rrl · 2016-10-02 · the soil and results in the soil's supporting action being represented...
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TRANSPORT AND ROAD RESEARCH LABORATORY Department of Transport R R L
Cont rac to r Repor t 229
Bur ied f lex ib le p ipes: 2 The ana ly t ica l me thod deve loped by G u m b e l for T R R L
by G N Smith (Consulting Engineer)
The work reported herein was carded out under a contract placed on G N Smith (Consulting Engineer) by the Transport and Road Research Laboratory. The research customer for this work is Highways Engineering Division, DTp.
This report, like others in the series, is reproduced with the author's own text and illustrations. No attempt has been made to prepare a standardised format or style of presentation.
Copyright Controller of HMSO 1991. The views expressed in this Report are not necessarily those of the Department of Transport. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.
Ground Engineering Division Structures Group Transport and Road Research Laboratory Old Wokingham Road Crowthorne, Berkshire RG11 6AU
1991
ISSN 0266-7045
Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on 1 st April 1996.
This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposesl provided the source is acknowledged.
CONTENTS Page
I. Introduction
2. The design model
2.1 Basic assumptions 2.2 Non linearity 2.3 Anisotropy 2.4 Elasticity 2.5 Advantage of a continuum analysis 2.6 Effects of longitudinal bending
3. The design problem
v 3.1 Performance criteria 3.2 External loads applied to the system
4. Static response of the pipe ring
4.1 Notation and sign convention 4.2 General description of system response
5. Evaluation of the system response to Pz and py
5.1 Structural properties of the system 5.2 First order response to Pz 5.3 First order response to py 5.4 The ranges of system behaviour 5.5 Second order response to Pz 5.6 Ring bending moment 5.7 Ring buckling 5.8 Yield buckling interaction
8
8 9
ii 12 13 13 13 14
6. Simplification of theoretical expressions for design 15
6.1 Interface slippage assumptions 15 6.2 Treatment of load boundary conditions 16 6.3 Treatment of the value of Poisson's ratio for soil 17 6.4 Effects of hoop compressibility 18
7. Preparation of Deflection/Buckling charts 19
7.1 Dimensionless design equations 7.2 Description of a typical design chart
19 20
8. Acknowledgements 23
9. References 24
Appendix I The arching factor = 26
Appendix II The distortional thrust and deflection 28 coefficients B and
Appendix III The second order coefficients ~y2 and Ny2 31
Appendix IV A new formula for buckling pressure, Pb 33
Appendix V
Figures
Preparation of deflection/buckling charts 37
40
NOTATION
A
D E,Ep E' m'' Ep* Es Es* F H I K My N Ny Nyz Ny2 Nz R Ri Sc Sf Y Z
Cross sectional area of pipe wall per unit length, normal atmospheric pressure Outside diameter of pipe Young's modulus of pipe material Spangler's modulus of soil ( = ks/R ) Specific stiffness of pipe per unit length ( = EI/D 3) Plane strain modulus of pipe material Young's modulus of soil, Modulus of soil reaction Plane strain modulus of soil Factor of safety against buckling Depth of cover to crown of pipe Moment of inertia of pipe wall per unit length ( = t3/12) Lateral pressure ratio Maximum value of bending moment in wall Uniform hoop thrust in pipe wall created by Pz Equilibrium value of distortional hoop thrust (Ny + Ny2 ) Peak value of hoop thrust created by py Second order component of hoop thrust due to Pz and py Mean hoop thrust (uniform component of N) External pipe radius Internal pipe radius Compression stiffness of pipe ring Flexural stiffness of pipe ring Flexural stiffness ratio Compression stiffness ratio
fb
f c fy ks n n¢ P Pb Pv,Ph
P y , P z r r m t u , v w
Mean compressive hoop stress that will cause buckling of pipe wall Critical hoop stress Yield stress of pipe wall material Soil spring constant for radial ring displacements Buckling mode (number of sinusoidal waves) Critical buckling mode External pressure on pipe Value of external pressure on pipe to cause buckling Vertical and horizontal components of pressure acting on the pipe-soil system Distortional and uniform pressure components of Ph and Pv Radial co-ordinate Maximum radius of curvature (occurs at crown and invert). Pipe wall thickness Radial and tangential displacements of elastic soil medium Radial displacement of pipe wall
ii
V
Q..
a,, z B Bfs
l~ns
~a ~v,~h ~y ~yl , ~y2
e ~= ~2
vp vp * Vs Vs*
P
Arching factor Uniform thrust coefficient Distortional thrust coefficient Theoretical value of B assuming full slippage at pipe-soil interface. Theoretical value of B assuming no slippage at pipe-soil interface Ring deflection (relative change of pipe diameter) Allowable maximum value of 5 Relative shortening of vertical and horizontal diameters Distortional (out-of-round) component of ring deflection First and second order elastic components of ~y Uniform component of ring deflection created by P= Angular co-ordinate (measured from horizontal pipe axis) Load magnification factor Reduction factor on buckling pressure due to yield- buckling interaction Poisson's ratio of pipe material Modifed plane strain Poisson's ratio for the pipe material Poisson's ratio of surrounding soil Modified plane strain Poisson's ratio for the soil Elastic deflection coefficient Out-of-roundness correction factor in buckling formula
iii
BURIED FLEXIBLE PIPES
THE ANALYTICAL METHOD DEVELOPED BY GUMBEL FOR TRRL
1 INTRODUCTION
This Contractor's Report is the second in a series of four reports
that consider the modern design of buried flexible pipes. The first
report (Smith and Young, 1991) discusses the need for a new design
approach and describes how TRRL commissioned Mott Hay and Anderson,
Consulting Engineers, in 1977 to examine the situation regarding
buried flexible pipe design and to attempt to develop a more
rational approach. Due to this initiative a design method for
buried flexible pipes has been evolved by Gumbel (1983).
The possible design applications of Gumbel's method have already
been discussed by Gumbel and Wilson (1981) and by Gumbel, O'Reilly,
Lake and Carder (1982). The intention of this present report is to
describe, as simply as possible, the proposed design method and to
present the theory involved in its development.
2 THE DESIGN MODEL
2,,~ Basic assumDtions
The analytical model used in the development of the method is that
of a long thin-walled pipe embedded in a uniform isotropic soil
that reacts with it in a linear elastic manner to form a composite
structural unit. Both the external loading and the static response
of the pipe ring are assumed to be two-dimensional, acting
symmetrically about the vertical and horizontal axes in the plane
of the cross-section of the pipe; the cover to the pipe crown, H,
is taken as not less than the diameter of the pipe, D.
As will be illustrated, for soil-pipe systems with H ~ D, this
basic idealisation is valid in practice for diameter/thickness
ratios, D/t, greater than 20 and can be used to predict the first
and second order reponses of different pipe systems from which
design charts can be prepared.
As it is well known that the stress-strain behaviour of a soil is
neither linear, isotropic nor elastic, the suitability of the
proposed model for practical buried pipe design will first be
examined.
2,2 Non-lin~arity
The effect of soil non-linearity can be assessed by use of the
finite element method. Non-linear stress-strain relationships are
allowed for in the model by assuming that the load is applied in a
series of small increments. For each increment of strain the
corresponding different values of the soil tangent modulus are
inserted into the calculations. See, for example, Smith (1971).
However, with the use of a simple relationship between the soil
modulus and the mean vertical effective stress, Katona (1978)
showed that the predicted response of a buried pipe in a non-linear
backfill is of the same order as that predicted by a linear model.
Katona also showed that, if the fill is built up in layers, a
prediction using the value of the secant modulus corresponding to
the mean fill height is very similar to that obtained when non-
linearity is allowed for. This result has been confirmed from good
quality field data by Chang et al. (1980) who concluded that, in
view of the uncertainties involved, a linear model is as good as
any other for the practical design of a buried flexible pipe.
2,3 AnisotroDv
In the buried pipe literature dealing with the structural response
2
of soil there is little information on the effect of anisotropy. In
an anisotropic soil any difference in soil stiffness between the
vertical direction and the horizontal direction must have some
effect on pipe distortion but it is unlikely that this effect could
ever be distinguished from distortional effects caused by the
inevitable variability of the backfill surrounding the pipe. A
qualitative analysis of the effect of anisotropy can be obtained
from a finite element approach but it was considered, in the model
analysis, that any anisotropic effects would be more than allowed
for in the value of lateral loading assumed to act on the pipe.
