The Analysis of H shaped Horizontal Lifeline Fall Protection System

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UNIVERSITY OF CINCINNATI
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hereby submit this as part of the requirements for the
degree of:
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in:
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The Analysis of H-shaped Horizontal Lifeline Fall Protection System 
 
 
A thesis submitted to the 
 
Division of Research and Advanced Studies 
of the University of Cincinnati 
 
in partial fulfillment of the 
requirements for the degree of 
 
MASTER OF SCIENCE 
 
in the Department of Civil & Environmental Engineering 
of the College of Engineering 
 
2001 
 
by 
 
Shiqiao Zhang 
 
B.S., Harbin Architecture and Engineering Institute, 1986 
 
 
Thesis Committee: 
 
Dr. Frank E. Weisgerber, Chair 
Dr. Michael T. Baseheart 
Dr. James A. Swanson 
Abstract 
 
 
 To protect workers at elevated positions from falling, many fall protection systems are 
used. Generally, fall protection systems are divided into two categories: fall restraint systems and 
fall arresting systems. In this thesis, emphasis is given to the H-shaped horizontal lifeline 
(HHLL) fall protection system, which belongs in the fall arresting category. 
Various fall protection systems are first introduced and the advantages and disadvantages 
are compared. Explanations about the competitiveness of the HHLL system and the importance 
of giving attention to the HHLL system are stated followed by the basic cable theories, which 
constitutes the background of the thesis. 
The force and displacement analysis of the HHLL system proceeds in several stages. In 
the first stage, single-span HHLL systems with different configurations are analyzed with a 
simplified consideration of the cable self-weight. In the second stage, a refined method is 
followed because the self-weight is found to have much effect on the analysis. In the refined 
method, the self-weight is accurately calculated and the practice of setting up systems on the site 
is considered for detailed analysis. In the third stage, simple multi-span systems and four selected 
configurations of overlapped-cable multi-span systems are analyzed and compared. Finally, a 
system configuration is recommended based on the evaluation of the overall system 
performance. 
 
 
Acknowledgements 
 
 
I would like to express my gratitude and appreciation to my advisor Dr. Frank 
Weisgerber for his priceless guidance and advice, especially for his encouragement and help 
during and after my family difficulty. Without him, I would have never finished this thesis. 
I would also like to extend my thanks to my committee member Dr. Michael Baseheart 
and Dr. James Swanson who were always ready to give me valuable help for the thesis 
completion. Particular thanks are given to many classmates and colleagues of mine who helped 
me during my research. It is a great pleasure to work with them. 
Last but not least, deep appreciation is given to my beloved parents and my brother for 
their strong support during the thesis startup as well as my angel mom’s selfless contemplation 
and my brother’s encouragement later on which inspired me all the time. Especially, this is 
dedicated to my dad who passed away during my thesis research. He burnt his lifetime to lighten 
me. May his soul in the heaven rest! 
 
 
Shiqiao Zhang 
January 17, 2002 
 
i 
Table of Contents 
 
 
List of Tables iii 
List of Figures iv 
List of Symbols vi 
Chapter 1 Introduction 1 
Chapter 2 Overview of Fall Protection Systems 4 
 2.1 Fall Restraint System 4 
 2.2 Fall Arresting System 5 
Chapter 3 Problem and Research Statement 9 
 3.1 Problem Statement 9 
 3.2 Research Significance and Objective of the Thesis 12 
Chapter 4 Background of the Study 13 
 4.1 Basic Concepts and Theoretical Assumptions 13 
 4.2 Cables Subject To Vertical Concentrated Loading 15 
 4.3 Cables with Horizontal Chord Subject to Uniformly Distributed Loading 17 
 4.4 Cable System with Single Load at Mid-span 21 
Chapter 5 Developing Methodology 23 
 5.1 Additional Assumptions in the HHLL system Analysis 23 
 5.2 HHLL System Parameters 23 
 5.3 Research Procedures and Findings 24 
 5.3.1 Single-span HHLL system with simplified method 24 
 5.3.2 Single-span HHLL system with refined method 31 
ii 
 5.3.3 Multi-span HHLL system 45 
Chapter 6 Conclusions and Recommendations 67 
 6.1 Conclusions 67 
 6.2 Recommendations 67 
References 69 
Appendix A Calculation for the Single-span HHLL System with Simplified Method 71 
Appendix B Calculation for the Single-span HHLL System with Refined Method 98 
Appendix C Calculation for the Multi-span HHLL System with Refined Method 125 
iii 
List of Tables 
 
 
 No Name Page 
Table 5.1 The Configurations in the Selected Systems 26 
Table 5.2 Summary of the Calculation Results for the Single-span HHLL System 
 with Simplified Method 30 
Table 5.3 Summary of the Calculation Results for the Single-span HHLL System 
 with Refined Method (Known Unstressed Length) 40 
Table 5.4 Summary of the Calculation Results for the Single-span HHLL System 
 with Refined Method (Sag Control) 42 
Table 5.5 Comparison of the Results for the Single-span HHLL System with 
 Refined Method 44 
Table 5.6 Summary of the Calculation Results for the Multi-span HHLL System 
 with Refined Method (Overlapped Cable, Clamped, Outer Span Fall) 62 
Table 5.7 Summary of the Calculation Results for the Multi-span HHLL System 
 with Refined Method (Overlapped Cable, Clamped, Inner Span Fall) 63 
Table 5.8 Summary of the Calculation Results for the Multi-span HHLL System 
 with Refined Method (Overlapped Cable, Free, Outer Span Fall) 64 
Table 5.9 Summary of the Calculation Results for the Multi-span HHLL System 
 with Refined Method (Overlapped Cable, Free, Inner Span Fall) 65 
Table 5.10 Comparison of the Results for the Multi-span HHLL System with 
 Refined Method 66 
iv 
List of Figures 
 
 
 No Name Page 
Figure 3.1 Swing Fall in the VLL System 9 
Figure 3.2 The One-dimensional HLL System 10 
Figure 3.3 The Swing Fall in the One-dimensional HLL System 11 
Figure 3.4 The HHLL System 12 
Figure 4.1 The Cable System Under Concentrated Load 14 
Figure 4.2 The Cable System Under Distributed Load with Catenary Shape 14 
Figure 4.3 The Cable System Under Distributed Load with Parabolic Shape 14 
Figure 4.4 General Cable System Subject to Concentrated Loadings 16 
Figure 4.5 The Cable System Subject to Uniformly Distributed Loading 18 
Figure 4.6 The Cable System Subject to Concentrated Loading 22 
Figure 5.1 Single-span HHLL System Layout 24 
Figure 5.2 Single-span HHLL System Before and After the Fall 25 
Figure 5.3 The Across Cable Calculation 27 
Figure 5.4 The Side Cable Calculation 28 
Figure 5.5 Establishing the Parameter Relation 31 
Figure 5.6 The Across Cable in the Initial State (Single-span) 33 
Figure 5.7 The Across Cable in the Final State (Single-span) 34 
Figure 5.8 The Side Cable in the Initial State (Single-span) 36 
Figure 5.9 The Side Cable in the Final State (Single-span) 37 
Figure 5.10 The Simple Multi-span System 46 
vNo Name Page 
Figure 5.11 The Overlapped-cable Multi-span System 47 
Figure 5.12 The Across Cable in the Initial State (Multi-span) 48 
Figure 5.13 The Across Cable in the Final State (Multi-span) 49 
Figure 5.14 The Side Cable in the Initial State (Multi-span) 51 
Figure 5.15 Outer Span Fall in the Clamped Case 52 
Figure 5.16 Inner Span Fall in the Clamped Case 54 
Figure 5.17 Outer Span Fall in the Free Case 56 
Figure 5.18 Inner Span Fall in the Free Case 58 
 
