a full coupled numerical analysis approach for buried structures subjected to subsurface blast

Computers and Structures 83 (2005) 339–356

www.elsevier.com/locate/compstruc

A full coupled numerical analysis approachfor buried structures subjected to subsurface blast

Zhongqi Wang a, Yong Lu a,*, Hong Hao b, Karen Chong c

a School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singaporeb Department of Civil and Resource Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

c Defense Science and Technology Agency, Ministry of Defense, 1 Depot Road, Singapore 109679, Singapore

Received 5 September 2003; accepted 31 August 2004

Available online 26 November 2004

Abstract

The physical processes during an explosion in soil and the subsequent response of buried structures are extremely

complex. Combining all these processes into a single analysis model involves several numerical difficulties but such a

model will enable more realistic reproduction of the underlying physical processes. This paper presents a full coupled

numerical analysis approach, in which the SPH (smooth particle hydrodynamics) method is adopted to model the near

field medium to cater for large deformation, while the conventional FEM is used to model the intermediate and the far

field soil medium and the structural response. A robust three-phase soil model developed by the authors is employed to

model the soil mass. The numerical model is verified against empirical predictions and the comparison shows a favor-

able agreement.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Buried structure; Subsurface blast; SPH–FEM coupled method; Stress wave; Structural response; In-structure shock

1. Introduction

Underground reinforced concrete structures are used

for essential installations protected against the effects of

conventional weapons. Usually such structures are box

shaped, partially or fully buried. The physical processes

that govern the response of the underground structure

are very complex, involving dynamic interactions among

the explosive, the soil and the underground structure.

Major phenomena include the formation of the crater

0045-7949/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.compstruc.2004.08.014

* Corresponding author. Tel.: +65 6790 5272; fax: +65 6791

0676.

E-mail address: [email protected] (Y. Lu).

or camouflet by the explosion; the propagation of the

shock wave and elastic–plastic wave in the soil; and

the interaction between soil and the structure. The non-

linear properties and large deformation of the soil and

reinforced concrete make the whole physical process

highly nonlinear, both in terms of the material and

geometric nonlinearities. Consequently, a numerical

approach is necessary in order to fully describe the entire

process.

Two kinds of numerical methods are usually used to

analyze the response of an underground structure under

blast loading, namely the �uncoupled method� and the

�coupled method�. In the �uncoupled method�, the main

physical process is divided into several consecutive

phases; the output of one phase is the input of the next

ed.

mailto:[email protected]

340 Z. Wang et al. / Computers and Structures 83 (2005) 339–356

phase. In this respect, the problem under consideration

can be divided into three phases: (1) the detonation of

charge and the formation of crater or camouflet; (2)

the propagation of blast wave; and (3) the response of

the structure. The �coupled method� can be divided into

two categories, namely the �partial coupled method�and the �full coupled method�. In the �partial coupledmethod�, the aforementioned three phases are reduced

into two phases, with either the first two or the last two

phases being merged. The �full coupled method� com-

bines all three phases together in a single model.

Many research works on the numerical analysis of

blast loaded underground structures have been reported

[1–6]. Most of these studies are based on either the

�uncoupled method� or the �partial coupled method�. Inthese methods a fundamental question lies on the ade-

quacy of defining the loads on the structure. In the

�uncoupled method�, the histories of stress or velocity

in the free field are calculated first. These time histories

are then applied on the structure as boundary conditions

for analyzing the response of the structure. As such, the

interaction between the soil and the structure cannot be

considered in a realistic manner. Hamdan and Dowling

[7] pointed out that the uncoupled method may result in

an unsafe structure if resonance at the interface occurs.

Henrych [8] also suggested that the coupling effect could

be significant, especially when the structure is in a dense

medium (e.g. water, soils). In addition, the interfacial

effects, such as slippage, separation, and rebond, are also

important factors that influence the response of the

structure. To take these effects into account, several cou-

pled analysis techniques emerged. Nelson [9] used a �soilisland� approach to analyze the wall of buried structures,

whereby a small portion of the soil in front of the struc-

ture was modeled and the stresses were input onto the

free surface of the soil island. Stevens and Krauthammer

et al. [1,10] adopted a hybrid approach which merges the

finite difference technique (FDT) with the finite element

method (FEM), such that the soil is modeled by FDT

which is suited for analyzing wave propagation in con-

tinuous nonlinear media, while the structure is modeled

by the FEM. These coupled approaches considered the

dynamic interaction and the coupling effect between

the soil and structure, but the blast loading still needs

to be defined in terms of stress or pressure time histories.

While it may be considered appropriate to define the

blast loading for relatively simple and symmetric situa-

tions, it becomes questionable in cases where the shape

of the structure is not regular or the ground surface

effect becomes significant as in the case of shallow buried

structures. In these situations, a full coupled approach

including the explosion source is desired [6].

Few studies can be found from the existing literature

to have incorporated the explosion source. Besides the

problem with high computational cost, the difficulties

associated with modeling the dynamic interaction be-

tween the explosion and the soil medium is a major bar-

rier. As the stress condition varies drastically in the near

field of the charge, it is very difficult to model the behav-

iour of soil in this region, especially in view of the multi-

phase nature of the soil medium.

The present study aims to establish a fully coupled

numerical approach for analyzing the response of under-

ground structures subjected to blast loading. In this

approach, the explosion source, the propagation of

stress wave in the soil and the interaction between soil

and the structure are integrated into a single model.

The state-of-the-art hydrodynamic numerical techniques

and material models are adopted. In the aspect of

numerical techniques, the smooth particle hydrody-

namic method (SPH) is merged with the Lagrangian fi-

nite element method (FEM), whereby the SPH is used

to model the near field response and the FEM is used

to model the intermediate and far field ground move-

ment and the structural response. On the material mod-

eling, a robust three-phase soil model developed by the

authors for shock loading [11,12] is employed to model

the soil mass. For the buried structure, the Riedel–

Thoma–Hiermaier (RHT) concrete model [13] is applied

for modeling the concrete, while an elastic–plastic hard-

ening model is used for the steel. The JWL (Jones–

Wilkins–Lee) equation of state is adopted for simulating

the detonation of the charge.

A numerical example is given to demonstrate the

implementation of the proposed approach. The numeri-

cal results are verified against some empirical and engi-

neering observations.

