EngineeringMechanical EngineeringMedium

Thermal Stress

Also known as:Thermally Induced StressTemperature Stress

Thermal stress is the internal stress developed in a material when its thermal expansion or contraction is restrained by external constraints or non-uniform temperature distribution. It arises because materials naturally expand when heated and contract when cooled, and any restriction to this dimensional change generates internal forces. Thermal stresses are critical in the design of pipelines, bridges, engine components, and structural elements exposed to temperature fluctuations.

Key Formula

sigma_T = E * alpha * Delta_T

LaTeX: \sigma_T = E \alpha \Delta T

Symbol	Meaning	Unit
\sigma_T	Thermal stress	Pa (Pascal)
E	Young's modulus of elasticity	Pa
\alpha	Coefficient of linear thermal expansion	/°C or /K
\Delta T	Change in temperature	°C or K

Worked Example

Problem

A steel rod (E = 200 GPa, α = 12 × 10⁻⁶ /°C) is rigidly fixed at both ends. If the temperature rises by 80°C, find the thermal stress developed in the rod.

Solution

Step 1: Identify given values. E = 200 × 10⁹ Pa, α = 12 × 10⁻⁶ /°C, ΔT = 80°C. Step 2: Apply the thermal stress formula. σ_T = E × α × ΔT σ_T = 200 × 10⁹ × 12 × 10⁻⁶ × 80 Step 3: Calculate. σ_T = 200 × 10⁹ × 9.6 × 10⁻⁴ σ_T = 192 × 10⁶ Pa

Answer

Thermal stress = 192 MPa (compressive)

Coefficients of Thermal Expansion and Typical Thermal Stresses for Common Engineering Materials

Material	E (GPa)	α (×10⁻⁶ /°C)	Thermal Stress per 100°C (MPa)	Application
Structural Steel	200	12	240	Bridges, buildings
Aluminium	70	23	161	Aircraft, automotive
Copper	120	17	204	Electrical wiring, pipes
Concrete	30	10	30	Pavements, dams
Glass	70	9	63	Windows, optical devices

Interactive Tools

WolframAlpha Thermal Stress Calculator

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Khan Academy – Thermal Expansion

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NIST Material Properties Data

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Wikimedia Commons, CC BY-SA

Related Terms

Physics

Thermal Expansion

Thermal expansion is the tendency of matter to increase in length, area, or volume when its temperature rises, due to the increased average separation between particles as they vibrate more vigorously. Most solids, liquids, and gases expand on heating, with gases expanding far more than liquids or solids. Thermal expansion has important engineering implications — railway tracks have expansion gaps, bridges have expansion joints, and thermometers exploit the expansion of mercury or alcohol — while water's anomalous contraction on cooling from 4 °C to 0 °C is critical to aquatic life in cold climates.

From Greek "therme" (heat) and Latin "strictus" (drawn tight, from "stringere"). The concept was formally analysed by French engineer Henri Navier and British physicist George Gabriel Stokes in the early 19th century as part of the theory of elasticity.

thermal stressmechanical engineeringthermomechanicselasticitystructural analysis