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An experimental and theoretical investigation of the behavior of cables under point impact loading

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An experimental and theoretical investigation of the behavior of cables under point impact loading
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Ledesma, Michael Scott
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English
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xv, 160 leaves : ; 28 cm

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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 )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 157-160).
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Department of Civil Engineering
Statement of Responsibility:
Michael Scott Ledesma.

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|University of Colorado Denver
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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 (£ =2%).................. 41
3.14 Additional Tension Response for Impulsive Load at 0.5L (£ =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 (£ =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 (£ =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 (£ =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
£ Fraction of critical damping
A2 Dimensionless cable parameter
Vertical component of mode shape
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) = {x)elult, h(t) = helwt, and the
equation of motion may be rewritten as
dx2
hmg
~H~
eluJt = mi2u>2 16


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
9
^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 (§) =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 (L) = 0 are applied, and the constants are deter-
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
(x)
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) = '^2n(x)qn(t) (3.13)
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)
r=l r=l
where dots (") and accents ( ) indicate derivatives with respect to time and length, respec-
tively. Next, multiply both sides of the equation by the cable. Note that the summation and integration operations are interchanged at this step
as well
oo -L oo -L
Hy^qr (t) / " (x) (j)n (x) dx mV qr (t) / T (x) 4>n (x) dx
r=l Jo r=l Jo
-/i(t) / Jo Jo
mg,
H
24


Because of the orthogonality of q (t) and (x) [8], all the terms of the summation vanish except
for r = n. Thus, Equation 3.14 becomes
pL pL
Hqn(t) / n(x)4>n(x)dx-Tnqn(t) / 4>l(x)dx
Jo Jo
= 1jphn(t) J 4>n(x)dx + J p (x, t) Equation 3.4 can be rewritten as
(*)
mg
H2
hn
nibj
H
~4>n {x)
(3.16)
and substituted into 3.15
pL ryy-^ TWjP" /"^
Hqn (t) j rj^hn---------~lfL(t>n ^" (X) dx miin (<) J 4>l ix) dX
= rjphn(t) J n(x)dx + J p(x,t)4>n{x)dx (3.17)
After some rearrangement, and recognizing that hn (t) = hnqn (t),
qn (f) -Jfhn J 4>n (x) dx qn (t) J mwfyl (z) dx mqn (t) J 4% (x) dx
- qn(t)^-hn J 4>n(x)dx +J p(x,t)4>n(x)dx (3.18)
The two terms with hn cancel, leaving the following expression
pL pL pL
mqn{t) 4>l(x)dx + qn(t)mLjl 4>l{x)dx = p(x,t)4>n(x)dx (3.19)
Jo Jo Jo
Dividing all terms by m 4>n (z) dx results in the differential equation for the nth mode
normal coordinates
f_ n (x. t.) d)_ lx) d.x.
(3.20)
.. s . 2 _ fo P (x 1 t) tfin ix) dx
Qn (t) + qn {t) mfo 4>l{x)dx
It can be seen that this derivation did not contain a qn (t) term or address the effect of
25


damping. Cable systems often have damping values that are extremely low. Values may range
from 0.5% to 4% or greater. Their magnitude is influenced by cable construction type, cable
geometry, and the ambient environment. Research has suggested that the primary means of
energy dissipation is by rubbing between the individual strands in a helically wound cable. It
has also been shown that damping can be increase significantly is the cable is slackened.
Irvine [6] accounts for viscous damping by adding a term to the normal coordinate
equation. Thus, the final normal coordinate equation is given by
Qn (f) d- 2qn (f) U^nCn d" Qn (0
foL P (x, t) n (X) dx
mIo (X)dx
(3.21)
where (n is the damping ratio for the nth mode of vibration.
Up to this point, no consideration has been given to the nature of p(x,t). For the
purposes of this work, the load function must be defined such that the result is a point load
at some location along the cable span. To accomplish this, the load function is written as
p (x, t) = P0 (t) 6 (x xq)
(3.22)
where <5 (x xo) is the Dirac delta function and Pq (t) is the magnitude of the point load
applied at xq with respect to time. The normal coordinate equation becomes
Qn (0 d 2Qn (t) L^nCn d Qn (^) ^n
Po (t) f0L S(x Xo) n (l) dx
m Jo 4 n (X ) dx
(3.23)
The numerator of the right side can be integrated directly, resulting in
Qn (t) d- 2Qn (t) u>nC,n + Qn (t) 77T"TTT7_ (3-24)
mJo Kx)dx
This equation may be solved for qn (t) using a time-stepping numerical analysis such as New-
marks Method. Point load magnitudes may be input from load history data, a time function
defined. In this work, a load history was recorded from experimental measurements, and used
to compute a corresponding theoretical response.
26


gEggasag^^ ........piwuipiui t^a
Figure 3.6: Screenshot of Rapid-Q Integrated Development Environment
3.3 Numerical Analysis
3.3.1 Description
To perform the modal analysis required to obtain a solution to the forced vibration
problem, a computer program was written using Rapid-Q, version 1.0.0. Rapid-Q is a BA-
SIC compiler written by William Yu, and is freely available. Rapid-Q uses syntax similar to
QBASIC, with modifications to allow the manipulation of windows, forms, and other interface
elements. Since the task at hand was primarily computational in nature, a sophisticated user-
interface was not necessary; however, it was convenient to be able to select input files, etc.
using standard file selection dialogs instead of requiring the user to type them in manually.
Because of its slowness relative to languages such as C, BASIC is not usually associated
with mathematical analyses. However, the rapid program development possible with high-
level languages in this case far outweighed the compromises in computational speed. Figure
3.6 shows a screenshot of the Rapid-Q integrated development environment (IDE).
The program itself can be broken into three parts:
(1) Input of problem parameters, and preliminary computations
27


