Liner Buckling Study

GEOSYNTHETICS INTERNATIONAL • 2001, VOL. 8, NO. 4 303

Technical Paper by A.S. Dhar and I.D. MooreLINER BUCKLING IN PROFILEDPOLYETHYLENE PIPES

ABSTRACT: Thermoplastic pipes are often manufactured with profiled walls to max-imize the flexural stiffness of the pipe for a given amount of polymer. Thin elements inthe profile can buckle under the influence of large earth pressures associated with deepburial or other extreme loading conditions. Earth load tests have been conducted onhigh density polyethylene pipes with a number of different wall profiles. Two high-pressure pipe test cells have been used to conduct these tests. Observations of localbuckling in the internal liners of these products have been examined and compared tostability assessments based on the conventional equation for buckling in stiffened platestructures (following modification of that equation to an equation that defines criticalstrain instead of critical stress). The strain levels that develop in the liner are, however,dependent on three-dimensional bending within the pipe profile. Provided the effects ofthree-dimensional bending in the pipe profile are considered, the modified Bryan equa-tion appears to be a useful tool for quantifying liner stability and should be consideredfor inclusion in limit-state design procedures for these structures.

AUTHORS: A.S. Dhar, Graduate student, Department of Civil and EnvironmentalEngineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9,Telephone: 1/519-661-2111, Ext. 88517, Telefax: 1/519-661-3942, E-mail:[email protected]; and I.D. Moore, Professor and Canada Research Chair, Depart-ment of Civil Engineering, Queen’s University, Kingston, Canada K7L 3N6, Tele-phone: 1/613-533-3160, Telefax: 1/613-553-2128, E-mail: [email protected].

KEYWORDS: Thermoplastic pipe, HDPE, Culvert, Deep burial, Local buckling,Plate-buckling model, Limit state design.

PUBLICATION: Geosynthetics International is published by the Industrial FabricsAssociation International, 1801 County Road B West, Roseville, Minnesota 55113-4061, USA, Telephone: 1/612-222-2508, Telefax: 1/612-631-9334. GeosyntheticsInternational is registered under ISSN 1072-6349.

DATE: Original manuscript submitted 18 July 2000, revised version received 9 April2001, and accepted 12 April 2001. Discussion open until 1 April 2002.

REFERENCE: Dhar, A.S. and Moore, I.D., 2001, “Liner Buckling in Profiled Poly-ethylene Pipes”, Geosynthetics International, Vol. 8, No. 4, pp. 303-326.

DHAR AND MOORE • Liner Buckling in Profiled Polyethylene Pipes

304 GEOSYNTHETICS INTERNATIONAL • 2001, VOL. 8, NO. 4

1 INTRODUCTION

Large diameter thermoplastic pipes are manufactured with a variety of different wallgeometries in order to attain effective utilization of the polymer material in resistingbending. Decentralization of the material from the neutral axis of the wall rendershigher pipe stiffnesses with reduced use of material. However, high compressive strainacross pipe sections, especially in deeply buried pipe, may cause local buckling on var-ious components of the profile that can compromise the pipe’s structural integrity. Thepresent paper focuses on the prediction of local buckling on the liner (the inner unsup-ported wall) of a number of different thermoplastic pipes. Liner buckling can beexpected to affect the hydraulic properties of the pipe. Liner buckling may also influ-ence the structural integrity of profiles where loss of liner stiffness leaves little or nomaterial providing stiffness in the inner surface of the pipe wall.

Four commonly used profiles for large-diameter, high density polyethylene(HDPE) pipes are shown in Figure 1. Figures 1a and 1b are twin-walled annular pipeswith an internal diameter of 610 mm, the former with a deeper profile and smallerpitch (i.e., more closely spaced corrugations). Figures 1c and 1d are helically woundprofiles with helix angles of 2o and 6o, respectively. The third has a rectangular boxedprofile and a 710 mm internal diameter. The fourth possesses a tubular profile, with a1,060 mm internal diameter. Tests have been conducted on each of these pipes usingthe two pipe test vessels developed at The University of Western Ontario to model typ-ical deep burial conditions (Brachman et al. 2000). Tests on pipes have been performedunder both biaxial and axisymmetric loading conditions.

The present paper reviews the literature on local buckling, describes the pipe test-ing program, presents analyses of the test data, and examines the use of elastic buck-ling solutions for prediction of liner buckling.

2 LITERATURE

In the past several decades, various workers have undertaken studies to improve ther-moplastic pipe design. Local buckling of individual structural elements in profiledHDPE pipes first received attention by Hashash (1991) and DiFrancesco (1993), whoobserved ripples during field and laboratory tests on twin-wall HDPE pipe, respec-tively. Moore and Hu (1995) demonstrated that these ripples were due to local buck-ling in the inner wall, i.e., the liner. Moore (1996) derived solutions for liner bucklingusing the linear buckling theory for a stiffened cylindrical shell (Moore 1990). Thesolution was expressed graphically in terms of critical compressive hoop strain in theliner. Moser (1998) agreed that local buckling is a critical performance limiting factorin tests conducted on profile-wall pipes.

