1. Mechanical Behavior of
Materials
Topic: Theories of Failure
Submitted To: Submitted
By:
Dr. Sanjeev Kumar Mandeep
Kumar
Prof. in MED SID: 13209024

2. Requirement of TOF
 Theories of failure are used to determine
the safe dimension of a component
when it is subjected to combined
stresses due to various loads.
 Theories of failure are used in design by
establishing a relationship between
stresses induced under combined
loading conditions and properties
obtained from tension test like Syt & Sut

3. Various Theory Of Failure
• Maximum principal stress theory (Rankine’s
Theory)
• Maximum principal strain theory (St.
Venant’s theory)
• Maximum strain energy theory (Haigh’
Theory)
• Maximum Distortion energy theory (Von-
mises and Henky’s Theory)
• Maximum shear stress theory (Guest and
Treska’s Theory)

4. Maximum Principle Stress
Theory:
 Condition For Failure:
σ1 > Syt or Sut
 Condition for safe design:
σ1 < Syt/N or Sut/N
Where N is factor of safety.
 For ductile material:
Syc > Syt > Sys
 For Brittle material:
Syc > Sys > Syt
cont.

5. Maximum Principle Stress
Theory
 This theory is suitable for the safe design of
machine component made up of brittle
material, because brittle material are weak in
tension
 This theory is not good for design of ductile
material because shear failure may occur
 But this theory is also suitable for ductile
material under following condition :
1. uniaxial state of stress condition.
2. Under biaxial state of stress when are like in
nature.
3. Hydrostatic stress condition.

6. Graphical representation of MPST

7. Maximum shear stress theory
 Condition For Failure:
 Condition for safe design:

8. For triaxial state of stress
condiition
For biaxial state of stress: σ3=0
When σ1 & σ2 are like in nature:
σ1 < Sty/N
When σ1 & σ2 are unlike :
σ1- σ2 < Syt/N

9. Graphical representation of MSST

10.  MPST and MSST will give same
result under biaxial state of stress
when principle stresses are like in
nature.
 M.S.S.T. is not valid under hydrostatic
stress condition (because every plane
passing through the point is principle
plane hence absolute shear stress is
zero)
 M.S.S.T. Gives over safe design for
ductile material .(safe and

11. Maximum principal strain theory:
 Condition For Failure:
 Condition for safe design:

12. For Biaxial state of stress: σ3 = 0
So

13. Maximum strain energy theory:
 Condition For Failure:
Total strain energy/vol > [(S.E/vol)yp]TT
 Condition for safe design:
Total strain energy/vol < [(S.E/vol)yp]TT
Under triaxial loading condition:

14. By putting the values in safe design condition, we get:
For Biaxial state of stress, σ3=0
• It is a equation of ellipse whose graphical
representation is shown in figure.

15. • For hydrostatic stresses, it is the best theory.

16. Maximum Distortion energy theory:
 Condition For Failure:
Max. D.E. / volume > [(D.E./vol.)yp]TT
 Condition for safe design:
Max. D.E. / volume < [(D.E./vol.)yp]TT
 Max. D.E. / vol = Total S.E. /vol – volumetric S.E. / vol

17. By inserting the values of energies in D.E.equation
We got:
By inserting the values of energies in Safe design condition
We got:

18. For biaxial state of stress:
This is a equation of ellipse with
Semi major axis = 1.414 Syt
Semi minor axis = 0.816 Syt

19. Comparison of TOF
• As area bounded by curve increases ,
• failure stress increases,
• dimensions of part decreases,
• so safety and cost decreases.

20.  Area of MDETcurve > area of MSST curve
 So dimension of MDET < dimension of
MSST
 MPST, MSST, MDET will give same result
under biaxial state of stress of same
nature i.e. σ1=σ2= σ and σ3=0

21. Conclusion:
 MPST:
 Best TOF for brittle material design.
 Suitable for ductile material under three
case.
 MSST:
 Gives safe design for ductile.
 MDET:
 Best TOF for ductile material design.

22. Thank You