1. ( BDA 3033 ) CHAPTER 6 Theories of Elastic Failures
SOLID MECHANICS II
( BDA 3033 )
CHAPTER 6
Theories of Elastic Failures

2. TEST 2: 14 DEC, PM, F2 ATAS
ASSIGNMENT 2 SUBMIT TODAY
LAST TOPIC TODAY
GROUP PROJECT TODAY. BASED ON LAB EXPT.

3. Chapter VI Theories of Failure
Maximum Normal Stress Theory
Maximum Normal Strain Theory
Maximum Shear Stress Theory (Tresca Yield Criterion)
Strain Energy Theory
Maximum Distortion Energy Theory
Out of these five theories of failure, the maximum normal stress theory and maximum normal strain theory are only applicable for brittle materials, and the remaining three theories are applicable for ductile materials.

4. Topics Introduction Maximum Normal Stress Theory (Rankine Theory)
Maximum Normal Strain Theory (St-Venant Theory)
Maximum Shear Stress Theory (Tresca Yield Criterion)
Strain Energy Theory (Haigh Theory)
Maximum Distortion Energy Theory (Von Misses)

5. Introduction ( Contd..)
Stress-Analysis is performed on a component to determine
The required “size or geometry” (design)
an allowable load (service)
cause of failure (forensic)
For all of these, a limit stress or allowable stress value for the component material is required.
Hence, a Failure-Theory is needed to define the onset or criterion of failure

6. Introduction (Contd..) Failure
Occurs if a component can no longer function as intended.
Failure Modes:
yielding: a process of global permanent plastic deformation.
Change in the geometry of the object.
low stiffness: excessive elastic deflection.
fracture: a process in which cracks grow to the extent that the
component breaks apart.
buckling: the loss of stable equilibrium. Compressive loading can lead to bucking in columns.

7. Introduction (Contd..) Failure Prediction
The failure of a statically loaded member in uni-axial tension or compression is relatively easy to predict.
One can simply compare the stress incurred with the strength of the material.
When the loading conditions are Complex (i.e. biaxial loading, sheer stresses) then we must use some method to compare multiple stresses to a single strength value.
These methods are known failure theories

8. Introduction (Contd..) Need for Failure Theories
To design structural components and calculate margin of safety.
To guide in materials development.
To determine weak and strong directions.

9. Maximum normal stress theory
this theory postulates, that failure will occur in the structural component if the maximum normal stress in that component reaches the ultimate strength, u obtained from the tensile test of a specimen of the same material.
Thus, the structural component will be safe as long as the absolute values of the principle stresses 1 and 2 are both less than u:
1 = U and 2 = U
This theory deals with brittle materials only.
The maximum normal stress theory can be expresses graphically as shown in the figure. If the point obtained by plotting the values 1 and 2 of the principle stress fall within the square area shown in the figure, the structural component is safe.
If it falls outside that area, the component will fail. 2 u 1
-u u
-u

10. Maximum normal strain theory
This theory alson known as Saint-Venant’s Theory
According to this theory, a given structural component is safe as long as the maximum value of the normal strain in that component remains smaller than the value u of the strain at which a tensile test specimen of the same material will fail.
As shown in the figure, the strain is maximum along one of the principle axes of stress if the deformation is elastic and the material homogenous and isotropic.
Thus denoting by 1 and 2 the values of the normal strain along the principle axes in plane of stress, we write
1 = u and 2 = u

11. Maximum normal strain theory (cont)
Making use of the generalized Hooke’s Law, we could express these relations in term of the principle stresses 1 and 2 and the ultimate strength U of the material.
We would find that, according to the maximum normal strain theory, the structural component is safe as long as the point obtained by plotting 1 and 2 falls within the area shown in the figure where  is Poisson’s ration for the given material. 2 U 1
-U U
-U

12. Maximum shearing stress theory
This theory is based on the observation that yield in ductile materials is caused by slippage of the material along oblique surfaces and is due primarily to shearing stress.
A given structural component is safe as long as the maximum value max of the shearing stress in that component remains smaller than the corresponding value of the shearing stress in a tensile test specimen of the same material as the specimen starts to yield.
For a 3D complex stress system, the max shear stress is given by:
max = ½ (1-2)
On the other hand, in the 1D stress system as obtained in the tensile test, at the yield limit, 1= Y and 2=0, therefore:-
max= ½ Y

13. Maximum shearing stress theory (cont.)
Thus,
Graphically, the maximum shear stress criterion requires that the two principal stresses be within the green zone as shown in the figure.

14. Maximum distortion energy theory
This theory is based on the determination of the distortion energy in a given material, i.e. of the energy associated with changes in shape in that material (as opposed to the energy associated with changes in volume in the same material).
A given structural component is safe as long as the maximum value of the distortion energy per unit volume in that material remains smaller than the distortion energy per unit volume required to cause yield in a tensile test specimen of the same material.
The distortion energy per unit volume in an isotropic material under plane stress is:

15. Maximum distortion energy theory (cont)
In the particular case of a tensile test specimen that is starting to yield, we have:-
This equation represents a principal stress ellipse as illustrated in the figure
Von Mises criterion also gives a reasonable estimation of fatigue failure, especially in cases of repeated tensile and tensile-shear loading

16. Problem 1
The solid shaft shown in Figure has a radius of 0.5 cm and is made of steel having a yield stress of 360 MPa. Determine if the loadings cause the shaft to fail according to Tresca and von mises theories.
32.5 Nm
15 kN
 1 cm

17. Solution Calculating the stresses caused by axial force and torque
The Principal stresses
191.1MPa
165.5 MPa

18. Solution ( Contd..) Applying Maximum Shear stress theory
So shear failure occurs
Applying Maximum distortion theory
No Failure

19. Problem 2
40 MPa
The state of plane stress shown occurs at a critical point of a steel machine component. As a result of several tensile tests, it has been found that the tensile yield strength is Y=250 MPa for the grade of steel used. Determine the factor of safety with respect to yield, using:
(a) the maximum shearing stress theory
(b) the maximum distortion energy theory
80 MPa
25 MPa

20. Solution:

21. Solution (cont)

22. FRONT
Lab 3
Lab 1
Lab 5
Lab 4
Lab 2