TRANSPORT AND ROAD RESEARCH LABORATORY Department of Transport R R L

Contractor Report 228

Bur ied f lexible pipes: 1 Design methods present ly used in Br i ta in

by G N Smith and O C Young. Consultants

The work reported herein was carried out under a contract placed on G N Smith and O C Young Consultants 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 authors' 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 purposes; provided the source-is acknowledged:

CONTENTS

i. Introduction

Page

1

2. Buried pipes - basic design approaches

2.1 Rigid pipes 2.2 Flexible pipes 2.3 Deflection time lag factor 2.4 Buckling of a circular pipe 2.5 Ring compression theory

3. The CIRIA design procedure for buried flexible pipes

3.1 Required data 3.2 External loads on pipes 3.3 Maximum D/t ratio 3.4 Pipe specific stiffness 3.5 Critical buckling pressure 3.6 Deflection calculations 3.7 Minimum pipe stiffness for handling stresses

2 4

13 13 16

18

18 19 22 23 23 24 26

4. The Water Research Centre design procedure for buried flexible pipes

4.1 Backfillmaterial selection 4.2 Pipe deflections 4.3 Buckling

2

33 35 36

5. Comments on design methods presently used in Britain

6. Acknowledgements

7. References

Appendix I

Appendix II

Appendix III

Example 1

Example 2

Example 3 (Spangler's approach)

Example 4 (Barnard's approach)

Example 5 (W.R.C.'s approach)

Figures

39

39

40

43

44

50

45

46

26

28

37

51

i

NOTATION

A

B

D DR Db DL E, Ep E' E ' ' Es F F S H I K

Ka Kb Ko L

Mmax Me P P0 R Ri T

We Z

Cross sectional area of pipe wall per unit length, normal atmospheric pressure Width of assumed uniformly loaded footing (= 0.766D), (Barnard) Outside diameter of pipe Re-rounding factor Buckling reduction factor Deflection lag factor Young's modulus of pipe material Spangler's modulus of soil (= ks/R ) Specific stiffness of pipe per unit length (= EI/D s) Young's modulus of elasticity of soil Shape factor (Barnard) Factor of safety Depth of cover to crown of pipe Moment of inertia of pipe wall per unit length Lateral pressure ratio (ratio of lateral to vertical pressures). Coefficient of active earth pressure Bedding factor Coefficient of earth pressure at rest Effective length of strut • Effective length of•equivalent earth column (= 1.25D). (Barnard) Maximum value of bending moment in wall Bending moment in pipe wall at coordinates r and 8 Concentrated load Force per unit length of pipe normal to a diameter External pipe radius Internal pipe radius Thrust in pipe wall per unit length of pipe, Euler's critical load Vertical crown load per unit length of pipe Limiting height of cover

a b c fa

fb fy h ks P Pb Pd Pe

Pi

PL

Constant Constant Dimension from concentrated live load to pipe crown Allowable circumferential (i.e. tensile or compressive) stress in pipe wall Stress in pipe wall due to the buckling pressure, Pb Yield stress of pipe wall material Height of ground water level above pipe invert Soil spring constant for radial ring displacements External vertical pressure on pipe at crown level Value of external pressure on pipe to cause buckling External pressure on pipe due to backfill External pressure applied uniformly around the circumference of the pipe Internal pressure applied uniformly around the circumference of the pipe External pressure on pipe due to concentrated load

ii

Pq Ps Pt Pv Pw r t w w a

W h

Wp

w V Wvd WvL Z 0

Decrease in absolute pressure in pipe below atmospheric Vertical pressure due to any surcharge loadings Total external pressure on pipe Vertical pressure at crown due to backfill External pressure on pipe due to ground water Variable radius within thickness of pipe wall Pipe wall thickness Radial displacement of pipe wall, Unit weight of fill Vertical pressure supported by active soil pressure, (Barnard) Value of horizontal contact pressure at axis of pipe (Barnard) Vertical pressure supported by pipe alone when av = .02D, (Barnard) Total vertical pressure on pipe crown, (Barnard) Dead load pressure on pipe crown, (Barnard) Live load pressure on pipe crown, (Barnard) Depth at which full weight of fill assumed to act

G&

7w ~p

~W c ~s Ewh e

¢3 ~0

~r 70

vp v s AL av ax

Half of the angle subtended by the bedding width Unit weight of water Change in vertical pressure Change in crown load Soil compressive strain Strain (%) corresponding to Wh, (Barnard) Angular co-ordinate (measured from horizontal pipe axis) Major principal stress Minor principal stress Hoop stress acting at right angles to radius at point (r,0) Radial compressive stress acting at radius r Compressive stress induced in pipe wall Angle of shearing resistance of fill Poisson's ratio of pipe material .Poisson's ratio of surrounding soil Shortening of equivalent earth column, (Barnard) Change in vertical diameter of pipe Change in horizontal diameter of pipe

iii

BURIED FLEXIBLE PIPES

DESIGN METHODS PRESENTLY USED IN BRITAIN

1 INTRODUCTION

The present methods most widely used for the design of buried

flexible pipes have been reviewed in two publications, one by the

American Highway Research Board, Progress Report No. 116,

prepared by Krizek et al. (1971) and the other by the Construction

Industry Research and Information Association (CIRIA), Report

No.78, prepared by Compston et al. (1978).

It has long been felt that these methods have a common shortcoming

in that they all consider the various criteria used for the

structural performance of the pipe separatelY so that, with each

criterion based on a different set of assumptions, the designer

cannot readily assess which one is the most critical for his

design.

In 1977 the Transport and Road Research Laboratory (TRRL)

commissioned Mott, Hay and Anderson, Consulting Engineers, to

examine this unsatisfactory situation and to attempt to develop a

more rational approach to the design of buried flexible pipes.

The work, backed up by both research, model and prototype tests,

has now been completed and limited details of this new design

method, which provides a unified approach to the prediction of

values of deflection, stresses and buckling, have been published

(Gumbel & Wilson, 1981; Gumbel, O'Reilly, Lake and Carder, 1982).

The project is at the start of the next stage, the preparation of a

series of four reports that will set out the principles of the

proposed design model and eventually may lead to the production of

a design guide.

This first report in the series gives a brief review of the present

position regarding the design of buried flexible pipes in Britain.

Most of the material presented has been abstracted from the CIRIA

Report No. 78 and from Gumbel's treatise on the analysis and design

of buried flexible pipes (1983).

The text is written on the assumption that the reader has a working

knowledge of the terminology and theory used in buried flexible

pipe design but, if this is not the case, reference can be made to

the appendices which give an outline of this material and are

placed at the back of the Report.

2 BURIED PIPES - BASIC DESIGN PROCEDURES

The object of this section is to discuss the current design

approaches used for buried flexible pipes but, for the sake of

completeness, an introductory section dealing with rigid pipes

has also been included.

2,1 Riaid DiDes

A guide fir the design of such pipes, has been prepared by Young

and O'Reilly (1983) and published by the Transport and Road

Research Laboratory.

A rigid pipe, buried in soil, is subjected to soil pressures that

can rarely be considered as equivalent to an external hydraulic

pressure hence, in most situations, bending moments will be induced

in the pipe walls. If the equilibrium distribution of the contact

pressures between the pipe and its surrounding soil is known or can

be assumed, then the moments and thrusts in the pipe ring can be

determined from statics alone with no knowledge of the material

properties of the pipe.

Consider a rigid pipe subjected at its crown level to a uniformly

distributed vertical pressure p, (Figure 5A). If we ignore the

weight of the soil over the depth of the pipe then we can say that

the pipe will also be subjected to a uniform horizontal pressure Kp

where K is the ratio of horizontal to vertical soil stresses.

The moment per unit length of pipe, Me, at a point on the pipe's

circumference, defined by the polar co-ordinates R and 8, can be

obtained from the equation quoted by Bulson (1985):-

pR 2

M 0 - (2cos2e + 2Ksin2e - K - I) ......... (6) 4

The maximum value of bending moment to which the cross section

will be subjected, Mmax, occurs at the springing, i.e. when e = 0 °

Some designers prefer to think in terms of Wc, the load acting on

the crown of the pipe and equation (6) becomes::-

WcR M 0 - (2cos2e + 2Ksin20 - K - i) ........ (6A)

8 where W c = 2pR

Charts for the determination of W c for a rigid pipe buried in a

trench or under an embankment, have been prepared by Young and

O'Reilly (1983).

A plot of Mma x values obtained from equation (6) for K values from

0 to 1.0 is shown in Fig.5B.

If the pipe is so stiff that horizontal deflections will be

negligible then K will tend to equal the coefficient of earth

pressure at rest, K 0. A typical K 0 value for a granular fill is

0.45 giving a maximum bending moment, M, = 0.138pR 2 = 0.069WcR.

Because of the difficulty of ensuring good compaction of the side

fill for a pipe placed in a trench the effect of horizontal

pressures acting on a trenched rigid pipe is usually ignored. In

effect K is taken as equal to 0 and the maximum bending moment

acting on the cross section as equal to 0.25pR 2 = 0.125WcR. If the

pipe is flexible enough to deflect horizontally outwards into the

soil then the value of K will increase leading to a consequent

reduction in the bending moment to which the pipe is subjected.

When K = 1.0 then M = 0 and the condition of an applied hydrostatic

pressure has been achieved.

2,2 Fl~xibl# DiDes

The basic problem considered in the current design approaches is

the two dimensional plane strain response of the circular buried

pipe cross section, otherwise termed the "pipe ring" Possible

detrimental longitudinal effects are briefly discussed in section

1.6 of the second report of this series, Smith (1991).