Moore and Laidlaw (1997) examined local buckling elsewhere in the corrugation,utilizing the stiffened plate model from the “Structural Plastics Design Manual”(ASCE 1984) for prediction purposes. The Manual makes use of the Bryan (1891)equation commonly used to quantify metal plate buckling. The Euler formula for thinplate buckling, as proposed by Bryan (1891), is:



(1)

where: σcr = critical buckling stress; k = edge support coefficient; W = plate width;t = plate thickness; E = elastic modulus; and ν = Poisson’s ratio.

Moore and Laidlaw (1997) expressed the critical buckling criterion in the form ofcritical hoop strain to avoid problems associated with the nonlinear, time-dependent

σcrkπ2E

12 1 ν2–( ) W t⁄( )2--------------------------------------------=

Figure 1. Types of pipe profiles: (a) lined corrugated (profile depth, 55.15 mm): (b) linedcorrugated (profile depth, 55.15 mm); (c) rectangular box profile; (d) lined tubularprofile.

(a)

(b)

(c)

(d)

Crest

Valley

Web

Liner

Liner

Rib

Outer wall

Liner 1

(1) (2)

Liner 2

Wall 2Wall 1(3)

Joint(4) (4)



material modulus for the polyethylene. Moore and Laidlaw expressed the critical hoopstrain as a percentage of the diameter decrease, yielding:

(2)

where: W = plate width; t = plate thickness; D = pipe diameter; ∆D = diameterdecrease; and Cb = edge restraint factor. Recommendations were made for edgerestraint factors, Cb , for corrugation sidewall, valley, and crest, based on the results oflaboratory “hoop compression” cell tests. Moore and Laidlaw proposed design valuesfor Cb that account for the potential of the soil to penetrate into the corrugation valleyand provide “plate edge” resistance to translation and rotation.

McGrath and Sagan (2000) reviewed the state of the art of local buckling for corru-gated polyethylene pipe in a report for the National Cooperative Highway ResearchProgram (NCHRP) Project 4-12. The plate buckling equations for metals (Bryan 1891)were examined for use in estimating the local stability of the corrugated profiles.McGrath and Sagan used the methodology developed by Winter (1946) to quantify theinelastic behavior of metal plates. Winter proposed an “effective width approach” thattakes into account the post-buckling capacity of plate elements. The theory is based onthe assumption that the central portion of the plate becomes ineffective when it bucklesand that the ultimate capacity is reached when the edge segments reach the yield stressof the material.

3 TEST PROGRAM

3.1 Introduction

Two pipe test cells were used to study the limit states of buried pipe in the laboratory.The biaxial test cell at the University of Western Ontario (UWO) models the biaxialgeostatic stress field expected under real burial conditions. The hoop cell tests thepipes under axisymmetric loading conditions. Details of the test procedures and mea-surements are briefly outlined in the following sections.

3.2 Biaxial Cell Test

The test cell (Brachman et al. 2000), is a high-strength, steel box with dimensions 2 m× 2 m in plan and 1.6 m in height. Pipes tested within the cell are placed horizontallyon a bed of soil and backfilled within the rectangular prism of soil. Uniform pressuresare applied at the top surface of the soil using an air bladder placed beneath the stiff lidof the cell, simulating the overburden stresses, σv . Special sidewall treatment isemployed to reduce the sidewall friction and to ensure that most of the surface loadsreach the pipe. The sides of the box are stiff enough to restrain lateral deformation suchthat approximate plane strain conditions (Ko lateral pressure) are attained, according tothe following expression:

εcr∆DD

--------

cr

Cb tW

------------

2= =



(3)

Placement of each pipe in the cell is illustrated in Figure 2. The pipe is backfilledwith poorly graded granular soil (uniformity coefficient, Cu = 3.4, curvature coefficient,Cc = 1.1). The soil is compacted to a density of 1,600 to 1650 kg/m3. The mid-rangevalue of 1,625 kg/m3 corresponds to 85% of the maximum standard Proctor density.

Instrumentation in the cell measures soil stresses (at the springline level and at thetop of the cell) and soil settlements at the springline. Figure 2 illustrates the details ofthe cell instrumentation.

3.3 Hoop Cell Test

Although the biaxial test cell is able to simulate the idealized field conditions, the max-imum pipe size that can be tested is limited to minimize the influence of the bound-aries. However, pipes with a large diameter can be tested under axisymmetric loadingconditions in a hoop compression cell. The axisymmetric component of the radialearth pressures (σv + σh)/2 is important for deeply buried pipe, generating a hoop thrustthat leads to circumferential shortening in most profiled thermoplastic pipes.

The UWO hoop cell is a 12.7 mm-thick, vertical steel cylinder of internal diameter

σh Ko σv=

Figure 2. Schematic diagram of pipe installation.

