Thermal Strain and Stress in Solids

Overview

This lecture explains thermal strain and thermal stress in solids, focusing on a steel rod and railway rails, and derives and applies key formulas.

Thermal Expansion: Basic Concept

• When temperature of a solid increases, its length increases; when temperature decreases, its length decreases.
• Change in temperature is denoted by T, where T = T₂ − T₁.
• For a bar of original length L, free change in length due to temperature change T is:
  • ΔL = α T L
• α is the coefficient of linear expansion.

Case 1: Free Expansion of a Steel Rod

• One end of rod fixed, other end free; rod is allowed to expand freely.
• Original length = L; temperature increases from T₁ to T₂ (rise = T).
• Free expansion:
  • ΔL = α T L
• When a material is allowed to expand freely:
  • No thermal stress is developed.
  • Only change in length occurs; no internal resistance.

Key Points (Free Expansion)

• Free expansion formula applies whenever no external restraint opposes expansion.
• Under free expansion:
  • Thermal strain and thermal stress are considered zero because no load is induced.

Case 2: Restrained Expansion of a Steel Rod

• Steel rod of original length L held between two rigid supports (at A and B).
• Temperature increase T tends to increase length by α T L.
• Supports prevent expansion, so actual extension = 0.
• Concept: “extension prevented” = amount the bar wants to expand but cannot.

Thermal Strain (Restrained Rod)

• Extension prevented = α T L
• Thermal strain, e:
  • e = (extension prevented) / (original length)
  • e = (α T L) / L = α T
• This is called thermal strain.
• Nature:
  • For temperature rise with restraint, this thermal strain is compressive.

Thermal Stress (Restrained Rod)

• From Hooke’s law:
  • E = stress / strain = σ / e
  • So, σ = E e
• Substituting e = α T:
  • σ = E α T
• This is thermal stress.
• Nature:
  • For temperature rise with fixed supports, thermal stress is compressive.
  • If temperature decreases and contraction is prevented, thermal stress becomes tensile.

Summary Table: Free vs Restrained Expansion

Important Thermal Formulas (Rod Under Restraint)

• Free expansion:
  • ΔL = α T L
• Thermal strain (when expansion/contraction is fully prevented):
  • e = α T
• Thermal stress (full restraint):
  • σ = E α T

These three formulas are used for many objective and conventional problems in thermal stress and strain.

Numerical Problem: Steel Railway Rails

Given (for all parts unless changed in part statement):

• Material: Steel rails
• Initial temperature: T₁ = 24 °C
• Length of each rail: L = 32 m
• Coefficient of linear expansion: α = 11 × 10⁻⁶ per °C
• Young’s modulus: E = 205 GPa = 205 × 10⁹ Pa

Part 1: Stress at 80 °C with No Allowance for Expansion

• Final temperature: T₂ = 80 °C
• Temperature rise:
  • T = T₂ − T₁ = 80 − 24 = 56 °C
• Rails are laid with no gap; no allowance for expansion.
• So expansion is completely prevented → full thermal stress develops.
• Formula:
  • σ = E α T
  • σ = (205 × 10⁹) × (11 × 10⁻⁶) × 56
• Calculated result:
  • σ ≈ 12.628 × 10⁶ Pa ≈ 12.63 MPa
  • (Lecturer writes 12 6.28 × 10⁶ Pa and quotes ≈ 12.6 MPa type magnitude)
• Nature:
  • Thermal stress is compressive because the rails try to expand but are restrained.

Part 2: Stress at 80 °C with 8 mm Expansion Allowance

• There are two rails with a gap between their ends:
  • Expansion allowance (gap) = 8 mm
• Each rail length: 32 m
• Temperature rise: still T = 56 °C
• The rail can first expand into the 8 mm gap without any stress.
• Only the remaining portion of free expansion is prevented, which causes stress.

Thermal Strain with Partial Allowance

• Free expansion that would occur without restraint:
  • ΔL_free = α T L
• Actual “expansion prevented”:
  • (α T L − 8 mm)
• Thermal strain:
  • e = (expansion prevented) / L
  • e = (α T L − 8) / L
• Using Hooke’s law:
  • e = σ / E
  • So, σ / E = (α T L − 8) / L

Substitution

• σ / (205 × 10⁹) = [11 × 10⁻⁶ × 56 × 32 − 8 × 10⁻³] / 32

• Solving:

  • σ ≈ 75.3 × 10⁶ Pa = 75.3 MPa

• Comparison:

  • Stress with no allowance (Part 1) ≈ about 12.6 MPa (as per the values written).
  • Stress with 8 mm allowance ≈ 75.3 MPa using the given working.
  • Statement emphasis: allowing expansion reduces thermal stress compared with a case of zero gap for the same temperature change.

Part 3: Expansion Allowance for No Stress at 80 °C

• Aim: find expansion allowance (ΔL) so that no stress develops at 80 °C.
• Condition for no thermal stress:
  • Rail must be allowed to expand freely by the full free expansion amount.
• Free expansion:
  • ΔL = α T L
• Substitute:
  • ΔL = (11 × 10⁻⁶) × 56 × 32
  • ΔL ≈ 0.01971 m
  • Convert to mm:
    • ΔL ≈ 19.71 mm
• Interpretation:
  • If an allowance of about 19.71 mm is provided and temperature rises from 24 °C to 80 °C,
  • then rails can expand without restraint and no thermal stress will be generated.

Part 4: Maximum Temperature for No Stress with 8 mm Allowance

• Given:
  • Expansion allowance: ΔL = 8 mm = 8 × 10⁻³ m
  • Length L = 32 m
  • α = 11 × 10⁻⁶ per °C
  • Initial temperature: T₁ = 24 °C
• Condition for no stress:
  • Allowance exactly equals free expansion: ΔL = α T L
• Free expansion relation:
  • 8 × 10⁻³ = α (T₂ − T₁) L
  • 8 × 10⁻³ = (11 × 10⁻⁶) (T₂ − 24) (32)
• Solve for T₂:
  • T₂ ≈ 46.72 °C
• Rise in temperature allowed without stress:
  • T = T₂ − T₁ ≈ 46.72 − 24 ≈ 22.72 °C

Numerical Results Summary

Practical Insight: Railway Track Gaps

• Rails are joined with plates and bolts; a small gap is intentionally left between rail ends.
• This gap is the expansion allowance to:
  • Let rails expand in hot weather.
  • Reduce thermal stresses to safe levels.
• Without gaps (allowance), high compressive thermal stresses can develop, leading to track buckling.

Key Terms & Definitions

• Thermal expansion: Increase in length or volume of a material due to temperature rise.
• Coefficient of linear expansion (α):
  • Constant relating change in length to temperature change for a given material.
• Free expansion: Expansion when a body is not restrained; no stress develops.
• Thermal strain (e):
  • Strain in a body due to temperature change when expansion or contraction is prevented.
  • For full restraint: e = α T.
• Thermal stress (σ):
  • Stress induced due to prevention of thermal expansion or contraction.
  • For full restraint: σ = E α T.
• Compressive thermal stress:
  • Stress produced when expansion is prevented during temperature rise.
• Tensile thermal stress:
  • Stress produced when contraction is prevented during temperature decrease.
• Expansion allowance:
  • Intentional gap provided to permit some thermal expansion without developing stress.

Action Items / Next Steps

• Memorize and practice using:
  • ΔL = α T L
  • e = α T
  • σ = E α T
• Practice similar numericals involving:
  • Different materials and lengths.
  • Various expansion allowances and temperature ranges.
• Prepare for next lecture on:
  • Composite or compound bars consisting of two or more materials.
  • Thermal stresses in such combined systems.