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Numerical study on the response of pile groups under lateral loading Fayyazi, Mohammad Sajjad 2015

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Numerical study on the response of pile groups underlateral loadingbyMohammad Sajjad FayyaziBachelor of Civil Engineering, University of Tehran, 2007Master of Structural Engineering, University of Tehran, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Civil Engineering)The University of British Columbia(Vancouver)August 2015c©Mohammad Sajjad Fayyazi, 2015AbstractWhen piles act in a group, soil–pile interaction reduces the lateral resistance of theindividual piles. A practical approach to characterize the group behavior in differ-ent pile groups is using appropriate factors such as p–multiplier or group reductionfactor. The experimental studies on pile groups are usually carried out on smallpile groups with close spacings and free-head condition. These limitations are dueto the difficulty and high cost of full scale testing particularly in larger pile groups.These limitations justify using three–dimensional numerical simulations to studylateral response of pile groups. This research focuses on group reduction factorsand p–multipliers to characterize the group effects in a wide range of pile groups.In order to systematically study the group reduction factors, a numerically derivedbenchmark database is established using a continuum approach to simulate the re-sponse of the pile groups. The capability of the numerical model in predicting thepile group behavior is first evaluated by three–dimensional continuum modeling ofthree field tests on actual pile groups. Then the continuum model is used to gener-ate benchmark database. The calculated group reduction factors compare well withavailable experimental data, which are typically extracted from small pile groups.Current study also covers a wide range of pile groups with different numbers ofpiles, various pile spacings and pile head condition for which there is no experi-mental data available in the literature. Furthermore, this study gives greater insightinto the interaction between piles based on their row position in the pile groupswith different layouts. To this end, carried load at the pile head and bending mo-ment profiles for different piles are compared based on their row position in thegroup when they are pushed simultaneously. The p–multipliers are also calculatedto quantify the contribution of different rows to the lateral resistance of the group.iiThe study shows that design guidelines such as AASHTO and FEMA P-751 over-estimate the group reduction factors and p–multipliers, hence the lateral resistance,in larger pile groups or pile groups with larger spacings, especially for fixed pilehead conditions.iiiPrefaceIn 2010, the Natural Sciences and Engineering Research Council of Canada (NSERC)funded a strategic project at the University of British Columbia (UBC) to study the“Soil-Structure Interaction in Performance Based Design of Bridges”. The projectbelonged in the strategic target area of Safety and Security and the proposed re-search topics dealt with Risk and Vulnerability and Resiliency of Systems. Profes-sors Ventura, Taiebat, and Finn, from the department of Civil Engineering at UBCwere leading the project. As a part of this project and an NSERC Discovery projecton “Advanced Computational Geomechanics for Analysis of Earth Structures” ledby Professor Taiebat, the present research aims to study the response of pile groupsunder lateral loading and to evaluate the widely used design guidelines in practice.The outputs of this thesis aid the advancement of the state of practice in this area.I, Mohammad Sajjad Fayyazi, am the principal contributor to all chapters ofthis thesis. I was responsible for all major areas of concept formation, data collec-tion and analysis, and writing the chapters. Some parts of the findings of this thesishave been published in a journal and two conferences so far.• M. S. Fayyazi, Taiebat, M., and Finn, W. D. L., Group reduction factors foranalysis of laterally loaded pile groups, Canadian Geotechnical Journal, vol.51, no. 7, pp. 758-769, 2014.This paper includes a version of chapter 4 and also some sections in chapters3 and 6. I conducted all the numerical analyses and wrote the first draft ofthe manuscript. Professors Taiebat and Finn were the supervisory authors onthis project and were involved throughout the project in concept formationand manuscript edits.iv• M. S. Fayyazi, Taiebat, M., Finn, W. D. L., and Ventura, C. E., Evaluationof group reduction factors for analysis of pile groups, Proceedings of theFifteenth World Conference on Earthquake Engineering. Lisbon, Portugal,9 pages, 2012This paper includes a version of chapter 4. Professors Taiebat, Finn, andVentura were the supervisory authors on this project and were involved through-out the project in concept formation and manuscript edits.• M. S. Fayyazi, Taiebat, M., Finn, W. D. L., and Ventura, C. E., Evaluation ofp-multiplier method for performance-based design of pile groups, Proceed-ings of the Second International Conference on Performance-Based Designin Earthquake Geotechnical Engineering. Taormina, Italy, 11 pages, 2012This paper includes a version of chapter 5. Professors Taiebat, Finn, andVentura were the supervisory authors on this project and were involved through-out the project in concept formation and manuscript edits.Additional papers are under preparation to publish the remainder of the thesis find-ings.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives and features . . . . . . . . . . . . . . . . . . . . . . . 31.3 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . 52 Soil–Pile Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Single pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Strain wedge method . . . . . . . . . . . . . . . . . . . . 82.2.2 p− y method . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Continuum method . . . . . . . . . . . . . . . . . . . . . 132.3 Pile group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13vi2.3.1 p-multiplier . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Group reduction factor . . . . . . . . . . . . . . . . . . . 152.4 Previous studies on pile groups . . . . . . . . . . . . . . . . . . . 152.4.1 Previous numerical studies . . . . . . . . . . . . . . . . . 162.4.2 Previous full scale tests . . . . . . . . . . . . . . . . . . . 172.4.3 Previous centrifuge tests . . . . . . . . . . . . . . . . . . 272.5 Gap in knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Model Development and Methodology for Analysis of Pile Groups . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.1 FLAC3D computer program . . . . . . . . . . . . . . . . 403.2.2 Continuum model development . . . . . . . . . . . . . . 413.2.3 Continuum model validation – primary validation . . . . . 473.3 p–y model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.1 GROUP program . . . . . . . . . . . . . . . . . . . . . . 533.3.2 p–y model development . . . . . . . . . . . . . . . . . . 533.4 Calculating p-multiplier and group reduction factor . . . . . . . . 543.4.1 Equivalent soil parameters in continuum and p–y models . 553.4.2 Adopted group reduction factor vs. average p-multiplier . 563.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Effects of Different Parameters on Group Reduction Factor . . . . . 804.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Investigated parameters . . . . . . . . . . . . . . . . . . . . . . . 814.3 Calculated group reduction factors . . . . . . . . . . . . . . . . . 814.4 Extended sensitivity analysis . . . . . . . . . . . . . . . . . . . . 834.4.1 Shear modulus distribution along the depth . . . . . . . . 834.4.2 Larger pile spacing . . . . . . . . . . . . . . . . . . . . . 844.4.3 Larger pile groups . . . . . . . . . . . . . . . . . . . . . 844.5 Importance of using appropriate group reduction factor . . . . . . 854.6 Comparison with experimental studies – secondary validation . . . 86vii4.6.1 Free-head pile groups . . . . . . . . . . . . . . . . . . . . 864.6.2 Fixed-head pile groups . . . . . . . . . . . . . . . . . . . 884.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 Evaluation of Interaction Between Rows of Piles . . . . . . . . . . . 1005.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Free-head pile groups . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.1 Lateral load distribution . . . . . . . . . . . . . . . . . . 1025.2.1.1 Bending moment profile . . . . . . . . . . . . . 1045.2.2 p-multipliers . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 Fixed-head pile groups . . . . . . . . . . . . . . . . . . . . . . . 1075.3.1 Lateral load distribution . . . . . . . . . . . . . . . . . . 1075.3.2 Bending moment distribution . . . . . . . . . . . . . . . 1085.3.3 p-multipliers . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Assessment of Design Guidelines . . . . . . . . . . . . . . . . . . . . 1386.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2 Available recommendations . . . . . . . . . . . . . . . . . . . . . 1396.2.1 AASHTO (2012) . . . . . . . . . . . . . . . . . . . . . . 1396.2.2 FEMA P-751 (2012) . . . . . . . . . . . . . . . . . . . . 1406.2.3 Reese and van Impe (2010) . . . . . . . . . . . . . . . . . 1406.3 Group reduction factors . . . . . . . . . . . . . . . . . . . . . . . 1436.4 p-multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467 Conclusions, Implications for Practice, and Recommended FutureResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.1 Conclusions and contributions . . . . . . . . . . . . . . . . . . . 1567.2 Implications for practice . . . . . . . . . . . . . . . . . . . . . . 1587.3 Recommended future research . . . . . . . . . . . . . . . . . . . 159Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161viiiA Calculating p–multipliers for Fixed Head Pile Groups . . . . . . . . 169ixList of TablesTable 2.1 Beam theory equations (Ting, 1987) . . . . . . . . . . . . . . 31Table 2.2 p-multipliers and group reduction factors from previous exper-imental studies . . . . . . . . . . . . . . . . . . . . . . . . . . 32Table 3.1 Properties of the soil profile in the continuum model for theWalsh (2005) pile group test . . . . . . . . . . . . . . . . . . . 58Table 3.2 Properties of the soil profile inside the pile group in the contin-uum model of the Christensen (2006) test . . . . . . . . . . . . 58Table 3.3 Properties of the soil profile outside the pile group in the con-tinuum model of the Christensen (2006) test . . . . . . . . . . 58Table 4.1 Simulated pile groups . . . . . . . . . . . . . . . . . . . . . . 91Table 4.2 Equivalent soil parameters for different friction angles of soil . 91Table 4.3 Influence of not using appropriate group reduction factor for3×3 fixed-head pile group (total applied load = 1500 kN, ϕ=35◦) 92Table 4.4 Influence of not using appropriate group reduction factor for6×6 fixed-head pile group (total applied load = 6000 kN, ϕ=35◦) 92Table 6.1 p-multipliers suggested in AASHTO (2012) . . . . . . . . . . 148Table 6.2 Laterally loaded pile groups studies used as the basis for recom-mendations of AASHTO (2012) for quantifying group effects(Hannigan et al., 2006) . . . . . . . . . . . . . . . . . . . . . 149Table 6.3 An example for calculation of p-multipliers and group reduc-tion factors based on AASHTO (2012) guidelines for 4×4 pilegroups with different spacing . . . . . . . . . . . . . . . . . . 149xTable 6.4 An example for calculation of p-multipliers and group reduc-tion factor based on Reese and van Impe (2010) guidelines fora 4×4 pile group with S/D of 3 . . . . . . . . . . . . . . . . . 149Table 6.5 Consideration of various design guidelines for some importantfactors related to group effects in lateral loading of square pilegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150xiList of FiguresFigure 2.1 Strain Wedge model concept (after Dodds and Martin, 2007) . 33Figure 2.2 Representation of (a) pile and nonlinear springs in p–y method,and (b) corresponding p–y curves . . . . . . . . . . . . . . . 34Figure 2.3 Graph presented in API (2007) for coefficients C1, C2, and C3 34Figure 2.4 Graph presented in API (2007) for initial modulus of subgradereaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.5 Illustration of shadowing effect and edge effect . . . . . . . . 35Figure 2.6 Definition of p-multiplier (Pm) . . . . . . . . . . . . . . . . . 36Figure 2.7 Bending moment profiles for two different piles from test (afterHolloway et al., 1982) . . . . . . . . . . . . . . . . . . . . . 36Figure 2.8 Distribution of load in a pile group for different cycles of load-ing (after Brown et al., 1988) . . . . . . . . . . . . . . . . . . 37Figure 2.9 Design curve for pile groups in sand (after Rollins et al., 2005) 37Figure 2.10 Laterally loaded pile groups in sand: a) effect of relative den-sity on group capacity, b) load distribution, total and for in-dividual rows, c) effect of pile spacing on total lateral resis-tance, d) influence of acceleration level during driving (lateralloads and displacements plotted at prototype scale) (after Mc-Vay et al., 1994) . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.1 Schematics of different approaches for modeling soil–pile in-teraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.3 Plan view for soil pile connection modeling with (a) 4, (b) 8and (c) 12 rigid connection beams . . . . . . . . . . . . . . . 59xiiFigure 3.2 Finite difference model of the 3×3 pile group . . . . . . . . 60Figure 3.4 Comparison of the pile head load-deflection curve for a singlepile with 4, 8, and 12 rigid connection beams at each node . . 60Figure 3.5 Different levels of mesh refinement for a 3×3 pile group . . . 61Figure 3.6 Comparison of the total group load-deflection curves for dif-ferent levels of mesh refinement . . . . . . . . . . . . . . . . 62Figure 3.7 Finite difference mesh of a pile group . . . . . . . . . . . . . 62Figure 3.8 Comparison of the total group load-deflection curves for dif-ferent dimensions of the model. (X: loading direction, Y: per-pendicular to the loading direction , D: Pile diameter) . . . . . 63Figure 3.9 Comparison of the total group load-deflection curves for dif-ferent loading rates . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.10 Analysis progress in 200min using different number of CPUs . 64Figure 3.11 Plan view of the full scale pile group test by Brown et al. (1988) 65Figure 3.12 Measured and computed load–deflection curves for the singlepile test by Brown et al. (1988) . . . . . . . . . . . . . . . . . 65Figure 3.13 Finite difference model of the test by Brown et al. (1988) . . . 66Figure 3.14 Comparison of the measured and computed total group load-deflection curves for different rows of pile for the full scale testby Brown et al. (1988) . . . . . . . . . . . . . . . . . . . . . 66Figure 3.15 Soil profile for the test; adopted from Walsh (2005) . . . . . . 67Figure 3.16 Profile of qc values from CPT tests close to the pile group;adopted from Walsh (2005) . . . . . . . . . . . . . . . . . . . 68Figure 3.17 Plan view of the full scale pile group test by Walsh (2005) . . 68Figure 3.18 Instrumentation for load and deflection (Walsh, 2005) . . . . . 69Figure 3.19 Finite difference model of the test by Walsh (2005) . . . . . . 69Figure 3.20 Computed and measured total group load-deflection curves forthe test by Walsh (2005) . . . . . . . . . . . . . . . . . . . . 70Figure 3.21 Computed and measured bending moment for center pile ofdifferent rows at the head deflection of 6 mm for the test byWalsh (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 70xiiiFigure 3.22 Computed and measured bending moment for center pile ofdifferent rows at the head deflection of 19 mm for the test byWalsh (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.23 Computed and measured bending moment for center pile ofdifferent rows at the head deflection of 38 mm for the test byWalsh (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.24 Plan view of the full scale pile group test by Christensen (2006) 72Figure 3.25 Profile of qc values for the upper layers adapted from Chris-tensen (2006) and selected values for simulation in this study . 73Figure 3.26 Finite difference model of the test by Christensen (2006) . . . 74Figure 3.27 Computed and measured total group load-deflection curves forthe test by Christensen (2006) . . . . . . . . . . . . . . . . . 75Figure 3.28 Computed and measured bending moment for center pile ofdifferent rows at the head deflection of 6 mm for the test byChristensen (2006) . . . . . . . . . . . . . . . . . . . . . . . 75Figure 3.29 Computed and measured bending moment for center pile ofdifferent rows at the head deflection of 51 mm for the test byChristensen (2006) . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.30 Soil layers input in GROUP program . . . . . . . . . . . . . 77Figure 3.31 Methodology for calculating p-multiplier using continuum andp–y models (Pm: p-multiplier) . . . . . . . . . . . . . . . . . 77Figure 3.32 Methodology for calculating group reduction factor using con-tinuum and p–y models (P¯m : Group reduction factor) . . . . . 78Figure 3.33 Dimensions of the single pile used for obtaining equivalent soilproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 3.34 Methodology for obtaining equivalent soil parameters in con-tinuum and p–y models . . . . . . . . . . . . . . . . . . . . . 79Figure 3.35 Comparison of load–deflection curves (a), and bending mo-ment profiles at different pile head deflections (b), for a singlepile in soil with ϕ = 35◦ computed by the p–y model and thecontinuum model . . . . . . . . . . . . . . . . . . . . . . . . 79xivFigure 4.1 Calculated group reduction factors for different pile group set-tings: (a) 3 3; (b) 4 4; (c) 5 5; (d) 6 6 pile groups . . . . . . 93Figure 4.2 Corresponding profiles of Vs with uniform or parabolic distri-butions of shear modulus G in depth (for the case of φ = 35◦) . 94Figure 4.3 Comparison of load–deflection curves (a), and bending mo-ment profiles at different pile head deflections (b), for a singlepile in soil with φ = 35◦ calculated by the p–y model and thecontinuum model with parabolic distribution of G . . . . . . . 94Figure 4.4 Calculated group reduction factors for 3×3 pile groups usingsoil profiles with φ= 35◦, and with uniform or parabolic distri-butions of G in depth . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.5 Calculated group reduction factors for 3×3 pile groups withS/D values ranging from 3 to 10 (φ= 35◦ and uniform G) . . . 95Figure 4.6 Calculated group reduction factors for 10×10 pile groups (φ=35◦ and uniform G) . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.7 Comparison between calculated group reduction factors andprevious experimental works for 3×3 free-head pile groups insand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.8 Comparison between calculated group reduction factors andprevious experimental works for 4×4 free-head pile groups insand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 4.9 Comparison between calculated group reduction factors andprevious experimental works for 5×5 free-head pile groups insand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 4.10 Comparison between calculated group reduction factors andprevious experimental works for 3× 3 fixed-head pile groupsin sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.11 Comparison between calculated group reduction factors andprevious experimental works for 4× 4 fixed-head pile groupsin sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.12 Comparison between calculated group reduction factors andprevious experimental works for 5× 5 fixed-head pile groupsin sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99xvFigure 4.13 Comparison between calculated group reduction factors andprevious experimental works for 6× 6 fixed-head pile groupsin sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 5.1 Schematic representation of the simulated pile groups in thisstudy where n=3, 6 and 10, and S/D =3 and 6; (a) plane view(b) side view . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.2 Pile positions in regards to the loading direction in the 6× 6pile group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 5.3 Load distribution at the deflection of 5 cm for piles based ontheir position in different rows in a 6×6 free-head pile groupwith S/D = 3 (a), and S/D = 6 (b) . . . . . . . . . . . . . . . 114Figure 5.4 Average pile head loads for different rows in 3× 3 free-headpile groups with S/D = 3 (a) and S/D = 6 (b) . . . . . . . . . 115Figure 5.5 Normalized average pile head loads over the entire group fordifferent rows in 3×3 free-head pile groups with S/D = 3 (a)and S/D = 6 (b) . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 5.6 Average pile head loads for different rows in 6× 6 free-headpile groups with S/D = 3 (a), and S/D = 6 (b) . . . . . . . . 117Figure 5.7 Normalized average pile head loads over the entire group fordifferent rows in 6×6 free-head pile groups with S/D = 3 (a),and S/D = 6 (b) . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 5.8 Average pile head loads for different rows in 10×10 free-headpile groups with S/D = 3 (a), and S/D = 6 (b) . . . . . . . . 119Figure 5.9 Normalized average pile head loads over the entire group fordifferent rows in 10× 10 free-head pile groups with S/D = 3(a), and S/D = 6 (b) . . . . . . . . . . . . . . . . . . . . . . 120Figure 5.10 Average bending moment profiles for different rows in 3× 3free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pilehead deflection: 5cm) . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.11 Average bending moment profiles for different rows in 6× 6free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pilehead deflection: 5cm) . . . . . . . . . . . . . . . . . . . . . . 122xviFigure 5.12 Average bending moment profiles for different rows in 10×10free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pilehead deflection: 5cm) . . . . . . . . . . . . . . . . . . . . . . 123Figure 5.13 Calculated p-multipliers for 3× 3 free-head pile groups withS/D = 3 (a), and S/D = 6 (b) . . . . . . . . . . . . . . . . . 124Figure 5.14 Calculated p-multipliers for 6× 6 free-head pile groups withS/D = 3 (a), and S/D = 6 (b) . . . . . . . . . . . . . . . . . 125Figure 5.15 Calculated p-multipliers for 10×10 free-head pile groups withS/D = 3 (a), and S/D = 6 (b) . . . . . . . . . . . . . . . . . 125Figure 5.16 Calculated p-multipliers versus available experimental data for3×3 free-head pile groups with S/D = 3 (a), and S/D = 6 (b) 126Figure 5.17 Average pile head loads for different rows in the 3× 3 free-and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b) . 127Figure 5.18 Normalized average pile head loads over the entire group fordifferent rows in the 3×3 free- and fixed-head pile groups withS/D = 3 (a) and S/D = 6 (b) . . . . . . . . . . . . . . . . . . 128Figure 5.19 Average pile head loads for different rows in the 6× 6 free-and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b) . 129Figure 5.20 Normalized average pile head loads over the entire group fordifferent rows in the 6×6 free- and fixed-head pile groups withS/D = 3 (a) and S/D = 6 (b) . . . . . . . . . . . . . . . . . . 130Figure 5.21 Average pile head loads for different rows in the 10×10 free-and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b) . 131Figure 5.22 Normalized average pile head loads over the entire group fordifferent rows in the 10× 10 free- and fixed-head pile groupswith S/D = 3 (a) and S/D = 6 (b) . . . . . . . . . . . . . . . 132Figure 5.23 Average bending moment profiles for different rows in the 3×3 free- and fixed-head pile groups with S/D = 3 (a) and S/D =6 (b) (pile head deflection: 5cm) . . . . . . . . . . . . . . . . 133Figure 5.24 Average bending moment profiles for different rows in the 6×6 free- and fixed-head pile groups with S/D = 3 (a) and S/D =6 (b) (pile head deflection: 5cm) . . . . . . . . . . . . . . . . 134xviiFigure 5.25 Average bending moment profiles for different rows in the 10×10 free- and fixed-head pile groups with S/D = 3 (a) and S/D =6 (b) (pile head deflection: 5cm) . . . . . . . . . . . . . . . . 135Figure 5.26 Calculated p-multipliers for 3× 3 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b) . . . . . . . . . 136Figure 5.27 Calculated p-multipliers for 6× 6 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b) . . . . . . . . . 136Figure 5.28 Calculated p-multipliers for 10× 10 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b) . . . . . . . . . 137Figure 6.1 Different pile group configurations with respect to loading di-rection: two side by side piles (a), two in-line pile (b), threepiles in a row (c), and 2x2 pile group (d) . . . . . . . . . . . 151Figure 6.2 Comparison between calculated group reduction factors in thisstudy, AASHTO (2012), FEMA P-751 (2012), and Reese andvan Impe (2010) recommendations . . . . . . . . . . . . . . . 152Figure 6.3 Comparison between calculated p-multipliers in this study, AASHTO(2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 3×3 pile groups . . . . . . . . . . . . 153Figure 6.4 Comparison between calculated p-multipliers in this study, AASHTO(2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 6×6 pile groups . . . . . . . . . . . . 153Figure 6.5 Comparison between calculated p-multipliers in this study, AASHTO(2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 10×10 pile groups . . . . . . . . . . . 154Figure A.1 Methodology for calculating p–multiplier for fixed head pilegroups using continuum and p–y models (Pm : p–multiplier) . 170Figure A.2 Demonstration of the challenge for calculating p–multipliersusing GROUP:(a)Needed fixity for calculating p–multipliers(b)Pile to the cap fixity in the GROUP . . . . . . . . . . . . . 170Figure A.3 Solution for modeling a row of fixed head piles in GROUP . . 171Figure A.4 Verifying the GROUP solution using SAP2000 . . . . . . . . 171xviiiAcknowledgmentsI am deeply grateful to my supervisor, Dr. Mahdi Taiebat. Not only for his out-standing guidance in research and breadth of geotechnical knowledge but also forhis immeasurable support throughout my time at University of British Columbia(UBC). He helped me to drive my research by linking the small details to the largerpicture. The completion of this thesis would not have been possible without hisinsightful comments and valuable feedback. His encouragement and high level ofcompetence inspired me to explore the subject.I would like to express my gratitude to Dr. Liam Finn. His constant pursuit ofknowledge inspired me to strive for a thorough understanding of my research dataand its implications. His valuable comments and guidance for my research werecrucial to achieve this final result. I would also like to acknowledge Dr. CarlosVentura for his guidance in my research work and his committed collaboration inthis research.I am grateful to be a part of the Theoretical and Applied Geomechanics (TAG)research group at UBC during my studies. They have been both my colleagues andfriends throughout my years at UBC. I am grateful to have had the opportunity toshare ideas during meetings.Supports to conduct this study which is provided by the Natural Sciences andEngineering Research Council of Canada (NSERC) and the Office of GraduateStudies at the UBC are gratefully acknowledged.Throughout my years at UBC I was very fortunate to have the company offriends Amin Rahmani, Elnaz Amirzehni, Abbas Javaherian and Ehsan Esfahanian.I will remember fondly our time together.I have reserved the last part for the most important people in my life; my family.xixI cannot thank my parents, Esmaeil and Tahereh, enough from whom I have beenconstantly away. Special thanks to my lovely brothers, Saleh and Saber, who havebeen taking care of them and tried to fill my missing spot. I owe my dear wife,Sahar, a depth of gratitude for her patience, understanding and sacrifices duringthe past years and continuous support and encouragements, which enabled me toconfront challenges that came to the path.xxDedicationTo my parents,for their unconditional love and supportAnd to my wife,for her incredible support, understanding, and companionship.xxiChapter 1Introduction1.1 OverviewScientists study the world as it is; engineers create the world that hasnever been. — Theodore von KarmanPiles are mainly designed to sustain vertical loads, but often significant lat-eral loads may be present and must be accounted for in the design. These lateralloads can come from different sources such as wind forces, collisions, wave or iceimpacts, slope failure, or earthquakes.The lateral response of piles is typically analyzed using the beam on the non-linear Winkler springs model. The nonlinear springs are usually based on the p–ycurves recommended by the American Petroleum Institute (API, 2007). Piles foun-dations are sometimes used as single piles but most often used in groups. Whenpiles act in a group, the individual responses of piles within the group are influ-enced by the presence and actions of neighboring piles such that the group willgenerally exhibit less lateral capacity than the sum of the lateral capacities of theindividual piles. API (2007) p–y curves are proposed only for single piles. In thedesign of pile groups under lateral loading, characterizing the group interactionand its effect on the lateral resistance of the foundation is one of the substantialchallenges.1One of the most common methods of accounting for the group effects in theWinkler model is to modify the single pile p–y curves using a p-multiplier, assuggested by Brown et al. (1988). In this approach, the soil resistance, p, is reducedby a constant factor, Pm. The p-multipliers in pile group design depend on the rowspacing in the loading direction. The p-multiplier for a leading row is higher thanthe p-multiplier for a trailing row.In an alternative approach, rather than definingp-multipliers row by row, an average p-multiplier for all piles in the group is used,which gives the same pile cap load–deflection curve (Brown et al., 2001). Thisaverage p-multiplier is called the group reduction factor. Use of a group reductionfactor is convenient for seismic and cyclic loading because the direction of loadingchanges repeatedly and often unpredictably during the loading event. Each loadreversal converts a leading row, with a high p-multiplier, to a trailing row, with alow p-multiplier, instantaneously. This makes using an average p-multiplier for allpiles reasonable.The p-multipliers can be obtained using experimental studies such as field tests.However, field data are limited, given that the full scale testing of large pile groupsis not feasible. The capacity of the loading equipment also limits the size of thepile group that can be tested. Therefore, full scale tests are usually carried out onsmall pile groups with close spacings. Model tests in the laboratory or centrifugehave offered some insight, but these tests are few in number, group configurationsare still limited in size, and interpretation of results is marred by scaling issues.Most of the pile group experiments were performed on 3× 3 free-head pilegroups with center to center spacing of three pile diameters (D) and pile head de-flections of up to 5 cm. Because p-multipliers were typically derived from free-head pile group tests, there are some uncertainties regarding their applicability forfixed-head conditions, which are more routinely encountered in engineering prac-tice where a pile cap is used (Rollins and Sparks, 2002). Free- and fixed-headconditions refer only to the rotational degree of freedom at the pile head. Trans-lational degree of freedom at the head is free for all pile groups discussed in thisstudy. Lack of a comprehensive study on larger pile groups, various pile spacings,pile head conditions, and soil properties as well as the limitations in the availableexperimental database justify using numerical simulations to study the effects ofthese different parameters on the lateral response of the pile group.2The fundamental nature of the laterally loaded pile problem is three-dimensional;therefore in order to properly assess behavior of the pile groups the use of three-dimensional numerical models is appropriate. Previous numerical studies such asWakai et al. (1999), Yang and Jeremic (2003), Comodromos and Pitilakis (2005),and Dodds and Martin (2007) indicate that such a numerical approach is capableof providing a realistic assessment of pile-soil behavior, thus allowing a rationalmeans for assessing the mechanics at play.1.2 Objectives and featuresAn extensive literature review was conducted, and the lack of a comprehensivestudy on the lateral response of the pile groups with different layouts promptedthe numerical approach undertaken here. Given the prevalence of p–y curves inpractice, emphasis on determination of appropriate factors like p-multipliers andgroup reduction factors for different pile groups is considered as the most practicalway to account for the group behavior. This research focuses on p-multipliers andgroup reduction factors to characterize group effects and the influence of interac-tion between piles on the lateral resistance of the pile groups. Square pile groupsof vertical piles equally spaced in each direction are investigated.More specific components and features of this research are as follows:• The capability of the numerical model to predict the pile group behavior isevaluated by three-dimensional continuum modeling of three field tests onactual pile groups and comparing the calculated pile head load-deflectioncurves and bending moment profiles with the measured values (primary val-idation).