fall protection.pdf

UNIVERSITY OF CINCINNATI

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I,______________________________________________,hereby submit this as part of the requirements for thedegree of:

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in:

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It is entitled:

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________________________________________________

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Approved by:________________________________________________________________________________________________________________________

Time History Analysis of the Dynamic Response of

Horizontal Lifelines

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

2003

by

Xiaohua Annie Yu

M.S., Tongji University, 1996 B.S., Ningbo University, 1993

Thesis Committee:

Dr. Frank E. Weisgerber, Chair Dr. T. Michael Baseheart

Dr. James A. Swanson

Abstract

Workers at elevated positions must be protected from falling or from hazardous

consequences of falls. A horizontal lifeline system (HLL) is a commonly used fall arrest system

(FAS) that provides protection such that when a fall does occur, the fall would be stopped

promptly in a manner that prevents injury.

Although the HLL systems have been in use for several decades, the design of these

systems is generally completed using simple methods. One popular design method is based on a

simplified energy balance analysis, which is to predict the maximum force and the maximum

displacement for a fall of one person. This method may be applied to the special case of multiple

persons falling only with the restrictive assumption that all people in the system must fall

precisely simultaneously. It is generally regarded that this assumption is implausible and

furthermore, the validity of consequent solution has not been verified because of limitation of the

analysis method. The present research shows the simultaneous-fall assumption may result in an

unconservative solution.

In this paper, a numerical time-history method is introduced. Based on this method, two

computer programs are developed: one for the single-person fall and the second for the two-

person fall. Satisfactory agreement has been found in comparison with previous research results.

Using these programs, an extensive parametric analysis has been conducted on the configuration

of the HLL system. Suggestions for design optimization are provided.

Acknowledgements

I owe my thanks and gratitude to my advisor Dr. Frank Weisgerber for being such a

wonderful mentor and friend. He is always being encouraging, understanding, and ready to help.

He not only shared with me his great insights and provided timely guidance; but also put

remarkable efforts improving readability of the entire paper. Without him this work would not be

possible.

I would like to thank Dr. Baseheart for being my co-advisor. He was an excellent

supervisor for my TA work and quite exhilarating to work for. I registered for his STRUCTURE

DYNAMICS and it turned out to be a dynamic and enlightening class. I wish I have taken the

TIMBER AND MASONRY class by him, so that the preparation of my PE exam could be

accelerated.

Thank Dr. Swanson for serving on my committee. I feel so lucky that I had Dr. Swanson

to be my instructor for the THEORY OF STEEL STRUCTURES class. In my memory he is

always ready and happy to answer questions, with his famous big smile on. I benefit so much

from his class and I am grateful.

I am thankful to my dearest sister Xiaoxia for being my best friend with her

unconditional love and support. As well I thank all my family and friends, whether they are

living cross the ocean or in Cincinnati. They are who make my life meaningful and wonderful. I

would like to dedicate the paper to them.

Table of Contents

List of Figures .................................................................................................................... iii

List of Symbols .................................................................................................................. vi

Chapter 1 INTRODUCTION

1.1 Why fall-protection? ................................................................................................1

1.2 Types and Applications of Fall-Protection ............................................................. 2

1.3 Components of an HLL system .............................................................................. 9

1.4 Design issues of HLL systems...............................................................................12

1.5 Objective ................................................................................................................13

Chapter 2 RESEARCH BACKGROUND........................................................................14

Chapter 3 METHODOLOGY DEVELOPMENT

3.1 Illustration of the problem .....................................................................................17

3.2 Numerical analysis method....................................................................................24

3.3 Algorithm of computer simulation ........................................................................29

3.4 Errors involved in numerical integration of dynamic motions problems of HLL system

......................................................................................................................................35

3.5 Cable Setup Condition ..........................................................................................36

3.6 Assumptions of the Method...................................................................................39

i

Chapter 4 PARAMETRIC STUDIES

4.1 Analytical result for a typical one-person fall........................................................43

4.2 Parametric analysis of one-person fall ...................................................................48

4.3 Analytical result for a typical two-person fall .......................................................57

4.4 Parametric analysis of two-person fall...................................................................65

Chapter 5 Conclusions ......................................................................................................75

References..........................................................................................................................78

ii

List of Figures

No. Name Page

Figure 1.2.1 A simple FAS System � Fixed Point Anchorage System 3

Figure 1.2.2 A Potential Hazardous Swing Fall 3

Figure 1.2.3 A HLL Fall-Protection System 4

Figure 1.2.4 HLL System Eliminating the Swing Fall in Lifeline Direction 5

Figure 1.2.5 Multi-Span HLL System Theorized as a Single-Span System 7

Figure 1.2.6 A Special Case Where Needs the H-shaped HLL System 8

Figure 1.2.7 H-shaped HLL System Theorized as a Single-Span System 9

Figure 1.3.1 HLL Components 10

Figure 3.1.1 Single -Person HLL System Before and After fall 18

Figure 3.1.2 Illustration of Models 19

Figure 3.1.3 A Cable System with Energy Absorbers 20

Figure 3.1.4 Free Body Diagram of a Cable System 20

Figure 3.1.5 Bisection Method 23

Figure 3.1.6 Flow Chart of Resisting Force Calculation 25

Figure 3.1.7 Resisting Force of an HLL system 26

Figure 3.2.1 Models of Systems 27

Figure 3.2.2 Analytic Methods 29

Figure 3.3.1 One Degree Dynamic System 31

iii

Figure 3.5.1 Cable Subject to Uniformly Distributed Load 39

Figure 3.5.2 Cable with Flexible Anchorages 40

Figure 3.5.3 Equivalent Lump Weight of Cable 40

Figure 3.6.1 Equilibrium Analysis at the Beginning of a Fall 43

Figure 3.6.2 Cable Weight Approximation 44

Figure 4.1.1 Time-History of Displacements 46

Figure 4.1.2 Time-History of Forces 47

Figure 4.1.3 Time-History of Velocities 47

Figure 4.1.4 Time History of Energy and Work 49

Figure 4.2.1 Displacements Comparison 51

Figure 4.3.1 Two-Person HLL System Before and After fall 60

Figure 4.3.2 Time-History of Displacements (Two Masses, ∆t=0.0) 62

Figure 4.3.3 Time-History of Forces (Two Masses, ∆t=0.0) 62

Figure 4.3.4 Time-History of Velocities (Two Masses, ∆t=0.0) 63

Figure 4.3.5 Time History of Energy and Work (Two Masses, ∆t=0.0) 64

Figure 4.4.1 Time-History of Displacements (Two Masses, ∆t=0.2s) 69

Figure 4.4.2 Time-History of Forces (Two Masses, ∆t=0.2s) 69

Figure 4.4.3 Time-History of Velocities (Two Masses, ∆t=0.2s) 70

Figure 4.4.4 Time History of Energy and Work (Two Masses, ∆t=0.2s) 70



iv







v

List of Symbols

A Cable effective cross-section area

E Cable effective elasticity modulus

EA Energy absorber

EAF Energy absorber threshold force

EAHLL Energy Absorber built into the cable

EAVLL Energy Absorber built into the lanyard

F Sag of the cable

FAS Fall arrest system

fs Resisting force in a dynamic system

FFD Free fall distance

H Horizontal component of the cable tension

HLL Horizontal lifeline

Kc Equivalent stiffness of the Columns or stanchions, i.e. anchorage stiffness

L Stressed length or curved length of the cable

L0 Unstressed length or length of the cable at zero tension

M, m Mass of a fall object

MAF Maximum arresting force of a fall arrest device

MAL Maximum Anchorage Loads

n Cable sag ratio, ration of sag to span

vi

P Force of the lanyard

S Nominal span of the cable

T Tension of the cable

TFD Total fall distance

V Vertical component of the cable tension

VLL Vertical lifeline

W Cable weight per unit length

x Anchorage spring deformation under anchorage force

vii

1

Chapter 1

INTRODUCTION

1.1 Why fall-protection?

Elevated fall hazard is an essential concern in every industry. According to the National

Safety Council Accident Facts (1997), falls to lower levels were the third leading cause of fatal

occupational injuries [1]. And in the field of construction, for example, statistics show even

higher rates of fall casualties for construction workers, with one in every six construction

workers suffering an occupational injury or illness on the average of about once a year. Falling is

the number one cause of construction accidents and deaths [3].

Every year, occupational accidents cost billions of dollars in economic loss. According to

The Business Roundtable survey in 1982 [2], direct and indirect costs of accidents account for a

staggering 6.5% of the cost in industrial, utility and commercial construction.

Besides the considerable economic loss of accidents, the toll in human waste, misery and

suffering through occupational injuries, sickness and deaths is immeasurable. Therefore, OSHA

(Occupational Safety and Health Administration) and other organizations such as NIOSH

(National Institute of Occupational Safety and Health) have drafted strict regulations on the issue

of fall protection, requiring that any worker in an elevated position must be protected from

falling or from the hazardous consequence of falls.

Both OSHA and ANSI (The American National Standards Institute) standards identify

the various circumstances under which fall protection is required: when the potential fall height

2

exceed 6 feet (1.8m), (OSHA 1926.502(d)(16)(iii)) for the construction industry, and 4 feet for

general industry.

Some common fall protection applications are listed as below:

o Structural framing

o Building rooftops

o Bridge construction

o Railcar and truck loading

o Industrial crane runways

o Pipe racks

o Aircraft hangars

o Arena rigging

o Exposed walkways

o Drilling

o Mining

1.2 Types and Applications of Fall-Protection

There are two main forms of fall protection: Fall Prevention and Fall Arrest. Fall

prevention devices for industrial use include fences, guardrails and warning lines, etc., and

restraint systems that prevent the workers from reaching any point from which they can fall. Fall

arrest devices include safety nets and a variety of fall arrest systems (FAS), which are able to

arrest a fall in progress while limiting fall distance and impact force to prevent harmful

3

consequences. Fall arrest systems are typically more appropriate for infrequently visited

locations where fall prevention systems are not feasible.