2. Basic considerations on SPH–FEM coupled analysis

To incorporate the explosion source in the numerical

analysis of structural response is beyond the capacity of

common structural analysis codes because these codes

usually do not include the energy conservation consider-

ation. The hydrocodes (or �wave codes�) are suited for

simulating such complex processes as the present case

which involves the explosion and blast wave propa-

gation in soil, the soil–structure interaction and the

response of the structure. The explosion product ex-

pands enormously. The soil in the vicinity of the charge

undergoes large deformation. Large deformation can

also occur in the structure if it is located close to the

charge. On the other hand, the response of the structure

depends on the interfaces and boundaries and the effects

associated with them.

There are two major ways of describing the contin-

uum media based on the relative movement between

the material particles and the mesh; one is the Eulerian

description, the other is the Lagrangian description. In

the Eulerian description, the mesh is fixed in space and

different material particles move through it. In the

Z. Wang et al. / Computers and Structures 83 (2005) 339–356 341

Lagrangian description, the mesh and the material par-

ticles coincide. The Eulerian description is suited for sit-

uations where the mesh may be highly distorted; but

modeling of the material boundary conditions such as

slippage and contact surface using Eulerian method is

very difficult. The Lagrangian description is more suit-

able for situations where the deformation is not large

but the effects of interface and free boundaries are signif-

icant. To take advantage of both descriptions, the Arbi-

trary Lagrangian Eulerian description (ALE) has been

put forward, such that the analyst can choose if the

mesh should follow the material (i.e. Lagrangian) or

be fixed (i.e. Eulerian). This approach involves a compli-

cated rezoning technique, the rezoning procedure often

requires interventions from experienced users [14]. Fur-

thermore, it does not totally prevent the problem with

severe mesh distortion and the subsequent sharp reduc-

tion of the time step, which in turn reduces the compu-

tational efficiency.

The difficulty in combining the Eulerian and Lagrang-

ian methods is primarily due to the mesh. Once a mesh is

produced, the �elements� or �grids� that represent the

physical region cannot be changed easily. For a Lagrang-

ian mesh, large deformations can result in a severe mesh

distortion and hence reduce the accuracy and the time

step sharply. In the Eulerian mesh mixed cells could

appear, whereby two or more than two kinds of materials

come together. The mixed cell will blur interface and

boundary among materials, so the Eulerian mesh is diffi-

cult to be coupled with the Lagrangian mesh.

To get rid of the difficulties arising from the mesh,

some meshless methods have been put forward. One of

the most important meshless methods is the SPH

(smooth particle hydrodynamics) method. The advan-

tage of the SPH method is that there is no need to track

the materials interface (in this sense the SPH can be

looked upon as a special Lagrangian method), and

hence avoids the aforementioned difficulty with the

Eulerian method. The calculation can continue regard-

less of the amount of turbulence in the solution, so it

can deal with large deformation. It also avoids the diffi-

culty with severe mesh distortion as in the Lagrangian

mesh because there is no real mesh in the SPH method.

Currently the SPH technique has been incorporated

into several hydrocodes [15,16]. Although most problems

can be modeled by the SPH method, certain limitations

exist. As mentioned by Attaway et al. [17], modeling thin

walled structures by smooth particles is inefficient since

many small particles would be required and the time step

would become very small. In this regard, coupling the

SPH and the Lagrangian method appears to be an effec-

tive solution, so that part of a problem such as a structure

can be modeled by solid or shell elements, while other

parts can be simulated using smooth particles. This cou-

pling approach is expected to be highly effective for the

type of problems under consideration.

3. Computational framework

3.1. Conservation equations

In three-dimensional realm, the conservation equa-

tions of mass, momentum and energy are expressed as

[18]

Mass : q ¼ q0V 0

V¼ m

Vð1Þ

Momentum : q _ui ¼ rij;j þ qfi ð2ÞEnergy : q _e ¼ rij _eij þ qfiui ð3Þ

where q is density, V is the volume, the subscript �0� indi-cates the initial value, m is the mass, rij are the stresses,

eij are the strains, u is the spatial velocity, e is the energy,

f is the body force, the mark �Æ� is the first derivative of

time, i and j range from 1 to 3.

The strain is expressed by the deformation,

eij ¼1

2

oW j

oX iþ oW i

oX j

� �ð4Þ

where X is spatial coordinates, W = Wi(Xj) is position

vector.

In general the stresses and strains can be separated

into two parts, a hydrostatic component and a devia-

toric component. The former corresponds to the volume

deformation, the latter is related to the shear

deformation.

rij ¼ sij þ1

3rkkdij ð5Þ

eij ¼ eij þ1

3ekkdij ð6Þ

dij ¼1 i ¼ j

0 i 6¼ j

�

where sij is deviatoric stress, eij is deviatoric strain.

The constitutive relations which relate the stress and

the strain can also be divided into two part, the strength

model and the equation of state (EOS), describing

respectively the shear deformation and the volume

deformation. The boundary conditions are either the

specific displacements or traction,

xiðX ; tÞ ¼ giðX ; tÞ on Cx; rijnj ¼ si on Cs ð7Þ

where x is the current coordinate of a point, X is the ref-

erence coordinate, t is time, n is the exterior normal, g is

the specific displacement function, Cx or Cs denotes the

surface where the displacement or traction boundary

condition are applied.

3.2. The smooth particle hydrodynamic (SPH) method

SPH is a meshless Lagrangian technique which orig-

inated for an application in astrophysics in 1977 [19].


The main advantage of the method is to bypass the

requirement for a numerical grid to calculate spatial

derivatives. This avoids the severe problems associated

with mesh tangling and distortion which usually occur

in Lagrangian analyses involving large deformation

impact and explosive loading events. Although the name

includes the term ‘‘hydrodynamic’’, in fact the material

strength can be incorporated [16].

In SPH methodology, the material is represented by

fixed mass particles to follow its motion. Unlike the grid

based methods, such as the Lagrangian or the Eulerian,

which assumes a connectivity between nodes to construct

spatial derivatives, SPH uses a kernel approximation,

which is based on randomly distributed interpolation

points, with no assumptions about which points are

neighbors, to calculate the derivatives.

