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A NONLINEAR DYNAMIC ANALYSIS OF THE TRAFFIC SIGNAL CABLE SYSTEM By ADEOLA KEHINDE ADEDIRAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997 To him who is, and who was and is to come the Almighty. "For I know the plans I have for you, declares the Lord, "plans to prosper you and not to harm you, plans to give hope and a future. " Jeremiah 29:11 ACKNOWLEDGMENTS The author would like to express profound gratitude to Dr. Marc Hoit and Dr. Ronald Cook for their unwavering support throughout the length of this doctoral program. Their affable personalities maintained an invaluable open door policy to students that has been most beneficial to this author. Special thanks also go to Dr. Fernando Fagundo, Dr. John Lybas and Dr. James Hobert for their various contributions to this dissertation and their professional guidance. It has indeed been an honor to have worked with such a distinguished caliber of lecturers. Other people that have proven to be a tremendous inspiration and support throughout this dissertation include Dr. Duane Ellifritt for, among other things, helping to obtain some needed reference articles; Petros Christou, for his help with programming; Dr. Glagola for all the friendly advice and Mrs. Clara Osinulu for the idea and encouragement to go for a doctorate degree. The author will also like to thank fellow graduate students for the moral support, the Engineering Minority office and the American Association of University Women for financial aid at various times in the course of this doctoral program. Finally and more important, the author would like to express gratitude to God, the author's mother, Adenihun, sisters, Adeoti and Bolaji and brother, Bayo for all their loving support. TABLE OF CONTENTS ACKNOWLEDGEMENT ........................................ iii LIST OF FIGURES ............................................. vii LIST OF TABLES .............................................. ix ABSTRACT .................................................... xi CHAPTERS 1. INTRODUCTION .............................................. 1 Problem Definition ......................................... 1 The Structure ........................................ 1 Theory ............................................ 3 The Computer Program .................................. 6 Overview .................................................. 6 Content ............................................. 7 Scope ............................................. 8 2. LITERATURE BACKGROUND ................................ 10 Introduction .............................................. 10 Fundamentals of Cable Analysis ............................... 10 Cable Dynamics .......................................... 18 Closed Form Solutions ................................. 19 Finite Element Method ....................................... 22 Formulation of Element Properites .............................. 23 Beam Properties ................................... 23 Truss Element Properties ................................ 27 Cable Element Properties .............................. 29 Connector Element .................................... 36 Transformation Matrix ................................. 46 Damping Matrix ................ ..................... 50 Dynamic Solution M ethods ................................... 53 Explicit Methods ...................................... 54 Implicit Methods ...................................... 55 Newmark Beta Method ....................................... 56 3. THEORETICAL DEVELOPMENT ............................... 59 Introduction ................................................ 59 Two Node Nonlinear Cable Element ................... ...... 61 Equation of M otion .................................... 73 Modified Newmark Beta Method ......................... 77 Modified Newton Raphson ............................. 81 Solution Algorithm ........................................ 86 Solution Flowchart ........................................ 89 4. LOADING ............................................... 91 Introduction .................................... .......... 91 Variable Area W ind Load .................................... 92 Variable Velocity Dynamic Wind Load ......................... 95 Autoregressive Filters ....................................... 103 Test Data ................................................. 107 5. PROGRAM VERIFICATION FOR STATIC LOADS ................ 113 Static Problem Definition .................................... 113 Single Cable Structure Example ....................... 114 Static Test System Configuration .................... ... 116 Loading Sequence of Tests Performed ............... 120 Test Results and Predicted Values .................. 123 Discussion of Results ............................ 130 6. PROGRAM VERIFICATION FOR DYNAMIC LOADS .............. 144 The Structure's Configuration ............... .............. 144 Coordinates of Wind Load Test System ................... 146 Wind Test Setup ........................................ 147 Description of Wind Load Test ......................... 148 Comparison of Results ................................. 149 Test Data ....................................... 150 Wind Load Modeling .................................. 153 7. CONCLUSION ................ .............................. 156 Summary ................................................ 156 Review of Analysis Method Developed ................... 157 Verification of Analysis Method ......................... 159 Advantages and Disadvantages of Solution Method ............... 159 Recommendations for Further Research ......................... 160 APPENDICES A. CABLE ELEMENT DERIVATION ............................. 162 B. USER'S MANUAL FOR DYNASS ............................ 179 LIST OF REFERENCES ........................................ 221 BIOGRAPHICAL SKETCH ..................................... 225 LIST OF FIGURES Figure page 1.1 Traffic Signal System ................... ....... 2 1.2 Static load model ..................... .......... 4 1.3 Mean and time varying components of wind Speed .......... 5 2.1 Cartesian Coordinate... ....... ... ............. 11 2.2 Displacement of a cable segment . ... .. 12 2.3 Shape of cable due to selfwt.. . ..... .... 14 2.4 Notations for Eqn. 212 ................. ........ 21 2.5 Element degrees of freedom for a beam element . .. 24 2.6 Degrees of freedom for a truss element . ... 27 2.7 Truss element showing direction cosines . ... 28 2.8 Straight nonlinear truss element used for cables . ... 30 2.9 Deformed length due to displacements . ..... 31 2.10 Joint displacement and corresponding end forces . .... 37 2.11 Relative deformation and associated forces . ..... 38 2.12 Rigid body and natural modes for large displacement beams .. 39 2.13 Rigid body rotations .............. .......... .. 41 2.14 Relative nodal rotations . . . 43 2.15 Euler angles for transformation . . 48 3.1 CableElement. ................ ........ ...... 61 3.2 Newton Raphson Iteration . . .. 82 3.3 Flowchart of solution algorithm . . 90 4.1 Resolved wind load on signals . .. .. 92 4.2 Lift and Drag coefficients .. ....... ...... ..... .. 94 Figure page 4.3 Distribution of sample generated . .. .. 98 4.4 Spectral density of fluctuating part of wind . .. .. 101 4.5 Numerical filteration process .. ..... ..... .. 105 4.6 Wind speed generated using the spectral density function in Eqn. 410 .. 106 4.7 Wind load test systems with final cable tensions ..... 108 4.8 Spectral Density (All wind speed ranges) . .. . 109 4.9 Spectral Density (Low wind speeds). . ..... 110 4.10 Spectral Density (Medium wind speeds) . .... 111 4.11 Spectral Density (High wind speeds) . . 112 5.1 Single Cable Problem ..................... .......... 114 5.2 Static Load Test System Dimensions and Node Numbers. ..... 118 5.3 Cable tensions Static Test # 1 . . .. 133 5.4 Cable tensions Static Test # 2 . ..... 135 5.5 Cable tensions Static Test # 3 . . 137 5.6 Dynamic Structural Response at Catenary Node 6 (Test 1) ...... 141 5.7 Dynamic Structural Response at Catenary Node 6 (Test 2) .. .... 142 5.8 Dynamic Structural Response at Catenary Node 6 (Test 3) .. ...... 143 6.1 Wind Load Test System Dimension and Node Numbering .. ..... 145 viii LIST OF TABLES page Table 51 52 53 54 55 56 57 58 59 510 511 512 513 514 515 516 517 518 519 61 Dimensions and node coordinates for Fig. 6.1 . Displacement Comparison for Suspended Cable . ... 115 Stress Comparison for Suspended Cable ........ .. 115 Section Properties for Members shown in Fig. 5.2 .. ... 117 Dimensions and node coordinate for Fig. 5.2 .. ..... 119 Loading Sequence for test # 1 . . 120 Loading Sequence for test # 2 . .. 121 Loading Sequence for test # 3 ........... ..... 122 Catenary Cable Results Comparison for Test # 1 . 124 Messenger Cable Results Comparison for Test # 1 .. ... 125 Catenary Cable Results Comparison for Test # 2 .. 126 Messenger Cable Results Comparison for Test # 2 .. ... 127 Catenary Cable Results Comparison for Test # 3 .. .. 128 Messenger Cable Results Comparison for Test # 3 .. ... 129 Test 1 Cable Tensions . . 132 Test 2 Cable Tensions . .. . 134 Test 3 Cable Tensions . . 136 Displacement Comparisons for Test 1 ..... . 138 Displacement Comparisons for Test 2 .... 139 Displacement Comparisons for Test 3 .. . ... 140 . 146 62 Section Properties for Members shown in Fig 6.1 ... .. 148 63 Wind Data for Low Wind Speed Test ........ .. .. 150 64 Wind Data for MediumWind Speed Test ... .. 151 65 Wind Data for High Wind Speed Test ....... .. 152 66 Wind Speeds and Resulting Drag Forces used in Dynamic Analysis. 153 67 Cable Tensions for Wind Load Test # 1 ....... 