Fig 1.2.1 A simple FAS System – Fixed Point Anchorage System

Fig 1.2.2 A Potential Hazardous Swing Fall

Obstruction

4

A simple fall arrest system (FAS), which is called “fixed point anchorage system”,

consists of a harness worn by the worker attached by a fixed length lanyard to a single point

anchorage, as shown in Fig. 1.2.1. Another available simple fall arrest system is vertical lifeline

system (VLL). A VLL has a vertical rope to which a lanyard is attached by way of a rope grab. It

can be observed that, whether regarding design or installation, simplicity is the major advantage

of these systems. Unfortunately the fixed point systems and VLL system are sometimes

impractical as they greatly restrict the area that the worker can reach. Another deficiency for the

fixed point anchorage system is a high potential for a swing fall hazard when the worker moves

horizontally beyond the anchorage point, as shown in Fig 1.2.2. Therefore, those simple systems

are more applicable and efficient when workers merely need to move vertically within a small

horizontal range. A typical application is for workers on electric transmission towers.

Fig 1.2.3 A HLL Fall-Protection System

5

Beyond the fixed point anchorage system and VLL system there are some more

complicated fall arrest systems. The horizontal lifeline (HLL) is a widely-used one, which

combines a horizontal lifeline with a fall arresting lanyard, as shown in Fig. 1.2.3 and Fig. 1.2.4.

Since the hazard of the swing fall in the lifeline direction is eliminated, this system is especially

effective for cases when workers need to travel parallel to edges of a precipice, or the work

location is laterally distant from the available anchorage points. With a high degree of mobility

and safety when properly designed, the HLL system has been widely utilized in both industrial

and construction circumstances.

Fig 1.2.4 HLL System Eliminating the Swing Fall in Lifeline Direction

In this paper, the combination of the HLL rope and the connecting lanyard(s) are referred

to as the HLL system. The term “horizontal lifeline” is used either referring to the cable alone or

the HLL system.

The general HLL system has various forms including the single span HLL, multi-span

HLL, H-shaped HLL and a few other special forms.

The single span HLL is the most commonly used HLL system and contains the

fundamental concepts of design of HLL system. With only two anchorage points, the single span

HLL is limited to a straight line. Since long spans result in large deflections, accurate analysis is

Working area

6

needed to prevent the worker from hitting the ground before the fall is completely arrested. In

most applications, the single span HLLs are limited to about 100 feet in length and even then the

maximum dynamic deflection may be on the order of 20 feet.

The multi-span HLL system is designed based on the same general principles as used for

the single-span HLL system. Such a system has a lightly tensioned cable with its two extreme

ends fixed and running through and supported by several intermediate brackets. This enables a

longer extent of working area and a bent line route instead of just a straight line. The sliding

connector attaching the harnessed person via lanyards to the cable is often designed to freely

pass across the intermediate brackets, thus permitting the workers free movement over the whole

length of the cable. The advantage of the multi-span HLL is obvious. For example, a principle

advantage of the multi-span HLL, in comparison to multiple single-span HLLs, is that the

intermediate supports require far less strength than the end supports. Furthermore, at some

construction sites, efficiency can be observed by using only one multi-span HLL system to

protect workers working on more than one face of a building.

In calculation, the multi-span HLL system can be approximated as a single-span system

with lateral springs attached to the end anchorages, as shown in Fig. 1.2.5.

7

Mass

Intermediate bracketsEnd Anchor End Anchor

(a) Before falling

(a) Before Fall

Fallen Mass

Intermediate bracketsEnd Anchor End Anchor

(b) After falling

(b) After Fall

Anchor

Equivalent stiffness Kright Equivalent stiffness Kleft

(c) Theorized as a Single-Span System

Fig 1.2.5 Multi-Span HLL System Theorized as a Single-Span System

8

The H-shaped HLL system is rarely used due to having a relatively complex behavior.

This system is applicable to the situation where no anchorage point is feasible right above the

work site, for example, for a work area illustrated in Fig. 1.2.6. As shown, there are two side

cables attached to four anchorage points at four corners; 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. The H-shaped HLL system provides a broader work area than the ordinary

HLL system; however, it costs much more to design and install.

Fig. 1.2.6 A Special Case Where Needs the H-shaped HLL System

To analyze the H-shaped HLL system, it can be approximated as a single span system

with special anchorage condition as shown in Fig 1.2.7. The stiffness of the “equivalent” springs

is not linear, however, and must be related to the force-displacement characteristics of the side

cables.

9

equivalent stiffness Kleft

Anchor Point

equivalent stiffness Kright

Fig 1.2.7 H-shaped HLL System Theorized as a Single-Span System

In summary, the design of all HLL systems can be viewed as being based on the theory of

the single span HLL. In this paper, behavior of the single span HLL, with either one person or

two persons falling, is investigated.

1.3 Components of an HLL system

In general, an HLL system consists of all or some of the following components:

o Anchorages

o Tensioning device (not shown)

o Cable, or “lifeline” (wire rope, synthetic rope or a rigid rail)

o Energy absorbers (for the HLL and lanyard)

o Lanyard (fixed-length lanyard or SRL)

o Harness (full body)

o Connecting hardware (D-rings, turnbuckles and so forth)

10

Shock Absorber

Anchor Point CableAnchor Point

Shock Absorber

Lanyard

Sliding Connector

Fig 1.3.1 HLL Components

The cable and the lanyard may be regarded as the main components of an HLL system.

There are other components, such as shock absorbers, harness, turnbuckle, D-ring, which are

used to ensure that the HLL system works safely. Some of these components are optional and

may add complexity to the design of the HLL.

Generally there are two types of lanyard, fixed-length lanyard and self-retracting lanyard

(SRL). For the fixed-length lanyard, the length should be specified to maintain a free fall

distance not exceeding 6 feet (OSHA 1910.66 Appendix C). The self-retracting lanyard (SRL) is

a device connecting the worker to the cable, which acts mechanically similar in effect to those

straps in automobile seat belts. The lanyard in a SRL can be pulled out and retracted back easily

to avoid a slack line. But it has a brake that is activated to self-lock when a sudden pull indicates

that a fall occurs. Most self-retracting lanyards are able to limit free fall distance to 2 feet or less.

For both kinds of lanyard, a minimum breaking strength of 5000 lbs is required when tested

statically. (OSHA 1910.66 Appendix C, ANSI A10.14 (4.3.4.1)).

11

If a falling worker is connected to an anchorage point with a simple rope, for different

free fall distances, the maximum arrest force (MAF) would vary substantially and may reach a

high amount to injury the worker. Therefore, to reduce the force applied to the falling worker,

the lanyard usually includes an energy-absorbing (or shock-absorbing) device. Both OSHA

1910.66 Appendix C and ANSI Z359.1(3.2.4.7) require that the MAFs of the dynamically tested

energy absorbers shall not exceed 900 lbs, otherwise a high abrupt impact force may cause injury

to the human body as well as possible damage to the fall arrest system (FAS). The available

shock absorbers for lanyards in the industrial practices have a threshold force of 650 lbs to 900

lbs depending on model selected, and have a maximum deployment capacity of up to 3.5ft.

Energy absorbers are also introduced for the horizontal lifeline to reduce the hazard of

abruptly breaking the cable due to some high tension. Usually the energy absorber built into the

horizontal lifeline has a higher threshold force, up to thousands of pounds, while it has a lower

deployment capacity, sometimes only a few inches, because of the unfavorable associated

increase in deflection.

There are several different kinds of energy absorbers. The tear webbing energy absorbers,

for example, comprises a continuous length of webbing stitched to form a loop. When a

sufficient force is applied, the stitching tears which limits the arresting force until all of the

stitching has been torn and the webbing extended into a single straight length. Although

experimental results show that the force during stitch tearing fluctuates near the nominal

threshold force, the force is approximated to be constant in design calculations.

The turnbuckle may be used as an anchorage connector as well as a tensioning device.

The turnbuckle, as any other component of a HLL system, must have sufficient strength to

ensure that a minimum factor of safety of 2 exists for the entire system.

12

As a device connecting the person to the lanyard, the body belt has been abandoned by

OSHA and ANSI for any personal fall arrest system. A full body harness that is proven to

distribute force more evenly to the human body is required to be worn by the person.

Multiple persons may work using one same HLL system protection; however ANSI

A10.14 (4.3.3.5) limits users to a maximum of two at one time between supports.

1.4 Design issues of HLL systems

To prevent any harmful consequence when and if a fall occurs, it has to been affirmed

that the fall protection system is properly anchored and will prevent the worker from hitting an

obstruction or lower level before the fall has been completely arrested. Therefore two primary

factors must be considered in designing a horizontal lifeline:

o The forces that are applied to the anchorages during a fall arrest situation must be known

to design or verify the strength of the anchorages.

o The deflection of the lifeline during a fall arrest situation must be known to ensure the

worker will not contact an obstruction or lower level.

The deflection and forces are closely related. They are a function of several factors which

are the configuration of the HLL system, which includes initial setup condition of the lifeline,

number and type of energy absorbers, anchorage stiffness, number of workers connected, etc., as

well as loading factors such as the weight of the falling persons and the free fall distances.

In this paper, the anchor force of the lifeline, whether rigid or flexible, is studied. The

flexibility due to anchor flexibility, multiple rope spans or other effects, is simulated by a lateral

spring at the anchorage with a stiffness denoted as Kc.

13

1.5 Objective

Although HLL systems have been in use for decades, the design of systems has generally

been based on simplified and, it is hoped, conservative, analysis methods. A simplified energy

balance analysis method is often used, but this is only to predict the maximum forces and

displacement for falls of single person. Special cases of falls of multiple persons can be

considered if one takes the restrictive assumption that all falls of the multiple persons are

precisely simultaneous so that the maximum downward deflections of all the falling persons

occur at the same time. From a realistic point of view, simultaneous falls are not only extremely

unlikely, but the results of the assumption may not represent the worst case scenario.

In this research, the numerical time-history integration is applied to the dynamic

equations of motion. Based on this method, two computer programs are written in C++ to

simulate the entire falling process for the single-person fall and for the two-person fall

respectively. A number of calibration analyses are conducted to verify the algorithm and

accuracy of the programs. The results are compared with results from the energy balance

method.

Using the generated computer programs, an extensive parametric analysis is conducted

on the configuration of the HLL system. Suggestions for design optimization are provided.

14

Chapter 2

RESEARCH BACKGROUND

Although fall arrest systems have been in use for forty years or more, initial published

work on such systems first began to appear since 1970s. Such work concentrated on two areas:

prediction of the mechanical response of the system and the response of human bodies to shock

load. The former work includes prediction of the maximum forces applied to a falling worker,

the maximum forces incurred in the HLL components, and the dynamic deflection of the

systems. The latter work mainly focuses on medical studies of the human body under the load,

economical cost analysis, etc.