The type of structural failure experienced by a buried pipe is

affected by its degree of flexibility. A rigid pipe, with its

thick walls, usually experiences a structural failure caused by a

form of bending failure of the cross section. A buried flexible

pipe can be subjected to one of three possible failure modes;-

i. Excesssve deflections - the maximum deflection of the pipe ring must not exceed some specified allowable value.

2. Buckling failure - buckling of a pipe can occur in various ways, which are discussed later in this section. The compressive hoop stress within the pipe ring must be restricted to such a value that buckling failure will not occur.

3. 0verstressing of the pipe walls - due to the value of the compressive hoop stress becoming excessive. It should be noted that with flexible pipes this failure mode is generally less critical than (i) and (2).

Present practice is to check each condition and then choose a pipe

that will withstand the worst case.

2.2.1Spangler's deflection theory.

This theory is fully discussed in the CIRIA Report No. 78 and only

a brief resume will be given here.

Spangler (1941) used a discrete spring (Winkler) model to

represent the horizontal stiffness of the backfill soil and

suggested that the form of the pressure distribution around a

buried flexible circular pipe could be obtained by the following

assumptions:-

i) The vertical load/unit length, We, acting on the crown

is assumed to be uniformly distributed over an area equal to the

diameter of the pipe. W c is the summation of the effect of the

weight of the backfill soil above the pipe and the effect of any

vertical surcharge. The effect of the soil weight is determined

from Marston's theory (1913, 1930) which, in the case of

pipes in a trench, allows for the transference of part of the,

weight of the backfill to the undisturbed soil at the sides of

the trench. As is discussed in section 3.2.1 of this report,

CIRIA conservatively recommends that for most practical

situations this reduction of load can be ignored and the full

weight of the soil above the pipe assumed to act at crown level.

ii) The invert reaction is assumed to be vertical and to be

uniformly distributed over the width of the bedding of the pipe.

iii) The horizontal soil pressure is parabolically distributed

over the arc subtended by the central i00 degrees, varying from

zero to a maximum value of ksax/2 where ax is the total horizontal

deflection of the pipe ring. k s is generally known as the soil

stiffness or as the modulus of passive resistance although CIRIA

Report 78 refers to it as Spangler's modulus of subgrade reaction.

The assumed pressure distribution is illustrated in Fig.6.

5

Spangler developed the following equation for ax:-

KbWcR 3 ax = . .......... (7)

EI + 0.061ksR 4

where K b = the bedding factor whose value depends upon

= half the value of the bedding angle (in degrees) - See Fig.6

R = radius of unloaded pipe

I = Mom. of inertia of pipe wall per unit length (= t3/12

k s = Soil spring stiffness

E = Young's Modulus of the pipe material

Spangler developed a relationship between ~ and the bedding

factor Kb:-

0 15 22.5 30 45 60 90 K b 0.Ii 0.108 0.105 0.102 0.096 0.090 0.083

A linear regression analysis of these values gives the following

approximate formula:- K b : 0.11134 -0.00032~

With a smooth walled pipe the value of I per unit length is equal

to t3/12 where t = the wall thickness. However when a pipe wall is

corrugated its I value is considerably increased and this value

should be obtained from manufacturers' tables.

In order to establish a value for ks, CIRIA Report No. 78 uses the

approximate relationship E' = ksR where E' = Spangler's Modulus for

the soil.

Substituting E' = ksR and D = 2R, Eqn.(7) can be re-arranged as:-

KbWc

~x =

8EI/D 3 + 0.061E'

0.083W¢ or, when ~ = 90°: ~x =

8EI/D 3 + 0.061E'

EI/D 3, which only occurs in these modified forms of Eqn.(7),

involves the properties of the pipe and its value is called the

specific stiffness of the pipe and given the symbol E''

EI Hence E . . . . the specific stiffness of

D 3 the pipe per unit length

Substituting 8E'' for 8EI/D s in the modified equations leads to:-

0.083W¢ ~x = . ......... (7A)

8E'' + 0.061E'

Equation (7A) is the expression suggested by CIRIA and has the

general form:-

Factor x vertical load/unit length ~x =

Pipe stiffness factor + soil stiffness factor

The denominator therefore involves two constants, 8E'', the

structural stiffness of the pipe ring when its sides are

unsupported and 0.061E' the increase in stiffness with side

support from the backfill. It is interesting to look more closely

at these two parameters.

At the present time the upper limit of specific stiffness, E'', is

taken as ll,000N/m 2 and therefore, with the thickest recommended

flexible pipe, giving the maximum value of E'', it is seen that

8E'' cannot be greater then 88,000N/m 2

For backfills the lowest value of E' considered to be an

acceptable limit for work with buried pipes is 2 x 106N/m 2 so

that the smallest possible value for 0.061E' is 122,000N/m 2

These two values represent the extreme values at either end of the

range of possible values. Obviously, at all times, the stiffness of

the backfill is of considerably more significance than the

structural stiffness of the pipe ring.

Although the simple representation of external loading as a

vertical load acting on the crown leads to a simple design

approach, it means that only the horizontal soil pressures

created by static vertical pressures are considered and the

effects of any horizontal components of earth pressure created

during the placing and compaction of the backfill are ignored.

This implies that the soil modulus, E', is taken as a constant

whose value can be either assumed or can be determined from

laboratory tests.

In fact the value of E' can be extremely variable as it is heavily

dependent upon the value of lateral pressure created during

backfilling.

A further point worthy of mention is that Spangler's approach

assumes a unique distribution of the equilibrium soil pressures

acting on the pipe cross section. In fact both the distribution

and the magnitude of these pressures are statically indeterminate

as they depend upon the properties of the pipe-soil system.

In summary it can be said that to apply the Marston-Spangler

theory realistically is not at all simple as neither the loads on

the pipe nor the properties of the backfill material are uniquely

defined. Nevertheless it must be stated that the direct application

of the Spangler method for the estimation of deflection values of

buried flexible pipes is widely used and considered satisfactory by

many engineers.

2.2.2 Barnard's method for pipe deflections. (1957)

The design method proposed by Barnard for buried flexible pipes is

used by the American Water Works Association and is described in

in their design manual, AWWA Manual MII, (1964). The method is

discussed in the CIRIA Report No. 78, where it is presented as an

alternative to the Spangler approach. The soil is considered to be

be elastic and the value of its Young's Modulus, Es, can be found

from the graphs, .prepared by Barnard, of typical axial stress/

strain relationships obtained from triaxial tests carried out on

different soil types, (Fig. SA).

The calculation procedure has the real advantage that it considers,

step by step, the way in which a steel pipe can resist earth loads.

Barnard considered three different cases of pipe resisting action:-

case I - When the wall thickness and the diameter of the pipe

selected to meet internal pressure and other service requirements

is such that the pipe is strong enough to carry all live and soil

loads without undue deflection.

Case II - When the pipe ring strength is sufficient to carry part

of the live and earth loads, but not all without undue deflection

To limit deflection values some side support must be provided by

the surrounding soil.

Case III- When the pipe is so weak that its cross section is a

completely flexible ring which can carry very little live and dead

load on its own without undue deflection. If acceptable values of

pipe ring deflection are to be achieved there must be full

mobilisation of the resistance that can be provided by the

surrounding soil. The resulting enveloping forces are essentially

radial, the pipe ring being confined and subjected to compressive

stress only. Barnard suggested that, for such situations, the

limiting value of compressive stress in the steel pipe ring

should be taken as 7,5001b/in 2, i.e. 52 x 106N/m z

As this report deals with flexible buried pipes Case III will be

the one most considered.

2.2.2.1 Barnard's shape factor

A buried flexible pipe that is perfectly round is not subjected to

a bending moment and 70, the value of the compressive stress

induced in the wall, is given by:-

pR T e - where p = the pressure at the

t crown of the pipe

As this same stress must prevail in every portion of the pipe

periphery, the confining pressure in the soil must therefore

maintain the same compressive stress at every point of the pipe

ring if the pipe is not to change shape. This condition is only

possible if the confining pressure acts radially and is equal to

p. However, if the cross section of the pipe suffers a vertical

compression, together with a horizontal expansion, the contact

pressure on the horizontal axis must be greater than that on the

vertical axis.

Barnard allowed for this effect by introducing a shape factor and

showed that, for 2% deflections of the horizontal and vertical

diameters the shape factor equals 1.127.

2.2.2.2.Equivalent earth column

In order to calculate the horizontal pipe deflection Barnard used

an analogy with the method of calculating settlement under a

foundation and considered the central i00 ° arcs on either side of

the pipe as uniformly loaded strip footings, each of width B =

0.766D, and bearing against vertical soil surfaces. These vertical

surfaces are shown in Fig. 6, a diagram of Spangler's assumed

pressure distribution.

The "sideways settlement", i.e. the horizontal deflection ax/2 on

either side of the pipe is assumed to be equal to the shortening of

an equivalent earth column, (Fig. 7). The value of the horizontal

stresses varies throughout the length of the earth column having a

maximum value at the "footing" and reducing to a negligible value

at some distance from the pipe. Because of this variation ax/2 is

i0

taken as equal to aL, the shortening of a uniformly stressed earth

column of length L = 1.25D.

Barnard used the symbols w h for the value of the uniform horizontal

contact pressure assumed to act on the earth columns and ~wh for

the corresponding strain.

