Mohr’s Circle in 3-DStarting with 2-D

MAE 322

Stress Elements

Rotate Coordinates for Stress

We can find 2-D Principal Stressesand Max Shear Stress

Typical Mohr’s Circle

Finding 2-D Principal Stresses

Let’ use numbers as an example 2D

τmax= (σ1-σ2)/2

1 Mohr Circle for each of the 3 Orthogonal Planes

Now think of 3 Orthogonal 2-D Planes for 3-D

τmax= (σ1-σ3)/2

3-D Plane Stress Example, 3 Planes

Rotating 3 times, from any 3-D stress state we can get 3-Principal Stresses

Mathematically this is an Eigenvalue Problem to get 3-Principal Stresses

2-D Plane Stress is Always a 3-D Stress State: σ3=0

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

Case 1: Mohr’s Circle σ1> σ2 >σ3=0

Case 2: Mohr’s Circle σ1> 0 >σ2 >σ3

Case 3: Mohr’s Circle σ1= 0 >σ2 >σ3