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SEISMIC DESIGN OF CIRCULAR TUNNELS: NUMERICAL VALIDATION OF CLOSED FORMED SOLUTIONS Vasilis AVGERINOS1. Bending moment and thrust in the tunnel lining, when it is assumed that there is full-slip between the soil. (Hashash, et al 2001, Kontoe et al, 2008, Park et al.
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
AN EARLY-STAGE DESIGN PROCEDURE FOR CIRCULAR TUNNEL LINING UNDER SEISMIC ACTIONS 1
1
1
2
E. Bilotta , G. Lanzano , G. Russo , F. Santucci de Magistris and F. Silvestri
1
1
2
DIGA, University of Napoli Federico II, Naples, SAVA, Structural and Geotechnical Dynamics Laboratory, University of Molise, Campobasso, Italy Email: [email protected]
ABSTRACT : The increments of the internal forces induced by an earthquake in the transverse section of a tunnel lining can be ascribed to the ovalisation of the section, induced by soil shear straining in the vertical plane. They can be assessed with several procedures at different levels of complexity. In this paper, two kind of analysis were per-formed on idealised geometry and soil conditions, considered representative of soil classes specified by Euro-code 8: pseudo-static analysis, where the seismic input was reduced to an equivalent peak strain amplitude, computed through a free-field pseudo-static analysis of the ground and then considered acting on the tunnel lining in static conditions; and full dynamic analysis, where the soil and tunnel responses were mechanically coupled and modelled by using FEM. Both were performed considering the soil as an equivalent linear medium. On the basis of the comparison of the results of both approaches, modification factors of the usual pseudo-static formulae are proposed, which take into account the kinematic interaction between the tunnel and the ground during shaking. The method, based on the use of simple charts, can be easily adopted for early-stage design.
KEYWORDS:
tunnels, lining, seismic actions, pseudo-static, dynamic
th
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
1. INTRODUCTION Civil infrastructures and lifelines in seismic areas need to be designed to support the extra loading produced by earthquakes. Some indications for such design can be found in Owen & Scholl (1981), JSCE (1992), AFPS/AFTES Guidelines (2001), ISO TC 98 (2003). Design rules for tunnels are not introduced in Eurocode 8 (EN 1998-5, 2003). This maybe because earthquake effects on underground structures were deemed to be negligible, in spite of the different evidences from several case-histories (see for instance Lanzano et al., 2008). Research activities are in progress in Italy to refine the design methods for tunnels under seismic actions (e.g. Bilotta et al., 2007). The shear waves propagating during an earthquake perpendicularly to the tunnel axis, result in a distortion of the cross-section of the structure: in this paper a procedure to calculate the forces induced by ground shaking in the tunnel lining in simple subsoil conditions is illustrated. Such simplified procedure incorporates the results of finite elements dynamic analyses, which consider the kinematic interaction between the tunnel lining and the ground, in the framework of the pseudo-static approach commonly adopted for early-stage design. Three idealized ground conditions (Fig. 1) were considered: a 30 m thick layer of soft clay, medium dense sand or gravel, overlying a compliant rock bedrock (Vr= 800 m/s, γ=22 kN/m3, D0=0.5%). The tunnel has the following characteristics: circular shape with reinforced concrete lining (variable thickness from 0.1 to 1.3 m, diameter D=6 m); axis depth z0=15 m; VS (m/s) 0
200
400
600
0
5
12 m 10
z(m)
t = 0.3 m 6m
argilla clay (D)(D) sabbia sand (C)(C) ghiaia (B) (B) gravel
15
20
(riv. Cls Rbk concrete lining450 Rckkg/cmq) =45 MPa
12 m 25
30
bedrock rock Bedrocksoft roccia tenera
VS = 800 m/s
3
γ = 22 kN/m
D0 = 0.5%
Figure 1 - Ground conditions
The values of small strain soil parameters have been chosen according to literature empirical relationships linking the shear modulus (G0) and the damping ratio (D0) to the lithostatic stress, the void ratio and intrinsic soil properties, such as particle size and plasticity index IP (Santucci de Magistris, 2005; d’Onofrio & Silvestri, 2001). The profiles of VS with depth adopted for each soil type are shown in Fig. 1, where the dashed lines represent the value of the so called ‘equivalent velocity’ VS,30 (EN 1998-1, 2003). Table 1 summarizes the geotechnical parameters and the ground type according to EC8.
