Numerical simulation of dynamic response of operating metro tunnel induced by ground explosion

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1 Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): Numerical simulatin f dynamic respnse f perating metr tunnel induced by grund explsin Yubing Yang 1, Xingya Xie 1*, Rulu Wang 2 1 Key Labratry f Getechnical Engineering, Department f Getechnical Engineering, Tngji University, Shanghai, 292, China 2 Shanghai Metr Operatin C., Ltd., Shanghai, 23, China Received 1 Octber 21; received in revised frm 28 Octber 21; accepted 1 Nvember 21 Abstract: T evaluate the effects f pssible grund explsin n a shallw-buried metr tunnel, this paper attempts t analyze the dynamic respnses f the perating metr tunnel in sft sil, using a widely applied explicit dynamic nnlinear finite element sftware ANSYS/LS-DYNA. The blast induced wave prpagatin in the sil and the tunnel, and the vn Mises effective stress and acceleratin f the tunnel lining were presented, and the safety f the tunnel lining was evaluated based n the failure criterin. Besides, the parametric study f the sil was als carried ut. The numerical results indicate that the upper part f the tunnel lining crss-sectin with directins ranging frm t 22.5 and hrizntal distances t 7 m away frm the explsive center are the vulnerable areas, and the metr tunnel might be safe when tunnel depth is mre than 7 m and TNT charge n the grund is n mre than 5 kg, and the selectin f sil parameters shuld be paid mre attentins t cnduct a mre precise analysis. Key wrds: grund surface explsin; numerical simulatin; metr tunnel; dynamic respnse 1 Intrductin Recently, the wrldwide terrrism attacks are becming intensive and mre frequent. Vehicle bmb is the main way t implement the terrrist activities due t its massive charge pwer, high success rati and serius destructin [1]. Frm the blasting events ccurring in recent years such as the Wrld Trade Centre f New Yrk in 21, and thse in Chechnya (22) and Lndn (25), the terrristic blast raid will nt nly result in the damages f building structures, huge lsses f lives and prperties, but als prbably threaten the safety f an perating metr tunnel with the increase f TNT equivalence fr vehicle bmbs. Three types f methds can be used t analyze the Di: /SP.J *Crrespnding authr. Tel: ; xiexingya@tngji.edu.cn Supprted by the Natinal Natural Science Fundatin f China (487474, ), the Natinal High Technlgy Research and Develpment Prgram (863 Prgram) f China (26AA11ZAA8), and Shanghai Science and Technlgy Develpment Funds (7ZR14117) dynamic behavirs f structures under blast lad: theretical slutins, experimental study and numerical methds. Generally, the fllwing subsystems are invlved: (1) the prpagatin f blast induced waves in air, rck r sil; (2) the dynamic respnse behavirs f structures; and (3) the material damage analysis f structures [2, 3]. Fr the prpagatin f blast induced waves in the air and rcks, a lt f studies have been dne [4 6]. With respect t the undergrund structures such as metr tunnels, a clsed-frm slutin fr this cmplicated prblem is almst unavailable at present, because f the necessary simplificatin and subdivisin in the theretical mdel and a large amunt f calculatins. As fr the experimental study, labratry mdel tests and field prttype investigatins may be the tw pssible chices. Unfrtunately, until nw n reprts n the field prttype experiment can be fund in China. In Russia, a castal surface explsin experiment with a 1 -tn TNT charge was perfrmed n August 25, 1987, apprximately 1 m away frm the castal line. Mre details can be fund in Ref.[7]. Varius limitatins, such as the apprpriate selectin f the structure mdel, the nnlinearity f

2 374 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): rcks r sils, the input f wave lads, the high cst f tests, have als impeded the prgress f the experimental study [8]. Recently, the rapid develpments f numerical methds have prvided strng supprts fr the behavir study f undergrund structures under dynamic lads, amng which the finite element methd (FEM) may be the researchers favurite chice fr its ability in dealing with nnlinear and anistrpic prblems. As t these unique advantages, the FEMs have been widely used in this field. A lt f researches [9 11] have been cnducted t analyze the effect f an internal explsin n tunnels. Fr the explsin effect n the grund surface, Lu et al. [12] analyzed the dynamic respnses f the tunnel fr surface explsins f 1 and 3 kg TNT charges, respectively, accrding t the features f Nanjing subway tunnel in sandy sils. Fan et al. [13] discussed the lad characteristics n a shallw-buried cncrete structure under a grund explsin by means f numerical simulatins. Hwever, there is little wrk fcusing n the respnses f metr tunnels in sft sil induced by grund explsin. The studies fr the prpagatin f blast induced waves in the sft sil, the dynamic respnses f a metr tunnel and its safety are carried ut in the paper, using a widely applied explicit dynamic nnlinear finite element sftware ANSYS/LS-DYNA. Firstly, the backgrund f mdelling is presented briefly, including the size selectin f the mdel, the sil type, the material cnstitutive mdels and parameters, etc.. Secndly, a case withut tunnel structure in the sft sil is intrduced t analyze the blast induced wave prpagatin in the sft sil. Then six cases f different explsive charges and tunnel depths are cnsidered t analyze the dynamic respnses f the tunnel. The prpagatin f the blast induced waves in the tunnel, the vn Mises effective stress and the acceleratin f the tunnel lining are presented. The safety f the tunnel lining based n the failure criterin is evaluated. Finally the parametric study f sft sil is dne t analyze the effect n the calculating results. 2 Numerical mdel 2.1 Backgrund and finite element mdel The prject f the Shanghai metr line N.1 is cnsidered in this paper [14]. The tunnel lining has a circular shape, with 5.5 m in inner diameter and 6.2 m in uter diameter. The typical tunnel depths frm Caba Rad Statin t Shanghai Railway Statin range frm 7 t 15 m. Mrever, accrding t the typical Shanghai stratigraphic distributin, the sil layer arund the tunnel cnsidered in analysis cnsists f 6 types f sils frm the tp dwn, i.e. the miscellaneus fill, the yellwish dark brwn silty clay, the gray silty clay, the gray mucky silty clay, the gray clay, and the yellwish dark brwn clay. Their physic-mechanical parameters can be fund in Ref.[15]. Fr many previus studies n systems laded explsively, equivalent time-histry pressures were used t simulate the lads. Time-histry pressures generated with high explsives tend t exhibit large variatins, even with identical charges. Obviusly, whether the respnses f the structure can be predicted strngly depend n the ability t generate accurate lad functins [16]. In this study, the explsive was mdeled explicitly using a ANSYS/LS-DYNA material specifically designed fr simulating a high explsive detnatin. It is assumed that the explsin will take place at the mst unfavrable psitins such as the interface f air and sil, abve the metr tunnel. Cnsidering the symmetries f the tunnel structure and the blast lad, fr the sake f saving cmputatin time, a 1/4 symmetrical gemetrical mdel with a size f 25 m 25 m 3 m was established based n the Alekseenk test [17] (Fig.1). Fig.1 Ttal and 1/4 symmetrical gemetrical mdels. In the finite element mdel (Fig.2), the eight-nde element f SOLID 164 is adpted fr the 3D explicit analysis. In rder t prevent the element distrtin in large defrmatin and nnlinear structural analyses, an arbitrary Lagrangian-Eularian (ALE) algrism is used in this paper. The TNT charge, the air and the sil are mdeled with ALE multi-material meshes, but the tunnel lining with Lagrangian meshes, while the minimal time step is cntrlled by the smallest element size in the explicit integral methd, and the glbe unifrm mesh size is set t be 5 cm.