(2) Computation of normal coordinates using Newmarks Method
(3) Output of problem solution data
The program created comma-delimited text files containing the solution data, which were
imported into a spreadsheet for review and presentation.
Input files were simple ASCII text files containing certain problem parameters. See Ap-
pendix B for the input and output file formats. One input file, containing a INP file extension,
contained basic values such as span length and cable weight per unit length. Another file, con-
taining a .HST file extension, contained a series of values representing point load magnitudes
at regular time slice intervals, i.e., a time-history of loading.
3.3.2 Summary of Computation Algorithm
Figure 3.7 shows a flow-chart of the basic computational algorithm. The program
follows a fairly linear progression from start to finish.
The program reads in the user-supplied input parameters from a *. INP file. Required
parameters are:
span length (L),
cable cross-sectional area (A),
weight per unit length (w),
cable modulus of elasticity (E),
midspan sag (s),
point load location (xO),
maximum number each of symmetric and antisymmetric modes (maxmodes),
time stepping increment (timeinc),
damping ratio for each mode.
28


Figure 3.7: Flow-chart for CableCalc Program
29


In addition, three parameters used in the calculations are derived from the above values. These
are
mass per unit length, m = w/g (m),
static horizontal tension, H = (H),
cable length, Le ~ L + 8 (s2/L) (Le),
the independent cable parameter, A2 = (lafflbda-squared).
After determining the basic problem parameters, the program proceeds to compute the re-
quested antisymmetric and symmetric natural frequencies. It does so using the Get_Antisym-
metric_Frequencies() and Get_Symmetric_Frequencies() functions, respectively. The an-
tisymmetric frequencies are calculated using Equation 3.5 in a straightforward manner and
stored in an array Freq_A(l to maxmodes). The symmetric frequencies are a little trickier.
Because the non-zero roots of Equation 3.9 must be solved numerically, the function Newton_-
Raphson_Root() is implemented, which finds the nearest root of Equation 3.9 about a given
seed value. Because Equation 3.9 can have large slopes, especially for higher modes, Get_-
Symmetric_Frequencies() calls Newton_Raphson_Root() for a large number of values over
a defined range of frequencies (arbitrarily set from zero to 60 rad/sec). Though rather crude
and computationally wasteful, this ensures that all the possible roots are found. Because
the vast majority of roots found this way will be duplicates, the program only records roots
if they have not been found previously. Though the computations performed for this work
limited the considered modes to five, the program can handle more modes. Because of the
difficulty in (or outright impossibility of) obtaining large roots Equation 3.9, the program will
only compute the first five frequencies with that expression. For higher modes, the expression
to maxmodes).
After determining the frequencies, the function Sort_Freqs() is called. This simply
sorts the frequencies to ensure that they are listed in ascending order. Though not really
an issue for antisymmetric modes, because of the way symmetric modes are calculated, it is
Symmetric natural frequencies are stored in the array Freq_S(l
30


possible that higher modes are found before lower modes. It should be noted that symmetric
modes and antisymmetric modes are not combined, and are kept segregated.
The mode shapes themselves are implemented as functions. Antisymmetric_Shape_-
FunctionO and Symmetric_Shape_Fuiiction() return the value of the antisymmetric or sym-
metric mode shape, respectively, at a given point along the span. For antisymmetric modes,
Equation 3.6 is used. For symmetric modes, Equation 3.10 is used for the first five modes.
Should greater than five modes be requested, the expression (f>n (x) = An sin x is used
for those higher modes.
Though not strictly necessary, the mode shapes are normalized to a maximum mag-
nitude of unity. The scaling factors required are computed by the function Get_Scaling_-
FactorsO.
Load magnitude information is read from an external data file (*. HST). The format
of this file is quite simple. It consists of standard text with one load value per line. The
program interprets these sequential values as occurring at the previously input time increment
(timeinc). The time history data could be artificial or measured experimentally. Figure 3.8
shows a sample time history and a portion of the associated data file. The function Read_-
Load_History() handles reading in the load information. It requires the filename before the
extension to be identical to that used for the basic problem input file (i.e., ABC123.INP <>
ABC 123. HST.)
The function Newmarks_Method() is the workhorse of the entire program. This is where
the normal coordinates are computed using Equation 3.24. The program follows a fairly
standard implementation of the Newmark algorithm. Newmarks Method is performed for
each mode, and for symmetrical and antisymmetric modes, separately. The linear acceleration
method is used ((3 = \, A = \)- Preliminary calculations are as follows:
31


