LINER BUCKLING IN PROFILED POLYETHYLENE PIPES

Transcription

1 Technical Paper by A.S. Dhar and I.D. Moore LINER BUCKLING IN PROFILED POLYETHYLENE PIPES ABSTRACT: Thermoplastic pipes are often manufactured with profiled walls to maximize the flexural stiffness of the pipe for a given amount of polymer. Thin elements in the profile can buckle under the influence of large earth pressures associated with deep burial or other extreme loading conditions. Earth load tests have been conducted on high density polyethylene pipes with a number of different wall profiles. Two highpressure pipe test cells have been used to conduct these tests. Observations of local buckling in the internal liners of these products have been examined and compared to stability assessments based on the conventional equation for buckling in stiffened plate structures (following modification of that equation to an equation that defines critical strain 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 of three-dimensional bending in the pipe profile are considered, the modified Bryan equation appears to be a useful tool for quantifying liner stability and should be considered for inclusion in limit-state design procedures for these structures. AUTHORS: A.S. Dhar, Graduate student, Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9, Telephone: 1/ , Ext , Telefax: 1/ , asdhar@engga.uwo.ca; and I.D. Moore, Professor and Canada Research Chair, Department of Civil Engineering, Queen s University, Kingston, Canada K7L 3N6, Telephone: 1/ , Telefax: 1/ , moore@civil.queensu.ca. KEYWORDS: Thermoplastic pipe, HDPE, Culvert, Deep burial, Local buckling, Plate-buckling model, Limit state design. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 181 County Road B West, Roseville, Minnesota , USA, Telephone: 1/ , Telefax: 1/ Geosynthetics International is registered under ISSN DATE: Original manuscript submitted 18 July 2, revised version received 9 April 21, and accepted 12 April 21. Discussion open until 1 April 22. REFERENCE: Dhar, A.S. and Moore, I.D., 21, Liner Buckling in Profiled Polyethylene Pipes, Geosynthetics International, Vol. 8, No. 4, pp GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4 33

2 1 INTRODUCTION Large diameter thermoplastic pipes are manufactured with a variety of different wall geometries in order to attain effective utilization of the polymer material in resisting bending. Decentralization of the material from the neutral axis of the wall renders higher pipe stiffnesses with reduced use of material. However, high compressive strain across pipe sections, especially in deeply buried pipe, may cause local buckling on various components of the profile that can compromise the pipe s structural integrity. The present paper focuses on the prediction of local buckling on the liner (the inner unsupported wall) of a number of different thermoplastic pipes. Liner buckling can be expected to affect the hydraulic properties of the pipe. Liner buckling may also influence the structural integrity of profiles where loss of liner stiffness leaves little or no material 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 pipes with an internal diameter of 61 mm, the former with a deeper profile and smaller pitch (i.e., more closely spaced corrugations). Figures 1c and 1d are helically wound profiles with helix angles of 2 o and 6 o, respectively. The third has a rectangular boxed profile and a 71 mm internal diameter. The fourth possesses a tubular profile, with a 1,6 mm internal diameter. Tests have been conducted on each of these pipes using the two pipe test vessels developed at The University of Western Ontario to model typical deep burial conditions (Brachman et al. 2). Tests on pipes have been performed under both biaxial and axisymmetric loading conditions. The present paper reviews the literature on local buckling, describes the pipe testing program, presents analyses of the test data, and examines the use of elastic buckling solutions for prediction of liner buckling. 2 LITERATURE In the past several decades, various workers have undertaken studies to improve thermoplastic pipe design. Local buckling of individual structural elements in profiled HDPE pipes first received attention by Hashash (1991) and DiFrancesco (1993), who observed ripples during field and laboratory tests on twin-wall HDPE pipe, respectively. Moore and Hu (1995) demonstrated that these ripples were due to local buckling in the inner wall, i.e., the liner. Moore (1996) derived solutions for liner buckling using the linear buckling theory for a stiffened cylindrical shell (Moore 199). The solution was expressed graphically in terms of critical compressive hoop strain in the liner. Moser (1998) agreed that local buckling is a critical performance limiting factor in 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 thin plate buckling, as proposed by Bryan (1891), is: 34 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

