Research on the Connector Loads of a Multi-Module Floating Body with Hinged Connector Based on FMFC Model

Abstract

VLFSs (Very Large Floating Structures) are often assembled by multiple modules through connectors where the connector structure is prone to stress concentration. Therefore, the loads at the connectors have become a significant focus in VLFS design. In this paper, the FMFC (Flexible Module Flexible Connector) method, which can account for the elastic deformation of each module and the connector, is established in order to predict the coupling response of the connector and the platform. The finite element model of a dual-module transfer platform with a hinged connector is established and the modal shape characteristics of the entire structure are analyzed. The accuracy of the method is verified through a model test, where the model was scaled as 1:50. Furthermore, the RAOs (Response Amplitude Operators) of connector loads were calculated. It was found that the horizontal loads (along the x and y directions) were more sensitive to the wave direction. Furthermore, the wave direction where the maximum short-term significant value appears was not consistent with the direction of the maximum RAO. The loads response law of the VLFS with a hinged connector is summarized in this paper and provides a reference for the design of connectors.

1. Introduction

It is known that seaports are nodes that are integrated into logistical and transportation networks [1]. However, in the deep sea, where there are no seaports, offshore floating platforms play the same role as mobile seaports. With offshore floating platforms developing towards large-scale and multi-modularization, the connector structure plays a vital role in the design of the VLFS. In general, connectors can be sorted into three types: rigid joint, hinged joint and flexible joint [2]. Through model tests, it has been found that there are great design loads on the rigid joint connectors in extreme sea conditions. In addition, the flexible joint connector structure is relatively more complicated. Therefore, hinged joint connectors are more widely used in order to avoid the shortcomings of the other connectors. The hinged joint structure, such as the Articulated Concrete Block Mattress (ACB Mat), be applied as a revetment on the breakwater body or for shoreline protection [3].

At present, there are two methods that are used to calculate the connector loads according to whether the deformation of the floating structure is considered: the RMFC (Rigidity module flexible connector) model and the FMFC model. Wang [4] first propose the RMFC method for calculating the response of very large floating structures based on the three-dimensional hydroelasticity theory, where the stiffness of the connector is much smaller than that of the floating body and the elastic deformation of the connector structure is almost non-elastic in waves, so the elastic deformation comes from the connector. The RMFC method is widely used because it consumes less time and involves simple modeling. Riggs [5] simplified the flexible connector into a linear spring with three-directional stiffness for hinged connectors and converted the force of the connection point to the centroid of the module structure for solution purposes when studying MOBs (Mobile Offshore Bases). Then, based on Riggs’ study, Yu [6,7] limited the relative translation of the connector and released the relative rotation in order to study the motion and connector load of MOBs under different wave environments and with different connector stiffness, which was verified by a model test conducted at a 1:100 scale. Li [8] took the seabed as the second fixed body in the diffraction theory and used the RMFC method to study the connector loads of VLFSs under varying water depths. Yang [9] and Ding [10] further established the hydroelastic analysis method based on the Boussinesq equation and the three-dimensional potential flow theory, which considers the influence of topography and non-uniform waves. Then, the connector loads between modules of VLFSs were studied, and it was found that ignoring the influence of the terrain would lead to underestimations of the connector load.

However, with the development of the hydroelasticity theory [11,12,13,14,15] and the improvement of computer calculation abilities, the hydroelasticity method has become more widely used. Watanabe [16] summarized the hydroelasticity analysis method of VLFSs and pointed out that the application of the hydroelasticity analysis method in VLFSs is the future direction of development. Wu [17] considered that the elastic deformation of VLFSs should not be ignored in practice, so both the floating body and the connector structure should be considered elastic bodies. Therefore, the FMFC method is proposed, considering the elastic deformation of the platform and the connector at the same time based on the hydroelastic analysis method, which considers the floating bodies and the connector structure as a whole.

Fu [18] analyzed the general characteristics of the hydroelastic response of a dual-module platform with a hinged connector structure. As a special case, the response of the dual-module interconnection structure with very high connector stiffness was found to be in good agreement with the experimental results of equivalent continuous structures. Loukogeorgaki [19] developed a natural frequency iterative program for modes and analyzed and demonstrated the influence of the stiffness of breakwater connectors on the generalized response, the effectiveness and the hydroelastic response of free, flexible and sheet-like floating breakwater (FB) using the hydroelastic method. Over the past few years, the FMFC method has been widely used because of its accuracy in solving connector loads.

