´ otechnique otechnique 55, No. 3, 189–199 Franzius, J. N., Potts, D. M. & Burland, J. B. (2005). Ge´
The influence of soil anisotropy and K 0 on groun ground d sur surfac facee mo movem vement entss result res ulting ing from tunn tunnel el exc excaava vatio tion n J . N . F R A NZ N Z I U S *, * , D . M . P O TT T T S † a n d J . B . B U R L A N D† D† ´ e a` L’analyse d’e´ le´ me ment ntss fin finis is (F (FE) E) es estt so souv uven entt ut utili ilise se pre´ sen sentt dan danss la pra pratiq tique ue ind indust ustriel rielle le pou pourr mod mode´elis liser er le ´ par les tun tassem tas sement ent de sur surfac facee de sol pro provoq voque ue tunnel nels. s. ´ gi Pou ourr le less re gime mess de co cont ntra rain inte te in init itia iaux ux,, ave vecc un fo fort rt coefficient de pression terrestre late´rale au repos K 0 , il a e´ te´ montre´ par div divers erses es e´ tud tudes es que l’a l’aug ugee de tas tassem sement ent transvers trans versal al pre´ dit par une ana analys lysee FE bid bidime imensi nsionn onnell ellee (2D) est trop large par rapport a` ce qui se passe sur le terrain. Il a e´te´ sugge´re´ que les effets 3D et /ou d’anisotropie tro pie du sol pou pourra rraien ientt exp expliq liquer uer cet ´ecart. Cet expos exposee´ pre´ se sent ntee un unee su suit itee d’ d’an anal alys yses es FE en 2D et 3D de la constr con struct uction ion d’u d’un n tun tunnel nel dan danss de l’a l’argi rgile le de Lon Londre dres. s. ` ´ ´ Nous employons deux modeles elastiques non lineaires de pre´ -e´ co coul ulem emen ent, t, un is isot otro rope pe et un an anis isot otro rope, pe, et no nous us montro mon trons ns que que,, meˆ me po pour ur un de degr gree´ e´ leve´ d’anisotropie de so sol, l, l’ l’au auge ge de ta tass ssem emen entt tr tran ansv sver ersa sall re rest stee tr trop op pe peu u prof pr ofon onde de.. En co comp mpar aran antt au aux x do donn nnee´es de te terr rrai ain n le less ` partir profil pr ofilss de tas tassem sement ent lon longit gitudi udinal nal obt obten enus us a partir des analys ana lyses es 3D, nou nouss de´ montr montrons ons que l’auge longitudinale longitudinale ´ s’eten tend d tro trop p loi loin n dan danss la dir direct ection ion lon longit gitudi udinal nalee et que, par conse´quent, il est difficile d’e´tablir des conditions de ` re la face du tass ta ssem emen entt de re´ gim gimee per perman manent ent der derrie rie ´ gi tunn tu nnel el.. De Dess co cond ndit itio ions ns de re gime me pe perm rman anen entt on ontt e´ te´ obtenues obten ues uniqu uniquement ement en appliq appliquant uant un degr degree´ anormalement me nt e´ leve´ d’anisotropie combine´ a` un re´gim gimee de fai faible ble provo voque que une per perte te de vo volum lumee ano anorma rmalem lement ent K 0 , ce qui pro e´ leve´ e.
Finite ele Finite elemen mentt (FE (FE)) ana analys lysis is is no now w oft often en use used d in eng engiineering practice to model tunnel-induced ground surface settle set tlemen ment. t. For ini initia tiall str stress ess reg regime imess wit with h a hig high h coe coeffifficient cie nt of lat latera erall ear earth th pr press essur uree at res rest, t, K 0 , it has been been shown sho wn by sev severa erall stu studie diess tha thatt the tra transv nsvers ersee set settle tlemen mentt trough predicted by two-dimensional (2D) FE analysis is too to o wi wide de wh when en co comp mpar ared ed wi with th fie field ld da data ta.. It ha hass be been en sugges sug gested ted tha thatt 3D eff effect ectss and and/or /or soi soill ani anisot sotro ropy py cou could ld account for this discrepancy. This paper presents a suite of both 2D and 3D FE analyses of tunnel construction in London Lond on Cla Clay y. Both isotr isotropic opic and aniso anisotrop tropic ic non-l non-linear inear elasti ela sticc pre pre-yi -yield eld mod models els ar aree emp emplo loyed yed,, and it is sho shown wn that th at,, ev even en for a hi high gh de degr gree ee of so soil il an anis isot otro rop py, th thee transv tra nsvers ersee set settle tlemen mentt tr troug ough h re remai mains ns too sha shallo llow w. By comparing compa ring longit longitudina udinall settle settlement ment profi profiles les obtain obtained ed from 3D ana analys lyses es wit with h fiel field d dat data a it is dem demons onstra trated ted that the longit lon gitudi udinal nal tro trough ugh ext extend endss too far in the lon longit gitudi udinal nal direction, and that consequently it is difficult to establish steady-state settlement conditions behind the tunnel face. Steady-state conditions were achieved only when applying an unr unreal ealist istical ically ly hig high h deg degree ree of ani anisot sotrop ropy y com combin bined ed with wit h a lo loww- K 0 regime, regime, leadin leading g to an unre unrealisti alistically cally high volume loss.
