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.

v

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

vi

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.

vii

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.

viii

Table of Contents

List of Tables………………………………………………………………..….xvi

List of Figures……………………………………..………………………….…xx

CHAPTER 1: Introduction………………………...……………………………1

1.1 General Background………………………………………………..…………1

1.2 Research Objectives………………………………………………..………….1

1.3 Scope of Research..……………………………………….………..………….1

CHAPTER 2: Background and Literature Review………………………...….1

2.1 Background………………………...……………..……………………….

2.1.1Steel Wire Ropes…………………………………………………

2.1.2 Synthetic-Fiber Ropes …………………………………………

2.2 Literature Review ………………………………………………………….1

2.2.1 Steel Wire Ropes Models ……………………………………….

2.2.1.1 Fiber Models…………………………………………..

2.2.1.2 Helical Rod Models……………………………………

2.2.1.3 Semi-Continuous Models……………………………….

2.2.2 Synthetic-Fiber Ropes Models ….…...………………………….

2.3 Common Types of Rope Constructions …………………………………

2.3.1 Overview………………………..……………………………….

2.3.2 Common Synthetic-Fiber Rope Constructions... ……………….

2.3.2.1 Three-Strand Laid Construction…………………………..

2.3.2.2 Eight-Strand Construction……………………………..

2.3.2.3 Double Braid Construction…………………………….

2.3.2.4 Parallel Yarn/Strand Construction……………………

2.3.2.5 Wire Rope Construction………………………………

2.4 Summary…...................................................................................................

xii

xiii

1

1

4

5

7

7

7

9

10

10

11

12

14

14

18

18

19

19

20

20

21

21

22

ix

CHAPTER 3: Mathematical Modeling………………………...……………….

3.1 Kinematics of a Rope. …………………………………………………………

3.1.1 Structure of a Synthetic-Fiber Rope. …………………………….

3.1.2 Initial and Deformed Configuration of a Rope…………………….

3.1.2.1 Curvature of a Rope..……………………………………

3.1.2.2 Torsion of a Rope…………………………………………

3.1.2.3 Formulae of Frenet……………………………………

3.1.2.4 Axial Deformation of Rope Elements…………………

3.2 Constitutive Models…………………………………………………………

3.3 Cross-Section Modeling.…………………………………………………..

3.3.1 Packing Geometry……………………………………………….

3.3.2 Wedging Geometry..............................................……………….

3.3.3 Cross-Section Update……………………………………………

3.3.3.1 Constant Cross-Section Model…………………………..

3.3.3.2 Constant Volume Model……………………..………..

3.3.3.3 Poisson Effect Model……………………………..…….

3.4 Equilibrium Equations and Friction Models………………………………….

3.4.1 Differential Equations of Equilibrium…………………………..

3.4.2 Reduced Equilibrium Equations…………………………………

3.4.3 Force and Moment Resultants in a Rope Element………………

3.4.4 Friction Model……………………………………………………

3.5 Damage Model………………………………………………………………..

3.5.1 General Background……………………………..............................

3.5.2 Damage Evolution Rule…………………........……………….……1

3.5.3 Basic Concepts of Continuum Damage Mechanics………………..

3.6 Summary…......................................................................................................

24

24

24

26

28

31

32

35

35

36

37

37

37

38

38

38

39

39

44

45

47

50

50

51

54

56

x

CHAPTER 4: Rope Degradation and Static Rope Response……………….

4.1 Overview………………………………………………………………..…

4.2 Damage Measurement …………………………………………………….

4.2.1 Secant Moduli Ratio………………………………………………

4.2.2 Strain Energy Deviation…………………………………………..

4.2.3 Softening Behavior…………………………………………………

4.3 Numerical Implementation………………………………………………….

4.4 Finite Element Analysis………………………………………………………

4.4.1 Finite Element Model……………………………………………

4.4.2 Finite Element Analysis Results……………………………………

4.4 Summary……………………………………………………………………

CHAPTER 5: Numerical Simulation of Damage Localization in

Synthetic-Fiber Mooring Ropes………………………………….……….….

5.1 Overview……………………………………………………...……………

5.2 Calculation of normal Forces and Recovery Length Concept………………

5.2.1 Case 1: Packing Rope Geometry……………………………………

5.2.2 Case 2: Wedging Rope Geometry…………………………………

5.2.3 Case 3: Mixed Rope Geometry……………………………………

5.3 Calculation of the Recovery Length………………………………………….

5.3.1 Numerical Algorithm……………………………………………….1

5.3.2 Special Case: Linear friction Model……………………………….

5.3.3 Numerical Examples of Recovery Length Values………………….

5.3.4 Effect of the Rope Jacket on the Recovery Length Values……….

5.4 Numerical Analyses of Damaged Ropes……………………………………

5.5 Numerical Examples……………………………………...............................

5.6 Summary……………………………………………....................................

CHAPTER 6: Summary and Conclusions………………………………………

6.1 Summary and Conclusions.…………...………………………………………1

57

57

59

59

65

70

75

83

84

86

89

91

91

94

98

107

113

117

118

120

122

130

136

140

167

139

146

169

169

xi

6.2 Future Studies…………………………………………………………………1

APPENDIX: Softening Behavior Modeling…………………………………….

References…………………………………………...…………………………388

VITA………………………………………………...…………………………404

175

177

181

190

xii

List of Tables

Table 3.1 Directions cosines………………………………………………

Table 4.1 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 Coefficients to compute the normal force exerted on a

component of a rope element with mixed cross-section

geometry……………………………………………………………

Table 5.2 Summary of the models used to predict damaged rope

behavior ……………………………………………………………

Table 5.3 Summary of the initially undamaged rope capacities…………

Table 5.4 Summary of the damaged rope capacities (10% of initial

damage) with specified breaking strength equal to 35 Tonnes

(77.2 Kips)………………………………………………………….

Table 5.5 Summary of the damaged rope with specified breaking

strength equal to 700 Tonnes (1543.2 Kips)……………………….

41

77

81

117

141

161

164

166

xiii

List of Figures

Fig. 1.1 Strength of steel and synthetic-fiber ropes…………………..….

Fig. 1.2 Strength-to-weight ratio of steel and synthetic-fiber ropes……..

Fig. 1.3 Types of mooring systems: Taut-leg and catenary…………….

Fig. 1.4 (a) Stress-strain curve for polyester rope; (b) rope pitch

distance(p); (c) rope radius (r)…………….……………………

Fig. 2.1 Longitudinal view and typical types of steel rope

construction……………………………………………………..

Fig. 2.2 Cross-section of a four-level rope………………………………

Fig. 2.3 Types of synthetic-fiber rope constructions……………………

Fig. 3.1 Hierarchy ranking of rope elements……………………………..

Fig. 3.2 Rope elements cross-section at different rope levels…………….

Fig. 3.3 Generic point P of a curve C…………………………………….

Fig. 3.4 Rope geometry………………………………………………….

Fig. 3.5 Local coordinate system for a rope element…………………….

Fig. 3.6 Kinematic of a rigid body B…………………………………….

Fig. 3.7 Developed view of a helical rope element………………………

Fig. 3.8 Rope cross-sectional modeling………………………………….

Fig. 3.9 Loads acting on a line circular helix element…………………….

Fig. 3.10 Centerline of a line element looking down (no moments)

(a) x3 axis and (b) x2 axis………………………………………….

Fig. 3.11 Centerline of a line element looking down (no forces)

(a) x3 axis and (b) x2 axis…………………………………………..

Fig. 3.12 (a) Cross-section for bending moment; (b) Cross-section for

torsion………………………………………………………….

Fig. 3.13 Normal force acting on a rope element (wedging geometry)….

Fig. 3.14 Effect of frictional forces on rope force resultants……………..

2

2

3

5

8

16

20

25

25

43

46

48

49

26

28

30

34

35

37

40

42

xiv

Fig. 3.15 Principle of strain equivalence…………………........................

Fig. 4.1 Response of a one-level, two-layer rope under monotonic

strain history and cross-section and geometric parameters

of the rope………………………………………………………..

Fig. 4.2 Attempt to describe damage evolution based on Eq. 3.48b

for different values of the threshold strain εt……………………

Fig. 4.3 (a), (b), (c) Damage index evolution for different values

of εt and εb = 0.124……………………………………………….

Fig. 4.4 Variation of the damage parameters α1 and β for different

values of εt........................................................................................

Fig. 4.5 Strain Energy Deviation (SED) definition ………………………..

Fig. 4.6 Attempt to describe damage evolution based on

energy-based method………………………………………………

Fig. 4.7 Damage index evolution for different values of εt and

εb = 0.124 using the energy-based method……………………

Fig. 4.8 Overlap domain………………………………………….……..

Fig. 4.9 One-level rope response using energy-based damage model …

Fig. 4.10 One-level rope response using secant moduli ratio damage

model …………………………………………………………..

Fig. 4.11 Response of a one-level rope under monotonic strain

history: experimental, virgin response and damaged

simulation curves………………………………………………….

Fig. 4.12 Cross-section and geometrical properties of a two-level,

one- layer rope…………………………………………………….

Fig. 4.13 Response of a two-level rope under monotonic strain history….

Fig. 4.14 Response of a two-level rope under monotonic strain history….

Fig. 4.15 Predicted responses of one-level initially damaged rope (33%)

Fig. 4.16 Predicted responses of one-level initially damaged rope (11%) ..

55

58

62

64

26

76

78

78

79

80

82

83

65

67

69

71

74

76

xv

Fig. 4.17 (a) Three-strand rope cross-section and (b) finite elements

mesh of the rope…………………………………………………

Fig. 4.18 Two-level rope response predictions, including finite

element analysis, and experimental data……………………….

Fig. 4.19 Finite element model: (a) undeformed rope geometry and

(b) undeformed-deformed rope geometry………………………

Fig. 4.20 Axial stress distribution at a rope axial strain of 12%....................

Fig. 5.1 Recovery length concept of a broken rope element……………..

Fig. 5.2 Pattern of inter-layer/inter-element contact forces in an

axially loaded rope element………………………………………

Fig. 5.3 Contact forces in axially loaded rope element with packing

and wedging geometry………………………………………….

Fig. 5.4 Gradual increase in tension load Ti in a broken rope element…….

Fig. 5.5 Three-level packing geometry rope and normal forces acting

on the rope elements at each rope………………………………

Fig. 5.6 Three-level wedging geometry rope and normal forces

acting on rope elements at each rope level……………………….

Fig. 5.7 Three-level mixed geometry rope and normal forces acting

on the rope elements at each rope level…………………………..

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……………………………………………………………

Fig. 5.9 Normal forces acting on rope elements of a damaged

one-level two-level rope assuming packing geometry…………….

Fig. 5.10 Recovery length values of broken strands of a two-level

damaged rope as a function of the axial rope strain………………..

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

85

87

88

89

92

96

97

98

100

109

114

124

125

127

xvi

different values of rope axial strain……………………………..

Fig. 5.12 Rope jacket and sub-ropes of a parallel rope construction………..

Fig. 5.13 (a) Normal forces acting on the core; (b) compressive stress

acting on the core……………………………………………….

Fig. 5.14 Effect of the rope jacket on the recovery length values:

(a) outer rope yarn; (b) core……………………………………….

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……………………………………………………………..

Fig. 5.16 Two-noded axial-torsional “sub-rope” element………………….

Fig. 5.17 Initially damaged one-level rope responses: (a) one out nine

rope yarns cut and (b) three out nine rope yarns cut…………….

Fig. 5.18 Initially damaged two-level rope cross-section and analyses

of rope response…………………………………………………

Fig. 5.19 Damaged cross-section (10%) of DPJR 1……………………….

Fig. 5.20 Experimental data and predicted responses of DPJR 1………….

Fig. 5.21 Variation of the (a) failure strain and (b) failure load of the

DPJR 1 with respect to L/d ratio values………………………….

Fig. 5.22 Damaged cross-section (10%) of DPJR 2………………………….

Fig. 5.23 Experimental data and predicted responses of DPJR 2……….

Fig. 5.24 Damaged cross-section (10%) of DPJR 3……………………….

Fig. 5.25 Experimental data and predicted responses of DPJR 3………..

