Experimental Characterization of Stiffness of a Polyester Mooring Rope for a CFPSO

Abstract

Synthetic ropes are increasingly favored for use in offshore systems. Accurate predictions of the coupled hydrodynamic performance and structural response of offshore structures depends on a thorough understanding of the nonlinear characteristics of fiber materials. The objective of this study is to experimentally characterize the stiffness of a polyester mooring rope for a cylindrical floating production storage and offloading system. The quasi-static stiffness of the ropes and aged ropes after installation and the dynamic stiffness in various loading conditions were computed based on sub-rope tests following the guidelines from the American Bureau of Shipping. The quasi-static stiffness curves exhibited a linear decrease in values as the logarithm of the loading period (in minutes) increased. The dynamic stiffness was, in general, much higher than the quasi-static stiffness. The dynamic stiffness under various loading conditions revealed the complexities of the mechanical properties of polyester rope. Parameters such as the mean load, load amplitude, loading period, and loading cycles all had a notable impact. More tests are required to have a better understanding of their effects.

1. Introduction

Offshore environments, whether utilized for oil and gas extraction or renewable energy generation, require robust mooring systems to ensure stability, functionality, and safety. In the oil and gas industry, mooring systems are critical for anchoring offshore platforms, including floating production, storage, and offloading (FPSO) vessels, as well as semi-submersibles and other drilling units. These systems must withstand extreme environmental conditions, such as high waves and strong currents, to ensure the safety of personnel and equipment. The renewable energy sector, particularly the floating offshore wind and wave energy industries, relies heavily on mooring systems to ensure the survivability and economic viability of offshore installations. The commercialization of wave energy faces challenges, with one significant hurdle being the higher levelized cost of energy (LCOE) compared to other renewable sources [1]. Similarly, floating offshore wind turbines require robust mooring systems to withstand harsh marine environments and to optimize power generation.

Mooring design, which depends on the type of structure and the water depth, can have different levels of complexity. The longevity and proper functioning of floating structures often depend on the appropriate mooring design and their impact on the motion of the structure. Mooring system components and associated installation processes are a significant cost and technical challenge to floating systems. The anchor layout can also become more challenging due to the potential for interference between adjacent units’ moorings.

Traditionally, mooring design optimizations are primarily aimed at enabling operations in deeper and harsher environments, with a lesser focus on reducing capital expenditure (CAPEX). Failures in mooring lines due to snap loads have been observed in various offshore platforms and vessels, underscoring the importance of studying the ultimate load capacity of mooring lines, especially in extreme conditions [2,3,4]. Additionally, the mooring failure accidents of floating wind turbines have gained attention, and mooring line snap tensions have also been observed when taking the aerodynamic coupling into consideration [5,6,7]. The coupling effect of the mooring line snap tension and the out-of-plane loading that causes the failure of the residual mooring system was investigated by Xu et al. [8]. In extreme environmental conditions, Howey et al. [9] observed that individually moored wave energy converters (WECs) demonstrated lower mooring loads, whereas interconnected arrays frequently encountered snap loads.

Recent advancements in mooring technology have seen the emergence of synthetic fiber ropes as a preferred alternative to traditional chains in both oil and gas platforms and offshore renewable energy installations. These ropes offer substantial economic and operational advantages, including lower submerged mass, reduced cost [10,11] and potentially lower peak loads [12,13,14]. Experimental model tests by Xu et al. [15,16] showed that nylon and polyester rope are favorable materials for the station keeping of a point absorber WEC. Elastic mooring cables used on the multi-float WEC M4 reduced snap force by a factor up to about six below inelastic in the study of Stansby et al. [17]. Innovative solutions such as shared mooring/anchor systems and hybrid moorings, incorporating fiber ropes, have been explored to enhance safety and reliability [18,19].

Numerical simulations play a pivotal role in assessing mooring dynamics, but the highly nonlinear and time-dependent properties of synthetic ropes present challenges. The stiffness curve from static analysis may result in high uncertainties of mooring line reliability and fatigue of synthetic materials. A clear characterization of the mechanical behavior is essential for a reliable and economical mooring design of floating systems. Research efforts have aimed at developing empirical dynamic stiffness models based on material tests to improve the accuracy of numerical modeling [20]. Understanding the nonlinear characteristics of fiber materials is crucial for accurate predictions in coupled hydrodynamic performance and structural responses [21].