• To extend the available data for group reduction factors, comprehensive setsof p–y and continuum analyses are employed to develop group reductionfactors for pile groups in sand with different properties. These factors arecalculated for a wide range of pile groups.– Number of piles: 3× 3, 4× 4, 5× 5, 6× 6, and 10× 10 pile groupconfigurations are considered to cover both small and large pile groups.3– Center to center pile spacing: For each pile group configuration spac-ings of 3D, 4D, 5D, and 6D are considered to investigate the effect ofpile spacing. To obtain a better insight into this effect at a very largepile spacing, pile spacing of 10D is also considered for the 3× 3 pilegroup.– Pile head fixity: Because almost all the available experimental data arederived from pile groups with free-head conditions, each pile group issimulated with both free-head and fixed-head conditions to investigatethe effect of pile head fixity on the response of pile groups.– Soil parameters: Soil is simulated using an elastic – perfectly plasticsoil model with the non-associative flow rule. Three uniform soil pro-files with different friction angles are considered: 30◦, 35◦, and 40◦.• After the group reduction factors for different cases are calculated, theyare compared with previous experimental works to established an additionallevel of validation on the numerical modeling approach (secondary valida-tion). For this purpose wide range of experimental studies in sand with thesame or nearly the same pile arrangements, pile head conditions, and alsogeneral soil type are selected from previously gathered data in the literaturereview.• To get a better insight into the interaction between piles in different rows, thelateral response of piles in the pile groups is investigated in terms of– Load distribution between different rows– Bending moment profile along the piles– p-multipliers for different rowsAfter consideration of the huge amount of simulated data for each pile groupconfiguration, a limited number of configurations are selected to study theinteraction between rows in the pile group. The results for pile groups with3×3, 6×6, and 10×10 configurations with center to center pile spacing of3D and 6D are presented to show the response trends in the pile groups. Thisrange is selected to cover a range of small pile groups to large pile groups.4Center to center spacings of 3D and 6D are considered for each pile group,which represents pile groups with close spacing and wide spacing, respec-tively. For each configuration both free-head and fixed-head conditions areconsidered. For each pile group p-multipliers are calculated and comparedwith the available recommendations to achieve a better understanding aboutcontribution of piles in different rows to the total lateral resistance.• Calculated p-multipliers are compared with available experimental data as asecondary validation for the results of the study.• Calculated group reduction factors and p-multipliers in this study cover awide range of pile groups. However, available recommendations and designguidelines are based on a very limited database. Calculated values for differ-ent pile group configurations are compared with available recommendationssuch as AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe(2010). The differences between calculated results and each recommenda-tion are discussed to point out the deficiencies in current practice.1.3 Organization of thesisThis dissertation consists of the following chapters:Chapter 2 reviews the literature comprehensively. This chapter starts with dif-ferent methods for analysis of a single pile under lateral loading. The group effectsin pile groups are introduced afterwards. The p-multiplier and the group reductionfactor are explained; these are the common tools used to account for group effectsin pile groups. Previous numerical and experimental studies on the lateral loadingof pile groups are also reviewed in this chapter.Chapter 3 presents the tools and methodologies used in this research to fulfillthe objectives listed in section 1.2. First, the numerical models (continuum and p–y) are described, and the processes for simulation of a pile group in the continuumand p–y models are explained. Then, the continuum model is validated againstthree full scale tests to make sure that we have the right tool (primary validation).After initial validation, the methodologies for calculating the p-multiplier and thegroup factor using these validated numerical models are described.5Chapter 4 offers an evaluation of the influence of different parameters on theoverall response of the pile groups through the calculation of group reduction fac-tors for different pile group configurations. The group reduction factor can charac-terize the group effects in the pile group; also, in practice, use of the group reduc-tion factor is a very common method for analysis of laterally loaded pile groups. Inorder to systematically study the group reduction factors in various pile group prob-lems, a numerically driven benchmark database is established using a continuummodeling approach to simulate the response of pile groups. Then this numericallybased database of calculated group reduction factors is compared with availableexperimental data on pile groups to validate the results (secondary validation).Chapter 5 provides an insight on the interaction between piles inside the pilegroup. For this purpose, continuum simulations of different pile group configura-tions subjected to static lateral loading are used to study the response pile groups.Response of piles is demonstrated based on their row position in the group. Lateralload distribution, bending moment profiles, and p-multipliers for different rows arediscussed for each pile group.In Chapter 6, group reduction factors calculated in Chapter 4 and p-multipliersfor different pile groups calculated in Chapter 5 are compared with recommenda-tions by AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe (2010),which are common design guidelines for pile group analysis. Shortcomings of theavailable recommendations are discussed based on the results of this study.Finally, Chapter 7 presents a summary of the thesis, the conclusions drawnfrom this research, and directions for some future research on the lateral responseof pile groups.6Chapter 2Soil–Pile InteractionEngineering is the profession in which a knowledge of themathematical and natural sciences gained by study, experience, andpractice is applied with judgment to develop ways to utilize,economically, the materials and forces of nature for the benefit ofmankind. — - Engineers Council for Professional Development(1961/1979)2.1 IntroductionPile foundations are widely used in special structures like bridges, high-rise build-ings and towers. In practice, piles are sometimes used as single piles but they areusually put into pile groups. In designing a pile group, vertical loads are importantbut in addition to vertical loads significant lateral loads may be present and must betaken into account. When a pile group is subjected to the lateral load, the inducedinteraction between the piles and the surrounding soil can be very complicated.The nonlinear nature of soil response is an obvious source of complexity andthe interaction of pile behavior on the surrounding soil increases this complex-ity. Presence of other piles in the vicinity of each pile provides further complexitythrough pile–soil–pile interaction. Understanding single pile behaviour under lat-eral loading helps to comprehend general soil–pile interaction issues as well asproviding a basis for assessing the lateral response of the pile groups. Therefore,single pile behavior is reviewed in the current chapter. Afterwards, a review of pile7groups and group effect occurring in the pile groups are discussed. Also, previousstudies on pile groups are discussed in this chapter to reveal the gap in knowledgein analysis of pile groups under lateral loading.2.2 Single pileThere are some methods like those suggested by Brinch-Hansen (1961) and Broms(1964a,b) for estimating the ultimate lateral resistance of single piles directly.However, they are not widely used because these methods do not provide the cor-responding information about the lateral deflections (Tehrani, 2009). The mainapproaches for analyzing soil–pile interaction are the strain wedge (SW) method,p–y method, and continuum method. Each one of these approaches are describedbriefly in the following sections.2.2.1 Strain wedge methodThe strain wedge (SW) method has been developed to predict the lateral responseof a flexible pile (Ashour and Norris, 2000, Ashour et al., 1998, Norris, 1986). TheSW model parameters are related to the passive wedge of soil developing in frontof the pile. The SW model relates stress-strain-strength behavior of the layered soilin the wedge to Beam on Elastic Foundation (BEF) parameters. Therefore, the SWmodel is able to provide a theoretical link between the more complex 3D soil–pileinteraction and the simpler one dimensional BEF characterization.Figure 2.1 shows the concept of SW model. The geometry of the passive failurewedge is characterized by the mobilized effective friction angle of the soil (φm), thedepth of pile deflection under lateral load (h), and mobilized base angle βm= 90-θm,where θm= 45◦ -φm. For the calculation of soil reaction at each stress level (SL)in the soil, the initial pile deflection pattern is assumed to be linear (being zero atdepth h and reaching to a nominal value at the surface) and an iterative procedureis used to determine φm and h for a given head load. Iteration terminates whenequilibrium between mobilized geometry of the passive wedge and the deflectedpattern of the pile is satisfied (refer to Ashour et al. 1998 for more details).82.2.2 p− y methodThe p–y method is the most common analysis tool for pile foundations in practice.In this approach resistance of the surrounding soil is represented by a series ofnonlinear springs which are attached to the pile as shown in Figure 2.2. The prop-erties of these nonlinear springs are defined by the so called p–y curves. Figure2.2 also shows that p–y curves get stiffer progressively with depth. The pile in thismethod is modeled with beam elements. Table 2.1 shows the basis for obtainingsoil resistance in the field using beam theory equations.For a laterally loaded pile, soil loads are deducted from bending moment in-duced in the pile:p = d2Mpdz2= EpIpd4ydz4(2.1)where Mp = EpIp(d2y/dz2)is moment in pile at the depth of z, y is lateral deflec-tion of the pile at depth of z, EpIp is flexural rigidity of pile and p is soil reactionat depth of z.McClelland and Focht (1958) introduced the procedure for obtaining p and yfor laterally loaded piles. They used bending moment profiles at different stagesof their lateral load test and derived corresponding p–y relations according to theirtest. Matlock (1970) and Reese et al. (1974) also have developed the concept ofp–y curves for solving laterally loaded pile problems. Since the initial work ofMcClelland and Focht (1958) numerous studies have been undertaken to calibratethe p–y curves for different depths, soil types or pile geometries.API (2007) p–y curvesThe most commonly used p–y curves in engineering practice are those provided byAmerican Petroleum Institute (API, 2007). These recommendations are based onthe works of Murchison (1983) and Matlock (1970). The API suggests differentp–y curves for sand, soft clay and stiff clay which are also referred as standard p–ycurves.Sand API suggestions for sand are based on a work by Murchison (1983). Theultimate lateral bearing capacity for sand has been found to vary from a9value at shallow depths (pus) to a value at deep depths determined (pud). Ata given depth the equation giving the smallest value of pu should be used asthe ultimate bearing capacity:pus = (C1×H +C2×D)× γ×H (2.2)pud = C3×D× γ ′×H (2.3)wherepu: ultimate resistance (force/unit length), lbs/in (kN/m) (s = shallow, d =deep),γ ′: effective soil weight, lb/in3 (kN/m3),H: depth, in (m),C1,C2,C3: coefficients determined from Figure 2.3 as function of φ ′,φ ′: angle of internal friction of sand, deg,D: average pile diameter from surface to depth, in (m).The lateral soil resistance-deflection (p–y) relationship for sand is:p = A× pu× tanh[k×HA× pu× y](2.4)whereA: factor to account for cyclic or static loading condition:A = 0.9 for cyclic loadingA = 3.0–0.8(H/D) ≥ 0.9 for static loadingpu: ultimate bearing capacity at depth H (minimum of pus and pud), lbs/in(kN/m)k: initial modulus of subgrade reaction, lb/in3 (kN/m3). Determine fromFigure 2.4 as function of angle of internal friction, φ ′ , or relative density DrH: depth, inches (m)y: lateral deflection, inches (m)10Clay The p–y curves derived from a full scale test by Matlock (1970) are used inAPI (2007) for soft clays. To determine p–y curves for laterally loaded pilesin soft clays the following relation is recommended in API:p =0.5pu (y/yc)1/3 y/yc < 8pu y/yc ≥ 8(2.5)wherey: lateral deflection, in (m)yc: 2.5εcD, in (m),εc: strain which occurs at one-half the maximum stress on laboratory uncon-solidated undrained compression tests of undisturbed soil samplespu: ultimate resistance, psi (kPa) (for obtaining pu please refer to API (2007))API (2007) does not have any relationship for stiff clays but it refers to thework of Reese et al. (1975) for stiff clays.Limitations of p− y methodAlthough p–y method has proven to be a popular and effective method for analysisof pile groups under lateral loading, it has some shortcomings. The standard p–ycurves were derived from a number of field tests that reflect a limited set of con-figurations. The reliability of standard p–y curves was questioned by Murchisonand O’Neill (1984) and Gazioglu and O’Neill (1984) using data from multiple fieldtests. Some concerns related to standard p–y curves are listed below:• Discrete approach: In this method each layer of soil responds independentlyfrom adjacent layers. This is primarily due to the fact that the set of discretesprings that are used to represent the soil-pile interaction cannot representthe continuum nature of this phenomenon. This assumption ignores the sheartransfer between layers of soil.• Installation effect: The soil surrounding a pile experiences changes due tothe installation process but there is no specific consideration in API (2007)11regarding the installation effect. According to Broms (1964a,b) driven pilesaffect the soil within a distance of approximately one pile diameter aroundthe pile, causing increases in the relative density of cohesionless soils andimmediate decreases of stiffness and strength in cohesive soils. Huang et al.(2001) and also Brown et al. (2001) performed lateral load tests on bored anddriven pile groups for the Taiwan High Speed Rail Authority. They reportedevident differences between p–y curves of driven piles and bored piles.• Pile head condition: In current practice p–y curves are considered to beindependent of the pile head condition. Matlock (1970) conducted lateralloading tests on fixed-head and free-head steel piles in soft clay and reportedthe same p–y curves. Jamiolkowski and Garassino (1977) discussed the pufor p–y curves and noted the suggestion of de Beer that the value of pu maystrongly depend on pile rigidity and boundary conditions at the top of thepile. Ashour and Norris (2000) evaluated effects of different parameters onp–y curves using Strain Wedge model and showed that the effect of pile headcondition is one of the significant factors that determines the p–y curve.• Pile material: Piles are built in manufactured processes and in most casesthey are idealized as an elastic element in the soil-pile modeling. For steelpiles, this is generally the case but for reinforced concrete piles, the issue ofnonlinear behavior must be considered because of the cracking phenomenonwhich can cause changes in the pile response. Using Strain Wedge model,Ashour et al. (2001) showed that the nonlinear pile material has some effectson the lateral response and capacity of the pile. These effects depend onthe level of bending moment (level of loading). The p–y curves for crackedsection are affected by the change of bending moment, reduced bending stiff-ness, and the change of deflection pattern of the pile. Particularly in confinedconcrete piles, this nonlinearity has a significant impact on the p–y curves.122.2.3 Continuum methodThe major weakness of the p–y method is the independence of each spring fromothers. However, in reality soil should be considered as continuum media. FiniteDifference (FD) and Finite Element (FE) methods are two numerical approachesfor discretisation and solution of the problem which consider the soil as continuummedia.The main advantage of the continuum approach compared to other approachesis that less simplifications need to be made in simulation of a real problem. Otherfeatures include the ability to apply any combination of axial, torsion, and lateralloads, the capability of considering the nonlinear behavior of structure and soil, andthe capability to model soil–pile-structure interactions. The continuum approachpotentially provides the most powerful means for performing Soil–Structure In-teraction (SSI) analyses, but it has not yet been fully realized as a practical tool.Challenges to successful implementation of numerical technique lie in the selec-tion of appropriate constitutive models for materials for the soil and the pile, andthe accurate assignment of the parameter values for these models. The selectionor development of constitutive models and quantification of their parameters arestill open areas of research. The degree of uncertainty associated with the spec-ification of model parameters as well as laborious mesh generation and interpre-tation of results, often renders the continuum approach a secondary option (to thep–y approach) in engineering practice (Tehrani, 2009). Also, performing three-dimensional continuum analysis requires considerable time for generating inputand interpreting results. Therefore, continuum approach has been mostly used forresearch on pile group behavior, rarely for design.2.3 Pile groupIn practice, piles are sometimes used as single piles to transmit a column load to adeeper and stronger soil layer, but they are generally used in groups. The responseof a single pile can be drastically different from the response of a similar pile in apile group. When piles act in a group, pile–soil–pile interaction reduces the lateralresistance of the individual piles so that the group will generally exhibit less lateralresistance than the sum of the lateral resistances of the individual piles.13In the pile group, each pile pushes against the soil in front of it, creating ashear zone in the soil. These shear zones begin to enlarge and overlap as the lateralload increases. More overlapping occurs if the piles are spaced close to each other.In this context the term edge effect is used to describe the effect of overlappingzones of influence between piles in the same row. The term shadowing effectis used to describe the effect of overlapping zones of influence between piles indifferent rows (Larkela, 2008). Figure 2.5 describes the edge effect and shadowingeffect in a laterally loaded pile group. Leading row and trailing rows are definedbased on the direction of loading as shown in Figure 2.5. Group effect reducesthe lateral resistance of each individual pile in the pile group compared to a singlepile. Because of the group effect in the pile group, soil surrounding each pile hasless resistance than soil surrounding a single pile therefore it allows the pile groupto deflect more for the same load per pile. More deflection in the group causeslarger maximum bending moment in a pile within a group compared to a singlepile. Test measurements indicate that the leading row of the piles in the group willcarry higher loads than the piles in the trailing rows at the same deflection. Groupeffect is expected to become less significant as the spacing between piles increasesbecause there is less overlap between adjacent zones of influence.2.3.1 p-multiplierOne of the most common methods of accounting for the group effects in Winklersapproach is to modify the single pile p–y curves using a p-multiplier, as suggestedby Brown et al. (1988). In this approach, the soil resistance, p, is reduced by aconstant factor, Pm, as shown in Figure 2.6. Pm can not be bigger than 1. Basedon available recommendations like AASHTO (2012) and FEMA P-751 (2012),defining p-multipliers for pile group design relies on the row spacing in the loadingdirection. The p-multiplier for a leading row is higher than the p-multiplier for atrailing row because of the shadowing effect. Given the relative prevalence of p-y curves, emphasis on determination of appropriate p-multipliers for a given pilespacing is considered as the most appropriate means to analyze pile group responseunder lateral loading.142.3.2 Group reduction factorLeading row has higher p-multiplier than trailing rows. In an alternative approach,rather than defining p-multipliers row by row, an average p-multiplier is used forall piles in the group which gives the same pile cap load–deflection curve (Brownet al., 2001). This average p-multiplier is called the group reduction factor. Use ofa group reduction factor is convenient for seismic and cyclic loading because thedirection of loading changes repeatedly and often unpredictably during the loadingevent, and each load reversal converts a leading row, with high p-multiplier, to atrailing row, with low p-multiplier, instantaneously.2.4 Previous studies on pile groupsTo find the gap of knowledge, previous numerical and experimental research per-formed by different researchers are reviewed in this section. Several field andlaboratory tests have been performed by different researchers to evaluate the groupeffect, obtain parameters like p-multipliers or validate analytical methods. Theexisting lateral load tests on pile groups can be divided into these categories:• Full scale tests: These tests are conducted in the field. They have advantagesof real piles, real soil and realistic soil–pile condition. It is, however, verydifficult and expensive to perform a full scale test on a pile group. Thecapacity of the loading equipment also limits the size of the pile group thatcan be tested. Therefore, full scale tests are usually carried out on small pilegroups with close spacings.• Centrifuge tests: These tests are conducted in the laboratory. Centrifugetests are more economical and easier to modify and repeat for parametricstudies in comparison with full scale tests. The main advantage of centrifugetesting is that the gravitational stress field in the model can replicate the pro-totype. This consideration is crucial when testing materials such as sand forwhich the stress–strain behavior is a function of confining pressure (Mey-mand, 1998).Later on, in this study, many of the experimental works will be used for vali-dation and comparison. Also, design guidelines are based on some of these exper-15iments. Therefore, a brief summary of some of the studies that has been done onlaterally loaded pile groups is provided in the next section.2.4.1 Previous numerical studiesSome analytical models like the ones by Poulos and Randolph (1983) have uti-lized elasticity-based relationships for modelling pile–soil–pile interaction and alsomodification of p-y curves (O’Neill et al., 1977). However, experimental researchby Brown et al. (1987), Brown et al. (1988), Holloway et al. (1982), and Mei-mon et al. (1986)) has indicated that the elasticity-based analytical models do notreproduce the degree of nonlinearity observed in pile–soil–pile interaction.Brown and Shie (1990) performed a series of numerical experiments on onerow of piles subjected to lateral loading. To model clay soils, an elastic- per-fectly plastic (Von Mises) model was used which provides a constant yield strengthenvelope. Sand was modeled using a modified Drucker-Prager model with non-associated flow. They observed that group effects are most significantly influencedby row position and also center to center pile spacing. Brown and Shie (1991) usedthe same simulation to evaluate the group effects using p-y curves derived from pilestresses. Pile spacing effects were determined in terms of p-multipliers and relatedy-multiplier that were applied to p-y curves of single pile. First, p-multipliers weredetermined, and then appropriate y-multipliers were selected to best fit the slopedportion at the beginning of the p-y curve.Trochanis et al. (1991) conducted numerical analysis on in-line pairs of pilesto assess the main features of axial and lateral response of piles to monotonic andcyclic loading. They concluded that the assumption of purely elastic behaviour forsoil can substantially overestimate the degree of interaction in realistic situations.Finn and Wu (1994) showed that by relaxing some of the boundary conditionsassociated with a full 3D continuum analysis, it is possible to get reliable solutionsfor nonlinear response of pile foundations including both kinematic and inertialinteraction with reduced computational effort. The method is incorporated in thecomputer program PILE-3D. An effective stress version of this program also hasbeen developed by Thavaraj and Finn (1999). Wakai et al. (1999) presented the 3Delasto-plastic Finite Element (FE) simulations of model tests on 3×3 pile groups.16Yang and Jeremic (2003) also simulated centrifuge tests on 3× 3 to 4× 3 pilegroups which were conducted by McVay et al. (1998). Yang and Jeremic (2003)showed their FE simulations can predict the behaviour of pile groups with goodaccuracy.Finn (2004) explored the reliability of approximate methods for estimating therotational and translational stiffnesses of pile foundations in the 3D nonlinear anal-yses of a superstructure. Comodromos and Pitilakis (2005) carried out parametricthree-dimensional non-linear numerical analysis for different arrangements of pilegroups. They used Finite Difference (FD) simulations to evaluate the influence ofthe interaction between piles on the lateral resistance of pile groups. They alsoproposed a relationship to predict the response of fixed-head pile groups usingresponse of a single pile. Dodds and Martin (2007) examined large pile groupsunder lateral loading using a three-dimensional FD approach. They focused onlocal soil–pile interaction using p-y curves as the primary assessment tool and p-multipliers to characterize group effects. In their research, two piles with in-lineconfiguration and a single pile with periodic boundaries represented typical lead-ing and immediately trailing piles, and internal piles, respectively. Ultimately theyprovided p-multipliers to be applied to p-y curves for large pile groups in clay andsand conditions.2.4.2 Previous full scale testsFeagin (1937) - The first pile group lateral load tests were conducted by Feagin(1937), who performed full scale test on groups of 32 ft long timber piles insand. The tests were performed in Mississippi River sand on pile groups offour, twelve, and twenty piles with a pile head fixity provided by a pile cap.The focus was to “secure data on the movement of timber and concrete pilegroups of various sizes when subjected to lateral loads”. The arrangement forall pile groups was 2× n configuration with about 3 pile diameters spacingbetween piles. Pile diameter was 0.356 m.Observations:• Increasing the number of piles in a pile group caused less resistanceper pile.17• Group effects were only significant at large deflections. In this study,group effects did not affect response of piles for deflections less than 6cm.Kim and Brungraber (1976) - Three fixed-head pile groups with arrangement of2× 3 and pile spacings of 3.6D and 4.8D were tested. The H–shaped pileswere driven in cohesive soil. They compared performance of individual pilesin the pile group (fixed-head condition) with single reference pile (free-head)and computed pile group efficiencies of more than one, contrary to conven-tional notation. The pile cap was at the ground surface, which introduceda potential for pile cap base frictional contribution to lateral resistance. Inresponse to critics, Kim et al. (1979) published the results of the second se-ries of tests where 100 mm of soil under the pile caps had been excavated torelieve any potential frictional resistance.Observations:• The pile cap base friction contribution was negligible for battered pilegroups, but significant for vertical pile groups. The removal of thesoil resulted in deflections and maximum bending moments that werenearly twice what was observed with the soil contact.