Fig. 5.26 Variation of the (a) failure strain and (b) failure load of the

DPJR 3 with respect to L/d ratio values………………..………

Fig. 5.27 Damaged cross-section (5%) of DPJR 4…………………………

Fig. 5.28 Experimental data and predicted responses of DPJR 4…………..

Fig. 5.29 Damaged cross-section (10%) of DPJR 5………………………

Fig. 5.30 Experimental data and predicted responses of DPJR 5………..

129

131

132

135

137

137

142

146

149

149

150

151

151

152

152

153

154

154

155

155

xvii

Fig. 5.31 Variation of the (a) failure strain and (b) failure load of the

DPJR 5 with respect to L/d ratio values……………………………

Fig. 5.32 Damaged cross-section (10%) of DPJR 6……………………….

Fig. 5.33 Experimental data and predicted responses of DPJR 6…………

Fig. 5.34 Damaged cross-section (10%) of DPJR 7………………………….

Fig. 5.35 Experimental data and predicted responses of DPJR 7…………

Fig. 5.36 Variation of the (a) failure strain and (b) failure load of the

DPJR 7 with respect to initial rope damage state………………

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……………………………………….

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……….

156

157

157

158

158

159

160

165

1

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 synthetic-

fiber 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.

2

Fig. 1.1: Strength of steel and synthetic-fiber ropes

0

1000

2000

3000

4000

PE PP NY PET ST AR HMPE

Type of Rope

Stre

ngth

(Mpa

)

0

100

200

300

400

PE PP NY PET ST AR HPME

Type of Rope

Stre

ngth

-to-W

eigh

t Rat

io (c

N/T

ex)

Fig. 1.2: Strength-to-weight ratio of steel and synthetic-fiber ropes

3

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 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.

Fig. 1.3: Types of mooring systems: Taut-leg and catenary (Kennedy, 2004)

Catenary

Taut-leg

Anchor

4

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

5

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.

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.

Fig. 1.4: (a) Stress-strain curve for polyester rope; (b) rope pitch distance (p);(c) rope radius (r)

(c)(b)

r AA

p

View A-A

0

40

80

120

160

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Axial Strain

Axi

al S

tres

s (ks

i)

(a)

6

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. In addition, opportunities for future

research are identified.

7

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

8

and Maclaren, 1997), and the typical cross-sectional arrangement of the elements include

the spiral strand, six strand and multi-strand geometries (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

Fig. 2.1: Longitudinal view and typical types of steel rope constructions (Schrems and Maclaren, 1997; Chaplin, 1998)

Spiral Strand

Six Strand

Multi-Strand

9

(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

10

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.

11

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

12

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

13

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

14

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

15

analysis of the mechanics is not easy to develop. This difficulty in modeling synthetic-

fiber 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).

16

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

Fibers

Rope Yarn

Strand

Rope

Fig. 2.2: Cross-section of a four-level rope

17

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 time-

related 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

18

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 cross-

sections 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

19

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 eight-

strand ropes.

20

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

Fig. 2.3: Types of synthetic-fiber rope constructions (Leech, 1987)

Wire rope

21

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 sub-

rope. 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

22

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 load-

bearing 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.

23

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.

24

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

25

Fig. 3.2. Note that the components appear elliptical in cross-section due to the projection

of the helix angle onto the section shown.

Rope Core

Rope cross-section

asr

Sub-rope cross-section (level 1)

Sub-rope comp. helix radius: as

Strand cross-section(level 2)

rs

Strand radius :rs

Sub-rope core:Strand

Sub-rope helix radius: asr

Sub-rope

Fig. 3.2: Rope elements cross-section at different rope levels

as

(Sub-ropes)

Fig. 3.1: Hierarchy ranking of rope elements (Leech, 2002)

26

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 cross-

sections 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).

P

z

yx

O

u C

Fig. 3.3: Generic point P of a curve C

27

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

)cos()( φφ ax = (3.1a)

)sin()( φφ ay = : 0 ≤ φ ≤ pLπ2 (3.1b)

πφφ

2)( pz = (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

28

= 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:

paπθ 2)tan( = (3.1d)

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

A A

L

θ

p

Section A-A

Ra

Sub-rope

Fig. 3.4: Rope geometry

29

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:

∫

+

+

=

1

0

222)()()()(

φ

φφφ

φφ

φφφ

ddz

ddy

ddxs (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:

φπ

φ2

2

2)(

+=pas (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

dssds )()( ux1 = (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).

30

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

dssd

dssd

s)(

)(

)(1

1

2 x

x

x = (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

)(sdsd

21 xx κ= (3.6)

Fig. 3.5: Local coordinate system for a rope element

u(s)

x3

x2

x1 R

S N

z

y

x

31

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

dssd

dssd )()( ϕκ == 1x (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

22

2

)( aLaφ

φκ+

= (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).

32

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

)()( sds

sd2

3 xx ξ−= (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

)()( sds

sd2

3 xx−=ξ (3.10)

For the case of a helix curve, the torsion ξ is given by

22 )( aLLφ

φξ+

= (3.11)

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).

33

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

)()()( ssds

sd31

2 xxxξκ +−= (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:

−−=

)()()(

000

00

)(

)(

)(

sss

dssd

dssd

dssd

3

2

1

3

2

1

xxx

x

x

x

ξξκ

κ (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 three-

dimensional 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

34

curvature, when a point moves along C with constant velocity 1, is given by the

expression

31 xxd κξ += (3.14)

where vector d is also called the vector of Darboux, whose absolute value |d| is the

angular velocity of rotation”.

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

)tan(θξκ= (3.15)

where θ is the helix angle (Fig. 3.4).

B

|d|

d

Fig. 3.6: Kinematic of a rigid body B

35

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)

1)(

)(2

0020

22−

+

+=

aLaL

erφφ

ε (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.

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

L

φa

Centroidalaxis of rope

comp.

θ

Core axis

Fig. 3.7: Developed view of a helical rope element

36

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):

5

5

4

4

3

3

2

21

+

+

+

+

=

bbbbbb

aaaaaεε

εε

εε

εε

εε

σσ (3.17a)

5

5

4

4

3

3

2

21

+

+

+

+

=

bbbbbb

bbbbbγγ

γγ

γγ

γγ

γγ

ττ (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).

37

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

Sub-rope

Packing Geometry Wedging Geometry

Sub-rope (sector)

Fig. 3.8: Rope cross-sectional modeling

38

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

0rr = (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

er

rrε+

=1

0 (3.18b)

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

)1(0 errr νε−= (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.

39

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).

40

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.

z

y

x w1, m1

w3, m3 w2, m2

V3, M3

V2, M2 V1, M1

x3

x2

x1

Center of Curvature

Fig. 3.9 Loads acting on a line circular helix element

41

Table 3.1 Direction Cosines

Direction cosine V1+dV1 V2+dV2 V3+dV3

x1 cos(α1’1) ≈ 1 -sin(α2’1) ≈ -κ3ds -sin(α3’1) ≈ -κ2ds

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

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

0233211 =+−+ dsVdsVdswdV κκ (3.19)

which becomes, upon normalizing by ds,

0233211 =+−+ κκ VVw

dsdV

(3.20)

Similarly, a summation of forces in the x2 and x3 directions yield

033122 =−++ ξκ VVw

dsdV

(3.21)

and

022133 =+−+ ξκ VVw

dsdV

(3.22)

42

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

0233211 =+−+ κκ MMm

dsdM

(3.23)

Similarly, a summation of moments about the x2 and x3 axes yield

(a)

(b)

x2

ds

V2+dV2

V1+dV1

x’2

x’1 x1

V2

V1

w2

V3

x’3

x’1

x3

x1

w3

V1 V1+ dV1

V3+ dV3

ds

Fig. 3.10: Centerline of a line element looking down (no moments) (a) x3 axis and (b) x2 axis

43

0333122 =−−++ VMMm

dsdM

ξκ (3.24)

and

0222133 =++−+ VMMm

dsdM

ξκ (3.25)

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).

x2

ds

M2+dM2

M1+dM1

x’2

x’1 x1

M2

M1

m3

M3

x’3

x’1

x3

x1

m2

M1 M1+ dM1

M3+ dM3

ds

Fig. 3.11: Centerline of a line element looking down (no forces): (a) x3 axis and (b) x2 axis

(a)

(b)

44

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

0=++− XTV eκξ (3.26)

0=−+− VMM tebe κξ (3.27)

03121 ==== wwVm (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:

θθ sincos VTT es += (3.29)

θθ sincos tebebs MMM −= (3.30)

45

aVTMMM ebetets )cossin(sincos θθθθ −++= (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:

∫=A

e dAT )(εσ (3.32)

∫=A

be dAM )(εζσ (3.33)

∫=A

te dAM )(γρτ (3.34)

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

46

of a rope element and belong to the interval [-r, r], where r is the radius of the cross-

section of the rope element as shown in Fig. 3.12.

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:

)( 0κκζεε −+= er (3.35)

)( 0ξξργ −= (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.

x2

x3 ζ

2r

dA

ρx3

x2

2r

dA

(a) (b)

Fig. 3.12: (a) Cross-section for bending moment; (b) Cross-section for torsion

47

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’ cross-

sections. 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

b

nca pff )(µ+= (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.

48

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

ψθ sincos2Xpnc = (3.38)

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).

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.

2β

pc pcX

ψsin2Xpc = βπψ −=

2

pc

pnc

rope element longitudinal axis

Fig. 3.13: Normal force acting on a rope element (wedging geometry)

Wedging Geometry

x1 x3

θ

49

The vertical (Vf) and horizontal (Hf) contribution of the frictional force f to the

rope force resultants are given by

θsin2 dfVf = (3.39)

and

dfH f −= (3.40)

where d = 2asinψ, a is the helix radius and θ is the helix angle of the element,

respectively.

rope longitudinal axis

rope elementlongitudinal axis

A A

pnc

f

d

f

f f

A AVf

Hf

θtand ff

θ

Just frictional forces

θ pnc

View A-A

Fig. 3.14: Effect of frictional forces on rope force resultants

50

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

51

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

321 1321

βββ

ααεεε

α

+

+

−+= ∑

i ife

fa

b

tmI NW

WDD (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

52

[ ] kkkkfa

kfa slipffWW ∆++= −− 11

21 (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

)sin()sin(2)sin()sin(2 111

kkkkk

kk aa

llslip θψθψ −=∆ −−− (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

100 1i

ibbfe i

alAW εσ (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

k

icNi e

−

=∆ 0σσ (3.45)

53

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

),,( tUVsFDei = (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

),,,( tttFD fcccr σ= (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.

54

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

De −=

1σσ (3.48)

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:

,...),( 2εεσ fe = (3.49a)

,...),( 2εεσ g= (3.49b)

55

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:

,...),()1(,...),( 22 εεεε fDg −= (3.50a)

or

,...),(,...),(1 2

2

εεεε

fgD −=

(3.50b)

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

σ

σ

ε

σ

σ

σe

σe

VirginD

Material Models Virgin Damaged Equivalent virgin (virtual) ε = F(σ,σ2,…) ε = G(σ, σ2,…) ε = F(σe, σe

2,…)

Fig. 3.15: Principle of strain equivalence (Lemaitre and Chaboche,1990)

56

(Rungamornrat et al., 2002). An equivalent expression for Eq. 3.50b, based on Eqs. 3.49a

and 3.49b, is given by

e

Dσσ

−=1 (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

)()(

1εε

sv

sd

EE

D −= (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.

57

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

1

1

β

εεε

α

−+=

b

tmIDD (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

58

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.

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

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Predicted Response.Exp. Data

Fig. 4.1: Response of a one-level, two-layer rope under monotonic strain history and cross-section and geometric parameters of the rope

Radius of rope yarn (mm): 1.016Pitch of rope yarn (mm) : 81.28 Rope length (mm) : 609.6

Rope Yarn

Rope Cross-Section

59

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

60

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

61

compute the evolution of D throughout the strain history is derived using Eq. 3.48b (or

Eq. 3.48c) and is given by

H

dD tk

k∑≥=

ε

εε

εε

)()( (4.2)

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

62

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.

