Applied Ocean Research 45 (2014) 22–32
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Contents lists available at ScienceDirect
Applied Ocean Research
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p o r
n experimental investigation on nonlinear behaviors of synthetic
ber ropes for deepwater moorings under cyclic loading
aixiao Liu
a , * , Wei Huang
a , Yushun Lian
a , Linan Li b
School of Civil Engineering, Tianjin University, Tianjin 300072, China School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
r t i c l e i n f o
rticle history:
eceived 15 March 2013
eceived in revised form 13 November 2013
ccepted 23 December 2013
eywords:
aut-wire mooring system
ynthetic fiber rope
olyester
ramid
MPE
onlinear behavior
ynamic stiffness
mpirical expression
xperimental investigation
a b s t r a c t
The nonlinear mechanical behaviors of synthetic fiber ropes including polyester, aramid and HMPE under
cyclic loading are of vital importance to the dynamic response and fatigue life of taut-wire mooring systems.
In the present work, important topics including how the stiffness develops and how the main factors influence
the evolution of dynamic stiffness as well as the nonlinear tension–elongation relationship are systematically
investigated utilizing a specially designed experimental system. The similarity criterion for the dynamic
stiffness of fiber ropes is derived from the dimensional analysis and verified by experiments. The empirical
expressions of dynamic stiffness, which are currently used, are examined by the measured data. It is observed
that the mean load is a main factor that significantly affects the dynamic stiffness; not only the effect of strain
amplitude on the stiffness can not be ignored, but also the influence of loading cycles is of vital importance
to the dynamic stiffness. Based on the measured data, an empirical expression that takes into account both
the mean load, strain amplitude and number of loading cycles is proposed, which is the only one that can
evaluate the evolution of dynamic stiffness under long-term cyclic loading. c © 2013 Elsevier Ltd. All rights reserved.
. Introduction
Since 1980s, people have made an effort to use synthetic fiber
opes as the main component of mooring lines for station keeping of
he floaters like MODU, FPSO, Spar and Semi. Indubitably, synthetic
ber ropes have been proved to be the most suitable substitute for
teel wire ropes or chains in deepwater mooring applications. The
ost widely used fiber rope is of polyester since the first engineering
pplication by Petrobras in 1997 [ 1 ]. In deep waters, the mooring
ines are quite long and the diameter of the polyester rope is usually
everal-hundred millimeters for the demand of breaking strength,
hich bring in a big challenge for the storage ability of the installation
essels. Therefore, researchers have been trying to find other better
aterials than the polyester [ 2 ]. In fact, about 30 years ago, the aramid
nd HMPE which have higher modulus than the polyester have been
pplied in the moorings [ 3 ]. Some reported failure cases [ 4 –6 ] and
heir disadvantages restricted their wide application.
Other than the steel wire ropes or chains, the mechanical prop-
rties of synthetic fiber ropes are generally nonlinear and time-
ependent and exhibit viscoelasticity and viscoplasticity. Bitting [ 7 ]
tudied the dynamic stiffness and hysteresis of nylon and polyester
ouble braid lines 1.5 and 2 in. in diameter based on a series of lab-
ratory tests. Del Vecchio [ 8 ] proposed an empirical expression for
* Corresponding author. Tel.: + 86 2227401510; fax: + 86 2227401510.
E-mail address: [email protected] (H. Liu).
141-1187/ $ - see front matter c © 2013 Elsevier Ltd. All rights reserved.
ttp://dx.doi.org/10.1016/j.apor.2013.12.003
determining the rope modulus at constant temperature as a func-
tion of the testing parameters including the mean load, load am-
plitude and loading period, which was the first attempt to define a
formula of the dynamic modulus for synthetic fiber ropes. Fernandes
et al. [ 9 ] performed a comprehensive set of tests of actual full-scale
polyester mooring cables with diameter of 0.127 m, and detected a
weak dependence of the dynamic stiffness on the frequency. Bosman
and Hooker [ 10 ] carried out experimental studies of dynamic mod-
ulus characteristics of the polyester subrope with breaking strength
of 11.25 tons and the full-size rope with breaking strength of 150
tons, which demonstrated that good predictions of the modulus can
be made from small-scale tests to full-size tests. Casey and Banfield
[ 11 ] performed an investigation into the dynamic axial stiffness of
polyester ropes in the size range 600–1000 tons, and pointed out that
the strain amplitude does exist as a variable for the dynamic stiffness.
It was further demonstrated that, for stochastic loading, the effect
of strain amplitude was sufficiently small which could be neglected
when considering dynamic stiffness for mooring design under low
strain amplitude up to 0.3%; at higher strain amplitude around 0.6%,
the effect should not be ignored. Based on the public domain data,
Wibner et al. [ 12 ] utilized the upper and lower bound stiffnesses to
describe the dynamic stiffness of the polyester rope, in which only the
mean load was taken into account. Davies et al. [ 13 ] performed exper-
imental studies of synthetic fiber ropes including polyester, aramid
and HMPE ones in a dry condition, and the effects of the mean load,
load range and loading frequency on stiffness were investigated. The
H. Liu et al. / Applied Ocean Research 45 (2014) 22–32 23
Table 1
Parameters of synthetic fiber ropes.
