Research ArticleDynamic Response Analysis of Cable of Submerged FloatingTunnel under Hydrodynamic Force and Earthquake

ZhiwenWu and Guoxiong Mei

Key Laboratory of Disaster Prevention and Structural Safety of Ministry of EducationGuangxi Key Laboratory of Disaster Prevention and Structural Safety College of Civil Engineering and ArchitectureGuangxi University Nanning 530004 China

Correspondence should be addressed to Guoxiong Mei meiguox163com

Received 21 June 2017 Accepted 28 August 2017 Published 15 October 2017

Academic Editor Hugo Rodrigues

Copyright copy 2017 Zhiwen Wu and Guoxiong Mei This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A simplified analysis model of cable for submerged floating tunnel subjected to parametrically excited vibrations in the oceanenvironment is proposed in this investigation The equation of motion of the cable is obtained by a mathematical method utilizingthe Euler beam theory and the Galerkin method The hydrodynamic force induced by earthquake excitations is formulated tosimulate real seaquake conditions The random earthquake excitation in the time domain is formulated by the stochastic phasespectrummethod An analytical model for analyzing the cable for submerged floating tunnel subjected to combined hydrodynamicforces and earthquake excitations is then developed The sensitivity of key parameters including the hydrodynamic earthquakeand structural parameters on the dynamic response of the cable is investigated and discussed The present model enables apreliminary examination of the hydrodynamic and seismic behavior of cable for submerged floating tunnel and can provide valuablerecommendations for use in design and operation of anchor systems for submerged floating tunnel

1 Introduction

Submerged floating tunnel is an innovative underwater trans-portation system to avoid water traffic and weather at adepth of usually 20ndash50m The tunnel floats in water and itsposition is restrained at a certain distance from the sea bedby means of suitable anchoring systems such as cables orbars Cables for submerged floating tunnel are lightweightvery flexible and lightly damped the features of which makethem particularly prone to vibration Frequent vibration maylead to fatigue damage of cable where fatigue cracks couldbe formed on the cable surface to destroy the anticorrosionsystem Eventually the bearing capacity of cable will be lostcompletely with the propagation of fatigue cracks The cablefor submerged floating tunnel experiences complex dynamicforces under various ocean conditions which is the key to thesafe operation

Many attentions have been received to the vibration ofsubmerged floating tunnel cables Sun et al [1] analyzed the

nonlinear response of cables subjected to parametric excita-tions Sun and Su [2] investigated the parametric vibration ofsubmerged floating tunnel cable under random excitationsLu et al [3] studied the slack phenomenon and the snapforce in the cable for submerged floating tunnels under waveconditions Seo et al [4] conducted a series of simplifiedanalyses to estimate the behavior of submerged floatingtunnel cables in waves and compared their calculationswith experimental measurements Cifuentes et al [5] imple-mented a numerical model to analyze the coupled dynamicresponse of a submerged floating tunnel with mooring linesin regular waves

It is noted by Duan et al [6] that 85 of the totalamount of earthquakes occurs in the oceanThe Bohai Straitthe Qiongzhou Strait and the Taiwan Strait in China arethree potential areas to build submerged floating tunnelswhich are also in the Circum-Pacific seismic belt Hencethe performance of subsea cables in seismic zones becomesone of the significant academic and engineering attractions

HindawiShock and VibrationVolume 2017 Article ID 3670769 14 pageshttpsdoiorg10115520173670769

2 Shock and Vibration

L

u

x

y

z

Sea bed

Cable

Submerged oating tunnel

oEarthquake wave

Wave current

Tp cospt

Hz

Figure 1 Definition sketch of the submerged floating tunnel model and its cable coordinate system

in offshore engineering In addition the effect of seaquakeis much more complex which can induce hydrodynamicpressures acting on the cable [7 8] The combined fluidand seismic motion induced by ocean hydrodynamic forcesmust be considered rigorously in the analysis of cables forsubmerged floating tunnel during earthquakes Thereforeit is of significance to evaluate how parametrically excitedvibrations influence the behavior of cable for submergedfloating tunnel under combined hydrodynamic force andearthquake

Although earthquakes have significant effect on the per-formance of cables for submerged floating tunnel few studieshave been reported to assess the seismic response of cablesexcept those from [9ndash11] In these previous studies the effectsof earthquake on the development of hydrodynamic forcewere usually neglected and the seismic design spectrumwas taken from design guidelines for buildings (eg theChinese code for seismic design of buildings (GB 50011-2010) [12] directly) However the seismic design spectrumfor buildings is not always suitable for marine structuresthe distinctiveness of marine site classifications of offshorestructures in marine environment cannot be considered andthe correctness of this approach becomes questionable

The present study aims to establish a mathematicalmethod to explore the dynamic response of cable for sub-merged floating tunnel under simultaneous hydrodynamicforces and earthquake excitations A simplified analysismodel to estimate the behavior of cable for submergedfloating tunnel under the ocean environmental excitation isproposedThe equation ofmotion of the cable is obtained by amathematicalmethodutilizing the Euler beam theory and theGalerkinmethodThe hydrodynamic force induced by earth-quake excitations is formulated to simulate real seaquakeconditionsThe random earthquake force in the time domainis calculated by the stochastic phase spectrum methodAn analytical model for analyzing the cable for submergedfloating tunnel subjected to combined hydrodynamic forcesand earthquake excitations is then developed The sensitivityof key parameters including the hydrodynamic earthquake

and structural parameters on the dynamic response of thecable is investigated and discussed

2 Method

21 Equation of Motion for the Cable Following previousstudies [1 3 11 13] a simplified analysis model is proposedto evaluate the motion of cables for submerged floatingtunnel subjected to ocean environmental excitations Forsimplification the cable for submerged floating tunnel ismodeled as a Bernoulli-Euler beam which only allows thein-plane motion of the cable Furthermore a conservativeassumption is adopted that the response of the cable undercombined effects of hydrodynamic forces and earthquakeexcitations ismaximizedwhen the two actions are in the samedirection A submerged floating tunnel and the direction ofearthquake wave and wavecurrent force are schematicallyshown in Figure 1 where the cable is allowed to move onlyin the inline direction It should be emphasized that thehorizontal earthquake excitation is acting on the anchoringpoint of the cable

Figure 1 shows the schematic illustration of a submergedfloating tunnel and cable In this simplified analysis modelthe three-dimensional behavior is not taken into considera-tion Although this assumption simplifies the physical phe-nomenon the two-dimensional model enables a preliminaryexamination of the hydrodynamic and seismic behavior ofcables The origin of the coordinate system is set at theanchored point of the cable on the sea bed The parameter119871 is the undeformed length of the cable 119867119911 shows thevertical height above the sea bed 119906 represents the dynamicdisplacement at the mid-span of the cable and 120579 denotes theinclination angle of the cable with respect to the sea bed

The effect of the tunnel tube on the cable is simplifiedas a parametric excitation 119879119901 cos120596119901119905 and the ends of thecable are hinged [2 14] where 119879119901 is the dynamic tensionacting on the cable induced by the motion of the tube and120596119901 is the parametric excitation frequency For simplicity theinitial tension geometry stiffness and material property ofthe cable are assumed to be constant along the cable length

Shock and Vibration 3

Thus the governing equation of the motion of the cablecan be written as [2 3 11 13]

119864eq1198681205974119906 (119911 119905)1205971199114 minus 120597

120597119911 (1198790 + 119879119901 cos120596119901119905)120597119906 (119911 119905)120597119911

+ 1198981205972119906 (119911 119905)1205971199052 + 119862119904 120597119906 (119911 119905)120597119905 = 119865119871 minus 119865119863 minus 119876119904 sin 120579(1)

where 119906(119911 119905) is the dynamics lateral displacement of cable1198790119898 119864eq and 119868 are the initial tension mass per unit lengththe equivalent modulus of elasticity and the cross-sectionalmoment of inertia of the cable respectively 119862119904 is the viscousdamping coefficient 119865119871 is the vortex-induced lift force 119865119863 isthe acting drag force applied by water per unit length whenthe cable oscillates and119876119904 is the earthquake excitation force

While considering the cable sag effect the equivalentelastic modulus of cable can be obtained by the followingequivalent method [11]

119864eq = 119864(1 + 119864119864119891) 119864119891 = 121205903

[12057421 (119871 cos 120579)2] 120590 = 1198790

119860119888 (2)

where 119864 is the modulus of elasticity of cable 119864119891 is themodulus of elasticity induced by sag of cable 1205741 is the buoyantunit weight of the cable 120590 is the stress of cable and 119860119888 is thecross-sectional area of cable