RestrainedBoundaries

200 mm

640 mm

Earth pressure cells

Settlement plates (A,B)

760 mmFrictiontreatment

2,000 mm



of 1,500 mm and height of 1,450 mm. An inflatable polymer bladder is placed on theinside surface of the metal test cylinder. The instrumented pipe is placed upright andconcentrically within the cylindrical test cell. The space between the pipe and the airbladder is then backfilled to form a ring of soil. Two, 19.5 mm-thick circular steelplates cover the ends of the test vessel and are bolted in place using flanges at the endof the steel cylinder. The top plate is fitted with a 390 mm-diameter sleeve to allowaccess to the instrumentation within the pipe.

The end plates resist the longitudinal (axial) pipe movements during the test as theair bladder is used to apply radial pressures around the outside of the soil annulus. Thecylinder located on the inside boundary of that soil ring is thereby placed in a state ofaxisymmetric hoop compression.

Figure 3 presents the schematic view of the test arrangement of the hoop cell test.The same poorly graded granular backfill was used in the tests and was compacted toprovide the required soil density and stiffness (based on nuclear density measurements).

3.4 Pipe Samples

Table 1 provides a list of the four pipe samples used in the seven pipe loading tests.Samples A (Figure 1a) and B (Figure 1b) are lined corrugated pipe, manufactured tohave annular geometry. Pipe Samples C (Figure 1c) and D (Figure 1d) were formed byhelically winding a strip of the pipe wall, so that wall elements are oriented at 2 and 6o

to the pipe circumference, respectively.

To DA

Bolts

Air supply line

Sand

Pipe

Airbladder LVDT

Instrumentation

Steelcylinder

Figure 3. Schematic diagram of the UWO hoop test cell, after Laidlaw (1999).



3.5 Pipe Instrumentation

Pipe instrumentation includes a linear variable differential transducer (LVDT) to mea-sure the change in pipe diameter and electronic strain gauges to measure the wallstrains on different components of the profile (Figure 4). Both the horizontal and thevertical deflections were measured in the biaxial cell tests. Strain gauge measurementswere taken at the midpoint of each element of the profile (e.g., the liner, the web, thecrest, and the valley of the corrugation). The circumferential strains at these locationsare judged to be the most critical with respect to local buckling. The axial strain is lessimportant, although values at the connection of the different profile elements (e.g.,between the liner and the valley) may be important with respect to limit states such aslocal bending.

For profiled pipes manufactured to feature annular geometry, the principal straindirections are known in advance and biaxial strain gauges were used to measure thecircumferential and the axial strains. Strains on the helically profiled pipes were mea-sured using strain gauge rosettes, since the major and minor principal strain directionswere unknown. Instrumentation was placed at two sections to check the reproducibil-ity of the test results and to ensure that the data are collected in the event that some ofthe gauges failed to operate successfully. For samples prepared for testing in the biax-

Table 1. Definition of the tests.

Label Test no. Pipe Test type Internal diameter (mm) Profile type

AB 1 A(Figure 1a) Biaxial 610 Lined corrugated pipe with

profile depth = 55.15 mm

BB 2 B(Figure 1b) Biaxial 610 Lined corrugated pipe with

profile depth = 58.70 mm.

AH 3,4 A(Figure 1a) Hoop 610 Lined corrugated pipe with


BH 5 B(Figure 1b) Hoop 610 Lined corrugated pipe with


CH 6 C(Figure 1c) Hoop 760 Rectangular box profile

DH 7 D(Figure 1d) Hoop 1,060 Lined tubular profile

(c)(b)(a)

(a) Lined corrugated (b) Box profile (c) Tubular profileFigure 4. Location of the strain gauges: (a) lined corrugated; (b) box profile; (c) tubularprofile.



ial pipe test cell, strain gauges were placed at the crown, invert, and both springlines.Strain gauges were placed on two diametrically opposed sections at each axial (verti-cal) position for samples prepared for testing in the hoop compression cell.

3.6 Observations of Local Buckling

A camera, capable of moving along the pipe axis, was mounted inside each pipe torecord the appearance of the inner pipe wall during both hoop compression and biaxialloading tests. The camera was connected to a video system to record and display theimage continuously on a television screen. The camera was also free to rotate about itsaxis, therefore, it could be directed to observe every point around the inner peripheryof the pipe. The inside surface of the pipe was marked around the circumference, toidentify each position being photographed during the test.

Loads were applied with the air bladder system in increments of 25 kPa. After theapplication of each incremental load, 20 minutes was allowed to pass to allow mea-surements to stabilize. The video images of the inner pipe wall were recorded at eachload level. Increments of load were continued until the ripples were observed on theinner wall (the liner) or the test cell capacity was reached.

4 TEST RESULTS

4.1 Introduction

The pipe responses during the biaxial and hoop cell tests are presented in this section.The focus of attention is observations relating to liner buckling. Values of circumferen-tial strain are discussed since these compressive strains govern the development ofpipe liner buckling.

