A MODEL FOR THE RESPONSE OF SINGLE TIMBER FASTENERS AND PILES UNDER CYCLIC AND DYNAMIC LOADING by Nil KWASHIE ALLOTEY B.Sc. University of Science and Technology, Kumasi, Ghana, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1999 © Nii Kwashie Allotey, 1999 In presenting this degree at the thesis in partial fulfilment of University of British Columbia, I agree freely available for reference copying of department this or publication of and study. this his or her representatives. C l ^ L 6MGrH H £ 6 £ | A ( C - The University of British Columbia Vancouver, Canada Date DE-6 (2/88) l ^ / O G f that the may be It thesis for financial gain shall not permission. Department of requirements I further agree thesis for scholarly purposes by the is that an advanced Library shall make it permission for extensive granted by the understood be for that allowed without head of my copying or my written Abstract ii ABSTRACT A model is presented for the cyclic and dynamic analysis of timber fasteners and pile foundations. Both systems are, in that they are beam-like structures that are embedded in a flexible medium. In the past, many different numerical methods, ranging from very elaborate finite element models, to closed-form solutions for beams on elastic foundations have been used to analyze these systems. Among these, in the development of computer software, the use of a 1-D finite element implementation of beams on a nonlinear foundation has been shown to be the most promising, mainly due to the relative accuracy and simplicity of the model. In this thesis, such a model is developed for the analysis of both systems. The model is based on one previously developed, which used a beam element on a nonlinear flexible foundation that takes into account the formation of gaps between the fastener/pile and medium. In this thesis, the effect of interface friction, and the initial confining pressure surrounding the fastener/pile has been included in the model. Due to the inclusion of interface friction, the model allows, in the case of piles, the study of problems with horizontal and/or vertical acceleration inputs. The model is the basis for the development of a cyclic analysis code for single timber fasteners, and cyclic and dynamic analysis codes for single piles in cohesionless soil. Various numerical examples of timber fasteners and piles are used to show that the predicted results of the model are consistent with expected response behavior. Since previous models have not taken either friction or initial confining pressure into account, the effect of both of these on the predicted response of both systems has been investigated. It is seen from this study that the amount of initial confining pressure around a timber fastener does not significantly influence the hysteretic response under lateral cyclic loading, but rather influences the amount of withdrawal in the case of driven fasteners. For estimated practical values of the coefficient of friction and tangential stiffness of the interface, friction is seen to have a significant influence on the amount of withdrawal, and not much influence on the hysteretic response. In the case of piles, neither the initial confining pressure nor friction are observed to have any considerable effect on the hysteretic response under lateral cyclic loading Table of Contents iii TABLE OF CONTENTS Abstract " Table of Contents »'» List of Tables Viii List of Figures t List of Symbols xii Acknowledgements x v l ? i Dedication 1. 2. x xvii Chapter 1: Introduction 1 1.1. Introduction 1 1.2. Objectives and Scope of Study 2 1.3. Outline of Study 3 Chapter 2: Background 5 2.1. 5 2.2. Review of Analytical Methods for Beams on Flexible Foundations 2.1.1. The Modulus of Foundation Reaction Method 5 2.1.2. The Boundary Element Method 6 2.1.3. The Finite Element Method 7 Friction 9 2.2.1. Definition of Coulomb's Laws of Friction 9 2.2.2. Coulomb's Friction against other Theories of Friction 9 2.2.3. Plasticity Model of Friction 2.3. Contact Elements in ANSYS Commercial FE Package 2.4. Mechanically Fastened Timber Joints 11 12 • 16 2.4.1. Types of Timber Fasteners 16 2.4.2. Monotonic Response 16 2.4.3. Cyclic Response 17 Table of Contents 2.4.4. 2.5. 2.6. 3. lv Modeling of Embedment Pressure Piles 18 18 2.5.1. Types of Piles 18 2.5.2. Monotonic Response 19 2.5.3. Cyclic Response 21 Concluding Remarks 21 Chapter 3: Analytical Models 23 3.1. Introduction 23 3.2. Foundation Pressure 24 3.2.1. Modeling Foundation Pressure 26 3.2.1.1. Components of Foundation Pressure 28 3.2.1.2. Modeled Response of Medium 29 3.2.1.3. Different Types of Foundation Responses 30 3.2.2. Embedment Pressure Relationships 35 3.2.2.1. Embedment Curve for Timber Fastener System 36 3.2.2.2. Embedment Curve for Cohesionless Soil-Pile System 37 3.2.3. Determination of Initial Confining Pressure 38 3.2.3.1. Estimation of Initial Confining Pressure around Timber Fastener 38 3.2.3.2. Estimation of Initial Confining Pressure around Pile 39 3.3. Modeling of Frictional Forces 39 3.4. Formulation of Static Loading Governing Equations and FE Implementation 40 3.4.1. Statement of Problem 40 3.4.2. Derivation of Governing Equilibrium Equations 41 3.4.2.1. Kinematics 41 3.4.2.2. Constitutive Equations 42 3.4.2.3. Equilibrium Equations 42 3.4.3. FE Implementation and Solution Procedure 43 Table of Contents 3.5. 3.6. 3.7. 4. v 3.4.3.1. Displacement Field 43 3.4.3.2. FE Representation of Virtual Work Equation 43 3.4.3.3. Solution Procedure 44 Formulation of Dynamic Analysis Equations and FE Implementation 45 3.5.1. Statement of Problem 45 3.5.2. Derivation of Dynamic Equilibrium Equations 46 3.5.3. FE Implementation and Solution Procedure 47 3.5.3.1. Time-Integration Procedure 47 3.5.3.2. FE Representation of Virtual Work Equation 48 Enhancing Solution Algorithm 49 3.6.1. Efficiency of Solution Algorithm 49 3.6.2. Line Search Method 50 Concluding Remarks 51 Chapter 4: Results of Static Analysis 53 4.1. Introduction 53 4.2. Description of Programs 53 4.3. Timber Fastener Examples 57 4.3.1. Variables under Study 58 4.3.1.1. Coefficient of Static Friction between Wood and Steel 59 4.3.1.2. Effective Elastic Tangential Stiffness of Interface 59 4.3.1.3. Initial Confining Pressure 61 4.3.2. Input Displacement History 63 4.3.3. Single Bolt Connecting Three Wood Layers 63 4.3.3.1. Specifications of Problem 63 4.3.3.2. Cyclic Response 63 4.3.4. 4.3.4.1. Single Nail/Dowel in Wood Specifications of Problem 67 68 Table of Contents vi_ 4.3.4.2. 4.3.5. 4.4. 5. 75 Variables under Study 80 81 4.4.1.1. Coefficient of Static Friction between Pile and Soil 81 4.4.1.2. Effective Elastic Tangential Stiffness 84 4.4.1.3. Coefficient of Lateral Earth Pressure 85 4.4.2. Input Displacement History 86 4.4.3. Cyclic Response 87 4.4.4. Discussion of Results 91 Concluding Remarks 93 95 5.1. Introduction 95 5.2. Description of Program 95 5.3. Numerical Example 96 5.3.1. Specifications of Problem 97 5.3.2. Input Acceleration Record 98 5.3.3. Dynamic Response 99 5.3.4. Discussion of Results Concluding Remarks 103 105 Chapter 6: Conclusions and Recommendations 106 6.1. 106 6.2. 7. Discussion of Results Chapter 5: Results of Dynamic Analysis for a Pile 5.4. 6. 68 Single Pile Example 4.4.1. 4.5. Cyclic Response Conclusions 6.1.1. Computer Programs Developed 107 6.1.2. Predicted Response from the Model 107 6.1.3. Effect of Friction and Initial Confining Pressure 108 Recommendations Bibliography 108 U° Table of Contents A B C Appendix A: Listing of HYST, Sample Input Files, Plots and Worksheets vii 116 A.l Listing of HYST 116 A.2 Sample Input File 131 A.3 Sample Plots 133 A. 4 Sample Worksheets 135 Appendix B: Listing of PILE, Sample Input File, Plots and Worksheets 138 B. l Listing of PILE 138 B.2 Sample Input File 150 B.3 Sample Plots 152 B. 4 Sample Worksheets 155 Appendix C: Listing of EPILE, Sample Input File and Worksheets 157 C. l Listing of EPILE 157 C.2 Sample Input File 169 C.3 Sample Worksheets 171 List of Tables LIST OF TABLES Table 4.1 Recommended Values of the Coefficient of Friction between Wood and Steel (From Wood Handbook [86]) Table 4.2 Estimated Initial Confining Pressure around Nail based on Experimental Results of Nail Withdrawal Test (From Wood Handbook [86]) Table 4.3 Estimated Initial Confining Pressure around Nail based on Nail Withdrawal Design Loads (From NDS [2]) Table 4.4 Recommended Values of the Angle of Wall Friction for Given Soil-Pile Interface Conditions (From Bowles [10]) Table 4.5 Recommended Values of the Angle of Wall Friction for Given Soil-Pile Interface Conditions (From API Design Code [3]) Table 4.6 Recommended Values for the Coefficient of Lateral Earth Pressure around Piles Proposed by Different Researchers Table 4.7 Summary of the Effect of Friction and Initial Confining Pressure on the Cyclic Response of Pile List of Figures LIST OF FIGURES Figure 2.1 Non-Localized Distribution of Stress on Contact Surface 10 Figure 2.2 Shear Force-Displacement Curve (After Johnson [42]) 11 Figure 2.3 General Force-Displacement Curve for COMBIN39 Element 13 Figure 2.4 Schematic Diagram of COMBIN40 Element 13 Figure 2.5 Schematic Diagram of CONTAC12 Element 14 Figure 2.6 Schematic Diagram of CONTAC48 Element 15 Figure 2.7 Recommended API P-y Curves for Sands and Clays 20 Figure 2.8 Recommended API Q-z Curves 21 Figure 3.1 Basic Generic Beam-Foundation System 23 Figure 3.2 Variation of Initial Confining Pressure with Time due to Viscoelastic Properties of Wood 27 Figure 3.3 Different Methods of Shifting Embedment Curve to account for Initial Confining Pressure....28 Figure 3.4 Stress Distribution and Corresponding Resultant Forces around Beam Element 31 Figure 3.5 Schematic Plan View of a Typical Response 34 Figure 3.6a Force-Displacement Response for a Case (i) Type of Foundation Response 35 Figure 3.6b Force-Displacement Response for a Case (ii) Type of Foundation Response 36 Figure 3.6c Force-Displacement Response for a Case (iii) Type of Foundation Response 37 Figure 3.7 Wood Embedment Response (After Foschi [30]) 31 Figure 3.8 P-y Curve for Cohesionless Soils (After Yan and Byrne [94]) :31 Figure 3.9 Forces acting on Fastener 34 Figure 3.10 Relationship between Tangential Frictional Force and Tangential Slip 39 Figure 3.11 Free-Body Diagram of an Elemental Length of the Beam 41 Figure 3.12 Schematic Diagram of Mass supported by Generic Beam-Foundation System 45 Figure 3.13 Free-Body Diagram of Mass supported by Generic Beam-Foundation System 46 Figure 3.14 Case where N-R Solution Procedure proceeds in a Circular Manner 50 Figure 4.1 Schematic Diagram of Algorithm used to Compute New Input Displacement 57 List of Figures x Figure 4.2 Schematic Diagram of Nail/Dowel 58 Figure 4.3 Schematic Diagram of Three Wood Layer Bolted Connection 58 Figure 4.4 Plot of Experimental Data from a Nail Withdrawal Test (From Wood Handbook [86]) 60 Figure 4.5 Reversed Cyclic Input Displacement History used in Timber Fastener Example 63 Figure 4.6 Effect of Coefficient of Friction on the Hysteretic Response of Bolted Connection 65 Figure 4.7 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Bolted Connection 66 Figure 4.8 Effect of Initial Confining Pressure on the Hysteretic Response of Nail/Dowel in Wood 69 Figure 4.9 Effect of Coefficient of Friction on the Hysteretic Response of Nail/Dowel in Wood 70 Figure 4.10 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Nail/Dowel in Wood 71 Figure 4.1 la Effect of Initial Confining Pressure and Coefficient of Friction on the Tip Displacement Response of Nail/Dowel in Wood 72 Figure 4.1 lb Effect of Effective Elastic Tangential Stiffness on the Tip Displacement Response of Nail/Dowel in Wood 73 Figure 4.12 Summary of the Effect of Friction on the Cyclic Response of Timber Fastener 79 Figure 4.13 Assumed Variation of the Angle of Wall Friction for Given Pile Problem 84 Figure 4.14 Reversed Cyclic Input Displacement used in Pile Example 87 Figure 4.15 Effect of Coefficient of Friction on the Hysteretic Response of Pile 88 Figure 4.16 Effect of Coefficient of Lateral Earth Pressure on the Hysteretic Response of Pile 89 Figure 4.17 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Pile 90 Figure 5.1 Initial Stiffness of Example Soil-Pile System in the Vertical and Horizontal Directions 98 Figure 5.2 Horizontal Component of Joshua Tree Station Acceleration Record in E/W Direction 99 Figure 5.3a Vertical and Horizontal Displacement of Mass for Peak Accelerations of 0.284g and 0.426g in Vertical and/or Horizontal Directions Figure 5.3b Vertical and Horizontal Displacement of Mass for Peak Accelerations of 0.568g and 0.7 lOg in Vertical and/or Horizontal Directions Figure 5.4 100 101 Amount of Vertical Displacement at High Levels of Acceleration for Acceleration in Vertical Direction only 104 List of Figures Figure A . l Shape of Nail/Dowel and Hole at the 400 Displacement Step 13 Figure A.2 Shape of the Lower Portion of Nail/Dowel and Hole at the 400 Displacement Step 13 Figure B. 1 Cyclic Response of Pile Example for Relatively Small Input Displacements 15 Figure B.2 Shape of Pile and Hole at the 750 Displacement Step 15 Figure B.3 Shape of the Lower Portion of Pile and Hole at the 750 Displacement Step th th th th 15 List of Symbols LIST OF SYMBOLS a , Horizontal and vertical ground accelerations a Vector of degrees of freedom gh C/, C , C C Boundary condition constants for Hetenyi's beam-column solution in Eq. 2.2 C/,, C„ Horizontal and vertical damping constants d Diameter of fastener/pile 2 3i 4 d, d Inner and outer diameters of fastener/pile 0 Di, D Gap developed on the left and right sides between fastener/pile and medium r D Displacement corresponding to maximum embedment pressure in Eq. 3.1 max D Relative density of soil E Young's Modulus of Elasticity rs E Maximum Young's Modulus of soil F Lateral load G Specific gravity of wood max Hi, H Hole developed on the left and right sides between fastener/pile and medium r i,j Nodes of beam element / Moment of inertia about y-y axis k Given number of iterations k Elastic foundation modulus kj Coefficient of lateral earth pressure k„ Coefficient of lateral earth pressure at rest K Initial Stiffness of Foschi's wood embedment curve Kj Effective elastic tangential stiffness of interface e KJI, Kfl, Kp K eci,, Ksec2,, K Estimated elastic tangential stiffnesses of interface S K, sec3 Kffy„ Assumed secant vertical stiffnesses of nail withdrawal experiment Static and dynamic global stiffness matrices List of xili Symbols Linear and geometric stiffness matrices Foundation and Frictional stiffness matrices Mass and Damping stiffness matrices K, Dummy global stiffness matrix L Length of fastener/pile L,U Lower and upper triangular stiffness matrices u,w Axial and Lateral displacement of cross-sectional centroid M Concentrated mass supported by beam-foundation system N Axial load P Foundation force per unit length of beam Pe Embedment pressure Pi Initial confining pressure P,,Pr Foundation pressure on the left and right sides of fastener/pile Qo, Q, Q* Q , Q4 Constants in Foschi's embedment curve, Eq. 3.1 R Residual force vector used in Eq. 3.44 3 Computed constants for Hetenyi's beam-column solution in Eq. 2.2 t Given time Tf Tangential frictional force per unit length of beam Frictional force on the right and left sides of fastener/pile u Displacement vector V Volume of fastener/pile V„em V , V New, previous and current input displacements V Vector representing previous displacement vector p c W/,W Lateral displacement of left and right sides of fastener/pile wu Horizontal and vertical relative velocities of given point along centroidal axis. wu Horizontal and vertical relative accelerations of given point along centroidal axis, w Intermediate vector in Cholesky decomposition r List of Symbols xW x' x' Horizontal and vertical total accelerations of mass supported by system x' x' Horizontal and vertical relative velocities of mass supported by system x x Horizontal and vertical relative accelerations of mass supported by system x x Horizontal and vertical relative displacements of mass supported by system x Direction along the fastener/pile y Direction perpendicular to fastener/pile h v h v h v h v a,, J3 Constants for Yan and Byrne's P-y curve in Eq. 3.2 S (Xf Force tolerance factor f5 Displacement tolerance factor 8 Angle of wall friction Se Virtual strain S, Measured tangential displacement along interface d u SWj„ 8W , Internal and external virtual work da Variation of vector of degrees of freedom <5ii Virtual displacement vector h ex A,A Current and previous tangential displacements along interface A Imposed displacement of middle layer in three layer bolted connection At Time step A Ai Tangential displacement on the right and left sides of fastener/pile Au Vector of incremental displacements Av Vector representing w, w' or w incremental displacement vectors 0 m n e, e 0 Current and previous strains at a given point in fastener/pile <j> Total angle of internal friction 0' Effective angle of internal friction y Total unit weight of soil List of Symbols y' Effective unit weight of soil 7] Parameter for line search ju Coefficient of friction z Tangential shear stress around fastener/pile a, a Current and previous stresses at a given point in fastener/pile a <r Normal stress around fastener/pile r a>h, OK *P~, ^fdyn f¥f *F/ ? ¥ Horizontal and vertical natural frequencies of system Static and dynamic global out-of-balance force vectors Foundation and Frictional force vectors a Linear and nonlinear internal force vectors faciei Inertia and damping force vectors % Dummy global out-of-balance force vector Q Added force vector £h £v Horizontal and vertical damping ratios of system R Acknowledgements xvii ACKNOWLEDGEMENTS Proverbs 3:1...5,6,7 (KJV) My son, forget not my law; but let thine heart keep my commandments. Trust in the Lord with all thine heart; and lean not on your own understanding. In all your ways acknowledge and he shall direct your him, path. Be not wise in your own eyes; fear the Lord, and depart from evil. I wish to express my sincere thanks and gratitude to my thesis supervisor, Dr. Ricardo Foschi of the Department of Civil Engineering at UBC. His expertise in the development of mathematical models was the backbone to the success of this thesis. I would also like to acknowledge financial support from his Forest Renewal British Columbia (FRBC) research grant, which enabled us work on this topic. His comments and suggestions in the preparation of this thesis are also sincerely appreciated. I finally wish to thank him for all the emotional support he gave me during my stay in Vancouver; especially during my initial days. I wish to thank Dr. Reza Vaziri, Dr. Frank Lam and Dr. Peter Byrne for the respective contributions they gave to me at different points in time. I wish to especially thank Dr. Helmut Prion for reading the thesis and offering very important suggestions. I also wish to express my sincere appreciation to the following graduate students for assisting me in various ways: Marlen Buitelaar, Ernie Wong;. Hong Li; Tomas Horyna; David Moses; and Tuna Onur. The help given me by Dr. Don Anderson in my preparation to come to Vancouver is also deeply appreciated. I would like to thank all the members of the International Student Ministry of the Inter-Varsity Christian Campus (IVCF) for the various forms of support they gave me as I worked on this thesis. In the same regard, I wish to thank all members of the Ghanaian community here in Vancouver, and also all members of Charisma Foundation. Lastly, I wish to express my deepest gratitude to my parents and sisters, for the love they have for me, and the unfailing prayer they offer on my behalf each day. Dedication xvii: To all who await for the coming of the Lord Jesus Chapter 1: Introduction 1 CHAPTER 1: 1.1 1 INTRODUCTION Introduction The continuous improvements in the speed and capacity of the present generation of relatively inexpensive personal computers has, completely revolutionized many professions today. Business, Science, Engineering and even the Arts are currently relying heavily on the computer revolution. In Engineering Mechanics, the advent of these computers has made numerical methods of analysis, which in the early parts of this century were not possible, the way of the future. After World War I, significant contributions to classical continuum mechanics were made by, among others, Timoshenko, Flugge, Von Karman, etc. [52]. Most of their contributions were in the area of closed-form solutions to elastic engineering problems. In the ensuing period, the study of more complex nonlinear engineering problems also became a very active field. For many of the elastic problems, the analysis easily got very mathematically intensive; nevertheless, they were less difficult to solve than the resulting equations obtained for most nonlinear engineering problems. With the arrival of the main frame computer came the possibility of using numerical methods to solve complex equations with a large number of degrees of freedom. The Finite Element Method, as a result, became the most popular form of analysis. The evolution from the expensive main frame computer to the inexpensive and fast new personal computer has, over the last few decades, virtually removed all limitations to the types of engineering problems that can be studied and simulated with numerical modeling. In line with the above, the Structural and Geotechnical Earthquake Engineering fields have benefited tremendously, due to the relative ease of conducting nonlinear static and dynamic analysis, with new sophisticated computer software with friendly user-interfaces. The introduction of the design of structures using capacity design principles has led to the post-yield nonlinear response of structures being an area of great interest to researchers. The ability of a structure to deform under adverse loading conditions, in a ductile manner, without significant strength Chapter 1: Introduction 2 degradation, is currently seen to be not only desirable, but also extremely important, if early collapse is to be prevented. Consequently, the energy dissipation capability of structures is currently viewed to be a subject of utmost importance. In view of this, there is currently a lot of research going on in the development of efficient and robust algorithms, for nonlinear static and dynamic analysis. This study concentrates on developing such a model for timber fasteners and vertical piles. Timber structures are made of wood, which is essentially a brittle material. They gain most of their ductility and energy dissipation capabilities from their mechanically fastened joints. As a result, in the development of analytical tools for timber structures, the modeling of the cyclic and dynamic response of these timber joints is of utmost importance and has received a great deal of attention in recent years. The response of pile foundations to both lateral and vertical loading is a very complex nonlinear problem. For many years, the design of piles to resist lateral and vertical loading relied on empirical methods and local experience. However, over the last couple of decades, numerical methods have enabled some of these approaches to be substantiated, and others to be replaced. These numerical approaches have also enabled the prediction of the complete nonlinear response of the soil-structure system, under static and dynamic loading, for some very important structures. The basic analytical model, upon which the analysis of timber joints and piles is based, is that of a beam on a flexible foundation (in this case, a nonlinear foundation). This study is part of ongoing research into the development of a numerical model for the nonlinear behavior of timber joints and piles. It focuses on the effect of friction and initial confining pressure on the nonlinear response of both systems. 1.2 Objectives and Scope of Study Recent research at the University of British Colombia (UBC) has focussed on the development of a computational code for the finite element computation of the response of timber fasteners and piles Chapter 1: 3 Introduction under cyclic and dynamic loading. This study is a continuation of this previous work and has the following objectives: • to develop a method to include the initial confining pressure around single timber fasteners and piles into an existing model and to study its effect on the cyclic response of laterally loaded single timber fasteners and piles; • to extend the existingfiniteelement model for single timber fasteners and piles to include frictional forces and to study the effect of the frictional forces on the cyclic response of laterally loaded single timber fasteners and piles; and • to develop a corresponding dynamic model of the above for piles, and verify its applicability under horizontal and/or vertical earthquake ground excitation. In this model, the frame of reference is assumed not to change; as such, the coordinates of the system are never updated. Amplification of displacements due to P-delta effects is taken into account in the model. However, there is a limit to the amount of displacement for which the model is well suited. This model is also limited to materials that undergo relatively small strains; as such, this model does not cover hyperelastic materials. The current model assumes an elastic-perfectly-plastic fastener (pile), but strain-hardening of the fastener (pile) could also be included, if desired. 