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Koneshwaran, Sivalingam, Thambiratnam, David, & Gallage, Chaminda(2015)Response of segmented bored transit tunnels to surface blast.Advances in Engineering Software, 89, pp. 77-89.

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https://doi.org/10.1016/j.advengsoft.2015.02.007

* corresponding author: tel: 61 7 31381467; fax: 61 7 3138 1170; e-mail: d.thambiratnam@qut.edu.au 1

Abstract Increasing threat of terrorism highlights the importance of enhancing the resilience of underground tunnels to all hazards. This paper develops, applies and compares the Arbitrary Lagrangian Eulerian (ALE) and Smooth Particle Hydrodynamics (SPH) techniques to treat the response of buried tunnels to surface explosions. The results and outcomes of the two techniques were compared, along with results from existing test data. The comparison shows that the ALE technique is a better method for describing the tunnel response for above ground explosion with regards to modeling accuracy and computational efficiency. The ALE technique was then applied to treat the blast response of different types of segmented bored tunnels buried in dry sand. Results indicate that the most used modern ring type segmented tunnels were more flexible for in-plane response, however, they suffered permanent drifts between the rings. Hexagonal segmented tunnels responded with negligible drifts in the longitudinal direction, but the magnitudes of in-plane drifts were large and hence hazardous for the tunnel. Interlocking segmented tunnels suffered from permanent drifts in both the longitudinal and transverse directions. Multi-surface radial joints in both the hexagonal and interlocking segments affected the flexibility of the tunnel in the transverse direction. The findings offer significant new information in the behavior of segmented bored tunnels to guide their future implementation in civil engineering applications. Keywords: segmented bored tunnel, explosion; finite element method, arbitrary Lagrangian Eulerian method, smooth particle hydrodynamics, dry soil.

1. Introduction Underground transit tunnel systems provide a quick and cost effective alternative to laying surface rails and roads and play an essential role in addressing transportation needs in many cities. Increasing blast attacks highlight that these underground transit tunnels are vulnerable to potential terrorist attacks with possible failure causing loss of lives, transit network interruptions and considerable financial implications. Transit tunnels must therefore be designed to withstand ground shocks transmitted from credible explosions. It is desirable that if such an explosion did occur, the tunnel should be able to return to service as soon as possible with minor repairs. The central part of a rapid transit network in cities is usually built in tunnels. These tunnels are mostly bored tunnels constructed using tunnel boring machines (TBM) with the support of permanent linings. The principle of bored tunnel construction has been known for a long time. The tunnels constructed in the beginning of the nineteenth century are still in use in

Response of segmented bored transit tunnels to surface blast.

Sivalingam Koneshwaran, David P. Thambiratnam*, Chaminda Gallage

Science & Engineering Faculty, Queensland University of Technology, Australia.

2

many cities. They have a direct relationship with the identity of the city as they illustrate its history, culture, and its economic, political and social states. This also highlights the importance of protecting such structures. Bored tunnel consist of prefabricated reinforced concrete segments placed together with bolts in both the longitudinal and transverse directions. In order to prevent water from entering the tunnel through the joints, the segments are provided with an inside groove to accommodate a watertight gasket. The primary load resisted by the segments is circumferential (hoop) stress induced by external pressure from the surrounding ground acting on the circumference of the tunnel. The segments are generally designed to resist the axial bearing loads and buckling from the TBM thrust loads. Under geostatic conditions, the segments transfer load across the joints without damage in the concrete segments. However, the response of the segments under blast loads is more complex as the tunnel system employs the flexibility of its segmented linings to resist the blast load. In segmented tunnels, the segments resist the blast load by allowing the joints to rotate, slide and dissipate energy in order to achieve equilibrium before the concrete segments are damaged. Structural analysis of segmented tunnel under static and earthquake loads has been the subject of several studies. However, there is inadequate information on the blast response of bored tunnels. Nasri Munfah[1] described that thin precast segmented tunnel linings are more vulnerable to blast loads than thick cast in place concrete tunnels. In this research field, explosive tests with real physical models are extremely risky and expensive to investigate the tunnel response. A limited number of studies however have been conducted using scaled-down centrifuge modeling techniques to investigate the tunnel response under surface blast loading. The centrifuge modeling is useful for scale modeling of large-scale nonlinear problems in geotechnical engineering. Studies [2-7] have shown the successful implementation of centrifuge modeling to simulate the blast response of buried structures. De et al.[2, 3] described a recent series of centrifuge tests to examine the surface blast effect on a buried copper pipe in dry sand. During this process, the gravitational acceleration increases with radial distance along the rotating arm of the centrifuge and hence the gravitational field is not constant across the depth of the model in the test bucket. This limitation controls the centrifuge testing to smaller models. Scaled-down modeling of large structures such as bored tunnels with segments may be impossible due to the size limitation. Moreover, it may not be feasible to investigate the effect of contact joints using the scaled-down models. The possible alternative therefore is to use numerical modeling techniques which can provide valuable data in a timely and cost effective manner to enable the development of design tools and retrofit measures. Several studies [3, 8-12] have treated the simulation of the blast response of transit tunnels. De[3] used the coupled fluid-structure interaction (FSI) approach in Arbitrary Lagrangian Eulerian (ALE) algorithm to study the surface blast induced tunnel response using Autodyn. Eulerian meshes were used to model the air and explosive while the soil and the tunnel were modeled with Lagrangian meshes. Yang et al.[9] studied the blast response of a metro tunnel in Shanghai using an advanced general purpose multi-physics computer software LS-DYNA[13]. The study also used ALE method, but the interface between Eulerian soil meshes and Lagrangian tunnel meshes was merged at the common nodes. The modeling was unable to simulate the ground-lining interaction and subsequent separation, re-contact and sliding at the contact interface. Bessette[14] simulated the Conventional Weapon Effect Backfill test [15] using FSI approach in ALE to investigate the

