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Research ArticleModelling Blast Effects on a Reinforced Concrete
Bridge
Markellos Andreou,1 Anastasios Kotsoglou,1 and Stavroula
Pantazopoulou2
1Department of Civil Engineering, Laboratory of Reinforced
Concrete, Democritus University of Thrace,V. Sofias 12, 671 00
Xanthi, Greece2Department of Civil Engineering, The Lassonde
Faculty of Engineering, York University, 4700 Keele Street,Toronto,
ON, Canada M3J 1P3
Correspondence should be addressed to Stavroula Pantazopoulou;
[email protected]
Received 21 April 2016; Revised 27 June 2016; Accepted 28 June
2016
Academic Editor: Chiara Bedon
Copyright © 2016 Markellos Andreou et al. This is an open access
article distributed under the Creative Commons AttributionLicense,
which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properlycited.
The detailed investigation of blast phenomena and their
catastrophic effects on existing structures are the main objectives
of thepresent paper. It is well known that blast phenomena may be
characterized by significant complexity, often involving
complicatedwave propagation effects as well as distinguishable
material behaviors. Considering the above and in an attempt to
provide asimplified modelling approach for the simulation of blast
effects, a novel procedure is presented herein based on
well-establishedmethodologies and common engineering practices. In
the above framework, firstly, the “predominant” deformation shape
of thestructure is estimated based on elastic finite element
simulations under blast loads and then the structural response of
the system isevaluated as a result of common computational
beam-element tools such as displacement-based pushover analysis.
The proposedmethodology provides an immediate first estimation of
the structural behavior under blast loads, based on familiar
engineeringprocedures. A two-span reinforced concrete bridge was
thoroughly investigated and the results provide insightful
informationregarding the damage patterns and localization.
1. Introduction
Although blasts are considered phenomena of significantseverity
and potential socioeconomic impact, only recentlydid the
authorities realize the necessity for the formulation ofan
integrated design and assessment protective framework.Modeling the
effects of these phenomena on structures isvery demanding,
requiring highly sophisticated simulationsincluding advanced
constitutive material models. These pro-cedures are resource- and
time-consuming. On the otherhand, there is a lack of simplifying
procedures that could beimplemented by practicing engineers through
the utilizationof common computational tools. Herein, a simplifying
proce-dure is proposed based on common analytical and
computa-tional tools that would provide a preliminary but yet
reliableestimation of the blast impact to the structural integrity
ofbridges.
Blasts are short duration dynamic events that generatedynamic
pressure waves which propagate radially from thesource in space,
exciting dynamic response in the structures
that are encountered in their path. The pressures acting onthe
affected surfaces are impulsive loads that impart signifi-cant
amount of potential energy which sets damage-causingvibrations in
the structure.
The various loads that act on a structure during its
lifetime(natural or man-made) are characterized [1] by their
rangeof frequency content and intensity (Figure 1(a)).
Dynamicpressures exerted by blast explosions are considered
amongthe most critical loads owing to their high intensity
andfrequency content, which falls within the range of values
asso-ciated with fundamental eigenvalues of common buildings.Being
a conveyer of potential energy, the blast wave evolvesdilating in
space (Figure 1(b)) as soon as it emerges. Duringthis process it
gets reflected on any surfaces it encounterswhen colliding with
objects or structures or the ground (Fig-ure 1(b)). In the case of
surface explosions (where the source isat ground level), the
reflection takes place almost simultane-ouslywith the genesis of
thewave. A schematic representationof the translation and dilation
of the blast wave front causedby a surface explosion on a given
structure are illustrated in
Hindawi Publishing CorporationAdvances in Civil
EngineeringVolume 2016, Article ID 4167329, 11
pageshttp://dx.doi.org/10.1155/2016/4167329
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2 Advances in Civil Engineering
Very low Low Medium High Very high
Low
Med
ium
Hig
h
Range of frequencies
Load
inte
nsity
Blast
Earthquake
Win
dMechanical
vibrationSound
(a)
Reflection surfaces
Source of the blast, t = 0
t3 t2
t3
t1
t2
t3
(b)
Figure 1: (a) Frequency range of different dynamic loads
(according to [1]). (b) Collision and reflection of aerial blast
waves (𝑡𝑖are arrival
times of the expanding wave front).
Refle
cted
pres
sure
Incidentalpressure
Figure 2: Schematic representation of the evolution of a
surfaceblast wave in space; reflected and main pressure blast wave
on anadjacent structure.
Figure 2. Note that reflected pressure waves affect
surfacesperpendicular to the direction of the blast wave, whereas
theincidental pressure wave affects the sides and back-face of
thestructure.
The effects of explosions are studied in this paper basedon a
simplifying procedure, with particular emphasis onone type of
structure whose operation is vital in emergen-cies, namely,
reinforced concrete highway overpasses. In thesimplifying
framework, the structure is approximated usingconcepts from
generalized single-degree-of-freedom systems(Clough and Penzien
[2]), where the dynamic response of thecontinuous structure is
examined in a predominant shape ofresponse. The proposed two-step
methodology utilizes well-established structural engineering
procedures as a prelimi-nary design and assessment tool for the
estimation of theimpact of blast effects on structures. According
to the above,the predominant deformation shape under blast loading
isfirstly evaluated based on ordinary elastic FE analyses andthen a
displacement-based pushover analysis is conducted to
a simplified beam-element lumped plasticity model in orderto
evaluate the nonlinear behavior of each structural compo-nent for
the induced deformation pattern of the previous step.With the above
proposed procedure, the complicated blastproblem is reduced to a
simplified two-step procedure, whichmay be further simplified if
certain characteristic deforma-tion patterns for design purposes
would be instantly providedin the future, based on the structural
configuration and theblast initiation point.