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
Table 1: Ground parameters and classification according to EC8 Ground type φ’ IP γ D0 VS,30 (°)
(%)
(kN/m3)
(%)
(m/s)
Clay
D
25
30
18
2.5
125
Sand
C
35
-
20
1.0
240
Gravel
B
45
-
21
1.0
400
1
25
0.8
20
0.6
0.4
clay stiffness sand stiffness gravel stiffness clay damping sand damping gravel damping
15
10
0.2
0 0.0001
damping ratio, D (%)
normalised shear stiffness, G/Go
Soil non-linearity and cyclic energy dissipation were taken into account through an equivalent linear approach which considers the variation of the shear modulus G and the damping ratio D with the shear strain γ. Therefore, the curves G(γ)/G0 and D(γ) for the three materials (Figure 2) have been assumed according to literature indications (Vucetic & Dobry, 1991; Stokoe, 2004).
5
0 0.001
0.01 shear strain, γ (%)
0.1
1
Figure 2 - Variation of shear modulus and damping with shear strain level
2. 2 PSEUDO-STATIC VS FULL DYNAMIC CALCULATION In the usual simplified methods the kinematic soil-structure interaction is neglected as the free-field displacements are applied to the tunnel boundary (e.g. Hashash et al., 2001) and the seismic force increments in the lining are calculated by means of the closed-form elastic solutions (Wang, 1993) for a tunnel surrounded by a homogeneous and isotropic half-space, using the average shear deformation γPS of the ground at the tunnel depth as input: 1 π N (θ) = ± K 2Gm Dγ PS cos 2 θ + 2 4 M (θ) = ± where:
1 π K1Gm D 2 γ PS cos 2 θ + 12 4
(1a) (1b)
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
K1 =
12(1 −ν m ) 2 F + 5 − 6ν m
(2a)
1 2 F[(1 − 2ν m ) − (1 − 2ν m )C] − (1 − 2ν m ) + 2 2 K2 =1 + 5 F[(3 − 2ν m ) + (1 − 2ν m )C] + C − 8ν m + 6ν m2 + 6 − 8ν m 2
(2b)
The dimensionless parameters F and C represent the relative soil/tunnel stiffness and refer to a tunnel with diameter D, lining thickness t, and elastic parameters El and νl, in an elastic ground (Gm and νm):
(
)
(
)
G m 1 − ν l2 D 3 2El t 3 G 1 −ν l2 D C= m E l t (1 − 2ν m ) F=
(3) (4)
Based on the equilibrium of a deformable soil column from the surface to a given depth z, several procedures can be adopted to evaluate the average shear deformation γPS, as discussed in Bilotta et al. (2007). A first class of methods is based on the specification of a vertical profile of peak acceleration amax. Then, the maximum shear stress τmax is computed by integration as: z
τ max ( z ) = ∫ ρamax ( z )dz
(5)
0
Another class of methods follows an approach similar to those adopted for the evaluation of liquefaction susceptibility based on simplified procedures to define the seismic induced shear stress profile. Hence, the shear stress distribution with depth is calculated according to the following equation:
τ max ( z ) = rd ( z )
a max, s g
σ v (z)
(6)
In Eq. (6), σv is the total vertical stress, and rd is a reduction parameter which takes into account the deformability of the soil column. Several empirical relationships (Iwasaki et al., 1978; Liao & Whitman, 1986; Power et al., 1996; Idriss & Boulanger, 2004) to define rd are reported in literature. All the considered pseudo-static methods require a preliminary evaluation of the peak acceleration at surface. In the paper this value has been computed as:
a max, s = S ⋅ a g
(7)
where a g is the peak acceleration on outcropping rock site and S the site response factor. Its value has been originally specified by EC8 part 1 (EN 1998-1, 2003 ), followed by several proposed of updating (i.e. Italian OPCM 3274, 2003; ETC12, 2006) for each ground type (Table 2). In this paper, the average value has been assumed. Alternatively, a non-linear response factor can be used, varying with the ground motion amplitude, as proposed by Ausilio et al. (2007).