1.3 Outline of Study This thesis contains six chapters. Chap. 1 is an introduction to the thesis and outlines the main objectives of this study. Chap. 2 reviews current background information on friction, and the various analytical methods used for the analysis of beams on foundations. Chap. 3 briefly discusses the already existing analytical models, and introduces modifications needed to include the effects of friction and initial confining pressure. A brief mention of the line search method is also included. Chap. 4 then discusses the cyclic analysis programs for timber fasteners and piles: HYST and PILE, respectively. In this chapter, the effect of friction and initial confining pressure is studied using a number of examples. Chap. 5 then deals with the verification of the dynamic model with an example Chapter 1: Introduction 4 of a pile under horizontal and/or vertical ground acceleration. Summary and conclusions can then be found in Chap. 6. Further areas of research are also identified in this chapter. Chapter 2: 5 Background 2 CHAPTER 2: BACKGROUND 2.1 Review of Analytical Foundations Methods for Beams on Flexible Beams on flexible foundations are structures, or parts of a structure, in which a deformable beam bears on an infinite or semi-infinite deformable medium. Typical practical examples of such structures are: a soil-pile system, a shallow foundation-soil system, cross-ties on railroad tracks, the floating hull of a ship and mechanical fasteners in soft materials like wood. Early studies into problems of this nature were initiated by the classical work of beam-columns on elastic foundations by Hetenyi [37]. Since the pioneering work of Hetenyi, many other different types of analysis have been used to solve these types of problems. This ranges from the closed-form solutions of Hetenyi to current sophisticated nonlinear computer models. The types of analysis used are: • the modulus of foundation reaction method; • the boundary element method; and • the finite element method. 2.1.1 The Modulus of Foundation Reaction Method This method is based directly on the classical work of Hetenyi. It is the oldest and perhaps the simplest and most versatile method of all the above. In this method, the supporting medium is modeled as a series of uncoupled Winkler springs. The simplest of these is an elastic beam resting on an elastic foundation. The solution to this was given by Hetenyi [38]. The governing differential equation is: EI—i--N—^+ kw dx* dx e 2 = 0 (2.1) Chapter 2: 6 Background where w is the lateral displacement, N is the axial load, E is the Young's Modulus, / is the moment of inertia of the beam and k is the elastic modulus of the foundation. The general solution of the above e equation, given by Hetenyi, is: w = (c e ax ] where + C e' )cosfJx ax 2 a = jx +-^2 V 4EI + (c^e * + C ^e' )sin fjx 0 J3 = JA —— 2 V 4£7 001 (2.2) with A = */—. \4EI The different constants above are computed by substituting the appropriate boundary conditions of the given problem. The fundamental assumption of p = k w presupposes that the foundation is elastic and e is essentially discontinuous. This poses limitations on the above and makes its range of applicability small. To make this method more general, generalized nonlinear load-displacement curves are now used to represent the behavior of the supporting medium. With this approach, however, closed-form solutions cease to exist and numerical methods (mainly the Finite Element Method) are thus used to obtain solutions [67]. The nonlinear load-displacement curves are related to the elastic-plastic properties of the supporting medium. This is further generalized for inelastic beams by Foschi [31 ] and Vazinkhoo et al [87]. Embedded beams in the foundation have been proposed in certain cases to bridge the gap between the completely discontinuous spring foundation and the completely continuous supporting medium [37]. This idea, however, has never been popular since it complicates the model. 2.1.2 The Boundary Element Method The Boundary Element Method is an example of a continuum approach, in which the actual stress and strain characteristics of the medium are taken into account. It is a reinterpretation of the Boundary Integral Equations (BIE), in a manner that can be used to solve nonlinear problems [11]. Stresses, displacements and other variables are primary results obtained from a Boundary Element (BE) analysis. Thus, it can be seen as a mixed formulation, in which results for different sets of variables are produced with the same degree of accuracy. The method requires only a discretization of the surface of a given body. In applying the Boundary Element method (BEM), only one element is used Chapter 2: Background 7 to model the medium. This is often taken as a half-space, and a given analytical solution is used to represent the displacements in the medium. In piles, Poulos [61] used Mindlin's solution for soil displacements due to a point load within a homogenous medium, to characterize the displacements in his linear elastic half-space boundary element. In this case, the pile was modeled as a vertical strip with a rigidity of EI. Various researchers such as Hsiung [39] and Lee [48], have also used very similar approaches in modeling the response of both single piles and pile groups, under axial and lateral loading. The elastic half-space used by Poulos has been also extended by him to account for an elastic-perfectly-plastic continuum. Mamoon and Ahmad [49] have also developed a new Green function corresponding to the displacement field due to a dynamic point force, for use with the BEM technique, assuming the surrounding soil as a viscoelastic half-space. In other areas, the BEM was used by Wang and Gao [90] to model the interaction between frictional bolts and rock mass, by formulating the frictional forces on the bolts in terms of the displacements in the rock mass. In this study, the action of the bolts on the rock mass was reduced to a line integral along the bolt length. Tanaka et al [78] used this method to solve the bending problem in a circular plate, as it applies to a bolted cover plate of a pressure vessel. Tanaka and Horie [80] also used the technique with Melan's solution, for the numerical treatment of bolted connections, in which a clamped plate is a T-flange, and there is a flat unstressed surface. Lin-Chien and Lin-Chuen [47] have used this method with a fundamental solution derived by Rizzo and Shippy, to analyze bolted joints of orthotropic composite plates, under uniform loading. 2.1.3 The Finite Element Method The Finite Element Method is currently the most widely used type of analysis for beams on elastic or inelastic foundations. Even in the Modulus of Foundation Reaction approach, when different sources of nonlinearity are being considered, the Finite Element Method (FEM) is often the numerical approach used. A full Finite Element (FE) analysis of this type of problem involves discretizing both Chapter 2: 8 Background the medium and the beam into finite elements. Such FE analyses are also forms of a continuum analysis. Another numerical approach used in such cases is the Finite Difference Method. The FEM is currently used widely in virtually all fields of engineering stress-strain analysis. Due to its extensive coverage in the literature, it would not be prudent here to give an extensive summary of existing work in this area. In a full FE analysis of a beam on a flexible foundation, the beam-foundation system is broken up into elements of finite size. The displacements within the respective elements are linked to the displacements at the nodes by simple polynomial shape functions. Continuum elements are generally used to represent the medium whiles beam elements are used to represent the beam. Sophisticated stress-strain models such as strength degradation models, etc., are often used to represent the deformations in each element, depending on the type of problem being studied. For the soil-pile system, a wide range of formulations exists in the literature. Faruque and Desai [27] performed a true nonlinear 3-D analysis. Desai [22], Baguelin et al [6] and Randolf [65] are examples of researchers who have also used a quasi-3-D analysis to study the problem. In the quasi 3-D analysis, advantage is taken of symmetry in the problem, by expanding the displacement field in terms of a Fourier series. In the study of timber fasteners, major contributions in 2-D FE analysis of timber joints have been at the University of Wisconsin. Published work on the above can be found in Wilkinson [91,92], Rahman et al [63,64] and Rowlands et al [72]. Recently, Moses and Prion [54] at UBC, used an anisotropic plasticity model for Douglas Fir and Structural Composite Lumber, within a 3-D FE model, to predict the load-displacement, maximum strength and mode of failure of a single bolt connection. Arnold et al [4], have also used a FE analysis to investigate the possible modeling techniques of a composite bolted joint made of Carbon Fibre Reinforced Plastic (CFRP). Their results show that the difference between using either a 2-D model or 3-D model could be quite significant due to the complex nonlinear behavior. Most of these FE studies are performed using commercial software like ABAQUS, DYNA-3D and ANSYS. These packages are usually limited to given types of analysis and Chapter 2: Background 9 as such, user-specific modules for performing different tasks must be implemented within the framework of these commercial packages. Most of the above models are highly sophisticated and require considerable care in modeling and substantial CPU time and memory. Due to this, full FE analysis is popular only for elastic applications. 2.2 Friction 2.2.1 Definition of Coulomb's Laws of Friction When two bodies in contact are subjected to applied forces which tend to produce relative sliding motion, frictional stresses develop on the interface that tend to oppose the motion. In the classical Coulomb's laws of dry friction, the resultants of the stresses on the contact surface are a compressive normal force and a frictional shear force. The classic laws of friction are summarized as: • the frictional force at the onset of sliding and during sliding is proportional to the normal contact force, the coefficient of proportionality being known as the coefficient of friction (static and kinetic); • the coefficient of friction is independent of the apparent area of contact; • the static coefficient is greater than the kinetic coefficient; • in limited velocity ranges the kinetic coefficient may be taken as independent of the sliding velocity; and • when tangential motion occurs, the friction force acts in the same direction of the relative velocity, but opposite in sense. 2.2.2 Coulomb's Friction against other Theories of Friction In developing FEM codes for contact problems, it has been argued that Coulomb's laws are only capable of describing friction between bodies that are effectively rigid, and have the gross sliding of a body relative to another. As a result, the basis for the point-wise application of Coulomb's law has been questioned [59]. Considerable effort has thus been devoted to developing new theories of Chapter 2: Background 10 friction. An example of one of these new frictional laws is the Non-local and Nonlinear Friction Law [59]. Stated below are two tenets upon which these new frictional laws are based. • Normal stresses are distributed (smoothed out) over the contact surface in a non-local manner, i.e., due to lack of smoothness of the contact surface, there is mollification of the normal contact pressure on the contact boundary. Due to this, impending motion is proportional to the weighted measure of the normal stresses, in the neighborhood of the point in question. Fig. 2.1, is an illustration of the above. / / Distribution of stress around contact point Undeformed asperities Deformed asperities. Points not directly under load affected Insert A Figure 2.1 Non-Localized Distribution of Stress on Contact Surface • There are elastic and inelastic deformations of the junctions between the two surfaces in contact, and sliding does not occur until this junction fractures. This means that the stiffness of the interface against lateral displacement under static conditions is not infinity as stated by Coulomb's law. Fig. 2.2 shows results from an experiment by Johnson [42] showing this phenomenon. Chapter 2: 11 Background Shear Force Shear force vrs. Displacement diagram for Fig 2.1 as seen from experiments. Displacements (micro-mm) Figure 2.2 Shear Force-Displacement Curve (After Johnson [42]) FEM implementation of contact problems based on the above can be found in Oden et al [58,60] and Wriggers et al [93] The micromechanics of friction between two bodies in contact involves a complex phenomenon, sensitive to numerous factors. Examples of these are: the constitution and morphology of the interface; inertia and thermal effects; roughness of the surfaces; history of sliding; presence of impurities such as lubricants; and the general failure of the interface. Given these complexities, regardless of all the arguments posed against it, Coulomb's laws of friction are still the model widely used in many FEM codes in existence. To cater for the second point stated against Coulomb's laws, junctions are modeled with an elastic stiffness, and are thus capable of deforming before the commencement of sliding. This has been called Elastic Coulomb friction, in contrast to classical Coulomb friction, now called Rigid Coulomb friction. 2.2.3 Plasticity Model of Friction Much similarity is seen to exist between plasticity and friction. The literature thus contains a great deal of analytical models for friction, based on theories of plasticity. Researchers who have contributed significantly to this area are: Giannakopolous [35], Torstenfelt [84,85], Refaat and Meguid [70], etc. Analogous to the yield criterion and yield surface in plasticity are the slip criterion and slip surface in friction. In 3-D, a regular cone is used to describe the slip surface for isotropic Coulomb-type friction. In 1-D however, the simple bilinear elastic-perfectly-plastic model, is suitably used to model friction. Chapter 2: Background 12 Examples of 1-D applications in concrete and timber can be found in Mehlhorn et al [53] and Chui et al [17] respectively. Because friction is a phenomenon that dissipates energy, it is highly path-dependent; as such, incremental forms of FEM solutions are adopted in its implementation. The non-convergence of problems including friction is an issue of major concern, and in contact problems, when too many contact conditions change in a given iteration, there is always a risk of losing control over the solution process. To mitigate against all this, the choice of very efficient integration algorithms of the friction constitutive equations, and the use of very small load/displacement increments is very important. 2.3 Contact Elements in the ANSYS Commercial FE Package The FE commercial package, ANSYS, has elements that are capable of modeling contact between two bodies. There are a total of ten such elements, of which, the seven relevant ones are: COMBIN39, COMBIN40, CONTAC12, CONTAC26, CONTAC48, CONTAC49, and CONTAC52. Each of these is briefly discussed below. • COMBIN39 Nonlinear Spring This is a unidirectional element with a generalized force-deflection capability, having one degree of freedom at each node. Any unidirectional force-deflection curve could be specified for the element. The element allows unloading along the same curve or along a line parallel to the initial stiffness. It does not account for any form of gapping. Fig. 2.3 is an example of such a force-deflection curve. Chapter 2: Background 13 Force Up to 20 discrete points of force vrs. deflection could be used to define the curve f / Deflection 2 WvVvV 2 Figure 2.3 General Force-Displacement Curve for COMBIN39 Element COMBIN40 Combination Element This is a unidirectional element with a combination of a spring-slider and damper in parallel, coupled to a gap in series. It has only one degree of freedom at each node. The element has a bilinear force-deflection relationship and could work both in tension and compression. A schematic diagram of the element is shown in Fig. 2.4. 2.1.1 - V W V V W 2*~ gap WXAAAAAAr K 2 damper Figure 2.4 Schematic Diagram of COMBIN40 Element • CONTAC12 2-D Point -to-Point Contact Element This is an element that models contact between two surfaces, which may maintain or break contact (develop a gap), and may slide relative to each other. The element is capable of supporting only compression in the direction normal to the surfaces and shear (Coulomb friction) in the tangential direction. The element has two translational degrees Chapter 2: Background 14 of freedom per node. Upon contact, the normal force responds only as a linear spring. A sketch of this element is shown in Fig. 2.5. < 2.1.. Both surfaces currently in contact Normal force ' t *~ ^ ^ ^ ^ — i T a n p e n r i a l force. Tangential slip < *2.L. Figure 2.5 Schematic Diagram of CONTAC12 Element • CONTA C26 2-D Point-to-Ground Contact Element This element is similar to CONTAC12 but exists between a node on a body and the ground. Upon contact, the normal force also responds as a linear spring. • CONTAC48 2-D Point-to-Surface Contact Element This is a 3-node element that is used to represent contact and sliding between two surfaces in two dimensions. The element has two translational degrees of freedom at each node. Contact is between a node on one surface and a target line on another. Penetration of the target line is not allowed, and this is taken into account by using the Penalty function method only, or a combination of the Penalty method and the Lagrange multiplier method [7,40]. This is generally for contact problems, in which the target is rigid, and as such, does not deform, e.g. base plate of a structure. A Coulomb friction slip curve is used to model frictional contact. Fig. 2.6 is an illustration of this element. The figure shows the 3 nodes of the element, with the target line between nodes / and J. Chapter 2: Background 15 Figure 2.6 Schematic Diagram of CONTAC48 Element • CONTAC49 3-D Point-to-Surface Contact Element This is the 3-D counterpart of the above. A coulomb friction slip surface is used in this case. • CONTAC52 3-D Point -to-Point Contact Element This is the 3-D counterpart of CONTAC12. With the capabilities of all the above elements, it would be difficult to use ANSYS to efficiently model the problem being studied. To study the problem with ANSYS, taking into account the nonlinear foundation embedment, tangential frictional forces and the formation of gaps, requires either a complex combination of the above elements, or calls for the development of a new user element. Since the elements have been developed separately and problems controlled by friction usually face convergence problems, care must be taken in choosing any combination of contact elements. Noting that the modulus of foundation reaction approach is already a simplification of a full FE analysis, a combination of contact elements would in effect tend to complicate the relatively simple model. Chapter 2: Background 16 2.4 Mechanically Fastened Timber Joints 2.4.1 Types of Timber Fasteners The Canadian Wood Design Manual [13] lists nine different types of wood fasteners. These are nails and spikes, bolts, drift pins, lag screws, glulam rivets, shear plates and split rings, truss plates, joist hangers and framing anchors. Some of these fasteners can be loaded laterally and/or loaded in withdrawal. Most timber joints usually comprise of multiple fasteners, acting together as a unit. To correctly model these analytically, there is a need to understand clearly the behavior of a single timber fastener. 2.4.2 Monotonic Response In the study of these joints, their behavior under monotonic, cyclic and earthquake excitation is very important. Early attempts to analytically predict the load-displacement response of single fastener timber joints assumed the beam to act like a beam on an elastic foundation [37], This linear approximation predicted well the initial stiffness, but not the maximum load capacity, ductility or residual deformation, because the true fastener behavior is nonlinear. Foschi [33] and Foschi and Bonac [32] were the first to use 1-D FE analysis to study the loaddisplacement characteristics of laterally loaded nails in single shear. The model included the nonlinear bearing behavior of wood under the nail shank, and the yielding of the nail in bending. From the above research, it became evident that 1-D FE analysis of mechanical timber joints gives a satisfactory prediction of the load-displacement behavior of the joints. Smith [75], at the Timber Research and Development Association (TRADA) in England, developed FE programs based on the Foschi's work, to analyze two and three-piece non-symmetric joints. He developed small displacement theory and large displacement theory models. The programs took into account frictional forces along the entire length of the fastener and restraining at its end. The frictional resistance was calculated as the product of the foundation force and the coefficient of static friction between the fastener and wood. The Chapter 2: Background 17 foundation force was computed as the sum of the embedment force and the foundation residual stress per unit length. On withdrawal, the frictional resistance was computed as the product of the foundation force and the coefficient of sliding friction for wood to steel. Erki [24,25] further developed the large displacement model of Smith, by introducing different embedment curves for driven nails, and taking in account the rotation of the fastener head. She noted that in the modeling of fasteners with a head, modeling the head as either fully fixed or freely rotating, created differences in the results. Her results showed that the fixed head condition enabled a better prediction of the initial stiffness and the ultimate capacity. 2.4.3 Cyclic Response Prior to Foschi's [62] analytical model, which was an extension of Foschi's monotonic model, most studies done on the performance of timber joints to both one-way cyclic and reversed cyclic loading, were experimental. The experimental work focussed on the influence of various joint parameters on performance characteristics, such as ductility, strength, shape of hysteresis loops and strength degradation. Examples of these include: Soltis and Mtenga [76], Dolan and Madsen [23], and Ni and Chui [57] for single-fastener nailed joints; Gutshell et al [36], and Daneff et al [21] for bolted joints; and Ceccoti and Vignoli [14] for moment-resisting joints. In general, it is now known that only joints with small diameter fasteners are capable of failing under reversed cyclic load, in a ductile manner. As an outcome of these experiments, empirical hysteresis models, composed of a set of loading and unloading rules, have been developed and introduced in some of the commercial structural engineering software. One of the latest developments was by Baber et al [5], with later modifications by Foliente [29]. In [62] Foschi took into account the development of gaps in reversed cyclic loading. The results of their work showed a good correlation between the analytical and experimental results. The characteristic pinching effect of timber connections, because of the degradation of the connection due to the formation of gaps, is seen to be evident in this study. Chui et al [19], improved on Foschi's model by arguing on the basis of Smith and Erki's monotonic results, that it was important to consider Chapter 2: Background 18 large displacement effects, shear deformation in the fastener, fastener head rotation, and friction between the fastener and wood, if a reverse cyclic analytical model was to be complete. However, in contrast to Foschi's model, in which the cyclic load-embedment response was automatically computed by the program, Chui et al developed an empirical set of embedment rules for loading and unloading. This empirical set of rules embodies the formation of the gaps. 2.4.4 Modeling of Embedment Pressure The load-embedment properties of a joint have been seen to vary with wood properties and fastener diameter. These properties are obtained from experiments. Most embedment tests involve pushing the length of a connector directly against a wood foundation. In [33], a wood-bearing test, called the standard bearing test was used to define the load-embedment properties of a joint. For driven fasteners, a modified response called the modified bearing curve of the wood was proposed in [32]. This was to account for the reaction of the wood displaced by the driven shank. Foschi and Bonac observed that for common wire nails, the modified curve gave a better prediction of the initial stiffness and ultimate behavior, while for the intermediate behavior, the standard curve gave a better prediction. Smith also used a similar embedment curve as the above. For driven fasteners, he modeled the initial confining pressure (due to the volume changes in the medium), as foundation residual stresses. Erki also proposed a new fourth root equation for predicting the response of driven fasteners. In [62], an embedment curve similar to the monotonic model was used, however, beyond a displacement w = Dmax, the nonlinear spring was assumed to soften along a straight line. In a recent paper by Foschi [30], the straight line portion beyond D max was replaced with an exponentially decaying function. 