3

blast response of a reinforced concrete box structure buried in various backfill conditions. In this study, the test structure was modeled using Lagrangian meshes while the other three materials, soil, air and explosive, were modeled as Eulerian meshes. Wang et al.[16] used a fully coupled procedure involving the Smooth Particles Hydrodynamics (SPH) method and the Finite Element Method (FEM) for analyzing the response of buried cut-and-cover tunnel subjected to blast loading using Autodyn.SPH particles were used to model the explosive and near field soil while Lagrangian meshes were used to model the rest of the soil and the tunnel. The techniques discussed above have the capability to simulate the sequences of phases, such as explosion, crater formation, shockwave propagation and the tunnel response. However, the numerical techniques need to be thoroughly validated in order to investigate a real problem. This paper first compares ALE and SPH numerical techniques to investigate the above ground explosion and the subsequence tunnel response with regards to the modeling aspects of numerical prediction and computational efficiency reported in Koneshwaran et al.[17]. This study identified the better numerical technique which was then employed to treat the blast response of segmented bored tunnels buried in dry sand. This particular study was extended from previous investigation reported in Koneshwaran et al.[17]. The commercially available non-linear finite element software package LS-DYNA is used in this study. Tunnels with different types of segments were further compared in the present study to investigate the flexibility and drifting effects of the segmented tunnels.

2.Numerical simulations Numerical simulations divide the system into finite elements, a process called discretization which occurs with respect to time (temporal) and space (spatial). The temporal discretization uses the explicit method which calculates the state of a system at a later time as a function of time step from the current state of the system. To describe any activities within an element, the time step should comply with the Courant-Friedrich-Levy (CFL) function such that the time step ( t ) is less than the period for sound to travel across the smallest element. For blast problems in LS-DYNA, it is recommended to use a Safety Factor (SF) of 0.67. This function can be generally described as below:

0

lt N

c (1)

where N is the safety factor, l is the least element size and c is the speed of sound through the element. 2.1. ALE method Computer hydrocodes include two types of spatial discretization solvers which are the Largrangian and Eulerian solvers. In the Lagrangian solver, the elements move with the material during the distortion. This is mainly used in structural mechanics where the distortion is represented by the mesh distortion. The Lagrangian solver provides easy tracking of free surfaces and interaction between different materials. This solver often suffers severe element distortion during large deformation which can result in very small time steps and grid tangling. The Eulerian solver, in which the mesh is fixed in space while the material flows freely through the mesh, is broadly used in fluid dynamics. Eulerian solver can also be used for solid materials to handle large distortions, but it is unable to define the material boundary conditions involving surface slippage in contact [16].

4

Arbitrary Lagrangian-Eulerian (ALE) approach was developed combing the best features of the above solvers, while reducing their respective weaknesses. ALE is capable of solving problems in fluid dynamics, solid mechanics and coupled problems describing fluid-structure interaction (FSI). Coupled FSI in ALE is a multi-physics simulation process for solving highly non-linear problems with large distortions such as those resulting from an explosion. It allows modeling the explosive and its surrounding using ALE meshes, in which deformable structures are modeled using Largrangian meshes. Firstly, the computation searches for the intersections of the ALE with Lagrangian meshes. When the Lagrangian surface is detected inside the ALE mesh, the coupling algorithm initiates the computation of the penetration of the ALE material across the Lagrangian surface. The interaction forces are calculated during every computational step for their resultant penetration of both materials. The ALE algorithm satisfies the governing equations describing the conservation of mass, momentum and energy[18]. 2.2. SPH method SPH is a meshless computational Lagrangian hydrodynamic particle method developed for astrophysics problems [19, 20] in 1977. It initially dealt with modeling of interacting fluid masses in vacuum without boundaries. It was then improved as a deterministic meshless particle method and implemented to continuum solid and fluid mechanics [21, 22]. SPH is free from mesh tangling encountered in large deformation problems. It is based on interpolation theory of kernel approximation of a function[18], which is adequately smooth for higher order derivatives to deliver stable and accurate results. The coupled SPH-FEM approach reduces high computational demand and the FEM meshes act as boundaries for SPH particles [23]. There are methods, such as tied contact, hybrid element coupling and nodes-to-surface contact, allowing the coupling interaction between SPH particles and FEM meshes. Penalty based automatic ‘Node-to-surface’ coupling method is employed in the present study. This is a computationally efficient method than the tied contact method. As shown in Figure 1, SPH particles are constructed close to the solid elements.