154 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A NONLINEAR DYNAMIC ANALYSIS OF THE TRAFFIC SIGNAL CABLE SYSTEM By ADEOLA KEHINDE ADEDIRAN May 1997 Chairperson: Marc I. Hoit Cochairperson: Ronald A. Cook Major Department: Civil Engineering Department Finite element analysis of linear structures is an increasingly popular tool of choice for analysis by engineers. Only recently has this tool been applied to the case of nonlinear cable structures with a sufficient degree of veritable accuracy. Work in this dissertation provides a finite element approach to the analysis of cable structures with verification of the approach done for a traffic signal cable structure. There has been a preponderance of work done on the dynamic analysis of single cables and cable networks but very little is known about dynamic analysis of nongridlike cable elements in conjunction with other elements such as beams and trusses. The approach of analysis presented in this dissertation xi provides a valuable bridge in available theories by creating a program that dynamically analyzes a cable in context of other types of elements also present within the structure. Using the principle of virtual work, the cable element developed in this dissertation is based upon a total Lagrangian formulation. The cable element models geometric nonlinear capability. Also modeled in the program developed is a nonlinear loading capability. A realistic wind load generation was accomplished by the use of autoregressive numerical filters. With a complete finite element model sufficiently exhibiting the elastic and kinematic properties of the true structure, dynamic analysis was accomplished by numerical integration of the equation of motion. An incremental form of the general equation of motion is used because of a piecewise linear assumption. The numerical integration scheme presented in this dissertation is the Newmark's Beta method. The Newmark's Beta method reduces the equation of motion to a set of nonlinear algebraic equations. These equations are then solved using a modified Newton Raphson. The finite element approach presented in this dissertation is programmed into a software, DYNASS. Verification of the program DYNASS and subsequently the solution approach is done by comparing predictions from DYNASS to test results obtained from recent traffic signal experiments performed at the Civil Engineering Department, University of Florida. CHAPTER 1 INTRODUCTION Problem Definition The use of cables as loadcarrying elements is common for structures such as bridges, cable reinforced membrane roofs, traffic signal systems, spatial cable nets and mooring lines. The set of forces seen in each cable application is different but there is a common trend in the behavior of these structural systems. This behavior is nonlinear. One source of nonlinearity is the large displacement the cables undergo. This type of nonlinear behavior is known as geometric nonlinearity. Several theories have been proposed to handle this problem. One such theory is presented in this report. The Structure The structure investigated here is the traffic signal system. The choice of this type of cable stayed structure was a result of the recent work done for the Florida Department of Transportation. The Civil Engineering Department at the 2 University of Florida developed the software program, ATLAS, to perform a static analysis of a twopoint connection traffic signal system as shown in Fig. 1.1. The restrictions encountered during developing this program spurred the need for another analysis approach. The typical system considered consists of a catenary cable (slack cable configuration), a messenger cable (taut cable configuration), the signal heads and signs (points of dead and wind load application), and the prestressed concrete poles (as the support system). Catenary Cable Prestressed concrete pole Connection hanger { ... ...... . _< _ Disconnect box A A Messenger cable Signal head Fig. 1.1 Traffic Signal System The traffic signal system considered includes that shown in Fig. 1.1 but is not limited to the twopoint connection traffic signal system shown. The method presented in this research is capable of handling virtually any system of cables and frame elements, for example, the onepoint connection traffic signal system and the suspended box system. The Theory The approach presented here uses the theories of structural dynamics to model the behavior of cables subjected to both dynamic wind loads and static loads. The static load case can be considered a special case of the general dynamic model. The static load is modeled as a constant load with duration equal to the full time of analysis. The structural response often of interest in the static load case is the damped out response, i.e. the stresses, moments, forces and deflections at die down. For this response the initial rate of application of load is inconsequential. It is, however, good practice to maintain a slow ascent to full load in order to minimize impact effects of the load and to obtain a dynamic response more reflective of a stationary load. P / pp/ o / t> 3 mins time Fig. 1.2 Static load model The wind load is modeled as both static and dynamic. The equivalent static wind load as determined by AASHTO (American Association of State Highway Officials, "Standard Specifications for Structural Supports for Highway Signs, Luminaries and Traffic signals") is applied as a horizontal force (called the drag force) and a vertical force (called the lift force). An aspect of nonlinear behavior of traffic signal cable is the nonlinear wind loading. The research work presented provides a dynamic wind model comprising a steady wind pressure and a variable wind component. The variable wind component models wind gust effects. A wind gust is a short duration wind turbulence often characterized by its frequency content and its duration. Fig. 1.3 shows a typical wind gust representation. Vw W t L ,t Fig. 1.3 Mean and time varying components of wind speed IM 6 The dynamic theory presented uses geometric and time discretization. The geometric discretization or discrete element technique employed is the finite element method. Numerical integration is used in the solution of the equations of motion in the discrete time domain. Generally a nonlinear response analysis involves a significant amount of computation as it was in this case. Consequently, a computer program was written. The Computer Program The resulting program is called DYNASS. This program performs a nonlinear dynamic analysis of the traffic signal cable system. This program is written in Fortran with some wind generation routines in C. The results of this program are compared with results from other programs where possible and with test results from full size traffic signal system tests. Overview Each chapter in this report discusses topics or themes built upon in subsequent chapters. Topics discussed start from the theory proposed to the program written, to the comparison of the results from the program with test 7 results and finally to the typical applications and conclusion. Figures and formulas are numbered consecutively throughout each chapter. Content Chapter two presents a literature background. This includes an overview of existing work done on this topic and an explanation of the fundamentals of structural dynamic theories and cable analysis theories. Chapter three contains a detailed presentation of the theories and methods developed in this research. These include the various assumptions made and the equations derived. This chapter is essentially subdivided into two parts. The first part is the theoretical development just mentioned, and the second part discusses the algorithm used in the program, DYNASS. Chapters four discusses the wind load models developed plus a general description of the other loads on the traffic signal system. There are two wind models formulated, the variable area wind load and the variable velocity wind load. Both models are discussed in some detail in chapter four. Finally, chapter four presents a brief discussion of the wind load data obtained from dynamic experiments performed. These experiments were conducted previously for a Florida Department of Transportation sponsored research. Chapter five and six present comparisons of the results from DYNASS to other software, where available, and to test data obtained from experiments done. Chapter five compares results for static load cases only and chapter six deals with the comparison of results for dynamic wind load cases. Each chapter is concluded with a discussion of strengths and limitations of either the theory proposed and/or the program developed, for each load case. The final chapter, chapter seven, deals with conclusions observed from the research and recommendations. Areas needing further research are also identified and noted. Also included are Appendices A and B. Appendix A shows a detailed derivation of the 2node cable element developed in this research. Appendix B gives a user's manual of the computer program developed. In appendix B, defaults assumed in the program are explained and for portions of the program not developed by the author, references are provided. Scope The scope of the research presented is limited by the inherent assumptions made in the cable analysis. The assumptions made, common to other types of cable analysis, are as follows: + The cables are perfectly flexible and possess no bending stiffness, only axial stiffness. * Cable materials are of hookean materials. Ferrous materials are commonly used for cables. * Only small strains are involved, i.e. no yielding of cables ever reached. By the nature of the type of structure investigated (traffic signal systems), the loads are essentially predefined. For