LOCAL BUCKLING OF TUBES IN ELASTIC CONTINUUM

By James A. Cheney,1 Fellow, ASCE

ABSTRACT: A theory for the local buckling of a buried, flexible, perfect tube that utilizes linear buckling theory and an elastic continuum model for the ground is developed, with only radially inward displacement permitted. Comparisons are made with the Winkler spring support model for local buckling and multiwave solutions of both Winkler and continuum support. It is evident that the accuracy of predic-tions depends upon the knowledge of the localized behavior of the surrounding soil. In this solution the spring constant is taken as a function of the mode number in buckling. A graphical procedure for solving for the eigenvalues is presented. The solution represents an upper bound on local buckling of buried flexible tubes that may also be affected by imperfections in geometry and residual internal stresses.

INTRODUCTION

In a previous paper (Cheney 1971), the writer showed that an eigenvalue solution exists for local buckling of a long tube surrounded by soil and loaded by external pressure with the postulate that outward displacement is prohib-ited. The assumption was made that the soil behaved like a Winkler foun-dation, having a radial spring constant of wall support, k. This makes the support of the soil a linear function of the tube local displacement, u.

The normal stress on a cylindrical wall in an elastic continuum owing to a sinusoidal displacement, however, is distributed sinusoidally and the ratio of stress to displacement is a function of the wavelength of the sinusoidal displacement (Cheney 1976). This approach leads to a different critical pres-sure than that arrived at by the Winkler assumption and appears to predict the correct trend in the experimental data (Moore 1989).

In a discussion of the paper (Moore 1989), it was pointed out that a theory for local buckling (single wave), using an elastic continuum model for the surrounding soil similar to that used for the multiwave buckling theories, is needed. The purpose of this paper is to provide such a solution in the context of the previous paper of the writer (Cheney 1971), wherein only inward radial displacement of the tube is permitted. This assumption leads to a buck-ling instability that does not require initial imperfections or prebuckling de-formation in order to occur.

Derivation of Basic Equations The assumption is made that, prior to buckling, the wall (soil) moves

inward with the tube and prohibits all but inward deflections upon buckling. As the buckle forms locally, the intergranular soil pressure will decrease until the limit of the active state is reached. Although the force-deflection relationship is nonlinear, an effective radial spring may be assumed, which approximates the force-deflection relationship for small inward displace-ments. In the previous derivation (Cheney 1971), a constant spring constant

'Prof, of Civ. Engrg., Dept. of Civ. Engrg., Univ. of California, Davis, CA 95616. Note. Discussion open until June 1, 1991. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on October 25, 1989. This paper is part of the Journal of Engineering Mechanics, Vol. 117, No. 1, January, 1991. ASCE, ISSN O733-9399/91/00Ol-O2O5/$l.OO + $.15 per page. Paper No. 25409.

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FIG. 1. Local Buckling of Soil-Surrounded Tube

kx was postulated. If the surrounding medium is assumed to be an elastic continuum, a much more accurate approximation can be made for the sup-port provided by the soil.

The equations of equilibrium governing the buckling of a circular arch or ring may be derived by the use of the principle of minimum potential energy. The arch is considered to be deformed by a radial displacement prior to buckling under the action of the external pressure. The energy of distortion into a buckled shape is then minimized to determine the equations governing equilibrium in the buckled state.

The following coordinates will be used: (1) Coordinate x measured in the plane of the cross section, radially inward from the centroid; (2) coordinate y normal to the plane of the ring; and (3) coordinate z = RQ along the cen-troidal axis of the arch. As usual, these coordinates participate in the de-formations in such a manner that the xy plane is always normal to the de-formed arch or ring axis.

The displacements, described in Fig. 1, are described by the following displacement components: (1) u and v, which are, respectively, in the x- and y-directions of the undeformed arch or ring; and (2) w, which is a curvilinear displacement along the z-axis. For the problem at hand, displacements in the y-direction will be precluded, therefore, v = 0, and u, and w are taken as the additional displacements during buckling.

