An experimental investigation on the hydrodynamic performance of floating bridge foundation with taut mooring system

Abstract

Floating bridges with floating foundations have wide application prospect in deep-water transportation infrastructures. The dynamic properties of floating foundation would deeply affect the structural characteristics of the whole floating bridges. However, limited research works, especially experimental studies, have been conducted to investigate the dynamic behaviors of floating bridge foundations. In this paper, a series of hydrodynamic model tests (including still water tests, regular wave tests, and environmental loading tests) on a floating bridge foundation composed of a floating platform and the mooring system combining tension legs and incline cables, are presented. The test campaign was conducted under the guidance of similarity criteria of Froude and Strouhal numbers, and focused on the structural hydrodynamic responses regarding platform motions and mooring tensions under wave and current loadings. Results show that, the hydrodynamic characteristics of the floating foundation could be quite different in bridge-transversal and -longitudinal directions. The platform dynamic motion is dominated by horizontal displacements, and slight coupling effects between rotational and lateral responses were found. Compared to the wave only conditions, the superposed currents in wave-current combination cases would mitigate the horizontally dynamic responses, and enhance the vertically heave oscillations of the floating foundation. The presented study provides beneficial references for further research on the design and construction of floating bridges in deep-water environment.

Keywords

floating bridge floating foundation hydrodynamic model test dynamic behavior spectral analysis

Introduction

As the constructions of transportation infrastructures are continuously developed and facing the application scenarios of wide and deep water environments (such as the straits and fjords), the floating bridge structures have gained the attentions of the engineers and researchers around the world, and various concepts have been proposed (Moan and Eidem, 2019; Watanabe, 2003; Lin et al., 2022). Generally, the floating bridge is a kind of bridge structure that carries the self-weight and live loads by the buoyancy provided by the floating foundations. It can avoid the construction of fixed foundations for the traditional bridge structures, which is quite challenging in the aforementioned engineering conditions with regard to the technological and financial aspects. Nowadays, a number of research works have focused on the structural safety of deep-water bridges subjected to environmental loadings (Guo et al., 2015; Wei et al., 2018; Fang et al., 2022; Han et al., 2022), and it is believed that with the continuous development of bridge engineering, the deep water conditions will inevitably be faced by engineers and researchers. In this context, the floating bridge is thought to be a promising structural concept for water-crossing engineering, and several investigations regarding to the structural performance of floating bridges have been published in the past few years. Sha et al. (2018) performed a time domain dynamic analysis for a curved multi-span floating bridge, in which the effects of wind load, first-order and second-order wave loads on the dynamic responses of the investigated bridge were evaluated. The author has further investigated the local and global responses of the structure under the ship-girder collisions by using numerical technique (Sha et al., 2019). Considering the inhomogeneous wave conditions, Cheng et al. (2018a, 2018b, 2019) studied the structural response characteristics of the same bridge concept, and further investigated the load effects of the combined environmental wind and current. Also for this concept, the effects of wave directionality on bridge extreme response were numerically studied by Viuff et al. (2019), and results indicate that the structural response is less sensitive to the natural occurring short-crested waves and the main wave directions within 15° from beam sea. Viuff et al. (2020a, 2020b) also conducted the investigations on the numerical modelling of floating bridges, which improved the understanding of numerical simulation techniques for such type of structures under environmental loadings. Based on numerical time domain analysis, Wei et al. (2019) investigated the effects of the wave inhomogeneity on the hydro-elastic responses of a multi-span floating bridge with 19 pontoons, during which different wave spectra were defined for divided regions along the bridge axis. Wang et al. (2018) performed a time domain analysis on a three-span suspension bridge with two floating towers considering the actions of wind and wave, using the finite element software ABAQUS. In this study, the authors have developed a specialized subroutine for calculating the radiation forces acting on the floating towers. Xu et al. (2018a, 2018b) also conducted an investigation on the the dynamic behavior of a floating suspension bridge under the excitations of wind and wave, where the radiation forces on the floating towers and the aerodynamic forces acting on the bridge girder, were calculated by introducing the theory of state-space model. Considering wave and wind loads, Wei et al. (2021) studied the influence of the sag-to-span ratio on the dynamic response of a long-span suspension bridge with two floating towers, and results show that the natural period of the floating bridge decreases with the reduction in sag-to-span ratio. Wan et al. (2021) conducted a numerical study to evaluate the influences of bridge radius, cross-sectional rigidity on the structural performance of floating bridge with three pontoons, where static loads and wave load were considered. Dai et al. (2020, 2021) studied the hydroelastic response of a straight side-anchored floating bridge under inhomogeneous wave conditions based on numerical simulation, and further investigated the influence of wave inhomogeneity on the fatigue damage of mooring lines. Based on a numerical time domain approach, Xiang et al. (2022) investigated the effects of spatial inhomogeneity of waves on the dynamic behaviors of multi-span floating bridges by considering typical wave height distributions.

