Effects of marine sediment on the response of a submerged floating tunnel to P?wave incidence

Heng Lin · Yiqiang Xiang · Zhengyang Chen · Ying Yang

Abstract Submerged floating tunnels (SFTs) are a novel type of traffic structure for crossing long straits or deep lakes. To investigate the dynamic pressure acting on an SFT under compression (P) wave incidence, a theoretical analysis model considering the effect of marine sediment is proposed. Based on displacement potential functions, the reflection and refraction coefficients of P-waves in different media are derived. Numerical examples are employed to illustrate the effects of the thickness of the sediment layer, the incident P-wave angle, the tether stiffness and spacing, and the permeability of the sediment on the dynamic pressure loading on the SFT. The results show that the dynamic pressure is related to the saturation of the sediment and affected by its thickness. Partially saturated sediment will amplify the dynamic pressure loading on the SFT, and the resonance frequency increases slightly with fully saturated sediment. Besides, increasing the tether stiffness or decreasing the tether spacing will decrease the dynamic pressure. Locating the SFT at greater depth and reducing the permeability of the sediment are effective measures to reduce the dynamic pressure acting on the SFT.

Keywords Submerged floating tunnel · Dynamic pressure · Porous medium · P-wave · Displacement potential function

1 Introduction

The submerged floating tunnel (SFT), also termed the Archimedes bridge, is an innovative traffic structure for crossing long waterways. Under the united action of surplus buoyancy and an anchoring system, an SFT can be installed at the desired depth in the ocean. Compared with traditional bridges and tunnels, SFTs offer better spanning capacity and lower longitudinal slope [1]. High-speed trains and passenger vehicles can travel in an SFT at faster velocity, shortening crossing times and increasing convenience. Besides,SFTs have little effect on the environment and can operate in all weather conditions under the sea. Therefore, they are considered to be the most competitive sea-crossing structures for long straits and lakes in the 21st century [2].

Earthquakes are a kind of critical excitation that determines the safety of SFTs. Many studies have been devoted to the seismic responses of SFTs. Considering the fluid–structure and soil–structure interactions, Fogazzi and Perotti [3] developed a nonlinear numerical analysis procedure for seabed-anchored floating tunnels under extreme seismic excitation. A geometrically nonlinear finite-element method was refined by Di Pilato et al. [4], who also established a full three-dimensional (3-D) model for the response of the Messina Strait crossing due to seismic and wave loads.The seismic response of SFTs was investigated by Martire et al. [5] by considering synchronous and multisupport excitation. Moreover, Martinelli et al. [6] analyzed the nonlinear dynamic behavior of an SFT based on the pseudo-acceleration elastic response spectrum. In the cited works, the seismic effect was equivalent to the displacement or acceleration excitation acting directly on the structure. However, in the process of seismic propagation, the hydrodynamic pressure around the SFT will also change. Ignoring such variation of the hydrodynamic pressure may underestimate the seismic effect on the SFT.

The hydrodynamic pressure acting on a structure plays an important role in seismic analysis. In contrast to ground motion transmitted via the structure supports, hydrodynamic pressure is caused by propagation of seismic motion of the seabed through seawater, which is known as a seaquake. Some cases of SFTs subjected to an earthquake or earthquake plus seaquake were investigated by Martinelli et al. [7] using the finite element method. Higher stress in the tunnel and greater deformation in the mooring tethers were observed in the cases considering the seaquake effect. Based on two-dimensional (2-D) wave potential theory, Morita et al. [8] developed a numerical model and estimated the hydrodynamic force on an SFT subjected to vertical seismic excitation. Adopting 2-D and 3-D velocity potential functions, Mirzapour et al. [9] considered the fluid–structure interaction in seismic analysis. In addition, Jin et al.[10] rigorously considered the compressibility of seawater and studied the seismic response of an SFT system using a numerical model. The cited numerical models have enabled deeper understanding of the seismic behavior of SFTs subjected to seaquakes. However, the influence of different types of earthquake and wave propagations on the hydrodynamic pressure has not been clearly revealed.

