2. What are theories of failure?
• Material strengths are determined from uni-axial tension tests.
• Thus, the strengths obtained from those tension tests cannot be directly used for component design since, in
actual scenarios components undergo multi-axial stress conditions.
• Hence, to use the strengths determined from tension tests to design mechanical components under any
condition of static loading, theories of failure are used.
3. How many theories are there?
• Maximum Principal Stress Theory (Rankine’s Theory).
• Maximum Shear Stress Theory (Tresca-Guest Theory).
• Maximum Principal Strain Theory (Venant’s Theory).
• Total Strain Energy Theory (Haigh’s Theory).
• Maximum Distortion Energy Theory (Von-Mises and Hencky’s Theory).
4. Maximum Principal Stress Theory
• Designed for brittle materials.
• Can be used for ductile materials in special occasions:
• Uni-axial or bi-axial loading (if principal stresses are of similar nature).
• Hydrostatic stress condition (no shear stresses).
• Condition for safe design:
→ Maximum principal stress (𝜎1) ≤ Permissible stress (𝜎 𝑝𝑒𝑟, shown below).
→ 𝜎1 ≤
𝑆 𝑦𝑡
𝑁 or 𝑆 𝑢𝑡
𝑁
Where:
• 𝑆 𝑦𝑡 and 𝑆 𝑢𝑡 are yield strength and ultimate strength, respectively.
• N is the factor of safety.
6. Maximum Shear Stress Theory
• Suitable for ductile materials as these are weak in shear.
• Gives over safe design which could be uneconomic sometimes.
• Condition for safe design:
→ Max. shear stress ( 𝜏 𝑚𝑎𝑥) ≤ Permissible shear stress ( 𝜏 𝑝𝑒𝑟).
→ 𝜏 𝑚𝑎𝑥 ≤
𝑆 𝑦𝑡
2𝑁.
→ For 3D stresses, larger of {|𝜎1 - 𝜎2|, |𝜎2 - 𝜎3|, |𝜎3 - 𝜎1|} ≤
𝑆 𝑦𝑡
𝑁.
→ For biaxial state, 𝜎3 = 0, |𝜎1| ≤
𝑆 𝑦𝑡
𝑁 when 𝜎1, 𝜎2 are like in nature, else,
|𝜎1 − 𝜎2| ≤
𝑆 𝑦𝑡
𝑁.
8. Maximum Principal Strain Theory
• Condition for safe design,
→ Max. Principal Strain (∈1) ≤ Permissible Strain (∈ 𝑌.𝑃
𝑁). Where ∈ 𝑌.𝑃 is strain at yield.
→ ∈1 ≤ 𝑆 𝑌.𝑇
𝐸𝑁. Where 𝑆 𝑌.𝑇 is yield strength,
→
1
𝐸
[𝜎1 - µ(𝜎2 + 𝜎3 )] ≤ 𝑆 𝑌.𝑇
𝐸𝑁 . E is Youngs modulus,
→ 𝜎1 - µ(𝜎2 + 𝜎3 ) ≤ 𝑆 𝑌.𝑇
𝑁 . N is factor of safety.
9. • Safe zone : Area inside the rhombus
• For biaxial state of stress , condition for safe design,
• 𝜎1 - µ(𝜎2)] ≤ 𝑆 𝑌.𝑇
𝑁
Maximum Principal Strain Theory
10. Total Strain Energy Theory
• Condition for safe design,
→ Total Strain Energy per unit volume (TSE/vol) ≤ Strain energy per unit volume at yield point.
→ 1
2 [𝜎1 ∈1+ 𝜎2 ∈2+ 𝜎3 ∈3] ≤ 1
2 𝜎 𝐸.𝐿 ∈ 𝐸.𝐿. - (1) Where E.L elastic limit,
∈1 = 1
𝐸[𝜎1 - µ(𝜎2 + 𝜎3)] ; similarly ∈2 and ∈3 . - (a), (b), (c)
→ Using (a), (b), (c) in (1), we get
→ [LHS] = TSE/vol = 1
2𝐸 [𝜎1
2
+ 𝜎2
2
+ 𝜎3
2
-2 µ(𝜎1 𝜎2+ 𝜎2 𝜎3 + 𝜎3 𝜎1) ] - (2)
→ [RHS] = For [TSE/vol] at yield point we can use, 𝜎1 = 𝜎 = 𝑆 𝑌.𝑇
𝑁 , 𝜎2 = 𝜎3 = 0. in (2),
to get TSE/vol at yield = 1
2𝐸 (
𝑆 𝑦𝑡
𝑁)2
• Therefore, condition for safe design : 𝜎1
2
+ 𝜎2
2
+ 𝜎3
2
-2 µ(𝜎1 𝜎2+ 𝜎2 𝜎3 + 𝜎3 𝜎1) ≤ (
𝑆 𝑦𝑡
𝑁)2
11. • Safe zone : Area inside the ellipse
• For biaxial state of stress ( 𝜎3= 0), condition for safe design,
• 𝜎1
2
+ 𝜎2
2
-2 µ𝜎1 𝜎2 ≤ (
𝑆 𝑦𝑡
𝑁)2
Total Strain Energy Theory
12. Total Distortion Energy Theory
• Condition for safe design,
→ Max Distortion Energy per unit volume (DE/vol) ≤ Distortion energy per unit volume at yield point (DE/Vol @ yield).
→ Now, TSE/vol = Volumetric SE/Vol + DE/Vol
→ DE/Vol = TSE/vol - Volumetric SE/Vol ; Here we already know TSE/Vol from equation 2 in slide 7.
→ Volumetric SE/Vol = ½ (Avg. Stress)(Volumetric Strain)
= 1
2 ( 𝜎1+ 𝜎2+ 𝜎3
3 )[(1 −2µ
𝐸 ) (𝜎1 + 𝜎2 + 𝜎3 )]
→ Substituting the above we get, DE/Vol = 1+ µ
6𝐸 [(𝜎1− 𝜎2)2
+ (𝜎2− 𝜎3)2
+ (𝜎3− 𝜎1)2
] - (1)
At yield point 𝜎1 = 𝜎 = 𝑆 𝑌.𝑇
𝑁 , 𝜎2 = 𝜎3 = 0 - (2)
→ Using (2) in (1), DE/Vol @ yield = 1+ µ
3𝐸 ( 𝑆 𝑌.𝑇
𝑁 )
2
• Therefore, condition for safe design: [(𝜎1− 𝜎2)2
+ (𝜎2− 𝜎3)2
+ (𝜎3− 𝜎1)2
] ≤ 2 ( 𝑆 𝑌.𝑇
𝑁 )
2
13. • Safe zone : Area inside the ellipse
• For biaxial state of stress ( 𝜎3= 0), condition for safe design,
• 𝜎1
2
+ 𝜎2
2
- 𝜎1 𝜎2 ≤ ( 𝑆 𝑌.𝑇
𝑁 )
2
Total Distortion Energy Theory
14. References
• The Gate Academy: http://thegateacademy.com/files/wppdf/Theories-of-failure.pdf
• Book: Introduction to Machine Design by VB Bhandari
• NPTEL: https://nptel.ac.in/course.html