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
Table 2: Response factors in pseudo-static methods EC8 OPCM ETC12 average 3274 Clay 1.35 1.35 1.1 1.27 Sand 1.15 1.25 1.15 1.18 Gravel 1.2 1.25 1.3 1.25 Soil
The maximum shear strain at a depth z is therefore calculated from the maximum shear stress, τmax(z), according to the Ramberg & Osgood (1943) model:
τ (z) τ (z) γ max (z) = max + C max G0 G0
R
(8)
where the parameters C and R have been calibrated on the curves of Fig. 2. On the other hand, in the full dynamic analysis of the coupled ground-tunnel system undergoing shaking, the incremental internal forces in the lining are computed using a numerical model. The finite elements software Plaxis v8 (Brinkgreve, 2002) was used to perform two-dimensional free-field and soil-structure interaction dynamic analyses. A set of input acceleration time histories was selected from a database of records of Italian seismic events (Scasserra et al., 2006). All the signals have been scaled to the same conventional value of ag (0.35g) and applied at the base of the model. The bedrock has been assumed as a rigid boundary, whereas lateral mesh boundaries were located at a distance about 8 times the layer thickness (Visone et al., 2008) and were modelled with dampers according to the Lysmer & Kuhlemeyer (1969) formulation (Fig. 3). Ground conditions and soil behaviour have been modelled according to Figs. 1 and 2. As the FE analyses were performed with a linear elastic model for the soil, the dependency of the soil stiffness and damping ratio on the strain level has been first considered by a secant equivalent approach. Therefore, preliminary one-dimensional SSR analyses have been performed by means of the code EERA (Bardet et al., 2000), which operates in the frequency domain. The material properties calculated as output from the SSR analysis were hence used as input to the FE analyses. The continuum was divided into the same number of sublayers as specified in EERA and different materials were defined for each sublayer. The soil damping was modelled through the Rayleigh formulation, according to the double frequency method, assuming an almost constant damping ratio between the first natural frequency of the deposit and a frequency n times larger; n is the first odd integer which approximates by excess the ratio between the fundamental frequency of the seismic signal and the first natural frequency of the deposit (Lanzo et al., 2004).
Figure 3 - Sketch of the mesh used for FE Plaxis analyses
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
3. ANALYSIS OF THE RESULTS Depending on the selected method, the pseudo-static analyses allowed the maximum hoop force Nmax,PS and bending moment Mmax,PS to be calculated neglecting the kinematic interaction. On the other hand, the FE analyses allowed calculating the maximum hoop force Nmax,DYN and bending moment Mmax,DYN accounting for kinematic interaction. The results of the pseudo-static and full dynamic analyses where combined together and the following parameters were defined, having the dimensions of a compliance:
k N* = k M* =
N max, PS γ DYN , FF
(9a)
DYN G m γ PS N max M max, PS γ DYN , FF
(9b)
DYN G m γ PS M max
In Eqs (9) the values:
N max, PS G m γ PS M max, PS G m γ PS
=
1 K2D 2
(10a)
=
1 K1 D 2 12
(10b)
are in fact representative of the relative stiffness between soil and lining. In the same equations the ratios N max, dyn / γ DYN , ff and M max, dyn / γ DYN , ff are factors which quantifies the effects of kinematic soil-tunnel interaction in the numerical analyses.
kN* [MPa-1]
0.05 0.04 0.03 method 1 method 2 method 3 method 4
0.02 0.01 0 0
0.5
1
1.5
t (m) Figure 4 - Kinematic interaction parameter k*N vs lining thickness
th
-1
kM* [MPa ]
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
method 1 method 2 method 3 method 4
0
0.5
1
1.5
t (m) Figure 5 - Kinematic interaction parameter k*M vs lining thickness
The free-field estimations of the shear strain γPS, computed by pseudo-static methods were obviously different from the corresponding finite element solution γDYN,FF. The following dimensionless parameter was defined to quantify such a difference:
α=
γ PS
(11)
γ DYN , FF
In Table 3 the values of α are shown, as computed by Eq. (11) using the average shear strain of each of the four method proposed by Bilotta et al. (2007). For each method, the values of α for sand and gravel are very close. Different is the case of clay, for which the average values are about twice as larger.
method 1 method 2 method 3 method 4
Table 3: Average values of ratio α Gravel Sand 2.5 2.2 2.3 2.1 1.4 1.2 0.4 0.5
Clay 5.8 5.4 3.5 0.9
By means of any of the above mentioned pseudo-static methods, the following expressions may be used to evaluate the maximum bending moments and hoops, taking into account the possible kinematic interaction:
1 π K D cos 2 + γ θ 2 PS 4 2 ⋅ α ⋅ k N* 1 π M (θ ) = ± K 1 D 2 γ PS cos 2θ + * 4 12 ⋅ α ⋅ k M N (θ ) = ±
(12a) (12b)
where α is given for each method in Table 3 and the modification factor for a given lining thickness t, k N* and
k M* , can be obtained (for sand) from the charts in Figs.10 and 11.