For prediction of response of FAS system, Sulowski and Miura [4,5] developed equations

to determine the lanyard force and fall distance using the energy balance method. Prior to a fall,

the shape of the HLL between supports is cantenary and therefore the classical solution of cable

was applied. According to the conservation law of energy, the total energy prior to a fall, which

is the potential of the mass at a height, would equal the sum of kinetic energy and strain energy

and energy dissipated by energy absorbers, dampers and frictions. If there is no energy

dissipated, when the falling mass reaches the lowest point, it will reach zero velocity so then

there is no kinetic energy. The total potential of the system will transfer solely to strain energy of

the cable and lanyard. From equating energy before the fall to energy after the fall, the position

of mass at zero velocity and the maximum force are directly related variables and can be

determined.

The energy balance solution is acceptable when these assumptions are met: that the

lanyard and HLL have negligible mass, that the lanyard and HLL have linear elastic properties,

15

that no energy is lost to the support structures or friction or damping. Using the energy balance

method, the maximum forces and the lowest point of the fall are determined at the instant when

the falling person first reaches the lowest point in the first arrest sequence. The results are proven

to have relatively good agreement with experimental results.

However, because there is only one independent equation involved in the energy balance

method, there is only one independent unknown allowed for this method. As a matter of fact the

energy balance method is limited to solve the one-mass fall problem for the HLL system. In

order to address the multiple-mass fall problem with the energy balance method, an assumption

has been made that the worst load case of the problem is to have multiple persons fall

simultaneously in a HLL system. By assuming multiple persons fall at the same time, multiple

sets of independent unknowns are made identical and therefore solvable by the method. But does

the solution for the forces and displacements under this assumption represent the worse case

scenario? More research is needed to shed light on this issue.

Concerned with the limitation of the energy balance method, Drabble and Brookfield [10]

developed a numerical analysis technique for predicting the forces occurring in each component

of an FAS during a fall. The HLL was simulated as a series of lumped masses coupled by spring

elements. Two multi-span HLL systems were tested, one system with a single-dummy and one

with four dummies dropped in the tests. Experimental results were compared against the energy

balance method and other analytical results.

Much work of the medical research area is concerned with the maximum arrest force

(MAF). MAF must not exceed the force that would cause significant injury to the falling person.

Research was carried out to determine the tolerance of the human body to forces due to falls.

Synder, et al, [11] conducted a major statistical survey which strongly justified the use of FAS in

16

terms of the injuries to be incurred by unprotected falling persons. Hearon and Brinkley [12]

surveyed falls of workers wearing harness, human suspension tests and USAF tests on parachute

opening injuries and occupational falls. This research shows that forces as low as 2 kN (453 lbs)

and 3 kN (680 lbs) applied via a harness may cause some level of injuries. Orzech and Wilkerson

[13] conducted experiments to evaluate capabilities of the different fall protection harnesses. As

a result, one configuration of harness was recommended for improving body support by

distributing loads over the human bony structures. The importance of rapid rescue of fallen

workers is also concluded from these experiments. Additional works [14] are conducted to build

expert systems for fall accident analysis, fall protection analysis, and accident cost and scenario

analysis.

17

Chapter 3

METHODOLOGY DEVELOPMENT

3.1 Illustration of the problem

For a single-person HLL system as shown in Fig 3.1.1, the dynamic motion problem can

be theorized as a single-degree dynamic system with a nonlinear stiffness K, as shown in Fig

3.1.2. The equivalent stiffness K is a function of mass position y at any arbitrary point in time.

For example, during the time period of a person’s free fall, stiffness K equals zero; after the

cable and lanyard become straight, K varies with respect to the person’s downward

displacement. And after a certain point, when the shock absorber in the lanyard starts to deploy,

K equals zero once again. Therefore it is essential to find the function of the equivalent stiffness

K.

Shock Absorber

Shock Absorber

Mass

Fig 3.1.1 Single-Person HLL System Before and After fall

18

M

mg

y

K=f(y)

(a)

cy

M

mg

y

fs

mÿ

(b)

.

(a) Equivalent Single-degree System; (b) Free-Body Diagram

Fig 3.1.2 Illustration of Models

To find the equivalent stiffness K is to find the system resisting force, fs, at a given

position y of mass M. It is important to note that the deployment of the energy of absorbers,

either in cable or in lanyard, is not recoverable, therefore the resisting force fs is route-

dependent, or time-dependent.

Resisting force fs equals the cable load as shown in Fig 3.1.3. Once fs reaches a certain

amount (EAVLLF), the energy absorber EAVLL in the lanyard begins to open. Therefore the

mass M undergoes a constant resisting force EAVLLF while it has further downward

displacement but there is no increment of sag of the cable. The mass position is related to sag

directly only when resisting force fs does not equal zero or EAVLLF. That is, cable and lanyard

are not slack, yet the internal force of the lanyard has not reached EAVLLF. For this case, to

solve the relationship between cable sag and applied load, an equilibrium analysis is employed.

Fig. 3.1.4 illustrates the forces in equilibrium.

19

fs

sag

x

EAHLL

EAVLL

y

Kc

Span

Fig 3.1.3 A Cable System with Energy Absorbers

Fig 3.1.4 Free Body Diagram of a Cable System

The equilibrium equations are:

(3.1.2) )(5.0

(3.1.1)

xSpansagHV

xKcH

−⋅⋅=

⋅=

Cable tension is the resultant of the horizontal and vertical component:

(3.1.3) 22 VHT +=

20

And the tension is related to the elastic elongation of the cable as:

(3.1.5) )4)((

(3.1.4) 2

)(2

0

022

02

2

0

LLsagxSpanAE

LL

AET

LsagxSpanLLL

−+−⋅⋅=∆

⋅=

−+

−

=−=∆

It is also seen that:

(3.1.6) 2Vfs =

In the above equations:

1. E, A, L0, Span are known, either as cable properties or by the system setup

conditions. The value of sag is given for each step in a solution and then the

unknowns are x and fs.

2. After the tension of the cable T reaches EAHLLF, the activating force of

energy absorber EAHLL, energy absorber EAHLL begins to deploy until it

reachs the maximum deployment capacity or the deployment ceases because

motion is reversed.

3. Energy absorbers, EAHLL and EAVLL are not retractable, which means once

a pull-out happens, this part will not be retracted back. As a result, the pull-out

will be added to the length of cable or lanyard. Therefore, tracking and

recording of the change of lengths of cable and lanyard for each step is

necessary.

Substituting the expressions for H, V into Eqn. 3.1.3, there is only one independent equation Eqn

3.1.8 as seen below:

21

(3.1.7b) )4)((

(3.1.7a) )(5.0

1

0

022

222

LLsagxSpanAE

T

xSpansagxKcVHT

−+−⋅⋅=

−⋅

+⋅⋅=+=

Therefore

(3.1.8) )4)((

)(5.01

0

0222

LLsagxSpanAE

xSpansagxKc

−+−⋅⋅=

−⋅

+⋅⋅

Eqn. 3.1.8 is a correlation function between x and sag. In other words, for any given

designated sag, there is a specific spring deformation x and vice visa. To solve the correlation

equation Eqn 3.1.8, the bisection method is used to find the solution of x for a given sag.

It can also be observed that for rigid anchorage, x, the deformation of the anchorage

spring equals zero. When x=0 substituted into Eqn. 3.1.7b, the tension force T can be obtained

instantly.

a0

b0

a1 b1

t

f(t

f(a0)

f(b0)

Fig 3.1.5 Bisection Method

22

The bisection method is a powerful and easy-to-use numerical method to find a single

root for a continuous function. The theoretical foundation of the bisection method is: so long as

the function f(t) whose root is sought is continuous, there must be at least one root between two

guesses, say a0 and b0, that give results f(a0) and f(b0). The bisection method locates such a root

by repeatedly narrowing the distance between the two guesses.

At any point of the simulation, the average of the positive and negative guesses, which is

displayed in Fig 3.1.5 as the t-coordinate of the bisection point, will be an approximation to a

root of f(t). Typically, one considers the midpoint m0 = )(21

00 ba + , and evaluates f (m0). If f (m0)

< 0, one sets a1 = m0 and b1 = b0. If f (m0) > 0, one sets a1 = a0 and b1 = m0. (If f (m0)=0, the

process is already done.) The situation in the present case is that f (a1) and f (b1) have opposite

signs, but the length of the interval [a1, b1] is only half of the length of the original interval [a0,

b0]. Repeating the simulation steps as shown above, the more steps that the simulation performs,

the better the approximation will be.

Eqn 3.1.8 may be rewritten as:

(3.1.9) 0 )4)((

)(5.01f(x)

0

0222

=−+−⋅⋅

−

−⋅

+⋅⋅=L

LsagxSpanAExSpan

sagxKc

For the correlation function between x and sag (Eqn 3.1.9), the initial guesses of the x

can be set as zero and Span.

Once Eqn 3.1.9 is solved for x, x is substituted back into Eqn.3.1.1 to 3.1.5 to get support

reaction forces H, V, cable tension T, and the resisting force of the cable system, fs.

A flow chart to solve the resisting force of the cable system is shown in Fig 3.1.6.

23

START STEP i

GET MASS POSITION y(i)

FREE-FALL OR EA2 ACTIVATED?

fs(i)=fs(i-1)

YES

EA1 ACTIVATED?

T=EAF1 SOLVE x, UPDATE L0

SOLVE x

CORRESPONDENT Sag

CALCULATE fs(i)

YES

NO

NO

Fig 3.1.6 Flow Chart of Resisting Force Calculation

A sample solution of the resisting force of the cable system with respect to the person

position is shown in Fig. 3.1.7. The free fall distance (FFD) in this case is 6 feet.

24

0

100

200

300

400

500

600

700

800

900

1000

0 2 4 6 8 10 12 14Person Fall Position (ft)

Lany

ard

Forc

e, C

able

Res

ista

nt F

orce

(lb)

Free Fall

EAHLL Deploying

EAVLL Deploying

Unloading

Fig 3.1.7 Resisting Force of an HLL system

3.2 Numerical analysis method

The numerical analysis required in this problem is to solve of the differential equations of

motion by arithmetic procedures. The one-mass fall problem in an HLL system can be viewed as

motion in a one-degree dynamic system. A one-degree system can be determined at any instant

by the single coordinate, as shown in Fig. 3.2.1, where the mass can move in a vertical direction

under the external force F(t).