The first section of the study deals with the
mathematicaldefinition of the excitation function exerted by the
explosionand its numerical representation in the framework of
estab-lished structural analysis software, considering both the
tem-poral and spatial definition of the evolving pressure
profilesacting on the exposed structural surfaces. The
investigationuses a selected bridge case study subjected to a
passing blastwave for illustration of concepts. A linear 3D finite
elementmodel of the bridge structure was combined with a
consistenttime history simulation of the explosion pulse, placing
partic-ular emphasis on the sequence of contact of the pressure
wavewith the structure, which depends on the physical distanceand
the location of the source relative to the structure. Anessential
ingredient for assessment of deformation demandsand damage
potential caused by the event is the deformedshape assumed by the
structure through the time historyof the event; this is extracted
from the calculated responseresults. Next, using this pattern of
deformations, the structurewas analyzed based on the implementation
of a simplifiedordinary beam-element structural model so as to
enable theuse of the results of a nonlinear pushover analysis of
thebridge under transient pressure profiles that simulate
theexplosive loads.This simplification alleviated partly the
com-plexity of the problem associated with the time dependencyof
the constitutive properties of the materials, whereas itwas
possible to identify the tendency for damage localizationthroughout
the structure. In this way, it is possible to takeadvantage of the
nonlinear modelling technology that is
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Advances in Civil Engineering 3
stable and convergent when dealing with linear prismaticelements
while avoiding convergence problems that would beowing to brittle
failures in the continuous 3D finite elementmodel.
2. Effects of Blast Waves on Structures andMaterial Response
2.1. Effects of Blast Waves on Structures. At any single
point,the pressure wave has a time history of the type shown in
Fig-ure 3.The primary effect of a blast wave on a structure
occursduring the positive phase, where pressure values are high;the
negative phase has lesser consequences due to the atten-uated
pressure magnitudes and is usually neglected (Agrawaland Yi
[1]).
An event is classified as blast if, compared with
othercatastrophic events such aswind and earthquake, the
pressuremomentum is several orders of magnitude greater than thatof
other phenomena (FEMA 426 [3]), whereas the pressurecaused by the
explosion attenuates quickly with increasingdistance from the
source affecting only a small part of thestructure and causing
great localized damage. Owing to theshort duration of the blast,
the mass participating in dynamicresponse is engaged usually at a
later time, that is, until afterthe impulse has delivered its
potential energy to its surround-ings (positive phase). So, given
that the loading has expiredby the time the mass is mobilized in
dynamic motion, it isextremely difficult to resonate the
systemwhile the load is act-ing. This definition underscores the
difference between moreconventional dynamic loads such as
earthquakewhere partialresonance of eigenmodes is of primary
significance. In thesame context, due to the short duration of the
loading exertedby the blast wave which typically ends within few
millisec-onds, it is possible to excite several additional modes
(highermode contribution) in the free vibration phase that
follows,usually neglected in conventional seismic design. Of
coursethis is a theoretical postulate; nonlinearity takes hold of
theresponse as its intensity builds up, damping out quickly
theparticipation of higher modes near failure.
The size and distribution of the applied pressures onthe
structure depend on the amount and type of releasedenergy (which
depend on the explosives used), the positionof the source relative
to the structure, and the magnitude andpossible amplification of
the resulting blast pressure owingto its interactions with objects
encountered during dilationand propagation of the wave (Birhane
[4]). The explosivewave exerts pressure on every exposed point of
the structureencountered during its translation. The pressure
depends onthe arrival time 𝑡
𝑎of the wave front at the point of the
structure considered. This time includes the time of transferof
the released energy from the explosive material to theenvironment
(Martin [5]). The pressure rises from an initialvalue 𝑝
𝑜(the atmospheric or ambient pressure) to the peak
normal pressure 𝑝𝑠𝑜(peak overpressure) instantaneously and
it is subsequently reduced till its value attenuates to return
tothe initial pressure value. The durations of the positive
andnegative phases of the wave front are denoted by 𝑡
𝑑and 𝑡𝑛
(Figure 3): 𝑡𝑑is the time of pressure attenuation from the
peak
value to the initial value, whereas 𝑡𝑛is the subsequent time
Duration of Duration of
Positive phase
Negative phase
p(t)
pso
po
pmin
ta ta + td
t
positive phase td negative phase tn
Figure 3: Diagram of pressure variation as a function of
time.
interval up to the occurrence of the minimum negative
value,𝑝min, in the negative phase.The pressure intensity versus
timeattenuation relationship is approximated by the
followingexpression, known as the Friedlander equation (Baker
[6]).The positive phase of the phenomenon concerns the mainpressure
wave of the blast wave (Figure 3):
𝑝 (𝑡) = 𝑝𝑜 + 𝑝𝑠𝑜 ⋅ (𝑡 − 𝑡𝑎
𝑡𝑑
) ⋅ 𝑒−𝑏((𝑡−𝑡
𝑎)/𝑡𝑑);
𝑝𝑠𝑜=1
𝑍⋅ [1772
𝑍2−114
𝑍+ 108] (kPa) ,
(1)
where 𝑍 = 𝑅/(𝑊)1/3 is the scaled distance in m/kg1/3(obtained
after pertinent normalizing; Ngo et al. [7]), 𝑊 isthe amount of the
explosivematerial in [kg], and 𝑏 is the coef-ficient of degradation
which defines the slope of the attenua-tion curve (Martin [5]):
𝑏 = 5.2777 ∗ 𝑍−1.1975. (2a)
Variable 𝑡𝑑
(the positive phase duration) is defined by(Pandey et al.