this research, loads examined are restricted to the dead load from the weight of the signal heads and signs, and wind loads. Other assumptions discussed in chapter four further define the application of the theory and subsequently the program. CHAPTER 2 LITERATURE BACKGROUND Introduction Several authors have dealt with the problem of the geometric nonlinearity of cables and have provided some solutions to the problem. A brief summary of pertinent results of their work is given in this chapter as they provide a foundation to the author's work. Also presented is a general theoretical background of cable analysis and dynamic analysis. The work covered here reflects only the solutions provided for cable dynamics. It is, however, congruent to consider the static cable problem as a special case of the cable dynamics problem hence extending these solutions to cover static cable analysis. Other methods strictly for static solutions include the force density method [1]. Fundamentals of Cable Analysis Consider a uniform cable which hangs in static equilibrium in a vertical plane suspended between two supports. The forces acting on a differential element of the cable is dependent of the profile assumed by the cable atrest position and the cable final deformed position. Let any point on this cable be described in the 3D Cartesian coordinates shown in Fig. 2.1. z (w) Y (v) x (u) Fig. 2.1 Cartesian Coordinate Let the cable ordinate after deformation be denoted x, y, z as shown in Fig. 2.2. X=x+u y=y+v Z=Z+W The cable segment shown in Fig. 2.2 below can be described by a vector along the chord joining both ends of the cable segment. L= t+ Yi Y + j k ds ds ds The terms xj xi Yi and zj z are referred to as the direction cosines for ds ds ds the element. 00< ' I uj ,, i, Fig. 2.2 Displacement of a cable segment The relationship between the nodal forces and displacements at any point along the cable relative to the next is defined as follows, du uj = ui + s (21) This says that the displacement at nodej is the sum of the displacement at node i and the differential strain in the cable multiplied by the arc length of cable. Similarly for the displacements in the y and z direction, the relationship is dv v = v + 7ds (22) d s d w w = W + d s (23) The above displacementstrain relationship is referred to as the linear strain  displacement relationship. The relationship between the forces at node i andj both along the cable can similarly be summarized as follows, Tj = Ti + dT (24a) Fxj = Fxi + dFx (24b) Fyj = Fy + dF (24c) Fzj = Fzi + dF (24d) For static analysis the forces acting on a finite length of a cable are T = Cable tension Fx, Fy, Fz= equivalent nodal loads due to for example dead weight of cable and/or cladding or applied nodal load. In the "unloaded configuration" otherwise referred to as the null state, the applied load is taken as zero. The tension in the undeformed slack cable is computed from the cable profile. The basic shape of a slack cable subjected to its own weight is a catenary profile. However, if the slope of the cable is everywhere small, the profile adopted by the cable can be accurately described by a parabola. It is advantageous as a means of simplification to describe cable shape in two dimensions. Here the xz plane is used. w" SV2 L" Ti 1 x L Fig. 2.3 Shape of cable due to self wt. According to Ref. [2] the parabolic profile is given by Eqn. 25. 4hx z(x) = xtan(e) (Lx) (25) The tension in the cable at any point x along the span of the cable is as given in Eqn. 26. WL2 [i 4h+8hxl (26) T(x) w + tan(0)+ 8h L ,2 S = length of the cable along the sag (Fig 2.3). z (x) = vertical distance of any point along the cable relative the starting node. tan (0) = slope of the chord joining both ends of the cable. h = distance from the chord at midspan of the cable. This is used to define cable sag. w = weight of the cable per horizontal distance. L = span of the cable. T (x) = tension in the cable (null state) at any point along the cable. sag = h Comparing the above profile with the catenary profile in Ref. [3], the following equations form the catenary profile equivalent of Eqns 25 and 26 above. Note that the catenary shape is more difficult to define and indeed is only defined by an iterative process of trial and error. H x z(x)=H cosh(E)cosh(20 e) (27) tan (0) E = sinh [ tan( )+ 0 (28) wL S=2H (29) S WL tan(0) T (x)=H 1+ sinh sinh 2H wL wx (210) sinh wL 2H H 2 2H I H = horizontal component of the tension in the cable. This is invariant along the cable. The other variables retain their definitions given on previous page. There are two unknowns in the above equations, z(x) and H. The value of H is 17 often obtained by approximately setting z(x) to sag x span (L) at x = L/2. The rest of the configuration is obtained by using the H computed. It is apparent that it is more convenient to use the parabolic approximation. The error introduced by the approximation is of the order of 1% 2% for slack cables with small sag. For example, if the tension at the first third point and the z coordinate were required of the cable with the following properties: sag = 10%, w = 0.12 plf, L = 3500 ft, take tan(0) as zero i.e. both cable ends are on the same level. The results will be the following. Assuming the cable shape is parabolic  z (x = /3L) = 311.111 ft. And T (x = /3L) = 529.646 lbf. Using the catenary equations above the results  z (x = 1/3L) = 311.557 ft. And T (x = 1/3L) = 536.839 lbf. An error of just 1.4%. For the remainder of this dissertation, the parabolic (null) profile will serve as the reference state for the slack catenary cable. The null state shape is important in the derivation of the stiffness matrix of the cables as well as in determining the initial tensions in the cable. Any assumption about this initial configuration could significantly influence analysis results. Cables, like trusses, are idealized as not having any bending stiffness (in reality they possess some bending stiffness). This introduces equation of conditions in the equilibrium analysis of cables making them inherently unstable. In addition, they can undergo large displacements and cannot resist compression. These complexities invalidate the more conventional linear theories often based on the assumption of small displacements and unconditional stability. To solve the problem of inherent cable instability it is necessary to introduce additional resisting forces that can keep the cable in equilibrium. The problem of large displacement is best dealt with by applying the load in small enough increments as to achieve small enough displacements in each increment. Lastly, the inability of cables to sustain compression can be taken into account by solving the problem in steps and keeping track of the stresses in the cable. The cable elements are rendered inactive when in compression and active when in tension. A very logical method of solving a cable problem, therefore, is to use the theories of structural dynamics. It is advantageous to use this technique because it introduces inertia and damping forces that can be used to stabilize the cable and it provides a time step solution which allows the application of load in small increments and supplies a framework for keeping track of cable stresses. Cable Dynamics The solutions to the cable dynamics problem can be discussed in two major categories, closed form solutions otherwise referred to as the "exact" solution, and the finite element solutions i.e. explicit solutions such as the Euler's method, Central Difference method and RungeKutta method and the more common implicit methods such as the Wilson Theta method and the Newmark's Beta method. Closed Form Solutions Most closed form solutions, for taut cables and cables with small sag to span ratios, are based on a linear theory (i.e. small displacement theory). Irvine and Caughey [4], Uwe Starossek [5] and Veletsos and Darbre [6] have all presented semiempirical solutions based on this linear theory. Of importance is the experimental results of Irvine and Caughey [4]. Their work revealed the observed dynamic behavior of cables with small sag subjected to free vibration. When a cable vibrates it does so in two major motions. The transverse horizontal motion and the inplane motion. Both motions are observed to be uncoupled. The transverse motion is essentially a swinging motion as the cable rocks in and out of its atrest plane. The inplane motion consists of two components, the symmetric and antisymmetric components. The antisymmetric components consist of in line motion similar to a wave traveling along the length of the cable and inplane transverse motions which are unsymmetrical inplane deflections of the cable. The symmetrical components are analogous to simple harmonic motions occurring 20 along the cable. For small sag to span ratios (< 1:8) these components of motion constitute the modal shapes that could then be analyzed. The results of most of the aforementioned authors establish the mode shapes of cable vibration and their corresponding frequencies. The limitation of the linear theory is that for catenary cables with large sag the theory ceases to be accurate The theory also becomes problematic when the cable is considered as part of a structure and not standing alone. The equation of motion from linear theory is given in Eqn. 212. From Uwe Starossek [6], m g12 (211) H = s f H = Horizontal component of the cable tension m = mass of the cable per unit length g = acceleration due to gravity I = cable span f = sag of the cable The inplane equation of motion is for sag 1/20; d2w d2z d2w dw (212) H + hI = m + c dx2 dx 3t2 t dx ds X, U z, w I z dz T+dT Fig. 2.4 Notations For Eqn. 212 for a cable segment = damping force per unit length = displacement in the