1. Transformation methods - Examples

2. Overview Transformations Mohr’s Circle Eigenvalues and Eigenvectors
Direct Approach
Transformation Matrix
Mohr’s Circle
Construction
Principal Stresses
Eigenvalues and Eigenvectors
Principal Scalar Invariants
Example Problems

3. Transformations
Transformation of axes:

4. Transformations
Transformation of axes:

5. Transformations 1. Direct Approach Balance forces in y’ direction:
Simple Example: uniaxial stretch
Balance forces in y’ direction:
Balance forces in τx’y’ direction:

6. Transformations
1. Direct Approach
Generalized Example:

7. Transformations
2. Transformation Matrix

8. Transformations
2. Transformation Matrix

9. Mohr’s Circle
Graphical representation of the transformation equations
General conventions
τxy σx σy
Shear stress (τxy) is positive if resulting distortion stretches in 1st and 3rd quadrant
Shear stress (τxy) is negative if resulting distortion stretches in 2nd and 4th quadrant
Tensile stress (σx or σy) is positive
Compressive stress is negative
Mohr’s conventions on σ-τ plot
Shear stress (τxy) is positive if surface stress rotates element clockwise
Shear stress (τxy) is negative ifsurface stress rotates element counter-clockwise
Tensile stress (σx or σy) is positive
Compressive stress is negative

10. Mohr’s Circle σy = -3 τxy = 4 σx = 5 σ τ
Graphical representation of the transformation equations
τxy = 4
σx = 5
σy = -3 σ τ

11. Mohr’s Circle σy = -3 τxy = 4 σx = 5 σ τ
Graphical representation of the transformation equations
τxy = 4
σx = 5
σy = -3 σ
Points on the circle represent this state of stress at any angle τ

12. Mohr’s Circle Rotation on Mohr’s circle = 2x angle of transformation
Graphical representation of the transformation equations
Rotation on Mohr’s circle =
2x angle of transformation

13. Mohr’s Circle
Principal Stress Calculation

14. Mohr’s Circle Principal Stress Calculation Principal Stresses
Maximum Shear Stress
Angle to Principal Planes:

15. Eigenvalues and Eigenvectors
Definition:
T is a tensor
a is a normalized vector which transforms under T into a vector parallel to itself
λ is a scalar
a is an eigenvector
λ is an eigenvalue

16. Eigenvalues and Eigenvectors
Solution:
(index notation)
(long form)

17. Eigenvalues and Eigenvectors
Characteristic Equation:
The eigenvalues of tensor T are the solution to the characteristic equation

18. Eigenvalues and Eigenvectors
Vector Solution:
The eigenvectors of tensor T are the solution to the following (for each λ):

19. Eigenvalues and Eigenvectors
Example:

20. Eigenvalues and Eigenvectors
Example:

21. Eigenvalues and Eigenvectors
Principal Values and Principal Directions
For symmetric tensors (i.e. stress tensor, strain tensor, rate of deformation tensor, etc.):
There exist at least three real eigenvectors, which are called the principal directions, which are mutually perpendicular
The corresponding eigenvalues are called the principal values
Stresses normal to these principal directions are principal stresses,
in which the normal stress vector is maximized

22. Principal Scalar Invariants
Characteristic Equation:
* Cubic equation in λ
Note: invariants are associated with tensor independent of coordinate system

23. Principal Scalar Invariants

24. Principal Scalar Invariants
Example:
Characteristic Equation: