The evaluation on shock absorption performance of buffer layer around the cross section of tunnel lining

Soil Dynamics and Earthquake Engineering 131 (2020) 106032

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The evaluation on shock absorption performance of buffer layer around the cross section of tunnel lining C.L. Xin a, b, Z.Z. Wang c, *, J. Yu c a

State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu, PR China College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, PR China c School of Civil Engineering, Dalian University of Technology, Dalian, PR China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Tunnel lining Buffer layer Multivariable nonlinear regression Shaking table test

The buffer layer around the cross section of lining structure would be the most simple and effective shock ab­ sorption measure, which can cut off the transferring paths of adverse effects from surrounding rock to lining structures. Meanwhile, it can reduce the seismic action intensity and minimize the extra pressure from sur­ rounding rock by changing seismic acting patterns. This paper presents outcomes by investigating the interaction between lining structure and surrounding rock during excitation as well as deriving influencing parameters with regard to shock absorption performances of buffer layer. Afterwards, series numerical calculations are system­ atically carried out with attention given to the seismic-induced deformation responses of rock-layer-lining sys­ tem. Finally, an elaborated evaluation method for assessing shock absorption performance of buffer layer with an evaluation expression is proposed, which is validated by series shaking table model tests and numerical simu­ lations. This evaluation expression would be able to guide the selection and construction of buffer layer ac­ cording to the geological conditions and construction circumstances by changing geometric and material parameters. The evaluation result would have uniqueness once all the parameters are confirmed. From this expression and its derivation process, the following findings can be drawn: (1) The cross section of tunnel has the identical particular deformation pattern in each constructing phase during excitation regardless of the existence of lining structure and buffer layer, that is, the two orthogonal diagonal diameters of cross section alternatively expand and contract. (2) The geometric and material parameters simultaneously determine the shock absorption performances of buffer layer, which grows in direct proportion to geometric variables but decreases in inversely proportion to material variables. (3) When the ratio of lining inside radius to buried depth is equal to 0.2 (i.e. r0/ H ¼ 0.2), the buried depth of tunnel engineering should be the ideal position for adopting buffer layer. Mean­ while, when the ratio of buffer layer thickness to lining inside radius is equal to 0.2 (i.e. tb/r0 ¼ 0.2), it can be confirmed the optimal thickness of buffer layer. (4) The evaluation expression of shock absorption performance is independent of seismic waveforms but involved in the excitation intensities and the destruction states of lining structure.

1. Introduction Tunnel engineering makes great progress with the development of high-speed railway, which motivates the scientific and technological advances of infrastructure construction in adverse geological zones [1]. Currently, it is possible that the high-speed railways with large curva­ ture radius are able to cross the mountains in high-intensity seismic zones due to the existence of reliable tunnel engineering [2]. However, once seismic hazards strike the tunnel engineering, there will be still hidden troubles affect the whole high-speed railway [3]. Moreover, once

the accidents of high-speed trains happen, it will be the focus spot of public opinion [5]. Generally, the reliability and durability of lining structures in tunnel engineering determine the security of the whole high-speed railway engineering, which simultaneously enjoy the positive effects (e.g. selfsupporting effect, compressive arch, loosen zone, etc.) and suffer the negative effects (e.g. seepage, creep and squeezing deformations, etc.) from surrounding rock [6]. There into, the blast-induced and seismic-induced effects acting on lining structures are the most adverse circumstances [7,8]. Among many anti-seismic and shock absorption

* Corresponding author. E-mail address: [email protected] (Z.Z. Wang). https://doi.org/10.1016/j.soildyn.2020.106032 Received 24 June 2019; Received in revised form 5 December 2019; Accepted 2 January 2020 0267-7261/© 2020 Elsevier Ltd. All rights reserved.

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

measures for tunnel structures, the buffer layer around the cross section of lining structure would be the most simple and effective one [4]. Actually, the buffer layer can play a vital role in shock-absorption per­ formance along both the transverse directions and longitudinal di­ rections of lining structures. Theoretically, anti-seismic and shock absorption measures are the two different concepts but they are usually confusing. Apparently, the buffer layer is belong to the latter [9]. In general, shock absorption would be the better method for tunnel structures than anti-seismic because the buffer layer would be the cho­ sen one to change the seismic interaction between surrounding rock and lining structures [10]. Specifically, the existence of buffer layer changes the rock-lining system into rock-layer-lining system, rock-lining-layer system, rock-layer-rock-lining system, etc. [11], that is, buffer layer can cut off the transferring paths of adverse effects from surrounding rock to lining structures. Meanwhile, it can reduce the seismic action intensity and minimize the extra pressure from surrounding rock by changing the seismic acting patterns. Further, the buffer layer can decrease the constraining forces of surrounding rock as well as absorb the circular relative displacements and discontinuous shearing de­ formations between surrounding rock and lining structures [12]. Un­ fortunately, buffer layer will not always bring beneficial effects. An over thick buffer layer could trigger high acceleration seismic responses and rigid body displacements along the longitudinal directions of lining structures, which will cause axial direction cracking or even staggering. In general, a new technology will not be applied in the practical engineering even it can effectively solve the problem but apparently raise the constructing costs. Therefore, it promotes many researchers to keep the balance between effects and costs [13]. For this reason, both simple plate-type and injection-type are the two common types of buffer layers in recent research [14]. Normally, the rubber is the representative material of the former while the foamed concrete represents the latter material [15]. The suitable material would make the buffer layer not only have low shearing elasticity and ideal deformable property but enough carrying capacity and sustainability. Furthermore, it is neces­ sary to finely research the geometric and material parameters of buffer layers, which would effectively assess the applicability and reduce the selection and construction cost of buffer layer. Ordinarily, lining structures usually have synchronous shock with surrounding rock due to the constraint effect of rock or the existence of buffer layer, thus the impact of inertia force on lining is quite limited [16]. Therefore, the research about seismic responses and post-earthquake damage mechanism of lining structure should focus on the adverse effects of earthquake-induced surrounding rock failure [17]. Consequently, an ideal buffer layer with sustainable serviceability should be able to carry the cracking surrounding rock and help it to rebuild itself. Herein, this research aims to propose an elaborated evaluation method to assess the shock absorption performance of buffer layer, then numerical simulations and shaking table tests validates this elaborated evaluation method. The results may provide useful refer­ ences for further research in terms of the shock absorption mechanism of other damping measures or renovation techniques for lining structures