 
 
vi 
List of Symbols 
 
 
A Total cable cross-section area 
Aa Total across cable cross-section area 
As Total side cable cross-section area 
E Effective cable modulus of elasticity 
f Sag of the cable 
fa Sag of the across cable 
faf Sag of the across cable in the final state 
fai Sag of the across cable in the initial state 
fs Sag of the side cable 
fsf Sag of the side cable in the final state 
fsi Sag of the side cable in the initial state 
fsm Sag of the side cable in the intermediate state 
ft Total sag of the system 
H Horizontal component of the cable tension 
Hai Horizontal component of the across cable tension in the initial state 
Hsi Horizontal component of the side cable tension in the initial state 
L Stressed length or curved length of the cable 
L0 Unstressed length of the cable 
La Stressed length of the across cable 
La0 Unstressed length of the across cable 
Laf Stressed length of the across cable in the final state 
Lai Stressed length of the across cable in the initial state 
Ls Stressed length of the side cable 
Ls0 Unstressed length of the side cable 
Lsf Stressed length of the side cable in the final state 
Lsf11 Stressed length of the left-portion cable in the left side cable in the final state 
Lsf12 Stressed length of the right-portion cable in the left side cable in the final state 
Lsf21 Stressed length of the left-portion cable in the right side cable in the final state 
vii 
Lsf22 Stressed length of the right-portion cable in the right side cable in the final state 
Lsi Stressed length of the side cable in the initial state 
Lsi11 Stressed length of the left-portion cable in the left side cable in the initial state 
Lsi12 Stressed length of the right-portion cable in the left side cable in the initial state 
Lsi21 Stressed length of the left-portion cable in the right side cable in the initial state 
Lsi22 Stressed length of the right-portion cable in the right side cable in the initial state 
Lsm Stressed length of the side cable in the intermediate state 
n Cable sag ratio 
nai Across cable sag ratio in the initial state 
nsi Side cable sag ratio in the initial state 
p Cable self-weight 
pa0 Unconverted across cable self-weight in the initial state 
pa1 Converted across cable self-weight in the initial state 
ps Side cable self-weight in the initial state 
P Working load under the fall 
Paf Concentrated load applied on the across cable in the final state 
Psf Concentrated load applied on the side cable in the final state 
Psm Concentrated load applied on the side cable in the intermediate state 
S Horizontal distance between the cable supports 
Sa Horizontal distance between the across cable supports in unstressed condition 
Sa0 Unconverted horizontal distance between the across cable supports in unstressed condition 
Sa1 Converted horizontal distance between the across cable supports in the initial state 
Smin Cable minimum break strength 
Ss Horizontal distance between the side cable supports 
T Tension of the cable 
Ta Tension of the across cable under load 
Taf Tension of the across cable in the final state 
Tai Tension of the across cable in the initial state 
Ts Tension of the side cable under load 
Tsf Tension of the side cable in the final state 
Tsf11 Tension of the left-portion cable in the left side cable in the final state 
viii 
Tsf12 Tension of the right-portion cable in the left side cable in the final state 
Tsf21 Tension of the left-portion cable in the right side cable in the final state 
Tsf22 Tension of the right-portion cable in the right side cable in the final state 
Tsi Tension of the side cable in the initial state 
Tsi1 Tension of the left side cable in the initial state 
Tsi2 Tension of the right side cable in the initial state 
Tsm Tension of the side cable in the intermediate state 
V Vertical component of the cable tension 
f∆ Differential change of sag 
L∆ Differential change of cable length 
S∆ Differential change of span 
θ Deflection angle 
θa Deflection angle of the across cable 
θaf Deflection angle of the across cable in the final state 
θai Deflection angle of the across cable in the initial state 
θsf Deflection angle of the side cable in the final state 
θsf11 Left deflection angle of the left side cable in the final state 
θsf12 Right deflection angle of the left side cable in the final state 
θsf21 Left deflection angle of the right side cable in the final state 
θsf22 Right deflection angle of the right side cable in the final state 
θsi Deflection angle of the side cable in the initial state 
1 
Chapter 1 Introduction 
 
 
Falls from elevated positions may occur at any time in nearly every industry. But certain 
industries have higher rates of incidents and the construction industry is one of these. In 
construction, many operations and work are performed at elevated locations and workers are 
frequently subject to the risk of falling to the lower level, incurring injuries or fatalities. 
According to OSHA statistics, in 1995, violations of fall protection regulations resulted in 
employers paying penalties of $7,784,357 and this was the most frequently cited type of OSHA 
violation. 
The adverse outcome of a fall protection violation is not simply paying fines. Every 
injury or fatality due to a fall will bring, to some extent, significant misfortune to the worker 
himself/herself as well as his/her family. In the future, workers may demand high compensation 
for ‘dangerous operations’ or, fearing to endanger themselves, refuse to work at high places 
without proper protection. Unions may protest and call workers to strike or even enjoin the 
employer in a lawsuit. In the case of the occurrence of fall, OSHA will send people to conduct 
investigations, fully document the accident and, if the accident is serious, stop the operation or 
work. Expenses will be incurred because the employer will need to take some time to train new 
worker(s) to fill the vacancy and the worker(s) will need a period of adjustment to become as 
proficient as the previous worker(s). Even more significantly, with even one serious accident, the 
employer’s insurance rates will rise drastically and remain high for years. 
All these direct and indirect costs would be much higher than the cost of applying 
adequate fall protection measures and avoiding the fall accidents. Thus, keeping workers safe, 
2 
minimizing production cost and maintaining continuous operation become the driving forces for 
fall protection. 
Both OSHA and ANSI identify the various circumstances and require that fall protection 
be applied when the potential fall height exceeds 6 feet for the construction industry and 4 feet 
for general industry. In OSHA 1926, Subpart M, Subpart X, Subpart R and Subpart L deal with 
general fall protection, ladders, steel erection and scaffolding, respectively, in the construction 
industry. In OSHA 1910, Subpart I, Subpart D deal with personal protective fall equipment and 
walking/working surfaces, respectively, in general industry. Also, ANSI A10.14 and ANSI Z-
359.1 describe requirements for the fall protectionpractice in the construction industry and 
general industry, respectively. 
There are many schemes that can fulfill the need for fall protection. The most effective 
scheme is to eliminate fall hazard potentials or reduce them to the lowest level. For example, 
during the design of production equipment, various meters and valves can be placed near the 
ground or as low as possible to avoid or decrease the need for operators climbing and the 
probability of a fall. The next most effective way is to use engineering controls to block workers 
from approaching hazards. For roof construction, rails or fences can be placed to bar workers 
from reaching the edge and falling accidentally. The third scheme is to use personal protection 
equipment (PPE) such as safety net, lifeline, etc. However, some PPEs need workers’ active 
engagement in order for them to be effective. For example, when a worker at the edge of the roof 
tries to use a lifeline to protect himself from a fall, he must first securely attach the cable to his 
body. If he fails to do so, the scheme fails. The least effective scheme is to use warnings near a 
dangerous zone. Warnings are easily disregarded by workers and lose their effectiveness if the 
employer does not adopt a strict policy of enforcement. 
3 
In this thesis, an overview of fall protection systems is provided and basic cable theories 
are introduced. Main concentration is given to the H-shaped horizontal lifeline (HHLL) system 
with the objective to develop an analysis method of the HHLL system subject to the load caused 
by a falling worker and provide an outline of the effective system configurations. 
 
 
 
 
4 
Chapter 2 Overview of Fall Protection Systems 
 
 
 Based on their goals and working mechanisms, fall protection systems can be divided 
into two categories: fall restraint systems and fall arresting systems. In general, the fall arresting 
system requires more complex engineering and more specially designed components. Workers 
require more training to use fall arresting systems and rescue plans are necessary because the 
fallen worker could end up in a potentially difficult location below the elevated work position 
and above the lower floor. 
 