Material testing plays a crucial role in characterizing the mechanical properties of synthetic ropes, including stiffness, strength, and fatigue resistance. These tests provide valuable data for designing and optimizing mooring systems for offshore applications. Extensive material tests have been conducted to generate stiffness data for mooring line analysis, contributing to a better understanding of the mechanical behavior of synthetic ropes [22,23,24,25,26,27,28,29,30]. The Syrope model, which represents the complex behavior of synthetic ropes, was developed based on data from polyester sub-ropes [31] and recommended by DNV [32]). Sørum et al. [33] assessed nylon and polyester ropes for the moorings of floating wind turbine platforms by modeling their nonlinear stiffness with the Syrope model. The Syrope model for nylon ropes was adapted to fit the test results.

In recent years, there has been a growing interest in cylindrical floating production storage and offloading (CFPSO) units, primarily due to their better sea-keeping characteristics, especially their feature of insensitivity to the incident wave direction because of their axisymmetric geometry [34,35,36]. These include reduced yaw motion, as well as mitigated hogging and sagging loads, which contribute to enhanced stability and operational efficiency. Moreover, CFPSO units have demonstrated higher cost efficiency, particularly in terms of fatigue resistance and power consumption, making them increasingly attractive options for offshore operations. Deng et al. [35] investigated the hydrodynamics of a CFPSO moored with a spread mooring system that consisted of 12 mooring lines. A polyester rope was utilized as one main component; however, constant stiffness was considered the mechanical property.

To further investigate the effects of the nonlinear behavior of polyester rope on the performance of the CFPSO unit and moorings, the present study aims at characterizing the mechanical properties of the sub-rope of mooring rope, including the quasi-static stiffness at different stages and the dynamic stiffness in various test cases. The elongation and stiffness tests were conducted on six sub-rope samples following the American Bureau of Shipping guidance (ABS 90-2011) [37]. The quasi-static and dynamic stiffnesses were calculated based on the tested data. The effects of the mean load, load amplitude, loading period, loading cycles, and mean strain of the cyclic loading on the dynamic stiffness are discussed. The performances of the empirical equations in the literature for polyester mooring ropes are evaluated. A detailed analysis of the axial stiffness of the polyester used in a CFPSO system and the experimental data of the loading procedures are available for further investigation.

2. Description of the Tests

2.1. Test Samples

The full rope SINOROPE is constructed by Zhejiang four brother rope company (Taizhou, China) [38]. The tested sample was a 12-strand, braided, and single-leg sub-rope with a diameter of 68 mm, as shown in Figure 1. The minimum breaking strength (MBS) of the sub-rope was computed from the given MBS of the full-size rope (i.e., 22,600 kN) divided by the number of sub-ropes (=12). Therefore, the sub-rope’s MBS equaled 1883 kN. This value of the MBS was used in the rope tests.

2.2. Test Facility

The rope tests were conducted on a 300 T capacity tensile testing machine in Zhejiang Four Brothers Rope Co., Ltd., Taizhou, China. Figure 2 shows some photographs of the test facility in the laboratory. The maximum breaking load of the testing machine was 300 T and it was equipped with 160 mm-diameter loading pins. The machine operated with one fixed and one moving crosshead. Several output parameters like applied force, time, and total elongation (i.e., the movement of the hydraulic ram in the testing machine) were logged continuously during the test. The sample was tested wet and underwent water irrigation cooling during the test. The rope test setup is illustrated in Figure 3, which includes the total length of the samples, the measuring length with the extensometer, and the configuration of the loading pins. The initial length of the sample was 5 m, as marked using the “Gauge length”. In the results analysis, the elongation of the sample was calculated as the ratio relative to this initial length.

2.3. Test Procedures

In total, the elongation and stiffness tests of the six samples were tested following the ABS guidance. A graphical illustration of the test procedure is shown in Figure 4, which covers the four phases of testing: the installation pre-loading test, the quasi-static stiffness for post-installation test, the quasi-static stiffness for aged rope test, and the dynamic stiffness test. The horizontal axis is the time duration for each step in seconds, and the vertical axis means the loading force as the percentage of the MBS. The loading procedures for the first three samples are shown in Figure 4a, while the other three are shown in Figure 4b.

