Copyright
by
Juan Felipe Beltran
2006
The Dissertation Committee for Juan Felipe Beltran certifies that this is the
approved version of the following dissertation:
COMPUTATIONAL MODELING OF SYNTHETIC-FIBER ROPES
Committee:
________________________________
Eric B. Williamson, Supervisor
________________________________
Karl H. Frank
________________________________
Loukas F. Kallivokas
________________________________
Mark E. Mear
________________________________
John L. Tassoulas
COMPUTATIONAL MODELING OF SYNTHETIC-FIBER ROPES
by
Juan Felipe Beltran, B.Eng., M.S.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
August 2006
DEDICATION
I would like to dedicate this dissertation work to my family. Without their
love, patience, and support I would not have made it through.
A special feeling of gratitude to my loving parents, Juan Francisco and
Carmen Gloria, who have always believed in me and continue to do so, offering their
encouragement and support.
To my brothers Cristian and Matias, thanks for your friendship and support.
I will never forget my two grandmothers, Julia (“mi abuelita grande”) and
Maria (“mi abuelita chica”). Although the cancer took their lives, their spirits have
always been with me.
Finally to my wife Oriana and my two lovely children, Juan Baltazar and
Francisca Carolina, who mean everything to me.
Acknowledgements
I would like to acknowledge the members of my dissertation committee,
Drs. Karl H. Frank, Loukas F. Kallivokas, Mark E. Mear, and John L. Tassoulas.
Your time and support are truly appreciated.
A special thanks to my mentor, Dr. Eric B. Williamson for his guidance,
and support throughout this research. Dr. Williamson was always there for me
throughout this process, whether responding question by e-mails or having long
chats in his office. I not only got a degree, but also I got a friend.
This
dissertation could not be completed without all the ideas and suggestions given by
him.
I also wish to thank the Offshore Technology Research Center for
providing financial support through their Cooperative Agreement with the
Minerals Management Service.
The contribution of Jaroon Rungamornrat to the development of the
computational model is greatly appreciated.
To my friends for their support, inspiration and encouragement.
Juan Felipe Beltran
August, 2006
v
COMPUTATIONAL MODELING OF SYNTHETIC-FIBER ROPES
Publication No.______________
Juan Felipe Beltran, Ph.D.
The University of Texas at Austin, 2006
Supervisor: Eric B. Williamson
The objective of this research is to develop a computational model to
predict the response of synthetic-fiber ropes under both monotonic and cyclic
loads. These types of ropes are believed to offer a better alternative to more
traditional mooring systems for deepwater applications. Of particular interest for
this study are the degradation of rope properties as a function of loading history
and the effect of rope element failure on overall rope response.
A computational tool developed specifically for this research accounts for
the change in rope properties as it deforms and the change in configuration of a
rope cross-section due to the failure of individual rope components. The software
includes both geometric and material nonlinearities, and it incorporates a damage
index so the strength and stiffness degradation of the rope elements can be
modeled. Following the failure of rope elements, the software considers the
possibility that the failed rope elements can resume carrying their proportionate
share of axial load as a result of frictional effects.
vi
Using the computational model developed under this research, several
rope geometries are studied. Virgin (i.e., undamaged) ropes and initially damaged
ropes are considered. In all cases, experimental data for a monotonically
increasing load are available for comparison with the analytical predictions. For
most of the cases analyzed, the proposed numerical model accurately predicts the
capacity of the damaged ropes, but the model overestimates the rope failure axial
strain. A three-dimensional finite element analysis is performed for a particular
rope geometry to validate some of the assumptions made to develop the proposed
computational model.
If failed rope elements resume carrying their proportionate share of axial
load, numerical simulations demonstrate the existence of strain localization
around the failure region. Based on the damage model, damage localization
occurs as well. This damage localization can cause the premature failure of rope
elements and reduce the load capacity and failure axial strain of a damaged rope.
Possible extensions to the computational model include the treatment of
variability in rope properties and a lack of symmetry of the cross-section. Such
enhancements can improve the accuracy with which damaged rope response is
predicted. With the availability of validated software, engineers can reliably
estimate the performance of synthetic-fiber moorings so that the use of these
systems can be used with confidence in deepwater applications.
vii
Table of Contents
xii
List of Tables………………………………………………………………..….xvi
xiii
List of Figures……………………………………..………………………….…xx
1
CHAPTER 1: Introduction………………………...……………………………1
1.1 General Background………………………………………………..…………11
4
1.2 Research Objectives………………………………………………..………….1
5
1.3 Scope of Research..……………………………………….………..………….1
7
CHAPTER 2: Background and Literature Review………………………...….1
2.1 Background………………………...……………..……………………….
7
2.1.1Steel Wire Ropes…………………………………………………
7
2.1.2 Synthetic-Fiber Ropes …………………………………………
9
2.2 Literature Review ………………………………………………………….1 10
2.2.1 Steel Wire Ropes Models ………………………………………. 10
2.2.1.1 Fiber Models………………………………………….. 11
2.2.1.2 Helical Rod Models…………………………………… 12
2.2.1.3 Semi-Continuous Models………………………………. 14
2.2.2 Synthetic-Fiber Ropes Models ….…...…………………………. 14
2.3 Common Types of Rope Constructions ………………………………… 18
2.3.1 Overview………………………..………………………………. 18
2.3.2 Common Synthetic-Fiber Rope Constructions... ………………. 19
2.3.2.1 Three-Strand Laid Construction…………………………..19
2.3.2.2 Eight-Strand Construction…………………………….. 20
2.3.2.3 Double Braid Construction……………………………. 20
2.3.2.4 Parallel Yarn/Strand Construction……………………
21
2.3.2.5 Wire Rope Construction………………………………
21
2.4 Summary…................................................................................................... 22
viii
CHAPTER 3: Mathematical Modeling………………………...……………….
24
3.1 Kinematics of a Rope. …………………………………………………………
24
3.1.1 Structure of a Synthetic-Fiber Rope. ……………………………. 24
3.1.2 Initial and Deformed Configuration of a Rope…………………….26
3.1.2.1 Curvature of a Rope..…………………………………… 28
3.1.2.2 Torsion of a Rope…………………………………………31
3.1.2.3 Formulae of Frenet……………………………………
32
3.1.2.4 Axial Deformation of Rope Elements………………… 35
3.2 Constitutive Models………………………………………………………… 35
3.3 Cross-Section Modeling.………………………………………………….. 36
3.3.1 Packing Geometry………………………………………………. 37
3.3.2 Wedging Geometry..............................................………………. 37
3.3.3 Cross-Section Update…………………………………………… 37
3.3.3.1 Constant Cross-Section Model………………………….. 38
3.3.3.2 Constant Volume Model……………………..……….. 38
3.3.3.3 Poisson Effect Model……………………………..……. 38
3.4 Equilibrium Equations and Friction Models………………………………….39
3.4.1 Differential Equations of Equilibrium…………………………..
39
3.4.2 Reduced Equilibrium Equations………………………………… 44
3.4.3 Force and Moment Resultants in a Rope Element……………… 45
3.4.4 Friction Model…………………………………………………… 47
3.5 Damage Model………………………………………………………………..50
3.5.1 General Background……………………………..............................50
3.5.2 Damage Evolution Rule…………………........……………….……1
51
3.5.3 Basic Concepts of Continuum Damage Mechanics………………..54
3.6 Summary…......................................................................................................56
ix
CHAPTER 4: Rope Degradation and Static Rope Response………………. 57
4.1 Overview………………………………………………………………..… 57
4.2 Damage Measurement ……………………………………………………. 59
4.2.1 Secant Moduli Ratio……………………………………………… 59
4.2.2 Strain Energy Deviation………………………………………….. 65
4.2.3 Softening Behavior…………………………………………………70
4.3 Numerical Implementation…………………………………………………. 75
4.4 Finite Element Analysis………………………………………………………83
4.4.1 Finite Element Model…………………………………………… 84
4.4.2 Finite Element Analysis Results……………………………………86
4.4 Summary…………………………………………………………………… 89
CHAPTER 5: Numerical Simulation of Damage Localization in
Synthetic-Fiber Mooring Ropes………………………………….……….…. 91
5.1 Overview……………………………………………………...…………… 91
5.2 Calculation of normal Forces and Recovery Length Concept……………… 94
5.2.1 Case 1: Packing Rope Geometry……………………………………98
5.2.2 Case 2: Wedging Rope Geometry…………………………………107
5.2.3 Case 3: Mixed Rope Geometry……………………………………113
5.3 Calculation of the Recovery Length………………………………………….
117
5.3.1 Numerical Algorithm……………………………………………….1
118
5.3.2 Special Case: Linear friction Model……………………………….120
5.3.3 Numerical Examples of Recovery Length Values………………….
122
5.3.4 Effect of the Rope Jacket on the Recovery Length Values……….130
5.4 Numerical Analyses of Damaged Ropes……………………………………136
5.5 Numerical Examples……………………………………...............................140
5.6 Summary……………………………………………....................................167
CHAPTER 6: Summary and Conclusions………………………………………
169
139
6.1 Summary and Conclusions.…………...………………………………………1
169
146
x
6.2 Future Studies…………………………………………………………………1
175
APPENDIX: Softening Behavior Modeling…………………………………….
177
References…………………………………………...…………………………388
181
VITA………………………………………………...…………………………404
190
xi
List of Tables
Table 3.1
Table 4.1
Directions cosines……………………………………………… 41
Summary of values of damage parameters and predictions
of the damage models: one-level rope…………………… …
Table 4.2
Summary of values of damage parameters and predictions
of the damage models: two-level rope……………………….
Table 5.1
77
81
Coefficients to compute the normal force exerted on a
component of a rope element with mixed cross-section
Table 5.2
geometry……………………………………………………………
117
Summary of the models used to predict damaged rope
behavior ……………………………………………………………
141
Table 5.3
Table 5.4
Summary of the initially undamaged rope capacities………… 161
Summary of the damaged rope capacities (10% of initial
damage) with specified breaking strength equal to 35 Tonnes
(77.2 Kips)………………………………………………………….
164
Table 5.5
Summary of the damaged rope with specified breaking
strength equal to 700 Tonnes (1543.2 Kips)……………………….
166
xii
List of Figures
Fig. 1.1
Strength of steel and synthetic-fiber ropes…………………..…. 2
Fig. 1.2
Strength-to-weight ratio of steel and synthetic-fiber ropes…….. 2
Fig. 1.3
Types of mooring systems: Taut-leg and catenary…………….
Fig. 1.4
(a) Stress-strain curve for polyester rope; (b) rope pitch
3
distance(p); (c) rope radius (r)…………….…………………… 5
Fig. 2.1
Longitudinal view and typical types of steel rope
construction…………………………………………………….. 8
Fig. 2.2
Cross-section of a four-level rope……………………………… 16
Fig. 2.3
Types of synthetic-fiber rope constructions…………………… 20
Fig. 3.1
Hierarchy ranking of rope elements…………………………….. 25
Fig. 3.2
Rope elements cross-section at different rope levels…………….25
Fig. 3.3
Generic point P of a curve C……………………………………. 26
Fig. 3.4
Rope geometry…………………………………………………. 28
Fig. 3.5
Local coordinate system for a rope element……………………. 30
Fig. 3.6
Kinematic of a rigid body B……………………………………. 34
Fig. 3.7
Developed view of a helical rope element……………………… 35
Fig. 3.8
Rope cross-sectional modeling…………………………………. 37
Fig. 3.9
Loads acting on a line circular helix element…………………….40
Fig. 3.10
Centerline of a line element looking down (no moments)
(a) x3 axis and (b) x2 axis………………………………………….42
Fig. 3.11
Centerline of a line element looking down (no forces)
(a) x3 axis and (b) x2 axis…………………………………………..
43
Fig. 3.12
(a) Cross-section for bending moment; (b) Cross-section for
torsion…………………………………………………………. 46
Fig. 3.13
Normal force acting on a rope element (wedging geometry)…. 48
Fig. 3.14
Effect of frictional forces on rope force resultants…………….. 49
xiii
Fig. 3.15
Principle of strain equivalence…………………........................ 55
Fig. 4.1
Response of a one-level, two-layer rope under monotonic
strain history and cross-section and geometric parameters
of the rope………………………………………………………..58
Fig. 4.2
Attempt to describe damage evolution based on Eq. 3.48b
for different values of the threshold strain εt…………………… 62
Fig. 4.3
(a), (b), (c) Damage index evolution for different values
of εt and εb = 0.124……………………………………………….64
Fig. 4.4
Variation of the damage parameters α1 and β for different
values of εt........................................................................................
65
Fig. 4.5
Strain Energy Deviation (SED) definition ………………………..
67
Fig. 4.6
Attempt to describe damage evolution based on
energy-based method………………………………………………
69
Fig. 4.7
Damage index evolution for different values of εt and
εb = 0.124 using the energy-based method……………………
71
Fig. 4.8
Overlap domain………………………………………….…….. 74
Fig. 4.9
One-level rope response using energy-based damage model … 76
Fig. 4.10
One-level rope response using secant moduli ratio damage
26
model ………………………………………………………….. 76
Fig. 4.11
Response of a one-level rope under monotonic strain
history: experimental, virgin response and damaged
simulation curves………………………………………………….78
Fig. 4.12
Cross-section and geometrical properties of a two-level,
one- layer rope…………………………………………………….78
Fig. 4.13
Response of a two-level rope under monotonic strain history…. 79
Fig. 4.14
Response of a two-level rope under monotonic strain history…. 80
Fig. 4.15
Predicted responses of one-level initially damaged rope (33%) 82
Fig. 4.16
Predicted responses of one-level initially damaged rope (11%) .. 83
xiv
Fig. 4.17
(a) Three-strand rope cross-section and (b) finite elements
mesh of the rope………………………………………………… 85
Fig. 4.18
Two-level rope response predictions, including finite
Fig. 4.19
element analysis, and experimental data………………………. 87
Finite element model: (a) undeformed rope geometry and
Fig. 4.20
(b) undeformed-deformed rope geometry……………………… 88
Axial stress distribution at a rope axial strain of 12%....................89
Fig. 5.1
Recovery length concept of a broken rope element…………….. 92
Fig. 5.2
Pattern of inter-layer/inter-element contact forces in an
axially loaded rope element………………………………………96
Fig. 5.3
Contact forces in axially loaded rope element with packing
and wedging geometry…………………………………………. 97
Fig. 5.4
Gradual increase in tension load Ti in a broken rope element…….98
Fig. 5.5
Three-level packing geometry rope and normal forces acting
on the rope elements at each rope……………………………… 100
Fig. 5.6
Three-level wedging geometry rope and normal forces
acting on rope elements at each rope level……………………….
109
Fig. 5.7
Three-level mixed geometry rope and normal forces acting
on the rope elements at each rope level…………………………..
114
Fig. 5.8
Recovery length values of the core element of one-level
two-layer damaged ropes as a function of the axial rope
strain…………………………………………………………… 124
Fig. 5.9
Normal forces acting on rope elements of a damaged
one-level two-level rope assuming packing geometry…………….
125
Fig. 5.10
Recovery length values of broken strands of a two-level
damaged rope as a function of the axial rope strain………………..
127
Fig. 5.11
Profiles of the gradual increase in tension load for two broken
strands of a two-level rope: (a) core, (b) outer rope yarn for
xv
different values of rope axial strain……………………………..129
Fig. 5.12
Rope jacket and sub-ropes of a parallel rope construction………..
131
Fig. 5.13
(a) Normal forces acting on the core; (b) compressive stress
acting on the core………………………………………………. 132
Fig. 5.14
Effect of the rope jacket on the recovery length values:
(a) outer rope yarn; (b) core……………………………………….
135
Fig. 5.15
Damaged rope discretizations: (a) failure located at one end
of the rope and (b) failure located around the midspan of the
rope…………………………………………………………….. 137
Fig. 5.16
Two-noded axial-torsional “sub-rope” element………………….
137
Fig. 5.17
Initially damaged one-level rope responses: (a) one out nine
rope yarns cut and (b) three out nine rope yarns cut…………….142
Fig. 5.18
Initially damaged two-level rope cross-section and analyses
of rope response…………………………………………………146
Fig. 5.19
Damaged cross-section (10%) of DPJR 1……………………….149
Fig. 5.20
Experimental data and predicted responses of DPJR 1………….149
Fig. 5.21
Variation of the (a) failure strain and (b) failure load of the
DPJR 1 with respect to L/d ratio values………………………….150
Fig. 5.22
Damaged cross-section (10%) of DPJR 2………………………….
151
Fig. 5.23
Experimental data and predicted responses of DPJR 2………. 151
Fig. 5.24
Damaged cross-section (10%) of DPJR 3……………………….152
Fig. 5.25
Experimental data and predicted responses of DPJR 3……….. 152
Fig. 5.26
Variation of the (a) failure strain and (b) failure load of the
DPJR 3 with respect to L/d ratio values………………..……… 153
Fig. 5.27
Damaged cross-section (5%) of DPJR 4…………………………
154
Fig. 5.28
Experimental data and predicted responses of DPJR 4…………..
154
Fig. 5.29
Damaged cross-section (10%) of DPJR 5………………………155
Fig. 5.30
Experimental data and predicted responses of DPJR 5……….. 155
xvi
Fig. 5.31
Variation of the (a) failure strain and (b) failure load of the
156
DPJR 5 with respect to L/d ratio values……………………………
Fig. 5.32
157
Damaged cross-section (10%) of DPJR 6……………………….
Fig. 5.33
Experimental data and predicted responses of DPJR 6…………157
Fig. 5.34
158
Damaged cross-section (10%) of DPJR 7………………………….
Fig. 5.35
Experimental data and predicted responses of DPJR 7…………158
Fig. 5.36
Variation of the (a) failure strain and (b) failure load of the
DPJR 7 with respect to initial rope damage state……………… 159
Fig. 5.37
Variation of the (a) failure axial load and (b) failure axial
strain of the DPJR 7 with respect to L/d ratio for different
initial rope damage states………………………………………. 160
Fig. 5.38
Variation of the experimental (a) rope failure strain and
(b) rope failure load of the damaged ropes with specified
breaking strength of 35 Tonnes with respect to L/d ratio……….165
xvii
CHAPTER 1
Introduction
1.1 GENERAL BACKGROUND
Over the years, developments in the offshore industry have focused on deep and
ultra-deep water (depth greater than 1700 meters) in order to explore new reservoirs of oil
and gas. Conventional steel wire ropes and chains have been used in the past to moor
floating platforms. As the exploration and production of oil and gas move to deeper and
deeper water, conventional mooring systems are costly and may become unfeasible due
to the high magnitude of vertical load (self-weight) and size of the anchor footprint
needed.
One way to facilitate the exploration and production of oil and gas in (ultra) deep
water is the use of low-weight material to optimize the strength-to-weight ratio of the
mooring system. In this context, synthetic-fiber ropes appear to be a promising
alternative. Synthetic-fiber ropes offer several advantages over conventional mooring
systems, such as substantial installation costs savings, reduced dead-weight loads and
reduction of the size of the seafloor footprint.
In order to illustrate the advantage of using synthetic-fiber ropes as a mooring
system over conventional mooring systems, Figs. 1.1 and 1.2 show comparisons of
strength and strength-to-weight ratio for steel rope (ST) and several types of syntheticfiber ropes (PE: Polyethelyne, PP: Polypropelyne, NY: Nylon, PET: Polyester, AR:
Aramid, HMPE: High Modulus Polyethelyne).
Considering Fig. 1.1, it is important to note that the current technology is capable
of producing synthetic fibers stronger than steel, such as aramid and high modulus
polyethelyne. Fig. 1.2 shows that synthetic-fiber ropes have greater strength-to-weight
ratio than steel ropes; thus, they appear to be a valid alternative for use with mooring
systems for floating platforms in deep water.
1
Strength (Mpa)
4000
3000
2000
1000
0
PE
PP
NY
PET
ST
AR
HMPE
Type of Rope
Strength-to-Weight Ratio (cN/Tex)
Fig. 1.1: Strength of steel and synthetic-fiber ropes
400
300
200
100
0
PE
PP
NY
PET
ST
AR
HPME
Type of Rope
Fig. 1.2: Strength-to-weight ratio of steel and synthetic-fiber ropes
2
Synthetic-fiber ropes can either be used as taut-leg moorings or as insets in
catenary mooring systems (Fig. 1.3). Taut-leg moorings rely on the elasticity of the
mooring line to provide restoring forces that keep a platform in its desired position.
Catenary moorings, however, rely on the weight of the mooring line to provide the
restoring forces (Hooker, 2000).
Taut-leg
Catenary
Anchor
Fig. 1.3: Types of mooring systems: Taut-leg and catenary (Kennedy, 2004)
Taut-leg moorings, in comparison to catenary moorings, give rise to a smaller sea
floor footprint and can utilize shorter line lengths, but they require anchors that are
capable of withstanding vertical load. Catenary moorings with synthetic inserts require
long and heavy lengths of chain or steel wire rope near the seabed, but they can utilize
standard drag anchors (Hooker, 2000).
The use of synthetic mooring lines in recent applications has brought to light
several unknown aspects of behavior that can be categorized under the following
headings (Hooker, 2000): (a) durability and (b) extension and modulus.
3
The durability of synthetic-fiber ropes is vital to their future success in offshore
applications. Synthetic-fiber ropes in permanent mooring systems will need to exhibit
design lives in excess of 20 years. When assessing the suitability for long-term design
lives, it is important to first identify all the possible sources that could affect the
performance of a mooring system. Some of these technical challenges are: special
requirements during installation, complex loading history and associated time-dependent
deformation during service, potential strength degradation due to hysteresis effects,
seawater exposure and fiber abrasion. These challenges may be overcome with long-term
testing and physical deformation and failure prediction models coupled with additional
field demonstration projects (Karayaka, et al., 1999; Hooker, 2000).
Extension (strength) and modulus characteristics are fundamental to the design of
a mooring system because they determine the allowable loads and the vessel offset. The
stiffness properties of synthetic-fiber ropes are generally not constant or linear. They can
depend on time, tension history, temperature, moisture and humidity. The slope of the
load versus deformation plot becomes steeper with increasing tension. This nonlinear
property is due to the characteristics of the synthetic-fiber ropes. The polymer fibers
which make up a rope generally have nonlinear stress-strain behavior. Also, when
tensioned, a rope structure compacts, decreasing the rope radius (r) and increasing the
pitch distance p within the rope (Fig. 1.4). Typical synthetic-fiber rope curves can be
reduced to a polynomial to be used in static and dynamic analysis computer programs
(Flory, 2001).
1.2 Research Objectives
The main purpose of the current study is to develop a computational model to
predict the response of synthetic-fiber ropes under both monotonic and cyclic load,
considering a taut-leg mooring configuration. To investigate the effect of damage to a
rope cross-section on overall rope response, a computational framework that quantifies
the deterioration that takes place in a damaged rope throughout its loading history is
4
needed. The computational model must be capable of tracing the behavior of a rope from
its initial configuration to the onset of rope failure under the loads being considered.
(a)
160
(c)
(b)
View A-A
Axial Stress (ksi)
120
p
80
A
40
A
r
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Axial Strain
Fig. 1.4: (a) Stress-strain curve for polyester rope; (b) rope pitch distance (p);
(c) rope radius (r)
In order to include the deterioration experienced by ropes throughout their loading
history, a damage model that evolves as a function of loading history is proposed. The
failure of a synthetic-fiber rope is a complicated process that could depend on variety of
factors. In this research, the combination of these factors is captured by the computation
of a scalar quantity called the damage index parameter.
In addition to account rope properties deterioration, the effect of rope elements
failures on overall rope response is also addressed to the computational model. Using the
developed computational model, analytical studies have been carried out to simulate
damaged rope response.
1.3 Scope of Research
Additional background information and a literature review of rope modeling are
presented in Chapter 2.
Details of the most common rope cross-sections are also
described along with their most common uses.
5
In Chapter 3, the modeling of synthetic-fiber ropes is presented. Details on the
theoretical basis and formulation of the damage model to account for rope degradation
are provided.
Calibration of the proposed damaged model based on experimental data is
discussed in Chapter 4. Static response of synthetic ropes damaged and undamaged prior
to loading are presented. Modeling of softening behavior of synthetic ropes is also
explained in this chapter. A three-dimensional finite element analysis is performed for a
particular rope geometry to validate some of the assumption used to develop the proposed
computational model.
A numerical procedure to estimate the effect of rope elements failure on overall
rope response is presented in Chapter 5. Simulations of responses of damaged ropes prior
to loading along with comparisons with experimental data are also provided in this
chapter.
In Chapter 6, a summary of the results obtained from the current research is
presented. Conclusions are drawn regarding how synthetic-fiber ropes behave when
loaded under monotonically increasing load.
research are identified.
6
In addition, opportunities for future
CHAPTER 2
Background and Literature Review
This chapter consists of three sections. The first section provides an overview of
the applications for ropes and cables in mechanical systems along with the most common
materials used to construct them. The second section gives a general description of the
available mathematical models to describe the behavior of these types of elements. In the
third section, information about the most common type of rope constructions, including a
summary of how such ropes are used, is provided.
2.1 BACKGROUND
The essential characteristic of a rope is that it has high axial strength and stiffness,
in relation to its weight, combined with low flexural stiffness. This combination is
achieved in a rope by using a large number of elements, each of which is continuous
throughout the rope length. When loaded axially, each component within a rope provides
tensile strength and stiffness, but when deformed in bending, the components have low
combined bending stiffness provided their bending deformation is uncoupled. To
facilitate handling, it is necessary to ensure that a rope has some integrity as a structure,
rather than being merely a set of parallel elements. This characteristic is achieved by
twisting the elements together (Chaplin, 1998).
2.1.1 Steel Wire Ropes
A rope can be a critical component in many engineering applications, including
cranes, lifts, mine hoisting, bridges, cableways, electrical conductors, offshore mooring
systems and so on. Different classes of ropes, suited for different purposes, have a
different number and arrangement of rope elements within the rope cross-section, and
rope elements can be made of different materials. Fig. 2.1 shows a longitudinal view of a
steel wire rope. Its three basic components are the wire, the strand and the core (Schrems
7
and Maclaren, 1997), and the typical cross-sectional arrangement of the elements include
the spiral strand, six strand and multi-strand geometries (Chaplin, 1998).
Spiral Strand
Six Strand
Multi-Strand
Fig. 2.1: Longitudinal view and typical types of steel rope constructions
(Schrems and Maclaren, 1997; Chaplin, 1998)
The need to pull, haul, lift, hoist, hold or otherwise control objects has been a
necessity since the beginning of civilization. In the earliest days, ropes were simply an
assemblage of vines. Later, these assembles gave way to the use of vegetable or natural
fibers from plants, usually original to a particular region. For example, the pyramids of
Egypt and the Aztec dwellings, among others, could never have been built without these
natural fiber ropes, quite probably aided by some mechanical schemes. The names for
this assemblage of fibers twisted together in strands to form a strong, flexible and round
strength member became cordage and rope (Foster, 2002).
As civilization became more progressive, so did the rope manufacturing. As
McKenna et al. state “the Industrial Revolution (1800), which moved textile
manufacturing in Britain and later elsewhere from cottages to factories, changed the yarn
8
(fiber) production, including the preparatory stages of opening and cleaning, carding and
drawing, from hand-spinning to machine processing”. The most important consequence
of the Industrial Revolution for the rope industry, however, resulted not from the
machinery, but from the invention of steels that could be made into wires which could be
assembled into wire ropes and cables. The usual construction was to wrap successive
layers on top of each other, with alternating twist directions. Wire ropes came to
dominate the newer engineering applications such as bridge cables, mine hoists,
elevators, and cranes (McKenna et al., 2004).
The extensive use of steel wire ropes for load bearing elements is mainly due to
the strength offered by steel coupled with the flexibility of rope construction, rope
geometry and wire size that can be suited to the required application. Although a wire
rope is essentially an element for transmitting a tensile load, the rope construction is such
that the individual wires in a rope are subjected to bending and torsional moments,
frictional and bearing loads, as well as tension. The magnitude and distribution of the
stresses resulting from these loadings determine the overall rope response, which can be
expressed in terms of the extension and rotation of the rope (Utting and Jones, 1984).
Over the years, each field of wire rope application has developed a specific body
of knowledge, based on extensive testing and field experience, leading to empirical rules
for each particular application. Unifying these empirical rules under some general
mathematical and mechanical theory would allow a better understanding, and in the long
term, a better prediction of the mechanical behavior of wire ropes as well as reduce the
need for expensive tests under a variety of operating conditions. Thus, due to their
extensive use and the need to predict their behavior, several researchers have presented
analytical models that permit the calculation of wire rope response based on the wire
material and geometry (Cardou and Jolicoeur, 1997).
2.1.2 Synthetic-Fiber Ropes
The birth of nylon polyamide fiber in the late 1930s started the concept of
industrial-grade fibers. Fibers of higher tenacity (strength of a fiber of a given size) have
since been developed. Some of these other fibers include nylon, polyester, polypropylene
9
and polyethylene, making it possible to produce flexible tension members of much higher
strength and durability (Foster, 2002). The second generation of synthetic polymer fibers
consists of high-modulus fibers with low breaking extensions and tenacities more than
twice that of nylon and polyester. The first high-modulus, high-tenacity (HM-HT)
synthetic polymer fiber, which became available in the 1970s, was the para-aramid fiber
(Kelvar). Kevlar was followed in the 1980s by high-modulus polyethylene (HMPE) fiber
(Spectra, Dyneema), and more recently by fibers made of poly-para-phenylene
bisoxazole (Zylon). The development of these new types of synthetic fibers has given to
the cordage and rope industry the possibility to build high-strength members that can
potentially replace steel wire ropes (Foster, 2002; McKenna et al., 2004).
2.2 LITERATURE REVIEW
In this section, a general description of mathematical models used to describe
rope and cable behavior is given. The initial focus is on models that predict the behavior
of steel wire ropes. These models have been studied in great detail due to the extensive
use of steel wire ropes, and they are a starting point for developing a mathematical or
analytical model to study synthetic-fiber rope behavior (Chapter 3). Following this initial
discussion, a review of some experimental and analytical research conducted on
synthetic-fiber rope behavior is presented.
2.2.1 Steel Wire Ropes Models
Several mathematical models are currently available to predict the response of
twisted steel wire cables and aluminum conductor steel reinforced (ACSR) electrical
conductors under axisymmetric loading (Jolicoeur and Cardou, 1991). In order to develop
these mathematical models, some researchers such as Hruska (1951, 1952, and 1953),
Machida and Durelli (1973), McConnell and Zemke (1982), Knapp (1975, 1979),
Lanteigne (1985) and Costello (1990), among others, have used a discrete approach in
which equations are established for each individual wire. Other researches such as Hobbs
and Raoof (1982) and Blouin and Cardou (1989) have used a semi-continuous approach
in which each wire layer is replaced by a transversely isotropic layer.
10
The aim of this section is to classify current discrete models based on the
assumptions made in their development as well as by the manner in which they predict
global cable deformation (axial strain and twist) under applied axial loads. In the
literature, there are hundreds of works that could be classified as a discrete model, but
just some of them are mentioned in this section. It should be noted, however, Costello
(1978), Utting and Jones (1984) and Cardou and Jolicoeur (1997) have published detailed
historical surveys of the available models to predict cable response.
Based on the works by Jolicoeur and Cardou (1991) and Cardou and Jolicoeur
(1997), current discrete mathematical models can essentially be divided into two
categories, which are based on the types of hypotheses employed: (a) fiber models where
the wire ropes can develop only tensile forces, and (b) rod models where the wire ropes
can develop tensile and shear forces as well as bending and twisting moments.
2.2.1.1 Fiber Models
This type of model was developed by Hruska (1951, 1952, and 1953). It is based
on the simplest hypotheses: (a) no end conditions effects, although zero end rotation is
assumed; (b) contact mode between wires is purely radial; (c) radial contraction is
neglected; (c) wires are assumed to be subjected only to tensile forces, neglecting their
flexural and torsional stiffness; (d) frictional forces are neglected and (f) wire
deformations are small, obtained from purely geometrical considerations and expressed
in terms of the axial strain and twist per unit length of the cable. Hruska’s analysis leads
to expressions for tangential, radial and tensile forces in the wires. The resulting stiffness
matrix of the cable is linear, constant and symmetric. The work of Hruska was extended
by Leissa (1959) to the case of a rope made up of six strands, each of seven wires. Leissa
sought a relationship between the stresses in the single wires and the geometry of the
rope, the load applied and the stresses due to contact between the single wires and the
strands. Based on the work by Hruska (1952) and Leissa (1959), Starkey and Cress
(1959) developed a mathematical model to compute contact stresses between wires that
asserts that the usual mode of failure of a wire rope is fretting fatigue initiated at such
points of contact. The authors also performed experimental tests to substantiate the
11
predictions made by their mathematical model. Hruska’s equations were also rederived
and extended by Knapp (1975, 1979), as well as by Lanteigne (1985), addressing the case
of compressibility of the core and possibly large wire strains.
2.2.1.2 Helical Rod Models
These models are an extension of the fiber models. McConnell and Zemke (1982)
simply added the sum of the torsional stiffness of all individual wires to the cable
torsional stiffness, which is valid only for small ropes. A mores rigorous analysis was
performed by Machida and Durelli (1973). The authors took into account the bending and
torsion of the helical wires by determining the change in geometry of the original helices
under loading which are constrained by the core wire to maintain a constant helix radius.
The elementary theories of bending and torsion of a linear-elastic circular cross-section
bar are used to determine the stresses in the helical wires and core. The relations between
overall external forces on the strands and loadings on the individual wires are obtained
from equilibrium considerations. By using the assumption of small deformations, the
expressions of the geometry of the individual wires in their deformed configuration are
linearized to obtain linear expressions for the global cable deformation (axial strain and
twist) under applied axial loads. In a similar effort, Knapp (1975 and 1979) presented
geometrically nonlinear equations of equilibrium for straight rope wires subjected to axial
loads derived by the energy method. Linear-elastic wires were considered and the
increment in their strain energy is based on the wire axial, bending and twisting strains.
The author included in his model the potential variation of the core radius due to pressure
exerted by the layers. The nonlinear equilibrium equations were then linearized to obtain
a linear stiffness matrix for the rope. For rigid cores, Knapp’s equations reduce exactly to
Machida and Durelli’s equations.
Phillips and Costello (1973) adopted a more basic approach to investigate the
static behavior of cables. They treated the cables as groups of separate curved rods (Love,
1944) in the form of helices, leading to a set of nonlinear equilibrium equations. The
authors assumed that no core was present and that frictional forces were negligible. The
model considered the effects of variations of helix angle and wire radius (Poisson’s
12
effect) in addition to lateral contact between the wires. Huang (1978), using the same
theory of curved rods, studied the problem of finite extension of an elastic strand
including the radial constraint provided by the core to the helical wires, influence of
Poisson effects on individual wires and frictional forces between wires. Geometrical
nonlinearities due to wire deformation were included in the analysis. These resulting
nonlinear equations were linearized by Velinsky (1981) considering small helix angles
and still including radial contraction due to Poisson’s effect. Costello (1983) using this
linearized theory and a simplified bending theory (Costello and Butson, 1982), studied
the response of a multilayered cable subjected to axial, bending and torsional loads. Later
on, Velinsky et al. (1984) applied the linear theory to predict the static response of a wire
rope with complex cross-sections. Although, no closed-form solutions were given for the
cable stiffness matrix coefficients, these values were later obtained by Jolicoeur and
Cardou (1991) by matrix inversion. Velisky (1985) extended the nonlinear theory to
analyze multilayered cables. For the load range in which most of the wire cables are used,
the author concluded that the nonlinear theory showed no significant advantage over the
linear theory. Kumar and Cochran (1987) have also linearized Costello’s equations
including radial contraction, obtaining closed-form solutions for the analysis of elastic
cables under tensile and torsional loads. A different approach was taken by Jiang (1995)
to analyze multilayered wire ropes. The author developed a general linear and nonlinear
formulation where the rope is not treated as a collection of smooth rods (wires) as in
previous works, but wires, strands and ropes are considered ropes by themselves,
differing in their values of stiffness and deformation. Thus, according to this formulation,
a rope is made up of smaller ropes which can give rise to complex cross-sections.
As mentioned by Samras et al. (1974), linearized models should satisfy the
Maxwell-Betti reciprocity theorem. Accordingly, the stiffness matrix of a cable should
be symmetric. In their work, Jolicoeur and Cardou (1991) showed that fiber models
satisfy this condition while rod models deviate by a few percent. The origin of the lack of
symmetry in the rod models was identified by Sathikh et al. (1996). In response to this
finding, they developed a linear symmetric model based on the works by Wempner
(1981) and Ramsey (1988, 1990), and compared the relative significance of the
13
individual contributions of wire forces and moments to the stiffness coefficients of the
cable.
2.2.1.3 Semi-Continuous Models
This type of model was primarily developed for the study of strand bending.
Hobbs and Raoof (1982) developed the Orthotropic Sheet Model (OSM) in which each
wire layer is replaced with an orthotropic thin layer in a state of plane stress. In this
model, four elastic constants are necessary. As Cardou and Jolicoeur (1997) stated, two
of these constants are directly related to the properties of the wires considered as fibers.
The other two constants are related to the contact stiffness between adjacent wires in the
layer. The normal and tangential stiffness values are obtained from contact mechanics
equations, using parallel cylinder in contact results. Thus, this formulation is a mixed
contact model in which in-layer lateral contact is considered as primary. Friction between
adjacent wires is also included in the model, with a deformation process evolving
gradually from a no-slip to a full-slip stage.
Another semi-continuous model was first developed by Blouin and Cardou (1989)
and subsequently extended by Jolicoeur and Cardou (1994, 1996). This model also
consists of replacing each layer with a hollow cylinder of orthotropic, transversely
isotropic material. Cylinder thickness is arbitrary; thus the model is a three-dimensional
one requiring five elastic constants. The following three principal directions are defined
for the equivalent cylinder: radial, tangential and the longitudinal axis of the wires in the
corresponding layer. Standard linear elasticity equations are then applied, resulting for
the axial loading case in a set of ordinary differential equations (one for each layer).
Among the five elastic constants, some can be used as free adjustable parameters or be
estimated from the contact mechanics equations as in the case of OSM. This model has
been described in detail by Jolicoeur and Cardou (1996).
2.2.2 Synthetic-Fiber Ropes Models
Synthetic-fiber ropes, which, unlike steel wire ropes, contain millions of fibers in
their cross-sections, are extremely complicated structures, where the correct and detailed
14
analysis of the mechanics is not easy to develop. This difficulty in modeling syntheticfiber rope behavior arises mainly because of the varying nature of the fiber properties and
the complexity of the rope geometry. Fiber moduli depend on prior loading history,
currently applied load level, and changes in rope geometry that occur due to changes in
load.
Most of the textile industry still relies on the traditional empirical approach for
estimating rope behavior, but, in the last two decades, this situation has begun to change
because synthetic-fiber ropes have started being used in demanding applications such as
deepwater moorings. Much of the research on synthetic-fiber rope mechanics is based
upon earlier work on the mechanics of twisted yarns described by Hearle (1969). This
author, using an energy-based approach on the twisted continuous fibers following a
helical geometry, considering only the tensile stress-strain properties of the rope
components, established the equation to determine the axial stress in a yarn (formed by
fibers) based on the deformation of the constituent fibers.
A major contribution to the modeling of synthetic-fiber rope behavior has been
made by Leech (1987, 2002). The author assumes that the rope geometry has a
hierarchical structure: the lowest structure in a rope is the textile yarn assembled from
twisted fibers, and then the rope yarn is formed by twisting together a small number of
textile yarns. The next structure, the strand, is formed by twisting a group of rope yarns.
The final structure, the rope, is assembled from a small number of strands, often twisted
together. Sometimes a group of small ropes can be used to form bigger ropes either
twisting the small ropes together or arranging them in a parallel configuration. If this is
the case, small ropes are called sub-ropes. In Fig. 2.2, the cross-section of a four-level
rope is shown, where the structures considered are: fibers, rope yarn, strand and rope.
Strains at each rope structure (or level) are determined by imposed deformations to the
rope, using differential geometry concepts. External and internal forces are computed
based on the principle of virtual work in which the strain energy is based only on the
tensile stress-strain properties of the rope components. The model also addresses the
effects of frictional forces, heat generation due to fiber hysteresis and fatigue on rope
behavior (Banfield et al., 2001).
15
Strand
Fibers
Rope
Rope Yarn
Fig. 2.2: Cross-section of a four-level rope
In a similar way as Leech, Wu (1992) using an energy approach, developed an
analytical model for rope-strength prediction. The rope strain energy is based only on the
tensile stress-strain properties of the rope components. The external and internal axial
forces are computed based on the principle of virtual work for any given rope axial strain,
provided that load/strain curve of rope components be known. This model was used to
predict the static tensile strength of particular rope type: double braided (see Section
2.3.2), where friction was explicitly considered in the model. To corroborate his model,
the author performed experimental tests on nylon and polyester double braided ropes
whose results agree well with model prediction.
Some researchers, based on existing experimental data on yarns and ropes, have
derived useful design information for synthetic-fiber ropes. Kenney et al. (1985)
conducted experimental tests under variety of loading conditions on fibers, yarns and
small ropes made of nylon, polyester and aramid. It was found that the dominated failure
mode for all type of fibers was the creep rupture mode. Consequently, the authors
propose a creep rupture model to predict the residual strength and cyclic lifetimes of a
rope. Mandell (1987) proposed a model based on creep-rupture behavior of individual
fibers and yarns to predict fatigue and abrasion failure for nylon and polyester ropes. Wu
(1993) carried out experimental tests on double-braided (see Section 2.3.2) nylon and
polyester ropes to study the effect of frictional constraints. In addition, he also
16
investigated the slippage process between rope elements that can cause abrasion failure.
Banfield and Casey (1998) conducted experimental tests on aramid, polyester and HMPE
ropes to evaluate several mechanical properties such as axial stiffness, modulus,
hysteresis and elongation. Lo et al. (1999) established equations to account for the effects
of creep deformation, estimated fatigue life, evaluated residual strength and remaining
life, and considered long-term strength retention in polyester ropes subjected to a saline
(i.e., sea water) environment. Fernandes et al. (1999), based on the work by Del Vecchio
(1992), established an expression to deal with the nonlinear stiffness of polyester ropes,
which depends on the average load, load amplitude and excitation period. Davies et al.
(2000) presented regression analyses of tests on polyester ropes to evaluate their timerelated mechanical properties. Creep and relaxation tests were conducted at different rope
levels: fiber, yarn, rope yarn, sub-rope and ropes. Ghoreishi et al. (2004) presented a
linearized model to predict the axial stiffness of small synthetic wire ropes based on the
work presented by Labrosse et al. (2000) and Nawrocki and Labrosse (2000). These
authors also validated their model with experimental data and with a three-dimensional
finite element rope model. Bradon and Chaplin (2005) developed the Assured Residual
life Span (ARELIS) method for estimating residual creep life of polyester ropes. The
author developed a statistical model to quantify the uncertainties in the method, such as
the scatter in creep rupture test data and load sharing between sub-ropes. This model can
be used to determine the required test load, duration and number of test required to
guarantee a minimum creep life for a rope at its service load.
The development of mathematical models that can account for the degradation of
the mechanical properties or damage of synthetic ropes during their loading history has
also been a major focus of research. Seo et al. (1990) formulated models for prediction of
service life of synthetic-fiber double braided ropes (see Section 2.3.2). The authors, based
on experimental evidences, considered two mayor rope deterioration factors: tensile creep
fatigue and wear behavior of constituent fibers. Karayaka et al. (1999) presented a linear
and nonlinear damage accumulation model to predict the failure and residual strength of
synthetic ropes due the combined effects of creep and fatigue. As Seo et al. (1990), the
authors recognized that fiber-fiber abrasion and hysteric heating may also contribute to
17
the damage accumulation, thus the proposed damage accumulation law may need to be
modified to quantify contributions of these factors to failure. Li et al. (2002) conducted
experimental tests in small damaged synthetic-fiber ropes. The authors also presented a
simple model to predict the breaking load of the damaged specimens tested. This model
just neglects the contribution to rope response of the broken elements and is based on the
linearized theory presented by Costello (1983) to analyze wire ropes. Beltran et al. (2003)
and Beltran and Williamson (2004) validated the damage model proposed by
Rungamornrat et al. (2002) and presented a methodology to estimate the evolution of
damage in synthetic ropes under monotonically applied loads. Recently, Beltran and
Williamson (2005) presented a numerical model to estimate the effect of rope elements
failure on overall rope response.
2.3 COMMON TYPES OF ROPE CONSTRUCTIONS
2.3.1 Overview
In Section 2.1, it was stated that the most common materials to build ropes are
steel and synthetic fibers. In addition, Fig. 2.1 showed a longitudinal view of a steel wire
rope along with the most common types of rope construction to clarify the ideas and
concepts introduced. For additional details on steel wires rope constructions, readers are
referred to, for example, the works by Walton (1996) and Chaplin (1998). Because the
focus of the current study is on synthetic-fiber ropes, only these types of constructions
will be considered in the current section and in the remainder of the document.
Synthetic ropes can be considered as structures made of textile fibers. As already
discussed, ropes are defined as approximately cylindrical textile bodies whose crosssections are small compared to their lengths, and they are primarily used as tension
members. A synthetic rope structure contains large amounts of fibers in coherent,
compact and flexible configurations, usually to produce a selected breaking strength and
extensibility with a minimum amount of fibers (McKenna et al., 2004).
According to McKenna et al. (2004), synthetic-fiber ropes can be divided into two
general categories: (1) laid and braided ropes and (2) low twist ropes. Laid and braided
18
ropes are the most common structures for general purpose use. They are found in
industrial, marine, recreation and general utility service. They include everything from
small cords to large hawsers for mooring tankers. Conversely, low twist ropes are used
for specialized and demanding applications where high strength-to-weight ratios and low
extensibility are essential. Such situations include guys for tall masts, deep sea salvage
recovery ropes, mooring lines for floating platforms and hoist cables for deep mines.
2.3.2. Common Synthetic-Fiber Rope Constructions
Leech (1987) established that the most common types of rope constructions used
are 3-strand, 8-strand, double braided, parallel yarn/strand and wire ropes (Fig. 2.3). In
the following subsections, each of these types of rope construction is described in detail.
2.3.2.1 Three-Strand Laid Construction
The oldest and still the most widely used fiber rope structure consists of three
strands laid together by a twisting process. The strands are laid (right helix: Z) in the
opposite direction to the rope yarns (left helix: S) that make up the strand so that the rope
yarns are nearly aligned with the rope axis. This orientation is the optimum arrangement
for best external wear resistance (McKenna et al., 2004). Three-strand laid ropes are not
torque free. Thus, if an axial force is applied to such a rope, the rope will rotate and tend
to unwind as it elongates axially. Different types of fibers can be combined to construct a
rope of this geometry. For example, the center of each strand may be made with
polypropylene, and the rope yarns can be comprised of polyester. This arrangement has
the benefits of the abrasion resistance of polyester on the outside coupled with the
lightweight and lower cost of polypropylene on the inside. In general, three-strand laid
ropes have lower strength and higher elongation than braided ropes of the same size and
material. However, they have virtually identical strength/elongation properties as eightstrand ropes.
19
Wire rope
Fig. 2.3: Types of synthetic-fiber rope constructions (Leech, 1987)
2.3.2.2 Eight-Strand Construction
The eight-strand construction is a torque balanced rope that is made by a braiding
technique called plaiting. A braided structure has alternating strands laid in opposite
directions. This construction can be made from the full range of high and low modulus
fiber materials and can be used for general purposes as with three-strand ropes. Their
typical applications are ship moorings and general utility service.
2.3.2.3 Double Braid Construction
A double braid rope is made by braiding a cover rope (sheath) over a braided core
(Fig. 2.3). The tension in the rope is shared by both the core and cover. Most double
braids are made of either all nylon or all polyester fiber, but combinations using different
20
material may be found. High-modulus fibers in the cover and core result in inefficient
load sharing between them and are rarely encountered. It should be noted that jackets of
low-modulus fiber that do not support any load are often braided over a braided core of
high-modulus fiber to provide abrasion protection. Although they have the appearance of
a double braid construction, they are not considered as such (McKenna et al., 2004).
Typical applications of double braid ropes are ship moorings and general utility service.
2.3.2.4 Parallel Yarn/Strand Construction
For the case of parallel yarn rope construction, a collection of a large number of
parallel textile yarns, which may be enclosed in a jacket, form the rope structure.
Because they lack twist or a helical construction, it is more difficult to achieve good load
distribution among the fibers, particularly if high-modulus fibers are used, unless
accurately controlled tension is maintained on all yarns as they are bundled into a rope
(McKenna et al., 2004). Similarly to parallel yarn rope construction, strands in a parallel
strand rope are aligned along the axis of loading of the rope. Strands are formed by many
individual elements (rope yarns) held together by a braided jacket or twisted yarn
bundles. A strand can be considered by itself to be a small rope and is often called a subrope.
Strands can have a braided or laid structure. Parallel yarn rope construction, if
carefully made and properly terminated, has very good strength efficiency. It is on the
same order of magnitude as parallel strand and wire rope (see section 2.3.2.5)
constructions. Their typical applications are antenna guys and moorings.
2.3.2.5 Wire Rope Construction
Synthetic-fiber ropes are often described as wire ropes when their cross-section
resembles that of a steel wire rope. This rope construction was developed for fiber ropes
because it achieved good fiber-to-rope strength with high modulus fibers such as aramid.
Later, it was found to work well with low modulus fibers, especially polyester. Wire rope
construction has low twist levels and a long pitch distance in the strands in order to
achieve its high strength. A braided jacket is usually required to provide abrasion and
snag protection. These types of ropes provide excellent tension-tension fatigue resistance
21
and also perform well in bending fatigue. Their typical applications are moorings and
dynamic applications such as lifting.
2.4 SUMMARY
It is well recognized that steel wire ropes have been used extensively as loadbearing elements due to the strength of steel coupled with the flexibility of rope
construction, rope geometry and wire size that can be suited to the required application.
For applications requiring tension members with high strength-to-weight ratios, the
textile industry has been able to produce high-modulus fibers with high tenacities. Thus,
the development of these new types of synthetic fibers has given to the cordage and rope
industry the possibility to build high-strength members that can potentially replace steel
wire ropes.
Several mathematical models are currently available to predict the response of
twisted steel wire cables under axisymmetric loading. These models have been developed
in great detail due to the extensive use of steel wire ropes. Based on the assumptions
made in their development, the mathematical models can be classified as either discrete
models or semi-continuous models. In the discrete models, equations are established for
each individual wire. For the semi-continuous models, each wire layer is replaced by a
transversely isotropic hollow cylinder. Both types of models compare well with
experimental data, and their use depends on the problem under study.
For the case of synthetic-fiber ropes, few mathematical models have been
developed to predict rope response. This situation exists in part due to the complexity of
the rope geometry and the varying nature of the fiber properties coupled with the fact that
the textile industry still relies on the traditional empirical approach for estimating rope
behavior. However, this situation has begun to change because synthetic-fiber ropes have
started being used in demanding applications such as deepwater moorings. Experimental
tests on small and large ropes are being conducted along with the development of
analytical models to study the behavior of rope response. The major concern of these
studies is to evaluate and quantify the ability of synthetic ropes to withstand damage.
22
In the following chapter, the foundations of a mathematical model to predict
synthetic-fiber rope response are presented. This model relies on the previous research
conducted on steel wire ropes and synthetic-fiber ropes. The major contributions of the
proposed model is the inclusion of a damage index, which quantifies the level of
deterioration of rope properties when a rope is loaded and the estimate of the effect on
overall rope response of rope elements failure. The model predictions are compared with
available experimental data on virgin (i.e., undamaged) and on initially damaged ropes.
23
CHAPTER 3
Mathematical Modeling
In this chapter, an analytical model to predict the behavior of synthetic-fiber ropes
is presented. The general assumptions made to develop the model are explained including
the kinematics of deformation, constitutive models, cross-sectional behavior and
equilibrium equations of a rope. This chapter also describes the use of a damage model,
along with the use of a damage index, to account for strength and stiffness degradation.
3.1 KINEMATICS OF A ROPE
The mathematical description of the deformation of a rope element is described in
this section. With this information known, the response of individual rope elements can
be assembled to compute the response of an entire rope. Details are provided below.
3.1.1 Structure of a Synthetic-Fiber Rope
A synthetic-fiber rope is defined as a structural element constructed by twisting
all components in hierarchical order. A typical rope consists of many levels of
components. The diagram below and Fig. 3.1 show the typical hierarchy ranking from the
highest to the smallest level (Liu, 1989, 1995; Leech, 2000).
Rope → Sub-rope → Strand → Rope yarn → Textile yarn → Fiber
Thus, the rope itself is defined as level 1, and its components (e.g., sub-ropes and
strands) comprise the structure at the second and third levels, respectively. This naming
convention applies to all levels of a rope. Accordingly, level j includes all components in
level j+1, which in turn includes all components in level j+2, etc. For example, Fig. 3.2
shows a rope cross-section with two levels of helical geometry because each sub-rope
contains its own core element with components (strands) wound around the axis of the
sub-rope. A complete description of the cross-section of each level is also presented in
24
Fig. 3.2. Note that the components appear elliptical in cross-section due to the projection
of the helix angle onto the section shown.
(Sub-ropes)
Fig. 3.1: Hierarchy ranking of rope elements (Leech, 2002)
Rope cross-section
Sub-rope cross-section
(level 1)
Strand cross-section
(level 2)
as
rs
asr
Strand radius :rs
Sub-rope
Rope Core
Sub-rope core:
Strand
Sub-rope comp. helix radius: as
Sub-rope helix radius: asr
Fig. 3.2: Rope elements cross-section at different rope levels
25
3.1.2 Initial and Deformed Configuration of a Rope
A rope element is a structural element whose cross-section is small compared to
its length. The so-called plane-sections hypothesis is assumed: rope element crosssections that are plane before deformation remain plane after deformation, thus the
motion of a rope element is described in terms of parameters that are a function of only
its axial coordinate. It is assumed that in the deformed and initial configurations, the
geometry of a rope element, represented by its centerline (longitudinal or helix axis), can
be described by a circular helix curve. Curves in space can be seen as paths of a point in
motion. In a Cartesian coordinate system, every point P of a curve C can be uniquely
determined by its position vector OP = u(φ) = (x(φ), y(φ), z(φ)) where φ is a real variable
defined in a closed interval I: φ1 ≤ φ ≤ φ2. The position vector u(φ) is called the
parametric representation of the curve C, and the variable φ is called the parameter of this
representation (Fig.3.3).
z
P
u
C
O
y
x
Fig. 3.3: Generic point P of a curve C
26
Two assumptions are made on the parametric representation u(φ) of any curve C:
(1) The functions x(φ), y(φ) and z(φ) are r (≥ 1) times continuously differentiable in I
(where the value of r will depend on the problem under consideration); (2) For every
value of φ in I, at least one of the three functions dx(φ)/dφ, dy(φ)/dφ and dz(φ)/dφ is
different from zero. A parametric representation u(φ) of the a curve C satisfying these
conditions is called an allowable parametric representation (Kreyszig, 1991).
Based on the descriptions given in Figs. 3.1 and 3.2, rope geometry can be
represented by an mth order helix (Lee, 1991). A one-level rope can be represented by a
single helix, because rope components are wound around a vertical axis (rope core)
arranged in different layers. A two-level rope can be represented by a double helix. Rope
components of the second level can be represented by a single helix, but these
components form the rope components of the first level, which has its own helical
geometry. Accordingly, a two-level rope possesses a double helix structure. Ropes that
can be modeled with multiple levels generate multiple helix rope geometry.
The modeling of rope geometry follows a hierarchical approach with a defined
number of levels, which mirrors the manufacturing process. Thus, the geometry of a
multiple level rope can be analyzed considering each level as a single helix. The
parametric representation of the centerline of each rope component (single helix) is given
by
x(φ ) = a cos(φ )
y (φ ) = a sin(φ ) :
z (φ ) =
(3.1a)
0≤φ≤
2πL
p
(3.1b)
pφ
2π
(3.1c)
where a is the helix radius, measured from the core axis of the level under consideration
to the centerline of the rope component, φ is the swept angle, p is the pitch distance and L
is the projected length of the rope component on the core axis. Based on the above
parametric representation, the centerline of a rope component lies on the cylinder x2 + y2
27
= a2 and winds around it in such a way that when φ increases by 2π, the x and y
components return to their original value, while z increases by p, the pitch of the helix.
Based on Eqs. 3.1, three geometric parameters are needed to describe a rope
component in a single helix configuration: helix radius (a), projected length of the rope
component on the core axis (L) and pitch distance (p) as shown in Fig. 3.4. By definition,
a circular helix curve makes a constant angle (helix angle) with a fixed line in space. This
fixed line is the longitudinal axis of each component, and the helix angle (θ) is defined as
the angle between the axis of the component and the axis of the core component (Fig
3.4). The helix angle (θ) can be computed using the following expression:
tan(θ ) =
2πa
p
(3.1d)
θ
L
p
a
R
A
A
Section A-A
Sub-rope
Fig. 3.4: Rope geometry
3.1.2.1 Curvature of a Rope
To analyze a rope element in space, it is convenient to use a local coordinate
system at each point on its centroidal axis defined by the tangent (x1), normal (x2) and
28
binormal (x3) vectors at that point (Fig. 3.5). This naming convention is referred to as the
Frenet frame at that point (Struik, 1988). The arc length of a helix curve in the space
between two points is defined by the following expression:
φ1
s (φ ) =
∫
φ0
2
2
 dx (φ ) 
 dy (φ ) 
 dz (φ ) 

