Optimal Stiffness for Flexible Connectors on A Mobile Offshore Base at Rough Sea States

This paper investigates a simplified method to determine the optimal stiffness of flexible connectors on a mobile offshore base (MOB) during the preliminary design stage. A three-module numerical model of an MOB was used as a case study. Numerous constraint forces and relative displacements for the connectors at rough sea states with different wave angles were utilized to determine the optimized stiffness of the flexible connectors. The range of optimal stiffnesses for the connectors was obtained based on the combination and intersection of the optimized stiffness results, and the implementation steps were elaborated in detail. The percentage reductions of the optimized and optimal stiffness of the flexible connector were determined to quantitatively evaluate the decreases of the constraint force and relative displacement of the connectors compared with those calculated by using the original range of the connector stiffnesses. The results indicate the accuracy and feasibility of this method for determining the optimal stiffness of the flexible connectors and demonstrate the rationality and practicability of the optimal stiffness results. The research ideas, calculation process, and solutions for the optimal stiffness of the flexible connectors of an MOB in this paper can provide valuable technical support for the design of the connectors in similar semisubmersible floating structures.


Introduction
A very large floating structure (VLFS) is defined as a marine structure with geometric dimensions that are measured in kilometers (Lamas-Pardo et al., 2015). A VLFS is a typical built-up structure in that it requires specially designed connectors to combine multiple modules due to its very large size, and each module can operate in a self-propelled mode (Wang and Tay, 2011). VLFSs can be divided into two structural styles: box type (pontoon type) and semisubmersible type (Wang et al., 2008). The semi-submersible type has received more attention from scholars and researchers in the area of ocean engineering than the box type does due to its good hydrodynamic performance and its ability to adapt to rougher marine environments in deep water (Pham et al., 2009). The connectors between adjacent modules are the weak points of VLFSs, and very large constraint forces can be generated at the connectors due to excitation by external loads. When the constraint loads and fatigue cracks in the connectors exceed their width, the entire VLFS becomes dangerous. Therefore, it is necessary to research the optimal design of the connectors to ensure the safe operation of VLFSs.
The Office of Naval Research (ONR) (Remmers et al., 1998) and the McDermott Company (Zueck et al., 2001) in the United States reported that the connection type (rigid or flexible) among adjacent modules of a VLFS is the key factor in different constraint loads on the connectors. Brown and Root (Kriebel and Wallendorf, 2001) presented a conceptual design of a mobile offshore base (MOB) that used rigid connectors to join 6 rigid modules, and the resulting constraint forces on these rigid connectors were very large and exceeded the connectors' strength. The McDermott Company (Zueck et al., 2001) presented a new conceptual design called the Mc-MOB that used hinge connectors to connect 5 rigid modules, and the Mc-MOB could remain connected for normal operation under conditions less than Sea state 5 (SS5); however, for the sea states above SS7, the connectors must be disconnected due to the overloaded constraint forces. Derstine and Brown (2000) designed a flexible connector for a VLFS, and their design can effectively change the natural frequency of the structural motion and substantially reduce the constraint forces on the connectors. Similarly, Riggs et al. (1998) investigated the relationships between the hydrodynamic response, natural structural frequency and connector stiffness and concluded that the stiffnesses have a substantial influence on the amplitude of the motion of a floating structure and that the resonance phenomenon occurs more easily as the connector stiffness decreases. In addition, the constraint force on the connectors increases with the increasing stiffness assuming that the other conditions remain stable. Kim et al. (2007) discussed box-type VLFSs and validated the conclusions of Riggs et al. (1998). Yu (2004) and Ding et al. (2005) studied the hydrodynamic characteristics of the MOB connectors through physical model experiments, and the test data indicated that the constraint loads of flexible connectors were sensitive to changes in the stiffness. In summary, the stiffness has a significant influence on the constraint force on the connectors.
