PIPE-SOIL INTERACTION ASPECTS IN BURIED EXTENSIBLE PIPES by LALINDA WEERASEKARA B.Sc., The University of Peradeniya, 2002 M.A.Sc., The University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2011 © Lalinda Weerasekara, 2011 ii ABSTRACT The performance of buried pipelines in areas subjected to permanent ground displacements is an important engineering consideration in the gas distribution industry, since the failure of such systems poses a risk to public and property safety. Although, the ground movements and its variations over time can be detected and mapped with reasonable confidence, these data are of little use due to a lack of reliable models to correlate such displacements to the condition of the buried pipe. The objective of this thesis is to develop methods to estimate the pipe performance based on the measured ground displacement. An analytical method was developed to estimate the pipe performance when the pipe is subjected to tensile loading caused by the relative ground movements occurring along the pipe axis. As a part of the derivation, a modified interface friction model was developed considering the increase in friction due to constrained dilation of the soil, and the impact of mean effective stress on soil dilation. This interface friction model was combined with a nonlinear pipe stress–strain model to derive an analytical solution to represent the performance of the pipe. Using the proposed model, axial force, strain, and mobilized frictional length along the pipe can be obtained for a measured ground displacement can be obtained. Large-scale field pipe pullout tests were performed to verify the results of the proposed analytical model, in which good agreements were observed for tests conducted at different soil/burial conditions, displacement rates and pipe properties. Considering the similarities in the axial pullout mechanism, the analytical model was extended to explain the pullout response of geotextiles buried in reinforced soil structures. In this derivation, a new interface friction model was developed for planar members by considering the changes in normal stress due to constrained soil dilation. Another analytical model was derived for the case of a pipe that is subjected to combined loading from axial tension and bending when the initial soil loading is acting perpendicular to the pipe axis. With the direct account of the axial tensile force development, more realistic pipe performance behaviors were obtained as compared to the results obtained from traditional numerical formulations. iii PREFACE This dissertation contains details of the research program conducted at the University of British Columbia during the period from 2007 – 2011. All experimental, analytical, and numerical works presented in this dissertation were carried out by the author under the supervision of Dr. Dharma Wijewickreme. Author also acknowledges the constructive comments from the members of the supervisory committee to improve the readability and presentation of results. Following is a list of publications arisen from the studies presented in this dissertation. Also presented next to the publication is the corresponding chapter(s) of the dissertation in which the work is located. Author acknowledges the contributions from other co-authors (Dr. Dharma Wijewickreme and Mr. Gary Johnson). The original drafts of these publications were completed by the author and subsequently modified and improved by the co-authors. [J1]Weerasekara, L. and Wijewickreme, D. (2008) Mobilization of soil loads on polyethylene natural gas pipelines subject to relative axial ground displacements, Canadian Geotechnical Journal, 45(9):1237-1249. (Chapter 4) [J2]Weerasekara, L. and Wijewickreme, D (2010) An analytical method to predict the pullout response of geotextiles. Geosynthetics International. 17(4): 193-206 (Chapter 5) [C1]Wijewickreme, D, Weerasekara, L and Johnson, G.(2008) Soil Load Mobilization in Axially Loaded Buried Polyethylene Pipes, Proceedings of the 12th IACMAG Conference, 1-6th Oct, Goa, India (Chapter 4) [C2]Weerasekara, L. and Wijewickreme, D (2010) Response of buried plastic pipelines subject to lateral ground movement, Proc. of 8th Int Pipeline Conf., ASCE, Calgary, AB. (Chapter 6) [C3]Weerasekara, L. and Wijewickreme, D. (2010) Full-scale Model Testing of Buried Extensible Pipes Subject to Relative Axial Soil Loading, 7th International Conference on Physical Modeling in Geotechnics (ICPMG), June 28 - July 01, 2010, Zurich, Switzerland, (Chapter 3) [C4]Wijewickreme, D and Weerasekara, L. (2011) Analytical modeling of pipe subject to axial ground displacement, Proceedings of the 12th IACMAG Conference, May 9 - 11, 2011, Melbourne, Australia.(Chapter 3 and 4) iv TABLE OF CONTENTS ABSTRACT ............................................................................................................................. ii PREFACE.................................................................................................................................... iii TABLE OF CONTENTS ......................................................................................................... iv LIST OF TABLES ................................................................................................................... ix LIST OF FIGURES .................................................................................................................. x NOMENCLATURE .............................................................................................................xviii ACKNOWLEDGEMENTS .................................................................................................... xx 1. Introduction ....................................................................................................................... 1 1.1 Background……………………………………………………………………………… .. 1 1.1.1 Accidents in natural gas pipelines ............................................................................ 2 1.1.2 Accidents caused by ground movements .................................................................. 2 1.2 Determination of the pipe response due to ground movements…………………………. .. 3 1.2.1 Framework for understanding the pipe-soil interaction ............................................. 4 1.3 Pipe performance due to axial soil loading………………………………..........................6 1.4 Performance of geotextiles during pullout……………………………...............................7 1.5 Pipes subject to axial tension and bending due to abrupt ground displacement..................7 1.6 Objectives of the thesis……………………………………………………………………. 8 1.7 Scope of the thesis…………………………………..........................................................10 1.8 Thesis structure…………………………………………………………………………... 11 1.9 Organization of the thesis....................................................................................................12 2. Literature review ............................................................................................................. 15 2.1 Overview of the chapter………………………………………………………………….. 15 2.2 Performance of the pipes subject to axial soil loading…………………………………... 16 2.3 Experimental studies to determine response of buried pipes subject to axial soil loading ………………………………………………………………….......................……….…16 2.3.1 Field pipe pullout testing........................................................................................ 19 2.3.2 Field pipe monitoring............................................................................................. 20 2.4 Analytical models to determine pipe response from axial soil loading…….......................20 2.4.1 Interface friction between soil and pipe .................................................................. 21 2.4.2 Stress-strain behavior of the pipe material .............................................................. 31 v 2.5 Analytical and numerical methods to model response of axially loaded pipes………….. 39 2.6 Summary of key observations: Axial loading of pipes…………………………………... 41 2.6.1 Limitations in experimental studies ........................................................................ 41 2.6.2 Limitations in the current analytical models to estimate the pipe performance in axial soil loading .................................................................................................. 42 2.7 Analytical models to estimate the pullout response of geotextiles………………………. 43 2.8 Summary of key observations: Pullout response of geotextiles…………………………. 45 2.9 Performance of the pipes subject to tension and bending caused by ground movement …………………………………………………………………………............................46 2.9.1 Experimental studies to determine pipe performance when subject to abrupt ground deformations ......................................................................................................... 46 2.9.2 Analytical models to determined pipe performance subject to abrupt ground deformations ......................................................................................................... 47 2.10 Determining the lateral soil resistance per unit length of pipe (P)……………………… 47 2.10.1 Analytical approaches to determine the lateral soil resistance of pipe ..................... 48 2.10.2 Experimental approaches to determine the lateral soil resistance per unit length. .... 49 2.10.3 Numerical methods to model the lateral soil resistance .......................................... 52 2.10.4 Recommendations in current guidelines to calculate the lateral soil resistance ....... 54 2.11 Analytical models to estimate pipe performance when pipe is subject to tension and bending..................................................................................................................................55 2.11.1 Analytical solutions based on theory of beams on elastic foundation ...................... 56 2.11.2 Analytical methods based on cable-like profile for the pipe behavior ..................... 59 2.11.3 Analytical methods developed by combining the axial force and bending moment interaction........................................................................................ …………….61 2.12 Numerical modelling of performance of a pipe subject to ground movement…………..65 2.12.1 3D continuum finite element modeling .................................................................. 65 2.12.2 Soil-spring type analysis ........................................................................................ 66 2.13 Summary of key observations: analytical models to find response of pipes subject to axial loading and bending…………………………………………………………………. 67 2.14 Closure….....................................................................................................................…..68 3. Experimental aspects ...................................................................................................... 69 3.1 Characterization of material properties…………………………………………………… 69 3.1.1 Properties of backfill material (Fraser River Sand) ................................................. 69 3.1.2 Properties of the pipe material (MDPE) ................................................................. 71 3.2 Experimental details of field pipe pullout tests…………………………………………... 73 3.2.1 Trench and other supporting structures .................................................................. 74 3.2.2 Pulling mechanism and self-reaction frame ............................................................ 76 3.2.3 Instrumentation and data acquisition ...................................................................... 80 3.2.4 Preparation of test specimens ................................................................................. 82 vi 3.3 Test program…………………………………………………………………………….. 86 3.3.1 Density of soil backfill ........................................................................................... 89 3.3.2 Burial depth ........................................................................................................... 89 3.3.3 Rate of ground displacement .................................................................................. 91 3.4 Typical results from pipe pullout tests………………………………………………....... 92 3.4.1 Variation of pullout resistance and strain with displacement .................................. 92 3.4.2 Stress –strain behavior of the pipe material in pullout tests .................................... 94 3.4.3 Instrumentation to estimate mobilized frictional length from pipe pullout .............. 96 3.5 Closing remarks………………………………………………………………………….97 4. Analytical model for pipe response from axial soil loading ........................................... 99 4.1 Introduction……………………………………………………………………………… 99 4.1.1 Slide geometry ..................................................................................................... 100 4.2 Analytical model to determine pipe performance................................................................101 4.2.1 Development of analytical model for interface friction......................................... 101 4.2.2 Modified interface frictional resistance ................................................................ 108 4.2.3 Stress-strain behavior of the pipe material ............................................................ 108 4.2.4 Analytical formulation of the pipe–soil interaction response ................................ 111 4.3 Selection of input parameters...............................................................................................114 4.3.1 Determination of the interface friction angle (δ) ................................................... 114 4.3.2 Determination of thickness of shear zone (∆td) ..................................................... 115 4.3.3 Determination of shear modulus degradation (G) ................................................. 116 4.3.4 Determination of strain rate dependant hyperbolic constants ................................ 118 4.3.5 Determination of lateral earth pressure coefficient (K0) ........................................ 120 4.4 Comparison of results obtained from analytical solution and laboratory pullout tests......121 4.4.1 Experimental details ............................................................................................ 121 4.4.2 Selection of input parameters for the analytical solution....................................... 124 4.4.3 Axial pullout tests performed on dense sand ........................................................ 126 4.4.4 Axial pullout tests performed on loose sand ......................................................... 132 4.5 Comparison of results obtained from analytical solution and field pullout tests...............133 4.5.1 Selection of input parameters for the analytical solution....................................... 133 4.5.2 Analytical model estimations for field pullout Tests # 1, 2 and 3.......................... 135 4.5.3 Analytical model estimations for field pullout Test # 4 ........................................ 141 4.5.4 Analytical model estimations for field pullout Test# 5 ......................................... 145 4.5.5 Estimation of mobilized frictional length ............................................................. 150 4.5.6 Extension of the pipe in the exposed area of the pipe ........................................... 154 4.6 Numerical modelling of the axial soil loading of pipes..………………..........................154 4.6.1 Determinations of the axial soil-springs ............................................................... 155 4.6.2 Viscoelastic modeling of pipe stress-strain behavior ............................................ 155 4.6.3 Numerical model development using ABAQUS .................................................. 156 4.6.4 Numerical modeling of the laboratory pipe pullout tests....................................... 156 4.6.5 Numerical modeling of the field pipe pullout tests ............................................... 159 vii 4.7 General observations…………………………………………………………................165 4.7.1 Comparison between the new analytical approach and experimental results ......... 165 4.7.2 Comparison between numerical (ABAQUS) and experimental results ................. 165 4.8 Proposed interface friction model for cylindrical objects …..………………..................166 4.8.1 Normalization of the interface friction ................................................................. 168 4.9 Proposed analytical solution for the overall pipe performance….....................................170 4.9.1 Impact of different of pipe size and burial depth on pipe displacement capacity ... 171 4.9.2 Impact of different SDR values on pipe performance ........................................... 172 4.9.3 Redistribution of strain ........................................................................................ 174 4.9.4 Some comments on the effect of internal pressure on the pipe .............................. 176 4.10 Closing remarks…………............................................................................….………...176 5. An analytical model to estimate the pullout response of geotextiles ………………….178 5.1. Introduction……………………………………………………......................................178 5.2 . Development of the analytical solutions for determining the pullout response of geotextiles………………………………………………………………...........................179 5.2.1 Factors affecting the geotextile–soil interface behavior ........................................ 180 5.2.2 Stress-strain behavior of geotextiles ..................................................................... 183 5.2.3 Analytical formulation of the geotextile–soil interaction response ........................ 184 5.3 Validation of the analytical solution……………….........................................................188 5.3.1 Hyperbolic parameters to represent geotextile material stress-strain response....... 191 5.3.2 Soil-geotextile interface friction angle ( /s GSYφ′ ) ..................................................... 192 5.3.3 Comparison of results from the analytical approach with reported experimental observations........................................................................................................ 194 5.3.4 Performance charts for geotextiles ................................................... …………….202 5.4 Discussion…..………………………………………………….......................................205 5.4.1 Advantages in the proposed analytical solution .................................................... 206 5.4.2 Consideration for correct interpretation of data from pullout tests ........................ 208 5.4.3 Soil dilation in planar and cylindrical objects ....................................................... 209 5.4.4 Other factors influencing the pullout response of geotextiles ................................ 210 5.5 Closure…………………….……………………….………………….............................214 6. Pipes subject to axial tension and bending arising from ground displacement .......... 216 6.1 Introduction...…………………………………………………………………..…..........216 6.2 Derivation of analytical solution to determine pipe response from combined loading from tension and bending………………………………………………………........................217 6.2.1 Lateral soil resistance........................................................................................... 219 6.2.2 Axial soil resistance ............................................................................................. 224 6.3 Numerical modelling to determine pipe response due to combined loading from tension and bending…………………………………….………………………............................226 viii 6.4 Comparison between analytical and numerical results………………….........................227 6.4.1 Discussion of results obtained from analytical and numerical modeling ............... 231 6.5 Closure…………………………………………………………………...........................241 7. Summary and conclusions ............................................................................................ 242 7.1 Experimental research contributions……………………………………........................243 7.2 Development of an analytical solution to determine pipe response due to axial soil loading………………………………………....................................................................243 7.2.1 Validation of the proposed analytical model .. ……………………………………245 7.3 Development of an analytical solution to determine pullout response of a geotextile ……………………………………………………..........................................................246 7.4 Development of an analytical solution to determine the performance of a pipe subject to axial loading and bending………..……………………………………….........................247 7.5 Future work and recommendations…………………………..........................................249 REFERENCES ..................................................................................................................... 251 APPENDIX .......................................................................................................................... 266 ix LIST OF TABLES Table 2.1. Relaxation power law exponents for MDPE (after Stewart et al. 1999) ..................... 34 Table 3.1 Components of test setup ........................................................................................... 76 Table 3.2 Details of the strain gauges ........................................................................................ 81 Table 3.3 Testing dates/ durations for the field pipe pullout tests ............................................... 87 Table 3.4 Average soil densities measured in field pipe tests ..................................................... 89 Table 3.5 Summary of test parameters for field pipe pullout tests .............................................. 92 Table 4.1. Input parameters for the analytical solution to model the axial pullout tests performed in laboratory environment ................................................................................................ 125 Table 4.3 Mobilized frictional lengths back calculated from the analytical solution for each pullout test. ...................................................................................................................... 151 Table 4.4 Relative contribution to the interface friction from static overburden and soil dilation ........................................................................................................................................ 170 Table 5.1 Details of sand used in pullout tests ......................................................................... 189 Table 5.2 Details of geotextiles used in pullout tests ............................................................... 190 Table 5.3 Input parameters for the geotextile pullout tests ....................................................... 191 x LIST OF FIGURES Figure 1.1 Soil load represented as soil-spring loads acting on a pipe element in three directions. ............................................................................................................................................4 Figure 1.3 Schematic representation of the (a) axial soil loading and (b) soil loading arising from abrupt ground movement .....................................................................................................8 Figure 1.4 Components of the thesis. ......................................................................................... 12 Figure 2.1 Axial pullout resistance characteristics of 60-mm and 114-mm pipes buried in dense sand (after Weerasekara and Wijewickreme 2008). ............................................................ 19 Figure 2.2 Variation of normalized pullout resistance with displacement compared with the loads calculated from ASCE (1984) and ALA (2001) guidelines (after Wijewickreme et al. 2009) ................................................................................................................................ 24 Figure 2.3 Normalized axial pullout force and normalized displacement for 60-mm and 114-mm pipes in loose and dense sands (after Anderson, et al. 2004) ............................................. 29 Figure 2.4 Different regions of stress-strain behavior for Polyethylene ...................................... 33 Figure 2.5 Actual stress-strain relationships obtained at different strain rates ( at a temperature of 21˚C) and the respective Prony series predictions for MDPE (after Popelar et al. 1990) ... 36 Figure 2.6 Affected soil regions (soil wedges formed) for pipes buried in (a) deep and (b) shallow burial conditions ................................................................................................... 50 Figure 2.7 Proposed lateral soil resistance for buried pipes by Trautmann and O'Rourke (1983). .......................................................................................................................................... 55 Figure 2.8 Schematic diagram showing the different regions of PGD assumed by Miyajima and Kituara (1989) .................................................................................................................. 57 Figure 2.9 Schematic representation of the peak and post peak soil resistance in frost-heave or thaw-induced settlements (after Hawlader et al. 2006). .................................................... 58 Figure 2.10 Schematic diagram of the pipe deformed shape assumed by Kennedy et al. (1977). 62 Figure 2.11 Schematic diagram showing the different regions of interaction when subject to abrupt ground deformation occurring normal to the pipe axis. ........................................... 63 Figure 3.1 Grain size distribution of Fraser River Sand ............................................................. 70 Figure 3.2 Variation of the peak friction angle of FRS with confining stress (after Karimian 2006)................................................................................................................................. 71 xi Figure 3.3 Uniaxial compression tests performed on a 60mm pipe specimen. ............................ 72 Figure 3.4 Stress-strain responses obtained from the uniaxial compression tests at different strain rates (at a temperaure of 20 ˚C).......................................................................................... 73 Figure 3.5 Site layout for field pipe pullout testing .................................................................... 75 Figure 3.6. Pulling mechanism and wooden frame..................................................................... 76 Figure 3.7 The component of the pulling mechanism and the reaction frame (plan view) .......... 77 Figure 3.8 The key components of the pulling mechanism......................................................... 78 Figure 3.9 Instrumentation, coupling, gasket and the other arrangements at the pulling end of the pipe ................................................................................................................................... 80 Figure 3.10 Strain gauge arrangement in a MDPE pipe (Weerasekara 2007) ............................. 83 Figure 3.11 Trench used for the field axial pipe pullout tests with (a) pipe placed before backfilling and (b) after filling the trench with FRS ........................................................... 85 Figure 3.12 Arrangement of the two-piece split clamp ............................................................... 86 Figure 3.13 Precipitation during the testing period from June 21st to Aug 19th 2010 in Vancouver (extracted from http://vancouver.weatherstats.ca/) ............................................................. 88 Figure 3.14 Temperature fluctuations during the testing period from June 21st to Aug 19th 2010 in Vancouver (extracted from http://vancouver.weatherstats.ca/) ...................................... 88 Figure 3.15 Perspective views of the Test #3 after backfilling to a height of 0.98m ................... 90 Figure 3.16 Variation of leading end displacement with time for each field pipe pullout test ..... 91 Figure 3.17 Variation of pullout resistance with displacement at leading end for all field pipe pullout tests ....................................................................................................................... 93 Figure 3.18 Variation of axial strains measured from two strain gauges (SG1 and SG2) with the displacement at the leading end of the pipe (from Test #1)................................................. 94 Figure 3.19 Stress-strain behavior obtained from all five field pipe pullout tests ........................ 95 Figure 3.20 Variation of strain rate with time for Test #1 .......................................................... 96 Figure 3.21 Displacement measurements for the leading and trailing ends of the pipe in Tests#3. .......................................................................................................................................... 97 Figure 4.1 Example of the slide geometry simulated in the current full-scale tests and by the analytical solutions .......................................................................................................... 101 Figure 4.2 Comparison of IR values computed from Chakraborty and Salgado (2010) relationship and triaxial tests performed on dry FRS at low overburden stresses. ................................ 104 xii Figure 4.3 (a) Mobilized frictional lengths corresponding to different degrees of relative pipe displacements; (b) Assumed response of the unit interface friction (T) with relative displacement for MDPE–sand interface. .......................................................................... 106 Figure 4.4 Model predictions using linear viscoelastic – viscoplastic model and experimental results for the test in which the strain rate was changed from 10-3/s from 10-2/s and changed back to 10-3/s (reproduced after Chehab and Moore 2004).............................................. 110 Figure 4.5 Direct measurement of the thickness of the shear zone at the pipe-soil interface at MDPE-FRS interface (after Karimian 2006) ................................................................... 116 Figure 4.6 Hyperbolic fit to the shear modulus degradation curves obtained by Seed and Idriss (1970) and Idriss (1990). ................................................................................................. 117 Figure 4.7. Comparison of initial shear modulus obtained from the empirical solutions and the values used in the analytical solution ............................................................................... 118 Figure 4.8. Variation of pullout resistance versus axial strain for Test #4................................. 120 Figure 4.9 Soil chamber at University of British Columbia ...................................................... 122 Figure 4.10 Typical pipe configuration for axial pullout behavior of 60 and 114 mm pipes. ... 122 Figure 4.11 Variation of pullout resistance with displacement at leading end for the tests conducted by Weerasekara (2007) and Anderson (2004) ................................................. 123 Figure 4.12 Variation of pullout resistance versus displacement at the leading end of the pipe for pullout tests conducted in loose backfill conditions (after Anderson 2004) ...................... 124 Figure 4.13. Comparison of pullout resistance versus axial strain at leading end behaviors between analytical solution and pullout tests for 60-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 126 Figure 4.14. Comparison of pullout resistance versus axial strain at leading end behaviors between analytical solution and pullout tests for 114-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 127 Figure 4.15 Variation of axial strain rate with time, obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Weerasekara 2007) ............. 128 Figure 4.16 Comparison of pullout resistance versus displacement at leading end obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 129 xiii Figure 4.17 Comparison of pullout resistance versus displacement at leading end obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Anderson 2004) .............................................................................................................. 129 Figure 4.18 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 130 Figure 4.19 Comparison of axial strain versus displacement at leading end obtained from analytical solution and pullout tests performed on 60-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 131 Figure 4.20 Comparison of axial strain versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in dense sand (Weerasekara 2007) ......................................................................................................... 131 Figure 4.21. Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 60-mm pipe buried in loose sand (Anderson 2004) .............................................................................................................. 132 Figure 4.22. Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in loose sand (Anderson 2004). ............................................................................................................. 133 Figure 4.23 Comparison of pullout resistance versus displacement at leading end obtained from the analytical approach and Test#1 .................................................................................. 135 Figure 4.24 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#2.......................................................................................... 136 Figure 4.25 Comparison of pullout resistance versus displacement at leading end obtained from analytical approach and Test#3 ........................................................................................ 136 Figure 4.26 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#1.......................................................................................... 137 Figure 4.27 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#2.......................................................................................... 138 Figure 4.28 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#3.......................................................................................... 138 xiv Figure 4.29 Comparison of pullout resistance versus axial strainobtained from analytical solution and from Test#1 .............................................................................................................. 139 Figure 4.30 Comparison of pullout resistance versus axial strain obtained from analytical solution and from Test#2 ................................................................................................. 140 Figure 4.31 Comparison of pullout resistance versus axial strainobtained from analytical solution and from Test#3 .............................................................................................................. 140 Figure 4.32 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#4.......................................................................................... 141 Figure 4.33 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test #4......................................................................................... 142 Figure 4.34 Comparison of pullout resistance versus axial strainobtained from analytical solution and Test#4 ....................................................................................................................... 142 Figure 4.35 Variation of axial strain rate with time obtained from analytical solution and Test #4 ........................................................................................................................................ 143 Figure 4.36 Time variation of the pullout resistance at the leading end of the pipe measured from Test#4 and modeled using the analytical solution ............................................................ 144 Figure 4.37 Time variation of the axial strain at the leading end of the pipe measured from Test#4 and modeled using the analytical solution ............................................................ 144 Figure 4.38 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#5.......................................................................................... 146 Figure 4.39 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#5.......................................................................................... 146 Figure 4.40 Comparison of pullout resistance versus axial strain at leading end obtained from analytical solution and Test#5.......................................................................................... 147 Figure 4.41 Variation of pullout resistance for the entire duration of the test, calculated from the analytical solution and from Test#5. ................................................................................ 148 Figure 4.42 Variation of axial strain at the leading end of pipe for the entire duration of the test, calculated from the analytical solution and from Test#5. ................................................. 148 Figure 4.43 Variation of pullout resistance at the initial period of the testing (up to 8000 s), calculated from the analytical solution and from Test#5. ................................................. 149 xv Figure 4.44 Variation of axial strain at the initial period of the testing (up to 8000 s), calculated from the analytical solution and from Test#5 ................................................................... 149 Figure 4.45 Schematic illustration of the frictional mobilization along a pipe: (a) before and (b) after mobilizing the anchoring resistance. ........................................................................ 150 Figure 4.46 Variation of displacement at leading end with mobilized frictional length of the pipe for Test #1 ....................................................................................................................... 152 Figure 4.47 Variation of axial strain at the leading end of the pipe with mobilized frictional length Test #1 .................................................................................................................. 153 Figure 4.48 Variation of pullout resistance at leading end with mobilized frictional length of the pipe for Test #1 ............................................................................................................... 153 Figure 4.49 Variation of: (a) pullout resistance with displacement, (b) axial strain with displacement, and (c) pipe stress with strain for the axial pipe pullout test conducted on a 60-mm pipe buried in dense sand (Weerasekara 2007). .................................................... 157 Figure 4.50 Variation of: (a) pullout resistance with displacement, (b) axial strain with displacement, and (c) pipe stress with strain for the axial pipe pullout test conducted on a 114-mm pipe buried in dense sand (Weerasekara 2007) ................................................... 158 Figure 4.51 Variation of pullout resistance with displacement at leading end the pipe for: (a) 60- mm pipe, and (b) 114-mm pipe buried in loose sand (Anderson 2004). ............................ 159 Figure 4.52 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#1. ............................................... 160 Figure 4.53 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#2. ............................................... 161 Figure 4.54 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#3. ............................................... 162 Figure 4.55 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, (c) pullout force with strain, (d) pullout resistance with time, (e) axial strain with time and (f) displacement at leading end with time for Test#4. ................................. 163 Figure 4.56 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, (c) pullout force with strain, (d) pullout resistance with time, (e) axial strain with time and (f) displacement at leading end with time for Test#5. ................................. 164 xvi Figure 4.57 Comparison of interface frictional behavior obtained from current guidelines and the proposed method. ............................................................................................................ 167 Figure 4.58 Axial soil resistance measured from pullout test performed on a steel pipe showing the sharp drop in interface friction at initial pipe displacements (after Karimian 2006) .... 168 Figure 4.59 Variation of pullout resistance with displacement at the leading end of the pipe for Tests #2 and 3. ................................................................................................................ 170 Figure 4.60 Predicted response of maximum displacement at the leading end of the pipe to reach 5% pipe axial strain versus burial depth in dense sand. .................................................... 172 Figure 4.61 Mobilization of friction along the pipe for two pipes with different SDR values .. 173 Figure 4.62. Axial strain distribution along the pipe for two pipes with different SDR values. 174 Figure 4.63. Comparison between pullout resistance obtained from Test #1 and Test #5 ........ 175 Figure 5.1 (a) Mobilized frictional lengths corresponding to different degrees of relative displacements; (b) Assumed response of the unit interface friction (T) with relative displacement for geotextile–sand interface....................................................................... 185 Figure 5.2 Analytical estimationsfor pullout tests performed by Fannin and Raju (1993) ......... 195 Figure 5.3 Analytical estimations for pullout tests performed by Konami et al. (1996) ............ 197 Figure 5.4 Analytical estimations for pullout tests performed by Tzong and Cheng–Kuang (1987) ........................................................................................................................................ 198 Figure 5.5 Analytical estimations for pullout tests performed by Racana et al. (2003) ............ 199 Figure 5.6 Analytical estimations for pullout tests performed by Bakeer et al. (1998) ............. 200 Figure 5.7 Analytical estimations for pullout tests performed by Ali (1999) ........................... 201 Figure 5.8 Analytical estimations for pullout tests performed by Alobaidi (1997) .................... 202 Figure 5.9 Simplified chart for determining performance of buried geotextiles under pullout conditions. Note: u is measured in millimeters (mm) and remaining length measurements are in meters (m). ............................................................................................................ 204 Figure 5.10 Pullout characteristics of planar steel straps buried at four different overburden stresses (after Abremento 1993) ..................................................................................... 210 Figure 5.11 Predictions for pullout tests performed by Racana et al. (2003), (a) without considering the change in thickness of the geotextile (black), and (b) considering the change in thickness of the geotextile (red-dotted lines). ............................................................... 212 xvii Figure 5.12 Pullout response predicted for Racana et al. (2003) considering the impact of change in thickness due to overburden stress (shown in dotted line) and assuming no change in thickness (black continuous line) ..................................................................................... 214 Figure 6.1. Forces acting on a pipe element subject to lateral soil loading (plan view). ........... 217 Figure 6.2 (a) Bilinear lateral soil resistance per unit length of pipe (b) schematic diagram of the deformed shape of the pipeline showing the two analytical regions. (Note that “w” is the pipe deformation from the pipe, wf is the one-half ground deformation measured at the abrupt ground deformation) ............................................................................................. 220 Figure 6.4 Comparison of: (a) deformed shape; (b) bending moment; (c) shear force; and (d) axial force on the pipe obtained from the numerical and analytical approaches for 114- mm pipe (large-scale ground deformations). ........................................................................... 229 Figure 6.5 Comparison of maximum bending moment obtained from the numerical and analytical approaches. ...................................................................................................... 230 Figure 6.6 Comparison of maximum axial force obtained from the numerical and analytical approaches ...................................................................................................................... 230 Figure 6.7 (a) Pipe deformation; (b) bending moment; (c) shear force; and (d) axial force in the pipe, obtained from the numerical solution for three different magnitudes of axial soil resistances. ...................................................................................................................... 232 Figure 6.8 (a) Pipe deformation, (b) bending moment, (c) shear force and (d) axial force in the pipe, obtained from the analytical solution for three different magnitudes of axial soil resistances. ...................................................................................................................... 233 Figure 6.9 Idealized soil pressure distribution around the pipe during lateral movement of the pipe (coupling of axial and lateral soil resistances) .......................................................... 235 Figure 6.10 Assumed soil stiffness distributions around the abrupt ground movement, (a) varying soil stiffness and (b) constant soil stiffness. ......................................................... 237 Figure 6.11 (a) Pipe deformation, (b) bending moment, (c) shear force and (d) axial force in the pipe, obtained from the numerical solution for two different magnitudes of axial soil resistances. ...................................................................................................................... 238 Figure 6.12 (a) Pipe deformation; (b) bending moment; (c) shear force; and (d) axial force in the pipe for 60-mm and 114-mm pipes. ................................................................................. 239 xviii NOMENCLATURE Ap The cross sectional area of the pipe a, b, c,d Constants that can be determined by fitting the stress-strain responses obtained from uniaxial extension or compression tests ˆ ˆ,a b Constants to represent the variation of mobilized frictional length (non dimensional) a ) and b ) Hyperbolic constants to represent the degradation of non-dimensionalized shear modulus ratio (non dimensional) GTXb Width of the geotextile (m) 0 0,C C ′ Constants to be determined from boundary conditions in pre-peak zone (non dimensional) , n n C C ′ Constants to be determined from boundary conditions in post peak region, for the nth geotextile/pipe element (m). D Diameter of Pipe d50 The average grain size of the soil. Eini Initial modulus of the geotextile/pipe (N/m 2) G0 Initial/ small strain soil shear modulus (N/m2) )(γG Shear modulus of soil at a shear strain of γ (N/m2) H Burial depth of the geotextile or depth to pipe springline (m) ID Relative density of the soil (non dimensional) IR Relative dilatancy index (non dimensional) 0K Lateral earth pressure coefficient (non dimensional) P Pullout force measured in the geotextile/pipe (N) Q, R Constants used in Bolton’s (1986) relationship to link effective stress to dilatancy (non dimensional). T Frictional force developed along a unit length geotextile/pipe (N/m) xix 1T Peak frictional resistance developed along a unit length of geotextile/pipe (N/m) Td Net increase in friction along a unit length of geotextile due to soil dilation (N/m) GTXt Thickness of the geotextile (m) u Relative displacement of the geotextile/pipe at any given point x (m) eu Relative displacement in the pre-peak zone (m) 'u Tensile strain in the geotextile/pipe at any given point x (%) Wp Weight of the pipe and its content x Distance to any point along the geotextile/pipe, and represents the mobilized frictional length (m) z Constant used to simplify the expression for relative displacement (non dimensional) Γ Constant used to simplify the expression for interface friction (non dimensional) δ Interface Friction Angle γ Shear strain (%) γ Effective unit weight of soil (N/m3) ∆σdc Increase in normal stress due to soil dilation (N/m2) η Hyperbolic constant used to represent the nonlinear stress-strain response of geotextile/pipe (non dimensional) λ Constant that represents the interface and geotextile/pipe material properties (1/m) ν Poisson’s ratio of the soil (non dimensional) σ ′ Normal effective stress due to soil overburden (N/m2) uσ ′ Stress in geotextile/pipe corresponding to a strain level of 'u (N/m2) τ Shear stress at the geotextile- soil interface (N/m2) /S GSYφ′ Interface friction angle between soil and geotextile (°) maxφ ′ Peak friction angle of the soil (°) cvφ′ Constant volume friction angle (°) maxψ Maximum angle of dilation (°) xx ACKNOWLEDGEMENTS It is a pleasant aspect that now I have the opportunity to express my gratitude to all those who gave me unconditional support in completing this thesis. First and foremost, my sincere gratitude is extended to my supervisor, Dr. Dharma Wijewickreme for his unwavering guidance and encouragements towards the successful completion of this thesis. His detailed and constructive comments, understanding and enthusiasm paved the way for a pleasant research environment. Furthermore, I also greatly appreciate the support given by the supervisory committee members comprising of Drs. Jonathan Fannin, Donald Anderson and Michael Lee for providing valuable comments to improve the content. The project was funded by FortisBC (formerly known as Terasen Gas Inc., B.C). My warm thanks are due to Mr. Gary Johnson and Mujib Rahmen for providing the necessary pipe materials and technical details promptly. Their valuable comments and support was an immense help towards the completion of this thesis and in publishing technical papers on this area. The gratitude is also extended to the staff at Terasen Gas for their assistance and productive remarks in making this project a success. It is with great pleasure to recognize the valuable support given by the staff of the Civil Engineering workshop. I would like to express my deep and sincere gratitude to Mr. Blair Patterson and Mr. John Wong of the Civil Engineering workshop for manufacturing the required testing components, instrumentations and for sharing his technical expertise. Taking into xxi consideration the immense difficulty of the physical work required to perform field pipe pullout tests, my sincere gratitude is extended to Mr. David Chin and David Lee for their tireless efforts in assisting in test preparation. Last but not least, I would like to thank my wife, Ashanthi, and my parents for their support during this period. 