Blast and Impact Analyses of RC Beams Considering Bond-Slip Effect and Loading History of Constituent Materials
International Journal of Concrete Structures and Materials
Blast and Impact Analyses of RC Beams Considering Bond-Slip Effect and Loading History of Constituent Materials
MinJoo Lee 0
Hyo-Gyoung Kwak 0
0 Department of Civil and Environmental Engineering , KAIST, Daejeon 34141 , Republic of Korea
An improved numerical model that can simulate the nonlinear behavior of reinforced concrete beams subjected to blast and impact loadings is introduced in this paper. The layered section approach is based in the formulation, and the dynamic material behaviors of concrete and steel are defined with the use of the dynamic increase factor. Unlike the classical layered section approach that usually gives conservative structural responses because of no consideration of the bond-slip effect, the introduced numerical model takes into account the bond-slip between reinforcing steel and surrounding concrete by changing the bending stiffness EI of elements placed within the plastic hinge length. Since the bond-slip developed after yielding of reinforcing steel is dominant and accompanies fixed-end rotation, the equivalent bending stiffness to be used in the critical region can be evaluated on the basis of the compatibility condition. In advance, the consideration of the unloading and reloading histories of reinforcing steel and concrete makes it possible to exactly trace the structural behavior even after reaching the maximum structural response. Finally, correlation studies between analytical results and experimental data are conducted to establish the validity of the introduced numerical model, and the obtained results show that it is important to consider the bond-slip effect and the loading history of constituent materials.
blast; impact; layered section approach; high strain rate; bond-slip; loading history of constituent material
Due to an increase of accidental explosions and terrorist
acts of bombing worldwide, a lot of researches to reserve the
safety of structures under blast and impact loadings, which
were originally limited to military facilities, have been
conducted with considerable attention to civil structures
(Kwak and Gang 2015; Lin et al. 2014; Luccioni et al.
. Since explosions usually cause severe damage to
structures with loss of human lives, blast protection of
structures is strongly required and can be achieved through
accurate prediction of structural responses. In particular,
recent increases in the size and height of structures have
accelerated the need to secure structural resistance to blast
and impact loadings.
Reinforced concrete (RC) structures subjected to blast and
impact loadings exhibit remarkably different structural
behavior from that observed under a quasi-static loading
condition due to the change in the material properties of
concrete and reinforcing steel at a high strain rate condition
(Carta and Stochino 2013; Qu et al. 2016; Valipour et al.
. Thus, the accurate prediction of structural behaviors
in RC structures subjected to blast or impact loadings will be
possible through an exact implementation of the high strain
rate deformations in concrete and reinforcing steel that occur
during a short period of time and the nonlinearity such as the
bond-slip effect that dominantly affects the resisting capacity
of RC structures. Furthermore, consideration of the loading
history of constituent materials, which is expressed through
the description of loading, unloading, and reloading, is also
required to precisely trace the structural response even after
the application of blast and impact loadings.
In this context, many studies, from experiments to identify
the strain rate dependent material properties
(Cadoni et al.
2009; Cusatis 2011)
to numerical analyses of RC structures
under impact loading
(Fujikake et al. 2009; Tachibana et al.
, have been conducted. Nevertheless, it is also true that
few experimental results for RC structural members
subjected to blast and impact loadings can be found in the
literature due not only to national security reasons but also to
the fact that experiments are costly, time consuming and
difficult to carry out. To overcome these difficulties in
dynamic experiments, numerous analytical approaches from
a simple approach such as a single degree of freedom model
to a rigorous finite element analysis using
numerous solid elements
(Chen et al. 2012; Jones et al.
2009; Ozˇbolt and Sharma 2011)
have been proposed to
verify the structural responses of RC structures. The
obtained research results have been used in developing
design codes such as the CEB-FIP model code
and ACI 318-08
and ACI 2008)
, and also have been implemented into many
commercialized programs including LS-DYNA
(Hibbitt and Karlsson and Sorenson
to be used in tracing the nonlinear response of RC
structures subjected to blast and impact loadings.