In the aforementioned research works, the superstructure of floating bridges (including main girder and piers/pylons) are commonly supported by the floating foundations consisting of a floating platform and the mooring system. The floating platform, which can be made of pre-stressed concrete or steel, provides the buoyancy for bearing the whole structure, and is generally restrained by the mooring lines anchored to the bed. During the service period of bridge, the floating platforms are permanently subjected to the actions of wave and current. In the past few years, researchers have conducted preliminary numerical analyses regarding to the floating bridge foundations. Fredriksen et al. (2016, 2019) carried out a hydrodynamic analysis for evaluating the performance of floating foundations with the platforms in different geometric shapes, and the results shown that the elliptic cylindrical platform has the prior performance compared to the other considered geometries. Based on finite element simulation, Xiang et al. (2017) investigated the effect of the platform heave plate on the bridge responses, concluding that a properly designed heave plates on the floating platform can lead to a reduction of the maximum bending moment in bride girder. The effects of the heave plate on hydrodynamic characteristics of floating platform were further studied by Shao et al. (2019) based on both numerical simulation and model test. In the previous studies, taut mooring system or catenary mooring system with incline mooring cables are usually employed in the floating bridge foundations to provide lateral restrains for the whole structures. On the other hand, some researchers adopted tension legs to the conceptual design of floating bridges, aiming to achieve higher vertical stiffness and carrying capacity for the floating platform (Wang et al., 2018; Xu et al., 2018a, 2018b). However, calculated results indicated that the mooring system with only tension legs shows a compliance property in platform lateral movements, which would lead to significant bridge transversal displacements under current and wave loadings. Based on the present understanding of floating bridge foundation, this paper proposes a foundation system constituted by a floating platform and the taut mooring system combining incline cables and tension legs, and a scaled model test has been conducted to investigate its hydrodynamic characteristics under wave and current loads.

Water basin test is thought the most reliable research method for floating structures involving hydrodynamics; however, few experimental studies on floating bridge have been conducted and reported in literature, noting that the abovementioned research works were mostly relied on numerical tools. A technical challenge is that, since the floating bridges are relatively long in one direction (i.e., the bridge axis), few water basins are adequate for conducting a model test considering the whole bridge structure, with a reasonable and meaningful scaling factor. Moreover, the high financial cost is another handicap due to the complex rigid-elastic model and the measuring instruments, for such a mechanical scenario involving hydrodynamics, aerodynamics and multi-structure dynamics. However, the hydrodynamic model test regarding to single-floater systems is relatively mature considering both the theoretical and technical aspects, and quite many test campaigns have been carried out for the vessels, floating production platforms, and floating wind turbines, etc., in the previous studies (Cui et al., 2019; Jiao et al., 2018; Ruzzo et al., 2019; Xiao et al., 2017).

Based on the above considerations, the presented experiment focused on the hydrodynamic characteristics of a floating bridge foundation with taut mooring system, considering the wave and current loadings. Complying with the similarity criteria of Froude and Strouhal numbers, the tested model of floating foundation was designed based on a conceptual prototype which is composed of a concrete-made floating platform and the mooring system of polyester ropes. To take into account the effect of bridge superstructure, ballasts corresponding to the weight of the superstructure carried by each platform were adopted and arranged in the floating platform model. Still water tests, regular wave tests and the environmental loading tests were conducted for the tested model, and the hydrodynamic responses of the floating platform and mooring tensions have been finely measured. Based on statistical and spectral analyses, the hydrodynamic characteristics of the floating bridge foundation have been comprehensively investigated, which is beneficial for the research and design of deep-water floating foundations, and further the overall floating bridge structures.

Overview of model test

Scaling law

Generally, gravity and inertia force play the dominant roles in the hydrodynamic model tests that investigate the structural motions and mechanics under environmental loadings. As Froude number describes the relationship between inertial forces and gravity, it forms a basic scaling criterion (i.e., the Froude Similarity Criterion) for physical experiments in water waves and currents, which requires the Froude numbers of model and prototype to be equal, as expressed by equation (1) (Yang et al., 2008):

\frac{V_{m}}{\sqrt{(g L_{m})}} = \frac{V_{p}}{\sqrt{(g L_{p})}}

(1)

where L and V represent the linear length and velocity parameters, respectively; g refers to the gravity acceleration; the subscripts m and p denote the test model and the prototype, respectively.

Additionally, as the motion and force responses of floating body subjected to wave loading have the nature of periodicity, equality of Strouhal numbers between test model and prototype (i.e., the Strouhal Similarity Criterion) should be ensured as represented in equation (2) (Yang et al., 2008):

\frac{V_{m} T_{m}}{L_{m}} = \frac{V_{p} T_{p}}{L_{p}}

(2)

where T represents the time parameters (e.g., the period).

For a geometric scaling factor λ, to comply with the two mentioned similarity criteria, the conducted model test has to satisfy the scaling factors regarding various properties as list in Table 1. The symbol β in Table 1 is the density ratio of water in full-scale scenario to the water in test (usually are sea water and fresh water, respectively). Based on the conceptual design of prototype, the physical model of floating bridge foundation was designed under the guidance of scaling factors presented in Table 1. Considering the dimensions and device capability of the circulating water channel in SKLOE, as well as the test accuracy, scaling factor of λ=70 was adopted for the test campaign, which are within the range of 60 to 80 that recognized as the optimal scaling factors by the international ocean engineering circles (Yang et al., 2008).

Table 1.

Summary of scaling factors.

Properties	Symbols	Scaling factors
Length	L_p/L_m	λ
Velocity	v_p/v_m	λ ^0.5
Angle	φ_p/φ_m	1
Angular velocity	φ'_p/φ'_m	λ ^−0.5
Time	T_p/T_m	λ ^0.5
Frequency	f _p/ f _m	λ ^−0.5
Area	A_p/A_m	λ ²
Volume	V_p/V_m	λ ³
Density	ρ_p/ρ_m	βλ ²
Mass	M_p/M_m	βλ ³
Moment of inertia	J_p/J_m	βλ ⁵
Force	F_p/F_m	βλ ³

Description of scaled model

Figure 1 shows the elevation and plane views of the investigated floating foundation model. It can be seen that the floating platform of the model is shaped in an elliptic cylinder with outward heave plate at the bottom, of which the total height is 210 mm. The long and short axes of the ellipse section are 780 mm and 360 mm, respectively. A draft of 150 mm is adopted in the test, and the water depth was set as 1400 mm, corresponding to a full-sized deep-water environment of 98 m. The extended heave plate is 10 mm thick and 60 mm wide, attached around the bottom edge of platform. The physical properties of the floating platform model are given in Table 2, noting that the location of center of mass, abbreviated as CM, represents the height with respect to the bottom plane of floating platform. It is also worth to note that, according to the conceptual design of the floating bridge prototype, the mass of the platform model should be scaled down as 16.0 kg; however, additional 10.0 kg is incorporated by arranging ballasts in the model for considering the effect of the bridge superstructure supported by the platform. For connecting the mooring lines to the floating platform, metal rings were used to model the fairleads, assembled to the designed positions on the platform bottom plane (labeled by F1 to F4 in Figure 1).

