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The response of underground structures subjected to subsurface blast is an important topic in protective engineering. Due to various

structures, the modelling is complicated by the presence the model parameters are difficult to determine when the

Soil Dynamics and Earthquake EngE-mail address: cylu@ntu.edu.sg (Y. Lu).constraints, pertinent experimental data are extremely scarce. Adequately detailed numerical simulation thus becomes a desirable alternative.

However, the physical processes involved in the explosion and blast wave propagation are very complex, hence a realistic and detailed

reproduction of the phenomena would require sophisticated numerical models for the loading and material responses. In this paper, a fully

coupled numerical model is used to simulate the response of a buried concrete structure under subsurface blast, with emphasis on the

comparative performance of 2D and 3D modeling schemes. The explosive charge, soil medium and the RC structure are all incorporated in a

single model system. The SPH (smooth particle hydrodynamics) technique is employed to model the explosive charge and the close-in zones

where large deformation takes place, while the normal FEM is used to model the remaining soil region and the buried structure. Results show

that the 2D model can provide reasonably accurate results concerning the crater size, blast loading on the structure, and the critical response

in the front wall. The response in the remaining part of the structure shows noticeable differences between the 2D and 3D models. Based on

the simulation results, the characteristics of the in-structure shock environment are also discussed in terms of the shock response spectra.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Underground structure; Subsurface blast; Soil medium; Blast wave propagation; Numerical simulation; Coupled model

1. Introduction

The response of underground structures subjected to

blast loading is an important topic in protective engineering.

Usually such structures are box shaped concrete structures,

partially or fully buried in soil medium. The loading and

response of the underground structure involve different

mechanisms as compared to the above-ground structures.

Concerning the general response, the above-ground struc-

tures are often modeled using single-degree-of-freedom

(SDOF) system. This formulation offers an efficient method

of analysis for preliminary designs, optimization studies,

and concept evaluations. In the case of underground

of the surrounding soil. If the underground structure is to be

modeled using SDOF, the dominant mode of the response

should be identified; however, in reality the response of an

underground structure involves the structuresoil inter-

action (SSI). Consequently, the choice of an appropriate

SDOF model and loading function are complicated. Some

modifications on the SDOF methods were put forward to

consider the SSI effect [1,2]. These methods focused on the

modification of the free-field stress function or the

parametric values of the model to fit the experimental

data. But the usefulness of these modifications is limited

because they could give unreliable results in some ranges

[3]. Moreover, the modifications on the loading function andA comparative study of bur

to blast load using 2D an

Yong Lua,*, Zhongq

aSchool of Civil and Environmental Engineering, NanybDefense Science and Technology Agency, Ministry

Accepted 1

Abstractstructure in soil subjected

D numerical simulations

anga, Karen Chongb

echnological University, Singapore, Singapore 639798

fense, 1 Depot Road, Singapore, Singapore 109679

ruary 2005

ineering 25 (2005) 275288

www.elsevier.com/locate/soildynthe structure.

To overcome the abovementioned difficulties, the finite

element method (FEM) may be adopted, so that the structure

and soil can be modeled in a more realistic manner while

the structuresoil interaction can also be incorporated.0267-7261/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.soildyn.2005.02.007

Engineering, Nanyang Technological University, 50 Nanyang Avenue,

Singapore, 639798. Tel.: C65 6790 5272; fax: C65 6791 0676.initial conditions of the problem are unclear. Besides, a

SDOF does not provide detail response information within* Corresponding author. Address: School of Civil and Environmental

velocity function applied on the boundary of the compu-

tational domain. This simplified treatment may be accep-

time step would thus become very small. This difficulty can

thquatable in certain cases such as the structure being far away

from the explosion charge, but in more complicated

situations such as a short standoff distance explosion or a

shallowly buried structure case, the above simplification

may be prone to larger errors because the loading functions

are difficult to define accurately.

In order to model the whole physical process in a more

realistic manner, it is desirable that a fully-coupled method

be devised so that all the processes can be integrated in a

single numerical model. The main challenge to a fully-

coupled method is the needs to model large deformation

zone surrounding the charge while at the same time to be

able to model appropriately the complex geometry of the

buried structure. In the present model, the above problem is

solved by coupling the SPH (smooth particle hydrodyn-

amics) technique, which is superior in modeling large

deformation zone with relatively simple geometry, with the

Lagrange FE elements that are effective in modeling

complex structures.

In the coupled computation, the three-dimensional (3D)

analysis is the ideal approach as it resembles the actual

situation in a more straightforward manner. However, the

computational cost is high with a 3D model, and this can

become a big barrier especially when a large number of runs

are required. As an alternative, in many cases a simplified

2D axisymmetric model may be considered. In this respect,

it is meaningful to provide a general assessment on the

comparative performance of the 2D and 3D models for such

complex problems as predicting the responses of an

underground structure subjected to blast loading. For this

purpose, in this paper the 2D and 3D models are used to

analyze a typical case in which a side-burst is set on a buried

reinforced concrete structure. Through the comparison

between the 2D and 3D models, the response features and

the accuracy of using 2D model for the prediction are

examined. The general characteristics of the damage and the

in-structure shock environment are discussed.

2. Overview of the fully-coupled method

The subsurface blast effect on underground structures

involves many complicated physical processes, namely the

explosion of the charge, formation of the crater, propagation

of the shock wave and stress wave in soil, soilstructure

interaction, and the response of the structure. To handle thisSome coupled methods have been proposed for the analyses

of responses of underground structures under blast loading.