22 Earthquake Excitation in the Time Domain The earth-quake excitation in the time domain can be expressed as

119876119904 = 119898119892 (3)

where 119892 is the acceleration of the input ground motionwhich can be derived from the earthquake records obtainedfrom seismic stations at a given location or from an artificialstochastic process using the trigonometric series method

Since the random excitation of earthquakes is transientthe nonstationary characteristics of randomexcitation shouldbe considered Using the evolutionary theory of powerspectrum density [15] the nonstationary random process canusually be written as the product of a stationary random pro-cess and a deterministic slowly varying modulation function

119892 = 119886 (119905) 119908 (119905) (4)

where 119886(119905) is a stationary process and119908(119905) is the deterministicenvelope function with a maximum value of 10

Thedeterministic envelope function119908(119905) is written as [10]

119908 (119905) =

( 1199051199050)2

0 lt 119905 lt 11990501 1199050 lt 119905 lt 119905119899exp [minus119888 (119905 minus 119905119899)] 1199050 lt 119905

(5)

where 1199050 and 119905119899 are the initial time and the 119899th time respec-tively 1199050 minus 119905119899 is the significant duration of the accelerogramand 119888 is the nonuniformly modulated coefficient A typicalvalue of 119888 is adopted as 02

T (s)0 02 10

S a(T

)

CaSaＧＪ(02)

CSaＧＪ(10)

SaＭＣＮ (T) = (3T + 04)CaSaＧＪ(02)

SaＭＣＮ (T) = CSaＧＪ(10)T

Ts

Figure 2 ISO seismic design spectrum

A possible approach to define 119886(119905) is by assuming thatthe motion induced by earthquakes is a realization of astationary stochastic processThe time history of a stationaryearthquake excitation can be readily generated based on thespectral density as follows

119886 (119905) =119899

sum119896=1

radic(4 sdot Δ120596 sdot 119878 (120596119896)) cos (120596119896119905 + 120576119896) (6)

where Δ120596 is the constant difference between successivefrequencies 120596119896 = 119896 sdot Δ120596 is the 119896th wave frequency 119878(120596119896) isthe power spectrum of 120596119896 119899 is the total number of frequencypoints considered in the calculation and 120576119896 is the randomphase angle varying between 0 and 2120587

In engineering practice [15] the design target responsespectrum at a given site is more commonly available than theground motion power spectrum density function Thereforeit will be very useful to generate time histories of groundmotion that are compatible to a given design target responsespectrum

The design target acceleration response spectrum 119878119879119886 (120596)can be obtained in seismic design guidelines which cannotbe directly used by (6) Thus a conversion formula is neededto obtain the corresponding ground motion power spectrum119878(120596)

For a given design target acceleration response spectrum119878119879119886 (120596) the corresponding ground motion power spectrum119878(120596) can be estimated by [15]

119878 (120596) = 120577120587120596 [119878

119879119886 (120596)]2 1

ln [minus (120587120596119879119889) ln (1 minus 119875)] (7)

where 119879119889 is the time duration of earthquake 120577 is the dampingratio and 119875 is the exceeding probability In the present studythe value of the parameter 119875 is taken as 09

The design target acceleration response spectrum isnormally determined by basic parameters of seismic intensitysite classification peak acceleration and damping ratio Theseismic design spectrum curve as shown in Figure 2 canbe found in the ISO design guidelines (ISO 2006) [16] Inthe figure 119878119886 is the spectral acceleration 119878119886site(119879) is thesite spectral acceleration and 119878119886map is the bedrock spectralacceleration When the fundamental period of a structure is

4 Shock and Vibration

equal to 02 or 10 s the bedrock spectral acceleration can becalculated by 119878119886map(02) = 05119892 or 119878119886map(10) = 02119892 Theparameter 119879 is the fundamental period of the structure and119879119904 is the site characteristic period and can be calculated asfollows

119879119904 = 119862V119878119886map (10)119862119886119878119886map (02) (8)

where 119862119886 and 119862V are the site coefficients and can be obtainedfrom the design tables in ISO (2006) [16]

Assuming that the submerged floating tunnel investigatedin this paper is located in the Bohai Sea of China withanchoring systems on a shallow foundation the site class ofthe foundation is categorized as A For a site class AB anda shallow foundation the parameters 119862119886 and 119862V are bothassigned as 10

Using the above approach the generated time historiesof ground motion usually match well with the design targetresponse spectrum Iterations should be carried out to adjustthe power spectrumdensity function if the two spectra do notmatch satisfactorily [15]

23 Formulation of Hydrodynamic Force in Time DomainThe vortex-induced lift force can be expressed as [17]

119865119871 = 12120588119908119863119862119871 (V119871 sin 120579)

2 sin (120596V119905) (9)

where 120588119908 is the water density 119863 is the outer diameter of thecable 120596V is the vortex shedding frequency 119862119871 is the vortex-induced lift coefficient and V119871 is the vortex flow velocity andits typical value is taken as 10ms

The hydrodynamic force 119865119863 can be expressed by Mori-sonrsquos equation [14] as follows

119865119863 = 12120588119908119863119862119863 (V minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816V minus12059711990612059711990510038161003816100381610038161003816100381610038161003816 + 119862119872

1205874 120588119908119863

2 120597V120597119905

minus (119862119872 minus 1) 1205874 1205881199081198632 12059721199061205971199052

(10)

where 119862119863 is the drag coefficient 119862119872 is the inertia coefficient119906 is the displacement of the cable and V is the instantaneouscurrent velocity

The hydrodynamic force induced by earthquake excita-tion is formulated following four assumptions [7 18 19]

(1) As the earthquake excitation process is in a shorttime duration the current velocity around the cable duringearthquakes is assumed to be negligible

(2) When the ground motion is horizontal the currentvelocity is set as zero when the groundmotion is vertical thecurrent velocity is equal to the ground velocity

(3) The wave force is neglected as the cable is located indeep water

(4) The coupled effect between the motions in differentdirections is neglected

Thus the hydrodynamic force under earthquake excita-tion can be expressed by [7 18 19]

119865119863 = 12120588119908119863119862119863 (119892 minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816119892 minus12059711990612059711990510038161003816100381610038161003816100381610038161003816

+ 1198621198721205874 1205881199081198632119892 minus (119862119872 minus 1) 1205874 120588119908119863

2 12059721199061205971199052

(11)

24 Solution of the Equation Suppose that the oscillationmode of the cable is that of standard chord

119906 (119911 119905) =infin

sum119899=1

sin 119899120587119911119871 119906119899 (119905) (12)

To obtain an approximate solution of (1) Galerkinmethod is applied to transform the partial differential equa-tion into a set of ordinary ones

Substituting (12) into (1) multiplying each term bysin(119899120587119911119871) integrating over 119911 isin (0 119871) and invoking theorthogonality property of the normal modes give

119899 + [1205962119872119899 + 1205962119860119899 (1 + 120576 cos120596119901119905)] 119906119899 + 119862119904119899119898 119899

minus int1198710

2 (119865119871 minus 119865119863 minus 119876119904 sin 120579)119898119871 sin 119899120587119911119871 119889119911 = 0

(13)

with

1205962119872119899 = (119899120587119871 )4 119864eq119868119898 (14)

1205962119860119899 = (119899120587119871 )2 1198790119898 (15)

120596119899 = radic1205962119872119899 + 1205962119860119899 (16)

119862119904119899 = 2119898120596119899120577 (17)

where 120577 is the damping ratio of the cable120596119872119899120596119860119899 are the 119899thorder natural frequency of the system of bending vibrationand axial vibration respectively 120596119899 is the total 119899th ordernatural frequency of the system and 120576 is the ratio of dynamictension to static tension

The fourth-order Runge-Kutta method is used to solvethe differential equation (13) eachmode response of the cableunder excitation may be obtained By substituting the moderesponse into (12) the displacement response of the cablemaybe obtained

3 Numerical Results and Discussion

The cable of a submerged floating tunnel is analyzed in thepresent study as an illustrative example to show the efficacyof the proposed analytical solution that is programmed inMATLAB The physical and geometric parameters are listedin Table 1 following the work of Sun and Su [2] If the samevalues of 119862119872 = 10 and 119862119863 = 07 provided by Sun and Su[2] are used it can result in the loss of hydrodynamic added

Shock and Vibration 5

minus20

minus15

minus10

minus05

00

05

10

15

20a g

(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(a) Artificial earthquake (AB)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(b) El Centro earthquake (C)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(c) Taft earthquake (D)

5 10 15 20 25 300t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

(d) Loma Prieta earthquake (E)