2.5 Piles 2.5.1 Types of Piles Piles are structural members of timber, concrete and/or steel, used to transmit foundation loads to lower levels in the soil mass. Piles are placed either by inserting them in bored holes or by driving. Chapter 2: Background 19 For inserted piles, the piles are inserted in pre-drilled holes, or occasionally, in the case of concrete, the cavity is filled with concrete which, upon hardening, produces a pile. Driven piles are driven, using either a pile hammer, a vibratory device or by jacking [10]. Piles are designed to resist both lateral and axial loading. They resist load either by a vertical distribution of the load along the pile shaft, or by a direct application of load to a lower stratum, through the pile point. A vertical distribution of the load is made by using the friction or "floating" pile, and/or directly by direct end-bearing. In determining the load capacity of a pile, the maximum skin friction and maximum end bearing resistance are not directly additive, since both are not mobilized simultaneously. The American Petroleum Institute (API) design code [3] suggests depending only on soil adhesion for shaft resistance in cohesive soils. For cohesionless soils it suggests the use of only the lateral earth pressure for the shaft resistance. 2.5.2 Monotonic Response Early work on laterally loaded piles assumed a linear foundation response as developed by Hetenyi. Terzaghi [82] went further to introduce the subgrade reaction in terms of the soil pressure. The foundation response (called P-y curves) in this method varies linearly with depth. This approach gives similar results to Poulos' elastic boundary element approach with the only difference being the discontinuous soil medium in Terzaghi's model. For cohesionless soils, Reese et al [69], with later modification by Murchinson and O'Neill [55], proposed an empirical hyperbolic tangent function for P-y curves. This is currently the adopted method in the API design code. Based on FE analysis which were confirmed experimentally, Yan and Byrne [94] also proposed a new set of P-y curves which consist of an initial elastic segment joined to a parabolic segment. They realized that for depths up to one diameter, a different type of P-y response exists. For cohesive soils, the API design code adopts the nonlinear P-y curves proposed by Matlock [51] and Reese and Cox [68], for soft and stiff clays respectively. In the literature, the relationship between mobilized soil-pile shear transfer and local pile displacement at any depth is described using T-z curves. Similarly, the relationship between mobilized end bearing resistance and axial tip deflection is described using Q-z curves. T-z curves for clays and sands are Chapter 2: Background 20 provided by Vijayvergiya [89]. Similar curves are also proposed by Coyle and Suliaman [18] for sands. API also adopts similar curves to those proposed by the above researchers. These are shown in Fig. 2.7. The curves for sands are bilinear while those for clays are almost linear until a peak value of shear, after which, they plateau at a value slightly lower than the peak. The influence of friction can be clearly seen in the respective responses. For Q-z curves, relatively large pile movements are required to mobilize the full end bearing resistance. Fig. 2.8 shows API's adopted curve, which has an initial curved segment that plateaus at the ultimate capacity. Clay: z/D Range of t„ for clays s tre =0.7U s Slip Sand: z (inches) t/t 0.00 0.00 0.00 0.30 0.10 1.00 0.50 oo 1.00 0.75 z: Slip 0.90 D: Diam. ofpile 1.00 t: Shear force 0.70-0.90tma*: Max. shear force 0.70-0.90 t/tmax 0.00 0.0016 0.0031 0.0057 0.0080 0.0100 0.0200 CO max Figure 2.7 Recommended API P-y Curves for Sands and Clays z/D 0.02 0.013 0.042 0.073 0.100 End bearing Load (Q) z„ = 0.10D QtQ, 0.25 0.50 0.75 0.90 1.00 End displacement (z) Figure 2.8 Recommended API Q-z Curves Q: End load Q : Total end capacity D: Diameter z: Tip deflection p Chapter 2: Background 21 2.5.3 Cyclic Response The cyclic response of piles is a very complex phenomenon involving the interplay of many variables. There is both material and mechanical degradation of the system. Material degradation as stated here, refers to increase in pore pressures, changes in soil density and attendant changes in principal stress directions. Likewise, mechanical degradation refers to the development of gaps and the failing of the soil interface, especially in cohesionless soils. The cyclic response of piles embedded in clays was studied experimentally by Reese et al [66] and by Swane and Poulos [77]. They realized that after a given number of cycles of shaking, the degradation stabilized resulting in a state of permanent strains and residual stresses. Based on Yan's [95] experimental work on laterally loaded piles in cohesionless soils and Foschi's timber fastener model, Vazinkhoo [87,88] developed empirical relationships for deriving cyclic P-y curves for cohesionless soils. For cases where these cyclic P-y curves are not available, he proposed the use of Yan's monotonic P-y curves, coupled with the neglect of any tensile capacity of the soil. This approach is also used by Foschi [30,31]. 2.6 Concluding Remarks Based upon the above review, the following points are observed. • The use of nonlinear foundation reaction curves in a 1-D FE analysis of both timber fasteners and piles is an area that has gained popularity. Results from this method are seen to be in reasonable agreement with experimental results. They require less computer resources than a full two or three-dimensional FEM or BEM model. • The 1-D FE study of the problem of a beam on a nonlinear foundation could be modeled in a commercial FE package like ANSYS, however, either a combination of contact elements is necessary or a new user-element must be developed. • In the loading of piles and timber fasteners in reversed cyclic loading, the foundation undergoes degradation as a result of permanent deformation of the foundation. In accurately modeling the cyclic behavior, the gaps formed between the beam and the foundation due to this phenomenon must be taken into account. Chapter 2: • Background 22 Other theories of friction, apart from Coulomb's laws have been proposed when modeling friction on a point-wise basis. However, since friction depends on several factors, most researchers modeling friction on a point-wise basis still use Coulomb's laws. Chapter 3: Analytical 23 Models 3. CHAPTER 3: 3.1 Introduction ANALYTICAL MODELS The model developed by Foschi [31] forms the basis for modeling the response of the soil-pile and timber fastener systems. This model has been widely used by a number of researchers including, Khan [43], Frenette [34], Vazinkhoo [87], Prion and Foschi [62], etc. Here, the original model is extended to include the effects of friction and initial confining pressure. In this study, only a generic soil-pile or timber fastener system is being studied. Fig. 3.1 shows a diagram of such a basic generic system. For some specific fasteners and piles, certain modifications to this model would be necessary, e.g. for H-piles, lag screws, etc. As illustrated in Fig. 3.1, both in the soil-pile and timber fastener systems, the beam can be loaded laterally and axially. Different types of fasteners and piles respond to these loads in different ways. As an example, bolts respond mainly by bending and extending, while nails respond by bending, extending and withdrawing. Axial Load (N) Lateral Load (F) Foundation reaction! forces on sides oj beam (P(w)) Beam segments Frictional forces on sides of beam (T/u,w)) x(u) A Different layers of foundation Figure 3.1 Basic Generic Beam-Foundation System Chapter 3: Analytical 3.2 Models 24 Foundation Pressure As shown in Fig.3.1, a pressure is exerted by the medium on the beam, as it bears on the medium. At a point along the beam, the force-displacement response of the medium is modeled with an embedment curve (since our model is a 1-D model, unless otherwise stated, the foundation pressure refers to the force exerted by the medium, per unit length of the beam). These curves are specific to different systems and represent both the elastic and plastic deformations that occur in the medium. The embedment action is only compressive in nature, and this results in the subsequent formation of gaps between the beam and the medium. Certain beams are placed into a medium by driving. Examples of some of these are: driven fasteners in wood and driven piles in soil. In such situations, the fastener damages a portion of the medium and creates a hole equal in size to the external diameter of the beam. In addition to this, the driving process causes a portion of the medium to be displaced and densified with the consequence that, the beam experiences an initial confining pressure. The different factors that affect the initial confining pressure around fasteners in wood and piles in a soil mass are discussed below • Timber Fasteners Wood is a naturally occurring material that has its properties varying over wide ranges. Most of its engineering properties are obtained by testing and the resulting data analyzed using statistical and probabilistic methods. The initial confining pressure experienced by driven fasteners in wood is a property that exhibits this phenomenon. The following are some of the factors that influence it. • Viscoelastic Property of Timber Wood is a viscoelastic material and consequently, the elastic properties change with time. Due to this, the initial confining pressure around fasteners after driving also changes with Chapter 3: Analytical Models 25 time. Fig. 3.2 shows that after attaining a peak value, the pressure decreases asymptotically to a residual value with time. Pressure ' / N. Maximum Pressure / i Residual Pressure > Time Figure 3.2 Variation of Initial Confining Pressure with Time due to Viscoelastic Properties of Wood • Humidity and Moisture Content Humidity and the moisture content also contribute significantly to the confining pressure around the fastener at a given time. An increase in humidity causes a corresponding increase in the moisture content of wood, which then results in swelling. As a consequence of this, there is an increase in the confining pressure. The converse of the above statement is also true. Similarly, the confining pressure experienced by a fastener in wood that is green, is more than that experienced by the same fastener in wood that has been dried. In addition to this, the type of drying process and the corresponding rate of drying also affect the confining pressure. • Type of Fastener and Method of Driving The shape of a fastener influences to a great extent the amount of the wood that is displaced when the fastener is driven. The more volume of wood displaced, the larger is the confining pressure. The shape of the fastener, especially how the tip is formed, affects the way wood deforms as the fastener is driven. Damage is experienced in the form of splitting of the wood medium along planes of weakness. For close-fitting fasteners such as dowels, not much splitting of the wood occurs since smaller pre-drilled holes are used. In such cases, the size of Chapter 3: Analytical Models 26 the pre-drilled holes determines the volume of wood displaced and consequently, the confining pressure. • Piles Soil is a naturally occurring medium that is also very variable. Several factors influence the magnitude of the initial confining pressure experienced by piles in a soil mass. Some of these factors are also discussed below. • Type of Pile and Method of Driving The method used in driving piles, (dropping of hammer, vibrating or jacking) has a considerable effect on the resulting pressure distribution around the pile. Depending on the method used, cohesionless soils become denser, resulting in an increased initial confining pressure. For open-ended piles, whether they are driven unplugged or plugged has a considerable effect. Plugged piles push the soil under it to the sides and this results in more volume of soil being displaced, which in turn leads to larger confining pressures. Piles inserted in drilled undersized holes displace less volume of soil, and this leads to confining pressures that are relatively less than those driven without pre-drilled holes. • Consolidation and Creep of Soil When piles are driven into a soil mass they produce remolding of the soil in the immediate vicinity of the pile and at that instant, undrained soil-strength parameters are achieved. With time, however, these values approach remolded drained values. Due to the resulting high pore pressures in soft cohesive soils, consolidation of the soil immediately around the pile occurs, and this results in an increase in the confining pressure. In overconsolidated clays, soil creep also results in a dissipation of the lateral confining pressure with time [10]. 3.2.1 Modeling Foundation Pressure For both the soil-pile and timber fastener systems, researchers have proposed different empirical relationships to model the force-displacement response at a point along the beam. Because of the initial confining pressure surrounding driven fasteners/piles, most of these researchers have proposed Chapter 3: Analytical 27 Models different equations for driven and inserted fasteners/piles. As an example of this, Erki used different experiments to measure the force-displacement response of driven and inserted fasteners. After fitting curves to the experimental data, she proposed the use of a fourth-root equation to model driven fasteners, and the use of Foschi's standard embedment curve to model inserted fasteners. Because the main difference between driven fasteners/piles and inserted fasteners/piles is the initial confining pressure, it is desirable to have a general modeling technique that can be used for both cases. In line with this objective, it is proposed here, to model both the soil-pile and timber fastener systems with a single embedment curve, but with a superimposed initial confining pressure. With this approach, the initial confining pressure has the effect of shifting the specified embedment curve in a given direction. The figure below shows a sketch of the different ways in which the embedment curve could be shifted. Displacement corresponding to the initial pressure Displacement (w) Figure 3.3 Different Methods of Shifting Embedment Curve to account for Initial Confining Pressure The figure shows that the backbone curve could be shifted vertically or horizontally. Both of these approaches are discussed below. • Horizontal Shift Chapter 3: Analytical Models 28 For this case, the initial pressure is assumed to be due to an initial displacement that is imposed by the beam on the medium. Inherent in this approach is the assumption that the medium undergoes only an initial displacement, and as such has its initial state given by a point along the embedment curve. In other words, the model assumes that the medium has already been pre-compressed. From the figure, this approach is seen to lead to no increase in the medium's embedment capacity and also an early initiation of the softening of the medium. • Vertical Shift This approach models the problem by assuming that the beam exerts an initial pressure on the medium, which is independent of the specified force-displacement curve. From the figure, this approach leads to an increase in the medium's embedment capacity and no change in the displacement at which softening of the medium begins. Since the initial confining pressure experienced by the beam is due to both densificatication and displacement of the medium, the correct curve that models the embedment response is a curve bounded by the curves resulting from a shift of the backbone curve in the horizontal and vertical directions. In this study, it is desired to study the influence of the initial confining pressure on the cyclic response of timber fasteners and piles that are loaded in the horizontal direction. Since the approach of a vertical shift of the embedment curve produces a response that is an upper bound to the correct response, this is the approach used in this work. 3.2.1.1 Components of Foundation Pressure Consistent foundation forces are computed at every integration point (Gauss point) along the xdirection of Fig. 3.1. The pressure exerted by the foundation is the sum of the initial confining pressure as a result of driving the fastener and the reaction from the foundation due to the embedment of the beam into the foundation. Since the problem is being modeled in one dimension, the total forcedisplacement response at a given point along the beam can be represented by the sum of the forcedisplacement responses on the right and on the left sides of the beam. Each of these components is Chapter 3: Analytical 29 Models assumed to be an independent response of a beam bearing on a foundation. Fig. 3.4 shows an illustration of the stress distribution and the component resultant forces at a point along the beam on each side, based on the approach of vertically shifting the embedment curve to account for the initial confining pressure. 1 1 ' i Initial Stress distribution and resultant forces + 1 ^ ' ' P ( W r ) Stress distribution and resultant forces after movement of beam to the right. Stress distribution and resultant forces after movement of beam to the left. Figure 3.4 Stress Distribution and Corresponding Resultant Forces around Beam Element 3.2.1.2 Modeled Response of Medium From Fig 3.4, it can be seen that as the beam displaces to one side, the component foundation pressure on that part increases (loading occurs), while the other side decreases. Since the medium has elastic-plastic properties, when loading occurs, the medium experiences both elastic and plastic deformations. The nonlinear embedment curve represents these two types of deformations that take place in the medium. The ability of the medium to unload by rebounding is a result of its elastic properties. The point to which the medium rebounds is the edge of its unstressed (relaxed) state. Anytime the medium is loaded further, a new edge of the relaxed state develops. Plastic deformation results in permanent deformation of the medium, which in turn, results in the formation of gaps between a given side of the beam and the medium. Loading of each side proceeds along the embedment curve and the unloading of each side also occurs along a line parallel to the initial stiffness. Loading of the medium to the previously experienced pressure also occurs along a line parallel to the initial stiffness. Inherent in this approach, is the Chapter 3: Analytical 30 Models assumption that during the entire response, the elastic properties of the medium at a point do not change. 3.2.1.3 Different Types of Foundation Responses Based on the method used in modeling the initial confining pressure, three different types of foundation responses could occur. These different cases are: i. when no initial pressure exists at the beginning of the response and as such the beam bears on at most one side at a given time; ii. when an initial pressure exists at the beginning of the response, but a gap never develops. In this case the beam bears on both sides of the medium throughout the response; and iii. when an initial pressure exists at the beginning of the response and the beam bears on both sides and gaps later develop that results in the beam bearing on at most one side at a given time. Thefiguresbelow illustrate the different types of responses. Fig. 3.5 shows a schematic plan view of a typical response. The circular elements in the figure represent the beam's cross-section. Figs. 3.6a,b,c also illustrate examples of the force-displacement response of the aforementioned types of responses. The notation used in the figures is defined below. • Variables P,: Left side foundation pressure wf. Displacement at left side Df Left edge of relaxed medium P : Right side foundation pressure r w : Displacement at right side r D : Right edge of relaxed medium r Xj: This format is used to represent variables in different states: X. Variable in question s: Left or right side V. A number representing the state of the variable Chapter 3: Analytical D," Models D,' D° r D' D,° r Beam moves to the right wj>0 w,°=0 w,°=0 W=0 Beam moves back to the left, but not up to wi=0. w,°=0 w/ <0 Relaxed edges of medium inside the circle. Consequently, bearing of medium occurs on both sides, and an initial confining pressure exists. New relaxed edges £>/ and D, still lie in the circle, so bearing of medium still occurs on both sides. No gap exists on either side. Relaxed edges of medium do not change, since no loading on embedment curve occurs. They are still inside the circle, and as such, bearing of medium still occurs on both sides. Right side not in contact with edge and gap forms. Hence, the beam bears on only the left side. The relaxed edge on the right also moves to the positive side of w/=fl. Moving to initial position does not require loading on the embedment curve. Due to this, the relaxed edges do not change. It can be seen here that, because D? passed the wi=0 point, a permanent hole is created and the beam bears on only the right side. The initial position demarcates the size of the beam, and shows the hole created in the medium due to the presence of the beam. From this, it can be seen that if the relaxed edge becomes a positive value, the initial size of the hole increases. The increase in the size of the hole is given by: IF (D, <0) H, = 0 IF(D, > 0) Hi = A IF (D, <0) H = 0 IF(D > 0) H, =D, Total increase in hole H = Hi + H, r wf>0 r Figure 3.5 Schematic Plan View of a Typical Response Chapter 3: Analytical 32 Models Case (i) The right side unloads along a line parallel to the initial stiffness, and loading also occurs on the left side. New left relaxed edge forms, due to loading occurring on embedment curve. The beam bears on only the left side. The left side unloads along a line parallel to the initial stiffness. The right also loads long a line parallel to the initial stiffness, until it gets to the previously experienced pressure, where it then loads along the embedment curve. The beam bears on only the right side. Figure 3.6a Force-Displacement Response for a Case (i) Type of Foundation Response Chapter 3: Analytical 33 Models Case (ii) Initial pressure existing at beginning. Relaxed edges of the medium are each on the negative side (their position falls in the beam). The beam bears on both sides of the medium. Confining pressure still exists on both sides. Since loading on the right occurs on the embedment curve, a new relaxed edge forms on the right, Unloading on the left also occurs, but the left does not fully unload. The left loads along the line parallel to the initial stiffness, and later along the embedment curve. The right also unloads partially along a similar line. Bearing is still occurring on both sides. Both the left and the right unload and load respectively along a line parallel to the initial stiffness. Bearing still occurs on both sides. Figure 3.6b Force-Displacement Response of a Case (ii) Type of Foundation Response Case (iii) Chapter 3: Analytical 34 Models Pr • A Pr°^ / Df \ / ^ / W,(+) ^ P , ° \ . Further push to the right Small move to the right D° p. P, Confining pressure still exists on both sides. Since loading on the right occurs on the embedment curve, a new relaxed edge forms on the right. Unloading on the left also occurs, but the left does not fully unload. Initial pressure existing at beginning. Relaxed edges of the medium are each on the negative side (their position falls in the beam). Beam bears on both sides of the medium. \ '/ / > D w,r-> Wl(-) / '< Wi(+) Pr Small move back to the left ' ^ Pi Wr(-) Further move to the left P.' P, Further displacement to the right results in a gap developing on the left; as such, the left fully unloads. The new relaxed edge is now on the Dositive side. The left side loads up the line parallel to the initial stiffness, and the right side also unloads down a similar line. Bearing occurs on both sides at this moment Pr Pr ( Pr 4 \ >«</(+; •=> J ^ - / A / Move back to the right Pi The relaxed edge is now on the positive side after loading on the left embedment curve. The beam bears on only the left side. The response at this point is very similar to the case where no initial pressure exists. \ Di \ \ / ^ Pi The right side loads along the line parallel to the initial stiffness and then loads up on the embedment curve. The left side also unloads fully, and a gap forms. Bearing is thus, only on the right. Figure 3.6c Force-Displacement Response for a Case (iii) Type of Foundation Response Chapter 3: Analytical Models 35 Case (i) In this case, the resultant force-displacement response of the medium is exactly the same as the specified force-displacement curve. Case (ii) In this type of response, the resultant force-displacement response is the summation of the pressures on both the left and right sides of the beam. Case (iii) The initial response in this case is similar to (ii), however, after awhile, gaps begin to develop. The response then takes two forms. • When a gap develops, the beam bears on only one side of the medium. Since the positions of the relaxed edges are on the negative sides of the initial position, when it comes back to its initial position, it bears again on both sides. This type of response is a combination of (i) and (ii). • When the beam loads on the embedment curve, the new relaxed edges formed are situated on the positive side of the initial position; as such, they lie outside the beam's initial position. When this occurs, the response becomes similar to (i) and the beam bears on at most one side at a given time. Using this model, problems with or without initial confining pressure can be solved using the same specified force-displacement curve. For problems with an initial pressure in which relatively large displacements are experienced, the response follows the second form of case (iii). Conversely, for relatively small lateral displacements the response follows case (ii). For this case, further increase in the lateral displacement results in the response following the first form of case (iii). 