Parameter Polyester Aramid HMPE
D (m) 0.006 0.008 0.006 0.008 0.006 0.008
MBL
(kN)
6.8 9.8 9.7 34.1 27.8 49.7
Fig. 1. Experimental system for cyclic loading tests of synthetic fiber ropes.
stiffness and bending behavior of aramid and HMPE ropes for deep
sea handling operations were also investigated in their Subsequent
experiments [ 14 ]. Fran c ¸ ois and Davies [ 15 ] carried out experiments
on the polyester subrope samples with 70-ton breaking strength and
the full size rope with 800-ton breaking strength, in which a “quasi-
static stiffness” was defined to consider the visco-elastic response of
ropes to slow variations of mean load under the effect of changing
weather conditions; the upper and lower bound stiffnesses that only
took into account the effect of mean load were presented.
To sum up, most of the previous studies focused on the value of
the stable dynamic stiffness and seldom concerned about some im-
portant issues including how the stiffness develops and how the main
factors influence the evolution of dynamic stiffness and the nonlinear
stress–strain hysteresis loops. Therefore, four points of knowledge
can be summarized: firstly, although the synthetic fiber ropes show
obvious time-dependent behaviors, which are recognized to be of vi-
tal importance to the dynamic response and fatigue life of taut-wire
mooring systems, there are seldom systematic experiments focusing
on the time-dependent behaviors of the fiber ropes under cyclic load-
ing partially due to the complexity and uncertainty of them; secondly,
most researches focused on the stable dynamic stiffness and factors
based on the empirical expression proposed by Del Vecchio [ 8 ]; how-
ever there were different opinions on the effects of the mean load and
strain amplitude. To obtain a uniform and generally accepted empir-
ical expression of dynamic stiffness is still the final objective of these
researches; thirdly, there were seldom researches that involved the
stiffness changing with loading cycles. However under cyclic load-
ing, the stiffness would significantly change with the cycles. This is a
particular property due to the time-dependent behaviors of synthetic
fiber ropes; fourthly, because a higher storage ability of the installa-
tion vessel is demanded by polyester ropes, recently the aramid and
HMPE ropes become attractive again. The studies on the mechanical
properties of aramid and HMPE ropes are not only necessary but also
urgent.
In the present work, a specially designed experimental system
is employed to systematically investigate the nonlinear behaviors of
three types of synthetic fiber ropes under cyclic loading, including
polyester, aramid and HMPE ones. Important topics including how the
stiffness develops and how the main factors influence the evolution
of the dynamic stiffness as well as the nonlinear tension–elongation
relationship are investigated in detail. The similarity criterion for the
dynamic stiffness of fiber ropes is derived from the dimensional anal-
ysis and examined by experiments. The empirical expressions of dy-
namic stiffness, which are currently used, are also examined by the
measured data. Based on the measured data, an empirical expression
that takes into account both the mean load, strain amplitude and
number of loading cycles is proposed, which is the only one that can
evaluate the evolution of dynamic stiffness under long-term cyclic
loading.
2. Experimental system, synthetic fiber ropes and test cases
The experimental system mainly consists of four parts, i.e., the
loading elements, the equipment foundation, the measurement sys-
tem and the water cycling system, as shown in Fig. 1 . The loading el-
ements are constitutive of two divided parts, one is the static loading
element, and the other is the dynamic loading element. The maximum
capacity of static loading is 6 tons, while the one of dynamic loading
is 5 tons. The loading elements can provide two loading ways, i.e.,
the load-controlled and the displacement-controlled, which are re-
spectively provided by the static and dynamic loading elements. The
former is to apply a constant mean load, and the latter is to apply a
strain amplitude. The two ways are combined to approximately sim-
ulate the response of mooring lines in a taut-wire mooring system,
where a specific pre-tension is applied to the mooring line and the
floater moves periodically with a relatively unchangeable amplitude.
The equipment foundation was designed and jointed by the box irons
and rolled angles to be easily taken down. A water cycling system
was designed and fabricated utilizing double water tanks to simulate
the real water environment. With the help of two water pumps, the
water in the outer tank can be injected into the inner one. The tested
fiber rope will be always immerged in water as long as the flow of
the injected water is larger than the outlet flow. An elongation mea-
surement system was designed combined with the wire transducer,
axletree, sliding block and clamp. Two axletrees are parallel installed
above the water tanks. On the axletrees, there are two sliding blocks,
one of which is used as the base for the wire transducer and the other
is used to fix the wire. Two specially designed clamps installed un-
der sliding blocks are used to calibrate the test region, which can be
reliably fixed on the gauge marks of the rope even if the load is near
the breaking load of the rope. The terminations of the rope are made
with the eye splice method instead of the knot method. Therefore the
possible slip between contact surfaces at knots when subjected to a
large tension can be effectively eliminated.