 + 
 + 

 dφ 
 dφ 
 dφ 
2
(3.2)
where x(φ), y(φ) and z(φ) are defined from Eqs. 3.1a through 3.1c. The solution of Eq.
3.2 yields the following expression for the arc length s of a helix curve in terms of the
variableφ:
2
 p 
s (φ ) = a + 
 φ
 2π 
2
(3.3)
where a and p were defined in Fig. 3.4. Thus, the arc length may be used as parameter in
the parametric representation of the helix curve. Furthermore, u(s) satisfies the criteria
defined above for use as an allowable parametric representation. It should be noted that
the parameter s is often called the natural parameter (Kreyszig, 1991).
By definition, the unit tangent vector x1(s) to the helix curve at the point u(s) is
given by
x1 ( s ) =
du( s)
ds
(3.4)
All vectors at the point u(s) of the helix curve, which are orthogonal to the
corresponding unit tangent vector, lie in a plane. This plane is called the normal plane (N)
to the helix curve at u(s) (Fig. 3.5).
29
z
x3 R
N
x2
S
x1
y
u(s)
x
Fig. 3.5: Local coordinate system for a rope element
An orthogonal vector to the unit tangent vector x1(s) that measures the rate of
change of the tangent vector along the curve is called the curvature vector and is given by
dx1(s)/ds. The curvature vector is the intersection of the normal plane (N) of the helix
curve and the osculating plane (S), which is the plane spanned by vectors x1(s) and
dx1(s)/ds at the point under consideration (Struik, 1988). Thus, the unit principal normal
vector x2(s) to the helix curve at the point u(s) (Fig. 3.5) is given by
dx1 ( s)
x2 ( s) = ds
dx1 ( s)
ds
(3.5)
where |⋅| is the Eucledian norm or absolute value, defined for this particular case as a
mapping |⋅|: R3→R. A proportionality factor κ can be introduced to relate the curvature
vector dx1(s)/ds and the unit principal normal vector x2(s) such that
dx 1
= κx 2 (s )
ds
(3.6)
30
where κ is called the curvature of the curve at the point under consideration.
Alternatively, if two tangent vectors x1(s) and x1(s) + ∆x1(s + h) are compared, then x1(s),
∆x1(s + h) and x1(s) + ∆x1(s + h) form an isosceles triangle with two sides equal to 1,
enclosing the angle ∆φ, the angle of contingency (Struik, 1988). If the absolute value of
the vector ∆x1(s + h) is computed, |∆x1(s + h)|, and the result is linearized in terms of ∆φ,
it turns out that as ∆φ tends to zero, the following relation holds
κ=
d x 1 ( s ) dϕ ( s )
=
ds
ds
(3.7)
which is the usual definition of curvature κ in the case of a plane curve. For the case of a
helix curve, the curvature κ is given by
κ=
φ 2a
L2 + (φa) 2
(3.8)
where L is the projected length of the rope component on the core axis, φ is the swept
angle and a the helix radius.
3.1.2.2 Torsion of a Rope
As stated earlier, the intersection of the osculating plane (S) and the normal plane
(N) is the unit principal normal vector x2(s) (Fig. 3.5). All the vectors that lay in the
osculating plane (S) can be spanned by the vectors x1(s) and x2(s). The vector x3(s), called
the binormal vector (Fig. 3.5), can be obtained by the computing the cross product
between vectors x1(s) and x2(s). Thus, by definition of the cross product, the vector x3(s)
is normal to the osculating plane (S) and also is the intersection of the normal plane (N)
and the rectifying plane (R), which is plane spanned by the vectors x1(s) and x3(s) (Fig.
3.5). Due to the orthogonality of the vectors x1(s), x2(s) and x3(s), they can be taken as a
new frame of reference (Struik, 1988).
31
The rate of change of the vector defining the osculating plane (S) is given by
dx3(s)/ds. Differentiating the relation x3(s)⋅ x1(s) = 0 with respect to s, using the
expression given by Eq. (3.7) and considering the orthogonal property of the vectors
x1(s), x2(s) and x3(s), it can be shown that the vector dx3(s)/ds lies in the direction of the
unit principal normal vector x2(s). Thus, a proportionality factor ξ can be introduced to
relate unit principal normal vector x2(s) and the vector dx3(s)/ds such that
dx 3 ( s )
= −ξx 2 ( s )
ds
(3.9)
where ξ is called the torsion of the curve at the point under consideration. A scalar
multiplication of Eq. 3.9 by x2(s) yields the following expression for ξ
ξ =−
dx 3 ( s )
x2 (s)
ds
(3.10)
For the case of a helix curve, the torsion ξ is given by
ξ=
φL
L + (φa ) 2
(3.11)
2
where L is the projected length of the rope component on the core axis, φ is the swept
angle and a the helix radius.
3.1.2.3 Formulae of Frenet
The first derivates dx1(s)/ds, dx2(s)/ds and dx3(s)/ds of the unit vectors x1(s), x2(s)
and x3(s) can be taken as a linear combination of the vectors x1(s), x2(s) and x3(s). The
corresponding formulae are called the formulae of Frenet and have a kinematic
interpretation when the frame of reference x1(s)-x2(s)-x3(s) moves along a curve C
(Kreyszig, 1991).
32
An expression for the vector dx2(s)/ds in terms of the unit vectors x1(s), x2(s) and
x3(s) can be obtained by differentiating the identity x2(s)⋅ x2(s) = 1, which is the square of
the absolute value of the vector x2(s). Having done the differentiation, it turns out that if
the vector dx2(s)/ds is not the null vector, it is orthogonal to x2(s) and consequently can
be spanned by a linear combination of the unit vectors x1(s) and x3(s). Using the fact that
the vectors x1(s), x2(s) and x3(s) are orthogonal, and by differentiating with respect to s
the expressions x2(s)⋅ x1(s) = 0 and x2(s)⋅ x3(s) = 0, the vector dx2(s)/ds can be taken as
dx 2 ( s )
= −κx 1 ( s ) + ξx 3 ( s )
ds
(3.12)
Eq. 3.12, along with Eqs. 3.6 and 3.9, are the formulae of Frenet, which in matrix
notation has the following form:
 dx 1 ( s ) 


 ds   0
 dx 2 ( s )  =  − κ
 ds  
 dx 3 ( s )   0


 ds 
κ
0
−ξ
0   x1 ( s ) 


ξ  x2 ( s ) 
0  x 3 ( s ) 
(3.13)
The kinematic interpretation of the formulae of Frenet is based on Fig. 3.6.
Consider that the frame of reference x1(s)-x2(s)-x3(s) moves along a curve C with a
constant velocity of a unit magnitude. Vectors x1(s), x2(s) and x3(s) span a threedimensional space, each with a constant length equal to 1. Thus, these vectors can be
thought of as being inscribed on a rigid body B (Fig. 3.6), that performs the same motion
as the frame of reference. Therefore, the kinematic interpretation of formulae of Frenet
may be considered as a problem of rigid body kinematics. The study of the problem
presented in Fig. 3.6, is summarized in the following theorem (Kreyszig, 1991): “The
rotation vector d of the trihedron (frame of reference) of a curve C: u(s) of class r ≥ 3
(u(s) posseses continuous derivatives up to the order r, inclusive) with non-vanishing
33
curvature, when a point moves along C with constant velocity 1, is given by the
expression
d = ξx1 + κx 3
(3.14)
where vector d is also called the vector of Darboux, whose absolute value |d| is the
angular velocity of rotation”.
d
|d|
B
Fig. 3.6: Kinematic of a rigid body B
Based on Eq. 3.14, the torsion ξ and the curvature κ of a curve C are the
projections of the angular velocity |d| of the trihedron (frame of reference) on the unit
tangent vector x1(s) and unit binormal vector x3(s), respectively. If the curve C is a helix
curve, both kinematic parameters, the torsion ξ and the curvatureκ, are constant along the
curve (Eqs. 3.8 and 3.11), and the ratio between these two parameters is given by
κ
= tan(θ )
ξ
(3.15)
where θ is the helix angle (Fig. 3.4).
34
3.1.2.4 Axial Deformation of Rope Elements
The axial strain of a rope element can be obtained using the fact that, as
previously mentioned, the centerline of a rope element lies on the cylinder x2 + y2 = a2
and winds around it in such a way that when φ increases by 2π, the x and y components
return to their original value, while z increases by p, the pitch of the helix (Fig. 3.7).
Based on trigonometric relations and the engineering strain definition, the axial strain of a
rope element is given by (Costello, 1990)
ε er =
L2 + (φa) 2
L20 + (φ 0 a 0 ) 2
−1
(3.16)
where L0 is the projected length of the rope element on the core axis in the reference
configuration, L is the projected length of the rope element on the core axis in the current
(i.e., deformed) configuration, φ is the swept angle and a the helix radius.
φa
Core axis
θ
L
Centroidal
axis of
rope
comp.
Fig. 3.7: Developed view of a helical rope element
3.2 CONSTITUTIVE MODELS
The material behavior for synthetic-fiber ropes is assumed to be known at the
lowest hierarchical level of a rope element and behaves elastically. The stress-strain
35
relationship could be linear or nonlinear depending on the type of fiber used to construct
the rope under investigation. Time-dependent behavior of the material fibers is not
included in the current model. Any variability in fiber properties that could affect rope
strength is also neglected. Due to the lack of available test data, and in order to be
consistent with previous researchers (Liu, 1989, 1995; Leech, 1990; Fernandes et al.,
1999; Flory, 2001), both normal and shear stresses are expressed as polynomial functions
of the normal and shear strain, respectively, up to the fifth degree, having the following
form (Rungamornrat et al., 2002):
2
ε 
ε 
ε
σ
= a1   + a 2   + a3 
σb
εb 
 εb 
 εb
2
γ
γ 
γ 
τ
= b1   + b2   + b3 
τb
γb
γb 
γb 
3
4

