Vibration response of cable for submerged floating tunnel under simultaneous hydrodynamic force and earthquake excitations

Abstract

A simplified analysis model of cable for submerged floating tunnel subjected to simultaneous hydrodynamic force and earthquake-excited vibrations in an ocean environment is proposed in this study. The equation of motion of the cable is obtained by a mathematical method utilizing the Euler beam theory and the Galerkin method. The hydrodynamic force induced by earthquake excitations is formulated to simulate real seaquake conditions. Random earthquake records are generated in the time domain by the stochastic-phase spectrum method. The cable for submerged floating tunnel is then subjected to combined hydrodynamic force and earthquake excitations. A sensitivity study is performed to assess the influence of key parameters, including hydrodynamic, earthquake, and structural parameters, on the dynamic response of the cable. It has been observed that the ratio of parametric frequency to natural frequency, the direction and magnitude of earthquake excitation, the initial tension in the cable, and the damping ratio all have significant influences on the hydrodynamic and seismic response of the cable. Therefore, these effects must be considered rigorously during the design of anchor systems for submerged floating tunnel.

Keywords

cable seaquake simplified analysis model submerged floating tunnel vibration response

Introduction

Submerged floating tunnel is an innovative underwater transportation system to avoid water traffic and weather at a depth of usually 20–50 m. The tunnel floats in water, and its position is restrained at a certain distance from the sea bed by means of suitable anchoring systems, such as cables or bars. Cables for submerged floating tunnel are lightweight, very flexible, and lightly damped, the features of which make them particularly prone to vibration. Frequent vibration may lead to fatigue damage in the cable, where fatigue cracks could be formed on the cable surface to destroy the anti-corrosion system. The tensile capacity of the cable will eventually lose due to the propagation of fatigue cracks. The cable for submerged floating tunnel could experience complex dynamic forces under various ocean conditions, which is the key to the safe operation of the system.

Many attentions have been received to the vibration of cables for submerged floating tunnel. Sun et al. (2009) analyzed the nonlinear response of cables subjected to parametric excitations. Sun and Su (2011) investigated the parametric vibration of submerged floating tunnel cable under random excitations. Lu et al. (2011) studied the slack phenomena and the snap force in the cable for submerged floating tunnel under wave conditions. Seo et al. (2015) conducted a series of simplified analyses to estimate the behavior of submerged floating tunnel cables in waves and compared their calculations with experimental measurements. Cifuentes et al. (2015) implemented a numerical model to analyze the coupled dynamic response of a submerged floating tunnel with mooring lines in regular waves.

The submerged floating tunnel cable is placed in the ocean during its whole service life under complex environmental conditions, such as ocean waves, currents, and earthquakes. Therefore, the structural integrity of the tunnel is greatly affected by the coupled effect of the motion of the floating tube in water and the mooring cables. The environmental loads acting on the tunnel tube are prominent, where the existing methods (Kunisu, 2010; Lu et al., 2011; Seo et al., 2015) can provide satisfactory calculation results. However, the mechanism of the motion of cable is still not well understood, which requires further study. The mooring cable for submerged floating tunnel is vital to the safety of the whole structure, which could experience a wide range of complex dynamic forces under different environmental conditions. The suspended cable suffers a considerable extent of hydrodynamic excitations, which can last for almost the whole design life in the ocean environment. Vibrations can cause distress in the cable, eventually resulting in fatigue problems.

It is noted by Duan et al. (2010) that 85% of the total amount of earthquakes occur in the ocean. The Bohai Strait, the Qiongzhou Strait, and the Taiwan Strait in China are three potential areas to build submerged floating tunnels, which are also in the Circum-Pacific seismic belt. Hence, the performance of subsea cables in seismic zones becomes one of the most significant research topics in offshore engineering. In addition, the effect of seaquake is much more complex, which can induce hydrodynamic pressures acting on the cable (Lee et al., 2016; Qiao et al., 2008). The combined fluid and seismic motion induced by ocean hydrodynamic forces must be considered rigorously in the analysis of cables for submerged floating tunnel during earthquakes. Therefore, it is of significance to evaluate how parametrically excited vibrations influence the behavior of cable for submerged floating tunnel under combined hydrodynamic forces and earthquakes.

Although earthquakes are significantly affected on the performance of cables for submerged floating tunnel, few studies have been reported to assess the seismic response of cables except those from Di Pilato et al. (2008), Martinelli et al. (2011), and Su and Sun (2013a). In these previous studies, the effects of earthquake on the development of hydrodynamic force were usually neglected, and the seismic design spectrum was taken from design guidelines for buildings (e.g. the Chinese code for seismic design of buildings (GB50011-2010) directly. However, the seismic design spectrum for buildings is not always suitable for marine structures, and the correctness of this approach becomes questionable.