The seismic wave originating from the hypocenter can be mainly divided into a compression (P) wave and shear (S)wave. Because seawater has no shear resistance, only the P-wave propagates in seawater, imposing a pressure fluctuation on the structure. By considering the propagation of a seismic wave in water, Takamura et al. [11] deduced the incident wave potentials and investigated the response of a large floating structure to a seaquake, providing a reference for theoretical analysis of the effect of a seismic wave on marine structures. However, the P-wave is a mechanical wave and is thus affected by the properties of the medium during its propagation. Due to tidal scouring, seawater is always enriched with large amounts of sediment and silt.After a period of deposition, various layers of alluvium and sediment form on the sea bottom. The re is no doubt that the presence of such marine sediment will change the geological characteristics of the seabed and interfere with the propagation of seismic waves [12]. A porous medium theory was developed by Biot [13, 14] to consider the effect of sediment, which has received a great deal of attention for modeling sediment. Based on the theory of Biot, a number of studies [15, 16] have analyzed refraction and reflection of acoustic waves at the water–sediment interface. Indeed, the refraction and reflection of waves represent a form of energy transmission and directly determine the dynamic pressure in seawater. Therefore, determining the dynamic pressure with consideration of the sediment effect can have great practical significance for SFT safety design.

The objective of the work presented herein is to investigate analytical solutions for the dynamic pressure on a submerged floating tunnel due to P-wave incidence, considering a porous sediment layer in the theoretical model. Based on the displacement potential functions, the reflection and refraction coefficients of a P-wave in different media are derived. Using numerical examples, the effects of various parameters, such as the thickness of the sediment medium,the angle of incidence of the wave, the position of the tunnel structure, the stiffness and spacing of the tethers, and the permeability of the sediment medium, on the dynamic pressure acting on the SFT are discussed.

2 Governing equations

An SFT basically consists of a tubular structure floating at a certain depth in water, anchor tethers and devices fixed into the submarine rock or foundation, and a revetment structure for connection to the off shore, as shown in Fig. 1.The anchor tethers are installed symmetrically in a plane at each position along the longitudinal direction with different spacings Li(i = 1, 2, …, N). To analyze the dynamic pressure acting on the SFT theoretically, various assumptions are adopted to simplify the model. The sediment is modeled as a linear porous medium based on Biot theory. The seawater is assumed to be an ideal compressible fluid, and the seabed as a half-space filled with an elastic solid. The tubular structure is considered to be a rigid body, dividing the seawater into two parts named the upper and lower water layer. within the small-strain linear elastic hypothesis [17],the tethers can be regarded as elastic supports with certain stiffness. The full analysis model is established in the 2-D plane, as shown in Fig. 2.

Using a Cartesian system with x-axis rightward and z-axis upward, z = 0, z = h1, and z = h2represent the interfaces separating the elastic solid, porous medium, rigid body, and ideal water, respectively, where H is the depth of the seawater.The model is assumed to be infinitely long in the x-axis, and the interfaces are regarded as flat in the investigations described herein.

Fig. 2 Analytical model of the SFT with P-wave incidence

2.1 Porous medium

Based on Biot theory [13, 14], the porous medium is treated as an interacting two-phase elastic system; one phase is the porous solid skeleton, and the other phase is the fluid filling the voids. During seismic wave propagation, the high frequency parts of a wave will be attenuated quickly in the lithosphere, thus only seismic waves in the low frequency range will propagate into the seabed sediment, resulting in coupled solid–fluid motion in the porous medium. The equations of motion of a linear isotropic porous medium can be expressed in terms of the displacement vectors of the two phases as follows [18]

where umsand umware the displacement vectors of the solid skeleton and pore fluid, respectively, the over dot symbol denotes differentiation with respect to time, ? and ?2are the gradient and Laplace operators, respectively,pmwrefers to the pore fluid pressure,is the dissipation coefficient, g is the acceleration due to gravity,k is Darcy’s permeability coefficient (assumed to be frequency independent) [19, 20], and n is the porosity. A lso,pss=(1 ? n)pms+ pa,psf= ?pa,pff=npmw+ pa, in which pms,pmw,paare the solid skeleton density, pore fluid density, and added apparent mass, respectively. A, G, Q, and R are Biot’s elastic constants, which are determined by the bulk modulus of the pore fluid and the Lamé constants of the solid skeleton.

If the solid grains are assumed to be incompressible, Biot’s constants are given by Mei and Foda [21]in the formandin which λmsand μmsare Lamé constants of the solid skeleton, andis the effective bulk modulus of the pore fluid. For a fully saturated porous medium,Following Verruijt [22], the effective bulk modulus of the pore fluidis related to the degree of saturation s according to the formula

where s is the degree of saturation of the sediment and p0is the absolute pore fluid pressure.

The displacement vector umsdepends on the scalar and vector displacement potentials, i.e.,φmsand Ψms, as follows

Similarly, umwdepends on φmwand Ψmwas follows

where ?× is the curl operator.