In order to determine the value of Ewh the vertical overburden

pressure acting on the earth columns is considered analagous to a

triaxial cell pressure and w h as the axially applied compressive

stress. Barnard collected data of axial stress and strain values

obtained at different cell pressures from the results of triaxial

tests on various types of soil. In the original paper this

information was presented in graphical form and some of these

diagrams, converted to S.I. units, are reproduced in Fig.8A.

The shortening of one equivalent earth column, aL, is simply equal

to the product of its length and its strain;-

ax aL - = Ewh.L = 1.25D~wh

2

Barnard's fundamental expression for the total horizontal

deflection of the pipe ring, ax, is therefore:-

ax = 2 x 1.25D x ~wh .......... (8) °

The shape factor for 2% deflection equals 1.127 which, when

multiplied by the constant 1.25 in the above equation, gives an

overall factor, F, = 1.4.

Barnard produced a set of values for F varying from 1.25, for no

pipe deflection, to 1.70, for 5% pipe deflection. He concluded

that, for practical design, F can be taken as equal to 1.35.

Hence Barnard's general design expression for the horizontal

deflection of the pipe is:-

~x = 2 x 1.35~wh D = 2.7~whD

ii

which, as ~wh = P/Es, can be written as:-

2.7pD 2.7W ~x -

Es Es

c

........ (8A)

Remembering that Barnard assumed that the sideways settlement is

equal to the shortening of a soil column of length 1.25D, we can

express Barnard's value for the soil spring stiffness, ks, in terms

of Young's Modulus of the soil, E s.

E s 0.4E s ksR k s - = i.e. E s -

1.25D R 0.4

Substituting for E s in Eqn. 8A gives a further expression for ~x:-

0.4 x 2.7W c 1.08W c ~x = = . .......... (8B)

ksR ksR

Eqn. (8B) gives a means of comparing the predictions obtained from

Barnard to those obtained from Spangler. In section 2.2.1, it was

shown that the structural stiffness of the pipe ring has

considerably less influence on the pipe's deflection than has the

backfill. If we ignore the term 8E'' and take ~ as equal to 900

(Eqn 7), can be written as:-

1.36W c - . ........ (7B)

ksR

then Spangler's formula for ax,

0.083Wc R3 ~X =

0.061ks R4

Comparingequations 7B and 8B it is seen that, if the values of E s

and E' exactly correspond, Spangler's prediction will be some 25%

greater than Barnard's. However, in view of the uncertainty in the

estimation of the values of E' and Es, the two methods can be

regarded as giving similar predictions.

For computational purposes Barnard included F in the determination

of w h by using the expression --

w h = F(w v - Wp)

where w v = the total vertical load acting on the pipe crown

12

wp = the ring load carried by the pipe when the pipe deflection equals 2%.

This value of w h is used to determine Ewh (from Fig.8A) and then

the horizontal pipe deflection, ax, which is simply equal to 2EwhD.

2,$ D~flec~iQ~ ~ime l~g factor

The value of ax, as determined from either Spangler's or Barnard's

approach, approximates to the instantaneous, or end of

construction, value of the horizontal deformation of the pipe.

Various studies have shown that the deformation of a buried

flexible pipe can increase with time due to the consolidation of

the surrounding soil. The effect is most noticeable with a non

pressurised pipe, i.e. one that is not carrying fluid under

pressure, and can be allowed for by multiplying the value of ax

by a factor, DL, known as the deflection lag factor. With a pipe

subjected to an internal pressure at least equal to the vertical

pressure caused by the soil weight the consequent re-rounding of

the pipe ring tends to nullify time effects and the horizontal

deformation of the pipe remains sensibly constant throughout its

life, at a value of the order of ax.

The generally accepted rule in buried flexible pipe design is, in

the absence of soil tests, to take D L as equal to 1.5 for

non-pressurised pipes and as 1.0 for pressurised pipes. It should

be remembered that D L is only applied to the value of the

deflection caused by dead loading and not to any caused by

superimposed live loading.

2,4 Bucklinq Qf ~ circular DiDe

2.4.1 Pipe with no radial support

The classic ring buckling formula for an unsupported circular ring

subjected to external hydrostatic pressure only was derived by Levy

13

in 1884 and is quoted as Eqn. (5) at the end of Appendix II.

3EI Pb -

R 3

For a long pipe the expression becomes:-

3EI 24EI/D 3

Pb = = . ..... (9) (i- vp2)R 3 (I- vp 2)

where Pb = critical external pressure required to cause buckling.

vp = Poisson's ratio of the pipe material.

Eq. (9) is referred to as Timoshenko's buckling equation and is

simply the plane strain version of the plane stress equation (5).

2.4.2 Pipe with elastic radial support

The simplest model of buried pipe buckling is that of a ring

loaded hydrostatically and supported by radial springs of unit

stiffness, k s , as shown in Fig.9. Such a ring usually tends to

buckle by forming a series of symmetrical ripples, or waves,

around its circumference, of the form shown in Fig.10A, whilst

its cross section remains sensibly circular. The number of waves

created can vary from 2 upwards and they are essentially

sinusoidal in shape being of the form shown in Fig.10. Another

form of buckling is the formation of a localised single wave,

illustrated in Fig.10B.

The multi-wave mode of buckling is the simplest to analyse and is

also more critical than the single wave mode, provided it is

assumed that the soil spring stiffness is constant for both inward

and outward displacements of the pipe wall. Single wave buckling

involves a much larger deformation and is therefore sensitive to

any initial out-of-roundess of the pipe wall. It is for this reason

that soil-loaded pipes which intially buckle in a multi-wave mode

14

invariably collapse in a single wave mode which should really be

regarded as a post-buckling behaviour.

Many researchers have considered the problem of multi-wave mode

buckling, some of the more important being:- Link (1963),

Cheney (1963), Meyerhof and Baikie (1963) and Duns (1966).

For large values of n the plane strain expression for Pb is

generally accepted to be:-

/ ksREI Pb = 2 / ............. (i0)

~/ R3(I - vp 2 )

At the present time, there is not a design formula for single-wave

buckling that is comparable in simplicity to the one used for

multi-wave buckling, i.e. Eqn. (i0).

The major problem with the use of Eqn. (i0) is the determination of

a realistic value for k s as its value varies with both the soil

properties and the size of the loaded area, Terzaghi (1955).

Various writers Habib & Luong (1965, 1966); Luscher (1966);

Chelapati (1966) have proposed the following relationship for plane

strain:- m ,

ksR = (i- VS 2 )

where v s = Poisson's ratio for the soil.

Current practice is to use the same k s value for the buckling

calculation as is assumed for the deflection calculation which

means that the formula for buckling used in CIRIA Report No 78 is

Eqn. (I0) with E' substitued for ksR. There is no attempt to allow

for the reduction in buckling resistance due to any out of

roundness of the pipe's cross section caused by non-uniform

loading in the ground.

More recent investigations of buckling by other researchers have

led to other expressions for Pb- The problem is that all of them

15

depend heavily upon unreliable, and often unrelated, soil stiffness

measurements and it is almost impossible to use the difference

between calculated and measured results to determine whether there

are errors in the particular theory used or whether there are

errors in the values assumed for the soil parameters. This

difficulty has been overcome in the design method evolved by Gumbel

(1983) in which the interpretation of buried pipe buckling data

can be carried out without reference to the value of E'

2.4.3 Snap-through buckling

With some pipes excessive deflection at the crown can lead to a

sudden, single wave, snap-through buckling failure as illustrated

in Fig.ll, which, like single wave buckling, is really a type of

post buckling behaviour. This form of failure is not a feature of

reinforced plastic pipes where deformations tend to occur gradually

and although corrugated metal culverts can fail by snap-through

buckling the effect does not occur until the crown deflection is at

least 0.2D. The main risk of a snap-through failure is with smooth

thin-walled metal pipes where it can occur at relatively low values

of crown deflection. Design practice is to limit such deflections

to not more than 0.05D but it should be noted that full scale load

tests by Watkins & Moser (1969), Howard (1972) and Crabb and Carder

(1985) illustrated that a flexible pipe wall can fail by buckling

or crushing at deflections anywhere between 0.01D to 0.2D. As the

response is generally non-linear, a pipe that fails at 0.2D crown .

deflection will have a factor of safety considerably less than 4

when the crown deflection equals 0.05D.

2,2 Rinq compression theorv

It is iliustrated in Appendix II that a pipe subjected to a

16

uniform hydrostatic external pressure is not subjected to bending

effects. This led White and Layer (1960) to suggest a simple

method for the estimation of the strength of buried thin-wall

pipes. They maintained that deep-buried circular pipes are

subjected to a uniform radial soil pressure equal to the

overburden pressure (including any surcharges) at the pipe crown.

The value of the moment acting at a point on a pipe's

circumference divided by the compressive thrust is the distance

that the line of thrust is from the centre of the wall thickness.

By ignoring bending moments the line of thrust is tacitly assumed

to coincide with the centre of the wall and the value of

compressive thrust can be obtained from equation (3) listed in

Appendix II:- Pe R

~8 = t

with Pi, the internal pressure, replaced by Pe, the external

pressure acting on the pipe.

If the value of allowable compressive stress is known then a

suitable pipe can be determined. The theory does not consider

how the assumed equilibrium state comes about although some form of

pipe deformation must occur. However the deflections experienced by

a buried'flexible pipe during its installation can be considerably

affected by control of the compaction process as the work proceeds.

