A 4D FINITE ELEMENT MODEL FOR TUNNELLING UN MODÈLE FINI DE L'CÉlément 4D POUR PERCER UN TUNNEL G.J. Schotmejer 1, S.J.M. van Eekelen 1 1 GeoDelft, Delft, The Netherlands, smr@geodelft.nl ABSTRACT. Ths 4D model s capable of a contnuous smulaton of the process of sheld tunnellng. The use of the Arbtrary Lagrangean Eulerean (ALE) method makes t possble to smulate the large dsplacements of the tunnellng machne wthn one sngle calculaton. Specal attenton has been pad to the modellng of the tal vod, ncludng a complete descrpton of the grout njecton process. The sol around the tunnel s descrbed wth the MIT-S1 model (Pestana and Whttle, 1999). RÉSUMÉ. Ce modèle 4d smule sans nterrupton le processus du perçage d'un tunnel de boucler. L'utlsaton de la méthode lagrangéenne arbtrare d'eulerean (ALE) permet pour smuler les grands déplacements de la machne de perçage d'un tunnel à mons d'un calcul smple. Une partculère attenton a été prêtée à modeler du vde de queue, y comprs une descrpton complète du procédé d'njecton de couls. Le sol autour du tunnel est décrt avec le modèle MIT-S1 (Pestana and Whttle, 1999). 1 Introducton Recent developments n sheld tunnellng methods help to reduce the constructon tme and costs of tunnels. However, the deformatons around the tunnel have to be consdered carefully, as the constructon process appears to have most nfluence on ths. Emprcal and analytcal methods have too many lmtatons to study the complex characterstcs of tunnel constructon methods. Therefore, tunnellng engneerng s one of the areas n sol and rock mechancs n whch numercal analyss are frequently adopted. Goda et al. (1999) summares the developments and applcatons of the numercal analyss of tunnels. For example Abu-Farsakh et al (1999) presented an advanced 2D model that s capable to smulate the contnuous advance of the sheld and ncorporates the 3D deformaton of the sol around and ahead of the sheld face. Several other authors reported on 3D fnte element models to smulate tunnellng processes (for example Lee and Rowe, (1990a and b), Akag, (1994), De Buhan et al, (1999), Augarde and Burd (2001). Although many of these models gve promsng results, none of them s able to smulate the whole contnuous sheld tunnellng process, ncludng the nfluence of the most mportant aspect: the grout njecton nto the tal vod. Ths paper ntroduces a 4D model to smulate sheld tunnellng, n whch tme s the fourth dmenson. The model uses the ALE technque (Van Eekelen et al. 2002, Van den Berg, 1994 and 1996) to be able to smulate the large dsplacements of the TBM machne and the large deformatons n the grout layer durng one sngle calculaton. The paper starts wth a descrpton of the general dea of the 4D model. After that, the man characterstcs of the model wll be descrbed: (1) the descrpton of the tunnel face, (2) the detaled model for the grout njecton process, ncludng a specally developed consttutve model for grout behavour, (3) the sol behavour, for whch we have chosen to use the MIT-S1 model of Pestana and Whttle (1999).
2 A 4D model for sheld tunnellng The 4D model makes use of the ALE technque; the sol partcles are uncoupled from the element mesh. In the ALE model as proposed by Van den Berg (1994 and 1996), the materal s uncoupled from the mesh. The ntal stuaton from whch the tunnellng calculaton starts s dentcal to the stuaton n whch the TBM s close to the recevng pt. Small ncremental dsplacements are ntroduced at the left sde of the mesh (fgure 1), so that nstead of movng the TBM through the sol (lke n practce), the sol s movng, through the elements, nto and along the tunnel. Durng ths process, grout s njected through the njecton ponts to fll the tal vod. Tunnel face Fgure 1 A schematc vew of the 4D-tunnel model. The ALE method s a promsng method to smulate large deformaton processes. Several sol models are avalable wthn the model, lke the Mohr Coulomb model, a grout model (Van Eekelen et al, 2002) and the MIT-S1 model (Pestana and Whttle, 1999). The ALE model also provdes possbltes to model free surfaces and layered sol profles. The model has been appled before to other large deformaton problems, lke cone penetraton (Van den Berg, 1994, Van Eekelen et al., 2002), samplng (Van Eekelen et al., 1994) and mcro tunnellng (Van de Berg et al., 1999). In the followng we wll descrbe several aspects of the 4D model for sheld tunnellng, namely the tunnel face, the grout njecton process and the sol behavour. 3 Aspects of modellng sheld tunnellng 3.1 Tunnel Face In the analyses of face stablty a number of parameters are mportant. At the face a bentonte cake s buld up to control the stablty of the face. The excess groundwater pressure can nfluence ths stablty n front of and above the TBM. The movements of the sol n front of the TBM also effect the face stablty. In the current model the face stablty s smplfed. The pressure of the bentonte s modelled wth appled forces and not by an actual flud pressure. The sol moment n front of the TBM can be controlled by varyng these forces. In fgure 2, the sol moves from the left sde to the rght sde. The
