Thermal stress from restrained expansion: separating load and temperature components

What is calculated?

A compensation gauge on freely expanding material at the same temperature supplies the free thermal-strain signal in the adjacent arm. The bridge therefore shows only the difference between actual and free thermal strain plus the mechanical load strain. For a fully restrained part ε_T = 0 and the bridge reads −α·ΔT; the thermal stress is −E·α·ΔT – about −126 N/mm² for steel at 50 K. A measurement with known ε_L or extrapolation to ΔT = 0 per Keil separates load and thermal parts.

Equations

ε_th = α·ΔT (Keil Gl. 11.1)

ε = ε_L + ε_T − α·ΔT (Gl. 11.2) ⇒ ε_T = ε − ε_L + α·ΔT

σ_th = −E·(α·ΔT − ε_T) (einachsig, Hooke)

Behinderungsgrad = (α·ΔT − ε_T)/(α·ΔT)

Step by step

1. Balance at the initial temperature; compensation gauge on unloaded material at the same temperature.
2. Enter measured strain, ΔT and α; take ε_L from a load case without temperature change.
3. Assess degree of restraint and thermal stress, using biaxial Hooke's law for areal restraint if needed.

Typical mistake

Treating a self-compensating gauge without compensation gauge the same way – the residual temperature response must then be subtracted additionally.

Application and limits

Thermal stresses in piping, stacks, composite parts and tooling; assessment of restraints under temperature cycling; separation of load and thermal components in aerodynamically heated structures.

Uniaxial treatment, temperature-independent α and E, ideal compensation of the temperature response; temperature gradients require several measuring points.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 11.1 origin of thermal stresses, eq. (11.1); sec. 11.2; sec. 11.6 separation of strain components, eqs. (11.2)–(11.4), fig. 11.7.

The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.

Frequently asked questions

In the same cluster

• Strain-gauge basic equation: resistance change from strain and gauge factor
• Transverse sensitivity: correction for two perpendicular gauges
• Temperature response: apparent strain of a gauge under temperature change
• Bridge excitation voltage and gauge self-heating
• Wheatstone bridge: output voltage of the unbalanced bridge
• Basic strain-gauge bridge equation: signal from four strains

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