Euler's Buckling Formula: Complete Derivation from the Differential Equation of the Elastic Curve

Structural Mechanics · Elasticity

Euler's Buckling Formula
Complete Derivation

From the differential equation of the elastic curve to the critical load — every step, every assumption, fully explained.

Structural Engineering 20 min read Intermediate – Advanced

When a slender column is loaded in compression, it doesn't simply crush — at a critical load it suddenly bows sideways in a phenomenon called buckling. Leonhard Euler derived the exact formula for this critical load in 1744, and it remains one of the most elegant results in classical mechanics.

This post walks through the complete mathematical derivation starting from the differential equation of the elastic curve (the Euler-Bernoulli beam equation). No steps are skipped. By the end you will not only know the formula but understand every assumption baked into it.

§1 What Is Buckling?

Buckling is a stability failure, not a strength failure. A perfectly straight column carrying a purely compressive axial load $P$ is in equilibrium — but is that equilibrium stable? At loads below the critical value $P_{cr}$, any small lateral disturbance is restored and the column springs back. At $P = P_{cr}$, the column can remain in a slightly bent configuration — it has reached a bifurcation point.

Fig. 1 — Three stages of a pin-ended column under increasing axial compression.

§2 Assumptions of Euler's Theory

The derivation is exact only within the following idealisations:

• Perfectly straight column — no initial imperfections or eccentricity in loading.
• Homogeneous, isotropic, linearly elastic material — Hooke's Law holds; Young's modulus $E$ is constant throughout.
• Uniform cross-section — the second moment of area $I$ does not vary along the length.
• Axial load only — the load $P$ acts exactly through the centroid; no lateral forces or moments.
• Small deflections — allows linearisation: $\sin\theta \approx \tan\theta \approx \theta$, so curvature $\kappa \approx d^2y/dx^2$.
• Shear deformation neglected — the Euler-Bernoulli beam model (plane sections remain plane).
• Self-weight ignored — the column weight is negligible compared to $P$.

📐#128208; Important Note

These assumptions define the regime of validity. Real columns deviate — especially short/stubby columns where inelastic effects dominate (see Johnson's formula) and columns with imperfections (see Southwell plot).

§3 The Differential Equation of the Elastic Curve

The starting point is the Euler-Bernoulli moment-curvature relationship. For a beam in bending, the internal bending moment $M$ at any cross-section is related to the curvature $\kappa$ by:

$$M = EI \kappa = EI \frac{d^2 y}{dx^2}$$ (3.1)

where $y(x)$ is the lateral deflection, $x$ is measured along the column axis, $E$ is Young's modulus, and $I$ is the second moment of area about the axis of bending.

The small-deflection approximation $\kappa \approx d^2y/dx^2$ (discarding the exact curvature denominator $[1+(dy/dx)^2]^{3/2}$) is the key linearisation that makes the equation analytically tractable.

Fig. 2 — Free body diagram of the lower portion of a pin-ended column. The internal moment at the cut balances $P \cdot y$.

§4 Setting Up the Boundary Value Problem

Consider a pin-ended (simply supported) column of length $L$, loaded by an axial compressive force $P$ at both ends. Let the $x$-axis run along the undeformed column and $y(x)$ denote lateral deflection.

We isolate the lower portion of the column up to a cross-section at height $x$. Taking moments about the cut, the internal bending moment is:

$$M(x) = -P \cdot y(x)$$ (4.1)

The negative sign arises because the compressive load $P$ creates a restoring moment that is in the opposite sense to positive curvature convention. Substituting into Eq. (3.1):

$$EI \frac{d^2 y}{dx^2} = -P \cdot y$$ (4.2)

Rearranging and introducing the parameter $\lambda^2 = P / (EI)$:

$$\frac{d^2 y}{dx^2} + \lambda^2 y = 0 \qquad \text{where} \quad \lambda^2 = \frac{P}{EI}$$ (4.3)

🔍#128269; Recognition

Equation (4.3) is the simple harmonic oscillator ODE — one of the most important differential equations in physics and engineering. Its general solution is well known.

