NEW ALGORITHMS FOR THE SIMULATION OF THE SEQUENTIAL TUNNEL EXCAVATION WITH THE BOUNDARY ELEMENT METHOD (EURO:TUN 2007)

Transcription

1 ECCOMAS Themaic Conference on Compuaional Mehods in Tunnelling (EURO:TUN 2007) J. Eberhardseiner e.al. (eds.) Vienna, Ausria, Augus 27-29, 2007 NEW ALGORITHMS FOR THE SIMULATION OF THE SEQUENTIAL TUNNEL EXCAVATION WITH THE BOUNDARY ELEMENT METHOD (EURO:TUN 2007) Duenser C. 1 and Beer G. 1 1 Graz Universiy of Technology, Insiue for Srucural Analysis Lessingsrasse 25, 8010 Graz, Ausria {duenser, gerno.beer}@ugraz.a Keywords: Tunnelling, Sequenial Tunnel Excavaion, Boundary Elemen Mehod, Compuaional Mehods Absrac. The Boundary Elemen Mehod (BEM) is well suied for modelling problems in infine domains. This siuaion is presen in he case of sequenial excavaion in unnelling. The radiional way for he modelling of a sequenial excavaion process is he use of he muliple region BEM. Wihin an infinie domain numerous finie subdomains are embedded. Each of he finie subdomains represens he rock volume o be excavaed in an excavaion sep. Beween wo subsequen analysis seps, he problem geomery changes such, ha one or more of he finie subdomains become inacive, i.e., are excavaed. Siffness marices are calculaed for each subdomain and assembled o a global sysem of equaion. For each excavaion sep he whole equaion sysem has o be solved, including unknowns a inerface nodes belonging o regions of subsequen analysis seps. In he proposed new algorihms he surface of he excavaed par of he unnel is discreized only. The loading for he following excavaion sep is deermined by an inernal resul compuaion of he BEM. As i is well known, he deerminaion of inernal domain resuls wih he BEM exacly saisfies he condiion of equilibrium, his means no error is inroduced by his procedure. Thus, a single region problem only has o be solved as a consequence, and he efficiency of he calculaion will be increased in comparison o he mehod described above. The loading of a subsequen excavaion sep is deermined eiher by displacemen or sress calculaion a inernal poins. From inernal sresses racions a he excavaion surfaces can be deermined. If displacemen a inernal poins are evaluaed inermediae calculaions on finie regions are necessary. In his case previous calculaed inernal displacemens are he Dirichle boundary condiions on a single finie region. From hese calculaions he loading of he following excavaion sep is obained. The resuls of a sequenial unnel excavaion in 2D and 3D for he proposed new algorihms are presened and a comparison wih he radiional mehod menioned above and o Finie Elemen calculaions is done.

2 1 INTRODUCTION The New Ausrian Tunnelling Mehod is characerised by a complicaed sequence of excavaions and insallaion of suppor sysems. In his conex he modelling of he sequenial unnel excavaion in an elasic coninuum will be discussed in his paper. The numerical mehod which will be applied is he Boundary Elemen Mehod (BEM). The BEM is well suied for modelling problems in infine domains. This siuaion is presen in he case of sequenial excavaion in unnelling. The radiional way for he modelling of he sequenial excavaion process wih he BEM is he use of muliple regions[1]. The echnique of muliple regions is shown o be applied successfully for he modelling of he sequenial excavaion process[2, 3]. For he purpose of inroducion and recapiulaion of modelling he sequenial excavaion in unneling his mehod will be explained firs. Then wo compuaional mehods will be explained which are raher new. The main idea is o apply a single region boundary value problem for each excavaion sep insead of using he concep of muliple regions. The problem in his conex is he deerminaion of he excavaion loads of he curren load sep. One possibliy is he evaluaion of displacemens a ineriour poins which belong o he excavaion surface of he subsequen excavaion seps. Wih hese displacemens inermediae calculaions on a single region have o be performed o ge he soluion for he loading. Anoher possibiliy is he evaluaion of sresses a ineriour poins siuaed a he excavaion surface. The racions a he excavaion surface are calculaed easily by a ransformaion of he sresses obained before. This wo mehods will be explained in deail and resuls are shown for an example in 2D. 2 EXCAVATION PROCESS MODELLED BY THE MULTIPLE REGION CONCEPT In he following secions firs he heory of muliple regions in he BEM is explained briefly and hen an example in 3D is presened. 2.1 Theory of he Muliple Region Boundary Elemen Mehod (MRBEM) The displacemen boundary inegral equaion in is discreized form is given by [1]. c u (P i ) = E N E U e ni e n N T e ni ue n (1) U e ni and T e ni are he inegraed kernel shape funcion producs of he elemen e for a collocaion poin P i. u e n and e n are he displacemens and he racions respecively for he elemen e a elemennode n. Wriing Eq. 1 for all collocaion poins leads o a sysem of equaions of he form: T u = U (2) where u, are vecors conaining all displacemens and racions a he boundary, respecively, and T, U are coefficien marices assembled by gahering all elemen conribuions. If a mixed boundary value problem is given, i.e. u is prescribed on some porion of he boundary and on he remaining par, on boh sides of Eq. 2 knowns and unknowns are involved. Thus, Eq. 2 has o be rearranged o he form B x = f (3) 2

