Theories of Failure
Maximum Stress, Tresca, Von Mises — Which Theory to Use & When
Last Updated: March 2026
📌 Key Takeaways
- Failure theories predict when a material under complex (multi-axial) loading will fail, based on simple uniaxial test data.
- Brittle materials: Use Maximum Normal Stress Theory (Rankine) — failure when any principal stress reaches the ultimate strength.
- Ductile materials: Use Von Mises (Distortion Energy) or Tresca (Maximum Shear Stress) — both predict yielding accurately.
- Von Mises is the most widely used and most accurate theory for ductile materials.
- Tresca is simpler to apply and more conservative than Von Mises (predicts failure earlier).
- The choice of theory depends on the material type — ductile or brittle — not on the loading type.
1. Why We Need Failure Theories
Material properties (yield strength, ultimate strength) are measured in simple uniaxial tension or compression tests. But in real engineering, components experience multi-axial stress states — combinations of normal stresses and shear stresses acting simultaneously (e.g., a shaft under combined bending and torsion).
The question failure theories answer is: given a multi-axial stress state (σ₁, σ₂, σ₃) and the known uniaxial yield strength σy, will the material fail?
Each theory proposes a different criterion for what “causes” failure — maximum stress, maximum shear, maximum strain, or maximum distortion energy. No single theory works perfectly for all materials, but decades of experimental validation have shown which theories work best for which material types.
2. Maximum Normal Stress Theory (Rankine)
Failure Criterion
Failure occurs when the maximum principal stress reaches the material’s strength.
|σ₁| ≤ σy (or σu for brittle materials)
|σ₂| ≤ σy
|σ₃| ≤ σy
Best for: Brittle materials (cast iron, concrete, glass, ceramics) under static loading.
Limitation: Ignores the effect of the other principal stresses. Fails to predict yielding of ductile materials under combined loading accurately.
Failure envelope: Square on the σ₁-σ₂ plane.
3. Maximum Shear Stress Theory (Tresca)
Failure Criterion
Failure occurs when the maximum shear stress reaches the shear yield strength.
τmax = (σmax − σmin)/2 ≤ σy/2
Or equivalently: σmax − σmin ≤ σy
In 2D (σ₃ = 0), if σ₁ and σ₂ have the same sign: max(|σ₁|, |σ₂|) ≤ σy
If σ₁ and σ₂ have opposite signs: |σ₁ − σ₂| ≤ σy
Best for: Ductile materials. Simple to apply. Slightly conservative (predicts failure before it actually occurs, giving a built-in safety margin).
Failure envelope: Hexagon on the σ₁-σ₂ plane, inscribed within the Von Mises ellipse.
Shear Yield Strength
τy = σy/2 = 0.5σy (according to Tresca)
4. Distortion Energy Theory (Von Mises)
Failure Criterion
Failure occurs when the distortion energy reaches the distortion energy at yield in a uniaxial test.
Von Mises equivalent stress:
σVM = √[(σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²] / √2
Failure when: σVM ≥ σy
For 2D stress (σ₃ = 0):
σVM = √(σ₁² − σ₁σ₂ + σ₂²) ≤ σy
In terms of σx, σy, τxy:
σVM = √(σx² − σxσy + σy² + 3τxy²) ≤ σy
Best for: Ductile materials — the most accurate and widely used theory. Matches experimental data for ductile metals within 5%.
Failure envelope: Ellipse on the σ₁-σ₂ plane, circumscribing the Tresca hexagon.
Shear Yield Strength
τy = σy/√3 = 0.577σy (according to Von Mises)
Experimentally verified to be more accurate than Tresca’s τy = 0.5σy.
5. Maximum Strain Theory (Saint-Venant)
Failure Criterion
Failure occurs when the maximum strain reaches the yield strain.
ε₁ ≤ σy/E, where ε₁ = (σ₁ − ν(σ₂ + σ₃))/E
This gives: σ₁ − ν(σ₂ + σ₃) ≤ σy
Limited use: This theory is not widely used because it does not match experimental data well for most materials. It is included for completeness and because it occasionally appears in exam questions.