The boundary conditions for a pin-pin column are:

$$y(0) = 0 \qquad \text{and} \qquad y(L) = 0$$ (4.4)

Both ends are pinned — they cannot deflect laterally but are free to rotate. This is the simplest (and most fundamental) end condition.

§5 Solving the ODE

The general solution to $y'' + \lambda^2 y = 0$ is:

$$y(x) = A \sin(\lambda x) + B \cos(\lambda x)$$ (5.1)

where $A$ and $B$ are constants determined by the boundary conditions.

1

Apply BC at $x = 0$: $$y(0) = A\sin(0) + B\cos(0) = B = 0$$ Therefore $B = 0$.

2

Apply BC at $x = L$: With $B = 0$, $$y(L) = A\sin(\lambda L) = 0$$

Now there are two possibilities:

• Trivial solution: $A = 0$, giving $y(x) = 0$ everywhere — the column stays straight. This is always a valid equilibrium but tells us nothing about buckling.
• Non-trivial solution: $A \neq 0$, which requires $\sin(\lambda L) = 0$.

The non-trivial condition is:

$$\sin(\lambda L) = 0 \implies \lambda L = n\pi \quad \text{for } n = 1, 2, 3, \ldots$$ (5.2)

Therefore:

$$\lambda = \frac{n\pi}{L} \implies \lambda^2 = \frac{n^2\pi^2}{L^2}$$ (5.3)

§6 Finding the Critical Load

Recall $\lambda^2 = P/(EI)$. Substituting Eq. (5.3):

$$\frac{P}{EI} = \frac{n^2\pi^2}{L^2} \implies P_n = \frac{n^2 \pi^2 EI}{L^2}$$ (6.1)

This gives an infinite set of buckling loads corresponding to different mode shapes $y(x) = A\sin(n\pi x/L)$. However, the column will buckle at the lowest load it encounters, which corresponds to $n = 1$:

✦ Euler's Critical Buckling Load (Pin-Pin Column) ✦

The Euler Load

$$P_{cr} = \frac{\pi^2 E I}{L^2}$$

The corresponding buckled shape (the mode shape or eigenfunction) is a half sine wave:

$$y(x) = A \sin\!\left(\frac{\pi x}{L}\right)$$ (6.2)

📌#128204; Key Observation

Notice that the amplitude $A$ is indeterminate — the linear theory predicts the load at which buckling occurs but not the post-buckling deflection magnitude. A non-linear (large-deflection) analysis is needed to find the actual deflected shape after buckling.

Fig. 3 — Animated buckling mode shapes for $n = 1, 2, 3$. Only $n = 1$ is reachable in practice (lowest critical load).

§7 Effective Length and End Conditions

The derivation above assumed pin-pin end conditions. For other boundary conditions the shape of the buckled mode differs, but the governing ODE is the same. The elegant solution is to replace $L$ with an effective length $L_e = K \cdot L$, where $K$ is the effective length factor.

The generalised Euler formula becomes:

✦ General Euler's Formula ✦

Generalised for Any End Condition

$$P_{cr} = \frac{\pi^2 E I}{(KL)^2} = \frac{\pi^2 E I}{L_e^2}$$

End Conditions	K (Theoretical)	K (Design — conservative)	Physical Meaning
Pin – Pin	1.0	1.0	One half-wave; both ends rotate freely
Fixed – Fixed	0.5	0.65	Both ends fully restrained; inflection pts at L/4 and 3L/4
Fixed – Pin	0.699	0.80	One end fully fixed, one pinned
Fixed – Free (cantilever)	2.0	2.10	Quarter wave; free end displaces freely
Pin – Fixed (sway)	1.0	1.20	Sway permitted at top

💡#128161; Insight

A fixed-fixed column buckles at four times the load of a pin-pin column of the same length. Fixing the ends doubles the effective stiffness in both directions.