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

Fig. 4.2: Attempt to describe damage evolution based on Eq. 3.48b for different values of the threshold strain εt

0.05

0.07

0.09

0.11

0.13

0 0.2 0.4 0.6 0.8( ε - ε t )/ ε b

Dam

age

Eq. 3.48bεb = 0.124εt = 0.03

0.03

0.05

0.07

0.09

0.11

0 0.2 0.4 0.6( ε - ε t )/ ε b

Dam

age

Eq. 3.48bεb = 0.124εt = 0.05

63

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).

0

0.03

0.06

0.09

0.12

0 0.2 0.4 0.6 0.8 1η = (ε -ε t)/ε b

Dam

age

Eq. 4.2

Fitted curve (Eq. 4.1)

D = 0.1099η 0.5988

R2 = 0.9669ε t = 0.01

(a)(a)

64

0

0.03

0.06

0.09

0.12

0 0.2 0.4 0.6 0.8η = (ε -ε t)/ε b

Dam

age

Eq. 4.2

Fitted curve (Eq. 4.1)

D = 0.1154η 0.8412

R2 = 0.9934ε t = 0.03

0

0.03

0.06

0.09

0.12

0 0.2 0.4 0.6 0.8

η = (ε −ε t) /ε b

Dam

age

Eq. 4.2

Fitted curve (Eq. 4.1)

D = 0.1515η 0.9709

R2 = 0.9811ε t = 0.05

Fig. 4.3: (a), (b), (c) Damage index evolution for different values of εt and εb = 0.124

(c)

(b)

65

4.2.2 Strain Energy Deviation

The density of elastic energy ψe of a virgin system can be defined as

∫=ε

ρρεψ0

)()( dfe (4.3)

Fig. 4.4: Variation of the damage parameters α1 and β for different values of εt

0.06

0.08

0.1

0.12

0.14

0.16

0 0.02 0.04 0.06ε t

α 1

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06ε t

β 1

66

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

ρρεψ0

)()( dgd (4.4)

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

ρρρεψ0

)())(1()( dfFd (4.5)

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

ρρρε0

)()()( dfFSED (4.6)

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

67

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.

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

a

b

a

dxxqpdxxqxp )()()()( ζ (4.7)

σ

ε

Fig. 4.5: Strain Energy Deviation (SED) definition

―― f (ε): Virgin System ------ g (ε): Damaged System | | | | | SED (ε)

68

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:

∫= ε

ε

ε

ρρ

εζ

t

df

SEDF)(

)()( (4.8)

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

∫

∫=

b

t

t

df

df

Lε

ε

ε

εε

ρρ

ρρ

)(

)(

(4.9)

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

∫=

b

t

dfL

SEDF ε

εε

ε

ρρ

εζ)(

)()( (4.10)

69

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 energy-

based 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.

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8(ε - ε t )/ε b

Eq.

(4.1

0)

Fig. 4.6: Attempt to describe damage evolution based on energy-based method

εb = 0.124 εt = 0.03

εt = 0.05

70

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:

3

3

2

21

1

−+

−+

−+=

b

tm

b

tm

b

tmIFF

εεε

ωε

εεω

εεε

ω (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.

71

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)

Fig. 4.7: Damage index evolution for different values of εt and εb = 0.124 using the energy-based method

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1η = ( ε - ε t )/ ε b

Dam

age

(F)

F = 0.047η – 0.02η2 + 0.083 η3

F = 0.05η + 0.019η2 + 0.088 η3

F = 0.072η + 0.056η2 + 0.107 η3

εt = 0.03

εt = 0.05

εt = 0.01

72

0)()( 1112

2

11 =+− −βηβα

ηη

ηηθ

ddD

dDd

(4.12)

and

032)()( 23212

2

2 =+++− ηωηωωηη

ηηθ

ddD

dDd

(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 at η = ηu (4.14b)

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

L+++= )()()()( 22

10 ηθηθηη DDDD (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

73

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 one-

term 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

+−−++=−

111 )1()( 11θ

ηηββ ηαηαη

u

eDDD IuI

)( 111 111

1111

θηη

ββ ηβαηβαθu

eu

−−− − (4.16)

74

and the one-term composite expansion of Eq. 4.13 including a non-zero value of the

initial damage DI has the following form

[ ] 2)(1)( 33

221

33

221

θηη

ηωηωηωηωηωηωηu

eDDD uuuII

−

+++−++++= (4.17)

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)

)ln(tolub

iηη

θ−

≤ for i = 1, 2 (4.18)

Fig. 4.8: Overlap domain

y(t)

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2

Boundary Layer Outer Region

t

Overlap Domain

Inner Expansion

Outer Expansion

75

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

76

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).

Fig. 4.9: One-level rope response using energy-based damage model

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Exp. Data

Degradation Models

Jumps due to core and rope yarns failure

Fig. 4.10: One-level rope response using secant moduli ratio damage model

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Exp. Data

Degradation Models

Jumps due to core and rope yarns failure

77

Table 4.1: Summary of values of damage parameters and predictions of the damage

models: one-level rope

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.

Power Law Damage Evolution Polynomial Damage Evolution εt α1 β1 Max

Load [kips]

Strain at max Load

ω1 ω2 ω3 Max Load [kips]

Strain at max 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

78

Fig. 4.11: Response of a one-level rope under monotonic strain history: experimental, virgin response and damaged simulation curves

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Exp. Data

Energy-based Model

Secant moduli ratio Model

εt = 0.05 Jumps due to core and rope yarns failure

Strand 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

Rope Yarn

Fig. 4.12: Cross-section and geometrical properties of a two-level, one- layer rope.

79

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).

0

5

10

15

20

0 0.05 0.1 0.15

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Exp. Data

Secant moduli ratio Model

Energy-based Model

Failure

εt = 0.04

Fig.4.13: Response of a two-level rope under monotonic strain history

80

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.

Fig. 4.14: Response of a two-level rope under monotonic strain history

0

5

10

15

20

0 0.05 0.1 0.15

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Exp. Data

Secant moduli ratio Model

Energy-based Model

εt = 0.05 Failure

81

Table 4.2: Summary of values of damage parameters and predictions of the damage

models: two-level rope

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.

Power Law Damage Evolution Polynomial Damage Evolution εt α1 β1 Max

Load [kips]

Strain at max Load

ω1 ω2 ω3 Max Load [kips]

Strain at max 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

82

(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 load-

deformation curves.

Fig. 4.15: Predicted responses of one-level initially damaged rope (33%)

: Cut Rope Yarn

0

1

2

3

4

5

6

0.00 0.03 0.06 0.09 0.12 0.15

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope ResponseNet Area Effect & Degradation ModelDegradation ModelNet Area Effect & No Degradation ModelExp. Data 1Exp. Data 2

Rope Cross-Section

83

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).

Fig. 4.16: Predicted responses of one-level initially damaged rope (11%)

: Cut Rope Yarn

0

1

2

3

4

5

6

0.00 0.03 0.06 0.09 0.12 0.15

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Virgin Rope Response

Net Area Effect & Degradation Models

Degradation Model

Net Area Effect & No Degradation Model

Exp. Data 1

Exp. Data 2

Rope Cross-Section

84

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

85

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.

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 cross-

section is prevented from rotating.

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

(a) (b)

86

• 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).

87

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.

0

2

4

6

8

10

12

14

16

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Undamaged Rope SimulationRope Simulation: Degradation ModelExp. DataFinite Element Analysis

Fig. 4.18: Two-level rope response predictions, including finite element analysis, and experimental data

88

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.

(a) (b)

Undeformed Rope Geometry

Deformed Rope Geometry

Fig. 4.19: Finite element model: (a) undeformed rope geometry and (b) undeformed-deformed rope geometry

Rope Axial Deformation

89

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

Axial Stress (Kips/mm2)

Fig. 4.20: Axial stress distribution at a rope axial strain of 12%

90

(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.

91

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

92

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.

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

Rope element failure

Damage on rope cross-section

Fig. 5.1: Recovery length concept of a broken rope element

93

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 inter-

wire 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

94

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.

95

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.

96

However, there exists an inter-element type of contact between rope elements in the same

layer for ropes with wedging geometry.

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

gi-1,i

Xi

gi+1,i

pnc,ipnc,i Xi gi+1,i

pnc,i

gi-1,i

Fig. 5.2: Pattern of inter-layer/inter-element contact forces in an axially loaded rope element

97

b

a Nff )(µ+= (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

ii dfxT0

)()( ρρ (5.2)

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.

2βi

pnc,i pnc,i Xi

gi-1,i

Xi

gi+1,i

Packing geometry Wedging geometry

Fig. 5.3: Contact forces in axially loaded rope element with packing and wedging geometry

98

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)

iiiii ggN ,1,1 −+ += (5.3a)

iiiii Xgg += +− ,1,1 (5.3b)

iiii XgN += + ,10.2 (5.3c)

where the value of the line contact force gi+1,i can be estimated as (LeClair, 1991)

j

j

ik k

kml

ij jji

iijii X

pnpn

g

= ∏∑

−

+=+=+

1

11,1 cos

coscoscos

αα

αα

(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 lα and lα are defined as ( )( )lll rap −− π2/tan 1 and

( )( )1tan / 2π− +l l lp a r , 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

Ti+ dTi Ti

fi(ρ)

Fig. 5.4: Gradual increase in tension load Ti in a broken rope element

99

iii TX κ= (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 one-

level rope with packing geometry is given by

[ ] ρρκµ dTgxfxTbx

iiiiai ∫ ++= +0

,1 )(0.2)( (5.6)

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

xgn

fxTb

ac

+=

2

2,12

cos)(

θµ (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

100

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.

2 nd rope level-3 layers 3 rd rope level-4 layers

1st rope level-5 layers

Fig.5.5: Three-level packing geometry rope and normal forces acting on the rope elements at each rope level

13,2g

13,4g

13X

22,1g

22,3g

[ ]2

3,2X

32,1g

32,3g

[ ]3

3,2,2X

101

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

( ) 13,2

13,4

13 ggN += (5.8a)

( ) 13

13,4

13 0.2 XgN += (5.8b)

( ) [ ] [ ] [ ]3

3,2,22

3,23

3,2,213

13

11

4

5

4311

3

13

13

113 coscos

coscos

coscos

0.2 TXXpnpn

N j

j

k k

k

j jj

j θθκαα

αα

++

= ∏∑

−

==

(5.8c)

where 13X 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; 13κ is the curvature of an element that is

in layer three in the first level of the rope; [ ]3

3,2,2θ and [ ]2

3,2θ 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 [ ]3

3,2,2T 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

[ ]( ) 22,1

22,3

23,2 ggN += (5.9a)

[ ]( ) [ ]2

3,22

2,32

3,2 0.2 XgN += (5.9b)

[ ]( ) [ ] [ ] [ ]

[ ] [ ] [ ][ ]

[ ] [ ] [ ]

[ ] [ ] [ ]( ) [ ]

[ ][ ]

23,22

3,3

23,3

2132

3,32

3,32

3,2

23,2

23,2

23,32

3,323,3

23,3

23,2

23,2

23,2

23,32

3,2

coscoscos

0.2coscos

0.2 Xn

Npnpn

Xpnpn

N ++=θ

αα

αα

[ ] [ ] [ ]3

3,2,23

3,2,22

3,2 cosθκ T+ (5.9c)

102

where the expression for 22,3g 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 ( )13N in Eq. 5.9c, whose expression is given by Eq. 5.8c. The

normal line force ( )13N 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 [ ]( )23,3Y . The expression to estimate the value of [ ]( )2

3,3Y is given by

[ ][ ]

[ ]2

3,3

23,3

213

23,3

cosn

NYθ

= (5.9d)

where [ ]2

3,3θ 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 [ ]2

3,3n 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

[ ]( )23,2N is given by

[ ]( ) [ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ]

⋅++=

13

23,3

23,3

2

23,3

23,3

23,2

23,2

23,2

23,32

3,22

3,323,3

23,3

23,2

23,2

23,2

23,32

3,2

coscoscos

0.2coscos

0.2 Xnpn

pnXX

pnpn

Nθ

αα

αα

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ]

⋅⋅⋅+

+ ∏∑

−

==

132

3,3

23,3

2

23,3

23,3

23,2

23,2

23,2

23,31

1

4

5

4311

3

13

13

1 coscoscos

0.2coscos

coscos

0.2 κθ

αα

αα

αα

npnpn

Xpnpn

j

j

k k

k

j jj

j

[ ] [ ] [ ] [ ][ ]

33,2,2

33,2,2

23,2

23,2

33,2,2 coscoscos

T⋅

+⋅ θκθθ (5.9e)

103

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

[ ]( ) 32,1

32,3

33,2,2 ggN += (5.10a)

[ ]( ) [ ]3

3,2,23

2,33

3,2,2 0.2 XgN += (5.10b)

[ ]( ) [ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ][ ]

[ ] [ ] [ ]

[ ] [ ] [ ]⋅+

= ∏∑

−

==3

3,2,43

3,2,43

3,2,2

33,2,2

33,2,2

33,2,43

3,2,

1

33

3,2,

33,2,

4

33

3,2,3

3,2,3

3,2,2

33,2,2

33,2,2

33,2,3

3,2,2 coscos

0.2coscos

coscos

0.2αα

αα

αα

pnpn

Xpnpn

N j

j

q q

q

j jj

j

[ ]

[ ][ ]( ) [ ]

[ ][ ] [ ]

33,2,2

33,2,23

3,2,4

33,2,4

22

3,233,2,3

33,2,3 cos

coscos

Tn

N κθ

αα

+

(5.10c)

where, as in the analysis of the element in the second rope level, the expression for 3

2,3g 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 [ ]( )23,2N

(which also includes the term ( )13N ) in Eq. 5.10c, whose expression is given by Eq. 5.9c.

