---
title: "Arch Snap-Through Interactive Lab | Sandbox Physics"
description: "Why does a buckled strip jump to its mirror shape, and what sets how fast it goes?"
canonical_url: "https://sandboxphysics.com/labs/arch-snap-through/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/arch-snap-through/"
last_modified: "2026-10-31"
---

Geometric elasticity · Snap-through

# Arch Snap-Through

**Why does a buckled strip jump to its mirror shape, and what sets how fast it goes?** A thin strip clamped at both ends and pushed together buckles into an arch with two stable shapes, arched up and arched down. Push its middle down and the force first rises, then falls. Once holding the strip would take a pull, no balanced shape is left nearby, so it jumps to the inverted arch. A deep arch twists into an S shape before it jumps; a shallow one is pressed flat first.

Push the middle of a buckled strip clamped at both ends. The force rises, peaks and falls; when the blade would have to pull, the strip snaps to the inverted arch and rings. The force curve is set against the continuous Föppl–von Kármán solution, and the snap and the ringing against linear stability.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The strip follows the dynamic beam equation with midline stretching of Pandey, Moulton, Vella and Holmes (EPL, 2014), in their units: lengths in the strip length, forces in the bending stiffness over the length squared, and time in the bending time of the strip. It is 48 finite-difference segments with clamped, flat ends and a fixed end shortening. The blade is a frictionless point contact that can only push. The push is quasi-static: each frame is the equilibrium with the midpoint at the blade, and when the blade would have to pull, the strip is released into the dynamic equation. The uniform viscous damping is not in the paper; it stands in for air and material losses. Gravity, twist, the width of the strip and plasticity are left out. Deflections on the canvas are stretched vertically by the factor shown there.

Orange is the push from above and blue the push from below, both computed on the strip; cream dashes are the continuous solution and the predicted growth.

More numbers and the saved record

Largest force of the continuous solution $16.6\,B/L^{2}$

Largest force of the push from below $16.5\,B/L^{2}$

Growth rate from linear stability, with damping $21.2/t^{*}$

Antisymmetric vibration of the arch, with damping $44.1/t^{*}$

Symmetric vibration of the arch, with damping $59.5/t^{*}$

Rise of the unloaded arch $0.0891\,L$

Midpoint deflection when the strip leaves the blade $-2.19\times10^{-4}\,L$

Relative energy drift of a free vibration without damping $5.05\times10^{-6}$

Time step of the dynamic equation $1.29\times10^{-4}\,t^{*}$

The JSON record holds the parameters, both simulated pushes with their release, snap and ringing, and, kept apart, the continuous force curve, the linear-stability rates and the published reference values.

[Course: how thin things bend, buckle and snap](https://sandboxphysics.com/stories/geometric-elasticity/)

Physics tutorial

## How does a buckled strip snap through?

**Background** Hair clips, jumping-popper toys, the Venus flytrap and many switches use one shell or strip with two stable shapes. Pushing it from one shape to the other ends in a fast jump called snap-through.

**Why it matters** The blade moves slowly, yet the strip moves fast. The jump is not caused by pushing harder: it starts where the force has already fallen to zero.

Start with the essentials

- **Focus question**: Why does a buckled strip jump to its mirror shape, and what sets how fast it goes?
- **One-sentence intuition**: Under displacement control the strip follows a branch of balanced shapes until that branch stops being stable. There the nearest stable shape is the inverted arch, and the strip falls into it at a rate set by the unstable mode of the last shape it held.

### Core mathematical model

#### Dynamic beam with midline stretching

$$
\frac{\partial^{2}w}{\partial t^{2}}+\gamma\,\frac{\partial w}{\partial t}+\frac{\partial^{4}w}{\partial x^{4}}+\tau^{2}\,\frac{\partial^{2}w}{\partial x^{2}}=f\,\delta(x),\qquad -\tfrac{1}{2}<x<\tfrac{1}{2}
$$

Pandey et al. (2014), Eq. 1, with the damping $\gamma$ added here. Lengths are in $L$, forces in $B/L^{2}$ and time in $t^{*}=L^{2}(\rho_{\mathrm{s}}hb/B)^{1/2}$. The clamps hold $w=0$ and $\partial w/\partial x=0$ at both ends.

#### Compression from the end shortening

$$
S\,\tau^{2}=d-\frac{1}{2}\int_{-1/2}^{1/2}\Bigl(\frac{\partial w}{\partial x}\Bigr)^{2}dx,\qquad S=\frac{h^{2}}{12L^{2}}
$$

Hooke's law for the midline (Pandey et al., Eq. 2) integrated along the strip. The stretchability $S$ compares bending with stretching; the ratio $d/S$ alone decides the shape of the response.

#### Three regimes of the confinement

$$
\Bigl(\frac{d}{S}\Bigr)_{\mathrm{Euler}}=4\pi^{2}\approx 39.48,\qquad \Bigl(\frac{d}{S}\Bigr)_{\mathrm{twist}}=\tau_{\max}^{2}\approx 80.76
$$

Below the Euler value the strip stays straight. Between the two values the arch is pressed flat before it snaps; above the second it twists into an S shape first. $\tau_{\max}\approx 8.99$ solves $\tan(\tau/2)=\tau/2$ and is the load of the first antisymmetric clamped mode.