3 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): Table 2 Parameters f the TNT charge [19]. (g/cm 3 ) ν D (m/s) P CJ (MPa) A (MPa) B (MPa) R 1 R 2 V E ( J/m 3 ) Fig.2 The finite element mdel. Furthermre, the transitinal displacement f the ndes nrmal t the symmetry planes (XZ and YZ planes) is cnstrained. Nn-reflecting bundary cnditin is applied t the ther tw lateral surfaces and the bttm surface, and the free bundary cnditin is used fr the upper surface. 2.2 Material cnstitutive mdels and parameters Fur kinds f materials are invlved in this finite element mdel: air, TNT charge, tunnel lining and sil. The air is cmmnly mdeled by null material mdel with a linear plynmial equatin f state (EOS), which defines the pressure by the fllwing equatin: P C C1 C2 C3 ( C4 C5 C6 ) E (1) where the parameter is defined as / 1, is the current density, and is a nminal r reference density; C C 6 are the equatin cefficients; and the parameter E is the initial internal energy f reference specific vlume per unit. Table 1 gives the parameters used in the air mdel in Refs.[1, 18]. Table 1 Parameters f the air. (g/cm 3 ) C C 1 C 2 C 3 C 4 C 5 C 6 E (J/m 3 ) (g/cm 3 ) The TNT charge is mdeled by the high explsive material mdel and the Jnes-Wilkins-Lee (JWL) equatin f state [18]. The JWL equatin f state defines the pressure by the fllwing equatin: RV 1 R2V E p A 1 e B 1 e (2) RV 1 RV 2 V where ABR,, 1, R2, are the equatin cefficients and they all shuld be tested in an accurate blast analysis; and V is the initial relative vlume. Here we adpt the parameters selected frm Ref.[19], as shwn in Table 2. The tunnel lining is mdeled by the plastic kinematic mdel fr simplicity and applicability. It is a mixed mdel in which the hardening cefficient is used t adjust the cntributin prprtins f istrpic hardening and kinematic hardening. The main parameters in this mdel include mass density, Yung s mdulus E, Pissn s rati, yield stress v, tangent mdulus E tan, hardening parameter, failure strain fr erding elements f. Because the respnse f the reinfrced cncrete t the dynamic lad is always a cmplex nnlinear and rate-dependent prcess, quite a few mdels have been prpsed t describe its dynamic behavir [2]. In additin, the tunnel lining is usually reinfrced with cncrete. Fr simplicity, the steel bar and the cncrete are regarded as a whle accrding t the principle f equivalent stiffness EI in this study. Table 3 gives the parameters in the tunnel lining mdel. Table 3 Parameters f the tunnel lining. (g/cm 3 ) E (GPa) y (MPa) E tan (MPa) f The sil is mdeled by a sil and fam mdel put frward by Krieg in 1972 [21]. It is a simple mdel and perates in sme way like a fluid, and has been demnstrated t be useful fr sil mdeling [22]. The main parameters in this mdel include: mass density, shear mdulus G, bulk mdulus K u at unlading path, yield functin cnstants a, a 1 and a 2, pressure cutff fr tensile fractures p cut. Table 4 gives the main parameters in the sil and fam mdel based n Ref.[23]. It is assumed that the entire sil layer has the same parameters as the gray mucky silty clay fr cmputatinal simplicity. Table 4 Main parameters in the sil and fam mdel. (g/cm 3 ) G (MPa) K u (MPa) a a 1 a 2 p cut (MPa) Numerical results and discussins Kng et al. [1] presented that the scale f vehicle

4 376 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): bmbs culd range frm 2 kg TNT equivalence f a cmmn family car t 1 kg TNT equivalence f a small bx van, hence three cases f TNT charge are cnsidered in this paper: 3, 5 and 1 kg. Cnsidering the tunnel depths f 7 and 14 m, abut ne and tw times the tunnel diameter, respectively, under the situatin f free field (n tunnel structure in the sil), the fllwing seven calculatin cases are perfrmed (Table 5). (a) t =.12 s. (b) t =.32 s. Table 5 Calculatin cases. Case Tunnel depth (m) TNT (kg) Tw research paths are selected t analyze the dynamic respnses f the tunnel lining (Fig.3). Path 1 is alng the transverse directin, 5 typical pints are selected arund the crss-sectin; and path 2 is alng the lngitudinal directin, the hrizntal distance frm the explsin center ranges frm t 2 m. Fig.3 Tw research paths. 3.1 Prpagatin f the blast induced waves in the sil Figure 4 shws the wave prpagatin in the sil at different times in case. It demnstrates that the pressure waves prpagate in the sil in the frm f hemispherical waves. The area f wave frnt increases with the wave prpagatin. The affected regin f the sil in case is abut 14 m beneath the grund surface and the duratin is arund.12 s. The extruding part f the sil layer in each subfigure is rather remarkable. Actually, the explsin will cause a crater thrugh the ejectin f the sil (c) t =.64 s. (d) t =.1 s. Fig.4 Pressure cnturs in the sil at different times in case (unit: 1 2 GPa). away frm the blast. Simulating this prcess during which small parts mve away with a cntinuus finite element becmes very difficult. This mdel des nt prvide a gd slutin. We assume that the large defrmatin f the sil in vicinity f the blast center, the abve-mentined extruding part, represents that a crater will be frmed. Tw examples in reality are given here, accrding t which we can realize the huge destructive pwer f the grund surface explsin. Figure 5(a) shws the ruins at Ryngchn railway statin in Nrth Krea after a huge explsin (April 24, 24). It was reprted that this explsin was induced by a cllisin between tw trains, which carried il and gas and ammnium nitrate, respectively. Tw craters were frmed in the explsin center, and the bigger ne shwn in Fig.5(a) had a diameter f 3 m and a depth f 1 m. Figure 5(b) shws a crater in a factry in Anhui Prvince f China after an explsin (June 21, 29). The explsive hidden in a building withut authrizatin (a) The ruins at Ryngchn railway statin in Nrth Krea.

