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ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING
Vol. XI 2011 No. 3
Modelling of reinforced concrete structures
and composite structures with concrete strength degradation
taken into consideration
P. KMIECIK, M. KAMISKIWrocaw University of Technology, Wybrzee Wyspiaskiego 25, 50-370 Wrocaw, Poland.
Because of the properties of the material (concrete), computer simulations in the field of reinforced
concrete structures are pose a challenge. As opposed to steel, concrete when subjected to compression ex-hibits nonlinearity right from the start. Moreover, it much quicker undergoes degradation under tension.
All this poses difficulties for numerical analyses. Parameters needed to correctly model concrete under
compound stress are described in this paper. The parameters are illustrated using the Concrete Damaged
Plasticity model included in the ABAQUS software.
Keywords: numerical modelling, concrete degradation, stress-strain relation, reinforced concrete struc-
tures, composite structures, Abaqus, concrete damaged plasticity
1. Introduction
The two main concrete failure mechanisms are cracking under tension and crushing
under compression. However, concrete strength determined in simple states of stress (uni-axial compression or tension) radically differs from the one determined in complex states
of stress. For example, the same concrete under biaxial compression reaches strength of
between ten and twenty per cent higher than in the uniaxial state while in the hydro-static
state (uniform triaxial compression) its strength is theoretically unlimited. In order to de-
scribe strength with the equation for triaxial stress, its plane should be presented in a three-
dimensional stress space (since concrete is considered to be an isotropic material in a wide
range of stress). The states of stress corresponding to material failure are on this surfacewhile the states of safe behaviour are inside. Also the so-called plastic potential surface is
located inside this space. After the plasticity surface is crossed, two situations arise [9]:
an increase in strain with no change in stress (ideal plasticity), material weakening (rupture).
2. Strength hypothesis and its parameters
One of the strength hypotheses most often applied to concrete is the Drucker
Prager hypothesis (1952). According to it, failure is determined by non-dilatational
strain energy and the boundary surface itself in the stress space assumes the shape of a
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energy and the boundary surface itself in the stress space assumes the shape of a cone.
The advantage of the use of this criterion is surface smoothness and thereby no com-
plications in numerical applications. The drawback is that it is not fully consistent with
the actual behaviour of concrete 0.
Fig. 1. DruckerPrager boundary surface 0: a) view, b) deviatoric cross section
The CDP (Concrete Damaged Plasticity) model used in the ABAQUS software is
a modification of the DruckerPrager strength hypothesis. In recent years the latter has
been further modified by Lubliner 0, Lee and Fenves 0. According to the modifi-
cations, the failure surface in the deviatoric cross section needs not to be a circle and it
is governed by parameterKc.
Fig. 2. Deviatoric cross section of failure surface in CDP model 0
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Physically, parameter Kc is interpreted as a ratio of the distances between the
hydrostatic axis and respectively the compression meridian and the tension merid-
ian in the deviatoric cross section. This ratio is always higher than 0.5 and when it
assumes the value of 1, the deviatoric cross section of the failure surface becomesa circle (as in the classic DruckerPrager strength hypothesis). Majewski reports
that according to experimental results this value for mean normal stress equal to
zero amounts to 0.6 and slowly increases with decreasing mean stress. The CDP
model recommends to assume Kc= 2/3. This shape is similar to the strength crite-
rion (a combination of three mutually tangent ellipses) formulated by William and
Warnke (1975). It is a theoretical-experimental criterion based on triaxial stress
test results.
Similarly, the shape of the planes meridians in the stress space changes. Experi-
mental results indicate that the meridians are curves. In the CDP model the plastic
potential surface in the meridional plane assumes the form of a hyperbola. The shapeis adjusted through eccentricity (plastic potential eccentricity). It is a small positive
value which expresses the rate of approach of the plastic potential hyperbola to its as-
ymptote. In other words, it is the length (measured along the hydrostatic axis) of the
segment between the vertex of the hyperbola and the intersection of the asymptotes of
this hyperbola (the centre of the hyperbola). Parameter eccentricitycan be calculated
as a ratio of tensile strength to compressive strength [4]. The CDP model recommends
to assume = 0.1. When = 0, the surface in the meridional plane becomes a straight
line (the classic Drucker-Prager hypothesis).