The samples were soaked in fresh water for at least 4 h before starting the test. Extensometers were fixed on the samples to measure the elongation between the splices. The specific steps following the recommended procedures are described in

Table 1. Description of the test procedures for each test.

Test	Procedures
1. Installation Pre-loading Test	(1) A load of 1% of MBS shall be applied and held for 5 min. (2) Install extensometers and measure the gauge and pin-to-pin length. (3) Increase the tension to 12% MBS and hold for 2 h. (4) Increase tension to a preload tension of 40% (for three samples)/35% (for three samples) MBS and hold the load for 3 h. (5) If the load drops to less than 5% MBS, then the load should be increased back to 40%/35% MBS (set the load and hold, allowing it to drop naturally, then adjust it back to 40%/35%, and this should only occur two to three times during the hold period). (6) Decrease tension to a 12% MBS level and hold the load for 6 h.
2. Quasi-Static Stiffness for Post-Installation Rope	(1) Increase the tension from a pre-tension of 12% MBS to 30% MBS and then hold for 100 min. (2) Increase the tension from 30% to 45% MBS and hold for 100 min. (3) Increase the tension from 45% to 60% MBS and hold for 100 min. (4) Reduce the tension from 60% MBS to a pre-tension of 12% MBS and hold at this tension for at least 200 min.
3. Quasi-Static Stiffness for Aged Rope	(1) Increase the tension from 13% MBS to 65% MBS and hold at this tension for 100 min. (2) Apply 1000 cycles of dynamic load with a tension range of 35% to 65% MBS and a period of 12 to 35 s. (3) Reduce the tension from 65% MBS to 13% MBS and hold at this tension for 100 min. (4) The quasi-static stiffness test outlined in Step 2 should be repeated after the above loading.
4. Dynamic Stiffness	(1) Cycle the rope 10 times between a pre-tension of 12% MBS and 55% MBS with a period of 12 to 35 s and then return to the pre-tension of 12% MBS and hold for at least 100 min. (2) For each WF test case, cycle the rope between Tmin and Tmax 40 times, and then record the load and elongation with a frequency of at least 1 Hz. (3) For each LF test case, cycle the rope at the tension between Tmin and Tmax 20 times, and then record the load and elongation with a frequency of at least 0.25 Hz. (4) The sequence of the test cases can be selected to best facilitate the tests and perform the dynamic test cycles listed in Table 2.

. Four stages of tests were conducted: the installation pre-loading test, the quasi-static stiffness for post installation test, the quasi-static stiffness for aged rope test, and the dynamic stiffness test. The first test was used to determine the quasi-static stiffness of the pre-installation rope. In the quasi-static stiffness for post installation test, the samples should be loaded using pre-tension for at least 6 h. In the third stage, the sample should be first loaded with static and dynamic loadings for a full bedding in the rope, and then the quasi-static stiffness test in Stage Two should be repeated. In the dynamic stiffness test, different cyclic loadings are applied with various mean loads and amplitudes, as seen in

Table 2. Dynamic stiffness test matrix.

Test Case	Mean Load (%MBS)	Amp. (%MBS)	Min Load (%MBS)	Max Load (%MBS)	Loading Period (s)	Cycles
1	15%	5%	10%	20%	75	20
2	20%	3%	17%	23%	24	40
3	20%	3%	17%	23%	70	20
4	23%	16%	7%	39%	110	20
5	30%	16%	14%	46%	34	40
6	30%	16%	14%	46%	129	20
7	30%	28%	2%	58%	140	20
8	35%	8%	27%	43%	31	40
9	35%	8%	27%	43%	84	20
10	40%	30%	10%	70%	44	40
11	40%	30%	10%	70%	170	20
12	50%	20%	30%	70%	33	40
13	50%	20%	30%	70%	106	20
14	60%	10%	50%	70%	92	20

. The whole process should be continuous without significant interruption. If necessary, a short pause can be placed between each test case.