with the impact of dynamic effects. 2. Theoretical analysis for seismic-induced fundamental deformations of tunnel cross section 2.1. Mechanical model for rock-layer-lining system Konagai and Kim focused the buffer layer around tunnel structure nearly two decades ago [18,19]. After long-term observation, they found that the tunnel cross section alternatively expanded and contracted along its two orthogonal diagonal diameters (θ ¼ �45� ) during excita­ tion (shown in Fig. 1). Based on this phenomenon, they found that when the tunnel structure was concerned, the inertia interaction was often of less essential than kinematic interaction. Afterwards, they employed the multi-step method to isolate two primary causes of rock-tunnel inter­ action and to evaluate the shock absorption properties of lining structure without buffer layer. Two different interfaces were set between sur­ rounding rock and lining structure. The one was fixed tangential strain interface and the other was shear-stress free one. The former suggested a firm bond between lining structure and surrounding rock, whereas the latter was associated with an artificial slippery rock-lining interface reducing shearing stress on lining structure. There were six assumptions before deriving, i.e. (1) The surrounding rock was a two-dimensional homogeneous viscous-elastic medium of an infinite extent with harmonic motion. (2) The tunnel cross section had the particular deformations as Fig. 1. (3) The deformations of lining structure totally depended on the deformations of surrounding rock during excitation, that is, the hollow space had identical deformations as lining structure. (4) The buried depth of tunnel was nearly equal to a quarter of seismic wavelength and longer than the diameter of tunnel cross section. (5) The existence of lining structure cannot affect the deformation patterns of hollow and surrounding rock during excitation. (6) There were no slippage and secession on interface between lining structure and surrounding rock, besides, the shearing stiffness was high enough, while the shearing strain was low enough. Therefore, the governing equations of a viscous-elastic medium un­ dergoing harmonic motion can be expressed in the cylindrical co­ ordinates as: ðλs;b þ 2Gs;b Þ

∂Δeiωt ∂r

2Gs;b ∂Ωz eiωt ∂2 ur eiωt ​ ¼ ρs;b r ∂θ ∂t2

(1)

ðλs;b þ 2Gs;b Þ

∂Δeiωt ∂Ωz eiωt ∂2 uθ eiωt ​ þ 2Gs;b ¼ ρs;b r ∂θ ∂r ∂t 2

(2)

where, Δ¼

1 ∂ 1 ∂uθ ðrur Þ þ r ∂r r ∂θ

​ Ωz ¼

� 1 ∂ðruθ Þ 2r ∂r

∂ur ∂θ

(3) � (4)

Fig. 1. Fundamental deformations of lining cross section during excitation: (a) The first vibration mode; (b) The second vibration mode; (c) Alternate deformations. 2

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Hereon, the subscripts s and b stand for surrounding rock and buffer layer, respectively. λs,b and Gs,b are Lame’s constants whose imaginary parts describe viscosity of surrounding rock and buffer layer material, pffiffiffiffiffiffi respectively; ρ is density; i ¼ 1; ω is circular excitement frequency; t represents time; ur and uθ are displacements in r and θ directions, which are related to the potential functions ϕ and ψ as follows: The functions φm and ψ m can be written in Bessel function forms: � φm ¼ ðqrÞ⋅ðAK;m ⋅Km þ AI;m ⋅Im ÞðAC;m cos mθ þ AS;m sin mθÞ (5) ψ m ¼ ðsrÞ⋅ðBK;m ⋅Km þ BI;m ⋅Im ÞðBC;m cos mθ þ BS;m sin mθÞ

8 � � 4 > > þ qr I1 ðqrÞ Ur;CI ¼ 2I0 ðqrÞ þ > > > qr > > > > > 4 > > > < Uθ;CI ¼ 2I0 ðqrÞ qrI1 ðqrÞ > 4 > > > U ¼ 2I0 ðsrÞ þ I1 ðsrÞ > > r;DI sr > > > � � > > 4 > > þ sr I1 ðsrÞ : Uθ;DI ¼ 2I0 ðsrÞ sr

Afterwards, the normal and shear stresses components can be ob­ tained as: � � � � b σrr σ rr sin 2 θ ¼ (13) bτ rθ cos 2 θ τrθ

where, AK;m , AI;m , AC;m , AS;m , BK;m , BI;m , BC;m and BS;m are unknown constants, Km and Im are the first and second kind modified Bessel functions of order m and q​2¼

ρs;b ω2

(6)

λs;b þ 2Gs;b 2

ρs;b ω

s2 ¼

� for surrounding rock

b σ rr bτ rθ

�

�

(7)

Gs;b

It is assumed that the cross section of a circular hollow with a radius r deforms in such a particular pattern that its two orthogonal diagonal diameters (θ ¼ �45� ) alternatively expand and contract. This calls for m ¼ 2 and AC;m ¼ BC;m ¼ 0 in equation (5). Thus, the deformations should be:

and for buffer layer

b σ rr bτ rθ

� ¼

� �� � Gs Trr;AK ​ ​ Trr;BK ASK;2 BCK;2 r2 Trθ;AK ​ ​ Trθ;BK

� Gb Trr;CK Trr;CI Trr;DK Trr;DI ¼ 2 r Trθ;CK Trθ;CI Trθ;DK Trθ;DI

ur uθ

�

� ¼

b u r sin 2 θ b u θ cos 2 θ

(8)

(16)

b ur b uθ

�

� ¼

� � �� � ASK;2 Ur;AK �r ​ ​ Ur;BK �r BCK;2 Uθ;AK r ​ ​ Uθ;BK r

� � � � 8 � � λb 2 24 2 2 > > ðqrÞ ðqrÞ ðqrÞ ðqrÞ T I I1 ðqrÞ I þ 2ðqrÞ 6qr þ ¼ þ12 I rr;CI 0 1 0 > > qr qr Gb > > > � � > > > 24 > > > < Trθ;CI ¼ 12I0 ðqrÞþ 4qr þ qr I1 ðqrÞ � � > 24 > > > I1 ðsrÞ Trr;DI ¼ 12I0 ðsrÞ 4sr þ > > sr > > > � � > > � > > : Trθ;DI ¼ ðsrÞ2 þ12 I0 ðsrÞþ 4sr þ 24 I1 ðsrÞ sr (17)

(9)

8 9 C � � � � �> � � � = < SK;2 > b u Ur;CK � r Ur;CI �r Ur;DK �r Ur;DI �r CSI;2 and for buffer layer r ¼ D b uθ Uθ;CK r Uθ;CI r Uθ;DK r Uθ;DI r > > CK;2 : ; DCI;2 (10) where, 8 > > U ¼ Ur;CK ¼ > > r;AK > > > > > > > > > Ur;BK ¼ Ur;DK ¼ <