Figure 1: Coupled SPH-FEM.

3. Description of the experiment

5

In geotechnical investigations, centrifuge testing uses small-scale physical model to simulate the physical behavior of large-scale prototype model under different loading conditions. De et al. [3, 24] conducted a centrifuge test to investigate the performance of a buried copper tunnel subjected to surface explosion. He used a 70g centrifuge testing machine, where g is the gravitational acceleration. A scaled-down model was prepared by burying a copper tunnel in dry Nevada sand (with relative density Dr =60%) to simulate a depth of 3.6m in the prototype scale. As shown in the half symmetry Figure 2, a spherical shape explosive was symmetrically placed above the mid-span, directly over the centerline of the copper tunnel, such that the ground surface was tangent to the spherical surface of the explosive. During this process, readings from strain gauges mounted on the exterior surface of copper pipe were recorded. The present paper uses these results to compare the response predicted by the numerical simulations.

Figure 2: Experimental setup of centrifuge test (all dimensions are in prototype scale).

Centrifuge scaling laws explain how a physical model and its dynamic events are correlated in the centrifuge test, in which the scaled-down model is sufficiently raised to N times the gravitational acceleration. The centrifuge scaling laws [25] allow to convert the scaled model dimensions to the prototype model dimensions as shown in Table 1.

Table 1: Conversion to prototype measurements.

Model Parameters Scaled-down model dimension

Prototype model

dimension Copper pipe diameter 76 mm 5.62 m

Copper pipe thickness 2.5 mm 175 mm

Explosive weight of TNT 2.6 g 888 kg

4. Material Constitutive Models

6

This numerical simulation includes the following material models in LS-DYNA for modeling air, explosive, soil and copper tunnel: 4.1. Air The air is modeled as an ideal gas [13] using null material model with a linear polynomial Equation of State (EOS). The pressure P is expressed by: ECCCCCCCP 2

6543

32

210 (2)

where E is internal energy per unit initial volume, C0, C1, C2, C3, C4, C5, and C6 are constants

and 10

, where

0

is the ratio of current density to initial density.

The linear polynomial equation represents an ideal gas with the gamma law EOS, in which C0=C1=C2=C3= C6= 0 and C4 = C5= 1 , where is the ratio of specific heat at constant pressure per specified heat at constant volume. The pressure is then described by:

00

( 1)P E

(3)

where is an adiabatic constant for air behaving as an ideal gas (estimated value for =1.4), 31.29 kg m is the density and the initial internal energy per unit volume, 0E , is estimated

as 0.25MPa [9]. 4.2. Explosive The Jone-Wilkin-Lee (JWL) EOS [13] is used to describe the explosive as it is the most popular and easiest to calibrate. The JWL EOS defines the pressure Pas:

V

Ee

VRBe

VRAP VRVR

21

21

11 (4)

where V is the relative volume (or the expansion of the explosive) , E is the initial energy per volume, other parameters A, B, R1, R2 and ω are empirically derived constants for the explosive. Table 2 shows the material parameters used for TNT (Trinitrotoluene) [16] explosive.

Table 2: Material properties for TNT explosive[16]. ρ vD PCJ A B R1 R2 ω V E0

g/cm3 m/s GPa GPa GPa kJ/ m3

1.630 6930 21 373.77 3.747 4.15 0.90 0.35 1 6.0e+06

7

4.3. Soil LS-DYNA material library includes more than fourteen types of soil models for the soil material. MAT_FHWA_SOIL model developed by Federal Highway Administration (FHWA) is considered for the present study. The material model includes the effects of strain softening, kinematic hardening, strain rate, element deletion, excess pore water effects (although dry sand is used in this analysis) and stability with no soil confinement [26, 27]. The FHWA soil model is a modified version of Mohr-Coulomb yield criterion [28] based on the geo-technical parameters evaluated in laboratory tests. The modified yield surface is a smooth hyperbolic surface which provides accurate, robust and cost saving numerical simulations. This material model requires the main parameters of mass density, specific gravity, bulk modulus, shear modulus and moisture content. These soil parameters are generally determined through laboratory tests. Parameters required for defining strain softening, kinematic hardening, strain rate effects and pore water effects can be evaluated through laboratory tests and/or equations in the material manual [26]. Studies by Jayasinghe et al.[29], Lee [30] and Ortman et al. [31] illustrate that the FHWA soil model can be used for the blast simulation. Values suggested by Lee [30] were used in this study to include the strain softening, kinematic hardening and strain rate effects in the soil. Strain softening occurs in the vicinity of the blast. To avoid mesh sensitivity in the model a mesh sensitivity analysis was first used to select mesh size in this region. In addition the model uses an input parameter of “void formation (VDFM=1.0e+07Ncm)” as suggested by Lee [30] to significantly minimize mesh dependent results. At the outset of the Civil and Mechanical Systems Program of the National Science Foundation (NSF), Nevada sand (with a relative density Dr of 60%) was used for the centrifuge tests by De [3]. In 1992, Arulmoli et al. [32]conducted an extensive laboratory test for the Nevada sand with different Dr values including: 40% and 60% in the VELACS (Verification of Liquefaction Analyses by Centrifuge Studies) Program. This study provides the main soil properties such as mass density and specific gravity. Based on the initial void ratio, porosity of the sand was derived as 0.4. Anriban De [3] presented data for density ( ) versus sound speed (c) and this was used for back-calculation of shear modulus (G) as 56.0 MPa. The Bulk modules (K) was derived as 146.0 MPa from Poisson’s ratio of the Nevada sand ( ) = 0.33 [3]. As suggested by Lee [30], the pore-water parameters PWD1, PWD2 and Ksk were defined as zero in the input material card [26] to eliminate the pore-water effects. PWD1 defines the stiffness of the soil by adjusting the bulk modulus before the air voids collapse. PWD2 computes the pore-water pressure in the soil before the air voids collapse. Ksk