3.2.2 Embedment Pressure Relationships The relationships used to model the embedment response of a cohesionless soil-pile system and a timber fastener system are described below. Chapter 3: Analytical 3.2.2.1 Embedment 36 Models Curve for Timber Fastener System An embedment curve was proposed by Foschi [30], to model the force-displacement response of a beam bearing on wood. This relationship is a 6-parameter curve (K, Q , Qi, Q,2,Qi, D ), and is shown 0 max in Fig. 3.7. The force increases with an exponential function until a displacement D max% after which it decreases exponentially with further increase in displacement. Wood Foundation Figure 3.7 Wood Embedment Response (After Foschi [30]) K is the initial slope of the curve, Q and Q, are, respectively, the pressure axis intercept, and the slope 0 of the asymptote to the curve, and Q and Q control the decay after D . 2 3 max load is a fraction (Q2P x) at a displacement (QiD ). ma These are defined so that the These six independent parameters are calibrated max with a monotonically increasing displacement test. The equation of the curve is given by, - / ^ J Kw p ={— a P 1 where P max - ] V e -Ot^-A™) J mm </w<:£>max m a x z/w>£> 2 max'' =(Q +Q D ) 1.0 + e 0 '[ (3.1) max and Q =HQ )/[D (Q,-l.O)] . 2 4 2 imx Unloading of the medium from a point on the curve occurs along a line with a slope equal to the initial stiffness K. This is shown in the figure by the line AB. Chapter 3: Analytical 3.2.1.2 Embedment 37 Models Curve for Cohesionless Soil-Pile System Yan and Byrne's P-y curve for piles in cohesionless soils is the model used here for foundation soil pressure. This curve has an initial linear segment, which precedes a parabolic segment. The only soil properties in this formulation are the maximum Young's modulus (E ), which changes with depth, max and a , which is a parameter depending on the soil's relative density, (D ). Other variables in the s rs equation are d, the external diameter of the pile and fJ , a parameter commonly taken as 0.5 for s cohesionless soils. The equation of the curve with f3 taken as 0.5 is given by s 2 if w < a s where a = 0.5(Z)„.) s 0 8 d (3.2) . An illustration of this curve can be seen in Fig. 3.8. The soil unloads from a point on the embedment curve along a line parallel to the initial stiffness. This is represented by line AB Soil Foundation Force per unit length (PJ 4 w-ord Displacement AW Figure 3.8 P-y Curve for Cohesionless Soils (After Yan and Byrne [94]) Chapter 3: Analytical 38 Models 3.2.3 Determination of Initial Confining Pressure 3.2.3.1 Estimation of Initial Confining Pressure around Timber Fastener As previously stated, the initial pressure depends on several factors; and only an estimate of it would be feasible. In this model, the confining pressure is assumed to be constant along the length of the fastener. . To estimate the initial pressure for fasteners with smooth surfaces, a method based on withdrawal data may be used. The assumption used in this approach is that the beam is axially inextensible during the withdrawal response. For fasteners with threaded surfaces, a different approach would have to be used. AN L Figure 3.9 Forces acting on Fastener In Fig. 3.9, N is the withdrawal load, P is the confining pressure, d is the external diameter of the fastener and // is the coefficient of friction between wood and fastener material. Considering equilibrium of vertical forces, it can be derived from the above figure that N = TLTUI. Based on this, the initial pressure given by this method is: (force per unit length) (3.3) Chapter 3: Analytical Models 3.2.3.2 Estimation of Initial Confining 39 Pressure around Pile The initial confining pressure is taken as the static lateral earth pressure. From Rankine's earth pressure formula, the initial confining pressure is given as: P =k-,y'x i (force per unit length) (3.4) with kj being the coefficient of lateral earth pressure, y' being the effective unit weight of the soil medium and d being the external diameter of the pile. Thus, the initial confining pressure around the pile varies linearly with the depth (x). k, also changes depending on how the pile is driven, type of soil and other variables. 3.3 Modeling of Frictional Forces Coulomb's laws of friction are used here to model the frictional force along the length the beam. As in the case of foundation forces, the consistent tangential frictional forces are computed at every integration point along the length of the beam. These forces are computed for both the left and right sides of the beam. This method applies to the cohesionless soil-pile system and timber fastener system. In piles, the method is similar to Burland's P-method [12] for cohesionless soils, and Tomlinson's amethod [83] for cohesive soils, used for computing the skin resistance around piles. Modified versions of these two methods are adopted in the API design code. The difference between the two is that there is no adhesion in cohesionless soils, and as such, the frictional resistance depends only on the lateral pressure. The relationship between the tangential frictional force and the associated slip at a point on the beam, based on elastic Coulomb friction laws, is shown in Fig. 3.10 below. Chapter 3: Analytical 40 Models Frictional Force (Tj) -MP Slip (A) -MP Figure 3.10 Relationship between Tangential Frictional Force and Tangential Slip In the above figure, Kf is the effective elastic stiffness of the interface between the beam and the medium. It is linked to the roughness of the surfaces in contact and measures the resistance of the interface to tangential deformation before sliding begins. // is the coefficient of static friction and P is the foundation pressure at the given point. The equation defining the above curve is given by, T K (A-A ) ftt+ f 7>H if 0 K, +A <A < K r P if A > ^ + A -pf if A<^J^ +A M (3.5) D 0 where Tj- and A are the frictional force and slip at a prior state. 0 3.4 Formulation of Static Loading Governing Equations and FE Implementation 3.4.1 Statement of Problem It is desired to determine the governing equilibrium equations of the generic system shown in Fig. 3.1 for static loading, and to implement a finite element approximation to solve the resulting equations. In this model, the motion of the system is assumed to be small; accordingly, the frame of reference is never updated, i.e. the coordinates of the beam are not updated. Based on this, there is a limit to the Chapter 3: Analytical 41 Models amount of displacement for which this model is well suited. To enable large displacements to be taken into account, either an updated or total Lagrangian approach must be adopted in modeling the problem. In such a case, Lagrangian strains and 2 Piola Kirchoff stresses would be the appropriate stress and nd strain measures [17]. This formulation is directly based on that given by Foschi [30,31], and is modified slightly to account for the effects of friction and initial confining pressure. 3.4.2 Derivation of Governing Equilibrium Equations 3.4.2.1 Kinematics Because displacements are small, the beam is assumed to follow the Euler-Bernoulli hypothesis of cross-sections perpendicular to the centroidal locus before bending, remaining plane and perpendicular to the deformed locus after bending. Based on this assumption, the beam undergoes no transverse strains, and the only strain of importance is along the longitudinal axis. Fig. 3.11 shows a free-body diagram of an elemental length of the beam. < 1 d/2 > \ J PrM > * f T (A ) fr r Figure 3.11 Free-Body Diagram of an Elemental Length of the Beam Chapter 3: Analytical Models 42 From the figure, w = w, w = -w, A = u-w'd/2 and A = u + w'd/2. r t r t Von Karman's strain-displacement relationship given by d 2... w 1i du £• = V & —+ ' ax 2 /'a.A ~ 2 £ ox 2 I ( 3 - 6 ) is used to define the strain at each point in the beam. This equation accounts for moderate strains and leads to P-delta (P-5) effects being taken into account. Here, u and w are axial displacement and lateral deflection of the cross-sectional centroid of the fastener/pile. 3.4.2.2 Constitutive Equations The stress-strain behavior of the beam is defined by an elastic-perfectly-plastic model. Based on the kinematic model, the stress state is defined by (G,E), where a is the longitudinal stress and e is the longitudinal strain. The equation representing this model is given by Eq. 3.7. a +E(s-£ ) 0 if ^p = 0 er = < a? L + £ <£<^^0 +£ 0 if £>^p- + £ (3.7) 0 -C7 if y £< +£ £ 0 E is the Young's modulus, a is the yield stress and (a ,£ ) is the stress-strain combination at a prior y 0 0 state. 3.4.2.3 Equilibrium Equations Based on Fig. 3.11, the principle of virtual work is now used to formulate the weak form of the governing equilibrium equations. For a kinematically admissible variation in the displacement (Su), the internal and external virtual work are given by: 8W = [o-( )S£dv mi L L 8W =-\{P {w )8w +P {w )Sw )dx-|(r (A )JA exl (3.8) £ r r r l i l /r r r +T (A,)SA,)dx - N5u J1 x=L + FSw x=L (3.9) Chapter 3: Analytical 43 Models where v, is the volume of the member, and L is the length of the beam. From Fig. 3.11 substituting for w wi, A„ Ai, setting the internal virtual work to the negative of the external virtual work and moving n both the foundation pressure and friction terms to the internal virtual work side of the equation gives: L L ^oSedv + j(P Sw-P Sw)dx + ^{r 8{u-''/ w')+T ,S{u + <i/ w'^)dx = -NSu r l fr 0 2 j 2 x=L + F5w x=L (3.10) 0 3.4.3 FE Implementation and Solution Procedure 3.4.3.1 Displacement Field The length L of the beam, is discretized into beam elements. The displacements, u(x), w(x), within a beam element are represented respectively by cubic and fifth-order polynomials. Given that the vector of degrees offreedomfor a beam element with nodes /' and j is given by: a =(w ,w' ,w",u ,u' ,Wj,w'j,w",Uj,u'j) (3.11) r i i i i the finite element approximations to the displacements and their corresponding derivatives is: u(x) = N a , u'(x) =N' a,w(x) = M a w'(x) =M' a , w"(x) =M" a r r r T T (3.12) where N and M, represent the vector of shape functions. 3.4.3.2 FE Representation of Virtual Work Equation From Eq. 3.12, the strain at any point in the beam element with respect to the nodal degrees of freedom can be expressed as: £ =(N' -^M" )a + ia 'M'M' 'a r r 7 (3.13) 7 and correspondingly, the virtual strain is given by: Se = SH [(N -yM" )+M'M' a] T ,t T (3.14) r The beam internal virtual work is thus given by: \O-SE = <ya '{CT[(N' -jM" )+M'M a]^ 7 V r V The work done by the foundation pressure is given by: r ,7 (3.15) Chapter 3: Analytical 44 Models \{P 5w-P 8w)dx = Sa \{P -P,)Mdx (3.16) T r i r o o Similarly, the work done by the frictional force is given by: L L \(T 5{u-y w') T 8{u fr 2 + ]1 y w ))dx=Sa , + J[(T> + 7 > ) N + % ( 7 > - 7 > ) M ' ] < & T 2 0 (3.17) 0 The work done by the external axial and lateral loads is also given by: -NSu . 3.4.3.3 Solution = <5a (-NN +FSw x L r x=L X=L + FM = ) X L (3.18) Procedure Following standard finite element techniques, the Newton-Raphson (N-R) procedure is used to solve for the global solution vector. The solution procedure proceeds on an incremental basis, and the global solution vector for the (k+l)th iteration is given by: u t + , =u +Au (3.19) k The out-of-balance force vector is given by: = NN -FM X=L X=L (3.20) The expression for the linear portion of the internal force vector is: = jo-(w -yM" )dv r T (3.21) V Similarly, the expression for the nonlinear portion of the internal force vector is: x ¥ = j " o " M ' M ' a dv r a (3.22) V The foundation embedment force vector is also given by: L V = l(P -P,)M e r dx (3.23) o Finally, the frictional force vector is given by: V = ]\f ( N - f M') + Tj, (N+f M'Jfr f fr (3.24) Chapter 3: Analytical 45 Models The expression for the consistent tangent matrix is given by: K, =K +K +K +K ; f f e (3.25) / K/ is the linear stiffness matrix and is given by: K = / (3.26) 1{^)(N -yM" \N' -yM" )dv ,t T T T K is the geometric stiffness matrix and is given by: CT K = jaM'M' T CT (3.27) dv K is the foundation stiffness matrix and is given by: e e (3.28) M M ' dx 7 j\ dw dw) K/is the frictional stiffness matrix and is also given by: ydA r j (N-|M')(N-|M') R dA, K j (N+f M ' ) ( N + | - M ' ) 7 dx (3.29) To solve for the global displacement solution vector, *F = 0, in Eq. 3.20. The Newton-Raphson procedure is given by the equation: a = a +K,-'(-'F) 0 (3.30) where, a is a previous vector, and K, = d^/da is the corresponding consistent tangent stiffness 0 matrix. 3.5 Formulation of Dynamic Implementation Analysis Equations and FE 3.5.1 Statement of Problem It is desired to determine the governing dynamic equilibrium equations for a system composed of a mass supported by the generic beam-nonlinear foundation system, that is experiencing base excitation. A practical example of this is a pile in soil, supporting a given mass at the top. The mass of the beam is assumed to be negligible in comparison with the applied mass. It is assumed that the mass is Chapter 3: Analytical 46 Models concentrated; as such, its rotational inertia is neglected. Fig. 3.12 shows an illustration of such a system. This formulation is based directly on the model developed by Foschi [30], but considers the initial confining pressure around the beam, and friction along the sides of the beam. M Foundation Force Frictional Force Figure 3.12 Schematic Diagram of Mass supported by Generic Beam-Foundation System 3.5.2 Derivation of Dynamic Equilibrium Equations Fig. 3.13 shows afree-bodydiagram of the mass supported on the beam-foundation system. t \(u,u,ii) \ // y( w,w,w ) Figure 3.13 Free-Body Diagram of Mass supported by Beam-Foundation System Chapter 3: Analytical 47 Models From the figure, x =w , h x =u , x=L v x' =x +a x=L h h and x' =x +a . gh v v All kinematic relationships gv and constitutive laws valid for the static loading model in Sec. 3.4.2 are also valid for the dynamic model. The principle of virtual work is used in formulating the dynamic equation of equilibrium. From Fig. 3.13, for a kinematically admissible variation in the displacement (Su), the internal and external virtual work at a time t are given by: 8W mt SW = -Mx' Sx -C x Sx ext h h h h = JJJ(E)SE dv (3.31) - Mx' Sx -C x Sx h v v L v v -NSx v v L - \{P {w )8w +P,(w )Sw )dx-J(7> (A )<5A r r r l l r 0 r r + T (A,)5A )dx f! (2 32) l 0 where Q and C„ are the horizontal and vertical damping factors respectively. Thus, only the external work of the inertia and damping terms need to be added to the external work of the static loading model given in Eq. 3.9. Equating the internal virtual work the negative of the external virtual work and moving both the foundation pressure and friction terms to the internal work side of the equation gives: L I \cjSsdv + j(P Sw + P Sw)dx+ ^ s(u-^w')+T s(u + ^w']jdx = o o -M(x +a )5x -C x Sx - M(X +a )§x -C x Sx - N5x r h gh h l h h h fr fl v gv v v v v (3.33) v 3.5.3 FE Implementation and Solution Procedure The displacement field used to represent the displacements within a beam element is the same as in the static loading case. 3.5.3.1 Time-Integration Procedure Newmark's constant acceleration time-integration scheme is used to integrate the time-dependent equilibrium equation given in Eq. 3.33. This method is unconditionally stable for linear systems, but Chapter 3: Analytical 48 Models this is not the case for nonlinear systems [19,20]. Predictor/corrector time-integration methods have been proposed by Crisfield [19], as a solution to this problem. As a result of this, in using Newmark's constant acceleration method in the solution of this problem, very small time steps are necessary. The average acceleration method assumes a constant acceleration over a given time interval At. Based on this method, the expressions for the acceleration and velocity of a point at a given time t= t + At are given by: --'x, + - (3.34) rr-[x,+Ar -X,At-X ] t (At) 2 t+At ~ t X x . + \ t+Al t x x (3.35) ] x At where, x is the displacement of that point at a given time. 3.5.3.2 FE Representation of Virtual Work Equation The left side of Eq. 3.33 is the same as that of Eq. 3.10; as such in this section, only the right-hand side of Eq. 3.33 would be considered. Expressing the displacements at a point in a beam element, in terms of the generalized degrees of freedom at its nodes, at a time t = t + At, corresponding to a state i+l, leaves the right-hand side of the equation as: - M5a ( M T a, , + a ) M r M x=L + - MSa {p a T T M x=L gh - 5a C M r x = l M X=L M M +0^^-Sa C N a N T M a T h v x=L M X = L - T x=L (3.36) NSaJ N +l x=L Rearranging and substituting for the acceleration and velocity terms gives: - a i+i T ^ - f M ^ M L , + N , N ; J + ^ - ( c M (At) A/ 7 = i = A ; c = i M^ + C N , V = X = L ) 2 ^-(M M r 8a i+l (At) ( AM At x=L +N Nl_ )+^-(c M M A? +C N Nl ) T x=L x=L L h x=L ( M ^ M L i +N N )+(c M __ M T x=L a ^ M ^ M ^ +N 7 ; c = / j N x=L T x=L 7 c = i h x L )-M(a M g A x=L x = i v +CvN,=X=l + a^N )-NN X=L X=L x=L L j] + (3.37) Chapter 3: Analytical 49 Models In the above equation, the generalized nodal displacements for the state /' are known. Since we are solving from a previously known equilibrium state, the global displacements at state i+1 are unknown, as such, the a term is moved to the left hand of the equation. /+/ To solve the resulting equations, the Newton-Raphson algorithm stated in Eq. 3.30 is used. The out-ofbalance force vector in this case is given by: *V dyn = % + T +T + a + V e +Q aM (3.38) R with the same variables as in Eq. 3.20 defined in Sec. 3.4.3.3, and the new variables defined as below: AM add - „M M +C N fi T X=L (At) 4M (M M'' X=L x=l *'(^f * » , M [ M ( *=i '=i M X=L M M L i + N + N *=X= )+(c ,M i X=L N / L i ) + M(agh M V x=L •i+i x=Li (3.39) +X Hl )+^(c M Ml C N ^l ) x=L At w T x=L x = i X = L IvC + A G V =L h x=L =L+ v x +C K N „ ) T L v N X = L ) + ATM xmL x L L (3.40) X = L Correspondingly, the consistent tangent stiffness matrix (K^„ = 3vuv9a ) is given by: i+1 (3.41) K/, K K and K/ are as defined in Sec. 3.4.3.3. CT e is also defined as: (3.42) 3.6 Enhancing Solution Algorithm 3.6.1 Efficiency of Solution Algorithm Apart from the need for accuracy in the computed response from an iterative procedure such as the N R algorithm, the problem of the time it takes to achieve convergence for a given suitable tolerance level is very important. The number of iterations and the size of the global stiffness matrix, mainly Chapter 3: Analytical Models 50 control the time it takes to converge. The latter is considered by taking into account any form of symmetry in the problem being analyzed. The N-R algorithm has been shown to possess a quadratic rate of convergence, which is preserved if the consistent tangent matrix is used instead of other stiffness matrices, such as the continuum stiffness matrix, secant stiffness matrix, initial stiffness matrix, etc. [74]. By itself, the N-R procedure is generally capable of achieving quadratic convergence, however, during the course of certain complex deformation processes, situations occur where the straight application of the N-R method becomes insufficient. An example of such a case is when the solution oscillates between two values, making convergence slow or impossible. The figure below demonstrates this problem. — c i r c u l Solution proceeds in a r manner Figure 3.14 Case where N-R Solution Procedure proceeds in a Circular Manner To make the N-R method more robust and efficient, a line search method is implemented within the framework of the N-R procedure. 3.6.2 Line Search Method In this technique, a search is conducted along the direction of the computed incremental displacement vector, so as to investigate in which direction the solution resides. The solution vector \x , for the k+ (k+l)th iteration based on the above, is given by the expression: u t+i= i u + ? 7 A u where, TJ is an arbitrary parameter that can be chosen with different approaches. approaches used in estimating TJ are given below. ( - ) 3 43 Two of the Chapter 3: Analytical • 51 Models 77 is chosen such that the total potential energy at the end of a given iteration is minimized. This requirement means that the residual force at the end of each iteration should be orthogonal to the direction of advance of the incremental displacement vector. This is expressed mathematically as: A i / R ^ +^Au)=0 (3.44) where, R is the residual vector. Due to extreme nonlinearity in the above equation, it is solved iteratively by choosing a suitable tolerance level. Researchers such as Bonet [9], have proposed a polynomial function in 77 to represent the above equation. This function is minimized with respect to 77 to obtain the minimum value of 77. Comparing this to the standard N-R method, the time taken for a given iteration using this approach is slightly more. However, the number of iterations needed for convergence is always less than for the standard N-R. There is thus a trade-off between the two. As such, this method is faster for problems in which the standard N-R uses a large number of iterations to achieve convergence. For problems in which only a few iterations are used, the standard N-R method is faster. • An arbitrary value of n is chosen and this is usually left the same throughout the full analysis. Since no criterion is used in choosing the value for 77, this approach does not work in some given situations. To further enhance this approach, the standard N-R method is used for a specified number of iterations, after which, if convergence is not achieved, this approach is used. Numerically, this method has the advantage of requiring less time than the above method. 