3 (a) Crest Valley Liner Web (b) (c) Outer wall Liner Rib (d) (4) Joint (1) (2) Wall 1 (3) Liner 1 Wall 2 (4) Liner 2 Figure 1. Types of pipe profiles: (a) lined corrugated (profile depth, mm): (b) lined corrugated (profile depth, mm); (c) rectangular box profile; (d) lined tubular profile. kπ 2 E σ cr = (1) 12( 1 ν 2 )( W t) 2 where: σ cr = 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 of critical hoop strain to avoid problems associated with the nonlinear, time-dependent GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4 35

4 material modulus for the polyethylene. Moore and Laidlaw expressed the critical hoop strain as a percentage of the diameter decrease, yielding: D C ε cr b t = = (2) D cr W where: W = plate width; t = plate thickness; D = pipe diameter; D = diameter decrease; and C b = edge restraint factor. Recommendations were made for edge restraint factors, C b, for corrugation sidewall, valley, and crest, based on the results of laboratory hoop compression cell tests. Moore and Laidlaw proposed design values for C b that account for the potential of the soil to penetrate into the corrugation valley and provide plate edge resistance to translation and rotation. McGrath and Sagan (2) reviewed the state of the art of local buckling for corrugated polyethylene pipe in a report for the National Cooperative Highway Research Program (NCHRP) Project 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 the inelastic behavior of metal plates. Winter proposed an effective width approach that takes into account the post-buckling capacity of plate elements. The theory is based on the assumption that the central portion of the plate becomes ineffective when it buckles and that the ultimate capacity is reached when the edge segments reach the yield stress of 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 biaxial geostatic stress field expected under real burial conditions. The hoop cell tests the pipes under axisymmetric loading conditions. Details of the test procedures and measurements are briefly outlined in the following sections. 3.2 Biaxial Cell Test The test cell (Brachman et al. 2), 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 horizontally on a bed of soil and backfilled within the rectangular prism of soil. Uniform pressures are applied at the top surface of the soil using an air bladder placed beneath the stiff lid of the cell, simulating the overburden stresses, σ v. Special sidewall treatment is employed to reduce the sidewall friction and to ensure that most of the surface loads reach the pipe. The sides of the box are stiff enough to restrain lateral deformation such that approximate plane strain conditions (K o lateral pressure) are attained, according to the following expression: 36 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

5 σ h = K o σ v (3) Placement of each pipe in the cell is illustrated in Figure 2. The pipe is backfilled with poorly graded granular soil (uniformity coefficient, C u = 3.4, curvature coefficient, C c = 1.1). The soil is compacted to a density of 1,6 to 165 kg/m 3. The mid-range value of 1,625 kg/m 3 corresponds to 85% of the maximum standard Proctor density. Instrumentation in the cell measures soil stresses (at the springline level and at the top of the cell) and soil settlements at the springline. Figure 2 illustrates the details of the cell instrumentation. 3.3 Hoop Cell Test Although the biaxial test cell is able to simulate the idealized field conditions, the maximum pipe size that can be tested is limited to minimize the influence of the boundaries. However, pipes with a large diameter can be tested under axisymmetric loading conditions in a hoop compression cell. The axisymmetric component of the radial earth pressures (σ v + σ h )/2 is important for deeply buried pipe, generating a hoop thrust that 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 2, mm 2 mm Friction treatment Earth pressure cells 76 mm Settlement plates (A,B) 64 mm Restrained Boundaries Figure 2. Schematic diagram of pipe installation. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4 37

6 of 1,5 mm and height of 1,45 mm. An inflatable polymer bladder is placed on the inside surface of the metal test cylinder. The instrumented pipe is placed upright and concentrically within the cylindrical test cell. The space between the pipe and the air bladder is then backfilled to form a ring of soil. Two, 19.5 mm-thick circular steel plates cover the ends of the test vessel and are bolted in place using flanges at the end of the steel cylinder. The top plate is fitted with a 39 mm-diameter sleeve to allow access to the instrumentation within the pipe. The end plates resist the longitudinal (axial) pipe movements during the test as the air bladder is used to apply radial pressures around the outside of the soil annulus. The cylinder located on the inside boundary of that soil ring is thereby placed in a state of axisymmetric 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 to provide 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 to have annular geometry. Pipe Samples C (Figure 1c) and D (Figure 1d) were formed by helically winding a strip of the pipe wall, so that wall elements are oriented at 2 and 6 o to the pipe circumference, respectively. Instrumentation To DA Pipe Air supply line Air bladder LVDT Steel cylinder Sand Bolts Figure 3. Schematic diagram of the UWO hoop test cell, after Laidlaw (1999). 38 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

