Hydrodynamic pressures analysis of submerged floating tunnel under coaction of seismic SV wave and ocean wave

Abstract

Submerged floating tunnel (SFT) is an innovative traffic structure for crossing long waterway or deep straits. To investigate the hydrodynamic pressure acting on the SFT under coaction of seismic SV wave and ocean wave, a theoretical model is developed for considering the effect of marine sediment. According to the displacement potential functions and linear wave theory, the reflection and refraction coefficients of wave action in different media are derived. Numerical examples are illustrated to analyze the hydrodynamic pressure distribution. The results show that the hydrodynamic pressure has a spatial difference along the SFT longitudinal direction when the structure operating in the water subjects different incident phase ocean wave. Wave incidence frequency not only changes the loading action mode but also affects the amplitude of the hydrodynamic pressure. Besides, due to the ocean wave loading combined, the effect of saturation on the hydrodynamic pressure load on the structure is not obvious. And the dense arrangement of tethers is recommended to the structure construction.

Keywords

submerged floating tunnel hydrodynamic pressure porous medium SV wave airy wave

Introduction

With the development of modern society, the fast, smooth and convenient transportation facilities are expected to connect different economic hubs. The vast water fjords are the challenges that must be overcome in the construction systematic road network. Submerged floating tunnel (SFT) as a potential novel structure has been designed for crossing sea channels and large waterways. Compared to immerse tunnels and long span bridges over the sea, SFT does not require massive underground excavation and deep underwater piers foundation (Mazzolani, 2008). Besides, SFT has the advantages of stronger spanning capacity, lower unit engineering cost, and all weather operation ability, which has received the attention of engineering circles and scientific scholar from the all over the world (Xiang, 2016).

Due to the instantaneous suddenness and huge destructiveness, the seismic load is a key factor threatening the safety of the underwater structure. Lee et al. (2016) studied the two-dimensional seismic behaviors of a rectangular cross-section SFT and concluded that the energy absorption by the flexible seabed influence the structure response. Di Pilato et al. (2008) and Martinelli et al. (2011) once developed a full three-dimensional (3-D) model for spatial SFT to analyze the multi-support seismic input, where the soil springs are adopted to simulate the connections between the anchor cables and structure foundations. Xiang et al. (2015) proposed a torsion boundary structural model to analyze the transverse seismic response of SFT. Meantime, Jin et al. (2021) presented the optimization process to mitigate the vibration of the SFT under seismic action. However, as an underwater structure, SFT have to be affected by the frequent environmental waves. It is necessary to consider the wave influence in the structure behavior assessment. For the dynamic response of the SFT under the action of earthquake and ocean wave, Sun et al. (2011) formulated a numerical model for the tether nonlinear random vibration based on the sag-to-span ratio. Xie et al. (2021) considered the transmission effect of canyon water to the original horizontal seismic wave input and analyzed the dynamic response characteristics of SFT-Canyon water system under horizontal sloshing motion. Besides, Wu et al. (2018) took the SFT cable as research object and the parameter sensitivity study of the cable under hydrodynamic force and earthquake excitations has been carried. Akbarzadeh et al. (2021) and Mirzapour et al. (2017) assessed the spatial vibration and spectrum response of the SFT under asynchronous ground excitation. Actually, the excited water will exert hydrodynamic pressure on the structure under the seismic ground motion, which results in higher stress and larger deformation. This amplification effect has been predicted in the seaquake analysis carried by Martinelli et al. (2016) and Shekari et al. (2022). Although these cited models have explained and predicted some dynamic phenomena about the SFT, the mechanism of the seismic and ocean wave transmission is not developed in detail.

Hydrodynamic pressure caused by seismic wave plays an important role in structure dynamic analysis. Takamura et al. (2003) investigated the response of large-scale floating structures under seismic wave, which provided a reference for theoretical analysis. Taking wind-power-generating system as an object, Lee et al. (2015) deduced hydrodynamic pressure on the structure under the vertical ground motion. The seismic waves originate from the hypocenter and propagate in the medium layers as compressive waves (P waves) and shear waves (SV waves). Wang et al. (2004 & 2009) derived the reflection and refraction coefficients of different mediums on rigid dam under seismic P waves and SV waves. Due to tidal scouring, various layers of alluvium and sediment form on the sea bottom. Ye et al. (2015) once pointed out the possibility of the seabed liquefaction under earthquake loading, which results in a more severe threat to the underwater structure. Considering the marine sediment effect, Lin et al. (2019) calculated the hydrodynamic pressure acting on the SFT under seismic P waves. Feng et al. (2016 & 2017) improved traditional transmission and reflection matrix (TRM) method and studied the influence of sediment properties, such as sediment thickness, sediment layer number, and sediment real shear modulus, on the seismic wave transmission. However, the above-mentioned research only talks about the local position of the structure and has not combined the ocean wave effect in the hydrodynamic pressure evaluation. As for a spatial structure, analyzing the changes in the hydrodynamic pressure caused by the combined effect along the longitudinal direction of the SFT can help engineers design the structure.

