Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002430/00001
## Material Information- Title:
- An experimental and theoretical investigation of the behavior of cables under point impact loading
- Creator:
- Ledesma, Michael Scott
- Publication Date:
- 2005
- Language:
- English
- Physical Description:
- xv, 160 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Cables ( lcsh )
Cables -- Vibration ( lcsh ) Lateral loads ( lcsh ) Strains and stresses ( lcsh ) Cables ( fast ) Cables -- Vibration ( fast ) Lateral loads ( fast ) Strains and stresses ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 157-160).
- General Note:
- Department of Civil Engineering
- Statement of Responsibility:
- Michael Scott Ledesma.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 62879852 ( OCLC )
ocm62879852 - Classification:
- LD1193.E53 2005m L42 ( lcc )
## Auraria Membership |

Full Text |

AN EXPERIMENTAL AND THEORETICAL INVESTIGATION
OF THE BEHAVIOR OF CABLES UNDER POINT IMPACT LOADING by Michael Scott Ledesma B.S., Lawrence Technological University, 1997 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering This thesis for Master of Science degree by Michael Scott Ledesma has been approved by Brian Brady Chengyu Li Ledesma, Michael Scott (M.S., Civil Engineering) An Experimental and Theoretical Investigation of the Behavior of Cables Under Point Impact Loading Thesis directed by Professor Kevin L. Rens ABSTRACT An experimental and theoretical investigation was completed to study the behavior of flat- sag cables subjected to dynamically applied point loads. Numerical calculations using the linear theory of cable vibrations were performed using a computer program written to perform the required modal analyses. Also, a scale model was tested in a laboratory and subjected to different dynamic point loadings while recording the displacement and cable tension responses. The results of the study indicate that the linear theory of vibrations provides an accurate means of predicting cable response. This abstract accurately represents the content of the candidates thesis. I recommend its publication. m ACKNOWLEDGEMENTS The author would like to express his gratitude to several persons and institutions who have contributed, directly and indirectly, to making this research possible: Dr. Kevin Rens, for his advice and direction, and for his extreme flexibility as I struggled with a move to another state during the preparation of this thesis. The University of Colorado at Denvers Soils Testing Laboratory, for allowing me to carve out some space to perform the experimental testing reported in this work. Mr. David Blanchette and SM&RC, Inc., for kindly allowing me to work at his firm while keeping irregular hours, and for his advice and wisdom. Boyle Engineering Corporation, my employer for many years, and who was always very supportive of my educational goals, both as a scholar and as a practicing engineer. All of my family and friends, especially my parents, who have given me support and guidance throughout my life, and always encouraged my curiosity. Last, but certainly not least, my wife, Jen, whose love, support, encouragement, and patience is greatly appreciated. Without your pushes when I didnt want to move, without your optimism when things got frustrating, this work could never have come to fruition. I love you! CONTENTS List of Figures................................................................ xii List of Tables ................................................................xiii List of Symbols xv Chapter 1. Introduction 1 1.1 Cable Analysis........................................................... 1 1.2 Cable Behavior........................................................... 2 1.3 Summary of Work.......................................................... 4 2. Literature Review 5 2.1 Overview ................................................................ 5 2.2 Early Work............................................................... 6 2.3 Twentieth Century........................................................ 8 2.4 Recent Work............................................................. 11 2.5 Numerical Methods and Computers 13 3. Theoretical Analysis......................................................... 15 3.1 Overview 15 3.2 Linear Theory of Cable Dynamics 15 3.2.1 General Theory 15 3.2.2 Free Vibration and Modal Behavior ............................... 16 v 3.2.3 Forced Vibration Response and Modal Analysis......................... 24 3.3 Numerical Analysis......................................................... 27 3.3.1 Description.......................................................... 27 3.3.2 Summary of Computation Algorithm..................................... 28 3.3.3 Input Parameters..................................................... 35 3.4 Results.................................................................... 36 4. Experimental Analysis 49 4.1 Introduction............................................................... 49 4.2 Objective.................................................................. 49 4.3 Instrumentation............................................................ 49 4.3.1 Data Acquisition Unit................................................ 50 4.3.2 Displacement Gages................................................... 51 4.3.3 Tension Gages ....................................................... 53 4.3.4 Applied Force Gage 57 4.3.5 Strain Gages......................................................... 58 4.4 Cable Model 59 4.5 Tests...................................................................... 64 4.5.1 Static............................................................... 65 4.5.2 Dynamic 65 4.6 Results.................................................................... 67 5. Discussion of Results 80 5.1 Introduction............................................................... 80 5.2 A Comparison of Displacement Responses..................................... 80 5.3 A Comparison of Additional Tensions ....................................... 86 5.4 The Effects of Damping..................................................... 