1 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 Sustainable Marine Structures https://ojs.nassg.org/index.php/sms Copyright © 2022 by the author(s). Published by Nan Yang Academy of Sciences Pte Ltd. This is an open access article under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) License. (https://creativecommons.org/licenses/ by-nc/4.0/). *Corresponding Author: Chungkuk Jin, Department of Ocean Engineering, Texas A&M University, Texas, the United States; Email: jinch999@tamu.edu DOI: http://dx.doi.org/10.36956/sms.v4i2.492 ARTICLE Comparison of Potential Theory and Morison Equation for Deformable Horizontal Cylinders Chungkuk Jin* Department of Ocean Engineering, Texas A&M University, Texas, the United States ARTICLE INFO ABSTRACT Article history Received: 18 February 2022 Accepted: 10 March 2022 Published Online: 30 March 2022 This study investigates the hydro-elastic behaviors of fully submerged horizontal cylinders in different regular waves. Two methods were proposed and compared in this study. The first method was based on potential theory in frequency domain and the discrete-module-beam (DMB) method, which discretizes a floating elastic structure into a sufficient number of rigid bodies while simultaneously representing the elastic behavior from beam elements with Euler-Bernoulli beam and Saint-Venant torsion. Moreover, the Morison method in time domain was employed; this method estimates wave forces from the semi-empirical Morison equation, and the elastic behavior is embodied by massless axial, bending, and torsional springs. Various parametric studies on cylinder diameter, submergence depth, and wave direction were conducted. Wave forces, dry/wet mode shapes/natural frequencies, and dynamic motions are presented and analyzed. Keywords: Wave force Discrete-module-beam method Potential theory Morison equation Horizontal cylinder 1. Introduction Underwater cylindrical structures have been suggest- ed for various reasons. Many of them lie on the seabed, and typical examples include submarine pipelines that transport water, oil, and natural gas, submarine cables for electricity transmission, and immersed tunnels for public transportation. Even if these structures have accumulated engineering practices, several critical limitations still exist. These structures are susceptible to seismic activities [1,2]; the seabed should be even for installation; scours, the re- moval of sediment around the pipeline, can occur and are critical for vibration and fatigue damage associated with seabed movement [3]. Also, the deepwater application is very challenging for submarine pipelines and cables due to not only high hydrostatic pressure and temperature dif- ference but also high costs associated with a longer length and additional intervention work for the seabed [4]. Submerged floating structures have been proposed as novel alternatives to the above structures, and repre- sentative examples are submerged floating pipelines and submerged floating tunnels. These structures float at a submergence depth and keep their location with mooring 2 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 systems if needed. They have several advantages: (1) they can be safe from wave and seismic actions by properly selecting design parameters, (2) the deepwater applica- tion is much more straightforward without considerable hydrostatic pressure and temperature difference, as in the deepwater location, and (3) structural health monitoring is much easier [4-9]. Their considerable dynamic and structural behavior and the resulting large mooring tension are still consid- ered severe under various environmental conditions such as waves, currents, earthquakes, and tsunamis. Many stud- ies adopted the Morison equation [10] for force estimation of the horizontal cylinder [11-13] due to low computational cost. However, the Morison equation has limitations in that wavelength should be more than five times the char- acteristic length ( / 0.2D L ≤ where D is cylinder diameter and L is wavelength), and the radiation effect is neglect- ed. In addition, when the structure is close to free surface, the free-surface effect can be substantial, which results in a considerable variation of the scattering wave force compared with the deeply submerged structure cases [14]. The resulting mooring tension can also be different to a great extent. In this regard, validation work of the Mor- ison equation for fully submerged floating structures is essential with respect to different diameters, submergence depths, and wave directions. Several papers discussed the validity of the Morison equation by