This study aims to establish a mathematical method to explore the dynamic response of cable for submerged floating tunnel under simultaneous hydrodynamic forces and earthquake excitations. A simplified analysis model to estimate the behavior of cable for submerged floating tunnel under the ocean environmental excitation is proposed. The equation of motion of the cable is obtained by a mathematical method utilizing the Euler beam theory and the Galerkin method. The hydrodynamic force induced by earthquake excitations is formulated to simulate real seaquake conditions. The random earthquake excitation in the time domain is formulated by the stochastic-phase spectrum method. An analytical model for analyzing the cable for submerged floating tunnel subjected to combined hydrodynamic forces and earthquake excitations is then developed. The sensitivity of key parameters including the hydrodynamic, earthquake, and structural parameters on the dynamic response of the cable is investigated and discussed.

Method

Equation of motion of the cable

Following previous studies (Lu et al., 2011; Su and Sun, 2013b; Sun and Su, 2011), a simplified analysis model is proposed to evaluate the motion of cables for submerged floating tunnel subjected to ocean environmental excitations. If the current velocity is assumed to be small and vortex induced vibration is not an issue (Lei et al., 2014, 2017; Sun and Su, 2011), the vibration of the riser in the cross-flow direction can be discounted. The cable can move only in the in-line direction and the lateral deflection is considered to be small. For simplification, the cable for submerged floating tunnel is modeled as a Bernoulli–Euler beam, which only allows the in-plane motion. Furthermore, a conservative assumption is adopted that the response of the cable under combined effects of hydrodynamic forces and earthquake excitations is maximized when the two actions are in the same direction. A submerged floating tunnel and the direction of earthquake wave and wave/current force are schematically shown in Figure 1, where the cable is allowed to move only in the in-line direction. It should be emphasized that the horizontal earthquake excitation is acting on the anchoring point of the cable.

Figure 1.

Schematic illustration of a submerged floating tunnel and cable.

Figure 1 shows the schematic illustration of a submerged floating tunnel and cables. In this simplified analysis model, the three-dimensional behavior is not taken into consideration. Although this assumption simplifies the physical phenomena, the two-dimensional model enables a preliminary examination of the hydrodynamic and seismic behavior of cables. The origin of the coordinate system is set at the anchored point of the cable on the sea bed. The parameter $L$ is the undeformed length of the cable, $H_{z}$ shows the vertical height above the sea bed, $u$ represents the dynamic displacement at the mid-span of the cable, and $θ$ denotes the inclination angle of the cable with respect to the sea bed.

The tube is assumed to be rigid in the analysis as the deflection of the tube is much smaller compared to the cable (Seo et al., 2015; Su and Sun, 2013b; Xiang and Chao, 2012). Due to the combined effects of mooring constraints and mutual constraints of adjacent tubes, the horizontal motion of the tube is assumed to be very small, so that the tube only oscillates in the vertical direction. For general sea states, the displacement of the structure is small, and as such, the fluid force is generally calculated at the initial position.

Hydrodynamic loads on the tube structure are expressed by the sum of inertia force and drag force as follows

F_{T} = C_{M} ρ_{w} V \frac{dv}{dt} + C_{D} ρ_{w} Av | v |

(1)

where $V$ is the volume of the structure, $A$ is the projected area of the structure, and $v$ is the velocity of water particle surrounding the tube.

Based on the previous studies (Islam and Ahrnad, 2003; Mousavi et al., 2013; Muhammad et al., 2017; Su and Sun, 2013b), the velocity of water particle surrounding the tube is less affected by the seaquake in relatively deeper sea. For a simple calculation, the velocity and acceleration of the flow at the center of the structure can be assumed to be constant along the body (Muhammad et al., 2017; Seo et al., 2015; Su and Sun, 2013b). Thus, the effect of the tunnel tube on the cable can be simplified as a constant parametric excitation during the dynamic analysis.

The main contribution of parametric excitation is the hydrodynamic loads. The heave motion is dominated by one frequency, which is determined by the physical characteristics of the floater, and can thus be expressed in the harmonic form. In fact, the frequency is closely related to the Strouhal number and current velocity (Su and Sun, 2013b; Xiang and Chao, 2012).

In this study, the effect of the tunnel tube on the cable is simplified as a parametric excitation of $T_{p} \cos ω_{p} t$ , and the ends of the cable are hinged (Lei et al., 2014; Sun and Su, 2011), where $T_{p}$ is the dynamic tension acting in the cable induced by the motion of the tube and $ω_{p}$ is the parametric excitation frequency. For simplicity, the initial tension, geometry, stiffness, and material property of the cable are assumed to be constant along the cable length.

