Stress Linearization
Stress linearization is a procedure in which the stress distribution along a line through the thickness in a solid is approximated with an equivalent linear stress distribution, similar to what would be the result of an analysis using shell theory. The line is commonly referred to a stress classification line, SCL. This type of evaluation is common in the analysis of pressure vessels. It is also useful for the design of reinforcements for concrete structures and for fatigue analysis of welds.
To perform a stress linearization, you add one Stress Linearization node for each SCL.
Orientation of Stress Components
The stresses along the SCL are represented in a local orthonormal coordinate system, x1-x2-x3. The x1 direction is oriented along the SCL, so it is mainly the stresses in the second and third directions that are of interest.
In 3D, you must specify the x2 direction and thus implicitly the x3 direction. You specify the orientation either by selecting a point in the x1-x2 plane or by defining an orientation vector in an approximate x2 direction. In either case, the actual x2 direction is chosen so that it is perpendicular to the SCL and lies in the plane you have specified. The x3 orientation is then taken as perpendicular to x1 and x2. As long as you are only interested in a stress intensity, the choice of orientation is arbitrary.
In 2D, the x3 direction is the out-of-plane direction, and the x2 direction is perpendicular to the SCL in the XY-plane.
In 2D axial symmetry, the x3 direction is the azimuthal direction, and the x2 direction is perpendicular to the SCL in the RZ-plane.
Creating the Stress Classification Line
The most straightforward way to create an SCL is to include it in the geometry, and then select it in a Stress Linearization node. This corresponds to using the Line linearization type where the SCL is defined by Edge.
Figure 2-35: Four stress classification lines in a transition region at a pressure vessel nozzle.
There are, however, some situations where this direct approach is less convenient:
In the two above cases an alternative is to define an SCL that is not included in the geometry by connecting two arbitrary points with a straight line. This corresponds to using the Line linearization type where the SCL is defined by Two points. The introduction of the SCL will in this case not make it necessary to update the geometry and rerun the analysis. Since the SCL is disconnected from the geometry and mesh, this option will also make meshing easier. The downside is then the you cannot control the mesh quality along the SCL.
In some cases it is difficult to a priori determine where to create the SCL or the critical location may change depending on, for example, loading conditions or time. For such cases, it is also possible to define it in a distributed manner such that an infinite number of SCLs are created in a domain. The starting point of the SCLs are given by a boundary selection, and they extend along the normals of the boundary through the selected domains. This corresponds to using the Distributed linearization type.
Studies and Solutions
Stress linearization is a pure postprocessing operation. The Stress Linearization node will only create a number of variables, which can be evaluated under Results. It is thus possible to add such nodes after the main analysis has been performed. In order to make the new variables available for postprocessing, you must then run an Update Solution.
Results
When you have included one or more Stress Linearization nodes in a model, a number of datasets and an extra predefined plot are generated.
One edge dataset is created for each SCL. These datasets are named Linearization Line <n>, where n is an integer number.
Figure 2-36: Generated datasets in a model with five SCL.
The plot contains graphs for the 22 component of the actual stress, the membrane stress, and the linearized stress. The first Linearization Line dataset is selected. By changing edge dataset in the plot group, you can easily move between the different stress classification lines.
Figure 2-37: Plot along a stress classification line.
For the Distributed linearization type, the created Linearization Line dataset corresponds to a single SCL in selected domain. The starting point is specified in the Postprocessing section of the feature, and must lie on the selected boundaries. The endpoint is computed internally such that the cut line extends through the selected domains along the boundary normal of the point.
Variable Names
Each Stress Linearization node adds a number of variables. Many of these variables exist with two different scopes, physics scope and feature scope. The physics scope is the name of the physics interface, having the default value ‘solid’. The feature scope contains also the tag of the Stress Linearization node, for example, ‘sl1’.
As an example, the variable solid.Sm22 and the variable solid.sl1.Sm22 have the same value. The variables with physics scope make it more convenient to create expressions using postprocessing. You could, for example, make a line plot of solid.sb22 and get all edges having a stress linearization colored by their individual results.
σij
ij = 11, 12, 13, 22, 23, 33
Smij
σm,ij
ij = 11, 12, 13, 22, 23, 33
σb(max),ij
ij = 11, 12, 13, 22, 23, 33
Sbij
σb,ij
ij = 11, 12, 13, 22, 23, 33
Smbij
σmb,ij
ij = 11, 12, 13, 22, 23, 33
Spsij
σp(start),ij
ij = 11, 12, 13, 22, 23, 33
Speij
σb(end),ij
ij = 11, 12, 13, 22, 23, 33
σint
σint
σint
Nij
Nij
ij = 22, 23, 33
Mij
Mij
ij = 22, 23, 33
Qi
i = 2, 3
X1
 
Stress Linearization in the Structural Mechanics Theory chapter.