Numerical simulation of dynamic response of operating metro tunnel ...

Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

Numerical simulation of dynamic response of operating metro tunnel induced by ground explosion Yubing Yang1, Xiongyao Xie1*, Rulu Wang2 1

Key Laboratory of Geotechnical Engineering, Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

2

Shanghai Metro Operation Co., Ltd., Shanghai, 200003, China

Received 10 October 2010; received in revised form 28 October 2010; accepted 10 November 2010

Abstract: To evaluate the effects of possible ground explosion on a shallow-buried metro tunnel, this paper attempts to analyze the dynamic responses of the operating metro tunnel in soft soil, using a widely applied explicit dynamic nonlinear finite element software ANSYS/LS-DYNA. The blast induced wave propagation in the soil and the tunnel, and the von Mises effective stress and acceleration of the tunnel lining were presented, and the safety of the tunnel lining was evaluated based on the failure criterion. Besides, the parametric study of the soil was also carried out. The numerical results indicate that the upper part of the tunnel lining cross-section with directions ranging from 0 to 22.5° and horizontal distances 0 to 7 m away from the explosive center are the vulnerable areas, and the metro tunnel might be safe when tunnel depth is more than 7 m and TNT charge on the ground is no more than 500 kg, and the selection of soil parameters should be paid more attentions to conduct a more precise analysis. Key words: ground surface explosion; numerical simulation; metro tunnel; dynamic response

1

Introduction



Recently, the worldwide terrorism attacks are becoming intensive and more frequent. Vehicle bomb is the main way to implement the terrorist activities due to its massive charge power, high success ratio and serious destruction [1]. From the blasting events occurring in recent years such as the World Trade Centre of New York in 2001, and those in Chechnya (2002) and London (2005), the terroristic blast raid will not only result in the damages of building structures, huge losses of lives and properties, but also probably threaten the safety of an operating metro tunnel with the increase of TNT equivalence for vehicle bombs. Three types of methods can be used to analyze the Doi: 10.3724/SP.J.1235.2010.00373 *Corresponding author. Tel: +86-13501782144; E-mail: [email protected] Supported by the National Natural Science Foundation of China (40874074, 50950110347), the National High Technology Research and Development Program (863 Program) of China (2006AA11ZAA8), and Shanghai Science and Technology Development Funds (07ZR14117)

dynamic behaviors of structures under blast load: theoretical solutions, experimental study and numerical methods. Generally, the following subsystems are involved: (1) the propagation of blast induced waves in air, rock or soil; (2) the dynamic response behaviors of structures; and (3) the material damage analysis of structures [2, 3]. For the propagation of blast induced waves in the air and rocks, a lot of studies have been done [4–6]. With respect to the underground structures such as metro tunnels, a closed-form solution for this complicated problem is almost unavailable at present, because of the necessary simplification and subdivision in the theoretical model and a large amount of calculations. As for the experimental study, laboratory model tests and field prototype investigations may be the two possible choices. Unfortunately, until now no reports on the field prototype experiment can be found in China. In Russia, a coastal surface explosion experiment with a 1 000-ton TNT charge was performed on August 25, 1987, approximately 100 m away from the coastal line. More details can be found in Ref.[7]. Various limitations, such as the appropriate selection of the structure model, the nonlinearity of

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rocks or soils, the input of wave loads, the high cost of tests, have also impeded the progress of the experimental study [8]. Recently, the rapid developments of numerical methods have provided strong supports for the behavior study of underground structures under dynamic loads, among which the finite element method (FEM) may be the researchers’ favourite choice for its ability in dealing with nonlinear and anisotropic problems. As to these unique advantages, the FEMs have been widely used in this field. A lot of researches [9–11] have been conducted to analyze the effect of an internal explosion on tunnels. For the explosion effect on the ground surface, Luo et al. [12] analyzed the dynamic responses of the tunnel for surface explosions of 100 and 300 kg TNT charges, respectively, according to the features of Nanjing subway tunnel in sandy soils. Fan et al. [13] discussed the load characteristics on a shallow-buried concrete structure under a ground explosion by means of numerical simulations. However, there is little work focusing on the responses of metro tunnels in soft soil induced by ground explosion. The studies for the propagation of blast induced waves in the soft soil, the dynamic responses of a metro tunnel and its safety are carried out in the paper, using a widely applied explicit dynamic nonlinear finite element software ANSYS/LS-DYNA. Firstly, the background of modelling is presented briefly, including the size selection of the model, the soil type, the material constitutive models and parameters, etc.. Secondly, a case without tunnel structure in the soft soil is introduced to analyze the blast induced wave propagation in the soft soil. Then six cases of different explosive charges and tunnel depths are considered to analyze the dynamic responses of the tunnel. The propagation of the blast induced waves in the tunnel, the von Mises effective stress and the acceleration of the tunnel lining are presented. The safety of the tunnel lining based on the failure criterion is evaluated. Finally the parametric study of soft soil is done to analyze the effect on the calculating results.

