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Simplified Damage Plasticity Model for Concrete
Article in Structural Engineering International · February 2017
DOI: 10.2749/101686616X1081
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Simplified Damage Plasticity Model for Concrete
Milad Hafezolghorani, PhD Candidate; Farzad Hejazi, Senior Lecturer; Ramin Vaghei, PhD; Mohd Saleh Bin Jaafar, Prof.;
Keyhan Karimzade, Msc, Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, Selangor, Malaysia.
Contact: farzad@upm.edu.my
DOI: 10.2749/101686616X1081
Abstract
The past several years have witnessed an increase in research on the nonlinear
analysis of the structures made from reinforced concrete. Several mathematical
models were created to analyze the behavior of concrete and the reinforcements. Factors including inelasticity, time dependence, cracking and the interactive effects between reinforcement and concrete were considered. The
crushing of the concrete in compression and the cracking of the concrete in tension are the two common failure modes of concrete. Material models were
introduced for analyzing the behavior of unconfined concrete, and a possible
constitutive model was the concrete damage plasticity (CDP) model. Due to
the complexity of the CDP theory, the procedure was simplified and a simplified concrete damage plasticity (SCDP) model was developed in this paper.
The SCDP model was further characterized in tabular forms to simulate the
behavior of unconfined concrete. The parameters of the concrete damage plasticity model, including a damage parameter, strain hardening/softening rules,
and certain other elements, were presented through the tables shown in the
paper for concrete grades B20, B30, B40 and B50. All the aspects were discussed in relation to the effective application of a finite element method in the
analysis. Finally, a simply supported prestressed beam was analyzed with
respect to four different concrete grades through the finite element program.
The results showed that the proposed model had good correlation with prior
arts and empirical formulations.
Keywords: concrete damage plasticity; concrete failure; unconfined concrete;
finite element analysis; crack; crush.
Introduction
Finite element method based analysis
of reinforced concrete structures has
significantly developed since 1970.
Researchers have tried to analyze the
behavior of concrete and have published several reports and technical
reports on this subject. However, the
behavior of concrete is complex, and
many parameters must be considered
for its analysis. Concrete is composed
of qualitatively and quantitatively different types of materials. These materials exhibit different properties in
terms of tension and compression.
The structural mechanics of concrete
structures is quite important, and concrete identification parameters including the non-linear stress–strain
relation of the concrete under imposing stress conditions and strain
Peer-reviewed by international experts and accepted for publication
by SEI Editorial Board
Paper received: April 8, 2015
Paper accepted: December 17, 2015
68
Scientific Paper
hardening/softening make the behavior of concrete more complicated.
Hence, it becomes difficult to determine damage in concrete. Constitutive
models are used for this purpose, and
an example of these models is the
concrete damage plasticity (CDP)
model. This model uses the flow theory of plasticity and damage mechanics
to
analyze
the
concrete
structures.1–4
Previous Studies
Concrete damage plasticity is widely
recognized as a precise and practical
constitutive model to simulate concrete behavior. Carol et al.5 analyzed
different combinations of plasticity
and damage. Gatuingt and PijaudierCabot,6 and Kratzig and Polling7 used
isotropic damage in their research,
and elaborated on several types of
plasticity combinations.
Models involving the development of
stress related plasticity in the effective
stress space were introduced by some
researchers.8–10 Lubliner et al.,11
Ananiev and Ozbolt,12 and Imran and
also
considered
Pantazopoulou13
models that involved the formulation
of plasticity in the nominal stress
space.
Grassel and Jirasek14 introduced different combinations of plasticity and
damage applied to the concrete failure
models. They analyzed the local
uniqueness conditions of two combinations of strain scalar damage and
stress based plasticity types. Furthermore, the triaxial damage plastic
model accounted for the failure of
concrete. A three dimensional interface model was presented by Grassl
and Rempling.15 The combination of
damage mechanics and the theory of
plasticity were used as a basis for this
model, and it enabled the researchers
to vary the ratio of the permanent
and total inelastic displacements.
Yu et al.16 introduced a modified plastic damage model. The theoretical
framework of the CDPM was used as
a basis for this model, and the
confined concrete was modeled by
ABAQUS with the conditions of nonuniform confinement. The Lubliner
yield criterion was used for the triaxial
compression stress states by Zhang
et al.17 The Lubliner criterion was
reworked, improving its limits, and
hence many stress states in engineering structures could be accounted for
by this revised version of the Lubliner
criterion. Taddei et al.18 proposed
three-dimensional finite element models for unreinforced and reinforced
walls panels based on concrete damage plasticity constitutive law. Grassel
et al.19 utilized a constitutive model
for concrete structures subjected to
multiaxial and rate-dependent loading
by combining an effective stress based
plasticity model with an isotropic
damage model based on plastic and
elastic strain measures.
Larsson et al.20 examined the laterally
loaded lime-cement columns in a
shear box and used a damage plasticity model to numerically analyze the
columns. The model accounted for
the stiffness degradation of the columns. Ming21 analyzed the basic
Structural Engineering International Nr. 1/2017
concepts and properties of the concrete damage plasticity model using
inferential reasoning and analysis.