Three types of materials are employed in the present study, in-
cluding 3 strand construction polyester ropes, 16 × 2 double braided
aramid ropes, and 12 strand construction HMPE ropes. For each type
of materials, there are still two sizes, i.e., 0.006 and 0.008 m in diame-
ter, as shown in Fig. 2 . The parameters of materials are listed in Table
1 , in which D denotes the diameter of the rope and MBL means the
minimum breaking load of the rope. All the sample ropes were pro-
vided by Jiuli Ropes Company Limited. The fiber of HMPE is Dyneema
SK75, which was produced by DSM Dyneema. The fiber of aramid is
Kevlar49, which was produced by DoPont. The fiber of polyester is
polyethylene terephthalate, which was produced by Jiuli Ropes Com-
pany Limited. The total length of the specimen including splices is 1.7
m, but the effective length in measurement is 0.8 m. Before cyclic load-
ing tests, the static tension–elongation curve was measured through
static loading tests, which repeat five times for each case. Note that
enough bedding-in process was performed for all sample ropes before
static loading tests. The static tension–elongation curves of polyester
ropes are presented in Fig. 3 , which indicate that the normalized elon-
gations at rupture of 0.006 and 0.008 m ropes are 14.0% and 14.6%,
respectively. The static tension–elongation curves of aramid ropes are
presented in Fig. 4 , which indicate that the normalized elongations at
24 H. Liu et al. / Applied Ocean Research 45 (2014) 22–32
Table 2
Test cases.
Case Polyester Aramid HMPE
D (m) L m (%MBL) ε a (%) D (m) L m (%MBL) ε a (%) D (m) L m (%MBL) ε a (%)
1 0.006 8.3 0.1 0.006 19.6 0.1 0.006 15.1 0.1
2 0.006 24.7 0.1 0.006 58.8 0.1 0.006 45.3 0.1
3 0.006 8.3 0.3 0.006 19.6 0.3 0.006 15.1 0.2
4 0.006 9.7 0.1 0.006 10.2 0.1 0.006 13.0 0.1
5 0.008 9.7 0.1 0.008 10.2 0.1 0.008 13.0 0.1
Fig. 2. Three types of fiber ropes and their constructions. (a) Construction and (b) two
sizes.
r
T
F
0
2
a
a
t
s
3
t
ii
Fig. 3. Static tension–elongation curves of polyester ropes with different size.
Fig. 4. Static tension–elongation curves of aramid ropes with different size.
Fig. 5. Static tension–elongation curves of HMPE ropes with different size.
upture of 0.006 and 0.008 m ropes are 3.0% and 2.0%, respectively.
he static tension–elongation curves of HMPE ropes are presented in
ig. 5 , which indicate that the normalized elongations at rupture of
.006 and 0.008 m ropes are 2.9% and 2.2%, respectively.
The experimental cases in cyclic loading tests are listed in Table
, in which L m
denotes the mean load, and ε a denotes the strain
mplitude. Cases 1–3 were designed to investigate the effects of L m
nd ε a on the cyclic tension–elongation and hysteresis properties and
he stiffness evolution. Cases 4 and 5 were designed to investigate the
ize effects on the stiffness evolution.
. Cyclic tension–elongation relationship and hysteresis
behaviors
The cyclic tension–elongation relationship, hysteresis behaviors, he stiffness and residual evolution, and the effects of the parameters
ncluding the mean tension and the strain amplitude are investigated
n the present work. The loading period is 4 s, the data acquisition
speed is 9 points per second and there are 36 points per cycle, and
1000 cycles are performed for each test. The directly measured data
H. Liu et al. / Applied Ocean Research 45 (2014) 22–32 25
Fig. 6. Cyclic tension–elongation relationship in Case 3 of polyester ropes. (a) Cyclic
tension–elongation relationship and (b) last 10 cycles of tension–elongation relation-
ship.
include the tension and the gauge elongation �l . Note that the nor-
malized tension and elongation can be expressed as T / MBL and �l / l ,respectively, then the normalized stiffness K r can be expressed as
K r =
E A
MB L
=
(T
p n − T t n −1
)/ MB L
ε p n − ε t n −1
(1)
where, EA denotes the stiffness; T p
n and ε p n are the peak tension and
the peak strain of the n th tension–elongation hysteresis loop, respec-
tively; T t n −1 and ε t n −1 are the trough tension and strain of the ( n − 1)th
tension–elongation hysteresis loop, respectively; n ≥ 2.
The cyclic tension–elongation curves as well as the last 10 cycles
in Case 3 of polyester, aramid and HMPE ropes are presented in Figs.
6 –8 . Through systematically examining these curves, several findings
can be summarized as follows:
(1) The tension–elongation relationship is strongly nonlinear un-
der cyclic loading. The hysteresis loop is formed because of
nonlinear viscoelastic and viscoplastic properties. The change
in length is relevant to time, so any form of stiffness which
is used to express the tension–elongation relationship is an
approximate method to some extent.
(2) The hysteresis loop obviously changes with cycles. Each cycle
of hysteresis loops gradually coincides with each other, and
becomes nearly stable after 1000 cycles. The area of the hys-
teresis loop represents the energy dissipation which induces
the generation and accumulation of the heat of fiber ropes. In
the present experiments, the temperature of fiber ropes does
not obviously increase. This is attributed to the water cycling
system which effectively eliminates the heat accumulation.
(3) The stiffness of synthetic fiber ropes shows obvious hardening.
The stiffness increases with cycles and gradually becomes sta-
ble due to the time dependent property. Therefore, using the
empirical expression that represents the nearly stable dynamic
stiffness is reasonable. However, the empirical method is only
conditionally acceptable because in some cases the evolution
of dynamic stiffness under long-term cyclic loading is notable,
as analyzed in the subsequent section.
(4) The residual strain after each cycle also increases and accumu-
lates with loading cycles, and gradually becomes stable.
(5) The mean load and strain amplitude are main factors which
influence the cyclic tension–elongation, hysteresis properties
and stiffness. However, the effects differ due to different ma-
terials and constructions of fiber ropes. This means that for a
specific type of mooring ropes, specific studies are needed to
explore the effects.