The compaction of the sidefill tends to ellipse the pipe upwards

which, with the subsequent placing of backfill above the pipe,

reverses. An experienced crew can therefore bury a flexible pipe

in such a manner that after construction there is very little

difference in the pipe's horizontal and vertical diameters. The use

of strutting, before any backfill is placed, can also be used to

cause the pipe to be ellipsed upwards although some engineers

17

consider the practice to be unnecessary or even undesirable.

However it should be noted that, in some situations, not to

consider possible buckling and/or deflections of the pipe ring can

lead to a serious over estimation of the pipe strength.

3 THE CIRIA DESIGN PROCEDURE FOR BURIED FLEXIBLE PIPES

At the present time the publications mainly referred to in the

U.K. for the design of buried flexible pipes are the CIRIA Report

No.78 (1978), the American Waterworks Association's manual MII

(1964), and, more recently, the Water Research Centre's Manual

(1988) .

A summary of the design procedures suggested in the CIRIA Report

Report now follows. The Report uses plane stress formulae and

defines thin-walled pipes as those having specific stiffnesses,

EI/D ~, within the range 400 to ll,000N/m 2 which, for mild steel

pipes, with E = 0.21 x 1012N/m 2, means a diameter to thickness

ratio, D/t, ranging from 350 to 115.

Although the Report acknowledges that a draft British standard is

considering the use of plastic pipes with specific stiffnesses as

low as 250N/m 2 it stresses that the design procedure it suggests is

based on practical experience obtained from buried pipes with

specific stiffnesses of not less than 2000N/m 2 and is therefore

only applicable to such pipes. This means that, for mild steel

pipes, the D/t ratio should not be greater than 200 or so.

5,1 ReQuiro~ data

The following information is required:

The pipe diameter, the pipe material and its properties.

Depth of cover and properties of backfill.

18

Superimposed loadings and the maximum and minimum internal

pressures to which the pipe will be subjected.

Maximum and minimum groundwater levels

3,2 External loads Qn bides

The total pressure, Pt, acting at the crown of a buried pipe is

equal to the weight of the backfill plus the effects of any uniform

surcharges and concentrated loads, both dead and superimposed,

that act above the pipe plus the value of the ground water pressure

acting at the invert level of the pipe.

3.2.1 Soil weight

The weight of the backfill is taken to be equal to wH where H

is the height of soil above the pipe crown and w is the unit

weight of the fill. For a cover depth between 0.25 to 1.0D an

equivalent height of soil, H + 0.107D, should be used. (See

Appendix III). H should normally not be less than 1.0m unless

load-relieving slabs are used or other precautions taken.

The CIRIA Report points out that, for large depths of cover, it is

possible to use Marston and Anderson's theory (1913) to allow for

alleviation in the weight of the soil on the pipe due to arching

effects. A simple approximation to this theory is to assume that

the full weight of the fill acts to a certain depth, z0, and that,

at further depths of cover, the pressure remains constant and equal

to wz 0. z 0 is obtained from the formula:-

D Z 0 =

2Katan#

where K a = The coefficient of active earth pressure

# = Angle of shearing resistance of fill

For a set of typical soil values z o : 2.6D and the CIRIA Report

19

recommends that, for most practical flexible pipe situations, the

total vertical pressure of the soil should be taken as equal to wH

and assumed to act at crown level, which is the same procedure as

that proposed by Barnard (1957).

3.2.2 Uniform surcharges

Uniform surcharges of great extent and applied at the ground

surface are assumed to be transferred unaltered to the pipe. If

such a surcharge is to be a permanent feature then it is regarded

as dead loading but if it is temporary, as may occur during

construction, then its transitory affect should be allowed for in

the design.

If a uniform surcharge is of limited extent then the pressure that

it will exert on the buried pipe will tend to diminish with depth.

This effect can be determined by the use of influence factors such

as those proposed by Fadum (1941).

3.2.3 Concentrated loads

Concentrated loads are assumed to spread in accordance with the

Boussinesq theory (1885).

The all round pressure applied to the pipe at a point on its crown,

PL, due t 9 a concentrated load P acting at the ground surface, is

taken to be:-

3PH 3

PL - 2~C 5

where c is the distance from the point of application of P

to the point on the pipe crown.

3.2.4 Traffic and other transient surcharge loads

Surcharge loads on buried flexible pipes are normally assumed to be

the same as those on buried rigid pipes and can therefore be

20

obtained from published charts or tables.

Young and O'Reilly (1983) prepared sets of vehicle load charts to

cover road loads (three classes), railway loads (two classes) and

construction traffic loads. The chart that deals with road loadings

is reproduced in Fig. 12. Simplified tables using these charts have

since been published, (Young, Brennan and O'Reilly, 1986).

It will be obvious to the reader that the chart selected for

design should be the one that best represents the worst traffic

loading conditions that the buried pipe will experience. In this

connection it should be noted that Trott & Gaunt (1976) found

that, on large civil engineering works, buried pipes may well be

subjected to their highest life-time loads from contractor's

plant during construction.

3.2.5 Groundwater

Groundwater pressure will only affect the pipe if the water

table is above the invert level. If the water table is above the

crown of the pipe then, due to submergence, there is a reduction in

the apparent weight of the soil below ground water level and a

resultant reduction in the value of Pt, to which must be added the

value of the ground water pressure.

The ground water pressure, Pw, varies around the pipe with a

maximum value at the invert level equal to ~wh where h is the

height of the water table above the invert. This maximum value of

Pv should be added to the reduced value of Pt if submergence is

allowed for. However the procedure involves the determination of

the precise level of the permanent water table and the effect of a

temporary lowering of the water table during installation should

also be checked.

The approach suggested in the CIRIA Report is that submergence

21

effects should be ignored and that the soil's contribution to the

value of Pt taken as equal to wH, where w is the saturated unit

weight of the soil. This approach is justified when it is

considered that any theoretical increase in the value of Pt, and

hence in the theoretical value of the applied horizontal stress,

due to an effective stress analysis, is likely to be of the same

order as the uncertainty in the value asumed for the horizontal/

vertical pressure ratio.

This practice of ignoring submergence and simply regarding the soil

as saturated has been adopted by both the American Water Works

Association in their Design Manual MII (1964) and by the British

Water Research Centre (See Section 4). Gumbel (1983) also uses

this approach.

3.2.6 Vacuum conditions

If the absolute pressure in the pipe can reduce to a value

(A - pq) where A is the normal atmospheric pressure then the

pressure pq is treated as a uniform live load applied at the

ground surface, i.e. pq should be added to the value of Pt-

3.3 Determination Q~ m~ximDmdiameter/w~ll thickness ratio. D/~

A buried pipe is designed for one of two possible situations. It

will either be subjected to internal hydraulic pressure as it

carries a fluid under pressure, or it will be open to the

atmosphere, such as a culvert used as a pedestrian underpass or

when a watermain is drained.

For a pipe subjected to internal hydraulic pressure the maximum D/t

ratio is obtained from eqn. (3) in the form:-

D 2f a

t Pi

22

where fa = allowable circumferential (in this case tensile) stress

in the pipe walls.

Pi = internal fluid pressure within the pipe

The pipe is then checked to ensure that it will be safe to

withstand the circumferential compressive stress induced in its

wall by the action of the total, all-round, external pressure, Pt-

Buried pipes which will only be subjected to internal atmospheric

pressure are designed to withstand the action of the external

pressure, Pt, only and the maximum D/t ratio is obtained from the

expression:- D 2f a

t Pt

where fa = allowable circumferential (i.e. compressive) stress.

3.4 Determination Qf DiDe sDecific stiffness

The formula for the specific stiffness of the pipe, E'', is:-

EI m t l

D 3

As already discussed, the design procedure suggested by the CIRIA

Report No.78 is intended to cover buried pipes of specific

stiffnesses ranging from 2000 to ll000N/m 2. .

3.5 Determin@~ion Q~ the Gri~iG~l buGkling Dressure

The critical buckling pressure is the value of externally applied

hydraulic pressure that will cause buckling of the pipe walls and

can be obtained from Eqn.(10). The plane stress equivalent of this

equation is:-

/ ksREI pb = 2 /

~/ R 3

and putting E' = ksR and E , , _

8El

R 3

gives:-

23

Pb = 2 4 8E'E'' which is the formula suggested in the CIRIA

Report.

The soil modulus, E', is a variable with a value that depends upon

the type of soil, its density and the in-situ effective vertical

stress which varies with depth. CIRIA suggests that, for

preliminary design, a minimum value of i0 x 106N/m 2 can be used for

E' Higher values for E' are justified when the soil is well

compacted and can be taken as equal to the elastic modulus of the

fill, as determined from triaxial tests carried out on 100mm

diameter samples compacted to the expected insitu density.

If the pipe wall were unyielding then fb, the compressive stress

created by Pb, would be equal to PbD/2t and the CIRIA Report

therefore suggests that an expression for fa, the allowable

compressive stress in the pipe wall, is:-

1 fy x fb fa - x

FS fY + fb

Where FS = the factor of safety against buckling (usuallytaken as 3)

fy = the yield stress of the pipe material

It is this formula for fa which effectively limits the use of the

suggested design procedure to pipes with specific stiffnesses

between 2,000 and ii,000 N/m 2 The formula is not acceptable for

very flexible pipes whose high values of slenderness ratio

dictate that the risk of buckling is critically dependent upon

any initial imperfections that may have caused departures from

the circular shape of the cross section of the pipe.