5 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): (b) A crater in a factry in Anhui Prvince f China. Fig.5 Tw examples in reality. by the factry s wner was said t be 5 7 tns, and the crater had a diameter f 1 m and a depth f 5 m. Figure 6 describes the cmpressive waves in the sil at different depths belw the explsin center (4, 7, 1, 14 and 2 m). Accrding t the definitin in the Alekseenk test [17] (Fig.7), these psitins are lcated in the central zne, and the cmpressive waves in the sil are mainly the grund shck wave. Pressure (MPa) Pint A. 4mA (4 m) Pint B. 7m(1D) B (7 m) Pint C. 1mC (1 m) Pint D. 14m(2D) D (14 m) Pint E. 2mE (2 m) Time (s) Fig.6 Cmpressive waves in the sil at different depths (case ). Central zne Surface zne Fig.7 Successive lcatins f wave frnt [17]. We can see the prpagatin and attenuatin f the cmpressive waves in the sil. Each cmpressive wave, except the ne at pint A, has nly ne peak value induced by the direct grund shck, which exactly crrespnds t the results in the Alekseenk test [17]. The discrepancy at pint A may be explained by its clse distance away frm the surface zne. The first peak value can be caused by the grund wave, the secnd may be caused by the air shck wave, and the third may be the reflected wave at the sil interface. 3.2 Cmparisn between numerical results and predictin by the manual (TM ) Figure 8 shws the vertical acceleratin f these five pints. The peak values f pints A, B and C are , 129.8, 57.7 m/s 2, respectively. Vertical acceleratin (m/s 2 ) Pint A. 4mA (4 m) Pint B. 7m(1D) B (7 m) Pint C. 1mC (1 m) Pint D. 14m(2D) D (14 m) Pint E. 2mE (2 m) Time (s) Fig.8 Vertical acceleratin in the sil at different depths (case ). The US Army Crps f Engineers Manual (TM ) has been widely used t estimate the grund shck parameters. It adpts the cube-rt scaled distance t predict grund shck parameters. The fllwing equatins are prvided in the manual (TM ) t predict the peak values f pressure and acceleratin, respectively [24]: n R Pp.47 f c 1/3 W (3) a fc R W W p 1/3 1/3 ( n 1) (4) where P p is the peak pressure (Pa); f is a cupling factr, which is dependent n the scaled depth f the 1/3 explsin and is given by d / W, d is the depth f the centrid f the explsive charge; c is the acustic impedance; c is the seismic velcity; R is the distance frm the surce; W is the charge weight; n is

6 378 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): an attenuatin cefficient; and a p is the peak acceleratin. Based n the seismic velcity f Shanghai sft sil, the sil in this mdel may be described as the type 4 in the manual (TM ) (Table 6). Table 6 Sil prperties fr calculating grund shck parameters [24]. Sil type Seismic Acustic Density, Attenuatin velcity, c impedance, c (kg/m 3 ) cefficient, n (m/s) (1 6 Pa s/m) (1) Heavy saturated > clays and clay shale (2) Saturated sandy clays and sands with air vids < 1% (3) Dense sand with high relative density, wet sandy clay with air vids > 4% (4) Sandy lam, less, dry sands and backfills (5) Lse, dry sands and gravels with lw relative density Using the parameters f the sil type 4 in Table 6, and assuming that f is equal t 1., we can cmpare the numerical results btained in this paper with the predicted nes using the manual (TM ), as shwn in Tables 7 and 8, respectively. The difference f the peak pressures is expressed as (predictin numerical result)/ numerical result. Table 7 Cmparisn f peak pressures between numerical results and predictins with the manual (TM ). W (kg) 25 Peak pressure (MPa) R R/W 1/3 Difference (m) (m/kg 1/3 ) Predictins ( c = Numerical (%) GPa s/m, n = 2.75) results Frm Tables 7 and 8, it can be seen that the predictins with the manual (TM ) are higher than the numerical results fr this mdel. Hwever, we can als find that the