The connectors' stiffness is one of the vital factors that affect the constraint force. Therefore, the optimal stiffness of the connectors should be determined during the design of an MOB and its connectors. Few international studies described the optimal stiffness of flexible connectors. Michailides et al. (2013) presented a frequency-domain numerical analysis algorithm to estimate the internal loads on flexible connectors in a modular box-type floating structure (BTFS) and to identify its optimum configuration under regular waves based on the hydroelastic response theory. Hu et al. (2012) researched an optimal design scheme for a new type of subsea pipeline connectors using ANSYS, and a dimensional optimization of the connectors was used to obtain the minimum loading force for optimum sealing performance based on the zero-order optimization method. Peng et al. (2015) discussed a deepwater connector that was a key component of a subsea production system. For a hub seat mouth alignment structure with a 0.3-m-diameter horizontal sleeve-type connector in 1500 m of water, the maximum deflection rectifying capacity of the mouth structure was taken as the optimization objective, and the fseminf function of the MATLAB optimization toolbox was used to optimize the parameters. The results provided a theoretical basis for the optimal design of a deepwater sleeve-type connector mouth structure.
This paper investigates an evaluation method for determining the optimal range of the stiffnesses of flexible connectors for a representative conceptual design of an MOB structure at rough sea states with a constant number of connectors between two adjacent MOB modules at the preliminary design stage. The idea of this design is that connectors with the optimal stiffness can reduce the constraint forces and relative displacements at rough sea states in order to ensure the security of the entire MOB structure. The evaluation method and the solutions for this typical conceptual design of an MOB structure may provide valuable technical support for studies of similar components and structures.

MOB and connectors
2.1 Conceptual design of the MOB and flexible connectors Fig. 1a shows the conceptual design of the MOB. An MOB typically consists of individual modules that are interlinked by the connectors. A single module includes a top platform, columns, and pontoons. The rigid modules and flexible connectors (RMFC) model is the most widely used numerical coupling model for VLFS (Remmers et al., 1998;Zueck et al., 2001). This paper uses the RMFC model to estimate the optimal stiffness of flexible connectors in the MOB. A hypothetical design of a flexible connector was considered to restrain the linear displacement of the module but allow angular rotations through the installation of a linkage (springs with a stiffness are set into a linkage) in three translational directions. Fig. 1b shows a complete set of flexible connectors that includes multiple linkages in the x, y, and z directions, which are the vital components of flex- ible connectors. As shown in Fig. 1c, a single linkage consists of a spring canister with ball joints at the two ends. This key component mainly includes a spring, piston, canister and stretchable rod. The rationale of the linkage is the use of two-force rods, which only exhibit axial forces and no bending moment. Fig. 1c shows that two linked springs are connected with a stretchable rod and piston within the canister. When the stretchable rod moves because of external forces, the rod drives the spring's motion. To solve the constraint force and relative displacement of the flexible connectors by external environmental loads, this paper generalizes the flexible connectors as a simplified spring model in the x, y, and z directions based on the RMFC assumption and the conceptual design of the flexible connectors. The simplified spring model is shown in Fig. 1d.

Geometric dimensions of a single MOB module
The geometric dimensions of a single MOB module are based on a classical conceptual design by Yu et al. (2003) (Fig. 2). Two connectors are installed between two adjacent modules of the MOB, and both are located on the side of the top platform (Fig. 1). This paper regards the three-module MOB model as a case study with which to investigate the optimal stiffness of flexible connectors.

Rough sea states
Generally, sea states can be classified into four categories according to their risk magnitude, and a "rough sea state" is the third of these four categories (Huang, 1998). A "rough sea state" can be defined as equivalent to the 6th grade (or Sea state 6; SS6) or higher. This paper considers Sea states 6, 7 and 8 (SS6, SS7, and SS8) as the rough sea states based on the statistical parameters of waves in the open ocean. The wave parameters of SS6, SS7, and SS8 are as: (1)  where H s is the SWH, T p is the wave peak spectral period, and f is the wave frequency range. These data and the wave spectral density function were utilized to simulate the irregular wave field.