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
4. CONCLUSIONS The simplified procedure proposed in the paper was derived from the comparison of the results of a series of full dynamic analyses of circular tunnels embedded in a schematic subsoil with four pseudo-static methods proposed in a previous conference paper (Bilotta et al., 2007). It allows a simple modification to improve the accuracy of widely used closed form elastic solutions to calculate the increment of internal forces on a tunnel lining due to a seismic action (Wang, 1993). The improvement is based on a simplified way to introduce the kinematic interaction between the tunnel and the ground into a pseudo-static approach. Efforts are currently performed in order to improve the reliability of such procedure on the basis of the results of more sophisticated numerical models calibrated on centrifuge tests. This might lead also to produce simplified methods to estimate the magnitude of tunnel displacements during seismic loading.
5. ACKNOWLEDGEMENTS This work is a part of a Research Project funded by ReLUIS (Italian University Network of Seismic Engineering Laboratories) Consortium. The Authors wish to thank the coordinator, prof Stefano Aversa, for his continuous support and the fruitful discussions. The strong motion database used in this study was developed as part of an ongoing joint project involving researchers from the University of Rome La Sapienza and the University of California, Los Angeles, with support from the Pacific Earthquake Engineering Research Center. Preliminary results from this group were reported by Scasserra et al. (2006), but the data utilized here have not been published yet.
REFERENCES AFPS/AFTES. (2001) Guidelines on earthquake design and protection of underground structures Ausilio E., Silvestri F., Troncone A., Tropeano G. (2007). Seismic displacement analysis of homogeneous slopes: a review of existing simplified methods with reference to Italian seismicity. IV ICEGE, Thessaloniki Bardet J. P., Ichii K., and Lin C. H. (2000). EERA a Computer Program for Equivalent-linear Earthquake site Response Analyses of Layered Soil Deposits. Univ. of Southern California, Dep. of Civil Eng. Bilotta E., Lanzano G., Russo G., Santucci de Magistris F., Aiello V., Conte E., Silvestri F., Valentino M. (2007 ). Pseudo-static and dynamic analyses of tunnels in transversal and longitudinal directions, IV ICEGE, Thessaloniki. Brinkgreve R.B.J. (2002). Plaxis 2D version8. A.A. Balkema Publisher, Lisse. d’Onofrio A. and Silvestri F. (2001). Influence of micro-structure on small-strain stiffness and damping of fine grained soils and effects on local site response. Proc. IV Int. Conf. on ’Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics’. San Diego, Paper 1.19, EN 1998-1. (2003 ). Eurocode 8: Design of structure for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings. CEN European Committee for Standardisation, Bruxelles, Belgium. EN 1988-5. (2003) Eurocode 8 - Design of structures for earthquake resistance - Part 5: Foundations, retaining structures and geotechnical aspects CEN European Committee for Standardisation, Bruxelles, Belgium. Idriss I.M. and Boulanger R. (2004). Semi-empirical procedures for evaluating liquefaction potential during earthquakes, Proc. V ICSDEE & III ICEGE, Berkeley, USA (1) 32 -56. ISO TC 98/SC 3 N229 (2003). Bases for design of structures – Seismic actions for designing geotechnical works. Hashash, Y.M.A., Hook, J.J., Schmidt, B. and Yao, J.I-C.( 2001). Seismic design and analysis of underground structures, Tunnelling and Underground Space Technology, 16, 247-293. Iwasaki T., Tatsuoka F., Tokida K. and Yasuda S. (1978). A practical method for assessing soil liquefaction potential based on case studies at various sites in Japan, Proc. II Int. Conf. on Microzonation, San Francisco. Lanzano G., Bilotta E., Russo G. (2008). Tunnels under seismic loading: a review of damage case histories and
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
