Displacement Response of Submerged Floating Tunnel Tube Due to Single Moving Load Procedia Engineering 166 ( 2016 ) 143 – 151 Available online at www sciencedirect com 1877 7058 © 2016 The Authors Pub[.]
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 166 (2016) 143 – 151 2nd International Symposium on Submerged Floating Tunnels and Underwater Tunnel Structures Displacement response of submerged floating tunnel tube due to single moving load ZHANG Yuana,b, DONG Man-shenga,b,**, DING Haoa, YANG Long-changb a China Merchants Chongqing Communications Technology Research&Design Institute Co.,Ltd, Chongqing 400067, P.R.China b School of Transportation Engineering, Hefei University of Technology, Hefei 230009, P.R.China Abstract In this paper, the submerged floating tunnel(SFT) tube is simplified as an elastically supported beam with two springs and dampings at each end The kinematic equation for SFT tube under a single moving load is established to investigate kinematic problems By using the Galerkin method, the kinematic equation at the mid-span section of SFT tube can be solved, and the midspan displacement of SFT tube is gained by numerical simulation method The influence of anchor stiffness, moving load and moving velocity on SFT tube can be learned about by analyzing the simulation results The present results indicate that the anchor stiffness plays a significant role in influencing the SFT tube’s displacement At the same time, it also decides the vibration frequency of SFT tube Moreover, the magnitude and velocity of the single moving load can also have an obvious impact on the tube’s mid-span displacement The research results would provide as a theoretical reference for further study and futural construction of SFT © 2016 2016The TheAuthors Authors Published by Elsevier © Published by Elsevier Ltd Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of SUFTUS-2016 Peer-review under responsibility of the organizing committee of SUFTUS-2016 Keywords: submerged floating tunnel; single moving load; displacement response; parameter design; numerical simulation Introduction The submerged floating tunnel(SFT) has provided a new traffic method for water crossing It maintains the balance and stability by a combined effect of deadweight, buoyancy and anchoring system As a economy and ecofriendly structure, the SFT has a very broad prospect in application and that can be the reason why it has attracted so many researchers to study *Corresponding author.Tel.: +86-181-1967-0189; Email address: dongms@hfut.edu.cn 1877-7058 © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of SUFTUS-2016 doi:10.1016/j.proeng.2016.11.577 144 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 Since last century, many foreign researchers have carried out research study on SFT The dynamic response of different types SFT under effect of seismic force and wave force has been researched by Brancaleoni [1] for finding out the environment effect on SFT By finite element modeling and analysis with Navier-Stokes equation, Remseth [2] adopted numerical simulation method to study fluid/structure interaction and dynamic response of SFT Entering the 21st century, the researched have diversified Sato [3] considered that the SFT can be regarded as a beam on discrete elastic supports if its tension legs are separately spaced along its length and the elasticity of the tension legs is taken into consideration Lu [4] simulated the slack/taut alternate state of mooring tethers by bilinear oscillator and analyzed effect of buoyancy-weight ratio and inclined mooring angle on SFT dynamic response for preventing the occurrence of tether slacking and snap force The wave force characteristics acting on the SFT with different structure and size has been investigated by Kunisu [5] , and Morison’s equation was applied to evaluate the role of drag force Xiang took the coupling vibration of SFT tube and cable into consideration, and studied nonlinear Vortex-induced dynamic response by discussing mid-span displacement and time curves under different working conditions and parameters In these years, a lot of researchers have put more focus on dynamic response of SFT under moving loads Such as Tariverdilo [7] , who has proposed that the dynamic response caused by moving loads can be controlled by increasing anchor stiffness On the base of the research by Yang [8] and combining with structural characteristics of SFT, the