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 � that are within a given distance (usually as-

sumed to be 2h) from it. The distance h is called the

smooth length. The interaction is weighted by the func-

tion W(x � x 0,h) which is called the smoothing (or ker-

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

continuous function, or it�s derivative, can be estimated

at any particle �I � based on known values at the sur-

rounding particles �J � using the following kernel

estimates:

f ðxÞ �Z

f ðx0ÞW ðx� x0; hÞdx0 ð8Þ

r � f ðxÞ �Z

r � f ðx0ÞW ðx� x0; hÞdx0 ð9Þ

where f is a function of three-dimensional position

vector x, dx 0 is a volume.

Fig. 1(a) illustrates the concept of a kernel estimate.

Full details of the mathematical derivation of the kernel

approximation can be found in [20]. One of the com-

monly used symmetric formulation for Eq. (9) is

2h

J

I

x-x′

(a) Neighboring particles of a kernel estimate

Fig. 1. SPH approximation and coupl

r � f ðxIÞ � 1

qI

XNJ¼1

mJ ðf ðxIÞ � f ðxJ ÞÞ � rW ðxI � xJ ; hÞ

ð10Þ

where the gradient $W is with respect to xJ, m is the

mass, q is the density. Function f can be any variants

in the computation, e.g., the density, stress, or strain

etc. Note that no connectivity or spatial relation of the

interpolation points is assumed in the derivation of the

SPH equations, and this avoids the mesh tangles. An-

other important point is that the SPH nodes can use

the same constitutive models as used for the FEM

element.

3.3. The coupled SPH and FEM method

Accurate SPH simulations require large number of

particles throughout the SPH region. Hence if high accu-

racy is sought or some special geometry is required, such

as thin walls etc., large run time can become a problem.

The joining of SPH to Lagrange FEM is a potentially

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 computa-

tional demand. Fig. 1(b) shows the basic concept on

how the SPH particles can be embedded into a tradi-

tional Lagrange FEM mesh.

There are two different ways that the SPH particles

can be coupled with the FEM elements. When they are

attached to the FEM elements, the SPH particles and

the FEM element will be joined together, then the force

from other SPH particles as well as from the FEM ele-

ments act on the particle for the equations of motion.

If the SPH particles and FEM element are not attached,

they will slide along the surface of the FEM element, in

this case, a special sliding interface algorithm must be

used [15]. In the present study, the SPH particles are

joined together with the FEM elements because the

SPH particles herein represent the near-field soil med-

(b) Coupled mesh of SPH particles and FEM elements

ing of SPH and FEM elements.


ium; the interface between the SPH mesh and the FEM

mesh is not a material interface.

4. Material models

There are four kinds of materials involved in the

problem under investigation, namely the soil mass, the

concrete and reinforcing steel in the structure, and

the high energy charge. The models used to describe

these materials are as follows.

4.1. Three-phase soil model

A number of distinctive approaches have been pro-

posed for modeling the static and dynamic response of

soils, including elasticity model, endochronic model,

plasticity with rate-independent and rate-dependent

models, viscoplasticity model, critical state model, etc.

[21]. But for concerns of explosion in soils and the sub-

sequent blast wave propagation, the range of variation

of stress in soils is much larger than what is usually

encountered in the common soil dynamics. The pressure

in the vicinity of a charge can reach several GPa (giga

pascal) and it attenuates rapidly with the increase of

the distance from the charge. As soil is a multi-phase

mixture composed of solid mineral particles, water and

air, the deformation mechanism and the contribution

of different phases vary with abrupt change of the stress

condition. Therefore, to model a blast event in soil, a

robust soil model is required to cater for the whole range

of loading condition; to this end, a realistic reflection of

the deformation mechanisms is necessary. Unfortu-

nately, none of the established soil models seems to meet

the above requirements. To fill in this gap, the authors

recently developed a three-phase soil model for simulat-

ing blast wave propagation in soils [11,12].

In this model, which stems from the conceptual

model introduced in [8], the soil is considered as an

assemblage of solid particles with different sizes and

Solid particles Void

Bond

(a) Conceptual model

Fig. 2. Concept of the three-phase

shapes that form a skeleton and their void are filled with

water and air. The solid particles, water, air as well as

the skeleton formed by the solid particles deform under

different laws when external load acts on the soil mass.

Fig. 2 illustrates the basic idea of the three-phase soil

model and its mathematical representation, where ele-

ments A, B, C correspond to the deformation of the

solid particles, water and air respectively, and elements

D, E describe the friction and the resistance of the bond

connection between the solid particles. The bonds be-

tween the solid particles are represented by a series

of filaments. The model formulation can be roughly

divided into two main parts; the equation of state and

the strength model. The volumetric ratios of the solid,

air and water phases are assumed to be a1, a2, a3, respec-tively. The following gives an overview of the three-

phase soil model formulation.

4.2. The equation of state (EOS)

To satisfy the continuity requirements, the total vol-

ume change of a multi-phase system must be equal to the

sum of volume changes associated with each phase, i.e.

DVV 0

¼ DV w

V 0

þ DV g

V 0

þ DV s

V 0

ð11Þ

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

total volume of the element, Vw is the volume of water,

Vg and Vs are volumes of air and soil particles, respec-

tively. Denote the volume of voids as Vp, Vp = Vg + Vw,

and hence V = Vs + Vp.

The pressure load causes deformation in each phase,

as well as friction between the solid particles and defor-

mation of the bond between the solid particles. The fric-

tion force and the force due to the bond are all exerted

on the solid phase. Satisfying the equilibrium leads to

dp� dV �oV s

opdp

� �oV g

opbþoV w

opb

� ��1

þ opaoV p

þ opcoV p

" #¼ 0

ð12Þ

P

Solid particlesA

B

ba

c

CD

EWater

Air Elastobrittlelinkagebetweenblocks

Frictionbetweenblocks

(b) Mathematical model

soil model for shock loading.


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

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

by the friction between the solid particles, pb is the pres-

sure borne by 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 skele-

ton which is equal to the sum of pa and pc.

Eq. (12) describes the volumetric deformation under

the hydrostatic pressure, in which oV s

op ;oV g

opb; oV w

opb;

opaoV p

; opcoV p

can be obtained from their independent equa-

tions of state or stress–strain relationship.

The following equation of state is adopted for water

[22]:

pw ¼ pw0 þqw0c

2w0

kw

qw

qw0

� �kw

� 1

" #ð13Þ

where pw, pw0 are the current and initial pressure of

water, respectively; cw0 is the initial sound speed of

water, qw0 is the initial density of water, qw is the current

density; and kw is a constant.