KEYWORDS KEYW ORDS:: gr groun ound d mo move vemen ments; ts; num numeri erical cal mod modell elling ing and analysis; analy sis; settle settlement; ment; tunne tunnels ls
INTRODUCTION Finite Fi nite ele elemen mentt (FE (FE)) ana analy lysis sis of tun tunnel nel con constr structi uction on in sof softt soil has become widely adopted over recent years. Most of the analyses are performed employing a plane strain model. It has been noted by several authors that the surface settlementt tro men trough ugh in the transver transverse se dir directi ection on to the tunnel axis obtai ob taine ned d fr from om su such ch an anal alys yses es is to too o wi wide de wh when en co comp mpar ared ed with field data if the initial stress profile is described by a high value of the coefficient of lateral earth pressure at rest, K 0 . al.. (1997) pres Addenbrooke et al presen ente ted d a su suit itee of tw twoodimens dim ension ional al (2D (2D)) FE ana analys lyses es inc includ luding ing bot both h lin linear ear elas elastic tic and non non-li -linea nearr ela elasti sticc pre pre-yi -yield eld mod models els,, com combin bined ed wit with h a Mohr–Coulomb yield surface. By modelling the construction of th thee Ju Jubi bile leee Li Line ne Ex Exte tens nsio ion n be bene neath ath St Ja Jame mes’ s’ss Pa Park rk,, . Lond Lo ndon on,, th they ey co conc nclu lude ded d th that at fo for r K 0 1 5 th thee pr pred edict icted ed surface settlement trough was too wide when soil parameters appr ap prop opri riat atee fo forr Lo Lond ndon on Cl Claay we were re in incl clud uded ed in th thee so soil il models.. Their study also sho models showed wed that modelling soil anisotropy did not significantly improve the results when realistic soill par soi parame ameter terss we were re ado adopte pted. d. Sim Simila ilarr con conclu clusio sions ns we were re
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Manuscript received 13 June 2004; revised manuscript accepted 29 November 2004. Disc Di scus ussi sion on on th this is pa pape perr cl clos oses es on 3 Oc Octo tobe berr 20 2005 05,, fo forr fu furth rther er details see p. ii. * Geo Geotec techni hnical cal Con Consul sultin ting g Grou Group, p, Lon London don;; form formerl erly y Impe Imperia riall College Colle ge of Scien Science, ce, Technol echnology ogy and Medici Medicine, ne, Londo London. n. † Imperial Imperial College of Scien Science, ce, Techno echnology logy and Medic Medicine, ine, London London..
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al.. (1996) drawn dra wn by Gunn (1993 (1993). ). In con contra trast, st, Simpson et al presented results of a plane strain FE study modelling the Heathrow Express trial tunnel in which they concluded that soil anisotropy gives better surface settlement predictions for overcons ove rconsolidat olidated ed clay clay.. They compared resul results ts from a linear and a non non-li -linea nearr tra transv nsvers ersely ely ani anisot sotrop ropic ic soi soill mod model el wit with h those of a non-linear isotropic model. However, only limited details about the applied soil model were given. Tunn unnel el con constr struct uction ion cle clearl arly y is a thr three-d ee-dimen imensio sional nal pro pro- blem, and one would expect that 3D FE analysis would improve impro ve the surfac surfacee settlem settlement ent predic predictions, tions, compared with 2D modelling. Such a conclusion was drawn by Lee & Ng (2002)) who compared results of a 3D study in which both (2002 the deg degree ree of soi soill ani anisot sotrop ropy y and K 0 were were var varied ied wit with h the results of Addenbrooke et al. (1997). The surface settlement trough tro ughss fro from m the 3D ana analys lyses es by Lee Lee & Ng (2 (200 002) 2) were much more sensitive to changes in the ratio of horizontal to E h9 = E v9 ) th vertical vertic al Young’s modulu moduluss (defin (defined ed as n9 than an al.. (1997). obse ob serve rved d in the 2D st stud udy y by Addenbrooke et al However, Lee However, Lee & Ng (2002) (2002) adopted a linear elastic perfectly plastic soil model in contrast to the non-linear elastic perfectly plastic constitutive model adopted by by Addenbrooke Addenbrooke et al al.. (1997). (1997). Mor Moreo eover ver,, the tun tunnel nel dia diamet meter er D and and th thee tunnel depth z 0 were different in the two studies. The statement by Lee by Lee & Ng (2002) (2002) that 3D FE modelling leads lea ds to bet better ter sur surfac facee set settle tlemen mentt pre predic dictio tions ns tha than n corr correesponding 2D analyses is in sharp contrast to the findings of sever se veral al oth other er aut author hors. s. Gued Guedes es & Sa Sant ntos os Pe Pere reir iraa (2 (200 000 0) presented a suite of FE studies (adopting an elastic soil
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model) that showed that, for both K 0 0.5 and 1.5, 3D and 2D analyses give similar transverse surface settlement troughs. Similar conclusions were drawn by Dolezˇalova´ (2002), who used both a linear elastic perfectly plastic and a non-linear elastic perfectly plastic constitutive model for the soil. Vermeer et al. (2002) presented results from linear elastic perfectly plastic analyses that showed that the transverse settlement profile obtained from 3D analysis is similar to that from a corresponding 2D study. They also showed that in the longitudinal direction the tunnel has to be constructed over a certain length in order to achieve a steady-state settlement condition behind the tunnel face (i.e. settlement caused by the immediate undrained response). For their particular analysis with K 0 0.66 the tunnel had to be constructed over 80 m (10 D) to achieve a sufficient distance from the vertical start boundary. Steady-state settlement developed approximately 5 D behind the tunnel face. This paper investigates the differences between 3D and 2D modelling on the prediction of tunnel-induced ground surface settlement troughs when varying soil anisotropy and K 0 . The analyses presented here are similar to the plane strain study presented by Addenbrooke et al. (1997), which compared results from numerical analysis with field observations from the Jubilee Line Extension at St James’s Park, London. For K 0 1.5 both isotropic and transversely anisotropic non-linear elastic perfectly plastic soil models have been adopted. The transverse surface settlement troughs from the 3D analyses are compared with corresponding 2D results and field data. Finally a K 0 0.5 initial stress regime is incorporated in the study to highlight the significance of the magnitude of the coefficient of lateral earth pressure at rest on the predicted shape of the ground surface settlement trough.
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DETAILS OF ANALYSIS All FE analyses presented in this paper were carried out using the Imperial College Finite Element Program (ICFEP).
Reduced integration was used, with an accelerated modified Newton– Raphson solution scheme with an error-controlled substepping stress point algorithm for solving the non-linear FE equations. All analyses were performed undrained, although the soil models applied are formulated in terms of drained parameters. Undrained conditions were enforced by using a high bulk modulus for the pore water (Potts & Zdravkovic´, 1999, 2001).