25

M

F(t)

y

K

(a)

cỳ

M

F(t)

y

Ky

mÿ

(b)

(a) Equivalent Single-degree System; (b) Free-Body Diagram

Fig 3.2.1 Models of Systems

Applying Newton’s second law of motion, the equation of motion for a linear elastic

system can be written as

(3.2.1) )(tFkyycyM =++ &&&

Where ky is the elastic resisting force, yc& is the damping force, F(t) is the external force.

For a nonlinear system the equation of motion is

(3.2.2) )(),( tFyyfsycyM =++ &&&&

Where ),( yyfs & is the nonlinear resisting force at time t.

For an HLL system, the resisting force of the system is obviously geometrically

nonlinear even though the materials are in the elastic range. Furthermore, the excitation

force function is a constant value, which equals the weight of the person (including the

equipments the person carries).

(3.2.3) ),( MgyyfsycyM =++ &&&&

26

The initial conditions are

(3.2.4) 0 0 00 == yy &

For the one mass problem, resisting force fs=fs(y), which is solely determined by

the mass position y. For the two-mass problem, resisting force ),( yyfs & for each mass is a

function of the velocities of both masses as well as of the positions of the masses.

The process of numerical integration is a procedure by which the differential

equation of motion is solved step by step, starting at zero time when the displacement and

velocity are presumably known. The time scale is divided into discrete intervals, and one

progresses by successively extrapolating the displacement from one time station to the

next. In this research, the lumped-impulse procedure (also known as constant-velocity

procedure) is adopted.

The lumped-impulse procedure is essentially a constant-acceleration method. It is

assumed that over the small interval of time, ∆t, the acceleration of the system is

constant. For the lumped-impulse method, instead of assuming acceleration ÿ(s) at the

beginning of the interval is constant throughout time station s to s+1, the acceleration is

assumed constant from mid-point of s-1 and s to the mid-point of s to s+1. It is

categorized as a time-stepping procedure based on assumed variation of acceleration.

Fig.3.2.2 illustrates the lumped-impulse method in comparison to the constant-

acceleration method with acceleration assumed at the start of the time step.

27

ts-1 s s+1 ts-1 s s+1

y

: :

y

(a) (b)

(a) Lumped-impulse Method (b) Constant-acceleration Method

Fig 3.2.2 Analytic Methods

To explain the process, one may suppose the displacement y(s) at time station s

and y(s-1) at the preceding time station s-1 had been previously determined. The

acceleration at time station s can then be determined using the equation of motion. The

problem is to determine the next displacement y(s+1) by extrapolation:

y(s+1)=y(s)+vavg·∆t (3.2.5)

where vavg is the average velocity between time station s and s+1, and ∆t is the time

interval between two stations.

The average velocity may be expressed by the following approximate formula:

(3.2.6) v )(1)(

avg tytyy s

)(ss

∆+∆−

=−

&&

where the first term is the average velocity in the time interval between s-1 and s, and the

second term is the increase in the velocity between the two time intervals, assuming that

ÿ(s) is an average acceleration throughout that time period.

28

Substituting Eqn 3.2.6 into Eqn 3.2.5, the following recurrence formula is

obtained:

y(s+1)=2y(s)-y(s-1)+ÿ(s)·(∆t)2 (3.2.7)

With this equation, one is able to extrapolate to find the displacement at the next

time station s+1 based on results at station s and s-1. From Eqn 3.2.3, ÿ(s) may be

determined since it depends upon the displacement and velocity at station s.

The recurrence formula given by equation Eqn 3.2.7 is obviously approximate,

but it gives sufficiently accurate results provided that the time interval ∆t is taken small

in relation to the variations in acceleration. In fact, as ∆t goes to zero, the solution

becomes exact.

Using the recurrence formula, one is able to simply begin at time zero and

proceed step by step to determine the displacements at time station selected. Since at t =

0, no value of y(s-1) is available, it is necessary to set up for the first two steps.

For this specific problem, when t=0,

y(0)=0 (3.2.8)

When t= ∆t, step 1:

y(1)=1/2·ÿ ·( ∆t)2 (3.2.9)

when the initial condition is free fall, ÿ = g.

29

3.3 Algorithm of computer simulation

3.3.1 One person fall problem

M

mg

y

K=f(y)cỳ

M

mg

y

fs

mÿ

Fig 3.3.1 One-Degree Dynamic System

As discussed in section 3.2, the mechanics of a single person fall HLL system can

be written as a single degree of freedom dynamic motion problem.

(3.3.1) ),( MgyyfsycyM =++ &&&&

For each time step i, the displacement y (i), velocity y& (i) and acceleration y&& (i)

are known in a dynamic motion problem. As long as the resistant force fs(i) at each time

step is determined, the variables at step (i+1) can be obtained through numerical

integration. From section 3.1, it is known that during a fall the cable length and the

lanyard length are subject to change because of the built-in energy absorbers. Therefore

fs(i) is not only a function of the fall person’s position, but also a function of past

deployment of the energy absorbers. In order to determine fs(i), it is necessary to

understand and define phases during a fall.

30

There are three essential variables, which are sag position, lanyard length and

velocities of the persons and at cable mid-span, in order to determine a phase during a

fall. The cable’s sag position, i.e. HLL deformation is not only to establish the resistant

force function, but also necessary to record the zero-force position to determine at which

time step when the mass starts or finishes free fall. Lanyard length, with the built-in

energy absorber (EAVLL), is a time-dependent variable; it will remain constant as long

as the EAVLL is not activated. Once the EAVLL is activated to deploy, lanyard length

equals the original length with addition of the EAVLL deployment. Lanyard length is a

variable used to determine when a free fall is started or terminated. When the person is in

free fall or the energy absorber EAVLL is deploying, velocity of cable is zero, i.e. the

shape of cable remains constant. Otherwise, the velocity at the mid-point of the cable

equals the velocity of the mass.

The mass starts to free-fall from an elevation with zero displacement and zero

velocity. The lanyard is initially slack between the mass and horizontal cable. When the

distance between the mass and the cable’s sag zero-force position is less than the lanyard

length, the mass is still in free fall, and the cable remains still at the zero force position.

When the distance between the mass and the cable reaches the lanyard length, the

mass finishes free fall. Since then, HLL cable follows the mass movement with the same

velocity as that of the mass. Since the cable sag has deviated away from the zero force

position, a resistant force is generated and applied in turn to the falling mass.

Once the force in the lanyard, i.e. the resistant force from the cable, reaches the

threshold force of the EAVLL, the energy absorber EAVLL is activated and begins to

deploy. As a result of a constant resistant force, the cable stays still at the position with

31

the velocity of the cable back to zero. The lanyard length is extended with the increment

of EAVLL deployment.

As the resistant force fs remains at FEAVLL, the velocity of the mass is gradually

slowed down to zero then the direction is changed to the opposite, i.e. the mass starts to

rebound up. With the upward movement of the mass, cable sag position can no longer

stay still so a rebounding also occurs on the cable sag. The dependent variable of the

cable sag, resistant force fs, is hence decreased. Since the feature of the energy absorber

is only to deploy at a force no less than FEAVLL, so the EAVLL ceases deployment

right at the rebounding. The extended length of the lanyard is therefore recorded for the

future need in order to identify the free fall position of the mass in the next bouncing

cycle.

It is summarized that the motion of the cable (HLL) in the one-mass problem is

determined by the mass displacement history, resistant force history, velocity of the mass

and velocity of motion at the cable mid-span.

The motion phases of the mass can be categorized into 3 stages. First one is

“Free-Fall”. “Free Fall” is the mass fall with a slack lanyard and zero resistant force fs(i).

This phase is defined for when the mass falls at the beginning and when mass rebounds

freely in the air after fall. The second phase is for when lanyard is tight with a force

which is larger than zero but less than energy absorber EAVLL deploying force. The

second phase is denoted as “Lanyard-Straight” in this paper. “Lanyard-Straight” includes

when the mass was just finished free fall as well as when the mass is rebounding while

the lanyard is not slack yet. The last phase is “EAVLL Deploying,” which is for when

EAVLL is in deployment. During “EAVLL Deploying” the lanyard has a constant

32

resistant force (EAVLLF) and the lanyard length is prolonged, however the cable sag

position will remain still and the mass is having a downward velocity.

It can also be observed that when a mass in phase of “Free-Fall” or “EAVLL

Deploying”, the motion of the mass does not affect the cable’s deformation during the

time step. In other words, in these phases, the mass moves as though the cable has zero

stiffness.

3.3.2 Two-mass fall problem

The first mass, denoted as mass A, starts with free-fall from an elevation with

zero displacement and velocity. The lanyard is initially slack between the mass and

horizontal cable.

Before the second mass, denoted as mass B, joins mass A in falling, mass A’s fall

is identical to one-mass problem. By using the above criteria, the phases of mass A’s fall

and correspondent cable deformation can be obtained.

At an arbitrary time point mass B starts to fall as free-fall from an elevation with

zero displacement and velocity. The elevation may differ from mass A’s original

elevation. The lanyard connecting mass B to the HLL is initially slack as well.

Mass B will keep free falling until the slack portion of lanyard length is run out.

The slack portion of the lanyard length of mass B may vary with respect of time, because

the cable may be in a movement along with mass A. As long as the mass B is in “Free-

Fall”, the HLL system is a one-mass fall problem. Hence at each time step, cable sag

position corresponds with the status of the fall of mass A. The distance between the mass

33

B and the cable sag position is monitored and used to determine when mass B finishes

free fall.

After mass B finishes free fall, there are a few possible situations.

If mass A is in “Free-Fall”, the cable will follow mass B’s motion. Therefore

mass A keeps free-falling, mass B changes status from “Free-Fall” to “Lanyard-Straight”.

The correspondent resistant force from cable deformation goes solely to the mass B.

If mass A is in the phase of “Lanyard-Straight”, the distribution of the resistant

forces between the two masses depends on the velocity of the masses. When mass B has

a downward velocity faster than the mass A, mass B will take the entire resistant force so

that mass A instantly loses the force from lanyard and become “Free-Fall”. Vice versa if

the mass B’s velocity is lower than the mass A’s, mass A will continue to have the entire

resistant force and mass B will be “Free-Fall”. If the two mass happen to have same

velocity, the resistant force from the cable deformation will be divided by two and each

of them will have one half force. However the chance for this happening can virtually be

ruled out.

