Analysis of the Interaction between the Lining of a TBM Tunnel and the Ground Using the Convergence-Confinement Method

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1 Ameican Jounal of Applied Sciences Oiginal eseach Pape Analysis of the Inteaction between the Lining of a TBM Tunnel and the Gound Using the Convegence-Confinement Method Piepaolo Oeste Depatment of Envionmental, Land and Infastuctual Engineeing, Politecnico di Toino (Italy, Coso Duca degli Abuzzi 4, I-09 Toino Italy Aticle histoy eceived: evised: Accepted: piepaolo.oeste@polito.it Abstact: The suppot used when a tunnel is excavated by a shielded TBM is complex because it consists of two diffeent mateials: A concete segmental lining and a fille annulus between the lining and the tunnel wall. A detailed analysis of the behavio of such a suppot system equies a thee-dimensional numeical modeling and should also conside the pesence of the TBM machine. This aticle pesents an analytical calculation pocedue that allows to assess the inteaction between the suppot system and the tunnel using the convegence-confinement method and Vlachopoulos-Diedeichs method. Futhemoe, it is defined the oveall stiffness of the suppot system stating fom the detailed analysis of the stesses and stains developed in the two constituent mateials. Fom the analysis of the evolution of the adial displacements in the longitudinal section, it was then possible to evaluate anothe fundamental paamete: The adial displacement of the tunnel wall at the point of installation of the suppot system. The computational pocedue was then applied to a specific case, showing the influence of the stiffness of the filling mateial on the final loads acting on the segmental lining. Keywods: Tunnel Stability, Deep Tunnel, Cicula Tunnel, Convegence- Confinement Cuve, Tunnel Suppot, Stess and Stain Aound the Tunnel, TBM, Shielded TBM, Segmental Lining, Filling Mateial, eaction Line, Suppot Stiffness Intoduction Often nowadays tunnels ae excavated though the use of machines TBMs (Benato and Oeste, 05. The constuction of a tunnel with shielded TBM machines equies the ealization of a concete lining (segmental lining in the tail of the machine, at a cetain distance fom the excavation face (0- m. Such a lining is put into opeation within the shield and the hollow space between the tunnel wall and the lining extados is geneally filled with gavel mateial, sometimes cemented. The suppot system is, theefoe, epesented by a less igid oute ing (ing composed by natual o cemented gavel and an inne ing of concete (the segmental lining (Fig.. The assessment of the oveall stiffness of the suppot system is vey impotant because this stiffness influences the loads acting on the suppot itself: The highe the stiffness, the geate the loads which act on the lining. Futhemoe, it appeas to be vey impotant to know the adial displacement u,in of the tunnel wall in the moment in which the lining is inseted. This adial displacement also has effect on the final applied loads. A thoough analysis of the inteaction between a tunnel excavated by shielded TBM and segmental lining can be developed with the numeical modeling, pefeably of thee-dimensional type (Do et al., 04a; 04b. This modeling tuns out to be quite complex and computation times ae typically high. Fo this eason, in this study an analytical pocedue fo the assessment of the inteaction between the lining and the tunnel, when a shielded TBM machine is used, is pesented. The analytical methods ae widespead in tunnelling and help solve poblems of geat impotance in elation to the stability aound the tunnel and on the excavation face and to the design of the suppot and einfocement stuctues (Oeste, 009; 0; Osgoui and Oeste, 007. The pocedue is based 05 Piepaolo Oeste. This open access aticle is distibuted unde a Ceative Commons Attibution (CC-BY.0 license.