The main techniques include FEM, FDM (finite difference

method) and the so-called hybrid method, and some

successful applications have been reported [46]. But in

these methods the explosion process is not included, and

usually the blast loading has to be simplified as a pressure or

Y. Lu et al. / Soil Dynamics and Ear276complex problem, the traditional finite element methodsbe avoided through the incorporation of the SPH method

with traditional FEM approach [9]. The coupled SPH-FEM

approach benefits from the advantages of both methods, and

can be efficient in dealing with the aforementioned

complexities. The following gives an overview of the

computational framework of the fully-coupled model.

2.1. Conservation equation

The conservation equations of mass, momentum and

energy can be expressed as [10]

Mass : r Zr0V0

VZ

m

V(1)

Momentum : r _ui Z sij;j Crfi (2)

Energy : r _e Z sij _3ij Crfiui (3)

where r is density, V is the volume, the subscript 0

indicates the initial value, m is the mass, sij are the stresses,

3ij are the strains, u is the spatial velocity, e is the energy, f

is the body force, the mark $ is the first derivative of time,i and j range from 1 to 3.

The boundary conditions are either specified displace-

ments or traction

xiX; t Z giX; t on Gx; sijnj Z ti on Gt (4)where x is the current coordinate of a point, X is the

reference coordinate, t is time, n is the exterior normal, g(FEM) will meet many difficulties, in particular the large

deformation of soil near the charge. The large deformation

will result in severe distortion of the mesh and thus interrupt

the computation [7]. Some researchers have tried to use the

hybrid method, which integrates the advantage of the finite

element method and finite difference method to overcome

the mesh distortion. But it is not easy to model the interface

condition with this method and also not easy to apply it in

3D analysis. The primary difficulty with these kinds of

traditional FEM or hybrid methods comes from the mesh

they rely on. Once a mesh is produced, the elements or

grids that represent the physical region cannot be changed

easily.

To get rid of the difficulties arising from the mesh

method, some meshless methods have been put forward.

One of the most important meshless methods is the SPH

(smooth particles hydrodynamics) method [8]. The SPH

method avoids the large distortion of the elements, and can

track the material boundary easily. One of the primary

limitations with the current SPH method is the inefficiency

when modeling thin wall structure. This is because the

thickness of a thin wall is very small comparing to other

dimensions, hence small particles would be required and the

ke Engineering 25 (2005) 275288is the specific displacement function, Gx or Gt denotes

the surface where the displacement or traction boundary

condition are applied.

2.2. The smooth particle hydrodynamic (SPH) method

The main advantage of SPH, as a meshless technique, is

to bypass the need of a numerical grid to calculate spatial

derivatives. This avoids the problems associated with mesh

tangling and distortion, which usually occur in FEM

analyses for large deformation impact and explosive loading

events. Although the name includes the term hydrodyn-

hydrodynamic, in fact the material strength can be

incorporated [9].

In SPH methodology, the material is represented by fixed

mass particles to follow its motion. Unlike the grid based

methods, which assume a connectivity between nodes to

construct spatial derivatives, SPH uses a kernel approxi-

mation that is based on randomly distributed interpolation

points. The particles carry material quantities such as mass

m, velocity vector v, position vector x, etc. and form the

computational frame for the conservation equations. In this

method, each particle I interacts with all other particles J

Y. Lu et al. / Soil Dynamics and EarthquaCoupled mesh of SPH - FEMthat are within a given distance from it (see Fig. 1a). The

interaction is weighted by the so-called smoothing (or

kernel) function. Using this principal, the value of a

continuous function, or its derivative, can be estimated at

any particle I based on known values at the surrounding

particles J using the kernel estimates [8]. Another

important point is that the SPH nodes can use the same

constitutive models as used for the FEM element.

2h

J

I

x-x

Neighboring particles of a kernel estimateFig. 1. SPH and coupling with FEM elements.3. Constitutive models

The material systems of the problem include soil mass,

concrete/reinforcing steel in the structure, and the high

energy charge. In the present fully-coupled analysis,

sophisticated material models are applied. An overview of

these material models is given in this section. More details

can be found from the respective previous publications

[11,12].

3.1. Three-phases soil model

Soil is a multi-phase mixture composed of solid mineral

particles, water and air, and the deformation mechanism

varies with the stress condition. In the process of explosion

in soils and the subsequent blast wave propagation, the

spatial variation and time variation of the stress in soil are

very large. To cater for this drastic change of stress

conditions requires a robust soil model. Recently the authors

have formulated a numerical three-phase soil model which

is capable of simulating explosion and blast wave

propagation in soils [11,12]. The idea stems from a

conceptual model described in [13]. As illustrated in

Fig. 2, in this model the soil is considered as an assemblage

of solid particles that form a skeleton, while the voids are

filled with water and air. In the figure, the element A, B and

C, respectively, represent the deformation of the solidMore details about the SPH technique can be found in

related publications [9].

2.3. Coupling of SPH with FEM

Accurate SPH simulations require large number of

particles throughout the SPH region. Hence if high accuracy

is sought or some special geometry is required, such as thin

walls, etc. large run time can become a problem. The

combination of SPH and Lagrange FEM is a good solution

to this problem. The materials in the low deformation

regions can be modeled using the FEM element. The size of

the particles in the SPH region can also be graded, thus

reducing the overall computational demand. Fig. 1b

schematically illustrates how the SPH particles can be

embedded into a traditional Lagrangian FEM mesh.