sol that s eaten by the TBM also dsappears n the calculaton. It just dsappears nto the empty hole that smulates the TBM and the tunnel. Fgure 2 The modellng of the pressures at the face 3.2 Tal vod effects The shape of a TBM s concal; the dameter at the front of the TBM s larger than at the end. Ths overcuttng creates a free space between the lnng and the sol. Ths space, the vod, s njected wth grout (Fgure 3). Fgure 3 The tal vod. Groutng s one of the most mportant aspects to control the tunnellng process. If the grout s njected at a too low pressure the vod wll not be flled completely. Ths stuaton can result n major settlements and a not fully supported lnng. Injectng at a too hgh pressure can lead to sol heave and hydraulc fracturng. Changng the locaton or the number of njecton ponts, can be an mportant controllng mechansm for the tunnellng process. The grout njecton can nfluence the pressure dstrbuton n sol and lnng, the deformatons around the tunnel and the progress of the tunnellng. Ths model can gve full predctons of the consequences of such choces. Grout s njected n a lqud state at one or more njecton ponts. Dfferent postons of the njecton ponts result n dfferent pressure dstrbutons n sol and lnng. In the 4D model the grout s njected at node level. The fnte elements of the grout layer are deformable and for the behavour of the grout an especally developed consttutve model s used. We wll descrbe the last two aspects n more detal n the next sectons.
Fgure 4 The grout njecton ponts n the calculaton 3.2.1 Deformable elements of the grout layer: free surface descrpton In an Eulerean approach the materal s uncoupled from the element mesh. Ths means that the defnton of boundares between dfferent sols n an Eulerean analyss requres specal attenton and can be consdered as free surfaces. Here, the deformatons of the grout n radal drecton are mportant. Wsselnk proposed a formulaton for such free surfaces. Wth hs formulaton, the deformatons n radal drecton are treated n a Lagrangean approach. That means that the mesh deforms n radal drecton and remans the same shape n axal drecton (Fgure 5). Usng two approaches n one analyss s called the Arbtrary Lagrangean approach. Fgure 5 The modellng of the Lagrangean behavour of the grout n radal drecton For the flexble grout layer thckness, the 4D model uses an ALE-formulaton that uses the Lax-Wendroff convecton scheme. The surface (boundary) nodes keep ther ntal co-ordnate n axal drecton, the y- and z-co-ordnates are calculated wth the convecton scheme: The scheme needs the poston of three surface nodes, two upwnd nodes and one downwnd node. Fgure 6 Nodes for the Lax-Wendroff convecton scheme The ntermedate y-co-ordnate scheme: y nt s calculated wth the Lax-Wendroff convecton
y nt = y m - C( y m - y -1 m 1 é + 1-1 ) - C(1 - C) y ( r )( - ) - ( )( - ) ù + 1 ym ym y r - 1 ym ym (1) 2 êë 2 2 úû The parameter C s the courant number, whch s determned from the materal dsplacements n the x-drecton (axal drecton). = xg - xm -1 x - m xm C (2) The developed formulaton s stable, robust, effcent and descrbes the doman accurately. Although some dffuson occurs n calculatons wth very large curvatures n the flow drecton. 3.2.2 Consttutve behavour of grout We have developed a consttutve model for the behavour of grout, whch s applcable for a range of tme. Freshly made grout behaves as a vscous lqud. Its vscosty wll ncrease n tme untl the grout has become a sold. The grout model splts the behavour of grout nto two phases. The frst phase s the vscous phase, ths a vscous Bnghamlke model wth a tme-dependent vscosty to model the changes n grout stffness due to the chemcal settng reactons. Durng ths phase, the grout remans rgd n the case the shear stress s smaller than the yeld stress t 0 and flows lke a flud when the shear stress exceeds t 0. The second phase s the sold phase; ths s modelled wth the Mohr Coulomb model. In the model the grout behavour develops smoothly from the vscous phase nto the sold phase. Van Eekelen dscusses ths grout model more n detal (Van Eekelen et al., 2002). 