9 8 C �> = < SK;2 > CSI;2 ​ ​ ​ D > > CK;2 ; : DCI;2 (15)

�

� for surrounding rock

(14)

where,

� � � � 8 � � λb 2 24 > > K1 ðqrÞ Trr;AK ¼ Trr;CK ¼ ðqrÞ2 K0 ðqrÞþ K1 ðqrÞ þ 2ðqrÞ2 þ12 K0 ðqrÞþ 6qr þ > > qr qr Gb > > > � � > > > 24 > > > < Trr;BK ¼ Trr;DK ¼ 12K0 ðsrÞþ 4sr þ sr K1 ðsrÞ � � > 24 > > > Trθ;AK ¼ Trθ;CK ¼ 12K0 ðqrÞ K1 ðqrÞ 4qr þ > > qr > > > � � > > � > 24 > : Trθ;BK ¼ Trθ;DK ¼ ðsrÞ2 þ12 K0 ðsrÞ 4sr þ K1 ðsrÞ sr

�

(12)

Hereon, ASK;2 , BCK;2 , CSK;2 , CSI;2 , DCK;2 , DCI;2 are unknown constants. Finally, b σ rr and bτ rθ are expressed in terms of b u r and b u θ as: � � � �� � Gb Srr ​ ​ Srθ b b ur σ rr ¼ ​ (18) bτ rθ b uθ r0 Sθr ​ ​ Sθθ

�

2K0 ðqrÞ 2K0 ðsrÞ

� 4 þ qr K1 ðqrÞ qr � � 4 K1 ðsrÞ sr

> 4 > > > U ¼ Uθ;CK ¼ 2K0 ðqrÞ þ K1 ðqrÞ > > θ;AK qr > > > � � > > > 4 > : Uθ;BK ¼ Uθ;DK ¼ 2K0 ðsrÞ þ þ sr K1 ðsrÞ sr

(11)

8 � Srr ¼ ð Trr;AK Uθ;BK þ Trr;BK Uθ;AK > > �Þ S > > < Srθ ¼ ðTrr;AK Ur;BK Trr;BK Ur;AK Þ S � where; Sθr ¼ ð Trθ;AK Uθ;BK þ Trθ;BK Ur;AK �Þ S > > > > Sθθ ¼ ðTrθ;AK Ur;BK Trθ;BK Ur;AK Þ S : S ¼ Ur;AK Uθ;BK Uθ;AK Ur;BK

(19)

Therein, Srr , Srθ , Sθr and Sθθ are components of the stiffness matrixes of surrounding rock and buffer layer materials, respectively. 3

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Substituting equations (26) and (27) into equation (18) yields sur­ rounding rock reactions to the lining structure as: � � �� � � Gs Srr ​ ​ Srθ b b μl b μs σ rr ¼ (28) b bτ rθ μ l =2 b μs r0 Sθr ​ ​ Sθθ Since the lining structure should be assumed as a circular Bernoulli Euler beam, its deformation parameter b μ l is expressed in terms of b σ rr and bτ rθ as: � � r4 b σ b u l ¼ 0 ½2 ​ ​ 1� rr (29) bτ rθ 18EI where, EI is bending stiffness of lining structure. Combining equations (28) and (29), the transferring factor equation is finally obtained: b ul ξð2Srr þ 2Srθ þ Sθr þ Sθθ Þ ​ ¼ b u s G ξð2Srr þ Srθ þ 2Sθr þ Sθθ Þ

where, ξs ¼ Gs r30 =18EI and G ¼ Gs =Gb . u s is indispensable in these derivations by The transferring factor b ul =b assessing the stiffness of buffer layer in the outer surrounding rock. Besides, the transferring factor is a function with respect to the following six non-dimensional parameters regardless of the relative slippage be­ tween lining structure and buffer layer or surrounding rock during excitation: (1) Ratio of lining inside radius to buried depth r0 =H. (2) Relative surrounding rock stiffness ξs . (3) Ratio of shear stiffness G. (4) Poisson’s ratio of surrounding rock νs . (5) Poisson’s ratio of buffer layer νb . (6) Ratio of buffer layer thickness to lining inside radius t=r0 .

Fig. 2. Mechanical model of rock-layer-lining system.

2.2. The evaluation coefficients of buffer layer Based on the above derivations, the rock-lining system can be divided into the inner lining structure and the outer surrounding rock including a buffer layer. As illustrated in Fig. 1 (c), the radial and tangential deformations ur;s and uθ;s were tentatively approximated by the static solutions for a cylindrical hollow inclusion in an unbounded medium with alternate shearing action, therefore, ur;s ffi b u r;s sin 2 θ

(20)

uθ;s ffi b u θ;s cos 2 θ

(21)

3. Numerical calculation for elaborated evaluation method on shock absorption performances of buffer layer The following series numerical calculations are based on the above theoretical analysis, which integrates all the non-dimensional variables and focuses on establishing the evaluation expression to assess the shock absorption performances of buffer layer. It is obvious that the rock-layer-lining system has remarkable nonlinear properties, which can be described in three parallel di­ mensionalities. The first one is nonlinear dynamic properties of lining structures, buffer layer and surrounding rock materials. The second one is the nonlinear geometry problems in interaction between lining structure, buffer layer and surrounding rock. The last one is the nonlinear dynamic contact relationship. The commercial calculating software ABAQUS is famous for its nonlinear analysis ability, which can supply reliable solutions and accurate results for computing the rocklayer-lining system interaction. To reveal the propagation paths of seismic motions in far field and the site effect of irregularity terrain in near field, Bielak put forward Domain Reduction Method (DRM) [20], which was a multiscale modelling and calculating method. The DRM can create artificial boundaries of irregularity terrain sub-model by precisely transforming fault-triggering seismic loads in far field, which should be able to effectively reduce the numerical calculating workload in large-scale conventional seismic responses analysis. Further, the DRM focused on the solution of internal wave motion problems rather than external ones, thereby simplifying the processes of seismic motion input and seismic responses analysis in near field by merely solving deformation field equation instead of solving stress field equation in free field. Besides, the DRM had admirable compatibility with several common absorbing boundaries (e.g. viscous boundary, viscous-spring boundary, infinite element boundary, etc.), thus, the DRM can effectively raise the speed and precision of seismic responses analysis by narrowing the calculating model range. Actually, the DRM would be able to generate accurate or reference solutions for seismic dynamic problems in real tunnel projects, since it can provide precise and reasonable seismic wave motion field for numerical calculating models in finite field [21].

