1. DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
UNIVERSITY OF NAIROBI
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
ENGINEERING DESIGN II
FME 461
PART 9
GO NYANGASI
November 2008

2. DISTORTION ENERGY THEORY
A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS

3. STATEMENT OF THE THEORY
When Yielding occurs in any material,
The distortion strain energy per unit volume
At the point of failure
Equals or exceeds
When yielding occurs in the tension test specimen.

4. DISTORTION ENERGY THEORY
Is based on yielding
Applies to ductile materials

5. STRAIN ENERGY AT A LOCATION OF THE ELEMENT
SEGREGATED INTO THREE CATEGORIES:
Total strain energy per unit volume of the stressed element, arising from the three principal stresses
Strain energy per unit volume arising from the hydrostatic stress that causes change of volume only, and which is uniform in all three directions
Strain energy per unit volume arising from stresses causing distortion of the element, and this can be expressed as the difference between category (1) and (2).

6. THREE DIMENSIONAL STRESS
General Case

7. TRI-AXIAL STRESS SITUATION

8. ELASTIC STRESS-STRAIN RELATIONS
UNI-AXIAL STRESS
This is the case of a single principal stress
Principal strains are then given in terms of principal stresses by the expressions in next slide

9. ELASTIC STRESS-STRAIN RELATIONS: Uni-Axial stress
The variables are:
Principal strain in the direction of the principal stress
Poisson’s ratio for the material
Modulus of elasticity for the material
Principal stress

10. Uni-Axial Stress One dimensional (Normal/Shear)

11. ELASTIC STRESS-STRAIN RELATIONS: Bi-Axial stress
In this case the stress situation consists of two principal stresses,
The strains[1] are given by in terms of the two principal stresses as shown in next slide [1] Mechanical Engineering Design; Shigley, Joseph, pg 124, McGraw Hill, Seventh Edition, 2004

12. STRAINS IN BI-AXIAL STRESS
Stress situation consists of two principal stresses and strains are given by the expressions

13. Bi-Axial Stress Two Dimensional (Plane)

14. ELASTIC STRESS-STRAIN RELATIONS
Tri-Axial Stress
This is the case of three principal stresses
The most general case
Three strains in the directions of the principal stresses
Given by in terms of the three principal stresses as shown in next slide

15. STRAINS IN TRI-AXIAL STRESS
Strains are given by the expressions

16. Tri-Axial Stress Three Dimensional stress

17. ENERGY PER UNIT VOLUME Tri-Axial Stress
Total strain energy
The total strain energy is the strain energy caused by the three principal stresses. It is given by the expressions

18. TOTAL STRAIN ENERGY
Substituting for elastic strains

19. Tri-Axial Stress Three Dimensional stress

20. STRAIN ENERGY DUE HYDROSTATIC STRESS
Hydrostatic stress is the stress that causes change of volume only
Hydrostatic stress may be considered as the average of the three principal stresses and derived and expressed as

21. HYDROSTATIC STRAIN ENERGY
Using the equation for total strain energy yields an expression for hydrostatic strain energy:

22. HYDROSTATIC STRAIN ENERGY
Simplifying for hydrostatic strain energy

23. DISTORTION STRAIN ENERGY
This is the difference between total strain energy and the hydrostatic strain energy

24. Tri-Axial Stress Three Dimensional stress

25. DISTORTION STRAIN ENERGY

26. CASE OF SIMPLE TENSION
When yielding occurs in simple tension test

27. Uni-Axial Stress One dimensional (Normal/Shear)

28. DISTORTION ENERGY THEORY
For the general three dimensional stress situation
When Yielding occurs in any material,
The distortion strain energy per unit volume
At the point of failure, Equals or exceeds
When yielding occurs in the tension test specimen.

29. THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
Comparing three dimensional case with simple tension

30. Tri-Axial Stress Three Dimensional stress

31. THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
Equating the two conditions

32. EQUIVALENT STRESS
Left hand side of equation referred to as the Equivalent, or Von-Mises stress

33. APPLICATION OF DESIGN EQUATION
Principal stresses are
Determined by stress analysis.
Stress analysis describes the principal stresses as a function of
Load carried,
Geometry and dimensions of the machine or structural element.

34. APPLICATION OF DESIGN EQUATION
Left hand side of design equation
Equivalent stress in terms of Loads and Dimensions of machine or structural element,
Right hand side of design equation
Indicator of strength expressed as Working, (design, allowable) stress a function of strength of the material, and a factor of safety.