Floating solar platforms have recently been evaluated as eco-electric plants in ocean engineering areas. Many engineers have tried the multi-body type with connectors for the floating solar platform instead of the whole ship type because it reduces costs. In addition, the design feasibility of multi-body solar platforms has been studied widely (Hong et al., 2018; Kim et al., 2020; Xu and Wellen, 2022; Song et al., 2023; Kim and Lee, 2024). Time domain analysis to calculate the stress and fatigue life of floating body connectors in waves is a key step in the hydrodynamic design of multi-body floating solar platforms. Generally, those systems have many floating bodies, connectors, and mooring lines, and hydrodynamic analysis will require considerable computation effort. For example, the 3.3 MW floating solar platform in Fig. 1 has 990 floating bodies, 1320 H-beam connectors, and 252 mooring lines. In that case, time domain hydrodynamic analysis of floating bodies, connectors, and mooring lines will be unrealistic if all the connectors are modeled using shell elements. A total beam analysis instead of a total shell analysis will be a more practical method. Nevertheless, the shell effects, such as the stress concentration at the fixed part, cannot be obtained by a beam analysis because the stress distribution from the loading point to the fixed point is linear according to linear beam theory. On the other hand, the real stress distribution is exponential around the fixed parts, and shell analysis can catch the nonlinear distribution. This paper proposes a modified procedure for the total beam analysis by adding local shell analysis steps. The time domain connector stresses were calculated for the total system with a beam model, and the target connector with maximum stress and section forces were determined. Local shell analysis was conducted by applying section forces to the target connector, and the revision factor was obtained by comparing the maximum local shell stress and maximum beam stress. The time series of the stress was modified by the revision factor, and the fatigue life of the connector was calculated from the revised time series.

The two methods were compared by analyzing a 3.3 MW floating solar platform with 990 floating bodies, 1320 H-beam connectors, and 252 polyester mooring lines as a numerical example, as shown in Fig. 1. The target site of the example structure was the Saemangeum area in South Korea, and the significant wave heights for extreme and mild waves were H_s =1.5 m and 0.74 m, respectively. The modal period was T_p = 4 s. The wave forces at floating bodies, added mass, and hydrodynamic damping were calculated using the higher-order boundary element method (HOBEM) (Choi et al., 2001; Hong et al., 2005). Total beam analysis, local shell analysis, and mooring line analysis were conducted using the finite element method (FEM) (Donea and Lamain, 1987; Allman, 1988; Kim and Kim, 2019). The fatigue evaluation is a key step in the design of ocean structures, and many studies have been done on the fatigue of ocean structures. Blagojevic and Domazet (2002) presented a simplified procedure for ship fatigue assessment. Bhowmik (2019) and Repalle et al. (2020) applied machine learning to evaluate the fatigue life of the jacket or riser. Kim et al. (2022) proposed an average sea-state method for a rapid fatigue evaluation and applied it to the sea-fastening structure of a deck transportation ship. Two-dimensional horizontal motion analysis of two floaters was performed considering the nonlinear behavior of internal flow using the higher-order multi-modal method performed by Kim and Choi (2023). The fatigue damage and life of the 3.3 MW floating solar system in this study were also computed using the rain-flow algorithm (Downig and Socie, 1982; Singh and Ranganath, 2007). The stresses and fatigue lives of the time domain connectors were calculated using the proposed total beam and local shell method. The results were compared with those using only a total beam analysis. The comparisons were made by changing the occurrence probability of extreme and mild waves to evaluate the differences between the two methods according to the changes in sea state.

This section summarizes the proposed procedures for the total beam and local shell analyses to calculate the time-domain stress and fatigue life of floating multi-body connectors. The first step in the proposed method was to set up beam models for the connectors of the total system and solve the following equation of motion for floating bodies, beams, and mooring lines to obtain the time domain connector stresses in waves.

where [M ], [C ] and [K ] are the mass, damping and stiffness matrices for beam models for connectors and mooring lines. They were formulated using the FEM beam elements (Kim and Kim, 2019). {u} is the displacements vector of beams and mooring lines. [M_B] and [K_B] are the matrices for mass and restoring force coefficients of floating bodies. [M_add], [C_B] and {f_wave(t)} are the added mass matrix, hydro-dynamic damping matrix and wave force vector at floating bodies which were formulated using the HOBEM (Choi et al., 2001; Hong et al., 2005). The time domain beam stresses and section forces for the connectors were obtained by solving Eq. (1). The Modified Newmark method (Chung and Hulbert, 1993) was applied for time marching.

