local z-direction is obtained by crossing a unit vector in the local x-direction with a unit vector in the local y-direction.

• If the positive sense of both the local x-direction and absolute Y-direction oppose one another, the local y-direction is taken to be the positive absolute X-direction. The local z-direction is obtained by crossing a unit vector in the local x-direction with a unit vector in the local y-direction.

In the Stress Linearization dialog box, in the Bending Stress Components group, you select which stress components to include in the stress linearization.

To obtain the average value of a stress component over the SCL, the stress component, {\sigma _{ij}}, is integrated over the SCL and normalized with respect to the length of the SCL, t. Thus, the average value is given by the following formula:

\sigma {ij}^m = \frac{1}{t}\int\limits{ - t/2}^{t/2} {{\sigma _{ij}}dx}

where \sigma _{ij}^m is the average value of the stress component over the SCL. For the normal stresses in the y- and z-directions, the average values represent membrane stresses. For all of the other stress components, the average value simply represents the average value of the component over the SCL.

The first moment of the actual distribution of a stress component about x = 0 is as follows:

\int\limits_{ - t/2}^{t/2} {{\sigma _{ij}}} xdx

Assume the stress component is distributed linearly along the SCL. Thus, the functional relationship for the stress distribution can be written as follows:

{\sigma _{ij}}\left( x \right) = \sigma _{ij}^m + \left( {\frac{{2\sigma _{ij}^b}}{t}} \right)x

where \sigma _{ij}^b is the magnitude of the linearly varying portion of the stress distribution for a stress component at the ends of the SCL. That is, at x = \pm t/2.

Thus, the first moment of this distribution about x = 0 is:

\int\limits_{ - t/2}^{t/2} {x{\sigma {ij}}dx = \int\limits{ - t/2}^{t/2} {x\left[ {\sigma _{ij}^m + \left( {\frac{{2\sigma _{ij}^b}}{t}} \right)x} \right]} } dx = \frac{1}{6}\sigma _{ij}^b{t^2}

Equating the first moment of the actual distribution to the first moment of the linear distribution and solving yields the following result:

\sigma {ij}^b = \frac{6}{{{t^2}}}\int\limits{ - t/2}^{t/2} {{\sigma _{ij}}} xdx

where for normal stresses in the y- and z-directions, \sigma _{ij}^b represents the bending stress. For all other stress components, \sigma _{ij}^b represents the magnitude of the linearly varying portion of the stress distribution for the stress component at the ends of the SCL.

When you display the results in the Information window, any values calculated using the averaging formula are listed as membrane stresses, and any values calculated using the formula for the stress at the ends of the SCL are listed as bending stresses.

How do I

Perform stress linearization

Learn more

Stress linearization

Stress classification lines

Stress linearization formulas for axisymmetric models

Quick links

Command reference

Pre/Post video examples

Bulk Entry Descriptions

Simcenter 3D tutorials

Browse Simcenter 3D help by product area

Stress linearization formulas for 3D models, Simcenter 3D 2021.1 Series

© 2020 Siemens

window.mainLanguage="en_US"

window.delivId=""

window.projectId=""

MathJax.Hub.Config({ TeX: { extensions: ["autoload-all.js"] }, tex2jax: { displayMath: [ ] }, "SVG": { scale: 125 } });