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RESEARCH NOTE: DEFLECTION EQUATION FORTHE BUCKLING OF AN ELASTIC COLUMNSUBJ ECTED TO SURFACE PRESSURED. E. Newland*

The deflection equation for the buckling of an initially straight elastic column subjected to external or internalpressure is derived for the case when the pressure and the area of the column may vary along its length. Appar-

ently this equation has not been reported in the literature previously.

IntroductionTHE EFFECT of pressure on he buckling of elastic columnshas been studied for at least the last thirty years. Haringxconsidered the instability of thin-walled cylinders underinternal pressure ( x ) t in his analysis of the behaviour ofcorrugated bellows expansion joints under internal pres-sure. Nagel (2) and Peterson (3) were concerned withaerospace applications where slender columns are sub-jected to axial load and lateral pressure. Other authorshave considered the theoretical aspects (Handelman (4))and reported the results of experimental tests (Mills (5)).The deflection equation used by these authors (see, forinstance, equation (1) of (3)) is

d4Y d2YEZ--,+(P-pA), =0 . . (1)d# dxwhere y is the lateral deflection of the neutral axis of aninitially straight elastic column of constant external cross-sectional area A (symmetrical about the neutral plane) andbending stiffness EZ when subjected to an external (com-pressive) pressure p at section x and a central compressiveend load P. The same equation applies for a hollow tubesubjected to internal pressure provided that P is still thecompressive load applied directly to the ends of the tube,A is now the internal cross-sectional area, and p becomesthe hydrostatic tension (negative pressure) applied to theinternal walls. All but one of the authors considered onlythe case when the applied pressure is constant along thelength of the column. However, Handelman (4) considereda pressure distribution varying linearly from one end ofthe column to the other end.

The purpose of this note is to show that equation (1) isonly correct when the applied pressure and area are con-stant along the length of the column, and that, whenpressure and area vary the deflection equation should be

where P s now the axial compressive load in the columnat section x . The pressure p ( x ) is assumed to be a functionof x only so that surfaces of constant pressure are perpen-dicular to the axis of the straight undeflected column.This equation, which is derived below, is shown to pro-Th e MS. f this researc h note was received at the Institution on 14th

April 1972 and accepted fo r publication on 8th August 1 972. 2Professor, Department of Mechanical Engineering, University ofShefield, S t. Georges Square, Ma ppin Street, Sh&eld SI 3JD.Fellow of the Institution.t References are given at the end of this note.

duce results in agreement with those obtained by theauthor by another method when studying the effect ofvarying external pressure on the instability of a cantileverbeam (6). It is believed that the results given in Handel-mans paper (4) require amendment to take account of theterm --A(dp/dx)(dy/dx) which occurs in equation (2) butis absent from equation (1).Derivation of the deflection equationThe column will be assumed to be a slender member forwhich Eulers equation

relates the applied bending moment M at section x to the(approximate) curvature day/ka. Consider the equili-brium of a small element of the column of length dx (Fig.1). For convenience the column is assumed to have arectangular section of width b (constant) and depth 2h(x).The internal stresses acting on the cut sections of thecolumn element are represented by the resultant axialforce P , shear force V and moment M, as shown. The ex-ternally applied lateral pressure is taken to be locallyhydrostatic so that the pressure forces are always normalto the external surface. For equilibrium of the elementshown in Fig. 1, resolving forces axially,

dP dA* (4)d r = p d x - * *-

resolving forces laterally,dVdx=o . . . . * (5)

Deflected positiono f column element

I I IX x+dxFig. 1

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74 RESEARCH NOTEand taking moments about 0

(2 )YdM+P-dx-Vdx-pb -dx ( 2 h ) = Odxor

-+P--V-pA-=OM dy dY . . (6)dx dx dxDifferentiating equation (6) to eliminate the unknownshear force V gives

The dP/& term may be eliminated from equation (4), andusing equation (3) hen gives

dA dy-P---x d x - O

Finally, eliminating like terms of opposite sign, the de-flection equation becomes

as already stated.This analysis may be extended to the case of a columnof non-rectangular section by considering the equilibriumof a thin, parallel sided slice through the non-rectangularcolumn element, and then summing the results for all theslices of the element to verify equations (4)-(6). Alterna-tively, the following independent approach may be used.Alternative derivationIn (6) the deflection equation for a cantilever beam sub-jected to external pressure has been obtained directly inthe integrated form (equation (4) of (6))

X = P

wherey is the deflection at section x and r ] is the deflectionat section f (Fig. 2). Th e differential deflection equation(2) may also be obtained by differentiating this equationtwice.The standard formula (7) for differentiating an integralinvolving a variable parameter (in this case x) can be used,

is here a function of 8 so that its par-X - ltial derivative with respect to the limiting value x is zero.

According to the formula, differentiating equation (7)once gives

X - l

and then differentiating again gives

x - lFrom the theory given in reference (6), the integral inequation (8) may be related to the axial force at section x.A fluid column will only be in equilibrium if there is a

distributed body force A- x acting to the right (inFig. 2). The axial force in the column at section x is thenjust

P = p AHowever in the absence of such body forces, the axialcompressive force at x must be

( 2 )

P=pA+/:(A$)df . . ( 9 )x = l

in order to hold the column in place against the pressureforces pushing it to the left. This heuristic argument is ex-plained more fully in (6 ) and may be verified mathe-matically by the methods described in the reference.Substituting from equation (9 ) into equation (8) thengives

thus confirming the previous derivation.ExampleConsider the caseof a straight cantilever beam of constantcross-sectional area A and length 1 subjected to a pressurefield varying linearly from zero at the fixed end to po at thefree end (Fig. 3).

The pressure at section x is p = p o - and the axial force1P =poA, which, on substituting into (2), givesEZa+poA(I-7)G--- day POA dY - .4Y I dx-In (6) it has been shown that this problem is identicalwith the problem of a vertical flagpole deflecting under its

Fig. 2Journal Mechanical Engineering Science

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RESEARCH NOTE 75own weight when th e weight per u nit length of the pole isq =poA/Z. Th e deflection equation for a flagpole is, from(8)2

d3Y dYEZ--,+q(Z-X)- =0 . . (11)dx dxPutting q =poA/Z and differentiating equation (11) gives

thus providing an independent check of equation (10).REFERENCES

(I) HARINGX, J. A. The instability of thin-walled cylinders sub-jected to internal pressure, Philips Res. Rep. 1952 7 , 112.

(2) NAGEL,. E. Column instability of pressurized tubes, 3.Aerospace Sci. 1956 23, 608-609.(3) PETERSON,. P. Axially loaded column subjected to lateralpressure, A m . Inst. ueronuut. u s t r m u t .31. 1963 1, 1458.(4) WELMAN,. H. Buckling under locally hydrostaticpressure, 3. appl. Mech., Trans. Am . SOC .mech. Engrs 194668,198-200.(5) MILLS,B.D., JR. The fluid column, A m .3. Phys. 1960 28,353-356.(6)NEWLAND,. . Whirlingof a cantilever elastic shaft sub-jected to external pressure,J. mech. Engng Sci. 1972 14 (No.(7) HILDEBRAND,. B. Methods of applied mathematics 1952 383( 8 ) TIMOSHENKO,.P. nd GERE,. M. Theory of elastic stability

l), 11-18.(Prentice-Hall Inc. Englewood Cliffs, New Jersey).1961 101 (McGraw-Hill Book Co. Inc., New York).

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