There are two different ways that the SPH particles can

be coupled with the FEM elements, one is to join the SPH

particles and the FEM elements, the other is not to join them

but to allow the SPH particles to slide along the surface of

the FEM element; in this case, a special sliding interface

algorithm must be used. In the present study, the SPH

particles are joined with the FEM elements because here the

SPH particles are used for the near-field soil medium (see

Fig. 5); the interface between the SPH mesh and the FEM

mesh is not a material interface.

ke Engineering 25 (2005) 275288 277particles, water and air, and elements D and E describe

2thquathe friction and resistance of the bond connection between

the solid particles. The model formulation can be roughly

divided into two main parts; the equation of state and the

strength model. In the equation of state, the contribution of

each phase is considered. The damage of soil skeleton is

also included, taking into account the strain rate effect [14].

Satisfying the continuity requirements, in the three-phase

soil system there should be

DV

V0Z

DVwV0

CDVgV0

CDVsV0

(5)

where V is the volume of a soil element, V0 is the initial

volume of the element, Vw is the volume of water, Vg and Vsare volumes of air and soil particles, respectively.

Denote the volume of voids as Vp, VpZVgCVw, andhence VZVsCVp.

The pressure load causes deformation in each phase, as

well as friction between the solid particles and deformation

of the bond between the solid particles. The friction force

and the force due to the bond are all exerted on the solid

phase. Satisfying the equilibrium it follows that

dpK dV KvVsvp

dp

vVg

vpbC

vVwvpb

K1C

vpavVp

CvpcvVp

Z0

(6)

where p is the total hydrostatic pressure, ps is the pressure

exerted on the solid phase, pa is the pressure borne by the

friction between the solid particles, pb is the pressure borne

Fig. 2. Concept of three-phase soil model for shock loading.

Y. Lu et al. / Soil Dynamics and Ear278by the water and gas, or the pore pressure, pc is the

pressure borne by the bond between the solid particles, and

pe is the pressure carried by the soil skeleton which is equal

to the sum of pa and pc.

Eq. (6) describes the volumetric deformation under the

hydrostatic pressure, in which vVs/vp, vVg/vpb, vVw/vpb,vpa/vVp, vpc/vVp can be obtained from their independentequations of state or stressstrain relationship, respectively.

The continuum damage model is applied to describe the

damage of the soil skeleton. The bonds between the solid

particles can be represented by a series of elastic brittle

filaments. The resisting stress in each filament obeys the

Hookes law until the filament breaks. Introducing a damage3eff Z3

31 K322 C 32 K332 C 33 K3121=2:

On the other hand, the friction between the solid

particles, pa, is dependent on the normal stress between

the particles and hence can be assumed to be proportional to

the deformation of the soil skeleton

pa Z fKpDVp (9)

where f is the friction coefficient of the solid particles, Kp is

the coefficient of proportionality, DVp is the incrementalvolume of voids.

In the soil model, the viscosity of the water and air is

neglected, so the total shear stress is borne by the soil

skeleton formed by the solid particles. To include the effect

of hydrostatic stress on the shearing resistance of the soil,

the modified Drucker-Prager yield criterion [15] is adopted,

as follows

f ZJ2

pKaI1 Kk Z 0 (10)

in which a and k are material constants related to the

frictional and cohesive strengths of the material, respect-

ively; and I1, J2 are the first and deviatoric stress invariant,

respectively. Taking into account the strain rate enhance-

ment, the yield function becomes

f ZJ2

pK aI1 Kk 1 Cb ln _3eff_30

Z 0 (11)

where _30 is the reference effective strain rate, b is the slopeof the strength against the logarithm of strain rate curve, _3effis the effective strain rate defined as

_3eff Z

2

3d_3ijd_3ij

r:

3.2. Concrete model

The response of the concrete under shock loading is a

complex nonlinear and rate-dependent process. A variety of

constitutive models for the dynamic and static response of

concrete have been proposed in the past. This study adopts

the RHT model developed by Riedel, Hiermaier and Thoma

[16]. This model contains many features known to influence

the behaviour of brittle materials, namely pressure

hardening, strain hardening, strain rate hardening, thirdvariable D, it has

pc Z E01 KDDVp=Vp (7)and

D Z 1 Kexp K1

hb3effh

(8)

where B, h are constants related to the properties of the soil,

b is a constant, 3eff is the effective strainp

ke Engineering 25 (2005) 275288invariant dependence for compressive and tensile

the EOS model and the strength model. The strength model

by Carroll and Holt [18] to yield

p Z1

af

v

a; e

(14)

where the factor 1/a was included on the basis of an

argument that the pressure in the porous material is nearly

1/a times the average pressure in the matrix material. In the

present study, a polynomial form is adopted for the

functions f, as

f Z A1m CA2m2 CA3m

3 with m Zv0vs

K1R0

f Z T1m CT2m2 with m Z

v0vs

K1!0(15)

and

a Z 1 C ainitial K1 ps Kpps Kpe

n(16)

where A1, A2, A3, T1, T2, are constants, v0 is initial specific

volume of solid material, ainitial is initial porous rate, pe is

The model defines the yield stress Y as

rthquake Engineering 25 (2005) 275288 279uses three strength surfaces (Fig. 3); an elastic limit surface,

a failure surface and the remaining strength surface for the

crushed material. Usually there is a cap on the elastic

strength surface.

Following the hardening phase, additional plastic strain-

ing of the material leads to damage and strength reduction.