Figure 1.

Elevation and plane views of tested model (Unit: mm).

Table 2.

Physical properties of floating platform model.

Item	Unit	Measured values
Mass of pontoon	kg	26.0
Vertical position of CM	mm	96.6
Roll inertia	kg·mm²	1119020
Pitch inertia	kg·mm²	337395
Yaw inertia	kg·mm²	1213401

In this study, a taut mooring system with polyester ropes, which commonly perform straightly under water due to the light-weight property, are designed for the prototype structure. The layout of mooring system of the foundation model is illustrated in Figure 1. The models of mooring lines are constituted by a steel wire rope and a bottom spring. The tensile deformations of wire ropes are negligible and the elasticities of the mooring lines are modeled by adapting the spring lengths. To measure the mooring forces during testing, tension sensors were positioned at the top of each mooring line, linking the wire ropes to the fairleads fabricated on the floating platform. Since the mooring lines are light in weigh, placing the sensors at the lines’ ends could minimize the effect of sensors’ weights on the dynamic responses of the cables and further the whole structure. The tensile stiffness of all mooring line models were measured in advance and the averages were calculated as 0.341 N/mm and 0.468 N/mm respectively for incline cables and tension legs, and all the mooring lines shows a good linearity in the tested range. Before loading test, the pretension forces in mooring lines have to be adapted in accordance with target values. The measured pretensions in all mooring lines are listed in Table 3 with the division of transversal and longitudinal loadings, since the test model has to be installed twice in the test campaign, individually for the two directional cases. From Table 3 it can be seen that, the relative differences (i.e., the ratio between standard deviation and average) of the lines in each category were no greater than 4%, which was regarded to be acceptable for the test.

Table 3.

Pretensions in the mooring lines (Unit: N).

Fairleads	Bridge-transversal loading		Bridge-longitudinal loading
Fairleads	Incline cables	Tension legs	Incline cables	Tension legs
F1	12.9	11.3	13.0	10.7
F2	13.9	10.5	13.0	10.4
F3	14.2	10.7	13.1	10.8
F4	13.8	10.5	13.5	11.4
Average	13.7	10.7	13.1	10.8

Test rig and load cases

The presented hydrodynamic model test was conducted in the circulating water channel of State Key Laboratory of Ocean Engineering (SKLOE) in Shanghai Jiao Tong University. The schematic of test rig is presented in Figure 2(a) local coordinate system is also labeled in the zoom figure, marking out the 6 degrees of freedom (DOFs) of the floating platform. As shown in the schematic, the test model is placed in the test area of water channel supported by an aluminum frame. Anchor points were prefabricated on the bottom of the support frame, to which the lower ends of mooring lines would be moored during the test. The high-precision wave maker locates upstream the test area with a distance sufficient to assure the full development of generated waves. Besides, in order to mitigate the reflection effect of waves as much as possible, a wave absorber was placed at the downstream side of the test area for absorbing the incoming waves. Water flows can be produced in the channel by the water circulating system to simulate the currents in real sea state.

Figure 2.

Experimental rig of the floating foundation model test: (a) Schematic view; (b) Photographic view.

Figure 2(b) shows the real image of the experimental setup in laboratory. Non-contact displacement measurement system (a commercial product provided by Qualisys), which is composed of high-speed cameras and markers, is utilized in the test to measure the motion responses of floating platform. Three markers were attached to the upper plate of platform so that the rigid body displacements in all 6 DOFs can be recorded. Additional industrial cameras were also used to capture the dynamic images of both floating platform and mooring lines for further analysis. Beside motion measurement, tension forces in mooring lines were synchronously surveyed during the loading test by the tension sensors as introduced above. Regarding to environmental condition, two wave height gauges and a current meter were arranged in a channel cross-section 1060 mm upstream to the test model as depicted in Figure 2(a), which was found hardly affect the wave and current conditions form at the position where the floating platform located.

For both bridge-transversal and -longitudinal loadings, three parts of experiments have been implemented for the floating foundation system, which are still water tests, regular wave tests, and environmental loading tests. The still water tests were primarily conducted to verify the horizontal stiffness of mooring system and obtain the natural frequencies of foundation system. The regular wave tests studied the response amplitude operators (RAOs, calculated as the ratio of response amplitude to the wave amplitude (Det Norske Veritas, 2014)) of floating foundation, in which regular waves of frequencies range from 0.5 to 1.8 Hz and wave heights within 20∼30 mm, were applied. For the applied regular waves, intensive frequency values have been selected near the natural frequencies identified from still water tests, aiming to obtain finely representative RAO curves.

Regarding to the environmental loading tests, 1-year and 100-years environmental conditions combining wave and current were adopted in accordance with real sea states, noting that the considered waves follow the JONSWAP spectrum (Det Norske Veritas, 2014) and the current were modeled by uniform water flow in the conducted test. With the purpose of comprehensively investigating the dynamic characteristics of floating bridge foundation, load cases of current only and wave only were also defined for comparative analysis. The considered load cases corresponding to real sea states have been scaled down to the model dimension according to similarity criteria, and defined as listed in Table 4, where the main environmental parameters (including current velocity v_c, significant wave height H_s, spectrum peak period T_p, and the peak enhancement factor γ) for each cases are also presented. In this study, both 1-year and 100-years conditions have been conducted in the test category of bridge-transversal loading, while for bridge-longitudinal loading, only the 1-year condition has been considered. The structural responses of floating foundation system under wave loading would be emphasized, and attentions would also be paid to the loading effects of the combined current.