Computational techniques were used
to determine the plastic factors for the
concrete damage in ABAQUS.
Zhang and Li22 introduced calibration
techniques for concrete, under uniform and non-uniform confinement,
based on Lubliner theory through
three dimensional simulation. Grassl
et al.23 used a combination of plasticity and damage mechanics, to create a
constitutive model that was employed
to study the failure of concrete structures. The model was designed to analyze the properties of the failure
process in concrete structures experiencing multi-axial loading. The plasticity model based on effective stress
and the plastic and elastic strain measures based damage models were used
for this purpose. Vaghei et al.24 proposed the finite element model to
develop a three dimensional version
addressing precast walls and the
connection.
Shang et al.25 studied the torque of the
RC girder. They applied the finite element analysis software ABAQUS to
concrete damaged plasticity model as a
constitutive model for the analysis of
concrete material. The outcomes from
other experiments were used to validate the results of the finite element
analysis. Tao et al.26 utilized a single,
robust finite element (FE) model to
study the bond behavior subjected to
shear. Tiwari et al.27 investigated
underground tunnels with curved alignment in the longitudinal direction subjected to blast loading. He observed
that stress, deformation and damage
responses of tunnel lining through
three dimensional finite element simulations using concrete damage plasticity
theory.
Extant research indicates that it is quite
complicated to represent reinforced
concrete behavior using concrete
damaged plasticity models as a constitutive law. This also may not be fully
understood by prospective researchers.
In the present study, the procedure of
concrete damage plasticity theory was
simplified and characterized in tabulated forms. The findings were formulated and presented in a tabular format
for four common concrete grades,
while concrete parameters could be
extended to other concrete grades.
Numerical analysis was then conducted
to investigate the SCDP model, which
was implemented in the ABAQUS
finite element software in order to easily understand the mechanical behavior
of concrete. The SCDP model was then
verified against the current concrete
damage plasticity theory and empirical
formulations.
Damage Plasticity Constitutive
Model
The isotropic damaged elasticity and
the isotropic tensile and compressive
plasticity were used in the concrete
damaged plasticity model to study the
behavior of concrete in a non-elastic
manner. The total strain tensor ε was
comprised of the elastic part εel and
the plastic part εpl.
ε = ε el + ε pl
σ = D el : ε − ε pl
σ = Del0 : ε −ε pl
ð1Þ
ð3Þ
D el = ð1 −dÞDel0
ð4Þ
ð2Þ
The nominal stress with the degraded
elastic tensor from (4) could be
rewritten as follows:
ð5Þ
σ = ð1 −dÞDel0 : ε −ε pl
The damage plasticity constitutive
model was based on the following
stress–strain relationship:
σ = ð1 −dÞ:σ ! σ = ð1 −dt Þσ t + ð1 −dc Þσ c
ð6Þ
where dt and dc were two scalar damage variables, ranging from 0 (undamaged) to 1 (fully damaged).23 The
damage model used for concrete was
based on plasticity and considered the
failure process of tensile cracking and
compressive crushing.
Isotropic hardening variables were
expressed by inelastic compression
, h and cracking strain εck, h ,
strain εin
t
c
which include the plastic hardening
strain εpl,h plus the residual strain due
to damages.
pl, h ε pl, h = εtpl, h ; ε pl = h ε pl, h , σ : ε_ pl ,
εc
ε_ = ε_ el + ε_ pl
ð7Þ
Hardening variables were used to
control the development of the yield
or failure of the surface. These variables were connected to the processes of tension and compression
loading.
Structural Engineering International Nr. 1/2017
The behavior of the concrete was
explained by the assumption that concrete damage plasticity
utilized the
yield function, f ε pl, h , σ , which represented the yield surface in effective
stress space to determine the states of
damage or failure.11
In the concrete damage plasticity
model, the flow rule was defined as
follows:
_
ε_ pl = λ:
∂Gðσ Þ
∂σ
ð8Þ
In the concrete damage plasticity
model, the flow rule was non-associated. This
meant
that the yield function f ε pl, h , σ
and the plastic
potential gp did not coincide, and,
therefore, the direction of the plastic
flow ∂G∂σðσ Þ was not normal to the yield
surface. The plastic potential was
defined in effective stress space.
However, in this study, due to the
complex degradation mechanism of
the uniaxial cyclic behavior of concrete (opening and closing of formed
micro-cracks), the uniaxial response
of concrete was investigated.