4.2 Tests AB and BB

Figure 5 shows the pipe deflections recorded during the biaxial Tests AB and BB. Ini-tiation of a wavelike pattern on the liner was first noticed at a pressure of 275 kPa inTest AB. The buckling became more pronounced at higher load levels. Progression oflocal buckling with different levels of load for the test is shown in Figure 6. Whitelines in the Figure 6 are drawn around the pipe circumference on each liner segment.Liners were numbered from 1 to 10 for identification. Circumferential positions of thepipe are defined as 0 and 180o at the two springlines, 90o at the crown, and 270o at theinvert. The notation C5180 denotes liner number 5 at 180o (the springline). The verticaland the horizontal diameter changes at the maximum level of load were –48.3 mm (–7.9%) and +18.8 mm (+3.1%), respectively.

In the Test BB, pipe diameter changes at the maximum stress were –57.7 mm (–9.4%)and +15.0 mm (+2.38%) in the vertical and horizontal directions, respectively. Linerbuckling was apparent at a stress level of 425 kPa and became more pronounced at higherstress. Figure 7 shows liner buckling during Test BB at a cell pressure of 500 kPa.

The distributions of circumferential wall strain recorded during both tests are



illustrated in Figure 8. Strains were measured at the center of each of the profilecomponents (i.e., the liner, the corrugation valley, the corrugation sidewall, and cor-rugation crest). The maximum circumferential compression on the mid-liner devel-ops at the springline of the pipe. Liner buckling initiated at the springline in both thetests. Springline strains are, therefore, the strain values of interest for the analysis ofliner buckling.

Measurements of circumferential springline strains are plotted in Figure 9. Mea-sured strains have been divided by 0.7 prior to plotting, since the strain gauges areknown to stiffen the HDPE pipe wall and strain readings must be corrected in thismanner to account for that strain gauge stiffening (Brachman (1999) provides detailsof this calibration). Figure 9 (also, Figure 8) reveals that the circumferential strains onthe liner are much less (approximately 20%) than those on the valley. This is caused bylocal bending that occurs on the liner due to the three-dimensional (3-D) structuralform of the profiled pipe (Moore and Hu 1995). Moore and Hu (1995) demonstratedthat the circumferential strain is a minimum at the center of the liner.

The circumferential strains in both the liner and valley elements are compared inFigure 9 with those predicted by conventional two-dimensional (2-D) ring theory(Flugge 1960). The ring theory utilizes the pipe deflections to calculate circumferentialstrains; the theory significantly over-predicts the strains.

Liner buckling is presumed to be fully developed at the point where the strain levelin the liner no longer increases as further increments of load were applied to the system(i.e., the liner reaches its load capacity and further loads are redistributed elsewhere inthe pipe profile). Circumferential strains of the liner, corresponding to the elastic buck-ling, are 0.76 and 1.05% for the Tests AB and BB, respectively.

-30

-20

-10

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600

Cell pressure (kPa)Def

lect

ion

(mm

)

Dv: Test ABDv: Test BBDh: Test ABDh: Test BB

Point of elasticbuckling

Note. Dv: Change in vertical diameter Dh: Change in horizontal diameter

Dv , Test ABDv , Test BBDh , Test ABDh , Test BB

Figure 5. Vertical and horizontal pipe deflection in Tests AB and BB.Note: Dv = change in vertical diameter; Dh = change in horizontal diameter.



At 200 kPa At 275 kPa

At 350 kPa At 425 kPa

Liner C5 near 180o before buckling

Liner C5 near 180o after buckling

Liners

Liner

Valley

Figure 7. Liner buckling observed during Test BB.

Figure 6. Development of liner buckling in Test AB.



4.3 Tests AH and BH

Tests AH and BH are the axisymmetric tests on the two lined corrugated pipe profiles.The axisymmetric responses of the pipes are illustrated in Figure 10. Two tests of TypeAH were performed (Figure 10a). Two completely different samples were instru-mented and tested, and discrepancies between the two sets of measurements resultfrom the differences of wall stiffening by strain gauging and also from small inconsis-tencies between the profile geometries.

The liner strains in the hoop tests are found to be approximately 62% of the interiorvalley strains due to the local bending discussed in Section 4.2. Comparison of thispercentage with the biaxial test results implies that the effects of local bending under

-6

-4

-2

0

-6

-4

-2

0

Valley (exterior)Valley (interior)CrestLinerWall

Stra

in (%

)

-6

-4

-2

0

-6

-4

-2

0

Valley (exterior)Valley (interior)CrestLinerWeb

Stra

in (%

)

Figure 8. Circumferential strain distributions: (a) pipe strains for Test AB, 425 kPapressure; (b) pipe strains for Test BB, 500 kPa pressure.

(a)

(b)



biaxial loading are more significant than under axisymmetric loading. A separate com-ponent of this research project involves study of local bending within different wallprofiles under both biaxial and axisymmetric loading conditions. Moore and Hu(1995) showed that local bending can be evaluated using 3-D finite element analyses; astudy using this approach has been conducted and will be reported elsewhere (Dharand Moore 2001).