Applied Force (N)
0 1280853c5
0.123085365
0.12?u8c365
0. 128035 365
-0.60*393/7
-0.SO 539337
-fj 18307582*
50.0
40.0
30.0
20.0
10.0
0.0
I
0.0 0.1 0.2 0.3 0.4
-0.60*393/7
-0.494237016
0 43924172
0.43924172
-1.11656456
-0.183075825
0.75040291
1.663876909
1.995037999
3.S61?90633:
5.417786913,
10.70749813;
20.6645595 i
37,77835244
49.60238095
4.795469369'
5.417786913.
-1.427710914
-0.494237016
-0.4942370X6
Tt
ic[5Uv7
Figure 3.8: Sample Time History and Load Data Format
32


1
k
a
b
k +
7 c
+
/3A t /J(A*)2
1 / 7C
a;"*+7
where cn = 2tjn£n, kn = and m' = 1. The program then loops through each time interval
performing the following calculations
Apt

A?i
Agt
Qi+1
9i+l
LF (Pj+i P) + ag* + biji
Api
1 A 1 1 ..
/3(At)2 /3At9i 2/}9i
qi + A 9i + Agi
g'i + Agii4
A little explanation regarding the first equation is in order. Pi and Pi+i are values from the
load history and are fairly self-explanatory. LF, however, needs some clarification. This is a
load factor that has a value of
________(Sp)
mfo It arises from the form of Equation 3.24. The program contains functions for evaluating the
integral term, based on the Trapezoidal Rule of integration.
Once the program has computed normal coordinates for each mode (both symmetric
and antisymmetric) and for all time increments, it outputs the final solution data. These
include modal displacements, total displacements as various points, modal and total additional
horizontal tension, mode shapes, and the normal coordinates themselves. These are output
33


into the following files, assuming an input file ABC123.INP:
ABC123_TENSI0N.CSV
ABC123_DISP.CSV
ABC123.N0RM.CSV
ABC123_M0DE_SHAPES.CSV
ABC123_BASIC.TXT
(Another file, ABC123_VIS.CSV is also produced, but is only a debugging aid used by the
author and may be ignored.) As their names imply, they contain the various computed data,
described below. Files with a .CSV extension are comma-delimited text files, which are easily
imported into spreadsheets or other programs while still maintaining a columnar format.
The _BASIC. TXT text output file contains a restatement of the input problem parameters
and is intended as a check to ensure the program has read the input files correctly. Also
contained in this file are the values of the computed natural frequencies.
Tension values are computed using the expression
for the nth mode and the ith time increment. Though this computation is performed for both
symmetric and antisymmetric modes, the nature of antisymmetric mode shapes should always
make the integral evaluate to zero. Computing this serves as a good way to debug the program.
The .TENSION. CSV files contain columns of data, the first of which is time. Subsequent columns
contain the additional horizontal tension for each mode. The final column contains the sum
of all the modal tensions for a particular time increment.
To compute total displacement response at x for time increment i the following is
computed
maxmodes
^ ^ ^(9n,z0n(^))
antisymmetric

symmetric
n=1
34


The _DISP. CSV files start with a column of time values, followed by total displacement response
at locations corresponding to the displacement gages used in the experimental study (see
Chapter Four). These locations are hard-coded in the program, but could be changed with
relative ease.
The _MODE_SHAPES.CSV simply contain the normalized mode shapes used in the com-
putations. These are organized in rows instead of columns (the first line contains mode 1,
second line contains mode 2, etc.)
The _N0RM. CSV files contain the computed normal coordinates for each mode. The first
column is time, the subsequent columns are the normal coordinates for the requested modes. It
is useful to review these to see the relative impact various modes have on a particular response.
3.3.3 Input Parameters
The input parameters used for the theoretical analyses were selected to correspond to
the model developed for the experimental program (see Chapter Four.) The intention was to
produce a theoretical response that matched the observed response.
Specific details of the model are reported in Chapter Four and are not discussed here.
However, the input parameters for the computer program are as follows:
span length = 8484 mm (334 in),
cable cross-sectional area = 4.5 mm2 (0.007 in2),
weight per unit length = 0.0164 N/mm (0.0967 lbs/in),
cable modulus of elasticity = 170 GPa (24,650,000 lbs/in2),
midspan sag = 381 mm (15 in),
maximum number each of symmetric and antisymmetric modes = 5,
time stepping increment = 0.005 sec,
damping ratio for each mode = 2%.
35


While numerous tests were performed, it was decided that reporting on four individual sets of
input parameters would be sufficient to illustrate the salient characteristics without drowning
the reader in graphs and figures. These included two different load application locations, and
two different load types.
The two locations that were reported were 0.5L and 0.2L. It was felt that these two load
points would illustrate the behaviors of both symmetrical and antisymmetric modes. Loads
applied at midspan primarily excite the symmetric modes, while loads applied at 0.2L primarily
excite antisymmetric modes.
The two load types investigated were a suddenly-applied persistent point load, and an
impulsive point load that was not persistent. The suddenly-applied load was simply a weight
that was allowed to drop from very nearly zero height above the unloaded cable. It was firmly
connected to the cable by a string, and so continued to affect the vibration response. The
impulsive load was also applied by dropping a weight from the cable, but in this case a clip
release was provided that allowed the weight to disengage from the system after a certain load
threshold was achieved. In this way, no persistent effect on the response would occur.
Therefore, the four conditions that were analyzed were as follows:
(1) a suddenly applied point load at midspan (0.5L),
(2) an impulsive point load applied at midspan (0.5L),
(3) a suddenly applied point load at 0.2L, and
(4) an impulsive point load applied at 0.2L.
The same basic parameters listed previously were used in all analyses.
3.4 Results
The results of the linear modal analyses are presented in the figures below. Each plot
contains displacements plotted against time, or additional horizontal tensions against time.
The load history data used in the analysis is reproduced on the plots for convenience.
36