~,~ D@fl@G~ion calcula~iQn~

The expressions for the total horizontal deflection, ax, by

24

Barnard and Spangler are both recommended by CIRIA and have been

discussed in sections 2.2.1 and 2.2.2. It should be noted that,

the value for ax calculated for dead loading should be increased

by multiplying by 1.5 (the deflection lag factor) unless the pipe

is to be subjected to an internal pressure at least equal to the

vertical soil pressure acting at the crown of the pipe, see

section 2.3.

The load to which a buried pipe will be subjected will be

applied in stages and it is convenient to modify the deflection

formula to use the term ~p where ~p is the change in the pressure

acting at the crown due to a change in the crown load, ~W c.

Now 5p.D = ~W c so the expression becomes:-

ax 0. 083 ~p - . . . . . . . . . . . ( Z l )

D 8E'' + 0.061E'

where ax/D is known as the horizontal diametral change.

Two separate deflection calculations are necessary for a buried

flexible pipe.

The first one is to estimate the amount of horizontal diametral

change that will occur during the placing and backfilling of the

pipe. The compaction of the side fill tends to compress the pipe

horizontally and to increase its vertical diameter whilst the

placing of backfill above the pipe tends to return it to its

original circular cross section. For this stage of construction the

diametral change can be estimated from eqn. (ll) with ~p replaced by

wH.

With corrugated steel pipes it is generally not necessary to make

this calculation. For other types of pipes, such as pipes with a

brittle protective coating, the limits of allowable deflections can

be obtained from the manufacturers' tables.

25

The second calculation is the estimation of the cycle of diametral

change that the pipe will experience under the action of live

loadings. The procedure is to determine a relevant value for ~p,

from published tables, etc. and to determine ax/D from Eqn. (ii).

Data presently available indicates that values of ax/D less than

0.03 do not affect either brittle pipe coatings, such as cement

mortar, or the compacted fill in which a pipe is encased.

9,7 MinimDm DiDe stiffness Qr h~n~linq stresses

Experience in the construction of corrugated steel plate culverts

indicate that for pipes of 3m diameter or less the specific pipe

stiffness should be such that E''D is greater than 4,000 N/m unless

some form of bracing is employed.

Example ~ - Spangler's approach

Data Pipe material - unlined mild steel

Diameter = 3.0m

Internal water pressure = 1,500kN/m a

Cover depth = 3.5m

Unit weight of granular fill = 20kN/m s

Ground water level - below invert of pipe

_ The pipeline will cross under a dual carriageway which will be subjected to main road loading.

Steel properties

E = 0.21 x 10*2N/m 2

f¥ = 240 x 106N/m 2

Backfill properties

E' = i0 x 106N/m 2

M~ximDm D/t ratio

Allowable tensile stress = fa = fy/2 = 120 x 106N/m 2

26

D 2f a 2 x 120 x 106 Maxm.

t P i 1.5 x 106 = 160

Hence: minimum t - 3.0

160 - 0.0188m = 20mm

I =

t 3 0.023

12 1 2 = 6.667 x i0- m 3

EI m f t --

D 3 3.03

0.21 x 10 *2 x 6.667 x 10 -7 = 5,185N/m 2

This value is within the range 2,000 to ll,000N/m 2 and is

therefore acceptable.

Ext~rn~l Dressur@ Pt

Soil load = wH = 20 x 103 x 3.5 = 70 x 103N/m 2

Uniform surcharge equivalent to main road loadings with H = 3.5m (From Fig.12) = 24 x 103N/m 2

Vacuum conditions (worst possible = -i atmosphere) = i00 x 103N 2

Pt = total = 194 x 103N/m 2

Buckling Dressure

Pb = 24 8E'E'' = 2 4 8 x i0 x 106 x 5185

= 1.288 x 106N/m 2

This corresponds to a wall stress value, fb:-

PbD fb =

2t

1.288 x 3 x 106

2 x 0.02

= 96.6 x 106N/m 2

NOW allowable compressive stress, fa - 1

FS

fy x fb

fY + fb

i.e. fa = 1 [240

3

x i0 e x 96.6 x i0 e]

] (240 + 96.6)106

= 23.0 x 106N/m 2

NOW Pb = 2fat/D which means that the maximum value of radial

27

pressure that can be withstood by the pipe is equal to:-

2 x 23.0 x 0.02 x 106

3.0

= 306 x 10SN/m 2

Pt = 194 x 103N/m 2 :- the chosen pipe is satisfactory.

Han¢linq stresses

E'' = 5,185N/m 2 so E''D = 5185 x 3.0 = 15,555N/m, which is

greater than 4,000 and therefore O.K.

D@fleGtion values

Considering the weight of the soil only:-

~p = 20 x 3.5 = 70kN/m 2 = 70 x 10SN/m 2

E'' = 5,185N/m 2 and E' = i0 x 106N/m 2

Hence 8E'' + 0.061E' = 8 x 5185 + 0.061 x i0 x 106

= 651 x 10SN/m 2

0.083 x 70 x i0 s x 3 x I0 s ax = = 26.7mm

651 x i0 s

Cyclic change in diameter due to live loading:-

~p = 24kN/m 2 = 24 x 10SN/m 2

and ax = 9.2

Totab. pipe deflection = 26.7 + 9.2 = 35.9 : 36mm

If the pipe was coated these values of deflection would now be

checked against the manufacturer's tables as to their suitability.

It should be noted that as the internal water pressure is

considerably greater than the vertical soil pressure acting on the

crown of the pipe, deflection lag effects can be ignored, i.e. the

value of D L has been taken as 1.0.

Ex~mpl~ ~ - Barnard's approach

The suitability of the pipe section selected by Spangler's method

28

in Example 3 will now be checked using Barnard's approach.

Data Pipe material - unlined mild steel

Diameter = 3m

Wall thickness = 20mm

Internal water pressure = 1,500kN/m 2

Cover depth = 3.5m

The pipeline will cross under a dual carriageway which will be subjected to main road loading.

Ground water level - below invert of pipe.

Steel properties

Yield stress, fy = 240 x 1012N/m 2

B~ckfill proDer~i~$

Granular fill

Unit weight = 20kN/m z '

Stress/strain relationships from Fig.8A

W~ll thickness

Thickness of wall required for internal water pressure.

Allowable tensile stress, fa, = fy/2 = 120 x 106N/m 2

For minimum wall thickness, t: fa

Required t = 1500 x 3

2 x 120

P i D

2t

= 18.75mm

Actual value used = 20mm ....... O.K.

SteD l

The total vertical pressure, Wv, acting on the top of the pipe is

the sum of the dead load pressure, w v and live load pressure, WvL '

In example 3 the value of WvL was found from Young & O'Reilly's

chart (Fig.12) to be = 24kN/m 2'

w v = 20 x 3.5 + 24 = 94kN/m 2

29

Step

The limiting height of cover H which the structural stiffness of

the pipe can support with 2% deflection without side support is

obtained from Fig.8B and equals 0.47m. This height is now used to

determine the value of vertical pressure, wp, that the pipe can

support.

If wp > w v then the pipe can carry the vertical loads on its own

and satisfies the conditions of Case I.

If wp < w v then aid from the active pressure of loose fill is

necessary and the pipe must be considered as Case II.

wp = 20 x 0.47 = 9.4kN/m 2

wp (9.4kN/m 2) < w v (94kN/m 2) : active pressure must be considered.

SteD

Wa, the active pressure value determined, is that acting at the

level of the pipe's horizontal axis and is equal to the vertical

pressure, due to dead load only, times a factor (either 0.5 for

clay or 0.33 for sand).

If (wp + Wa) > w v then the pipe can carry its vertical load with

the aid of loose fill around it and satisfies the conditions of

Case II.

If (wp +-wa) < w v then the pipe requires more aid from the side

fill and must be considered as Case III.

Vertical pressure at pipe axis = 20(3.5 + 1.5) = 100kN/m 2

Sidefill is sand, hence w a = 0.33 x i00 = 33kN/m 2

(wp + Wa) = 9.4 + 33 : 42kN/m 2 < w v (94kN/m 2)

The pipe must be considered as a Case III situation.

Step

If the limiting compressive stress is taken as 52 x 103kN/m 2

(See 2.2.2 Case III) then the limiting height of cover, Z, is

30

obtained from the expression:-

52 x 103 x 2t 20 x 103 x Z =

D

If H > Z £hen the assumed pipe is too weak and must be rejected.

If H < Z then the assumed pipe is satisfactory.

52 x i000 x 2 x 20 Z = = 34.7m

20 x 3 x i000

H = 3.5 hence the assumed pipe is satisfactory.

SteD

The contact pressure, wh, is determined from the formula:-

w h = F(w v - wp)

Considering dead load only, with F = 1.35:-

Vertical pressure at crown of pipe = wv = 70kN/m 2

Hence w h = 1.35(70 - 9.4) = 81.8kN/m 2

Considering dead load + surcharge, with F = 1.35:-

Vertical pressure at crown of pipe = w v = 70 + 24 = 94kN/m 2

Hence w h = 1.35(94 - 9.4) = ll4.2kN/m 2

Deflection values

SteD

a) Deflection of pipe due to dead loads only

With dead load only: w h = 81.8kN/m 2

Vertical pressure at hoz. axis of pipe = 20(3.5+1.5) = 100kN/m 2

which is regarded as the minor principal stress, ~3, whilst w h

(81.8kN/m 2) iS regarded as the deviator stress, (~z - ~3)-

The value of ~wh that corresponds to these stress values can be

estimated from Fig.8A (for well graded sand) and is appoximately

0.7%. Hence the deflection of the pipe due to dead load is in the

order of 2 x .007 x 3000 = 42mm.