The circumferential strain given by 2-D ring theory is plotted along with the strainmeasurements in Figure 10. Simple ring theory indicates that the percentage of diameterdecrease is equal to the hoop strain where the response is purely axisymmetric. Figure10 indicates that ring theory over-predicts strains for axisymmetric loading conditions.

-5

-4

-3

-2

-1

0

1

0 100 200 300 400 500

Cell pressure (kPa)

Liner strainValley strainRing theory

Circ

umfe

rent

ial s

train

(%)

-6

-5

-4

-3

-2

-1

0

1

0 100 200 300 400 500 600

Cell pressure (kPa)

Liner strainValley strainRing theoryC

ircum

fere

ntia

l stra

in (%

)

Figure 9. Circumferential strains on the interior surface at the springline: (a) springlinestrains (Test AB); (b) springline strains (Test BB).

(a)

(b)



Almost identical critical strains were obtained in each type of test (biaxial and axi-symmetric): 0.76 and 0.85% for Sample A under biaxial and hoop compression, respec-tively. The corresponding strains for Sample B are 1.1 and 1.2%, respectively. Figure11 presents photographs of liner buckling in Tests AH and BH (the image from Test BHshows initiation of local buckling, which is difficult to discern in the printed version ofthe video image).

4.4 Test CH

Buckling on the liner was apparent in Test CH (hoop test) at bladder pressure of 375kPa. Figure 12 shows the liner buckling observed during the test. Measurements of

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 100 200 300 400

Cell pressure (kPa)

Liner (Test 1)Valley (Test 1)Liner (Test 2)Valley (Test 2)Ring theory (Test 1)Ring theory (Test 2)

Circ

umfe

rent

ial s

train

(%)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 100 200 300 400

Cell pressure (kPa)

LinerValleyRing theoryC

ircum

fere

ntia

l stra

in (%

)

Figure 10. Axisymmetric response in Tests AH and BH: (a) Test AH; (b) Test BH.

(a)

(b)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 100 200 300 400

Cell pressure (kPa)

LinerValleyLinerValleyRing theory (for Test 1)Ring theory (for Test 2)

Circ

umfe

rent

ial s

train

(%)



strains along with the ratio ∆D/D are plotted in Figure 13. It shows the hoop strain inthe liner is approximately 80% of the ∆D/D ratio. For this profile, it is evident fromFigures 13a and 13b that the circumferential liner strain is much higher than the strain

Rib Liner

Buckling inthe liner

(a)

(b)Local buckling inthe liner

Figure 12. Local buckling in Test CH.

Figure 11. Liner buckling: (a) Test AH; (b) Test BH.



on the rib (rib strain is approximately 60% of the strain on the liner). This may becaused by the helical geometry. Hoop strain at the initiation of liner buckling isassessed as being 2.4%.

4.5 Test DH

This helically wound tubular profiled pipe is manufactured by spirally winding andwelding together a succession of units of four tubes. At each welded joint, there is anincrease in the material present in the pipe wall and, consequently, an increase in localstiffness. Strain gauges were placed to measure strains on each of the tubes of the unit.Figure 14 shows the local buckling observed in the middle two tubes (Tubes 2 and 3 inFigure 1d) during Test DH. Circumferential strains on the element that buckled arepresented in Figure 15. Buckling was initiated at a hoop pressure of 180 kPa. The crit-ical liner strain is assessed as 0.8%.

-4

-3

-2

-1

0

1

2

0 100 200 300 400

Cell pressure (kPa)

Stra

in (%

)Hoop

Diagonal

Axial

Diameter decrease

-1.2

-0.9

-0.6

-0.3

0

0.3

0.6

0 100 200 300 400

Cell pressure (kPa)

Stra

in (%

)

AxialDiagonalHoop

(a)

Figure 13. Interior pipe strains for Test CH: (a) linear strains; (b) rib strains.

(b)



Joints Tubes 1 and 4

Tubes 2 and 3

Figure 14. Local buckling in Test DH.

Figure 15. Internal strain in Test DH.

-4

-3

-2

-1

0

1

0 100 200 300 400

Cell pressure (kPa)

Liner strainOuter wall strainPercent diameter decrease

Circ

umfe

rent

ial s

train

(%)



5 EVALUATION OF PLATE BUCKLING MODEL

Bryan’s plate buckling equation (Equation 1) was evaluated for prediction of linerbuckling. The key parameters in Equation 1 are the material modulus, E, the width tothickness ratio of the “plate” element, W/t, and the edge support coefficient, k. Follow-ing the approach of Moore and Laidlaw (1997), Equation 1 is expressed in terms ofcritical strain for use with thermoplastic pipe. Assuming the liners to be acting underplane strain conditions, σcr = (E/(1 - ν2 ))εcr , Equation 1 yields:

(4)

The Structural Plastics Design Manual (ASCE 1984) provides guidelines for edgesupport coefficients for plate buckling. McGrath and Sagan (2000) recommended theuse of a value of 4 for elements intersecting approximately at right angles with otherelements and 0.43 for free-standing elements. McGrath and Sagan obtained values byfitting the model to provide a lower bound to results of column compression tests onpipe segments. McGrath and Sagan employed simple beam theory to calculate thestrains on the pipe wall, since the wall strains were not measured during the tests(likely reasonable for a condition involving simple axial compression along the hoopdirection of the pipe wall).

