RFEM 5

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RFEM 5

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8.34 Solids - Stresses

To control the graphical display of solid stresses, select the Solids check box in the Results navigator. Table 4.34 shows the stresses of solids in numerical form.

The results in the table refer to the grid points of the boundary surfaces, which means that the table does not list any stresses available inside the solid. However, stresses within the solid can be represented graphically on the interior FE mesh points: In the Results navigator, select the Values on Surfaces → Settings → On FE mesh points option. To display the values specifically, use a clipping plane (see Chapter 9.9.2).

Figure 8.69 Results navigator: Solids → Stresses
Figure 8.70 Table 4.34 Solids - Stresses

The table shows the solid stresses sorted by surfaces. The results are listed in reference to the grid points of each surface.

Grid Point

The numbers of the grid points are listed by surface. For more information about grid points, see Chapter 8.13.

Grid Point Coordinates

Table columns C to E show the coordinates of grid points in the global coordinate system XYZ.

Basic Stresses / Shear Stresses / Principal Stresses

Unlike surface stresses, solid stresses cannot be described by simple equations. The Basic Stresses σx, σy, and σz as well as the Shear Stresses τxy, τyz, and τxz are directly determined by the analysis core.

If a cube with the edge lengths dx, dy, and dz is cut from a 3D object with multiaxial loading, the stresses in each cubic surface can be split into normal and shear stresses. If neither the spatial force nor stress differences on parallel surfaces are considered, the stress condition in the cube's local coordinate system can be described by nine stress components.

Figure 8.71 Solid element with stress components

The matrix of the stress tensor is the following:

S=σxτxyτxzτyxσyτyzτzxτzyσz 

The Principal Stresses σ1, σ2, and σ3 result from the eigenvalues of the tensor according to the following formula:

Principal stresses:

det(S-σE)=0 

where

E : 3x3 unit matrix

The maximum Shear Stress τmax is determined according to Mohr's circle:

Maximum shear stress:

τmax=12(σ1-σ3) 

The trajectories of the principal stresses can be displayed graphically by selecting the σ123 navigator entry.

Equivalent stresses

The equivalent stress σv according to von Mises can be expressed by the following homologous equations:

Equivalent stress from principal stresses according to von Mises:

σv=12(σ1-σ2)2+(σ1-σ3)2+(σ2-σ3)2 

Equivalent stress from basic stresses according to von Mises:

σv=σx2+σy2+σz2-σxσy-σxσz-σyσz+3(τxy2+τxz2+τyz2) 

For determining the Equivalent Stress σv according to Tresca, RFEM analyzes the differences from the principal stresses in order to determine the maximum value with them.

Determination of the equivalent stress according to Tresca:

σv=maxσ1-σ2, σ2-σ3, σ3-σ1  

The Equivalent Stress σv according to Rankine is determined from the maximum absolute values of the principal stresses.

Determination of the equivalent stress according to Rankine:

σv=maxσ1, σ2, σ3 

For determining the Equivalent Stress σv according to Bach, RFEM analyzes the principal stress differences while taking Poisson's ratio ν into account in order to determine the maximum value with them.

Determination of the equivalent stress according to Bach:

σv=maxσ1-ν(σ2+σ3), σ2-ν(σ3+σ1), σ3-ν(σ1+σ2)