Finite element analysis of tunnel–soil–pile interaction
using displacement controlled model
C.Y.Cheng 1,G.R.Dasari
*,2,
Y.K.Chow,C.F.Leung
Department of Civil Engineering,National University of Singapore,Singapore
Received 11January 2006;received in revid form 8August 2006;accepted 12August 2006
Available online 2October 2006
Abstract
Significant additional loads could be induced in pile foundations adjacent to new tunnels.Accurate prediction of magnitude and shape of the ground displacements,which define curvature changes,is crucial for the computation of tunnelling induced bending and axial stress in pile foundations.The finit
e element simulation of tunnelling by removing forces corresponding to initial stress-state,tend to predict incorrect shape of ground displacement profiles,hence incorrect forces in pile foundations adjacent to tunnels.To overcome this difficulty,this paper describes the development and application of a simple and uful displacement controlled model (DCM)to pre-dict the effects of tunnel excavation on adjacent pile foundations.The DCM simulates tunnelling by applying displacements to the tunnel boundary.A method to determine magnitude and direction of tunnel boundary displacements,bad on convergence patterns obrved in field and centrifuge test results,is propod.Back analys of numerous greenfield tunnel ca histories using the DCM indicate good agreement between computed displacement profiles and field/test data.The suitability of the DCM in modelling tunnel–soil–pile inter-action problems is demonstrated through back analysis of a centrifuge test and a field ca study.Ó2006Elvier Ltd.All rights rerved.
Keywords:Tunnelling;Pile foundation;Displacement profiles;Bending moment;Axial force
1.Introduction
Ground movements due to tunnelling activity can be reasonably well predicted using empirical methods (Peck,1969;Attewell and Woodman,1982),or quasi-analytical solutions (Loganathan and Po
ulos,1998).The applicability of the solutions is limited to tunnelling in greenfield sites only and may not be suitable to analy the effects of tun-nelling in denly built-up areas with existing structures,where tunnel–soil–structure interaction plays an important role.Recent studies on the modelling of tunnel–soil–build-
ing interaction (Potts and Addenbrooke,1997;Burd et al.,2000)show significant differences between ttlements that would have been obtained at greenfield sites and at sites with the structure in place.Such complex interaction can be simulated using the finite element (FE)method,in the-ory.However,it is well known that commonly ud finite element modelling techniques,henceforth called force con-trolled models (FCM),which simulate tunnelling by removing nodal forces corresponding to the initial soil stress-state predicts incorrect shape of displacement pro-files (Simpson et al.,1979;Dasari,1996;Leca,1996;Stalle-brass et al.,1996).The predicted ttlement profiles tend to be shallower and wider than field ar field movements are under predicted while far field move-ments are over predicted.This shortcoming can be partly improved by using advanced soil constitutive models,Lee and Rowe (1989),Simpson (1992),Gunn (1993),Bolton et al.(1994),Stallebrass et al.(1994),Dasari et al.(1996),Addenbrooke et al.(1997).As reported by Stallebrass
0886-7798/$-e front matter Ó2006Elvier Ltd.All rights rerved.doi:10.1016/j.tust.2006.08.002
*
Corresponding author.Tel.:+17134314844;fax:+17134316115.E-mail address:chngyih.cheng@ (C.Y.Cheng), (G.R.Dasari),cvechow@nus.edu.sg (Y.K.Chow),cvelcf@nus.edu.sg (C.F.Leung).1
Prently with Maunll Consultants (S)Pte Ltd.2
Prently with ExxonMobil Upstream Rearch Company,Houston,Texas,USA.
/locate/tust
名著阅读Tunnelling and Underground Space Technology 22(2007)
学生成绩怎么查
450–466
Tunnelling and
Underground Space Technology
incorporating Trenchless Technology Rearch
et al.(1996)and Addenbrooke et al.(1997)for plane strain modelling,and for three dimensional NATM tunnelling studies by Dasari et al.(1996),the u of advanced soil models has only resulted in limited success,especially with shape,as farfield displacements were consistently over pre-dicted.Addenbrooke et al.(1997)also reported that,in order to match ground displacement due to tunnelling, unrealistic material anisotropy needs to be assumed.ge
The shape of ground displacement profiles,which deter-mines the degree of curvature,is a critical factor in obtain-ing realistic bending moments in piles adjacent to new tunnels.The limitation of the FCM in predicting the cor-rect shape of surface and subsurface displacement profiles makes it unsuitable for the analysis of tunnel–soil–pile interaction problems.Analytical solutions(Loganathan and Poulos,1998;Park,2004)have been propod to over-come this issue.The analytical solutions as
sume linear elas-ticity and do not take the soil non-linearity into account. Therefore,there is a need for an alternative simple FE modelling technique for modelling the effects of tunnelling.
