Estimate axial thermal stress, free thermal strain, and free length change for a straight elastic member under uniform heating or cooling.
Advanced options
How to use our Thermal Stress Calculator
- Choose Temperature input mode. Use initial and final temperatures if you know both values, or use direct temperature change if you already know the difference.
- If helpful, pick a Material preset to autofill Young's modulus (GPa) and Coefficient of linear thermal expansion (1/K). You can still edit those numbers.
- Enter Young's modulus (GPa). This is the material stiffness, so it must be greater than 0.
- Enter Coefficient of linear thermal expansion (1/K). This must also be greater than 0 for the materials covered here.
- Enter either Initial temperature (deg C or K) and Final temperature (deg C or K), or enter Temperature change (K or deg C). Positive change means heating and negative change means cooling.
- Choose Stress output unit for the displayed thermal stress.
- If you want extra context, open Advanced options and enter Member length (m) for free length change, and adjust Restraint factor (0 to 1) if the member is not fully restrained.
- Click Calculate and read the signs carefully: negative Thermal stress means compressive stress from heating under restraint, while positive means tensile stress from cooling under restraint.
- Sanity-check the result: if Restraint factor (0 to 1) is 0, the Thermal stress should be 0 while Free thermal strain and Free length change can still be nonzero.
Definitions
Temperature input mode: Chooses whether you enter two temperatures or enter the temperature change directly.
Young's modulus (GPa): A measure of stiffness. Larger values mean the material resists stretching or squeezing more strongly.
Coefficient of linear thermal expansion (1/K): How much the material wants to change length for each degree of temperature change.
Temperature change: The difference between final and initial temperature. Positive means heating and negative means cooling.
Free thermal strain: The length change per unit length the member would have if nothing restrained it.
Free length change: The actual amount of length increase or decrease the member would have with no restraint, based on Member length (m).
Thermal stress: Axial stress caused when thermal expansion or contraction is restrained. In this calculator, negative means compressive from heating and positive means tensile from cooling.
Restraint factor (0 to 1): A simple scaling factor where 0 means no restraint and 1 means full axial restraint.
Material preset: A quick fill for typical reference values of stiffness and thermal expansion for common materials [1][1].
Common mistakes and quick fixes
Mistake: Mixing temperature scales between Initial temperature (deg C or K) and Final temperature (deg C or K) .
Fix: Use the same scale for both fields. The calculator only treats the difference as valid when both inputs use one consistent scale.
Mistake: Entering a positive Temperature change (K or deg C) when the member actually cooled.
Fix: Use a negative value for cooling. Then Thermal stress should become positive for restrained cooling, which means tensile stress.
Mistake: Typing Young's modulus (GPa) in MPa or Pa instead of GPa.
Fix: Enter the value in GPa exactly as labeled. For example, steel is about 200 in this field, not 200000.
Mistake: Entering Coefficient of linear thermal expansion (1/K) as 12 instead of 12e-6.
Fix: Use scientific notation or a decimal form such as 0.000012 so Free thermal strain and Thermal stress are realistic.
Mistake: Expecting Member length (m) to change Thermal stress .
Fix: Length only affects Free length change . For this axial model, stress comes from stiffness, expansion coefficient, temperature change, and Restraint factor (0 to 1) .
Mistake: Using Restraint factor (0 to 1) outside the allowed range.
Fix: Keep it between 0 and 1. Use 0 for free expansion, 1 for full restraint, and a value in between for an estimate of partial restraint.
Limitations & Key Assumptions / Boundary Conditions
- This calculator is for a straight member with uniform temperature change along its length and cross-section.
- It assumes linear elastic behavior, so the relation between stress and strain is taken as proportional over the range used.
- The Restraint factor (0 to 1) is an estimate for partial restraint, not a full structural analysis of supports, connections, or surrounding members.
- Thermal stress is axial only. The model does not include bending from temperature gradients, buckling, creep, yielding, cracking, or residual stress.
- Free length change is shown only when Member length (m) is provided and greater than 0.
- Material presets are typical values for quick estimates. Real properties vary with alloy, mix, temperature, and test conditions [1][2].
- For temperature differences, a change of 1 K equals a change of 1 deg C, but absolute Celsius and kelvin zero points are not interchangeable.
Methodology
Core equations
The calculator first finds the temperature change used in all later steps. If you choose initial and final temperatures, it subtracts the initial value from the final value. If you choose direct temperature change, it uses that entered value as-is.
ΔT = T_final - T_initial
Next it computes the free thermal strain, which is the strain the member would have if it could expand or contract without restraint.
ε_th = α x ΔT
If Member length (m) is provided, the calculator finds the free change in length.
ΔL = α x L x ΔT
Then it estimates axial thermal stress using the restraint factor. Young's modulus entered in GPa is converted to Pa for the calculation, then the result is converted to the chosen stress unit for display.
σ = -k x E x α x ΔT
The minus sign sets the displayed sign convention: heating under restraint gives negative stress, which is compressive, and cooling under restraint gives positive stress, which is tensile.
Unit handling
Young's modulus (GPa) is multiplied by 1,000,000,000 to convert GPa to Pa. Stress display then converts from Pa into MPa, GPa, psi, or ksi as selected. The calculator also treats a temperature change of 1 K as the same size as a temperature change of 1 deg C.
1 MPa = 1,000,000 Pa
1 GPa = 1,000 MPa
1 psi = 6894.757293168 Pa
1 ksi = 1000 psi
Worked mini-example
Suppose a steel-like member has Young's modulus (GPa) = 200, Coefficient of linear thermal expansion (1/K) = 12e-6, Initial temperature (deg C or K) = 20, Final temperature (deg C or K) = 80, Member length (m) = 2, and Restraint factor (0 to 1) = 1.
ΔT = 80 - 20 = 60 K
ε_th = 12e-6 x 60 = 0.00072
ΔL = 12e-6 x 2 x 60 = 0.00144 m
σ = -1 x 200 GPa x 12e-6 x 60 = -144 MPa
That result means the member wants to expand, but full restraint turns that tendency into compressive axial stress.
Assumptions used in the result
This method is meant for a straight member under uniform heating or cooling with elastic material response. It is a quick axial estimate, so real structures can differ if restraint is not purely axial, if temperature varies through the part, or if the material yields, cracks, creeps, or changes properties significantly with temperature. Typical preset values are reference values and can vary by material type [1][2].