Nevertheless, most numerical methods still accompany
some problems when applied to RC beams. The use of solid
elements gives mesh-dependent numerical results
and Gang 2015)
and necessitates choosing one of the three
dimensional failure criteria for concrete even though these
criteria cannot simulate the concrete cracking behavior
(Tu and Lu 2009; Wu et al. 2012)
. A single
degree of freedom (SDOF) model also has some limitations
in simulating the nonlinear response of RC beams, even
though it has been popularly adopted in design practice
because of ease-of-use. Since the SDOF method not only
adopts a lot of approximations but also cannot exactly take
into account the nonlinear behavior in a RC section induced
from the cracking of concrete and yielding of reinforcing
steel, the use of the SDOF method may be inappropriate
when a precise evaluation of the structural response is
The use of beam elements is also not exceptional. The
beam model cannot consider the bond-slip effect between
reinforcing steel and surrounding concrete because the strain
compatibility has been based upon the perfect bond
(Bicanic et al. 2011)
. This restriction makes it
more difficult to take into account the fixed-end rotation
which occurs after yielding of the main reinforcement. The
nonlinear analyses of RC beams consequently may give
different results by ignoring the bond-slip effect, and in
advance, the accuracy of the simulation results may not be
guaranteed. Nevertheless, the bond-slip effect is still
excluded in the numerical formulation of RC beams subjected to
blast and impact loadings
(Yao et al. 2016)
To address these limitations in the numerical analyses of
RC beams, this paper presents a numerical model developed
to consider the bond-slip effect in a beam element. The
layered section method is based in the formulation, and the
dynamic material behaviors of concrete and steel are defined
with the use of the dynamic increase factor (DIF). The very
different feature of the introduced numerical model is the
implementation of the bond-slip effect, which cannot be
considered in the classical layered section approach because
of the difficulty in defining the relative slip along the
reinforcing steel. In order to take into account the influence of
bond-slip, the proposed numerical model suggests using the
equivalent bending stiffness EIeq within the plastic hinge
length, upon the assumption that a large portion of the
bondslip will be concentrated, due to the anchorage slip, within
the plastic hinge length where the yielding of reinforcing
steel is subjected. Based on the compatibility condition, the
equivalent bending stiffness EIeq to be used in the critical
region is evaluated.
In advance, the consideration of the loading histories of
reinforcing steel and concrete makes it possible to exactly
trace the structural behavior even after reaching the
maximum structural response. Since the residual structural
response beyond the maximum structural response is due to
the unloading and reloading behavior of constituent
materials, its exactness will be directly related to the
consideration of the loading history for reinforcing steel and concrete.
Finally, the validity of the proposed numerical model is
confirmed by the comparison of analytical predictions with
experimental data. Furthermore, the effects of bond-slip and
loading history of the constituent materials are discussed
through parametric studies, and the obtained results show the
importance of considering both effects in the nonlinear
dynamic behavior of RC beams subjected to blast and
2. Material Properties
Since the equilibrium equation of a RC beam element,
which is divided into imaginary layers to represent the
different material properties, is constructed on the basis of the
constitutive relationships in a layer, the behavior of RC
beams subjected to external loads is highly dependent on the
used material model and the magnitude of stress. Among the
various available mathematical models currently used in the
numerical analyses of RC structures, the monotonic
envelope curve proposed by
Kent and Park (1971)
Scott et al. (1982)
is adopted in this paper
because of its simplicity and computational efficiency (see
Fig. 1a). More details of the stress–strain relation to define
the envelope curve can be found elsewhere
For the tensile region, concrete is linearly elastic up to the
tensile strength. Beyond that, the tensile stress decreases
along a linear softening branch with increasing principal
tensile strain. It is assumed that ultimate failure take places
by cracking, when the strain exceeds the value of et0 where b
is the length of elements and Gf denotes the fracture
toughness of concrete, as shown in Fig. 1b
After defining the monotonic envelope curve, it is
necessary to exactly define the unloading–reloading behavior in
order to describe the hysteretic response of concrete.
However, the monotonic envelope curves are obtained on the
basis of experimental studies, whereas the definition of an
accurate cyclic stress–strain relation is very limited since it is
difficult to carry out experiments for concrete subjected to
cyclic loadings. Only a few of cyclic constitutive models
have been proposed through experimental results
(Konstantinidis et al. 2007)
Since the exact definition of the unloading–reloading paths
in cracked concrete is complex, while these nonlinear paths
have a minor effect in describing the hysteretic behavior of
RC structures, simplified relations are usually adopted to
define the cyclic stress–strain relation of concrete. This paper
also adopts a straight unloading–reloading relation to be
used in the compression region [see Eq. (1)], as proposed by
Karsan and Jirsa (1969)
and later extended by
Taucer et al.
to remove unreasonable behavior under high
compressive strain conditions, because it has been popularly
used in the dynamic analyses of RC structures
et al. 2011; Valipour et al. 2009)
where er is the strain at which unloading starts to a point ep
on the strain axis and eo is the strain corresponding to the
maximum stress in compression. On the other hand, the
unloading–reloading paths in the tension region are assumed
to always pass the origin regardless of the loading history
because their application is limited to RC beams in which
the bending behavior is dominant. However, it is also true
that the used unloading–reloading path does not account for
the cyclic damage of concrete, but the importance of this
effect on the hysteretic behavior of RC beams is beyond the
scope of this paper.