The normal line force [ ]( )23,2N 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 [ ]( )33,2,4Y . The expression to estimate the value of [ ]( )3

3,2,4Y is given by

[ ] [ ][ ]

[ ]2

3,2,4

33,2,4

22

3,23

3,2,4

cosn

NYθ

= (5.10d)

where [ ]3

3,2,4θ 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

104

layers, respectively; and [ ]3

3,2,4n 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 [ ]( )33,2,2N is given

by

[ ]( ) [ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ][ ]

[ ] [ ] [ ]

[ ] [ ] [ ]⋅+

= ∏∑

−

==3

3,2,43

3,2,43

3,2,2

33,2,2

33,2,2

33,2,43

3,2,

1

33

3,2,

33,2,

4

33

3,2,3

3,2,3

3,2,2

33,2,2

33,2,2

33,2,3

3,2,2 coscos

0.2coscos

coscos

0.2αα

αα

αα

pnpn

Xpnpn

N j

j

q q

q

j jj

j

[ ]

[ ]

[ ]

[ ]

[ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

⋅++⋅

⋅ 2

3,32

3,32

3,2

23,2

23,2

23,32

3,22

3,323,3

23,3

23,2

23,2

23,2

23,3

33,2,4

33,2,4

2

33,2,3

33,2,3

coscos

0.2coscos

0.2cos

coscos

αα

ααθ

αα

pnpn

XXpnpn

n

[ ]

[ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

⋅+

+ ∏∑

−

==3

3,2,43

3,2,43

3,2,2

33,2,2

33,2,2

33,2,41

1

4

5

4311

3

13

13

1132

3,3

23,3

2

coscos

0.2coscos

coscos

0.2cos

αα

αα

ααθ

pnpn

Xpnpn

Xn j

j

k k

k

j jj

j

[ ]

[ ]

[ ]

[ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

[ ][ ] [ ] +

⋅⋅⋅

2

3,23

3,2,2132

3,3

23,3

2

23,3

23,3

23,2

23,2

23,2

23,3

33,2,4

33,2,4

2

33,2,3

33,2,3 coscos

coscoscos

0.2cos

coscos

θθκθ

ααθ

αα

npnpn

n

[ ] [ ][ ] [ ]

33,2,2

33,2,2

33,2,2

23,2 cos

T

+

κ

θκ (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 [ ]3

3,2,2T 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

105

tti

t

s

tti

t

sq

qqqI

sssI

tti

tti ThAN )(,

1

1)(,

1)(),()(),()(,)(, JJJJJJ

+

+= ∑ ∏

−

= +=

κκ (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; sssI )(),( Jκ 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 ttiA )(,J ,

whose value does not depend on the current value of the tensile load ttiT )(,J of the element,

is given by the following recursive relation:

[ ] ttml

ttmlt

ttIt

ttIt

tit

tit

ti nWACBA

)(,

)(,2

1)1(),1(

1)1(),1()(,)(,)(,

cos

J

JJJJJJ

θ−−−

−−− ++= t ≥ 2 (5.12)

where the values of the constants 1iA , t

tiW )(,J , ttiB )(,J , and t

tiC )(,J depend on the position of

the layer i in level t: if the element belongs to the outermost layer, i = ttml )(,J ; if the

element belongs to the first layer, i = 1; and if the element belongs to an intermediate

layer, 2)(, ≥> iml ttJ . Therefore, the expressions to compute the previously mentioned

constants are:

[ ]

=⇔+

≥⇔=

+

1cos

20.2

121

12,31

2

12

1,1

1

maxiXg

n

igA

t

ss

i δθ

(5.13a)

106

[ ]

=⇔

≥⇔=

1)(cos

2)(

)(,2)(,2

)(,2

)(),(

)(),(tIX

ntIX

Wt

ttt

tt

tttI

tttI

JJ

J

J

J

θ

(5.13b)

=⇔

+

≥⇔

=

∏∑

∏∑

−

==

−

+=+=

1coscos

coscos

cos

2coscos

coscos

0.2

)(,2)(,

1

3 )(,

)(,

3 )(,)(,)(,2

)(,2)(,2)(,

)(,2

)(,2

)(,

1

1 )(,

)(,

1 )(,)(,)(,

)(,)(,)(,

)(,

max

)(,

)(,

iXXpnpnn

iXpnpn

Bt

tttt

ts

s

qt

tq

ttq

ml

st

tst

tst

t

tt

tt

tts

tt

tt

tts

s

iqt

tq

ttq

ml

ist

tst

tst

ti

tti

tti

tts

tti t

t

tt

JJJ

J

JJJ

JJJ

J

J

JJ

J

JJJ

JJJ

JJ

J

δαα

αα

θ

αα

αα

(5.13c)

=⇔

≥>⇔

=⇔

=

∏

∏

−

=

−

+=

1coscos

coscos

cos

2coscos

coscos

0.2

0.2

1

3 )(,

)(,

)(,)(,)(,2

)(,2)(,2)(,

)(,2

)(,2

)(,

1

1 )(,

)(,

)(,)(,)(,

)(,)(,)(,

)(,

)(,

)(,

)(,

ipnpnn

imlpn

pn

mli

C

tt

tt

ml

qt

tq

ttq

ttml

ttml

tt

tt

tt

ttml

tt

tt

tt

ml

iqt

tq

ttq

ttml

ttml

tti

tti

tti

ttml

tt

tti

J

J

J

J

JJJ

JJJ

J

J

JJ

J

JJJ

JJJ

J

J

αα

αα

θ

αα

αα

(5.13d)

where tmax is the level where the broken element is located and max1tδ and

maxttδ are the

Kronecker delta functions. The expression for the constants qqqIh )(),( J 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:

=⇔

≥>⇔

⋅

⋅

=⇔

=

∏

∏

−

=

−

+=

1)(cos

coscoscos

coscos

cos

2)(cos

coscoscos

coscos

0.2

)(cos

0.2

)(,

)(,2

)(),(

1

3 )(,

)(,

)(,)(,)(,2

)(,2)(,2)(,

)(,2

)(,2

)(,)(,

)(,2

)(),(

1

1)( )(,

)(,

)(,)(,)(),(

)(),()(),()(,

)(,)(,

)(,3

)(),(

)(,

)(,

qInpn

pnn

qImln

pnpn

mlqIn

h

qqml

qqmlq

qqI

ml

sq

qs

qqs

qqml

qqml

qq

qq

qq

qqml

qq

qq

qqq

qml

qqml

qqqI

ml

qIsq

qs

qqs

qqml

qqml

qqqI

qqqI

qqqI

qqml

qqq

qml

qqml

qqqI

qq

qq

J

JJ

J

J

JJJ

JJJ

J

J

JJ

J

JJ

J

JJJ

JJJ

JJ

J

J

J

J

θθ

αα

αα

θ

θ

θαα

αα

θ

(5.13e)

107

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 ttiT )(,J 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), is given by

ρρκκµ dThAxfxTb

xt

ti

t

s

tti

t

sq

qqqI

sssI

ttia

tti ∫ ∑ ∏

+

++=

−

= +=0)(,

1

1)(,

1)(),()(),()(,)(, )()( JJJJJJ (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)

ii

iinc

Xp

ψθ sincos0.2, = (5.15a)

Therefore, the normal line force Ni exerted on a rope element located in layer i is

given by

inci pN ,0.2= (5.15b)

ii

ii

XNθψ cossin

= (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

108

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

ρθψ

ρκµ d

TxfxT

bx

ii

iiai ∫

+=

0 cossin)(

)( (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

109

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”.

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

1st rope level-2 layers 2 nd rope level-3 layers 3 rd rope level-4 layers

12X

12,ncp1

2,ncp

122β

[ ]2

2,3X3

4,ncp23,ncp

[ ]2

2,32β

34,ncp

23,ncp

[ ]3

2,3,42β

[ ]3

2,3,4X

Fig. 5.6: Three-level wedging geometry rope and normal forces acting on the rope elements at each rope level

[ ]2

2,3Y[ ]

32,3,4Y

110

12

12

121

2, sincos0.2 ψθXpnc = (5.17a)

( ) 12,

12 0.2 ncpN = (5.17b)

( ) [ ] [ ] [ ]12

12

22,3

32,3,4

32,3,4

12

121

2 cossincoscos

θψθθκ TX

N+

= (5.17c)

where 12X 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; 12κ is the curvature of an element that is

in layer two in the first level of the rope; [ ]3

2,3,4θ and [ ]2

2,3θ 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 [ ]3

2,3,4T 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

[ ] [ ]

[ ] [ ]2

2,32

2,3

22,3

22,32

3, sincos0.2 ψθYX

pnc

+= (5.18a)

[ ][ ]

[ ]2

2,3

22,3

212

22,3

cosn

NYθ

= (5.18b)

[ ]( ) 23,

22,3 0.2 ncpN = (5.18c)

[ ]( ) [ ]

[ ] [ ] [ ] [ ]

[ ]

[ ]+

+= 2

2,3

22,3

2

22,3

22,3

12

12

12

22,3

22,3

22,32

2,3

cos)cos)(sincos(sincossin n

XXNθ

θψθψθψ

111

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ] [ ]

[ ] [ ][ ]

32,3,42

2,32

2,3

32,3,4

22,3

22,3

22,3

2

22,3

22,3

12

12

32,3,4

22,3

12

)cos(sincoscos

)cos)(sincos(sincoscos

Tn

+

θψθκθ

θψθψθθκ

(5.18d)

where [ ]2

2,3Y (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

[ ] [ ]

[ ] [ ]3

2,3,43

2,3,4

32,3,4

32,3,43

4, sincos0.2 ψθYX

pnc

+= (5.19a)

[ ] [ ][ ]

[ ]2

2,3,4

32,3,4

22

2,33

2,3,4

cosn

NYθ

= (5.19b)

[ ]( ) 34,

32,3,4 0.2 ncpN = (5.19c)

[ ]( ) [ ]

[ ] [ ] [ ] [ ]

[ ]

[ ]+

= 3

2,3,4

32,3,4

2

32,3,4

32,3,4

22,3

22,3

22,33

2,3,4

cos)cos)(sincos(sin n

XNθ

θψθψ

[ ] [ ] [ ] [ ]

[ ]

[ ]

[ ]

[ ]+

3

2,3,4

32,3,4

2

22,3

22,3

2

32,3,4

32,3,4

22,3

22,3

12

12

12 coscos

)cos)(sincos)(sincos(sin nnX θθ

θψθψθψ

[ ] [ ]

[ ] [ ] [ ] [ ]

[ ]

[ ]

[ ]

[ ]+

3

2,3,4

32,3,4

2

22,3

22,3

2

32,3,4

32,3,4

22,3

22,3

12

12

32,3,4

22,3

12 coscos

)cos)(sincos)(sincos(sincoscos

nnθθ

θψθψθψθθκ

[ ] [ ]

[ ] [ ] [ ] [ ]