3.7 Concluding Remarks In this chapter an analytical model has been developed for the static and dynamic loading of a beamnonlinear foundation system. The model takes into account the presence of an initial pressure surrounding the beam and also the influence of frictional forces on the sides of the beam. Chapter 3: Analytical Models 52 Expressions for the finite element discretization of the resulting equilibrium equations for both the static and dynamic loading cases have been obtained, and the N-R iterative algorithm for solving these resulting equations has also been discussed. Factors affecting the magnitude of the surrounding initial pressure in either a timber fastener system or a soil-pile system have also been discussed. As a result of this, a general algorithm has been developed to model cases with and without initial confining pressure. It is assumed in this model that the initial confining pressure has the effect of shifting the backbone curve in the vertical direction. Chapter 4: Results of Static 4 CHAPTER 4: 4.1 Analysis 53 RESULTS OF STATIC ANALYSIS Introduction Based on the mathematical models developed in the previous chapter, static analysis computational codes have been developed for both the timber fastener and the cohesionless soil-pile system. These stand-alone programs are called HYST and PILE respectively. A subroutine version of HYST; called SHYST, has also been developed. This subroutine (module) can be included in multiple fastener joint programs that account for the force-displacement response of each connector. A listing of HYST and PILE can be found in Appendix A . l and Appendix B . l , respectively. In this chapter, the cyclic response of timber fasteners and piles is simulated with the above programs, for different examples. The effect of friction and initial confining pressure on both of these systems is also investigated. 4.2 Description of Programs HYST and PILE are programs written in Fortran 77 that have been under development at UBC. These existing programs were based on analytical models that did not take either friction or initial confining pressure into account. The existing versions of HYST and PILE have as such been modified to take both friction and initial confining pressure into account. Microsoft Fortran Power Station® version 4.0, was used to develop the new versions of HYST and PILE. CYCPILE is a similar program to PILE that has also been previously developed at UBC [88]. The basic structure of HYST and PILE is exactly the same. The difference between them is mainly in the computation of the foundation pressure. As discussed in Chap. 3, the embedment response of wood that is used in HYST is Foschi's standard embedment curve. The embedment pressure response used in PILE is Yan and Byrne's P-y curve for cohesionless soil. Chapter 4: Results of Static 54 Analysis The following points highlight the basic structure of the programs. • The maximum number of medium layers and beam elements that could be specified in a given problem, are 10 and 50 respectively. This could be increased, if desired. • The nodes and layers are numbered from the bottom to the top. • The solution procedure is based on displacement control, i.e. incremental displacements are imposed and corresponding forces are computed. Either a nodal or layer displacement can be imposed in HYST; while only a nodal displacement can be imposed in the case of PILE. • The line search method, as explained in Chap. 3, is implemented within the framework of the Newton-Raphson iterative procedure. A flag controls the option to use the method in a given problem. • The Cholesky decomposition algorithm is used to solve the resulting linear system of equations. This involves the decomposition of the stiffness matrix K . into lower and 4 upper triangular matrices, L and U respectively. For the resulting linear system of equations, substituting LU for K„ and rearranging will give: K,u = (LU)u = L(Uu) = T s (4.1) The above equation can then be expressed as Lw = *F, (4.2) Uu = w (4.3) with The solution to Eq. 4.3 then gives the incremental displacement vector u. The decomposition of the stiffness matrix shown in Eq. 4.1 fails when the stiffness matrix is singular, i.e. when the rank of the matrix is not equal to the number of equations. The stiffness matrix is found to be singular, when any of the following conditions occur. i. The beam develops a sufficient number of plastic hinges to enable a mechanism be formed. Chapter 4: Results of Static ii. 55 Analysis The beam is not appropriately restrained laterally. This could be due to either of the following. • Along most of its length, the beam does not bear on the medium on either side. This is due to the hole developed due to the crushing of the wood, being larger than the width of the beam. • The medium surrounding the beam becomes "plastic", i.e., there is sustained lateral deformation of the medium at constant load, e.g., stiff pile in loose soil. iii. There is rigid body movement (sliding) of the beam relative to the medium. This is as a result of the limiting frictional force being exceeded along the entire length of the beam. • Both displacement and force criteria are used to check for convergence of the solution. Convergence of the displacements and forces are checked as below. i. A relative criterion is used to check the convergence of the displacements. The Euclidean norm of the nodal w, w' and u incremental displacement vectors should be less than the product of the displacement tolerance factor and the previous Euclidean norm of the respective displacement vector. For the (k+l)th iteration, this can be expressed mathematically as: ||Av ||<A,||v || t+1 t (4.4) where fj is the displacement tolerance factor and v represents the previous vector d of w, w', or u displacements. ii. In checking the convergence of the resultant force, an absolute convergence criterion is used. • For nodal displacement problems, convergence of the resultant force is achieved when the difference between the current computed resultant force and the previous, is less than the force tolerance factor. Mathematically, for the (k+l)th iteration, this can be represented as: Chapter 4: Results of Static Analysis 56 M~ k F (4.5) <F F a where, F is the resultant force and a is the force tolerance factor. For problems involving layer displacements, Eq. 4.5 takes the form (4.6) < an displaced displaced orstationary orstationary layers layers A further requirement for convergence is that the forces in the displaced and stationary layers should be in equilibrium. For the (k+l)th iteration, this is also represented mathematically as: displaced \layers (4.7) < at stationary layers Upon convergence not being achieved in a specified number of iterations, the bisection method is used to compute a new value of the input displacement. This is represented by the equation below: v +v p (4.8) c where, v is the previous input displacement for which a solution has been obtained, and p v is the current input displacement for which convergence has not been achieved. An c illustration of this is shown in Fig. 4.1i. The bisection method is applied repetitively until either \v - v | is less than a specified value, or convergence is achieved for v . In the new event of \v p lim/ new - v | being less than the specified value, v is skipped and the next input p displacement chosen. Upon convergence for a value of v c mw ±v ,v c p is set to v„ ew and with this new value of v , convergence is tried to be achieved at v again. An illustration of the p above can be seen in Fig. 4.1ii. c Chapter 4: Results of Static Analysis 57 Figure 4.1 Schematic Diagram of the Algorithm used to Compute New Input Displacement 4.3 Timber Fastener Examples HYST was used to simulate the cyclic response of two types of laterally loaded wood connections. In both of these examples, the effect of friction and initial confining pressure was studied. The two examples used are: • a single driven fastener (nail, dowel) in wood; and • a single bolt connecting three wood layers (double shear bolted connection). Illustrations of the nail/dowel connection and the bolted connection are shown in Fig. 4.2 and Fig. 4.3, respectively. Both of these problems have been previously studied by Foschi [30] and his results were used to verify the new version of HYST, by comparing them to those for the case of no friction and initial confining pressure. Both results were found to match exactly. Chapter 4: Results of Static Analysis 58 Figure 4.2 Schematic Diagram of Nail/Dowel in Wood Figure 4.3 Schematic Diagram of Three Wood Layer Bolted Connection 4.3.1 Variables under Study From the mathematical model in Chap.3, the effect of friction and initial confining pressure on the cyclic response of the above examples, is controlled by the following variables: • the coefficient of static friction between wood and steel (fu); • the effective elastic tangential stiffness of the interface (Kj), and • the initial confining pressure (PJ. Experimental work to determine the range of values for the above variables was not part of this study; as such, the range of values used for the different variables was chosen based on information obtained from other sources. Chapter 4: Results of Static 4.3.1.1 Coefficient Analysis 59 of Static Friction between Wood and Steel The coefficient of friction depends on the moisture content of the wood and the roughness of the surface. From the Wood Handbook [86] it has been observed not to vary much for different species of wood, except for those species that contain large amounts of oily or waxy extractives, such as lignum vitae. The Wood Handbook gives the coefficient of friction the following values for the different conditions listed below. Table 4.1 Recommeded Values of the Coefficient of Friction between Wood and Steel (From Wood Handbook [86]) Description of Surfaces Coefficient of Static Friction (p) Unpolished steel on dry wood 0.70 Unpolished steel on green wood 0.40 Unpolished steel in dry wood (lignum vitae) 0.20 Unpolished steel in green wood (lignum vitae) 0.34 In the literature, researchers who have studied different types of timber fastener problems and have included friction, have used very different values for the coefficient of friction for wood to steel. For example, Wilkinson [92] used a value of 0.7 in his 2-D FEM analysis of a bolted joint; while, based on Erki's work, Chui et al [17] used a value of 0.1 in their FE computation of the cyclic response of nails in wood. Based on the above, in studying the effect of the coefficient of friction on the response of the two examples, the range chosen for the coefficient of friction was 0.0 - 0.70. The specific values of the coefficient of friction were selected as: 0.0, 0.2, 0.4, and 0.7. 4.3.1.2 Effective Elastic Tangential Stiffness of the Interface As discussed in Chap. 2, elastic and inelastic tangential deformations of the interface occur before sliding commences. As an approximation of this, an effective elastic tangential stiffness was used in this model to approximate the tangential stiffness of the interface between the two bodies in contact, Chapter 4: Results of Static Analysis 60 before sliding commences. Oden and Pires [58] studied its influence on the elastic contact stress distribution between a deformable half-space and a rigid foundation over a wide range of values. They realized that the tangential stress on the contact surface decreased as Kj decreased. They attributed this to larger tangential displacements occurring for the same maximum tangential stress, in the case of relatively smaller values of Kf. Contact elements in the ANSYS FE commercial package, that take friction into account, require the input of Kf if elastic Coulomb friction is used. The method given in the ANSYS manual [78] for estimating Kf is the division of the maximum expected force by the maximum expected surface displacement. In the event of no value being entered, this defaults to the elastic normal stiffness. Since this is a difficult quantity to measure and there is virtually no information in the literature on accurate methods of estimating it; estimates were obtained from a nail withdrawal test. Fig. 4.4 shows a plot of the experimental data of a nail withdrawal test, taken from the Wood Handbook. The parameters of the test are diameter of nail = 2.51mm (0.098"); depth of penetration = 31.75mm (1.25"); species of timber = Douglas Fir; moisture content = 12%; and, specific gravity of wood = 0.54. Figure 4.4 Plot of Experimental Data from a Nail Withdrawal Test (From Wood Handbook [86]) The different lines shown in the graph represent different assumed vertical secant stiffnesses KKC2, K ) sec3 (K sech of the nail-wood system. To estimate the average effective elastic tangential stiffness at a Chapter 4: Results of Static Analysis 61 point along the beam, it was assumed that the fastener was inextensible and that the frictional force per unit length did not change over the entire length of the fastener. These assumptions are similar to those used in estimating the initial confining pressure in Sec. 3.2.2.2. Based on these assumptions, the expression for the resultant tangential force at a point, on one side of the beam is, 7) = N/(2L). Correspondingly, the effective elastic tangential stiffness is given by the equation: N (4.9) 2I<5„ where S is the tangential displacement. w Using Eq. 4.9, for K , K , and K : K , = 15360kN/m , K = 8480kN/m and K = 23400kN/m . 2 secl sec2 sec3 f 2 2 j2 p Because the values computed above are just estimates of Kf, a wide range of values of Kf was used to study its influence on the response. From the above, substantive values of Kj were presumed to lie within the range: 10 - 10 kN/m . As such, the specific values of Abused in the study were 10 kN/m , 3 6 2 3 2 10 kN/m , 10 kN/m , and 10 kN/m . 4 2 5 4.3.1.3 Initial Confining 2 6 2 Pressure For the test mentioned above and shown in Fig. 4.4, Table 4.2 gives estimates of the initial confining pressure around the nail, based on Eq. 3.3 for different values of the coefficient of friction. Table 4.2 Estimated Initial Confining Pressure around Nail based on Experimental Results of Nail Withdrawal Test (From Wood Handbook [86]) Coefficient of Static Friction p = 0.1 Initial Confining Pressure (kN/m) 66.9 H = 0.3 p = 0.5 ^ = 0.7 22.3 13.4 9.5 Table 4.3 also gives the estimated initial confining pressure, computed with Eq. 3.3, for different combinations of the friction coefficient, fastener diameter and specific gravity, based on withdrawal load values taken from the American National Standard's, National Design Specification® (NDS) for Chapter 4: Results of Static Analysis 62 Wood Construction [2]. All design withdrawal loads are based on 5* percentile strength. The confining pressures shown below are resultant forces per unit length in kN/m. Table 4.3 Estimated Initial Confining Pressure around Nail based on Nail Withdrawal Design Loads (From NDS [2]) Diameter (d) Specific d =2.51mm Gravity (0.099") (G) d =3.76mm d = 6.19mm d =9.52mm (0.148") (0.244") (0.375") H=0.1 H =0.7 ju=0.1 H=0.7 ju=0.1 H=0.7 n=o.i H=0.7 0.73 34.5 4.9 51.8 1A 85.3 12.2 131.5 18.8 0.67 28.9 4.2 43.4 6.21 71.3 10.2 110.0 15.7 0.58 19.5 2.8 30.0 4.14 47.9 6.8 74.1 10.6 0.54 17.2 2.5 25.6 3.7 42.4 6.1 64.7 9.2 0.51 13.9 2.0 21.2 3.0 37.3 5.3 53.5 7.6 0.49 12.8 1.8 18.9 2.7 31.8 4.5 48.5 6.9 0.46 11.1 1.6 16.2 2.3 26.8 3.8 43.4 6.2 0.40 7.8 1.1 11.7 1.7 18.9 2.7 30.0 4.1 For a diameter of 2.51mm (0.099"), the range of the confining pressures based on the experimental results shown in Fig. 4.4 is 9.5 - 66.9kN/m. For the same size of nail and specific gravity of wood, the pressure range based on the NDS is 2.5 - 17.2kN/m. This discrepancy can be attributed to the withdrawal load values in the NDS being based on the 5* percentile of the data. Since the aim of this parametric study was to investigate the influence of the initial confining pressure on the response for any connector, the values in Table 4.2 were assumed to be a lower bound on the initial confining pressure. As a result of this, the range of the confining pressures chosen for a given fastener diameter was assumed to be approximately twice that shown in Table 4.9. As such, for a fastener diameter d = 9.52mm, which is extremely large for a nail, and wood of specific gravity G = 0.67, the range of the initial confining pressure chosen was 0 - 250kN/m. Noting that the size of the nail chosen is very large, an initial pressure of 250kN/m is most likely very extreme, however, it was used in the analysis to show the influence of the initial confining pressure at extreme values. Chapter 4: Results of Static Analysis 63 4.3.2 Input Displacement History The input displacement history used in both examples was a varying amplitude reversed cyclic displacement history, with a maximum displacement of 15mm on both the negative and positive sides. This relatively large maximum amplitude was chosen to ensure that considerable inelastic deformation of the fastener would be observed. The input history was divided equally into 640 displacement steps. Fig. 4.5 shows a plot of the input displacement history. 20 0 100 200 300 Time steps 400 500 600 Figure 4.5 Reversed Cyclic Input Displacement History used in Timber Fastener Examples 4.3.3 Single Bolt Connecting Three Wood Layers In this problem, a hole of the same size as the bolt is pre-drilled in the wood. As a result, the bolt does not experience any initial confining pressure. The bolt is clamped at both ends and typically experiences an initial axial stress. The lateral imposed displacement (AJ is applied to the middle layer, and this causes the bolt to deform, as shown in Fig. 4.3. 4.3.3.1 Specifications of Problem Since the bolt does not experience initial confining pressure the variables that are of interest are the coefficient of static friction (ft), and the effective elastic tangential stiffness (Kf). The specifications used in this example were the same as those in Foschi [30]. These specifications are listed below. Chapter 4: Results of Static • Analysis 64 Wood: Parallel Strand Lumber (Parallam®) Thickness of each layer = 50mm, Specific gravity (G) = 0.67 Parameters for Embedment Curve: Qo = 500kN/m Q = 1.5 3 • Q, = 1500kN/m 2 D max = 7.5mm Q = 0.5 2 K = 40xl0 kN/m 4 2 Bolt: Mild steel bolt Length (L) = 150mm Yield stress (a ) = 250MPa Diameter (d) = 9.525mm Young's Modulus (E) = 200GPa y Because of the symmetry existing in the problem, only half of the bolt was discretized using a total of 6 elements; 2 in the central layer and 4 in the outside layer. 4.3.3.2 Cyclic Response The simulated hysteretic response of the three-wood-layer bolted connection for different combination of variables is shown in the following figures. Fig. 4.6 shows the influence of the coefficient of friction on the hysteretic response of the system for two values of the elastic tangential stiffness (A/ = 10 kN/m and 10 kN/m ). Fig. 4.7 also shows the influence of the effective elastic tangential stiffness 4 2 5 2 on the hysteretic response for a coefficient of friction of 0.2 and 0.7. Figure 4.6 Effect of Coefficient of Friction on the Hysteretic Response of Bolted Connection Figure 4.7 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Bolted Connection Chapter 4: Results of Static Analysis 67 From both figures, the characteristic pinching of the hysteresis loops, which is due to the formation of gaps [62,29] is evident. The basic shapes of the hysteresis loops also do not change with a variation in any of the variables. With respect to the effect of friction and initial confining pressure on the hysteretic response of the connection, the following points can be observed. i. From Fig. 4.6, it can be seen that the coefficient of friction has a noticeable effect on the hysteretic response of the system for the case when A^=10 kN/m . In the case of A^-=10 kN/m 5 2 4 2 the difference is not significant, and only a slight difference in the resultant force is observed for large displacements. ii. From Fig. 4.7, it can be seen that for [i = 0.2 and ft = 0.7, K ha% a significant effect on the f hysteretic response of the system, depending on its value. For smaller values of KfX\o notable change in the response is observed. However, for larger values of Kf the resultant force is seen to increase with an increase in Kj, especially at larger displacements. This increase is more pronounced for JX = 0.7. 4.3.4 Single Nail/Dowel in Wood In this problem, it is assumed that the nail or dowel is driven into the wood without any pre-drilling. As such, the nail/dowel experiences an initial confining pressure, which mobilizes tangential frictional forces that keep the nail/dowel in place. The fastener has no fixed boundary conditions and only the frictional force on the sides of the nail/dowel provide the needed fastener fixity. Due to this configuration, the tip of the nail/dowel is able to displace vertically, i.e., the nail/dowel is able to withdraw from the wood (the tip of the nail/dowel refers to the bottom). It is assumed in this example that the fastener does not have a head, consequently, in the case of nails, the embedment of the nail head in the wood is not taken into account. Since this model is based on small displacement theory, only relatively small axial displacements can be accurately predicted. The lateral displacement is applied to the top of the nail/dowel as shown in Fig. 4.2. Chapter 4: Results of Static 4.3.4.1 Specifications Analysis 68 of Problem All the variables considered in Sec. 4.3.1 are of interest in this problem, i.e., the coefficient of static friction (ju), the effective elastic tangential stiffness (Kj) and the initial confining pressure (PJ. In this example, the length of the nail/dowel is equal to 63.5mm and the diameter, which is also equal to 9.525mm, is the same as in the bolted connection example. The chosen diameter is not a typical nail size, however, it was chosen to represent an extreme example of a nail. Since the nail is very long, the ratio between the bending and shear stiffnesses is similar to that of a typical nail. The specifications for all other variables are also the same as in the bolted connection example. The nail/dowel was discretized into 5 elements. 4.3.4.2 Cyclic Response For this problem, the cyclic response of the nail/dowel is shown in the following figures. Fig. 4.8, Fig. 4.9 and Fig. 4.10 show the hysteretic response of the system for different combinations of the variables under study. The corresponding tip displacements of the nail/dowel for each of the above cases, are also shown in Figs. 4.1 la,b. Figure 4.8 Effect of Initial Confining Pressure on the Hysteretic Response of Nail/Dowel in Wood Figure 4.9 Effect of Coefficient of Friction on the Hysteretic Response of Nail/Dowel in Wood Figure 4.10 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Nail/Dowel in Wood Chapter 4: Results of Static 72 Analysis 1a)/J = 0.2, Kf = 10000 0 1 00 200 300 Time steps 400 500 600 200 300 Time steps 400 500 600 200 300 Time steps 400 500 600 1b) fj =0.7, Kf= 10000 0 5.0 100 2a) Pi = 50, Kf= 10000 E E £ E 2.5 F 0.0 100 Figure 4.11a Effect of Initial Confining Pressure and Coefficient of Friction on the Tip Displacement Response of Nail/Dowel Chapter 4: Results of Static 73 Analysis 3a)iJ = 0.2, Pi= 100 E e c E a. — 100 200 300 Time steps 500 400 600 3b) fj = 0.7, Pi =100 5.0 Kf • 1000 E Kf = 10000 E Kf = 100000 c c E Kf = 1000000 2.5 A Q. 300 Time steps Figure 4.1 lb Effect of Effective Elastic Tangential Stiffness on the Tip Displacement Response of Nail/Dowel 1) The influence of the various variables on the hysteretic response of the system as observed from the above figures, is discussed below. i. From Fig. 4.8, for Kj = 10 kN/m the initial confining pressure is seen to significantly 4 2 influence the hysteretic response of the system. In both plots, the resultant force increases with the initial confining pressure. There is no noticeable difference between the responses, for both coefficients of friction. The initial lateral stiffness of the system is also observed to increase with the confining pressure, ii. In Fig. 4.9, a variation in the coefficient of friction is not seen to affect considerably the hysteretic response. This is similar to the case of K = 10 kN/m in the bolted connection 4 f 2 Chapter 4: Results of Static Analysis 74 example. As in that example, only a slim difference in the resultant force at relatively large displacements is observed, iii. From Fig. 4.10, an increase in K \s seen to yield a corresponding increase in the resultant f force. This increase is more pronounced for fi = 0.7. For ju = 0.2 an increase in Kf from 10 kN/m to 10 kN/m does not result in a change in the hysteretic response. Unlike the 5 2 6 2 case of the initial confining pressure an increase in Kf is not seen to influence the initial lateral stiffness of the system. 