1 1 INTRODUCTION 1.1 Background Plastic pipes have been employed in gas, oil, potable water, sewer, marine, landfill, electrical and telecommunication lines due to their many advantages over rigid steel or concrete pipes. One of the main attractions is the cheaper material cost of plastic pipes compared to metallic pipes. Additionally, the lightweight and flexibility offered by the plastic pipes are likely to reduce the costs relating to pipe installation. It has been suggested that, compared to steel pipes, plastic pipes require less maintenance during their operation life-time if the pipes are properly designed and installed (PIPA 2001). A greater deformation tolerance and stress-relaxation in plastic pipes is another key advantage when considering the plastic pipes response to external loading. Owing to these benefits, plastic pipes have been employed in rugged terrain, in the presence of aggressive chemicals and in extreme climates. Owing to the aforementioned advantages, polyethylene (PE) pipes in particular have become vastly popular in natural gas distribution industry since its introduction in late 1960’s. In North America, more than 90% of the natural gas distribution systems use plastic pipes, of which 99% are PE pipes (PIPA 2001) in which MDPE (Medium Density Polyethylene) pipes account for 2/3rdof the usage in gas distribution industry. MDPE pipes have the added advantage of having a higher ductility and fracture toughness together with long-term strength and stiffness that is comparable to that of HDPE (Stewart et al., 1999). These gas pipes are manufactured in different sizes ranging from 12.5 mm (½”) conduits to 610 mm (24”) diameter pipes, and are available in wide range of pipe thicknesses. In contrast to these small diameter plastic pipes used in the distribution systems, large diameter steel pipes are used in gas transmission pipelines. These high–capacity transmission lines are designed to transmit natural gas from the source to refineries and distribution locations. 2 1.1.1 Accidents in natural gas pipelines In many applications, the pipe failure is defined as the loss of containment integrity of the pipeline. Unlike other utilities, the performance of pipelines in natural gas and oil industry is a key concern since the failures of the systems can cause property damage and even human losses, in addition to the invariably associated business disruption. Regardless of the smaller diameter in comparison to transmission lines, PE pipes are designed to carry significant gas pressures (e.g., maximum operating pressures around 700 kPa); thus, in an event of a pipe failure, the damage may become significant. According to Pipeline Hazardous Material Safety Administration (PHMSA 2010), the number of injuries and loss of human lives associated with accidents in distribution pipelines are much higher in comparison to those in transmission pipelines, despite the monetary loss that is relatively similar in both types of pipeline systems. This is in part due to the fact that gas distribution lines are located proximity to residential areas, whereas, the transmission lines are often placed in remote areas, away from locale or properties. Furthermore, distribution pipes cover a considerably longer pipe length compared to transmission pipes, thus increasing the exposure to natural hazards. 1.1.2 Accidents caused by ground movements According to PHMSA (2010), the post-failure analyses revealed that the majority of distribution pipe failures are attributed to excavation damages. However, when the excavation and intentional damages to the pipeline are taken aside, the more than 10% of the accidents are identified to be caused by ground movements. This is almost the half the amount when all the natural causes (including heavy rains, floods, and severe temperature fluctuations) are considered. Unlike other natural hazards that are uncertain (e.g., heavy rains, floods), in most instances, the ground movement hazard can be identified prior to the actual occurrence of the pipe failure. 3 Furthermore, if the ground movement is occurring at a slow rate, it may provide adequate time to undertake preventive measures as the cumulative displacement gradually increases. Thus, the natural hazard arising from the ground movement is “preventable” or “manageable” in most instances, as the ground movement is detected with fair and general certainty. 1.2 Determination of the pipe response due to ground movements With the modern technology, the ground movements and its variations over time can be detected and mapped with reasonable confidence. However, these precise ground movement data are of little use due to the lack of reliable models to correlate such displacements to the condition of the buried pipe (typically the strain in the pipe). One of the direct options of determining the pipe condition is to unearth the pipe or install strain measuring instruments on the pipe. However, this is a lengthy and expensive task and rarely undertaken in gas distribution pipelines. For these reasons, it is important to develop analytical and numerical methods to correlate the pipe performance to measured ground displacement. When the ground movement spans a wide area, initially, the vulnerable regions are identified from on a regional-level pipeline risk assessment. This is followed by detailed macro-level pipe- soil interaction analysis on pipe segments identified from the regional-level assessment (Wijewickreme et al. 2005). In such macro-level analysis, four basic pipe loading mechanisms can be identified depending on the orientation of the pipe with respect to the direction of the ground movement. (a) Pipe is subject to axial tensile loading; (b) Pipe is subject to axial tensile loading and pipe bending; (c) Pipe is subject to axial compressive loading; and (d) Pipe is subject to axial compressive loading and pipe bending. For each situation, the pipe performance can be estimated by using analytical or numerical methods. Closed-form analytical models can be derived considering the equilibrium and compatibility conditions at a pipe element-level. This thesis presents analytical models for 4 loading situations (a) and (b); however, the same theoretical derivations can be extended to explain the pipe performance in the remaining two situations as well. Alternatively, these loading situations are often analyzed numerically using traditional soil- spring analysis. In such situations, soil loading is represented by soil-springs in three basic directions: (a) along the pipe axis (axial); (b) perpendicular to the pipe axis (lateral); and (c) upward or downward directions (Figure 1.1). With the selected pipe material properties, the pipe performance (e.g., strain) is obtained by solving the equilibrium and compatibility equations using in-built algorithms in the computer program. Figure 1.1 Soil load represented as soil-spring loads acting on a pipe element in three directions. From the above discussion, it is evident that for deriving both analytical and numerical models, it is important to know the soil loading characteristics in each direction (i.e., soil-spring values). To obtain such soil loading characteristics, often pullout tests are performed with pipe buried at a specific orientation. For example, axial soil springs are obtained by pulling a pipe along the pipe axis whereas; the lateral soil-spring is obtained by pulling the pipe perpendicular to the axis. 1.2.1 Framework for understanding the pipe-soil interaction To comprehend the analytical and numerical formulations in this thesis, the author wishes to discuss the pipe-soil interaction problem using a three-component framework. The follow up Axial Lateral Vertical 5 discussions in this thesis will also be based on this framework. The steps in the framework are illustrated as follows. 1. Determination of the external soil loads on the pipe due to relative ground movements (e.g. in case of axial loading- friction at the pipe-soil interface T, or in case of lateral loading- the lateral soil resistance P); 2. Determination of the stress-strain behavior of the pipe material; 3. Combining the soil loading and the stress-strain behavior to derive an overall pipe performance model based on the element-level equilibrium and compatibility conditions. 1.2.1.1 Model-scale and element-level tests on pipes At this point, it is also important to illustrate the difference between model-scale tests and element-level tests. A pipe pullout test can be classified as an element-level test if the pipe is considered to be rigid. Thus, most of the pipe pullout tests (axial and lateral directions) conducted in steel pipes can be considered as element-level tests as the loading observed within the boundaries of soil-chamber environment is not sufficient to create significant strains or deformations in the pipe. In these element-level tests, the axial soil-spring is simply obtained by dividing the measure pullout resistance by the length of the buried length. In contrast, a pipe pullout test can be classified as a model-scale test when measurable strains or deformations are observed in the pipe before mobilizing any end boundary conditions within the test domain. As a result, a model–scale test results are influenced by the external load and the stress-strain properties of the pipe material. In brief, according to the framework presented previously, the overall response of the pipe can only be determined from model-scale tests, whereas, element- level tests are only sufficient to determine the external soil load acting on an object. 6 1.3 Pipe performance due to axial soil loading When considering the axial soil loading, the external load is applied in the form of frictional resistance along the pipe. The current practice is to use bilinear soil-spring model to represent the frictional force at the pipe –soil interface (ASCE 1984, ALA 2001, PRCI 2006). Although, these assumptions may yield reasonably good results for steel pipes, significant shortcomings are expected when such models are used for extensible plastic pipes. Numerous experimental evidence supports the fact that the frictional behavior is highly nonlinear and may significantly influenced by the increase in normal stress due to soil dilation and subsequent decrease in the normal stress (i.e., termed as frictional degradation). These factors are generally ignored in the current practice of determining the pipe performance. In steel pipes, the stress-strain behavior is assumed to be linear elastic or bilinear at its most complex form. In contrast, in viscoelastic material such as MDPE, the stress-strain behavior is highly nonlinear and will depend on the strain rate and the temperature. However, often these pipes are modeled as a linear elastic material (Katona 1990, Moore 1995 and Moser 1997). The exact modeling of the viscoelasticity and the relaxation of stress are important aspects as the rate of ground displacement is likely to vary during different seasons of the year (Keefer and Johnson, 1983; Kalaugher et al., 2000). For example, the rate of ground movement is likely to surge during the periods of high precipitation; Sweeney et al. (2004) notes that two-thirds of the pipe failure occurred during the months of heaviest rainfallin an investigation conducted in Colombian Andes. After establishing the actual interface frictional characteristics and the stress-strain behavior for the pipe material, it is important to derive an analytical solution for overall pipe performance by considering the element level equilibrium and compatibility conditions. Although, there are few analytical and numerical models exist, none of these models have been validated by comparing with real “model-scale” tests. This is partly due to problems associated with conducting large- scale model tests without the interferences from end boundary constraints arising due to limited test space. Besides the independent validation of the pipe response model, it is also important to 7 experimentally validate the interface frictional model and the material stress-strain response model. 1.4 Performance of geotextiles during pullout The determination of the pullout response of geosynthetics is an important aspect in determining the internal stability of reinforced soil walls, slopes and embankments. Considering the similarities in the pullout mechanism in axial pullout behavior of pipes and planar geotextiles, the same analytical formulations can be extended to simulate the pullout response observed in geotextiles. Unlike in pipes, numerous “model-scale” pullout tests have already been performed on geotextile over the past few decades, thus these experimental results can be employed to validate the analytical solutions for wide range test conditions. These solutions would provide the necessary analytical framework to determine the suitable thickness, strength and lengths for geotextiles in reinforced structures. 1.5 Pipes subject to axial tension and bending due to abrupt ground displacement The exact frictional force development along the pipeline is an important aspect in attempting to determine the pipe response when the ground movement results in bending and axial tension in the pipe [see Figure 1.2(b)]. In this thesis, it is also attempted to develop a new analytical model to evaluate the pipe performance when the initial soil loading is acting perpendicular to the pipe axis by combining the model developed for axial soil loading. 8 (a) (b) Figure 1.2 Schematic representation of the (a) axial soil loading and (b) soil loading arising from abrupt ground movement The early analytical solutions to assess the pipe performance in above mentioned loading situations were derived based on the theories of beams on elastic foundation. These methods are generally valid for pipes with smaller allowable strain levels and undergoing small deformations. However, extensible pipes with larger allowable strains can subject to large deformations before reaching their performance limits. At large strain levels, the sections of the pipe closest to the ground displacement offset will likely to form a cable-like profile thereby reducing the bending moment and increasing the dependency on axial load carrying capacity of the pipe. Nonetheless, the pipe sections further away from the ground displacement offset will carry the lateral soil load by bending. Therefore, it is essential to formulate an analytical solution combining both tension and bending; and representative of the entire pipeline behavior. The model should also consider the nonlinear stress-strain response of the pipe material. 1.6 Objectives of the thesis From the above discussion, it is evident that there is a need to conduct experimental and analytical research to understand the performance of buried MDPE pipelines subjected to ground movement. The outcome from such research would provide the key input to developing the 9 needed criteria or guidelines for design and performance evaluation of pipeline systems. Preferably, such design methodologies should be based on simpler and well-established analytical principles, thus easily implemented by design engineers. Accordingly, the following are identified as the detailed objectives of this thesis: 1. Develop an analytical solution to model the friction at the pipe-soil interface, incorporating the influence of soil dilation and frictional degradation. These factors need to be accounted through proper analytical models to calculate the frictional resistance at pipe element level. 2. Determine a stress-strain model to simulate the strain rate dependant nonlinear stress- strain behavior for the pipe material and validate using independent experimental findings. 3. Develop an analytical model to represent the overall pipe performance by combining the interface frictional forces with nonlinear stress-strain behavior of the pipe material. 4. Conduct a numerical model (soil-spring based) to simulate the pipe response using the proposed frictional resistance model and a viscoelastic model to represent the stress- strain behavior of the pipe material. 5. Perform large-scale field axial pipe pullout tests to overcome the limitations (e.g., limited burial length) in laboratory-scale pullout tests. The experiments should be designed to investigate the response of pipes at different rates of loading, the impact of stress- relaxation in viscoelastic pipes, the impact of the overburden stress and the pipe response at large strain levels. Compare the experimental results obtained from the field and laboratory pullout tests with the results obtained from the analytical and numerical models. 10 6. Extend the analytical solution to explain the observed pullout response of planar geotextiles in reinforced earth structures, and verify the analytical approach by comparing with published results from different scholars. In this analytical model for geotextiles, it is required to develop a separate interface frictional model to account for the soil dilation and frictional degradation aspects in planar members. Develop an analytical approach to determine the pipe response accounting for the combined effect of tension and bending moments in pipes when the initial ground movement is perpendicular to the pipe axis. 1.7 Scope of the thesis To accomplish the aforementioned research objectives, the following scope of work was conducted in this research project. 1. Conduct a field pipe pullout testing program to study axial pullout response of pipes. This included the design and construction of the self-reacting loading mechanism and wooden shores to retain the trench material. As a part of this work, conduct of five axial pipe pullout tests with different burial depths, pullout rates and loading regimes. The pipe performance is directly measured through strain gauges, string potentiometers and the load cells. 2. Develop of a new analytical model to determine the response of a pipe due to axial soil loading. The model is derived using an advanced interface friction model to account for the soil dilation and interface friction aspects and combined with the nonlinear stress-strain response for the pipe material. This also includes model validation by comparing the strain, pullout resistance, displacement and mobilized frictional length with experimental data obtained from laboratory and the field pullout tests [a total of ten pullout tests]. A numerical model based on soil-spring analysis is also compared with the experimental findings. 11 3. Develop a similar analytical solution to explain the pullout response observed in planar geotextiles. The analytical model includes a new interface friction model for planar geotextiles. The proposed model is validated by comparing with twenty four pullout tests performed by nine different scholars. A simplified performance chart and equations are proposed to estimate the strain and mobilized frictional length along a geotextile. 4. Develop an analytical solution to explain the pipe performance when the pipe is subject to bending and tensile loading arising due to ground movement occurring perpendicular to the pipe axis. 1.8 Thesis structure Based on thesis objectives and scope listed above, Figure 1.3 shows an illustration of the basic components of this thesis. The boxes shown in green color represent the external soil loads acting on the pipe (axial or lateral soil load) and the pink color boxes represent the material stress-strain behavior of the pipe. The work leading to original contributions by the author is shown in solid boxes. The boxes bordered in dotted lines shows the contribution stemming from other researchers which largely contributed to this thesis. 12 Figure 1.3 Components of the thesis. 1.9 Organization of the thesis Chapter 1: Introduction. This is the present chapter, and it presents the statement of the problem and the relevance of this research topic. A brief description of the soil loading arising from ground movement and the analytical techniques to determine the pipe performance is included. Furthermore, thesis structure is described together with the objectives and scope of the thesis. 13 Chapter 2: Literature review. This chapter presents a review of published information to describe the current understanding of the pipe-soil interaction aspects. A description of previous experimental studies (e.g., full-scale pipe pullout tests), and the resulting analytical techniques for analyzing the pipe response under axial and lateral soil loading are included in this chapter. Owing to the lack of research conducted on plastic pipes, previous studies on steel pipes are also discussed under the topics of model tests, numerical modeling, analytical methods, field testing and monitoring, since this information is directly relevant for the understanding of the subject problem. Chapter 3: Experimental aspects. This chapter will discuss the parameter selection for the tests with regard to burial depth, rate of displacement and boundary condition of the branch pipe and the resulting test matrix for the field pipe pullout tests. The properties of soil and MDPE pipes are discussed in details. Furthermore, this chapter will include a description on the test setup, preparation, instrumentation and testing method. Chapter 4: Analytical model to determine the pipe response from axial soil loading In this chapter, an analytical model to explain the pipe performance due to axial soil loading is presented. This also includes details on the methods to account for the nonlinear interface frictional behavior and material behavior of MDPE pipes. Furthermore, the proposed model is validated by comparing with over twenty laboratory and field axial pipe pullout tests. Chapter 5: Analytical model to determine the pullout response of planar geotextiles As an extension of the analytical model derived for pipes in Chapter four, this chapter presents the derivation of a similar analytical model estimate the pullout response of planar geotextiles. The model is then employed to model over twenty two pullout tests performed by nine different scholars. 14 Chapter 6: Response of a pipe subject to tension and bending caused by ground movement This chapter presents details on the derivation of a new analytical solution to estimate the pipe performance when the pipe is subject to tensile loading and bending. The results from the analytical solution are compared with the results obtained from traditional soil-spring analysis, and respective merits of each approach are discussed. Appendix This contains a summary of the equations presented in Chapter 4 and 6. 15 2 LITERATURE REVIEW 2.1 Overview of the chapter Over the last few decades, several studies have been undertaken to determine the response of the buried pipelines subject to ground movement. These studies include laboratory tests that spans from full-scale pipe pullout tests to centrifuge tests, field pipe testing, and pipe system monitoring. Based on these experimental results, several numerical and analytical models have been developed to determine the response of buried pipes subject to ground movements. However, most of the research has been focused on large diameter steel pipes, and it is believed that the findings on steel pipes can contribute to the understanding of more complex interaction in plastic pipes. In addition, considering the similarities in the interaction aspects, the studies conducted on piles, soil nails and planar geotextiles will further facilitate the understanding of buried pipes under different soil loading conditions. First half of this chapter will focus on the previous research conducted on axial pullout behavior of buried pipes and geotextiles. A brief overview of the axial pullout tests performed on pipes is presented. Afterwards, several existing analytical models developed to understand the pipe response under axial soil loading is presented with emphasis on the interface frictional models and pipe stress-strain models adopted in each of these analytical approaches. Similar discussion on geotextiles is also presented following the section on axial soil loading on pipes. The second half of the chapter includes details of the existing analytical methods to determine the pipe response when the pipe is subject to combined tensile and bending moment caused by ground movement. Finally, the limitations identified in each area are summarized. 16 2.2 Performance of the pipes subject to axial soil loading Determination of the pipe performance subject to axial soil loading is an important aspect not only when the ground movement is along the pipe axis, but also when pipe is subject to soil loadings is acting in other directions. However, only a limited number of experimental studies have been performed on axially loaded pipes in comparison to situations in which the soil loading is acting perpendicular to the pipe axis (i.e., lateral soil loading). This is partly due to the notion that longitudinal resistance is about an order of magnitude smaller than the lateral soil resistance, thus, deemed more important to understand the lateral soil resistance compared to axial frictional resistance. However, with the use of pipes made of relatively high flexibility, the pipes may undergo large deformations; as such, the response of the pipe can be significantly influenced by the axial soil resistance with the formation of cable-like profile even when the initial soil loading is perpendicular to the pipe axis. More details on this subject matter will be discussed in Chapter 6. For these reasons, a thorough understanding of the axial soil loading is paramount when determining the response of pipe, irrespective of the initial soil loading direction. Several experimental and analytical studies have been performed by a number of researchers to understand the frictional force development along buried pipes subjected to relative axial movement. The following sections intend to discuss some of the previous key studies related to this subject in detail. 2.3 Experimental studies to determine response of buried pipes subject to axial soil loading Thus far, only a very few experimental studies have been performed to evaluate the performance of plastic pipes subject to axial soil loading. To gain a fundamental understanding of the frictional force development along the pipe, it is typical to perform axial pullout tests on buried pipes. To the best of author’s knowledge, the experimental research on plastic pipes has been 17 centered only at two research institutions, i.e., Cornell University, Ithaca, NY and University of British Columbia, Vancouver, B.C. The following is aimed at presenting the details of the pullout test performed at these institutions. 2.3.1 Pullout tests performed at Cornell University The first recognized axial pullout tests on plastic pipes were performed at Takeo Mogami Geotechnical Laboratory at Cornell University to determine the pullout resistance of high density polyethylene (HDPE) pipes at different temperatures. These axial pullout tests were conducted in a box of 1.22mm in length and 0.914min width. The axial pullout test results from this study were reported by Stewart et al. (1999) and later by Bilgin et al. (2007). The investigation was also aimed at determining the interface frictional resistance when subject to temperature-induced cyclic loading. The knowledge of interface frictional resistance at different temperature fluctuations is required to estimate the loads transmitted to cast iron pipes that are connected to plastic piping when the pipeline is subject to thermal fluctuations. The thermal expansion coefficients in PE is about 15 times larger than in cast iron. The PE pipe pullout tests at Cornell University were performed in the temperature control chamber at temperatures ranging from (-20 to 60 °C). For all the tests, pipes having a nominal diameter of 150 mm with standard dimension ratio (diameter to thickness ratio) of 11 were used. An important aspect of these experiments was the measurement of the changes in pipe diameter with the temperature. Bilgin et al. (2007) observed a linear relationship between the pipe diametric reduction and the interface shear resistance, thus it is hypothesized that the diametric reduction results in decrease in normal stresses acting on the pipe. With this assumption, in large diameter pipes, the change in resistance would be large due to proportional diametric change resulted from the drop in temperature. For example, extrapolating these observations to large diameter pipes, Bilgin and Stewart (2009) stated that a pipe of diameter 300 mm, subject to a temperature change of 12°C would result in zero axial frictional resistance. This hypothesis will be investigated subsequently in Chapter 4 on basis of interface friction model developed in this thesis. 18 In repeated pulling and pushing tests (i.e., cyclic tests), Bilgin and Stewart (2009) observed that the soil resistance degrades considerably in each cycle. Furthermore, additional tests revealed that there is only a very small recovery of shear resistance due to aging after cyclic loading, although some resistance can be recovered through surface vibrations. 2.3.2 Pullout tests performed at University of British Columbia (UBC) To expand the knowledge base on the performance of plastic pipes, a full-scale pipe pullout testing facility was constructed in 2002 as a part of the research collaboration with FortisBC (earlier known as Terasen Gas Inc), Surrey, B.C. The test configuration comprise of a large soil chamber, hydraulic actuators and a data acquisition system. The soil chamber has a width and height of 2.5 m. The compartmental construction of the soil chamber allows the length of the soil chamber to be changed. The initial tests conducted by Anderson (2004) were performed with a length of 5 m and the subsequent tests conducted by Weerasekara (2007) were performed with a box length of 3.8 m. More details of these soil test chamber and the pullout tests are presented in Anderson (2004) and Weerasekara (2007). In each test, the buried pipes were instrumented with strain gauges and string potentiometers to measure displacements at selected locations. The pullout resistance was measured using a load cell placed at the pulling end (also called the leading end herein). Typical variations of pullout resistance with displacement are shown in Figure 2.1 (Weerasekara 2007) using data from tests conducted on 60-mm and 114-mm diameter pipes. As may be noted, it was observed that the peak axial load would develop at an axial displacement of about 20 mm (peak resistances are shown in arrows) in these tests on PE pipes. Similar pullout test results were observed by Anderson (2004) when performing pullout tests on pipes having similar diameters. Besides axial pipe pullout tests performed on plastic pipes, few pipe pullout tests have been performed on steel pipes in laboratory environment, (e.g., Karimian 2006, Paulin et al. 1998). The displacement value to reach peak axial pullout load in PE pipes are significantly large 19 compared to results from pullout tests on steel pipes, where the peak axial resistance would develop at a relatively smaller displacement (i.e. 2-3 mm). Further details of these test results will be discussed in the following sections. Figure 2.1 Axial pullout resistance characteristics of 60-mm and 114-mm pipes buried in dense sand (after Weerasekara and Wijewickreme 2008). 2.3.3 Field pipe pullout testing Field pullout testing on buried pipes can be considered of significant merit as the tests are conducted essentially under “real-life” conditions. Another advantage in field testing is that these tests can be conducted over a longer period, without being subjected to space-time constraints that are more common in laboratory environments. Therefore, the field tests are preferred when the focus is towards understanding the effects of time related behaviors and large-scale behaviors. Nevertheless, only a very few field pipe pullout tests have been performed due to logistic problems and costs associated with such testing. 20 Audibert and Nyman (1977) performed a lateral pullout test on a steel pipe with a diameter of 230 mm buried in carver sand. Rizkalla et al. (1991) and Cappelletto et al. (1998) reported eight axial pullout tests on buried steel pipes with different backfill materials. As a part of the investigation to determine the impact of thermal variation on pipelines, Bilgin et al. (2007) performed two field pipe pullout tests using 6”(150 mm) nominal diameter cast iron pipe buried in dense and loose sands. Some of the important findings arising from these experiments are presented in the following sections. 2.3.4 Field pipe monitoring Continuous monitoring of critical gas pipeline sections sometimes forms an important part in assessing the performance of buried pipelines, especially when certain pipeline segments are considered critical, despite the difficulties in long-term monitoring, accessibility in remote areas and large cost associated with the process. Notably, Bruschi et al. (1996) and Bughi et al. (1996) reported details of such field monitoring of a pipeline located in Italian mountainous areas that is subject to slow ground movement. The data obtained from the investigation was subsequently used in numerical modeling of the pipe-soil interaction mechanism. Bruschi et al. (1996) noted the difficulties in numerical modeling due to the lack of knowledge in representing the large slide-front unlike the tests conducted in a controlled environment where the slide geometry is directly identifiable or pre-defined. 2.4 Analytical models to determine pipe response from axial soil loading The following briefly discusses the existing analytical and numerical approaches to determine the response of pipes subject to axial ground movement. When discussing these analytical formulations, it is important to identify the components of the framework described in Chapter 1. Accordingly, the analytical models will be discussed based on the following topics: (1) interface friction between pipe and soil; (2) stress-strain properties of the pipe material; and (3) pipe performance equations derived by combining above two aspects. 21 2.4.1 Interface friction between soil and pipe When axial loading of the buried pipe is concerned, the friction between the pipe and soil is the primary source of loading on the pipe. Therefore, it is prudent to closely inspect the key parameters that impact the pipe–soil interface behavior. The frictional resistance at the interface of the soil and steel pipes is generally modeled as bilinear soil-spring with maximum interface shear resistance per unit length (T) computed as follows, 0(1 ) tan 2 DH KT pi γ δ+= (2.1) Where D is the pipe diameter, H is the burial depth of the pipe, γ is the density of the surrounding soil, Ko is the lateral earth pressure coefficient at rest, and δ is the interface friction angle between pipe and soil. This equation assumes an idealized soil pressure distribution around the pipe. Although, the calculation of the frictional resistance using Equation 2.1 is simple and straight forward, the calculated value would depend on the accuracy of the selected input parameters as well as the accuracy of the assumptions that Equation 2.1 is based upon. Equation 2.1 is widely used in pipeline performance assessments. Most notably, Newmark and Hall (1975) used this equation to calculate axial soil loads on pipes subjected to strike slip fault movement. In a similar analytical model for pipe crossing a strike slip fault, Kennedy et al. (1977) used this formula with the assumption that frictional resistance is the primary mode of load acting when a pipe is subject to large deformations and eventually forming into a cable- like profile closer to the fault crossing (details will be discussed in Section 2.9). Most importantly, this equation has been recommended for the computation of axial soil loads in ASCE (1984) “Guidelines for the Seismic Design of Oil and Gas Pipeline Systems”, American Lifeline Alliance (2001) “Guidelines for the Design of Buried Steel Pipe”, and 22 PRCI (2009) “Guidelines for the Seismic Design and Assessment of Natural Gas and Liquid Hydrocarbon Pipelines”. Besides Equation 2.1, few other equations have been proposed for the estimation of axial soil loads. McAllister (2001)suggested using the following equation for determining the axial frictional resistance, in which the weight of the pipe is considered. 2 ( ) tan 2u p DT D H Wγ δ = − + (2.2) Where Wp is the weight of the pipe which can be considered negligible for the light weight plastic pipes such as PE. Danish Submarine Pipeline Guidelines (1985) proposed Equation 2.3 to estimate the frictional force per unit length of pipe. This equation was derived based on the integration of shear stresses around the pipe. 0 0 0 4 tan ( ) (1 ) (2 ) (2 ) tan 2 2 3 3 p u WD D DT H K K Kγ γϕ pi δ pi = + + + + − + (2.3) 2.4.1.1 Comparison of interface friction models with actual pullout test results With this knowledge on interface friction, it is important to confirm the validity of these models for interface friction by comparing with few known “element-level” axial pipe pullout test results. Equation 2.1 has been widely used in practice and the following comparisons are limited only to this equation. According to the argument presented in Chapter 1 with regard to model- scale and element-level tests, the best method of determining the interface friction is to conduct pullout tests on a buried rigid pipe (i.e., essentially inextensible pipe), in which the impact of the pipe stress-strain behavior has little influence. Because of the high rigidy, a pullout test conducted on such scale can be considered as an ‘element-level’ test. 23 Paulin et al. (1998) conducted full-scale testing on a 324-mm steel pipe buried in loose and dense sand in a 5.2-m-long soil chamber at C-CORE, Newfoundland. When comparing the frictional resistance obtained from these test, Paulin et al. (1998) stated that the value obtained by Equation 2.1 will under-predict the frictional resistance when the pipe is buried in loose sand. However, it was noted that the peak axial frictional resistance in pipes buried dense sand is much larger than the value calculated from Equation 2.1 using the best-estimated input parameters. Similar observations were made by Wijewickreme et al. (2009) and Karimian (2006) from the axial pullout tests performed on steel pipes having diameters of 457 mm. Figure 2.2 shows the axial pullout tests results obtained by Wijewickreme et al. (2009) for steel pipes buried in dense and loose sands compared with ASCE (1984) and ALA (2001) recommendations. The tests comprehensive set of test details relating to Tests AB-3, AB-4 and AB-6 are provided in Wijewickreme et al. (2009). Note that in Figure 2.2, the measured axial pullout resistances (FA) from three different pullout tests are normalized with respect to pipe diameter, overburden stress and pipe length (L) as follows: ' A A FF H DLγ pi = (2.4) 24 Figure 2.2 Variation of normalized pullout resistance with displacement compared with the loads calculated from ASCE (1984) and ALA (2001) guidelines (after Wijewickreme et al. 2009) 2.4.1.2 Factors influencing the interface friction With the discrepancies in the estimated and the actual frictional resistance observed from the axial pullout tests as described above, it is advisable to investigate the factors influencing the interface friction and the relevance of Equation 2.1. (i) Shear–induced dilation of soil There is evidence from pullout and interface shear tests performed on pipes (e.g., Karimian 2006; Wijewickreme et al. 2009), piles (e.g., Lehane 1992), soil nails (Luo, et al. 2000) and in planar soil reinforcements such as geotextiles (e.g., Schlosser and Elias 1978) that under constrained volume or constant stiffness conditions, the frictional resistance may increase by several orders. For example, Jardine and Lehane (1993) observed that the normal stresses on 25 the pile could rise as much as 50% due to soil dilation, and the increase in normal stress is inversely proportional to the pile radius. In a soil with relatively high density, the shear–induced volumetric expansion (i.e. dilation) of soil is expected to occur predominantly within a thin annular shear zone around the pipe. Since the outward movement of the soil particles is constrained by the surrounding soil mass, the normal stress on the pipe will increase. This increase in normal stress has been further established through direct pressure measurements at the interfaces of driven piles (e.g., Lehane 1992) and in pullout tests on rigid steel pipes (e.g., Wijewickreme et al. 2009). Karimian (2006) performed some steel pipe pullout tests with pressure transducers mounted on the pipe surface. Based on the normal stress measurements from these pressure transducers, it was evident that the normal stress will increase during pipe pullout. Karimian (2006) attributed this increase in normal stress to soil dilation, and further stated that the use of conventional earth pressure coefficient “at rest” (K0) in Equation 2.1 does not adequately represent the actual normal stress distribution on the pipe, in particular during pipe pullout. With absence of other parameters to attribute this increase in normal stress in Equation 2.1, Wijewickreme et al. (2009) suggested using a much larger K value in place of K0 for pipes buried in dense soils. For example, Karimian (2006) observed that for a 457 mm (18”) diameter steel pipe buried in dense sand with a H/D ratio of 2.5, the K value of about 2.5 is required to match the observed peak frictional resistance. This value for K is much larger than the typical K0 values that ranges around 0.5. The experimental findings from Karimian (2006) were directly incorporated in PRCI (2009) guidelines for axially loaded pipes. In spite of the noted increase in normal stress due to shear induced dilation, the following shortcomings are identified. 1. There is lack of guidelines to determine the proper K value and often the value is determined empirically. As stated in PRCI (2009), K value varies from the value from “at rest” conditions for loose soil to values as high as 2 in dense dilative soils. In the K- 26 value approach, the increase in normal stress cannot be determined simply based on known analytical theories of soil mechanics. Moreover, the increased K value, i.e., ratio of lateral to vertical effective stress has little physical meaning when representing the actual increase in normal stress around the pipe. 2. The method considers only the peak interface frictional resistance, whereas the pipe-soil interaction is affected by the frictional resistance in the entire displacement range (i.e., post-peak interface frictional resistance). Furthermore, frictional resistance–displacement behavior is not bilinear as observed even in the pullout tests performed on steel pipes. As a result, the above proposed K-value approach lacks the analytical explanation to describe the entire friction–displacement behavior. (ii) Frictional degradation From large displacement interface shear tests, it is also known that after the initial soil dilation, the normal stress will reduce with increase in displacement. This, termed as frictional degradation behavior, has been observed in soil element tests such as large displacement ring shear tests (e.g. Tan et al. 1998), buried geotextiles (e.g., Mak and Lo 2001), buried pipes (e.g., Weerasekara and Wijewickreme 2008) and in pile driving (e.g. White and Lehane 2004). For example, the strain gauges attached to the pile have shown a reduction in local skin friction from pile toe to head (Randolph et al. 1994: Foray et al., 1998). The reduction in diameter due to Poisson’s effect is not sufficient to explain this stress reduction in pullout tests (de Nicola and Randolph 1993; Wijewickreme et al. 2008). This behavioral phenomenon can be effectively explained as the contraction associated with wear and tear of grain asperities leading to particle rearrangement, particularly under large shear displacements (Johnson et al. 1987, Foray et al. 1998, Luo et al. 2000, Zeghal and Edil, 2002). Price (1988) stated that with the increasing displacement, the finer particles may fill the space between larger particles, in turn increasing the contact area. The increase in contact area will reduce the inter-particle forces. With this mechanism, it is also argued that the rate of particle crushing will be reduced with the increasing displacement. 27 Although the frictional degradation behavior has been widely observed in large–displacement frictional tests, very few studies have been performed to evaluate this behavior in detail (Randolph et al. 1994). It is known that the frictional degradation behavior will depend on the average grain size, shape, level of stress, density, mineralogy of the soil particles (crushability) and roughness of the interface (Boulon and Nova 1990; Al-Douri and Poulos 1991). A few soil-pipe interaction models exist to model the above frictional degradation aspects. However, it is not feasible to account for all these aspects in formulating constitutive laws to account for frictional degradation, especially in terms of the particulate–level features of the interface, e.g. degree of roughness, grain crushability, etc. (Zeghal and Edil 2002). As a result, some of the models have been developed using semi-analytical approaches (e.g. Selvadurai and Boulon 1992). Alternatively, using the results obtained from constant normal stiffness direct shear tests performed at the University of Grenoble, Hoteit (1990) proposed a set of experimental based formulae to evaluate the frictional degradation behavior for different relative densities, average particle sizes, initial effective stresses and normal stiffness values. As indicated earlier, frictional degradation arises mainly due to particle crushing and particle rearrangement. Particle crushing could take place due to large overburden pressures in excess of about 800 kPa. Since the stress levels corresponding to burial depths encountered in typical buried pipes are significantly low (i.e., 10 to 50 kPa), the increase in normal stress due to dilation may not be large enough to initiate particle crushing. Nevertheless, these large displacements can cause wear and tear of grain asperities. The results of constant volume and constant stress direct shear tests conducted on loose and dense sands has shown that grain crushing is related to the amount of plastic work and independent of stress path and confining stress (Zeghal and Edil 2002). 28 2.4.1.3 Comparison of existing analytical models to determine the response of PE pipes Anderson (2004) and Weerasekara (2007) have performed axial pipe pullout tests on polyethylene pipes buried in dense and loose soil. In attempt to explain these test results, Weerasekara and Wijewickreme (2008) distinguished the differences between the “element- level” axial pullout response of rigid pipe and the “model scale” axial pullout response of the extensible MDPE pipes. Simply stated, the tests on extensible MDPE pipes cannot be treated as element-level tests to directly obtain the respective axial soil-spring values, where the force per unit length would be calculated by dividing the total axial force by the length of buried pipe. An axial pullout test performed on an extensible plastic pipe may be considered as an element-level test only if the test is conducted with a short burial length for the pipe or with a pipe having a large diameter/thickness – i.e., in cases where the axial deformation of the pipe will be significantly small and the pipe can be treated as rigid. In absence of proper analytical framework to explain the model-scale pipe pullout response, in Figure 2.3 Anderson et al. (2004) attempted to compare the magnitudes of the frictional resistance (T) obtained from the pipe pullout tests with the frictional force estimated from current guidelines (e.g., ASCE 1985;ALA 2001) assuming rigid pipe response (i.e., element-level). In Figure 2.3, shows two tests conducted on loose sand (60aL and 114aL) and two tests conducted in dense sand (60aD and 114aD). Also note that the number in the test identification refers to the pipe diameter in millimeters. 29 Figure 2.3 Normalized axial pullout force and normalized displacement for 60-mm and 114- mm pipes in loose and dense sands (after Anderson, et al. 2004) 2.4.1.4 Determining the displacement corresponding to peak frictional resistance In a bilinear soil-spring model, the knowledge of displacement at which the maximum force is mobilized (xmax) is also an important aspect. In current practice, this value has been recommended based mainly on data from experimental observations. After a series of tests on steel pipelines, Singhal (1980) observed that the peak interface frictional resistance is mobilized after 0.1” (2.5 mm) to 0.2” (5 mm) of pipe displacement. The experimental finding on steel pipes performed by Paulin et al. (1998) and Karimian (2006) were also similar to this observation. Based on these experimental findings, in most guidelines (e.g., ASCE 1984), it is recommended that xmax to be taken as 2 to 3 mm for steel pipes in dense sand, whereas in loose sand this value would be around 5 mm. In these guidelines (e.g., ALA 2001, 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 2.0 4.0 6.0 8.0 Normalized SP Displacement, x/D N o rm al iz ed Fo rc e, T/ γγ γγH D L 115aD 60aD 115aL 60aL Predicted tu Dense Predicted tu Loose 30 PRCI 2009), a specific value for xmax is not recommended for plastic pipes. However, considering that the smooth surface in a plastic pipe will require a larger displacement to mobilize its peak frictional resistance, Stewart et al. (1999) employed a xmax value of 5mm for the axial soil-spring model. In model-scale axial pullout tests performed on plastic pipes (e.g., Anderson 2004), it has been observed that the mobilized interface frictional length along the pipe increases progressively with the increasing leading end displacement of the pipe. As a result, xmax would depend on pipe properties (pipe cross sectional area, pipe length tested), material properties (stress-strain behavior) and soil characteristics (burial depth, density, friction angle, lateral earth pressure coefficient). Therefore, as explained by Weerasekara (2006), defining a unique xmax value as in steel pipe is not appropriate for pipes which would experience significant elongations during pullout; instead, as per the delayed peak observed in pipe pullout tests in model-scale tests, the value of xmax would be larger than those for the steel pipes (Figure 2.2). 2.4.1.5 Interface frictional behavior of pipes buried in fine-grained materials Although the focus of the present research is on coarse-grained soils, it important to recognize the performance of pipes buried in fine-grained soils as well. To author’s knowledge, there have not been any axial pullout tests performed on pipes buried in fine-grained soils. This is mainly due to difficulties in handling the material in a soil chamber. However, few field pipe pullout tests were performed on steel pipes by Rizkalla et al. (1991) using a fine-grained backfill. Eight axial pipe pullout tests were conducted on pipe buried pipes having diameters of 219 mm and 600 mm. The test results have revealed that the use of the undrained shear strength (Su) of soil to estimate the axial soil loads on pipe would overestimate the friction when the pipe is subject to slow axial ground deformations. As the shear zone is confined to a very small annular area around the pipe, Cappelletto et al. (1998) stated that using an effective stress model provides more reasonable evaluation of the soil loads than a total stress model even for the fine-grained soils. Their studies further noted that 31 soil aging would likely to increase the dilatancy of the surrounding soil, result in an increase in pullout resistance. In comparison to coarse-grained soils, it has been noted that the effect of soil aging will be particularly important in fine-grained soils where the peak resistance could increase by a factor of 1.8-2.0 (Cappelletto et al. 1998). 