Figure 1 indicated that the uniaxial
compressive and tensile response of
concrete was assumed to be influenced by damaged plasticity, and this
assumption formed the basis of the
model. The uniaxial compressive and
tensile responses of concrete with
respect to the concrete damage plasticity model subjected to compression
and tension load were given by:
pl, h
ð9Þ
σ t = ð1 − dt ÞE0 εt − εt
, h
σ c = ð1 − dc ÞE0 εc − εpl
ð10Þ
c
Given the nominal uniaxial stress, the
effective uniaxial stress σ t and σ c were
derived as follows:
σt
pl, h
ð11Þ
= E 0 ε t − εt
σt =
ð1 −dt Þ
σc
, h
= E0 εc − εpl
σc =
ð12Þ
c
ð1 −dc Þ
where compressive strain εc equalled
, h + εel , and tensile strain ε equalεpl
t
c
c
, h + εel .
led εpl
t
t
Simplified Concrete Damage
Plasticity
The values of the hardening and softening variables were used for the
Scientific Paper
69
(a)
(b)
c
c
t
B
cu
t0
cu
c0
E0
(1 – dt) E0
(1 – dc) E0
0.5cu
C
0.2cu
E0
A
E0
E0
"c
in,h
"c
"
"
pl,h
c
el
c
"t
ck,h
"t
"tel
pl,h
t
"
F ig . 1: Response of concrete to a uniaxial loading condition: (a) Compression, (b)
Tension.11 (Units: [–])
determination of the cracking and
crushing trends, respectively. They
were responsible for the loss of the
elastic stiffness and the development of
the yield surface. The damage states in
compression and tension were characterized independently by two hardening variables. These were indicated by
εcpl, h and εtpl, h , which referred to
equivalent plastic strains in tension
and compression, respectively.
where σ c and εc were nominal
compressive stress and strain, respectively, and σ cu and ε’c were ultimate
compressive strength and the strain of
In concrete damage plasticity models,
the plastic hardening strain in compression εcpl, h played a key role in
finding the relation between the damage parameters and the compressive
strength of concrete (see Fig. 1a) as
follows:
σ c = ð1 −dc ÞE0 εc − εcpl, h
ð13Þ
8
σc
in, h
>
< εc = εc − E
0
σc
1
>
pl
,
h
: εc = εc −
E0 1 − dc
ð14Þ
dc σ c
ð1 −dc Þ E0
ð15Þ
εcpl, h = εcin, h −
Generally, uniaxial compressive behavior could be characterized by either
experimental tests or existing constitutive models, such as those proposed by
Hognestad28 and Kent et al.29 for
unconfined concrete. However, the
present study employed the Kent and
Park parabolic constitutive model for
unconfined concrete, which was
expressed by the following equation:
" #
εc
εc 2
ð16Þ
σ c = σ cu 2 ’ − ’
εc
εc
70
Scientific Paper
"0 .5
"
"0.8
F i g. 2: Kent and Park model for confined
and unconfined concrete..29 (Units: [–])
the unconfined cylinder specimen,
respectively.
Park30 reported that ε’c equaled 0.002,
and this value was also assumed in
this study. Figure 2 indicated a parabolic increasing trend (A–B) for the
hardening stage, while a linear behavior (B–C) was observed for the
Plasticity parameters
Material’s
parameters
B20
Concrete elasticity
E (GPa)
Uniaxial Compressive Behavior
0.002
D
21.2
0.2
Concrete compressive behavior
Yield stress (MPa)
Inelastic strain
Dilation angle
31
Eccentricity
0.1
fb0/fc0
1.16
K
0.67
Viscosity parameter
0
Concrete compression damage
Damage parameter
C
Inelastic strain
10.2
0
0
0
12.8
7.73585E-05
0
7.73585E-05
15
0.000173585
0
0.000173585
16.8
0.000288679
0
0.000288679
18.2
0.000422642
0
0.000422642
19.2
0.000575472
0
0.000575472
19.8
0.00074717
0
0.00074717
20
0.000937736
0
0.000937736
19.8
0.00114717
0.01
0.00114717
19.2
0.001375472
0.04
0.001375472
18.2
0.001622642
0.09
0.001622642
16.8
0.001888679
0.16
0.001888679
15
0.002173585
0.25
0.002173585
12.8
0.002477358
0.36
0.002477358
10.2
0.0028
0.49
0.0028
7.2
0.003141509
0.64
0.003141509
0.81
0.003501887
3.8
0.003501887
Concrete tensile behavior
Yield stress (MPa)
2
0.02
Cracking strain
Concrete tension damage
Damage parameter
T
Cracking strain
0
0
0
0.000943396
0.99
0.000943396
Table 1: Material properties for concrete with SCDP model in class B20
Structural Engineering International Nr. 1/2017
Plasticity parameters
Material’s
parameters
B30
Concrete Elasticity
E (GPa)
26.6
0.2
Concrete compressive behavior
Yield stress (MPa)
Inelastic strain
Dilation angle
31
Eccentricity
0.1
fb0/fc0
1.16
K
0.67
Viscosity parameter
0
Concrete compression damage
Damage parameter
C
Inelastic strain
(modulus of elasticity, E0) due to the
damage when plastic strains increased
in brittle materials (such as concrete
and concrete-like material) as shown in
Fig. 1a. The damage parameter (dc)
was 0 at the maximum compressive
stress, and thereafter, it began to
decrease and continued decreasing until
0.8 was reached with respect to 20%
remaining strength in large strains.