Damage is accumulated via

D ZX D3pl

3failurep(12)

3failurep Z D1p KpspallD2 R3minf (13)

where D1 and D2 are damage constants, 3minf is the minimum

strain to reach failure, 3pl denotes the plastic strain, p* is the

pressure normalized by fc, and pspall Zp

ft=fc, where ft andmeridians, and cumulative damage (strain softening). The

RHT model for concrete has been evaluated successfully in

the modeling of concrete perforation under shock loading,

and systematic parameters have been obtained for several

kinds of concrete.

The RHT model can be generally divided into two parts,

Uniaxial Compression

Failure Surface

Elastic Limit Surface

Residual Surface

Uniaxial Tension

P

Tensile Elastic Strength

Compressive Elastic Strength

fc

ft

Y

Fig. 3. Three strength surfaces for concrete.

Y. Lu et al. / Soil Dynamics and Eafc are tensile and compressive strength, respectively.

The equation of state in the RHT concrete model is P-a

type. The basic Herrmanns P-a model [17] is a phenom-

enological approach, which emphasizes on a correct

behavior at high stresses, but at the same time it also

attempts to provide a reasonable description of the

compaction process at low stress levels. The principal

assumption is that the specific internal energy for a porous

material is the same as that of the same material at solid

density under the same pressure and temperature. Define the

porosity as aZv/vs, where v is the specific volume of theporous material and vs is the specific volume of the material

in the solid state with the same pressure and temperature. If

the equation of state of solid material is given by: pZf(vs,e),then the equation of state of the porous material simply

takes the form: pZf(v/a,e). This equation has been modifiedY Z Y0 CB3np1 CC log 3p 1 KTmH (17)

Air

Soil

Blast

D R

Target point

(a) Free field without buried structure

Air

Soil

Underground structure Blast

D R

(b) Coupled field with buried structure the pressure when the skeleton of the porous material starts

to collapse, ps is the pressure when the porous material is

fully compacted, n is a constant.

3.3. Elasticstrain hardening plastic model for steel

Under blast loading, the reinforcing steel may be subject

to strain hardening, strain rate hardening and heat softening

effects. In this study, the John-Cook model [10] is adopted

to model the response of the steel bars in the concrete. The

John-Cook model is a rate-dependent, elasticplastic model.Fig. 4. Schematic description of model configurations.

thquake Engineering 25 (2005) 275288Y. Lu et al. / Soil Dynamics and Ear280where Y0 is the initial yield strength, 3p is the effective plastic

strain, 3p is the normalized effective plastic strain rate, B, C, n,m are material constants. TH is homologous temperature,

TH Z T KTroom=TmeltKTroom, with Tmelt being the meltingtemperature and Troom the ambient temperature.

3.4. JWL equation of state for explosive charge

The Jones-Wilkens-Lee (JWL) equation of state [19]

models the pressure generated by the expansion of the

detonation product of the chemical explosive, and it has

been widely used in engineering calculations. It can be

written in the form

P Z C1 1 Ku

R1v

expKr1v

CC2 1 Ku

R2v

expKr2vC ue

v(18)

Fig. 5. Numerical models for 2D and 3D free-field analyses.

Table 1

Parameters used in the three-phase soil model for numerical calculations

Soil Air phase

Solid particles a1Z0.58 Initial densityrg0Z1.2 kg m

3

Water a2Z0.38 Initial sound speedcg0Z340 m/s

Air a3Z0.04 Constant kgZ1.4

Initial density r0Z1.92!103 kg m3 Soil skeleton

Solid particles phase Shear modulus GZ55 MPaInitial density rs0Z2.65!10

3 kg m3 Bulk modulus KpZ165 MPaInitial sound speed cs0Z4500 m/s fZ0.56Constant ksZ3 aZ0.25

Water phase kZ0.2Initial density rw0Z1.0!10

3 kg m3 _3Z1%=minInitial sound speed cw0Z1500 m/s bZ0.1Constant kwZ7 hZ1.0

E0Z20 MPabZ5.0

modeled using regular FEM elements with completely

joined surface.

Fig. 5 shows the final meshes that produce stable results

with a tolerable computation time. A SPH zone is arranged

for the area surrounding the charge (including the charge

itself). The element size for the FEM region is about 0.5 m.

For the 3D model, only half of the field is modeled

considering the symmetry about the yZ0 plane. Thetransmission boundary condition is applied at all the

artificial boundaries to minimize the stress wave reflection

at these computational boundaries. The parameters used in

the models for the soil and the charge are listed in Tables 1

and 2 based on existing literature [13,19].

Several target points are arranged along the radial

direction from the charge at embedded depth of 4.8 m to

record the propagation of the stress wave in the soil.

Theoretically speaking, the situation can be very well

simulated as an axis-symmetrical problem so the 2D and 3D

models are expected to produce practically the same results,

given appropriate model settings.

Fig. 6(a) and (b) show the computed shape of the crater

e0 (MJ mK3) VOD (m sK1) r0 (kg m

K3)

2 6.0!103 6.93!103 1.63!103

rthquake Engineering 25 (2005) 275288 281analyses, the computational domain is chosen to be on

the order of 50 m wide and 30 m deep. The explosive

charge is chosen to be 50 kg (TNT equivalent),

embedded at 4.8 m deep.