[8])
𝑡𝑑
𝑊1/3
=
980 ⋅ [1 + (𝑍/0.54)10]
[1 + (𝑍/0.02)3] ⋅ [1 + (𝑍/0.74)
6] ⋅ √1 + (𝑍/6.9)
2
.
(2b)
For hemispherical explosions, that is, explosions thatoccur on
ground surface, it is recommended that the variableterm in (1) be
multiplied by a factor equal to 1.8 to accountfor the reflection of
the blast wave which occurs during theexplosion (Lam et al.
[9]):
𝑝 (𝑡) = 𝑝𝑜 + 1.8 ⋅ 𝑝𝑠𝑜 ⋅ (𝑡 − 𝑡𝑎
𝑡𝑑
) ⋅ 𝑒−𝑏((𝑡−𝑡
𝑎)/𝑡𝑑)(kPa) . (3)
2.2.Material Response under Strain Rate Loading. High
strainrates affect the mechanical properties of the materials
andthus the mechanisms of degradation of the various elements
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4 Advances in Civil Engineering
of the structure. The effect of high deformation rate onmaterial
strengths is quantified by the Dynamic IncreaseFactor (DIF), which
is defined as the ratio of the dynamicto static strength (Javier
and Allen [10]). Blast loading causesexcessively high rates of
deformation in the order of 102/s–107/s, whereas for usual loads
the rate of deformation rangesbetween 10−7/s and 100/s (for
reference, note that the ratesassociated with Creep and Relaxation
phenomena are in therange of 10−8/s to 10−5/s; pseudostatic loads
occur at strainrates from 10−8/s to 10−4/s; earthquake loads occur
at strainrates from 10−5/s to 10/s. Explosions and collisions
occurat strain rates from 10/s to 107/s, whereas even higher
ratescorrespond to astrophysical phenomena, CEB-FIP, Bulletin56
[2]).
For reinforced concrete structures subjected to blast load-ing,
the strength of concrete and reinforcing materials mayexperience a
significant increase due to the rate effects. Theincrease may
exceed 50% for the reinforcing steel, whereasit may exceed 100% for
concrete in compression and morethan 600% for concrete in tension
(Javier and Allen [10]). Animplication of this disproportional
strength increase of thetwomaterials is an alteration in practice
of the intended hier-archy of failure modes: thus, a design
controlled by flexuralyielding under relatively low rates of
loading may becomecontrolled by shear or by a combined
shear-flexural mode offailure at higher rates (Ngo et al. [7]).
Design charts are avail-able to estimate the DIF for the two
materials based on thestrain rate (CEB-FIP, Bulletin 55 [2]);
however, the phenom-ena may be also approximated by the following
equations.
2.2.1. Rate Effects on Concrete Strength in Compression.
Con-sider
DIF =𝑓𝑐,𝑖𝑚𝑝,𝑘
𝑓𝑐𝑚
= (̇𝜀𝑐
̇𝜀𝑐𝑜
)
0.014
, for ̇𝜀𝑐≤ 30 s−1 (4a)
DIF =𝑓𝑐,𝑖𝑚𝑝,𝑘
𝑓𝑐𝑚
= 0.012 ∗ (̇𝜀𝑐
̇𝜀𝑐𝑜
)
1/3
,
for ̇𝜀𝑐> 30 s−1,
(4b)
where 𝑓𝑐,𝑖𝑚𝑝,𝑘
is the dynamic compression strength for load-ing rate ̇𝜀
𝑐, 𝑓𝑐𝑚
is the static compressive strength at loadingrate ̇𝜀𝑐𝑜, ̇𝜀𝑐is
the rate of deformation taking values in the range
from 30 × 10−6 s−1 to 3 × 102 s−1, and ̇𝜀𝑐𝑜= 30 × 10−6 s−1 is
the
rate of deformation under pseudostatic compression loading.
2.2.2. Rate Effects on Concrete Strength in Tension.
Consider
DIF =𝑓𝑐𝑡,𝑖𝑚𝑝,𝑘
𝑓𝑐𝑡𝑚
= (̇𝜀𝑐𝑡
̇𝜀𝑐𝑜
)
0.018
, for ̇𝜀𝑐𝑡≤ 10 s−1 (5a)
DIF =𝑓𝑐𝑡,𝑖𝑚𝑝,𝑘
𝑓𝑐𝑡𝑚
= 0.0062 ∗ (̇𝜀𝑐𝑡
̇𝜀𝑐𝑡𝑜
)
1/3
,
for ̇𝜀𝑐𝑡> 10 s−1.
