Principal Stresses and Maximum Shear Stresses

Note that as the stress element is rotated, the transformed normal and shear stresses vary with θ.

This graph is for a special case of σ_y = 0.2 σ_x and τ_xy = 0.8 σ_x, but realize that in general the stresses cycle between maximum and minimum values, which occur at certain values of θ.  Note that the stresses are also zero for other distinct values of θ.  For design purposes, we will need to determine these maximum and minimum stress values and where they occur.

 

principal stresses, σ_1,2

the max. and min. normal stresses

• generally, these stresses will be termed in-plane principal stresses

 

principal planes

the planes on which the principal stresses act

• the principal stresses will occur on mutually perpendicular planes
• the shear stresses are zero on the principal planes

maximum stresses in beams

 

3D Mohr's plot, cube, arrows  (1, 2) Cosserat brothers, 1909 Helices in nature fruit and vegetables wood: 1, 2  (by Blaine Medlock) wood: 1  (by Daniel Deposki, taken at Grand Teton National Park near Lake Jenny) ice helicola  (by Robert Ginzburg) periodic table: 1, 2 higher-order helices in lattices  (pp.87-91) heliospheric current sheet molecules bone annulus fibrosis

 

principal angles, θ_p

the angles the principal planes are rotated through relative to the original coordinate system

• there will be two angles -- you must determine which angle corresponds to σ₁ and which one corresponds to σ₂

 

shear angles, θ_s

the angles the max. and min. shear stress planes are rotated through relative to the original coordinate system

• the planes of maximum shear stress occur at 45^o to the principal planes

θs = θp ± 45o

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