numerical results f the peak pressure almst have the same rder f magnitude as the predictins. This phenmenn can be elucidated in the fllwing tw aspects: Table 8 Cmparisn f peak acceleratins between numerical results and predictins with the manual (TM ). W (kg) 25 R R/W 1/3 Peak acceleratin (m/s 2 ) Difference (m) (m/kg 1/3 ) Predictins Numerical (%) (c = 34.8 m/s, n = 2.75) results (1) The predictins with the manual (TM ) are based n the assumptin that the buried-depth f TNT charge is big enugh t frm a full cntainment explsin (f = 1.). Hwever, the case in this paper, which ccurs at the interface f sil and air, i.e. the buried-depth f TNT charge is zer and the cupling cefficient f will be smaller than 1., cannt cincide well with the assumptin. Hence, it s natural that the numerical results are lwer than the predictins. (2) The Alekseenk test in 1967 indicated that, if the charge was buried in the sil and its upper surface was at the same level as the grund surface, such as the case in this study, the prprtin f energy absrbed by the air and the sil wuld be 53% and 47%, respectively. That means mre than half energy disperses in the air. This cnclusin can explain why the cmputed results are smaller than the predictins. In ttal, the numerical results mentined abve suggest that this finite element mdel, t sme extent, is ratinal t simulate the dynamic respnses f undergrund structures under explsin lads n the grund surface. Figure 9 presents the wave prpagatin in the sil at different times fr case 3. (a) t =.12 s. (b) t =.32 s. (c) t =.64 s. (d) t =.1 s. Fig.9 Pressure cnturs in the sil at different times fr case 3 (unit: 1 2 GPa).

7 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): In Fig.9, we can see the affected regin f the sil in case 3 just reaches the middle part f the tunnel, abut 1 m away frm the grund surface, and the duratin is abut.12 s. Obviusly, the difference between the tw cases (cases and 3) is the affected regin in the sil. The existence f the tunnel structure in the sil prevents the blast induced waves frm migrating t a deeper sil layer. The energy f the blast induced waves is transferred frm the sil t the tunnel lining. 3.3 Prpagatin f the blast induced waves n the tunnel Figure 1 shws the wave prpagatin n the tunnel at different times in case 3. It presents the whle prcess f the prpagatin f blast induced waves n the tunnel: expansin, migratin and dissipatin. The respnse f tunnel ccurs at t =.32 s and ends at arund t =.5 s. directins, unlike thse in the sil, and they have n fixed frms and nticeable wave frnts. It is quite clear that the lngitudinal prpagatin is faster than the circumferential ne. Figure 11 shws the prpagatin and the attenuatin f blast induced waves alng path 1 f the tunnel in case 3, where the pressure is psitive in cmpressin and negative in tensin. Pressure (MPa) Pint A. A (º) Pint B B (22.5º) Pint C. 45 C (45º) Pint D. 9 D (9º) Pint E. 18 E (18º) Time (s) Time / s 4 (a) t =.32 s. (b) t =.44 s. (c) t =.64 s. (d) t =.1 s. (e) t =.16 s. (f) t =.25 s. (g) t =.35 s. (h) t =.5 s. Fig.1 Pressure cnturs n the tunnel at different times fr case 3 (unit: 1 2 GPa). The pressure waves prpagate n the tunnel alng bth the lngitudinal and the circumferential Fig.11 Blast induced waves at five different pints alng path 1 in case 3. It demnstrates that the upper lining (pints A, B and C) is cmpressed but the lwer lining (pints D and E) is tensined under the explsive lad. Every pint has a peak value at t =.1 s and then the value decreases with time gradually, and finally clse t zer. The maximum f all 5 peak values ccurring at pint A is 9.93 MPa, and the rest are 7.87, 2.53, 3.2 and 3.95 MPa at pints B, C, D and E, respectively. Cmpared with the pint A, they decrease by 2.7%, 74.5%, 67.8% and 