Geometry Figure 1 shows the 3D finite element mesh for the Jubilee Line Extension westbound tunnel beneath St James’s Park, London. This was the first of the twin tunnels to be constructed beneath the St James’s Park greenfield measurement site (Standing et al., 1996). The tunnel diameter D was 4.75 m and the tunnel depth z 0 was approximately 30.5 m. The subsequent construction of the eastbound tunnel was not included in the analyses described here.
Modelling of tunnel excavation For the 3D analyses the tunnel was excavated in the negative y-direction, starting from y 0 m (see Fig. 1). Only half of the problem was modelled, as the geometry is symmetrical. On all vertical sides of the mesh normal horizontal movements were restrained, whereas for the base of the mesh movements in all directions were restricted. The mesh shown in Fig. 1 consisted of 10 125 20-node hexadron elements which had 45 239 nodes. For the 2D analyses the mesh in the x-z plane was adopted, and the soil was modelled by 8-node quadrilateral elements. In the two-dimensional analyses tunnel excavation was modelled using the volume loss method (see Potts & Zdravkovic´, 2001). The volume loss V L quantifies the amount of over-excavation and is defined as the ratio of the difference between the volume of excavated soil and the tunnel volume (defined by the tunnel’s outer diameter)
¼
Longitudinal profile ( x 0 m)
Transverse profile (y 50 m)
30·5 m
x
y z
5 5 m
Tunnel diameter
4·75 m
1 0 0 m
m 8 0
Fig. 1. FE mesh for 3D analyses of tunnel excavation beneath St James’s Park greenfield monitoring site
GROUND SURFACE MOVEMENTS FROM TUNNEL EXCAVATION divided by the tunnel volume. Under undrained conditions V L can also be obtained by dividing the volume (per running metre) of the surface settlement trough by the tunnel volume (per running metre). In London Clay values of V L between 1% and 2% have been reported by several authors (Attewell & Farmer, 1974; O’Reilly & New, 1982). However, the volume loss measured during construction of the Jubilee Line Extension beneath St James’s Park was higher. Standing et al. (1996) r eported a value of V L 3.3% for the construction of the westbound tunnel, which is addressed in this paper. In the 2D analyses the tunnel was excavated over 15 increments. This was done by evaluating the stresses that act on the tunnel boundary within the soil and applying them in the reverse direction over the 15 increments. Elements within the tunnel boundary were not included in the analyses during this procedure. After each increment the volume loss was calculated and results were taken from that increment in which the desired volume loss (i.e. V L 3.3%) was achieved. The step-by-step approach (Katzenbach & Breth, 1981) adopted in the 3D analyses was not volume loss controlled. In this approach, excavation is modelled by successive removal of elements in front of the tunnel while successively installing lining elements behind the tunnel face. The tunnel lining was modelled by elastic shell elements (Schroeder, 2003) with a Young’s modulus of 28 3 106 kN/m2 , a Poisson’s ratio of 0.15, and a thickness of 0.168 m. As noted above, in contrast to the 2D analyses there was no volume loss control in the 3D analyses. Specification of a certain volume loss to be achieved during the tunnel excavation would require further assumptions, such as modelling the tunnelling technique in more detail. In the excavation model presented in this paper the magnitude of V L depends on the excavation length Lexc , which is the length over which soil elements within the tunnel boundary are removed in each excavation step. Over this length the soil around the tunnel boundary remains unsupported. Increasing Lexc leads to higher values of V L . The choice of Lexc has serious consequences for the computational resources (storage and time) needed. For a given tunnel length to be modelled a reduction in Lexc not only increases the number of elements within the entire mesh but also requires more excavation steps. For the analyses presented here the excavation length was set to Lexc 2.5 m, and therefore 40 excavation steps were required to model the tunnel construction over a length of 100 m (21.0 D), as shown in Fig. 1. The distance in the longitudinal direction from the tunnel face to the remote vertical boundary after the last excavation step was 55 m (11.5 D). The mesh dimensions and the value of Lexc were chosen after performing a number of analyses in which these measures were varied (see Franzius, 2004). From this study it was concluded that the dimensions used in this paper provided the best compromise between mesh size and a reasonable computational time (typical analyses took 15 days to run on a Sun SF880 server). Furthermore, by varying Lexc between 2.5 m and 2 m Franzius (2004) demonstrated that similar surface settlement and horizontal surface strains were obtained when normalising results against volume loss. Noting that a Lexc 2.0 m analysis took approximately 50% more calculation time than the 2.5 m analysis, it was decided that it was not possible to model a more realistic (i.e. smaller) value of Lexc , and it was therefore not attempted to simulate the actual tunnelling technique applied during the construction of this part of the Jubilee Line Extension. In practice the value of Lexc is unlikely to be less than the width of the segmental lining (1000 mm), but its actual value is likely to be larger, and depends on workmanship, as
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an open face excavation procedure was adopted. Nyren (1998) reports a maximum reach of the backhoe in advance of the tunnel shield cutting edge of 1.9 m.
Initial stress profile The ground profile consisted of London Clay with a saturated bulk unit weight of ª 20 kN/m3 . This is a simplification compared with the work of Addenbrooke et al. (1997), who modelled the layers of Thames Gravel and Sand overlaying the clay. Comparison plane strain analyses were performed to assess the influence of this change in soil layering on the surface settlement behaviour. It was found that modelling the gravel and the sand led to slightly narrower transverse settlement troughs. A constant value of the coefficient of lateral earth pressure at rest of K 0 1.5 was applied during the first analyses. Later, K 0 0.5 was also considered. The value of K 0 1.5 was given as an upper bound value by Hight & Higgins (1995) for London Clay at a depth between 10 m and 30 m below the ground surface. At St James’s Park layers of Thames Gravel and Sand have been deposited on top of the London Clay in recent geological time. This has the effect of reducing K 0 at the top of the soil profile. Addenbrooke et al. (1997) modelled a lower value of K 0 0.5 in the gravel and the sand. Comparing the different initial stress profiles in a set of plane strain analyses showed only marginal influence in the choice of K 0 in the top layer. It is the value of K 0 at tunnel depth that has the major influence on the surface settlements. Consequently the simplification of adopting a constant value of K 0 is unlikely to have a major influence on the results. A hydrostatic pore water pressure distribution was prescribed, with a water table 2 m below the ground surface. Above the water table pore water suctions were specified. In all analyses the soil was modelled to behave undrained, by specifying effective stress soil parameters and a high value of the bulk stiffness of the pore water.
