Time-Dependent Behavior of Ropes Under Impact Loading

Research ArticleDOI: 10.1002/jst.62

Time-dependent behavior of ropes under impact loading: adynamic analysisIgor Emri�, Anatoly Nikonov, Barbara Zupanc̆ic̆ and Urska Florjanc̆ic̆

Center for Experimental Mechanics, University of Ljubljana, and Institute for Sustainable Innovative Technologies,Ljubljana, Slovenia

In this paper, we present new methodology based on a simple, non-standard

falling-weight experiment, which allows for the examination of the

functionality and durability of ropes beyond the findings from Union

Internationale des Associations d’Alpinisme experiments. The

experimental–analytical–numerical treatment allows for the examination of

the time-dependent viscoelasto-plastic behavior of ropes exposed to arbitrary

falling-weight loading conditions. Developed methodology allows for the

prediction of the impact force and the jolt (the derivative of the acceleration/

deacceleration acting on the climber); the viscoelasto–plastic deformation of

the rope; stored, retrieved, and dissipated energy during the loading and

unloading of the rope; and the modification of the stiffness of the rope within

each loading cycle. By means of parametric error analysis, we showed that

the relation between the error of calculated data and the error of input data is

extremely non-linear. This demands careful and precise experiments. It was

shown that the accuracy of prediction of all sought-after physical quantities

could be obtained within the acceptable limits, which confirms that the

proposed experimental–analytical methodology may be used for analyses of

the functionality and durability of ropes and the safety of climbers. & 2008

John Wiley and Sons Asia Pte Ltd

1. INTRODUCTION

Climbing is becoming one of the fastest growing extreme

sports. In this sport, ropes are probably the most critical part

of the equipment. Climbing ropes are designed to secure

climbers, and for that reason, they are dynamic; this means

that they are designed to stretch under a high load so as to

absorb the shock force. This protects the climber by reducing

fall forces. In comparison, static ropes are more durable and

resistant to abrasion and cutting, but they lack the necessary

protection against shock loads produced in a climber fall. For

that reason, they are used only in situations where such shock

loads would never occur (e.g. rappelling, canyoneering, and

spelunking) [1].

Ropes should have good mechanical properties, such as

high-breaking strength, large elongation at rupture, and good

elastic recovery. The Union Internationale des Associations

d’Alpinisme (UIAA) has established standard testing proce-

dures to measure, among other things, how ropes react to se-

vere falls [2,3]. The international standard test for climbing

ropes is based on a standard dynamic drop test . Ropes are

drop tested with a standardized weight and procedure simu-

lating a climber fall. This tells us how many of these hy-

pothetical falls the rope can withstand before it ruptures.

Different rope categories have different norms, but the stan-

dard requires climbing ropes to withstand a minimum of five

such test falls. Virtually all of the ropes on the market can

withstand the minimum number of test falls, while some are

*Center for Experimental Mechanics, Faculty of Mechanical Engineer-

ing, University of Ljubljana, Pot za Brdom 104, 1000 Ljubljana,

Slovenia

E-mail: [email protected]

Keywords:. ropes. impact. viscoelasticity. time-dependent behavior. jolt. energy dissipation

& 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 4–5, 208–219208

Time-dependent behavior of ropes

https://www.researchgate.net/publication/296433836_Materials_in_sports_equipment?el=1_x_8&enrichId=rgreq-41a9d203de55290ff195ae24798631ff-XXX&enrichSource=Y292ZXJQYWdlOzI1NDM2ODk5NjtBUzoxMDM0ODM4OTgzMzUyMzVAMTQwMTY4Mzg5MjIyOA==

rated to withstand a much higher number. The second thing that

a standard drop test measures is the amount of force that is

transmitted to the falling climber. For all of the tests, these forces

must stay within a certain range. The standard also rates factors,

such as rope stiffness, sheath slippage, and rope stretch under

body weight. Different simplified testing procedures are pre-

scribed for each of these properties. For example, rope stiffness

in a standardized test is measured by tying an overhand knot,

exposing the knot to a 10-kg load, and then measuring the size of

the hole in the knot. This test is known as the knot-ability test,

which indicates the handling and suppleness of a rope. Ac-

cording to the procedure prescribed by the standard, the hole

must measure less than 1.1 times the rope diameter. By all

means, these are very practical ways for rapid testing; however,

they provide no information on the underlying mechanisms that

govern the time-dependent behavior of ropes.

The standard says little about the durability of ropes,

which is more difficult to define or assess with simplified pro-

cedures commonly used by rope manufacturers. Durability in

this case does not mean just failure of the rope, but rather,

deterioration of its time-dependent response when exposed to

an impact force. The experiments prescribed by the existing

standard are not geared to analyze the time-dependent de-

formation process of the rope, which causes structural changes

in the material, and consequently affects its durability.

In this paper, we present a comprehensive dynamic analysis

of a simple, non-standard falling-weight experiment, which

allows for the examination of the time-dependent viscoelas-

to–plastic behavior of ropes exposed to arbitrary falling-weight

loading conditions. Developed analytical treatment is subse-

quently examined by using the ‘‘synthetic experimental data’’.

By means of the parametric error analysis, we determine the

required precision of all measured physical quantities used in

the derived analytical equations for physical quantities that

determine the durability of ropes and the safety of climbers.

2. THEORETICAL TREATMENT

The time-dependent response of a rope under dynamic

loading generated by a falling mass may be retrieved from the

analysis of the force measured at the upper fixture of the rope.

This force is transmitted through the rope and acts on the

falling weight (mass), as schematically shown in Figure 1. In

such experiments, a mass is dropped from an arbitrary height,

hp2l0, where l0 is the length of the tested rope.

Force measured as function of time, FðtÞ, may be expressed

as a set of N discrete data pairs:

FðtÞ ¼ fFi; ti; i ¼ 1; 2; 3; � � � ;Ng ð1Þ

An example of such measured force is schematically shown in

Figure 2. The diagram is subdivided into three distinct phases:

A, B, and C.

In phase A, the weight (mass) is dropped at t5 0, and it

falls freely until t ¼ t0 ¼ffiffiffiffiffiffiffiffiffiffi2h=g

p, where h indicates the height

from which the mass was initially dropped. Here the rope

becomes straight, which is indicated in Figure 2 as point T0. If

we neglect the air resistance, the velocity of the mass at point

T0 is v0 ¼ffiffiffiffiffiffiffiffi2gh

p. Point T0 represents the end of the free-falling

phase of the mass, and the beginning of phase B, which is the

beginning of the rope deformation process.

