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ADVANCED MATERIAL MODELLING OF CONCRETE IN ABAQUS
Conference Paper · July 2016
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2. ADVANCED MATERIAL MODELLING OF
CONCRETE IN ABAQUS
M Vilnay
Abertay University
L Chernin D Cotsovos
University of Dundee Heriot Watt University
United Kingdom
ABSTRACT. Abaqus is a complex finite element (FE) package widely used in civil
engineering practice. In particular, it is used for modelling of reinforced concrete structures.
One of the concrete models incorporated in Abaqus is the brittle cracking model. The main
shortcoming of this model is that it assumes linearly elastic behaviour in compression. This
paper proposes to eliminate this shortcoming through the use of the user subroutine
VUSDFLD. This subroutine allowed to add the nonlinear compressive behaviour into the
brittle crack model by introducing the dependency of the modulus of elasticity of concrete on
strain. Additionally, the concrete material is modelled to be able to develop damage defined
by the maximum strain and damaged elements are deleted from the FE model. The extended
brittle crack model is used to examine the strain rate effects and to simulate three benchmark
cases with static and blast type loading regimes. The limits of the model applicability are
examined. The FE simulation results favourably compared with those observed in
experiments. Overall, the extended brittle crack model offers a robust reliable way for
modelling of concrete.
Keywords: Concrete; Numerical modelling; ABAQUS; User subroutine.
Margi Vilnay is a lecturer in structural engineering at the School of Science, Engineering
and Technology of Abertay University.
Dr Leon Chernin is a lecturer in civil engineering at the School of Science and Engineering
of the University of Dundee.
Dr Demetrios Cotsovos is a lecturer in structural engineering at the School of Energy,
Geoscience, Infrastructure and Society of Heriot Watt University.

3. 2 Vilnay et al
INTRODUCTION
Abaqus is often used by scientists and engineers for modelling of reinforced concrete (RC)
structures, e.g., [1-5]. The choice of material models of concrete is limited in Abaqus to the
smeared cracking model, the brittle cracking mode and the damaged plasticity model [6].
Each model is designed for a particular type of usage. The smeared cracking model can
handle only monotonic loading and low confining pressures. This sufficiently limits the range
of its applicability. The damaged plasticity model is by far most complex concrete model
incorporated in Abaqus that can be used in any loading regime. However, it is not ‘user
friendly’, includes multiple parameters and its calibration can be very challenging.
Additionally, this model does not allow damaged elements to be deleted form the finite
element (FE) analysis, which can lead to numerical instability of the solution algorithms. The
brittle cracking model can be used in any loading regime and is very ‘user friendly’ and easy
to calibrate. The main disadvantage of this model is that it assumes linear elastic material
behaviour in compression. As a result, the model can be reliably used only in the cases where
the concrete behaviour is dominated by the tensile failure.
The limited choice of the built-in concrete models combined with their shortcomings often
resulted in new models introduced in Abaqus through user-defined subroutines, e.g., [1, 2]. In
this paper, the brittle cracking model is extended to include the nonlinear compressive
behaviour using the user subroutine VUSDFLD. The new material model is compared with
the original brittle cracking model and the damaged plasticity model. It is then used to
examine strain rate effects [7, 8] and also to simulate a number of benchmark cases including
a three point bending test [9], a standard brittle failure test [10] and an RC column under blast
[11]. The limitations of model application are examined.
EXTENDED BRITTLE CRACKING MODEL
The brittle cracking model is built to work in Abaqus with the explicit time integration
scheme [6]. It is an elastic cracking model with concrete between cracks considered as an
isotropic linearly elastic material. In this model, the initiation and evolution of individual
cracks is not tracked. Instead, a smeared crack method is utilised to present the material
discontinuities. The constitutive calculations are independently performed at each material
point of an FE element. The presence of cracks affects the stress and material stiffness
associated with the material point. A simple Rankine criterion is used to detect crack
initiation. Thus, a crack forms when the maximum principal tensile stress exceeds the tensile
strength of the concrete. The crack surface is oriented in the direction normal to the
maximum principal tensile stress. Once a crack is formed at a point, its orientation is stored
for subsequent calculations. A new crack can form at the same point only in a direction
orthogonal to the direction of an existing crack. Therefore, this model is called a fixed
orthogonal crack model. Cracks are modelled as irrecoverable. They may close and reopen,
but remain throughout the rest of the analysis. The tension softening in the direction normal
to a crack is described based on the Hillerborg cohesive crack model [12], in which a stress-
displacement curve is adopted from CEB-FIP Model Code 2010 [13]. Additionally, the effect
of the amount of crack opening on the shear response of concrete is formulated using the
shear retention model. In this model, the post-cracked shear stiffness is defined as a power
function of the strain across an opening crack, reducing as the crack opens.
The nonlinear behaviour of concrete in compression is incorporated into the brittle cracking
model by using the user subroutine VUSDFLD [6]. This subroutine allows to redefine
material properties at a material point as a function of a field variable such as stress, strain,
temperature, etc. The field variable is updated at each analysis step and the value of the