where, b u r;s ¼ b u θ;s ¼ b u s ¼ 2γr0 ð1 νÞ and γ is the applied shear strain, ν is Poisson’s ratio. The b u r;s and b u θ;s are identical to each other regardless of ν. Thus, b u s is a representative deformation parameter of surrounding rock. The surrounding rock deformations act on the tunnel lining through buffer layer (shown in Fig. 2), which forcibly deforms the lining struc­ ture in the following manner: ur;l ffi b u r;l sin 2 θ

(22)

uθ;l ffi b u θ;l cos 2 θ

(23)

The subscript l represents lining structure. Based on the BernoulliEuler hypothesis, it is assumed that the lining structure has a perfect bond with surrounding rock, whose stiffness is too high in its tangential direction to generate tangential strain (i.e. εθθ;l ¼ 0). Thus, the lining deformations ur;l and uθ;l eventually satisfy the following equation at r ¼ r0 :

εθθ;l ¼ r0 :

1 ∂uθ;l ur;l þ ¼0​ r ∂θ r

(24)

Substituting equations (22) and (23) into equation (24) yields at r ¼

b u r;l ¼ 2b u θ;l ¼b ul

(25)

The rock-lining interaction thus causes the deformation parameters of the stress-free cylindrical hollow to deviate from b u r;s and b u θ;s by b u r and b u θ , respectively, where, b ur ¼ b ul b uθ ¼ b u l =2

(26)

b us b us

(30)

(27)

4

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Fig. 3. Acceleration time history of sinusoidal seismic motion.

3.1. Selection and adjustment of the input motion Currently, the majority seismic response research on practical tunnel engineering focuses on the action mechanism of shearing seismic motion with vertical propagation directions rather than the impact of point wave source [22], especially in the shaking table model tests. As for the seismic motions, any complicated excitation signals are the recombi­ nation of several sinusoidal waveforms with different amplitudes and frequencies, whereas the amplitude, frequency spectrum and duration can demonstrate a particular seismic motion. Since this research focused on the fundamental shock absorption mechanism of buffer layer, a si­ nusoidal waveform with 0.6 g amplitude, 18 s duration and 2.5 Hz frequency (plotted in Fig. 3) was chosen to make the variation of vari­ ables easy to control and the research results can be easily applied to practical engineering [23]. Hereinto, the amplitude, duration and main frequency of input motion in numerical calculation were identical with the Wolong ground motion from Wenchuan Great Earthquake, which would be applied in shaking table model tests for validation [24]. As for the frequency spectrum, 0.1–30 Hz is the common variation range of seismic motion frequencies, further, 1–2 Hz is the general variation range of dominate frequencies. However, depending on the stations positions for recording seismic motions, the dominate frequency range of Wolong seismic motion was 2–3 Hz, which determined the sinusoidal input motion in numerical calculation with a dominate frequency of 2.5 Hz. Based on the propagation mechanism of seismic motion in litho­ sphere, this sinusoidal motion was input at the inner bottom boundary of finite elements along the horizontal direction according to the DRM method to generate the seismic shear wave excitation like the shaking table model tests.

Fig. 4. Modelling for numerical calculation: (a) The global calculating model; (b) The lining structure and buffer layer calculating models with moni­ toring positions.

lining structure, the corresponding dimensions of global calculating model were labelled in Fig. 4 (a). Specifically, the bold full line was the demarcation line of finite and infinite elements while the bold dash line was the input boundaries of equivalent load for DRM. Correspondingly, Fig. 4 (b) plotted the lining structure and buffer layer in the calculating model as well as the monitoring positions on the outer surface of lining structure. The height of global calculating model should be changed with the variation of tunnel buried depth. In addition, there was no primary supports in the model, since they were usually regard as the enhancement of surrounding rock intensity. According to a huge number of geometric dimensions of practical tunnel engineering and considering the application scopes of buffer layer, the variation ranges of buried depth, lining thickness, lining inside radius and buffer layer thickness were confirmed as 0–3000 m, 0–1.5 m, 1–20 m, and 0–1.2 m, respec­ tively. The 4-node bilinear elements in ABAQUS element library were applied as finite elements. The arbitrary Lagrangian-Eulerian (ALE) adaptive mesh division technique was employed to distribute the lining

3.2. Modelling and parameters assignments A two dimensional input method of seismic shear wave was imple­ mented into ABAQUS code by a self-developed Python program. The viscous-spring artificial boundary (VSAB) acted as both sides and bot­ tom absorbing boundaries of finite elements in calculating model, while the top was free-field boundary. The infinite element boundaries were set at the outside of VSAB to make the specific deformation pattern of rock-layer-lining system in theoretical analysis accurately reappear [25]. Based on the geological and construction circumstances of several tunnel projects along Lhasa to Nyingchi section of the Szechwan-Tibet railway, a two-dimensional surrounding rock, buffer layer and lining structure models were established in ABAQUS code. A 10.6 m span and 8.7 m height horseshoe-shaped cross section in the general design dia­ gram of tunnel projects was employed as the numerical calculating model while the buffer layer was set between the surrounding rock and 5

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structure, which allowed the meshes to move independently from the material, thus, the meshes can guarantee a high-quality result in a large deformation analysis. The maximum size of the elements was measured to be smaller than one-tenth of the lowest excitation wavelength. Spe­ cifically, the maximum elements size was 5.0 m in this numerical calculation model, which satisfied the calculating precision. To clarify the interaction mechanism of surrounding rock, buffer layer and lining structure with the seismic excitation, the concrete damage plasticity (CDP) model in the ABAQUS material library was employed to simulate the lining structure. The CDP model was able to express the nonlinear dynamic behavior of multiphase material like concrete, which should be suitable to analyze the seismic-induced damage on concrete lining as well. Additionally, it was of equal importance to capture the seismic nonlinear responses of surrounding rock since the seismic motion of surrounding rock applied on lining structure or buffer layer directly, thus, the surrounding rock should undergo plastic failure according to the Drucker-Prager failure criterion. Moreover, the linear elastic model was employed in the infinite elements area. The corresponding physical and mechanical parameters of lining structure and surrounding rock in the present calculations are listed in Table 1 and Table 2. Considering the buffer layer would have high volumetric compress­ ibility and strain-ratio dependence, the curable foam model in ABAQUS material library acted as buffer layer, whose elliptical yield surface in the p-q stress plane was written as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 þ α2 ðp p0 Þ2 B ¼ F ¼ 0 (31)