15.3
0
0
0
Uniaxial Tensile Behavior
19.2
4.8249E-05
0
4.8249E-05
22.5
0.000119844
0
0.000119844
25.2
0.000214786
0
0.000214786
27.3
0.000333074
0
0.000333074
28.8
0.000474708
0
0.000474708
29.7
0.000639689
0
0.000639689
30
0.000828016
0
0.000828016
29.7
0.001039689
0.01
0.001039689
28.8
0.001274708
0.04
0.001274708
In concrete damage plasticity models,
the plastic hardening strain in tension
εtpl, h was derived (see Fig. 1b as
follows:
pl, h
ð19Þ
σ t = ð1 − dt ÞE0 εt − εt
8 ck, h
σt
>
= εt −
< εt
E0
ð20Þ
σt
1
pl, h
>
: εt = εt −
E0 1 − dt
27.3
0.001533074
0.09
0.001533074
25.2
0.001814786
0.16
0.001814786
22.5
0.002119844
0.25
0.002119844
19.2
0.002448249
0.36
0.002448249
15.3
0.0028
0.49
0.0028
10.8
0.003175097
0.64
0.003175097
0.81
0.003573541
5.7
0.003573541
Concrete tensile behavior
Yield stress (MPa)
3
0.03
Cracking strain
pl, h
εt
Concrete tension damage
Damage parameter
T
Cracking strain
0
0
0
0.001167315
0.99
0.001167315
Table 2: Material properties for concrete with SCDP model in class B30
softening stages of confined and
unconfined concretes. The softening
phase continued until 20% of the
unconfined
cylinder
compressive
strength (Point C) was reached; that
is, the stress value was not allowed to
continue to decrease, and perfect plastic behavior was assumed following
the softening trend (C–D). For simplicity, the entire constitutive model
was assumed to be a parabolic curve.
Equation (16) assumed a nonlinear
behavior for concrete from the beginning to the end. However, defining
the behavior of concrete up to 40% of
its strength in the elastic phase was
important in determining the effective
elastic modulus. In other words, the
constitutive model came into effect
when the compressive strength was
60% of the concrete compressive
strength. According to Figs. 1a and 2,
inelastic hardening strain in compression, εcin, h was derived as follows:
εcin, h = εc −
σc
E0
ð17Þ
Cyclic behavior contributed to concrete behavior, which was defined by
effective parameters, including damage in compression and damage in
tension. Compression damage (dc)
was based on inelastic hardening
strain in compression εcin, h that controlled the unloading curve slope.
Given that dc increased with respect
to an increase in εcin, h , it could be
expressed as follows:
dc = 1 −
σc
σ cu
ð18Þ
The tangent of the curve decreased
with respect to the initial tangent
Structural Engineering International Nr. 1/2017
ck, h
= εt
−
dt σ t
ð1 −dt Þ E0
ð21Þ
Although concrete had many constitutive models in the tension phase, there
were no significant differences in their
results due to the brittle behavior of
concrete. Engineers seldom define
the tension behavior of concrete in
numerical modeling, mainly because
the interaction between reinforcement
and concrete is simplified and because
certain changes in the stress–strain
relation of concrete in the tension
phase must be made to consider bar
slips in concrete. Among the constitutive models for the tension phase considering tensile strength, 7% to 10%
of maximum compressive strength σ cu
was chosen as a tensile strength, σ t0,
and in this study, the maximum value
was taken (i.e. σ t0 = 0.1σ cu). In this
paper, 1% of the tensile strength was
considered during the analysis regardless of the realistic condition to prevent numerical instability. In contrast,
correspondence strain value, where
stress is 1% of the ultimate tensile
strength, was taken as 10 times the
percentage of the strain, in which
stress was equal to ultimate tensile
strength.
Figure 1b showed that with a further
increase in the hardening cracking
ck, h
the tension damage constrain, εt
tinued to increase, and this could be
expressed as follows:
dt = 1 −
σt
σ t0
ð22Þ
Scientific Paper
71
Application of SCDP Parameters
in FE Programs
Poisson’s ratio range for concrete was
between 0.1 and 0.2. Poisson’s ratio,
elasticity modulus, and stress–strain
curve of concrete in compression and
tension were related to loading history, which resulted in strain differences. The dilation angle was equal to
volume strain over shear strain. The
dilation angle for concrete was usually
20 to 40 , which affected material
ductility. Consequently, the dilation
angle had considerable effects on the
entire model. An increase in the dilation angle increased the system flexibility. From a practical viewpoint, the
internal dilation angle depended on
certain parameters, including plastic
strain and confined pressure. An
increase in plastic strain and confined
pressure decreased the internal dilation angle. The material has a constant dilation angle for a large range
of pressure stresses used for the confinement of the material. The default
flow potential eccentricity was ε = 0.1,
and by raising the value of , the curvature of the flow potential increased. If
the default flow of potential eccentricity had a value much lower than the
default value, there could be convergence problems when the confining
pressure is not high. The ratio of initial equibiaxial compressive yield
stress to initial uniaxial compressive
yield stress was given by fb0/fc0, and
its default value was 1.16 [34]. The
ABAQUS software31 used a null
default viscosity parameter so that the
viscoplastic regularization did not
occur. This parameter enhanced the
convergence rate of the model when
the softening process occurred, and it
gave good results. In the FE program,
the tension recovery parameter equal
to zero denoted that this parameter
contributed to the cyclic behavior and
monitored the modulus of elasticity
when compression behavior changed
to tension behavior and vice versa. A
value of zero for tension recovery
implied that material tangent in the
tension phase was completely affected
by compression damages. However,
compression recovery was taken as
1, which meant that tension damage
did not affect the material tangent.
