8.16 Surfaces - Principal Internal Forces

To control the graphical display of the principal internal forces, select Surfaces → Principal Internal Forces in the Results navigator. Table 4.16 shows the principal internal forces of surfaces in numerical form.

Image 8.43 *Results* navigator: Surfaces → Principal Internal Forces

Image 8.44 Table 4.16 *Surfaces - Principal Internal Forces*

The table shows the principal internal forces 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 4.15 Surfaces - Basic Internal Forces.

Moments / Shear Forces / Axial Forces

The Basic Internal Forces described in the previous chapter refer to the more or less freely defined coordinate system xyz of a surface. Conversely, Principal Internal Forces represent the extreme values of the internal forces in a surface element. For this purpose, the basic internal forces are transformed in the directions of both principal axes. The principal axes 1 (maximum value) and 2 (minimum value) are arranged orthogonally.

The principal internal forces are determined from the basic internal forces:

Table 8.9 Principal internal forces
m₁	Bending moment in direction of principal axis 1 $\frac{1}{2} (m_{x} + m_{y} + \sqrt{(m_{x} - m_{y})^{2} + 4 m_{x y}^{2}})$
m₂	Bending moment in direction of principal axis 2 $\frac{1}{2} (m_{x} + m_{y} - \sqrt{(m_{x} - m_{y})^{2} + 4 m_{x y}^{2}})$
α_b	Angle between local axis x (or y) and principal axis 1 (or 2) $\frac{1}{2} [arctan (\frac{2 m_{x y}}{m_{x} - m_{y}})]$
m_τ,max,b	Maximum torsional moment $\frac{\sqrt{(m_{x} - m_{y})^{2} + 4 m_{x y}^{2}}}{2}$
v_max,b	Maximum resulting shear force from bending components $v_{max, b} = \sqrt{v_{x}^{2} + v_{y}^{2}}$
β_b	Angle between principal shear force v_max,b and local axis x $β = arctan \frac{v_{y}}{v_{x}}$
n₁	Axial force in direction of principal axis 1 $\frac{1}{2} (n_{x} + n_{y} + \sqrt{(n_{x} - n_{y})^{2} + 4 n_{x y}^{2}})$
n₂	Axial force in direction of principal axis 2 $\frac{1}{2} (n_{x} + n_{y} - \sqrt{(n_{x} - n_{y})^{2} + 4 n_{x y}^{2}})$
α_m	Angle between axis x and principal axis 1 (for axial force n₁) $\frac{1}{2} [arctan (\frac{2 n_{x y}}{n_{x} - n_{y}})]$
v_max,m	Maximum shear force from membrane components $\frac{\sqrt{(n_{x} - n_{y})^{2} + 4 n_{x y}^{2}}}{2}$

The principal axes directions α_b (for bending moments), β_b (for shear forces), and α_m (for axial forces) can be displayed as trajectories in the work window.

Image 8.45 Trajectories of the principal axes

Displaying the angle α_b, for example, also shows the size of the respective principal moments because the trajectories are scaled to the values of the moments m₁ and m₂.

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8 Results