The objective of this work is to theoretically investigate the hydrodynamic pressure acting on the SFT under seismic SV wave and ocean wave. The porous medium layer is adopted to simulate the marine sediment, and the whole model is established in the plane. By using the displacement potential function, the reflection and refraction coefficients of a SV-wave in different media are derived. Numerical examples are used to show the distribution characteristics of hydrodynamic pressure along water depth. Besides, the thickness, degree of saturation, and the incident wave frequency on the hydrodynamic pressure applying to the structure have been discussed.

Governing equations

As shown in Figure 1, a typical SFT maintains itself in a certain depth underwater by the buoyancy of the tube structure and the tension of the anchor tethers. Both ends of the tube structure are connected to the offshore bank by the revetment structure. The other side of the anchor tether is fixed to the submarine rock with the anchor foundation. The anchor tethers are installed symmetrically in the cross section to provide stable vertical support for the tube structure. The distances between the anchor tethers along the longitudinal direction are L_i (i=1, 2, …, N) respectively. The entire structural model is placed in an ocean wave field and subjected to the seismic action. To theoretically analyze the hydrodynamic pressure acting on the SFT, some further assumptions are given to simplify the model. It supposes that the seawater is the ideal compression fluid, and the half-space seabed is a linear and impermeable elastic solid. Based on Biot theory, the linear porous medium is adopted to simulate the sediment on the seabed. Meantime, the tube structure of the SFT is regarded as a rigid body, which divides the seawater into two parts named the upper and the lower water layer. According to the isotropic nature of hydrodynamic pressure, the ocean wave occurs on the free surface of the upper water layer and affects the hydrodynamic pressure on the structure. The anchor tethers are assumed as a series of elastic springs in the lower water layer. Due to the small-strain linear elastic hypothesis, the sag effect of the anchor tethers is not considered, and the elastic stiffness of the tether is regarded as a constant. The whole theoretical analysis model is established in the 2-D plane, as shown in Figure 2.

Figure 1.

Schematic diagram of the Submerged floating tunnel.

Figure 2.

Theoretical analysis model of the Submerged floating tunnel under coaction of seismic SV wave and ocean wave.

The origin of the coordinate system is set at the water and sediment interface, and the x-axis and z-axis point the rightward direction and the upward direction respectively. The depth of the seawater is H, where z = 0, z = h₁, and z = h₂ represent the interfaces separating the elastic solid, porous medium, rigid body, and ideal water, respectively. The centerline of the ocean wave coincides with the free surface. The model is assumed to be infinitely long in the x-axis, and the interfaces are regarded as flat in the investigations described herein.

Linear elastic rock foundation

The governing equation for linear elastic rock foundation can be written as

(λ_{1} + μ_{1}) \nabla \nabla \cdot u_{1} + μ_{1} \nabla^{2} u_{1} = ρ_{1} {\ddot{u}}_{1}

(1)

where λ₁ and μ₁ are Lame’s constants of the elastic rock foundation; ρ₁ denotes the solid density;

u_{1}

and

{\ddot{u}}_{1}

are the displacement and acceleration vectors, respectively. ∇ and ∇² are the gradient and Laplace operators.

Based on the Helmholtz theorem, the displacement vectors for the elastic solid $u_{1}$ is of the form:

u_{1} = \nabla ϕ_{1} + \nabla \times ψ_{1}

(2)

where

ϕ_{1}

ψ_{1}

are scalar and vector displacement potentials, respectively.

Linear porous medium (fully or partially saturated)

Following Biot theory (1956a & 1956b), the motion equations of the linear porous medium can be written in terms of the displacement vectors of the solid skeleton and the pore fluid as

G \nabla^{2} u_{2} + \nabla [(A + G) \nabla \cdot u_{2} + Q \nabla \cdot U_{2}] = ρ_{2 s s} {\ddot{u}}_{2} + ρ_{2 s f} {\ddot{U}}_{2} + b ({\dot{u}}_{2} - {\dot{U}}_{2})

(3)

\nabla [Q \nabla \cdot u_{2} + R \nabla \cdot U_{2}] = ρ_{2 s f} {\ddot{u}}_{2} + ρ_{2 f f} {\ddot{U}}_{2} - b ({\dot{u}}_{2} - {\dot{U}}_{2})

(4)

- n p_{2} = Q \nabla \cdot u_{2} + R \nabla \cdot U_{2}

(5)

Where

u_{2}, {\dot{u}}_{2}, {\ddot{u}}_{2}, U_{2}, {\dot{U}}_{2}

, and

{\ddot{U}}_{2}

denote the displacement, velocity, acceleration vectors of the solid skeleton and the pore fluid, respectively; p₂ is the pressure of the pore fluid; ρ_2ss = (1−n)ρ_2s + ρ_2a, ρ_2sf = −ρ_2a, ρ_2ff = nρ_2f + ρ_2a, in which ρ_2s, ρ_2f, and ρ_2a denote the solid skeleton density, pore fluid density and added apparent mass, respectively; Besides, b = ρ_2fgn²/k, g is the gravity acceleration; n is the porosity of the porous medium; k is Darcy’s permeability coefficient (assumed to be frequency independent)(Bougacha, 1991). When the solid grain is assumed to be incompressible, Biot’s constants can be obtained from following equations, R = nK₂, Q = (1 − n)K₂, G = μ₂ and A = λ₂ + (1 − n)₂K₂/n, in which λ₂ and μ₂ are Lame’s constants of the skeleton, and K₂ is the bulk modulus of the pore fluid. In the case of partial saturation, the effective bulk modulus of pore fluid can be obtained from the following formula (Verruijt, 1969).