If mass A is in “EAVLL Deploying”, i.e. the energy absorber (EAVLL) in

lanyard of mass A is deploying and therefore the cable sag is remaining still, the

distribution of the resistant forces between the two masses as well depends on the

velocity of the masses. If mass A’s velocity is higher, motion of mass A will keep being

“EAVLL Deploying”, the cable sag will follow the mass B’s velocity and increase the

deformation. With energy absorber EAVLL deploying, mass A will get the same resistant

force as the previous time step. The increment of the resistant force from the cable

increased deformation will go to mass B; as a result the status of motion of mass B will

34

change from “Free-Fall” into “Lanyard-Straight”. If mass A’s velocity is lower than mass

B, mass B will take over the resistant force and its EAVLL will be activated instantly,

provided that the same energy absorbers are installed for the two masses in this case.

The motion phase of mass B will change from “Free-Fall” to “EAVLL Deploying”. With

a deploying EAVLL, the lanyard of mass B is equivalent to zero stiffness contributing to

the cable. Since the mass A is still having downward velocity, the cable sag will follow

the mass A’s velocity and increase its deformation, the increment of the resistant force

then will go to mass A. Thus the status of the motion of mass A will change from

“EAVLL Deploying” into “Lanyard-Straight”.

During the fall, there are 6 cases for the two masses interacting with each other.

1) Both in “Free-Fall”

2) One in “Free-Fall”, one in “Lanyard-Straight”

3) One in “Free-Fall”, one in “EAVLL Deploying”

4) Both in “Lanyard-Straight”

5) One in “Lanyard-Straight”, one in “EAVLL Deploying”

6) Both in “EAVLL Deploying”

The first three cases have been discussed above. The fourth case with both masses

in “Lanyard-Straight”, the mass with the higher velocity will have all resistant force so

that the other mass will have zero resistant force and therefore change status to “Free-

Fall”. The cable sag moves following the mass with the higher velocity.

35

The fifth case is one in “Lanyard-Straight”, one in “EAVLL Deploying”. The

mass in “EAVLL Deploying” will have a known force of MAF; the other mass will take

the rest of the force. The cable sag will move following the mass in “Lanyard-Straight”.

The sixth case is both in “EAVLL Deploying”. Each of the masses will have a

resistant force of MAF. The cable sag will be still until one of the masses finishes

“EAVLL Deploying” with changing of the direction of its velocity. Once the direction of

velocity is changed to upward, the status of the mass changes from “EAVLL Deploying”

to “Lanyard-Straight”. Then as described in the fifth case, the cable sag will move

upward following the mass in “Lanyard-Straight”.

3.4 Errors involved in numerical integration of dynamic motions

problems of HLL system

The errors introduced into the numerical integration, because the procedure, in

fact, provides a solution of a finite difference approximation of the equation of motion

rather than of the original difference equation, are common to both linear and nonlinear

systems. These errors will not be eliminated but can be reduced by increasing the number

of time steps and using a small time interval.

For nonlinear systems, there are additional sources of errors. These additional

errors can be classified as those arising due to (1) use of tangent stiffness in place of

secant stiffness, and (2) delay in detecting the transitions in the force-displacement

relationship.

The errors due to the use of tangent stiffness can be minimized by using an

iteration process, where using a method of checking equilibrium condition after each new

36

step, then applying the residual force repetitively until convergence is achieved. This

iteration process is in fact the full Newton Raphson Method.

The errors involved in numerical integration during transitions in the force-

displacement relationship can be minimized by using a sub-increment of time to carry out

the integration during an interval in which a transition is detected.

The criteria for the selection of the time step used in integration for a linear

system are related to the natural period of the system. For the nonlinear HLL systems,

since the stiffness changes drastically, the criteria used for the linear systems are not

appropriate. Therefore, in the present research, the time step is chosen based on (1)

stability and convergence of the results and (2) comparison of numerical solution to the

other calculations which have been verified with previous solutions by others and with

experimental results.

The accuracy of the numerical integration process may be verified by performing

the response calculation with two different time steps that are close to each other. If the

response obtained with the shorter time step is not significantly different from that

obtained with the larger time step, the process may be taken to have converged to the true

solution.

3.5 Cable Setup Condition

For industrial application, it is common practice to erect the HLL with a target

“sag ratio”, the ratio of initial sag to cable span, as a control condition for the erection.

37

p

S

fH

TV

H

VT

Fig 3.5.1 Cable Subject to Uniformly Distributed Load

From the basic theory of cable mechanics (Scalzi, [15]), for the cables with

horizontal chord subject to uniformly distributed load p, and when both ends of the cable

are rigidly anchored, where sag ratio n=f/S:

(3.5.4) 1618

(3.5.3) 8

(3.5.2) 21

(3.5.1) ...]n5

32-n38 1[

22

22

2

42

nf

pSHVT

fpSH

pV

SL

+=+=

=

=

++=

At erection, the external load p is the self-weight of cable, w.

38

p

S-x

f

H

T V

H

V T

x

Kc

Fig 3.5.2 Cable with Flexible Anchorages

If the cable is anchored with springs that represent the effect of flexibility of the

anchorage point, the bisection method may be used to solve for the length of the cable L

and the tension force T along the cable.

Fig 3.5.3 Equivalent Lumped Weight of Cable

The initial tension force T is due to the self-weight of cable. Using the value of T

in the cable and the initial setup condition, there is a concentrated equivalent force at

mid-span of the cable that causes the same tension force T and hence approximately the

same extension within the cable. Therefore the force is an equivalent lump weight of

Equivalent cable weight

w

T T T T

39

cable. The equivalent weight is a function of self-weight of cable w and set-up sag ratio

n. For the time-stepping method developed in this thesis, the equivalent weight is used to

approximate the set-up condition and simplify the procedure so that the effect of the

catenary shape of the cable can be neglected. This is justified in the present application

because the concentrated force due to a falling mass is far greater than the total cable

weight.

3.6 Assumptions of the Method

3.6.1 Damping effect neglected.

In fall-protection problems, the greatest interest is given to the maximum

displacement response and the maximum resisting force, which generally happen within

the first cycle of the vibration motion. Therefore damping effect is neglected for practical

and conservative reasons, while the margin of conservativeness is small. First, examining

the nature of the falling, the excitation of the dynamic motion is actually a rectangular

pulse with an extended period. According to the theory of dynamics, for pulse

excitations, the effect of damping on maximum response is usually not significant unless

the system is highly damped. Second, most structural engineering systems have a very

small damping ratio (usually estimated as 5% in structural calculations), especially in a

cable system. Third, during the first cycle of motion, the energy amount dissipated by

damping is very small in the limited time.

40

3.6.2 Infinite stiffness of lanyard

Since the stiffness of a lanyard is by far larger than that of the horizontal lifeline

(cable), in this research, the lanyard is assumed to have an infinite stiffness for

simplification in simulation. The omitted displacement of mass caused by the elongation

of lanyard is less than 0.1 inch, which is negligible in comparison to total fall distance.

3.6.3 Mid-span deformation of cable

Although the fall may happen when the person(s) is not necessarily right at or

even near the mid-span of the horizontal lifeline, it is assumed that after fall, the sliding

connector which connects the lanyard to the HLL, will slide the hanging person to the

mid-span of the cable. This has been confirmed by observation of experimental results.

From the equilibrium analysis as shown in Fig.3.6.1, the left-side HLL tension equals the

right-side HLL tension according to basic theory of cable mechanics. The resulting

lateral force is therefore always pointing to mid-span, which results in pushing the falling

mass to mid-span.

Accordingly, it is assumed that the lateral motion of mass due to the sliding from

the person’s fall point to the mid-point of cable would not strongly affect the falling

motion, though the resulting offset leads to a swing motion in x direction.

41

mg

sag

x

EAHLL

EA2

Kc

Span

y

x

T T

mg

θ1 θ2 T(-sinθ1+sinθ2)

mg-T(cosθ1+cosθ2)

Fig 3.6.1 Equilibrium Analysis at the Beginning of a Fall

It is also believed that to assume falls to occur near mid-span is conservative

overall. Experimental results showed that with a drop location closer to the end of the

span, generally the maximum anchorage loads (MAL) and total fall distance (TFD)

become smaller and shorter respectively. Concerns with safety requires interest in the

worst case scenario.

3.6.4 Some weight not collected

The weight of the energy absorbers, sliding connector and other small

miscellaneous masses are not taken in account.

42

The distributed weight of the cable is approximated as an equivalent lumped

weight at mid-span that introduces the same initial tension in the cable cross-section. This

simplification is reasonable so long as the falling mass is much larger than the mass of

the cable. Then the effect of the catenary shape of the cable can be neglected and it can

be assumed to be straight lines.

Equivalent cable weight

Fig 3.6.2 Cable Weight Approximation

3.6.5 No flutter effect of vibration considered

A possible flutter effect of cable and lanyard vibration due to a sudden impact is

common in the cable problem. The influence on the solution of maximum response is not

included here. The flutter effect would be significant to the predominant behavior only if

the concentrated force was small.

43

Chapter 4

PARAMETRIC STUDIES

4.1 Analytical result for a typical one-person fall

A stretched cable having a weight per unit length w = 0.46 lbs/ft, a span S = 60 ft and

initial sag ratio n = f/S = 0.1 in/ft was analyzed. The cable’s Young’s modulus is taken as E =

1.6E07 psi, and the effective area of cross-section A = 0.120 in2. The cable is assumed to be set

up between two cantilever columns for temporary use. The equivalent stiffness of two 7.5ft

W8x21 columns is Kc = 6.92 E3 lb/in. There are energy absorbers both for cable and for lanyard.

The threshold force for the energy absorber EAHLLF = 2000 lbs, and the extension capacity is

EAO1= 5 in. The threshold force for the energy absorber EAVLLF = 900 lbs, and the extension

capacity is EAO2= 5.0 ft. The weight of the falling person is M = 310 lbs, which is the

maximum allowable weight of workers by OSHA and ANSI standards for standard equipment.

The free fall distance is FFD = 6 ft, which is the maximum free fall distance permitted by OSHA

and ANSI.

Cable nominal span S = 60 ft

Cable initial sag (calibrated at set-up) f = 6 in

Cable initial length (zero tension) L0 = 59.9932 ft

Total weight of cable W0 = 27.600 lbs

Cable set-up tension due to self-weight T = 414.162

Equivalent lump weight at midspan W1 = 15.943 lbs

44

From erection condition of the cable, above are calculated using the method described in

Section 3.4.