2.1 Fall Restraint System 
The fall restraint system provides protection by setting restraints on workers to prevent 
the fall from happening. Thus, the system works in a ‘preventive’ way. This system can also be 
subdivided into two types: the passive fall restraint system and the active fall restraint system. 
The passive fall restraint system, as its name suggests, is passively activated immediately 
when danger exists for a worker at work and the protection need not come from the worker’s 
special action. Examples of such system are: rail, fence barrier, handrail, ladder cage, etc. The 
system will be erected at the fall hazard area and can keep the worker within a safe zone. 
The active fall restraint system requires that a worker either set up the system in advance 
in order for the system to function later, or activate the system himself/herself before 
commencing work. Examples of such system are: single-point tether, two-point tether, 
temporarily placed guardrail, etc. Before work, the worker needs to attach his/her body belt or 
harness to the tether or place the guardrail at a proper place to stop the initiation of the fall. 
 
5 
2.2 Fall Arresting System 
The fall arresting system is designed to function when a worker is in the process of 
falling. It provides protection by catching the worker during the fall and fully stopping the fall 
before the worker hits the lower level or an obstacle. Care will be taken to limit the impact upon 
the worker to avoid the injury caused by the system itself. There are several types of such 
systems. 
Safety net system. The safety net is basically a net placed within the working area 
between the higher and lower level and can catch the worker when the fall happens. 
Generally, nets are used for long-term projects. The system can be used where many 
workers work. Also it can be used in large open areas or long leading edges that expose workers 
to height hazards. Normally, the net should be as close to the working level as possible and must 
expand outward a certain distance from the edge of the working area. As the net is erected prior 
to the commencement of work and the worker is not directly involved with wearing or attaching 
anything, this type of system needs little worker training for the system to be effective. 
Fixed anchorage system. This system connects the worker, via his/her harness, to a 
fixed point close to and generally above the work position. The length of the line includes a 
personal energy absorber (EAP) and this arrangement catches the worker as the fall proceeds. 
The EAP removes kinetic energy from the fall event and limits the force upon the worker to a 
tolerable magnitude (often 900 pounds) as that upward force arrests the fall. 
The system is often selected for short-term work for workers working at a fixed location. 
The anchorage point could be on a truss, a beam or a column, and it could also be a specially 
designed and fabricated point. Due to the simplicity of the system, the system is easy to use but 
requires some training and anchor points. 
6 
Climbing protection system. This system is specially designed for protection while 
climbing. Usually a taut cable or rigid rail is anchored securely from the top to the bottom along 
the surface of such structures as poles, ladders, towers, antennas and rigs. A special grabbing 
device attaches the worker’s harness to the cable or rail. This device can move freely up and 
down during normal climbing but lock up instantly when a fall is sensed and thus the fall can be 
stopped. 
The system with rail has a structural attachment at every few feet. It is more reliable than 
the vertical cable and can allow several workers to climb at the same time. The system using a 
cable connects the cable to a bracket at the top and bottom of the structure. Tightening devices 
can keep the cable taut to avoid wind vibration damage to the cable. The cable assemblage is 
simple and economically installed but weathering may affect the cable strength, the reliability of 
the upper fixture point and the performance of the grabbing devices. 
Vertical lifeline (VLL) system. The vertical lifeline system consists of vertical rope or 
cable fixed to a point above the work position, a device which grabs the vertical line at a variable 
position, and a lanyard which connects the worker’s harness to the grabbing device. This lanyard 
typically includes an EAP to limit forces on the worker in the event of a fall. This system is 
similar to the fixed anchorage system but the anchorage point need not be right above the worker 
or may be set to a needed point as the worker sets the grab. Also the VLL gives the worker more 
lateral movement flexibility. 
The VLL system can be used for long-term or short-term work when the system with the 
moving anchorage point is mobile. Additional protection is needed when the worker sets up the 
anchorage point. Because of its flexibility and simple installation, the system is an economical 
solution for the worker who frequently needs to move vertically within a certain area. The VLL 
7 
may also permit longer free fall distance than that in the climbing protection system, which gives 
more impact on the worker when the fall stops. Thus, EAP is also used to decrease the arresting 
force at the expense of adding more fall distance. However, the system must still guarantee to 
fully stop the fall before the worker hits the lower level. 
Horizontal lifeline (HLL)system. The HLL system includes one or more taut horizontal 
cables on which a connector can slide freely. The position of the rail or cable must be above the 
waist-height of the users and preferable above their heads. The worker’s harness is then attached 
to the sliding connector via a lanyard which is a short, flexible rope or strap with a shock 
absorber in it. 
The system can be a permanent or temporary system. Span length between supports 
varies from 10 feet to over 100 feet. One HLL typically has one or two persons attached, but 
systems have been designed for up to 5 workers in a single span. The HLL system allows 
workers to move in a band parallel to the span within the working area while providing fall 
protection throughout this area. 
The permanent system lasts as long as the structure to which it is attached. The cable or 
rail is attached to the structure at regular intervals and the system can be designed to 
accommodate several workers simultaneously (with one sliding connector for each worker). 
Special designs can allow the connector to move continuously around horizontal corners. Special 
support attachments permit the worker to maintain connection while passing intermediate 
supports of multi-span HLL systems. Special items causing increased costs of erection are 
appropriate only for permanent or long-term systems and permit the user to range over longer 
work areas without the need for disconnect-reconnect actions. 
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The temporary system is intended for several days’ to several weeks’ use and synthetic 
rope or wire rope is usually adopted. The system often has a simplified configuration to allow 
quicker and easier installation. 
 H-shaped horizontal lifeline (HHLL) system. The HHLL system is an extension of the 
HLL system where, instead of anchoring the horizontal cable in the HLL system to fixed points, 
the two cable ends are attached via separate sliding connectors to two other parallel horizontal 
cables, which are then anchored to fixed points. The two parallel cables, called the side cables, 
are usually placed along the two parallel sides of the work area while the cable in between, called 
the across cable, will slide along the side cables. This design allows the sliding connector on the 
across cable to stay right above the worker in the work area and removes the potential of swing 
fall that exists in the HLL system and thus provides more effective fall protection within the total 
work area. 
HHLL system is usually erected for short-term use during construction. The height of the 
horizontal cables in the HHLL system should be above the user’s head. As more details need to 
be taken care of, the HHLL system installation usually takes more time and its costs are higher 
than the HLL system. 
 
 
9 
Chapter 3 Problem and Research Statement 
 
 
3.1 Problem Statement 
As described in Chapter 2, the lifeline system, including VLL system and HLL system, 
gives workers more freedom to move within the workspace while providing sufficient fall 
protection. In addition, the VLL system can be easily designed, has low installation and 
maintenance cost as well as a rapid installation time. 
However, nothing is perfect. One serious problem of the VLL system is the potential of 
the swing fall hazard. Due to the restriction of the anchorage point location or due to the 
worker’s movement, the anchorage point may be above the worker but not directly overhead, as 
shown in Figure 3.1. If at this time, the fall happens, the worker will act like the pendulum in an 
antique clock and swing back and forth. This is the swing fall. If there is an obstruction on the 
path of the swing arc, the worker will collide with it and be in danger as the movement along the 
arc of the swing produces just as much energy at the bottom point as a free-fall through the same 
vertical distance. 
 
 
 anchorage point 
 
 
 
 
 
 obstruction free-fall distance 
 
 
 
 
Figure 3.1 Swing Fall in the VLL System 
10 
 The HLL system is a significant improvement over the VLL system regarding swing fall 
while it retains the same or more freedom of movement. The HLL system widely used today is 
one-dimensional, i.e. the horizontal cable is in plane XY, as shown in Figure 3.2. When a worker 
moves in plane XY, the sliding connector is always right above him/her and the likelihood of the 
swing fall in plane XY is eliminated. 
 