Thus, the governing equation of motion of the cable can be written as (Lu et al., 2011; Su and Sun, 2013b; Sun and Su 2011)

\begin{matrix} E_{eq} I \frac{\partial^{4} u (z, t)}{\partial z^{4}} - (T_{0} + T_{p} \cos ω_{p} t) \frac{\partial^{2} u (z, t)}{\partial z^{2}} \\ + m \frac{\partial^{2} u (z, t)}{\partial t^{2}} + C_{s} \frac{\partial u (z, t)}{\partial t} \\ = F_{D} - Q_{s} \sin θ \end{matrix}

(2)

where $u (z, t)$ is the dynamic lateral displacement of the cable; $T_{0}$ , $m$ , $E_{eq}$ , and $I$ are the initial tension, the mass per unit length, the equivalent modulus of elasticity, and the cross-sectional moment of inertia of the cable, respectively; $C_{s}$ is the viscous damping coefficient; $F_{D}$ is the drag force applied by water per unit length when the cable oscillates; and $Q_{s}$ is the earthquake excitation force.

In order to consider the cable sag effect, an equivalent elastic modulus of the cable can be obtained following the method of Su et al. (2013) as follows

E_{eq} = \frac{E}{1 + E / E_{f}}

(3)

E_{f} = \frac{12 σ^{3}}{[γ_{1}^{2} {(L \cos θ)}^{2}]}

(4)

σ = \frac{T_{0}}{A_{c}}

(5)

where $E$ is the modulus of elasticity of the cable, $E_{f}$ is the modulus of elasticity induced by the cable sag effect, $γ_{1}$ is the buoyant unit weight of the cable, $σ$ is the stress in the cable, and $A_{c}$ is the cross-sectional area of the cable.

Hydrodynamic force under earthquake excitation

The hydrodynamic force $F_{D}$ can be expressed by Morison’s equation (Lei et al., 2014) as follows

F_{D} = \frac{1}{2} ρ_{w} D C_{D} (v - \frac{\partial u}{\partial t}) | v - \frac{\partial u}{\partial t} | + C_{M} \frac{π}{4} ρ_{w} D^{2} \frac{\partial v}{\partial t} - (C_{M} - 1) \frac{π}{4} ρ_{w} D^{2} \frac{\partial^{2} u}{\partial t^{2}}

(6)

where $C_{D}$ is the drag coefficient, $C_{M}$ is the inertia coefficient, $u$ is the displacement of the cable, and $v$ is the instantaneous flow velocity.

The hydrodynamic force induced by earthquake excitation is formulated based on the following assumptions (Datta and Mashaly, 1988; Lee and Kim, 2015; Qiao et al., 2008):

As the earthquake excitation process occurs in a short time duration, the current velocity around the cable during earthquakes is assumed to be negligible.

The wave force is neglected as the cable is located in deep water.

The coupled effect between motions in different directions is neglected.

It is reasonable to assume that sea water particles remain stationary. Under the combined effect of hydrodynamic force and earthquake excitation, the drag and inertia forces will be modified by replacing $v$ by $v + {\overset{\cdot}{x}}_{g}$ and $\partial v / \partial t$ by $(\partial v / \partial t) + {\overset{\cdot\cdot}{x}}_{g}$ , where $\partial v / \partial t$ and $\partial^{2} v / \partial t^{2}$ are the velocity and acceleration of the cable, respectively. Thus, Morison’s equation can be modified (Islam and Ahrnad, 2003; Mousavi et al., 2013; Muhammad et al., 2017; Su and Sun, 2013a) as follows

F_{D} = \frac{1}{2} ρ_{w} D C_{D} (v + {\dot{x}}_{g} - \frac{\partial u}{\partial t}) | v + {\dot{x}}_{g} - \frac{\partial u}{\partial t} | + C_{M} \frac{π}{4} ρ_{w} D^{2} (\frac{\partial v}{\partial t} + {\overset{\cdot\cdot}{x}}_{g}) - (C_{M} - 1) \frac{π}{4} ρ_{w} D^{2} \frac{\partial^{2} u}{\partial t^{2}}

(7)

In equation (7), the first term on the right-hand side denotes the drag force, the second term denotes the inertia force, and the third term denotes the added mass effect. The first term in equation (7) represents the nonlinear drag force due to the relative motion between fluid and the structure, which acts to dampen the motion of the structure. Morison’s equation cannot be utilized in the form of equation (7) for linear spectral analysis because of its suitability for nonlinear problems. In practice (Su and Sun, 2013a; Xiang and Chao, 2012; Xiang and Yang, 2017; Yang and Xiao, 2014), the water velocity is assumed to be negligible with respect to the velocity of the structure, and as such, the nonlinear drag force can be linearized by removing the term of water velocity for simplicity.

It is noted that the parameters $v$ and $u$ contain variables of $z$ and $t$ , that is, vary along the depth of the water. The sea water tends to have particle velocity, which decays exponentially with the depth of the sea. A submerged floating tunnel could be located 30 m below the sea surface, and the water depth could reach as high as about 170 m or even deeper. Since the analysis is exclusively concerned with the seismic effects, the water particle kinematics are neglected, especially in deep water (Islam and Ahrnad, 2003; Mousavi et al., 2013; Muhammad et al., 2017; Su and Sun, 2013a).