fitting parameters. Based on the experimental results, the parameters for buffer layer numerical model are presented in Table 3. Finally, although there was synergistic reaction between lining structure and buffer layer in theoretical analysis, it was necessary to consider the relative sliding, squeezing and separating between lining and buffer layer during seismic excitation in numerical calculation. Thus, the penalty function and Lagrange multiplier method should be employed to establish the dynamic contact relationship. The dynamic mechanical transferring features should be achieved by defining the master and slave surfaces as lining structure and buffer layer surfaces, respectively. Then, the compressive stress transferring interaction con­ trols the normal direction contact relationship, that is, the element nodes on the contact surface obeys the Hook’s law and harmonious displacement circumstances. Meanwhile, the Coulomb’s friction law controls the tangential contact relationship, which has closed state, bonded state and sliding state. After beyond the ultimate shear stress, the tangential stress on contact surface would lead to relative sliding deformation. The Coulomb’s friction law equation demonstrates the ultimate shearing stress:

where, μ and f represent the coefficients of friction and the normal contact pressure, respectively. Taking into account the calculating convergence and accuracy, the interface friction coefficient should be confirmed as 0.36 between lining structure and buffer layer. From Table 1, Tables 2 and 3, the density, elasticity modulus and Poisson’s ratio were selected as the three fundamental variables in accordance with the results of theoretical analysis and materials in this numerical calculation model.

where, p is the confining stress, p ¼

σ =3; σkk is the principal stress ffiffiffiffiffiffiffiffi qffiffiffikk tensor; q is the von Mises stress, q ¼ 1:5S2ij ; Sij is the deviation stress pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tensor; α ¼ 3k= ð3kt þ kÞð3 kÞ, k ¼ σ 0c =p0c , kt ¼ pt =p0c and pt ¼ 0:05pc ; σ0c and p0c are the initial yield stresses under uniaxial compres­ sion and hydrostatic pressures, respectively; p0 ¼ ðpc pt Þ= 2, pt and pc are the three dimensional uniform tensile and compressive yield stresses, respectively, here pt ¼ 0:05pc ; B is the minor axis of the ellip­ tical yield surface. The flow potential and plastic strain ratio can be written as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ¼ q2 þ 4:5p2 (32) pl

ε_ pl ¼ ε_

∂G ∂σ

3.3. Calculating conditions and monitoring schemes According to the selected fundamental geometric and material pa­ rameters, there were nine dimensionless combinational variables designated to evaluate the shock absorption performance of buffer layer (i.e. r0/H, tb/r0, tl/r0, ρb/ρs, El/Es, Eb/Es, νl, νb and νs) [26]. All the vari­ ation ranges of variables were set from 0 to 1 or a certain interval in it. According to the geological survey reports of tunnel projects on Szechwan-Tibet railway, the constants of variables should be set as r0 ¼ 5 m、H ¼ 50 m, tb ¼ 0.5 m, tl ¼ 0.5 m, ρb ¼ 1000 kg m 3, ρl ¼ 2500 kg m 3, ρs ¼ 2500 kg m 3, Eb ¼ 1 GPa, El ¼ 30 GPa, Es ¼ 30 GPa, νb ¼ 0.2, νl ¼ 0.2 and νs ¼ 0.2 to confirm the variation of ratios. These constants can help to assess the shock absorption performance of buffer layer with respect to some particular variations or even obtain an evaluation expression to analyze the shock absorption sensibilities in terms of these particular variations. During the numerical calculation, the ABAQUS code recorded the maximum deformations on the outer surface of lining structure with and without a buffer layer. Before seismic nonlinear calculation, it was necessary to calculate the equilibrium state of initial stress field, which can accurately simulate the after-tunneling mechanical states of sur­ rounding rock around the tunnel structure.

(33)

where, G is the plastic flow potential function; ε_ pl is the equivalent pffiffiffiffiffiffiffiffi pl pl 2=3, ε_ axial is the axial plastic strain ratio. plastic strain ratio; ε_ ¼ ε_ pl axial In the crushable foam model, the plastic volumetric strain was selected in the strain-hardening criterion as follow: � � �� � �� � � �� � pl �� pl � � 2 pt þ σ c εpl 3 pc εpl α þ σ c εplvol 9 þ pt 3 vol ¼ σ c εvol σ c εvol vol

(34)

where, the σ c is the uniaxial compressive strength. The quasi-static strain ratio was determined as 10 5s 1, and the relationship between strength and strain ratio was given as:

3.4. Calculating results analysis and comparisons

(35)

σ c ð_εÞ = ½σc ð_ε0 Þ� ¼ xlgð_ε = ε_ 0 Þ þ y where, ε_ 0 is the initial strain ratio treated as 10

5

(36)

τcrit ¼ μf

In order to obtain the evaluation expression of shock absorption performances in terms of those nine variables, the first step was to

s 1, x and y are the

Table 1 Physical and mechanical parameters of lining structure. Material

Density

ρ/kg⋅m Concrete

3

2000–3000

Elasticity modulus

Poisson’s ratio

Dilatancy angle

Flow strain rate

Yield compressive stress

Ultimate compressive stress

Yield tensile stress

E/GPa

ν

ψ /�

k

σc0/MPa

σcu/MPa

σt0/MPa

20–40

0.1–0.3

36

0.1

13.6

23.3

2.80

6

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Table 2 Physical and mechanical parameters of surrounding rock. Material

Density

ρ/kg⋅m Surrounding rock

Elasticity modulus 3

2000–3000

Poisson’s ratio

Cohesion force

Frictional angle

E/GPa

ν

c/MPa

ϕ/

1–60

0.1–0.5

0.6

39

�

Dilatancy angle

Flow strain rate

ψ/

k

40

0.8

�

Table 3 Physical and mechanical parameters of buffer layer. Material

Density

ρ/kg⋅m Foamed concrete

3

400–1600

Elasticity modulus

Poisson’s ratio

Compression yield stress ratio

Hydrostatic yield stress ratio

Uniaxial compressive strength

E/GPa

ν

r

ri

Rc/MPa

0.3–1.2

0.1–0.5

0.3

0.05

2.3

investigate the variation laws of shock absorption performances con­ cerning a certain variable. Afterwards, the sensitivities of shock ab­ sorption performances with regard to variables needed to be analyzed to form the fitting functions. Then, the insensitive variables should be eliminated to establish the entire explicit expression for assessing the buffer layer with respect to all variables by multivariable regression analysis method.

absorption performances of buffer layer grow in direct proportion to geometric variables (e.g. r0/H and tb/r0) but decrease in inversely pro­ portion to material variables such as Eb/Es, νb and νs. When r0/H � 0.2, the gradient of fitting function curve would not change any more, which means the buried depth should be the best position at r0/H ¼ 0.2 for adopting the buffer layer. Actually, although the initial stress cannot directly affect the stability of lining structure due to the dividing func­ tion of buffer layer, the dynamic strength of deep rocks increases with the increasing depth or the ratio of horizontal-to-vertical initial stress [27]. Thus, the shock absorption effects are more dependent on the performances of buffer layer in deep surrounding rock rather than the combination performances of shallow surrounding rock and buffer layer, thereby decreasing the shock absorption performances with the increasing buried depth. On the other hand, when tb/r0 � 0.2, the gradient variation of fitting function curve can be neglected, that is, the optimal thickness of buffer layer can be confirmed at tb/r0 ¼ 0.2. Besides, the buffer layer should be better have certain static and post-earthquake carrying capacities for tunnel lining structures.

