Parametric Vibration Analysis of Submerged Floating Tunnel Tension Legs

According to the characteristics of submerged floating tunnel anchored by tension legs, simplifying the tube as point mass and assuming that the tension leg is a nonlinear beam model hinged at both ends, the nonlinear vibration equation of the tension leg is derived. The equation is solved by the Galerkin method and Runge-Kutta method. Subsequently, numerical analysis of typical submerged floating tunnel tension leg is carried out. It is shown that, the parametric vibration response of the submerged floating tunnel tension leg is related to the amplitude and frequency of the end excitation. Without considering axial resonance and transverse resonance, it is reasonable that higher order modes are abandoned and only the first three modes are considered. The axial resonance amplitude of the second or third order mode is equivalent to the first order mode axial resonance amplitude, which should not be ignored.


Introduction
Submerged floating tunnels are a promising alternative to long-span bridges for crossing sea straits, fjords or inland waters (Di Pilato et al., 2008a). It is mainly composed of the revetment section connecting both sides, the anchorage device limiting the excessive displacement of the tube and the suspended tube. Compared with immersed tunnel and submarine tunnel, submerged floating tunnel (SFT) has little impact on environment and landscape (Remseth et al., 1999). Submerged floating tunnel has become the most competitive cross-river and sea traffic structure in the 21st century due to its advantages of being less affected by bad weather and crossing deep water at low unit cost (Su and Sun, 2013;Chen et al., 2012).
However, there is no existing SFT in the world due to various scientific and technical difficulties. Numerous researches have been carried out to verify structural safety in wave and seismic excitations on the SFT. Based on computational fluid dynamics and ABAQUS program, Mandara et al. (2016) studied the vibration characteristics of submerged floating tunnel in variable speed flow. Long et al. (2009) investigated the effects of structure parameters (including buoyancy-weight ratio, stiffness coefficients of the tether systems, tunnel net buoyancy and tunnel length) on dynamic response of SFT under hydrodynamic loads. Lee et al. (2016) studied the influence of sea water on the seismic performance of SFT system. The hydrodynamic pressure was applied to the structure considering the interaction of fluid structure. Di Pilato et al. (2008b) carried out a coupled dynamic analysis to investigate the effect of wave and seismic excitations. Jin and Kim (2018) carried out global dynamic analysis of a 700-m-long SFT section planned in the South Sea of Korea for the survival random wave and seismic excitations. Muhammad et al. (2017) evaluated the displacements and internal forces of submerged floating tunnel under hydrodynamic and three-dimensional seismic excitations to check the global performance of an SFT in order to conclude on the optimum design. Chamelia et al. (2015), considering the interaction of current and wave on the tunnel, studied the influence of mooring angle on the dynamic response of submerged floating tunnel. Most of the above researches mainly addressed the global performance of the SFT tube. However, despite that the support stiffness provided by tethers or tension legs acting on the tube was included in most of these models, the vibration of the tether or tension leg itself was neglected. Dong et al. (2013) analyzed the vibration characteristics of tethers of SFTs under random earthquake based on displacement power spectrum, in which the random earthquake input was simulated by virtual excitation method. Mai et al. (2004) studied the vortexinduced vibration of tension legs of submerged floating tunnel subjected to wave and current loadings and analyzed the influence of parameter excitation frequency on displacement response, dynamic shear force and dynamic bending moment of tension leg. Chao et al. (2016) carried out an experimental study on the vortex-induced vibration characteristics of tethers of SFT in Qiandao Lake under uniform inflow excitation. Luo et al. (2015) compiled program to forecast the vortex-induced vibration fatigue damage of floating tunnel based on the theory of vortex-induced vibration and modal superposition method. Cantero et al. (2017) investigated the effect of parametric excitation on the response of taut mooring lines and concluded that the tension values are barely affected by the parametric resonance. From the above, experts and scholars worldwide have done a lot of researches on the vibration characteristic of tension leg SFT. However, currently existing research on parametric vibration of tension leg still needs further digging.
As an important structural component of the submerged floating tunnel, the tension leg bears the main external load of the tunnel. The dynamic performance of the tension leg is very important to submerged floating tunnel. Tension legs are characterized by large flexibility, small damping and light weight. Under the external loads, they are prone to various forms of vibration. Studying the vibration characteristics of tension leg is not only of great significance in theory, but will also provide experience for practical engineering. In this paper, the tube of submerged floating tunnel is simplified to point mass. Assuming that the tension leg is a non-linear beam structure hinged at both ends, the parametric vibration of the tension leg is preliminarily studied.

