Thermal Stresses

Why it matters

Thermal stresses are crucial in engineering because they can lead to structural failure if not properly managed. They occur when temperature changes cause materials to expand or contract, leading to stress if the material is constrained. Understanding thermal stresses helps in designing structures that can withstand temperature variations without damage.

Key ideas

• Thermal Expansion: Materials expand when heated and contract when cooled. The amount of expansion or contraction depends on the material's coefficient of thermal expansion.
• Thermal Stress: When a material is constrained and cannot freely expand or contract, thermal stresses develop.
• Coefficient of Thermal Expansion (α): A material property that indicates how much a material expands per degree of temperature change.
• Temperature Change (ΔT): The difference in temperature that causes expansion or contraction.
• Elastic Modulus (E): A measure of a material's ability to withstand changes in length when under lengthwise tension or compression.

Restraint and sign

For a uniaxial linear-elastic bar, total strain is ε_total = σ/E + αΔT, taking tension positive. A uniform freely expanding bar has σ = 0. Complete axial restraint gives ε_total = 0 and σ = −EαΔT: heating creates compression for positive α. Partial restraint requires compatibility with the supports. The formula assumes uniform temperature, constant properties, elastic response, and no buckling or yielding.

Formulas

• Thermal Strain: ε = α·ΔT
  • ε: Thermal strain (dimensionless)
  • α: Coefficient of thermal expansion (1/°C)
  • ΔT: Temperature change (°C)
• Fully restrained stress magnitude: |σ| = E·|α·ΔT|
  • σ: Thermal stress (Pa)
  • E: Elastic modulus (Pa)
  • α: Coefficient of thermal expansion (1/°C)
  • ΔT: Temperature change (°C)

Worked example

Given: A straight, uniform bar is initially stress-free and fully restrained against axial extension by rigid supports. It is heated uniformly; assume linear elasticity and no buckling.

• Coefficient of thermal expansion, α = 12 × 10^-6 /°C
• Elastic modulus, E = 200 GPa
• Temperature change, ΔT = 50°C

Steps:

1. Calculate thermal strain using ε = α·ΔT:
   • ε = 12 × 10^-6 /°C × 50°C = 600 × 10^-6
2. Calculate thermal stress using σ = E·α·ΔT:
   • σ = 200 × 10^9 Pa × 12 × 10^-6 /°C × 50°C
   • σ = 120 × 10^6 Pa

Final Answer: 120 MPa compression (σ = −120 MPa). The calculated 600 × 10⁻⁶ is the free thermal strain; actual total axial strain under full restraint is zero.

Common mistakes

• Forgetting to convert units, especially when dealing with GPa and MPa.
• Not considering constraints that prevent free expansion or contraction.
• Ignoring the material's coefficient of thermal expansion.

For GATE ME

Questions often involve calculating thermal stresses in constrained bars or beams. Practice problems that require understanding the relationship between temperature change, material properties, and resulting stresses.

Quick check

1. What is thermal stress?
2. How does the coefficient of thermal expansion affect thermal stress?
3. What happens to a material's length when it is heated?

Answers: 1. Stress due to temperature change in a constrained material. 2. It determines how much stress develops for a given temperature change. 3. A freely supported material with positive α expands; full axial restraint prevents the length change and produces stress.