NAFEMS LE5: Torsion of a Z-Section Cantilever Beam

Benchmark Specification and Theoretical Background

What LE5 Sets Out to Verify

NAFEMS LE5 is a verification problem for shell elements: a cantilever of Z-section loaded at its free end by a torsional couple. What makes it interesting is that although the load is torsion, the pass/fail judgement is made on the axial stress near the root. An open section such as a Z has extremely low Saint-Venant torsional stiffness, so it carries torsion through the restraint of warping — the mechanism in which the flanges are bent in opposite directions. The result is a large axial membrane stress in the flanges. LE5 tests whether the membrane behaviour of shell elements can reproduce this key piece of structural mechanics: the restrained torsion of an open section.

Problem Setup

Why It Is a Good Shell Problem

Procedure and Key Points of Reading the Result

The Standard Procedure

1. Modelling — build the shell model on the mid-surface (the thickness of 0.1 m is supplied as a property). Confirm that the connection lines between the flanges and the web are shared
2. Load — apply 0.6 MN in opposite directions to the free-end edges of the upper and lower flanges, as a shear distributed uniformly along the edge (an edge load). Always check the resultant force and the resultant moment
3. Restraint — fix every degree of freedom on the root end face
4. Mesh series — start from the division recommended in the specification, then refine it by factors of 2 and 4 to confirm the convergence of the membrane stress at point A
5. Reading the result — extract the axial component of the mid-surface (membrane) stress at point A and report the convergence towards −108 MPa

The Most Common Mistake — Confusing Top, Bottom and Mid-Surface

Shell stress output comes in three layers: top, bottom and mid-surface (mid). The LE5 target value is the membrane stress (mid) — read top or bottom instead and you pick up the plate-bending component as a systematic offset. Establishing which layer your tool writes out by default, and which side counts as the top, that is, the direction of the shell normal, is the first hurdle of this problem. In real work as well, which layer a shell stress refers to is a mandatory entry in the report, and LE5 is the problem that drills that discipline into you.

Good Practice in Substituting an Equivalent Load

Applying the torsional moment of 1.2 MN·m as a torque at a single point is wrong; it has to be applied as the specification states, as a couple of uniformly distributed shears on the flange end edges. A concentrated torque brings local deformation and a singularity with it, and that reaches the stress at the root as well. As the check on the substitution, confirm in the load summary that the resultant of the applied load is zero (a pure couple) and that the resultant moment is 1.2 MN·m. The habit of checking resultant force and resultant moment whenever a load is replaced by an equivalent pays off right across practice, as the load-side counterpart of boundary condition verification.

Putting It to Work in Practice

An Acceptance Test for Shell Elements

LE5 is ideal for comparing element types and formulations. Line up first-order reduced-integration shells, first-order fully integrated shells and second-order shells at the same mesh density, compare the convergence of the stress at point A, and you have quantified the real capability of your standard element — along with its habits on a coarse mesh. This lightweight problem also suits regression checking after a solver update or a change to element settings such as drilling stiffness or hourglass control parameters. Set it beside T4 and LE11, and the recommended configuration is to make that trio of heat, thermal stress and shells the basic frame of the verification register.

A Lesson for Structural Design — Open Sections Are Weak in Torsion

The physics of LE5 connects straight to an important lesson for real design. An open section (Z, C or L) has a Saint-Venant torsional stiffness tens to hundreds of times smaller than a closed section, and because it carries torsion through the axial stress of warping restraint, it behaves in a way best summarised as twist it and it fails in bending stress. Why closing the section, turning it into a box, is so effective in the torsional design of thin-walled members, and why stress concentrates where warping is restrained, at the root and at diaphragms — the experience of solving LE5 turns both into quantitative intuition. Benchmarks exist not only for verification; they are also living teaching material for structural mechanics.

Cross-Checking Against Beam Theory (Vlasov)

Restrained torsion can be worked out by hand with Vlasov's thin-walled beam theory (the warping function and the bimoment), and setting that against the shell solution gives a mutual verification of shell FEM against classical theory. The region near the root is where Saint-Venant's principle does not apply, since the effect of warping restraint dominates, so the agreement with theory will never be perfect — but it is quite sufficient for checking the order of magnitude of the stress and the trend of its distribution. That gives LE5 further value as an exercise in checking an FEM answer against a different theoretical framework.

Setup Essentials Tool by Tool

Setting Comparison

Cautions When Building Equivalent Nodal Forces by Hand

When a uniformly distributed load on an edge is reduced to nodal forces, follow the rule of consistent nodal forces: for first-order elements, equal shares at the interior nodes and half at the two end nodes (for second-order elements the weighting shifts onto the mid-side nodes). Giving every node the same value, an equal split, breaks the distribution at the ends and distorts the local stress. This is largely within the reach of the resultant force and moment check on the substituted load, but a distorted distribution shape slips through that check, so the distribution rule itself has to be kept in mind.

Shell Verification Today

The Lineage of Shell Formulations and the Role of Benchmarks

Shell elements are the history of the remedies for shear locking and membrane locking — reduced integration with hourglass control, the mixed interpolation of the MITC family, the enhanced assumed strain method — and the standard benchmark set that includes LE5 (the pinched cylinder, the Scordelis-Lo roof, the twisted plate and the rest of the compulsory shell tests) has acted as the referee throughout. Running through this classical set is the convention of the element development community whenever a new element or solver is evaluated, and on the user side the same set reveals the lineage and the habits of the element you rely on.

The Trend Towards Solid Shells and Dispensing With the Mid-Surface

Reluctance to spend the effort of extracting mid-surfaces from CAD solids has strengthened the move towards solving thin-walled parts directly with solid-shell elements, or with solids one element thick through the thickness. A benchmark such as LE5 also serves as a yardstick for confirming whether that simplification and that choice of element preserve the accuracy of the membrane behaviour, which makes it usable in decisions about modelling policy. An in-house study that solves LE5 once with mid-surface shells and once with solid shells makes a persuasive basis for changing direction.

Building It Into Automated Verification

As with T4 and LE11, LE5 is light enough to suit CI and regression testing. Updates to the element library and changes to default parameters can affect membrane behaviour quietly, so installing an automatic pass/fail check at a tolerance of ±1%, as the reference standard for shells, transforms how much confidence you can place in an environment update.

Troubleshooting

Diagnostic Table for When the Answer Is Not −108 MPa

Completing the Basic Trio of the Verification Register

Related reading: NAFEMS T4, NAFEMS LE11, index of NAFEMS benchmarks, how to read analysis results correctly.