ε
 + a 4 

 εb

ε
 + a5 

 εb



3
4
5
γ

 + b4 
γb

γ

 + b5 
γb




5
(3.17a)
(3.17b)
where εb and γb are the normal and shear strain, respectively, at which an element reaches
its maximum normal (σb) and shear (τb) stress under monotonic loading. The coefficients
ai and bi are constitutive parameters and are chosen to provide a best fit to measured data
of rope components that belong to the lowest hierarchical level of a rope.
3.3 CROSS-SECTION MODELING
In order to model the cross-section of a rope element, two types of arrangements
of the components are considered: packing and wedging geometry (Fig. 3.8), which
represent the extreme cases of transverse deformation of the cross-section for real ropes
(Leech, 2002).
36
Sub-rope
Sub-rope
(sector)
Wedging Geometry
Packing Geometry
Fig. 3.8: Rope cross-sectional modeling
3.3.1 Packing Geometry
It is assumed that all components of a rope element are initially straight and
circular in cross-section and transversely stiff, and a twist of a specified number of turns
is imparted to the central component. Contact between components in the same level is
assumed to be only in the radial direction, and slip at those points is prevented due to the
assumptions made regarding the kinematics of deformation (Leech, 2002).
3.3.2 Wedging Geometry
In the wedging geometry, the components in the same level are allowed to deform
transversely and change their shape into a wedge or truncated wedge, which is the shape
that would develop for deformable components. It is assumed that there is circumferential
contact pressure and friction acting along the length of components due to axial slip
between contiguous components (Leech, 2002). Radial contact, however, is assumed to
be negligible in comparison to the circumferential contact and is ignored for
computational purposes.
3.3.3 Cross-Section Update
In general, the total deformation of a rope element is attributable to the following
three effects: rope elongation, rope rotation, and a radial deformation of the rope element.
The first two effects were already analyzed in Section 3.1. In this sub-section, the
37
following three models are used to compute the radius r of a rope element in its deformed
configuration: constant cross-section, constant volume and Poisson effect.
3.3.3.1 Constant Cross-Section Model
In this model, it is assumed that the cross-section of a rope element remains
constant through its entire loading history. Thus, the value of r is given by
r = r0
(3.18a)
3.3.3.2 Constant Volume Model
In this model, it is assumed that the material of the rope element is
incompressible, which means that its volume is preserved throughout its entire loading
history. By equating the initial and current volume, the value of r in the deformed
configuration is given by
r=
r0
(3.18b)
1 + ε er
3.3.3.3 Poisson Effect Model
In this model, the transverse strain of a rope element is related to its axial strain
by Poisson’s ratio. Thus, the value of r is given by
r = r0 (1 − νε er )
(3.18c)
where ν is the Poisson’s ratio of the material rope element, εer is given by Eq. 3.16 and r0
is the radius of the rope element in its initial configuration.
38
3.4 EQUILIBRIUM EQUATIONS AND FRICTION MODELS
The following subsections describe the governing equilibrium equations for a
element within a rope. First, general equations are provided that account for bending,
twisting, axial, and shear deformations. Next, these expressions are simplified based on
assumptions of the stress state in an individual rope element. Finally, friction models that
account for the interaction of rope components within a given rope cross-section are
described.
3.4.1 Differential Equations of Equilibrium
In order to compute the stresses acting in a rope, each element that belongs to the
lowest hierarchical level of a rope component is treated as a helical rod. As explained in
Section 2.2.1.2, a helical rod has axial, shear, flexural and torsional stiffness. The
tractions associated with a deformed configuration of an element are statically equivalent
to three mutually orthogonal forces acting at the centroid of the cross-section along with
couples around each axis (Fig. 3.9), where Vi are the forces in the i direction; Mi are the
moments about the i axis; and wi and mi are the contact forces and distributed moment per
unit length, respectively, in the i direction. Thus, equilibrium equations for an element are
established along its centerline (defining a line (thin) element) in space, assuming
incrementally small deformations.
Due to the assumptions made concerning the construction of a rope, axial
elongation leads to the development of both tensile and torsional forces. Because of the
geometric restrictions made on the kinematics of deformation, the state of stress and
strain associated with axial elongation is constant along the centerline of each helical
rope component. Therefore, stresses and loads in rope components can be described by
the stresses and loads on a single transverse cross-section of a helical rope. The general
differential form of the equilibrium equations of a line element is given considering an
orthogonal local coordinate system x1- x2- x3 (Fig. 3.8), where x1 is the tangent vector, x2
is the normal vector and x3 is the binormal vector that form a Frenet frame (Section
3.1.2.1).
39
Center of Curvature
x1
z
x2
V2 , M 2
V1 , M 1
V3 , M 3
x3
w2, m2
x
y
w3, m3
w1, m1
Fig. 3.9 Loads acting on a line circular helix element
In order to obtain the equilibrium equations for a line element in space, Fig. 3.10
shows two planar views of length ds of a line element along with the forces (no moments)
acting on it. Fig. 3.10a shows a view looking down the x3 axis, whereas Fig. 3.10b shows
a similar view looking down the x2 axis. The direction cosines of the forces V1+dV1, V2+
dV2 and V3+dV3 with the axes x1, x2 and x3 are linearized using a Maclaurin series
expansion. Thus, cos(αk’l) ≈ 1 and sin(αk’l) ≈ αk’l with αk’l (k’, l = 1, 2, 3) the angle of
rotation of the vectors x’k relative to the vectors xl (Fig. 3.10). Based on the definitions
given for the curvature Eq. 3.7 and torsion Eq. 3.10 of a curve, the required direction
cosines are listed in Table 3.1.
40
Table 3.1 Direction Cosines
Direction cosine
V1+dV1
x1
cos(α1’1) ≈ 1
x2
sin(α1’2) ≈ κ3ds
cos(α2’2) ≈ 1
sin(α3’2) ≈ -ξds
x3
sin(α1’3) ≈ κ2ds
sin(α2’3) ≈ ξds
cos(α3’3) ≈ 1
V2+dV2
V3+dV3
-sin(α2’1) ≈ -κ3ds -sin(α3’1) ≈ -κ2ds
where κ2 and κ3 are the curvatures of the line element in the rectifying and osculating
plane respectively, and ξ is the torsion of the line element. Based on Fig. 3.10, a
summation of forces in the x1 direction, neglecting second and higher order terms, yields
dV1 + w1ds − V2κ 3 ds + V3κ 2 ds = 0
(3.19)
which becomes, upon normalizing by ds,
dV1
+ w1 − V 2 κ 3 + V3κ 2 = 0
ds
(3.20)
Similarly, a summation of forces in the x2 and x3 directions yield
dV2
+ w2 + V1κ 3 − V3ξ = 0
ds
(3.21)
and
dV3
+ w3 − V1κ 2 + V2ξ = 0
ds
(3.22)
41
x’2
x2
V2+dV2
w2
V1
x1
(a)
V1+dV1
ds
x’1
V2
(b)
V3
ds
x1
V1
V1+ dV1
w3
V3+ dV3
x3
x’1
x’3
Fig. 3.10: Centerline of a line element looking down (no moments) (a) x3
axis and (b) x2 axis
Following the procedure used to obtain the equilibrium equations for the forces,
Fig. 3.11 shows the same line element of length ds loaded just with the moments that act
on the element. The moments M1+dM1, M2+dM2 and M3+dM3 make the same angles αk’l
(k’, l = 1, 2, 3) with respect to the axes x1, x2 and x3 as do the forces V1+dV1, V2+ dV2 and
V3+dV3. Thus, the direction cosines given in Table 3.1 can be used to establish the three
moment equilibrium equations. Neglecting second order effects, a summation of
moments about the x1 axis yields
dM 1
+ m1 − M 2κ 3 + M 3κ 2 = 0
ds
(3.23)
Similarly, a summation of moments about the x2 and x3 axes yield
42
dM 2
+ m 2 + M 1κ 3 − M 3ξ − V3 = 0
ds
(3.24)
and
dM 3
+ m3 − M 1κ 2 + M 2ξ + V2 = 0
ds
(3.25)
x2
M2+dM2
m3
M1
(a)
x’2
x1
M1+dM1
ds
x’1
M2
(b)
M3
ds
x1
M1
M1+ dM1
m2
M3+ dM3
x3
x’1
x’3
Fig. 3.11: Centerline of a line element looking down (no forces): (a) x3 axis
and (b) x2 axis
Equations 3.20 through 3.25 are the six differential equations of equilibrium for
the line element loaded as shown in Fig. 3.9 (Love, 1944).
43
3.4.2 Reduced Equilibrium Equations
The set of six differential equations of equilibrium previously described can be
simplified using the assumptions made to describe the response of a rope element
(Costello, 1990):
•
A rope element develops constant stresses along its length, which means that any
variation in a stress resultant with respect to the arc length s vanishes (d()/ds = 0).
The rope is long enough to ignore clamping effects/conditions.
•
Based on Eq. 3.14, the curvature κ of a rope element is referred to the binormal
axis x3, which means that κ3 = κ and κ2 = 0.
•
A rope element is not subjected to bending moments per unit length, that is, m3 =
0 and m2 = 0.
By letting Te= V1, V = V3, Mte = M1, Mbe = M3 and X = w2, in which Te is the
tension force, V is the shear force in the binormal direction, Mte is the twisting moment,
Mbe is the bending moment about the binormal direction and X is the contact force per
unit length in the normal direction, the equilibrium equations of a rope component reduce
to
− Vξ + Teκ + X = 0
(3.26)
− M beξ + M teκ − V = 0
(3.27)
m1 = V2 = w1 = w3 = 0
(3.28)
The contribution of the element stress resultants (axial force, shear force, twisting
and bending moments) to the next higher level resultants can be determined as follows:
Ts = Te cos θ + V sin θ
(3.29)
M bs = M be cos θ − M te sin θ
(3.30)
44
M ts = M te cos θ + M be sin θ + (Te sin θ − V cos θ )a
(3.31)
where Ts, Mbs and Mts are the axial force, bending and twisting moments on the next
higher level, respectively, and a and θ are the helix radius and the helix angle of the rope
element respectively, defined at the level of the rope element considered. It should be
noted that if the arrangement of the elements in each layer is symmetric, the value of Mbs
must vanish. However, there may be some situations in which the value of Mbs does not
vanish, and an unbalanced bending moment can act on a rope. Some possible situations
that can lead to an unbalanced bending moment include variation in material properties in
rope elements, variation in cross-sectional geometry in rope elements, variation in the
degree of damage experienced by rope elements, etc.
3.4.3 Force and Moment Resultants in a Rope Element
Once the deformed configuration of a rope element is known, and using the
constitutive models proposed in Section 3.2, the internal stress resultants acting on the
cross-section of a rope element can be computed. By enforcing the equilibrium
conditions on the cross-section of a rope element between the internal and external load
actions, the following well known expressions are obtained:
Te = ∫ σ (ε )dA
(3.32)
M be = ∫ ζσ (ε )dA
(3.33)
M te = ∫ ρτ (γ )dA
(3.34)
A
A
A
where Te is the tension force, Mte is the twisting moment, Mbe is the bending moment
about the binormal direction, and σ(ε) and τ(γ) are the normal and shear stresses given by
Eqs. 3.17a and 3.17b, respectively. The distances ζ and ρ are measured from the centroid
45
of a rope element and belong to the interval [-r, r], where r is the radius of the crosssection of the rope element as shown in Fig. 3.12.
x2
x2
(a)
dA
x3
(b)
dA
ρ
ζ
x3
2r
2r
Fig. 3.12: (a) Cross-section for bending moment; (b) Cross-section for torsion
It is assumed that both the normal strain due to bending and shear strain due to
torsion vary linearly through the cross-section of a rope element, thus the following
expressions hold:
ε = ε er + ζ (κ − κ 0 )
(3.35)
γ = ρ (ξ − ξ 0 )
(3.36)
where εer is the axial strain of a rope element due to an extension and/or rotation of the
rope given by Eq. 3.16 and κ0 and ξ0 are the curvature and torsion in the initial
configuration of the rope element, respectively. Based on Eqs. 3.35 and 3.36, Eqs. 3.32
through 3.34 can be solved using a numerical integration scheme (e.g., Gaussian
Quadrature), and the resulting values of Te, Mte and Mbe are used with Eqs. 3.26 and 3.27
to obtain the values of the shear force V and contact force per unit length X that act on a
rope element.
46
3.4.4 Friction Model
In any fiber rope structure where there is no bonding between rope elements, the
results of any deformation must result in a slip of contiguous rope elements due to the
assumption of geometry preservation and finite dimensions of the rope elements’ crosssections. Due to the helical geometry of the rope elements, the applied external action on
the rope (loads or displacements) results in bearing pressures at the contact regions. The
actual slip magnitude in the contact regions is a fraction of the rope elements’ diameter,
whereas bearing forces are functions of the rope geometry. Due to the relative slip and
the presence of bearing pressures between contiguous rope elements, frictional forces can
be developed. The slip at contact regions and the bearing pressure combine to give the
work done by friction in opposing rope deformation. Although the magnitude of the
contribution of frictional forces to rope force resultants is small, the number of slipping
location and the repeated action of loading and unloading could have significant effects
on the long-term performance of a rope. Thus, the energy loss through the accompanying
friction hysteresis would be substantial and become an important factor in studying the
deterioration of ropes during their loading history. Several modes of slip can be identified
during rope deformation (Leech, 2002). In this study, however, only the two most
important modes of slip identified by Leech (2002) are considered: slip due to stretch and
slip due to change of the helix angle.
In order to estimate the magnitude of frictional forces, a simple model has been
established, based on the classical slip-stick model, where frictional forces act in the
direction opposite the relative slip direction between contiguous rope elements (Leech,
2002). The frictional force f is given in terms of the normal contact force, pnc, as
f = f a + µ ( p nc ) b
(3.37)
where fa and b are friction parameters and µ is the coefficient of friction of the rope
material. Based on the assumptions made to model the cross-sectional behavior of rope
elements (Section 3.3), relative slip between elements vanishes for the packing geometry
configuration because the contact region is assumed to be only in the radial direction.
47
Consequently, frictional forces do not arise between rope elements. For the case of the
wedging geometry configuration, the contact region is assumed to be in the
circumferential direction and relative slip between contiguous rope elements is not
prevented. Thus, the contact force, pnc, normal to the contact region, is given by
p nc =
X
(3.38)
2 cos θ sinψ
where X is the radial contact force obtained from Eq. 3.26, ψ is one half of the subtended
angle of the wedge and is equal to π/n, with n the number of rope elements in the layer
under study, and θ is the helix angle given by Eq. 3.1d. (Fig. 3.13).
Wedging Geometry
rope element
longitudinal axis
x3
2β
pc
X
pc
x1
pnc
θ
pc =
X
2 sinψ
ψ=
π
2
pc
−β
Fig. 3.13: Normal force acting on a rope element (wedging geometry)
The contribution of the frictional forces to the force resultants of the rope can be
obtained by adding the projected components of the frictional forces to Eqs. 3.29–3.31 as
shown in Fig. 3.14.
48
The vertical (Vf) and horizontal (Hf) contribution of the frictional force f to the
rope force resultants are given by
V f = 2 f d sin θ
(3.39)
and
Hf = −f d
(3.40)
where d = 2asinψ, a is the helix radius and θ is the helix angle of the element,
respectively.
rope element
longitudinal axis
rope longitudinal
axis
f
pnc
A
d tan θ
A
θ
f
Just frictional forces
f
f
A
pnc
f
θ
Hf
A
Vf
View A-A
f
d
Fig. 3.14: Effect of frictional forces on rope force resultants
49
3.5 DAMAGE MODEL
The following subsections describe the proposed damage model to account for the
deterioration of rope element properties as a function of the loading history. First, a
background of the possible factors that can cause damage of synthetic ropes is provided.
Next, the proposed evolution law that describes the damage accumulation in rope
elements throughout their loading history is presented. Finally, invoking principles of
continuum damage mechanics, the effects of damage accumulation on rope strength and
stiffness are described.
3.5.1 General Background
The deterioration of rope properties during loading has been observed in previous
investigations (Liu, 1989, 1995; Lo et al., 1999; Banfield and Casey, 1998; Karayaka et
al., 1999; Mandell, 1987; Van Den Heuvel, et al., 1993). The failure of a rope element is
a complicated process that could depend on a variety of factors such as strain range,
abrasion, number of loading cycles, installation procedures, environmental interaction,
etc. In the current model, the following three processes are considered: strain range,
abrasion (due to frictional forces) and number of load cycles at a given stress range.
The damage of materials takes place when atomic bonds break at the
microestructural level. The phenomenon of damage represents surface discontinuities in
the form of microcracks or volume discontinuities in the form of cavities. In this study,
the hypothesis of isotropic damage is assumed, which implies that the microcracks and
cavities (defects) are uniformly distributed in all directions. Thus, the damaged state is
completely characterized by the scalar damage index parameter D. This parameter
evolves during the response history and modifies rope behavior when any of the rope
elements experience damage (Rungamornrat et al., 2002; Beltran et al., 2003).
The degradation of the mechanical properties of a system is, in general, an
irreversible process. As the damage index parameter captures the damage accumulation
of the system, it should be represented by a continuous and monotonically increasing
function with values that can vary from 0 to 1. A value of 0 corresponds to the
undamaged state of a component, and D =1 indicates its complete rupture. In this
50
particular study, the evolution rule of damage D is obtained experimentally, which
includes curve fitting and eventually statistical analysis.
3.5.2 Damage Evolution Rule
As previously mentioned, Rungamornrat et al. (2002) proposed a damage
evolution model in which the damage index parameter depends on the strain range,
abrasion (due to frictional forces) and number of load cycles at a given stress range. This
model estimates the surface discontinuities due to the breakage of bonds that exist
between long change of molecules and its formulation is independent of the rope element
length. Accordingly, the evolution rule of the damage index parameter D is given by
 ε −εt
D = DI + α 1  m
 εb
β
1
W

 + α 2  fa
W

 fe




β2

1
+ α 3  ∑
 i Ni



β3
(3.41)
The first term in the above equation represents the initial damage DI that an
element could have before being loaded. The second term relates to the maximum strain
(εm) that an element experiences over its entire loading history. The term εt represents the
threshold strain that must be exceeded before damage occurs, and εb is the strain at which
an element reaches its maximum stress under monotonic loading. The threshold strain
value εt represents the value of the strain at which the bonds between the molecules that
form the polymer start to break. Consequently, the molecular weight lowers and the
molecular weight distribution narrows, generating local stress concentrations,
intensifying further accumulation of molecular breakdown, leading to the ultimate
rupture of the rope element (Van den Heuvel et al., 1993).
The third term is introduced to account for the effects of abrasion due to frictional
forces induced from the slip between elements during loading. The accumulated work
done due to friction (Wfa) is utilized to measure the degree of abrasion. The toughness of
an element under monotonic loading (Wfe) is used to normalize Wfa. Using a linear
approximation between two consecutive load steps, the value of (Wfa) for the kth load step
is given by
51
k
W fa = W fa
k −1
+
[
1
f
2
k −1
]
+ f k ∆slip k
(3.42)
where (Wfak-1) is the accumulated work done by frictional forces up to the (k-1)th load
step, f
k-1
and f k are the frictional forces in the (k-1)th and kth load step respectively, and
∆slipk is the relative slip between two contiguous elements at the kth load step given by
∆slip =
lk
k
l
k −1
2a k −1 sin(ψ ) sin(θ k −1 ) − 2a k sin(ψ ) sin(θ k )
(3.43)
where l k-1 and l k, a k-1 and a k and θ k-1 and θ k are the length, helix radius and helix angle
of the rope element in the (k-1)th and kth load step, respectively.
The toughness of an element (Wfe) is defined as the required work needed to break
the element in pure tension. Thus, the value of (Wfe) is given by
 5 a 
W fe = A0 l 0σ b ε b ∑ i 
 i =1 i + 1
(3.44)
where A0, l0 are the initial area and initial length of the element under consideration, σb
and εb are the normal stress and axial strain, respectively, at the onset of element failure,
and ai are the constitutive parameters of the constitutive model for normal stress (Eq.
3.17a).
The fourth term corresponds to the effect of stress range experienced by an
element in the entire loading history and is used to account for fatigue effects under
cyclic loads. The parameters Ni represent the minimum number of cycles necessary to
break the element under the stress range of the ith cycle. The relation between Ni and the
stress range ∆σi in the proposed damage rule is given by
∆σ i = σ 0 e cN i
−k
(3.45)
52
where c and k are model constants, and σ0 is the endurance limit of the element. The
coefficients αk and βk are damage parameters and are selected based on experimental data.
It is important to point out that although in the current model three processes are
considered to account for damage (Eq. 3.41), additional terms can be included to consider
other potential sources of damage such as environmental interaction and creep effect. For
the case of the damage term related to environmental interaction Dei, this term can be
considered to be a function of salt content, ultra-violet light exposure, and time. This
relationship can be expressed as
Dei = F ( s,UV , t )
(3.46)
where s is a term that depends on salt content, UV is a term that depends on ultra-violet
light exposure and t is time.
Similarly, to account for damage due to creep (Dcr), this term can be considered to
be a function of constant axial stress, time at which constant stress is applied, time at
which an element fails due to creep effect, and time. This relationship can be expressed
as
Dcr = F (σ c , t c , t f , t )
(3.47)
where σc is a term that depends on the value of the constant stress applied, tc is the time at
which the constant stress is applied, tf is the failure time at a given constant stress and t is
time.
The introduction, however, of the two processes just described into the damage
model, would necessitate the modification of the material constitutive model to account
for time effects.
53
3.5.3 Basic Concept of Continuum Damage Mechanics
Synthetic-fibers are produced from polymers. Thus, they are made of molecules
tied together in long repeating chains. Damage in polymers can be defined as the
existence of distributed broken bonds between long chains of molecules, whereas the
process of breaking the bonds between molecules that causes progressive material
degradation through strength and stiffness reduction is called damage evolution
(Chaboche, 1988; Lemaitre and Chaboche, 1990). Among the various definitions of the
damage variable D is the ratio of the damaged area to the total (undamaged or virgin)
area at a local material point (Kachanov, 1986). Based on this definition, the effective
stress (σe) that the damaged medium experiences is defined as
σe =
σ
(3.48)
1− D
where σ is the actual or nominal stress.
An expression for D can be found relating the undamaged and damaged medium.
In this particular study, the principle of strain equivalence is used. This principle states
that the deformation behavior of a material is only affected by damage in the form of the
effective stress. Lemaitre and Chaboche, (1990) define this principle such that “any
deformation behavior, whether uniaxial or multiaxial, of a damaged material is
represented by the constitutive laws of the virgin material in which the nominal stress is
replaced by the effective stress.” A schematic representation of the principle of strain
equivalence is presented in Fig. 3.15.
Consider the constitutive response for the virtual and damaged materials as shown
in Fig. 3.15. Based on the previous assumptions, it is assumed that the following relations
hold:
σ e = f (ε , ε 2 ,...)
(3.49a)
σ = g (ε , ε 2 ,...)
(3.49b)
54
If for each strain ε the difference between the above expressions is computed
(based on the principle of strain equivalence), and the definition of σe given by Eq. 3.48 is
applied, the following expression relating the constitutive functions in terms of the
damage index D is obtained:
g (ε , ε 2 ,...) = (1 − D) f (ε , ε 2 ,...)
(3.50a)
or
D = 1−
g (ε , ε 2 ,...)
f (ε , ε 2 ,...)
(3.50b)
σ
σ
ε
D
σ
σe
σ
Virgin
σe
Material Models
Virgin
ε = F(σ,σ2,…)
Damaged
ε = G(σ, σ2,…)
Equivalent virgin (virtual)
ε = F(σe, σe2,…)
Fig. 3.15: Principle of strain equivalence (Lemaitre and Chaboche,1990)
The expression given by Eq. 3.50a is used to obtain the constitutive equations for
a damaged system once the evolution of D is known, where f(ε, ε2, …) can be either the
normal stress σ or shear stress τ given by Eqs. 3.17a and 3.17b, respectively. Similarly,
g(ε, ε2, …) can be either the normal stress σd or the shear stress τd of the damaged system
55
(Rungamornrat et al., 2002). An equivalent expression for Eq. 3.50b, based on Eqs. 3.49a
and 3.49b, is given by
D = 1−
σ
σe
(3.50c)
Based on Eq. 3.50c, the definition of secant modulus Es = σ(ε)/ε, and the principle
of strain equivalence previously explained, Eq. 3.50c can be written as
D = 1−
E sd (ε )
E sv (ε )
(3.50d)
where Esd(ε) and Esv(ε) are the secant moduli of the damaged and virgin system,
respectively, for a particular value of the axial strain ε.
3.6 SUMMARY
In this chapter, the foundations of a proposed mathematical model to predict the
behavior of synthetic-fiber ropes under a variety of loads conditions have been presented.
This model is an extension of the discrete helical rod model to study steel wire ropes and
the model developed by Leech (2002) to analyze synthetic-fiber ropes. The kinematics of
a rope, constitutive models, cross-section modeling, equilibrium equations and friction
models described in this chapter are based on concepts from the two models previously
mentioned. The major contribution that the proposed mathematical model addresses is the
inclusion of a damage index to account for the effects of strength and stiffness
degradation of rope elements. The calibration of the proposed damage model based on
available experimental data is described in the next chapter. In addition, the static
response of small ropes with different rope configurations is also presented. For a
particular rope configuration, a three-dimensional finite element analysis is performed to
validate some of the assumptions made to develop the proposed computational model.
56
CHAPTER 4
Rope Degradation and Static Rope Response
4.1 OVERVIEW
The current study is focused on polyester ropes loaded monotonically and
subjected to a prescribed strain history. Under the assumption that frictional forces play a
small role in affecting the breaking load of a rope under monotonic loading, and using the
damage model evolution described in Section 3.5.2, deterioration of the mechanical
properties of a rope is assumed to depend solely on the strain range of the response.
Hence, based on Eq. 3.41, the evolution of the damage index D is given by
 ε −εt
D = DI + α 1  m
 εb