[ ]

[ ]

[ ]

[ ] [ ][ ]

32,3,43

2,3,43

2,3,4

32,3,4

32,3,4

32,3,4

2

32,3,4

32,3,4

22,3

22,3

32,3,4

22,3

)cos(sincos

)cos)(sincos(sincos

Tn

+

θψκθ

θψθψθκ

(5.19d)

112

where [ ]3

2,3,4Y (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 [ ]3

2,3,4T 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 [ ]mt ,,1L∈ and i = ml(t) for t ≥ 2, can be

computed as

tti

t

qt

qs

sssI

t

qs

sssml

qqqI

t

qt

qs

sssI

t

qs

sssml

qqqI

tti T

w

e

w

cXN )(,

1)(),(

1)(),(

)(),(

1

1)(),(

1)(),(

)(),()(, J

J

J

J

J

J

JJ

+

= ∑∏

∏∑

∏

∏=

=

+=−

=

=

+= κ (5.20)

where the constants, sssmlc )(),( J , s

ssIw )(),( J and sssmle )(),( J are computed from the following

expressions:

sssml

sssmls

ssml nc

)(),(

)(),(2

)(),(

cos

J

JJ

θ= for s ≥ 2 (5.21a)

sssI

sssI

sssIw )(),()(),()(),( cossin JJJ θψ= for s ≥ 1 (5.21b)

113

>⇔

≤⇔=

ts

tsne s

ssml

sssml

sssml

1

cos

)(),(

)(),(3

)(),( J

J

J

θ (5.21c)

The gradual increase in tension ttiT )(,J 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

ρρκµ dTw

e

w

cXxfxT

b

xt

ti

t

qt

qs

sssI

t

qs

sssml

qqqI

t

qt

qs

sssI

t

qs

sssml

qtqIa

tti ∫ ∑

∏

∏∑

∏

∏

+

+==

=

+=−

=

=

+=

0)(,

1)(),(

1)(),(

)(),(

1

1)(),(

1)(),(

)(),()(, )()( J

J

J

J

J

J

JJ (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

114

the third rope level has four layers. The rope elements under study at each rope level are

identified with an “X”.

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)

( ) [ ] [ ] [ ]3

3,3,42

3,33

3,3,413

13

11

4

5

4311

3

13

13

113 coscos

coscos

coscos

0.2 TXXpnpn

N j

j

k k

k

j jj

j θθκαα

αα

++

= ∏∑

−

==

(5.23)

where all the variables presented in Eq. 5.23, were defined following Eq. 5.8c.

1st rope level-5 layers

13,2g

13,4g

13X

3 rd rope level-4 layers

34,3g

[ ]3

3,3,4X

2 nd rope level-3 layers

[ ]2

3,3X2

3,ncp

[ ]2

3,32β

23,ncp

[ ]3

3,3,4Y

Fig.5.7: Three-level mixed geometry rope and normal forces acting on the rope elements at each rope level

[ ]2

3,3Y

115

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:

[ ]( )[ ] [ ]( )

[ ]

[ ]+

+

⋅= ∏∑

−

==

13

11

4

5

4311

3

13

13

1

23,3

23,3

2

23,3

23,3

23,3 cos

coscoscos

0.2cos

cossin1 XX

pnpn

nN j

j

k k

k

j jj

j

αα

ααθ

θψ

[ ]

[ ] [ ]( ) [ ] [ ][ ]

[ ][ ] [ ] [ ]

33,3,4

33,3,4

23,32

3,3

23,3

22

3,33

3,3,4132

3,32

3,3

23,3 cos

coscoscos

cossin1 T

nX

+⋅+

+θκ

θθθκ

θψ(5.24)

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:

[ ]( ) [ ]

[ ] [ ]( ) [ ]

[ ]

[ ]

+

+

⋅= ∏∑

−

==

13

11

4

5

4311

3

13

13

1

23,3

23,3

2

33,3,4

23,3

23,3

33,3,4

23

3,3,4 coscos

coscos

0.2cos

cossincos0.2

XXpnpn

nnN j

j

k k

k

j jj

j

αα

ααθ

θψθ

[ ] [ ]

[ ] [ ]( ) [ ][ ] [ ]

[ ]

[ ][ ] [ ]

+

+⋅+

3

3,3,42

3,323,3

23,3

22

3,33

3,3,4133

3,3,42

3,32

3,3

333,,4

223,3 cos

coscoscos

cossincos0.2

θκθ

θθκθψθ

nnX

[ ][ ]

33,3,4

333,4 T

κ (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 333,4T

of the element under investigation. Therefore, the normal force exerted on a component

116

of a rope element with multi-level hierarchical structure and mixed cross-section

geometry can be described as

tti

t

s

tttI

tti

t

sq

qqqI

sssI

tti

ttmi TvvMN )(,

1

1)(),()(,

1

1)(),()(),()(,)(, JJJJJJJ

+

+= ∑ ∏

−

= +=

− κκ (5.26)

where ttiM )(,J represents the line normal forces independent of the value of the tensile

load ttiT )(,J 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 qqqIv )(),( J , for q = 1,..., t-1, are given by the

following expression:

1)(),1(1

)1(,

1)1(,

2

1)(),( coscos

21 +++

+

++

+

= q

qqIqqml

qqml

qqq

qqI nccv J

J

JJ θ

θ (5.27)

where the values of the coefficients c1q and c2q+1 are given in Table 5.1 along with the

values of tttIv )(),( J .

For clarity of the expressions presented in Table 5.1, the constants [ ] 1)1(),1(

+++

qqqIpk J and

qqqIw )(),( J are defined as follows:

[ ]

= ∏

−

++= +

++

++

++

+++

+++

+++

+++

++

++ 1

1)1( )1(,

1)1(,

1)1(,

1)1(,

1)1(),1(

1)1(),1(

1)1(),1(

1)1(,1

)1(),1(

1)1(,

coscos

coscos

0.2q

qml

qIsq

qs

qqs

qqml

qqml

qqqI

qqqI

qqqI

qqmlq

qqI pnpn

pkJ

J

J

JJJ

JJJJ α

ααα

(5.28a)

qqqI

qqqI

qqqIw

)(),()(),()(),( cossin

1

JJJ θψ

= (5.28b)

117

The gradual increase in tension ttiT )(,J 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

ρρκκµ dTvvMxfxTb

xt

ti

t

s

tttI

tti

t

sq

qqqI

sssI

ttia

tti ∫ ∑ ∏

+

++=

−

= +=

−

0)(,

1

1)(),()(,

1

1)(),()(),()(,)(, )()( JJJJJJJ (5.29)

Table 5.1 Coefficients to compute the normal force exerted on a component of a

rope element with mixed cross-section geometry

5.3 CALCULATION OF THE RECOVERY LENGTH

In the previous section, the expressions for the gradual increase in tension ttiT )(,J

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

Packing Geometry Wedging Geometry

Interior Layers Outermost Layer

For level t (tmax) t

ttIv )(),( J 1.0 1.0 1/ tttIw )(),( J

If (q +1) = tmax

c1q 1.0 1.0 1/ qqqIw )(),( J

c2q+1 [ ] 1)1(),1(

+++

qqqIpk J 2.0 1/ 1

)1(),1(+

++q

qqIw J

If (q +1) < tmax

c1q 1.0 1.0 1/ qqqIw )(),( J

c2q+1 [ ] 1)1(),1(

+++

qqqIpk J 2.0 1.0

118

rope element for a known value of the rope axial load, which is used to establish the

gradual increase in tension load ttiT )(,J 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 ttiT )(,J . Thus, a general expression for the gradual increase in tension t

tiT )(,J of a

broken rope element has the following form:

[ ] ρρµ dTCCxfxTbx

tti

tti

ttia

tti ∫ ++=

0)(,)(,2)(,1)(, )()( JJJJ (5.30)

where the coefficients ttiC )(,1 J and t

tiC )(,2 J are constants, and Eq. 5.30 represents a

nonlinear integral equation for the variable ttiT )(,J . As previously mentioned, if the value

of ttiT )(,J 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 ttiT )(,J . 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 ttiT )(,J can be obtained from Eq. 5.30.

5.3.1 Numerical Algorithm

To simplify the algorithm description, let ttiT )(,J (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:

119

jT

jj

GxQQ

dxxdT

=∆−

= 12)( (5.31a)

where jQ2 and jQ1 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:

[ ] 12121 )(21)(

1

RxPPdTCC jjx

x

bj

j

=∆+≈+∫−

ρρ (5.31b)

where jP2 and jP1 are the values of the integrand at the ends of the interval j.

• For any interval j, Eq. 5.30 can be written as

11 RxfTT ajj µ+∆+= − (5.31c)

where Tj-1 = jQ1 .

• For the first interval (i.e., j = 1), the following may initially be assumed:

x0 = 0; 11Q = 0; 1

1P = bC1 ; x1 = ∆x; 12P = bC1 (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 12Q = T1, the value of

12P is computed to update the values of R1 (Eq. 5.31b), 1

TG (Eq. 5.31a), and T1 (Eq.

5.31c). Finally, the value of 12Q is set equal to T1 to compute 1

2P . The iteration process

is repeated until the calculated values of T1 and 12Q 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 22Q and 2

1Q can be

assumed:

120

12

21 QQ = (5.31e)

xGQQ T∆+= 121

22 (5.31f)

• Using the values of 22Q and 2

1Q , the value of R1 is computed using Eq. 5.31b, and the

value of T2 is computed from Eq. 5.31c. After setting 22Q = T2, 2

TG 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 22Q 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

xjrl ∆= (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 )()(, xT tti J of a broken rope element given by Eq.

5.30 reduces to

121

∫+=x

tti

tti

tti

tti dTCxCxT

0)(,)(,2)(,1)(, )()( ρρµ JJJJ (5.33)

where the coefficients ttiC )(,1 J and t

tiC )(,2 J are constants whose values depend on the rope

cross-section modeling. In this equation, ttiC )(,1 J , in terms of the variables of Eq. 5.30,

equals µ ttiC )(,1 J + 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

tti dTx

0)(, )()( ρρλ J (5.34a)

Accordingly, the first order non-homogeneous differential equation associated

with Eq. 5.33 is given by

xCxCdx

xd tti

tti )(,1)(,2 )()(

JJ =− λµλ (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

[ ] xC

CeC

Cx tti

ttixC

tti

tti t

ti

)(,2

)(,12

)(,2

)(,1 1)( )(,2

J

J

J

J J

µµλ µ −

−= (5.34c)

for a value of the friction coefficient µ greater than zero. An equivalent relationship

between the functions λ(x) and )()(, xT tti J presented in Eq. 5.34a is given by

122

dxxdxT t

ti)()()(,

λ=J (5.34d)

Thus, using Eqs. 5.34c and 5.34d, the gradual increase in tension )()(, xT tti J of a broken

rope element has an exponential evolution given by

−= 1)( )(,2

)(,2

)(,1)(,

xCt

ti

ttit

ti

ttie

CCxT J

J

JJ

µ

µ x ≥ 0 (5.35)

If the value of the tensile load )()(, xT tti J 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

+= 1ln1

)(,1

)(,2

)(,2t

ti

tti

tti C

TC

Crl

J

J

J

µ

µ (5.36)

Therefore, based on the theory developed in Section 5.2, the nonlinear integral

equation that describes the gradual increase in tension )()(, xT tti J 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 two-

level 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

123

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 force-

strain 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

values are infinite because of the zero value of the constant ttiC )(,1 J (Eq. 5.33). The value

of the constant ttiC )(,1 J ( 1

21C 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

124

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

κ2 T2 (Eq. 5.5), making the value of the constant ttiC )(,1 J vanish.

It is important to emphasize that in Eq. 5.33, the constant ttiC )(,1 J 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 )()(, xT tti J 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

if the constant ttiC )(,1 J 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.

