8.19 Surfaces - 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)

${\sigma }_{1,+}=\frac{1}{2}\left({\sigma }_{x,+}+{\sigma }_{y,+}+\sqrt{({\sigma }_{x,+}-{\sigma }_{y,+}{)}^2+4{\tau }_{xy,+}^2}\right)$

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)

${\sigma }_{2,+}=\frac{1}{2}\left({\sigma }_{x,+}+{\sigma }_{y,+}-\sqrt{({\sigma }_{x,+}-{\sigma }_{y,+}{)}^2+4{\tau }_{xy,+}^2}\right)$

α₊

Angle between local axis x (or y) and principal axis 1 (or 2) for the stresses on the positive side of the surface

${\alpha }_{+}=\frac{1}{2}\text{atan2}\left(\frac{2 {\tau }_{xy,+}}{{\sigma }_{x,+}-{\sigma }_{y,+}}\right) \in (-90°, 90°]$

σ_1,−

Stress in the direction of principal axis 1 on the negative side of the surface

${\sigma }_{1,-}=\frac{1}{2}\left({\sigma }_{x,-}+{\sigma }_{y,-}+\sqrt{({\sigma }_{x,-}-{\sigma }_{y,-}{)}^2+4{\tau }_{xy,-}^2}\right)$

σ_2,−

Stress in the direction of principal axis 2 on the negative side of the surface

${\sigma }_{2,-}=\frac{1}{2}\left({\sigma }_{x,-}+{\sigma }_{y,-}-\sqrt{({\sigma }_{x,-}-{\sigma }_{y,-}{)}^2+4{\tau }_{xy,-}^2}\right)$

α₋

Angle between local axis x (or y) and principal axis 1 (or 2) for the stresses on the negative side of the surface

${\alpha }_{-}=\frac{1}{2}\text{atan2}\left(\frac{2 {\tau }_{xy,-}}{{\sigma }_{x,-}-{\sigma }_{y,-}}\right) \in (-90°, 90°]$

σ_1,m

Membrane stress in the direction of principal axis 1

${\sigma }_{1,m}=\frac{1}{2}\left({\sigma }_{x,m}+{\sigma }_{y,m}+\sqrt{({\sigma }_{x,m}-{\sigma }_{y,m}{)}^2+4{\tau }_{xy,m}^2}\right)$

σ_2,m

Membrane stress in the direction of principal axis 2

${\sigma }_{2,m}=\frac{1}{2}\left({\sigma }_{x,m}+{\sigma }_{y,m}-\sqrt{({\sigma }_{x,m}-{\sigma }_{y,m}{)}^2+4{\tau }_{xy,m}^2}\right)$

α_m

Angle between local axis x and principal axis 1 for the membrane stresses

${\alpha }_m=\frac{1}{2}\text{atan2}\left(\frac{2·{\tau }_{xy,m}}{{\sigma }_{x,m}-{\sigma }_{y,m}}\right) \in (-90°, 90°]$

τ_max

Maximum shear stress perpendicular to surface

${\tau }_{\text{max}}=\sqrt{{\tau }_{xz}^2+{\tau }_{yz}^2}$