is the volumetric strain factor. Table 3 presents the main material parameters for the soil.

Table 3: Material parameters for soil[33]. Parameters Dry Nevada sand

Density (g/cm3) 1.60

Specific gravity 2.67

Shear modulus (MPa) 56.0

Bulk modulus (MPa) 146.0

Cohesion (MPa) 6.20e-03

Friction angle 35o

8

4.4. Copper The copper tunnel is modeled using MAT_PLASTICIY_KINEMATIC material model which incorporates both non-linear material behavior and high strain rate effects due to the ground shock. The strain rate effect is incorporated by using the Cowper Symonds strain rate relationship given by Eq. 2.

1

01P

C

(5)

where, is the dynamic flow stress at a uniaxial plastic strain rate , 0 is the associated

flow stress. The appropriate values for the strain rate parameters C and P can be found in [34]. The main parameters include mass density ( ), Young’s modules (E), Poisson’s ratio ( ),tangent modules (Etan), hardening parameter ( ). The material properties for the copper pipe are described in Table 4.

Table 4: Material parameters for copper[35]. ρ Es v σ Etan β C P

g/cm3 GPa MPa MPa s-1

8.93 117 0.35 400 100 0 1.346e+06 5.286

5. Numerical model The problem described in Section 3.0 was simulated using two different numerical approaches:

i. ALE simulation: Coupled FSI in ALE ii. SPH simulation: Coupled SPH-FEM

Each approach simulates the same model dimensions and material parameters as described in Section 4. As the explosive was detonated over the centerline of the tunnel, as shown in Figure 2, symmetric modeling capabilities were adopted in both simulations by considering quarter symmetry-geometrical numerical models to reduce the computation demand. 5.1. Coupled FSI in ALE using Lagrangian meshes for the soil The experiment was first simulated using coupled FSI in ALE. The numerical model was composed of four components describing the explosive, air, soil and the tunnel as shown in Figure 3(b). The fluid space for the air was represented using the eight-node hexagonal solid elements with a grid of Eulerian solver while the eight-node hexagonal solid elements with Lagrangian solver were used to represent the soil and tunnel (structural mesh). The spherical explosive within the air domain was defined into the air mesh using INITIAL_VOLUME_FRACTION_GEOMETRY by specifying its radius and detonation point. A mesh consistency condition was achieved through a series of cases with different meshes to capture the analytical solution in the limit of a mesh refinement process. The soil was refined with a gradual increase in mesh size in both X and Y directions from the region adjacent to

9

the explosive. The smallest element size in the near field of the explosive was 12.5cm x 12.5cm x 12.5cm, whereas the larger element size of 40cm x 40cm x 40cm was used for the far field region of the soil. Three elements were created across the thickness of the tunnel to facilitate a nonlinear stress distribution. During the refinement process, it was identified that the mesh alignment along the curvature interface significantly affected the accuracy and robustness of the model in the circumferential direction where the penetrating nodes and crossed edges interlocked the system. Therefore, a procedure was developed so that nodes on the soil side were coincident with nodes on the tunnel side. This procedure significantly improved the prediction of the tunnel response from that in the previous study [17]. The mesh size was gradually increased in the tunnel, away from the explosive. The smallest element size was 10.25cm x 12.5cm x 5.83cm.

Figure 3: Coupled FSI in ALE.

The contact interface between the soil and copper tunnel was defined using CONTACT_AUTOMATIC_SURFACE_TO_SURFACE. The translational displacements of symmetry boundaries XZ and YZ plans were constrained in the normal direction. Infinite domains were bounded by non-reflecting boundaries which allowed the blast wave to leave the domain without reflection. The base was fixed in all directions to represent the bedrock. Minimizing the computational cost is essential in the numerical modeling which relates to a time-ordered sequence of interrelated phases describing the entire simulation. As such, LS-DYNA’s restart feature enables dividing the entire simulation into three stages such as stress initialization, coupled FSI in ALE and deletion of ALE background mesh. Prior to the explosion, gravity load was applied to the structural mesh using a ramped load curve in the form of a time-dependent mass damping DAMPING_GLOBAL to impose near-critical damping until the gravity was established. The blast loading was applied after obtaining stability under soil gravity load. This simulation was carried out without the background air mesh as shown in Figure 3(a). Upon initializing the structural mesh, the background mesh was inserted into the preloaded model as illustrated in Figure 3(b). This enabled the essential process of a fully coupled FSI approach combing both Lagrangian and Eulerian meshes using CONSTRAINED_

(b) Coupling with air. (a) A quarter symmetrical model.