SFT tube is simplified as an elastically supported beam with two springs and dampings at each end The kinematic equation for SFT tube under a single moving load is established to investigate kinematic problems in this paper By using the Galerkin method, the kinematic equation at the mid-span section of SFT tube can be solved, and the mid-span displacement of SFT tube is gained by numerical simulation method The influence of anchor stiffness, moving load and moving velocity on SFT tube can be learned about by analyzing the simulation results Physical structure and simplified model An innovative concept design of SFT is proposed in this paper, which is called ‘biplanar mooring SFT’, see Fig.1 The traditional SFT structure system includes tube, tension legs, tube section connecting device, anchoring system, etc These anchoring systems are usually set symmetrically in the plane perpendicular to tunnel axis with equal distances, consisting of or anchor cables or tension legs on each plane The new one has been improved to conquer the disadvantages of traditional structure It includes joint structures between tubes, tube segments and two sets of symmetric anchor structures set respectively to the tube at the flank of the joint The new one could not only have the advantages of foregoing SFT structure, but also avoid its possible disadvantages The precast segmental construction technique is usually used for SFT building because the length of SFT can be hundreds meters or even kilometers After carried by barges to the designated place, these precast tube sections are installed in situ underwater One of them is discussed and analyzed in the paper (a) (b) c1 Tube c1 Tension leg c1 c1 Anchor pile Energy dissipation connecting device Fig.1 The non-simplified model of SFT.(a)The schematic diagram of SFT model and (b)The schematic cross section of SFT on the support 145 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 The SFT with tension legs includes transverse and vertical tension leg forces As the transverse forces is canceled out by its self balance, only the effect of vertical force is taken into account The vertical anchor stiffness of tension legs is simulated with spring K , and the damper c1 will be used to simulate the energy dissipating connecting device between tubes The non-simplified model usually makes the research about displacement response difficult because of its complex structure Some special joins are set between SFT tubes, and they can divided into three different types according to the connection stiffness: rigid connection, semi-rigid connection and flexible connection The energy dissipating connecting device discussed in this paper is belong to the flexible connection The SFT can be regarded as a beam on discrete elastic supports if its tension legs are separately spaced along its length and the elasticity of the tension legs is taken into consideration [3] For studying conveniently, the structure of SFT will be simplified as an elastically supported beam with two springs and dampings at each end, see Fig And the following assumptions will be made: (1)The tension legs along SFT tube are simplified as springs at each end (2)The spring vertical stiffness of simplified model is K and the damping coefficient of connecting device is c1 , whose masses are neglected (3)The vibration of SFT is approximated to a superposition of the first mode shape of vibration of an elastic beam and a rigid beam section y o v z P c1 K c1 c1 c1 K x Fig The simplified model of SFT subjected to a single moving load Kinematic equation It can be seen from Fig 2, for researching the dynamic behavior of SFT, one of SFT tube sections is chosen to be studied without loss of generality The displacement response of the tube under a single moving load is mainly investigated in this paper, and its kinematic equation can be expressed as EI w4w w2w ww ms c s wz wt wt PG ( z vt ) f D G ( z ) PC G ( z l ) PC (1) Where EI is flexural rigidity of the beam, w is vertical displacement of the SFT tube, cs is SFT viscous damping coefficient, ms is the linear mass, P , v denote the magnitude and velocity of the single moving load, G is Dirac function, l is the length of the tube section, f D denotes sum of added inertia force and damping force per unit length, which are caused by the transverse vibration of tension legs, PC is damping force at bearing According to Morison equation, f D can be expressed