At point T0 in phase B, where t ¼ t� t0 ¼ 0, the falling

mass starts to deform the rope. Neglecting the air resistance,

and the wave propagation in the rope, the equation of motion

of the moving mass between points T0 and T7 may be written

as:

m €xðtÞ ¼ mg� FðtÞ ð2Þ

Here, m is the mass of the weight, and g is the gravitational

acceleration; €xðtÞ denotes the second derivative of the weight

displacement, xðtÞ, measured from point T0. Thus, xðtÞ re-presents the time-dependent deformation of the rope. The

solution of equation 2 gives the displacement of the weight as

the function of time, which is equal to the viscoelasto–plastic

deformation of the rope:

xðtÞ ¼gt2

2�

1

m

Z t

0

Z l

0

FðuÞ du� �

dlþ C1tþ C2 ð3Þ

Constants C1 and C2 may be obtained from the initial condi-

tions at point T0:

xðt ¼ 0Þ ¼ 0; and _xðt ¼ 0Þ ¼ v0 ¼ffiffiffiffiffiffiffiffi2gh

pð4Þ

Therefore:

C2 ¼ 0 ð5Þ

and

C1 ¼ v0 þ1

m

Z t

0

FðlÞ dl� �

t¼0¼ v0 ð6Þ

l 0

mg

mm

F(t)

F(t)

F(t)

F(t)

m

h

t = t0

l 0

mg

mm

F(t)

F(t)

F(t)

F(t)

mm

h

t = t0

Figure 1. Schematics of the rope exposed to the falling mass. Here m

is the mass of the falling weight, h is the height from which the mass is

initially dropped. F(t) is the measured force that is generated in the

rope, l0 is the initial length of the rope, t 5 t0 is the time when the rope

becomes straight, g is gravitational acceleration. Reproduced from [3]

by kind permission of Taylor & Francis.

Sports Technol. 2008, 1, No. 4–5, 208–219 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 209


Displacement of the weight, which is equal to the deformation

of the rope, may be expressed now as:

xðtÞ ¼gt2

2�

1

m

Z t

0

Z l

0

FðuÞ du� �

dlþ v0t ð7Þ

Since the deformation of the rope and the displacement of the

weight are the same, we may now calculate the velocity, the

acceleration/deacceleration, and the jolt acting on the weight,

that is, the climber, respectively:

vðtÞ ¼ _xðtÞ ¼ gt�1

m

Z t

0

FðlÞ dlþ v0 ð8Þ

aðtÞ ¼ €xðtÞ ¼ g�FðtÞm

ð9Þ

jðtÞ ¼ _€xðtÞ ¼ �1

m

dFðtÞdt ð10Þ

At point T1, where t ¼ t1, the force acting on the rope

becomes equal to the weight of the mass, Fðt1Þ ¼ mg. At this

point, the velocity of the weight reaches its maximum value:

vmax ¼ vðt1Þ ¼ gt1 �1

m

Z t1

0


The location of T1, where t ¼ t1 may be found numerically

from

dvðtÞdt¼ g�

Fðt1Þm¼ 0 ð12Þ

At T2, the jolt will reach its negative extreme value,

t ¼ t2 ¼ tðj ¼ jminÞ, where:

jmin ¼ jðt2Þ ¼MIN �1

m

dFðtÞdt

� �ð13Þ

The force acting on the rope and on the weight has its max-

imum at T3, where: t ¼ t3 ¼ tðF ¼ FmaxÞ, and

Fmax ¼ Fðt3Þ ¼MAXfFi; i ¼ 1; 2; 3; . . . ;Ng ð14Þ

The deformation of the rope at this point is:

sðt3Þ ¼ xðF ¼ FmaxÞ ¼ xðt3Þ

¼gt232�

1

m

Z t3

0

Z l

0

FðuÞ du� �

dlþ v0t3 ð15Þ

If the properties of the rope would be elastic, the location of

the maximum force should coincide with the location of the

maximal deformation; however, because of the viscoelastic

nature of the rope, its maximal deformation, smax, will be de-

layed and will take place at t ¼ t4, that is, at point T4, where

the velocity of the weight is equal to zero:

v4 ¼ vðt4Þ ¼ gt4 �1

m

Z t4

0

FðlÞ dlþ v0 ¼ 0 ð16Þ

The time, t4, may be retrieved numerically from equation 16.

The maximum deformation of the rope is then:

smax ¼ xðt4Þ ¼gt242�

1

m

Z t4

0

Z l

0

FðuÞ du� �

dlþ v0t4 ð17Þ

Now we can calculate the viscoelastic component of the rope

deformation by subtracting equation 15 from 17:

sve ¼smax � sðt3Þ ¼ xðt4Þ � xðt3Þ

¼gðt24 � t23Þ

2þ v0ðt4 � t3Þ �

1

m

Z t4

t3

Z l

0

FðuÞ du� �

dlð18Þ

The unloading phase of the rope starts at point T4. The elastic

component of a rope’s deformation will be retrieved and will

Time - t

For

ce-

F(t

)

T4

T6

t90 t0 t4t1

T1mg

First loading cycle Second loading cycle

CA B

t6

T0

T4

T7

Fmax

T3

T9

t2 t7

T2 T5

T8

0

t3 t5 t8

1τ 2τ 3τ 4τ 5τ 6τ 7τ 8τ 9τ

Figure 2. Schematics of the force measured during the falling mass experiment (phases A–C). ti, absolute time of individual events in deformation

process of the rope; ti, relative time of individual events in deformation process of the rope; Fmax, maximum force in the rope; m, mass of the falling

weight; g, gravitational acceleration;T0, beginning of the loading phase of the rope; T1, the moment when the force in the rope is equal to the weight

of the mass; T2, the moment of the extreme negative value of the jolt; T3, the moment of the maximum force in the rope; T4, the moment of the

maximum deformation of the rope when the velocity of the weight is equal to 0; T5, the moment of the positive extreme value of the jolt; T6, the

moment when force in the rope is equal to the weight of the load; T7, the moment when the force in the rope is equal to 0 and the weight starts to fly

in upwards; T8, the moment when the weight reaches the maximum upper point of its free fly in the vertical direction; T9, the beginning of the second

loading cycle.

www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 4–5, 208–219210

Research Article I. Emri et al.