6.2%, respectively. Therefre, the tp f the tunnel (pint A) is liable t be destryed, but the middle-upper part (pint C) is the safest. The prpagatin and the attenuatin f blast induced waves alng path 1 in ther five cases are nt given, as their trends are similar t thse in case 3, but the peak values are smaller. In Fig.12(a), the peak pressures in all cases alng path 1 are cmpared, frm which we can cnclude that the maximum pressure in all cases ccurs at pint A (º), but fr different TNT charges and tunnel depths, the peak value varies. When the tunnel depth is 7 m, as the TNT charge increases frm 3 t 1 kg (cases 1, 2 and 3), the peak values at pint A are 4.15, 5.86 and 9.93 MPa, respectively; while the tunnel depth is 14 m, the crrespnding values (cases 4, 5 and 6) are

8 38 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): Peak pressure (MPa) / Mpa Peak pressure (MPa) / Mpa Case 1 Case 2 Case 4 Case 5 Case Lcatin ( ) Lcatin / (a) Alng path Case 1 Case 2 Case 4 Case 5 Case Hrizntal 4 distance 8frm the explsive 12 center 16 (m) 2 Hrizntal diastance frm the explsive center / m (b) Alng path 2. Fig.12 Cmparisn f peak pressure n the tunnel alng different paths (t =.64 s fr cases 1, 2 and 3; t =.1 s fr cases 4, 5 and 6)..98, 1.58 and 2.54 MPa, respectively. The decreasing amplitudes reach 76.4%, 73.% and 74.4%, respectively. Hence, the effect f deep-buried tunnel (abut 2D in this paper, D is the tunnel diameter) under blasting lads is reduced by 75% than that f the shallw-buried tunnel (abut 1D). Figure 12(b) shws the peak pressure f six cases alng path 2, frm which we can knw that the maximum pressure is 1.6 MPa, which ccurs in case 3 at a hrizntal distance abut 2.5 m away frm the explsin center. That means it will prbably fail at this psitin in case 3 if the failure f the lining happens. 3.4 Failure evaluatin based n the effective stress In sme ways, we knw that the vn Mises effective stress can be regarded as a uniaxial equivalence i f a multi-axial stress state and used in many failure r yield criteria. Thus, if a material is knwn t fail in a uniaxial cmpressin test (with 1 the nly nnzer stress cmpnent) when 1 crit, it will fail under multi-axial lading. Figure 13 shws the effective stresses n the tunnel alng path 1 in case 3. Every pint has ne peak at t =.1 s, and the maximum f 5 peak values is 26.2 MPa at pint A, the rest are 21.5, 6.15, 1.3 and 11.3 MPa at pints B, C, D and E, respectively. Cmpared with the value at pint A, they decrease by 17.9%, 76.5%, 6.7% and 56.9%, respectively. Effective stress (MPa) / Mpa Pint A. O A ( ) Pint B B (22.5 ) O Pint C. 45C O (45 ) Pint D. 9D O (9 ) Pint D. 18 E (18 ) O Time (s) Time / s Fig.13 Effective stress at five different psitins alng path 1 in case 3. In Fig.14(a), we can see that the maximum effective stress alng path 1 is btained at pint A (º) and the minimum at pint C (45º) in each case, which may be determined by the circular shape f the tunnel lining. When the tunnel depth is 7 m, as the TNT charge increases frm 3 t 1 kg (cases 1, 2 and 3), the peak values at pint A are 1.8, 15.4 and 26.2 MPa, respectively; while the tunnel depth is 14 m, the crrespnding values (cases 4, 5 and 6) are 2.69, 4.29 and 6.91 MPa with the decreasing amplitudes f 75.1%, 72.1% and 73.6%, respectively. Figure 14(b) demnstrates that the effective peak stress decreases alng the lngitudinal tunnel far away frm the explsin center, and the maximum value is 28. MPa at a hrizntal distance abut 2.5 m away frm the explsive pint, which is als btained in case 3. The strength grade f the cncrete f tunnel lining used in this analysis is C5, whse uniaxial cmpressin strength is 32.4 MPa fr a standard value