2

Numerical model

2.1 Background and finite element model The project of the Shanghai metro line No.1 is considered in this paper [14]. The tunnel lining has a circular shape, with 5.5 m in inner diameter and 6.2 m in outer diameter. The typical tunnel depths from

Caobao Road Station to Shanghai Railway Station range from 7 to 15 m. Moreover, according to the typical Shanghai stratigraphic distribution, the soil layer around the tunnel considered in analysis consists of 6 types of soils from the top down, i.e. the miscellaneous fill, the yellowish dark brown silty clay, the gray silty clay, the gray mucky silty clay, the gray clay, and the yellowish dark brown clay. Their physico-mechanical parameters can be found in Ref.[15]. For many previous studies on systems loaded explosively, equivalent time-history pressures were used to simulate the loads. Time-history pressures generated with high explosives tend to exhibit large variations, even with identical charges. Obviously, whether the responses of the structure can be predicted strongly depend on the ability to generate “accurate” load functions [16]. In this study, the explosive was modeled explicitly using a ANSYS/LS-DYNA material specifically designed for simulating a high explosive detonation. It is assumed that the explosion will take place at the most unfavorable positions such as the interface of air and soil, above the metro tunnel. Considering the symmetries of the tunnel structure and the blast load, for the sake of saving computation time, a 1/4 symmetrical geometrical model with a size of 25 m × 25 m × 30 m was established based on the Alekseenko test [17] (Fig.1).

Fig.1 Total and 1/4 symmetrical geometrical models.

In the finite element model (Fig.2), the eight-node element of SOLID 164 is adopted for the 3D explicit analysis. In order to prevent the element distortion in large deformation and nonlinear structural analyses, an arbitrary Lagrangian-Eularian (ALE) algorism is used in this paper. The TNT charge, the air and the soil are modeled with ALE multi-material meshes, but the tunnel lining with Lagrangian meshes, while the minimal time step is controlled by the smallest element size in the explicit integral method, and the globe uniform mesh size is set to be 50 cm.

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375

Table 2 Parameters of the TNT charge [19].  (g/cm )

νD (m/s)

PCJ (MPa)

A (MPa)

B (MPa)

1.0

5 500

1.0  104

7.41  105

0.18  105

R1

R2



V

E0 ( J/m3)

5.56

1.65

0.35

1.0

5.56

3

Fig.2 The finite element model.

Furthermore, the transitional displacement of the nodes normal to the symmetry planes (XZ and YZ planes) is constrained. Non-reflecting boundary condition is applied to the other two lateral surfaces and the bottom surface, and the free boundary condition is used for the upper surface. 2.2 Material constitutive models and parameters Four kinds of materials are involved in this finite element model: air, TNT charge, tunnel lining and soil. The air is commonly modeled by null material model with a linear polynomial equation of state (EOS), which defines the pressure by the following equation: P  C0  C1   C2  2  C3  3  (C4  C5   C6  2 ) E0 (1) where the parameter  is defined as    / 0  1 ,  is the current density, and 0 is a nominal or reference density; C0–C6 are the equation coefficients; and the parameter E0 is the initial internal energy of reference specific volume per unit. Table 1 gives the parameters used in the air model in Refs.[10, 18]. Table 1 Parameters of the air.  (g/cm3)

C0

C1

C2

C3

C4

C5

C6

E0 (J/m3) 0 (g/cm3)

1.29

0

0

0

0

0.4

0.4

0

2.5 × 105

1.0

The TNT charge is modeled by the high explosive material model and the Jones-Wilkins-Lee (JWL) equation of state [18]. The JWL equation of state defines the pressure by the following equation:      R1V    R2V  E0 p  A 1  (2)  B 1   e e R1V  R2V  V   where A, B, R1 , R2 ,  are the equation coefficients and they all should be tested in an accurate blast analysis; and V is the initial relative volume. Here we adopt the parameters selected from Ref.[19], as shown in Table 2.