95 5.5 Parametric Study...........................................................100 5.6 Conclusions................................................................109 vi 5.6.1 Limitations and Future Work..........................................110 Appendix A. The Mathematics of Cables.......................................................112 A.l Introduction..............................................................112 A.2 Static Response Under Self-Weight.........................................113 A.3 Static Response Under Uniform Loading.....................................117 A.3.1 Introduction .........................................................117 A.4 Static Response Under Point Load..........................................120 A.4.1 Introduction .........................................................120 A.4.2 Strings Subjected to Point Loads......................................120 A.4.3 Cable Response to a Point Load .......................................122 A. 5 Dynamic Behavior of Cables................................................127 A.5.1 Introduction .........................................................127 A.5.2 Modes of Vibration....................................................128 B. CableCalc Program Listings .....................................................132 B. l Program Code .............................................................132 B.2 Input Files ..............................................................147 B.3 Load History Files........................................................148 C. Instrument Calibrations.........................................................149 D. Test Programs ..................................................................155 D.l Static Tests..............................................................155 D.2 Dynamic (Impact) Tests....................................................158 Bibliography........................................................................160 vii LIST OF FIGURES Figure 1.1 Classification of Cable Systems.......................................... 3 3.1 Coordinate System Definition Diagram for Equilibrium Equations 16 3.2 First Three Antisymmetric Mode Shapes.................................... 18 3.3 First Three Symmetric Mode Shapes (A2 < 4n2) 21 3.4 First Three Symmetric Mode Shapes (A2 > 47r2) 22 3.5 Symmetric and Antisymmetric Frequency Behavior over a Range of Values of A2 23 3.6 Screenshot of Rapid-Q Integrated Development Environment................. 27 3.7 Flow-chart for CableCalc Program......................................... 29 3.8 Sample Time History and Load Data Format................................. 32 3.9 Displacement Response for Suddenly Applied Load at 0.5L (( =2%)........... 37 3.10 Displacement Response for Suddenly Applied Load at 0.5L (C =2%)........... 38 3.11 Additional Tension Response for Suddenly Applied Load at 0.5L (( =2%) ... 39 3.12 Displacement Response for Impulsive Load at 0.5L (C =2%).................. 40 3.13 Displacement Response for Impulsive Load at 0.5L (Â£ =2%).................. 41 3.14 Additional Tension Response for Impulsive Load at 0.5L (Â£ =2%).......... 42 3.15 Displacement Response for Suddenly Applied Load at 0.2L (( =2%)........... 43 3.16 Displacement Response for Suddenly Applied Load at 0.2L (Â£ =2%)........... 44 viii 3.17 Additional Tension Response for Suddenly Applied Load at 0.2L (( =2%) ... 45 3.18 Displacement Response for Impulsive Load at 0.2L (Â£ =2%).................. 46 3.19 Displacement Response for Impulsive Load at 0.2L (( =2%).................. 47 3.20 Additional Tension Response for Impulsive Load at 0.2L (Â£ =2%)........... 48 4.1 Campbell Scientific CR5000 Electronic Data Logger ....................... 50 4.2 Displacement Gage Schematic................................................ 52 4.3 Displacement Gages on Cable Model........................................ 54 4.4 Tension Gage Schematic..................................................... 55 4.5 Typical Installed Tension Gage............................................. 56 4.6 Applied Force Gage Schematic .............................................. 57 4.7 Applied Force Gage Showing Strain Gage..................................... 58 4.8 Typical Strain Gage (Not to Scale)......................................... 59 4.9 Terminal Input Module ................................................... 60 4.10 Cable Anchorage (West End)............................................... 61 4.11 Model Geometry and Instrumentation Layout................................ 62 4.12 Additional Weights on Cable.............................................. 63 4.13 Impact Drop Mechanism.................................................... 66 4.14 Displacement Response for Suddenly Applied Load at 0.5L 68 4.15 Displacement Response for Suddenly Applied Load at 0.5L ................... 69 4.16 Additional Tension Response for Suddenly Applied Load at 0.5L............ 70 4.17 Displacement Response for Impulsive Load Applied at 0.5L................... 71 4.18 Displacement Response for Impulsive Load Applied at 0.5L................... 72 4.19 Additional Tension Response for Impulsive Point Load Applied at 0.5L 73 IX 4.20 Displacement Response for Suddenly Applied Load at 0.2L ................ 74 4.21 Displacement Response for Suddenly Applied Load at 0.2L ................ 75 4.22 Additional Tension Response for Suddenly Applied Load at 0.2L.......... 76 4.23 Displacement Response for Impulsive Load Applied at 0.2L................ 77 4.24 Displacement Response for Impulsive Load Applied at 0.2L................ 78 4.25 Additional Tension Response for Impulsive Point Load Applied at 0.2L 79 5.1 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.5L ((,Theor =2%)............................. 81 5.2 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.5L (CTheor =2%).............................. 82 5.3 Comparison of Theoretical and Experimental Displacement Responses Due to Impulsive Load at 0.5L (CTheor =2%)..................................... 84 5.4 Comparison of Theoretical and Experimental Displacement Responses Due to Impulsive Load at 0.5L (Cr/ieor =2%).................................... 85 5.5 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.2L (CTheor =2%).............................. 87 5.6 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.2L (Cr/ieor =2%)............................. 88 5.7 Comparison of Theoretical and Experimental Displacement Responses Due to Impulsive Load at 0.2L (Cr/ieor =2%).................................... 89 5.8 Comparison of Theoretical and Experimental Displacement Responses Due to Impulsive Load at 0.2L (CTheor =2%)..................................... 90 5.9 Comparison of Theoretical and Experimental Additional Tension Responses Due to Suddenly Applied Load at 0.5L (Cr/ieor =2%) ......................... 91 5.10 Comparison of Theoretical and Experimental Additional Tension Responses Due to Impulsive Load at 0.5L (CTheor =2%).................................. 