MULTIBODY DYNAMICS METHOD FOR IMMERSED TUNNEL SUBJECTED TO LONGITUDINAL SEISMIC LOADING

th World Congress on Computatonal Mechancs (WCCM XI) 5th European Conference on Computatonal Mechancs (ECCM V) 6th European Conference on Computatonal Flud Dynamcs (ECFD VI) E. Oñate, J. Olver and A. Huerta (Eds) MULTIBODY DYNAMICS METHOD FOR IMMERSED TUNNEL SUBJECTED TO LONGITUDINAL SEISMIC LOADING ZHONGYUAN SHEN *, YONG YUAN, HAITAO YU AND RUI CHAI * Department of Geotechncal Engneerng, Tongj Unversty, Shangha, Chna, 0009, 0shenzhongyuan@tongj.edu.cn State Key Laboratory of Dsaster Reducton n Cvl Engneerng, Tongj Unversty, Shangha, Chna, 0009, yuany@tongj.edu.cn Department of Geotechncal Engneerng, Tongj Unversty, Shangha, Chna, 0009, yuhatao@tongj.edu.cn Hong Kong-Zhuha-Macao Brdge Authorty, Zhuha, Chna, 59095, cr@hzmbo.com Key Words: MULTIBODY DYNAMICS, SEISMIC RESONSE, DISCRETE TIME TRANSFER MATRIX METHOD, IMMERSED TUNN Abstract. The longtudnal dynamc response analyss of mmersed tunnel s crucal for sesmc desgn of mmersed tunnel. The modelng theory and numercal method of multbody dynamcs are ntroduced to the dynamc response analyss of mmersed tunnel. An mmersed tunnel can be consdered as a multbody system whch s composed of a seres of renforced concrete segments connected wth flexble jonts. In ths paper, a chan dynamc model of multbody system,.e. a mult-rgd body, elastc dampng hnge, and dampng hnge model, s developed for dynamc analyss of mmersed tunnel under longtudnal sesmc exctatons. The mathematcal model s mplemented to solve the dynamc response of the system. The dynamc equatons of each element and the system are deduced usng dscrete tme transfer matrx method n multbody dynamcs (). As an applcaton, an example s conducted wth the presented dynamc model and mathematcal model. The solutons are acqured usng Matlab. Results obtaned from are n a good agreement wth the tradtonal fnte element method (). Results show that the dynamc and mathematcal models of mutltbody system are sutable for asesmc desgn of mmersed tunnel. The multbody dynamc modelng process for mmersed tunnel under earthquake s smple and flexble. Ths research provdes a powerful tool for sesmc response analyss of mmersed tunnel, especally for those ultra-long mmersed tunnel located n sesmcally actve regons. INTRODUCTION Immersed tunnel belongs to underwater tunnel composed of mmersed tubes. A large deformaton of GINA gasket subjected to longtudnal sesmc loadng may lead to leakage and hence jeopardzng the watertghtness []. Thus the longtudnal sesmc response analyss

