Thermal Stresses

Why it matters

Thermal stresses occur in materials when they are subjected to temperature changes while being constrained. This is crucial in civil engineering as structures like bridges, buildings, and pavements experience temperature variations, which can lead to expansion or contraction, potentially causing structural damage if not properly accounted for.

Key ideas

• Thermal Expansion: Materials expand or contract when subjected to temperature changes. The extent of this change 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. These stresses can lead to cracking or failure if not properly managed.
• Coefficient of Thermal Expansion (α): A material property that indicates how much a material will expand per degree of temperature change.
• Temperature Change (ΔT): The difference in temperature that a material experiences.
• Elastic Modulus (E): A measure of a material's stiffness, which affects how it responds to thermal stresses.

Formulas

• ΔL = α·L·ΔT
  • ΔL: Change in length (m)
  • α: Coefficient of thermal expansion (1/°C)
  • L: Original length (m)
  • ΔT: Temperature change (°C)
• σ = -E·α·ΔT
  • σ: Thermal stress (Pa)
  • E: Elastic modulus (Pa)
  • α: Coefficient of thermal expansion (1/°C)
  • ΔT: Temperature change (°C)

The stress formula assumes uniform temperature change, complete axial restraint, constant properties and linear elasticity, without buckling or yielding. Free uniform expansion creates strain but no stress in an otherwise unconstrained homogeneous member.

Worked example

Given:

• A steel rod of length 2 m is constrained at both ends.
• Coefficient of thermal expansion for steel, α = 12 × 10⁻⁶ /°C
• Elastic modulus for steel, E = 200 GPa
• Temperature increase, ΔT = 30°C

Steps:

1. Calculate the change in length if the rod were free to expand.
   • Formula: ΔL = α·L·ΔT
   • Calculation: ΔL = 12 × 10⁻⁶ /°C × 2 m × 30°C = 0.00072 m
2. Calculate the thermal stress developed due to constraint.
   • Formula: σ = -E·α·ΔT
   • Calculation: σ = -200 × 10⁹ Pa × 12 × 10⁻⁶ /°C × 30°C = -72 × 10⁶ Pa

Final Answer: 72 MPa compression (σ = -72 MPa with tension positive)

Common mistakes

• Ignoring the constraints that prevent free expansion or contraction.
• Using incorrect units for the coefficient of thermal expansion.
• Forgetting to convert the elastic modulus to the correct unit (e.g., from GPa to Pa).

For GATE CE

Questions often involve calculating thermal stresses in constrained members, requiring an understanding of material properties and temperature changes. Practice problems involving different materials and varying constraints to strengthen your understanding.

Quick check

1. What is the effect of temperature increase on a constrained material?
2. How does the coefficient of thermal expansion affect thermal stress?
3. Why is it important to consider thermal stresses in bridge design?

Answers: 1. It induces thermal stress. 2. Higher α increases thermal stress. 3. To prevent structural damage due to temperature changes.