The calculations are carried out using the programme4. Numerical model

The response of an underground structure depends on

the input load. The input load is transmitted from the soil

medium and may be measured in terms of ground shock

or stress wave. Therefore when different analytical

approaches are considered, such as 2D or 3D model, it

is important to first examine the prediction of the stress

wave in the free-field soil before going into the soil

structure coupled analysis. Fig. 4(a) and (b) show

schematically the scenarios without or with a buried

structure. Considering the example structure size (to be

described later) and the observations from some trialwhere v is the specific volume, e is specific energy. The

values of constants C1, R1, C2, R2, u for many common

explosives have been determined from dynamic

experiments.

3.5. Soilconcrete interface

According to the experimental results from Huck et al.

[20], in general the soilstructure (concrete) interface

strengths may be described by Coulomb failure laws. On a

smooth soilconcrete interface failure is initiated when

the shear stress parallel to the surface exceeds the failure

law. On a rough soilconcrete interface, failure is initiated

when the maximum soil shear stress exceeds the failure law.

The experimental results from Mueller [21] indicate that the

strength properties of the interface are close to the strength

properties of the soil. For this reason, in the present study

the interface between the (rough) concrete and soil is

3.738!10 3.747 4.15 0.9 0.35

e0, the initial C-J energy per volume; VOD, the C-J detonation velocity.Table 2

JWL parameters used for modeling TNT in the present study

C1 (GPa) C2 (GPa) R1 R2 u

Y. Lu et al. / Soil Dynamics and EaAutodyn [22]. Specific material models, such as the three-

phase soil model, are implemented by incorporating user

subroutines.

4.1. Free field analysis

In order to minimize the effect of mesh sizes, three

different meshes were tried for both the 2D and 3D models.formed in the soil at about 100 ms after the detonation of the

charge. The radius of the crater is about 2.2 m. The shapeFig. 6. Computed craters in soil.

and the size of the crater from the 2D and 3D models are

comparable. The estimated range of the radius of the crater

according to Henrych [13] is about 1.53.5 m for different

shapes of charge and properties of the soil, which are

consistent with the current simulation results.

Fig. 7 shows the computed pressure time histories at a

target of 10 m away from the charge. The results from the

2D and 3D analysis are in good agreement.

The attenuations of the peak pressure, peak particle

velocity (PPV) and peak particle acceleration (PPA) in

soil with increasing distance from the charge are

compared in Fig. 8. The respective empirical equations

recommended by TM5 [2] are also shown in the graphs

(straight lines) for a comparison, namely

Function 1 peak pressure : PP 6! RW3

p K2:4

MPa

Function2 peakparticlevelocity : PPVZ7 RW3

p K2:4

m=s

Function 3 peak particle acceleration :

PPA Z 600RW3

p K3:1

g

in which R is the distance away from the charge, W is

the weight of the charge.

The 2D and 3D results both agree well with the above

empirical equations.

It is also interesting to compare the computing time

needed in running the 2D and 3D analysis for problems of

this scale. With a PC of P4-CPU 2.66 GHz, RAM 1G and

Hard disk 88 GB, the computing times needed for an

Time /ms0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

Pres

sure

/MPa

Target at 10m2D model3D model

Fig. 7. Typical pressure time history in soil at target of 10 m from the

charge.

100

a

3D mesh 2D mesh

100 2D model 3D model

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288282Scaled distance (R/W1/3) /mkg-1/30.4 0.6 0.8 1 2 4 6

0.1

1

10

Function 1Pea

k Pr

essu

re /M

P0.4 0.6 0.8 11

10

100

1000

10000

Function 3PPA

/g

Scaled Distanc

Fig. 8. Attenuation of the peak pressure, peak particle velocity (PPV) and p0.4 0.6 0.8 1 2 4 6

0.1

1

10

Function 2

PPV

/m/s

Scaled Distance (R/W1/3)/mkg-1/3

2 4 6

e (R/W1/3) /mkg-1/3

2D model 3D modeleak particle acceleration (PPA) in soil (straight lines based on TM5).

analysis of duration 130 ms are approximately 45 min for

the 2D model and about 13 h for the 3D model. With the

inclusion of the buried structure as will be described in

Section 4.2, the computing time is about 1 h 30 min for the

2D analysis and about 20 h for the 3D analysis.

(a) 2D view of structure and target points

(b) 3D view of structure (halved) and target points

23m 10m

0.5m

5.0m

Steel rebar

1

52

346 0.6m

1.3m

23m

1.3m 1

52 3 4

5m

0.6m0.5m

Fig. 9. Buried structure configuration in 2D and 3D models.

Table 3

Parameters used in the RHT model for concrete

Initial density

r0 (kg mK3)

2.314!103 Specific heat Cv(J/kg K)

6.54!102

Reference den-

sity rs (kg mK3)

2.75!103

The RHT strength model

fc (Mpa) 35 3pl(elasticplastic) 1.93!10K3

A 1.6 B 1.6

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288 283Fig. 10. Numerical model meshes for 2D and 3D analyses.4.2. Coupled analysis with buried structure

A representative buried structure is considered for the

analysis. The structure has a box shape of overall dimensions

of 23 m (length)!20 m (width)!5 m (depth), and is buried inthe soil so that the roof is on the ground surface level. Fig. 9

shows the details of the structure. The charge profile is the

same as in the earlier free-field analysis (50 kg at embedded

depth of 4.8 m), and the charge standoff distance to the front

wall is 10 m (scaled distance y2.7 m/kg1/3)As the structure is relatively wide in the third direction,

the problem may be simplified as a 2D axis-symmetric

problem. Fig. 10 shows the coupled models for the 2D and

3D analysis, respectively. In the coupled models the charge

and soil are modeled with the same mesh as in the free-field

analysis, while the structure is modeled using fine FE

elements. It is noted that in the RC structure the reinforcing

N 0.61 M 0.61

ft (MPa) 3.5 D1 0.04

n1 0.036 D2 1.0

n2 0.032 3minf 0.01

Q2,0 0.6085 Pe (MPa) 2.33!101

Ginitial (MPa) 1.67!104 Ps (MPa) 6.0!10

3

Gresidual (MPa) 2.17!103

P-a EOS

A1 (MPa) 3.527!104 T1 (MPa) 3.527!10

4

A2 (MPa) 3.958!104 T2 (MPa) 0.0

A3 (MPa) 9.04!103 n 3.0steel is modeled by equivalent shell elements with the same

steel volume content. Complete binding condition is

imposed at the interface between the soil and structure.