2 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp on the convegence-confinement method and uses the esults of the studies developed by (Vlachopoulos and Diedeichs, 009 fo the evaluation of the longitudinal pofile of the adial displacements of the tunnel wall. The convegence-confinement method is an analytical calculation method widespead in the tunneling field (ibacchi and iccioni, 977; Panet, 995; AFTES, 99. It allows to study the behavio of a tunnel in a simple and intuitive way, obtaining the evaluation of the elationship between the applied intenal pessue and the adial displacement of the wall. The method appeas to be fast and analyzes the inteaction between the tunnel and the suppot though the intesection of the chaacteistic cuve of the tunnel with the eaction line of the suppot. Pecisely fo the speed of calculation and fo the accuacy of the esults, this method was also used in the past fo studies of pobabilistic (Oeste, 005 and back-analysis type. The eaction line of the suppot is, theefoe, essential to be able to analyze the behavio of a segmental lining in tunnels excavated by a shielded TBM. Fo this eason will be evaluated in the following the stiffness of the composite suppot system consisting of the segmental lining and of an oute ing of fille mateial. Futhemoe, in ode to position with some pecision the eaction line with espect to the chaacteistic cuve of the tunnel, it is necessay to undestand the evolution of the adial displacements of the wall in the longitudinal section. The theoetical pofile of (Vlachopoulos and Diedeichs, 009 will be modified to take into account the pesence of the TBM shield. Though the developed pocedue will be possible to aive at estimating the load acting on the segmental lining and, theefoe, define the value of its thickness and the steel einfocement necessay to ensue the stability of the tunnel. Mateials and Methods The convegence-confinement method expects to obtain the intesection between the chaacteistic cuve of the tunnel (convegence-confinement cuve and the eaction line of the suppot (Fig.. The slope of the eaction line with espect to the hoizontal is a function of the stiffness of the suppot system sostegno k sys (tgϑ = k sys. In ode to popely evaluate the stiffness of the composite system consisting of the segmental lining and the oute ing of fille mateial, it is necessay to analyze in detail the stesses and stains developed in the two mateials, when the tunnel wall has adial displacements subsequent to the lining installation (u -u,in (Fig.. Fig.. The chaacteistic cuve of the tunnel (convegence-confinement cuve and the eaction line of the suppot. The intesection point is used to detemine the final load acting on the suppot (σ,eq and the final displacement of the tunnel wall (u,max. Key: u,in : Displacement of the tunnel wall at the time of installing the suppot; σ : adial pessue applied on the tunnel wall; u : adial displacement of the tunenl wall; p 0 : Lithostaticstess to the depth of the tunnel; ϑ: Slope of the eaction line of the suppot with espect to the hoizontal (tgϑ = k sys, with k sys the stiffness of the suppot system 77

3 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp Fig.. Composite suppot system when the tunnel is excavated by a shielded TBM. The oute ing is fomed by the filling mateial (mateial, the inne ing is the segmental lining (mateial. The extados of the oute ing coincides with the tunnel wall, having a adius ; t g : Thickness of the oute ing; t c : Thickness of the inne ing Fo the adial symmety, the adial displacement u in the two ings is expessed by the following geneal fomulation, in function of the geneic distance fom the cente of the tunnel (ibacchi and iccioni, 977: B u = A + ( Because the shea defomation ε ϑ is the atio u/ and the adial defomation ε is the deivative of u with espect to (ε = du/d, we have: B ε ϑ = A + ( B ε = A ( Futhemoe, fo the adial symmety in a field of plan defomations, one can wite the following elations between stesses and stains: σ = C ε + D ε (4 ϑ ϑ σ = C ε + D ε ϑ (5 Whee: ν ν C = E D = E ( ν ( + ν ( ν ( + ν E = The elastic modulus of the mateial; ν = The Poisson atio of the mateial. (6 The following bounday conditions of the suppot system ae posed: adial displacement on the extados of the oute ing (mateial equal to: (u -u,in fo =, whee u,in is the displacement of the tunnel wall at the time when the lining is installed and u is the geneic displacement of tunnel wall adial stess σ at the inteface between