-
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
rri2 mi
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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