Suitability of the edge support coefficient of 4 for buried pipes is evaluated in thepresent study. Strains measured in the laboratory tests have been utilized in this ver-

εcrkπ2

12 W t⁄( )2-------------------------=

W

t

W

t t

W

Figure 16. Width of the plate, W, and plate thickness, t, for the liners.



ification. Width and thickness of the inner wall elements are defined as shown in Fig-ure 16. Table 2 summarizes the liner buckling test results and provides information onthe liner geometry (including the ratio of liner span between web elements to averagethickness). The buckling solution, Equation 4, is presented graphically in Figure 17together with the test observations by plotting hoop strain versus the W/t ratio. Themodel results are plotted in Figure 17 for k = 4, as well as for k = 3 and k = 5.

Table 2. Summary of liner buckling.

Test type Pipe W (mm) t (mm) W/t Critical strain (%)

BiaxialLined corrugated

62.0 2.46 25.2 0.76

50.0 2.36 21.1 1.05

Hoop

Lined corrugated

62.0 2.46 25.2 0.85

62.0 2.46 25.2 1.1

50.0 2.36 21.1 1.2

Box profile 46.0 4.42 10.4 2.4

Tubular profile 43.5 2.21 19.8 0.80

Figure 17. Comparison of the liner buckling model with laboratory test results.

0

1

2

3

4

5

0 10 20 30 40 50

W/t

Biaxial test data

Hoop test data: annular pipes

Hoop test data: helical woundpipesBryan (1891)'s model, k=4

Bryan (1891)'s model, k=5

Bryan (1891)'s model, k=3

Circ

umfe

rent

ial s

train

(%)

Biaxial test data

Hoop test data: annular pipes

Hoop test data: helical wound pipes

Bryan (1891) model, k = 4





With the exception of one data point, it appears that the use of the modified form ofBryan’s model of elastic buckling with an edge coefficient, k = 4, forms a lower boundto the test observations of critical strain (Figure 17). Critical strains for tests underbiaxial and hoop compression are largely identical, since the wavelength of the buck-ling pattern is small enough such that one or more of the buckles fit within the regionof maximum hoop strain at the springline.

The single point that lies below the line represents a relatively thick liner element.It may be that the liner element has reached a material, rather than geometrical (i.e.,buckling), strength limit for that element; McGrath and Sagan (2000), discuss theimplications of material capacity for thick liner elements.

6 COMPARISON WITH OTHER TEST OBSERVATIONS

Other buckling observations for lined corrugated pipes have been reported in the liter-ature (Selig et al. 1994; Li and Donovan 1994). Selig et al. (1994) used visual inspec-tion and simple diametral contraction to quantify liner buckling in three hoopcompression tests. External radial pressure was applied in increments of 35 kPa. Linerbuckling was perceived to have developed between the third and fourth increments ofload. Diameter shortening at the end of the third and the fourth increments of pressurewere 7 and 11 mm, respectively. Pipe deflection corresponding to elastic buckling isassumed in the present study to be 9 mm (the mid-value). Li and Donovan (1994)reported the pipe deflection corresponding to the initiation of buckling from one fur-ther hoop test using Selig’s apparatus. Table 3 provides details of these hoop test con-ditions and results.

Moser (1998) noted liner buckling on tubular profiled pipe under non-axisymmet-ric loading. Three tests were conducted on 1,215 mm diameter pipes with differentdensities of soil. The pipes possessed a liner span of 47.5 mm, a liner thickness of 1.7mm, and a profile depth of 71.5 mm. Buckling was viewed at the springline of thepipe. Horizontal and vertical deflections of the pipe were measured at the developmentof liner buckling. Test conditions and results are presented in Table 4.

Notes: * Selig et al. (1994), + Li and Donovan (1994).

Table 3. Observations of liner buckling in a lined corrugated pipe.