the two mateials ( = -t g, equal fo the two mateials adial displacement u at the inteface between the two mateials ( = -t g, equal fo the two mateials Nil adial stessσ at the intados of the second mateial: σ = 0 fo = (-t g -t c Fo condition one can wite the following elation (fom Equation : B u = ( u u, in = A + (7 Fom which: ( in B = u u, A (8 Fo the second condition one can wite the following elation (fom Equation -5: B B C A + D A + ( tg ( tg B B = C A + D A + ( tg ( tg (9 whee, A and B, A and B : A and B paametes, espectively fo mateial andmateial ; C and D, C and D : C and D paametes, espectively fo mateial and mateial ; t g is the thickness of extenal annulus (the gavel statum, mateial ; this thickness is 78

4 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp typically equal to the thickness of the shield in the tail zone t sh, but may be highe when thee is a significant ove beak by the TBM head (this aspect will be discussed in moe detail below. Fo condition one can wite the following elation (fom Equation : χ ω ρ A = ξ + ρ u u ( in, ω ξ C χ ω ρ A = u u ( in, ω ξ C (5 (6 B B A ( tg + = A ( tg + (0 t t ( g ( g Fom which, substituting Equation 8 to B : ( g ( g ( in B = A t A t + u u, ( Fo Condition 4 one can wite the following elation (fom Equation -5: B B C A + D A + = 0 ( ( tg tc ( tg tc whee, t c is the thickness of the inne annulus (concete lining, mateial. Substituting Equation in Equation, afte a few steps we get: ( in A = ξ A + ρ u u (, Whee: ( D C D C C D ρ ( ( α β ( ( α β α β ξ = = D C D C ( t g β ( tg tc ( tg tc α = = eplacing Equation 8 and in Equation 9, afte some steps we get: ( ω A = C A + u u (4 Whee: χ, in ( ( ( ( ω = C + η + D η + C η D η η χ = ( D C + C D η = t ( g Solving the system composed of the Equation and 4 in the two unknowns A and A, we get: The stiffness of the suppot system K sys is given by the atio between the adial stess σ on the tunnel wall ( = and the displacement at that point: k σ sys = = = B B C A D + A + ( u u, in ( u u, in ( D C ( C + D A + B ( u u, in (7 Fom which, by eplacing the Equation 5 and 8 in Equation 7, afte a few steps we get: χ ω ρ ksys = C ξ + ρ + D C ( ω ξ C (8 The othe fundamental paamete in ode to daw the eaction line of the suppot of Fig. is the adial displacement of the tunnel wall at the moment in which the suppot is installed (u,in (Oeste, 009; 0. Accoding to Vlachopoulos and Diedeichs (009 adial displacements u have the following evolution (cuve in Fig. : pl ( σ, eq x 0.5 u = u,max e e fo negative ahead of the excavation face ( ( plσ x, eq 0.5 pl ( σ, eq u = u,max e e fo positive (behind the excavation face (9 (0 whee, pl (σ,eq : Plastic adius of the tunnel in the pesence of an intenal pessue equal to the final pessue (σ,eq applied by the suppot stuctue to the tunnel wall. In the absence of ove beak, the pesence of a shielded TBM involves blocking of the adial displacements immediately behind the TBM head, in coespondence of the excavation face. Fo the whole length L of the TBM shield adial displacements of the tunnel wall ae stationay equal to the u,in value which is pesent in coespondence of the excavation face (cuve of Fig. : 79

5 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp u, in = u,max e ( σ, pl eq 0.5 ( The distance d fom the face at which the tunnel wall comes into contact with the shield is given by the following Equation 4: It can be assumed that fo x>l the tend of u emains the same as the cuve, shifted by L along the x axis. In pactice, the effect of the shielded TBM is to feeze the adial displacements along the whole of the TBM shield (fom x = 0 to x = L and to tansfe the position of the face at the installation point of the lining. δ in Fig. epesents the pevented displacement by the pesence of the TBM shield: ( plσ x, eq 0.5 pl ( σ, eq δ = u,max e e ( To this displacement coesponds a adial pessue p acting on the extados of the shield: This pessue can be evaluated by applying the displacement δ to the chaacteistic cuve of the tunnel (Fig. 4. Since both δ and p vay along the shield (Fig., the value of p along x (fom x = 0 to x = L can be integated in ode to evaluate the total fictional foce on the inteface shield-tunnel wall that it is necessay to win duing the TBM advancement. Instead, if the TBM head ealizes an ove beak, i.e., the tunnel adius is geate than the shield adius sh ( = - sh, the pofile of the adial displacements is given by the cuve of Fig. and the initial displacement u,in is given by the following equation, which