protection methods. Strategy for Reduction of the Seismic Risk (Fabbrocino & Santucci de Magistris eds.). pp 65-74. ISBN 88-88102-15-3. Lanzo G., Pagliaroli A., D’Elia B. (2004). Influenza della modellazione di Rayleigh dello smorzamento viscoso nelle analisi di risposta sismica locale, Proc. XI Italian National Conference on “L’ingegneria Sismica in Italia. (in Italian) Liao S.S.C. and Whitman R.V. (1986). Overburden correction factors for SPT in sand, Journal of Geotechnical Engineering, ASCE, 112(3):373-377. Lysmer J. and Kuhlemeyer R.L. (1969). Finite dynamic model for infinite media, J. of Eng. Mech. Div., ASCE: 859-877. OPCM 3274. (2003). Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica, GU Repubblica Italiana, 105-8/5/03 (in italian) Owen G.N., and Scholl R.E. (1981). Earthquake engineering of large underground structures, Report no. FHWA/RD-80/195. Federal Highway Administration and National Science Foundation. Penzien J. (2000). Seismically induced racking of tunnel linings, Int. J. Earthquake Eng. Struct. Dynamics 29, pp. 683–691. Power M.S., Rosidi D. and Kaneshiro J. (1996).Vol.III -Strawman: screening, evaluation and retrofit design of tunnels, Report Draft, Nat. Center for Earthquake Engineering Research, Buffalo, New York. Ramberg W. and Osgood W.R. (1943). Description of stress strain curves by three parameters, Technical Note 902, National Advisory Committee for Aeronautics, Washington, D.C. Santucci de Magistris F. (2005). Fattori di influenza sul comportamento meccanico dei terreni, App. B in 'Aspetti geotecnici della progettazione in zona sismica - Linee Guida AGI', Associazione Geotecnica Italiana, Patron Editore, Bologna. (in Italian) Scasserra G., Lanzo G., Mollaioli F., Stewart J.P., Bazzurro P. and Decanini L.D. (2006). Preliminary comparison of ground motions from earthquakes in Italy with ground motion prediction equations for active tectonic regions, Proc. 8th U.S. National Conference on Earthquake Engineering, San Francisco. Stokoe K.H. (2004). Comparison of Linear and Nonlinear Dynamic Properties of Gravel, Sand, Silts and Clays, Proc. V ICSDEE & III ICEGE, Berkeley, USA JSCE - The Japanese Society of Civil Engineers. (1992). Earthquake resistant design for civil engineering structures in Japan. Visone C., Bilotta E., Santucci de Magistris F. (2008). Remarks on site response analysis by using Plaxis dynamic module, Plaxis Bulletin, Bulletin 23, p. 14-18. http://www.plaxis.nl/upload/bulletins/PLAXIS%20Bulletin23%20MARCH2008.pdf Vucetic M. and Dobry R. (1991). Effects of the soil plasticity on cyclic response, Journal of Geotechnical Engineering, ASCE, 117(1):89-107. Wang J. (1993). Seismic Design of Tunnels: A Simple State-of-the-art Design Approach, Monograph 7, Parsons, Brinckerhoff, Quade and Douglas Inc, New York.
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
AN EARLY-STAGE DESIGN PROCEDURE FOR CIRCULAR TUNNEL LINING UNDER SEISMIC ACTIONS 1
1
1
2
E. Bilotta , G. Lanzano , G. Russo , F. Santucci de Magistris and F. Silvestri
1
1
2
DIGA, University of Napoli Federico II, Naples, SAVA, Structural and Geotechnical Dynamics Laboratory, University of Molise, Campobasso, Italy Email: [email protected]
ABSTRACT : The increments of the internal forces induced by an earthquake in the transverse section of a tunnel lining can be ascribed to the ovalisation of the section, induced by soil shear straining in the vertical plane. They can be assessed with several procedures at different levels of complexity. In this paper, two kind of analysis were per-formed on idealised geometry and soil conditions, considered representative of soil classes specified by Euro-code 8: pseudo-static analysis, where the seismic input was reduced to an equivalent peak strain amplitude, computed through a free-field pseudo-static analysis of the ground and then considered acting on the tunnel lining in static conditions; and full dynamic analysis, where the soil and tunnel responses were mechanically coupled and modelled by using FEM. Both were performed considering the soil as an equivalent linear medium. On the basis of the comparison of the results of both approaches, modification factors of the usual pseudo-static formulae are proposed, which take into account the kinematic interaction between the tunnel and the ground during shaking. The method, based on the use of simple charts, can be easily adopted for early-stage design.