3 wih all unknown values placed on he lef hand side and all known values on he righ hand side. In he case of sequenial excavaion, which has o consider he saged advance of he excavaion, a special soluion mehod is needed. A he inerfaces beween regions, boh he displacemens and racions are unknown. Therefore he number of unknowns is increased and addiional equaions are necessary. These equaions can be obained from he condiions of equilibrium and compaibiliy a he region inerfaces. Various mehods o solve muliple region problems have been proposed(see [5, 6, 7, 8, 9]). In his work he so called siffness marix assembly, developed by Beer [1], is used. This approach is similar o he approach aken by he finie elemen mehod. A siffness marix is compued for each region. The coefficiens of his marices are he nodal inerface racions due o uni displacemens. The siffness marices of all regions are assembled in he same way as in he FEM. This mehod is also suiable for coupling boundary elemens o finie elemens. A muliregion problem may be fully or parially coupled. In he firs case all nodes of a sysem are reaed as coupled. In he second case here is a differeniaion beween nodes which are coupled o oher regions and nodes which are free nodes. In boh mehods he siffness marix is calculaed for coupled nodes. In he fully coupled mehod all nodes of he sysem are reaed as coupled nodes and he size of he assembled siffness marix nearly remains he same during he seps of excavaion. In he parially coupled mehod a change from coupled nodes o free nodes is involved by removing regions hroughou he excavaion process. The size of he assembled siffness marix reduces every sep of excavaion. The parially coupled mehod is preferred o he fully coupled mehod because of he reducion of he size and he effor for solving he equaion sysem. Consider for example a unnel excavaion which is performed in Region 1 Region 2 Inerface beween Region 1 and 2 Figure 1: Parially coupled problem op heading and bench excavaion, which is shown in Fig. 1. The acual load case consiss of a Neumann boundary condiion which is specified a he surface of he removed op heading maerial as a resul of he iniial sress sae. As can be seen in Fig. 1 only some of he nodes of region 1 are conneced o region 2. 3

4 Obviously i would be more efficien o consider only he inerface nodes for he calculaion of he siffness marix, i.e. only hose nodes ha are belonging o more han one region. The nodes which are no conneced o anoher region (free nodes) are assigned he given boundary condiions, in his case racions a he free surface (see Fig. 2). Region 1 Region 2 Figure 2: Fixed inerface nodes and given boundary condiions a free nodes For region 1 and 2 he following equaions can be wrien: { } B N N c0 u N = f0 N (4) f0 { } B N N cn u N = fn N n = 1, 2... Ndofc (5) fn where N is he region number, B N is he assembled lef-hand side. f0 N in Eq. 4 is he righhand side obained from he given boundary condiions, i.e. he racions a he free nodes due o excavaion loading and zero values of displacemens a he coupled nodes. The unknown vecors in Eq. 4 conain racions a he coupled nodes N c0 and displacemens a he free nodes u N f0, respecively. The same equaion is shown in Eq. 5, bu here he loading consiss of uni displacemens. Uni displacemens are applied a every inerface node for every degree of freedom (n = 1, 2... Ndofc). The soluion vecors of racions N cn, belonging o he inerface, are assembled o he siffness marix K N and he vecors of displacemens a he free nodes u N fn o he marix A N (shown in Eq. 7). The final soluion for N c, he racions a he inerface nodes, and u N f, he displacemens a he free nodes, can be expressed in erms of u N c, he displacemens a coupled nodes, by superposiion { } { } { } N c N = c0 K N u N + f0 A N u N c (6) u N f where he marices K N and A N for region N are defined by: K N = [ c1, c2,... cndofc ] A N = [u f1, u f2,...u fndofc ] (7) 4