Thus, the drag and inertia forces in equation (7) are simplified by removing the term of water velocity as

\frac{1}{2} ρ_{w} D C_{D} (v + {\dot{x}}_{g} - \frac{\partial u}{\partial t}) | v + {\dot{x}}_{g} - \frac{\partial u}{\partial t} | ≃ \frac{1}{2} ρ_{w} D C_{D} ({\dot{x}}_{g} - \frac{\partial u}{\partial t}) | {\dot{x}}_{g} - \frac{\partial u}{\partial t} |

(8)

C_{M} \frac{π}{4} ρ_{w} D^{2} (\frac{\partial v}{\partial t} + {\overset{\cdot\cdot}{x}}_{g}) \approx C_{M} \frac{π}{4} ρ_{w} D^{2} {\overset{\cdot\cdot}{x}}_{g}

(9)

The hydrodynamic force under earthquake excitation can then be simplified (Datta and Mashaly, 1988; Lee and Kim, 2015; Qiao et al., 2008) by

F_{D} = \frac{1}{2} ρ_{w} D C_{D} ({\dot{x}}_{g} - \frac{\partial u}{\partial t}) | {\dot{x}}_{g} - \frac{\partial u}{\partial t} | + C_{M} \frac{π}{4} ρ_{w} D^{2} {\overset{\cdot\cdot}{x}}_{g} - (C_{M} - 1) \frac{π}{4} ρ_{w} D^{2} \frac{\partial^{2} u}{\partial t^{2}}

(10)

Based on the previous studies (Islam and Ahrnad, 2003; Mousavi et al., 2013; Muhammad et al., 2017; Su and Sun, 2013a), the difference between the complete hydrodynamic force under earthquake excitation expressed in equation (7) and the simplified one in equation (10) was not significant. Therefore, the simplified solution of hydrodynamic force can be used in the preliminary design of submerged floating tunnel.

Earthquake excitation in the time domain

The earthquake excitation in the time domain can be expressed as

Q_{s} = m {\overset{\cdot\cdot}{x}}_{g}

(11)

where ${\overset{\cdot\cdot}{x}}_{g}$ is the acceleration of the input ground motion, which can be derived from the earthquake records obtained from seismic stations at a given location or from an artificial stochastic process using the trigonometric series method.

Since the random excitation of earthquakes is transient, the non-stationary characteristics of random excitation should be considered. Using the evolutionary theory of power spectrum density (Bi and Hao, 2012), the non-stationary random process can usually be written as the product of a stationary random process and a deterministic slowly varying modulation function

{\overset{\cdot\cdot}{x}}_{g} = a (t) w (t)

(12)

where $a (t)$ is a stationary process and $w (t)$ is the deterministic envelope function with a maximum value of 1.0.

The deterministic envelope function $w (t)$ is written (Martinelli et al., 2011) as

w (t) = {\begin{matrix} (t / t_{0})^{2} 0 < t < t_{0} \\ 1 t_{0} < t < t_{n} \\ \exp [- c (t - t_{n})] t_{0} < t \end{matrix}

(13)

where $t_{0}$ and $t_{n}$ are the initial time and the nth time, respectively; $t_{0} - t_{n}$ is the significant duration of the accelerogram; and $c$ is the non-uniformly modulated coefficient. A typical value of $c$ is adopted as 0.2.

A possible approach to define $a (t)$ is by assuming that the motion induced by earthquakes is a realization of a stationary stochastic process. The time history of a stationary earthquake excitation can be readily generated based on the spectral density as follows

a (t) = \sum_{k = 1}^{n} \sqrt{(4 \cdot Δ ω \cdot S (ω_{k}))} \cos (ω_{k} t + ε_{k})

(14)

where $Δ ω$ is the difference between successive frequencies, $ω_{k} = k \cdot Δ ω$ is the kth wave frequency, $S (ω_{k})$ is the power spectrum of $ω_{k}$ , $n$ is the total number of frequency points considered in the calculation, and $ε_{k}$ is the random phase angle varying between $0$ and $2 π$ .

In engineering practice (Bi and Hao, 2012), the design target response spectrum at a given site is more commonly available than the ground motion power spectrum density function. Therefore, it will be very useful to generate time histories of ground motion that are compatible to a given design target response spectrum.

For a given design target acceleration response spectrum $S_{a}^{T} (ω)$ , the corresponding ground motion power spectrum $S (ω)$ can be estimated (Bi and Hao, 2012) by

S (ω) = \frac{ζ}{π ω} {[S_{a}^{T} (ω)]}^{2} \frac{1}{\ln [- \frac{π}{ω T_{d}} \ln (1 - P)]}

(15)

where $T_{d}$ is the time duration of earthquake, $ζ$ is the damping ratio, and $P$ is the exceeding probability. In this study, the value of the parameter $P$ is taken as 0.9.