Fig. 3. Hyperbolic surface of plastic potential in meridional plane 0
Another parameter describing the state of the material is the point in which the
concrete undergoes failure under biaxial compression. b0/c0(fb0 /fc0) is a ratio of the
strength in the biaxial state to the strength in the uniaxial state. The most reliable in
this regard are the experimental results reported by Kupler (1969). After their approxi-
mation with the elliptic equation, uniform biaxial compression strength fccis equal to
1.16248fc0. The ABAQUS users manual specifies default b0/c0= 1.16.
The last parameter characterizing the performance of concrete under compound
stress is dilation angle, i.e. the angle of inclination of the failure surface towards the
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P. KMIECIK, M. KAMISKI626
hydrostatic axis, measured in the meridional plane. Physically, dilation angle is in-
terpreted as a concrete internal friction angle. In simulations usually = 36 0,0 or
= 40 0 is assumed.
Fig. 4. Strength of concrete under biaxial stress in CDP model 0
Table 1. Default parameters of CDP model under compound stress
Parameter name Value
Dilatation angle 36
Eccentricity 0.1
fbo /fco 1.16
0.667
Viscosity parameter 0
The unquestionable advantage of the CDP model is the fact that it is based on pa-rameters having an explicit physical interpretation. The exact role of the above pa-
rameters and the mathematical methods used to describe the development of the bound-
ary surface in the three-dimensional space of stresses are explained in the ABAQUS
users manual. The other parameters describing the performance of concrete are deter-
mined for uniaxial stress. Table 1 shows the models parameters characterizing its per-
formance under compound stress.
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3. Stress-strain curve for uniaxial compression
The stress-strain relation for a given concrete can be most accurately described on
the basis of uniaxial compression tests carried out on it. Having obtained a graph fromlaboratory tests one should transform the variables. Inelastic strains inc
~ are used in the
CDP model. In order to determine them one should deduct the elastic part (correspond-
ing to the undamaged material) from the total strains registered in the uniaxial compres-
sion test:
,~ 0elcc
inc = (1)
.0
0
E
celc
= (2)
Fig. 5. Definition of inelastic strains 0
When transforming strains, one should consider from what moment the material
should be defined as nonlinearly elastic. Although uniaxial tests show that such be-
haviour occurs almost from the beginning of the compression process, for most nu-
merical analyses it can be neglected in the initial stage. According to Majewski, a lin-
ear elasticity limit should increase with concrete strength and it should be rather as-
sumed than experimentally determined. He calculated it as a percentage of stress to
concrete strength from this formula:
.80
exp1lim
= c
fe (3)
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P. KMIECIK, M. KAMISKI628
This ceiling can be simply arbitrarily assumed as 0.4 fcm. Eurocode 2 specifies the
modulus of elasticity for concrete to be secant in a range of 00.4 fcm. Since the basic
definition of the material already covers the shear modulus and the longitudinal
modulus of concrete, at this stage it is good to assume such an inelastic phase thresh-old that the initial value of Youngs modulus and the secant value determined accord-
ing to the standard will be convergent. In most numerical analyses it is rather not the
initial behaviour of the material, but the stage in which it reaches its yield strength
which is investigated. Thanks to the level of 0.4fcmthere are fewer problems with so-
lution convergence.
Having defined the yield stress-inelastic strain pair of variables, one needs to de-
fine now degradation variable dc. It ranges from zero for an undamaged material to
one for the total loss of load-bearing capacity. These values can also be obtained from
uniaxial compression tests, by calculating the ratio of the stress for the declining part
of the curve to the compressive strength of the concrete. Thanks to the above defini-tion the CDP model allows one to calculate plastic strain from the formula:
( ),
1
~~
0Ed
d c
c
cinc
plc
= (4)
whereE0stands for the initial modulus of elasticity for the undamaged material. Know-
ing the plastic strain and having determined the flow and failure surface area one can
calculate stress cfor uniaxial compression and its effective stress c .