5 Using he condiions for equilibrium and compaibiliy a he inerface, which are I c + II c = 0 u I c = uii c he siffness marices K N and he vecors N c0 of all regions can be assembled ino a global sysem of equaions, which can be solved for he unknown inerface displacemens u c. (8) c0 + K u c = 0 (9) K is he assembled siffness marix and c0 is he assembled vecor of racions a he inerface nodes due o given boundary condiions a he free nodes. The remaining unknowns, displacemens a he free nodes and racions a he coupled nodes can be evaluaed reurning back from sysem o region level, by using Eq Example of a unnel excavaion in 3D In he following example he sequenial excavaion of a unnel is analyzed wih he muliple region boundary elemen approach. The resuls from his calculaion are compared wih he resuls from a finie elemen calculaion. The finie elemen mesh can be seen in Fig. 3 (lef). The mesh consiss of abou quadraic brick elemens. Figure 3: Finie elemen mesh (lef) - boundary elemen mesh (righ) The saged advance of he unnel is assumed as a full face excavaion. The maerial parameers are he following: Young s modulus: E = 5000 MN/m 2 Poisson s raio: ν = 0.3 Specific weigh: γ = MN/m 3 Laeral pressure: k 0 = 0.5 The virgin sress field is linearly disribued, whereas on he ground surface he sress is zero and in a deph of m he virgin sress is: σ xx = 5.00 MN/m 2 σ yy = 5.00 MN/m 2 σ zz = MN/m 2 τ xy = 0.00 MN/m 2 τ yz = 0.00 MN/m 2 τ zx = 0.00 MN/m 2 (10) 5

6 In Fig. 3 he boundary elemen mesh (righ) can be seen. Linear boundary elemens are used. The BE-mesh consiss of 2418 elemens and 27 regions (1 infinie region and 26 finie regions). The cross secion of he unnel is divided in op heading and bench. Therefore 13 regions of op heading and 13 regions of benches exis. In he following calculaion a each load case a region of op heading and a region of bench is excavaed simulaneously. A deailed comparison of BEM and FEM resuls of verical displacemens a he crown of he unnel for each loadsep is shown in Fig. 4. I can be seen ha he BEM-resuls and he FEM-resuls agree very well and he shape of he displacemen curves of he BEM calculaion compared wih he FEM calculaion is almos he same for each load case. E I F =? A A J & $ * -. -! $ ' # & " %!!!! $! '? D = E = C A F = A I J H = E I K J E Figure 4: z-displacemens a he crown of he unnel for 13 load cases 3 NEW MODELLING STRATEGIES OF EXCAVATION The mehod discussed before requires a predefiniion of he complee geomery of he unnel excavaion problem. From beginning he calculaion has o deal wih all he regions, inerfaces, ec., which in subsequen load cases will be par of he excavaion process. This means, ha he size of he equaion sysem is deermined by he number of all degrees of freedom a he enire geomery and i nearly remains he same for every load sep of calculaion. The idea now is o use only a single boundary elemen region, represening he acual excavaion surface. Displacemens and sresses are calculaed a poins inside he domain, provided ha he boundary condiions are known. For he respecive load sep excavaion loads are deermined wih his approach. In he following secions wo modelling sraegies of sequenial excavaion are discussed. 3.1 New modelling sraegy - inernal displacemen approach The algorihm is explained bes for an example in 2D. In Fig. 5 he geomery of a unnel is shown in a longiudinal secion. The excavaion sequence is shown and he excavaed pars of each sep are indicaed by hached areas. Firs 5 seps of op heading excavaion are modelled, followed by he bench excavaions. In sum 10 load seps are calculaed. Modelling a 2D excavaion in such a way means an excavaion of infinie exen ou of plane, which of course is no a real unnel excavaion, bu his model is appropriae for he demonsraion of he algorihm. Wih his mehod i is necessary o discreize he excavaed unnel surface only. A single bound- 6