Figure 3 Time histories of four selected earthquake records

Table 1 Basic characteristics of the analysis case

Parameters ValuesDensity of water 120588119908 (kgm3) 1028Depth of water 119889 (m) 170Vertical height119867119911 (m) 140Unstretched length of cable 119871 (m) 16166Outside diameter of cable119863 (m) 0489Density of cable 120588119904 (kgm3) 7850Modulus of elasticity of cable 119864 (Pa) 21011986411Initial tension of cable 1198790 (N) 25721198647Mass per unit length of cable119898 (kgm) 147423Damping ratio 120577 00018Inertia coefficient 119862119872 20Drag coefficient 119862119863 10Inclination angle 120579 (∘) 60

mass Thus the values of 119862119872 and 119862119863 are taken as 20 and 10respectively following the work of Martire et al [20]

In the present study four earthquake records in differentrepresentative sites are selected For comparison the peakacceleration of these accelerograms is adjusted to 02119892corresponding to earthquakes with an intensity factor of 8 asshown in Figure 3 and Table 2 [16 21]

The artificial earthquake is generated by using the afore-mentioned theory in Section 22 and the other three earth-quakes are the classical earthquake records obtained fromcorresponding site classifications The time histories in timedomain the amplitude and energy in frequency domainof four earthquakes are different matching different siteclassifications

A sensitivity analysis is conducted to evaluate the influ-ence of key parameters to provide recommendations foruse in design and construction of the anchor system for asubmerged floating tunnel under hydrodynamic force andearthquake The effects of key environmental and structural

6 Shock and Vibration

Su and Sun (2013) method (under El Centro earthquake

20 40 60 80 1000t (s)

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

000

005

010

FFT

ampl

itude

(a) 120596119901 = 151205961

20 40 60 80 1000t (s)

minus3

minus2

minus1

0

1

2

3

u(z

t) z

=05L

(m)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

00

02

04

06

08

10

12FF

T am

plitu

de

(b) 120596119901 = 21205961

Figure 4 Mid-span displacement of the cable subjected to El Centro earthquake

parameters on the dynamic responses of the cable for sub-merged floating tunnel are investigated

31 Evaluation of the Proposed Method In order to evaluatethe effectiveness of the developed method results evaluatedusing the developed approach are compared with thosecalculated using the method of Chen et al (2008) [22] andSu and Sun (2013) [11]

In Figure 5 the overall trends of the results obtainedby the present method generally agree with the evaluationscalculated from the other methods The present methodcaptures the peak value well at resonance condition (ie1205961199011205961 = 1 2) as well as the subsequent attenuation trendOverall the calculation by the present method is acceptableand reasonable

In Figures 4 and 5 the dynamic responses of the cableunder hydrodynamic force and earthquake excitation can begreatly amplified by the present method Especially underthe super-resonance condition (ie 1205961199011205961 = 2) it can beobserved that the FFT amplitude component obtained by thepresent method has more abundant high-frequency contentsand excites wider peak frequency region than that of thosecalculated using the method of Su and Sun (2013) A possibleexplanation for the difference in calculations could be dueto the reason that the seismic design spectrum for marinegeotechnical geology is used in the present approach whichismore reasonable and accurate than other strategies by usingseismic design spectrum for overland building Anotherpossible explanation could be the fact that the coupledeffect of hydrodynamic force and earthquake excitation is

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

Thus the governing equation of the motion of the cablecan be written as [2 3 11 13]

119864eq1198681205974119906 (119911 119905)1205971199114 minus 120597

120597119911 (1198790 + 119879119901 cos120596119901119905)120597119906 (119911 119905)120597119911

+ 1198981205972119906 (119911 119905)1205971199052 + 119862119904 120597119906 (119911 119905)120597119905 = 119865119871 minus 119865119863 minus 119876119904 sin 120579(1)

where 119906(119911 119905) is the dynamics lateral displacement of cable1198790119898 119864eq and 119868 are the initial tension mass per unit lengththe equivalent modulus of elasticity and the cross-sectionalmoment of inertia of the cable respectively 119862119904 is the viscousdamping coefficient 119865119871 is the vortex-induced lift force 119865119863 isthe acting drag force applied by water per unit length whenthe cable oscillates and119876119904 is the earthquake excitation force

While considering the cable sag effect the equivalentelastic modulus of cable can be obtained by the followingequivalent method [11]

119864eq = 119864(1 + 119864119864119891) 119864119891 = 121205903

[12057421 (119871 cos 120579)2] 120590 = 1198790

119860119888 (2)

where 119864 is the modulus of elasticity of cable 119864119891 is themodulus of elasticity induced by sag of cable 1205741 is the buoyantunit weight of the cable 120590 is the stress of cable and 119860119888 is thecross-sectional area of cable

22 Earthquake Excitation in the Time Domain The earth-quake excitation in the time domain can be expressed as

119876119904 = 119898119892 (3)

where 119892 is the acceleration of the input ground motionwhich can be derived from the earthquake records obtainedfrom seismic stations at a given location or from an artificialstochastic process using the trigonometric series method

Since the random excitation of earthquakes is transientthe nonstationary characteristics of randomexcitation shouldbe considered Using the evolutionary theory of powerspectrum density [15] the nonstationary random process canusually be written as the product of a stationary random pro-cess and a deterministic slowly varying modulation function

119892 = 119886 (119905) 119908 (119905) (4)

where 119886(119905) is a stationary process and119908(119905) is the deterministicenvelope function with a maximum value of 10

Thedeterministic envelope function119908(119905) is written as [10]

119908 (119905) =

( 1199051199050)2

0 lt 119905 lt 11990501 1199050 lt 119905 lt 119905119899exp [minus119888 (119905 minus 119905119899)] 1199050 lt 119905

(5)

where 1199050 and 119905119899 are the initial time and the 119899th time respec-tively 1199050 minus 119905119899 is the significant duration of the accelerogramand 119888 is the nonuniformly modulated coefficient A typicalvalue of 119888 is adopted as 02

T (s)0 02 10

S a(T

)

CaSaＧＪ(02)

CSaＧＪ(10)

SaＭＣＮ (T) = (3T + 04)CaSaＧＪ(02)

SaＭＣＮ (T) = CSaＧＪ(10)T

Ts

Figure 2 ISO seismic design spectrum

A possible approach to define 119886(119905) is by assuming thatthe motion induced by earthquakes is a realization of astationary stochastic processThe time history of a stationaryearthquake excitation can be readily generated based on thespectral density as follows

119886 (119905) =119899

sum119896=1

radic(4 sdot Δ120596 sdot 119878 (120596119896)) cos (120596119896119905 + 120576119896) (6)

where Δ120596 is the constant difference between successivefrequencies 120596119896 = 119896 sdot Δ120596 is the 119896th wave frequency 119878(120596119896) isthe power spectrum of 120596119896 119899 is the total number of frequencypoints considered in the calculation and 120576119896 is the randomphase angle varying between 0 and 2120587

In engineering practice [15] the design target responsespectrum at a given site is more commonly available than theground motion power spectrum density function Thereforeit will be very useful to generate time histories of groundmotion that are compatible to a given design target responsespectrum

The design target acceleration response spectrum 119878119879119886 (120596)can be obtained in seismic design guidelines which cannotbe directly used by (6) Thus a conversion formula is neededto obtain the corresponding ground motion power spectrum119878(120596)

For a given design target acceleration response spectrum119878119879119886 (120596) the corresponding ground motion power spectrum119878(120596) can be estimated by [15]

119878 (120596) = 120577120587120596 [119878

119879119886 (120596)]2 1

ln [minus (120587120596119879119889) ln (1 minus 119875)] (7)

where 119879119889 is the time duration of earthquake 120577 is the dampingratio and 119875 is the exceeding probability In the present studythe value of the parameter 119875 is taken as 09

The design target acceleration response spectrum isnormally determined by basic parameters of seismic intensitysite classification peak acceleration and damping ratio Theseismic design spectrum curve as shown in Figure 2 canbe found in the ISO design guidelines (ISO 2006) [16] Inthe figure 119878119886 is the spectral acceleration 119878119886site(119879) is thesite spectral acceleration and 119878119886map is the bedrock spectralacceleration When the fundamental period of a structure is

4 Shock and Vibration

equal to 02 or 10 s the bedrock spectral acceleration can becalculated by 119878119886map(02) = 05119892 or 119878119886map(10) = 02119892 Theparameter 119879 is the fundamental period of the structure and119879119904 is the site characteristic period and can be calculated asfollows

119879119904 = 119862V119878119886map (10)119862119886119878119886map (02) (8)

where 119862119886 and 119862V are the site coefficients and can be obtainedfrom the design tables in ISO (2006) [16]