β1
(4.1)
where the internal variable εm and the damage parameters DI, εt, εb, α1, β1 are described in
Section 3.5.2. The above model is based on the work by Van den Heuvel et al. (1993)
which suggests that polyester yarns experience damage for strains greater than 8%.
To calibrate the damage index evolution D given by Eq. 4.1, experimental data
reported by Li et al. (2002) are used. In Fig. 4.1, the behavior of a one-level, two-layer
rope loaded monotonically, along with its geometrical properties, is presented. In this
rope construction, the rope yarns do not contain their own core with components wound
around it. As such, it is defined as having just one level. The rope yarns are laid around
the rope core giving rise to two layers (including the core). In this figure, two different
curves are presented — simulation of rope response without any source of damage
(considering a Poisson’s ratio v equal to 0.25 (Li et al., 2002)), and experimentally
measured data. The response of the rope without the effects of damage was computed
using the governing equations presented in Chapter 3. The simulation of the rope
response provides an upper bound for the experimental curve along the entire strain
57
history. The differences between the predicted response and the experimental data can be
related to many factors such as damage experienced by the rope, variability in rope
properties, assumptions included in the computational model, etc.
Rope Axial Load (Kips)
6
Predicted Response.
5
Rope Cross-Section
Exp. Data
4
3
Rope Yarn
2
Radius of rope yarn (mm): 1.016
Pitch of rope yarn (mm) : 81.28
Rope length (mm)
: 609.6
1
0
0
0.02
0.04
0.06
0.08
0.1
Rope Axial Strain
0.12
0.14
Fig. 4.1: Response of a one-level, two-layer rope under monotonic strain
history and cross-section and geometric parameters of the rope
In general, the analysis of a system is commonly performed assuming
deterministic material properties. This approach disregards uncertainties with respect to
the internal stresses, spatial variation of the mechanical properties, inhomogeneities of
the material and other effects that lead to a random mechanical response of a system
(stochastic response). For a realistic analysis, it is necessary to consider not only average
values but also variances of the material parameters (stiffness, breaking stress, breaking
strain, etc.). In order to consider stochastic material behavior, various experimental data
are needed to determine the set or sets of parameters that can be used for the simulation
of the random response of a system, which are associated with mean values, variances
and probability density functions (Reusch and Estrin, 1998).
In this particular study, the extent of experimental data is not sufficient for a
detailed statistical analysis to be performed. Accordingly, only the average values of the
58
material parameters such as breaking stress, stiffness and breaking strain are used as input
data to simulate rope response. In addition, at least in portions of a rope that are
sufficiently far away from the terminations, the geometry of a rope does not present
geometrical irregularities or discontinuities such as notches, holes or sharp corners that
could affect the stress distribution. If irregularities or discontinuities were present,
stresses near these locations would become concentrated in which small variations in the
material properties may be accentuated and cause pronounced deviations of the overall
stress values from the expected average behavior (Reusch and Estrin, 1998). For the
reasons just mentioned, and in order to verify the assumptions made in developing the
computational model including calibration of the proposed damage evolution equation,
the differences between the predicted rope response and the experimental data are
assumed to be related solely to the damage experienced by the rope (Fig. 4.1a).
Using experimental data, the values of εb and εt can be obtained directly. The
value of εb is the strain at which a rope element (rope yarn) reaches its maximum load,
and εt is the strain at which the experimental curve starts deviating from the one
determined by simulation. In order to compute the values of the damage parameters α1
and β1, concepts from continuum damage mechanics (CDM) introduced in Section 3.5.3
are invoked.
4.2 DAMAGE MEASUREMENT
In this section, two methods to quantify the evolution of damage experienced by
rope elements are presented: secant moduli ratio and strain energy deviation. These
methods are based on the comparison of the virgin (i.e, undamaged) curve and
experimental curve (damaged) shown in Fig. 4.1.The formulation of both methods relies
on the effective stress concept and principle of strain equivalence (Section 3.5.3)
4.2.1 Secant Moduli Ratio
As discussed earlier, the differences between the predicted response and the
experimental data are assumed to be solely related to the damage experienced by the
rope. According to the concepts presented in Section 3.5, it is expected that the damage
59
index evolution be represented by an increasing monotonic function with values varying
between 0 and 1. In order to achieve this behavior, based on the damage evolution given
by Eq. 3.48b (or Eq. 3.48d), the following three conditions must be satisfied: (1) f(ε, ε2,
…), g(ε, ε2, …) > 0, (2) f(ε, ε2, …) ≥ g(ε, ε2, …) and (3) d( f(ε, ε2, …)) ≥ [f(ε, ε2, …)/ g(ε,
ε2, …)]·d(g(ε, ε2, …)) for every value of ε on [εt, εb], where d() is the total derivate with
respect to ε. In this particular application, the functions f(ε, ε2, …) and g(ε, ε2, …) are
associated with the predicted response and experimental data curves, respectively (Fig.
4.1a). The value of ε is the axial rope strain, and it is assumed, due to the small size of the
rope tested, that all rope elements (rope yarns) experience the same axial strain as the
rope. It is also assumed that the initial damage DI (Eq. 4.1) is equal to zero.
It is important to note that the value of D, when computed using Eq. 3.48b,
reaches a value approximately equal to 0.1 at the onset of failure based on the model
selected for its evolution. Thus, before failure, rope elements have experienced a
relatively low level of damage according to the model. This observation suggests that
under increasing monotonic load, rope components have a quasi-brittle behavior and a
rapid increase in the value of the damage index must occur just prior to complete failure.
For each value of the strain ε, Eq. 3.48b is applied to obtain the evolution of D based on
the curves presented in Fig. 4.1a. The evolution obtained for D using this approach,
however, does not satisfy the condition of being an increasing monotonic function
because it fluctuates (i.e., local rate of change of damage is negative), especially for
values of the threshold strain εt smaller than 0.04, where the oscillatory behavior of
damage is more pronounced (Fig. 4.2).
As stated before, the simulation curve is always an upper bound for the
experimental data, and, unlike the value for the breaking strain εb, the value of the
threshold strain εt is not well defined. In addition, using Eq. 3.48b to describe the damage
index D does not satisfy the condition of zero damage for a value of a strain ε equal to εt
(Fig. 4.2). In order to match the conditions of being an increasing monotonic function and
having a zero value for strain ε equal to εt, a modified expression is used to compute the
evolution of D, which is an empirically-based equation. The modified equation to
60
compute the evolution of D throughout the strain history is derived using Eq. 3.48b (or
Eq. 3.48c) and is given by
ε
D(ε ) =
d (ε
∑
ε ε
k≥ t
k
)
(4.2)
H
where d(εk) = 1-σd(εk)/ σ(εk) and εt ≤ εk ≤ ε, for every ε such that εt ≤ ε ≤ εb and H is a
normalizing constant. The values of σ(εk) and σd(εk) are the axial stress of the undamaged
and damaged medium, respectively, for a particular value of the axial strain εk.
As previously mentioned, the direct use of Eq. 3.48b is not appropriate for
obtaining the evolution of the damage index for the case currently under investigation.
Based on the results shown in Fig. 4.2, the difference between the predicted rope
response and the experimental data, measured by the use of Eq. 3.48b, is not only due to
damage. However, for values of the normalized variable (ε - εt)/ εb greater than 0.2 for εt
= 0.03, and greater than 0.1 for εt = 0.05, the use of Eq. (3.48b) results in values that meet
the requirements to be considered as a valid damage model (i.e., values between 0 and 1
and increasing behavior as the value of the strain ε increases). The role of Eq. 4.2 is to
smooth the values of the variable d(εk) and construct an allowable function to measure the
damage experienced by the rope for the general case.
The form of Eq. 4.2 satisfies two basic requirements of CDM principles: it
ensures that the evolution of the damage index parameter D(ε) is monotonically
increasing and its value varies from 0 to 1. The first requirement is satisfied by
accumulating the variable d(εk), and the second one by using the normalizing constant H.
It is important to note that the form of H could be more complex than a constant and may
be described by a generalized function of the strain ε, and in general, will depend on the
problem under study. This possibility is currently under investigation by the author.
Some precautions must be taken when Eq. 4.2 is used however. As stated above, the
value of the threshold strain defines when the damage starts being accumulated. Near the
origin of the σ – ε plane, there are small nonlinearities due to the rearrangement of rope
61
elements (the so called “bedding in” effect) that are not considered to be a source of
damage. For these conditions, it is best to ignore the small variations between the test and
simulation values by defining a minimum value for the threshold strain.
Damage
0.13
Eq. 3.48b
ε b = 0.124
ε t = 0.03
0.11
0.09
0.07
0.05
0
Damage
0.11
0.2
0.4
( ε - ε t )/ ε b
0.6
0.8
Eq. 3.48b
ε b = 0.124
ε t = 0.05
0.09
0.07
0.05
0.03
0
0.2
0.4
( ε - ε t )/ ε b
0.6
Fig. 4.2: Attempt to describe damage evolution based on Eq. 3.48b for
different values of the threshold strain εt
Using Eq. 4.2 and the curves plotted in Fig. 4.1a, the evolution of D(ε) throughout
the strain history is computed. For this particular application, the normalizing constant H
has a value such that D will reach a maximum value equal to 0.1. This value was chosen
because it is the value of the variable d(εj) just prior to the onset of failure for individual
elements within a rope. The use of this normalization scheme implies that the evolution
62
of the damage index must vary rapidly to reach a value of one at the moment of rope
element failure. Accordingly, this formulation suggests that under increasing monotonic
load, rope elements have a quasi-brittle behavior, which is consistent with observations
made from polyester ropes loaded to failure.
After the value for H is selected, a curve of the form of Eq. 4.1, setting the initial
damage DI equal to zero, is fitted to D(ε) in order to obtain the damage parameters α1 and
β1 so that the evolution of the damage index is represented as a continuous process..
According to the experimental data reported by Li et al. (2002), the value of εb is well
defined; however, as stated earlier, such is not the case for the value of εt. For this reason,
evolution of the damage index D(ε) for different values of threshold strain εt are
presented in Fig. 4.3.
In all cases, the evolution of D(ε) has a high value of coefficient of correlation
(R2) with a curve of the form of Eq. (4.1). However, for values of εt less than 0.03, the
slope of the damage function is greater for small values of the variable (ε - εt)/εb than it is
for higher values of the variable (ε – εt)/εb. This behavior can be related to nonlinear
effects rather than damage. The values of α1 and β1 depend on the value chosen for the
threshold strain εt as is shown in Fig. 4.4. The damage index evolution becomes more
linear as the value of the threshold strain εt increases (see Fig. 4.3).
(a)
Eq. 4.2
0.12
Fitted curve (Eq. 4.1)
Damage
0.09
0.06
D = 0.1099 η
0.5988
2
0.03
R = 0.9669
ε t = 0.01
0
0
0.2
0.4
0.6
η = (ε -ε t)/ε b
63
0.8
1
Damage
(b)
0.12
Eq. 4.2
0.09
Fitted curve (Eq. 4.1)
0.06
D = 0.1154 η
0.8412
2
0.03
R = 0.9934
ε t = 0.03
0
0
0.2
0.4
η = (ε -ε t )/ε b
0.6
0.8
Damage
(c)
0.12
Eq. 4.2
0.09
Fitted curve (Eq. 4.1)
0.06
D = 0.1515 η
0.9709
2
R = 0.9811
ε t = 0.05
0.03
0
0
0.2
0.4
0.6
0.8
η = (ε −ε t )/ε b
Fig. 4.3: (a), (b), (c) Damage index evolution for different values
of εt and εb = 0.124
64
0.16
0.14
α1
0.12
0.1
0.08
0.06
0
0.02
0
0.02
εt
0.04
0.06
0.04
0.06
1.2
1
β1
0.8
0.6
0.4
0.2
εt
Fig. 4.4: Variation of the damage parameters α1 and β for different values of ε t
4.2.2 Strain Energy Deviation
The density of elastic energy ψe of a virgin system can be defined as
ε
ψ e (ε ) =
∫ f ( ρ ) dρ
(4.3)
0
65
where σ = f(ε) represents the constitutive law for the virgin material and ε is the current
strain of the system. For the case of the damaged system, the density of the elastic energy
ψd can be defined as
ε
∫
ψ d (ε ) = g ( ρ )dρ
(4.4)
0
where σd = g(ε) represents the constitutive law for the damaged system.
Based on the concept of effective stress and the principle of strain equivalence
described in Section 3.5.3, the density of the elastic energy ψd for the damaged system
can be described as
ε
ψ d (ε ) = ∫ (1 − F ( ρ )) f ( ρ )dρ
(4.5)
0
where F(ρ) is the damage index that, based on the assumption made in Section 4.1 for the
case of monotonic loading, solely depends on the strain range of the system.
Defining the density of the elastic strain energy deviation (SED) as the difference
between the density of the elastic energy of the virgin system and damaged system
(Najar, 1987), the evolution of SED, using the property of linearity of the integral
operator and the principle of strain equivalence, can be described as (Fig. 4.5)
ε
SED (ε ) = ∫ F ( ρ ) f ( ρ )dρ
(4.6)
0
If it is assumed that the behavior of the virgin system f(ε) and damaged system are
known, the value of SED(ε) can be computed for any value of ε. Thus, Eq. 4.6 becomes a
linear integral equation for the unknown function F(ε). This equation could be solved, for
example, by replacing the integral ∫F(ρ) f(ρ)dρ over the interval [0, ε] by a numerical
66
integration rule. Thus, the interval of integration [0, ε] is discretized into points such as 0
≤ ε1 ≤ ··· ≤ ε and the integral is replaced by a quadrature on these points. This procedure
leads to a system of linear equations whose solution gives the values of F(ε) at the points
in which the interval [0, ε] was discretized (Linz, 1979). If Eq. 4.6 is solved using the
procedure just described, the evolution obtained for the function F(ε) would be the same
as the one obtained using Eq. 3.48b, and consequently Eq. 4.2 would be used again to
obtain an allowable damage evolution expression F(ε). An alternative procedure to obtain
damage evolution F(ε), based on Eq. 4.6, is used in which the generalized mean value
theorem for integrals is invoked.
σ
―― f (ε): Virgin System
------ g (ε): Damaged System
| | | | | SED (ε)
ε
Fig. 4.5: Strain Energy Deviation (SED) definition
According to Bartle and Sherbert (1982), the generalized mean value theorem for
integrals states: “if p(x) is continuous and q(x) is integrable on [a, b] such that q(x) ≥ 0 (or
q(x) ≤ 0) on [a, b], then
b
b
a
a
∫ p( x)q( x)dx = p(ζ )∫ q( x)dx
(4.7)
67
for some ζ on [a, b]”. Thus, the value p(ζ) is the weighted-average of the function p(x)
over the interval [a, b], with q(x) the weighting function.
Considering that the function f(ρ) in Eq. 4.6 is continuous on [εt, εb], and by
hypothesis the function F(ρ) is also continuous and positive on [εt, εb], the above theorem
can be applied to the definition of SED given by Eq. 4.6. The following expression
results:
F (ζ ε ) =
SED(ε )
(4.8)
ε
∫ f ( ρ )dρ
εt
Thus, the damage of the system F(ε) at a strain ε, such that εt ≤ ε ≤ εb, can be
estimated by its weighted-average value F(ζε), where ζε is some point on [εt, ε]. Provided
that the damage index F is an increasing function, the following relation holds: F(ζε) ≤
F(ε) for every ε on [εt, εb]. It is important to note that the denominator in above equation
is the accumulated density of strain energy of the virgin system from εt through ε. Using
the fact that an integral is a linear operator, the scalar factor Lε can be introduced such as
ε
Lε =
∫ε f ( ρ )dρ
(4.9)
t
εb
∫ f ( ρ )dρ
εt
Thus, the factor Lε is a measured of the stored density of strain energy at strain ε (ε ≤ εb)
with respect to the stored density of strain energy at strain ε = εb (Uv) of the virgin system
and Lε ≤ 1. Then, Eq. 4.8 can be written as
SED(ε )
F (ζ ε ) =
(4.10)
εb
Lε
∫ε f ( ρ )dρ
t
68
for every ε on [εt, ε]. The use of Eq. 4.10 generates a series of increasing values for F(ζε)
less than 1 provided the following three conditions are satisfied: (1) the values of Lε· Uv
and SED(ε) are positive on [εt, εb], (2) Lε· Uv ≥ SED(ε) on [εt, εb]
and (3) [(Lε·
Uv)/SED(ε)]·d(SED(ε)) ≥ d(Lε· Uv) for every value of ε on [εt, εb], where d() is the total
derivate with respect to ε. By definition, the first two conditions are satisfied, but the third
condition could not be satisfied for every value of ε on [εt, εb] because, as stated in
Section 4.2.1, damage is not the only cause of the deviation of the experimental curve
from the virgin curve (Fig. 4.1), especially for small values of strain ε. In Fig. 4.6, Eq.
4.10 is used to obtain the evolution of F(ζε) on [εt, εb] for two values of εt, 0.03 and 0.05,
as used in Fig. 4.2 (Section 4.2.1). The evolution obtained for F(ζε) using this energybased approach does not satisfy the condition of being an increasing monotonic function.
In fact, the evolution obtained is similar to the evolution obtained using the secant moduli
ratio method (Eq. 3.48b) presented in Fig. 4.2, but smoother. Thus, the energy-based
method smoothes the damage evolution obtained by the secant moduli ratio method.
Eq. (4.10)
0.15
εt = 0.03
εb = 0.124
0.1
εt = 0.05
0.05
0
0.2
0.4
(ε - ε t )/ε b
0.6
0.8
Fig. 4.6: Attempt to describe damage evolution based on energy-based
method
69
If Eq. 4.10 is used with the value of Lε equals one for every value of strain ε on
[εt, εb], an increasing lower bound evolution for F(ζε) on [εt, εb] can be obtained. This
evolution meets the CDM requirements to be considered as an allowable damage
function, in which the value of F(ζε), given by Eq. 4.10, at ε = εb is achieved. The
procedure described in Section 4.2.1 is again used here: the evolution of F(ζε) is
computed for different values of the threshold strain εt and then a curve is fit to the data
points obtained. The use of this method, based on the strain energy deviation concept,
leads to a polynomial evolution of the damage index F(ζε), having the following form:
 ε −εt
F = FI + ω1  m
 εb

 ε −εt
 + ω 2  m

 εb
1
2

 ε −εt
 + ω3  m

 εb



3
(4.11)
where the internal variable εm and the damage parameters FI, εt, εb were described in
section 3.5.2, being FI the initial damage, and the parameters ω1, ω2, and ω3 are obtained
by fitting a curve of the form of Eq. 4.11 to F(ζε) obtained based on Eq. 4.10 with the
value of the factor Lε equals to one for every value of strain ε on [εt, εb]. As before, the
value of the initial damage FI is set equal to zero. Evolution of the damage index F(ζε) for
different values of threshold strain εt are presented in Fig. 4.7.
4.2.3 Softening Behavior
Based on the concepts of CDM, at the moment of failure, the load carrying
capacity of the system should be zero. Therefore, at a certain value of strain εu (ultimate
strain), the system has no stress. According to the system behavior shown in Fig. (4.1),
after the rope reaches its maximum load, the system should undergo a softening behavior
in which the stiffness of the system decreases rapidly as strain ε increases. This effect
cannot be captured for reasonable values of strain using the evolution laws of the damage
index parameter proposed in Eqs. 4.1 and 4.11.
70
0.1
εt = 0.01
Damage (F )
0.08
εt = 0.03
εt = 0.05
0.06
F = 0.047η – 0.02η2 + 0.083 η3
0.04
F = 0.05η + 0.019η2 + 0.088 η3
0.02
F = 0.072η + 0.056η2 + 0.107 η3
0
0
0.2
0.4
0.6
0.8
1
η = ( ε - ε t )/ ε b
Fig. 4.7: Damage index evolution for different values of εt and εb = 0.124
using the energy-based method
To simulate the softening behavior of the system, it is assumed that after the
maximum load is reached, the system experiences softening in a very small region of the
axial force-strain plane (in comparison to the region in which the system develops its
maximum capacity). The reason why this assumption is made is given later when
numerical values of the damage index are obtained. In this small region, the rate of
change of the slope of the damage index parameter D(ε) (Section 4.2.1) or F(ζε) (Section
4.2.2) (F(ζε) will be also referred as D(ε) in the subsequent discussion because both
measures the damage of the system) governs the behavior of the system. Accordingly,
the evolution of the damage index parameter D(ε) can be expressed as a continuous
process along the strain history of the system using the asymptotic expansion technique
(perturbation method) as described below.
The kinetic equations that describe the evolution of the damage index parameter
D(ε) presented in Eqs. 4.1 and 4.11 along the entire strain history are (Beltran and
Williamson, 2004)
71
d 2 D(η ) dD(η )
−
+ α1β1η β1 −1 = 0
θ1
2
dη
dη
(4.12)
and
θ2
d 2 D(η )
dη
2
−
dD (η )
+ ω1 + 2ω 2η + 3ω3η 2 = 0
dη
(4.13)
respectively, where θ1 and θ2 are constants much smaller than 1 (θ1, θ2 << 1), and η is a
dimensionless variable defined as (ε - εt )/ εb . The boundary conditions that Eqs. 4.12 and
4.13 must satisfy are
D(η) = DI at η = 0
(4.14a)
D(η) = 1
(4.14b)
at η = ηu
In order to obtain uniform solutions (approximations) valid throughout the entire
domain of these equations (ηt = 0 to ηu), a perturbation method technique is used in which
the uniform solutions of Eqs. 4.1 and 4.11 are expressed as a power (perturbation) series
in θ (Logan, 1997; Lagerstrom, 1988; Holmes, 1995) of the form
D(η ) = D0 (η ) + θD1 (η ) + θ 2 D 2 (η ) + L
(4.15)
where the functions D0, D1, D2 … are to be determined by substitution of Eq. (4.15) into
Eqs. 4.12 and 4.13. The term D0 in the perturbation series is called the leading-order term
and it is the solution of the unperturbed problem (θi = 0, i = 1, 2.). In this particular study,
Eqs. 4.1 and 4.11 are the leading-order terms of Eqs. 4.12 and 4.13, respectively.
By construction, Eqs. 4.12 and 4.13 are singular equations because the
unperturbed problem (θi = 0, i = 1, 2) cannot satisfy the two boundary conditions
specified (Eq. 4.14). The singularity occurs in the vicinity of ηu where the approximate
72
solutions of Eqs. 4.12 and 4.13 develop the so-called boundary layer region, which is a
narrow interval where D(η) changes very rapidly. Conversely, the function D(η) varies
more slowly in the larger interval called the outer region away from the ultimate strain
(ηu).
The uniform approximations of Eqs. 4.12 and 4.13 are formed by an outer and
inner approximation. The outer approximation is valid only in the outer region and
satisfies the boundary condition at η = 0 (Eq. 4.14a). Conversely, the inner approximation
is valid only in the boundary layer and satisfies the boundary condition at η = ηu (Eq.
4.14b). Both approximations have the form of Eq. 4.15, but because of the significant
changes in D(η) that take place in the boundary layer, a new strain scale on the order of
some function of θ is introduced when the inner approximation is obtained.
To obtain a composite expansion that is uniformly valid throughout the interval
[0, ηu], Van Dyke’s matching rule is used (Nayfeh, 1973). This rule is based on the fact
that there is an overlap domain that is intermediate between the boundary layer and outer
region where both expansions, inner and outer, are valid. This matching condition is
illustrated in Fig. 4.8 for a function y(t) that develops a boundary layer near the origin.
For this particular study, consider a two-term composite expansion of Eqs. 4.12 and 4.13
does not give more information about the evolution of the damage index than the
information given by the one-term composite expansion (Appendix). Thus, just a oneterm composite expansion of Eqs. 4.12 and 4.13 is used to capture the evolution of the
damage index. For the case of Eq. 4.12, the composite or uniform solution for the damage
index, including a non-zero value of the initial damage DI, is given by
D (η ) = D I + α 1η
θ 1 (α 1 β 1η
β1 −1
β1
+ (1 − α 1η u
− α 1 β 1η u
β1 −1
e
β1
η −η u
θ1
− D I )e
η −η u
θ1
)
+
(4.16)
73
and the one-term composite expansion of Eq. 4.13 including a non-zero value of the
initial damage DI has the following form
[
]
D(η ) = DI + ω1η + ω 2η + ω 3η + 1 − ( DI + ω1η u + ω 2η u + ω 3η u ) e
2
3
2
3
η −η u
θ2
(4.17)
Inner Expansion
3
2.5
2
y(t)
Outer Expansion
1.5
1
Overlap Domain
0.5
0
0
0.2
0.4
Boundary Layer
0.6
0.8
1
1.2
Outer Region
t
Fig. 4.8: Overlap domain
The values of the dimensionless parameters θ1 and θ2 are obtained by requiring
Eqs. 4.1 and 4.16, and 4.11 and 4.17 match, within a prescribed tolerance, in the outer
region of the corresponding damage index evolution D(η). Using this approach, the
following condition is used to determine the value of θi (i = 1, 2)
θi ≤
η b −η u
ln(tol )
for i = 1, 2
(4.18)
74
where the variable tol is the tolerance defined to guarantee that Eqs. 4.16 and 4.17
represent the damage index evolution D(η) along the entire strain history.
Near the vicinity of ηu, both damage models (Eqs. 4.16 and 4.17) predict similar
behavior for D(η) because, in this region (boundary layer), the behavior of D(η) is
governed by the term e(η-ηu)/θ, which is a common factor for both models. As the value of
θ gets smaller, the boundary layer region gets smaller as well, making the function D(η),
discontinuous in the limit as θ → 0 (Appendix).
4.3 NUMERICAL IMPLEMENTATION
Once the evolution of the damage index is completely determined, damaged rope
behavior can be simulated using either the model described by Rungamornrat et al.
(2002) or the polynomial damage model presented in this study (Eq. 4.17). In Figs. 4.9
and 4.10, three types of curves are presented: simulation of the rope response without any
source of damage, experimentally measured data, and simulation of damage-dependent
rope response. The first two types of curves were already described and plotted in Fig.
4.1a. For the case of the damaged rope response curves, the evolution of D(η) computed
by the energy-based method (Fig. 4.9) and secant moduli ratio method (Fig. 4.10) are
used. In both cases, the value of the variable tol was set equal to 1E-5, the ultimate strain
εu was arbitrarily considered to be 3% greater than the value of the strain at which the
rope components reach their maximum capacity (εb) (just for the sake of the example),
and the value of threshold strain εt was treated as a parameter and allowed to vary from
0.03 to 0.05. The damage parameters associated with each model are given in Table 4.1.
For the cases shown in Figs. 4.9 and 4.10, simulation curves (considering
degradation model and softening behavior) agree well with the experimental curve for
values of axial strain greater than 0.05. In addition, predictions made using both damage
models predict the maximum axial load with very good accuracy (error varies from 0.6%
to 1.5% in stress and from 1.7% to 2% in strain, with respect to the experimental data).
The simulated curve (virgin rope response) that ignores the effects of damage
overestimates the breaking load by 10 %. The first jump in the simulation curves is
75
related with the failure of the core, and the second one is related with the failure of the
remaining elements (Figs. 4.9 and 4.10).
Rope Axial Load (Kips)
6
Virgin Rope Response
5
Exp. Data
4
Degradation Models
3
Jumps due to
core and rope
yarns failure
2
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rope Axial Strain
Fig. 4.9: One-level rope response using energy-based damage model
Rope Axial Load (Kips)
6
Virgin Rope Response
Exp. Data
5
Degradation Models
4
Jumps due to
core and rope
yarns failure
3
2
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rope Axial Strain
Fig. 4.10: One-level rope response using secant moduli ratio damage model
76
Table 4.1: Summary of values of damage parameters and predictions of the damage
models: one-level rope
Power Law Damage Evolution
Polynomial Damage Evolution
Max Strain
ω2
ω3
Max Strain
ω1
εt
α1
β1
Load at max
Load at max
[kips] Load
[kips] Load
0.03 0.12 0.84 5.217 0.1264 0.05 0.019 0.088 5.248 0.1261
0.04 0.12 0.87 5.251 0.1264 0.058 0.036 0.096 5.254 0.1261
0.05 0.15 0.97 5.221 0.1264 0.072 0.056 0.107 5.255 0.1261
Max. load of virgin rope response: 5.68 [kips]
Max. load experimental data: 5.19 [kips]
Strain at max. load experimental data: 0.124
A more detailed analysis is presented in Fig. 4.11. For a value of εt equal to 0.05,
the predicted curves derived from the energy-based and secant moduli ratio methods are
compared relative to the experimental data. As stated before, simulation curves agree
quite well with the experimental curve and the jumps in them reflect first the core failure
and then the failure of the remaining elements (rope yarns). However, the predicted curve
given by the energy-based method can be considered as an upper bound of the rope
response, because it slightly overestimates the predicted curve given by the secant moduli
ratio method and the experimental data.
In Fig. 4.12, the cross-section of a two-level, one-layer rope, along with its
geometrical properties, is presented. This rope is composed of three rope elements
(strands) of the type shown in Fig. 4.1b wound around a soft core (not shown), and each
rope element is formed by nine rope yarns.
77
Rope Axial Load (Kips)
6
Virgin Rope Response
5
Exp. Data
4
Energy-based Model
Secant moduli ratio Model
3
Jumps due to
core and rope
yarns failure
εt = 0.05
2
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rope Axial Strain
Fig. 4.11: Response of a one-level rope under monotonic strain history:
experimental, virgin response and damaged simulation curves
Radius of rope yarn (mm): 1.016
Pitch of rope yarns (mm) : 81.28
Pitch of strand (mm)
: 149.86
Rope length (mm)
: 609.6
Strand
Rope Yarn
Fig. 4.12: Cross-section and geometrical properties of a two-level,
one- layer rope.
78
In Fig. 4.13 and 4.14, three types of curves are presented with respect to the rope
cross-section shown in Fig. 4.12: simulation of the rope response without any source of
damage, experimentally measured data, and simulation of rope response including
damage. The difference between Fig. 4.13 and 4.14 is the value of the threshold strain εt
chosen to compute the damaged rope response. The simulation curve without any source
of damage overestimates the maximum capacity of the rope by approximately 15%. The
simulation curves that include damage were computed using the damage parameters
presented in Table 4.1 for each of the models used. In this particular analysis, a value of
εt equal to 0.04 and 0.05 were chosen, the variable tol was set equal to 1E-5, and the
ultimate strain εu was arbitrarily considered 1% greater than the value of the strain at
which the rope components reach their maximum capacity (just for the sake of the
example).
20
Rope Axial Load (Kips)
Virgin Rope Response
Exp. Data
15
Secant moduli ratio Model
Energy-based Model
10
Failure
εt = 0.04
5
0
0
0.05
0.1
0.15
Rope Axial Strain
Fig.4.13: Response of a two-level rope under monotonic strain history
79
20
Rope Axial Load (Kips)
Virgin Rope Response
Exp. Data
15
Secant moduli ratio Model
Energy-based Model
10
Failure
εt = 0.05
5
0
0
0.05
0.1
0.15
Rope Axial Strain
Fig. 4.14: Response of a two-level rope under monotonic strain history
As in Fig. 4.11, and based on Figs. 4.13 and 4.14, the simulation curves obtained
using the damage models based on the deviation strain energy and secant moduli ratio
provide similar rope response along the whole strain history. For the case of the damage
model based on the deviation of strain energy, these curves overestimate the maximum
capacity of the breaking load by approximately 8% and in the corresponding strain by
nearly 1.5%. The simulation curves based on the secant moduli ratio damage model
overestimate the maximum capacity of the breaking load by approximately 6%. As with
the previous method, the corresponding strain at the breaking load is overestimated by
1.5%. Similarly to Figs. 4.10 and 4.11, rope response curves show two jumps — the first
one is related to the core failure of each strand and the second one to the failure of the
remaining components (rope yarns) of each strand (Figs. 4.13 and 4.14). Table 4.2
summarizes the damage parameters used and the predictions of the damage models.
80
Table 4.2: Summary of values of damage parameters and predictions of the damage
models: two-level rope
Power Law Damage Evolution
Polynomial Damage Evolution
Max
ω2
ω3
Max Strain
Strain
ω
εt
α1
β1
1
Load at max
Load at max
[kips] Load
[kips] Load
0.04 0.12 0.87 14.92 0.1276 0.058 0.036 0.096 15.02 0.1276
0.05 0.15 0.97 14.87 0.1276 0.072 0.056 0.107 15.03 0.1276
Max. load of virgin rope response: 16.01 [kips]
Max. load experimental data: 13.95 [kips]
Strain at max. load experimental data: 0.126
In the previous examples, computed results were compared against data used to
calibrate the damage models. As expected under these conditions, the predicted response,
obtained by the use of both damage models, matches the measured values very well. To
evaluate the performance of the proposed damage models, it is necessary to compare the
results from simulations to data obtained from other tests. Computations were carried out
using the results shown in Figs. 4.9, 4.10, 4.11, 4.13 and 4.14 and summarized in Tables
4.1 and 4.2. For the subsequent analyses, only the damage model based on the secant
moduli ratio is considered. Therefore, the computational model, including the damage
evolution obtained by using Eq. 4.2, is now used to predict the response of a rope that is
damaged prior to testing. The damage parameters used to predict the damaged rope
response were α1 = 0.12 and β1 = 0.87, associated with a value of εt equal to 0.04. These
values were selected from the results obtained in the calibration study described in
Section 4.2.1.
In Figs. 4.15 and 4.16, five types of curves are presented to simulate the response
of 1-level, 2-layer ropes that were damaged prior to loading: simulation of virgin rope as
a reference, simulation of rope response accounting degradation of its mechanical
properties (based on the proposed damage model), experimental data reported by Li et al.
81
(2002) and, simulations of rope response neglecting the contribution of cut strands (net
area effect) including and not including degradation of its mechanical properties. Also,
the cross-sections of the damaged ropes are shown, indicating the rope elements that were
cut prior to testing. For initial conditions, the cut rope elements are assigned a value of
the initial damage DI equal to 1 (i.e., complete failure). The geometrical parameters of the
ropes are: rope yarns radius: 1.016 mm, pitch of the rope yarns: 81.28 mm and rope
length: 609.6 mm. The damaged simulation curves, considering the degradation model,
agree quite well with both experimental data curves along the entire strain history and
predict with good accuracy the maximum capacity of the damaged ropes. If only the net
area effect is considered without considering the degradation model, not only the
maximum capacities of the damaged ropes are overestimated but also the loaddeformation curves.
Virgin Rope Response
6
Rope Axial Load (Kips)
Net Area Effect & Degradation Model
Degradation Model
5
Rope Cross-Section
Net Area Effect & No Degradation Model
4
Exp. Data 1
Exp. Data 2
3
2
1
: Cut Rope Yarn
0
0.00
0.03
0.06
0.09
0.12
0.15
Rope Axial Strain
Fig. 4.15: Predicted responses of one-level initially damaged rope (33%)
82
Rope Axial Load (Kips)
6
Virgin Rope Response
Rope Cross-Section
Net Area Effect & Degradation Models
5
Degradation Model
4
Net Area Effect & No Degradation Model
Exp. Data 1
3
Exp. Data 2
2
: Cut Rope Yarn
1
0
0.00
0.03
0.06
0.09
0.12
0.15
Rope Axial Strain
Fig. 4.16: Predicted responses of one-level initially damaged rope (11%)
4.4 FINITE ELEMENT ANALYSIS
In Chapter 2, the fundamental principles of some of the available analytical
models to predict the response of steel cables and synthetic-fiber ropes under a variety of
loading conditions were presented. These models differ from each other in the number
and types of assumptions made about various problem parameters including end
conditions, contact modes, helix angles, rope element cross-section variations, internal
forces and moments, friction between rope elements, and material constitutive equations.
Predicted cable/rope responses obtained through use of some of the previously
mentioned analytical models have been compared with experimental data and finite
element models (Le and Knapp, 1994; Jiang et al., 1999; Jiang and Henshall, 1999;
Nawrocki and Labrosse, 2000; Jiang et al., 2000; Ghoreishi et al., 2004). These
comparisons have allowed for the identification of the limitations and the ranges of
applicability of the analytical models in predicting rope behavior (e.g., load-deflection
curve, contact stresses, end terminations effects, etc).
83
The proposed computational model to predict synthetic-fiber rope response relies
on the assumptions presented in Chapter 3. Many of these assumptions are the same as
the ones used by previous researchers and are related to the kinematics of deformation of
the rope elements, constitutive properties of the rope material, and contact between rope
components. With the assumptions made, it is possible to predict the response of a rope
using a two-dimensional model that is based on the cross-sectional properties of the rope
being analyzed. In this section, a three-dimensional finite element model is used to help
validate some of the assumptions made in developing the proposed computational model.
Thus, in order to demonstrate that a simplified two-dimensional analysis is sufficiently
accurate to predict overall rope response, analyses of the polyester three-strand rope
presented in Figs. 4.13 and 4.14 are compared with a three-dimensional finite element
study using a commercial finite element program (ABAQUS).
4.4.1 Finite Element Model
For modeling purposes, the cross-section of the rope presented in Fig. 4.12 was
modeled as a two-level rope in which the strands represent the first rope level and the
rope yarns represent the second rope level. The predicted response of this rope under
monotonic loading using the proposed computational model was compared with
experimental data in Figs. 4.13 and 4.14.
To simplify the finite element model of the three-strand rope, its cross-section is
modeled as a one-level, two-layer rope, where the strands are the only hierarchical rope
structure for modeling purposes. The strands are wound around a small core, and their
sections are discretized using 48 three-dimensional linear finite elements (12 linear
triangular prisms (6-noded elements) and 36 linear bricks elements (8-noded elements))
as shown in Fig. 4.17. This figure also includes some geometrical properties of the rope.
The two types of finite elements used to model the rope have three degrees of freedom at
each node, namely the translation in the x, y and z directions. Each strand was discretized
into 100 intervals (4800 finite elements) along its length, resulting in an aspect ratio of
the finite elements approximately equal to 4. Preliminary tests performed for models of
lengths between two and five pitch distances have demonstrated that the overall axial
84
response is not influenced by end effects. Hence, the results detailed in the next section
have been obtained for a finite element model that considers only two pitch distances as
the rope length.
(b)
(a)
Strand
Pitch of strand (mm) : 150.0
Radius of strand (mm) : 3.048
Fig. 4.17: (a) Three-strand rope cross-section and (b) finite elements mesh of
the rope
The radius of the small core (first rope layer in Fig.4.17) was selected such that
the strands (second rope layer in Fig. 4.17) are just in contact in the undeformed rope
configuration. The small core is discretized using 24 three-dimensional finite elements
(12 linear triangular prisms (6-noded elements) and 12 linear brick elements (8-noded
elements)). In the longitudinal direction, the core is discretized in 2880 finite elements.
The applied boundary conditions to the rope model are defined as follows:
•
One end section of the rope is fully clamped.
•
At the other end, an axial displacement history is specified and the crosssection is prevented from rotating.
85
•
Circumferential and radial contacts are defined for all of the rope elements. A
surface-to-surface contact formulation is used with Coulomb friction sliding
model.
For the purposes of the finite element model, it is assumed that the rope material
(polyester) is isotropic and has a hyperelastic behavior with some degree of
compressibility, assuming a Poisson’s ratio equal to 0.20. The rope response is computed
based on an associated strain energy function, which depends on the principal strains and
some material parameters. These material parameters are calculated internally by the
program based on the results of test data provided by the user (Abaqus Manual, 1998).
For this particular study, uniaxial test data are specified for the strands, which are, as
previously mentioned, the only hierarchical rope structure considered in the finite
element model. These data are obtained by predicting the strand response using a
monotonic strain history without any source of damage. This analysis is similar to the one
presented in Fig 4.1 (Predicted Response curve) with the exception that a Poisson’s ratio
equal to 0.20 is specified.
4.4.2 Finite Element Analysis Results
The predicted response of a polyester two-level rope and experimental data are
presented in Fig. 4.18. Three different predictions of the rope response, using a Poisson’s
ratio equal to 0.20, are given in the figure: rope response considering a degradation model
with damage parameters equal to α1 = 0.12 and β1 = 0.87; rope response without
considering a degradation model (i.e., virgin rope response); and rope response given by
a three-dimensional finite element model. All of the curves were obtained by performing
a displacement control analysis in which the rope is strained monotonically up to 12%.
As shown in Fig. 4.18, the curve obtained from the finite element model provides
an upper bound for the experimental data curve and the predicted curve that considers the
degradation model, but it correlates quite well with the predicted curve that does not
include any source of damage (undamaged rope simulation).
86
16
Undamaged Rope Simulation
Rope Simulation: Degradation Model
Exp. Data
Finite Element Analysis
Rope Axial Load (Kips)
14
12
10
8
6
4
2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rope Axial S train
Fig. 4.18: Two-level rope response predictions, including finite element
analysis, and experimental data
In Fig. 4.19, the undeformed and deformed rope geometries of the finite element
model are presented. The undeformed geometry of the rope is obtained by assuming that
the rope elements (strands) geometry can be described by a circular helix curve. The
deformed rope geometry presented in Fig. 4.19b, corresponds to an axial deformation of
12%. The figure indicates that the strands in their deformed configuration preserve their
initial geometry. Accordingly, rope elements in their undeformed and deformed
configurations can be represented by a circular helix curve, which is consistent with the
assumption made in the proposed computational model. Based on the comparison of the
predicted curves obtained by the proposed computational model (undamaged rope
simulation) and the three-dimensional finite element model (Fig. 4.18) and Fig. 4.19b,
this assumption appears to be reasonable.
87
(a)
(b)
Rope Axial
Deformation
Deformed Rope
Geometry
Undeformed Rope
Geometry
Fig. 4.19: Finite element model: (a) undeformed rope geometry and (b)
undeformed-deformed rope geometry
A contour plot of the axial stress distribution through the cross-section and along
the rope length for a rope axial strain of 12% is presented in Fig. 4.20. This figure shows
that the stress distribution varies through the cross-section but is uniform along the
length. The variation of the stress distribution through the cross-section is mainly due to
the effect of the contact forces between rope elements. This effect is ignored in the
development of the proposed computational model. However, the uniform distribution of
the axial stress along the rope length is consistent with assumption made in the proposed
computational model.
88
Axial Stress (Kips/mm2)
Fig. 4.20: Axial stress distribution at a rope axial strain of 12%
4.5 SUMMARY
In this chapter, damage evolution functions based on a polynomial and on a power
law have been used to account for degradation of synthetic-fiber ropes under increasing
monotonic load. Both models rely on the concept of effective stress and the principle of
strain equivalence. The damage parameters involved in both models are calibrated using
available experimental data from small polyester ropes. Concepts of secant moduli ratio
and strain energy deviation are used to calibrate the power law and polynomial damage
laws, respectively. Softening behavior of rope elements is simulated using the
perturbation method technique. Thus, damage models evolve as a continuous process
from an initial value DI through the complete failure of the rope elements (D =1).
Numerical simulations compare well with available experimental data for virgin ropes
89
(i.e., undamaged ropes, DI = 0 for all rope elements) and initially damaged ropes prior
loading (i.e., DI > 0 for some rope elements). Results obtained from three-dimensional
finite element analyses validate the predicted rope response obtained by the proposed
computational model A numerical procedure to estimate the effect of rope elements
failure (D =1) on overall rope response is presented in the next chapter.
90
CHAPTER 5
Numerical Simulation of Damage Localization in
Synthetic-Fiber Mooring Ropes
In this chapter, an analytical model to estimate the effects of damage on overall
rope response is presented. Damage can be represented through a degradation of the
properties of individual rope elements, and it can also include the complete rupture of one
or more elements. The general assumptions made to estimate the length over which
damage propagates along the rope length and the numerical procedure developed to
account for this length are explained in this chapter. In addition, numerical predictions of
polyester rope response are compared with experimental data obtained from static
capacity tests.
5.1 OVERVIEW
In the previous two chapters, a computational model to predict the behavior of
synthetic-fiber ropes was presented, addressing the issue of strength and stiffness
degradation through the use of a damage index. For rope elements that have a damage
index less than 1 (i.e., D < 1), the damage-dependent response can be computed using the
equations described in Section 3.5.3. In this chapter, a numerical procedure is proposed to
account for the effect of rope element(s) failure (i.e., D = 1) on overall rope response.
This feature of the computational model is considered to be essential given the results of
observations made from recent experimental studies on large-scale damaged ropes (Ward
et al., 2005).
Several studies have shown that if friction develops along rope elements, a broken
rope element is capable of supporting its total share of a load within a relatively short
length of the rupture location. This length is called the recovery length (Raoof, 1991) and
is defined as the distance measured from the failure end of a broken rope element to the
point at which this element resumes carrying its proportionate share of axial load. The
frictional forces depend on the presence of contact forces between rope elements (due to
91
their helical nature and/or rope jacket confinement) and the surface characteristics (i.e.,
coefficient of friction) of the elements in contact.
This concept is illustrated in Fig. 5.1 in which the broken element is represented
along the rope length by its centerline location, and the rope cross-sections at the failure
location (the broken rope element identified by an “X”) and at locations away from the
rupture site where the recovery length is fully developed are presented. In the regions
between the failure location and where the recovery length is fully established, the rope
cross-section experiences a variable degree of damage, depending on the contribution of
the broken rope element to the rope response. The extreme cases, corresponding to full
contribution and no contribution of the broken rope element to rope response, are
sketched in Fig. 5.1.
Rope element
failure
Damage on rope cross-section
Fig. 5.1: Recovery length concept of a broken rope element
The value of the recovery length, and thus the effect of rope element failure on
overall damaged rope response, depends on the type of rope construction as well as the
position and number of broken rope elements distributed around a rope cross-section. If
multiple rope elements fail in a relatively short length of a rope, strain localization around
the failure zone could develop, as can damage localization. Aside from the load carrying
capacity, the rope breaking strain and stiffness are dependent upon the number of broken
92
rope elements, their failure locations along the rope length, and whether or not their
recovery lengths are fully developed.
Some efforts have been made in order to estimate the values of the recovery
lengths of broken rope elements for different types of rope construction. On the
experimental side, Hankus (1981), Chaplin and Tantrum (1985), Cholewa and Hansel
(1981), Cholewa (1989), Evans et al. (2001), among others, have carried out tests for
measuring the recovery lengths of individual wires in axially loaded cables (strands and
ropes) and studying the effects of wire breaks on other wires (unbroken) at the same rope
(strand) cross-section. They have also tried to quantify the relationships between the
number and distribution of broken rope elements, their recovery length, and the rope
strength loss. These references have recommended a range of values for the recovery
length expressed as a percentage of the rope diameter or the strand pitch of the rope.
On the theoretical side, Chien and Costello (1985) published an analytical model
for estimating the recovery length in cables. They used the rigid-plastic Coulomb friction
model and invoked Saint-Venant’s principle to investigate the recovery length of the
central (straight) core wire of a seven-wire helical strand, and they also studied the
recovery length associated with the outer layer of six-strand ropes. Raoof (1991)
extended the theoretical model of Chien and Costello (1985) to include the transition
between the full-slip to no-slip friction interaction along the core wire of an axially
preloaded and multi-layered helical strand. This model was also used by Raoof and
Huang (1992) to address the problem of estimating the recovery length in parallel wire
cables prestressed by external wraps or intermittent bands. This later problem was
originally studied by Gjelsvik (1991), but only the case of full slippage along the entire
recovery length was considered in this initial study. As with the previous work by Chien
and Costello (1985), Gjelsvik’s (1991) work relied upon the rigid-plastic Coulomb
friction model.
Raoof and Kraincanic (1995 and 1998) presented a theoretical model for
determining the recovery length in helical wires in any internal layer of an axially loaded,
multi-layered spiral strand. Their model accounts for the effects of hoop and radial interwire contact, and they suggested a direct procedure for determining the recovery length
value of a broken wire. Subsequent theoretical parametric studies over a wide range of
93
cable configurations, including changes in wire diameters and helix angles, led to
recommendations concerning the determination of the required minimum length of test
specimens to be used for axial fatigue tests in order for the results to represent the actual
behavior of longer cables under service conditions.
For the current research, the works by Chien and Costello (1985) and Raoof and
Kraincanic (1995 and 1998) are extended for the case of synthetic-fiber ropes to compute
the recovery length value of a failed rope element. The resulting model considers the rope
cross-section modeling presented in Chapter 3 (Section 3.3) to compute the normal forces
acting on rope elements and the fact that a rope cross-section can consist of many levels
of components (Section 3.1). The numerical algorithm presented by Raoof and
Kraincanic (1998) to compute the value of the recovery length is also extended to
consider the case of a nonlinear relationship between frictional forces and contact forces,
as the one given by Eq. 3.37 if the parameter b is different than 1.
5.2 CALCULATION OF NORMAL FORCES AND RECOVERY LENGTH CONCEPT
In general, there are two basic approaches for quantifying the effects of broken
rope elements on overall rope response: (1) ignore the contribution of the broken rope
elements after failure (i.e., the so-called “net area” effect) or (2) include the contribution
of the broken rope elements assuming that they are capable of developing admissible
recovery length values. In essence, however, both of these models can be considered as
depending upon the estimated length over which damage propagates along the length of a
rope. For the case of the net area effect model, rope damage is assumed to propagate over
a distance greater than rope length (i.e., the recovery length required for a failed rope
element to resume carrying load exceeds the length of the rope). Accordingly, broken
rope elements do not contribute to rope response anymore after their failure. With this
approach, the stiffness and strength loss of the rope are proportional to the area that the
broken rope elements represent with respect to the rope area, and the rope breaking strain
remains unchanged. This effect on rope response is captured by the computational model
presented in Chapters 3 and 4, neglecting the potential lack of symmetry of the rope
cross-section.
94
Contrary to the net area effect concept, theoretical investigations and tests on
actual ropes suggest that a broken element eventually can start contributing to rope
response. The ability of a broken element to resume carrying load will depend upon the
value of its recovery length and whether or not it is physically admissible. For example, if
the failure region of a broken rope element is located at the mid-length position, the value
of the recovery length of this broken element is admissible if it is smaller than L/2, with L
defined as the total rope length. If such is the case, a strain localization region around the
failure site could develop, and based on the damage model, damage localization may also
occur. This damage localization can cause the premature failure of rope elements,
reducing the maximum load and maximum strain that the damaged rope is capable of
resisting.
In order to estimate the value of the recovery length of a broken rope element, the
normal forces exerted on this element must first be computed. In general, a rope element
is subjected to normal forces in two directions: radial (inter-layer contact forces) and
circumferential (inter-element contact forces). Thus, assuming a friction model based on
the normal forces, the gradual increase in tension in a broken rope element is partly due
to the radial forces exerted on the broken rope element from the inner and outer
neighboring layers and partly due to the circumferential forces exerted on the broken rope
element from the unbroken rope elements in the same layer. In Fig. 5.2, the pattern of
inter-layer/inter-element contact forces in an axially loaded rope element is presented.
The gi+1,i and gi-1,i are the line inter-layer contact forces exerted on a rope element located
in layer i by rope elements of layer i+1 and i-1, respectively; pnc,i is the line inter-element
contact force exerted on a rope element located in layer i by the other elements of the
same layer i; and Xi is the is the radial line body force necessary to preserve the helical
geometry of a rope element.
Based on the cross-section modeling of rope elements presented in Chapter 3
(Section 3.3), two types of patterns of inter-layer/inter-element contact forces in an
axially loaded rope element are considered. In packing geometry, contact between rope
elements in the same level is assumed to be only in the radial direction (inter-layer
contact forces). With wedging geometry, radial contact is assumed to be negligible in
comparison to the circumferential contact and is ignored for computational purposes.
95
However, there exists an inter-element type of contact between rope elements in the same
layer for ropes with wedging geometry.
pnc,i
gi+1,i
Xi
pnc,i
gi-1,i
Xi
pnc,i
gi+1,i
gi-1,i
Fig. 5.2: Pattern of inter-layer/inter-element contact forces in an axially
loaded rope element
In Fig. 5.3, the contact forces acting on a rope element located in layer i,
considering the cross-sectional modeling just described are presented. In this figure, the
angle βi is the complement of the subtended angle ψi, which is defined as π/ni, where ni is
the number of rope elements in layer i. Accordingly, one can note that the value of the
contact normal forces exerted on a broken rope element, its gradual increase in tension
for a given friction model, and the value of its recovery length depend on the modeling of
the rope cross-section for the particular case being evaluated.
In the following, the derivation of the equations that govern the gradual increase
in tension in a broken rope element for each type of contact pattern between rope
elements shown in Fig. 5.3 is presented for a one-level rope. These equations allow
computing the value of the recovery length of a broken rope element in layer i for a given
value of a tensile load Ti. After this presentation, these equations are extended for the
general case of a multi-level rope. Similarly to Eq. 3.37, friction is assumed to be based
on a slip-stick model given by
96
f = f a + µ (N )b
(5.1)
where fa, b, µ are friction parameters, and f is the frictional force per unit length due to the
normal line force N. Based on Fig. 5.4, the gradual increase in tension Ti of a broken rope
element that belongs to layer i and is subjected to the frictional line force fi is given by
x
Ti ( x ) =
∫ f ( ρ ) dρ
(5.2)
i
0
where the boundary condition Ti(0) = 0 is assumed at the location where rupture has
occurred, and the position x coincides with the longitudinal axis of the rope element. The
above general expression is valid for the two types of contact patterns shown in Fig. 5.3,
considering the appropriate expression for the normal line force N acting on the broken
element.
gi+1,i
2βi
Xi
pnc,i
Xi
pnc,i
gi-1,i
Packing geometry
Wedging geometry
Fig. 5.3: Contact forces in axially loaded rope element with packing and
wedging geometry
97
Ti
Ti+ dTi
fi(ρ)
Fig. 5.4: Gradual increase in tension load Ti in a broken rope element
5.2.1. Case 1: Packing Rope Geometry
For the case of a one-level rope with packing geometry, the normal line force Ni
exerted on a rope element located in layer i is given by (Fig. 5.3)
N i = g i +1, i + g i −1, i
(5.3a)
g i −1, i = g i +1, i + X i
(5.3b)
N i = 2 .0 g i +1, i + X i
(5.3c)
where the value of the line contact force gi+1,i can be estimated as (LeClair, 1991)
g i +1,i =
n j pi cos α i  j −1 cos α k
∏
∑

