The Stress menu
Figure 1. The Stress
commands
To visualize the stress field over the cross-section of a beam, the following steps must be completed first.
- Load the problem you want to run and perform the finite element analysis, and
- Select a static loading case.
Once these steps are completed, you are ready to visualize the stress field. Select the menu Stress, to reveal the options shown in fig. 1. If you prefer to list the components of the stress tensor for one single element of the model, use the commands under the Element menu.
The stress field
The stress field comprises six stress components, which can be divided into two groups.
- The out-of-plane stress components. The axial stress component, σ_{11} (x_{2}, x_{3}), the two transverse shear stress components, τ_{12} (x_{2}, x_{3}) and τ_{13} (x_{2}, x_{3}).
- The in-plane stress components. The two in-plane direct stress components σ_{22} (x_{2}, x_{3}) and σ_{33} (x_{2}, x_{3}), and the in-plane shear stress component τ_{23} (x_{2}, x_{3}).
The usual sign conventions are used for these various stress components. Note that in Euler-Bernoulli or Timoshenko beam theory, the in-plane stress components are assumed to vanish, i.e., σ_{22} (x_{2}, x_{3}) ≈ 0, σ_{33} (x_{2}, x_{3}) ≈ 0, and τ_{23} (x_{2}, x_{3}) ≈ 0.
Visualization of the axial stress field (Stress→Axial stress)
Command Stress→Axial stress represent the axial stress field, σ_{11}(x_{2}, x_{3}), by a set of vectors pointing in the direction normal to the plane of the cross-section. The length of each vector is proportional to the magnitude of the axial stress component, σ_{11}, at that point. This option depicts the axial stress component only; the other stress components are ignored.
Figure 2. Distribution of axial stress over a flattened tubeFigure 3. Distribution of axial stress over a circular arc
Figure 2 shows the distribution of axial stress, σ_{11}, over a flattened tube subjected to a bending moment M_{2}. Figure 3 shows the distribution of axial stress, σ_{11}, over a circular arc subjected to a bending moment M_{3}. Because the axial stress vector acts in the direction normal to the plane of the cross-section, it was necessary, in both cases, to rotate the cross-section using the Graphics→Rotate commands.
If the lengths of the stress vectors are too long or too short, use the Graphics→Data size + or Graphics→Data size - commands to adjust their length appropriately.
Visualization of the shear stress field (Stress→Shear stress)
Command Stress→Shear stress represents the transverse shear stress vector field, τ_{12} b_{2} + τ_{13} b_{3}, by a set of vectors pointing in the plane of the cross-section. The length of each vector is proportional to the magnitude of the transverse shear stress vector, √ τ_{12}^{2} + τ_{13}^{2}, at that point. This option depicts the transverse shear stress components only; the other stress components are ignored.
Figure 4. Distribution of shear stress over a flattened tubeFigure 5. Distribution of shear stress over a circular arc
Figure 4 shows the distribution of transverse shear stress, τ_{12} b_{2} + τ_{13} b_{3}, over a flattened tube subjected to a torque M_{1}. Figure 5 shows the distribution of axial stress, τ_{12} b_{2} + τ_{13} b_{3}, over a circular arc subjected to a torque M_{1}. Because the transverse shear stress vector acts in the plane of the cross-section, it was not necessary, in either cases, to rotate the cross-section.
If the lengths of the stress vectors are too long or too short, use the Graphics→Data size + or Graphics→Data size - commands to adjust their length appropriately.
Visualization of individual stress components
Individual stress components of the stress field, σ_{11}, σ_{22}, σ_{33}, τ_{23}, τ_{13}, and τ_{12} can be visualized by invoking commands Stress→Sigma_11, Stress→Sigma_22, Stress→Sigma_33, Stress→Tau_23, Stress→Tau_13, and Stress→Tau_12, respectively.
When visualizing individual stress components, the stress field is represented through color mapping: the magnitude of the stress components is associated with a given color select from a color palette. This representation depicts one stress component only; all other stress components are ignored.
Figure 6. Distribution of stress component σ_{11} over a flattened tubeFigure 7. Distribution of stress component σ_{11} over a circular arc
Figure 6 shows the distribution of stress component σ_{11} over a flattened tube subjected to a bending moment M_{2}. This figure plots the same data as that shown in figure 1, but the representation is different. Figure 7 shows the distribution of stress component σ_{11} over a circular arc subjected to a bending moment M_{3}. This figure plots the same data as that shown in figure 2, but the representation is different.
Visualization of the reserve factor (Stress→Reserve factor)
The previous options have focused on the representation of a single stress component. It is often desirable to visualize the field of reserve factors over the cross-section. Indeed, reserve factors take into account all the stress components at a point. Reserve factors are associated with a failure criteria, which is defined for each material the cross-section is made of.
ShapeBuilder is a pre-processor to SectionBuilder. It enables the parametric definition of simple cross-sectional shapes that are used in mechanical and aerospace engineering. In ShapeBuilder, each zone of the cross-section is associated with a material. For instance, the I-section allows the definition of up to four different materials for the various zones of the cross-section. When defining the physical properties of a material, a failure criterion is associated with the material and this failure criterion leads to the concept of reserve factor.
Command Stress→Reserve factor represents the reserve factor field through color mapping: the magnitude of the reserve factor is associated with a given color select from a color palette.
Figure 7. Distribution of reserve factor over a triangular sectionFigure 8. Distribution of reserve factor over a tube with fins
Figure 7 shows the distribution of reserve factor over a triangular section subjected to a horizontal shear force F_{2}. Figure 8 shows the distribution of reserve factor over a tube with fins subjected to a torque M_{1}.