After Eszter Fehér, András Á. Sipos, Péter L. Várkonyi,“Quasi-static shape control of structures with topologically complex equilibrium sets”, European Journal of Mechanics - A/Solids (2026), doi:10.1016/j.euromechsol.2025.106000.
- Against the exact arc and our own runs
- 12 of 12 checks pass· see the checks
- Against the paper’s figures
- 8 of 13 checks pass· see the checks
Research, not a design tool. Nothing here is an engineering result.
Turn the two clamps of an actively bent strip under a fixed load and take it from an arch to its inverted shape without a snap.
The strip and its two actuators
A flat strip of length 1 is bent into two clamps placed sin 1 apart. Each clamp is a rotary actuator. The end tangents are θ(A) = Θ + Δ/2 and θ(B) = Δ/2 − Θ: Θ sets how much the strip arches and Δ how much it tilts. A dead load q·Q(s) pushes down. Section 3 of the paper treats the strip as a planar, inextensible elastica, θ″/(4π²) + qρ cos θ + λ sin θ − μ cos θ = 0, where ρ(s) is the load carried between A and s and λ, μ are the reactions at A. With a bending stiffness of 1/(4π²), λ = 1 is the Euler load of the clamped strip.
Two shapes for the same clamp angles
At Θ = 1, Δ = 0 and q = 2.5 the strip has two stable shapes: an arch that carries the load in compression (shape 1) and an inverted shape in tension (shape 5). Section 3.4 moves from one to the other along (Θ, Δ) = (1, 0) → (1.3, 0) → (1.3, 1.3) → (1, 1.3) → (1, 0). The rectangle goes around the cusp of the region where both shapes exist, so the strip changes shape without losing stability on the way.
The five shapes of Figure 9(c) on our path: · · · ·
How we reproduce it
We model the strip with 60 members of Stabileo’s production 2D co-rotational frame (EA·L²/EI = 8·10⁵π², so the axial strain stays below 10⁻⁴). For this experiment the engine gained two things: prescribed end rotations and displacements in geometrically nonlinear analysis, and analyses that continue from a converged state. We reach the first state the way the paper does: the clamps turn while B slides, then B is clamped and moved to sin 1, then the load is applied. After that the actuators move in steps of 0.02 rad. Every state is checked for stability: its tangent stiffness must be positive definite.
How Θ and Δ map to the end tangents was settled against the figures. Θ is half the difference of the tangents and Δ their sum: θ(A) = Θ + Δ/2, θ(B) = Δ/2 − Θ. With this reading the shapes of Figure 9(c) fall on the corners of the rectangle, where Figure 9(b) labels them; with Δ as the half sum they would fall halfway along its sides.
What the run shows
Every state along the path is stable and none jumps from its neighbour: the midpoint height goes smoothly from y(1/2) = 0.226 on the arch to −0.150 on the inverted shape, at the same clamp angles.
The rectangle has to go around the cusp, near (Θ, Δ) = (1.2, 1.2). With its far corner at Δ = 0.6 the strip comes back as an arch. With Θ = 1.05 the path crosses the fold where the arch stops existing, and no equilibrium is found nearby: the strip would snap there.
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Limitations
The model is planar, as in the paper. Unstable branches are not traced, because the production arc-length solver does not take prescribed rotations yet; the equilibrium surface of Figure 9(a) is not reproduced, and a fold shows up as the point where no nearby equilibrium exists. The co-rotational frame is slightly extensible and discretized. This is a numerical comparison with the paper’s figures and text, not a laboratory test.
Comparison with the paper
Every row compares one number. For the shapes of Figure 9(c) it is the midpoint height y(1/2), at the clamp angles where Figure 9(b) places each shape; the paper’s value is measured on its curve, digitized from the preprint, at half its length. The whole curves also lie on our path, within 0.002 to 0.005 of the strip length. The values of Section 3.2 are read from the text, which prints the limit loads as negative numbers; the table gives their size. ✓ marks agreement within 0.008 for heights and 5% for the other values.
| Quantity | Paper | Stabileo | Difference |
|---|---|---|---|
| Figure 9(c): shapes along the path | |||
| Shape 1: midpoint height y(1/2)Θ = 1.00 · Δ = 0.00 | 0.223 | 0.226 | +0.003 |
| Shape 2: midpoint height y(1/2)Θ = 1.30 · Δ = 0.00 | 0.188 | 0.189 | +0.001 |
| Shape 3: midpoint height y(1/2)Θ = 1.30 · Δ = 1.30 | 0.107 | 0.108 | +0.001 |
| Shape 4: midpoint height y(1/2)Θ = 1.00 · Δ = 1.30 | −0.076 | −0.065 | +0.011 |
| Shape 5: midpoint height y(1/2)Θ = 1.00 · Δ = 0.00 | −0.155 | −0.150 | +0.005 |
| Section 3.2: limit loads and Table 1 | |||
| Limit load, uniform loadΘ = 0.90 · Δ = 0.00 | ≈ 4.2 | 4.25 | +1.1 % |
| Limit load, uniform loadΘ = 0.90 · Δ = 0.04 | ≈ 3.9 | 3.95 | +1.3 % |
| Limit load, load on half the spanΘ = 0.90 · Δ = 0.00 | ≈ 2.5 | 2.37 | −5.4 % |
| Largest limit load, load on half the span, Θ = 0.66paper at Δ = 1.36, Stabileo at Δ = 1.16 | ≈ 4.0 | 3.74 | −6.5 % |
| Largest curvature at q = 2.5, uniform loadΘ = 0.96 · Δ = 0.00 | 2.10 | 2.13 | +1.5 % |
| Largest curvature at q = 2.5, load on half the spanΘ = 0.96 · Δ = 0.00 · reached after a snap | 7.60 | 14.07 | +85.2 % |
| Largest curvature at q = 2.5, uniform loadΘ = 0.64 · Δ = 1.16 · reached after a snap | 11.70 | 11.38 | −2.7 % |
| Largest curvature at q = 2.5, load on half the spanΘ = 0.64 · Δ = 1.16 | 3.35 | 3.76 | +12.3 % |
| Exact solution | |||
| Unloaded strip at Θ = 1largest distance to the circle of radius 1/2, over the radius | 0 (exact) | 3.4 × 10⁻⁵ | 3.4 × 10⁻⁵ |
The rows marked ✗ are open. Shape 4 sits just past the cusp, where the height changes fast with the clamp angles: our path reaches the paper’s height about 0.08 lower in Δ. With the load on half the span our limit loads come out 5 to 7% lower, and the bifurcation curve where the paper finds its optimum lies about 0.2 lower in Δ in our runs, which also puts the Table 1 point (0.64, 1.16) right on it. For the half-span load at (0.96, 0) the snap lands on a stable state with curvature 14.1, not the paper’s 7.6. The inverted shape at the end of the path, which matches the figure, has a largest curvature of about 10.6, above the threshold of 6 that the paper sets for the path. These are open questions for the authors.
Preprint (arXiv:2509.12916v1) · Journal version
Cite this
Stabileo Research, “Steering a bent strip between two shapes”, reproduction of Eszter Fehér, András Á. Sipos, Péter L. Várkonyi (2026), doi:10.1016/j.euromechsol.2025.106000. Implementation: research/stability/feher-sipos-varkonyi-2026 at commit cb671ed319d1. Engine: the Stabileo engine at the same commit.