Assuming that the submerged floating tunnel investigatedin this paper is located in the Bohai Sea of China withanchoring systems on a shallow foundation the site class ofthe foundation is categorized as A For a site class AB anda shallow foundation the parameters 119862119886 and 119862V are bothassigned as 10

Using the above approach the generated time historiesof ground motion usually match well with the design targetresponse spectrum Iterations should be carried out to adjustthe power spectrumdensity function if the two spectra do notmatch satisfactorily [15]

23 Formulation of Hydrodynamic Force in Time DomainThe vortex-induced lift force can be expressed as [17]

119865119871 = 12120588119908119863119862119871 (V119871 sin 120579)

2 sin (120596V119905) (9)

where 120588119908 is the water density 119863 is the outer diameter of thecable 120596V is the vortex shedding frequency 119862119871 is the vortex-induced lift coefficient and V119871 is the vortex flow velocity andits typical value is taken as 10ms

The hydrodynamic force 119865119863 can be expressed by Mori-sonrsquos equation [14] as follows

119865119863 = 12120588119908119863119862119863 (V minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816V minus12059711990612059711990510038161003816100381610038161003816100381610038161003816 + 119862119872

1205874 120588119908119863

2 120597V120597119905

minus (119862119872 minus 1) 1205874 1205881199081198632 12059721199061205971199052

(10)

where 119862119863 is the drag coefficient 119862119872 is the inertia coefficient119906 is the displacement of the cable and V is the instantaneouscurrent velocity

The hydrodynamic force induced by earthquake excita-tion is formulated following four assumptions [7 18 19]

(1) As the earthquake excitation process is in a shorttime duration the current velocity around the cable duringearthquakes is assumed to be negligible

(2) When the ground motion is horizontal the currentvelocity is set as zero when the groundmotion is vertical thecurrent velocity is equal to the ground velocity

(3) The wave force is neglected as the cable is located indeep water

(4) The coupled effect between the motions in differentdirections is neglected

Thus the hydrodynamic force under earthquake excita-tion can be expressed by [7 18 19]

119865119863 = 12120588119908119863119862119863 (119892 minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816119892 minus12059711990612059711990510038161003816100381610038161003816100381610038161003816

+ 1198621198721205874 1205881199081198632119892 minus (119862119872 minus 1) 1205874 120588119908119863

2 12059721199061205971199052

(11)

24 Solution of the Equation Suppose that the oscillationmode of the cable is that of standard chord

119906 (119911 119905) =infin

sum119899=1

sin 119899120587119911119871 119906119899 (119905) (12)

To obtain an approximate solution of (1) Galerkinmethod is applied to transform the partial differential equa-tion into a set of ordinary ones

Substituting (12) into (1) multiplying each term bysin(119899120587119911119871) integrating over 119911 isin (0 119871) and invoking theorthogonality property of the normal modes give

119899 + [1205962119872119899 + 1205962119860119899 (1 + 120576 cos120596119901119905)] 119906119899 + 119862119904119899119898 119899

minus int1198710

2 (119865119871 minus 119865119863 minus 119876119904 sin 120579)119898119871 sin 119899120587119911119871 119889119911 = 0

(13)

with

1205962119872119899 = (119899120587119871 )4 119864eq119868119898 (14)

1205962119860119899 = (119899120587119871 )2 1198790119898 (15)

120596119899 = radic1205962119872119899 + 1205962119860119899 (16)

119862119904119899 = 2119898120596119899120577 (17)

where 120577 is the damping ratio of the cable120596119872119899120596119860119899 are the 119899thorder natural frequency of the system of bending vibrationand axial vibration respectively 120596119899 is the total 119899th ordernatural frequency of the system and 120576 is the ratio of dynamictension to static tension

The fourth-order Runge-Kutta method is used to solvethe differential equation (13) eachmode response of the cableunder excitation may be obtained By substituting the moderesponse into (12) the displacement response of the cablemaybe obtained

3 Numerical Results and Discussion

The cable of a submerged floating tunnel is analyzed in thepresent study as an illustrative example to show the efficacyof the proposed analytical solution that is programmed inMATLAB The physical and geometric parameters are listedin Table 1 following the work of Sun and Su [2] If the samevalues of 119862119872 = 10 and 119862119863 = 07 provided by Sun and Su[2] are used it can result in the loss of hydrodynamic added

Shock and Vibration 5

minus20

minus15

minus10

minus05

00

05

10

15

20a g

(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(a) Artificial earthquake (AB)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(b) El Centro earthquake (C)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(c) Taft earthquake (D)

5 10 15 20 25 300t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

(d) Loma Prieta earthquake (E)

Figure 3 Time histories of four selected earthquake records

Table 1 Basic characteristics of the analysis case

Parameters ValuesDensity of water 120588119908 (kgm3) 1028Depth of water 119889 (m) 170Vertical height119867119911 (m) 140Unstretched length of cable 119871 (m) 16166Outside diameter of cable119863 (m) 0489Density of cable 120588119904 (kgm3) 7850Modulus of elasticity of cable 119864 (Pa) 21011986411Initial tension of cable 1198790 (N) 25721198647Mass per unit length of cable119898 (kgm) 147423Damping ratio 120577 00018Inertia coefficient 119862119872 20Drag coefficient 119862119863 10Inclination angle 120579 (∘) 60

mass Thus the values of 119862119872 and 119862119863 are taken as 20 and 10respectively following the work of Martire et al [20]

In the present study four earthquake records in differentrepresentative sites are selected For comparison the peakacceleration of these accelerograms is adjusted to 02119892corresponding to earthquakes with an intensity factor of 8 asshown in Figure 3 and Table 2 [16 21]

The artificial earthquake is generated by using the afore-mentioned theory in Section 22 and the other three earth-quakes are the classical earthquake records obtained fromcorresponding site classifications The time histories in timedomain the amplitude and energy in frequency domainof four earthquakes are different matching different siteclassifications

A sensitivity analysis is conducted to evaluate the influ-ence of key parameters to provide recommendations foruse in design and construction of the anchor system for asubmerged floating tunnel under hydrodynamic force andearthquake The effects of key environmental and structural

6 Shock and Vibration

Su and Sun (2013) method (under El Centro earthquake

20 40 60 80 1000t (s)

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

000

005

010

FFT

ampl

itude

(a) 120596119901 = 151205961

20 40 60 80 1000t (s)

minus3

minus2

minus1

0

1

2

3

u(z

t) z

=05L

(m)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

00

02

04

06

08

10

12FF

T am

plitu

de

(b) 120596119901 = 21205961

Figure 4 Mid-span displacement of the cable subjected to El Centro earthquake

parameters on the dynamic responses of the cable for sub-merged floating tunnel are investigated

31 Evaluation of the Proposed Method In order to evaluatethe effectiveness of the developed method results evaluatedusing the developed approach are compared with thosecalculated using the method of Chen et al (2008) [22] andSu and Sun (2013) [11]

In Figure 5 the overall trends of the results obtainedby the present method generally agree with the evaluationscalculated from the other methods The present methodcaptures the peak value well at resonance condition (ie1205961199011205961 = 1 2) as well as the subsequent attenuation trendOverall the calculation by the present method is acceptableand reasonable

In Figures 4 and 5 the dynamic responses of the cableunder hydrodynamic force and earthquake excitation can begreatly amplified by the present method Especially underthe super-resonance condition (ie 1205961199011205961 = 2) it can beobserved that the FFT amplitude component obtained by thepresent method has more abundant high-frequency contentsand excites wider peak frequency region than that of thosecalculated using the method of Su and Sun (2013) A possibleexplanation for the difference in calculations could be dueto the reason that the seismic design spectrum for marinegeotechnical geology is used in the present approach whichismore reasonable and accurate than other strategies by usingseismic design spectrum for overland building Anotherpossible explanation could be the fact that the coupledeffect of hydrodynamic force and earthquake excitation is

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

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RotatingMachinery

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Volume 2014

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

equal to 02 or 10 s the bedrock spectral acceleration can becalculated by 119878119886map(02) = 05119892 or 119878119886map(10) = 02119892 Theparameter 119879 is the fundamental period of the structure and119879119904 is the site characteristic period and can be calculated asfollows

119879119904 = 119862V119878119886map (10)119862119886119878119886map (02) (8)

where 119862119886 and 119862V are the site coefficients and can be obtainedfrom the design tables in ISO (2006) [16]

Assuming that the submerged floating tunnel investigatedin this paper is located in the Bohai Sea of China withanchoring systems on a shallow foundation the site class ofthe foundation is categorized as A For a site class AB anda shallow foundation the parameters 119862119886 and 119862V are bothassigned as 10

Using the above approach the generated time historiesof ground motion usually match well with the design targetresponse spectrum Iterations should be carried out to adjustthe power spectrumdensity function if the two spectra do notmatch satisfactorily [15]