3.4.1. Evaluation coefficient of buffer layer Rb stood for the evaluation coefficient of shock absorption perfor­ mance for buffer layer, which was adopted to deal with the maximum deformation around the outer surface of lining structure. It was a ratio of deformation difference between lining structure covered a buffer layer to the deformation of lining structure without a buffer layer, i.e. � Rb ¼ 1 ul;b ul;s (37) where, ul;b and ul;s are the maximum deformations on outer surface of lining structure with the impact of buffer layer or surrounding rock during excitation, respectively. The relationships between each variables and Rb are listed in Table 4 and Table 5. Apparently, the variations of several variables lead to the sharp increase or decrease of Rb , which means that the Rb is quite sen­ sitive to these variables, which needs to be finely researched.

3.4.3. Regression analysis of multivariable According to equation (38)–(42) and fitting functions in Fig. 5, the evaluation expression of shock absorption performance for buffer layer should be constructed in terms of multiple factors as: Rb ¼ X 1 þ X 2 e

3.4.2. Sensitivity analysis of variables The data in Tables 4 and 5 are selected to draw the sub-diagrams in Fig. 5, which are fit by the functions as follows: (38)

Rb1 ¼ 0:819

0:942e

29:92r0 =H

Rb2 ¼ 0:890

0:828e

21:14tb =r0

(39)

223:41Eb =Es

(40)

Rb3 ¼ 0:328 þ 0:679e Rb4 ¼ 0:761

0:00217e11:34νb

(41)

Rb5 ¼ 0:767

0:010e8:42νs

(42)

29:92r0 =H

where, r0 =H ¼ e

þ X3 e

29:92r0 =H

21:14tb =r0

þ X4 e

, tb =r0 ¼ e

223:41Eb =Es

21:14tb =r0

þ X5 e11:34νb þ X6 e8:42νs (43)

, Eb =Es ¼ e

223:41Eb =Es

, νb ¼

e11:34νb andνs ¼ e8:42νs . Then, there is a relationship between Rb and r0 =H,

tb =r0 , Eb =Es , νb , νs . Thus, the above coefficients can be obtained by multivariable regression analysis method. After that, the evaluation expression of shock absorption performance for buffer layer can be expressed as: Rb ¼ 0:713 0:0004e

0:187e 11:34νb

29:92r0 =H

0:166e

21:14tb =r0

þ 0:135e

223:41Eb =Es

(44)

8:42νs

0:002e

The equation (44) can be applied to guide the selection and instal­ lation of buffer layer for tunnel structures in high intensity seismic zones. It is obvious that the buffer layer materials with acceptable compressibility, low Poisson’s ratio and elasticity modulus can generate ideal shock absorption performances. As for the surrounding rock, which

The interdependence coefficients R2 in equation (38)–(42) are 0.990, 0.991, 0.986, 0.998 and 0.994, respectively, which have the satisfied fitting results. From Table 4, Table 5 and Fig. 5, one can see that the shock Table 4 Relationships between geometric variables and Rb . Variables

Values of geometric variables and Rb

r0/H Rb tb/r0 Rb

0.01 0.109 0.01 0.198

0.02 0.304 0.02 0.357

0.03 0.453 0.03 0.486

0.05 0.618 0.05 0.583

0.1 0.714 0.1 0.782

7

0.2 0.818 0.2 0.852

0.3 0.826 0.3 0.897

0.4 0.831 0.4 0.903

0.5 0.835 0.5 0.905

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Table 5 Relationships between material variables and Rb . Variables

Values of material variables and Rb

Eb/Es Rb

0.001 0.893 0.1 0.765 0.1 0.732

νb

Rb

νs

Rb

0.002 0.736 0.15 0.752 0.15 0.724

0.005 0.544 0.2 0.734 0.2 0.718

0.01 0.423 0.25 0.721 0.25 0.694

0.02 0.368 0.3 0.693 0.3 0.664

0.03 0.337 0.35 0.638 0.35 0.597

0.04 0.324 0.4 0.559 0.4 0.456

0.05 0.312 0.45 0.416 0.45 0.327

0.1 0.302 0.5 0.128 0.5 0.116

Fig. 5. The impacts of geometric and material parameters on sensitivities of shock absorption performances: (a) The impact of r0/H; (b) The impact of tb/r0; (c) The impact of Eb/Es; (d) The impact of νb; (e) The impact of νs.

should be better have favorable compressibility and low Poisson’s ratio as well.

numerical simulations with real seismic motion excitation were rigor­ ously conducted.

4. Validations for optimized buffer layer by shaking table model tests and numerical simulations

4.1. Similitude relations confirmation and derivation The numerical calculating model in section 3 was deemed as the prototype to determine the similitude relations between prototype and models in shaking table model tests. Due to the restrictions of shaking table (e.g. maximum load capacity and overall dimensions), a model

To validate the equation (44) and assess the shock absorption per­ formances of buffer layer with different materials (e.g. rubber sponge and ebonite), series shaking table model tests and the corresponding 8

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Table 6 Similitude ratios for shaking table model tests. Name Similitude ratios

Geometry

Elasticity modulus

Density

Cohesive strength

Friction angle

λL ¼ Lm/Lp

λE ¼ Em/Ep

λρ ¼ ρm/ρp

λC ¼ λm/λp

Φ( )

λA ¼ Am/Ap

λt ¼ tm/tp

1/25

1/32.5

1/1.3

1/42

1/1.0

1/1.0

1/5

scale of 1:25 was adopted as geometry similitude ratio in these vali­ dating model tests. Taking into account the actual seismic responses of surrounding rock, buffer layer and lining structure, three fundamental independent similitude ratios (i.e. geometry, density and elasticity modulus) should sternly obey the similitude relations, which were derived based on Buckingham-π theorem with an equation as follow [28]:

Acceleration

Time

relations. It tool 7–8 h to let the settlement and compaction happen in the surrounding rock material before excitation, which would veritably reproduce the seismic responses as the surrounding rock around the practical tunnel engineering. Currently, rubber sponge and ebonite usually act as shock absorption and isolation materials in majority ground structures. Thus, they were also employed into model tests and numerical simulations to validate the evaluation expression of shock absorption performance for buffer layer. Fig. 6 (a) shows the rubber sponge with a thickness of 5 cm while Fig. 6 (b) illustrates the ebonite with a thickness of 2 cm as the buffer layer around the outer surfaces of lining models. All the materials pa­ rameters in model tests and numerical simulations are listed in Table 7.