eplaces the Equation : pl ( σ, eq 0.5 u, in = u,max e + ( d = pl u ( σ, ln eq ( σ,,max e pl eq 0.5 (4 Then, in the pesence of an ove beak is detected: An incease in the value of u,in ; it has the effect of educing the final load σ,eq on the lining (Fig. A eduction in the value of δ; it esults in a eduction of the pessue applied by the tunnel wall to the shield extados If then, due to a high ove beak, d tuns out to be geate than the length L, the tunnel wall doesn t come in contact with the shield and on it is not applied any adial pessue (except the eaction to the own weight; the aveage thickness of the oute ing of the filling mateial t g is no longe simply given by the thickness of the shield in the tail zone t sh, but by the following Equation 5: pl ( σ, eq 0.5 e tg = tsh + u,max L pl ( σ, eq e (5 Fig.. Longitudinal pofile of the adial displacements of the tunnel wall. Key: : Cuve obtained fom the equations of Vlachopoulos and Diedeichs (009 (Equation 9 and 0; : Modified cuve to account fo the pesence of a shielded TBM in the absence of ove beak of the head; : Cuve modified to take into account the pesence of a shielded TBM with ove beak of the head equal to ; δ: Displacement pevented by the shield TBM which has as a consequence that a adial pessue is applied on the oute suface of the shield; L: Length of the TBM shield 80

6 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp Fig. 4. Evaluation of the pessue p that the tunnel wall applies to the TBM shield duing its advancement, stating fom the pevented displacement δ of Fig.. Key: F: Point of the tunnel chaacteistic cuve which has a adial displacement u,in In this case, u,in is obtained by the following equation, which eplaces the Equation o : u, in = u,max e pl 0.5 ( σ, eq ( pl σ L, eq 0.5 pl ( σ, eq + u,max e e (6 The incease of the aveage thickness of the oute ing of filling mateial (t g involves a eduction of the total stiffness of the suppot system k sys (Equation 8, with additional effects on the eduction of the final load on the concete lining. Since both u,max and σ eq depend on u,in and also u,in (Equation, o 6 depends in tun on u,max and σ eq, it is necessay to poceed iteatively, stating fom an initial value of u,in equal to 0, to obtain the convegence of the following thee values: u max, σ eq, u,in. Once obtained u,max and u,in one can detemine the load acting diectly on the concete lining (σ at = -t g in the following way: eplacing u,max to u in Equation 5 and 8, obtaining the A and B paametes The adial stess σ at = -t g is then given by the following equation (stating fom Equation 5, and : σ B B = t = C A + D A + ( tg ( tg ( g (7 esults The pocedue descibed in paagaph has been applied to the case of a cicula tunnel of adius m, excavated in a ock mass of medium geomechanic quality with the following chaacteistic paametes: Elastic modulus E = 844 MPa, Poisson's atio ν = 0., cohesion c = 0.9 MPa, fiction angle ϕ = 5, dilatancy Ψ = 5. The lithostaticstess p 0 to the depth of the tunnel is 5 MPa. The thickness of the shield in the tail zone t sh is equal to 0 cm. The thickness of the concete lining t c is equal to 0. m. The mechanical chaacteistics of the used concete ae the following: E = 8000 MPa, ν = 0.5. It is assumed the use of natual o injected gavel as a fill of the oute ing of the suppot system, with an elastic modulus anging fom 00 MPa to 4000 MPa (Poisson's atio of the filling mateial was assumed equal to 0.. It is deemed null the ove beak ( = 0 and, theefoe, the tunnel wall comes into contact with the shield immediately behind the excavating head (d = 0. Figue 5 shows the tend of the oveall stiffness of the suppot system (k sys vaying the elastic modulus of the fill mateial. In Figue 6 is shown the tend of the load acting on the segmental lining (σ at = -t g vaying the elastic modulus of the fill mateial. The chaacteistic cuve which is obtained fom the calculation fo an elastic modulus of the filling mateial equal to 000 MPa (k sys = MN/m, is shown in Fig. 7. 8

7 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp Fig. 5. Tend of the oveall stiffness of the suppot system (ksys vaying the elastic modulus of the mateial constituting the filling of the oute ing to the segmental lining Fig. 6. Tend of the adial load on the extados of the segmental lining ( = -tg vaying the elastic modulus of the mateial constituting the filling of the oute ing to the segmental lining Fig. 7. Tunnel convegence-confinement cuve (chaacteistic cuve of the tunnel and eaction line of the suppot system, in the studied case and fo elastic modulus of the fille mateial equal to 000 MPa (k sys = MN/m 8