23 Formulation of Hydrodynamic Force in Time DomainThe vortex-induced lift force can be expressed as [17]

119865119871 = 12120588119908119863119862119871 (V119871 sin 120579)

2 sin (120596V119905) (9)

where 120588119908 is the water density 119863 is the outer diameter of thecable 120596V is the vortex shedding frequency 119862119871 is the vortex-induced lift coefficient and V119871 is the vortex flow velocity andits typical value is taken as 10ms

The hydrodynamic force 119865119863 can be expressed by Mori-sonrsquos equation [14] as follows

119865119863 = 12120588119908119863119862119863 (V minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816V minus12059711990612059711990510038161003816100381610038161003816100381610038161003816 + 119862119872

1205874 120588119908119863

2 120597V120597119905

minus (119862119872 minus 1) 1205874 1205881199081198632 12059721199061205971199052

(10)

where 119862119863 is the drag coefficient 119862119872 is the inertia coefficient119906 is the displacement of the cable and V is the instantaneouscurrent velocity

The hydrodynamic force induced by earthquake excita-tion is formulated following four assumptions [7 18 19]

(1) As the earthquake excitation process is in a shorttime duration the current velocity around the cable duringearthquakes is assumed to be negligible

(2) When the ground motion is horizontal the currentvelocity is set as zero when the groundmotion is vertical thecurrent velocity is equal to the ground velocity

(3) The wave force is neglected as the cable is located indeep water

(4) The coupled effect between the motions in differentdirections is neglected

Thus the hydrodynamic force under earthquake excita-tion can be expressed by [7 18 19]

119865119863 = 12120588119908119863119862119863 (119892 minus

120597119906120597119905 )

10038161003816100381610038161003816100381610038161003816119892 minus12059711990612059711990510038161003816100381610038161003816100381610038161003816

+ 1198621198721205874 1205881199081198632119892 minus (119862119872 minus 1) 1205874 120588119908119863

2 12059721199061205971199052

(11)

24 Solution of the Equation Suppose that the oscillationmode of the cable is that of standard chord

119906 (119911 119905) =infin

sum119899=1

sin 119899120587119911119871 119906119899 (119905) (12)

To obtain an approximate solution of (1) Galerkinmethod is applied to transform the partial differential equa-tion into a set of ordinary ones

Substituting (12) into (1) multiplying each term bysin(119899120587119911119871) integrating over 119911 isin (0 119871) and invoking theorthogonality property of the normal modes give

119899 + [1205962119872119899 + 1205962119860119899 (1 + 120576 cos120596119901119905)] 119906119899 + 119862119904119899119898 119899

minus int1198710

2 (119865119871 minus 119865119863 minus 119876119904 sin 120579)119898119871 sin 119899120587119911119871 119889119911 = 0

(13)

with

1205962119872119899 = (119899120587119871 )4 119864eq119868119898 (14)

1205962119860119899 = (119899120587119871 )2 1198790119898 (15)

120596119899 = radic1205962119872119899 + 1205962119860119899 (16)

119862119904119899 = 2119898120596119899120577 (17)

where 120577 is the damping ratio of the cable120596119872119899120596119860119899 are the 119899thorder natural frequency of the system of bending vibrationand axial vibration respectively 120596119899 is the total 119899th ordernatural frequency of the system and 120576 is the ratio of dynamictension to static tension

The fourth-order Runge-Kutta method is used to solvethe differential equation (13) eachmode response of the cableunder excitation may be obtained By substituting the moderesponse into (12) the displacement response of the cablemaybe obtained

3 Numerical Results and Discussion

The cable of a submerged floating tunnel is analyzed in thepresent study as an illustrative example to show the efficacyof the proposed analytical solution that is programmed inMATLAB The physical and geometric parameters are listedin Table 1 following the work of Sun and Su [2] If the samevalues of 119862119872 = 10 and 119862119863 = 07 provided by Sun and Su[2] are used it can result in the loss of hydrodynamic added

Shock and Vibration 5

minus20

minus15

minus10

minus05

00

05

10

15

20a g

(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(a) Artificial earthquake (AB)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(b) El Centro earthquake (C)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(c) Taft earthquake (D)

5 10 15 20 25 300t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

(d) Loma Prieta earthquake (E)

Figure 3 Time histories of four selected earthquake records

Table 1 Basic characteristics of the analysis case

Parameters ValuesDensity of water 120588119908 (kgm3) 1028Depth of water 119889 (m) 170Vertical height119867119911 (m) 140Unstretched length of cable 119871 (m) 16166Outside diameter of cable119863 (m) 0489Density of cable 120588119904 (kgm3) 7850Modulus of elasticity of cable 119864 (Pa) 21011986411Initial tension of cable 1198790 (N) 25721198647Mass per unit length of cable119898 (kgm) 147423Damping ratio 120577 00018Inertia coefficient 119862119872 20Drag coefficient 119862119863 10Inclination angle 120579 (∘) 60

mass Thus the values of 119862119872 and 119862119863 are taken as 20 and 10respectively following the work of Martire et al [20]

In the present study four earthquake records in differentrepresentative sites are selected For comparison the peakacceleration of these accelerograms is adjusted to 02119892corresponding to earthquakes with an intensity factor of 8 asshown in Figure 3 and Table 2 [16 21]

The artificial earthquake is generated by using the afore-mentioned theory in Section 22 and the other three earth-quakes are the classical earthquake records obtained fromcorresponding site classifications The time histories in timedomain the amplitude and energy in frequency domainof four earthquakes are different matching different siteclassifications

A sensitivity analysis is conducted to evaluate the influ-ence of key parameters to provide recommendations foruse in design and construction of the anchor system for asubmerged floating tunnel under hydrodynamic force andearthquake The effects of key environmental and structural

6 Shock and Vibration

Su and Sun (2013) method (under El Centro earthquake

20 40 60 80 1000t (s)

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

000

005

010

FFT

ampl

itude

(a) 120596119901 = 151205961

20 40 60 80 1000t (s)

minus3

minus2

minus1

0

1

2

3

u(z

t) z

=05L

(m)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

00

02

04

06

08

10

12FF

T am

plitu

de

(b) 120596119901 = 21205961

Figure 4 Mid-span displacement of the cable subjected to El Centro earthquake

parameters on the dynamic responses of the cable for sub-merged floating tunnel are investigated

31 Evaluation of the Proposed Method In order to evaluatethe effectiveness of the developed method results evaluatedusing the developed approach are compared with thosecalculated using the method of Chen et al (2008) [22] andSu and Sun (2013) [11]

In Figure 5 the overall trends of the results obtainedby the present method generally agree with the evaluationscalculated from the other methods The present methodcaptures the peak value well at resonance condition (ie1205961199011205961 = 1 2) as well as the subsequent attenuation trendOverall the calculation by the present method is acceptableand reasonable

In Figures 4 and 5 the dynamic responses of the cableunder hydrodynamic force and earthquake excitation can begreatly amplified by the present method Especially underthe super-resonance condition (ie 1205961199011205961 = 2) it can beobserved that the FFT amplitude component obtained by thepresent method has more abundant high-frequency contentsand excites wider peak frequency region than that of thosecalculated using the method of Su and Sun (2013) A possibleexplanation for the difference in calculations could be dueto the reason that the seismic design spectrum for marinegeotechnical geology is used in the present approach whichismore reasonable and accurate than other strategies by usingseismic design spectrum for overland building Anotherpossible explanation could be the fact that the coupledeffect of hydrodynamic force and earthquake excitation is

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

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Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

minus20

minus15

minus10

minus05

00

05

10

15

20a g

(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(a) Artificial earthquake (AB)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(b) El Centro earthquake (C)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

5 10 15 20 25 300t (s)

(c) Taft earthquake (D)

5 10 15 20 25 300t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

a g(t)

(mmiddotＭminus2)

(d) Loma Prieta earthquake (E)

Figure 3 Time histories of four selected earthquake records

Table 1 Basic characteristics of the analysis case

Parameters ValuesDensity of water 120588119908 (kgm3) 1028Depth of water 119889 (m) 170Vertical height119867119911 (m) 140Unstretched length of cable 119871 (m) 16166Outside diameter of cable119863 (m) 0489Density of cable 120588119904 (kgm3) 7850Modulus of elasticity of cable 119864 (Pa) 21011986411Initial tension of cable 1198790 (N) 25721198647Mass per unit length of cable119898 (kgm) 147423Damping ratio 120577 00018Inertia coefficient 119862119872 20Drag coefficient 119862119863 10Inclination angle 120579 (∘) 60

mass Thus the values of 119862119872 and 119862119863 are taken as 20 and 10respectively following the work of Martire et al [20]