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec)
Time (sec)
1.5
Time (sec)
Figure 3.9: Displacement Response for Suddenly Applied Load at 0.5L (C =2%)
37


Applied Force (N) Displacement (mm) Displacement (mm)
Figure 3.10: Displacement Response for Suddenly Applied Load at 0.5L (£ =2%)
38


Figure 3.11: Additional Tension Response for Suddenly Applied Load at 0.5L (£ =2%)
39


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 1 2 3 0 1 2 3
Time (sec) Time (sec)
Figure 3.12: Displacement Response for Impulsive Load at 0.5L (£ =2%)
40


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
30 25 20 E 15
AAAM/w fc 1 u c 5 E 0 0 -5 (0 a--10
D5 b -15 -20 D6
012: -25 -30 3 >-4. 112 3
Time (sec)
Time (sec)
Figure 3.13: Displacement Response for Impulsive Load at 0.5L (£ =2%)
41


Additional Cable Tension (N)
Figure 3.14: Additional Tension Response for Impulsive Load at 0.5L (( =2%)
42


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
Time (sec) Time (sec)
Time (sec)
Time (sec)
Time (sec)
Figure 3.15: Displacement Response for Suddenly Applied Load at 0.2L (£ =2%)
43


Applied Force (N) Displacement (mm) Displacement (mm)
Figure 3.16: Displacement Response for Suddenly Applied Load at 0.2L (£ =2%)
44


Additional Cable Tension (N)
Figure 3.17: Additional Tension Response for Suddenly Applied Load at 0.2L (£ =2%)
45


Figure 3.18: Displacement Response for Impulsive Load at 0.2L (( =2%)
Applied Force (N)
W W ^ Ul O)
o o o o o o o
Displacement (mm) Displacement (mm)
w ro ro i ro ro co corbro-^--^. ro ro co
ocnocnocnooiocnocno ocnocnocnocnocnocno


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec)
1 2
Time (sec)
Time (sec)
Figure 3.19: Displacement Response for Impulsive Load at 0.2L (£ =2%)
47


Additional Cable Tension (N)
Time (sec)
Time (sec)
Figure 3.20: Additional Tension Response for Impulsive Load at 0.2L (C =2%)
48


4.
Experimental Analysis
4.1 Introduction
In order to evaluate the theoretical analyses, an experimental analysis was completed
in which a scale model of a simple suspended cable system was constructed, instrumented, and
observed under a variety of static and dynamic loadings.
The scope of this experimental program was limited to the behavior of a single cable
system geometry. In addition, only the response to single static and dynamic point loads was
investigated. The dynamic loads were impulsive in nature, generated by masses falling from
various heights. The masses were connected to the cable, therefore the response reduced to
the static response after damping reduced the transient effects to zero.
4.2 Objective
The objective of this experimental analysis was to observe and measure the response of
a suspended cable to dynamically applied point loads. In addition, the static response of the
cable was also measured to provide confirmation of the basic cable behavior.
The response of the cable was determined by measuring the displacements at up to
eight locations, and the change in tension at eight locations. All measurements were made
with respect to time. Also, the applied load was also measured with respect to time.
4.3 Instrumentation
The basic data acquisition strategy was to use an electronic data logger to collect
data from a variety of sensors. After tests were completed, data was downloaded to a laptop
computer and imported into a spreadsheet for further processing and manipulation. In general,
numerical conditioning such as zeroing of measurements and converting into proper units was
49


not done on the datalogger. This was done primarily in the interest of achieving the fastest
data sampling frequency. Rather, such conditioning was done in a spreadsheet.
4.3.1 Data Acquisition Unit
A Campbell Scientific CR5000 data logger was used to collect data from the sensors.
This unit is capable of reading up to 20 differential measurement channels. As this project only
used 17 devices, this proved to be adequate. The accuracy of measurement is rated at 0.05%
over the range of 0 to 40 degrees Celsius. (Since testing was done in a laboratory at essentially
room temperature, effects of temperature variation were not considered to be significant.) The
fastest reliable data sampling rate achieved was 5 milliseconds per scan. Figure 4.1 shows the
CR5000 data logger.
The data logger uses a serial interface (RS-232) to communicate with a computer. In
this research, a laptop with a 1.8 GHz microprocessor and 256 Mbytes of RAM was used.
50