2.4 Elasticitv
To assume that a material acts in an elastic manner is to assume
that strain energy is conserved over a loading cycle. This is
particularly relevant to modern buckling theories which consider
the strain energy stored in the displacements of the pipe-soil
interface. A possible solution is to assume different values of
radial stiffnesses for loading and unloading but, when this
procedure is compared with a true plasticity model as provided by
critical state theory (Schofield & Wroth, 1968), it is seen to be
both crude and arbitrary. Bearing in mind that even the behaviour
of a simple elastic model is not yet completely understood, the
introduction of further complications appear to offer little
advantage.
The true test of the validity of the assumption of elastic
behaviour is whether it provides a reasonable qualitative
description of the structural response for which elastic soil
parameters can be defined. Possible limitations in the use of
elastic theory for the predicted behaviour of buried flexible pipes
will be examined in further reports.
2,~ Th@ advantaqe Qf ~ GQn~in~um ~n~ivsis
British design methods are based on either the approach by
Marston-Spangler or by Barnard and involve the assumption that the
behaviour of the soil surrounding the pipe can be modelled by a
series of springs (see Smith and Young, 1991). Such a discrete
spring model considers the pipe to be the main structural component
and therefore becomes concerned with the form of the distribution
of the soil pressure around the pipe's perimeter. A solution to
this statically indeterminate problem is not possible and the
designer is forced to assume a distribution, a procedure that is a
major potential for inaccuracy in present design methods. The
problem does not arise with continuum models which allow the soil
and the pipe to be analysed as a composite structure.
Although the spring analogy has been used successfully in the past
for the preparation of safe, but possibly uneconomical, foundation
designs it is now generally recognised that it is a poor physical
model and can give rise to erroneous results (Institution of
Structural Engineers, 1978).
The spring analogy neglects the effect of shearing action within
the soil and results in the soil's supporting action being
represented by a single independent parameter, either ks, the soil
stiffness, or E$, the modulus of soil reaction.
Possibly the main disadvantage in the use of a continuum theory in
soil mechanics can be the necessity to evaluate vs, Poisson's ratio
for the soil. As will be demonstrated this problem is of little
significance in the proposed approach which, by using a systematic
definition of external loads and working with plane strain elastic
parameters, considerably reduces the sensitivity of the predicted
response of the pipe ring to the value of v s .
4
2,~ The ~ Qf iQnqitudinal bendinQ
The plane strain analyses on which the new design method has been
based all assume that the deformations that occur in the cross
section of the pipe are due solely to lateral effects.
However longitudinal effects can be important, as will be obvious
to anyone who has seen a small diameter pipe being reeled off a
drum where, in the length of suspended pipe, there is an obvious
risk that a kink may develop resulting in snap-through buckling of
such severity that the pipe may even become closed.
The risk is much less with larger diameter pipes and if, as will
usually be the case, the pipe is constructed in sections on firm
ground then the possibility of longitudinal bending and the
associated buckling defects will tend to be almost negligible.
Nevertheless the risk of longitudinal bending is always present
and, in soft ground, any possible pipe buckling effects caused by
longitudinal bending should be guarded against. The procedure
adopted in the proposed method is to determine a suitable diameter
and wall thickness for the pipe by considering plane strain effects
only and to then check that the chosen section can withstand any
possible detrimental effects caused by longitudinal bending.
3 THE DESIGN PROBLEM
9,1 Performance ~riteria
With the possible exception of restrictions on the surface
settlement of the backfill, all performance criteria relate to the
response of the pipe ring, Gumbel (1983). These are:-
Ring deflection : In this report the ring deflection, ~, is defined as the diametral strain, i.e. the relative change of the pipe diameter. The value of ring deflection must be limited to some allowable value, ~a, usually related to serviceability considerations.
Ring buckling : defined as a reversal of curvature or snap- through at any point on the circumference of the pipe due to the action of the ring compressive stress. Buckling is treated as a failure condition whether or not actual collapse of the pipe occurs. To prevent failure the compressive stress must be limited to a suitable value.
Ring strength : Overstressing of the pipe wall due to combined hoop thrust and bending moment is another potential failure condition although it is usually less critical than deflection or buckling in thin-wall flexible buried pipes.
Because of the interactions between deflection, buckling and yield
of the pipe wall the above three criteria must generally be
considered together.
~,2 Ex~@rnal loads aDDlied to th@ system
The total external loading applied to the model is represented by
uniform vertical and horizontal pressures Pv and Ph (= KPv ) where
K = the lateral pressure ratio (See Fig. !). For deeply buried
pipes Pv and Pb are equivalent to the free-field soil total
stresses at pipe mid-height, i.e. the stresses which would
prevail in the ground at the same level if no pipe were present.
The make-up of Pv and Ph due to backfill weight, ground water
pressure, uniform and concentrated surcharges may be calculated by
the standard techniques used in soil stress analysis as outlined
in Smith and Young (1991) and reference should be made to section
3.2.5 of the report for a description of the method usually
adopted to allow for ground water effects.
Fig. 2 shows how Pv and Ph can be split into two components: a
uniform, or average pressure, Pz, equal to 0.5(p v + Ph) and a
distortional or out-of-balance pressure, p¥, equal to 0.5(p v - ph)-
The significance of this division stems from Gumble's assertion
that pipe ring deflection is caused primarily by the action of p¥
6
whereas buckling depends upon the value of the mean hoop thrust,
which is governed by the action of Pz.
4 STATIC RESPONSE OF THE PIPE RING
4,1 NQtation and $iqn convention
The notation adopted for pipe ring response is illustrated in Fig.3
which adopts the sign convention that compressive hoop stress,
hogging moments and inward displacements are positive.
4.2 General ~@scriDtion of system ~@sDonse
The first and second order responses of the pipe to the actions of
Pz and py are illustrated in Fig. 4. The uniform compressive
pressure, Pz, when acting alone, produces a uniform hoop thrust,
denoted by Nz, and a uniform inward displacement, w, resulting in a
uniform ring deflection ~z = 2w/D = w/R where R is the initial
outside radius. No bending is involved. The distortional pressure,
py, when acting alone produces bending of the pipe wall resulting
in an out-of-round deflection 6y,cos20 and a hoop thrust of value
Ny,cos20. (The suffix 'i' denotes first order responses)
When Pz and py act together the values of both the hoop thrust and
the pipe deflection are affected by second order distortional
increments. This is because py deforms the pipe ring from its
original circular shape and the deformed shape, when acted upon by
Pz, results in second order changes, Ny 2 in the compressive hoop
thrust and ~y2 in the pipe deflection. These second order
components combine with Ny I and 6y, to give the final equilibrium
values of hoop thrust and pipe deflection, Ny and ~y, caused by the
combined action of Pz and py.
Additional distortional terms can be caused by initial out-of-
roundness and long-term deformations of the pipe and these will be
7
considered later. At this stage it is assumed that the final
expressions for the ring deflection, ~, the hoop thrust, N, and the
bending moment, M, have the general form:-
Final value = Component due to Pz + component due to py
= ~z + ~yCOS2e ........ (i)
N = N z + NyCOS20 ........ (2)
M = MyCOS2e ............. (3)
5 EVALUATION OF THE SYSTEM RESPONSE TO Pz AND py
The definitions given above for external loads do not specify
where they will be applied to the system. In order to include for
backfill weight, ground water and surcharges two alternative
boundary conditions are considered, viz:
Case 1 : Loads applied at a distant soil boundary.
Case 2 : Loads applied at the pipe-soil interface.
These conditions are illustrated in Fig.5.
Closed-form solutions for the response of the system have been
prepared by various authors, notably Flugge (1962), Burns and
Richards (1964) and Hoeg (1968). Gumbel (1983) prepared simplified
derivations using plane-strain coefficients. In order to use any of
the above solutions the structural properties of the system must
first be established.
$,1 Structural DroDerties Qf ~h~ svstem
The Young's Modulus and Poisson's ratio for the pipe and the soil
are Ep, vp, E s and v s respectively. As plane strain conditions are
assumed in the analysis it is convenient to adopt the approach of
Duns and Butterfield (1971) and to convert the foregoing parameters
into two-dimensional elastic constants, as set out below.
