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Thermal Stress Calculator

Estimate stress from prevented thermal expansion with full, partial, or prescribed-strain restraint modes.

Thermal Stress inputs

Result section

warningfully-restrained

Restrained thermal stress

-120 MPa

Secondary results

Free thermal strain: 600 microstrain
Mechanical strain: -600 microstrain
Thermal stress: -120 MPa
Free expansion: 0.6 mm

Unit conversion summary

Unit-normalization summary: 1.20000e-5 1/K; 50 K; -1.20000e+8 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Positive normal stress is tension. Negative normal stress is compression. Positive shear follows the displayed stress-element convention.
Temperature-dependent material properties, yielding, creep, gradients, contact, and supports are not modeled.

Assumptions

Linear thermal expansion and linear-elastic material response.
Positive temperature change under full restraint produces compressive stress with the default sign convention.

Calculation steps

Calculate free thermal strain0.0006
Calculate mechanical strain-0.0006
Calculate thermal stress-1.20000e+8 Pa

Thermal strain and restrained thermal stress

epsilon_th = alpha DeltaT; sigma = E epsilon_mech

epsilon_th = alpha DeltaT; sigma = E (epsilon_total - epsilon_th) or -E r alpha DeltaT.

epsilon_th = 1.20000e-5 1/K x 50 K

sigma = 2.00000e+11 Pa x -0.0006

Technical visual

Engineering interpretation

Restrained expansion converts prevented thermal strain into mechanical strain and stress.

What is the thermal stress calculator?

Use this calculator to understand how thermal expansion turns into stress when movement is restrained.

How to use the thermal stress calculator

Select the mode that matches the stress, material, section, or thermal data you already have.
Enter values with units; the calculator normalizes inputs to SI internally.
Review the formulas, normalized values, assumptions, and warnings before transferring the result to another check.
Select free expansion when movement is unconstrained, or restrained modes when supports prevent some or all thermal strain.

Input explanations

Thermal strain is epsilon th = alpha DeltaT.
Free expansion is Delta L = alpha DeltaT L and creates zero stress in the idealized free case.
Full restraint uses mechanical strain = -epsilon th.
Partial restraint scales the prevented strain by a user-entered factor from 0 to 1.

Notation

Formula notation
Symbol	Meaning	Unit
alpha	Coefficient of thermal expansion	1/K
Delta T	Temperature change	K
epsilon th	Free thermal strain	dimensionless
sigma	Restrained thermal stress	Pa

Worked examples

Steel with E = 200 GPa, alpha = 12 microstrain/K, and Delta T = 50 K under full restraint gives about -120 MPa.

Free expansion with the same inputs gives strain and displacement but zero stress.

Partial restraint factor 0.5 gives half of the fully restrained stress in this idealized model.

Cross-tool workflow

The Thermal Stress Calculator supports free expansion, fully restrained expansion, partial restraint, and prescribed total strain. It reports thermal strain, mechanical strain, stress, and free expansion.

Use thermal stress in Factor of Safety Calculator

Common mistakes

Treating a temperature gradient as uniform temperature change.
Ignoring temperature-dependent material properties, creep, yielding, or support flexibility.
Using the wrong sign convention for heating and cooling.
Assuming thermal stress exists when expansion is completely free.

Limitations

Only uniform temperature change and linear thermal expansion are modeled.
Temperature gradients, nonlinear material data, creep, plasticity, contact, and detailed supports are not included.
The result is a preliminary mechanics check, not a thermal-structural FEA solver.

References

Roark's Formulas: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.
Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill.
Mechanics of Materials: Hibbeler, Mechanics of Materials, Pearson.
Theory of Elasticity: Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill.

What is thermal stress?

It is stress created when thermal expansion or contraction is restrained.

Does free expansion create stress?

In the idealized free-expansion case, no. It creates strain and displacement without stress.

Why is heating under restraint compressive?

With the default sign convention, prevented positive thermal strain creates negative mechanical strain and compressive stress.

Can I model partial restraint?

Yes. Enter a restraint factor between 0 and 1 to scale the prevented thermal strain.

Does this solve thermal gradients?

No. Uniform temperature change only is included.

Does this calculator prove that a part is safe?

No. It reports an educational preliminary stress or factor comparison. Real design decisions need applicable standards, validated models, material data, and engineering review.

Can these results replace commercial FEA software?

No. ScholarTool provides transparent hand-calculation support and postprocessing checks, not a replacement for validated FEA solvers.

Professional support from ScholarEase Consultancy Services LLP

Professional FEA, simulation setup, solid-mechanics calculations, technical documentation, and research support are available through the connected services website. These calculators remain free to use without purchasing services.

Visit ScholarEase

Engineering disclaimer

This tool is intended for educational, estimation, and preliminary solid-mechanics support. Always verify critical engineering calculations with applicable standards, validated software, material data, and qualified professional judgment.

Need Professional Support?

ScholarTool is powered by ScholarEase Consultancy Services LLP. Professional engineering, simulation, research, and technical documentation support is available through the connected services website.

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Last reviewed: 2026-06-21

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Thermal Stress inputs

Result section

warningfully-restrained

Restrained thermal stress

-120 MPa

Secondary results

Free thermal strain: 600 microstrain
Mechanical strain: -600 microstrain
Thermal stress: -120 MPa
Free expansion: 0.6 mm

Unit conversion summary

Unit-normalization summary: 1.20000e-5 1/K; 50 K; -1.20000e+8 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Positive normal stress is tension. Negative normal stress is compression. Positive shear follows the displayed stress-element convention.
Temperature-dependent material properties, yielding, creep, gradients, contact, and supports are not modeled.

Assumptions

Linear thermal expansion and linear-elastic material response.
Positive temperature change under full restraint produces compressive stress with the default sign convention.

Calculation steps

Calculate free thermal strain0.0006
Calculate mechanical strain-0.0006
Calculate thermal stress-1.20000e+8 Pa

Thermal strain and restrained thermal stress

epsilon_th = alpha DeltaT; sigma = E epsilon_mech

epsilon_th = alpha DeltaT; sigma = E (epsilon_total - epsilon_th) or -E r alpha DeltaT.

epsilon_th = 1.20000e-5 1/K x 50 K

sigma = 2.00000e+11 Pa x -0.0006

How to use the thermal stress calculator

Select the mode that matches the stress, material, section, or thermal data you already have.
Enter values with units; the calculator normalizes inputs to SI internally.
Review the formulas, normalized values, assumptions, and warnings before transferring the result to another check.
Select free expansion when movement is unconstrained, or restrained modes when supports prevent some or all thermal strain.

Symbol

Meaning

Unit

alpha

Coefficient of thermal expansion

1/K

Delta T

Temperature change

epsilon th

Free thermal strain

dimensionless

sigma

Restrained thermal stress

References

Roark's Formulas: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.

Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill.

Mechanics of Materials: Hibbeler, Mechanics of Materials, Pearson.

Theory of Elasticity: Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill.

Thermal Stress inputs

Material properties

Thermal loading

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Thermal strain and restrained thermal stress

Technical visual

Engineering interpretation

What is the thermal stress calculator?

How to use the thermal stress calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?

Thermal Stress inputs

Material properties

Thermal loading

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Thermal strain and restrained thermal stress

Technical visual

Engineering interpretation

What is the thermal stress calculator?

How to use the thermal stress calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?