#### Force on the twisted branch

$$
f=-207.75\,w_{0},\qquad f_{\mathrm{c}}=-129.53\,\bigl(d-80.76\,S\bigr)^{1/2}
$$

Pandey et al., Eq. 3: once the strip twists, the force falls in proportion to the midpoint deflection and reaches zero when the midpoint does. Solving the same equations again here gives 207.745 and 129.530.

#### How fast the snap grows

$$
\lambda=-\frac{\gamma}{2}+\sqrt{\frac{\gamma^{2}}{4}+\sigma^{2}},\qquad \sigma\approx 24.11\ \text{from the twisted state}
$$

A small departure $a\,e^{\lambda t}$ from the last shape the blade held obeys $\ddot{a}+\gamma\dot{a}=\sigma^{2}a$. Pandey et al. find $\sigma\approx 24.113$ for the twisted state; this strip of 48 segments gives 24.04.

### Common difficulties

#### The largest force is not where it snaps

**Typical misconception** The strip snaps when the blade pushes hardest.

**Better mental model** With the blade position controlled, the strip passes the force peak and keeps a balanced shape. It snaps only where the force falls to zero, because beyond that point the blade would have to pull. Under a fixed dead load the jump would instead start at the force peak.

#### The speed is not just the wave speed

**Typical misconception** A snap is as fast as sound crossing the strip.

**Better mental model** The time scale is the bending time of the strip, and the rate within it comes from the unstable mode of the last balanced shape. Close to the buckling threshold that mode is weak, so the jump is slow.

#### What the model leaves out

**Typical misconception** A real plastic strip behaves exactly like this one.

**Better mental model** The strip follows the dynamic beam equation with midline stretching of Pandey, Moulton, Vella and Holmes (EPL, 2014), in their units: lengths in the strip length, forces in the bending stiffness over the length squared, and time in the bending time of the strip. It is 48 finite-difference segments with clamped, flat ends and a fixed end shortening. The blade is a frictionless point contact that can only push. The push is quasi-static: each frame is the equilibrium with the midpoint at the blade, and when the blade would have to pull, the strip is released into the dynamic equation. The uniform viscous damping is not in the paper; it stands in for air and material losses. Gravity, twist, the width of the strip and plasticity are left out. Deflections on the canvas are stretched vertically by the factor shown there.

### Run the experiment

1. 01

   #### Watch a deep arch twist

   Start from Deep arch twists, then snaps and watch the first push.

   **What to observe:** The force rises to about 16.5, then the arch leans into an S shape and the force falls in a straight line to zero as the midpoint reaches the clamps' level. The strip then snaps, with a growth rate of 21.1 against 21.2 from linear stability.

2. 02

   #### Make the arch shallow

   Choose Shallow arch flattens, then snaps, where the confinement is 60.

   **What to observe:** The arch stays symmetric and is pressed flat. The peak force is only 1.95 and the snap starts from the flat strip, growing at 13.9 against 14.0 predicted.

3. 03

   #### Get close to buckling

   Choose Barely buckled: a slow snap, with the confinement at 45, just above the Euler value of about 39.5.

   **What to observe:** The jump grows at only 7.18, a third of the deep arch. Near the threshold the flat state is barely unstable, so the strip leaves it slowly.

4. 04

   #### Let it ring

   Choose Little damping: it rings.

   **What to observe:** After the jump the inverted arch rings for many cycles. The measured angular frequency, 46.1, sits close to the 44.2 of the first antisymmetric vibration mode of the arch.

5. 05

   #### Push it yourself

   Drag the top blade down slowly, then drag the lower blade up quickly.

   **What to observe:** Your push draws its own yellow force curve over the computed one. Whatever the speed, the strip leaves the blade and jumps once the force reaches zero.

Physical conclusion

A bistable strip under displacement control snaps where its balanced branch loses stability, and the unstable mode of that last shape sets how fast it goes.

## Related experiments and articles

Geometric elasticity · Reading path

### Geometric elasticity: how thin things bend, buckle and snap

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a thin sheet bends instead of stretching**](https://sandboxphysics.com/stories/geometric-elasticity/why-thin-sheets-bend-instead-of-stretch/)
2. [Lab **Non-Euclidean Plate** Grow a flat disc into a dome](https://sandboxphysics.com/labs/non-euclidean-plate/)
3. [Article **A clamped strip buckles into an arch at the Euler load**](https://sandboxphysics.com/stories/geometric-elasticity/euler-buckling-of-a-clamped-strip/)
4. [Article **Pushing an arch until it snaps**](https://sandboxphysics.com/stories/geometric-elasticity/pushing-an-arch-until-it-snaps/)
5. [Lab **Kresling Origami Cell** Press a folded Kresling tube](https://sandboxphysics.com/labs/kresling-origami/)
6. [Article **How fast a snap happens**](https://sandboxphysics.com/stories/geometric-elasticity/how-fast-a-snap-happens/)
7. [Article **Reading the snapping-beams paper**](https://sandboxphysics.com/stories/geometric-elasticity/reading-the-snapping-beams-paper/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)