j =i +1ni p j cos α j  k =i +1 cos α k
ml

X
 j

(5.4)
where ml is the number of layers in the rope and nl, pl and Xl are the number of rope
elements, the pitch distance, and the radial line body force of a rope element in layer l,
respectively. The angles α l
and α l
are defined as tan −1 ( pl / 2π (al − rl )) and
tan −1 ( pl / 2π ( al + rl ) ) , respectively, where al is the helix radius and rl is the radius of a
rope element in layer l (Phillips et al., 1980). The value of the radial line body force Xi
can be computed using Eq. 3.26. Thus, Xi can be estimated as
98
X i = κ i Ti
(5.5)
where κi is the curvature of the rope element located in layer i. For simplicity, in Eq. 5.5,
it is assumed that the rope element only develops axial force, and the effects of the shear,
bending, and torsional stiffness can be ignored. Thus, considering Eqs. 5.2, 5.3 and 5.5,
the gradual increase in tension Ti(x) of a broken rope element located in layer i of a onelevel rope with packing geometry is given by
x
b
Ti ( x) = f a x + µ ∫ [2.0 g i +1,i + κ iTi ( ρ )] dρ
(5.6)
0
The above formulation is valid for rope elements located from layer 2 to ml. If
the broken rope element is the core of the rope (layer 1), its gradual increase in tension
Tc(x) can be estimated, considering full slippage along the entire recovery length (Chien
and Costello, 1985; Raoof, 1991), by
b

 n2 g1, 2  
Tc ( x) =  f a + µ 
 x

 cos θ 2  
(5.7)
where Tc is the core axial force, θ2 is the helix angle of the rope elements that lay in the
second layer, n2 is the number of elements that lay in the second layer of the rope, and
g1,2 is the line contact force exerted by the core on one rope element of the second layer.
The equations established to compute the normal line force Ni exerted on a rope
element located in layer i and its gradual increase in tension Ti(x) for the case of one-level
rope with packing geometry can be extended for the case of a rope with a multi-level
hierarchical structure. In order to illustrate this process, the analysis of a three-level rope
(Fig. 5.5) is performed, and the equations obtained are then extended for the case of an
m-level rope. For clarity purposes, each rope level is sketched separately in Fig. 5.5, and
99
each rope element is assumed to have the same hierarchical structure. The first rope level
has five layers, the second rope level has three layers and the third rope level has four
layers. The rope elements under study at every rope level are identified with an “X”.
Accordingly, it is first assumed that an element is broken in the third level of layer two.
Because of the geometrical configuration in the rope, this broken element is also part of
the second and first levels, belonging to layers three and two in those levels, respectively.
In the following, the superscripts of the equations refer to the level to which the element
belongs and the subscripts refer to the layers where the element is located, except for the
variable gi+1,i in which the subscripts refer to layers in the same rope level (as defined
earlier). The range of the subscripts and superscripts start from the level in consideration
and consider all hierarchical levels.
1st rope level-5 layers
2 nd rope level-3 layers
g 14,3
X 31
g 12,3
g 32, 2
X 22,[3]
g12, 2
3 rd rope level-4 layers
g 33, 2
X 23,[2,3]
g13, 2
Fig.5.5: Three-level packing geometry rope and normal forces acting on
the rope elements at each rope level
100
Thus, based on Fig. 5.5 and Eqs. 5.3, 5.4 and 5.5, the normal line force acting on
an element of the first level, which contains the broken element in the third level of a
three-level rope, is given by
(N ) = g + g
(N ) = 2.0 g + X
1
3
1
4,3
1
3
1
2,3
1
4,3
5
1
j
j =4
1
3
1
3
(5.8b)
cos α 31  j −1 cos α k

1
3 ∏
cos
α
j
j  k = 4 cos α k
(N ) = 2.0∑ nn pp
1
3
(5.8a)
1
3
 1
 X j + X 13 + κ 31 cos θ 23,[2,3] cos θ 22,[3]T23,[2,3]


(5.8c)
1
where X 3 is the radial line body force of an element that is in layer three of the first level
of the rope, ignoring the contribution of the broken element that is in level three, which is
part of the element under study in the first level; κ 31 is the curvature of an element that is
in layer three in the first level of the rope; θ 23,[2,3] and θ 22,[3] are the helix angles of elements
that belong to layer two in the third level and layer two in the second level, respectively,
which form elements of the second and first level of layers two and three, respectively;
and T23,[2,3] is the axial force of an element in layer two in the third level that forms
elements of the second and first levels, belonging to layers two and three, respectively.
Similarly, considering radial equilibrium in the rope element of the second rope
level presented in Fig.5.5, the normal line force acting on an element of the second level
located in layer two, which contains the broken element in the third level of a three-level
rope, is given by
(N [ ] ) = g + g
(N [ ] ) = 2.0 g + X
2
2, 3
2
2, 3
2
3, 2
2
1, 2
2
3, 2
(5.9a)
2
2 ,[3 ]
α
(N [ ] ) = 2.0 nn [ ] pp [ ] cos
cos α
2
2, 3
2
3, 3
2
2 ,[3 ]
2
2, 3
2
3,[3 ]
(5.9b)
2
2 ,[3 ]
2
3,[3 ]
X 32,[3] + 2.0
n32,[3] p 22,[3] cos α 22,[3]
n 22,[3] p32,[3] cos α 32,[3]
+ κ 22,[3]T23,[2,3] cos θ 23,[2,3]
(N ) cosn θ
2
1
3
2
3,[3 ]
2
3,[3 ]
2
+ X 2,[3]
(5.9c)
101
where the expression for g 32, 2 not only considers the effect of the upper layers of the same
level (layer three of the second level in this example), but it also accounts for the effect of
the contact forces exerted on the rope element of the first level (element in layer three in
the first level) of which the rope element of the second level is a component. This effect
is reflected by the term
( )
normal line force N 31
(N ) in Eq. 5.9c, whose expression is given by Eq. 5.8c.
1
3
The
is distributed uniformly over the elements of the second level that
belong to the outermost layer (layer three in this example), resulting in the normal line
force (Y32,[3] ) . The expression to estimate the value of (Y32,[3] ) is given by
2
3,[3 ]
Y
=N
1
3
cos 2 θ 32,[3]
(5.9d)
n32,[3]
where θ 32,[3] is the helix angle of an element of the second level located in the third layer,
which is part of an element of the first level located in the third layer; and n32,[3] is the
number of elements of the second level located in the third layer which are part of an
element of the first level located in the third layer. Therefore, the normal line force
(N [ ] ) is given by
2
2, 3
(N
2
2 , [3 ]
1
n32,[3] p22,[3] cos α 22,[3] 2
n32,[3] p22,[3] cos α 22,[3] cos 2 θ32,[3]  X 3
2
) = 2.0 n2 p 2 cosα 2 X 3,[3] + X 2,[3] + 2.0 n2 p 2 cosα 2 ⋅ n2 
2 , [3 ] 3, [3 ]
3, [3 ]
2 , [3 ] 3, [3 ]
3, [3 ]
3, [3 ]

n1j p31 cos α 31  j −1 cos α k

+ 2 .0 ∑ 1 1
3 ∏
j = 4 n3 p j cos α j  k = 4 cos α k
5
 1   n32,[3] p22,[3] cos α 22,[3] cos 2 θ 32,[3] 1
 X j  +  2 .0
⋅
⋅ κ3 ⋅
   n22,[3] p32,[3] cos α 32,[3]
n32,[3]
  
⋅ cosθ 23,[2,3] cosθ 22,[3] + κ 22,[3] cosθ 23,[2,3]  3
 ⋅ T2,[2,3]

(5.9e)
102
Finally, following the same procedure as before, the normal line force acting on
the broken element in the third level located in layer two of a three-level rope is given by
(N
(N
(N
)= g + g
] ) = 2 .0 g + X
3
2 , [2 , 3 ]
3
2 , [2 , 3
3
2 ,[2 , 3]
3
3, 2
3
1, 2
3
3, 2
(5.10a)
3
2 , [2 , 3 ]
3
j , 2,3]
3
2 ,[2 , 3]
) = 2.0∑ nn [
4
j =3
(5.10b)
p23,[2,3] cos α 23,[2,3]  j −1 cos α q3,[2,3]  3
n43,[2,3] p23,[2,3] cos α 23,[2,3]
+
⋅
X
2
.
0
∏
j ,[2 , 3 ]
p 3j ,[2,3] cos α 3j ,[2,3]  q=3 cos α q3,[2,3] 
n23,[2,3] p43,[2,3] cos α 43,[2,3]


2 3
 cos α 33,[2,3] 

 N 2 cos θ 4,[2,3] + κ 3 T 3
2 , [2 , 3 ] 2 , [2 , 3 ]
 cos α 33,[2,3]  2,[3] n43,[2,3]


(
)
(5.10c)
where, as in the analysis of the element in the second rope level, the expression for
g 33, 2 not only considers the effect of the upper layers of the same level (layers three and
four of the third level in this example), but also accounts for the effect of the contact
force exerted on the rope element of the first level (element in layer three in the first
level) and on the rope element of the second level (element in layer two) of which the
rope element of the third level is a component. This effect is reflected by the term (N 22,[3] )
( )
(which also includes the term N 31 ) in Eq. 5.10c, whose expression is given by Eq. 5.9c.
The normal line force (N 22,[3] ) is distributed uniformly over the elements of the third level
that belong to the outermost layer (layer four in this example), resulting in the normal line
force (Y43,[2,3] ). The expression to estimate the value of (Y43,[2,3] ) is given by
3
4 ,[2 , 3 ]
Y
=N
2
2 ,[3 ]
cos 2 θ 43,[2,3]
(5.10d)
n42,[2,3]
where θ 43,[2,3] is the helix angle of an element of the third level located in the fourth layer,
which is part of elements of the first and second levels located in the third and second
103
layers, respectively; and n 43,[2,3] is the number of elements of the third level located in the
fourth layer, which are part of an element of the first and second levels located in the
third and second layers, respectively. Therefore, the normal line force (N 23,[2,3] ) is given
by
(N
3
2 ,[2 , 3 ]
)
n 3j ,[2,3] p 23,[2,3] cos α 23,[2,3]  j −1 cos α q3,[2,3]  3
n 43,[2,3] p 23,[2,3] cos α 23,[2,3]
= 2.0∑ 3
+ 2. 0 3
⋅
X
3
3
3
∏
 j , [2 , 3 ]
n 2,[2,3] p 43,[2,3] cos α 43,[2,3]
j = 3 n 2 ,[2 , 3] p j ,[2 , 3] cos α j ,[2 , 3 ] q = 3 cos α q ,[2 , 3 ]


4
 cos α 33,[2,3]  cos 2 θ 43,[2,3]

⋅
 cos α 33,[2,3]  n43,[2,3]


 n32,[3] p 22,[3] cos α 22,[3]
n32,[3] p 22,[3] cos α 22,[3]
2
2
⋅ 2.0 2
⋅
X 3,[3] + X 2,[3] + 2.0 2
n2,[3] p32,[3] cos α 32,[3]
n2,[3] p32,[3] cos α 32,[3]

5 n 1 p 1 cos α 1  j −1
cos 2 θ 32,[3]  1
cos α k
3 
j 3

X
3 + 2.0∑
2
1 1
3 ∏
n3,[3] 
j = 4 n3 p j cos α j  k = 4 cos α k

 cos α 33,[2,3]  cos 2 θ 43,[2,3]

⋅
 cos α 33,[2,3] 
n43,[2,3]


 1    n43,[2,3] p 23,[2,3] cos α 23,[2,3]
 X j   + 2.0
⋅
    n23,[2,3] p 43,[2,3] cos α 43,[2,3]
  
 n32,[3] p 22,[3] cos α 22,[3] cos 2 θ 32,[3]
2.0 2
⋅
⋅ κ 31 cos θ 23,[2,3] cos θ 22,[3] +
2
2
2
n3,[3]
 n2,[3] p3,[3] cos α 3,[3]
 3
3
 + κ 2,[2,3] T2,[2,3]


κ 22,[3] cos θ 23,[2,3] 
(5.10e)
The expressions just derived to compute the line normal force acting on particular
elements of the example presented in Fig. 5.5 from levels one to three in a three-level
rope with packing geometry can be generalized for the case of an m-level rope. This
generalization process is done by induction over the number of rope levels and over the
number of layers in each level. It is important to note that Eqs. 5.8c, 5.9e and 5.10e have
a common pattern: the line normal force exerted on a rope element is the sum of a
dependent and independent term of the current tensile load T23,[2,3] of the element under
investigation. Hence, the line normal force acting on an element in layer i of the t level of
an m-level packing rope geometry, such that t ∈ [1… m], can be computed as
104
 t −1

 t

N it, J ( t ) = Ait, J (t ) + ∑ κ Is( s ),J ( s )  ∏ hIq( q ),J ( q )  + κ it, J ( t ) Ti ,tJ ( t )
 s =1

 q = s +1

(5.11)
where J(t) stores the information of the layers of the higher levels (i.e., t-1, t-2,…,1) for
which the element under study belongs; κ Is( s ),J ( s ) is the curvature of an element that is
included in layer I(s) of the s level, being part of elements in higher levels, (s-1, s-2,…,1),
located in layers J(s) in those levels. The expression to compute the constant Ait, J (t ) ,
whose value does not depend on the current value of the tensile load Ti ,t J (t ) of the element,
is given by the following recursive relation:
t
i , J (t )
A
=B
+C
t
i , J (t )
t
i , J (t )
[A
t −1
I ( t −1), J ( t −1)
t −1
I ( t −1), J ( t −1)
+W
]
cos 2 θ mlt , J ( t )
t
nml
, J (t )
t≥2
(5.12)
where the values of the constants Ai1 , Wi ,t J ( t ) , Bit, J ( t ) , and Cit, J ( t ) depend on the position of
the layer i in level t: if the element belongs to the outermost layer, i = ml,tJ (t ) ; if the
element belongs to the first layer, i = 1; and if the element belongs to an intermediate
layer, ml,tJ ( t ) > i ≥ 2 . Therefore, the expressions to compute the previously mentioned
constants are:
2.0 g 1s +1, s ⇔ i ≥ 2

Ai1 =  n12
g 31, 2 + δ 1tmax X 21 ⇔ i = 1

1
cos
θ
2

[
(5.13a)
]
105
WIt( t ),J ( t )
 X tI ( t ),J ( t ) ⇔ I (t ) ≥ 2

=  n2t , J ( t )
X 2t , J ( t ) ⇔ I (t ) = 1

t
 cos θ 2,J (t )
[
(5.13b)
]
Bit, J ( t )
 ml,tJ ( t ) nt p t cos α t  s −1 cos α t

i , J (t ) 
q, J (t ) 
2.0 ∑ ts , J ( t ) ti , J (t )
X st , J ( t ) ⇔ i ≥ 2
∏
t
t


cos
α
cos
α
n
p
 s = i +1 i , J (t ) s , J ( t )
s , J ( t ) q = i +1
q, J (t )



(5.13c)
=
t
ml , J ( t ) t
t
t
t
t


1
−
s


 n2, J ( t )
n
p
cos α 2, J ( t ) 
cos α q , J ( t )  t
 ∑ st , J ( t ) 2t , J (t )
X s , J ( t ) + δ tt max X 2t , J ( t )  ⇔ i = 1

∏
t
t
t


cos
θ
n
p
cos
α
cos
α


2, J ( t )
q , J (t )
s , J (t ) q =3



 s = 3 2, J ( t ) s , J ( t )

Cit, J ( t )


2.0 ⇔ i = ml,tJ ( t )

t
t
t

 ml,tJ ( t ) −1 cos α qt

nml
, J (t ) 
, J ( t ) pi , J ( t ) cos α i , J ( t ) 
= 2.0 t
⇔ ml,tJ ( t ) > i ≥ 2
∏
t
t
t


n
p
α
α
cos
cos
i , J ( t ) ml , J ( t )
q , J (t )
ml , J ( t )  q = i +1



t
t
t
t
 ml,J ( t ) −1 cos α qt , J ( t ) 
 n2t , J (t ) nml
, J ( t ) p2 , J ( t ) cos α 2 , J ( t ) 
 ⇔ i =1

∏
t
t
t
t
t

 cosθ 2, J ( t ) n2, J ( t ) pml , J (t ) cos α ml , J ( t )  q = 3 cos α q , J ( t ) 
(5.13d)
where tmax is the level where the broken element is located and δ1t max and δ tt max are the
Kronecker delta functions. The expression for the constants hIq( q ), J ( q ) in Eq. 5.11 also
depend on the position of the layer I(q) in level q. These constants are given by the
following expressions:
hIq( q ),J ( q )
  cos3 θ mlq ,J ( q ) 
q
2.0
 ⇔ I (q) = ml,J ( q )
q
n
 
ml , J ( q )


q
q
q
 nml ,J ( q ) pI ( q ),J ( q ) cos α Iq( q ),J ( q )  ml,J ( q ) −1 cos α sq,J ( q ) 
2
.
0
cosθ Iq( q ),J ( q ) ⋅
∏

(5.13e)
q
q
q
q


n
p
cos
cos
α
α
I ( q ),J ( q ) ml , J ( q )
ml , J ( q )
s ,J ( q )

 s = I ( q )+1

=
2 q
  cos θ ml ,J ( q ) 
q
⋅
 ⇔ ml,J ( q ) > I (q) ≥ 2
  nq
ml ,J ( q )

 
 q
q
q
q
 ml,qJ ( q ) −1 cos α sq,J ( q ) 
 cos2 θ mlq ,J ( q ) 
 n2,J ( q ) nml ,J ( q ) p2,J ( q ) cos α 2,J ( q ) 
 cos θ q
 ⇔ I ( q) = 1
I ( q ),J ( q ) 
q
q
q
q
 cosθ 2q,J ( q ) n2q,J ( q ) pml
 ∏

nml
s =3 cos α s , J ( q )

,J ( q )
, J ( q ) cos α ml , J ( q )





106
Having obtained an expression to compute the normal line force acting on any
rope element of a multi-level rope with packing geometry, and considering the friction
model previously adopted (Eq. 5.1), the gradual increase in tension Ti ,Jt (t ) of a broken
rope element located in layer i of level t, belonging to layers J(t) in higher levels (t-1, t2,…,1), is given by
b


 t −1

 t

Ti ,tJ (t ) ( x) = f a x + µ ∫  Ait, J (t ) + ∑ κ Is( s ),J ( s )  ∏ hIq( q ),J ( q )  + κ it, J ( t ) Ti ,tJ ( t ) ( ρ ) dρ

 s =1

0
 q = s +1


x
(5.14)
5.2.2. Case 2: Wedging Rope Geometry
In Chapter 3 (Section 3.4.2) it was established that the contact force pnc,i (Fig.
5.3) normal to the contact region of a rope element located in layer i is given by (Eq.
3.38)
p nc ,i =
Xi
2.0 cos θ i sinψ i
(5.15a)
Therefore, the normal line force Ni exerted on a rope element located in layer i is
given by
N i = 2.0 p nc ,i
Ni =
(5.15b)
Xi
sin ψ i cos θ i
(5.15c)
where the subtended angle ψi is defined as π/ni and is the complement of the angle βi
(Fig.5.3), ni is the number of rope elements in layer i, and θi is the helix angle of a rope
element in layer i. Similarly to the case of a packing rope geometry, the value of the
radial line body force Xi can be estimated by Eq. 5.5, and based on the friction model
107
given by Eq. 5.1, the gradual increase in tension Ti(x) of a broken rope element located in
layer i of a one-level rope with wedging geometry is given by
b
 κ i Ti ( ρ ) 
Ti ( x) = f a x + µ ∫ 
 dρ
sinψ i cos θ i 
0
x
(5.16)
The above equation is valid for elements from layer two to ml, where ml is the
maximum number of layers in level one. This equation is also valid for elements that
belong to the first layer (core) if the number of elements within the core (i.e., layer n1) is
greater than one and has a helix angle different than zero. If the value of n1 is equal to
one, Eq. 5.16 is not valid because no normal forces are exerted on the core due to the
assumption of the contact pattern for wedging geometry (Fig. 5.3).
As in the case of a multi-level rope with packing geometry in all rope levels, the
equations previously established for one-level ropes with wedging geometry can be
extended to compute the normal line force Ni exerted on a rope element located in layer i
and its gradual increase in tension Ti(x) for the case of a rope with a multi-level
hierarchical structure. It is important to point out that the subsequent analysis is valid
only if the broken rope element that belongs to level t is located in the outermost layers of
the elements that are part of levels t, t-1,…,2. In level one, the broken element can be part
of a rope element located in any layer. This last assertion is because of the
circumferential contact pattern assumed between rope elements of a rope with wedging
geometry. Hence, if the broken rope element that belongs to level t is not located in the
outermost layer (i.e, it is located in one of the interior layers), the total line normal force
exerted on that element entirely depends on the value of its current tension force T.
Accordingly, the analysis of this broken element located in level t reduces to the analysis
of an element of the first level, and Eq. 5.16 can be used to obtain its gradual increase in
tension T(x).
Similarly to the process used to obtain Eq. 5.11, and using the same notation and
variables such that the superscripts of the equations refer to the level to which the
element belongs and the subscripts refer to the layers where the element is located, the
108
analysis of a three-level rope with wedging geometry in all levels is considered. After, the
equations obtained are extended for the case of an m-level rope. As in the example of a
three-level rope with packing geometry, each rope level is sketched separately in Fig. 5.6,
and each rope element is assumed to have the same hierarchical structure: the first rope
level has two layers, the second rope level has three layers and the third rope level has
four layers. The rope elements under study in each rope level are identified with an “X”.
1st rope level-2 layers
2 nd rope level-3 layers
3 rd rope level-4 layers
Y32,[2 ]
2β 21
p1nc , 2
X
1
2
Y43,[3, 2 ]
2 β 32,[2 ]
p1nc , 2
pnc2 ,3
X 32,[2 ]
2 β 43,[3, 2 ]
pnc2 ,3
3
pnc
,4
X 43,[3, 2 ]
3
pnc
,4
Fig. 5.6: Three-level wedging geometry rope and normal forces acting on
the rope elements at each rope level
Based on the equilibrium equation in the radial direction (Fig. 5.6) and Eq. 5.15,
the normal line force acting on an element of the first level and layer two, which contains
the broken element in the third level of a three-level rope, is given by
109
p1nc , 2 =
X 21
2.0 cosθ 21 sinψ 21
(5.17a)
(N ) = 2.0 p
1
2
1
nc , 2
(5.17b)
1
(N )
=
1
2
X 2 + κ 21T43,[3, 2 ] cos θ 43,[3, 2 ] cos θ 32,[2 ]
(5.17c)
sinψ 21 cos θ 21
1
where X 2 is the radial line body force of an element that is in layer two of the first level
of the rope, ignoring the contribution of the broken element that is in level three, which is
part of the element under study in the first level; κ 21 is the curvature of an element that is
in layer two in the first level of the rope; θ 43,[3, 2 ] and θ 32,[2 ] are the helix angles of elements
that belong to layer four in the third level and layer three in the second level, respectively,
which form elements of the second and first level of layers three and two, respectively;
and T43,[3, 2 ] is the axial force of an element in layer four in the third level that forms
elements of the second and first levels, belonging to layers three and two, respectively.
With these equations established, the normal line force acting on an element of
the second level, which contains the broken element in the third level of a three-level
rope, is given by
pnc2 ,3 =
2
3 ,[2 ]
Y
X 32,[2 ] + Y32,[2 ]
2.0 cos θ 32,[2 ] sinψ 32,[2 ]
=N
1
2
cos 2 θ 32,[2 ]
(N )
2
3, [2 ]
(5.18b)
n32,[2 ]
(N [ ] ) = 2.0 p
2
3, 2
(5.18a)
2
nc , 3
(5.18c)
2
1
 cos 2 θ 32,[2 ] 
X 3, [2 ]
X2
=
+