KEYWORDS:
tunnels, lining, seismic actions, pseudo-static, dynamic
th
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
1. INTRODUCTION Civil infrastructures and lifelines in seismic areas need to be designed to support the extra loading produced by earthquakes. Some indications for such design can be found in Owen & Scholl (1981), JSCE (1992), AFPS/AFTES Guidelines (2001), ISO TC 98 (2003). Design rules for tunnels are not introduced in Eurocode 8 (EN 1998-5, 2003). This maybe because earthquake effects on underground structures were deemed to be negligible, in spite of the different evidences from several case-histories (see for instance Lanzano et al., 2008). Research activities are in progress in Italy to refine the design methods for tunnels under seismic actions (e.g. Bilotta et al., 2007). The shear waves propagating during an earthquake perpendicularly to the tunnel axis, result in a distortion of the cross-section of the structure: in this paper a procedure to calculate the forces induced by ground shaking in the tunnel lining in simple subsoil conditions is illustrated. Such simplified procedure incorporates the results of finite elements dynamic analyses, which consider the kinematic interaction between the tunnel lining and the ground, in the framework of the pseudo-static approach commonly adopted for early-stage design. Three idealized ground conditions (Fig. 1) were considered: a 30 m thick layer of soft clay, medium dense sand or gravel, overlying a compliant rock bedrock (Vr= 800 m/s, γ=22 kN/m3, D0=0.5%). The tunnel has the following characteristics: circular shape with reinforced concrete lining (variable thickness from 0.1 to 1.3 m, diameter D=6 m); axis depth z0=15 m; VS (m/s) 0
200
400
600
0
5
12 m 10
z(m)
t = 0.3 m 6m
argilla clay (D)(D) sabbia sand (C)(C) ghiaia (B) (B) gravel
15
20
(riv. Cls Rbk concrete lining450 Rckkg/cmq) =45 MPa
12 m 25
30
bedrock rock Bedrocksoft roccia tenera
VS = 800 m/s
3
γ = 22 kN/m
D0 = 0.5%
Figure 1 - Ground conditions
The values of small strain soil parameters have been chosen according to literature empirical relationships linking the shear modulus (G0) and the damping ratio (D0) to the lithostatic stress, the void ratio and intrinsic soil properties, such as particle size and plasticity index IP (Santucci de Magistris, 2005; d’Onofrio & Silvestri, 2001). The profiles of VS with depth adopted for each soil type are shown in Fig. 1, where the dashed lines represent the value of the so called ‘equivalent velocity’ VS,30 (EN 1998-1, 2003). Table 1 summarizes the geotechnical parameters and the ground type according to EC8.
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
Table 1: Ground parameters and classification according to EC8 Ground type φ’ IP γ D0 VS,30 (°)
(%)
(kN/m3)
(%)
(m/s)
Clay
D
25
30
18
2.5
125
Sand
C
35
-
20
1.0
240
Gravel
B
45
-
21
1.0
400
1
25
0.8
20
0.6
0.4
clay stiffness sand stiffness gravel stiffness clay damping sand damping gravel damping
15
10
0.2
0 0.0001
damping ratio, D (%)
normalised shear stiffness, G/Go
Soil non-linearity and cyclic energy dissipation were taken into account through an equivalent linear approach which considers the variation of the shear modulus G and the damping ratio D with the shear strain γ. Therefore, the curves G(γ)/G0 and D(γ) for the three materials (Figure 2) have been assumed according to literature indications (Vucetic & Dobry, 1991; Stokoe, 2004).