The tunnel lining is modeled by the plastic kinematic model for simplicity and applicability. It is a mixed model in which the hardening coefficient  is used to adjust the contribution proportions of isotropic hardening and kinematic hardening. The main parameters in this model include mass density  , Young’s modulus E, Poisson’s ratio , yield stress  v , tangent modulus Etan, hardening parameter  , failure strain for eroding elements  f . Because the response of the reinforced concrete to the dynamic load is always a complex nonlinear and rate-dependent process, quite a few models have been proposed to describe its dynamic behavior [20]. In addition, the tunnel lining is usually reinforced with concrete. For simplicity, the steel bar and the concrete are regarded as a whole according to the principle of equivalent stiffness EI in this study. Table 3 gives the parameters in the tunnel lining model. Table 3 Parameters of the tunnel lining.  (g/cm )

E (GPa)



y (MPa)

Etan(MPa)



f

2.65

39.1

0.25

1.0  102

4.0  103

0.5

0.8

3

The soil is modeled by a soil and foam model put forward by Krieg in 1972 [21]. It is a simple model and operates in some way like a fluid, and has been demonstrated to be useful for soil modeling [22]. The main parameters in this model include: mass density  , shear modulus G, bulk modulus Ku at unloading path, yield function constants a0, a1 and a2, pressure cutoff for tensile fractures pcut. Table 4 gives the main parameters in the soil and foam model based on Ref.[23]. It is assumed that the entire soil layer has the same parameters as the gray mucky silty clay for computational simplicity. Table 4 Main parameters in the soil and foam model.  (g/cm3) 1.8

3

G (MPa) 1.2

Ku (MPa) 130

a0 3.3 × 10

11

a1

a2

pcut (MPa)

0.0

0.0

0.0

Numerical results and discussions Kong et al. [1] presented that the scale of vehicle

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Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

bombs could range from 200 kg TNT equivalence of a common family car to 1 000 kg TNT equivalence of a small box van, hence three cases of TNT charge are considered in this paper: 300, 500 and 1 000 kg. Considering the tunnel depths of 7 and 14 m, about one and two times the tunnel diameter, respectively, under the situation of free field (no tunnel structure in the soil), the following seven calculation cases are performed (Table 5).

(a) t = 0.012 s.

(b) t = 0.032 s.

(c) t = 0.064 s.

(d) t = 0.1 s.

Table 5 Calculation cases. Case

Tunnel depth (m)

TNT (kg)

0

—

1 000

1

7

300

2

7

500

3

7

1 000

4

14

300

Fig.4 Pressure contours in the soil at different times in case 0

5

14

500

(unit: 102 GPa).

6

14

1 000

Two research paths are selected to analyze the dynamic responses of the tunnel lining (Fig.3). Path 1 is along the transverse direction, 5 typical points are selected around the cross-section; and path 2 is along the longitudinal direction, the horizontal distance from the explosion center ranges from 0 to 20 m.

Fig.3 Two research paths.

3.1 Propagation of the blast induced waves in the soil Figure 4 shows the wave propagation in the soil at different times in case 0. It demonstrates that the pressure waves propagate in the soil in the form of hemispherical waves. The area of wave front increases with the wave propagation. The affected region of the soil in case 0 is about 14 m beneath the ground surface and the duration is around 0.12 s. The extruding part of the soil layer in each subfigure is rather remarkable. Actually, the explosion will cause a crater through the ejection of the soil

away from the blast. Simulating this process during which small parts move away with a continuous finite element becomes very difficult. This model does not provide a good solution. We assume that the large deformation of the soil in vicinity of the blast center, the above-mentioned extruding part, represents that a crater will be formed. Two examples in reality are given here, according to which we can realize the huge destructive power of the ground surface explosion. Figure 5(a) shows the ruins at Ryongchon railway station in North Korea after a huge explosion (April 24, 2004). It was reported that this explosion was induced by a collision between two trains, which carried oil and gas and ammonium nitrate, respectively. Two craters were formed in the explosion center, and the bigger one shown in Fig.5(a) had a diameter of 30 m and a depth of 10 m. Figure 5(b) shows a crater in a factory in Anhui Province of China after an explosion (June 21, 2009). The explosive hidden in a building without authorization

(a) The ruins at Ryongchon railway station in North Korea.

Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

Fig.5 Two examples in reality.

by the factory’s owner was said to be 5–7 tons, and the crater had a diameter of 10 m and a depth of 5 m. Figure 6 describes the compressive waves in the soil at different depths below the explosion center (4, 7, 10, 14 and 20 m). According to the definition in the Alekseenko test [17] (Fig.7), these positions are located in the central zone, and the compressive waves in the soil are mainly the ground shock wave. 2.2 2.0

A. 4mA (4 m) Point B. 7m(1D) Point B (7 m) C. 10m Point C (10 m) D. 14m(2D) Point D (14 m) E. 20m Point E (20 m)

1.8

Pressure (MPa)

1.6 1.4

We can see the propagation and attenuation of the compressive waves in the soil. Each compressive wave, except the one at point A, has only one peak value induced by the direct ground shock, which exactly corresponds to the results in the Alekseenko test [17]. The discrepancy at point A may be explained by its close distance away from the surface zone. The first peak value can be caused by the ground wave, the second may be caused by the air shock wave, and the third may be the reflected wave at the soil interface. 3.2 Comparison between numerical results and prediction by the manual (TM5-855-1) Figure 8 shows the vertical acceleration of these five points. The peak values of points A, B and C are 5 093.08, 129.8, 57.7 m/s2, respectively.

2 000

0

2 000

4 000

6 000 0.00

1.2

Point A. 4mA (4 m) Point BB (7 B. 7m(1D) Point (7 m) m) Point C (10 m) C. 10m D. 14m(2D) Point D (14 m) E. 20m Point E (20 m)

4 000

Vertical acceleration (m/s2)

(b) A crater in a factory in Anhui Province of China.

377

0.02

0.04

0.06

0.08

0.10

Time (s)

1.0

Fig.8 Vertical acceleration in the soil at different depths (case 0). 0.8 0.6 0.4 0.2 0.0 0.0

0.02

0.04

0.06

0.08

0.10

Time (s)

Fig.6 Compressive waves in the soil at different depths (case 0).

The US Army Corps of Engineers Manual (TM5-855-1) has been widely used to estimate the ground shock parameters. It adopts the cube-root scaled distance to predict ground shock parameters. The following equations are provided in the manual (TM5-855-1) to predict the peak values of pressure and acceleration, respectively [24]:  R  Pp  0.407 f c  1/3  W 



Surface zone

Central zone

Fig.7 Successive locations of wave front [17].

ap 

39.8 fc  R  W 1/3  W 1/3 

n

(3)

 ( n 1)

(4)

where Pp is the peak pressure (Pa); f is a coupling factor, which is dependent on the scaled depth of the explosion and is given by d / W 1/3 , d is the depth of the centroid of the explosive charge; c is the acoustic impedance; c is the seismic velocity; R is the distance from the source; W is the charge weight; n is

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Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

an attenuation coefficient; and ap is the peak acceleration. Based on the seismic velocity of Shanghai soft soil, the soil in this model may be described as the type 4 in the manual (TM5-855-1) (Table 6). Table 6 Soil properties for calculating ground shock parameters [24]. Seismic Density,

Soil type

velocity, c impedance, c

 (kg/m3)

(m/s) (1) Heavy saturated

Acoustic

1 920–2 080 > 1 524

(106 Pa·s/m)

Attenuation

33.9–40.68

1.5

(2) Saturated sandy 1 760–1 984

1 524

29.38

2.25–2.5

1 744

487.68

9.944

2.5

1 920–2 000 548.64

10.848

2.5

304.8

4.972

2.75

1 440–1 600 182.88

2.712

3–3.25

air voids < 1% (3) Dense sand with high relative density, wet sandy clay with air voids > 4% (4) Sandy loam, loess,

1 984

dry sands and backfills (5) Loose, dry sands and gravels with low

W

R

R/W1/3

(kg)

(m)

(m/kg1/3)

250

Peak acceleration (m/s2)

Difference

Predictions

Numerical

(c = 304.8 m/s, n = 2.75)

results

(%)

4

0.64

10 260.6

5 093.08

102

7

1.12

1 239.7

129.8

858

10

1.60

325.36

57.7

463

14

2.24

91.53

—

—

20

3.20

24.01

—

—

coefficient, n

clays and clay shale clays and sands with

Table 8 Comparison of peak accelerations between numerical results and predictions with the manual (TM5-855-1).

relative density

Using the parameters of the soil type 4 in Table 6, and assuming that f is equal to 1.0, we can compare the numerical results obtained in this paper with the predicted ones using the manual (TM5-855-1), as shown in Tables 7 and 8, respectively. The difference of the peak pressures is expressed as (prediction − numerical result)/ numerical result. Table 7 Comparison of peak pressures between numerical results and predictions with the manual (TM5-855-1).