\frac{1}{{K^{'}}_{2}} = \frac{1}{K_{2}} + \frac{1 - s}{p_{0}}

(6)

where

K^{'}

is the effective bulk modulus for partially saturated condition, s is the degree of saturation, and p₀ is the absolute pressure of the pore fluid.

The displacement vector $u_{2}$ depends on the scalar and vector displacement potentials, i.e., ϕ_2s and ψ _2s, as

u_{2} = \nabla ϕ_{2 s} + \nabla \times ψ_{2 s}

(7)

Similarly, U ₂ depends on ϕ_2f and ψ _2f, as

U_{2} = \nabla ϕ_{2 f} + \nabla \times ψ_{2 f}

(8)

where ∇× is the curl operator.

Ideal compressible fluid

For the ideal compressible fluid considered in this system, only compressive seismic waves can be transmitted. Hence, the motion equation of the ideal compressible fluid is

K_{w} \nabla \nabla \cdot U_{w} = ρ_{w} {\ddot{U}}_{w}

(9)

where K_w is the bulk modulus of the ideal fluid, ρ_w is the fluid density,

U_{w}

and

{\ddot{U}}_{w}

denote the displacement and acceleration vector, respectively.

Known from Figure 2, the seawater is separated by the SFT into two parts in the ZOX plane. The displacement vectors of the upper and the lower water layer are written as

U_{3} = \nabla ϕ_{3}

(10)

U_{4} = \nabla ϕ_{4}

(11)

where ϕ₃ and ϕ₄ are the scalar displacement potential of the lower and the upper water layer, respectively.

In addition, the hydrodynamic pressure caused by seismic wave in these two parts is

p_{3} (z, x, t) = - ρ_{w} \frac{\partial^{2} ϕ_{3}}{\partial t^{2}}

(12)

p_{4} (z, x, t) = - ρ_{w} \frac{\partial^{2} ϕ_{4}}{\partial t^{2}}

(13)

where t is the time, z is the depth of the seawater, and x is the positive distance from the coordination origin.

The ocean wave theory

To simply the theoretical model, the linear ocean wave theory (Airy wave) is adopted to analyze the hydrodynamic pressure acting on the structure. The Airy wave velocity potential can be expressed as

Φ_{w} = \frac{g H_{w}}{2 ω_{w}} \frac{ch k_{w} (z_{w} + d)}{ch k_{w} d} \sin (k_{w} x - ω {}_{w}t)

(14)

where H_w is the Airy wave height, z_w is the underwater depth from static sea leval, z_w = z-H, k_w is the angular wavenumber in radians per meter, related to the wavelength λ_w, k_w = 2π/λ_w. d is the depth of the ideal fluid, d=H-h₁. Meantime, the angular frequency ω_w, the angular wavenumber k_w, and the fluid depth d satisfy the linear dispersion relation.

ω_{w}^{2} = {gk}_{w} \tanh k_{w} d

(15)

According to Bernoulli’s equation, the pressure at any location in the ocean wave field can be written as

p_{w} = - ρ_{w} g z_{w} + ρ_{w} g \frac{H_{w}}{2} \frac{\cosh k_{w} (z_{w} + d)}{\cosh k_{w} d} \cos (k_{w} x - ω_{w} t)

(16)

Known from equation (16), the pressure of the ocean wave consists of the hydrostatic pressure and the hydrodynamic pressure. Because the hydrostatic pressure only affects the alignment and the initial equilibrium position of the SFT, only the hydrodynamic pressure of the ocean wave is considered in the total coaction.

Boundary conditions

Figure 2 shows the multi-layered media of compressible ideal fluid, rigid tube, porous medium, and elastic half space of rock foundation. With the Cartesian system of x-axis right and y-axis upward, the thickness of porous medium layer is h₁, and the heights of the upper and lower water layers are h₃, h₂, respectively. Besides, the rigid body is supported by the several elastic springs spaced equidistant L along the longitudinal direction, and the elastic stiffness of these springs is K. The displacement continuity and interaction balance equations are applying to the analysis model for adapting to boundary conditions.

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

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

\frac{\partial ϕ_{1}}{\partial z} - \frac{\partial ψ_{1}}{\partial x} = \frac{\partial ϕ_{2 s}}{\partial z} - \frac{\partial ψ_{2 s}}{\partial x} = \frac{\partial ϕ_{2 f}}{\partial z} - \frac{\partial ψ_{2 f}}{\partial x}

(17)

where

ψ_{1}

ψ_{2 s}

, and

ψ_{2 f}

are the only nonzero components of the displacement vector

ψ_{1}

ψ_{2 s}

,and

ψ_{2 f}

, respectively.

