Thermal Stress Calculator for Restrained Expansion

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)

Assumptions and typical mistake

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

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

What you enter – and where the values come from

Measured strain ε (with compensation gauge) [µm/m]

The strain the amplifier displays when the active gauge on the part and a compensation gauge on a freely expanding, unloaded piece of the same material at the same temperature form the half bridge. This arrangement does not show the thermal expansion itself, only its restraint plus the mechanical load. Source: reading after balancing at the initial temperature.

Known mechanical load strain ε_L (0 if unloaded) [µm/m]

Strain component from an external mechanical load such as axial force or internal pressure, separate from temperature. Source: comparison measurement at the initial temperature with unchanged load, or calculation from the known load. Enter 0 if no additional mechanical load acts during heating.

Thermal expansion coefficient α [10⁻⁶/K]

Thermal expansion coefficient of the part material: by how many µm/m it would expand freely per kelvin. Source: material table; steel ≈ 12, aluminium ≈ 23, concrete ≈ 10–12 (·10⁻⁶/K).

Temperature change ΔT [K]

The temperature change is the difference between the temperature at bridge balancing and the temperature during measurement. Only this difference produces temperature-related errors; if balanced at measuring temperature it is zero. Source: thermocouple or resistance thermometer at the measuring point or cable.

Young's modulus E [N/mm²]

Young's modulus describes how stiff the part's material is – how much stress it takes to produce a given strain (σ = E·ε). It is needed to turn measured or computed strains into stresses and vice versa. Source: material tables or datasheet; steel ≈ 210 000 N/mm², aluminium ≈ 70 000 N/mm², titanium ≈ 110 000 N/mm². E drops at elevated temperature.

What this calculator is for

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.

What comes next

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.

How to read the results

• The bridge with compensation gauge does not show the thermal strain itself, only its restraint; hence 0 µm/m at ΔT ≠ 0 means free expansion without thermal stress.
• Degree of restraint and σ_th together tell whether a restraint is critical; σ_tot is the actual stress at the measuring point.

What each value means

Free thermal strain α·ΔT [µm/m]

Free thermal strain: how much the part would lengthen or shorten from temperature alone if no support restrained it. It is the reference for actual and prevented expansion.

Actual thermal strain ε_T [µm/m]

Thermal strain that actually occurred. Equal to α·ΔT for free expansion, zero for full restraint.

Prevented strain α·ΔT − ε_T [µm/m]

The part of the thermal expansion prevented by restraint – it produces the thermal stress.

Degree of restraint (1 = full)

0 = free to expand (no thermal stress), 1 = fully restrained (maximum thermal stress). Values in between indicate compliant restraint.

Thermal stress σ_th = −E·(α·ΔT − ε_T) [N/mm²]

Thermal stress: negative (compression) for prevented expansion on heating, positive (tension) for prevented contraction on cooling.

Total stress σ_th + E·ε_L [N/mm²]

Total normal stress at the measuring point from thermal stress plus mechanical load stress E·ε_L.

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 Resistance Change Calculator
• Strain Gauge Transverse Sensitivity Calculator
• Strain Gauge Apparent Strain Calculator
• Strain Gauge Excitation Voltage Calculator
• Wheatstone Bridge Output Voltage Calculator
• Strain Gauge Bridge Equation Calculator

All strain-gauge calculators