2.2 Half?space of linear elastic solid

The equation of motion for the elastic solid can be written as

where λsand υsare Lamé constants of the elastic solid,usandare displacements and accelerations, respectively, and psdenotes the solid density.

Due to the Helmholtz theorem, the displacement vectors for the elastic solid ustake the form

where φsand Ψsare the scalar and vector displacement potentials, respectively.

2.3 Ideal compressible water

For the ideal compressible fluid considered in this system,only compressive seismic waves can be transmitted. Hence,the governing equation takes the form

where Kwis the bulk modulus of the ideal fluid, ρwis thefluid density, and Uwand are the displacement and acceleration vector, respectively.

According to the model depicted in Fig. 2, the seawater in the ZOX plane is divided into two parts by the SFT. The displacement vectors of the two parts are expressed as

In addition, the dynamic pressure in the two parts is

where t is the time,pwis the seawater density, z is the depth of the seawater, and x is the location of P-wave incidence.

3 Boundary conditions

As seen from Fig. 2, the thickness of the sedimentary layer is h1, and the heights of the upper and lower water layers are h3and h2, respectively. In addition, the tubular structure is moored by elastic springs spaced equidistant L along the longitudinal direction, and the elastic stiffness of the tethers is K.The effect of sediment on the tether stiffness is ignored. The boundary conditions of the analysis model can be given based on continuity and equilibrium of interaction.

3.1 Interface between the half?space elastic solid and the porous medium (z = 0)

(1) The continuity of the normal displacements of the porous sediment and the solid is given by

where usz,umsz, and umwzare the z-components ofums, and umw,respectively. Based on Eqs. (5), (6), and (8),usz,and umwzcan be expressed in terms of the displacement potentials as

where Ψs,and Ψmware the only nonzero components f the displacement vectorand Ψmw, respectively.

(2) The continuity of the tangential solid displacement and the skeleton displacement is given by

where usxand umsxare the x-components of usand ums, and can be expressed as

(3) The equilibrium of the normal traction on the solid and the total normal traction on the porous medium is written as

where òzz,sand òzz,mare the normal stress of the solid and porous medium, given by

(4) The equilibrium of the normal traction on the solid and total traction on the porous medium is written as

where òzx,sand òzx,mare the shear stress of the solid and the porous medium.

3.2 Interface between the porous medium and the lower water layer (z = h1)

(1) The compatibility between the normal pore fluid movement in and out of the skeletal frame and the normal fluid displacement is given by

(2) The equilibrium of the total normal traction on the porous sediment and the fluid pressure is given by

(3) The equilibrium between the lower water layerand the pore fluid pressure pmwis given by

or

(4) The equilibrium of the tangential traction on the porous skeleton is given by

3.3 Interface between the lower and upper water layer at the SFT position (z = h1 + h2)

(1) The equilibrium of the normal traction on the lower water layer and the normal traction on the upper water layer is written as

where αcis the tether distribution coefficient. For an SFT with equidistant tethers,αcequals 1. Because the SFT is regarded as a rigid body, the displacement of the SFT is equivalent to the deformation of the tethers, written as

(2) The continuity of the normal displacements of the lower and upper water layer is given by

3.4 Free surface of the upper water layer (z = H)

The equilibrium of fluid pressure

3.5 Interface between the elastic solid and the lower water layer (h1 = 0 and z = 0)

(1) The continuity of the normal solid displacement and the lower water layer displacement is given by

(2) The equilibrium of the normal traction on the solid and the normal traction on the lower water layer pressure is given by

(3) The equilibrium of the tangential traction on the solid is written as

4 Formulation of the system

Figure 2 shows a P-wave with angular frequency ω and angle of incidence θiwhich crosses the whole model from the half-space elastic solid, generating reflected P- and S-waves in the half-space elastic solid, a P1-wave (the first kind of compressible wave), a P2-wave (the second kind of compressible wave), and an S-wave propagating upward and downward in the porous medium. In addition, only the P-wave propagates upward and downward in the lower and upper water layers. Based on Snell’s law, all wave numbers in the x-direction are the same apart from the upper water layer. Since rigid-body displacement of the SFT is considered, the upper water layer only vibrates in the z-direction.

The displacement potentials of the multi layered media are described as follows.