Table 4.

Definition of load cases.

	Load case	v_c (mm/s)	H_s (mm)	T_p (s)	γ ^a	Remark
1-year condition	LC1.1	—	36	1.05	3.3	Irregular wave only
	LC1.2	200	—	—	—	Current only
	LC1.3	200	36	1.05	3.3	Current and irregular wave
100-years condition	LC2.1	—	54	1.16	3.3	Irregular wave only
	LC2.2	300	—	—	—	Current only
	LC2.3	300	54	1.16	3.3	Current and irregular wave

^aγ is the peak enhancement factor in the theoretical formula of JONSWAP wave spectrum, which is related to the concentration of wave energy in frequency (Det Norske Veritas, 2014).

Preliminary calibrations

Calibration of wave and current environments

The qualities of generated wave and current environments were calibrated prior to the installation of model to the water channel. Wave elevations and current speeds were respectively measured at the position where the model would be placed. The wave heights and frequencies of the generated regular waves were measured to be steady in time history, and were highly consistent with the target values set in the wave maker, reflecting the high quality of the produced regular waves. For the considered irregular wave conditions, comparisons between measured data and target wave spectra are shown in Figure 3. As can be seen in Figure 3, the spectra of generated waves agree well with the required specifications regarding to both 1-year and 100-years conditions. Current environments produced by the circulating system in the water channel were measured by current meters positioned at the test area, and good consistence between the system settings and measured data can also be confirmed with regard to current speed. The measurements also indicated that a satisfied uniformity of sectional speed distribution were created for the current environments within the test area.

Figure 3.

Comparisons of measured and target spectra for irregular wave conditions: (a) 1-year wave condition; (b) 100-years wave condition.

Verification of mooring system

A horizontal stiffness test has been conducted once the floating foundation system was installed and moored to the support frame in water channel, aiming to verify the restoring behaviors of mooring system in the surge and sway directions. The test results of offset distances and the corresponding restoring forces are plotted in Figure 4, in which the lines of theoretically calculated stiffness coefficients are also presented for comparison. Generally, good consistencies between test data and the calculated coefficients can be seen in Figure 4, noting that the slightly greater stiffness found in test results is thought to be the artificial operation error since slight pitch displacement could occur during the test. The horizontal restoring stiffness in the sway direction is understandably greater than those in surge direction, since the restoring forces contributed by the incline cables could be more significant in sway direction, considering their spatial layout shown in Figure 1.

Figure 4.

Verification of horizontal stiffness for mooring system.

Natural vibration properties of test model

To identify the natural frequencies of floating foundation system, free decay tests have also been conducted in calm water in advance of the loading test, and the tested natural frequencies (denoted by f_n) for platform motions in all 6 DOFs are listed in Table 5. It should be noted that, the vibrations of the system were produced by artificially applying an initial displacement/rotation to the floating platform and releasing it suddenly, and the trials with unacceptable coupling motions in other DOFs would be rejected. To reduce the test error, an average of three effective trials are taken as the final results for each motion mode. From the results it can be noticed that, the natural frequency in surge motion is relatively low compared to that in sway motion, which is reasonable since the stiffness in surge direction is smaller than that in sway direction. The natural frequencies of the designed floating foundation remain well away from the frequency range of high wave energy, thus beneficial for avoiding resonances and strong motions.

Table 5.

Natural frequencies of floating foundation (Unit: Hz).

Motion modes	f _n
Surge	0.26
Sway	0.61
Heave	1.19
Roll	1.45
Pitch	1.30
Yaw	0.48

Test results and discussions

Regular wave test

Loading in transverse bridge direction

Regular waves of frequencies range from 0.5 to 1.8 Hz were applied for testing the RAOs of the floating bridge foundation. Figure 5 shows the tested RAOs of platform sway, heave, and roll motions under bridge-transversal loading, noting that the responses in surge, pitch, and yaw were read relatively slight due to symmetry. It can be seen that, for the three motion types, significant RAO values concentrate near the frequency of 0.6 Hz that corresponds to the sway natural frequency, indicating that the heave and roll motions, to a certain extent, are coupled to the sway response for this system. However, for the rest frequencies, limited RAO values are read as depicted in Figure 5. The absence of curve peaks corresponding to the heave and roll natural frequencies indicates that the vertical movements of floating platform have been well restrained by the mooring system. Besides, since that the RAOs in sway motion (up to 8.84 for resonance) is dramatically superior to those of heave motion, and that the roll RAOs are generally low, one can believe that sway motion would play a dominant role in the platform dynamic responses.

Figure 5.

Response amplitude operators of floating platform motions (bridge-transversal loading): (a) Sway; (b) Heave; (c) Roll.

The RAOs of tension forces in the mooring lines are further presented in Figure 6 with divisions of incline cables and tension legs. Results of both upstream and downstream lines (respectively represented by mooring lines corresponding to the fairleads F4 and F1) are plotted together for comparison. It can be seen that, under the excitation of regular waves, the response characteristics of upstream and downstream mooring lines are highly similar to each other. Meanwhile, the resonances of platform motions revealed in Figure 5 are also reflected accordingly in the tension RAO curves, as resonance responses at frequency of 0.6Hz are prominently displayed in Figure 6.

Figure 6.

Response amplitude operators of tensions in mooring lines (bridge-transversal loading): (a) Incline cables; (b) Tension legs.

