Nonlocal Formulation for Numerical Analysis of Post-Blast Behavior of RC Columns
International Journal of Concrete Structures and Materials
Nonlocal Formulation for Numerical Analysis of Post-Blast Behavior of RC Columns
Residual axial capacity from numerical analysis was widely used as a critical indicator for damage assessment of reinforced concrete (RC) columns subjected to blast loads. However, the convergence of the numerical result was generally based on the displacement response, which might not necessarily generate the correct post-blast results in case that the strain softening behavior of concrete was considered. In this paper, two widely used concrete models are adopted for post-blast analysis of a RC column under blast loading, while the calculated results show a pathological mesh size dependence even though the displacement response is converged. As a consequence, a nonlocal integral formulation is implemented in a concrete damage model to ensure mesh size independent objectivity of the local and global responses. Two numerical examples, one to a RC column with strain softening response and the other one to a RC column with post-blast response, are conducted by the nonlocal damage model, and the results indicate that both the two cases obtain objective response in the post-peak stage.
strain softening; RC column; residual axial capacity; blast loading; nonlocal damage
Reinforced concrete (RC) columns are important
loadbearing elements to prevent progressive collapse of the
reinforced concrete structures. However, in the design of the
reinforced concrete structures, the extreme loads generated
by terrorist attacks or accidental explosions were usually
ignored that made the structures particularly vulnerable
when they are subjected to these loading conditions. When a
specific blast load is applied, RC columns usually suffer
localized failure (spalling) and exhibit global deformation
due to shear and flexural bending that depend on load
characteristics and structural parameters (Crawford and
Magallanes 2011). In addition, certain residual axial load
carrying capacity may still remain, which provides the main
basis for assessing of damage level and collapse risk for
structures under blast loading (Brunesi et al. 2015; Parisi
2015; Petrone et al. 2016; Russo and Parisi 2016).
Therefore, it is quite essential to develop a more precise numerical
model for post-blast response of RC columns.
Currently, only a few experiments for post-blast residual
axial capacity of RC columns have been reported. Li et al.
(2012) obtained the residual axial capacity of limited seismic
(LS) and non-seismic (NS) RC columns under quasi-static
loads, while the blast damage is applied through the predicted
residual lateral deflection calculated by a numerical model.
Roller et al. (2013) took a deep investigation on residual axial
capacity of conventional and hardened RC columns subjected
to contact or close-range detonation loading, and the blast
damage is mainly due to material losses. With great progress
on numerical method in recent years, a lot of research work has
been carried out to study the dynamic response of RC
components and structures under blast loading (Shi et al. 2008;
Bao and Li 2010; Jayasooriya et al. 2011; Park et al. 2014;
Brunesi et al. 2015; Li et al. 2016; Lim et al. 2016). In most of
these numerical studies, a numerical model was firstly
developed and validated by the experimental results. Then,
successive parametric studies were carried out with this
validated numerical model. However, the convergence test of
post-blast response is not included in these papers, as a
consequence, once the element size changes, the accuracy of
postblast residual capacity may be compromised.
As it is well known, the stress strain relation of concrete
materials contains a descending branch, which is called
strain softening, such that the finite element results always
become mesh-dependent (Jira´sek and Bazˇant 2002). In
classical continuum mechanics, the introduction of strain
softening generally leads to erroneous results. From a
mathematical point of view, the governing partial differential
equations lose hyperbolicity in dynamics and ellipticity in
quasi-statics. Due to the lack of a length scale in the
constitutive model, the localization zone approaches to zero
volume where energy dissipation vanishes in the fracture
process since a finite amount of dissipated energy per unit
volume is defined through the strain softening law, which is
unacceptable from a physical point of view. In the numerical
sense, results show an extreme sensitivity to the spatial
discretization, and deformation localizes into an infinitely
narrow band upon mesh refinement.
This sensitivity of the mesh size on the numerical results
has attracted enormous attention of researchers in the field of
solid mechanics, and different solutions have been proposed
to deal with the ill-posed problem (Bazˇant and Oh 1983;
Pijaudier-Cabot and Bazˇant 1987; Bazˇant and Jira´sek 2002;
Armero and Ehrlich 2006). Among which, the crack band
model is practically the simplest but a crude method to avoid
the pathological sensitivity to mesh refinement by scaling of
the constitutive law. And it has been incorporated into the
constitutive model of concrete in some commercial finite
element software, e.g., LS-DYNA (Hallquist 2007).