5
0 0.001
0.01 shear strain, γ (%)
0.1
1
Figure 2 - Variation of shear modulus and damping with shear strain level
2. 2 PSEUDO-STATIC VS FULL DYNAMIC CALCULATION In the usual simplified methods the kinematic soil-structure interaction is neglected as the free-field displacements are applied to the tunnel boundary (e.g. Hashash et al., 2001) and the seismic force increments in the lining are calculated by means of the closed-form elastic solutions (Wang, 1993) for a tunnel surrounded by a homogeneous and isotropic half-space, using the average shear deformation γPS of the ground at the tunnel depth as input: 1 π N (θ) = ± K 2Gm Dγ PS cos 2 θ + 2 4 M (θ) = ± where:
1 π K1Gm D 2 γ PS cos 2 θ + 12 4
(1a) (1b)
th
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
K1 =
12(1 −ν m ) 2 F + 5 − 6ν m
(2a)
1 2 F[(1 − 2ν m ) − (1 − 2ν m )C] − (1 − 2ν m ) + 2 2 K2 =1 + 5 F[(3 − 2ν m ) + (1 − 2ν m )C] + C − 8ν m + 6ν m2 + 6 − 8ν m 2
(2b)
The dimensionless parameters F and C represent the relative soil/tunnel stiffness and refer to a tunnel with diameter D, lining thickness t, and elastic parameters El and νl, in an elastic ground (Gm and νm):
(
)
(
)
G m 1 − ν l2 D 3 2El t 3 G 1 −ν l2 D C= m E l t (1 − 2ν m ) F=
(3) (4)
Based on the equilibrium of a deformable soil column from the surface to a given depth z, several procedures can be adopted to evaluate the average shear deformation γPS, as discussed in Bilotta et al. (2007). A first class of methods is based on the specification of a vertical profile of peak acceleration amax. Then, the maximum shear stress τmax is computed by integration as: z
τ max ( z ) = ∫ ρamax ( z )dz
(5)
0
Another class of methods follows an approach similar to those adopted for the evaluation of liquefaction susceptibility based on simplified procedures to define the seismic induced shear stress profile. Hence, the shear stress distribution with depth is calculated according to the following equation:
τ max ( z ) = rd ( z )
a max, s g
σ v (z)
(6)
In Eq. (6), σv is the total vertical stress, and rd is a reduction parameter which takes into account the deformability of the soil column. Several empirical relationships (Iwasaki et al., 1978; Liao & Whitman, 1986; Power et al., 1996; Idriss & Boulanger, 2004) to define rd are reported in literature. All the considered pseudo-static methods require a preliminary evaluation of the peak acceleration at surface. In the paper this value has been computed as:
a max, s = S ⋅ a g
(7)
where a g is the peak acceleration on outcropping rock site and S the site response factor. Its value has been originally specified by EC8 part 1 (EN 1998-1, 2003 ), followed by several proposed of updating (i.e. Italian OPCM 3274, 2003; ETC12, 2006) for each ground type (Table 2). In this paper, the average value has been assumed. Alternatively, a non-linear response factor can be used, varying with the ground motion amplitude, as proposed by Ausilio et al. (2007).
th
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
Table 2: Response factors in pseudo-static methods EC8 OPCM ETC12 average 3274 Clay 1.35 1.35 1.1 1.27 Sand 1.15 1.25 1.15 1.18 Gravel 1.2 1.25 1.3 1.25 Soil
The maximum shear strain at a depth z is therefore calculated from the maximum shear stress, τmax(z), according to the Ramberg & Osgood (1943) model:
τ (z) τ (z) γ max (z) = max + C max G0 G0
R
(8)
where the parameters C and R have been calibrated on the curves of Fig. 2. On the other hand, in the full dynamic analysis of the coupled ground-tunnel system undergoing shaking, the incremental internal forces in the lining are computed using a numerical model. The finite elements software Plaxis v8 (Brinkgreve, 2002) was used to perform two-dimensional free-field and soil-structure interaction dynamic analyses. A set of input acceleration time histories was selected from a database of records of Italian seismic events (Scasserra et al., 2006). All the signals have been scaled to the same conventional value of ag (0.35g) and applied at the base of the model. The bedrock has been assumed as a rigid boundary, whereas lateral mesh boundaries were located at a distance about 8 times the layer thickness (Visone et al., 2008) and were modelled with dampers according to the Lysmer & Kuhlemeyer (1969) formulation (Fig. 3). Ground conditions and soil behaviour have been modelled according to Figs. 1 and 2. As the FE analyses were performed with a linear elastic model for the soil, the dependency of the soil stiffness and damping ratio on the strain level has been first considered by a secant equivalent approach. Therefore, preliminary one-dimensional SSR analyses have been performed by means of the code EERA (Bardet et al., 2000), which operates in the frequency domain. The material properties calculated as output from the SSR analysis were hence used as input to the FE analyses. The continuum was divided into the same number of sublayers as specified in EERA and different materials were defined for each sublayer. The soil damping was modelled through the Rayleigh formulation, according to the double frequency method, assuming an almost constant damping ratio between the first natural frequency of the deposit and a frequency n times larger; n is the first odd integer which approximates by excess the ratio between the fundamental frequency of the seismic signal and the first natural frequency of the deposit (Lanzo et al., 2004).