Calculated conditions
This paper selects 108 combinations of conditions based on Riggs et al. (1998) to solve the hydrodynamic constraint forces and relative displacements of the flexible connectors. These 108 conditions include three sea states (SS6, SS7, SS8), six wave angles (0°, 45°, 75°, 80°, 85°, 90°), and six connector stiffnesses (K 1 -K 6 ). When the wave incidence direction is parallel to the ox axis in Fig. 1, the wave angle is 0°. Similarly, when the wave incidence direction is parallel to the oy axis, the wave angle is defined as 90°. The stiffness values of the connectors (K 1 -K 6 ) in the x, y, and z directions are the same (K x =K y =K z ), and the stiffness values are as follows: K 1 =1.0×10 6 N/m, K 2 =1.0×10 7 N/m, K 3 =1.0×10 8 N/m, K 4 =1.0×10 9 N/m, K 5 =1.0×10 10 N/m, and K 6 =1.0×10 11 N/m. The Bretschneider spectrum is adopted to simulate the random irregular wave field at different sea states.

Implementation procedures for the optimal stiffness
4.1 Hydrodynamic response of the MOB MOB is a built-up structure that requires specially designed connectors to connect multiple semi-submersible modules, and the potential flow theory (PFT) is generally used to estimate the hydrodynamic response of the MOBs (Clauss et al., 2003;Palo, 2005;Nematbakhsh et al., 2015). However, the theoretical derivation process of the PFT method for simulating the wave force is relatively complex, and the computational time of the PFT depends greatly on the precision of the mesh of the hydrodynamic model of the MOB. During the preliminary design stage, we should rapidly and efficiently calculate the hydrodynamic responses of the MOB for numerous conditions. Therefore, this paper adopts a simplified algorithm to evaluate the hydrodynamic responses of the MOBs during the preliminary design stage at rough sea states.
The algorithm proposed by this paper highlights a vital geometrical characteristic in which different components within an MOB structure, such as columns and pontoons, can be regarded as small-scale components by comparing them with the wavelengths at rough sea states, and the revised Morison theory for floating bodies is adopted to evaluate the wave forces on the MOB structures. Thus, it is not necessary to apply the relatively complex PFT to estimate the wave force. The revised Morison method can improve the computational efficiency because it avoids the meshing problem. The expression of the revised floating body Morison theory can be viewed in Wu (2015) and Wu et al. (2016Wu et al. ( , 2017. As a wave propagates in one direction, the shielding effects due to the interactions between multiple components within the MOBs are non-negligible (Liu, 2013). Therefore, the proposed algorithm also considers the shielding effects of different columns and pontoons located next to each other, which is similar to the method that is used to evaluate the effects of wave loads on pile groups referred from Bonakdar and Oumeraci (2015).
In summary, the MOB structure with its connectors can be regarded as a multiple degree of the freedom system in structural dynamics, and the dynamic balance equation of the entire structural system based on the D'Alembert theory can be expressed as: where m s is the structural mass matrix, m f is the added mass matrix, c f is the damping coefficient matrix, k s is the stiffness matrix for the entire structural system, k f is the restoring force coefficient matrix, P w is the external stimulating force matrix, which represents the random and irregular wave force at rough sea states, , and represent the acceleration, velocity and displacement, respectively, of a single MOB module under wave force excitation, and t is time. However, direct evaluation of the hydrodynamic response of the entire structural system is relatively complex due to its very large size, so the isolation method is adopted to analyze the hydrodynamic motion of a single module. Each module is isolated at the positions of the connectors, and the dynamic equation of a single MOB module can be expressed as: (2) where k s =0 due to the lack of connection of the entire structure. In addition, the flexible connectors' constraint force F c (t) changes from an internal force to an external force, and it appears on the right side of Eq. (2). The hydrodynamic response of each module in the six motion directions (surge, sway, heave, roll, pitch, and yaw) can be solved using Eq. (2) for the 108 conditions. The derivation process and expressions of the hydrodynamic coefficient matrixes (m s , m f (t), c f (t), k f (t)), wave force vectors P w (t) and constraint force vectors F c (t) in the six directions from Eq. (2) are specifically given in Wu (2015) and Wu et al. (2016Wu et al. ( , 2017. 