Origami / geometry under load Independent research · Flexible-fold experiment 02
A fold gives a sheet height.
What lets it keep that height under a load?
A single ridge can look strong in a bending calculation, yet spread and flatten under an object. An accordion gives the material a different set of ways to move.
Here the creases can open, the panels can bend, and the loading base touches only the parts it actually meets. Press three arrangements of the same sheet and watch where the resistance comes from—and where it falls away.
O–02 / Recorded nonlinear equilibrium
Loading recorded equilibria…
The press moves downward by a prescribed distance; its reaction force is computed. This is not a falling weight or an impact. Playback interpolates between solved configurations. Dashed lines show the unloaded sheet; gold dots identify contact nodes. The base is shown translucent so the folds remain visible.
01 / The missing motion
The first version held each cross-section in shape. That made the deep single ridge look especially stiff. But it had removed precisely the motion you can see when a folded card spreads under a cup.
This version gives that motion back. Elastic springs resist rotation at the creases. Separate bending terms let each panel flex, while membrane elasticity resists stretching. A ridge is free to surrender its height instead of behaving like a permanently braced beam.
In these fixtures, the fine accordion resists the initial press more than the single ridge. Continue pressing: the force can peak and then fall as the shape flattens. The answer depends on the stage of loading.
Inspect the earlier fixed-fold benchmark →Three arrangements / same press travel
Each row uses the same scale. Gray: unloaded. Teal: deformed. Gold: underside of the loading base. Initial height differs between patterns.
02 / Follow the force, not the crease count
At the same downward travel, a larger reaction means more resistance in this test. It does not mean that the corresponding weight is a safe payload. A descending part of a curve can be followed by a controlled press even when it would not be followed smoothly by a freely resting weight.
The dots are solved states. The lines connect those samples; they do not add a separate fitted model.
— One crease— Three creases— Seven creases
| Arrangement | Reaction force | Crease rotation | Fold energy | Panel-bending energy |
|---|
Energies are in N·mm. Crease rotation is the largest change from the initial fold angle anywhere along the sheet. Values between saved states are interpolated and identified as such.
03 / What the solver is allowed to do
The sheet is triangulated. Each triangle can stretch and shear; neighboring triangles can rotate. Edges on prescribed creases use one stiffness, and edges inside panels use a separate bending stiffness. The solver minimizes elastic and contact energy at each press position.
Energy = membrane + crease rotation
+ panel bending + contact penalties
Equilibrium: free-node force residual ≈ 0
Two frictionless end supports prevent downward penetration but permit lifting and lateral spreading. The left end is held longitudinally; one center node at each end is guided laterally to remove drift. An 80 × 80 × 8 mm rigid block stays horizontal and contacts the sheet through its underside or sides.
Method reference: Liu & Paulino, nonlinear mechanics of non-rigid origami (2017). This is an independently implemented triangulated membrane-and-hinge model, not a reproduction of that paper’s calibrated examples.
These are reference parameters, not measurements of a particular paper. Small contact penetration is permitted numerically and measured in the verification record. No plastic crease damage, tearing, self-contact or material failure law is included.
04 / Ask the result to survive a harder test
Loading the verification record…
Independent finite differences check the energy gradient and assembled tangent matrix. Rigid-body rotations must leave internal energy unchanged. Contact must be non-adhesive and continuous at the block edge. The reaction at the supports must balance the load on the base.
Loading mesh-refinement results…
Loading crease-stiffness sensitivity…
Loading contact sensitivity…
05 / The experiment this prepares
The useful next measurement is a force–travel curve for actual folded sheets, with the paper direction, crease preparation, supports and base dimensions recorded. Measure the crease resistance separately, then ask whether the model predicts the deformation and the point where resistance begins to fall.
That is the research question I want to carry forward: which geometric relationships preserve a load path, and which allow the structure to move out of it?
The implemented model can express a mechanism the fixed-section calculation excluded. Its solutions can be inspected and checked for equilibrium. The sampled initial ordering is tested against numerical and parameter changes in the record.
It does not establish a universal best fold pattern, a safe cup weight or a physical breaking load. Self-weight, paper anisotropy, crease plasticity and manufacturing imperfections still need to enter a calibrated experiment.
06 / Open the record
Independent computational exploration · 26 October 2026 · Flexible-fold experiment 02