 Y 
 
 
 sliding connector horizontal cable 
 
 
 
 
 lanyard 
 
 
 working area 
 
 O X 
 
 
 
 Z 
 
Figure 3.2 The One-dimensional HLL System 
 
Nevertheless, the swing fall in the plane YZ is still possible. If the worker moves out of 
plane XY and the fall happens, the worker will swing in the plane YZ and endanger 
himself/herself, as is illustrated in Figure 3.3. 
 
11 
 
 Y 
 
 sliding connector horizontal cable 
 
 
 lanyard 
 
 
 working area 
 
 
 
 O X 
 
 
 obstruction 
 
 
 Z 
 
 
Figure 3.3 The Swing Fall in the One-dimensional HLL System 
 
The obvious choice is to set the HLL system two-dimensional, i.e. whichever direction 
the worker moves horizontally, the sliding connector will be always right above him/her and thus 
swing fall cannot happen any more. This produces the concept of the H-shaped horizontal 
lifeline (HHLL) system. As shown in Figure 3.4, two side cables are attached to four fixed 
anchorage points; the across cable is connected to the side cables via sliding connectors; and the 
worker’s lanyard is connected to the across cable via a sliding connector. 
12 
 
 sliding connectors 
 
 
 
 
side cable 
 across cable 
 
 
 side cable 
 lanyard 
 
 
 
Figure 3.4 The HHLL System 
 
 
3.2 Research Significance and Objective of the Thesis 
Most of the emphasis of past research and the current practice regarding fall protection 
with lifeline systems relate to VLL and one-dimensional HLL systems. 
For the HHLL system, many factors such as cable selection, cable length, cable 
connection, system span and system installation may affect the behavior ofthe system and have 
not yet been sufficiently studied. Also, the cost of HHLL system, which depends upon the details 
of the system configuration, should be minimized. This leads to the necessity of research on the 
HHLL system. 
The primary objective of this thesis is to develop and illustrate a method to analyze the 
HHLL system subject to the load caused by a falling worker and provide an outline of the 
effective system configurations. 
13 
Chapter 4 Background of the Study 
 
 
This chapter presents a summary of the basic concepts and fundamental mechanics 
related to cable systems on which the analysis methodology developed in this thesis is based. 
These facilitate a better understanding of the behavior of the systems with simple configuration 
and help predict the response of the complex systems. 
 
4.1 Basic Concepts and Theoretical Assumptions 
The simplest cable system can be set up by attaching two ends of a cable to fixed 
anchorage points. The system under different types of loads is shown in Figure 4.1 - 4.3. In 
Figure 4.1, the concentrated load leads to the cable of V-shape. In Figure 4.2, the uniform load is 
distributed along the curved cable length and results in the cable of catenary shape. In Figure 4.3, 
the uniform load is distributed along the horizontal projection of the cable chord and results in 
the cable of parabolic shape. 
In the cable system, the straight line joining the two supports, i.e. line AB, is called the 
cable chord. The chord length, for a cable with small sag, is nearly equal to the initial cable 
length (unstressed or stressed) in the calculation when no load is applied. Under load, the cable 
will be elongated due to its elasticity and the length of the cable ACB is called the stressed cable 
length or, in the case of the system subject to distributed load, the curved cable length. The 
distance between the farthest point on the deflected cable and the cable chord is called the cable 
sag. This is the maximum deflection of the cable from its unloaded state. Usually for curved 
cables, parabolic or catenary, this sag is calculated from the deflection at mid-chord point. The 
angle between the deflected cable and the cable chord at the support is called the deflection 
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angle. The ratio of cable sag to the horizontal distance between the cable supports is the sag 
ratio. This ratio reflects the performance of the cable system. 
 
 
 A θ B 
 
 P f 
 
 C 
 
Figure 4.1 The Cable System Under Concentrated Load 
 
 
 
 
 p 
 
 A θ B 
 f 
 C 
 
 curved cable length L 
 
Figure 4.2 The Cable System Under Distributed Load with Catenary Shape 
 
 
 
 p 
 
 A θ B 
 f 
 C 
 
 curved cable length L 
 
Figure 4.3 The Cable System Under Distributed Load with Parabolic Shape 
 
 
Throughout this thesis, the system supports are assumed to be perfectly rigid. The 
relation between cable force and elongation is assumed to be linear along the stressed cable 
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length. The distributed load may be assumed to be uniformly distributed along the chord length, 
giving rise to the system with parabolic shape with little loss of accuracy if the sag ratio is 
relatively small, say less than 0.10. 
 
4.2 Cables Subject To Vertical Concentrated Loading 
For most cases of vertical loading in the structural cable problem with the assumptions 
noted in the above section, simple analytical solutions can be derived by applying the principles 
of statics. 
Shown in Figure 4.4 is a general case of a cable which is well anchored at two points A 
and B and acted upon by vertical loads P1, P2, …, Pn. Assuming the cable is perfectly flexible 
with respect to flexure, then the bending moment at any point on the cable must be zero. A 
general point m on the cable is considered, which is of distance y measured vertically from the 
cable chord AB. 
Taking moments about point B gives, 
Hh + VA S - Σ MB = 0 
or 
VA = S
M B∑ - 
S
Hh (4.2.1) 
The structure is split into two parts at m. Selecting the left part and taking moments about 
point m gives, 
H ( 
S
hx - y ) + VA x - Σ Mm = 0 (4.2.2) 
where: 
 Σ MB – sum of moments of all the loads P1, P2, …, Pn about support point B; 
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 Σ Mm – sum of the moments about point m of loads P1, P2, …, Pj that act on the cable 
to the left of m. 
 TB 
 VB 
 
 
 B H 
 
 
 
 y Pn 
 
 hx Pj h 
 S m 
 VA hx-Sy 
 S P4 
TA 
 
 A 
 H P2 P3 
 P1 
 x 
 
 S 
 
 
 Figure 4.4 General Cable System Subject to Concentrated Loadings 
 
Substituting VA from (4.2.1) into (4.2.2) gives, 
Hy = 
S
x Σ MB - Σ Mm (4.2.3) 
In Equation 4.2.3, the right side is equal to the bending moment that would occur at point 
m if the same loads as that applied to the cable in Figure 4.4 were to be applied to a beam of span 
17 
S and if m were a point on this beam whose distance from the left support is x. Thus, the 
following general cable theorem is implied: 
“At any point on a cable supporting gravity loads only, the product of the horizontal 
component of cable tension and the vertical distance from that point to the cable chord equals the 
bending moment which would occur at that section if the gravity loads were acting on a beam of 
the same horizontal span as that of the cable.” (Utku, 1948) 
 
4.3 Cables with Horizontal Chord Subject to Uniformly Distributed Loading 
The following derivation of equations for cable mechanics is based on Scalzi(1969). 
For the cable shown in Figure 4.5, the fictitious bending moment at any point x and 
maximum bending moment at mid-span are, respectively, 
Mx = 2
px ( S – x ) 
M = 
8
1 pS2 
hence, 
 Hf = 
8
1 pS2 
or 
H = 
f
pS
8
2
 (4.3.1) 
Substituting the above expressions for H and Mx into the general cable theorem, Hy = Mx, 
gives, 
f
pS
8
2
 y = 
2
px ( S – x ) 
18 
or 
y = 2
4
S
f ( Sx – x2 ) (4.3.2) 
 
 
 p 
 
 
 V S/2 S/2 V Tmax 
 
 H H 
 θ X 
 Y f 
 Y X 
 C 
 f 
 
 
 
Figure 4.5 The Cable System Subject to Uniformly Distributed Loading 
 
 
 
The absolute value of the maximum slope for this equation of a parabola occurs at the end points 
and is, 
Sxxdx
dy
==



,0
= 2
4
S
f ( S – 2x ) 
 = 
S
f4 (4.3.3) 
The cable tension at any point along the cable is given by 
T = H 
2
1 