92 5.11 Comparison of Theoretical and Experimental Additional Tension Responses Due to Suddenly Applied Load at 0.2L (Crheor =2%) .......................... 93 x 5.12 Comparison of Theoretical and Experimental Additional Tension Responses Due to Impulsive Load at 0.2L (Cr/ieor =2%)................................... 94 5.13 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.5L (CTheor 7%)................................ 96 5.14 Comparison of Theoretical and Experimental Displacement Responses Due to Suddenly Applied Load at 0.5L (CTheor =7%)................................ 97 5.15 Comparison of Theoretical and Experimental Additional Tension Responses Due to Suddenly Applied Load at 0.5L (Cr/ieor =7%) ........................... 98 5.16 Load Functions Used in Parametric Study..................................101 5.17 Dynamic Response at 0.1L .................................................103 5.18 Dynamic Response at 0.2L..................................................104 5.19 Dynamic Response at 0.3L..................................................105 5.20 Dynamic Response at 0.4L..................................................106 5.21 Dynamic Response at 0.5L..................................................107 A.l Definition Diagram for Cable Subjected to Self-Weight.....................113 A.2 Definition Diagram for Cable Subjected to Uniform Load....................118 A.3 Definition Diagram for Taut-String Analysis...............................121 A.4 Definition Diagram for Cable Subjected to Point Load......................122 A.5 Displacement Diagram for Differential Cable Element Subjected to Point Load 125 A.6 Modes of Vibration of a Hanging Cable ......................................129 A.7 Definition Diagram for Dynamic Behavior....................................130 C.l Calibration Data for Displacement Gages D1 to D4..........................150 C.2 Calibration Data for Displacement Gages D5 to D8..........................151 C.3 Calibration Data for Tension Gages T1 to T8...............................152 C.4 Calibration Data for Tension Gages T5 to T8...............................153 xi C.5 Calibration Data for Applied Tension Gage 154 xii LIST OF TABLES Table 5.1 Static Displacements and Additional Tension..............................102 xm LIST OF SYMBOLS Cable cross-sectional area Cable modulus of elasticity Acceleration due to gravity, 9.81 m/s2 (386.4 in/s2) Additional horizontal component of tension in cable Initial horizontal component of tension in cable Span length of cable Total length of cable Mass per unit length of cable Normal coordinate Point load magnitude Distance along cable profile Initial midspan sag Time (s) Initial tension in cable due to dead load Additional longitudinal displacement Additional out-of-plane displacement w Additional vertical displacement ivg Weight per unit length of cable x Horizontal distance along cable span xq Location of point load z Vertical distance describing initial cable profile Â£ Fraction of critical damping A2 Dimensionless cable parameter r Additional tension in cable due to incident loads u> Natural circular frequency of vibration (rad/s) xv 1. Introduction The study of cables and their behavior is one of the older classes of structural problems to be rigorously investigated, but it remains an important subject of study. Cable systems are extremely efficient at carrying loads with minimal material, because of their ability to transmit loads in tension only. Their use in such notable structures as suspension bridges, long-span stadium roofs, and tall transmission towers has continually advanced the state of the art in analysis and construction of cable systems. With the increasing realization that the threat of terrorism must be addressed during the design of key structures, it is important to understand the response of the various classes of structures, including cable structures, to potential blast loadings. While blast design has been studied by researchers since the 1960s, to the authors knowledge, little has been done regarding the response of cables to blast loadings. This concept behind this work has its origins in the question: How would a suspension bridge respond to an explosive discharge on its deck? While the answer to that question is beyond the scope of the this work, the research presented herein seeks to investigate one aspect of that question. The present work focuses specifically on the behavior of a simple suspended cable subjected to impulsive point loads at different locations along the span. Blast theory, suspension bridge design, and other factors that, while important to designers, obscure the basic question at hand, will not be addressed. 1.1 Cable Analysis The behavior of hanging cables, chains, strings, and the like has been subject of much scrutiny over the years. With the advent of calculus, it became possible to analyze cables in a quantitative, rather than qualitative, manner. One of the reasons cables defy simple analysis 1 is because of their ability, even necessity, to adjust their shape to accommodate an equilibrium state. Most basic structural analyses are first-order. In other words, the geometric changes due to applied loads are assumed small. For example, a point load applied to the midspan of a rigid beam will cause the beam to deflect a small amount, but not enough to change the original geometry in any significant manner. Were the beam to deflect significantly, the geometry would be significantly different, and the distribution of stresses and strains would be different than originally assumed. Another iteration of the calculation would be required, based on the new geometry. This would need to continue until an equilibrium state was found, or failure occurred. In other words, a higher order analysis must take into consideration the changes in geometry when determining static or dynamic equilibrium. Because a cables profile can change significantly, a higher order analysis must be used. However, the mathematics underlying cable analysis becomes very complex. Though the differential equations of equilibrium may be written, it becomes difficult or even impossible to solve them exactly. Therefore, researchers resort to simplified methods to find usable solutions to problems. One result has been the linear theory of cables. Essentially, higher order terms are disregarded in the solution. It has proved useful in the analysis of cable systems. One of the limitations of this method is that it is not appropriate for systems in which large changes in geometry are possible; for example, where large initial sags are present. However, when sags are relatively small (sag-to-span ratios of 1:8 or less) the theory is quite accurate. As most cable systems found in real structures meet this limitation, the theory has practical application. 