comparing it with theory-based approaches. Typically, the inertia coefficient MC is fixed at 2 accord- ing to the slender body assumption and can vary for the non-slender body. For example, Chakrabarti and Tam [15] estimated the effective inertia coefficient for the Morison equation for bottom-mounted surface-piecing vertical cylinder and proved that the effective inertia coefficient significantly decreases as /D L increases. Chung [16] stat- ed that when an object is close to free surface, the inertia coefficient can be a function of frequency, and radiation damping can be empirically added. In other words, its validity may be weakened when the free-surface effect exists to a great degree. Chang et al. [17] showed that the Morison equation reasonably estimates wave force up to second order for the hinged vertical cylinder at small wave steepness. However, they also mentioned that the Morison equation underestimates the peak wave force and overestimates trough wave forces at high wave steepness. Varying inertia coefficients along the submergence depth are proposed as in Ref. [18]. Regarding horizontal cylin- ders, Li et al. [19] conducted experimental studies for in- line responses for a submerged horizontal cylinder and showed that the inertia coefficient is not changing signifi- cantly with respect to the Keulegan-Carpenter number as opposed to vertical cylinders. Chen et al. [20] compared the Morison equation with CFD simulation and showed that the Morison equation underestimates the wave force on a partially submerged horizontal cylinder by up to 50% rel- ative errors. This study compared the potential theory with the Mori- son equation for fully-submerged horizontal cylinders. Both ends were fixed with the fixed-fixed boundary condition in which there are no displacements and angles at both ends. The multibody-based hydro-elasticity method with the po- tential theory in frequency domain—referred to as the dis- crete-module-beam (DMB) method—was compared with the lumped mass method with Morison equation—referred to as the Morison method. The Froude-Krylov, scattering wave, and radiation damping forces in the DMB method were com- pared with the inertia force of the Morison equation in the Morison method. Dry/wet natural frequencies/mode shapes and horizontal/vertical motions/forces were systematically presented and analyzed. 2. Theory and Formulation Two different approaches were selected to evaluate dynamic behaviors and wave forces for a fully submerged horizontal cylinder. The first method was the DMB meth- od, in which wave forces are estimated by 3-dimensional (3D) potential theory in frequency domain and Euler-Ber- noulli beam and Saint-Venant torsion are employed for representing elastic behaviors. The second method is the Morison method, in which the Morison equation estimates wave forces with representative added mass and inertia coefficients in time domain while beam elements are modeled by the lumped mass method with massless axial, bending, and torsional springs. The theory and formula- tion of the two methods are explained in this section. De- tailed formulations regarding the two methods and their validations can be found in Refs [12,21,22]. 2.1 Discrete-Module-Beam Method in Frequency Domain The DMB method is based on 3D potential theory for the multibody in frequency domain; a large deformable structure consists of M rigid bodies and 1M − connecting beams [23,24], as shown in Figure 1. The DMB method uses the right-handed Cartesian coordinate system. Figure 1. DMB method [25]. 3 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 Laplace equation is governing equation for the poten- tial theory when fluid is incompressible, irrotational, and inviscid as: 2 0 i te ωϕ − ∇ Φ = Φ = (1) where Φ is the total time-dependent velocity potential, which can be decomposed into the total time-independent velocity potential ϕ and the time-dependent term i te ω− based on the assumption of harmonic excitation with wave angular frequency ω and time t . ϕ can further be decom- posed into diffraction and radiation components with the incident, scattered, and radiation potentials Iϕ , Sϕ , and ( )m Rϕ as: ( ) ( ) ( ) ( ) 1 6 1 M m I S R m m m m jR jR j i ϕ ϕ ϕ ϕ ϕ ω ξ ϕ = = = + + = − ∑ ∑ (2) where m represents the mth rigid body and ( )mjξ is six de- grees of freedom (6DOF) displacements of the mth rigid body, i.e., surge, sway, heave, roll, pitch, and yaw mo- tions. First-order Iϕ can be written as: ( ) ( )cos sincosh cosh ik x yw I k z HigA e kH θ θϕ ω ++= (3) where wA is the amplitude of an incident wave, k is the wavenumber, H is water depth, θ is the wave direction, and