All these assumptions were also
adopted in reality.
According to the aforementioned
damage plasticity formulation derived
in this section, four different concrete
72
Scientific Paper
grades (20, 30, 40, and 50) were
implemented within the framework in
tabular format, that is, Tables 1, 2, 3,
and 4, respectively. Accordingly, the
hardening and softening rule as well
as the evolution of the scalar damage
variable for compression and tension
were presented for concrete grades
20, 30, 40, and 50. The general framework of the damage plasticity formulation was clearly stated and could be
extended to other concrete grades
between B20 and B50.
Numerical Computation
The performance of the proposed
constitutive model was further evaluated through the structural analysis of
a simply supported partially prestressed beam, which was subject to
self-weight and superimposed loading.
Figure 3 shows a partially prestressed
concrete beam that measured 16 m ×
1 m × 0.6 m, and was modeled in
ABAQUS. Twelve pieces of steel
reinforcements sized D = 20 mm with
Grade 460 Mpa and E = 200 GPa
were placed in two rows with 100 mm
spacing, horizontally and vertically.
A prestressed tendon of grade 270
with seven wire strands (270 ksi =
1861 MPa; E = 196.5 GPa; and area
of 1600 mm2) was placed at 200 mm
eccentricity from the center.
In this study, effective prestressing
force (Pe = 2667.2 kN), self-weight, and
superimposed dead load were applied
to the simply supported partially prestressed beam. A concrete block and
tendon were modeled by a threedimensional cube element (C3D8), and
the steel reinforcement elements were
3D truss (T3D2).
Plasticity parameters
Material’s
parameters
B40
Concrete elasticity
E (GPa)
30
0.2
Concrete compressive behavior
Yield stress (MPa)
Inelastic strain
Dilation angle
31
Eccentricity
0.1
fb0/fc0
1.16
K
0.67
Viscosity parameter
0
Concrete compression damage
Damage parameter
C
Inelastic strain
20.4
0
0
0
25.6
2.66667E-05
0
2.66667E-05
30
0.00008
0
0.00008
33.6
0.00016
0
0.00016
36.4
0.000266667
0
0.000266667
38.4
0.0004
0
0.0004
39.6
0.00056
0
0.00056
40
0.000746667
0
0.000746667
39.6
0.00096
0.01
0.00096
38.4
0.0012
0.04
0.0012
36.4
0.001466667
0.09
0.001466667
33.6
0.00176
0.16
0.00176
30
0.00208
0.25
0.00208
25.6
0.002426667
0.36
0.002426667
20.4
0.0028
0.49
0.0028
14.4
0.0032
0.64
0.0032
0.81
0.003626667
7.6
0.003626667
Concrete tensile behavior
Yield stress (MPa)
4
0.04
Cracking strain
Concrete tension damage
Damage parameter
T
Cracking strain
0
0
0
0.001333333
0.99
0.001333333
Table 3: Material properties for concrete with SCDP model in class B40
Structural Engineering International Nr. 1/2017
Plasticity parameters
Material’s
parameters
B50
Concrete elasticity
E (GPa)
33.4
0.2
Concrete compressive behavior
Yield stress (MPa)
Dilation angle
31
Eccentricity
0.1
fb0/fc0
1.16
K
0.67
Viscosity parameter
0
Concrete compression damage
Inelastic strain
Damage parameter
C
Inelastic strain
25.5
0
0
0
32
5.73819E-06
0
5.73819E-06
37.5
4.13628E-05
0
4.13628E-05
42
0.000106874
0
0.000106874
45.5
0.000202271
0
0.000202271
48
0.000327555
0
0.000327555
49.5
0.000482726
0
0.000482726
50
0.000667782
0
0.000667782
49.5
0.000882726
0.01
0.000882726
48
0.001127555
0.04
0.001127555
45.5
0.001402271
0.09
0.001402271
42
0.001706874
0.16
0.001706874
37.5
0.002041363
0.25
0.002041363
32
0.002405738
0.36
0.002405738
25.5
0.0028
0.49
0.0028
18
0.003224148
0.64
0.003224148
0.81
0.003678183
9.5
0.003678183
Concrete tensile behavior
Yield stress (MPa)
5
0.05
beam failed when the damage in tension reached 0.99. Accordingly, the
maximum compression stress (S,
Mises) in concrete and the mid-span
displacement of the beam subjected
to self-weight, superimposed dead
load, and prestressing load were
determined.