The design target acceleration response spectrum is normally determined by basic parameters of seismic intensity, site classification, peak acceleration and damping ratio. The seismic design spectrum as shown in Figure 2 can be found in the ISO design guidelines (ISO, 2006). In Figure 2, $S_{a}$ is the spectral acceleration, $S_{a, site} (T)$ is the site spectral acceleration, and $S_{a, map}$ is the bedrock spectral acceleration. When the fundamental period of a structure is equal to 0.2 or 1.0 s, the bedrock spectral accelerations can be calculated by $S_{a, map} (0.2) = 0.5 g$ or $S_{a, map} (1.0) = 0.2 g$ . The parameter $T$ is the fundamental period of the structure and $T_{s}$ is the site characteristic period in the following form

T_{s} = \frac{C_{v} S_{a, map} (1.0)}{C_{a} S_{a, map} (0.2)}

(16)

where $C_{a}$ and $C_{v}$ are the site coefficients and can be obtained from the design tables in ISO (2006).

Figure 2.

ISO seismic design spectrum.

Assuming that the submerged floating tunnel investigated in this article is located in the Bohai Sea of China with anchoring systems on a shallow foundation, the site class of the foundation is categorized as A. For a site class A/B and a shallow foundation, the parameters $C_{a}$ and $C_{v}$ are both assigned as 1.0.

Using the above approach, the generated time histories of ground motion usually match well with the design target response spectrum. Iterations should be carried out to adjust the power spectrum density function if the two spectra do not match satisfactorily (Bi and Hao, 2012).

Solution of the equation

The oscillation mode of the cable is supposed to follow the standard form as

u (z, t) = \sum_{n = 1}^{\infty} \sin \frac{n π z}{L} u_{n} (t)

(17)

To obtain an approximate solution for equation (2), the Galerkin method is applied to transform the partial differential equation into a set of ordinary ones.

Substituting equation (17) into equation (2), the equation is multiplied by $\sin (n π z / L)$ and integrated over $z \in (0, L)$ , from which the orthogonality of the normal mode gives

{\overset{\cdot\cdot}{u}}_{n} + [ω_{M n}^{2} + ω_{A n}^{2} (1 + ε \cos ω_{p} t)] u_{n} + \frac{C_{s n}}{m} {\dot{u}}_{n} - \int_{0}^{L} \frac{2 (F_{D} - Q_{s} \sin θ)}{m L} \sin \frac{n π z}{L} d z = 0

(18)

with

ω_{Mn}^{2} = {(\frac{n π}{L})}^{4} \frac{E_{eq} I}{m}

(19)

ω_{An}^{2} = {(\frac{n π}{L})}^{2} \frac{T_{0}}{m}

(20)

ω_{n} = \sqrt{ω_{Mn}^{2} + ω_{An}^{2}}

(21)

C_{sn} = 2 m ω_{n} ζ

(22)

where $ζ$ is the damping ratio of the cable; $ω_{Mn}$ and $ω_{An}$ are the $n th$ order natural frequency of the system due to lateral and axial vibrations, respectively; $ω_{n}$ is the $n th$ order natural frequency of the system; and $ε$ is the ratio between dynamic tension and static tension.

The fourth-order Runge–Kutta method is used to solve the differential equation of equation (18), where the response of the cable of each mode under excitation can be derived. By substituting the mode response into equation (17), the displacement response of the cable can be calculated. For a Bernoulli–Euler beam, the maximum dynamic response generally occurs at its mid-span, which has been reported by different researchers based on numerical simulations and model-scale experiments (Chao, 2013; Xiang and Yang, 2017; Yang and Xiao, 2014). Thus, the mid-span displacement of the cable is considered in this study.

The seismic response of a cable is often solved in the frequency domain. The spectral response analysis method (Lee et al., 2016; Wu et al., 2016) can be utilized to provide an insight into the response characteristics of the cable under hydrodynamic forces and earthquakes in the frequency domain.

Numerical results and discussion

The cable of a submerged floating tunnel is analyzed in this study as an illustrative example to show the efficacy of the proposed analytical solution (i.e. being programmed in MATLAB). The physical and geometric parameters are listed in Table 1 following the work of Sun and Su (2011). If the same values of $C_{M} = 1.0$ and $C_{D} = 0.7$ provided by Sun and Su (2011) are used, it can result in the loss of hydrodynamic added mass. Thus, the values of $C_{M}$ and $C_{D}$ are taken as 2.0 and 1.0, respectively, following the work of Martire et al. (2010).

Table 1.

Basic characteristics of the analysis case following the work of Sun and Su (2011).