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LAGRANGE_IN_SOLID. Degree of refinement of ALE air mesh coupling the Lagrangian soil mesh determines the accuracy. Reducing the ALE mesh size increases the computational cost. For adequate accuracy in the solution, the ALE mesh flow passage above the tunnel was made nearly the same size as the Lagrangian soil meshes. The coupled FSI in ALE is suited for relatively short duration problems and conducting the entire blast simulation using this approach is quite expensive. In the present simulation, considerably short duration is required to transfer the blast energy from the background air mesh to the structural mesh [17]. When the kinetic energy of the background air mesh was sufficiently reduced to zero, the redundant background air mesh was hence removed from the simulation. 5.2. Coupled SPH-FEM The experiment was further simulated using the coupled SPH-FME approach. As shown in Figure4(b), a portion of the soil experiencing large deformations and the explosive were modeled with SPH particles while the rest of the geometry was modeled with Lagrangian meshes. No attempt was made to model the interior volume of the tunnel. The surrounding outside space of the explosive was assumed to be a vacuum which ignored the later interaction process between the explosion-produced gas and surrounding atmosphere.

Figure 4: Coupled SPH-FEM.

First, a number of models were developed to determine the optimal size of a box filled with SPH soil particles. For a quarter symmetric model, the optimum size of the box was determined as 350cm x 350cm x 276cm. The SPH particles were 10cm in diameter with equal inter-particle distance of 10cm. Beyond the SPH region, the soil and the tunnel were replicated with the same mesh resolution as described in the ALE simulation. The coupling interaction between the SPH and Lagrange FEM is formed by the penalty based contact CONTACT_AUTOMATIC_NODES_TO_SURFACE. Though the boundary conditions were identical to the ALE model, a special symmetry boundary BOUNDARY_SPH_SYMMETRY_PLANE was applied to those SPH particles at the symmetry planes. The simulation was considered with two stages such as stress initialization

(a) A quarter symmetrical model. (b) Insertion of explosive.

11

and blast analysis. A model described in Figure4(a) was used for stress initialization with a time-dependent mass damping. Upon initializing the model, as illustrated in Figure 4(b), the explosive SPH particles were added into the preloaded model for the blast analysis.

6. Results and Discussion Numerical simulations were conducted using LS-DYNA R7.0.0 (2013). This section compares the numerical results with those from an experiment which was described in Section 3. Before performing the blast (transient) analysis, the stress initialization phase brought the numerical model to a steady-state preload in 1500ms. The blast load was applied to the models by detonating the explosive at 1500ms. 6.1 Shockwave propagation Immediately after the detonation, spherical shockwaves propagated in the air from the charge epicentre which was 50.5cm above the ground. As the shock waves reached the soil, they transferred into the ground in the form of hemispherical waves while forming a crater in the ground. Figure 5 shows that the area of wave front expanded in the soil and impacted the tunnel crown after approximately 7ms of explosion.

Figure 5: Shockwave propagation in soil.

6.2 Tunnel response The stabilized tunnel started to respond significantly when the shock wave impacted the tunnel surface. Figure6 illustrates the sequence of shock wave propagation through the tunnel during the ALE simulation. As shown in Figure6(a), the tunnel response commenced at t = 1507ms due to the strike of direct shock waves on the tunnel crown. The shock waves propagated in both longitudinal and transverse directions. It can be seen that the shock waves propagation in the longitudinal direction is faster than that in the circumferential direction while positive and negative phases of stress contours changed with time.

(a) Shockwave in ALE simulation. (b) Shockwave in SPH simulation.

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Figure 6: Pressure contours on the tunnel.

In order to compare the dynamic performance of the tunnel obtained from the numerical simulations with the results from the centrifuge test, four measuring points in the centrifuge model were simulated in the numerical model. As illustrated in the half symmetry model of the prototype structure in Figure 7(a), measuring points AS1 and AS2 were arranged along the surface of tunnel crown to record the axial strains while measuring points CS1 and CS2 on either side of the springline at mid-span recorded the circumferential strains. By considering the symmetry, as shown in Figure 7(b), three gauge points were considered in the numerical model where Gauge 1 and Gauge 2 represented strain gauges AS1 and AS2 respectively and Gauge 3 simulated the experimental readings at corresponding points CS1 and CS2.

Figure 7: Arrangement of measuring points.

Figure 8 shows a comparison of axial and circumferential strain histories at two locations (Gauge 1 and Gauge 2) obtained from the two different numerical approaches. Figure 8(a) shows that the peak axial strain in the SPH simulation is about 7 % more than that in the ALE simulation. General trend of circumferential strain histories are similar as shown in Figure 8(b). However, the peak values from the SPH simulations are slightly more than those from the ALE simulations in both the positive and negative phases. After the first peak, there were inconsistent peaks and valleys in the positive phase. This could have resulted from the existing air background mesh in the ALE simulation. Both curves display some fluctuations in strains after 1575ms. Repeated reflection of shock waves caused this fluctuation which continued until the shock waves completely attenuated in the soil.

