1. UNIVERSITY OF NAIROBI
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
ENGINEERING DESIGN II
FME 461
PART 5
GO NYANGASI
November 2008

2. MAXIMUM SHEAR STRESS THEORY
A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS

3. STATEMENT OF THE THEORY
When Yielding occurs in any material
The maximum shear stress at the point of failure
Equals or exceeds
The maximum shear stress when yielding occurs
In the tension test specimen.

4. STATEMENT OF THE THEORY
The theory applies to ductile materials only
Because it is based on yielding.

5. THREE-DIMENSIONAL (TRIAXIAL) STRESS SITUATION.
In the three-dimensional stress situation,
State of stress at a particular location is fully defined by three principal stresses

6. THREE DIMENSIONAL STRESS

7. TRI-AXIAL STRESS SITUATION

8. MAXIMUM SHEAR STRESS AT A LOCATION OF THE ELEMENT
The extreme values of shear stresses , in each of the three principal planes are then given by the expressions:

9. MAXIMUM SHEAR STRESS AT A LOCATION OF THE ELEMENT
Expressing the principal stresses in the order of magnitude and sign

10. MAXIMUM SHEAR STRESS AT A LOCATION OF THE ELEMENT
Then the maximum shear stress is given by

11. MAXIMUM SHEAR STRESS AT A LOCATION OF THE ELEMENT
THE CASE OF SIMPLE TENSION TEST WHEN YIELDING OCCURS
The principal stresses are

12. MAXIMUM SHEAR STRESS AT A LOCATION OF THE ELEMENT
THE CASE OF SIMPLE TENSION TEST WHEN YIELDING OCCURS
The maximum shear stress then becomes

13. THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
The maximum shear stress theory of failure states:
When Yielding occurs in any material
The maximum shear stress at the point of failure
Equals or exceeds
The maximum shear stress when yielding occurs
In the tension test specimen.

14. THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
Maximum shear stress is then compared with the case of simple tension to obtain

15. THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
The equation in slide 11 implies that when yielding occurs in simple tension
Maximum shear stress equals Shear yield strength of the material
Maximum shear stress equals half the tensile yield strength

16. DESIGN EQUATION: MAXIMUM SHEAR STRESS THEORY
This is derived by adjusting the shear yield strength of the material with an appropriate factor of safety
The design equation then becomes:

17. DESIGN EQUATION: MAXIMUM SHEAR STRESS THEORY
For plane stress situation, when principal stresses are of OPPOSITE sign
The maximum shear stress can be expressed in terms of the plane stress elements, as shown below

18. DESIGN EQUATION: MAXIMUM SHEAR STRESS THEORY
Design equation in slide 13 employs principal stresses to determine maximum shear stress at the location
Design equation in slide 14 employs plane stress elements to determine maximum shear stress on the plane
When principal stresses in the plane are of opposite sign, maximum shear stress at the location is in the plane
When Principal stresses in the plane are of same sign, maximum shear stress at the location is not in the plane

19. APPLICATION OF THE DESIGN EQUATION
The principal stresses are first determined by stress analysis.
Such analysis describes the principal stresses as a function of the load carried, and the geometry and dimensions of the machine or structural element.

20. APPLICATION OF THE DESIGN EQUATION
The maximum shear stress in the design equation is expressed in terms of the load and dimensions of the machine or structural element
Right hand side of design equation is the design, or allowable shear stress, a function of the tensile yield strength of the material.
The tensile yield strength of the material is used because it is more easily determined from laboratory experiments than shear strength.

21. APPLICATION OF THE DESIGN EQUATION
The factor of safety is simply a number chosen by the designer.
The factor of safety together with the tensile yield strength of the material, gives the working[1] (design, allowable) stress expected in the machine part.
The solution to the design equation then gives the minimum dimensions required to avoid failure of the element by yielding.
[1] Working Stress, page 527,Handbook, Metals Engineering –Design, American Society of Mechanical Engineers (ASME)