Parameters	Values
Density of water $ρ_{w}$ $(kg / m^{3})$	1028
Depth of water $d$ $(m)$	170
Vertical height $H_{z}$ $(m)$	140
Undeformed length of cable $L$ $(m)$	161.66
Outside diameter of cable $D$ $(m)$	0.489
Density of cable $ρ_{s}$ $(kg / m^{3})$	7850
Modulus of elasticity of cable $E$ $(Pa)$	2.10E11
Initial tension in the cable $T_{0}$ $(N)$	2.572E7
Mass per unit length of cable $m$ $(kg / m)$	1474.23
Damping ratio $ζ$	0.0018
Inertia coefficient $C_{M}$	2.0
Drag coefficient $C_{D}$	1.0
Inclination angle $θ$ (∘)	60
Fundamental naturalfrequency of cable $ω_{1}$ (rad/s)	2.4137

In this study, four earthquake records in different representative sites are selected. For comparison, the peak accelerations of these accelerograms are adjusted to $0.2 g$ , corresponding to earthquakes with an intensity factor of 8 as shown in Figure 3 and Table 2 (ISO 2006; Wu et al., 2016).

Figure 3.

Time histories of four selected earthquake records: (a) Artificial earthquake (A/B); (b) El Centro earthquake (C): (1) horizontal earthquake and (2) vertical earthquake; (c) Taft earthquake (D); and (d) Loma Prieta earthquake (E).

Table 2.

Site classifications of earthquake records.

Ground motion	Site classifications	Site condition	Time duration (s)	Time interval (s)
Artificial earthquake	A/B	Rock	30	0.02
El Centro earthquake in the EWdirection (vertical direction)	C	Very dense hard soil and soft rock	30	0.02
Taft Kern County (Taft for short)earthquake in the N21E direction	D	Stiff to very stiff soil	30	0.02
Loma Prieta Oakland outer wharf(Loma for short) earthquake	E	Soft soil	30	0.02

The Artificial earthquake is generated using the aforementioned theory in section “Earthquake excitation in the time domain,” and the other three earthquake records are selected considering the compatibility with the design target acceleration response spectrum. The time histories in the time domain, and the amplitude and the energy in the frequency domain of four earthquakes are different, matching different site classifications.

A sensitivity analysis is conducted to evaluate the influence of key parameters to provide recommendations for use in design and construction of anchoring systems for a submerged floating tunnel under hydrodynamic forces and earthquakes. The effects of key environmental and structural parameters on the dynamic responses of the cable for submerged floating tunnel are investigated.

Effect of the ratio of parametric frequencyto natural frequency

Figure 4(a) to (c) shows three typical examples of time histories of the dynamic response of the cable subjected to combined parametric excitation and El Centro horizontal earthquake. The dynamic responses exhibit interesting and different features in terms of the pattern and the shape of the time history.

Figure 4.

Time histories of mid-span displacement of the cable subjected to El Centro horizontal earthquake. (a) $ω_{p} = 0$ , (b) $ω_{p} = 1.5 ω_{1}$ , and (c) $ω_{p} = 2 ω_{1}$ .

Parametric excitations satisfy the condition that can excite resonance of a system (i.e. the primary resonance condition $ω_{p} = ω_{1}$ ) or the super-resonance condition (i.e. $ω_{p} = 2 ω_{1}$ ). Thus, it can be easily seen from Figure 4(c) that the amplitudes of motion due to parametric excitations are obviously larger and more unstable than those observed in Figure 4(a) and (b) for motions without parametric excited resonance. When the hydrodynamic force induced by earthquake excitation is considered, the maximum mid-span displacement calculated using this method at $ω_{p} = 0$ , $ω_{p} = 1.5 ω_{1}$ , and $ω_{p} = 2 ω_{1}$ is 0.57, 0.93, and 2.98 m, respectively.

Figure 5 indicates that two local peak amplitudes of the dynamic response occur at the ratio between parametric frequency and natural frequency of $ω_{p} / ω_{n} = 1 and 2$ , respectively. The peak root-mean-square (RMS) values of mid-span displacement of the cable at $ω_{p} / ω_{n} = 1 and 2$ are calculated using this method considering the influence of earthquake excitation on hydrodynamic force as 1.13 and 1.62 m, respectively. Particularly, the largest response amplitude occurs at the frequency ratio of $ω_{p} / ω_{1} = 2$ . In other words, the largest dynamic response in the first mode occurs when the parametric frequency of the tunnel tube is twice as much as the natural frequency, that is, $ω_{p} = 2 ω_{1}$ . It can be seen that the dynamic response of the cable at other frequencies is relatively small. The results suggest that parametric excitation has a significant effect on the dynamic response of the cable.

Figure 5.

Variations of RMS of mid-span displacement of the cable with $ω_{p} / ω_{n}$ calculated using different methods.

Therefore, the fundamental frequency of the cable for submerged floating tunnel subjected to combined hydrodynamic force and earthquake should be controlled carefully to avoid the occurrence of parametric resonance. Based on equations (19) to (21), the key parameters that influence the fundamental frequency of the cable are $m$ , $E_{eq}$ , $I$ , $T_{0}$ , and $L$ , while the parametric excitation frequency $ω_{p}$ is mainly dependent on the magnitude of ocean environmental excitations acting on the tunnel tube.

