TIME-DEPENDENT BEHAVIOR OF ROPES UNDER IMPACT LOADING
A. Nikonov, M. Udovč, B. Zupančič, U. Florjančič, B.S. von Bernstorff1, S. Burnik, and I. Emri
Center for Experimental Mechanics, University of Ljubljana,
Cesta na Brdo 85, Ljubljana, Slovenia, [email protected]
1
BASF Aktiengesellschaft, Ludwigshafen, Germany
ABSTRACT
The experimental-analytical methodology, based on a simple non-standard falling weight experiment, which allows
examination of the time-dependent elasto-visco-plastic behavior of ropes exposed to arbitrary falling weight loading conditions,
was developed for mechanical characterization of climbing ropes. In this paper we show that the newly developed simple
procedure can be successfully applied for calculation of several important characteristics, such as the impact force on the
rope; derivative of the (de)acceleration; maximum deformation of the rope; dissipated energy during the loading and unloading
of the rope; modification of the stiffness of the rope within each cycle of loading. All those parameters can be determined just
from a single dynamic response of the rope exposed to impulse loading. Three different commercial manufacturers were
selected for comparative analysis of the mechanical behavior of climbing ropes. The ropes were exposed to the same loading
conditions. Calculated mechanical properties of all three types of ropes were then compared using the developed
methodology. The obtained results indicate that ropes, which according to existing UIAA standard belong to the same quality
class and are declared to have the same UIAA standard characteristics, actually exhibit significantly different behavior when
exposed to the same loading conditions by using our experimental-analytical methodology.
Introduction
The climbing ropes are designed to secure a climber. They are designed to stretch under high load so as to absorb the shock
force. This protects the climber by reducing fall forces. Ropes should have good mechanical properties, such as high breaking
strength, large elongation at rupture and good elastic recovery [1, 2].
The UIAA (Union Internationale des Associations d'Alpinisme) has established standard testing procedures to measure,
among other things, how a rope reacts to severe falls [3]. Ropes are drop tested with a standardized weight and procedure
simulating a climber fall. This tells us how many of these hypothetical falls the rope can withstand before it ruptures. Virtually
all the ropes on the market can withstand the minimum number of test falls, while some are rated to a much higher number.
The second thing the standard drop test measures is the amount of force which is transmitted to the falling climber.
The standard says little about the durability of the rope, which is more difficult to define or assess with a simplified procedures.
Ropes are produced from polyamide fibers, which exhibit viscoelastic behavior. Thus 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 UIAA standard are not geared to analyze the time-dependent deformation process of the rope, which causes
structural changes in the material and consequently affects its durability. Time-dependent response of the rope also governs
the evolution of all physical quantities that are responsible for the safety of a climber, e.g., first derivative of climber
(de)acceleration.
In this paper we present comprehensive dynamic analysis of a simple non-standard falling weight experiment, which allows
examination of the time-dependent elasto-visco-plastic behavior of ropes exposed to arbitrary falling weight loading conditions.
Developed analytical treatment is subsequently examined on commercial dynamic climbing ropes.
Theoretical Treatment
The time-dependent response of the rope under dynamic loading generated by a falling weight may be retrieved from the
analysis of the force measured at the upper fixture of the rope, as schematically shown in Figure 1a. In such experiment a
weight, m , is dropped from an arbitrary height, h . The length of a tested rope is l 0 . Force measured as function of time,
F (t ) , may be expressed as a set of N discrete datum pairs,
{Fi , ti ; i = 1,2,3,", N} .
Such diagram of measured force is
schematically shown in Figure 1b. The diagram may be subdivided into three distinct phases, indicated as A, B, and C.
(a)
(b)
First loading cycle
F ( t)
A
F ( t)
B
T2
F max
T3
Force - F(t)
m
F ( t)
h
l0
Second loading cycle
C
T4
F ( t)
m
t = t0
T1
mg
m
T4
T0
0
mg
t0 t1
T5
t2 t3
t4 t5
T6
t6
Time - t
Figure 1. Schematics of the rope exposed to the falling weight (a) and force measured during the falling weight experiment (b)
Phase A. In this phase the weight is dropped at t = 0 , and it falls freely until at t = t0 = 2h g the rope becomes straight,
which is indicated in Figure 1b as point T0. Here h indicates the height from which the weight was initially dropped. If we
neglect the air resistance, the velocity of the weight at this point is v 0 = 2gh . The point T0 represents the end of the freefalling phase of the weight, and beginning of the phase B.