31

b) Deflection of pipe under dead and superimposed loads

With dead load + surcharge: w~ = ll4.2kN/m 2

Vertical pressure acting at centre of pipe (due to dead load only)

= 100kN/m 2, which is regarded as ~3 whilst w h (ll4.2kN/m 2) is

regarded as (~I - ~3)- The value of ~wh that corresponds to these

values is found from Fig.8A and equals 0.9%.

The total horizontal deflection of the pipe is therefore:-

ax = 2 x .009 x 3000 = 54mm.

c) Cyclic change in diameter

Due to the superimposed loading the cyclic change in diameter

= 54 - 42 = 12mm.

4 WATER RESEARCH CENTRE DESIGN PROCEDURE FOR BURIED

FLEXIBLE PIPES

In 1988 the Water Research Centre published a manual which deals

with the design of buried, rigid, semi-rigid and flexible water

carrying pipes. As with the design methods recommended by CIRIA,

the methods proposed by WRC are mainly concerned with the design of

the pipe ring although possible longitudinal effects are discussed. °

The Manual stresses that it is only concerned with water carrying

pipes and also that, whilst its design philosophy is of general

application, the assumed performances of materials, quality levels

and installation procedure are based on UK practice and may not be

applicable in non UK situations. A summary of the manual's

recommendations for buried flexible pipe design is given below.

i) A trial pipe section is selected by a consideration of the

proposed internal water pressure value and possible handling

stresses during installation, (As in both examples 3 and 4). Sudden

32

stoppages in the water flow, due to the operation of a valve, etc.,

can produce surge pressures which can be either positive or

negative and these should be allowed for.

ii) The design limits for the pipe are determined. The value of

the critical buckling pressure is calculated whilst the value of

allowable deflection is either taken to be 3% or is established

from manufacturers' tables. As has been discussed, bending effects

are rarely considered for flexible steel pipes.

iii) Values for DL, the deflection lag factor, and DR, the

re-rounding factor, are established.

iv) A suitable backfill material is selected.

v) The buckling stability is examined.

vi) The initial and long-term pipe deflections are determined,

and checked against their permissible values.

4,~ Backfill materi~l selection

The Water research Centre's manual uses E', Spangler's soil

modulus, for its calculations and tabulates a set of conservative

values for design work. Those values applicable to flexible buried

pipes are given in Table i. It should be noted that, in the table,

BSma isthe maximum dry density achieved in the 2.5kg rammer

compaction test described in BS1377: Part 2 (1990).

Fine material is that material passing the 63um sieve.

33

Table 1 Suggested E' values for buried flexible pipes. From W.R.C. Manual (1988)

Type of backfill

Gravel - single size

Gravel - graded

Sand & coarse grained soil with less than 12% fines

Coarse grained soil with more than 12% fines

Fine grained soil with medium to no plasticity (LL < 50%) and more than 25% coarse particles

Fine grained soil with medium to no plasticity (LL < 50%) and less than 25% coarse particles

Modulus of soil reaction, E', (MN/m 2)

Loose

3

80% BSmax

7

5

3

1

85% BSmax

90% BSmax

i0

i0

7

5

95% BSmax

14

20

14

i0

Because of the difficulty of achieving 95% BS maximum compaction

in the field the maximum value of compaction should normally be

assumed to be limited to 90% BSma x and the maximum value for E'

used in calculations should be 10MN/m 2. This is more conservative o

than therecommendation given in the CIRIA Report No. 78 which

suggests a minimum value for E' of 10MN/m 2 for preliminary designs

(See section 3.5).

The WRC manual also considers that, for a buried flexible pipe,

fine grained soils of medium to high plasticity, LL > 50%, are

unsuitable as backfill and that soils with E' values less than

3.0MN/m 2 should also not be used as they cannot offer adequate

lateral support to the pipe.

34

4,2 Pipe deflections

Pipe deflections are calculated from an equation based on

Spangler's formula (Eqn 7).

KbW¢ av = x D R ...... (12)

8EI/D 3 + 0.061E'

where av = change in vertical diameter

K b = bedding factor

W c = vertical crown load/unit length of pipe

= Pt D = (DLPv + Ps + pq)D

Pv = vertical pressure at crown due to backfill

pq = value of vacuum pressure (if any)

Ps = vertical pressure due to any surcharge loads

D L = deflection lag factor (defined in section 2.3)

D R = re-rounding factor

EI/D 3 = pipe stiffness

E' = Spangler's modulus of backfill material

As explained in 2.2.1, the bedding factor, KB, has a value

ranging from 0.Ii to 0.083. The WRC Manual recommends that a value

of 0.083 is taken where the material on which the pipe sits is

gravel and 0.ii when the bedding material is sand.

Values for DL, the deflection lag factor, suggested by WRC are

given in Table 2. WRC maintains that the value of the re-rounding

factor, Dr, depends upon the value of the internal water pressure

and equals 1.0 for Pi = 300kN/m2 and 0.5 for Pi = 20,000kN/m2

However for cover depths greater than 2.5m D R should be taken as

1.0, irrespective of the value of the internal water pressure.

35

Table 2 Deflection lag factor, D L, after WRC , (1988).

Backfill Normal case Long time lapse - backfilling at material least one year after initial

pressurisation

I Degree of compaction

All I Heavy I Light

Gravel I 1.0 1.00 1.25

Sand I 1.0 1.25 2.00

Note. Backfill materials other than sands or gravels may produce

deflection lag factor values of up to 3.0.

It should be noted that Spangler's equation is for ax, the change

in the horizontal diameter, whereas WRC use av, the change in the

vertical diameter. Generally ax and av are equal and opposite but,

in the case of non-elliptical pipe deformations, this may not be

the case.

4,~ BDcklina

If the designer cannot be certain that there will be constant soil

support throughout the life of the pipe it is recommended by WRC,

for water mains with less than 1.5m cover, that soil support should

be ignored and the buckling resistance determined by using

Timoshenko's buckling equation, Eqn.(9):

24EI/D 3

p5 = (I - vp 2)

where Pb = critical external pressure required to cause buckling.

EI/D 3 = E'' = the pipe stiffness

vp = Poisson's ratio of the pipe material

For pipes buried at cover depths greater than 1.5m, or shallower

pipes where soil support can safely be assumed, the value of P5

should be calculated from Luscher's formula, based on the work of

%d

36

Luscher and Hoeg (1964).

Pb = ~ 32E'EI/D~

As the buckling resistance of the pipe ring is reduced by any

out-of-roundness of the pipe caused by the deflection achieved just

before bucking failure WRC suggests that a reduction factor, Db,

should normally be applied to the value of Pb- A formula for D 5 is:

3ax D b = 1

D

As buckling can occur after a period of time the value of ax used

in the formula can be the long term value, i.e including DL, the

lag deflection factor, but not including the re-rounding factor,

D R •

Example

The suitability of the pipe section discussed in Examples 3

and 4 will now be examined using WRC's design approach with the

properties of the steel pipe and the backfill assumed to be the

same as in Example 3.

Selection of trial PiPe sectiQn

As in both examples 3 and 4 the thickness of the pipe wall (20mm)

was obtained by allowing for the internal water pressure of value

1,500kN/m 2 .

Buckling stability

Spangler's modulus for the backfill, E' = i0 x 106N/m 2

Moment of inertia of pipe ring, t 3 0.02 s

12 12 = 6.667 x 10-Tm 3

Pipe stiffness - EI 0.210 x 6.67

D 3 3.0 3 x l0 s = 5185N/m 2

37

Note In order to have the same pipe loading conditions in this

example as in examples 3 and 4 a vacuum surge pressure, pq, of

100kN/m 2 has been assumed.

Hence total vertical pressure at crown, Pt, = Pv + P$ + Pq

i.e. Pt = 3.5 x 20 + 24 + I00 = 194kN/m 2

From section 4.3 it is seen that, as the cover depth exceeds 1.5m,

the value for Pb is obtained from Luscher's formula:

P5 = ~ 32E'EI/D~ = ~ 32 x i0000 x 5.185 = 1288kN/m 2

Pt = 194 so that FS, the factor of safety against buckling,

= 1288/194 = 6.6. The risk of buckling is negligible.

Deflection values

Deflection values are obtained by the use of Eqn.(12):

KbWc

av = x D R 8EI/D 3 + 0.061E'

For this example we can assume that av = ax

Considering only the weight of backfill:

W c = pvD = 3.5 x 20 x D = 70D

ax K5 x 70 Hence - x D R

D 8EI/D 3 + 0.061E'

For deptks greater than 2.5m, D~ = 1.0

~x 0.083 x 70

D 8 x 5.185 + 0.061 x 10000

= 0.0089 = 0.89%

Hence ax = .0089 x 3000 = 26.8mm

With live loading Pt = 70 + 24 = 94kN/m 2 and ax = 36mm

Cyclic change in diameter due to live loading : 9mm

Total pipe deflection = 36mm

38

5 COMMENTS ON DESIGN METHODS PRESENTLY USED IN BRITAIN

As can be seen from the comments contained in section 2.2 the

design approaches suggested both in the CIRIA Report 78 and in the

WRC Manual are mainly semi-empirical and are based on the

interpretation of test results and long experience. This approach

has undoubtedly led to safe designs but it can be conservative and,

with the increasing use of extremely thin-walled pipes, may well be

uneconomical. There is a need for a new design approach that will

take into account the new types of flexible pipes now available.