Thefirst part of the paper prents details of the Dis-placement Controlled Model(DCM),which can predict both realistic soil displacement magnitude and more importantly the correct shape of ground displacement pro-files due to tunnelling.In the DCM,nodes on the tunnel boundary are‘‘pulled’’,to simulate the effects of stress relief,to a predeterminedfinal configuration depending on tunnel cover(distance between ground surface and tun-nel crown)to diameter ratio(C/D t)and anticipated volume loss.The applied displacements in the DCM are computed bad on obrvations of convergence patterns around tun-nels from centrifuge tests andfield cas.The model has been verified by back analysing many centrifuge andfield tunnel ca studies.The potential of the DCM has been explored by back analysing a tunnel–soil–pile interaction centrifuge test.The DCM has also been ud to back ana-ly afield ca study to determine induced axial force and bending moment in a pre-existing instrumented pile due to excavation of twin tunnels.
2.Displacement controlled model
Tunnelling infinite element models is usually simulated by applying forces corresponding to a fraction
of the initial stress-state,to the nodes on the tunnel boundary.The models hitherto will be referred to as the force controlled models(FCM).When the FCMs are ud to simulate tun-nelling,in plane strain or in3D conditions,they commonly result in wider surface ttlement troughs accompanied with higher farfield ttlements thanfield or centrifuge test data.The reason for such prediction lies in the resultant deformation mechanism of the excavated tunnel boundary. It has been obrved that the FCM tend to result in higher than measured invert displacements(Dasari et al.,1996; Stallebrass et al.,1996;Leca,1996).This excess invert heave provides an avenue for soil below the tunnel spring line to experience higher movements.The higher move-ments cau soil in the farfield to be drawn towards the region below the tunnel spring line to satisfy volumetric constancy as shown in Fig.1.This results in excessive far field ttlements,and for a given volume loss,maximum surface ttlement(S max)would be small.The predicted ttlement profiles would be shallower and wider than the field or test measurement.Although displacement patterns can be improved using advanced soil models or by other means,there is a need for a simple approach to predict tun-nelling induced ground movements as discusd in the introduction.
The key for a different approach lies in the displacement convergence pattern around a deforming tunnel boundary. Upon excavation,soil around the unsupported tunnel con-verges inwards in a radial
fashion towards a point on the tunnel vertical line of symmetry.Previously,this pattern of convergence has been ideally assumed to be uniform in the analytical solutions propod by Sagata(1987)as a means of simplifying mathematical derivations.However, it is expected that the tunnel convergence is highly non-uni-form with more crown ttlement and less invert heave. Fig.2shows displacement vector plots of soil deformation around the excavated tunnel for plane strain centrifuge experiments conducted by Mair(1979).The displacement vectors in the tests clearly show large crown ttlement with very little invert heave.Centrifuge tests by Hagiwara et al. (1999)andfield measurements at the Heathrow trial tunnel by Deane and Bastt(1995)also show that the area clo to tunnel invert experienced very little movement compared to the crown.The above obrvations lead to thefirst assumption in the displacement controlled model(DCM) that convergence is non-uniform.Loganathan and Poulos (1998)and Park(2004)reported that such non-uniform convergence profiles lead to realistic predictions of ground displacements due to tunnelling.
The cond assumption for the DCM is that deformed tunnel shape is similar to the original excavated shape. Such an assumption is justified as deformations are usu-ally small compared to tunnel size under working condi-tions.The third assumption for the DCM is that there exists a single point on the tunnel vertical line of symme-try to which all nodes on the excavated tunnel boundary co
书面表达nverge to.There have been numerous studies,which propo that soil displacement vectors of the excavated tunnel boundary converge to the tunnel centre(Attewell et al.,1978;O’Reilly and New,1982),0.175/0.325z o below tunnel axis level(Taylor,1995)or towards a point on the tunnel invert(Deane and Bastt,1995).The lat-ter is propod bad onfield data while the former two are derived bad on the following well established empirical relations:
S¼S max e
Àx2
2i2
ð1Þi¼Kðz oÀzÞð2ÞS h¼
x
z o
Sð3Þ
C.Y.Cheng et al./Tunnelling and Underground Space Technology22(2007)450–466451
Various symbols in the above equations are defined under notations at the end of the paper.Grant and Taylor(2000) reported that if ground movements converge to a single
focus point and constant volume conditions apply,the movements must theoretically converge to the interction of the tangent to the distribution of i with depth and the vertical centre line of the tunnel as prented in Fig.3. Studies by Mair et al.(1993),Taylor(1995),and Grant and Taylor(2000)show that the variation of trough width
e bookFig.2.Displacement vectors of(a)Mair’s(1979)and(b)Hagiwara et al.