accelerate the weight in the opposite (upward) direction. At

t ¼ t5, indicated as point T5, the jolt will reach its positive

extreme value: t5 ¼ tðj ¼ jmaxÞ, where

jmax ¼ jðt5Þ ¼MAX �1

m

dFðtÞdt

� �ð19Þ

At T6, where t ¼ t6, the force acting on the rope again be-

comes equal to the weight of the load, Fðt6Þ ¼ mg. At this

point, velocity will obtain its extreme value in the opposite

(negative) direction:

vmin ¼ vðt6Þ ¼ gt6 �1

m

Z t6

0


Time t6 may be again easily determined numerically from

Fðt6Þ ¼ mg. At point T7, where the force acting on the rope

becomes equal to zero, Fðt7Þ ¼ 0, the weight will start its free

fly in the upward (vertical) direction. The velocity of the weight

at point T7 may be calculated with equation 8:

v7 ¼ vðt7Þ ¼ gt7 �1

m

Z t7

0


We can also calculate the elastic part of rope deformation, sel,

which is equal to the weight displacement during the unloading

of the rope that takes place between points T4 and T7:

sel ¼ xðt4Þ � xðt7Þ

¼1

m

Z t7

t4

Z l

0

FðuÞ du� �

dl�gðt27 � t24Þ

2� v0ðt7 � t4Þ ð22Þ

Furthermore, we can calculate the visco–plastic deformation of

the rope, svp, by subtracting the recovered elastic deformation,

sel, from the rope’s maximum deformation, smax. Therefore:

svp ¼smax � sel ¼ xðt7Þ ¼gt272

�1

m

Z t7

0

Z l

0

FðuÞ du� �

dlþ v0t7

ð23Þ

By subtracting the visco–plastic (equation 23) and the viscoelastic

(equation 18) components, we can calculate the plastic compo-

nent of rope deformation:

spl ¼ svp � sve ¼ xðt7Þ þ xðt3Þ � xðt4Þ ð24Þ

In phase C, point T7 represents the beginning of phase C, in

which the weight has no interaction with the rope, that is,

Fðt7Þ ¼ 0, and starts to fly upwards with the initial velocity: v7,

v7 ¼ vðt7Þ ¼ gt7 �1

m

Z t7

0


It then returns back at point T9 to start the second cycle of the

rope deformation process. From the velocity, v7, we can calculate

the time of the weight vertical flight:

tu ¼v7

g¼ t7 �

1

mg

Z t7

0

FðlÞ dlþv0

gð26Þ

Furthermore, we are also able to calculate the height, sb, to which

the weight will be bounced:

sb ¼ v7tu �gt2u2

ð27Þ

At point T9, the second loading cycle of the rope starts, which

may be analyzed with the same set of equations derived for

phases B and C.

2.1. Force–Deformation Diagram of the Rope Deformation

Process: Energy Dissipation

Energy dissipation during the rope deformation process,

that is, between points T0 and T7, is one of the most important

rope characteristics, and should be used for comparing the

quality of ropes. Force, FðtÞ, measured during the loading and

unloading of the rope in phase B, may be expressed as the

function of the rope deformation, F ¼ FðsÞ, as schematically

shown in Figure 3. Notations used in the Figure are later ex-

plained.

The discrete form of F ¼ FðsÞ interrelation may be

obtained by calculating the isochronal values of the rope

deformation corresponding to each discrete value of the

measured force between points T0 and T7:

Fi ¼ FðtiÞ; si ¼ xðtiÞ ¼gt2i2

�

�1

m

Z ti

0

Z l

0

FðuÞ du� �

dlþ v0ti; 0ptipt7; i ¼ 1; 2; . . . ;M

�ð28Þ

Here, M is the number of measured force data points within

the time interval ½0; t7�.

For

ce –

F(s

)

T0

Fmax

T1 T6

T7

smax

selsvp

Wdis

kendkinit

T3

T4

mg

s1

s6

Deformation - s

Figure 3. Force deformation diagram of the rope loading and

unloading phase (phase B). Fmax, maximum force in the rope; m,

mass of the falling weight; g, gravitational acceleration; T0, beginning

of the loading phase of the rope; T1, the moment when the force in the

rope is equal to the weight of the mass; T3, the moment of the

maximum force in the rope; T4, the moment of the maximum

deformation of the rope when the velocity of the weight is equal to

0; T6, the moment when force in the rope is equal to the weight of the

load; T7, the moment when the force in the rope is equal to 0 and the

weight starts to fly in upwards; s1 and s6, deformations of the rope

when the force in the rope becomes equal to the weight of the mass;

smax, maximum deformation of the rope; svp, viscoplastic part of

deformation of the rope; sel, elastic part of deformation of the rope;

kinit, stiffness of the rope at the beginning of loading cycle; kend,

stiffness of the rope at the end of loading cycle; Wdis, dissipated

energy of the process.



The deformation energy of the rope at any stage of

deformation may be expressed as:

WðtÞ ¼Z sðtÞ

0

FðxÞ dx ¼Z t

0

FðlÞ@xðlÞ@l

dl

¼Z t

0

FðlÞ gl�1

m

Z l

0

FðuÞ duþ v0

� �dl

ð29Þ

and should be equal to the sum of the kinetic, WkðtÞ, and the

potential energy, WpðtÞ, of the falling weight at any time:

WðtÞ ¼WkðtÞ þWpðtÞ ð30Þ

We are particularly interested in the stored energy, which is the

only source of energy absorption (neglecting the air resistance),

and consequently the reduction of the force acting on

the climber:

Wstore ¼Z smax

0

FðxÞ dx ¼Z t4

0

FðlÞ@xðlÞ@l

dl

¼Z t4

0

FðlÞ gl�1

m

Z l

0

FðuÞ duþ v0

� �dl

ð31Þ

Since the stored energy must be equal to the total potential

energy of the weight, then:

Wstore ¼mgðhþ smaxÞ

¼mg hþgt242�

1

m

Z t4

0

Z l

0

FðuÞ du� �

dlþ v0t4

� � ð32Þ

During the unloading phase, the elastic component of the rope

deformation is retrieved and it accelerates the weight in an

upward direction:

Wret ¼Z smax

svp

FðxÞ dx ¼ �Z t7

t4FðlÞ

@xðlÞ@l

dl

¼�Z t7

t4FðlÞ gl�

1

m

Z l

0

FðuÞ duþ v0

� �dl

ð33Þ

The retrieved energy must be equal to the kinetic energy of the

mass at point T7. Thus:

Wret ¼mv272þmg ½xðt4Þ � xðt7Þ�

¼m

2gt7 �

1

m

Z t7

0

FðlÞ dlþ v0

� �2

þmg ½xðt4Þ � xðt7Þ�

ð34Þ

The dissipated energy within a loading and unloading cycle,

represented as the shaded area in Figure 3, can be expressed as:

Wdiss ¼Wstore �Wret ¼Z t4

0

FðlÞ gl�1

m

Z l

0

FðuÞ duþ v0

� �dl

� �Z t7

t4FðlÞ gl�

1

m

Z l

0

FðuÞ duþ v0

� ��

¼Z t7

0

FðlÞ gl�1

m

Z l

0

FðuÞ duþ v0

� �dl

ð35Þ

Alternatively:

Wdiss ¼ mgðhþ smaxÞ �mv272�mg ½xðt4Þ � xðt7Þ� ð36Þ

2.2. Increase of the Rope Stiffness

An important parameter for comparing the performance of

different ropes could be the modification of their stiffness within

each loading cycle. The rope becomes stiffer in each loading

cycle, which means that the performance of the rope is de-

creasing. Thus, an indicator of the quality and rope durability

could be the ratio of the stiffness at the beginning, kinit, and at

the end, kend, of the rope deformation process. Therefore:

w ¼kinit

kendp1 ð37Þ

Stiffness, kinit and kend, may be calculated from the slope of the

force-displacement diagram FðsÞ at points T1 and T6, as sche-

matically shown in Figure 3:

kinit ¼dFðxÞdx

x¼s1

ð38Þ

Table 1. Physical quantities representing the functionality and durability of ropes.

n Physical quantity Symbol Corresponding equation

1 Maximum force Fmax 14

2 Maximum deformation smax 17

3 Elastic part of rope deformation sel 22

4 Visco–plastic part of rope deformation svp ¼ smax � sel 23

5 Viscoelastic part of rope deformation sve 18

6 Plastic part of rope deformation spl ¼ svp � sve 24

7 Stored energy Wstore 31 or 32

8 Retrieved energy Wret 33 or 34

9 Dissipated energy Wdiss ¼Wstore �Wret 35 or 36

10 Stiffness of the rope at the beginning of deformation kinit 38

11 Stiffness of the rope at the end of deformation kend 39

12 Ratio of the stiffness w ¼ kinit=kend 37

13 Jolt j 10



and

kend ¼dFðxÞdx

x¼s6

ð39Þ

where s1 and s6 are rope deformations at corresponding points

T1 and T6, indicating the beginning and the end of the rope

deformation process beyond the deformation caused by the

weight of the falling mass. The stiffness of both is indicated in

Figure 3.

3. PARAMETRIC ERROR ANAYLIS

Based on the measured force, FðtÞ, acting on a rope and a

climber during the falling-weight experiment, we derived a

variety of different physical quantities that may be used as

criteria in the evaluation of the functionality and the durability

of climbing ropes and the safety of climbers. These physical

quantities are summarized in Table 1.

Preliminary experimental investigations [4,5] showed that

calculated physical quantities (listed in Table 1) are very sen-

sitive to the precision of the input data, that is, the mass of the

falling weight, height from which we drop the weight, length of

the rope, measured force, time at which measurements

were performed (sampling rate), and number of significant

digits in gravitational acceleration. To evaluate the effect

of the input data precision on the accuracy of the

calculated physical quantities, we will use a synthetic

‘‘error free’’ reference signal, FðtÞ, which closely mimics the

measured signals:

FðtÞ ¼ F½1� cosð20tÞ� HðtÞ �H t�p10

�h i½N� ð40Þ

where HðtÞ is the Heaviside (step) function, that is,

Hðto0Þ ¼ 0, and HðtX0Þ ¼ 1. The reference signal is shown

in Figure 4. In addition, we used

F ¼4000 N; m ¼ 80 kg h ¼ l ¼ 3:263 m

t9 ¼1:2 sec; and g ¼ 9:80665 m=s2ð41Þ

3.1 Calculation of the ‘‘Error-Free’’, Sought-After Physical

Quantities

For the parametric error analysis of the characteristic

physical quantities (Table 1), we will first calculate the

reference error-free values. We will first need to determine

the characteristic times: t3, t4, and t7. From equation 40,

it is easy to see that the maximum force, Fmax ¼ F, will

appear at t3 ¼ p=20 sec. We determine the location of

the maximum deformation at T4 by combining equations 40

and 16:

g�F

m

�t4 þ

200

msinð20t4Þ þ v0 ¼ 0 ð42Þ

whereas the location of T7 may be found directly from the

chosen reference signal, equation 40. Thus:

t3 ¼ p=20 sec t4 ¼ 0:176026 sec and t7 ¼p10

sec ð43Þ

Introducing equation 40 into equations 7 and 8, we obtain the

evolution of the rope deformation process, and the

corresponding velocity of the weight:

xðtÞ ¼gt2

2þ v0t�

F

m

t2

2þ

1

400ðcosð20tÞ � 1Þ

� �

� HðtÞ �H t�p10

�h i

�20pm½20t� p�H t�

p10

�;

ð44Þ

Figure 4. Synthetic reference signal F(t). Here ti denotes relative time

of individual events in deformation process of the rope; Fmax,

maximum force in the rope; m, mass of the falling weight; g,

gravitational acceleration; T1, the moment when the force in the rope is

equal to the weight of the mass; T2, the moment of the extreme

negative value of the jolt; T3, the moment of the maximum force in the

rope; T4, the moment of the maximum deformation of the rope when

the velocity of the weight is equal to 0; T5, the moment of the positive

extreme value of the jolt; T6, the moment when force in the rope is

equal to the weight of the load; T7, the moment when the force in the

rope is equal to 0 and the weight starts to fly in upwards.