In the past few years there has been a considerable increase in the

amount of investigation into the behaviour of buried flexible

pipes. One field of research has been the analysis of pipe-soil

interaction and there has been a move towards the assumption that

the soil acts as an isotropic elastic medium. Although the stress-

strain curve of a soil medium is not linear, Katona (1978) has

shown that little error is involved if linearity is assumed and it

is now generally felt that the assumption that the soil is an

isotropic linear elastic medium is more realistic than the Winkler

model assumed by earlier workers. This, and other work relevant to

the proposed new design method, will be described in the succeeding

reports of the series.

6 ACKNOWLEDGEMENTS

This report is one of four and forms part of a research contract

with the Ground Engineering Division (Division Head Dr M.P.

O'Reilly) of the Structures Group of the Transport and Road

Research Laboratory. The material on which this report is based has

39

been taken from several sources and particularly from J.E. Gumbel's

PhD thesis (1983). To all these sources (see list of references)

the authors wish to acknowledge their indebtedness.

7 REFERENCES

American Waterworks Association (1964) Steel Pipe Manual MII.

Barnard, R.E. (1957) "Design and deflection control of buried steel pipe supporting earth loads and live loads". Proc. Am. Soc. Testing Materials, Voi.57, pp 1233-1258.

Boussinesq, J. (1885) "Application des potentiels a l'etude de l'equilibre et du mouvement des solides elastiques". Gauthier-Villars (Paris).

British Standards BS 1377 (1990) "Methods of test for soils for civil engineering purposes. Part 2, Classification tests"

Bulson, P.S. (1985) "Buried strustures - static and dynamic strength" Hall Ltd., London.

Chapman and

Chelapati, C.V. (1966) "Critical pressures for radially supported cylinders" Tech. Note N-773, U.SD. Naval Civ. Engng. Lab., Port Hueneme, Calif., Jan.

Cheney, J.A. (1963) "Bending and buckling of thin-walled open section rings".Jour. Engng. Mech. Div. Proc. Am. Soc. Civ. Engrs., Vol. 86, No. EM5, Oct., pp 17-44.

Compston, D.G., Gray, P., Schofield, A.N. and Shan, C.D. (1978) "Design and construction of buried thin-wall pipes" Construction Industry Research and Information Association, Report No.78, July.

Crabb, G.I. and Carder, D.R. (1985) "Loading tests on buried flexible pipes to validate a new design model" Research Report 28, Transport & Road Research Laboratory, Dept. of Transport, Crowthorne, Berks.

Duns, C.S. (1966) "The elastic critical load of a cylindrical shell embedded in an elastic medium". Report CE/I0/66, Univ. of Southampton.

Fadum, R.E. (1941) "Influence values for vertical stresses in a semi-infinite solid due to surface loads". School of Engineering, Harvard University.

40

Gumbel, J.E. (1983) "Analysis and design of buried flexible pipes" University of Surrey.

PhD Thesis,

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.

Habib, P. & Luong, M.P. (1965) "Comportement des tuyaux souples enterres" (Behaviour of flexible conduits). Proc. 6th Int. Conf. SMFE, Montreal, Vol. II pp373-376.

Habib, P. and Luong, M.P. (1966) "Etude theorique et experimentale de la stabilitie des tuyaux et buses cylindriques places dans les remblais" (Theoretical and experimental study of the stability of pipes and cylindrical conduits placed in fill). Annales de l'Institut du Batiment et des Travaix Publics, No. 218.

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.

Krizek, R.J., Parmelee, R.A., Kay, J.N. and Elnaggar, H.A. (1971) "Structural analysis and design of pipe culverts" Nat. Cooperative Highw. Res. Prog. Report 116, Washington D.C.: Highw. Res. Board.

Levy, M. (1884) "Memoire sur un nouveau cas integrable du probleme de l'eastique et l'une de ses applications" (Memoir on a new integrable case of the problem of elasticity and one of its applications) J. Math. Pure et Appl. (Liouville), Series 3, Vol. i0, pp 5-42.

Link, H. (1963) "Beitrag zum Knickproblem des elastich gebetteten Kreisbogentragers".(A contribution on the problem of buckling of an elastically embedded circular arch). Der Stabhlbau, 32 (7), pp 199-203.

Luscher, U. (1966) "Buckling of soil-surrounded tubes" J. Soil Mech. Found. Div., Proc. Am. Soc. Civ. Engrs., Voi.92, No. SM6, Nov. pp 211-228.

Luscher, U. & Hoeg, K. (1964) "The interaction between a structural tube and the surrounding soil." US Air Force Weapons Lab., Kirtland Air Force Base, Rprt. RT TDR-63-3109.

Marston, A. (1930) "The theory of external loads on closed conduits in the light of recent experiments" Iowa Engng. Expt. Stn., Bulletin No. 96.

41

Marston, A. & Anderson, A.O. (1913) "The theory of loads on pipes in ditches and tests of cement and clay drain tile and sewer pipe", Iowa State University Engineering Research Institute, Bulletin No.31, Ames, Iowa.

Meyerhof, G.G. and 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.

Morley, A. (.1943) "Strength of Materials" Longmans, Green and Co., London.

Smith, G.N. (1991) "Buried Flexible Pipes - The analytical method developed by Gumbel for TRRL". Contractor Report 229, Transport & Road Research Laboratory, Dept. of Transport, Crowthorne, Berks.

Spangler, M.G. (1941) "The structural design of flexible pipe culverts" Iowa Engng. Expt. Stn., Ames, Iowa.

Bulletin No.153,

Terzaghi, K. (1955) "Evaluation of coefficients of subgrade reaction". Geotechnique, Vol.5, No. 4, Dec. pp 279-326

Trott, J.J. & GAUNT, J. (1976) "Experimental pipelines under a major road: performance during and after road construction". Dept. of the Environ., Transport & Road Research Laboratory, Report 692, Crowthorne, Berks.

Wang, C.T. (1953) "Applied elasticity", McGraw-Hill, New York.

Water Research Centre (1988) "Pipe materials selection manual - Water mains, U.K. Edition, Appendix 1 - Structural design of pipelines". W.R.C., Marlow, Bucks.

White, H~L. and Layer, J.P. (1960) "The corrugated metal conduit as a compression ring" Highw. Res. Board, Voi.39, pp 389-397.

Proc.

Young, O.C. and O'Reilly, M.P. (1983) "A guide to design loadings for buried rigid pipes" Transport and Road Research Laboratory, Department of Transport, London.

Young, O.C., Brennan, G. & O'Reilly, M.P. (1986) "Simplified tables of external loads on buried pipelines" Transport and Road Research Laboratory, Department of Transport, HMSO, London.

42

APPENDIX I

TERMINOLOGY

The basic pipe terms used in this text are listed below and are

illustrated in Fig.l.

Crown - the uppermost part of the pipe ring.

Haunch - the side of the pipe between the springing and the invert

Invert - the lowermost part of the pipe ring.

Shoulder - the side of the pipe between the crown and the springing.

Springing - the level of the horizontal axis of the pipe ring.

t - wall thickness

R - external radius of pipe

R i - internal radius of pipe

r - variable radius within wall thickness

43

APPENDIX II

BASIC PIPE THEORY

AII,I PiDes $Dbj~cted ~Q in~@rn~l flDi~ Dressur@,

AII.I.I Thick walled pipe.

A thick cylinder subjected to an internal fluid pressure is acted

upon by three principal stresses:- a circumferential tensile

stress, a radial compressive stress and a tensile stress acting

parallel to the axis of the cylinder caused by the fluid pressure

acting on the ends of the cylinder. With a pipe there are no ends

so the axial tensile stress is generally absent, although axial

tensile stresses can be induced by the action of turning valves on

or off along the length of the pipe. Fig.2A shows the cross

section of a thick pipe, of internal radius R i and external radius

R and subjected to an internal pressure, Pl, and an external

pressure, Pe, both applied uniformly around the circumferences of

the pipe.

Let

d e = the circumferential tensile stress, or hoop stress, acting at radius r within the thickness of the pipe and at right angles to the radius.

~r = the radial compressive stress acting at r, any variable radius.

A pipe is assumed to operate in a state of plane strain and the

study of its failure mechanisms involves the analysis of the cross "'

section of the pipe. If it is assumed that cross sections

originally plane will remain plane then it can be shown that:-

b d e - + a ...... (i)

r 2

b and ~r - a ...... (2)

2 r

44

where a and b are constants which depend upon the dimensions of the

pipe and the magnitude of the applied pressures. The formulae for a

and b are derived in most strength of materials text books, for

example Morley (1943), and are:-

Pe R2 - PiRi 2 Ri2R2(pe - Pi)

a = and b = R 2 _ Ri 2 R 2 - Ri 2

The values of the hoop tension and the radial pressure are not

constant but vary over the cross section of the pipe. The forms of

these variations are illustrated in Fig.2B (see example i below).

ExamPle !

A water main has internal and external diameters of 150 and 200mm

respectively and is subjected to a uniform water pressure of

7500kN/m 2. Prepare a plot showing the variation of the

circumferential tension, ce, and the radial pressure, ~r, along

the length of a radius.