For the region z < 0

where A and B denote the amplitude of P- and S-waves,respectively, k is the wavenumber, ing is the amplitude ratio of the solid skeleton potential to the pore fluid increment potential, the subscripts “i”, “r”, and “t” denote incidence, reflection, and transmission, respectively; the subscripts “P”, “S”,“P1”, and “P2” denote P, S, P1, and P2waves, respectively;the subscripts “s”, “m”, “l(fā)”, and “u” indicate the solid, the porous medium, the lower water layer, and the upper water layer, respectively; the subscripts “x” and “z” denote the x and z-components, respectively.

The wavenumber relationships between the incident P-wave and its x- and z-components are expressed as

where ksP= w∕is the velocity of the P-wave in the elastic solid,vsP=In addition, the wavenumber relationships in the other layered media are written as

where vsSis the velocity of the S-wave in the elastic solid,vsS=is the velocity of the P-wave in the water,and Kwis the bulk modulus of the seawater.

Due to the results of Deresiewicz and Skalak [23] are given as

Substituting Eqs. (41)–(48) into Eqs. (14)–(40) for the boundary conditions, the following equations for unknown constantsAuP1, and AuP2are obtained

where AiPis the know n constant, a is a 12 × 12 square matrix, and f is a column matrix with 12 elements.

When the thickness of the sedimentary layer h1= 0, the simultaneous equations for unknown constants AsP,BsS, AlP1,AlP2,AuP1, and AuP2are obtained as

where a? is a 6 × 6 square matrix, and f? is a column matrix with six elements.

By solving Eqs. (59) and (60), the amplitudes of wave reflection and transmission at the elastic solid, the porous medium, and the ideal water can be obtained. The dynamic pressure on the SFT due to P-wave incidence can the n be calculated analytically. The details of the matrices a, f, a?, and f? are provided in “Appendix A.”

5 Numerical example and analysis

No actual SFT projects exist anywhere in the world, so the parameters of the SFT were selected from some conceptual design study cases [24, 25]. The SFT is built in a strait with depth of H = 120 m. The anchor tethers installed on the SFT are equidistant along the longitudinal direction.Each pair of tethers is spaced L = 2.5H apart, and has the same geometric and material properties. The support stiffness of each pair is 8.0 × 108N/m. Based on the works of Wang et al. [26], Domínguez and Gallego [27], and Cheng[28], the material properties of the ideal fluid, the elastic half-space solid, and the porous medium, are as follows:the seawater has a bulk modulus Kw=2.0×109Pa and density pw=1000 kg/m3. Lamé constants of the elastic half-space solid are taken as λs=15.30×109Pa and υs=7.65×109Pa, with density ps=2483 kg/m3. The porous medium has a porosity n = 0.6 and a permeability coefficient k = 0.001 m/s, and the Lamé constants of the solid skeleton are λms= 17.975 × 106Pa and μms= 7.704 × 106Pa,the density of the skeleton is pms=2640 kg/m3, the density of the pore fluid is pmw=1000 kg/m3, and the added apparent mass pa=0. Two cases of saturation are considered for the porous medium layer s = 1.0 and s = 0.995. Assuming the solid skeleton grains to be incompressible, the corresponding bulk modulus of the pore fluid and Biot’s constants can be obtained.

In the analysis, the constant Aipof the incident P-wave is assumed to bewhich corresponds to the incident acceleration amplitude of g. The dimensionless hydrodynamic pressure(where pwgH is the hydrostatic pressure) and the dimensionless depth of the seawater z/H are applied to describe the variation of the dynamic pressure in different cases. In addition, the dimensionless frequencyw∕w1is employed to present the range of the incident wave frequency, in which/ w1is the first natural frequency of the seawater,w1= πvw(2H).

In the case where the position of the SFT is h3/H = 0.3 and the depth of the porous medium h1/H = 0.1, Fig. 3 shows the distribution of dynamic pressure along the seawater depth (z/H) under P-wave incidence. It is seen that the dynamic pressure of the system generally decreases as the seawater depth decreases. Affecting the solid skeleton of the porous medium, the dynamic pressure on the interface (h1) between the porous medium and lower water layer shows a mutation on the curve. This mutation causes a change in the dynamic pressure, which will propagate into the seawater. Meantime, the re is also a mutation in the dynamic pressure at the position of the SFT. Table 1 lists the results for the dynamic pressure approaching the interface between the porous medium and lower water layer and the position of the SFT. The dynamic pressure acting on the SFT is caused by the pressure difference between the lower and upper water layers. It is seen that the marine sediment affects the P-wave propagation in the seawater and changes the dynamic pressure acting on the SFT. Ignoring the marine sediment would cause deviation of the calculation results and underestimate the dynamic pressure.