Phase B. At point T0 the weight starts to deform the rope. Neglecting the air resistance the equation of motion of the weight
between the points T0 and T5, may be written as,
(1)
mx(t ) = mg − F (t ) .
Here m is the mass of the weight, g is the gravitational acceleration, x (t ) denotes the second derivative of the weight
displacement, x (t ) , measured from the point T0 on. Thus, the x (t ) represents the time-dependent deformation of the rope.
Taking into account the initial conditions at point T0: x (t = t0 ) = 0 , and x (t = t0 ) = v 0 = 2gh , solution of the Eq. (1) gives the
displacement of the weight as function of time, which is equal to the elasto-visco-plastic deformation of the rope,
x(t ) =
t τ
⎤
gt 2 1 ⎡
− ∫ ⎢ ∫ F (ϑ ) dϑ ⎥ dτ + v0 t ,
2
m t0 ⎢⎣t0
⎥⎦
(2)
and velocity of the weight is expressed as,
t
v(t ) = x (t ) = gt −
1
F (τ )dτ + v 0 .
m t∫0
(3)
At point T1, where t = t1 , the force action on the rope becomes equal to the weight of the load, F (t1 ) = mg . At this point the
velocity of the weight reaches its maximum value, v max = v (t1 ) . The force acting on the rope reaches its maximum at point T2,
when t = t2 , or Fmax = F (t 2 ) . Due to the viscoelastic nature of the rope, its maximum deformation will be delayed, and will take
place at t = t3 , i.e., at point T3, where the velocity of the weight is equal to zero. The maximum deformation of the rope is then,
t3
t3
τ
⎡
⎤
1
s max = ∫ v(τ )dτ = ∫ ⎢ gτ − ∫ F (ϑ )dϑ + v0 ⎥dτ .
m t0
⎢
t0
t0 ⎣
⎦⎥
(4)
The unloading phase of the rope starts at the point T3. The elastic component of the rope’s deformation will be retrieved and
will accelerate the weight in the opposite (upward) direction. At t = t 4 , indicated as point T4, the force acting on the weight
becomes again equal to the weight of the load, F (t 4 ) = mg . At point T5, where the force acting on the rope becomes equal to
zero, F (t5 ) = 0 , the weight will start its free-fly in the upward (vertical) direction. The velocity of the weight at the point T5 may
be calculated with the help of Eq. (3) at t = t5 .
Phase C. Point T5 represents the beginning of the phase C in which weight has no interaction with the rope. The weight starts
to fly upwards with the initial velocity, v 5 , and returns back at point T6, to start the second cycle of the rope deformation
process.
At the point T6 starts the second loading cycle of the rope, which may be analyzed with the same set of equations derived for
the phase B and C.
Force-displacement diagram of rope deformation process – energy dissipation.
Energy dissipation during the rope deformation process is one of the most important rope characteristics, and should be used
for comparing the quality of ropes. Force, F (t ) , measured during the loading and unloading of the rope, phase B, may be
expressed as function of the rope deformation, F = F (s ) , as it is schematically shown in Figure 2. The discrete form of this
interrelation may be obtained by calculating the isochronal values of the rope deformation corresponding to each discrete
value of the measured force between the points T0 and T5, {Fi = F (t i ), si = x(ti ); t0 ≤ t i ≤ t5 , i = 1, 2, ", M} , where M is the
number of measured force datum points within the time interval [t0, t5].
T2
Fmax
Force – F(s)
T3
Wdis
T1
mg
T4
kinit
kend
T5
T0
Deformation - s
s1
smax
s4
svp
sel
Figure 2. Force-displacement diagram of the loading and unloading phase
The deformation energy of the rope, at any stage of deformation, may be expressed as
t
W (t ) = ∫ F (τ )
t0
t
τ
⎡
⎤
dx(τ )
1
dτ = ∫ F (τ ) ⎢ gτ − ∫ F (υ )dυ + v0 ⎥dτ ,
dτ
m t0
t0
⎣⎢
⎦⎥
(5)
and should be equal to the sum of the kinetic, Wk (t ) , and the potential energy, Wp (t ) , of the falling weight at any time. We are
particularly interested in the stored energy which is the only source of energy absorption (neglecting the air resistance) and
consequently reduction of the force acting on the climber,
t3
τ
⎡
⎤
dx(τ )
1
= ∫ F (τ )
dτ = ∫ F (τ ) ⎢ gτ − ∫ F (υ )dυ + v0 ⎥dτ .