Fig. 4.1.1 shows the time-history result of the displacement of the falling person and

position of the mid-span of the HLL. Fig. 4.1.2 shows the internal force of the lanyard and HLL.

Fig. 4.1.3 shows velocity the falling person and velocity of the motion at the HLL mid-span.

12.785612.0147

4.09493.5252

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (s)

Dis

p (ft

)

Person Disp Cable Disp

Fig 4.1.1 Time-History of Displacements

45

414.2

71.7

2000.0

3384.9

0

500

1000

1500

2000

2500

3000

3500

4000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Time (s)

Forc

e (lb

)

P1 Lanyard Force

Cable Tension

Fig 4.1.2 Time-History of Forces

-100

-50

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

Velo

city

(in/

s)

V (HLL) V (Person)

Fig 4.1.3 Time-History of Velocities

46

Summary of Analytical Result:

Results are compared as below with those of energy balance. Considering the energy

balance method usually omit the effect of the anchorage flexibility, a case with no anchorage

flexibility was studied.

Results Energy Balance

Method Time History

Method Error

Maximum Tension of HLL 3492 lbs 3487.2 lbs 0.1%

Maximum sag of HLL 3.96 ft 3.975 ft 0.3%

Final Pull of Energy Absorber in HLL 5 in 5 in 0.0%

Final Pull of Energy Absorber in Lanyard 3.24 ft 3.297 ft 1.7%

As a further confirmation of the validity of the numerical analysis method and program,

the potential, kinetic and strain energies and work done in the energy absorbers are calculated at

each time step. Fig. 4.1.4 shows these quantities plotted with respect of time. The zero height

position is assigned to the end of free fall distance. In this case zero height position is 6 ft lower

than the person start point. The figure suggests that, although the individual components change

with time, the total energy, which is the sum of the potential, kinetic and strain energies and

Free Fall Distance 6 ft

Maximum Tension of HLL 3384.94 lbs

Total Fall Distance of the person 12.79 ft

Maximum sag of HLL 4.094 ft

Final Pull of Energy Absorber in HLL 5 in

Final Pull of Energy Absorber in Lanyard 3.27 ft

47

work done, remains constant with negligibly small numerical error. Thus conservation of energy

is satisfied and the numerical analytical results are verified.

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (seconds)

Ener

gy (l

b-in

)

Gravitational

KE

EAVLL

EAHLL

HLL Strain

Anchorage Spring E

Fig 4.1.4 Time History of Energy and Work

48

4.2 Parametric analysis of one-person fall

To investigate the influence of parametric variation on the HLL system, the following

factors are studied. A comparison of results is made with respect to the case listed in section 4.1.

4.2.1 With or without energy absorber in HLL

Energy absorber EAHLL in HLL is not a necessary component of the system. The main

function of this energy absorber is to reduce the force in the HLL and thereby improve the safety

of anchorages and columns.

The erection conditions of the two compared cases are identical.

Cable nominal span S = 60 ft

Cable initial sag (calibrated at set-up) f = 6 in




Equivalent lump weight at midspan W1 = 15.943 lbs

From the comparison between results from the two methods, it is observed that the force

in the HLL is significantly larger when there is no HLL energy absorber. This could be

decreased by either choosing a shorter cable span or decreasing the initial cable tension (by

increasing the initial sag ratio at erection), or by the most effective way – adding an HLL energy

absorber in the cable. From the comparison, it is observed that the HLL energy absorber

effectively reduces the force in cable but meanwhile increases the total fall distance and therefore

requires a clearance increment of 1.55 ft.

49

Comparison of the Analytical Results:

Results With EAHLL Without EAHLL

Maximum Tension in HLL 3384.94 lbs 5321.65 lbs

Maximum HLL Midspan Displacement (Sag) 4.095 ft 2.589 ft

Total Fall Distance of the Person 12.79 ft 11.24 ft

Final Deployment of Energy Absorber in HLL 5 in NA

Final Deployment of Energy Absorber in Lanyard 3.27 ft 3.23 ft

Maximum velocity after first cycle 54.13 in/sec 77.83 in/sec

Rebound distance 0.77 ft 1.86 ft

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time(s)

Dis

p. (f

t)

Person Disp. With EAHLL

Person Disp. Without EAHLL

HLL Disp. Without EAHLL

HLL Disp. With EAHLL

Fig 4.2.1 Displacements Comparison

It also can be observed that there is no significant difference for the final deployment

length of the energy absorber in lanyard. However both kinetic energy and rebound distance of

50

the person in the second case are increased over 100%. As a result, the discomfort suffered by

the falling worker increases severely if the HLL energy absorber is not used.

4.2.2 Anchorage flexibility

As a matter of fact, anchorage stiffness in real sites usually varies significantly. For

example, the equivalent stiffness of two 7.5 ft W8x21 columns is Kc = 6.92 E3 lb/in; while for a

pair of cantilever columns with section of W10x19 and a height of 6.5ft, Kc = 1.082E4 lb/in, or,

if the columns are stanchions with braces at both ends, Kc = 8.653E4 lb/in.

For simplification, most energy balance methods omit the effect of flexibility of support

columns by assuming fixed-end anchorages, while the time-history method developed in this

research is competent in tracing the difference resulting from the variation of the anchorage

stiffness.

In order to study the effect of anchorage stiffness, three different sets are investigated

with Kc = 5,000 lb/in, Kc = 50,000 lb/in, and infinite stiffness, which represent anchoring to

flexible columns, stanchions, or permanent large columns or braced stanchions.

From comparison of the results, it can be observed that the effect of anchorage stiffness

does not play a significant role in the problem. However when using the energy balance method

with fixed-end anchorage assumption, the clearance calculation will be unconservative with an

error of several inches. On the other hand, the resultant maximum anchorage force found by the

energy balance method is conservative and acceptable.

Initial Condition of the HLL due to Erection:

Kc 5,000 lb/in 50,000 lb/in Infinite Kc

51

Cable Nominal Span S = 60 ft S = 60 ft S = 60 ft

Cable initial sag (calibrated at set-up) f = 6 in f = 6 in f = 6 in

Cable initial length (zero tension) L0 (ft) 59.9913 59.9913 59.9982

Cable set-up tension due to self-weight T (lbs) 414.135 414.221 414.229

Equivalent lump weight at midspan W1 (lbs) 15.9429 15.9429 15.9429

Comparison of Analytical Results:

Kc 5,000 lb/in 50,000 lb/in Infinite Kc

Maximum Anchorage Loads (lbs) 3352.03lbs 3473.2 3487.18

Maximum HLL Displacement (ft) 4.13471 3.99138 3.97497

Total Fall Distance of the person (ft) 12.8187 12.7085 12.6974

Final Pull of Energy Absorber in HLL (in) 5 in 5 in 5 in

Final Pull of Energy Absorber in Lanyard (ft) 3.25607 3.29172 3.29727

Maximum velocity after first cycle (in/sec) 56.0924 48.8515 47.8624

Rebound distance (ft) 0. 8314 0.6214 0.5952

4.2.3 Span variation

The impact of different spans of the cable is studied: 30 ft, 60 ft, and 90 ft. All HLLs

have the same set-up condition, e.g., control sag ratio n as 0.1 in/ft and have energy-absorbers in

HLL and lanyard.

Initial Condition of the HLL due to Erection

52

Cable Nominal Span 30 ft 60 ft 90 ft

Initial set-up sag (in) 3.0 6.0 9.0

Cable initial length (zero tension) L0 (ft) 30.0023 59.9932 89.9875

Total weight of cable (lbs) 13.801 27.600 41.3942

Cable set-up tension due to self-weight T (lbs) 207.115 414.162 621.344

Equivalent lump weight at midspan W1 (lbs) 7.97135 15.9429 23.9132

Comparison of Analytical Results:

Cable Nominal Span 30 ft 60 ft 90 ft

Maximum Anchorage Loads (lbs) 2623.64 3384.94 4041.54



Final Pull of Energy Absorber in HLL (in) 5 5 5



Rebound distance (ft) 0.1734 0.7711 1.1724

It can be observed that long span in HLL problem is unfavorable because it increases the

anchorage force, clearance requirement, human body’s uncomfortable level greatly. In this case

the multi-span HLL system should be taken into design consideration.

4.2.4 Free fall distance

53

The free fall distance (FFD) is basically the length of the slack part of the lanyard, which

is determined by total length of the lanyard, the working deck elevation, elevation of the person

harness, elevation of sliding connector on the cable, and activation distance for energy absorber.

Therefore it may vary for different designs, different work sites and different workers, from the

maximum free fall distance that is limited to 6 ft by OSHA and ANSI, to less than 2 ft for some

self-retracting lanyards.

3 different cases are investigated with a free fall distance (FFD) of 2ft, 4ft and 6ft with

same initial HLL set-up conditions. Fixed anchorage and built-in energy-absorbers in HLL and

lanyard are assumed.


Free Fall Distance 2 ft 4ft 6 ft



The comparison shows below that the free fall distance directly affect the clearance

requirement of a work site. Therefore at a work site with a limited fall clearance, the self-

retracting lanyard is highly preferred over the conventional HLL systems.

Comparison of Analytical Results

Free Fall Distance 2 ft 4ft 6 ft





54


Rebound distance (ft) 0.5966 0.5970 0.5952

4.2.5 Falling person’s weight

It has been long established in the drop tests for HLL systems to use 220 lbs (100kg)

dummies, however in OSHA and ANSI standards the upper limit of the worker’s weight is 310

lbs (140kg). The 220 lbs test weight has been established by some empirical results that the

human body in harness, with the fleshiness and springiness, is able to absorb impact energy up to

40%.

Further research is needed to verify the ability of a human body to absorb impact energy.

Furthermore every individual worker weight varies. To study the influence of the falling person’s

weight, 3 cases with same initial set-up condition are calculated with weight varying from 220

lbs, 265 lbs to 310 lbs.


Falling person weight (lbs) 220 265 310



Comparison of Analytical Results (with EAHLL, fixed anchorages):

Falling person weight (lbs) 220 265 310


55





Rebound distance (ft) 0.8395 0. 6971 0.5952

It has been argued recently that if it is reasonable to use dummies of 220 lbs (100kg) in

tests to represent harnessed workers of maximum weight of 310 lbs. Since analysis has shown

here that although the weight varies only 40% from 220 lbs to 310 lbs, the deployment of energy

absorber increases by 104%, which implies that a drastic increase of clearance is involved.