In this paper, in order to predict the connector characteristics, the FMFC analysis model, which considers the influence of the elastic deformation of each module and the connector, is established. The accuracy of this method was verified by testing the model. Furthermore, the connector loads and the short-term prediction results were calculated. Then, the response characteristics of the connector under different wave directions are discussed and the load response law of the VLFS is summarized, which provides reference for the design of VLFSs and connectors.

2. Theory

According to the FMFC method, the finite element model of the floating structure connector should first be established, and then the generalized elastic displacement ${\overrightarrow{u}}^0{}_r$ of each node can be obtained by modal analysis. Based on the three-dimensional hydroelastic theory [11], it is assumed that the fluid around the floating body is inviscid and incompressible. The fluid perturbation velocity potential is decomposed into the incident wave velocity potential ${\varphi }_0$, the diffraction velocity potential ${\varphi }_D$ and the radiation velocity potential ${\varphi }_R$. The radiation velocity potential can be further decomposed into the following parts:

$${\varphi }_R={\displaystyle \sum _{r=1}^m{\varphi }_r}={\displaystyle \sum _{r=1}^m{\phi }_r{p}_r{e}^{i{\omega }_{e}t}}$$

(1)

where m represents the number of modes selected for calculation, ${\varphi }_r$ represents the r-order radiation velocity, ${\phi }_r$ and $p_r$ represent the amplitude of r-order velocity and displacement, respectively, and ${\omega }_e$ is the encounter frequency.

Since the velocity potential of the incident wave is known, it can be directly calculated using an analytical formula. However, the diffraction velocity potential and the radiation velocity potential should be solved numerically because they are coupled with the motion of the floating body itself. The diffraction velocity potential and the radiation velocity potential should satisfy the same control equation (Laplace equation $[L]$), free surface condition ($[F]$) and far-field condition ($[R]$). In order to avoid repetition, Formula (2) uses $\varphi$ to replace the two velocity potentials.

$$\begin{array}{lll}[L]& {\nabla }^2\varphi (x,y,z,t)=0& \mathrm{in} \mathrm{fluid}\\ [F]& {\left(\frac{\partial }{\partial t}-U_0\frac{\partial }{\partial x}\right)}^2\varphi +g\frac{\partial \varphi }{\partial z}=0& z=\zeta (x,y,t)\\ [B]& \underset{\mathrm{z}\to \infty }{\lim }\nabla \varphi =0 or\underset{\mathrm{z}\to -\mathrm{H}}{\lim } \frac{\partial \varphi }{\partial n}=0& \mathrm{on} \mathrm{bottom} \\ \left[R\right] & \underset{R\to \infty }{\lim }\sqrt{R}\left(\frac{\partial \varphi }{\partial R}-ik{\varphi }_D\right)=0& \mathrm{far} \mathrm{filed} \mathrm{radiation} \mathrm{condition}\end{array}$$

(2)

In addition to the above conditions, the diffraction velocity potential should also satisfy the boundary conditions, and the radiation velocity potential should also satisfy the radiation surface boundary, as follows:

$$\begin{array}{ll}\frac{\partial {\varphi }_D}{\partial n}=-\frac{\partial {\varphi }_0}{\partial n}& \mathrm{on} \mathrm{wet} \mathrm{surface} S_b\\ \frac{\partial {\varphi }_r}{\partial n}=n_r{\dot{p}}_r(t)+m_r{p}_r(t)& \mathrm{on} \mathrm{wet} \mathrm{surface} S_b\end{array}$$

(3)

where k is the incident wave number, $U_0$ is the speed of the floating body, $R$ is the distance to the coordinate origin, $n_r$ is the r-order normal component where the normal vector points to the inside of the floating body, $m_r$ is the r-order component related to the steady ship wave and ${\dot{p}}_r(t)$ and $p_r(t)$ represent r-order velocity and displacement, respectively.

After the complete solution boundary conditions are established, the Hess–Smith method can be used to solve the flow field velocity potential around the structure. The three-dimensional hydroelastic equation of the floating body connector structure can be derived by combining the calculated hydrodynamic coefficients related to the fluid velocity potential with the structural dynamic equation, which can be expressed as:

$$\left[a+A\right]\left\{\ddot{p}\right\}+\left[b+B\right]\left\{\dot{p}\right\}+\left[c+C\right]\left\{p\right\}=\left\{F\right\}+\left\{R^{\prime }\right\}$$