2.4.2 Stress-strain behavior of the pipe material The stress – strain behavior of the pipe material is a key factor that affects the overall response of the pipe, besides the exact knowledge of the interface behavior. In most guidelines, accurate representation of the pipe stress-strain behavior in the computations is seldom addressed. Instead, the common practice has been to assume a linear elastic stress-strain behavior, which is a reasonable representation of the stress-strain behavior of steel pipes. Bilinear or multi-linear models are also adopted in certain analytical solutions to represent yield-hardening of steel pipes. However, in plastic pipes, the stress-strain behavior is known to be nonlinear and governed by the viscoelastic properties of the polymer. The work undertaken in this study mainly focuses on the response of buried MDPE pipes, since the pipes made of MDPE are widely used in gas distribution industry. As such, the following discussion is mainly focused on the mechanical (stress-strain) properties of MDPE material, since its direct impact on the pipe performance. It is however noted that the new framework arising from this study is applicable to other polymer based pipes that are used in various other applications. 2.4.2.1 Mechanical behavior of MDPE (stress-strain) It is important to understand the molecular structure of the polymers to as a part of comprehending the mechanical response of polymeric materials. Polyethylene is produced by polymerization of ethylene molecules. This manufacturing process creates a molecular structure that consists of crystalline and amorphous molecular formations. Crystalline long-chain molecules form the backbone of the structure. The amorphous short branches extending from 32 these molecules form the cross-links with the adjoining crystalline molecules (Powel 1983). The short and long-term performance depends on the response of these amorphous and crystalline structures under loading. As a result, the different properties of high, medium and low density PE can be explained from their respective molecular structure. A typical stress–strain characteristic of MDPE can be schematically shown as in Figure 2.4. Accordingly, the stress-strain behavior of MDPE can be categorized into three regions. The different behavior in each region depends on the molecular response in each stage of loading. The crystallinity of the molecular structure will result in elastic-like behavior, while amorphous structure would cause polymers to behave as a viscous fluid. The stress-strain curve can be interpreted as linear elastic behavior at small strains (instantaneous recoverable strains) followed by viscoelastic (recoverable strains but delayed recovery) and visco-plastic (inelastic response with irrecoverable strains) characteristics. However, in reality, it is impossible to clearly demarcate these regions during the straining process of MDPE. In general, the elastic region is significantly small and viscoelastic responses can be observed even at small strain levels. Polymers such as MDPE can stretch more than 400% of the tensile yield stress before breaking. Nevertheless, from a user/operator’s point of view, the preference is not to allow the strains to exceed the viscoelastic region. 33 Figure 2.4 Different regions of stress-strain behavior for Polyethylene The stress-strain behavior of MDPE pipe is further complicated by the dependence on temperature and strain-rate. As discussed by Stewart et al. (1999), the modulus of PE is affected by the variation of temperature, e.g. between 0-49ºC, the modulus can vary by a factor of two with a higher modulus at lower temperatures. In addition, the stress-strain properties of a viscoelastic material will also depends on the strain rate. When a polymer based material is subject to a higher strain rate, a higher deformation modulus is observed. It has been suggested that the molecules have less time to orient in the direction of loading, resulting this observed stiffer stress-strain response. Similarly, a softer response can be expected for the stress-strain response when the pipe is subject to slower strain rate. 2.4.2.2 Viscoelastic behavior of MDPE The following section is aimed at briefly discussing the viscoelasticity and some of the analytical models to represent the viscoelastic behavior. Ignoring the temperature dependency of the viscoelastic materials, the stress –strain relationship can be expressed in the following form (Powel 1983): 34 ( , )f tσ ε= (2.5) At smaller strain levels, this behavior can be further simplified as follows. ( )f tσ ε= (2.6) This is termed as “linear” viscoelasticity and according to Bilgin et al. (2007), the linear viscoelastic assumptions can be successfully employed at small strains (less than 1%). Based on this assumption, the modulus of the pipe at a given time can be written as 0 0 ( )( ) ntE t E tσ ε − = = (2.7) Where the modulus and the corresponding strain at time, t = 1 min is given as 0E and 0ε , respectively. The power law exponent representing the rate of stress relaxation is denoted by n. Table 2.1 lists some of n values obtained for MDPE by few scholars Table 2.1. Relaxation power law exponents for MDPE (after Stewart et al. 1999) Nominal Temperature Power Law Exponent, n Husted and Thompson (1985) 70 °F (21°C) 0.105 Janson (1985) 70 °F (21°C) 0.081 Keeney (1999) 20 to 120 °F (-7 to 49°C) 0.085 ± 0.01 35 2.4.2.3 Models to determine the stress-strain behavior of a viscoelastic model Modeling of the viscoelastic characteristics of the polymer based material is an extensive research topic. Several methods have been developed to model the viscoelastic material behavior of selected plastics (e.g., Popelar et al. 1990; Hasan and Boyce 1995; Zhang and Moore 1997; Boyce et al. 2001). As the principle objective of herein is not on the stress-strain behavior of viscoelastic materials, a detailed discussion on this matter is not undertaken. Only the following models that are relevance to the context of this thesis have been selected for the discussion. (i) Prony series modeling of viscoelasticity Prony series can be considered as one of the most simplistic forms of modeling the stress-strain behavior of a viscoelastic material (e.g., Powel 1983). Prony series expansion for reference relaxation modulus (ER) can be written as / ) 0 1 ( ) (1 i N t R i i E t E E e τ− = = + −∑ (2.8) Where iEN ,, ,τ are material constants. The characteristic relaxation times are taken as .sec10 2−= iiτ Hence, the coefficients 0E [= (0)RE ] to NE are generally determined from stress- relaxation or creep tests data of a particular polymer. Despite the simplicity in the formulation, Popelar et al. (1990) noted that Prony series is not capable of predicting stress-strains for a wide range of the strain rates. Note that in Figure 2.5, the Prony series estimations are shown in red dotted lines. 36 Figure 2.5 Actual stress-strain relationships obtained at different strain rates ( at a temperature of 21˚C) and the respective Prony series predictions for MDPE (after Popelar et al. 1990) (ii) Hyperbolic model for viscoelastic materials The hyperbolic model is another simple approach to model viscoelasticity and commonly presented in the following form. 1ini E εσ ηε = + (2.9) Where Eini and η are hyperbolic constants. Considering the strain rate dependant behavior of the polymeric materials, Suleiman and Coree (2004) proposed a modification to the hyperbolic model to account for the strain rate dependant behavior. In this approach, the strain rate dependant hyperbolic constants are determined by curve-fitting the experimental observations. 0 5 10 15 20 25 0 5 10 15 20 25 Strain (%) St re ss (M Pa ) 10-1 (%/sec) 10-2 (%/sec) 10-5 (%/sec) 10-4 (%/sec) 10-3 (%/sec) 37 The following strain rate dependant hyperbolic constants were obtained by matching the uniaxial compression tests performed by Zhang and Moore (1997) on HDPE pipe material. ( ) 1207.055.2932 ε&=iniE (2.10) ( ) )ln(526.1396.39 55.2932 1207.0 ε εη & & + = (2.11) Whereε& is the strain rate. Hyperbolic model provides a simple framework for accurate modeling of the stress-strain behavioral patterns of a viscoelastic material. However, it to be noted that such models are not capable of modeling the plastic (permanent) deformations and the unloading responses. Suleiman and Coree (2004) proposed another approach named “focus point approach” to obtain more precise strain rate dependant stress-strain characteristics using the basic hyperbolic format. Merry and Bray (1996) proposed a similar strain rate dependant stress-strain model using a modified form of the hyperbolic equation as follows max ( ) ( ) ( ) fR E ε σ ε εβ ε σ ε = + & & & (2.12) Where fR is the ratio of the maximum stress to a fictitious ultimate stress which is larger than the maximum stress. ( )E ε& is the strain dependant secant modulus and ( ) ini E E εβ = & The parameters fR and β are determined from curve-fitting experimental data, once the rate dependant parameters (i.e., ( )E ε& and max( )σ ε& ) are determined from uniaxial compression or 38 extension tests. For example, Merry and Bray (1996) found fR and β to be 0.92 and 0.65, respectively for HDPE geomembranes tested under uniaxial tensile tests. In an independent study, Wesseloo et al. (2004) proposed a method similar to Merry and Bray (1996) approach by defining a transition point in the stress-strain plot. This method assumes that the stress-strain response under tensile loading is hyperbolic in shape up to a transition strain and becomes linear at large strain levels. However, this method requires a subjective judgment in selecting the transition strain. According to Wesseloo et al. (2004) this transition strain was determined to be 16%. Wesseloo et al. (2004) also observed that the transition strain is independent of the strain rate. Therefore, it can be inferred that if the relevant stress-strain behavior is less than the transition strain (say ~16%), the hyperbolic model is adequate for modeling stress-strain behavior of the material. In most engineering applications, the strain is seldom allowed to exceed 10%. In Chapter five, it is attempted to extend the analytical model developed for axial loading in Chapter four to model the pullout response of planar geotextiles (see Section 2.6). As a result, it is also important to investigate the stress-strain properties of the geotextiles as most of the geotextiles are manufactured from polymers that tend to show nonlinear stress-strain behaviors similar to the plastic pipe materials discussed above. Merry and Bray (1996) and Wesseloo et al. (2004) approaches above have been proposed to simulate the stress-strain response of HDPE geotextiles. Besides the above referenced methods, several other strain rate dependant stress-strain models have been proposed by a number of other researchers to model, e.g., “hyperbolic tangent functions” by Prager (1955), and “n-order polynomial method” by Giroud (1994). However, the hyperbolic format is more efficient with less number of input parameters than polynomial functions. 39 2.4.2.4 General comments on viscoelastic models The strain rate dependant modeling of the stress-strain behavior is an important aspect in modeling the pipe response under soil loading. For example, even if a constant rate of ground displacement is maintained, the strain rate can even change by two orders of magnitude. This aspect will be demonstrated in Chapter four when the actual strain gauge readings obtained from pullout testing are presented. Most importantly, the typical displacement rates experienced in practice are likely to be several orders slower than the rates observed in experiments conducted in a laboratory environment. Thus, it is likely that the actual strain rates are also several orders smaller than in a pullout test conducted in a control environment. Therefore, a stress-strain model that would suitable capture the strain-rate-effects is required to for the effective modeling of the response of buried MDPE pipes subjected to soil loadings. With this background, it is also worthwhile to note that the model predictions at higher strain rates have been observed to be difficult even with rather complex viscoelastic model (e.g. Chehab and Moore 2004). As the mechanical response of these polymers are affected by the strain rate, stress level and loading direction (compression or tension), it is not easy to develop a model which simulates all the loading situations successfully. 2.5 Analytical and numerical methods to model response of axially loaded pipes After establishing a suitably representative behavior for the interface behavior and the stress- strain properties for the pipe material, the next step is to combine these components to obtain the governing equations for pipe response. This type of analysis can also be performed using soil- spring type models with the aid of a computer program. Alternatively, analytical solutions have been developed considering the equilibrium and the compatibility of pipe elements. Over the last few decades, a few analytical models have been proposed in attempt to determine the pipe response from axial soil loading, e.g., O’Rourke and Nordberg (1992), Trigg and Rizkalla (1994), Rajani et al. (1995) and Chan and Wong (2004). Although the formulation 40 in these methods is similar, slightly different assumptions for the pipe stress-strain behavior and frictional behavior were employed in each method. The assumptions on the selection of boundary conditions, slide geometry and the deflected shape will also influence the solution approach in each method. When developing models for axially loaded pipes, it is important to identify the slide geometry of the landslide. For example, in the method proposed by Trigg and Rizkalla (1994) and Rajani et al. (1995) two analytical approaches were proposed to determine the axial response of the pipe depending on the slide geometry. The first approach considers the soil to be sliding along the pipe and employs a limit equilibrium type analysis. Thus, with the known frictional coefficient, the stresses on the pipe can be determined. Sliding along the pipe is only possible if the soil mass is unstable at the pipe level. In the second approach, the soil mass was assumed to be stable in the soil block containing the pipe segment. This type of loading is expected in most slow landslides, in which the surrounding soil is stable but the actual sliding is occurring at a weak soil layer below the pipe level. In this approach, the relative displacement arises from the elongation of the pipe triggered from the frictional development along the pipe. The governing pipe performance equations are derived based on the force equilibrium of pipe elements. Extending these analytical derivations for soil block movements, Chan and Wong (2004) derived an analytical solution for deep-seated slip failures having circular arch profile. In all the above approaches, the interface frictional resistance is assumed to be either elastic or elastic-perfectly plastic. The necessary input parameters for the frictional resistance are obtained from the respective guidelines (e.g., Equation 2.1). Furthermore, certain models (e.g., Rajani et al. 1995; Trigg and Rizkalla 1994) employed a bilinear stress-strain model for steel pipe that accounts for the yield hardening of the pipe material. Thus, the pipe modulus of steel pipe is assumed to be of two values, with initial modulus E1 and post yield modulus of E2 (E1>E2). From this analysis, Rajani et al. (1995) and Trigg and Rizkalla (1994) have shown that, elastic assumptions for the pipe and soil will cause the axial strain to be overestimated. However, the steel pipes are capable of tolerating strain levels which exceed the strain corresponding to the strain-hardening point (~0.1%). Therefore, by using a smaller pipe modulus beyond the yield point can avoid excessive strain estimations for the pipe. This aspect is of special interest in 41 MDPE pipes which are known to exhibit nonlinear stress-strain behavior as discussed previously. In summary, these analytical results highlight the need to address the nonlinear behaviors of the frictional resistance and pipe stress-strain behavior. 2.6 Summary of key observations: Axial loading of pipes As noted in the above literature review, there are a number of limitations that arises when attempting to estimate the pipe response from axial ground movement. The following section summarizes these limitations with respect to experimental and analytical studies on axially loaded pipes. 2.6.1 Limitations in experimental studies It is evident that only limited number of experimental studies has been performed so far on both steel and plastic piping to investigate axial soil loading on pipes. In particular, testing conducted under controlled conditions, at least under model-scale level, is desirable to obtain a reasonable understanding of the buried pipe performance under axial soil loading conditions. The length of piping that can be accommodated in the laboratory testing facilities has limited the level of pipe strains that can be imparted during testing, in turn, constraining the ability understand the response of pipe at relatively large strain levels. Furthermore, investigations on time-dependant behavior of the plastic pipes are rarely undertaken due to time constrictions in laboratory environment; however, it is to be appreciated that accounting for time-dependant behavior has considerable merit especially when it is known that the ground movements experienced by buried pipes in real-life can be intermittent. It is believed that large-scale field pipe pullout tests can be employed to overcome some of the above limitations. 42 2.6.2 Limitations in the current analytical models to estimate the pipe performance in axial soil loading The limitations in the analytical approaches can be highlighted with regard to the following two topics: (a) Limitations in modeling the interface frictional behavior (b) Limitations in the models representing the stress-strain properties 2.6.2.1 Limitations in modeling the interface frictional behavior It is evident from the literature review, that the main obstacle in the modeling the pipe response is the lack of understanding in interface frictional behavior as per below: (a) Although very few methods recognize the increase in soil normal stress on the pipe due to constrained soil dilation (Section 2.4.1.2), none of the methods account for this increase in soil normal stress on the pipe based on an analytical framework. (b) None of the methods have recognized the gradual decrease in interface frictional resistance in pipe after reaching its peak interface frictional resistance. Even if this aspect is not essential in rigid steel pipes with smaller allowable strain limit, this is an important factor in extensible pipes with the potential to undergo large displacement before failure. (c) Other aspects such as the impact of mean effective stress on soil dilation should also be accounted using an analytical approach when investigating the impact of pipe burial depth, soil density etc. 43 2.6.2.2 Limitations in modeling stress-strain behavior of the pipe material The following limitations relating to stress-strain modeling were identified when exploring the current analytical tools. (a) Most analytical solutions are based on linear elastic stress-strain assumption for the pipe material. At most, more sophisticated solutions will employ a bilinear stress-strain behavior for the pipe material. Although such assumptions are representative of the steel pipes, it is not sufficient to model the highly nonlinear stress-strain response of most plastic pipes. (b) The stress-strain behavior of plastic pipes is viscoelastic and depends on the strain rate. None of the analytical solutions employ a stress-strain model to account for the strain rate dependant behavior of pipe material. 2.7 Analytical models to estimate the pullout response of geotextiles The pullout mechanism in plastic pipes is fundamentally similar to the pullout behavior of geosynthetics. As indicated earlier, in the present study, an attempted is also made to extend the analytical model developed for axial loading of MDPE pipes (Chapter 4) to model the pullout response of planar geotextiles (see Chapter 5). Unlike the axial soil loading on pipes, a number of pullout tests have been already performed on geotextiles by different researchers and gives an opportunity to validate the analytical solutions for wide range of test conditions. The following section presents some of the analytical solutions that were proposed to determine the pullout response of geotextiles. However, the author has intentionally limited the discussion on this subject as the main focus of this thesis is to address the soil loading on pipes. During the last few decades, several analytical methods have been derived based on continuum mechanics by combining the interface responses and the material properties to explain the 44 pullout response of relatively extensible reinforcements (i.e., geotextiles). Abramento and Whittle (1995) proposed a method based on shear-lag analysis assuming a linear elastic stress– strain behavior of the geotextile and the soil. Their method was in good agreement with the pullout test results at a small displacement levels corresponding to the linear range of the stress– strain behavior of the geotextile. Sobhi and Wu (1996) proposed an analytical model that was based on rigid plastic shear stress mobilization, and the method was employed to predict the pullout test results obtained by Tzong and Cheng-Kuang (1987). In a similar analytical model, Madhav et al. (1998) used a bilinear model to characterize the non-linear interface shear behavior, while Gurung and Iwao (1999) and Milligan and Tei (1998) employed a hyperbolic models that is computationally efficient compared to a bilinear models used to represent the interface friction. Besides these models, several other models have been derived with different mathematical derivations and interface behaviors, e.g. Konami et al. (1996), Alobaidi et al. (1997), Gurung et al. (1999), Racana et al. (2003). Despite the best effort to represent the non-linear behavior of the interface response, all these methods were based on the assumption that the stress–strain behavior of the geotextile material is linear elastic. However, as most of the geotextiles are made out of polymers which are likely to exhibit nonlinear stress-strain behavior, it is acknowledged that it is not justifiable to represent the entire stress-strain behavior with linear elastic assumptions. Recognizing these shortcomings, Perkins and Cuelho (1999) proposed an analytical model in which the force- strain relationship of the geotextile material is assumed to be hyperbolic in shape. This was combined with the interface friction relationship proposed by Juran and Chen (1988). The resulting analytical solution was solved using a finite difference approach. Despite the better agreement between the experimental and the analytical results in some of these methods, it is admitted that the values used in the models are different from the actual values obtained from independent experiments, e.g., interface friction angle, soil dilation angle (Palmeira 2009). In the absence of direct account of the soil dilation, frictional degradation and nonlinear stress –strain behavior of the geotextiles, in order to match the experimental result, some of the input parameters (e.g. interface friction angles)have been assigned with unrealistic values (Perkin and Cuelho 1999). Alternatively, the curve fitting can only be achieved using 45 input parameters that lack physical meaning and which are not determined from independent tests. For example, ks (interface stiffness), α, β and Ertr were changed by Gurung and Iwao (1999) to match the pullout responses. However, in their approach the entire pullout – displacement behavior was not able to model using one consistent set of input parameters. Perkin and Cuelho (1999) changed values for Gi, pψ and rψ until good match was obtained for the pullout tests. 2.8 Summary of key observations: Pullout response of geotextiles The following limitations are identified after examining the existing analytical methods to determine the pullout response of planar geotextiles. 1. Some of the oversimplified interface friction models are not sufficient to capture the complex soil-geotextile interface friction development. The factors such as soil dilation and frictional degradation need to be taken into account in an appropriate analytical framework. 2. Although, the nonlinear stress-strain behavior is exhibited in most polymer based geotextiles, often this behavior has been simplified with bilinear or linear stress-strain assumptions. 3. An analytical model needs to be derived by combining the nonlinear interface friction and stress-strain model, in which the input parameters are determined from independent tests. 4. The analytical solution needs to be validated by comparing with different pullout tests conducted by various researchers. Ideally, a consistent and realistic set of input parameters should be employed in the validation process to model multiple pullout tests. 46 2.9 Performance of the pipes subject to tension and bending caused by ground movement When the ground movement is not along the pipe axis, a given buried pipe will be subject to a combination of bending and tensile/compressive loading. The bending moments in the pipe are mainly generated as a result of the lateral soil resistance to the pipe movement. The lateral soil resistance is often an order of magnitude larger than the frictional resistance experienced by the pipe. Therefore, it is commonly believed by pipeline engineers that the prominence should be given to precise determination of the lateral soil loading. As a result, emphasis has been more on the lateral soil loading on buried pipes as opposed to axial soil loading. Experimental and analytical models derived to determine the pipe performance subject to combined axial loading and bending moment are presented in the following sections. 2.9.1 Experimental studies to determine pipe performance when subject to abrupt ground deformations Most of the experimental research has been focused on determining the lateral soil resistance on a unit length of a pipe (i.e., element-level tests to estimate the external soil load on the pipe). Furthermore, it is impossible to perform “model-scale” laboratory tests to simulate the full effect of abrupt ground deformations that occur in pipe sizes that are employed in practice. The main reason is that very large pipe lengths should be tested to account for the soil-pipe interaction that would take place on either sides of the ground offset (i.e., to ensure that the boundary conditions are not influencing the test results). An alternative method to overcome this problem would be to perform centrifuge tests using a smaller scale tests. Ha et al. (2008) performed few centrifuge tests using the split soil chamber and the abrupt ground displacements are simulated by offsetting the soil chambers. However, even with a split soil box of length of 0.6 m on either side, appreciable loads were recorded at the load cells placed at the two ends of the pipe. This implies that the boundary conditions were in effect even after a very small offset. In other words, with the deformation of the pipe, the axial force in the pipe will increase due to anchoring 47 that caused by lack of mobilized length along the pipe. Thus, limitations exist when attempting to use such data for validating any numerical or analytical model. In addition, in tests involving bending of the pipe, it is difficult to set up instruments to capture the critical strains with local strain measurement devices (e.g. strain gauges). For example, when the pipe is subject to bending from lateral soil loading, the location of the maximum bending moment will depend on the level of relative pipe displacement. With the increase in deformation of the pipe, the location of the maximum bending movement will move along the pipe (Bruschi et al. 1996). As a result, unlike in axial soil loading, it is difficult to locate the point of the maximum bending movement with maximum strain in the pipe. 2.9.2 Analytical models to determined pipe performance subject to abrupt ground deformations Similar to axial soil loading on pipes, the development of analytical model to assess the pipe performance subject to abrupt ground deformations will follow the same framework as mentioned in Chapter 1. First, the externally applied soil loads (lateral and axial directions) need to be determined. Next, an accurate stress-strain model for the pipe material response is required. As the last step in the formulation, the above two components needs to be combined to obtained the overall pipe performance equations considering the element level equilibrium and compatibility conditions. In Section 2.4 and 2.4.2 the development of the frictional resistance along the pipe axis and the stress-strain behavior of the pipe material have been discussed in detail. Therefore, the following discusses the previous studies on determining the lateral soil resistance per unit length of pipe and the derivation of analytical solutions for overall pipe performance. 2.10 . Determining the lateral soil resistance per unit length of pipe (P) When investigating pipes subject to bending, it is important to determine the external soil resistance acting on a pipe element when subject to lateral ground movement. Numerous 48 analytical and laboratory tests have been performed over the years to determine the lateral soil resistance over a unit length of pipe. It is noted that the accurate determination of the lateral soil resistance is not part of the scope of this thesis; as such, considering the magnitude of research performed on this area, discussion on this aspect has been purposely limited by the author to selected studies. 2.10.1 Analytical approaches to determine the lateral soil resistance of pipe The early analytical approaches to calculate lateral soil load on pipes were developed based on the observed behavioral patterns in vertical anchors and retaining walls. As such, the lateral soil loads on buried anchors is calculated based on the assumption that active soil pressure is developed at the back of the anchor plate and passive soil pressure in front of the anchor. The first well-known analytical solution was developed by Hansen (1961) for buried vertical anchors, in which the failure surfaces were assumed to be straight and extended to the ground surface. Accordingly, Hansen observed that the approach would yield good results for anchors buried at rather shallow depths (i.e., burial depth/width of anchor plate up to 2). However, when the model results are compared with the experimental results from deep buried anchors, the soil resistances were observed to be overestimated in Hansen’s method. Although many factors could contribute to this overestimation, it is to be noted that Hansen’s method did not account for the vertical load equilibrium. Furthermore, this approach assumes only the horizontal movement of the buried structure. In most pullout tests on pipes and vertical anchors, it is observed that the object may also move upward as long as the upward soil resistance is smaller than the lateral soil resistance, e.g., in shallow burial conditions. Hence, additional restrictions on the vertical movement in the analytical model would likely to result in higher pullout resistances. With the intention of investigating the buried anchors at intermediate burial depths, Ovesen (1964) performed several pullout tests on vertical anchor. Based on these experimental findings, Ovesen (1964) derived an analytical solution for vertical anchor plates by accounting for the curvature in the active soil wedge and the upward movement of the anchor plate. With these 49 factors taken into consideration, Ovesen (1964) method resulted in a smaller lateral soil resistance than suggested by Hansen (1961) method. 2.10.2 Experimental approaches to determine the lateral soil resistance per unit length The early approaches to determine the lateral soil resistance per unit length of pipes were mainly based on the experiments conducted on vertical anchor plates. Considering the similarity of failure surfaces in the lateral movement of pipes and vertical anchor plates, it has been argued that the experimental results on anchors can be employed to determine the lateral soil loads on pipes after applying correction factors for the aspect ratio and shape. 2.10.2.1 Lateral pullout tests performed on vertical anchor plates Ovesen (1964) performed a series of pullout tests on plate anchors buried in both loose and dense sand. The height of the vertical anchors was 15 cm and a sufficiently wide width was selected to represent the plane strain conditions. With these selected dimensions for the anchor plate, tests were conducted with overburden ratios (i.e., ratio between burial depth to height of the anchor plate) ranging from 1 to 10. These experimental findings were employed in the analytical solution mentioned in the previous section. Neely et al. (1973) and Das and Seeley (1975) conducted similar investigation to estimate the effect of aspect ratio (i.e., ratio between width and height of the anchors) on soil resistance in vertical anchor plates. Das and Seeley (1975) observed that the measured resistance per unit width of the anchor would decrease with the increasing aspect ratio. This finding is of relevance to pipe since it shows the importance of proper account of the geometry of the object in determining lateral soil resistance. Besides the aspect ratio of the anchor plate, the overburden ratio is also an important factor affecting the lateral soil resistance. It was observed from experimental studies that the failure mechanism in front of the object will be different in shallow and deep burial conditions. In shallow burial conditions, distinct active and passive failure surfaces are visible. In contrast, at 50 deep burial depths, the bearing type of failure can be expected in front of the object, although a clear soil wedge is not formed. The different failure mechanisms for shallow and deep burial conditions are shown in Figure 2.6 by using a pipe as the object. Following a series of investigation on the effect of overburden ratio and the associated failure mechanism, Akinmusura (1978) concluded that the transition from shallow to deep failure mechanism for loose sand occurs at an overburden ratio of about 6.5. (a) (b) Figure 2.6 Affected soil regions (soil wedges formed) for pipes buried in (a) deep and (b) shallow burial conditions Besides these observations on the magnitude of the soil resistance, the displacement corresponding to the peak soil resistance is also an important aspect when developing bilinear soil-springs. From the tests conducted on vertical anchors buried on loose sand, Neely et al. (1973) observed that for tests with presumed plain strain conditions (i.e., aspect ratio larger than 5), the displacement at failure varied from 0.1 to 0.2 times the height of the anchor plate when the overburden ratio ranges from 1 to 5. Despite the observed variability in magnitude of the peak pullout resistance and the corresponding displacement, Das and Seeley (1975) showed that the dimensionless force- 51 displacement curves can be approximated by a single rectangular hyperbola of the form given in Equation 2.13. Subsequently, this idea has been widely used by scholars in the defining the shape of the respective soil-springs in lateral, upward and downward directions. 0.15 0.85 YP Y = + (2.13) / uP P P= / uY Y Y= P = Lateral soil resistance on the anchor uP = Maximum lateral soil resistance Y = Horizontal displacement of the anchor uY = Horizontal displacement corresponding to maximum lateral soil resistance of anchor Despite some of the similarities, there are obvious limitations when attempting to use these analytical results derived for buried anchors without considering the curvature of pipe, mostly leading to potential overprediction of soil loads. 2.10.2.2 Lateral pullout tests performed on pipes Considering that the stress distributions around the pipe and anchor plates are different, the importance of conducting separate physical model tests on pipes was recognized. To the best of the author’s knowledge, the first known tests on pipes were performed by Audibert and Nyman (1977) on steel pipes having diameters of 25 mm, 60 mm and 114 mm. In this investigation, the overburden ratios ranged from 1.5 to 24.5 with both loose and dense sand used as the backfill. Audibert and Nyman (1977) concluded that the lateral soil resistance values obtained from the pullout tests were in good agreement with the analytical solution proposed by Hansen (1961). In addition, Audibert and Nyman (1977) observed an increase in normalized displacement at the peak lateral soil resistance with the decrease in sand density and pipe size. 52 Trautmann and O’Rourke (1983) performed a series of lateral pipe pullout tests on steel pipes of diameter 102 mm and 324 mm. The tests were conducted at different H/D ratios ranging from 1.5 to 22 at three levels of soil densities characterized by three different peak friction angles (i.e., 31, 36, and 44°). Trautmann and O’Rourke (1983) observed that the lateral soil resistance values are in good agreement with the values obtained from the analytical model derived for vertical anchors by Ovesen (1964). In relation to displacement corresponding to peak lateral soil resistance (Yu), Trautmann and O’Rourke (1983) observed that Yu is equal to 0.13H, 0.08H, and 0.03H for loose, medium, and dense sand, respectively. Besides the above mentioned experiments, several other laboratory-scale pipe pullout tests have been conducted by several scholars. For example, Hsu (1993) investigated the rate of loading on the lateral soil resistance. In addition, Hsu et al. (1996, 2001) conducted series of pullout tests to investigate the effect of oblique soil loading on pipes, and developed interaction charts for vertical –horizontal planes and horizontal-axial planes. Paulin et al. (1998) conducted pipe pullout tests on large-diameter (324-mm) steel pipes under different burial conditions, however, the exact magnitude of the pullout resistances were not reported. Calvetti et al. (2004) conducted small-scale pullout tests on 20-mm to 50-mm diameter steel pipes and the results were used to calibrate the discrete finite element model developed using PFC2D. Turner (2004) conducted lateral pipe pullout tests on pipes buried in moist sand and later concluded that the moisture content has only a small impact on the lateral soil resistance. 2.10.3 Numerical methods to model the lateral soil resistance In 1980’s, with the advent of computers and their applications in engineering, numerical models were developed to investigate the pipe-soil interaction aspects. Using a finite element model, Rowe and Davis (1982), investigated the behavior of anchor plates buried in sand. The investigation includes the parametric study on the effects of initial soil stress conditions, burial depth, friction angle of the soil, and the surface roughness of the vertical anchor plates. The findings revealed the importance of accounting for the soil dilatancy at different overburden 53 stress levels. Furthermore, Rowe and Davis (1982) observed that the roughness of the anchor plate is a significant factor in shallow burial depths. Upon validating the numerical method by comparing with pullout test on vertical anchor plates, Rowe and Davis (1982) developed a series of charts to calculate the lateral soil resistance of anchors plates at different conditions. In these charts, several correction factors were introduced to account for different surface roughness values and soil dilatancy. Trautmann and O’Rourke (1983) observed that the magnitude of the lateral soil resistance measured for dense and medium-dense sand was in close agreement with results obtained by Rowe and Davis (1982). Nevertheless, the resistance measured in loose sand exceeded the values estimated from Rowe and Davis (1982) model and found to be closer to the soil resistances calculated for medium-dense sand. Guo and Stolle (2005) conducted a numerical analysis using ABAQUS (Hibbits et al. 2006) to explain the significantly different values obtained for lateral soil resistance by different scholars. First, the numerical model was calibrated and validated by comparing with the pullout tests performed by Popescu (2002). The analyses were conducted to investigate the effects of burial depth, overburden ratio, soil dilatancy, strain hardening and scale effect. In particular, it is shown that using the scale effect (ratio between pipe diameter with respect to reference pipe diameter), pipe size and burial depth, the wide variability observed in Nh versus H/D relationship can be explained. For example, the measured dimensionless soil loads on a pipe with small diameter can be significantly larger than the dimensionless soil load measured in a larger diameter pipe. When investigating the impact of soil dilation on the lateral soil resistance, Guo and Stolle (2005) observed that soil resistance increases with the soil dilation, and the rate of increase in lateral soil resistance depends on the burial depth and H/D ratio. Yimsiri et al. (2004) performed numerical analysis to investigate the lateral and upward soil resistances at deep burial conditions. It was found that the extrapolation of the soil resistances from the shallow burial conditions would result in overestimation of the soil resistance leading to conservative pipe designs. At shallow burial depths, the formation of the active and passive soil wedges will relieve some of the soil resistances, while at deep burial depths the local shear 54 failure would result in peak dimensional forces which are independent of the burial depth. After calibrating the finite element numerical model using the experimental results of Trautmann and O’Rourke (1983, 1985) for tests performed in sand with overburden ratios of 2 to 11, Yimsiri et al. (2004) used the numerical model to investigate deep burial conditions with overburden ratios as much as 100. Karimian (2006) also conducted 2D numerical modeling using the computer code FLAC2D (Itasca 2002) to investigate the lateral pullout behavior. The parametric investigations supported the conclusions made by Guo and Stolle (2005). Furthermore, Karimian (2006) investigated a trench effect when the pipe is moved in lateral direction. Besides these investigations, few other numerical studies have been performed by Ng (1994), Zhou and Harvey (1996), Yang and Poorooshasb (1997), Guo and Popescu (2002) and Popescue et al. (2002). These investigations were employed to explain the experimental findings obtained from lateral pipe pullout tests and demonstrate the importance of accounting the soil dilation, effect of pipe sizes, etc. Detailed discussion of this subject is not presented in this section as finding of lateral soil resistance is not a primary objective of this thesis. 2.10.4 Recommendations in current guidelines to calculate the lateral soil resistance In most state-of-practice guidelines, the ultimate lateral soil resistance per unit length ( uP ) for soils with no apparent cohesion is calculated using the following equation. u qP N HDγ= (2.14) where qN is the dimensionless horizontal bearing capacity factor. ALA (2001) guidelines recommends Hansen (1961) analytical model to calculate qN value which is in agreement with the experimental findings from Audibert and Nyman (1977). In contrast, PRCI (2009) recommends the charts derived by Turner (2004) to estimate qN values which is in agreement with the experimental findings from Trautmann and O’Rourke (1983) and the analytical results 55 from Ovesen (1963). Considering the observed variations in experimental findings, ASCE (1984) guideline presents both Hansen (1961) and Trautmann and O’Rourke (1983) charts without any guidance on the selection of the method. The relationship proposed by Trautmann and O’Rourke (1983) to obtain qN values based on the friction angle and the H/D ratio is presented in Figure 2.7. Figure 2.7 Proposed lateral soil resistance for buried pipes by Trautmann and O'Rourke (1983). 2.11 Analytical models to estimate pipe performance when pipe is subject to tension and bending With the knowledge of the lateral soil resistance per unit length and the stress-strain properties of the pipe material, the governing equations for the pipe performance can be derived considering Nq 56 the equilibrium and the compatibility aspects of a pipe element. The following sections present some of the analytical models derived by several scholars with the intention of evaluating the pipe response. All these analytical models were derived with the aim of analyzing rigid steel pipes, thus based on the assumption of linear elastic or bilinear stress-strain behavior for the pipe material. In these analytical solutions, often the pipe is treated as a beam on an elastic foundation or as a cable carrying a uniform lateral soil load. Even if the basic element level assumptions (beam or truss) are similar, each analytical solution is different from the other depending on the treatment of boundary conditions, soil and pipe properties. 2.11.1 Analytical solutions based on theory of beams on elastic foundation Most analytical solutions were derived using the basic theory of “beams on elastic foundations” (Hetenyi 1941) with the pipe being modeled as a beam. In these methods, the solution is derived from the basic equation of the form given below. 4 1 24 d wEI K w K dx + = (2.15) Where, I is the second moment of the area of the pipe cross-section and w is the deformation of the pipe in lateral direction from its original position. K1 and K2 are constants that depend on the formulation of the analytical solution. Trigg and Rizkalla (1994), Rajani et al. (1995) and Chan and Wong (2004) presented analytical models to determine the pipe response when the pipe is subject to abrupt ground displacement. These models account for the elastic perfectly-plastic (bilinear) lateral soil resistance. For example, in Equation 2.15, when the soil medium is in elastic state K1= ks and K2= 0 (ks is the soil stiffness in the elastic region of the pipe). However, when the maximum lateral soil resistance is reached, a separate set of equations can be derived based on K1 = 0 and 57 K2= -Pu. For these two situations, a closed-from solution can be obtained by solving Equation 2.15 and the constants in the solution are determined by considering the appropriate boundary conditions. In the method proposed by Miyajima and Kituara (1989), the soil resistance is considered for two regions depending on the location of the PGD (Figure 2.8). According to the assumed deformation pattern for the PGD, the constants 1K and 2K in Equation 2.15, are obtained as follows (1) 1K = 1k and 2 1 1 sin xK k W piδ = − for << 2 0 Wx (2) 1K = 1k and 2 0K = for 2 W x ≥ Figure 2.8 Schematic diagram showing the different regions of PGD assumed by Miyajima and Kituara (1989) In the above methods, the steel pipe was assumed to be linear elastic, although the strain hardening behavior is observed in steel after the yield point. Furthermore, in these analytical solutions, the development of tensile force in the pipe is not considered in the formulations. However, it is sometimes admitted that tension at large displacements is important (Rajani et al. 1995). Depending on the complexity of the equations, some methods require numerical techniques to obtain the solutions from these equations (e.g. Miyajima and Kituara 1989). x W/2 k2 k1 δ 58 Nyman (1983) and Hawlader et al. (2006) developed analytical methods to determine the performance of buried pipes that are affected by frost-heaving or thaw-induced settlement. The loading mechanism in pipes subject to lateral ground displacement is similar to these frost- heaving or thaw-induced settlements despite the disparity in the loading direction. Therefore, the theories of beams on elastic foundations are employed and the basic equation has a similar form to that of Equation 2.15. In these methods, a nonlinear soil resistance is assumed with three stages representing the peak and post peak reduction in soil resistance (see Figure 2.9). This type of behavior is commonly observed in frost heave or thaw induced settlement type loading. For each region, the respective constants, 1K and 2K in Equation 2.15 can be determined if the shape and magnitude of the nonlinear soil resistance are known (ks, k’s, we and wp, etc). Figure 2.9 Schematic representation of the peak and post peak soil resistance in frost-heave or thaw-induced settlements (after Hawlader et al. 2006). Hawlader et al. (2006) investigated the effects of magnitude of the post-peak reduction, the rate of post-peak degradation and the initial stiffness of the soil spring. The results revealed that the rate of post-peak degradation has a smaller impact on the pipe response, whereas the magnitude of the reduction will significantly affect the maximum bending moment. In addition, the initial stiffness of the soil resistance found to have a moderate impact on the pipe performance. So il re si st a n ce w we wp Region A Region B Region C ks k’s 59 Most importantly, Hawlader et al. (2006) employed a hyperbolic bending moment – curvature relationship to represent the nonlinear behavior of the pipe material. Nevertheless, the development of axial force in the pipe is not accounted in Hawlader et al. (2006) method. 2.11.2 Analytical methods based on cable-like profile for the pipe behavior Aforementioned analytical solutions are developed based on small deformation theory, and assuming that the lateral soil stresses are solely carried by pipe bending. As the pipe deforms, large tensile or compressive forces will be developed along the pipe. At small displacements, the effect of any axial force developed on the pipe performance will be small, whereas the impact is likely to increase as the pipe deformation increases. For example, in the case of significant strike-slip fault movement, the pipe crossing the fault tends to develop considerable tensile force closer to the point of abrupt ground movement. As some of the scholars hypothesized, a cable- like profile can be formed if excessive deformation of the pipe is allowed (e.g., Newmark and Hall 1975). In such situations, the lateral soil resistance is exclusively carried by the axial tension in the pipe and the theories based on pipe bending will no longer be appropriate. The following analytical methods have been developed with the assumption that the lateral soil resistance is entirely carried by the tensile force in the pipe. 2.11.2.1 Newmark and Hall (1975) approach After recognizing the importance of the development of axial forces in the pipe, Newmark and Hall (1975) presented a simplified design procedure, with the assumption that the most critical situation would arise when the pipe is predominantly subject to tensile force when crossing a fault zone. Consequently, the pipe is modeled as a cable in which the bending stiffness of the pipe and the lateral interaction is ignored. As stated, the transverse component is said to have small effect except for local flexural strains, if the anchoring points are sufficiently away from the fault crossing. As the theory assumes failure from tension, the development of friction along the pipe is identified as a critical factor. Thus, it is expected that the capacity of the pipe to 60 undergo large deformations will increase with the increased mobilized frictional length and lower frictional resistance. 