(5b)
Term 𝑓𝑐𝑡,𝑖𝑚𝑝,𝑘
in (5a) and (5b) is the dynamic tensile strengthfor loading rate
̇𝜀
𝑐𝑡, 𝑓𝑐𝑡𝑚
is the static tensile strength for
loading rate ̇𝜀𝑐𝑡𝑜, ̇𝜀𝑐𝑡is the rate of tensile deformation in
the
range from 1 × 10−6 s−1 to 3 × 102 s−1, and ̇𝜀𝑐𝑡𝑜
= 1 × 10−6 s−1(the rate of deformation at pseudostatic tensile
loading con-ditions). Apart from material strength, the high
deformationrate affects all other mechanical properties. For
example,by definition, the modulus of elasticity of concrete
underhigh deformation rate may be estimated from the
followingequation:
𝐸𝑐,𝑖𝑚𝑝
𝐸𝑐𝑖
= (̇𝜀𝑐
̇𝜀𝑐𝑜
)
0.026
, (6)
where ̇𝜀𝑐is the rate of deformation under dynamic loading,
𝐸𝑐,𝑖𝑚𝑝
is the dynamicmodulus of elasticity, and𝐸𝑐𝑖is themod-
ulus of elasticity of concrete. Combining the above
equations(and considering the familiar Hognestad’s parabola for
thestress strain response in uniaxial compression), the strain
atpeak stress considering the modulus and strength
dynamicenhancement are estimated from
𝜀𝑐1,𝑖𝑚𝑝
𝜀𝑐1
= (̇𝜀𝑐
̇𝜀𝑐𝑜
)
0.02
, (7)
where 𝜀𝑐1,𝑖𝑚𝑝
is the deformation at peak stress for a load ratė𝜀𝑐and 𝜀𝑐1is
the deformation at peak stress under pseudostatic
load.
2.2.3. Coefficient of Dynamic Amplification for Steel. Javierand
John [11] studied the strength increase of reinforcementthrough
experimental testing under high strain rates. Resultsfollowed a
nonlinear relationship, where the rate of changeof the dynamic
amplification increased linearly with strainrate. It was concluded
that the logarithm of DIF is at a linearrelationship with the
logarithm of the strain rate, ̇𝜀. Therelationship that was used,
both for the yield stress and for theultimate strength, is (Javier
and John [11])
DIF = ( ̇𝜀10−4)
𝑎
. (8)
Parameter 𝑎 used for the calculation of the yield and
ultimatereinforcement stress is obtained from the following
relation-ships:
𝑎 = 𝑎𝑓𝑦
, where 𝑎𝑓𝑦
= 0.074 − 0.04 ⋅ (
𝑓𝑦
414) (9a)
𝑎 = 𝑎𝑓𝑢
, where 𝑎𝑓𝑢
= 0.019 − 0.009 ⋅ (
𝑓𝑦
414) . (9b)
𝑓𝑦is the yield stress of the reinforcement and 𝑓
𝑢is the
ultimate strength of the reinforcement in MPa.This model applies
to reinforcing steel with yield stress
ranging between 290 and 710MPa and for strain rates
rangingbetween 10−4 and 225 s−1. The strength increase
estimatedaccording to (9a) and (9b) is considered only for passive
rein-forcement (i.e., it is not valid for prestressing streel.
Dynamicamplification is not considered when dealing with
thestrength of prestressing cables).
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Advances in Civil Engineering 5
(a)
1.00
5.50
1.73
3.75 1.50
3.88
0.30
13.68
15.84
0.14
1.400.19
(b)
1.12 0.20
0.66 41.02
7.19
5.18
0.32
1.00
5.50
4.75
76.00
77.52 0.66 33.02 0.66
0.20 1.12
(c)
Figure 4: (a) Schematic representation of the bridge. (b)
Geometric characteristics (transverse cross section). (c)
Lengthwise cross section ofthe bridge.
3. Investigating the Effects of Blast Waves ona Highway
Overpass
Blast effects on bridges are studied on a model bridgehighway
overpass, as illustrated in Figure 4(a). The bridgesuperstructure
selected for study has the geometric propertiesof an actual bridge
system (Kotsoglou and Pantazopoulou[12]) with monolithic connection
at the central bent and theedge abutments (Figure 4(b)). The bridge
has two unequalspans with a two-column central bent. The deck
comprisesa box section of multiple cells. Span lengths are 42.00m
and34.00m (Figure 4(c)).The box cross section is divided to
cellsthrough 5 beams of I cross section, having a web width of0.3m,
whereas the superstructure height is 1.73m.
The central bent columns have a circular cross section of1.50m
diameter supported by separate pile caps resting onpile groups;
each foundation block was a 3.75m × 4.75m ×1.00m rectangular block.
Clear height of the columns was5.50m.The portal frame bent was
centered at themidpoint ofthe deck width with no eccentricity,
whereas the clear trans-verse distance between columns was 3.88m.
Columns weremonolithically connected with a bent cap beam
(transversebeam) which in turn was connected monolithically with
theadjacent deck superstructure. In the analysis of the bridgecase
study, the following parameter values were assumed:C30/37 concrete
class (Eurocode 2, 2004)with a characteristic
compressive strength 𝑓𝑐𝑘
= 30MPa and a nominal elasticmodulus 𝐸
𝑐= 32GPa, whereas S500B steel was taken for all
loose reinforcement (characteristic steel yield strength 𝑓𝑦𝑘
=500MPa). Self-weight and support reactions were estimatedfrom
the available architectural drawings of the bridge.
3.1. Variation of Simulated Blast Pressure on the Bridge.
Theproblem considered in this study concerns the dynamicresponse of
the bridge described above, owing to a surfaceexplosion at near
distance. A key issue to resolve first is thepressure profile
occurring throughout the bridge structureand how the pressure wave
propagates in space and time.
At the moment of the explosion the main pressure waveis
transmitted uniformly in all directions. Over the durationof the
transmission process the pressure varies with time: thepeak
pressure that occurs in each point in space depends onthe distance
from the source and the amount of the explosivematerial (Figure 5).