• With increasing the lateral load, compressive stresses increased in theleading row piles and decreased in the trailing row piles. It was de-termined that this was a result of the fixed-head condition of the pilegroup and that the leading row piles were resisting rotation and thushad increased compressive stresses.Holloway et al. (1982) - The first field pile group load test program that clearlyshowed group effects was performed by Holloway et al. (1982), who used thesame site as Feagin’s. They installed timber piles with the same constructiontechniques as originally used in the 1930s, and tested a 2× 4 pile group tothe failure point.Observation:• The key result was experimental evidence of pile group shadowing, i.e.larger carried load by piles in the leading row in comparison with the18trailing rows. This load distribution can be concluded from the bendingmoment profiles for different piles in Figure 2.7.Meimon et al. (1986) - In this study full scale tests were conducted on box-shapedsteel piles. The pile group configuration was 3×2 pinned head pile groupswith pile spacing of three pile widths in the loading direction and two pilewidths perpendicular to the load direction in cohesive soil. The piles widthwas about 0.27 m and depth was 7.5 m.Observations:• They confirmed what had been observed by Feagin (1937): soil resis-tance per pile decreased in larger deflections.• They noted what Kim and Brungraber (1976) had observed: the frontrow piles had greater resistance and also developed greater momentsfor a given pile group deflection.• Neither Feagin (1937) nor Kim and Brungraber (1976) directly mea-sured load on each pile. Meimon was the first to do this, which lead theway for further establishments in load distribution.Brown et al. (1987) and Brown et al. (1988) - Brown et al. (1987) performed cycliclateral load tests on 3×3 steel pipe pile groups in stiff clay. The pile diam-eter was 0.273 m and spacing in both directions was 3D. Another test wasperformed by Brown et al. (1988) on the previous pile group. This timemedium dense sand replaced and compacted to the depth of 2.9 m underlaidby very stiff clay.Observations:(Figure 2.8)• Carried load by each pile in the leading row is almost the same as thecarried load by a single pile for the same deflection. The leading rowcarries the most load within the group, while trailing row piles carrysubstantially lower values.• It was observed that the pile group deflected significantly more than thesingle pile when they loaded to a similar average load per pile.19• Bending moments of the pile group exceeded those for single piles, andwere shifted to higher depth.• Maximum soil resistance in the pile group was reduced relative to sin-gle piles under both static and cyclic loads.• Brown et al. (1987) concluded that the elasticity-based methods did notaccurately predict the distribution of load within a pile group and thatempirical modification factors were necessary.• Brown et al. (1988) introduced p-multipliers to account for the reduc-tion in resistance in each row. These back-calculated p-multipliers,along with other p-multipliers suggested from other tests, can be foundin Table 2.2.• The shadowing effect was more significant in the sand compared to theclay.• In cyclic loading, group effects were still significant in the sand. This iscontrary to the reduced significance of shadowing with cyclic loadingthat was observed in clay.• Pile group cyclic loading effects contributed to degradation of soil re-sistance for the stiff clay soil, but densification for the sand soil pre-vented loss of capacity.Ruesta and Townsend (1997) - An isolated single pile and a group of 16 pre-stressed concrete piles spaced at 3 diameters in sand were subjected to astatic lateral loading. They performed full scale tests in sand in order tocompare the results with full scale tests conducted by Brown et al. (1988) andcentrifuge tests conducted by McVay et al. (1995). Obtained p-multipliersare presented in Table 2.2.Observations:• The tests conducted by Ruesta and Townsend were the first full scaletests to have more than three rows and were unique because they hadalso four piles in each row.• The piles were much larger than other tests and were made of rein-forced concrete rather than steel.20• Their conclusions suggested p-multipliers that are similar to those sug-gested by the other tests (Table 2.2).• They also saw that outside piles took more load than inside piles withina row and attributed this to edge effects and pile driving sequence.• The Reese et al. (1974) recommendations for p-y curves in sandy layersare very reasonable for the calculation of ultimate soil resistance usingthe estimated friction angles. However, it appears that the estimatedcoefficient of subgrade reaction is conservative.Rollins et al. (1998) - They investigated the group effects with lateral load testingof a 3×3 group (and a single pile) consisting of 0.324 m diameter concretefilled steel pipe piles with spacing of 3D and driven 9 m into cohesive soil.Observations:• The pile group deflected over two times more than the single pile underthe same average load.• Shadowing effect resulted in the maximum load being distributed tothe front row of piles, and that more load was distributed to the backrow than the interior piles, in contrast to Browns findings.• Maximum bending moments in the piles of the pile group were 50 to100% higher than in the single pile at the same average load level,particularly at higher load levels.• They proposed p-multipliers for this soil ranging from 0.6 for the frontrow piles to 0.4 for the interior and back piles (Table 2.2).Weaver et al. (1998) - They conducted statnamic lateral loading tests on the same3×3 pile groups used by Rollins et al. (1998), with both fixed and free-headconditions, and with and without pile cap embedment.Observations:• The dynamic resistance was 30 to 50% higher than the static resistancefor a free-head pile and for the fixed-head group without cap embed-ment.21• The dynamic resistance was 100 to 125% higher for the fixed-headgroup with cap embedment.They attributed the increased resistance mainly to damping and acknowl-edged that further work needs to be done in interpreting statnamic test re-sults.Huang et al. (2001) - They performed full scale testing on both bored and drivenprecast piles groups in sand to investigate the effects that pile installation hadon pile group response. The bored pile group was consist of 2×3 concretepiles with diameter of 1.5 m and length of 35 m. The pile spacing was 3D.The 3× 4 precast pile group was consist of round and hollow, prestressed,and centrifugally cast in 17 m long segments in a factory. The bottom seg-ment had a closed, pointed steel shoe. The outer diameter was 0.8 m andinner diameter was 0.56 m. Pile head condition for both pile groups wasfixed-head.Observations:• Huang et al. (2001) found that installation procedure can have a sig-nificant effect on lateral soil resistance. Driven pile installation causesthe soil to become denser and increases group interactions, while boredpile installation loosens the soil and decreases group interactions.• They performed their test on large diameter piles and obtained p-multiplierswhich were significantly higher than other p-multipliers proposed byother researchers as it can be seen in Table 2.2.Rollins and Sparks (2002) - A monotonic lateral load test was performed on afull scale 3× 3 pile group driven in saturated low plasticity silts and clays.The steel pipe piles were attached to a concrete pile cap which created afixed-head condition. Pile diameter was 0.324 m and the spacing was 3D. Agravel backfill was compacted in place on the backside of the cap. Lateralresistance was therefore provided by soil-pile interaction, as well as basefriction and passive pressure on the cap.Observations:22• In this study, passive resistance contributed about 40% of the total re-sistance.• The p-multipliers developed for the free-head pile group provided rea-sonable estimates of the soil-pile resistance for the fixed-head pile group.This is in contrast with Huang et al. (2001) findings.• Gaps adjacent to the piles in cohesive soils can significantly reduce thelateral capacity provided by soil-pile interaction. Lateral soil resistanceshould be eliminated or significantly reduced when analysis indicatesthat the pile is being reloaded to deflections less than that produced byprevious loadings.Rollins et al. (2003a) - They did further testing on the same site tested in 1998with two new pile groups that were driven at the site, spaced at 2.8 and 5.65pile diameters.Observations:• They concluded that p-multipliers are higher in the pile group spaced at5.65 diameters than at 2.8 diameters and that pile interaction decreasedwith spacing as expected.• Group interaction was still fairly significant at 5.65 diameters and itwas concluded that interaction remained significant up to spacing ofabout 6.5 pile diameters, after which group effects can be neglected.Rollins et al. (2003b) - They did further full scale testing in clay at a differentsite in Salt Lake City on four pile groups spaced at 3.0, 3.3, 4.4 and 5.6pile diameters. The configuration of free-headed pile groups ranged from a3× 3 to a 3× 5 pile groups. Piles of different diameters were tested. Onepile group had piles with diameter of 0.61 m at 3D spacing and the otherremaining groups had piles with diameter of 0.324 m.Observations:• The results of this these tests suggested similar p-multipliers to thosepreviously suggested.23• It was observed that, contrary to what Ruesta and Townsend (1997)observed in sand, outside piles within a row in clay did not resist moreload than inside piles.• Similar to what McVay et al. (1998) concluded, they concluded that inpile groups with greater than three rows the trailing row carried higherloads than previous rows.• The maximum bending moments in the trailing rows tended to be higherfor a given load and occurred at a lower depth due to group effects.Rollins et al. (2005) - They performed a full scale test on a 3×3 pipe pile groupin loose to medium dense sand at 3.3D spacing.Observations:• Their study observed that within a row the outside piles consistentlyresisted more load than the inside piles. The same effect was observedin sand by Ruesta and Townsend (1997), but Rollins et al. (2003b)didn’t observed that in clay by.• The results from tests in sand suggest that the method of pile instal-lation (open-end driving, closed-end driving, or uniform compaction)had relatively little effect on the p-multipliers for rows within the group.In addition, the relative density of the sand does not appear to cause alarge variation in the back calculated p-multipliers.• More p-multipliers were suggested in this study as listed in Table 2.2.Figure 2.9 shows the design line that was constructed by Rollins et al.(2005) for p-multipliers in sand at different pile spacings. This designcurve is based on the p-multipliers that were found in the Rollins et al.(2005) report and shows an estimate of what the p-multipliers for pilegroups in sand should be. Some other commonly used design curvesare shown in the figure for comparison.Rollins et al. (2006) - To investigate group interaction effects as a function of pilespacing, full-scale cyclic lateral load tests were performed on pile groupsfrom Rollins et al. (2003b) test in cohesive soil. Tests were conducted on24three pile groups spaced at 3.3, 4.4 and 5.6 pile diameters with as many asfive rows of piles. Pile groups configurations were the same as Rollins et al.(2003b).Observations:• For a given deflection, the leading row piles carried the greatest load,while the second and third row piles carried successively smaller loads.Fourth and fifth row piles carried about the same load as the third rowpiles.• The lateral resistance of the piles in the group was found to be a func-tion of row location within the group, rather than location within a row.Contrary to expectations based on the elastic theory, the piles locatedon the edges of a row did not consistently carry more load than thecenter piles for a given deflection. This is consistent with observationsfrom other full scale lateral pile group tests in clay (Brown et al., 1987,Meimon et al., 1986, Rollins et al., 1998), but conflicts with some fullscale test (Rollins et al., 2005, Ruesta and Townsend, 1997) and cen-trifuge pile group tests in sands (McVay et al., 1998).• Cyclic loading reduced the peak load at the same deflection by about15% after 15 cycles and about half of this reduction occurred after onlyone cycle. However, at deflections less than the peak, the reduction inlateral resistance was considerably greater due to gap formation.• After 15 cycles of loading, the distribution of load within the groupwas essentially the same as for the virgin loading once the pile movedthrough the gap and again engaged the surrounding soil. For deflectionsless than the gap width, load distribution was relatively uniform withinthe pile group.• Cyclic loading also led to increases of 14 to 30% in the maximumbending moment for a given load with the smallest increases in thesingle pile and lead row piles and the greatest increases in the trailingrow piles.• Extrapolation of the test results suggests that group effects can be ne-glected for spacings greater than about 6.5 diameters for leading row25piles and 7 to 8 diameters for trailing row piles.• These equations have been developed to compute the p-multipliers fordifferent spacings:• Recommendations for p-multipliers provided in GROUP program over-estimate the lateral resistance for closely spaced pile groups.first (lead) row piles: fm = 0.26× ln(S/D)+0.5≤ 1 (2.6)Second row piles: fm = 0.52× ln(S/D)≤ 1 (2.7)Third or higher row piles: fm = 0.60× ln(S/D)−0.25≤ 1 (2.8)Where S = center to center spacing between piles in the direction of loadingand D = width or outside diameter of pile.Lemnitzer et al. (2010) - A 3× 3 bored pile group consisting of nine fixed cast-in-drilled-hole reinforced concrete shafts and a comparable single-shaft weresubjected to reversed cyclic, lateral head loading to investigate group inter-action effects across a wide range of lateral displacements. All the piles hadthe same diameter of D=0.61 m and similar soil conditions. All piles ex-tended 7.6 m below ground into overconsolidated silty clays.Observations:• Group efficiency factors were found to vary significantly with displace-ment level, being nearly unity near zero displacement (< 0.004×D),0.8 at middisplacements (0.010.02×D), and approaching 0.9 at largerdisplacements (< 0.04×D).• FE analyses of the pile group were conducted to investigate the sen-sitivity of the load-displacement response to the presence of cap rota-tion. Results revealed no significant differences in the load-deflectionresponse with and without rotation.• Recommended p-multipliers from Huang et al. (2001) for fixed-headlarge diameter pile groups provide a good estimate of average observedefficiency.262.4.3 Previous centrifuge testsMcVay et al. (1994) - They performed laterally loaded centrifuge tests on 3× 3pile groups in loose and dense sand. Their tests compared very well withBrown et al. (1988) tests on full scale pile group in sand. They evaluatedeffects of a) soil relative density, b) load distribution among rows, c) pilespacing, and d) in flight installation. Their evaluation results are shown inFigure 2.10.Observations:• Driving or placing the piles at 1 g is not recommended since it resultsin lower group capacity. This may be a result of the sand looseningto a lower Dr state during installation since high geostatic stresses areabsent during the installation process.• The density of soil affects both the group capacities as well as the indi-vidual row contributions for a nine-pile group at spacing of 3D.• An influence of pile spacing on the total lateral capacity of the groupwas observed. As the pile spacing increased from 3D to 5D, the group’scapacity also increased; however, the distribution among the individualrows within a group became less pronounced.McVay et al. (1995) - Centrifuge tests were conducted on single pile and 3× 3pile groups at 3D and 5D spacings. In all of the tests, the piles were drivenand laterally loaded in flight without stopping the centrifuge. The piles sim-ulated 0.430 m diameter by 13 m long pipe piles founded in medium loose(Dr = 33%) and medium dense (Dr = 55%) sands.Observations:• Results of the tests showed that the ratio of lateral resistance of a groupto a single pile, i.e. efficiency, was independent of soil density. Thegroup efficiency at 3D spacing was 0.74, whereas at 5D spacing thegroup efficiency was 0.94.• The p–y curves by Reese et al. (1974) resulted in excellent predictionsof the prototype single pile response for both relative densities.27• The p-multipliers were back computed using the single pile responseand the measured total and individual row contributions within thegroup. It was found that the multipliers did not vary significantly at5D spacing but did vary at 3D spacing.McVay et al. (1998) - Lateral load tests of 3× 3 to 3× 7 pile groups were con-ducted in both loose and medium dense sands. Square piles with width of0.429 m were tested. Pile length was 13.7 m in in loose (Dr = 36%) andmedium dense (Dr = 55%) sands.Observations:• From the test results of variable size groups in loose and medium densesands, it was found that an individual row’s contribution to the lateralresistance of pile group did not change with size of the group, only withits row position.• The p-multipliers for all pile groups were calculated.(Table 2.2)• It was found that the percentage of load carried by an individual rowdid not change with soil density. Therefore the p-multipliers are foundto be independent of soil density and only a function of the pile groupgeometry.2.5 Gap in knowledgeLiterature review on previous numerical studies showed that there is a lack of infor-mation on large pile groups with large pile spacing. Also, this gap of knowledge ismore pronounced in the experimental studies. Table 2.2 shows a summary of previ-ous experimental studies on p-multipliers for pile groups of 3×3 and larger. Thistable shows that most of available experimental data are related to pile groups witharrangement of 3×3 and the center-to-center pile spacing for most of them is 3 pilediameters. There are very few works performed on larger pile groups with largerpile spacings. Another important fact is that free-head condition is the prevalenthead condition in these tests. Although, most of the pile groups in practice havepile cap and their piles are fixed in the cap.282.6 SummaryDifferent methods for analysis of single pile under lateral loading are discussed.The p-y method is explained in detail for single pile analysis. The p-multiplier andthe group reduction factor which are common methods of accounting for the groupeffects in pile groups are explained.Some of the previous studies on the lateral loading of pile groups are reviewed.Most of these studies are conducted on smaller pile groups (3×3) with close spac-ing (3˜D) and free-head condition. There is a lack of information regarding theresponse of pile groups with higher number of piles, larger pile spacings, differentpile head conditions, and different soil properties. Main conclusions from availabletests are listed below:1. Pile groups deflect significantly more than single piles when subjected to thesame average load per pile.2. Group effects are relatively insignificant at small deflections but become con-siderable at higher loads or higher deflection levels.3. Group effects are significant for piles in a group spaced at about three diam-eters. These effects lessen with increased spacing.4. The leading row tends to have a response similar to a single pile. However,carried load decreases in subsequent rows. Group effects relative to locationwithin a row are negligible as most of the research show.5. Bending moments in closely spaced pile groups tend to be greater than bend-ing moments in a single pile for the same load per pile.6. The p-multiplier concept introduced by Brown et al. (1988) to match thetotal group load as well as row distribution. Some researchers claim that p-multipliers are independent of cyclic effects, pile head fixity, and soil density.7. The p-multipliers suggested in computer programs such as GROUP do notprovide accurate prediction of field response suggesting the need for modifi-cations.298. GROUP program with user-defined p-multipliers that are determined fromdesign curves based on full-scale tests can effectively predict pile perfor-mance for small pile groups.30Table 2.1: Beam theory equations (Ting, 1987)Displacement Slope Moment Shear Loading(2nd integral) (1st integral) (1st derivative) (2nd derivative)y ⇐= dydz⇐= EpIpd2ydz2=⇒ EpIpd3ydz3=⇒ EpIpd4ydz431Table 2.2: p-multipliers and group reduction factors from previous experimental studiesReference Soil ϕ Test Pile Pile D S/D Pile head Reported p-multiplier for row Grouptype type configuration type (cm) condition #1 #2 #3 #4 #5 #6 #7 reduction factorBrown et al. (1987)* Stiff clay – Full scale 3×3 Steel pipe 27.3 3 Free 0.7 0.6 0.5 - - - - 0.6aStiff clay – Full scale 3×3 Steel pipe 27.3 3 Free 0.7 0.5 0.4 - - - - 0.53bBrown et al. (1988)* Medium dense sand 38.5◦ Full scale 3×3 Steel pipe 27.3 3 Free 0.8 0.4 0.3 - - - - 0.5Morrison and Reese (1988) Medium dense sand 38.5◦ Full scale 3×3 Steel pipe 27.3 3 Free 0.8 0.4 0.3 - - - - 0.5McVay et al. (1995)* Medium loose sand 30◦c Centrifuge 3×3 Steel pipe 43 5 Free 1 0.85 0.7 - - - - 0.85Medium dense sand 33◦c Centrifuge 3×3 Steel pipe 43 5 Free 1 0.85 0.7 - - - - 0.85Medium loose sand 30◦c Centrifuge 3×3 Steel pipe 43 3 Free 0.65 0.45 0.35 - - - - 0.48Medium dense sand 33◦c Centrifuge 3×3 Steel pipe 43 3 Free 0.8 0.4 0.3 - - - - 0.5Ruesta and Townsend (1997)* Loose sand 32◦ Full scale 4×4 Square concrete 76 3 Free 0.8 0.7 d 0.3 0.3 - - - 0.52 (0.45)dMcVay et al. (1998) Sand 30◦, 33◦e Centrifuge 3×3 Square steel 42.9 3 Fixed 0.8 0.4 0.3 - - - - 0.5Sand 30◦, 33◦e Centrifuge 3×4 Square steel 42.9 3 Fixed 0.8 0.4 0.3 0.3 - - - 0.45Sand 30◦, 33◦e Centrifuge 3×5 Square steel 42.9 3 Fixed 0.8 0.4 0.3 0.2 0.3 - - 0.4Sand 30◦, 33◦e Centrifuge 3×6 Square steel 42.9 3 Fixed 0.8 0.4 0.3 0.2 0.2 0.3 - 0.37Sand 30◦, 33◦e Centrifuge 3×7 Square steel 42.9 3 Fixed 0.8 0.4 0.3 0.2 0.2 0.2 0.3 0.34Rollins et al. (1998)* Clay and silt – Full scale 3×3 Steel pipe 40 3f Free 0.6 0.38 0.43 - - - - 0.47fHuang et al. (2001) Silty caly – Full scale 2×3 RC 150 3 Fixed 0.93 0.7 0.74 - - - - 0.79gSilty caly – Full scale 3×4 Precast RC 80 3 Fixed 0.89 0.61 0.61 0.66 - - - 0.69Rollins and Sparks (2002) Silt and clay – Full scale 3×3 Steel pipe 32.4 3 Fixed 0.6 0.38 0.43 - - - - 0.47hSnyder (2004) Soft clay – Full scale 3×5 Steel pipe 32.4 3.92 Free 1 0.81 0.59 0.71 0.59 - - 0.74Walsh (2005) Sand 40◦ Full scale 3×5 Steel pipe 32.4 3.92 Free 1 0.5 0.35 0.3 0.4 - - 0.51Rollins et al. (2005) Sand 38◦ Full scale 3×3 Steel pipe 32.4 3.3 Free 0.8 0.4 0.4 - - - - 0.53Christensen (2006) Sand 38◦ Full scale 3×3 Steel pipe 32.4 5.65i Free 1 0.7 0.65 - - - - 0.78Rollins et al. (2006) Stiff clay – Full scale 3×5 Steel pipe 61 3i Free 0.82 0.61 0.45 - - - - 0.62Stiff clay – Full scale 3×3 Steel pipe 32.4 5.65i Free 0.95 0.88 0.77 - - - - 0.87Stiff clay – Full scale 3×4 Steel pipe 32.4 4.4i Free 0.9 0.8 0.69 0.73 - - - 0.78Stiff clay – Full scale 3×5 Steel pipe 32.4 3.3i Free 0.82 0.61 0.45 0.45 0.51 - - 0.57* These tests are the basis for AASHTO (2012) suggested p-multipliers.a These values are reported in Brown and Shie (1991) for 3 cm deflection of the test performed by Brown et al. (1987).b These values are reported in Brown and Shie (1991) for 5 cm deflection of the test performed by Brown et al. (1987).c Friction angle is not reported; therefore, it is obtained using the API (2007) relationship between relative density and friction angle.d It was noted that the second row response reported in this test was not very sensitive to the p-multiplier assigned and a value of 0.4 was acceptable (McVay et al., 1998).e Tests were performed on both medium dense and medium loose sand. The reported results were the same for all cases (McVay et al., 1998). Friction angle is notreportedtherefore it is obtained using the API (2007) relationship between relative density and friction angle.f Reported in Rollins et al. (2006): spacing 2.82D, p-multipliers: 0.6, 0.4, 0.4.g Applied displacement was very small compared with pile diameter (0.02D).h p-multipliers were not derived from experiment. Suggestions of Rollins et al. (1998) were used and good agreement reported.i Spacing normal to the direction of loading was 3.3D.32 52 Using (2.23) as a theoretical link between wedge distortion and triaxial test distortion, the deflected shape of the flexible pile that accompanies the wedge distortion is linearized in the manner shown in Figure 2-35(b), in order to assign appropriate values of stress change hσΔ  to each sublayer (as illustrated in Figure 2-35c).   Figure 2-34:  Strain wedge model concepts (after Norris, 1986) 'mφmβ'mφ'mφ'mφhσΔmβ245'mmφθ −=' 0vσ' 0vσhσΔhv σσ Δ+' 0' 0vσ ' 0vσhσΔεFigure 2.1: Strain Wedge model concept (after Dodds and Martin, 2007)33Lateral loadSoil resistance, pDeflection, yy y yLateral load(a)(b)Figure 2.2: Representation of (a) pile and nonlinear springs in p–y method,and (b) corresponding p–y curves70 API RECOMMENDED PRACTICE 2A-WSDbetween 8c and 12c. Due to rapid deterioration under cyclicloadings the ultimate resistance will be reduced to somethingconsiderably less and should be so considered in cyclic design.6.8.5 Load-Deflection (p-y) Curves for Stiff ClayWhile stiff clays also have non-linear stress-strain relation-ships, they are generally more brittle than soft clays. In devel-oping stress-strain curves and subsequent p-y curves forcyclic loads, good judgment should reflect the rapid deterio-ration of load capacity at large deflections for stiff clays.6.8.6 Lateral Bearing Capacity for SandThe ultimate lateral bearing capacity for sand has beenfound to vary from a value at shallow depths determined byEq. 6.8.6-1 to a value at deep depths determined by Eq. 6.8.6-2. At a given depth the equation giving the smallest value ofpu should be used as the ultimate bearing capacity.pus = (C1 × H + C2 × D) × γ × H (6.8.6-1)pud = C3 × D × γ × H (6.8.6-2)wherepu = ultimate resistance (force/unit length), lbs/in. (kN/m) (s = shallow, d = deep),γ = effective soil weight, lb/in.3 (KN/m3),H = depth, in. (m),φ´ = angle of internal friction of sand, deg.,C1, C2, C3 = Coefficients determined from Figure 6.8.6-1 as function of φ´,D = average pile diameter from surface to depth, in. (m).6.8.7 Load-Deflection (p-y) Curves for SandThe lateral soil resistance-deflection (p-y) relationships forsand are also non-linear and in the absence of more definitiveinformation may be approximated at any specific depth H, bythe following expression:(6.8.7-1)whereA = factor to account for cyclic or static loading condi-tion. Evaluated by:A = 0.9 for cyclic loading.A =  ≥ 0.9 for static loading.pu = ultimate bearing capacity at depth H, lbs/in. (kN/m),P A pu×  tanh k H×A pu×------------- y××=3.0 0.8 HD---–⎝ ⎠⎛ ⎞Figure 6.8.6-1—Coefficients as Function of φ´Figure 6.8.7-1—Relative Density, %54321020 25 30 35 401009080706050403020100Values of Coefficients C 1 and C2Values of Coefficients C 3C2C1C3Angle of Internal Friction, F´, deg028 29 30 36 40 4520Sand abovethe watertable40 60 80 100300250200150100500k (lb/in3 )F´, Angle of Internal FrictionRelative Density, %VeryLoose LooseMediumDense DenseVeryDenseSand belowthe watertableFigure 2.3: Graph presented in API (2007) for oefficients C1, C2, and C33470 API RECOMMENDED PRACTICE 2A-WSDbetween 8c and 12c. Due to rapid deterioration under cyclicloadings the ultimate resistance will be reduced to somethingconsiderably less and should be so considered in cyclic design.6.8.5 Load-Deflection (p-y) Curves for Stiff ClayWhile stiff clays also have non-linear stress-strain relation-ships, they are generally more brittle than soft clays. In devel-oping stress-strain curves and subsequent p-y curves forcyclic loads, good judgment should reflect the rapid deterio-ration of load capacity at large deflections for stiff clays.6.8.6 Lateral Bearing Capacity for SandThe ultimate lateral bearing capacity for sand has beenfound to vary from a value at shallow depths determined byEq. 6.8.6-1 to a value at deep depths determined by Eq. 6.8.6-2. At a given depth the equation giving the smallest value ofpu should be used as the ultimate bearing capacity.pus = (C1 × H + C2 × D) × γ × H (6.8.6-1)pud = C3 × D × γ × H (6.8.6-2)wherepu = ultimate resistance (force/unit length), lbs/in. (kN/m) (s = shallow, d = deep),γ = effective soil weight, lb/in.3 (KN/m3),H = depth, in. (m),φ´ = angle of internal friction of sand, deg.,C1, C2, C3 = Coefficients determined from Figure 6.8.6-1 as function of φ´,D = average pile diameter from surface to depth, in. (m).6.8.7 Load-Deflection (p-y) Curves for SandThe lateral soil resistance-deflection (p-y) relationships forsand are also non-linear and in the absence of more definitiveinformation may be approximated at any specific depth H, bythe following expression:(6.8.7-1)whereA = factor to account for cyclic or static loading condi-tion. Evaluated by:A = 0.9 for cyclic loading.A =  ≥ 0.9 for static loading.pu = ultimate bearing capacity at depth H, lbs/in. (kN/m),P A pu×  tanh k H×A pu×------------- y××=3.0 0.8 HD---–⎝ ⎠⎛ ⎞Figure 6.8.6-1—Coefficients as Function of φ´Figure 6.8.7-1—Relative Density, %54321020 25 30 35 401009080706050403020100Values of Coefficients C 1 and C2Values of Coefficients C 3C2C1C3Angle of Internal Friction, F´, deg028 29 30 36 40 4520Sand abovethe watertable40 60 80 100300250200150100500k (lb/in3)F´, Angle of Internal FrictionRelative Density, %VeryLoose LooseMediumDense DenseVeryDenseSand belowthe watertableFigure 2.4: Graph presented in API (2007) for initial modulus of subgradereactionLateral LoadEdge EffectShadowing EffectLeading rowTrailing row 1Trailing row 2Figure 2.5: Illustration of shadowing effect and edge effect35Horizontal Displacement, yHorizontal Resistance, p   Pm ×  ps psSingle pileA pile in the  pile groupFigure 2.6: Definition of p-multiplier (Pm)160Figure 4.5 - Field Pile Group Load Test Results Indicating Preferential Load Distributionto Leading Piles (after Holloway et al., 1982) The first field pile group load test program that clearly delineated group effectswas performed by Holloway et al. (1982), who revisited Lock and Dam No. 26 in a studyof rehabilitation schemes for the facility.  They installed timber piles with the sameconstruction techniques as originally used in the 1930’s, and tested a 2x4 pile group tofailure.  One of the key results was experimental evidence of pile group “shadowing”, i.e.the preferential load carrying capacity of piles in front of the line of loading, therebyreducing load on piles at the rear of the line of loading.  