It is shown that the dynamic compressive and tensile
strengths of concrete under rapid loading increase
significantly due to the lateral inertia confinement effect and the
change of the crack pattern
(Yan and Lin 2006)
experimental studies provide a more detailed description of
strain rate effects on concrete
many mathematical models have also been proposed, which
express an increase in strength and critical strain depending
on the strain rate
(Hao et al. 2012; Shkolnik 2008)
. In spite
of many accurate numerical models for the consideration of
the strain rate effect, however, the simple relations
introduced by Saatcioglu et al. (2011) are adopted in this paper
for computational convenience.
This model introduced a dynamic increase factor (DIF) to
take into account the strain rate effect, and the compressive
and tensile strength under a high strain rate condition can be
determined by multiplying the DIF corresponding to the
developed strain rate. Nevertheless, the strain at peak stress
and the shape of the descending branch were assumed to be
constant regardless of the change in the strain rate (see
Fig. 1). This model requires only the strain rate e_ to compute
the dynamic strength increase in concrete and can be
expressed as follows
(Saatcioglu et al. 2011)
DIF ¼ 0:03 ln e_ þ 1:30 1:0 for e_\30 s 1, and DIF ¼
0:55 ln e_ 0:47 for e_ 30 s 1. The same equations are used
in compression and tension for the computational
convenience because DIF values for compression and tension do
not show a large difference in a structure subjected to
general blast loading which accompanies the relatively small
A bilinear stress–stain relation with the yield strength fy,
which assumes a linear elastic and linear strain hardening
behavior, is usually used for reinforcing steel. On the other
hand, the yield stress of the reinforcing bar surrounded by
concrete needs to be reduced to fn, as shown in Fig. 2. When
the steel stress at the cracked section reaches the yield
strength of the bare bar, the average steel stress at a cracked
element still will be less than the yield strength, because of
the tension stiffening effect in concrete. The concrete matrix
located between cracks is still partially capable of resisting
tensile forces, owing to the bond between the concrete and
reinforcement. Thus, in the analysis of RC beams under
cyclic loading which accompanies relatively large
deformation, the use of the average stress–strain relation is
DIF × f n
DIF × f n
(Belarbi and Hsu 1994)
, and this paper adopts the
linearized average stress–strain relation proposed by
and Hsu (1994)
as the revised monotonic envelope curve of
rs ¼ Es
rs ¼ fy ð0:91
0:02 þ 0:25A es
where ey and fy denote the yield strain and stress of a bare
steel bar, and es and rs are the average strain and stress,
respectively. The average stress rs is a linear function of the
parameter A ¼ ðft=fyÞ1:5=q and is limited by the boundary
strain en ¼ eyð0:93 2AÞ for the yielding of steel, where q
represents the percentage of the steel ratio and must be
greater than 0.5%. More details of the average stress–strain
relation of steel can be found elsewhere
(Belarbi and Hsu
The strain rate effect is considered by introducing a DIF
(see Fig. 2), as mentioned in connection to Fig. 1. The
relation of DIF as used by
Saatcioglu et al. (2011)
under the same assumption to define the dynamic stress–
strain relation of reinforcing steel, and the corresponding
equation is presented as follows:
DIF ¼ 0:034 ln e_ þ 1:30 1:0.
