The global energy sector is transitioning towards renewable sources, driven primarily by the urgent need to address environmental concerns and reduce carbon emissions. This shift is particularly evident when adopting renewable energy technologies, with wind power emerging as a leading alternative to conventional fossil fuels because of its minimal environmental impact and sustainability (Greiner et al., 2014; Mohamad et al., 2023). Offshore wind technology, especially for the floating offshore wind turbine (FOWT), represents a substantial advance over traditional fixed-type wind turbine installations, offering unique advantages in harnessing wind energy in deeper waters where wind resources are typically more abundant (Firouzi et al., 2022). FOWTs are designed to operate in deeper waters, which allows them to access more consistent wind resources than fixed-type wind turbines (Viselli et al., 2015). Deploying these turbines at greater distances from shore enhances the energy production potential and alleviates the concerns related to noise pollution and visual impact, making them more acceptable to coastal communities (Ma et al., 2014). On the other hand, the offshore environment presents distinct engineering challenges, particularly the complex interactions of wave and wind loads acting on the FOWT rather than fixed-type platforms (Lackner and Rotea, 2011).

Conventional approaches to assessing the structural performance and safety of FOWTs rely primarily on time-domain simulations. These methods are computationally demanding because they must account for intricate and time-dependent interactions between the structural dynamics and environmental loads (Robertson et al., 2014). According to IEC 61400-3, one-hour Integrated Load Analyses (ILAs) should be performed for at least six random seeds (IEC, 2009). On the other hand, at least three hours of simulations are needed to achieve statistically reliable results for moored structures under a short-term sea state. A long-term assessment of the floating structure is necessary to ensure the fatigue performance of mooring lines. The American Petroleum Institute (API) suggests running simulations spanning five to ten three-hour durations with varying seeds for floating systems (API, 2005). Specifically, in the time-domain approach, a single one-hour simulation typically requires 24 hours of computation time (Karimirad and Moan, 2012; Lerch et al., 2018).

Researchers have explored alternative analysis methods to address the computational challenges of time-domain simulations. Frequency response analysis using numerical tools has become a promising approach. Hydrodynamic analysis tools, such as ANSYS-AQWA and WAMIT, are used widely to analyze wave-induced motions and forces on floating platforms (ANSYS Inc., 2020; WAMIT Inc., 2023). Building upon these tools, frequency response hydrodynamic analysis was conducted using the commercial potential flow software ANSYS-AQWA to determine the acceleration response amplitude operator (RAO) (Park and Choung, 2023). This analysis aimed to predict the dynamic responses of the structure under various wave conditions, evaluating the effects of wave frequencies and incident angles to provide critical data on the motion and acceleration characteristics of the FOWT. This study excluded the hydrodynamic pressure acting on the structure and did not fully consider the coupling effects between the aerodynamic and hydrodynamic loads.

In other studies, the hydrodynamic pressure effects were examined in greater detail. For example, a fully coupled time-domain structural analysis was applied to evaluate the combined effects of aeroelastic, hydrodynamic, and inertial loadings on FOWTs. This paper highlights the computational intensity of this approach and proposes a lodal response-based method to enhance the computational efficiency significantly (Lee et al., 2023). Similarly, a response-based time-domain analysis method was proposed to synthesize the stress responses for fatigue damage evaluation, successfully addressing the loadings from waves, wind, and mooring systems (Kyoung et al., 2019, 2020). Furthermore, an advanced methodology that combined pseudo-spectral response synthesis with pre-processed lodal responses was introduced to enhance the efficiency of time-domain structural assessments. This approach revealed its utility in buckling and strength assessments for FOWTs, significantly reducing computational time (Moon et al., 2022). On the other hand, these studies calculated the radiation pressure using the six-degree of freedom (DOF) modes of motion results from WAMIT rather than adopting a single transfer function approach. In addition, they considered concentrated loads, such as aerodynamic forces and mooring line tensions, in a time-series format, which increased the computational time and resources.