g is the gravity acceleration. Moreover, Sϕ and ( )mRϕ can be obtained by considering the following bounda- ry conditions on the free surface FS , bottom BS , body ( )1, 2, ... ,kS k M= , and far-field S∞ as: ( ) ( ) ( ) ( ) ( ) 2 0 on 0 0 on on , on 0, lim 0 on F B S I k m m jR j kk r g S z z S z H z S n n n k m S n k m r ik S r ω ϕ ϕ ϕ φ ϕ ϕ × × × ∞ →∞ ∂  − + = = ∂  ∂ = = − ∂ ∂ ∂ = − ∂ ∂ ∂ = =  ∂ ≠ ∂  − = ∂  (4) where r is the radial distance from the origin. ϕ × can be Sϕ or ( )m jRϕ . ( )mjn is the inward unit normal vector for the j th DOF on the m th body surface with respect to the body- fixed coordinate system. Sϕ and ( )m jRϕ are obtained by the 3D boundary element method. Each body’s coordinate center is its center of gravity. The wave excitation force that is the sum of the Froude-Krylov and scattered wave forces for the j th DOF on the m th rigid body in the body-fixed coordinate system is given in Equation (5): ( ) ( ) ( ) m m m I S jjW S F i n dSρω ϕ ϕ= +∫∫ (5) The radiation force for the j th DOF on the m th rigid body induced by the k th DOF of the n th rigid body in the body-fixed coordinate system is given in Equation (6): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 m m mn mn n mn n n n jR jkR jk k jk k k kR m S F A i B dS n ϕ ω ξ ω ξ ρω ξ ϕ ∂ = + = ∂∫∫ (6) where ( )mnjkA and ( ) mn jkB denote the added mass and radiation damping coefficients of the j th DOF of the m th rigid body induced by the k th DOF of the n th rigid body. In addition, the hydrostatic restoring force of the m th rigid body for the j th DOF is given in Equation (7): ( ) ( ) ( ) ( )( ) ( )53 4 m m m m m m m m jjH S F g y x n dSρ ξ ξ ξ= − + −∫∫ (7) After the hydrodynamic forces and coefficients have been introduced, the equation of motion for M rigid bod- ies can be written with the mass, added mass, radiation damping, hydrostatic restoring coefficient matrices, M , A , B , and K , and wave-excitation-force vector WF as: ( )( ) ( )( ) ( ) ( )E Wω ω ω+ + + + + =RM A ξ B B ξ K K ξ F  (8) where upper dot means time derivative and RB is the structural damping matrix. In Equation (8), the external stiffness matrix EK is introduced to consider the elastic behavior of a deformable floating structure. EK is con- structed with Euler-Bernoulli beam and Saint-Venant’s torsion theory for 1M − beam elements. The 12 by 12 sub-stiffness matrix eK for the e th beam element in the local coordinate system is given in Equation (9): 3 3 2 2 3 2 3 3 2 3 2 2 . 12 0 12 0 0 0 0 0 6 4 0 0 0 6 4 0 0 0 0 0 0 0 0 0 12 6 12 0 0 0 0 0 12 6 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 2 6 4 0 0 0 0 0 0 0 c e z e y e x e y y e e z z e e e c c e e z z z e e e y y y e e e x x e e y y y e e e EA symm l EI l EI l GI l EI EI l l EI EI l l EA EA l l EI EI EI l l l EI EI EI l l l GI GI l l EI EI EI EI l l l − = − − − − − − K 2 2 6 2 6 4 0 0 0 0 0 0 0 0 y e z z z z e e e e l EI EI EI EI l l l l                                                                    −   (9) where E and G are Young’s and shear moduli, cA is the cross-sectional area, el is the element length, and xI , yI , and zI are the torsional, vertical, and lateral second moments of area about x, y, and z axes, respectively. EK can be expressed for 1M − beam elements with the 6 by 6 sub-stiffness matrices of eK as: 4 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 (11) (12) 1 1 ( 21) ( 22) (11) (12) 1 1 2 2 ( 21) ( 22) (11) 2 2 3 ( 22) 1 ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 E M ii ij e e e ji jj e e −     +   = +          =     K K K K K K K K K K K K K K K K          (10) 2.2 Morison Method in Time Domain The Morison method was also modeled in time domain [26]. In this method, a deformable floating structure consists of finite elements with nodes and segments. Key physical properties are included in the nodes while segments pres- ent structure’s deformability with massless axial, bending, and torsional springs. The Morison equation evaluates the hydrodynamic forces for a moving body at its instantane- ous node position. The computational time of this method is lower than the frequency-domain DMB method since this method is dependent on the Morison equation for wave-force estimation. It is worth utilizing this method if accurate global performance estimation is possible with a cheaper computational cost. The equation of motion for a deformable floating structure in time domain can be ex- pressed as: ( )1 2 n n n n n n A M DC C C Aρ + + = + = − ∆ + ∆ + − − R S M M MX B X K X F w F X η η X η X       (11) where SK are the structural stiffness matrix, MF is the hydrodynamic force vector based on the Morison equa- tion, w is the wet-weight vector (i.e., buoyancy minus dry weight), AC , MC , and DC are the added mass, inertia, and drag coefficients, ∆ and A are the displaced mass and drag area, ρ is the seawater density, η and η 4 are the velocity and acceleration vectors of fluid particles, and n denotes the normal direction. Structural deformability is represented by KS with axial, bending, and torsional springs. The drag term of the Morison equation, i.e., 3rd term in Equation (11), was neglected since the Keule- gan-Carpenter number of the presented problem is low (inertia dominant). 