7 Table 1. Definition of the tests. Label Test no. Pipe Test type AB 1 BB 2 AH 3,4 BH 5 CH 6 DH 7 A (Figure 1a) B (Figure 1b) A (Figure 1a) B (Figure 1b) C (Figure 1c) D (Figure 1d) Internal diameter (mm) Biaxial 61 Biaxial 61 Hoop 61 Hoop 61 Profile type Lined corrugated pipe with profile depth = mm Lined corrugated pipe with profile depth = 58.7 mm. Lined corrugated pipe with profile depth = mm Lined corrugated pipe with profile depth = 58.7 mm Hoop 76 Rectangular box profile Hoop 1,6 Lined tubular profile 3.5 Pipe Instrumentation Pipe instrumentation includes a linear variable differential transducer (LVDT) to measure the change in pipe diameter and electronic strain gauges to measure the wall strains on different components of the profile (Figure 4). Both the horizontal and the vertical deflections were measured in the biaxial cell tests. Strain gauge measurements were taken at the midpoint of each element of the profile (e.g., the liner, the web, the crest, and the valley of the corrugation). The circumferential strains at these locations are judged to be the most critical with respect to local buckling. The axial strain is less important, 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 as local bending. For profiled pipes manufactured to feature annular geometry, the principal strain directions are known in advance and biaxial strain gauges were used to measure the circumferential and the axial strains. Strains on the helically profiled pipes were measured using strain gauge rosettes, since the major and minor principal strain directions were unknown. Instrumentation was placed at two sections to check the reproducibility of the test results and to ensure that the data are collected in the event that some of the gauges failed to operate successfully. For samples prepared for testing in the biax- (a) (b) (c) Figure 4. Location of the strain gauges: (a) lined corrugated; (b) box profile; (c) tubular (a) Lined corrugated (b) Box profile (c) Tubular profile profile. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4 39

8 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 (vertical) 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 to record the appearance of the inner pipe wall during both hoop compression and biaxial loading tests. The camera was connected to a video system to record and display the image continuously on a television screen. The camera was also free to rotate about its axis, therefore, it could be directed to observe every point around the inner periphery of the pipe. The inside surface of the pipe was marked around the circumference, to identify each position being photographed during the test. Loads were applied with the air bladder system in increments of 25 kpa. After the application of each incremental load, 2 minutes was allowed to pass to allow measurements to stabilize. The video images of the inner pipe wall were recorded at each load level. Increments of load were continued until the ripples were observed on the inner 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 circumferential strain are discussed since these compressive strains govern the development of pipe liner buckling. 4.2 Tests AB and BB Figure 5 shows the pipe deflections recorded during the biaxial Tests AB and BB. Initiation of a wavelike pattern on the liner was first noticed at a pressure of 275 kpa in Test AB. The buckling became more pronounced at higher load levels. Progression of local buckling with different levels of load for the test is shown in Figure 6. White lines in the Figure 6 are drawn around the pipe circumference on each liner segment. Liners were numbered from 1 to 1 for identification. Circumferential positions of the pipe are defined as and 18 o at the two springlines, 9 o at the crown, and 27 o at the invert. The notation C5 18 denotes liner number 5 at 18 o (the springline). The vertical and the horizontal diameter changes at the maximum level of load were 48.3 mm ( 7.9%) and mm (+3.1%), respectively. In the Test BB, pipe diameter changes at the maximum stress were 57.7 mm ( 9.4%) and +15. mm (+2.38%) in the vertical and horizontal directions, respectively. Liner buckling was apparent at a stress level of 425 kpa and became more pronounced at higher stress. Figure 7 shows liner buckling during Test BB at a cell pressure of 5 kpa. The distributions of circumferential wall strain recorded during both tests are 31 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