Solution

Pi = 7500kN/m2", Pe = 0;

Hence a = 7500 x .1502

0.22 - 0.152

R i = 0.150m; R = 0.2m

7500 x .152 x .22 = 9642.86; b =

0.22 - 0.152 = 385.71

By selecting various values for r the corresponding values for ~e

and cr can be obtained, as shown tabulated below.

r (m) .15 .16 .17 .18 .19 .20 ~e (kN/m2) 26800 24700 23000 21500 20300 19300 ~r (kN/m2) 7500 5420 3700 2260 1040 0

The resulting plots are shown in Fig.2.B.

AII.I.2 Thin walled pipe

Consider the case of a large diameter pipe, with a relatively thin

wall thickness, t, subjected to an internal hydrostatic pressure,

Pi, (Fig.3A). Due to the thinness of the wall, the tensile hoop

45

stress, ~e, tends to be uniform and to act at the centre of the

wall thickness, at radius r, whereas the radial stress is reduced

to a negligible value. Under these circumstances a simple way to

determine the value of the hoop stress becomes possible.

Consider the equilibrium of the element of cross section contained

within the angle 80 and a unit length of the wall, between the

parallel sections AA and BB, (Fig.3B):-

Let the radial force acting on the element due to Pi be ~P0-

Then 8Pc = Pir~e and the vertical component of this force is

Pir~esine. Hence the total force normal to a diameter is:-

O P0 = I Pirsined0 = 2Pir

o 0

Resisting force due to hoop stress = 2¢et (See Fig.3A) and, for

equilibium, 2cet = 2Pir

i.e. ~e Pi r

t

r is the' mean radius and equal to 0.5(R i + R) but, for most

thin walled pipe calculations the outside radius, R, can be used

in place of r and the equation written as:-

Pi R = . . . . . . . . (3)

t

Determine the tensile hoop stress in a three metre diameter pipe

which has a wall thickness of 25mm and is subjected to a uniform

internal pressure of 1500kN/m 2.

Solution pi R 1500 x 3

~e - =

2t 2 x 0.025 = 90MN/m 2

46

AII,2 PiDes subjected to ~ fluid

If a pipe of circular cross section is subjected to an external

hydraulic pressure then the symmetry of loading ensures that no

bending moments are induced in the pipe walls, provided that the

pipe ring maintains its circular shape as it undergoes compression.

For a thick pipe if the external hydraulic pressure, Pe, is the

only pressure acting on the pipe then the value of Pi = 0 and,

substituting into equations (I) and (2), we obtain the following

expressions for Ce and Cr, the hoop and radial compressive

stresses:-

[ i2 I Pe 2 [ i2 1 - + 1 and ¢r = 1 cTe 2 2 R2 2 r 2 R 2 - R i r - R i

It is seen that when R i is put equal to r then Cr = 0

and, as R ~ m and ¢8 ~ 2Pe:-

2Pe R2

Ce -- R 2 _ Ri 2

It can be shown, by the theory of elasticity, see Wang (1953) for

example, that the inward displacement , w, caused by the elastic

compression of the pipe ring, at any point on the inner surface of

the pipe is found from the equation:-

W =

2RiR2pe

E(R 2 - Ri 2 ) where E = Young's Modulus of

the pipe material.

Hence w - ~eRi

E ....... (4)

For a thin pipe the compressive hoop stress, ~e, will be almost

uniform and, by comparing with equation (3), the formula for this

stress, for a pipe of radius R, is seen to be:-

Pe R

Ge - t

47

and the inward displacement, w, at any point on the circumference

can be obtained by modifying eqn.(4):-

~0R w =

E

AII.2.1 Buckling effects

There is a significant difference in the behaviour of thick and

thin walled pipes when the applied pressure is external instead of

internal. If its walls are thick enough a thick walled pipe will

maintain its circular cross section until the magnitude of its

compressive hoop stress reaches its limiting value and failure by

crushing is imminent. A thin walled pipe behaves in a manner

similar to an axially loaded strut and may collapse by either

buckling or by crushing.

As established in the previous section, the formula for the

compressive circumferential stress, ~e, in a thin walled pipe of

radius R, is:-

Pe R ~e -

t

Consider a unit length of the pipe. Then the total circumferential

length of the wall = 2~R and the area of the rectangular cross

section acted upon by the compressive stress = t x unit length = t,

whilst its moment of inertia, I, = ts/12.

The total thrust, T, acting on the cross section of the pipe is

equal to the compressive stress times the area. Hence:-

PeRt T = ce x t x 1 = = pe R

t

An approximation to the critical value of T, i.e. the value that

could produce buckling failure, can be deduced by analogy to

Euler's rules if it is assumed that the form of collapse will be -o

48

symmetrical. The simplest form of collapse, proved by experiment,

is illustrated in Fig.4 and has four points of contraflexure

(points of change over from increased to decreased curvature) A,B,C

and D. These four points divide the cross section into four equal

arcs, each of length ~R/2.

Euler's formula for the critical load, T, acting on a strut of

length L and hinged at both ends, is given by the expression:-

~2EI T -

L 2

where I = Moment of inertia and E = Young's Modulus.

If we assume that the length L is the distance between two

adjacent points of contraflexure, say AD, then L = ~R/2. If Pb is

the value of Pe that will cause buckling then:

4~2EI pbR = T =

~2R2

4 E I or Pb -

R 3

Now I = t3/12 so that another expression for Pb is;--

P b ~ . . . . .

3 R 3 D

The final expression glves an approximate value for the critical

external pressure. It should be noted that this value has been

determined by analogy and not by rigorous analysis. A more rigid

mathematical analysis leads to the classic ring buckling formula

due to Levy (1884):-

3EI

Pb - R 3

. . . . . . . . . . . . . . ( 5 )

49

APPENDIX III

Determination Qf ~h~ averaae GQver Qv~r ~h@ pipQ width

For depths of cover H between 0.25D and D the depth of soil above

the pipe is taken as equal to the average cover over the pipe

width. (See Para 3.2.1). The argument below explains why this value

of average cover is taken as equal to H + 0.107D.

The cross section of a buried pipe is illustrated below.

D/2

H

I L D

Ground level

Crown level

Pipe axis

H = cover = height of soil above pipe crown

D = outside diameter of pipe

Area of the half pipe above = ! ~D 2 = 0.3927D 2 its horizontal axis 2 4

Area of soil and pipe between = D.D = 0.5D 2 pipe axis and crown level 2

Area of soil between axis = (0.5 - 0.3927)D 2 = 0.107D 2 and crown level

Average cover of soil = H + 0.107D 2 = H + 0.107D over pipe width D

50

J

FIGURES

51

Crown

h0u lde r

-t-- Springing

i aunch

g. Pipe Fi ferminotogy

(_A/ - °

0",, + ~ ~

R adiu s(mm)l

E - 10000

~, -20000 u~

,- -30000 .,¢..

or)

Fig . 2 Examp[e 1

1 '5 0 O0 ( f i l l I l / , I I1" , . I / I / ' 1 / ' "

~ t st ress d r

d e HooI~ stress

(8)

X

e~ (A) o-et

Fig. 3 Thin-walLed pipe

A B

A

%

u

[

IB) 8

A

'\~ I h /

o " ~ -+- - - .7-c

.,,,t

Kp~

Fig. /. Simpte buck[ing

P I

0"25

p•P- Kp 0 • 2 "4 .6 "8 1-0 K

F ig . SA

I P

Loading on thick watled pipe B V a r i a f i o n of M max

with K

Fig 6 Wc/D c osd,.

'S Sp[angler assumed pressure distribufi on

Fig.7

Barnard's equivalent earth column

o : I I " -

~..i ~ " ' , ,

"<~ ,'~.' 0

Well gr~leQ ~ n d

'=°1 i i l / / __ | AolSlie~l Lateral. / I A Lean claw Sand with clay bin(~er

a ~ . I - - P ~ , , , . , a - ~ N ' m ' 2 q 7 - 1 - zoo . " I I I ~ / E i , ~ z ' . / l ~ I .°o,,.,.,-, 1 3 '°°1 = J I I

---- ~"~' [ - - ' ~ " Pressunl--kN/m: 200 ' : ' ooo %!!

0 I 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4

Axial strain. ~,,,.n (%} Axial strain, ewn (%| Axiai s1~i. , ~wn (%)

(,4.) A x i a l s t ra ins in t y p i c a l soi ls .- 2 0

O

- 2 O "S

°

Fig.8

Diagrams relating to calculation of pipe deflection (based on AWWA Manual) and reproduced from CIRIA

Report No.78.

10

0.5

0.2. 2 0 0 5 0 0 1 0 0 0 2 0 0 0 5 0 0 0

P~I dia. (ram)

B) Theoretical height of cover for stee: pipe with 2% deflection.

(Unit w e i g h t o f s o i l = 20kN/m ~)

"- ks

Fig,9 Winkter mode[ soit s t i f fness

of i

f ~

/ \

(A) Mu[ti-v~ave mode (B) Single ,dave mode

Fig_.fO Genera[ and local ring buckling i

I m

J

Fig l t . Excessive d e f l e c t i o n causing snap-through • buck[i ng

i

After £ I R! A) 1978

Z

t , -

D

I

100

9O

7 0

6O

5 0 - -

40

30

20

10 9

8

i t I I I

-

Fields

I I I

Main roads

Light roa~i

I I I i

m

!

!

m

Wc$ u = (7 B c Inclusive of relevant dynamic faclors due to ~mpact

(See sect,on 2.6.1.4)

I I | I | I I

0.5 0.6 0.7 0.80.91.0 I I 2 3

Depth, H (m|

4 5 6 7 8 9 10

Fig.12 Equivalent pipe loads "due to v~nicle" loads

(From Young and o'Reilly, (1983)