4. Evolution of stiffness and residual strain
The nonlinear cyclic tension–elongation properties lead to evolu-
tion of the stiffness and residual strain. The evolution of stiffness with
loading cycles in Cases 1 and 2 is shown in Figs. 9 –11 . The evolution
of residual strain with loading cycles in Cases 1 and 2 is presented in
Figs. 12 –14 . In order to quantitatively know the evolution of stiffness,
the relative increment of stiffness is calculated, which is defined as
�K r = 100 × K rn − K rc
K rc (2)
where, �K r denotes the relative increment of stiffness of the n th
cycle compared with the reference cycle; K rn denotes the stiffness
of the n th cycle; K rc denotes the stiffness of the reference cycle. For
polyester ropes, the 200th cycle is chosen as the reference cycle; for
aramid and HMPE ropes, the 100th cycle is chosen as the reference
cycle. The calculated results of Cases 1–3 are presented in Figs. 15 –17 .
Several findings can be summarized from these figures as follows:
(1) For any material, the mean load significantly affects the stiff-
ness. The stiffness increases with increasing mean load. Being
a preliminary and simplified analysis, it is reasonable to evalu-
ate the stiffness only considering the mean load while ignoring
other factors such as the strain amplitude and loading period.
(2) Even for smaller strain amplitude, the effect of strain amplitude
on stiffness is obvious. It was proposed by Casey and Banfield
[ 11 ] that, for the stochastic loading, the effect on dynamic stiff-
ness can be ignored if the strain amplitude is smaller than
0.3%. Bosman and Hooker [ 10 ] and Fran c ¸ ois et al. [ 15 , 19 ] stated
that the effect of strain amplitude should be ignored. How-
ever, the present experiments demonstrate that for any type
of fiber ropes, the influence of strain amplitude on stiffness is
significant and cannot be ignored even for the strain amplitude
smaller than 0.3%. This means that only considering the effect
of mean load on stiffness in the mooring analysis is not suffi-
cient; the effect of strain amplitude should also be taken into
account in a precise analysis.
(3) The residual strain increases with loading cycles, but the in-
creasing trend becomes inconspicuous. This means that the
plastic strain accumulates gradually with the loading process.
Although the final hysteresis loops can be approximated as
stable, the gradually changing strain brings in uncertainties to
the long-term properties of synthetic fiber ropes. The specific
effects on long-term mechanical properties of synthetic fiber
ropes still need to be further investigated.
(4) For the polyester rope, the stiffness significantly increases be-
fore the 200th cycle; for the aramid and HMPE ropes, the hard-
ening process before the 100th cycle is obvious. Comparing
26 H. Liu et al. / Applied Ocean Research 45 (2014) 22–32
Fig. 7. Cyclic tension–elongation relationship in Case 3 of aramid ropes. (a) Cyclic
tension–elongation relationship and (b) last 10 cycles of tension–elongation relation-
ship.
Fig. 8. Cyclic tension–elongation relationship in Case 3 of HMPE ropes. (a) Cyclic
tension–elongation relationship and (b) last 10 cycles of tension–elongation relation-
ship.
Fig. 9. Evolution of stiffness with loading cycles in Cases 1 and 2 of polyester ropes.
with the reference cycle, the stiffness changes differently for
different case and material, as shown in Figs. 15 –17 . In Cases
1 and 3 of polyester, the relative increments of stiffness of the
1000th cycle are averagely 5% and 7%, respectively; in Cases
2 and 3 of aramid, the relative increments of stiffness of the
1000th cycle are averagely 3% and 7%, respectively; in Case 1
of HMPE, the relative increment of stiffness of the 1000th cycle
is averagely 9%. Even in the cases where the stiffness becomes
nearly stable like Case 2 of polyester, Case 1 of aramid and
Cases 2 and 3 of HMPE, the number of cycles corresponding to
the nearly stable stiffness is different. For example, in Case 2
of polyester, the number is about 600; in Case 1 of aramid, the
number is about 400; in Cases 2 and 3 of HMPE, the number is
about 500. All these indicate that the stiffness of fiber ropes is
time dependent but the evolution remains complexity.
(5) The number of cycles corresponding to the nearly stable
stiffness is relevant to loading conditions. For example, for
polyester and HMPE ropes, the stiffness becomes nearly sta-
ble after fewer cycles if the mean load is larger; for aramid
ropes, the stiffness reaches nearly stable after fewer cycles if
the mean load is smaller; for HMPE ropes, the stiffness becomes
nearly stable after fewer cycles under larger strain amplitudes;
while for aramid ropes, under smaller strain amplitudes, the
stiffness reaches nearly stable after fewer cycles. These also
indicate that the effects of the mean load and strain amplitude
on the evolution of stiffness are different for different type of
materials.
Note that Cases 1–3 were not designed to compare the three mate-
rials to each other because the mean load and strain amplitude were
not uniform corresponding to the same case. Figs. 15 –17 indicate
that for each type of fiber ropes, the evolution of dynamic stiffness
is notable under some loading conditions. However, a general rule
for synthetic fiber ropes cannot be gained presently because some
complex factors such as the material type, construction, bedding-in
process and loading condition may influence the evolution of dynamic
stiffness. Further investigation is still needed on this topic.