( ) ( ),~1 0 plcccc Ed = (5)
( ) ( ).~
10
plcc
c
cc E
d
=
= (6)
3.1. Plotting stress-strain curve without detailed laboratory test results
On the basis of uniaxial compression test results one can accurately determine the
way in which the material behaved. However, a problem arises when the person run-ning such a numerical simulation has no such test results or when the analysis is per-
formed for a new structure. Then often the only available quantity is the average com-
pressive strength (fcm) of the concrete. Another quantity which must be known in order
to begin an analysis of the stress-strain curve is the longitudinal modulus of elasticity
(Ecm) of the concrete. Its value can be calculated using the relations available in the lit-
erature 0:
( ) ,1.022 3.0cmcm fE = (7)
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where:
fcm[MPa],
Ecm[GPa].
Other values defining the location of characteristic points on the graph are strain c1at average compressive strength and ultimate straincu0:
( ) ,7.0 31.01 cmc f= (8)
cu= 3.5 . (9)
The formulas (89) are applicable to concretes of grade C50/60 at the most.
On the basis of experimental results Majewski proposed the following (quite accu-
rate) approximating formulas:
( ) ( )[ ],140.0exp024.0exp20014.01 cmcmc ff = (10)
( )[ ].0215.0exp10011.0004.0 cmcu f= (11)
Knowing the values of the above one can determine the points which the graph should
intersect.
Fig. 6.Stress-strain diagram for analysis of structures, according to Eurocode 2
The curve can be also plotted on the basis of the literature 0, 0, 0,0,0. The most
popular formulas are presented in Table 2, but the original symbols have been re-
placed with the uniform denotations used in Eurocode 2.
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Choosing a proper formula form to describe relation c c one should note
whether the longitudinal modulus of elasticity represents initial value Ec(at stress
c = 0) or that of secant modulus Ecm. Most of the formulas use initial modulus Ec
which is neither experimentally determined nor taken from the standards. Anotherimportant factor is the functional dependence itself. Even though the Madrid pa-
rabola has been recognized as a good relation by CEB (Comit Euro-International
du Bton), this function is not flexible enough to correctly describe the perform-
ance of concrete.
Table 2. Stress-strain relation for nonlinear behaviour of structure
Formula name/
sourceFormula form Variables
Madrid parabola
= 12
1
1c
c
ccc E
( )1, ccc Ef =
Desay
& Krishnan
formula
2
1
1
+
=
c
c
cc
c
E
( )1, ccc Ef =
EN 1992-1-1( )
21
2
+
=
k
kfcmc
cm
ccm
fEk 105.1
= ,1c
c
=
( )1,, ccmcmc fEf =
Majewski
formula
ccc E = if cmc felim
( )( )
( )( ) ( )
cmc
cm
c
ccm
c
ccmc
feif
e
ef
e
ef
e
ef
lim
lim
2
lim
1lim
2
lim
2
1lim
2
lim
1412
2
14
2
>
+
+
=
( )lim2 ef
Ec
cm
c =
,
elimin formula (3)
( )1,, ccmcc fEf =
Wang & Hsuformula
=
2
11
2c
c
c
ccmc f
if 1
1
c
c
=
2
1
1/2
1/1
cccmc f
if 11
>c
c
( )1, ccmc ff =
Senz formula 32ccc
cc
DCBA
+++=
symbols in formula (12)
=
11,
,,,
cuc
cucmc
c
ffEf
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Fig. 7. Property of 2nd order parabola
The 2nd order parabola has this property that the tangent of the angle of a tangent
passing through a point on its branch, measured relative to the horizontal axis passingthrough this point, is always double that of the angle measured as the inclination of the
secant passing through the same point and the extremum of the parabola, relative to
the same horizontal axis.