In general, under bridge-transversal loading, the tested floating foundation system shows significant responses at its sway natural frequency regarding both platform motions and mooring tensions, while fairly limited responses can be found for other frequencies. This result implies that the dynamic responses of the tested floating bridge foundation could be largely controlled by the sway motions under bridge-transversal loading.

Loading in longitudinal bridge direction

Figure 7 plots the RAO results of platform surge, heave, and pitch motions for bridge-longitudinal loading. Different from bridge-transversal loading, the RAO curves of platform motions show distinctive features between each other. For surge motion, the RAOs decrease gradually as the wave frequency increases. Since the natural frequency of surge is read at 0.26 Hz (smaller than the frequencies of test regular waves), it is reasonable that as wave frequency getting away from the natural frequency, the surge response would be less excited. For RAOs of heave and pitch responses as shown in Figure 7(a) and 7(b), curve peaks can be found at the frequencies of 1.2 Hz and 1.3 Hz, which are corresponding to the heave and pitch natural frequencies (as given in Table 5), respectively. Moreover, it should be noticed that, though the RAOs away from natural frequencies are generally smaller than the curve peak value, the RAO values corresponding to low frequencies are commonly greater than those at higher frequencies. The relatively remarkable heave and pitch responses within the low frequency range might be the related motions of the excited surge response.

Figure 7.

Response amplitude operators of floating platform motions (bridge-longitudinal loading): (a) Surge; (b) Heave; (c) Pitch.

Figure 8 illustrates the RAOs of tension forces in mooring lines. From the RAO curves, similarity between the responses in upstream and downstream lines can generally be observed, for both incline cables and tension legs. Furthermore, a downward trend analogous to the curves of surge motions can be seen for the RAOs of incline cables, which demonstrates again that the platform lateral response is mainly restrained by the incline cables. Regarding to tension legs, relatively high values are read near the heave and pitch natural frequencies, since the tension legs primarily provide the vertical constraint for floating platform. Thus, generally speaking, the resonances observed in platform motions have been correspondingly reflected in the RAO curves of mooring forces.

Figure 8.

Response amplitude operators of tensions in mooring lines (bridge-longitudinal loading): (a) Incline cables; (b) Tension legs.

Combined wave and current tests

In this section, the random dynamic response of the floating foundation system is analyzed based on the power spectral density (PSD) and the statistic values including mean value, standard deviation (STD), and the maximum and minimum values. It should be noted that, the PSD curves provided are acquired based on the Fast Fourier Transform (FFT) processing with a time series of 8192 data points (corresponding to a duration of 81.92 s), by using the data-processing software OriginPro,version 2016 (OriginLab Corporation, 2016). It should be noted that the consistency of the PSD curves throughout the entire recorded time histories has been confirmed during the analysis. The original spectra were further smoothed through the second-order binomial smoothing method proposed by Marchand and Marmet (1983), which has also been incorporated in OriginPro and is widely used in the field of signal processing.

Loading in transverse bridge direction

Motions of floating platform

Figure 9 compares the statistics of platform motions under bridge-transversal loadings of 1-year and 100-years conditions. The relevant tested data are also presented in Table 6. It can be seen that:

(1) Under current only loadings, the system responded almost statically since negligible standard deviations are read. It is understandable that the adopted current velocity is relatively low so as the occurring condition of vortex-induced vibration has not been reached. Besides, the loaded platform shown backward and downward offsets respectively in sway and heave directions, as well as a positive roll-directional rotation. The backward displacement is undoubtedly due to the drag force of currents, and the positive roll motion may be resulted from a positive moment induced by the currents, of which the values are related to the underwater geometric shape of the floating platform. The negative heave displacement is thought to be attributed to the Bernoulli Effect which causes a downward force due to the unbalanced hydrodynamic pressures in vertical direction.

(2) Considering the dynamic responses of platform sway and roll, decreases in standard deviations can be found when comparing the load cases of combining wave and current to those of wave only (i.e., LC1.3 to LC1.1 for 1-year condition, LC2.3 to LC2.1 for 100-years condition). Similar finding can be drawn from the differences between maximum and minimum values, especially in the case of 100-years condition. Such phenomenon indicates that the combined currents have weakened the dynamic responses of platform motions in these two DOFs, which might be attributed to the additional viscous damping induced by the incoming current. It is worth to note that, due to the reductions caused by the combined currents, the maximum values of sway and roll responses would occur in the wave only case of 100-years condition (i.e., LC2.1).

(3) Comparing the standard deviation of platform heave in LC1.3 to LC1.1 (for 1-year condition), and LC2.3 to LC2.1 (for 100-years condition), relative increments of 16.2% and 9.2% are calculated, respectively. Considerable enlargement can also be found in the differences between maximum and minimum values. The observed enhancements in heave vibrations might be interpreted by the fact that, an additionally oscillating force in the vertical direction has been generated as the incident water flow acting on the rolling platform, and due to which the dynamic responses in heave motion have been increased.

Figure 9.

Statistics of floating platform motions under bridge-transversal loading: (a) Sway; (b) Heave; (c) Roll.

Table 6.

Statistics of floating platform motions under bridge-transversal loading.

		Sway (mm)	Heave (mm)	Roll (deg)
LC1.1	Mean	0.67	0.28	0.05
LC1.1	STD	9.37	1.30	0.44
LC1.2	Mean	1.86	−0.43	0.11
LC1.2	STD	0.10	0.04	0.01
LC1.3	Mean	3.02	0.45	0.12
LC1.3	STD	8.30	1.51	0.41
LC2.1	Mean	0.72	0.22	0.06
LC2.1	STD	22.53	2.49	0.85
LC2.2	Mean	3.82	−0.87	0.25
LC2.2	STD	0.36	0.09	0.03
LC2.3	Mean	5.45	-0.01	0.31
LC2.3	STD	15.78	2.72	0.69

From Figure 9 it can also be observed that, except for the heave motion, the mean responses of platform under the wave-current cases correspond approximately to the current only cases, and the standard deviations and extreme values are in accord with the wave only cases. The presence of the distinct feature in heave response should be due to the randomness of time histories, since the mean values of heave motion are very close to zero.

