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

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

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

Unlike surface stresses, solid stresses cannot be described by simple equations. The Basic Stresses σ_x, σ_y, and σ_z as well as the Shear Stresses τ_xy, τ_yz, and τ_xz are directly determined by the analysis core.

If a cube with the edge lengths d_x, d_y, and d_z 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.

The matrix of the stress tensor is the following:

$S=\left[\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_y& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_z\end{array}\right]$

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

Principal stresses:

$\text{det}(S-\sigma E)=0$

where

E : 3x3 unit matrix

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

Maximum shear stress:

${\tau }_{\text{max}}=\frac{1}{2}({\sigma }_1-{\sigma }_3)$

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

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

Equivalent stress from principal stresses according to von Mises:

${\sigma }_{\mathrm{eqv}}=\sqrt{\frac{1}{2}\left[\right({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_1-{\sigma }_3{)}^2+({\sigma }_2-{\sigma }_3{)}^2]}$

Equivalent stress from basic stresses according to von Mises:

${\sigma }_{\mathrm{eqv}}=\sqrt{{\sigma }_x^2+{\sigma }_y^2+{\sigma }_z^2-{\sigma }_x{\sigma }_y-{\sigma }_x{\sigma }_z-{\sigma }_y{\sigma }_z+3({\tau }_{xy}^2+{\tau }_{xz}^2+{\tau }_{yz}^2)}$

For determining the Equivalent Stress σ_eqv 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:

${\sigma }_{\mathrm{eqv}}=\text{max}(\left|{\sigma }_1-{\sigma }_2\right|, \left|{\sigma }_2-{\sigma }_3\right|, \left|{\sigma }_3-{\sigma }_1\right|)$

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

Determination of the equivalent stress according to Rankine:

${\sigma }_{\mathrm{eqv}}=\text{max}(\left|{\sigma }_1\right|, \left|{\sigma }_2\right|, \left|{\sigma }_3\right|)$

For determining the Equivalent Stress σ_eqv 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:

${\sigma }_{\mathrm{eqv}}=\text{max}[\left|{\sigma }_1-\nu ({\sigma }_2+{\sigma }_3)\right|, \left|{\sigma }_2-\nu ({\sigma }_3+{\sigma }_1)\right|, \left|{\sigma }_3-\nu ({\sigma }_1+{\sigma }_2)\right|]$