Theories of Failure

When a mechanical component is subjected to complex loading (like combined bending, torsion, and axial loads), a multi-axial state of stress exists. Theories of Failure are used to predict the failure of a material under such conditions by comparing the complex stress state to the material's strength properties, which are typically determined from a simple uniaxial tensile test.

For design purposes, a Factor of Safety (F.S.) is introduced to ensure that the component operates well below its failure point.

1. Maximum Principal Stress Theory (Rankine's Theory)

This theory is primarily recommended for brittle materials.

Principle: Failure occurs when the maximum principal stress (σ₁) at any point in the component reaches the ultimate strength (σ_u) of the material. For ductile materials, failure is considered as yielding, so the yield strength (σ_y) is used instead.

σ₁ ≤ σ_allowable = σ_yield / F.S.

σ₁: Maximum Principal Stress in the component.
σ_yield: Yield strength of the material from a uniaxial test.
F.S.: Factor of Safety.

This theory does not consider the effect of other principal stresses and is not safe for ductile materials, especially under pure shear.

2. Maximum Shear Stress Theory (Tresca or Guest's Theory)

This theory is recommended for ductile materials and gives conservative (safer) results.

Principle: Yielding in a component occurs when the maximum shear stress (τ_max) at any point reaches the maximum shear stress at the yield point in a simple tension test. The maximum shear stress at yield in a tension test is σ_y / 2.

τ_max = (σ₁ - σ₃) / 2 ≤ σ_y / (2 * F.S.)

This simplifies to:

σ₁ - σ₃ ≤ σ_y / F.S.

τ_max: Absolute maximum shear stress in the component.
σ₁, σ₃: Maximum and Minimum Principal Stresses, respectively.
σ_y: Yield strength of the material.
F.S.: Factor of Safety.

3. Maximum Distortion Energy Theory (Von Mises-Hencky Theory)

This is the most widely accepted theory for ductile materials as its results are in best agreement with experimental tests.

Principle: Failure by yielding occurs when the distortion strain energy per unit volume at a point in the component equals the distortion strain energy per unit volume at the yield point in a simple tension test.

The theory is based on a "Von Mises Stress" (σ_v), which is a single equivalent stress value for the complex stress state.

σ_v = &sqrt;[ ( (σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)² ) / 2 ] ≤ σ_y / F.S.

For a 2D state of stress (σ₃ = 0):

σ_v = &sqrt;(σ₁² - σ₁σ₂ + σ₂²) ≤ σ_y / F.S.

σ_v: Von Mises equivalent stress.
σ₁, σ₂, σ₃: The three principal stresses.
σ_y: Yield strength of the material.
F.S.: Factor of Safety.

4. Maximum Principal Strain Theory (Saint-Venant's Theory)

This theory gives reliable results for brittle materials, especially under tensile loading.

Principle: Failure occurs when the maximum principal strain (ε₁) at a point in the component reaches the strain at the yield point in a simple tension test. The yield point strain is σ_y / E.

From the generalized Hooke's Law, the maximum principal strain is given by:

ε₁ = (1/E) * [σ₁ - ν(σ₂ + σ₃)]

According to the theory, for no failure:

(1/E) * [σ₁ - ν(σ₂ + σ₃)] ≤ (σ_y / E) / F.S.

This simplifies to:

σ₁ - ν(σ₂ + σ₃) ≤ σ_y / F.S.

ε₁: Maximum Principal Strain.
E: Young's Modulus of the material.
ν (mu): Poisson's Ratio of the material.
σ₁, σ₂, σ₃: The three principal stresses.
σ_y: Yield strength of the material.

5. Total Strain Energy Theory (Haigh's Theory)

This theory is applicable to ductile materials but is not preferred over the Von Mises theory as it can be unsafe in cases of hydrostatic pressure.

Principle: Failure occurs when the total strain energy per unit volume at a point in the component equals the total strain energy per unit volume at the yield point in a simple tension test.

Total strain energy per unit volume (U) in a 3D stress system is:

U = (1 / 2E) * [σ₁² + σ₂² + σ₃² - 2ν(σ₁σ₂ + σ₂σ₃ + σ₃σ₁)]

At the yield point in a simple tension test, U_yield = σ_y² / 2E. For no failure:

σ₁² + σ₂² + σ₃² - 2ν(σ₁σ₂ + σ₂σ₃ + σ₃σ₁) ≤ (σ_y / F.S.)²

U: Total strain energy per unit volume.
E, ν: Young's Modulus and Poisson's Ratio.
σ₁, σ₂, σ₃: The three principal stresses.