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SOIL MODEL Isotropic model The London Clay was represented by a non-linear elasto plastic model that was also included in the study presented by Addenbrooke et al. (1997). The model described by Jardine et al. (1986) was used to model the non-linear elastic pre-yield behaviour, and the yield surface and the plastic potential were described by a Mohr– Coulomb model. The non-linear elastic model accounts for the reduction of soil stiffness with strain. Trigonometric expressions describe G / p9 and K / p9 as a function of shear strain E d and volumetric strain v respectively in the non-linear region. G is the tangent shear modulus, K is the tangent bulk modulus, and p9 is the mean effective stress. The non-linear region is defined by maximum and minimum values of shear and volumetric strain. More details of this model and the soil parameters used are given in Appendix 1.
Anisotropic model A new constitutive model was implemented into ICFEP to combine the transversely anisotropic stiffness formulation of Graham & Houlsby (1983) with the non-linear stiffness behaviour described above. It has been shown that transversely anisotropic material behaviour is fully described by five independent material parameters (Pickering, 1970): E v , the vertical Young’s modulus; E h , the horizontal Young’s modulus; vh , the Poisson’s ratio for horizontal strain due to vertical strain; hh , the Poisson’s ratio for horizontal strain
FRANZIUS, POTTS AND BURLAND
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due to horizontal strain in the orthogonal direction; and G vh , the shear modulus in the vertical plane. Graham & Houlsby (1983) showed that only three of these material parameters can be obtained from conventional triaxial tests, as no shear stress can be applied to the sample. They introduced a material model that uses only three parameters, namely E v , hh , and an anisotropic scale parameter Æ that relates the remaining material properties as:
Æ
¼
r ffiffi ffiffi E h E v
¼ ¼ G G hh
hh
vh
vh
(1)
n
¼ E E
h v
and m
¼ G E
vh
(2)
v
are often used. Similar expressions are defined for drained stiffness properties. A relation between the drained and undrained ratios is given by Hooper (1975). Lee & Rowe (1989) showed that m influences the shape of the transverse settlement trough. For K 0 0.5 they concluded that a ratio of m 0.2– 0.25 produced a reasonable match between FE results and centrifuge tests. Field data for London Clay summarised by Gibson (1974), however, give a ratio of approximately m 0.38. In the same publication a ratio of undrained Young’s moduli of n 1.84 is reported. Addenbrooke et al. (1997) included a transversely anisotro pic model in their 2D study. The anisotropic parameters in that study were chosen to match field data reported by Burland & Kalra (1986) giving drained ratios of n9 1.6 and m9 0.44 with vh 0:125 and hh 0:0. The 9 9 equivalent undrained ratios can be calculated to be n 1.41 and m 0.3 using the relations given by Hooper (1975). Two parameter sets referred to as ‘set 1’ and ‘set 2’ were applied to the anisotropic model. Table 1 summarises the ratios n9, m9, n and m for these sets. They were calculated for small strains (i.e. E d , E d,min ), as they change with strain level. The material parameters (listed in Table 3) were derived from the isotropic model (Table 2) such that E v9 reduces with increasing Æ, compared with the Young’s modulus used in the isotropic model. The parameters were chosen in order to obtain similar stress–strain curves when simulating triaxial extension tests with the different materials (as outlined below). Of the two parameter sets, the first one represents a degree of anisotropy that is appropriate for London Clay. In contrast, the second set incorporates an extremely high degree of anisotropy, and is therefore more of academic interest than for use in engineering practice. In the first set, Æ 1.265 was chosen to match the drained ratio of n9 adopted by Addenbrooke et al. (1997). A Poisson’s ratio of hh 0:4 was adopted in order to 9 achieve a ratio of m9 0.46, which is close to that used by Addenbrooke et al. (1997) (0.44). Owing to the coupling of the anisotropic stiffness parameters through the scale factor Æ (see equation (1)) the ratios n and m derived for set 1 differ slightly from those applied in the study of Addenbrooke et al. (1997), where E v9 , E h9 , hh 9 , 9vh and G vh were independent. The undrained ratio of n 1.18 is lower
¼
¼
¼
where G hh is the shear modulus in the horizontal plane. To implement this form of anisotropy into ICFEP, it was combined with a small-strain formulation reducing the vertical Young’s modulus E v with increasing deviatoric strain (within the strain limits E d,min and E d,max defining the nonlinear range). As an additional option the value of Æ can be varied linearly with strain level from its anisotropic value at E d,min to the isotropic case of Æ 1.0 at E d,max . Appendix 2 presents more details of this soil model. Although this model describes stiffness anisotropy, the strength parameters were kept isotropic. Two parameter sets, referred to as ‘set 1’ and ‘set 2’ and summarised in Table 1, were included in the analyses presented here. A further set (‘set 2v’) adopted a variable Æ. The different parameter sets will be discussed in the following section.