Plane strain modulus for the pipe material, Ep* = Ep/(l-vp 2)
8
Plane strain modulus for the soil, E s = Es/(l-vs 2)
Modified plane strain Poisson's ratio for the pipe material:
vp* = Vp/(l-vp)
Modified plane strain Poisson's ratio for the soil:
vs* = vs/(l-v,)
The following extra elastic parameters are also required:-
Compression stiffness of pipe ring, S c = Ep*A
D
Flexural stiffness of pipe ring, Sf = Ep*I
D 3
Where A, I and D are respectively the cross-sectional area of the
pipe wall per unit length, the moment of inertia of the wall per
unit length (= t3/12) and the outside diameter of the pipe.
It is interesting to compare the expression for flexural stiffness
with that for pipe specific stiffness (E'' = EI/D 3) which is given
in the first report of this series (Smith and Young, 1991).
The relationships between the parameters of the soil and the pipe
can be expressed by two ratios, referred to as the pipe-soil
interaction parameters:-
Flexural stiffness ratio, Y = Es*/S f
Compression stiffness ratio, Z = Es*/S ¢
The algebraic expressions for the uniform and distortional response
components that immediately follow have been developed in the
first instance by drawing from the elastic theory solutions
prepared by the various workers referred to above. Except where
otherwise stated the expressions quoted are for load case i.
2,2 First order /_e~Ip_Q~ to uniform ~ Pz
As discussed the symbols N z and ~z are respectively used to denote
the uniform compressive hoop thrust and the uniform ring deflection
9
created by Pz- Hence:-
Uniform hoop thrust, Nz, = =pz R
Uniform pipe ring deflection, 5z,
. . . . . . . . . . (4)
= P z = . ..... (5)
2S c
where = is a coefficient, known as the "arching factor" and
represents the proportion of Pz that actually acts on the pipe and
produces compression in the pipe ring.
If a uniform compressive pressure, Pz, is applied directly at the
soil-pipe interface then the value of compressive hoop stress
produced within the pipe wall will be less than if the uniform
pressure had been applied at a boundary some distance from the
pipe. Frictional conditions at the interface have no effect on this
phenomenon. To allow for this response to external uniform pressure
the following mathematical approach is adopted:-
If Pz is applied directly at the interface then the value of ~ is
taken as equal to =z whereas if Pz is applied at a distant boundary
its value is taken as equal to =z~z, where Iz is a magnification
factor. It is seen that = involves two elastic components and the
expression for its evaluation is:-
= kz=z .......... (6)
~z is the factor by which uniform pressure applied at a distant
boundary is magnified on reaching the pipe-soil interface. Its
formula (7) is given in Appendix I. By substituting various values
of vs* into equation 7, it is seen that ~z can range from a
maximum value of 1.5, when vs* = 0.3, to a minimum value of 1.0,
when vs* = 1.0. For uniform loading applied directly at the
interface ~z reduces to a value of 1.0.
=z is known as the uaiform thrust coefficient and its value heavily
depends upon the value of Z, the compression stiffness ratio.
10
=z is therefore a measure of how much a relatively flexible pipe
will shed its load into the surrounding soil and varies in value
from 0, when Z is infinite, to 1.0, when Z = 0.
The formula for ~z, (8), is given in Appendix I and its variation
with Z, for plane strain, is plotted in Fig. 6. A description of
how this plot can be obtained is given in Appendix I.
5.3 First order ~ to ~istortional D~Q~JAT_~ Py
The peak value of thrust due to py is given the symbol Ny1 and the
peak value of deflection caused by py is given the symbol ~¥I- The
first order expressions for these values have the general form:-
Ny, = B.pyR ............. (9)
~¥I = ~-P¥ .............. (10)
Es*
~, the distortional thrust coefficient, and ~, the elastic
deflection coefficient, are both functions of the system flexural
stiffness ratio, Y, but they are also dependent upon the friction
conditions at the interface. The expressions for B and ~ for both
full slippage, fs, and no slippage, ns, are given in Appendix II
together with a description of how plots of the variation of Bfs
and Bns, and of ~fs and ~ns, with Vs* ,Y and Z were prepared.
The plots of the variation of Bfs and B,s are shown in Fig. 7 and
are seen to exhibit two clear trends:-
(i) As Y increases, i.e. as the system becomes more flexible, falls in value and, in the case of full slippage,
reaches zero.
(ii) Interface friction has a profound effect upon the value of which can have any value lying within the outermost
lines of Fig.7.
The plots of the variation of ~fs and ~,s are shown in Fig.8 and
from them it is apparent that:-
ii
(i)
(ii)
As Y increases ~ tends towards a constant value indicating that, as the system becomes more flexible, ~.. becomes y~
proportional to py/Es* and essentially independent of the pipe stiffness.
Interface friction has only a moderate influence on the value of ~ so that a design assumption that full slippage conditions prevail will lead to safe, but not unduely conservative, predictions of deflection.
~,4 Th@ ranaes Qf svstem b@h~ViQDr
If the curves of B and ~, shown in Figures 7 and 8, are examined
it is seen that they tend to divide into three separate regions of
system response.
If Y is less than i0 or greater than i000 both the fs and ns values
of B and ~ are more or less constant and, if the value of Y lies
between i0 and I000, the values vary with the value of Y.
It is therefore possible to define ranges of system behaviour in
terms of Y, as shown in Table i.
TABLE 1 DEFINITION OF RANGES OF SYSTEM BEHAVIOUR
Y Proportion of distortional load carried by bending action of the pipe ring
I System behaviour
Less than i0 I more than 90% I Rigid
i0 to I000 I 10% to 90% I Intermediate
More than I000 1 less than 10% I Flexible
Ranges of behaviour of various pipe-soil combinations are
illustrated in Fig. 9 where the limits of diameter/thickness ratio
shown correspond to those of currently manufactured pipes. The
diagram illustrates the erroneous simplicity of the traditional
distinction between rigid and flexible pipes. A traditionally rigid
pipe, such as one made of concrete, can exhibit a behaviour almost
approaching flexible when buried in a stiff soil whereas a
corrugated steel pipe, generally regarded as fully flexible, may
12
act as a rigid pipe when buried in a weak backfill.
5.5 Second order ~ to Pz
The action of py causes deformations of the cross section of the
pipe ring . When the different vertical and horizontal projected
areas of the pipe ring are acted upon by the uniform pressure Pz
secondary distortions occur, ay2, an additional pipe ring
deflection and Ny 2 an increase in the hoop thrust. The expressions
for ~Y2 (14) and Ny 2 (16) are given in Appendix III. At
equilibrium, i.e. when Pz and py act together, the final ring
deflection is 5y and the final value of hoop thrust is Ny.
Expressions for 6y and Ny, (15A and 16A), are also in Appendix III.
5.6 Rinu bendinu moment
The bending moment M / unit length generated by the changes in the
curvature of the pipe wall is proportional to the net out-of-round
deflection, ~yCOS20, with a peak value, My, when 0 = 0 °
Gumbel et. al (1982) quote the following expression for My:-
My = 6Ep*I . 8y ........... (17)
D
With flexible thin-walled pipes the bending moments generated are
generally negligible but, when there is a chance of the bending
stresses becoming significant, they can be controlled by reducing
the allowable ring deflection.
5.7 Rinu bucklina
The present generally accepted equation for Pb, the value of
external hydraulic pressure that would produce buckling in a
flexible pipe operating under plane strain conditions, is quoted
as Eqn. 10 on page 15 in the first report of this series, Smith and
Young (1991).
13
J ksREI Pb = 2 R3( 1 _ vP 2)
The major problem with the use of the equation is the determination
of a realistic value for ks, the unit stiffness of the soil.
Nevertheless, as pointed out by Smith and Young (1991), the plane
stress version of this equation is used to derive the buckling
formula suggested in the CIRIA Report No. 78:-
Pb = 24 8E'E''
where E' = ksR and E'' = 8EI/R 3
In order to incorporate buckling effects into the proposed design
method it is necessary to go back to the more basic formula for
the buckling pressure, derived by Link (1963), Cheney (1963) and
Luong (1964). The approach, described in Appendix IV, produces new
formulae for Pb (Eqns. 23 and 26).
