UNIVERSITY OF CINCINNATI August 14 03 _____________ , 20 _____
Xiaohua Annie Yu I,______________________________________________, hereby submit this as part of the requirements for the degree of:
Master of Science ________________________________________________
in: Civil Engineering ________________________________________________
It is entitled: Time History Analysis of the Dynamic Response of ________________________________________________ Horizontal Lifelines ________________________________________________
________________________________________________ ________________________________________________
Approved by: Frank E. Weisgerber, PhD ________________________ T. Michael Baseheart, PhD ________________________ James A. Swanson, PhD ________________________ ________________________ ________________________
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 twoperson 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
Figure 4.4.5
Time-History of Displacements (Two Masses, ∆t=0.4s)
72
Figure 4.4.6
Time-History of Forces (Two Masses, ∆t=0.4s)
72
iv
Figure 4.4.7
Time-History of Velocities (Two Masses, ∆t=0.4s)
73
Figure 4.4.8
Time History of Energy and Work (Two Masses, ∆t=0.4s)
73
Figure 4.4.9
Time-History of Displacements (Two Masses, ∆t=0.6s)
75
Figure 4.4.10 Time-History of Forces (Two Masses, ∆t=0.6s)
75
Figure 4.4.11 Time-History of Velocities (Two Masses, ∆t=0.6s)
76
Figure 4.4.12 Time History of Energy and Work (Two Masses, ∆t=0.6s)
76
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
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
1
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
2
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
Obstruction
Fig 1.2.2 A Potential Hazardous Swing Fall
3
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
4
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.
Working area
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
5
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.
6
Intermediate brackets End Anchor
End Anchor
(a) Before falling Mass
(a) Before Fall
Intermediate brackets End Anchor
End Anchor
(b) After falling
Fallen Mass
(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 7
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.
8
Anchor Point
equivalent stiffness Kright
equivalent stiffness Kleft
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)
9
Shock Absorber Anchor Point
Sliding Connector
Cable Anchor Point
Shock Absorber Lanyard
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)).
10
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.
11
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.
12
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.
13
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,
14
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
15
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.
16
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
17
fs mÿ
. cy
K=f(y)
M
M
y y mg
mg (b)
(a)
(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 routedependent, 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.
18
Span x Kc
EAHLL sag
EAVLL fs
y
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: H = Kc ⋅ x V =H⋅
(3.1.1)
sag 0.5 ⋅ ( Span − x)
(3.1.2)
Cable tension is the resultant of the horizontal and vertical component: T = H 2 +V 2
(3.1.3)
19
And the tension is related to the elastic elongation of the cable as: ( Span − x) 2 ∆L = L − L0 = 2 + sag − L0 2
(3.1.4)
E ⋅ A ⋅ ( ( Span − x) 2 + 4 sag 2 − L0 ) E⋅A T= ∆L = L L0
(3.1.5)
2
It is also seen that:
fs = 2V
(3.1.6)
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:
20
sag T = H + V = Kc ⋅ x ⋅ 1 + 0.5 ⋅ ( Span − x) 2
2
2
(3.1.7a)
E ⋅ A ⋅ ( ( Span − x) 2 + 4sag 2 − L0 ) T= L0
(3.1.7b)
Therefore 2
E ⋅ A ⋅ ( ( Span − x) 2 + 4 sag 2 − L0 ) sag = Kc ⋅ x ⋅ 1 + L0 0.5 ⋅ ( Span − x)
(3.1.8)
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.
f(t f(b0)
a0
a1
b1 t b0
f(a0) Fig 3.1.5 Bisection Method
21
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 1 root of f(t). Typically, one considers the midpoint m0 = (a 0 + b0 ) , and evaluates f (m0). If f (m0) 2 < 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: 2
E ⋅ A ⋅ ( ( Span − x)2 + 4 sag 2 − L0 ) sag f(x) = Kc ⋅ x ⋅ 1 + =0 − L0 0.5 ⋅ ( Span − x)
(3.1.9)
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.