+
dx
dy (4.3.4) 
19 
As the 
dx
dy is equal to maximum of 
S
f4 at x = 0 or x = S and n = 
S
f (sag ratio), the maximum 
cable tension occurs at the end points and is, 
Tmax = 22 HV + 
 = H
2
1 


+
H
V 
 = H
2
2
8
21








+
f
pS
pS
 
 = H 
241 


+
S
f 
 = H 2161 n+ (4.3.5) 
The maximum deflection angle θ is, 
θ = tan-1 


S
f4 (4.3.6) 
For easier derivation, the origin of the coordinates ( x , y ) may be set at the mid-point of 
the cable curve. Then, 
x = x + 
2
S and 
y = f - y 
Substituting Equation 4.3.2 into the above equation gives, 
y = f - 2
4
S
f ( Sx – x2 ) 
20 
 = f - 2
4
S
f (
4
2S - x 2 ) 
 = 2
4
S
f x 2 (4.3.7) 
Thus, the curved length of the cable may be derived as, 
L = ∫
L
ds
0
 
 = 2 ∫ 



+
2/
0
2
1
S
xd
xd
yd (4.3.8) 
Differentiating Equation 4.3.7 and substituting 
xd
yd into Equation 4.3.8 gives, 
L = 2 xd
S
xfS
2/1
2/
0
4
22641∫ 



+ (4.3.9) 
Integrating Equation 4.3.9 and expanding the integrand using binomial expansion gives, 
L = S 


+−+ ...
5
32
3
81 42 nn (4.3.10) 
Generally, the first two terms in the bracket are quantitatively accurate enough for L. Therefore, 
for practical accuracy, 
L = S ( 2
3
81 n+ ) (4.3.11) 
Due to axial stress, the cable will be lengthened. The elongation is, 
L∆ = ∫
L
AE
dssT
0
)( 
21 
 = ∫ 




 

+


 

+
S
AE
dx
dx
dy
dx
dyH
0
2/122/12
11
 (4.3.12) 
Differentiating Equation 4.3.2, substituting 
dx
dy into Equation 4.3.12 yields, 
L∆ = 
AE
H dx
dx
dyS∫ 


 


+
0
2
1 
 = 
AE
HS 


+ 2
3
161 n (4.3.13) 
In some cases, it is useful to have the relations between L∆ and S∆ as well as f∆ . The 
following is derived by differentiating Equation 4.3.10 but considering only the first two terms: 
L∆ = S∆ + 


 ∆
−
∆
2
22
3
8
S
Sf
S
ff 
 = Sn ∆


−
2
3
81 + fn∆
3
16 (4.3.14) 
 
4.4 Cable System with Single Load at Mid-span 
When a concentrated load is applied to the cable system at mid-span and the load far 
exceeds the total distributed load, the distributed load may be neglected with little loss of 
accuracy. This case is assumed in Figure 4.6. 
From simple geometry and equilibrium: 
V = P / 2 
H = 
L
S T and 
T = 22 VH + 
bruno
Highlight
22 
 
 V S/2 S/2 V 
 T 
 H A θ B H 
 
 P f 
 
 L/2 C 
 
 
 
Figure 4.6 The Cable System Subject to Concentrated Loading 
 
 
From the basic principles of statics and triangle geometry, the following relations can be found: 
f
SVH
2
= (4.4.1) 
T = 
f
VL
2
 (4.4.2) 
L = 2 2
2
2
fS +

 (4.4.3) 
f = 
22
22



−

 SL (4.4.4) 
AE
TLL =∆ (4.4.5) 
bruno
Highlight
23 
Chapter 5 Developing Methodology 
 
 
Single-span, simple multi-span and overlapped-cable multi-span HHLL systems with 
different configurations are considered in this thesis. Cable tensions and sags are calculated and 
used to judge the performances of these systems under the working load which would arise if a 
worker should fall while attached to the H-shaped Horizontal Lifeline system (HHLL). 
 
5.1 Additional Assumptions in the HHLL System Analysis 
As previously mentioned, one sliding connector connects the worker to the across cable 
and other two sliding connectors connect the across cable to the side cables (See Figure 3.4). As 
the friction on the movement of connectors is typically ineffective, it is neglected while still 
reaching reasonably accurate results. It is also assumed that the across cable will translate along 
the side cables when under significant load, i.e. the across cable is perpendicular to the side 
cables when a fall occurs. 
 
5.2 HHLL System Parameters 
The HHLL systems with the following parameters are examined: 
a. Cable size: half-inch 6 X 19 ( IWRC Galv’d IPS ), 
b. Cable cross-section area: 0.135 in2, 
c. Cable self-weight: 0.46 lbs/ft, 
d. Effective cable modulus of elasticity: E= 16 X 106 psi, 
e. Cable minimum break strength: Smin = 20700 lbs, 
24 
f. Side cable or across cable length: 25’ or 50’, 
g. Number of cables for side cable or across cable: 1 or 2, 
h. Working load applied at the middle of the span: P = worker + accessories + dynamic factors 
= 900 lbs. (This load is the maximum possible when the worker uses a personal energy 
absorber (EAP) which deploys at 900 pounds). 
 
5.3 Research Procedures and Findings 
To correctly interpret the HHLL system behavior, the analysis is developed in the 
following three stages. 
 
5.3.1 Single-span HHLL system with simplified method 
The single-span HHLL system layout is shown in plan view in Figure 5.1 and the system 
states before and after the fall are shown in Figure 5.2. Thirteen system configurations which 
combine different side cable lengths, across cable lengths, number of side cables and number of 
across cables are selected and analyzed. The parameters used for each of the configurations are 
given in Table 5.1. 
 
 25’ or 50’ 
 side cable (1 or 2)across cable (1 or 2) 
 
 25’ or 50’ 
 
 
Figure 5.1 Single-span HHLL System Layout
 
25
 
 
 
 
 
be
fo
re
 th
e 
fa
ll 
 
af
te
r t
he
 fa
ll 
 
Fi
gu
re
 5
.2
 
Si
ng
le
-s
pa
n 
H
H
LL
 S
ys
te
m
 B
ef
or
e 
an
d 
A
fte
r t
he
 F
al
l
 
 26 
 
Table 5.1 The Configurations in the Selected Systems 
 
No Layout Number Length (ft) Number Length (ft)
S1 1 50 1 50
S2 2 50 2 50
S3 1 25 1 25
S4 1 50 1 25
S5 1 25 1 50
S6 2 50 1 50
S7 2 25 1 25
S8 2 50 1 25
S9 2 25 1 50
S10 1 50 2 50
S11 2 25 2 25
S12 1 25 2 25
S13 1 50 2 25
Side Cable Across Cable
 
 
 27 
 
There are many simplified assumptions which may make the relatively complex analysis 
of the HHLL system more tractable. However, at the start, it is not known how accurate a 
solution will be which employs these assumptions. Therefore, a simplified method is first tried 
which ignores the cable self-weight. As shown in Figure 5.3, the across cable will take a V-shape 
in the vertical plane. Looking only at the across cable and supposing Ta is known, the unstressed 
cable length Sa equals the span, and fs is known, gives, 
aL = aS + AE
ST aa (5.3.1) 
af = 
22
cos
22



−−


αs
aa fSL (5.3.2) 
aT = 
a
a
f
PL
4
 (5.3.3) 
Ta is recursively adjusted until Ta converges in the foregoing set of 3 equations. 
 