H. Liu et al. / Applied Ocean Research 45 (2014) 22–32 27
Fig. 10. Evolution of stiffness with loading cycles in Cases 1 and 2 of aramid ropes.
Fig. 11. Evolution of stiffness with loading cycles in Cases 1 and 2 of HMPE ropes.
Fig. 12. Evolution of residual strain with loading cycles in Cases 1 and 2 of polyester
ropes.
Fig. 13. Evolution of residual strain with loading cycles in Cases 1 and 2 of aramid
ropes.
Fig. 14. Evolution of residual strain with loading cycles in Cases 1 and 2 of HMPE ropes.
Fig. 15. Relative increment of stiffness of polyester ropes.
Fig. 16. Relative increment of stiffness of aramid ropes.
Fig. 17. Relative increment of stiffness of HMPE ropes.
28 H. Liu et al. / Applied Ocean Research 45 (2014) 22–32
5
5
t
o
[
w
i
δ
s
λ
w
i
t
s
e
d
c
e
K
h
a
t
s
o
o
ρE
m
l
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩F
K
B
e
o
λ
C
r
t
m
E
t
e
Fig. 18. Stiffness of polyester ropes with different size.
Fig. 19. Stiffness of aramid ropes with different size.
Fig. 20. Stiffness of HMPE ropes with different size.
. Dynamic stiffness
.1. Dimensional analysis
Fernandes and Rossi [ 16 ] studied the relationship between the dis-
orted model and the prototype in applying the empirical expression
f dynamic stiffness adopted by Del Vecchio [ 8 ] and Fernandes et al.
9 ], which is written as
E
ρ= α + βL m
− γ L a − δ L og ( T ) (3)
here, E /ρ is the specific modulus; L m
is the mean load ( %MBL); L a s the load amplitude ( %MBL); T is the excitation period; α, β , γ and
are coefficients.
Based on the dimensional analysis, the similarity criterion of the
pecific modulus can be expressed as [ 16 ]
E/ρ =
( E/ρ) p
( E/ρ) m
=
λF b
λd = 1 (4)
here, the subscript p stands for prototype and m for model; λE /ρ
s the scale factor of the specific modulus; λF b is the scale factor of
he breaking load; λd is the scale factor of the linear density. This
imilarity criterion was adopted by Vlasblom et al. [ 17 ] in the creep
xperiments of HMPE fibers and ropes.
The dimensional analysis is also adopted in the present work to
erive the similarity criterion of the normalized stiffness K r . The pro-
edure is as follows:
Firstly, based on the knowledge of dynamic stiffness, K r can be
xpressed as the function f 1 of several parameters, that is
r = f 1 ( MB L , T m
, εa , N, D, T , l, d, ρ, . . . ) (5)
ere it is necessary to emphasize the definition of these parameters
gain, i.e., MBL denotes the minimum breaking load (kN); T m
denotes
he mean tension (kN); ε a denotes the strain amplitude (nondimen-
ional); N is the loading cycles (nondimensional); D is the diameter
f the fiber rope (m); T is the loading period (s); l is the initial length
f the fiber rope (m); d is the linear density of the fiber rope (kg / m);
is the density of the fiber rope (kg / m
3 ). Note that ε a is adopted in
q. (5) to substitute the effect of L a .
Secondly, choose D , d , T as the basic dimensions, and the nondi-
ensional parameters can be obtained based on π theorem. The fol-
owing nondimensional parameters are considered in analysis:
π1 =
E/ρ
MB L /d π2 =
MB L · T 2
ρ · l 2
π3 =
T m
· T 2
ρ · l 2 π4 = ε a
π5 = N
(6)
inally, K r can be further expressed as the function f 2 as the following:
r = f 2 ( π1 , π2 , π3 , π4 , π5 ) (7)
ased on the similarity parameters adopted in Eqs. (6) and (7) , and
mploying Eq. (4) , the similarity criterion of dynamic stiffness can be
btained as
K r =
( K r ) m
( K r ) p =
λE/ρλd
λMBL = 1 (8)
The present measured data of polyester, aramid and HMPE ropes in
ases 4 and 5 are employed to examine Eq. (8) . The stiffnesses of fiber
opes with different size are presented in Figs. 18 –20 . It is observed
hat the normalized dynamic stiffnesses generally agree well. This
eans that the derived similarity criterion of dynamic stiffness, i.e.,
q. (8) , is reasonable. Eq. (8) is very important in practice because
he dynamic stiffness obtained from small size ropes can be easily
xtrapolated to large size ones.
5.2. Comparison between the empirical expression and the measured
data
The present measured data of dynamic stiffness are also used to
compare with the empirical expressions. There are several types of
expressions. Even in the same type, the coefficients in the expression
are usually different to others. Due to that polyester ropes are most
widely used, there are more studies and expressions for this type of
material.
5.2.1. Polyester
The factors affecting the dynamic stiffness of polyester ropes in-
clude the mean load, the strain or load amplitude, the load history,
and the load period, etc. The effect of load period is considered to
H. Liu et al. / Applied Ocean Research 45 (2014) 22–32 29
Table 3
Coefficients in empirical expressions of dynamic stiffness for polyester ropes.
