The Effects of Moving Load on the Hydroelastic Response of a Very Large Floating Structure

The hydroelastic response of a very large floating structure in regular waves suffering an external moving point load is considered. The linearized velocity potential theory is adopted to describe the fluid flow. To take into account the coupled effects of the structure deformation and fluid motion, the structure is divided into multiple segments and connected by an elastic beam. Then through adding a stiffnessmatrix arising from the elastic beam into themultiple bodies coupledmotion equations, the hydroelastic response is considered. By applying the Fourier transform to the obtained frequency domain coefficients, themotion equation is transformed into the time domain and the external point load is further considered. The accuracy and effectiveness of the proposed method are verified through the comparison with experimental results. Finally, extensive results are provided, and the effects of the moving point load on the hydroelastic response of the very large floating structure are investigated in detail.


Introduction
Very large floating structure (VLFS) can be used as floating airports and bridges and for many other purposes.Due to the large dimension of the structure compared with waves, the elastic deformation should be considered when dealing with wave-structure interaction.Therefore, hydroelasticity method has been developed to account for coupling between hydrodynamic loading and structural deformation for floating flexible structures in waves.
Traditional hydroelasticity methods may be categorized into two different approaches, i.e., direct method and modesuperposition method.For the former, the equation of motion for a flexible structure is solved directly using conceptually full modes of the discretized system [1][2][3].While for the latter, the hydroelastic response is obtained by dividing it into diffraction and radiation problems for each (flexible) mode of the flexible structure and then superposing all modal equations together to finally solve them.The modesuperposition method can be referred to Senjanović et al. [4] and Michailides et al. [5].
Different from the direct and mode-superposition methods, Lu et al. [6] proposed a frequency domain discrete-module-beam-bending based hydroelasticity method for a continuous flexible structure in waves.One advantage of this approach is that it avoids the need for predetermination of the flexible modes, which may be difficult for complicated geometric (or connection) features of the flexible structure.Sun et al. [7] extended the method proposed by Lu et al. [6] to be applicable for the hydroelastic response of a hinged VLFS in waves (in frequency domain).Other applications of this method can be referred to Xu et al. [8] and Zhang et al. [9].
For floating bridges and floating airport, the passage of vehicles or landing and take-off of an airplane will introduce unsteady external loading, which may affect the safety of the structure.Therefore, it is of practical importance to investigate at the design stage the transient hydrodynamic response of VLFSs.Traditionally, the hybrid frequency-time domain method based on mode-superposition approach was adopted by Kashiwagi [10,11] to investigate the effects of unsteady external loading on VLFSs.Also the direct time domain method was used, for example, by Qiu [12] and Cheng et al. [13].
In the present study, the discrete-module-beam-bending based hydroelasticity method [6,7,9] is used to investigate 2 Mathematical Problems in Engineering the hydroelastic response of a VLFS under unsteady external loading in waves.The effects of mass and moving velocity of the point loading on the hydroelastic response of the VLFS in waves are investigated in detail.

Governing Equation for the Multiple Floating Structures.
To solve the problem, we assume that the fluid is inviscid, incompressible, and homogeneous, and the fluid flow is irrotational.Then the velocity potential Φ can be introduced to describe the fluid motion.Under the further assumption that the amplitude of the wave motion is small compared to its length and the dimension of the floating structure, we can apply the linearized velocity potential theory to describe the fluid flow.We now consider the wave interactions between multiple floating structures, as shown in Figure 1.For sinusoidal fluid motion in time  with radian frequency , we may write the total velocity potential Φ into the following form: with where Re indicates taking the real part,  I ,   , and   are, respectively, the complex incident potential, diffracted potential, and the radiation potential.In fluid domain, the mass conservation requires that the velocity potential should satisfy the Laplace equation: throughout the fluid domain Ω.On the free surface   , the combined kinematic and dynamic boundary condition gives On the seabed   , the impermeable condition states where the seabed is assumed to be flat.On the wetted structure surface   ( = 1, . . ., ), the nonpenetration condition should be satisfied, i.e., for the diffracted velocity potential, and for the radiation velocity potential.Here,  indicates the q-th floating structure.In the far field  ∞ , the radiation condition should be also enforced to close the problem, where  denotes the wave number.