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Anisotropic material parameters When using anisotropic soil parameters, the undrained ratios Table 1. Stiffness ratios for the two sets of anisotropic soil parameters
Isotropic model
n9 m9 n m
Anisotropic model Set 1 1.60 0.46 1.18 0.33
1.00 0.55 1.00 0.33
Set 2 6.25 1.14 1.66 0.28
¼
¼ ¼
¼
¼
¼ ¼
¼ ¼ ¼
¼
Table 2. Input parameters for isotropic pre-yield model (M1) A
C : %
B
373.3
338.7
R
1.0 3 104 T : %
S
549.0
506.0
1.0 3 103
ª
1.335
0.617
2.069
0.42
E d,min : %
8.66025 3 104
v,min : % 5 3 103
E d,max : %
G min : kPa
0.69282
2333.3
v,max : % 0.15
K min : kPa
3000.0
Table 3. Input parameters for anisotropic pre-yield model (M2)
Parameter set 1 Aa
373.3
Ba
C : %
338.7
1.0 3 104
1.335
ª 0.617
1.335
ª 0.617
E d,min : %
E d,max : %
E v,min : kPa
8.66 3 104
0.69282
5558.8
E d,min : %
E d,max : %
E v,min : kPa
8.66 3 104
0.69282
5558.8
Æ 1.265
9hh 0.4
Æ 2.5
9hh 0.1
Parameter set 2 and set 2v Aa
308.8
Ba
C : %
280.2
1.0 3 104
GROUND SURFACE MOVEMENTS FROM TUNNEL EXCAVATION than n 1.41, which was calculated from their parameters. It is also below the value of n 1.84 given by Gibson (1974). In contrast the ratio m 0.33 is higher than in the work of Addenbrooke et al. (1997) and closer to the value of 0.38 reported by Gibson (1974) for London Clay. The second set was chosen in order to reduce the undrained ratio m close to a value adopted by Lee & Rowe (1989) and bringing n close to the ratio reported by Gibson (1974). This was achieved by increasing the anisotropy factor to Æ 2.5. Such a high value for London Clay is not supported by any literature. Both Simpson et al. (1996) and Jovic˘ic´ & Coop (1998) r eported ratios of approximately G hh / G vh Æ 1.5. The results of this parameter set can be seen, however, as an extreme example of how anisotropy affects tunnel-induced settlement predictions. If, in the anisotropic model, the vertical Young’s modulus E v9 was set to a similar magnitude as in the isotropic model (calculated from K 9 and G for very small strains), E h9 would be increased by Æ2 , hence leading to an overall stiffer behaviour. In order to obtain a similar overall stiffness response of the anisotropic model compared with the isotropic one, E v9 has to be reduced. The input parameters Aa and Ba listed in Table 3 for the anisotropic model were chosen in order to give a similar response to the isotropic model when used to simulate the behaviour of an ideal undrained triaxial extension test. These tests were simulated in a single-element FE analysis by prescribing vertical displacement at the top of the element (i.e. they were strain controlled). The initial stress state within the sample was isotropic, with p9 750 kPa. Fig. 2 presents stress–strain curves from these tests. The deviatoric stress, defined as
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¼
¼
¼
¼ ¼
193
corresponds to the upper limit of non-linear elastic behaviour of E d,max 0.69%. Different curves are given for deviatoric stress q and excess pore water pressure u. It can be seen that for ax < 0.3% the isotropic analysis and that based on the anisotropic parameter set 1 are in very good agreement with the laboratory data, whereas that based on parameter set 2 slightly over-predicts the data. The pore water pressure curves in this plot highlight the anisotropic behaviour of parameter sets 1 and 2 (note: no laboratory data were available). Higher excess pore pressure is generated owing to the coupling of deviatoric and volumetric strains, in contrast to isotropic behaviour, where volumetric strain and therefore changes in pore water pressure are induced only by changes in p9 whereas deviatoric strain is caused only by changes in q. This coupling is further illustrated in Fig. 3, which shows the stress paths obtained during the single element analyses for the different soil models. For the isotropic model the path is vertical (showing no change in mean effective stress p9) until it reaches the Mohr– Coulomb yield surface. For the anisotropic model p9 changes during the tests, leading to inclined stress paths before the yield surface is reached. The figure also includes the stress path for an additional analysis with the same input parameters as set 2 but with Æ being varied linearly with deviatoric strain from 2.5 at E d,min to 1.0 at E d,max . It can be seen that the stress path from this analysis (referred to as set 2v) initially shows an anisotropic response but then becomes vertical.
¼
¼
q
¼ ax
(3)
r
where ax and r are the axial and radial stress in the sample respectively, is plotted against the axial strain ax . Note that the values of q in the plots are negative, as triaxial extension has been considered. The results from the isotropic model are referred to as ‘M1’, and ‘M2’ denotes the anisotropic results. Laboratory data reported by Addenbrooke et al. (1997) are also shown. Figure 2(a) shows the strain up to ax 0.01%. For this strain range the results of the anisotropic parameter set 2 coincide with the those from the isotropic model. The results of the anisotropic parameter set 1 are in slightly better agreement with the laboratory data. Fig. 2( b) presents the results for a strain range up to ax 0.4%, which
Fig. 3. Stress paths for isotropic (M1) and anisotropic (M2, constant and variable) soil models
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¼
Fig. 2. Stress– strain curves for isotropic (M1) and anisotropic (M2) soil models subjected to triaxial extension tests: (a) strain range up to ax 0.01%; (b) strain range up to ax 0.4%
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ISOTROPIC ANALYSES Settlement profiles from the 3D analyses are presented along a longitudinal monitoring profile above the tunnel centre line and along a transverse monitoring profile at 50 m, as indicated in Fig. 1. Initially a 3D analysis y was performed with the isotropic soil model and K 0 1.5. The development of the longitudinal settlement profile as the tunnel heading is advancing is shown in Fig. 4. Different curves are given for every 10 m of tunnel progress. The position of the tunnel face for each curve is indicated by an arrow. This figure demonstrates that initially the settlement trough has a similar shape to that of the cumulative error curve, which is often applied to describe the longitudinal settlement behaviour (Attewell & Woodmann, 1982). However, with further tunnel excavation, and in particular when the tunnel face reaches approximately y 80 m, the settlement does not continue to follow this anticipated trend. Instead the profile indicates that some reverse curvature is developing at approximately y 60 m (as indicated in the figure). However, the main concern that is evident from Fig. 4 is that for all stages of excavation additional settlement occurs over the whole mesh length. One would expect that, once the tunnel heading had reached a certain y-position, there would be no further settlement remote from the tunnel face as a result of any additional tunnel excavation (note: only the short-term response is considered in the undrained FE analysis). It can be seen in Fig. 4 that such a steady-state settlement condition is not established during the analysis. There is still additional settlement at the y 0 m boundary when the tunnel is excavated from y 90 m to 100 m. Also, the remote boundary at y 155 m settles over the whole analysis, although settlements for the first few increments are negligible on an engineering scale. The additional settlement over the whole mesh length obtained in the last increments of the analysis indicates that the longitudinal distance from the last excavation step to both the start and remote vertical boundaries is too small to obtain steady-state conditions. An increase in the longitudinal dimension of the mesh, however, would lead to excessive computational time, and therefore could have been achieved only in combination with a drastic increase of element size within the transverse mesh plane, leading to a substantial loss in accuracy. It should be noted that the longitudinal dimension of the FE mesh used here and shown in Fig. 1 is considerably larger than in most 3D studies recently published by other authors (e.g. Dolezˇalova´, 2002; Lee & Ng, 2002; Vermeer et al., 2002). Figure 5 presents transverse settlement profiles normalised against maximum settlement, and compares the results with
