BIZOMAT MAKEOS · Strength
Restrained Thermal Stress
What stress and force build up if this member is prevented from expanding?
Stress and force in a bar or pipe whose thermal expansion is fully or partly restrained, compared with the yield strength.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Elastic modulus | GPa | Elastic modulus from the material data sheet or an approved materials database (steel is typically about 200 GPa). |
| Expansion coefficient | µm/(m·K) | Linear expansion coefficient over the temperature range, from material data. |
| Temperature change | K | Change from the temperature at which the member was fixed (positive for heating). |
| Degree of restraint | % | 100% if the ends cannot move at all; less if supports allow some movement. |
| Cross-section area | mm² | Cross-section area of the member. |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification, not a typical value. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the member’s material, from the material data sheet.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thermal stress rises in proportion to the modulus.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Expansion coefficient
- What it is
- The coefficient of linear thermal expansion of the material: how much a unit length grows per degree of temperature rise, over the relevant temperature range.
- Units and format
- Enter a number in µm/(m·K): micrometres per metre per kelvin. The form accepts 0.000001 µm/(m·K) to 100 µm/(m·K). Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 µm/(m·K) as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thermal stress rises in proportion to it.
- Get it right
- Enter µm/(m·K), which is the same number as ×10⁻⁶ per K.
What this means: Temperature change
- What it is
- The change in temperature from when the member was fixed in place to the condition being checked. Positive for heating, negative for cooling.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts -1000 K to 3000 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress rises in proportion; heating gives compression and cooling gives tension.
- Get it right
- Use the difference from the installation temperature, not the operating temperature itself.
What this means: Degree of restraint
- What it is
- How much of the free thermal expansion is prevented: 100 % when the ends cannot move at all, less when the supports let the member move partly.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stress and restraint force rise in proportion to it.
- Get it right
- Type the percentage itself (100 for full restraint), not a fraction.
What this means: Cross-section area
- What it is
- The cross-section area of the member, used to turn the thermal stress into the force on its anchors or restraints.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0.000001 mm² or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The force on the restraints rises in proportion; the stress does not change.
- Get it right
- Use the actual area of material, in mm².
What this means: Yield strength
- What it is
- The minimum yield strength of the member’s material, from the material certificate or specification. The thermal stress is compared with it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the value at the operating temperature if it differs from room temperature.
Method and formulas
- Unrestrained strain ε = α ΔT.
- Stress σ = E α ΔT × restraint (compressive when heated).
- Force on restraints = σ A.
- Example: steel heated 60 K with full restraint reaches 144 MPa and pushes on its anchors with 288 kN.
Limitations
- Uniform temperature through the section; temperature gradients create additional stresses.
- Slender members may buckle before reaching the calculated compressive stress (check with the column buckling tool).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check rails, pipes and frames fixed at both ends
- Estimate anchor loads from heating
- Decide whether an expansion joint is needed
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Thermal stress
- Axial force on the restraints
- Unrestrained expansion strain
- Safety factor against yield
Interpreting the result
- If the stress approaches yield or the member is slender, allow movement or add an expansion joint.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 375–376, 578. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Use motion, thermal and flow studies as load sources
In Design Simulation by the Numbers
What it covers: Derive loads from a motion study, check them by hand, apply temperature with the right restraint and decide when a flow study earns its place in the plan.
Method explained here · BIZOMAT original book
Compatibility, thermal restraint and locked-in stress
In Stress Transformation by the Numbers
What it covers: Why stresses need compatibility as well as equilibrium: restrained thermal expansion and buckling, a fit between different metals as temperature changes, and locked-in stress.
Method explained here · BIZOMAT original book
Thermal Expansion by the Numbers
What it covers: Calculate how parts, pipes, liquids and gases change with temperature: growth, measuring at 20 °C, thermal stress, pipe loops, shrink fits, expansion vessels, gas pressure and machine design.
Method explained here · BIZOMAT original book
Restraint, residual stress and annealing
In Yield and Plastic Flow by the Numbers
What it covers: How restrained thermal strain makes stress, why welds always leave yield-level residual stress, how a heated bar can buckle, and how stress relief and annealing undo plastic flow.
Worked example · BIZOMAT original book
Temperature, alloys and apparent strain
In Strain Gauges and Load Cells by the Numbers
What it covers: Why gauge alloys barely change resistance with temperature, how expansion mismatch causes apparent strain, and how matched gauges, bridges and shielding keep temperature out of readings.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.