4. Abaqus Modelling of Concrete 3
relevant material property is recalculated. In this study, the nonlinear compressive behaviour
of concrete is introduced into the brittle crack model by formulating the modulus of elasticity
of concrete (Ec) as a function of strain (εc). To define the Ec – εc function, the stress-strain
(σc – εc) relationship describing the uniaxial compression behaviour of concrete is adopted
from CEB-FIP Model Code 2010 [13]
η
η
η
σ
⋅
−
+
−
⋅
−
=
)
2
(
1
2
k
k
fcm
c
for lim
c
c ,
ε
ε < (1)
where 1
c
c ε
ε
η = , εc1 is the strain at the maximum compressive stress fcm, εc,lim is the strain at
crushing of concrete in compression, 1
c
ci E
E
k = is the plasticity number, Eci is the initial
modulus of elasticity of concrete and Ec1 is the secant modulus obtained by connecting the
diagram origin to the curve peak, i.e., (εc1, fcm). The Ec – εc relationship can be obtained from
Eq. (1) taking into account that c
c
c
E ε
σ
= and 1
1 c
cm
c f
E ε
=
1
)
2
(
1
c
c E
k
k
E ⋅
⋅
−
+
−
−
=
η
η
for lim
c
c ,
ε
ε < (2)
In Eq. (2), Ec is the secant modulus obtained by connecting the diagram origin to a point on
the σc – εc curve. The σc – εc and Ec – εc curves yielded by Eqs. (1) and (2) are schematically
shown in Figure 1.
Figure 1 The σc – εc and Ec – εc curves describing Eqs. (1) and (2)
Additional advantage of the original and extended brittle cracking models is that they can be
used together with an algorithm that removes failed elements from the FE model. The failure
criterion is defined in this algorithm as the maximum compressive/tensile strain.
VALIDATION OF THE MATERIAL MODEL
In all FE simulations discussed hereafter, concrete was modelled using 8-node linear brick
elements (C3D8R) with reduced integration to prevent over-stiff elements and enhanced
hourglass control to avoid spurious deformation mode in the model mesh [6]. The elements
were controlled during the analysis to prevent excessive distortion of the mesh.
σ
c
/
f
cm
&
E
c
/
E
ci
COMPRESSIVE STRAIN εc
εc,lim
εc1
Ec1
Eci
1.0
Ec – εc curve
σc – εc curve

5. 4 Vilnay et al
Single Finite Element
The efficiency of the extended brittle cracking model was initially examined using a single
finite element model representing a 1 m × 1 m × 1 m concrete block. The bottom face of the
block was constrained against the vertical movement. Additional constrains where applied at
three corners of the block to prevent its rotation and movement in the horizontal plane. Such
boundary conditions allowed to avoid the development of an arching effect in the block. A
vertical displacement load was applied to the top face of the block in order to stabilise the
procedure of the numerical solution. The concrete block had the following material
properties: Eci = 30 GPa, Poisson’s ratio ν = 0.2 and the density ρ = 2400 kg/m3
.
Three FE models using the extended brittle cracking model, the original brittle cracking
model and the damaged plasticity model were compared. The material parameters used in the
models are given in Table 1. These parameters correspond to the Abaqus benchmarked
solution of a three point bending test [6].
Table 1 Material parameters
COMPRESSION TENSION
Yield Stress
(MPa)
Inelastic
Strain
Yield
Stress
(MPa)
Displacement
(m)
Damage
Parameter
20 0 3.33 0 0
30 0.015 0.333 7.447e-5 0.9
Note that only the tensile properties are needed for the brittle cracking models. Additional
compressive properties (i.e., the Ec – εc curve) necessary for the extended brittle crack model
were obtained based on the σc – εc curve generated by the damaged plasticity model. This
was done in order to exclude the influence of the input data on the material model
performance.
Figure 2 Reaction-displacement curves for compression load