For solid particles, a similar equation of state is rec-

ommended by Lyakhov (in Henrych [8]) with the sub-

scripts w replaced by s,

ps ¼ ps0 þqs0c

2s0

ks

qs

qs0

� �ks

� 1

" #ð14Þ

When a pressure wave propagates in soil the air bub-

bles are compressed suddenly, thus, the equation of state

for a polytropic gas can be used to model air in the voids

[8],

pg ¼ pg0qg

qg0

!kg

ð15Þ

where pg0 is the initial pressure of air; qw0 is the density

of air at initial pressure, qg is the density of air at pres-

sure pg, and kg is the isentropic exponent.

In the skeleton of soil, the friction between the solid

particles, pa, is dependent on the normal stress between

the particles. Generally, it can be assumed that the nor-

mal stress is proportional to the deformation of the soil

skeleton. Hence,

pa ¼ fKpDV p ð16Þ

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

is the coefficient of proportionality, DVp is the incremen-

tal volume of voids in the soil, DVp = Vp � Vp0, with Vp,

Vp0 being the current and initial volume of voids,

respectively.

The bonds between the solid particles, on the other

hand, can be represented by a series of elastic brittle fil-

aments. The resisting stress in each filament obeys the

Hooke�s law until the filament breaks. Introducing a

damage variable D, we have

pc ¼ E0ð1� DÞDV p=V p ð17Þ

where E0 is the initial modulus of the bonds.

With the above definitions and the initial condition

p(V0) = p0, the pressure p at any time instant can be

obtained from Eq. (12).

4.3. Damage for soil

The continuum damage model is applied to describe

the damage of the soil skeleton. Based on the filament

breaking model, the damage can be defined as

D ¼ 1� exp � 1

gðbeeffÞg

� �ð18Þ

where B, g are constants related to the properties of the

soil, b is a constant, eeff is the effective strain,

eeff ¼ffiffiffi2

p

3e1 � e2ð Þ2 þ e2 � e3ð Þ2 þ e3 � e1ð Þ2

h i1=2ð19Þ

It should be pointed out that in the present model the

nonlocal effect due to the heterogeneous microstructure

of the material is not included. This issue is to be inves-

tigated when pertinent experimental data on soil mass

under shock loading become available.

4.4. The strength model for soils

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 von Mises� yield criterion [23] is

adopted, as follows:

f ¼ffiffiffiffiffiJ 2

p� aI1 � k ¼ 0 ð20Þ

in which a and k are material constants related to the

frictional and cohesive strengths of the material, respec-

tively; and I1, J2 are the first and deviatoric stress invari-

ant, respectively.

Under shock loading, the strain rate is a very impor-

tant factor to the strength of the soil. A number of inves-

tigators have reported that the undrained shear strength

of the soil increases linearly with the increase of the log-

arithm of the strain rate [24]. To take the strain rate ef-

fect into account, the yield function is modified as

f ¼ffiffiffiffiffiJ 2

p� ðaI1 � kÞ 1þ b ln

_eeff_e0

� �¼ 0 ð21Þ

where _e0 is the reference effective strain rate, b is the

slope of the strength against the logarithm of strain rate

curve, _eeff is the effective strain rate defined as

_eeff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3d _eijd _eij

rð22Þ


The plastic potential function for soils is different

from the yield function as the direction of the plastic

strain increment depij is not normal to the yield surface.

Most hydrocodes employ the nonassociated flow rule.

The plastic potential function employed in this study

is the Prandtl–Reuss type

QffiffiffiffiffiJ 2

p� �¼

ffiffiffiffiffiJ 2

p� Y ¼ 0 ð23Þ

where Y is the yield limit defined by the yield function.

4.5. Concrete model

The response of the concrete under shock loading is a

complex nonlinear and rate-dependent process. A vari-

ety of constitutive models for the dynamic and static

response of concrete have been proposed over the years.

The RHT model is adopted in the present study.

The RHT model is a new model for general brittle

materials, developed by Riedel, Hiermaier and Thoma

[13], and it contains many features common to various

similar constitutive models found in the literature,

namely pressure hardening, strain hardening, strain

rate hardening, third invariant dependence for compres-

sive and tensile meridians, and cumulative damage

(strain softening). It can be used in conjunction

with the existing tensile crack softening algorithm. In

this model the P–a equation of state for volumetric

Uniaxial ComUniaxial Tension

Tensile Elastic Strength

CompresElastic St

fc

ft

Y

(a)

(b)

σ2σ3

σ1

Compressive meridian

Tensile meridian

0.12 =Q

5.02 =Q

Y

elaY

Three strength

The stress plane π

Fig. 3. The RHT materi

compaction is also included, which will be discussed

later.

The material model uses three strength surfaces: an

elastic limit surface, a failure surface and the remaining

strength surface for the crushed material. Often there is

a cap on the elastic strength surface. Fig. 3(a) shows

these strength surfaces.

4.6. The failure surface

The failure surface Y is defined as a function of pres-

sure p, the lode angle h and strain rate _e,

Y fail ¼ Y TXCðpÞ � R3ðhÞ � F RATEð_eÞ ð24Þ

where Y TXC ¼ fc½Aðp� � p�spallF RATEð_eÞÞN �, with fc being

the compressive strength, A the failure surface constant,

N the failure surface exponent, p* the pressure normal-

ized by fc, p�spall ¼ p�ðft=fcÞ. F RATEð_eÞ is the strain rate

function. R3(h) defines the third invariant dependency

of the model as a function of the second and third stress

invariants and a meridian ratio Q2. Fig. 3(b) illustrates

the tensile and compressive meridian on the stress pplane.

4.7. The elastic limit surface and strain hardening

The elastic limit surface is scaled from the failure

surface,

pression

Failure Surface

Elastic Limit Surface

Residual Surface

P

siverength

(c)

ε

ε )( softprepl

pl

*Y

fail

stic

surfaces

The stain hardening

al strength model.


Y elastic ¼ Y fail � F elastic � F CAPðpÞ ð25Þ

where Felastic is the ratio of the elastic strength to failure

surface strength, FCAP(p) is a function that limits the elas-

tic deviatoric stresses under hydrostatic compression,

and it varies in the range of (0,1) for pressure between

initial compaction pressure and the solid compaction

pressure.