This load distribution isillustrated in Figure 4.5. Brown et al. (1987, 1988) performed cyclic lateral load tests on 3x3 pile groups instiff clay and sand, and provided detailed evidence of pile group effects.  Theirconclusions can be summarized as follows, and are reflected in Figures 4.6a and b.· Pile group deflections exceed single pile deflections under equivalent per pile load.· Pile group bending moments exceed those for single piles, and are shifted deeper.· Pile group maximum soil resistance is reduced relative to single piles under bothstatic and cyclic loads, and is most pronounced at depth.Figure 2.7: Bending moment profiles for two different piles from test (afterHolloway et al., 1982)36161· The greatest portion of shear on the pile group is distributed to piles in the front row,and variations in load within the group are approximately 20 %. Pile group cyclic loading effects were found to contribute to degradation of soilresistance at the stiff clay soil site, but densification at the sand soil site prevented anyloss of capacity.(a) (b)Figure 4.6 - Field Pile Group Load Test Results Depicting; a) Cyclic Degradation ofResistance; b) Distribution of Load by Row (after Brown et al., 1987)  Abacarius (1990) statically loaded pile groups to failure that supported two bentsof the then demolished Cypress Freeway that had catastrophically collapsed in the 1989Loma Prieta earthquake.  The two bents were founded on 60 ft long concrete filled steelFigure 2.8: Distribution of load in a pile group for different cycles of loading(after Brown et al., 1988) (a) 1st Row P-Multipliers0.00.20.40.60.81.01.22 3 4 5 6 7 8Pile Spacing (c-c)/Pile Diam.P-MultiplierReese et al (1996)Full-Scale TestsCentrifuge TestsDesign LineAASHTO (2000)(b) 2nd & 3rd Row P-Multipliers0.00.20.40.60.81.01.22 3 4 5 6 7 8Pile Spacing (c-c)/Pile Diam.P-MultiplierReese et al (1996)Full-Scale TestsCentrifuge TestsDesign LineAASHTO (2000)Figure 2.4  Design curve for pile groups in sand (Rollins et al., 2005). 21 (a) First row p-multipliers (a) 1st Row P-Multipliers0.2.4.60.8.01.22 3 4 5 6 7 8Pile Spacing (c-c)/Pile Diam.P-MultiplierReese et al (1996)Full-Scale TestsCentrifuge TestsDesign LineAASHTO (2000)(b) 2nd & 3rd Row -Multipliers02460 8.01.22 3 4 5 6 7 8Pile Spacing (c-c)/Pile Diam.P-MultiplierReese et al (1996)Full-Scale TestsCentrifuge TestsDesign LineAASHTO (2000)Figure 2.4  Design curve for pile groups in sand (Rollins et al., 2005). 21 (b) Second and third row p-multipliersFigure 2.9: Design curve for pile gro s i sand ( fter Rollins et al., 2005)371.2.2 Centrifuge testsMcVay et al. (1994) - He performed laterally loaded centrifuge tests on 3× 3 pile group in looseand dense sand. Their test compared very well with Brown et al. (1988) tests on full scale pilegroup in sand. They evaluated effects of a)soil relative density, b)load distribution among rows, c)pile spacing, and d) in flight installation. Their evaluation results are shown in Figure 5.197(a) (b)(c) (d)Figure 4.26 -  Centrifuge Modeling of Laterally Loaded Pile Groups in Sand: a) Effect ofRelative Density on Group Capacity; b) Load Distribution By Rows; c) Effect of PileSpacing on Total Lateral Resistance; d) Influence of Acceleration Level During Drivingon Total Lateral Resistance (after McVay et al., 1994)The centrifuge testing facilities at U.C. Davis have been the site of a series ofresearch projects dealing with SSPSI.  Chacko (1993) described model tests of singlepiles embedded in remolded Bay Mud in a hinged container on the small centrifuge, andanalyzed the results with the free-field and pile response computer codes SRANG andNONSPS from Kagawa (1980, 1983).  The analyses showed only fair agreement with theFigure 5: Laterally Loaded Pile roups in Sand: a) Effect of relative density on group capacity, b)Load distribution, total and for individual rows; c) Effect of pile spacing on total lateral resistance;d) Influence of acceleration level during driving (lateral loads and displacements plotted at prototypescale) (after McVay et al., 1994)22(a)1.2.2 Centrifuge testsMcVay et al. (1994) - He performed laterally loaded centrifuge tests on 3× 3 pile group in looseand dense sand. Their test compared very well with Brown et al. (1988) tests on full scale pilegroup in sand. They evaluated effects of a)soil relative density, b)load distribution among rows, c)pile spacing, and d) in flight installation. Their evaluation results are shown in Figure 5.197(a) (b)(c) (d)Figure 4.26 -  Centrifuge Modeling of Laterally Loaded Pile Groups in Sand: a) Effect ofRelative Density on Group Capacity; b) Load Distribution By Rows; c) Effect of PileSpacing on Total Lateral Resistance; d) Influence of Acceleration Level During Drivingon Total Lateral Resistance (after McVay et l., 1994)The centrifuge testing facilities at U.C. Davis have been the site of a series ofresearch projects dealing with SSPSI.  Chacko (1993) described model tests of singlepiles embedded in remolded Bay Mud in a hinged container on the small centrifuge, andanalyzed the results with the free-field and pile response computer codes SRANG andNONSPS from Kagawa (1980, 1983).  The analyses showed only fair agreement with theFigure 5: Laterally Loaded Pile roups in Sand: a) Effect of relative density on group capacity, b)Load distribution, total and for individual rows; c) Effect of pile spacing on total lateral resistance;d) Influence of acceleration level during driving (lateral loads and displacements plotted at prototypescale) (after McVay et al., 1994)22(b). . Centrifuge testsc ay et al. (1994) - He performed laterally loaded centrifuge tests on 3× 3 pile group in looseand dense sand. Their test compared very well with Brown et al. (1988) tests on full scale pilegroup in and. They evaluated effects of a)soil relative density, b)load distribution among rows, c)pile s acing, and d) in flight installation. Their evaluation results are shown in Figure 5.197(a) (b)(c) (d)Figure 4.26 -  Centrifuge Modeling of Laterally Loaded Pile Groups in Sand: a) Effect ofRelative Density on Group Capacity; b) Load Distribution By Rows; c) Effect of PileSpacing on Total Lateral Resistance; d) Inf uence of Acc le ation Level During Drivingon Total Lateral Resistance (after McVay et al., 1994)The centrifuge testing facilities at U.C. Davis have been the site of a series ofresearch projects dealing with SSPSI.  Chacko (1993) described model test  of singlepiles embedded in remolded Bay Mud in a hinged container on the small centrifuge, andanalyz d the results with the free-field and pile response comput r codes SRANG andNONSPS from Kagawa (1980, 1983).  The alys s showed only fair agreement with theFigure 5: Laterally Loaded Pile roups in Sand: a) Effect of relative density on group capacity, b)Load distribution, total a d for individual rows; c) Effect of pile spacing on total lateral resistance;d) Influence of acceleration level du ing driv ng (lateral loads and displacements plotted at prototypescale) (after McVay et al., 1994)22(c)2f l t r ll l e ce trifuge tests on 3 3 pile group in looser e l it ro n et al. (1988) tests on ful scale pilel ff f )s il r ti e e sity, b)lo d distribution a ong rows, c)fli i ll i . eir e l atio results are sho n n F gure 5.197(b)(c) (d)Fi r  .  -  trif  li  f L t r ll  L  Pile roups in Sand: a) Effect ofl tiv  sity  r  city; ) L  istri t on y o s; c) Effect of Pileaci  o  Total Lateral esista ce; ) I fl e ce of cc leration Level During Drivingon Total Lateral esist nce (after c ay et al., 1994)The centrifuge esting f cilities at .C. avis have been the site of a series ofresearch projects dealing ith SSPSI.  Chacko (1993) described odel tests of singlepiles e bedded in re olded Bay ud in a hinged container on the s all centrifuge, andanalyzed the results with the free-field and pile response co puter codes SRANG andNONSPS fro  Kagawa (1980, 1983).  The analyses showed only fair agree ent with theigure 5: Laterally Loaded ile Groups in Sand: a) ffect of relative density on group capacity, b)Load distribution, total a d for individual ro s; c) Effec of pi e spacing on total lateral resistance;d) Influence of acceleration level during driving (lateral loads and displace ents plotted at prototypescale) (after cVay et al., 1994)22(d)Figure 2.10: Laterally loaded pile groups in sand: a) effect of relative densityon group capacity, b) load distribution, t tal and for individual rows,c) effect of pile spacing on total lateral resistance, d) influence of ac-c eratio leve during dr ving (lateral loads nd di placements plottedat prototyp scal ) (afte McVay t l., 1994)38Chapter 3Model Development andMethodology for Analysis of PileGroupsReality is more complex than it seems.3.1 IntroductionThe literature review in previous chapter shows that most of the experiments havebeen performed on pile groups with arrangement of 3× 3, center-to-center pilespacing of 3D, and free-head conditions. As was shown in Table 2.2, there is alack of information regarding pile groups with large number of piles, large pilespacings or fixed-head condition. In this study, various pile groups are simulatedand p-multipliers and group reduction factors are calculated to cover the gap ofknowledge about wide range of pile groups.The current chapter presents the tools and methodologies used in this researchto fulfill the objectives previously listed in Chapter 1. First, the numerical ap-proaches are described which includes continuum and p–y models. The proce-dures for creating a pile group in the continuum and p–y models are also explained.Then the continuum model is validated against three full scale tests to make sure39that it is reliable enough to evaluate the lateral response of pile groups. In thenext step, methodologies for calculating p-multiplier and group reduction factorare described, their challenges are illustrated followed by the suggested solutionsfor those challenges.3.2 Continuum modelThe use of full-scale tests to establish design parameters is desired choice in pilegroup design. However, the significant scattered and insufficient data from exper-iments indicates obvious difficulties in being able to clearly assess soil–pile inter-action in the complex environment that nature presents. Continuum approacheshave resolved this problem to some extent, being able to provide a well definedenvironment for pile group from which observations of soil–pile interaction can beinferred.Using mathematical models to idealize physical problems in engineering re-search is well established. Depending on the geometry and loading condition ofthe problem one dimensional (1D), two dimensional (2D), or three dimensional(3D) numerical tools can be used to simulate the physical problem. In the laterallyloaded pile groups, each pile affects the response of other piles (Section 2.3) andstress and strain zones spread in all three dimensions therefore three dimensionalmodeling is required from a research perspective to simulate related mechanismsin the pile group properly.3.2.1 FLAC3D computer programIn this research continuum models are built in FLAC3D (Fast Lagrangian Analysisof Continua in 3 Dimensions) computer program (Itasca, 2012). This programis a three-dimensional, nonlinear, explicit finite difference program that can beused to simulate static and dynamic soil-structure interaction. In addition to its3D formulation, the ability to account for material nonlinearities makes FLAC3D aproper choice for this research. The finite difference method is a well-establishednumerical method that has been utilized in various three-dimensional studies oflaterally loaded piles (e.g. Comodromos and Pitilakis, 2005, Dodds and Martin,2007)40FLAC3D is capable of simulating the behavior of three-dimensional structuresbuilt in soil, rock or other materials that undergo plastic flow when their yieldlimits are reached. Materials are descritized by polyhedral elements within a three-dimensional grid that is adjusted by the user to fit the shape of the object to be mod-eled. Each element behaves according to a prescribed linear or nonlinear stress/s-train law in response to applied forces or boundary conditions. The material canyield and flow, and the grid can deform (in large-strain mode) and move with thematerial that is represented.An explicit solution scheme is used in FLAC3D. Explicit schemes can followarbitrary nonlinearity in stress/strain laws in almost the same computer time aslinear laws, whereas implicit solutions can take significantly longer to solve non-linear problems.Furthermore, it is not necessary to store any matrices, which meansa large number of elements may be modeled with a modest memory requirement.Also, a large-strain simulation is hardly more time-consuming than a small-strainrun, because there is no stiffness matrix to be updated.One of the drawbacks of the explicit formulation is that the loading steps shouldbe very small to get reliable results. Smaller loading step leads to slower analy-sis. In finite difference method the element size can drastically change the results,therefore element size and loading steps must be chosen carefully in FLAC3D. Lin-ear simulations run more slowly with FLAC3D than with equivalent finite elementprograms. FLAC3D is most effective when applied to nonlinear or large-strainproblems, or to situations in which physical instability may occur.3.2.2 Continuum model developmentSoil modelSoil behavior has lots of complexities and using a comprehensive soil model whichcan capture these complexities is the ideal solution for research and study. For eachproblem some simplifications are needed to find the proper model depending on thenature and aspects of the problem. The Mohr-Coulomb model is a classic modelused to represent shear failure in soils and rocks. The Mohr-Coulomb model sim-ulates elastic-perfectly plastic behavior. The elastic behavior is linear. When soil41yields, the behavior becomes perfectly plastic. The soil is sand in the current studywith zero tension and it is modeled using the non–associative flow rule by adoptinga dilation angle of 0◦ for the Mohr–Coulomb model. This assumption means thatstrength parameters remained constant regardless of volumetric and shear strainmagnitudes.Soil–pile connectionThere are several methods to simulate the soil–pile system in a continuum model,and each one with some advantages and some disadvantages. Main methods arelisted below:Method 1 – The simplest way is simulating a pile using beam elements and con-necting each node of the pile directly to the soil grid. This method requiresless computational effort in comparison with other methods. Figure 3.1(a)shows a schematic view for this approach.Method 2 – Using pile element provided by FLAC3D for modeling the pile. Thisis actually a beam element with integrated interface springs between pilenodes and soil grids. This element is defined with more parameters as ithas interface parameters. This method and method 1 don’t account for thevolume pile occupies in the soil.Method 3 – Simulating pile using beam elements with considering the pile vol-ume in the soil. Each pile node is connected to the soil grids using rigidconnection beams as shown in Figure 3.1(b). Connection beams are pinnedto the pile element to avoid inducing additional bending moment in the pile.Method 4 – Figure 3.1(c) shows the simulation of a single pile using method 4. Inthis method, the pile is simulated as a continuum and interface elements areused between soil and pile. This method does account for the volume of thepile and also has the interface elements but it is computationally expensive.Also, interpreting the bending moment and shear forces in the pile is moredifficult in comparison with other methods.42It should be noted that methods 2 and 4 have the interface elements. Primary anal-ysis show that the response of the piles in these methods are very sensitive to theassigned parameters for interface elements and it can change drastically with theseparameters. Therefore, using these elements needs extra cautions. Unfortunately,there is no clear recommendation for selecting proper values for all of these pa-rameters.Method 3 is selected to model the piles in this study. For this purpose struc-tural beam elements are used. The volume of each pile is accounted for in thenumerical model, as it can be important especially for closely-spaced piles. To thisend, the soil zones in the region occupied by a pile are removed from the model,and the pile nodes at each elevation are connected to the soil grids with rigid beamelements as shown in Figure 3.2. These rigid beams are used to enforce deflec-tion compatibility between the pile and the soil nodes (Rahmani et al., 2014). Theconnections between these rigid beams and the pile nodes are pinned. For a morerealistic analysis of soil–pile interaction under severe demands, instead of usingsuch connections at the interface of soil and structure, one would need to simu-late the possible slippage and gapping mechanisms by using advanced interfaceelements. However, selecting proper values for interface elements is a challenge.Sensitivity analysis on the proper number of rigid connection beams are per-formed using simulation of a single pile. Steel pipe pile is modeled with outerdiameter of 30 cm and wall thickness of 1 cm. The properties of the pile, boundaryconditions, and soil parameters are the same in all simulations. Only the numberof rigid connection beams is changed for each model as shown in Figure 3.3. Thisfigure shows the plan view for all investigated soil pile interaction models whichincludes 4, 8 and 12 rigid connection beams for the single pile model. The lateralboundaries should be placed at a location where the three dimensional effects dueto presence of the pile are negligible. Boundaries are placed at a distance of 3 mfrom the single pile in each direction. Depth of the soil is 10 m. Mohr–Coulombconstitutive model is adopted for the continuum analyses. The friction angle ofϕ=35◦ is used for soil, a dilation angle of 0◦, a Poisson’s ratio (ν) of 0.3, and shearmodulus of 21,154 kN/m2 for soil. The reason for selecting this shear modulus willbe discussed later in Section 3.4.1 as here the focus is on the soil–pile connection.43The pile head load-deflection curves for single pile with different number ofrigid connection beams are shown in Figure 3.4. This figure shows that the loaddeflection curve for 8 and 12 rigid connection beams are very close, therefore 8rigid connection beams are used for soil–pile connection.Mesh sizeIn using a finite difference method to solve (or, more generally, approximate the so-lution of) a problem, first the domain of the problem must be discretized. Selectinga proper mesh size is very important because it considerably affects the accuracy ofthe results. It is important to have a high density mesh in regions of high stress orstrain gradients. Sizing the mesh for accurate results, but with a reasonable numberof zones, can be complicated. Mesh size significantly affects the analysis time. Forselecting proper mesh size, these factors should be considered:1. Finer mesh leads to more accurate results because it provides a better repre-sentation of high stress gradients.2. Accuracy increases as zone aspect ratios approaches unity.3. If different zone sizes are needed, the results are more reliable if zoneschange more gradually from the smallest to the largest one.Sensitivity analyses are performed to find proper mesh size. These sensitivity anal-yses are conducted on a 3×3 free-head pile group with the spacing of 3D to cap-ture the effect of interaction between piles. The pile diameter is 30 cm. Soil gridsare connected to pile nodes using 8 rigid beams (as described previously). Mohr–Coulomb constitutive model is adopted for the continuum analysis. Soil parametersare the same as previously described model. Mesh size around the pile group is dif-ferent in each analysis, with finer mesh near the pile group. Lateral boundaries areplaced at a distance of 3 m from the exterior piles in the direction of loading (DL)and 2 m perpendicular to the direction of loading (DW ). The depth of soil is 10 m(H).Figure 3.5 shows different levels of refinement for the 3× 3 free-head pilegroup. Figure 3.6 depicts the total group load-deflection curves for different levels44of mesh refinement. Total group load is the sum of the all pile head forces in thegroup. According to this figure, Mesh B with the width of about 10 cm is theoptimum one for zones around the pile and zones finer than that won’t result indifferent response. It should be noted that the important depth of soil which hasthe most effect on the response of pile is 10 pile diameters, therefore after the depthof 10D (3 m), the height of the mesh is doubled to avoid unnecessary fine meshsize at higher depths.Boundary conditionsFixed displacement condition is assumed for the soil boundaries. To ensure thatthe stress zone around the pile group is not affected by the boundaries, they shouldbe far enough from the piles. Sensitivity analyses were carried out to determineappropriate dimensions of the continuum model. Trial push over analyses are per-formed for the continuum models of a 3×3 free-head pile group with the spacingof 3 pile diameters. with the soil domain dimensions of 5.4 m×4.2 m ×10 m, 7.8m×4.8 m ×10 m, and , 10.8 m×7.8 m× 10 m (in directions x, y, and z shown inFigure 3.7) which represent 6D, 4D, 10D, 5D, 15D, and 10D distance in the di-rection of loading and perpendicular to the loading direction, respectively for eachmodel (where D is the pile diameter). The properties of the soil and pile are thesame as previous models for sensitivity analyses.Figure 3.8 shows the total group load-deflection curves for models with differ-ent dimensions. Results of sensitivity analyses suggest that the boundary condi-tions should be placed at least at distances of 10D and 5D from the exterior piles inthe direction of loading and perpendicular to the direction of loading, respectively.These dimensions are sufficient to diminish the boundary effects on the responseof pile group.Loading rateThe displacement is applied at the pile head as input and the corresponding forceat the pile head is measured. In order to apply a given displacement at the pilehead, it is necessary to prescribe the velocity or loading rate for a given number45of steps. Velocity is defined as displacement at each step (m/step). If the desireddisplacement is d, a velocity V over N steps (where N = d/V ) may be applied.The velocity should be kept small and N large, in order to minimize shocks to thesystem being modeled. It should be noted that using a very small velocity willincrease the analysis time. There is an optimum value for loading rate in eachproblem, using values smaller than that for the loading rate would not increase theaccuracy of the response.To find the proper velocity or loading rate, deflection of 5 cm is applied withdifferent loading rates at the pile heads. The pile group configuration was 3× 3with center-to-center pile spacing of 3D. The properties of the pile group modelwere described in Section 3.2.2. Different loading rates of 1E-6, 1E-7, 1E-8 and1E-9 m/step are used to apply total displacement of 5 cm. Figure 3.9 shows theforce-deflection curves for these loading rates. This figure shows that loading ratecan affect the results drastically. The response for the rates of 1E-6 and 1E-7 m/stepare completely different but values of 1E-8 and 1E-9 m/step result in very similarresponse, therefore 1E-8 m/step is the optimum value.Computational demandFLAC3D has the parallel processing ability to distribute the analysis burden onmultiple CPUs to increase the analysis speed. In the Theoretical and Applied Ge-omechanics (TAG) group at UBC, there are two high end servers available forcomputational needs of the group. Each server has eighty 2.4 GHz CPUs with 64GB of RAM on Windows Server 2012 64bit Operating System. Analysis of thecontinuum model would be very slow if only one CPU were used. No matter howefficient the parallel algorithm is, there is an upper limit on the usefulness of addingmore CPUs. This is referred to as Amdahls law (Amdahl, 1967). For finding theoptimal number of CPUs to use, the single pile model described previously is an-alyzed with loading rate of 1E-8 m/step for 200 min. This single pile is analyzedusing different numbers of allocated CPUs and for each analysis the progress of theanalysis is recorded. Figure 3.10 shows analysis progress versus number of allo-cated CPUs. This figure shows that 20 is the optimum number of allocated CPUs,46as the incremental benefit from using more than 20 CPUs is very small.3.2.3 Continuum model validation – primary validationThe study is based on numerical analyses; therefore, the validity of the numericalmodel should be evaluated thoroughly. Validation is the procedure of determiningthe degree to which a model is an accurate representation of the real world fromthe perspective of the intended uses of the model (Oberkampf et al., 2002, Royand Oberkampf, 2011, Taiebat, 2008, Tasiopoulou et al., 2015). Quantifying theconfidence in the capability of the model in predicting the response by comparisonwith experimental data is the goal of validation. There are two sets of validation inthis study, primary validation and secondary validation. In the primary validationthe continuum model of the pile group is validated by simulating three specific pilegroups and comparing the results of the pushover analyses with the correspondingdata available from the corresponding field tests. Then, the validated continuummodel is used to create a numerically derived database. In the secondary valida-tion the numerically derived database is compared with available experimental datagathered from the literature.The primary validation is described in the current section and the secondaryvalidation will be explained in the next chapter. For all of these tests the sand ismodeled using the non–associative flow rule by adopting a dilation angle of 0◦for the Mohr–Coulomb model. Appropriate mesh size and model dimensions areselected on the basis of sensitivity analyses described in previous sections. Allpiles are fixed at the base of the model. These simulations show that the relativelysimple continuum modeling approach adopted in the present study is capable ofcapturing most characteristics of the pile group response under static loading forselected range of deflection.Full scale test by Brown et al. (1988)The first test which is used for validation of the continuum model is a set of testsby Brown et al. (1988). Full scale tests on a group of steel pipe piles and also on anisolated single pile were performed by Brown et al. (1988). Piles were subjectedto two way cyclic lateral loadings. Pile group was a 3× 3 piles in a submerged47dense sand that was placed and compacted around the piles to have friction angleof 38.5◦ (ϕ) and density of 1580 kg/m3. The steel pipe piles had an outer diameterof D = 0.273 m and thickness of 0.00927 m. Embedded length of the piles was4.87 m. Piles were arranged in a 3×3 group with spacing of 3D center to center inthe direction of loading and also perpendicular to the loading direction as shown inFigure 3.11. The piles were attached to the loading frame through pinned connec-tion. A loading cell at each connection measured the load on each pile. Load wasapplied on the piles at 0.3 m above the ground level. Load was applied as a push ora pull using a double-acting hydraulic cylinder. At each preselected load, the loadwas applied in either the compression or tension direction and the group deflectionwas measured. The same procedure was used in loading the isolated single pile.The first cycle of pushing is considered for simulation of the single pile and alsothe pile group.Equivalent shear modulus is obtained using the results of the single pile test.Friction angle of 38.5◦ is reported for this full scale test. The single pile test issimulated using the continuum model with a Poisson’s ratio of ν = 0.3, and aninitial uniform shear modulus G with depth. Calculated load-deflection curve iscompared with the single pile test data. The shear modulus of the soil in the con-tinuum model is changed iteratively until the load deflection curve matches the fullscale test data with maximum 10 % difference at each deflection. The soil grids atthe base of the mesh are fixed in vertical direction, and the soil grid displacementat the four lateral boundaries are fixed in the corresponding direction but it is freein two other directions.Figure 3.12 shows the measured and computed load-deflection curve of thesingle pile for shear modulus of 45,261 kN/m2. This figure shows there is a matchbetween measured and computed values.After finding equivalent shear modulus from the single pile simulation, the 3×3 pile group is simulated. Figure 3.13 shows the continuum model for Brown’s test.Figure 3.14 shows computed and measured pile head load-deflection curves fordifferent rows of the pile group. This comparison shows that continuum simulationcan capture the main characteristics of the pile group.48Full scale test by Walsh (2005)Another test selected for validation of the continuum modeling approach in thepresent study is from the experimental study of Walsh (2005). The test site is lo-cated near the Salt Lake City International Airport. The testing program includeda full scale 3×5 pile group subjected to cycles of lateral loading to different max-imum pile head deflections.Rigorous geotechnical site investigation was conduced for the site and reportedin Peterson (1996). The site investigation provided considerable information aboutthe subsurface soil including data of cone penetrometer (CPT) testing, standardpenetration (SPT) testing, pressuremeter (PMT) testing, and vane shear (VST)testing. Additional laboratory testing including particle size distribution, Atter-berg limits, soil classification (USCS), shear strength and consolidation tests werealso reported in the study of Walsh (2005). For the pile group testing program,the surface soil, which was consisting of clay deposit, was excavated to depth of2.44 m (8 ft) from around the piles. The entire excavated depth was then replacedwith washed concrete sand. According to Brown et al. (1988) the response of pilessubjected to lateral loadings is dominated by the soil layers down to depth of about10 pile diameters. In this test, the pile diameter was 0.324 m; therefore the top 3 mof the soil profile, i.e. mainly sand, is expected to govern the lateral response of thepiles, and deeper layers of the soil profile are not expected to have significant ef-fect on the response. A summary of the site characterization, adopted from Walsh(2005) are reported in Figures 3.15 and 3.16.The full scale 3× 5 pile group, shown in Figure 3.17, consisted of steel tubepile with outer diameter of D = 0.324 m and thickness of 0.0095 m. The piles werespaced at distance of 3.92D (1.27 m) center to center in the direction of loading.The pile spacing perpendicular to the loading direction was 3.29D (1.07 m) centerto center. Embedded length of the piles was 12.8 m and the water table was