Figure 8 shows the value of the concrete stress and mid-span displacement of the simply supported partially
prestressed beam. The plots demonstrate that raising the concrete grade
increases the maximum compressive
stress in concrete as well as the midspan displacement. The compressive
stress of concrete for different classes,
B20, B30, B40, and B50, reached
92.35%, 84.53%, 81.475%, and
77.94%, respectively, of their corresponding ultimate concrete strength.
Furthermore, the mid-span displacement shows a gradual increase from
53.8 mm in B20 to 74.64 mm in B50.
The force versus displacement graphs
obtained in the service stage due to
self-weight and superimposed dead
loads are shown in Fig. 9. The capacity of the partially prestressed concrete beam in B20 is 28.20452,
whereas the capacities of B30, B40
and B50 indicated increases of 48.2%,
119% and 172.7%, respectively.
Concrete tension damage
Cracking strain
Damage parameter
T
Cracking strain
0
0
0
0.001494322
0.99
0.001494322
(a)
(b)
600 m m
1m
Table 4: Material properties for concrete with SCDP model in class B50
1000
mm
200 mm
16 m
6 ϕ20 @ 100 mm
0.6 m
Fi g. 3: Schematic view of the partially prestressed concrete beam: (a) Isometric view, (b)
Cross section
The effect of the SCDP model on the
concrete behavior was numerically
investigated for four different prestressed concrete grades, namely, B20,
B30, B40, and B50. Three key features, including damage in tension,
Von Mises stress, and maximum displacement, were recorded and compared to determine the effect of the
SCDP method. The results were
plotted to interpret the effect of the
SCDP method on the aforementioned
partially prestressed concrete classes.
The damage in tension, stress distribution, and mid-span deflection of the
partially prestressed concrete beams
are shown in Figs. 4–7 for the four
concrete classes B20, B30, B40, and
B50, respectively. As demonstrated in
these figures, the prestressed concrete
Structural Engineering International Nr. 1/2017
Validation of Simplified and
Tabulated Concrete Damage
Plasticity Model
Validation of the simplified and tabulated concrete damage plasticity
model was achieved by comparing the
empirical formulations given in previous research with that presented in
this paper. Table 5 indicated the
results of the empirical formulation as
well as the SCDP model for a simply
supported beam in the service stage
subjected to self weight and super
imposed dead load. The results
showed a good correlation between
these two approaches. The discrepancy was observed in the analysis
techniques as the Newton–Raphson
method was incorporated in the finite
element modeling, while the linear
technique was employed in the empirical formulations.
where C, I and W were the distance
of bottom fiber to the neutral axis,
moment of inertia and service load at
failure, respectively. Deflection due to
prestress load and service load were
sequentially depicted by ΔP, ΔSL.
Scientific Paper
73
(a)
DAMAGET
(Avg: 75%)
+9.900e–01
+8.910e–01
+7.920e–01
+6.930e–01
+5.940e–01
+4.950e–01
+3.960e–01
+2.970e–01
+1.980e–01
+9.900e–02
+0.000e+00
(b)
S, Mises
(Avg: 75%)
+1.847e+01
+1.663e+01
+1.479e+01
+1.295e+01
+1.111e+01
+9.277e+00
+7.439e+00
+5.601e+00
+3.763e+00
+1.925e+00
+8.737e–02
(c)
U, U2
+0.000e+00
–5.385e+00
–1.077e+01
–1.616e+01
–2.154e+01
–2.693e+01
–3.231e+01
–3.770e+01
–4.308e+01
–4.847e+01
–5.385e+01
F ig . 4: The results of the analysis on concrete B20: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
(a)
DAMAGET
(Avg: 75%)
+9.900e–01
+8.910e–01
+7.920e–01
+6.930e–01
+5.940e–01
+4.950e–01
+3.960e–01
+2.970e–01
+1.980e–01
+9.900e–02
+0.000e+00
(b)
S, Mises
(Avg: 75%)
+2.536e+01
+2.283e+01
+2.031e+01
+1.778e+01
+1.526e+01
+1.274e+01
+1.021e+01
+7.687e+00
+5.163e+00
+2.639e+00
+1.147e–01
(c)
U, U2
+0.000e+00
–5.986e+00
–1.197e+01
–1.796e+01
–2.394e+01
–2.993e+01
–3.591e+01
–4.190e+01
–4.789e+01
–5.387e+01
–5.986e+01
F ig . 5: The results of the analysis on concrete B30: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
(a)
DAMAGET
(Avg: 75%)
+9.900e–01
+8.910e–01
+7.920e–01
+6.930e–01
+5.940e–01
+4.950e–01
+3.960e–01
+2.970e–01
+1.980e–01
+9.900e–02
+0.000e+00
(b)
S, Mises
(Avg: 75%)
+3.259e+01
+2.935e+01
+2.610e+01
+2.286e+01
+1.961e+01
+1.637e+01
+1.312e+01
+9.877e+00
+6.632e+00
+3.387e+00
+1.420e–01
(c)
U, U2
+0.000e+00
–6.983e+00
–1.397e+01
–2.095e+01
–2.793e+01
–3.492e+01
–4.190e+01
–4.888e+01
–5.587e+01
–6.285e+01
–6.983e+01
F i g. 6: The results of the analysis on concrete B40: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
74
Scientific Paper
Structural Engineering International Nr. 1/2017
(a)
DAMAGET
(Avg: 75%)
+9.900e–01
+8.910e–01
+7.920e–01
+6.930e–01
+5.940e–01
+4.950e–01
+3.960e–01
+2.970e–01
+1.980e–01
+9.900e–02
+0.000e+00
(b)
S, Mises
(Avg: 75%)
+3.897e+01
+3.509e+01
+3.121e+01
+2.733e+01
+2.344e+01
+1.956e+01
+1.568e+01
+1.180e+01
+7.924e+00
+4.044e+00
+1.638e–01
(c)
U, U2
+0.000e+00
–7.464e+00
–1.493e+01
–2.239e+01
–2.986e+01
–3.732e+01
–4.478e+01
–5.225e+01
–5.971e+01
–6.718e+01
–7.464e+01
Fi g. 7: The results of the analysis on concrete B50: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
18.47
Subscripts “H” and “FE” denoted the
contributions made by the empirical
formulation and finite element modeling, respectively.