(d) t = 1519ms

(c) t = 1514ms (b) t = 1509ms (a) t = 1507 ms

(e) t = 1524ms (f) t = 1529ms

(a) A half-symmetry prototype model. (b) A quarter-symmetry numerical model.

13

-1800

-1500

-1200

-900

-600

-300

0

300

1500 1550 1600 1650

Mir

cros

trai

n

Time elapsed, ms

Coupled FSI in ALE

Coupled SPH-FEM

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500 1550 1600 1650

Mir

cost

rain

Time elapsed, ms

Coupled FSI in ALE

Coupled SPH-FEM

Figure 8: Comparison of axial and circumferential strains.

Figure 9 shows the comparison of the numerical strain histories at Gauge 2 with the results from the centrifuge test [2]. The ALE predictions are closer to the experimental results compared to the SPH predictions which were somewhat conservative across most part of the duration. This could be due to the assumption in the SPH simulation that the surrounding of the explosive SPH particles was considered to be a vacuum. This assumption ignored the importance of the interaction of the SPH explosive particles with the air and hence the energy imparted from the explosive into the soil was significantly larger in the SPH simulations than that in the ALE simulations.

Figure 9: Comparison of axial strain between numerical and experiment.

Figure 10 compares the magnitudes of peak axial strains at Gauge 1 and the peak circumferential strain at Gauge 3 with respect to the equivalent scaled distance of R/W1/3 to the explosive. Results for these strains obtained from both ALE and SPH simulations are compared with those from the test data and overall the results from both simulations agree reasonably well with each other and with the test result. Figure10(a) shows that the SPH prediction is slightly (about 7%)higher than the ALE prediction which is closer to the test data. There is however a small discrepancy in the peak axial strain between the test data and the ALE simulation. This was also observed in the numerical simulations conducted by De[3]. This could be due to the experimental limitation of the confinement effect of the test-bucket which accommodated the test materials. A real tunnel in an infinite soil domain has no

-500

-250

0

250

500

750

1500 1550 1600 1650

Mic

rost

rain

Time elapsed, ms

Centrifuge test

Coupled FSI in ALE

Coupled SPH-FEM

(a) Axial strain at Gauge 1. (b) Circumferential strain at Gauge 3.

14

movement restrictions, but the four sides of the test-bucket constrained the motion of the soil and the (copper) pipe ends.

Figure 10: Comparison of peak axial and circumferential strains.

The circumferential strains at Gauge 3 obtained from the numerical simulations are compared with the test data, as illustrated in Figure10(b). The comparisons show that both numerical best-fit lines are very near and they fall within the range of test data at CS1 and CS2.The test data CS1 and CS2 should be the same for this case with symmetry, but the magnitude of CS2 is 25% more than that of CS1. This discrepancy may have resulted from either the displacement of the explosive from its initial orientation or rotation of the (copper) pipe about its axis. This could also be another factor that affected the experimental peak axial strain at gauge AS2. 6.3 Comparison of computational efficiency Computer simulations were conducted using nine parallel processors on a standard high performance workstation. Each simulation involved two stages of stress initialization and blast analysis. Table 5 shows the comparison of the number of elements and the computational time of both ALE and SPH simulations. The ALE simulation is much faster for the stress initialization than the SPH as the ALE simulation used the Lagrangian structural mesh alone. For agreed mesh resolution, the ALE simulation took slightly more CPU time than the SPH to simulate the blast problem for a period of 180ms.

(a) Comparison of peak axial strains. (b) Comparison of peak circumferential strains.

15

Table 5: Comparison of computational efficiency.

ALE simulation SPH simulation

Initialization Blast-analysis Initialization Blast-analysis Nos. of Lagrangian elements

220255 220255 213472 213472

Nos. of Eulerian elements

- 307530 - -

Nos. of SPH particles - - 34300 34438

Simulation duration (ms) 1500 180 1500 180

Timestep ( s ) 1.06e+01 5.89e+00 1.06e+01 4.72e+00

Total CPU time (h:m:s) 10:51:51 164:10:11 96:31:49 142:54:41

7. Blast analysis of segmented bored tunnels In order to assess the blast response of a segmented bored tunnel, the coupled FSI in ALE technique was applied. The present study simulates a shallow bored tunnel constructed in dry sand and the ground exerted a surface surcharge of 75kPa [36]. As shown in Figure 11(a), the most used ring type for modern railway tunnel was considered. Each ring includes six identical rectangular segments (RS) spanning the circumference of the tunnel. In this tunnel, segments are rotated from ring to ring by 30o in order not to line up along the longitudinal direction. Every single segment is attached to its neighboring segments by bolts in both radial and circumferential directions, in which joints were considered as flat surfaces, as described in Figure 11(b).

Figure 11: Rectangular Segments (RS).