6. Abaqus Modelling of Concrete 5
The performance of the three concrete models under uniaxial tension and compression was
then examined. All the models behaved similarly under tension with a slight difference
between the damaged plasticity model and both the brittle cracking models developing in the
part of the curves corresponding to the crack opening. On the other hand, the behaviour of the
models highly diverged under compression. Figure 2 shows that the original brittle cracking
model exhibited a purely elastic response. The two remaining models behaved similarly until
the designated maximum strain of 0.015 (corresponding to the displacement of 0.015 m in the
concrete block), when the extended brittle cracking model failed. The damaged plasticity
model failed at the strain just under 0.02 (the displacement of 0.02 m in the block). Following
its failure, the damaged plasticity model exhibited an unstable response with a series of sharp
partial recoveries and failures (see Figure 2). As a result, this model may not be entirely
reliable in simulating the post-failure behaviour of concrete structures.
Strain Rate
The extended brittle cracking model does not explicitly include the effect of the rate of load
application. The sensitivity of this material model to the strain rate was examined using a
standard concrete prism with the height of 253 mm and the cross-section of 100 mm by
100 mm [7]. Each edge of the prism cross-section was discretised into 5 elements, while the
prism was discretised into 13 elements along its height. This gave 125 elements with the
dimensions of 20 mm × 20 mm × 19.5 mm. The uniaxial compressive strength of concrete
was assumed to be fcm = 30 MPa, Poisson’s ratio equal to ν = 0.2 and the density to
ρ = 2400 kg/m3
. The bottom face of the prism was fixed and the load was applied to the top
face at different rates.
Figure 3 Effect of tensile strain rate
Initially, the effect of the tensile strain rate was examined using a displacement load. The
displacement load was selected to stabilise the numerical solution during concrete failure in
tension. Six different displacement rates between 10 mm/sec and 20,000 mm/sec
(corresponding to the strain rates between 0.0005 sec-1
and 3 sec-1
, respectively) were
considered. The increase in the tensile stresses was observed with the growing strain rate.

7. 6 Vilnay et al
Figure 3 shows the analysis results plotted together with the existing strain rate experimental
data [7]. The abscissa of the diagram in the figure is in a logarithmic scale, and the ordinate is
the maximum dynamic reaction force, Rm,d, at the top face of the prism normalised by the
maximum static reaction force, Rm,s. As can be seen, the numerical results fall within the
experimental scatter, and the Rm,d / Rm,s ratio increases more rapidly for the strain rates larger
than 0.1 sec-1
. It is also necessary to note that the displacement loads with the rates larger
than 20,000 mm/sec (corresponding to the strain rate of 3 sec-1
) caused distortion of the finite
elements, rendering the results unreliable.
The effect of the compressive strain rate was examined using the pressure load with the rates
between 10,000 MPa/sec and 4,000,000 MPa/sec (corresponding to the strain rates between
0.01 sec-1
and 70.8 sec-1
, respectively). Figure 4 shows the results of numerical simulations as
well as the existing experimental data [7]. The abscissa of the diagram in the figure is in a
logarithmic scale, and the ordinate is the maximum dynamic pressure, Pm,d, normalised by the
maximum static pressure, Pm,s. It is evident that the growing strain rate leads to the increase
of the Pm,d / Pm,s ratio and this increase becomes more rapid for the strain rates larger than
0.3 sec-1
. In addition, the numerical results fall within the experimental scatter.
Figure 4 Effect of compressive strain rate
It can therefore be concluded that since the strain rate effect is not incorporated into the
extended brittle cracking model, the observed increase in the material strength can only be
attributed to the inertia effects occurring at the structural level.
SIMULATION OF BENCHMARK CASES
The extended brittle cracking model is implemented in this section for simulation of three
standard benchmark cases including a notched concrete beam, an RC beam and column.
Notched Concrete Beam
The 3 point bending tests on a notched unreinforced concrete beam [9] was chosen for the
examination of the efficiency of the brittle cracking model in simulating the tensile structural