3. Case Description Figure 2 shows the configuration of a concrete hollow cylinder. Its density and Young’s modulus were 2300 kg/m3 and 30 GPa. The length of the concrete cylinder was fixed at 500 m, whereas various diameters of 10 m–20 m, sub- mergence depths of 20 m–60 m, and wave directions of 30°–90° were selected as design parameters. In this study, the cylinder’s buoyancy-weight ratio was set at 1.0, which means its buoyancy is the same as its dry weight. Then, the inner diameter and axial, bending, and torsional stiff- ness were reversely estimated based on the given buoy- ancy-weight ratio. The fixed-fixed boundary condition at both ends was designed; in other words, displacements and angles at both ends do not change. Water depth was fixed at 100 m. 4. Results and Discussions 4.1 Wave Force The wave excitation force WF in potential theory is of- ten correlated with the inertia force term IF in the Morison equation, i.e., 2nd term in Equation (11). IF is with the in- ertia coefficient ( )1M AC C= + , and 1 and AC are associated with contributions from the Froude-Krylov force and scat- tered wave force. The validity of the Morison equation is then assessed by comparing WF and IF . Figures 3–4 show the comparison of horizontal/vertical WF and IF (wave force per unit length). Wave direction was fixed at 90° Figure 2. Configuration of a cylinder with fixed-fixed boundary condition (D denotes diameter; SD stands for submergence depth). 5 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 with respect to the positive x-axis, and IF was calculated from the fixed body. At a low submergence depth of 20 m, which is close to free surface, trends and magnitudes of WF are somewhat different from those of IF , as shown in Figure 3(a) and (c). The Morison model tends to over- estimate the wave force at high frequencies of 0.7 rad/s or more. However, WF and IF are very close at higher sub- mergence depths, regardless of the wave frequency range, as shown in Figure 3(b) and (d), which means that the Morson method can estimate the wave force accurately for deeply submerged cylinders even if the cylinder diameter is large. Moreover, the higher the submergence depth, the lower the importance of high-frequency wave forces since the wave kinematics decreases with submergence depth as a function of ( )cosh k z H+ for intermediate water and kze for deep water. As shown in Figure 4, the difference between the two forces is significantly reduced at a small diameter of 10 m even if the submergence depth is 20 m. The previous results show that the Morison equation accurately estimates the wave inertia force at the large submergence depth and small cylinder diameter. It is well known that the Morison equation has a limitation, i.e., / 0.2D L ≤ . At the given water depth of 100 m and a cylin- der diameter of 20 m, the wave frequency should be less than around 0.8 rad/s to be included in this condition. The scattered wave force SCF is mainly associated with this limitation since the scattered wave force has a tendency to be important as the wave frequency increases. Figure 5 shows the comparison of the Froude-Krylov force FKF and SCF in the DMB method with half of IF in the Morison method ( )0.5 I = MFF F that corresponds to FKF in the DMB method. Wave kinematics for the Morison equation is es- timated at the geometric center, while that for FKF is based on the cylinder surface. At the given cylinder diameters, regardless of submergence depths, MFF is similar to FKF , which means that kinematics estimation at the geometric center in the Morison method is still valid for this horizon- tal cylinder at the given frequency range. Then, as noticed in Figure 5(a), SCF induces the difference between WF and IF only when submergence depth is low. In this case, the existence of free surface influences the calculation of SCF . No free-surface effect is detected when the submergence depth is 60 m due to a large distance between free surface and the cylinder. (a) (b) (c) (d) Figure 3. Comparison of horizontal/vertical WF and IF