locatedat the depth of 2.1 m. The test represented free pile head conditions, where the pileheads were horizontally pushed to prescribed pile head deflections by a jackingsystem placed 0.48 m above the ground level. Each pile in the group had a tie rodthat attached the pile to the load frame. Each tie rod had a pair of strain gagesattached to it that measured the load resisted by each pile and canceled out any49bending strains. Figure 3.18 shows this set-up which load cell attached to a pileand connected to the load frame.The testing program also included lateral loading of a reference single pile.Using the LPile program and by characterizing the soil-pile interaction using p–ycurves, Walsh (2005) simulated the lateral response of the single pile test. Startingfrom a proposed idealized soil profile informed from the existing field and lab-oratory testing, the p–y curves were calibrated by adjusting the parameters untilthe computed response matched the measured response for the single pile. Adjust-ments primarily involved the friction angle and modulus of subgrade reaction of thesand backfill in the soil profile. At the end Walsh (2005) provided the correspond-ing profiles of subgrade modulus and friction angle or undrained shear strength forthe soil layers used for the analysis with LPile.As part of the validation process for the continuum modeling approach, the fullscale 3×5 pile group of Walsh (2005) was simulated.The soil layering used in the validation process was informed by the reportedsoil profile from the LPile analysis of Walsh (2005) with the following details.In terms of layering, unit weight, and shear strength, the same soil layering asthat of Walsh (2005) was considered. This includes the friction angle for sandylayers and undrained shear strength for the clayey layers. In terms of stiffness,Young’s modulus (E) for the layers of the soil was derived from available CPT testresults using the empirical correlation of E = 7qc (Bowles, 1996), where qc is conepenetration resistance of soil. Figure 3.16 shows the CPT profile from three testsconducted close to the pile group. According to this figure there is no field dataavailable for the top 2.44 m which was replaced with washed concrete sand. TheYoung’s modulus for this top layer was back calculated from the subgrade modulusused in the LPile simulation of the test by Walsh (2005) using the Vesic correlationprovided in Kulhawy and Mayne (1990):ks =0.65B× ((E×B4)E f × I f)1/12× E1−ν2 (3.1)where, ks is the subgrade modulus of soil, B is the diameter of pile, E f is theYoung’s modulus of the pile material which for this case it is steel and I f is themoment of inertia of the pile section, and ν is the Poisson’s ratio for the soil.50This equation can be used for both shallow and deep foundations. Table 3.1 listsproperties of the soil profile used in the continuum model simulation. Figure 3.19shows the continuum model for Walsh’s test.Figure 3.20 shows computed and the measured total group load versus pilehead deflection for defections up to 50 mm. Figures 3.21, 3.22, and 3.23 show thecomputed and measured bending moment at the pile head deflection of 6, 16, and38 mm for the center pile in different rows. These cover the range of deflectionswith reported bending moment data from Walsh (2005). All four figures showreasonably good agreement between the measured and computed values.Full scale test by Christensen (2006)The last test used for validation of the continuum model is a full scale test on a3×3 pile group by Christensen (2006) which was performed at same site as Walsh(2005). Same as the case of Walsh (2005), the first 2.4 m layer of the soil wasreplaced with washed concrete sand.The steel tube pile had an outer diameter of D = 0.324 m and a thickness of0.0095 m. The piles were spaced at a distance of 5.65D (1.83 m) center to center inthe direction of loading, as shown in Figure 3.24. The pile spacing perpendicularto the loading direction was 3.29D (1.07 m) center to center. Embedded length ofthe piles was 12.8 m and the water table was located at the depth of 2.1 m. The testset up was the same as what discussed in previous section.For the continuum simulation, the soil layering used in the validation processwas informed by the reported soil profile from the LPile analysis of Christensen(2006) with the following details. According to Christensen (2006), in the case ofthis experiment two different soil profiles are necessary in order to model the topsoil around the pile group. This was necessary here because of the differences incompaction of the top layer of the soil inside the group versus outside the group.In particular, the upper sand layer outside of the pile group was compacted verydensely, while the sand inside the pile group was not compacted to the same densityas the sand outside the pile group. Therefore different values were reported for thefriction angles and CPT profiles as shown in Figure 3.25.In terms of layering, unit weight, and shear strength, the same soil layering51as that of Christensen (2006) was considered. This includes the friction angle forsandy layers and undrained shear strength for the clayey layers. Christensen (2006)reported different values for the first 2.4 m of soil inside and outside the group but athigher depth the soil profiles inside and outside were the same. Reported values areused for simulation of soil profile inside and outside of the pile group. In terms ofstiffness, Young’s modulus (E) for the layers of the soil was derived from availableCPT test results the same as previous section. Different values are considered forinside and out side of the group for the top layer based on Figure 3.25. The lowerlayers were the same as Walsh (2005) therefore Figure 3.16 is used. Tables 3.2 and3.3 list properties of the soil profile used in the continuum model simulation for thesoil inside and outside of the pile group. Figure 3.26 shows the continuum modelfor Christensen’s test.Figure 3.27 shows computed and the measured total group load versus pilehead deflection. Figures 3.28 and 3.29 show the computed and measured bendingmoment at the pile head deflection of 6 and 51 mm for the center pile in differentrows. These cover the range of deflections with reported bending moment datafrom Christensen (2006). All four figures show reasonably good agreement be-tween the measured and computed values.523.3 p–y model3.3.1 GROUP programGROUP program version 8 (Reese et al., 2010) has been well–received as a usefultool in practice for analyzing the behavior of piles in a group subjected to axial,lateral, and torsional loadings. The piles may be installed vertically or on a batter,and pile heads may be fixed, pinned, or elastically restrained (with springs) to thepile cap. The cap is assumed to act as a rigid body, and may settle, translate, androtate. GROUP models the pile as a beam and the soil resistance as non-linearsprings. The program internally generates the nonlinear response of the soil, in theform of t–z (frictional spring) and q–w (pile tip resistance spring) curves for axialloading, t–r curves for torsional loading, and p–y curves for lateral loading (Reeseet al., 2010).3.3.2 p–y model developmentp–y curvesGROUP program uses non-linear p–y curves that are associated with the inputsoil properties to model the lateral stiffness of each soil layer. Figure 3.30 showsan example of soil layers input in GROUP program. Recommendations of API(2007) are used for p–y curves. These curves are described in the previous chapter.Pile elementThe stiffness of the pile is calculated using the modulus of elasticity (E) and themoment of inertia (I) of the pile. The piles can be linear or non-linear depending onthe user input. A sensitivity analysis on the size and number of the elements alongthe pile was conducted. Based on the results, each pile is divided to incrementswith the length of 0.2 m. Each increment has one set of p–y springs.53Assigning p-multiplierGROUP program has three options when using p-multipliers. The first option isto allow the p-multipliers to default to 1.0, ignoring the group effects. The secondoption is to allow the program to calculate the p-multipliers according to input soilspacing and recommendation made by Reese et al. (2010) and the last option is toassign the p-multipliers by user. In this study we are assigning the p-multipliers.Recommendations of Reese will be evaluated in Chapter 6.3.4 Calculating p-multiplier and group reduction factorp-multiplierThe method for obtaining p-multiplier follows the procedure described by Rollinset al. (2006) and also Christensen (2006), except that they used data from a fieldtest rather than data from analysis, as in this study. The load–deflection curve foreach row of the piles in the group is computed using the continuum model. Theload corresponding to a prescribed deflection of this model for each row is thenapplied to the p–y model of that row and the deflection under this load is cal-culated using the GROUP program. The required p-multiplier to be used in theGROUP program is initially unknown; a starting value of 0.5 was assumed in thep–y model. The analysis for the p–y model is then repeated using different valuesof the p-multiplier until the pile head deflection from the p–y model of that rowof piles matches the pile head deflection from the continuum model analysis. Thedifference of less that 10 % is considered to be acceptable match in this study. Thisprocedure is repeated for each row to obtain p-multipliers for all the rows in thegroup. Figure 3.31 illustrates the procedure for free-head pile groups. In fixed-head pile groups the procedure is the same but there is a limitation in the GROUPprogram regarding the fixity of just one row of piles. The problem and how it wasaddressed is explained in Appendix A.54Group reduction factorThe process for obtaining the group reduction factor at a prescribed pile head de-flection is similar to the procedure for calculating p-multiplier and it is depictedin Figure 3.32. The load–deflection curve for the pile group in Figure 3.32(a) iscomputed using the continuum model. The load corresponding to a prescribeddeflection of this model is then applied to the p–y model in Figure 3.32(b), andthe deflection under this load is calculated using the computer program GROUP(Reese et al., 2010). Since the required group reduction factor to be used in theGROUP program is initially unknown, a starting value of 0.5 was assumed in thep–y model. The analysis for the p–y model is then repeated using different val-ues of the group reduction factor until the pile head deflection from the p–y modelmatches the pile head deflection from the continuum model analysis satisfactorily(difference less that 10 %).The main goal is to evaluate response of pile groups and also to compare theresulting p-multipliers and group reduction factors against those recommended indifferent design guidelines such as AASHTO (2012), FEMA P-751 (2012), andReese and van Impe (2010). These recommendations are based on various exper-imental studies, most of them are listed in Table 2.2. The pile head deflectionsin these tests are in the range 3-5 cm. Based on these observations, it is decidedto obtain the group reduction factors for pile head deflections of 3 cm, 4 cm, and5 cm to cover the similar range in the corresponding experimental studies that arethe basis for the design guidelines. The calculated results in this range of pile headdeflections do not appear to have significant variations; therefore, the average foreach pile group is reported.3.4.1 Equivalent soil parameters in continuum and p–y modelsOne of the key issues for calculating p-multiplier and group reduction factor ishaving equivalent soil parameters in the continuum and p–y models. In this study,the equivalent properties are determined using simulation of a free-head single pilein the continuum and p–y models. Dimensions of this single pile are depictedin Figure 3.33. The API (2007) p–y curves are defined by ϕ . The continuummodel is characterized by the same friction angle ϕ as used to develop the p–y55curves, a dilation angle of 0◦, a Poisson’s ratio ν of 0.3, and an initial uniformshear modulus G in depth. Pile head load-deflection curves (up to a deflection of 5cm) and associated bending moment profiles at different pile head deflections arecomputed in both p–y and continuum models, and the results are compared. Theshear modulus of the soil in the continuum model is changed iteratively until theload deflection curves and the bending moment profiles from the two models agreewithin 10 % (Figure 3.34). As an example, the matched load–deflection curvesand bending moment profiles for both models with friction angle of 35◦ for soil areshown in Figure 3.35. Obtained equivalent shear modulus for the friction angle of35◦ is 21,154 kN/m2.3.4.2 Adopted group reduction factor vs. average p-multiplierIn the current study, the group reduction factor is evaluated following the processillustrated in Figure 3.32. Another approach which can be used is to average p-multipliers of different rows.Ruesta and Townsend (1997) performed full scale test on a 4×4 pile group andreported p-multipliers of 0.8, 0.7, 0.3 and 0.3 for different rows in the pile group.The largest p-multiplier is associated with the leading row. They also reported 0.55as group reduction factor which is very close to 0.52, the average of p-multipliers.It can be concluded that the average of p-multipliers is almost the same ascalculated group reduction factor using described methodology in Figure 3.32.3.5 SummarySensitivity analyses on different parameters of the continuum model have been per-formed. In the present study piles are modeled with beam elements. The volume ofeach pile is accounted for in the continuum model, as it can be important especiallyfor closely-spaced piles in a group. Therefore, the soil zones in the region occu-pied by a pile are removed from the model, and the pile nodes at each elevationare connected to the soil grids with 8 rigid beam elements to enforce deflectioncompatibility between the pile and the soil grid. It is important to have a highdensity mesh in regions of high stress gradients. Therefore finer mesh around thepiles are used. The width of the mesh around the piles is 0.1 m. To ensure that the56stress zone around the pile group is not affected by the boundaries in current study,they should be placed at least at distances of 5D and 10D from the exterior piles inthe direction of loading and perpendicular to the direction of loading, respectively.Analyses show that the loading rate has a significant effect on the results, differentloading rate will completely change the response. For pile group of this study, theloading rate of 1E-8 m/step is the optimum value.Three full scale tests by Brown et al. (1988), Walsh (2005), and Christensen(2006) on 3× 3 and 3× 5 pile groups are simulated to validate the continuummodel. Each one of these tests had different spacings from 3 to 5.65 pile diame-ters. These simulations show that the simple Mohr–Coulomb model is capable ofcapturing most characteristics of the pile group response under monotonic load-ing. Therefore, the continuum model appears to be reliable to use for exploring theeffects of different parameters on the response of pile groups under lateral loading.The methodology for calculating p-multiplier and group reduction factor usingvalidated continuum model and p–y model are described. Achieving equivalent pa-rameters in continuum and p–y models is a challenge for calculating p-multipliersand group reduction factors. The methodology to achieve equivalent parametershave been described in this chapter. Numerical models and methodologies de-scribed in this chapter will be used in the next chapters to create a comprehensivedatabase of pile groups and to study their response.It should be noted that in numerical modeling there is always space for im-provement and this study is not an exception. There are always some simplifica-tions depending on what is expected from the model. Finding the best and simplestapproach which does the job is always a challenge. Depending on what is expectedfrom the model, these simplifications may or may not be acceptable. The valida-tion process is to build confidence on the prediction capability of the model for acertain problem.57Table 3.1: Properties of the soil profile in the continuum model for the Walsh(2005) pile group testDepth Unit weight Shear modulus Poisson’s Friction angle Dilation angle Undrained shear(m) (kN/m3) (kN/m2) ratio (degrees) strength (kN/m2)0.0–2.1 16.7 10,579 0.3 40 0 -2.1–2.4 16.8 15,800 0.3 40 0 -2.4–2.7 19.1 7,000 0.3 - - 412.7–3.7 19.1 7,000 0.3 - - 503.7–4.6 19.1 8,100 0.3 - - 404.6–6.3 18.1 17,800 0.3 38 0 -6.3–8.0 19.1 6,500 0.3 - - 578.0–12.8 16.7 22,600 0.3 33 0 -Table 3.2: Properties of the soil profile inside the pile group in the continuummodel of the Christensen (2006) testDepth Unit weight Shear modulus Poisson’s Friction angle Dilation angle Undrained shear(m) (kN/m3) (kN/m2) ratio (degrees) strength (kN/m2)0.0–0.9 17.4 13,460 0.3 39 0 -0.9–2.4 16.8 10,770 0.3 35 0 -2.4–2.7 19.1 7,000 0.3 - - 412.7–3.7 19.1 7,000 0.3 - - 503.7–4.6 19.1 8,100 0.3 - - 404.6–6.3 18.1 17,800 0.3 38 0 -6.3–8.0 19.1 6,500 0.3 - - 578.0–12.8 16.7 22,600 0.3 33 0 -Table 3.3: Properties of the soil profile outside the pile group in the contin-uum model of the Christensen (2006) testDepth Unit weight Shear modulus Poisson’s Friction angle Dilation angle Undrained shear(m) (kN/m3) (kN/m2) ratio (degrees) strength (kN/m2)0.0–2.1 16.7 16,150 0.3 40 0 -2.1–2.4 16.8 12,100 0.3 40 0 -2.4–2.7 19.1 7,000 0.3 - - 412.7–3.7 19.1 7,000 0.3 - - 503.7–4.6 19.1 8,100 0.3 - - 404.6–6.3 18.1 17,800 0.3 38 0 -6.3–8.0 19.1 6,500 0.3 - - 578.0–12.8 16.7 22,600 0.3 33 0 -58(a) Methods 1 and 2 (b) Method 3(c) Method 4Figure 3.1: Schematics of different approaches for modeling soil–pile inter-actionThree translational degrees of freedom (Free rotation)Rigid beam connectionSoil zonePile node(a)(b)(c)Figure 3.3: Plan view for soil pile connection modeling with (a) 4, (b) 8 and(c) 12 rigid connection beams59Three translational degrees of freedom (Free rotation)Rigid beam connectionSoil zonePile node10D5DS112.8 mS1S2S25D10DLoadingdirectionFigure 3.2: Finite difference model of the 3×3 pile group0 0.01 0.02 0.03 0.04 0.05050100150Pile head deflection (m)Pile head load (kN)  4 connection beams8 connection beams12 connection beamsFigure 3.4: Comparison of the pile head load-deflection curve for a singlepile with 4, 8, and 12 rigid connection beams at each node600.9 m0.9 mD LD LD WD W0.9 m0.9 m H(a) Mesh A - 6720 zones0.9 m0.9 mD LD LD WD W0.9 m0.9 m H(b) Mesh B - 15960 zones0.9 m0.9 mD LD LD WD W0.9 m0.9 m H(c) Mesh C - 23520 zonesFigure 3.5: Different levels of mesh refinement for a 3×3 pile group610 0.01 0.02 0.03 0.04 0.050100200300400500600700Pile head deflection (m)Total group load (kN)  Mesh AMesh BMesh CFigure 3.6: Comparison of the total group load-deflection curves for differentlevels of mesh refinementYXZLoading directionFigure 3.7: Finite difference mesh of a pile group620 0.01 0.02 0.03 0.04 0.050100200300400500600700Pile head deflection (m)Total group load (kN)  Boundaries: X: 6D, Y: 4DBoundaries: X: 10D, Y: 5DBoundaries: X: 15D, Y: 10DFigure 3.8: Comparison of the total group load-deflection curves for differentdimensions of the model. (X: loading direction, Y: perpendicular to theloading direction , D: Pile diameter)0 0.01 0.02 0.03 0.04 0.050100200300400500600700Pile head deflection (m)Total group load (kN)  Loading rate : 1E−6 m/stepLoading rate : 1E−7 m/stepLoading rate : 1E−8 m/stepLoading rate : 1E−9 m/stepFigure 3.9: Comparison of the total group load-deflection curves for differentloading rates631 5 10 20 40020406080100Allocated CPUsAnalysis progress in 200 min (%)Figure 3.10: Analysis progress in 200min using different number of CPUs64S= 3DS = 3DLoading DirectionD=0.273 mt=9.27mmRow #1Row #2Row #3Loading FrameLoading CellPileFigure 3.11: Plan view of the full scale pile group test by Brown et al. (1988)0 0.01 0.02 0.03 0.04 0.05020406080100120140Pile head deflection (m)Pile head force (kN)  ContinummMeasuredFigure 3.12: Measured and computed load–deflection curves for the singlepile test by Brown et al. (1988)656 m5D3D3D10DLoading direction10D3D3DFigure 3.13: Finite difference model of the test by Brown et al. (1988)0 0.01 0.02 0.03 0.04 0.05020406080100120140Pile head deflection (m)Pile head force (kN)  Leading row − ContinummLeading row − MeasuredMiddle row − ContinummMiddle row − MeasuredTrailing row − ContinummTrailing row − MeasuredFigure 3.14: Comparison of the measured and computed total group load-deflection curves for different rows of pile for the full scale test byBrown et al. (1988)6640          Figure 3.11  Idealized soil profile modified from test results for drill hole DH-96-W (Peterson, 1996).   Figure 3.15: Soil profile for the test; adopted from Walsh (2005)6732        Figure 3.6  Comparison of measured results from two CPT soundings performed in 1998.  01234567891011121314150 5000 10000 15000 20000 25000Tip Resistance, qc (kPa)Depth Below Ground Surface (m)CPT-98-WCPT-98-E0 50 100 150 200 250 300Sleeve Friction, fs (kPa)0 2 4 6 8 10Friction Ratio, Fr (%)-200 0 200 400 600 800Pore Water Pressure, u (kPa)Figure 3.16: Profile of qc values from CPT tests close to the pile group;adopted from Walsh (2005)Loading Directiont=0.0095mRow #3Row #4Row #5Loading FrameLoading CellS1 = 3.92DS2 = 3.29 DD=0.324 mt=0.0095mRow #1Row #2PileFigure 3.17: Plan view of the full scale pile group test by Walsh (2005)68 Figure 5.5   Instrumentation for load and deflection (Walsh 2005)  was never exact agreement, however the difference was usually within 1% indicating that the loads measured were indeed very accurate. 5.3.2 Deflection Instrumentation All deflections were measured using string potentiometers.  These string potentiometers were accurate to 0.25 mm (0.01 in).  A total of 11 string potentiometers were used, one attached to each pile in the pile group and one attached to each drilled shaft.  Deflection measurements were accomplished using an independent fixed reference frame with supports located about 1.5 m outside the pile group.  This fixed frame was placed slightly above the load frame and remained stationary as the load 86 Figure 3.18: Instrumentation for load and deflection (Walsh, 2005)Figure 3.19: Finite difference model of the test by Walsh (2005)690 20 40 600500100015002000Group head deflection (mm)Total group load (kN)  ComputedMeasuredFigure 3.20: Computed and measured total group load-deflection curves forthe test by Walsh (2005)−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #1−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #2−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #5  ComputedMeasuredFigure 3.21: Computed and measured bending moment for center pile of dif-ferent rows at the head deflection of 6 mm for the test by Walsh (2005)70−50 0 50 100 1500246810Bending moment (kN.m)Depth (m)Row #1−50 0 50 100 1500246810Bending moment (kN.m)Depth (m)Row #2−50 0 50 100 1500246810Bending moment (kN.m)Depth (m)Row #5  ComputedMeasuredFigure 3.22: Computed and measured bending moment for center pile of dif-ferent rows at the head deflection of 19 mm for the test by Walsh (2005)−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #1−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #2−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #5  ComputedMeasuredFigure 3.23: Computed and measured bending moment for center pile of dif-ferent rows at the head deflection of 38 mm for the test by Walsh (2005)71S1= 5.65 DS2 = 3.29 DLoading DirectionD=0.324 mt=0.0095mRow #1Row #2Row #3Loading FrameLoading CellPileFigure 3.24: Plan view of the full scale pile group test by Christensen (2006)72Inside ‐selectedOutside ‐ selectedFigure 3.25: Profile of qc values for the upper layers adapted from Chris-tensen (2006) and selected values for simulation in this study73Figure 3.26: Finite difference model of the test by Christensen (2006)740 20 40 60020040060080010001200Group head deflection (mm)Total group load (kN)  ComputedMeasuredFigure 3.27: Computed and measured total group load-deflection curves forthe test by Christensen (2006)−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #1−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #2−50 0 50 1000246810Bending moment (kN.m)Depth (m)Row #3  ComputedMeasuredFigure 3.28: Computed and measured bending moment for center pile of dif-ferent rows at the head deflection of 6 mm for the test by Christensen(2006)75−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #1−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #2−100 0 100 2000246810Bending moment (kN.m)Depth (m)Row #3  ComputedMeasuredFigure 3.29: Computed and measured bending moment for center pile of dif-ferent rows at the head deflection of 51 mm for the test by Christensen(2006)76Figure 3.30: Soil layers input in GROUP programΔCalculating the force for each rowApplying the calculated force to one row with the same pile head conditionp x Pp yChanging P  till Δ = Δ(a) Continuum model(b) p-y model1Δ 212m mFigure 3.31: Methodology for calculating p-multiplier using continuum andp–y models (Pm: p-multiplier)77 Calculating the forceApplying the calculated forcep x Pp yChanging P  till   =  (a) Continuum model (b) p-y model1 21 2mmFigure 3.32: Methodology for calculating group reduction factor using con-tinuum and p–y models (P¯m : Group reduction factor)0.5 m10 mD=0.3 mDt=0.01mFigure 3.33: Dimensions of the single pile used for obtaining equivalent soilproperties78Force Pile head force-displacement curveContinuum modelChanging G in the continuum model to match the force-displacement curves and bending moment profilesp-y modelPile head force-displacement curve(Reference)  = 0.3G   = 35°   = 35°Figure 3.34: Methodology for obtaining equivalent soil parameters in contin-uum and p–y models0 0.01 0.02 0.03 0.04 0.05 0.06020406080100120140Pile head deflection (m)Pile load (kN)  p−y modelContinuum model(a)0 50 100 150 2000246810Bending Moment (kN.m)Depth (m)  (b)∆ : Pile head deflectionp−y modelContinuum model∆ = 1 cm ∆ = 3 cm∆ = 5 cmFigure 3.35: Comparison of load–deflection curves (a), and bending momentprofiles at different pile head deflections (b), for a single pile in soilwith ϕ = 35◦ computed by the p–y model and the continuum model79Chapter 4Effects of Different Parameterson Group Reduction FactorThe ideal engineer is a composite ... He is not a scientist, he is not amathematician, he is not a sociologist or a writer; but he may use theknowledge and techniques of any or all of these disciplines in solvingengineering problems. — - N. W. Dougherty, 19554.1 IntroductionAs it is discussed in previous chapters, limited studies are conducted on pile groupsunder lateral loading. Comprehensive literature review showed the lack of infor-mation on the behavior of piles in various pile groups. In this chapter the overallinfluence of the group effects on the response of pile groups will be evaluated.This evaluation will be performed thorough calculating group reduction factors forvarious pile groups. As it was described previously, use of group reduction factoris convenient for seismic and cyclic loading analysis of pile groups. Group re-duction factor can characterize influence of interaction between piles in the group.Using group reduction factor is a common method for analysis of laterally loadedpile groups in practice. This chapter evaluates the influence of various parameters,including pile spacing, pile head condition, and the friction angle of soil, on theinteraction between piles in the pile groups by calculating group reduction factors.804.2 Investigated parametersFollowing the method presented in Chapter 2 for calculating group reduction factora comprehensive parametric study has been conducted. Group reduction factorsare calculated for square pile groups with different numbers of piles (3×3 to 6×6), different S/D ratios (3 to 6) where S is the center to center spacing betweenpiles and D is the pile diameter, different soil friction angles (30◦ to 40◦), anddifferent pile head conditions (free and fixed), as listed in Table 4.1. This tablelists a total of 96 pile groups. As it was explained in previous chapter, equivalentsoil parameters in the continuum and p–y models must be used for group reductionfactor calculations. Equivalent soil parameters for each soil are calculated usingthe methodology described in Section 3.4.1. Calculated equivalent soil parametersfor soils with friction angle of 30◦, 35◦ and 40◦ are listed in Table 4.2.4.3 Calculated group reduction factorsCalculated group reduction factors from the parametric study on the pile groupslisted in Table 4.1 are shown in Figure 4.1. Each one of the subfigures shows theresults for specific arrangement of piles in the group. The horizontal axis showsS/D ratio in the pile group. Vertical axis shows the calculated group reduction fac-tor. Solid lines in the subfigures represents the calculated group reduction factorsfor free-head pile groups simulated in soils with different friction angles. Dashedlines represent the values for fixed-head pile groups. Figure 4.1(a) shows calcu-lated group reduction factors for 3×3 pile groups. With increasing S/D ratio, thegroup reduction factor increases which means group effect decreases in the pilegroup. Group reduction factors for free-head and fixed-head pile groups are closeto each other at lower S/D but as S/D increases the difference between free-headand fixed-head pile groups increases. The group reduction factors for fixed-headpile groups are less than free-head ones which means there is more group effectinvolved in the fixed-head pile groups. As the friction angle of the soil increasesthe group effect increases which results in lower group reduction factor. For free-head pile groups, group reduction factor ranges from a minimum of 0.44, for a pilegroup with S/D = 3 and φ = 40◦, to a maximum of 0.89, for a pile group withS/D = 6 and φ = 30◦. For fixed-head pile groups, group reduction factor changes81from 0.4 for a pile group with S/D = 3 and φ = 40◦ to 0.67 for a pile group withS/D = 6 and φ = 30◦.Calculated group reduction factors for 4× 4 pile groups are shown in Fig-ure 4.1(b). The trends for 4× 4 pile groups are the same as 3× 3 pile groups,however group reduction factors for 4×4 pile groups are generally less than groupreduction factors for 3×3 pile groups. For free-head pile groups group reductionfactor ranges from a minimum of 0.36, for a pile group with S/D = 3 and φ = 40◦,to a maximum of 0.84, for a pile group with S/D = 6 and φ = 30◦. For fixed-headpile groups, the group reduction factors change from 0.35 for a pile group withS/D = 3 and φ = 40◦ to 0.62 for a pile group with S/D = 6 and φ = 30◦. It canbe seen that the group reduction factors for different pile groups are close to eachother at lower S/D regardless of their head condition or soil properties.Calculated group reduction factors for 5× 5 pile groups are shown in Fig-ure 4.1(c). The overall trends are the same as 3×3 and 4×4 pile groups, howevergroup reduction factors for 5×5 pile groups are less than group reduction factorsfor smaller pile groups. For free-head pile groups, group reduction factor rangesfrom a minimum of 0.33, for a pile group with S/D = 3 and φ = 40◦, to a maximumof 0.79, for a pile group with S/D = 6 and φ = 30◦. For fixed-head pile groups, thegroup reduction factors change from 0.29 for a pile group with S/D = 3 and φ =40◦ to 0.58 for a pile group with S/D = 6 and φ = 30◦.Figure 4.1(d) shows calculated group reduction factors for 6× 6 pile groups.For free-head pile groups, group reduction factor ranges from a minimum of 0.29,for a pile group with S/D = 3 and φ = 40◦, to a maximum of 0.78, for a pile groupwith S/D = 6 and φ = 30◦. For fixed-head pile groups, the group reduction factorschange from 0.26 for a pile group with S/D = 3 and φ = 40◦ to 0.58 for a pilegroup with S/D = 6 and φ = 30◦.According to Figure 4.1, the group reduction factor decreases with increasingnumber of piles in the pile group. The group reduction factor also decreases withincreasing friction angle of the soil. The reason is that the fan angle Ashour et al.