-
ied
d 3
i W
ang T
of De
2 Feb
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: [email protected]
(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
the blast wave propagation in the soil medium. With a buried
structure, the 2D model can predict the blast loading and
the
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;
117(1):197212.
[6] Stevens DJ, Krauthammer T, Chandra D. Analysis of
blast-loaded,
buried arch response. Part II. Application. J Struct Eng ASCE
1991;
117(1):21334.
[7] Benson DJ. Computational methods in Lagrangian and
Eulerian
hydrocodes. Comput Methods Appl Mech Eng 1992;99:235394.
[8] Hayhurst CJ, Clegg RA. Cylinderically symmetric SPH
simulations of
hypervelocity impacts on thin plates. Int J Impact Eng
1997;20:
33748.
[9] Johnson GR, Petersen EH, Stryk RA. Incorporation of a SPH
option
into the EPIC code for a wide range of high velocity impact
computations. Int J Impact Eng 1993;14:38594.
[10] Meyers MA. Dynamic behavior of materials. New York: Wiley;
1994.
[11] Wang Z, Lu Y. Numerical analysis on dynamic deformation
mechanism
of soils under blast loading. Soil Dyn Earthq Eng
2003;23:70524.
[12] Wang Z, Hao H, Lu Y. The three-phase soil model for
simulating
stress wave propagation due to blast loading. Int J Numer
Anal
Methods Geomech 2004;28:3356.
[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
undrained strength derived from pressuremeter tests.
Geotechnique
1989;39(4):61524.
[15] Drucker DC, Prager W. Soil mechanics and plastic analysis
or limit
design. Q Appl Math 1952;10(2):15764.
[16] Riedel W, Thoma K, Hiermaier S. Numerical analysis using a
new
macroscopic concrete model for hydrocodes. In: Proceedings of
ninth
international symposium on interaction of the effects of
munitions
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
test results for WES flume sand against steel and grout: Report
3.
USAE WES, Technical Report, SL-86-20; 1986.
[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