(45)

λε ⋅ λE ¼ λL λρ ⋅ λA

�

where, λL, λρ and λE are the similitude ratios of geometry, density and elasticity modulus, respectively, which are employed to generate the authentic seismic responses, thereby deriving other similitude ratios. Further, both the similitude ratios of gravity (λA) and strain (λε) should be equal to one. The former resulted from the identical gravitational field while the latter guaranteed the comparability between the results of shaking table model tests and numerical simulations. The similitude relations employed in these validating model tests were listed in Table 6.

4.3. Test schemes and monitoring instruments The E-D component of Wolong pulse-like ground motion recorded from Wenchuan Great Earthquake was adopted as the real seismic input motion for both of model tests and numerical simulations [29], which was constructed and plotted in Fig. 7 according to the similitude relations. To validate the evaluation expression of shock absorption perfor­ mance for buffer layer, series model tests were conducted by applying the same seismic input motion with the identical durations but under different intensities [30]. Thereby, the input peak accelerations (IPAs) were scaled to five levels as shear waves (from 0.2 g to approximately 1.0 g with a common difference of 0.2 g, where, g ¼ 9.81 m/s2). Fig. 8 exhibits test schemes and monitoring instruments in shaking table model tests. Fig. 8 (a) shows a three-dimensional scene of lining model and the strain gauges layout. There are two strain monitoring cross-section on the outer and inner surfaces of lining model to acquire the deformations and internal forces. Thus, thirty-two strain gauges are installed on each lining model in total. As the introduction of in­ struments, the letters on strain monitoring positions refer to the prop­ erties of different items (e.g. the first letters such as L and R mean left and right. The second letters such as C, S, W, A and I mean crown, spandrel, sidewall, arch springing and invert, respectively. Letters in brackets like I and O mean inner and outer surfaces of lining model, respectively.). Fig. 8 (b) illustrates the lining models and accelerometers in trans­ verse profile, which guides the comparison of lining model deformations with and without a buffer layer. Specifically, model A represents the lining model without a cover while model B shows the lining model with

4.2. Surrounding rock and tunnel lining models To generate the realistic seismic mechanical behavior and monitor deformations on outer surface of lining structure, the sieved cement, aggregate and water with the weight ratio of 1:6:1 made up of lining model with a thickness of 1.8 cm and a length of 80 cm. Further, the sieved cement was ordinary Portland cement with 1% fineness while the aggregate was medium sand with the 0.15–0.25 mm particle diameter. Additionally, steel wire mesh with a diameter of 1.06 mm and a spacing of 25.4 mm was installed in the middle of lining model to simulate the rebar, since it was necessary to reappear the lining structure in model level to generate the veritable seismic responses. With the similar con­ dition in numerical calculation, there was no primary supports around the lining model. Moreover, a mixture of 31% river sand, 57% fly ash and 12% engine oil was acted as the aggregate, cement and accessory materials, respectively in synthetic surrounding rock model to produce the favor­ able coherence with the governing similitude relations. Considering the dominated deformations of surrounding rock during excitation, it was necessary to accurately reproduce the surrounding rock in the model container with the above material. Thus, the surrounding rock material was equably spread into the model container around the lining models layer by layer. Each layer was tempted to 10 cm to guarantee the ac­ curate and consistent material parameters according to similitude

Fig. 6. Simulation materials of buffer layer in shaking table model tests. (a) Rubber sponge; (b) Ebonite. 9

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Table 7 Mechanical parameters for prototype and model of rock-layer-lining system. Physical Quantity

Cohesion c (kPa)

Friction Angle ϕ(� )

Elasticity modulus E (MPa)

Density ρ (g/cm3)

Poisson’s ratio ν

Compression Strength σ (MPa)

Rock prototype Rock model Lining prototype Lining model Rubber sponge Ebonite

50–200 1.19–4.76 – – – –

20–27 20–27 – – – –

1.3e3-6e3 40–185 28e3 860 1.38 5.0

1.7–2.0 1.21–1.43 2.5 1.94 0.06 0.85

0.25 0.25 0.2 0.2 0.3 0.45

– – 12.5 0.3 – –

Fig. 7. Wolong ground motion from Wenchuan Great Earthquake. (a) Acceleration time history of E-W component; (b) Fourier spectrum of E-W component.

a rubber sponge or an ebonite buffer layer. Lining models, surrounding rock and shaking table are instrumented with 27 accelerometers in total. Accelerometer A0 is fastened on the shaking table to monitor the pre­ cision of input motions and guarantees the comparability between the accelerations recorded by the accelerometers in and out of the model container. A1 and A2 are placed at the bottom surface of model container whilst A25 and A26 are fixed on the surface of surrounding rock material. Further, there are two vertical lines across symmetry axes of lining models, the one strings together with accelerometers A1, A3, A11, A7, A23 and A25 while the other is formed by the a bunch of ac­ celerometers as A2, A4, A19, A15, A24 and A26. The time history graphs at crowns (A7 and A15), spandrels (A8, A14, A16 and A22), sidewalls (A9, A13, A17 and A21), arch springing (A10, A12, A18 and A20) and inverts (A11 and A19) of each lining model are recorded by eight ac­ celerometers. Besides, A5 and A6 act as monitors for checking mitigation measures of boundary effect.

comparing the results of model tests and numerical simulations. The constant values of Rb are marked as red dash lines in Fig. 12 (a) and (b). Apparently, both of model tests and numerical simulations can suc­ cessfully validate the reliability of evaluation expression of shock ab­ sorption performance regardless of the buffer layer parameters, although the results of model tests have a certain deviation due to some uncontrollable factors in conducting process (e.g. progressive destruc­ tion process of lining structures and permanent plastic deformations of surrounding rock). Specifically, the Rb values of model tests sharply drop when the IPA is beyond 0.6 g because the lining models start to break, thereby, the buffer layer cannot play a role of protection any more, which is out of the range in this research as well. Apparently, both shock absorption performances of rubber sponge and ebonite as buffer layers are almost identically validated both in shaking table model tests and in numerical simulations. In summary, equation (44) is able to elaborately evaluate the shock absorption performance of buffer layer before the lining structure in the seismic induced breakage phase.