The energy expression is given by the sum of the internal strain energy from extension and bending of the centroidal axis minus the work done by the external pressure and boundary forces and moments. The writer derived this expression in a previous publication (Cheney 1963) to be

V = - \ \ EAel + EIK2 - - l(u')2 - u2] + kxu2 \RdQ

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M Nzw + Nxu + (w + u')

*2

CD

in which E = Young's modulus; A = cross-sectional area of ring or arch; / = moment of inertia of cross section; R = radius of ring; Nx, N2 = force stress resultants at the boundaries in x- and z-directions, respectively. M = moment stress resultant at the boundaries in the y-direction; q = external pressure; kx = radial spring constant of wall support; 4>i, 2 = values of 8 at boundaries ( ) ' = d{ )/dQ and

1 e0 = - (vc' - K)

R (2)

K = (" + U) R2

(3)

The minimization of the potential energy is accomplished by setting the first variation of V equal to zero, then applying the fundamental lemma of the calculus of variations (Hildebrand 1958) to obtain two equilibrium equa-tions and three natural boundary conditions. The first variation yields

r*2 q

EAeoSeo + 7K8K (u'hu' ubu) + kxubu RdQ R AV

hi M

Nzbw + Nxu + (Sw + 8K') R

2

= 0 (4)

Substitution of Eqs. 2 and 3 and successive use of integration by parts results in

8V "/{ ~EA EI . q (vc' - u) + (uw + 2u" + u) + - (u" + u) + kxu R R R

EA (W - ')

EI R1

8vv }Rd(l + EI M (u" + u) R3 R

bu'

- ('" + u') - qu' - Nx 8K + EA M (w' u) Nz R R

8vv

8K

*2

(5)

The coefficients of the variationals inside the integral must be zero and comprise the equilibrium conditions. The coefficient of 8vv set to zero yields

(w" - K') = 0 . EA R

and the coefficient of 8M yields

EA EI q (w' - K) + ("" + 2K" + K) + - (" + K) + kxu = 0 R R

(6)

(V)

The appropriate boundary conditions are that at the juncture between buck-led and unbuckled zones, i.e., 6 = cj> K = 0 (8)

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u' = 0 .

u" = 0 . (9) (10)

w' + ix (j>

= 0 (11)

Eq. 11 comes from the continuity of stress at 6 = . The differential equation and boundary conditions define an eigenvalue

problem, which will be solved herein by a graphical procedure. The sym-metric solution of the differential equation is

u = C\ + C2 cos mfi + C3 cos m26 (12) C2 C3

w = C49 H sin mfi H sin m26 (13) nil m2

wherein the mode numbers m, and m2 are obtained by solution of the char-acteristic equation, obtained by substituting Eqs. 12 and 13 into Eq. 7, i.e.

EA EI q - (C4 - C.) + - C, + I C, + *XC,

+ c, (w, - 2m, + 1) - - (m, - 1) + ^ cos m,9 L4

" /

To satisfy the differential equations for all admissible constants of integration

+ C3 ( ; - 2m| + 1) - ~ (m22 -l) + kx cos m26 = 0 , (14)

I qR kxR' C4 = C, | 1 + - + + - i -

AR2 EA EA and

/ a (m* - 2m] + 1) - - (m2 ~ 1) + K = 0 j = 1 or 2 i?

(15)

(16)

To this point in the derivation there is no difference between a Winkler spring assumption or a continuum assumption. The difference will lie in the choice of kx.

Choice of Soil Constant kx In the Winkler spring constant approach, one may obtain a spring constant

by solving the elastic equations for a uniform radial displacement of a cir-cular boundary in an infinite elastic medium. Taking the stress function (Ti-moshenko and Goodier 1951)

d2 _ A

dr2 ~ ~?' ffe = -7T = - ~ (1 9)

wherein a r = radial stress; CT8 = tangential stress; and r = radius. The radial strain is given by

1 A (1 + vs) er = (a r - v sa9) = (20)

Es Es r2

wherein Es = soil modulus; vs = soil Poisson's ratio. The radial displacement may be obtained by

f " A ( l + v . ) . A(l + v.) = ; dr = (21)

JR ES r2 ESR

T h e spring constant

V Es kXo = - = ; (22)

u R(l + v.) If kXo is taken as the value for the spring constant k, then the Winkler

spring constant solution for local buckling can be obtained (Cheney 1971). In the present formulation, kXo is appropriate for the coefficient of CI in Eq. 14, which corresponds to a uniform radial displacement, but it is not ap-propriate for a sinusoidal displacement associated with cosine terms in Eq. 14. For those forms a stress function

(C D\ $ = + r cos mO (23) V" r"-1/ should be used (Timoshenko and Goodier 1951).