Linear hardening is used prior to the peak load. Dur-

ing hardening, the current yield surface (Y*) is scaled be-

tween the elastic limit surface and the failure surface via

Y � ¼ Y elastic þepl

eplðpre-softeningÞðY fail � Y elasticÞ ð26Þ

where epl, epl(pre-softening) are the current and pre-soften-

ing plastic strain. Fig. 3(c) shows this relationship

schematically.

4.8. Residual failure surface

A residual (frictional) failure surface is defined as

Y �resid ¼ B � p�M ð27Þ

where B is the residual failure surface constant, M is the

residual failure surface exponent.

4.9. Damage for concrete

Following the hardening phase, additional plastic

straining of the material leads to damage and strength

reduction. Damage is accumulated via

D ¼X Depl

efailurep

ð28Þ

efailurep ¼ D1ðp� � p�spallÞD2 P emin

f ð29Þ

where D1 and D2 are damage constants, eminf is the min-

imum strain to reach failure.

The post-damage failure surface is then interpolated

via

Y �fracture ¼ ð1� DÞY �

failure þ DY �residual ð30Þ

and the post-damage shear modulus is interpolated via

Gfracture ¼ ð1� DÞGinitial þ DGresidual ð31Þ

where Ginitial, Gresidual, Gfracture are the shear moduli.

4.10. P–a equation of state

Herrmann�s P–a model [25] is a phenomenological

approach to devising a model which gives the correct

behavior at high stresses but at the same time it provides

a reasonably detailed description of the compaction pro-

cess at low stress levels. The principal assumption is that

the specific internal energy for a porous material is the

same as the same material at solid density under the

same conditions of pressure and temperature.

Defining the porosity by a = v/vs, where v is the spe-

cific volume of the porous material and vs is the specific

volume of the material in the solid state and at the same

pressure and temperature. a becomes unity when the

material compacts to a solid. If the equation of state

of solid material is given by

p ¼ f ðvs; eÞ ð32Þ

Then the equation of state of the porous material is

simply

p ¼ fva; e

ð33Þ

Carroll and Holt [26] modified the above equation to

yield

p ¼ 1

af

va; e

ð34Þ

where the factor 1/a was included to allow for their argu-

ment that the pressure in the porous material is nearly 1/

a times the average pressure in the matrix material. The

function f can be any form of equations of state, in this

paper the polynomial form of equation is adopted, as

p ¼ A1lþ A2l2 þ A3l

3 with l ¼ v0v� 1 P 0

p ¼ T 1lþ T 2l2 with l ¼ v0

v� 1 < 0

ð35Þ

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

specific volume of the porous material.

The RHT model for concrete has been evaluated suc-

cessfully in the modeling of concrete perforation under

shock loading, and systematic parameters have been

obtained for several kinds of concrete [27].

4.11. Elastic–strain hardening plastic model for steel

Under blast loading, the reinforcing steel may be sub-

ject to strain hardening, strain rate hardening and heat

softening effects. In this study, the John–Cook model

[18] is adopted to model the response of the steel bars

in the concrete. The John–Cook model is a rate-depen-

dent, elastic–plastic model. The model defines the yield

stress Y as

Y ¼ ½Y 0 þ Benp�½1þ C log e�p�½1� TmH� ð36Þ

where Y0 is the initial yield strength, ep is the effective

plastic strain, e�p is the normalized effective plastic strain

rate, B, C, n, m are material constants. TH is homolo-

gous temperature, TH = (T � Troom)/(Tmelt � Troom),

with Tmelt being the melting temperature and Troom

the ambient temperature. The expression in the first set

of brackets gives the effect of strain hardening. The

expressions in the second and third sets of brackets rep-

resent the effects of strain rate and temperature,

respectively.

Nodal velocities & displacement

Element volumes & strain rates

Nodal accelerationElement Pressure & stresses

Integration

Conservation equations

Materialmodel

Deformation strain relation

Boundary forces

Fig. 4. Illustration of the computational cycle.

Fig. 5. Configuration of the numerical example.


4.12. JWL equation of state for explosive

The Jones–Wilkens–Lee (JWL) equation of state [28]

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 ¼ C1 1� xR1v

� �expð�r1vÞ

þ C2 1� xR2v

� �expð�r2vÞ þ

xev

ð37Þ

Fig. 6. The SPH–FEM co

where v is the specific volume, e is specific energy. The

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

explosives have been determined from dynamic

experiments.

upled model zoning.

Table 1

Soil profile considered in the numerical simulation

Soil description Dry density

(kg/m3)

Density

(kg/m3)

Air-filled

void (%)

Seismic

velocity (m/s)

Acoustic impedance

(MPa/m/s)

Attenuation n

Sandy clay �1530 �1920 �4 1400 2.14 2–2.5

Table 5

JWL parameters used for modeling TNT in the present study

C1 (GPa) C2 (GPa) R1 R2 x

3.738e2 3.747 4.15 0.9 0.35

e0: the initial C–J energy per volume; VOD: the C–J detonation velo

Table 2

Parameters used in the three-phase soil model for numerical

calculations

Soil Air phase

a1 = 0.58 qg0 = 1.2kg/m3

a2 = 0.38 cg0 = 340m/s

a3 = 0.04 kg = 1.4

q0 = 1.92 · 103kg/m3 Soil skeleton

Solid particles phase G = 55MPa

qs0 = 2.65 · 103kg/m3 Kp = 165MPa

cs0 = 4500m/s f = 0.56

ks = 3 a = 0.25

Water phase k = 0.2

qw0 = 1.0 · 103kg/m3 _e0 ¼ 1%=min

cw0 = 1500m/s b = 0.1

kw = 7 g = 1.0

E0 = 20MPa

b = 5.0

Table 3

Parameters used in the RHT model for concrete

Initial density q0 (kgm�3) 2.314e3

Reference density qs (kgm�3) 2.75e3

The RHT strength model

fc (MPa) 35

A 1.6

N 0.61

ft (MPa) 3.5

n1 0.036

n2 0.032

Q2,0 0.6085

Ginitial (MPa) 1.67e4

Gresidual (MPa) 2.17e3

P–a EOS

A1 (MPa) 3.527e4

A2 (MPa) 3.958e4

A3 (MPa) 9.04e3

Table 4

Parameters used for modeling reinforcement steel bar

Reference density q0 (kgm�3) 7.896e3 B (MPa) 2.75e2

Bulk modulus K (MPa) 2.0e5 C 0.022

Specific heat Cv (J/kgK) 4.52e2 n 0.36

Shear modulus G (MPa) 8.18e4 m 1.0

Y0 (MPa) 3.5e2 Troom (K) 3.0e2

Tmelt (K) 1.811e3


4.13. The interface model

An accurate representation of the interface between

the structure and the surrounding medium is crucial to

a successful analysis of the structural response. Accord-

ing to the experimental results from Huck et al. [29], the

soil/structure (concrete) interface strengths may be de-

scribed by Coulomb failure laws. On a smooth soil–con-

crete interface failure is initiated when the shear stress

parallel to the surface exceeds the failure law; whereas

e0A (MJm�3) VOD (ms�1) q0 (kgm�3)

6.0e3 6.93e3 1.63e3

city.