(4)

where $\left[a\right]$, $\left[b\right]$ and $\left[c\right]$ are the inertia matrix, structural damping matrix and structural elastic restoring force matrix, respectively. $\left[A\right]$, $\left[B\right]$ and $\left[C\right]$ are the generalized fluid added mass matrix, damping coefficient matrix and restoring force matrix in water, respectively. $\left\{F\right\}$ and $\left\{R^{\prime }\right\}$ represent the generalized fluid wave excitation force and the generalized concentration force, respectively. The generalized hydrodynamic added mass matrix, additional damping coefficient, restoring force matrix and generalized wave excitation force can be obtained by the following formulae, respectively:

$$\left\{\begin{array}{l}{A}_{rk}=\frac{\rho }{{\omega }_e{}^2}\mathrm{Re}\left\{{\displaystyle \underset{\overline{S}}{\iint }\overrightarrow{n}\cdot {\overrightarrow{u}}_r^0\left(i{\omega }_e+\overrightarrow{W}\cdot \nabla \right){\phi }_k\left({\omega }_e\right)d{S}_b}\right\}\\ B_{rk}=-\frac{\rho }{{\omega }_e{}^2}\mathrm{Im}\left\{{\displaystyle \underset{\overline{S}}{\iint }\overrightarrow{n}\cdot {\overrightarrow{u}}_r^0\left(i{\omega }_e+\overrightarrow{W}\cdot \nabla \right){\phi }_k\left({\omega }_e\right)d{S}_b}\right\}\\ C_{rk}=-\rho {\displaystyle \underset{\overline{S}}{\iint }\overrightarrow{n}\cdot {\overrightarrow{u}}_r^0\left[g{w}_k^0+\frac{1}{2}\left({\overrightarrow{u}}_k^0\cdot \nabla \right){\left|\overrightarrow{W}\right|}^2\right]d{S}_b}\\ F_r=\rho {\displaystyle \underset{\overline{S}}{\iint }\overrightarrow{n}\cdot {\overrightarrow{u}}_r^0\left(i{\omega }_e+\overrightarrow{W}\cdot \nabla \right)\left({\varphi }_0+{\varphi }_D\right)d{S}_b}\end{array}\right.$$

(5)

where ${\phi }_k$ is the k-order radiation velocity potential amplitude (the same as the above definition), $\overrightarrow{W}$ is the velocity of steady flow relative to the equilibrium coordinate system and $w_k^0$ represents the vertical displacement of the k-order mode.

The principal coordinates of the floating body can be solved in the frequency domain by substituting Formula (5) into the three-dimensional hydroelastic Equation (4). Then, according to the principle of structural dynamics, the loads can be obtained by the modal superposition method using Formula (6).

$$F^i=\sqrt{{\left({\displaystyle \sum _{r=1}^m{p}_r\cdot F_r^i}\cdot \cos {\theta }_r\right)}^2+{\left({\displaystyle \sum _{r=1}^m{p}_r\cdot F_r^i}\cdot \sin {\theta }_r\right)}^2}$$

(6)

where $F^i$ is the load in the direction of I, ${\theta }_r$ refers to the phase of the r-order response and m is the number of modes selected for calculation (the same as the above definition).

3. Method Verification

In this paper, the dual-module platform was selected as the research object whose main parameters are shown in

Table 1. Main parameters of the single module.

Physical Quantity	Value
Length (m)	200
Breadth (m)	50
Depth (m)	9.5
Draught (m)	5
Center of gravity (m)	(100, 0, 6.17)

. The platform is symmetrical with respect to the central longitudinal section and the midship section. The origin of the platform coordinates is located at the intersection of the baseline and the after perpendicular, with the x axis pointing to the bow, the y axis pointing to the port side and the z axis pointing vertically upward. There were two hinged connectors used to constrain the relative roll and relative yaw, which released the relative pitch between the two modules. The finite element information of the dual-module platform structure is shown in Figure 1, where the connector is connected to the platform by MPCs (multi-point constraints).

3.1. Modal Analysis

The first 15 order modes were selected for modal analysis. In addition to the six rigid body modes, the first four order vertical bending modes, the first three order torsional modes and the first two order horizontal bending modes were obtained. The mode shapes selected in this paper are shown in Figure 2, Figure 3 and Figure 4, and the natural frequencies are shown in

Table 2. Natural frequency.

Mode Type		Frequency (Hz)
Vertical bending mode	2-node	0.00
	3-node	0.53
	4-node	0.80
	5-node	1.45
Torsional mode	1-node	0.53
	2-node	1.50
	3-node	1.83
Horizontal bending mode	2-node	0.58
	3-node	1.75

.

