Dynamic Analysis of Guyed Towers Subjected to Wind
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DYNAMIC ANALYSIS OF GUYED TOWERS SUBJECTED TO WIND LOADS INCORPORATING NONLINEARITY OF THE GUYS by ROHIT KAUL, B.E. A THESIS IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN CIVIL ENGINEERING Approved August, 1999 ACKNOWLEDGEMENTS First of all, I would like to express my sincere gratitude to Dr. C. V.Girija I Vallabhan, chairman of my committee, whose depth of knowledge and enthusiasm introduced me to the field of nonlinear mechanics and inspired me to put in my best. His exceptional perception for accurate structural modeling and analysis has been invaluable to me throughout my research. I would also like to thank Dr. Vallabhan for closely working with me and sharing with me his insights for more suitable solutions. I would like to thank Dr. K. C. Mehta for funding my research. I would also like to thank him for providing guidance throughout the course of my research and for providing valuable suggestions that helped in shaping my thesis. In particular, Dr. Mehta's willingness to help at all times was greatly appreciated. I extend my appreciation to Dr. P.P. Sarkar for providing insights in the wind engineering field and for his help in the review of the thesis manuscript. I would like to thank Mr. John Schroeder and other colleagues for providing the field wind data and their cooperation. My sincere appreciation goes to my wife, Rajeswari for her patience and help in the thesis documentation. Finally, I would like to dedicate my thesis to my parents, who have always encouraged me to set high academic goals. n TABLE OF CONTENTS ACKNOWLEDGEMENTS n LIST OF TABLES VI LIST OF FIGURES vn CHAPTER 1. INTRODUCTION 1 1.1 Overview of Guyed Masts 1 1.2 Literature Review 3 1.3 Analysis Procedures 4 1.3.1 Mast Analysis 4 1.3.2 Cable Analysis 6 1.4 Objectives 9 1.5 Plan of Development 11 2. WIND LOADS ON GUYED TOWERS 12 2.1 Introduction 12 2.2 Wind Characteristics 12 2.3 Transformation of Wind Speeds to Wind Loads 13 3. CABLE ANALYSIS 18 3.1 Classical Solution of Cables 18 3.1.1 Analysis Procedure 18 3.1.2 Description of Sample Cables and Results of Classical Analysis 11 HI 3.2 Cable Analysis using Finite Element Method 31 3.2.1 Continuum Formulation 31 3.2.2 Linearized Approximation for Cable Analysis 33 3.2.3 Nonlinear Cable Analysis 37 3.2.4 Comparison of Classical and Finite Element Solutions for Cables 38 4. ANALYSIS OF GUYED MAST SYSTEM 43 4.1 Analysis of Mast as a 3-D Truss 43 4.2 Static Analysis of Guyed Masts 46 4.3 Dynamic Analysis of Guyed Masts 56 4.3.1 Response to 3 Second Mean Wind Speed Gust 59 4.3.2 Steady State Response to Turbulent Wind 62 5. COMPUTATIONAL TECHNIQUES 71 5.1 Equation Solvers 71 5.1.1 Modified Half-Band Solver 72 5.1.2 Modified Skyline Solver 74 5.1.3 Linked List Based Solver 75 5.2 Solution for Nonlinear Iterative Problems 78 5.3 Object-Oriented Programming 81 5.3.1 Class Definition 82 5.3.2 Classes Developed and Solution Methodology 83 6. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 85 6.1 Summary 85 iv 6.1.1 Analysis of Cables 86 6.1.2 Analysis of Mast 86 6.1.3 Cable Mast Interaction 87 6.1.4 Object Oriented Programming 87 6.2 Conclusions 88 6.2.1 Analysis of Cables 88 6.2.2 Analysis of Mast 89 6.2.3 Cable Mast Interaction 90 6.2.4 Object Oriented Programming 92 6.3 Recommendations 92 REFERENCES 94 APPENDIX A. LISTING OF SPECIFIC FUNCTIONS 100 B. LISTING OF CLASS DEFINITIONS 107 C. INFLUENCE OF VARIOUS PARAMETERS THAT AFFECT THE FORCE COEFFICIENTS FOR WIND LOADS ON GUYED TOWERS 115 LIST OF TABLES 3.1 Description of the Physical Properties of Cables 23 3.2 Calculated Initial Properties of the Cables: Ho, So and S 24 4.1 Member Properties of the Mast 49 4.2 Statistical Properties of Steady State Response, 10 m/s Wind at 10 m Height 68 4.3 Statistical Properties of Steady State Response, 25 m/s Wind at 10 m Height 70 VI LIST OF FIGURES 1.1 Schematic of a Typical Guyed Mast Tower 2 1.2 Patch Loads for Calculating Support Moments 5 1.3 Equivalent Spring-Mass Notation for a Cable 7 2.1 Mean Wind Speed profile 14 2.2 Wind Load on the Mast and Cable due to Mean Wind Speeds 17 3.1 Catenary Cable Profile Under Self-Weight 18 3.2 Variation of Horizontal Component of Cable Tension in 22 Each Iteration 24 3.3 u-w Displacements of a Cable in X-Z Plane 25 3.4 Horizontal Force Versus u-w Displacements at the Top End of Cables 27 3.5 Vertical Force Versus u-w Displacements at the Top End of Cables 29 