Because this machine did not have a serial port, the USB port was used along with a serial-
to-USB converter.
To control the CR5000, test program code is written in a language quite similar to
BASIC. In it, one defines data storage structures, sampling rates, trigger events and so forth,
such that the data logger knows what it must read, when it must read it, and where to store
the data. Fortunately, the control software is a proprietary interface from Campbell Scientific
called PC9000. It features a graphical interface that allows the user to create testing programs
in a structured and intuitive manner, using point-and-click functionality. The interface then
generates the actual code used by the data logger. Alternatively, editors are provided to
manually create or edit the test programs that require a higher degree of complexity. However,
the graphical interface was adequate for the needs of this research, and was the primary means
of generating testing programs. Testing programs were created for each type of test performed
on the cable model.
The PC9000 software also contains real-time control and visualization capabilities which
were used extensively to verify the correct operation of the data acquisition system, and the
model behavior itself. With the software, one may in real-time display a field of measured data
in a spreadsheet-like format or in a graphical oscilloscope-like manner.
4.3.2 Displacement Gages
To measure displacements, a displacement gage was designed that offered acceptable
resolution, minimal influence on the system being measured, and economy. A design utilizing
a rotary linear potentiometer actuated by a lever armature was developed. See Figure 4.2 for
a schematic of a typical displacement gage.
This displacement gage design functioned essentially uses a potentiometer (variable
resistor) as a voltage divider. The circuit is powered by two AAA-size batteries, which feeds
current through a 1 megaOhm linear taper potentiometer. Two leads from the potentiometer
provide a voltage difference to be measured by the data logger. This voltage difference is
proportional to the turn of the armature.
51



NOTE: ALL DMBJSION3 IN HILL METERS UNLESS OTNERWBE NOTED
Figure 4.2: Displacement Gage Schematic


The armature is simply an 11.1 mm (0.438 in) diameter wooden dowel. It is firmly
fastened to the spindle of the potentiometer. The armature extends beyond the spindle to
allow the addition of a counterbalance to minimize the effect of gage resistance on the cable
model itself, however, this was found to not be necessary.
The potentiometer is mounted to an L-shaped strip of aluminum which forms the body
of the gage. To this, a battery holder is also fastened. The gage is mounted by clamping the
aluminum base to any suitable surface.
The gage was mounted above the cable model such that the armature was approximately
horizontal, with the tip being directly above the cable. A thinner wooden dowel was used
to connect the cable to the end of the armature. This linkage-type mechanism allowed the
direct transfer of vertical motion while not offering resistance to the small horizontal motions
inevitable in the cable model.
Because the displacement gage is connected to the suspended cable with a rigid linkage,
a small degree of nonlinearity is introduced into the measurement because a vertical displace-
ment is being measured by a circular mechanism. In simple terms, the error is analogous to
the difference between an arc and its chord, or the classic approximation of an angle being
equivalent to the sine of an angle. However, given a sufficiently long lever arm (i.e., arc radius)
relative to the chord distance being measured, the error is minimal. The devices used in this
research have a radial distance of 305 mm (12 inches), while the displacements are generally
in the range of 25 mm (1 inch). The gages were calibrated over a range of 75 mm (3 in),
over which no significant nonlinearity of the gage was observed (see Appendix C).
Figure 4.3 shows the array of displacement gages in place on the model.
4.3.3 Tension Gages
The tension in the cable was measured with strain gages. However, a small diameter
braided steel cable was used for the cable model, making the direct application of strain
gages not feasible. Therefore, small aluminum cylinders were machined to provide a small flat
mounting area with small cross sectional area to which the strain gages could be attached.
53


Figure 4.3: Displacement Gages on Cable Model
54


NOTE: ALL DMENSONS IN MIUHETERS UNLESS OTHERWISE NOTED
12.0
7.7
/TT^
3.2# THREADED
HOLE

Figure 4.4: Tension Gage Schematic
These aluminum cylinders also had holes drilled in each side into which the cables could be
inserted and glued. Thus, the tension gage functions as an insert between segments of cable.
Figure 4.4 shows a drawing of the tension gage.
The flat area was sized to be just large enough to mount a strain gage, and thick enough
to provide minimal cross sectional area while not experiencing bending effects.
The holes in each side were drilled to be just large enough to accept the cable. These
holes were then threaded to provide a mechanically rough surface to bond to. An epoxy
adhesive with minimal creep characteristics was used to bond the gage to the cable (see Figure
4.5.)
55


Figure 4.5: Typical Installed Tension Gage
56


NOTE: ALL DHBISONS IN WLLMETERS UNLESS OTHERWISE NOTED
STEEL BAR (SUDES
FREELY M BLOCK
AND RESTS ON
DEFLECTION BAR)
STEEL DEFLECTION
.BAR (SNPLE SPAN
BIWN 2 SCREWS)
Figure 4.6: Applied Force Gage Schematic
4.3.4 Applied Force Gage
To measure the force applied to the cable by a falling weight, a device based on the
bending of a simply supported beam was devised (see Figure 4.6). This consisted of a block of
wood approximately 178 mm (7 in) with a strip of steel screwed to the bottom. A rectangular
hole was made in the block of wood and another shorter steel bar inserted perpendicular to
the first such that it rested on top of it. In this way, a load hung from the shorter bar bore
directly on the midspan of the longer bar, like a point load on a simply supported beam. The
wood block was hung from the cable, and the weights were suspended from the short steel bar.
57