9 Deflection (mm) Dv: D v, Test AB AB Dv: D v, Test BB BB Dh: D h, Test AB AB Dh: D h, Test BB BB Point of elastic buckling Cell pressure (kpa) Note. Dv: Change in vertical diameter Figure 5. Vertical and horizontal pipe deflection in Tests AB and BB. Dh: Change in horizontal diameter Note: D v = change in vertical diameter; D h = change in horizontal diameter. illustrated in Figure 8. Strains were measured at the center of each of the profile components (i.e., the liner, the corrugation valley, the corrugation sidewall, and corrugation crest). The maximum circumferential compression on the mid-liner develops at the springline of the pipe. Liner buckling initiated at the springline in both the tests. Springline strains are, therefore, the strain values of interest for the analysis of liner buckling. Measurements of circumferential springline strains are plotted in Figure 9. Measured strains have been divided by.7 prior to plotting, since the strain gauges are known to stiffen the HDPE pipe wall and strain readings must be corrected in this manner to account for that strain gauge stiffening (Brachman (1999) provides details of this calibration). Figure 9 (also, Figure 8) reveals that the circumferential strains on the liner are much less (approximately 2%) than those on the valley. This is caused by local bending that occurs on the liner due to the three-dimensional (3-D) structural form of the profiled pipe (Moore and Hu 1995). Moore and Hu (1995) demonstrated that the circumferential strain is a minimum at the center of the liner. The circumferential strains in both the liner and valley elements are compared in Figure 9 with those predicted by conventional two-dimensional (2-D) ring theory (Flugge 196). The ring theory utilizes the pipe deflections to calculate circumferential strains; the theory significantly over-predicts the strains. Liner buckling is presumed to be fully developed at the point where the strain level in 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 in the pipe profile). Circumferential strains of the liner, corresponding to the elastic buckling, are.76 and 1.5% for the Tests AB and BB, respectively. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

10 Liner C5 near 18 o before buckling Liners At 2 kpa At 275 kpa Liner C5 near 18 o after buckling At 35 kpa At 425 kpa Figure 6. Development of liner buckling in Test AB. Valley Liner Figure 7. Liner buckling observed during Test BB. 312 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

11 (a) Strain (%) -6 Valley (exterior) Valley (interior) Crest Liner Wall (b) Strain (%) -6 Valley (exterior) Valley (interior) Crest Liner Web Figure 8. Circumferential strain distributions: (a) pipe strains for Test AB, 425 kpa pressure; (b) pipe strains for Test BB, 5 kpa pressure. 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 1. Two tests of Type AH were performed (Figure 1a). Two completely different samples were instrumented and tested, and discrepancies between the two sets of measurements result from the differences of wall stiffening by strain gauging and also from small inconsistencies between the profile geometries. The liner strains in the hoop tests are found to be approximately 62% of the interior valley strains due to the local bending discussed in Section 4.2. Comparison of this percentage with the biaxial test results implies that the effects of local bending under GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

12 (a) Circumferential strain (%) Cell pressure (kpa) Liner strain Valley strain Ring theory (b) Circumferential strain (%) Cell pressure (kpa) Liner strain Valley strain Ring theory Figure 9. Circumferential strains on the interior surface at the springline: (a) springline strains (Test AB); (b) springline strains (Test BB). biaxial loading are more significant than under axisymmetric loading. A separate component of this research project involves study of local bending within different wall profiles under both biaxial and axisymmetric loading conditions. Moore and Hu (1995) showed that local bending can be evaluated using 3-D finite element analyses; a study using this approach has been conducted and will be reported elsewhere (Dhar and Moore 21). The circumferential strain given by 2-D ring theory is plotted along with the strain measurements in Figure 1. Simple ring theory indicates that the percentage of diameter decrease is equal to the hoop strain where the response is purely axisymmetric. Figure 1 indicates that ring theory over-predicts strains for axisymmetric loading conditions. 314 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

13 (a) (b) Circumferential strain (%) (%) Circumferential strain (%).5.5 Cell pressure (kpa) (kpa) Liner Liner(Test 1) Valley(Test 1) Liner(Test 2) Valley(Test 2) Ring Ring theory theory (Test (for Test 1) 1) Ring theory (Test (for Test 2) 2).5 Cell pressure (kpa) Liner Valley Ring theory Figure 1. Axisymmetric response in Tests AH and BH: (a) Test AH; (b) Test BH. Almost identical critical strains were obtained in each type of test (biaxial and axisymmetric):.76 and.85% for Sample A under biaxial and hoop compression, respectively. The corresponding strains for Sample B are 1.1 and 1.2%, respectively. Figure 11 presents photographs of liner buckling in Tests AH and BH (the image from Test BH shows initiation of local buckling, which is difficult to discern in the printed version of the video image). 4.4 Test CH Buckling on the liner was apparent in Test CH (hoop test) at bladder pressure of 375 kpa. Figure 12 shows the liner buckling observed during the test. Measurements of GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