3.6 Horizontal Secant Stiffness for Cables 33 3.7 First-Order Isoparametric Cable Element 39 3.8 Comparison of Finite Element Analysis with Classical Solution 41 3.9 Influence of Lateral Wind Loads on the Horizontal Cable Force 43 4.1 X-Braced Truss Configuration of Guyed Masts 44 4.2 Bar Element in Three Dimensional Coordinate System 47 4.3 Nodal Numbering for Tightly Banded System Matrices 47 4.4 Member Numbering for Tightly Banded System Matrices 4.5 Orientation of the mast in X-Y Plane 48 4.6 Member Forces in Mast Leg, with no Loading 49 vn 4.7 Member Forces in Mast Diagonals, with no Loading 50 4.8 Member Forces in Mast Posts, with no Loading 50 4.9 Mast Deflection under Static Wind Loads 51 4.10 Member Forces in Mast Legs 52 4.11 Member Forces in Mast Diagonals 52 4.12 Member Forces in Mast Posts 53 4.13 Mast Deflection under Static Wind Loads, Considering the Effect of Wind Forces on the Cables 54 4.14 Member Forces in Mast Legs, Considering Wind Forces on the Cables 55 4.15 Member Forces in Mast Diagonals, Considering Wind Forces on the Cables 55 4.16 Member Forces in Mast Posts, Considering Wind Forces on the Cables 56 4.17 Dynamic Response of the Mast to a 3-Second Gust 59 4.18 Wind Speed Field Data 63 4.19 Dynamic Response to 900 Second Turbulent Wind 64 4.20 Steady State Response of the Mast to Turbulent Wind 67 4.21 Steady Sate Response at 295.13 m Level 69 5.1 Data Storage in Modified Half Band Solver 73 5.2 Data Storage in Modified Skyline Solver 74 5.3 Comparison of Storage Techniques using Half Band, Modified Half Band and Modified Skyline Solvers 75 5.4 Data Storage in Linked List Format 76 5.5 Sample Matrix Stored in a Linked List Format 77 vni 5.6 Linked List Element 77 5.7 Subroutine for Initializing Linked List Array to a Quantity 78 Equal to 'value' 79 5.8 Guyed Mast Stiffness Matrix with Efficient Nodal Numbering 84 5.9 Hierarchy of the Member Classes Developed 84 5.10 Member Objects of the Class Cmast IX CHAPTER 1 INTRODUCTION 1.1 Overview of Guyed Masts Guyed masts are unique civil engineering structures, structurally efficient, self- supporting lattice towers. High structural efficiency of guyed towers is achieved by the use of pre-tensioned cables and a skeletal design. The height of guyed masts can exceed 600m (Sparling, 1995), therefore they are extensively used by telecommunication industry. Guyed towers also have the highest failure rate. Since 1959, there have been 100 confirmed collapses of guyed towers in United States (Madugula, 1998). Failure of guyed tower results in significant economic loss and human inconvenience. This report emphasizes on the study of guyed towers subjected to dynamic wind loads. A schematic of guyed tower is shown in Figure 1.1. Typically the guyed mast is constructed as a triangular space truss with warren or cross-braced configuration. The mast is pinned or fixed at the base while the top usually supports an antenna. Pre- tensioned cables, radiating symmetrically from the mast at several elevations, provide lateral support to the mast. Traditional techniques for the analysis of guyed towers rely on pseudo-static analysis and are insufficient because of significant secondary effects and nonlinear behavior of cables. Due to overall flexibility, slendemess and lightweight, guyed masts are susceptible to large deflections and also exhibit high dynamic sensitivity to turbulent winds. As a result, dynamic analysis is considered imperative for calculating the peak axial forces in the mast. Other environmental factors like icing and snow accumulation on 1 cables can significantly enhance the mast response, sometimes resulting in structure failure. Figure 1.1. Schematic of a Typical Guyed Mast Tower. 1.2 Literature Review Significant time and effort has been spent in the study of guyed towers, especial 1> towards the study of nonlinear interaction of guys with the mast. In this report, the word 'guys' is also referred to as 'cables'. Irvine (1981) has summarized both static and dynamic analysis of cables. A number of authors have also addressed the problem of cable dynamics (Triantafyllou, 1981; Veletsos and Darbre, 1983; Starossek, 1991). Details of experimental work on nonlinear cable behavior are also available (Zui. Shinke and Namita, 1996; Russell and Lardner, 1996; Tan and Pellegrino, 1997). In recent times, notable progress has been made to study the wind effects on guyed towers and cable mast interactions (Nakamoto, 1985; Gerstoft and Davenport, 1986; Issa and Avent.