Coefficients Eq. (10) Eq. (9) Eq. (11)
1 2 3 4 1 2 3 1 2
α 11.477 10.019 11.042 10.830 15 18.5 22 17 23
β 0.153 0.175 0.161 0.100 0.25 0.33 – 0.2 0.25
γ −5.429 −5.035 −5.462 −4.700 – – – – –
Conditions 3 strand rope with 6 ton MBL Full scale
rope with
750 ton MBL
Lower bound Upper bound L m ≤10%MBL
Lower bound Upper bound
Table 4
Comparison between the measured and calculated stiffnesses of polyester ropes.
Polyester
Measured
data Eq. (10) Eq. (9) Eq. (11)
1 2 3 4 1 2 1 2
Case 1 10.14 12.20 10.97 11.83 11.19 22.00 – 18.66 25.08
Relative
error (%)
16.91 7.55 14.30 9.38 53.91 – 45.66 59.56
Case 2 16.87 14.71 13.84 14.47 12.83 21.18 26.65 21.94 29.18
Relative
error (%)
14.66 21.91 16.57 31.49 20.33 36.70 23.11 42.18
Case 3 8.72 11.12 9.96 10.74 10.25 22.00 – 18.66 25.08
Relative
error (%)
21.57 12.46 18.81 14.93 60.36 – 53.27 65.23
Table 5
Comparison between the measured and calculated stiffnesses of aramid ropes.
Aramid Measured data Eq. (12) Relative error (%)
Case 1 66.28 56.37 17.58
Case 2 80.57 79.10 3.14
Case 3 53.97 56.37 8.96
be not obvious and can be ignored [ 10 , 12 , 13 , 18 ]. However, different
researchers had different opinions on the effects of other factors. It
was proposed by Fernandes and Del Vecchio [ 8 , 9 ] that the mean load
and load amplitude should be taken into account in order to eval-
uate the dynamic stiffness more precisely. But Bosman and Hooker
[ 10 ] thought that the influence of load amplitude could be ignored.
The expression of dynamic stiffness proposed by Fran c ¸ ois and Davies
[ 1 , 15 ] and Fran c ¸ ois et al. [ 19 ] is written as
{K r = α + βL m
K r = 22 f or L m
≤ 10% MB L
(9)
Another form of expression was proposed by Casey et al. [ 20 ] as
the following:
K r = α + βL m
+ γ ε a (10)
While the expression utilized by Wibner et al. [ 12 ] is written as
K r = α + βL m
(11)
The coefficients α, β and γ in the above expressions are listed in
Table 3 .
Cases 1–3 of polyester ropes in Table 2 are used in comparison be-
tween the calculated and measured dynamic stiffnesses. Generally,
after more than several-hundred loading cycles, the stiffness reaches
nearly 90% of the so-called stable dynamic stiffness. Therefore, the
stiffness of the 1000th cycle is adopted as the measured one. The rel-
ative error of the measured data to the calculated stiffness is analyzed
and the results are listed in Table 4 .
In general, the difference between the calculated and measured
stiffness is significant, which may be caused by various factors such as
experimental conditions, measurement methods, and different fibers
and structures of the rope. Especially when the mean tension is lower
than 10%MBL (Cases 1 and 3), the relative error between the mea-
sured value and the calculated result from Eq. (9) is above 50%. This
means that the recommended value 22 significantly overestimates
the dynamic stiffness under lower tension levels. When the mean
tension equals 24.7%MBL, the measured value is generally closer to
the lower bound of the calculated result, but the difference is still
obvious, which may lead to overestimation of the tension response
of mooring ropes. Similar to Eq. (9) , Eq. (11) also overestimates the
dynamic stiffness, and generally the measured stiffness is closer to
the lower bound of the calculated result. Table 4 demonstrates that
the calculated stiffness from Eq. (10) is closest to the measured value.
5.2.2. Aramid
Davies et al. [ 13 ] proposed an empirical expression for evaluating
the dynamic stiffness of aramid ropes, as the following:
K r = 45 . 0 + 0 . 58 L m
(12)
Cases 1–3 of aramid ropes in Table 2 are used in comparison between
the calculated and measured dynamic stiffnesses. The relative error
of the measured data to the calculated stiffness is analyzed and the
results are listed in Table 5 . It is observed that the difference between
the measured and calculated stiffnesses is small. However, the effect
of strain amplitude is not taken into account in Eq. (12) . As can be
clearly indicated, the measured dynamic stiffness in Case 1 is 66.28,
while in Case 3 it decreases to 53.97. It is obvious that Eq. (12) is too
simple to capture the variation induced by strain amplitude.
5.2.3. HMPE
Davies et al. [ 13 ] also proposed an empirical expression for evalu-
ating the dynamic stiffness of HMPE ropes, as the following:
K r = 59 . 0 + 0 . 54 L m
(13)
Cases 1–3 of HMPE ropes in Table 2 are used in comparison between
the calculated and measured dynamic stiffnesses. The relative error
30 H. Liu et al. / Applied Ocean Research 45 (2014) 22–32
Table 6
Comparison between the measured and calculated stiffnesses of HMPE ropes.
HMPE Measured data Eq. (13) Relative error (%)
Case 1 60.11 67.15 10.49
Case 2 75.08 83.46 10.04
Case 3 53.23 67.15 20.73
Table 7
Coefficients in the empirical expression taking into account loading cycles.