at a diameter of 20 m and different submergence depths. (a) (b) Figure 4. Comparison of horizontal/vertical WF and IF at a diameter of 10 m and a submergence depth of 20 m. 6 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 4.2 Modal Analysis Next, modal analysis was conducted, and dry/wet mode shapes/natural frequencies were obtained. Wet mode shapes and natural frequencies require the added mass and hydrostatic restoring coefficient matrices. In the Morison equation, AC is typically selected as 1 based on previous practices for the slender body [27]. However, the added mass is frequency-dependent, and wet mode shapes and natural frequencies can be changed. Tables 1-2 give dry and wet natural frequencies at different diameters and submergence depths. Morison and DMB models calculate similar dry natural frequencies with a maximum difference of 0.3%. Note that elements number for DMB and Morison methods for this example are 22 and 50, respectively. Since the Mor- ison method is based on the lumped mass method, more elements are generally required than the high-order beam- based DMB method. Twenty-two finite elements in the DMB method are sufficient to represent the elastic behavior. For the wet mode, while both approaches are well matched at a small diameter of 10 m or a large submergence depth of 60 m, the large difference is observed when diameter and submergence depth are both 20 m. In this case, free surface plays some roles in the modification of wet natural frequen- cies by changing added mass. The maximum difference is increased to be 5.0%. In this regard, when a cylinder is large and close to free surface, special care should be made to evaluate wet natural frequencies. Figure 6 shows the representative wet mode shapes of two methods at the diameter and submergence depth of 20 m. Even if there is a maximum difference of 5% in wet natural frequencies, there is no noticeable difference in the wet mode shape up to 3rd mode. (a) (b) Figure 5. Comparison of FKF , SCF , and MFF at different diameters and submergence depths. Table 1. Dry natural frequencies at different diameters. D(m) Direction Mode# Morison (rad/s) DMB (rad/s) Difference (%) 20 Horizontal 1 2.01 2.01 0.0 2 5.51 5.51 -0.1 3 10.73 10.73 -0.3 Vertical 1 2.01 2.01 0.0 2 5.51 5.51 -0.1 3 10.73 10.73 -0.3 10 Horizontal 1 1.01 1.01 -0.3 2 2.77 2.76 -0.2 3 5.40 5.38 -0.3 Vertical 1 1.01 1.01 -0.3 2 2.77 2.76 -0.2 3 5.40 5.38 -0.3 7 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 4.3 Global Behavior In previous sections, there were two distinct differences between the DMB and Morison methods; at the diameter and submergence depth of 20 m, the existence of free surface changes the scattered wave force and added mass, which results in differences in total wave force and wet natural frequencies. In this section, global behaviors are further checked at different wave directions. The structur- al damping matrix based on Rayleigh damping RB was added to deal with unrealistic resonant motions at natural frequencies. This study assumes a representative damp- ing ratio of 5% for concrete structures at the fundamental horizontal and vertical natural frequencies. Figure 7 shows the amplitude envelopes of horizontal and vertical displacements at the diameter and submer- gence depth of 0.2 m and wave direction of 90°. It turns out that the lowest mode shape is dominant for the given frequency range. In other words, the lowest horizontal and vertical natural frequencies for the DMB method are around 1.50 rad/s and 1.49 rad/s, and thus the lowest nat- ural frequency is dominant for the given frequency range of 0.3 rad/s–1.3 rad/s. Since the Morison method overesti- mates the wave force at the high-frequency region, highly overestimated motions are seen, especially when wave frequency is close to the fundamental natural frequency. Table 2. Wet natural frequencies at different diameters and submergence depths. D(m) SD(m) Direction Mode# Morison (rad/s) DMB (rad/s) Difference (%) 20 20 Horizontal 1 1.42 1.50 5.0 2 3.91 4.04 3.4 3 7.62 7.88 3.4 Vertical 1 1.42 1.49 4.9 2 3.91 4.06 3.7 3 7.62 7.94 4.1 20 60 Horizontal 1 1.42 1.42 0.1 2 3.91 3.93 0.5 3 7.62 7.70 1.1 Vertical 1 1.42 1.42 -0.3 2 3.91 3.91 0.2 3 7.62 7.68 0.8 10 20 Horizontal 1 0.71 0.72 0.3 2 1.96 1.98 1.0 3 3.82 3.86 1.0 Vertical 1 0.71 0.72 0.3 2 1.96 1.98 0.9 3 3.82 3.85 0.7 (a) (b) Figure 6. Wet mode shapes at the diameter and submergence depth of 20 m. 8 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 Since the center has the most considerable movement re- lated to the fixed-fixed boundary condition at both ends, results at the mid-length are