5.8 Yield-bucklinq interaction
If the mean ring compressive stress, fb, to cause elastic buckling
approaches the yield stress of the pipe wall, f¥, then some
reduction in the buckling resistance may be expected due to the
development of plastic strains. To allow for this Meyerhof and
Baikie (1963) proposed that the value of fc, the critical value of
the ring compressive stress, be obtained from a formula originally
formulated by Southwell (1915):-
fb fy fc --
fb + fy
This formula is known to be very conservative and is therefore
generally considered to include an allowance for the effects of any
original out-of-roundness of the pipe wall.
Gumbel (1983) proposed the use of a reduction factor, ~2, to be
14
applied to the buckling pressure, Pb- The value of u 2 reduces as
the ratio of fb/fy increases, as can be seen from eqn. (27A).
Gumbel took the limiting value of ~2 to apply when the unadjusted
value of fb = 2fy, i.e. ~2 = 0.4375
For fb ~ fy/2, U2 = 1.0
For fb > fy/2, U 2 = 1 fb
f¥ -I I ...... (27A)
4f b _I where fb is the mean elastic compressive hoop stress given by:-
fb Pb D (aPz) c r D
2t 2t ............. (27B)
It is seen that no reduction in buckling pressure is implied until
fb = f¥/2. In the limit, when fb = 2fy, ~2 is taken as equal to
0.4375 so that fb is reduced to 0.4375 x 2 x f¥ = 0.875f¥
6 SIMPLIFICATION OF THEORETICAL EXPRESSIONS FOR DESIGN
6.1 Interface sliDDaae assumptions
As has been discussed the worst cases of both deflection and
buckling occur under full slippage conditions so that the design
assumption of full slippage appears sensible. In fact, if a
frictional bond does develop between the pipe and the soil,
neither the resulting overestimation of deflection nor the
underestimation of buckling resistance can be greater than 20%
which is not unduly conservative.
However any interface reaction produces a significant increase in
in the distortional thrust coefficient, B, (see Fig. 7) so that the
assumption of no slippage can become unduly conservative with
pipes made from ductile material. Because of this both the full
slippage and the no slippage B-curves have been retained for
design.
15
$,2 Treatment Qf load boundary conditions
The theoretical system response for both distant boundary and
interface loading have been discussed in section 5. Obviously a
single set of load boundary conditions would be more convenient for
design purposes. Before this can be determined we must first
examine the effective points of application of the various loads to
which the pipe will be subjected.
Backfill weight : in accordance with elastic theory, in which gravity acts on the composite structure, self- weight forces are equivalent to loads acting at distant boundaries.
Distributed : as this form of loading is equivalent to an surcharge additional upper layer of fill it must clearly
act at a distant boundary.
Fluid pressure : external groundwater and internal vacuum pressures act directly at the pipe-soil interface.
Concentrated : this surcharge is assumed to spread through the load soil and to subject it to elastic stresses which
decrease in valuewith depth. Logically the value of these stresses at the level of the centre of the pipe, should be assumed to be applied at the soil-pipe interface.
In order to achieve a single set of loading conditions Gumbel
(1983) proposed the following strategy:-
i. Apply all distortional load components at a distant boundary.
2. Apply all uniform load components at the pipe-soil interface.
Fluid pressure is essentially uniform so that its point of
application is correctly represented by this approach. The apparent'
conservative treatment of the distortional effect of a concentrated
surcharge is not inappropriate when one considers the degree of
approximation involved in representing such a point load as a
uniform pressure.
In theory the response to the uniform component of the backfill
weight and any uniformly distributed surcharge could be
16
underestimated by the proposed strategy. Response in this mode is l
indicated by the value of the arching factor, =, (see section 5.2)
which, for interface loading reduces to:-
= ~z .................. (28)
By analysing hoop thrust data from a large number of published
experiments it has been found that for pipes buried in granular
soil and subject to such loadings the value of ~/~z rarely exceeds
unity. This finding suggests that the load concentration factor,
~z, is offset in practice by frictional arching within the soil and
tends to be equal to unity. Thus, on the evidence presently
available, it can be assumed that ~ = ~z for all load components
and that ~ can be used in all predictions of combined loading
effects.
6.3 Treatment of the value Qf Poisson's ratio for the ~Qil
As is discussed in Appendix I, free draining soils have vs* values
that range from 0.3 to 0.7 whereas undrained soils have a constant
vs* value approaching unity.
From Figs. 6, 7 and 8 it is seen that the coefficients ~, Bfs~.and
~fs are quite insensitive to variations in the value of vs*
within the stated range and, for design purposes, their values can
be determined by assuming that vs* has a constant value of 0.5.
Similarly, setting vs* = 0.5 in the buckling formula (23) leads to
a maximum theoretical error of only 3%.
It should be noted that the coefficient Bns is relatively
sensitive to the value of vs*, as can be seen from Fig. 7, so that
if vs* is taken as equal to 0.5 there appears to be a risk that the
thrust component, N¥, will be underestimated in a fully bonded, low
Y system where the value of vs* is actually close to 1.0. However
this condition is really only possible in conditions where the
17
backfill is essentially a fluid and the pressure, py, then tends to
be zero. In practice the error in the predicted value of Ny created
by assuming vs* = 0.5 is less than 5% and will reduce further if
some slippage does actually occur.
A final point worthy of mention is that the values of both ~y2 and
Ny2 can be 15% in error with the assumption that Vs* = 0.5. However
these two terms are of second order and contribute only a small
percentage of the total deflection and thrust response of the pipe
ring so that 15% errors in their predictions are of little
consequence.
With the above arguments it has been shown that the Poisson's ratio
of the soil for plane strain need no longer be regarded as a
significant variable but can be assumed to have a constant value of
0.5. This removes a major obstacle to the practical application of
two-dimensional elastic theory in buried flexible pipe design.
6,4 Effects Qf hQQP comDreSsibilitv
The system stiffness ratios Y and Z are not independent and it is
often convenient to replace Z with the diameter/thickness ratio,
D/t. For a pipe with a rectangular wall section of thickness t,
I = t3/12 and Area = t. From section 5, we see that:-
E*pI E*pt 3 E*pA E*pt Sf - - and S c - -
D 3 12D 3 D D
Z Sf t 3 D t 2 1
Y S c 12D 3 t 12D 2 12(D/t) 2
Fig. 12 shows the coefficients =z and B replotted from Figs. 6 and
7 as functions of Y and D/t for vs* = 0.5. This figure shows that
the value of =z is sensibly constant for Y values less than 10 4 .
When the Y value is greater than 10 4 then the value of =z tends to
reduce significantly. Fig. 9 shows that most pipe-soil systems have
18
Y values less than 104 and one can therefore conclude that
significant reductions in wall thrust due to hoop compressibility
can only occur over a small range of practical pipe-soil
combinations.
7 PREPARATION OF DEFLECTION-BUCKLING CHARTS
7,1 Dimensionless ~esian euuations
. Collecting equations i0, 12A, 15B, 22, 23 and 26, setting v s =0.5
and rearranging, leads to a set of non-dimensional design
equations for out-of-round deflection and elastic buckling. The
procedure is described in Appendix V and leads to the five design
equations set out below.
8y I = 4Y py
108+Y E s
................. (A)
Y
P
Pb
S f
y l
0.625~y, 1 -
(Py/=Pz)
1 - Y
(i + ~¥) 2
= 8(n2-i) p3 +
................ (B)
.................. (C)
Y
4 (2n+0.5) ........ (D)
Pb (=Pz) cr (py/Sf)
Sf Sf EPy/(=Pz) ¢r
.... (E)
With the use of equations (A) to (E) the interrelationships between
the dimensionless parameters:
py , py , ~y and Y
~Pz Sf
can be combined and plotted as design charts.
19
In order to cover the complete range of the possible buckling and
deflection behaviour of any buried pipe system it is necessary to
have at least ten such charts, covering a range of values for the
load distribution parameter Py/=Pz, i.e. 0.05 to 0.8. The use of
these charts are described in the third report of this series
(Smith, 1991) but, at this stage, discussion will be restricted to
how these charts can be prepared.
7,2 Description Qf ~ tyDiCal desiQn chart
Fig. 13 shows the chart for Py/=Pz = 0.4. The set of parallel and
diagonal lines on the diagram indicate the load-deflection
response for pipe-soil systems of various values of Y. The
influence of second-order distortions is demonstrated by the
curvature of these lines at large deflections. Intersecting these
deflection curves is the buckling limit line which delineates an
area of the chart within which the pipe wall is stable.