(1) The predictions with the manual (TM5-855-1) are based on the assumption that the buried-depth of TNT charge is big enough to form a full containment explosion (f = 1.0). However, the case in this paper, which occurs at the interface of soil and air, i.e. the buried-depth of TNT charge is zero and the coupling coefficient f will be smaller than 1.0, cannot coincide well with the assumption. Hence, it’s natural that the numerical results are lower than the predictions. (2) The Alekseenko test in 1967 indicated that, if the charge was buried in the soil and its upper surface was at the same level as the ground surface, such as the case in this study, the proportion of energy absorbed by the air and the soil would be 53% and 47%, respectively. That means more than half energy disperses in the air. This conclusion can explain why the computed results are smaller than the predictions. In total, the numerical results mentioned above suggest that this finite element model, to some extent, is rational to simulate the dynamic responses of underground structures under explosion loads on the ground surface. Figure 9 presents the wave propagation in the soil at different times for case 3.

Peak pressure (MPa) W

R

R/W1/3

(kg)

(m)

(m/kg1/3)

250

Difference Predictions (c = 4.972

Numerical

GPa·s/m, n = 2.75)

results

(%)

4

0.64

6.66

2.02

229

7

1.12

1.43

0.38

276

10

1.60

0.54

0.15

260

14

2.24

0.21

0.03

600

20

3.20

0.08

0

—

From Tables 7 and 8, it can be seen that the predictions with the manual (TM5-855-1) are higher than the numerical results for this model. However, we can also find that the numerical results of the peak pressure almost have the same order of magnitude as the predictions. This phenomenon can be elucidated in the following two aspects:

(a) t = 0.012 s.

(c) t = 0.064 s.

(b) t = 0.032 s.

(d) t = 0.1 s.

Fig.9 Pressure contours in the soil at different times for case 3 (unit: 102 GPa).

Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

directions, unlike those in the soil, and they have no fixed forms and noticeable wave fronts. It is quite clear that the longitudinal propagation is faster than the circumferential one. Figure 11 shows the propagation and the attenuation of blast induced waves along path 1 of the tunnel in case 3, where the pressure is positive in compression and negative in tension. 10

A. 0Ao (0º) Point o B. 22.5 Point B (22.5º) o C. 45 Point C (45º) o D. 90 Point D (90º) o E. 180 Point E (180º)

8

6 Pressure (MPa)

In Fig.9, we can see the affected region of the soil in case 3 just reaches the middle part of the tunnel, about 10 m away from the ground surface, and the duration is about 0.12 s. Obviously, the difference between the two cases (cases 0 and 3) is the affected region in the soil. The existence of the tunnel structure in the soil prevents the blast induced waves from migrating to a deeper soil layer. The energy of the blast induced waves is transferred from the soil to the tunnel lining. 3.3 Propagation of the blast induced waves on the tunnel Figure 10 shows the wave propagation on the tunnel at different times in case 3. It presents the whole process of the propagation of blast induced waves on the tunnel: expansion, migration and dissipation. The response of tunnel occurs at t = 0.032 s and ends at around t = 0.5 s.

379

4

2 0 0.0 0 2

0.2 0.2

0.4 0.6 0.4 Time (s) 0.6

0.8 0.8

1.0 1

Time / s

4

Fig.11 Blast induced waves at five different points along path 1 in case 3. (a) t = 0.032 s.

(c) t = 0.064 s.

(b) t = 0.044 s.

(d) t = 0.1 s.

(e) t = 0.16 s.

(f) t = 0.25 s.

(g) t = 0.35 s.

(h) t = 0.5 s.

Fig.10 Pressure contours on the tunnel at different times for case 3 (unit: 102 GPa).

The pressure waves propagate on the tunnel along both the longitudinal and the circumferential

It demonstrates that the upper lining (points A, B and C) is compressed but the lower lining (points D and E) is tensioned under the explosive load. Every point has a peak value at t = 0.1 s and then the value decreases with time gradually, and finally close to zero. The maximum of all 5 peak values occurring at point A is 9.93 MPa, and the rest are 7.87, 2.53, ―3.20 and 3.95 MPa at points B, C, D and E, respectively. Compared with the point A, they decrease by 20.7%, 74.5%, 67.8% and 60.2%, respectively. Therefore, the top of the tunnel (point A) is liable to be destroyed, but the middle-upper part (point C) is the safest. The propagation and the attenuation of blast induced waves along path 1 in other five cases are not given, as their trends are similar to those in case 3, but the peak values are smaller. In Fig.12(a), the peak pressures in all cases along path 1 are compared, from which we can conclude that the maximum pressure in all cases occurs at point A (0º), but for different TNT charges and tunnel depths, the peak value varies. When the tunnel depth is 7 m, as the TNT charge increases from 300 to 1 000 kg (cases 1, 2 and 3), the peak values at point A are 4.15, 5.86 and 9.93 MPa, respectively; while the tunnel depth is 14 m, the corresponding values (cases 4, 5 and 6) are