¼
¼
¼
¼
¼
¼
¼
Fig. 5. Transverse normalised settlement profiles for different stages of isotropic 2D and 3D analyses together with field data
field data from the St James’s Park monitoring site (Standing et al., 1996; Nyren, 1998). Field data are given for the tunnel face being just beneath the monitoring profile (referred to as ‘set 22’) and for a distance between tunnel face and monitoring section of 41 m (‘set 29’). Nyren (1998) reports no further short-term settlement after this survey. These results therefore represent the end of the immediate settlement response. Comparing the two sets of field data shows that the shape of the settlement trough does not change as the tunnel face moves away from the monitoring section. The 3D FE results included in this figure are taken from the increment when the tunnel face was just beneath the transverse monitoring section at y 50 m and for the largest possible distance between tunnel face ( y 100 m) and monitoring section of 50 m corresponding to the last increment of the analysis. The figure also includes results from a similar 2D analysis where the results were taken from increments where volume losses were close to those calculated from the field data (as listed in the figure). It can be seen that all curves obtained from the FE analyses, regardless of whether 2D or 3D, are very similar, indicating that 3D effects cannot account for the discrepancy between plane strain results and field data. The results also show that in the FE analysis the shape of the settlement trough does not change with V L (or face position) over a certain range of V L. Similar conclusions were drawn by Potts & Addenbrooke (1997) when analysing tunnel-induced building deformation.
¼
¼
ANISOTROPIC ANALYSES Initially a set of 2D FE analyses were performed with the anisotropic soil model. In all cases an initial stress regime
Fig. 4. Longitudinal settlement profiles from isotropic 3D analysis
GROUND SURFACE MOVEMENTS FROM TUNNEL EXCAVATION
Fig. 6. Transverse normalised settlement profiles for isotropic (M1) and anisotropic (M2) soil models
with K 0 1.5 was adopted. Fig. 6 shows the normalised transverse ground settlement troughs for these analyses taken from increments that achieved a volume loss of approximately V L 3.3% (as listed in Fig. 6). This amount of volume loss was obtained from the field data for a tunnel face position approximately 41 m beyond the monitoring section (which were previously included in Fig. 5 and labelled ‘set 29’). The figure also lists the increments in which the desired volume loss was achieved. These increment numbers show that, for the anisotropic analyses, volume losses in the range 3.1– 3.6% were obtained in increment 10 (of 15 excavation steps) for constant Æ and increment 9 for variable Æ. In contrast, results for the isotropic analysis are presented for increment 12. This indicates that, when comparing same stages in the analysis (i.e. same percentage of unloading), the anisotropic model predicts higher values of V L . Values of maximum settlement are also listed in the figure. The maximum settlement obtained for parameter set 1 is 10 mm and lies close to the maximum settlement (approx. 12 mm, V L 3.2%) obtained by Addenbrooke et al. (1997) in their anisotropic analysis with similar ratios of n9 and m9. Comparing the settlement curve for the isotropic model with those for the anisotropic parameter sets 1 and 2 (with a constant value of Æ ) shows that the surface settlement trough becomes narrower with increasing degree of anisotropy. Adopting a variable Æ (parameter set 2v) improves the settlement curve further. The reason for this behaviour is the fact that the anisotropic parameters were chosen such that E v9 was reduced (compared with the isotropic Young’s modulus for very small strains) whereas E h9 increased according to
¼
¼
¼
195
the choice of Æ. This parameter choice also leads to a lower value of G vh (compared with the isotropic shear modulus), whereas G hh increases with the degree of anisotropy. Parameter sets 2 and 2v show initially the same stiffness properties. However, as the deviatoric strain increases from E d,min to E d,max , the reduction of Æ leads to an additional decrease of E h9 and G hh with deviatoric strain compared with that obtained for the constant-Æ case. When the material with variable Æ becomes isotropic at E d,max it shows softer stiffness properties than the corresponding isotropic material (M1). During tunnel excavation the largest strains occur close to the tunnel. The material with variable Æ therefore behaves as softer in this region, which explains the narrow settlement trough. The figure demonstrates that even the narrowest settlement trough obtained from the FE analysis is still too wide when compared with field data. As parameter set 2v gave the best results of all plane strain analyses shown in Fig. 6 it was adopted in a 3D greenfield analysis. The boundary conditions of the 3D mesh were the same as described previously for the isotropic 3D model. Figure 7 shows the development of longitudinal settlement profiles during this 3D analysis ( K 0 1.5). Comparing this figure with the isotropic results of Fig. 4 it can be seen that the anisotropic 3D analysis yielded settlement values that are nearly one order of magnitude higher than those obtained with the isotropic model. Such behaviour is consistent with the 2D results presented in Fig. 6, in which results were presented for earlier increments than in the isotropic analyses in order to achieve similar volume losses (i.e. the variation in percentage of unloading, indicated by the different increments in Fig. 6, is consistent with the observed trend of the variation of volume loss in the 3D analyses). However, as discussed earlier, the higher volume losses are unlikely to affect the normalised shapes of the settlement troughs. The general shape of the longitudinal settlement troughs during the initial stages of tunnel excavation is similar for both isotropic and anisotropic analyses. The reverse curvature behind the tunnel face developing from a face position of approximately y 60 m is magnified in the anisotropic analysis. Previously it was shown that a high degree of anisotropy leads to a narrower transverse settlement trough. A similar effect would be expected for the longitudinal curve, and comparison of Figs 4 and 7 indicate that this is so. With a narrower and steeper trough steady-state conditions should develop earlier than for the wide trough obtained from the isotropic analysis. However, no steady-state conditions are reached in Fig. 7.