Although this model ensures correct energy dissipation in
the localized damage band, it is only appropriate for mode-I
fracture. However, as pointed by Ozˇbolt et al. (2011), the
failure mode tends to change from mode-I to mixed modes
as the loading rate increases. In order to achieve a
thoroughly regularization, the nonlocal theory is widely adopted
in the analyses related to mesh sensitivity strain softening
problems since the pioneering work by Bazˇant
(PijaudierCabot and Bazˇant 1987). The essential characteristic of the
nonlocal model is the incorporation of a length scale, by
which the constitutive law at a material point of a continuum
depends on not only this point but also its certain
neighborhood. In fact, concrete has a complicated internal
microstructure, the change of which is crack bridging, and
can be captured by the enriched continuum model through
the characteristic length. Moreover, the embedded strong
discontinuity approach developed in recent years
successfully overcame the mesh-dependent problem by
incorporating a localized dissipative mechanism through introduction
of strong discontinuity in kinematics and defining a
softening cohesive law (Armero and Ehrlich 2006; Jukic´ et al.
2014). However, it is not straightforward to implement into
the general finite element codes such as LS-DYNA.
This paper aims to eliminate the pathological mesh size
dependent problem of post-blast residual axial capacity for
RC columns, which will result in a more precise dynamic
simulation of RC structures subjected to blast loads. Firstly,
a comprehensive analysis of the post-blast residual axial
capacity of a RC column is conducted using the currently
available concrete models in LS-DYNA that would
demonstrate the deficiencies of these models. Then, a
nonlocal integral formulation is incorporated into the Mazars
damage model (Mazars and Pijaudier-Cabot 1989) to
maintain the objectivity of post-peak responses for RC
columns. Furthermore, the applications of two numerical
examples demonstrate the feasibility of the nonlocal damage
model in dealing with the pathological mesh size dependent
responses involving concrete strain softening. It is worth
noting that the dynamic response of structural components to
blast loading strongly change with the type of explosion, and
only the type of detonation loading is considered in this
2. Assessment of Post-Blast Responses in Considering Concrete Strain Softening
RC columns typically suffer certain damages when the RC
structures are subjected to blast loading. Its subsequent
response of the structure mainly depends on the post-blast
residual axial capacity of critical RC columns. Therefore, it
is of importance to get an appropriate post-blast residual
axial capacity such that correct dynamic response of RC
structures can be generated. Generally, a high-fidelity
physics-based (HFPB) model was developed in LS-DYNA to do
some research work related to blast analysis of RC columns
as several concrete constitutive models in the material library
are available and proved to be suitable for high strain rate
and intensely nonlinear behavior of concrete (Shi et al. 2008;
Bao and Li 2010; Park et al. 2014; Li et al. 2016), especially
the Karagozian & Case concrete (KCC) model
(*MAT_072R3) and the continuous smooth cap (CSC)
model (*MAT_159) (Hallquist 2007). In this section, two
constitutive models are firstly introduced. Then, comparative
studies are carried out to investigate the influence of the
different mesh sizes on the post-blast residual axial capacity
of a RC column. Corresponding discussions are given based
on these comparative studies.
2.1 Karagozian & Case Model
As known, the volumetric and deviatoric responses of
concrete are decoupled in the KCC model, in which a user
defined equation of state (EOS) defines volumetric strain
versus pressure relationship and a movable failure surface
limits the deviatoric stress. In order to govern the evolution
of the failure surface, three fixed independent failure
surfaces termed as yield, maximum and residual are defined.
For hardening behavior, the failure surface is interpolated
between the yield and maximum failure surface based on a
damage variable, and for softening behavior, a similar
interpolation is performed between the maximum and
residual failure surfaces. The strain rate dependent behavior
is considered through a user defined dynamic increase factor
(DIF) curve (Hallquist 2007).
Three parameters are used in the KCC model to control
the damage evolution, they are b1, b2 and x for
compression damage, tensile damage and volume expansion,
respectively. It is a common sense that the numerical
results will not be objective upon mesh refinement if the
strain softening behavior of the constitutive model is not
associated with a localization limiter or characteristic
length. To eliminate the pathological mesh size dependent
phenomenon, the crack band method is incorporated into
this model by forcing the area under the stress–strain curve
to be a constant value Gf/h, where Gf is the fracture energy
and h the element characteristic size. Gf can be obtained
through experiment (Lee and Lopez 2014). However, this
method is only valid for tensile softening and becomes
invalid when the element size exceeds 250 mm or the
element size is smaller than the localization width
(Magallanes et al. 2010).