Figure 10 presents the PSDs of platform motions under LC1.1 and LC1.3, comparing the dynamic responses under wave only and wave-current combination conditions. In general, the spectra show complex motion combinations containing wave excitation components and the resonances at natural frequencies. Specifically, the following observations can be found:

(1) The sway response of the floating platform is mainly contributed by sway resonance and the wave excitation, as the spectra peaks are observed at their sway natural frequencies (tested in still water) and the peak frequency of the incident wave. Both components are found to be comparable, while the responses under LC1.3 are generally smaller than that under LC1.1, which interprets the reduction of platform dynamic responses as described in previous text.

(2) The heave response of floating platform is mainly contributed by the heave resonance and wave excitation. Since the platform is well restrained by both the incline cables and tension legs in vertical direction, the heave response is fairly small compared to the sway displacement. Compared to LC1.1, the responses contributed by wave loading is much more prominent in LC1.3, which corresponds to the relatively significant standard deviations and extreme values as illustrated in Figure 9(b).

(3) The platform roll motion is composed of sway resonance and wave loading components as shown in Figure 10(c). As considerable response component contributed by sway resonance has been recognized in the roll spectra, one can believe that a coupling effect of platform sway and roll is existing. However, since the roll motion is relatively minor, the coupling effect was observed to be slight. The mitigation in platform roll motion due to the superposed current is also reflected in the spectra.

Figure 10.

Spectra of floating platform motions under LC1.1 and LC1.3 (bridge-transversal loading): (a) Sway; (b) Heave; (c) Roll.

Figure 11 presents the samples of the time histories of platform’s sway under LC1.1 to LC1.3. Generally, it can be seen that the response under LC1.1 is relatively more significant than that under LC1.3, which agrees with the previous findings.

Figure 11.

Samples of time histories of platform sway under 1-year load cases.

Figure 12 further shows the PSDs of platform motions under LC2.1 and LC2.3 corresponding to 100-years environmental condition. Dynamic response components for each motion types are found similar to those in 1-year condition (shown in Figure 10). However, it should be noted that the heave resonance makes little contribution to the heave response in the 100-years condition, and the wave excitation component plays the sole contributor to the platform heave motion. What’s more, though the peak values of spectra between LC2.1 and LC2.3 are comparable for sway and roll motions as shown in Figure 12, responses of narrower frequency range can be seen in the spectra of LC2.3 near the wave peak frequency, which results in the relatively weak dynamic responses as concluded previously based on the standard deviations and extreme values.

Figure 12.

Spectra of floating platform motions under LC2.1 and LC2.3 (bridge-transversal loading): (a) Sway; (b) Heave; (c) Roll.

Tensions in mooring lines

Figure 13 presents the statistics of tension forces in mooring lines under environmental conditions, in which the responses of mooring lines corresponding to fairleads F4 and F1 are taken for representing the upstream and downstream lines, respectively. The corresponding data are also listed in Table 7 for comparisons. Based on statistical analysis, the following observations should be remarked:

(1) Under current only condition, the upstream incline cables were further stretched while the tensions in downstream incline cables have been decreased. On the other hand, the tensile forces of upstream and downstream tension legs were reduced and enhanced, respectively. The quasi-static responses in mooring lines are obviously due to the drag force and the roll-directional moment induced by the currents as discussed above. Figure 14 further illustrates the force changes of mooring lines under current only conditions. Obviously, the total tension forces in mooring lines have decreased since negative increments are found more significant than the positive ones, as shown in Figure 14. The interesting observation again supports the suspicion of the existence of Bernoulli Effect.

(2) Due to the superposed current, the dynamic responses of incline cables are generally mitigated in the wave-current conditions compared to those in wave only conditions, regarding both standard deviations and extreme values; however, insignificant changes are found for the tension legs. These response characteristics are consistent with the platform motions as the sway directional motion is mainly resistant by the incline cables, and the tension legs provide the vertical restrictions for the floating platform. Beside the above analysis, it is also worth to note that the extreme tension forces are generally occurred in LC2.1 (that represents the 100-years wave only condition) for both incline cables and tension legs, which should be paid attentions in the structural design practice.

Figure 13.

Statistics of mooring forces under bridge-transversal loading: (a) Incline cables; (b) Tension legs.

Table 7.

Statistics of mooring forces under bridge-transversal loading (Unit: N).

		Incline cables		Tension legs
		Upstream	Downstream	Upstream	Downstream
LC1.1	Mean	13.53	12.80	10.41	11.12
LC1.1	STD	1.98	1.88	1.18	1.16
LC1.2	Mean	14.08	12.42	10.10	11.82
LC1.2	STD	0.10	0.09	0.03	0.03
LC1.3	Mean	13.73	12.37	9.92	11.29
LC1.3	STD	1.76	1.69	1.24	1.10
LC2.1	Mean	13.56	12.82	10.46	11.18
LC2.1	STD	4.68	4.53	2.39	2.10
LC2.2	Mean	14.36	12.05	9.62	11.97
LC2.2	STD	0.25	0.25	0.10	0.07
LC2.3	Mean	14.05	11.88	9.46	11.58
LC2.3	STD	3.40	3.13	2.06	1.91

Figure 14.

Changes of static mooring forces due to current loading (bridge-transversal loading).