For concrete with a cylinder compressive strength of

35 MPa and reinforcing steel of yield strength 350 MPa,

the parameters used in modeling the concrete and reinfor-

cing steel materials are summarized in Tables 3 and 4 based

on [16,10]. Several target points are arranged throughout the

2D and 3D structure to record the response of the structure

for analysis and comparison, as indicated in Fig. 9.

Table 4

Parameters used for modeling reinforcement steel bar

Reference density r0(kg mK3)

7.896!103 B (MPa) 2.75!102

Bulk modulus K (MPa) 2.0!105 C 0.022Specific heat Cv (J/kg K) 4.52!10

2 n 0.36

Shear modulus G (MPa) 8.18!104 m 1.0

Y0 (MPa) 3.5!102 Troom (K) 3.0!10

2

Tmelt (K) 1.811!103

4.2.1. Damage of the structure

Fig. 11 shows the damage in the front part of the structure

facing the detonation. It can be seen that the damage pattern

of the 2D model is similar to that of the 3D model at the

middle section; however, the maximum value of damage

from the 2D analysis is notably less than that of the 3D

results, especially at the top and bottom connection regions.

The above difference may be attributable to the fact that,

by the 2D axis-symmetric model, the structure is treated

effectively as a circular ring. This introduces some artificial

constraint in the circumferential direction as compared to

the actual box-shaped structure, leading to certain under-

estimation of the critical damage to the front wall of the

structure. In this respect, the use of the 3D model certainly

produces more realistic damage distribution, including that

in the transverse direction. The maximum damage in the 3D

model for this case is about 0.20, which corresponds to a

state with some minor cracking in the concrete.

Fig. 11. Computed damages to buried structure (front part facing the charge).

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288284-0.5

0.0

0.5

1.0

1.5

2.0

Pres

sure

/MPa

Center of Front Wall 2D model 3D model0 20 40 60 80 100 120 140-1.0

Time /ms

0 20 40 60

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Pres

sure

/MPa

Tim

Fig. 12. Computed blast pressure on th0.0

0.5

1.0

1.5

2.0

Pres

sure

/MPa

Corner of the section 2D model 3D model0 20 40 60 80 100 120 140

-0.5

Time /ms

80 100 120 140e /ms

Center of ground plate 2D model 3D model

e front wall/bottom plate centre.

4.2.2. Computed blast loading on structure

Fig. 12 shows the computed reflected pressure on the

front wall and the bottom plate. As can be seen, the results

for the front wall, whether at the centre or near the lower

corner, are very similar between the 2D and 3D analyses.

Marked difference is observed in the pressure on the bottom

plate, where the 3D results exhibit much larger peak

pressure than from the 2D analysis.

In fact, the stress wave impinging the structure will

experience complex reflection processes which depend on

the profile of the structure, particularly around the edge and

corner areas. Since in the 2D axis-symmetric model the

structure is treated as a ring in which only the cross-section

geometry is preserved, the blast wave path is somewhat

altered, and consequently the loading conditions on other

parts of the structure (except the front wall) are affected.

4.2.3. In-structure shock

The in-structure shock environment will affect the safety

and functionality of the equipment in the structure and

hence is an important aspect to be evaluated from the

analysis. The time histories of the acceleration and velocity

obtained from the 2D and 3D models at representative target

points are plotted in Figs. 13 and 14. From these curves the

following may be observed: (a) The 2D and 3D predictions

of the shock response at the front wall centre are

comparable. The peak values of the acceleration and

velocity in the horizontal (x) direction agree reasonably

well between the two models. The difference in the

z-direction (vertical) is larger. (b) The difference of the

responses at the bottom plate centre between the 2D and 3D

models is more obvious, particularly in the vertical

direction. This echoes the observation of markedly larger

pressure at the bottom plate from the 3D model as compared

to the 2D model. Besides the difference in the blast pressure,

the support condition of the bottom plate in the idealized

ring shape in the 2D model, as well as the distribution of

the load along y-direction that is not represented in the 2D

model, all affect the accuracy of the computed bottom plate

response in the vertical direction.

Based on the results from the 3D model, some general

observations regarding the in-structure shock environment

can be deduced:

(i) The maximum acceleration and velocity response

occur around the center of the front wall. For

Target 5, x-dir

-40

-20

0

20

40

60

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

2D model3D model

Target 5, z-dir

-30

-20

-10

0

10

20

30

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

2D model3D model

ont w

ont w

om p

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288 285-60

2D model3D model

2D model3D model

(a) Fr

Target 2, x-dir

-30

-20

-10

0

10

20

30

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

(b) Fr

Target 3, x-dir

-15

-10

-5

0

5

10

15

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

(c) Bott

Fig. 13. Computed acceleration time h2D model3D model

2D model3D model

Target 2, z-dir

-30

-20

-10

0

10

20

30

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

all corner

Target 3, z-dir

-15

-10

-5

0

5

10

15

0 20 40 60 80 100 120 140 160

Time (ms)Acce

lera

tion

(g)

late centre all ceistorientre s at selected target points.

