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..