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1010 88

Peak / Mpa Peak pressure pressure (MPa)

of a multi-axial stress state and used in many failure or yield criteria. Thus, if a material is known to fail in a uniaxial compression test (with  1 the only nonzero stress component) when  1   crit , it will fail under multi-axial loading. Figure 13 shows the effective stresses on the tunnel along path 1 in case 3. Every point has one peak at t = 0.1 s, and the maximum of 5 peak values is 26.2 MPa at point A, the rest are 21.5, 6.15, 10.3 and 11.3 MPa at points B, C, D and E, respectively. Compared with the value at point A, they decrease by 17.9%, 76.5%, 60.7% and 56.9%, respectively.

Case Case11 Case Case22 Case Case33 Case Case44 Case Case55 Case Case66

66 44 22 00 -22 -44 0.0 0

22.5 22.5

45.0 45

67.5 90.0 112.5 135.0 157.5 67.5 90 112.5 135 157.5 Location ()o

180.0 180

3030

A. 0OA (0) Point O B. 22.5 Point B (22.5 ) C. 45CO (45) Point D. 90DO (90) Point O D. 180 Point E (180)

Location /

(a) Along path 1.

25 25 1212 1010

Peak / Mpa Peak pressure pressure (MPa)

88

Effective / Mpa Effective stress stress (MPa)

Case Case11 Case Case22 Case Case33 Case Case44 Case Case55 Case Case66

66

2020

15 15

10 10

44

55

22 00

00

-22 -44

0.0 0

0.2 0.2

0.4 0.4

Time (s)

0.6 0.6

0.8 0.8

1.0 1

Time / s 0 0

16 12 8 4 4 distance8 from the explosive 12 16 (m) Horizontal center

20 20

Horizontal diastance from the explosive center / m (b) Along path 2.

Fig.12 Comparison of peak pressure on the tunnel along different paths (t = 0.064 s for cases 1, 2 and 3; t =0.1 s for cases 4, 5 and 6).

0.98, 1.58 and 2.54 MPa, respectively. The decreasing amplitudes reach 76.4%, 73.0% and 74.4%, respectively. Hence, the effect of deep-buried tunnel (about 2D in this paper, D is the tunnel diameter) under blasting loads is reduced by 75% than that of the shallow-buried tunnel (about 1D). Figure 12(b) shows the peak pressure of six cases along path 2, from which we can know that the maximum pressure is 10.6 MPa, which occurs in case 3 at a horizontal distance about 2.5 m away from the explosion center. That means it will probably fail at this position in case 3 if the failure of the lining happens. 3.4 Failure evaluation based on the effective stress In some ways, we know that the von Mises effective stress  i can be regarded as a uniaxial equivalence

Fig.13 Effective stress at five different positions along path 1 in case 3.

In Fig.14(a), we can see that the maximum effective stress along path 1 is obtained at point A (0º) and the minimum at point C (45º) in each case, which may be determined by the circular shape of the tunnel lining. When the tunnel depth is 7 m, as the TNT charge increases from 300 to 1 000 kg (cases 1, 2 and 3), the peak values at point A are 10.8, 15.4 and 26.2 MPa, respectively; while the tunnel depth is 14 m, the corresponding values (cases 4, 5 and 6) are 2.69, 4.29 and 6.91 MPa with the decreasing amplitudes of 75.1%, 72.1% and 73.6%, respectively. Figure 14(b) demonstrates that the effective peak stress decreases along the longitudinal tunnel far away from the explosion center, and the maximum value is 28.0 MPa at a horizontal distance about 2.5 m away from the explosive point, which is also obtained in case 3. The strength grade of the concrete of tunnel lining used in this analysis is C50, whose uniaxial compression strength is 32.4 MPa for a standard value

Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

3030 Case 1 Case 2 Case 33 Case Case 44 Case Case 55 Case Case 66 Case

Peak effectivepressure pressure / Mpa Peak effective (MPa)

2525

2020

1515

1010

55

00 0.0 0

22.5 22.5

45.0 45

67.5 90.0 112.5 135.0 157.5 180.0 67.5 90 112.5 135 157.5 180 Location (/o) Location (a) Along path 1.