Fig. 3 Distribution of dynamic pressure along seawater depth under P-wave incidence at a θ = 0° and b θ = 45°

6 Discussion on the factors influencing the dynamic pressure

6.1 Influence of porous medium thickness

Figure 4 shows the effects of the porous medium thickness on the dynamic pressure p at the position of the SFT(h3/H) for an angle of incidence of θ = 0°. Three different types of porous media with different sedimentary thicknesses are discussed. As the thickness h1/H is increased from 0.05 to 0.15, the response peak of the dynamic pressure increases from 1.360 to 2.711 with the partially saturated porous medium. In contrast, the presence of a fully saturated porous medium slightly decreases the resonance peak value, from 0.741 to 0.660. In addition, the partially saturated porous medium causes a shift of the first natural frequency w∕w1from 0.777 to 0.581 as the thickness is increased, while the first natural frequency for the fully saturated porous medium increases slightly.It can be found that the thickness of the sediment has a greater impact on the system with the partially than fully saturated sediment.

Fig. 4 Effect of porous medium thickness on dynamic pressure(θ = 0°, h3/H = 0.3), a h1/H = 0.05, b h1/H = 0.10, c h1/H = 0.15

Table 1 Dynamic pressure results at the interface (h1) and the position of the SFT (h3)

6.2 Influence of upper fluid layer depth

The effect of the upper fluid layer depth on the dynamic pressure p at the position of the SFT (h3/H) is shown in Fig. 5. For the same angle of incidence (θ = 0°) and sedimentary thickness (h1/H = 0.1), the dynamic pressure acting on the SFT decreases significantly as the SFT is located deeper. The resonance peaks for the fully and partially saturated porous media reduce from 0.817 to 0.524 and 2.187 to 1.704, respectively. In fact, the P-wave comes from the lower fluid layer and is applied to the full structure through the incompressible fluid. Due to the hydrostatic pressure of the upper water layer, a deeper position of the SFT results in a smaller dynamic resonance peak value. Thus, for the cases of fully and partially saturated porous media, the numerical results show almost the same variation trend. Besides, comparing these results with Fig. 4, the first natural frequencies w∕ w1with the fully and partially saturated porous media change little. This means that the effect of the thickness of the sediment layer is more remarkable than that of the depth of the upper water layer.

Fig. 5 Effect of thickness of upper water layer on dynamic pressure(θ = 0°, h1/H = 0.1), a h3/H = 0.1, b h3/H = 0.3, c h3/H = 0.5

6.3 Influence of P?wave incident angle

Figures 6 and 7 show the effects of the P-wave incident angle on the dynamic pressure at the first natural frequency in different cases. As the angle of incidence increases, the dynamic pressure acting on the SFT gradually decreases,because most of the P-wave is reflected at the solid–porous medium interface when the angle of incidence of the P-wave is large, so only a small fraction of the P-wave propagates into the seawater to cause dynamic pressure on the structure.According to the results of these numerical simulations, the dynamic pressure reduces smoothly as the angle of incidence of the P-wave changes from 0° to 90°.

Fig. 6 Effect of incident angle on dynamic pressure at first natural frequency (saturation), a h1/H = 0.1, b h3/H = 0.3

Comparing the results in the different cases, the rate of decline of the dynamic pressure is clearly different. According to Fig. 6, the deeper the position of the SFT, the slower the rate of decline of the dynamic pressure. In addition, the thickness of the saturated porous medium has only a limited impact on the rate of decline of the pressure. The three rates of decline of the pressure are almost the same as the angle of incidence is increased to 60°. However, the results are totally opposite to the partially saturated porous medium.Figure 7 shows a higher rate of decline of the pressure for the thicker partially saturated porous medium. According to the se results, it can be further explained that the effect of the sediment thickness is more prominent than that of the depth of the upper water layer.

Fig. 7 Effect of incident angle on dynamic pressure at first natural frequency (partially sat. 99.5 %), a h1/H = 0.1, b h3/H = 0.3

6.4 Influence of tether stiffness

The tethers are critical components of the whole structure,being used to support the structure and limit large displacements of the SFT. Both the cross-section and material of the tethers will change the tether stiffness and affect the dynamic behavior of the SFT. Thus, the tether stiffness and spacing were investigated in various cases. Figure 8 shows the cases with tether stiffness of 6.0 × 108N/m,8.0 × 108N/m, 1.0 × 109N/m, and 1.2 × 109N/m. As the tether stiffness is increased, the resonance response peaks of both the fully and partially saturated porous media increase, from 0.516 to 1.094 and 1.436 to 3.231,respectively. Moreover, the first natural frequency reduces from 1.002 to 0.961 and 0.678 to 0.627, respectively, as the tether stiffness is increased. In some previous studies [24, 29], it was reported that structural vibrations can be controlled by increasing the tether stiffness. However,higher tether stiffness will increase the stiffness of the overall system and thereby amplify the dynamic pressure loading on the SFT. It is seen that higher tether stiffness is not good for the SFT under P-wave incidence.