Establishment of Equation
Based on the submerged floating tunnel anchored by tension legs shown in Fig. 1, the vibration model of the tension leg is illustrated in Fig. 2. By referring to the z and x directional coupled nonlinear beam model for tensionlegged offshore platform (Han et al., 2000a(Han et al., , 2000b, the vibration model of tension leg of submerged floating tunnel is established, in which the tube is simplified to point mass, the tension leg to non-linear beam, and the non-linear coup-ling effect in the x-direction and z-direction is considered. With Hamilton principle and Kirchhoff hypothesis (Xu et al., 2009), the nonlinear vibration equation of tension leg is derived.
The potential energy PE of the tension leg is written as: where, L, E, A and I are the length, modulus of elasticity, cross section area, and cross-sectional moment of inertia of tension leg, respectively. u and v are displacements in the x and z directions, respectively. "′" is used for derivatives with respect to z. The kinetic energy KE of the tension leg is given by in which, m is the quality of tension leg unit length, ρ is the density of tension leg, and "·" is used for derivatives with respect to time.
, according to Hamilton's principle, the following formula is obtained.
where and are the initial time and the end time, respectively. Virtual work done by combining external forces can be expressed as: When the value of L/D (D is the external diameter of tension leg) is large, the influence of moment of inertia can be neglected (Xu et al., 2008). Thus, the influence of moment of inertia is neglected in the control equation. By considering the effect of material damping, the nonlinear vibration control equation of tension leg is finally obtained.
In Eqs. (5) and (6), and are the distribution of transverse and axial unit length respectively, is the difference between buoyancy and gravity of unit length and ; is the density of sea water; is the cross-sectional area of tension leg discharging sea water; g is the acceleration of gravity.
According to Morison formula (Sun et al., 2011) f in which, is the additional mass coefficient, and is the drag coefficient. Suppose that , (Zhu, 1991).

Boundary condition
Through simplifying the tube of submerged floating tunnel as a point mass (Cantero et al., 2016;Chamelia et al., 2015), assuming tension leg as a hinged non-linear beam structure (Patel and Park, 1995;Park and Jung, 2002), boundary conditions are given as follows: In Eq. (8) is the initial displacement of the top of the tension leg under the initial pre-tension; is the initial pre-tension of the top of tension leg; and are the axial and transverse motions of the tube of the underwater suspension tunnel, respectively. and are shown as follows (Xu et al., 2008): where and are the transverse and axial vibration amplitudes of submerged floating tunnel tube respectively. and are the circular frequencies of transverse and axial vibration of submerged floating tunnel tube, respectively.

Numerical method
By ignoring the influence of moment of inertia in Eqs. (1) and (2), according to Galerkin method, the following formulas are obtained.
in which, and .