4.2 Constraint force and relative displacement of the connectors (1) Simplified theoretical formulae The geometrical relationship of the two modules adjacent to the connectors before and after the motion, combined with the hydrodynamic responses of the two modules, can be used to obtain the relative displacements (Δ c ) of the flexible connectors in the x, y, and z directions. The con- straint forces (F c ) in three directions can then be determined using Hooke's Law. Fig. 1 shows that there are two connectors between two adjacent modules. The formulas for evaluating the constraint forces on C 1 and C 2 are shown in Eqs. (3) and (4), where (F c1x , F c1y , F c1z ), (Δ c1 x, Δ c1 y, Δ c1 z), and (K c1x , K c1y , K c1z ) are the constraint forces, relative displacements and stiffnesses, respectively, of C 1 in the x, y, and z directions, (F c2x , F c2y , F c2z ), (Δ c2 x, Δ c2 y, Δ c2 z), and (K c2x , K c2y , K c2z ) are the equivalent quantities for C 2 , R, r, Rr, , , and are the geometrical parameters of a single MOB module, which can be obtained by actual measurements, ( (1) , (1) , (1) ) and ( (2) , (2) , (2) ) represent the translational displacements of the two adjacent modules (M 1 and M 2 ) in the x, y, and z directions, and ( (1) , (1) , (1) ) and ( (2) , (2) , (2) ) are the rotational angles of M 1 and M 2 , respectively, around the x, y, and z axes. The theoretical derivations of Eq. (3) and Eq. (4) are provided in Wu (2015) and Wu et al. (2016Wu et al. ( , 2017. Note that C 1 , C 2 , M 1 , and M 2 are generalized concepts, and they refer to the two connectors between two random adjacent modules. Thus, Eq. (3) and Eq. (4) can also be utilized to calculate the constraint forces and relative displacements of two arbitrary connectors, such as C 3 and C 4 , where the two adjacent corresponding modules are M 2 and M 3 .
(2) Algorithm flow for the connectors' relative displacement and constraint force The relative displacements and constraint forces of the flexible connectors for the 108 conditions can be solved by a program based on the principles and formulas described previously. The algorithm is shown in Fig. 3.
(3) Verification of the results Because PFT is considered to be an accurate and reliable numerical method for evaluating the hydrodynamic responses of the MOB structures, the statistics of the constraint force results of the MOB calculated using the proposed algorithm are compared with those from the results of the PFT method under the same conditions (i.e., geometric dimensions of the MOB modules, stiffnesses of the connectors, wave angles, wave spectra, sea states, and shielding effects between multiple components) to verify the accuracy of the proposed algorithm of this paper. The geometric dimensions of the MOB module are shown in Fig. 2, and K 4 (Section 3.2) is selected as the stiffness of the flexible connector. SS6, SS7, SS8 and wave angles of 0°, 45°, 75°, 85°, and 90° are considered.
The single symmetrical composite potential method is used to mesh the elements on the wetted surface of a threemodule hydrodynamic MOB model assuming structural symmetry during the PFT process. The hydrodynamic model consists of quadrilateral elements, and the single MOB module includes 976 elements and 1024 nodes. The added mass, damping, restoring force and wave loads of a single MOB module in the frequency domain are first calculated using the PFT under random and irregular wave stimulation, and then are converted to the time domain using the fast Fourier transform (FFT) method (Zhao et al., 2014). The MOB motion equations in the time domain (Eq. (2)) are solved, and the hydrodynamic responses of the MOB modules are determined. Finally, the hydrodynamic constraint loads of the connectors in the time domain can be obtained using Eqs. (3) and (4). These results for the connectors are based on the PFT.
Connectors C 1 and C 3 are treated as the representatives to show the results using the two methods (the proposed algorithm and PFT) due to the symmetry of the MOB structure. The results for the other connectors (C 2 and C 4 ) are similar to those for C 1 and C 3 . The significant constraint forces for C 1 and C 3 within the simulation time (t=7200 s) are statistically calculated in the x, y, and z directions. Fig. 4 shows a comparison between the computational results of the proposed algorithm and the PFT method. SS6-Simp and SS6-PFT denote the constraint forces obtained using the simplified algorithm and the PFT method, respectively, and the other symbols are similar. As shown in Fig. 4, the magnitudes of the constraint forces on C 1 and C 3 and their variations under different conditions calculated by the proposed algorithm of this paper agree very well with those calculated with the PFT, which validates the methodologies developed in this paper.