9 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): Peak effective pressure (MPa) / Mpa Peak effective pressure (MPa) / Mpa Case 1 Case 2 Case 4 Case 5 Case Lcatin ( ) / (a) Alng path 1. Case Case1 1 Case Case2 2 Case Case3 3 Case Case4 4 Case Case5 5 Case Hrizntal distance frm the explsive center (m) Hrizntal distance frm the explsive center / m (b) Alng path 2. Fig.14 Cmparisn f peak effective stress n the tunnel alng different paths (t =.64 s fr cases 1, 2 and 3; t =.1s fr cases 4, 5 and 6). and 23.1 MPa fr design. When cnsidering the effective stress caused by ther lads such as sil and water pressures, the ttal effective stress in case 3 maybe exceed the standard value and cause the reinfrced cncrete t fail at the psitins ranging frm º t 22.5º f the crss-sectin and the hrizntal distance t 7 m away frm the explsin center. Lu et al. [12] pinted ut that the tp and the bttm blcks f the tunnel fr the Nanjing subway tunnel were the mre damaged znes and the subway tunnel was safe when 1 kg TNT was detnated at a height f 1.5 m. The Nanjing subway tunnel has the same diameter and depth as the Shanghai metr tunnel. The difference f the results in this paper maybe lie in the fllwing aspects: (1) The discrepancy f tunnel supprt cnditins makes the damaged znes different. The tunnel mdel f Nanjing subway has an anchrage area surrunding the tunnel lining at the arch crwn and the sidewall, but this paper des nt cnsider the reinfrcement area. (2) The difference f sil types. The Shanghai sft clay bviusly has different dynamic respnses frm the liquescent Nanjing sand under explsin lads. 3.6 Acceleratin f tunnel lining Figure 15 shws the acceleratin respnse f five pints alng path 1 in case 3, which reflects the features f high amplitude, shrt duratin and fast attenuatin under the blast induced waves. Vertical Vertical acceleratin acceleratin (m/s / m/s 2 ) Pint A. A O ( ) Pint B. B 22.5 (22.5 ) O Pint C. C 45(45 ) O Pint D. D 9(9 ) O Pint E. 18 E (18 ) O Time /(s) s Fig.15 Acceleratin values vs. time at five different psitins alng path 1 in case 3. In Fig.16, we knw the maximum acceleratin is Peak vertical acceleratin (m/s / m/s 2 ) Case 1 Case 2 Case 4 Case 5 Case Lcatin 9 ( ) Lcatin / Fig.16 Cmparisn f peak acceleratin f six cases alng path 1.

10 382 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): m/s 2 at the tp f the tunnel in case 3, and the minimum is 2.98 m/s 2 at the bttm in case 4, indicating that the peak acceleratin nearly has n effect n the tunnel lining. 4 Parametric study f sil The dynamic defrmatin f sil has a strng nnlinear characteristic that the larger the shear strain is, the smaller the shear mdulus is, and the larger the damping rati is. In this cntext, the selectin f the sil parameters will have a great effect n the results, s fur extra cases are taken int accunt t analyze this effect. It is wrthy t be mentined that the maximum value f the dynamic shear mdulus is MPa fr the third layer f the Shanghai sft sil, and fr the layers 4 and 5, the crrespnding values are 32.8 and MPa, respectively, accrding t the relevant getechnical investigatin reprt. The ther five calculatin cases are presented in Table 9. In case 3, the shear mdulus crrespnds t a large strain defrmatin ranging frm 1 2 t 1 1, and in cases 7 and 8, the shear mdulus crrespnds t a medium ne ranging frm 1 3 t 1 2. Furthermre, cases 9, 1 and 11 are used t discuss the effect f bulk mdulus K u at unlading path n the calculating results. Table 9 Calculatin cases. Case Tunnel depth (m) TNT charge (kg) G (MPa) K u (MPa) Figure 17 shws the cmparisns f cases 3, 7 and 8 alng path 1. It can be seen frm Figs.17(a) and (b) that, when the shear mdulus G increases t five and ten times the initial value, the peak pressure and the peak effective stress shw a slight increase. Fr example, the peak effective stress at pint A ( ) grws t 29.6 and 27.3 MPa