The stresses are

1 rf4> 1 d2$ ar = + - (24)

r dr r2 362

a2i>

at r = . Substitution of Eq. 23 into Eqs. 24 and 26, then into Eqs. 27 and 28, yields

ov = -Cm(m + 1) D(m - l)(m + 2)

/ ? m + 2 /r . Cm(ffl + 1) Dm(m - \)

COS OT0 = B COS OT0

/?" sin m6 = 0 .

Solving

BRm+2 C =

D =

Thus

2(m + 1) BR'"

2(m - 1)

(30)

(31)

(32)

(33)

Vr = B\ (m + 2 / T m R"

go! EI

2,500

2,000 -

1,500

'i\ V V v ^

ESR3 - p = 10,000-^,

ESR3

EI

// tl . fj.

/

rri2 mi

= 10,000

FIG. 3. m2/m, versus m, for Two Values of Parameter ESR3/EI

C2m\ cos W) +

sin mify

+ C- cos m2 + sin m2

m 2 ( i r - ). 0

(45)

(46)

Successive elimination of constants of integration lead to two simultaneous equations in mu m2, and , i.e. mx cot mx = m2 c o t w2 (47) mx cot Wit)) 1 + mjair cot m ^ = 0 (48) wherein a = I/AR2(l + qR3/EI + kXoR4/El) (Cheney 1971).

Note that in Eq. 48, k is taken as the estimated spring constant for a uniform radial displacement kXc (Eq. 22), because it is associated with the constant C, in the displacement equation (Eq. 12).

The values of m,/m2, which satisfy Eq. 47, are obtained by plotting m cot /rt. For every value of Wi chosen, a corresponding value of m2 may be found that satisfies the condition of Eq. 47. The ratio m2/m1 = m2/ mx is plotted in Fig. 4 versus m^. The eigenvalues, however, must also satisfy Eq. 48, which may be rearranged to read

1 mi4> c o t Wi ir cot mi (49)

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~1

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mi

FIG. 4. m2/m! versus m,c|>

The right-hand side of Eq. 49 may be plotted as a function of m,4> as shown in Fig. 5. For each mfy in the plot, m2/ml can be taken from Fig. 4, then W] is obtained from Fig. 3. To calculate a, the term qR3/EI is obtained from Fig. 2 for each value of mx and combined with the other terms that come from given values for ESR3/EI, R/t and vs. The crossing of the two curves yields the critical value of m^, which can be traced through the plots to Fig. 2 to give the critical buckling load in terms of qR3/EI.

mia

3 = 10,M_ = 1,ooo

1 EI

5 = 100, ^ L = 10,000 -t EI

Critical m-i

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

mi

FIG. 5. Graphical Solution for Critical /H,4>

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Examples Two examples are given here, one for the case where ESR3/EI = 1,000

and R/t = 10 and the other where ESR3/EI = 10,000 and R/t = 100. For the first case various values of /n,(f> are assumed and a and ml is determined from the graphs. By plotting w,((> as , in Fig. 5, a critical value of m, of 1.45 is read. This corresponds to a critical ratio m2/mi = 3.2 and a critical m, = 3.2 from which qR3'/EI = 148. The wave length of buckle is $ = 26.

For the second case a similar plot on Fig. 5 yields a critical m^ = 1.25, this leads to m2/ml = 3.8 and mx = 6.3. This mx corresponds to qR3/EI = 760 and 4> = 10.9.