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. s crtcal to asesmc desgn of mmersed tunnel. Currently, the mechancal models of longtudnal sesmc response analyss of mmersed tunnel nclude beam-sprng model [], partcle-sprng model [3] and the equvalent masssprng model [4]. The beam-sprng model assumes the tunnel as a contnuous beam supported on elastc foundaton, whle the other models consder the tunnel as a partcle system connectng wth sprng jonts. Sol-structure nteracton s smulated by sprng and dampng components. The mathematcal models are proposed usng equatons of moton on the elastc foundaton beam and structural dynamcs equaton, respectvely. The solutons are computed wth the analytcal soluton [5] and dynamc fnte element method, respectvely [6]. Analytcal method can get accurate results, but ts expresson s complex. For a general multbody system, t s dffcult to get the analytcal soluton. Therefore the method s only sutable for solvng the less degree of freedom system. Wth the development of computer technology, the overall dynamc analyss method of the earthquake response for mmersed tunnel s common based on the D and 3D fnte element analyss [3].The calculaton precson s mproved usng fnte element mesh of refnement, however, the modellng s usually complex and the calculaton demand s larger [7]. How to calculate the longtudnal earthquake response of the mmersed tunnel quckly and effcently s the crtcal problem for the sesmc researches. Multbody system dscrete tme transfer matrx method () s a dynamcs method used for tme varyng, nonlnear, large moton of the general multbody system [8]. Based on the object-orented mult-body system dynamcs, the multbody system s dvded nto several parts and the mechancal propertes of each part can be ndcated as matrx. Accordng to the boundary condton and ntal condton, the dynamc response of the system can be ganed easly [9]. Ths method combnes the advantages of transfer matrx method, multbody system transfer matrx method, transfer matrx method and numercal ntegraton procedure [0]. It has hgh computatonal effcency and smaller memory storage requrement. It can realze rapd analyss and mprove dynamc performance for large-scale complex multbody systems of modern project. It has been appled n avaton, aerospace, weapons, and other felds [0]. In the feld of buldng structure, t s also n the condton of applcaton of exploratory []. But so far there are not cases reported n the feld of underground structures. Accordng to the natural attrbute of bodes and hnges of the system, any complex engneerng structure can be dvded nto a certan number of subsystems, whch can be represented by varous elements ncludng bodes and hnges [9]. So accordng to the geometrcal characterstcs of the mmersed tunnel and the characterstcs of the structure, consderng sol-structure nteracton, ths paper establshed the mmersed tunnel longtudnal sesmc response analyss of mult-body dynamcs model, that s a mult-rgd-body - elastc dampng hnge - dampng hnge model; Introducton of multbody system dynamcs of MS- DT-TMM method, the mathematcal model are derved for mmersed tunnel subjected to longtudnal earthquake exctaton. Fnally based on MS- DT-TMM method, a numercal example s analysed and compared wth the computatonal results of tradtonal fnte element method (), a good agreement shows the feasblty and valdty of the method.

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. MULTI-BODY DYNAMICS MODEL OF IMMERSED TUNNEL. Fundamental Assumpton.. Mult-Rgd-Body Model Dscretzaton of the Immersed Tunnel Structure Mult-rgd-body dscretzaton method s a separaton of structure elastcty to establsh the mechancal model of mult-rgd-body systems. It s unnecessary to meet the deformaton contnuty condtons between the unts and has no lmtaton n the process of stress deformaton and movement of the structure []. The method developed by Wnge and Huston [3] n 976 has been appled to the cables power [3], dynamc characterstcs of elastc beam bucklng analyss [4], dynamc response analyss of curve box grder brdge [5], sesmc response analyss of shear structure [6], sesmc response analyss of renforced concrete frame structure [7]and of shear wall structure []. Immersed tunnel structure conssts of numbers of renforced concrete elements connected wth several GINA flexble jonts, as shown n Fgure. Based on mult-rgd-body dscretzaton model, mmersed tunnel can be modeled as a chan mult-rgd-body system, consderng each element as a rgd body. The dscrete model s shown n Fgure... Elastc Dampng Hnge Model of Jont Elastc dampng hnge refers to a hnge composed by sprngs and parallel dampng hnge movng n the translatonal drecton [0]. Flexble jont wth GINA gasket and steel shear keys s wdely adopted n mmersed tunnel. GINA gasket s a knd of hyperelastc materal [8] and lastcty need to be consdered n steel shear keys. Consderng the effects of energy consumpton, the jont s smplfed as an elastc dampng hnge model, as shown n Fgure...3 Dampng Jont Model of Sol-Structure Interacton The man dfference between the mult-body system of mmersed tunnel and other multbody system s that mmersed tunnel s bult n soft sol wth a certan restrcton on tunnel s movements []. Wang and Cao [9] beleve that the sol-structure nteracton s a vscous boundary and the vscous dampng force s proportonal to ts speed [0]. Consderng a relatve slp between the sol and mmersed tunnel [], ths paper smplfy the sol-structure nteracton nto a dampng hnge. The dampng force between each element and sol s gven by f c x c () where f s the dampng force between the th element and the sol, c s the correspondng dampng parameter, x c s the longtudnal velocty of the center of the th mass...4 Earthquake Loadng Sesmc loadng s a knd of hghly rregular short-term cyclc loadng and can be decomposed nto a seres of smple harmonc component. Therefore research on the dynamc 3