(1999)plane strain centrifuge tests.
Fig.3.Variation of characteristic i distance and convergence point with
depth(Grant and Taylor,2000).
452 C.Y.Cheng et al./Tunnelling and Underground Space Technology22(2007)450–466
parameter(K)with depth is reasonably reprented by Eq.
(4),thus indicating that the soil mass displacement vectors above the tunnel crown converge to a point0.175/0.325z o below tunnel axis level
K¼0:175þ0:325ð1Àz=z oÞ
1Àz=z o
ð4Þ
However,Eq.(4)is not valid within an estimated0.5D t dis-tance above the tunnel crown and0.5D t distance below the ground surface as obrved in centrifuge studies by Grant and Taylor(2000),e lines reprented by‘a’and‘c’in Fig.3.At depths clor to the tunnel axis level,it was ob-rved that the gradient of i with depth(d i/d z)decreas as illustrated in Fig.3.This would imply that the focus point is moving upwards,on the vertical centreline,from0.175/ 0.325z o below tunnel axis level to the tunnel centre,as the depth increas from the line‘c’in Fig.3.Bad on the above derivations,it would be realistic to assume that the point of convergence for soil on the excavated tunnel boundary is located somewhere between tunnel centre and the invert as shown in Fig.4.
For a tunnel excavated in an infinite very deep tunnel),the point of convergence for soil around the tunnel boundary is the tunnel centre.As the tunnel becomes ver to tunnel diameter(C/D t) ratio decreas,the convergence point is expected to shift downwards within the bounds of the tunnel centre and the invert.The point of convergence may not be below the invert as this would mean there is ttlement rather than heave at the invert.Thus the restriction of the conver-gence point to lie between tunnel centre and invert is justi-fied.The location of the convergence point in this study is controlled by an assigned beta(b)value which when mul-tiplied by tunnel radius(R),reprents the vertical distance between the tunnel centre and the propod convergence
point.The distance b R influences the direction of tunnel boundary displacement vectors(Fig.4).A b parameter value equal to zero implies that the focus point is at the tunnel centre(deep tunnel),and b value equal to1.0means the focus point is at invert(shallow tunnel).The value of b determines the directions of excavated boundary displace-ments,which in turn influence the surrounding ground dis-placements.Bad on the back analys of veral tunnels, the variation of b with C/D t ratio is found to be consistent and is prented later in this paper.
叫醒服务3.Analysis procedure and soil constitutive models
One of the key input parameters for the DCM is volume loss,which is usually measured and also can be estimated bad on tunnel construction method,quality of workman-ship and soil properties.The volume loss is distributed non-uniformly around the tunnel boundary due to the high crown to invert displacement ratio assumed.First,the direction of nodal displacements is determined from an assumed value of b,which defines the direction of displace-ment vectors.Using the volume loss and the known direc-tions of displacement vectors,the nodal displacement magnitudes are then computed.Lastly,the computed dis-placements are applied to nodes along the excavated tunnel boundary to simulate the convergence due to stress relief.
In all the analys prented here,invert displacement has been assumed to be zero,which is justified bad on the results in Fig.2.All analys reported in this paper were carried out using the FE software ABAQUS version 5.8.Undrained analys were performed for all the ca
C.Y.Cheng et al./Tunnelling and Underground Space Technology22(2007)450–466453
studies as only short-term displacements were back ana-lyd.The initial vertical stress in the model was obtained using the lf-weight of soil and horizontal stress were computed using K0.The soil model ud for all analys is non-linear elastic with no failure criterion.The u of such constitutive model is justifiable as deformations in most tunnels are small and soil does not reach failure con-dition(Dasari,1996).The model accounts for the non-linear stiffness degradation with strain.A simple power function,Eq.(5),which requires two input parameters,A and n was ud to model the degradation of shear stiffness (G)with deviatoric strain(e q).