x, z directions respectively. dx hr = 't ds S = dynamic part of the total cable tension. This is invariant along the cable. An alternative to the exact solution is the discrete idealization of the continuum. There are two major techniques of discrete idealization of a structural c U, W continuum, the lumped mass method and the finite element method. The finite element method is more commonly used because, among other reasons, the numerical modeling has a clearer physical interpretation. With the finite method, the dynamic solution for cable vibrations are either explicit solutions or implicit solutions. Finite Element Method Due to the nonlinear behavior of cable systems, direct superposition of loads and displacements as would be necessary with the typical modal superposition analysis is not strictly valid. However, according to Leonard [7] and others, if an incremental load approach is adopted and the load increment is small enough, then within each step it is reasonable to assume linearity of the superposed dynamic response. An additional condition for a linearized solution is that the prestressed cable configuration prior to load application be known and that the cable system possess a significant stiffness. Using finite element analysis the traffic signal system is idealized as follows; the strain poles as beam elements, the cables as nonlinear cable elements and the connectors as large displacement beam elements. A common practice is the use of steel bracing for the poles. These braces could be modeled as linear truss elements. Formulation of Element Properties The properties of the linear beam elements are given below, their derivations can be found in most structural analysis books. The typical geometrically nonlinear truss element often used for cable analysis is also given in this chapter. The cable element adopted for this dissertation, however, is derived in chapter 3. Beam Properties According to Ref. [8] the stiffness matrix for the 3D beam element is given in Eqn. 213. ff' f6 j f1 i F y4 4 Zf. ff Fig. 2.5 Element degrees of freedom for a beam element. EAx/1 0 0 0 0 0 EAx// 0 0 0 0 0 0 12EI3/13 0 0 0 6EI3/12 0 12E13/13 0 0 0 6E13/12 0 0 12E12//13 0 6E12/2 0 0 0 12EI2/13 0 6EE12/2 0 0 0 0 GJ/1 0 0 0 0 0 G J/1 0 0 0 0 6E 1/12 0 4EI2// 0 0 0 6E12/12 0 2EI2/1 0 = 0 6E13/12 0 0 0 4EI3/1 0 6EI3/12 0 0 0 2EI3// K= EAx// 0 0 0 0 0 EAx/l 0 0 0 0 0 0 12EI3/13 0 0 0 6E13/12 0 12EI3/13 0 0 0 6E3//12 0 0 12EI2/13 0 6E12/12 0 0 0 12E2/13 0 6E12/12 0 0 0 0 G J/1 0 0 0 0 0 G J// 0 0 0 0 6EI2/12 0 2E12// 0 0 0 6E12//2 0 4EI2// 0 0 6E13/12 0 0 0 2EI3// 0 6E3/12 0 0 0 4EI3/l (213) E = Elastic Modulus (Young's Modulus) 12 = Moment of inertia about the strong axis 13 = Moment of inertia about the weak axis S = Length of the beam element The consistent mass matrix for the above 3D beam element is as follows [9]: [M] pAL 420 g 140 0 0 0 0 0 70 0 0 0 0 0 0 156 0 0 0 22L 0 54 0 0 0 13L 0 0 156 0 22L 0 0 0 54 0 13L 0 0 0 0 140/A 0 0 0 0 0 707 0 0 0 0 70 0 0 22L 0 54 22L 0 0 0 0 0 0 0 4L2 0 0 0 0 4L2 0 13L 0 0 140 0 0 13L 0 156 13L 0 0 0 0 0 0 0 3L2 0 0 0 0 3L2 0 22L 0 0 0 0 0 0 0 13L 54 0 13L 0 0 7A 0 0 13L 0 3L2 0 0 0 0 3L2 0 0 0 0 0 0 0 22L 156 0 22L 0 0 140 0 0 22L 0 4L2 0 0 0 0 4L2 (214) J = Polar moment of inertia A = Cross sectional area p = Density of beam material (lb/ft3) g = Acceleration due to gravity (32.2 ft/sec2) Truss Element Properties The degree of freedom for a typical truss element is shown in Fig. 2.6. f V Fig. 2.6 Degrees of freedom for a truss element The stiffness matrix for the truss is given in Eqn 215 [10]. This stiffness matrix is in the element local coordinate. 1 0 0 1 0 0 EA 0 0 0 0 0 0 [K] EAx 0 0 0 0 0 0 (215) 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 In the structure's global coordinate, the stiffness matrix is given in Eqn. 216. t Az // Ax Fig. 2.7 Truss element showing direction cosines C2 a cacr Ca Cr C2 a Ca Cy cnc r2 Ca cc ya SCP Cy C Ya C2 1r c2 a C C8 a Ca Cy C C cc c c Pia c2 O Cr AE [K] AE L c C ya qC2 1' (216) Ax L C Ay L 1' Az L L = J(Ax)2 + (A y)2 + (Az)2 The mass matrix in global coordinates is also given below [9]. [M] m 6 (217) (218) (219) = mass per unit length for the element = length of the element Cable Element Properties A straight truss element capable of large displacement is commonly used for cables. The derivation [3] of the stiffness matrix for this element is briefly 30 discussed here so as to be differentiated from that in chapter 3. Fig. 2.8 presents the straight element nonlinear truss in its local axis. u _+ su ds (220 I dsds s Fig. 2.8 Straight nonlinear truss element used for cables From Eqn. 21, the component of horizontal displacement is, du du (220) ui + sds ui cs vi +ds vi ds (221) wi + ds w ds (222) as ds The resultant axial strain along the member due to the above displacements can be computed by determining the deformed element length Ms. The deformed length due to displacement in the x direction is simply, du ds + ds ds (223) The deformed length due to displacements in the y and z directions are shown in Fig. 2.9. ds asds [(ds) +( ds)2 ds awds ads as (ds) +(aw s) Fig. 2.9 Deformed length due to displacements The axial strain from the above in the x, y and z directions is given in Eqns 224, 225 and 226. du ds + dsu x s du xd ds as dv ds2 + ds ds = = 1d+ ) Sds ds Applying the binomial expansion 1+a2 = l+la2 4 2 8 and neglecting the higher order terms Eqns 225 and 226 become S2 (+2 Y 2 TS 12 dv = l+ s 1 =2 Y 2 Os2 s (224) (225) (226) (227) E = l+, 1 = S= 2+ d21= I 2(228) z 2 (sJ 2 ds The total axial strain in the element is the summation of the strain expressions in Eqns 226, 227 and 228. This summation is referred to as the large deflection straindisplacement equation. 2 2du E = + L + l 2 (229) ds 2 ds) 2 ds If the nodal displacements are known for a finite length L of the nonlinear truss element, Eqn 229 becomes, lvV._i '2 i W Wi 2 E = u iv w, L (230) L 2 L 2 L Using Castigliano's theorem for the forcedisplacement relationship, the expression for the strain energy in the element is given in Eqn 231. U = 2JTA = 2 ds (231) 2 2 Substituting the expression for strain and expanding the brackets we obtain U AE +u) du v2+ du (dw2 1 (v4 1(Ov (w2 1 dw 4 + 2 + s ds + ++ ds 2 ds ss jds Es 4 ds 2das s +4 J ds (232) 34 Ignoring the fourth order terms and substituting the notation used in Eqn 230, we obtain a new expression for U. E j i u Ui Vj Vi Uj Ui ] 2i 2 (233) S2 (LL L L sL Note that, T AE (u i (234) Where T is the tension force in the truss element. Solving the integral we get Eqn 235. U = u) + A( i j vi) + "(Uj u wi)2 (235) Differentiating the Eqn 235 with respect to the nodal displacements gives the stiffness coefficient associated with that displacement. The expression for tension, from Eqn 234 above is substituted into Eqn 235. The assumption is made that the tension in the cable is approximately a constant. dU E (u,_ u) (236) du L (uu dU T i ) (237) d vi L U T Wi wj (238) dwi L U = AE ( i) (239) duj L ( dU = Ij ,) (240) dvj L 'i dU T(wj i) (241) dwj L .I In matrix form, the geometric stiffness of a nonlinear truss can be written as given in Eqn 242. The first matrix term below is the conventional elastic stiffness of a truss element. The second term is referred to as the geometric stiffness term. This geometric stiffness accounts only for initial cable stress and not for nonlinear strain displacement relationship. 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 EA 0 0 0 0 0 0 T0 0 1 0 0 1 [K]  +  L 1 0 0 1 00 L 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 00 0 01 0 0 1 (242) Connector Element The load on the traffic signal system is applied by the traffic signal heads and the traffic signs. These are attached to the system via connectors. The connectors have translational and rotational degrees of freedom. In this they are identical to the beam element discussed previously. The connectors, however, undergo large displacements. These displacements affect how the stiffness of the connector is added to the structure's overall stiffness. The connector stiffness matrix presented next is a beam element stiffness accounting for large displacements. The element presented here and adopted for this dissertation was derived by Chen and Agar [11]. The element stiffness is broken into the elastic component and the geometric component. The elastic stiffness is as given in the earlier described beam element, the geometric stiffness is set up in terms of the natural displacements (deformation) of the beam shown in Fig 2.11. The derivation of the geometric stiffness is given in detail in Ref. [11]. The derivation presented in Ref. [11] rely predominantly on work done by Bathe and Bolourchi [12]. In this dissertation, only the definition of the coefficients of the geometric stiffness and its application is described. The degrees of freedom for the large displacement beam element used as connectors is identical to that of regular beams. The resulting beam end forces corresponding to these degrees of freedom are shown in Fig. 2.10. To accurately determine the end forces another set of forces referred to as the natural forces or basic beam forces must be defined. The natural beam forces are the forces developed in the beam due to relative deformation of the beam ignoring rigid body motions. These forces are shown in Fig. 2.11. M\ Mz Myj V. Deformed Position Vyj Mxj \i ;Mzi J I Mv'W M V'i \w Sui Original Position Fig. 2.10 Joint displacements and corresponding end forces \/ Weak Axis Bending z 'o J k P M, p Mi 0' Myj y Mi / Strong Axis Bending Fig. 2.11 Relative deformations and associated forces From Fig. 2.10, the nodal degrees of freedom are, dIT = f "i VI W. xi yi zi Uj v wj Xj y j zj (243) { = { 1 I 9.9 u. w. .9 'J' (243) The nodal displacements