7 Figure 5: Longiudinal unnel secion - sequence of all load cases ary elemen region is sufficien for he discreizaion of he respecive load sep which can be seen in Fig. 5. The crucial par is he deerminaion of excavaion racions for he curren load case. This is explained nex for load case LC4. In Fig. 6 he excavaion seps from LC1 unil LC4 are shown. The region o be excavaed in LC4 (indicaed by hached areas) is shown in he skeches for each of he previous loadcases. The resulan displacemens and racions for LC4 are he sum of incremenal resuls of all previous load cases. This implies ha all he previous load cases have o be considered for he evaluaion of he loading for LC4. For he evaluaion of excavaion racions inernal displacemens are calculaed on nodes be- Figure 6: Excavaion of LC4 longing o he region o be excavaed. In he BEM Somigliana s ideniy is used for he compuaion of displacemens in he ineriour domain of a region. In Eq. 11 Somigliana s Ideniy is shown in is discreized form. u (P i ) = E N E U e ni e n N T e ni ue n (11) 7

8 u e n and e n are he known values of displacemens and racions a he boundary, respecively, U e ni and Te ni are he inegraed kernel shape funcion producs for node n a elemen e. For LC1 and LC2 all he nodes are inernal nodes, bu for LC3 only a par of hem. Thus, based on he finie region o be excavaed (shown in Fig. 6 by hached areas) wo inermediae calculaions have o be done dependen on he prevailing boundary condiions. This is shown in Fig. 7. For LC1 and LC2 inernal displacemens are calculaed for all nodes of he finie region and hey are he Dirichle boundary condiions of he inermediae calculaion. For LC3 no all nodes of he finie region are inernal nodes and he boundary condiions are mixed (Dirichle and Neumann). Fig. 7 shows he given boundary condiions and he soluion values for he wo inermdediae calculaions. Figure 7: Inermediae calculaion for LC1, LC2 and LC3 The resulend excavaion racions for LC4 are he sum of racions of he wo inermediae calculaions compleed by he racions due o he virgin sress field. These excavaion racions are applied a he freed elemens of he single region for LC4 (shown in Fig. 5 and Fig. 6). Now all boundary condiions for a single region boundary elemen calculaion are known and he soluion for he curren load sep is achieved by using Eq Resuls - inernal displacemen approach For he example shown in Fig. 5 a virgin sress field is assumed, wih a horizonal sress and a verical sress of consan value of 5.00 MN/m 2. The Young s modulus is E = 5000 MN/m 2 and he Poisson s raio is ν = 0.3. The size of each finie region is 3m in lengh and 5m in high. Resuls of verical displacemens along he crown of he unnel are shown in Fig. 8 for each load case. These resuls (LC? NEW) are compared wih he reference soluion (LC REF) and as can be seen hey agree well. 3.3 New modelling sraegy - inernal sress approach Compared o he mehod explained before he excavaion racions for his approach are calculaed wih inernal sress evaluaion. The sress a an inernal poin is calculaed similar as 8