The best way to illustrate the preparation of a design chart is
by a numerical example and the following two sections, 7.2.1 and
7.2.2, show the procedure for a stiffness system with Y = i00.
7.2.1 Load-deflection curves
The first step in the procedure is to decide which chart the curve
is being prepared for, i.e. the value of Py/=Pz- Let us assume that
Py/~Pz = 0.4.
Taking p¥/Sf = i0 and remembering that Y = I00:-
4 x i0 ~y, = = 0.192 ....... From (A)
108 + i00
~y = 0.192
1 - ,625x.192 0.4
= 0.275 .... From (B)
The complete curve, shown in Fig. 13, can be obtained by selecting
20
a suitable range of values for py/Sf and determining the
corresponding values for ~y as illustrated below.
py/Sf ~¥
0.i 0.002 0.5 0.010 1.0 0.020 2.5 0.052 5.0 0.113 7.5 0.186
10.0 0.275
The full curve can be seen in Fig.13. It should be noted that
deflection values less than 0.003 are not plotted and that the
values of py/Sf are plotted to a logarithmic scale.
7.2.2 The buckling limit line
The procedure to obtain this line involves iterations in both n and
~y, in order to minimise the buckling equation (D). The work can be
carried out with the aid of a microcomputer using the simple basic
program listed in Appendix V.
As mentioned in Appendix V, and illustrated in Figs ii and 13, for
systems with Y <i000 the buckling failure mode has n = 2 whilst
for Y values >i000 the value of n increases with the value of Y.
With n = 2 the value of (py/Sf)cr = 5.42 with a deflection, ~¥, =
0.124.
The fact that the value of n only increases above 2 when the value
of Y becomes greater than i000 can be illustrated by computing the
critical value of py/Sf with Y = i00 but taking n as equal to 3.0.
In this case (py/Sf)cr is found to be equal to 7.1 with a
corresponding deflection value of 0.174. The 7.1 value is of course
greater than 5.42 which illustrates that, when Y = I00, (py/Sf)cr
equals 5.42 and occurs with the number of waves, n, = 2.0
As a further example consider a soil-pipe system with Y = 3000.
In order to determine the most critical value of py/Sf it is
21
necessary to determine the value of (p¥/Sf)cr for several
different values of n. The lowest value obtained for (py/Sf)cr ks
obviously the buckling limit for Y = 3000. When this procedure is
carried out the following results are obtained:-
For Y = 3000 and Py/=Pz = 0.4
n py/Sf ~y
2 70.4 .106 3 58.2 .085 4 58.0 .084 5 62.5 .092 6 68.9 .103
The minimum value of py/Sf is seen to be 58.0 with a failure mode
of n = 4 and a corresponding ring deflection of 0.084. This can be
checked by examining Fig.13.
Although equation (23) is intended for use with large values of n
it is interesting to note, that whilst the iteration gives n = 4
for 6y = 0.84, using equations 22 and 23 from Appendix V, and with
~¥ = 0.84, the value obtained for ncr = 3.67 which is fairly close
to the iterative value of 4.
A better illustration of how the effectiveness of the computer
program increases with larger values of n is to determine the value
of ncr for a relatively flexible system (i.e. a high Y value, say Y
= 30,000) and to then compare this value with the one obtained by
the formulae. When Y = 30,000 the iterative procedure gives n = 7.0
with ~¥ = 0.044 and, using this value for ~y in eqn.(22), leads to
a value ncr= 7.028 from eqn. (23).
7.2.3 Lines of constant value of p¥/Es*
Es* Py Py Bearing in mind that Y - we see that -
Sf Es* YSf
With this information it becomes possible to have an alternative
representation of system static response by using the loci of
22
constant values of p¥/Es* These curves are shown as dashed lines
in Fig. 13 and can be particularly useful in design as they
illustrate the effect of varying the pipe stiffness, Sf, for given
soil and loading conditions.
7.2.4 Use of the charts
The selection of suitable design parameters and the use of the
deflection-buckling charts is described in the third report of this
series (Smith, 1991).
8 ACKNOWLEDGEMENTS
This report forms part of the programme of research of the Ground
Engineering Division (Division Head Dr M.P.O'Reilly) of the
Structures Group of the Transport and Road Research Laboratory. The
author would particularly like to put on record his indebtedness to
Dr J.E. Gumbel and Messrs Mott, Hay and Anderson who developed the
proposed design method.
Whilst acknowledging that any mistakes or misinterpretation of
other people's work, are entirely due to himself, the author would
like to express his thanks to Dr O'Reilly and Mr O.C. Young for
their guidance and assistance during the preparation of this
report.
23
9 REFERENCES
ANDERSON, R.H. & BORESI, A.P. (1962) "Equilibrium and stability of rings under non-uniformly distributed loads" Proc. 4th Nat. Congr. Appl. Mech., Am. Soc. Mech. Engrs., Vol. i, pp 459-467.
BURNS, J.Q. & RICHARD, R.M. (1964) "Attenuation of stresses for buried cylinders" Proc. Symp. on Soil-Structure Interaction, Univ. of Arizona, Tucson, pp 378-392.
CHANG, C.S., ESPINOZA, J.M. & SELIG, E.T. (1980) "Computer analysis of Newton Creek Culvert".J. Geotech. Eng. Div., Proc. Am. Soc. Civ. Engrs., Vol. 106, No. GT5, May, pp 531-556.
CHELAPATI, C.V. (1966) "Critical pressures for radially supported cylinders" Tech. Note N-773, U.S. Naval Civ. Eng. Lab., Port Hueneme, Calif., January.
CHENEY, J.A. (1963) "Bending and buckling of thin-walled open section rings" J. Eng. Mech. Div., Proc. Am. Soc. Civ. Engs., Vol. 86, No. EM5, Oct., pp 17-44.
C.I.R.I.A. (1978) "Design and construction of buried thin-wall pipes" Construction Industry Research and Information Association, Report 78.
DUNS, C.S. (1966) "The elastic critical load of a cylindrical shell embedded in an elastic medium". Report CE/I0/66, Univ. of Southampton.
DUNS, C.S. & BUTTERFIELD, R. (1971) "Flexible buried cylinders. Part III: Buckling behaviour". Int. Jour. Rock Mech. Min. Sci., Vol. 8, No. 6, Nov. pp 613-627.
FLUGGE, W. (1962) "Stresses in shells" Berlin: Springer-Verlag.
FORRESTAL, M.J. & HERMANN, G. (1965) "Buckling of a long cylindrical shell surrounded by an elastic medium". Int. J. Solids Struct., Vol. i, pp 297-310.
GAUBE, E., HOFER, H. & FALCKE, F. (1974) "Statische Berechnung von Abwasserrohren aus Polyathylen hart". [The statics of rigid polyethylene drainpipes] Kunstoffe 64 (4), April, pp 193-196.
GUMBEL, J.E. & WILSON, J. (1981) "Interactive design of buried flexible pipes - a fresh approach from basic principles" Ground Eng., Vol. 14, No. 4, May, pp 36-40.
GUMBEL, J.E., O'REILLY, M.P., LAKE, L.M. & CARDER, D.R. (1982) "The development of a new design method for buried flexible pipes" Paper 8, Europipe '82 Conf. Basle, Switzerland.
24
GUMBEL, J.E. (1983) "Analysis and design of buried flexible pipes". PhD Thesis, Univ. of Surrey.
HOEG, K. (1968) "Stresses against underground cylinders". J. Soil Mech. Div., Proc. Am. Soc. Civ. Engrs., Vol. 94, No. SM4, July, pp 833-858.
INSTITUTION OF STRUCTURAL ENGINEERS (1978) "Structure-soil interaction. A state of the art report". London, April.
KATONA, M.G. (1978) "Analysis of long-span culverts by the finite element method". Transp. Res. Rec. No. 678, pp 59-66, Washington D.C.: Transp. Res. Board.
LINK, H. (1963) "Beitrag zum Knickproblem des elastisch gebetteten Kreisbogentragers". [A contribution on the buckling problem of an elastically embedded circular arch]. Der Stahlbau, 92 (7), pp 199-103.