2.11.2.2 Sweeney et al. (2004) approach In a similar method, Sweeney et al. (2004) assumed that the contribution from the pipe bending is insignificant to the load carrying capacity of the pipeline at large displacement levels, hence; the pipe was modeled as a cable. The following second-order differential equation was derived by considering the element-level equilibrium of the pipe. This is in contrast to the fourth-order differential equation (Equation 2.15) derived with pipe bending assumptions, 02 2 =− N P dx wd (2.16) where, the soil resistances per unit length of pipe in lateral and axial directions are given as P and N, respectively. In this derivation, the deformation shape of the pipe is assumed to be a parabola. It is also assumed that the axial force is generated from two mechanisms: (a) the elongation of the pipe caused by the frictional resistance acting along the pipe; and (b) additional elongation required by the pipe meet the deformed profile of the pipeline, if the elongation from the interface friction is not sufficient. Therefore, the corresponding equations are solved using iterative methods. The main drawback of these methods is that, the derivation has led to the assumption that the maximum strain will always occur at the point of abrupt ground movement. This is not an accurate assumption in many instances, where the bending moment has a dominant effect. Furthermore, the full cable-like action will only develop after considerable amount of pipe deformation, and some pipes may not have the capacity to reach such levels of deformation. 61 2.11.3 Analytical methods developed by combining the axial force and bending moment interaction Despite the simplified nature of the solutions derived by Newmark and Hall (1975) and Sweeney et al. (2004), it is unrealistic to assume that the entire pipe would be subjected to tensile loading (or deform as a cable). In reality, the pipe sections away from the abrupt ground deformation are likely to carry the lateral soil loading by pipe bending. Therefore, the pipe sections are likely to subject to complex interaction between axial tension and bending moment and the following analytical methods were derived to address these shortcomings. 2.11.3.1 Kennedy et al. (1977) approach Extending the Newmark and Hall (1977) approach, Kennedy et al. (1977) proposed an analytical method which accounts for the lateral soil interaction. In this method, the assumed deformed profile of the pipe is shown in Figure 2.10. As a simplification, it was assumed that the pipe will deformed with constant curvature between an anchoring point and the point of ground offset. Beyond the anchoring points, the pipe is assumed straight. With these assumptions, the bending strain ( bε ) is calculated in the following manner. 2b c D R ε = (2.17) Where cR is the constant curvature assumed for the pipe segment closest to the abrupt ground movement. This assumption of constant curvature is required to determine the axial force in the pipe segment. It is assumed that the axial tensile force is essentially independent of the curvature if the bending is less than 80% of the axial strain. However, in reality, the pipe curvature is likely to change gradually as moving away from the location of abrupt ground deformation; as such the pipe sections away from the abrupt ground displacement will resemble a beam on an elastic foundation. Furthermore, in this method, the flexural rigidity is assumed zero for the pipe segment closer to the abrupt ground deformation. The models proposed by Kennedy et al. 62 (1977) and Newmark and Hall (1975) are referenced in ASCE (1984) guidelines for strike-slip and normal fault movements. Figure 2.10 Schematic diagram of the pipe deformed shape assumed by Kennedy et al. (1977). 2.11.3.2 Wang and Yeh (1985) approach Considering some of the drawbacks in Kennedy et al. (1977) method, Wang and Yeh (1985) attempted to incorporate the bending stiffness of the pipe segment closest to the abrupt ground deformation. The bending stiffness of the pipe cannot be ignored unless the pipe undergoes very large deformations. In this method, the pipe is divided into two distinct regions on either side of the abrupt ground deformation (see Figure 2.11). The outside regions (AA’ and CC’) are modeled based on the beams on elastic foundation theory, while the pipe sections closest to the fault crossing (AB and BC) are assumed to be behaving as a cable with constant curvature. However, when considering the force equilibrium of these pipe segments, the end bending moment and shear forces transmitted at A and C from the pipe segments AA’ and CC’ are considered. Thus, the flexural stiffness for the pipe segments AB and AC are accounted in the analytical formulation. Although the bending stiffness is considered in this manner, the Rc Lc Pu w/2 63 reduction in bending moment resulting from the increased axial force is overlooked in this derivation (Karamitros et al. 2007). Figure 2.11 Schematic diagram showing the different regions of interaction when subject to abrupt ground deformation occurring normal to the pipe axis. The radius of curvature in the tension dominated region is calculated by considering the required deformation to ensure continuity of the pipe segments. Based on these derivations, Wang and Yeh (1985) suggested calculating two factors of safeties at locations denoted by A (or C) and B. At A or C, the calculated bending moment is compared with the ultimate moment capacity of the pipe. At the point of ground separation (at B), the factor of safety is calculated by comparing the ultimate strain capacity of the pipe with the axial strain calculated from the assumed cable-like behavior. This, in turn, implies that the critical points are at the boundaries of these two regions (point A, B and C). However, in most occasions the critical locations were observed to be located within the pipe segment AB or AC. 64 2.11.3.3 Karamitros et al. (2007) approach The method proposed by Karamitros et al. (2007) is fundamentally similar to the method proposed by Wang and Yeh (1985), but contains certain improvements to the solutions approach. Similar to Wang and Yeh (1985) approach, the pipe was divided into four regions as shown in Figure 2.11. In a similar fashion, the shear force, bending movement, rotation and displacement at the pipe section AA’ and CC’ were obtained using the theories of beams on elastic foundations. Furthermore, the pipe segment AC is also treated as a beam and the necessary boundary conditions (rotational stiffness and the bending moment) are obtained considering behavior of the pipe segment AA’ and CC’. Besides the direct account of the bending stiffness for the pipe section AC, the axial force at the point of abrupt ground deformation is determined by calculating the force required to elongate pipe segment to match the deformed shape of the pipe segment AB. It is assumed that the maximum axial force at point B is decreasing linearly along the pipe when moving towards point A or C. For relatively large deformations, Karamitros et al. (2007) considered the geometrical second-order effects, which were calculated based on the constant curvature assumption for the pipe segment AB and AC. Compared to previous analytical methods, Karamitros et al. (2007) considered the actual strain distribution at a pipe cross-section so that the actual reduction in bending moment due to increasing axial force is accounted. In the above methods, it is required to calculate the elongation of the pipe based on the deformed profile of the pipe segment closer to the abrupt ground displacement. However, the pipe elongation is few orders smaller in magnitude than the curvature of the pipe. Therefore, it is extremely difficult to calculate the elongation of the pipe segment in a precise manner, despite the simplifying assumption that the radius of curvature is constant for pipe segments closest to the abrupt ground deformation. 65 2.12 Numerical modelling of performance of a pipe subject to ground movement Owing to the complexity of some of the analytical solutions, often the pipe-soil interaction analysis is performed using numerical methods. A comprehensive numerical model could avoid costly or even sometimes unfeasible large-scale model testing to investigate different pipe configurations, soil failure mechanism, and soil conditions. Furthermore, these numerical methods are equipped with features to account for highly nonlinear behavior of the pipe materials, pipe/soil interface behavior, pipe distortions, etc. Two basic numerical approaches are available to model the pipe-soil interaction aspects: (1) the state-of-the-art, three-dimensional continuum finite element or boundary element methods; and (2) the state-of-practice, soil-spring structural models. 2.12.1 3D continuum finite element modeling In most situations, the pipe soil-interaction problems are considered three-dimensional. In finite element framework a three dimension models can be generated and solved to determine the performance of certain pipe sections. However, a number of difficulties can be identified when attempting to execute these 3D finite element models (1) It is extremely difficult to model the interface frictional behavior using continuum finite element applications. Overlapping of the two meshes is one of the common problems associated with friction modeling and convergence is difficult to achieve with finer meshes. (2) When modeling pipes subject to lateral soil loading, as explained earlier, very long pipe sections are required on either side of the ground deformation. This is to avoid any influence from the end boundary conditions. Therefore, very large amount of elements are required to model the pipe and the surrounding soils. For example, Nobahar and Popescu (2001) stated that within the shear zone, the thickness of the element should be at least same or smaller than the thickness of the shear band. This will lead to very large 66 number of elements when attempting to maintain the proper aspect ratio for the elements located in the shear zone. (3) The in-built constitutive models are sometimes are not capable of capturing the realistic soil and pipe behavior, thus often the models require calibration of certain parameters to match the experimental data. In addition, more information is required than a simple soil-spring model. (4) In most finite element applications, the amount of pipe displacement that can be modeled is limited to by the amount of mesh distortion which in turn causes convergence problem even in FE programs with re-meshing options. (5) In finite element models, it is impossible to model the soil flow around the pipe, soil wedges and the associated fractures in soil mass. 2.12.2 Soil-spring type analysis In comparison to continuum finite element approaches, soil-spring models have been widely employed to estimate the pipe performance from abrupt ground displacements. The pipe elements are modeled as series of inelastic beam elements, and rather sophisticated material models can be assigned for the stress-strain behavior of the pipe (e.g., nonlinear strain hardening etc). The soil resistance is generally modeled as a series of uncoupled nonlinear springs, and the ground movements are applied to the base of the soil spring elements. Although the soil-spring analysis is computationally less demanding than continuum FE models, soil resistances represented by nonlinear springs lacks the physical significance, and limitations exists when modeling the soil plasticity, dilation and hysteretic behavior. 67 2.13 Summary of key observations: analytical models to find response of pipes subject to axial loading and bending Analytical modeling of the pipe response from abrupt ground deformation is a difficult task considering the number of input parameters and the complexity of the pipe-soil interaction mechanism. It has been argued that many improvements can be suggested for the existing analytical techniques (Karamitros et al. 2007). However, the following shortcomings are identified after examining the existing analytical methods, which are relevant in relation the context of this thesis. (a) Most analytical solutions employ a constant curvature assumption for the pipe segment closer to the abrupt ground displacement to ease the analysis; instead the curvature is likely to change gradually when moving away from the point of abrupt ground deformation. (b) When using the theories of beams on elastic foundations, the direct account of the axial force is not considered. In general, the pipe deformation is calculated based on the bending theories, ignoring the axial force development of the pipe. Afterwards the axial force is incorporated based on the deformed shape of the pipeline. This process is likely to yield a higher bending moment and a different deformation pattern; thereby the overall calculation of the axial force would be erroneous. (c) Simplified models are assumed for the development of axial force along the pipeline, as such in most instances, the frictional development is assumed to be linear. (d) These models are developed for steel pipes with the assumption of linear or bilinear stress-strain behaviors for the pipe material. (e) All of the above-mentioned pipe models lack the flexibility to simulate the variable soil resistances and material properties. Note that certain closed-form solution type 68 approaches give little flexibility in modeling different soil sections with different soil resistances. 2.14 Closure The literature survey conducted above has revealed several limitations in previous analytical models and experimental undertakings with respect to soil loading on pipes. The limitations with respect to axial and lateral soil loadings are addressed in Sections 2.6 and 2.13, respectively. Furthermore, the limitations in the current practice of determining the pullout response of geotextiles are addressed in Section 2.8. In brief, limitations were highlighted in modeling the (1) external loading acting on the pipe/geotextile (2) material stress-strain behavior and (3) overall performance of the pipe/geotextile. These proposed analytical models are required to be validated by comparing with model-scale experiments. Although such database is available for geotextiles, extensive experimental undertaking is required to determine the response of buried pipes subject to different loading conditions. In recognition of these limitations, an experimental and analytical research program was undertaken at University of British Columbia, Vancouver, Canada. This thesis will describe the findings from this study in detail. The research is also aimed at generating information for potential improvements for the current design guidelines and criteria used by the pipeline industry. Furthermore it is intended to highlight the deficiencies in the current practice when attempting to model the response of buried pipes and geotextiles. 69 3 EXPERIMENTAL ASPECTS This chapter describes the experimental aspects of the field axial pullout tests conducted on MDPE pipes, including the details on: (a) field pipe pullout test setup; (b) testing methodology and test preparation; and (c) instrumentation (i.e., instrumentation for the measurement of load, displacement, and strains on piping during pullout testing). The pullout resistance, axial strain and displacement readings will be obtained from these pullout tests. This chapter also includes details on experimentation undertaken to characterize the relevant properties of the pipe material (stress- strain properties) and interface friction (soil density, moisture content, etc). The detailed results obtained from laboratory and field tests are presented in Chapter 4, and these results were used in validating the proposed analytical and numerical solutions for axial soil loading. 3.1 Characterization of material properties 3.1.1 Properties of backfill material (Fraser River Sand) In all tests, Fraser River Sand (FRS) was used as the soil backfill, which has been extensively used in laboratory research at UBC over the past 10 years. Furthermore, in standard pipe installation practice requires sand to be used as the backfill material. The mineral composition of FRS was found to be 40% quartz, 11% feldspar, 45% unaltered rock fragments and 4% other mineral (Vaid and Thomas, 1995). The sand grains are angular to sub-rounded in shape (Wijewickreme et al., 2005). The grain size distribution shows a minimum and an average grain size of 0.074 mm and 0.5mm, respectively (see Figure 3.1). The coefficient of uniformity (Cu) was found to be 1.5 with a specific gravity (Gs) of 2.70 (Anderson, 2004). It is further assumed that the grain size distribution was not altered significantly due to repeated compaction of sand. This assumption has already been shown reasonable based on the comparison of grain size distributions obtained before and after testing (Karimian 2006). The minimum and maximum void ratios of FRS are 0.62 and 0.94, respectively (Anderson, 2004). 70 Figure 3.1 Grain size distribution of Fraser River Sand 3.1.1.1 Friction angle of sand The strength deformation properties for Fraser River Sand were available from Karimian (2006). In his study, several drained triaxial tests were performed to find the friction angle of dry FRS at low confining stress levels. Figure 3.2 shows the variation of friction angle for two soil densities, 1665 and 1575 kg/m3 at different overburden pressures. It is highlighting that it is extremely difficult to perform triaxial tests under a confining stress as low as 10 to 15kPa. As a result, extrapolation of curves shown in Figure 3.2 is required to obtain a friction angle corresponding to typical burial conditions (i.e., overburden of about 10kPa) . For example, for overburden pressures of 10 kPa the soil friction angle was determined to be about 46.5°. Uthayakumar (1996) and Sivathayalan (2000) have reported that the constant volume friction angle of FRS as 32-34º. 0 10 20 30 40 50 60 70 80 90 100 0.001 0.01 0.1 1 10 Pe rc e n ta ge fin e r (% ) Grain size (mm) 71 Figure 3.2 Variation of the peak friction angle of FRS with confining stress (after Karimian 2006). 3.1.2 Properties of the pipe material (MDPE) The field pipe pullout tests were performed on 60-mm diameter pipes having a standard dimension ratio (ratio between pipe diameter to pipe wall thickness) of 11. These pipes are designed to operate at a maximum operating pressure of 690kPa (100 psi). 3.1.2.1 Characterization of stress-strain properties of MDPE pipes In contrast to the pipe material properties, the stress-strain behavior of plastic pipes have a significant impact on the pipe response obtained from the pullout tests. However, the characterization of the stress-strain properties in plastic pipes is complicated due to nonlinearity, strain rate and temperature dependency. To determine the viscoelastic stress-strain behavior of the pipe material, uniaxial compressive tests were performed on 60-mm diameter pipe 72 specimens. Each pipe specimen was cut to a length of 150 mm and placed under a loading frame as shown in Figure 3.3. Afterwards, the test specimens were subjected to axial compression loading at different compression rates. Knowing the rate of axial displacement, the axial strain rate can be calculated for each test. Figure 3.4 shows the stress-strain responses obtained for five different axial strain rates. Furthermore, stress-strain behavior obtained from these tests exhibitsa highly nonlinear behavior of these plastic materials. However, tensile tests were not performed due to difficulties in clamping the pipe specimens at the two ends. Figure 3.3 Uniaxial compression tests performed on a 60mm pipe specimen. Note that, at small strain levels (less than 10%) the “engineering” and “true” stress-strain behaviors are essentially the same (Merry and Bray 1996). As a result, all the discussions and calculations in thesis are based on “engineering” stresses and strains since the maximum strains measured in these tests are less than 10% (in compression tests) or 5% (in pipe pullout tests). 73 Figure 3.4 Stress-strain responses obtained from the uniaxial compression tests at different strain rates (at a temperaure of 20 ˚C) . 3.2 Experimental details of field pipe pullout tests The following sections discuss the test details relating to five field pipe pullout tests performed in this study. The details of these field pipe pullout tests can be discussed in relation the following key topics: 1. Trench and other supporting structures; 2. Pulling mechanism and self-reaction frame; 3. Instrumentation and data acquisition 74 3.2.1 Trench and other supporting structures 3.2.1.1 Test site All five field pipe pullout tests were performed at UBC Botanical Garden Nursery located at the south end of the UBC Point Grey Campus. This location was considered suitable for undertaking long-term and large-scale pipe pullout tests, considering the uninterrupted access, vicinity to facilities and technical support. The test area has a flat topography, and the excavation revealed a soil stratigraphy consists of a glacial till layer of about 0.6 m thick overlaying a stiff sandy silt deposit. Initially, a trench was excavated using a backhoe to a length of 8.5 m. Two pits (2.5 long) were added on either end of the trench to facilitate the instrumentation and other pulling mechanisms. The width of the trench was 1 m and excavated to a depth of 0.9 m. The trench was located at the far end of the Botanical Garden Nursery, and a schematic diagram of the site plan and trench layout is shown in Figure 3.5. The soil in the test zone of the trench was retained using two wooden frames on either side of the trench. These wooden frames were fabricated using a 12.5 mm (½”) thick plywood board supported by 3-m-long two 100 mm x 100 mm (4” x 4”) lumber cross beams placed horizontally at the bottom and the top of the frame. The height of the wooden frame was 1 m. In addition, four vertical wooden beams of similar size supported the 3m-long plywood board. These beams were placed at 1 m interval. Additional two vertical beams were fixed on to the frame that was placed at the pulling end to support the additional load exerted by the pulling mechanism. These pre-fabricated wooden frames were then lowered into the trench and supported using eight wooden shores on either side. Although, it is observed that the frictional resistance between the frame and unexcavated backfill alone is sufficient to provide the required anchoring resistance to hold the frame in place, these wooden braces would provide additional support. These 75 mm x 25 mm (3” x 1”) wooden braces were about 3 m in length. Figure 3.6 shows the wooden frame and pulling mechanism. 75 Figure 3.5 Site layout for field pipe pullout testing 76 Figure 3.6. Pulling mechanism and wooden frame 3.2.2 Pulling mechanism and self-reaction frame A layout of the key components in the pulling mechanism is shown in Figure 3.7 and the respective components are identified in Table 3.1. Table 3.1. Components in the test setup No. Component No. Component 1 C-Channel section 6 Wooden frame 2 DC gear electric motor 7 MDPE pipe 3 Linear worm gear actuator 8 Steel legs of reaction frame 4 Load cell 9 Reduction gear box 5 Pipe clamp 77 Figure 3.7 The component of the pulling mechanism and the reaction frame (plan view) In this test setup, instead of anchoring the pulling mechanism to the ground, a self-reaction frame was employed (see Figure 3.8). Accordingly, the pulling arrangement is connected to the wooden frame via the reaction frame, thus the reactionary thrust created during the pullout is transmitted through the steel legs of the reaction frame on to the wooden frame. As the wooden frame is securely anchored to the native ground, no movement of the wooden frame was observed as the pipe pulling was in progress. However, vertical supports for the reaction frame were provided to support the heavy weight of the reaction frame, thus it is ensured that no overturning moments are applied on the wooden frame from the weight of the overhanging pulling mechanism. As shown in Figure 3.8, the reaction frame consists of a steel C-channel section welded to four steel legs which are connected to the wooden frame. The steel C-channel section is fabricated from a C8x11.5 section and the connecting steel legs are made out of 30mm square hollow steel 78 sections having a wall thickness of 2mm. A base plate is welded to each steel leg, and the base plate is bolted to the wooden frame using two screws (see Figure 3.9). Parts of the web in the C- channel section were removed to reduce the weight of the reaction frame. Figure 3.8 The key components of the pulling mechanism 3.2.2.1 Electric motors and the gear arrangement The thrust required for the pullout testing is provided by a 93.2 W (1/8 hp) permanent magnet DC gear motor. In addition, a gear box and a linear worm gear actuator were required to reduce Couplings 79 the output speed of the motor. A DC speed controller was also used to control the speed, thus allowing the flexibility to operate at wide range of pulling rates without making changes to the components in the pulling mechanism. The assembly of the DC motor, gear arrangement and the controllers are shown in Figure 3.8. These components are mounted on to the steel C-channel section of the reaction frame. Before using this pulling mechanism in the actual pipe pullout testing, trial tests were performed by pulling a heavy vehicular load to ensure that the pulling capacity is sufficient. The details of the DC gear motor and the accompanying gear arrangements are given as follows. Permanent magnet DC gear motor Manufacturer: Dayton Electri MFC, Chicago. Gear ratio: 1 to 11 Torque: 4.8 Nm (43 lbin ) Input motor power = 93.1W (1/8 hp) Gear reducer Manufacturer: Boston Gear works, Quincy, Mass., US. Gear ratio: 1 to 10 Worm gear actuator Manufacturer: Simple Uni-Lift , Templeton Kenly and Co., Broadview JLI, Maximum load in tension : 17.7kN (2 tons) Gear ratio: 48 turns per inch DC speed controller Manufacturer: Dayton Electri MFC., Chicago. 80 3.2.3 Instrumentation and data acquisition In these pullout tests, the pullout resistance, displacement and the strain in the pipe were measured using load cells, string potentiometers and strain gauges, respectively. Figure 3.9 shows these instrumentation and further details relating to these instrumentations will be discussed in the following sections. Figure 3.9 Instrumentation, coupling, gasket and the other arrangements at the pulling end of the pipe 3.2.3.1 Measurement of pullout resistance using load cells The pullout resistance was measured using a 22 kN (5000 lb) capacity load cell attached to the pulling end of the pipe. The details of the load cell is given as follows Manufacturer: Transducers, Inc, Santa fe Spring , California, USA Capacity: 22 kN (5000lb) Model: BTL-FF62-CS-5K 81 3.2.3.2 Measurements of pipe displacement using string potentiometers The displacement was measured at the pulling end (i.e., clamped end) and the trailing end (i.e., embedded end) of the pipe using two string potentiometers having a displacement range of 480 mm (19”) and a resolution of ±1.5 mm. A 0.5-mm diameter high-strength steel wire connected the string potentiometer to the pipe. The excitation voltage for the string potentiometers and load cell was 10 V. The string potentiometers were manufactured by UniMeasure Inc (model: NEMA 1) 3.2.3.3 Measurement of pipe strain using strain gauges In all tests, an array of strain gauges was placed closer to the pulling end of the pipe (about 100mm from the edge of the pipe) to measure the axial and radial strain. Additional strain gauges were used to check the accuracy and the consistency of the strain gauge readings. However, the strain gauges were not placed in the buried length of the pipe, as it would create additional frictional resistance to pullout (Weerasekara 2007). Kyowa KFEL-5-120-C1 type strain gauges were selected based on the previous experience in strain gauging MDPE pipes. The details of strain gauges are given in Table 3.2. The excitation voltage for strain gauges was 5V. Table 3.2 Details of the strain gauges Manufacturer Kyowa Type KFEL-5-120-C1 Gauge Length 5 mm Strain Range ± 15% Gauge Resistance 119.8 ± 0.2. Gauge Factor (24˚C, 50% RH) 2.13 ± 1.0 % Temperature Coefficient of Gauge Factor + 0.015 % /˚C 82 3.2.3.4 Data acquisition system The test data relating to pullout resistance (load cell), displacement (string potentiometer) and strain in the pipe (strain gauges) were collected using a data acquisition system (made in-house) that comprise of a 16-channel signal conditioner and an analog to digital converter (digitizer). When performing the experiments, the pullout resistance, displacement and strains were monitored in real-time using the graphical user interface provided by the commercially available software, DASYLAB (Data Acquisition System Library, 2009). The data was recorded at a rate of 0.5 Hz to 1 Hz, with slower recording rates employed in tests performed at slow pullout strain rates. In some tests, further preprocessing was required by filtering the data to a manageable size. 3.2.4 Preparation of test specimens In this section, the following key aspects relating to test preparation are discussed in detail. (a) Mounting of strain gauges on the pipe (b)Placement of pipe and soil compaction 3.2.4.1 Mounting of strain gauges on the pipe To mount the strain gauges on to the pipe, initially, the selected locations of the MDPE pipe surface were cleaned with a degreaser to remove any dirt and other contaminants present. A soft cloth was used to wipe the degreaser, and the surface was allowed to air dry for about 10 min. Afterwards, a coat of Loctite 770 primer was applied on the dried surface and allowed to dry for another half an hour. The strain gauges were then positioned and aligned in the required direction using cellophane tapes. Anderson (2004) and Weerasekara (2007) have demonstrated that strain gauge cement such as Loctite 414 is sufficient to provide a good bond between the strain gauge and the smooth surface of the MDPE pipe. The strain gauge cement was applied sparingly by pealing one end of the cellophane tape. Using thumb pressure applied on the gauge for about 1 min to ensure adequate bonding is achieved and excess bonding agent between the strain gauge and pipe is displaced. Afterwards, the gauges were secured by placing a rubber pad on top of the strain gauge and by strapping them to the pipe with an electrical tape to maintain a nominal pressure during setting. After 24 hours, the pads were removed and a thin coat of “M- 83 Coat A” (air-drying polyurethane coating) was applied on top of the strain gauges and allowed to dry for few minutes. The terminals of the gauges and lead wires were soldered to a soldering pad which was glued to the MDPE pipe using the same adhesive. The wires and the strain gauge arrangement were further secured to the pipe by applying a rapid setting epoxy on top. Excessive amounts of epoxy should be avoided since the epoxy could locally strengthen the pipe (Briassoulis and Schettini 2002). In these experiments, excess strain gauge cement has been squeezed out using the thumb pressure; in addition, numerical analysis conducted by Anderson (2004), on pipe configurations tested at UBC, has shown that the error due to this local strengthening is less than 5%. The epoxy coating protects the solder pad and strain gauges during the installation of the pipes. Once the epoxy has dried, 290 mm-long plastic cable ties were used to further secure the wires in place (see Figure 3.10). A stress-relief loop was formed in the wires to avoid transmitting accidental stresses to the delicate connection at the soldering pad. Figure 3.10 Strain gauge arrangement in a MDPE pipe (Weerasekara 2007) Strain Gauge Stress relief loop Plastic ties Plastic ties 84 3.2.4.2 Placement of pipe and soil compaction For each test, a sand volume of about 10 to 16 m3 was required depending on the test configuration. Initially, the sand was filled into the trench and compacted using a 2000-lb vibratory plate tamper until the level of sand is same as the required level of the pipe invert. The strain gauged pipe is then placed on the compacted surface and aligned with the axis of the linear worm gear actuator. Afterwards, the sand was spread uniformly as possible in ~ 125 mm lifts and each lift was compacted using the vibratory tamper with four passes along the trench. Considering the risk of damaging the pipe from direct contact with the 2000-lb vibratory tamper, the area besides the pipe springline was compacted using a hand tamper having a width of about 125mm. Taking into account the variability of the actual operating pressure in the gas distribution systems and safety concerns, the pipes were not pressurized. Figure 3.11 shows the placement of the pipe in the trench and after back filling the trench to its full height. The piezometers were installed along the pipe length to observe the level of the water table, and the readings from these piezometers revealed that the water table was below the pipe level. 85 (a) (b) Figure 3.11 Trench used for the field axial pipe pullout tests with (a) pipe placed before backfilling and (b) after filling the trench with FRS Couplings: In the axial pullout tests, two piece split-clamps (made in-house) were employed to connect the pipe to the shaft extending from the linear worm gear actuator (see Figure 3.7). At this connection, a tapered cylindrical probe connected to the shaft of the linear worm gear actuator supported the pipe from inside (see Figure 3.12). When the anticipated strains were large, there was a possibility that the pipe may slip from the coupling due to the reduction in the pipe thickness. In such situations, it was required to regularly tighten the nuts of the spilt-clamp to ensure no slippage. Piezometers 86 Figure 3.12 Arrangement of the two-piece split clamp Gaskets: In the pipe pullout tests, the pipes are required pass in and out of the trench through the opening(s) on the wooden frames (Figure 3.7). The hard-rubber gasket at the entry and exit locations of the trench allowed the passing of the pipe and to prevent sand from escaping the trench. A separate pullout test was conducted to find the gasket friction. In this test, the maximum frictional resistance at the gasket was found to be about 0.05 kN. This value was approximately similar to the noise level of the load cell, thus it was judged reasonable to assume that the friction at the gasket is negligible. 3.3 Test program A test matrix for the field pipe pullout testing was developed considering the objectives and the scope of work stated in Chapter 1. Accordingly, five field pullout tests were performed during the period from June 2010 to Sept 2010 as shown in Table 3.3. Considerable time is required for the initial test preparation, mainly due to the large scale of the experiments. Furthermore, it was 87 decided to accommodate all the pullout tests during the summer period to avoid interruptions and damages to the delicate instruments from rain. Note that pullout tests are numbered to facilitate the discussion of the test results and not related to the sequence at which the tests are performed. Table 3.3 Testing dates/ durations for the field pipe pullout tests Test no Testing date (or period) TEST# 1 Aug 5th, 2010 TEST# 2 June 28th, 2010 TEST# 3 July 20th, 2010 TEST# 4 July 5th,2010 TEST# 5 Aug 9th – 19th, 2010 Figure 3.13 shows the corresponding precipitation levels for the period that the tests were conducted. In fact, the variability of the moisture contents measured in each tests can be related to the precipitation pattern as shown in the following section. In turn, the moisture content in the soil is likely to affect the level of compaction, and influences the resistance to pullout. Note that the high moisture content measured in Test#2 (the first tests to perform) can be linked to the rather high precipitation observed at the end of spring. Note that moisture content readings are shown in Table 3.4. Furthermore, as pointed out earlier, the stress-strain response of the pipe material depends on the operating temperature, e.g., lower temperature will give rise to higher stiffness modulus and lower creep deformations. The field pipe pullout tests were performed at an average temperature range of 14-23 ºC (Figure 3.14), and it is assumed that the temperature fluctuations were not significant enough to produce considerable changes in the stress-strain response of the pipe material. Furthermore, it is assumed that the ambient air temperature is not significantly different from the ground temperature. 88 Figure 3.13 Precipitation during the testing period from June 21st to Aug 19th 2010 in Vancouver (extracted from http://vancouver.weatherstats.ca/) Figure 3.14 Temperature fluctuations during the testing period from June 21st to Aug 19th 2010 in Vancouver (extracted from http://vancouver.weatherstats.ca/) 89 3.3.1 Density of soil backfill All pullout tests were conducted using a dense sand backfill. The soil density was primarily measured using a nuclear densometer (Manufacturer: Troxler Inc., and Model: 3440) at a depth of about 50 mm from the surface. In each pullout test, at least eight density readings were obtained at different surface locations of the trench. In some tests, the additional density measurements were taken by pushing a sharp cylindrical sampler (diameter 75.9 mm, height 70.3 mm and wall thickness 1.5 mm) into the sand. The density was calculated by dividing the weight of the sand extracted from the sampler by the volume occupied. It was observed that these density readings were in close agreement with the density readings obtained from the nuclear densometer. Moreover, these density readings were consistent with the values reported by Anderson (2004), Karimian (2006) and Weerasekara (2007) for the same sand and with similar compaction effort. Table 3.4 shows the average soil unit weights and moisture content measurements obtained using the nuclear densometer. Table 3.4 Average soil densities measured in field pipe tests Test No Moisture content (%) Unit weight (kN/m3) TEST #1 0.53 16.0 TEST #2 1.07 16.1 TEST #3 0.48 15.8 TEST #4 0.56 15.8 TEST #5 1.32 16.1 3.3.2 Burial depth The standard practice is to install the distribution pipes with a soil cover ranging between 0.3-1.5 m, to prevent exposure to UV radiation and construction damages. With this knowledge, the burial depth was selected to be about 0.56 m for four of the pullout tests. In Test #3, the burial depth was increased to 0.98 m with the intention of exploring the impact of the large overburden stress. With the extra soil fill in Test #3, two additional wooden frames were fabricated and 90 assembled on top of the existing wooden frames (Figure 3.15). The width of the soil fill at the top surface was measured to be 1.5m. Figure 3.15 Perspective views of the Test #3 after backfilling to a height of 0.98m ~0.4 Existing wooden frames 1.5m Additional wooden frames 91 3.3.3 Rate of ground displacement In these field pipe pullout tests, the tests were performed at different pulling rates at the leading end of the pipe ranging from 0.6 mm/min to 2.2 mm/min. At these displacement rates, different pipe responses were anticipated, as the stress-strain behavior of the pipe would depend on the applied strain rate. Table 3.5 presentsthe displacement rates and corresponding duration for each rate of displacement. Figure 3.16 also presents that variation of the measured displacement with time for the five axial pipe pullout tests. The variations in the displacement rates in Tests #4 and #5 are evident from Figure 3.16 and Table 3.5. Figure 3.16 Variation of leading end displacement with time for each field pipe pullout test The key test parameters for the five pullout tests discussed in Section 3.1 to 3.3 are summarized in Table 3.5. 0 20 40 60 80 100 120 140 160 0 2000 4000 6000 8000 10000 12000 14000 16000 Time (sec) D is pl ac e m en t a t l ea di n g en d (m m ) Test 2, 3 Test 1 Test 5 Test 4 92 Table 3.5 Summary of test parameters for field pipe pullout tests Input parameter Test number Test #1 Test #2 Test #3 Test #4 Test #5 Soil density at pipe level (kN/m3) 16.0 16.1 15.8 15.8 16.1 Burial depth (m) 0.56 0.57 0.98 0.55 0.55 Displacement rate (mm/min) Stage 1 Stage 2 Stage 3 0.601 2.145 2.185 0.601 0.151 2.16 0.729 0 0.729 Test duration (min) Stage 1 Stage 2 Stage 3 225 32 41 40 162 6 68 15770 117 After the pipe installation, the pullout test was commenced by applying the selected displacement rates to the leading end of the pipe while recording the pullout resistance, strain and displacement at the instrumented locations. Upon completing the test, the sand is completely removed from the trench before installing another pipe in the trench. 3.4 Typical results from pipe pullout tests The following will present some of the typical results obtained from the pipe pullout tests to illustrate the data generation ability and quality from the experiments. The detailed test results are presented in Chapter 4. 3.4.1 Variation of pullout resistance and strain with displacement The variation of pullout resistance and strain with the relative ground displacement are important aspects when considering the fact that in most instances the ground displacement is measured with reasonable accuracy and the challenge will be to determine the corresponding stress or strain in the pipe. Figure 3.17 shows the variation of pullout resistance with the leading end displacement for all field pipe pullout tests. The observed pullout response is highly nonlinear and with variable displacement rates resulting even more complex behavior for pullout resistance. The reasons for the observed behavior will 93 be discussed in Chapter 4 in detail. As shown in Table 3.5, the test conditions in Test# 1 and 5 are similar in initial stages of loading. Accordingly, the pullout resistance versus displacement behaviors was also observed to be in good agreement for these two tests, showing the repeatability of test results. Figure 3.17 Variation of pullout resistance with displacement at leading end for all field pipe pullout tests Figure 3.18 shows the strain gauge readings obtained from two different strain gauges (SG1 and SG2) mounted at leading end of the pipe, but at different points along the circumference in Test #1. Examining these strain gauge readings, it is evident that consistent strain gauge readings were obtained from multiple strain gauges mounted on the pipe. 94 Figure 3.18 Variation of axial strains measured from two strain gauges (SG1 and SG2) with the displacement at the leading end of the pipe (from Test #1). In Test #2 (the first pullout test performed), the strain gauge channels were saturated after reaching a strain value of about 20000µε (2%). In the subsequent pullout tests, the gain in the strain gauge channels was adjusted such that the maximum achievable strain was increased to about 200000µε (20%). It is observed that even with the higher gain factor, the level of fluctuations in the strain values was small for interpreting test results. 3.4.2 Stress –strain behavior of the pipe material in pullout tests The knowledge of the stress-strain response of the pipe material is an important aspect since this behavior reflects the actual viscoelastic stress-strain response of the pipe material. Figure 3.19 shows the stress- strain behavior obtained at the leading end of the pipe for all pipe pullout tests. Furthermore, even if a constant displacement rate maintained, a significant variation of strain rate is observed over the duration of the pullout test. This is an important consideration as the 0 5000 10000 15000 20000 25000 0 10 20 30 40 50 60 Displacement at leading end (mm) Ax ia l s tra in a t l ea di n g e n d ( µε ) SG1 SG2 60 mm pipe 95 stress-strain response of a viscoelastic material is directly dependant on the strain rate. Figure 3.20 shows the measured variation of the strain rate with time for Test#1. Figure 3.19 Stress-strain behavior obtained from all five field pipe pullout tests -2000 0 2000 4000 6000 8000 10000 12000 0 10000 20000 30000 40000 S tr e ss ( k P a ) Axial strain (µε) TEST#1 TEST#2 TEST#3 TEST#5 TEST#4 96 Figure 3.20 Variation of strain rate with time for Test #1 3.4.3 Instrumentation to estimate mobilized frictional length from pipe pullout Besides the pipe response characteristics defined by the axial force (i.e., pullout resistance) and strain in the pipe, the knowledge of the mobilized frictional length along the pipe is an important consideration when evaluating the pipe response. The progressive mobilization of the friction with the cumulative displacement at the leading end of the pipe cannot be measured continuously through direct measurements, and requires several displacement measuring devices (e.g. string potentiometers) to be placed along the burial length of the pipe to track the movement of different locations of the pipe when pullout is in progress. In this test program, it was decided not to attach any string potentiometers to the buried length of the pipe due to additional frictional resistance created from their presence. Therefore, only two string potentiometers were placed at the pulling and trailing ends of the pipe to measure the displacement at these two locations. Figure 3.21 shows the displacement measured at the trailing end of the pipe plotted against the displacement of the pulling end of the pipe for Test#3. It is noted that until a displacement of about 95mm is achieved at the leading end, no movement was observed at the trailing end of the 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 2000 4000 6000 8000 10000 12000 14000 Time (sec) St ra in ra te (% /s ec ) 97 pipe. With this information, it can be argued that the trailing end begins to move only when the friction along the entire buried length of the pipe is mobilized. Although direct continuous measurement of the mobilized frictional length was impossible, this provided an opportunity to estimate the mobilized frictional length (i.e., same as the burial length of the pipe) by noting the instance when the trailing end begins to move. Figure 3.21 Displacement measurements for the leading and trailing ends of the pipe in Tests#3. 3.5 Closing remarks Experimental details relating to the field axial pipe pullout tests are presented in this chapter. This includes details on test setup, preparation, instrumentation, characterizations for the backfill and pipe material. A total of five axial pullout tests were performed, and in each test, the pullout resistance, strain, displacement and time were directly monitored. Although the mobilized frictional length along the pipe is not measured directly, by tracking the movement of the pipe at the trailing end of the pipe, the mobilized length can be estimated when it is equal to the burial length of the pipe. In addition, other soil properties such as soil density, moisture content, and pipe burial depth were recorded. 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Displacement at leading end (mm) Di sp la cm en t o f l e ad in g an d tra ilin g e n ds (m m ) SP1 SP2 Trailing end starting to move 98 To the best of author’s knowledge, the laboratory and field axial pipe pullout tests reported in this thesis were the first known model-scale tests performed on pipes, sometimes reaching an axial strain value of about 5%. Note that, this strain is achieved without any influence from the boundary conditions of the test area. These pipe performance parameters obtained from these real “model-scale” pipe pullout tests provide a valuable opportunity to validate both analytical and numerical solutions. Furthermore, the experiments performed to characterize the pipe and backfill material behaviors provide independent input parameters required for validation of the analytical/numerical approaches. Details of the analytical/numerical methods and test results of the axial pipe pullout tests are presented in the following chapter. 99 4 ANALYTICAL MODEL FOR PIPE RESPONSE FROM AXIAL SOIL LOADING 4.1 Introduction The determination of the pipe-soil interaction aspects due to axial soil loading is an important aspect when attempting to explain the pipe response when the ground movement is along the pipe axis. The knowledge of the frictional loading mechanism at the pipe-soil interface becomes an integral part of understanding this interaction, even in attempting to develop models to determine the pipe response when the soil loading is at an angle to the pipe axis (e.g., Newmark and Hall 1975, Kennedy et al. 1977). More details on the latter aspect will be discussed in Chapter 6. Based on the discussions in Chapter 2, it is recognized that the overall pipe response will depend on pipe properties (e.g., pipe cross-sectional area, pipe material stress-strain behavior) and soil characteristics (e.g., pipe burial depth, soil density, internal friction angle of soil, interface friction angle between pipe and soil, stiffness of soil, lateral earth pressure coefficient). The engineering challenge will be to incorporate all these aspects in an analytical framework. In this chapter, a new interface friction model is developed to describe soil loads developed from relative axial soil movement after identifying the key factors influencing the friction on a pipe element. This frictional model will be aimed at overcoming the shortcoming in bilinear frictional models that are commonly adopted in pipeline engineering guidelines. Furthermore, the nonlinear viscoelastic properties of the plastic pipes will be accounted in the formulation as opposed to typical linear-elastic or bilinear stress-strain responses adopted in practice. Afterwards, a new analytical solution is derived to determine the pipe response arising from axial soil loading by combining the aforementioned interface friction model and nonlinear stress-strain behavior of the pipe material. 100 The results obtained from this analytical approach are further verified by comparing with “model-scale” pipe pullout tests performed in laboratory and field environments. The numerical model results for these pullout tests using the proposed interface friction model (i.e., axial soil- spring”) are also presented. 