in Eqn 243 consist of the two major components the rigid body component and the natural displacements (i.e. relative deformation) component. From Ref. [13] six rigid body modes and six natural displacement modes completely defines any displacement that may be seen in a large displacement beam. The identified modes are as given in Fig 2.12. Rigid Body Modes Sequence of modes H V dN x d z Y do, / Natural Modes z Y Xd V2 dV dNd V2 du X X. x dJ / A dx Md^y vLd^ Fig. 2.12 Rigid body and natural modes for large displacement beams 40 From Fig. 212 it can be seen that natural mode 1 can be defined by the change in length of the member i.e. axial deformation. The rest of the modes of deformation are bending deformations. These can be defined by the end rotations. The natural displacements (i.e. relative deformations) that are necessary for the computation of the end forces can therefore, be summarized as given in Eqn 244. Notations are shown in Fig. 2.11. U 0' . dN} = (244) zj 0' t The computation of { d } in Eqn. 244 is not intuitively obvious. For illustration of the process of computing the relative rotations, an assumption is made here that the connector is initially straight as shown in Fig. 2.13. The relative rotational angles are the total nodal rotations obtained from analysis, i.e. static or dynamic analysis minus any member rigid body chord rotations. In Fig. 2.13, it can be seen that the rigid body rotations about the y, x and z global axes are as follows: 1 d 1 dx S= sin Oy = sin d "dy2 +dz2 dx2 +dz2 0 projection 4  ed chord Position Position X Z plane projection  Fig. 2.13 Rigid Body Rotations = X, X = y, yi = Zj Z = dx2 + dy2 + dz2 d x d y dz I' 42 Using the cartesian coordinate system shown in Fig 2.1, clockwise rotation about the y axis is considered positive and counterclockwise rotation about the x axis is positive. Therefore when dx is positive (,) gets a positive sign but when dy is positive .x must be assigned a negative sign. The relative rotational deformation is then computed as shown in Eqn. 245 below. u I I yi yi y YJ (245) 6'z. 6 .O x 'zj zj x 6' 8  t xI xJ Eqn 245 above assumes that the nodal rotational displacements (with the nodal designation i orj) have been resolved to the local coordinate system but the chord rotations (() are in global reference. Nodal rotations are measured relative to the original orientation of the beam. Relative rotations are measured relative to the chord of the beam. T is the deformed length of the beam and I is the original length of the beam. To check the validity of Eqn. 245, consider the chord rotation and bending deformation of the next connector example. In Fig. 2.14, for ease of illustration the assumption is made that deformation exist only in the 2  dimensional, x z plane. Chord of deformed beam y axis y axis Undeformed Position ' / Deformed beam o' Rigid body chord rotation 8' Relative deformation 8 Nodal rotation Fig. 2.14 Relative Nodal Rotations Note that the nodal rotation at node (i) is clockwise hence positive, at node (/) the nodal rotation is counterclockwise hence negative. The chord rotation is positive at both ends. The resultant relative rotation at node (i) is smaller in magnitude but with the same sign as the nodal rotation, i.e. clockwise. The resultant relative rotation at node (j) is bigger in magnitude but with the same sign as the nodal rotation, i.e. counterclockwise. This conforms with results from 44 applying Eqn. 245. The natural end forces of the beam element, as shown in Fig 211, can be obtained by multiplying the above relative deformation vector by the basic beam stiffness, [kNE]. {Pu} [=NE{ d]N (246) Substituting the matrix terms into AE A 0 0 0o L 4EI 2EI 0 Y Y 0 L L 2El 4El 0 LY Y 0 IP = L  L \' N 4 4El 0 0 0 " L 2El 0 0 0  L 0 0 0 0 Eqn. 246, Eqn 247 is obtained. 0 0 0 2EI z L 4EI z L 0 0 0 0 0 0 GJ L o L, u 0' . yi 8' 8' . 8' zj t The end force vector computed from Eqn 247 can be seen in Fig. 2.11. P M. yL zN} i (248) M. zi M (247) The coefficients of geometric stiffness are obtained from the basic end forces. The geometric stiffness matrix for a large displacement beam according to Chen and Agar [11] is shown in Eqn. 249 below. k] = 70 0 0 a 0 a 0 e f g i 0 a 0 b  e f 0 a 0 0 a c f e 0 0 a SYMM. 0 k A I d g h 0 k l d 0 e f g n m 0 e f g i 0 f e h m n 0 f e h J i (249) The terms in the matrix above are as follows: e= Mt/l, f= P/10 = (Mzi+Mzj/6, h=(Myi+Myj/6, i= 2P1/15, j= Mt (IzIy) 2= y k= My jl, z y I= M i/l, m= Mt/2, n= Pl/30, (250) a=6P/51, b= Myil, c= Mzi/ 1, d= P(ly + z/Ax, 46 Note that I is the length of the beam and the other variables are as defined in Eqn 248. The final expression for the stiffness of the connector element with reference to the element local axes is, kT] = [ke]+[kg] (251) The above stiffness in global coordinates can be obtained by applying the standard coordinate transformation. Transformation Matrix The translational transformation matrix can be obtained by using the k node method. This method is used in the SSTAN program developed at the civil engineering department, University of Florida. The transformation matrix is a 3x3 matrix. The terms of the transformation matrix [T] is obtained as follows. The first 1 x 3 row entries represent a unit vector lying along the member local axis, {dx dy dz 1 = length of the member dx, dy and dz are similar to Ax, Ay, Az in Fig. 2.7 and similar to those given on page 40. Another unit vector must be defined along a node in the plane of the element but not along the member. This node is referred to as the knode. dxk dyk dzk, 1k 1k k' The second 1 x 3 row in the transformation matrix is obtained from the cross product of the knode unit vector and the member unit vector. The third 1 x 3 row is obtained by getting the cross product of the previous member unit vector (in the first row of the transformation matrix) and the unit vector gotten in the first cross product (now in the second row of the transformation matrix). The resultant 3 x 3 matrix [T] is used to transform nodal displacements from global to local coordinate systems. For the case of the beam with large displacement, in addition to the initial transformation matrix [T] described above, it is often advantageous to use another transformation matrix [R] based on Euler angles. This is because this transformation matrix formulation can better keep track of the twisting of the element than the former. A combination of both transformation matrices could be adopted with the first transformation [T] used to transform element properties from the local coordinate system to the initial member global position. The second transformation [R] can then be used to transform from the initial global position to the member's deformed position. The transformation matrix, [R], used in the context just described is described next. The Euler transformation matrix [R] is also explained in the context of a numerical time step solution, which is reflective of its usage in chapter 3. I + Fig. 2.15 Euler angles for transformation I = original length of the beam. ii = axial relative displacement. v = relative displacement in the y direction. w = relative displacement in the z direction. The Euler angles in Fig. 2.15 can be computed as shown in Eqn. 252 and Eqn. 2 53. Note that u, v and w above are computed using the cumulative nodal displacements in the case of dynamic time step solution. 1 v = tan + (252) 1 W / = tan1 (253) (+ 2 +2 ( The final Euler angle that is needed is the angle of twist, y. The angle of twist can be obtained from the rotational displacement about the local x axis. It is assumed that the nodal incremental displacements have been transformed to the element's initial local coordinate system using just the initial knode transformation [T]. The angle of twist is cumulated at each iteration using the change in nodal rotations. y = tAt + I{cosfcosa(AO .+A x+cos3sina AO .+AO +sinflAO .+AO .) 2 x xI X) ( yi yj zz zy (254) Y' = the angle of twist at time t yt +A = the angle of twist at time t + At A {axis)(node) = change in nodal rotation about the axis {x, y or z) at node (i orj}. a and P are as computed from Eqn. 252 and Eqn. 253. The final transformation matrix [R] is presented in Eqn. 255. 50 cos(p)cos() cos(p)sin(a) sin(p) [R] = cosy)sin(a) sin(y)sin(3)cos(a) cos(y)cos(a) sin(y)sin(j8)sin(a) sin(y)cos(p) sin(y)sa) cs(y)sin(P)cos(a) sin(y)coa) cos(y)sin(P)sin(a) cos(y)cos(3) (255) The final transformation matrix for the large displacement beam is therefore: [A] = [R][T] [A] = final transformation. Applying the transformation matrix, the stiffness of the member in terms of the global nodal displacements is A o T A O [KT]G = A A A(256) 0 A O A Damping Matrix The last property that needs to be defined is the damping matrix. The damping matrix for a structural system is easier defined proportional to either the mass, the stiffness or both. There are two classical damping matrices. The 51 Rayleigh damping and the Caughey damping [14]. The Rayleigh damping matrix is described below. With Rayleigh damping, the damping matrix is defined as follows. [C] = ao[M] + al[K] (257) ao and a, are damping constants depending on the controlling modal frequencies and the corresponding specified modal damping ratio. 