as fD ww ww w2w C m ʌD U C D UD wt wt wt (2) Where U is fluid density, D is diameter of the tube, C D , C m denote drag force coefficient and added mass coefficient, whose values are C D 0.7 , C m 1.0 [9] As can be seen by Eq (2), f D includes two parts and the front part can be reasonably simplified as ˄(2ʌS t ww ww / D)JUD ˅[10] , the simplified kinematic equation can be rewritten as wt wt 146 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 EI Here, c f w4w w2w ww ( m s m f ) (c s c f ) wz wt wt (2 ʌS t PG ( z vt ) G ( z ) PC G ( z l ) PC ww / D)JUD denotes fluid damping, it can be obtain with J wt (3) 0.8 S t 0.2 [11], w is average displacement of the tube, its value can be simplified as w (2l / ʌ l ) q(t ) according to Eq (10), which is only associated with time variable As a nonlinear item, c f is solved by equivalent linearization, Making the equivalent linear value as close as possible to the exact solution Substituting the values of J and St for simply calculation, wq 0.64ʌUD˄2l / ʌ l 2˅ The initial value is taken as wq / wt and substituted into Eq (6) Solution are wt solved by numerical iteration soft When the accuracy condition N n N n1 10 -3 is satisfied, iteration will be cf stopped and equivalent linear solution of c f can be obtained m f is added fluid mass, m f ʌUD C m / According to the above assumptions, the SFT is a superposition of an elastic beam and a rigid beam section, the solution of SFT displacement is as follow with Galerkin method w( z , t ) ¦I ( z)q (t ) n n (4) Where In (z ) is the shape function of the N-th order vibration mode, q n (t ) denotes the generalized coordinate As the action time of moving loads on beam is very short, the vibration response of the beam is basically a transient problem in this case Only the first mode of vibration of the tunnel need be considered, while the higher modes can be neglected without losing much accuracy [12] Therefore, only the first order mode I (z ) and q (t ) are taken into consideration in this paper as the premise of the qualitative conclusion would not be affected I ( z ) sin( ʌz / l ) N [8,13], N ( EIS ) /( Kl ) denotes the ratio of the flexural rigidity of the beam to the stiffness K of the elastic bearing Substituting Eq (4) into Eq (3), multiplying both sides of the equation by I (z ) and integrating on length of the tube Let m ms m f , c c s c f , the simplified equation can be written as w q c wq Z 2q wt m wt 2P 8N Svt N 8N (1 2N ) 1 (sin N) (1 2N ) 1 [ PC (0, t ) PC (l , t )] l ml S ʌ ml (5) ww wq c1N wt wt Where c1 is the damping coefficient of energy dissipating connecting device between tube sections, the frequency Here, PC (0, t ) PC (l , t ) c1 of vibration Z is Z Z0 ( ʌ 4N ) /( ʌ 8N 2ʌN ) , Z indicates the frequency of vibration of tunnel as a beam with simple supports, Z0 ( ʌ/l ) EI / m In order to verify the reliability of the model, the simplified calculation method of natural frequency [14] is used to compare with the calculated results of the model above Though the differences between these two models can make the results of this paper little bit less than the results given in [14], the error is still within the acceptable range Further simplified, the expression can be rewritten as 147 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 w 2q wq A Bq wt wt Where A F (sin ʌvt N) l (6) 8N c (1 2N ) 1 c1N , B ʌ m ml Z2 , F 2P 8N (1 2N ) 1 ʌ ml As the tube in this paper is taken with a flexible connection, the boundary condition and initial condition are given on the basis of SFT simplified physical model built above EI w2w w2w (0, t ) EI (l , t ) wz , wz , w( z , t ) 0, ww( z,0) wt EI w3w w3w (0, t ) - Kw(0, t ) EI (l , t ) wz , wz Kw(l , t ) (7) (8) On the basis of Eq (7) and Eq (8), the generalized coordinate q(t ) is A t q(t) e (F1 cos Zt F2 sin Zt) F3 cos Zd t F4 sin Zd t FN B Where Z d is drive frequency caused by the single moving load, Z d F2 [ FZ d ( B Z d ) A ] / Z , F3 F1 2 ( B Z d ) A 2Z d AFZ d , F4 ( B Z d2 ) A 2Z d2 (9) Sv / l , F1 AFZ d FN , 2 B ( B Z d ) A 2Z d F ( B Z d2 ) ( B Z d2 ) A 2Z d2 Substituting Eq (9) into Eq (4), the first order mode of vertical vibration displacement w( z, t ) at any position of SFT under single moving load can be obtain as w( z, t ) q(t )(sin Sz l N) (10) Numerical simulation 4.1 Parameters selection As there is none built SFT around the world, the basic parameters selection of SFT in this paper refer to the design parameters