4.1.1 Slide geometry Identifying the actual slide geometry of the field soil mass is of significant importance, in order to use any pipe-soil interaction model in a real-life situation. Most of the past research has been concentrated on the soil loading arising from relatively quick moving soil liquefaction type ground displacements. As mentioned in Chapter 1, the dominant focus of this research is mainly focused on the slow moving landslides, which gives ample time to undertake necessary precautionary measures to avoid catastrophic accidents. The approach however can also be applied to situations under soil liquefaction type ground displacements. In a landslide, tensile stresses can be developed in the pipe section at the ground separation zone when the sliding soil block is separated from the stable ground (Figure 4.1). The block type slide movement is more likely to occur if a weak soil layer is situated below the pipe level. The slide geometry replicated in the full-scale tests is also similar to that of a landslide scenario depicted in Figure 4.1 and this was selected for the validation purposes, Even if more complex loading patterns can exist in practice, the element-level formulations developed in this study can be utilized in finite element framework. 101 Figure 4.1 Example of the slide geometry simulated in the current full-scale tests and by the analytical solutions 4.2 Analytical model to determine pipe performance It is also important to begin by expressing that any numerical or analytical model to capture the pipe performance should address the following three concerns: (i) determination of the mobilized soil load on the pipe due to the relative ground movement by a systematic expression of pipe-soil interaction (e.g., in case of axial loading, friction at the pipe-soil interface T); (ii) realistic representation of the stress-strain behavior of the pipe material; (iii) combination of the soil loading and stress-strain behavior to derive an overall pipe performance model. The following sections are intended to discuss the above considerations and, in turn, provide a basis for subsequent assessment of the testing outcomes. 4.2.1 Development of analytical model for interface friction The current practice is to model the interface friction as a bilinear model in which the peak interface friction per unit length (T) is calculated from Equation 2.1 (rewritten as Equation 4.1 for reader’s convenience). Stable ground Moving soil mass Strain distribution 102 0(1 ) tan 2 DH KT pi γ δ+= (4.1) As noted in Chapter 2, significant limitations exist when attempting to use this equation to model the interface friction. As noted, some of the key aspects affecting the pipe-soil interface friction, such as soil dilation and frictional degradation are not accounted in Equation 4.1. The following sections describe an analytical framework to account for these aspects in appropriate manner. 4.2.1.1 Increase in normal stress due to soil dilation (∆σd) For cylindrical objects, Boulon and Foray (1986) and Johnston, et al. (1987) suggested that the increase in normal soil stress due to soil dilation (∆σd) can be derived from elastic cavity expansion theory. A slightly different method to that of traditional elastic cavity expansion theory was proposed by Luo et al. (2000) for soil nails by considering a saw–tooth type model to represent soil dilation. According to this method, the increase in normal stress at peak dilation can be expressed as follows: max 2 ( ) tandd G t D γ σ ψ∆∆ = (4.2) where, G(γ) is the shear modulus of the soil dependant on the shear strain (γ), ∆td is the thickness of the dilation zone and ψmax is the maximum angle of soil dilation. 4.2.1.2 Impact of the mean effective stress on soil dilation It is also known that an increase in the mean effective stress will reduce the amount of soil dilation (e.g., Bolton 1986). Considering the importance of this aspect in determining the friction acting on soil nails buried in shallow and deep conditions, Luo et al. (2000) proposed a method to account for the impact of the mean effective stress on the dilation angle. In this method, initially, the following equation (Bolton 1986) was employed to relate the relative dilatancy index (IR) and the mean effective stress σ’: 103 ( )ln 'R DI I Q Rσ= − − (4.3) Where ID is the relative density and Q and R constants that depends on the type of soil. Owing to the difficulties in obtaining reliable test results under low stress conditions, Bolton (1986) suggested to limit IR to less than4, which in turn limits the dilation angles to 12°, until good test data become available for low stress conditions. Recently, based on the triaxial and plane strain test performed at overburden stresses (9kPa – 400kPa), Chakraborty and Salgado (2010) proposed the following relationship between the peak and constant volume friction angles (i.e. max'φ and 'cvφ ) for both triaxial and plane strain conditions: max max' ' 3.8cv RIψ φ φ= − = (4.4) With the assumption of R = 1, Chakraborty and Salgado (2010) proposed the following relationship to calculate Q at different overburden stresses under plane strain conditions: 7.1 0.75 lnQ σ ′= + (4.5) Figure 4.2 shows the IR values calculated from Chakraborty and Salgado (2010) relationship compared with IR values computed from triaxial tests performed on dry Fraser River sand by Karimain (2006). Note that these triaxial tests were performed at low overburden stresses ranging from 15kPa - 50kPa. The good agreement between these two relationships shows the applicability of Chakraborty and Salgado (2010) relationship to calculate the amount of dilation around the pipes buried in low overburden stresses. 104 Figure 4.2 Comparison of IR values computed from Chakraborty and Salgado (2010) relationship and triaxial tests performed on dry FRS at low overburden stresses. Combining Equations 4.3 to 4.5 will yield the following relationship for the peak dilation angle. ( )maxtan tan 3.8 7.1 0.25 ln 3.8DIψ σ ′ = − − (4.6) Now, Equation 4.6 can be substituted in Equation 4.2 to obtain the following equation for the increase in normal stress due to soil dilation. [ ]8.3))ln(25.01.7(8.3tan)(2 −′−∆=∆ σγσ Ddd ID tG (4.7) In this expression, increase in ∆σd is proportional to the increase in shear modulus and shear zone thickness, whereas a decrease in ∆σd is observed when the pipe size is increased. These deductions from the analytical formulation are also supported by experimental findings by Lehane and Jardine (1994) and Milligan and Tei (1998) who showed that ∆σd would increase 4 4.5 5 5.5 6 4 4.5 5 5.5 6 I R = (φ m a x - φ cv ) / 3. 8 I R= I D (10 - ln σ')-1 15 kPa 50 kPa 35 kPa 25 kPa 20 kPa 105 with increase in soil density and sharply reduces with increasing diameter of the pile or soil nail. Similar test observations have been reported in pipe pullout tests (Weerasekara 2007 and Wijewickreme et al. 2008). 4.2.1.3 Frictional degradation It is known from large displacement interface shear tests that after the initial soil dilation, the normal stress will reduce with the increase in displacement. Evaluations of buried, welded–steel gas pipelines with good–quality girth welds typically permit longitudinal tensile strains to reach 3% to 5% (Wijewickreme et al. 2005). In comparison, buried plastic pipes can also experience even larger elongations when subject to soil loadings induced from ground movements. This would typically lead to the progressive development of friction along the pipe (Weerasekara and Wijewickreme 2008) since each pipe section will undergo different levels of interface shear displacements. For example, at a given instant of pipe movement, a pipe section located at the “leading end” of an axial soil loading situation will have experienced larger pipe-soil interface displacements than those felt by a pipe section located away from the axial soil movement. As such, thus it is unreasonable to assign the peak frictional forces at these large displacement levels. As a result, the impact of frictional degradation is considered of increased significance in the case of extensible pipes undergoing large displacements before reaching the performance limits. In this analytical derivation, it is argued that the decrease in the normal stress after the initial soil dilation can be effective captured using the commonly known degradation of the shear modulus (G) with shear strain ( γ ) as in Equation 4.7. The resulting change in normal stress on the pipe will directly influence the interface frictional force acting on a pipe element. For the ease of discussion, the degradation of G can be examined and formulated for two regions of pipe displacement; namely, segment of pipe subjected to small and large interface soil displacements (Figure 4.3). 106 Figure 4.3 (a) Mobilized frictional lengths corresponding to different degrees of relative pipe displacements; (b) Assumed response of the unit interface friction (T) with relative displacement for MDPE–sand interface. (i) Small displacement behavior It is well known that the shear modulus of soil will decrease with increasing shear strain. This, in turn, can influence the magnitude of the soil normal stress on the pipe; in other words, a very Small displacement region xn xn+1 (a) x = 0, u=0 xL Large displacement region Leading end of the pipe Large displacement region ue un Tn+1 un+1 Relative pipe displacement (u) (b) Tn Small displacement region T 107 significant reduction in the soil normal stress on the pipe can be expected at very small displacements due to the reduction in shear modulus. The following hyperbolic relationship proposed by Hardin and Drnevich (1972)can be used to represent the shear modulus degradation in an analytical framework. )ˆˆ( 1)( 0 baG G + = γ γ (4.8) where, G0 is the initial shear modulus, a ) and b ) are constants that can be obtained from the published shear modulus degradation curves for sands (e.g., Seeds and Idriss 1970). In absence of direct measurement of G0, there are number of empirical relationships (e.g., Hardin and Drnevich 1972, Iwasaki et al. 1978, Tatsuoka and Shiuya 1992; Jardine et al. 1984) to obtain G0 in relation to the effective confining stress and void ratio. With the knowledge of thickness of the shear zone (∆td), shear strain (γ) for use in Equation 4.8, for a selected pipe displacement (u) is obtainable using γ = u/∆td. From this information, G at a given shear strain level can be calculated from Equation 4.8, which in turn, can be used in Equation 4.7 to calculate ∆σd. (ii) Large displacement behavior Tatsuoka and Shibuya (1991) claimed that the hyperbolic relationship cannot be used to represent the small strain and large strain behavior of G simultaneously as a slower rate of degradation than suggested by Equation 4.8 is observed at large displacements. Similar observations are also noted by Wijewickreme et al. (2011) in pullout tests performed on steel pipes. Considering this aspect, beyond the threshold value of γ = 2.5%, the shear modulus was assumed to degrade linearly with the displacement, such that G of the surrounding soil mass will become negligible at a predefined displacement. This displacement at which the impact of soil dilation becomes negligible is often obtained from experimental observations. For example, Scarpillai and Wood (1982) observed that soil dilation 108 will become negligible when the displacement exceed about 100 times d50 (average grain size of the soil). Stone and Muir Wood (1992) and Vardoulakis et al. (1981) stated that this length to be about 176 and 120 times d50, respectively. As per above, in the present analytical formulation, the entire reduction in normal stress in the post-peak region is attributed to the reduction in shear modulus. However, particle level observations suggest that excessive particle wear and tear can also lead to reduction in thickness of the shear zone (Scarpillai and Wood 1982); any such reduction in thickness of the shear zone would also contribute to a reduction of the soil normal stress. Nevertheless, it is admitted that more research is needed to characterize the frictional degradation behavior for the entire displacement range since only limited studies have been performed to investigate this post-peak behavior (Wood 2002). 4.2.2 Modified interface frictional resistance Combining the changes in normal stress on pipe, described in Sections 3.1.1 and 3.1.2, a more detailed interface friction model can be formulated as follows: ( ) tandcT Dσ σ pi δ′= ∆ + (4.9) where 0(1 ) 2 K Hγ σ + ′ = and ∆σdc is the net normal pressure increase due to initial soil dilation and the subsequent frictional degradation. By accounting for the potential changes in normal soil stress, Equation4.9 presents an improved version of Equation 4.1. Note that in absence of ∆σdc term, Equation 4.9 will be similar to Equation 4.1. 4.2.3 Stress-strain behavior of the pipe material A reasonably accurate representation of the stress-strain behavior of the pipe material is a key factor when attempting to develop an analytical solution to represent the overall pipe response. 109 As stated in Chapter 2, MDPE pipes used in the gas distribution industry are known to have viscoelastic characteristics. Hence, besides the highly nonlinear stress-strain behavior, these pipes are known to be strain rate and temperature dependant. Considering the various analytical models to present the stress-strain behavior of a viscoelastic material, the hyperbolic model was selected due to its simplicity. Although, many other viscoelastic models exist, those models cannot be incorporated in a derivation of another analytical solution due their complexities. 1 Eini ε σ ηε = + (4.10) Eini and η are the hyperbolic model constants. To account for the strain-rate dependant behavior of these viscoelastic materials, the above hyperbolic constants can be obtained by representing in the following format (Suleiman and Coree 2004). ( )biniE a ε= & (4.11) ( ) ln( ) b a c d εη ε = + & & (4.12) whereε& is the strain rate, and a, b, c and d are constants that can be determined by fitting the stress-strain responses obtained from uniaxial extension or compression tests. In the absence of independent tensile test data, these constants can be determined from axial pipe pullout tests if the stress and strain are measured independently at a selected location. The intention of using these strain rate dependent hyperbolic constants is to modify the stress- strain properties for the next calculation step based on the current strain rate values. To employ such approach, the stress-strain behavior of this material needs to be independent of the stress- strain history. Based on the uniaxial compression tests performed by Zhang and Moore (1997) 110 at strain rates ranging from 10-5/s to 10-1/s, it has been observed that HDPE is essentially independent of the strain rate history. In other words, if the strain rate is changed during a tensile test, the stress-strain behavior for the new strain rate would be similar to stress-strain behavior attained if the new strain rate had been used from the beginning (see Figure 4.4). As a material having similar molecular and stress-strain behavioral characteristics, it is reasonable to postulate that MDPE is also independent of the strain path. Therefore, it is acceptable to assign the current strain rate value in Equation 4.11 and 4.12, irrespective of the strain rates experienced previously by the pipe segment. Figure 4.4 Model predictions using linear viscoelastic – viscoplastic model and experimental results for the test in which the strain rate was changed from 10-3/s from 10-2/s and changed back to 10-3/s (reproduced after Chehab and Moore 2004). Although the temperature effects are not explicitly accounted in the above formulation, the same analytical framework can be used with modified hyperbolic constants to account for the temperature dependency. As noted by Stewart et al (1999), the modulus change of about 15% is expected when temperature changes from 18 – 23C. 0 5 10 15 20 25 30 0 0.05 0.1 0.15 St re ss ( M P a ) Strain Variable strain rate 10e-3 /sec 10e-2 /sec 111 4.2.4 Analytical formulation of the pipe–soil interaction response With the knowledge of the pipe-soil interface frictional behavior and pipe material stress-strain behavior, an analytical model can be formulated to represent the overall pipe response. In the analytical formulation, considering the equilibrium of a pipe element, second order differential equation can be derived with the knowledge of the externally applied frictional load and the material properties as follows. p d duEA T dx dx = (4.13) Where, E is the Young’s modulus of the pipe material and Ap is the cross sectional area of the pipe. Substituting the hyperbolic material properties in Equation 4.13, the following condensed from of differential equation can be obtained. Note that, the nonlinearity of the stress – strain response of the pipe material described in Equation 4.14 is considered in this derivation. 3 2 2 d1d dx ddx 1 dx n ini p u Tu uE A η η + = − (4.14) Note that according to Figure 4.3, for a given displacement (u) of the leading end, the elemental pipe lengths between x = 0 and the leading end of pipe (x = xL) after having experienced the peak interface resistance will now have post–peak frictional behavior (see Figure 4.3(a)). As shown in Figure 4.3, the interface frictional force Tnis assumed to be mobilized over the pipe segment between x = xnand x = xn+1. The corresponding solution for Equation 4.14, i.e., the relative displacement (un) is obtained as follows. 112 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (4.15) where, n n ini p T E A ηλ = , Cn and C΄n are constants that depend on the pipe and interface properties; and ( ) 124 1 4n n n n nz x Cλ λ= − + − (4.16) The strain along the pipe is obtained by differentiating Equation 4.15. (1 )1 ' 1 2 ( ) n n n n n z u x Cη λ − = − + (4.17) The corresponding pullout resistance at any given location of the pipe can be obtained by substituting these strain and pipe modulus relationships in the following '( ) n n p nN E x A u= × × (4.18) 4.2.4.1 Solution approach and boundary conditions The objective is to use Equation 4.15 to compute the values of T along the pipe lengths where interface friction mobilized while accounting for∆σdas appropriate and then use those values in Equation 4.15. In Equation 4.15, if T is assumed to be a constant, the solution will become closed–form, thus Equation 4.15 can be solved for two known boundary conditions. However, as described in Section 4.2.1, the interface frictional force, T is dependent on relative displacement, u. Therefore, to solve this pipe – soil interaction problem, the pipe length was divided into small pipe element lengths; a step–by–step computational method for u (i.e. u = un, un+1 etc.) was adopted using the basic solutions of the differential equation. In this method, with T assumed constant for a small finite length of pipe (and called Tn), the basic solution is solved for known 113 boundary conditions of the adjacent pipe element. These computations can be easily performed using a spreadsheet or any other programming tool. The unknowns (Cn, Cn΄, xn) in Equation 4.15 to Equation 4.18) can be obtained by considering the following boundary conditions for nth pipe segment: (i) at x = xn, un+1 = un (the displacement is continuous in n+1and nth pipe segment); (ii) at x = xn, u'n+1 = u'n (strain is continuous across the boundary between n+1and nth pipe segment); (iii) at x = xL, N = NL (force at the leading end of the pipe). To initiate this step-by-step analytical approach, ux=0 and u'x=0 were selected to be zero at x = 0 in the small displacement region. The coefficients Cn, Cn΄ can be obtained as follows +++−−−= − − − −− − −− )log( 2 1)(tanh11' 1 1 1 11 1 11 nn n n nn n nnn CxzzxuC λλλη (4.18) [ ] [ ] [ ] 041)'21( 2')('4)( 11 2 1111 1 2 1 2 1111 2 1 2 1 2 1 2 =+−− +−+ −−−−−− −−−−−−−−− nnnnnn nnnnnnnnnnn xxu xuuCuC λλη ηληηλ Although the analytical formulation is presented for both small displacement and large displacement zones to illustrate the entire interaction behavior, the overall pipe response is dominated by the interaction of the large displacement region. By examining the strain development along the pipe length from actual pipe pullout tests, the maximum strain generated from the interaction in the small-displacement region is practically insignificant i.e., axial strain is about 400 µε with a mobilized frictional length of 0.5 m in dense sand, and it is about 200 µε with a mobilized frictional length of 0.4 m in loose sand. Furthermore, the displacement levels corresponding to the small displacement region (i.e. about 0.5 mm) is inconsequential in practical applications. Therefore, it was noted that the equations for the large displacement zone may be solved by using these approximate values for strain and mobilized frictional length as the initial conditions. This allowed simplifying the formulation by avoiding any calculations in the small displacement region. In addition, such simplifications could avoid modeling difficulties associated with the very large drop in the soil resistances expected in the small displacement region due to very rapid degradation of shear modulus. 114 4.3 Selection of input parameters When using Equation 4.15 to 4.18) the values of burial depth (H), soil unit weight (γ), outside pipe diameter (D) and pipe cross-sectional area (Ap) can be measured directly. The following presents details on the selection of values for the remaining input parameters. 4.3.1 Determination of the interface friction angle (δ) Over the years, several interface friction tests have been performed on polymers mainly using direct shear and ring shear test devices. For example, O’Rourke et al. (1990) reported over 400 direct shear tests performed on three different polymers (HDPE, MDPE and PVC) and with different types of sands. In their tests, a peak was not observed for friction angle in tests performed with MDPE and HDPE specimens. O’Rourke et al. (1990) reported a friction angle of 19 degrees at an overburden stress of 3-70kPa. Martin et al. (1984) performed direct shear tests on HDPE geomembranes at an overburden stress of 10-100kPa and reported that the residual friction angle to be 20 to 24 degrees for Concrete sand and Ottawa sand, respectively. Williams and Houlihan (1987) reported values of 27 and 19 for Concrete and Ottawa sand buried in similar overburden stresses. Similar direct shear tests have been reported by Saxena and Wong (1984) and Akber et al. (1985) with normal stress ranging from 120kPa to 200 kPa, and the corresponding the friction angles were in the range of 19-25 degrees. In the proposed analytical derivation, one of the key aspects in selecting the interface friction angle is to obtain an interface friction angle unaffected by the soil dilation. Therefore, it is judged that one of the best ways of estimating δ which is unaffected by the soil dilation would be from large-displacement ring shear tests. Negussey et al. (1989) performed ring shear tests to find the interface friction between smooth HDPE surface and two types of sands- i.e., Ottawa and Concrete sand (Earl’s creek). From these experiments, they observed that the large- displacement interface friction angle is about 16.5° when HDPE is in contact with Concrete sand having an angular grain shape, and about 15° when in contact with rounded Ottawa sand. Similar ring shear test performed on smooth HDPE geomembranes were reported by Vaid and Rinne (1995) at normal stresses less than 100kPa. In tests performed with Ottawa sand and 115 smooth HDPE, the residual friction angle was measured to be as low as 14 degrees whereas a residual friction angle of 22 degrees was measured when a smooth geomembrane is in contact with Target sand. Furthermore, Tan et al. (1998) observed that the interface shear friction angle is not significantly affected by the rate of shearing. With this available information, it was judged that a large-displacement interface friction angle of 16° would be reasonable to represent large- displacement interface frictional characteristics for all the pipe pullout tests considered herein. 4.3.2 Determination of thickness of shear zone (∆td) Several studies on sand–steel interfaces have shown that the dilation is confined to a narrow zone of shearing. The thickness of this shearing zone at these large displacements has been observed to be 10–20 times d50 (Roscoe 1970, Scarpillai and Wood 1982, Usegui et al. 1988, Luo, et al. 2000). Furthermore, Wood (2002) stated that the stress level showed no consistent influence on the thickness of the shear band. Karimian (2006) used an axial pipe pullout test to directly measure the thickness of the shear zone at the MDPE-FRS interface. In this test, the thickness of the shear zone was determined to be about 3 mm by directly measuring the thickness of the disturbed zone closer to the pipe (Figure 4.5) using colored sand. With a d50 value of 0.23 mm, this measurement was in line with the other experimental results presented previously. From these experimental observations, ∆td for MDPE – FRS interface was selected to be 3 mm for all field and laboratory pullout tests. 116 Figure 4.5 Direct measurement of the thickness of the shear zone at the pipe-soil interface at MDPE-FRS interface (after Karimian 2006) 4.3.3 Determination of shear modulus degradation (G) As mentioned in Section 4.2.1.3, the shear modulus degradation is discussed and analyzed separately for small and large displacement regions considering the behavioral differences in these regions. In small displacement region, the constants a) and b ) in the hyperbolic degradation relationship are determined to be a) = 15 and b ) = 1 considering published shear modulus degradation relationships for sands (Seed and Idriss 1970; Idriss 1990). Figure 4.6 shows the fitted hyperbolic relationship for the experimentally obtained shear modulus degradation curves by Seed and Idriss (1970) and Idriss (1990). Colored sand MDPE pipe 117 Figure 4.6 Hyperbolic fit to the shear modulus degradation curves obtained by Seed and Idriss (1970) and Idriss (1990). Considering the approximation presented in Section 4.2.4.1, for the small displacement region, it is required only to select a value for the shear modulus at 2.5% shear strain [G(γ=2.5%)] to calculate the pipe pullout response using the interface friction behaviour for the large displacement region. Figure 4.6 reports G(γ=2.5%) values selected for the analytical formulation to obtain the best match between the experimental and analytical results. In the absence of direct measurement of shear modulus, it is interesting to compare these values used in the analytical solution with the G(γ=2.5%) values calculated from empirical relationships that have been proposed for G0(e.g., Hardin and Drnevich 1972, Iwasakai et al. 1978, Yu and Richart 1984). Although G0 values can be obtained from these empirical relationships based on the soil density (or void ratio) and overburden stress, in typical geotechnical applications, there is some uncertainty when attempting to determine the initial shear modulus and its degradation (e.g., Lehane and White 2004). As noted in Figure 4.6, there is a slight discrepancy observed between the G(γ=2.5%) values used in the analytical solution and the values obtained from empirical equations, and such variations are admissible in typical geotechnical applications. 0 0.2 0.4 0.6 0.8 1 0.0001 0.001 0.01 0.1 1 10 G /G 0 Shear strain (%) Seed and Idriss (1970) Idriss (1990) Series1 118 Figure 4.7. Comparison of initial shear modulus obtained from the empirical solutions and the values used in the analytical solution For the large displacements, the soil dilation is assumed to be linearly decreasing until the impact of soil dilation becomes negligible. This displacement corresponding to zero soil dilation is selected to be 50 mm considering the experimental observation presented by Scarpillai and Wood (1982), Stone and Muir Wood (1992) and Vardoulakis et al. (1981). It is admitted that a larger soil dilation would require a longer displacement to diminish the effects soil dilation, whereas in slightly less dense soils (i.e., smaller dilation), this displacement is likely to be smaller. However, in the absence of precise information on frictional degradation behavior at large displacements, a constant value is selected for the sake of maintaining a consistent set of input parameters. 4.3.4 Determination of strain rate dependant hyperbolic constants The strain rate dependant hyperbolic constants (a, b, c and d) can be determined from fitting the stress-strain responses obtained from uniaxial extension or compression tests. As mentioned in Chapter 3, author performed a number of uniaxial compression tests on the MDPE pipe material. 119 However, these data is little significance when attempting to model the tensile loading of the pipe material. It was considerably difficult to perform uniaxial extension tests as the clamping was proved difficult at the two ends of the pipe. In the absence of independent tensile test data, these constants can be determined directly from the pipe pullout tests if the stress (or force) and strain are measured independently at a particular location. As noted earlier, with direct strain and force measurements on the pipe, it is possible to obtain the stress-strain plots for pipe pullout tests. With this information, the respective strain dependant hyperbolic constants can be obtained by matching the respective stress-strain plots. These strain-rate dependant constants to simulate the stress-strain responses for laboratory and field pullout tests are given in Table 4.1 and Table 4.2. For example, Figure 4.8 shows stress-strain behavior modeled for Test#4, which was subject to three different displacement rates at the pulling end. Similar good agreement between analytical model and experimental results was observed all remaining pullout tests, and these results are presented in following sections. Note that a consistent set of values were assigned for these rate dependant hyperbolic constants in all pullout tests despite the wide range of strain rates observed in these tests. 120 Figure 4.8. Variation of pullout resistance versus axial strain for Test #4 4.3.5 Determination of lateral earth pressure coefficient (K0) The lateral earth pressure coefficient (K0) values computed using the constant volume friction angle Jaky’s formula (1944) for dense and loose sands are about 0.45 and 0.5 (using the constant volume friction angle). Northcutt (2010) measured lateral earth pressure in Fraser River sand for different placement techniques and fabrics. It is observed that for dense sand with relative density of about 85%, the lateral earth pressure changes from 0.47-0.53 in normally consolidation state and K0 ranges from 0.59- 0.65 in sand with overconsolidation ratio of 2.0. However, in certain documented experiments (e.g., Filz and Duncan 1996) have reported that compaction induced lateral soil resistance can be considerably higher than the K0 values calculated using Jaky`s formula (1944). Furthermore, using direct soil stress measurements and pressure transducers attached to the pipe surface, Karimian (2006) has shown that K0 is in the range of 0.45 to 0.55 in dense and loose backfills, respectively. The sensitivity analysis performed using the analytical solution has revealed that the variation of K0 has only a small impact on the overall response obtained from the analytical solution. As such, K0 was selected as 121 0.5 for both dense and loose sands. However, the selection of K0 should be based on case-by- case basis with proper account of soil conditions around the pipe. The selected input parameters for laboratory scale tests are presented in Table 4.1and field pipe pullout tests are presented in Table 4.2. 4.4 Comparison of results obtained from analytical solution and laboratory pullout tests The following section is aimed at comparing the pipe pullout responses obtained in the laboratory environment with the analytical solution. For this discussion, two pipe pullout tests conducted by Weerasekara (2007) and three pipe pullout tests conducted by Anderson (2004) were considered. First, the experimental details relating to these pipe pullout tests are presented for better understanding of the test results. 4.4.1 Experimental details All the tests conducted by Anderson (2004) and Weerasekara (2006) were performed using a testing apparatus that was designed and constructed at the University of British Columbia. The main components of the test setup comprise of a soil chamber of 3.8m (or 5m) in length, 2.5m in width, 2.5m in height, a hydraulic actuator and a data acquisition system (see Figure 4.9). The compartmental construction of the soil chamber allows changing the length of the soil chamber. In these axial pullout tests, the buried pipe was aligned parallel to the longer direction of the box and pulled from one end (referred to as the leading end) using the double–acting hydraulic actuator maintaining a constant displacement rate (Figure 4.10). Note that in Figure 4.10, the abbreviations “SG” and “SP” stands for strain gauges and string potentiometers respectively. The axial pullout resistance and displacement of the pipe at predetermined sections were monitored and recorded using load cells and string potentiometers, respectively. Furthermore, Weerasekara (2007) directly measured the axial strain using strain gauges mounted at the leading end of the pipe. 122 Figure 4.9 Soil chamber at University of British Columbia Figure 4.10 Typical pipe configuration for axial pullout behavior of 60 and 114 mm pipes. SP2 Load Cell Strain Gauges SP1 Actuator 60 mm or 114 mm pipe Leading end of the pipe 3.82 m 2 .5 m 123 For the tests performed by Weerasekara (2007), the applied rate of displacement at the leading end of the pipe was 36 mm/hr which is the slowest rate of displacement achievable using the hydraulic actuator system. In comparison, Anderson (2004) employed a pulling rate of 10 mm/hr using an electric motor as the pulling device. The pullout tests were performed on 60mm and 114 mm diameter MDPE pipes with SDR value of 11. These pipes are extensively used in natural gas distribution systems. All the tests were performed at an ambient temperature of 20±1ºC. Anderson (2004) also performed pullout tests using loose soil backfills. The loose soil state was achieved by dumping the sand from a height of about 200mm. This resulted in a soil density of about 1450 kg/m3 with 43% relative density. The pullout resistance – displacement plots obtained from the axial pipe pullout tests conducted in dense and loose sand backfills are shown in Figure 4.11 and Figure 4.12, respectively. In these plots, the pullout resistances are plotted only up to the instance where trailing end of the pipe begins to move. The tests conditions for the two pipe pullout tests performed with 60mm diameter pipe buried in dense sand were very similar. As noted in Figure 4.11, the results for these two tests are very similar, showing the repeatability of experimental findings. Figure 4.11 Variation of pullout resistance with displacement at leading end for the tests conducted by Weerasekara (2007) and Anderson (2004) 124 Figure 4.12 Variation of pullout resistance versus displacement at the leading end of the pipe for pullout tests conducted in loose backfill conditions (after Anderson 2004) 4.4.2 Selection of input parameters for the analytical solution All the details relating to pullout tests are summarized in Table 4.1 together with the input parameters used for the analytical solution. The resulting analytical model estimations for the these laboratory pipe pullout tests are presented in the following sections -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 2 4 6 8 10 12 P ul lo ut r es is ta n ce ( kN ) Displacement at leading end(mm) 60mm -loose (Anderson 2004) 114mm -loose (Anderson 2004) 125 Table 4.1. Input parameters for the analytical solution to model the axial pullout tests performed in laboratory environment Input parameter Test number Weerasekara (2007) Anderson (2004) 60D 114D 60D 60L 114L Interface friction on pipe ∆td (mm) 3 3 3 N/A N/A G (γ=2.5%) (kNm-2) 800 600 800 N/A N/A γ (kN/m-3) 16.0 15.8 16.0 14.2 14.2 H (m) 0.6 0.6 0.72 0.72 0.72 K0 0.5 0.5 0.5 0.5 0.5 δ (degrees) 16 16 16 16 16 Displacement at zero dilation (mm) 50 50 50 N/A N/A Relative density (%) 88 82 88 23 23 Pipe properties Pipe outside diameter (mm) 60.3 114 60.3 60.3 114 Pipe thickness (mm) 5.5 11 5.5 5.5 11 Rate dependant hyperbolic constants* a 2340 2340 2340 2340 2340 b 0.109 0.109 0.109 0.109 0.109 c 49.24 49.24 49.24 49.24 49.24 d 1.53 1.53 1.53 1.53 1.53 Ground movement conditions Pullout rate at clamp (mm/hr) 36 36 10 10 10 Parameters that are not part of the analytical solution Burial length (m) 3.8 3.8 5.0 5.0 5.0 Note: *The stress is measured in MPa and the strain rate measured in (1/s). 126 4.4.3 Axial pullout tests performed on dense sand The behavioral patterns of axial pullout resistance with strain for 60-mm and 114-mm pipes obtained from the analytical solution are compared with those from axial pullout test experiments in Figure 4.13and Figure 4.14. These plots essentially express the stress-strain plot for the pipe material, and the observed agreement between the experimental and analytical results confirms the ability to model the stress-strain behavior at variable strains rates using the hyperbolic model. In all comparison presented in following sections, the experimental data is only presented up to point where trailing end of the pipe begins to move, since the comparisons between analytical model and experimental data will only be valid up to this point. Figure 4.13. Comparison of pullout resistance versus axial strain at leading end behaviors between analytical solution and pullout tests for 60-mm pipe buried in dense sand (Weerasekara 2007) 127 Figure 4.14. Comparison of pullout resistance versus axial strain at leading end behaviors between analytical solution and pullout tests for 114-mm pipe buried in dense sand (Weerasekara 2007) When validating the above stress-strain properties for the pipe material, it is interesting to explore the strain rate variation with time. It is recalled that the actual viscoelastic properties of the pipe material will depend on the strain rate. Thus, in order to obtain a good agreement for stress-strain behavior, the analytical model should be capable of estimating the strain rate variation with reasonable accuracy. Figure 4.15 plots the measured strain rate (average obtained for a period of 50 s) for the pullout test performed on 60-mm pipe (Weerasekara 2007). It is observed that the strain rate drops with increase in time asymptotically by several orders since the start of the pullout test. The time variation of the strain rate calculated from the analytical solution is also superimposed in Figure 4.15, and a good agreement between the analytical and experimental results is observed. 128 Figure 4.15 Variation of axial strain rate with time, obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Weerasekara 2007) Figure 4.16 and Figure 4.17 compare the pullout resistance – displacement responses obtained from the analytical solution and the large–scale pullout tests performed by Weerasekara (2007) and Anderson (2004) on 60 mm diameter pipes buried in dense sand. Note that these two tests were performed with different burial depths, pulling rates and buried lengths. Figure 4.18 shows the pullout resistance- displacement plot for 114-mm pipe buried in dense sand. -0.001 0 0.001 0.002 0.003 0.004 0.005 0 500 1000 1500 2000 Time (sec) St ra in ra te (% /s ec ) Analytical solution Experimental 129 Figure 4.16 Comparison of pullout resistance versus displacement at leading end obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Weerasekara 2007) Figure 4.17 Comparison of pullout resistance versus displacement at leading end obtained from the analytical solution and pullout test performed on 60-mm pipe buried in dense sand (Anderson 2004) 0 2 4 0 5 10 15 20 Displacement at leading end (mm) Pu llo u t r e sis ta n ce (kN ) Experimental Analytical solution 0 2 4 0 5 10 15 20 25 Displacement at leading end (mm) Pu llo u t r es ist an ce (kN ) Analytical solution Experimental 130 Figure 4.18 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in dense sand (Weerasekara 2007) Figure 4.19 and Figure 4.20 show the plots of axial strain–displacement behaviors for 60-mm and 114-mm pipes (Weerasekara 2007). Again, very good agreement between the analytical and experimental results can be noted. In absence of direct strain measurements, it was not possible to compare the strain-displacement relationship for tests conducted by Anderson (2004). 0 2 4 6 8 0 2 4 6 8 10 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) Experimental Analytical solution 131 Figure 4.19 Comparison of axial strain versus displacement at leading end obtained from analytical solution and pullout tests performed on 60-mm pipe buried in dense sand (Weerasekara 2007) Figure 4.20 Comparison of axial strain versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in dense sand (Weerasekara 2007) 0 2000 4000 6000 8000 10000 12000 0 5 10 15 20 Displacement at leading end (mm) Ax ia l s tra in at le ad in g en d ( µε ) Experimental Analytical solution 0 2000 4000 0 2 4 6 8 10 Displacement at leading end (mm) Ax ia l s tra in a t l e ad in g en d ( µε ) Experimental Analytical solution 132 4.4.4 Axial pullout tests performed on loose sand Figure 4.21 and Figure 4.22 show the pullout resistance-displacement characteristics obtained from the analytical approach and pullout tests performed on loose sand for 60-mm and 114-mm diameter pipes, respectively. Note that, it was not possible to compare the strain-displacement relationship since the pipe strain was not measured in these two tests. Figure 4.21. Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 60-mm pipe buried in loose sand (Anderson 2004) 133 Figure 4.22. Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and pullout tests performed on 114-mm pipe buried in loose sand (Anderson 2004). 4.5 Comparison of results obtained from analytical solution and field pullout tests The following compare the estimations from the analytical solution with the field pipe pullout test results. The details of the field pipe pullout tests are described in Chapter 3. In these field pipe pullout tests, a considerably longer burial length was employed compared to the burial length in laboratory pipe pullout tests. As a result, larger axial strains were achieved from field pullout tests, before mobilizing the end boundary conditions of the trench. Therefore, with these test results, it is possible to investigate the applicability of the analytical solution at these large strain levels, which are generally closer to the maximum strains allowed in typical operating conditions. 4.5.1 Selection of input parameters for the analytical solution Similar to the pullout tests performed in the soil chamber, it is important to determine the input parameters used for the analytical model. The process of the selecting the input parameters is 134 described previously in Section 4.3 and the values of these input parameters are listed in Table 4.2. Table 4.2 Input parameters used in the analytical approach to model the field pipe pullout tests Input parameter Test number Test#1 Test#2 Test#3 Test#4 Test#5 Interface friction on pipe ∆td (mm) 3 3 3 3 3 G (γ=2.5%) (kNm-2) 800 900 600 600 900 γ (kNm-3) 16.0 16.1 15.8 15.8 16.1 H (m) 0.56 0.57 0.98 0.55 0.55 K0 0.5 0.5 0.5 0.5 0.5 δ (degrees) 16 16 16 16 16 Displacement at zero dilation (mm) 50 50 50 50 50 Relative density (%) 88 92 82 82 92 Pipe properties Pipe outside diameter (mm) 60.3 60.3 60.3 60.3 60.3 Pipe thickness (mm) 5.5 5.5 5.5 5.5 5.5 Rate dependant hyperbolic constants* A 2020 2020 2020 2020 2020 B 0.109 0.109 0.109 0.109 0.109 C 43.35 43.35 43.35 43.35 43.35 D 1.37 1.37 1.37 1.37 1.37 Ground movement conditions Displacement rate of soil block (mm/min) 0.601 2.145 2.185 0.601 0.151 2.16 0.729 0 0.729 Note: *The stress is measured in MPa and the strain rate measured in (1/s). 135 4.5.2 Analytical model estimations for field pullout Tests # 1, 2 and 3 Figure 4.23 to Figure 4.25 show the variation of pullout resistance with the displacement at the leading end of the pipe obtained for Test# 1 through to #3. The analytical results are also superimposed in these figures. A good agreement is observed between the analytical and experimental results. The slight discontinuity in the analytical estimation around 50mm in Test #1 is due to the assumption of that the dilation ceases at 50mm. With slightly different value for this parameter, a better match can be achieved. However, for the sake of maintaining a consistent set of input parameters, the displacement for the dilation to cease is assumed to be 50 mm for all the pipe pullout tests. Figure 4.23 Comparison of pullout resistance versus displacement at leading end obtained from the analytical approach and Test#1 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 Displacement at leading end (mm) Pu llo u t r es is ta n ce (kN ) Experimental (Test#1) Analytical solution 136 Figure 4.24 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#2 Figure 4.25 Comparison of pullout resistance versus displacement at leading end obtained from analytical approach and Test#3 0 1 2 3 4 5 6 7 8 0 20 40 60 Pu llo ut re sis ta nc e (kN ) Displacement at leading end(mm) Experimental Analytical 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 Displacement at leading end (mm) Pu llo u t r es is ta n ce (kN ) Experimental (Test #3) Analytical 137 Figure 4.26 to Figure 4.28 plots the variation of axial strain with displacement measured at the leading end of the pipe. In general, a good agreement is observed between the analytical estimations and the experimental results. Compared to the pullout tests conducted in the laboratory environment, significantly large axial strains were obtained with the longer burial length. Note that in Test #2, the strain gauge was saturated after reaching 20000µε strain. Figure 4.26 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#1 0 10000 20000 30000 40000 0 20 40 60 80 100 120 140 160 Displacement at leading end (mm) Ax ia l s tra in ( µε ) Experimental (Test #1) Analytical 138 Figure 4.27 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#2 Figure 4.28 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#3 0 10000 20000 30000 0 20 40 60 Ax ia l s tra in (µε ) Displacement at leading end (mm) Experimental Analytical 0 20000 40000 60000 0 20 40 60 80 100 120 140 Displacement at leading end(mm) Ax ia l s tra in ( µε ) Experimental (Test #3) Analytical 139 It is also interesting to investigate the force –strain relationship for the pipe material in these three tests. Figure 4.29 and Figure 4.30 show good agreement between the force-strain relationship for Test #1 and #2, although the pullout resistance is slight over-predicted for Test #3 in Figure 4.31. Note that considering the simplicity of the simulated ground displacement behavior (i.e., constant rate of displacement), the time variation of the pullout resistance and axial strain are not presented for Test #1 through #3. Figure 4.29 Comparison of pullout resistance versus axial strainobtained from analytical solution and from Test#1 0 1 2 3 4 5 6 7 0 5000 10000 15000 20000 25000 30000 35000 40000 Axial strain at leading end (µε) Pu llo ut re si st an ce (kN ) Analytical Experimental 140 Figure 4.30 Comparison of pullout resistance versus axial strain obtained from analytical solution and from Test#2 Figure 4.31 Comparison of pullout resistance versus axial strainobtained from analytical solution and from Test#3 0 2 4 6 8 0 10000 20000 30000 40000 Axial strain at leading end (µε) Pu llo u t r es ist an ce (kN ) Experimental Analytical 141 4.5.3 Analytical model estimations for field pullout Test # 4 The soil loading simulated in Test #4 is more complex than the other pullout test due to the variable pullout rates applied at the leading end of the pipe. As noted in Table 4.2, the pullout test was conducted at three different displacement rates. Figure 4.32 to Figure 4.34 show the variations of the pullout resistances, axial strain and displacement at the leading end of the pipe. Although, the pullout resistance in the second phase of loading was slightly under-predicted by the analytical approach, the general trend of the pullout response was satisfactorily captured by the proposed analytical method. Figure 4.32 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#4 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 Displacement at leading end (mm) Pu llo u t r es is ta n ce (kN ) Experimental Analytical 142 Figure 4.33 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test #4 Figure 4.34 Comparison of pullout resistance versus axial strainobtained from analytical solution and Test#4 0 5000 10000 15000 20000 25000 0 10 20 30 40 50 60 70 80 Displacement at leading end (mm) Ax ia l s tra in ( µε ) Experimental Analytical 143 With the variable displacement rates, it is also expected that the strain rate will also show distinct variations in relation to the changes in the displacement rate. As shown in Figure 4.35, these variations in strain rate were also captured by the analytical solution. Figure 4.35 Variation of axial strain rate with time obtained from analytical solution and Test #4 Figure 4.36 and Figure 4.37 presents the variations of the pullout resistance and axial strain with elapsed time, obtained from the analytical solution and Test#4. Most importantly, these complex loading patterns were captured using the analytical solution just by changing the pulling rates after reaching a particular displacement. -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 2000 4000 6000 8000 10000 12000 14000 Time (sec) St ra in ra te (% / s ec ) Analytical Experimental 144 Figure 4.36 Time variation of the pullout resistance at the leading end of the pipe measured from Test#4 and modeled using the analytical solution Figure 4.37 Time variation of the axial strain at the leading end of the pipe measured from Test#4 and modeled using the analytical solution 145 4.5.4 Analytical model estimations for field pullout Test# 5 The field pullout Test #5 was performed with conditions similar to Test#1 (see Table 4.2). However, after 67 mins of initial displacement, the displacement was paused for 10 days before commencing the pullout at the same rate as in the initial stage. The aim of this pullout test was to investigate the impact of stress- relaxation in plastic pipes. Figure 4.38 shows the variation of pullout resistance with displacement obtained from the analytical solution and Test#5. The drop in measured pullout load can be attributed to the stress- relaxation aspects observed in plastic pipes. Although, the proposed analytical solution is not capable of capturing the unloading response of the stress-strain behavior, the drop in the pullout resistance can be simulated by assuming a very small displacement rate for the second stage of loading. This small rate of displacement will lead to a very small strain rate in the pipe, in turn, a small modulus value for the pipe material is obtained according to Equation 4.11 and 4.12. Consequently, according to Equation 4.18), a smaller pullout resistance is obtained as shown in Figure 4.38. The axial strain-displacement and force-strain relationships for Test#5 are depicted in Figure 4.39 and Figure 4.40, respectively. It is interesting to note that the axial strain decreased slightly over the period of 10 days during which the loading was paused. The mechanism to explain this behavior is proposed in Section 4.9.3. However, it is not possible to model this drop in strain using the analytical solution as observed in Figure 4.39 and Figure 4.40. 