20; M 2 o = i aJ a (258) a0 +i+j = Zi+ j = modal damping ratio i, = modal frequency corresponding to mode i. j = modal frequency corresponding to modej. Controlling modes are often the higher modes corresponding to the number of degrees of freedom for the system. For nonlinear analysis the eigenvalues are never obtained as part of the solution. There needs to be a way to estimate the natural frequency of the system prior to a rigorous nonlinear analysis. For the structural system defined, an estimate of the natural frequency of the system can be taken as either the natural frequency of the cables on infinitely rigid supports or the natural frequency of the poles without the cables. Both estimates are given below. For the catenary cable subjected to self weight, no initial pretension and a specified sag, the natural frequency [3] is S= n n = 1, 2,3,... (259) n = modes to investigate g = acceleration due to gravity f = sag at midspan of the cable 1 = span of the cable For cables where the sag is assumed to be zero the above equation ceases to be a good estimate. Therefore for a messenger cable with initial pretension and a near zero sag the natural frequency can be estimated as nn irTg (260) rn = l nm For the poles, taken as cantilever beams, the natural frequency can be estimated as follows [9]: n= x2 m (261) n 2 m 53 Matrix representation of the element properties have now been developed. The solution techniques that will be used to determine the time history response of the discretized structure is considered next. Dynamic Solution Methods The equation of motion [14] for a typical structure is [M]{d} + [C]{d} + [K]{d} = {P} (262) [ M ] = assembled mass matrix of the structure. [C ] = assembled damping matrix of the structure. [ K ] = assembled stiffness matrix. {d} = acceleration vector. {d} = velocity vector. {d} = displacement vector. { P } = applied load vector. For nonlinear analysis direct time integration is used. This means that the solution to the equation of motion is determined at discrete times (ti i=0,1,2...). The initial 54 conditions at the beginning of the time steps are always known and it is assumed that some variation in, and self consistent relationship between, the displacements, velocities and accelerations occur during the time step. The methods of solution available for dynamic analysis vary depending on the assumptions made for the variation in displacements, velocities and accelerations within the time step. As earlier mentioned these methods are classified as either explicit or implicit. Explicit Methods A disadvantage of using explicit method is that they are conditionally stable. The more common explicit solutions are Central Difference method and RungeKunta method. In explicit methods it is assumed that the displacement and velocity at time t + At are independent of the acceleration at t + At. Consequently, the displacement at the end of the time step is projected from the equation of motion at the beginning of the time step. In particular, the elastic forces and damping forces within the time step are computed using displacements and velocities at the beginning of the current time step and the beginning of the previous time step. For details of the available explicit methods, the writer recommends Ref. [7], [9], [14]. For the cable analysis it is important to apply the load in small increments, it is, therefore, more convenient to perform an incremental analysis. Implicit methods of analysis allows us to rewrite the equation of motion in terms of the change in the displacement, velocity and acceleration. Implicit Methods The incremental steps are measured relative to a reference state at the beginning of the time step. The equation of motion given in Eqn 262 can then be rewritten. [M]{Ad(t)} + [C]{Ad (t)} + [K]{Ad(t)}= {AP(t)} (263) The new shape of the cable at any time (t) can be obtained as follows d(t + At)= d(t)+ Ad(t) (264) The solution of Equation 263 above can be obtained by a number of different implicit methods. The more common methods are the AdamStoermer method, Newmark's Beta method and Wilson Theta method. These methods actually all have the same form and can be easily interchanged with a manipulation of the parameters associated with each method. The Newmark's Beta is used by the writer and is discussed next. Newmark Beta Method Newmark Beta method [14] is based on the following equations written in terms of incremental parameters. {Ad} = A {d ()} + 3){ d(t)} + 3{d(t + At)}] (265) {A d} = [{d(t)+ d ( + A t (266) Solve Eqn. 265 for {d(t + At)} and Eqn. 267 is obtained. {d(t +)} A = 1 {Ad} d(t)} 2 ) (267) {d(+A)} At2 P]3 0 From Eqn. 267 the change in acceleration within the time step can be expressed as follows: Ad} = {d(r+Ar)} {()} = 1 {Ad} d(t)} {d()} (268) d = A A 2fAtP Substituting Eqn. 267 into Eqn. 266 and the expression for the change in velocity can be rewritten as, 2 dI {d() 2 1 t t) (269) 1 2 Ad} d(t)} 2 (269) d 2At 2]f Ar {d(P)} Substituting Eqn. 265, 266 and 268 into 263 and rearranging terms we have the final equations for the Newmark's Beta method. [K(t)]{Ad} = AF(t)} (270) [(t)] = [M] + [(t)] + [K(t)] (271) SA[ 2 At(2 AP)} = {AP(t)} + [M][ d(r)} + (+) + + [c()] {) + ~ At {d(t)}J (272) The solution procedure using Newmark's Beta method may then be summarized as follows: * From initial conditions, calculate effective stiffness (Eqn. 271) and effective incremental load vector (Eqn. 272). * Solve for incremental displacement (Eqn. 270) and cumulative displacement (Eqn. 264). * Solve for incremental velocity (Eqn. 269) and velocity at t + At. * Impose total equilibrium at the end of the time step to obtain the acceleration at the end of the time step. (t + At)} = {P(t + At)} fd (t + At) (t + At) (273) [M] fd = damping force vector fe = elastic force vector. Repeat the process for the next time step. The parameter P reflects the variation of the acceleration within the time step. There are various values that could be used for P. For P = 1/6 we have the linear acceleration method. For P = 1/4 we have the constant acceleration method. And finally for P = 1/8, the acceleration is constant, equal to the beginning value over the first half of the time step and constant, equal to the ending value over the last half of the time step. When P is taken as zero the Newmark's Beta method degenerates to the Euler integration scheme. Similarly if P is taken as /2 Newmark's method becomes identical to the central difference integration and to backward difference when P is unity. The constant acceleration method is unconditionally stable This means that the solution will not become unbounded as the integration progresses. With the numerical integration, there are inaccuracies due to roundoff or due to time increment being too large to represent the forcing function or the structural response. The errors can be redistributed at each time step by using the NewtonRaphson method. CHAPTER 3 THEORETICAL DEVELOPMENT Introduction In chapter 2, the typical cable element commonly found in literature was discussed. The formulation of a different 2 node straight cable element adopted for this dissertation is presented in this chapter. This formulation includes the equation of motion and solution to the equation for the traffic signal system. The equations derived in this chapter are based on the Cartesian coordinate system shown in Fig. 2.1. The deformation parameters u in the x direction, v in the y direction and w in the z direction are deformations relative to the initial (null) state. The initial states for the cables are taken as either taut or slack. The taut cable has an initial pretensioned cable state. The load acting on this cable at the initial configuration (time t = 0) is the self weight of the cable and the initial tension. The cable with this null state (condition of zero deformation), in the case of the traffic signal system, is referred to as the messenger cable. The messenger cable null state has an initial axial strain but an assumed zero sag. The catenary cable, in the case of the traffic signal system, has a slack null state. The catenary cable in this case is presumed to have zero pretension (except for tension from self weight) and a specified sag. The default sag if none is specified is 5 %. Two additional reference configurations are of importance. These are the present configuration at time t and the next configuration at time t + At. The incremental displacement at the nodes, therefore is the difference between the coordinates of the nodes at time t and at time t + At. The cumulative displacement at the beginning of the time step is the difference between the coordinates at time t and the null state. Similarly the cumulative displacement at the end of the time step is the difference between the coordinates at time t + At and the null state. The null configuration is the base reference configuration. Both the messenger and catenary cable are idealized by the 2node nonlinear cable element derived in the following section. The main difference between this cable element and that derived in chapter 2 is the inherent assumption in the strain displacement relationship. Two Node Nonlinear Cable Element The shape function of the element adopted is given in Eqn. 32. 6 3 L4 Fig. 3.1 Cable Element Fig. 3.1 Cable Element The element properties for this cable element are derived directly in the global coordinate system. An explicit derivation of a 2 node cable element from first principles using some novel manipulations of typical energy expression is presented next. The derivation presented is based on the work presented by Henghold and Russell [16] and Haritos and He [17]. Ref. [17] presented a total Lagrangian formulation of the incremental equation of motion for cable movement. This writer has adopted this approach with a deviation in that the cable tension used for the geometric stiffness is cumulated in each time step in an updated Lagrangian fashion. This is observed to be more convenient in terms of numerical implementation of the element. Ref. [16], however, uses the method of perturbation of the virtual work expression for a cable element to derive the equation of motion of the cable. Though this is very different from the approach adopted in this dissertation, the writer has chosen to adopt the reference's approach of defining the element's displacement field (shape function) in such a way as to derive the element stiffness directly in terms of the deformed global coordinates as opposed to the more classical approach of using a local coordinate system with a local to global transformation. The shape function is defined to relate the coordinates of any generic point along the cable element to the nodal coordinates. xl Y2 Y5 .Z6 In terms of a convenient notation Eqn. 31 can be rewritten as shown in Eqn. 32. {ds = [N]dn} (32) ds = coordinate of an arbitrary point s along the cable d, = coordinates of the nodal points. n is the degrees of freedom shown in Fig. 3.1. N = shape function The interpolation function for a linear displacement field is given below [18]. s I o o L 0 1s 0 L 0 0 1 [N]T= s L (33) s L s L Applying the principle of virtual work, the expression for total work done for a single element at the time t + At is as follows. 