G¼Ap0e n
q
ð5ÞThe parameter n governs the rate of stiffness degrada-tion with deviatoric strain beginning from the value
e qðG
maxÞto e qðG
minÞ
.A more negative n value implies a steeper
drop in shear stiffness with deviatoric strain.The parameter A and mean effective stress(p0)governs the magnitude of the shear modulus.Shear modulus data from triaxial tests fitted with bender elements and local strain gauges have been ud to derive the material parameters.London clay (Jardine et al.,1991)and Kaolin clay(Viggiani and Atkin-son,1995)data have been ud as shown in Fig.5to obtain a reprentative n value.Bad on extensive experimental work on the small strain stiffness of kaolin clay by Viggiani and Atkinson(1995),an average G max/p0ratio of550was lected for Kaolin clay.Addenbrooke et al.(1997)com-piled shear modulus data for London clay and reported that G max/p0is about400.It is also assumed that shear modulus is constant at G max below the minimum deviatoric strain of0.001%.Using the G max/p0and the minimum devi-atoric strain,the value of A in Eq.(5)can be computed and is shown in Table1.The shear modulus above1%devia-toric strain was assumed to be constant and equal to the value at1%strain.
4.Back analysis of tunnelling ca studies
Two tunnel ca histories,one each in London Clay and Kaolin clay,are reported here to asss the suitability of the DCM in predicting the ground displacementfield around tunnels of varying C/D t ratio
s.Volume loss per-centages ud in the analysis of the respective ca studies are the same as the values from thefield obrvations. Details of the ca histories are given in Table2.The com-puted displacements are compared withfield or test data and previous studies(analytical or numerical)whenever possible.复仇revenge
Table1
Values of soil constants ud for analys
长裙的英文Soil e qðG
max Þ
e qðG
min
速度与激情7字幕文件Þ
A n
London clay0.001%1%0.40À0.5
Kaolin clay0.001%1%0.55À0.6 454 C.Y.Cheng et al./Tunnelling and Underground Space Technology22(2007)450–466
4.1.Ca I:Heathrow trial tunnel
Prior to construction of the Heathrow express tunnel, three types of NATM trial tunnels were constructed in stiffLondon Clay to asss the nsitivity of ground displace-ments to excavation quence in order to minimi effects on major structures.Here,the Type2construction quence(excavation of right hand drift after left hand drift)has been back analyd.The oval shaped tunnel was idealid as a circular excavation with equivalent area of60m2.The reported undrained volume loss of1.06% was ud for the analysis which translates to46mm of tun-nel crown displacement.The b value of0.lor to tunnel invert)was obtained by trial and error,and this value is consistent with obrvations by Deane and Bastt (1995),who showed that the convergence point is clo to the invert.
Computed surface ttlement profiles are compared with field data in Fig.6.There is a good agreement betweenfield data and computed results in terms of magnitude and shape of the profile.Although the maximum ttlement is slightly under predicted,farfield ttlements are negligible coupled with a realistic narrow trough width.Results com-puted using the quasi-analytical expression by Loganathan and Poulos(1998)are also shown in Fig.6.Although the same volume loss of1.06%was ud in Loganathan and Poulos expression,volume swept by their predicted surface ttlement profile is less than1.06%.This suggests that undrained conditions were not satisfied in the
Loganathan and Poulos solution.The major advantage of the DCM over Loganathan and Poulos(1998)analytical solution is that pile foundations can be included in thefinite element model,and realistic soil–pile interaction can be modelled as shown later.
Horizontal displacement profiles at different offts from the tunnel centre(Fig.7)also yield a reasonably goodfit withfield data above the tunnel spring line.Maximum computed horizontal movement compares well withfield obrvations and the location is slightly above the tunnel spring line.Below the tunnel spring line,computed dis-placements are obrved to diminish at a rapid rate with depth.Horizontal displacements computed using the
Table2
Geometrical and volume loss details for back analyd tunnels
Ca Volume loss
V l(reported)Volume loss
V l(this study)
Diameter
D t(m)
Cover to diameter
ratio C/D t
References
I.Heathrow Trial Tunnel (Type2)1.06% 1.06%8.74m 1.92Deane and Bastt(1995),
Atzl and Mayr(1994)
II.Loganathan’s Centrifuge
(Test3)
1%1%6m3Loganathan et al.(2000)
C.Y.Cheng et al./Tunnelling and Underground Space Technology22(2007)450–466455
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