It can also be observed that for a relatively lighter worker, the phenomenon of

rebounding is severe; hence the degree of discomfort is increased. As an extreme case, a falling

with the worker weighs at 130 lbs is examined. As a results, it is found that the energy absorber

in lanyard has barely deployed with a merely 0.329587ft, while the rebounding distance up to

1.42118ft, maximum velocity up to 97.4581 in/sec. Though the fall arrest system still protects

the worker to a force within 900 lbs, it is not as effective to stabilize and “stop” the fall as for the

heavier workers. Hence an interesting question has emerged: “Does the ANSI standard need to

request for a minimum weight of a worker?”

4.2.6 Energy Absorber Threshold Force

For industrial common practice, there are two energy absorbers in terms of the threshold

force with 750 lbs and 900 lbs respectively. When a typical HLL system with a nominal span S =

60 ft and initial sag ratio n = f/S = 0.1 in/ft was analyzed, supported by two 7.5ft W8x21

56

columns (Kc = 6.92 E3 lb/in), it has been found in Section 4.2.1 that without energy absorber in

cable (EAHLL), the maximum anchorage force increases to 5321.65 lbs. Under this

circumstance, a 750 lbs energy absorber in lanyard could be a choice to solve the conflict. By

increasing the deployment of the energy absorber by 39.3%, the maximum anchorage force is

controlled within the legal limit.

Despite the increased clearance requirement, 750 lbs energy absorbers are recommended

to be used for circumstances where there are smaller free fall distances or lighter-weighted

workers involved.


Energy Absorber Threshold Force (MAF) 900 lbs 750 lbs

Cable Nominal Span S = 60 ft S = 60 ft

Cable initial sag (calibrated at set-up) f = 6 in f = 6 in

Energy Absorber in HLL NA NA

Comparison of Analytical Results (without EAHLL):

Energy Absorber Threshold Force (MAF) 900 lbs 750 lbs

Maximum Anchorage Loads (lbs) 5321.65 4729.48

Maximum HLL Displacement (ft) 2.589 2.436

Total Fall Distance of the person (ft) 11.24 12.37

Final Pull of Energy Absorber in Lanyard (ft) 3.23 4.51

Maximum velocity after first cycle (in/sec) 77.83 60.90

Rebound distance (ft) 1.86 1.42

57

4.3 Analytical result for a special case of two-person fall

In order to compare with the results from the energy balance method, a special case of

two persons fall is studied. In this case the two persons fall simultaneously with the same free

fall distance and reach the lowest point at the same time.

Mass2

Shock Absorber EAHLL

Shock AbsorberEAVLL1, EAVLL2

Mass1 Mass2

Shock Absorber EAHLL

Shock AbsorberEAVLL1, EAVLL2

Mass1

Fig 4.3.1 Two-Person HLL System Before and After fall

A similar setup condition for the HLL system is used as in Chap.3.1, except there are two

masses connected to the HLL. The horizontal cable has a weight per unit length w = 0.46 lbs/ft,

with a span S = 60 ft and set-up sag ratio n = f/S = 0.1 in/ft. The Young’s modulus of cable E =

1.6E07 psi and effective area A = 0.120 in2. The cable is set up between two cantilever columns

for temporary use. The equivalent lateral stiffness of the two 7.5ft W8x21 columns Kc = 6.92 E3

lb/in. There are energy absorbers EAHLL for cable and EAVLL for each of the lanyard

connecting person to the cable. The threshold force for the energy absorber built in the cable

EAFHLL = 2000.0 lbs, the extension capacity is 5.0 in. The threshold forces for the energy

58

absorbers built in both lanyards (EAFVLL) are 900.0 lbs with the extension capacity 5.0 ft. The

weight of the falling persons are identical, M1 = M2 = 310 lbs. 310 lbs is the maximum

allowable weight of the worker by OSHA and ANSI standard. The free fall distances for both

persons are (FFD) 6 ft, which is the maximum free fall distance allowed by OSHA and ANSI.

From erection condition of the cable, the following are calculated using the method

provided in Section 3.4.




Equivalent lump weight at mid-span W1 = 15.943 lbs

Fig. 4.3.2 shows the time-history result of displacement of the falling person and the mid-

span of the HLL. Fig. 4.3.3 shows the internal force of the lanyards and HLL. Fig. 4.3.4 shows

the velocity of the falling persons.

59

3.5247

4.5055

13.8182

12.5419

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (s)

Dis

p (ft

)

Cable DispPerson1 DispPerson2 Disp

Fig

4.3.2 Time-History of Displacements

900.1

414.272.2

2000.0

6107.1

0

1000

2000

3000

4000

5000

6000

7000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

Forc

e (lb

)

P1 Lanyard Force

P2 Lanyard Force

Cable tension

60


-100

-50

0

50

100

150

200

250

300

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time (s)

Velo

city

(in/

s)

V (HLL)V - Person1V - Person2


Summary of Analytical Result

Free Fall Distance 6 ft


Maximum Fall Distance of the person 13.82 ft

Maximum sag of HLL 4.51 ft

Final Pull of Energy Absorber in HLL 5 in

Final Pull of Energy Absorber in Lanyard 3.89 ft

61

Results are compared as below with those of energy balance. Considering the energy

balance method usually omit the effect of the anchorage flexibility, a case with no anchorage

flexibility was studied.

Results Energy Balance

Method Time History

Method Error

Maximum Tension of HLL 6402 lbs 6389.9 lbs 0.2%

Maximum sag of HLL 4.29 ft 4.307 ft 0.3%

Final Pull of Energy Absorber in HLL 5 in 5 in 0.0%

Final Pull of Energy Absorber in Lanyard 3.90 ft 3.914ft .3%


-40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (seconds)

Ener

gy (l

b-in

)

Potential p1

EAVLL1

Kinetic p1

HLL strain E

EAHLL

Potential p2

Kinetic p2

EAVLL2

Anchorage Spring E

62

Since the numerical analysis method idealize the HLL system as a closed system by

neglecting the energy worn by damping and air friction, at each time step the law of energy

conservation shall be satisfied as well. By verifying the energy conservation, the accuracy of the

program is established.

ET = EP + EK

ET: Total mechanical energy

EK: Kinetic energy

EP: Potential energy, the energy stored as the result of the position of an object.

General potential energy EP includes gravitational potential energy and elastic potential

energy. Strain energy is one type of elastic potential energy. Anchorage spring energy is elastic

potential energy.

For the system of the HLL and the attached persons, total energy are in forms of

gravitational potential of person 1, gravitational potential of person 2, kinetic energy of person 1,

kinetic energy of person 2, elastic potential energy of anchorage spring, energy absorbed by the

Energy Absorbers (EA) in each lanyard and cable, strain energy of the cable. Since the lanyards

are assumed non-extensible, the lanyards do not store or absorb any energy. The above

components of energy are calculated at each time step.

Fig. 4.3.5 shows the energy components plotted with respect of time. Zero height position

is assigned to the end of free fall distance. In this case zero height position is 6 ft lower than the

person start point. Table 4.3.1 lists the numerical results at some typical time steps. It can be

seen that, although the individual energy components vary with time, the total energy, which is

the sum of the quantity of the components, is a constant value with some negligible numerical

63

error less than 1.5%. Thus conservation of energy is satisfied and the numerical analytical results

are verified.

EPgrav,1 = m1gh1

EPgrav,2 = m2gh2

EK1 = 0.5m1v12

EK2 = 0.5m2v22

Estrain = 0.5T∆L = 0.5T * (TL/EA) for cable

Espring = 0.5 P∆L = 0.5 k * (∆L) 2 for anchorage spring

EEA = (Threshold Force) * (Deployment) for Energy Absorbers

64

Time (sec)

EPgrav,1 (lb.in)

EK1 (lb.in)

EEAVLL1 (lb.in)

Estrain (lb.in)

EAHLL(lb.in)

EPgrav,2 (lb.in)

EK2 (lb.in)

EEAVLL2 (lb.in)

Espring (lb.in)

Sum (lb.in)

0.0 22320.0 0.0 0.0 0.0 0.0 22320.0 0.0 0.0 0.0 44640.0

0.1 21721.1 600.1 0.0 0.0 0.0 21721.1 600.1 0.0 0.0 44642.4

0.2 19924.3 2398.1 0.0 0.0 0.0 19924.3 2398.1 0.0 0.0 44644.8

0.3 16929.7 5393.9 0.0 0.0 0.0 16929.7 5393.9 0.0 0.0 44647.2

0.4 12737.3 9587.5 0.0 0.0 0.0 12737.3 9587.5 0.0 0.0 44649.6

0.5 7347.0 14979.0 0.0 0.0 0.0 7347.0 14979.0 0.0 0.0 44652.0

0.6 758.9 21568.3 0.0 0.0 0.0 758.9 21568.3 0.0 0.0 44654.4

0.7 -6953.7 27529.8 0.0 717.8 2857.7 -6953.7 27529.8 0.0 287.1 45014.7

0.8 -15200.5 26385.8 1679.6 6960.1 9975.1 -15200.5 26385.8 1679.6 2635.2 45300.3

0.9 -22013.3 13421.4 21458.6 6960.1 9975.1 -22013.3 13421.4 21458.6 2635.2 45303.8

1.0 -26545.8 4797.5 34617.6 6960.1 9975.1 -26545.8 4797.5 34617.6 2635.2 45309.0

1.1 -28798.2 514.3 41156.7 6960.1 9975.1 -28798.2 514.3 41156.7 2635.2 45316.0

1.2 -28778.4 511.7 41943.0 5792.2 9975.1 -28778.4 511.7 41943.0 2196.8 45316.7

1.3 -26999.4 1895.7 41943.0 1151.9 9975.1 -26999.4 1895.7 41943.0 448.5 45254.0

1.4 -25132.6 795.4 41943.0 -31.2 9997.3 -25132.6 795.4 41943.0 0.4 45178.1

1.5 -24349.9 13.9 41943.0 -31.2 9997.3 -24349.9 13.9 41943.0 0.4 45180.4

1.6 -24765.1 430.3 41943.0 -31.2 9997.3 -24765.1 430.3 41943.0 0.4 45182.8

1.7 -26353.0 1764.1 41943.0 369.7 9997.3 -26353.0 1764.1 41943.0 152.5 45227.6

1.8 -28360.9 1109.0 41943.0 4287.4 9997.3 -28360.9 1109.0 41943.0 1631.0 45298.0

1.9 -29048.4 67.0 41951.1 6806.7 9997.3 -29048.4 67.0 41951.1 2577.6 45321.0

2.0 -27715.7 1679.1 41951.1 2499.4 9997.3 -27715.7 1679.1 41951.1 957.4 45283.3

65

4.4 Parametric analysis of two-person fall

It is important to study how the assumption of “simultaneous fall” in Energy

Balance method affects the result of two-person fall cases. Hereby a couple of two-mass

falls with different time lapse are studied.