Test type Size(mm)

Liner span, W (mm)

Thickness, t(mm)

∆D(mm)

∆D / D(%)

Corrected strain (%)

Hoop

614* 77.0 2.74 9 1.46 0.73

614* 78.4 2.72 9 1.46 0.73

614* 78.1 2.42 9 1.46 0.73

604+ 51.7 2.50 10 1.66 0.83



One difficulty with using the above-mentioned tests is that the strains on the linerwere not measured during the tests. Therefore, the ring theory was used to calculate thecircumferential strains from the measured pipe deflection. However, it is obvious fromSection 4 that simple ring theory often significantly over-predicts the liner strains. InFigure 18, the measured liner strains from four of the new hoop tests are plottedagainst strain estimates based on the ring theory. For the point corresponding to linerbuckling, the ratio of the measured strain to that calculated using the ring theory isapproximately 0.5 (e.g., 2% prediction for ring theory corresponds to measured valuesof 0.8 and 1.1% for Tests AH1 and AH2, respectively). Thus, a correction factor of0.50 has been selected for the tests referenced in Table 3 to factor down the estimatesof liner strains from those obtained using ring theory. The liner strain is approximately70% of the percent diameter decrease for the tubular profiled pipes at liner buckling(Figure 18). The factor will be less at the springline of a biaxially loaded pipe due tomore significant effects of local bending. As discussed in Sections 4.2 and 4.3, theliner strain for corrugated pipe was approximately 62% of the valley strain in the hoop

Table 4. Observations of local buckling in lined tubular pipe.

Test type Test no. W/t ∆Dv /D(%)

∆Dh /D(%)

Strain (%)(ring theory) Corrected strain (%)

USU cell(Moser 1998)

1 27.8 -10.2 +8.0 4.22 0.97

2 27.8 -4.9 +3.1 2.27 0.52

3 27.8 -3.5 -0.2 2.42 0.55

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0-4-3-2-10

Strain using ring theory (%)

Test AH1Test AH2Test BHTest DH

Mea

sure

d st

rain

(%)

Approximated forlined corrugatedpipe

Approximatedfor tubularpipe

Figure 18. Liner strain versus ring strain.



test, whereas the proportion in the biaxial test was approximately 20% (i.e., one-thirdof the axisymmetric result). The ring theory does provide a rational estimate of the val-ley strain. To estimate the liner strain using the ring theory for the biaxially loadedtubular pipe, a correction factor of one-third of the proportion (0.70) of diameterdecrease encountered in the hoop test was used (i.e., 0.23). Table 4 shows the test con-ditions and the estimated strains for the tubular profiled pipe.

The buckling model discussed in Section 5 is compared to the additional linerbuckling data (Tables 3 and 4) in Figure 19. The model with the edge support coeffi-cient of 4 is plotted in Figure 19. The additional data points also indicate that Equation3 with k = 4 generally provides a reasonable lower-bound estimate of the critical buck-ling strain.

7 DISCUSSION AND CONCLUSIONS

Local buckling, an important limit state for profile-walled thermoplastic pipe, has beenignored in current design practice. Tests have been performed to study the develop-ment of local buckling under biaxial and axisymmetric loading conditions. The Bryan(1891) solution for plate stability has been examined for use in predicting local buck-ling in profiled high-density polyethylene pipes. Comparisons of critical strain havebeen made between the theory and laboratory observations. Estimates of local buck-

0

1

2

3

4

5

0 10 20 30 40 50

W/t

Bryan (1891)'s model

Biaxial and hoop cell tests

Lined corrugated (Selig et al. 1994)

Lined tubular (Moser, 1994)

Li and Donovan, 1994

Circ

umfe

rent

ial s

train

(%)

Bryan (1891) model

Biaxial and hoop cell tests

Lined corrugated (Selig et al. 1994)

Lined tubular (Moser 1998)

Li and Donovan (1994)

W / t

Figure 19. Comparison of buckling model with other test data.



ling using Bryan’s equations were undertaken using end support coefficients of 3, 4,and 5 to obtain a suitable value to be used in the design for liner buckling.

Observations of liner buckling were recorded using a camera mounted inside thepipe. Strains were measured at the center of the profile components. It appears that thecritical buckling strain is not significantly affected by the method of loading. However,local bending within the profile reduces the hoop strains in the critical elements belowvalues that correspond to simple ring theory. These reductions are greater for the biax-ial loading condition. The test observations of critical hoop strain reveal that theBryan’s model with an edge support coefficient of 4 generally provides a useful lowerbound to the critical buckling strains; however, the stability of wider profile elementsmay be influenced by material yield.

Additional data points for liner buckling have been included from the literature.While these tests did not feature measurements of local strain, estimates of critical strainwere made that also appear to be bounded by the modified form of Bryan’s equation.

The investigation implies that the plate stability theory of Bryan (1891) provides auseful tool for characterizing the local buckling of profile thermoplastic pipe. Furtherresearch would be useful to assess the implications of nonuniform element thickness,the post-buckling behaviour of these elements, and the relationship between linerinstability and the overall performance of the pipe profile.

ACKNOWLEDGEMENTS

Both buried-pipe test cells were developed using support from the Natural Sciencesand Engineering Research Council of Canada. The contributions of R.W.I Brachman,T.C. Laidlaw, R.K. Rowe, and A. Tognon to the development of these facilities aregratefully acknowledged. The research on thermoplastic pipes is being conducted aspart of NCHRP Project 4-26 for the Transportation Research Board, Washington, DC,USA, in collaboration with T.J. McGrath. Any opinions, findings, conclusions, or rec-ommendations expressed in the present paper are those of the authors and do not nec-essarily reflect the views of the sponsors.