5 T n T 0 I T T  SW= JO 3ds qds+ Sd F S d ds OL A eT a ds =0 j=1 (34) 6W = sum of the internal and external virtual work done on the element. q, = gravity and inertia forces acting along the arc of the cable. Fj = applied nodal load j is the degrees of freedom. (j = 1...n). fd = damping force vector. 6d = vector of small virtual variations in coordinates. Note that, though for convenience the writer has chosen to omit the brackets { } used as vector notation, the displacement and forces in Eqn. 34 and subsequent equations are vectors. Note also that the straight line overstrike ,, indicate values or properties at time t + At. The superscript T, indicate the transpose of the vector. The last term in the Eqn. 34 is the equivalent internal virtual work due to stress o at time t + At. This is referred to as the virtual strain energy. The variation in the coordinate at time t + At is equivalent to the variation in the difference between the coordinates at time t and time t + At. It is, therefore useful to express the coordinate at time, t + At, as follows. {dst+At =dst + {Ads (35) {Ads} = [N]{Adn} (36) = change in nodal displacement vector. Ad, 65 Ad, = change in displacement vector of any generic point along the cable. Substituting Eqn. 32 into Eqn. 34 yields Eqn. 37. W=JoL[N]TS5AdT sds+ [N ]TAdnT F L[N]T AdnTfd ds L AT ads (37) o = Kirchhoff stress. e = Lagrangian strain of deformed cable. A nonlinear straindisplacement relationship is adopted, therefore the strain is as given in Eqn. 38. s 2,s 1 (38) 5s = stretched length of a cable segment at time t + At aso = unstressed length of a cable segment at time t = 0 In terms of the shape function expression in Eqn. 33, Eqn. 38 may be rewritten as, E 1 dd T dds (39) s 2 ds ds es = { {}dn}T[N,]T[N"]idn} (310) 1 S0 0 L 0 0 L 0 0 [N']T= L (311) 0 0 L 0 0 00 L 0 0 L Note that N' is the differential of the shape function with respect to the generic distance, s, along the cable. The analysis approach described in chapter 2 is that of incremental displacement. It is therefore important to define the corresponding incremental strain. Subtracting the nodal coordinates at time t from the corresponding coordinates at time t + At, the incremental nodal displacement is obtained. Similarly, subtracting the expression for strain (Eqn. 38) at time, t, from a corresponding expression at time t + At yields the change in strain. 1 ( s dds 1 ( ds Tds Ae = ff s1 It .ds (312) 2 s s s2 = coordinate of an arbitrary point s along the cable at time t + At Ads = d ds (313) Solving for din Eqn. 313 and substitute into Eqn. 312 results in Eqn. 314. A 2 a s ) T Ads j 2 As d + dds T(dAds ds d s ) (314) a+ s J da Js Substituting the notation used in Eqn. 310 into 314, Eqn. 315 is obtained. Ae = I{AdT[N']T[N']Ad +d T[N'T[N']Ad,+AdnT[N'T[N'] d} (315) The last two terms in Eqn. 315 are equal, therefore the above equation can be simplified further as shown in Eqn. 316. Ae = !{AdnT[N']T[N']Ad}+ dnN'IT[N'] Adn The virtual work of internal forces as given by the last term of Eqn. 34 is given below. (316) L T (317) 8U = AO 6E ads (3 Applying the unitload method [19], the virtual strain energy expression in Eqn. 3 17 can be rearranged to be in terms of the stiffness for the cable element. WU = AEJL5eTEds (318) The strain in the cable element at time t + At is equivalent to the strain at time t plus the change in strain given in Eqn. 316. The variation in strain at time t + At is equivalent to the variation of the change in strain. 6 3T = Ae (319) Substituting Eqn. 316 into 319 and then into 318 the equation given in Eqn. 3 20 is obtained. 3U=AE jL( {AdT[N']T[N'] }SAdn + SLAdT[N']T[N'] }Ad + dnT[N']T [N']SAdnr eds (320) 6Adn = variation of incremental nodal displacement vector. Again, note that the first two terms in Eqn. 320 are equivalent. sU =AE L( AdnT[N'IT[N'] }Adn + dnT[N']T[N']5Adn ) 1ds (321) From Eqn. 321, it can be noted that the term, 6Ad,, is common to all the terms of Eqn. 34, and hence cancels out. Substituting the term for strain,e, Eqn. 321 reduces to an expression in terms of stiffness. e = e + AE (322) {fs} = AEJ {Ad T[N ]T ['] + d T [N ]T [N x e + (i{Ad T [N']T [N']Ad} + dT [NjT [N']Adn ds (323) Rearranging and collecting subexpressions in Eqn. 323, Eqn. 324 is obtained. fs = KEAdn + KL Ad + E (324) L T K = A E J E [N [N ']ds (325) 0 KL = AE ({AYT[N'] [N'] + {dT[N'][N]} x {Ad7[N][N']}+ {dnfT[N' [N'}d (326) L RE = A E J [N ]T [N '] dn ds (327) 0 KE is the referred to as the initial stress matrix and RE is the force vector due initial stress. Note that the term AEe is equivalent to tension in the cable at the beginning of the time step. For t = 0, this term is the pretension in the element. For subsequent time steps the tension term can be computed as follows, T = T + AEAtAt (328) Note that KL is dependent on unknown displacements. This is the nonlinear 70 component of the element stiffness, [K]. This can not be computed at the start of analysis and must be substituted by an initial approximate stiffness. Let this approximation be L dT[N]T[NI xdT[N']T N'Ids (329) The expression in Eqn. 329 was achieved by simply setting Ad, in Eqn. 326 to zero. If the shape functions for the 2node element is substituted into the Eqn. 3 25, 326, 327 and 329, the following expressions for KE, KL and KA on the next page are obtained. See Appendix A for complete derivation of the expression. The following notations are used. xi ui Yi Vi Szi = 1...3 Awi i = 1...3 S 'j = 4...6 dj j = 4...6 yj Vj zJ. w. SxjX i du ui S= Yj Yi dv = v S Zi dw w wi The expression for KE is as follows: E t T + AE AE SE I rL 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 (330) = initial cable element length. [KAi AL [KLi AE 2L3 dx2 dx dy dx dz d2 dy dx dzdx dxdy dy2 dydz dyd dy2 dydz dxdz dydz dz2 dzdx dydz dz2 dx2 dydx dzdx dx dxy dxdz dy dx dy2 dydz dxdy dy2 dydz dzdx dvdz dz2 dxdz dvdz dz2 (331) (dA+du)(2d +du) (d + u)(2dy + dv) (d + du)(2 + dw) ( + du)(2d + ) (dr+ d)(2dy+dv) ( +du)(2 +dw) (y+ dv) (2d + d) (dy + dv) (2dy + dv) (dy + dv) (2c + dw) (dy + dv)(2 + du) dy + dv)(2dy + dv) (dy + dv)(2 + dw) ( + dw)(2dr + ) ( + dw)(2dy + dv) ( + dw) (2 + dw) ( + dw) (2d + d) (k + dw)(2dy + dv) ( + dw)(2, + dw) ({+dau)(2d+du) (ad +u)(2dy+dv) {dr+ u)(2 + dw) (dc + au)(2d + du) (d + u)(2dy + dv) (d + du)(2 + dw) (y+dv)(2d+du) (dy + dv)(2dy + dv) (dy + dv)(2t + dw) (dy + v)(2 + (dy + dv) (2y+dv) (dy + dv) (2 +dw) (t+dw)(2d+diu) { +dw)(2dy+dv) {, +dw)(2c+c+dw) (c+ dw)(2d + d) ( + dw)(2dy + dv) ( + dw)(2 + dw) (332) Eqn. 333 can be deduced from Eqns. 325 and 327. {RE} = [KE{d,} (333) dx tAt+AEAE d RE= L dx (334) dy dz Equation of Motion The above portion calculated the strain energy portion of Eqn. 34. It is now necessary to consider the other terms in Eqn. 34. The kinetic energy term is the first term in Eqn. 34. This is in terms of the variable q, defined in Eqn. 335 below. qs = gs + fi (335) q, = distributed load along the element. gs = gravity load due to self weight. fi = inertial forces. Rewriting the first term in Eqn. 37 yields Eqn. 336. L L gs6K = fJwsN]Thds fms[NT [N]{dn}ds (336) 0 0 ws = cable self weight per unit length. m, = cable mass per unit length. 8K = kinetic energy due to inertia forces. { = nodal acceleration. h, is an index array based on the coordinate system adopted. It is used to determine how the weight of the element is added to the structure. The self weight acts in the negative z direction. h ={0 0 1} 2 0 0 1 0 0 iui 0 2 0 0 1 0 SK = L 0 0 2 0 0 1 w i (337) 6g 1 0 0 2 0 0 iu 0 1 0 0 2 0 v 0 0 1 0 0 2 w g = acceleration due to gravity. 