2) From a cursory look at Fig. 4.11, it can be seen that the cyclic input history produces a cumulative withdrawal of the nail/dowel from the wood medium. Each of the plots in Fig. 4.11 is discussed in detail in the following paragraphs. i. From graphs la) and lb), which represent the change in the tip displacement response with initial confining pressure, the tip displacement is seen to generally increase with the number of cycles. Superimposed on the general trend are periodical localized decreases in the tip displacement. For P, = O.OkN/m, maximum slips of approximately 15mm and 12mm are observed for both coefficients of friction. These are rather on the high side and may include a greater margin of error. For the other confining pressure values, an increase from 50kN/m to 250kN/m results in a decrease in the maximum tip displacement from 3.6mm to 1.1mm for ju = 0.2, and from 2.2mm to 1.0mm for pi = 0.7. ii. The same trend in the tip displacement response as stated in (i) is also seen in graphs 2a) and 2b). The tip displacement is seen to decrease with an increase in the coefficient of friction. This decrease is more conspicuous for the case of P = 50kN/m. For P, = t lOOkN/m, an increase in the coefficient of friction from 0.2 to 0.7 results in only a decrease in the maximum tip displacement from 1.57mm to 1.41mm. iii. The same general pattern of the tip displacement response is seen in graphs 3a) and 3b). Nonetheless, for large values of Kf the periodical decrements diminish and the tip displacement increases somewhat monotonically. The tip displacement increases as Kf increases, however, at large values of Kf it is seen to decrease with further increase in Kf. Chapter 4: Results of Static Analysis 75 In the case of /J = 0.7 the same tip displacement response is observed for the entire displacement history, for K = 10 and 10 kN/m . 3 4 2 f Surprisingly, for K = 10 kN/m , a 6 2 f comparison of both tip displacement responses shows that the tip displacement for p = 0.7 is greater than that for // = 0.2. Sample plots of the shape of the nail/dowel at a given displacement step can be found in Appendix A.3. (The shape of the nail is shown for the given displacement step. The hole developed on the other hand is for the duration of the response up to that displacement step.) 4.3.5 Discussion of Results In making any inferences from the results obtained with this model and in discussing the effect of friction and initial confining pressure on the cyclic response of the system, the following are assumed. i. Within the input displacement range, the wood medium does not fail in a brittle manner, by splitting along any given plane of weakness (especially, splitting along the grain as a result of perpendicular-to-grain tensile stresses). ii. The rotations and displacements of the fastener are moderately large, and as such are adequately predicted without the coordinates of the system being updated at each time step. Predicted Response • It is seen that the general shape of the hysteretic response does not change significantly with a change in any of the variables under study. The shape is also seen to be very similar to that obtained with the basic model that took neither friction nor initial confining pressure into account. In the case of the nail/dowel example, there is only a slight reduction in the amount of pinching as the initial confining pressure is increased to 250kN/m. Since the approach used in modeling the initial confining pressure predicts an upper bound response, it could be concluded that both the initial confining pressure and friction do not significantly change the general shape of the hysteretic response. Chapter 4: Results of Static • Analysis 76 In addition to the above, from the nail/dowel example the model predicts a cumulative withdrawal of the nail/dowel from a wood medium. To ascertain the cause of the periodical decrease in the tip displacement, data from the hysteretic and tip displacement responses were matched. It was realized that the decrease was due to the unloading of a given side of the medium, with the subsequent increase being due to the loading of the other side of the medium. For relatively large effective elastic tangential stiffnesses, this effect diminishes since the rotations and displacements of the lower portion of the fastener decrease significantly. The phenomenon of nails withdrawing from shear walls undergoing horizontal earthquake excitation is one that is usually observed in the experimental testing of wood shear walls. Even though the size of the nail used in this example is not typical to shear wall construction, the model would still predict a similar phenomenon in the case of such a nail. Since the model predicts this behavior, it could be concluded that the model has the capability of predicting the withdrawal of laterally loaded driven timber fasteners. However, since the coordinates of the fastener are not updated, only relatively small amounts of tip displacement can be correctly predicted. In summary, it can be concluded that for moderately large displacements, the predicted cyclic response of laterally loaded fasteners is consistent with the expected trend. However, the predicted response is an upper bound to the true response. Influence of Friction and Initial Confining Pressure In this section, the effects of friction and initial confining pressure on the hysteretic and tip displacement responses as seen from the previous section are discussed. • Effect of Initial Confining Pressure It was observed in the previous section that the amount of energy dissipated, or the area enclosed by the hysteresis loops increases with increasing confining pressure, while the amount of tip displacement also decreases with increasing confining pressure. Chapter 4: Results of Static Analysis 11 From the different graphs, an increase in the initial confining pressure results in a corresponding increase in the ultimate load and initial stiffness of the system. Since the approach used in modeling the initial confining pressure assumes that it produces only a vertical shift of the embedment curve, the response of the system shown for these different confining pressures is an upper bound to the true response. From Table 4.3, which is based upon 5 percentile strength values, most of the estimated th confining pressures are below 50kN/m. Based on this, there is a 7% increase in the amount of energy dissipated by the system when compared to the case of no initial confining pressure. For a high initial confining pressure of the order of lOOkN/m, the predicted increase in the amount of energy dissipated is 10%, and for an extreme confining pressure of 250kN/m, the increase is 27%. Since an initial confining pressure of 250kN/m is extreme, in drawing any inferences about the influence of the confining pressure, this value is disregarded. Based on the predicted percentage increases in the amount of energy dissipated, and noting that the predicted responses are upper bounds to the true response, it can be concluded that the initial confining pressure does not significantly influence the amount of energy dissipated by the system. With regard to the tip displacement, there is a marked drop in the tip displacement response as the initial confining pressure increases from OkN/m to 50kN/m. Since most estimates of the confining pressure shown in Table 4.3 fall within this range, it can be inferred that the initial confining pressure plays an important role in reducing the tip displacement of a nail/dowel. Even though this inference is based on an upper bound response, it would most probably be the case for the true response. For P, = 0, however, the maximum tip displacement could be in error, since it is in the neighborhood of 20% of the length of the nail/dowel. Nevertheless, in the case of a large displacement model, a larger tip displacement response would be predicted, since the lateral Chapter 4: Results of Static Analysis 78 stiffness would reduce considerably. In practice, this would represent the case where after long-term effects a nail becomes very loose and experiences very little or no confining pressure. • Effect of Friction Since our frictional model is represented by two variables, namely, the coefficient of friction (ft) and the effective elastic tangential stiffness (Kj), both variables are considered together in discussing the influence of friction the response. From the previous sections, depending on the relative magnitude of both of these variables, friction was seen to affect the response of the system in both example problems. Fig. 4.12 summarizes the effect of friction on the response of the nail/dowel and bolted connections. Graph a) is a plot of the tip displacement at the 200 and 600 displacement th th step against the logarithm of the elastic tangential stiffness, for different coefficients of friction. The input displacement is the same for both displacement steps. Graphs b) and c) also show the amount of energy dissipated by the system versus the logarithm of the elastic tangential stiffness, for both examples From Sec. 4.3.1.2, the range of the estimated values for Allies within the region where the amounts of energy dissipated and tip displacement increase with Kj. The amount of energy dissipated is more for a larger coefficient of friction. For example, in Fig. 4.13, for Kf = 10 kN/m in graphs b) and c), the amount of energy dissipated for ft = 0.7 is respectively, 5% 5 2 and 3% more than that for ft = 0.2. The increase in the amount of energy dissipated is due to an increase in the ultimate load. The concept of an increase in the ultimate load as a result of friction is in line with the results obtained by Rodd [71], in his experimental work on the effect of normal and frictional forces on the interface. The tip displacement also increases with Kj due to smaller values of Kf requiring larger displacements to produce a given tangential force. In contrast to this, an increase in the coefficient of friction results in a significant decrease in tip displacement. The decrease in the tip displacement could be Chapter 4: Results of Static Analysis 79 attributed to more sliding occurring at different points along the fastener for lower coefficients of friction. For smaller values of K friction does not have any noticeable effect on both the amount of fi energy dissipated and the amount of tip displacement. a) Tip displacement for nail/dowel example 5.0 •g 4.0 E. S E 3.0 8 f -•- v =0.2, time step = 200 -•- M =0.7, time step = 200 - • — C = 0.2, time step = 600 — • — P = 0.7, time step = 600 2.0 a F 1.0 0.0 1.0E+3 -i 1 1 1—i—i—i—r 1.0E+4 1.0E+5 Effective elastic tangential stiffness (kN/m 2) A 0.80 # b) Energy dissipated by system in nail/dowel example — • — VJ = 0.2 -•- p = 0.7 0.60 c w 0.40 -I 1 1—I—I—I—t- 1.0E+3 -1 1 1—1—I— I I -l 1 1 1—I—i I 1.0E+4 1.0E+5 Effective elastic tangential stiffness (kN/m 2) 1.0E+6 A 0.90 c) Energy dissipated by system in bolted connection example • M = 0.2 M = 0.7 0.80 0) 0.70 —I—I—| 1.0E+3 1 1 1 1—I—I I I | -1 1 1—i—i— l l 1.0E+4 1.0E+5 Effective elastic tangential stiffness (kN/m 2) A Figure 4.12 Summary of the Effect of the Friction on the Cyclic Response of Timber Fastener 1.0E+6 Chapter 4: Results of Static Analysis 80 At larger values of K the rotations and displacements along the lower section of the beam fi decrease significantly and this results in a reduction in the amount of tip displacement. When Kj is large, sliding occurs at a number of points on the fastener at very small lateral displacements and this has the effect of limiting the ultimate load attainable. Due to this, when A"yis large, an increase in the coefficient of friction results in a considerable increase in the amount of energy that the system can dissipate. From all the above, it can be concluded that based on the estimated values of Ay and fi, friction does not have a significant influence on the amount of energy dissipated; however, it significantly influences the amount of tip displacement that occurs. Nevertheless, for cases where Kf is reasonably large, it has a considerable effect on the amount of energy dissipated. 4.4 Single Pile Example The prediction of the response of piles in a soil mass is based on a number of variables that are generally not deterministic. Some of these are: in-situ soil properties (these are very difficult to determine and even change after pile installation); soil variability in both the lateral and vertical directions; and, complex soil-pile interaction. Most of these factors are related and create a highly formidable problem to analyze. In spite of these relationships, only the effect of friction and initial confining pressure were considered in this example. The problem was studied using a tubular mild steel pile in a homogenous single layer of sandy soil. The pile was assumed to be floating; as such, load transfer by end bearing was assumed to be insignificant. The method used in this analysis is similar to Burland's /?-method for computing the skin resistance of friction piles [10,12]. The specifications of the problem are as listed below: • Soil: Dense sandy soil Number of layers = 1, Soil Properties: Depth of layer = 30m Chapter 4: Results of Static Analysis 81 Effective soil unit weight (f) = 20kN/m 2 Relative Density (D ) = 75% rs • Pile: Tubular mild steel pile Length (L) = 30ni Yield stress (a ) = 250MPa Inner diameter (dj = 336mm Outer diameter (d„) = 356mm y Young's Modulus (E) = 200GPa The same problem, without friction and initial confining pressure, was previously studied by Foschi [30 ]. To provide appropriate boundary conditions in his solution, the vertical displacement of the tip of the pile was not permitted. As such, the problem studied by Foschi is similar to a pile carrying most of its vertical load by end bearing, and sitting on a rock mass which does not experience any significant deformation. To verify the new version of PILE, it was used to solve the same problem studied by Foschi. The results compared very well with those obtained by him. 4.4.1 Variables under Study The variables that control the effect of friction and the initial confining pressure on the response of the soil-pile system are: • the coefficient of static friction between the pile and cohesionless soil (/u); • the effective elastic tangential stiffness of the interface (KJ; and, • the coefficient of lateral earth pressure (kj). As in the case of the timber fastener examples, the range of values used for the different variables was based on information obtained from other sources. 4.4.1.1 Coefficient of Static Friction between Pile and Soil In geotechnical literature, this is also referred to as the angle of wall friction (8). Since our model is for cohesionless soils only its properties in cohesionless soils are discussed in this section. 8 represents Chapter 4: Results of Static Analysis 82 the angle between the resultant and normal soil forces, acting on the wall. // and 8 are linked by the relation H = \<m{8) (4.10) 8 is related to the angle of internal friction ((/>), which has been seen in cohesionless soils to depend on the confining pressure and relative density (D ). <p is seen to increase with D , however, due to the rs rs high levels of confining pressure from the overburden at great depths in sand, it is seen to decrease with increasing confining pressure [10]. Computation of the estimated angle of friction at different depths based on an equation proposed by Zeitlen and Paikowsky [96], can be found in Appendix B.4. Based on the above, Acar et al [1] have proposed a decrement of 2° in the angle of wall friction in sandy soils, with every lOOkN/nr increase in soil confining pressure. 8 has also been identified not to depend only on the soil properties, but also on the amount and direction of movement of the wall (in this case, a pile) [73]. This is however, not accounted for in the present model. Since 8 is a difficult quantity to measure and changes significantly during the lifetime of the pile, considerable engineering judgement is necessary in obtaining realistic values for it. As an approximation, 8 is usually assumed to range between 0.5 - 0.7^'. The tables below show recommended values for the angle of wall friction between cohesionless soils and different surfaces taken from Bowles [10] and the API design code [3]. The API values are general values recommended for cases in which no other form of data is available. The values given by Bowles are based on a normal soil confining pressure of 100kN/m . 2 Table 4.4 Recommended Values for the Angle of Wall Friction for Given Soil-Pile Interface Conditions (From Bowles [10]) Description of Surfaces Angle of Wall Friction (S),° Mass concrete or masonry on Clean sound rock 35 Chapter 4: Results of Static Analysis 83 Clean gravel, gravel sand mixtures, course sand 29-31 Clean fine to medium sand, silty medium to coarse sand 24-29 Clean fine sand 19-24 Formed concrete of concrete sheetpiling against Clean gravel, gravel-sand mixture 22-26 Clean sand, silty sand-gravel mixture, hard rock fill 17-22 Silty sand, gravel or sand mixed with silt 17 Fine sandy silt, nonplastic silt 14 Steel sheet piles against Clean gravel, gravel-sand mixture 22 Clean sand, silty sand-gravel mixture, hard rock fill 17 Silty sand, gravel or sand mixed with silt 14 Fine sandy silt, nonplastic silt 11 Wood against soil 14-16 Table 4.5 Recommended Values for the Angle of Wall Friction for Given Soil-Pile Interface Conditions (From API Design Code [3]) Soil Description Density Limiting Skin Friction Angle of Wall Friction (kN/m ) (S),° 2 Gravel Dense 114.8 35 Sand Very loose 47.8 15 Loose 67.0 20 Mediumly dense 81.3 25 Dense 95.7 30 Very dense 114.8 35 Loose 47.8 15 Mediumly dense 67.0 20 Dense 81.3 25 Very dense 95.7 30 Mediumly dense 47.8 15 Dense 67.0 20 Sand-silt Silt Based on the above, for the given problem being studied, the range of values chosen for the angle of wall friction was 14 - 30°. This was assumed to be based on a soil confining pressure of 100kN/m . 2 Chapter 4: Results of Static Analysis 84 Due to the depth of the sand medium, it was divided into 3 sub-layers, with a different angle of wall friction for each layer. The figure below represents how the angle of wall friction was assumed to change with depth. Figure 4.14 Assumed Variation of the Angle of Wall Friction for Given Pile Problem Based on the above, the different combinations of the angle of wall friction used in the study are: 30°, 28°, 26°; 23°, 21°, 20°; 17°, 15°, 13°; and 14°, 12°, 10°. The pile was discretized into 20 elements, 10 in the first layer, 6 in the middle layer and 4 in the bottom layer. 4.4.1.2 The Effective Elastic Tangential Stiffness Evans et al [26] used vibration testing in estimating the vertical stiffness of piles embedded in layers of crushed rock and alluvium. The test was conducted on Japanese Daido piles (prestressed 500mm tubular pile, with a wall thickness of 100mm). For the two Daido piles of length 21m and 15m, the initial vertical stiffness of both piles was 383kN/mm and 761kN/mm respectively. A plot of the vertical load-displacement response for the 21m pile gave an initial vertical stiffness of 730kN/mm in a pile uplift test, which degraded to 185kN/mm as the frictional bond began to yield. Using Eq. 4.9 with the same assumptions as in the timber fastener example; for vertical stiffnesses of 383kN/mm, 730kN/mm and 761kN/mm, the corresponding values of Kf are: 9342kN/m , 17805kN/m and 2 25366kN/m . 2 2 Chapter 4: Results of Static Analysis 85 Evans et al also performed uplift tests on four steel tubular piles of lengths between 15-21m, founded in the same soil medium as the Daido piles. They estimated the average skin friction on the four piles of outer diameter 406mm and 610mm as, 37kN/m , 38kN/m , 54kN/m , and 40kN/m . Vertical load2 2 2 2 displacement plots of the uplift test were, however, not given. In Table 4.5 the API design code gives limiting values of skin friction for different types of cohesionless soils. Based upon Fig. 3.5 and for an assumed pile of diameter 500mm, estimated values of Kf based on Table 4.5 are: 14780kN/m , 2 20718kN/m , 25490kN/m , 29592kN/m , and 35500kN/m . 2 2 2 2 As stated in the timber fastener example, these are all estimates and cannot be fully relied on. As such, the values of Kj chosen in studying the effect of friction on the response of the soil-pile system were 10 kN/m , 10 kN/m , 10 kN/m , and 10 kN/m . 3 2 4.4.1.3 Coefficient 4 2 5 2 of Lateral Earth 6 2 Pressure This is a coefficient representing the ratio between the effective horizontal and vertical stresses acting at a given depth in a soil mass. The value depends on a number of factors, some of which are the amount of volume displacement and the initial soil density. Many researchers have proposed different values for the coefficient of lateral pressure acting around a pile. The most commonly used value is the lateral earth pressure at rest (for cohesionless soils: k„ = 1 sin^'), since it is assumed that over time, the stresses in the soil mass stabilize, and the strains become zero. Bhushan [8] suggested the use of Eq. 4.11, which relates the coefficient of lateral pressure around a pile in cohesionless soil, to the relative density. kj =0.50+0.008£>„. (4.11) The table below also shows values of k based on extensive pile test programs that have been proposed by Mansur and Hunter [50], Tavenas [81], and Ireland [41]. Chapter 4: Results of Static Analysis 86 Table 4.6 Recommended Values for the Coefficient of Lateral Earth Pressure around Piles Proposed by Different Researchers Type of Pile Coefficient of Lateral Earth Pressure (kj Mansur & Hunter Tavenas Pipe piles 12-1.3 H piles 1.4-1.9 0.5 Precast square concrete piles 1.45-1.6 0.7 Timber piles Ireland 1.25 Tapered timber piles 1.25 Step-taper piles 1.11-3.64* Tension tests (all pile types) 0.4-0.9 * suspected to be wrong The API design code recommends k, = 0.8 for unplugged pipe piles in either tension or compression, and 1.0 for full displacement piles. It can be seen from the above that there is not a good agreement as to what should be used for k Bowles [10] summarizes all the above and specifies the range for kj as, h A„ - 1.75. In this study, the effect of the confining pressure was studied based on the range of the coefficient of lateral pressure proposed by Bowles; with an assumed </>' = 35°, i.e., a range of 0.4 - 1.75. The specific values of k, that were used are 0.4, 0.8, 1.2, and 1.75. 4.4.2 Input Displacement History The input displacement history used in this example was a varying amplitude reversed cyclic displacement history, with a maximum displacement of 266mm on both the negative and positive sides. As in the case of the timber fastener example, a relatively large maximum amplitude was chosen so as to ensure that considerable inelastic deformation of the pile was observed. Sample plots of the hysteretic response of the pile for smaller amplitudes can be found in Appendix B.3. The input history was divided equally into 1500 displacement steps. Fig. 4.15 shows a plot of the input displacement history. Chapter 4: Results of Static Analysis 87 300 ~ E E. 1 150 o <D O ro Q. W b -150 i i i i i 250 500 750 Time steps 1000 1250 15 00 -300 C Figure 4.14 Reversed Cyclic Input Displacement History used in Pile Example 4.4.3 Cyclic Response The following figures show the hysteretic response of the pile for different combinations of the variables under study. Chapter 4: Results of Static Analysis Figure 4.15 Effect of Coefficient of Friction on the Hysteretic Response of Pile 88 Chapter 4: Results of Static Analysis Figure 4.16 Effect of Coefficient of Lateral Earth Pressure on the Hysteretic Response of Pile Chapter 4: Results of Static Analysis Figure 4.17 Effect of Effective Elastic Tangential Stiffness on the Hysteretic Response of Pile Chapter 4: Results of Static Analysis 91 The following points below can be observed from each of the above figures. i. It can be seen from Fig. 4.15 that the angle of wall friction does not significantly affect the hysteretic response. The only difference between the responses is at relatively large lateral displacements, where the difference is even only slim. This difference is noticed at only large values of Kf. ii. Fig. 4.16 shows that the coefficient of lateral pressure hardly influences the hysteretic response of the system. displacements. The only difference between the responses is seen at small lateral This region represents the point in the response, and range of lateral displacements, where a relatively large length of the pile would not bear on either side of the soil medium. iii. In Fig. 4.17, some difference in the hysteretic responses can be seen. The resultant force increases with Kf and this is seen to occur especially at large lateral displacements. The increase in the amount of energy dissipated by the system is, however, not that considerable. 