1.2 Cable Behavior Displacements in suspended cables arise from two mechanisms: profile adjustment, and cable stretch. Because a cable is assumed to have no resistance to bending, it is possible for a cable to assume a number of geometric profiles without increasing its total length. When one portion of a cable is depressed from an equilibrium state, other portions are elevated, for example. This is simply a manifestation of the conservation of gravitational potential energy. 2 Not Very Deformable Figure 1.1: Classification of Cable Systems Cable systems carrying heavy uniform dead loads tend to resist changes in geometry due to applied loads, while light dead loads allow more. Cable stretch arises from the fact that loads applied transversely to a suspended cable induce tension in the cable. Indeed, this is what makes them so remarkable as a structural element. Because the material that comprises a cable is elastic, it elongates in response to a strain. The lengthening of the cable and profile adjustment both must be considered in determining the equilibrium state. However, the relative importance of each of these mechanisms varies from system to system. Some suspended cables achieve their equilibrium positions primarily through cable stretch, while others do so through changes in cable profile. Figure 1.1 shows the range of possible cable systems. Cable extensibility is plotted on the horizontal axis, while cable deformability, or how easy it is for a cable to assume a different profile, is plotted on the vertical axis. As can be seen, at one extreme is the classical taut string with zero sag. Small deformations are associated with very large increases in tension. At the other extreme is the highly elastic, high sag cable. The increase in tension in the cable is approximately limited to 3 the added load, while the cable profile changes drastically. A suspension bridge, for example, has relatively inextensible cables and a fairly large dead load, but also has moderate sag. In Figure 1.1, it would perhaps lie in the vicinity of the star. It is important to recognize that there are several behaviors at work here. In later chapters, a parameter, A2, will be introduced and used extensively in the static and dynamic analyses. This dimensionless parameter is analogous to the Reynolds number or Froude num- ber of fluid dynamics. It accounts for the initial sag, cable weight, and cable elasticity and compactly classifies the system. 1.3 Summary of Work This work presents the results of a theoretical and experimental investigation in which a simple suspended cable was subjected to impulsive point excitations. Chapter Two presents a review of the available literature pertaining to cable analysis, especially as it applies to point loadings. Chapter Three describes the application of the linear theory of cable dynamics to the problem of impulsive point load response. It also documents the development of a computer program to perform a modal analysis for a given cable system and load history, from which a history of displacement and additional cable tension response history was computed. Chapter Four documents an experimental study in which a scale model of a suspended cable was subjected to impulse loadings. The results were recorded, and these data were compared in Chapter Five to theoretical calculations to verify the accuracy of the linear theory of cables in predicting response to impulsive loads. After verifying the ability of the linear theory to predict responses, a parametric study was performed using the computer program to compare the response of cable systems to a variety of dynamic point loadings, which were compared with static loads of similar magnitude. In this way, these results could then be evaluated in a manner consistent with classical dynamic analysis. 4 2. Literature Review 2.1 Overview The response of cables to point loads and other forces has been the subject of consider- able research and discussion over the past few centuries. Early mathematicians were primarily interested in determining the profile assumed by a hanging chain or string under its own weight. As applications such as suspension bridges became feasible, cable theory was advanced further as engineers sought new techniques to analyze their designs. In the mid-1800s, it became clear to researchers that the response of a cable to a point load was non-linear; that is, for each successive increment of load added, the deflection response was smaller and smaller. However, engineers lacked a coherent means of analyzing this behavior. Consequently, engineers designed suspension bridge decks as large stiff trusses so that concentrated loads were more uniformly distributed and did not cause excessive deflections. By the late 1800s, researchers had developed a theory that accounted for the static non-linear response of cables to point and other loads. Using Deflection Theory, as it was known, engineers were able to take advantage of the fact the dead load of the structure had a stiffening effect on live load response. Consequently, suspension bridges were constructed with less material and achieved longer spans. When the Tacoma Narrows suspension bridge collapsed in 1940, clearly due to a wind- induced dynamic resonance, the engineering community was forced to reevaluate their methods. This necessitated an appreciation that aerodynamic effects were of critical to understanding the response of such long and slender structures. The result was increased interest in the dynamic response of cable systems. The remainder of the 20th century saw research in such topics as support excitation, non-linear vibrations, and inelastic response. In particular, the behavior of cable systems under 5 seismic excitation saw considerable attention. With the advent of digital computers, purely numerical methods became of increasing importance and allowed researchers to study systems what would be difficult or impossible using classical methods. More recently, research has been focused on the problem of vibration in the cables used in cable-stay bridges. This has led to the use of viscous dampers attached to the cables to reduce vibrations. Interestingly, since some of these systems actively adjust to the state of vibration in real-time, it is important to have a means of solution not dependent on numerical methods like the finite element method, due to their relatively long computation times. Instead, many of these research efforts rely on the results obtained decades earlier. The remainder of this chapter is devoted to a review of the existing literature relevant to the analysis of cables and cable systems, specifically their response to point loads. 