The difference of the results in this paper maybe lie in the following aspects: (1) The discrepancy of tunnel support conditions makes the damaged zones different. The tunnel model of Nanjing subway has an anchorage area surrounding the tunnel lining at the arch crown and the sidewall, but this paper does not consider the reinforcement area. (2) The difference of soil types. The Shanghai soft clay obviously has different dynamic responses from the liquescent Nanjing sand under explosion loads. 3.6 Acceleration of tunnel lining Figure 15 shows the acceleration response of five points along path 1 in case 3, which reflects the features of high amplitude, short duration and fast attenuation under the blast induced waves. 200 200

2525

Case Case Case111 Case Case Case222 Case Case Case333 Case Case Case444 Case Case555 Case Case Case666 Case

2020

O Point A. A 0 (0) O Point (22.5 ) B. B 22.5 O C. C 45(45 Point ) O D. D 90(90) Point O E. 180 Point E (180)

150 2 Vertical (m/s ) 2 Verticalacceleration acceleration / m/s

Peak effectivepressure pressure / Mpa Peak effective (MPa)

381

1515

1010

100 50 0 -50 50

-100 100

-150 150

55

-200 200

0.0 0

00

0 0

8 12 16 4 4 8 12 16 Horizontal distance from the explosive center (m)

20 20

Horizontal distance from the explosive center / m

0.2 0.2

0.4

0.4

0.6

0.6

0.8 0.8

1.0 1

Time /(s) Time s

Fig.15 Acceleration values vs. time at five different positions along path 1 in case 3.

(b) Along path 2.

100 100

Peak Peakvertical vertical acceleration acceleration(m/s / m/s )

2

and 23.1 MPa for design. When considering the effective stress caused by other loads such as soil and water pressures, the total effective stress in case 3 maybe exceed the standard value and cause the reinforced concrete to fail at the positions ranging from 0º to 22.5º of the cross-section and the horizontal distance 0 to 7 m away from the explosion center. Luo et al. [12] pointed out that the top and the bottom blocks of the tunnel for the Nanjing subway tunnel were the more damaged zones and the subway tunnel was safe when 100 kg TNT was detonated at a height of 1.5 m. The Nanjing subway tunnel has the same diameter and depth as the Shanghai metro tunnel.

In Fig.16, we know the maximum acceleration is

50 50

2

Fig.14 Comparison of peak effective stress on the tunnel along different paths (t = 0.064 s for cases 1, 2 and 3; t = 0.1s for cases 4, 5 and 6).

00

50 -50

Case Case11 Case Case22 Case33 Case Case44 Case Case55 Case Case66 Case

100 -100

-150 150

200 -200 0.0 0

22.5 22.5

45.0 67.5 90.0 112.5 135.0 157.5 180.0 45 67.5Location 90 (112.5 135 157.5 180 o)

Location /

Fig.16 Comparison of peak acceleration of six cases along path 1.

382

Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

−192.1 m/s2 at the top of the tunnel in case 3, and the minimum is 2.98 m/s2 at the bottom in case 4, indicating that the peak acceleration nearly has no effect on the tunnel lining.

The dynamic deformation of soil has a strong nonlinear characteristic that the larger the shear strain is, the smaller the shear modulus is, and the larger the damping ratio is. In this context, the selection of the soil parameters will have a great effect on the results, so four extra cases are taken into account to analyze this effect. It is worthy to be mentioned that the maximum value of the dynamic shear modulus is 28.75 MPa for the third layer of the Shanghai soft soil, and for the layers 4 and 5, the corresponding values are 32.08 and 45.21 MPa, respectively, according to the relevant geotechnical investigation report. The other five calculation cases are presented in Table 9. In case 3, the shear modulus corresponds to a large strain deformation ranging from 10−2 to 10−1, and in cases 7 and 8, the shear modulus corresponds to a medium one ranging from 10−3 to 10−2. Furthermore, cases 9, 10 and 11 are used to discuss the effect of bulk modulus Ku at unloading path on the calculating results.

Peak / Mpa Peakpressure pressure (MPa)

Parametric study of soil

Case Case33 Case77 Case Case88 Case

1515

1010

55

00

-55 0.0 0

45.0 67.5 90.0 112.5 135.0 157.5 180.0 45 67.5 90 112.5 135 157.5 180 Location Location /(o)

22.5 22.5

(a) Peak pressure. 30 30

Peakeffective effective pressure Mpa Peak pressure /(MPa)

4

2020

25 25

Case 3 Case 7 Case 8

20 20

15 15

1010

55

0.0

22.5

0

22.5

45.0 45

67.5 67.5

90.0 90

112.5 135.0 157.5 180.0 112.5

Location Location /(o )

135

157.5

180

(b) Peak effective stress.