It has therefore, for each explosion, thesame value for all the
points that are located at specificdistance from source (spherical
distance from the source asillustrated in Figure 2). In this
representation 𝑝
𝑠𝑜1, 𝑝𝑠𝑜2
, 𝑝𝑠𝑜3
,and 𝑝
𝑠𝑜4are the peak values of the blast wave at distances 𝑅
1,
𝑅2,𝑅3, and𝑅
4from the source for the respective arrival times
𝑡1, 𝑡2, 𝑡3, and 𝑡
4. Note that the values of the peak pressure
are inversely proportional to the normalized distance, 𝑍.For
small values of normalized distance the pressures that
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6 Advances in Civil Engineering
Blast wavepropagation
Peak overpressure
Wave front
pso1
pso2
pso4R1
R2 R3 R4
t1
t2
t3t4
Distance, R
Figure 5: Attenuation of peak overpressure with distance
fromsource: blast wave propagation.
develop are much higher relative to the peak pressures atpoints
at a greater normalized distance.
Furthermore, the pressure of the blast wave does notattenuate at
the same rate in all points in space since itdepends on the
coefficient of degradation 𝑏 which is a func-tion of𝑍. It is easy
to demonstrate that the value of 𝑏 increases,while the normalized
distance decreases; that is, for smalldistances from the source and
large amounts of explosivematter the coefficient is large, and the
pressure decays quickly.This is the physical significance of 𝑏,
whereas its calculation isrelated to the impulse (potential energy
release) of the blastwave.
The effect of the pressure on the structure begins at theinstant
when the blast wave arrives at a specific point on itsexposed
surface, denoted henceforth as the “arrival time.”Pressure value is
maximum for any given spatial point inconsideration at the arrival
time. The pressure magnitudeattenuates from that peak value
according to (1).
The duration of the time period over which the pressurevalues
exceed the atmospheric pressure also depends on 𝑍.The so-called
positive phase begins at the time of arrival andends at the instant
where the pressure becomes equal to theatmospheric value for the
first time. The instantaneous equi-librium that takes place heralds
the beginning of the secondphase of the phenomena, referred to as
“negative phase.”Thenegative phase has a significantly longer
duration as com-paredwith the positive phase; however, its
implications on thestructure are significantly lower andoften are
neglected,whilethe values assumed by the negative pressure are
compara-tively much lower than those of the positive phase. The
windsuction that follows the blast wave during itsmotion is
relatedto the dynamic pressure, with the value always being equal
toor greater than the atmospheric one (i.e., it does not
assumenegative values).
Regarding the bridge under consideration, the source ofthe
explosion was assumed to be located on ground surface.Thermal
radiation released upon the explosion and its effectson the
material response were ignored in the present study.Surfaces that
lie on the path of the wave front receive thereflected pressure,
whereas the others receive the incidentalpressure (Figure 2). To
calculate the reflected pressure a coef-ficient equal to 1.8 was
used on peak pressure in the Friedlan-der equation (see (6)).
Material properties used in dynamicanalysis of the FE model of the
bridge were adjusted to
account for the effects of dynamic amplification; note thatfor
blasts of moderate and high intensity (i.e., for strain rateshigher
than the reference value of 3 × 102 s−1) the DIF maybe estimated
from (4a)–(9b). For the case study consideredherein the assumed
explosive material was 500Kgs ΤNΤ ata transverse distance of 3m
from the bridge’s central bent(Figure 6(a)).
4. Methodology in Modelling the Blast Wave
In the present section, a two-step simplifying methodologyfor
the evaluation of blast effects on structures is pro-posed and
implemented. The main scope is to provide anapproximate yet
reliable tool for instant estimations of theabove effects based on
well-established engineering designand assessment strategies.
According to the methodology,a corresponding predominant
deformation shape is initiallyestimated based on a simplified
elastic analysis to be used as adisplacement loading pattern in a
detailed lumped plasticitybeam-element computational model of the
bridge [13]. Thelatter is considered to be a displacement-based
pushoveranalysis in that instead of a fixed pattern of loads of
graduallyincreasing intensity, the structure is subjected to a
fixedpattern of displacements of gradually increasing intensity.The
target displacement intensity level is that attained in theelastic
analysis when accounting explicitly for the pressurefront
evolution.The above simplification is advantageous, notonly because
many engineers worldwide are familiar to the“pushover analysis”
concept, but also because it reduces amultiparametric problem into
a simplified structural engi-neering problem. Furthermore,when
comparedwith detailedFE inelastic simulation models, a wide range
of computa-tional shortcomings may be eliminated, such as
convergencedue to ill-conditioning of material stiffness after
brittle crack-ing, time and resource consuming problems, and so
forth.Herein, the corresponding deformation shape of the
system,under blast loads, was evaluated based on an elastic,
time-dependent FE model of the case study bridge [14], but notethat
simplified, rational deformation shape approximationscould be used
instead.
Therefore, based on the deformation shape patternobtained in the
initial step, a displacement-based pushoveranalysis is conducted on
a detailed beam-element compu-tational simulation. Lumped
plasticity or even brittle failure[13] properties (e.g.,
moment-curvature and shear failure cri-teria) were incorporated to
each structural element in orderto account for the induced
damage.