9 Figure 8: Verical displacemens a unnel crown displacemens. Somigliana s Ideniy in he formulaion of sress is shown in Eq. 12 σ (P i ) = E N E R e ni e n N S e ni ue n (12) Eq. 12 is described in discreized formulaion, where σ(p i ) is he sress a Poin P i, R e ni and S e ni are he inegraed kernel shape funcion producs for node n a elemen e, e n and u e n are he racion and displacemen values a he boundary, respecively. The procedure is explained for he same example of he previous chaper and again for LC4 (shown in Fig. 6). The sress is calculaed a he same inernal poins for load cases 1 o 3. For load case 1 and 2 all poins are inernal poins as shown in Fig. 6. There is no difficuly in he evaluaion of he sress. For load case 3 some of he poins of he excavaed volume are boundary poins. Because of he sharp corners a poin A and B (shown in Fig. 9) he sress is infinie and a calculaion direcly a he poin is no possible. To overcome his problem he sress is evaluaed inside he adjacen elemen very near o he boundary, a an inrinsic coordinae of value ξ = 0, 90. The sress is exrapolaed o he boundary according o he parabolic shape funcion of he elemen. This is shown in Fig. 9. The racion vecor is calculaed using Eq. 13 in which he sress ensor σ is muliplied by he ouward normal vecor o he excavaion surface n. = σ n (13) The resuling racion a he excavaion surface for LC4 is he sum of racions obained by inernal sress evaluaion for LC1 o LC3 compleed by he racions due o he virgin sress field. Once he loading racions are found he soluion for he curren load sep is evaluaed by a single region boundary elemen calculaion using Eq. 3. 9

10 Figure 9: Treamen of singular poin 3.4 Resuls - inernal sress approach The resuls of verical displacemens along he crown of he unnel are shown in Fig. 10 for each load case (LC? NEW). These resuls are compared wih he reference soluion (LC? REF). As can be seen in Fig. 10 i seems ha some loading is los from load case o load case. The reason for his is he inaccurae evaluaion of he racions near o he singular poins A and B shown in Fig. 9. Figure 10: Verical displacemens a unnel crown 4 CONCLUSIONS Two new mehods for he sequenial excavaion are shown. The main advanage of boh mehods agains he convenional mehod of domain decomposiion (MRBEM) is ha a single boundary elemen region is necessary only o solve he problem. The simulaion of excavaion sars wih a very small BEM mesh and his will exend from excavaion sep o sep. Theoreically he number of seps is unlimied and has no o be defined prior o he calculaion. The 10

11 algorihm of inernal sress evaluaion is prefered o he one of inernal displacemens. Wih his mehod an inermediae calculaion is no necessary and he excavaion racions can be calculaed direcly from he sress. Addiional effor has o be spen for he correc evaluaion of he sress disribuion near o he boundary, where singular values of sress appear. Currenly he implemenaion for 3D ino he compuercode BEFE++ is an ongoing ask. A reasonable comparison of he efficiency o he convenional mehod of MRBEM only makes sense for a 3D example. 5 ACKNOWLEDGMENT The work repored here is suppored by he European Commission, under is 6h framework program, wihin he inegraed projec TUNCONSTRUCT. REFERENCES [1] G. Beer: Programming he Boundary Elemen Mehod, John Wiley & Sons, [2] C. Duenser, G. Beer: Boundary elemen analysis of sequenial unnel advance. Proceedings of he ISRM regional symposium Eurock, , [3] C. Duenser: Simulaion of sequenial unnel excavaion wih he Boundary Elemen Mehod, Verlag der Technischen Universiä Graz, [4] M. Noronha, C. Duenser, G. Beer: Acceleraing Sraegies o he Numerical simulaion of Large-Scale Models for Sequenial Excavaion. Inernaional Journal for Numerical and Analyical Mehods in Geomechanics, 30, [5] X.W. Gao and T.G. Davies, 3D muli-region BEM wih corners and edges, In. J. Solids Srucures, 37, 2000, [6] X.W. Gao and T.G. Davies, Boundary Elemen Programming in Mechanis, Camebridge Universiy Press, [7] P.K. Buerfield and R. Buerfield, Boundary Elemen Mehods in Engineering Science McGraw-Hill, Maidenhead, UK, [8] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Elemen Techniques Springer Verlag, Berlin, [9] J.H. Kane, B.L.K. Kumar and S. Saigal, An arbirary condensing, noncondensing soluion sraegy for large scale, muli-zone boundary elemen analysis, CMAME, 79, 1990,