Material α β γ δ κ
Polyester 7.16 0.41 6.30 1.78 0.010
Aramid 64.07 0.37 62.35 8.87 0.020
HMPE 53.56 0.62 50.70 8.24 0.013
o
r
g
t
c
1
t
5
w
fi
u
c
b
b
t
c
K
w
s
r
W
d
c
b
n
b
t
2
e
y
C
a
t
2
b
t
fi
c
fi
6
d
Fig. 21. Comparison between the measured data and empirical expression for
polyester ropes.
f the measured data to the calculated stiffness is analyzed and the
esults are listed in Table 6 , which also shows that the measured value
enerally agrees well with the calculated result. Similar to Eq. (12) ,
he effect of strain amplitude is not taken into account in Eq. (13) . As
an be observed in Table 6 , the measured dynamic stiffness in Case
is 60.11, while in Case 3 it decreases to 53.23. Obviously Eq. (13) is
oo simple to capture the variation induced by strain amplitude.
.3. The empirical expression taking into account loading cycles
The present experiments demonstrate that the stiffness changes
ith loading cycles due to the time-dependent properties of synthetic
ber ropes. Under complicated ocean environments, there are more
ncertainties on the variation of dynamic stiffness. In order to pre-
isely describe the dynamic stiffness, the loading cycles should also
e taken into account in the empirical expression of dynamic stiffness
esides other key factors including the mean load and strain ampli-
ude. A new empirical expression that takes into account the loading
ycles N is herein proposed as the following:
r = α + βL m
− γ ε a − δe −κ N (14)
here α, β , γ , δ and κ are coefficients related to the material and
tructure of fiber ropes, in which δ and κ are specially introduced to
eflect the effect of long-term cyclic loading on the dynamic stiffness.
ith increase of loading cycles, the effect of the item δe −κN would
ecrease and become stable. The property of the expression is ac-
ordant with the evolution of dynamic stiffness that the stiffness is
eing hardened during the former period of loading and then become
early stable with increasing loading cycles.
The measured results of Cases 1–3 in Table 2 are adopted to cali-
rate the coefficients in Eq. (14) by fitting the stiffness data. Values of
he coefficients for the three materials are listed in Table 7 . Figs. 21 –
3 present the comparison between the measured data and empirical
xpressions, which show a generally good agreement. The error anal-
sis indicates that, for polyester ropes the average relative errors of
ases 1–3 are 2.0%, 2.6% and 2.2%, respectively; for aramid ropes the
verage relative errors of Cases 1–3 are 3.4%, 1.3% and 1.9%, respec-
ively; for HMPE ropes the average relative errors of Cases 1–3 are
.8%, 2.4% and 2.7%, respectively.
Eq. (14) is the only empirical expression that takes into account
oth the mean load, the strain amplitude and the loading cycles, and
o some extent reflects the time-dependent properties of synthetic
ber ropes. It can be simply combined in the mooring analysis to
onveniently evaluate the evolution of dynamic stiffness of synthetic
ber ropes under long-term cyclic loading.
. Concluding remarks
The nonlinear mechanical behaviors of synthetic fiber ropes un-
er cyclic loading are of vital importance to the dynamic response and
fatigue life of taut-wire mooring systems. In the present work, a spe-
cially designed experimental system is employed to systematically
investigate the nonlinear behaviors of three types of synthetic fiber
ropes under cyclic loading, including polyester, aramid and HMPE
ones. Important topics including how the stiffness develops and how
the main factors influence the evolution of the dynamic stiffness as
well as the nonlinear tension–elongation relationship are investigated
in detail. It is observed that the mean load is a main factor that sig-
nificantly affects the dynamic stiffness; not only the effect of strain
amplitude on the stiffness can not be ignored, but also the influence
of the loading cycles is of vital importance to the dynamic stiffness.
The similarity criterion for the dynamic stiffness of fiber ropes
is also derived based on the dimensional analysis and preliminarily
H. Liu et al. / Applied Ocean Research 45 (2014) 22–32 31
Fig. 22. Comparison between the measured data and empirical expression for aramid
ropes.
Fig. 23. Comparison between the measured data and empirical expression for HMPE
ropes.
verified by experiments. Through utilizing the similarity criterion,
the dynamic stiffness obtained from small size ropes can be easily
extrapolated to large size ones. It should be emphasized that the
similarity criterion was only examined by two sizes of fiber ropes.
Further work is meaningful to examine the efficiency of the similarity
criterion with more cases.
The empirical expressions of dynamic stiffness, which are cur-
rently used, are examined by the measured data. For polyester ropes,
in general, the difference between the calculated and measured stiff-
ness is significant, which may be caused by various factors such as
experimental conditions, measurement methods, and different fibers
and structures of the rope. Compared with Eqs. (9) and (11) , which
were proposed by Fran c ¸ ois et al. [ 1 , 15 , 19 ] and Wibner et al. [ 12 ], re-
spectively, the calculated results from Eq. (10) proposed by Casey et
al. [ 20 ] are closer to the measured data. For aramid and HMPE ropes,
the measured data generally agree well with the calculated results
from Eqs. (12) or (13) proposed by Davies et al. [ 13 ]. However, the
comparison also indicates that Eqs. (12) and (13) are too simple to
capture the variation induced by strain amplitude.