Figures 15 and 16 further depict the response spectra of tension forces in the mooring system under 1-year and 100-years conditions, respectively. In the figures, results for upstream and downstream sides are still taken from mooring lines corresponding to fairleads F4 and F1. Based on comparative analysis, the following conclusions can be drawn:

(1) The dynamic responses of incline cable tension are mainly due to the sway resonance and wave excitation, which is similar to the platform sway motion as illustrated in Figures 10 and 12. The consistency of the dynamic response component of incline cable tensions and the platform sway fully demonstrates that, as the vertical motions (including heave and roll) of floating platform has been effectively restrained by the tension legs, the primary function of the incline cables in the floating foundation system is to resistant the lateral displacement of the platform.

(2) For tension leg forces, the wave excitation is identified as the primary contribution to the dynamic responses, in addition to which the relatively weak response components due to sway and heave resonances can also be found in the spectra. These observations are coherent to the platform motions that have been indicated in Figures 10 and 12, convincingly demonstrating again that the dynamic response of the system is dominated by the wave loading and sway resonance.

Figure 15.

Spectra of mooring forces under LC1.1 and LC1.3 (bridge-transversal loading): (a) Incline cables; (b) Tension legs.

Figure 16.

Spectra of mooring forces under LC2.1 and LC2.3 (bridge-transversal loading): (a) Incline cables; (b) Tension legs.

Loading in longitudinal bridge direction

Motions of floating platform

1-year condition loadings in bridge-longitudinal direction were conducted for the floating bridge foundation. Figure 17 shows the statistics of platform responses in surge, heave and pitch directions, the mean values and standard deviations of these motions are further given in Table 8. From the statistics, the following characteristics can be observed:

(1) A quasi-static responses of platform displacement under current only condition (i.e., LC1.1) can still be observed, and the displacement is similar to the case of bridge-transversal loading, that is, a backward horizontal displacement, combined with a rotation corresponding to downward displacement at upstream side and upward displacement at downstream side. However, the horizontal and rotational displacements under this directional loading are generally greater compared to those under bridge-transversal loading. The more remarkable horizontal displacement is definitely attributed to the relatively low surge directional stiffness. On the other hand, the greater pitch motion should be due to the relatively weak restriction in platform pitch direction, as the moment arm for restoring forces (i.e., the distance of 250 mm between fairleads F1 and F2, or, F3 and F4) is relatively short compared to that in roll direction (i.e., the distance of 540 mm between fairleads F1 and F4, or, F2 and F3).

(2) Focusing on the dynamic responses, a reduction in horizontal dynamic response (i.e., the surge motion) due to the superposed current can also be found with regard to the standard deviations (which reduced by 21.7% comparing LC1.3 to LC1.1). The reason can be interpreted similarly to the sway response under bridge-transversal loading as due to the additional viscous damping induced by the applied current. What’s more, a different response feature between the two directional loadings lies in that, with the action of the combined current, the rotational response under bridge-longitudinal loading (i.e., pitch motion) would be enlarged, while the corresponding motion under transversal loading (i.e., roll motion) was found decreased. Such distinction can also be explained with the relatively weak restriction on platform pitch motion, as analyzed above.

Figure 17.

Statistics of floating platform motions under bridge-longitudinal loading: (a) Surge; (b) Heave; (c) Pitch.

Table 8.

Statistics of floating platform motions under bridge-longitudinal loading.

		Surge (mm)	Heave (mm)	Pitch (deg)
LC1.1	Mean	−1.37	−0.22	0.00
LC1.1	STD	9.65	2.07	0.27
LC1.2	Mean	−16.31	−2.44	1.24
LC1.2	STD	1.33	0.08	0.05
LC1.3	Mean	−17.49	−2.65	1.27
LC1.3	STD	7.56	2.74	0.41

Figure 18 presents the PSDs of platform motions under LC1.1 and LC1.3, loading in bridge-longitudinal direction. Regarding to surge motion, the responses in LC1.1 and LC1.3 are both composed of surge resonance component and wave excitation component. However, in the wave only case (i.e., LC1.1), the contribution of surge resonance is found the primary factor in the platform response, while in wave-current case (i.e., LC1.3), the surge resonance has little influence compared to the wave loading component. For heave motion response, contributions of wave loading and heave resonance are both non-negligible as shown in the response spectra, and it is found that the wave loading component is more significant in LC1.3 compared to LC1.1, which is similar to the heave response under bridge-transversal loading described previously. In pitch motion spectra, it is interesting that the response contributed by platform pitch resonance is highly limited, while the contributions of wave loading and surge, heave resonances are relatively prominent. This observation illustrates that the natural pitch mode has not been excited, but the pitch motion was coupled to the surge and heave responses of the floating platform. Moreover, compared to the pitch response under wave only case, the wave excitation component is much more remarkable under the case of combining wave and current (i.e., LC1.3). Since the platform pitch is coupled with the heave motion, this peculiarity is reasonable as analogous characteristic can be recognized in the heave spectra.

Figure 18.

Spectra of floating platform motions under LC1.1 and LC1.3 (bridge-longitudinal loading): (a) Surge; (b) Heave; (c) Pitch.

Figure 19 further shows the time series of the platform surge under LC1.1 to LC1.3. Consistent with the observations concluded from Figure 18(a), one can intuitively find that the motion of platform under LC1.1 presents a more apparent low-frequency feature compared to the motion under LC1.3, especially for the response after 20 s as the surge resonance has matured. Additionally, it can be found that the variation corresponding to LC1.2 is more remarkable than the sway response under transversal loading (as shown in Figure 11), of which the reason lies in that the mooring stiffness in the surge direction is relatively low and thus the system is sensitive to the low-frequency variation of current speed.