This paper presents a novel approach to evaluate the stress responses of the FOWT in the frequency domain, giving a solution to overcome the above limitations. The hydrodynamic solver examined in this study included pressure calculation modules for the frequency and time domains, where frequency-domain pressure was computed as a transfer function and time-domain pressure was determined through the impulse response function (IRF) Cummins (1962) suggested. The proposed methodology enabled calculations of the motion responses, hydrodynamic pressures, fairlead tensions, and rotor thrust in the frequency domain, significantly reducing the computational requirements while maintaining accuracy. The accuracy of the calculated motion responses, fairlead tensions, and rotor thrust was validated by a comparison with the OpenFAST simulation results. Furthermore, the proposed methodology was verified using the commercial finite element analysis software ABAQUS through comparative validation against conventional structural analysis approaches in the time and frequency domains. The incident and diffraction pressures were calculated based on wave trains, while the radiation pressures were derived from structural motion responses. Structural analysis was conducted using transfer functions representing incident, diffraction, and radiation pressures. In addition, the transfer function for radiation pressure was calculated by distinguishing the effects of waves and wind.

The approach can be simplified for conventional offshore structures where wave forces dominate environmental loading to focus exclusively on wave-induced responses. For FOWTs, however, where wind forces are predominant, the methodology incorporates the effects of radiation pressure to derive comprehensive structural responses. The significance of this methodology lies in its ability to enhance computational efficiency while maintaining engineering accuracy in FOWT structural analysis. This approach can be extended to fatigue analysis and initial design stage analysis of FOWTs, providing a versatile framework applicable to various offshore installations.

The subsequent analyses focus on panels near the waterline, where hydrodynamic effects are most dominant. This analytical approach is based on two fundamental hydrodynamic principles. First, the incident and diffraction pressures induced by waves are most prominent near the surface, as the orbital motion of water particles attenuates with depth. Second, the radiation pressure, resulting from structural motion, reaches its peak magnitude near the waterline because of the maximum moment arm from the center of gravity. Therefore, the structural analysis results are presented for these waterline regions. Fig. 10 shows the stress distribution across all structural and the locations of selected elements for structural analysis. The stress is distributed unevenly across the elements (Fig. 10(a)), with significant stress concentrations observed particularly at the waterline. The thickness of the structure was 0.0372 m, and the ballast was 49.416 m. The steel material properties were used with Young’s modulus E = 206 GPa, density ρ = 7,850 kg/m³, and Poisson’s ratio ν = 0.3. The spar was modeled using 4,128 finite shell elements. A coarse mesh modeling approach was adopted instead of fine mesh analysis to validate the proposed methodology.

The proposed method analyzed the model across 180 discrete frequencies, ranging from 0.01 to 1.8 rad/s, with 0.01 rad/s increments. The conventional method used a time series of 10,800 seconds (three-hour duration) for the simulation. Fig. 11 shows the validation process comparing the proposed and conventional methods. The incident and diffraction pressure transfer functions were derived from the first term of Eq. (2), while their corresponding pressure time series were calculated using Eq. (4). In the proposed methodology, stress transfer functions are obtained through structural analysis of incident and diffraction pressure transfer functions. The stress spectrum was then calculated by multiplying these transfer functions with the wave spectrum of Case #11 from Table 4. In contrast, the conventional methodology conducts structural analysis on the pressure time series to derive the stress time series. These stress time series are subsequently converted to the stress spectrum. Fig. 11(a) presents the verification process, comparing the incident and diffraction stress spectrum obtained from the two methodologies. The validation process for the radiation stress spectrum in Fig. 11(b) followed a similar procedure, with the additional step of motion calculations. Figs. 6 and 9 present the motion RAO induced by wave and wind, respectively. The motion results in the time domain were obtained using OpenFAST. The radiation pressure transfer functions were obtained from the second term of Eq. (2), and their corresponding pressure time series were calculated using Eq. (6). The radiation stress spectrum was obtained by multiplying the transfer functions with their corresponding environmental spectrum. For the environmental effects, the wave spectrum was used for wave-induced radiation stress, while the wind spectrum was applied for wind-induced radiation stress.