 
 Sa 
 
 side cable support 
 connection point 
 side cable 
 fa 
 fs across cable 
 P 
 La /2 
 
 
 
 P/2 Ta 
 α = 



⋅ aT
P
2
arcsin 
 
 
Figure 5.3 The Across Cable Calculation 
 
 28 
 
 
Now looking only at the side cable, which assumes a V-shape in some plane, as shown in 
Figure 5.4, and supposing Ts is known while Ta comes from Equation 5.3.3 gives, 
sL = sS + AE
ST ss (5.3.4) 
sf = 
22
22



−

 ss SL (5.3.5) 
sT = 
s
sa
f
LT
4
 (5.3.6) 
 
 Ss 
 
 fs 
 
 Ls/2 side cable 
 Ta 
 
Figure 5.4 The Side Cable Calculation 
 
Ts is recursively adjusted until it converges. However, the fs obtained from Equation 5.3.5 
is needed for finding Ta. Thus the adjustment of Ta or Ts will cross affect each other but 
convergence can be realized finally. Microsoft Excel is applied to carry out the repetitive task. 
To evaluate the significance of neglecting the cable self-weight, the above steps are 
repeated except assuming that half the total of the cable self-weight is applied at the center point 
of the across cable along with the working load. The resulting calculations can be found in 
 
 29 
 
Appendix A and the results are summarized in Table 5.2. After comparison, it is found that the 
cable self-weight does have an effect on tension and sag in some cases and ignoring cable self-
weight in the analysis doesn’t seem to be a good shortcut. Not only taking the cable self-weight 
into consideration but also analyzing it in an accurate way should be the correct direction to 
follow. 
 
 
 30 
 
Table 5.2 Summary of the Calculation Results for the Single-span HHLL System 
with Simplified Method 
 
No Load
(lbs)
Cable
self-weight
(lbs)
Layout
Difference
(to no s-w)
(%)
Taf 1176.23 Taf 1215.42 3.33
S1 900 34.58 Tsf 7213.94 Tsf 7373.58 2.21
ft 124.22 ft 124.94 0.58
Taf 1288.59 Taf 1374.06 6.63
S2 900 69.15 Tsf 9653.69 Tsf 10076.47 4.38
ft 111.80 ft 113.06 1.12
Taf 1176.23 Taf 1195.85 1.67
S3 900 17.29 Tsf 7213.94 Tsf 7294.08 1.11
ft 62.11 ft 62.29 0.29
Taf 909.90 Taf 935.50 2.81
S4 900 28.81 Tsf 6077.53 Tsf 6191.14 1.87
ft 85.35 ft 85.78 0.50
Taf 1542.28 Taf 1576.23 2.20
S5 900 23.05 Tsf 8644.97 Tsf 8771.63 1.47
ft 91.52 ft 91.87 0.38
Taf 1286.35 Taf 1357.44 5.53
S6 900 57.63 Tsf 9642.49 Tsf 9994.92 3.65
ft 112.03 ft 113.09 0.95
Taf 1286.35 Taf 1321.96 2.77
S7 900 57.63 Tsf 9642.49 Tsf 9819.82 1.84
ft 56.01 ft 56.28 0.48
Taf 989.17 Taf 1038.95 5.03
S8 900 57.63 Tsf 8091.91 Tsf 8361.44 3.33
ft 76.63 ft 77.30 0.88
Taf 1691.00 Taf 1746.60 3.29
S9 900 57.63 Tsf 11573.78 Tsf 11826.45 2.18
ft 82.82 ft 83.30 0.58
Taf 1177.73 Taf 1230.05 4.44
S10 900 46.10 Tsf 7220.11 Tsf 7432.74 2.94
ft 124.04 ft 124.99 0.76
Taf 1288.59 Taf 1331.44 3.32
S11 900 34.58 Tsf 9653.69 Tsf 9866.74 2.21
ft 55.90 ft 56.22 0.57
Taf 1177.73 Taf 1203.93 2.22
S12 900 23.05 Tsf 7220.11 Tsf 7326.97 1.48
ft 62.02 ft 62.26 0.39
Taf 910.35 Taf 941.09 3.38
S13 900 34.58 Tsf 6079.55 Tsf 6215.84 2.24
ft 85.30 ft 85.81 0.60
 Cable tension & sag
(w/o cable self-weight)
(lbs, inch)
Cable tension & sag
(w cable self-weight)
(lbs, inch)
 
 
 31 
 
5.3.2 Single-span HHLL system with refined method 
Considering the practice of setting up systems on the site, two cases are examined. Case 
A (Known unstressed length) assumes that the unstressed across cable length before setting up 
the system is known. Case B (Sag control) assumes that the known sag under the cable self-
weight, often 1/120 of the unstressed across cable length, is applied as a control to set up the 
system before use. This stage is to analyze the single span systems with above configurations and 
to pre-select several configurations with better performance. Cable self-weight is taken into 
consideration and accurately calculated. 
Prior to the analysis, calculation of the cable sag under self-weight for a single cable is 
established. The model is shown in Figure 5.5. 
 
 S 
 
 p 
 
 
 f 
 
 L 
Figure 5.5 Establishing the Parameter Relation 
 
From Section 4.3, it is known, 
H = 
f
pS
8
2
 
n = 
S
f 
L = S 


+ 2
3
81 n and 
 
 32 
 
L = L0 + ∆L 
 = L0 + 


+ 2
3
161 n
AE
HS 
So it is derived, 
2
0
2
3
16
3
8 n
AE
HS
AE
HSLSnS ++=+ 
2
233
02
2
83
16
83
8
S
f
fAE
pS
fAE
pSLS
S
fS ++=+ 
( ) 03241664 40223 =−−+− pSfLSAESfpSAEf (5.3.7) 
In the following, three states of the HHLL system are considered. In the initial state, 
components of the HHLL system, i.e. side cable and across cable, are treated as independent 
systems subject toself-weight loading only. In the intermediate state, the HHLL system is still 
subject to the self-weight loading only but the self-weight of the across cable is calculated as the 
external force exerted on the side cables. In the final state, the HHLL system is subject to the 
working load as well as the self-weight loading and analyzed as a whole. 
 
1. The across cable in the initial state: 
This initial state is represented in Figure 5.6. Two cases are considered: Case A when the 
initial unstressed length is known; Case B when the initial sag is known. 
 
Case A: Known unstressed length 
In Case A, Sa0 is known but the span of the parabolic curved cable, Sa1, depends on the 
sag of the side cable. i.e., 
 aisaa fSS θcos201 −= 
 
 33 
 
The effective weight per foot over the span is, 
 
1
00
1
a
aa
a S
Sp
p = 
Assuming θai is known and substituting the above into Equation 5.3.7 gives, 
 03cos481664 4111
22
11
3
=−−− aaaisaiaaiaaai SpffAESfSpAEf θ (5.3.8) 
 
 
 Sa0 
 
 Sa1 
 
 pa1 
 
 ≈ θai 
 
connection point fai 
 θai 
 fsi Lai 
 
 
Figure 5.6 The Across Cable in the Initial State (Single-span) 
 
From Equation 5.3.8, fai and then θai can be calculated using equations in Section 4.3. The 
assumed θai is compared with the calculated θai, and a new θai is assumed. The steps are repeated 
until θai converges (The assumed equals the calculated). 
 
Case B: Sag control 
In Case B, Sa0 and fai 


=
120
0aS are known. Sa and pa1 are calculated in the same way as 
Case A, 
 
 34 
 
 aisaa fSS θcos201 −= 
 
1
00
1
a
aa
a S
Lp
p = 
Assuming θai is known and substituting the above into Equation 5.3.7 gives, 
 ( ) 03241664 41101122113 =−−+− aaaiaaaaiaaai SpfLSAESfSpAEf (5.3.9) 
From Equation 5.3.9, fai and then θai can be calculated. Similar to those in Case A, the steps are 
repeated until θai converges. 
 
2. The across cable in the intermediate state: 
This intermediate state is basically the same as the above initial state (A) and the 
calculation procedures are no different. 
 