of the proposed SFTs at home and abroad [15,16], and the current general specifications for design of highway bridges and culverts in China is also used for reference [17] The specific values of parameters can be seen in Table The value of spring vertical stiffness K and damping coefficient of connecting device between tubes c1 refer to research results of bridge damping device According to these researches, if the damping efficiency is maintained at a high level, the reasonable range of spring vertical stiffness K should be granter than 1u 10 N/m [18] and damping coefficient c1 is between 0.3 and 0.5, whose value can be adjusted appropriately [19] In this paper, the damping coefficient is c1 u 106 N/(m/s) 148 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 Table Basic parameters of SFT Property Symbol Length l m 100 Modulus of elasticity E N/m2 3.2h1010 External diameter D m 14.26 Internal diameter D’ m 11.40 Damping coefficient cs N/(m/s) Density of SFT Fluid density Unit Value 2018 1028 kg/m ȡ’ kg/m ȡ 0.0001 In order to study the displacement response of SFT under the single moving load, the numerical simulation is carried out by designing three different working conditions, in which the damping coefficients are same and the values of spring vertical stiffness are K 1u 10 N/m , u 10 N/m and u 10 N/m , respectively Finally, the displacement-time curves at mid-span of SFT tube with different P and v in different working conditions can be obtained The values of P and v refer to the current Highway Technical Standard (JTG B01-2003) [20] for higher engineering reference value Because of the 55t maximum load limit and 80km/h tunnel speed limit, the specific values are P 150kN , 350kN , 550kN and v 45km/h , 60km/h , 75km/h , 90km/h 4.2 Results and analysis The time-history curves of displacement response in different working conditions have the similar change tendency during numerical calculation and simulation Therefore, only the time-history curves of displacement response in different working conditions with P 550kN are given in this part, see Fig (a) (b) 149 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 (c) Fig When P=550kN, the time-history curves of mid-span displacement response with different v (a) K u 107 N/m ; (b) K (c) K u 107 N/m u 107 N/m ; Regarding to the time-history curves of mid-span displacement response, the displacement and the vibration period at mid-span decrease with the increase of vertical stiffness of tension legs For example, when P 550kN , v 75km/h and c1 u 10 N/(m/s) , the maximum displacements of mid-span with different vertical stiffness K 1u 10 N/m , u 10 N/m , u 10 N/m are 5.21mm , 2.19mm , 1.10mm Apparently, the inhibitory effect of vertical stiffness K on tube displacement response should not be ignored In order to study the vertical stiffness which is suitable for the SFT designed in this paper, the values of K are continued to increase to 1u 10 N/m , u 10 N/m and 1u 10 N/m As can be seen from Fig 4, the values of maximum displacement decrease to 0.57mm , 0.14mm , 0.086mm In view of the SFT structure designed in the paper, the value range of K should be granter than 1u 10 N/m for ensuring the stability of structure Besides, serious attention should be paid to the increase vibration frequency which is caused by the increase of vertical stiffness of tension legs Though K can control the mid-span displacement effectively, its optimal design should take traffic volume, driving speed, etc into consideration in practical engineering to make sure the stability and safety of SFT during its future operation Through analyzing the time-history curves in three different working condition, it can also be found that in the range of K 1u 10 ~ u 10 N/m , these curves under the same load reflect the same trend with the increase of moving velocity What’s more, the vibration periods are similar while the increase of mid-span displacement is not very obvious Such as the Fig 3(b), When K u 10 N/m and P 550kN , with the increase of v from 45km/h to 90km/h , the load action time decreases from 8s to 4s and the maximum displacements of mid-span are respectively 2.171mm , 2.178mm , 2.185mm and 2.191mm In fact, as it has been presented in Eq 9, the shape of the time-history curve under the effect of the single moving A t load is mainly influenced by two parts: one is e (F1 cos Zt F2 sin Zt) consisting of the natural vibration frequency Z of