146 Figure 4.38 Comparison of pullout resistance versus displacement at leading end obtained from analytical solution and Test#5 Figure 4.39 Comparison of axial strain versus displacement at leading end obtained from analytical solution and Test#5 147 Figure 4.40 Comparison of pullout resistance versus axial strain at leading end obtained from analytical solution and Test#5 Figure 4.41and Figure 4.42 plot the variation of pullout resistance and axial strain for the entire time span of testing. Since the time variations of these parameters are not clearly identifiable in these plots, Figure 4.43 and Figure 4.44 show the variation of these parameters only at the initial stages of loading. 148 Figure 4.41 Variation of pullout resistance for the entire duration of the test, calculated from the analytical solution and from Test#5. Figure 4.42 Variation of axial strain at the leading end of pipe for the entire duration of the test, calculated from the analytical solution and from Test#5. 149 Figure 4.43 Variation of pullout resistance at the initial period of the testing (up to 8000 s), calculated from the analytical solution and from Test#5. Figure 4.44 Variation of axial strain at the initial period of the testing (up to 8000 s), calculated from the analytical solution and from Test#5 150 4.5.5 Estimation of mobilized frictional length Apart from the parameters discussed above, the accurate determination of the mobilized frictional length is also an important consideration especially in distribution pipe networks. As shown by Weerasekara et al. (2006), a branch pipe connection would create considerable anchoring resistance when mobilized. In these instances, the strain gauges located at the vicinity of the pipe connection recorded a large localized increase in strains making the pipe connection vulnerable (Figure 4.45). Based on these observations, depending on the location of the pipe connection relative to the location of ground separation point, the failure of the pipe could occur at the point of ground separation or closer to the pipe connection. Thus, an accurate determination of the mobilized frictional length allows assessing whether the anchoring resistance of the pipe connection is mobilized, and in turn, estimating the location of probable failure in a more accurate manner. (a) (b) Figure 4.45 Schematic illustration of the axial strain along a pipe subject to axial loading: (a) mobilized length (x) less than Land (b) mobilized length exceeding or close to L. Axial strain (relatively uniform across the pipe cross- section Pipe connection Mobilized frictional length (x) Maximum strain Pipe pulling direction Mobilized frictional length (x) Localized axial strain concentration (not necessarily uniform across pipe cross- section Axial strain (relatively uniform across the pipe cross- section L 151 The direct continuous measurement of the mobilized frictional length from pipe pullout tests is difficult. However, in pipe pullout tests, it is correct to state that the trailing end of the pipe will only move when the friction is fully mobilized along the buried pipe length. Knowing the displacement at the leading end corresponding to the instance that trailing end begins to move, the analytical solution can be used to back-calculate the mobilized frictional length corresponding to this specific instance of initiation of displacement at trailing end. These back- calculated mobilized frictional lengths from the analytical solution are presented in Table 4.3 for comparison with the burial lengths used in the pullout tests. Despite the slight under-prediction in most cases, it is observed that the mobilized frictional lengths obtained from the analytical solution are in reasonable agreement with the burial lengths in each pullout test, except for Test #5. The reason for the mismatch in frictional length for Test #5 is further explained in Section 4.9.3. Table 4.3 Mobilized frictional lengths back calculated from the analytical solution for each pullout test. Test No. Burial length (m) Mob. fric. length from analytical solution (m) † Weerasekara (2006) 60D 3.81 4.0 114D 3.81 4.5 Anderson (2004) 60D 5.0 4.5 60L 5.0 4.8 114L 5.0 5.1 Field pullout tests Test#1 8.5 8.4 Test#2 8.5 6.8 Test#3 8.5 7.4 Test#4 8.5 7.8 Test#5 8.5 5.9 †The back calculated mobilized frictional length when the trailing end of the pipe begins to move. 152 Figure 4.46 to Figure 4.48 plot the variation of the displacement, axial strain and pullout resistance with the mobilized frictional length along the pipe for Test #1. Most importantly, it can be noted that the variation of displacement and strain at a particular location is highly nonlinear with respect to the mobilized frictional length. Figure 4.46 Variation of displacement at leading end with mobilized frictional length of the pipe for Test #1 153 Figure 4.47 Variation of axial strain at the leading end of the pipe with mobilized frictional length Test #1 Figure 4.48 Variation of pullout resistance at leading end with mobilized frictional length of the pipe for Test #1 154 4.5.6 Extension of the pipe in the exposed area of the pipe In pullout tests, it is observed that when the pipe is displaced, a pipe section equivalent to the amount of displacement at the leading end will be exposed to air. The axial strain in the exposed section will be as same as the strain at the end of the buried section of the pipe. This is expected since no external frictional force is acting on the exposed section of the pipe. However, this will lead to an additional elongation of the pipe, and the analytical approach presented in this thesis does not account for this elongation of the pipe. The addition elongation of the exposed pipe section can be computed by knowing the length of the exposed section (i.e. displacement of the leading end) and the strain at the leading end (Equation 4.19). This additional elongation computed from this method is then added to the displacement calculated from the analytical solution to obtain the total displacement. uu aa ×= ε (4. 19) Subject to the ranges of strains measured in the pullout tests, it is observed that the additional displacement due to the exposed section is significantly small (1 to 2 mm). Therefore, considering the negligible impact and to preserve the simplicity of the analytical approach, above change to the format of the solution approach is not incorporated to the analytical model. 4.6 Numerical modelling of the axial soil loading of pipes Numerical analysis of the pipe pullout tests was also performed in order to investigate the applicability of soil-spring models to simulate the axial pipe pullout behavior. The soil-spring analyses were performed using ABAQUS/ standard general purpose software (Hibbit et al., 2006). The following sections present the modeling details relating to the soil-spring analyses. 155 4.6.1 Determinations of the axial soil-springs The axial soil-springs (interface frictional resistance) for the numerical model were obtained from the proposed interface friction model presented by Equation 4.15. In other words, the same values used for the interface friction in the analytical solution are employed in the numerical method as well. Accordingly, the impact of soil dilation, mean normal stress on soil dilation and frictional degradation are accounted in these axial soil-springs. 4.6.2 Viscoelastic modeling of pipe stress-strain behavior The viscoelastic behavior of the pipe material was modeled using the Prony series (see Section 2.4.2.1). This is a convenient approach to model the viscoelastic behavior if the stress-relaxation or creep test data are available for the pipe material. ABAQUS allows calculating the Prony series constants in Equation 2.7 from the stress-relaxation test data. With linear viscoelastic assumptions, the stress relaxation behavior can be obtained from the following equation with the knowledge of relaxation coefficient (n). [Note that this equation is presented as Equation 4.20, in Chapter 2, and repeated herein for reader’s convenience]. ( ) n i E t t E − = (4.20) However, as shown in Figure 2.5, significant differences between the actual and the predicted stress-strain responses were observed when using a single set of Prony series constants to model the stress-strain behavior for wide range of strain rates. Therefore, in these soil-spring analyses, the value of the relaxation coefficient (n) was varied until a good agreement is obtained between the numerically and experimentally obtained stress-strain behaviors. 156 4.6.3 Numerical model development using ABAQUS The pipe elements were modeled using PIPE31 beam elements. To model the pipe burial length of 3.81 m and 5.0 m in laboratory pullout tests, 140 and 185 elements were employed, respectively. Similarly, the 8.5-m long pipe in field pipe pullout tests was modeled using 400 elements. The pipe-soil interaction is modeled using similar number of PSI34 soil-spring elements in each test. The other end of the soil-spring is connected to RB3D2 rigid element, and a velocity similar to the actual rate of displacement applied in the pullout test was assigned to this rigid element in direction of the pipe axis to simulate the ground movement. 4.6.4 Numerical modeling of the laboratory pipe pullout tests The following presents results from numerical modeling for the following laboratory pipe pullout tests: (a) Two axial pipe pullout tests performed on 60-mm [Figure 4.49] and 114-mm [Figure 4.50] pipes buried in dense sand by Weerasekara (2007). (b) Two axial pipe pullout tests performed on 60-mm [Figure 4.51a] and 114-mm [Figure 4.51b] pipes buried in loose sand (Anderson 2004). Note that in Figure 4.51, the plots relating to the axial strain in the pipe are not presented due to absence of strain measurements. A good agreement was observed for the variations of pullout resistance and axial strain with the displacement at the leading end of the pipe. Further details relating to these observations are discussed in latter sections. 157 (a) (b) (c) Figure 4.49 Variation of: (a) pullout resistance with displacement, (b) axial strain with displacement, and (c) pipe stress with strain for the axial pipe pullout test conducted on a 60- mm pipe buried in dense sand (Weerasekara 2007). 0 1 2 3 4 0 5 10 15 P u ll o u t re si st a n c e ( k P a ) Displacement at leading end (mm) Numerical Experimental 0 2000 4000 6000 8000 10000 0 5 10 15 A xi a l s tr a in ( µε ) Displacement at leading end (mm) 0 1000 2000 3000 4000 0 5000 10000 St re ss ( kP a ) Axial strain (µε) 158 (a) (b) (c) Figure 4.50 Variation of: (a) pullout resistance with displacement, (b) axial strain with displacement, and (c) pipe stress with strain for the axial pipe pullout test conducted on a 114-mm pipe buried in dense sand (Weerasekara 2007) 0 2 4 6 8 0 5 10 P u ll o u t re si st a n ce ( kP a ) Displacement at leading end (mm) Numerical Experimental 0 1000 2000 3000 4000 0 2 4 6 8 10 12 A x ia l st ri a n ( µε ) Displacement at leading end(mm) 0 500 1000 1500 2000 2500 0 1000 2000 3000 4000 5000 St re ss ( kP a ) Axial strain (µε) 159 (b) Figure 4.51 Variation of pullout resistance with displacement at leading end the pipe for: (a) 60-mm pipe, and (b) 114-mm pipe buried in loose sand (Anderson 2004). 4.6.5 Numerical modeling of the field pipe pullout tests Figure 4.52 to Figure 4.54 show the numerical model estimations for Tests #1 to 3, and good agreement between the numerical and experimental results were observed. Note that these tests were conducted at a constant pullout rate. Therefore, the time variation of the pullout resistance and axial strain are similar in shape when these parameters are plotted with respect to the leading end displacement. As a result, a similar good agreement was observed for variation of these parameters with time (these model predictions are not presented herein for brevity). 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 P u ll o u t re si st a n ce ( k P a ) Displacement at leading end (mm) Numerical Experimental 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 P u ll o u t re si st a n ce ( k P a ) Displacement at leading end (mm) Numerical Experimental 160 (b) (c) Figure 4.52 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#1. 161 (a) (b) (c) Figure 4.53 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#2. 162 (a) (b) (c) Figure 4.54 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, and (c) pullout force with strain for Test#3. Figure 4.55 and Figure 4.56 present different plots of pullout resistance, axial strain, displacement and time for Tests # 4 and #5, respectively. Note that, unlike other pullout tests presented above, these pipe pullout tests were conducted at variable displacement rates to simulate complicated soil loading characteristics. 163 (b) (c) (d) (e) (f) Figure 4.55 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, (c) pullout force with strain, (d) pullout resistance with time, (e) axial strain with time and (f) displacement at leading end with time for Test#4. 164 (b) (c) (d) (e) (d) Figure 4.56 Variation of (a) pullout resistance with displacement (b) axial strain with displacement, (c) pullout force with strain, (d) pullout resistance with time, (e) axial strain with time and (f) displacement at leading end with time for Test#5. 165 4.7 General observations 4.7.1 Comparison between the new analytical approach and experimental results In general, a good agreement was observed between the newly developed analytical solution and experimental results obtained from the pipe pullout tests performed in laboratory and field environments. It is important to note that these good agreements were observed for tests conducted with different pipe diameters (60-mm and 114-mm), soil density conditions (loose and dense), burial depths (0.54 m to 0.98 m), burial lengths (3.8 m, 5.0 m, and 8.5m) and wide range of displacement rates. Most importantly, to obtain these plots, a consistent set of input parameters have been employed in the analytical solution. These input parameters can be obtained with reasonable accuracy through independent laboratory experiments or direct field measurements. The noted good agreement between the force -strain relationship for the pipe material assures the robustness of the hyperbolic model for simulating the viscoelastic model behavior at different strain rates. Despite the simplicity and rather easy implementation using typical programming tool (e.g., use of spreadsheets), one of the main drawbacks of the hyperbolic model is the inability to model the true viscoelastic material behavior. As a result, the hyperbolic model is not capable of successfully modeling the unloading behavior and the stress-relaxation aspects when the pipe is stationary. This aspect is particularly evident in Test #5, in which the reduction in axial strain was not captured using the analytical solution. Further details relating to Test #5 are discussed in Section 4.9.3. 4.7.2 Comparison between numerical (ABAQUS) and experimental results When examining the numerical model estimations presented in Figure 4.49 to Figure 4.56, a good agreement was observed between the numerical and experimental results obtained from laboratory and field pipe pullout tests. However, unlike in the analytical model, it was difficult to replicate the different stress-strain behaviors at different strain rates using a single relaxation coefficient (or single set of Prony series coefficients). It is to be stressed that the exact modeling 166 of the viscoelastic stress-strain response for wide range of strain rates and temperatures is challenging even with the use of more sophisticated models. Nevertheless, the good agreement for the remaining pipe response behaviors (pullout response with displacement, and axial strain with displacement) indirectly confirms the accuracy of the axial soil-springs employed for the numerical modeling. Furthermore, similar to the analytical modeling, some limitations were observed when attempting to model the stress-relaxation behavior observed in Test #5. In particular, both analytical and numerical models failed to capture the drop in strain when the pipe is stationary. 4.8 Proposed interface friction model for cylindrical objects It is also important to highlight that the good agreement between the analytical and experimental results is partly due to the new improved interface friction model employed in these solutions (i.e., Equation 4.15). In this new interface friction model, characteristics such as soil dilation, impact of mean effective stress on soil dilation and frictional degradation are accounted using an appropriate analytical framework. The fundamental differences in interface friction characteristics obtained from the proposed analytical approach and the current recommended practice for axial soil-springs (ASCE 1984) are illustrated in Figure 4.57. Note that the differences are more pronounced in dense sand as the impact from soil dilation is considerable. 167 Figure 4.57 Comparison of interface frictional behavior obtained from current guidelines and the proposed method. The evidence to further support the validity of the proposed interface friction model can be obtained from the axial pullout resistance – displacement responses observed for steel pipes buried in dilative soil. Figure 4.58 shows the results obtained by Karimian (2006) for an axial pullout test performed on a steel pipe having a diameter of 324mm. As mentioned previously, a 3.8-m long steel pipe in a soil-pipe chamber test can be considered to perform as a rigid member during a pullout test (i.e., the back end moves in harmony with front end). As such, the pullout tests performed on steel pipes can be readily normalized with respect to the pipe length (divide by the pipe length) and essentially be considered as interface frictional tests (or element level tests). The initial peak in the measured frictional resistance and the subsequent sharp decrease in pullout resistance in the test shown in Figure 4.58 are indicative of the degradation associated with the shear modulus as mentioned in Section 4.2.1.3. The sharp drop observed in interface friction cannot be explained using any other known geotechnical parameters. Author believes that these observations further support the validity of the interface friction model employed in the analyses. Fr ict io n a l f o rc e (T) ASCE (1984) Proposed method Displacement (x) 168 Figure 4.58 Axial soil resistance measured from pullout test performed on a steel pipe showing the sharp drop in interface friction at initial pipe displacements (after Karimian 2006) 4.8.1 Normalization of the interface friction Equation 4.9 for the interface friction can be rewritten in the following form with two normalized normal stress coefficients, )()( nd uk and k ( . ( ) ( ) ' dn n T k u k T = + ( (4.21) Chapter 1 Where, T' is given by tan 2 HDpiγ δ , the normal stress correction factor for soil dilation and frictional degradation is given by ( ) 2( )d d n k u H σ γ ∆ = and 01k K= + ( , respectively. It is evident from this formulation that the common practice of dividing the pullout resistance (N) by γ HD to obtain the “normalized” pullout resistance is erroneous since the frictional resistance (T) is not 0 5 10 15 20 25 30 0 50 100 150 200 250 300 Fr ic tio n a l r e si st a n ce pe r u n it le n gt h (kN /m ) Displacement at the leading end (mm) 169 independent of the pipe diameter; instead, Equation 4.21 can be effectively used to normalize the interface frictional resistance measured from “element-level” axial pullout tests on pipes. Using Equation 4.21, the relative significance of soil dilation [ ')()( Tuk nd × ] and static loading [ 'Tk × ( ] on the pipe response can be explored. Most importantly, it is evident that ')()( Tuk nd × is independent of the pipe diameter, and a constant for a given burial depth and soil density. Furthermore, by comparing the values of ')()( Tuk nd × and 'Tk × ( , it is observed that the contribution from the soil dilation is most significant in small diameter pipes and in shallow burial depths. This aspect is also evident when examining the pullout resistances obtained from Tests #2 and #3. It is also interesting to note that the pullout resistance and strains measured in Test #2 is much larger than in Test #3 despite the fact that those were conducted at a similar pullout displacement rate. Note that the variations of pullout resistance with the displacement at the leading end of the pipe for Tests # 2 and 3 are reproduced in Figure 4.59 for comparison. The computed effective overburden stress at pipe springline level ( γ H) for Test #3 is 15.5 kNm-2, and this is considerably larger than 9.0kNm-2 obtained for Test #2. Therefore, if Equation 4.1 (or 'Tk × ( ) is employed according to current guidelines, a higher pullout resistance will be predicted for Test #3 than for Test# 2, in contrast to the experimental findings. On the other hand, the proposed interface friction model (Equation 4.21) that accounts for the increase in normal stress due to soil dilation is able to satisfactorily capture the observed pullout behaviors as shown in Section 4.5. From these test results, it is evident that the increase in burial depth [or 'k T× ( ] by almost 100% is not significant compared to the increase in friction from soil dilation [ ( )( ) 'd nk u T× ]. To illustrate these aspects, Table 4.4 compares the frictional contribution from static overburden ( 'k T×( ) and from the soil dilation ( ( )( ) 'd nk u T× ) for Tests #2 and 3. It is evident from these values that the contribution from the soil dilation is much larger compared to the contribution from the static overburden. As a result, a higher frictional resistance is experienced by Test # 2, despite the smaller overburden stress. Furthermore, the larger soil unit 170 weight alone in Test # 2 is not sufficient to explain the larger pullout resistance compared to Test #3. Table 4.4 Relative contribution to the interface friction from static overburden and soil dilation Test #2 Test #3 ( )( ) 'd nk u T× (kN) 1.174 0.722 'k T× ( (kN) 0.349 0.589 Figure 4.59 Variation of pullout resistance with displacement at the leading end of the pipe for Tests #2 and 3. 4.9 Proposed analytical solution for the overall pipe performance While the interface friction model (Equation 4.21) shows the relative impact of factors influencing the interface friction (H, D, G(γ), etc) , the proposed analytical model combining the interface frictional model with the pipe stress-strain model considering the overall pipe Chart Title 0 1 2 3 4 5 6 7 8 9 0 20 40 60 80 100 Displacement at leading end (mm) Pu llo ut re si st a nc e (kN ) Test #3 Test #2 171 performance (Equation 4.15 to 4.18) would be required to compute the capacity of a pipe to withstand soil displacement (i.e., the relative pipe displacement at which the pipe suffers unacceptable strains). The following sections are intended to discuss some of the findings obtained from this analytical solution which are considered relevant to the gas distribution industry. 4.9.1 Impact of different of pipe size and burial depth on pipe displacement capacity A wide range of pipe diameters are encountered in gas distribution industry. Therefore, it is important to investigate the impact of the pipe diameter and able to select the optimum pipe size for a given situation. For example, often the question arises whether it is possible to install two smaller diameter pipes in place of a larger diameter pipe depending on the performance of relative pipe sizes. Figure 4.60 shows the maximum displacement at the ground separation point corresponding to a strain level of 5% for different sizes of pipes buried at various burial depths. It is worth noting that axial strain of 5% was selected only to form a basis for comparison, and it does not imply a suggested (or recommended) allowable strain level for MDPE pipes. In this example, the pipe is assumed to be buried in dense sand (80% relative density) and subject to rate of ground movement of 36 mm/hr, similar to the rate of axial displacement of the axial pullout tests performed by Weerasekara (2007) using the laboratory soil-pipe chamber. Furthermore, all the pipes are assumed to have a standard dimension ratio (diameter to thickness ratio) of 11. [Note that, this example presented in Figure 4.60 cannot be considered as a generalization for all the pipe and soil conditions.] 172 Figure 4.60 Predicted response of maximum displacement at the leading end of the pipe to reach 5% pipe axial strain versus burial depth in dense sand. According to Figure 4.60, having a large diameter pipe at shallower depth significantly increase the capacity of the pipe to undergo axial ground movement, however at deep burial depths, this advantage is less pronounced. For example, the ratio between the maximum allowable displacement for 219-mm and 60-mm pipes is 6.7 at a burial depth of 0.5 m, whereas the ratio reduces to 4.4 at a burial depth of 3 m. This aspect is further evident when comparing the contribution from the soil dilation [ ')()( Tuk nd × ] for each pipe diameters as discussed above. 4.9.2 Impact of different SDR values on pipe performance As defined previously, the standard dimension ratio (SDR) is the ratio between the pipe diameter to pipe wall thickness. Figure 4.61 and Figure 4.62 show displacement and strain distribution along the mobilized frictional length for two pipes with SDR of 11 and 7. As expected, the strain in the pipe with SDR of 7 (pipe with thicker wall) is much smaller than the pipe with SDR of 11 173 (Figure 4.62). In spite of the smaller strain in the pipe with smaller SDR, according to Figure 4.61, a longer frictional length will be mobilized for the same amount of ground displacement at the leading end. This aspect is an important consideration in gas distribution pipes as discussed previously. The tests performed on branch pipes by Weerasekara (2007) showed that if the anchoring resistance of branch pipe connection is mobilized, a considerable amount localized strain (or stress concentration) is observed closer to the connection rendering pipe segment closer to the connection vulnerable. Therefore, despite the higher strength in the pipe, the pipe with smaller SDR may be at risk if the anchoring resistance at a pipe connection is mobilized as a result of the longer mobilized frictional length. Note that the respective SDR values in Figure 4.61 and Figure 4.62 are selected for the purpose of illustrating the impact of SDR and do not relate to the actual SDR values used in practice. Figure 4.61 Mobilization of friction along the pipe for two pipes with different SDR values 174 A similar response to that of SDR can be expected when discussing the impact of different axial stiffness values for the pipe. For example, a HDPE pipe with a higher strength and axial stiffness will behave similar to a pipe with low SDR value, in turn, mobilizing a longer frictional length. In contrast, a MDPE pipe with smaller pipe stiffness will mobilize a smaller frictional length compared to a HDPE pipe (higher pipe stiffness) for the same amount of displacement at the leading end. Figure 4.62. Axial strain distribution along the pipe for two pipes with different SDR values. 4.9.3 Redistribution of strain In field Tests #1 and #5, with almost identical test conditions, similar pullout responses were observed up to the second phase of loading in Test # 5 (Table 4.2). However, after the relaxation phase, when the pipe is reloaded, within a small displacement (about 80 mm), the trailing end of the pipe started to move. In contrast, in Test #1, the corresponding displacement at the leading end was 125mm when the trailing end of the pipe started to move. 0 10000 20000 30000 40000 50000 60000 0 5 10 15 20 Ax ia l s tra in (µε ) Mobilized frictional length (m) SDR 11 SDR 7 175 Considering the drop in strain observed in Test #5 (e.g., Figure 4.39), it is hypothesized that the strain was redistributed along the pipe during the relaxation phase. During this process, it can be argued that more pipe length is required to distribute the frictional stresses along the pipe, even when the front end of the pipe is stationary. Subsequently, when the pipe is reloaded, the pipe response is influenced by the longer mobilized frictional length. Examining the rate of increase in pullout resistance in the third phase of loading, it is apparent that the pullout resistance is increasing at a faster rate corresponding to a longer mobilized frictional length (see the dotted lines in Figure 4.63). This further supports the notion that the mobilized frictional length would have increased when the pipe remained stationary. However, this behavior cannot be modeled using the analytical approach or the numerical model, since these methods assume an unchanged mobilized frictional length when the pipe is stationary. This process of redistribution of strain is considered desirable considering the reduction in maximum strain at the pulling end; however, in some instances, this may likely to create increased strain during reloading as a result of the longer mobilized frictional length. Moreover, additional resistance may also develop if the increased frictional length is long enough to mobilize the anchoring resistance of a neighboring pipe connection. Figure 4.63. Comparison between pullout resistance obtained from Test #1 and Test #5 Estimated trend of pullout force 176 4.9.4 Some comments on the effect of internal pressure on the pipe In this study the direct impact of internal pressure on the pipe performance was not investigated. However, it is apparent that the pipe performance will be affected by the internal pressure in two different ways. First, with the internal pressure inside the pipe, the pipe may expand its diameter, thus potentially having a similar impact as the increase in normal pressure due to soil dilation. However, the magnitude of the increase in normal stress on the pipe will heavily depend on the construction practice and operating conditions. Secondly, the stress-strain properties evaluated in this study considers only the uniaxial loading; however, with the internal pressure, the stress- strain properties resemble a biaxial loading situation. With proper material characterization subject to biaxial stress conditions, the analytical solution can be reevaluated to explore the impact of internal pipe pressure. 4.10 Closing remarks The proposed analytical solution provides a framework to link the displacement, axial strain, pullout resistance and mobilized frictional length along the pipe. Most importantly, with this analytical solution, it is possible to relate the measured ground displacement to the pipe strain and mobilized frictional length to assist the decision making on the pipe system rehabilitation or replacement aspects. These pullout tests were performed at different soil densities, pipe diameters, burial depths, displacement rates and burial lengths. As a result, pipe pullout responses were obtained for wide range of test conditions, and these pullout responses were modeled using a consistent set of input parameters. These input parameters can be obtained from independent experiments or direct measurements. The modeling of the nonlinear pipe responses observed in these ten model-scale pullout responses was possible, only because of the proper consideration of the nonlinearity in interface friction and stress-strain behaviors for the pipe material. 177 Using the analytical solution as the basis, the relative impact of interface frictional characteristics (burial depth and soil density) and pipe characteristics (pipe diameters, thickness and stiffness) were explored. With respect to interface friction, the impact of soil dilation is found to be significant in small diameter pipes and in shallow burial conditions. In a similar parametric analysis, a pipe with small SDR or large pipe stiffness is likely to mobilize a longer frictional length for the same amount of ground displacement. These findings are considered important when selecting the pipes for different burial conditions. Apart from analyzing the situations involving ground movement along the pipe axis, the proposed analytical solution can be incorporated in other analytical solutions to simulate the axial force development in instances where the initial soil loading is acting at an angle to the pipe axis. This aspect will be further discussed in Chapter 6. Knowing that the proposed analytical approach is capable of capturing the axial force development when subject to pullout, the same basic principles are applicable to explain the pullout response of planar geotextiles as shown in Chapter 5. 178 5 AN ANALYTICAL MODEL TO ESTIMATE THE PULLOUT RESPONSE OF GEOTEXTILES 5.1 Introduction Over the years, geosynthetics have become popular as a soil reinforcement to improve the tensile capacity of the soil. From an engineering point of view, the interaction between the reinforcement and soil is a key consideration in effective design of mechanically stabilized earth (MSE) walls, slopes, embankments and in foundation soils. In essence, the interface frictional resistance between the reinforcement and soil causes the stress from the soil to transfer into the reinforcements as tensile forces; this, in turn, creates a composite soil mass with an equivalent shear resistance significantly higher than that generated from soil alone. To understand the fundamentals of soil-geotextile interaction, pullout tests and direct shear tests have been used extensively as the primary tools in generating the needed experimental data. The test results and the accompanying analytical models have been developed to select the reinforcement with appropriate strength, stiffness, thickness, and surface properties for a given application. The interface direct shear tests are considered to provide a reasonable understanding of the development of interface frictional forces between the soil and reinforcement, and the results from such tests have been commonly used in estimating the pullout response of relatively inextensible soil-reinforcement (e.g., AASHTO, 1990). The argument herein has been that the mechanism that exists in interface direct shear tests at element level is fundamentally not different from the interaction that would take place between the buried reinforcement and soil mass in a mechanically stabilized soil structure. However, as noted by many scholars (e.g., Mallick and Zhai 1996), the frictional resistance from interface direct shear test is independent of the stiffness and thickness of the geotextile, whereas the pullout response of the geotextiles is influenced by these factors. In essence, the direct shear test results are representative of the pullout tests, only if the reinforcing material is relatively inextensible when compared with the 179 magnitude of relative displacement at the soil-reinforcement interface. This factor introduces a serious limitation when applying the direct shear test results to interpret the response of a given geotextile pullout test. The overall pullout response is a reflection of the combined behavior of both the properties of the geosynthetic material as well as the geosynthetic-soil interface behavior. In the last few decades, several analytical models have been developed to explain the complex interaction between soil and geotextiles. As mentioned in Chapter 2, it is important to overcome the shortcomings identified in these analytical models and to develop a new analytical solution with proper consideration given to the actual stress-strain response of the geotextile and the interface frictional characteristics. From a fundamental point of view, there are significant similarities between buried polyethylene (PE) pipes subjected to relative axial soil movement and pullout of buried extensible reinforcement such as geotextiles. In this chapter, it is shown that an analytical solution developed to capture the mechanical response of extensible polyethylene pipelines subject to relative axial soil movement can be extended to characterize the pullout characteristics of the geotextiles. The new analytical method developed on this basis is verified by demonstrating its ability to predict the response derived from geotextile pullout tests conducted by a number of previous experimental researchers. 5.2 Development of the analytical solutions for determining the pullout response of geotextiles The following section presents the derivation of the proposed analytical solution. The following discussion is limited to: (a) Pullout tests and findings on geotextiles (exclude discussions on geogrids) (b) Experiments and derivations arising from tests on geotextiles buried in soils with no apparent cohesion (discussions on geotextiles buried in cohesive soils are not included) 180 Similar to the framework presented in Section 1.2.1 for pipes subject to axial soil loading, the same three-component framework is adopted for developing the following analytical solution for geotextile. Initially, a model for frictional behavior at the soil-geotextile interface due to the relative ground movement was developed. Secondly, the stress-strain behavior of the geotextile material was modeled. Finally, the interface friction model and stress-strain model for the geotextile were combined to derive an overall performance model for the geotextile. The following sections will present detailed discussion on above aspects. 5.2.1 Factors affecting the geotextile–soil interface behavior The geotextile–soil interface behavior can be expressed in the most simplistic form, characterized as a frictional force per unit length, as follows (Jewel et al., 1984): /2 tanGTX s GSYT b Hγ φ′= (5.1) where GTXb is the width of the geotextile, H is the burial depth of the geotextile, γ is the density of the surrounding soil, and /s GSYφ′ is the interface friction angle between the geotextile and soil. As discussed below, this simplicity of Equation 5.1 is arrived at primarily because the equation does not account for some salient phenomena such as the effect of soil dilation and the frictional degradation aspects at large displacements that take place at the interface shear zone. 5.2.1.1 Effect of shear–induced dilation on planar members Similar to the response of buried extensible pipes subject to axial ground movements discussed in Chapter 4, there is evidence from pullout and interface shear tests performed on planar soil reinforcements that the frictional resistance at the reinforcement-soil interface may be several magnitude greater than the friction derived with the interface shearing takes place under constant normal stress conditions (e.g., Schlosser and Elias 1978, Ingold 1982, Alfaro and Pathak 2001 For example, in relatively compact to dense soil conditions, considerable shear-induced volumetric expansion (i.e. dilation) of soil particles can be expected within the relatively thin shear zone at the interface when buried elements are subjected to pullout type loadings. Such 181 outward expansion of the shear zone would be constrained by the surrounding soil mass, in turn, causing an increase of the normal stress on the object. In actual practice, the density in the backfill is always medium to high even in the absence of significant compaction (Schlosser and Elias 1978), thus the soil dilation is likely to be a significant factor affecting the interface friction between geotextile and soil. Few methods have been already derived to calculate this increase in normal pressure from constrained soil dilation. The elastic cavity expansion theory used in Chapter 4 for cylindrical objects cannot be employed for planar objects, thus a different analytical model is required to determine the increase in normal soil load due to soil dilation (∆σd). Wang and Richwien (2002) proposed a method to calculate ∆σd for planar members, based on elasticity and the constant volume conditions of the soil. In turn, ∆σd was expressed in terms of Poisson’s ratio of soil (ν ), the average shear stress on the surface (τ ), the lateral earth pressure at rest (K0) and peak angle of dilation (ψmax) as follows: max 0 2 (1 ) tan(1 2 )(1 2 )d K ν σ τ ψ ν +∆ = − + (5.2) Thus, the shear stress arising from the friction (τ ) can be written as follows /( ) tand S GSYHτ γ σ φ ′= + ∆ (5.3) From Equation 5.2 and 5.3, the initial pullout resistance per unit length (T) can be derived in the following form: / 0 / max 2 tan 1 [2(1 ) /((1 2 )(1 2 ))] tan tan GTX s GSY S GSY Hb T K γ φ ν ν φ ψ ′ = ′ − + − + (5.4) This equation is different from Equation 5.1 presented above and considers the impact of soil dilation at initial stages. In other words, the initial pullout resistance per unit length can be written as an extension to Equation 5.1: 182 /2 tand GTX s GSYT T Hbγ φ ′= + (5.5) where, the net increase in friction from soil dilation (Td) can be expressed as follows. /2 tan 1 GTX s GSY d HbT γ φ ′Γ= − Γ (5.6) where, / max 0 2(1 ) tan tan (1 2 )(1 2 ) s GSY K ν φ ψ ν ′+Γ = − + . Note that, however, this initial increase in frictional resistance is expected to gradually decrease with the increase in displacement due to particle wear and tear. As a result, the normal stress experienced by a particular section of the geotextile would gradually decrease (termed frictional degradation), and this aspect will be further discussed in Section 5.2.1.3. 5.2.1.2 Impact of the mean effective stress on soil dilation It is also known from experimental studies (e.g., Abdelouhab et al. 2010) that the confinement from the burial depth or the increase in the mean effective stress will constrain the amount of soil dilation. With a similar approach adopted as in Section Chapter 4 for pipes, the following equation can be obtained for the maximum angle of dilation. Note that, due to the similarity of the derivation, the complete derivation of the equation is not presented herein. ( )maxtan tan 3.8 7.1 0.25 ln 3.8DIψ σ ′ = − − (5.7) This expression for maxtanψ is substituted in Equation 5.4 to obtain an expression for the increase in pullout resistance due to soil dilation. 183 5.2.1.3 Frictional degradation Similar to the argument presented for pipes, geotextiles can experience relatively large relative displacements at the interface during pullout depending on the extensibility of the geotextile material. This would lead to progressive development of friction along the geotextile, and different parts of the geotextile (i.e., segments along the length of a buried geotextile extending into a soil mass) will undergo different levels of interface shear displacements. At a given instance during pullout, because of the extensibility, the geotextile segments in the vicinity of the pulling end will undergo larger displacements than those at the trailing end; thus, it is unreasonable to attribute the peak frictional forces to those portions of the geotextile that have already experienced large displacement levels. Therefore, accounting for the impact of frictional degradation is considered of increased significance in the case of extensible soil reinforcements having the potential to undergo large displacements. 5.2.2 Stress-strain behavior of geotextiles As discussed in Chapter 2, the pullout response of geotextiles is highly influenced by the stress- strain relationship. Since most polymer-based materials exhibit viscoelastic material properties, the stress-strain response of the geotextiles is expected to be nonlinear, strain rate and temperature dependant (Koutsourais et al. 1998). Similar to the axial pullout tests on MDPE pipes, the hyperbolic stress-strain model is used to represent the nonlinear stress ( uσ ′ ) versus strain ( u′ ) behavior as follows. 1u uEini u σ η′ ′ = ′+ (5.8) where Eini is the initial tangential modulus of the soil reinforcement and η is a material constant that influences the shape of the stress–strain response. However, unlike MDPE pipes, in which considerable number of detailed experiments has been performed to find its stress-strain characteristics, very few studies have been performed to date to characterize the stress-strain 184 behavior of geotextiles. In addition, the strain rate dependant behaviors are rarely published for geotextiles used in the pullout testing. As a result, for the following derivation, it was decided to exclude the strain rate dependant hyperbolic parameters from the derivation, instead Eini and η are assumed to be constant for a particular test. As noted in Section 3.4.2, the strain rate in a pullout test is likely to change even if the pullout is performed at a constant pullout rate. Nevertheless, a parametric study performed on MDPE pipes showed that the sensitivity of the pullout response to strain rate variation occurring during the pullout process is not significant, thus the error associated with this assumption is small. If the strain rate dependant hyperbolic constants are known, these could be incorporated in the derivation as presented in Section 4.2.3for the pipe pullout tests as appropriate. 5.2.3 Analytical formulation of the geotextile–soil interaction response This section presents the details relating to the development of analytical solution to determine the response of a buried geotextile subjected to pullout. The soil-geotextile interface behavior and the non-linear stress-strain behavior discussed in Sections 5.2.1 and 5.2.2 would provide the basis for the analytical solutions derived herein. Considering the level of relative displacement and the response at the interface at a given instance, the portion of the geotextile along which interface friction has already mobilized can be considered to comprise of two zones as shown in Figure 5.1a and Figure 5.1b. The geotextile element lengths in the first zone (pre-peak zone) have just begun to experience movement relative to the soil mass (u). In other words, this corresponds with the pre–peak zone of mobilized soil interface friction (see Figure 5.1). In the second zone (post-peak zone), geotextile element lengths experience post–peak values for the soil interface friction. 185 Figure 5.1 (a) Mobilized frictional lengths corresponding to different degrees of relative displacements; (b) Assumed response of the unit interface friction (T) with relative displacement for geotextile–sand interface 5.2.3.1 Pre–peak zone Considering force–equilibrium of a unit length of the geotextile in the pre–peak zone, where the geotextile displacement (u) is less than the pre–peak displacement (i.e. ue), the following second order differential equation can be derived. 1 d d dx dxini e u uE A T u = (5.9) 186 A is the cross-sectional area calculated as the product of width ( GTXb ) and thickness ( GTXt ) of the geotextile. As observed in interface shear tests (O’Rourke et al. 1990), it is reasonable to assume that the pre–peak displacement (ue) is generally limited to about 0.3 - 0.8 mm. This displacement corresponding to the peak will depend on the roughness of the surface, normal stress, size and shape of the soil particles. In smooth surfaces and at low overburden stresses, this peak is observed at a smaller displacement (Hong et al. 2003). However, it is found that analytical results are not significantly affected by the value selected for ue, especially when modeling large displacements. It is also reasonable to assume that the tensile modulus of the geotextile material within this zone to be constant, since the induced strains are not significant at these small displacement levels (i.e., use initial tangential modulus (Eini) to represent the deformation characteristics of the geotextile under small displacement levels). The corresponding solution for Equation 5.9, the relative displacement (u0) is given by 0 0 0 x xu C e C eγ γ−′= + (5.10) where, eu λγ η = , 1 ini T E A ηλ = and C0, C΄0 are constants. As shown in Figure 5.1b, T1 is the peak value of unit interface friction developed in a small segment of geotextile that falls within the pre–peak zone, and the location of T1 corresponds to x = 0 (in Figure 5.1a). The value of T1, which accounts for the initial soil dilation, can be determined using Equation 5.4. 5.2.3.2 Post–peak zone For a given displacement (u) of the leading end, the length between x = 0 and the leading end of geotextile will experience the post–peak frictional behavior (see Figure 5.1b). Moreover, the stress–strain response of the geotextile within this zone can be nonlinear owing to the large strain levels. 187 As shown in Figure 5.1, for a given displacement (u>ue), the interface frictional force Tnis assumed to be mobilized over the geotextile segment between x = xnand x = xn+1. Similar to the formulation of the pre-peak region, the following equation can be derived considering the force- equilibrium of an element length within the post–peak zone. Note that this fundamental equation is similar to the equation derived for pipes. 3 2 2 d1d dx ddx 1 dx n ini u Tu uE A η η + = − (5.11) The corresponding solution which is the relative displacement (un) is obtained as follows 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (5.12) Where Cn and C΄n are constants that depend on the geotextile and interaction properties selected for the model and ( ) 124 1 4n n n n nz x Cλ λ= − + − . The strain along the geotextile is obtained by differentiating Equation 5.12 with respect to x. (1 )1 1 2 ( ) n n n n n z u x Cη λ − ′ = − + (5.13) The corresponding force at the leading end of the geotextile, when u = uncan be obtained by substituting these strain and modulus relationships in the following '( )n n nN E u A u′= × × (5.14) 5.2.3.3 Solution approach and boundary conditions The objective herein is to use Equation 5.4 to compute the values of T as appropriate, and then use these values in Equations 5.12 through 5.14. The same step-by step solution procedure (i.e. u 188 = un, un+1 etc.) presented in Section 4.2.4.1is used to obtain the key performance parameters by using the Equations 5.12 to 5.14 as the basis. Accordingly, the unknowns (Cn, Cn΄, xn) in Equation 5.12 can be obtained by considering the following boundary conditions for the nth element: (i) at x = xn, un+1 = un (the displacement is continuous in (n+1)th and nth element); (ii) at x = xn, u'n+1 = u'n (strain is continuous in (n+1)th and nth element); (iii) at x = xL, N = NL (force at the pulling end of the geotextile). Although the analytical formulations are presented for both pre-peak and post-peak zones to illustrate the entire interaction behavior, the overall pullout response is dominated by the interaction in the post–peak zone. By examining the strain development along the geotextile from pullout tests, the strain generated from the interaction in the pre-peak region is practically insignificant. Therefore, it can be argued that it is reasonable to solve the equations of the post- peak zone by using approximate values for strain and mobilized frictional length as the boundary conditions for the first element in the post-peak zone. This will simplify the formulation by avoiding any calculations in pre-peak zone. 5.3 Validation of the analytical solution Already published data available from several geotextile pullout tests performed by eight different researchers provided an opportunity to validate the above analytical solution. The reported properties of the soil and geotextile are given in Table 5.1 and Table 5.2, respectively. 