The second step was to obtain the stresses and section forces of the connectors from the solution of Eq. (1) and to find the time series of stress, σ(t), at the target connector where the stress is the largest. The third step was to choose the maximum stress, σ_beam from σ(t) and to find section forces that induce σ_beam.

The fourth step was a local shell analysis of the target connector by applying the section forces in step 3 to the local shell model by solving the following equation:

where [K_shell] is the stiffness matrix of the local shell model and it was formulated using the FEM shell elements (Donea and Lamain, 1987; Allman, 1988). {u_shell } is the displacement vector of the local shell. {f_section } is a force vector by section forces. Then, maximum shell stress, σ_shell was obtained at a fixed part.

The fifth step was to calculate the revision factor as follows:

where α is the revision factor.

The sixth step was to revise the time series of stress at target connector stress in step 2 by

where σ_revised (t) is the revised time series of stress at the target connector.

The last step was to calculate the fatigue damage and life from σ_revised (t). The fatigue life was calculated using the following equations:

where L is the fatigue life in years and D is the total damage. d_ij is the damage for S_ij. S_ij is the i^th stress range in the j^th sea-state. N_S is the number of stress ranges and N_state is the number of sea-states. n_ij is the number of cycles of S_ij per year. p_j is the probability of the j^th sea-state. T_d is the time duration of the stress signal. The N_ij is the number of cycles to failure by S_ij. K is the intercept of the S-N curve of the connector material and M is the slope of the S-N curve. S_ij^j and N_S were obtained from data analysis for σ_revised (t) using the rain-flow algorithm (Downig and Socie, 1982; Singh and Ranganath, 2007). Steps 3–6 were skipped if a total beam analysis alone was applied. Table 1 lists the steps for the total beam analysis and the total beam and local shell analyses.

A 3.3 MW floating solar platform designed by the Korea Research Institute of Ships & Ocean Engineering (KRISO) was analyzed as a numerical example to compare the total beam analysis method and the total beam and local shell analyses method. Time domain connector stresses and fatigue lives using those two methods were analyzed, and their results were compared. Fig. 1 shows the geometric shapes of the example floating solar platform. The total breadth and length of the example structure were 198 m × 180 m, respectively. The draft was 1.005 m, and the total weight was 3034.51 t. The model was composed of 110 units, with each unit containing nine floaters and 12 connectors. Hence, the total structure consisted of 990 floating bodies and 1320 connectors. The units were connected with a hinge link. The hinge link was modeled numerically as follows. A degree of freedom (DOF) control method for the internal hinge link (Arbabi, 1991) was applied. In the method, two nodal points were assigned at the same hinge point. In addition, one slavery nodal point followed the same equation numbers as other master nodal points for the translational DOF. Nevertheless, the rotational DOF at the common nodes had different equation numbers. Hence, translational motions moved together at the hinge point, and the rotational motions were released. Therefore, the moments were also released at the hinge point. The floating body connector was an H-beam made of aluminum alloy 6082. Two hundred and fifty-two mooring lines were attached to the platform, and the material for the mooring line was polyester rope. Table 2 lists the design properties. The desired target site for the example structure was the Saemangeum area in South Korea. Table 3 lists the design wave conditions. Irregular waves for mild and extreme cases were considered.

The numerical analyses were conducted based on the procedure in section 2. Figs. 2,–8 and Tables 4–5 present the results. Fig. 2 shows the numerical model for the time domain total beam analysis. The total number of HOBEM nodes at the floating bodies was 25,740, and the number of FEM nodes for the beams and mooring lines was 35,442. Wave period ranges of T = 1.6 to 60 s were considered in the JONSWAP spectrum for the irregular calculation in this study. Therefore, the shortest wavelength was λ = 4 m at T = 1.6 s. In addition, the chosen mesh resolution in the floating body was ΔL < λ/4, which covers at least the shortest wavelength. Fig. 3 presents snapshots of the deformed shape of the example structure by the total beam analysis. Fig. 4 shows the time series of the stresses at the target connectors by the total beam analysis. The maximum stresses were chosen from the time series. Table 4 lists the results with the section forces that induce maximum stresses. The local shell analyses for the target connectors were conducted using the section forces in Table 4. Fig. 5 shows the FEM model for the local shell analysis. The connection part between the floating body and the connector was a fixed end. The connection part on the other side is a loading region for section forces. The number of FEM shell nodes was 12,261. Figs. 6 and 7 show the local shell analysis results. Fig. 6 shows the stress contour, and Fig. 7 presents the longitudinal distribution of the shell stresses and a comparison with the beam results. The stress concentration is shown at the fixed end (x = 0 m) in the local shell analysis. The stresses obtained from the beam and shell analyses were different at the point. The maximum shell stresses were chosen from the local shell analyses. Fig. 8 and Table 5 summarize the results. The revision factors were derived by comparing the maximum beam and shell stresses, and the results are summarized in Table 5. The revision factors were α = 1.513 to 1.729. The connector stresses by beam analysis were modified from the revision factors; Fig. 9 presents the revised time series.