4.4.4 Discussion of Results Predicted Response From the different graphs, the predicted hysteretic response is seen to be similar to the results obtained with the previous model that did not take friction or initial confining pressure into account. The general shapes of the loops are the same, the only difference being a change in the computed resultant force. The maximum calculated load reported by Foschi in his solution of the same problem, but with the vertical displacement of the pile tip not allowed was 437kN. From the graphs in the previous section, the maximum calculated load predicted by this model ranges between 450-500kN. Based on the above, if the result of the previous model is taken as a standard, it can be inferred that the predicted response is consistent with the expected results. As in the case of timber fasteners, this predicted response would be an upper bound to the true response. However, from Fig. 4.17, the coefficient of lateral earth pressure is seen not to have any noticeable effect on the response. As such Chapter 4: Results of Static Analysis 92 based on the assumptions made, it can be concluded the predicted cyclic response of the model for laterally loaded piles in cohesionless soil is good agreement with the results expected. From the literature study in Chap. 2, it was seen that a large amount of previous work has been done in the numerical modeling of laterally loaded piles. Most of the models in the literature, however, do not account for either friction or initial confining pressure. For example, in the modulus of subgrade finite element model developed by Vazinkhoo [87], he noted that his model did not take into account the effect of the frictional forces on the sides of the pile. In his concluding remarks he recommended the study of the effect of frictional forces on the response as an area for further research. With this new model, which is also similar to Vazinkhoo's model, it is possible to assess the effect of both friction and initial confining pressure on the hysteretic response. Influence of Friction and Initial Confining Pressure The table below summarizes all the figures in the previous section by giving the maximum percentage increase in the amount of energy dissipated by the system, based upon the values used for the different variables under study. Table 4.7 Summary of the Effect of Friction and Initial Confining Pressure on the Hysteretic Response of Pile Variation in k Variation in 8 K = 10 k = 0.8 K = 10 k = 0.8 8=35,33,31 K = 10 0.07% 4.5% 3.5% 4 f 6 f 4 r Variation in Kf k=1.75 8 = 23,21,19 K = 10 8=23,21,19 4 f 3.5% 4.3% k = 0.8 8=23,21,19 4.3% From these results, it can be seen that based upon the values assumed for the different variables, the increase in the amount of energy dissipated by the system, as a result of a variation in any of the variables is less than 5%. Noting that the range of values chosen for each variable was significantly large, it can be concluded that none of these variables has as significant effect on the hysteretic response of laterally loaded piles. Chapter 4: Results of Static Analysis 93 Since the approach used in modeling the initial confining pressure results in an upper bound response, it can be inferred from the above inference that the initial confining pressure around piles in cohesionless soil hardly affects the cyclic response under lateral loading. This could probably be attributed to the initial confining pressure, which is due to static earth pressure, being relatively low when compared to the embedment pressure. For the computed estimates of Kf in Sec. AAA.2, only a maximum increase of 3% in the amount of energy dissipated was computed from the plots in the previous section. As such, the use of approximate values of Kf'm a given problem does not significantly affect the answer. In summary, it can be concluded that based on the assumptions made, the frictional forces on the sides of a pile and the initial confining pressure around it do not have any noticeable effect on the hysteretic response of laterally loaded piles in cohesionless soil. This would most probably be true in the case of the dynamic response of piles under lateral load. As such, this ratifies the assumption in previous models, of not taking these factors into account. This is in line with the results obtained by Vazinkhoo whose model did not account for these factors, but was still in good agreement with experimental results. 4.5 Concluding Remarks In this chapter, two computer programs for both single timber fasteners and piles that are based on the static loading model derived in Chap. 3 have been discussed. To validate the model, the programs were used to solve different example problems. These problems were also used to investigate the influence of friction and initial confining pressure on the cyclic response of laterally loaded single timber fasteners and piles. A summary of the results obtained in this chapter is given below. • For moderately large displacements, the predicted cyclic response of the model for single timber fasteners and piles in cohesionless soil and under lateral cyclic loading is consistent with the results expected. Chapter 4: Results of Static • Analysis 94 The initial confining pressure around driven timber fasteners does not have a significant effect on the hysteretic response of laterally loaded fasteners, but plays an important role in reducing the amount of withdrawal under cyclic loading. • Based on the estimated values of the coefficient of friction and the frictional tangential stiffness of the interface, friction does not significantly influence the hysteretic response but significantly influences the amount of withdrawal of laterally loaded fasteners under cyclic loading. • Neither friction nor the initial confining pressure surrounding piles significantly affects the hysteretic response of laterally loaded piles under cyclic loading. Chapter 5: Results of Dynamic Analysis for a Pile 5. CHAPTER 5: 95 RESULTS OF DYNAMIC ANALYSIS FOR A PILE 5.1 Introduction This chapter discusses the dynamic analysis program for piles that has been developed using the finite element model proposed in Chap. 3. EPILE, as it is called, was developed by modifying a previous program written by Li & Foschi [44], in their study of the inverse reliability method and its application in engineering design. As in the case of HYST and PILE, a listing of EPILE can be found in Appendix C. 1. Two versions of EPILE were developed; one in which the mass supported by the pile is a variable and the ground acceleration is constant, and another in which the mass is constant but the peak acceleration is varied. EPILE was written in Fortran 77, and was developed using Microsoft Fortran Power Station version 4.0®. In this chapter, an example problem similar to the one used in the previous chapter is used to test the suitability of this model, in predicting the dynamic response of a cohesionless soil-pile system, under both horizontal and/or vertical earthquake excitations. 5.2 Description of Program EPILE was developed by modifying the previous version to base it on the structure of PILE. The different features that characterize the program are discussed below. • The program is basically structured in the same manner as PILE; and all the variables used in PILE, are also used in EPILE. As a result of this, the description of the structure of PILE given in Sec. 4.1 also applies in the case of EPILE. An important difference between the two programs is the input of an acceleration record, and hence a force in the case of EPILE, as opposed to the input of a displacement record in the case of PILE. Chapter 5: Results of Dynamic Analysis for a Pile • 96 It is assumed that all the soil layers share the same ground acceleration and displacement. This assumption infers that the soil layers are assumed not to undergo any significant deformation, but rather move with the bedrock, as stress waves travel from the bedrock to the surface. In practice, however, this is not the case, nevertheless, to account for the deformability of the soil medium in the horizontal direction, this program could be linked with a program developed by Khan [43], that gives lateral displacements of the soil medium, as a shear wave travels from the bedrock to the surface. To model the relative deformation of the soil medium in the vertical direction, a similar program could also be developed for compression waves. • Viscous damping is catered for in the program by specifying corresponding damping ratios for both the horizontal and vertical directions. Since the present model does not take the deformation of the soil medium into account, the damping ratios to be specified, represent the effective amount of distributed damping in the superstructure that is experienced by a single pile. The damping force is computed based on the initial stiffness of the system, and this is assumed not to change during the entire response. This is an assumption, since damping is known to increase with increased damage (permanent deformation) of a structure. • The program does not compute the natural frequency of the system at any point in time; as such, the time step is never updated in the program. Due to this, the input time step is divided into an equal number of smaller time steps. The size of the step is chosen so that the solution procedure would be stable and accurate, for the entire duration of the response. 5.3 Numerical Example In this section, the program is used to simulate the dynamic response of a pile under earthquake ground motion in the horizontal and/or vertical directions. To fully test the model, the program is used to predict the dynamic response of the system, for different amplifications of the basic earthquake excitation. The soil-pile system used in this example is similar to the example used in the previous chapter. 97 Chapter 5: Results of Dynamic Analysis for a Pile 5.3.1 Specifications of Problem The specifications of the different parameters of the problem are listed below. • Soil: Dense sandy soil Number of layers = 1, Depth of layer = 30m Soil Properties: Effective soil unit weight (y) = 20kN/m 2 Relative density (D ) = 75% rs Coefficient of lateral earth pressure (k) = 0.8 • Pile: Tubular mild steel pile Length (L) = 30m Yield stress (a ) = 250MPa Inner diameter (dj) = 336mm Outer diameter (dj = 356mm y Young's Modulus (E) = 200GPa • Interface Properties: Angle of wall friction at 100kN/m confining pressure (8) = 23° 2 Effective elastic tangential stiffness (Kf) = 50xl0 kN/m 4 • 2 Damping Properties: Damping ratio for vibration in the horizontal direction ( ^ = 5% Damping ratio for vibration in the vertical direction (Q ) = 5% v • Concentrated weight/mass supported by pile (M): Weight/Mass = 750kN/76.5tonnes • From the force-displacement response of the system in both the horizontal and vertical directions, shown in the figure below; the initial stiffnesses of the system are 63kN/mm and 460kN/mm respectively. The corresponding initial natural frequencies are: Horizontal direction (ci\) = 5.0Hz Vertical direction (o^) = 12.3Hz 98 Chapter 5: Results of Dynamic Analysis for a Pile The graph below was obtained by running PILE in the horizontal and vertical directions, for an input displacement range in which the whole system stayed elastic. 10 ^ - " Horizontal direction — — 0.000 — Vertical direction 0.005 0.010 0.015 0.020 Displacement (mm) Figure 5.1 Initial Stiffness of Example Soil-Pile System in Vertical and Horizontal Directions • The discretization of the pile into finite elements was handled in the same manner as in the example used in Sec. 4.4.1.1 of the previous chapter. 5.3.2 Input Acceleration Record The input acceleration record used in the analysis was the Joshua Tree Fire Station record, from the magnitude 7.3 California Landers earthquake in 1992. The specific record used was the horizontal component of the ground record in the E/W direction. The same acceleration record was used to excite the system in both the horizontal and vertical directions. This earthquake record is characterized by two acceleration peaks and is well-suited for investigating the response of the system to a second acceleration peak, after it has lost a considerable amount of stiffness, due to the damage produced by the first acceleration peak. The acceleration record was given for a time step of 0.02secs, and out of 4000 possible data points, 2000 data points were used. This gives the total duration of the excitation used in the analysis as 40secs. Each time step was divided into 4 equal parts; as such, the time step used in the analysis was 0.005secs. The peak acceleration of the record is 0.284g. 99 Chapter 5: Results of Dynamic Analysis for a Pile From a response spectrum plot of the excitation shown in Appendix C.3, the dominant frequencies in the excitation are 0.5 - 0.75Hz, and 1.25 - 1.50Hz. Other intermediate spectral peaks occur between 0.25 - 0.4Hz and 2.5 - 3.25Hz. A plot of the acceleration record is shown in the figure below. 0.3 | Time (sec) | Figure 5.2 Horizontal Component of Joshua Tree Station Acceleration Record in E/W Direction 5.3.3 Dynamic Response The program was used to predict the response of the system for the following peak accelerations: 0.284g; 0.426g; 0.568g; and 0.7 lOg. Plots of the simulated dynamic response are shown in the following figures for three cases of the excitation being in the horizontal direction only, the vertical direction only and both the horizontal and vertical directions simultaneously. For the combined case of the horizontal and vertical excitation, the same peak acceleration is applied in both directions, e.g., in the case of 0.568g the same acceleration record with a peak ground acceleration of 0.568g is applied in both directions. Chapter 5: Results of Dynamic Analysis 100 for a Pile Horiz. only Vert, only 7a; Lateral displacement 60 Hociz. & Vert. of mass - Peak acc. = 0.284g 30 0 si -30 -60 20 Time (sec) 1b) Vertical displacement of mass - Peak acc. = 0.284g -4 H 20 Time (sec) 2a> Lateral displacement of mass - Peak acc. = 0.426g 200 100 O a -100 -200 10 2b) Vertical displacement I - 1 0 20 Time (sec) 30 of mass - Peak acc. = 0.426g -i -15 20 Time (sec) Figure 5.3a Vertical and Horizontal Displacement of Mass for Peak Accelerations of 0.284g and 0.426g in Vertical and/or Horizontal Directions 40 101 Chapter 5: Results of Dynamic Analysis for a Pile Horiz. only Vert, only 3a) Lateral displacement Horiz. & Vert of mass - Peak acc. = 0.568g 200 100 -1CO -200 10 3b) Vertical displacement 8 5 20 Time (sec) 30 40 30 40 of mass - Peak acc. = 0.568g -20 -40 10 4a) Lateral displacement 600 20 Time (sec) of mass - Peak acc. = 0.710g 300 5 •o xo -300 H -600 0 10 4b) Vertical displacement 20 Time (sec) of mass - Peak acc. 3 40 =0.710g o -50 -1CO -150 5 -200 10 20 Time (sec) Figvore 5.3b Vertical and Horizontal Displacement of Mass for Peak Accelerations of 0.568g and 0.710g in Vertical and/or Horizontal Directions Chapter 5: Results of Dynamic Analysis for a Pile 102 The following paragraphs discuss observations made from the different plots shown on the preceding pages. i. From the plots corresponding to a peak acceleration of 0.286g, it can be seen that there is no difference in the horizontal displacement response of the system for both the horizontal-only and combined horizontal-vertical cases. As expected there is no horizontal displacement observed for the vertical-only case. For the vertical displacement, all the three cases initially vibrate about a negative equilibrium displacement (« 6mm), which represents the amount the soil medium settles, as a result of the axial load imposed by the supported mass. In the vertical-only case, this phenomenon is maintained throughout the duration of the response. For the other cases, a slight amount of additional settlement can be observed at the two acceleration peaks. For the portion of the response in-between these two points, the system vibrates about the new respective equilibrium positions. For these two cases, there is no significant difference between their vertical displacement responses. As such from these two graphs, it can be inferred that at this level of acceleration, the vertical acceleration does not have a significant influence on the response. ii. A similar response to (i) is observed in graphs 2a) and 2b). In comparison to (i), a more significant amount of settlement is seen to occur at the second acceleration peak for the horizontal-only and combined horizontal-vertical cases. Between these two, a greater amount of settlement is observed for the combined horizontal-vertical case. This is most likely due to the system being less stiff in the vertical direction as a result of more sliding at points along the beam. Under the supported mass, this then results in a greater amount of settlement as seen in the figure. A residual horizontal displacement is also observed in 2a) for both these cases. iii. In graphs 3a) and 3b), the residual horizontal displacement is seen to increase by a marked amount. This marked increase is attributed to second-order ( P-8) effects. The amount of vertical settlement is also seen to increase in a similar manner to that of the horizontal Chapter 5: Results of Dynamic Analysis for a Pile 103 displacement. The difference in the vertical settlement between the horizontal-vertical case and that of the horizontal-only case is also seen to become significant after the first acceleration peak. At this level of acceleration, both the horizontal and vertical displacement responses of the vertical-only case are still the same as in (i). iv. For a peak acceleration of 0.7 lg, the horizontal and vertical displacements are seen to increase considerably, and practically, failure of the system could be assumed to occur at this acceleration level. 5.3.4 Discussion of Results From the observations made in the previous section, the predicted dynamic response of the system is seen to be consistent with the expected results. This point is buttressed by the fact that the results generally show that, in comparison to the vertical acceleration, the horizontal acceleration has greater impact on the system. Due to the fact that the levels of acceleration used, and the mass supported by the pile are within a practical range, it may be concluded that the predictions from the model represent what occurs in practice. Other observations that reinforce this inference are: • at increased levels of acceleration, the model is seen to predict the amplification of the displacements as a result of second-order effects; and • the influence of the vertical acceleration on the predicted response is seen to become more significant at higher levels of acceleration. For the case of the acceleration in only the vertical direction, the predicted response is seen not to be significantly influenced by changes in the level of acceleration. To investigate this further, simulations for this case were run at high levels of acceleration. Fig. 5.4 shows a plot of the results obtained for peak acceleration levels of the order of 2g. Chapters: Results of Dynamic Analysis for a Pile ^ ^ ^ ^ I 104 Peak acc. = 2.556g 0 10 20 Time (sec) 30 40 Figure 5.4 Amount of Vertical Displacement at High Levels of Acceleration for Acceleration in Vertical Direction only From this graph, it can be seen that for these high levels of vertical shaking, a considerable amount of settlement takes place. Similar to the case of the horizontal displacement, the vertical displacement also increases by marked amounts as the system looses stiffness. As such, it can be inferred that the influence of the vertical acceleration becomes significant only at relatively high levels of vertical acceleration. This is also seen to be consistent with what is expected, and as such validates the model. It is interesting to note that for a peak acceleration of 2.556g near the first acceleration peak, the model predicts some positive vertical displacements. When the program was run for relatively large peak accelerations of the order of l.Og in the horizontal direction, and 3.0g in the vertical direction, the displacements began to increase considerably. The simulated response of the system at these levels of acceleration was not seen to be consistent with expected results. In view of this, it can be inferred that the model may not be suitable for problems that involve large displacements. As a result, in solving problems of this kind that involve large displacements, a large displacement model should be used. In addition to this, in order to ensure that significant amounts of numerical error are not introduced into the computed response, the time step should be updated periodically. Chapter 5: Results of Dynamic Analysis for a Pile 105 Based on all of the above, it can be concluded that for moderately large displacements, the predicted dynamic response of piles in cohesionless soil undergoing horizontal and/or vertical earthquake excitation is consistent with the expected trend. 5.4 Concluding Remarks In this chapter, the dynamic analysis program for piles; EPILE, was discussed and run for an example problem. The results obtained were consistent with the expected results, and the model was seen to predict the expected dynamic response of piles in cohesionless soil under horizontal and/or vertical earthquake excitation. Chapter 6: Conclusions and 6. CHAPTER 6: Recommendations 106 CONCLUSIONS AND RECOMMENDATIONS This thesis is part of ongoing research at UBC in the development of efficient and robust computational tools for the numerical modeling of the response of both timber structures and pile foundations, under dynamic and psuedo-dynamic loading. The use of efficient, robust and accurate algorithms and models cannot be over-emphasized, since the results are to be used as input data to other programs for reliability analysis, risk assessment and general civil and structural engineering design. Timber structures and pile foundations are made up of individual fasteners or piles, respectively. To obtain a reliable prediction of the response of either of these systems, it is necessary to accurately model the response of each of the individual elements. As such, this thesis dealt with the development of a new model to predict the response of single timber fasteners and piles under both cyclic and dynamic loading. It is an improvement of previous models developed at UBC, which did not account for the effect the initial confining pressure around driven timber fasteners and piles, and friction along the interfaces of both the wood and fastener, and soil and pile. In addition to developing the model, a sensitivity study was conducted to investigate the error introduced into the predicted response of the previous models as a result of not taking the above into account. 