22
START STEP i
GET MASS POSITION y(i)
FREE-FALL OR EA2 ACTIVATED?
YES
NO fs(i)=fs(i-1) CORRESPONDENT Sag
YES EA1 ACTIVATED?
T=EAF1 SOLVE x, UPDATE L0
NO SOLVE x
CALCULATE fs(i)
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.
23
1000
EAVLL Deploying
Lanyard Force, Cable Resistant Force (lb)
900 EAHLL Deploying
800 700 600 500 Free Fall
400 300 200 100 0 0
2
4
6
8
10
Person Fall Position (ft)
12
14 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).
24
Ky cỳ
K
mÿ
M M
y y F(t)
F(t) (b)
(a)
(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 M&y& + cy& + ky = F (t )
(3.2.1)
Where ky is the elastic resisting force, cy& is the damping force, F(t) is the external force. For a nonlinear system the equation of motion is M&y& + cy& + fs ( y, y& ) = F (t )
(3.2.2)
Where fs ( y, y& ) 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). M&y& + cy& + fs ( y, y& ) = Mg
25
(3.2.3)
The initial conditions are y0 = 0
y& 0 = 0
(3.2.4)
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 fs ( y, y& ) 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 constantacceleration method with acceleration assumed at the start of the time step.
26
:
: y
y
s-1
s
s+1
s-1
t
(a)
s
s+1
t
(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: v avg
y ( s ) − y (s −1 ) = + &y&( s ) ∆t ∆t
(3.2.6)
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.
27
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 when the initial condition is free fall, ÿ = g.
28
(3.2.9)
3.3 Algorithm of computer simulation 3.3.1 One person fall problem fs cỳ
K=f(y)
mÿ
M
M
y y mg
mg
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.
M&y& + cy& + fs( y, y& ) = Mg
(3.3.1)
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.
29
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
30
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
31
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 “FreeFall”, 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
32
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
33
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 “FreeFall”. The cable sag moves following the mass with the higher velocity.
34
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 35
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 forcedisplacement 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.
36
p T
V
V
H
T H
f S
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: 8 32 L = S[1 + n 2 - n 4 + ...] 3 5 1 V = p 2 pS 2 H= 8f pS 2 T = V +H = 1 + 16n 2 8f 2
2
At erection, the external load p is the self-weight of cable, w.
37
(3.5.1) (3.5.2) (3.5.3) (3.5.4)
p T
T
V
V
H
H f Kc x
S-x
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. w T
T
T
T
Equivalent cable weight
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
38
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.
39
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.
40
Span x
Kc
EAHLL sag
EA2 mg
T
θ1
θ2
T
T(-sinθ1+sinθ2)
x y
mg
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.
41
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.
42
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
43
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.
14 12.7856 12.0147
12 Person Disp
Disp (ft)
10
Cable Disp
8 6 4.0949
4
3.5252
2 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Fig 4.1.1 Time-History of Displacements
44
1.6
1.8
2
4000 P1 Lanyard Force 3384.9
3500 Cable Tension 3000
Force (lb)
2500 2000.0 2000
1500
1000
414.2
500
71.7 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (s)
Fig 4.1.2 Time-History of Forces
300 V (HLL)
V (Person)
250
Velocity (in/s)
200 150 100 50 0 0
0.2
0.4
0.6
0.8
1
1.2
-50 -100 Time (s)
Fig 4.1.3 Time-History of Velocities
45
1.4
1.6
1.8
2
Summary of Analytical Result: 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
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
3492 lbs
3487.2 lbs
0.1%
3.96 ft
3.975 ft
0.3%
5 in
5 in
0.0%
3.24 ft
3.297 ft
1.7%
Maximum Tension of HLL Maximum sag of HLL Final Pull of Energy Absorber in HLL Final Pull of Energy Absorber in Lanyard
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
46
work done, remains constant with negligibly small numerical error. Thus conservation of energy is satisfied and the numerical analytical results are verified. 40,000 EAVLL
30,000 KE
Energy (lb-in)
20,000
EAHLL 10,000 HLL Strain 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Anchorage Spring E -10,000 Gravitational -20,000
-30,000 Time (seconds)
Fig 4.1.4 Time History of Energy and Work
47
1.6
1.8
2
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
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
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.