2.2 Continuous Smooth Cap Model
The CSC model is originally developed for use in roadside
safety simulations, and is also frequently used to model
concrete materials subjected to impact and blast loads. Its
yield surface combines a shear surface with a hardening
compaction surface smoothly and continuously through the
multiplicative formulation, which avoids the numerical
complexity of dealing with ‘‘corner’’ regions. As softening is
a remarkable behavior of concrete in tension and low
confinement compression, the strain-based damage formulations
are incorporated into this model, and brittle damage and
ductile damage are distinguished. In order to consider the
strain rate effect for impact and blast analysis, the
viscoplastic formulation is adopted. Details of the description of
the model can be referred to the reports (Murray et al. 2007;
2.3 Validation of the Numerical Model
In all the HFPB simulations, concrete is modeled by
8-noded solid element with single integration point. The
reinforcing steel is modeled with 2-noded beam element.
And perfect bond between concrete and reinforcement is
assumed by sharing nodes. The KCC or CSC model is used
for concrete material and an elasto-plastic material
(*MAT_PICEWISE_LINEAR_PLASTICITY) is used to
model steel. The erosion technique is implemented to
consider the severely damaged concrete elements.
In order to validate the reliability of the numerical model,
two numerical analyses of the CTEST20 column (Crawford
et al. 2012) and B40 beam (Magnusson et al. 2010a) under
blast loading are conducted and the numerical results are
compared with experimental data.
2.3.1 CTEST20 Column
The geometry and reinforcement details of the column are
shown in Fig. 1. The unconfined compressive strength of
concrete is 41.4 MPa. The yield stress and hardening
modulus of reinforcement are 475 and 751 MPa, respectively.
The convergence tests with 12.7, 25.4 and 50.8 mm mesh
sizes show that the mesh size of 25.4 mm is appropriate for
the numerical analysis.
The mid-height deflection time histories of the column are
shown in Fig. 2a, as can be observed in this figure, the
numerical model with both KCC and CSC models present
reliable peak deflection as compared with the measured value.
However, the predicted residual deflection by the numerical
model is higher than the experimental result. This might be due
to the erosion of concrete material at the two ends and the
simplified modeling of blast load by adopting
LOAD_BLAST_ENHANCED in LS-DYNA (Hallquist 2007).
Figure 2b shows the calculated final deformation state of the
column by KCC and CSC models, there is a slight difference
between the results although the same erosion criterion is used.
The most possible reasons are that the automatic generated
material parameters are adopted for CSC model such that the
dynamic increase factor (DIF) might be different from that of
KCC model. Moreover, different damage definitions are
adopted in the two models. Since strong nonlinear behavior
existed in the tested RC column under blast loading, the
numerical predictions conducted in this section could be
considered within an acceptable level.
2.3.2 B40 Beam
In the authors’ opinion, the deviations between the
simulated and tested results of CTEST20 column mainly generate
from simplified modeling of blast load. As a result, a RC beam
(type B40) tested by Magnusson et al. (2010a) that also
presented the accurate blast load-time history, is simulated in this
section. Some primary geometry parameters of the beam are
shown in Fig. 3, and more detailed information can be found
in the related references (Magnusson et al. 2010a, b). In the
finite element model, the concrete compressive strength and
reinforcement yield strength are defined as 43 and 604 MPa,
respectively. And the blast load with reflecting pressure of
1250 kPa and impulse density of 6.38 kPa s is adopted in this
study as reported by Magnusson et al. (2010a).
The dynamic response of the RC beam is simulated by
both KCC and CSC models. As can be observed from
Fig. 4, the calculated results of mid-span displacement time
histories agree well with the experimental data, and the
ultimate deflections are all about 17.5 mm in the experiment
and our simulations. Moreover, the calculated residual
Fig. 1 CTEST20 column configuration and reinforcement details.
14 inch (355.6 mm)
#3 (9.5 mm)@324 mm
Fig. 3 Configuration and reinforcement details of B40 beam.
displacement is about 16.5 mm in both cases due to shear
failure of the RC beam, which is obviously shown by the
calculated damage distribution and final deformation mode
in Fig. 5. And the shear failure mode is just the same as that
reported by Magnusson et al. (2010a).
2.4 Post-Blast Response Analysis of A RC
The KCC and CSC models have been frequently used in
dynamic response analysis of RC columns under blast
loading, and it is generally considered to be converged when
the displacement response does not vary a lot upon mesh
refinement. However, there is limited information on the
studies that considered the mesh size influence on the
postblast residual axial capacity to date.