Upon the definition of the monotonic envelope for
reinforcing steel, the nonlinear model of
Menegotto and Pinto
, which was modified by
Filippou et al. (1983)
include isotropic strain hardening, was selected to define the
unloading and reloading behavior because of its advantages
such as easy estimation of parameters through comparison
with experimental data and good representation of the
Bauschinger effect. The unloading–reloading path can be
defined as follows:
r ¼ be þ ð1 bÞe
ð1 þ e RÞ1=R
where r ¼ ðr rrÞ=ðro rrÞ, e ¼ ðe erÞ=ðeo erÞ, ro
and eo are the stress and strain at the point where the two
(ε r ,σ r )2
(ε o ,σ o )2
A(ε o ,σ o )1
B(ε r ,σ r )1
asymptotes of the branch under consideration meet (see
point A in Fig. 3); similarly, rr and er are the stress and
strain at the point where the last strain reversal with stress of
equal sign took place (see point B in Fig. 3); b is the
strainhardening ratio between slope ES1 and ES2, and R is a
parameter that influences the shape of the transition curve
and represents the Bauschinger effect. More details related to
this model can be found elsewhere
(Kwak and Kim 2006)
3. Consideration of Bond-Slip Effect
The perfect bond assumption usually adopted in the
analysis of RC beams is reasonable only in uncracked
regions where bond stress transferred along the interface
between reinforcing steel and surrounding concrete is
(Monti and Spacone 2000)
. The influence of
bond-slip, however, is particularly noticeable in a cracked
region, and the bond-slip will be remarkably enlarged with
the yielding of reinforcing steel
(Kwak and Kim 2010)
Therefore, consideration of the bond-slip effect is required to
simulate the structural behavior more exactly.
In this regard, many studies have been conducted to
consider this effect
(Oliveira et al. 2008; Santos and
, and the bond-slip models such as the bond-link
element and the bond-zone element have been introduced to
take into account the bond-slip effect (Lowes et al. 2004). In
these models, the relative slip between concrete and
reinforcement is evaluated by using a double node. However, in
a beam element defined by both end nodes along the length
direction, it is impossible to use the double node at each end
node. To address this limitation in adopting the bond model,
a numerical algorithm that includes the bond-slip effect is
proposed in this paper.
Since the critical region where the bending moment is
larger than the yielding moment is usually located in the
vicinity of the beam mid-span or both clamped ends of the
beam, the bond-slip may be concentrated in this region. In
particular, cracking in this critical region accompanies
fixedend rotation hfe in a RC beam, which has been induced from
slippage of the main reinforcing steel (d in Fig. 4) and
cannot be simulated by any mechanical model (see Fig. 4).
In advance, this rigid body deformation may increase with
an increase of the deformation, which is about 50% of the
total deformation. Accordingly, the fixed-end rotation, which
is induced by the slippage of the main reinforcement at the
critical region, needs to be considered.
A half of a simply support RC beam can be considered as
a free body diagram, as shown in Fig. 5a, because the critical
region will be placed at the mid-span, where Lp is the plastic
hinge length where the plastic deformation is concentrated
and EIeq represents the reduced bending stiffness caused by
the concentrated bond-slip. For plastic hinge length, the
relatively simple equation of Lp ¼ xh proposed by
and Sheikkh (1997)
is used, where x is an experimentally
determined parameter ranging from 0.9 to 1.0 and h is the
section depth. As shown in Fig. 5a, if a point load P is
applied at the mid-span of an RC beam, the maximum
deflection D1 ¼ P ðEIeqL13 þ EILpð3LpL1 þ
3L21ÞÞ=ð3EIeqEI Þ can be obtained by the moment area
In addition, the beam can also be idealized by using the
equivalent rotational stiffness Kh, as shown in Fig. 5b,
because additional rigid body rotation, which causes a
reduction of the bending stiffness, will be accompanied by
slippage of the main reinforcement and can be simulated by
introducing the end rotational stiffness. In this case, Kh can
be obtained by the ratio of the moment to the fixed-end
rotation (Kh ¼ My=hfe) and, the fixed-end rotation is
determined by the relation of hfe ¼ d=ðd cÞ, where d denotes
the bond-slip of the reinforcing steel (see Fig. 4), and d and
c are the effective depth in an RC section and the distance
from the extreme compression fiber to the neutral axis,
If the same load P acts on the beam with the rotational
stiffness Kh at the mid-span, the mid-span deflection of the
beam D2 can be evaluated in Fig. 5b as
D2 ¼ PL3=3EI þ PL2=2Kh, where the first term accounts for
the deformation due to bending and the second represents
the contribution by the rigid body rotation. Then, from the
equality between D1 and D2, the equivalent bending stiffness
EIeq can be determined as the following relation.
where b ¼ að1 a þ 1=3a2Þ and a ¼ Lp=L. The
proportional constant b is dependent on the boundary condition,
and the obtained expression for the proportional constant can
be applied to a simply supported or cantilevered beam. The
same derivation procedure can also be applied to RC beams
with other boundary conditions, and the same expression as
in Eq. (4) is obtained. The only difference is the value of b,
which has the expression of b ¼ að1 2a þ 4=3a2Þ for a
both clamped boundary condition. In advance, even with
different loading type such as a uniformly distributed load by
blast loading, the same derivation procedure can be applied,
and the obtained expression for the constant b in a simply
supported beam has the expression of
b ¼ að1 1=2a 1=3a2 þ 1=4a3Þ.