Solution Procedure for the Very Large Floating Structures.
The boundary value problem described in (1) to ( 9) can be solved through the standard boundary element method.
Here, the focus is put on the extension of the solution for multiple structure interaction problem to the hydroelastic response of very large floating structures.After the velocity potential has been solved, we can obtain the hydrodynamic pressure through the linearized Bernoulli equation.
In the frequency domain, the wave exciting force  →   corresponding to the incident and diffracted velocity potentials can be given as where  is the density of water and  0 represents the average wetted surface.The hydrodynamic force arising from the radiation potential can be expressed through the added mass   and damping coefficients   , i.e., After the hydrodynamic forces are obtained, the equation of motion of freely floating multibody system can be given according to Newton second law.Here, it may be noticed that the multibody system comprises  modules and each module has a six degree-of-freedom motion, as shown in Figure 1.For incident wave with unit amplitude, we have for  =  (12), the constraint of displacement due to the existence of connections is not considered.
When considering the elastic response of the very large structure, (12) becomes where [] is the term due to the connection between the submodules, and details can be seen in Sun et al. [7].

Motion Equation of Flexible Structure in Time Domain.
To consider the effects of moving load on the hydroelastic response of the very large floating structure, the solution should be constructed in the time domain.To do this, in our work the hybrid frequency/time domain method (namely the time domain method based on impulse response function (IRF), which was first introduced by Cummins (1962)), is adopted to establish the motion equations.Here we consider a flexible structure in waves.The motion equations of the structure (which is divided into m submodules) are established using the SAM approaches as follows: where [  (∞)] is the added mass matrix of the i-th module caused by the motion of the j-th module at frequency of infinity.
[  ()] is the kernel function of the i-th module caused by the motion of the j-th module, which is related to the radiation damping matrix [  ()].The expressions of [  (∞)] and [  ()] are given as follows: where   is an arbitrarily chosen frequency.The total stiffness matrix [  ] is composed of two parts, i.e., the hydrostatic stiffness of the structure [] and the structural stiffness matrix (considering the deformation) [𝐾].The expression of [  ] is given as follows: = 2.4.Consideration of a Moving Load.For a floating bridge or airport, there are vehicles moving on the upper surface of the structure.Usually the size of vehicles is small compared with the floating structure, which means that the vehicle can be regarded as a point load.Thus the problem of a vehicle moving on the floating bridge (or airport) can be simplified as a point load moving on the upper surface of a flexible structure.In this section, the time domain method is adopted to investigate the dynamic response of a flexible structure with moving point loads on the upper surface in waves (see Figure 2).The linear assumption for the wave-structure interaction still remains here.Besides, we assume that the existence of the moving point load does not cause large structural deformation.Thus the floating system shown in Figure 2 is still a linear system.The point mass is assumed to move in a uniform horizontal velocity   .The force due to the point load on the structure is denoted as {  ()} 6×1 .As the motion equation is established on the center of each submodule of the structure.Thus the force {  ()} should be transformed to the ones on the center of submodules, which is denoted as {  ()} 6×1 = ({ 1 ()} . . .{  ()})  .
In order to obtain the relationship between {  ()} and {  ()}, we first consider a (static) point load acting on a flexible structures in calm water.The motion equation (can be obtained from (14) after some manipulation) is given as follows: It should be noted that in calm water, the whole structure is divided into  1 submodules. 1 is a large number ( 1 = 150 adopted in this section) and is much larger than the number of submodule used in hydroelastic response,  ( = 8 in the present analysis).The reason is that by setting a series of submodules (or elements) of very small size, the point load can be regarded as acting on the center of one particular submodule (for example, the  ℎ 1 submodule shown in Figure 3).Then the equivalent force satisfies {  } = 0 ( ̸ =  1 ) and {  } = {  ()} ( =  1 ).By solving (18), we obtain the distribution of displacement along the structure in calm water caused by the static point load at a given position on the structure.Then the displacement at the center of submodules (the number is ), {} 6×1 for hydroelastic calculation (see ( 14)), is also obtained.Thus we can obtain the equivalent force {  } 6×1 as follows: In calm water, we can obtain the equivalent force {  } 6×1 for static point loads acting on any position of the structure.Finally, the hydroelastic response equation can be written as [  to  = 71.7 is out of the computed time range, thus only a trend to the large trough is observed in the figure.