presented in the later section. Figure 8 shows the horizontal and vertical motions at the diameter of 20 m, different submergence depths of 20 m and 60 m, and wave direction of 90°. Again, the motion trend coincides with wave force. Overevaluated motions by the Morison method at the submergence depth of 20 m are observed, particularly in the high-frequency region. Since fundamental wet natural frequency for the Morison method is lower than that for the DMB method, resonant induced motion associated with high wave force is detect- ed for the Morison method. On the other hand, the Mori- son method accurately estimates displacements for deeply submerged cylinders for the given frequency range since wet natural frequencies and wave forces between the two approaches coincide. Finally, Figure 9 shows the horizontal and vertical mo- tions at the diameter and submergence depth of 20 m and dif- ferent wave directions of 30° and 60°. As wave direction is not perpendicular to the cylinder, there is an arrival time lag of input waves along the length, which can induce a phase cancellation effect where wave force in the differ- ent locations can be canceled out. The phase cancellation effect can be dominant in the high-frequency region. In this example, the lower the wave direction, the smaller the high-frequency motions. As shown in Figure 3, the DMB model estimates the higher wave excitation force for the range of 0.3 rad/s–0.7 rad/s; thus, the DMB model tends to produce higher motion than the Morison method for the given wave directions. Also, the magnitude of motions at the mid-length tends to decrease as the wave direction de- creases. (a) (b) Figure 7. Amplitude envelopes of horizontal (a) and vertical (b) displacements (diameter of 0.2 m; submergence depth of 0.2 m; wave direction of 90°). (a) (b) Figure 8. Amplitude of displacements at the mid-length and submergence depths of 20 m (a) and 60 m (b) (diameter of 0.2 m; wave direction of 90°). (a) (b) Figure 9. Amplitude of displacements at the mid-length and wave directions of 30° (a) and 60 ° (b) (diameter of 0.2 m; submergence depth of 0.2 m). 9 Sustainable Marine Structures | Volume 04 | Issue 02 | July 2022 5. Conclusions This study focuses on the hydro-elastic behaviors of a fully submerged horizontal cylinder in regular waves. The frequency-domain discrete-module-beam (DMB) method was developed; a continuous elastic floating structure is discretized into a certain number of rigid bodies while structural flexibility is modeled by beam elements with Euler-Bernoulli beam and Saint-Venant torsion; the hydro- dynamic coefficients and wave forces were obtained from 3D potential theory. The time-domain Morison method was also employed; the Morison equation estimates wave forces and elastic behaviors are represented by massless axial, bending, and torsional springs. Various parametric studies on the cylinder diameter, submergence depth, and wave direction were conducted, and wave forces, dry/wet mode shapes/natural frequencies, and dynamic motions were analyzed. The results derived from this study are as follows: (1) The presence of free surface tends to change the scattered wave force in the DMB model; the larger the di- ameter and the smaller the submergence depth, the greater the change in the scattered wave force. The overestimation of wave force in the high-frequency region is observed in the Morison model. (2) Consideration of frequency-dependent added mass in the DMB method slightly changes the wet natural frequencies, whereas wet mode shapes of the two ap- proaches are almost the same. (3) The difference of wave forces and wet natural frequencies results in significant changes in the overall magnitudes of the dynamic motions. In this example, the Morison model tends to significantly overestimate dynam- ic motions around the fundamental wet natural frequency. (4) Dynamics motions decrease with decreased wave direction, and high-frequency motion is reduced mainly due to the arrival time lag of incoming waves along the length, which induces phase cancellation of wave force. Conflict of Interest Author declares that there is no conflict of interest. References [1] Anastasopoulos, I., Gerolymos, N., Drosos, V., et al., 2007. Nonlinear response of deep immersed tunnel to strong seismic shaking. Journal of Geotechnical and Geoenvironmental Engineering. 133(9), 1067-1090. [2] Chandrasekaran, S., Nannaware, M., 2014. Response analyses of offshore triceratops to seismic activities. Ships and Offshore Structures. 9(6), 633-642. [3] Zhao, E., Dong, Y., Tang, Y., et al., 2021. 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