Theories of Failure

Why it matters

Understanding the theories of failure is crucial for civil engineers to ensure the safety and reliability of structures. These theories help predict when materials will fail under complex stress states, allowing engineers to design structures that can withstand various loads and conditions.

Key ideas

• Failure Theories: These are criteria used to predict the failure of materials under different stress conditions. The main theories include:
  • Maximum Principal Stress Theory (Rankine's Theory): Assumes failure occurs when the maximum principal stress reaches the material's ultimate tensile strength.
  • Maximum Shear Stress Theory (Tresca's Theory): Suggests failure occurs when the maximum shear stress in the material reaches half the uniaxial tensile yield strength for a ductile material.
  • Distortion Energy Theory (von Mises Theory): Proposes that failure occurs when the distortion energy per unit volume reaches a critical value.
  • Maximum Principal Strain Theory: Assumes failure occurs when the maximum principal strain reaches the ultimate strain of the material.
  • Coulomb-Mohr Theory: A brittle-material criterion accounting for different tensile and compressive strengths.
• Application: These theories are applied based on the type of material (ductile or brittle) and the nature of the loading conditions.

Formulas

• Maximum Principal Stress Theory: σ1 = σ_ult
  • σ1: Maximum principal stress (Pa)
  • σ_ult: Ultimate tensile strength (Pa)
• Maximum Shear Stress Theory: τ_max = σ_y/2
  • τ_max: Maximum shear stress (Pa)
  • σ_y: Uniaxial tensile yield strength (Pa)
• Distortion Energy Theory: σ_v = σ_y
  • σ_v: von Mises stress (Pa)
  • σ_y: Yield strength (Pa)
• Maximum Principal Strain Theory: ε1 = ε_ult
  • ε1: Maximum principal strain (dimensionless)
  • ε_ult: Ultimate strain (dimensionless)

Worked example

Given: A steel bar is subjected to a plane-stress state with in-plane principal stresses of 150 MPa and -50 MPa, and third principal stress zero. The yield strength of the steel is 250 MPa. Determine if the bar will fail according to the von Mises theory.

1. Calculate von Mises stress:
   • Formula: σ_v = sqrt(σ1^2 - σ1·σ2 + σ2^2)
   • Substitute values: σ_v = sqrt((150)^2 - 150*(-50) + (-50)^2)
   • Calculation: σ_v = sqrt(22500 + 7500 + 2500)
   • σ_v = sqrt(32500)
   • σ_v = 180.28 MPa
2. Compare with yield strength:
   • Since σ_v = 180.28 MPa < σ_y = 250 MPa, the criterion does not predict first yield.

Final Answer: Von Mises does not predict first yield for this stress state; this does not rule out buckling, fatigue, fracture or other failure modes.

Common mistakes

• Confusing the application of different failure theories for ductile and brittle materials.
• Incorrectly calculating principal stresses or von Mises stress.
• Forgetting to check units consistency in calculations.

For GATE CE

• Questions often involve calculating stresses and determining failure using different theories.
• Practice problems on identifying suitable failure theories for given materials and loading conditions.

Quick check

1. What is the main assumption of the Maximum Principal Stress Theory?
2. Which failure theory is most suitable for ductile materials?
3. How is von Mises stress calculated?

Answers: 1. Failure occurs when the maximum principal stress reaches the ultimate tensile strength. 2. Distortion Energy Theory (von Mises Theory). 3. σ_v = sqrt(σ1^2 - σ1·σ2 + σ2^2).