Figs. 12 and 13 show the pressure transfer functions and corresponding time histories for incident and diffraction, wave-induced radiation, and wind-induced radiation, respectively.

Figs. 14 and 15 present the stress responses derived from structural analysis as transfer functions and time histories, respectively.

The stress spectrum was obtained by multiplying the stress transfer functions with the spectrum from Case #11 in Table 4. The total stress spectrum (Fig. 16) combines the incident and diffraction, wave-induced radiation, and wind-induced radiation components calculated using the methodologies presented in Fig. 11.

The total stress spectrum analysis between the proposed and conventional methodologies showed agreement, with area-based spectral differences of 2%. The previous results were derived from the values at the waterline. On the other hand, additional validation was performed at other positions because wave excitation forces, motions, damping, and added mass vary depending on the location. Three panels were selected along the vertical length of the spar platform (panels #1, #2, and #3), as shown in Fig. 17; Fig. 18 presents the corresponding stress spectra.

The stress spectrum at different locations obtained using the proposed methodology quantitatively showed agreement with those from the conventional approach. On the other hand, the differences in peak values between the conventional and proposed methods arise from the limited consideration of nonlinear effects because the proposed method analyzes the linearized wave exciting forces, rotor thrust, and structural motions.

This study developed and validated frequency-domain structural analysis methodology by linearizing the transfer functions of the rotor thrust, mooring line tension, and structural motion. The effectiveness of the proposed approach was validated through numerical analyses of the OC3-Hywind spar platform. The rotor thrust was linearized using a probabilistic approach by decoupling the wind spectrum influence from thrust data obtained from OpenFAST. The transfer function of the rotor thrust was derived specifically in the surge direction. For mooring line tension, linearization was implemented by applying the static offset test to obtain the appropriate coefficients. The structural motion transfer function, expressed as a motion RAO, was constructed using these coefficients and validated against OpenFAST simulation results. The natural frequency differences for the wave-induced motion RAO were 2%, 4%, and 1% in the surge, heave, and pitch modes, respectively. The wind-induced motion RAO exhibited 2% differences in the surge and pitch modes.

The developed frequency-domain pressure module was used to calculate the transfer functions of the incident, diffraction, and radiation pressures. These transfer functions were then applied as inputs to a commercial structural analysis program, ABAQUS, to calculate the corresponding stress transfer functions. The stress spectra were then derived by post-processing the stress transfer functions with environmental loading conditions. For validation purposes, the conventional methodology calculates the pressures using the developed time-domain pressure module, with motion time series derived from OpenFAST. Similarly, the stress time series were obtained through ABAQUS and transformed into spectral form for validation and comparison.

Comparative analysis between the conventional and proposed methodologies showed agreement for total stresses, showing differences of 2%. A similar agreement was observed at other locations, with less than 4% differences. The proposed methodology showed significant improvements in computational efficiency. The analysis time was reduced from 12 hours to one hour, representing a 92% reduction in computational costs. In addition, this approach provides substantial cost advantages by enabling the derivation of rotor thrust, mooring line tension, and structural motion transfer functions without reliance on commercial software, enhancing its scalability and applicability for large-scale analyses.

The frequency-domain structural analysis methodology is particularly effective for fatigue and initial design stage analysis of ships and offshore structures in wave-dominated environments. For FOWTs, where the wave and wind effects are significant, this methodology is highly effective for initial design stage analysis and proves efficient for fatigue analyses involving multiple case studies. Nevertheless, as with other frequency-domain approaches, this methodology inherently cannot account for nonlinear dynamic effects by linearizing wave exciting forces, rotor thrust, and structural motions. Accordingly, this methodology provides a framework for advancing the structural analysis and design of FOWTs by addressing the challenges associated with complex environmental loads while ensuring high accuracy and efficiency.