As shown in Figure 12, the problem was modeled using similar mesh resolution and material parameters as before. However, the segments were modeled using MAT_PLASTICIY_

(b) Joint details. (a) Tunnel cross section.

16

KINEMATIC material model as a smeared concrete as also assumed by others [9]. Table 6 describes those concrete parameters. Bolts (M24 bolt grade 8.8) were simulated as beam elements which were merged to the solid concrete elements at the common nodes [14]. Spherical TNT explosives were placed on the ground surface directly above the crown of the tunnel.

Table 6: Material parameters for concrete [9]. ρ Es v σ Etan β C P

g/cm3 GPa MPa MPa s-1

2.65 39.1 0.25 100 4.0e+03 0.5 0 0

Figure 12: Configuration of the coupled numerical model.

Prior to the blast, both gravity and surcharge loadings were applied to preload the model. As shown in the displacement vs. time plot in Figure 14 (a), the tunnel was brought to the geostatic equilibrium state where the maximum displacement was within the allowable deflection of 1% internal diameter. A number of blast cases (1 - 5) were considered by varying the explosive from 250 to 1250kg of TNT, by equal amounts of 250kg, for a given tunnel depth of 6.35m. Figure 13(a) illustrates the blast load transfer mechanism in the tunnel. When the shock waves impacted the tunnel, the segments along the tunnel crown were compressed in the longitudinal direction. However, as a result of soil confinement, energy dissipation of joint interfaces and shock wave attenuation, this effect was localized within a limited numbers of segments from the explosive, as shown in Figure 13(b). At the invert level, blast induced tension force was redistributed to adjoining segments through the bolted circumferential joints. Although out of plane movements in the tunnel are naturally restrained along the longitudinal tunnel axis, the blast induced tension triggered the yielding of bolts. It was found that gaps opened between segments were insignificant (maximum of 1.2mm for 1250kg of TNT) due to the fact that the bolts were strong in tension.

Figure 13: Tunnel response in longitudinal direction.

(a) Blast load transfer mechanisms. (b) Numerical simulation.

17

-2

-1.5

-1

-0.5

0

0 300 600 900 1200

Dis

pla

cem

ent,

cm

Time, ms

0

0.5

1

1.5

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1200 1210 1220 1230 1240 1250

Dri

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ovem

ent,

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Time elapsed, ms

Case1 (250 kg)

Case2 (500 kg)

Case3 (750 kg)

Case4 (1000 kg)

Case5 (1250 kg)

In order to assess the in-plane response of the ring immediately below the explosive, displacement history of closest point on the tunnel crown was considered as shown in Figure 14(b). As expected, the vertical deformation increased with explosive. For 250kg TNT (Case1), the tunnel crown experienced more than four times its geostatic displacement. For load cases 2 - 5, the tunnel exceeded the deflection limit. Although the exceeded deflections were transient and lasted for few milliseconds, it can be a hazardous for the operational envelop which accommodates train and other services.

Figure 14: Tunnel crown response.

Figure 15(a) demonstrates that the segments moved relative to each other (in the form of drifting response). This mainly occurred perpendicular to the tunnel axis between the rings along the longitudinal direction caused by the vertical component of the blast load redistributed to the adjoining rings through the bolted joints. These were weak in shear and in the vicinity of the explosion, many bolts failed in shear in the common planes and not through global deformation of the rings. There were no significant offsets or drifts between segments in the transverse direction. Figure 15(b) demonstrates that load cases 2 - 5 generated a permanent drifting response. It may affect the alignment of gasket which is very essential in water tightness of the tunnel lining.

Figure 15: Drift movement between lining rings.

The simulations indicated that the drifting response resulted from the continuous vertical joints between the rings. In order to prevent the development of drift, two types of segments were considered. The first one is the precast concrete hexagonal segmental (HS) system

(a) Displacement vs. time –initialization. (b) Displacement vs. time - blast analysis.

(a) Drifting response. (b) Drift vs. time elapsed.

18

0

0.5

1

1.5

2

2.5

1200 1210 1220 1230 1240 1250

Dri

ft m

ovem

ent,

cm

Time elapsed, ms

Case1 (250 kg)

Case2 (500 kg)

Case3 (750 kg)

Case4 (1000 kg)

Case5 (1250 kg)

which has the advantage of non-continuous ring joint. It is offset by a half segment width between the neighboring segments along the longitudinal direction as shown in Figure 16(a). As there could be construction difficulties associated with the handling of the hexagonal segments by the conventional TBM which uses horizontal segment erection method, the authors considered an interlocking segment (IS) system where a part of the segment is stepped and interlocked with neighboring segments as shown in Figure 16(b).

Figure 16: Different types of segments.

Under geostatic conditions, responses of both HS and IS are similar to RS system, but the blast responses are significantly varied due to the behavior of joints. In the HS system, drifting response along the tunnel axis is essentially improved and converged to a very minimal residual displacement as shown in Figure 17(a). However, as shown in Figure 17(b), the peak crown displacements (of immediate segment) were similar to those described in the RS system for load cases 1 and 2. For high explosive cases, the crown displacements were slightly reduced because of the adopted joint system.