3. The across cable in the final state: 
This final state is represented in Figure 5.7. If the concentrated load P is much greater 
than the total distributed load, the final shape is sufficiently represented by a V-shape and the 
curvature is neglected. Similar to the above initial state (A), either the unstressed length or the 
sag may be known. 
 Sa0 
 
 fsf 
 Laf/2 
 
 faf θaf 
 
 
 connection point P 
 
Figure 5.7 The Across Cable in the Final State (Single-span) 
 
 35 
 
Case A: Known unstressed length 
As the applied concentrated load is 900 lbs, the solution may be well approximated using 
the load plus the self weight assumed to act at a point. Then, 
00900 aa SpP += 
and 




=
−
af
af T
P
2
sin 1θ 
Assuming Taf is known gives, 
( )
AE
STT
LL aaiafaiaf
0−+= (5.3.10) 
2
0
2
cos
22



−−



= afsf
aaf
af f
SL
f θ (5.3.11) 
af
af
af f
LP
T
2
2
= 
 
af
af
f
PL
4
= (5.3.12) 
Here, Taf may be estimated, then Equation 5.3.10 to 5.3.12 are solved to obtain a new Taf, which 
is, in turn, used in Equation 5.3.10. The steps are repeated until Taf converges. 
 
Case B: Sag control 
 If the sag, rather than the unstressed length, is known, 
00900 aa LpP += 
and 
 
 36 
 




=
−
af
af T
P
2
sin 1θ 
Assuming Taf is known gives, 
( )
AE
LTT
LL aaiafaiaf
0−
+= (5.3.13) 
2
0
2
cos
22



−−



= afsf
aaf
af f
SL
f θ (5.3.14) 
af
af
af f
LP
T
2
2
= 
 
af
af
f
PL
4
= (5.3.15) 
Similar to Case A, Taf may be estimated, then Equation 5.3.13 to 5.3.15 are solved to obtain a 
new Taf, which is, in turn, used in Equation 5.3.13. The steps are repeated until Taf converges. 
 
4. The side cable in the initial state: 
This initial state is represented in Figure 5.8 and the calculation for the side cable in this 
state is the same whenever the across cable is in Case A or Case B. 
 
 Ss 
 
 ps 
 
 θsi 
 fsi 
 
 Lsi 
 
Figure 5.8 The Side Cable in the Initial State (Single-span) 
 
 
 37 
 
 
 
Simply applying SL =0 to Equation 5.3.7 gives, 
 031664 4223 =−− sssisssi SpfSpAEf (5.3.16) 
Then, fsi can be calculated from Equation 5.3.16. 
 
5. The side cable in the intermediate state: 
This intermediate state is basically the same as the final state except the loading is only 
the self-weight. The calculation procedures are no different from those in the final state (F) 
below. So to avoid repetition, only the calculations for the final state are presented. 
 
6. The side cable in the final state: 
This final state is represented in Figure 5.9 and the calculation for the side cable in this 
state is the same whenever the across cable is in Case A or Case B. If the concentrated load P is 
much greater than the total distributed load, the final shape is sufficiently represented by a V-
shape and the curvature is neglected. 
 
 
 Ss 
 
 θsf fsf 
 Lsf /2 P 
 
Figure 5.9 The Side Cable in the Final State (Single-span) 
 
 
 
 
 38 
 
The solution may be accurately approximated using the load plus the self weight assumed 
to act at a point. Then, 
afssaf SpTP θsin+= 
Assuming Tsf is known gives, 
( )
AE
STT
LL ssisfsisf
−
+= (5.3.17) 
22
22



−



=
ssf
sf
SL
f (5.3.18) 
sf
sf
sf f
LP
T
2
2
= 
 
sf
sf
f
PL
4
= (5.3.19) 
Here, Tsf may be estimated, then Equation 5.3.17 to 5.3.19 are solved to obtain a new Tsf, which 
is, in turn, used in Equation 5.3.17. The steps are repeated until Tsf converges. 
 
Again, Microsoft Excel is applied to carry out the repetitive task and the calculations can 
be found in Appendix B. It is noted that the adjustments of parameters may cross affect each 
other but convergence can be reached finally. 
The results are summarized in Table 5.3 and Table 5.4 and the comparison of the results 
for single-span system by the refined method is shown in Table 5.5. 
From the comparison, it is observed that doubling the side cable only or doubling the 
across cable only has the same effect on the sag, but the former decreases the tension per cable 
which otherwise approaches design strength limit. Besides, in Case A (known unstressed length), 
 
 39 
 
not only the strength of cable is not fully used, but also the sag is too much to be of any practical 
use. Therefore, it is decided to choose the systems with double side cables and single across 
cable which seem to have better performance. Four layouts, S6, S7, S8, S9, with sag control will 
be further adopted in the later analysis. 
 
40
 
 
 
Ta
bl
e 
5.
3 
 S
um
m
ar
y 
of
 th
e 
C
al
cu
la
tio
n 
R
es
ul
ts
 fo
r t
he
 S
in
gl
e-
sp
an
 H
HLL
 S
ys
te
m
 w
ith
 R
ef
in
ed
 M
et
ho
d 
 
(K
no
w
n 
U
ns
tre
ss
ed
 L
en
gt
h)
 
To
ta
l
N
o
La
yo
ut
Sa
g
 L
s0
L s
i
f si
T
si
L s
m
f sm
T
sm
L s
f
 f s
f
T
sf
L a
0
L a
i
 f a
i
T
ai
L a
f
f af
T
af
θ a
f
f t
S1
60
0.
00
60
0.
10
09
4.
76
36
36
3.
13
60
0.
21
04
7.
94
55
75
7.
50
60
2.
04
24
.7
7
73
52
.0
6
60
0.
00
60
0.
00
84
57
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5
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35
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7
S4
60
0.
00
60
0.
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09
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76
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36
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0.
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62
05
52
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97
60
1.
71
22
.6
9
61
67
.9
2
30
0.
00
30
0.
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2
12
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22
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30
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9
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7
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33
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13
12
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7
0.
35
92
11
2.
66
S7
30
0.
00
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0.
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44
30
0.
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0.
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30
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.0
8
97
48
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9
30
0.
00
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0.
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.9
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18
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6
30
0.
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52
.6
4
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99
.5
7
0.
35
83
56
.1
7
In
it
ia
l s
ta
teA
cr
os
s 
C
ab
le
Fi
na
l s
ta
te
Si
de
 C
ab
le
In
it
ia
l s
ta
te
In
te
rm
ed
ia
te
 s
ta
te
Fi
na
l s
ta
te
 
 
 
41
 
 
 
Ta
bl
e 
5.
3 
 S
um
m
ar
y 
of
 th
e 
C
al
cu
la
tio
n 
R
es
ul
ts
 fo
r t
he
 S
in
gl
e-
sp
an
 H
H
LL
 S
ys
te
m
 w
ith
 R
ef
in
ed
 M
et
ho
d 
(K
no
w
n 
U
ns
tre
ss
ed
 L
en
gt
h)
 (C
on
t'd
) 
To
ta
l
N
o
La
yo
ut
Sa
g
 L
s0
L s
i
f si
T
si
L s
m
f sm
T
sm
L s
f
 f s
f
T
sf
L a
0
L a
i
 f a
i
T
ai
L a
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af
θ a
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5
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S1
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60
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0.
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60
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7
62
15
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6
30
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30
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38
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37
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24
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8
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06
74
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92
9.
51
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51
96
85
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0
Si
de
 C
ab
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cr
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C
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le
In
it
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te
rm
ed
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te
 s
ta
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Fi
na
l s
ta
te
In
it
ia
l s
ta
te
Fi
na
l s
ta
te
 
 
 
42
 
 
 
Ta
bl
e 
5.
4 
 S
um
m
ar
y 
of
 th
e 
C
al
cu
la
tio
n 
R
es
ul
ts
 fo
r t
he
 S
in
gl
e-
sp
an
 H
H
LL
 S
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te
m
 w
ith
 R
ef
in
ed
 M
et
ho
d 
 