A specific RC column is taken as an example here to
calculate the post-blast residual axial capacity by varying the
element sizes. Figure 6 shows the scheme of the RC column,
the clear height of which is 2750 mm. And the section width
and height are both 300 mm with a concrete cover of
40 mm. A total of four U20 mm longitudinal rebar are
uniformly distributed along the perimeter of the section, and
the stirrup reinforcement with a diameter of 10 mm is spaced
200 mm along the length. The uniaxial compressive strength
for the concrete is 30 MPa. The yield and ultimate strength
of steel reinforcement are 335 and 480 MPa, respectively.
The loading scheme consists of three stages, as shown in
Fig. 6. In stage 1, an initial axial load of 821 kN is applied
on the top end to produce an initial stress state in the column.
After a steady state is reached, an idealized blast load with
peak overpressure of 10 MPa and duration time of 1 ms is
applied to the column along the full height and certain
damages will be generated till the dynamic response is
damped out in stage 2. Finally, the displacement control
loading scheme is used to calculate the post-blast residual
axial capacity in stage 3.
Four different element sizes, i.e., 50, 25, 12.5 and
6.25 mm, are used in this study. Both KCC and CSC models
are adopted for the concrete material. And the piecewise
linear isotropic plasticity model is employed to model the
steel reinforcement. The mid-span transverse displacement
time history curves are described in Fig. 7 for different mesh
sizes. As can be observed, for the results calculated by both
KCC and CSC models, the 50 mm element size seems a
little large for the results to achieve convergence, while the
displacement time history curves are almost identical for
cases with element sizes of 25, 12.5 and 6.25 mm before
loading stage 3. In stage 3, the displacement curves diverge a
lot, which indicates different post-blast response of the RC
column for analysis results with different mesh sizes. The
post-blast residual axial capacity can be easily obtained from
the axial load time history curves as shown in Fig. 8. It can
be observed that the residual axial capacity has a
pathological dependence on the mesh size. This is mainly
because that the concrete material models are not equipped
with a complete regularization formulation, hence the
damages generated in the local areas still have a mesh dependent
character even if the constant fracture energy method is
introduced. Figure 9 shows the damage distribution of the
RC column at the end of stage 2, the local damages
generated during blast loading are obviously very different, and
leading to different residual axial capacity. It is no doubt that
the different residual axial capacity will affect the dynamic
response of RC structures under blast loading. Therefore, it
is essential to take measures eliminating the mesh dependent
issue in order to ensure reliable analysis results for RC
structures under blast loading.
3. Proposed Numerical Model
As described in Sect. 2, the KCC and CSC models with
the constant fracture energy method still fail to obtain
objective results of post-blast residual axial capacity. To
solve this issue, an enrichment to the Mazars damage model
is proposed by means of introduction of a length scale, in
which the local equivalent strain is replaced by the nonlocal
equivalent strain through a nonlocal formulation. And the
enriched Mazars damage model is incorporated into the
general finite element code LS-DYNA so that the numerical
study of post-blast response for RC columns and even RC
structures subjected to blast loads can be quite convenient.
Because the three dimensional concrete constitutive model
with nonlocal formulation is computational demanding even
for RC components, the uniaxial form of the Mazars damage
model is adopted for the computations of the
mesh-dependent problem in this study.
3.1 Description of Mazars Damage Model
Damage mechanics has been proved to be efficient for
modelling concrete behavior in recent years (Heo and
Fig. 7 Mid-span transverse displacement time histories of different mesh sizes a KCC, b CSC.
Fig. 8 Axial force time histories of different mesh sizes a KCC, b CSC.
Fig. 9 Damage distribution of the RC column after blast loading (CSC model).
Kunnath 2013; Ren et al. 2015). Mazars damage model
belongs to one of the first models based on this framework
and has been widely used in simulating the damage behavior
of concrete (Mazars and Pijaudier-Cabot 1989). It is
described in Fig. 10 by the relation
where r is the stress tensor, e is the strain tensor, C is the
stiffness matrix of material and D is the scalar damage which
varies from 0 to 1 with 0 represents the intact material and 1
for fully failure of the material. And the damage criterion
inspired from the St. Venant criterion of maximum principal
strain is adopted, its equation is:
Fig. 10 Scheme of Mazars damage model.
where j represents the current damage value which is the
maximum value of equivalent strain ever reached in time
history, and the initial value of j is j0, which indicates the
initiation of damage.