It can be seen from Figure 2 that the modal shapes of the dual-module platform with a hinged connection structure are quite different from those of the single-module platform. Because the connector releases the relative pitch between the two modules, the two-node vertical bending mode (shown in Figure 2a) actually represents the relative pitch of the platform, whose natural frequency is 0 Hz. Similarly, due to the relative pitch released by the connector, the four-node vertical bending mode (shown in Figure 2c) exhibits a ‘double U’ shape, and there is almost no transition in this mode shape at the joint between the two modules, which is in the form of ‘sharp corner’. It can be seen from Figure 3 and Figure 4 that the horizontal and torsional modal shapes are similar to those of the single-module platform. This result means that releasing one DOF (degree of freedom) will have a great influence on the corresponding modal shapes, whereas it has little influence on the other modal shapes.

Furthermore, by comparing the natural frequencies in Table 2, it can be seen that the horizontal bending stiffness is greater than the torsional stiffness and the torsional stiffness is greater than the vertical bending stiffness. Among the first 15 order modes, the number of vertical modes is the most, whereas the number of horizontal bending modes is the least, which conforms to the distribution law of flat barge structural modes.

3.2. Model Experiment

In order to verify the accuracy of the method established in this paper, a model test was carried out in a tank with an L-shaped wave maker shown in Figure 5. The length of the tank was 44.5 m and the width of the tank was 40 m. Based on factors such as the size of the tank, the wave-making capacity and the connector model size, the scale ratio of the test model was set to 1:50. The stiffness of the model connector in X, Y and Z directions was obtained through scale conversion. It is known that the stiffness of the model connector is relatively small after scale reduction. Therefore, selecting an appropriate material to simulate the stiffness of the model connector was a key challenge in the experiment. In this experiment, an equivalent connector structure was designed as shown in Figure 6, where nylon and steel were used. The main body of the model connector was made of nylon, whereas the pin in the middle was made of steel.

The connector cannot be directly connected to the platform because of the requirement to satisfy the height from the platform baseline. Connecting the model connectors with the platform model presents another problem. In this experiment, the connector was directly connected to the force sensor, and the force sensor was connected to the measuring beam through the flange to ensure that the measuring structure remained rigid and that measurement error caused by the deformation of the base was prevented. The specific connection way is shown in Figure 7.

Due to space limitations, only the experimental results of the RAO (Response Amplitude Operator) in 0° and 45° are compared with the calculated results, as shown in Figure 8. Overall, the theoretical calculation was in good agreement with the test results, and the FMFC model established in this paper accurately reflected the load characteristics of connectors. Specifically, the calculated results were in good agreement with the experimental results at low frequencies, but there were relatively large fluctuations at high frequencies, which may be related to the quality of high frequency waves in the tank. Figure 9 shows the relative error results, which reveals interesting information, including that the relative error of vertical shear was about 15% or lower, such as Fz in 0°. However, at some frequency points, the error for horizontal shear in the y direction was greater than 20%. Through analysis, we found that the reason for this problem was that a soft spring was used to limit the horizontal displacement of the test model in order to prevent the model from drifting with the wave during the experiment. Although the spring stiffness was as small as possible, the tension level of the soft spring still affected the measured horizontal force of the connector in the model test.

4. Simulation and Discussions

4.1. Load Response Analysis of Connectors

Because connectors $C_1$ and $C_2$ were symmetrically arranged, only the calculation results of connector $C_1$ are given (Figure 10), where the RAO under different directions is shown.

Generally speaking, the wave load on the connector reaches its maximum at low frequencies, particularly near the wave direction of 75 degrees. Furthermore, the maximum value of the three-direction load decreases in the order of Fx, Fz and Fy. Therefore, special attention should be paid to the strength of the axial and vertical directions when designing the connector. Specifically, the axial force of the connector showed a clear increase as the wave direction increased, and the maximum value was 1.36 × 10⁷ N at 75 degrees (Figure 10a). When the wave direction was small, the horizontal shear force was small, which can be ignored when compared with the maximum value (1.63 × 10⁶ N) under 60° (Figure 10b). The vertical shear force distribution exhibited little change with changes in the wave direction, and the maximum value appeared at 75°, which was 1.28 × 10⁷ N (see Figure 10c).

In terms of high-frequency loads, almost no obvious wave-induced vibration phenomenon was observed within the calculation frequency range for the load along the y axis, whereas the first-order resonance frequency of the load along the x axis was around 3.5 rad/s. In addition, the first-order resonance frequency along the z axis was between 1.5 rad/s and 2.0 rad/s, which was close to the natural wave frequency range. Therefore, more attention should be paid to the wave-induced vibration phenomenon of the connector in subsequent checks of connector strength in the z direction.

In order to study the influence of modes on the connector, the load of the first three modes with the greatest influence on the connector load under 75 degrees were taken, as shown in Figure 11. It can be seen from Figure 11a that the load of the connector in the x direction was mainly determined by the two-node horizontal bending mode. Furthermore, it can be seen from Table 2 that the natural frequency of two-node horizontal bending was 3.64 rad/s (0.58 Hz), which is consistent with the peak frequency of the first-order resonance spectrum in Figure 10a.