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

\frac{\partial ϕ_{1}}{\partial x} + \frac{\partial ψ_{1}}{\partial z} = \frac{\partial ϕ_{2 s}}{\partial x} + \frac{\partial ψ_{2 s}}{\partial z}

(18)

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

λ_{1} \nabla^{2} ϕ_{1} + 2 μ_{1} (\frac{\partial^{2} ϕ_{1}}{\partial z^{2}} - \frac{\partial^{2} ψ_{1}}{\partial x \partial z}) = A \nabla^{2} ϕ_{2 s} + 2 G (\frac{\partial^{2} ϕ_{2 s}}{\partial z^{2}} - \frac{\partial^{2} ψ_{2 s}}{\partial x \partial z}) + Q \nabla^{2} ϕ_{2 s} + (Q + R) \nabla^{2} ϕ_{2 f}

(19)

(4) The equilibrium of the tangential traction on the solid and on the porous skeleton

μ_{1} (2 \frac{\partial^{2} ϕ_{1}}{\partial x \partial z} + \frac{\partial^{2} ψ_{1}}{\partial z^{2}} - \frac{\partial^{2} ψ_{1}}{\partial x^{2}}) = G (2 \frac{\partial^{2} ϕ_{2 s}}{\partial x \partial z} + \frac{\partial^{2} ψ_{2 s}}{\partial z^{2}} - \frac{\partial^{2} ψ_{2 s}}{\partial x^{2}})

(20)

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

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

(1 - n) (\frac{\partial ϕ_{2 s}}{\partial z} - \frac{\partial ψ_{2 s}}{\partial x}) + n (\frac{\partial ϕ_{2 f}}{\partial z} - \frac{\partial ψ_{2 f}}{\partial x}) = \frac{\partial ϕ_{3}}{\partial z}

(21)

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

(A + Q) \nabla^{2} ϕ_{2 s} + 2 G (\frac{\partial^{2} ϕ_{2 s}}{\partial z^{2}} - \frac{\partial^{2} ψ_{2 s}}{\partial x \partial z}) + (Q + R) \nabla^{2} ϕ_{2 f} = ρ_{w} \frac{\partial^{2} ϕ_{3}}{\partial t^{2}}

(22)

(3) The equilibrium between the pore fluid pressure and the lower water layer

Q \nabla^{2} ϕ_{2 s} + R \nabla^{2} ϕ_{2 f} = n ρ_{w} \frac{\partial^{2} ϕ_{3}}{\partial t^{2}}

(23)

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

G (2 \frac{\partial^{2} ϕ_{2 s}}{\partial x \partial z} + \frac{\partial^{2} ψ_{2 s}}{\partial z^{2}} - \frac{\partial^{2} ψ_{2 s}}{\partial x^{2}}) = 0

(24)

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

α_{c} \frac{K_{c}}{L} (\frac{\partial ϕ_{3} (x, h_{1} + h_{2}, t)}{\partial z} - \frac{\partial ϕ_{3} (x, h_{1}, t)}{\partial z}) = ρ_{w} \frac{\partial^{2} ϕ_{3} (x, h_{1} + h_{2}, t)}{\partial t^{2}} - ρ_{w} \frac{\partial^{2} ϕ_{4} (x, h_{1} + h_{2}, t)}{\partial t^{2}}

(25)

where α_c is the tether distribution coefficient. For an SFT with equidistant tethers, α_c equals 1. Due to the SFT is assumed as a rigid body, the displacement of the SFT is equivalent to the deformation of the tethers.

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

\frac{\partial ϕ_{3} (x, h_{1} + h_{2}, t)}{\partial z} = \frac{\partial ϕ_{4} (x, h_{1} + h_{2}, t)}{\partial z}

(26)

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

The equilibrium of the fluid pressure (Taking the centerline of the ocean wave as the baseline)

\frac{\partial ϕ_{4} (x, H, t)}{\partial z} = 0

(27)

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

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

\frac{\partial ϕ_{1}}{\partial z} - \frac{\partial ψ_{1}}{\partial x} = \frac{\partial ϕ_{3}}{\partial z}

(28)

(2) The equilibrium of the normal traction on the solid and on the lower water layer

λ_{1} \nabla^{2} ϕ_{1} + 2 μ_{1} (\frac{\partial ϕ_{1}}{\partial z^{2}} - \frac{\partial^{2} ψ_{1}}{\partial x \partial z}) = ρ_{w} \frac{\partial^{2} ϕ_{3}}{\partial t^{2}}

(29)

(3) The equilibrium of the tangential traction on the solid

μ_{1} (2 \frac{\partial^{2} ϕ_{1}}{\partial x \partial z} + \frac{\partial^{2} ψ_{1}}{\partial z^{2}} - \frac{\partial ψ_{1}}{\partial x^{2}}) = 0

(30)

Formulation of the system

Known from Figure 2, a plane SV wave with an angular frequency ω and an angle of incidence θ crosses the whole model from the half-space elastic solid. The reflected P wave and SV wave occur in the region where z < 0. In the region 0 < z < h₁, the transmitted P_I wave, P_II wave and SV wave as well as the reflected P_I wave and SV wave are generated. Besides, only the P wave propagates upward and downward in the lower and upper water layers. All wave numbers in the x direction are the same apart from the upper fluid layer. Since rigid body motion of SFT is considered, the upper fluid layer only propagates in the z direction.

