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Euler's famous equation for critical buckling load is rather cumbersome to derive, relying as

it does on solving a second order differential equation of the form:

EI(d2y / dx

2) + Py = 0.

The simplified and approximate derivation proposed here is not intended to replace Euler'sbrilliant work, but rather is offered as a supplementary analysis that may lend some insight

into the behavior of axially-loaded columns. To anticipate the results: one can get within 3%

of Euler's critical buckling load for a pin-ended column of length, L, simply by dividing the

mid-span moment of a uniformly-loaded simply-supported beam of the same length by its

mid-span deflection. Here's why.

Buckling occurs when a second state of equilibrium becomes possible for a given load on a

column; i.e., when in addition to the obvious condition (Fig. 1-a) of axial loading, we get a

second possible state of equilibrium (Fig. 1-b) with a deflected, or buckled, shape.

Fig. 1 Alternate states of equilibrium: (a) for straight axially-loaded column; and (b) forbuckled column.

The basis for this second possible state of equilibrium is that the resisting moments in the

deflected column are consistent with the moments caused by the axial load in relation to the

deflected shape. One way of looking at this is to imagine that the column is first pulled into a

deflected shape by some supernatural power (or just by our powers of imagination: see Fig.

2).

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Fig. 2 Column pulled into a deflected position (with apologies to Michelangelo).

The column, having been stretched and compressed elastically, would attempt to regain(spring back to) its initial undeflected shape were it not for our ability to temporarily hold it

in its deflected position. Now, if we quickly replace our own "supernatural" force with an

axial force on the column, but leave the deflected shape as it was, we generate a bending

moment due to the misalignment of the applied axial force with the internal axial forces (Fig.

3). If this moment due to the axial force is smaller than the moment caused by our initial

(supernatural or imaginary) misalignmentsmaller because the axial load, P, is not big

enough to match the resisting momentequilibrium cannot be maintained and the column

will spring back elastically to its original (straight) position. However, if the moments caused

by the applied axial load are consistent with those moments originally applied by the prior

(supernatural) loading, then equilibrium is maintained, and the column can be said to have

buckled.

Fig. 3 Moment caused by deflection of axially-loaded column.

The axial load, P, that establishes this equilibrium is the critical buckling load, since any

small misalignment will move the column inexorably from the ideal axial condition (Fig. 1-

A) to the buckled state (Fig. 1-B). Based on the approximate method outlined here (as well as

Euler's), this load is consistent with any deflection, so that the safety of such a column would

be immediately threatened when that load is approached.

Fig. 4 Column uniformly loaded: (a) like a simply-supported beam; and (b) free-body

diagram of column cut at mid-span.

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To calculate the critical buckling load, Pcr, the axial load, P, is replaced with a uniformly-

distributed load, w, acting along the column's length as if it were a vertical beam (Fig. 4-a).

Under this loading, the mid-span moment and deflection are as follows (Fig. 4-b):*

This deflected shape is then held in place while the uniformly-distributed load, w, is removedand "quickly" replaced with an axial load, P, just big enough to maintain the mid-span

deflection that we had under uniformly-distributed loading. Now, if we approximate the

critical buckling load by assuming that the shape of the buckled column under axial loading

is the same shape taken by the "beam" under transverse uniform loadingdefined by a

fourth-degree polynomial or quartic equationwe get a value that is extremely close to the

Euler buckling load (within 1/10 of 1%). However, this method,** while approximate, is not

particularly easy.

Instead, our approximate method uses the well-known mid-span moment and deflectionvalues for a uniformly-loaded, simply-supported beam to compute the critical buckling load

of the column. It should be noted that such an approximation presumes a critical load,

moment, and deflection for which no curved shape exists. In other words, there is no curve

whose strain energy of bending is consistent with the work done by a critical buckling load

whose magnitude is less than the value determined by Euler. As Timoshenko and Gere point

out in their Theory of Elastic Stability, any curve used to approximate the critical buckling

load must result in a value greater than the value determined when the sine curve of the actual

buckled shape is used. Looked at another way, if Euler's critical buckling load causes a mid-

height deflection equal to that of the column viewed as a simply-supported beam with a

uniform (transverse) load, w, its mid-height moment would be somewhat greater than that

associated with the uniform load, w. This can be seen (Fig. 5) by examining the rotationalequilibrium of a free-body diagram on which Euler's critical load buckles a column with the

uniformly-loaded beam's mid-span deflection of 5wL4

/ (384EI). Solving for the moment, we

get M = P x = wL2

/ 7.78, a value slightly larger than the beam's moment under uniform

load, wL2

/ 8. Reducing this "buckling moment" below the value that triggers bucklingas

is the case for our approximate methodmeans that buckling cannot actually occur and

therefore no approximate curve can be found consistent with internal strain energy and

external work.

Fig. 5 Free-body diagram for buckled column.

That being said, we can still find the approximate buckling load corresponding to these

"beam" values, even if no compatible deflected shape can be found. As before, rotationalequilibrium for the critical buckling load (Fig. 3) is defined by Pcr= M / .

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Substituting the familiar "beam" values for M and (Fig. 4-b) into this equation, we get:

This is quite close (and slightly conservative) compared to Euler's equation:

Because all applied loads have cancelled out of these equations, the deflection associated

with Pcr is indeterminate: in other words, since any deflection is consistent with this critical

buckling load, the safety of such a column is fatally compromised once this load is reached.

It should be noted that even Euler's famous equation is not exact. His exact equation for

buckling, rarely used in practice, shows that a point of catastrophic and sudden bucklingnever actually occurs.*** Rather, equilibrium may be maintained with increasing loads

corresponding to increasing deflections (beyond the critical buckling load of the approximate

equation), up until the point where the material's ultimate strength is exceeded and failure

occurs under combined axial and bending stresses. However, deflections do increase quite

rapidly once the approximate critical buckling load is reached, so that these approximate

equations serve as an entirely rational indicators of the "point of no return" for the initiation

of buckling.

* The mid-span moment can be found by examining the rotational equilibrium of the free-

body diagram shown in Fig. 5. The derivation of the deflection equation for a uniformly-

loaded, simply-supported beam is somewhat more complex, but still easier than for a buckled

column. The general strategy, worked out in detail in Timoshenko, Strength of Materials,

Part I, Van Nostrand Reinhold Co., New York, 1958, pp. 140-141, starts with the basic

moment-deflection (y-axis) relationship, i.e., EI d2y/dx

2= -M; and the basic moment-distance

(x-axis) relationship, i.e., M = (wLx - wx2)/2. Substituting the latter value for M into the first

equation, and integrating twice (with the constants of integration found based on our

knowledge of mid-span slope being zero and deflection at the supports being zero), we can

find the deflection, y, at mid-span.

** The so-called "energy method" involves equating the strain energy of bending to the work

done by the axial load as it moves vertically through a distance resulting from the curved(buckled) shape of the column.

*** For a discussion of Euler's elastica, the deflected shape of an axially-loaded column

based on exact differential equations, see Timoshenko and Gere, Theory of Elastic Stability,

McGraw-Hill, New York, 1961.

First posted 29 February 2008. Last updated: 2 April 2010