In the present study four earthquake records in differentrepresentative sites are selected For comparison the peakacceleration of these accelerograms is adjusted to 02119892corresponding to earthquakes with an intensity factor of 8 asshown in Figure 3 and Table 2 [16 21]

The artificial earthquake is generated by using the afore-mentioned theory in Section 22 and the other three earth-quakes are the classical earthquake records obtained fromcorresponding site classifications The time histories in timedomain the amplitude and energy in frequency domainof four earthquakes are different matching different siteclassifications

A sensitivity analysis is conducted to evaluate the influ-ence of key parameters to provide recommendations foruse in design and construction of the anchor system for asubmerged floating tunnel under hydrodynamic force andearthquake The effects of key environmental and structural

6 Shock and Vibration

Su and Sun (2013) method (under El Centro earthquake

20 40 60 80 1000t (s)

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

000

005

010

FFT

ampl

itude

(a) 120596119901 = 151205961

20 40 60 80 1000t (s)

minus3

minus2

minus1

0

1

2

3

u(z

t) z

=05L

(m)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

00

02

04

06

08

10

12FF

T am

plitu

de

(b) 120596119901 = 21205961

Figure 4 Mid-span displacement of the cable subjected to El Centro earthquake

parameters on the dynamic responses of the cable for sub-merged floating tunnel are investigated

31 Evaluation of the Proposed Method In order to evaluatethe effectiveness of the developed method results evaluatedusing the developed approach are compared with thosecalculated using the method of Chen et al (2008) [22] andSu and Sun (2013) [11]

In Figure 5 the overall trends of the results obtainedby the present method generally agree with the evaluationscalculated from the other methods The present methodcaptures the peak value well at resonance condition (ie1205961199011205961 = 1 2) as well as the subsequent attenuation trendOverall the calculation by the present method is acceptableand reasonable

In Figures 4 and 5 the dynamic responses of the cableunder hydrodynamic force and earthquake excitation can begreatly amplified by the present method Especially underthe super-resonance condition (ie 1205961199011205961 = 2) it can beobserved that the FFT amplitude component obtained by thepresent method has more abundant high-frequency contentsand excites wider peak frequency region than that of thosecalculated using the method of Su and Sun (2013) A possibleexplanation for the difference in calculations could be dueto the reason that the seismic design spectrum for marinegeotechnical geology is used in the present approach whichismore reasonable and accurate than other strategies by usingseismic design spectrum for overland building Anotherpossible explanation could be the fact that the coupledeffect of hydrodynamic force and earthquake excitation is

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Su and Sun (2013) method (under El Centro earthquake

20 40 60 80 1000t (s)

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

000

005

010

FFT

ampl

itude

(a) 120596119901 = 151205961

20 40 60 80 1000t (s)

minus3

minus2

minus1

0

1

2

3

u(z

t) z

=05L

(m)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

Su and Sun (2013) method (under El Centro earthquake

Present method

and hydrodynamic force excitation)

20 40 60 80 1000 (rads)

00

02

04

06

08

10

12FF

T am

plitu

de

(b) 120596119901 = 21205961

Figure 4 Mid-span displacement of the cable subjected to El Centro earthquake

parameters on the dynamic responses of the cable for sub-merged floating tunnel are investigated

31 Evaluation of the Proposed Method In order to evaluatethe effectiveness of the developed method results evaluatedusing the developed approach are compared with thosecalculated using the method of Chen et al (2008) [22] andSu and Sun (2013) [11]

In Figure 5 the overall trends of the results obtainedby the present method generally agree with the evaluationscalculated from the other methods The present methodcaptures the peak value well at resonance condition (ie1205961199011205961 = 1 2) as well as the subsequent attenuation trendOverall the calculation by the present method is acceptableand reasonable

In Figures 4 and 5 the dynamic responses of the cableunder hydrodynamic force and earthquake excitation can begreatly amplified by the present method Especially underthe super-resonance condition (ie 1205961199011205961 = 2) it can beobserved that the FFT amplitude component obtained by thepresent method has more abundant high-frequency contentsand excites wider peak frequency region than that of thosecalculated using the method of Su and Sun (2013) A possibleexplanation for the difference in calculations could be dueto the reason that the seismic design spectrum for marinegeotechnical geology is used in the present approach whichismore reasonable and accurate than other strategies by usingseismic design spectrum for overland building Anotherpossible explanation could be the fact that the coupledeffect of hydrodynamic force and earthquake excitation is

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

Table 2 Site classifications of earthquake records used for this study

Ground motion Site classifications Site condition Time duration(s)

Time interval(s)

Artificial earthquake AB Rock 30 002El Centro earthquake in the EWdirection C Very dense hard soil and soft

rock 30 002

Taft Kern County (Taft for short)earthquake in the N21E direction D Stiff to very stiff soil 30 002

Loma Prieta Oakland outer wharf(Loma for short) earthquake E Soft soil 30 002

Chen et al (2008) method

Su and Sun (2013) method (under El Centro earthquake

Present method

00

02

04

06

08

10

12

14

16

18

00 05 10 15 20 25 30 35 40minus05

(only under hydrodynamic force excitation)

and hydrodynamic force excitation)

uＬＧ

Ｍ(zt) z

=05L

(m)

pn

Figure 5 Variations of RMS of mid-span displacement of the cablewith 120596119901120596119899 calculated using different methods

incorporated in the present approach while this coupledeffect and the hydrodynamic added mass are neglected inother strategies Similar results of underestimation by theexisting analytical models [11 22] compared to experimentalmeasurements and numerical simulations were reported[8 19]

32 Dynamic Response of the Cable under Only VortexExcitation Vortex-induced vibration of cables is caused bycomplex interactions between cylindrical structures and thecorrelated wake vortex and it is an important source offatigue damage to the cables Figure 6 shows the dynamicresponse of the cable under only vortex excitation Two casesare investigated inwhich no oscillationwas excited by vortex-shedding in type of 120596V = 011205961 and vortex-induced vibrationin type of 120596V = 1205961 which are showed in Figures 6(a) and6(b) respectively In Figure 6(a) the dynamic response ofthe cable shapes as a sinusoidal curve and its amplitude issmall And in Figure 6(b) the action of exciting system tovibrate will happen as the lift force falls into the lock-in range

that is beat phenomenon obviously occurs Besides due tothe self-limitation of damping the time history of vortex-induced vibration shown in Figure 6(b) decreases rapidly inthe first 200 s but the decreasing pattern is roughly linearand reaches a steady-state response quickly And it can beobserved that the amplitude of dynamic responses of the cablein the condition of lock-in case120596V = 1205961 is signally larger thanthe other case of 120596V = 1205961 Figure 6(c) shows similar resultsthe largest dynamic response occurs when vortex-inducedfrequency is consistent with the structural natural frequencythat is120596V = 1205961 Besides the value of dynamic response of thecable in the other frequency is relatively small

33 Dynamic Response of the Cable under Combinationsof Vortex Excitation and Parametric Excitation In practicefor the cable of submerged floating tunnel suffering bothparametric excitation and vortex excitation its structuralconfiguration is more complex rather than merely an Eulerbeam alone Moreover it is more complicated if two excita-tion frequencies of both top-end tube and vortex-induced liftforce are involved

Figures 7(a)ndash7(d) show four typical examples of the cableunder combinations of vortex excitation and parametricexcitation and these dynamic responses exhibit interestingand different features of the pattern and shape of the timehistory In Figure 7(c) a combined resonance occurs owingto both parametric excitation and vortex-shedding excitationwhere both excitations satisfy the condition that can exciteresonance of a system under only parametric or vortexshedding excitation and its motion amplitude is obviouslylarger and more unstable than those of the other cases inFigures 7(a) 7(b) and 7(c) Figure 7(a) shows the time historyof the cable when in case of 120596V = 1205961 and 120596119901 = 011205961as vortex excitation dominates parametric excitation beatphenomenon still occurs in this case and has the smallestmotion amplitude In Figures 7(b) and 7(d) as parametricexcitation dominates vortex excitation beat phenomenon isbroken in those cases the pattern of each time history is akind of ldquostanding waverdquo where each has a repeating wave-unit but with mutually different profiles and the numberof wave-units is approximatively ranging from 25 to 35and the profiles become more irregular Therefore it canbe concluded that the dynamic response of the cable undercombinations of vortex excitation and parametric excitationbecamemore complicated especially in combined resonancecondition

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

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RotatingMachinery

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

100 200 300 4000t (s)

minus0004

minus0002

0000

0002

0004

u(z

t) z

=05L

(m)

