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Prof. Ramesh Singh Outline • Principal stresses • Mohr’s circle in 3D • Strain tensor • Principal strains
Prof. Ramesh Singh
Outline
• Principal stresses• Mohr’s circle in 3D• Strain tensor• Principal strains
Prof. Ramesh Singh
Principal Stresses in 3D• 3-D Stresses can be represented by in usual
notation
We will use a concept from continuum mechanics
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Traction vectorForce/area
Prof. Ramesh Singh
Principal Stresses
Prof. Ramesh Singh
Principal Stresses in 3D
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Prof. Ramesh Singh
Principal Stresses in 3D
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Prof. Ramesh Singh
3D Stress – Principal Stresses
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The three principal stresses are obtained as the three real roots of the following equation:
where
1
2 2 22
2 2 23 2
x y z
x y x z y z xy xz yz
x y z xy xz yz x yz y xz z xy
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s s s s s s t t t
s s s t t t s t s t s t
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= + + - - -
= + - - -
I1, I2, and I3 are known as stress invariants as they do not change in value when the axes are rotated to new positions.
Prof. Ramesh Singh
Principal Stress
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In[3]:= Eigensystem[{{0, -240, 0}, {-240, 200, 0}, {0, 0, -280}}]Out[3]= {{360, -280, -160}, {{-2, 3, 0}, {0, 0, 1}, {3, 2, 0}}}
Prof. Ramesh Singh
Principal Stresses in 3-D
Prof. Ramesh Singh
Linear Strains
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Linear strain formulation:
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Prof. Ramesh Singh
Shear Strain
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Prof. Ramesh Singh
Strain Tensor
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Prof. Ramesh Singh
Strain Transformation
www.efunda.com
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Prof. Ramesh Singh
Mohr’s Circle for Strain
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Prof. Ramesh Singh
Principal Strains
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