4.1. Calculating the Load of the Blast Wave Front. Let time
bedefined with the reference starting point set at the instant
ofthe explosion (𝑡
𝑖= 0); at that instant the wave begins to dilate
in space evolving in all directions (Figures 1 and 6(a)). At
time𝑡𝑎the blast front reaches the nearest point of the
structure
(the point at shortest travel distance from the source;
Figures6(b) and 6(c)). In the bridge case study and for the blast
eventunder consideration, this point is at the base of the
centralbent on the exposed side. At time 𝑡
1= 𝑡𝑎during which the
blast wave has come in contact with the structure, the
columnbase at the central pier support is the only point subjected
to
-
Advances in Civil Engineering 7
(a) (b) (c)
(d) (e) (f)
Figure 6: Schematic representation of contact of the blast wave
with the bridge superstructure.
normal pressure.The intensity of this pressure applied at
time𝑡1on this point is equal to 𝑝
𝑠𝑜1(Figure 5), whereas the rest of
the structure remains unloaded.During time 𝑡
2> 𝑡1, the wave has dilated and comes
in contact with a new surface (loading surface) which isdefined
by the common points that lie on the cross sectionof the sphere
that represents the wave front and the structure(Figure 6(d)) and
are also at a distance greater than that of theinitial point of
contact. This distance depends on the speedof evolution of the
blast front. At this instant of time thisspecific surface is loaded
by pressure 𝑝
𝑠𝑜2(Figures 6(c)–6(f)).
Simultaneously the initial point is loaded by pressure
𝑝1<
𝑝𝑠𝑜1
, where𝑝1is calculated according to the curve of Figure 3,
for 𝑡 = 𝑡2. Such surfaces, as they have not been met by
the blast front at the time in consideration, remain
unloaded.The same thing holds at time instances, 𝑡
3, 𝑡4, . . . , 𝑡
𝑛, when
the peak pressures occur, 𝑝𝑠𝑜3, 𝑝𝑠𝑜4, . . . , 𝑝
𝑠𝑜𝑛, for 𝑛 surfaces
as they come progressively in contact with the wave
front,whereas surfaces through which the wave has already passedare
loaded by attenuated pressure magnitudes. These areestimated by the
model of pressure variation as shown inFigure 3 after introducing
properly the parameter values.Thus each new surface of the
structure which is intersectedby the spherical wave front is loaded
by the peak pressureat the instant of contact, whereas afterwards
it continues tobe loaded by the pressure estimated from the
attenuationrelationships until it becomes eliminated.
The set of contact points which are loaded simultaneouslyis
evaluated automatically by calculating their normalizeddistance
from the source based on the assumption that theblast wave evolves
at constant speed radially (the geomet-ric distance and the amount
of the explosive matter arecombined to calculate the normalized
distance; this is then
introduced in the corresponding relationships for calculationof
pressure at every point in a given time). As illustrated inFigures
6(c)–6(f), there are surfaces in different elements ofthe structure
that are loaded simultaneously.
The speed of the wave front is calculated from thefollowing
equation (Ngo et al. [7]):
V𝑠= 𝑎𝑜⋅ (6𝑝𝑠𝑜+ 7𝑝𝑜
7𝑝𝑠𝑜
)
1/2
, (10)
where 𝑝𝑠𝑜
is the peak positive pressure, 𝑝𝑜is the ambient
atmospheric pressure before the explosion, and 𝑎𝑜is the
speed
of sound at the ambient conditions.The magnitude of the pressure
felt by each surface
depends on its position relative to the direction of evolutionof
the wave front. Surfaces located at the forefront of the
wavetransmission receive the reflected pressure, whereas all
othersurfaces receive the incidental pressure (Figure 2).
Calculating in this manner the pressure that acts on thevarious
points of the structure in time, the problem is reducedto a
classical problem of structural dynamics which may besolved
numerically using established procedures (e.g., withfinite
elements); the boundary of each finite element lyingon the
perimeter of the structure is loaded by a time-varyingpressure
function depending on its distance from the source.The volume of
data that must be calculated is proportional tothe size and
complexity of the structure, and thus significantcomputing capacity
is required for a usual structure. To dealwith this problem, in the
present investigation, a collec-tion of geometric loci representing
the intersection of theadvancing hemispherical wave front with the
structure weredetermined at no loss of accuracy. Based on the
consecutive,
-
8 Advances in Civil Engineering
Figure 7: Separation of the loading surfaces of column and
bridge deck to model the spherical propagation of the blast
wave.
time-dependent contact patterns with the bridge depictedin
Figure 7, discrete loading surfaces were defined on eachcritical
structural component (i.e., columns and deck sur-faces) in order to
simulate the wave propagation with time.From the diagram of
pressure variation with time depictedin Figure 3 and considering
the wave propagation pattern ofFigure 6, each loading surface is
characterized by a discretearrival time 𝑡
𝑎which depends on the distance from the source
and the speed of the wave as estimated by (10). Figure
7illustrates the loading surfaces defined on the bridge,
whereaspressure was taken as constant within each step.The
pressurevalue was set equal to the mean, that is, the value
associatedwith the average distance of the points on the surface of
thebridge to the source. For the column, surfaces had a smallerarea
as initially, after arrival, pressure shows a faster rate
ofattenuation, whereas for the deck these surfaces were
sub-stantially larger as, with increasing time, the rate of
pressurereduction drops as illustrated in Figure 3. Based on
theabove, deck and column surfaces were defined symmetricallyaround
the initial contact point and were discretized into fiveand eight
segments, respectively (Figures 7 and 8(a)). Notethat reflected
pressure waves affect the surfaces (Figure 8(b))at the forefront
(i.e., front-face of the columns, bottom, andfront-face of the
bridge deck), whereas the incidental pressurewave affects the sides
and back-face of the structure (back-face of the columns, upper
surface of the superstructure).
4.2. Proposed Simplifying Methodology. In the present sec-tion,
a simplifying methodology is proposed for a fast evalu-ation of
blast effects based on the utilization of common ana-lytical tools.