3. Analysis Methods for the Tested Data

In general, the stiffness of a fiber rope is expressed as follows:

$$EA={\displaystyle \frac{\Delta F}{\Delta \epsilon }},$$

(1)

where EA is the stiffness, ΔF is the change in load, and Δε is the change in strain. A non-dimensional stiffness is expressed as follows:

$$K_r={\displaystyle \frac{EA}{MBS}},$$

(2)

where MBS is the minimum breaking strength of the rope sample. The non-dimensional static and dynamic stiffness are denoted as Krs and Krd, respectively. Considering the cyclic loading condition, the non-dimensional stiffness is expressed as follows:

$$K_r={\displaystyle \frac{\left(F_n^p-F_{n-1}^t\right)/MBS}{{\epsilon }_n^p-{\epsilon }_{n-1}^t}},$$

(3)

where $F_n^p$ and ${\epsilon }_n^p$ are the peak tension and strain of the nth tension–elongation hysteresis loop, respectively; and $F_{n-1}^t$ and ${\epsilon }_{n-1}^t$ are the trough tension and the strain of the (n − 1)th tension–elongation hysteresis loop, respectively.

The quasi-static stiffness was determined for each load level during installation, post-installation, and after ageing using the following formula:

$$K_{rs}={\displaystyle \frac{F_{creep}-F_0}{L_r+c\times \log \left(t\right)\times 0.01}},$$

(4)

where F_creep is the tension in the creep phase (%MBS), F₀ is the tension at Lr (%MBS), c is the creep rate, t is the duration in the creep stage, and Lr is the difference between the elongation and the initial elongation under a certain load (%MBS).

An empirical equation was recommended for dynamic stiffness [39]:

$$K_{rd}=\alpha +\beta L_m+\gamma L_a+\delta \log \left(P\right),$$

(5)

where Lm is the mean load as the % of the MBS, La is the load amplitude as the % of the MBS, and p is the loading period in seconds.

When neglecting the influence of the load amplitude [40], Wibner et al. [22] predicted the dynamic stiffness of a polyester rope using a two-parameter empirical equation:

$$K_{rd}=\alpha +\beta L_m.$$

(6)

A similar equation was proposed by Francois et al. [23] as follows:

$$K_{rd}=\left\{\begin{array}{c}\hspace{1em}\alpha +\beta L_m\hfill \\ \hspace{1em}\hspace{1em}22\hspace{1em}\hspace{1em}\hspace{1em}for\begin{array}{cc}& \end{array}Lm<10\%MBS\hfill \end{array}\right..$$

(7)

Some researchers have also taken the effect of strain amplitude on the dynamic stiffness of polyester mooring ropes, such as Casey et al. [41] and Liu et al. [26]. Casey et al. [41] proposed the following empirical equation:

$$K_{rd}=\alpha +\beta L_m+\gamma {\epsilon }_a,$$

(8)

and Liu et al. [26] applied the following expression:

$$K_{rd}=\alpha +\beta L_m-\gamma {\epsilon }_a-\delta \exp (-\kappa N),$$

(9)

where α, β, γ, δ, and k are the coefficients fitted by the rope test data; ε_a is the strain amplitude; and N is the number of loading cycles. A similar equation was applied and fitted by Xu et al. [42] using the tests of a braided, 16-strand polyester rope; however, the strain amplitude was replaced with the load amplitude La:

$$K_{rd}=\alpha +\beta L_m-\gamma L_a-\delta \exp (-\kappa N)$$

(10)

4. Results and Discussions

According to the tests, the elongation vs. time and the elongation vs. loading relationships were presented for the four tests listed in Table 1 first, and then the permanent elongation of the sub-ropes after installation was determined. The quasi-static stiffness and dynamic stiffness was then determined and compared for the six samples.

Figure 5a shows the elongation vs. time relationship for the first three samples, while Figure 5b presents the results for the latter three samples, during the whole process of the tests, where A and B represents the start and end points of the installation pre-loading test, respectively. The elongation of the samples at Points A and B were obtained and are listed in

Table 3. Elongation of the samples during the installation pre-loading test.

Elongation (%)	Point A	Point B
S1	0	3.23%
S2	0	3.34%
S3	0	3.29%
S4	0	3.50%
S5	0	3.57%
S6	0	3.58%
Average	0	3.42%

. It was seen that the average of the sample elongation was 3.42% of the initial length.

Figure 6 shows the elongation vs. loading during the installation pre-loading test for S2 and S5. This test was conducted to simulate the installation pre-loading and pre-tensioning sequence, which removes as much permanent elongation as possible during installation, as well as to increase the stiffness of the rope. The results can be used to determine the relationship between the as-installed length at pre-tension and the manufactured rope length (which is typically at 2% MBS). The data from this test can also be used to determine the quasi-static stiffness of the pre-installation rope.

