BIZOMAT MAKEOS · Strength
Restrained Bar Heating (Thermal Stress, Yield and Buckling)
When this bar or pipe is held between fixed supports and heats up, does the thermal stress stay below yield, and will it buckle before then?
Check of a round bar or member held at both ends while its temperature changes: free thermal strain, restrained thermal stress and the force on the supports (as the thermal stress tool), the yield safety factor, the buckling load of the member by Euler or Johnson for its effective length (as the column buckling tool), the buckling safety factor against the thermal force, and the temperature rise at which yielding and buckling would begin.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Bar diameter | mm | Diameter of the round bar (solid). |
| Length between restraints | mm | Length between the supports that stop it growing. |
| Effective length factor | Effective length factor for the end fixity (0.5 fixed at both ends, 1 pinned). | |
| Elastic modulus | GPa | Elastic modulus. |
| Expansion coefficient | µm/m·K | Linear expansion coefficient. |
| Temperature rise | K | Rise in temperature from the fitting temperature. |
| Degree of restraint | % | Share of the free growth the supports prevent (100% for rigid anchors). |
| Yield strength | MPa | Yield strength at temperature. |
| Safety factor required (yield and buckling) | Safety factor you require on both yield and buckling. |
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: Bar diameter
- What it is
- The diameter of the solid round bar held between the restraints. It sets the cross-section area and the radius of gyration for buckling.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thermal force rises with the square of the diameter; the stress does not change.
What this means: Length between restraints
- What it is
- The length of bar between the supports that stop it from growing, which is also the length that can buckle.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Effective length factor
- What it is
- The effective length factor for the end fixity of the bar in buckling: it turns the actual length into the equivalent pinned length.
- Units and format
- Enter a plain number with no unit. The form accepts 0.5 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher factor lengthens the effective length and lowers the buckling load.
- Get it right
- Choose it from the real end conditions; fixed ends that can slip or rotate behave closer to pinned.
What this means: Elastic modulus
- What it is
- Young’s modulus of the bar material at the working temperature, from the material data.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 400 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
- Thermal stress rises in direct proportion to E.
- Get it right
- Enter it in GPa, not MPa: 1 GPa = 1000 MPa, so divide a data-sheet value given in MPa by 1000.
What this means: Expansion coefficient
- What it is
- The linear thermal expansion coefficient of the bar material: how much each metre grows per kelvin of temperature rise, from the material data.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0.000001 µm/m·K to 40 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
- Thermal stress and force rise in direct proportion.
- Get it right
- µm/m·K is the same as 10⁻⁶ per kelvin; enter the number without the ×10⁻⁶.
What this means: Temperature rise
- What it is
- How much the bar’s temperature rises above the temperature at which it was fitted between the restraints.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0 K to 1000 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Thermal stress and force rise in direct proportion.
- Get it right
- A rise from the fitting temperature, not the working temperature itself.
What this means: Degree of restraint
- What it is
- The share of the bar’s free thermal growth that the supports prevent: 100 % for supports that do not move at all, less if they yield or slide.
- 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
- Thermal stress and force are directly proportional to the degree of restraint.
What this means: Yield strength
- What it is
- The yield strength of the bar material at the working temperature. It sets the yield check and the Johnson buckling stress for stocky bars.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 3000 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.
What this means: Safety factor required (yield and buckling)
- What it is
- The safety factor you require, applied to both yielding and buckling, from your design code or practice.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Thermal stress σ = E α ΔT × restraint; force = σ A (as the thermal stress tool).
- Buckling stress by Euler for slender bars or Johnson for intermediate ones, with slenderness K L ÷ r (as the column buckling tool); buckling factor = P_cr ÷ force.
- Temperature rise at yield = S_y ÷ (E α × restraint); at buckling = σ_cr ÷ (E α × restraint).
- Example: 40 K gives 96 MPa and 120.64 kN; the bar buckles at 71.185 K and yields at 104.17 K.
Limitations
- Uniform temperature along the bar; local hot spots and bowing from uneven heating are not covered.
- Real supports are not fully rigid: the actual restraint is often less.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a pipe or rod anchored at both ends
- Find the temperature rise a tie rod can take
- Decide whether an expansion joint is needed
Who it is for
- Design engineers
- Piping engineers
- Maintenance engineers
What you get
- Thermal stress, force and yield factor
- Buckling load and factor
- Temperature rise at yield and at buckling
Interpreting the result
- Long slender members buckle long before they yield: a guide or an expansion loop is then the answer.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 375–376, 281–287. 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
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
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.
Useful alongside · BIZOMAT original book
Heat, restraint and buckling
What it covers: How a member that carries no external load can buckle when it is heated between restraints: thermal force, the degree of restraint, the temperature rise at buckling and a hot outlet pipe.
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