Figure 4.7: Applied Force Gage Showing Strain Gage
A strain gage was applied to the long bar to measure the flexural tension in the bar.
Provided yielding was avoided, this tension was proportional to the applied load. The advan-
tage of this method was that the sensitivity was significantly amplified over that of a plain
bar in direct tension. The disadvantage is that the device tended to be somewhat bulky and
unwieldy (Figure 4.7.)
4.3.5 Strain Gages
Strain gages were metal foil-type devices purchased from Vishay Micro-Measurements
(type CEA-13-125UW-120). These gages are composed of constantan alloy resistance elements
in a cast polyamide backing material with copper solder tabs. They had a nominal resistance
of 120 Ohms. The gage length was 3.2 mm (0.125 in), while the overall length and width were
10.7 mm and 6.9 mm, respectively (0.42 in and 0.27 in). The gage factor for the gages used in
this research was 2.100 0.5% at 24 degrees Celsius.
58


Figure 4.8: Typical Strain Gage (Not to Scale)
The strain gages are connected to the data logger via a full Wheatstone bridge, which
enables the variation in resistance to be read by the data logger as a voltage differential. In
addition, a third wire paralleling the signal wires from the instrument can be used to cancel
out the effects of wire resistance. The bridges used in this research were small prepackaged
units from Campbell Scientific called TIMs (terminal input modules.) They contain all the
necessary circuitry and completion resistors while being packaged small enough to plug into
the data logger input terminals directly (see Figure 4.9.) The strain gages were wired directly
into the TIMs, which also handle powering the strain gage circuit. The TIMs are specific to
the strain gage resistance; i.e., a 120 Ohm TIM must be used with a 120 Ohm strain gage.
Model number 4WFB120 TIMs were used in this project.
Wire used was 22 gage braided copper wire in a three-conductor configuration, color-
coded in black, white, and red for convenience.
4.4 Cable Model
The cable model was essentially a braided steel cable suspended between two masonry
walls in the west end of the Soils Testing Laboratory in the North Classroom building on
the Auraria Campus, Denver, CO. The sag of the cable was adjusted through the use of a
turnbuckle on one end of the cable. The distance between the walls was 8585 mm (338 in), but
59


Figure 4.9: Terminal Input Module
60


Figure 4.10: Cable Anchorage (West End)
the clear span of the cable model itself (accounting for the size of the anchorages) was 8484
mm (334 in).
The cable was a helically-wound steel cable of 3.2 mm (0.125 in) nominal diameter,
with a 7x7 wire pattern, purchased at a local hardware store. It was measured to have a wire
diameter of 0.33 mm (0.013 in) for a total cable cross-sectional area of 4.5 mm2 (0.007 in2). It
was anchored to the walls using plate-mounted steel eyes, cable thimbles, and rope clips (see
Figure 4.10.) A turnbuckle was provided on the east end of the cable to adjust sag.
The cable was constructed of segments of cable connected together by tension gages.
Figure 4.11 shows the geometry of the cable and the locations of the gages. Eight tension
gages were installed on the cable at approximately equal spacing. One was provided at the far
west end of the cable; no corresponding gage was located at the east end.
Though the research is not specifically focused on bridges, the cable model was cali-
brated to behave like a suspension bridge cable. The static and dynamic behavior of a cable
61


Figure 4.11: Model Geometry and Instrumentation Layout
62


Figure 4.12: Additional Weights on Cable
can be described using a single parameter, A2 (herein referred to as the dimensionless parame-
ter) which is equal to v ^LEA where w is the weight per unit length, L is the span length, E
is the effective modulus of elasticity, A is the cross sectional area, H is the horizontal compo-
nent of tension in the cable, and Le is the actual length of the cable [6]. Gimsing [4] reports
that the values of the independent parameter for suspension bridges range from 140 to 350.
Accordingly, the model was constructed have a dimensionless parameter in this range.
One problem with this is that because of the scale of the model, it becomes difficult to
simulate a large weight of cable without introducing significant flexural rigidity into the model,
which undermines one of the basic assumptions of cable theorycables of zero flexural rigidity.
Therefore, it was necessary to add weight using attached masses. After some experimentation,
it was decided that plastic bottles of sand suspended from the cable at 305 mm (12 in) intervals
would be adequate to approximate a uniformly distributed load (see Figure 4.12).
A sag of 381 mm (15 in) was selected. It should be noted that sag S is related to the
63


