NAFEMS LE11: Thick Cylinder Thermal Stress
Benchmark Specification and Theoretical Background
What LE11 Sets Out to Verify
NAFEMS LE11 is the standard verification problem for thermal stress analysis. There is no mechanical load whatsoever: the stress is produced by the temperature field alone. A single problem therefore exercises every stage of the thermal-stress chain — importing the temperature field, forming the thermal strain \( \varepsilon_{th} = \alpha \Delta T \), and developing stress through restraint. Even when verification under mechanical loading (the other problems of the LE series) passes, thermal stress carries bugs peculiar to this chain — interpolation of the temperature, the handling of α, the reference temperature — which is exactly why a dedicated benchmark exists for it.
Problem Setup
| Item | Content |
|---|---|
| Geometry | Axisymmetric solid (a thick-walled body of revolution in which a cylinder, a taper and a spherical surface join smoothly) |
| Material | \( E = 210 \) GPa, \( \nu = 0.3 \), \( \alpha = 2.3\times10^{-4} \) /°C |
| Temperature field | Prescribed directly as a function of position: \( T(x, y, z) = \sqrt{x^2 + y^2} + z \) (in axisymmetric coordinates, \( T = r + z \)) |
| Restraint | Axial displacement fixed on the end planes at both ends (plus the axisymmetric condition). No external load |
| Target value | Axial stress at point A (the specified point on the inner surface) \( \sigma_{zz} = -105 \) MPa |
Why It Is a Deviously Good Problem
You apply a temperature and read off the stress — that is all, isn't it? Where is the difficulty?
There are three traps built into it. First, the temperature field is given as a continuous function of position — it cannot be expressed as a uniform temperature or as a linear ramp between two points, so the test is whether you can drive your tool's temperature-field input (analytical fields, field mapping) correctly. Second, what you evaluate is a stress, and on the inner surface of a curved body at that — extrapolation, averaging and the choice of coordinate component (the axial one) feed straight into the number. Third, thermal stress is decided by the balance between restraint and expansion, so getting a single face of the restraint wrong moves the answer a long way. Temperature, restraint and stress recovery are the three classic accident sites of thermal stress analysis, and this problem is designed to walk you through all of them at once.
Procedure and Convergence Check
The Standard Procedure
- Axisymmetric modelling — use 2-D axisymmetric elements (second-order 8-node quadrilaterals recommended). Confirm against your tool's convention which axis is the axis of revolution
- Applying the temperature field — impose \( T = r + z \) on the nodes directly as a function of position. Set the reference temperature (the strain-free temperature) to 0
- Restraint — fix the axial displacement of both end faces. The radial direction is handled automatically by axisymmetry
- Mesh series — confirm convergence of \( \sigma_{zz} \) at point A over three levels. Note that the geometric approximation of the curved boundary (how closely element edges follow the arc) also improves with refinement
- Evaluation — read the stress at point A both unaveraged and averaged, and report the convergence towards −105 MPa
Acceptance Criteria
Healthy behaviour is to land within ±2–3% of the target value on a reasonable mesh of second-order elements, and to converge to roughly ±1% under refinement. Because the evaluation point sits on a curved surface, the number moves by a few percent depending on whether the stress is extrapolated from the element to the nodes or taken at the integration points, and on the extent of the averaging — so stating explicitly which definition of stress you are reporting is precisely the general rule laid out in result interpretation.
Applying the Temperature Field Without Polluting It
The heart of this problem is how the temperature is supplied. Mapping across the result of a coarse thermal analysis, or approximating the field with a handful of isothermal regions, feeds that temperature error straight into the stress error (thermal strain is linear in temperature, and restraint amplifies it into stress). Since LE11 gives the temperature field in closed form, the correct answer is to supply it to machine precision through a function input, an expression or a coordinate-dependent table. Checking that route — how the temperature actually gets across — is the dress rehearsal for verifying the coupled analyses of real work, where a thermal solution is mapped onto a structural model.
Putting It to Work in Practice
An Acceptance Test for the Thermal-Stress Chain
The practical value of LE11 lies in serving as an acceptance test for the thermal-to-structural chain: regression checking after a solver update; verification of the temperature mapping route when the thermal and structural analyses live in different tools (reproduce the LE11 temperature field on the thermal side, map it across, and see whether the stress lands on −105 MPa); and training for new engineers, who absorb the ideas of reference temperature, α and restraint from a single problem. Script the input file together with its expected value and enter it in the verification register, and T4 on the thermal side plus LE11 on the thermal-stress side cover the basic chain from heat transfer through to thermal stress.
Lessons That Pay Off in Practice — α and the Reference Temperature
α is a constant in LE11, which keeps things simple, but working the problem through brings the pitfalls of real projects into view. First, the reference temperature, the temperature at which strain is zero — the stress is governed by \( T - T_{ref} \), and getting the assembly or stress-free temperature wrong is the leading cause of thermal-stress accidents. Second, temperature-dependent α — two definitions are in circulation, the instantaneous α and the secant α (averaged from a datum temperature), and mistaking which one your tool expects produces large errors at high temperature. Third, the realism of the restraint — full fixity is the idealisation that maximises the stress, and with a realistic support stiffness the stress relaxes considerably, which leads on to the sensitivity checks of boundary condition verification.