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

% Failure Axial Rope Strain

Rec

over

y L

engt

h/p

2

Cross-Section A

Cross-Section B

Cross-Section A

Cross-Section B

: Cut Rope Yarn

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

Radius of rope yarn (mm) : 1.016Pitch of rope yarn (p2) (mm) : 81.0 Failure rope strain : 0.124

125

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

the packing geometry assumption, the value of the constant ttiC )(,1 J ( 1

21C 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

tiC )(,1 J accounts for the effect of the normal forces acting on the core, the normalized

g2,1

g2,1

g2,1 g2,1

g2,1

g2,1 g2,1

g2,1

g2,1

g2,1g2,1

g2,1

g2,1

g2,1

g1,2

X2

Core cross-section A Core cross-section B Outer element cross-section A

Fig. 5.9: Normal forces acting on rope elements of a damaged one-level two-level rope assuming packing geometry

126

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 cross-

section 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).

127

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

(second layer) do not develop a finite recovery length value because the constant ttiC )(,1 J

(Eq. 5.33) has a value of zero. In this rope configuration (Fig. 5.10), the value of the

constant ttiC )(,1 J ( 2

1,21C and 21,11C in this example) does not vanish for any broken rope yarn

Rope diameter (mm) : 14Radius of rope yarn (mm) : 1.016 Pitch of rope yarn (p2) (mm) : 81.0 Failure rope strain : 0.126 Pitch of strand (p1) (mm) :150

Strand 2 Strand 1

Rope Yarn

Fig. 5.10: Recovery length values of broken strands of a two-level damaged rope as a function of the axial rope strain

: Cut Rope Yarn

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0

% Failure Axial Rope Strain

Rec

over

y L

engt

h/p

1

Core-Strand 2

6

8

10

12

14

0.0 0.2 0.4 0.6 0.8 1.0

% Failure Axial Rope Strain

Rec

over

y L

engt

h/p

2

Outer Rope Yarns-Strand 2

Outer Rope Yarns-Strand 1

128

(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

129

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).

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

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

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8x /p 1

T(x

)/Tmax

0.10.30.50.70.9

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14

x /p 2

T(x

)/Tmax

0.10.30.50.70.9

(a)

(b)

130

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

131

to estimate the contact forces exerted by the rope jacket on rope elements according to

rope cross-section modeling under consideration.

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

dTf 2

= (5.37)

where d is the rope diameter. The stress f would be the radial stress if the rope cross-

section were solid, i.e., without the voids between rope elements. If there are voids

Sub-rope Rope jacket

Fig. 5.12: Rope jacket and sub-ropes of a parallel rope construction (Berryman et al., 2002)

132

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

TddP c

=

32 (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.

PP

P

P

P

PCORE

P

P P

dc

cd3

2

f

Fig. 5.13: (a) Normal forces acting on the core; (b) compressive stress acting on the core

(a) (b)

30° 30°

133

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

mlP exerted by the rope jacket on every rope element of the outermost layer (ml) of the

rope is given by

( )12

1

322

2

coscoscos

cosg

pn

pnP

ml

k k

kml

mlmlml

=

∏−

= ααα

α (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 mlP exerted by the

rope jacket on every rope element of the outermost layer (ml) of the rope can be

estimated as

mlml n

TP π2= (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.

134

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

leads to a non-zero value of the constant ttiC )(,1 J (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

tiC )(,1 J (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

135

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.

Fig. 5.14: Effect of the rope jacket on the recovery length values: (a) outer rope yarn; (b) core

Rope Cross-Section

Rope diameter (mm) : 6.096Pitch rope yarn (p2) (mm) : 81.0 ×: Cut rope yarn

Rope Jacket

Rope Yarn

0

4

8

12

16

20

0 0.02 0.04 0.06

Jacket Thickness/Rope Diameter

Rec

over

y L

engt

h/p

2 Packing Geometry

Wedging Geometry

0.0

0.5

1.0

1.5

2.0

2.5

0 0.2 0.4 0.6 0.8 1% Failure Rope Axial Strain

Rec

over

y L

engt

h/p

2

No Rope Jacket

Rope Jacket

Rope Jacket

×: Cut rope yarn (core)

(a)

(b)Rope Cross-Section

136

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 “sub-

rope” element shown in Fig. 5.16, the element forces associated with the essential set are

T3 and M4.

137

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

LS2 LS1

u, φu, φ

LS2LS3 LS1

u, φ u, φ

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

(a)

(b)

T3, M4 T1, M2

Node 1 Node 2

u1, 2φ u3, 4φ

LSi

Fig. 5.16: Two-noded axial-torsional “sub-rope” element

138

entire rope. For each “sub-rope” element, a 2-by-2 stiffness matrix is computed having

the following form:

[ ]i

i kkkk

=

ζζζε

εζεεsrk (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:

[ ]

i

i GG

GG

∂∂

∂∂

∂∂

∂∂

=

ζε

ζε22

11

srk (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:

139

−−+

=∂∂

=δε

ζδεεζδεεεεε 2

),(),(1 TTGk (5.43a)

−−+

=∂∂

=δε

ζδεεζδεεεζε 2

),(),(2 MMGk (5.43b)

−−+=

∂∂

=δζ

δζζεδζζεζεζ 2

),(),(1 TTGk (5.43c)

−−+=

∂∂

=δζ

δζζεδζζεζζζ 2

),(),(2 MMGk (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-by-

2 “sub-rope” element stiffness matrix [ksr]i computed, [Ksr]i can be calculated as follows:

[ ] [ ] [ ][ ] [ ]

−

−=

ii

iii

srsr

srsrsr kk

kkK (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

140

(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

[ ]{ } { }PuK ddr = (5.42)

where Kr is the rope tangent stiffness, ud 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 one-

level 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.

141

Table 5.2 Summary of the models used to predict damaged rope behavior

Model Outline of the model

Strain Localization and

Degradation Model

(Including Net Area

Effect)

The recovery length of each broken rope element is computed.

If this value is admissible, the rope is discretized into “sub-

rope” elements, and the numerical procedure outlined in

Section 5.4 is used to analyze the rope. Degradation of each

“sub-rope” element’s properties is accounted for through use

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

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.

Theoretical Upper-

Bound Response

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

(Lower-Bound

Response)

The rope is considered initially undamaged and degradation of

each rope element’s properties is accounted for by using Eq.

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).

142

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

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

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Degradation Model

Theoretical Upper Bound Response

Strain Localization & Degradation Model (Packing Geometry)

Net Area Effect& Degradation Model (Wedging Geometry)

Net Area Effect--No Degradation Model

Exp. Data 1

Exp. Data 2

Rope Cross-Section (a)

Radius of rope yarn (mm) : 1.016Pitch of rope yarn (mm) : 81.0 Rope Length (mm) : 610.0

: Cut Rope Yarn

0

1

2

3

4

5

6

7

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Degradation Model

Theoretical Upper Bound Response

Strain Localization & Degradation Model (Packing Geometry)

Net Area Effect& Degradation Model (Wedging Geometry)

Net Area Effect--No Degradation Model

Exp. Data 1

Exp. Data 2

Rope Cross-Section (b)

: Cut Rope Yarn

143

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 cross-

section 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

144

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.

145

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 cross-

section 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.

146

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 cross-

sectional 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

Fig. 5.18: Initially damaged two-level rope cross-section and analyses of rope response

Damage inflicted at rope midspan

0

4

8

12

16

20

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data 1Exp. Data 2Exp. Data 3Strain Localization & Degradation ModelsNet Area Effect & Degradation ModelsTheoretical Upper Bound ResponseDegradation ModelNet Area Effect--No Degradation Model

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]

Damaged rope cross-section

Strand

Rope yarn

: Cut Rope Yarn

147

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 non-

uniform 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

148

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.

149

Cut Strand

Fig. 5.19: Damaged cross-section (10%) of DPJR 1

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

Fig. 5.20: Experimental data and predicted responses of DPJR 1

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (No Initial Damage)

Upper Bound Response

Net Area Effect & Degradation Models

Strain Concentration & Degradation Models(L/d = 1000) Degradation Model

Exp. Data (L/d = 40)

Exp. Data (L/d =290)

Exp. Data (L/d = 1000)

150

Fig. 5.21: Variation of the (a) failure strain and (b) failure load of the DPJR 1 with respect to L/d ratio values

85

90

95

100

105

0 200 400 600 800 1000 1200

L /d ratio

Dam

aged

/Und

amag

ed

Rop

e Fa

ilure

Str

ain

(%) Exp. Data

Numerical Model

80

85

90

95

100

105

110

0 200 400 600 800 1000 1200

L /d ratio

Dam

aged

/Und

amag

ed

Rop

e Fa

ilure

Loa

d (%

) Exp. DataNumerical Model

(a)

(b)

151

Cut Strand

Fig. 5.22: Damaged cross-section (10%) of DPJR 2

24 parallel unjacketed sub-ropes3 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

Fig. 5.23: Experimental data and predicted responses of DPJR 2

0

500

1000

1500

2000

2500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (1)(L/d = 40)

Epx. Data (2) (L/d = 40)

Upper Bound Response

Net Area Effect & Degradation Models (1)

Net Area Effect & Degradation Models (2)

Strain Localization & Degradation Models(1) (L/d = 40)Strain Localization & Degradation Models(2) (L/d = 40)

(1): 5% of initial damage (2): 10% of initial damage

152

Cut Strand

Rope jacket

18 parallel unjacketed sub-ropes3 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)

Fig. 5.24: Damaged cross-section (10%) of DPJR 3

Fig. 5.25: Experimental data and predicted responses of DPJR 3

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (No Initial Damage)

Upper Bound Response

Net Area Effect & Degradation Models

Strain Concentration & DegradationModels (L/d = 1000) Degradation Model

Exp. Data (L/d = 40)

Exp Data (L/d = 290)

Enp. Data (L/d = 1000)

153

Fig. 5.26: Variation of the (a) failure strain and (b) failure load of the DPJR 3 with respect to L/d ratio values

90

95

100

105

0 200 400 600 800 1000 1200L /d ratio

Dam

aged

/Und

amag

ed R

ope

Failu

re S

trai

n (%

)

Exp. DataNumerical Model

NS

M

NS

M: Strands cut near rope midspanNS: Strands cut near rope splice

75

80

85

90

95

0 200 400 600 800 1000 1200L /d ratio

Dam

aged

/Und

amag

edR

ope

Failu

re L

oad

(%)

Exp. DataNumerical Model

M

NS

M: Strands cut near rope midspanNS: Strands cut near rope splice

(a)

(b)

154

Cut Strand

Rope jacket

20 parallel unjacketed sub-ropes3 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)

Fig. 5.27: Damaged cross-section (5%) of DPJR 4

Fig. 5.28: Experimental data and predicted responses of DPJR 4

0

400

800

1200

1600

2000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (No Initial Damage)

Exp. Data (5% Initial Damage)

Exp. Data (10% Initial Damage)

Upper Bound Response

Net Area Effect & DegradationModelsStrain Concentration &Degradation Models (L/d = 40)Degradation Model

All the simulated damaged rope responses consider 10% of initial cross-section damage

155

Cut Strand

Rope jacket

10 parallel unjacketed sub-ropes3 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)

Fig. 5.29: Damaged cross-section (10%) of DPJR 5

0

20

40

60

80

100

120

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (No Initial Damage)

Upper Bound Response

Net Area Effect & DegradationModelsStrain Concentration &Degradation Models (L/d = 1000) Degradation Model

Exp. Data (L/d = 40)

Exp. Data (L/d = 560)

Exp. Data (L/d = 1000)

Fig. 5.30: Experimental data and predicted responses of DPJR 5

156

Fig. 5.31: Variation of the (a) failure strain and (b) failure load of the DPJR 5 with respect to L/d ratio values

80

85

90

95

100

105

110

0 200 400 600 800 1000 1200

L /d ratio

Dam

aged

/Und

amag

ed R

ope

Failu

re S

trai

n (%

) Exp. DataNumerical Model

NS

M

NS

M

M: Strands cut near rope midspan NS: Strands cut near rope splice

85

90

95

100

0 200 400 600 800 1000 1200L /d ratio

Dam

aged

/Und

amag

ed

Rop

e Fa

ilure

Loa

d (%

) Exp. DataNumerical Model

NS

M

M: Strands cut near rope midspan NS: Strands cut near rope splice

(a)

(b)

157

Cut Strand

Damaged rope cross-section

Rope jacket

10 parallel unjacketed sub-ropes3 parallel strands per sub-rope 1859 yarns per strand Rope diameter (d): 36 [mm] Specified Breaking Strength: 700 Tonnes (1543.2 Kips)