dτ
m t0
⎢⎣
⎥⎦
t0
t0
t3
Wstore
(6)
During the unloading phase the elastic component of the rope deformation is retrieved and it accelerates the weight in the
upward direction,
Wret =
smax
t5
svp
t3
∫ F (s)ds = ∫ F (τ )
t5
τ
⎡
⎤
dx(τ )
1
dτ = ∫ F (τ ) ⎢ gτ − ∫ F (υ )dυ + v0 ⎥dτ .
dτ
m t0
t3
⎣⎢
⎦⎥
(7)
The dissipated energy within a loading and unloading cycle, represented as a shaded area in Fig. 2, can be expressed as,
t3
t5
τ
τ
⎡
⎤
⎡
⎤
1
1
Wdiss = Wstore − Wret = ∫ F (τ ) ⎢ gτ − ∫ F (υ )dυ + v0 ⎥dτ − ∫ F (τ ) ⎢ gτ − ∫ F (υ )dυ + v0 ⎥dτ .
m t0
m t0
⎥⎦
t0
t3
⎣⎢
⎦⎥
⎣⎢
(8)
All three energies may be calculated also from the sum of the kinetic and the potential energy of the weight at different phases
of the rope deformation process. Thus, the stored energy, Wstore , should be equal to the sum of the kinetic and the potential
energy at the point T3 where the rope is stretched maximally.
Rope stiffness
An important parameter for comparing the performance of different ropes could be 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 decreasing.
Thus, an indicator of the quality and rope durability could be the ratio of the stiffness at the beginning, k init , and the end, kend ,
of the rope deformation process. Hence,
χ=
kinit
≤ 1.
kend
(9)
Stiffness k init and kend may be calculated from the slope of the force-displacement diagram F (s ) at points T1 and T4, as
schematically shown in Figure 2,
kinit =
dF ( s )
, and
ds s = s1
kend =
dF ( s )
,
ds s = s4
(10)
where s1 and s4 are rope deformations at corresponding points T1 and T4, which indicate the beginning and the end of rope
deformation process.
Force impulse and de-acceleration
The experience from the car crash experiments teaches us that the time-variation of the de-acceleration to which a person has
been exposed is much more important for its safety than the magnitude of de-acceleration itself. Thus, the maximum of the
absolute value of the derivative of the de-acceleration may be used as one of the criterions for judging the quality of climbing
ropes,
⎡ d 2v(t ) ⎤
⎡ 1 dF (t ) ⎤ .
= max ⎢
2 ⎥
dt
⎣ m dt ⎥⎦
⎣
⎦
ψ = max ⎢
(11)
Thus, ropes with the smaller values of ψ are better (safer) than those with the larger ψ .
Another important parameter for judging the quality of ropes could be the impulse of force, I , to which a climber is exposed
during different phases of the fall arrest. There are two distinct phases of the fall arrest, see Figure 2, the de-acceleration
phase, between the points T0 and T3, and the following acceleration phase, which takes place between the points T3 and T5,
t3
t5
t0
t3
I1 = ∫ F (t )dt , and I 2 = ∫ F (t )dt .
(12)
For the purely elastic material the two impulses would be equal, I1 = I2 , while for the elasto-visco-plastic material I1 > I2 . The
ratio between the two, ξ =
I2
, is therefore proportional to the amount of viscoelastic deformation of the rope within a loading
I1
cycle, and will be always less or equal to one, ξ ≤ 1 . Smaller ξ indicates larger component of the viscoelastic deformation.
Thus, the evolution of ξ (n ) as function of loading cycles could be another criterion of durability and for comparison of different
ropes.
Experimental
Experimental setup is schematically presented in Figure 3. The console is fixed at the height of 6 m above the floor. The force
sensor is placed on the console. Signals from the force sensor pass through the carrier amplifier prior to being collected in
digital format by the data acquisition system (DAQ). The rope is connected to the force sensor with one end and to the weight
with another in such way that both ends of the rope are on the same level.
Free fall tests were conducted on specimens of three different commercial manufactures. Thickness of each rope was the
same, i.e., 9.8 mm. From each of three ropes four specimens were prepared. The rope was first cut in four pieces, having the
same length. Each specimen was then treated as such that nooses were sewed up on both ends of the specimen as shown in
Figure 4. Each specimen had the same initial length, l0, i.e., 3.38±0.04 m. The specimens were subjected to the same room
temperature conditions. Tests were conducted at temperature 26±2oC and at normal outside atmospheric pressure.
Figure 3. Schematic apparatus layout.
Figure 4. Specimen with nooses.
Before each test, the length of the specimen was measured. At certain time the weight, which was fixed at the end of the rope,
was dropped and the specimen was exposed to the impact loading. The mass of the weight was 43.85±0.02 kg. For each
specimen we repeated 10 falls consequently, with a waiting time between two falls of 5 minutes. The length of the specimen
was measured after each fall. The measured response, i.e., force versus time, F(t), was then saved for further analysis.