Displacement potential function

For the half-space elastic solid (z ≤ 0)

ϕ_{1} = A_{1 r} exp [i l_{1 P z} z+ i (ω t - l_{x} x)]

(31)

ψ_{1} = [B_{1 i} exp (- i l_{1 S z} z) + B_{1 r} exp (i l_{1 Sz} z)] \exp [i (ω t - l_{x} x)]

(32)

For the porous medium (0 < z ≤ h₁)

ϕ_{2 s} = [A_{2 t 1} \exp (- i l_{{2 P}_{I} z} z) + A_{2 r 1} \exp (i l_{{2 P}_{I} z} z) + A_{2 t 2} \exp (- i l_{{2 P}_{II} z} z) + A_{2 r 2} \exp (i l_{{2 P}_{II} z} z)]] \exp [i (ω t - l_{x} x)]

(33)

ψ_{2 s} = [B_{2 t} \exp (- i l_{2 S z} z) + B_{2 r} \exp (i l_{2 Sz} z)] \exp [i (ω t - l_{x} x)]

(34)

ϕ_{2 f} = {δ_{2 P_{I}} [A_{2 t 1} \exp (- i l_{{2 P}_{I} z} z) + A_{2 r 1} \exp (i l_{{2 P}_{I} z} z)] + δ_{2 P_{II}} [A_{2 t 2} \exp (- i l_{{2 P}_{II} z} z) + A_{2 r 2} \exp (i l_{{2 P}_{II} z} z)]} \exp [i (ω t - l_{x} x)]

(35)

ψ_{2 f} = δ_{2 S} [B_{2 t} \exp (- i l_{2 S z} z) + B_{2 r} \exp (i l_{2 S z} z)] \exp [i (ω t - l_{x} x)]

(36)

For the lower water layer (h₁<z≤h₂)

ϕ_{3} = [A_{3 t} \exp (- i l_{3 P z} z) + A_{3 r} \exp (i l_{3 P z} z)] \exp [i (ω t - l_{x} x)]

(37)

For the upper water layer (h₂<z≤h₃)

ϕ_{4} = [A_{4 t} \exp (- i l_{3 P} z) + A_{4 r} \exp (i l_{3 P} z)] \exp [i (ω t - l_{x} x)]

(38)

where l denotes the wave number, A and B represent the amplitude of P wave and SV wave, respectively, δ is the amplitude ratio of the solid skeleton potential to the pore fluid potential; the subscripts “i”, “r”, and “t” are the incidence, reflection, and transmission, respectively; the subscripts “P”, “S”, “P_I”, and “P_II” denote P, SV, P₁, and P₂ waves in the multilayered media; the subscripts “1”, “2”, “3”, “4” indicate the solid, the porous medium, the lower water layer, and the upper water layer; the subscripts “x” and “z” denote the x- and z-components, respectively. The SV wave number has the relation about l_1Sz=l_xcot θ.

Due to the results of Deresiewicz and Skalak (1963) and Wang (2013), δ_PI, δ_PII, δ_S are given as

δ_{{2 P}_{I}} = - \frac{Q l_{2 P_{I}}^{2} - ρ_{2 s f} ω^{2} - i b ω}{R l_{2 P_{I}}^{2} - ρ_{2 f f} ω^{2} + i b ω}

(39)

δ_{{2 P}_{II}} = - \frac{Q l_{2 P_{II}}^{2} - ρ_{2 s f} ω^{2} - i b ω}{R l_{2 P_{II}}^{2} - ρ_{2 f f} ω^{2} + i b ω}

(40)

δ_{2 S} = \frac{- ρ_{2 s f} ω + i b ω}{ρ_{2 f f} ω - i b ω}

(41)

The meanings of other parameters in the formula are as follows

l_{2 P_{I}}^{2}, l_{2 P_{II}}^{2} = \frac{- d \mp \sqrt{d^{2} - 4 c e}}{2 c}

(42)

l_{2 S}^{2} = \frac{(ρ_{2 s s} ρ_{2 f f} - 2 ρ_{2 s f}^{2}) ω^{3} - i b (ρ_{2 s s} + ρ_{2 f f} + 2 ρ_{2 s f}) ω^{2}}{G (ρ_{2 f f} ω - i b)}

(43)

c = (A+ 2 G) R - Q^{2}

(44)

d = - [(A + 2 G) ρ_{2 f f} + R ρ_{2 s s} - 2 Q ρ_{2 s f}] ω^{2} + i b (A + 2 G + R + 2 Q) ω

(45)

e = (ρ_{2 s s} ρ_{2 f f} - ρ_{2 s f}^{2}) ω^{4} - i b (ρ_{2 s s} + ρ_{2 f f} + 2 ρ_{2 s f}) ω^{3}

(46)

Coefficient equation calculation

Based on the displacement potential function and the boundary conditions, the coefficient equation can be acquired by substituting equations (31)∼(38) into Equations (17)∼(30). The amplitude constants of P- and SV- waves A_1r, B_1r, A_2t1, A_2t2, B_2t, A_2r1, A_2r2, B_2r, A_3t, A_3r, A_4t, and A_4r are obtained

a {[A_{1 r}, B_{1 r}, A_{2 t 1}, A_{2 t 2}, B_{2 t}, A_{2 r 1}, A_{2 r 2}, B_{2 r}, A_{3 t}, A_{3 r}, A_{4 t}, A_{4 r}]}^{T} = B_{li} f

(47)

where B_1i is the known constant, a is a 12×12 square matrix, and f is a column matrix with 12 elements.