Target 5, x-dir

-2

-1

0

1

2

0 20 40 60 80 100 120 140 160

Time (ms)Velo

city

(m/s)

2D model3D model

Target 5, z-dir

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140 160

Time (ms)Velo

city

(m/s)

2D model3D model

2D model3D model

2D model3D model

2D model3D model

2D model3D model

(a) Front wall centre

Target 2, x-dir

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140 160

Time (ms)Vel

ocity

(m/s)

Target 2, z-dir

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140 160

Time (ms)Vel

ocity

(m/s)

(b) Front wall lower corner

Target 3, x-dir

-0.4

-0.2

0

0.2

0.4

0 20 40 60 80 100 120 140 160

Time (ms)Vel

ocity

(m/s)

Target 3, z-dir

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140 160

Time (ms)Vel

ocity

(m/s)

(c) Bottom plate centre

Fig. 14. Computed velocity time histories at selected target points.

bottom plate, x-dir

-20

-10

0

10

20

Time (ms)Acce

lera

tion

(g)

target 2target 3target 4

bottom plate, z-dir

-20

-10

0

10

20

0 20 40 60 80 100 120

Time (ms)Acce

lera

tion

(g)

target 2target 3target 4

(a) Bottom plate

Front wall, x-dir

-60

-40

-20

0

20

40

0 20 40 60 80 100 1200 20 40 60 80 100 120

0 20 40 60 80 100 120

Time (ms)Acce

lera

tion

(g)

target 2target 3target 4

Front wall, z-dir

-20

-10

0

10

20

Time (ms)Acce

lera

tion

(g)

target 2target 3target 4

(b) Front wall

Fig. 15. Spatial variation of acceleration within the structure.

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288286

nZ2nZ2struc

over

C

abov

estim

evalu

nom

the e

respo

respo

show

load

respo

wall and the bottom plate in the case of a side burst. As a

result, the vibration amplitudes on these components are

significantly higher. Nevertheless, at locations where the

vibration response is less significant such as at the corners of

the structure (e.g. target point 2 in Figs. 1315), the

acceleration and velocity amplitudes tend to show reason-

able agreement with the empirical evaluation results.

To provide another perspective of the shock environment

concerning their effects on equipment in the structure,

Fig. 16 shows the shock response spectra for the accelera-

tions computed from the 3D model at different locations of

the structure. It can be seen that the shock response

spectrum at the centre of the front wall is critical in the

entire frequency range of interest, although in the vertical

direction the trend is not that clear. Furthermore, the spectral

acceleration is sharply reduced at the lower frequency

range. Therefore, for acceleration sensitive equipment

devices, the possible shock damage may be much alleviated

if appropriate isolation measures are taken so that the

natural frequency of the individual equipment installations

are reduced to below a certain level, for example below

30 Hz for the case under consideration.

5. Conclusions

In this paper, the response of underground structure

80

erat

i Rear wall cornerRoof centreFront wall edge corner

structure.

rthquaerical calculations). This is not surprising as the

ation according to TM-5 is only an indication of the

inal free-field ground shock over the space occupied by

mbedded structure and it does not reflect the dynamic

nse of the structure. In fact, as can be seen from the

nse histories at different locations on the structure

n in Figs. 1315, under the excitation of the blast

ing the structure exhibits considerable vibrationthe

numZ6.45 g (acceleration).ompared with the computer simulation results, the

e empirical evaluation appears to significantly under-

ate the in-structure shock amplitudes, especially for

front wall (comparing to 58 g and 1.3 m/s from0.31

aavg2e Vavg1, aavg1 correspond to the attenuation coefficient

, Vavg2, aavg2 correspond to the attenuation coefficient

.5. The average velocity and acceleration of the

ture, estimated from the integration of these functions

the span of the structure, are found to be Vavg1Zm/s, Vavg2Z0.09 m/s (velocity), and aavg1Z6.45 g,the present case where the scaled standoff distance

(from the charge to the front wall) is about 2.7 m/kg1/3,

the maximum acceleration reaches 58 g (horizontal)

with a primary frequency around 100 Hz, which

reflects the fundamental natural frequency of the

front wall. The maximum velocity reaches 1.2 m/s

(horizontal direction).

(ii) The maximum x-direction acceleration and velocity of

the bottom plate are 10.2 g and 0.3 m/s, respectively.

The x-direction response of the bottom plate can be

considered as representing the motion of the whole

structure in the horizontal direction.

(iii) There exists noticeable spatial variation of the shock

environment throughout the structure, as also shown in

Fig. 15. This is primarily attributable to the propa-

gation of the shock wave in the structure (within the

plane of each side) and the dynamic response of the

structural components.