horizontal component of tension H by the expression H = .
Gimsing [4] also reports that helically-wound cables experience a 15% to 20% reduction
in elastic modulus due to twisting of the wires under tension. Therefore, a value of E equal to
170 GPa (24,650,000 psi) was used.
Given w=0.0164 N/mm, A=4.5 mm2, E=170Gpa, S=381 mm, and L=8484 mm, the
independent parameter may be evaluated. It can be shown that Le ~ L 1 + 8 (f )2j, there-
tore, Le=8621 mm. Finally, A = ---------(3863)(862i)------= 253, which is in the appropriate
range for simulating suspension bridge cable behavior.
Tension gages were placed at regular intervals along the span. This was done more
for redundancy; the horizontal component of tension in cables is more or less constant at any
given moment. The gages can be seen in Figure 4.11, and are labeled T1 through T8.
Seven displacement gages labeled D1 through D7 were located along the span of the
cable model as shown in Figure 4.11. The locations were selected based on the expected mode
shapes for this model such that maximum response would be measured for symmetric and
antisymmetric modes of vibration. An eighth displacement gage, D8, was a movable gage used
for determining response at the point of application of loading, which in most cases did not
occur at the locations of other displacement gages.
4.5 Tests
Testing was performed at five locations on the bridge model along one-tenth span inter-
vals: 0.1L, 0.2L, 0.3L, 0.4L and 0.5L (midspan). However, in the interest of avoiding clouding
the issue with enormous amounts of data, only the testing at 0.2L and at 0.5L is discussed.
Loading at 0.5L primarily excites symmetric modes, while loading at 0.2L primarily excites
antisymmetric modes.
Two regimes of testing were performedstatic and dynamic testing. Static testing was
performed to establish the basic behavior of the cable model due to point loadings. Impact
testing was performed to observe the models response to suddenly applied and impulsive point
loads.
64


4.5.1
Static
Static loading consisted of incrementally adding weights to a point along the span and
measuring the response (tension and displacement) due to each increment of load. Steel weights
were manually suspended from the applied force gage while data was collected at a rate of 10
scans per second. As successive weights were added, displacements and cable tension increased.
The weights were then removed, and the data logger stopped. This procedure was repeated
for each loading location.
The steel weights were three 9.81N weights. A shackle which was used as a fastener
and contributed to the load was measured to be 9.44N. The weight of the shackle was added
first, followed by the three weights. These were then removed in the reverse order. In total,
the sequence of load was zero, 9.44N, 19.25N, 29.06N, 38.87N, and then back to zero.
The data logger was programmed to continuously collect data at the rate of ten scans per
second until manually halted, at which point the data was downloaded from the data logger
to the laptop PC. It should be noted that, because sampling rate was not critical for this
test, the zeroing routines included in the capabilities of the data logger were used. Rezeroing
could be done by tripping status flags in the control software. The data logger program file,
STATICTEST.CR5, can be seen in Appendix D.
4.5.2 Dynamic
Of interest in this project is the response of suspended cables to suddenly applied point
loads. Therefore, the dynamic portion of the testing program focused on using dropped masses
to generate impulsive forces. Two different types of applied load were investigated. The first
was a suddenly applied (relatively constant) point load which was persistent throughout the
duration of the test. The second was a pure impulse load which acted for only a short duration.
The mass used for the suddenly applied load was the shackle mentioned previously, a 0.96 kg
(0.005 ^-) mass, and was dropped from essentially zero height above the cable. For the
impulsive force, it was found that a heavier 5 kg (0.028 ^-) allowed to drop 100 mm (4 in)
was effective at producing a clean impulsive loading. In both cases, the mass was suspended
65


Figure 4.13: Impact Drop Mechanism
from a heavy string on a platform and held into position with a spring clip (see Figure 4.13).
The drop mechanism was configured so that the drop height could be varied from zero to over
300 mm (12 inches). For persistent loads, the mass was fastened directly to the cable, while
for impulsive loads, the mass was suspended by a clip that released when sufficient force was
applied.
Because the fastest possible sampling rate was desired, a different data logger program
was created. When the user tripped a status flag in the control software, the data logger col-
lected 30 seconds of data at 200 scans per second, then stopped automatically. The automatic
zeroing capabilities were not utilizedconditioning the measurements took place entirely in
the spreadsheet. The data logger program file, IMPACTTEST.CR5, can be seen in Appendix D.
66


4.6
Results
The results of the dynamic testing are presented in the figures below. Each plot contains
displacements plotted against time, or additional measured tension against time. The measured
applied load with respect to time is reproduced on the plots for convenience. These measured
results are analogous to the results of the numerical analyses of Chapter 3.
67


Applied Force (N) Displacement (mm) Displacement (mm)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time (sec)
Figure 4.14: Displacement Response for Suddenly Applied Load at 0.5L
68


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 12 3
Time (sec)
D7
Time (sec)
Figure 4.15: Displacement Response for Suddenly Applied Load at 0.5L
69


Additional Cable Tension (N)
Time (sec)
Time (sec)
Figure 4.16: Additional Tension Response for Suddenly Applied Load at 0.5L
70


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec) Time (sec)
Time (sec)
Figure 4.17: Displacement Response for Impulsive Load Applied at 0.5L
71


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec)
Figure 4.18: Displacement Response for Impulsive Load Applied at 0.5L
72


Additional Cable Tension (N)
Time (sec)
Time (sec)
Figure 4.19: Additional Tension Response for Impulsive Point Load Applied at 0.5L
73


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 1 2 3 0 1 2 3
Time (sec) Time (sec)
1 :
Time (sec)
1
Time (sec)
Figure 4.20: Displacement Response for Suddenly Applied Load at 0.2L
74


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 1 2 3 0 1 2 3
Time (sec) Time (sec)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 12 3
Time (sec)
Time (sec)
Figure 4.21: Displacement Response for Suddenly Applied Load at 0.2L
75