Fig. 8. Relation c
cfor Madrid parabola depending on longitudinal modulus of elasticity
The consequence of this property of the parabola is either the exceedance of the
concretes strength for a correct initial modulus value or the necessity to lower the
value in order to reach a specific stress value in the extreme. Figure 8 shows relation
c cfor the Madrid parabola for grade C16/20 concrete. The following batch deno-
tations were assumed:
Ecm Ec =Ecm = 28608 MPa was assumed as the initial modulus, calculated ex-tremumfcm= 26.81 MPa;
Ec/Ecu= 2 the doubled tangent of the angle of the secant passing through point(c1,fcm), amounting toEc = 25602 MPa, calculated extremumfcm= 24.00 MPa (correct);
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0.4fcm the value of initial modulusEc = 31808 MPa matched so that the curveintersects point (c, 0.4fcm), calculated extremumfcm= 29.81 MPa;
Ec=Ecm a straight line describing the elastic behaviour of the concrete up to(c, 0.4fcm).As one can see, when initial modulusEcis assumed to amount toEcm, the strength of
the concrete is much overrated despite the fact that the initial modulus is still underrated
(numericallyEcmis not the highest value). In the case of parabolic relations one should
artificially lower modulusEcin order for the graph to intersect the correct valuefcm.
A sufficiently detailed description of relation cchas been proposed by Senz. The
function with a 3rd order polynomial in the denominator (Table 2) depends on the vari-
ables:
( )( )
.
1
1
1,
,
1,
12
2
,
1
12
1
234
13
2
1
1
13
4
13
4
3
43
==
==
=
=
+
==
PP
PPP
f
EP
f
fPP
fP
PD
fP
PC
fP
PP
BEA
cm
cc
cu
cm
c
cu
ccmccm
cmc
(12)
Fig. 9. Comparison of curves c-cbased on table 2 relations for grade C16/20 concrete
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Modelling of reinforced concrete structures and composite structures... 633
The above notation allows one to shape the function graph so that it intersects
points: (c1,fcm) and (cu,fcu). The relation proposed by Wang and Hsu is an interesting
notation. These are two functions describing the curves ascending and descending
part. They also include coefficient representing the reduction in compressive stressof concrete resulting from locating reinforcing bars in the compressed zone. In figure
9 = 1.0 (no reinforcement taken into account). It is worth noticing that Wang and
Hsu relation, the Majewski relation and the Madrid parabola almost coincide. The
same applies to the Desay and Krishanan relation and the Senz relation, but in the
latter case the same point (cu,fcu) which followed from the Desay and Krishanan for-
mula was assumed since a lower value of function fcu would result in an improper
shape of the curve. The standard relation yields intermediate results.
4. Stress-strain curve for uniaxial tension
The tensile strength of concrete under uniaxial stress is seldom determined through a di-
rect tension test because of the difficulties involved in its execution and the large scatter of
the results. Indirect methods, such as sample splitting or beam bending, tend to be used [2]:
( ).30.0 3/2ckctm ff = (13)
Fig. 10. Definition of strain after cracking tension stiffening 0
The term cracking strainck
t~ is used in CDP model numerical analyses. The aim is
to take into account the phenomenon called tension stiffening. Concrete under tension
is not regarded as a brittle-elastic body and such phenomena as aggregate interlocking
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P. KMIECIK, M. KAMISKI634
in a crack and concrete-to-steel adhesion between cracks are taken into account. This
assumption is valid when the pattern of cracks is fuzzy. Then stress in the tensioned
zone does not decrease sharply but gradually. The strain after cracking is defined as the
difference between the total strain and the elastic strain for the undamaged material:
,~ 0eltt
ckt = (14)
.0c
telt
E
= (15)
Plastic strain plt~ is calculated similarly as in the case of compression after defin-
ing degradation parameter dt.
In order to plot curve t tone should define the form of the weakening function.
According to the ABAQUS users manual, stress can be linearly reduced to zero, start-
ing from the moment of reaching the tensile strength for the total strain ten times
higher than at the moment of reachingfctm. But to accurately describe this function themodel needs to be calibrated with the results predicted for a specific analyzed case.