74.64
69.83
59.7774
53.8059
38.97
32.59
25.36
Mid Span Displacement (mm)
B20
Maximum Principal Stress (Mpa)
B30
B40
B50
Fi g . 8: Concrete stress and mid-span displacement of the simply supported prestressed
concrete beam with different concrete grades
2000
B20
B30
B40
B50
1800
Reaction force (kN)
1600
1400
1200
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
80
Mid span displacement (mm)
Fi g . 9: Pushover curve of different concrete strength classes
ΔP*
(mm)
ΔSL**
(mm)
ΔH***
(mm)
ΔFE,SCDP
(mm)
σH****
(Mpa)
σFE,
(Mpa)
Grade
C
(mm)
I
(mm4)
B20
488.1
5.032 × 1010
78
−16.0
62.4
46.4
53.8059
24.3
18.47
B30
490.5
5.029 × 1010
101.5
−12.8
64.7
51.9
59.7774
31.9
25.36
B40
491.5
5.028 × 1010
128.8
−11.3
72.9
61.6
69.83
40.7
32.59
492.4
5.027 × 10
154.2
−10.2
78.3
68.1
74.64
48.9
38.97
B50
W
(N/mm)
Furthermore, SCDP in tabular form
was verified through comparison with
finite element models of the available
CDP models given in extant literature. Two comparisons were carried
out for concrete strength classes of
30 MPa and 50 MPa. Von Mises
stress, damage in tension and midspan displacement were obtained by
utilizing nonlinear finite element techniques of a partially prestressed concrete beam as presented in Figs. 10
and 10. The FE analysis incorporated
the concrete damage plasticity model
proposed by Tiwari27 and Jankowiak32 for concrete strength class of
30 MPa and 50 MPa. It should be
noted that the model proposed by
Tiwari assumed that the simple supported beam failed either the damage
in tension reaching 0.82 or the concrete exceeded the maximum compressive stress.
10
SCDP
2
*ΔP = − PeL
8EI
5WL4
**ΔSL = 384EI
***ΔH = ΔP + ΔSL
****
P Pe
σH = A
− Zt + MZSLt
Table 5: concrete stress and mid-span displacement in the SCDP and empirical formulation
Structural Engineering International Nr. 1/2017
Scientific Paper
75
DAMAGET
(Avg: 75%)
+8.254e-01
+7.428e-01
+6.603e-01
+5.778e-01
+4.952e-01
+4.127e-01
+3.302e-01
+2.476e-01
+1.651e-01
+8.254e-02
+0.000e+00
(a)
S, Mises
(Avg: 75%)
+2.640e+01
+2.377e+01
+2.114e+01
+1.852e+01
+1.589e+01
+1.326e+01
+1.063e+01
+7.997e+00
+5.367e+00
+2.738e+00
+1.080e-01
(b)
U, U2
+0.000e+00
–6.526e+00
–1.305e+01
–1.958e+01
–2.610e+01
–3.263e+01
–3.915e+01
–4.568e+01
–5.221e+01
–5.873e+01
–6.526e+01
(c)
F ig . 10 : The results of the analysis on concrete B3027 : (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
(a)
DAMAGET
(Avg: 75%)
+9.900e–01
+8.910e–01
+7.920e–01
+6.930e–01
+5.940e–01
+4.950e–01
+3.960e–01
+2.970e–01
+1.980e–01
+9.900e–02
+0.000e+00
(b)
S, Mises
(Avg: 75%)
+3.231e+01
+2.910e+01
+2.588e+01
+2.266e+01
+1.944e+01
+1.623e+01
+1.301e+01
+9.791e+00
+6.573e+00
+3.355e+00
+1.378e–01
(c)
U, U2
+0.000e+00
–6.807e+00
–1.361e+01
–2.042e+01
–2.723e+01
–3.404e+01
–4.084e+01
–4.765e+01
–5.446e+01
–6.126e+01
–6.807e+01
F ig . 11 : The results of the analysis on concrete B5032: (a) damage in tension, (b) stress distribution (MPa), and (c) displacement
distribution (mm)
With respect to the results obtained
and summarized in Table 6, it was
observed that the compressive stress
and mid-span displacement values of
the available concrete damage plasticity models and SCDP model had good
agreement.