Specific heat Cv (J/kgK) 6.54e2

epl(elastic–plastic) 1.93e�3

B 1.6

M 0.61

D1 0.04

D2 1.0

eminf 0.01

Pe (MPa) 2.33e1

Ps (MPa) 6.0e3

T1 (MPa) 3.527e4

T2 (MPa) 0.0

n 3.0

Fig. 7. Computed formation of crater in soil.

Ground surface

Charge

Target points

Group B

Group A

0.2 0.4 0.6 0.8 1 2 4 60.1

1

10

100

1000

Function 2

Function 1

Group A targetsGroup B targets

Stre

ss/M

Pa

Scaled Distance/m.kg-1/3

(a) (b)Arrangement of field target points Attenuation of stress

Fig. 8. Arrangement of target points in soil and calculated attenuation of stress.

0 5 10 15 20 25 30-2

0

2

4

6

8

10

12

14

16

Stre

ss/M

Pa

Time/ms

Distance from charge = 4m Group A target Group B target

Distance from charge = 4m Group A target Group B target

0 5 10 15 20 25 30

0123456789

101112

Abso

lute

Vel

ocity

/m/s

Time/ms(a) (b)Stress histories Velocity histories

Fig. 9. Comparison of typical stress and velocity time histories between Group-A and Group-B targets.


1 2

7

73 5

6

9

8

Reinforced bar

Target points

In-1

In-2

In-3

Out-1

Out-2

Out-3

(a) (b)Target points in concrete Target points in reinforcing bars

Fig. 10. Arrangement of target points within the structure.

0 10 20 30 40 50 60 70

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Stre

ss/M

Pa

Time/ms

Interface stress (at target 9)

Fig. 11. Interface stress at middle-height of the front wall.


on a rough soil–concrete interface failure is initiated

when the maximum soil shear stress exceeds the failure

law. The experimental results from Mueller [30] 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) con-

crete and soil is modeled using regular FEM elements

with completely joined surface, whereby the nodes from

�soil grid� are joined (‘‘fused’’) to the nodes from the

�structure grid� at the interface between them. The nodes

will remain joined throughout the calculation. Another

numerical interface method is the so-called slide surface,

in which a contact interface is formed between the �soilgrid� and the �structure grid�. The slide surface allows

for separation, recontacting, and sliding (with friction

by a friction coefficient) of the two surfaces, using a

complex contact algorithm [31]. If the interface is

smooth, the slide surface will be more appropriate.

4.14. The computational cycle

The conservation equations, the material models as

well as the boundary conditions describe the whole

physical problem. These equations are solved to

update the solution in successive time steps. The oper-

ational procedure will not be presented in detail but a

general computational cycle is described in what fol-

lows. Fig. 4 shows the series calculations that are car-

ried out in each incremental time step. Starting at the

bottom of the figure the boundary forces are updated

and combined with the forces for the element com-

puted during the previous time cycle. Then the acceler-

ations, velocities and positions are computed from the

momentum conservation equations and a further inte-

gration. From these values the new element volumes,

strain and strain rate are calculated. With the use of

the material model together with the energy equation

the element pressure, stresses and energies are calcu-

lated, providing forces for use at the start of the next

computational cycle. The majority part of the proce-

dure for the FEM mesh is the same as that for the

SPH mesh. Only the formulas for calculation of the

volume, strain and strain rate from the velocities and

position of the element (in FEM mesh) or particle

(in SPH mesh) are different.

5. Numerical example

A numerical example is presented to demonstrate the

implementation of the proposed full coupled approach

combining the SPH and FEM methods. The example

scenario is a shallow-buried generic reinforced concrete

box structure subjected to a side burst. From the calcu-

lation, it will be shown that the approach is efficient and

several potential numerical difficulties are avoided. The

proposed numerical approach can be applied both in

2-D and 3-D analyses. Since many practical problems

can be simplified into 2-D axisymmetric cases and a 2-

D analysis is sufficient to test the numerical model, the

example problem is analyzed using 2-D axisymmetric

model. The calculations are performed using a commer-

cial hydrocode Autodyn [32] with necessary user

subroutines.

0 10 20 30 40 50 60 70

0

2

4

6

8 Target 3

Target 1

Dis

plac

emen

t R/m

m

Time/ms0 10 20 30 40 50 60 70

0

5

10

15

20

25

Dis

plac

emen

t Z/m

m

Time/ms

Target 1 Target 3

0 10 20 30 40 50 60 70-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Target 1-3

Def

orm

atio

n dr

/mm

Time/ms0 10 20 30 40 50 60 70

-4

-2

0

2

4

6

8

10

12

Target 5-4D

efor

mat

ion/

mm

Time/ms

Target 3-4

(a)

(b)

Total displacement on front wall (horizontal, vertical)

Lateral deflections (front wall-horizontal, floor slab-vertical)

Fig. 12. Displacements and deflections in front wall and floor slab.


5.1. Buried structure configuration and numerical model

setup

A common class of underground structures can be

simplified into a generic �box� type structure. Fig. 5 de-

picts the example box unit subjected to a subsurface side

burst.

In the numerical model, the SPH mesh is used to

model the area surrounding the explosion charge where

severe deformation of soil occurs. The FEM mesh is

used to model the intermediate and far field soil med-

ium, as well as the soil–structure interface and the

underground structure itself. Fig. 6 shows the zoning

of the mesh, where AB is the axis of revolution, BC

and CD are transmission boundaries which help mini-

mize the wave reflection effect.