Figure 5. Evolution of the rope deformation process (solid line), and

the velocity of weight (dashed line). Here ti denotes relative time of

individual events in deformation process of the rope; t3, the moment

of the maximum force in the rope; t4, the moment of the maximum

deformation of the rope when the velocity of the weight is equal to 0;

t7, the moment when the force in the rope is equal to 0 and the weight

starts to fly in upwards; smax, maximum deformation of the rope; svp,

viscoplastic part of deformation of the rope; sel, elastic part of

deformation of the rope; sve, viscoelastic part of deformation of the

rope.



vðtÞ ¼gtþ v0 �F

mt�

1

20sinð20tÞ

� �

� HðtÞ �H t�p10

�h i�

400pm

H t�p10

� ð45Þ

The two relations are shown in Figure 5 , where the solid line

represents the deformation of the rope, and the dashed line

represents the corresponding velocity of the weight. In the

same Figure, the characteristic components of the rope de-

formation, smax, sve, sel, and svp, are also shown. These quan-

tities and spl may be obtained from equations 17, 18, 22–24,

respectively:

smax ¼t242

g�F

m

�þ v0t4 �

F

400 mðcos 20t4 � 1Þ ð46Þ

sve ¼ g�F

m

� t24 � t232

� �þ v0ðt4 � t3Þ

�F

400 mðcos 20t4 � cos 20t3Þ

ð47Þ

sel ¼ g�F

m

� t24 � t272

� �þ v0ðt4 � t7Þ

�F

400 mðcos 20t4 � cos 20t7Þ

ð48Þ

svp ¼ smax � sel ð49Þ

and

spl ¼ svp � sve ð50Þ

Taking into account values in equations 41 and 43, we find

their ‘‘true’’ (error-free) values:

smax ¼1:0266 m; sel ¼ 0:4968 m;

svp ¼0:5298 m sve ¼ 0:0159 m

spl ¼0:5139 m spl ¼ 0:5139 m

ð51Þ

Using equations 40 and 44, we can now calculate the corre-

sponding force and displacement data points:

fFi ¼ FðtiÞ; si ¼ xðtiÞ; 0ptipt7; i ¼ 1; 2; . . . ;Mg ð52Þ

We are also able to express the force as function of deforma-

tion, F ¼ FðsÞ. This relation is shown in Figure 6, where we

also show the elastic, sel, and the visco–plastic, svp, part of rope

deformation, and the stiffness of the rope at the beginning,

kinit, and at the end of deformation, kend.

The stiffness of both and their ratios, w, may be calculated

from:

kinit ¼dF

ds

t¼t1

¼dFdtdsdt

t¼t1

¼200F sinð20t1Þ

g� Fm

� t1 þ 200

msinð20t1Þ þ v0

ð53Þ

kend ¼dF

ds

t¼t6

¼dFdtdsdt

t¼t6

¼200F sinð20t6Þ

g� Fm

� t6 þ 200

msinð20t6Þ þ v0

ð54Þ

and

w ¼kinit

kend¼

sinð20t1Þsinð20t6Þ

�g� F

m

� t6 þ 200

msinð20t6Þ þ v0

g� Fm

� t1 þ 200

msinð20t1Þ þ v0

ð55Þ

where t1 and t6 are given with the relations:

t1 ¼1

20Arccos 1�

mg

F

�¼ 0:03185 sec ð56Þ

and

t6 ¼p10�

1

20Arccos 1�

mg

F

�¼ 0:28231 sec ð57Þ

Figure 7. Synthetic curve of the deformation energy as a function of

time, t. Here ti denotes relative time of individual events in

deformation process of the rope; T4, the moment of the maximum

deformation of the rope when the velocity of the weight is equal to 0;

T7, the moment when the force in the rope is equal to 0 and the weight

starts to fly in upwards; Wstor, stored energy of the process; Wdis,

dissipated energy of the process; Wret, retrieved energy of the process.

Figure 6. Force acting on the rope as function of its deformation. m,

mass of the falling weight; g, gravitational acceleration; T1, the

moment when the force in the rope is equal to the weight of the mass;

T3, the moment of the maximum force in the rope; T4, the moment of

the maximum deformation of the rope when the velocity of the weight

is equal to 0; T6, the moment when force in the rope is equal to the

weight of the load; T7, the moment when the force in the rope is equal

to 0 and the weight starts to fly in upwards; smax, maximum

deformation of the rope; svp, viscoplastic part of deformation of the

rope; sel, elastic part of deformation of the rope; kinit, stiffness of the

rope at the beginning of loading cycle; kend, stiffness of the rope at the

end of loading cycle.



The deformations of the rope at these two points are

sðt1Þ ¼ 0:2589 m, and sðt6Þ ¼ 0:6813 m.

Introducing equations 40 and 44 into equation 29, we can

obtain the relation describing the evolution of the rope

deformation energy, which is shown in Figure 7. Therefore:

WðtÞ ¼ F g�F

m

� t2

2�

t20

sinð20tÞ� �

þF

400gð1� cosð20tÞÞ

� FF

800msin2 ð20tÞ þ

v0

20sinð20tÞ � v0t

n oð58Þ

In the same Figure, we show also the corresponding stored,

Wstore ¼Wðt4Þ, dissipated, Wdiss ¼Wðt7Þ, and retrieved

(elastic) energy, Wret ¼Wstore �Wdiss. Their ‘‘true’’ values are

given as:

Wstore ¼ F g�F

m

� t242�

t420

sinð20t4Þ� �

þF g

400½1� cosð20t4Þ�

� FF

800msin2 ð20t4Þ �

v0

20sinð20t4Þ þ v0t4

h i;

ð59Þ

Wdiss ¼ F g�F

m

� t272�

t720

sinð20t7Þ� �

þF g

400½1� cosð20t7Þ�

FF

800msin2 ð20t7Þ �

v0

20sinð20t7Þ þ v0t7

h ið60Þ

and

Wret ¼ F g�F

m

� ðt24 � t27Þ2

�t420

sinð20t4Þ �t720

sinð20t7Þ ��

þg

400ðcosð20t7Þ � cosð20t4ÞÞ �

5

m½sin2 ð20t4Þ � sin2

� ð20t7Þ� �v0

20½sinð20t4Þ � sinð20t7Þ� þ v0ðt4 � t7Þ

o: ð61Þ

The numerical values for the three energies are Wstore ¼3365:35 Nm, Wdiss ¼ 2119:11 Nm, and Wret ¼ 1246:24 Nm,

respectively. According to the law of conservation of energy,

the sum of the kinetic and the potential energy of the falling

mass, WmðtÞ ¼WkðtÞ þWpðtÞ, and the deformation energy of

the rope, WðtÞ, should be constant at all times (neglecting the

dissipation due to the air resistance): WmðtÞ þWðtÞ ¼ const.