4.4. Validating results analysis and comparisons

5. Discussion and conclusions

Figs. 9 and 10 and Fig. 11 illustrate the maximum deformations of surrounding rock and lining structures with and without a buffer layers. The numerical simulation method and process are in accordance with section 3 but input Wolong seismic motion, then Rb value can be uniquely confirmed by the given simulating parameters. Thus, the simulation results can be utilized to obtain Rb values to compared with the results of shaking table tests. From the above figures, one can see that the deformation patterns of lining structures are identical with the result of theoretical analysis, that is, the two orthogonal diagonal diameters (θ ¼ �45� ) of cross section alternatively expand and contract during excitation. Therefore, all the maximum deformations occur at lining spandrels regardless of the ex­ istence of buffer layer. However, the existence of buffer layer can effectively reduce the relative deformation amplitude of lining struc­ tures and the surrounding rock near tunnels, which can even make the deformations gentler and more reasonable. Apparently, the rubber sponge as buffer layer has the better shock absorption performance due to its better geometric and material parameters. The Rb should have a constant value once the geometric and material parameters are confirmed, thus, the equation (44) would be validated by

Several influencing parameters with respect to shock absorption performances of buffer layer were derived via investigating the inter­ action between lining structure and surrounding rock during excitation. Afterwards, series numerical calculations was systematically conducted with attention given to the seismic-induced deformation responses of rock-layer-lining system. Then, an evaluation expression for elaborately assessing shock absorption performance of buffer layer was presented by multivariable regression analysis. Finally, series shaking table model tests and numerical simulations validated the reliability and accuracy of this evaluation expression, which would be able to guide the selection and construction of buffer layer according to the geological conditions and construction circumstances by changing geometric and material parameters. Although this research did not consider the progressive destruction process of lining structure, seismic performance of primary supports and sustainable serviceability of buffer layer, these factors would not directly affect shock absorption performances of buffer layer in terms of conventional mechanics except for introducing more irrelevant vari­ ables. Therefore, this research generates several key findings that are 10

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Fig. 8. Test schemes and instruments layouts for shaking table model tests (Unit: cm). (a) Three-dimensional scene of lining model; (b) Transversal profile of model tests schemes.

Fig. 9. The maximum deformations of simulation models with rubber sponge as buffer layer (IPA ¼ 0.6 g): (a) The maximum deformations of surrounding rock; (b) The maximum deformations of lining structure.

11

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

Fig. 10. The maximum deformations of simulation models with ebonite as buffer layer (IPA ¼ 0.6 g): (a) The maximum deformations of surrounding rock; (b) The maximum deformations of lining structure.

Fig. 11. The maximum deformations of simulation models without a buffer layer (IPA ¼ 0.6 g): (a) The maximum deformations of surrounding rock; (b) The maximum deformations of lining structure.

Fig. 12. The results comparisons of model tests and numerical simulations for assessing the evaluation expression of shock absorption performance: (a) Rubber sponge; (b) Ebonite.

summarized herein:

(2) Both of the geometric and material parameters determine the shock absorption performance of buffer layer, which grows in direct proportion to geometric variables but decreases in inversely proportion to material variables. (3) When the ratio of lining inside radius to buried depth is equal to 0.2 (i.e. r0/H ¼ 0.2), the buried depth of tunnel engineering should be the ideal position for adopting buffer layer. Meanwhile, when the ratio of buffer layer thickness to lining inside radius is

(1) The unlined tunnel and lining structure have the identical particular deformation pattern during excitation, that is, their two orthogonal diagonal diameters (θ ¼ �45� ) of cross section alternatively expand and contract, thereby causing the spandrel on lining cross section to be the most dangerous position.

12

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Soil Dynamics and Earthquake Engineering 131 (2020) 106032

equal to 0.2 (i.e. tb/r0 ¼ 0.2), it can be confirmed the optimal thickness of buffer layer. (4) The evaluation expression of shock absorption performance is independent of seismic waveforms but with respect to the exci­ tation intensities and the seismic induced destruction states of lining structures.

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Declaration of competing interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. CRediT authorship contribution statement C.L. Xin: Data curation, Writing - original draft, Formal analysis, Validation, Writing - review & editing. Z.Z. Wang: Conceptualization, Methodology, Supervision, Funding acquisition. J. Yu: Software. Acknowledgements The National Natural Science Foundation of China (Grant No.51778109) as well as the Miaozi Engineering Key Project from Sci­ ence and Technology Department of Sichuan Province (Grant No. 20MZGC0158) supported this research. All supports are gratefully acknowledged. References [1] Gao Y, Xu F, Zhang Q, et al. Geotechnical monitoring and analyses on the stability and health of a large cross-section railway tunnel constructed in a seismic area. Measurement 2018;122:620–9. [2] Su LJ, Liu HQ, Yao GC, et al. Experimental study on the closed-cell aluminum foam shock absorption layer of a high-speed railway tunnel. Soil Dyn Earthq Eng 2019; 119:331–45. [3] Wang ZZ, Jiang YJ, Zhu CA, et al. Shaking table tests of tunnel linings in progressive states of damage. Tunn Undergr Space Technol 2015;50:109–17. [4] Chen ZY, Shen H. Dynamic centrifuge tests on isolation mechanism of tunnels subjected to seismic shaking. Tunn Undergr Space Technol 2014;42:67–77. [5] Shan B, Zheng Y, Liu CL, et al. Coseismic Coulomb failure stress changes caused by the 2017 M7.0 Jiuzhaigou earthquake, and its relationship with the 2008 Wenchuan earthquake. Sci China Earth Sci 2017;60(12):2181–9. [6] Cheng XS, Li D, Ren Y, et al. Seismic stability analysis of subsea tunnels under the effects of seepage and temperature. Mar Georesour Geotechnol 2017;35(6): 808–16. [7] Deng XF, Zhu JB, Chen SG, et al. Numerical study on tunnel damage subject to blast-induced shock wave in jointed rock masses. Tunn Undergr Space Technol 2014;43:88–100.

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