The coupling of SPH mesh with the FEM mesh is

illustrated in Fig. 6(a) and (b). Within the SPH mesh,

the size of the SPH particles is graded, with the smallest

particles around the charge center. In the FEM mesh,

smaller elements are used in the region adjacent to the

SPH zone as well as for the structure, while larger size

elements are used in the remaining region. Within the

RC structure walls, the reinforcing steel grids are simpli-

fied into thin steel shell with equal volume of steel and

assuming a perfect bond between the steel and concrete

elements at the node points. Preliminary trial runs were

performed in choosing adequate mesh sizes.

The material models for the soil, concrete, reinforc-

ing steel bars and TNT charge described in the preceding

section are implemented in the calculation. The relevant

parameter values for these material models are summa-

rized in Tables 1–5. The parameters about the soil pro-

file and the soil skeleton (Table 1) are based on

experimental data [8], the parameters on the solid parti-

cle, water and air phase (Table 2) are recommended val-

ues from literature [8]. The data for the concrete (Table

3), reinforcing steel bars (Table 4) and the TNT charge

are also based on relevant literature [13,18,28].

A series of cases with different charge profiles were

analyzed. For illustration purpose, only the results from

one particular case are presented and discussed here.

The weight of the charge is 100kg TNT, embedded at

a depth of 0.5m. The distance from the charge to the

nearest edge of the structure is 6m. The top of the struc-

ture is leveled at the ground surface. The structure unit

has a length of 4m, height of 1m and a uniform wall

thickness of 100mm.

0 10 20 30 4 0 50 6 0 70-8

-6

-4

-2

0

2

4

6

8

10

12

14

Target 1 Target 2 Target 3

Stre

ssσ zz

/MP

a

0 10 2 0 30 40 5 0 60 7 0

-6

-3

0

3

6

9

12

Ta rget 1 Ta rget 2 Ta rget 3

Stre

ss σ

rr /

MP

a

(a)

(b)

0 10 2 0 30 40 5 0 60 70-0.0012

-0.0011

-0.0010

-0.0009

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

Lon

g. S

trai

n ε zz

Time/ms

Target O ut-1 Target O ut-2 Target O ut-3

0 10 2 0 30 4 0 5 0 60 70

-0.0002

-0.0001

0.00 00

0.00 01

0.00 02

0.00 03

0.00 04

0.00 05

0.00 06

0.00 07

0.00 08

0.00 09

Lon

g. S

trai

n ε zz

Time/ms

Time/ms Time/ms

Ta rget I n -1 Ta rget I n -2 Ta rget I n -3

Concrete stresses (front wall)

Reinforcing bar longitudinal strains (outer layer,inner layer)

Fig. 13. Typical concrete stress and reinforcing strain histories.

Fig. 14. Distribution of cumulative damage in concrete.


5.2. The crater formation

A shallow explosion in soils can form a crater while a

deep explosion usually forms a camouflet. In the process

of the crater formation, the soil will be ejected away

from the blast. As mentioned earlier, simulating this

process is very difficult with the continuous Lagrangian

FEM mesh, as the mesh will tangle due to large defor-

mation while the time step can reduce sharply. Now with

the SPH mesh this problem no longer exists. Fig. 7

shows the computed formation of the crater during the

explosion.

5.3. The propagation of blast wave in soil

To monitor the propagation of the blast wave in the

soil, two rows of target points are arranged as shown in

Fig. 8(a). Group-A targets are arranged parallel to the

ground surface, while Group-B are located along a 45�inclined line to capture the ‘‘free field’’ wave propaga-

tion. Fig. 8(b) shows the attenuation of the peak

pressure as a function of the scaled distance for the

group-A and group-B targets, respectively. Shown in the

figure are also two straight lines, denoted as Function 1

0 10 20 30 40 50 60 70-2

-1

0

1

2

Velo

city

Vzz

/m/s

Time/ms

Target 1,2,3

0 10 20 30 40 50 60 70-1

0

1

2

3

4

Target 3

Velo

city

Vrr /

m/s

Time/ms

Target 1,2

0 10 20 30 40 50 60 70

-100

0

100

200

300

Acce

lera

tion

a zz /

g

Time/ms

Target 1,2,3

0 5 10 15 20 25

-600

-400

-200

0

200

400

600

800

Acce

lera

tion

a r /g

Time/ms

Target 1,2 Target 3

(a)

(b)

Velocity (Vertical, horizontal)

Acceleration (Vertical, horizontal)

Fig. 15. Velocity and acceleration time histories of the front wall.


and Function 2, which correspond to the TM-5 [33]

empirical equations of free field pressure for the kind

of soils considered in this example,

Function 1 : p ¼ f � 15 � ðR=ffiffiffiffiffiW3

pÞ�2:5 ð38Þ

Function 2 : p ¼ f � 22 � ðR=ffiffiffiffiffiW3

pÞ�2 ð39Þ

where R is the distance from the charge centre, W is the

weight of charge in kilograms, and f is the coupling fac-

tor which reflects the effect of the buried depth of the

charge. For the current example, f = 0.6.

These two functions can be looked upon as the upper

and lower limits within which the particular type of soil

used in the current example should fall. From the figure

it can be seen that the attenuation law of group B tar-

gets, which represent the free field, lies between these

two limit curves. The attenuation curve from Group A

targets shows a notably quicker attenuation rate. This

is reasonable because these targets are near the ground

surface and it reflects the influence of the ground surface

on the propagation of the blast wave.

Fig. 9 shows a comparison of the typical stress and

velocity time histories between Group-A and Group-B

targets at the same distance from the charge.

5.4. Response of the structure

A number of target points are arranged to record the

response of the structure and the interface load, as

shown in Fig. 10.

Fig. 11 shows the computed interface stress at the

mid-height of the front wall (target point 9). The reload-

ing phenomenon [34] is clearly seen in the simulation re-

sult, and it reflects reasonably the characteristics

observed from experiments [1,10].

The displacement time histories computed on the

front wall (targets 1, 2, 3) are shown in Fig. 12(a).

As can be seen, marked permanent displacements of

the structure occur in both horizontal (order of 5–

10mm) and vertical directions (order of 15–30mm).