6.1 Conclusions The model developed as part of this study is a 1-D finite element model based on the modulus of foundation reaction method that uses nonlinear embedment curves to model the response of the respective medium. The initial confining pressure and friction were included in the new model in the following ways. • The initial confining pressure at a point along the fastener/pile was accounted for in model by vertically shifting the embedment curve of the respective system by an Chapter 6: Conclusions and Recommendations 107 amount equivalent to the initial confining pressure experienced at that point. This method was chosen since it predicts an upper bound response of the system. The initial confining pressure around timber fasteners was assumed to be uniform along the entire length of the fastener, while that around piles was taken as equal to the static lateral earth pressure, and as such, varied linearly with depth. • Friction was accounted for in the model by using elastic Coulomb friction laws to model the force-displacement response of the vertical tangential forces on the sides of the fastener/pile. 6.1.1 Computer Programs Developed The model was the basis of the development of three different computer programs. These are listed below. HYST: A static analysis program for the modeling of the cyclic response of single timber fasteners. PILE: A static analysis program for the modeling of the cyclic response of single piles in cohesionless soil. EPILE: A dynamic analysis program for the modeling of the dynamic response of single piles in cohesionless soil. 6.1.2 Predicted Response from the Model To verify the model, the above computer programs were used to simulate the response of a number example problems. In general, it was seen that the predicted cyclic and dynamic response for single timber fasteners, and piles in cohesionless soil was consistent with the expected behavior. Chapter 6: Conclusions and Recommendations 108 6.1.3 Effect of Friction and Initial Confining Pressure In light of the assumptions made in developing the model, the effect of friction and initial confining pressure on the predicted response was also investigated. Below is a summary of the conclusions arrived at. • The initial confining pressure around driven timber fasteners does not have a significant effect on the hysteretic response under lateral cyclic loading, but plays an important role in reducing the amount of withdrawal. • For the estimated values of the coefficient of friction and the frictional tangential stiffness of the interface, friction does not significantly influence the hysteretic response of fasteners under lateral cyclic loading, however, it has a significant influence on the amount of withdrawal. • Neither friction nor the initial confining pressure surrounding piles significantly affects the hysteretic response of laterally loaded piles. 6.2 Recommendations Considering the fact that research in this area is ongoing at UBC, a number of areas for future research are outlined in this section i. To enable modeling of relatively large displacements, the basic framework of the model should be used to develop a large displacement model. ii. To make the model applicable to other geometries, different methods are needed to estimate the initial confining pressure for these geometries. Experimental work is also needed to provide estimates of the initial confining pressure around different fasteners under different conditions. iii. A detailed experimental study is needed to accurately determine the coefficient of static friction and the effective elastic tangential stiffness, for wood under different conditions. Chapter 6: Conclusions iv. and Recommendations 109 To ensure that the programs are sufficiently robust, they should be tried in the solution of other different types of problems. In this regard, different acceleration records and input displacement histories should be used. v. To make EPILE more general, it should be linked with the program written by Khan [43], to compute the displacement of the soil medium, as a result of the passage of a shear wave. This could also be extended to compression waves. vi. Experimental studies are also needed to verify the predictions of the programs. Bibliography 110 BIBLIOGRAPHY 1. Acar, Y.B. et al, "Interface Properties of Sand", JGED, ASCE, Vol. 108, pp. 648-654, 1982. 2. America Forest and Paper Association, National Design Specification for Wood Construction, America Forest and Paper Association, Washington D.C., 1993. 3. American Petroleum Institute, Recommended Practice for Planning Designing and Constructing fixed Offshore Platforms, API-RP2A - WSD, American Petroleum Institute, Washington D.C., 1993 4. 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Appendix A: Listing of HYST, Sample Input Files, Plots and Worksheets A APPENDIX A: LISTING OF HYST, SAMPLE INPUT FILES, PLOTS AND WORKSHEETS A.1 Listing of HYST HYST is made up of the following modules (subroutines) • Main Program • PSUP module (Computes the foundation pressure at a point along the fastener) • DECOMP module (Performs Cholesky decomposition) • SOLV module (Solves resulting decomposed equations) • SEARCH module (Computes a new value for line search parameter) • FRICT module (Computes the frictional force at a point along the fastener) • STRESS module (Computes the stress at a point in the beam) • SHAPES module (Computes value of shape function at a given integration point) • GAUSS module (Provides Gauss integration point values) The listing of all these modules apart from GAUSS is given in the following pages. Most of these modules are also shared with PILE and EPILE and as such, a listing for those would not be given in Appendices B and C. 116 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n + O2a + + + a 2o 2D2Dn n SO2 > tr m 13 Pi PI tr- Cl C O LO C O toto C O Itt SOP SO Q 13 2 tn P32 50 13 C H t o O 0 3 H o Otoo o tr o o OOSOo W Ln H 2 O PO0o> o r IO OiOo o o o -3n X C > 1 3 c o c o c o o o 2 2 LP C OLO C O C O O3 13LO 2 O C O tO 2 C Lno p]o o o aP3 E LO O H o co o o L0 p i o " a C O C O t s ) ti pi L 2 on o L on o o o a •o to n 2C OIDG E C tr* -<1 tH r tco L O t r " 2 2 Dto O13 oC o O C O o 2 o3 C C DL n OD1 C D O C o LO co- —-o-o p t-^]P I -a C O o z + PI O O z z a **] a z O z z o z 3 r PI Z O z z PI n PI Z LO Z O Z Z 4 n H PI o z H PI X 1 IS) PJ o H ID r CD ID a X CD CD a o a n o a z PI z a u a s w f z c C O 2 to IrC D PI PJ o o z n o a a £ z ID 3 3 CD 3 a a 3 0 •< 3 3 0 3 3 3 H ^ <D C D ?3 X ( O i-* <D •O j=.N I rtot rt) Io M. rt tsc> ' sl> C O O t-t. mn 3 0 > MI PJ 3 3 d o 3 0 0 3 (D 3 3 H - T3 I H- K DJ n 3 3* 3 a o 3 3 3 0 a 3 H- W. 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Title Three l a y e r s / Fastener .750000E+02 .' Type of 1 Length Cross-section .' Diameter .952500E+01 .' Mod. Elast. .200000E+03 Yield Stress .250000E+00 .' Number of Layers 2 1 Number of Types of Layers .' Embedment Propeties of given Layer Medium ! QO Ql Q2 Q3 .500000E+00 .150000E-02 .500000E+00 .150000E+01 .' Coeff. Frict Stiffness .200000E+00 .100000E-03 of XK .400000E+00 Dmax .750000E+01 Interface .' Layer 1 Properties ! Layer number First Element 1 1 2 .25000E+02 Last Element Layer Thickness .' Layer 2 Properties ! Layer number First Element 1 3 6 .50000E+02 Last Element Layer Thickness .' Number of Gauss Points ! X-Direct. Y-Direct. 9 16 .' Axial Load .OOOOOOE+00 .' Number of Nodes with 1 .' Fixed BC data ! Node Number 1 2 2 4 BCs Number of BC at node .' Number of Layers 1 .' Layer 1 Fixed being Displaced Numbers for the Layers .' Displ. Tol. Factor .100000E-02 .100000E-03 List Force being Displaced Tol. Factor of Codes for all BCs at node Appendix A: Listing of HYST, Sample Input Files, Plots and Worksheets SINGLE NAIL OR DOWEL IN WOOD 132 PROBLEM (Units: Forces - kN; Distances - mm) ! Title Single .' F a s t e n e r .635000E+02 Length .' Type of 1 • Cross-section .' Diameter .952500E+01 .' Mod. Elast. .200000E+03 Yield Stress .250000E+00 .' Number of Layers 1 1 Number of Types of Layers .' Embedment Propeties of given Layer Medium ! QO Ql Q2 Q3 .500000E+00 .150000E-02 .500000E+00 .150000E+01 .' Coeff. Frict Stiffness .700000E+00 .100000E-01 of .' L a y e r 1 Properties ! Layer number First Element 1 1 5 .635000E+02 XK .400000E+00 Interface Last Element Layer Thickness .' Number of Gauss Points ! X-Direct. Y-Direct. 9 16 .' Axial Load .OOOOOOE+00 .' Number of Nodes with 0 .' Number of Layers 0 Fixed BCs being .' Node Number for Input 6 1 Displaced Disp. .' Displ. Tol. Factor .100000E-02 .100000E-03 Force Code for Direction Tol. Factor of Disp. Dmax .750000E+01 Appendix A.3 A: Listing of HYST, Sample Input Files, Plots and 1 Worksheets Sample Plots a) Kf = 1, Pi = 100,IJ = 0.2 b) Kf= 1, Pi= 100,ii = 0J 60 60 40 40 a> 20 -20 a> 20 -10 0 10 Lateral displacement (mm) 20 -20 c) Kf = 10000000, Pi = 100, /j = 0 60 40 40 a) 20 m 20 -10 0 10 Lateral displacement (mm) 20 d) Kf = 10000000, Pi = 100, IJ = 0.7 60 -20 -10 0 10 Lateral displacement (mm) 20 -20 -10 0 10 Lateral displacement (mm) 20 f) Kf= 1000, Pi = 0,iJ = 0.7 e)Kf = 10000, Pi = 0,ij = 0.2 60 60 40 40 o S 20 o> 20 shape of hole shape of beam -20 -10 0 10 Lateral displacement (mm) 20 -20 -10 0 10 Lateral displacement (mm) Figure A . l Shape of the Nail/Dowel and Hole at the 400 Displacement Step th 20 Appendix A: Listing of HYST, Sample Input Files, Plots and a) Kf = 1, Pi= 100, y = 0.2 134 Worksheets b) Kf= 1, Pi= 100, jj = 0.7 60 60 E E. I 40 40 r S 20 <°. 20 i - 2 - 1 — n 1 1 0 1 Lateral displacement (mm) 1 r -1 0 1 Lateral displacement (mm) c) Kf= 10000000, Pi = 100, \J = 0.2 d) Kf = 10000000, Pi = 100, IJ = 0.7 60 60 40 40 v 20 £ 20 1 1 i i r -1 o 1 Lateral displacement (mm) - 2 - 1 0 1 Lateral displacement (mm) f) Kf= 1000, Pi=0,ti = 0.7 e) Kf = 10000, Pi = 0,fj = 0.2 60 60 40 40 o S 20 h u 20 h shape of beam shape of hole r -1 o 1 Lateral displacement (mm) i ' 1 r -1 0 1 Lateral displacement (mm) Figure A.2 Shape of the Lower Portion of the Nail/Dowel and Hole at the 400 Displacement Step th Appendix A: Listing of HYST, Sample Input Files, Plots and A.4 Worksheets 135 Sample Worksheets Energy Dissipation Computation Sheet Variation in Tangential Stiffness and Coefficient of Friction - Bolt a) Setting units mm := 1 0 " m kN := 10 3 gkg kN 9.807 « 1 0 3 •kgnr^s "E:/users/nii/bolt/m0k2c.dat" "E:/users/nii/bolt/m0k3c.dat" "E:/users/nii/bolt/m0k4c.dat" "E:/users/nii/bolt/m2k2c.dat" "E:/users/nii/bolt/m2k3c.dat" "E:/users/nii/bolt/m2k4c.dat" "E:/users/nii/bolt/m2k5 c. dat" "E:/users/nii/bolt/m2k6c.dat" "E:/users/nii/bolt/m2k7c.dat" filenames := "E:/users/nii/bolt/m4k2c.dat" "E:/users/nii/bolt/m4k3c.dat" ''E:/users/nii/bolt/m4k4c.dat'' "E:/users/nii/bolt/m7k2c.dat" "E:/users/nii/bolt/m7k3c.dat" "E:/users/nii/bolt/m7k4c.dat" "E:/users/nii/boltym7k5c.dat" "E:/users/nii/bolt/m7k6c.dat" "E:/users/nii/bolt/m7k7c.dat" b) Read and compute the energy for different files Energy := for k e 0.. 17 data«— READPRN -data -data (filenames k <o> < 1 > •last(A) 1 energydissp 2 | •<i= 0 Dissip <- energydissp k Dissip - A. ( F i + . + F i kN mm 2 Appendix A: Listing of HYST, Sample Input Files, Plots and C) View the Results 0.74344 0.74344 0.74344 0.7435 0.74389 Energy 0.74984 0.7675 0.76509 0.76998 0.74353 0.74388 0.74905 0.74351 0.74388 0.74843 0.79037 0.83306 0.84606 Worksheets Appendix A: Listing of HYST, Sample Input Files, Plots and Worksheets 137 Energy Dissipation Computation Sheet Variation in Coefficient of friction - Nail a) Setting units mm := 10" m kN := 10 -gkg kN = 9.807.10 "E:/users/nii/nail/cofrict/m0plk4c.dat" "E:/users/nii/nail/cofrict/m2pl k4c.dat" "E:/users/nii/nail/cofrict/m4plk4c.dat" filenames: "E:/users/nii/nail/cofrict/m7pl k4c.dat" "E:/users/nii/nail/cofrict/m0p5k4c.dat" "E:/users/nii/nail/cofrict/m2p5k4c.dat" "E:/users/nii/nail/cofrict/m4p5k4c.dat" "E:/users/nii/nail/cofrict/m7p5k4c.dat" b) Read and compute the energy for different files Energy := for k e 0.. 7 data * - R E A D P R N (filename^) A«-data < l> F«-data <2> n<-last(A) n- 1 energydissp ( 2 | <i=0 Dissip <- energydissp k Dissip C) View the Results 0.48782 0.49585 0.49539 0.49526 Energy = 0.46509 0.47207 0.47198 0.47187 •kN.m F ^ i)] F i + •kN mm 3 .kg.nrs 2 Appendix B: Listing of PILE, Sample Input File, Plots and Worksheets 138 B APPENDIX B: LISTING OF PILE, SAMPLE INPUT FILE, PLOTS AND WORKSHEETS B.1 Listing of PILE PILE is made up of the following modules (subroutines) • Main Program • PSUP module (Computes the foundation pressure at a point along the fastener) • DECOMP module (Performs Cholesky decomposition) • SOLV module (Solves resulting decomposed equations) • SEARCH module (Computes a new value for line search parameter) • FRICT module (Computes the frictional force at a point along the fastener) • STRESS module (Computes the stress at a point in the beam) • SHAPES module (Computes value of shape function at a given integration point) • GAUSS module (Provides Gauss integration point values) The listing for all these modules apart from the Main Program and PSUP, are the similar to that given in Appendix A . l . 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H J M M H > Co M O O — o P IaZ oT oJ t" CGZ Z TJoo co 1-3 n n n a z n G o H H CM rt]pj < -PIX > < - rt] Z Otr--3-i O — —ZI z H H ITao oo :25 'CT Z I P i XX3 TJ CM D PI H Z O PJ CO O Z Pi -3 Z O PI >-J CO PPJ PI — H CD PJ — PI PJ PI H H 1 D TJ PJ PI PJ PJ — — M M P] PJ P3 T] , Z P fJPIa > r3 -3 X O 50o— • O 50 O, to fD d g IQ a Q i «Q rt £ fD 3" O < Ci Q i C D o M C rr 0) C O Co 0J TJ 3 PI PJ PJ PI PJ - TJ — • D n a o n n n pjconnnn-nsci o o o o o o s w O T)PIloPM I PW I HMPSPIH I -J Oi L" —. * l-l ~ , H — I -3 z n H I-I H O Z no < z n o D o o o i- o c PJ O OCOCn O t i oz cT Jtonoo t" Z n § no— i n H H Dn n cn o PlZDMUirnnin G - PI O Z tf pi m * n n J - x «: n P II PJ > P J Z z — P I -< O O JO oo — 23, I : H M to s PJ 1 H 3 O l-l TJ I—' H H M >-3 „ „ H I-H M M n PI - X H- M M >"< M M X I-H — X X — ' I-H X — X — II II II D O O Z TJ Z Z • 55- I-H M M M M M X M Z I II OJ M K* CD •-< g H X M Z t~Z ^ ~q^> co o — — c, PI -" PJ c o cnH< O — 4- -P II - J o ! I 5 I I 8 s PJ Pi PZ J CJPJTJ tZ U OT J to 50 50 D TJ n -3 PJ TJ CtJSOS— O O Z II • • Z OP Ci H OD I •O• O 1Q i D l-f P* Z o o <J rP fTJf MH' -3 H TJ O II 50 - •H fo D ftra p-OJ IQ P P aroCT n> p ( Hi s; TJ D z < a rt rt rtoiTro h-J ro O ro rt n rt i- OpO ro i pt r pDJ H- H 1 3* HI 3 H- •< o J—' H. 01 H- 3" DJ 01 3 Appendix B.2 B: Listing of PILE, Sample Input File, Plots and Worksheets 150 SAMPLE INPUT FILE PILE UNDER CYCLIC LOAD PROBLEM (Units: Forces - kN; Distances - mm) ! Title Pile .' Pile Length .300000E+05 .' Type of 2 Cross-section ! Inner Diameter Outer Diameter .336000E+03 .356000E+03 .' Mod. Elast. Yield Stress .200000E+03 .250000E+00 .' Number of Layers 3 3 Number of Types of Layers .' Different Embedment Propeties of Layers ! Type 1 ! Unit Weight Rel. Density Coeff. Lat Press. .200000E-07 .750000E+02 .1200000E+01 .' Coeff. Frict Stiffness .700020E+00 .100000E-01 .' Type 2 ! Unit Weight .200000E-07 Rel. Density .750000E+02 .' Coeff. Frict Stiffness . 649400E+00 .100000E-01 .' Type 3 ! Unit Weight .200000E-07 of Coeff. Lat Press. .1200000E+01 of Rel. Density .750000E+02 .' Coeff. Frict Stiffness .600860E+00 .100000E-01 Interface Interface Coeff. Lat Press. .1200000E+01 of Interface .' Layer 1 Properties ! Layer number First Element 3 1 4 .50000E+04 Last Element Layer Thickness .' Layer ! Layer 2 5 2 Properties number First Element 10 .10000E+05 Last Element Layer Thickness .' Layer ! Layer 1 11 3 Properties number First Element 20 .15000E+05 Last Element Layer Thickness .' Number of Gauss Points ! X-Direct. Y-Direct. 5 16 .' Axial Load .000000E+00 Appendix B: Listing of PILE, Sample Input File, Plots and .' Number of Nodes with 0 2 .' Node Number for Input 1 1 2 151 Worksheets Fixed BCs Disp. .' Displ. Tol. Factor .100000E-02 .100000E-03 Code for Direction Force Tol. Factor of Disp. Code for Units of Dist. Appendix B.3 B: Listing of PILE, Sample Input File, Plots and 152 Worksheets Sample Plots 300 Angles of Wall Friction = 23, 21, 19 Coeff of lateral earth pressure = 0.8 Effect, elastic tang, stiffness = 10000 Yield Point = 250 Displacement (mm) •30.0 Displacement (mm) •0.30 Figure B.l Cyclic Response of the System for Relatively Small Input Displacements Appendix B: Listing of PILE, Sample Input File, Plots and a) Stiff=10000,Cff. lat. press.=0.8,Frict ang.=35,33,31 30000 Worksheets 1 ) Stff=10000,Cff. lat. press.=0.8,Frict.ang. = 14,12,10 D 30000 ? E, E E E CO CD -200 -100 0 100 Lateral displacement (mm) 200 -200 c) Stiff. = 1'000000,Cff. lat. press.=0.8,Frict. ang.=23,21,19 30000 -200 -100 0 100 Lateral displacement (mm) 200 d) Stiffs WOO.Cff. lat. press.=0.8,Frict.ang.=23,21,19 30000 -100 0 100 Lateral displacement (mm) 200 -200 e) Stiff = 10000.Cff. lat. press.=0.4,Frict.ang.=23,21,19 30000 E, ¥ ro 0 200 f) Stiff=10000,Cff. lat. press.=1.75,Frict.ang.=23,21,19 30000 ? E. E ro ? -100 0 100 Lateral displacement (mm) Y 0) <*— o o JZ shape of hole shape of beam -200 1 1 I 1 I -100 0 100 Lateral displacement (mm) 1 200 -200 I -100 0 100 Lateral displacement (mm) Figure B.2 Shape of the Pile and Hole at the 750 Displacement Step 200 Appendix B: Listing of PILE, Sample Input File, Plots and a) Stiff=10000,Cff. lat. press.=0.8,Frict ang.=35,33,31 1 b) Stff=10000,Cff. lat. press =0.8,Frict.ang =14,12,10 30000 h 30000 h -1 Lateral displacement (mm) c) Stiff=1000000,Cff. lat. press.=0.8,Frict. ang =23,21,19 30000 -1 Worksheets 0 Lateral displacement (mm) 1 d) Stiff=1000,Cff. lat. press =0.8,Frict.ang.=23,21,19 30000 0 Lateral displacement (mm) 1 e) Stiff =10000,Cff. lat. press =0.4,Frict.ang =23,21,19 30000 -1 0 Lateral displacement (mm) f) Stiff.=10000,Cff. lat. press.=1.75,Frict.ang.=23,21,19 30000 0 Lateral displacement (mm) 1 0 Lateral displacement (mm) Figure B.3 Shape of the Lower Portion of the Pile and Hole at the 750 Displacement Step th Appendix B.4 B: Listing of PILE, Sample Input File, Plots and Worksheets 1 Sample Worksheets Energy Dissipation Computation Sheet Variation in the Angle of Wall Friction - Pile a) Setting units mm := 10" -m kN := 10 g k g 3 kN = 9.807 ^ l O 3 ^kg-m's "E:/users/nii/pilecyc/coeff/m4k4r8c.dat" "E:/users/nii/pilecyc/coeff/m3k4r8c.dat" "E:/users/nii/pilecyc/coeff/m2k4r8c.dat" "E:/users/nii/pilecyc/coeff/mlk4r8c.dat" filenames "E:/users/nii/pilecyc/coeff/m4k6r8c.dat" "E:/users/nii/pilecyc/coeff/m3k6r8c.dat" "E:/users/nii/pilecyc/coeff/m2k6r8c.dat" "E:/users/nii/pilecyc/coeff/ml k6r8c.dat" b) Read and compute the energy for different files Energy := for k s 0.. 7 data «— READPRN A«—data (filenames k < 0> F^data* ' ' n«—last(A) n - 1 energydissp Z IK i= 0 Dissip «— energydissp k Dissip C) View the Results 556.2825 556.48653 556.23913 Energy = 555.81743 562.94311 566.22613 572.93733 588.75679 •kN-m ( F , + . + F i •kN mm 2 Appendix B: Listing of PILE, Sample Input File, Plots and Worksheets 156 Computation of change in the angle of internal friction ) for sands, based on the formula given by Zeitlan and Paiskowskv Reference effective pressure Reference angles of friction Effective pressure at given depth: 100 45 q :=100 200 30 o 1 25 j:= 1..4 T 0 J,> - 5.5 log. • en. Coulmns represent decrease in angle of friction with depth. 45 30 25 20 43.344 28.344 23.344 18.344 41.689 26.689 21.689 16.689 40.72 25.72 400 600 20 i := 1.. 4 en' 20.72 15.72 Appendix C: Listing of EPILE, Sample Input File and Worksheet C APPENDIX C: LISTING OF EPILE, SAMPLE INPUT FILE AND WORKSHEET C.1 Listing of EPILE 157 EPILE is made up of the following modules (subroutines) • Main Program • PSUP module (Computes the foundation pressure at a point along the fastener) • DECOMP module (Performs Cholesky decomposition) • SOLV module (Solves resulting decomposed equations) • SEARCH module (Computes a new value for line search parameter) • FRICT module (Computes the frictional force at a point along the fastener) • STRESS module (Computes the stress at a point in the beam) • SHAPES module (Computes value of shape function at a given integration point) • GAUSS module (Provides Gauss integration point values) The listing for all these modules apartfromthe Main Program and PSUP, are the similar to that given in Appendix A. 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IX—PITJ HI PIH HH H PI M O 0 0 0 0 0 0 2 O — 0 U 3 D D H Hi PI a• XT O < <T> - 3 I- CD TJ Q> H 1 G PJi Z **• T »• H II « ra ci n z H to o o ra s o HI n o ra H M > ra M s tr TJ Cl >-3 — H — * HI tn II £ ^ - 2 ra M Hi ii tn a H n u i w a - z H* H M i • it. Ln — ra cn X to o to ra — pg rq ra o cn — G— HOi OX HIra HI a ra M S • * — I I O2 O M5; ?S 2 H D H-3 ra + - i - Ln ! 3 PJ Ln - n r o ra no o o z z M M o 3 - m — ra - s HIraLn X ;) — oOz• LO Appendix C.2 C: Listing of EPILE, Sample Input File and Worksheet SAMPLE INPUT FILE PILE UNDER DYNAMIC LOAD PROBLEM (Units: Forces - kN; Distances - mm) ! Title Pile .' Pile Length .300000E+05 .' Type of 2 Cross-section .' Inner Diameter Outer Diameter .336000E+03 .356000E+03 .' Mod. Elast. .200000E+03 Yield Stress .250000E+00 .' Number of Layers 3 3 Number of Types of Layers / Different Embedment Propeties of Layers ! Type 1 ! Unit Weight Rel. Density Coeff. Lat Press. .200000E-07 .750000E+02 .8000000E+00 .' Coeff. Frict .424440E+00 Stiffness .500000E-01 / Type 2 ! Unit Weight .200000E-07 Rel. Density .750000E+02 .' Coeff. Frict .383860E+00 Stiffness .500000E-01 .' Type 3 '. Unit Weight .200000E-07 Rel. Density .750000E+02 .' Coeff. Frict .363970E+00 Stiffness .500000E-01 of Interface Coeff. Lat Press. .8000000E+00 of Interface Coeff. Lat Press. .8000000E+00 of Interface .' Layer 1 Properties ! Layer number First Element 3 1 4 .50000E+04 Last Element Layer Thickness .' Layer ! Layer 2 5 2 Properties number First Element 10 .10000E+05 Last Element Layer Thickness .' Layer ! Layer 1 11 3 Properties number First Element 20 .1S000E+05 Last Element Layer Thickness .' Number of Gauss Points ! X-Direct. y-Direct. 5 16 .' Axial Load .750000E+03 .981000E+04 Appendix C: Listing of EPILE, Sample Input File and Worksheet 170 • Initial Stiffnesses of the System in both Directions and Corresp. ! Horiz. Stiff. Damp. Ratio Vertical Stiff. Damp. Ratio .630000+02 .500000E-01 .460000E+03 .500000E-01 .' Number of Nodes with Fixed BCs 0 .' Node Number for Input 21 1 2 Disp. .' Displ. Tol. Factor Force .100000E-02 .100000E-03 Code for Direction Tol. Factor of Disp. Damping Code for Units Ratios of Dist Appendix C.3 C: Listing of EPILE, Sample Input File and Worksheet Sample Worksheet Frequency Spectrum of Earthquake Excitation Landers Joshua Tree Station Record Input parameters: Original Signals: N:= 4000 accin:= "E:\users\nii\piledyn\extra\josh2.txt" a := READPRN(accin) N=4M0 n:=l., N At := 0.02 At = 0.02 3 tim^ := At(n- 1) Accjjri^ := a Extend Acc ori to power of 2: n .= N +• 1.. Npts P Npts := 2 Acc_or^ := 0.0 1 Af:=— T2 T=80 n 2 := ceil FFT of the original acceleration: frequency resolution: T := AtN T2 := NptsAt Npts = 4.096-10 p2 3 T2 = 81.92 AoFFT := cffi( Acc_ori) Af= 0.012 n.-l.. Npts Npts k:=—— +1 2 f:=Af(n-l) " Absolute values of FFt accelerations: AoFFT 0 1 2 3 4 f n 5 6 7 8
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A model for the response of single timber fasteners and piles under cyclic and dynamic loading Allotey, Nii Kwashie 1999
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Title | A model for the response of single timber fasteners and piles under cyclic and dynamic loading |
Creator |
Allotey, Nii Kwashie |
Date Issued | 1999 |
Description | A model is presented for the cyclic and dynamic analysis of timber fasteners and pile foundations. Both systems are, in that they are beam-like structures that are embedded in a flexible medium. In the past, many different numerical methods, ranging from very elaborate finite element models, to closed-form solutions for beams on elastic foundations have been used to analyze these systems. Among these, in the development of computer software, the use of a 1-D finite element implementation of beams on a nonlinear foundation has been shown to be the most promising, mainly due to the relative accuracy and simplicity of the model. In this thesis, such a model is developed for the analysis of both systems. The model is based on one previously developed, which used a beam element on a nonlinear flexible foundation that takes into account the formation of gaps between the fastener/pile and medium. In this thesis, the effect of interface friction, and the initial confining pressure surrounding the fastener/pile has been included in the model. Due to the inclusion of interface friction, the model allows, in the case of piles, the study of problems with horizontal and/or vertical acceleration inputs. The model is the basis for the development of a cyclic analysis code for single timber fasteners, and cyclic and dynamic analysis codes for single piles in cohesionless soil. Various numerical examples of timber fasteners and piles are used to show that the predicted results of the model are consistent with expected response behavior. Since previous models have not taken either friction or initial confining pressure into account, the effect of both of these on the predicted response of both systems has been investigated. It is seen from this study that the amount of initial confining pressure around a timber fastener does not significantly influence the hysteretic response under lateral cyclic loading, but rather influences the amount of withdrawal in the case of driven fasteners. For estimated practical values of the coefficient of friction and tangential stiffness of the interface, friction is seen to have a significant influence on the amount of withdrawal, and not much influence on the hysteretic response. In the case of piles, neither the initial confining pressure nor friction are observed to have any considerable effect on the hysteretic response under lateral cyclic loading |
Extent | 10627551 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064140 |
URI | http://hdl.handle.net/2429/9249 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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