¼
¼
Fig. 7. Longitudinal settlement profile from anisotropic 3D analysis
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The zone of reversed curvature behind the tunnel face forms a settlement crater at the vertical starting boundary (at y 0 m). This arises because, owing to the symmetry condition at the start boundary, tunnel construction essentially commences simultaneously in both the negative and positive y-directions (although only the negative y-part is modelled). A similar longitudinal settlement profile, including this settlement crater, was presented by Vermeer et al. (2002), who performed 3D analyses for a K 0 0.65 situation. Applying an isotropic elasto-plastic soil model, they showed that steady-state conditions were established approximately 5 D behind the face after the tunnel was constructed over a length of approximately 10 D (with a tunnel diameter D 8 m and z 0 16 m). The present analysis indicates that a much greater length of tunnel is required to reach steadystate conditions in a high- K 0 regime.
¼
¼
¼
¼
INFLUENCE OF K 0 The anisotropic 3D analysis was repeated with K 0 0.5 in order to investigate at which state of the analysis steadystate behaviour develops in a low- K 0 regime. Fig. 8 shows the development of longitudinal surface settlement profiles. It can be seen that points close to the start boundary settle during the first excavation steps but then show small values of heave. It is interesting to note that no settlement crater as observed for the K 0 1.5 case developed in this analysis. More importantly, a horizontal plane of vertical settlement emerges at about y 40 m after a face position of 80 m was reached. As tunnel construction continues, y
¼
¼ ¼
¼
the settlement at this point does not change: that is, steadystate settlement develops. However, comparing the magnitude of settlement developing during this analysis with the settlement values obtained in the anisotropic K 0 1.5 analysis reveals high values of settlement. At steady-state conditions the settlement above the tunnel centre line is 85.8 mm. This is high compared with field data, where a steady-state settlement of approximately 20 mm developed. The volume loss for the anisotropic K 0 0.5 is therefore significantly higher than that observed in field measurements: V L 18.1% compared with 3.3%. It is clear that such a high volume loss is unacceptable engineering practice. However, the results highlight the two effects that a reduction of K 0 has on tunnel-induced surface settlement. It has been noted by other authors that in a low- K 0 regime the transverse settlement trough is narrower (Dolezˇalova´, 2002) owing to smaller lateral stress at tunnel depth. Such an effect is also evident when normalising the longitudinal settlement profiles of the different analyses and comparing them with normalised field data. Fig. 9 shows such a plot. The FE results are normalised against the settlement at y 50 m, for which Fig. 8 indicated steady-state conditions for the anisotropic K 0 0.5 analysis. For consistency the same normalisation was applied to the corresponding results from the other 3D analyses (also including an isotropic K 0 0.5 case), although no steady-state settlement conditions were achieved there. The field data from St James’s Park, London (Nyren, 1998), are normalised against maximum settlement and are plotted such that the face position corresponds to 100 m to compare them with the FE results. The figure y
¼
¼
¼
¼
¼
¼
¼
Fig. 8. Longitudinal settlement profile from anisotropic 3D analysis with
K 0
0.5
Fig. 9. Normalised longitudinal settlement profile at end of analyses compared with field data
GROUND SURFACE MOVEMENTS FROM TUNNEL EXCAVATION
197
shows that the normalised anisotropic results for K 0 0.5 have a shape similar to that of the normalised field data. All other settlement troughs are too wide, with the isotropic K 0 1.5 analysis giving the widest trough. A low value of K 0 , however, influences not only the shape but also the magnitude of settlement, as listed in Fig. 9. Decreasing K 0 reduces the mean effective stress p9 around the tunnel. Both the isotropic and the anisotropic non-linear pre-yield models normalise the soil stiffness against p9. A low K 0 value therefore shows a reduction of soil stiffness compared with high- K 0 situations. This reduction of stiffness around the tunnel results in an increase in volume loss. Combining K 0 0.5 with an extreme (and unrealistic) anisotropic scale factor of Æ 2.5 leads to a longitudinal settlement profile that shows steady-state settlement behind the tunnel face, but also results in an unsatisfactory degree of volume loss, which is further demonstrated in Fig. 10. This graph compares the transverse settlement troughs at the end of the 3D analyses (from the anisotropic model M2, set 2v, K 0 0.5, the isotropic model M1 with K 0 1.5) with their 2D counterparts and with the field data (set 29). The graph indicates that the 2D analysis with the anisotropic model and with K 0 0.5 coincides well with the field measurements. However, this analysis is volume loss controlled, and, as pointed out above, this parameter set is not realistic for London Clay. Furthermore, this model is not applicable in 3D analysis as it results in an unrealistic value of volume loss, and hence exceeds the maximum measured settlement by more than four times. Applying an isotropic model with K 0 1.5, in contrast, predicts too small a maximum settlement that is only one third (3D analysis) or half (2D) of the field measurements. The results of the other analyses are not shown in the diagram but lie within the range of settlement curves presented in Fig. 10. The wide range in volume losses observed in this study shows that both 3D modelling and anisotropy do not resolve the problem that numerical analysis predicts settlement troughs that are too wide when compared with field data. It furthermore demonstrates the difficulty in modelling 3D tunnelling. As pointed out earlier, the high volume losses are also a result of the chosen excavation length Lexc 2.5 m. Owing to computational resources at the time of the analyses it was not possible to reduce this value of Lexc . With increasing computational power a refined 3D tunnelling model should be addressed in future research.