4.4.1 Two masses fall with 0.2 second time difference

Though from the initial condition the two persons start to fall at a same height,

because the second person starts at 0.2 second later, the free fall distances (FFD) for the

two persons are different. When the second person falls, the cable position has been

pulled away from the zero force position. Hence the cable is to arrest the second person at

a lower position, i.e., the second person experiences more free fall distance (FFD) than

that of the first person.


First Person Free Fall Distance (FFD) 6.0 ft

Second Person Free Fall Distance (FFD) 9.55 ft


Maximum HLL Sag 4.51 ft

Maximum Fall Distance of Person 1 13.1 ft


Final Pull of Energy Absorber in HLL 5.0 in

Final Pull of Energy Absorber in Person 1 Lanyard 3.2 ft


66

4.51

13.1

14.5

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (s)

Dis

p (ft

)

Cable DispPerson1 DispPerson2 Disp


6102.7

414.2

2789.7

2000.0

3384.9

-1000

0

1000

2000

3000

4000

5000

6000

7000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Time (s)

Forc

e (lb

)

0

1000

2000

3000

4000

5000

6000

7000

8000

Forc

e (lb

) - P

erso

n 2

P1 Lanyard Force

Cable Tension

P2 Lanyard Force


67


-40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (seconds)

Ener

gy (l

b-in

)

Potential p1

EAVLL1

Kinetic p1

HLL strain E

EAHLL

Potential p2

Kinetic p2

EAVLL2

Anchorage Spring E


-100

-50

0

50

100

150

200

250

300

350

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time (s)

Velo

city

(in/

s)V (HLL)V - Person1V - Person2

68


As stated in previous example, because the second person starts later, the cable

position is away from the zero force position when it arrests the second person; as a result

the second person has more free fall distance (FFD) than the first person’s.











69

4.094.51

13.2

14.4

0

2

4

6

8

10

12

14

16

0 0.4 0.8 1.2 1.6 2 2.4

Time (s)

Dis

p (ft

) Cable DispPerson1 DispPerson2 Disp


3384.9

414.2

6020.0

2000.0

3384.9

-1000

0

1000

2000

3000

4000

5000

6000

7000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time (s)

Forc

e (lb

)

0

1000

2000

3000

4000

5000

6000

7000

8000

Forc

e (lb

) - P

erso

n 2

Force1

Cable Tension

Force2


70

-100

-50

0

50

100

150

200

250

300

350

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time (s)

Velo

city

(in/

s)



-40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Time (seconds)

Ener

gy (l

b-in

)

Potential p1

EAVLL1

Kinetic p1

HLL strain E

EAHLL

Potential p2

Kinetic p2

EAVLL2

Anchorage Spring E


71


When the second person starts to fall after 0.6 second elapses, the first person has

finished his/her first cycle of bouncing and the cable has been back to the zero force

position. Because of the deployment of the EAHLL in cable, the zero force position of

the cable is lower than initial condition. In this case, the second person still has more free

fall distance (FFD) than the first person’s.











72

4.094.51

12.7813.4714.16

14.5

0

2

4

6

8

10

12

14

16

0 0.4 0.8 1.2 1.6 2 2.4Time (s)

Dis

p (ft

)Cable Disp

Person1 Disp

Person2 Disp


6103.1

414.2

3385.7

2000.0

3384.9

-1000

0

1000

2000

3000

4000

5000

6000

7000

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Time (s)

Forc

e (lb

)

0

1000

2000

3000

4000

5000

6000

7000

8000

Forc

e (lb

) - P

erso

n 2

P1 Lanyard Force

Cable Tension

P2 Lanyard Force


73

-100

-50

0

50

100

150

200

250

300

350

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time (s)

Velo

city

(in/

s)



-40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Time (seconds)

Ener

gy (l

b-in

)

Potential p1

EAVLL1

Kinetic p1

HLL strain E

EAHLL

Potential p2

Kinetic p2

EAVLL2

Anchorage Spring E


74

4.4.4 Discussion

One of the primary factors must be considered in designing a horizontal lifeline is

the deflection of the fall person. During a fall arrest situation enough clearance has to

provide to ensure the worker will not contact an obstruction or lower level. From the

comparison of above cases, it can be seen that the maximum deflection of the fall person

can reach 14.5ft. If use energy balance method, where the simultaneous fall assumption

has to apply, the maximum deflection is only 13.82 ft and results in an unconservative

solution for design.

75

Chapter 5

CONCLUSION

5.1 Effect of energy absorber in HLL

When there is no HLL energy absorber, the force in HLL is significantly larger. This

could be decreased by either choosing a shorter cable span or decreasing the initial cable

tension by increasing the initial sag ratio at erection, or by the most effective way – adding an

HLL energy absorber in the cable. From the comparison, it is observed that the HLL energy

absorber effectively reduces the force in the cable but meanwhile increases the person’s

displacement and therefore requires a clearance increment of 1.55 ft.

5.2 Effect of Anchorage flexibility

The effect of anchorage stiffness does not play a significant role in the HLL problem.

However when using the energy balance method with fixed-end anchorage assumption, the

clearance calculation will be unconservative with an error of a few inches. On the other hand

the resultant maximum anchorage force is conservative and acceptable.

5.3 Effect of Span

It can be observed that long span HLLs are unfavorable because of increased

anchorage force, clearance requirement, and the human body’s discomfortable level greatly.

In this case, the multi-span HLL system should be taken into design consideration.

5.4 Effect of free fall distance

76

Parametric study shows that the free fall distance directly affects the clearance

requirement of a work site. Therefore at a work site with a limited fall clearance, the self-

retracting lanyard is highly preferred over the conventional HLL systems.

5.5 Effect of Falling Person’s Weight

For a relatively lighter worker, the phenomenon of rebounding is severe; hence the

degree of discomfort is increased. As an extreme case, a falling with the worker weighs at

130 lbs is examined. As a result, it is found that the energy absorber in lanyard has barely

deployed with merely 0.329587ft, while the rebounding distance is up to 1.42118ft, and

maximum velocity goes up to 97.4581 in/sec. Though the fall arrest system still protects the

worker to a force within 900 lbs, it is not as effective to stabilize and “stop” the fall as for the

heavier workers. Hence an interesting question has emerged: “Does ANSI standard need to

provide a limit for the minimum weight of a worker?”

5.6 Effect of Energy Absorber Threshold Force

Low threshold force energy absorbers are recommended to be used for circumstances

where there are smaller free fall distances or lighter-weighted workers involved.

5.7 Error and Limits involved in energy balance method

There are two primary factors must be considered in designing a horizontal lifeline.

The first factor is the forces that are applied to the anchorages during a fall arrest situation.

The second is the maximum deflection of the fall person to ensure during a fall arrest

situation enough clearance is provided that the worker will not contact an obstruction or

lower level. From the parametric analysis, it can be seen that the energy balance method is

77

adequate for one-mass HLL system. But for the two-mass HLL system, the energy balance

method results in an unconservative solution in the prediction of the maximum deflection of

the fall person.

78

References

1. Ellis, J. N., Introduction to Fall Protection, 2nd Edition, American Society of Safety

Engineers, Des Plaines, IL, 1994.

2. The Business Roundtable. Construction Industry Cost Effectiveness. Report A3. The

Business Roundtable, New York (1982).

3. Sulowski, Andrew C.; Brinkley, James W. “Measurement of maximum arrest force in

performance tests of fall protection equipment”, Journal of Testing & Evaluation. v. 18 issue

2, 1990, p. 123-127.

4. Sulowski, Andrew C., “Assessment of Maximum Arrest Force”, National Safety News. v.

123, n4. Apr 1981, p 50-53.

5. Miura, M.; Sulowski, A. C., Introduction to horizontal lifelines. In Fundamentals of Fall

Protection (Ed. A. C. Sulowski), 1991, p. 217-283, International Society for Fall Protection,

Canada.

6. American National Standard Institute, ANSI Standards, A10.14-1991 Requirements for

safety belts, harness, lanyards, and lifelines for construction and demolition use, June 1991.

7. American national Standard Institute, ANSI Standards, Z359.1-1992 Requirements for

personal fall arrest systems, sub-systems and components, September 1992.

8. Occupational Safety & Health Administration, U.S. Department of Labor, OSHA

Regulations (Standards – 29 CFR), Part 1926 Safety and Health Regulations for

Construction, Subpart M, Subpart X, Subpart R and Subpart L.

79

9. Occupational Safety & Health Administration, U.S. Department of Labor, OSHA

Regulations (Standards – 29 CFR), Part 1910 Occupational Safety and Health Standards,

Subpart I, Subpart D.

10. Drabble, F.; Brookfield, D. J., Safety of fall arrest systems: A numerical and

experimental study. In Proceedings of the Institution of Mechanical Engineers, 2000, Part C:

Journal of Mechanical Engineering Science v 214 n 10 p.1221-1233, 2000.

11. Snyder, R.G., Foust, D. R. and Bowman, B. M. Study of impact tolerance through free-

fall investigations. In Fundamentals of Fall Protection (Ed. A. C. Solowski), 1991, pp. 105-

122 (international Society for Fall Protection, Canada).

12. Hearon, B. and Brinkley, J. W. Fall arrest and post-fall suspension: Literature review and

directions for further research. US Air Force AFAMRL-TR-84-021, Air Force Medical

Research Laboratory, Wright-Patterson AFB, Ohio, 1984.

13. Orzech, Mary A.; Wilkerson, Terri L.; “Evaluation of full body harnesses during

prolonged motionless suspension of volunteers”, Proceedings – Annual SAFE symposium

(Survival and Flight Equipment Association) 25th, 1987, p. 1-6.

14. Chen, Jen-Gwo; Fisher, Deborah J.; Krishnamurthy, K. Development of a computerized

system for fall accident analysis and prevention, Computers & Industrial Engineering v 28 n

3 1995 p.457-466.

15. Scalzi, J.B., Podolny, W. Jr. and Teng, W.C. Design Fundamentals of Cable Roof

Structures, United States Steel Corporation, 1969.

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