Following the ABS guidelines, the three key points, “2A”, “2B”, and “2C”, are marked in Figure 6. Point 2A is the end of the process with 35% or 40% MBS loading, Point 2B is the end of the process with 12% MBS loading and Point 2C represents the one after the loading dropping to 2% MBS. The corresponding elongation of the samples are summarized in

Table 4. Elongation of the samples at significant points for the three samples with 35% MBS loading.

Point	Loading (% MBS)	S1	S2	S3	Average
2A	35%	4.81%	4.91%	4.86%	4.86%
2B	12%	3.23%	3.34%	3.29%	3.29%
2C	1%	2.47%	2.59%	2.53%	2.53%

and

Table 5. Elongation of the samples at significant points for the three samples with 40% MBS loading.

Point	Loading (% MBS)	S4	S5	S6	Average
2A	40%	5.49%	5.52%	5.55%	5.52%
2B	12%	3.50%	3.57%	3.58%	3.55%
2C	1%	2.72%	2.80%	2.81%	2.78%

for the three samples with 35% MBS loading and the ones with 40% MBS loading, respectively. It was seen that the deviation of the elongation of the samples in the same loading condition was quite small. When comparing the average values between the two scenarios, the elongations of the samples with 35% MBS were always lower than those of the ones with 40% MBS loading, even when the loading rate was the same (see the values at 1% and 12% MBS loading).

Figure 7 and Figure 8 show the elongation vs. loading during the test of quasi-static stiffness for post-installation and for the aged ropes of Sample 2. The quasi-static stiffness in installation, post-installation, and the aged ropes were calculated according to Equation (4). The results are presented with the values obtained at 1 min, 10 min, and 100 min. To illustrate the steps clearly, the details of the post-installation stiffness calculation for S2 are presented first. As listed in

Table 6. Parameters used for the post-installation quasi-static stiffness calculation for S2.

Time (s)	Loading (%MBS)	Elongation (%)	Note
40,086.199	11.99877854	3.3400	Initial elongation (Point B)
40,117.578	30.00069039	4.1298	The moment when 30% of MBS loading starts
40,177.609	30.0000000	4.2560	1 min after 30% of MBS loading
40,717.391	30.00026553	4.3822	10 min after 30% of MBS loading
46,117.719	30.00053107	4.4960	100 min after 30% of MBS loading

, the initial elongation of the sample was 3.34%, and the initial loading T₀ = 11.99877854%. The loading in the creep phase was Tcreep = 30% at 1 min, Tcreep = 30.00026553% at 10 min, and Tcreep = 30.00053107% at 100 min. The difference between the elongation and the initial elongation at a certain load was Lr = (4.2560 − 3.3400)/(3.34 + 100) * 100 at 1 min, Lr = (4.3822–3.3400)/(3.34 + 100) * 100 at 10 min and Lr = (4.4960 − 3.3400)/(3.34 + 100) * 100 at 100 min. In Equation (4), log (t) is the logarithmic calculation of the load time, with log (1) = 0, log (10) = 1, and log (100) = 2; and c is the creep load, which is determined by linearly fitting the log (t) with the relative Lr.

By applying the data presented in

Table 7. The data used for linear fitting.

Time	Log(t)	Lr_relative
1 min	0	0
10 min	1	0.1210482
100 min	2	0.2302026

, the creep rate was obtained by the linear fitting, as shown in Figure 9, where the dotted line represents the linearized data, and the solid one means the original values. Through replacing the values in Equation (4), the quasi-static stiffness was calculated as Kr1 min = 20.29, Kr10 min = 17.96, and Kr100 min = 16.11.

Using the same calculation procedures, the quasi-static stiffness during installation for the six samples was calculated and is presented in Figure 10. As was seen for all the samples, the stiffness increased as the loading period increased. Compared with the three samples at the same loading value, the deviation of the stiffness was quite small. The average values of the quasi-stiffness Krs for the first three samples (S1S2S3) were 8.07%, 7.19%, and 6.48% at 1 min, 10 min and 100 min, respectively, while the ones for the latter samples (S4S5S6) were 7.95%, 7.23%, and 6.63%, respectively.