The differential displacements between different targets

indicate the deformation of the structure. Fig. 12(b) de-

picts the front wall deflection (horizontal) and the bot-

tom plate deflection (vertical) time histories. The

maximum deflection in the front wall reaches about

0.2% of its height, while in the bottom plate the

maximum deflection is about 1% of its length. The

residual deflections indicate the occurrence of plastic

deformation.


Regarding the material response, Fig. 13 shows

typical time histories of stress in concrete and strain

in the reinforcing bars at several target points. As

can be expected from the displacement response, the

material stress and strain histories vary significantly

from one target to another within the structure. The

longer period oscillations that follow the initial pulses

indicate the participation of the structural response.

Unfortunately the experimental data on stress and

strain are scarce. Stevens et al. [10] reported some time

histories of strain in reinforcing bars. The characteris-

tics of the measured steel strain histories exhibit favor-

able agreement with the computed ones shown in

Fig. 13.

Fig. 14 depicts the cumulative damage status at a

particular time. Apparently the effect of the ground sur-

face results in a shift of the most serious damage zone

from the center to the lower part of the front wall.

5.5. In-structure shock

An important aspect of the buried structure response

to blast loading is the in-structure shock, which consti-

0 10 20 30 40 50 60 70-2

-1

0

1

2

3

Target 5Target 4

Veloc

ity V

zz /m

/s

Time/ms

Target 3

0 10 20 30 40 50 60 70-100

-50

0

50

100

150

200

Acce

lera

tion

azz

/g

Time/ms

Target 4

(a)

(b)

Velocity (Vertica

Acceleration (Vert

Fig. 16. Velocity and acceleration

tutes the basis for evaluating the survivability and func-

tionability of the equipment housed in the structure. The

in-structure shock environment can be described by the

velocity and acceleration at different locations around

the structure. In the current example, the response of

the front wall and the bottom plate are expected to be

more critical than the remaining part of the structure,

so only the computed shock histories on these two

components are discussed here.

Fig. 15 shows the computed velocity and acceleration

histories on the front wall. As expected, the vertical mo-

tion at different target points are almost identical, while

the horizontal motion at the mid-height of the wall (tar-

get 2) exhibit a marked difference from that at the corner

(target 3), indicating significant response of the wall

panel. The computed velocity and acceleration histories

on the bottom plate are shown in Fig. 16. As can be ex-

pected from the bending response of plate and the effect

of wave propagating rightward, significant difference

can be observed in the vertical motion among targets

3, 4 and 5. Even the horizontal motion within the plate

plane exhibit some difference and time delay from target

3 onward to target 5.

0 10 20 30 40 50 60 70

0

1

2

Velo

city

Vrr

/m/s

X Axis Title

Target 3 Target 4 Target 5

0 10 20 30 40 50 60 70-300

-200

-100

0

100

200

300

400

Acce

lerat

ion a

rr /g

Time/ms

Target 4

l, horizontal)

ical, horizontal)

time histories of the floor.


The code TM-5 [33] provides a simple method for a

crude estimation of the in-structure acceleration and

velocity based on free-field ground shock. Using the

empirical formulas for the free-field ground shock, the

average value of acceleration can be derived by integrat-

ing the acceleration-range function over the span of the

structure. This average acceleration is regarded as the

nominal in-structure shock acceleration, and similarly

is the velocity. For the kind of soil considered in the cur-

rent example, the empirical formulas for the free-field

velocity and the acceleration, according to TM-5, are

in the range of

V avg1 ¼ f � 7:38 � RffiffiffiffiffiW3

p� ��2

V avg2 ¼ f � 4:62 � RffiffiffiffiffiW3

p� ��2:5

aavg1 ¼ f � 2338 � RffiffiffiffiffiW3

p� ��3

aavg2 ¼ f � 1464 � RffiffiffiffiffiW3

p� ��3:5

where Vavg1, aavg1 correspond to the attenuation coeffi-

cient n = 2, Vavg2, aavg2 correspond to the attenuation

coefficient n = 2.5. The average velocity and acceleration

of the structure, estimated from the integration of these

functions over the span of the structure, are found

to be Vavg1 = 1.59m/s, Vavg2 = 0.78m/s (velocity), and

aavg1 = 311g, aavg2 = 154g (acceleration).

Comparison of the above estimates and the average

peak values from the numerical results show a reason-

able agreement. The numerical results allow for a detail

characterization of the in-structure shock environment

for design considerations.

5.6. Conclusions

A full coupled numerical approach for simulating the

response of underground structure subjected to blast

loading is presented in this paper. The combined SPH

and FEM method overcomes many difficulties that are

known to be associated with other coupling methods.

In the proposed approach, the large movement and

deformation in the near-field soil medium around the

charge is modeled by the SPH mesh, whereas the FEM

mesh is used for intermediate and farther field soil med-

ium as well as for the RC structure. This combination

incorporates the merits from the two distinctive methods

in representing different physical processes. Besides the

computational considerations, the proposed approach

makes use of various state-of-the-art material models,

particularly noteworthy are the three-phase soil model

and the RHT concrete model, to enhance the reliability

of the simulation results. The numerical example shows

that the proposed approach is capable of reproducing

the physical processes in a realistic manner and the

numerical execution is smooth. With this full coupled

model, a wide range of problems related to subsurface

blast can be investigated numerically. The model can

also be used for parametric studies and verification of

practical models, and in special design situations where

great details of the responses are required.

For the underground structure considered in the

numerical example, the results reveal significant struc-

tural responses that affect the distribution of deforma-

tion and damage within the structure as well as the

in-structure shock environment. A full characterization

of these response features can be obtained through sys-

tematic numerical calculations using the proposed full

coupled model.

References

[1] Stevens DJ, Krauthammer T. Analysis of blast-loaded,

buried RC arch response. Part I: Numerical approach.

J Struct Eng, ASCE 1991;117(1):197–212.

[2] Hinman EE. Effect of deformation on the shock response

of buried structures subject to explosions. In: Structures

under shock and impact. Elservier; 1989. p. 455–65.

[3] Zhang YD, Fang Q, Liu JC. Experimental and numerical

investigations into responses of buries RC frames subjected

to impulsive loading. In: Structures under shock and

impact VII. Elservier; 2002. p. 69–78.

[4] O�Daniel JL, Krauthammer T. Assessment of numerical

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a full coupled numerical analysis approach for buried structures subjected to subsurface blast

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