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. response of the system to smple harmonc loadng s fundamental ssue wth respect to structural sesmc response []. Ths paper only studes the longtudnal response of tunnel subjected to sne harmonc earthquake exctaton. Assumng harmonc sesmc loadng s expressed as F sn( t ) () where F s the earthquake loadng, s the ampltude, s the frequency, s the ntal phase.. Fundamental Assumpton Based on the assumptons above, a mult-body dynamcs model of the longtudnal sesmc response, namely a mult-rgd-body - elastc dampng hnge - dampng hnge model s presented n Fgure 3. Accordng to the prncple of body and hnge defned by mult-body system transfer matrx method, the seral number of rgd elements are, 3,, n; The number of elastc dampng hnge are, 4,, n-; The seral number of dampng hnge between sol and tunnel structure are n+, n+,, n-; as shown n Fgure 3. The transfer drecton s from left to rght. The nput pont and output pont for the th segmental rgd body are I and O, respectvely. C s the center of mass for the th j segmental rgd body. s the connectng jont between the th and jth element. ox s the nertal coordnate system and ox( =, 3,, n) are the body-fxed coordnate system of the th segmental rgd body, respectvely. If the dampng hnge of the sol-structure nteracton sformulated by dampng force, the mechancal model can also be shown n Fgure 4. The followng calculaton mechancal models are all Fgure 4. 3 MATHEMATICAL MODEL AND FORMULAE 3. State Vectors The state vectors of the connecton pont among any rgd bodes and hnges are defned as T x, Z, j x, q, j (3) where x s the poston coordnates of the connecton pont wth respect to the nertal reference system and q x s the correspondng nternal force n the same reference system. 3. Transfer Matrces of Typcal Elements 3.. Transfer Matrx of A Segment Rgd Body Based on mechancs of rgd body, the external force poston can be equvalent to the mass pont of a segment. The mechancal analyss of a segmental rgd-body s shown n Fg. 5. 4

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. In the nertal coordnate system, the poston coordnate of nput pont I and output pont O are x I and x O, respectvely. And the poston coordnate of ts center of mass C s x C.The subscrpts denotes the th of the system. In the body-fxed coordnate system, the poston coordnate of nput pont I and output pont O are x and x, respectvely. And the poston coordnate C of ts center of mass s x. C So geometrcal equatons can been obtaned as follows: I O xi x (4) I x x x O I (5) O x x x (6) C I c Usng Newton-Euler method, the dynamcs equatons of the rgd body can be ganed as m x q q F + f (7) C xi xo where, m s the mass the th segmental rgd body, and x C s the column matrx of correspondng acceleraton of ts center of mass, qxi and q xo are the column matrces of nternal forces of nput pont I and output pont O, respectvely, f s the column matrces of external force acted on the center of mass of the th segmental rgd body. Accordng to numercal ntegraton procedures [9], the moton parameters of multbody system xt and x t at the tme nstant t are expressed as x A x B (8) t t t t x C x D (9) t t t t Where xt and x t represent column matrces of acceleraton and velocty of the center of mass at tme t, respectvely. To substtute Equaton (), (6), (8) and (9) nto (7), we can obtan q m A c C x q m B c D F (0) xo ( ) I xi Equatons (5) and (0) can be rewrtten n the form of transfer equaton as follows x 0 x O x q x m A cc m B c D F q x 0 0 O I () Therefore we can obtan the transfer matrx of a segmental rgd body, that s 5