The Effects of Moving Load Mass.
We then consider the effects of the mass of the moving point load on the hydroelastic response of the very large floating structure.The incident wave length is taken to be 300, and the incident wave angle is chosen as zero or heading waves.Three different moving point masses are considered, i.e., 200, 400, 600, and 800.For each case, the mass point load is moving with the same velocity or 5/.
In Figure 6, we show the typical contour plot of the vertical displacement and bending moment response of the VLFS in regular wave with a moving point mass 600, against time and longitudinal position of the structure.In both Figures 6(a) and 6(b), the trajectory of the moving point load is also given.It can be seen from Figure 6(a) that the absolute value of the displacement at the two end sides of the VLFS is usually larger than those for the point within the VFLS.This is not unexpected because due to that both the two end sides are free to move.From the figure, we can also see that the largest vertical displacement happens when the moving point load is located at the two end sides of the VLFS, i.e., approximately at 40 and 100, and the value is always negative.This is because the moving point load is always negative due to the gravity.Compared with Figure 6(a), it can be seen from Figure 6(b) that the bending moment response of the VLFS around the middle point is always larger than those at the two end sides of the VLFS, which is coincided with the free beam theory.Especially, from these two figures we can see that the effects of the moving point load on the bending moment response of the VLFS are much larger than those on the vertical displacement, as shown for the region (40, 100).
Computations are then carried out for the vertical displacement and bending moment responses of the VLFS in regular waves undergoing a moving point load with different masses, as shown in Figures 7(a) and 7(b), respectively, for the vertical displacement response and bending moment response.From Figure 7(a) we can see that with the increase of the moving point load mass, the absolute values of the peaks become much smaller while the absolute values of the troughs become much larger.This is for that the overall effect of the moving load is inducing a negative displacement.In addition, since the time histories are for the point located at / = 0.5, thus within (60, 80) the effects of the moving load are most obvious.From Figure 7(b) it can be seen that the absolute values of the peaks and troughs for the case with a moving load are always larger than those without a moving load.Compared with Figure 7(a), we can see that once the moving load is on the VLFS, its effects should be always considered.In addition, it is interesting to see that compared with the vertical displacement response, which varies with a single frequency, there are more frequency components in the bending moment response, which indicates that the hydroelastic effects must be taken into account to obtain a reasonable result.To see the effects of the moving load mass more clearly, in Figures 8(a) and 8(b) we plot the maximum vertical displacement and bending moment against the load mass, both of which increase with load mass linearly.

The Effects of Moving Load
Velocity.Finally, we investigate the effects of the moving load velocity on the vertical displacement and bending moment response of the VLFS.The incident wave length and wave angle are taken to be 300 and zero degree, respectively.The moving point mass is chosen as 600.Here, four moving velocities are considered, i.e., 5/, 10/, 15/, and 20/.In Figures 9(a  equations into the time domain ones via Fourier transform, the effects of the moving point load are taken into account.The comparison with experiment results shows that the method is accurate enough for engineering application.The extensive results indicate that both the vertical displacement and bending moment response will increase with the moving point load mass.However, there is no evidence to show that the vertical displacement or the bending moment increases or decreases with the moving velocity.

Figure 1 :
Figure 1: Definition of fluid and structure boundaries.

Figure 2 :Figure 3 :
Figure 2: A schematic of a flexible structure with moving point load on the upper surface in waves.

Figure 4 :
Figure 4: Plan view of the flexible VLFS prototype.

5 Figure 5 :
Figure 5: Time history of vertical displacement of the flexible structure in calm water.

Table 1 :
Parameters of the model.

Figure 6 :
Figure 6: Typical contours of the vertical displacement and bending moment responses of the VLFS in regular wave with a moving point mass as 600.

=300mFigure 7 :Figure 8 :
Figure 7: Time histories of the vertical displacement and bending moment responses of the VLFS in regular wave for a moving point load with different masses.
) and 9(b) we show the typical contour plots of the vertical displacement and bending moment response, respectively, in which the moving point load velocity is 15/.The comparison of Figures9(a) and 9(b) shows that when the moving point load leaves the VLFS, its effects on the vertical displacement disappear quickly, while its effects on the bending moment response will maintain for a long time.Also, from the comparison of Figure9(b) with Figure6(b) we can see that

=300mFigure 11 :
Figure 11: The maximum vertical displacement and bending moment responses of the VLFS in regular wave for a moving point load with different velocities.