2.2 Early Work The earliest theoretical investigations into the nature of hanging chains seems to have begun with Galileo. Leibniz [12] noted that Galileo incorrectly conjectured that a perfectly flexible and inextensible string hanging under its own weight assumed the shape of a parabola. In 1691, Jacob Bernoulli issued a challenge to other scholars to derive the correct expression. Solutions were given by Huygens, Leibniz, and Johann Bernoulli that correctly identified the shape as having the form of a hyperbolic cosine, y = C\ cosh Ci, also called the catenary (the constant Ciin this formulation is equal to the horizontal component of tension divided by the weight per unit length, ^.) However, Galileo was not far from the mark. The parabola is the form assumed by a hanging cable under a uniformly distributed vertical load. Most structures have applied uniform loads that greatly exceed the self-weight of the cable, making the parabolic profile all the more appropriate. Thus, for most practical problems, the assumption of a parabolic profile is justified and simplifies the problem greatly. Interestingly enough, many of the early investigations around cables centered on vi- brational rather than static behavior. The relationship between mass, length and tension of a taut string were known qualitatively to the ancient Greeks by way of musical theory. In 6 the mid 1600s, interest in swinging pendulums and taut strings sparked the development of mathematical techniques necessary for their analysis. By the mid 1700s, general vibrational theory had been firmly established. Indeed, in 1755, Daniel Bernoulli showed that a general vibrational behavior could be broken into individual component modes, which is the basis of modern dynamics. In 1820, Poisson gave the partial differential equations of motion of a cable element under the action of a general force system. While this was a significant achievement in cable theory, the only geometries that had been solved were ones where sag was either zero (taut string) or infinite (free-hanging cable). Solutions for problems of finite sag proved elusive. (The above synopsis is adapted from Irvine [6].) By the early nineteenth century, the increasing use of cables and chains in bridge design sparked interest in the static behavior of cables. It was fairly well understood that cables exhibit a stiffening effect, i.e., deflection does not increase linearly with applied load. This is especially apparent with point loadings. Rankine developed a theory of suspension bridge static behavior that stated that dead load of the entire structure was carried by the cables [16]. Live loads were assumed to be distributed by the deck to the cables in a uniform manner; i.e., the deck was of sufficient stiffness. This analysis essentially imposes the condition that the cable profile will not change in any significant manner. As a result, most of the early suspension bridge designs incorporated a rather large stiffening truss to the deck structure. The Brooklyn Bridge by John Roebling is a good example. In 1888, Josef Melan presented the so-called Deflection Theory, which was published later in 1906 [18]. In contrast to previous theory, deflection theory was a second-order analysis method that took into account the fact that the dead loads of the structure cause deflections, and that the dead loads have a tendency to reduce the response of the structure to live loads. This theory more accurately predicted suspension bridge response than previous theories. One result of this was that designers could justify smaller deck trusses, thereby significantly reducing costs and materials. 7 2.3 Twentieth Century The first four decades of the 20th century saw increased application and development of Melans Deflection Theory. Most of the advances in cable theory during this time were refine- ments to Deflection Theory, such as accounting for stretch and leaning of the suspender cables [13]. Timoshenko also devotes a chapter to the subject in his classic Theory of Structures textbook [18]. However, with the collapse of the Tacoma Narrows suspension bridge in 1940, and a new appreciation for the dynamic nature of wind, a flurry of research occurred around the topic of dynamic behavior of cables. Mostly, these focused on the nature of free vibrations in suspended cables. Furthermore, these investigations were often limited to flat-sag cables; that is, cables in which the sag is relatively small compared to the span. It should be remembered that, though the equations of motion had been formulated a century earlier, for practical cases they had not been solved. A number of researchers, prompted by the disaster, developed theories pertaining to suspension bridge vibrations; however, they typically were not general in nature. Also, because of their limited scope, they often did not have an appreciation of factors such as large versus small sags, not to mention cable elasticity. Building on earlier work that approximated a solution to the cable vibration problem by assuming a cycloidal cable profile, Pugsley undertook a theoretical and experimental review of the problem [15]. He presented an approximate solution for the first three modes of vibration. In addition, he presented data from laboratory tests which verified some of the results of his analysis. Pugsley drew four conclusions from the experimental results: (1) the natural frequencies were independent of the mass per unit length of chain; (2) the natural frequencies were practically independent of the span of the chain; (3) the natural frequencies varied approximately inversely as the square root of the sag in the chain; (4) the ratios of the natural frequencies in successive modes were roughly 1.0, 1.5, and 2.0. 