189 Table 5.1 Details of sand used in pullout tests Type of sand (description) Friction angle (degrees) Unit weight (kN/m3) Relative density (%) Fannin & Raju (1993) Rounded silica sand with Cu = 1.5, d50 = 0.9 mm 28 (const. vol) 17.2* 85-90 Konami et al. (1996) Decomposite granite sandy soil 37.2 (assumed peak) 20.6 NR Tzong & Cheng- Kuang (1987) Uniform Ottawa sand with Cu = 1.43, Gs = 2.65 37 (peak) 16.8 70 Racana et al. (2003) Fill sand † 33 (assumed peak) NR NR Bakeer et al. (1998) Compacted clean river sand† NR NR NR Ali (1999) Sabkha sand with Cu = 1.88,Cc= 0.83 and d50 = 0.27mm 46 (peak) and 39 (const. vol) 17.1* 75 Alobaidi et al. (1997) Leighton Buzzard sand with Gs = 2.65 NR 19.2 NR *Calculated from given data NR: Not reported 190 Table 5.2 Details of geotextiles used in pullout tests Type of geotextile Mass per unit area (g/m2) Stress-strain related data Interface friction angle Fannin & Raju (1993) HDPE geotextile (rough surface) NR 29 kN/m (yield), 8kN/m (break strength) 45 (calculated from pullout tests) Konami et al. (1996) Polyester geotextiles with polyester coating † NR 344 MPa at 2% 633 MPa at 5% 190 MPa (nom. strength) 38.2 (from direct shear tests) Tzong & Cheng-Kuang (1987) Needle punched Polyester NR 1.16 kN/m at 85% strain NR Racana et al. (2003) Non-woven Polyester 330 20.2 kN/m at 30% strain NR Bakeer et al. (1998) Woven Polyester (high strength) 2034 508 kN/m (assumed as strength) 28 (calculated from pullout tests) Ali (1999) Needle punched non-woven Polypropylene 400 20 kN/m at 70% strain 46-28 (calculated from pullout tests) Alobaidi et al. (1997) Woven Polypropylene (extruded) 120 17.3 kN/m at 28.5% strain 29-13 (calculated from pullout tests) *Calculated from given data NR: Not reported The input parameters selected for the proposed analytical model are listed in Table 5.3. Considering the parameters listed in 0, estimates for the parameters listed below were obtained using rational assumptions/approaches, since this information was not directly provided by many researchers in their test programs: (i) parameters Eini and η to describe geotextile material stress- strain behavior as per Equation 5.8; and (ii) the friction angle for the soil-geotextile interface ( /s GSYφ′ ). The rationale for the approaches used for the selection of input parameters is further elaborated in Sections 5.3.1and 5.3.2. 191 Table 5.3 Input parameters for the geotextile pullout tests Overburden (kPa) Thickness (mm) Width (m) Length (m) Eini (MPa) η /S GSYφ′ (degrees) Fannin & Raju (1993) MT12 MT08 MT04 12 8 4 2 0.5 0.965 400 11 21 Konami et al. (1996) PW10@4m PW5@2.4m PW3@1.6m 82 50 33 5 3 2 0.09 0.09 0.085 5.5 5.5 3.5 640 0.01 21 Tzong & Cheng-Kuang (1987) 30 3.18 0.45 0.3 4 0.01 21 Racana et al. (2003) 2.8 11.8 17.8 22.8 27.8 1.9 0.09 1 220 4.2 21 Bakeer et al. (1998) 24.7 3 2.06 7.32 1100 0.01 21 Ali (1999) 4.5 8.0 16.0 4 0.1 0.38 7.5 0.01 21 Alobaidi et al. (1997) 20 50 100 200 0.3 0.2 0.2 85 0.01 21 5.3.1 Hyperbolic parameters to represent geotextile material stress-strain response Hyperbolic parameters, Eini and η to describe the stress–strain behavior of geotextiles were not available from the reported information on the pullout testing programs considered herein. Despite the fact that modeling of failure is not the primary interest for many engineering applications, as may be noted from 0, only the ultimate tensile strength is reported in all the previous cases,. For those cases where data are available, the hyperbolic parameters were selected such that estimated curve will encompass the reported stress-strain values for the respective geotextile (e.g., Konami et al. 1996). In the absence of reliable direct stress-strain data to describe the geotextile response, the hyperbolic parameters were selected such that modeled pullout behavior would closely match one of the actual pullout test results from the 192 given series of pullout tests. Once these basic input parameters are selected, predictions for the remaining pullout tests were obtained only by changing the variables that represent a change in condition or dimension (e.g. overburden pressure, thickness and width of geotextile, etc). Furthermore, when selecting hyperbolic parameters, consideration was given to typical strength values of the geotextile material (e.g. polypropylene, polyester, etc). 5.3.2 Soil-geotextile interface friction angle ( /s GSYφ′ ) Table 5.2 shows some of the interface friction angles reported for the pullout tests considered for this study. These reported friction angles had been directly back-calculated from the pullout test data considering an average friction distribution along the geotextile. However, a careful examination of this data revealed that, for the reasons presented below, extraction of a large strain friction angle from this data was not possible. As alluded earlier, the pullout resistance obtained from a pullout test is not a sole representation of the friction angle, but it is the combined response of the geotextile material behavior and the interface friction response reflected in the pullout response. For example, for the same interface friction values, a thinner geotextile material will give a smaller pullout resistance whereas a thicker geotextile will give rise to a larger pullout resistance. Similar response can be observed when considering geotextiles with different stiffness values. Even if the interface frictional response is measured using a rigid element to negate the impact of the geotextile stress-strain properties, the friction angles measured in these tests may even be overestimated due to potential increase in normal stress due to soil dilation. Unless the normal stress is directly measured at the interface, it is difficult to distinguish the components contributing to the increase in normal stress and the interface friction angle. As described earlier, in the proposed analytical model, the intention is to account for the increase in normal stress acting on the geotextile due to soil dilation as a separate term in the equation for computing axial load. As such, it is important that the interface friction angle used in the equation represents only the interface frictional component. 193 These considerations implied that it would be prudent to use a large-strain interface friction angle for the present analysis – i.e., use an interface friction angle that would not be influenced by the impact from soil dilation. With this thinking, it was recognized that data derived from ring shear tests would provide one of the best ways of estimating the large-displacement interface friction angle values. Rinne (1985) performed several ring shear tests on smooth and rough HDPE geomembrane surfaces. Tan et al. (1998) performed both ring shear and direct shear tests on eight different geotextiles. From these experiments on soils with no apparent cohesion, they observed that the large-displacement interface friction angle may range from about 15 degrees at smooth surfaces (e.g. smooth steel-Ottawa sand interface), and greater than 30 degrees in very rough surfaces. Based on the tests performed by Tan el al. (1998), for most geotextiles, the interface friction angle with sand was found to be around 25°, regardless of the nominal mass of the geotextile and rate of shearing. In some experiments by Tan et al. 1998, a slightly larger interface friction angle was observed at low overburden stresses; however, this was considered to be contributed by the increase in normal stress due to soil dilation. Considering the possibility that even the values may have been subject to some degree of increase in normal stress due to soil dilation, it was judged that a slightly smaller friction angle of 21° would be suitable for the current analytical modeling. Although the large-displacement interface friction angle may depend on the angularity of the sand particles, density of the soil, type of geotextile, authors believe that the selected interface friction angle of 21º is a reasonable estimate for the pullout tests considered herein. When considering the other input parameters, the coefficient of lateral earth pressure (K0) required in Equation 5.4was estimated using the reported internal friction angle of backfill soil in Jaky’s (1944) formula. The Poisson’s ratio (ν ) of the soil was selected to be 0.33 unless otherwise stated. However, it was noted that the predicted performance during pullout testing is not significantly sensitive to the values of K0 and ν . The other information required for the computations is the value of the additional normal stress due to soil dilation (∆σdc) at any given displacement. Similar to the behavioral pattern assumed for pipes, the initial dilation at the interface is assumed gradually diminish within about 50 mm 194 of axial displacement. Accordingly, the additional normal stress due to soil dilation (∆σdc) at any given displacement is calculated by linearly interpolating between following two ∆σdc values: (i) when u = ue, the value of Τd is at the peak value as calculated from Equation 5.6, and (ii) when u = 50 mm, the value of Τd is 0. The values of these parameters Eini, η, and /s GSYφ′ estimated using the above approaches for the eight test programs are presented in Table 5.3. 5.3.3 Comparison of results from the analytical approach with reported experimental observations Validation I – Experiments by Fannin and Raju (1993) Fannin and Raju (1993) has reported results from three pullout tests performed using a 1.30 m x 0.64 m x 0.6 m box with rounded silica sand as the backfill soil. The relative densities of the backfill ranged from 85 to 90%. The tests were conducted with geotextiles having a length of 0.965 m. The pullout resistance–displacement relationships reported by Fannin and Raju (1993) from their measurements are shown in Figure 5.2 Analytical estimationsfor pullout tests performed by Fannin and Raju (1993) for tests MT12, MT08 and MT04. The predicted pullout resistance-displacement relationship from the analytical solution are also superimposed using “lines” in Figure 5.2 for tests MT12, MT08 and MT04. It is worth noting that, once the parameters Eini, η, and /s GSYφ′ have been decided as discussed earlier, the predictions for the remaining two tests were obtained by only changing the overburden stress which is the solitary variable among these tests. As may be noted, a very good agreement between the test observations and predictions can be noted. In a pullout test, when the interface friction has mobilized along the full length of the geotextile, the trailing end of the geotextile would begin to move. Moreover, the mobilized frictional length at this point would be equal to the length of the geotextile. Once the trailing end of the geotextile had begun to move, the total pullout resistance will begin to drop with increasing relative displacements at the geotextile-soil interface. In actual pullout tests, it is difficult to keep track 195 of the mobilized frictional length using direct measurements, thus above explanation provides an opportunity to compare the values obtained for mobilized frictional length from analytical and experimental observations at the peak pullout resistance. It is noteworthy that Mallick et al. (1997) observed that in the pullout tests, the trailing end of the geosynthetic begins to move exactly at the point when the pullout resistance records a peak value, further validating the above explanation. Figure 5.2 Analytical estimationsfor pullout tests performed by Fannin and Raju (1993) Fannin and Raju (1993) conducted their tests until the trailing end of the geomembrane had started moving; as such, the mobilized frictional length at this point would be equal the length of the geotextile. With this knowledge, the computations using the analytical solution for this case were made until the computed frictional length was equal to the length of the geotextile - i.e., of 0.965 m. This means that the coordinates of the end points of the predictions shown in Figure 5.2for each of the tests would represent the total pullout resistances and leading end displacements corresponding to a maximum frictional length of 0.965 m. As may be noted, there 0 2 4 6 8 10 0 5 10 15 20 25 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) MT12 MT08 MT04 196 is a good agreement between the end points of the actual pullout test and the end of the predicted curve. This ability to capture the forces and displacements at the maximum mobilized frictional length further suggests that the proposed analytical approach has the ability to capture the overall response of the buried geotextile during pullout. Validation II – Experiments by Konami et al. (1996) Konami et al. (1996) performed a series of field pullout tests on geotextile reinforcements, named PW-3, PW-5 and PW-10. Unlike for Fannin and Raju (1993), the thickness and width of the polymer strips were different between tests as shown in Table 5.3. The overburden pressures varied between 33 - 82 kPa as the geotextiles were buried in Masado sand (decomposed granite sandy soil). In absence of reported data, the relative density was assumed to be 75%. Figure 5.3shows the prediction of pullout resistance–displacement response for three different tests presented by Konami et al. (1996) for different overburden stresses. In each test, the pullout resistance and displacements were obtained by substituting the appropriate thickness, width and overburden stresses without any manipulation of the remaining input variables. Besides the above three tests, Konami et al. (1996) also reported two other tests: PW3 geotextile buried at 0.8 m and PW5 geotextile buried at 3.2 m depth. However, the pullout responses of these two tests were similar to PW3 geotextile tested at1.6 m depth and PW5 geotextile tested at 2.4 m, respectively. Since obtaining almost the same pullout resistance for polymer strips buried at two different depths is inconsistent, these latter two tests were not considered accurate, and their results were not considered for validations. According to the analytical solution, the computed mobilized frictional length (corresponding to the leading end displacement) at the end of the test was 3.94 m in PW10 test, and 3.78 m for PW5 test. Since these mobilized friction lengths are less than the total length of the geotextile (i.e. 5.5 m), as explained in the previous section, no peak is expected – i.e., the full length of the friction is not mobilized even at the end of the test. In contrast, the length of the geotextile was 3.5 m in PW3 test and a peak in pullout resistance was observed at a leading end displacement of 110 mm. The mobilized frictional length obtained from the analytical solution corresponding to 197 this displacement is about 3.3m. This again suggests that the predictive approach seems to be capturing the soil-geotextile pullout mechanism well. Figure 5.3 Analytical estimations for pullout tests performed by Konami et al. (1996) Validation III – Experiments by Tzong and Cheng-Kuang (1987) Tzong and Cheng-Kuang (1987) performed pullout tests on needle-punched non-woven geotextiles buried in Ottawa sand of relative density 70% (unit weight of 16.8 kN/m3). The tests were performed in a soil box of length of 1.2 m, width of 0.6 m and height of 1.45 m. The predictions from the analytical solution (made using data presented Table 5.3for the only reported pullout test) are presented in Figure 5.4. As may be noted very good agreement between the prediction and measured values can again be found. 0 5 10 15 20 25 30 0 50 100 150 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) PW-10 @ 4m PW-5 @ 2.4m PW-3 @ 1.6m 198 Figure 5.4 Analytical estimations for pullout tests performed by Tzong and Cheng–Kuang (1987) Validation IV – Experiments by Racana et al. (2003) Racana et al. (2003) performed five pullout tests in a soil box of length of 1 m × width of 1 m × height of 0.75 m. All tests were performed using the same nonwoven polyester geotextiles buried in a sand backfill but at five different overburden stresses. In the absence of any soil density data, a relative density of 70% is assumed for the sand backfill. In Figure 5.5, the predictions from the analytical solution for all five tests were obtained by changing only the overburden stress and plotted up to the peak pullout resistance. For the tests with overburden stresses ranging from 11.8 to 27.8 kPa, the mobilized frictional lengths obtained from the analytical solutions ranged from 0.95 –1.00 m, in close agreement with the actual length of the soil box of 1.0 m. However, in the test with 2.8 kPa, the mobilized frictional length corresponding to the peak was 1.33 m. 0.0 0.5 1.0 1.5 2.0 2.5 0 5 10 15 20 25 30 35 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) Tzong and Cheng (1987) 199 Figure 5.5 Analytical estimations for pullout tests performed by Racana et al. (2003) Validation V – Experiments by Bakeer et al. (1998) Bakeer et al. (1998) performed few large scale pullout tests on geotextiles made out of woven high strength polyester geotextiles. The length of the geotextile was 7.32 m and buried in compacted clean sand at a depth of 1.37 m. In absence of density data for backfill, the unit weight was assumed to be 18 kN/m3 considering that the sand was well compacted. The corresponding relative density was assumed to be 85%. Figure 5.6shows the prediction for the pullout resistance and the displacement up to the peak or until the friction is mobilized in the entire length. At this point, the frictional length obtained from the analytical solution was 7.43 m compared to the actual mobilized frictional length of 7.32 m. Bakeer et al. (1998) reported that the strain at this level of displacement was about 5%, whereas, the predicted strain from the analytical model was 8%. This example indicates that the analytical solution can be employed in situations involving very large lengths of geotextiles. The test was conducted to a significantly 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 Displacement at leading end (mm) Pu llo ut re si st a n ce (kN ) 2.8 kPa 11.8 kPa 17.8 kPa 22.8 kPa 27.8 kPa 200 large displacement, in excess of 300 mm. It is worth mentioning that the impact of the frictional degradation is clearly evident in such tests involving large displacements. Figure 5.6 Analytical estimations for pullout tests performed by Bakeer et al. (1998) Validation VI – Experiments by Ali (1999) Ali (1999) performed some pullout tests on sand – nonwoven needle punched geotextile interface using a box of 0.5 m in length and 0.1 m in width. Although, three geotextiles were used, Figure 5.7 shows only the test results obtained for A-400 geotextiles. The other two geotextiles have shown a large amount of necking during pullout, therefore excluded from the present comparison. Figure 5.7 shows the pullout resistance-displacement relationships at three different overburden pressures. Although, the results indicate reasonably good agreement, it was noted that the mobilized frictional lengths obtained by the analytical solution were almost double in length when the peak resistance was mobilized. It should be noted that these pullout tests were performed with a steel plate attached to the geotextile at the pulling end. Since the steel plate was also embedded in sand, the resistance of the plate was subtracted from the total pullout resistance. It is possible that such adjustments may have led to a smaller pullout resistance at 0 100 200 300 400 500 600 700 0 50 100 150 200 250 300 350 Displacement at leading end (mm) Pu llo u t r e si st a n ce T (kN ) Bakeer et al (1998) 201 initial displacement levels as observed in Figure 5.7 and impacted the mobilization of the friction along the geotextile. Figure 5.7 Analytical estimations for pullout tests performed by Ali (1999) Validation VII – Experiments by Alobaidi (1997) Alobaidi et al. (1997) analyzed the pullout test results obtained by Eltayeb (1986) using a box of 0.75 m in length × 0.37 m in width × 0.5 m in depth. Leighton Buzzard sand was used as the backfill for the two types of geotextiles, named as Geotextiles A and B. Figure 5.8 presents the test results only for Geotextile B, and it was impossible to predict the test results for Geotextile A since the thickness of the respective geotextile was not given. Four tests on Geotextile B were performed at overburden stresses of 20, 50, 100 and 200 kPa. The relative density of the soil was assumed to be 75%. At very large overburden stress levels (e.g., 200 kPa) and with a geotextile of thickness 0.3 mm, it is judged that there is more opportunity for anomalies, especially if the soil particles protrude into the geotextile. In such 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 40 50 60 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) 16 kPa 8 kPa 4.5 kPa 202 situations, the predicted behavior for the pullout response would deviate from the actual test results when the same material properties are used as in other three tests to predict the pullout response. Furthermore, Alobaidi (1997) noted that in the test with 200 kPa overburden, the geotextile was at break point. The pullout responses were plotted only up to a mobilized frictional length of 0.2 m. As in the previous cases, a close match was observed between the peak points of the pullout responses and the predictions up to the length of 0.2 m. Figure 5.8 Analytical estimations for pullout tests performed by Alobaidi (1997) 5.3.4 Performance charts for geotextiles Using the proposed analytical solution as the basis, simplified geotextile performance charts were developed to estimate the axial strain in the geotextile for a known relative displacement (u) between the soil and geotextile. The charts were derived considering the potential use by practitioners as a simple tool; as such, charts were derived assuming typical values for the input variables that have smaller impact on the performance of the geotextile. For example, the values selected to describe the frictional degradation behaviour,K0 and Poisson’s ratio of the soil are 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 5 10 15 20 25 30 Displacement at leading end (mm) Pu llo u t r e si st a n ce (kN ) 20 kPa 50 kPa 100 kPa 200 kPa 203 found to have significantly smaller impact compared to other input parameters. In that context, to represent the frictional degradation, the increase in normal stress due to soil dilation was assumed to be linearly degrading and become negligible at a displacement of 50mm. The Poisson’s ratio of the soil was assumed to be 0.33 and the lateral earth pressure coefficient was assumed to be 0.5. However, it should be noted that the detailed analytical method presented in 5.2 considers all the influencing parameters in geotextile-soil interaction, and any desired value can be assigned these input parameters. Using these simplifications in the proposed analytical formulation, it is observable that a unique relationship between uλ and axial strain can be obtained for a given value of η as shown in Figure 5.9. For a selected η value, this relationship remains unique up to a strain level of about 5% for variety of input parameters. As observed in the performance chart, λ is recognized as a key parameter that governs the pullout resistance behavior along the geotextile. The value of 1 ini T E A ηλ = has been already defined earlier in Equation 5.12, and it accounts for the peak interface frictional resistance (T1) and material properties of the geotextile (Eini, η and A). The performance chart is derived for λ corresponding to the peak value of T calculated from Equation 5.4. 204 Figure 5.9 Simplified chart for determining performance of buried geotextiles under pullout conditions. Note: u is measured in millimeters (mm) and remaining length measurements are in meters (m). It is equally important to know the mobilized frictional length (x) to decide whether the geotextile anchoring length is sufficient to accommodate a given relative displacement of the geotextile. The following simplified equation based on the detailed analytical solution is proposed to obtain the mobilized frictional length along the geotextile for a known relative displacement. ˆˆ ( )bax uλλ= (5.15) Where, ˆ ( 0.007 0.713) 0.003 0.082a η λ η= − + + + ˆ (0.128 4.105) 0.007 0.518b η λ η= − − + 205 This equation was derived by curve fitting the results obtained for x and u from the detailed analytical approach (Equation 5.12). If u is known, it is difficult to obtain x directly from Equation 5.12. Therefore, Equation 5.15 is proposed as an alternative to Equation 5.12. Note that in Figure 5.9 and Equation 5.15, u is measured in millimetres (mm) and the remaining length measurements are in meters (m). The author is of the opinion that the proposed charts and equations would be readily usable by practicing engineers since it would eliminate the detailed computations involved in the analytical solution. In summary, the following steps are to be followed to evaluate the performance of the geotextile subject to allowable displacement of the reinforced structure. (a) First, the peak interface friction, T1 is calculated from Equation 5.4 based on the soil properties and the burial conditions. (b) With the knowledge of T1 calculate 1 ini T E A ηλ = for the selected geotextile material. (c) From Figure 5.9, for a selected (or allowable) serviceable displacement of the reinforced structure (u), find the strain in the geotextile corresponding to the value of η selected for the geotextile. This will indicate whether the selected geotextile has the strength capacity to accommodate the serviceable displacement requirement of the reinforced structure. (d) To obtain the mobilized frictional length of the pipe, first, calculate the constants a andb for the corresponding value of η. Then, calculate the mobilized frictional length (x) for the selected serviceable displacement limit (λu) from Equation 5.15. Depending on the stiffness of the geotextile, this will provide the minimum anchoring length required for the geotextile to accommodate the selected serviceable displacement. 5.4 Discussion The proposed analytical approach provides a framework to link the displacement, strain, pullout resistance and the mobilized frictional length of the geotextile. With the knowledge of any single 206 parameter, the remaining three parameters can be obtained from the solutions presented in Equation 5.12 to 5.14. Based on the analytical solution, it is also observed that the pullout response is significantly affected by the properties of the geotextile material (i.e. extensibility, stress-strain modulus, and the load carrying capacity or the thickness). Besides these properties, the overburden stress will also have a significant influence on the performance of the geotextile. However, it is not possible to consider each parameter and analyze to obtain the optimal value for these parameters for every field situation, since the overall behavior is governed by the interaction of all these parameters. 5.4.1 Advantages in the proposed analytical solution The following key improvements were noted in the proposed analytical solution, when compared with the existing analytical methods: (a) The proposed analytical solution employs a more refined interface friction model to overcome some of the shortcomings in the current practice. With the consideration of the increase in normal stress due to soil dilation, impact of mean effective stress on soil dilation and the frictional degradation aspects, the proposed analytical model represents a more realistic interface friction behavior. Based on logical reasoning, it can be stated that if the pullout resistance is computed without considering the frictional degradation aspects, it will lead to an overestimation of the pullout resistance. In a similar manner, if the soil dilation is not accounted, this will also lead to underestimation of the pullout resistance. (b) In the proposed analytical solution, a consistent set of input parameters were employed to predict the entire pullout response behavior of the geotextile. For example, once the basic input behavioral parameters have been selected, predictions for multiple tests (e.g., Fannin and Raju 1993; Racana et al. 2003, Konami et al. 1996, Alobiadi et al. 1997 and Ali 1999) are obtained only by changing the variables that represent a change in condition or dimension (e.g., overburden stress, thickness and width of the geotextile). For example, considering the pullout tests performed by Racana et al. (2003), after obtaining a reasonable match between the pullout 207 resistance - displacement relationship for one pullout test, the remaining four pullout behaviors were obtained just by changing the overburden pressure. The ability to model multiple tests with one consistent set of input parameters gives further confidence on the selected values as input parameters. (c) All the input variables associated with the analytical solution proposed herein has a physical meaning and directly obtainable using experimental measurements, as appropriate(e.g. thickness, initial Young’s modulus, K0 etc). Therefore, any value used for the simulation of the pullout test results can be validated through independent experiments (not essentially pullout tests). As discussed previously, in analytical models that are based on the assumption of linear elastic material behavior and simplified interface characteristics, a good agreement between the actual tests and model was only achievable by using varying input parameters depending on the level of displacement. It had been difficult to model multiple pullout tests with consistent set of input parameters. For example, values for ks (interface stiffness), α, β and Ertr were selected by Gurung and Iwao (1999) to match the pullout responses. However, in their approach the entire pullout – displacement behavior was not able to model with one consistent set of input parameters. Perkin and Cuelho (1999) changed values for Gi, pψ and rψ until good match was obtained for the pullout tests. (d) The good agreement between the predicted and the actual mobilized frictional lengths in the present model is also an indication of the accuracy of the analytical model (exception: tests conducted by Ali 1999). This is also an essential aspect in estimating the anchoring lengths for geotextiles. (e) A step-by-step solution approach is developed herein using the closed-form solution as the basic equation. The solution is easier to implement in a spreadsheet environment, compared to other methods that relies on numerical methods (e.g. finite difference) to obtain the solutions for the basic differential equation e.g. Gurung and Iwao (1999), Madhav et al. (1998), Perkin and Cuelho (1999).. 208 5.4.2 Consideration for correct interpretation of data from pullout tests As noted in the introductory section, many ambiguities exist when attempting to interpret the data observed from geotextile pullout tests. Similar to the situation in pipes, many of difficulties in interpretation of the pullout test results can be reconciled based on the fundamental differences that exist between the “model-scale” and “element-level” tests. For example, pullout tests conducted using rigid geotextiles can be considered equivalent to element level tests, and the overall response can be argued to represents only the interface frictional behavior. In contrast, if the geotextile undergoes considerable axial deformation, it cannot be regarded as an element level test; rather, it would be a model-scale test and where both interface and geotextile stress- strain properties will influence the response. For example, based on the pullout tests conducted on extensible geotextiles, Schlosser and Elias (1978) stated that apparent friction increases with increasing length of the geotextile. From a fundamental point of view, the interface friction behavior should remain same at element level; however, a longer mobilized length could increase the pullout resistance measured at the pulling end. Unlike in inextensible geotextiles, the friction along an extensible geotextile will develop progressively, with the front end of the geotextile reaching very large strains while the rear end may not even feel the presence of the pullout effect (Mak and Lo 2001; Long et al. 1997). Thus, selecting an entire length of the geotextile to calculate the unit frictional force developed would be erroneous. Mallick et al. (1997) and Lo S-C (1998) stated that the there is an ambiguity on how the pullout resistance would change with the thickness and the stiffness of the geotextile. This confusion is a direct result of the assumption that the thickness or stiffness of the geotextile would not influence the element-level pullout tests performed on rigid geotextiles. It is important to note that the stress in the geotextile will affect the pullout response; as such, the thickness and stiffness of the geotextile will significantly influence the response of model-scale pullout tests. Sobhi and Wu (1996) noted that the stiffness of the reinforcement would affect the displacement and the mobilized frictional length along the reinforcement in pullout mode. The proposed analytical solution provides the analytical background to explain why a stiffer material mobilizes a longer 209 length while a softer extensible material requires greater displacement to mobilize the friction up to the same length, and the reason for two significantly different strain distributions in these two situations. In addition, Boyle et al. (1996) reported that the rate of displacement can also have an impact on the pullout response adding another dimension to the complexity of the soil-reinforcement interaction. However, according to the model presented herein, it can be argued that the rate dependant behavior is caused by the viscoelastic stress-strain response observed in polymer based geotextiles. This behavior can be modeled using a similar approach to that presented for pipes in Chapter 4. 5.4.3 Soil dilation in planar and cylindrical objects It is also interesting to note that the axial friction model derived for the planar members (Equation 5.4) is considerably different from the cylindrical members described in Section 4.2.2 (Equation 4.9). Most notably, it is evident that Equation 5.4is independent of the shear modulus, whereas the frictional resistance of a cylindrical object is dependent on the shear modulus of the soil. It is worth to remind that a sharp drop in frictional resistance was observed in pullout tests performed in rigid cylindrical objects, which can be attributed to the shear modulus degradation observed at small strain levels. In absence of shear modulus term in Equation 5.4, similar sharp drop in frictional resistance is not expected to be observed in pullout tests conducted on rigid planar members. For example, Figure 5.10 shows the geotextile response obtained by Abremento (1993) for the pullout tests performed on buried steel straps. 210 Figure 5.10 Pullout characteristics of planar steel straps buried at four different overburden stresses (after Abremento 1993) 5.4.4 Other factors influencing the pullout response of geotextiles The following sections are presented to address the impact of: (a) change in thickness of geotextile due to axial strain; and (b) change in thickness of geotextile due to overburden stress. These factors can be accounted in the proposed analytical formulation with slight modifications to the solutions approach. 5.4.4.1 The impact of the axial strain on the variation of the thickness of geotextile In the proposed analytical solution, the pullout response is sensitive to the thickness of the geotextile. However, at large strain levels, the reduction in thickness of the geotextile due to Poisson’s effect may become significant. Thus, the actual pullout force will likely to be smaller than the force estimated from constant thickness assumption for all the strain levels. As the axial strain increases, the change in thickness ( GTXt∆ ) of the geotextile can be calculated as follows GTX a GTX t t νε ∆ = (5.16) 211 If the Poisson’s ratio (ν ) is assumed to be 0.4 , Figure 5.11 shows the impact of accounting for the change in thickness of the geotextile for the pullout tests performed by Racana et al. (2003). The modified pullout response behaviors are shown in red dotted line. It is noticed that impact of the change in thickness due to Poisson’s effect is not significant except at very large strain levels. For example, at an overburden of 27.8 kPa and a displacement of ~45mm, the geotextile will experience an axial strain of about 20%, which is generally beyond any allowable strain limit for a given geotextile. However, within the typical serviceable strain limits, the impact of change in thickness from axial strain can be ignored. Similar trends can be observed for other pullout tests, but not presented for brevity. Although, the proposed analytical model can be modified to account for the variable thickness at different axial strains, to preserve the simplicity of the analytical solution it is decided not to include such effects. 212 Figure 5.11 Predictions for pullout tests performed by Racana et al. (2003), (a) without considering the change in thickness of the geotextile (black), and (b) considering the change in thickness of the geotextile (red-dotted lines). 5.4.4.2 The impact of the vertical stress on the thickness of the geotextile Considering the influence of the geotextile thickness on the pullout response, it is interesting to investigate the effect of change in thickness due to overburden stress. In all pullout tests used for the validation, the thicknesses were measured before applying any overburden stress. However, it is likely that this thickness will be decreased due to the applied overburden stress when the geotextile is buried in soil. Weggel and Gontar (1992) experimentally measured the thickness of geotextile at different vertical stresses. The tests were conducted on non-woven and needle-punched geotextiles. Accordingly, the thickness at a particular overburden stress is given by ( )GTXt t t e tα− ∞∞ ∞= − + (5.17) 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 Displacement at leading end (mm) Pu llo u t r es is ta n ce , T (kN ) 2.8 kPa 11.8 kPa 17.8 kPa 22.8 kPa 27.8 kPa 213 Where, GTXt is the initial thickness of the geotextile and α is an exponential coefficient. Examining the typical values given for a geotextile, t ∞ can be calculated from the following equation [ ]0 1 1 0.9090(1 / )s s m t m tρ ρ∞ = − − (5.18) Where m is the mass per unit area of the geotextile and sρ is the density of the geotextile. As the change in thickness is most significant in non-woven geotextiles, the pullout tests performed by Racana et al. (2003) is selected to investigate the impact of change in thickness on pullout response. Thus, with the given information, t ∞ is calculated to be 1.16mm. Considering an average value of 5.5 as n, the change in thickness can be calculated due to the overburden. For example, for the largest overburden stress in Racana et al. (2003), the change in thickness was calculated to be 0.087mm. Figure 5.12 shows the predicted variation after considering the change in thickness of the geotextile (shown in red-dotted line). No significant change is observed and in most cases, the estimations are similar to original estimation in which the change in thickness is not accounted. Furthermore, the current standard tests to determine the tensile properties do not apply a confining stress on to the geotextile (e.g. grab tensile strength). Therefore, the author believes that further advancement in material characterization with the proper account of confining stress is required for meaningful accounting of the variable thickness of the geotextile. 214 Figure 5.12 Pullout response predicted for Racana et al. (2003) considering the impact of change in thickness due to overburden stress (shown in dotted line) and assuming no change in thickness (black continuous line) In summary, it is observed that the impact of change in thickness due to Poisson’s effect and the overburden stress are not significant. As such, author decided to keep the formulation in its original form to preserve its simplicity. 5.5 Closure Understanding of the pullout mechanism of geotextiles is a key consideration in analyzing the stability of geotextile–reinforced soil structures. The analytical models developed based on simplified behavioral assumptions for the geotextile material stress–strain response and interface friction behaviors have significant limitations in capturing the highly non-linear pullout response observed in relatively extensible geotextiles. 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 Displacement at leading end (mm) Pu llo u t r es is ta n ce , T (kN ) 2.8 kPa 11.8 kPa 17.8 kPa 22.8 kPa 27.8 kPa 215 A more refined interface model was developed for the planar geotextiles. The interface behavior was modeled considering the changes in normal stress on the planar geotextile due to constrained dilation of soil and the subsequent frictional degradation behaviors at the interface. When considering the behavior geotextiles buried at different overburden stresses, the model also considers the impact of overburden stress on the soil dilation. The nonlinear stress-strain behavior of the geotextile is represented using a hyperbolic model. The analytical model was then developed by combining the non-linear responses of the geotextile and soil–geotextile interface characteristics. The analytical solution provides a framework for linking the pullout resistances, strain, displacement and mobilized frictional length along the geotextile. The validity of the solution was verified by predicting the pullout resistance versus displacement responses reported by seven researchers from experimental pullout tests conducted on buried geotextiles. The results from these tests performed under different confining stresses and geotextile types were well captured by the proposed model. In this chapter, based on the proposed analytical model, a geotextile performance chart is also presented to calculate the strain in the geotextile for a given level of relative displacement. Furthermore, simplified alternative equation is also proposed to calculate the mobilized frictional length along the geotextile at a given displacement. It is believed that the proposed detailed and simplified analytical approaches have the potential to become useful tools in designing mechanically stabilized earth retaining structures, particularly in determining the suitable thickness for a given geotextile, and the burial lengths required to comply with the design requirements. 216 6 PIPES SUBJECT TO AXIAL TENSION AND BENDING ARISING FROM GROUND DISPLACEMENT 6.1 Introduction The response of the pipe due to combination of bending and axial stress is a key consideration, unless the ground movement is entirely along the pipe axis. Early analytical methods to analyze pipes subject to lateral soil loading were based on the assumption that the entire lateral soil load component would be resisted by pipe bending. These methods were based on the theories of beams on elastic foundation and generally valid for pipes undergoing relatively small deformations. However, as discussed in Chapter 2, for pipes with larger strain or deformation capacities, these approaches tend to have significant limitations – e.g., at large deformations, the sections of the pipe closest to the abrupt ground displacement are likely to transform into a cable- like profile with increased dependency on the axial load carrying capacity of the pipe, while the pipe sections further away from the abrupt ground movement would mainly carry the lateral soil load by bending. An analytical solution to determine the response of plastic pipelines subject to combined loading from axial tension and bending is presented herein. It is important to note that apart from the “element level” tests conducted on steel pipes to determine the lateral soil resistance per unit length, it is not practically feasible to conduct meaningful “model-scale” experiments on plastic pipes due to the large lengths required to simulate the pipe-soil interaction aspects without any boundary effects. As a result, unlike in axial soil loading on pipes and geotextiles, it is not possible validate any numerical or analytical models by comparing with real pipe performance data. Alternatively, in this chapter, the results obtained from this analytical approach are compared with results obtained from a numerical modeling that are based on soil-spring analysis. However, as the soil-spring analysis is also based existing analytical approaches (e.g., beams on elastic foundation theories), the comparisons are not intended to validate the analytical solution, 217 but only highlight the differences between the proposed approach and the current analytical techniques. 6.2 Derivation of analytical solution to determine pipe response from combined loading from tension and bending The following presents, an analytical solution that was derived using the principles of continuum mechanics. Although, the following derivation is similar to the analytical derivation for a beam on an “elastic” foundation theory proposed by Hetenyi (1941), some of the aspects relating to the derivation are modified to incorporate relatively more realistic behaviors of soil reaction and axial force development. Most importantly, the solution is mainly governed by the constant lateral soil resistance exerted on the pipe, as opposed to the linearly increasing “elastic” soil resistance considered by Hetenyi (1941). Figure 6.1. Forces acting on a pipe element subject to lateral soil loading (plan view). As shown in Chapter 4, the stress – strain behavior of the pipe material is a key factor that affects the overall response of the pipe. However, owing to the difficulties in incorporating the Original position of the pipe axis N Qv + dQv w(lateral deformation) M + dM M P Qn Qv θ 218 nonlinear material stress-strain behaviors in the above analytical solution, in the above formulation, the pipe material stress-strain behavior is assumed to be linear elastic. Consider the equilibrium of the pipe element shown in Figure 6.1. For the vertical equilibrium, the following equation can be obtained. cos sinv nQ Q Nθ θ= + (6.1) Where, N is the axial force acting on the pipe element. Approximating Equation 6.1 for small angles, v n dwQ Q N dx = + (6.2) The one-half of the total ground offset measured at the location of the abrupt ground movement is denoted by w as shown in Figure 6.2b. Accordingly, the following equations can be written for the change in bending moment (M) in the element n dM Q dx = (6.3) and P dx dQv = (6.4) Where P is the lateral soil resistance per unit length of pipe. From Equation 6.3 and 6.4 2 4 2 4 n dQ d M d wEI dx d x dx = = − (6.5) Combining Equation 6.2 and 6.5, and rewritten in the following form 219 02 2 4 4 =+− βα dx wd dx wd (6.6) With EI N =α and P EI β = . and N is the axial force on the pipe. If N = 0, the above equation will be same as the basic differential equation for the beams on elastic foundations (Hetenyi 1941). 6.2.1 Lateral soil resistance When using the above set of equations, it is required to determine the external load acting on the pipe element. According to the current guidelines, a bilinear behavior is assumed for the lateral soil resistance per unit length (P), in which the ultimate lateral soil resistance per unit length (Pu) for a granular soil is given by u qP HN Dγ= (6.7) Pu is mobilized after a lateral displacement of we (Figure 6.2a). Nq is the peak dimensionless force which is generally a function of the soil friction angle and H/D ratio. These values can be obtained from guidelines such as ASCE (1984), ALA (2001) and PRCI (2009). To solve the basic differential equation given by Equation 6.6, the entire pipe length is divided into two sections, region A and B as shown in Figure 6.2b. The lateral soil resistance per unit length (P) is kw and Pu for region A and B, respectively. With this lateral soil loading behavior, it is required to divided the pipe section into two regions and solve the basic differential equation (Equation 6.6) for these two regions separately. 220 (a) (b) Figure 6.2 (a) Bilinear lateral soil resistance per unit length of pipe (b) schematic diagram of the deformed shape of the pipeline showing the two analytical regions. (Note that “w” is the pipe deformation from the pipe, wf is the one-half ground deformation measured at the abrupt ground deformation) we Pu La te ra l s o il re si st an ce , P Lateral displacement, w k region B region A 221 6.2.1.1 For Region A In region A, the maximum pipe deformation is limited to a lateral pipe deformation of we. Thus, the basic solution for the pipe deformation can be obtained by substituting P = kw in Equation 6.6, where k is the soil stiffness. 1 2 3 4cos sin cos sin mx mx mx mxw C e nx C e nx C e nx C e nx− −= + + + (6.8) where 4 4 kl EI = , 2 4 m l α= + , 2 4 n l α= − and iC (i { }4,3,2,1∈ ) are constants. Considering the far-field boundary conditions ( −∞→x ;, 0w = ), the following boundary conditions should be satisfied, 2C = 4C = 0. With this boundary condition, Equation 6.8 can be simplified as follows 1 3cos sin mx mxw C e nx C e nx= + (6.9) Accordingly, the slope (θ), bending moment (M) and shear force (V) can be obtained from the first, second and third derivative of Equation 6.9 as follows. ( )1 1 3 3cos sin sin cosmxdw e C m nx nC nx C m nx nC nxdx θ= = − + + (6.10) 2 2 22 1 1 1 3 2 2 3 3 cos 2 sin cos sin ... ... 2 cos sin mx C m nx mnC nx n C nx m C nxd wM EI EIe dx mnC nx n C nx − − + = − = − + − (6.11) 3 2 2 33 1 1 1 1 3 3 2 2 3 3 3 3 3 cos 3 sin 3 cos sin ... ... sin 3 cos 3 sin cos mx C m nx C m n nx C mn nx C n nxd wV EI EIe dx C m nx C nm nx C mn nx C n nx − − + = − = − + + − − (6.12) 222 To accommodate different regions of soil resistance, and to improve the accuracy of the solution procedure, a step-by step solution approach, similar to the procedure mentioned in Chapter 4 and 5 was implemented with pipeline divided into several segments. Based on this step-by-step approach, the coefficients 1C and 3C for each pipe segment can be derived considering the continuity of shear force, bending moment, slope and displacement from the adjacent pipe segment. However, to initiate the step-by-step solution procedure and thereby to determine 1C and 3C for the pipe segment adjacent to A-B boundary, it is required to know two initial boundary conditions. First, at A-B boundary, the displacement is limited to we which is the maximum displacement allowed for region A as obtained from the bilinear soil-spring. The second boundary condition required to solve this equation is explained in the following section. 