The relative errors of the hydrodynamic constraint forces for C 1 and C 3 between the proposed simplified algorithm and the PFT method in different situations are calculated as: denotes the relative error, and F c (Simp) and F c (PFT) are the values of the constraint forces for flexible connectors using the simplified algorithm and PFT, respectively. Fig. 5 shows that the relative errors are all within ±10%, further validates the accuracy, feasibility and reliability of the algorithm developed in this paper.

Constraint forces and relative displacements
In Sections 4.1 and 4.2, the constraint force F c and relative displacement Δ c for the MOB flexible connectors were solved for 108 conditions. The stiffness of the connectors is optimized based on these solutions (F c , Δ c ). Due to the bisymmetry of the MOB structure, C 1 is considered as a representative example to illustrate the stiffness optimization process. The constraint forces and relative displacements of C 1 at SS6 are shown in Fig. 6, and the curves of the relative displacement appear to trend downward when the stiffness reaches K 3 . These mutational points are treated as the relevant characteristics in the following optimization. However, Fig. 3. Flow chart of the algorithm for the connectors' relative displacement and constraint force.  Fig. 6 shows that the original results that relate the constraint force to the stiffness (F c -K) cannot be directly utilized to obtain functions with a good fit due to the increase in the stiffness by 10 orders of magnitude. Therefore, the original values are improved using an ingenious method-"logarithmetics"-in which the log-constraint force is defined as =log 10 (F c ), the log-relative displacement is defined as =log 10 (Δ c ), and the log-stiffness is defined as =log 10 K. For convenience in the following descriptions, the symbol "log 10 " is simply expressed as the "log". The original results for C 1 under different conditions can be recalculated using "logarithmetics". To sum up, Fig. 7 shows the relationships of and for C 1 in the x, y, and z directions with varying wave angles at SS6, SS7 and SS8.
5.2 Specific steps for the optimized stiffness of the connectors Fig. 7 shows that the variations in log(F c ), log(Δ c ) and logK are more significant than those of the original values in Fig. 6. The curves of log(F c ) and log K increase almost linearly. Similarly, the distinct inflection points occur at log K =8 (K 3 ) in the curves that relate log(Δ c ) and logK. The log(Δ cx ) curves have a Z-shape, and log(Δ cx ) decreases significantly in the range of logK = [8,9]; beyond this range, the log(Δ cx ) values are relatively irregular. When logK >8, the values of log(Δ cy ) and log(Δ cz ) all decrease. The original range of the connector stiffnesses can be determined during the optimization process; they are K=10 8 -10 9 N/m (K 3 -K 4 ) in the x direction and K=10 8 -10 11 N/m (K 3 -K 6 ) in the y and z directions. To ensure the reliability of the optimized stiffness results, three additional stiffness values of the flexible connectors in the x direction (approximately 2.5×10 8 , 5×10 8 , and 7.5×10 8 N/m) are added to the fitting functions. In the x direction, five stiffness values within the range of K 3 -K 4 are confirmed: K 3 =1.0×10 8 N/m, K 3.25 =2.5×10 8 N/m, K 3.5 =5×10 8 N/m, K 3.75 =7.5×10 8 N/m and K 4 =1.0×10 9 N/m.  688 WU Lin-jian et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 683-695 The specific steps for the stiffness optimization are described as follows: The linear fitting functions for and should be determined based on different stiffness ranges in the x, y, and z directions, where =log(F c ), =log(Δ c ), and = logK. The fitting function for general conditions is expressed as follows: a 1 and a 2 are the slopes of and , respectively, and b 1 and b 2 are the intercepts. In addition, By combining Eq. (6) with Eq. (7), the optimized stiff-  ness expression for the connectors is: In terms of this derivation process, Fig. 8 shows the fitting expressions and for C 1 in the x, y, and z directions at SS6 for a wave angle of 45°. The results for the other connectors under different conditions are similar to this case. The linear fitting curves in Fig. 8 validate the feasibility of the proposed optimization process and the specific procedures. The optimized stiffness of the flexible connectors can be solved using Eq. (8).