+
sinψ 32,[2 ] cosθ 32,[2 ] (sinψ 21 cosθ 21 )(sinψ 32,[2 ] cosθ 32,[2 ] )  n32,[2 ] 
110

 cos 2 θ 32,[2 ] 
κ 21 cosθ32,[2 ] cosθ 43,[3, 2 ]
κ 32,[2 ] cosθ 43,[3, 2 ]  3

T4,[3, 2 ]

+
2
2
2
1
1
2
2
 (sinψ 2 cosθ 2 )(sinψ 3,[2 ] cosθ 3,[2 ] )  n3,[2 ]  (sinψ 3,[2 ] cosθ3,[2 ] ) 
(5.18d)
where Y32,[2 ] (Fig. 5.6 and Eq. 5.18b) is the effect of the normal line force that acts on the
element of the first level belonging to the second layer, that has been redistributed
uniformly over the elements of the second level that belong to the outermost layer (layer
three in this example).
Finally, following the same procedure as before, the normal line force acting on
the broken element in the third level located in the fourth layer of a three-level rope is
given by
X 43,[3, 2 ] + Y43,[3, 2 ]
2.0 cosθ 43,[3, 2 ] sinψ 43,[3, 2 ]
3
pnc
,4 =
Y43,[3, 2 ] = N 32,[2 ]
(N
3
4 , [3, 2 ]
(N
3
4 ,[3, 2 ]
(5.19a)
cos 2 θ 43,[3, 2 ]
) = 2 .0 p
(5.19b)
n42,[3, 2 ]
3
nc , 4
) = (sinψ
(5.19c)
2
 cos 2 θ 43,[3, 2 ] 
X 3,[2 ]

+
2
2
3
3
3
n

3,[2 ] cos θ 3,[2 ] )(sinψ 4 ,[3, 2 ] cos θ 4 ,[3, 2 ] ) 
4
,
3
,
2
[
]

1
 cos 2 θ 32,[2 ]   cos 2 θ 43,[3, 2 ] 
X2


+
(sinψ 21 cos θ 21 )(sinψ 32,[2 ] cos θ 32,[2 ] )(sinψ 43,[3, 2 ] cos θ 43,[3, 2 ] )  n32,[2 ]   n43,[3, 2 ] 

 cos 2 θ 32,[2 ]   cos 2 θ 43,[3, 2 ] 
κ 21 cosθ 32,[2 ] cosθ 43,[3, 2 ]



+
1
1
2
2
3
3
2
3
 (sinψ 2 cosθ 2 )(sinψ 3,[2 ] cosθ 3,[2 ] )(sinψ 4,[3, 2 ] cosθ 4,[3, 2 ] )  n3,[2 ]   n4,[3, 2 ] 
 cos 2 θ 43,[3, 2 ] 
 3
κ 32,[2 ] cosθ 43,[3, 2 ]
κ 43,[3, 2 ]
+


T4,[3, 2 ] (5.19d)
(sinψ 32,[2 ] cos θ 32,[2 ] )(sinψ 43,[3, 2 ] cosθ 43,[3, 2 ] )  n43,[3, 2 ]  (sinψ 43,[3, 2 ] cosθ 43,[3, 2 ] ) 
111
where Y43,[3, 2 ] (Fig. 5.6 and Eq. 5.19a) is the effect of the normal line force that acts on the
element of the second level belonging to the third layer (including the effect of the
normal line force that acts on the element of the first level), that has been redistributed
uniformly over the elements of the third level that belong to the outermost layer (layer
four in this example).
Just like for the case of packing rope geometry, the expressions just derived to
compute the line normal force acting on particular elements that exist in levels one
through three in a three-level rope can be generalized for the case of an m-level rope. As
before, this generalization process is done by induction over the number of rope levels
and over the number of layers in each level by noting that Eqs. 5.17c, 5.18d and 5.19d
have a common pattern: the line normal force exerted on a rope element is the sum of a
dependent and independent term of the current tensile load T43,[3, 2 ] for the element under
investigation. Thus, the line normal force acting on a an element in layer i of the t level
of an m-level wedging rope geometry, such that t ∈ [1, L , m] and i = ml(t) for t ≥ 2, can be
computed as
N it, J ( t )
 t s
 
 t s

c
eml ( s ), J ( s )  
∏
∏
 t
ml ( s ), J ( s ) 


t −1
q
 + ∑ κ Iq( q ), J ( q )  s =tq +1
 Ti ,t J ( t )
= ∑ X I ( q ), J ( q )  s =tq +1


 q =1


s
s
q =1
 ∏ wI ( s ), J ( s )  
 ∏ wI ( s ), J ( s )  
 s=q
 
 s=q

(5.20)
s
s
s
where the constants, cml
( s ), J ( s ) , wI ( s ), J ( s ) and eml ( s ), J ( s ) are computed from the following
expressions:
s
ml ( s ), J ( s )
c
=
cos 2 θ mls ( s ), J ( s )
s
nml
( s ), J ( s )
wIs( s ), J ( s ) = sinψ Is( s ), J ( s ) cos θ Is( s ), J ( s )
for s ≥ 2
(5.21a)
for s ≥ 1
(5.21b)
112
s
eml
( s ), J ( s )
 cos3 θ mls ( s ), J ( s )
⇔ s≤t

s
=  nml
( s ), J ( s )

1 ⇔ s > t
(5.21c)
The gradual increase in tension Ti ,tJ ( t ) of a broken rope element located in layer i
of level t, belonging to layers J(t) in higher levels (t-1, t-2,…,1) with i = ml(t) for t ≥ 2, is
given by
b


 t s
 
 t s

c
eml ( s ),J ( s )  



∏
∏
J
(
),
(
)
ml
s
s



x t −1
t
q
s = q +1
s = q +1
t
q
t







+ ∑ κ I ( q ),J ( q ) t
Ti ,J ( t ) ( x) = f a x + µ ∫ ∑ X I ( q ),J ( t ) t
Ti ,J (t ) ( ρ ) dρ



  q =1


s
s
0 q =1


 ∏ wI ( s ),J ( s )  
 ∏ wI ( s ),J ( s )  
 s =q
 
 s =q



(5.22)
5.2.3. Case 3: Mixed Rope Geometry
As previously stated, the cross-section of a rope element can have a multi-level
hierarchical structure. If such is the case, for modeling purposes, it is possible that a rope
can have different arrangements (packing and wedging geometry) at different levels. In
order to compute the total normal force exerted on a component of a rope element with
mixed cross-sectional geometry, the expressions previously developed for rope elements
with only packing or only wedging geometry can be utilized. Similarly to the process
used to obtain Eqs. 5.11 and 5.20, and using the same notation, in which the superscripts
of the equations refer to the level to which the element belongs and the subscripts refer to
the layers where the element is located, the analysis of a three-level rope with packing
geometry in the first level, wedging geometry in the second level, and packing geometry
in the third level is considered. After, the equations obtained are extended for the case of
an m-level rope.
As in the previous examples of a three-level rope, each rope level is sketched
separately in Fig. 5.7, and each rope element is assumed to have the same hierarchical
structure: the first rope level has five layers, the second rope level has three layers, and
113
the third rope level has four layers. The rope elements under study at each rope level are
identified with an “X”.
1st rope level-5 layers
2 nd rope level-3 layers
Y32,[3]
g 14,3
g
Y43,[3,3]
2 β 32,[3]
X 31
pnc2 ,3
1
2,3
X
2
3,[3 ]
3 rd rope level-4 layers
X 43,[3,3]
pnc2 ,3
g33, 4
Fig.5.7: Three-level mixed geometry rope and normal forces acting on the
rope elements at each rope level
Based on the equations developed for the case of a multi-level rope with packing
geometry (Section 5.2.1), the normal line force acting on an element of the first level
located in layer three, which contains the broken element in the third level of a three-level
rope, is given by (Eq. 5.8 and Fig. 5.7)

j −1

k =4
cos α
 ∏ cos α
(N ) = 2.0∑ nn pp cos
α  cos α
1
3
5
j =4
1
j
1
3
1
3
1
j
1
3
3
j
k
k
 1
 X j + X 13 + κ 31 cosθ 43,[3,3] cosθ32,[3]T43,[3,3]


where all the variables presented in Eq. 5.23, were defined following Eq. 5.8c.
114
(5.23)
With the Eq. 5.23 established, the normal line force acting on an element of the
second level located in layer three, which contains the broken element in the third level of
a three-level rope, is given, based on Eq. 5.17 and Fig. 5.7, by the following expression:
(N ) (
2
3,[3 ]
1
=
2
sinψ 3,[3] cos θ 32,[3]
)
+ X 32,[3] 
1
+
2
2
 sinψ 3,[3] cosθ 3,[3]
(
 cos 2 θ 2
3,[3 ]
⋅
2
 n3,[3]

)

5 n 1 p 1 cos α 1  j −1
cos α k
3 
2.0∑ 1j 13
3 ∏
 j = 4 n3 p j cos α j  k = 4 cos α k

 1
 X j + X 31  +



 1
 3
cos 2 θ 32,[3]
3
2
2
3
+
κ
cos
θ
κ 3 cosθ 4,[3,3] cosθ3,[3] ⋅
T4,[3,3] (5.24)
3
,
3
4
,
3
,
3
[
]
[
]
n32,[3]


Finally, following the same procedure as before, the normal line force acting on
the broken element in the third level in layer four of a three-level rope is given, based on
Eq. 5.10 and Fig. 5.7, by the following expression:
(N
θ [ ]
) = (sinψ2.0 cos
[ ] cosθ [ ] )n [
2
3
4 , [3, 3 ]
2
3, 3
3
4 , 3, 3
2
3
3, 3
4 , 3, 3 ]
 cos 2 θ 2 
5 n1 p1 cos α 1  j −1
cos α k
3
j 3
3, [3 ]

⋅
2
.
0
∑
2
1 1
3 ∏
 n3,[3]  j = 4 n3 p j cos α j  k = 4 cos α k



 1
 X + X 1 +
3
 j




 1
2.0 cos 2 θ 43,[,33]
cos 2 θ32,[3]
X 32,[3]  
3
2
+
κ
cos
θ
cos
θ
⋅
+ κ 32,[3] cosθ 43,[3,3]  +
 3
4 , [3, 3 ]
3, [3 ]
2
2
3
2
n3,[3]
  sinψ 3,[3] cosθ 3,[3] n4,[3,3] 

(
)
κ 43,[33] 
3
T4,[3,3]

(5.25)
As for the cases of multi-level ropes with packing and wedging geometries in all
levels, the previous equations can be generalized for the case of an m-level rope with
mixed geometry. As before, this generalization process is done by induction over the
number of rope levels and over the number of layers in each level and is based on a
common pattern identified in Eqs. 5.23 to 5.25: the line normal force exerted on a rope
element is the sum of a dependent and independent term of the current tensile load T43,33
of the element under investigation. Therefore, the normal force exerted on a component
115
of a rope element with multi-level hierarchical structure and mixed cross-section
geometry can be described as
 t
 t −1 s
 t q −1

t
t
t
t


N mi
M
κ
v
κ
v
=
+
+

∑ I ( s ), J ( s )  q∏
i , J (t )
I ( q ), J ( q ) 
i , J ( t ) I ( t ), J ( t ) Ti , J ( t )
, J (t )
 = s +1


 s =1
(5.26)
where M it,J ( t ) represents the line normal forces independent of the value of the tensile
load Ti ,Jt (t ) and can be computed based on the expressions for the line normal forces for
the case of a rope element with packing or wedging geometry (Eqs. 5.12 and 5.20,
respectively). The values of the coefficients v Iq( q ),J ( q ) , for q = 1,..., t-1, are given by the
following expression:
 cos 2 θ mlq +,1J ( q +1) 
q +1
vIq( q ), J ( q ) = c1q c 2q +1 
 cosθ I ( q +1), J ( q )
q +1
 nml , J ( q +1) 
(5.27)
where the values of the coefficients c1q and c2q+1 are given in Table 5.1 along with the
values of v It ( t ),J ( t ) .
For clarity of the expressions presented in Table 5.1, the constants [ pk ]I ( q +1),J ( q +1) and
q +1
wIq( q ),J ( q ) are defined as follows:
[ pk ]
q +1
I ( q +1), J ( q +1)
wIq( q ), J ( q ) =
q +1
q +1
q +1
 ml ,qJ+(1q + 1) −1 cos α q +1

nml
s , J ( q +1) 
, J ( q +1) p I ( q +1), J ( q +1) cos α I ( q +1), J ( q +1) 
= 2.0 q +1
q +1
q +1
 s = I∏
nI ( q +1), J ( q +1) pml
cos α sq, J ( q +1) 
, J ( q +1) cos α ml , J ( q +1)
 ( q +1) +1

sinψ
1
cosθ Iq( q ), J ( q )
(5.28a)
(5.28b)
q
I ( q ), J ( q )
116
The gradual increase in tension Ti ,Jt ( t ) of a broken rope element located in layer i
of level t (for t ≥ 2), belonging to layers J(t) in higher levels (t-1, t-2,…,1), of a rope with
mixed geometry is given by
b



 t −1
 t

Ti ,tJ (t ) ( x) = f a x + µ ∫  M it, J (t ) + ∑ κ Is( s ), J ( s )  ∏ v Iq(−q1), J ( q )  + κ it, J (t ) v It (t ), J ( t ) Ti ,tJ ( t ) ( ρ ) dρ (5.29)


 s =1
0 
 q = s +1


x
Table 5.1 Coefficients to compute the normal force exerted on a component of a
rope element with mixed cross-section geometry
Packing Geometry
Interior Layers
Wedging Geometry
Outermost Layer
For level t (tmax)
v It ( t ),J ( t )
1.0
1.0
1/ wIt (t ),J ( t )
If (q +1) = tmax
c1q
1.0
1.0
1/ wIq( q ),J ( q )
c2q+1
[ pk ]qI (+q1+1),J ( q +1)
2.0
1/ wIq(+q1+1),J ( q +1)
If (q +1) < tmax
c1q
1.0
1.0
1/ wIq( q ),J ( q )
c2q+1
[ pk ]qI (+q1+1),J ( q+1)
2.0
1.0
5.3 CALCULATION OF THE RECOVERY LENGTH
In the previous section, the expressions for the gradual increase in tension Ti ,Jt (t )
of a broken rope element located in layer i of level t, belonging to layers J(t) in higher
levels (t-1, t-2,…,1), of a rope with packing, wedging and mixed geometry were derived.
These expressions can be used to compute the value of the recovery length of a broken
117
rope element for a known value of the rope axial load, which is used to establish the
gradual increase in tension load Ti ,Jt ( t ) for individual rope components. The expressions
for the total line normal force exerted on a rope element for all the cases studied have a
similar form: the sum of an independent and dependent term of the value of the tensile
load Ti ,Jt (t ) . Thus, a general expression for the gradual increase in tension Ti ,Jt ( t ) of a
broken rope element has the following form:
x
t
i ,J (t )
T
[
b
( x) = f a x + µ ∫ C
t
1i ,J (t )
+C
]
( ρ ) dρ
t
t
2 i ,J (t ) i ,J (t )
T
(5.30)
0
t
t
where the coefficients C1i , J ( t ) and C 2 i , J (t ) are constants, and Eq. 5.30 represents a
nonlinear integral equation for the variable Ti ,tJ (t ) . As previously mentioned, if the value
of Ti ,tJ (t ) is known, Eq. 5.30 can be used to compute the interval [0, x] which corresponds
to the value of the recovery length of a broken rope element for a particular value
of Ti ,tJ (t ) . Raoof and Kraincanic (1998) presented a numerical algorithm to compute the
recovery length value from a nonlinear integral equation similar to Eq. 5.30, but they
considered only the Coulomb friction model in its development (i.e., fa = 0 and b =1).
Using this numerical algorithm and the more general friction model considered in the
present study, the value of the interval [0, x] (recovery length) for a given value of the
tensile load Ti ,tJ (t ) can be obtained from Eq. 5.30.
5.3.1 Numerical Algorithm
To simplify the algorithm description, let Ti ,tJ (t ) (x) be equal to T(x). The numerical
algorithm to solve Eq. 5.30 is as follows:
•
Subdivide the pitch distance p into n segments of length ∆x with an assumed linear
variation of T(x) in the interval.
•
For each interval j, the following holds:
118
dT ( x) Q2j − Q1j
=
= GTj
dx
∆x
(5.31a)
where Q2j and Q1j are the assumed tensile forces at the ends of the interval j.
•
The integral in Eq. 5.30 can be evaluated using a numerical quadrature as follows:
xj
∫ [C
1
+ C 2 T ( ρ ) ] dρ ≈
b
x j −1
1 j
( P1 + P2j )∆x = R1
2
(5.31b)
where P2j and P1 j are the values of the integrand at the ends of the interval j.
•
For any interval j, Eq. 5.30 can be written as
T j = T j −1 + f a ∆x + µR1
(5.31c)
where Tj-1 = Q1j .
•
For the first interval (i.e., j = 1), the following may initially be assumed:
x0 = 0; Q11 = 0; P11 = C1b ; x1 = ∆x; P21 = C1b
•
(5.31d)
Using the expressions of Eq. 5.31d, the value of R1 can be computed using Eq. 5.31b,
and the value of T1 can be computed from Eq. 5.31c. By setting Q21 = T1, the value of
P21 is computed to update the values of R1 (Eq. 5.31b), GT1 (Eq. 5.31a), and T1 (Eq.
5.31c). Finally, the value of Q21 is set equal to T1 to compute P21 . The iteration process
is repeated until the calculated values of T1 and Q21 in the last iteration are sufficiently
close, based on a prescribed tolerance.
•
In the second interval (i.e., j = 2), the following initial values of Q22 and Q12 can be
assumed:
119
Q12 = Q21
(5.31e)
Q22 = Q12 + GT1 ∆x
(5.31f)
•
Using the values of Q22 and Q12 , the value of R1 is computed using Eq. 5.31b, and the
value of T2 is computed from Eq. 5.31c. After setting Q22 = T2, GT2 is computed from
Eq. 5.31a, the value of R1 is updated using Eq. 5.31b, and all other variables are
updated according the expressions above. As for the case of the first interval, this
iteration scheme is repeated until the calculated values of T2 and Q22 in the last
iteration are sufficiently close, based on a prescribed tolerance.
•
Once convergence is achieved in the second interval, the same iteration scheme is
used in the subsequent intervals until in one of them the magnitude of Tj is found to
be greater than the prescribed value of T(x). The value of the recovery length (rl),
which is the length of the interval [0, x] in Eq. 5.30, is then given as
rl = j∆x
•
(5.32)
The number of the subdivisions n of the pitch distance p is modified (e.g., the number
of assumed subdivisions is doubled from the initial value), and the process is repeated
so that an alternative value of the recovery length is found. This iteration scheme is
stopped when, based on a prescribed tolerance, two sufficiently close estimates of
recovery length are found.
5.3.2 Special Case: Linear Friction Model
If a model in which frictional forces have a linear dependence on the normal
forces (i.e., b = 1 in Eq. 5.1) is assumed to compute the response of a broken rope
element, the gradual increase in tension Ti ,tJ ( t ) ( x) of a broken rope element given by Eq.
5.30 reduces to
120
t
i , J (t )
T
( x) = C
t
1i ,J (t )
x
x + µ ∫ C 2 i , J ( t )Ti ,tJ (t ) ( ρ )dρ
t
(5.33)
0
t
t
where the coefficients C 1i , J ( t ) and C 2 i , J (t ) are constants whose values depend on the rope
t
cross-section modeling. In this equation, C 1i , J ( t ) , in terms of the variables of Eq. 5.30,
t
equals µ C1i , J ( t ) + fa. Because of the particular form of Eq. 5.33, this nonlinear integral
equation can be reduced to a first order non-homogeneous differential equation (Tricomi,
1985). For convenience, the function λ(x) is defined such that
x
λ ( x) = ∫ Ti ,tJ (t ) ( ρ )dρ
(5.34a)
0
Accordingly, the first order non-homogeneous differential equation associated
with Eq. 5.33 is given by
dλ ( x)
t
t
− µC 2 i , J (t ) λ ( x) = C 1i , J (t ) x
dx
(5.34b)
where λ(x) = 0 at x = 0 is the boundary condition to be satisfied by the function λ(x). The
solution of Eq. 5.34b is given by
λ ( x) =
t
C 1i ,J (t )
[µC
t
]
2
t
2 i ,J (t )
t
e µC2i , J ( t ) x − 1 − C 1i , J ( t ) x

 µC t
2 i ,J (t )
(5.34c)
for a value of the friction coefficient µ greater than zero. An equivalent relationship
between the functions λ(x) and Ti ,tJ ( t ) ( x) presented in Eq. 5.34a is given by
121
Ti ,tJ (t ) ( x) =
dλ ( x)
dx
(5.34d)
Thus, using Eqs. 5.34c and 5.34d, the gradual increase in tension Ti ,tJ ( t ) ( x) of a broken
rope element has an exponential evolution given by
t
Ti ,tJ ( t ) ( x) =
C 1i , J (t )  µC2ti , J ( t ) x 
e
−1

µC 2 ti , J (t ) 
x≥0
(5.35)
If the value of the tensile load Ti ,tJ ( t ) ( x) is prescribed to be equal to T, the recovery
length (rl) of the broken rope element, associated with the specified level of tensile load,
can be obtained from Eq. 5.35 and is given by
rl =
1
µC 2 ti , J (t )
 µC 2 ti , J ( t )T 
ln 
+ 1
t
 C 1i , J ( t )

(5.36)
Therefore, based on the theory developed in Section 5.2, the nonlinear integral
equation that describes the gradual increase in tension Ti ,tJ ( t ) ( x) of a broken rope element
due to frictional effects, and consequently its recovery length value for a particular level
of the tensile load, have closed-form solutions if the frictional forces are linearly
dependent on the normal forces.
5.3.3 Numerical Examples of Recovery Length Values
The recovery length values of rope elements that belong to one-level and twolevel ropes are computed over a wide range of rope axial strain. For this purpose, for each
value of the rope axial strain specified, the elements under consideration are initially
assumed unbroken to obtain the tensile force they carry and then it is assumed that they
suddenly fail. These assumptions are consistent with the damage model evolution
122
presented in Chapter 3 (Section 4.2.3), in which the rope element after reaching its
maximum load capacity, experiences softening in a very small region of the axial forcestrain plane (in comparison to the region in which the rope element develops its
maximum capacity). Finally, the equations derived in the preceding sections are used to
obtain the recovery length values of such elements. The Coulomb friction model is
assumed to compute the frictional forces in these analyses (Eq. 5.1) using the following
friction parameters: fa = 0, µ = 0.1 and b = 1.
In Chapter 4, Section 4.3, two one-level two-layer ropes with different damage
levels (broken/cut rope elements) were used to evaluate the performance of the proposed
damage model (Figs. 4.15 and 4.16). For that purpose, the contribution of the broken rope
elements to rope responses was ignored throughout the entire strain history. These broken
elements, however, based on the recovery length concept, eventually may start carrying
load again. To investigate this possibility, the recovery length values of the broken rope
elements, were computed using the equations derived previously. Results from these
analyses are given in Fig. 5.8 in which the computed recovery length, normalized by the
pitch distance p2 of the elements of the second layer, is presented as a function of the rope
axial strain. The values of the rope axial strain are normalized by the rope breaking strain
(0.124). In this particular application, for both ropes studied, the cross-section is modeled
using the packing geometry assumption. Thus, the only rope element (rope yarn) that
develops finite recovery length values is the core (i.e., the element in the first layer) due
to the radial pressure exerted by the unbroken rope elements wound around it (Eq. 5.7).
For the case of the broken rope elements located in the second layer, their recovery length
t
values are infinite because of the zero value of the constant C 1i , J ( t ) (Eq. 5.33). The value
t
1
of the constant C 1i , J ( t ) ( C 1 2 in this example) is zero because the frictional parameter fa is
chosen to be zero, and no radial forces from upper layers are being exerted on the rope
elements of the second layer.
In Fig. 5.9, the normal forces acting on the unbroken elements shown in Fig. 5.8
are presented. For the broken cores of cross-sections A and B, there are six and eight
contact forces (g21) acting on them, respectively. These contact forces do not depend on
123
the value of the actual tensile load of the core. Conversely, the normal force g12 exerted
on an outer broken element (second layer) of cross-section A depends on the value of its
actual tensile load T2. By radial equilibrium, g21 = X2, and the value of X2 is estimated by
t
κ2 T2 (Eq. 5.5), making the value of the constant C 1i , J (t ) vanish.
2.5
Cross-Section
A
R ecovery Length/ p 2
Cross-Section A
Cross-Section
B
Cross-Section B
2
1.5
Radius of rope yarn (mm)
: 1.016
Pitch of rope yarn (p2) (mm) : 81.0
Failure rope strain
: 0.124
1
0
0.2
0.4
0.6
0.8
1
: Cut Rope Yarn
% Failure Axial Rope Strain
Fig. 5.8: Recovery length values of the core element of one-level two-layer
damaged ropes as a function of the axial rope strain
t
It is important to emphasize that in Eq. 5.33, the constant C 1i , J (t ) represents the
frictional line force acting on a broken element whose value does not depend on the
actual tensile load level of such an element. If this constant vanishes, the value of the
recovery length given by Eq. 5.36 approaches infinity for any value of T, or equivalently,
the gradual increase in tension Ti ,tJ ( t ) ( x) of the broken rope element under study given by
Eq. 5.35 is zero for all x ≥ 0. This last conclusion can also be obtained from the fact that
t
if the constant C 1i , J (t ) vanishes, the first order non-homogeneous differential equation
given by Eq. 5.34b becomes homogeneous, having a solution of the form λ(x) = 0 for all
x, considering the boundary condition λ(0) = 0.
124
g2,1
g2,1
g2,1
g2,1
g2,1
g2,1
Core cross-section A
g2,1
g2,1
g2,1
g2,1
X2
g2,1
g2,1
g2,1
g1,2
g2,1
Core cross-section B
Outer element
cross-section A
Fig. 5.9: Normal forces acting on rope elements of a damaged one-level twolevel rope assuming packing geometry
If the rope cross-section shown in Fig. 5.8 is modeled as a wedging geometry,
none of the broken rope elements develop finite recovery length values. As in the case of
t
1
the packing geometry assumption, the value of the constant C 1i , J ( t ) ( C 1 2 in this example)
(Eq. 5.33) is zero for all rope elements because of the friction model assumed and the
contact pattern assumption for wedging geometry (i.e., circumferential contact between
contiguous elements in the same layer). The contact forces exerted on the elements of the
second layer depend on the actual level of their tensile loads (Eq. 5.15), and, as
mentioned in Section 5.2.2, Eq. 5.33 is not applicable for the core element because the
core is formed by one element and the assumption of the contact pattern for wedging
geometry.
The plot shown in Fig. 5.8 suggests that the normalized recovery length values of
the core element increase with increasing levels of axial rope strain (and thus, increasing
levels of axial load). Although the confinement (normal) forces exerted on the core also
increase with increasing levels of axial rope strain, this effect does not compensate the
increment of the axial load levels carried by the core with increasing levels of axial rope
strain. Hence, based on the form of Eq. 5.36 in which T is the axial force in the core and
t
C 1i , J ( t ) accounts for the effect of the normal forces acting on the core, the normalized
125
recovery length values of the core element increase with increasing levels of axial rope
strain. The dependence of the recovery length values on the axial rope strain levels shown
in Fig. 5.8, is only valid for the example presented in the mentioned figure. In general,
the normalized recovery length values of a broken rope element are problem-dependent.
These values may either increase or decrease with increasing levels of axial rope strain
depending on the location of the element inside the rope, the rope geometry and the
contact pattern between rope elements assumption (Beltran, 2004). The variations of the
recovery length values presented in Fig. 5.8, however, are found to be no more than 12%
of the mean values for the both cases studied over a wide range of axial rope strain.
Although both cross-sections have the same element arrangement, cross-section A has
two broken outer elements, which generates less radial pressure on the core than in crosssection B, resulting in greater recovery length values for the core (see Eq. 5.7).
In Fig. 5.10, the recovery length values of broken rope elements of a two-level
rope as a function of the rope axial strain are presented. As in the previous example, the
recovery length values are normalized by the pitch distance p of the broken elements. The
predicted response of this rope was computed and presented in Chapter 4, Section 4.3 for
the case in which there were no broken rope elements (initially undamaged rope: Figs.
4.13 and 4.14). This rope is composed of three rope elements (strands) of the type shown
in Fig. 5.8 wound around a soft core (not shown), and each rope element is formed by
nine rope yarns. For the current example, nine total rope yarns are cut (broken),
distributed in two rope elements (4 in one rope element (strand 1) and 5 in a second rope
element (strand 2)) as shown in Fig. 5.6. For modeling purposes, the first rope level is
modeled as a wedging geometry and the second rope level is modeled as a packing
geometry if the strand (first level) is damaged and as a wedging geometry if the strand is
undamaged. This choice of the cross-sectional modeling is based on comparisons
between the predicted rope responses and experimental data for this two-level rope that
are presented in the following section (Section 5.4).
126
Recovery Length/ p 2
14
Rope Yarn
12
10
Outer Rope Yarns-Strand 2
8
Outer Rope Yarns-Strand 1
6
0.0
0.2
0.4
0.6
1.0 Strand 2
0.8
Strand 1
Recovery Length/ p 1
% Failure Axial Rope Strain
Rope diameter (mm)
Radius of rope yarn (mm)
Pitch of rope yarn (p2) (mm)
Failure rope strain
Pitch of strand (p1) (mm)
0.8
0.6
: 14
: 1.016
: 81.0
: 0.126
:150
: Cut Rope Yarn
0.4
Core-Strand 2
0.2
0.0
0.2
0.4
0.6
0.8
1.0
% Failure Axial Rope Strain
Fig. 5.10: Recovery length values of broken strands of a two-level damaged
rope as a function of the axial rope strain
As in the previous example, the plots in Fig. 5.10 suggest that the recovery length
values of the broken rope yarns increase with increasing levels of axial rope strain. In the
analysis of a one-level rope (Fig. 5.8), it was established that the outer broken elements
t
(second layer) do not develop a finite recovery length value because the constant C 1i , J ( t )
(Eq. 5.33) has a value of zero. In this rope configuration (Fig. 5.10), the value of the
t
2
2
constant C 1i , J ( t ) ( C 1 2,1 and C 11,1 in this example) does not vanish for any broken rope yarn
127
(second rope level) because of the contact between rope elements in the first level of the
rope, resulting in finite recovery length values for all the broken rope yarns. The
variations of the recovery length values are found to be no more than 12% for the core
and no more than 10% for the outer rope yarns (second layer) of the mean values for the
cases studied over the entire range of the axial rope strain considered.
The broken rope yarns of strand 1 (Fig. 5.10) develop smaller recovery length
values than the outer broken rope yarns of strand 2 because the normal force exerted on
strand 1 is greater than the normal force exerted on strand 2. These normal forces depend
on the tensile load level of the rope elements (Eq. 5.15), and strand 1 has more unbroken
rope yarns (5) than strand 2 (4), contributing to its tensile load level. Therefore, the
normal force exerted on this element is higher than the normal force on strand 2. For the
case of the core rope yarn of strand 2, its recovery length value is not only smaller than
the ones of the outer rope yarns, but it is also smaller than the recovery length values of
the core of cross-section B (Fig. 5.5).
As previously stated, for the purposes of computing the recovery length values of
a broken rope element, the rope axial strain is specified, the element under consideration
is initially assumed unbroken to obtain the forces developed and then it is assumed that it
suddenly fails. These forces and the geometrical parameters of the element under study
allow for the recovery length to be computed using Eq. 5.36. Once the value of the
recovery length is obtained, Eq. 5.35 can be used to obtain the profile of the gradual
increase in tension load of a broken rope element for a specific value of the rope axial
strain. In Fig. 5.11, profiles of the gradual increase in tension load for two broken rope
yarns shown in Fig. 5.10 for a wide range of rope axial strain are presented. These broken
rope yarns are the core and one outer broken rope yarn of strand 2 (Fig. 5.10). In each
plot, the axial load accumulated by the broken element T(x) is normalized by the
maximum axial load (breaking load) of that element Tmax (0.61 Kips) and the axial rope
strain is normalized by the rope breaking strain (0.127). The domain of the variable x is
[0, rl], where 0 refers to the failure position of the rope element and rl is the recovery
length value for a given value of the rope axial strain (Fig. 5.10). The values of the
128
variables x are normalized by p1 (5.9 in.) for the case of the core of strand 2 and by p2
(3.2 in.) for the case of outer yarn of strand 2 (Fig. 5.10).
1.0
T (x )/Tmax
(a)
0.1
0.8
0.3
0.5
0.7
0.6
0.9
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
x /p 1
1.0
(b)
0.1
0.3
0.5
T (x )/Tmax
0.8
0.6
0.7
0.9
0.4
0.2
0.0
0
2
4
6
8
10
12
14
x /p 2
Fig. 5.11: Profiles of the gradual increase in tension load for two broken
strands of a two-level rope: (a) core, (b) outer rope yarn for different values of
rope axial strain
For the case of the core of strand 2, its recovery length values for given rope axial
strain values are smaller than the value of the pitch distance p1 (Fig. 5.10). Hence, for
each value of the rope axial strain, the recovery length value is fully developed in a
129
relatively short distance, generating a linear profile of the gradual increase in the tension
load (see Eq. 5.35 and Fig. 5.11a). Conversely, for the case of an outer rope yarn of
strand 2, its recovery length values are greater than the value of the pitch distance p2 and
than the recovery length values of the core over a wide range of rope axial strain (Fig.
5.10). The recovery length values are fully developed in a long enough distance to
generate nonlinear profiles, with increasing slopes, of the gradual increase in the tension
load (see Eq. 5.35 and Fig. 5.11b).
In the following section (Section 5.4), a numerical model to predict the damaged
rope response is presented. This model relies of the ability of the broken rope elements to
pick up their appropriate axial load after fully developing their recovery length values.
For simplicity and based on the plots presented in Fig. 5.11, in the development of this
model the gradual increase in load carried by broken rope elements is ignored. This
assumption is made because the recovery length values are relatively short or because
most of the tension load is accumulated toward the end of the interval [0, rl] (nonlinear
profile).
5.3.4 Effect of the Rope Jacket on the Recovery Length Values
In the previous sections of this chapter, an estimate of the recovery length in
synthetic-fiber ropes was given. Admissible recovery length values can develop because
of the inter-layer/inter-element contact forces that are induced between rope elements
because of their helical nature. It is also important to note, however, that contact forces
between rope elements can be induced by the confinement exerted by a rope jacket. In
general, a typical rope jacket has a double braid construction (Winkler and McKenna,
1995) as shown in Fig. 512.
In order to represent the effect of the rope jacket on the radial pressure exerted on
rope elements, the rope jacket is considered as a thin-walled tube for the purposes of
modeling. Gjelsvik (1991) and Raoof and Huang (1992) studied the problem of
determining the recovery length values of a fractured wire in a parallel-wire strand
experiencing external hydrostatic pressure by prestressed wrapping or intermittent cable
bands. In the current study, the works previously mentioned are extended and then used
130
to estimate the contact forces exerted by the rope jacket on rope elements according to
rope cross-section modeling under consideration.
Sub-rope
Rope jacket
Fig. 5.12: Rope jacket and sub-ropes of a parallel rope construction
(Berryman et al., 2002)
For modeling purposes, the rope jacket can be assumed to produce a uniform
transverse compression in the rope. To illustrate the procedure established to estimate this
effect, consider a one-level two-layer rope. In the first layer of this rope, there is one
element (core), and in the second layer, there are six elements wound around the core
(Fig. 5.13a), where the core is subjected to the same six compressive contact forces P
per unit length. It is assumed that the rope jacket experiences normal and circumferential
stresses according to the thin-walled tube assumptions. If T is the circumferential line
force in the rope jacket, the nominal compressive radial stress f in the core is given by
f =
2T
d
(5.37)
where d is the rope diameter. The stress f would be the radial stress if the rope crosssection were solid, i.e., without the voids between rope elements. If there are voids
131
between rope elements (packing geometry), Gjelsvik (1991) computed the compressive
contact forces P exerted on the core based on an equivalent cross-section that makes the
cable solid, i.e., without the voids between rope elements. The equivalent cross-section is
obtained by intersecting the tangent lines at every contact point between the core and the
elements wrapped around it.
For the rope cross-section shown in Fig. 5.13a, the
equivalent cross-section is a hexagon that circumscribes the core cross-section (Fig.
5.13b). Thus, by equilibrium in the vertical direction (Fig. 5.13b), the compressive
contact forces P is given by
P=
2  dc 
 T
3 d 
(5.38)
where dc is the diameter of the core element. The procedure just described can be used to
obtain the value of the compressive contact forces P for a different number of rope
elements wrapped around the core.
P
(a)
P
P
P
P
P
f
P
P
30°
30°
CORE
(b)
dc
2
dc
3
P
Fig. 5.13: (a) Normal forces acting on the core; (b) compressive stress acting
on the core
132
According to the notation used in Fig. 5.3, the contact force P is equal to g21, and
based on the LeClair (1991) work and Eq. 5.4, the value of the compressive contact force
P ml exerted by the rope jacket on every rope element of the outermost layer (ml) of the
rope is given by
Pml =
n2 pml cos α ml
 ml −1 cos α
k
nml p2 cos α 2  ∏
 k =3 cos α k