As extensions play a significant role in the damage
behavior of concrete, Mazars (1986) suggested the
equivalent strain evaluating the local intensity of extensions, which
is defined as:
where ei are the principal strains and h i is the Macaulay
In order to describe the different behaviors of tension and
compression, two separated damage variables denoted as Dt
and Dc are defined, and the total scalar damage D is
calculated from the weighted combination of them, namely:
where, Ai and Bi are characteristic parameters of the concrete
material identified from test data. In Eq. (4), the weighting
coefficients at and ac are functions of the strain state (at ¼ 0
for pure compression and ac ¼ 0 for pure tension, at and ac
vary from 0 to 1 in other cases).
3.2 Nonlocal Formulation
The post-blast residual axial capacity of RC columns is
closely related to the blast damages of concrete material. As
stated in Sect. 2, the blast damages show pathological
sensitivity to the mesh size, which directly results in the
meshwhere, R is the characteristic length that limits the interaction
zone, which makes the function exactly zero for s R.
3.3 Numerical Implementation
LS-DYNA is a general finite element code which provides
abundant interfaces for users developing their own
subroutines to achieve research objectives (Hallquist 2007). In this
study, before invoking the local constitutive model, the local
equivalent strain must be replaced by the nonlocal
counterpart of the current time step. The scheme of the beam
element formulation is shown in Fig. 11, only one integration
point is used for one element to save computational time in
explicit analysis. The values of the nonlocal equivalent strain
must be traced at every individual integration point, whose
coordinates are denoted as xk (k ¼ 1; 2; . . .; N; N represents
the total number of integration points). The numerical
algorithm can be determined as
dependent residual axial capacity. The integral type nonlocal
model provides a fully regularization for the strain softening
problem through introducing a characteristic length in the
local constitutive laws (Pijaudier-Cabot and Bazˇant 1987),
which is achieved by weighted spatial averaging of a
suitable state variable. Generally, state variables such as the
damage energy release rate, damage value, total strain and
equivalent strain are chosen for applying the nonlocal
formulation. In this paper, the equivalent strain in Mazars
damage model is taken as the nonlocal state variable, and
with its nonlocal counterpart replacing eeq in Eq. (5), which
is defined as
where e and ~e are the nonlocal and local equivalent strains,
respectively. V is the spatial domain of interest, which is
decided by the characteristic length. And aðx; nÞ is the
nonlocal weight function that defines the interaction
coefficients between point xR and n. The weight function is
typically normalized by V aðx; nÞ dn ¼ 1 for all x, then the
normalized weight function can be obtained:
where a0ðsÞ is a function which depends on the distance s
between point x and n. In this study, the Bell-type weight
function is adopted (Jira´sek and Bazant 2002):
Fig. 11 Integration scheme and weighting function along the element.
where wl is the integration weight of the integration point
number l, and the value of which is 1 as single integration
point is used for all the beam elements. It is worth noting that
akl would be zero when the distance between k and l is larger
than the interaction radius R (characteristic length). It could
be calculated only once before the first step and then stored
as it does not vary during the simulation.
Figure 12 shows the numerical procedure of the nonlocal
formulation. The transformation matrix from local to
nonlocal equivalent strain is defined as b, in which bkl ¼ wlakl.
Then, before invoking the local constitutive model at every
new time step, the local equivalent strain is transformed to
the nonlocal one, and stresses of each integration point are
updated, the corresponding history variables are stored for
4. Numerical Examples
In this section, the numerical model of RC columns is
developed using Hughes-Liu beam element (Hallquist
2007). The cross section is discretized into concrete and
reinforcing steel fibers, and perfect bond between concrete
and steel is assumed. The severe damage of concrete or steel
Get coordinates of nodes
Get all the node and element ID
Compute matrix β
Update and save history variables
can be considerer by the definition of ‘‘failure strain’’ in the
4.1 Softening Response of A Tested RC Column
One of the tests on RC columns conducted by Tanaka and
Park (Tanaka 1990), labeled specimen 7, is analyzed in this
example due to the considerable axial load applied, which
makes the overall behavior entering post-peak region. The
geometry and section details are shown in Fig. 13. The
unconfined and confined compressive strength of the
concrete are fc0 ¼ 32 MPa and fc ¼ 39 MPa, respectively. A
bilinear stress–strain relationship is used for the
reinforcement with yield strength of 510 MPa and 1% strain
In order to have a deep insight into the influence of
concrete strain softening behavior on post-peak response of the
RC column with mesh refinement, the analysis is carried out
with 10, 20 and 40 elements and both the local and nonlocal
damage models are adopted for concrete. The main material
parameters for concrete are Ac ¼ 0:501, Bc ¼ 707:1, At ¼
0:0005 and R = 1100. The displacement control scheme is
applied on the top node of the column. The computed force–
displacement curves are shown in Fig. 14. As can be seen in
the figure, the post-peak stage of the curves become more
and more brittle as the number of elements increases for the
local damage model. This is mainly because that the inelastic
deformation localizes in one element, resulting in the
increasing strains in the extreme element as the column
enters the post-peak stage. However, the post-peak results
computed by the nonlocal damage model are almost the
same with mesh refinement, and match well with the
envelope of the cyclic response from the experiment.