(1998) of wedge-shaped influence zone in front of each pile increases with in-creasing friction angle of soil and causes larger influence zone and ultimately moreinteraction between piles. The group reduction factor increases with the spacingwhich means the group effect decreases in the pile group as the distance between82piles increases. Group reduction factors for fixed-head pile groups are close to thevalues for free-head pile groups at S/D = 3, but the difference between fixed-headand free-head group reduction factors increases in larger S/D ratios. In all of thecases, group reduction factors for fixed-head pile groups are lower than the valuesfor free-head pile groups. This means that group effects in fixed pile head groupsare more pronounced than in free-head pile groups. This can be attributed to thedifferent modes of deformation in free- and fixed-head piles at the same pile headdeflection; the latter has larger influence zone in soil compared to the former.4.4 Extended sensitivity analysisAlthough Figure 4.1 covers a wide range of pile groups in terms of properties of thesoil profile, number of piles, pile spacings, and pile head conditions, one could stillconsider some other possible scenarios. Limited additional sensitivity analysis isdone and briefly presented in this section to cover other factors such as distributionof the soil shear modulus in depth, larger pile spacings in the group, and largernumber of piles in the group.4.4.1 Shear modulus distribution along the depthThe assumption of uniform G in depth of the soil was done for the simplicity inthe analyses. However, it is important to also look at the effect of using a morerealistic variable G on the group reduction factors, and to compare the results withthose from the uniform G assumption. To this end a parabolic distribution of shearmodulus in depth is assumed based on the expression G = G0(p/pat)0.5, which issimilar to what suggested by Richards et al. (1970). Where p and pat representthe mean effective stress and the atmospheric pressure, respectively, and G0 is aparameter to be found. Friction angle of φ= 35◦ is selected for soil and the sameprocedure described in Figure 3.34 is followed to come up with the representativeprofile of G in the continuum model, with the difference that the iteration processdescribed in Figure 3.34 is now carried out on the value of G0. This process resultedin a value of G0 = 65,833 kN/m2. For easy comparison, the resulting uniform andparabolic distributions of G are presented in terms of the equivalent profiles ofshear wave velocities in Figure 4.2. The performance of the p–y model and the83continuum model with the parabolic distribution of G are compared in Figure 4.3(similar to what was done for the case of uniform G in Figure 3.35).After finding proper value for G0, group reduction factors are calculated for3×3 pile groups with different spacings and pile head conditions in the soil profilewith φ = 35◦ and parabolic distribution of G in depth. The results are compared tothe case with uniform G. Figure 4.4 shows that using parabolic distribution of Gdoes not considerably affect the resulting group reduction factors. This additionalsensitivity analysis strengthens the validity of the findings of this study beyond thelimited case of uniform G.4.4.2 Larger pile spacingWhen piles are placed far enough, the interaction between piles becomes negligibleand each pile behaves similar to a single pile regardless of its position. To furtherstudy the effect of center to center pile spacings on the group reduction factorsand find the spacing which fades out the group effect, in addition to the alreadypresented results in Figure 4.1 for various pile spacing scenarios of 3D to 6D,limited additional analyses are conducted that cover the group factors for both free-head and fixed-head 3× 3 pile groups with spacings of 8D and 10D. The resultsare presented in Figure 4.5 for the case of soil profile with φ= 35◦ and uniform G.The trend of results shows that the group reduction factor increases with increasingthe center to center pile spacing. This figure also shows that the group reductionfactor for the free-head condition almost reaches 1 for pile S/D of about 10.4.4.3 Larger pile groupsMost of available studies are performed on 3×3 pile groups. In this study 3×3 to6×6 pile groups with different center to center spacing are simulated. Compari-son of group reduction factors for these pile groups shows that the group reductionfactor decreases with increasing the number of piles. To investigate group fac-tor variation in larger groups, 10×10 pile groups with different spacings and pilehead conditions are also simulated in a soil profile with φ= 35◦ and uniform G.Calculated group reduction factors are presented in Figure 4.6. Comparing Fig-ures 4.1 and 4.6 shows that the group reduction factors for 10×10 pile groups are84even lower than those for 6×6 pile groups. To study large pile groups, Law andLam (2001) represented an infinite repeating pile pattern by using a simplifyingapproach of finite element analysis coupled with a periodic boundary condition.Using this modeling approach Dodds and Martin (2007) conducted a numericalstudy in FLAC3D and suggested group reduction factors ranging from 0.1 to 0.2for large pile groups with the spacing of 3D in sand and clays; this is close to thefindings of the present study for the spacing of 3D as shown in Figure 4.6.4.5 Importance of using appropriate group reductionfactorIn this section, influence of using a value other than the calculated one for groupreduction factor on the response of the group is evaluated. To this end, 3× 3 and6×6 pile groups with S/D of 3 and 6 with fixed-head condition are analysed in thep–y model. For each pile group simulation, both values of group reduction factorsfor free-head condition (what is being used in practice) and fixed-head condition(what is correct to be used) are used separately to analyse the pile group under aspecific lateral loading. In each pile group, based on its pile head condition, thefixed-head values are the correct one to use. Therefore, the difference betweencalculated pile head deflections using these two different group reduction factors isreported as error. This error shows the influence of ignoring the pile head conditionin the selection of the group reduction factor on the group response.Friction angle of φ= 35◦ is considered for the soil in all cases. A selected forceis applied on each configuration to explore different cases. For 3× 3 pile groupstotal load of 1500 kN is applied at the pile cap. The 3× 3 pile group has 9 pileswhile the 6× 6 pile group has 36 pile, therefore total load of 6000 kN is appliedon the 6× 6 pile group. The load is applied at the top to calculate the deflectionat the top. The deflection for each pile group is calculated twice; first using groupreduction factor for free-head condition and then using group reduction factor forfixed-head condition.Tables 4.3 and 4.4 show the results. The difference between group reductionfactors for free and fixed-head condition in pile groups with smaller S/D are lessthan the larger ones. Therefore, the effect of using wrong group reduction factor85regarding the pile head condition in pile groups with smaller S/D is less than thelarger ones. In larger pile groups the difference can increase to about 20%.4.6 Comparison with experimental studies – secondaryvalidationIn this section, the numerically derived database of calculated group reduction fac-tors are compared with available experimental data on pile groups in sand whichare listed in Table 2.2 for secondary validation of the results of this study. Availableexperimental data are really limited because of the cost and difficulty of perform-ing experimental tests on pile groups. It should be noted that the calculated groupreduction factors are compared with the average of reported p-multipliers for dif-ferent rows in each experiment.4.6.1 Free-head pile groupsIn this section calculated group reduction factors for pile groups with free-headcondition are compared with corresponding free-head pile groups in the experi-ments. Properties of each test are compared to our numerical model to investigatethe possible reasons for differences and similarities.3×3 pile groups – Figure 4.7 shows calculated group reduction factors and ex-perimental data on 3×3 free-head pile groups. This figure shows that thereare a couple of experimental data available for 3× 3 pile groups and mostof them have S/D of 3. Morrison and Reese (1988) and Brown et al. (1988)both performed full scale tests on a 3× 3 free-head pile group with S/D of3 in sand. Based on their reports, the average p-multipliers were the samein both experiments. They both reported friction angle of 38.5◦ for the soil.Figure 4.7 shows that their result lays well between calculated group reduc-tion factor for φ = 35◦ and φ = 40◦ in this study.McVay et al. (1995) performed 4 centrifuge tests on 3× 3 free-head pilegroups with S/D = 3 and S/D = 5 in sand with φ = 30◦ and also φ = 33◦.Results of these tests are in good agreement with calculated values as shownin Figure 4.7. It should be noted that McVay et al. (1995) reported the same86result for pile groups with φ = 30◦ and φ = 33◦ when S/D = 5 in the pilegroup.Rollins et al. (2005) performed full scale tests on 3×3 free-head pile groupswith S/D = 3.3 in sand with φ = 38◦. Also Christensen (2006) performedfull scale test on 3×3 free-head pile groups with S/D = 5.65 in sand with φ= 38◦. Figure 4.7 shows that their results are in between the calculated groupreduction factors for φ = 35◦ and φ = 40◦.Figure 4.7 shows a good agreement between experimental data and calcu-lated group reduction factors for 3× 3 free-head pile groups with differentfriction angles and different center to center pile spacings.4×4 pile groups – Figure 4.8 shows the group reduction factors for 4× 4 free-head pile groups. In Figure 4.8 there is just one experimental data froma full scale test performed on a 4× 4 pile group by Ruesta and Townsend(1997). The reported friction angle of soil for this experiment is 32◦ and theS/D was 3. Initially, they reported 0.7 for the first trailing row p-multiplierwhich results in group reduction factor of 0.52, but a year later McVay et al.(1998) reported that the second row response reported in this test was notvery sensitive to the p-multiplier assigned and a value of 0.4 was accept-able which results in revised group reduction factor of 0.45. According toFigure 4.8 this experimental data matches very well with calculated groupreduction factors.5×5 pile groups– Figure 4.9 shows the calculated group reduction factors for 5×5 free-head pile groups in soils with different friction angles. There is no testdata available for 5×5 pile groups, but Walsh (2005) studied a full scale 3×5pile group (5 rows) with S/D of about 4 in a site with the friction angle of40◦. The group reduction factor for his test was 0.51. The calculated groupreduction factor for a 5× 5 free-head pile group with spacing of 4D andfriction angle of 40◦ is 0.43. The higher group reduction factor for the 3×5pile group of Walsh (2005) in compare to the calculated value for 5×5 pilegroup in this study (although both have 5 rows in the direction of loading) canbe attributed to the fewer piles in the former group resulting in smaller edge87effect between piles in this pile group and therefore less overall interactionbetween the piles in this pile group.6×6 pile groups– Unfortunately there are no test data available for comparisonof group reduction factors for 6×6 free-head pile groups because it is veryexpensive and difficult to conduct experiments on large pile groups.4.6.2 Fixed-head pile groupsThe calculated group reduction factors for fixed-head pile groups in Figure 4.1are compared with experimental data on fixed-head pile groups in sand listed inTable 2.2. Available experimental data on fixed-head pile groups are even less thanfree-head ones because conducting a test on fixed-head pile group needs muchmore powerful equipments and settings.McVay et al. (1998) conducted centrifuge tests on fixed-head pile groups withdifferent numbers of piles to obtain p-multipliers. The number of piles in theircentrifuge tests changed from 3× 3 to 3× 7 (3 to 7 rows of piles in the directionof loading) and the S/D = 3 for all of the tests. They performed the tests on thepile groups in soils with friction angles of 30◦ and 33◦. According to their tests thegroup reduction factors in each set of pile groups were the same for both frictionangles.3×3 pile groups– Experimental data and calculated group reduction factors forthe 3×3 fixed-head pile groups are shown in Figure 4.10. This figure showsthat for the 3×3 pile groups the calculated group reduction factors at S/D =3 for φ = 30◦ is 0.48 which is very close to the corresponding data fromMcVay et al. (1998). Average of reported p-multipliers by McVay et al.(1998) is 0.5.4×4 pile groups– Figure 4.11 shows calculated group reduction factors for 4×4fixed-head pile groups along with the result of McVay et al. (1998) centrifugetest on 3×4 fixed-head pile group. Group reduction factor for this test is 0.45but calculated value for φ = 30◦ and S/D of 3 is 0.39. Therefore, the resultof McVay et al. (1998) test is higher than the calculated group reductionfactor. The difference can be attributed to the fact that McVay et al. (1998)88had less edge effect in his pile group because he had 3 piles in each row (incomparison with 4 piles in each row of the 4×4 groups in this study).5×5 pile groups– Figure 4.12 shows calculated group reduction factors for 5×5fixed-head pile groups along with the result of McVay et al. (1998) centrifugetest on 3×5 fixed-head pile group with S/D of 3. It can be seen that differ-ence of calculated group reduction factor and the result of the test is higherthan the case for 4×4 pile group. Group reduction factor for this centrifugetest is 0.4 but calculated group reduction factor for fixed-head 5× 5 pilegroup in soil with φ = 30◦ with the same S/D is 0.31. The reason for thisdifference is that in 3×5 pile group the edge effects are much less than 5×5pile group. Therefore the smaller interaction between piles are present in the3×5 pile group which results in higher group reduction factor.6×6 pile groups– Figure 4.13 shows calculated group reduction factors for 6×6fixed-head pile groups along with the group reduction factor from McVayet al. (1998) centrifuge test on a 3×6 fixed-head pile group with S/D of 3.The 3×6 pile group of the test has less edge effects than 6×6 pile group withthe same S/D in this study because it has less piles in each row. This resultsin less interaction between piles in the 3× 6 pile group and higher groupreduction factor. Calculated group reduction factor for 6×6 pile group withS/D of 3 is 0.28 and the group reduction factor for the centrifuge test on3×6 pile group with the same S/D is 0.37.4.7 SummaryThe validated continuum model is used to generate an extensive numerically drivenbenchmark database for the group reduction factors using the procedures describedin previous chapter. Pile groups with different numbers of piles (3× 3 to 6× 6),different S/D ratios (3 to 6), different soil friction angles (30◦ to 40◦), and differentpile head conditions (free and fixed) are considered. Although a wide range ofpile groups are analysed, limited additional sensitivity analysis is done and brieflypresented to cover other factors such as distribution of the soil shear modulus indepth, larger pile spacings in the group, and larger number of piles in the group.89Reliability of the numerical results is further validated by comparing the calculatedgroup reduction factors with the available experimentally derived factors.The calculated group reduction factors in the generated benchmark databaseshow the following trends. For a given pile spacing, the group reduction factordecreases with increasing number of the piles in the pile group, and also withthe increase of friction angle of soil. The group reduction factor increases withthe increase of the S/D ratios. The calculated fixed-head group reduction factorsappear to be generally smaller than the free-head group reduction factors.It was observed that in larger pile groups, the group reduction is very low.Dodds and Martin (2007) had similar observation using a different modeling ap-proach. Using parabolic distribution of G does not considerably affect the result-ing group reduction factors. This strengthens the validity of the findings of currentstudy beyond the limited case of uniform G. It was shown that not using appro-priate group reduction factor regarding the pile head condition of the group canintroduce up to 30% error in the prediction of pile head deflection. Therefore, thepile head condition should be taken into account in determination of the groupreduction factor.Most of available data are derived from the tests on 3×3 free-head pile groups.However, all of the available experimental data on free-head pile groups comparewell with calculated group reduction factors. McVay et al. (1998) conducted cen-trifuge tests on fixed-head pile groups with similar pile diameter with differentnumbers of piles to obtain p-multipliers. The number of piles in their centrifugetests changed from 3× 3 to 3× 7 (3 to 7 rows of piles in the direction of load-ing) and the S/D = 3 for all of the tests. Experimental data and calculated groupreduction factors for the 3× 3 fixed-head pile groups are in very good agreementbut as the number of piles increases the group reduction factors from the tests arebecoming higher than the calculated values in this study. Given that McVay et al.(1998) performed their tests on 3×4, 3×5, and 3×6 pile groups, these pile groupsexperience lesser edge effect than the one in the 4×4, 5×5, and 6×6 pile groups,respectively. Therefore, it is again expected that for the former ones to have highergroup reduction factors.90Table 4.1: Simulated pile groupsPile Friction S/D = 3 S/D = 4 S/D = 5 S/D = 6arrangement angle Pile head fixity Pile head fixity Pile head fixity Pile head fixity3×330◦ Free Fixed Free Fixed Free Fixed Free Fixed35◦ Free Fixed Free Fixed Free Fixed Free Fixed40◦ Free Fixed Free Fixed Free Fixed Free Fixed4×430◦ Free Fixed Free Fixed Free Fixed Free Fixed35◦ Free Fixed Free Fixed Free Fixed Free Fixed40◦ Free Fixed Free Fixed Free Fixed Free Fixed5×530◦ Free Fixed Free Fixed Free Fixed Free Fixed35◦ Free Fixed Free Fixed Free Fixed Free Fixed40◦ Free Fixed Free Fixed Free Fixed Free Fixed6×630◦ Free Fixed Free Fixed Free Fixed Free Fixed35◦ Free Fixed Free Fixed Free Fixed Free Fixed40◦ Free Fixed Free Fixed Free Fixed Free Fixed10×10 35◦ Free Fixed Free Fixed Free Fixed Free FixedTable 4.2: Equivalent soil parameters for different friction angles of soilFriction angle Shear modulus Poisson’s(degrees) (kN/m2) ratio30 16,154 0.335 21,154 0.340 26,923 0.391Table 4.3: Influence of not using appropriate group reduction factor for 3×3fixed-head pile group (total applied load = 1500 kN, ϕ=35◦)S/D Group reduction Calculated pile head displacement (m) Differenceratio factor * based on assumed group reduction factor (%)3Free-head: 0.51 0.06714.1Fixed-head: 0.45 0.0786Free-head: 0.85 0.0220Fixed-head: 0.65 0.025* From Figure 4.1(a)Table 4.4: Influence of not using appropriate group reduction factor for 6×6fixed-head pile group (total applied load = 6000 kN, ϕ=35◦)S/D Group reduction Calculated pile head displacement (m) Differenceratio factor * based on assumed group reduction factor (%)3Free-head: 0.33 0.04914Fixed-head: 0.28 0.0576Free-head: 0.71 0.02521.8Fixed-head: 0.53 0.032* From Figure 4.1(d)920.20.40.6 SimulationFree head Fixed head φ30°35°40°2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(a) 3 × 3 pile groups2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(b) 4 × 4 pile groups2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(c) 5 × 5 pile groups2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(d) 6 × 6 pile groupsFigure 4.1: Calculated group reduction factors for different pile group set-tings: (a) 3 3; (b) 4 4; (c) 5 5; (d) 6 6 pile groups930 100 200 300 4000246810Vs (m/s)Depth (m)  Uniform GParabolic Gvs = (G/ρ)0.5ρ = 1670 kg/ m3Figure 4.2: Corresponding profiles of Vs with uniform or parabolic distribu-tions of shear modulus G in depth (for the case of φ = 35◦)0 0.01 0.02 0.03 0.04 0.05 0.06020406080100120140Pile head Diplacement (m)Pile head load (kN)p−y modelContinuum model(Parabolic G)(a)0 50 100 150 2000246810Bending moment (kN.m)Depth (m)(b)∆ : Pile head deflectionp−y modelContinuum model(Parabolic G)∆= 5 cm∆= 1 cm ∆= 3 cmFigure 4.3: Comparison of load–deflection curves (a), and bending momentprofiles at different pile head deflections (b), for a single pile in soilwith φ = 35◦ calculated by the p–y model and the continuum modelwith parabolic distribution of G942 3 4 5 6 700.20.40.60.81S/DGroup reduction factor3 × 3 pile groups  Free head − Uniform GFree head − Parabolic GFixed head − Uniform GFixed head − Parabolic GFigure 4.4: Calculated group reduction factors for 3×3 pile groups using soilprofiles with φ= 35◦, and with uniform or parabolic distributions of Gin depth2 4 6 8 1000.20.40.60.81S/DGroup reduction factor3 × 3 pile groups  Free headFixed headFigure 4.5: Calculated group reduction factors for 3×3 pile groups with S/Dvalues ranging from 3 to 10 (φ= 35◦ and uniform G)952 3 4 5 6 700.20.40.60.81S/DGroup reduction factor10 × 10 pile groups  Free headFixed headFigure 4.6: Calculated group reduction factors for 10×10 pile groups (φ=35◦ and uniform G)2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor3 × 3 pile groups   Morrison et al.(1988)Brown et al.(1988)McVay et al.(1995)Rollins et al.(2005)Christensen(2006)φ = 38°φ = 30° and 33°φ = 38.5°φ = 38°3 data points2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.7: Comparison between calculated group reduction factors and pre-vious experimental works for 3×3 free-head pile groups in sand962 3 4 5 6 700.20.40.60.81S/DGroup reduction factor4 × 4 pile groups    Ruesta and Townsend(1997)φ = 32°2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile groupFigure 4.8: Comparison between calculated group reduction factors and pre-vious experimental works for 4×4 free-head pile groups in sand2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor5 × 5 pile groups  Walsh(2005)(3 × 5 pile group)φ = 40°2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.9: Comparison between calculated group reduction factors and pre-vious experimental works for 5×5 free-head pile groups in sand972 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(a) 3 × 3 pile groups   McVay et al.(1998)(3 × 3 pile group)2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.10: Comparison between calculated group reduction factors andprevious experimental works for 3×3 fixed-head pile groups in sand2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(b) 4 × 4 pile groups  McVay et al.(1998)(3 × 4 pile group)2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.11: Comparison between calculated group reduction factors andprevious experimental works for 4×4 fixed-head pile groups in sand982 3 4 5 6 700.20.40.60.81S/DGroup reduction factor5 × 5 pile groups    McVay et al.(1998)  (3 × 5 pile group)2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.12: Comparison between calculated group reduction factors andprevious experimental works for 5×5 fixed-head pile groups in sand2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor6 × 6 pile groups     McVay et al.(1998)   (3 × 6 pile group)2 3 4 5 6 700.10.20.30.40.50.60.70.80.91S/DGroup factor(a) 3 × 3 pile groups    φ= 30°φ= 35°φ= 40°McVay et al.(1998)(3 × 3 pile group)Figure 4.13: Comparison between calculated group reduction factors andprevious experimental works for 6×6 fixed-head pile groups in sand99Chapter 5Evaluation of InteractionBetween Rows of PilesThe engineer requires the imagination to visualize the needs ofsociety and to appreciate what is possible as well as the technologicaland broad social age understanding to bring his vision to reality. — -Sir Eric Ashby (1958)5.1 IntroductionThe interaction between piles reduces the lateral resistance of the individual pilesand this results in less lateral capacity than the sum of the lateral capacities of theindividual piles. In the previous chapter, the influence of the group effects on theoverall response of pile groups was evaluated by calculating group reduction fac-tors for different pile groups. To get a better insight on the group effects inside thepile groups, the influence of interaction between piles on their response is evalu-ated based on their row position in the group. To this end, continuum simulationsof pile groups subjected to static lateral loading are used to study the response pilegroups. Average load carried by the piles of each row and average induced bend-ing moment profiles along the piles are discussed based on their row position indifferent pile groups. For each pile group, p-multipliers are also calculated to ob-tain a better understanding about contribution of different rows to the total lateral100resistance of the pile group.Considering the amount of data for each pile group configuration, limited num-ber of configurations in the Table 4.1 are selected in this chapter to benchmark thecharacteristic features of response in various piles groups. Each pile group is lat-erally pushed to the maximum head deflection of 5 cm. The reason for selecting 5cm deflection is that the most of the experiments on the pile groups are about thisvalue. Mohr–Coulomb constitutive model is adopted for the continuum analyses.Figure 5.1 shows the schematic representation of all steel pipe pile groups ana-lyzed. The pile groups are in two general categories of free-head and fixed-headconditions. In each category, square pile groups with arrangements of 3×3, 6×6and 10×10, each one with two different spacings of S/D of 3 and 6 are considered.Results are presented in terms of average pile head load distributions and bendingmoment profiles for different rows in the group.In the continuum model the lateral sides of the mesh are taken far enough toavoid their influence on the response of piles as discussed in Chapter 3. Boundariesparallel to x− y plane are free to move in y and z direction but they are fixed in xdirection. Similarly, boundaries parallel to x− z plane are fixed in y direction butthey are free in x and z directions. All of the piles in the pile group are pushedsimultaneously. The friction angle of φ= 35◦ is used for soil with dilation angle of0◦ and Poisson’s ratio ν of 0.3. Details of continuum models of the pile groups aredescribed in Section 3.2.2.Calculating p-multipliers is one of the most practical tools for evaluating inter-action between different rows of piles. Therefore, p-multipliers for each pile groupis calculated using the procedure described in Section 3.4. They are calculated us-ing the continuum model in conjunction with p–y model. Having equivalent soilparameters in the continuum and p–y models is very important. Equivalent soilparameters are obtained following the procedure described in Section 3.4.1 whichresulted in shear modulus of 21,154 kN/m2 for soil with friction angle of φ= 35◦.It should be noted that within each row, piles behave differently as corner pilescarry more load than inner piles. However, this difference is not significant. Asan example, carried load by different piles in 6×6 pile groups with S/D ratios of3 and 6 are evaluated. Figure 5.2 shows the pile group configuration. Figure 5.3shows the load distribution between different piles of leading row, middle row and101trailing row for two pile groups. Applied deflection at the top is 5 cm. This figureshows that the variation within each row is not considerable. As S/D increases,this variation becomes less.5.2 Free-head pile groupsFree-head means that no rotational constraints are imposed on pile heads; the onlyimposed constraints are the equal pile head lateral deformations.5.2.1 Lateral load distributionIn this section the interaction between different rows in the free-head pile groupsis evaluated based on the average load carried by piles in each row.3×3 pile groups – Figures 5.4(a, b) show the average pile head loads for differentrows in the group (Hr) versus the corresponding pile head deflections, forthe 3×3 free-head pile groups with S/D of 3 and 6. The amount of carriedload by row 1 is almost the same for both pile groups. In the pile group withS/D of 3, average pile head load carried by row 3 at the head deflection of 5cm is almost half of the load carried by row 1 (leading row). When S/D isincreased to 6, the piles in the trailing rows carry considerably higher loads.Another approach for evaluating contribution of different rows to the lateralresistance is normalizing the average pile head load in each row (Hr) by theaverage pile head load for each pile (Hp). The Hp is obtained from dividingthe total applied load at the pile cap by the number of piles in the group.Figures 5.5(a, b) illustrate the normalized average load undertaken by differ-ent rows as a function of the pile head deflection for 3× 3 pile groups withthe S/D of 3 and 6. At the target deflection of 5 cm, Hr/Hp in row 1 is 1.4and 1.2 for pile groups with S/D of 3 and 6, respectively. This means thatwith increase of spacing contribution of row 1 to the total lateral load de-creases. In both pile groups contribution of row 1 increases with increasingthe pile head deflection. For row 2 the contribution doesn’t change with thepile head deflection. Contribution of row 3 to the lateral resistance of thegroup decreases as the pile head deflection increases.1026×6 pile groups – Figures 5.6(a, b) show the average pile head loads for differ-ent rows in the group (H) versus the corresponding pile head deflections, for6× 6 pile groups with S/D of 3 and 6. In case of S/D= 3, similar to the3× 3 pile group row 1 carries considerably higher loads than trailing rows.Figure 5.6(a) shows that average load