Since the bond-slip d can be evaluated on the basis of the
assumption that the crack width x caused by the bending
behavior is equivalent to two times the bond-slip (0:5x ¼ d)
at the considered position, and the crack width can directly
be evaluated by the formula introduced by
Gergely and Lutz
, the rotational stiffness Kh can be evaluated from
Kh ¼ My=hfe ¼ Myðd cÞ=d. Moreover, Eq. (4) not only
gives the equivalent bending stiffness EIeq but also takes into
account the strain rate dependent bond-slip effect indirectly,
and the modification of the bending stiffness of the elements
within the plastic hinge length will be followed.
4. Solution Procedure
To analyze RC beams subjected to blast and impact
loadings, the construction of an element stiffness is based on the
layered Timoshenko beam theory, which takes the shear
deformation into consideration, and this paper adopts the
Newmark method, in which a constant average acceleration
with Newmark coefficients of b ¼ 0:25 and c ¼ 0:5 is based.
More details related to the construction of the element stiffness
and the numerical evaluation of the dynamic response can be
(Ayoub and Filippou 1999)
In advance, to minimize the difference in numerical results
according to the finite element mesh size, as was mentioned
in a previous study
(Kwak and Gang 2015)
, all the RC
beams considered in this paper have been idealized by ten
elements through a convergence test. On the other hand, the
critical regions within the plastic hinge length are discretized
by the use of two elements to accurately estimate the plastic
deformation especially after yielding of reinforcing steel
because such a separate consideration of the critical region is
required to avoid overestimation of the ultimate resisting
capacity and underestimation of the developed lateral
5. Numerical Applications
In order to establish the validity and applicability of the
proposed model, correlation studies between analytical
results and experimental data are conducted using
displacement–time curve, in which the mid-span displacements
in specimens were measured by potentiometers attached at
mid-span. Among the numerous experimental results that are
available in the literature, seven simply supported RC beams
are investigated and discussed, because these specimens
represent typical structural behaviors according to various
effects such as the steel ratio and loading type. The first two
beams of B40_D1 and B40_D2 experimentally evaluated by
Magnusson and Hallgren (2000)
are considered to show the
influence of the bond-slip effect on the structural behavior,
and the next three beams of WE2, WE5 and WE6
experimented on by
are considered to show the
importance for considering the loading history of constituent
materials in the post-peak response of RC beams. The last
two specimens of SS3a and SS3b experimented on by
are considered to verify the exactness of the
introduced nonlinear dynamic algorithm. Especially, four
specimens of B40_D2 to WE6 in Table 1 led to the shear failure
in the experiments. However, since the main reinforcements
of these specimens were yielded when the specimens
developed the maximum displacements at mid-span, the
bond-slip effect and the loading history of constituent
materials are expected to deliver considerable influence on
the structural behavior.
The material properties of each specimen are summarized
in Table 1, and more details of the experimental setup can be
(Magnusson and Hallgren 2000; Saatci
2007; Seabold 1967)
. Moreover, three beams B40_D1,
B40_D2, and SS3a, among these specimens are also
analyzed by using the equivalent SDOF model on the basis of
the approach introduced at UFC-3-340-02
(US DoD 2008)
to compare the accuracy of the proposed numerical model.
To define the dynamic material properties of constituent
materials in the SDOF, differently from the introduced
numerical model, which considers the change of DIF
according to the strain rate, a constant value of DIF = 1.2,
which is corresponding to the minimum value among the
mainly used DIF values ranged from 1.2 to 1.4
2007; Fujikura and Bruneau 2011)
, is used, because the test
specimens considered in this paper are subjected to relatively
small blast loadings. To trace the post-peak nonlinear
behavior of RC beams after the maximum loading history,
the damping coefficient c is assumed to be about 3% of the
critical damping (C ¼ 0:03Ccr) because the experiments
considered in this paper did not give any damping value.