14 (a) (b) Local buckling in the liner Figure 11. Liner buckling: (a) Test AH; (b) Test BH. Rib Liner Buckling in the liner Figure 12. Local buckling in Test CH. strains along with the ratio D/D are plotted in Figure 13. It shows the hoop strain in the liner is approximately 8% of the D/D ratio. For this profile, it is evident from Figures 13a and 13b that the circumferential liner strain is much higher than the strain 316 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

15 (a) 2 Strain (%) Cell pressure (kpa) Hoop Diagonal Axial Diameter decrease (b).6 Strain (%) Axial Diagonal Hoop Cell pressure (kpa) Figure 13. Interior pipe strains for Test CH: (a) linear strains; (b) rib strains. on the rib (rib strain is approximately 6% of the strain on the liner). This may be caused by the helical geometry. Hoop strain at the initiation of liner buckling is assessed as being 2.4%. 4.5 Test DH This helically wound tubular profiled pipe is manufactured by spirally winding and welding together a succession of units of four tubes. At each welded joint, there is an increase in the material present in the pipe wall and, consequently, an increase in local stiffness. 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 in Figure 1d) during Test DH. Circumferential strains on the element that buckled are presented in Figure 15. Buckling was initiated at a hoop pressure of 18 kpa. The critical liner strain is assessed as.8%. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

16 Joints Tubes 1 and 4 Tubes 2 and 3 Figure 14. Local buckling in Test DH. Circumferential strain (%) Cell pressure (kpa) Liner strain Outer wall strain Percent diameter decrease Figure 15. Internal strain in Test DH. 318 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

17 5 EVALUATION OF PLATE BUCKLING MODEL Bryan s plate buckling equation (Equation 1) was evaluated for prediction of liner buckling. The key parameters in Equation 1 are the material modulus, E, the width to thickness ratio of the plate element, W/t, and the edge support coefficient, k. Following the approach of Moore and Laidlaw (1997), Equation 1 is expressed in terms of critical strain for use with thermoplastic pipe. Assuming the liners to be acting under plane strain conditions, σ cr = (E/(1 - ν 2 ))ε cr, Equation 1 yields: kπ 2 ε cr = (4) 12( W t) 2 The Structural Plastics Design Manual (ASCE 1984) provides guidelines for edge support coefficients for plate buckling. McGrath and Sagan (2) recommended the use of a value of 4 for elements intersecting approximately at right angles with other elements and.43 for free-standing elements. McGrath and Sagan obtained values by fitting the model to provide a lower bound to results of column compression tests on pipe segments. McGrath and Sagan employed simple beam theory to calculate the strains 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 hoop direction of the pipe wall). Suitability of the edge support coefficient of 4 for buried pipes is evaluated in the present study. Strains measured in the laboratory tests have been utilized in this ver- t t W W t W Figure 16. Width of the plate, W, and plate thickness, t, for the liners. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

18 ification. Width and thickness of the inner wall elements are defined as shown in Figure 16. Table 2 summarizes the liner buckling test results and provides information on the liner geometry (including the ratio of liner span between web elements to average thickness). The buckling solution, Equation 4, is presented graphically in Figure 17 together with the test observations by plotting hoop strain versus the W/t ratio. The model 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 (%) Biaxial Lined corrugated Lined corrugated Hoop Box profile Tubular profile Circumferential strain (%) Biaxial test data Hoop test data: annular pipes pipes Hoop test data: helical wound pipes pipes Bryan (1891)'s model, k=4 k = 4 Bryan (1891)'s model, k=5 k = 5 Bryan (1891)'s model, k=3 k = W/t Figure 17. Comparison of the liner buckling model with laboratory test results. 32 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

