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Mohr’s Circle in 3-DStarting with 2-D
MAE 322
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Rotate Coordinates for Stress
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We can find 2-D Principal Stressesand Max Shear Stress
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Finding 2-D Principal Stresses
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Let’ use numbers as an example 2D
τmax= (σ1-σ2)/2
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1 Mohr Circle for each of the 3 Orthogonal Planes
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Now think of 3 Orthogonal 2-D Planes for 3-D
τmax= (σ1-σ3)/2
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3-D Plane Stress Example, 3 Planes
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Rotating 3 times, from any 3-D stress state we can get 3-Principal Stresses
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Mathematically this is an Eigenvalue Problem to get 3-Principal Stresses
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2-D Plane Stress is Always a 3-D Stress State: σ3=0
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Back to Maximum Shear Stress Failure Theory• For any stress element, use Mohr’s circle to find the
maximum shear stress. Compare the maximum shear stress to Sy/2.
• Ordering the principal stresses such that s1 ≥ s2 ≥ s3,
max
/ 2ySn
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Case 1: Mohr’s Circle σ1> σ2 >σ3=0
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Case 2: Mohr’s Circle σ1> 0 >σ2 >σ3
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Case 3: Mohr’s Circle σ1= 0 >σ2 >σ3