This value is usually considered in the nonlinear dynamic
analyses of RC structures.
The SDOF analyses conducted in this paper are based on
the transformation factors of mass (KM ), load (KL) and
stiffness (KR), where the value of each factor is determined
according to the boundary conditions, strain ranges (either
elastic or plastic) and loading types. In this paper, average
values of KM ¼ 0:41 and KL ¼ KR ¼ 0:57 were used
because the structural response may be extended to the
inelastic behavior. This modification was made according to
. The finally
determined parameters for the SDOF model conducted by
the standard procedure introduced at UFC-3-340-02
are summarized in Table 2. The average
acceleration method was used to evaluate the dynamic response
of RC structures, and more details of the transformation of
the multi-degree of freedom system into an equivalent SDOF
system as well as the description of the average acceleration
method can be found elsewhere
(Biggs 1964; Craig Jr and
Kurdila 2006; US DoD 2008)
The first two specimens are RC beams B40_D1 and B40_D2,
and the geometric configuration and the description for the blast
loading are represented in Fig. 6. The test setup is shown in
Fig. 6a, and since the beams are placed within the shock tube at
a distance of 10 m from explosive charge, the beams may be
assumed to be exposed to a planer blast wave. As the blast wave
strokes the closed end of the shock tube with the assembled
beam, the wave reflects and a uniformly distributed load
spreads across the surface of the beam
only difference between both specimens is the magnitude of the
blast loading applied to the structure, and information for the
applied blast loadings used in the experiments can be found in
Table 3. Actual blast loading obtained from the experiment is
idealized as simple triangular loading (see Fig. 6b). More
details related to the experimental setup can also be found
(Magnusson and Hallgren 2000)
. In advance, to take
into account the plastic hinge length in the finite element
idealization, the specimens were modeled with an element length
Pl peak load, td time duration, Rm maximum resistance, M mass, K stiffness.
aTime up to peak impact load.
of l ¼ 140 mm for the two elements at the mid-span, which is
smaller than the calculated plastic hinge length of
lp ¼ 0:9 160 mm ¼ 144 mm, and with an element length of
l ¼ 152:5 mm for the other eight elements.
Figures 7 and 8 present a comparison of the experimental
results with the predicted results for the mid-span deflections
with time for two specimens B40_D1 and B40_D2,
respectively. As shown in these figures, a relatively close
agreement between analyses and experiments was obtained
in predicting the maximum displacement, and the evaluated
displacement histories up to the maximum displacement are
also almost coincident with those obtained from the
experiments. This means that the introduced numerical model can
effectively be used in the numerical analyses of RC beams
subjected to blast loadings. These figures also show the
significant difference between the numerical results
depending on whether the bond-slip effect is considered, and
this difference is gradually enlarged with an increase of the
maximum displacement due to the yielding of reinforcing
Moreover, the displacement time history for both
specimens is also obtained by the SODF analysis. As shown in
Figs. 7b and 8b, the SDOF model also shows a similar
displacement history to the experimental data in the case of
B40_D1 but a large difference from the experimental data in
the case of B40_D2. This appears to be induced by the
classical SDOF model not taking into account the bond-slip
effect, which suddenly increased with the yielding of
reinforcing steel. Accordingly, if the bond-slip effect can be
implemented in the equivalent stiffness of the SDOF model,
additional improvement in the numerical results can be
On the other hand, since the blast loading for the beam
B40_D1 is not large enough to develop yielding of tensile
reinforcements, the results with and without consideration of
the loading history of the constituent materials are almost the
same even at the unloading stage (see Fig. 7a). A slight
difference in numerical results according to the consideration
of the loading history of materials can be found in beam
B40_D2, which accompanies the yielding of tensile
Fig. 7 Mid-span deflection with time for B40_D1. a
Comparison with experiment and b comparison with SDOF
Fig. 8 Mid-span deflection with time for B40_D2. a
Comparison with experiment and b comparison with SDOF
reinforcements (see Fig. 8). However, comparison of the
displacement history limited to the first unloading behavior
appears to be insufficient to exactly evaluate the influence of
considering the loading history of the constituent materials.