48
Comparison of the Analytical Results: Results
With EAHLL
Without EAHLL
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
5 in
NA
3.27 ft
3.23 ft
54.13 in/sec
77.83 in/sec
0.77 ft
1.86 ft
Maximum Tension in HLL
Final Deployment of Energy Absorber in HLL Final Deployment of Energy Absorber in Lanyard Maximum velocity after first cycle Rebound distance
14 Person Disp. With EAHLL 12
10
Disp. (ft)
Person Disp. Without EAHLL 8
6 HLL Disp. With EAHLL 4
2 HLL Disp. Without EAHLL 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time(s)
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
49
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
50,000 lb/in
Infinite Kc
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
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
Final Pull of Energy Absorber in HLL (in)
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
51
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
Maximum HLL Displacement (ft)
2.63658
4.095
5.1791
Total Fall Distance of the person (ft)
11.2636
12.79
13.8566
5
5
5
Final Pull of Energy Absorber in Lanyard (ft)
2.91293
3.27
3.54124
Maximum velocity after first cycle (in/sec)
26.0869
54.1721
66.4651
Rebound distance (ft)
0.1734
0.7711
1.1724
Final Pull of Energy Absorber in HLL (in)
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
52
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. Initial Condition of the HLL due to Erection Free Fall Distance
2 ft
4ft
6 ft
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
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 selfretracting lanyard is highly preferred over the conventional HLL systems. Comparison of Analytical Results Free Fall Distance
2 ft
4ft
6 ft
Maximum Anchorage Loads (lbs)
3488.70
3487.85
3487.18
Maximum HLL Displacement (ft)
3.97342
3.97558
3.97497
Total Fall Distance of the person (ft)
6.59324
9.64455
12.6974
Final Pull of Energy Absorber in Lanyard (ft)
1.19504
2.24333
3.29727
53
Maximum velocity after first cycle (in/sec)
47.8833
47.916
47.8624
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.
Initial Condition of the HLL due to Erection Falling person weight (lbs)
220
265
310
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
Comparison of Analytical Results (with EAHLL, fixed anchorages): Falling person weight (lbs) Maximum Anchorage Loads (lbs)
54
220
265
310
3487.31
3488.13
3487.18
Maximum HLL Displacement (ft)
3.97532
3.9734
3.97497
Total Fall Distance of the person (ft)
11.0172
11.7973
12.6974
Final Pull of Energy Absorber in Lanyard (ft)
1.6163
2.39826
3.29727
Maximum velocity after first cycle (in/sec)
65.8099
55.8625
47.8624
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
55
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. Initial Condition of the HLL due to Erection 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
NA
NA
Energy Absorber in HLL
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
56
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. Shock Absorber EAHLL
Shock Absorber EAVLL1, EAVLL2 Mass1
Mass2
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 57
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. 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 Equivalent lump weight at mid-span
T = 414.162 W1 = 15.943 lbs
Fig. 4.3.2 shows the time-history result of displacement of the falling person and the midspan 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.