Fig. 5.32: Damaged cross-section (10%) of DPJR 6

Fig. 5.33: Experimental data and predicted responses of DPJR 6

0

500

1000

1500

2000

2500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Upper Bound Response

Degradation Model

Strain Concentration & DegradationModels (L/d = 40)Net Area Effect & Degradation Models

Exp. Data (No Initial Damage)

Exp. Data (10% Initial Damage)

Exp. Data (15% Initial Damage)

All the predicted damaged rope responses consider 10% of initial cross-section damage

158

Cut Strand

Damaged rope cross-section

Rope jacket

7 parallel unjacketed sub-ropes12 twisted strands per sub-rope 1148 twisted yarns per rope yarn Rope diameter (d): 144 [mm] Specified Breaking Strength: 700 Tonnes (1543.2 Kips)

Fig. 5.34: Damaged cross-section (10%) of DPJR 7

Fig. 5.35: Experimental data and predicted responses of DPJR 7

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Rope Axial Strain

Rop

e A

xial

Loa

d (K

ips)

Exp. Data (No Initial Damage)

Exp. Data (5% Initial Damage)

Exp. Data (10% Initial Damage)

Upper Bound Response

Net Area Effect & Degradation

Strain Concentration &Degradation Model (L/d = 40)Degradation Model

All the simulated damaged rope responses consider 10% of initial cross-section damage

159

All h

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12

% Rope Cross-Section Iniatilly Damaged

Failu

re A

xial

Str

ain

Exp. Data

Strain Concentration &Degradation Models

1000

1200

1400

1600

1800

2000

0 2 4 6 8 10 12% Rope Cross-Section Iniatilly Damaged

Failu

re A

xial

Loa

d (K

ips) Exp. Data

Strain Concentration &Degradation Models

Fig. 5.36: Variation of the (a) failure strain and (b) failure load of the DPJR 7 with respect to initial rope damage state

(a)

(b)

160

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

1100

1200

1300

1400

1500

1600

1700

0 1000 2000 3000 4000

L/d ratio

Failu

re A

xial

Loa

d (K

ips)

5% Initial Damage7% Initial Damage10% Initial Damage

0.105

0.11

0.115

0.12

0.125

0.13

0 1000 2000 3000 4000

L/d ratio

Failu

re A

xial

Str

ain

5% Initial Damage7% Initial Damage10% Initial Damage

(a)

(b)

161

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

Capacity

(Kips)

Exp. Data

Max. Rope Capacity

(Kips)

Numerical Model

with Degradation

(Predicted)

Predicted/Exp.

Data Value

(with

Degradation)

Max. Rope Capacity

(Kips)

Numerical Model

No Degradation

(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

162

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

163

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

164

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)

Exp. Data

Max. Rope Capacity (Kips)

Numerical Model

(Predicted) Rope

L/d L/d L/d L/d

40 290 560 1000

Strain

Localization and

Degradation

Model

Net Area Effect

and No

Degradation

Model

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 83.61 82.44 76.02 82.96

165

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

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

85

90

95

100

105

0 200 400 600 800 1000 1200

L /d ratio

Dam

aged

/Und

amag

ed

Rop

e Fa

ilure

Str

ain

(%)

Rope DPJR 5Rope DPJR 3Rope DPJR 1

(a)

50

60

70

80

90

100

110

0 200 400 600 800 1000 1200

L /d ratio

Dam

aged

/Und

amag

ed

Rop

e Fa

ilure

Loa

d (%

)

Rope DPJR 5Rope DPJR 3Rope DPJR 1

(b)

166

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)

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 cross-

sectional 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

Max. Rope Capacity (Kips)

Exp. Data Max. Rope Capacity (Kips)

Numerical Model (Predicted) Plot

Initial

Damage L/d L/d

40 290

Strain

Localization and

Degradation

Model

Net Area

Effect and No

Degradation

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

167

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

168

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.

169

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

170

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.

171

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

172

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 cross-

section 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 sub-

ropes 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 sub-

rope (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

173

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,

174

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.

175

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, non-

symmetry 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

176

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.

177

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:

)()1()( 111111 111

111111

θηη

ββθηη

ββ ηβαηβαθηαηαηuu

eeDDD uIuI

−−−

−

−+−−++= (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:

[ ] ++++−++++=−

2)(1)( 33

221

33

221

θηη

ηωηωηωηωηωηωηu

eDDD uuuII

+−+

−

2)32(32 232

2322

θηη

ηωηωηωηωθu

euu (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.

178

Fig. A.1 Power law evolution of the damage index

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1η

D( η)

one-term expansion

two-term expansion

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8η

D( η)

one-term expansion

two-term expansion

εt = 0.03 εb = 0.124 εu = 0.1302 θ1 = 0.0043

Onset of failure

Onset of failure

εt = 0.05 εb = 0.124 εu = 0.1302 θ1 = 0.0043

179

Fig. A.2 Polynomial evolution of the damage index

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

η

D( η

)one-term expansion

two-term expansion

εt = 0.03 εb = 0.124 εu = 0.1302 θ1 = 0.0043

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8η

D( η

)

one-term expansion

two-term expansion

Onset of failure

Onset of failure

εt = 0.05 εb = 0.124 εu = 0.1302 θ1 = 0.0043

180

As can be seen in Figs. A.1 and A.2, for both damage models considered, a two-

term composite expansion agrees quite closely with the results obtained from the one-

term 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.

181

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

Mooring,” Ocean Engineering, Vol. 25, No 10, pp. 861-879.

4. Banfield, S., Hearle, J.W.S., Leech, C.M. and Lawrence, C.A., 2001, “Fibre Rope

Modeller: A Cad Program for the Performance Prediction of advanced Cords and

Ropes Under Complex Loading Environments,” Techtextil.

5. Bartle, R. and Sherbert, D., 1982, Introduction to Real Analysis, John Wiley & Sons

Inc., New York, USA.

6. Beltran, J. F., Rungamornrat, J. and Williamson, E. B., 2003, “Computational Model

for the Analysis of Damaged Ropes,” 13th International. Offshore and Polar

Engineering. Conference, Honolulu, Hawaii, USA., pp. 705-710.

7. Beltran, J.F. and Williamson, E.B., 2004, “Investigation of the Damage-Dependent

Response of Mooring Ropes,” 14th International. Offshore and Polar Engineering.

Conference, Toulon, France.

8. Beltran, J.F., 2004, “Determination of Recovery Length in Spiral Strand,” Class

Report, CE 381T, The University of Texas at Austin.

9. Beltran, J.F. and Williamson, E.B., 2005, “Degradation of Rope Properties Under

Increasing Monotonic Load,” Ocean Engineering, Vol. 32, pp. 823-844.

10. Berryman, C.T., Dupin, R.M. and Gerrits, N.S., 2002, “Laboratory Study of Used

HMPE MODU Mooring Lines,” Offshore Technology Conference, Houston, Texas,

USA.

11. Blouin, F. and Cardou, A., 1989, “A Study of Helically Reinforced Cylinders Under

Axially Symmetric Loads and Application to Strand Mathematical Modelling,”

International Journal of Solid and Structures, Vol. 25, No 2, pp. 189-200.

182

12. Bradon, J. and Chaplin, C.R., 2005, “Quantifying the Residual Creep Life of

Polyester Mooring Ropes,” International Journal of Offshore and Polar Engineering,

Vol. 15, No 3, pp. 223-228.

13. Cardou, A. and Jolicoeur, C., 1997, “Mechanical Models of helical Strands,” Applied

Mechanics Review, Vol. 50, No 1, pp. 1-14.

14. Chaplin, C.R and Tantrum, N., 1985, “The Influence of Wire Break Distribution on

Strength,” Organisation Internationale pour l’Etude de l’Endurance des Cables

(OIPEEC), Round Table Conference, June, Glasgow.

15. Chaplin, C.R., 1998, “Torsional Failure of Wire Rope Mooring Line During

Installation in Deep Water,” Engineering Failure Analysis, Vol. 6, pp. 67-82.

16. Chien, C.H. and Costello, G.A., 1985, “Effective Length of a Fractured Wire in Wire

Rope,” Journal of Engineering Mechanics, Vol. 111, No 7, pp. 952-961.

17. Cholewa, W. and Hansel, J., 1981, “The Influence of the Distribution of Wire Rope

Faults on the Actual Breaking Load,” Organisation Internationale pour l’Etude de

l’Endurance des Cables (OIPEEC), Round Table Conference, June, Krakow, Poland.

18. Cholewa, W., 1989, “Wire Fracture and Weakening of Wire Ropes,” Wire Rope

Discard Criteria: Round Table Conference: Swiss Federal Institute of Technology

(ETH), Institute of Lightweight Structures and Ropeways, September, Zurich,

Switzerland.

19. Costello, G.A., 1978, “Analytical Investigation of Wire Rope,” Applied Mechanics

Review, Vol. 31, No 7, pp. 897-900.

20. Costello, G.A., 1983, “Stresses in Multilayered Cables,” Journal of Energy Resources

Technology, Vol. 105, pp. 337-340.

21. Costello, G.A. and Butson, G.J., 1982, “Simplified bending Theory for Wire Rope,”

Journal of Engineering Mechanics Division, Vol. 108, No 2, pp. 219-227.

22. Costello, G.A., 1990, Theory of Wire Rope, Springer-Verlag, New York, USA.

23. Davies, P., Grosjean, F., and Francois, M., 2000, “Creep and Relaxation of Polyester

Mooring Lines,” Offshore Technology Conference, OTC 12176, Houston, Texas,

USA.

183

24. Del Vecchio, C.J.M., 1992, “Light Weight Materials for Deep Water Moorings,” PhD

Thesis, University of Reading, UK.

25. Evans, J.J, Ridge, I.M.L. and Chaplin, C.R., 2001, “Wire Failures in Ropes and their

Influence on Local Wire Strain Behaviour in Tension-Tension Fatigue,” Journal of

Strain Analysis, Vol. 36, No 2, pp. 231-244.

26. Fernandes, A.C., Del Vecchio, C.J.M. and Castro, G.A.V., 1999, “Mechanical

Properties of Polyester Mooring Cables,” International Journal of Offshore and Polar

Engineering, Vol. 9, No 3, pp. 208-213.

27. Flory, J.F., 2001, “Defining, Measuring and Calculating the Change-in-Length

Properties of Synthetic Fiber Rope,” Ocean Conference Record (IEEE), Vol. 2, pp.

679-684.

28. Foster, G.P., 2002, “Advantages of Fiber Rope Over Wire Rope,” Journal of

Industrial Textile, Vol. 32, pp.67-75.

29. Ghoreishi, S., Messager, T., Cartraud, P. and Davies, P., 2004, “Assesment of Cable

Models for Synthetic Mooring Lines,” International Offshore and Polar Engineering

Conference, Toulon, France.

30. Gjelsvik, A., 1991, “Development Length for Single Wire in Suspension Bridge

Cable,” Journal of Structural Engineering, Vol. 117, No 4, pp. 1189-1201.

31. Hankus, J., 1981, “Safety Factor for Hosting Rope Weakened by Fatigue Cracks in

Wires,” Organisation Internationale pour l’Etude de l’Endurance des Cables

(OIPEEC), Round Table Conference, June, Krakow.

32. Hearle, J.W.S., 1969, “On the Theory of the Mechanics of Twisted Yarns,” Journal of

Textile Institute, Vol. 60, pp. 95-101.

33. Hobbs, R.E. and Raoof, M., 1982, “Interwire Slippage and Fatigue Prediction in

Stranded Cables for TLP Tethers,” Behavior of Offshore Structures, Vol. 2, pp. 77-

99.

34. Holmes, M., 1995, Introduction to Perturbation Methods, Springer-Verlag, New

York, USA.

35. Hooker, J.G., 2000, “Synthetic Fibre Ropes for Ultradeep Water Moorings in drilling

and Production Applications,” 2nd Offshore and Marine Technology, Singapore.

184

36. Hruska, F.H., 1951, “Calculation of Stresses in Wire Ropes,” Wire and Wire

Products, Vol. 26, pp. 766-767, 799-801.

37. Hruska, F.H., 1952, “Radial Forces in Wire Ropes,” Wire and Wire Products, Vol.

27, pp. 459-463.