8 Piepaolo Oeste / Ameican Jounal of Applied Sciences 05, (4: 76.8 DOI: 0.844/ajassp Discussion Fom the analysis of Fig. 5 can be detected as the elastic modulus of the mateial filling the inte space outside the segmental lining has a cetain influence on the oveall stiffness of the suppot system, in paticula fo elatively low elastic moduli. With an incease of the elastic modulus of the filling mateial, the incease in the oveall stiffness of the suppot system is attenuated, with a tendency to an asymptotic value. The elastic modulus of the fill, pesents, instead, a significant influence on the load acting on the segmental lining, even fo high values of the elastic modulus (Fig. 6. Inceasing the elastic modulus of the fill mateial, the load on the segmental lining tends to decease significantly. It can, theefoe, be vey convenient to be able to intevene on the filling mateial though the injection of cement mota, theeby inceasing its elastic modulus, in ode to contain the loads on the segmental lining. This would esult in a eduction of the thickness of the segmental lining, with significant opeational and economic advantages and/o of incidence of steel einfocing, which would be necessay to povide in its inteio. Conclusion The suppot of a tunnel excavated using a shielded TBM machine equies the pesence of two diffeent mateials: The concete segmental lining mounted in the tail of the TBM shield and a filling mateial inseted in the inte space between the segmental lining and the tunnel wall. In ode to effectively analyze the inteaction between this suppot system and the tunnel, a computational pocedue that is based on the convegence-confinement method and on the studies of Vlachopoulos and Diedeichs (009 was developed. It was possible, though a detailed analysis of the stesses and stains that develop in the two mateials of the composite suppot system, to obtain the oveall stiffness useful to daw the suppot eaction line of the convegence-confinement method. The study of the evolution of the adial displacements along the longitudinal section of the tunnel, in the pesence of a shielded TBM, allowed to obtain the equations able to calculate the u,in displacement in the wall at the point whee the suppot system is installed. The calculation elating to a specific case has allowed to undestand the impotance of the stiffness of the filling mateial on the oveall stiffness of the system and on the loads which affect the segmental lining. efeences AFTES, 99. Goupe de tavail n.7- Soutenementetevetement, Emploi de la méthode convegence-confinement, Tunnels et ouvagessouteains, Supplément au n.7, maj-juin. Benato, A. and P. Oeste, 05. Pediction of penetation pe evolution in TBM tunneling as a function of intact ock and ock mass chaacteistics. Int. J. ock Mechanics Min. Sci., 74: 9-7. DOI: 0.06/j.ijmms Do, N.A., D. Dias, P. Oeste and I. Djean-Maige, 04a. Thee-dimensional numeical simulation of a mechanized twin tunnels in soft gound. Tunnell. Undegound Space Technol., 4: DOI: 0.06/j.tust Do, N.A., D. Dias, P. Oeste and I. Djean-Maige, 04b. Thee-dimensional numeical simulation fo mechanized tunnelling in soft gound: The influence of the joint patten. Acta Geotechnica, 9: DOI: 0.007/s Oeste, P., 005. A pobabilistic design appoach fo tunnel suppots. Comput. Geotechn., : DOI: 0.06/j.compgeo Oeste, P.P., 009. Face stabilisation of shallow tunnels using fibeglass dowels. Poc. Institut. Civil Eng. Geotechnical Eng., 6: DOI: 0.680/geng Oeste, P., 0. Face stabilization of deep tunnels using longitudinal fibeglass dowels. Int. J. ock Mechanics Min. Sci., 58: DOI: 0.06/j.ijmms Osgoui,.. and P. Oeste, 007. Convegence-contol appoach fo ock tunnels einfoced by gouted bolts, using the homogenization concept. Geotechn. Geolog. Eng., 5: DOI: 0.007/s Panet, M., 995. Le Calcul Des Tunnels Pa La Méthode Convegence-Confinement. st Edn., Pesses de l écolenationale des Pontsetchaussées, Pais, ISBN-0: , pp: 78. ibacchi,. and. iccioni, 977. Stato di sfozo e defomazione intono ad una galleia cicolae. Galleie e Gandi Opee Sotteanee, 5: 7-8. Vlachopoulos, N. and M.S. Diedeichs, 009. Impoved longitudinal displacement pofiles fo convegence confinement analysis of deep tunnels. ock Mech. ock Eng., 4: -46. DOI: 0.007/s Acknowledgement This eseach was pefomed with use of the equipment of the DIATI Depatment of Politecnico di Toino. 8