0 0 ws L 1 s 2 o (338) 0 1 The second term in Eqn. 37 and Eqn. 34 is the applied nodal load. f2 FI f (339) f5 This is the computed nodal load due to dead weight of cladding and/or traffic signal heads and/or wind load. Since the cable element is a two node element, the nodal distribution of a body load is an even split between the two adjacent nodes. The third term in Eqn. 37 is the damping term. This is obtained as discussed in chapter 2. The equation of motion for the cable element is therefore given as shown in Eqn. 340. [MId}+[c]{Jd}+[K]{Ad} = {Fj}+{gs}RE} (340) The solution to Eqn. 340 above can be obtained by a myriad of numerical integration schemes mentioned in chapter 2. When the cable element is assembled with the other element types, i.e. beams and connectors The RE term in Eqn. 340 above can be replaced by a similar term FE. Where FE is the initial stress vector for all the elements in the structure. [M]( [c](J }+ [K]{A dn = {Fj}+{gs,( E (341) It should be noted that the self weight vector remains the same at the beginning of the time and at the end. The change in force due to the self weight is zero. When the wind load is considered the nodal force vector would include the wind force. At the beginning of each time step the incremental displacement vector is zero and the velocity and acceleration vector are as obtained from the previous time step analysis. The initial conditions of the dynamic analysis is obtained from setting both the displacement and velocity vectors to zero and obtaining the acceleration from equilibrium of the structure. { \ + g, } { E (342) [M I The use of the incremental displacement facilitates the use of implicit numerical integration schemes for solution to Eqn. 341. Chapter 2 discusses the Newmark Beta method. This method in its modified form was adopted by the writer. The modification adopted is presented in the following section. This modification is referred to as the alpha modification. Modified Newmark Beta Method Due to the inherent problem of "blow up" in the analytically unstable traffic signal cable system and the fact that the unconditionally stable Newmark Beta method possesses no intrinsic numerical damping, Wood et al [20] proposed the use of an alpha factor. This factor introduces a positive artificial damping. This modified Newmark's method with positive artificial damping of higher vibrational modes is observed not only to reduce the "blow up" effect but also reduce the cumulative error which inevitably occurs in a numerical integration procedure. Applying the alpha modification to Eqn. 340 results in an averaging effect of the inertia forces at the beginning and at the end of the time step. (Ia)[MuIj{ }+(a)[M]{ } + [C]{J } +[K]{Adn = (t+A (343) P} = {F I + Ig IF (344) Note that if a is zero Eqn. 343 reduces to Eqn. 340. Apply Newmark's method based on Eqns. 345 and 346 to Eqn. 344. For convenience the nodal designation is dropped for Eqn. 345 to Eqn 350. It should however be understood that these are nodal displacement, velocity and acceleration. dt+At = dt+[(ly)At]dt +(yAt)d;+At (345) dt + At =dt + (A t)t + [(2 )(A t)2 t + [ (A t)2 t + At (346) Solving Eqn. 346 for acceleration at time t + At yields Eqn. 347. d 1 Ad t I (347) t+At At2 ftAt2 t Substituting Eqn. 347 into 345 yields Eqn. 348. t+At + = A dt I d + At Jd (348) Following the steps described in chapter 2 and substituting Eqn. 347 and 348 into Eqn. 343, the final expressions for the effective stiffness and load vectors are obtained. 79 (,) Ad2 d 1d +a[M] t +[CJ Y Adt L) Id(t +At  + [KAdj = P+A} (349) Note that {A d,}= {d l, } {d,} (350) Collecting terms and rearranging Eqn. 349 gives Eqn. 351. [i]{Adn = [AP (351) ([K = (1 a)[Ml( + ([C + [K] (352) PAt2 pAt (353) When a is zero the above described Newmark's method reduces to that described in chapter 2. The values of P and y are as described in chapter 2. If the alpha modification is to be used, the values of the constants are given below. When P = 4 and y = /2 for the regular Newmark's method, the constant acceleration method is obtained. This method is a second order solution with zero artificial damping and unconditional stability. The intent of the alpha factor is to insert positive artificial damping while keeping the property of unconditional stability. For a second order method with positive artificial damping and unconditional stability the following condition must be satisfied. l> > & a< & a+y> (354) 2 4 2 2 Wood et al [18] in their paper suggest the use of the following as giving the best results. a =0.1 = 0.3025 y = 0.6 a = 0.1 =0.5 y = 0.6 a =0.1 = 0.3025 y = 0.6 The writer suggests the use of a = 0.1 P = 0.3025 y = 0.6 for the traffic signal cable system as it yields the best artificial damping effect for the system. When the incremental displacement is obtained from Eqn. 351, in order to maintain dynamic equilibrium, this computed incremental displacement needs to be refined. The reasons for the need of refinement are as follows; a) The stiffness matrix at time t + At is dependent on the recently computed displacements values. b) The { FE } term in the equation of motion contain values dependent on the nonlinear cable stiffness and nonlinear connector stiffness. The NewtonRaphson method is adopted to refine the computed incremental displacements. Application of the Newton Raphson is discussed next. Modified Newton Raphson Method A graphical representation of the steps involved in the modified Newton Raphson method is given in Fig. 3.2. There are two additional modifications made to the regular nonrecursive Newton Raphson procedure [9], [14]. One has to do with the nonlinear loading and the other the nonlinear stiffness of the cable elements. Note that both the loading and the stiffness are dependent on the incremental displacements being computed. The writer has taken these into account in modifying the Newton Raphson procedure adopted. Recall from Eqn. 341 that, {P} = {F,} {RE} (355) {Fj = Applied nodal loads. These consist of dead weight of cladding and/or wind loads applied at the nodal points. Discussion of loading on the traffic signal cable system is presented in chapter 4. {RE} = Equivalent nodal force vector for member stress at time t. This accounts for the stresses in the structure at the beginning of the time step. 82 Force (K1)se K ec K, AR K_ K A  I0 ( I (K2 )secI I 4 Ise Ad Ad Ad n Ad2 Ad S Displacement Fig. 3.2 Newton Raphson Iteration The complete procedure to refine Ado shown in Fig. 3.2 on the previous page is given to illustrate the modified NewtonRaphson method : 1. Initialize data. K =[i] From Eqn (352) P = PI From Eqn(353) AR = P o Ad = K 1AR From Eqn(351) In computing the above initial values, AP is assumed to be zero in Eqn. 353. It is assumed that if At is taken small enough the approximation error introduced by this assumption is negligible. 2. Compute the instantaneous cable stiffness at a displacement Ado and set it equal to k1. 3. The incremental resisting force: A 1 = (KI)sec A do (356) Where the secant stiffness (K1)s can be approximated by Eqn. 357. (K 0 + i (357) () sec 2 AR + Ad .. AF o 21 1 2 (358) 4. The residual force at the end of the first iteration is as shown in Eqn. 359 AR = P AF (359) 5. Adl is computed as in step 1. (360) Adl = KAlR1 The sign of the displacement computed in Eqn. 360 is negative because the residual force is negative. 6. Repeat step 2, compute k2. 7. Repeat step 3, compute new resisting force. . AF = 2 io(Ado+Ad)+ k2 Ado + Ad,) (361) 8. Repeat step 4, compute the residual force. A 2 = P AF2 9. Using Eqn. 360 above compute Ad2. The whole procedure is continued until the residual force is smaller than some tolerance value. The true displacement, therefore, is as given in Eqn. 362. Adn = Ad + Ad + Ad2 (362) There are many convergence checks that could be used. Some authors suggest a check on subsequent displacement computations; that is Ad < d i = 0,1,2 ... (363) Ad n where S is a prescribed tolerance. The problem faced when convergence is strictly based on the convergence of the nodal displacement is that of different rates of convergence for rotational displacements and translational displacements. The units and range of values for rotations differ greatly compared to translations. As a result the translational displacements may converge while the rotational displacements may not simultaneously converge. The writer has chosen a convergence criterion based on the work done by the residual forces. T {A_ R A id= 0,1,2... (364) {AP} Adn The use of the modified Newmark's method and the above described Newton Raphson method in the entire solution procedure is described next. Solution Algorithm Based on the theory developed in the previous sections of this chapter the solution to the equation of motion given in Eqn. 340 was obtained as follows. 1. Compute the effective load vector as given in Eqn. 353. Note that AP is given by the right hand side of Eqn. 341. For the first time step analysis AP is taken to be zero. At the end of each time step P,+ t is computed and the actual change in load, AP, for that time step is obtained. This change in load is used as the estimate for the next time step analysis. The acceleration at the beginning of each time step is, however, obtained by imposing total equilibrium at the end of the previous time step and using the true load in the equilibrium equation. 2. Compute the effective stiffness matrix using Eqn. 352. The initial effective stiffness matrix at time t = 0, uses [K] = [KE] + [KA], where [KE] and [KA] are as defined in Eqn. 325 and 328. At subsequent times, the effective stiffness matrix uses [K] = [KE] + [KJ. The nonlinear stiffness [KJ is computed as given in Eqn. 326. 3. Solve for change in displacements. {Ad} = [1 ]tl { }, (365) 4. The Newton Raphson iteration scheme is used to refine the incremental displacement obtained in Eqn. 365. The NewtonRaphson procedure gives a better estimate of { Ad }. This is done as shown in steps 1 9 on pages 83 85. 5. Compute the final condition at the end of the time step. (d} = {d" Ad (366) dt + At A t + 2 ntAt = #A + ( )nU t 2p ni (367) {P}t + At a[M]{d } [C](n }+ A [K]{Adn t {dnh+At (1a)[M] (368) 6. Set the incremental displacement of the next time step as follows and repeat the above steps Ad n= O0 88 The dynamic time step analysis is concluded when the freevibration response dies down. The complete analysis gives the full dynamic response history for the transient dynamic loads applied and/or the equivalent static solution of the dead load applied. For the static equivalent solution the "diedown" criteria is as follows. [K ]{A d}2 [K]{Ad,2 (369) (369) p q p This means that the strain energy of the structure is averaged over several time steps and compared to the average of the next several time steps. When the difference between the two strain energies computed is negligible, diedown is achieved. 1, p and q are beginning and ending time step cycles for strain energy computation. 
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