LUONG, M.P. (1964) "Stabilitie des tuyaux souples enterres" [Stability of buried flexible pipes]. These, Faculte des Sciences de l'Universite de Paris. ~
MEYERHOF, G.G. & BAIKIE,L.D. (1963) "Strength of steel culvert sheets bearing against compacted sand backfill". Highw. Res. Rec. No.30, pp 1-14 Washington D.C.: Highw. Res. Board.
SCHOFIELD, A.N. & WROTH, C.P. (1968) "Critical state soil mechanics". New York: McGraw-Hill Book. Co.
SMITH, G.N. (1971) "An introduction to Matrix and Finite Element methods in civil engineering". London: Applied Science Pubs. Ltd.
SMITH, G.N. (1991) "Buried Flexible Pipes - Application of the new design method". Contractor Report 230, Transport and Road Research Laboratory, Dept. of Transport, Crowthorne, Berks.
Smith, G.N. & YOUNG, O.C. (1991) "Buried Flexible Pipes - Design methods presently used in Britain" Contractor Report 228, Transport and Road Research Laboratory, Dept. of Transport, Crowthorne, Berks.
SOUTHWELL, R.V. (1915) "On the collapse of tubes by external pressure". Philos. Mag., Vol. 29, No. 169, pp 67-77.
WATKINS, R.K. (1979) "Design of buried pressurised flexible pipes". Nat. Meeting on Transp. Eng., Preprint 1259, Austin, Mass.: Am. Soc. Civ. Engrs.
25
APPENDIX I
AI,I Th@ archinq f~tor
AI.I.I Evaluation of
involves two elastic components and the expression for its
evaluation is:-
= ~z=z .......... (6)
where 2 ~z = . ........ (7)
1 + v s
2(l+vs*) and =z = . ..... (8)
2(l+vs*)+Z
~z is a magnification factor used to increase the magnitude of a
uniform pressure applied at a distant boundary on reaching the
pipe-soil interface. =z is known as the uniform thrust coefficient
and its value depends upon both the value of vs*, Poisson's ratio
for plane strain, and upon the value of Z, the system compression
stiffness ratio (see Section 5.2).
AI.I.2 Determination of a plot of =z against Vs* and Z
The values of both vs* and Z are themselves subject to variation.
For typical free draining backfill soils the value of vs* is within
the range 0.3 to 0.7 whilst, for saturated cohesive fills subjected
to undrained loading, it is possible for Vs* to be equal to 1.0. Z
can have any value between 0 and 5.
By selecting a suitable range of values for Z and Vs* and inserting
them into equation (8), a table showing the variation of =z with
different values of Z and Vs* can be prepared:-
26
vs* 0.3 0.5 0.7 1.0
Z =z
0 1 2 3 4 5
1.0 .72 .57 .46 .39 .34
1.0 .75 .60 .50 .43 .38
1.0 .77 .63 .53 .46 .41
1.0 .80 .67 .57 .50 .44
These tabulated values are plotted in Fig.6.
27
APPENDIX II
AII,I The ~istQrtional thrust and f ~ ~Q@fficients. ~ ~nd
AII.I.I Evaluation of B and
The expressions for B and ~ for both full slippage, fs, and for no
slippage, ns, are set out below:-
96
24(5-vs* ) + Y ................... (IIA)
4 [96(i+v$*) + Y] Bns . . . . . . (liB)
*) (l+vs*)+(3+vs*+Z/2)Y 96 (3-V s
4Y
24 (5-v$*) +Y .................... (12A)
4Y [2(l+vs*) ] ~n$ = .... (12B)
96(3-vs*) (l+vs*)+(3+vs*+Z/2)Y
AII.2 Determination Qf plots of ~ @nd ~ aaainst ~s~ ~nd X
By taking values for Z of 0, 2.5 and 5.0 and the two limiting
values for vs*, 0.3 and 1.0, tabulated values of = and B can be
obtained. These values can then be plotted to illustrate how the
values of these coefficients are affected by the value of the
system flexural stiffness ratio, Y, which is extremely variable and
can range from 0.i to 107 It should be noted that, for the
condition of full slippage, the expressions for = and B do not
contain the term Z.
AII.2.1 Determination of plots of Bfs and Bns against vs* and Y
Taking the two extreme values of vs*, i.e. 0.3 and 1.0, the
expressions for Bfs and B,s can be simplified;-
28
v s = 0.3: Bfs = 96 112.8 + Y
Bns = 499.2 + 4Y 336.96+(3.3+Z/2)Y
V$ * = 1.0: Bfs = 96 96 + Y
Bns = 768 + 4Y 384 + (4+Z/2)Y
Values of Bfs for vs* = 0.3 and 1.0
Y
Vs* = 0.3 Vs* = 1.0
0.i 1.0 i0 102 103 104 105 i0 ~ 107
.850 .844 .782 .451 .086 .009 .000 .000 .000
.999 .990 .906 •490 .088 .010 .000 .000 .000
Values of Bns for vs* = 0.3
Y
Z= 0.0 Z = 2.5 Z = 5.0
0.i 1.0 i0 102 103 104 l0 s 106 107
1.48 1.48 1.46 1.35 1.24 1.21 1.21 1.21 1.21 1.48 1.47 1.41 1.14 0.92 0.88 0.88 0.88 0.88 1.48 1.47 1.37 0.98 0.73 0.69 0.69 0.69 0.69
Values of Bns for Vs* = 1.0
Y
Z = 0.0 Z = 2.5 Z = 5.0
0.i 1.0 I0 10 2 10 3 10 4 10 5 10 6 10 7
2.0 2.0 2.0
1.99 1.91 1.49 1.09 1.01 1.00 1.00 1.00 1.98 1.85 1.28 0.85 0.77 0.76 0.76 0.76 1.98 1.80 1.13 0.69 0.62 0.62 0.62 0.62
Plots of Bfs and Bns , for Z = 0.0 and 5.0, are shown in Fig.7.
AII.2.2 Determination of plots of ~fs
vs* = 0.3: ~fs = 4Y 112.8 + Y
~ns = 10.4Y 336.96+(3.3+Z/2)Y
and ~ns against Vs* and Y
Vs* = 1.0: ~fs = 4Y 96 + Y
~n s = 16Y 384 + (4+Z/2)Y
Values of ~fs for v s = 0.3 and 1.0
Y
v* = 0.3 $, V s = 1.0
0.1 1.0 10 10 2 10 3 10 4 10 s 10 6 10 7
.003 .035 .330 1.88 3•59 3.96 4.00 4.00 4.00 • 004 .041 .380 2.04 3.65 3.96 4.00 4.00 4.00
29
Values of ~n for Vs* = 0.3
Y
Z = 0.0 Z = 2.5 Z = 5.0
0.i 1.0 i0 102 103 104 105 106 107
.003 .031 .280 1.56 2.86 3.12 3.15 3.15 3.15
.003 .030 .270 1.31 2.13 2.27 2.28 2.29 2.29
.003 .030 .263 1.13 1.69 1.78 1.79 1.79 1.79
Values of ~ns for vs* = 1.0
Y Z = 0.0 Z = 2.5 Z = 5.0
0.i 1.0 i0 102 103 104 l0 s 106 107 .004 .041 .380 2.04 3.65 3.96 4.00 4.00 4.00 .004 .041 .370 1.76 2.84 3.03 3.05 3.05 3.05 .004 .041 .360 1.55 2.32 2.45 2.46 2.46 2.46
Plots of ~fs and ~ns, for Z = 0.0 and 5.0, are shown in Fig.8.
The expressions IIA, liB, 12A and 12B are for distant boundary
loading. These expressions can be converted into expressions for
interface loading by applying the factor i/X
1 3 - v s = . . . . . . . . . . ( 1 3 )
4 Y
where • - Y
30
APPENDIX III
AIII.I Second order ~Qmponents of DiDe rinq deflection. ~y2~
and hood thrust. ~¥2
The deforming action of p¥ on the cross-section of the pipe ring
gives rise to secondary deformations, caused by the action of the
uniform pressure Pz acting on the different vertical and horizontal
projected areas of the pipe ring. The magnitude of these secondary
effects can be forecast by the use of a procedure similar to that
used by Watkins (1979) to estimate the re-rounding of ellipsed
pipes under internal pressure.