Fig. 4 — Buckled shapes and effective lengths for four common end conditions. The dashed orange bracket shows the equivalent pin-pin length $L_e$.

§8 Critical Stress and the Slenderness Ratio

It is more useful to express the result as a critical compressive stress $\sigma_{cr}$ by dividing $P_{cr}$ by the cross-sectional area $A$:

$$\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E I}{(KL)^2 A}$$ (8.1)

The second moment of area $I$ is related to the area $A$ through the radius of gyration $r$:

$$I = A r^2 \implies r = \sqrt{I/A}$$ (8.2)

Substituting:

✦ Euler Critical Stress ✦

In Terms of Slenderness Ratio

$$\sigma_{cr} = \frac{\pi^2 E}{\left(\dfrac{KL}{r}\right)^2}$$

The dimensionless group $\lambda_s = KL/r$ is the slenderness ratio — arguably the most important parameter in column design. A high slenderness ratio means the column is long and thin (elastic buckling governs); a low slenderness ratio means short and squat (material yielding governs).

Fig. 5 — Column curve: critical stress vs slenderness ratio. Euler's formula governs only in the elastic (high slenderness) region.

§9 Limitations of Euler's Formula

Euler's formula underestimates the actual capacity of short/intermediate columns. The formula is invalid when $\sigma_{cr}$ computed from it exceeds the material's proportional limit $\sigma_{pl}$. The limiting slenderness ratio below which Euler is inapplicable is:

$$\left(\frac{KL}{r}\right)_{\min} = \pi\sqrt{\frac{E}{\sigma_{pl}}}$$ (9.1)

For structural steel ($E = 200\text{ GPa}$, $\sigma_y \approx 250\text{ MPa}$), this gives $KL/r \approx 89$. Columns with $KL/r < 89$ require inelastic buckling theories:

• Johnson's Parabolic Formula — widely used in machine design for intermediate columns.
• Tangent Modulus Theory (Engesser, 1889) — replaces $E$ with the tangent modulus $E_t$ for inelastic behaviour.
• Reduced Modulus Theory (von Kármán) — uses a double-modulus accounting for the elastic-plastic stress distribution.
• Shanley's Theory (1947) — reconciled the above two, showing the tangent modulus load is the true bifurcation load.

⚠#9888; Real-World Factors Not Captured

Initial geometric imperfections, eccentric loading, residual stresses from fabrication, creep at elevated temperatures, and local buckling of thin-walled sections all reduce the actual capacity below $P_{cr}$. Design codes incorporate these via column curves derived from extensive testing.

§10 Derivation Summary

1

Moment at cut: $M(x) = -Py(x)$ from equilibrium of the free body.

2

Beam equation: $EI\,y'' = M = -Py$ → the governing ODE is $y'' + \lambda^2 y = 0$, with $\lambda^2 = P/EI$.

3

General solution: $y = A\sin(\lambda x) + B\cos(\lambda x)$.

4

Boundary conditions: $y(0) = 0 \Rightarrow B = 0$; $y(L) = 0 \Rightarrow A\sin(\lambda L) = 0$.

5

Non-trivial solution: $\lambda L = n\pi$, $n = 1, 2, 3, \ldots$

6

Critical load ($n=1$): $P_{cr} = \pi^2EI/L^2$.

✅#9989; Final Result

The Euler buckling load for a pin-ended column of length $L$, Young's modulus $E$, and second moment of area $I$ about the weak axis is: $$P_{cr} = \frac{\pi^2 EI}{L^2}$$ Generalised for any end condition: $P_{cr} = \dfrac{\pi^2 EI}{(KL)^2}$, and in terms of critical stress: $\sigma_{cr} = \dfrac{\pi^2 E}{(KL/r)^2}$.

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"It is not enough that the mathematician should be able to calculate; he must also be able to think." — Leonhard Euler

Tags: Structural Mechanics · Column Buckling · Elasticity · Differential Equations · Euler-Bernoulli Beam