carried by row 1 at the deflection of 5cm is close to the value for row 1 in 3×3 pile group with the same pile spac-ing. The amount of load carried by the last trailing row is less than half ofthe corresponding value for the leading row. Also, this figure shows that thecarried load by rows 4, 5 and 6 are close to each other. Figure 5.6(b) showsthat for the S/D of 6 the difference between these trailing rows become moreconsiderable.Figures 5.7(a, b) represent the Hr/Hp in different rows of the group versusthe pile head deflection for 6×6 pile groups with S/D of 3 and 6. For S/Dof 3, piles in row 1 carry 1.7 times of the average load per pile in the group.Normalized lateral loads for row 2 in both pile groups with different S/Dare about the same and they are equal to one. Corresponding values for otherrows of the group with S/D of 3 are less than one. For deflections larger than3 cm, the normalized loads for different rows don’t change considerably.10×10 pile groups – Figures 5.8(a, b) show the lateral load distribution betweenrows versus pile head deflection for 10× 10 pile groups. These pile groupsare the largest pile groups in this study. For S/D of 3, similar to the previouscases the load carried by row 1 is considerably higher than that of the otherrows. Carried load by rows 4 to 10 are within the same range while rows 1to 3 carry higher loads. When S/D increases to 6, the responses of rows 4 to10 become considerably different from each other.Figures 5.9(a, b) illustrate the normalized load undertaken by the piles indifferent rows of the group versus the pile head deflection for the 10× 10pile groups with the S/D of 3 and 6. In the pile group with S/D of 3, thenormalized carried load by the piles in row 1 is about 2 times of the averageload per pile in the group. Contribution of the piles in row 2 is 1.25 times ofHm. Contribution of the piles in row 3 are equal to Hm. Other trailing rowscarry less load than the average load per pile in the group and they are all103within the same range.Figures 5.4 to 5.9 show that for close spacing the carried load by row 1 is aboutthe same regardless of the number piles in the group. Also, the carried load by thevery last trailing row in all of these pile groups is less than half of the correspondingvalue for the leading row. According to these figures the response of trailing rowsare similar to each other for the close spacing. The difference between response oftrailing rows increases with pile spacing.5.2.1.1 Bending moment profileHaving an insight on the induced bending moments in the piles is important be-cause maximum bending moment often controls the design of the pile group. Fur-thermore, in designing pile foundations for lateral loads, it is important to knowthe depth to which significant bending moments are produced. Below are discus-sions of average bending moment profiles for the piles in each row at the pile headdeflection of 5 cm.3×3 pile groups – Figure 5.10 shows the bending moment profiles for piles basedon their row position in the 3×3 pile groups with S/D of 3 and 6. For S/D of3, the maximum bending moment in the leading row is considerably higherthan the corresponding value in the trailing rows. When S/D increases to 6,maximum bending moment for the last trailing row is getting closer to thecorresponding value for leading row. Figure 5.10 shows that with increasingthe S/D in the 3× 3 pile group, bending moment of the last trailing rowincreases the most. This happens because there are less group effects as thespacing increases. Maximum bending moment occurs at the depth of 1.5 min the row 1 and it goes deeper as the row number increases. This phenomenais more noticeable in pile group with close spacing as there are more groupeffects.6×6 pile groups – Figure 5.11 illustrates average bending moment profiles fordifferent rows in the 6× 6 pile groups with S/D of 3 and 6. In case ofS/D= 3, the maximum bending moment in the leading row is about 100%more than the corresponding value for the piles in the last trailing row. With104increasing S/D to 6, group effect decreases in the group and the maximumbending moment in the last trailing row is getting closer to the correspondingvalue for leading row. In the pile group with S/D= 3, there is a considerabledifference between rows 1, 2 and 3 in terms of maximum bending momentwhile the response of row 4, 5 and 6 are in the same range. For S/D= 3, themaximum bending moment in the pile occurs at the depth of 1.5 m for theleading row while it occurs at depth of about 3 m in the last trailing row. ForS/D= 6, there are less group effects and the maximum bending moment inall rows occurs at the depth of 1.5 m to 2 for piles in the leading row to thelast trailing row.10×10 pile groups – Figure 5.12 shows bending moment profiles for the largestpile groups in this study. The S/D of 3 and 6 are considered for these pilegroups as well. In the pile group with S/D of 3, the maximum bendingmoment of the piles in the leading row is about 100% more than the pilesin the last trailing row. There is a considerable difference between bendingmoment profiles of the piles in the rows 1 to 3. Piles in the rows 4 to 10 havesimilar bending moment profiles. Maximum bending moment along the pileoccurs at the depth of 1.5 m for the piles in the leading row but for the lasttrailing rows this depth is about 3 m which is similar to the case of 3×3 and6× 6 pile groups with close spacing. When S/D increases 6, group effectsdecrease and the bending moment profiles of piles in the rows 4 to 10 aregetting different from each other.5.2.2 p-multipliersEvaluating the p-multipliers of a pile group is very useful for understanding therole of each row in a pile group under lateral loading. In this section, p-multipliersare calculated for different rows of different pile groups following the proceduredescribed in Section 3.4. Figures 5.13 to 5.15 show calculated p-multipliers ofdifferent rows for the 3× 3, 6× 6, and 10× 10 free-head pile groups, each onewith S/D of 3 and 6. In particular the calculated p-multipliers for the 3×3 groupsare depicted in Figure 5.13. The leading rows have the highest p-multipliers andthe trailing rows have smaller values. In the pile groups with S/D of 3, the p-105multipliers for the first rows are around 0.8-0.9; these values increase to 0.9-1 ofS/D= 6. This implies minimal to almost no group effect for the leading rows ofthese 3× 3 pile groups. Rows 2 and 3 in both pile groups experience more groupeffects. The group effects are more pronounced in the pile group with close spacing(S/D= 3).Figure 5.14 shows the calculated p-multipliers for the 6×6 pile groups. In caseof S/D= 3, the p-multiplier for the leading rows in the 6×6 pile groups are lowerthan the corresponding values in the 3×3 pile groups. At S/D of 6, similar to thecase of 3× 3 pile groups the p-multipliers for the leading rows of the 6× 6 pilegroups are close to one, implying almost no group effect in the leading rows. Forthe trailing rows of the 6×6 pile groups, the variation of the p-multipliers amongdifferent rows is more significant in the case of S/D= 6 compared to the case ofS/D= 3.The p-multipliers for 10×10 pile groups are depicted in Figure 5.15. Compar-ing the 3×3, 6×6, and 10×10 pile group, the p-multipliers are generally smallerin the 10×10 groups than the corresponding values in the smaller pile groups withthe same S/D. This is because with more piles in the group and at the same S/D,the average interaction between piles increases. Of course at S/D of 6 where theleading rows experience almost no group effect the corresponding p-multipliers re-main close to 1 regardless of the number of piles in the group. It is also interestingto observe that again in the 10×10 pile group with S/D of 3, the p-multipliers arealmost constant for 3rd and higher trailing rows, while this is not the case when theS/D increases to 6.To summarize the overall observations from Figures 5.13 to 5.15 the followingstatements can be drawn. In all of the pile groups, the leading rows have the highestp-multiplier and the trailing rows have smaller values. The trend observed for thep-multipliers of the trailing rows implies that when the piles are close to each otherand there is considerable group effect (which is the case for S/D= 3 presentedhere), most trailing rows experience similar group effect in large pile groups.Reliability of the calculated p-multipliers is evaluated using some availablefield test data as presented in Figure 5.16. This figure shows the calculated p-multipliers and experimental data for 3×3 free-head pile groups. There are moreexperimental data on 3× 3 pile groups with S/D of about 3 than S/D of about 6.106Brown et al. (1988) and Morrison and Reese (1988) performed experimental stud-ies on 3×3 free-head pile groups with S/D of 3 in soil with ϕ= 38.5◦ and reportedp-multipliers of 0.8, 0.4, and 0.3 for the leading row, first trailing row, and secondtrailing row, respectively. McVay et al. (1995) reported the same p-multipliers forsame arrangement of piles but in a soil with ϕ= 33◦. These values are close to thecalculated p-multipliers for free-head pile groups shown in Figure 5.16(a). McVayet al. (1995) also conducted the same experiment in soil with ϕ= 30◦ and reportedthe values of 0.65, 0.45 and 0.35 which are still in the acceptable range compar-ing to the calculated p-multipliers. Rollins et al. (2005) reported p-multipliers of0.8,0.4, and 0.4 for a 3× 3 pile group with S/D of 3.3. Figure 5.16(a) shows thatthere is an acceptable agreement between calculated p-multipliers and correspond-ing experimental data. There is no experimental study for 3× 3 free-head pilegroups with S/D of 6 for comparison; however there are some experimental stud-ies like Christensen (2006) which had S/D of 5.65. Christensen (2006) reportedp-multipliers of 1, 0.7 and 0.65 for the leading row, first trailing row and secondtrailing row respectively. McVay et al. (1995) also conducted centrifuge tests on3× 3 free-head pile groups with S/D of 6 in soils with ϕ of 30 and 33◦. Theyreported the same set of p-multipliers for both friction angels. These p-multipliersare close to the calculated values for free-head condition shown in Figure 5.16(b).There is not enough experimental data available for larger pile groups to com-pare with calculated data.5.3 Fixed-head pile groupsFixed-head means that pile-heads are restrained from rotating in addition to beingconstrained to move laterally all together.5.3.1 Lateral load distributionFigures 5.17 to 5.22 compare the lateral load distributions in the fixed-head pilegroups with those in the free-head ones. The trend of lateral load distributionsamong different rows appears to be very similar in both group head conditions.The first rows consistently carry the highest loads. The difference between the firstrow and other rows especially in smaller S/D levels is significant due to presence107of more group effects. Responses of the trailing rows are close to each other forS/D of 3 especially in larger pile groups; the difference becomes more noticeableas S/D increases to 6. Similar to the observation made for free-head condition, theresponse of the first row in all six fixed-head groups is in the same range. The maindifference between the free- and fixed-head pile groups is the amount of loads thatdifferent rows carry; the piles in the fixed-head groups carry considerably higherloads than free-head pile groups. This is due to the considerably higher lateralstiffness of the fixed-head pile groups.5.3.2 Bending moment distributionThe distributions of bending moment profiles in different rows of the fixed-headpile groups are very different from those in the free-head pile groups; this is ex-pected due to the difference of pile head boundary conditions. For the same pilehead deflection, the maximum bending moments in the fixed-head condition aremuch higher than those in the free-head conditions. The maximum bending mo-ments in the fixed-head pile group occur at the pile head while in the free-head pilesgroups they occur somewhere along the depth. Piles in the leading row have higherbending moments compared to those in the trailing rows, regardless of the pilehead condition. Below, the average bending moment profiles for different rows, allat the pile head deflection of 5 cm, are compared between the free- and fixed-headpile groups.3×3 pile groups – Figure 5.23 compares the average bending moment profilesof different rows in the 3×3 fixed-head pile groups with those in the free-head ones. In the fixed-head condition, similar to the free-head condition,the bending moment profiles for the piles in the leading rows are about thesame; this is the case in both S/D ratios of 3 and 6.6×6 pile groups – Figure 5.24 illustrates average bending moment profiles fordifferent rows in 6×6 free- and fixed-head pile groups with S/D of 3 and 6.In fixed-head pile groups, the difference between bending moment profilesof the leading row in 6× 6 pile groups with different S/D ratios is morenoticeable in compare to 3×3 pile groups.10810×10 pile groups – Figure 5.25 shows the bending moment profiles for 10×10pile groups. The S/D of 3 and 6 are considered for these pile groups as well.In the fixed-head pile group with S/D of 3, similar to the case of free-headpile group, the maximum bending moment along the piles in the leading rowis considerably higher than last trailing row. In both S/D ratios, there areconsiderable differences between bending moment profiles of the piles inthe rows 1 to 3. Rows 4 to 10 have similar bending moment profile. In thepile group with S/D of 6, there are less group effects between trailing rowswhich results in considerably higher bending moments than correspondingrows in 10×10 pile group with S/D of 3.5.3.3 p-multipliersFigures 5.26 to 5.28 show calculated p-multipliers of different rows for the 3×3,6× 6, and 10× 10 fixed-head and free-head pile groups, each one with S/D of 3and 6. In particular the calculated p-multipliers for the 3× 3 groups are depictedin Figure 5.13. The leading rows have the highest p-multipliers and the trailingrows have smaller values. Fixed-head p-multipliers have similar trends to free-head values. In all of the pile groups, the leading row has the highest p-multiplierand the trailing rows have lesser values. The fixed-head pile groups show smallerp-multipliers than the free-head pile groups. The difference for p-multipliers offree- and fixed-head pile groups can be significant in trailing rows of pile groupswith large spacing. Also, the difference is smaller in case of S/D= 3 and morenoticeable in case of S/D= 6.McVay et al. (1998) performed experimental studies on 3× 3 fixed-head pilegroups with S/D of 3. They reported the same p-multipliers for fixed-head con-dition as their free-head condition test in McVay et al. (1995). They reportedp-multipliers of 0.8, 0.4, and 0.3 for the leading row, first trailing row, and secondtrailing row, respectively. These p-multipliers compare well with the calculatedp-multipliers for fixed-head condition in Figure 5.13(a).There is not enough experimental data available for larger pile groups to com-pare with calculated data. McVay et al. (1998) performed a centrifuge test on 3×6fixed-head pile group and reported the values of 0.8, 0.4, 0.3, 0.2, 0.2 and 0.3.109These values are higher than the calculated p-multipliers for fixed-head conditionshown in Figure 5.14(a); this can be contributed to the fact that there were only3 piles in each row in the test by McVay et al. (1998) which means they had lessgroup effects in their pile group.5.4 SummaryInfluence of interaction between different rows of piles on the lateral resistanceof pile groups is evaluated in this chapter. Based on the results of the parametricstudy, the response of different rows in the group is evaluated in terms of carriedlateral load and bending moment. Contribution of different rows to the lateralresistance is also quantified. Then, p-multipliers are calculated for each pile groupto characterize group effect in the pile group.The continuum simulations of pile groups subjected to static lateral loadingare used to study the response of 3× 3, 6× 6, and 10× 10 pile groups to covera range of small pile groups to large pile groups. For each pile group, center tocenter spacings of 3 and 6 pile diameters are considered. Fixed-head and free-headcondition are considered separately.The average load each pile carries in the leading row is higher than average loadin trailing rows. In large pile groups with close spacing, carried load by the secondtrailing row and higher is about half of the leading row. As pile spacing decreasesin the pile group, the difference between response of leading row and other rowsdecreases. Also, when spacing increases, the contribution of the leading row to thetotal lateral resistance of the pile group significantly decreases.In large pile groups with close spacing, the response of the second trailing rowand higher rows are almost the same in terms of lateral resistance and bendingmoment but for large spacing, there is a considerable difference between secondtrailing row and higher rows. Carried load by piles in the fixed-head pile groupsare considerably higher than free-head pile groups for the same head deflection.However, there are similar trends in the response of different rows in free-head andfixed-head pile groups.In pile groups with close spacing, p-multipliers for second trailing row andhigher are almost the same. However, in pile groups with larger spacing calcu-110lated p-multipliers are considerably different for each trailing row. Fixed-headpile groups have lower p-multipliers than free-head pile groups. The differencefor p-multipliers of free-head and fixed-head pile groups is significant specially intrailing rows of pile groups with close spacing.According to this study, for deflections larger than 3 cm (∼1 in) an individualrow’s contribution to the lateral resistance is constant which confirms findings ofMcVay et al. (1998). It also shows that in trailing rows maximum bending momentoccurs at greater depths compared with the leading rows which confirms Brownand Shie (1990, 1991) observations.111Row #1Row #2Row#n-1Row #nLoadSSD= 0.3 mt= 0.01 m(a)Load Pinned (free) or rigid (fixed)SD= 0.3 m10 m0.5 m(b)Figure 5.1: Schematic representation of the simulated pile groups in thisstudy where n=3, 6 and 10, and S/D =3 and 6; (a) plane view (b) sideview112#4#5#6Load#1#2#3 (b)A B C D E FFigure 5.2: Pile positions in regards to the loading direction in the 6×6 pilegroup113 A B C D E F  050100150(a)Pile columnPile head load (kN)   A B C D E F  050100150(b)Pile columnPile head load (kN)  Leading row (#1)Middle row (#3)Trailing row (#6)Figure 5.3: Load distribution at the deflection of 5 cm for piles based on theirposition in different rows in a 6×6 free-head pile group with S/D = 3(a), and S/D = 6 (b)1140 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #30 0.01 0.02 0.03 0.04 0.05020406080100120Pile head deflection (m)Average pile head load, Hr (kN) S/D= 3(a)0 0.01 0.02 0.03 0.04 0.05020406080100120Pile head deflection (m)Average pile head load, Hr (kN)  S/D= 6(b)Figure 5.4: Average pile head loads for different rows in 3×3 free-head pilegroups with S/D = 3 (a) and S/D = 6 (b)1150 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #30 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 6  (b)Figure 5.5: Normalized average pile head loads over the entire group for dif-ferent rows in 3×3 free-head pile groups with S/D = 3 (a) and S/D = 6(b)1160 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #3Row #4Row #5Row #60 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 3Pile head deflection (m)Average pile head load, H (kN)(a)0 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, Hr (kN)  (b)Figure 5.6: Average pile head loads for different rows in 6×6 free-head pilegroups with S/D = 3 (a), and S/D = 6 (b)1170 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #3Row #4Row #5Row #60 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/HpS/D= 6  (b)Figure 5.7: Normalized average pile head loads over the entire group for dif-ferent rows in 6×6 free-head pile groups with S/D = 3 (a), and S/D = 6(b)1180 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #100 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 3Pile head deflection (m)Average pile head load, Hr (kN)(a)0 0.01 0.02 0.03 0.04 0.05020406080100120Pile head deflection (m)Average pile head load, Hr (kN)  S/D= 6(b)Figure 5.8: Average pile head loads for different rows in 10× 10 free-headpile groups with S/D = 3 (a), and S/D = 6 (b)1190 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #100 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 6  (b)Figure 5.9: Normalized average pile head loads over the entire group for dif-ferent rows in 10× 10 free-head pile groups with S/D = 3 (a), andS/D = 6 (b)1200 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #30 50 100 15002468Bending moment (kN.m)Depth (m)S/D= 3(a)0 50 100 15002468Bending moment (kN.m)Depth (m)S/D= 6(b)Figure 5.10: Average bending moment profiles for different rows in 3× 3free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pile headdeflection: 5cm)1210 0.01 0.02 0.03 0.04 0.05020406080100120 S/D= 6Pile head deflection (m)Average pile head load, H (kN)  Row #1Row #2Row #3Row #4Row #5Row #60 50 100 15002468Bending moment (kN.m)Depth (m)  S/D= 3(a)0 50 100 15002468Bending moment (kN.m)Depth (m)  S/D= 6(b)Figure 5.11: Average bending moment profiles for different rows in 6× 6free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pile headdeflection: 5cm)1220 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #100 50 100 15002468Bending moment (kN.m)Depth (m)S/D= 3(a)0 50 100 15002468Bending moment (kN.m)Depth (m)S/D= 6(b)Figure 5.12: Average bending moment profiles for different rows in 10× 10free-head pile groups with S/D = 3 (a), and S/D = 6 (b) (pile headdeflection: 5cm)123 1 2 3  00.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 3(a) 1 2 3  00.20.40.60.81Row numberp−multiplier  3 × 3 pile group, S/D = 6(b)Figure 5.13: Calculated p-multipliers for 3× 3 free-head pile groups withS/D = 3 (a), and S/D = 6 (b)124 1 2 3 4 5 6  00.20.40.60.811.2Row numberp−multiplier  6 × 6 pile group, S/D = 3(a) 1 2 3 4 5 6  00.20.40.60.81Row numberp−multiplier 6 × 6 pile group, S/D = 6(b)Figure 5.14: Calculated p-multipliers for 6× 6 free-head pile groups withS/D = 3 (a), and S/D = 6 (b) 1 2 3 4 5 6 7 8 9 10  00.20.40.60.81Row numberp−multiplier10 × 10 pile group, S/D = 3(a) 1 2 3 4 5 6 7 8 9 10  00.20.40.60.811.2Row numberp−multiplier10 × 10 pile group, S/D = 6(b)Figure 5.15: Calculated p-multipliers for 10×10 free-head pile groups withS/D = 3 (a), and S/D = 6 (b)1250 1 2 3 4 500.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 3  CalculatedMorrison et al.(1986)Brown et al.(1988)McVay et al.(1995) − φ =30°McVay et al.(1995) − φ =33°Rollins et al.(2005) − S/D=3.34 datapoints(a)0 1 2 3 4 500.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 6  CalculatedMcVay et al.(1995) − S/D=5Christensen (2006) − S/D=5.65(b)Figure 5.16: Calculated p-multipliers versus available experimental data for3×3 free-head pile groups with S/D = 3 (a), and S/D = 6 (b)1260 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head0 0.01 0.02 0.03 0.04 0.05050100150200250300Pile head deflection (m)Average pile head load, Hr (kN)S/D= 3(a)0 0.01 0.02 0.03 0.04 0.05050100150200250300Pile head deflection (m)Average pile head load, Hr (kN)  S/D= 6(b)Figure 5.17: Average pile head loads for different rows in the 3×3 free- andfixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)1270 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 6  (b)Figure 5.18: Normalized average pile head loads over the entire group fordifferent rows in the 3×3 free- and fixed-head pile groups with S/D =3 (a) and S/D = 6 (b)1280.009 0.01 0.011 0.012 0.013 0.014 0.015 0.0160.350.40.450.50.550.6Pile head deflection (m)Normalized lateral load H/H m*  Row #1Row #2Row #3Row #4Row #5Row #6Free−headFixed−head0 0.01 0.02 0.03 0.04 0.05050100150200250300Pile head deflection (m)Average pile head load, H (kN)S/D= 3(a)0 0.01 0.02 0.03 0.04 0.05050100150200250300 S/D= 6Pile head deflection (m)Average pile head load, Hr (kN)  (b)Figure 5.19: Average pile head loads for different rows in the 6×6 free- andfixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)1290.009 0.01 0.011 0.012 0.013 0.014 0.015 0.0160.350.40.450.50.550.6Pile head deflection (m)Normalized lateral load H/H m*  Row #1Row #2Row #3Row #4Row #5Row #6Free−headFixed−head0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/HpS/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/HpS/D= 6  (b)Figure 5.20: Normalized average pile head loads over the entire group fordifferent rows in the 6×6 free- and fixed-head pile groups with S/D =3 (a) and S/D = 6 (b)1300 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head0 0.01 0.02 0.03 0.04 0.05050100150200250300 S/D= 3Pile head deflection (m)Average pile head load, Hr (kN)(a)0 0.01 0.02 0.03 0.04 0.05050100150200250300Pile head deflection (m)Average pile head load, Hr (kN)  S/D= 6(b)Figure 5.21: Average pile head loads for different rows in the 10× 10 free-and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)1310 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 3(a)0 0.01 0.02 0.03 0.04 0.0500.511.52Pile head deflection (m)Normalized pile head load, Hr/Hp S/D= 6  (b)Figure 5.22: Normalized average pile head loads over the entire group fordifferent rows in the 10× 10 free- and fixed-head pile groups withS/D = 3 (a) and S/D = 6 (b)1320 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)S/D= 3(a)−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)S/D= 6(b)Figure 5.23: Average bending moment profiles for different rows in the 3×3free- and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)(pile head deflection: 5cm)1330.009 0.01 0.011 0.012 0.013 0.014 0.015 0.0160.350.40.450.50.550.6Pile head deflection (m)Normalized lateral load H/H m*  Row #1Row #2Row #3Row #4Row #5Row #6Free−headFixed−head−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)S/D= 3  (a)−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)  S/D= 6(b)Figure 5.24: Average bending moment profiles for different rows in the 6×6free- and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)(pile head deflection: 5cm)1340 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.20.40.60.811.21.41.61.82Pile head deflection (m)Normalized lateral load H/H m*S/D= 6  Row #1Row #2Row #3Row #4Row #5Row #6Row #7Row #8Row #9Row #10Fixed−headFree−head−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)S/D= 3(a)−400 −300 −200 −100 0 100 20002468Bending moment (kN.m)Depth (m)S/D= 6(b)Figure 5.25: Average bending moment profiles for different rows in the 10×10 free- and fixed-head pile groups with S/D = 3 (a) and S/D = 6 (b)(pile head deflection: 5cm)135 1 2 3  00.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 3(a) 1 2 3  00.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 6(b)Figure 5.26: Calculated p-multipliers for 3× 3 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b) 1 2 3 4 5 6  00.20.40.60.81Row numberp−multiplier6 × 6 pile group, S/D = 3(a) 1 2 3 4 5 6  00.20.40.60.81Row numberp−multiplier6 × 6 pile group, S/D = 6(b)Figure 5.27: Calculated p-multipliers for 6× 6 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b)136 1 2 3 4 5 6 7 8 9 10  00.20.40.60.81Row numberp−multiplier10 × 10 pile group, S/D = 3(a) 1 2 3 4 5 6 7 8 9 10  00.20.40.60.81Row numberp−multiplier10 × 10 pile group, S/D = 6(b)Figure 5.28: Calculated p-multipliers for 10× 10 free- and fixed-head pilegroups with of S/D = 3 (a), and S/D = 6 (b)137Chapter 6Assessment of Design GuidelinesScientists study the world as it is; engineers create the world that hasnever been. — Theodore von Karman6.1 IntroductionThere are different recommendations available to consider the influence of groupeffect on the lateral response of pile groups. The related design guidelines by theAmerican Association of State Highway and Transportation Officials (AASHTO)and Federal Emergency Management Agency (FEMA) are the most common onesused in practice for pile groups. Another commonly used method to account forgroup effects is the one proposed by Reese and van Impe (2010) and implementedin the GROUP program (Reese et al., 2010) to be used as the default value for thep-multipliers for lateral springs. The GROUP program is very popular in practicefor its simplicity and capabilities in pile group analysis under various types ofloading. In this chapter, the most recent versions of the above design guidelines,namely AASHTO (2012) and FEMA P-751 (2012), and the equations of Reeseand van Impe (2010) are first described. Then the resulting p-multipliers and groupreduction factors from these design guidelines and recommendations are comparedwith the corresponding calculated values in chapters 4 and 5 of this study and thedifferences are discussed in details for each case.1386.2 Available recommendationsThis section presents details of three commonly used design guidelines for consid-eration of group effects in pile groups subjected to lateral loading.6.2.1 AASHTO (2012)Table 6.1 shows the p-multipliers recommended by the American Association ofState Highway and Transportation Officials (AASHTO) in 2012. These valuesare deduced from several field and laboratory tests listed in Table6.2 (US FederalHighway Administration (FHWA) manual – Hannigan et al. (2006)). For clarity,these studies are also marked with ∗ symbol in Table 2.2. Here are some importantobservations in these tests that form the AASHTO reference database:• Except for the centrifuge tests by McVay et al. (1995) which had S/D of 5,all other tests had S/D= 3.• All these tests were on free-head pile groups.• Except for the field test on a 4×4 pile group by Ruesta and Townsend (1997),all other tests were performed on 3×3 pile groups.• The database covers tests in stiff clays, clayey silts, and dense and loosesands.Going back to Table 6.1 it can be observed that AASHTO suggests p-multipliersonly for rows 1, 2, and 3 and higher for pile groups with S/D of 3 and 5. It suggestsusing interpolation to establish p-multipliers for other pile spacings when S/D isbetween these values. It has no specific recommendations for S/D > 5. To estab-lish the p-multipliers for S/D of 6, similar to proposed procedure in Rollins et al.