At what bar slenderness does the 3D eigenbuckling model agree with the slender-beam closed form?

What it found

A stylised, preliminary scan of one steel bar cross-section (8×2 mm, tip shear, clamped end) at l/w = 3, 6, 12 and 24 on a fixed element size. The finite-element critical load sits 14.5%, 7.4%, 3.8% and 2.1% above the Timoshenko–Gere closed form, so the gap shrinks roughly in proportion to w/l and is consistent with a short-bar effect rather than mesh error. One cross-section and one cross-section mesh; no refinement at the longer lengths.

Paper, code and results

• README and method
• Experiments and results
• Code
• Tests

More in Simulation-first design

• How much stiffer than a uniformly thinned plate is a topology-optimised part, and how does that depend on volume fraction?How much stiffer a thresholded topology-optimised part is than a uniformly thinned plate across volume fractions (stylised): 26–31% lower compliance at v = 0.3–0.5, shrinking to 14% at 0.8, and disconnected designs below that on coarse grids.
• At what rate does the 3D eigenbuckling model converge under uniform mesh refinement?Observed convergence order of the 3D eigenbuckling model on a short bar (stylised): three uniform meshes give order ≈ 2.6 and a Richardson limit FE/closed-form ratio 1.137, so the gap to the closed form is not discretisation error.
• What does it cost to replace the w/b ≤ 8 rule of the laser-cut bracket with a buckling-factor guard?Replacing the w/b ≤ 8 slenderness rule of the laser-cut bracket with a lateral-torsional buckling factor ≥ 3 lets the optimiser use the 1.5 mm gauge at w ≈ 18.7 mm, 31.3 g instead of 35.9 g (about 12% lighter), and the guard does not bind at that optimum; the result depends on the assumed factor (no feasible design for factors ≥ 5 at that gauge); stylised, preliminary.
• When does the buckling guard of the laser-cut bracket start to bind as load and arm length change?Sweeping tip load (50–400 N) and arm length (60–120 mm) of the laser-cut bracket: the buckling-factor ≥ 3 guard binds only at 120 mm arms (unguarded optimum has factor 1.9; 69 g vs 59 g), while the old w/b ≤ 8 rule costs 14–25% in mass in five of six cases (stylised, two seeds).
• How sharp is the feasible edge of the 2 mm laser-cut bracket web at 0.1 mm resolution, and is it mesh noise?A 0.1 mm scan of the feasible edge of the gauge-2 bracket web (nine mesh/radius cases): peak stress is not monotone in w, with residual jitter of 0.9–2.4 MPa about a quadratic trend, the same size as the 0.1–1.7 MPa gap to the 100 MPa limit just below each edge, so the edge is a band of about 0.5 mm and the earlier refinement shift is not separated from mesh noise; stylised, preliminary.
• Do wider webs of the 2 mm laser-cut bracket stay feasible on fine meshes when the slenderness rule is relaxed?With the assumed slenderness limit raised from 8 to 12.5, gauge-2 bracket webs of 16–17 mm and wider are feasible at h = 1.5, 1.0 and 0.75 mm; refinement from 1.5 to 1.0 mm moves the lightest feasible width up 0.5 mm (about 1 g, 3%); stylised, preliminary.

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.