8 It should be noted that this work was semi-empirical in nature, though based on reasonable mechanical principles. The objective was to develop a rational means of estimating the natural frequencies of the lower modes. Only inextensible cables were investigated. Saxon and Cahn presented a method of solving for natural frequencies even for relatively deep cable sags [17]. Their methodology is somewhat involved, and the solutions are asymptotic in nature. This analysis, too, was limited to inextensible cables. In 1974, Irvine and Caughey presented a coherent theory of cable statics and dynamics, valid for low sag cable systems [7]. It reduced to previously known results, and resolved an embarrassing result of the theories of the time. All of the theories of cables with finite sag could not reduce to the theory of taut cables, which was well-known and tested. Previous theories for low sag cables, based on the assumption of inextensibility, define first symmetric natural frequency of vibration for systems with minimal sag as the first nonzero root of tan f f, which can be computed to be uj\ = 2.867T. However, taut string theory defines it as the first root of cos ^ = 0, or =7r (these expressions are presented here in their non-dimensionalized form, uj = -7^= ) As Irvine and Caughey pointed out, this discrepancy of almost 300% cannot V TO be resolved except by accounting for cable extensibility. In what is perhaps his most significant contribution to the study of cables, Irvine identi- fied a dimensionless parameter, which he called A2, that accounts for the relative contribution of cable displacements and cable stretch. This parameter is fundamental in describing the static and dynamic response of most, if not all, cable systems. It is defined as A2 = (#)a* \ EA ) (2.1) While this can be written in many different forms, this one, used by Irvine, illustrates the various components in play. A value of A2 equal to zero indicates a taut string. The only way a taut string can adopt a new profile is by stretching. By contrast, a cable with relatively large sag, and relatively inextensible (suspension bridges, for example) will have comparably high values of A2. It has greater freedom to deform without incurring additional tensions. One of the additional benefits using the nondimensional parameter is that effects such 9 as support flexibility and inclination are easily accounted for. Temperature effects may also be accounted for by manipulating the formulation of A2. Irvine and Caughey also reported on the interesting phenomenon of frequency crossover. For values of A2 greater than 47r2, the first symmetrical natural frequency is greater than the first antisymmetric frequency; for values less than 47r2, the first antisymmetric natural fre- quency is greater. A similar behavior is observed for the second symmetric and antisymmetric natural frequencies, at a cutoff of 167r2. This is significant because a cable system constructed with values of A2 near these cutoff values, if excited near its fundamental frequency, could shift from an antisymmetric mode to a symmetric one, and vice versa, in a fairly unpredictable manner. It is interesting to note that a similar behavior can occur in a global manner in a suspension bridge. In fact, one of the factors that doomed the Tacoma Narrows bridge is that its torsional vibration frequency was close to its vertical vibration frequency, and crossover was possible. Because the linearized derivation neglects certain higher-order terms in the differential equations, it is not appropriate for very high sag-to-span ratios. However, it produces quite acceptable results for most types of cable systems found in engineering problems, say, with sag-to-span ratios of 1:8 or less. Irvine expanded his work into a textbook that remains the definitive work on static and dynamic behavior of cable systems to this day [6]. Irvine also extended his work to include energy considerations [5]. He showed that the linear theory can be utilized in terms of extensional and gravitational energy and work done. Though the present work will not specifically address energy methods, it is perhaps important to note that the linear theory can be shown to be well-rounded. This paper has focused its attention on cable systems commonly found in bridges. However, many of the problems that motivated the development of cable theory were re- lated to electrical transmission lines, underwater mooring cables, tower guy cables, and other non-bridge applications. Underwater applications, such as oil rig supports, buoys, and tow- lines, present an especially rich field unto itself. Though deriving from the same physics as above-ground applications, underwater cable structures must contend with the peculiarities 10 of hydrodynamic and viscosity effects. Even so, much cross-pollination has occurred. Indeed, Irvine devotes a significant amount of attention on mooring and buoy anchorage problems, to name a few. 2.4 Recent Work While the body of literature pertaining to seismic analysis of cable structures is con- siderable and is an active field of study today, it is somewhat tangential to the present work. Seismic investigations have focused intently on such topics as support excitation, out-of-plane vibrations, and numerous other aspects of the subject. However, seismic loads are typically inertial in nature, and almost by definition distributed. Point load effects are perhaps not as significant in seismic analysis of cable structures. A significant amount of recent research efforts has been focused on another type of cable used in bridges the inclined stay-cables used in cable stay bridges. In the late 20th century, it was observed that these cables, which are long and have very low sag and aerodynamic weight, were experiencing excessive vibrations [4]. Further research into this phenomenon has lead to the theory of rain-wind-induced vibration. In essence, rivulets of rain flow down the inclined cable, creating a slightly altered aerodynamic cross-section. It affects the wind flow pattern around the cable, but it also responds to it, creating a situation in which resonance can occur at critical wind velocities lower than would be expected for a circular cross-section. The number of researchers who have worked on this topic are too numerous to mention here. Most have focused on the fact that the damping inherent in stay-cables is very low and look to find a way to increase damping. Three basic strategies have been proposed: (1) Tie the adjacent together with transverse cables. This is highly effective at control- ling vibrations, but can have a major and detrimental effect on the aesthetics of the structure. (2) Modify the aerodynamic profile of the cable. Though not often feasible for existing 11 structures, new structures can be built with cables jacketed or otherwise modified to provide a more optimal geometry. (3) Install transverse dampers to the cables near the supports. When designed appropri- ately, these can be very effective at mitigating vibrations. However, if not controlled properly, they can be ineffective. Of the three, the