Additional Cable Tension (N)
Time (sec)
Figure 4.22: Additional Tension Response for Suddenly Applied Load at 0.2L
76


Applied Force (N) Displacement (mm) Displacement (mm)
Figure 4.23: Displacement Response for Impulsive Load Applied at 0.2L
77


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 1 2 3 0 1 2 3
Time (sec) Time (sec)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0 12 3
Time (sec)
Figure 4.24: Displacement Response for Impulsive Load Applied at 0.2L
iA/yvv
D7
____
78


Additional Cable Tension (N)
Time (sec)
Figure 4.25: Additional Tension Response for Impulsive Point Load Applied at 0.2L
79


5.
Discussion of Results
5.1 Introduction
An experimental and theoretical study of flat-sag cable response to dynamic point loads
was completed. Chapter Three describes the modal analysis based on the linear theory of cable
vibrations and presents the displacement and additional cable tension responses computed
for several cases of loading. Chapter Four describes an experimental program in which a
scale model of a suspended cable wras subjected to dynamic point loading and the response
measured. This chapter is devoted to the comparison and interpretation of those results, as
well as a discussion of some of the various behaviors observed in the course of the theoretical
and experimental program.
5.2 A Comparison of Displacement Responses
The following figures show the theoretical and experimental displacements at seven
locations on the cable span. It can be seen from the graphs that, in general, the theoretical
calculations predict displacement response with reasonable accuracy. The first four graphs
are for loads applied at midspan, which primarily excited symmetric modes of vibration. The
following four graphs are for loads applied at 0.2L span, which primarily excited antisymmetric
modes. Of course, it is difficult, if not impossible, to excite one mode without exciting others,
so each graph will contain a blending of symmetric and antisymmetric components, regardless
of the location of the load.
Figures 5.1 and 5.2 show the displacement response at the seven points common to
the theoretical and experimental analyses, corresponding to displacement gages D1 through
D7. The load is a suddenly applied load applied at midspan. As shown by the load history
plot, it approximates a step load of approximately 8 N (1.8 lbs), though a spike occurs at the
80


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
Time (sec) Time (sec)
----Experimental
D4
Time (sec)
1 2
Time (sec)
Time (sec)
Figure 5.1: Comparison of Theoretical and Experimental Displacement Responses Due to
Suddenly Applied Load at 0.5L (Qrheor =2%)
81


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec) Time (sec)
Time (sec)
Time (sec)
Figure 5.2: Comparison of Theoretical and Experimental Displacement Responses Due to
Suddenly Applied Load at 0.5L (CTheor =2%)
82


beginning of loading. This spike is due to the small amount of free-fall experienced by the
dropped weight, despite efforts to minimize this.
Since the load is symmetrical, it would be natural to expect the responses to exhibit
symmetry. This is in fact the case. The responses of the pairs D1 and D7, D2 and D6, and D3
and D5 show very similar behaviors to one another. Also, the maximum response is shown by
gage D4, which is at midspan and at the point of loading. Gages D2 and D6 show very little
response because they were located near the node points of the first symmetric in-plane mode
shape.
While this test shows a general qualitative correlation between the experimental and
theoretical values, it is quite evident that the magnitudes of the the experimental response
decay significantly faster than the predicted values. This suggests that a damping value greater
than the 2% of critical assumed in the theoretical calculations. It will be shown later that this
particular behavior is unique to this test alone. This issue will be discussed in greater detail
in a subsequent section of this chapter.
Similar to the previous case, Figures 5.3 and 5.4 show the displacement response at
the seven points common to the theoretical and experimental analyses, only this time for an
impulsive short-duration load. Again, the load is applied at midspan. As shown by the load
history plot, it approximates an impulse load of approximately 50 N (11.2 lbs). The signature
was much cleaner due to the fact that the excitation mass released itself from the cable during
its fall, therefore avoiding the cable-mass interactions unavoidable in the previous test. Again,
a symmetric response is expected, and this is observed in the data. As before, the responses of
the pairs D1 and D7, D2 and D6, and D3 and D5 show very similar behaviors to one another.
What is not seen in these data is the high damping of the experimental response relative to
the theoretical. Indeed, the previous case was the only one to show the discrepancy.
One thing that is discernible in the data is that for some signals, the theoretical response
overestimates some of the peaks. The experimental signal appears truncated. A possible reason
for this is that more modes than were used in the calculations are necessary to match the
experimental data. However, preliminary computation iterations indicated that five symmetric
83


Applied Force (N) Displacement (mm) Displacement (mm)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
Theoretical 30 25 Theoretical
Experimental 20 Experimental
E 15 E 10 c 5 E 0
D1 3 '5 b -15 D2
-20 -25 -30
0 1 2 3 0 1 2 3
Time (sec)
Time (sec)
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
Time (sec) Time (sec)
Time (sec)
Figure 5.3: Comparison of Theoretical and Experimental Displacement Responses Due to
Impulsive Load at 0.5L (Crkeor =2%)
84


Applied Force (N) Displacement (mm) Displacement (mm)
Time (sec)
Time (sec)
Time (sec)
Time (sec)
Figure 5.4: Comparison of Theoretical and Experimental Displacement Responses Due to
Impulsive Load at 0.5L (Cr/ieor =2%)
85