As for the y direction load of the connector, it can be seen from Figure 11b that the three-node torsion and horizontal bending mode have a great influence on shear force in the y direction, but this is mainly determined by the three-node torsion mode. Furthermore, the two-node torsion mode has little contribution to the y direction load. Compared with Figure 4b, it can be seen that the lateral relative displacement of the two-node torsion at the connector was small, resulting in less force. As for the z direction load, it can be seen from Figure 11c that it was mainly determined by the first-order torsion mode at low frequencies and the three-node vertical bending mode at medium–high frequencies.

4.2. Short-Term Prediction

Based on the load transfer function calculated above, the short-term forecast was made selecting the JONSWAP spectrum, which can be expressed as follows:

$$S\left(\omega \right)=487\left[1-0.287\ln \left(\gamma \right)\right]\frac{H_s^2}{T_p^4}{\omega }^{-5}\exp \left(\frac{-1948}{T_p^4}\right){\gamma }^{\exp \left(\frac{-{\left(0.159\omega T_p-1\right)}^2}{2{\sigma }^2}\right)}$$

(7)

where $\gamma$ and $\sigma$ represent the spectral peak lifting factor and the spectral width parameter, respectively. When $\gamma$ is equal to 1, the above formula degenerates into the P–M spectrum. Based on statistical data of the sea state where the platform operates, the wave parameters were selected as follows: significant wave heigh $H_s$ = 2.5 m, period $T_p$ = 9.0 s and peak lifting factor $\gamma$ = 2.0.

It can be seen from

Table 3. Significant values of connector loads.

	Type	C1-Fx (MN)	C1-Fy (MN)	C1-Fz (MN)
Wave Direction
0.0		1.18	0.02	1.83
15.0		1.28	0.10	1.92
30.0		2.88	0.21	2.12
45.0		6.47	0.43	2.56
60.0		8.26	0.69	6.62
75.0		13.69	1.63	7.89
90.0		2.14	0.11	0.04

that the maximum of Fx and Fz appeared near 75°, which is consistent with the direction of the maximum RAO. Furthermore, the maximum of Fy also appeared near 75°, whereas the direction of the maximum RAO appeared near 60°. By analyzing the shape of RAO under different wave directions (Figure 10b), it was found that the spectral width of RAO increases rapidly with increases in the wave direction, but the peak value of 60 degrees was close to 75 degrees, resulting in an increasing trend in the variance of the transfer function, defined by $m={\displaystyle {\int }_0^{\infty }S\left(\omega \right)d\omega }$. According to the relationship between the significant value $x_{1/3}$ and variance m in the Rayleigh distribution ($x_{1/3}=2\sqrt{m}$), the maximum significant value of the load finally appeared at 75°. It can be concluded that the wave direction of the maximum short-term significant value of the platform was not the same as the direction where the maximum of RAO appeared. It is suggested that all the wave directions should be searched in order to obtain an accurate maximum short-term significant value when calculating the connector loads.

5. Conclusions

In this paper, the FMFC model was established in order to predict the connector loads of multi-module VLFSs with hinged connectors. Compared with the traditional RMFC method, this method combines multi-module structures with the hydroelasticity theory, so the elastic deformation of each module and connector can be considered. Moreover, it is more convenient to obtain the relative displacement between modules with this method because the relative pitch mode can be identified through modal analysis. A model test was carried out and the results show that the established method is correct. We also found that the model mooring greatly influenced the horizontal loads of the connector when comparing the simulation results with the experimental results. This serves as a reminder that special attention must be paid to the selection of mooring springs before initiating the experiment and during data processing after experiment. A lot of calculation work has been carried out for connector loads. We conclude that the loads in all three directions reached their maximum values near 75 degrees, where the value of Fy was relatively smaller. In addition, the horizontal loads (along the x and y directions) were more sensitive to changes in the wave direction. Therefore, we should pay more attention to the Fx and Fz when designing connectors. Furthermore, an important conclusion for designers is that the wave direction at which the maximum short-term significant value appears is not consistent with the wave direction where the maximum transfer function is located. Therefore, all the wave directions should be searched in order to obtain an accurate maximum short-term significant value before conducting a structural performance analysis.

In this paper, the influence of connector stiffness on connector loads was not referred to. However, parameter sensitivity analyses are planned for future work. All in all, the FMFC method established in this paper can be used to calculate the loads of VLFSs, and the laws summarized in this paper provide reference for future strength calculations and connector designs.