8. Abaqus Modelling of Concrete 7
failure. The simply supported beam had the span equal to 2 m, the depth to 0.2 m and the
width to 0.05 m. The midspan notch had the depth of 0.1 m and the width of 0.04 m. The
beam was loaded by a knife (line) load at midspan. The concrete had the following material
properties: Eci = 30 GPa, ν = 0.2, the tensile strength ftm = 3.33 MPa, the Mode I fracture
energy GI
f = 124 N/m and ρ = 2400 kg/m3
. This benchmark problem was also used for
verification of the damaged plasticity model [6], which allowed comparison between the two
material models.
Figure 5 FE model of noted beam
Taking advantage of symmetry, only half of the notched beam was modelled (see Figure 5).
The mesh consisted of 1120 three-dimensional elements of the type C3D8R [6]. The mesh
around and above the notch was refined to overcome mesh sensitivity due to the possibility of
cracking in the out of plane direction.
Figure 6 Reaction vs. midspan displacement of noted beam
The analysed beam was expected to fail in a brittle mode with a sudden drop in its load
carrying capacity. This behaviour can generally lead to an increase in the kinetic energy
content of the numerical response of the beam. Since the FE solution was carried out using
Displacement
load
Velocity
load

9. 8 Vilnay et al
the explicit time integration scheme, the beam had to be kept in the static regime by loading it
slowly enough to eliminate significant inertia effects. The static loading regime was achieved
by applying a velocity load (see Figure 5) that increased linearly from 0 to 0.06 m/s over a
period of 0.05 seconds, which led to the final displacement of 1.5 mm at the beam midspan.
This type of loading ensured a quasi-static solution in a reasonable number of time
increments, while the kinetic energy in the beam was small throughout the numerical
solution. The results of the FE simulations together with a comparison to the experimental
data are shown in Figure 6. As can be seen, both material models provide peak and failure
responses that agree well with the experimental observations [9]. Although, the extended
brittle crack model is slightly more accurate in the failure part of the curve. Small oscillations
of the reaction-displacement curves still develop due to the inertial effects before cracking of
concrete occurs. The amplitude of the oscillations becomes larger during the failure phase
due to amplification of the inertia effect by cracking.
Reinforced Concrete Beam
The efficiency of the extended brittle cracking model in
simulating the behaviour of a RC beam is examined using the
beam C-2 from the experimental series [10]. The simply
supported C-2 beam was 4572 mm long with the cross-section
presented in Figure 6. The concrete had the following material
properties: Eci = 22.924 GPa, ν = 0.2, fcm = 24.13 MPa and
ρ = 2400 kg/m3
. The longitudinal reinforcement consisted of 4
bottom #9 bars (28.65 mm diameter) and 2 top #4 bars (12.7 mm
diameter). The shear reinforcement consisted of #2 (6.25 mm
diameter) stirrups at spacing of 208 mm centres. The concrete
cover was equal to 41.3 mm. The material properties of
reinforcement are given in Table 2.
The FE model of the whole C-2 beam included 4510 three-
dimensional elements of the type C3D8R. All reinforcing bars
were modelled using 1178 Timoshenko beam elements (B31) and classic metal plasticity [6].
The reinforcing bar elements were embedded in the concrete elements. This formulation
assumed perfect, unfailing bond between steel bars and concrete.
The crack patterns developed in the C-2 beam during a 3 point bending test and the results of
the FE simulation are depicted in Figure 7. The FE results show the strain distribution in the
concrete. As can be seen, the test beam and the FE model underwent excessive cracking in
the same zones. The midspan deflection of the test beam and the FE model are shown in
Figure 8. The curves in the figure follow very similar paths till the FE model fails at the 10.1
mm deflection due to numerical instabilities introduced by excessive cracking of concrete. In
the case of brittle failure the underestimation of the structural capacity is preferable to its
overestimation. Therefore, the extended brittle cracking model provides safe prediction of the
beam response.
Table 2 Reinforcement material properties [10]
STEEL
TYPE
ELASTIC MODULUS
(GPa)
YIELDING STRESS
(MPa)
ULTIMATE STRESS
(MPa)
#9 205.46 551.58 932.8
#4 201.33 345.42 603.98
#2 189.6 325.43 429.54
Figure 6 C-2 beam
cross-section [10]
(all dimension in mm)
560
155
63.5
63.5
2 #4
4 #9
63.5