58
16
14
13.8182 12.5419
12
10
Cable Disp Person2 Disp
8
6 4.5055 4 3.5247 2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Fig 4.3.2 Time-History of Displacements
7000 P1 Lanyard Force 6000
6107.1
P2 Lanyard Force Cable tension
5000
Force (lb)
Disp (ft)
Person1 Disp
4000
3000 2000.0 2000
900.1
1000 414.2
72.2 0 0
0.2
0.4
0.6
0.8
1 Time (s)
59
1.2
1.4
1.6
1.8
2
Fig 4.3.3 Time-History of Forces 300
V (HLL) V - Person1 V - Person2
250
200
Velocity (in/s)
150
100
50
0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-50
-100 Time (s)
Fig 4.3.4 Time-History of Velocities
Summary of Analytical Result Free Fall Distance
6 ft
Maximum Tension of HLL
6107.7 lbs
Maximum Fall Distance of the person
13.82 ft
Maximum sag of HLL
4.51 ft
Final Pull of Energy Absorber in HLL Final Pull of Energy Absorber in Lanyard
60
5 in 3.89 ft
1.8
2.0
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
6402 lbs
6389.9 lbs
0.2%
4.29 ft
4.307 ft
0.3%
5 in
5 in
0.0%
3.90 ft
3.914ft
.3%
Maximum Tension of HLL Maximum sag of HLL Final Pull of Energy Absorber in HLL Final Pull of Energy Absorber in Lanyard
50,000
40,000
30,000
Energy (lb-in)
20,000
10,000
0 0 -10,000
Potential 0.2 p1
0.4
0.6
0.8
1
1.2
1.4
EAVLL1 Kinetic p1
-20,000
HLL strain E EAHLL
-30,000
Potential p2 Kinetic p2
-40,000
EAVLL2 Anchorage Spring E
Time (seconds)
Fig 4.3.5 Time History of Energy and Work
61
1.6
1.8
2
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
62
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
63
Time (sec)
EPgrav,1 (lb.in)
EK1 (lb.in)
EEAVLL1 (lb.in)
Estrain (lb.in)
EAHLL EPgrav,2 (lb.in) (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
64
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. Summary of Analytical Result First Person Free Fall Distance (FFD)
6.0 ft
Second Person Free Fall Distance (FFD)
9.55 ft
Maximum Tension of HLL
6102.7 lbs
Maximum HLL Sag
4.51 ft
Maximum Fall Distance of Person 1
13.1 ft
Maximum Fall Distance of Person 2
14.5 ft
Final Pull of Energy Absorber in HLL
5.0 in
Final Pull of Energy Absorber in Person 1 Lanyard
3.2 ft
Final Pull of Energy Absorber in Person 2 Lanyard
5.0 ft
65
16 14.5 14 13.1 12
Disp (ft)
10
Cable Disp Person1 Disp Person2 Disp
8
6 4.51 4
2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Fig 4.4.1 Time-History of Displacements 8000
7000 6102.7
P1 Lanyard Force 6000
7000
Cable Tension P2 Lanyard Force
5000
Force (lb)
4000 3384.9 2789.7
3000
4000
2000.0 3000
2000
2000
1000 414.2
1000
0
0
-1000 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time (s)
Fig 4.4.2 Time-History of Forces
66
1.6
1.8
2.0
Force (lb) - Person 2
6000
5000
350
300
V (HLL) V - Person1 V - Person2
250
Velocity (in/s)
200
150
100
50
0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
-50
-100 Time (s)
Fig 4.4.3 Time-History of Velocities 60,000 50,000 40,000
Energy (lb-in)
30,000 20,000 10,000 0 0 -10,000
0.2 p1 Potential
0.4
0.6
0.8
1
1.2
1.4
EAVLL1 Kinetic p1
-20,000
HLL strain E EAHLL
-30,000
Potential p2 Kinetic p2
-40,000
EAVLL2 Anchorage Spring E
Time (seconds)
Fig 4.4.4 Time History of Energy and Work 67
1.6
1.8
2
4.4.2 Two masses fall with 0.4 second time difference 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. Summary of Analytical Result First Person Free Fall Distance (FFD)
6.0 ft
Second Person Free Fall Distance (FFD)
9.4 ft
Maximum Tension of HLL
6105.2 lbs
Maximum HLL Sag
4.51 ft
Maximum Fall Distance of Person 1
13.2 ft
Maximum Fall Distance of Person 2
14.4 ft
Final Pull of Energy Absorber in HLL
5.0 in
Final Pull of Energy Absorber in Person 1 Lanyard