For a rough verification of the computed magnitude of

the in-structure shock, the simplified approach in code

TM-5 [2] is used to provide an empirical estimation of the

in-structure acceleration and velocity. This estimation is

based on free-field ground shock. By integrating the

empirical acceleration-range function over the span of the

structure, an estimate of the average acceleration is

obtained. The estimate of the velocity can be found in a

similar manner. For the kind of soil considered in the current

example, the empirical formulas for the free-field velocity

and the acceleration, according to TM-5, are in the range of

Vavg1 Z 7:38RW3

p Kn

; Vavg2 Z 4:62RW3

p Kn

;

aavg1 Z 651RW3

p KnK1

; aavg2 Z 406RW3

p KnK1

wher

Y. Lu et al. / Soil Dynamics and Eanse all over the structure, particularly on the front0

20

40

60

0 100 200 300 400 500 600Frequency (Hz)

Spec

tral a

ccel

(a) Horizontal (radial) direction

0

10

20

30

40

50

0 100 200 300 400 500 600Frequency (Hz)

Spec

tral a

ccel

erat

ion

(g) Front wall centre

Front wall central cornerbottom plate centreRear wall cornerRoof centreFront wall edge corner

(b) Vertical direction

Fig. 16. Shock response (acceleration) spectra at different locations of the100

120

on (g

) Front wall centreFront wall central cornerbottom plate centre

ke Engineering 25 (2005) 275288 287subjected to explosion in soil is analyzed using 2D and 3D

fully-coupled numerical models. The models incorporate

the SPH technique with FEM to form an efficient

combination, where the SHP is employed for its superior

ability in handling large deformations while the FEM is

suited for modeling the buried structure and the relatively

low deformation soil regions. The general response

characteristics are discussed. The computational efficiency

and the accuracy of using 2D model as compared to the 3D

actual 3D shape of the structure, especially around the edge

and corner areas. As a result, the loading conditions and the

[3] Hinman EE. Single degree of freedom solution of structuremedium

interaction. In: Proceedings of international symposium on the

interaction of conventional weapons with structures, March 913,

vol. 1. Mannheim, West Germany: Federal Minister of Defense; 1987,

p. 4459.

[4] Nelson I. Numerical solution of problems involving explosive. In:

Proceedings of dynamic methods in soil and rock mechanics,

September 516, vol. 2. Rotterdam: A.A. Balkema; 1977, p. 23997.

[5] Stevens DJ, Krauthammer T. Analysis of blast-loaded, buried RC arch

Y. Lu et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 275288288response of the remaining part of the structure using the 2D

model are less satisfactory as compared to the 3D model.

Therefore, in situations where mainly the critical responses in

the structure are of major concerns, the 2D model can be

acceptable for the analysis of such buried structures subjected

to blast loading.

In general, for a side burst scenario considered in this study,

the maximum acceleration and velocity response are found to

take place around the center of the front wall. The maximum

damage also occurs on the front wall at top and bottom regions

as well as around the centre. For a side burst at a standoff

distance about 2.7 m/kg1/3, the maximum acceleration is

found to be about 58 g (horizontal) and the maximum velocity

is about 1.2 m/s (horizontal). Based on the acceleration time

histories, the in-structure shock environment is also depicted

by the shock response spectra within the structure. It is

observed that the peak spectral acceleration occur in the front

wall when the frequency of the oscillator is around 100 Hz.

The shock response spectra drop drastically when the

frequency of the oscillator is reduced to below a certain

level (e.g. 30 Hz for the case herein). This observation can be

useful in the design of equipment installation for reducing the

in-structure shock hazard to the equipment.

References

[1] Biggs JM. Introduction to structural dynamics. New York: McGraw-

Hill; 1964.

[2] TM5-855-1. Fundamental of protective design for conventional

weapons. US Army Engineer Waterways Experiment Station,

Vicksburg; 1984.model are examined.

The 2D model is shown to be able to predict satisfactorily

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response of the front wall of the structure with reasonable

accuracy. The blast wave reflection is complicated by theresponse. Part I. Numerical approach. J Struct Eng ASCE 1991;

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[6] Stevens DJ, Krauthammer T, Chandra D. Analysis of blast-loaded,

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[7] Benson DJ. Computational methods in Lagrangian and Eulerian

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[13] Henrych J. The dynamics of explosions and its use. Amsterdam:

Elsevier; 1979.

[14] Prapahanran S, Chameau JL, Holtz RD. Effect of strain rate on

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[15] Drucker DC, Prager W. Soil mechanics and plastic analysis or limit

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with structures, p. 31522.

[17] Herrmann W. Constitutive equation for the dynamic compaction of

ductile porous materials. J Appl Phys 1969;40(6):24909.

[18] Carrol MM, Holt AC. Static and dynamic pore collapse relations for

ductile porous materials. J Appl Phys 1972;43(4):1626 et seq.

[19] Lee EL, Hornig HC, Kury JW. Adiabatic expansion of high explosive

detonation products, UCRL-50422.: Lawrence Radiation Laboratory,

University of California; 1968.

[20] Huck PJ, Saxena SK. Response of soil-concrete interface at high

pressure. In: Proceedings of the Tenth International Conference on

Soil Mechanics and Foundation Engineering, June 1519, Stockholm:

A.A. Balkema; 1981, 2: 141144.

[21] Mueller CM. Shear friction tests support program; laboratory friction

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[22] AUTODYN Theory Manual, revision 3.0. Century Dynamics, San

Ramon, CA; 1997.

A comparative study of buried structure in soil subjected to blast load using 2D and 3D numerical simulationsIntroductionOverview of the fully-coupled methodConservation equationThe smooth particle hydrodynamic (SPH) methodCoupling of SPH with FEM

Constitutive modelsThree-phases soil modelConcrete modelElastic-strain hardening plastic model for steelJWL equation of state for explosive chargeSoil-concrete interface

Numerical modelFree field analysisCoupled analysis with buried structure

ConclusionsReferences