REFERENCES

ASCE, 1984, “Structural Plastics Design Manual - ASCE Manual of Practice No. 63”,prepared by the Task Committee on Design of the Structural Plastics ResearchCouncil of the Technical Council on Research of the American Society of CivilEngineers, New York, New York, USA, 1176 p.

Brachman, R.W.I., 1999, “Structural Performance of Leachate Collection Pipes”,Ph.D. Thesis, Department of Civil and Environmental Engineering, The Universityof Western Ontario, London, Canada, 340 p.

Brachman, R.W.I., Moore, I.D., and Rowe, R.K., 2000, “The Design of a LaboratoryFacility for Evaluating the Structural Response of Small Diameter Buried Pipes”,Canadian Geotechnical Journal, Vol. 37, No. 2, pp. 281-295.



Bryan, G.H., 1891, “On the Stability of a Plane Plate Under Thrusts in its Own Plane,with Applications to The Buckling of the Sides of a Ship”, Proceedings of the Lon-don Mathematical Society, Vol. 22, London, United Kingdom, pp. 55-67.

Dhar, A.S. and Moore, I.D., 2001, “Thermoplastic Culvert Deformations and Strains:Evaluation Using Three-dimensional Finite Element Analyses” to be submitted.

DiFrancesco, L.C., 1993, “Laboratory Testing of High Density Polyethylene DrainagePipes”, M.Sc. Thesis, Department of Civil Engineering, The University of Massa-chusetts at Amherst, Amherst, Massachusetts, USA, 146 p.

Flugge, W., 1960, “Stresses in shells”, Springer-Verlag, Berlin, 499 p.Hashash, N.M.A., 1991, “Design and Analysis of Deeply Buried Polyethylene Drain-

age Pipes”, Ph.D. Thesis, Department of Civil Engineering, The University ofMassachusetts, Amherst, Massachusetts, USA, 221 p.

Laidlaw, T., 1999, “Influence of Local Support on Corrugated HDPE pipe”. M.E.Sc.Thesis, The University of Western Ontario, London, Canada, 230 p.

Li, H. and Donovan, J.A., 1994, “Ring Bending and Hoop Compression Tests onBig‘O’ HDPE Pipe”, Technical report, Department of Mechanical Engineering,The University of Massachusetts, Amherst, Massachusetts, USA.

McGrath, T.J. and Sagan, V.E., 2000, “LRFD Specification for Plastic Pipe and Cul-vert”, Final Report, NCHRP Project 4-12, Transportation Research Board, March1999.

Moore, I.D., 1990, “Influence of rib stiffeners on the buckling strength of elasticallysupported tubes”, International Journal of Solids and Structures, Vol. 26, Nos. 5-6,pp. 539-547.

Moore, I.D., 1996, “Local Buckling in Profiled HDPE Pipes”, 1996 Annual Confer-ence of the Canadian Society for Civil Engineering, Edmonton, Alberta, Canada,May 1996, pp. 84-94.

Moore, I.D. and Hu, F., 1995, “Response of Profiled High-Density Polyethylene Pipein Hoop Compression”, Transportation Research Record, No. 1514, pp. 29-36.

Moore, I.D. and Laidlaw, T.C., 1997, “Corrugation Buckling in HDPE Pipes – Mea-surements and Analysis”, TRB Annual Conference, Washington, D.C., USA, 24 p.

Moser, A.P., 1998, “Structural Performance of buried Profile-Wall High-Density Poly-ethylene Pipe and Influence of Pipe Wall Geometry”, Transportation ResearchRecord, No. 1624, pp. 206-213.

Selig, E.T., DiFrancesco, L.C., and McGrath, T.J., 1994, “Laboratory test of buriedpipe in hoop compression”, Buried Plastic Pipe Technology, Eckstein., D., Editor,ASTM Special Technical Publication 1222, 119-132.

Winter, G., 1946, “Strength of thin Steel Compression Flanges”, Transactions of theAmerican Society of Civil Engineering, Vol. 111, pp. 527-554.



NOTATIONS

Basic SI units are given in parentheses.

Cb = edge restraint factors (dimensionless)

Cc = coefficient of curvature (dimensionless)

Cu = uniformity coefficient (dimensionless)

D = pipe diameter (m)

Dh = change in horizontal diameter (m)

Dv = change in vertical diameter (m)

E = modulus of elasticity (N/m2)

Ko = coefficient of lateral earth pressure (dimensionless)

k = edge support coefficient (dimensionless)

t = plate thickness (m)

W = width of plate (m)

∆D = diameter decrease (m)

εcr = critical buckling strain (dimensionless)

ν = Poisson’s ratio (dimensionless)

σcr = critical buckling stress (N/m2)

σh = horizontal earth stress (N/m2)

σv = vertical earth stress (N/m2)

Liner Buckling Study

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