Figure 19.

Samples of time histories of platform surge under 1-year load cases.

Tensions in mooring lines

The statistics of tension forces in mooring system are presented in both Figure 20 and Table 9, in which the results are taken from mooring lines corresponding to fairleads F2 and F1 for upstream and downstream sides, respectively. The following conclusions can be drawn:

(1) Due to the current loading (in both LC1.2 and LC1.3), the mean tensions in upstream incline cables have been increased, and the downstream ones have been reduced. By contrast, the mean forces in the upstream and downstream tension legs have become lower and greater, respectively. The variation characteristics in force responses in the mooring system are analogous to those in bride-transversal loading cases, which is apparently attributed to the horizontal drag force and the moment caused by the incident current.

(2) By comparing the standard deviations in LC1.3 to those in LC1.1, it can be seen that the dynamic responses have slightly reduced by 8.1% and 7.4%, respectively for upstream and downstream incline cables. Nevertheless, the responses in tension legs have been amplified by 52.1% and 8.3% for upstream and downstream sides, respectively. The enhancements of dynamic response in the tension legs correspond to the increases of platform heave and roll motions as shown in Figure 17.

(3) The maximum tension responses of both incline cables and tension legs under bridge-longitudinal loading occurred in the wave-current combination case (i.e., LC1.3), which differ from the tested results in bridge-transversal loading (read in wave only case). Considering the mooring system layout shown in Figure 1, it is understandable that greater changes of static forces in incline cables were needed for balancing the horizontal drag force, under bridge-longitudinal loading. On the other hand, due to the relatively large horizontal mean displacement of floating platform, the tension legs were stretched more significantly, leading to a greater mean values of mooring forces in the legs.

Figure 20.

Statistics of mooring forces under bridge-longitudinal loading: (a) Incline cables; (b) Tension legs.

Table 9.

Statistics of mooring forces under bridge-longitudinal loading (Unit: N).

		Incline cables		Tension legs
		Upstream	Downstream	Upstream	Downstream
LC1.1	Mean	13.00	12.77	10.17	10.70
LC1.1	STD	0.86	0.95	0.94	0.96
LC1.2	Mean	13.90	11.21	8.38	11.22
LC1.2	STD	0.14	0.14	0.07	0.07
LC1.3	Mean	13.96	11.09	8.28	11.18
LC1.3	STD	0.79	0.88	1.43	1.04

Correspondingly, Figure 21 demonstrates the PSDs of tension forces in the mooring lines under bridge-longitudinal loading. For incline cables, all response components identified from the motion spectra (shown in Figure 18) have been reflected in the tension spectra. What’s more, similar to the motion responses, surge and wave excitation components are found prominent under LC1.1, while the contribution of surge resonance declines under LC1.3. For tension legs, under the wave only condition (i.e., LC1.1), the responses of upstream and downstream legs are highly similar since the both responses are due to the wave excitation and the platform heave resonance. However, under LC1.3, significant difference can be seen between upstream and downstream tension legs, which could be resulted from the combined effect of current loading and the coupling of platform heave and pitch.

Figure 21.

Spectra of mooring forces under LC1.1 and LC1.3 (bridge-longitudinal loading): (a) Incline cables; (b) Tension legs.

Conclusions

This paper presents an experimental research on the hydrodynamic characteristics of floating bridge foundation under wave and current loadings. The test campaign was implemented in the circulating water channel of SKLOE, guided by the similarity criteria of Froude and Strouhal numbers. The tested floating bridge foundation model was designed based on a conceptual prototype, which composed of a concrete-made floating platform and the mooring system of polyester ropes, noting that both tension legs and incline mooring cables have been adopted in the mooring system of tested model. Still water tests, regular wave tests and the environmental loading tests were carried out for the floating foundation, and the response characteristics regarding platform motions and mooring tensions have been investigated based on statistical and spectral analyses. Based on the obtained results, the following main conclusions can be summarized:

(1) The RAOs of floating bridge foundation illustrate that, under bridge-transversal loading, significant responses occur at its sway natural frequency regarding both platform motions and mooring tensions, but fairly limited responses can be found for other frequencies. However, for platform motions under bridge-longitudinal loading, relatively large responses locate at the corresponding natural frequencies, and the resonances observed in platform motions are generally reflected in the RAO curves of mooring tensions.

(2) Regarding to the platform dynamic responses under irregular wave loadings, the horizontal displacements were found the most prominent for both bridge-transversal and -longitudinal loadings (corresponding to sway and surge motions, respectively). However, slight coupling effects between translational displacement and rotational responses can also be observed. The rotational motion under bridge-longitudinal loading (i.e., surge) was found more significant compared to those under transversal loading (i.e., sway) due to the relatively weak rotational constraint; however, these motions would be further limited provided the restriction of bridge superstructure.

(3) The floating foundation systems would behave statically under the current only condition, and the variations of mooring tensions are relatively slight compared to the pretension forces. For both bridge-transversal and -longitudinal loadings, the superposed currents in wave-current combination cases would mitigate the horizontally dynamic responses (sway or surge), and enhance the vertically heave oscillations of floating foundation, compared to the responses under wave only. Nevertheless, the effects of current on the rotational responses under the two directional loadings differ, which is relevant to the rotational stiffness provided by the mooring system.

In general, this paper gives an insight into the dynamic characteristics of floating bridge foundation with mooring system combining tension legs and incline cables, which provides beneficial references for further research on the design and construction of floating bridges in deep-water environment.

Footnotes

Acknowledgments

The supports are gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research work was supported by the National Natural Science Foundation of China (project No. 52278192, and No. 52208180) and the Program of Science and Technology Innovation Action Plan, Shanghai, China (project No. 20200741600).

ORCID iD

Bin Cheng

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