To supplement this limitation, the next three RC beams of
WE2, WE5, and WE6 subjected to the same uniformly
distributed blast loading were considered, and the geometric
configuration and the description for the blast loading are
represented in Fig. 9. As shown in Fig. 9a, the specimens
were tested by using the NCEL blast simulator, and a
uniformly distributed dynamic load was delivered on the top
surface of specimen through expansion of gases in the
. As shown in Fig. 9 and Table 1,
the only differences in each specimen are the slight change
in the compressive strength of concrete and the yield
strength of reinforcing steel, and the applied blast loading
was sufficient to cause the yielding of tensile reinforcements
placed in these beams. These specimens were modeled with
ten elements including two smaller elements at the mid-span
whose length is l ¼ 341 mm, which is less than the plastic
hinge length of lp ¼ 343 mm.
Figure 10 shows the obtained displacement histories with
time at the mid-span of the beams. In order to investigate the
influence of the loading history in the constituent materials on
the structural behavior, the numerical analyses were continued
for longer time duration than that measured at the experiments.
From the obtained results in Fig. 10, it is observed that the
proposed model, in which both the bond-slip effect and the
loading history of the constituent materials are considered, can
reasonably predict the initial stiffness and the maximum
displacement. In advance, the consideration of the loading history
in steel and concrete does not affect the displacement history
with time up to reaching the maximum displacement point
because the constituent materials will be on the loading phase.
However, after the maximum displacement point, the
postpeak response is dominantly affected by the loading history of
steel and concrete. In particular, since a large deformation of
RC beams is induced from the ductile behavior of reinforcing
steel after yielding, an exact prediction of the post-peak
displacement history may depend on the exactness in defining the
unloading and reloading behavior of reinforcing steel.
Moreover, the displacement history in Fig. 10 also shows
that ignoring the loading history of the constituent materials
lengthens the returning period in the displacement cycle.
Since ignoring the loading history of materials leads the
constituent materials to behavior along the loading path even
at the unloading stage after developing large deformation,
the stiffness of the structure will be underestimated
compared to the real stiffness. This underestimation of the
stiffness causes an increase of the returning period because
Fig. 9 Geometry and loading history for WE2, WE5, and WE6
. a Experimental setup and section details and
b configuration of WE2, WE5, and WE6 and blast loading.
the period is inversely proportional to the stiffness of the
structure. This result is induced by the fact that concrete and
reinforcing steel under the unloading path represent larger
stiffness, which is not substantially different from the initial
stiffness. These differences in following the unloading path
cause a large difference in the displacement history at the
post-peak structural response, as shown in Fig. 10.
The last two specimens are the same RC beams of SS3a and
SS3b subjected to different concentrated impact loadings
through a dropping mass, and the geometric configuration and
the description for the impact loading are given in Figs. 11 and
12, respectively. Beams SS3a and SS3b are subjected to a drop
weight of 211 kg and a heavier one of 600 kg at the mid-span,
respectively, and Fig. 12 shows the impact loading history
measured in experiment. Since the introduced numerical
model is based on the beam element which cannot take into
account the contact behavior induced by the interaction of the
dropping mass and the structure, the initial influence of the
dropping mass on the structure is indirectly implemented by
introducing the initial velocity in solving the nonlinear
dynamic analysis algorithm. From the conservation of
momentum law, the initial velocity of the structure v2 is
calculated by v2 ¼ m1v1=ðm1 þ m2Þ, where m1 and v1 represent
the mass and the velocity of the dropping mass, and m2 is the
total mass of beam. In advance, the velocity v1 is determined
on the basis of the energy conservation law and has a value of
v1 = 8.0 m/s. For the idealization of this beam, ten elements
including two smaller elements at the mid-span whose length
is l ¼ 360 mm, which is less than the plastic hinge length of
lp ¼ 369 mm, are used and an additional two elements with
l ¼ 940 mm are considered to idealize the overhang part.
Upon the determination of the initial velocity of beam and the
defined impact loading history in Fig. 12, a nonlinear dynamic
time history analysis was conducted.
As shown in Fig. 13, the numerical results obtained by
considering the bond-slip effect and the loading history of
the constituent materials show a good correlation with
experimental data and lead to accurate predictions compared
with those obtained when ignoring both effects. Moreover,
the comparison of the displacement history after the
postpeak response shows the importance of considering both
effects. These effects appear to be more remarkable in the
case of RC beams accompanying large displacements by the
yielding of reinforcing steel (see Fig. 13). However,
Fig. 13b shows that the obtained numerical results represent
a slight difference from the experimental data in the
displacement history. This difference seems to be caused by the
linear simplification of the impact loading history after 2 ms
(see the dashed line in Fig. 12b). Since the large deformation
usually leaves residual deformation in a structural member
while preventing the structure from returning to its original
position, an inaccurate prediction of the displacement history
in a structural member may cause a wrong evaluation of the
ultimate resisting capacity of the entire structure damaged by
the blast and impact loadings.