Optimized stiffness results
Following the process described above, the optimized stiffnesses of C 1 -C 4 in the x, y, and z directions can be solved at SS6, SS7, and SS8 for different wave angles. The final results are shown in Table 1. The optimized stiffnesses in the x direction are all within the defined range of K=10 8 -10 9 N/m (K 3 -K 4 ), while in the y and z directions, they are all smaller than 1.0×10 10 N/m (K 5 ). These results validate the optimized stiffnesses obtained using the procedures proposed in this paper.

Percentage reduction of optimized stiffness
To quantify the decreases of the constraint forces and relative displacements of the connectors determined using the optimized stiffnesses K o in Table 1 compared with the results obtained using the original range of the stiffnesses (in the x direction: K 3 -K 4 ; in the y and z directions: K 3 -K 6 ), the percentage reductions of the constraint force and relative displacement of the connectors are defined as: where D o (F c ) and D o (Δ c ) are the percentage reductions of the optimized stiffness K o for the constraint force and the relative displacement of the connectors, respectively, and max[F c (K i )] and max[Δ c (K i )] are the maximum constraint force and relative displacement of the connectors obtained using the original stiffness ranges (K 3 , K 3.25 , K 3.5 , K 3.75 , K 4 in the x direction, and K 3 , K 4 , K 5 , K 6 in the y and z directions), F c (K o ) and Δ c (K o ) denote the constraint force and relative displacement of the connectors that are calculated us-ing the optimized stiffnesses K o in Table 1. Also, C 1 and C 3 are regarded as representatives, and Eq. (9) is used to evaluate the percentage reductions of C 1 and C 3 between K b and the original stiffness range at different sea states and wave angles, as shown in Fig. 9. Fig. 9 illustrates that the optimized stiffness K o can reduce the constraint forces and relative displacements of the connectors by different amounts. In the x direction, the constraint force and relative displacement are reduced by 35%-65% and 20%-45% compared with using the original range of the stiffnesses. In the y direction, the constraint force and relative displacement using K o are reduced by 85%-95% and 30%-70%, respectively, and in the z direction, the constraint force and relative displacement are decreased by 85%-93% and 35%-60%, respectively. These results indicate that K o can reduce the constraint force and relative displacement of the connectors.
6 Optimal connector stiffness 6.1 Method for determining the optimal stiffness Table 1 presents the optimized stiffnesses of the flexible connectors for different sea states and wave angles. The values of K o under different conditions are variable. However, the stiffnesses of the flexible connectors on an MOB should be the same in the x, y, and z directions to more expediently design, produce and install them in actual projects. Thus, it is necessary to obtain the optimal stiffness (K b ) for these connectors in the x, y, and z directions based on the optimized stiffnesses in Table 1. The specific procedures for evaluating the optimal range of the stiffnesses (K br ) of the flexible connectors are as follows. (1) Stiffness range for different wave angles The optimized stiffness ranges of C 1 -C 4 in the x, y, and z directions are statistically determined for different wave angles at SS6, SS7 and SS8. For example, the data in Line 1 of Table 1 represent the optimized stiffnesses of C 1 in the x direction for the wave angles of 0°, 45°, 75°, 80°, 85°, and 90° at SS6. The lower value (LV) and upper value (UV) are 2.23×10 8 N and 3.07×10 8 N, respectively, from which the optimized stiffness range of C 1 in the x direction at SS6 is defined as [2.23×10 8 , 3.07×10 8 ] N/m. The results for the other conditions are similar. The statistical results are reported in the histograms in Figs. 10a-10c.
(2) Stiffness range for different sea states Based on the statistical results from Step (1), the optimized stiffness range of the flexible connectors in the x, y, and z directions at SS6, SS7 and SS8 (SS6-K or , SS7-K or , SS8-K or ) are calculated from the stiffness ranges for different wave angles. For instance, the stiffness range at SS6 in the x direction (SS6-K or -x) is determined as [ 10a-10c for SS6-K or , SS7-K or , and SS8-K or .