In general, such complicated phenomena as blasteffects would demand
modelling and analytical approacheswith significant computational
cost. The dynamic nature ofthe problem, in conjunction with the
advanced materialconstitutive properties, would drive to extremely
complicatedFEmodels with very large number of nodes, significant
com-putational cost, and questionable results (e.g.,
convergenceproblems during inelastic finite element analysis).
Based onthe above, a simplifying, easy to use methodology was
devel-oped herein in order to reduce the computational cost and
atthe same time provide reliable solutions based onwidely
usedsoftware packages. The core of the proposed methodology isto
define the predominant deformation shape of the structureunder the
blast wave and then to proceed with ordinarypushover analysis based
on simplified, ordinary structural
beam-element models with lumped properties. The entiremethod may
be summarized in the following steps:
(a) Based on the theoretical backgroundpresented herein(Section
2), evaluate all acting pressures on the exist-ing structural
elements and their distribution alongthe bridge deck and
substructure.
(b) Evaluate the structural behavior of all critical compo-nents
(columns and beams) through the implemen-tation of established
analytical methodologies (i.e.,moment-curvature section diagrams,
shear and axialtension/compression ultimate strength values).
Theabove-mentioned values will be used as an input tothe lumped
plasticity beam-element model and at thesame time will form the
failure/yielding criteria forthe selection of a valid deformation
pattern on the FEmodel of the next step. Strain rate phenomena
shouldbe also considered herein for concrete and steel, basedon the
provisions of Section 2.2.
(c) Generate a finite element model (simplified ordetailed) and,
through the implementation of elasticanalytical tools, find the
predominant deformationshape of the structure due to blast wave
effects atpeak dynamic displacement response. The detailed3D finite
element modelling approach presented inSection 4.1 is considered to
be the most reliablesolution, while other less time consuming
strategieswould be also acceptable (e.g., the use of 2D elementsin
widely used commercial software). Consideringthat the deformation
shape is intended for use duringa pushover analysis, the selection
should correspondto the failure/yielding state of the structure
based onappropriate yielding/fracture criteria defined for
eachcritical structural component in the second step.
(d) Using the evaluated predominant deformation shapeof the
previous step, proceed with the conventionaldisplacement-based
pushover analysis in a simplifiedbeam-element structural model with
lumped proper-ties. Detailed section properties and interactions
arenecessary to be incorporated in the model, based onordinary
structural modelling strategies (Step (b)). Inany case, the
resulting total force should be equal toor less than the maximum
estimated applied forceextracted from the FE model.
-
Advances in Civil Engineering 9
(a) (b)
(c) (d)
Figure 8: FE elastic analysis for a case with monolithic
connections in the abutments and central pier. (a and b) Loading
surfaces on thestructure to model the wave transition. (c and d)
Snapshots of the deformed state of the structure at the deck and
around the central bent atpeak displacement response.
4.3. Predominant Deformation Shape. A key issue for
theimplementation of the proposed methodology is the estima-tion of
a representative, predominant deformation shape ofthe structure,
under the induced blast loads. For this reasonthe “predominant”
deformation shape is defined herein as theelastic deformation state
of the structural system just beforebrittle failure or yielding of
the selected, important for the sta-bility, structural components.
Considering the severity of theinduced loads and the corresponding
instant brittle failures incritical structural components, the
above-mentioned defor-mation shape approximation usually provides
representativepatterns of force distribution along the structure.
Note thatlocal yielding of structural elements under blast loads is
rare,while brittle failuremodes are usually the dominant pattern
ofdamage. According to the preceding, based on the
anticipatedfailure characteristics of each critical structural
componentof the bridge (e.g., shear failure, axial force, or
momentcapacity), the elastic deformation shape is extracted from
theimplemented FE computational study. For the structural sys-tem
under examination (post-tensioned concrete bridge), thecritical
components that control the entire response are thepost-tensioned
beams as well as the central bent columns andthe dominant
deformation shape of the system is depicted inFigures 8(c) and
8(d).
4.4. Application to theModel of the Case Study and Results. Asit
has been already stated in Section 4.2, in order to minimizethe
computational cost, an elastic analysis (Figure 8(c)) wasfirst
conducted in order to determine the “predominant”deformation shape
assumed by the structure which was thenscaled to yielding in the
plastic hinge regions of the centralpier columns.The results of the
analysis are illustrated in Fig-ure 8 for the peak estimated
response.The structure is initiallyat rest as shown in Figures 8(a)
and 8(b), where the loading
surfaces to which the structure has been subdivided
areillustrated. Based on the results of the elastic analysis it
wasfound that the most critical pattern of deformation assumedby
the structure resembles the shape of the fundamentalmode for
shaking in the transverse direction (Figures 8(c) and8(d)). It was
found that this pattern occurs while the explosivewave moving from
the base pushes the entire structureupwards. In order to proceed
with the implementation of theproposed methodology and to extract
the response into thenonlinear range without excessive calculation
complexity, thesimplified nonlinear framemodel was assembled for
the anal-ysis of the structure. Actually, using the proposed
modellingapproach, it is possible to reduce the size of the
problem, soas to enable the assessment of blast effects on the
bridge whileusing simplified constitutive laws and material
properties.Based on the above, the structure was subjected to the
evalu-ated predominant deformation pattern shown in Figure
8(c)after this had been normalized to the peak value so as
torepresent a shape function (considering the shape shown inFigure
8(c), it may be said here that an analysis resemblingin principle
the standard “pushover” was attempted, withthe approach being
motivated from earthquake engineeringpractices and intended to be
used to blast events). Based onthe above defined deformation
pattern, the intensity of defor-mationwas gradually increased using
a step by step nonlinearstatic procedure. This enabled inelastic
state determinationat different intensities of displacement of the
deck whendeforming according to the pattern shown in Figure 8(c).