(S
ag
 C
on
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l) 
To
ta
l
No
La
yo
ut
Sa
g
 L
s0
L s
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f si
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 f s
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f
L a
0
L a
i
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θ α
φ
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S1
60
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00
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5
In
iti
al
 st
at
e
Ac
ro
ss
 C
ab
le
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na
l s
ta
te
Si
de
 C
ab
le
In
iti
al
 st
at
e
Fi
na
l s
ta
te
In
te
rm
ed
ia
te
 st
at
e
 
 
43
 
 
 
Ta
bl
e 
5.
4 
 S
um
m
ar
y 
of
 th
e 
C
al
cu
la
tio
n 
R
es
ul
ts
 fo
r t
he
 S
in
gl
e-
sp
an
 H
H
LL
 S
ys
te
m
 w
ith
 R
ef
in
ed
 M
et
ho
d 
 
(S
ag
 C
on
tro
l) 
(C
on
t'd
) 
To
ta
l
No
La
yo
ut
Sa
g
 L
s0
L s
i
f si
T s
i
L s
m
f sm
T s
m
L s
f
 f s
f
T s
f
L a
0
L a
i
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i
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θ a
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60
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10
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36
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25
60
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32
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27
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9
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2
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03
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5
0.
36
09
56
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1
Si
de
 C
ab
le
A
cr
os
s C
ab
le
In
iti
al
 st
at
e
In
te
rm
ed
ia
te
 st
at
e
Fi
na
l s
ta
te
In
iti
al
 st
at
e
Fi
na
l s
ta
te
 
 
 44 
 
 
Table 5.5 Comparison of the Results for the Single-span HHLL System with Refined Method 
Unit: lbs , in
No Layo ut Di fference(%)
Taf 1 2 0 1 .25 Taf 1 6 69 .6 2 3 8 .9 9
S1 Tsf 7 3 5 2 .06 Tsf 9 1 38 .7 3 2 4 .3 0
f t 1 2 4 .84 f t 86 .1 7 -3 0 .9 8
Taf 1 3 4 3 .57 Taf 2 0 81 .7 2 5 4 .9 4
S2 Tsf 10 0 0 6 .58 Tsf 1 3 3 41 .3 4 3 3 .3 3
f t 1 1 2 .85 f t 69 .8 2 -3 8 .1 4
Taf 1 1 8 8 .75 Taf 1 5 21 .2 6 2 7 .9 7
S3 Tsf 7 2 8 3 .17 Tsf 8 5 79 .0 9 1 7 .7 9
f t 6 2 .27 f t 47 .0 8 -2 4 .4 0
Taf 9 1 8 .81 Taf 1 1 83 .4 3 2 8 .8 0
S4 Tsf 6 1 6 7 .92 Tsf 7 2 79 .6 4 1 8 .0 2
f t 8 5 .69 f t 62 .5 2 -2 7 .0 4
Taf 1 5 7 5 .79 Taf 2 1 32 .2 7 3 5 .3 1
S5 Tsf 8 7 8 2 .56 Tsf 1 0 7 42 .3 3 2 2 .3 1
f t 9 1 .90 f t 66 .4 6 -2 7 .6 7
Taf 1 3 1 2 .77 Taf 1 8 12 .1 5 3 8 .0 4
S6 Tsf 9 8 5 4 .62 Tsf 1 2 1 73 .2 9 2 3 .5 3
f t 1 1 2 .66 f t 78 .9 2 -2 9 .9 5
Taf 1 2 9 9 .57 Taf 1 6 54 .9 7 2 7 .3 5
S7 Tsf 9 7 4 8 .79 Tsf 1 1 4 37 .7 7 1 7 .3 3
f t 5 6 .17 f t 42 .9 5 -2 3 .5 4
Taf 9 9 7 .72 Taf 1 2 85 .7 0 2 8 .8 6
S8 Tsf 8 2 5 2 .78 Tsf 9 7 20 .7 8 1 7 .7 9
f t 7 7 .03 f t 56 .8 1 -2 6 .2 4
Taf 1 7 2 7 .32 Taf 2 3 12 .1 0 3 3 .8 5
S9 Tsf 11 7 6 7 .04 Tsf 1 4 2 81 .0 7 2 1 .3 6
f t 8 3 .18 f t 61 .0 8 -2 6 .5 7
Taf 1 2 2 8 .92 Taf 1 9 22 .1 8 5 6 .4 1
S1 0 Tsf 7 4 6 3 .99 Tsf 1 0 0 34 .7 9 3 4 .4 4
f t 1 2 5 .12 f t 75 .9 1 -3 9 .3 3
Taf 1 3 1 6 .13 Taf 1 8 19 .2 8 3 8 .2 3
S1 1 Tsf 9 8 3 0 .99 Tsf 1 2 1 78 .8 1 2 3 .8 8
f t 5 6 .17 f t 39 .3 0 -3 0 .0 4
Taf 1 2 0 3 .37 Taf 1 6 74 .4 4 3 9 .1 5
S1 2 Tsf 7 3 4 2 .65 Tsf 9 1 44 .6 5 2 4 .5 4
f t 6 2 .29 f t 42 .9 5 -3 1 .0 5
Taf 9 2 9 .51 Taf 1 3 03 .8 5 4 0 .2 7
S1 3 Tsf 6 2 1 5 .26 Tsf 7 7 59 .8 8 2 4 .8 5
f t 8 5 .80 f t 56 .6 1 -3 4 .0 3
Sag contro l
Resul t
Diffe rence is re lative to the kno wn unstre ssed length.
Kno w n unstressed l eng th
Resul t
 
 
 45 
 
5.3.3 Multi-span HHLL system 
In reality, a working or operation area in many cases is very large. Although using a 
single-span system to cover a large area is theoretically possible, its performance and efficiency 
are not good enough to meet the need of fall protection. So multi-span HHLL systems are highly 
recommended in this situation. 
There may exist several ways of connecting cables in the multi-span HHLL systems. Two 
of them are analyzed here. One system is composed of several single-span units set up in the way 
that side cables are multi-spanned. Each single-span unit may work either dependently, i.e. side 
cables cross all spans and can slide over the supports; or independently, i.e. the side cables in 
each unit will transfer tensions to the supports directly and no force interaction of cables in 
neighboring units exists. Considering the fact in the former scheme that the fall in one unit will 
transfer tension to the cables in neighboring units and the cable elongations incurred in the 
neighboring units will add more sag to this unit, the latter scheme where single-span units work 
independently will be taken, As shown in Figure 5.10, the analysis of this system is not much 
different from that of the single-span system. 
The other system is also composed of several single-span units, but the side cables are 
connected in an overlapped manner, as illustrated in Figure 5.11. In this system, two cases are 
assumed. In case 1 (Clamped), the sliding connector will clamp on the two side cables during the 
fall and permit no sliding movement of itself relative to the cables. In case 2 (Free), the sliding 
connector can slide frictionlessly on the two cables at any time. 
 
46
 
 
 
Fi
gu
re
 5
.1
0 
 T
he
 S
im
pl
e 
M
ul
ti-
sp
an
 S
ys
te
m
 
 
 
 
47
 
 
 
Fi
gu
re
 5
.1
1 
 T
he
 O
ve
rla
pp
ed
-c
ab
le
 M
ul
ti-
sp
an
 S
ys
te
m
 
 
 48 
Calculation procedures for the multi-span system, especially for the across cable, are 
quite similar to those for the single-span system with sag control. 
 
1. The across cable in the initial state: 
This initial state is represented in Figure 5.12. Sag control is assumed and the initial sag 
is known. 
 
 Sa0 
 Sa1 
 pa1 
 ≈ θai 
connection point θai fai 
 fsi Lai 
 
Figure 5.12 The Across Cable in the Initial State (Multi-span) 
 
Here, Sa0 and fai 


=
120
0aS are known but the span of the parabolic curved cable, Sa1, 
depends on the sag of the side cable. i.e., 
 aisaa fSS θcos201 −= 
The effective weight per foot over the span is, 
 
1
00
1