3.3 ft
Final Pull of Energy Absorber in Person 2 Lanyard
4.6 ft
68
16 14.4
14
13.2 12
Disp (ft)
10 Cable Disp Person1 Disp
8
Person2 Disp
6 4.51 4.09
4
2
0 0
0.4
0.8
1.2
1.6
2
2.4
Time (s)
Fig 4.4.5 Time-History of Displacements 8000
7000 6020.0
Force1 6000
7000
Cable Tension Force2
5000
Force (lb)
4000 3384.9 3384.9
4000
3000 2000.0
3000
2000
2000
1000 414.2
1000
0
0
-1000 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (s)
Fig 4.4.6 Time-History of Forces
69
1.8
2.0
2.2
2.4
Force (lb) - Person 2
6000
5000
350
300
V (HLL) V - Person1 V - Person2
250
Velocity (in/s)
200
150
100
50
0 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
-50
-100 Time (s)
Fig 4.4.7 Time-History of Velocities 60,000 50,000 40,000
Energy (lb-in)
30,000 20,000 10,000 0 0 -10,000
0.2
0.4
Potential p1
0.6
0.8
1
1.2
1.4
1.6
1.8
EAVLL1 Kinetic p1
-20,000
HLL strain E EAHLL
-30,000 -40,000
Potential p2 Kinetic p2 EAVLL2
Time (seconds)
Anchorage Spring E
Fig 4.4.8 Time History of Energy and Work
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2
2.2
2.4
4.4.3 Two masses fall with 0.6 second time difference 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.
Summary of Analytical Result First Person Free Fall Distance (FFD)
6.0 ft
Second Person Free Fall Distance (FFD)
9.0 ft
Maximum Tension of HLL
6103.1 lbs
Maximum HLL Sag
4.51 ft
Maximum Fall Distance of Person 1
13.5 ft
Maximum Fall Distance of Person 2
14.2 ft
Final Pull of Energy Absorber in HLL
5.0 in
Final Pull of Energy Absorber in Person 1 Lanyard
3.5 ft
Final Pull of Energy Absorber in Person 2 Lanyard
4.2 ft
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16
14
14.16 13.47
Cable Disp Person1 Disp
12
12.78
Person2 Disp
Disp (ft)
10
14.5
8
6 4.51
4.09 4
2
0 0
0.4
0.8
1.2
1.6
2
2.4
Time (s)
Fig 4.4.9 Time-History of Displacements 8000
7000 6103.1
P1 Lanyard Force 6000
7000
Cable Tension P2 Lanyard Force
5000
5000
Force (lb)
4000 3385.7
3384.9
4000
3000 2000.0 2000
3000
1000
2000 414.2
0
1000
0
-1000 0.0
0.4
0.8
1.2
1.6
Time (s)
Fig 4.4.10 Time-History of Forces
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2.0
2.4
Force (lb) - Person 2
6000
350
300
V (HLL) V - Person1 V - Person2
250
Velocity (in/s)
200
150
100
50
0 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
-50
-100 Time (s)
Fig 4.4.11 Time-History of Velocities 50,000
40,000
30,000
Energy (lb-in)
20,000
10,000
0 0 -10,000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Potential p1 EAVLL1 Kinetic p1
-20,000
HLL strain E EAHLL
-30,000
Potential p2 Kinetic p2
-40,000
EAVLL2 Anchorage Spring E
Time (seconds)
Fig 4.4.12 Time History of Energy and Work
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1.8
2
2.2
2.4
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.
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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
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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 selfretracting 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
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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.
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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.
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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 freefall investigations. In Fundamentals of Fall Protection (Ed. A. C. Solowski), 1991, pp. 105122 (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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