in cases 7 and 8, respectively, with a grwth rati f 12.98% and 4.19%, respectively. In ther wrds, the pressure and the effective stress f the tunnel lining are nt very sensitive t the shear mdulus f the sil. Peak pressure (MPa) / Mpa Peak effective pressure / (MPa) Mpa Case Case7 7 Case Case Lcatin /( ) (a) Peak pressure Lcatin /( ) Peak vertical acceleratin (m/s / m/s 2 ) (b) Peak effective stress. Case 7 Case 8 Case 7 Case Lcatin /( ) (c) Peak vertical acceleratin. Fig.17 Cmparisn f three cases alng path 1. Figure 18 shws the cmparisns f cases 3, 9, 1 and 11 alng path 1. Accrding t the cmparisn f cases 3 and 9, it can be seen that, when the peak pressure, the effective stress and the peak vertical acceleratin shw a

11 Yubing Yang et al. / Jurnal f Rck Mechanics and Getechnical Engineering. 21, 2 (4): Peak pressure /(MPa) Mpa Peak Peak effective pressure pressure / (MPa) Mpa Lcatin /( ) Peak vertical acceleratin (m/s 2 ) Peak vertical acceleratin / m/s 2 (a) Peak pressure. Case 9 Case 1 Case Lcatin /( ) (b) Peak effective stress Lcatin /( ) (c) Peak vertical acceleratin. Case 9 Case 1 Case 11 Case 9 Case 1 Case 11 Fig.18 Cmparisn f fur cases alng path 1. significant increase when the bulk mdulus K u at unlading path decreases by a half. Fr example, the peak effective stress at pint A ( ) increases t MPa in case 9, with a grwth rati f 43.27%. The cmparisn f cases 9, 1 and 11 shws that, when the shear mdulus G increases by five times, the peak pressure and the peak effective stress shw a slight decrease. When it keeps increasing by tw times, the peak pressure almst keeps the same and the peak effective stress shws a slight decrease. It is nted that the variatins in the peak vertical acceleratins shwn in Figs.17(c) and 18(c) are mre bvius than thse in the peak pressure and the effective stress. 5 Cnclusins The dynamic respnse f the metr tunnel lining has been presented with the general cmmercial prgram ANSYS/LS-DYNA in this paper. We can get sme useful cnclusins as fllws: (1) The blast induced waves prpagate in the sil in the frm f hemispherical waves. The numerical simulatin results f the peak pressure and the peak acceleratin in the sil are cmpared with the predictins with the manual (TM ). The discrepancy between tw results is analyzed. (2) The distributin and magnitude f the stress field f the tunnel lining are influenced by the tunnel depth and TNT equivalence. Accrding t the vn Mises failure criterin, the upper part f the tunnel lining, ranging frm t 22.5 f the crss-sectin and the hrizntal distance t 7 m away frm the explsive center, is the unstable area. (3) The metr tunnel at the abve-mentined area maybe fail when the tunnel depth is 7 m and the TNT equivalence reaches 1 kg. In ther wrds, the metr tunnel in sft sil might be safer when the tunnel depth is mre than 7 m and the TNT charge f grund surface explsin is n mre than 5 kg. (4) Mre attentins shuld be paid t the selectin f sil parameters t perfrm a mre precise analysis, especially the bulk mdulus K u at unlading path. References [1] Kng Xinli, Jin Fengnian, Jiang Meirng. Analysis f way and scale f terrristic raid. Blasting, 27, 24 (3): (in Chinese). [2] Li Zhngxian, Du Ha, Ba Chunxia. A review f current researches n blast lad effects n building structures in China. Transactins f Tianjin University, 26, 12 (Supp.): [3] Lu Zhifang. Dynamic respnse and structure damage f the Yangtze River tunnel subjected t explsin lading. MS Thesis. Wuhan: Wuhan University f Technlgy, 26 (in Chinese). [4] Yang Xin, Shi Shaqing, Cheng Pengfei. Frecast and simulatin f peak verpressure f TNT explsin shck wave in the air. Blasting, 28, 25 (1): (in Chinese). [5] Bakken J, Slungaard T, Engebretsen T, et al. Attenuatin f shck

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