Table 9 Calculation cases. 00

Tunnel depth (m)

TNT charge (kg)

G (MPa)

3

7

1 000

1.2

130

7

7

1 000

6

130

8

7

1 000

12

130

9

7

1 000

1.2

65

10

9

1 000

6

65

11

7

1 000

12

65

Figure 17 shows the comparisons of cases 3, 7 and 8 along path 1. It can be seen from Figs.17(a) and (b) that, when the shear modulus G increases to five and ten times the initial value, the peak pressure and the peak effective stress show a slight increase. For example, the peak effective stress at point A (0°) grows to 29.6 and 27.3 MPa in cases 7 and 8, respectively, with a growth ratio of 12.98% and 4.19%, respectively. In other words, the pressure and the effective stress of the tunnel lining are not very sensitive to the shear modulus of the soil.

22 Peakvertical verticalacceleration acceleration(m/s / m/s Peak )

Ku (MPa)

Case

100 -100

Case 3 Case 7 Case 8

-200 200

-300 300

0.0 0

22.5

22.5

45.0 45

67.5 90.0

112.5 135.0 157.5 180.0

67.5 90 112.5 Location Location /(o)

135

157.5

180

(c) Peak vertical acceleration.

Fig.17 Comparison of three cases along path 1.

Figure 18 shows the comparisons of cases 3, 9, 10 and 11 along path 1. According to the comparison of cases 3 and 9, it can be seen that, when the peak pressure, the effective stress and the peak vertical acceleration show a

Yubing Yang et al. / Journal of Rock Mechanics and Geotechnical Engineering. 2010, 2 (4): 373–384

peak pressure and the peak effective stress show a slight decrease. When it keeps increasing by two times, the peak pressure almost keeps the same and the peak effective stress shows a slight decrease. It is noted that the variations in the peak vertical accelerations shown in Figs.17(c) and 18(c) are more obvious than those in the peak pressure and the effective stress.

16 16 14 14

Case 3 Case 9 Case 10 Case 11

Peak Peak pressure pressure /(MPa) Mpa

12 12 10 10

383

88 66 44 22 00 -2 2

5

-4 4

Conclusions

-6 6

0.0 0

22.5 22.5

45.0 45

67.5 67.5

112.5 90.0 90 112.5 135.0 135 Location /(o) Location

157.5 180.0 157.5 180

(a) Peak pressure.

Peak pressure /(MPa) Peak effective effective pressure Mpa

40 40

Case Case 33 Case Case 99 Case 10 Case 10 Case 11 Case 11

30 30

20 20

10 10

00

0.0 0

22.5 45.0 22.5 45

67.5 67.5

90.0 90

112.5 135.0 112.5 135

157.5 180.0 157.5 180

Location /(o )

(b) Peak effective stress.

Peak vertical acceleration (m/s2)2 Peak vertical acceleration / m/s

00

-200 200

Case 3 Case 9 Case 10 Case 11

-400 400 -600 600

The dynamic response of the metro tunnel lining has been presented with the general commercial program ANSYS/LS-DYNA in this paper. We can get some useful conclusions as follows: (1) The blast induced waves propagate in the soil in the form of hemispherical waves. The numerical simulation results of the peak pressure and the peak acceleration in the soil are compared with the predictions with the manual (TM5-855-1). The discrepancy between two results is analyzed. (2) The distribution and magnitude of the stress field of the tunnel lining are influenced by the tunnel depth and TNT equivalence. According to the von Mises failure criterion, the upper part of the tunnel lining, ranging from 0° to 22.5° of the cross-section and the horizontal distance 0 to 7 m away from the explosive center, is the unstable area. (3) The metro tunnel at the above-mentioned area maybe fail when the tunnel depth is 7 m and the TNT equivalence reaches 1 000 kg. In other words, the metro tunnel in soft soil might be safer when the tunnel depth is more than 7 m and the TNT charge of ground surface explosion is no more than 500 kg. (4) More attentions should be paid to the selection of soil parameters to perform a more precise analysis, especially the bulk modulus Ku at unloading path.

-800 800

References

1-1000 000

0.0 0

22.5 45.0 22.5 45

67.5 90.0 112.5 135.0 180.0 157.5 90 112.5 135 157.5 180 Location /(o)

67.5

(c) Peak vertical acceleration.

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