AIII.I.I Evaluation of ~¥2 and Ny 2
If, at equilibrium, the final out-of-round deflection is &y (See
Fig. 4) then the additional distortional pressure due to Pz acting
at the pipe-soil interface will be =pzay, also acting at the
interface. From equations (i0) and (13) the deflection produced by
this second-order distortional pressure will be:-
8y 2 =
~.(3-V$*) . =pzSy
4 E s . . . . . . . . . . ( 1 4 )
With Pz and py applied together the equilibrium deflection,
~y = 5yt + ~y2, will be equal to:-
Y
[ PY + (3-Vs*) =Pz~y I (15A)
L ] E$* 4
Remembering that ~ ~ ¥,
Es Py
We can rewrite the expression as:
31
Y (3-vs*) ~Pz ]
1 By, 4 py
............. (15B)
The expression for the second order component of hoop thrust can
can be found in a similar manner:-
N¥2 = B [ (3-vs4
and Ny = B [ py
. ~pz&y R ........... (16)
(3-vs*) +
4 ~pz~y I R ..... (16A)
32
APPENDIX IV
AIV,~ ~ new formula for the bucklinu Dressure. ~b
In order to incorporate buckling effects into the proposed design
method it is necessary to go back to the more basic formula for
the buckling pressure, derived by Link (1963), Cheney (1963) and
Luong (1964) all of whom started from the bending equations for
curved bars. The problem they analysed was of an inextensible pipe
ring subjected to plane stress and experiencing symmetric multi-
wave buckling. In this case multi-wave is taken to be when n a 2,
where n equals the number of waves.
AIV. I.I Evaluation of Pb
Chelapati (1966) obtained the corresponding expression for plane
strain, which will be used in this text:-
Pb = ( nz -- i)
= (n z - I)
EpIp ksR +
(l-vp 2) (n z - i)
EpIp ksR +
R 3 (n 2 - i) ............ (18)
The major disadvantage in the use of this formula is the evaluation
of k s . Forrestal & Herrmann (1965) analysed the problem by
representing the soil as an elastic continuum, and showed that k s
is a function of the buckling mode, n, which means that the
simplified expressions for k s mentioned in the first report of this
series, Smith and Young (1991), cannot be correct. Duns (1966) and
Duns and Butterfield (1971) produced the following expression for
k s (for values of n _> 2):-
E s (n 2 - I)
k s = __ R (l + Vs) [ (2n + 1) - 2Vs(n + i)]
When E,/(l-Vs 2) is substituted for Es* and Vs/(l-vs) for Vs* , the
33
expression becomes:- Es* (n 2 - 1)
k$ = R (2n+l-Vs*)
For design purposes an empirical factor of 1/4, originally
proposed by Duns (1966) and Duns and Butterfield (1971) is
applied to ks, giving: Es* (n 2 + i)
ks= 4R (2n+l+Vs*)
.... (19)
If this expression for k s is now substituted into eqn (18) the
formula for Pb becomes:-
E'pip Es* Pb = (n2-1) + .... (20)
R 3 4 (2n+l-Vs*)
where Pb is applied at the pipe-soil interface.
It should be noted that the factor of 4, although having some
theoretical backing, is really empirical and covers a number of
practical deviations from the assumptions used in the basic
buckling theory, such as non symmetrical response of the soil to
loading and unloading, local variations in effective soil
stiffness, local imperfections in the pipe wall shape.
The distortional effects of the hoop thrust, N¥, and the ring
deflection, ~¥, have an effect on the number of waves that will
occur in multi-wave buckling and this must be allowed for.
Much has been published on the effects of initial-out-of-
roundness of a buried pipe and, by considering the precedents set
in the literature, an intuitive allowance for the effect of ~y can
be obtained by substituting the maximum instantaneous radius of
wall curvature into the buckling formula, eqn (20), instead of the
mean pipe radius R. In order to do this the pipe is assumed to
deform as an ellipse in which rmax, the maximum radius of
curvature, occurs at the crown and the invert (See Fig.10).
34
The relationship between rma x and R has been established by Gaube
et al (1974):-
R (1- ~v) = p =
rma x (I - 5h) Z
............ (21)
and, as
Replacing rma x
~v = +~y and ~h = -~y :-
(i- ~y)
(I + 5y) 2 .......... (22)
for R in equation 20 leads to the following
modified formula for Pb, the value of the interface pressure at
buckling:-
Pb = 8(n2-1)S~ - p3 Es
+
4 (2n+l-vs*) ........ (23)
The value of n which yields the lowest value of Pb is known as the
critical buckling mode and is given the symbol ncr. The value-of
increases with the value of Y, the flexural stiffness ratio. ncr
Examples of buckling modes in low and Y systems are shown in Fig.
II. With high values of Y an expression for nor is:-
nor 1
4p
1/3
-- . ........... (24)
2
and the associated critical value of buckling pressure is:-
1/3 2/3 (Pb)cr = 0.945p (Sf) (Es*) .......... (25)
The use of equation 23 or 25 depends upon the interpretation of the
interface pressure, Pb, in a non-uniform loading situation. In most
of the design and research work listed in this report it is
assumed that buckling is governed by the peak value of Pb, or the
corresponding peak value of the compressive hoop thrust. In this
present analysis it is assumed that buckling is governed only by
the mean hoop thrust, N z, so that Pb is interpreted as the
35
critical value of the uniform component of the interface pressure,
i.e.:- Pb = (~Pz)cr ........... (26)
The reason for this is that Eqns (20), (23) and (25) involve the
tacit assumption of zero shear transfer, i.e. full slippage, at
the pipe-soil interface, which is the worst case for overall ring
stability.
Fig. 7 shows that the full slippage coefficient Bfs, and hence the
deviatoric thrust component N¥, tend to zero. In this situation
buckling can only depend upon the uniform thrust component N z.
In rigid and intermediate systems, i.e. Y less than i000, this
arguement is no longer valid because, as indicated in Fig. ii, the
initial buckling mode with such a system is generally elliptical,
(n = 2). However a direct parallel can be drawn with Anderson and
Boresi's analysis, (1962), which infers that buckling in rigid and
intermediate systems is again governed only by the uniform thrust
component, N z .
In summary, equations 23 and 25 provide design formulae for the
critical buckling pressure which, if used with equations 22 and 26,
include a semi-theoretical allowance for the effects of
distortional loading, having regard to both out-of-round deflection
and non-uniform thrust in the pipe ring.
36
APPENDIX V
AV, I Preparation Q~ ~@flection-bucklina charts
AV.I.I Relevant equations
The following equations are used in the preparation of the
deflection-buckling charts.
~z: = ~'Pz .............. (z0)
Es*
~f$ -- 4Y
24 (5-Vs*) +Y .................... (12A)
~¥ = 8¥z
(3-vs*) =Pz ] 1 ~Yl
4 py
............ (15B)
R (I - ~¥) = #) =
rma x (i + ~y) 2
Pb = 8(n2-1)Se p3 +
.......... (22)
Es*
4 (2n+l-vs*) ........ (23)
Pb = (=Pz)cr ........... (26)
AV.2 Dimensionless desiQn eauations
By setting vs* = 0.5 and rearranging the above equations, as shown
below, a set of non-dimensional design equations for out-of-round
deflection and elastic buckling can be produced.
AV.2.1 Expression for 5y2
For full slip conditions, the expression for ~, (12A), becomes:-
~fs = 4Y 108+Y
and, substituting for ~fs in equation (i0), gives:-
37
NOW E
6yl =
4Y p y
108+Y E s
* = YS so the expression for 5 s f
~Yl = 4 (py/S f)
(108+Y)
y, can be written as:-
AV.2.2 Expression for ~y
Equation 15B, with vs* = 0.5 is:-
Y
8 y l
Pz 1 - 0.625=--
Py
5 y l
0.625~y, 1 -
(Py/=Pz)
,yl]
AV.2.3 Expression for Pb
* and 0.5 is substituted for vs* When YSf is substituted for E s
Equation 23 becomes:-
Y Pb 3 -- 8 (n2-1) p +
Sf 4 (2n+0.5)
where p = 1 - ~y
(i + ~y) 2 (Equation 22)
From equation 26 we see that Pb = (=Pz)cr hence:-
Pb ( = P z ) c r P y ( = P z ) c r (py/Sf)
Sf Sf pySf [Py/(=Pz) cr]
AV.3 ~ ~n~ bucklinu Gh~rts fQr buried DiDes
the
The interrelationships between the dimensionless parameters:
py , py , 6y and Y
~Pz Sf
38
can be combined and presented in plotted form. The procedure is
described in the main text, (section 6) and a listing of the
necessary basic program is given below.