The fatigue damage and life of the connector were obtained by analyzing the revised time series. In fatigue analysis, the S–N curve for the connector was chosen from the aluminum design manual by the Aluminum Association (2010), as shown in Fig. 10. The H-beam was assumed to be made from extrusion without weld joints. Therefore, the category was assumed to be A in the aluminum S–N curve. The fatigue analyses were done by changing the occurrence probability of extreme waves to compare the fatigue lives using the total beam analysis method and those by the total beam and local shell analyses method for various sea-states. Tables 6–10 list the results. The fatigue lives by the total beam analysis were 30–1533 years and 1–47 years from the total beam and local shell analyses. The fatigue lives by the two methods were different, as shown graphically in Fig. 11. The fatigue life changed radically for the wave condition because the fatigue damage takes the powered form of stress as Eq. (9). Therefore, the situation will be similar in other sea sites. The radical change in fatigue damage for wave conditions can also be found elsewhere (Kim et al., 2022). The design life of the example structure was 25 years because the life of a solar panel was 25 years. In sea states with a 2% probability of extreme waves, the fatigue lives using two methods were greater than the design life, and the check for fatigue safety was acceptable for both methods. On the other hand, the fatigue lives by the total beam analysis were greater than the design life, and those by the total beam and local shell analyses were less than the design life when the probability of extreme waves was larger than 5%. In that case, the checks of the former and latter indicated acceptable and unacceptable, respectively. The safety check was different in the two methods. Therefore, the incorrect check for fatigue safety may be derived from beam analysis because beam analysis alone is insufficient in the stress calculation for fatigue damage. Therefore, the revision process, such as adding local shell analysis proposed in this study, appears mandatory in evaluating the time domain stress or fatigue life of the floating multi-body beam connectors. Table 11 lists the CPU times for the total beam analysis and the total beam and local shell analysis. The computation for local shell analysis was very small compared to the total time domain beam analysis. Therefore, the total CPU times of the two methods were similar.

This paper proposed a total beam and local shell analysis procedure to calculate the time-domain stress and fatigue life for the connectors of floating multi-bodies and compared the method with the total beam analysis alone. The procedure was summarized as follows. Time domain hydrodynamic analysis for floating bodies, connectors, and mooring lines in waves was conducted using the HOBEM and FEM with a total beam model for the connectors, and the time series of stress at the target connector was obtained. The maximum beam stress and section forces were chosen. FEM analysis of the local shell model for the target connector was conducted by applying the section forces to the connector, and the maximum shell stress was obtained. The revision factor was calculated from the maximum beam and shell stresses. The time series of the connector stress was modified using the revision factor, and the fatigue damage and life were obtained by analyzing the revised time stress using the rain-flow algorithm.

A 3.3 MW floating solar platform with 990 floating bodies, 1320 H-beam connectors, and 252 polyester mooring lines was analyzed as a numerical example, and the time domain connector stress and fatigue life determined using the total beam analysis method and the total beam and local shell analyses method were compared. The significant wave heights were 0.74 m and 1.5 m for mild and extreme waves, respectively. The analyses were conducted by changing the probability of extreme waves from 2% to 100% to compare the two methods for various sea-states. The maximum stresses by the local shell analysis were larger than the maximum beam stresses because of the stress concentration effect at the fixed parts, and the revision factors were 1.513–1.729.

Fatigue lives by the total beam analysis were 30–1533 years, and those by total beam and local shell analyses were 1–47 years. The fatigue lives by the two methods were different. In sea states with a 2% probability of extreme waves, the fatigue lives determined using the two methods were greater than the design life of 25 years. The check for fatigue safety was acceptable in both methods. On the other hand, the check for fatigue safety was different for the total beam analysis and the total beam and local shell analyses when the probability of an extreme wave was larger than 5% because the fatigue life by the total beam analysis was greater than the design life. In contrast, it was less than the design life in the total beam and local shell analyses. Therefore, the incorrect check for fatigue safety was derived from a beam analysis in more extreme cases. The beam analysis alone appeared insufficient in the stress calculation for fatigue damage. Therefore, a revision process, such as the local shell analysis proposed in this paper, is needed to evaluate the time domain stress or fatigue life for the floating multi-body beam connectors.