(a) 120596V = 011205961

minus003

minus002

minus001

000

001

002

003

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961

05 10 15 20 25 30001

0000

0005

0010

0015

0020

0025

uＧＲ(z

t)

(m)

(c) Maximum displacement versus 120596V1205961

Figure 6 Dynamic response of the cable under only vortex excitation

Figure 7(e) indicates that three largest amplitudes ofdynamic response are respectively at the ratio of parametricfrequency to natural frequency of 1205961199011205961 = 1 2 and 3and the peak values of the mid-span displacement of thecable under above cases are 04013m 08914m and 02891mrespectively Particularly the largest response amplitudeoccurs at frequency 1205961199011205961 = 2 Or we may say for mode1 the largest dynamic response might occur when top-endtube parametric frequency is twice as much as the naturalfrequency that is 120596119901 = 21205961 when parametric excitationsatisfies the condition which can excite parametric resonanceof a system under parametric excitation Besides the valueof dynamic response of the cable in the other frequencycase is relatively small The results suggest that parametricexcitation has a strong effect on the dynamic response of thecable especially combined with vortex shedding frequencyconsistent with the structural natural frequency that is 120596V =1205961

Thus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance Based on (14)ndash(16) the key parameters for decidingthe fundamental frequency of the cable are 119898 119864eq 119868 1198790and 119871 while the parametric excitation frequency 120596119901 mainlydepends on ocean environmental conditions acting on thetube

34 Effect of Motion of Top-End Tube The comparison ofmodal dynamic responses between the cases with top-endtubemoving andwithout top-end tubemoving is presented inFigure 8The dynamic responses of the cable are numericallysimulated while the top-end tube is moving at the cablersquosnatural frequencies of modes ranging from mode 1 to mode12 and the most dangerous cases are taken into account thatis 120596V = 120596119895 and 120596119901 = 2120596119895 here 120596119895 is the 119895 model naturalfrequency In Figure 8 as modal number increases the cablersquos

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

minus008

minus006

minus004

minus002

000

002

004

006

008

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 and 120596119901 = 011205961

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(b) 120596V = 1205961 and 120596119901 = 051205961

minus10

minus08

minus06

minus04

minus02

00

02

04

06

08

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 and 120596119901 = 21205961

minus03

minus02

minus01

00

01

02

03u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(d) 120596V = 1205961 and 120596119901 = 251205961

01

02

03

04

05

06

07

08

09

uＧＲ(z

t)

(m)

1 2 3 40p1

(e) Maximum displacement versus 1205961199011205961

Figure 7 Dynamic response of the cable under combinations of vortex excitation and parametric excitation

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

With movingWithout moving

2 4 6 8 10 120Modal order number

00

02

04

06

08

10

uＧＲ(z

t)

(m)

Figure 8 Comparison of modal dynamic responses between thecases with top-end tube moving and without top-end tube moving

Root mean square valueMaximum value

12 14 16 18 20 2210Motion amplitude

05

10

15

20

25

30

u(z

t) z

=05L

(m)

Figure 9 The effect of motion amplitude of top-end tube on thedynamic response of the cable

vibration either with top-end tube moving or without top-end tube moving attenuates rapidly to small value even zeroIt is noted that the first third-order mode responses aremostly dominated and they take up more than 80 of thewhole mode response Also it is shown that the maximumamplitudes of the cable with top-end tube moving are largerthan the cases without top-end tube moving and takingmode 1 as an example its maximum amplitude is about twicelarger than the case without top-end tube moving

Figure 9 shows the curves of the maximum value androot mean square value of the dynamic response of the cableas the motion amplitude of top-end tube varies from 10 to

minus02

minus01

00

01

02

03

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(a) 120596V = 1205961 120596119901 = 001205961 and under Taft earthquake

100 200 300 4000t (s)

minus10

minus05

00

05

10u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 101205961 and under Taft earthquake

Figure 10 The effect of earthquake on the dynamic response of thecable

22 where motion amplitude is expressed by the ratio ofdynamic tension to static tension 120576 In both the two casesthe variational laws of curves are similar and the maximumvalue of the displacement increases with the increase ofmotion amplitude which is very dangerous for the cablesystem And the root mean square value of the dynamicresponse of the cable rises almost linearly with the increaseofmotion amplitudeThusmotion amplitude of top-end tubeis the most important parameter in determining the dynamicresponse of the cable which necessitates careful considerationduring the early stage of design

35 Effect of Earthquake Figure 10 shows the dynamicresponse time history of the cable when in combinations ofvortex excitation and parametric excitation under Taft earth-quake When the cable is subjected to the earthquake themaximum value and the mean value in this case are signally

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

100 200 300 4000t (s)

minus06

minus04

minus02

00

02

04

06u(z

t) z

=05L

(m)

(a) 120596V = 1205961 120596119901 = 011205961 and under artificial earthquake

100 200 300 4000t (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

(b) 120596V = 1205961 120596119901 = 011205961 and under El Centro earthquake

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

(c) 120596V = 1205961 120596119901 = 011205961 and under Taft earthquake

100 200 300 4000t (s)

minus04

minus02

00

02

04u(z

t) z

=05L

(m)

(d) 120596V = 1205961 120596119901 = 011205961 and under Loma earthquake

Figure 11 Comparison of dynamic responses of the cable between the cases of four site classifications earthquakes

larger than the other two cases with only combinations ofvortex excitation and parametric excitation Beside due tothe randomness of the earthquake the curves of dynamicresponse of the cable becomemore irregular Compared withFigure 6(b) due to the effect of earthquake the maximumvalue appears at the time of 0sim50 s that is the time durationof earthquake after that themotion amplitude attenuates andthemaximum values of dynamic response in Figure 10(a) addfrom 00223m to 0221m When compared with Figure 7(c)the maximum values of dynamic response in Figure 10(b)add from 0883m to 0916m and the maximum value ofmotion response of the cable appears around from thetime point of 50 s to 160 s It is noted that the earthquakecan signally affect the motion amplitude and time pointsreach the maximum value of the cable Therefore dynamicanalysis under combined vortex parametric and earthquakeexcitations is necessary

From Figures 11(a) to 11(d) it can be seen that site classof earthquake wave affects the vibration response of cablelargely and the pattern andmotion amplitude of time historyof the cable are invariant The maximum value appears at thetime of 0sim50 s that is the time duration of earthquake afterthat the motion amplitude attenuates In other words theenergy of the dynamic response of the cable is focused on thetime duration of earthquake The difference of seismic wavespectrum characteristics results in the difference of maxi-mummid-span displacements of the cableThe peak values ofthe dynamic response of the cable under artificial earthquakeEl Centro earthquake Taft earthquake and Loma earthquakeare 0522m 0383m 0356m and 0406m respectively andoccur at the time that the earthquake wave reaches its energy-intensive zone Similarly the root mean square values of thedynamic response of the cable under the above earthquakewaves are 0236m 0156m 0131m and 0202m respectively

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

= 00009 = 00036 = 00072

minus10

minus05

00

05

10

u(z

t) z

=05L

(m)

100 200 300 4000t (s)

Figure 12 The effect of damping ratio on the dynamic response ofthe cable (120596V = 1205961 and 120596119901 = 21205961)

It is concluded that dynamic response of the cable underartificial earthquake which is suited to rock foundation isthe largest Oppositely the Taft earthquake which is suitedto medium soil foundation induces the smallest dynamicresponse values of the cable

36 Effect of Damping Ratio The influence of dampingratio on the dynamic response of the cable is also inves-tigated and three alternative levels of damping ratio 120577 =00009 00036 00072 are adopted As shown in Figure 12 themotion amplitude and energy of the dynamic response of thecable are decreased as the modal damping ratio is increasedAs the damping ratio increases to 00072 the wave crestsinduced by combinations of vortex excitation and parametricexcitation are smoothed after the time of 150 s The rate ofdescent of the maximum values between 120577 = 00009 and120577 = 00072 is reached up to 7282 This is an expectedresult because damping plays an important role in energydissipation and makes the cable system more stable [23 24]In preliminary design of the cable system it is recommendedthat the damping of cable should be increased to reducethe dynamic responses of the system and large-amplitudevibration of cable may be avoided by locating dampers on thecable or the tube