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. 0 x O U m A cc m B c D F 0 0 () 3.. Transfer Matrces of an Elastc Dampng Hnge The mechancal analyss of an elastc dampng hnge s shown n Fgure 6. Assumng that the nonlnear sprng force and dsplacement s q q K l K l (3) xo, k xi, k where, l s the elongaton of sprng, l xo xi l 0, l0 s the ntal length of sprng, K and K are the rgd parameters of sprng, respectvely. The dampng force of parallel massless sprng dampng hnge on both ends s equal, we can obtan q q c x x K x x l K x x l xo xi ( O I ) ( O I 0) ( O I 0) (4) To substtute Equaton (9) nto (4), we can get Let q c Cx D Cx D K x x l K x x l xi [( O xo ) ( I xi )] ( O I 0) ( O I 0) (5) Equaton (5) can be wrtten as x x (6) O I q c C c D c D K K l K ( l l ) (7) xi xo xi 0 0 0 Usng the quadratc term lnearzed Equaton (8), Equaton (9) s then obtaned. a( t ) b( t ) a( t ) b( t ) a( t ) b( t ) a( t ) b( t ) a( t ) b( t ) T (8) ( t ) ( t ) ( t ) ( t ) ( t ) T (9) Assumng ( t) s abbrevated to, and to substtute Equaton (9) nto (7), the followng equatons are then obtaned. [ c C K K l K ( t )] q c D c D K l K l K ( t ) K ( t ) T (0) 0 xi xo xi 0 0 Let 6

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. E c C K K l K ( t ) () 0 Q c D c D K l K l K ( t ) K ( t ) T () xo xi 0 0 Equaton (0) can be wrtten as That s E q Q (3) xi Q qxi E E (4) To substtute Equaton (6) nto Equaton (4), we can get Q xo xi qxi E E (5) That s Q xo xi qxi E E (6) Equaton (3) and (6) can be wrtten n matrx form, that s Q x E E x q x 0 0 q x 0 0 O I (7) Hence the transfer matrx of an elastc dampng hnge can be expressed as Q E E U 0 0 0 0 (8) 3.3 Transfer Equatons The transfer equaton of the frst element s defned as Z U Z (9), 0, 7

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. The transfer equaton of the second element s defned as Z U Z (30) 3,,... The transfer equaton of the nth element s defned as Z U Z (3) n, n n n, n Then the transfer equaton of the system can be assembled nto a sngle transfer equaton as Z U U... U... U U UZ (3) n, n n n 0, Where, U and U are the transfer matrces of the th element and the system, respectvely. 4 SOLUTIONS OF DYNAMIC RESONSE FOR THE SYSTEM To substtute the free boundary condtons on both ends and the ntal condtons nto Equaton (3), the unknown quanttes n the boundary state vectors can be computed. Then, the state vectors and the moton quanttes of each element at tme t can be computed by teraton of correspondng transfer equatons of element smlar to Equaton (3). The longtudnal dsplacement, velocty, acceleraton of each segmental rgd body, the longtudnal deformaton and the axal of the jont at tme t are then obtaned, respectvely. Then entre procedure can be repeated for tme t and so on. The overall transfer matrx of the chan multbody system for mmersed tunnel can be assembled easly usng the transfer matrces of elements. So the method descrbed above reduces the amount of calculaton of dynamc response for mmersed tunnel subjected to sesmc loadng. 5 NUMERICAL EXAMLE To valdate the aforementoned method, a numercal example of an mmersed tunnel wth two elements s presented based on a reference dynamc test. The mechancal model of the numercal example s shown n Fg. 7. The sne harmonc dynamc cycle loadng s assumed as F 0sn t. The structure parameters of the numercal example are lsted n Table and Table. Whle the ntal condtons of the system are lsted n Table 3. Table : arameters of the segmental rgd body Element number Length (m) Wdth (m) Mass (kg) Dampng hnge parameter 6 4 00 3 6 4 00 Table : arameters of the elastc dampng hnge c (kg/s) 8

Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. Element number Sprng stffness K ( N / m) Sprng stffness K ( N / m) Damper parameter c ( kg / m ) Intnal sprng length l ( m ) 0.5 3 0.3 0 Table 3: Intal condtons Element number oston coordnate (m) oston coordnate x (m) O Velocty x I (m/s) Velocty x O (m/s) 0 6 0 0 6 6.3 0 0 3 6.3.3 0 0 6 CONCLUSIONS Ths paper presents a novel and relable dscrete tme transfer matrx method for multbody system dynamcs () for estmatng the longtudnal sesmc response of mmersed tunnel. The mult-segmental tunnel s modeled as a mult-rgd-body - elastc dampng hnge - dampng hnge model based on ratonal hypothess. The formulae of the method as well as a numercal example based on a reference dynamc test to valdate the method are gven. Results obtaned from have a good agreement wth that got by the tradtonal fnte element method (). Comparng wth, s less complcated and hghly effcent. It has the modelng flexblty, smaller memory stroage requrement and the extended applcablty of numercal ntegraton procedures. Ths research provdes a more powerful tool for rapd and practcal sesmc response analyss of mmersed tunnel, especally for ultra-long mmersed tunnel located n sesmcally actveregons. Although ths research was estmated for longtudnal sesmc response analyss, the method presented may also apply n other condtons such as lateral and bendng sesmc shakng and the combnaton of them. ACKNOWLEDGEMENTS Ths research s supported fnancally by the Natonal Natural Scence Foundaton of the eople s Republc of Chna (Seral Number: 50896), Natonal Key Technology R&D rogram of the eople s Republc of Chna (Seral Number: 0BAG07B0), Natonal Key Technology R&D rogram of the eople s Republc of Chna (Seral Number: 0BAK4B00). REFERENCES x I [] Anastasopoulos, I., Gerolymos, N., Drosos, V., et al. Nonlnear response of deep mmersed tunnel to strong sesmc. J. Geotech.Geoenvron. Eng. (007), Vol. 33 (9), pp. 067-090. [] Yan, S. H., Gao, F., L, D. W., et al. Studes on some ssues of sesmc response analyses for submerged tunnel. Chn. J. Rock Mech. Eng. (004), Vol. 3(5), pp. 846-850. (n Chnese) 9

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Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. Fgure : Sketch of an mmersed tunnel Fgure : Mult-body dynamcs model of an mmersed tunnel F 0,, 3, 3,4 I o o x x 3 O( I) O( I3) O3( I4) C C x o3 3 n nn, n,n O ( ) n In o n xn n C n On n n n Fgure 3 : Mult-body dynamcs model I of mmersed tunnel 0,, 3, 3,4 3 I O( I) O( I3) O3( I4) C C F o x o3 x3 o x f f 3 n nn, n,n O ( ) n In o n xn n C n f n On Fgure 4: Mult-body dynamcs model II of mmersed tunnel q xi, I o x C F, O q xo f Fgure 5: Mechancal analyss of a segmental rgd body I c O q xi k, k q xo Fgure 6: Mechancal analyss of an elastc dampng hnge 0,, 3, 3,4 3 I O( I) O( I3) O3 C C F o x o3 x3 o x f f 3

Longtudnal deformaton of the jont (mm) Axal force of the jont (N) Longtudnal velocty of No. segment (m/s) Longtudnal velocty of No. 3 segment (m/s) Longtudnal dsplacement of No. segment (m) Longtudnal dsplacement of No.3 segment (m) Zhongyuan SHEN, Yong YUAN, Hatao YU and Ru CHAI. Fgure 7: Mechancal model of the test 0.0 0.0 0.5 0.009 0.0 0.006 0.05 0.003 0.00 0.000 0 3 4 5 6 0 3 4 5 6 Tme (s) Tme (s) Fgure 8 : Tme hstory of the longtudnal dsplacement of No. segment Fgure 9 : Tme hstory of the longtudnal dsplacement of No. 3 segment 0.0 0.006 0.05 0.004 0.00 0.00-0.05 0.000-0.0-0.00 0 3 4 5 6 0 3 4 5 6 Tme (s) Tme (s) Fgure 0 : Tme hstory of the longtudnal veloct of No. segment Fgure : Tme hstory of the longtudnal velocty of No. 3 segment Longtudnal acceleraton of No. segment (m/s ) 0.0 0.5 0.0 0.05 0.00-0.05-0.0 Longtudnal acceleraton of No.3 segment (m/s ) 0.004 0.00 0.000-0.00-0.004 0 3 4 5 6 0 3 4 5 6 Tme (s) Tme (s) Fgure : Tme hstory of the longtudnal acceleraton of No. segment Fgure 3 : Tme hstory of the longtudnal acceleraton of No. 3 segment 80 0.6 50 0 0.4 90 0. 60 30 0.0 0-0. -30 0 3 4 5 6 0 3 4 5 6 Tme (s) Tme (s) Fgure 4 : Tme hstory of the longtudnal deformaton of the jont Fgure 5 : Tme hstory of the axal force of the jont