last is of particular interest. Usually, the method involves some variety of viscous damper attached to a point near the anchorage of the stay-cable. However, the stiffness of the damper mechanism must be finely tuned or the benefits may be lost. A damper that is too loose wont have enough effect. Similarly, a damper that is too stiff will simply become a vibrational support. Krenk et al. presented a solution for determining the vibrational characteristics of a stay-cable with a viscous damper [11], Their methodology is based on the work of Irvine; i.e., the solutions obtained are valid for shallow sag and relatively small displacements. Starting with the basic linearized equation of motion, a term representing the reaction at the damper, which is a function of the velocity at that point, is included. The resulting expression is H d2w d2z dx2 dx2 d2w m-Q^-+cS(x a) dw dt where c is the damping parameter and 6 (x a)is the Dirac delta function. This expres- sion may be manipulated in much the same way as in [7]. One important twist is introduced the Dirac delta function for the point load at a indicates that the solution will be discontin- uous. Recall that cables are considered as having zero flexural stiffness; it therefore must have a kink in the profile. The solution is comprised of a left and right component which require a condition of compatibility to be enforced, namely that displacements are equal at the damper location. One interesting result of this analysis is that the antisymmetric modes, which would ordinarily not be dependent upon the elastic characteristics of the cable, become just that, though to a small degree. In addition to deriving the expressions for nearly symmetrical and nearly antisymmetric 12 modal frequencies, Krenk [11] presented an expression for determining the optimal value of c for tuning the damper for maximum damping effect. Using a similar technique, Johnson et al., investigated the damping effectiveness of semi-actively tuned viscous dampers on both taut cables and low-sag cables [9, 10]. Unlike the passive dampers used by Krenk et al., semi-active dampers can provide any required dissipative force (similarly, an active damper is one which can supply ANY required force.) While semi- active dampers can be more effective than passive dampers, they require real-time modulation to realize optimal performance. This in turn requires a model by which to evaluate the system. The goal was to develop a model which could be used to adjust the damping mechanism in real-time to respond to the changing load conditions. Pacheco et al. [14] had shown that a modal analysis using sinusoidal shape functions and equations of motion that incorporated a) dampers modeled as point loads, and b) a random noise loading function, could be used to predict optimal damper tuning. Unfortunately, the computation of hundreds of modes was required for sufficient accuracy, which would be too demanding for the damper controller systems. One of the main reasons a large number of modes were necessary was to approximate the kink in the deflected shape at the damper. Johnson showed that by replacing the first mode shape with a new shape consisting of the first mode sinusoidal shape function minus the normalized static deflection shape, one could significantly improve convergence, from several hundred modes to under ten. Like Krenk and Pacheco, Johnson also derived the equations of motion based on Irvines work [7, 6]. 2.5 Numerical Methods and Computers As may be surmised, the widespread availability of powerful personal computers has made numerical methods a powerful means of solving cable problems. The difficulty or even outright impossibility of deriving a closed-form solution often makes the computer the only recourse. As such, computers have become an indispensable tool for the researcher and prac- ticing engineer alike. While the scope of the present work is not to introduce new numerical 13 methodology, it bears noting that it does make use of techniques developed over the past forty years, and it is worth mentioning a few of them here. While this list is by no means exhaustive, it does touch on some of the more popular methods. One of the first practical, and still relevant, numerical methods has been Newmarks Method. Developed by Newmark in 1959 [3], it describes a whole family of time-stepping methods which use an assumed variation of acceleration to predict the accelerations, velocities, and displacements of a given system subjected to a given loading function. Though developed for linear systems, it can be extended to non-linear systems as well. The algorithm is easily implemented on a computer, and its stability and accuracy are quite good, which has made it especially popular with researchers. However, because its normal implementation requires fairly detailed manipulation of the raw equations of motion themselves, it may not be suitable for general analysis. Another method of analysis that has proven wildly popular is the finite element method. It and other matrix methods relying on multiple degree-of-freedom stiffness calculations, have shown themselves to be very adaptable and have become the method of choice for most com- mercial analysis software systems. However, analysing cables using the finite element method requires additional effort and typically results in iterative solution methods. One reason for this is that the basic finite element method cannot handle large changes in geometry. As such, the algorithm must incrementally apply load until convergence to the final profile has been achieved. This results in significant computation, as the problem must be solved for each load increment. It also can lead to stability problems in the solution. Another method that is increasing in popularity is the brute force writing and solving of the system of differential equations of motion simultaneously. With computers becoming ever more powerful, this is no longer only suitable for large, expensive supercomputers. This method is proving very effective in analyzing systems involving kinematic behaviors, such as collisions, blast loading, component interactions and so forth. 14 3. Theoretical Analysis 3.1 Overview To investigate the behavior of catenaries subjected to impulse point loads, a theoretical analysis was completed using linear cable theory. A standard modal approach for distributed mass systems was used, with normal coordinates computed using Newmarks method. 