Results evaluated using the developed approach are compared with those calculated using the method of Su and Sun (2013a). As can be seen in Figures 4 and 5, the dynamic responses of the cable under hydrodynamic force and earthquake excitation can be greatly amplified by the present method. Especially under the super-resonance condition (i.e. $ω_{p} / ω_{1} = 2$ ), the present analytical technique considers the contribution of earthquake excitation to hydrodynamic force, and the calculated results are nearly twice larger than those evaluated by the methods of Chen et al. (2008) and Su and Sun (2013a). A possible explanation for the difference in calculations could be due to the fact that the coupled effect of hydrodynamic force and earthquake excitation is incorporated in the present approach, which is more reasonable than other modeling strategies by neglecting the hydrodynamic added mass. Similar results of underestimation by existing analytical models (Chen et al., 2008; Su and Sun, 2013a) compared to experimental measurements and numerical simulations were reported (Lee et al., 2016; Lee and Kim, 2015). In addition, the dynamic amplitudes evaluated by the present method considering the influence of earthquake excitation on hydrodynamic force are much larger than the calculated results without considering the coupled effect of hydrodynamic force and earthquake excitation. This demonstrates that the seismic responses of the cable can be greatly influenced by the combined effect of hydrodynamic force and earthquake excitation.

Effect of the direction and magnitude ofearthquake excitation

In previous studies, earthquake waves are only considered in the horizontal direction. Both horizontally and vertically propagated earthquake waves are employed in this study to investigate the effect of the direction of earthquake excitation on the dynamic responses of the cable. The analysis method can account for the influence of vertically propagated earthquake wave in a similar manner.

Figure 6 illustrates the variations of RMS values of mid-span displacement as a function of peak ground acceleration (PGA) for the cable subjected to El Centro earthquake excitations in different directions. In general, the trend of the three curves is similar, where the RMS value increases with the increase in PGA. The rapid increase of the dynamic response is actually very dangerous for the cable system. Thus, the magnitude of PGA is an important parameter that determines the dynamic response of the cable, which necessitates a careful consideration during the preliminary design stage.

Figure 6.

The effect of the direction of earthquake excitation on the RMS values of mid-span displacement as a function of peak ground acceleration.

As shown in Figure 6, the RMS values of mid-span displacement for the cable are increased when it experiences a horizontal earthquake, a vertical earthquake, and a combination of horizontal and vertical earthquakes. The cable system becomes more unstable when the vertical component of earthquake excitation is considered. As expected, the difference in displacements increases when the PGA increases. The RMS values of mid-span displacement of the cable at a PGA of 0.5g are 1.64, 1.85, and 2.01 m, respectively, for horizontal, vertical, and combined earthquake excitations. It can be observed in Figure 7 that the vertical component has more abundant high-frequency contents that excite a wider peak frequency region than the horizontal component. This implies that parametric resonance of the cable can be turned easily (resulting in more dramatic dynamic response of cable) when the vertical ground motion is applied to the system. It can be concluded that the direction of earthquake excitation has a significant influence on the dynamic response of the cable, and the negligence of the combination of horizontal and vertical earthquakes in existing approaches (Martinelli et al., 2011; Su and Sun, 2013a) can easily lead to unconservative results. In the following section, the analytical model considering the combination of horizontal and vertical earthquake excitations is adopted.

Figure 7.

Fourier transforms of the horizontal and vertical components of El Centro earthquake.

Effect of earthquake site classificationsand damping ratio

As given in Figures 8 and 9, site classifications of earthquake records can alter the dynamic response of the cable largely. In Figure 8, the general trend of the four curves at different sites is similar, where an increase in damping ratio can result in a reduction of displacement of the cable. It should be noted that the amplitudes of ground motions at the four sites are distinctly different. For a damping ratio of $ζ = 0.001$ , the RMS values of mid-span displacement of the cable under Artificial earthquake, El Centro earthquake, Taft earthquake, and Loma earthquake are 1.25, 1.77, 2.04, and 1.83 m, respectively. It is noteworthy that when the damping ratio becomes greater than 0.006, the difference between earthquake intensities is reduced. This is probably because that the system is easier to be turned into a stable state by introducing a sufficient damping ratio. At a higher damping ratio of $ζ = 0.011$ , the RMS values of mid-span displacement of the cable under Taft earthquake and Loma earthquake are approximately identical. The decreasing rate of the dynamic amplitude as a function of damping ratio under Taft earthquake is larger than that under Loma earthquake. In general, the cable under Taft earthquake presents the highest amplitude of motion, followed by the results of the cable under Loma and El Centro earthquakes. The Artificial earthquake produces the smallest amplitude of motion for the cable.

Figure 8.

Effects of earthquake site classifications on the RMS values of mid-span displacement as a function of damping ratio.

Figure 9.

Displacement response spectra for four sites: (a) Artificial earthquake (A/B), (b) El Centro earthquake (C), (c) Taft earthquake (D), and (d) Loma Prieta earthquake (E).