It means the model will not be affected by the porous medium that the thickness of the porous medium h₁=0. So the simultaneous equations for the analytical model can be reduced into

a^{'} {[A_{1 r}, B_{1 r}, A_{3 t}, A_{3 r}, A_{4 t}, A_{4 r}]}^{T} = B_{l i} f^{'}

(48)

where a ^′ is a 6×6 square matrix, and f ^′ is a column matrix with six elements. By solving equations (47) and (48), the amplitude of wave reflection and transmission in different media can be solved.

Combined hydrodynamic pressure

The dynamic pressure loading on the SFT due to SV wave incidence should consider both ocean wave and seismic wave combined interaction. Because the phase difference of the hydrodynamic pressure wave produced by the seismic wave and the ocean wave is uncertain, a random phase difference ought to be considered during the calculation.For the lower water layer (h₁ < z ≤ h₂)

\begin{array}{l} p_{3} = ρ_{w} ω^{2} [A_{3 t} \exp (- i l_{3 P z} z) + A_{3 r} \exp (i l_{3 P z} z)] \exp [i (ω t - l_{x} x + φ_{0})] \\ + ρ_{w} g \frac{H_{w}}{2} \frac{\cosh k_{w} (z_{w} + d)}{\cosh k_{w} d} \cos (k_{w} x - ω_{w} t) \end{array}

(49)

For the upper water layer (h₂ < z ≤ h₃)

\begin{array}{l} p_{4} = [A_{4 t} \exp (- i l_{3 P} z) + A_{4 r} \exp (i l_{3 P} z)] \exp [i (ω t - l_{x} x + φ_{0})] \\ + ρ_{w} g \frac{H_{w}}{2} \frac{\cosh k_{w} (z_{w} + d)}{\cosh k_{w} d} \cos (k_{w} x - ω_{w} t) \end{array}

(50)

where φ₀ is the phase difference between the seismic wave and the ocean wave. Because φ₀ is a uniformly distributed random number belong to (0, 2π), the mean value π has been applied in the analysis without loss generality. Besides, the hydrostatic pressure is ignored here.

Numerical example and analysis

Because the SFT is still in the feasibility study stage, the basic structural parameters take reference from the related design study projects (Lin et al., 2019; Luo et al., 2021). Meanwhile, the linear Airy wave theory is adopted for analysis. The properties of different media in the theoretical model come from the work research results (Wang et al., 2009). The entire SFT structure underwater is affected by the combine action of seismic wave and ocean wave. In each case, the amplitudes of the potential for reflection and transmission wave are acquired by calculating the inverse matrixes of a or a ^′. The hydrodynamic pressure distribution underwater is known from equations (49) and (50).

Basic parameter information

The properties of the ideal fluid, the elastic half-space solid, and the porous medium, are as follows: the water bulk modulus K_w and density ρ_w are 2.0×10⁹ Pa and 1000 kg/m³, respectively. Lame’s constant of the elastic half-space that λ₁ is 1.53×10¹⁰ Pa and μ₁ is 7.65×10⁹ Pa, and density ρ₁ is 2483 kg/m³. Besides, the porous medium has a porosity n to be 0.6, a permeability coefficient k to be 0.001 m/s. Lame’s constant of the skeleton grains that λ₂ is 1.7975×10⁷ Pa and μ₂ is 7.704×10⁶ Pa, and density ρ_2s is 2640 kg/m³. The pore fluid density ρ_2f and the added apparent mass ρ_2a are 1000 kg/m³ and 0. Here, the saturation degree of porous medium s is 0.995, and the skeleton grains are assumed as incompressible, the corresponding bulk modulus of pore fluid and Biot’s constants can be obtained by Section 2.2, shown as Table 1.

Table 1.

Biot’s parameters of porous media.

No.	s	K₂ or K′₂(GPa)	R(GPa)	Q(GPa)	G(MPa)	A(GPa)
1	1.000	2.0000	1.2000	0.8000	7.7040	0.5513
2	0.995	0.2106	0.1264	0.0842	7.7040	0.0741
3	0.990	0.1126	0.0667	0.0448	0.7040	0.0476

The position of SFT is 0.3H below the seawater, which totally has 120m deep depth. The anchor tethers install on the SFT along the longitudinal direction with 1.0H equidistance. Taking this as a standard, the anchor cable distribution coefficient α_c is set to be 1.0. Each pair of tethers provides a support stiffness K_c to be 8×10⁸ N/m. As for the ocean wave field, the incidence wave height H_w and the incidence frequency f_w are 4m and 0.06 Hz respectively. It is just corresponding to the frequency incidence of seismic wave. And the incidence amplitude and angle of seismic wave equal to 0.25 g and 45°, respectively. Some other related parameters will be stated separately in the result discussion.