Numerical analysis and results
Since no submerged floating tunnel has been realized yet, basic parameters of tether in this paper (shown in Table 1) are chosen with reference to that of foreign proposed submerged floating tunnel (Faggiano et al., 2001). Eqs. (14) and (15) are solved by the fourth-order Runge−Kutta method in MATLAB (Ma and Lei, 2006).
In order to avoid the axial and transverse resonances, supposing that and , the first mode amplitude of the tension leg is 0.24 m, the second one is 0.0075 m and the third one is 0.0014 m, as shown in Fig. 3. It can be seen that the first order mode plays an absolute control role. So by selecting the first three order modes for analysis without considering the influence of axial resonance and transverse resonance, the calculation accuracy requirements can be met.
When the transverse excitation frequency of the tension leg is multiplied by the first natural frequency of the tension leg, the mid-span displacement of the tension leg is enlarged. When the ratio of the transverse excitation frequency to the natural frequency of the tension leg equals approximately one, the mid-span displacement of the tension leg increases most significantly, as shown in Figs. 4 and 5.
When the axial excitation frequency of the tension leg is multiplied by the first natural frequency of the tension leg, the mid-span displacement of the tension leg is also enlarged. When the ratio of the axial excitation frequency to the natural frequency of the tension leg equals approxim- Effect of transverse excitation amplitude on the first-order modal displacement of the tension leg is shown in Fig. 9. It can be seen from Fig. 9 that when the difference between the transverse excitation frequency and the natural frequency of the tension leg is too large to produce resonance, the amplitude of the first-order modal vibration of the tension leg increases with the increase of the excitation amplitude, which is approximately proportional to each other. When the transverse excitation frequency is equal to the first order natural frequency of the tension leg, the resonance occurs. With the increase of excitation amplitude, the first-order modal amplitude of tension leg also increases, forming a nonlinear relationship. When the excitation amplitude increases to a certain extent, the first mode amplitude of the tension leg will not increase with the increase of the excitation amplitude.   134 SUN Sheng-nan et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 131-136 Effect of axial excitation amplitude on the first-order modal displacement of tension leg is shown in Fig. 10. It can be seen from Fig. 10 that when the difference between the axial excitation frequency and the natural frequency of the tension leg is too large to produce resonance, the vibration amplitude of the tension leg almost remains unchanged with the increase of the excitation amplitude. When the ratio of the axial excitation frequency to the natural frequency of the tension leg is two and the excitation amplitude is too small to cause the axial resonance, the mid-span displacement of the tension leg increases slowly with the increase of the excitation amplitude. When the excitation amplitude causes axial resonance, the mid-span displacement of the tension leg increases sharply, resulting in axial resonance.
Effect of axial vibration of high order modes on the displacement of tension leg is shown in Figs. 11−13. It can be seen that when the axial excitation frequency is two times the second-order natural frequency or the third-order natural frequency, the higher order axial resonance will occur, and the second and third order modal vibration amplitude is also very large. When the amplitude of high order axial excitation causes axial resonance, the span displacement of tension leg increases sharply, the amplitude of higher order axial vibration is close to the amplitude of first order mode vibration, and the amplitude of high order mode axial resonance cannot be ignored.

Conclusions
Submerged floating tunnel (SFT) is a new structural solution to waterway crossings. In spite of valuable researches accomplished already, so far no SFT has been constructed in the real world. Tension leg is the main force bearing component of submerged floating tunnel, and its vibration characteristics deserve attention. In this paper, the multi-order modal vibration response of tension leg of submerged floating tunnel under axial and transverse excitation is analyzed. Based upon current findings, the following conclusions are drawn.
(1) Without considering axial resonance and transverse resonance, it is reasonable to ignore the higher-order modes and only consider the first three modes. In this way, the calculation will be simplified and the accuracy be guaranteed. Axial resonance amplitude of the second and third modes is quite similar to that of the first modes. Thus, they should not be ignored. Therefore, the occurrence of axial resonance    SUN Sheng-nan et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 131-136 should be avoided in practical engineering.
(2) When the transverse vibration cannot produce resonance, the vibration amplitude of tension leg increases with the increase of excitation amplitude, which is approximately proportional to each other. When the resonance of tension leg occurs, the first-order modal amplitude of tension leg increases substantially as the excitation amplitude increases, forming a nonlinear relationship. However, when the excitation amplitude increases to a certain extent, the first-order modal amplitude of tension leg tends to a stable value. It can be seen that the transverse amplitude of the tension leg is greatly affected by the excitation frequency and amplitude. Therefore, the transverse resonance should be avoided and the excitation amplitude should be minimized in practical engineering.
(3) When the axial resonance does not occur, the amplitude of axial excitation has almost no effect on the midspan displacement of tension leg. When the ratio of axial excitation frequency to natural frequency of tension leg is two, the amplitude of excitation is too small to cause axial resonance, and with the increase of excitation amplitude, the mid-span displacement of tension leg increases slowly. However, when axial resonance is caused by excitation amplitude, the mid-span displacement of the tension leg will increase sharply, resulting in axial resonance. Therefore, in practical engineering, the amplitude of axial excitation should be minimized so that it is not enough to cause axial resonance. The ratio of the frequency of axial excitation to the natural frequency of tension leg should avoid being close to one and two.