SFT, and the other F3 cos Zdt F4 sin Zdt is composed of the drive frequency Z d , which is caused by the moving load These two parts are controlled by their respective coefficients (i.e F1 , F2 , F3 and F4 ) and compete with each other However, these coefficients are mainly effected by the vertical stiffness K of tension legs When K is less than orders of magnitude, the response frequency of the mid-span displacement is mainly effected by the natural vibration frequency Z , that’s the reason why the time-history curves have the same trend in spite of the change of moving velocity When K is granter than 1u 10 N/m , as can be seen from Fig 4, with continuous increasing of K , the influence of moving velocity on displacement response is gradually revealed The frequency of time-history curve increases with increasing of moving velocity and its effect on maximum displacement is more and more obvious When K u 10 N/m , v 60km/h , the time-history curves of mid-span displacement response with different v is shown in Fig According to Fig 5, under the condition of K u 10 N/m , c1 u 10 N/(m/s) and v 60km/h , the maximum 150 Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 displacement of mid-span increases from 0.30mm to 1.10mm when P increases from 150kN to 550kN And the maximum oscillation amplitude increases from 0.28mm to 1.01mm It indicates that the magnitude of the moving load has a great effect on mid-span displacement response (i.e maximum displacement and change rate of displacement) under the same working condition and moving velocity (a) (b) (c) Fig When K ! u 107 N/m , P 550kN , the time-history curves of mid-span displacement response with different v (a) K K u 108 N/m and (c) K 1u 109 N/m Fig.5 When K u 107 N/m and v 1u 108 N/m , (b) 60km/h , the time-history of mid-span displacement response with different P Conclusion In this paper, the kinematic equation for SFT tube under a single moving load is established to investigate kinematic problems By using the Galerkin method, the kinematic equation at the mid-span section of SFT tube can be solved, and the mid-span displacement of SFT tube is gained by numerical simulation method Zhang Yuan et al / Procedia Engineering 166 (2016) 143 – 151 According to the results of numerical simulation analysis, it has been found that the vertical stiffness of tension legs has a significant effect on the suppression of SFT displacement The mid-span displacement decreases with increasing of vertical stiffness It’s also the main influencing factor of the tube vibration frequency by controlling the tube natural vibration frequency Furthermore, the influence of moving velocity is not obvious with low vertical stiffness But with the increase of vertical stiffness of tension legs, its effect on displacement response is gradually revealed Finally, through the simulation research, it has also been found that the magnitude of the moving load has a great influence on maximum displacement and change rate of displacement As there are clear provisions about load and speed limit in the current general specifications for design of highway bridges and culverts, consideration should be given to the velocity and magnitude of vehicles during the design process of SFT Based on the current specifications, the existing designs of SFT should be adjusted and more attention should be paid to the selection of vertical stiffness of tension legs Acknowledgements This research was jointly supported by the project of construction science and technology of the Ministry of Transport (No 2013318740050) and National Engineering and Research Center for Mountainous Highways (GSGZJ-2014-05) References [1] F Brancaleoni, A Castellani, P D’Asdia The response of submerged tunnels to their environment, J Engineering Structures 11(1989) 47-56 [2] S Remseth, B J Leira, K M Okstad, K M Mathisen, T Haukås Dynamic response and fluid/structure interaction of submerged floating tunnels,J Computers and Structures 72(1999) 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generality... time-history curves of mid-span displacement response with different v (a) K u 107 N/m ; (b) K (c) K u 107 N/m u 107 N/m ; Regarding to the time-history curves of mid-span displacement response, ... the magnitude of the moving load has a great effect on mid-span displacement response (i.e maximum displacement and change rate of displacement) under the same working condition and moving velocity