Besides, differently from experimental data in the beam
SS3b, which represents a rapid decrease of the returning
period from the second half of the first fluctuation in the
displacement history, the numerical model still sustains
almost the same returning period regardless of the number of
displacement cycles (see Fig. 13b). This means that the
beam model may have a limitation in simulating the local
effect induced by repetition of the crack opening and closing
after experiencing the large deformation because the
formulation of the beam element is basically based on the
continuum displacement field, which does not allow
discontinuity in the deformation.
Fig. 10 Mid-span deflection with time for WE beams. a WE2,
b WE5, and c WE6.
Fig. 11 Geometry of beams SS3a and SS3b
On the other hand, as shown in Fig. 14, the numerical
results obtained by considering only one effect represent
larger differences from the experimental data. In particular,
Fig. 14 represents the relative importance of considering the
loading history of the constituent materials in these RC
beams. After reaching the maximum deflection at the
midspan at the first loading phase, which was developed by the
yielding of steel (phase a, b in Fig. 14b), the structural
response accompanies the unloading behavior with a
decrease in the developed deflection (phase b, c in Fig. 14b).
In this stage, the absence of consideration of the loading
history of steel forces the unloading phase of steel to follow
the monotonic envelope curve, which has very small
stiffness at the yielding stage. Accordingly, because of the very
small stiffness of steel, no consideration of the loading
history shows a larger variation in the mid-span deflections, as
shown in Fig. 14. The same explanation can also be
considered for the WE series specimens subjected to blast
loadings, and the reason for an increase of the return period
will be the same as that described in the previous example.
The beam SS3a was also analyzed by the SDOF method,
and Fig. 15 compares the displacement histories obtained by
the introduced numerical model and the SDOF method with
the experimental results. The SDOF method gives very
satisfactory predictions of the displacement history. As
mentioned before, the SDOF method can effectively be used
when the structural response is relatively small before
yielding of reinforcing steel. However, as shown in Fig. 15b,
the most reasonable result was obtained when the damping
ratio was assumed as 9% of the critical damping ratio, but
this ratio is quite different from the usually used damping
ratio of 3–4% of the critical damping ratio in RC structures.
Moreover, an increase of the damping ratio causes a decrease
of the maximum displacement. This means that the use of
the SDOF method to trace the displacement history after
reaching the maximum displacement may have some
limitations because the application procedure of the SDOF
method not only contains a lot of approximation, which
makes it difficult to obtain reliable results, as was mentioned
by many researchers
(Carta and Stochino 2013; Guner
, but also has a limitation in application to multi-degree
of freedom structures composed of many structural
This paper introduces an improved layered section model
that can simulate the nonlinear dynamic analysis of RC
beams subjected to blast and impact loadings. Unlike the
classical layered section approach that usually gives
conservative structural responses because of the absence of
consideration of the bond-slip effect, the introduced
numerical model takes into account the bond-slip between
reinforcing steel and surrounding concrete by changing the
bending stiffness EI of elements placed within the plastic
hinge length, and the consideration of the bond-slip effect
leads to a remarkable improvement in the accuracy of the
numerical results. Moreover, the consideration of the loading
history in the constituent materials makes it possible to trace
the structural response after reaching the maximum
response. Through correlation studies between the numerical
results and experimental data, the following conclusions are
reached: (1) to improve the accuracy of the simulation
results in RC beams, the bond-slip effect and the loading
history of the constituent materials must be considered. In
particular, the importance of considering the bond-slip effect
needs to be emphasized; (2) the SDOF method has a
limitation in application to RC beams with large deformation
induced by the yielding of reinforcing steel; and (3) the
proposed model can be used effectively in predicting the
structural response of entire RC structures subjected to blast
and impact loadings; nevertheless, (4) it is also true that the
proposed model has a limitation in simulating the
sheardominant structural behavior.
This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korean
Government (MSIP) (No. 2017R1A5A1014883), and this
research was supported by a Construction Technology
Research Project (17SCIP-B128706-01) funded by the
Ministry of Land, Infrastructure and Transport.
This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unre
stricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original
author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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