The displacement response spectra of the cable subjected to four earthquakes are depicted in Figure 9(a)–(d). The peak values of spectral displacement of the cable under Artificial earthquake, El Centro earthquake, Taft earthquake, and Loma earthquake are 1.12, 1.41, 1.78, and 1.59 m, respectively, which occur at the approximate time when the earthquake record reaches its energy-intensive zone.

In Figure 9(a), there are three local peak values occurring in the frequency range of $ω = 1.8 - 2.8 rad / s$ , $ω = 3.0 - 3.5 rad / s$ , and $ω = 6.0 - 7.0 rad / s$ , and the maximum value occurs at about $ω = 2.0 rad / s$ . The response spectrum in Figure 9(a) can also indicate the frequency content where the energy (i.e. area under the spectrum) of the earthquake is attracted by the cable due to the compliant natural frequency of the structure. The maximum energy concentration occurs in the vicinity of the natural frequency of the cable. In Figure 9(b) and 9(c), there are many local peak values occurring in all frequencies, and the maximum value occurs at about $ω = 1.6 rad / s$ . The results in Figure 9(b) and (c) imply that more high-frequency responses are excited by earthquakes at sites C and D, and the energy of the displacement power spectrum becomes scattered. In Figure 9(d), the amplitude and the energy of the response spectrum are localized in a small frequency range within $ω = 1.0 - 3.0 rad / s$ , and the occurrence of peak displacement moves to a higher frequency component.

The influence of damping ratio on the response spectra of the cable is also investigated for four levels of damping ratio $ζ = 0.002, 0.004, 0.006, and 0.008$ . All plots in Figure 9 demonstrate that the amplitude and the energy of the response power spectrum are decreased when the modal damping ratio is increased. Taking Figure 9(a) as an illustrative example, the curve of $ζ = 0.002$ presents many local peak values occurring in all frequencies. However, the curve of $ζ = 0.008$ is comparatively smooth with only one peak value occurring in the energy concentration zone. The maximum spectral displacement decreased by up to 61.95% when the damping ratio increase from $ζ = 0.002$ to $ζ = 0.008$ . This is expected because damping plays an important role in energy dissipation and makes the cable system more stable (Wu et al. 2014; Wu et al. 2016b). Increasing the damping of the cable can reduce the structural responses of the system. Large-amplitude vibrations of the cable could potentially be avoided by installing dampers on the cable or the tunnel tube.

Effect of earthquake site characteristic period and initial tension

In Figure 10, it can be seen that the RMS values of displacement response of the cable are significantly decreased with the increase in initial tension, indicating that the cable becomes much safer as the initial tension in the cable is maintained in a higher level. The RMS values of displacement response of the cable increase with the increase in earthquake characteristic site period, and the difference in displacement values is reduced when the initial tension increases. Here, an increase in initial tension can lead to a larger stiffness coefficient for use in the equation of motion, which can therefore increase the stability of the cable to a certain extent. The observations in this study are consistent with the interpretations of experimental measurements and numerical simulations (Li et al., 2005; Wu et al., 2016).

Figure 10.

Effects of site characteristic period on the RMS values of mid-span displacement as a function of the magnitude of initial tension.

To keep the safety of the cable system for submerged floating tunnel, numerical simulations of the cable system in the complex marine environment are of significance during the preliminary design stage. The hydrodynamic, earthquake, and structural parameters have a considerable influence on the response of the cable and should be considered carefully.

Conclusion

The behavior of the cable system for submerged floating tunnel under simultaneous hydrodynamic force and earthquake excitations is studied in this article. A simplified analysis model is proposed to evaluate the cable for submerged floating tunnel under ocean environmental excitations. A mathematical method describing the dynamic motion of the cable is developed utilizing the Euler beam theory and the Galerkin method. The hydrodynamic force induced by earthquake excitations is formulated to represent real seaquake conditions. The stochastic-phase spectrum method is used to calculate stochastic earthquake forces in the time domain. An analytical model is proposed to analyze the cable system for submerged floating tunnel model subjected to combined hydrodynamic force and earthquake excitations. A parametric study demonstrates that the ratio of parametric frequency to natural frequency, the direction and magnitude of earthquake excitation, the initial tension in the cable, and the damping ratio have significant influences on the dynamic response of the cable for submerged floating tunnel under ocean environment excitations. All these key parameters should be taken into account to enable a preliminary examination of the hydrodynamic and seismic behavior of the cable and to provide guidance for design and operation of the anchor system for submerged floating tunnel.

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 National Natural Science Foundation of China (Grant Nos 51578164 and 41672296), the China Postdoctoral Science Foundation (Grant No. 2017M610578), the Systematic Project of Guangxi Key Laboratory of Disaster Prevention and Structural Safety (Grant No. 2016ZDX008), and the Innovative Research Team Program of Guangxi Natural Science Foundation (Grant No. 2016GXNSFGA380008).

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