Inany case, the resulting total force should be equal to or
lessthan the maximum estimated applied force extracted fromthe FE
model.
Considering that the problem under investigation wasreduced into
a model comprising typical frame elements,appropriate inelastic
hinge properties were evaluated for each
-
10 Advances in Civil Engineering
structural component of the model. Based on well-knownstructural
engineering evaluations, a wide range of possibleconstitutive
behaviors (brittle or ductile) were modelledanalytically for each
existing critical component (shear andaxial compression/tension
ultimate forces, moment-axialload interaction diagrams,
moment-rotation envelopes forlumpedplastic hinges, andplastic hinge
lengths andmoment-curvature relationships for member sections where
a modedetailed idealization was needed). Objective of this
approachwas to quantify the extent of damage in terms of values
ofestablished stress and strain resultants in the critical
elementsof the bridge (Figure 9).
From the pushover analysis results it was found that plas-tic
hinges form quickly in the structure, the most immediatebeing those
at the top and the base of the two pier columns.Upon further
examination of the estimated values it wasconcluded that columns
would fail by a combination of shearand axial tension (as a result
of the upwards pressure appliedin the underside of the deck) after
flexural yielding in the endsof the members. Considering the
brittleness of the estimatedmode of failure, the anticipated damage
in the column couldbe extensive depending on the intensity of the
blast wave.Beyond that point, a sequential formation of plastic
hingesat various points in the structure is observed, owing to
thevertical component of the blast wave throughout the span.Thedeck
beamsnear the abutments and the central pier attainshear failure
over an extensive portion of the span; no flexuralfailure is
observed at mid-span suggesting that shear failureprecedes flexural
modes leading directly to nonproportionaldamage and collapse.
For the case under consideration, it is evident that, in
con-trast with ordinary seismic events, blast actions induce
exces-sive loads which are distributed based on significantly
dif-ferent patterns along the superstructure and the
substructureelements. Early axial tension and shear brittle
failures domi-nate the response of the entire system, while
flexural yieldingphenomena, which are commonly considered to be
critical inearthquake design practices, seem to be ofminor
importance.
Based on the above, it is evident that common earthquakedesign
and detailing practices do not cover at all blast actions,as the
developed mechanisms are significantly different fromthose
considered in conventional load combinations includ-ing earthquake.
Therefore, it should be underscored thatexisting infrastructure
elements which are designed basedon ordinary structural/earthquake
design practices and pro-visions are vulnerable to significant
blast loads and thatadditional measures need to be taken (active or
passiveprotection) in order to secure the structural integrity of
thosebridge systems that would be considered critical for
continu-ous functionality in an emergency situation.
5. Conclusions
A simplified modelling procedure for the fast assessmentof the
effects of blast loads on bridges is proposed. Themethodology is a
versatile tool for first-order estimation ofthe effects of blast
explosions on structures, whichwould oth-erwise require an
extensive and complicated nonlinear timehistory analysis. First the
evolving pressure wave front was
Results from inelastic analysis of bridge case study
Early failure of the column joints as a result of axial tension
and shear
(1)
(2)
(3)
Figure 9: Analysis results for a case with fixed supports:
monolithicconnections in the abutments and central pier (obtained
usingnonlinear static analysis).
describedmathematically in space and time so as to define
itsloci with the exposed structural surfaces. This enabled
defi-nition of the pressure forcing functions on the
structure.Thiswas applied using linear dynamic analysis on a
detailed elasticfinite element model of the structure in order to
identify thepredominant displacement profile experienced by the
struc-ture. Deformations were normalized to the peak response soas
to develop a shape function for the bridge structure atthe extreme
displaced response. The identified displacementpattern was then
applied on a nonlinear frame model of thestructure. Through this
modelling option, nonlinear staticanalysis was possible at a much
reduced computational costas compared to 3D solid nonlinear finite
element modellingwhich is still, today, prohibitively
time-consuming and com-putationally inaccessible when used to
conduct time historyinvestigation of the complete structure; a
reason for this islack of pertinent nonlinear cyclic brick FE
models appro-priate for modelling 3D solid reinforced concrete
structuresof realistic complexity, whereas brittle nonlinearity
causesinsurmountable convergence problems in 3D solid FE
mod-els.The framemodel was subjected to the identified
displace-ment pattern, with the intensity being increased gradually
to
-
Advances in Civil Engineering 11
the displacement levels identified by the elastic analysis.Note
that the concepts used are extended from earth-quake engineering
where performance is established from adisplacement-based pushover
analysis up to the anticipatedlevel of displacement demands.
Nonproportional damage isthe characteristic consequence of surface,
near field explo-sions such as the example considered in the
present studyfor a two-span typical highway overcrossing. It was
foundthat axial tension as well as shear failure in the columns
anddeck owing to the excessive displacements caused by the
blasthaving the intensity examined would lead to catastrophic
col-lapse of the bridge. Clearly, blast actions induce severe
loadson the structure which are distributed based on different
pat-terns along the superstructure and the substructure
elements.Considering that conventional structural and
earthquakedesign fails to provide a solid framework against blast
actions,additional measures are necessary to be implemented
(e.g.,active or passive protection of the system). Based on
thesimplicity and the reduced computational cost, the
proposedmethod could form the basis for future investigations of
blastphenomena as it is intended for a fast assessment procedureof
structures subjected to accidental explosions.
Competing Interests
The authors declare that they have no competing interests.
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