As already observed from experiments, the dynamic stiffness also
changes with loading cycles due to time-dependent properties of syn-
thetic fiber ropes. To develop an empirical expression which com-
prises main influential factors is meaningful. Based on the measured
data, an empirical expression that takes into account both the mean
32 H. Liu et al. / Applied Ocean Research 45 (2014) 22–32
l
w
d
b
t
t
g
o
t
fi
fi
n
A
C
R
oad, strain amplitude and number of loading cycles is proposed,
hich is the only one that can to some extent reflect the time-
ependent properties of synthetic fiber ropes. This expression can
e simply combined in the mooring analysis to conveniently evaluate
he evolution of dynamic stiffness of synthetic fiber ropes under long-
erm cyclic loading. However, it should be noted that although the
eneral form of Eq. (14) applies to synthetic fiber ropes, the presently
btained coefficients in Eq. (14) , as listed in Table 7 , are suitable to
he present sample ropes. As defined earlier, values of the five coef-
cients α, β , γ , δ and κ are related to the material and structure of
ber ropes. For a specific type of fiber ropes, specific experiments are
eeded to calibrate these coefficients by fitting the stiffness data.
cknowledgement
Financial support from the National Natural Science Foundation of
hina (Grant nos. 51179124 and 50979070 ) is greatly acknowledged.
eferences
[1] Fran c ¸ ois M, Davies P. Fibre rope deep water mooring: a practical model for the analysis of polyester systems. In: Proc. of Rio oil & gas conf., Rio de Janeiro,
Brazil. 2000 . [2] Davies P, Lechat C, Bunsell A, Piant A, Fran c ¸ ois M, Grosjean F. Deepwater moor-
ings with high stiffness polyester and PEN fiber ropes. In: Proc. 40th annu. offshore technol. conf., OTC 19315, Houston, USA. 2008 .
[3] Banfield S, Casey N. Evaluation of fiber rope properties for offshore mooring.
Ocean Engineering. 1998;25(10):861–79 . [4] Riewald PG, du Pont de Nemours EI. Performance analysis of an aramid mooring
line. In: Proc. 18th annu. offshore technol. conf., OTC 5187, Houston, USA. 1986 . [5] Corbetta I, Sloan F. HMPE mooring line trial for Scarabeo III. In: Proc. 33rd annu.
offshore technol. conf., OTC 13272, Houston, USA. 2001 . [6] Chi CH, Lundhild EM, Veselis T, Huntley MB. Enabling ultra-deepwater mooring
with aramid fiber rope technology. In: Proc. 41th annu. offshore technol. conf.,
OTC 20074, Houston, USA. 2009 .
[7] Bitting KR. Dynamic modeling of nylon and polyester double braid line. In: Proc.
oceans conf. rec., San Diego, USA. 1985 . [8] Del Vecchio CJM. Light weight materials for deep water moorings. UK: Reading
University; 1992, [Thesis] . [9] Fernandes AC, Del Vecchio CJM, Castro GAV. Mechanical properties of polyester
mooring cables. International Journal of Offshore and Polar Engineering.
1999;9:208–13 . [10] Bosman RLM, Hooker J. The elastic modulus characteristics of polyester mooring
ropes. In: Proc. 31st annu. offshore technol. conf., OTC 10779, Houston, USA. 1999 .
[11] Casey NF, Banfield SJ. Full-scale fiber deepwater mooring ropes: advancing the knowledge of spliced systems. In: Proc. 34th annu. offshore technol. conf., OTC
14243, Houston, USA. 2002 . [12] Wibner C, Versavel T, Masetti I. Specifying and testing polyester mooring rope
for the Barracuda and Caratinga FPSO deepwater mooring systems. In: Proc.
35th annu. offshore technol. conf., OTC 15139, Houston, USA. 2003 . [13] Davies P, Francois M, Grosjean F, Baron P, Salomon K, Trassoudaine D. Synthetic
mooring lines for depths to 3000 meters. In: Proc. 34th annu. offshore technol. conf., OTC 14246, Houston, USA. 2002 .
[14] Davies P, Reaud Y, Dussud L, Woerther P. Mechanical behaviour of HMPE and aramid fibre ropes for deep sea handling operations. Ocean Engineering.
2011;38:2208–14 .
[15] Fran c ¸ ois M, Davies P. Characterization of polyester mooring lines. In: Proc. 27th int conf. offshore mech arct eng, OMAE 2008-57136, Estoril, Portugal. 2008 .
[16] Fernandes AC, Rossi RR. Distorted polyester lines for model testing of offshore moored platforms. Ocean Engineering. 2004;32(7):817–25 .
[17] Vlasblom MP, Boesten J, Leite S, Davies P. Development of HMPE fiber for perma- nent deepwater offshore mooring. In: Proc. 44th annu. offshore technol. conf.,
OTC 23333, Houston, USA. 2012 .
[18] Casey NF, Banfield SJ. Factors affecting the measurement of axial stiffness of polyester deepwater mooring rope under sinusoidal loading. In: Proc. 35th annu.
offshore technol. conf., OTC 17068, Houston, USA. 2003 . [19] Fran c ¸ ois M, Davies P, Grosjean F, Legerstee F. Modelling fiber rope load–
elongation properties – polyester and other fibers. In: Annu. offshore technol. conf., OTC 20846, Houston, USA. 2010 .
[20] Casey NF, Belshaw R, Paton AG, Hooker J. Short- and long-term property be-
haviour of polyester rope. In: Proc. 32th annu. offshore technol. conf., OTC 12177, Houston, USA. 2000 .