Fig. 12 Numerical procedure of the nonlocal formulation.
Fig. 13 Configuration of the tested RC column.
Fig. 14 Comparisons of the load–displacement responses
for the RC bridge pier.
Fig. 16 Comparisons of the axial force time history in
different characteristic length.
4.2 Recalculation of the RC Column in Sect. 2.4
In Sect. 2.4, the post-blast residual axial capacity of the
RC column is pathologically sensitive to the mesh size as the
constant fracture energy method used in the concrete
constitutive models fails to capture objective damage values
under blast loading. In this section, the RC column is used as
an example to demonstrate the feasibility of the nonlocal
damage model in calculating the post-blast residual axial
capacity. Finite element models with 20, 40 and 80 elements
are analyzed with the same load in Sect. 2.4. The main
material parameters for concrete are Ac ¼ 1:5154,
Bc ¼ 1767:77, At ¼ 0:1 and R = 600. And the strain rate
effect of concrete is accounted by the dynamic increase
factor (DIF) calculated from Malvar (Malvar and Ross
The computed axial force versus time history curves are
presented in Fig. 15. It is obvious that the curve calculated
by the finite element model with 20 elements shows a minor
difference with the rest two curves, which means that the
finite element model with 20 elements does not guarantee
the convergence of the numerical results. This is consistent
with the consensus that a sufficiently fine mesh is generally
required for nonlocal model. The results computed from the
finite element models with 40 and 80 elements are almost the
Fig. 15 Comparisons of the axial force time history.
same, indicating that the proposed numerical model can not
only capture identical damage values during blast loading
but also ensure objective post-blast residual axial capacity of
the RC column.
Furthermore, the finite element model with 40 elements is
herein taken for a series of calculations by varying the
characteristic length R in the nonlocal model. According to
Bazˇant (Pijaudier-Cabot and Bazˇant 1987), the material
characteristic length cannot be less than the beam depth h,
which is 300 mm in this example. As can be observed in
Fig. 16, when the characteristic length R equals to 200 mm,
the RC column loses its loading capacity under blast
loading. And it is obvious that adopting a smaller characteristic
length will lead to underestimate the dissipated energy
within the damage zones as well as the post-blast residual
axial capacity. The proper calibration of the nonlocal internal
length is not a trivial task. Although this quantity is usually
related to the theoretical width of the fracture process zone
(material characteristic length), its actual definition is not
completely clear. In this study, the simulation results seem to
match the experiment well when twice of the column height
is used for R. However, some specifically-designated
experiments are still needed to calibrate this important
parameter in the future.
Although the KCC and CSC models are equipped with the
constant fracture energy method to consider strain softening
behavior of concrete, the calculated post-blast residual axial
capacity of a RC column still shows strong sensitivity to
spatial discretization because the constant fracture energy
method in the concrete constitutive models fails to capture
objective damage values, which may significantly affect the
post-blast residual axial capacity of RC columns. As a
consequence, the nonlocal formulation is incorporated into
Mazars damage model to overcome the problem. The
validity of the nonlocal model is shown through two
numerical examples, one to a tested RC column with strain
softening response and the other to a RC column with
postblast response. And both the two cases obtain objective
results with mesh refinement. It is worth mentioning that the
characteristic length R in the nonlocal model plays a
dominant role in accounting for the microstructure of the concrete
material, which could significantly affect the numerical
results and needs to be calibrated by experiment in the
future. And the presented numerical model should be further
validated against other experimental results, exploring other
types and magnitudes of blast loading. Nevertheless, it is still
potential for analysis of RC structures when softening or
post-blast response is expected.
The authors gratefully acknowledge the financial support of
the National Natural Science Foundation of China under
Grant No. 51238007 and the National Key Research and
Development Program of China under grant number
2016YFC0701105 for carrying out this research
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