Comparison with Winkler Medium The Winkler medium is characterized by a spring constant that is not a

function of the wavelength of the buckled form. For comparison with these examples, Eq. 22 is used here for the Winkler spring constant. The char-acteristic equation for the eigenvalues in the Winkler case becomes (Cheney 1971) qR3 , ESR3 1 2 _ =

m 2 _ j + ; ( 5 0 ) EI 7(1 + v,) m2 - 1

For comparison with the continuum medium formulation, Eq. 50 is plotted with dashed lines in Fig. 2. For the lowest values of m (i.e., m = 2) the Winkler solution is greater than the continuum solution, but for all other values of m the continuum solution yields higher values of pressure param-eter qR1/EI. The actual local buckling pressure parameter depends upon the ratio I/AR2 of the tube also, as indicated by the graphical solution.

It also should be noted that, as in the case of the continuum elastic sup-port, the Winkler support local buckling parameter qR3 /EI is always greater than the multiwave solution, given the same elastic modulus Es. Thus, if the soil stiffness to inward displacement is the same as that for outward dis-placement, the multiwave solution gives the lowest value of loading param-eter qR3/EI and is the critical condition. However, the radial stiffnesses in the two directions are not always the same. Quite often the soil surrounding a tube is in a state of plastic equilibrium for which further load results in a stiffness that is much smaller than the stiffness upon unloading.

It has been shown (Cheney 1989) that a radially outward displacement on the surrounding soil can constitute an unloading of the shear stresses in the soil immediately surrounding the tube, and therefore yields a very large ef-fective stiffness. On the other hand, a radially inward displacement of the soil boundary produces an increase in shear stress in the soil, which is equiv-alent to an increase in loading, and results in a lower effective stiffness.

An approximation of this state is to assume the outward radial stiffness to be infinite, or, equivalently, permit only radially inward displacement, as formulated in this paper.

CONCLUSION

The proposed theory for local buckling of a tube subjected to external soil pressure fills a gap in the solutions to the buckling of a soil-surrounded tube

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that represents the soil to be acting as an elastic continuum. Although local buckling solutions utilizing a Winkler-type stiffness are available in the lit-erature, a similar development for local buckling with an elastic continuum support is not. It is found that the critical buckling load for local buckling parameter qR3/EI is sensitive to the R/t ratio of the tube as well as the stiffness parameter ESR3 /EI, and no single plot of qR1/EI versus ESR3/EI can be drawn. The solution represents an upper bound of the buckling of soil-surrounded tubes that may also be affected by local imperfections in the tube walls and internal residual states of stress.

APPENDIX I. REFERENCES

Timoshenko, S., and Goodier, J. N. (1951). Theory of elasticity, Second Ed., McGraw-Hill Book Co., New York, N.Y., 116, 55-60.

Moore, J. D. (1989). "Elastic buckling of buried flexible tubesA review of theory and experiment." J. Geotech. Engrg., ASCE, 115(3), 340-358.

Cheney, J. P. (1963). "Bending and buckling of thin-walled open section rings." J. Engrg. Mech., ASCE, 89(5), 17-44.

Cheney, J. A. (1971). "Buckling of soil-surrounded tubes." J. Engrg. Mech., ASCE, 97(4), 1121-1132.

Cheney, J. A. (1976). "Buckling of thin-walled cylindrical shells in soil." Supple-mentary Report 204, Transp. Res. Lab., Crowthorne, Berkshire, England.

Cheney, J. A. (1989). Discussion of "Elastic buckling of buried flexible tubesA review of theory and experiment." by J. D. Moore, J. Geotech. Engrg., 115(3), ASCE (in press).

Hildebrand, F. B. (1958). Methods in applied mechanics. Prentice-Hall, Inc., En-glewood Cliffs, N.J.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A C E Es I

K M

ml,m2 Nx N2 q R r

t u

V V w X

y z

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

cross-sectional area; arbitrary constants; Young's modulus of tube material; equivalent Young's modulus of soil; moment of inertia; radial spring constant of soil support; moment about y-axis; mode numbers; shear stress resultant; hoop stress resultant; external pressure; radius of tube; radius; thickness of tube; radial displacement (positive inward); potential energy; longitudinal displacement; circumferential displacement; radial coordinate; longitudinal coordinate; circumferential coordinate;

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a = parameter Eq. 48; 8 = variational; e = strain; 6 = circumferential angular coordinate; vs = Poisson's ratio of soil; = angular coordinate at the buckle boundary;

a ' = effective soil stress; and

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