The post-installation values of Krs are depicted in Figure 11, where the average results of the three samples are plotted for each loading period. The horizontal axis represents the logarithm of 10 for the loading period in minutes. The curves demonstrate nearly linear relationships, with the quasi-static stiffness decreasing linearly as the loading period in minutes increased. In comparing the values for all the samples, the highest ones were obtained at the first three samples (S1S2S3) when the loading was 30% MBS, which were 20.22%, 17.87%, and 16.00%. The lowest ones were from the same samples (except the case with a loading of 60% MBS), which were 16.70%, 15.58%, and 14.59%. In addition, except the case with a 40% MBS loading value, the stiffness after 100 min was 14.52%.

Figure 12 shows the quasi-static stiffness of the aged samples at different loading values. Similarly, the curves illustrate the linear relationships between the stiffness values (%MBS) and the logarithm of 10 of the loading periods in minutes. The rate of decrease reduced as the loading value increased. Except for the case with 30% MBS loading, all of the stiffness values of the aged samples were higher than the ones in the post-installation procedure.

Figure 13 and Figure 14 show the loading values and the elongation vs. loading curve during the dynamic stiffness test. The tested cases listed in Table 2 can be seen from Figure 13 where the various mean loads, load amplitude, and periods were applied. The dynamic stiffness for each case was calculated based on Equation (3) at each cycle using the peak and trough tensions and strains, as shown in Figure 15, where the value n means the cumulative number of the cycles from test Case 1 to Case 14. The number of cycles for each case was consistent with the values shown in Table 2. As was seen, the dynamic stiffness varied from case to case and was different at loading cycles. In general, the dynamic stiffness increased as the loading cycles increased, except for test Cases 7, 10, and 14. By checking the loading conditions in Table 2, it was shown that when the mean load Lm and amplitude La are relatively small (Case 1, Case 2, and Case 3), the changing rate was low. By comparing Case 4 and Case 5, it was found that the mean load had significant effects on the dynamic stiffness. When the mean load was higher, the value of Krd was much higher. In comparing Case 5 and Case 6, where the mean load and load amplitude were the same, it was seen that the loading period had neglectable effects for this case. The results from Cases 6 and 7 also show the significant effects of the load amplitude, demonstrating that, with a low load amplitude, the dynamic stiffness is much higher. Observing the results for Cases 8 and 9, Cases 10 and 11, and Cases 12 and 13, it is evident that the loading period affects the changing rate.

To better understand the temporal evolution of the loading and strain, Figure 16 shows the history of the loading procedure and corresponding strain of the sample, where n is the cumulative number of points in the time series, the red asterisk (*) means the crest and the green circle represents the trough in the time series. The loading history (Figure 16a) shows the consistent values of the mean load (30% MBS) and amplitude (28% MBS), as listed in Table 2. As shown in Figure 16b, the amplitude of the elongation increased gradually, indicating the creep phenomenon of the polyester. As for test Case 8 with a mean load of 35% MBS and an amplitude of 8% MBS (see Figure 17a), the amplitude of the elongation decreased as the loading cycle increased, resulting in an increased trend in the dynamic stiffness. This is the stress relaxation phenomenon of polyester.

Figure 18 plots the cyclic tension–elongation relationship at all the loading cycles from four test cases (7, 8, 9, and 13), The hysteresis loop was formed at every circle, and it shifted gradually downward or toward the right. In test Case 7, the hysteresis loop was clear and stable with a slight shift toward the right. For test Cases 9 and 11, the hysteresis loop was smaller but still stable, whereas for test Case 8, no clear loop was observed. All these hysteresis loops reflected the nonlinear properties of the polyester mooring rope, and the properties were dependent on the mean load, load amplitude, loading cycles, and loading period.

The same calculations of the dynamic stiffness were made for all six samples. Considering the last three cycles in the cyclic tests, the average values of the samples (the first three as S1S2S3 and the latter ones as S4S5S6) are presented in Figure 19. The deviations between the values from the first three and latter samples were negligible. The lowest dynamic stiffness was 19.48 in test Case 7 with Lm = 30% MBS, La = 28% MBS, and loading period = 140 s, while the highest one was 34.02 from test Case 14 with Lm = 60% MBS, La = 10% MBS, and loading period = 92 s. The former value was 57% of the latter. In general, compared with the quasi-static stiffness of the ropes and aged ropes after installation, the dynamic stiffness values were much higher.