CLS REM Prog. name "DEFLECTION" Deflection-Buckling Charts PRINT "Input a value for py/apz" INPUT x LPRINT" py/apz = ";x PRINT "Input a value for Y" INPUT y LPRINT" y = ";y LABEL top CLS PRINT "Input a value for py/Sf" INPUT k dyl = 4*k/(108+y) dy = dyl/(i- (. 625"dyi)/x) PRINT "For py/Sf = ";k;" dy = ";ROUND(dy,3) p = (l-dy)/(l+dy)^2 PRINT "p = ";p PRINT "Insert a value for n" INPUT n a = 8*(n^2-1)*pA3+y/(4*(2*n+0.5)) PRINT "a = ";ROUND(a,3) a = x*a PRINT "(py/Sf)crit = ";ROUND(a, 3) ;" original py/Sf = ";ROUND(k,3) PRINT "Difference = ";ROUND(ABS(k-a),2) PRINT "To continue iteration press return" INPUT f GOTO top
39
FIGURES
40
Surcharge
, ' ,b ,Y~\ \
GWL
\\,,~ J i i
Fig. Hypothetical soil element for calculation of system external loads
Pv p, + pv
TOTAL U N I F O R M + D I S T O R T I O N A L
Ph = K. Pv Pz = }(Pv+Ph )
= ~(l+K)p v
py = ½(pv-ph )
= ~ ( i - K ) ~ V
Fig. Total loading expressed in terms of uniform and distortional pressure components
Fig. 3
N M
/" / / f . . . . ~ ' ~ . ~__,. M
i f~ N
/
\ ,,/ ~ _ . ~
D
Notation for response of the pipe ring
External load on system
P=pe hoop thrust (compress,ve posit ive) I
Pipe r,ng def lec t ion (Per unit radius)
U N I F O R M + D ISTORTIONAL = TOTAL
°Pv Pl
~ H I I
TT T ~o~ I I t
Nv2 Nyl
C> Ny : Nv l -Nvz
' ) I 6v2 I 6v 1
, f ~ 6 Z .
6v = °vl " 6v2
i [ . (=P,'Pv)
!!i ':°'-°"
N = N~ • N v - C O S 2 t : I
@ Ov (=~,z • ~yi
Fig. 4 Uniform and distortional response components
I i
Case 1:
0
f I I I
Ph = K'Pv
Distant boundary loading
X~ ~ l ~ p h = K.Pv
Case 2: Interface loading
Fig. Load cases solved by closed-form elastic Continuum analysis . . . . . .
O. z
1.0
0 . 8
0.6
13.4
13.2
. I I .,VS =
1.0 0.7 0.5 0.3
13 1 2 3 4 5
Z
Fig. 6 Uniform thrust coefficient ~z f u n c t i o n o f Z and ~e
S
a s a
2 . 0
1.2
/3
0"8
0"4.
0
U..~'= 0 "3 i
/ ' ~ S
~ r s
0 " 1 !
Fig. 7 '
\
", \ \
"• ' ' " ~ ' ~ ~ c . _ _
I 0 I 0 2 I 0 3 I 0 4. / 0 5
Y Distortional thrust coefficient 13 as a function of Y v* Z and interface slippage
S ~
~,*-- o.3, z ~ o
/.O,Z=O
o.3, z-- 5
/.o,z:5
,..
/.,,'s#: 0.3, Z,= 5
/0 7
~ j ~ " /.o J
i . _ ns
2 . " _ ¢ .~
. b e c o m e i d e n t i c a l f o r v s = ',
0 0. / / I 0 /0 2 I0 3 I 0 ~ /ds /0 ~ /0~ y,
Fig. 8 Deflection coefficient ~ as a function of Y, u;, Z and interface slippage
:::::::::::::::::::::::::::::::::::::::::::::::::::
r Max 1 i::iiiii!!::iiii! P,pe material .i~!iiii!i!!iii L
Es _- I... Assumed range of 4 Es = /
1MN/m~ I backf i l l st i f fness 1 0 0 M N / m ~
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: !!!!i!i::::i:iiiiiii Vitrified Clay :iii!ii!!ii!ii!iiiii 12 ....... ....................-......... -;..-'.:-:-:.:-:-:-:-:.:
Concrete 18
26 | ::::::::::::::::::::::Ductile Iron :::::::::::::::::::::: i ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :i:::::
81
~5 I"~;!~:~::::.::.::~::.::~;::i::~::~::i!i:~iii!.i::::~; i:~estos Cement !?;i!~:;:~:~i.;.::::::.i::!::~?~i!i~ I so
10 4 1
20
16 ! i i i ! i ! i i i i i i i i i i i i i i'u" PVCi! iii ili::!i!::iiiiii~iiii::~::~iii!i!i!i 41
,o . . . . . . . . . . . . . . . . . . . . . . . . . , ~ o i ~ ; ~ ; ~ ~:::,̀~:~:~:~,:i::,~:::::'~ii~i~i:i'i:i:i~,ii:i:i:i11i:i:', ~, lil i ~,iii ~!!i!!i!i i: ~, ~iiiii;i;~,,,;,~,,,,~,~,~,,;~,J
I
10
Equivalent thickness t of corrugated " section = J 12'.I"
- - K
:~:~̀:~:i1~i~::~:/:~!:~::~:.i~i~:~::i:~::~:~?~:~::::::::::::::::::::::::::::::::::::::::::::!ii~:~!~:~::~!::i:;i~̀!~i~̀~::!~:~|
80 !i!i!ii!!::i::i::i':!::i::i::i::Re~nfo=ed P~ast~:!':iii::i:-i::!::!::~::~:: 2OO : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
I I I , I I 102 103 104 10 s 10:6
Flexural stiffness ratio, Y
FLEXIBLE SYSTEM BEHAVIOUR INTERMEDIATE I
I 1 0 ~
> Fig. Typical ranges of system behaviour for
different pipe materials
~5 ~i = - cS y
~R + #t 9 _ ~ . m e <1 ' d e , £ o r m e d a 'Aa ,Je
F,~ I0
.~o olD~ i '~ ,'.,D'~I o.f-oP roo.d~exs
Low Y"
~o(y/d o . d .~ .zer..n.~.~te s.y'sZe~ s
Y < ~ 1 0 0 0 . n-2
Large deflect ion mode (buckles by snap-through of pipe c r o w n )
r/e,~ , ;6/e .r'~c.r (e,,-n
Y ~ 4 5 0 0 0 n-7
•
Short wavelength mode (appears as local buckling of pipe wal l)
Theoretical bucklin 9 mode
Typical buckled shape in practice
Fig. l[ Examples of modes of buckling in low and high Y systems
NOTE: For pipes of non-rectangular wall section
J 12I equivalent thickness t = A
1 . 2
0 . 8
0 . 4
0 . 0
f
t ! ! t i t !
f f l
I I I l l t l I i t | | | t
I I0 i I0 2
I I I i l l l I
10 3 10 4 10 5 10 s 10 7
2.0
B ns
B
1.2
0.8
0.4
0.0
~fs
0 -i 1 i0 i
I .
\
t \ \\
\
I I I I T M J | 7 i i i l l l l " ~ " I " ~ - % ~ - - ~ - t ' ~ . i I I I ~ i P . . - ~ I .* . . , * • i
0 ~ 10 3 i0 4 i0 5 i0 6 10 7
RIGID
Y
j INTERMEDIATE 1 FLEXIBLE SYSTEM BEHAVlOUR 1
Fig. ]~. Hoop thrust coufficients for design
0.3
0.1
E o y
I-.. C .o • " 0.03 U
C~
0.01
p,,/c~ p, = 0.4 Lu
CO ~ co
rb --..
O. 04
O. 02
0.01
0.0C5
0. 002
Py
E ~ $
0. 003
i0-~ 10 1 I0 z i0 ~ Py
=,
S,
L o a d T e r m
i0 i0
O. 001
5
Fig. 13 Annotated deflection-buckling chart for
Py/~Pz = O. 4
L..