This is demonstrated in Figure 8, where the solid line re-

presents the evolution of the rope deformation energy, WðtÞ,and the dashed line represents the sum of the kinetic and the

potential energy of the falling mass WmðtÞ ¼WkðtÞ þWpðtÞ.For completeness, we also show, with thinner solid and dashed

lines, the kinetic, WkðtÞ, and the potential energy, WpðtÞ, re-spectively. In the same Figure, the corresponding characteristic

times t ¼ t0, t4, and t7, which correspond to t0 ¼ 0,

t4 ¼ t4 � t0, and t7 ¼ t7 � t0, respectively are also shown.

Similarly, we can find jolt

jðtÞ ¼20F

msinð20tÞ ð62Þ

which is shown in Figure 9.

The absolute values of the minimum and the maximum

jolts are the same:

jjmaxj ¼ jjminj ¼ 1000 m=s3 ð63Þ

The calculated true values of the characteristic physical

quantities will now be used in the parametric error analysis to

determine the accuracy of the calculated physical quantities,

which represent the functionality and the durability of the

tested rope and the safety of a climber.

3.2. Error Analysis

The goal of the parametric error analysis is to determine

the effect of the error of the input data on the accuracy of the

calculated physical quantities (Table 1).

Figure 9. Jolt as a function of time. Here ti denotes relative time of

individual events in deformation process of the rope; T2, the moment

of the extreme negative value of the jolt; T3, the moment when the

force in the rope reaches its maximum; T5, the moment of the positive

extreme value of the jolt; T7, the moment when the force in the rope is

equal to 0 and the weight starts to fly in upwards; jmin, the extreme

negative value of the jolt; jmax, the extreme positive value of the jolt.

Figure 8. Evolution of the rope deformation energy in relation to the

sum of the kinetic and potential energy of the falling mass. Here ti and

ti denote absolute and relative time of individual events in

deformation process of the rope; T0, beginning of the loading phase

of the rope; T4, the moment of the maximum deformation of the rope

when the velocity of the weight is equal to 0; T7, the moment when the

force in the rope is equal to 0 and the weight starts to fly in upwards;

W(t), the energy of the rope as a function of time; Wm(t), the energy of

the mass as a function of time; Wk(t), kinetic energy of the mass as a

function of time; Wp(t), potential energy of the mass as a function of

time.



It is easy to see that Fmax may always be determined di-

rectly from the source data of measured force,

Fmax ¼MAXfFi; ti; i ¼ 1; 2; 3; . . . ;Ng. Therefore, an error in

the predicted maximum force, Fmax, is given directly with the

accuracy of the force sensor, and the sampling rate of the data

acquisition. Error estimation of the rope deformation com-

ponents, smax, svp, spl, sel, and sve, is much more complex. It

depends on errors in numerical integration in equation 7, er-

rors in determining t3, t4, t7, and errors in the input data of g,

m, h, and FðtÞ. The same is true for Wstore, Wdiss, and Wret,

where we need to integrate equation 29. The physical quan-

tities, Wret, svp, and spl, are linear combinations of Wstore and

Wdiss, and smax, sve, and sel, respectively. Thus, we need to

analyze the influence of the error of the input data on the last

five quantities only.

Assuming that the accuracy of the measured time, ti, at themoment when we measure the force, Fi, may be considered as

error free (which is a reasonable assumption), then the mea-

sured force may be expressed as:

FðtiÞ ¼ Fi � FðtÞ ð64Þ

where Fi is the measured strength of the force, and FðtÞ is itserror-free time dependency. Consequently, the expressions for

smax, sve, sel, Wstore, and Wdiss may be rearranged as:

smax ¼ xðt4Þ ¼gt242þ t4

ffiffiffiffiffiffiffiffi2gh

p�F

m

Z t4

0

Z l

0

FðuÞ du� �

dl ð65Þ

sve ¼xðt4Þ � xðt3Þ ¼gðt24 � t23Þ

2þ ðt4 � t3Þ


p

�F

m

Z t4

t3

Z l

0

FðuÞ du� �

dl

ð66Þ

sel ¼xðt4Þ � xðt7Þ ¼gðt24 � t27Þ

2

þ ðt4 � t7Þffiffiffiffiffiffiffiffi2gh

p�F

m

Z t7

t4

Z l

0

FðuÞ du� �

dl

ð67Þ

Wstore ¼Wðt4Þ ¼FgZ t4

0

lFðlÞ dlþ Fffiffiffiffiffiffiffiffi2gh

p Z t4

0

FðlÞ dl

�F2

m

Z t4

0

FðlÞZ l

0

FðuÞ du� �

dl

ð68Þ

and

Wdiss ¼Wðt7Þ ¼ Fg

Z t7

0

lFðlÞ dlþ Fffiffiffiffiffiffiffiffi2gh

p Z t7

0

FðlÞ dl

�F2

m

Z t7

0

FðlÞZ l

0

FðuÞ du� �

dl

ð69Þ

The errors of calculated smax, sve, sel, Wstore, and Wdiss may

now be estimated from the sum of their partial derivatives with

respect to g, m, h, F, t3, t4, and t7. Therefore:

Dsmax ¼�@x

@g

� �t¼t4

Dg

þ @x

@m

� �t¼t4

Dm

(

þ@x

@h

� �t¼t4

Dh

: þ @x

@F

� �t¼t4

DF

þ @x

@t

� �t¼t4

Dt

)

ð70Þ

Dsve ¼ �

@x@g

�t¼t4� @x

@g

�t¼t3

� �Dg

þ @x

@m

� t¼t4� @x

@m

� t¼t3

h iDm

þþ @x

@h

� t¼t4� @x

@h

� t¼t3

h iDh

þ @x@F

� t¼t4

þ @x@F

� t¼t3

h iDFþ

þ @x@t

� t¼t4

þ @x@t

� t¼t3

h iDt

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð71Þ

Dsel ¼ �

@x@g

�t¼t7� @x

@g

�t¼t4

� �Dg

þ @x

@m

� t¼t7� @x

@m

� t¼t4

h iDm

þþ @x

@h

� t¼t7� @x

@h

� t¼t4

h iDh

þ @x@F

� t¼t7

þ @x@F

� t¼t4

h iDFþ

þ @x@t

� t¼t7

þ @x@t

� t¼t4

h iDt

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;ð72Þ

DWstore ¼ �

@W@g

�t¼t4

Dg

þ @W

@m

� t¼t4

Dm þ

þ @W@h

� t¼t4

Dh þ @W

@F

� t¼t4

DF þ @W

@t

� t¼t4

Dt

8>>><>>>:

9>>>=>>>;ð73Þ

and

DWdiss ¼ �

@W@g

�t¼t7

Dg

þ @W

@m

� t¼t7

Dm þ

þ @W@h

� t¼t7

Dh þ @W

@F

� t¼t7

DF þ @W

@t

� t¼t7

Dt

8>>><>>>:

9>>>=>>>;ð74Þ

where DF is defined as the maximal error in the measured force

throughout the experiment, and Dt is the maximal error in

determining t3, t4, and t7. Therefore:

DF ¼ �MAXfjDFi j i ¼ 1; 2; . . . ;Ng ð75Þ

and

Dt ¼ �MAXfjDt3j; jDt4j; jDt7jg ð76Þ

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1.0 1.5 2.0

n (%

)

κ

W

W

s

s

s

(%)

Figure 10. Relative error, n, of smax, sve, sel, Wstore, and Wdiss as a

function of the relative error, k, of g, m, h, F and tc. The symbols used

in the Figure denote the following physical quantities: smax, maximum

deformation of the rope; Sve, viscoelastic part of deformation of the

rope; sel, elastic part of deformation of the rope; Wdiss, dissipated

energy of the process; Wstore, stored energy of the process.



Equations 70–74 may be expressed in a matrix form as:

Dsmax

Dsve

Dsel

DWstore

DWdiss

2666666664

3777777775¼ �

a11 a12 a13 a14 a15

a21 a22 a23 a24 a25

a31 a32 a33 a34 a35

a41 a42 a43 a44 a45

a51 a52 a53 a54 a55

2666666664

3777777775

jDgj

jDmj

jDhj

jDFj

jDtj

2666666664

3777777775

¼ �D

jDgj

jDmj

jDhj

jDFj

jDtj

2666666664

3777777775

ð77Þ

Individual components, aij , of the matrix D are given in

Appendix I.

We still need to comment the errors in estimating the

stiffness at the beginning, kinit, and at the end, kend, of the

impact loading cycle, and the error in the calculation of the jolt

(derivative of the acceleration/deacceleration). These require

numerical derivation of the measured force for rope de-

formation and time, respectively. Numerical derivations may

often be troublesome; however, it is a standard, well-known

numerical problem, which has been properly addressed in

commercial mathematical softwares, such as Mathematica,

and does not need any additional comment.

3.2.1 Sensitivity of the error of calculated data to the error of

input data

Let us first assume that the relative error of all input

physical quantities, g, m, h, F, and tc 2 ft3; t4; t7g, is equal:

Dgg¼

Dmm¼

Dhh¼

DFF¼

Dtiti; i ¼ 3; 4; 7

� �ð78Þ

Of course, this assumption is not realistic. However, it will help

us to understand which of the calculated physical quantities,

Wstore and Wdiss, smax, sve, and sel is most sensitive to the error

of input data. The relative error of the calculated data is de-

fined as:

Z ¼DCCtrue

� 100½%� ð79Þ

where DCrepresents Dsmax, Dsve, Dsel, DWstore, and DWdiss, and

Ctrue is their corresponding error-free values, respectively.

Equivalently, we may define the relative error of the input data

g, m, h, F, and tc as:

k ¼D��true

� 100½%� ð80Þ

where D� represents Dg, Dm, Dh, DF, and Dt, whereas �true is

the error-free values of g, m, h, F, and tc. Here, tc again

represents t3, t4, and t7. Figure 10 shows the results

of these error analyses, shown as Z ¼ ZðkÞ for each of the

five sought-after physical quantities. From the Figure, it

can be seen that the accuracy of prediction of the viscoelastic

component of rope deformation, sve, is most sensitive to the

errors of input data, followed by sel, Wdiss, Wstore, and smax.

The most important observation is that the errors of the cal-

culated data are up to 100 times larger than the error of input

data. Thus, in order to utilize the derived theory for analyzing

the durability of ropes and the safety of climbers, we need to

carry out experiments very accurately.

3.2.2 Example for realistic measuring setup

Let us now turn to the analysis of a realistic situation,

which corresponds to the experimental setup used in our

laboratory. The errors of the input data in our experiments

are typically: Dg ¼ �0:00001 m=s2, Dm ¼ �0:02 kg,

Dh ¼ �0:01 m, DF ¼ �5N, and Dt ¼ �0:0001 s. According

to equation 77, this leads to the following absolute errors of

calculated data: Dsmax ¼ �0:002959 m, Dsve ¼ �0:001568 m,

Dsel ¼ �0:00639153 m, DWstore ¼ �11:0115 Nm, and

DWdiss ¼ �27:5671 Nm. The corresponding relative errors are

then: dsmax ¼ 0:29 %, dsve ¼ 9:88%, dsel ¼ 1:29%, dWstore ¼0:33%, and dWdiss ¼ 1:3%, respectively.

As predicted previously, the largest error appears

in the prediction of the viscoelastic component of rope

deformation, sve. However, the prediction is still within the

acceptable limit. Predictions of all other physical quantities

are very good, which confirms that the proposed experi-

mental–analytical methodology may be used for the analyses

of the functionality and durability of ropes and safety of

climbers.

4. CONCLUSIONS

We have presented the methodology based on a simple

non-standard falling-weight experiment, which allows for

the examination of the functionality and durability of ropes

beyond the experimental findings of the UIAA. The experi-

mental–analytical–numerical treatment allows for the

examination of the time-dependent viscoelasto–plastic

behavior of ropes exposed to arbitrary falling-weight loading

conditions. A developed methodology can be successfully

applied for calculating the following important physical

parameters: the impact force and jolt (the derivative of the

acceleration/deacceleration acting on the climber); the viscoe-

lasto–plastic deformation of the rope; stored, retrieved, and

dissipated energy during the loading and unloading of the

rope; and modification of the stiffness of the rope within each

loading cycle.

A developed analytical treatment was subsequently ex-

amined by using the ‘‘synthetic experimental data’’. By means

of the parametric error analysis, we analyzed the required

precision of all measured physical quantities used in the cal-

culation of physical quantities that determine the durability of

ropes and safety of climbers.