6. Side-by-Side Comparison
| Theory | Failure Criterion | Best For | τy prediction | Conservatism |
|---|---|---|---|---|
| Max Normal Stress | |σmax| ≥ σy | Brittle materials | σy (overestimates) | Unsafe for ductile |
| Tresca (Max Shear) | τmax ≥ σy/2 | Ductile materials | 0.5σy | Slightly conservative |
| Von Mises | σVM ≥ σy | Ductile materials | 0.577σy | Most accurate |
| Max Strain | εmax ≥ σy/E | Rarely used | — | Inaccurate |
7. Which Theory to Use?
| Material Type | Recommended Theory | Why |
|---|---|---|
| Ductile (steel, aluminium, copper) | Von Mises (preferred) or Tresca | Best match with experimental data for yielding under multi-axial stress |
| Brittle (cast iron, concrete, glass) | Maximum Normal Stress or Mohr’s theory | Brittle materials fail by fracture due to tensile stress, not shear yielding |
| GATE default (if not specified) | Von Mises for ductile, Max Normal Stress for brittle | These are the most commonly tested and most accepted choices |
Key rule: The choice depends on the material type, not the loading type. Ductile materials use Von Mises/Tresca regardless of whether the loading is bending, torsion, combined, or anything else.
8. Worked Numerical Examples
Example 1: Von Mises — Biaxial Stress
Problem: A ductile steel component (σy = 250 MPa) has principal stresses σ₁ = 180 MPa, σ₂ = −60 MPa. Is it safe according to Von Mises?
Solution
σVM = √(σ₁² − σ₁σ₂ + σ₂²) = √(180² − 180×(−60) + (−60)²)
= √(32,400 + 10,800 + 3,600) = √46,800 = 216.3 MPa
σVM = 216.3 < σy = 250 → Safe ✓
FOS = 250/216.3 = 1.16 (just barely safe — consider redesign for more margin).
Example 2: Tresca — Same Problem
Solution
Since σ₁ and σ₂ have opposite signs:
|σ₁ − σ₂| = |180 − (−60)| = 240 MPa
σy = 250 MPa
240 < 250 → Safe by Tresca (but barely — FOS = 250/240 = 1.04)
Notice Tresca gives a lower FOS (1.04) than Von Mises (1.16) — Tresca is more conservative.
Example 3: Pure Shear — Yield Prediction
Problem: A shaft made of steel (σy = 300 MPa) is under pure torsion. Find the shear stress at yielding using both Tresca and Von Mises.
Solution
Tresca: τy = σy/2 = 300/2 = 150 MPa
Von Mises: τy = σy/√3 = 300/1.732 = 173.2 MPa
Experiments show τy ≈ 0.55–0.58σy for most metals — closer to Von Mises (0.577) than Tresca (0.5).
9. Common Mistakes Students Make
- Using Von Mises for brittle materials: Von Mises and Tresca predict yielding — brittle materials don’t yield, they fracture. Use Maximum Normal Stress or Mohr’s theory for brittle materials.
- Confusing Tresca and Von Mises in pure shear: Tresca gives τy = 0.5σy; Von Mises gives τy = 0.577σy. The difference is 15% — this matters in design.
- Forgetting to consider the third principal stress: In 2D problems where σ₃ = 0, students sometimes forget that σ₃ = 0 is still a principal stress. For Tresca, τmax = (σmax − σmin)/2, and σmin might be σ₃ = 0, not σ₂.
- Not computing principal stresses first: Von Mises and Tresca criteria are expressed in terms of principal stresses. If given σx, σy, τxy, you must first find σ₁ and σ₂ using Mohr’s circle before applying the failure criterion.
10. Frequently Asked Questions
What is the Von Mises failure theory?
The Von Mises (Distortion Energy) theory states that yielding in ductile materials occurs when the distortion energy per unit volume reaches the value at yield in a simple tension test. It predicts failure when σVM = √[(σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²]/√2 ≥ σy. It is the most widely used and experimentally validated theory for ductile materials.
When should I use Tresca vs Von Mises?
Both are for ductile materials. Von Mises is more accurate (matches experiments within 5%). Tresca is simpler to compute and more conservative (predicts failure about 15% earlier than Von Mises). In practice, use Von Mises for design and analysis. Use Tresca when a quick, conservative estimate is acceptable or when the problem specifically asks for it.