How to Extend It in Stages
Once LE11 passes, three one-step extensions make practice that transfers directly to real work: generate the temperature field yourself with a thermal analysis of the T4 kind and pass it across (mapping verification); introduce temperature-dependent properties and move on to nonlinear thermal stress; and replace the restraint with spring supports to examine the sensitivity to support stiffness. Verify the features you actually use, one after another, by taking a single step out from a problem that has a reference solution — the philosophy of staged verification set out in the T4 article holds just as well for thermal stress.
How to Set It Up Tool by Tool
Entering a Coordinate-Dependent Temperature Field
| Tool | How the temperature field is applied | Note |
|---|---|---|
| Ansys Mechanical | Thermal Condition plus a function defined on the coordinates (or External Data) | Take care with the reference temperature, which is set through Environment Temperature |
| Abaqus | *TEMPERATURE with the expression supplied as an Analytical Field | Manage the initial temperature (*INITIAL CONDITIONS) as the reference state |
| Nastran family | TEMP/TEMPD (nodal temperatures) plus a generator script that evaluates the coordinate expression | TREF sits on the MAT1 card |
| COMSOL | Enter the expression as it stands (T = sqrt(x^2+y^2)+z) | The most direct route. The thermal stress interface assembles it as one package |
| CalculiX / open-source FEM | Generate the nodal temperatures from the coordinate expression and write them into the input deck | Doubles as verification of the input-generation script |
Watch the Conventions of Axisymmetric Models
Axisymmetric analysis carries conventions that differ from tool to tool: which axis is the axis of revolution (X or Y), and what the stress components are called (σzz is the axial direction in some tools and the hoop direction in others). The LE11 target value is the axial stress — establish which component that corresponds to in your own tool's notation with a simple hand check, such as a restrained cylinder under uniform heating, before you read the result, and you avoid the accident of picking up the wrong component.
Benchmark Culture and Thermal Stress Today
Where LE11 Sits Within the LE Series
The NAFEMS LE series is a collection of linear-elastic problems built on the principle of one problem, one feature, and LE11 is the member that covers thermal loading. Combined with LE5 (twisting of a Z-section, that is, warping-restrained torsion of a thin-walled open section) and LE10 (bending of a thick plate), filling in the matrix of element type against load type with verified benchmarks is the systematic way to verify an analysis environment. Making the blank cells visible as features that remain unverified is the point already made in the T4 article.
Why Thermal-Stress Verification Still Matters
Electronics packaging (warpage during reflow: reflow analysis), metal additive manufacturing (residual stress from welding and deposition), thermal fatigue of power devices — thermal stress analysis today is growing harder in the direction of temperature fields that are complex, transient and laced with multi-material interfaces. That is exactly why the value of a simple problem such as LE11, which isolates and verifies the basic stage of the chain, is rising rather than falling. It belongs on the first line of the verification register as the fixed point you return to when a complicated real problem refuses to agree.
Refining the Reference Solution, and Automating the Check
As with T4, modern computing resources let you add digits to the LE11 reference value yourself with an ultra-fine mesh plus Richardson extrapolation. The problem is small enough that building it into CI — automatic pass/fail at a tolerance of ±0.5% on every commit — is equally easy, which turns it into an asset that runs a thermal-stress regression test in a few minutes.
Troubleshooting
Diagnostic Table for When the Answer Is Not −105 MPa
| Symptom | Likely cause | Fix |
|---|---|---|
| Stress almost zero | Restraint forgotten (free expansion carries no stress), or α or ΔT is zero | Check the axial restraint on both end faces, the reference temperature = 0, and the units of α |
| Sign reversed | Sign of the temperature field, or the reference temperature, taken the wrong way round | Check the sign of T−Tref |
| Tens of percent high or low | The temperature field was approximated as uniform or linear, or the units of α (10⁻⁴ against 10⁻⁶) | Apply it exactly, as a function of position. Note that α=2.3e-4 is the value specified for this problem and differs from real steel (about 1.2e-5) |
| Converges but stays a few percent off | Definition of the reported stress (averaged/unaveraged, extrapolation), or the evaluation point is out of place | Align the definitions and re-extract. Re-check the coordinates of point A against the specification |
| Value oscillates as the mesh is refined | Geometric approximation of the curved surface (the faceting of low-order elements) | Move to second-order elements and a mesh that follows the curvature faithfully |
| Solved in 3-D instead of axisymmetric and it does not match | Restraint deficiency peculiar to 3-D (rigid-body rotation modes), coarse circumferential mesh | Pass it in axisymmetric form first, then bring in the 3-D version as an extension for comparison |
A Closing Remark
LE11 passed to within ±1%! Does that mean I can now trust my thermal stress analysis?
The chain of linear behaviour, constant properties and a prescribed temperature field is verified — and that is a big step. What closes the gap to real work next comes in three parts: the route by which the temperature field arrives from your own thermal analysis (mapping); the management of temperature-dependent properties and the reference temperature; and the realism of the restraint. LE11 is the foundation on which you add those three one step at a time and check each one. A benchmark is not finished the moment it passes — it is the starting point from which you accumulate the differences against the real problem. Once you can use benchmarks that way, a culture of verification has as good as taken root.
Related reading: NAFEMS T4 (the reference standard on the thermal side), NAFEMS LE5, index of NAFEMS benchmarks.
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