Fig. 8 Effects of tether stiffness K on the dynamic pressure(h1/H = 0.1, h3/H = 0.3, θ = 0°, αc= 1), a saturation, b partially sat.99.5%

6.5 Influence of tether spacing

As the span length of SFTs is very long, the tether spacing is an important factor in the initial structural design.The tether distribution coefficient αcis defined to describe the tether spacing in the longitudinal direction. Figure 9 shows the dynamic pressure on the SFT for different tether distribution coefficients. For both the fully and partially saturated porous media, the response peak value decreases, from 0.893 to 0.490 and 2.583 to 1.360, with increasing tether spacing. Also, the first natural frequency increases slightly, from 0.974 to 1.004 and 0.645 to 0.681,respectively. According to Eq. (33), the tether stiffness and spacing play similar roles in the model by affecting the stiffness of the total system. Therefore, adjustment of the tether spacing or stiffness could be flexibly employed to reduce the dynamic pressure according to actual marine geological survey results.

6.6 Influence of porous medium permeability

To illustrate the influence of the permeability k of the sediment on the dynamic pressure p acting on the SFT, four values of k (0.1, 0.01, 0.001, and 0.0001 m/s) were analyzed while keeping the other parameters the same. Figure 10 shows the results for the cases with different permeability values, revealing a decrease with increasing angle of incidence of the P-wave. High permeability of the porous medium causes high dynamic pressure at low angle of incidence of the P-wave. However, at higher angle of incidence,the permeability effects differ for the two types of porous medium. The dynamic pressure with the fully saturated porous medium tends to be the same for the different permeability cases. In contrast, the dynamic pressure for the high permeability case is smaller than that with low permeability for the partially saturated porous medium at high angle of incidence of the P-wave. This means that partially saturated sediment with high permeability may accelerate the release of dynamic pressure. An increase in the permeability of the sediment results in a reduction of the dynamic pressure,except for the partially saturated sediment with a high angle of incidence of the P-wave.

Fig. 9 Effect of tether distribution coefficient αc on dynamic pressure(h1/H = 0.1, h3/H = 0.3, θ = 0°, K = 8.0 × 108 kN/m), a saturation, b partially sat. 99.5%

Fig. 10 Effect of permeability of porous medium on dynamic pressure (h1/H = 0.1, h3/H = 0.3), a saturation, b partially sat. 99.5%

7 Conclusions

Analytical solutions for the dynamic pressure acting on an SFT under P-wave incidence were investigated, paying particular attention to the effect of marine sediment in the theoretical model. Fully and partially saturated porous media were simulated in the analysis based on Biot’s the ory.The following conclusions can be drawn from the obtained results:

1. Marine sediment affects the P-wave propagation in the seawater and changes the dynamic pressure acting on the SFT. Ignoring the marine sediment will cause deviation of the calculations and underestimate the dynamic pressure. Partially saturated sediment will amplify the dynamic pressure loading on the SFT, and the resonance frequency will increase slightly with the fully saturated sediment. When the angle of incidence of the P-wave increases, the rate of decline of the dynamic pressure is faster with the partially than fully saturated sediment.

2. The influence of sediment on the dynamic pressure acting on the SFT depends on the position of SFT. The resonance peak of the dynamic pressure decreases with deepening SFT position. The results for both the fully and partially saturated porous media show the same variation law.

3. The tether stiffness and spacing, as important design parameters of the SFT, can be employed to improve the dynamic pressure acting on the SFT under P-wave incidence. Increasing the tether stiffness or decreasing the tether spacing will decrease the dynamic pressure value.

4. The permeability of the sediment affects the dynamic pressure acting on the SFT under P-wave incidence.Partially saturated sediment with high permeability may accelerate the release of dynamic pressure at high angle of incidence of the P-wave and result in lower pressure.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China (Grants 51541810 and 51279178) and the Fundamental Research Funds for the Central Universities (Grant 2018QNA4032).

Append ix A

The elements of the matrix a are listed below: