M026 · Solid mechanics

Column Buckling & Imperfection

Load a pinned rectangular column with controlled crookedness and eccentricity. Compare linear amplification with a separate displacement-controlled elastica branch, then infer the critical load from raw force and lateral readings.

Euler critical load $\text{Ready}$

Applied compression $\text{Ready}$

Peak lateral offset $\text{Ready}$

Maximum absolute fibre stress $\text{Ready}$

Geometric slenderness $\text{Ready}$

Maximum centreline slope $\text{Ready}$

End separation $\text{Ready}$

Bending strain energy $\text{Ready}$

Elastica arc closure $\text{Ready}$

Critical load from Southwell record $\text{Ready}$

Acquired stations $\text{Ready}$

Model state $\text{Ready}$

GEOMETRY / HISTORY / RAW RECORD

What fails first: shape or material?

Column length

L

1.00\,m

Section breadth

b

30.00\,mm

Bending thickness

h

3.00\,mm

Young modulus

E

70.00\,GPa

Elastic stress limit

\sigma_y

250.00\,MPa

Initial sine amplitude

a_0

2.00\,mm

Load line eccentricity

e

0.00\,mm

Compression to Euler load ratio

P/P_{cr}

0.95\,

Elastica end shortening fraction

\Delta/L

0.04\,

Branch: linear or perfect elastica

s

0.00\,

Elastica quadrature intervals

N

128.00\,

Lateral gauge station fraction

s_p/L

0.50\,

Sensor perturbation gain

\epsilon_n

0.00\,

Both ends rotate freely and stay on a common vertical chord. Bending is through the thickness. The blue solid is the current rectangular column; grey is its unloaded shape. All lengths share one metric scale, with no deflection magnification. The gold contact gauge selects a material station. Load arrows share one display gain.

Linear analysis superposes a sinusoidal initial crookedness and the secant solution for equal eccentric end loads. It neglects axial shortening and stops acquisition at the Euler load, slope above 0.2, stress above the selected elastic limit, or slenderness below forty. Invalid shapes are candidates, not certified equilibria.

The perfect elastica branch prescribes end shortening and evaluates elliptic integrals. It ignores initial crookedness, eccentricity, axial stretch and shear; it does not extend the imperfect linear branch through its singularity. The displayed positive first branch is selected; no dynamic snap or failure trajectory is simulated.

Acquire a subcritical sweep at one gauge station. Southwell fitting uses measured additional deflection divided by measured force versus additional deflection, with a free offset. It is available only for nonzero sine crookedness and zero eccentricity. Noise bounds per gain are 0.02 newtons and one micrometre; no confidence interval is claimed.

Slender elastic columns; linear predictions stop at excessive slope, yield or the Euler singularity. The perfect inextensible elastica is a separate branch, not an imperfect post-buckling solver.

Start with the essentials

Focus question: Can strength alone prevent buckling?
One-sentence intuition: Follow the same physical state through apparatus, readouts and the energy or stability record.

Core mathematical model

Euler load

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

The two pinned ends select the first half-sine mode. Geometric slenderness and elastic stress must support this idealization.

Imperfection amplification

\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 is a linear small-slope solution below the critical load, not a post-buckling trajectory.

Eccentric 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 eccentric end forces create end moments. The secant contribution superposes with the sine imperfection.

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}

First pinned inextensible branch, parameterized by prescribed end shortening. Complete elliptic integrals define its load and geometry.

Southwell inference

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

Fit a free offset and slope using raw additional displacement and force at one unchanged station; the reciprocal slope estimates the critical load.

Common difficulties

Model boundary

Typical misconceptionA smooth rendered curve proves physical validity.

Better mental modelUse explicit domain checks and acquired records; the teaching laws do not certify real specimens.

Run the experiment

01
Amplify a tiny defect
Increase compression with Small initial crookedness; inspect the side view.
What to observe: The unloaded shape is grey; the loaded offset grows before the Euler load.
02
Separate eccentricity
Choose eccentric compression, then acquire a sweep.
What to observe: An eccentric curve is measurable, but the sine-only Southwell estimate is withheld.
03
Cross a model boundary
Choose Beyond Euler load or a thicker, shorter column.
What to observe: Acquisition stops rather than hiding a singularity, yield or stocky-column limit.
04
Measure and compare branches
Acquire a sine-defect sweep; then select shortening-controlled elastica and refine integration.
What to observe: The raw regression estimates a critical force; the separate elastica branch closes its original arc length.