3.2.1 General Theory The equilibrium of an infinitesimal flat-sag cable element under the influence of a slight disturbance may be written as where u and w are the longitudinal and vertical components of displacement, respectively, and tension generated. See Figure 3.1 for a definition diagram of the differential element. Since a shallow cable profile (flat-sag) was assumed, a few simplifications may be in- troduced. After expanding the equations and substituting the equations of static equilibrium, second-order terms may be dropped. Also, the longitudinal component may be ignored. Fi- 1 Sec Appendix A for an overview of cable theory, including static behavior. 3.2 Linear Theory of Cable Dynamics1 (3.1) v is the out-of-plane component. T is the initial tension in the element, and r is the additional 15 X Figure 3.1: Coordinate System Definition Diagram for Equilibrium Equations nally, out-of-plane effects are not of interest here and may be ignored, expression TTd2w dPz d2w H~dx^ + hd^=mW What remains is the (3.2) This defines the force equilibrium of the element. The cable equation, which relates the geometric changes (i.e, stretch) of the cable to additional tension, is Mife)3 = du + dzdw EA dx dx dx or in integrated form hLe ~EA mg H (3.3) Equations 3.2 and 3.3 may be used to determine the in-plane of free vibrations of cables according to linear theory. 3.2.2 Free Vibration and Modal Behavior The modal frequencies and vibration shapes were determined using Equations 3.2 and 3.3. By using the substitutions w(x,t) = equation of motion may be rewritten as dx2 hmg ~H~ eluJt = mi2u>2 or + mw2(f> = 0 (3.4) axJ H Recall that antisymmetric modes are modes with an inflection point at midspan and gener- ate no additional tension, while symmetric modes are symmetrical about midspan and are associated with additional tension due to cable stretch. 3.2.2.1 Antisymmetric Modes Antisymmetric modes have the property that f wdx = 0, therefore h = 0, and the second term in 3.4 vanishes, resulting in ^dx^ + TrUjj2(t> = The solution to this second order linear differential equation is 0 (x) = Ci cos (3x + C2 sin (3x where j3 = Using the boundary conditions 0(0) = 0, and 0 (Â§) =0, the constant ci can be shown to be zero if a trivial solution is to be avoided. The constant C2 is an arbitrary non-zero scaling factor for the mode shape amplitude, resulting in the expression PL n C2 sin = 0 Since C2 must be non-zero to avoid a trivial solution, sin must be zero. Therefore, 0L Tin where n = 1,2,3____ Substituting for /?, L fm 17 Figure 3.2: First Three Antisymmetric Mode Shapes 2mr H TV m (3.5) This expression gives the natural circular frequency for the nth antisymmetric mode of vibra- tion. Similarly, , , . , . 27m 4>n (x) = Ansm -j-x (3.6) gives the mode shape for the nth antisymmetric mode of vibration, where Ais an arbitrary amplitude scaling factor. Figure 3.2 shows the first three antisymmetric modes shapes. 3.2.2.2 Symmetric Modes Because symmetric modes involve additional tensions caused by cable stretch, the h term in 3.4 cannot be ignored as it was in the case of antisymmetric modes. As a result, use of the cable equation3.3 is required to arrive at a solution. 18 Again, the equilibrium equation 3.4 is used. The complimentary solution to this differ- ential equation is 0C (x) = ci cos /3x + C2 sin /3x where f3 = A particular solution 0P = is chosen so that the complete solution is 0 (x) ci cos fix + C2 sin (3x + hg Hu2 The boundary conditions 0(0) = 0, and mined as c i = - hg Hw2 and hg f cos j3L 1 \ Hoj2 \ sin /3L sin (iL J resulting in the following expression for mode shape cos (3L sin (3L + ^_) sin PL J sin Px cos Px (3.7) The expression in parentheses can be simplified using trigonometric half-angle identities to tan resulting in 0 (x) = ^ ( 1 tan ^ sin Px cos Px) Hur \ 2 / The cable equation 3.3 may be rewritten as hLe ~EA mg H (3.8) and when 0(x)is integrated and substituted into 3.8, the h terms drop out, leaving mg2EA LH2lj2 PL + tan ^ cos PL tan ^ sin PL 19 This may be further simplified to j3L (3LeH2u>2 0L 2 2mg2EA *aU 2 When substituting for (3, and recognizing that A2 = L the following transcendental equation may be written This equation must be solved for lo numerically (in this work the Newton-Raphson method was used) to obtain the natural frequencies for symmetrical in-plane modes of vibration. Recalling Equation 3.7, it should be noted that the term is a constant multiplier that only affects amplitude of the mode shape, not the shape itself. Therefore, it may be replaced with an arbitrary multiplier for use in modal analysis. Thus, the mode shapes for symmetrical in-plane modes is given by 4>n (x) = An f 1 tan sina)r x coswr (3.10) It is worth noting that for cables with A2 < 47r2 the first symmetric frequency is less than the first antisymmetric frequency. The first symmetric mode shape contains no internal nodes. When A2 = 47r2, the first symmetric frequency equals that of the first antisymmetric frequency. As A2 increases beyond 47r2, the first symmetric frequency is greater than the first antisymmetric frequency, and the first symmetric mode shape has two internal nodes. Figures 3.3 and 3.4 show the first three symmetric mode shapes for values of A2 less than and greater than 47T2, respectively. Figure 3.5 shows the symmetric and antisymmetric natural frequencies over a range of values of A2. It can be seen that similar crossovers occur for higher modes. Though somewhat tangential to this work, it is worth discussing this behavior. The crossover phenomenon as it is known indicates that cable systems with A2 ss 4n2 which are excited in, say, an antisymmetric mode of vibration, can switch to a symmetric mode, often in 20 Figure 3.3: First Three Symmetric Mode Shapes (A2 < 47r2) 21 Figure 3.4: First Three Symmetric Mode Shapes (A2 > 47t2) 22 0.01 0.1 1 10 100 1000 10000 Values of A2 Figure 3.5: Symmetric and Antisymmetric Frequency Behavior over a Range of Values of A2 23 an unpredictable manner. Thus, it is important for designers of cable systems to be aware of this possible behavior. 3.2.3 Forced Vibration Response and Modal Analysis Irvine [6] gives the linearized equation of motion for a forced flat-sag cable as H d2w dx2 d2 w m dt2 mg H h(t) +p(x,t) (3.11) where h (t) = f w{x t}dx (3.12) H Le Jo The goal is to find a solution to 3.11 in the form of a modal expansion. Hence, a solution is written as OO w(x,t) = '^2 n= 1 where 4>n is the nth in-plane modal shape function, and qn is the nth associated normal coordinate. The normal procedure for modal analysis of distributed mass systems is then followed. First, Equation 3.11 is rewritten by substituting the assumed modal solution 3.13 OO OO (t) r (x) = ^h (t) + p (x, t) |