---
title: "Column Buckling & Imperfection Interactive Lab | Sandbox Physics"
description: "Does a loaded column fail first by buckling or by its material yielding? A slender column usually buckles first."
canonical_url: "https://sandboxphysics.com/labs/column-buckling/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/column-buckling/"
last_modified: "2026-10-27"
---

M026 · Solid mechanics

# Column Buckling & Imperfection

**Does a loaded column fail first by buckling or by its material yielding?** A slender column usually buckles first. Its buckling load depends on stiffness and length, not strength. As the load approaches it, any slight bend or off-center load grows quickly, so even a very strong material can buckle before it yields.

Compress a pin-ended rectangular column that starts slightly bent or is loaded off center. See how linear theory makes the bend grow, compare it with a perfect column bent by pushing its ends together, and estimate the critical load from the measured force and sideways deflection.

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

The columns are slender and elastic. The linear prediction stops when the slope gets too large, when the material would yield, or at the Euler load, where it becomes singular. The perfect, inextensible elastica is computed separately; it does not follow an imperfect column past buckling.

Physics tutorial

## Strength, crookedness and stability

**Background** A loading frame squeezes a column from both ends. The Lab ties the column's geometry to a stability model and records force and sideways deflection as separate measurements.

**Why it matters** Loading, unloading and careful control of the geometry let you tell competing explanations apart.

Start with the essentials

- **Focus question**: Can strength alone prevent buckling?
- **One-sentence intuition**: The picture of the column, the readouts and the energy or stability record all describe the same physical state.

### Core mathematical model

#### Euler load

$$
\begin{aligned}I&=\frac{bh^3}{12}\\P_{cr}&=\frac{\pi^2EI}{L^2}\end{aligned}
$$

With both ends pinned, the column buckles in its first mode, half a sine wave. The formula holds only if the column is slender enough and the stress stays elastic.

#### How an initial bend is amplified

$$
\begin{aligned}y_0(s)&=a_0\sin\frac{\pi s}{L}\\y(s)&=\frac{y_0(s)}{1-P/P_{cr}}\end{aligned}
$$

This linear solution assumes small slopes and holds only below the critical load. It does not describe the column after buckling.

#### Deflection under an off-center load

$$
y_e(s)=e\left[\frac{\cos\!\left(\sqrt{P/EI}(s-L/2)\right)}{\cos\!\left(\sqrt{P/EI}L/2\right)}-1\right]
$$

Equal off-center forces at both ends create bending moments at the ends. This secant deflection adds to the deflection caused by the sine-shaped initial bend.

#### Load and shortening of a perfect elastica

$$
\begin{aligned}P&=\frac{4EI\,K(k)^2}{L^2}\\\frac{\Delta}{L}&=2\left(1-\frac{\mathcal E(k)}{K(k)}\right)\end{aligned}
$$

This is the first buckled shape of a pinned, inextensible column, labeled by how far its ends are pushed together. Complete elliptic integrals give the load and the shape.

#### Southwell estimate of the critical load

$$
\begin{aligned}v&=y-y_0\\\frac{v}{P}&=\frac{v+a_0\sin(\pi s_p/L)}{P_{cr}}\end{aligned}
$$

At one fixed measuring point, fit a line with a free intercept to the measured extra deflection and force. One over the slope estimates the critical load.

### Common difficulties

#### Where the model stops working

**Typical misconception** If the curve on screen looks smooth, the physics must be right.

**Better mental model** Check whether the state lies inside the range where the model holds, and look at the recorded data. These teaching models do not certify any real specimen.

### Run the experiment

1. 01

   #### Watch a tiny bend grow

   Choose Small initial crookedness, increase the compression and look at the side view.

   **What to observe:** The grey line is the unloaded shape. The sideways offset of the loaded column grows even before the Euler load is reached.

2. 02

   #### Try an off-center load

   Choose Eccentric compression, then acquire a sweep.

   **What to observe:** You can still measure the curve, but no Southwell estimate appears, because that method assumes a pure sine-shaped initial bend.

3. 03

   #### Push past the model's limits

   Choose Beyond Euler load, or make the column thicker and shorter.

   **What to observe:** Acquisition stops, because the state has reached the Euler singularity, yielding, or the limit for short, stocky columns.

4. 04

   #### Measure, then compare with the elastica

   Acquire a sweep with a sine-shaped initial bend. Then choose Shortening-controlled elastica and refine the integration.

   **What to observe:** The fit to the measured data estimates the critical force. For the elastica, the computed arc length comes back to the column's original length.

Physical conclusion

A model solution, the range where it holds, and an estimate from measured data each answer a different question.

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)