19 With the exception of one data point, it appears that the use of the modified form of Bryan s model of elastic buckling with an edge coefficient, k = 4, forms a lower bound to the test observations of critical strain (Figure 17). Critical strains for tests under biaxial and hoop compression are largely identical, since the wavelength of the buckling pattern is small enough such that one or more of the buckles fit within the region of 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 (2), discuss the implications 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 literature (Selig et al. 1994; Li and Donovan 1994). Selig et al. (1994) used visual inspection and simple diametral contraction to quantify liner buckling in three hoop compression tests. External radial pressure was applied in increments of 35 kpa. Liner buckling was perceived to have developed between the third and fourth increments of load. Diameter shortening at the end of the third and the fourth increments of pressure were 7 and 11 mm, respectively. Pipe deflection corresponding to elastic buckling is assumed 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 further hoop test using Selig s apparatus. Table 3 provides details of these hoop test conditions and results. Moser (1998) noted liner buckling on tubular profiled pipe under non-axisymmetric loading. Three tests were conducted on 1,215 mm diameter pipes with different densities of soil. The pipes possessed a liner span of 47.5 mm, a liner thickness of 1.7 mm, and a profile depth of 71.5 mm. Buckling was viewed at the springline of the pipe. Horizontal and vertical deflections of the pipe were measured at the development of liner buckling. Test conditions and results are presented in Table 4. Table 3. Test type Hoop Observations of liner buckling in a lined corrugated pipe. Size (mm) Liner span, W (mm) Thickness, t (mm) D (mm) D / D (%) Corrected strain (%) 614* * * Notes: * Selig et al. (1994), + Li and Donovan (1994). GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

20 Table 4. Observations of local buckling in lined tubular pipe. Test type Test no. W/t USU cell (Moser 1998) D v /D (%) D h /D (%) Strain (%) (ring theory) Corrected strain (%) One difficulty with using the above-mentioned tests is that the strains on the liner were not measured during the tests. Therefore, the ring theory was used to calculate the circumferential strains from the measured pipe deflection. However, it is obvious from Section 4 that simple ring theory often significantly over-predicts the liner strains. In Figure 18, the measured liner strains from four of the new hoop tests are plotted against strain estimates based on the ring theory. For the point corresponding to liner buckling, the ratio of the measured strain to that calculated using the ring theory is approximately.5 (e.g., 2% prediction for ring theory corresponds to measured values of.8 and 1.1% for Tests AH1 and AH2, respectively). Thus, a correction factor of.5 has been selected for the tests referenced in Table 3 to factor down the estimates of liner strains from those obtained using ring theory. The liner strain is approximately 7% 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 to more significant effects of local bending. As discussed in Sections 4.2 and 4.3, the liner strain for corrugated pipe was approximately 62% of the valley strain in the hoop Measured strain (%) Approximated for tubular pipe Approximated for lined corrugated pipe Test AH1 Test AH2 Test BH Test DH Strain using ring theory (%) -4 Figure 18. Liner strain versus ring strain. 322 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

21 test, whereas the proportion in the biaxial test was approximately 2% (i.e., one-third of the axisymmetric result). The ring theory does provide a rational estimate of the valley strain. To estimate the liner strain using the ring theory for the biaxially loaded tubular pipe, a correction factor of one-third of the proportion (.7) of diameter decrease encountered in the hoop test was used (i.e.,.23). Table 4 shows the test conditions and the estimated strains for the tubular profiled pipe. The buckling model discussed in Section 5 is compared to the additional liner buckling data (Tables 3 and 4) in Figure 19. The model with the edge support coefficient of 4 is plotted in Figure 19. The additional data points also indicate that Equation 3 with k = 4 generally provides a reasonable lower-bound estimate of the critical buckling strain. 7 DISCUSSION AND CONCLUSIONS Local buckling, an important limit state for profile-walled thermoplastic pipe, has been ignored in current design practice. Tests have been performed to study the development of local buckling under biaxial and axisymmetric loading conditions. The Bryan (1891) solution for plate stability has been examined for use in predicting local buckling in profiled high-density polyethylene pipes. Comparisons of critical strain have been made between the theory and laboratory observations. Estimates of local buck- Circumferential strain (%) Bryan (1891)'s model Biaxial and hoop cell tests Lined corrugated (Selig et al. 1994) Lined tubular (Moser, 1998) 1994) Li Li and Donovan, (1994) Figure W W/t / t Comparison of buckling model with other test data. GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