6.2.1.2 For Region B As observed in many pullout tests (e.g., Audibert and Nyman 1977; Trautmann and O’Rourke 1985), the lateral soil resistance is not likely to increase continuously as suggested in the solution derived for Region A (Equation 6.9 through 6.12). In contrast, the soil resistance is likely to reach a constant value upon exceeding a specific displacement level and approximated as a bilinear soil-spring in the current practice (ASCE 1984, ALA 2001, PRCI 2009). In the region of constant soil resistance, the following relationships for the pipe deformation (w), slope (θ), bending moment (M) and shear force (V) can be obtained after substituting P = Pu in Equation 6.6. 2 1 2 3 4 ˆ ˆ ˆ ˆ 2 x xC C xw e e C x Cα α β α α α − = + + + + (6.13) 1 2 3 ˆ ˆ ˆx x C Cdw x e e C dx α α βθ αα α − = = − + + + (6.14) 223 2 1 22 ˆ ˆ x xd wM EI EI C e C e dx α α β α − = − = − + + (6.15) ( )3 1 23 ˆ ˆx xd wV EI EI C e C edx α αα −= − = − − + (6.16) With the knowledge of shear force, bending moment, slope and displacement at the A-B boundary, the constants ˆiC (i { }4,3,2,1∈ ) for each pipe segment belonging to region B can be determined as follows (see Equation 6.17 to 6.20). Note that, the constants of the nth pipe section from the A-B boundary can be obtained with the knowledge of shear forces, bending moment, slope and displacement of the n-1th pipe element. ( ) ( 1) ( 1) 1 ˆ 2 x n n n V M eC EIEI αβ αα − − = − − (6.17) ( ) ( ) ( 1) 2 1 ˆ ˆ nx x n n V C C e e EI α α α − − − = − (6.18) ( ) 1( ) 2( ) ( 1)3 ˆ ˆ ˆ n nx x nn C C xC e eα α βθ αα α − − = + − − (6.19) ( ) 2 1( ) 2( ) ( 1) 3( )4 ˆ ˆ ˆ ˆ 2 n nx x n nn C C xC w e e C xα α β α α α − − = − − − − (6.20) where, ( ) ( 1) 2 x xx x x − + = . Note that EI Pu =β for Region B and EI N =α The second boundary condition required to solve these set of equations are obtained by, considering the symmetry of the problem considered herein. Note that the bending moment at the point of abrupt ground offset will become zero (at a distance of x% from the A-B boundary - see Figure 6.2b). For a selected x% length, the corresponding shear force at A-B boundary can be back-calculated to meet the condition of zero bending moment at the point of abrupt ground 224 movement. The other unknown, i.e., one- half of the lateral displacement at the point of ground movement (wf) is obtained from Equation 6.13 with x = x% . Iterations can be performed until the target wf value is obtained for different x% . 6.2.2 Axial soil resistance In the analytical method presented previously, it is required to know the axial force acting on the pipe (N). The development of axial force in the pipe can be explained as follows. As the pipe deforms laterally, it must increase in length to maintain the deformed geometry of the pipe, in turn, developing tensile force in the pipe. As a result, the pipe will move axially, mobilizing friction along the pipe. The axial force generated due to interface friction can be obtained from the method described in Chapter 4,. Similarly, the axial elongation required in the pipe to match the deformed geometry of the pipe (compatibility) can be obtained as described as follows. Note that to achieve equilibrium of forces, the axial forces obtained from above two mechanisms should be the same. 6.2.2.1 Axial elongation due to frictional force development To determine the axial force development along the pipe, the same analytical solution presented in Chapter 4 was employed. The complete derivation is similar to the derivation presented in Chapter 4, thus it is not repeated herein. However, the following relationships for the axial elongation of the pipe (un) due to interface frictional force can be obtained as follows (Note that un is relative displacement of the nth pipe segment from the point in which the axial force begins to develop). 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (6.21) 225 where, Cn and C΄n are constants that depend on the pipe and interaction properties; and ( ) 124 1 4n n n n nz x Cλ λ= − + − . 6.2.2.2 Axial elongation required to match the deformed geometry of the pipe The axial elongation of the pipe to maintain the deformed shape of the pipe can be generally obtained from the following 2 0 1 2 nx n dw u dx dx = ∫ (6.22) Considering the complexity of the calculation and fact that the pipe deformation in region A is insignificant, thus, the elongation in region A is ignored in the following calculation. Therefore, the following solution can be obtained for region B, after substituting the expression for slope (i.e., Equation 6.14) in Equation 6.22. [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] [ ]( ) 2 22 2 2 2 2 1 1 23 2 1 1 2 2 2 15 2 2 3 2 2 3 1 2 3 3 1 2 1 2 2 1 ˆ ˆ ˆ ˆ( ) ( ) 4 ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) 6 2 2 n n n n n n n n n n n n n n n u C x C x C C C x x C x C x x C x C C C C C x x C x C xC C C x C x α β α α α β β α α α α = Η − Η − + − + Η − + Η − + Η − Η + − − + − Η − − Η − − + + + (6.23) Where ( ) e nxnx αΗ = and nx is selected to be the length along the pipe from the A-B boundary to the nth pipe element. Using Equation 6.21 and 6.22, iterative calculations are performed until the elongation due to frictional force is equivalent to the elongation required to match the deformed shape of the pipe. After matching the two elongations, the strain in the pipe can be obtained as follows. 226 (1 )1 ' 1 2 ( ) n n n n n z u x Cη λ − = − + (6.24) The corresponding axial force acting on the nth pipe segment can be obtained by substituting these strain and pipe modulus relationships in the following npnn uAxEN ')( ××= (6.25) The value obtained from Equation 6.25 is substituted in Equations 6.9 to 6.20 in place of Nn for each pipe segment, and new deformed shape, slope, bending moment and shear force in the pipe is obtained. 6.3 Numerical modelling to determine pipe response due to combined loading from tension and bending In absence of experimental results to validate the analytical solution, it is decided to compare the results from the analytical solution with the results obtained from soil–spring analysis performed using ABAQUS/ standard (Hibbit et al. 2006). ABAQUS is extensively used in oil and gas distribution industry to model the pipe response due to soil loading arising from ground movements. Any differences observed in results obtained from the two approaches would be a reflection of the differences in the analytical formulations adopted in ABAQUS and in the proposed analytical approach. To compare the results from the analytical approach and ABAQUS, a 30-m long pipe crossing a strike-slip fault was modeled. Initially, the pipe axis was assumed to be perpendicular to the fault crossing, thus, the pipe response will be symmetric about the axis of the fault. The pipe length was modeled using 400 PIPE31 elements, in which finer elements were employed closer to the abrupt ground movement and the element size was increased away from the ground offset. 227 The pipe diameter is assumed to be 114mm (with SDR = 11) and with an elastic modulus of 400MPa. The soil loading was modeled using nonlinear soil-springs (i.e., PSI34 elements). The respective values for axial and lateral soil-springs were obtained from Equation 4.9 and Equation 2.14, respectively. Assuming that the pipe is buried in dense Fraser River sand with a unit weight of 16 kNm-3 and buried depth of 0.6 m, the respective values for Pu and we for the lateral soil-spring can be obtained as 10 kNm-1 and 16 mm respectively for the 114-mm diameter pipe (ASCE 1984). In addition, the values for the nonlinear axial soil-spring were calculated from Equation 4.9, and the input values are similar to 114D test performed by Weerasekara (2007). The corresponding input parameters to calculate the axial soil-springs are given in Section 4.4.2, thus not repeated herein. In addition, a soil-spring was assigned for the upward direction, and the values for this soil- spring were selected according to ASCE (1984) recommendations. However, only a negligible amount of upward movement observed (e.g., maximum upward displacement less than 2mm) in the examples presented herein; thus, the contribution from the upward soil-spring is ignored in the following discussion and assumed that the pipe-soil interaction is predominantly occurring in the two-dimensional horizontal plane. 6.4 Comparison between analytical and numerical results With the above mentioned soil-spring values, Figure 6.3 show the comparison of the deformed shape of the pipe, bending moment, shear force and axial force in the pipe obtained from the numerical and analytical methods at three different levels of displacement at the point of ground offset. Note that the value for the pipe deformation at the location of the abrupt ground deformation is only one-half of the total ground displacement measured at the point of abrupt ground deformation. Furthermore, due to the symmetry about the location of abrupt ground movement, the behavioral patterns of one-side of the pipe were plotted. A close match between the numerical model and analytical approach was observed apart from the axial tensile force distribution along the pipe. In the current analytical approach, the compressive force acting 228 along a small pipe length is ignored in axial force calculation. This section of pipe is located at a sufficient distance away from the point of abrupt ground displacement, thus the impact of this assumption is minimal to the calculated forces in the pipe. Additional details relating to the observed discrepancies in axial force distribution are discussed in the following sections. (a) (b) (c) (d) Figure 6.3 Comparison of: (a) deformed shape; (b) bending moment; (c) shear force; and (d) axial force on the pipe obtained from the numerical and analytical approaches for 114- mm pipe (small to medium scale ground deformations). Figure 6.4 shows the comparison of pipe behavioral patterns obtained for relatively large ground offset values. In comparison to the small to medium-scale ground offsets (Figure 6.3), 229 significantly large differences were observed between the pipe responses obtained from the analytical and numerical solutions. (a) (b) (c) (d) Figure 6.4 Comparison of: (a) deformed shape; (b) bending moment; (c) shear force; and (d) axial force on the pipe obtained from the numerical and analytical approaches for 114- mm pipe (large-scale ground deformations). To illustrate the differences in the analytical and numerical model predictions, Figure 6.5 and Figure 6.6 plots the maximum values obtained from the analytical and numerical solutions for bending moment and axial force. Note that, compared to the analytical model, a larger bending moment and axial force were obtained from the numerical analysis at relatively large pipe deformations. In the following section, these behavioral patterns (shown in Figure 6.5 to Figure 230 6.6) and the reason for the discrepancies between analytical and numerical solutions will be discussed in detail. Figure 6.5 Comparison of maximum bending moment obtained from the numerical and analytical approaches. Figure 6.6 Comparison of maximum axial force obtained from the numerical and analytical approaches 231 6.4.1 Discussion of results obtained from analytical and numerical modeling First, with respect to the deformation, bending moment and shear force in the pipe, it is imperative to explore the reason for the good agreement observed between the analytical and numerical results at relatively small ground displacements and the discrepancy observed at relatively large ground offset values. In the numerical formulation, initially, the pipe deformation, bending moment and shear force are calculated omitting the secondary effects from axial force. The governing equation corresponding to this assumed behavior can be obtained with α = 0 in Equation 6.6. Afterwards, based on the deformed shape of the pipe, the axial strain/force is calculated by matching the elongation required to comply with the deformed shape of the pipe. As the calculation for the initial pipe deformation is independent of the axial force, the calculated pipe deformed shape, bending moment and shear force are unaffected by the input values of the axial soil-spring. To illustrate this aspect, Figure 6.7 shows the deformed shape, bending moment and shear force for three different magnitudes of axial soil resistances. The plot for the axial soil resistance (T) is similar to the axial soil resistance employed in the example shown in Figure 6.3. The second set of curves were obtained by using axial soil resistance values that is approximately three times larger than the axial soil resistance used in Figure 6.3. Thirdly, Figure 6.7 shows the plots for deformed shape, bending moment and shear force obtained for an axial soil resistance nearly equal to zero (T = α ≈ 0). Note that axial soil resistance equal to zero cannot be incorporated in the numerical formulation since infinite mobilized length is required to achieve an amount of elongation that is similar to the elongation required to satisfy the deformed geometry of the pipe profile. It is apparent from Figure 6.7, apart from the different axial force distributions along the pipe, minuscule change is observed for the deformed shape, bending moment and shear force distributions obtained from the numerical solution, despite the variations in magnitude of axial soil resistance. 232 (a) (b) (c) (d) Figure 6.7 (a) Pipe deformation; (b) bending moment; (c) shear force; and (d) axial force in the pipe, obtained from the numerical solution for three different magnitudes of axial soil resistances. On the other hand, with the direct account of secondary effects from the axial force in the pipe (α ≠ 0 in Equation 6.6), for each different magnitudes of axial frictional resistance, different responses were obtained for the pipe deformation, bending moment, and shear force as shown in Figure 6.8. As expected, the increase in axial force will reduce the slope of the pipe increasing the radius curvature - reducing the bending moment of the pipe. 233 ((a) (b) (c) (d) Figure 6.8 (a) Pipe deformation, (b) bending moment, (c) shear force and (d) axial force in the pipe, obtained from the analytical solution for three different magnitudes of axial soil resistances. As the axial force is not directly accounted in the governing equations in ABAQUS, the bending moment calculated from the numerical model will be over-estimated at large displacement levels as shown in Figure 6.5. Moreover, the larger deformation of the pipe (corresponding to a large bending moment) will lead to a larger axial force in the numerical results as shown in Figure 6.6. However, in the analytical model, with the increasing influence from the axial force in the pipe, the bending moment of the pipe will become less dominant at larger deformations. 234 From logical reasoning of the above behavioral patterns obtained from these two approaches, it is apparent that the pipe response estimated from the analytical solution is more realistic than the response obtained from the numerical solution for different axial soil resistance values. Note that a direct analogy can be observed between pre-stressed concrete beams and the results obtained from the proposed analytical solution. 6.4.1.1 Coupling of axial and lateral soil resistances With the lateral movement of the pipe, the normal stress on the pipe surface is likely to increase since the lateral soil resistance is few magnitudes larger than the typical axial soil resistance. Figure 6.9 illustrates this aspect of additional normal pressure created from the lateral movement of the pipe. As a result, the frictional resistance is likely to increase with the increased normal stress acting on the pipe. This is termed as the coupling of the lateral and axial soil resistance (e.g. Hsu et al. 1996). Note that the examples presented in Figure 6.3 and Figure 6.4 consider the uncoupled behavior between axial and lateral soil resistance. The analytical model can be modified to account for the increased axial soil resistance from the coupled behavior. Figure 6.9 shows an idealized soil pressure distribution around the pipe in which the static soil pressure due to overburden is acting uniformly on the pipe; however, the lateral soil pressure will not act on the entire circumferential area. If an incidence angle of 90° is assumed, the example illustrated in Figure 6.8, shows the different pipe responses for coupled and uncoupled behavior of the pipe. Note that, in Figure 6.8, with the increased lateral soil resistance, the interface frictional resistance (T) is approximately three times larger than the interface frictional resistance calculated from uncoupled behavior. As expected, the large axial force in the pipe leads to a smaller bending moment and shear force for the coupled behavior (Figure 6.8). 235 Figure 6.9 Idealized soil pressure distribution around the pipe during lateral movement of the pipe (coupling of axial and lateral soil resistances) Although, PRCI (2009) guidelines assume that the error associated with the assumption of independent soil-springs (uncoupled behavior) is generally small relative to the overall uncertainty associated with defining the soil strength properties in the equivalent soil-spring values, it is evident that in certain situations, the impact from coupling can be considerable and the proposed analytical approach formulated in a spreadsheet environment allows a convenient framework to account for coupling. 6.4.1.2 Impact of different soil stiffness values closer to the abrupt ground offset It is highly likely that zones near the ground offset may have varying values for soil stiffness due to disturbances arising from relative ground movements. Typically, with the soil disturbances, the soil stiffness may be several orders smaller than the stiffness of the undisturbed zone. For example, Takata et al. (1987) stated that in liquefied soils, the equivalent soil stiffness may range from 1/1000 to 1/3000 of the non-liquefied soil. O’Rourke (1994) stated that this range of soil stiffness is between 1/100- 5/100. Similar observations have been reported by Uematsu Increased soil pressure from lateral movement of the pipe Pipe Normal stress from the soil overburden 236 (1978), Matsumoto et al. (1987), Yasuda et al. (1987) and Tanbe (1988) with different estimations for the stiffness of the soil. Recognition of the variable soil stiffness in the zone of fault crossing is an important aspect due to its direct influence on the deformed shape of the pipe, bending moment, shear force and axial force. For example, the following example shows a comparison of two pipe responses obtained for two different soil stiffness profiles at the vicinity of the fault zone. In the first example, the stiffness is assumed to vary from a value of 1/50*Pu to Pu in the manner shown in Figure 6.10(a). The second example shows the pipe response if a constant soil stiffness of Pu is assumed for the entire area affected by the abrupt ground movement [Figure 6.10(b)], as typically assumed in most soil-spring analyses. The pipe deformation shapes, bending moment, shear force and axial force obtained from these two examples are shown in Figure 6.11. As expected, the smaller soil lateral resistance at the fault zone provides less resistance to change the alignment of the pipe axis, promoting more axial tension in the pipe. As a result, even with the smaller soil resistance, an increase in axial force is observed in the pipe together with a decrease in bending moment. Note that in traditional soil- spring analyses, it is difficult to assign gradually varying soil resistances (i.e., using different soil-springs) for different pipe sections, despite the options of modeling the nonlinear aspect of each soil-spring. However, with the formulation presented in the proposed analytical method, the soil loading arising from variable soil stiffness can be conveniently modeled in spreadsheet environment. 237 Figure 6.10 Assumed soil stiffness distributions around the abrupt ground movement, (a) varying soil stiffness and (b) constant soil stiffness. 238 (a) (b) (c) (d) Figure 6.11 (a) Pipe deformation, (b) bending moment, (c) shear force and (d) axial force in the pipe, obtained from the numerical solution for two different magnitudes of axial soil resistances. 6.4.1.3 Response of pipes subject to different burial and pipe conditions Similar to the parametric investigations conducted in Chapter 4 for axial soil loading on pipes, the proposed analytical solution can be employed to explore the impact of different pipe diameters, pipe stiffness, thicknesses (SDR values) etc. Most of the behavioral patterns can be expected from logical reasoning; therefore the detailed discussions on this subject are not 239 included in this thesis. For example, with the intention of comparing the responses from different pipe sizes buried in similar conditions, Figure 6.12 shows the pipe deformation, bending moment, shear force and axial force development for two pipe sizes (i.e., pipe diameters of 60 mm and 114 mm) for the same magnitude of ground offset at the location of abrupt ground movement. As expected, in the larger diameter pipe (i.e., 114mm), a longer pipe length is affected by the abrupt ground displacement. In addition, larger bending moment, shear force and axial forces are obtained for the larger diameter pipe. (a) (b) (c) (d) Figure 6.12 (a) Pipe deformation; (b) bending moment; (c) shear force; and (d) axial force in the pipe for 60-mm and 114-mm pipes. 240 6.4.1.4 Some of the advantages of the proposed analytical approach Most importantly, the proposed analytical method accounts for the direct influence of axial force in the pipe response calculation, as opposed to the indirect axial force computation in some of the numerical approaches. As shown previously for results obtained in ABAQUS, if the second- order effect of axial force in the pipe is not directly accounted in the formulation, the deformation shape, bending moment and shear forces will be unchanged irrespective of the axial force in the pipe. A more advanced interface friction model is employed in the proposed approach to account for the development of axial force along the pipe, which accounts for the increase in friction due to stress soil dilation, frictional degradation, etc. The new analytical approach also allows modeling of complex soil resistance distributions (section 6.4.1.2) around the point of ground offset, allowing more flexibility in calculation compared to traditional soil-spring analyses. 6.4.1.5 Limitations in the proposed analytical approach The soil-springs are defined with respect to a global coordinate system and the direction of the soil spring forces will not be maintained if the pipeline undergoes large rotations. As discussed in PRCI (2009) guidelines, the error associated with this misalignment is acceptable considering the other uncertainties inherent in the analysis. Analytical derivation is based on certain small deformation assumptions; therefore some limitations exist when applying this analytical approach to model large pipe deformations. It is also to be noted that both numerical and analytical models are not capable of capturing the localized buckling of the pipe wall, ovalization of the pipe cross-section or failure modes associated with the pipe failure. 241 6.5 Closure A new analytical solution was developed to evaluate the performance of pipes subject to combined loading of bending and axial tension arising due to abrupt ground movement. It is evident that although the initial soil movement is perpendicular to the pipe axis, the axial force in the pipe can have a significant impact on the overall pipe performance. To account for the axial force development along the pipe, the new analytical solution developed in Chapter 4 was employed which accounts for the impact of soil dilation and frictional degradation. In the absence of experimental data on the pipe performance subject to lateral ground displacements, a decision was made to compare the results from the new analytical solution with those results obtained from a numerical (ABAQUS) analysis. Although the results from the analytical and numerical methods are in good agreement for relatively small-medium level pipe deformations, large discrepancies were observed as the ground offset was increased. One of the key reasons for the mismatch is identified as the failure to account the direct influence of the axial tension in the numerical solution. However, the significance of direct account of the axial force in the pipe is evident when exploring the pipe responses obtained from the proposed analytical solution for different magnitudes of axial soil resistance. The proposed analytical solution can be modified to account for the coupling of the axial and lateral soil resistances after assuming an appropriate incidence angle for the increased lateral soil pressure. In addition, the proposed analytical model is capable of modeling the complex loading profiles (disturbed zones) closer to the abrupt ground movement. In such instances, the smaller lateral soil resistance at the soil disturbed zone is likely to promote the development of axial tension in the pipe, and reduce the dependency on pipe bending as the primary load carrying mode. 242 7 SUMMARY AND CONCLUSIONS In the past few decades, polymer plastics have been used as pipe material in gas, oil, electrical, potable water, sewer transmission pipelines, thereby replacing traditional steel and/or concrete pipes. In spite of the increased flexibility, the plastic pipes buried in areas prone to ground movement is an important engineering consideration for utility owners, since the failure of such systems poses a risk to public property and safety, in addition to the associated utility and customer business disruption. In particular, the potential pipe rupture caused by soil loading resulting from slow moving landslides is a significant concern in certain geographical areas. Although the ground movements and its variations over time in a given area can be detected and mapped with reasonable confidence using modern survey technology, the prediction of the distress of buried plastic pipelines located in such areas of ground displacements has become a difficult task due to alack of understanding of the underlying soil-pipe interaction problem. With this background, the research work presented in this thesis was undertaken with the main objective of studying the performance of polyethylene pipelines subjected to relative ground movements, and thereby developing analytical models to ascertain the pipe performance based on the measured ground displacements. This thesis focuses on buried PE pipelines subjected: (a) the axial tensile loading; and (b) combined axial tensile loading and pipe bending. Although no specific work was undertaken, it is believed that the analytical principles developed herein can be extended to model buried PE pipes subject to axial compressive loading situations. This chapter is intended to summarize the findings and conclusions arising from this research work. 243 7.1 Experimental research contributions With the objective of generating pipe performance data on axially loaded pipes, five large-scale field axial pipe pullout tests were performed. A pulling mechanism, self reaction frame and other supporting structures were designed and fabricated in order to perform these field pipe pullout tests. According to the author’s knowledge, this is the first instance that “model-scale” tests were performed on pipes subject to ground deformations reaching an axial strain of about 5%. The axial pipe pullout tests were designed to capture the different pipe responses at various burial, loading and pipe conditions encountered in practice. In each test, the pullout resistance, strain, displacement and elapsed time were directly recorded. Similar to the pullout tests performed in the soil chamber (Anderson 2004; Weerasekara 2007), highly nonlinear pipe responses were observed in these tests. The field pipe pullout tests, indicated that the overall pipe response will depend on pipe properties (e.g., pipe cross-sectional area, stress-strain behavior) and soil characteristics (e.g., burial depth, density, friction angle, lateral earth pressure coefficient). This in turn highlighted the need to incorporate these parameters any analytical models to capture the pipeline response. Apart from the field pipe pullout tests, experiments were conducted to characterize the pipe behavior (e.g., uniaxial compression tests) and to obtain specific independent input parameters (e.g., soil density) for the validation of the analytical/numerical approaches. 7.2 Development of an analytical solution to determine pipe response due to axial soil loading The work undertaken as a part of the thesis involved the development of an analytical solution to model response of PE pipes due to relative axial soil loading. As per the framework presented for this research program, the key activities and contributions pertaining to the derivation of the analytical solution involved the following. 244 7.2.1 Modeling of the soil-pipe interface friction during axial pullout A soil-pipe interface friction model was developed considering: (a) the increase in friction due to constrained soil dilation, (b) the impact of mean normal stress on soil dilation; and (b) subsequent decrease in friction (frictional degradation). Considerable differences were observed when comparing this model with the current recommended practice of bilinear axial soil-springs (ASCE 1984; ALA 2001; PRCI 2009), emphasizing the importance of accounting for above aspects. 7.2.2 Viscoelastic stress-strain behavior of the pipe material It was determined that the viscoelastic stress-strain behavior of the plastic pipe material could be effectively modeled using a hyperbolic formulation. A modified form of the hyperbolic model developed by Suleiman and Coree (2004) allowed the strain-rate dependant pipe material stress- strain behavior to be modeled using several constants. The pullout test responses obtained at different displacement rates emphasized the importance of considering the strain rate dependant behavior and the nonlinear aspects of the stress-strain behavior. 7.2.3 Overall pipe performance model After establishing the interface friction behavior and stress-strain behavior of the pipe material, a new analytical solution was derived considering the equilibrium and compatibility conditions at a pipe element-level. This analytical solution provides the framework to relate the displacement, axial strain, axial force and mobilized frictional length along the pipe. Most importantly, with the knowledge of the measured ground displacement, the solution provided a method to estimate the pipe strain and mobilized frictional length during axial soil loading. The capacity of this methodology would greatly assist the decision making processes on pipe system rehabilitation or replacement aspects. 245 7.2.4 Validation of the proposed analytical model The data from field pullout tests conducted as a part of this thesis combined with those from previous laboratory pullout testing formed a solid basis for validating the new analytical method. As such, the proposed analytical solution was validated by predicting with pullout responses obtained from a total of ten axial pipe pullout tests conducted in field and laboratory environments. It is of importance to highlight that all the validations were undertaken using a consistent set of input parameters used in the analytical solution. The input parameters were obtained with reasonable accuracy through independent laboratory experiments or direct field measurements. It was found that there was an excellent agreement between the analytical and experimental results for tests conducted with different pipe diameters (60mm and 114mm), soil density conditions (loose and dense), burial depths (0.54 m to 0.98m), burial lengths (3.8 m, 5.0 m, and 8.5m) and under a range of axial pullout displacement rates. This, in turn, confirmed that the analytical model is capable of capturing the complex pipe soil interaction prevalent during relative axial soil movements in buried PE pipes. In addition to the excellent predictive capabilities for pullout resistance, strain, displacement, and time measurements, the new method allowed a way of estimating the pipe length along which the soil-pipe interface friction would mobilize under a given pullout situation. The knowledge of this mobilized frictional length is of considerable importance in gas distribution pipes when attempting to determine whether the anchoring resistance of a pipe connection is mobilized. This information is required to determine the probable failure location more accurately (Weerasekara 2007). The satisfactory validation of the analytical solution provided a basis to undertake a parametric analysis to predict the axial pullout performance of buried pipelines under other conditions. For example, the relative impact of interface frictional characteristics (burial depth and soil density) 246 and pipe characteristics (pipe diameters, thickness and stiffness) were explored. The parametric analysis indicated that the impact of soil dilation at the soil-pipe interface would be significant in small diameter pipes and in shallow burial conditions. In addition, a pipe with smaller SDR (larger pipe wall thickness) or larger pipe stiffness is likely to mobilize a longer frictional length for the same amount of ground displacement. The ability to make these types of predictions is considered important when selecting the pipes for different burial conditions. 7.3 Development of an analytical solution to determine pullout response of a geotextile The interaction between the reinforcement and soil is a key consideration in design of mechanically stabilized earth (MSE) walls, slopes, embankments, and foundations. Over the years, numerous pullout tests have been performed on geotextiles to comprehend the fundamental interaction mechanism in these reinforced structures. Considering the similarities in the basic interaction mechanism, it was realized that the analytical solution developed to capture the axial pullout response of plastic pipelines could be extended/extrapolated to explain the response of buried geotextiles in MSE walls. The key contributions related to this analytical development are listed as follows. A more refined new interface model was developed for the planar geotextiles, considering the changes in normal stress on the planar geotextile due to constrained soil dilation and the subsequent frictional degradation behaviors at the interface. The new model also considers the impact of overburden stress on the soil dilation, which is an important aspect when analyzing the geotextiles buried at different depths. Accordingly, the interface friction behavior is a function of the burial depth, relative density, lateral earth pressure coefficient and Poisson’s ratio of the soil. It is assumed that the nonlinear stress-strain behavior of the geotextile can be represented using a hyperbolic model. Similar to plastic pipes, an analytical solution was derived by combining the non-linear stress-strain response of the geotextile and soil–geotextile interface characteristics. The solution provides an analytical framework for linking the pullout resistances, strain, displacement and mobilized frictional length along the geotextile. 247 The validity of the new solution was satisfactorily validated by back-analyzing the variation of pullout resistance with displacement for twenty pullout tests conducted by seven other researchers on buried geosynthetics (e.g., Fannin and Raju 1993; Racana et al. 2003, Konami et al. 1996, Alobiadi et al. 1997 and Ali 1999).The proposed method also allows to calculate the mobilized frictional length along the geotextile at a given displacement. The good agreement between the calculated and measured mobilized frictional length in a given pullout situation is also an indication of the success of the analytical model. Note that the mobilized frictional length is an essential parameter in deciding the anchoring lengths for geotextiles. In terms of implementation (i.e., to obtain the solutions for the geotextile-soil pullout problem from the basic differential equation), the closed form-solution format can be easily programmed compared to other methods that relies on numerical methods - e.g., finite difference methods (Gurung and Iwao 1999; Madhav et al. 1998; Perkin and Cuelho 1999). As an alternative to using the equations in the analytical method, a geotextile performance chart was derived to relate the strain in the geotextile to the relative displacement. It is believed that the proposed detailed and simplified analytical approaches have the potential to become useful tools in designing mechanically stabilized earth retaining structures, particularly in selecting the suitable thickness for a given geotextile and the burial length to comply with the design requirements. 7.4 Development of an analytical solution to determine the performance of a pipe subject to axial loading and bending When analyzing the pipes subject to ground movement, in most instances, the pipe will experience a combination of bending stresses and axial stresses, unless the ground movement is exclusively acting along the pipe axis. The discussion in this thesis considers the response of a pipe subject to abrupt ground displacement, with the initial soil loading acting perpendicular to 248 the pipe axis. In such situations, with the increasing pipe deformation, the sections of the pipe closest to the abrupt ground movement are likely to experience considerable amount of axial tensile force as the pipe transform into a cable-like profile. A detailed study was undertaken to address this consideration as a part of this thesis, and the key contributions related are summarized as follows: An analytical method was derived to account for the combined response axial tensile load and bending in the pipe using the principles of continuum mechanics. To account for the axial force development along the pipe, the analytical solution developed in Chapter 4 was employed, as such, the impact of soil dilation and frictional degradation are considered. The solution allows relating the measured ground offset at the location of abrupt ground movement to the bending moment, shear force and axial force at any location along the length of the pipe. The deformed shape of the pipe, bending moment, shear force, and axial force obtained from the analytical solution were compared with the results from soil-spring analyses conducted using ABAQUS. Although the results from the analytical and numerical methods are in good agreement for small pipe deformations (with relatively smaller axial force in the pipe), significant differences were observed at large pipe deformations. One of the main reasons for the disparity is identified as the failure to account for the direct influence of the axial tension in the numerical solution. The significance of direct account of the axial force in the pipe is evident when investigating the pipe responses at different magnitudes of axial soil resistances. The proposed analytical model is also capable of modeling the complex loading profiles (disturbed zones) closer to the abrupt ground movement. In such instances, the smaller lateral soil resistance at the soil disturbed zone is likely to increase the axial tension in the pipe, and reduce the dependency on pipe bending as the load carrying mode. 249 7.5 Future work and recommendations Although the thesis is mainly focused on the research findings on MDPE pipe which are commonly used in high- risk gas distribution industry, the theoretical derivations/concepts are equally applicable to other flexible and rigid pipes employed in different applications, e.g., PVC pipes in water mains or wastewater lines, steel pipes in gas and oil transmission lines, etc. Furthermore, with proper characterization of external soil loading (axial and lateral directions), the same principles can be employed to analyze the pipes or geotextiles buried in cohesive backfills. In addition to the contributions described above, as given below, the research undertaken herein has revealed a spectrum of additional work that can be undertaken to advance the knowledge on this topic further: Although the main focus of this thesis was on the tensile load development in pipes, the same analytical principles can also be extended to investigate the pipes subject to axial compression or combination of axial compression and bending. Further experimental studies can be performed to investigate the anchoring effect due to presence of pipe connections, and the resulting additional frictional force development along the pipe. This information would become helpful in analyzing the response of complex branch pipe configurations. The same analytical principles can be extended to determine the performance of geotextiles, if the boundary of the soil wedge is at proximity to the wall facing (a limited geotextile length is available between the wall facing and boundary of the soil wedge). As a result, it will be possible to model the additional tensile stresses that would be developed along the geotextile due to the anchoring effect at the wall facing, besides the friction induced tensile forces in the geotextile. The proposed method can be extended to pipes oriented at an angle to the ground movement (oblique soil loading). In such situations, unlike the symmetric boundary conditions assumed in 250 the current analysis, different boundary conditions can be employed on either side of the abrupt ground movement. The influence of the internal pipe pressure can be taken into consideration with proper characterization of the stress-strain behavior of the pipe material under biaxial stress conditions. 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Part I: Experimental investigation and model evaluation, Polymer Engineering and Science, 37 (2): 404-413. Zhou, J. Z. and Harvey, D. P. (1996) A model for dynamic analysis of buried and partially buried piping systems, Pressure Vessels and Piping Systems, ASME, pp. 21-29. 266 APPENDIX This section presents a summary equations presented in Chapter 4 to 6 for obtaining pipe and geotextile performance subject to relative soil movements occurring in axial and lateral directions, A.1. Analytical solution for axial loading of pipes (a) Analytical model for interface friction The interface friction may obtained from following equation ( ) tandcT Dσ σ pi δ′= ∆ + (A.1) where 0(1 ) 2 k Hγ σ + ′ = (A.2) And dσ∆ is obtained as follows [ ]8.3))ln(25.01.7(8.3tan)(2 −′−∆=∆ σγσ Ddd ID tG (A.3) In which, the variation of ( )G γ is obtained from the following hyperbolic relationship with the use of appropriate relationship to present the initial shear modulus ( 0G ) of the soil (e.g. Iwasaki et al. 1978). 0 ( ) 1 ( ) G G a b γ γ = + )) (A.4) (b) Stress-strain behavior for the pipe material The stress-strain behavior of the pipe material is represented using a modified form of the hyperbolic equation as follows 267 1 Eini ε σ ηε = + (A.5) where ( )biniE a ε= & (A.6) ( ) ln( ) b a c d εη ε = + & & (A.7) (c) Analytical solution for obtaining the overall pipe behavior The relative displacement of the pipe (at any location)can be obtained as follows. 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (4.8) where, n n ini p T E A ηλ = (A.9) ( ) 124 1 4n n n n nz x Cλ λ= − + − (A.10) The strain along the pipe is obtained as. (1 )1 ' 1 2 ( ) n n n n n z u x Cη λ − = − + (A.11) 268 The pullout resistance can be obtained by substituting above strain and pipe modulus relationships in the following '( ) n n p nN E x A u= × × (A.12) A.1.1.Summary of input parameters Table A.1 Summary of input parameters for axial soil loading on pipes Input parameter Interface friction on pipe ∆td (mm) G (γ=2.5%) (kNm-2) γ (kN/m-3) H (m) K0 δ (degrees) Displacement at zero dilation (mm) Relative density of soil Pipe properties Pipe outside diameter (mm) Pipe thickness (mm) Rate dependant hyperbolic constants* (a, b, c, d) Ground movement conditions Displacement rate of soil block (mm/hr) A.2. Analytical model for obtaining pullout resistance of geotextiles (a) Analytical model for interface friction of the geotextile The pullout resistance per unit length of planar geotextile (T) can be derived in the following form: 269 / 0 / max 2 tan 1 [2(1 ) /((1 2 )(1 2 ))] tan tan GTX s GSY S GSY HbT K γ φ ν ν φ ψ ′ = ′ − + − + (A.13) where ( )maxtan tan 3.8 7.1 0.25 ln 3.8DIψ σ ′ = − − (A.14) (c) Stress-strain behavior for the geotextile material The stress-strain behavior of the geotextile material can be obtained from the following hyperbolic equation (note that the strain dependant behavior is included herein). 1u uEini u σ η′ ′ = ′+ (A.15) (c) Analytical solution for obtaining the overall geotextile behavior The displacement is given by the following equation (at any given point in the geotextile) 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (A.16) where, 1 ini T E A ηλ = . ( ) 124 1 4n n n n nz x Cλ λ= − + − . The strain along the geotextile is obtained from. (1 )1 1 2 ( ) n n n n n z u x Cη λ − ′ = − + (A.17) 270 The corresponding force at a given location of the geotextile is given by '( )n n nN E u A u′= × × (A.18) A.2.1.Summary of the input parameters Table A.2.1.Summary of the input parameters for determining pullout resistance of geotextiles Input parameter Interface friction on pipe ∆td (mm) v γ (kN/m-3) H (m) K0 δ (degrees) Displacement at zero dilation (mm) Relative density of soil Pipe properties Pipe outside diameter (mm) Pipe thickness (mm) Rate dependant hyperbolic constants* (a, b, c, d) Ground movement conditions Displacement rate of soil block (mm/hr) A.3. Analytical solution for pipes subject to axial tension and bending (arising from ground movements) (a) External soil loads acting on the pipe (i) Interface friction behavior This is similar to the equations used to obtain the axial loading of pipes(Section A.1) (ii) Lateral soil resistance The lateral soil resistance per unit length is obtained as follows (e.g., ASCE 1984) 271 u qP HN Dγ= (A.19) (b) Pipe material stress- strain behavior Assumed to be linear elastic for this derivation (c) Analytical solution for obtaining the overall pipe behavior The analytical solution is obtained for two regions, i.e., region A and B (i) Analytical solutions for Region A 1 3cos sin mx mxw C e nx C e nx= + (A.20) ( )1 1 3 3cos sin sin cosmxdw e C m nx nC nx C m nx nC nxdx θ= = − + + (A.21) 2 2 22 1 1 1 3 2 2 3 3 cos 2 sin cos sin ... ... 2 cos sin mx C m nx mnC nx n C nx m C nxd wM EI EIe dx mnC nx n C nx − − + = − = − + − (A.22) 3 2 2 33 1 1 1 1 3 3 2 2 3 3 3 3 3 cos 3 sin 3 cos sin ... ... sin 3 cos 3 sin cos mx C m nx C m n nx C mn nx C n nxd wV EI EIe dx C m nx C nm nx C mn nx C n nx − − + = − = − + + − − (A.23) where 4 4 kl EI = , 2 4 m l α= + , 2 4 n l α= − (ii) Analytical solution for Region B 2 1 2 3 4 ˆ ˆ ˆ ˆ 2 x xC C xw e e C x Cα α β α α α − = + + + + (A.24) 1 2 3 ˆ ˆ ˆx x C Cdw x e e C dx α α βθ αα α − = = − + + + (A.25) 272 2 1 22 ˆ ˆ x xd wM EI EI C e C e dx α α β α − = − = − + + (A.26) ( )3 1 23 ˆ ˆx xd wV EI EI C e C edx α αα −= − = − − + (A.27) where, ( ) ( 1) 2 x x x x x − + = (a) Axial elongation in the pipe due to frictional force development. 1 '1 1 1tanh ( ) log( ) 2 n n n n n n n n n n z u x z x C C η λ λ λ − = − − + + + + (A.28) (a) Axial elongation required to match the deformed geometry of the pipe [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] [ ]( ) 2 22 2 2 2 2 1 1 23 2 1 1 2 2 2 15 2 2 3 2 2 3 1 2 3 3 1 2 1 2 2 1 ˆ ˆ ˆ ˆ( ) ( ) 4 ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) 6 2 2 n n n n n n n n n n n n n n n u C x C x C C C x x C x C x x C x C C C C C x x C x C xC C C x C x α β α α α β β α α α α = Η − Η − + − + Η − + Η − + Η − Η + − − + − Η − − Η − − + + + (A.29) Where ( ) e nx n x αΗ = After matching the two elongations, the strain in the pipe can be obtained as follows. (1 )1 ' 1 2 ( ) n n n n n z u x Cη λ − = − + (A.30) The corresponding axial force acting on the nth pipe segment can be obtained from the following 273 npnn uAxEN ')( ××= (A.31) A.3.1.Summary of the input parameters Table A.3 Summary of input parameters required to obtain a solution for combined axial force and pipe bending Input parameter Interface friction on pipe ∆td (mm) G (γ=2.5%) (kNm-2) γ (kN/m-3) H (m) K0 δ (degrees) Displacement at zero dilation (mm) Relative density of soil Lateral loading on pipe Lateral bearing capacity factors (Nq) Limiting displacement in region A (we) Pipe properties Pipe outside diameter (mm) Pipe thickness (mm) Elastic modulus of the pipe (MPa)
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Pipe-soil interaction aspects in buried extensible pipes Weerasekara, Lalinda 2011
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Title | Pipe-soil interaction aspects in buried extensible pipes |
Creator |
Weerasekara, Lalinda |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | The performance of buried pipelines in areas subjected to permanent ground displacements is an important engineering consideration in the gas distribution industry, since the failure of such systems poses a risk to public and property safety. Although, the ground movements and its variations over time can be detected and mapped with reasonable confidence, these data are of little use due to a lack of reliable models to correlate such displacements to the condition of the buried pipe. The objective of this thesis is to develop methods to estimate the pipe performance based on the measured ground displacement. An analytical method was developed to estimate the pipe performance when the pipe is subjected to tensile loading caused by the relative ground movements occurring along the pipe axis. As a part of the derivation, a modified interface friction model was developed considering the increase in friction due to constrained dilation of the soil, and the impact of mean effective stress on soil dilation. This interface friction model was combined with a nonlinear pipe stress–strain model to derive an analytical solution to represent the performance of the pipe. Using the proposed model, axial force, strain, and mobilized frictional length along the pipe can be obtained for a measured ground displacement can be obtained. Large-scale field pipe pullout tests were performed to verify the results of the proposed analytical model, in which good agreements were observed for tests conducted at different soil/burial conditions, displacement rates and pipe properties. Considering the similarities in the axial pullout mechanism, the analytical model was extended to explain the pullout response of geotextiles buried in reinforced soil structures. In this derivation, a new interface friction model was developed for planar members by considering the changes in normal stress due to constrained soil dilation. Another analytical model was derived for the case of a pipe that is subjected to combined loading from axial tension and bending when the initial soil loading is acting perpendicular to the pipe axis. With the direct account of the axial tensile force development, more realistic pipe performance behaviors were obtained as compared to the results obtained from traditional numerical formulations. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-ShareAlike 3.0 Unported |
DOI | 10.14288/1.0050760 |
URI | http://hdl.handle.net/2429/38050 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-sa/3.0/ |
AggregatedSourceRepository | DSpace |
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