To check the consistency of the computed dynamic stiffness of the polyester with the proposed empirical equations in the literature,

Table 8. The empirical equations and parameters for the dynamic stiffness of the polyester ropes.

Empirical Equations	α	β	ϒ	ẟ	k
Equation (5), Fernandes et al. [39]	12.058	0.152	−0.201	−0.473	x
Equation (5), ABS [37]	27.5	0.25	−0.59	−1.65	x
Equation (6), Wibner et al. [22]	18.5	0.33	x	x	x
Equation (8), Casey et al. [41]	11.477	0.153	−5.429	x	x
Equation (9), Liu et al. [26]	7.16	0.41	6.3	1.78	0.01

lists the expressions and the corresponding coefficients fitted in the references. For all the tested cases, the Lm was higher than 10% MBS, so Equation (7) is as same as Equation (5). In the present analysis, only two values were used for the number of the loading cycles, so the last term in Equation (10) was not meaningful. Therefore, these two equations are not discussed here.

The equations and the coefficients are different since they are fitted using the different data from various samples. For example, the parameters for Equation (8) were obtained from the tests on a three-strand rope with an MBS of 6 tons, while the ones for Equation (9) were from the three-strand 6 mm rope with 6.8 kN MBS. Even for the same equation, i.e., Equation (5), the coefficients from [39] are different from the ones from [37]. The parameters for Equation (5) in [39] were from the parallel rope with a diameter of 11.5 mm, whereas the sample structure and dimension from [37] are unclear.

Using the values in the test matrix shown in Table 2, the prediction of the dynamic stiffness of the rope at each test case was calculated. The calculations from different equations were compared with the present results in Figure 20. The dynamic stiffness from Equation (5) using the coefficients from Fernandes et al. [39], the values from Equation (8) using the parameters from Casey et al. [41] and the ones from Equation (9) using the parameters from Liu et al. [26], are all lower than the present results. Meanwhile, Equation (6) provides linearly increasing stiffness as the mean load and goes up since this expression only accounts for the effects of the mean load. This expression provides relatively different predictions on the dynamic stiffness since it only depends on one parameter. Equation (5), with the coefficients from ABS [37], works very well to predict the dynamic stiffness of the studied rope, though slight underestimation was observed. One of the possible reasons for the better agreement between the present analysis with the ones from ABS [37] is because the same test procedures were applied. Another reason could be the similar rope structure for the polyester rope.

5. Conclusions

Experimental tests were performed on six samples of a 12-strand, braided, and single-leg sub-rope of a polyester mooring rope following the ABS guidelines to characterize the stiffness at various loading conditions.

The results of the quasi-static tests show the linear relationships between the stiffness values (%MBS) and the logarithm of 10 of the loading periods in minutes. The rate of decrease reduced as the loading value increased. Except for the case with 30% MBS loading, all the stiffness values of the aged samples were higher than the ones in the post-installation procedure.

The tested results from the six samples showed negligible deviations in both the quasi-static and dynamic stiffness, thus showing low uncertainties in the experimental study. During the pre-installation step, three samples were preloaded with 40% MBS and the other three with 35% MBS for 3 h. The influences of these different pretensions on the calculated stiffness were limited.

Using 14 test cases with different mean loads, load amplitudes, loading periods, and loading cycles, the dynamic stiffness of the polyester sub-rope was estimated. The time series of the loading, elongation, and the hysteresis loop show the creep and stress relaxation phenomena. By comparing the predictions of the dynamic stiffness from various empirical equations from the literature, it was found that the expression in Equation (5) [39] with the coefficients from ABS [37] worked well for the present case.

In the dynamic stiffness test, a range of mean loads, load amplitudes, and loading periods was considered; however, only two loading cycles (20 or 40) were used. This was not enough to study the effect of the loading cycles. In future work, the same procedure for the dynamic stiffness tests can be conducted using various numbers of the loading cycle for each test case. In addition, to understand the effects of the rope structure and dimension, as well as the test procedures on the coefficients of the empirical equations, more tests could be performed using the same sample but with different procedures, or different samples but with the same procedure.