22 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 the pipe. Strains were measured at the center of the profile components. It appears that the critical 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 below values that correspond to simple ring theory. These reductions are greater for the biaxial loading condition. The test observations of critical hoop strain reveal that the Bryan s model with an edge support coefficient of 4 generally provides a useful lower bound to the critical buckling strains; however, the stability of wider profile elements may 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 strain were 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 a useful tool for characterizing the local buckling of profile thermoplastic pipe. Further research would be useful to assess the implications of nonuniform element thickness, the post-buckling behaviour of these elements, and the relationship between liner instability and the overall performance of the pipe profile. ACKNOWLEDGEMENTS Both buried-pipe test cells were developed using support from the Natural Sciences and 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 are gratefully acknowledged. The research on thermoplastic pipes is being conducted as part of NCHRP Project 4-26 for the Transportation Research Board, Washington, DC, USA, in collaboration with T.J. McGrath. Any opinions, findings, conclusions, or recommendations expressed in the present paper are those of the authors and do not necessarily 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 Research Council of the Technical Council on Research of the American Society of Civil Engineers, 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 University of Western Ontario, London, Canada, 34 p. Brachman, R.W.I., Moore, I.D., and Rowe, R.K., 2, The Design of a Laboratory Facility for Evaluating the Structural Response of Small Diameter Buried Pipes, Canadian Geotechnical Journal, Vol. 37, No. 2, pp GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4

23 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 London Mathematical Society, Vol. 22, London, United Kingdom, pp Dhar, A.S. and Moore, I.D., 21, 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 Drainage Pipes, M.Sc. Thesis, Department of Civil Engineering, The University of Massachusetts at Amherst, Amherst, Massachusetts, USA, 146 p. Flugge, W., 196, Stresses in shells, Springer-Verlag, Berlin, 499 p. Hashash, N.M.A., 1991, Design and Analysis of Deeply Buried Polyethylene Drainage Pipes, Ph.D. Thesis, Department of Civil Engineering, The University of Massachusetts, 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, 23 p. Li, H. and Donovan, J.A., 1994, Ring Bending and Hoop Compression Tests on Big O HDPE Pipe, Technical report, Department of Mechanical Engineering, The University of Massachusetts, Amherst, Massachusetts, USA. McGrath, T.J. and Sagan, V.E., 2, LRFD Specification for Plastic Pipe and Culvert, Final Report, NCHRP Project 4-12, Transportation Research Board, March Moore, I.D., 199, Influence of rib stiffeners on the buckling strength of elastically supported tubes, International Journal of Solids and Structures, Vol. 26, Nos. 5-6, pp Moore, I.D., 1996, Local Buckling in Profiled HDPE Pipes, 1996 Annual Conference of the Canadian Society for Civil Engineering, Edmonton, Alberta, Canada, May 1996, pp Moore, I.D. and Hu, F., 1995, Response of Profiled High-Density Polyethylene Pipe in Hoop Compression, Transportation Research Record, No. 1514, pp Moore, I.D. and Laidlaw, T.C., 1997, Corrugation Buckling in HDPE Pipes Measurements and Analysis, TRB Annual Conference, Washington, D.C., USA, 24 p. Moser, A.P., 1998, Structural Performance of buried Profile-Wall High-Density Polyethylene Pipe and Influence of Pipe Wall Geometry, Transportation Research Record, No. 1624, pp Selig, E.T., DiFrancesco, L.C., and McGrath, T.J., 1994, Laboratory test of buried pipe in hoop compression, Buried Plastic Pipe Technology, Eckstein., D., Editor, ASTM Special Technical Publication 1222, Winter, G., 1946, Strength of thin Steel Compression Flanges, Transactions of the American Society of Civil Engineering, Vol. 111, pp GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO

24 NOTATIONS Basic SI units are given in parentheses. C b = edge restraint factors (dimensionless) C c = coefficient of curvature (dimensionless) C u = uniformity coefficient (dimensionless) D = pipe diameter (m) D h = change in horizontal diameter (m) D v = change in vertical diameter (m) E = modulus of elasticity (N/m 2 ) K o = coefficient of lateral earth pressure (dimensionless) k = edge support coefficient (dimensionless) t = plate thickness (m) W = width of plate (m) D = diameter decrease (m) ε cr = critical buckling strain (dimensionless) ν = Poisson s ratio (dimensionless) σ cr = critical buckling stress (N/m 2 ) σ h = horizontal earth stress (N/m 2 ) σ v = vertical earth stress (N/m 2 ) 326 GEOSYNTHETICS INTERNATIONAL 21, VOL. 8, NO. 4