The characteristics of the rope as described above were then calculated by using software DAR which was developed at the
Center for Experimental Mechanics.
Results and discussion
The newly developed experimental-analytical methodology enabled us to analyse the time-dependent behavior of ropes under
impact loading. From the measured force during the fall, we have calculated the following characteristics: impact force,
maximum deformation of the rope, derivative of (de)acceleration, stiffness of the rope, ratio of the force impulses, and the
dissipated energy. By using this method we compared calculated characteristics of three different ropes, which according to
existing UIAA standard belong to the same quality class and are declared to have the same UIAA standard characteristics.
The ropes from three different commercial manufacturers were identified as R1, R2, and R3. The average values of rope’s
characteristics and standard deviation were calculated from 4 measurements for each type of rope.
The comparison analysis of the time-dependent behavior of ropes when exposed to impact loading for three different
commercial manufacturers is presented in Figs. 5 to 7. In diagrams calculated characteristics of ropes are presented as
functions of number of falls, N. Figure 5 shows the impact force, Fmax , and the maximum deformation of the rope, smax . The
derivative of (de)acceleration, ψ , and the stiffness of the rope at the beginning of rope deformation, k init , are presented in
Figure 6, while the ratio of impulses, ξ , and dissipated energy, Wdis , are shown in Figure 7.
(b)
5500
1.1
5000
1.05
smax[m]
Fmax[N]
(a)
4500
4000
1.0
0.95
3500
1
2
3
4
5
6
7
N - falls [/]
8
9
0.9
1
10
2
3
4
5
6
7
N - falls [/]
8
9
10
Figure 5. The impact force (a) and the maximum deformation of the rope (b)
as functions of number of falls: - rope R1,
- rope R2, - rope R3.
(b)
4800
1500
4600
1400
4400
1300
4200
kini[N/m]
ψΨ [m/s3]
(a)
1600
1200
4000
1100
3800
1000
3600
900
1
2
3
4
5
6
N - falls [/]
7
8
9
3400
1
10
2
3
4
5
6
N - falls [/]
7
8
9
10
Figure 6. The derivative of (de)acceleration (a) and the stiffness of the rope at the beginning of rope deformation (b)
as functions of number of falls: - rope R1,
- rope R2, - rope R3.
(a)
(b)
0.7
1500
0.68
1450
0.66
1400
0.64
1350
Wdis[Nm]
ξ ξ [/]
0.62
0.6
0.58
0.56
1300
1250
1200
0.54
1150
0.52
0.5
1
2
3
4
5
6
N - falls [/]
7
8
9
10
1100
1
2
3
4
5
6
N - falls [/]
Figure 7. The ratio of impulses (a) and the dissipated energy (b)
as functions of number of falls: - rope R1,
- rope R2, - rope R3.
7
8
9
10
Conclusions
From the diagrams presented above we may recognize significantly different time-dependent behavior of ropes from three
different commercial manufacturers when they are exposed to the same impact loading conditions.
The rope R2 has 15% bigger impact force, 10% smaller maximum deformation and 35% bigger maximum derivative of
(de)acceleration in comparison with the ropes R1 and R3 after the tenth fall. Therefore the rope R2 may be considered as
more dangerous for the climbers then the two others.
Stiffness at the beginning of the rope deformation is a little bigger for the rope R2 rather then for the ropes R1 and R3.
The ratio of impulses is less then 1 for all three ropes which indicates that all three ropes exhibit large amount of viscoelastic
deformation within a loading cycle. As indicated in Figure 6a, the component of the viscoelastic deformation for rope R1 is
being rather larger then for ropes R2 and R3. The dissipated energy for the rope R1 is bigger as well than for the ropes R2 and
R3.
The obtained results indicate that ropes, which according to existing UIAA standard belong to the same quality class and are
declared to have the same UIAA standard characteristics, actually exhibit significantly different behavior when exposed to the
same loading conditions by using our experimental-analytical methodology.
References
1.
2.
3.
Jenkins M. (ed.), Materials in Sports Equipment, Woodhead Publ. Ltd., Cambridge (2003)
McLaren A.J. Design and performance of ropes for climbing and sailing, Proc. ImechE, Part L: J. Materials: Design and
Applications, Vol.220, Number 1, 2006, pp. 1-12.
http://www.uiaa.ch/web.test/visual/Safety/UIAA101DynamicRopes07-2004.pdf