10. Abaqus Modelling of Concrete 9
Figure 7 Crack patterns: test beam [10] at the top and FE model at the bottom
Figure 8 Load-deflection curves
Reinforced Concrete Column
The efficiency of the extended brittle cracking model in capturing
the response of an RC column under a blast load is examined using
the Test 7 column from the experimental series [11]. In the tests,
the blast load was artificially created using an array of blast
generators, each of which was made of an impacting module and a
hydraulic actuator. Four blast generators were located over the
column height. They impacted the column imparting a controlled
blast-like impact. The column was casted with heavily reinforced
concrete blocks as its heading and footing. The column footing
was fixed to the ground, while a link system applied to the column
heading provided lateral and moment restraints while allowing
vertical movement only. The column was 3277 mm high and had a
356 mm × 356 mm cross-section. The reinforcement consisted of
eight #8 longitudinal bars and #3 hoops spaced at 324 mm centres.
The cover depth was 38 mm. The concrete had the following
material properties: Eci = 24 GPa, ν = 0.2, fcm = 40 MPa and
ρ = 2400 kg/m3
. The yield stress of longitudinal bars was equal to
fy = 335 MPa, while of hoops to fy = 235 MPa.
Figure 9 FE model of
Tests 7 column [11]

11. 10 Vilnay et al
Figure 10 Four link system: test setup [11] on the left and FE model on the right
Figure 9 shows the FE model of the column with the heading, footing and part of the
reinforcement exposed. In accordance with the experimental setup, each face of the footing
was restrained in the perpendicular direction creating a fixed support. The restraint system at
the top of the column was explicitly modelled by applying multipoint constraints between the
nodes on one face of the heading and four external nodes. The external nodes were restrained
from moving in the horizontal plane, but could move vertically. The experimental setup and
the FE model of the heading are shown in Figure 10. Since the footing and heading were
confined and heavily reinforced, they were assumed to be linearly elastic in the FE model.
The concrete column was modelled using 16224 elements of the type C3D8R. The
reinforcing bars were modelled using 1372 Timoshenko beam elements (B31) and classic
metal plasticity [6], and were embedded in concrete elements. The blast load was simulated
using the equivalent pressure measured in the experiment [11]. The pressure was applied
uniformly over a corresponding column face and its intensity was controlled through its
amplitude.
(a) (b) (c)
Figure 11 Test 7 experimental [11] and FE results at (a) 41.7 µsec (b) 84.3 µsec and (c) 558
µsec after blast load application

12. Abaqus Modelling of Concrete 11
Figure 11 shows the results of the results of test and FE simulations at different times after
blast load application. As can be see, in both cases shear cracks initially developed at both the
top and bottom ends of the column (see Figure 11a). This was followed by extensive crashing
of concrete (see Figures 11b and 11c). The column eventually failed in both the test and the
FE simulation due to the shear failure at both its ends. The peak deflections recorded in the
test at 41.7 µsec equalled 122 mm, while in the FE simulation it was equal at 43.2 µsec to
116.3 mm. The residual deflection recorded in the tests was equal to 85 mm, while in the FE
simulations to 92 mm. It can be concluded therefore that the numerical analysis closely
followed the test and provided an accurate prediction of the evolution of damage and
deflections in the column.
CONCLUSIONS
In this paper, a method for improving the brittle cracking material model incorporated in
Abaqus was suggested. This method allowed to introduce nonlinear compressive behaviour
into the model using a user subroutine VUSDFLD. This was done by formulating the
modulus of elasticity as a function of strain. Additionally, the user subroutine allowed to
eliminate failed elements from the finite element (FE) mesh which increased the stability of
numerical solution. The extended brittle cracking material model was implemented for
modelling concrete. Initially, the performance of the proposed model was compared with
other material models built-in in Abaqus using a single FE model. The advantages of the
proposed model were clearly outlined. The efficiency of the proposed material model to
simulate concrete under high rate loads was then examined using an FE model of a concrete
prism. Despite the fact that the strain rate effect was not incorporated into the model
formulation, the increase in the tensile and compressive material strength took place with the
increase in the rate of loading. The FE results were within the experimental scatter. This
behaviour was attributed to the inertia effects occurring at the structural level. Further, the
extended brittle cracking material model was implemented for simulation of three standard
benchmark cases including a notched concrete beam, a reinforced concrete (RC) beam and an
RC column. The beams were subjected to static loads while the column to a blast load. In all
examined cases, the extended brittle cracking model showed ability to accurately describe
failure modes and evolution of deflection and cracks in a structure.
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