(
)




g12
(5.39)
where g12 = (g21)cosθ2, θ2 is the helix angle of the rope elements of the second layer, and
the rest of the variables involved in Eq. 5.39 are described following the presentation of
Eq. 5.4.
For the case of a rope cross-section modeled as a wedging geometry, by
assumption, there are no voids between rope elements and there is no radial contact
between rope elements of contiguous layers. The circumferential line force T in the rope
jacket is equilibrated by the rope elements located in the outermost layer of the rope.
Thus, using the same notation as before, the compressive contact force P ml exerted by the
rope jacket on every rope element of the outermost layer (ml) of the rope can be
estimated as
Pml =
2Tπ
nml
(5.40)
where nml is the number of elements in the outermost layer of the rope.
To illustrate the effect of the rope jacket thickness on the value of the recovery
length of a rope element, along with the rope cross-section of a one-level two-layer rope,
is presented in Fig. 5.14a. As in the previous examples, the Coulomb friction model is
assumed to compute the frictional forces using the following friction parameters: fa = 0, µ
= 0.1 and b = 1.
133
For comparison purposes, the rope cross-section is modeled with both a packing
and a wedging geometry, and one rope element of the outermost layer is initially cut at
the rope midspan. For the sake of the example, the axial capacity of the rope jacket is
assumed to be some percentage of the rope axial capacity. Ayres (2001) mentions that for
undamaged ropes, the jacket can account for up to 8% of the overall rope axial strength.
In this case, the rope jacket is assumed to have around 5% of the overall rope axial
strength. In the example presented in Fig. 5.14a, the confinement effect of the rope jacket
t
leads to a non-zero value of the constant C 1i , J ( t ) (Eq. 5.33), resulting in a finite recovery
length value of the broken rope yarn located in the outermost layer of the rope for both
cross-sectional modeling approaches. For this particular example, and considering the
friction model assumed, if the rope jacket were not present, the broken rope yarn would
not have finite recovery length values because of the zero value of the constant
t
C 1i , J (t ) (Eq. 5.33).
In Fig. 5.8, the recovery length values as a function of the rope axial strain of a
one-level, two-layer rope with two different cross-section damage states were presented.
In Fig. 5.14b, the confinement effect of the rope jacket on the recovery length values is
included for the case of the broken core (Cross-Section B in Fig. 5.8). As in the previous
examples, the recovery length values and the rope axial strains are normalized by the
pitch distance p2 of the rope yarns of the second layer (Fig. 5.14) and rope breaking strain
(0.124), respectively. Based on the assumptions made about the contact pattern between
the rope elements of the same rope level, the rope cross-section is modeled as a packing
geometry. For the sake of the example, the rope jacket is assumed to have an axial
capacity in a range of 4% to 5% of the initially undamaged rope capacity and a thickness
equal to 0.0048 (in.), which corresponds to a 2% of the rope diameter.
The confinement effect of the rope jacket on the rope elements induces higher
normal forces exerted on the broken core than for the case in which the rope jacket is not
present. Accordingly, the recovery length values of the broken core as a function of the
rope axial strains decrease (Fig. 5.14b). As in the previous examples, however, the
increment in the normal forces exerted on the broken core with increasing levels of axial
134
rope strain and due to the confinement effect of the rope jacket does not compensate the
increment of the axial load levels carried by the core with increasing levels of axial rope
strain. Hence, based on the form of Eq. 5.7, the normalized recovery length values of the
core element increase with increasing levels of axial rope strain.
(a)
Recovery Length/p 2
20
Rope Yarn
Packing Geometry
16
Rope Cross-Section
Wedging Geometry
12
Rope Jacket
8
Rope diameter (mm)
: 6.096
Pitch rope yarn (p2) (mm) : 81.0
×: Cut rope yarn
4
0
0
0.02
0.04
0.06
Jacket Thickness/Rope Diameter
Recovery Length/ p 2
Rope Cross-Section
2.5
No Rope Jacket
2.0
Rope Jacket
(b)
1.5
1.0
Rope Jacket
0.5
×: Cut rope yarn (core)
0.0
0
0.2
0.4
0.6
0.8
1
% Failure Rope Axial Strain
Fig. 5.14: Effect of the rope jacket on the recovery length values: (a) outer
rope yarn; (b) core
135
5.4 NUMERICAL ANALYSIS OF DAMAGED ROPES
To evaluate and quantify the effect of rope element failure on overall rope
response, it is first necessary to determine if broken rope elements can take up their
appropriate share of axial load by considering the equilibrium equations that govern rope
behavior and by computing their recovery lengths. As previously stated, in order to
simplify the analysis, the gradual increase in load carried by broken rope elements is
ignored (see discussion of Fig. 5.11). Thus, only the full value of axial load at distances
outside admissible recovery length values are taken into consideration.
The recovery length of a broken rope element is computed for a prescribed
deformation level of the rope (ε, ζ), where ε is the rope axial strain and ζ is the rate of
twist of the rope cross-section. If this value is admissible (i.e., within the physical length
constraints of the rope being analyzed), the problem is linearized by discretizing the rope
into “sub-rope” elements (Si), with lengths LSi, based on the value of the recovery lengths
computed. In Fig. 5.15, two possible rope discretizations are presented along with the
“sub-rope” element cross-sections, where the broken rope elements have been identified
with an “X.” The values of u and φ are the rope axial displacement and the rope axial
rotation, respectively. The two possible discretizations are based upon the location of the
failed rope component: (a) the failure region is located near one end of the rope (i.e., at a
splice location), or (b) the failure region is located near the rope midspan.
Each “sub-rope” is modeled as a two-noded axial-torsional element of length LSi,
with two degrees of freedom at each node: axial displacement (ui) and axial rotation (φi).
Associated with these degrees of freedom are the corresponding axial force Ti and
torsional moment Mi (Fig. 5.16). This model captures the effect of having a weakened
cross-section acting over a localized region defined by the recovery length.
The “sub-rope” element response can be described in terms of only two
independent degrees of freedom (referred to as the “essential set”). Hence, based on the
degrees of freedom defined for the “sub-rope” element in Fig. 5.16, only the axial
deformation (u3 – u1) and axial rotational deformation (φ4 – φ2) are needed. For the “subrope” element shown in Fig. 5.16, the element forces associated with the essential set are
T3 and M4.
136
(a)
u,
φ
LS2
u,
φ
u,
φ
LS1
(b)
u,
φ
LS3
LS2
LS1
Fig. 5.15: Damaged rope discretizations: (a) failure located at one end
of the rope and (b) failure located around the midspan of the rope
Node 1
Node 2
T3, M4
T1, M2
u1, φ 2
u3, φ 4
LSi
Fig. 5.16: Two-noded axial-torsional “sub-rope” element
The tangent stiffness matrix of each “sub-rope” element Si is computed, and then
each tangent stiffness matrix is assembled to obtain the tangent stiffness matrix of the
137
entire rope. For each “sub-rope” element, a 2-by-2 stiffness matrix is computed having
the following form:
 kεε
kζε
[k sr ]i = 
kεζ 
kζζ  i
(5.41)
where no closed-form expressions exist to compute the stiffness coefficients due to the
nonlinear geometry and nonlinear constitutive laws of the rope elements. If there exists a
function Gi = (G1, G2)i such that (Ti, Mi) = Gi(εi, ζi), the stiffness coefficients can be
obtained by computing the gradient of the function Gi. Thus, the stiffness matrix (Eq.
5.41) of each “sub-rope” element would have the following form:
[ksr ]i
 ∂G1
 ∂ε
=
 ∂G2
 ∂ε
∂G1 
∂ζ 

∂G2 
∂ζ  i
(5.42)
As stated before, the function Gi does not have closed-form expressions for any
“sub-rope” element Si. Thus, the partial derivates of Eq. 5.42 are evaluated using a
numerical scheme by inducing small perturbations (δε, δζ)i to a prescribed level of
deformation (ε, ζ)i for each “sub-rope” element Si. These induced perturbations are
applied separately (i.e., small perturbations are assumed for one variable while holding
the other perturbation equal to zero), generating two different perturbed deformations. At
each perturbed deformation level, the strains and deformations, constitutive laws,
including the effects of damage, and the equilibrium equations of each rope element are
used to compute the corresponding axial force and twisting moments of the “sub-rope”
element Si. These computed loads, along with the loads associated with the unperturbed
deformation level (ε, ζ)i, are used to compute the stiffness coefficients. Accordingly, the
stiffness coefficients of the matrix [ksr]i can be evaluated, for any “sub-rope” element Si,
as follows:
138
k εε =
∂G1  T (ε + δε , ζ ) − T (ε − δε , ζ ) 
=