(3) Optimal stiffness range The complete optimized stiffness ranges in the x, y, and z directions from Figs. 10a-10c are summarized in the histogram in Fig. 11. The optimal stiffness ranges in the x, y, and z directions are obtained from the range of the optimized stiffnesses at different sea states. directions are similar to that for the x direction. The optimal stiffness ranges for the flexible connectors are obtained for the x, y, and z directions, and the final results are shown in Fig. 11. In summary, the optimal range of the stiffness values of the flexible connectors K b for the MOB in this paper can be obtained using Steps (1), (2), and (3). Fig. 11 shows the results: in the x direction: [2.58×10 8 , 2.67×10 8 ] N/m, in the y direction: [5.36×10 9 , 6.02×10 9 ] N/m, and in the z direction: [6.59×10 9 , 6.84×10 9 ] N/m.

Percentage reduction of the optimal stiffness
Similarly, to quantify the decreases of the constraint force and relative displacement of the connectors using the  Fig. 11. Optimal stiffness ranges for connectors K b in the x, y, and z directions. optimal stiffness K b compared with the results calculated using the original stiffness range, the percentage reductions of the constraint force and relative displacement are: where D b (F c ) and D b (Δ c ) are the percentage reductions of the constraint force and relative displacement of the connectors using the optimal stiffness K b , F c (K b ) and Δ c (K b ) denote the constraint force and relative displacement of the connectors evaluated using the optimal stiffness K b , and K b is the upper or lower value of the optimal stiffness range, show that using the optimal stiffness K b can reduce the constraint force and relative displacement of the connectors by different amounts.
(1) For K b (UV), the constraint force and relative displacement in the x direction are reduced by 16%-63% and 13%-52% compared with the results obtained by using the original range of the stiffnesses. In the y direction, the constraint force and relative displacement results using K b are reduced by 88%-93% and 30%-71%. In the z direction, the constraint force and relative displacement are decreased by 79%-96% and 15%-79%. For K b (LV), in the x direction, the constraint force and relative displacement are reduced by 18%-63% and 15%-51% compared with the results obtained by using the original range. In the y direction, the constraint force and relative displacement are reduced by 89%-93% and 26%-68%, and in the z direction, the results of the constraint force and relative displacement are decreased by 79%-96% and 15%-79%, respectively.
(2) The results of K b (UV) and K b (LV) show that the ranges of the percentage reductions of the constraint force and relative displacement for the connectors are similar in the x, y, and z directions due to the approximate values of K b (UV) and K b (LV). Therefore, regardless of whether the upper or lower values of K b are used, the constraint forces and relative displacements of the connectors all decrease by different amounts.
(3) The calculated results of D b show that the optimal stiffness K b evaluated by this paper's optimization method can reduce the constraint force and relative displacement of the connectors by different amounts within the original range of the stiffnesses. Additionally, the ranges of K b in the x, y, and z directions all remain the same to more expediently design, produce and install the connectors in actual projects.

Conclusions
This paper investigates a simplified method to determ- Fig. 12. Percentage reductions of the constraint forces and relative displacements of C 1 and C 3 between K b (UV) and the original range of stiffnesses.
ine the optimal stiffness of the flexible connectors in a three-module MOB model based on the constraint forces and relative displacements of the connectors at rough sea states with different wave angles. The optimized stiffnesses (K o ) for the flexible connectors under different conditions are obtained based on the relationships between the constraint force, relative displacement and stiffness using "logarithmetics" ( and ) in combination with Hooke's law for the flexible connectors. The ranges of optimal stiffnesses (K b ) for the flexible connectors in the x, y, and z directions are solved based on statistical combinations and intersections of the optimized stiffnesses.
The percentage reductions of the optimized and optimal stiffness of the flexible connectors were defined to quantify the decreases of the constraint force and relative displacement of the MOB connectors compared with the results obtained by using the original range of the stiffnesses. These parameters indicate the accuracy and feasibility of the proposed method for determining the optimal stiffness of the flexible connectors and the rationality and practicability of the optimal stiffnesses results. Ultimately, the range of optimal stiffnesses for the flexible connectors determined by this paper's method is significant for the design, production and installation of the connectors in actual projects, and the research ideas and implementation procedures provide technical support for the optimization of the connectors in similar structures.