Hydrodynamic pressure of SFT subjected to the combined action

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, Figure 3 show the hydrodynamic pressure distribution along the seawater depth under seismic SV wave and ocean wave combined action. The hydrodynamic pressure (P_SV) generated by the seismic SV wave gradually increases with water depth through Figure 3 (a). On the contrary, the hydrodynamic pressure (P_AW) results from the ocean wave decrease nonlinearly, which affects the total hydrodynamic pressure (P_TO) distribution. Known from Figure 3 (b), the hydrodynamic pressure (P_SV) is affected by the SFT and the porous medium. There are two mutations on the hydrodynamic pressure curve at the position of h₁ and h_2, respectively. Due to the conformability of deformation and the stress balance, the interface of the different mediums has the amplification phenomenon. Wang et al. (2009) and Lin et al. (2019) once captured the amplification phenomenon during the seismic wave transmission. Besides, some part of the hydrodynamic pressure also balanced by the SFT tether support force. The transmission route of the hydrodynamic pressure of SFT subjected to the seismic SV wave and ocean wave combined action is elaborated.

Figure 3.

The hydrodynamic pressure distribution of the different depths. (a) No Submerged floating tunnel and no porous medium (b) SFT and porous medium.

As well known, the horizontal position is an important factor affecting the ocean wave incident phase. Based on equations (49) and (50), the incident phase along the longitude direction of the SFT will further change the distribution of the hydrodynamic pressure. Figure 4 gives the hydrodynamic pressure distribution on the various horizontal positions and incidence frequencies cases. From Figure 4 (a) and 4(b), the hydrodynamic pressure caused by ocean waves with the same incident frequency differs along the SFT longitudinal direction. And this difference is greatly affected by the wave incident frequency through Figure 4(c) and 4(d).

Figure 4.

The hydrodynamic pressure distribution of various horizontal positions and incidence frequencies. (a) fw=0.06 Hz, x=0.25H (b) fw=0.06 Hz, x=0.75H (c) fw=0.08 Hz, x=0.25H (d) fw=0.08 Hz, x=0.75H.

As for the SFT spanning the entire water area, the various hydrodynamic pressure at the horizontal positions will change the internal force state of the structure. In the case where the incident frequency is 0.06 Hz, the hydrodynamic pressure of the SFT at the horizontal positions 0, 0.25H, and 0.75 Hz are 1.17×10⁵pa, 1.10×10⁵pa, and 0.68×10⁵pa, respectively. Compared with the hydrodynamic pressure at the same positions but with the incident frequency 0.08 Hz, the values are 2.20×10⁵pa, 1.71×10⁵pa, and −0.70×10⁵pa. The hydrodynamic pressures generated by seismic pressure are 4.83×10⁵pa and 4.84×10⁵pa at different incident frequency cases. It can be found that the influence of ocean wave on the structure hydrodynamic pressure is closely related to the incident frequency. Besides, the characteristic of the hydrodynamic pressure of the ocean wave and seismic wave varying with water depth can help to obtain the best water depth position for the SFT in the engineering sense.

Influence of the ocean wave action

In this section, the ocean wave height H_w varies from 2m to 8m. Figure 5 shows the effect of ocean wave height on the hydrodynamic pressure of the SFT in an H distance. Compared with the impact of seismic waves on the hydrodynamic pressure (P_{SV, A}) of the structure, the ocean wave amplitude is more obviously affects the hydrodynamic pressure loading on the structure. It means that the hydrodynamic pressure will be underestimated when ignoring the impact of ocean wave action. Besides, the ocean wave height does not change the hydrodynamic pressure distribution mode along the longitudinal direction of the SFT. Due to the mode of hydrodynamic pressure, the SFT has to bear the shear deformation along its length.

Figure 5.

Effect of ocean wave height on hydrodynamic pressure of the Submerged floating tunnel position.

Figure 6 shows the hydrodynamic pressure along the SFT longitudinal direction at different wave incidence frequencies. It can be observed that the wave incidence frequency not only changes the loading action mode but also affects the amplitude of the hydrodynamic pressure. According to equations (14) and (15), the incidence frequency is related to the wavenumber and the wavelength. So only a slight change will have a very different influence on the hydrodynamic pressure loading on the structure. For example, when the incident frequency increases from 0.06 Hz to 0.10 Hz, the hydrodynamic pressure grows more than 3 times. Meantime, a higher wave incident frequency means a more intensive ocean wave action, which also represents a more serious threat to the structure than the seismic wave action.

Figure 6.

Effect of wave incidence frequency on hydrodynamic pressure of the Submerged floating tunnel position.

Influence of the porous medium

To investigate the characteristics of the saturation degree of the porous medium, the thickness and wave incident frequency is fixed, and the saturated degree varies from 0.990 to 1. From Figure 7, the hydrodynamic pressure caused by the seismic SV wave has been more influenced in the porous medium. As the saturation of the porous medium increases, the seismic SV wave part of hydrodynamic pressure has grown a little. Combined with the results of Figure 3, the saturated porous medium has the biggest hydrodynamic pressure to be 5.01×10⁵ Pa at the interface between the elastic solid and porous medium (z = 0). The results of other saturated cases like 0.990 and 0.995 are 4.31×10⁵ Pa and 4.62×10⁵ Pa, respectively. Meantime, the hydrodynamic pressures are 4.82×10⁵ Pa, 4.83×10⁵ Pa, and 4.85×10⁵ Pa at the interface of the SFT position (h₂ = 0.7H) corresponding to the saturated degree 0.990, 0.995, and 1. Exactly, the physical reason for the pressure difference between the unsaturated medium and the saturated medium is the influence of porous fluid. The sediment would absorb the energy during the seismic wave propagation.