The parametric error analysis showed that that the errors of

calculated data are up to 100 times larger than the errors of input

data. Thus, in order to utilize the proposed methodology, one

needs to carry out experiments very accurately. When doing so, the

accuracy of prediction of all sought-after physical quantities are

within the acceptable limits, which confirms that the proposed

experimental–analytical methodology may be used for the analyses

of the functionality and durability of ropes and safety of climbers.

Acknowledgements

We would like to acknowledge the financial support provided by

the Slovenian Research Agency (http://www.arrs.gov.si/en/

dobrodoslica.asp). The contribution of our coworker Pavel

Oblak, University of Ljubljana, Slovenia in standardizing the

experimental procedures is also greatly appreciated.

5. APPENDIX I

5.1 Components of the Matrix D

a11 ¼@x

@g

� �t¼t4

¼ t24

2þ t4

ffiffiffiffiffih

2g

s

a12 ¼@x

@m

� �t¼t4

¼ F

m2

Z t4

0

Z t

0

FðlÞ dl� �

dt

a13 ¼@x

@h

� �t¼t4

¼ t4

ffiffiffiffiffig

2h

r

a14 ¼@x

@F

� �t¼t4

¼ � 1

m

Z t4

0

Z t

0

FðlÞ dl� �

dt

a15 ¼@x

@t

� �t¼t4

¼ gt4 þ


p�F

m

Z t4

0

FðlÞ dl

a21 ¼@x

@g

� �t¼t4

�@x

@g

� �t¼t3

¼ t24 � t23

2þ ðt4 � t3Þ

ffiffiffiffiffih

2g

s

a22 ¼@x

@m

� �t¼t4

�@x

@m

� �t¼t3

¼ F

m2

Z t4

t3

Z l

0

FðuÞ du� �

dl

a23 ¼@x

@h

� �t¼t4

�@x

@h

� �t¼t3

¼ ðt4 � t3Þ

ffiffiffiffiffig

2h

r

a24 ¼@x

@F

� �t¼t4

þ @x

@F

� �t¼t3

¼ �1

m

Z t4

0

Z l

0

FðuÞ du� �

dl

þ � 1

m

Z t3

0

Z l

0

FðuÞ du� �

dl

a25 ¼@x

@t

� �t¼t4

þ @x

@t

� �t¼t3

¼ gt3 þffiffiffiffiffiffiffiffi2gh

p�F

m

Z t3

0

FðlÞ dl

þ gt4 þ


p�F

m

Z t4

0

FðlÞ dl

a31 ¼

@x

@g

� �t¼t7

�@x

@g

� �t¼t4

¼ � t27 � t24

2� ðt7 � t4Þ

ffiffiffiffiffih

2g

s

a32 ¼@x

@m

� �t¼t7

�@x

@m

� �t¼t4

¼ � F

m2

Z t7

t4

Z l

0

FðuÞ du� �

dl

a33 ¼@x

@h

� �t¼t7

�@x

@h

� �t¼t4

¼ �ðt7 � t4Þ

ffiffiffiffiffig

2h

r

a34 ¼@x

@F

� �t¼t7

þ @x

@F

� �t¼t4

¼ �1

m

Z t7

0

Z l

0

FðuÞ du� �

dl

þ �

1

m

Z t4

0

Z l

0

FðuÞ du� �

dl

a35 ¼@x

@t

� �t¼t7

þ @x

@t

� �t¼t4

¼ gt7 þffiffiffiffiffiffiffiffi2gh

p�F

m

Z t7

0

FðlÞ dl

þ gt4 þ


p�F

m

Z t4

0

FðlÞ dl

a41 ¼@W

@g

� �t¼t4

¼ F

Z t4

0

lFðlÞ dlþ F

ffiffiffiffiffih

2g

s Z t4

0

FðlÞ dl

a42 ¼@W

@m

� �t¼t4

¼ F2

m2

Z t4

0

FðlÞZ l

0

FðuÞ du� �

dl

a43 ¼@W

@h

� �t¼t4

¼ F

ffiffiffiffiffig

2h

r Z t4

0

FðlÞ dl

a44 ¼@W

@F

� �t¼t4

¼ g

Z t4

0

lFðlÞ dl�2F

m

Z t4

0

FðlÞZ l

0

FðuÞ du� �

dl

þffiffiffiffiffiffiffiffi2gh

p Z t4

0

FðlÞ dl



a45 ¼@W

@t

� �t¼t4

¼Fgt4Fðt4Þ þ F


pFðt4Þ

�F2

mFðt4Þ

Z t4

0

FðtÞ dt

a51 ¼@W

@g

� �t¼t7

¼ F

Z t7

0

lFðlÞ dlþ F

ffiffiffiffiffih

2g

s Z t7

0

FðlÞ dl

a52 ¼@W

@m

� �t¼t7

¼ F2

m2

Z t7

0

FðlÞZ l

0

FðuÞ du� �

dl

a53 ¼@W

@h

� �t¼t7

¼ F

ffiffiffiffiffig

2h

r Z t7

0

FðlÞ dl

a54 ¼@W

@F

� �t¼t7

¼ g

Z t7

0

lFðlÞ dl�2F

m

Z t7

0

FðlÞZ l

0

FðuÞ du� �

dlþ


p Z t7

0

FðlÞ dl

a55 ¼@W

@t

� �t¼t7

¼Fgt7Fðt7Þ þ F


pFðt7Þ

�F2

mFðt7Þ

Z t7

0

FðtÞ dt

REFERENCES

1. Jenkins M, ed. Materials in sports equipment. Woodhead Publishing Limited:Cambridge, 2003. ISBN 1 85573 599 7.

2. EN 892:2004 (E). Mountaineering equipment. Dynamic mountaineeringropes. Safety requirements and test methods.The European Committee forStandardization, November 2004.

3. http://www.theuiaa.org/upload_area/cert_files/UIAA101_DynamicRopes.pdf.[15 March 2008]

4. Oblak P. Development of the methodology for dynamic characterization ofropes (Dissertation). University of Ljubljana: Ljubljana, 2007.

5. Emri I, Udovc M, Zupancic B, Nikonov AV et al. Examination of the time-dependent behaviour of climbing ropes. In: Fuss FK, Subic A, Ujihashi S, eds.The Impact of Technology on Sport II. Taylor & Francis: London, 2008; 695–700.

Received 1 May 2008

Accepted 9 June 2008

Published online 6 January 2009





Time-Dependent Behavior of Ropes Under Impact Loading

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Transcript of Time-Dependent Behavior of Ropes Under Impact Loading