37 Effect of Lift Coefficient Based on the work conducted byChen et al 2015 the lift coefficient is adopted as a constantvalue ranging usually from 08 to 12 As shown in Figure 13four cases of the cable system under combined excitation areexplored to determine the influence of the lift coefficient onthe dynamic response and the three different lift coefficientsare 119862119871 = 08 10 12 When the lift coefficient is rangedfrom 08 to 12 the lift coefficient is an important part of the

correlated lift force due to vortex shedding as a consequencelarger lift coefficient can cause larger dynamic responseThusthe dynamic response of the cable in case of lift coefficientin 08 is the smallest while comparing with the case ofother constant lift coefficients As a consequence larger liftcoefficient can cause more unstable dynamic response Inthe meanwhile the inclusion of parametric excitation canstrengthen the instability when it satisfies the special condi-tion which can cause parametric resonance of a cable systemAs the effect of self-excitation of nonlinear vortex-induced liftterms the excitation of resonance becomes weaker while thelift coefficient is getting lower In Figure 13(b) the mid-spandisplacement of the cable is largest when along axial locationand the maximum amplitude in the case of lift coefficient in12 is more than twice larger than the case of lift coefficient in08

4 Conclusions

The parametrically excited vibrations of cable of submergedfloating tunnel under hydrodynamic and earthquake excita-tion are studied in this paper The dynamics motion of thecable is built by Euler beam theory and Galerkin methodRandom earthquake excitation in time domain is formu-lated by trigonometric series method The hydrodynamicforce induced by earthquake excitations is formulated torepresent real seaquake conditions An analytical model isproposed to analyze the cable system for submerged floatingtunnel model subjected to combined hydrodynamic forceand earthquake excitations The effects of the hydrodynamicand earthquake parameters and structural parameters on theresulting dynamics response of the system are explored anddiscussed The sensitivity of key parameters on the anchorsystem for submerged floating tunnel is investigated to enablea preliminary examination of the hydrodynamic and seismicbehavior of the cable and provide guidance for design andoperation of the anchor system for submergedfloating tunnel

The following conclusions can be drawn(1) The overall trends of the results obtained by the

present method generally agree with the evaluations calcu-lated from the other methods Due to the seismic designspectrum for offshore structure and the coupled effect ofhydrodynamic force and earthquake excitation considered inthe present approach the dynamic responses of the cable aregreatly amplified and FFT amplitude component obtainedby the present method has more abundant high-frequencycontents and excites wider peak frequency region

(2) The dynamic response of the cable under combina-tions of vortex excitation and parametric excitation becomemore complicated especially in combined resonance condi-tionThus the fundamental frequency of cable of submergedfloating tunnel located in complicated marine environmentshould be thought over carefully to avoid parametric reso-nance

(3) Dynamic response of the cable under artificial earth-quake which is suited to rock foundation is the largestOppositely the Taft earthquake which is suited to mediumsoil foundation induces the smallest dynamic response valuesof the cable

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

100 200 300 4000t (s)

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

u(z

t) z

=05L

(m)

CL = 08CL = 10CL = 12

(a) Time history at 120596V = 1205961 120596119901 = 011205961

02 04 06 08 1000Axial location zL

CL = 08CL = 10CL = 12

00

01

02

03

04

uＧＲ(z

t)

(m)

(b) Maximum value of model 1 along axial location

Figure 13 The effect of lift coefficient on the dynamic response of the cable

(4) Damping plays an important role in energy dissipa-tion and makes the cable system more stable In preliminarydesign of the cable system it is recommended that thedamping of cable should be increased to reduce the dynamicresponses of the system and large-amplitude vibration ofcable may be avoided by locating dampers on the cable or thetube

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This work had been supported by the National Natu-ral Science Foundation of China (Grants nos 51578164and 41672296) the China Postdoctoral Science Foundation(Grant no 2017M610578) the Systematic Project of GuangxiKey Laboratory of Disaster Prevention and Structural Safety(Grant no 2016ZDX008) and the Innovative Research TeamProgram of Guangxi Natural Science Foundation (Grant no2016GXNSFGA380008)

References

[1] S-N Sun J-Y Chen and J Li ldquoNon-linear response oftethers subjected to parametric excitation in submerged floatingtunnelsrdquo China Ocean Engineering vol 23 no 1 pp 167ndash1742009

[2] S-N Sun and Z-B Su ldquoParametric vibration of submergedfloating tunnel tether under random excitationrdquo China OceanEngineering vol 25 no 2 pp 349ndash356 2011

[3] W Lu F Ge L Wang X Wu and Y Hong ldquoOn the slackphenomena and snap force in tethers of submerged floatingtunnels under wave conditionsrdquoMarine Structures vol 24 no4 pp 358ndash376 2011

[4] S-I Seo H-SMun J-H Lee and J-H Kim ldquoSimplified analy-sis for estimation of the behavior of a submerged floating tunnelin waves and experimental verificationrdquoMarine Structures vol44 pp 142ndash158 2015

[5] C Cifuentes S Kim M Kim and W Park ldquoNumericalsimulation of the coupled dynamic response of a submergedfloating tunnel with mooring lines in regular wavesrdquo OceanSystems Engineering vol 5 no 2 pp 109ndash123 2015

[6] MDuan YWang Z Yue S Estefen andXYang ldquoDynamics ofrisers for earthquake resistant designsrdquo Petroleum Science vol7 no 2 pp 273ndash282 2010

[7] R Dong X Li Q Jin and J Zhou ldquoNonlinear parametricanalysis of free spanning submarine pipelines under spatiallyvarying earthquake ground motionsrdquo China Civil EngineeringJournal vol 41 no 7 pp 103ndash109 2008

[8] J H Lee S I Seo and H S Mun ldquoSeismic behaviors of afloating submerged tunnel with a rectangular cross-sectionrdquoOcean Engineering vol 127 pp 32ndash47 2016

[9] M Di Pilato F Perotti and P Fogazzi ldquo3D dynamic response ofsubmerged floating tunnels under seismic and hydrodynamicexcitationrdquo Engineering Structures vol 30 no 1 pp 268ndash2812008

[10] L Martinelli G Barbella and A Feriani ldquoA numerical pro-cedure for simulating the multi-support seismic response ofsubmerged floating tunnels anchored by cablesrdquo EngineeringStructures vol 33 no 10 pp 2850ndash2860 2011

[11] Z-B Su and S-N Sun ldquoSeismic response of submerged floatingtunnel tetherrdquoChinaOcean Engineering vol 27 no 1 pp 43ndash502013

[12] Ministry of Construction ldquoCode for seismic design of build-ingsrdquo GB 50011-2010 Ministry of Construction Beijing China2010

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

[13] Z Su and S Sun ldquoStability of submerged floating tunnel tethersubjected to parametric excitationrdquo Journal of Central SouthUniversity vol 44 no 6 pp 2549ndash2553 2013

[14] S Lei W-S Zhang J-H Lin Q-J Yue D Kennedy and FW Williams ldquoFrequency domain response of a parametricallyexcited riser under random wave forcesrdquo Journal of Sound andVibration vol 333 no 2 pp 485ndash498 2014

[15] K Bi and H Hao ldquoModelling and simulation of spatiallyvarying earthquake ground motions at sites with varyingconditionsrdquo Probabilistic EngineeringMechanics vol 29 pp 92ndash104 2012

[16] International Organization for Standardization (ISO) 2006Petroleum and Natural Gas Industries-specific Requirementsfor Offshore Structures-Part 2 Seismic design procedures andcriteria Switzerland

[17] K Wang W Tang and H Xue ldquoTime domain approach forcoupled cross-flow and in-line VIV induced fatigue damage ofsteel catenary riser at touchdown zonerdquoMarine Structures vol41 pp 267ndash287 2015

[18] T K Datta and E A Mashaly ldquoSeismic response of buriedsubmarine pipelinesrdquo Journal of Energy Resources Technologyvol 110 no 4 pp 208ndash218 1988

[19] J H Lee and J K Kim ldquoDynamic response analysis of a floatingoffshore structure subjected to the hydrodynamic pressuresinduced from seaquakesrdquo Ocean Engineering vol 101 pp 25ndash39 2015

[20] G Martire B Faggiano F M Mazzolani A Zollo and T AStabile ldquoSeismic analysis of a SFT solution for theMessina Straitcrossingrdquo Procedia Engineering vol 4 pp 303ndash310 2010

[21] Z W Wu J K Liu Z Q Liu and Z R Lu ldquoParametricallyexcited vibrations of marine riser under random wave forcesand earthquakerdquoAdvances in Structural Engineering vol 19 no3 pp 449ndash462 2016

[22] J Chen S Sun and B Wang ldquoDynamic analysis for the tetherof submerged floating tunnelrdquoChinese Journal of ComputationalMechanics vol 25 no 4 pp 487ndash493 2008 (Chinese)

[23] W B Wu H Liu M H El Naggar G X Mei and G S JiangldquoTorsional dynamic response of a pile embedded in layeredsoil based on the fictitious soil pile modelrdquo Computers andGeotechnics vol 80 pp 190ndash198 2016

[24] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of