Figure 3 - Sketch of the mesh used for FE Plaxis analyses
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
3. ANALYSIS OF THE RESULTS Depending on the selected method, the pseudo-static analyses allowed the maximum hoop force Nmax,PS and bending moment Mmax,PS to be calculated neglecting the kinematic interaction. On the other hand, the FE analyses allowed calculating the maximum hoop force Nmax,DYN and bending moment Mmax,DYN accounting for kinematic interaction. The results of the pseudo-static and full dynamic analyses where combined together and the following parameters were defined, having the dimensions of a compliance:
k N* = k M* =
N max, PS γ DYN , FF
(9a)
DYN G m γ PS N max M max, PS γ DYN , FF
(9b)
DYN G m γ PS M max
In Eqs (9) the values:
N max, PS G m γ PS M max, PS G m γ PS
=
1 K2D 2
(10a)
=
1 K1 D 2 12
(10b)
are in fact representative of the relative stiffness between soil and lining. In the same equations the ratios N max, dyn / γ DYN , ff and M max, dyn / γ DYN , ff are factors which quantifies the effects of kinematic soil-tunnel interaction in the numerical analyses.
kN* [MPa-1]
0.05 0.04 0.03 method 1 method 2 method 3 method 4
0.02 0.01 0 0
0.5
1
1.5
t (m) Figure 4 - Kinematic interaction parameter k*N vs lining thickness
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-1
kM* [MPa ]
The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
method 1 method 2 method 3 method 4
0
0.5
1
1.5
t (m) Figure 5 - Kinematic interaction parameter k*M vs lining thickness
The free-field estimations of the shear strain γPS, computed by pseudo-static methods were obviously different from the corresponding finite element solution γDYN,FF. The following dimensionless parameter was defined to quantify such a difference:
α=
γ PS
(11)
γ DYN , FF
In Table 3 the values of α are shown, as computed by Eq. (11) using the average shear strain of each of the four method proposed by Bilotta et al. (2007). For each method, the values of α for sand and gravel are very close. Different is the case of clay, for which the average values are about twice as larger.
method 1 method 2 method 3 method 4
Table 3: Average values of ratio α Gravel Sand 2.5 2.2 2.3 2.1 1.4 1.2 0.4 0.5
Clay 5.8 5.4 3.5 0.9
By means of any of the above mentioned pseudo-static methods, the following expressions may be used to evaluate the maximum bending moments and hoops, taking into account the possible kinematic interaction:
1 π K D cos 2 + γ θ 2 PS 4 2 ⋅ α ⋅ k N* 1 π M (θ ) = ± K 1 D 2 γ PS cos 2θ + * 4 12 ⋅ α ⋅ k M N (θ ) = ±
(12a) (12b)
where α is given for each method in Table 3 and the modification factor for a given lining thickness t, k N* and
k M* , can be obtained (for sand) from the charts in Figs.10 and 11.
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The 14 World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China
4. CONCLUSIONS The simplified procedure proposed in the paper was derived from the comparison of the results of a series of full dynamic analyses of circular tunnels embedded in a schematic subsoil with four pseudo-static methods proposed in a previous conference paper (Bilotta et al., 2007). It allows a simple modification to improve the accuracy of widely used closed form elastic solutions to calculate the increment of internal forces on a tunnel lining due to a seismic action (Wang, 1993). The improvement is based on a simplified way to introduce the kinematic interaction between the tunnel and the ground into a pseudo-static approach. Efforts are currently performed in order to improve the reliability of such procedure on the basis of the results of more sophisticated numerical models calibrated on centrifuge tests. This might lead also to produce simplified methods to estimate the magnitude of tunnel displacements during seismic loading.
5. ACKNOWLEDGEMENTS This work is a part of a Research Project funded by ReLUIS (Italian University Network of Seismic Engineering Laboratories) Consortium. The Authors wish to thank the coordinator, prof Stefano Aversa, for his continuous support and the fruitful discussions. The strong motion database used in this study was developed as part of an ongoing joint project involving researchers from the University of Rome La Sapienza and the University of California, Los Angeles, with support from the Pacific Earthquake Engineering Research Center. Preliminary results from this group were reported by Scasserra et al. (2006), but the data utilized here have not been published yet.
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