3.3 The behavour of the sol : the MIT-S1model The MIT-S1 model (Pestana and Whttle, 1999) descrbes all-mportant features for sheld tunnellng both for the behavour of sand and clay. It s a numercally robust model that needs less parameters than ts predecessors or other, equally advanced models (13 for clays and 14 for sands). The model s applcable to a wde range of stresses or denstes. For the sand behavour, the densty s ntroduced as a separate state parameter, so that for one type of sand, one set of nput parameters s requred, ndependent on the ntal state of the sand. The model descrbes small stran behavour (hysterc behavour, plastcty below the yeld locus to descrbe cyclc behavour), and large stran behavour (crtcal state, stress path dependent ansotropc yeld locus). The MIT-S1 s especally sutable for the 4D-tunnel model because t wll be easer to smulate sol profles wth both sand- and clay layers wth a consttutve model that can descrbe both materals. Ths way, the large stffness dfferences between sand and clay gve less numercal problems. 4 Concluson A 4D fnte element model for the tunnellng process has been developed based on the ALE formulaton. The advantages of ths new model are: the whole tunnellng process
can be smulated wth one contnuous calculaton, the grout njecton process and the behavour of the grout layer n the tal vod can be descrbed detaled, ncludng an especally developed consttutve model for grout behavour an advanced The model uses the MIT-S1 model for the descrpton of the sol behavour. Ths s an elegant model that descrbes the behavour of sand ànd clay n one sngle framework. Acknowledgements. The research has been carred out wth the fnancal support of the Dutch Mnstry of Publc Works. 8. References Akag, H. (1994) Computatonal smulaton of sheld tunnelng n soft ground, Memors of the School of Scence and Engneerng, Waseda Unversty, 58, 85-119 Augarde, C.E. and Burd, H.J. (2001) Three-dmensonal fnte element analyss of lned tunnels Int. J. Numer. Anal. Meth. Geomech., 243-262 De Buhan, P., Cuvller, A., Dormeux, L., Maghous, S. (1999), Face Stablty of Shallow Crcular Tunnels Drven under the Water Table: A Numercal Analyss, Int. J. Numer> Anal. Meth. Geomech., 23, 79-95Van den Berg, P. (1994) Analyss of sol penetraton. Ph.D. Thess, Delft Unversty of Technology, Delft Van den Berg, Peter, de Borst, René, Huétnk, Han (1996) An Eulerean Fnte Element Model for Penetraton n Layered Sol Internatonal Journal for Numercal and Analytcal Methods n Geomechancs, vol. 20, 865-886 Van den Berg, P., Schotmejer, G.J., Hashmoto, T. and Shrakawa, J. (1999), Mcro tunnellng n urban areas, 3-dmensonal model and measurements. Proceedngs of European Conference on Sol Mechancs and Geotechncal Engneerng,Barends et al.(ed), Amsterdam, The Netherlands, Balkema 1979-1985. Van Eekelen, Suzanne J.M. and Bol, Vncent, Smulaton of cone penetraton n the geocentrfuge, Proceedngs of Numercal Methods n Geotechncal Engneerng, Mestat (ed.) 2002, Pars, France, Presses de lénpc/lcpc, Pars Van Eekelen, Suzanne J.M., Teunssen, Hans (J.A.M.) and Schotmejer, Gert-Jan, A Consttutve Model for Grout Behavour, Proceedngs of Numercal Methods n Geotechncal Engneerng, Mestat (ed.) 2002, Pars, France, Presses de lénpc/lcpc, Pars Van Eekelen, Suzanne J.M. and Van den Berg, Peter, (1994) Sample Dsturbance, Numercal Model and Experments Geotechncal Engneerng, Emergng Trends n Desgn and Practce, ed. Saxena, K.R., Oxford & IBH Publshng co. pvt.ltd., ISBN 81-204-0846-2, 1-32 Goda, G. and Swoboda, G., (1999) Developments and Applcatons of the Numercal Analyss of Tunnels n Contnuous Meda, Int. J. Numer. Anal. Meth. Geomech., 23, 1393-1405 Hll, R (1968), On consttutve nequaltes for smple materals. J. Mech. Phys. Solds, Vol 16, pp.229-224. Lee, K.M. and Rowe, R.K. (1990a) Fnte element modelng of the three0dmensonal ground deformatons due to tunnelng n soft cohesve sols: part I method of analyss, Comput. Geotec., 10, 87-109 Lee, K.M. and Rowe, R.K. (1990b) Fnte element modelng of the three0dmensonal ground deformatons due to tunnelng n soft cohesve sols: part II results, Comput. Geotec., 10, 111-138 Pestana, Juan M. and Whttle, Andrew J., Formulaton of a Unfed Consttutve Model for Clays and Sands, Internatonal Journal for Numercal and Analytcal Methods n Geomechancs, 23, 1215-1243