Figure 7.

The hydrodynamic pressure distribution of various saturated porous media. (a) s = 1.0 (b) s = 0.990.

In this section, the internal mechanism of the seismic SV wave transmits in different saturated porous media is revealed through the transmission coefficients. Taking the first natural frequency ω₁of the ideal fluid layer as a reference, the transmission coefficients at two different interfaces that vary from relative incidence frequency (ω/ω₁) can be obtained. From Figure 8(a), the case with saturation sediment at ω/ω₁=1 has a larger transmission value. That means the energy is not lost by the porous fluid. So the variation pressure of the saturated medium on the interfaces is smaller than that of the unsaturated. Besides, a sudden drop and steep rise phenomena will appear on the transmission coefficient curve at the incident frequency of the wave approaches 0.5ω₁ and 1.5ω₁.Due to the difference of the effective bulk modulus and Biot’s constants, the transfer behaviors in those saturated porous media show the difference. In contrast, the full saturated sediment has a higher efficiency in transmitting the seismic wave. The reason can be explained as the full saturated sediment at the interface has a greater hydrodynamic pressure. As for the impact on the structure, the low wave incident frequency has the larger transmission coefficient in the partially saturated sediment case. While the greater the incident frequency, the higher hydrodynamic pressure generated on the SFT with the fully saturated sediment. In addition, the non-smooth characteristics are also preserved, as shown in Figure 8(b).

Figure 8.

The transmission coefficients of SV wave in different saturated porous media with incidence frequencies. (a) Interface between the half-space elastic solid and the porous medium (z = 0) (b) Interface between the lower and upper water layer at the Submerged floating tunnel position (z = h₁ + h₂).

Influence of the structure tether distribution

Similar to the traditional cable-stayed bridge structure, the arrangement and spacing of tethers will affect the mechanical behavior of the SFT. In this section, the tether distribution coefficient α_c and the water depth of the SFT location are discussed. Figure 9(a) shows that the total hydrodynamic pressure loading on the structure decreases with deepening the SFT location. Taking the hydrodynamic pressure of the case with α_c=1.0 and h₃=0.3H as the benchmark, the case with the tether distribution coefficient α_c=1.2 always suffers the greater value. However, the seismic pressure of the case with smaller distribution coefficient is lower. For the whole system, the seismic effect on the structure can be quickly transmitted to the tethers and cut the hydrodynamic pressure on the structure. Meantime, this effect weakens with the depth of the structure location. Considering the actual construction condition, the lower tether distribution coefficient or the dense tether distribution is preferred for the SFT.

Figure 9.

Effect of structure tether distribution on the hydrodynamic pressure. (a) Effect of the Submerged floating tunnel location on the hydrodynamic pressure (b) Effect of the SFT location on the seismic pressure.

Conclusion

To investigate the hydrodynamic pressure acting on the SFT under seismic SV wave and ocean wave, a theoretical analysis model is proposed considering the marine sediment effect. The porous medium layer is adopted to simulate different saturated degree sediment, and the whole model is established in the plane. By using the displacement potential function and linear wave theory, the reflection and refraction coefficients of wave action in different media are derived. Through numerical examples, some conclusion can be derived.

(1) The hydrodynamic pressure has a spatial difference along the SFT longitudinal direction, when the structure operating in the water subjects different incident phase ocean wave. It is closely related to the wave incident frequency.

(2) Wave incidence frequency not only changes the loading action mode but also affects the amplitude of the hydrodynamic pressure. The higher ocean wave amplitude will cause larger hydrodynamic pressure loading on the structure.

(3) The saturation of the sediment will affect the transmission of seismic waves. Due to the ocean wave loading combined, the effect of saturation on the hydrodynamic pressure load on the structure is not obvious.

(4) The dense arrangement of tethers will cut the seismic influence on the structure tube, but this effect weakens with the depth of the structure location.

ORCID iD

Heng Lin https://orcid.org/0000-0002-9378-0303

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work had been supported by the Education Department of Zhejiang Province (grant no.Y202146432) and the National Natural Science Foundation of China (grant no. 52108341).

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.

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Hydrodynamic pressures analysis of submerged floating tunnel under coaction of seismic SV wave and ocean wave

Abstract

Keywords

Introduction

Governing equations

Linear elastic rock foundation

Linear porous medium (fully or partially saturated)

Ideal compressible fluid

The ocean wave theory

Boundary conditions

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

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

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

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

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

Formulation of the system

Displacement potential function

Coefficient equation calculation

Combined hydrodynamic pressure

Numerical example and analysis

Basic parameter information

Hydrodynamic pressure of SFT subjected to the combined action

Influence of the ocean wave action

Influence of the porous medium

Influence of the structure tether distribution

Conclusion

ORCID iD

Footnotes

Declaration of Conflicting Interests

Funding

Data availability statement

References

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

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

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