Figure 17: Immediate HS ring response.

Figure 18(a) shows that for load case 5, the segments along the tunnel crown were subjected to extreme drift movements perpendicular to the tunnel axis. As the shock waves progressed, the crown-segments were disengaged and slipped along the adjacent segments along the circumference upon failure of bolts in shear. While the drifting response developed and continued along the weakest plane as described by the zigzag lines, segments drifted relative to each other along the springline. This was a transient phenomenon and quickly returned to rest with a negligible residual displacement. However, the immediate crown-segment

(a) Hexagonal segments (HS). (b) Interlocking segments (IS).

(a) Drift vs. time elapsed. (b) Displacement vs. time - blast analysis.

19

-3

-2

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3

4

5

6

1200 1210 1220 1230 1240 1250

Dri

ft m

ovem

ent,

cm

Time elapsed, ms

Drift in crown segment

Drift in springline

exhibited a large residual drift of 5.0 cm in the transverse direction as shown in Figure 18(b). Considering Figures 17(b) & 18(b), it is evident that the localized drift in the crown segment dominated the displacement response by more than 25%. Apart from the localized drift response, the global response of the overall HS system is stiffer than RS system. This resulted from the “V” shaped (two-surface) radial joint in HS system affecting the rotational ability (degree of flexibility) of the tunnel even though the two-surfaces significantly increased the sliding contact area compared to RS system which has single surface radial joints (radial joints are parallel to the tunnel axis).

Figure 18: Hexagonal segmented tunnel response.

Finally, the IS system response is compared with RS system. The tunnel suffered permanent drifts between tunnel rings in all load cases, as shown in Figure 19(a), and the drifts were comparatively more than RS system. As shown in Figure 19(b), displacements were slightly more than RS system with unsmooth lines due to a serious of sudden drift movements. Although the radial joints are parallel to the tunnel axis, alignment of radial joints (multi-surface) in the interlocking segment affected the flexibility of the radial joints.

(a) Drifting response.

(b) In-plane drift vs. time.

20

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1200 1210 1220 1230 1240 1250

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Time elapsed, ms

Case1 (250 kg)

Case2 (500 kg)

Case3 (750 kg)

Case4 (1000 kg)

Case5 (1250 kg)

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1200 1210 1220 1230 1240 1250

Dri

ft m

ovem

ent,

cm

Time elapsed, ms

Drift in crown

Drift in springline-segment

Figure 19: Immediate IS ring response.

Figure 20 illustrates, for load case 5, that the tunnel rings suffered permanent drifts not only in the longitudinal direction, but also in the transverse direction (in-plane drifts) of the tunnel. As shown in Figure 20(a), nosing of the interlocking segments were subjected to a high concentrated stress due to the extreme hoop stress from the crushing of adjoining segments. This resulted in contact-element distortion. It can be clearly seen that the downward drift at the crown, as displayed in Figure 20(b), increased the crown-displacement response. Overall, the interlocking joint system affected the tunnel response in both longitudinal and transverse directions of the tunnel.

Figure 20: Interlocking segmented tunnel crown response.

(a) Drift vs. time elapsed. (b) Displacement vs. time - blast analysis.

(a) Drifting response.

(b) In-plane drift vs. time.

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8. Conclusion Two numerical modeling techniques (i) coupled FSI in ALE and (ii) coupled SPH-FEM have been developed and applied for treating the blast response of underground tunnels subjected to surface explosion using the software LS-DYNA. The modeling techniques were compared and validated using results from experiments. In terms of accuracy and computational efficiency, the coupled FSI in ALE outweighed the coupled SPH-FEM method for dealing with above ground explosion problems. The coupled FSI in ALE provides a comprehensive solution in a more efficient manner by handling the numerical model through different phases of stress initialization and blast analysis. This provides confidence in adopting the established techniques to treat the blast response of segmented bored tunnels. Performance of the segmented bored tunnels buried in dry sand was demonstrated for various blast load cases. The analysis showed that segments along the tunnel crown were compressed in the longitudinal direction while the segments in invert level were exposed to tension which generated a gap between the rings. The gap was insignificant compared to the compression induced effect at tunnel crown. However, the drifting response along the longitudinal axis is the main concern affecting the water tightness in the most used ring type modern tunnels. The hexagonal segmented tunnel system significantly reduced the longitudinal drifts, but it suffered in-plane drifts resulting from weak continuous joint alignments parallel to the tunnel axis. The ‘V’ shaped radial joints affected the flexibility of the tunnel as well. The interlocking segmented tunnel system responded slightly higher crown-deflections than the most used tunnel system. It also suffered from drifts in all directions.

Comparison of different segments demonstrated that the most used ring system with single phase radial joints displays the better performance with regard to the overall drift responses. In the transverse direction, this system is more flexible than other two systems without in-plane drifts. The drifts in the circumferential joints needs to be further investigated to improve the joint performance under different soil conditions.

The coupled modeling techniques developed and applied to treat the blast response of buried tunnels provided useful information. The results on drift responses and vertical displacements offer new and valuable information in the area of blast response of segmented tunnels and will provide guidance in future modeling and analysis in this area.

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