Figure 11 clearly demonstrated a reasonable correlation between the
CDP models
Tiwari
Jankowiak
SCDP
SCDP model and the studied FE
models. The maximum reaction force
of the beam with the Tiwari and
SCDP models in concrete class B30
showed a difference of approximately
4.6%. However, the discrepancy in
the concrete class of B50 between
Jankowiak model and SCDP model
was approximately 16%.
Compressive
stress
(MPa)B30
Compressive
stress
(MPa)B50
Mid span
displacement
(mm) B30
Mid span
displacement
(mm) B50
26.4
–
65.26
–
–
32.3
–
68.07
25.36
38.97
59.86
74.64
Table 6: Comparison of results between concrete stress and mid-span displacement
76
Scientific Paper
Conclusions
This paper developed the present
damage plasticity model (SCDP). This
model simplified the procedure of
existing damage plasticity model called
CDP. It combined a stress-based plasticity part with a strain-based damage
mechanics model for the unconfined
prestressed concrete beam based on a
tabular format for four different concrete grades (B20, B30, B40, and B50).
Accordingly, the findings in this paper
indicated the following conclusions
from the simplification provided by
the proposed tabulated concrete damage plasticity model:
• Due to its simplicity, the SCDP
model was a suitable solution to
Structural Engineering International Nr. 1/2017
Reaction force (kN)
2000
1800
1600
1400
1200
1000
800
600
400
200
0
B30
B30 (Tiwari 2015)
B50
B50 (JANKOWIAK 2005)
[2] Kang HD. Triaxial Constitutive Model for
Plain and Reinforced Concrete Behavior University of Colorado: Boulder, 1997.
[3] Grassl P, Lundgren K, & Gylltoft K. Concrete in compression: a plasticity theory with a
novel hardening law. Int. J. Solids Struct. 2002;
39(20): 5205–5223.
[4] Chen AC, & Chen WF. Constitutive relations for concrete. J. Eng. Mech. 1975; 101(4):
465–481.
0
10
20
30
40
50
60
70
80
Mid span displacement (mm)
Fi g. 12: Validation of force versus displacement curves of the SCDP and available CDP
models.27,32
model the cracking and the crushing of concrete. Thus, a 3D nonlinear FE model was developed to
assess the performance of concrete
with the SCDP model. All possible
nonlinearities, namely, material and
geometric, were considered in the
developed FE model.
• The presented 3D nonlinear FE
model was then used to model
different concrete classes. The
developed FE model successfully
predicted the concrete damage
caused by tension, compressive
stress in concrete, and mid-span displacement in the prestressed concrete beam.
• The model realistically described
the transition from tensile to
compressive failure. This finding
was achieved through the introduction of two separate isotropic
damage variables for tension and
compression.
• The proposed three-dimensional
FE model provided the framework
to develop a realistic model to
determine the behavior of the simply supported partially prestressed
beam. Then, the comparison of the
results between the SCDP model
and available CDP models including, the two different concrete
damage plasticity models, namely,
the Tiwari27 and the Jankowiak
and Lodygowski32 models, showed
that the present CDP model had a
good correlation with these CDP
models.
εpl
Del
Del0
σ
σ
d
dt
Nomenclature
This work was supported by the University
Putra
Malaysia
under
Putra
grant
No. 9438709. This support is gratefully
acknowledged.
E
Ec
Es
Ep
ν
εel
young’s modulus
concrete young’s modulus
steel young’s modulus
Prestressing steel young’s
modulus
poisson’s ratio
elastic strain tensor
dc
σ cu
ε’c
εpl
c
εpl
εpl
t
,h
εin
c
pl,h
ε
ck, h
εt
,h
εpl
c
pl, h
εt
:ε:
ε_ el
ε_ pl
:λ:
gp
ΔFE
σFE
plastic strain tensor
degraded elastic tensor
initial elastic tensor
nominal stress
effective stress
scalar damage variable
scalar
tension
damage
variable
scalar compression damage
variable
ultimate
compressive
strength
of
unconfined
cylinder
strain of unconfined cylinder
at
ultimate
compressive
strength
compressive plastic strain
tensor
plastic strain tensor
tensile plastic strain tensor
inelastic compression strain
plastic hardening strain
cracking strain
plastic hardening strain in
compression
plastic hardening strain in
tension
total strain rate
total strain rate in elastic part
total strain rate in plastic part
rate of plastic multiplier
plastic potential
mid span deflection due to
FE model
max concrete compressive
stress due to FE model
Acknowledgements
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