(2003b) extrapolation is performed based on the recommended values in Table 6.1with the restriction that p-multipliers cannot exceed 1. Group reduction factorscan be calculated by averaging the recommended p-multipliers for all rows in apile group. Table 6.3 shows an example of calculated p-multipliers and groupreduction factors using AASHTO recommendation for 4×4 pile groups with S/Dvalues of 3, 4, and 5.1396.2.2 FEMA P-751 (2012)According to FEMA P-751 (2012), “the lateral resistance is primarily a function ofrow location within the group, rather than pile location within a row”. The recom-mendation of FEMA P-751 (2012) for calculation of p-multipliers is based on theequations proposed by Rollins et al. (2006), which in turn were based on experi-mental data from four full scale tests in Salt Lake City. The tests were conductedon the following pile groups: (i) 3×3 with S/D= 5.65 and D = 0.324 m, (ii) 3×4with S/D= 4.4 and D = 0.324 m, (iii) 3×5 with S/D= 3.3 and D = 0.324 m, and(iv) 3× 3 with S/D= 3 and D = 0.61 m. All pile groups were installed in stiffclay, and the pile head condition was free, hence likely the proposed equations donot account for fifferent soil types or pile head conditions. The resulting equationsproposed by Rollins et al. (2006) for calculation of p-multiplier (Pm) are listedbelow:Pm =0.26ln(S/D)+0.5≤ 1 Leading row0.52ln(S/D)≤ 1 First trailing row0.60ln(S/D)−0.25≤ 1 Second and higher trailing rows(6.1)According to FEMA P-751 (2012), because the direction of loading varies dur-ing an earthquake, the average of p-multipliers is of primary interest. In that sensethe average Pm or the group reduction factor can be used for all piles in the group.6.2.3 Reese and van Impe (2010)Reese and van Impe (2010) proposed three equations for calculation of p-multipliersto account for the group effects between piles that are side-by-side, in-line, andskewed (neither in-line nor side-by-side), with respect to the direction of appliedload. These equations are obtained using experimental data from pile groups withjust one row (edge effect) or one column (shadowing effect) of piles. For obtainingthe p-multipliers for skewed piles, a simple mathematical expression based on su-perposition of side-by-side and in-line piles equation is suggested. The proposedprocedure for calculation of p-multipliers is described below using Figure 6.1.Note that in all cases the symbols D and S refer to the pile diameter and the centerto center spacing of piles.140Figure 6.1(a) shows two piles arranged side-by-side with respect to the appliedload. The p-multiplier of βa is proposed as follows for both piles:βa ={0.64(S12/D)0.34 for 1≤ S12/D < 3.751 for S12/D≥ 3.75(6.2)Figure 6.1(b) shows two piles arranged in-line with respect to the applied load.The p-multipliers βbl and βbt are proposed as follows for piles 1 (leading) and pile2 (trailing), respectively:βbl ={0.7(S12/D)0.26 for 1≤ S12/D < 41 for S12/D≥ 4(6.3)βbt ={0.48(S12/D)0.38 for 1≤ S12/D < 71 for S12/D≥ 7(6.4)Figure 6.1(c) shows three piles arranged in-line with respect to the applied load.The p-multiplier of p1, p2, and p3 are proposed as follows for piles 1,2, and 3,respectively. can be expressed as:p1 = βbl12βbl13 p2 = βbt21βbl23 p3 = βbt31βbt32 (6.5)where the related reduction factors βbl and βbt are:βbl12 = 0.7(S12/D)0.26 ≤ 1βbl13 = 0.7(S13/D)0.26 ≤ 1(6.6)βbt21 = 0.48(S12/D)0.38 ≤ 1βbl23 = 0.7(S23/D)0.26 ≤ 1(6.7)βbt31 = 0.48(S13/D)0.38 ≤ 1βbl32 = 0.48(S23/D)0.38 ≤ 1(6.8)For piles that are arranged skewed with respect to the applied load, with angle αbetween direction of loading and the line connecting two piles a reduction factor141βs is proposed as follows:βs =√(β 2b cos2 α +β 2a sin2 α) (6.9)Based on the cases described above, an example for calculation of p-multipliers ina square pile group is shown below for the 2×2 pile group of Figure 6.1(d). Herethe p-multipliers for pile 1 or 2, and the ones for piles 3 and 4 can be calculated as:for piles 3 and 4 are the same. The p-multiplier for pile 1 or 2 can be calculatedas:Pm1 = Pm2 = βa12βbl14βs13Pm3 = Pm4 = βa43βbt41βs42(6.10)where related reduction factors βa, βbl, βbt, and βs are:βa12 = 0.64(S12/D)0.34 ≤ 1βa43 = 0.64(S34/D)0.34 ≤ 1(6.11)βbl14 = 0.7(S14/D)0.26 ≤ 1βbt41 = 0.48(S41/D)0.38 ≤ 1(6.12)βs13 =√β 2bl14 cos2 α +β 2a12 sin2 αβs42 =√β 2bt41 cos2 α +β 2a43 sin2 α (6.13)In a more general case of large pile group, the effect of piles on each other mustbe considered as follows. Assume that βji characterizes the effect of pile j on pilei. In this case the p–multiplier (Pm) for any given pile i is:Pmi =∏βji (6.14)An example for application of this method for calculating the p-multipliers andgroup reduction factor for a 4× 4 pile group with S/D of 3 are shown below and142then summarized in Table 6.4:βa = 0.64(S/D)0.34 = 0.64×30.34 =0.9298βbl = 0.7(S/D)0.26 = 0.7×30.26 =0.9314βbt = 0.48(S/D)0.38 = 0.48×30.38 =0.7287βsl =√β 2bl cos2 α +β 2a sin2 α=√0.93142 cos2 45◦+0.92982 sin2 45◦ =0.9306βst =√β 2bl cos2 α +β 2a sin2 α=√0.72872 cos2 45◦+0.92982 sin2 45◦ =0.8353The p-multiplier for each individual pile is calculated using combinations ofabove factors and listed in Table 6.4. For example the p-multiplier for the side pilein the leading row (row 1) is combination of effect of one side-by-side pile (βa),one in-line pile (βl), and one skewed pile (βs):Pm1 = βa×βbl×βsl = 0.9298×0.9314×0.9306 = 0.8059' 0.81Table 6.5 lists the important factors that can influence the group effects in pilegroups based on the analyses done in the previous chapters. This table shows alsowhether or not each of the factors has been accounted for in the above designguidelines. According to this table, S/D is the only factor that all of listed designguidelines account for it in quantifying the group effects. None of the listed designguidelines account for the effect of pile head fixity or the soil properties. OnlyReese and van Impe (2010) account for the effect of total number of piles in thegroup. The presented benchmark numerical study in chapters 4 and 5 accounts forthe effects of S/D, pile group fixity, and the total number of piles in the group, onthe resulting group reduction factors and p-multipliers.6.3 Group reduction factorsIn practice, it is more common to use a single factor to account for the group effectin pile groups as it is more convenient specially when there is a seismic or cyclic143loading involved. This is the philosophy of group reduction factor for the entiregroup, as opposed to the row-specific p-multiplier. Based on results of the numer-ical study in this thesis, a relatively comprehensive database of group reductionfactors for different pile group settings was created in chapter 4. Figure 6.2 com-pares the calculated group reduction factors from chapter 4 with the derived valuesfrom recommendations of AASHTO (2012), FEMA P-751 (2012), and Reese andvan Impe (2010). For all of these design guidelines the group reduction factor isderived based on the average of proposed p-multipliers for each row/pile in thegroup. This figure covers the cases of 3×3, 4×4, 5×5, and 6×6 pile groups withS/D ranging from 3 to 6.The calculated group reduction factors for almost all four sizes of free-headpile groups at S/D= 3 are close to the ones recommended by all design guidelines,with the closest cases being for the 3×3 and 4×4 pile groups. With increasing theS/D the differences between the calculated and recommended values become morepronounced, and this is even more significant in larger groups. The trend shownin Figure 6.2 on overestimation of the group reduction factor implies that all threedesign guidelines underestimate the group effect when S/D > 3 and particularly inlarger pile groups. The differences between the calculated group reduction factorsand the ones from the design guidelines are even more significant in fixed-headpile groups.AASHTO has the closest values to the calculated ones. For example, the cal-culated group reduction factor for 3× 3 pile group with S/D = 5 in soil with afriction angle of 30◦ is 0.80 for free-head condition and 0.59 for fixed-head condi-tion; the recommended value from AASHTO is 0.85 that is closer to the calculatedone for free-head condition. For 6× 6 pile group with the same soil and spac-ing, a group factor of about 0.48 is calculated for the fixed-head condition whereas0.79 is recommended by AASHTO. The recommended group reduction factorsfrom FEMA P-751 (2012) are slightly higher than those from AASHTO (2012) insmaller spacings. With increasing S/D, the derived group reduction factors fromrecommendations of these two design guidelines become closer to each other. Thegroup reduction factors deduced from equations of Reese and van Impe (2010) ap-pear to be even higher than the preceding design guidelines when S/D > 3. Rollinset al. (2006) also had similar observation and showed that these recommendations144overestimate the lateral resistance for closely spaced pile groups.In conclusion, with increasing the number of piles or with changing the pilehead condition to fixed-head or increasing the pile spacing, the group factors rec-ommended by these three available design guidelines are considerably higher thanthe calculated results of this study. All these differences can be attributed to thelimitation of experimental database that was the basis for these design recommen-dations. More specifically, as it was described before the experimental databasewas mainly limited to small pile groups with close spacing and free pile condition.6.4 p-multipliersIn practice, p-multipliers for different rows represent the interaction and load dis-tribution between rows. Calculated p-multipliers for 3×3, 6×6, and 10×10 pilegroups, each one with S/D of 3 and 6 from chapter 5 are depicted along the p-multipliers from recommendations of AASHTO (2012), FEMA P-751 (2012), andReese and van Impe (2010) in Figures 6.3 to 6.5. In each figure there are two setsof calculated values representing the p-multipliers of both free-head and fixed-headpile groups. There is just one set of data for each recommendations as they haveno specific consideration regarding the pile head condition.Figures 6.3, 6.4, and 6.5 show that in smaller pile groups with close pile spac-ing there is an acceptable agreement between the calculated and recommendedp-multipliers; more with those from AASHTO and less with those from Reese andvan Impe (2010). When S/D increases to 6, the difference between calculated andrecommended values becomes considerable. The calculated and recommended p-multipliers for the leading row in all pile groups are close to each other, almostregardless of number of piles in the group, the S/D ratio, or pile head condition.Calculated p-multipliers for the first and the second trailing row in almost all pilegroups with S/D = 3 are close to the recommended values by design guidelinesspecially AASHTO. When the row number increases, the difference between cal-culated p-multipliers and recommended values increases. Recommendations ofAASHTO (2012), FEMA P-751 (2012), and Reese and van Impe (2010) almost donot change in after second trailing row (row 3) in all pile groups. For S/D of 3,this study also confirms the same trend. For S/D of 6, p-multipliers decrease as145the row number increases. The difference between calculated p-multipliers and therecommendations for trailing rows in the pile groups with S/D of 6 is considerable.Calculated p-multipliers for free-head and fixed-head pile groups are close toeach other at S/D of 3 for all pile groups but as the S/D increases to 6 the differ-ence becomes considerable. Recommendations of design guidelines are closer tothe calculated p-multipliers for free-head pile groups in compare to the fixed-headvalues. The reason is that all of these design guidelines are based on the experi-ments on the free-head pile groups.6.5 SummaryIn this chapter the calculated group reduction factors in chapter 4 and the calculatedp-multipliers in chapter 5 are compared with corresponding values deduced fromdesign recommendations of AASHTO (2012), FEMA P-751 (2012), and Reeseand van Impe (2010).There is a good agreement between calculated group reduction factors and theones derived from these three available design guidelines in smaller pile groupswith free head condition. With increasing the number of piles or with changingthe pile head condition to fixed-head or increasing the pile spacing, design guide-lines lead to considerably higher group reduction factors than the calculated values.AASHTO has the closest recommendations to the results of this study. However,the group reduction factors recommended by AASHTO for pile groups larger than4× 4 with pile spacings larger than 3D appear to be higher than the calculatedvalues from the simulations.Calculated p-multipliers for the first and the second trailing row in almost allpile groups with S/D = 3 are close to the recommended values. However, whenthe number of rows increases, the difference between calculated p-multipliers andrecommended values increases. In particular, the difference between the calculatedfactors for fixed-head pile groups and the recommendations is considerable. Eval-uated design guidelines recommend almost no change in p-multiplier after secondtrailing row in all pile groups. Results of this study also confirm the same trendfor pile groups with S/D of 3. In pile groups with larger pile spacing calculated p-multipliers are different for each trailing row. The recommendations of evaluated146design guidelines overestimate the p-multipliers especially for trailing rows. Therecommended values for the group reduction factors based on FEMA P-751 (2012)and Reese and van Impe (2010) are even higher than those in AASHTO (2012).In general this study shows that the recommendations of the above three designguidelines overestimate the lateral resistance for larger pile groups, pile groupswith larger pile spacing, and pile groups with fixed-head conditions which canraise serious concerns as it is not conservative. This could be due to the fact thatin these design guidelines the group reduction factors are essentially driven fromlimited experiments on smaller pile groups with close pile spacing and free-headconditions.147Table 6.1: p-multipliers suggested in AASHTO (2012)Pile spacing in the Row 1 Row 2 Row 3direction of loading and higher3D 0.8 0.4 0.35D 1 0.85 0.7148Table 6.2: Laterally loaded pile groups studies used as the basis for recom-mendations of AASHTO (2012) for quantifying group effects (Hanniganet al., 2006)Soil Test Center to center p-multipliers Deflection Referencetype type pile spacing for rows 1, 2 &3+ (mm)Stiff clay Field study 3D 0.70, 0.50 , 0.40 51 Brown et al. (1987)Stiff clay Field study 3D 0.70, 0.60 , 0.50 30 Brown et al. (1987)Clayey silt Field study 3D 0.60, 0.40 , 0.40 25-60 Rollins et al. (1998)Dense sand Field study 3D 0.80, 0.40 , 0.30 25 Brown et al. (1988)Dense sand Centrifuge model 3D 0.80, 0.40 , 0.30 76 McVay et al. (1995)Dense sand Centrifuge model 5D 1.0, 0.85 , 0.70 76 McVay et al. (1995)Loose sand Centrifuge model 3D 0.65, 0.45 , 0.35 76 McVay et al. (1995)Loose sand Centrifuge model 5D 1.0, 0.85 , 0.70 76 McVay et al. (1995)Loose sand Field study 3D 0.80, 0.70 , 0.30 25-75 Ruesta and Townsend (1997)Table 6.3: An example for calculation of p-multipliers and group reductionfactors based on AASHTO (2012) guidelines for 4× 4 pile groups withdifferent spacingPile spacing in the p-multiplier Group Reductiondirection of loading Row 1 Row 2 Row 3 Row 4 Factor3D 0.8 0.4 0.3 0.3 0.454D 0.9 0.625 0.5 0.5 0.635D 1 0.85 0.7 0.7 0.81Table 6.4: An example for calculation of p-multipliers and group reductionfactor based on Reese and van Impe (2010) guidelines for a 4× 4 pilegroup with S/D of 3Individual p-multiplier Row Group ReductionSide pile Middle pile Middle pile Side pile p-multiplier FactorRow 1 0.81 0.70 0.70 0.81 0.760.49Row 2 0.49 0.35 0.35 0.49 0.42Row 3 0.49 0.35 0.35 0.49 0.42Row 4 0.56 0.44 0.44 0.56 0.5149Table 6.5: Consideration of various design guidelines for some important fac-tors related to group effects in lateral loading of square pile groupsAASHTO (2012) FEMA P-751 (2012) Reese and van Impe (2010)S/D Yes Yes YesPile head fixity No No NoTotal number of piles in the group No No YesSoil properties No No No150DS1212(a)DS1212(b)DS12233S 12(c)S13S124SS121413 24α α(d)Figure 6.1: Different pile group configurations with respect to loading direc-tion: two side by side piles (a), two in-line pile (b), three piles in a row(c), and 2x2 pile group (d)1510.20.40.6 SimulationFree head Fixed head φ30°35°40°Recommendations and guidelinesAASHTO (2012)FEMA P−751 (2012)Reese and Impe (2010)2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(a) 3 × 3 pile groups(a)2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(b) 4 × 4 pile groups(b)2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(c) 5 × 5 pile groups(c)2 3 4 5 6 700.20.40.60.81S/DGroup reduction factor(d) 6 × 6 pile groups(d)Figure 6.2: Comparison between calculated group reduction factors in thisstudy, AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe(2010) recommendations1522 3 4 5 6 7 8 9 10 1100.20.40.60.81Spacing/Diameter ratioGroup factor SimulationFree head Fixed headRecommendations and guidelinesAASHTO (2012)FEMA P−751 (2012)Reese and van Impe (2010) 1 2 3  00.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 3(a) 1 2 3  00.20.40.60.81Row numberp−multiplier3 × 3 pile group, S/D = 6(b)Figure 6.3: Comparison between calculated p-multipliers in this study,AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 3×3 pile groups2 3 4 5 6 7 8 9 10 1100.20.40.60.81Spacing/Diameter ratioGroup factor SimulationFree head Fixed headRecommendations and guidelinesAASHTO (2012)FEMA P−751 (2012)Reese and van Impe (2010) 1 2 3 4 5 6  00.20.40.60.81Row numberp−multiplier6 × 6 pile group, S/D = 3(a) 1 2 3 4 5 6  00.20.40.60.81Row numberp−multiplier6 × 6 pile group, S/D = 6(b)Figure 6.4: Comparison between calculated p-multipliers in this study,AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 6×6 pile groups1532 3 4 5 6 7 8 9 10 1100.20.40.60.81Spacing/Diameter ratioGroup factor SimulationFree head Fixed headRecommendations and guidelinesAASHTO (2012)FEMA P−751 (2012)Reese and van Impe (2010) 1 2 3 4 5 6 7 8 9 10  00.20.40.60.81Row numberp−multiplier10 × 10 pile group, S/D = 3(a) 1 2 3 4 5 6 7 8 9 10  00.20.40.60.81Row numberp−multiplier10 × 10 pile group, S/D = 6(b)Figure 6.5: Comparison between calculated p-multipliers in this study,AASHTO (2012), FEMA P-751 (2012), and Reese and van Impe (2010)recommendations for 10×10 pile groups154Chapter 7Conclusions, Implications forPractice, and RecommendedFuture ResearchDevelopment of practical tools for foundation engineers has been a primary ob-jective in this research. In practice, p-multipliers and group reduction factors aretypically used with spring models of pile groups to account for group effects insoil–pile interaction analysis. Recommendations are available in several designguidelines for the values of these factors. These recommendations are essentiallyderived from limited available experiments on small pile groups with close pilespacing and free-head pile conditions. The primary reasons for these limitations arethe cost and the difficulty of conducting experiments on more general pile groups.A thorough literature review revealed an apparent lack of comprehensive study onthe pile groups. The current study covers small to large pile groups with variouspile spacing, pile head conditions, and soil properties. Investigations on the pilegroups involved detailed nonlinear three-dimensional continuum models as well asp–y models. The continuum model served as a verification and calibration tool andcomplemented available experimental data, as no data were available for a widerange of pile groups. Major outcomes of this research can be sorted into two cat-egories: (a) conclusions and contributions derived based merely on the databasecreated in this study and (b) implications and improvements for practice, which155is concluded by comparing results of this study with recommendations of designguidelines.7.1 Conclusions and contributionsThis study has led to the conclusions and contributions listed below:• The continuum model of the pile group, built in FLAC3D (Itasca, 2012),is validated (primary validation) by simulating three different field tests on3×3 pile groups conducted by Brown et al. (1988), Walsh (2005), and Chris-tensen (2006). The validated continuum platform provided in this researchmay be used in developing additional similar analysis.• An extensive numerically driven benchmark database is generated based onthe validated platform to study the pile groups. The database is used to ex-tend the knowledge on the group reduction factors. Also, the influence ofinteraction between rows of piles on the lateral resistance in various pilegroups is evaluated. To this end, the responses of different rows in the groupare investigated in terms of carried lateral load, bending moment, and p-multipliers. The benchmark database covers pile groups with different char-acteristics:– Number of piles: 3× 3, 4× 4, 5× 5, 6× 6, and 10× 10 pile groupconfigurations are considered.– Center to center pile spacing: For each pile group configuration spac-ings of 3D, 4D, 5D, and 6D are considered. Pile spacing of 10D is alsoconsidered for the 3×3 pile group to cover very large pile spacing.– Pile head fixity: Each pile group is simulated with both free- and fixed-head conditions.– Soil parameters: Three uniform soil profiles with different friction an-gles are considered: 30◦, 35◦, and 40◦. Also, an investigation is carriedout to determine how different shear modulus distributions along thedepth affect the responses of pile groups.156• Reliability of the numerical database was further validated by comparing thecalculated group reduction factors with the available experimentally derivedfactors (secondary validation). Fourteen comparable experimental studieson pile groups were found, and very good agreement between experimentaldata and calculated results was observed. Calculated p-multipliers are alsocompared with available experimental p-multipliers.• The calculated group factors in the generated benchmark database show thefollowing trends. For a given pile spacing, the group reduction factor de-creases with increasing number of piles in the pile group and also with theincrease of the friction angle of the soil. The group reduction factor increaseswith the increase of the S/D. The calculated fixed-head group reduction fac-tors appear to be generally smaller than the free-head group reduction fac-tors. The study suggests that the pile head condition should be taken intoaccount in determination of the group reduction factor.• When it comes to the response of individual rows in the group, the trend oflateral load distributions among different rows appears to be very similar inboth pile head conditions. The difference between the first row and otherrows, especially in smaller S/D levels, is significant because of the pres-ence of more group effects. Responses of the trailing rows are close to eachother for S/D of 3, especially in larger pile groups; the difference becomesmore noticeable as S/D increases to 6. In each pile head condition cate-gory, the response of the first row is in the same range. The p-multipliers forthe fixed-head pile groups are less than those for the free-head pile groups.The difference for p-multipliers of free- and fixed-head pile groups can besignificant in trailing rows of pile groups with large spacing.• Recommendations of AASHTO (2012), FEMA P-751 (2012), and Reese andvan Impe (2010) for analysis of pile groups are evaluated using the databasegenerated. The differences between calculated results and each recommen-dation are discussed to point out the deficiencies in current practice, espe-cially for large pile groups and pile groups with fixed-head condition. Thefollowing section summarizes the highlights of evaluation of design guide-157lines.7.2 Implications for practiceThe design guidelines by the American Association of State Highway and Trans-portation Officials (AASHTO), Federal Emergency Management Agency (FEMA),and the method proposed by Reese and van Impe (2010) are the most common onesused in practice for analysis of pile groups.Calculated group reduction factors for 3×3 and 4×4 pile groups with spacingsof about 3D in the present study are close to the deduced values from AASHTO(2012). Corresponding values from AASHTO for pile groups larger than 4×4 withpile spacings larger than 3D appear to be higher than the calculated values fromthe simulations. In particular, the difference between the calculated factors forfixed-head pile groups and deduced values from AASHTO is considerable. Use ofthe recommendations of AASHTO leads to overestimation of the group reductionfactors and, hence, the lateral resistance. The recommended values for the groupreduction factors based on FEMA P-751 (2012) and Reese and van Impe (2010)are even higher than those in AASHTO. The reason is that they are using lessexperimental data as the basis for their recommendations. For example, Reese andvan Impe (2010) use experimental studies on pile groups with just one row (edgeeffect) or one column (shadowing effect) of piles.To evaluate design guidelines more extensively, they are compared with calcu-lated p-multipliers for different rows in various pile groups. AASHTO (2012) andFEMA P-751 (2012) recommend different p-multipliers for the leading row, firsttrailing row, and second trailing row; however, for rows after that they consider thesame p-multiplier as the second trailing row. Also, p-multipliers resulting fromrecommendations of Reese and van Impe (2010) show similar trends. Accordingto this study, in large pile groups with close spacing the responses of the secondand higher trailing rows are almost the same in terms of lateral resistance, bendingmoment, and p-multipliers, which confirms the recommendations of design guide-lines, However, as the spacing increases each trailing row has a different response.In general, this study shows that the recommendations of the above three designguidelines overestimate the lateral resistance for larger pile groups and especially158for pile groups with fixed-head conditions, which in turn leads to underestimationof deflections at the pile heads under a certain demand from the superstructure.This could be because in these design guidelines the group reduction factors areessentially driven from experiments on 3×3 pile groups with free-head conditions.Therefore, using values calculated in this study for larger pile groups or pile groupswith different pile head conditions would be more reliable.7.3 Recommended future researchIt should be noted that in numerical modeling there is always space for improve-ment, and this study is not an exception. Each model works with some assumptionsand simplifications of the real physics. Depending on what is expected from themodel, these simplifications may or may not be acceptable. The validation processis to build confidence on the prediction capability of the model for a certain typeof problem.While this research has achieved greater insight into soil–pile interaction in thepile groups under lateral loading, further studies are still required on this subject.The following topics can be proposed as potential subjects of further research onpile foundations.• To expand the range of applicability of the simulations presented in this the-sis the following improvements can be explored: more realistic simulationof the pile, more realistic simulation of soil-pile interaction, accounting forthe effects of pile installation methods, more advanced constitutive modelsfor representing the stress-strain response of soil.• Performing centrifuge studies on some of the pile groups discussed, espe-cially larger groups with various pile head fixities, can provide additionalbasis for further validation of the employed numerical modeling approachand also strengthen the findings of this study.• Group reduction factors are typically calculated using monotonic loadingtests, but they are also used for cyclic and seismic loading of pile groups.Reliability of using these group reduction factors needs to be investigatedfor analysis of pile groups under cyclic and seismic loading.159• While this study covers only square pile groups with a circular profile sectionand diameter of 0.3 m, studying different arrangements of piles, the effect ofpile diameter, and the section geometry would be beneficial. Also, in all ofthe pile groups in this study and in most of the previous studies the pile capwas not embedded. Embedding the pile cap can change the response of thegroup drastically. Use of p-multipliers and group reduction factors for pilegroups with an embedded pile caps needs further investigation.• Batter piles provide a much higher resistance to lateral loads than verticalpiles. They can be very effective in resisting static lateral loads. There area few detailed studies on pile groups with batter piles, e.g., Kubo (1965),Awoshika and Reese (1971), and Zhang et al. (1999). These studies suggestmodifying p–y curves of a vertical pile by a factor to express the effect of thepile inclination. 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(1998), Lateral statnamic load testing andanalysis of a pile group, in ‘Proceedings of 3rd Conference on GeotechnicalEarthquake Engineering and Soil Dynamics’, Seattle, pp. 1319–1330. → pages21Yang, Z. and Jeremic, B. (2003), ‘Numerical study of group effects for pile groupsin sands’, International Journal for Numerical and Analytical Methods inGeomechanics 27, 1255–1276. → pages 3, 17Zhang, L., McVay, M. and Lai, P. W. (1999), ‘Centrifuge modelling of laterallyloaded single battered piles in sands’, Canadian Geotechnical Journal36, 1074–1084. → pages 160168Appendix ACalculating p–multipliers forFixed Head Pile GroupsThe methodology for calculating p–multipliers for fixed head pile groups is similarto the free head pile groups as depicted in Figure A.1. The only challenge is thatin GROUP software piles can only be fixed to the pile cap and they cannot be fixedin global directions. For calculating p–multiplier for fixed head pile groups, eachrow should be fixed against rotation in global directions. Figure A.2 describes theproblem.For resolving this issue, two rows of piles are modeled in GROUP and theyare connected rigidly to a pile cap as shown in Figure A.3 and applied force (F) ismultiplied by two. These two rows are placed far enough from each other (large L)to provide the fixity against rotation and also to have the same load on all piles. Toverify this solution, one row of fixed head piles is also simulated in SAP2000 CSI(2009). In SAP2000, each node can be fixed in each direction globally. The samevalues for springs stiffness are imported from GROUP to SAP2000. Describedmethod in Figure A.3 and the fixed head model in SAP2000 resulted in the sameresponse.169ΔCalculating the force for each rowApplying the calculated force to one rowp x Pp yChanging P  till Δ = Δ(a) Continuum model(b) p-y model1Δ 212m mFigure A.1: Methodology for calculating p–multiplier for fixed head pilegroups using continuum and p–y models (Pm : p–multiplier)Δ 2F(a) (b)Figure A.2: Demonstration of the challenge for calculating p–multipliers us-ing GROUP:(a)Needed fixity for calculating p–multipliers (b)Pile tothe cap fixity in the GROUP170L2FFigure A.3: Solution for modeling a row of fixed head piles in GROUPSAP2000SAP2000 v14.0.0 - File:fixed35Deg - Joint Loads (DEAD) (As Defined) - KN, m, C Units10/9/14 13:08:49  (a) SAP2000model of onerow of pilesSAP2000SAP2000 v14.0.0 - File:fixed35Deg - Deformed Shape (DEAD) - KN, m, C Units10/9/14 13:09:36  (b) Deformed shape ofthe piles in the SAP2000Figure A.4: Verifying the GROUP solution using SAP2000171

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