K 0 on the tunnel-induced ground settlement trough. The 3D excavation process was modelled by successive removal of elements in front of the tunnel while successively installing lining elements behind the tunnel face. In a first step the London Clay was modelled by a nonlinear elasto-plastic isotropic soil model and a coefficient of lateral earth pressure at rest of K 0 1.5. Comparing 3D with 2D results showed that 3D modelling has a negligible effect on the shape of the transverse surface settlement trough, which remained too wide when compared with field data. In the longitudinal direction the surface settlement trough did not develop steady-state settlement conditions. The curve extended too far when compared with field data. Settlement was obtained on the vertical boundaries during the entire analysis, although the total length of tunnel construction was chosen as 21.0 D. Such mesh dimensions are considerably larger than most FE models used in recently published studies. Soil anisotropy was included in the study to investigate whether this additional soil characteristic could improve results. Plane strain results showed little improvement in the transverse trough when a level of anisotropy appropriate for London Clay was adopted. The transverse settlement trough improved when an unrealistically high degree of anisotropy was included. With this high level of anisotropy a 3D analysis was carried out. The longitudinal profile of this analysis was still too wide when compared with field data, although it was steeper than the settlement curve obtained from the isotropic analysis. This analysis was then repeated with a low value of K 0 0.5. Only this combination of a high degree of anisotropy and a low K 0 , both unreasonable assumptions for London Clay, produced steady-state settlement conditions at the end of the analysis. The magnitude of settlement, however, was too high, as unrealistically high values of volume loss developed during the analysis. The study demonstrates that incorporation of 3D modelling and elastic soil anisotropy in the prediction of tunnelinduced ground surface settlement in London Clay does not significantly improve the settlement profile. Adopting realistic soil parameters brings only marginal improvements, and adopting an extreme case of soil anisotropy can lead to excessive values of volume loss. This indicates that neither 3D effects nor elastic soil anisotropy can account for the over-wide settlement curves obtained from FE tunnel analysis in a high- K 0 regime.
CONCLUSIONS A suite of 2D and 3D FE analyses was performed to investigate the influence of 3D effects, soil anisotropy and
APPENDIX 1: ISOTROPIC SOIL MODEL
¼
¼
¼
¼
¼
¼
¼
¼
¼
¼
¼
The non-linear elastic model (Jardine et al., 1986) describes the secant soil stiffness as depending on strain level using a trigonometric expression. To use this model in a finite element analysis, the secant expressions are differentiated and then normalised against mean effective stress, giving the following tangent values (Potts & Zdravkovic´, 1999):
G p9
Fig. 10. Transverse settlement profiles of isotropic and anisotropic 2D and 3D analyses compared with field data
ª
¼ A þ B cosð X Þ
B ª X ª1 sin X ª with X 2:303
ð
1
Y ¼ R þ S cos ðY Þ S 2:303
K 9 p9
Þ
¼ log
sin Y with Y
ð
Þ
¼ log
p ffiffi j j E d
10
10
3C (4)
v T
(5) where G and K 9 are the tangent shear and bulk moduli respectively, p9 is the mean effective stress, E d is the deviatoric strain invariant, and v is the volumetric strain. A, B, C , R, S , T , , , ª, and are constants, which are listed in Table 2. E d,max, E d,min , v,max and v,min define strain limits above or below which the stiffness varies only with p9 and not with strain. Minimum values of tangent shear and bulk moduli are given by G min and K min 9 respectively.
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APPENDIX 2: ANISOTROPIC SOIL MODEL The transversely anisotropic formulation by Graham & Houlsby (1983) was combined with a small-strain behaviour based on the isotropic model described in Appendix 1. The tangent vertical Young’s modulus E v9 is expressed as a
ª
a
ª 1
X ¼ A þ B cos ð X Þ B ª : 6 909
E 9v p9
a
sin X ª
ð
¼ log p ffi E 3ffi C d
10
(6)
where p9 is the mean effective stress and E d is the deviatoric strain invariant. The input parameters given in Table 3 are the same as used for the isotropic shear modulus in equation (4) apart from the values of Aa and Ba , which were chosen to be different from A and B. With the anisotropic scale parameter Æ and the Poisson’s ratio 9hh being further input parameters the remaining stiffness properties can be calculated from 2
¼ Æ E ¼ Æ
vh 9
9v
9hh
(7a)
2
G vh
¼ 2ð1Æþ E Þ 9 v
(7b)
9vh
(7c)
APPENDIX 3: MOHR–COULOMB MODEL The Mohr–Coulomb yield surface and plastic potential are expressed in terms of effective stress invariants (Potts & Zdravkovic´, 1999). The input parameters are the angle of shearing resistance 9, the cohesion c9 and the angle of dilation ł9, which are summarised in Table 4.
NOTATION A, Aa , B, Ba , C , R, S , T c D E d E d,min , E d,max E v E h G G vh G hh K K 0 Lexc m n p9 q S v V L x, y, z z 0
Æ , ª, , ª ax v v,min , v,max vh hh
constants for non-linear soil models cohesion outer tunnel diameter deviatoric strain deviatoric strain range of non-linear behaviour vertical Young’s modulus horizontal Young’s modulus shear modulus shear modulus in vertical plane shear modulus in the horizontal plane bulk modulus coefficient of lateral earth pressure at rest excavation length of step-by-step tunnel excavation G vh / E v E h / E v mean effective stress deviatoric stress vertical ground settlement volume loss coordinates of FE mesh tunnel depth (centre line to soil surface) anisotropic scale factor constants for non-linear soil models soil bulk unit weight axial strain volumetric strain volumetric strain range of non-linear behaviour Poisson’s ratio for horizontal strain due to vertical strain Poisson’s ratio for horizontal strain due to horizontal strain in orthogonal direction
¼ ¼
Table 4. Input parameter for Mohr–Coulomb model
9: deg 25
c9: kPa
5
9
axial stress radial stress angle of shearing resistance angle of dilation index denoting effective parameter
Þ
with X
E 9h
ax r ł
ł9: deg 12.5
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Proc. Int. Symp. Geotechnical Aspects of Underground Con struction in Soft Ground, London, 591–594. Standing, J. R., Nyren, R. J., Burland, J. B. & Longworth, T. I. (1996). The measurement of ground movement due to tunnelling at two control sites along the Jubilee Line Extension. Proc. Int. Symp. Geotechnical Aspects of Underground Construction in Soft Ground, London, 751–756. Vermeer, P. A., Bonnier, P. G. & Mo¨ller, S. C. (2002). On a smart use of 3D-FEM in tunnelling. Proc.8th Int. Symp. on Numerical Models in Geomechanics – NUMOG VIII, Rome, 361–366.
