8.19 Surfaces - Principal Stresses

To control the graphical display of principal stresses, select Surfaces → Stresses in the Results navigator (see Figure 8.48). Table 4.19 shows the principal stresses of surfaces in numerical form.

Image 8.50 Table 4.19 *Surfaces - Principal Stresses*

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

The Grid Point and Grid Point Coordinates table columns correspond to those of the previous results table 4.18 Surfaces - Basic Stresses.

Principal stresses

The basic stresses described in Chapter 8.18 refer to the coordinate system xyz of the surface. However, the principal stresses represent the extreme values of the stresses in a surface element. The principal axes 1 (maximum value) and 2 (minimum value) are arranged orthogonally.

It is possible to display the principal axis orientations α as trajectories in the work window (see Figure 8.45).

The principal stresses are determined from the basic stresses:

Table 8.12 Principal stresses
σ_1,+	Stress in the direction of principal axis 1 on the positive side of the surface (i.e. side in direction of positive surface axis z) $σ_{1, +} = \frac{1}{2} (σ_{x, +} + σ_{y, +} + \sqrt{(σ_{x, +} - σ_{y, +})^{2} + 4 τ_{x y, +}^{2}})$
σ_2,+	Stress in the direction of principal axis 2 on the positive side of the surface (i.e. side in direction of positive surface axis z) $σ_{2, +} = \frac{1}{2} (σ_{x, +} + σ_{y, +} - \sqrt{(σ_{x, +} - σ_{y, +})^{2} + 4 τ_{x y, +}^{2}})$
α₊	Angle between local axis x (or y) and principal axis 1 (or 2) for the stresses on the positive side of the surface $α_{+} = \frac{1}{2} atan2 (\frac{2 τ_{x y, +}}{σ_{x, +} - σ_{y, +}}) \in (- 90 °, 90 °]$
σ_1,−	Stress in the direction of principal axis 1 on the negative side of the surface $σ_{1, -} = \frac{1}{2} (σ_{x, -} + σ_{y, -} + \sqrt{(σ_{x, -} - σ_{y, -})^{2} + 4 τ_{x y, -}^{2}})$
σ_2,−	Stress in the direction of principal axis 2 on the negative side of the surface $σ_{2, -} = \frac{1}{2} (σ_{x, -} + σ_{y, -} - \sqrt{(σ_{x, -} - σ_{y, -})^{2} + 4 τ_{x y, -}^{2}})$
α₋	Angle between local axis x (or y) and principal axis 1 (or 2) for the stresses on the negative side of the surface $α_{-} = \frac{1}{2} atan2 (\frac{2 τ_{x y, -}}{σ_{x, -} - σ_{y, -}}) \in (- 90 °, 90 °]$
σ_1,m	Membrane stress in the direction of principal axis 1 $σ_{1, m} = \frac{1}{2} (σ_{x, m} + σ_{y, m} + \sqrt{(σ_{x, m} - σ_{y, m})^{2} + 4 τ_{x y, m}^{2}})$
σ_2,m	Membrane stress in the direction of principal axis 2 $σ_{2, m} = \frac{1}{2} (σ_{x, m} + σ_{y, m} - \sqrt{(σ_{x, m} - σ_{y, m})^{2} + 4 τ_{x y, m}^{2}})$
α_m	Angle between local axis x and principal axis 1 for the membrane stresses $α_{m} = \frac{1}{2} atan2 (\frac{2 \cdot τ_{x y, m}}{σ_{x, m} - σ_{y, m}}) \in (- 90 °, 90 °]$
τ_max	Maximum shear stress perpendicular to surface $τ_{max} = \sqrt{τ_{x z}^{2} + τ_{y z}^{2}}$

Parent section

8 Results