38. Hruska, F.H., 1953, “Tangential Forces in Wire Ropes,” Wire and Wire Products,

Vol. 28, pp. 455-460.

39. Huang, N.C., 1978, “Finite Extension of an Elastic Strand with a Central Core,”

Journal of Applied Mechanics, Vol. 42, No 4, pp. 852-857.

40. Jiang, W., 1995, “General Formulation on the Theories of Wire Ropes,” Journal of

Applied Mechanics, Vol. 62, pp. 747-755.

41. Jiang, W., Yao, M.S. and Walton J.M., 1999, “A Concise Finite Element Model for

Simple Straight Wire Rope Strand,” International Journal of Mechanical Sciences,

Vol. 41, pp. 143-161.

42. Jiang, W. and Henshall, J.L., 1999, “The Analysis of Termination Effects in Wire

Strand Using The Finite Element Method,” Journal of Strain Analysis, Vol. 34, No 1,

pp. 31-38.

43. Jiang, W., Henshall, J.L. and Walton J.M., 2000, “A Concise Finite Element Model

for Three-layered Straight Wire Rope Strand,” International Journal of Mechanical

Sciences, Vol. 42, pp. 63-86.

44. Jolicoeur, C., and Cardou, A., 1991, “A Numerical Comparison of Current

Mathematical Models of Twisted Wire Cables Under Axisymmetric Loads,” Journal

of Energy Resources Technology, Vol. 113, pp. 241-249.

45. Jolicoeur, C., and Cardou, A., 1994, “Analytical Solution for Bending of Coaxial

Orthotropic Cylinders,” Journal of Engineering Mechanics, Vol. 120, No 12, pp.

2556-2574.

46. Jolicoeur, C., and Cardou, A., 1996, “Semicontinuous Mathematical Model for

Bending of Multilayered Wire Strands,” Journal of Engineering Mechanics, Vol. 122,

No 7, pp. 643-650.

185

47. Karayaka, M., Srinivasan, S., Wang, S., 1999, “Advanced Design Methodology for

Synthetic Moorings,” Offshore Technology Conference, OTC 10912, Houston,

Texas, USA.

48. Kennedy, J., 2004, “Deepwater Floating Structures Face More Severe Forces, New

Phenomena,” Offshore Technology Research Center Supplement, May, pp. 10-12.

49. Kenney, M.C, Mandell, J.F. and McGarry, F.J., 1985, “Fatigue Behaviour of

Synthetic Fibres, Yarns and Ropes,” Journal of Materials Science, Vol. 20, No 6, pp.

2045-2059.

50. Knapp, R.H., 1975, “Nonlinear Analysis of a Helically Armored Cable with

Nonuniform Mechanical Properties,” IEEE Ocean Conference, San Diego, California,

pp. 155-164.

51. Knapp, R.H., 1979, “Derivation of a New Stiffness Matrix for Helically Armoured

Cables Considering Tension and Torsion,” International Journal for Numerical

Methods in Engineering, Vol. 14, pp. 515-529.

52. Kumar, K. and Cocran, J.E., 1987, “Closed-Form Analysis for Elastic Deformations

of Multilayered Strands,” Journal of Applied Mechanics, Vol. 54, pp. 898-903.

53. Labrosse, M., Nawrocki, A. and Conway, T., 2000, “Frictional Dissipation in Axially

Loaded Simple Straight Strands,” Journal of Engineering Mechanics, Vol. 126, No 6,

pp. 641-646.

54. Lagerstrom, P.A., 1988, Matched Asymptotic Expansions: Ideas and Techniques,

Springer-Verlag, New York, USA.

55. Lanteigne, J., 1985, “Theoretical Estimation of the Response of Helically Armored

Cables to Tension, Torsion and Bending,” Journal of Applied Mechanics, Vol. 52, pp.

423-432.

56. Le, T,T. and Knapp, R.H., 1994, “A Finite Element Model for Cables with

Nonsymmetrical Geometry and Load,” Journal of Offshore Mechanics and Artic

Engineering, Vol. 116, No 15, pp. 14-20.

57. LeClair, R., 1991, “Axial Response of Multilayered Strands with Compliant Layers,”

Journal of Engineering Mechanics, Vol. 117, No 12, pp. 2884-2903.

186

58. Leech, C.M., 1987, “Theory and Numerical Methods for the Modeling of Synthetic

Ropes,” Communications in Applied Numerical Methods, Vol. 3, pp.407-713.

59. Leech, C.M., 2002, “The Modeling of Friction in Polymer Fibre Ropes,” International

Journal of Mechanical Sciences, Vol. 44, pp. 621-643.

60. Leissa, A.W., 1959, “Contact Stresses in Wire Ropes,” Wire and Wire Products, Vol.

34, pp. 307-314.

61. Li, D., Miyase, A., Williams, J.G. and Wang S.S., 2002. “Damage Tolerance of

Synthetic-Fiber Mooring Ropes: Small-Scale Experiments and Analytical Evaluation

of Damaged Subropes and Elements,” Technical Report, CEAC-TR-03-0101,

University of Houston.

62. Linz, P., 1979, Theoretical Numerical Analysis: An introduction to Advanced

Techniques, Dover Publications, New York, USA.

63. Lo, K.H., Xu, H. and Skogsberg, L.A., 1999, “Polyester Rope Mooring Design

Considerations,” International Offshore and Polar Engineering Conference, Brest,

France.

64. Logan, J.D., 1997, Applied Mathematics, John Wiley & Sons Inc., New York, USA.

65. Love, A.E.H., 1944, Treatise on the Mathematical Theory of Elasticity, Dover

Publications, New York, USA, Chap. 18.

66. Machida, S. and Durelli, A.J., 1973, “Response of a Strand to Axial and Torsional

Displacements,” Journal of Mechanical Engineering Science, Vol. 15, pp. 241-251.

67. Mandell, J.F., 1987, “Modelling of Marine Rope Fatigue Behavior,” Textile Research

Journal, June, pp. 318-330.

68. McConnell, K.G. and Zemke, W.P., 1982, “A Model to Predict the Coupled Axial

Torsion Properties of ASCR Electrical Conductors,” Experimental Mechanics, Vol.

22, pp. 237-244.

69. McKenna, H.A., Hearle, J.S.W. and O’Hear, N., 2004, Handbook of Fibre Rope

Technology, Woodhead Publishing Ltd., Cambridge, England.

70. Najar, J., 1987, Continuum Damage Mechanics. Theory and Applications, Springer-

Verlag, New York, USA.

187

71. Nawrocki, A. and Labrosse, M., 2000, “A Finite Element Model for Simple Straight

Wire Rope,” Computers and Structures, Vol. 77, pp. 345-359.

72. Nayfeh, A.H., 1973, Introduction to Perturbation Techniques, John Wiley & Sons

Inc., New York, USA.

73. Phillips, J.W. and Costello, G.A., 1973, “Contact Stresses in Twisted Wire Cables,”

Journal of Engineering Mechanics Division, Vol. 99, pp. 331-341.

74. Ramsey, H., 1988, “A Theory of Thin Rods with Application to Helical Constituent

Wires in Cables,” International Journal of Mechanical Sciences, Vol. 30, No 8, pp.

559-570.

75. Ramsey, H., 1990, “Analysis of Interwire Friction in Multilayer Cables Under

Uniform Extension and Twisting,” International Journal of Mechanical Sciences, Vol.

32, No 9, pp. 707-716.

76. Raoof, M., 1991, “Wire Recovery Length in a Helical Strand under Axial-Fatigue

Loading,” International Journal of Fatigue, Vol. 13, No 2, pp. 127-132.

77. Raoof, M. and Huang, Y.P., 1992, “Wire Recovery Length in Suspension Bridge

Cable,” Journal of Structural Engineering, Vol. 118, No 12, pp. 3255-3267.

78. Raoof, M. and Kraincanic, I., 1995, “Recovery Length in Multilayered Spiral

Strands,” Journal of Engineering Mechanics, Vol. 121, No 7, pp. 795-800.

79. Raoof, M. and Kraincanic, I., 1998, “Determination of Wire Recovery Length in Stell

Cables and Its Practical Applications,” Computers & Structures, Vol. 68, pp. 445-

459.

80. Reusch, F. and Estrin, Y., 1998, “FE-analysis of Mechanical Response of Simple

Structures with Random Non-uniformity of Material Properties,” Computational

Materials Science, Vol. 11, No 4, pp. 294-308.

81. Rungamornrat, J., Beltran, J.F. and Williamson E.B., 2002, “Computational Model

for Synthetic-Fiber Rope Response,” 15th Eng. Mechanics Conference, ASCE, New

York, USA.

82. Samras, R.K., Skop, R.A. and Milburn, D.A., 1974, “Analysis of Coupled

Extensional-Torsional Oscillations in Wire Ropes,” Journal of Engineering for

Industry, Vol. 96, pp. 1130-1135.

188

83. Sathikh, S., Moorthy, M. and Krishnan, M. (1996), “Symmetric Linear Elastic Model

for Helical Wire Strands Under Axisymmetric Loads,” Journal of Strain Analysis,

Vol. 31, No 5, pp. 389-399.

84. Schrems, K. and Maclaren, D., 1997, “Failure Analysis of a Mine Hoist Rope,”

Engineering Failure Analysis, Vol. 4, No 1, pp.25-38.

85. Seo, M.H., Backer, S. and Mandell, J.F., 1990, “Modelling of Synthetic Fiber Ropes

Deterioration Caused by Internal Abrasion and Tensile Fatigue,” MTS Science and

Technology for a New Oceans Decade, Washington DC, USA.

86. Starkey, W.L. and Cress, H.A. (1959), “An Analysis of Critical Stresses and Mode of

Failure of a Wire Rope,” Journal of Engineering for Industry, Vol. 81, pp. 307-316.

87. Tricomi, F.G., 1985, Integral Equations, Dover Publications, New York, USA.

88. Utting, W.S. and Jones, N., 1984, “A survey Literature on the Behaviour of Wire

Ropes,” Wire Industry, pp. 623-629.

89. Van den Heuvel, C.J.M., Heuvel, H.M., Faasen, W.A., Veurink, J. and Lucas, L.J.,

1993, “molecular Changes of PET Yarns During Stretching Measured with Rheo-

Optical Infrared Spectroscopy and Other Techniques,” Journal of Applied Polymer

Science, Vol. 49, pp. 925-934.

90. Velinsky, S.A., 1981, “Nalysis of Wire Ropes with Complex Cross Sections,” PhD

Thesis, Department of Theoretical and Applied Mechanics, University of Illinois at

Urbana- Champaign, USA.

91. Velinsky, S.A., Anderson, G.L. and Costello, G.A., 1984, “Wire Rope with Complex

Cross Sections,” Journal of Engineering Mechanics, Vol. 110, No 3, pp. 380-391.

92. Velinsky, S.A., 1985, “General Nonlinear Theory for Complex Wire Rope,”

International Journal of Mechanical Sciences, Vol. 27, pp. 497-507.

93. Walton, J.M., 1996, “Developments in Steel Cables,” Journal of Constructions Steel

Research, Vol. 39, No 1, pp. 3-29.

94. Ward, E.G., Ayres, R.R., Banfield, S., O’Hear, N. and Smith, C.E., 2005,

“Experimental Investigation of Damage Tolerance of Polyester Ropes,” 4th

International Conference on Composite Materials for Offshore Operations, Houston,

Texas, USA.

189

95. Ward, E.G., May 2006. Personal Communication.

96. Wempner, G., 1981, Mechanics of Solids with Applications to Thin Bodies, Sijthoof

and Noordhoff, Rockville, Md.

97. Winkler, M.M. and McKenna, H.A., 1995, “The Polyester Rope Taut Leg Mooring

Concept: A Feasible Means for Reducing Deepwater Mooring Cost and Improving

Stationkeeping Performance,” 27th Annual Offshore Technology Conference,

Houston, Texas, USA.

98. Wu, H.C., 1992, “An Energy Approach for Rope-Strength Prediction,” Journal of

Textile Institute, Vol. 83, No 4, pp. 542-549.

99. Wu, H.C., 1993, “Frictional Constraint of Rope Strands,” Journal of Textile Institute,

Vol. 84, No 2, pp. 199-213.

190

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.