∂ε
2δε

(5.43a)
k ζε =
∂G2  M (ε + δε , ζ ) − M (ε − δε , ζ ) 
=

∂ε
2δε

(5.43b)
kεζ =
∂G1  T (ε , ζ + δζ ) − T (ε , ζ − δζ ) 
=

∂ζ 
2δζ

(5.43c)
kζζ =
∂G2  M (ε , ζ + δζ ) − M (ε , ζ − δζ ) 
=

∂ζ
2δζ


(5.43d)
where the centered differentiation formula is used to evaluate the stiffness coefficients, T
and M are the axial force and torsional moment, respectively, and the values of the
perturbations δε and δζ are found by performing an iterative process until two
consecutive values of the stiffness coefficients are sufficiently close based upon a
prescribed tolerance.
The 4-by-4 tangent stiffness matrix [Ksr]i of the “sub-rope” element Si, associated
with the degrees of freedom u1, φ2, u3 andφ4 (referred as to the “complete set”) described
in Fig. 5.16, can be obtained through consideration of equilibrium. Hence, with the 2-by2 “sub-rope” element stiffness matrix [ksr]i computed, [Ksr]i can be calculated as follows:
[K sr ]i
 [k sr ]i
=
− [k sr ]i
− [k sr ]i 
[k sr ]i 
(5.44)
A displacement control analysis is carried out to obtain overall rope response.
Increments in axial strain, dε, and axial rotation per unit length, dζ, are specified for the
entire rope. The aim of the analysis is then to obtain the values of the increments of axial
displacements dui and axial rotations dφi, as well as the increments in the external loads
139
(axial force and torsional moment), required to produce the specified increment in the
deformation level (dε, dζ) of the rope. These quantities are obtained by solving the
following linear system of equations
[K r ]{d u} = {dP}
(5.42)
where Kr is the rope tangent stiffness, d u is the increment in axial displacements and
axial rotations, and dP is the increment in axial forces and torsional moments.
Increments in axial strain and axial rotation per unit length (dεi and dζi,
respectively), for each “sub-rope” Sj are computed using the standard engineering strain
definition. Using the damage-dependant constitutive equations corresponding to the
current level of rope deformation, the increments in forces on the “sub-rope” elements
due to axial deformations are then computed. Because of the nonlinearity of the system
properties, iterations are performed until equilibrium is achieved.
5.5 NUMERICAL EXAMPLES
The analyses presented hereafter are carried out on polyester ropes that were
damaged prior to loading. The damage introduced to such ropes was made by cutting a
predetermined number of rope elements, either at the rope midspan or near one of the
rope splices (ends). When data are available, numerical predictions of rope response are
compared with experimental results obtained from static capacity tests. For the sake of
the analyses, axial rope displacements are specified and ropes are restrained to rotate due
to the lack of measurements of rope end rotations. In addition, ropes are modeled as onelevel and two-level ropes, and based on the analyses presented in Chapter 4 (Section 4.3)
to predict the response of small-scale one-level and two-level polyester ropes, the
following frictional and damage parameters were assumed for the analyses (Eqs. 3.37 and
4.1): fa = 0, µ = 0.1 and b = 1, α1 = 0.12, β1 = 0.84, εt = 0.04. For clarification purposes, a
description of the legends used in the upcoming plots and analyses is presented in Table
5.2.
140
Table 5.2 Summary of the models used to predict damaged rope behavior
Model
Outline of the model
The recovery length of each broken rope element is computed.
If this value is admissible, the rope is discretized into “subStrain Localization and
rope” elements, and the numerical procedure outlined in
Degradation Model
Section 5.4 is used to analyze the rope. Degradation of each
(Including Net Area
“sub-rope” element’s properties is accounted for through use
Effect)
of Eq. 4.1. The broken rope elements do not contribute to the
rope response in a distance equals to their recovery length
values.
Net Area Effect and
Degradation Model
Theoretical UpperBound Response
The broken rope elements do not contribute to the rope
response. Degradation of each rope element’s properties is
accounted for by using Eq. 4.1.
The rope response is obtained by summing up the response of
each rope element, ignoring the effects of degradation and the
helical nature of the rope geometry.
Degradation Model
The rope is considered initially undamaged and degradation of
(Lower-Bound
each rope element’s properties is accounted for by using Eq.
Response)
4.1.
In Fig. 5.17, the analyses of a one-level two-layer rope with two different degrees
of damage to its cross-section are presented: one and three rope elements (rope yarns) cut
at the rope midspan. As previously mentioned (Section 5.3.3), in both cases, when the
rope cross-section is modeled assuming wedging geometry, none of the broken rope
yarns develops an admissible value of recovery length. Thus, the analyses are performed
by just ignoring the contribution of these rope yarns to the rope response throughout the
entire strain history. As such, this response is adequately described using the net area
effect and degradation model (Table 5.2).
141
Rope Axial Load (Kips)
7
Degradation Model
(a)
Theoretical Upper Bound Response
6
Strain Localization & Degradation Model (Packing Geometry)
5
Net Area Effect& Degradation Model (W edging Geometry)
4
Rope Cross-Section
Net Area Effect--No Degradation Model
Exp. Data 1
3
Radius of rope yarn (mm) : 1.016
Pitch of rope yarn (mm) : 81.0
Rope Length (mm)
: 610.0
Exp. Data 2
2
1
: Cut Rope Yarn
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Rope Axial Strain
7
Rope Axial Load (Kips)
6
Degradation Model
Theoretical Upper Bound Response
(b)
Strain Localization & Degradation Model (Packing Geometry)
5
Net Area Effect& Degradation Model (W edging Geometry)
4
Net Area Effect--No Degradation Model
3
Rope Cross-Section
Exp. Data 1
Exp. Data 2
: Cut Rope Yarn
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Rope Axial Strain
Fig. 5.17: Initially damaged one-level rope responses: (a) one out nine
rope yarns cut and (b) three out of nine rope yarns cut
Conversely, when the rope cross-section is modeled using packing geometry, the
core of the rope develops an admissible value of its recovery length. As such, the rope is
142
subdivided is three “sub-rope” elements with different amounts of rope elements in each
of them, resulting in a strain concentration in the weaker “sub-rope” element. These
analyses result in a stiffer rope response but with smaller values of the rope breaking
strain than for the case when the rope cross-section is modeled using wedging geometry.
For the case where the degree of damage is one out of nine rope yarns cut, the
model that considers the rope cross-section as a wedging geometry compares with the
experimental data better than the packing geometry cross-section model. Conversely, for
the case where the degree of damage is three out of nine rope yarns cut, the model that
considers rope cross-section as a packing geometry compares better with the
experimental data than the wedging geometry up to rope strains equal to 0.07. For values
of rope strain greater than 0.07, the packing geometry model becomes stiffer than the
experimental data curves. This model performs well in predicting the value of the rope
breaking load, but it underestimates the value of the rope breaking strain. As in the
previous case, the model that considers the rope cross-section to be a wedging geometry
(net area effect and degradation model) does a good job in predicting the values of rope
breaking strain and breaking load, and it also performs well in estimating rope response,
especially for values of rope strains close to the onset of failure.
Based on the analyses presented in Fig. 5.17, the model that considers a packing
geometry assumption compares better with experimental data as the rope damage state
(number of rope yarns cut) increases. As the rope damage state increases (i.e., more rope
elements are severed), fewer rope elements are present at the location where damage has
been introduced. As a result, rope elements (rope yarns in these examples) that are
initially in radial and circumferential contact with their neighboring elements may
eventually start migrating to a new position in the layer, resulting in a new rope crosssection configuration. Accordingly, voids between rope elements of the same layer exist,
and the contact pattern between the rope elements is mainly in the radial direction, which
is consistent with the packing geometry assumption. It is important to recall that both
cross-sectional modeling approaches (packing and wedging) assumed in the development
of the computational model represent the extreme cases of cross-section response. For
packing geometry, contact between rope elements in the same level is assumed to be only
143
in the radial direction, and slip at those points is prevented. Conversely, for wedging
geometry, the rope elements are allowed to deform transversely and change their shape to
a wedge or truncated wedge. It is assumed that there is circumferential contact pressure
and friction acting along the length of the elements due to axial slip between contiguous
elements. Hence, both cross-sectional models are needed to bound rope cross-sectional
behavior.
Based on the comparison between the experimental data and the different
simulation curves presented in Fig. 5.17, the analytical model that predicts most
accurately the damaged one-level two-layer rope response, for both cases of assumed
initial damage, is the one that considers the net area effect and degradation of rope
properties. This observation indicates that the model chosen to represent the cross-section
behavior has a big impact on the predicted damaged rope response. If the packing
geometry assumption is used, some of the broken rope yarns have admissible recovery
length values, a strain localization region develops around the failure location, and the
predicted damaged rope response is stiffer than that indicated by the experimental data.
While the rope failure load is estimated reasonable well, the rope failure strain is
underestimated. Conversely, if the wedging geometry assumption is used, the computed
contact forces that are exerted on the broken rope yarns are not large enough to develop
admissible recovery length values, and the broken rope yarns do not contribute to the
rope response. Hence, the predicted damaged rope response is computed based on the net
area effect and degradation model (Table 5.2), and both the predicted rope failure load
and the predicted rope failure strain compare well with the experimental data. It is
important to note that the damaged rope responses are also obtained by using the net area
effect without considering the degradation model. As already discussed in Chapter 4
(Section 4.3-Fig. 4.15 and 4.16) and based on the available experimental data, if the
degradation model is not considered, the damaged rope capacities and load deformation
curves (stiffnesses) are overestimated. Accordingly, the damaged rope responses obtained
by using the net area effect and degradation model compare better with experimental data
than the responses obtained using just the net area effect.
144
Fig. 5.18 shows the damaged cross-section of a two-level rope along with some of
its geometrical parameters used for the present example. The figure also shows the
analysis results for the response of this rope. This damaged cross-section was previously
used in Section 5.3.3 to illustrate the dependence of the recovery length values on rope
strain (Fig. 5.10). For completeness and clarification purposes, this damaged crosssection is shown again in Fig. 5.18. For computational purposes, and based on the
previous analyses on one-level two-layer ropes, the first level of the rope (strand) is
treated as having a wedging geometry, and the second level of the rope is modeled using
wedging geometry if no initial damage is induced on the strand and packing geometry if
initial damage is induced on the strand.
Results computed with the net area effect and degradation models and the strain
localization and degradation models accurately predict the damaged rope response for
two of the three sets of experimental data (Li et al. (2002)) presented. Fig. 5.18 shows
that the model with strain localization effects results in a stiffer rope response that
corresponds better with the experimental data and provides a smaller value of the rope
breaking strain relative to the net area effect model. Both of these models, however, give
values of rope breaking strain and rope breaking load that are smaller than those indicated
by the test data. The difference between the theoretical upper bound response curve and
the other curves is mainly due to the effects of the degradation of each rope element’s
properties throughout the response history and the helical nature of the rope geometry.
As for the case of the damaged ropes analyses presented in Fig. 5.17, the response
of the damaged rope presented in Fig. 5.18 is also obtained by using the net area effect
without considering the degradation model. This curve is an upper bound for the curves
obtained by using the net area effect and degradation models and the strain localization
and degradation models along the entire strain history. Based on the experimental data
curves, this model predicts a stiffer damaged rope response and underestimates the value
of the rope breaking strain and rope breaking load.
145
Damaged rope cross-section
Damage inflicted at rope midspan
Strand
Exp. Data 1
Exp. Data 2
Exp. Data 3
Strain Localization & Degradation Models
Net Area Effect & Degradation Models
Theoretical Upper Bound Response
Degradation Model
Net Area Effect--No Degradation Model
Rope Axial Load (Kips)
20
16
12
Rope yarn
: Cut Rope Yarn
8
3 twisted strands
9 twisted rope yarns per strand
Rope diameter (d): 14 [mm]
Pitch of strand: 150 [mm]
Pitch of rope yarn: 81 [mm]
4
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Rope Axial Strain
Fig. 5.18: Initially damaged two-level rope cross-section and analyses of
rope response
The previous two analyses (Figs. 5.17 and 5.18) were performed on small-scale
polyester ropes. The following analyses are for large-scale damaged polyester-jacketed
ropes (DPJR) that have parallel sub-rope constructions, where each sub-rope is built by
twisting a certain number of strands together. For analysis purposes, the damage inflicted
to each rope prior to loading is at the strand level. This damage is simulated by cutting
(i.e., specifying a damage index D = 1) a certain number of strands to reduce the crosssectional area by a prescribed percentage. All damage is inflicted to adjacent sub-ropes
near the exterior of the rope to simulate surface damage. For most of the analyses,
damage is inflicted at the midspan location. If damage occurs near a rope splice, it is
identified in the corresponding plot. The amount of damage inflicted, its distribution
throughout the rope cross-section, and its location along the rope length are chosen such
146
that the predicted responses can be correlated with experimental studies. Four different
levels of initial damage (5%, 7%, 10%, 15% of net area loss from the rope cross-section)
are considered in the analyses along with different L/d ratios, where L is the rope length
and d is the rope diameter. Parameters are selected to study the potential influence of
length effects on overall damaged rope response. The numerical simulations of damaged
rope behavior are compared with experimental data for ropes with a specified breaking
strength of 35 Tonnes (77.2 Kips) and 700 Tonnes (1543.2 Kips).
As stated before, the following analyses are for jacketed ropes that have parallel,
unjacketed sub-rope constructions. For analysis purposes, to study the potential nonuniform strain distribution along the damaged rope length, the confinement exerted on
the rope by the jacket is considered using the thin-walled tube approximation described in
Section 5.3.4. The rope jacket of each rope analyzed is assumed to have an axial capacity
in a range of 4% to 5% of the initially undamaged rope capacity and a thickness around
1% of the rope diameter. As mentioned in Section 5.3.4, Ayres (2001) established that,
for undamaged ropes, the jacket can account for up to 8% of the theoretical rope strength.
Because the actual value is not known and has not been measured, a small value that is
believed to be representative of the rope jacket capacity has been approximated.
Furthermore, results from parameter studies using different values for the assumed axial
capacity of the jacket show small differences in the overall computed response of a rope.
Only for those cases in which the jacket strength is ignored or is selected to be an
unrealistically large value do the results deviate from those presented below.
For the analysis of each rope, four curves are presented: the theoretical upper-bound
response, the net area effect and degradation model, the strain concentration and
degradation model, and the degradation model. Assumptions associated with each of
these cases are outlined in Table 5.2. In addition, a detailed study is presented for the
effect of the L/d ratio on the residual strength and residual maximum axial strain of the
damaged rope. These two residual quantities are characterized as the ratio of the damaged
to undamaged value of each parameter and are compared with the ones obtained from the
experimental data to study their dependence on the L/d ratio. The experimental data are
obtained from a project organized through OTRC (Ward, et al., 2005) to test full-scale
147
polyester ropes. The purpose of these tests is to determine the impact of damage on the
strength of these ropes used as mooring lines for deepwater structures. Ropes from
different rope manufactures that are commercially available were tested to represent
different rope constructions. Three series of tests on damaged polyester rope samples
were conducted (Ward et al., 2005):
•
Length Effect Tests: The purpose of these tests was to determine the potential
influence (if any) of length effects on damaged polyester rope response. Different
L/d ratios were considered for ropes with two different specified breaking
strengths: 35 Tonnes (77.2 Kips) and 700 Tonnes (1543.2 Kips).
•
Damaged Full-Scale Rope Tests: The purpose of these tests was to quantify the
influence of damage on full-scale ropes. Ropes with a specified breaking strength
of 700 Tonnes (1543.2 Kips) were evaluated.
•
Verification Tests: The purpose of these tests was to verify the results of the
Damaged Full-Scale Rope Tests with a limited number of tests on longer samples.
Ropes with a specified breaking strength of 700 Tonnes (1543.2 Kips) were
tested.
Rope tests were completed at different test facilities; hence some variability in the
test data may be expected. For each rope tested, only the values of the rope breaking
strain and rope breaking load were provided. No information was given about the length
used to compute the values of the rope breaking strain based on the rope elongation
measurements.
All the ropes are made of polyester fibers which were provided by the same fiber
manufacturer. Thus, for computational purposes, the same polynomial function that
describes the tensile load as a function of the axial strain of the polyester fibers is used
for all the ropes under study.
Figs. 5.19-5.38 below show schematics of damaged rope cross-sections,
geometric properties of the ropes being analyzed, axial load response predictions, and
comparisons of the damage-dependant response to the undamaged case.
148
24 parallel unjacketed sub-ropes
3 twisted strands per sub-rope
70 twisted yarns per strand
Rope diameter (d): 32 [mm]
Specified Breaking Strength: 35 Tonnes (77.2 Kips)
Rope jacket
Cut Strand
Fig. 5.19: Damaged cross-section (10%) of DPJR 1
Exp. Data (No Initial Damage)
100
Upper Bound Response
Rope Axial Load (Kips)
Net Area Effect & Degradation M odels
80
Strain Concentration & Degradation M odels
(L/d = 1000)
Degradation M odel
60
Exp. Data (L/d = 40)
Exp. Data (L/d =290)
40
Exp. Data (L/d = 1000)
20
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Rope Axial Strain
Fig. 5.20: Experimental data and predicted responses of DPJR 1
149
Damaged/Undamaged
Rope Failure Strain (%)
105
Exp. Data
(a)
Numerical Model
100
95
90
85
0
200
400
600
800
1000
1200
L /d ratio
Damaged/Undamaged
Rope Failure Load (%)
110
(b)
Exp. Data
105
Numerical Model
100
95
90
85
80
0
200
400
600
800
1000
1200
L /d ratio
Fig. 5.21: Variation of the (a) failure strain and (b) failure load of the
DPJR 1 with respect to L/d ratio values
150
24 parallel unjacketed sub-ropes
3 twisted strands per sub-rope
22 twisted rope yarns per strand
70 twisted yarns per rope yarn
Rope diameter (d): 160 [mm]
Specified Breaking Strength: 700 Tonnes (1543.2 Kips)
Rope jacket
Cut Strand
Fig. 5.22: Damaged cross-section (10%) of DPJR 2
Exp. Data (1)(L/d = 40)
2500
Epx. Data (2) (L/d = 40)
Upper Bound Res pons e
Net A rea Effect & Degradation M odels (1)
2000
Rope Axial Load (Kips)
Net A rea Effect & Degradation M odels (2)
Strain Localization & Degradation M odels
(1) (L/d = 40)
Strain Localization & Degradation M odels
(2) (L/d = 40)
1500
(1): 5% of initial damage
(2): 10% of initial damage
1000
500
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.23: Experimental data and predicted responses of DPJR 2
151
0.14
18 parallel unjacketed sub-ropes
3 twisted strands per sub-rope
3 twisted rope yarns per strand
30 twisted yarns per rope yarn
Rope diameter (d): 32 [mm]
Specified Breaking Strength: 35 Tonnes (77.2 Kips)
Rope jacket
Cut Strand
Fig. 5.24: Damaged cross-section (10%) of DPJR 3
Exp. Data (No Initial Damage)
Rope Axial Load (Kips)
100
Upper Bound Response
Net Area Effect & Degradation M odels
80
Strain Concentration & Degradation
M odels (L/d = 1000)
Degradation M odel
60
Exp. Data (L/d = 40)
Exp Data (L/d = 290)
40
Enp. Data (L/d = 1000)
20
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.25: Experimental data and predicted responses of DPJR 3
152
0.14
Damaged/Undamaged
Rope Failure Strain (%)
105
Exp. Data
(a)
Numerical Model
NS
100
M
M: Strands cut near rope midspan
NS: Strands cut near rope splice
95
NS
90
0
200
400
600
800
1000
1200
L /d ratio
Damaged/Undamaged
Rope Failure Load (%)
95
(b)
90
M
Exp. Data
85
Numerical Model
NS
80
M: Strands cut near rope midspan
NS: Strands cut near rope splice
75
0
200
400
600
L /d ratio
800
1000
1200
Fig. 5.26: Variation of the (a) failure strain and (b) failure load of the
DPJR 3 with respect to L/d ratio values
153
20 parallel unjacketed sub-ropes
3 twisted strands per sub-rope
28 twisted rope yarns per strand
30 twisted yarns per rope yarn
Rope diameter (d): 147 [mm]
Specified Breaking Strength: 700 Tonnes (1543.2 Kips)
Rope jacket
Cut Strand
Fig. 5.27: Damaged cross-section (5%) of DPJR 4
Exp. Data (No Initial Damage)
2000
Exp. Data (5% Initial Damage)
Exp. Data (10% Initial Damage)
Rope Axial Load (Kips)
1600
Upper Bound Response
Net Area Effect & Degradation
Models
Strain Concentration &
Degradation Models (L/d = 40)
Degradation Model
1200
800
All the simulated damaged rope responses
consider 10% of initial cross-section damage
400
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.28: Experimental data and predicted responses of DPJR 4
154
0.14
10 parallel unjacketed sub-ropes
3 twisted strands per sub-rope
3 twisted rope yarns per strand
30 twisted yarns per rope yarn
Rope diameter (d): 36 [mm]
Specified Breaking Strength: 35 Tonnes (77.2 Kips)
Rope jacket
Cut Strand
Fig. 5.29: Damaged cross-section (10%) of DPJR 5
Exp. Data (No Initial Damage)
120
Upper Bound Response
Net Area Effect & Degradation
Models
Strain Concentration &
Degradation Models (L/d = 1000)
Degradation Model
Rope Axial Load (Kips)
100
80
Exp. Data (L/d = 40)
60
Exp. Data (L/d = 560)
Exp. Data (L/d = 1000)
40
20
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.30: Experimental data and predicted responses of DPJR 5
155
0.14
Damaged/Undamaged
Rope Failure Strain (%)
110
105
Exp. Data
NS
(a)
Numerical Model
100
M
95
M: Strands cut near rope midspan
NS: Strands cut near rope splice
90
M
85
NS
80
0
200
400
600
800
1000
1200
L /d ratio
Damaged/Undamaged
Rope Failure Load (%)
100
Exp. Data
(b)
NS
Numerical Model
M
95
M: Strands cut near rope midspan
NS: Strands cut near rope splice
90
85
0
200
400
600
800
1000
1200
L /d ratio
Fig. 5.31: Variation of the (a) failure strain and (b) failure load of the
DPJR 5 with respect to L/d ratio values
156
10 parallel unjacketed sub-ropes
3 parallel strands per sub-rope
1859 yarns per strand
Rope diameter (d): 36 [mm]
Specified Breaking Strength: 700 Tonnes (1543.2 Kips)
Rope jacket
Damaged rope cross-section
Cut Strand
Fig. 5.32: Damaged cross-section (10%) of DPJR 6
Upper Bound Response
2500
Rope Axial Load (Kips)
Degradation Model
2000
Strain Concentration & Degradation
Models (L/d = 40)
Net Area Effect & Degradation Models
1500
Exp. Data (No Initial Damage)
Exp. Data (10% Initial Damage)
Exp. Data (15% Initial Damage)
1000
All the predicted damaged rope responses
consider 10% of initial cross-section damage
500
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.33: Experimental data and predicted responses of DPJR 6
157
0.14
7 parallel unjacketed sub-ropes
12 twisted strands per sub-rope
1148 twisted yarns per rope yarn
Rope diameter (d): 144 [mm]
Specified Breaking Strength: 700 Tonnes (1543.2 Kips)
Rope jacket
Damaged rope cross-section
Cut Strand
Rope Axial Load (Kips)
Fig. 5.34: Damaged cross-section (10%) of DPJR 7
1800
Exp. Data (No Initial Damage)
1600
Exp. Data (5% Initial Damage)
1400
Exp. Data (10% Initial Damage)
1200
Upper Bound Response
1000
Net Area Effect & Degradation
Strain Concentration &
Degradation Model (L/d = 40)
Degradation Model
800
600
All the simulated damaged rope responses
consider 10% of initial cross-section damage
400
200
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Rope Axial Strain
Fig. 5.35: Experimental data and predicted responses of DPJR 7
158
0.14
0.14
(a)
Failure Axial Strain
0.12
0.1
0.08
0.06
0.04
Exp. Data
0.02
Strain Concentration &
Degradation Models
0
0
2
4
6
8
10
12
% Rope Cross-Section Iniatilly Damaged
Failure Axial Load (Kips)
2000
Exp. Data
1800
Strain Concentration &
Degradation Models
1600
(b)
All
h
1400
1200
1000
0
2
4
6
8
10
% Rope Cross-Section Iniatilly Damaged
12
Fig. 5.36: Variation of the (a) failure strain and (b) failure load of the
DPJR 7 with respect to initial rope damage state
159
Failure Axial Load (Kips)
1700
(a)
5% Initial Damage
1600
7% Initial Damage
10% Initial Damage
1500
1400
1300
1200
1100
0
1000
2000
3000
4000
L/d ratio
0.13
(b)
5% Initial Damage
Failure Axial Strain
0.125
7% Initial Damage
10% Initial Damage
0.12
0.115
0.11
0.105
0
1000
2000
3000
4000
L/d ratio
Fig. 5.37: Variation of the (a) failure axial load and (b) failure axial strain
of the DPJR 7 with respect to L/d ratio for different initial rope damage
states
160
Figs. 5.19-5.37 show different types of rope constructions and their corresponding
response for certain damage states. The analyses suggest that the estimate of the breaking
load (capacity) of initially undamaged ropes that is based upon the rope degradation
model (Chapter 4) compares well with the experimental data, except for the rope
response of Fig. 5.33 (DPJR 6) in which rope breaking load is overestimated by about
19%. A summary of the measured and predicted rope capacities shown in the previous
graphs are summarized in Table 5.3. In order to quantify the effect of using the
degradation model on the capacity of initially undamaged ropes, the capacity of these
ropes obtained without considering the degradation model are also included in Table 5.3.
In all the cases, the capacity of the initially undamaged ropes obtained based upon the
rope degradation model compare better with experimental data than the capacity values
obtained by ignoring the degradation model. These results validate the calibration of the
damage model using small-scale ropes as presented in Chapter 4.
Table 5.3 Summary of the initially undamaged rope capacities
Rope
Max. Rope
Max. Rope Capacity
Predicted/Exp.
Max. Rope Capacity
Capacity
(Kips)
Data Value
(Kips)
(Kips)
Numerical Model
(with
Numerical Model
Exp. Data
with Degradation
Degradation)
No Degradation
(Predicted)
(Predicted)
DPJR 1
72.61
76.91
1.06
83.35
DPJR 3
77.49
77.29
0.997
84.58
DPJR 4
1503.13
1581.18
1.052
1705.06
DPJR 5
83.4 - 89
84.45
0.979
92.18
DPJR 6
1542.86
1842.38
1.19
2014.74
DPJR 7
1422.57
1469.93
1.033
1604.92
161
The plots previously mentioned also suggest that the damaged rope response
curve is bounded by the model that ignores degradation of rope properties (upper bound)
and by the curve obtained through the use of the net area effect model (lower bound). If
the length over which damage propagates along the rope (“damage length”) approaches
that of the rope length, the damaged rope response approaches its lower bound.
Conversely, if the damage length is small compared to the rope length, the damaged rope
response approaches its upper bound. This last observation is due to the fact that the
value of the recovery length is a small percentage of the rope length, so the rope response
is only slightly reduced relative to the undamaged case. As the damage length increases
and broken rope elements develop admissible values for their recovery lengths, the
strains along the rope are non-uniform, generating a strain localization region around the
failure location. Based on the damage (deterioration) model, damage localization occurs
as well. This damage localization causes the premature failure of rope elements and
reduces the damaged rope breaking strength and maximum attainable deformation.
The predicted response of a damaged rope, including the strain localization effect,
depends on the estimate of the “damage length”. This length, computed based on the
recovery length of the broken rope elements, depends only on the rope cross-section
model and not on the rope length. Consequently, as the overall length of a damaged rope
increases (with the length over which damage occurs remaining constant), the damaged
rope response gets stiffer (approaching its upper bound curve), the rope strain at failure
for the damaged rope decreases, and the damaged rope strength is not noticeably affected.
In the analyses presented in Section 5.5, the predicted damaged rope responses are
compared with experimental data. Not only are the values of the maximum deformation
and capacity compared, their dependence on the values of the L/d ratios and percentages
of the initial damage are also provided. With the exceptions of the rope responses shown
in Figs. 5.17 and 5.18, the experimental data plot above the theoretical upper bound rope
response curve, which is the stiffest rope response that the numerical model can provide
using the material properties that were established previously from the small-scale rope
tests. The predicted rope axial failure strain overestimates the experimental value for all
162
cases studied, except for the predicted rope responses shown in Figs. 5.17 and 5.18. For
the numerical model, the predicted value of failure axial strain decreases in a nonlinear
fashion as the L/d ratio increases. Conversely, the experimental values do not show a
clear trend that depends upon the L/d ratio. In fact, for some ropes (DPJR 1and DPJR 5),
the experimental values of the failure axial strain decrease as the L/d ratio increases (Figs.
5.21 and 5.31, respectively), while for others (DPJR 3), the failure axial strain increases
with increasing L/d ratio (Fig. 5.26). This analysis is summarized in Fig. 5.38(a),
considering only the case in which the initial damage is inflicted at the rope midspan.
In Tables 5.4 and 5.5, the predicted and experimental values of the maximum
damaged rope capacity for ropes with specified breaking strengths equal to 35 Tonnes
(77.2 Kips) and 700 Tonnes (1543.2 Kips), respectively, for different values of the L/d
ratio and a 10% initial damage state are presented. For the case of ropes with specified
undamaged breaking strengths equal to 35 Tonnes, the maximum load capacities of ropes
DPJR 1 and DPJR 3 are accurately predicted by the numerical model (within a range of ±
5% of the experimental values) for values of L/d equal to 40 and 290. For these ropes, the
experimental data indicate that their load capacities decrease as the value of L/d increases
from 290 to 1000. This aspect of behavior is not captured by the numerical model (Figs.
5.21 and 5.26). The values of the predicted rope capacities do not show a strong
dependence on the L/d ratio. For the case of DPJR 5, the predicted maximum load
capacity correlates well with the experimental data for a value of L/d equal to 40. For
greater values of L/d, the maximum load capacity of the rope increases (by about 7%),
but it remains nearly constant for values of L/d equal to 560 and 1000 (Fig. 5.31). As for
the case of the failure axial strain, this analysis is summarized in Fig. 5.38(b).
As for the case of initially undamaged ropes, the capacity of the initially damaged
ropes obtained by just considering the net area effect without including the degradation
model are also presented in Tables 5.4 and 5.5. In most of the cases studied, not
considering the degradation model leads to overestimate the capacity of the damaged
ropes in a greater percentage than the case when the degradation model is included,
especially for the ropes with specified breaking strengths equal to 700 Tonnes (1543.2
Kips). This is not the case of DPJR 5. For this rope, for a value of L/d equal to 40, the
163
predicted value of the rope capacity including the degradation model compares better
with experimental data than the predicted value obtained when the degradation model is
ignored. Conversely, for values of L/d equal to 290 and 1000, the predicted values of the
rope capacity considering just the net area effect compare better with experimental data
than the predicted values obtained when the degradation model is considered.
As previously stated, only the values of the rope breaking strain and rope breaking
load were provided. The experimental data show much variability, several of the trends
are not consisted from one rope manufacturer to another, and in some cases experimental
data show unexpected behavior. For example, for the case of DPJR 1 and for a value L/d
equal to 290, the capacity of this rope is greater when its cross-section is damaged in 10%
than the initially undamaged case. More information about how the rope tests were done
and how the values of the breaking rope strain were computed based on the rope
elongation measurements is needed to establish the quality of the data.
Table 5.4 Summary of the damaged rope capacities (10% of initial damage) with
specified breaking strength equal to 35 Tonnes (77.2 Kips)
Max. Rope Capacity (Kips)
Max. Rope Capacity (Kips)
Numerical Model
Exp. Data
Rope
L/d
L/d
L/d
560
(Predicted)
Strain
Net Area Effect
Localization and
and No
Degradation
Degradation
1000
Model
Model
L/d
40
290
DPJR 1
71.49
72.61
63.7
69.11
75.24
DPJR 3
65.75
68.95
59.49
69.39
76.20
DPJR 5
77.12
82.44
76.02
82.96
83.61
164
105
Damaged/Undamaged
Rope Failure Strain (%)
(a)
100
95
Rope DPJR 5
Rope DPJR 3
Rope DPJR 1
90
85
0
200
400
600
800
1000
1200
L /d ratio
Damaged/Undamaged
Rope Failure Load (%)
110
(b)
100
90
80
70
Rope DPJR 5
Rope DPJR 3
60
Rope DPJR 1
50
0
200
400
600
800
1000
1200
L /d ratio
Fig. 5.38: Variation of the experimental (a) rope failure strain and (b)
rope failure load of the damaged ropes with specified breaking strength of
35 Tonnes with respect to L/d ratio
For the cases of ropes with specified breaking strengths equal to 700 Tonnes
(1543.2 Kips), the numerical model overestimates the measured damaged rope capacities
for all the cases studied. Based on reported experimental data (Ward, 2006), the damaged
165
capacity of certain ropes (DPJR 2, DPJR 4 and DPJR 6) shows some dependence on the
L/d ratio in which their load capacities decrease as the value of L/d increases. However,
such is not the case for the rope DPJR 7, whose load capacity remains nearly constant for
values of L/d equal to 40 and 290. As in the case of ropes with specified breaking
strengths equal to 35 Tonnes, the predicted rope capacities obtained by the numerical
model do not show a strong dependence on the L/d ratio.
Table 5.5 Summary of the damaged ropes capacities with specified breaking
strength equal to 700 Tonnes (1543.2 Kips)
Max. Rope Capacity (Kips)
Max. Rope Capacity (Kips)
Exp. Data
Numerical Model (Predicted)
Initial
Plot
Damage
L/d
L/d
40
290
Strain
Net Area
Localization and
Effect and No
Degradation
Degradation
Model
Model
DPJR 2
10%
1201.51
945.78
1536.95
1674.35
DPJR 4
10%
1278-1317.73
994.28
1415.60
1554.68
DPJR 6
15%
1315.52
1101.41
1562.55
1712.53
DPJR 7
10%
1146.40
1111.13
1329.77
1452.32
In Fig. 5.36, the variation of the failure axial load and failure axial strain with
respect to initial loss of rope cross-sectional area (0%, 5% and 10%) of a rope whose
cross-section was depicted in Fig. 5.34 (DPJR 7) is presented. The predicted values
overestimate the measured data over the whole range of initial loss of rope crosssectional area considered, for both parameters. However, both experimental and predicted
values for these quantities show the same trend – as the initial loss of rope cross-sectional
area increases, both parameters decrease. In Fig. 5.37, the predicted damaged rope
166
capacities and failure axial strains for each value of the initial loss of rope cross-sectional
area previously mentioned are plotted as a function of the L/d ratio. The predicted rope
capacities do not show a strong dependence on the L/d ratio for all the initial damage
states considered. The predicted failure axial strains decrease, however, showing a
decreasing rate of change as the L/d ratios increase. Eventually, the failure axial strains
reach a minimum value for large values of the L/d ratio.
5.6 SUMMARY
In this chapter, a numerical model to account for the effect of broken rope
elements on overall rope response was presented. This model relies on the ability of a
broken rope element to pick up its proportionate share of axial load over a distance
measured from the failure region. This distance is defined as the recovery length.
Numerical simulations were compared with experimental data obtained from
static capacity tests on different types of rope constructions and initial damage levels. For
the case of initially undamaged ropes, the proposed numerical model accurately predicted
maximum load capacity, except for the case of DPJR 6 (Fig. 5.32), whose load capacity
was overestimated by approximately 19%. For the case of initially damaged ropes, the
proposed numerical model does a good job predicting the capacity of the ropes with
specified undamaged breaking strengths equal to 35 Tonnes, but it overestimated the
capacity of the ropes with specified undamaged breaking strengths equal to 700 Tonnes.
For all of the cases analyzed, the proposed numerical model overestimated the rope axial
failure strains.
A detailed study was presented for the effect of the L/d ratio on the residual
strength and residual maximum axial strain of damaged ropes. These two residual
quantities were characterized as the ratio of the damaged to undamaged value of each
parameter and were compared with the ones obtained from the experimental data. The
numerical predictions of these two residual quantities showed a unique trend in the their
dependence on the L/d ratio: as the value of the L/d ratio increased, the value of residual
maximum axial strain decreased, and the value of the residual strength remained nearly
constant. The experimental data, however, did not show a clear trend. For some rope
167
constructions, as the L/d ratio increased, the value of the residual maximum axial strain
decreased, while for other ropes the maximum axial strain increased. In the case of the
residual strength, as the L/d ratio increased for some ropes, the value of the residual
strength decreased. For other ropes, however, the residual strength remained nearly
constant as a function of L/d ratio. Thus, because the ropes tested showed great variability
in their response, more data are needed to validate observed trends in the behavior of
damaged ropes. While the current model can be used to obtain a reasonable estimate of
rope behavior, improvements are still needed before it can be reliably used for detailed
design calculations.
Most of the experimental data, including that for undamaged ropes, plotted above
the theoretical upper bound, which is the stiffest rope response expected. Some of the
possible reasons of this inconsistency are the following: measurement error of some rope
variables, variability in the rope properties used as input in the computational model, or a
combination of these effects. As stated above, the numerical model overestimates the
rope axial failure strain and accurately predicts the capacity of the damaged ropes in most
cases. Thus, errors could have been made in computing the rope failure axial strains from
the information obtained from the tests. More detailed information about the
experimental data are needed to understand how rope strength capacities and rope failure
strains were measured.
168
CHAPTER 6
Summary and Conclusions
6.1 SUMMARY AND CONCLUSIONS
To study the behavior of synthetic-fiber ropes under a variety of loading
conditions, a computational model was proposed that accounts for the change in
structural properties and configurations of a virgin (i.e., undamaged) or initially damaged
rope. The developed model includes both geometric and material nonlinearities, and it
incorporates a damage index so that the strength and stiffness degradation of the rope
elements throughout the loading history can be modeled. The proposed model also
considers the capability of a broken rope element to keep contributing to rope response
after fully developing its recovery length.
The computational model developed under this research relies on the
displacement control method, and integration over the rope element cross-section is
carried out numerically using Gaussian quadrature. The assumption of geometry
preservation is employed to establish the kinematics of deformation for rope elements.
Thus, helical components are assumed to remain helical following the application of load.
Geometric nonlinearity is taken into account by using nonlinear strain/displacement
equations for each rope element in terms of the global strain/displacement quantities of
the entire rope. Rope geometry is updated at every load step. The nonlinear material
behavior of synthetic-fibers is considered by using a polynomial representation (up to the
fifth degree) of the normal and shear stresses in terms of their respective strain quantities.
The extreme cases of real rope cross-sectional geometry are considered in the
computational model development: packing and wedging geometries. The set of
linearized differential equilibrium equations is reduced to a system of linear algebraic
equations of order two using the assumptions related to the kinematics of deformation of
rope elements.
As mentioned above, the effects of strength and stiffness degradation are modeled
by means of a damage index D. The damage index is described by a linear combination
169
of three processes: maximum deformation, work done by frictional forces, and the stress
range experienced by a rope element. Each of these processes could have a nonlinear
dependence on the damage variables used to quantify them. The proposed damage model
that represents the evolution of the damage index throughout the load history was
calibrated based on available experimental data. When the value of damage index equals
one (i.e., D = 1), the damage length associated with the failed rope element is computed
based on the estimate of its recovery length. If the value of this damage length is greater
than the rope length, the failed rope element is assumed to no longer contribute to the
total response of the rope. Conversely, if the value of the damage length is admissible, the
failed rope element is assumed to contribute to rope response after fully developing its
recovery length value. If such is the case, the rope is discretized into “sub-rope” elements
based on the value of the recovery lengths computed. Each “sub-rope” is modeled as a
two-noded axial-torsional element with two degrees of freedom at each node: axial
displacement and axial rotation. This model captures the effect of having a weakened
cross-section acting over a localized region defined by the recovery length. Strains are
not uniformly distributed along the rope length, resulting in a potential strain localization
region around the failure site, and, based on the damage model, damage localization may
also occur. Any lack of symmetry in the rope cross-section due to the failure of rope
elements is neglected.
Using the computational model developed under this research, several rope
geometries were studied. Virgin (i.e., undamaged) ropes and initially damaged ropes
were considered. In all cases, experimental data for a monotonically increasing load were
available for comparison with the analytical predictions. The first example studied was a
one-level, two-layer polyester rope. This study was employed to calibrate the damage
model proposed by comparing the experimental data curve with the simulation curve
without any source of damage. A parametric study of the damage model was carried out
to study the effects of changes in the damage parameters on the damage evolution (and
consequently on rope response). The analysis results demonstrated that the proposed
exponential evolution of the damage index agrees well with the experimental data.
170
One drawback with the above damage model was that polyester rope elements
experienced low levels of damage near the onset of failure (i.e., values much less than
one). As previously mentioned, the failure of a rope element occurs when the damage
index equals one, and there is a need to capture the rapid change in the damage index
corresponding to the response just prior to component rupture. Accordingly, a
perturbation method was used because it allowed for the degradation of rope properties to
be represented as a continuous process. To verify the validity of the proposed damage
model, two different types of analysis were carried out on small ropes: (1) analysis of a
two-level, one-layer polyester rope, and (2) analyses of one-level, two-layer polyester
ropes with one and three rope elements initially damaged. For these analyses, the failed
components were assumed to not contribute to total rope response (net area effect model).
Computed results showed that for the case of the two-level, one-layer polyester rope, the
simulation curve overestimated by 6% the maximum rope capacity and by 1.5% the
corresponding strain. For the case of the one-level, one-layer polyester rope with one
rope element initially damaged (D = 1), the simulation curve overestimated by 2% the
maximum capacity of the rope and by less than 1% the corresponding strain. Finally, for
the case of the one-level, one-layer polyester rope with three rope elements initially
damaged (D = 1), the simulation curve overestimated by 1% the maximum capacity of
the rope and by less than 1% the corresponding strain.
In addition to using experimental data to evaluate the accuracy of the proposed
computational model, computed results were also compared against a three-dimensional
finite element analysis of the two-level, one-layer polyester rope described above.
Ignoring the effects of rope degradation, results from the two simulation approaches
compared quite well, demonstrating the validity of several of the assumptions made in the
formulation of the proposed computational model.
The one-level, two-layer initially damaged small polyester ropes mentioned above
were used to study the effect of failed rope elements on overall rope behavior. For this
purpose, two models were used to predict the damaged rope response: the net area effect
and degradation model and the strain localization and degradation model. As indicated
earlier, two initial damage states were considered for the one-level, two-layer rope: one
171
and three of the nine rope elements (representing 11% and 33% of the rope cross-section,
respectively) were cut prior loading. For both cases, the predicted rope response based on
the net area effect and degradation model correlated better with experimental data than
the strain localization and degradation model. The model with strain localization effects
predicted a stiffer rope response and underestimated the rope breaking strain relative to
the collected data, but it correlated better with experimental data as the initial crosssection damage increased from 11% to 33%. For the case of the two-level, one-layer
polyester rope, nine of the twenty seven rope elements (representing 33% of the rope
cross-section) were cut prior loading. Unlike for the previous case, the model with strain
localization effects corresponded better with experimental data and provided a smaller
value of the rope breaking strain relative to the net area effect model. Both of these
models, however, gave values of rope breaking strain and rope breaking load that were
smaller than those indicated by the test data.
The proposed damaged model was also verified with the analyses of large-scale
virgin and initially damaged polyester-jacketed ropes (DPJR). Most of the studies
considered a 10% loss of cross sectional area for the initial damage state. Experimental
data for ropes with a specified breaking strength (SBS) of 35 Tonnes and 700 Tonnes
were available. For analysis purposes, all of these ropes were modeled as two-level ropes.
At the first level, all ropes had parallel sub-rope constructions. Each sub-rope comprising
the second level was built by twisting a specified number of strands together. Ropes with
SBS values of 35 Tonnes were constructed by different rope manufacturers using the
following configurations: 24 sub-ropes and 3 strands per sub-rope (DPJR 1); 18 subropes and 3 strands per sub-rope (DPJR 3); and 10 sub-ropes and 3 strands per sub-rope
(DPJR 5). For the case of ropes with SBS values of 700 Tonnes, manufactured
configurations included the following: 24 sub-ropes and 3 strands per sub-rope (DPJR 2);
20 sub-ropes and 3 strands per sub-rope (DPJR 4); 10 sub-ropes and 3 strands per subrope (DPJR 6); and 7 sub-ropes and 12 strands per sub-rope (DPJR 7). In addition, for
ropes with SBS values of 35 Tonnes and 700 Tonnes, parametric studies were conducted
to evaluate the effect of different L/d ratios (rope length normalized by rope diameter) on
damaged rope response. The effect of initial damage states (0%, 5%, and 10% of rope
172
cross-section) on the values of the rope breaking strain and rope breaking load was
studied for a rope with an SBS value of 700 Tonnes for the case of 7 parallel sub-ropes
and 12 strands per sub-rope (DPJR 7).
Computed results showed that for the case of initially undamaged ropes, the
simulation values of the rope breaking axial load compared well with experimental data,
within a range of -2% to 6%, except for DPJR 6 whose breaking axial load was
overestimated by approximately 19%. The simulation values of rope breaking strain,
however, overestimated the experimental data values for all cases studied.
For the analysis of initially damaged polyester-jacketed ropes, the confinement
effect of the jacket on rope elements was considered to determine if the broken rope
elements contributed to rope response after fully developing their recovery length. For
modeling purposes, the rope jacket was considered as a thin-walled tube having an axial
capacity in a range of 4% to 5% of the axial capacity of the corresponding initially
undamaged rope. The predicted rope axial failure strain overestimated the experimental
values for all cases studied. For the numerical model that included degradation and strain
localization effects, the predicted value of axial failure strain decreased in a nonlinear
fashion as the L/d ratio increased. Conversely, the experimental values did not show a
clear trend that depended upon the L/d ratio. In fact, for some ropes (DPJR 1 and DPJR
5), the experimental values of the axial failure strain decreased as the L/d ratio increased,
while for others (DPJR 3), the axial failure strain increased with increasing L/d ratio.
Considering the maximum capacity of the damaged ropes, for some rope
constructions with SBS values of 35 Tonnes (DPJR 1 and DPJR 3) and values of L/d ratio
smaller than 290, the numerical model provided accurate predictions of response (within
a range of ± 5% of the experimental values). For these ropes, the test data showed that
their load capacities decreased as the value of L/d increased from 290 to 1000. This
aspect of behavior was not captured by the numerical model. In fact, the rope capacities
predicted by the numerical model showed very little dependence on the L/d ratio. For the
case of DPJR 5, the predicted maximum load capacity correlated well with the
experimental data for a value of L/d equal to 40. For greater values of L/d, the
experimental data showed that the maximum load capacity of the rope increased, but,
173
unlike some of the other ropes, it remained nearly constant for values of L/d equal to 560
and 1000 (10% greater than the value predicted by the numerical model).
For the case of ropes with SBS values of 700 Tonnes, the numerical model
overestimated the damaged rope capacity for all the rope constructions considered. For
some of these ropes (DPJR 2, DPJR 4 and DPJR 6), the test data showed that their load
capacities decreased as the value of L/d increased from 40 to 290. As with the analyses of
ropes with SBS values of 35 Tonnes, the numerical model did not capture this behavior.
For the case of DPJR 7, however, the maximum load capacity of the rope remained
nearly constant for values of L/d equal to 40 and 290 (overestimated by 15% by the
numerical model).
Based on the study of the dependence of the breaking axial strain and breaking
axial load on the L/d ratio of the rope, the test data showed much variability and several
of the trends are not consisted from one rope manufacturer to another. Even, in some
cases, the trends are different for ropes of the same manufacturer but with different SBS
values. Given this variability, the proposed computational model is believed to provide
reasonable predictions in performance of damaged ropes. Further testing is needed to
help better understand the response of damaged ropes and to improve the computational
model.
As previously mentioned, the effect of initial damage states (0%, 5%, and 10% of
the rope cross-section) on the values of the rope breaking strain and rope breaking load
was studied for a rope with an SBS value of 700 Tonnes (DPJR 7). Although the
experimental values of rope breaking strain and rope breaking load were overestimated
by the numerical model, both experimental and predicted values for these quantities
showed the same trend: as the initial loss of rope cross-sectional area increased, both
parameters decreased. In addition, for each damage state, the dependence of the breaking
axial strain and breaking axial load on the L/d ratio of the rope was studied. The rope
capacities predicted by the numerical model showed very little dependence on the L/d
ratio, and the predicted value of the breaking axial strains decreased in a nonlinear
fashion as the L/d ratio increased, converging asymptotically to a minimum value.
174
6.2 FUTURE STUDIES
The findings reported in this research are intended to demonstrate the need for
having computational tools that accurately predict the response of damaged ropes in order
to interpret and extend test data and to develop design guidelines. While this research
represents an advancement over previous approaches to analyzing damaged rope
response, many aspects of this study can be modified or extended to improve the
accuracy of the model. The following recommendations are given for further research on
this topic:
1. Experimental studies on large ropes to gain a better understanding of
response under increasing monotonic load – Information obtained from
these studies can be used to verify or eventually modify the current
damage model, the methods used to obtain damage evolution, and the
assumptions made to simulate softening behavior. In addition, new issues
that can affect rope behavior can be studied, including the following: rope
diameter-rope length ratio, partially or fully damaged rope elements, nonsymmetry in the cross-section, and variation in the arrangement of rope
elements and rope splices.
2. Incorporation of more accurate material models – The current constitutive
model ignores time-dependent behavior and approximates the known
viscoealstic response of synthetic fibers through the use of an empirically
fit fifth-order polynomial function. Thus, this approximate constitutive
model can be extended to consider intrinsic viscoelastic behavior of
synthetic fibers. A general nonlinear viscoelastic model can be developed
that allows studying long-term behavior of synthetic ropes. In addition, a
new term to the damage model can be added that considers creep or
relaxation failure. The model can also include the treatment of variability
in rope properties. This effect may be playing an important role in the
differences noted between the predicted and the measured rope behavior.
Variations in the stress-strain relationships and values of the breaking
175
loads and breaking strains of the rope elements play a significant role in
the computational model.
3. Evaluation and quantification of the effects of broken rope elements on
rope response – A new feature to the computational model can be added
that considers the loss of symmetry and remaining strength in a rope due
to the failure of individual rope elements.
4. Verification of the assumptions made to develop the current computational
model using three-dimensional finite element analyses – Analyses of
initially damaged and undamaged ropes with different cross-section
configurations and geometrical parameters using three-dimensional finite
element models can be performed to verify the assumptions made and the
range of applicability of the proposed computational model.
5. Detailed information about experimental data – Measurements of rope
end rotations are needed to accurately model boundary conditions.
Although such data are needed for the current computational model, this
information is rarely measured or reported.
176
APPENDIX
Softening Behavior Modeling
The analysis presented in this appendix is to demonstrate the validity of using just
a one-term composite expansion to solve Eqs. 4.12 and 4.13 by the perturbation method
technique. The two-term composite expansion of Eq. 4.12, including a non-zero value of
the initial damage DI, has the following form:
D (η ) = D I + α 1η
β1
+ (1 − α 1η u
β1
− D I )e
η −η u
θ1
+ θ 1 (α 1 β 1η
β 1 −1
− α 1 β 1η u
β1 −1
e
η −η u
θ1
)
(A.1)
and the two-term composite expansion of Eq. 4.13 including a non-zero value of the
initial damage DI has the following form:
[
]
D(η ) = DI + ω1η + ω2η + ω3η + 1 − ( DI + ω1ηu + ω2ηu + ω3ηu ) e
2
3
2
η −ηu


2
θ2
2
θ 2 2ω2η + 3ω3η − (2ω2ηu + 3ω3η u )e 


3
η −η u
θ2
+
(A.2)
where θ1 and θ2 are constants much smaller than 1 (θ1, θ2 << 1) determined by Eq. 4.18,
and η is a dimensionless variable defined as (ε - εt )/ εb . The Eqs. A.1 and A.2 are an
extension of the one-term composite expansions given by Eqs. 4.16 and 4.17,
respectively.
In Figs. A.1 and A.2, the evolution of the damage index D(η) considering a power
law and polynomial variation of their leading term, along with the damage parameters
used, are presented. These curves were computed considering one-term and two-term
composite expansions in order to evaluate the additional information (if there is any)
gained by performing a two-term composite expansion.
177
1.2
1
one-term expansion
two-term expansion
D (η )
0.8
εt = 0.03
εb = 0.124
εu = 0.1302
0.6
0.4
θ1 = 0.0043
0.2
Onset of failure
0
0
0.2
0.4
0.6
0.8
1
η
D (η )
1.2
1
one-term expansion
0.8
two-term expansion
εt = 0.05
εb = 0.124
εu = 0.1302
0.6
0.4
θ1 = 0.0043
0.2
Onset of failure
0
0
0.2
0.4
η
0.6
Fig. A.1 Power law evolution of the damage index
178
0.8
1.2
one-term expansion
1
two-term expansion
D (η )
0.8
εt = 0.03
εb = 0.124
εu = 0.1302
0.6
0.4
θ1 = 0.0043
0.2
Onset of failure
0
0
0.2
0.4
0.6
0.8
1
η
1.2
1
one-term expansion
two-term expansion
D (η )
0.8
εt = 0.05
εb = 0.124
εu = 0.1302
0.6
0.4
θ1 = 0.0043
0.2
Onset of failure
0
0
0.2
0.4
0.6
η
Fig. A.2 Polynomial evolution of the damage index
179
0.8
As can be seen in Figs. A.1 and A.2, for both damage models considered, a twoterm composite expansion agrees quite closely with the results obtained from the oneterm composite expansion. In addition, for each damage model, both composite
expansions predict the same level of damage at the onset of failure of the rope elements.
180
REFERENCES
1. Abaqus/Standard User’s Manual, 1998, Hibbitt, Karlsson & Sorensen, Inc.
2. Ayres, R., 2001, “Determining Interim Criteria for Replacing Damaged Polyester
Mooring Rope,” Final Report to BP Exploration, PN 6941-RRA.
3. Banfield, S. and Casey, N., 1998, “Evaluation of Fibre Rope Properties for Offshore
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VITA
Juan Felipe Beltran was born in Santiago, Chile on March 6, 1974, the son
of Juan Beltran and Carmen Morales. After completing his high school degree
from Instituto Alonso de Ercilla in 1991, he entered University of Chile. After
four years of study, he received the degree of Bachelor of Science in Civil
Engineering and in 1998 the degree of civil engineer. After his graduation in
December 1998, he joined the Department of Civil Engineering, University of
Chile, as a lecturer. In August 2000, he traveled to Austin, Texas, to pursue the
Doctoral Degree in Civil Engineering at The University of Texas at Austin. He
worked on a research project under the supervision of Professor Eric B.
Williamson and as a part of his graduate studies, he received the degree of Master
in Science in Civil Engineering in 2005. After his graduation, he plans to work at
the University of Chile use his knowledge and ideas gained during his studies to
benefit the university and enhance its reputation.
Permanent Address: Los Navegantes 2076
Providencia-Santiago
Chile
This dissertation was typed by the author.
190