Fig. 11. Modified Wang & Hsu formula for weakening function at tension stiffening for concrete C16/20
The proper relation was proposed by, among others, Wang and Hsu [11]:
,if
if
4.0
>
=
=
crt
t
crcmt
crttct
f
E
(16)
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where crstands for strain at concrete cracking. Since tension stiffening may consid-
erably affect the results of the analysis and the relation needs calibrating for a given
simulation, it is proposed to use the modified Wang & Hsu formula for the weakening
function:
,if crt
n
t
crcmt f
>
= (17)
where nrepresents the rate of weakening.
5. Conclusion
Problems with solution convergence may arise when full nonlinearity of the material
(concrete) with its gradual degradation under increasing (mainly tensile) stress is assumed.
Simple FE techniques, consisting in reducing the size of stress increment or increasing the
maximum number of steps when solving the problem by means of the Newton-Raphson
approach, may prove to be insufficient. Therefore the CDP model uses viscosity parameter
which allows one to slightly exceed the plastic potential surface area in certain suffi-
ciently small problem steps (in order to regularize the constitutive equations). Viscoplastic
adjustment consists in choosing such > 0 that the ratio of the problem time step to ap-
proaches infinity. This means that it is necessary to try to match the value ofa few times
in order to find out how big an influence it has on the problem solution result in
ABAQUS/Standard and to choose a proper minimum value of this parameter.The CDP model makes it possible to define concrete for all kinds of structures. It is
mainly intended for the analysis of reinforced concrete structures and concrete-con-
crete and steel-concrete composite structures. However, it is recommended that before
an analysis of the structure one should test the behaviour of the material itself, e.g. by
carrying out a numerical analysis of cylindrical samples under compression, in order
to compare it with the given stress-strain relation. Because of the character of concrete
failure, some quantities can be rather assumed than determined in laboratory tests.
Therefore the assumptions should be verified by comparing other parameters, e.g. the
deflection of the modelled structural component. This means that the model parame-
ters often need to be calibrated several times in the course of the numerical analysis.
Wrocaw Centre for Networking and Supercomputing holds a licence for the Abaqus software(http://www.wcss.wroc.pl), grant No. 56.
References
[1] ABAQUS:Abaqus analysis user's manual, Version 6.9, 2009, Dassault Systmes.[2] Eurocode 2: Design of concrete structures. Part 1-1: general rules and rules for buildings,
Brussels, 2004.
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P. KMIECIK, M. KAMISKI636
[3] Godycki-wirko T.: The mechanics of concrete, Arkady, Warsaw, 1982.[4] Jankowiak I., Kkol W., Madaj A.:Identification of a continuous composite beam numeri-
cal model, based on experimental tests, 7thConference on Composite Structures, ZielonaGra, 2005, pp. 163178.
[5] Jankowiak I., Madaj A.:Numerical modelling of the composite concrete-steel beam inter-layer bond, 8thConference on Composite Structures, Zielona Gra, 2008, pp. 131148.
[6] Kmita A., Kubiak J.:Investigation of concrete structures. Guide to laboratory classes,
Wrocaw University of Technology Publishing House, Wrocaw, 1993.[7] Lee J., Fenves G.L.:Plastic-damage model for cyclic loading of concrete structures, Journal
of Engineering Mechanics, Vol. 124, No. 8, 1998, pp. 892900.
[8] Lubliner J., Oliver J., Oller S, Oate E.: A plastic-damage model for concrete, Interna-tional Journal of Solids and Structures, Vol. 25, 1989, pp. 299329.
[9] Majewski S.: The mechanics of structural concrete in terms of elasto-plasticity, SilesianPolytechnic Publishing House, Gliwice, 2003.
[10] Szumigaa A.: Composite steel-concrete beam and frame structures under momentaryload, Dissertation No. 408, PoznaPolytechnic Publishing House, Pozna, 2007.[11] Wang T., Hsu T.T.C.:Nonlinear finite element analysis of concrete structures using new
constitutive models, Computers and Structures, Vol. 79, Iss. 32, 2001, pp. 27812791.
Modelowanie konstrukcji elbetowych oraz zespolonych
z uwzgldnieniem degradacji wytrzymaociowej betonu
Symulacje komputerowe w dziedzinie konstrukcji elbetowych swyzwaniem z uwagi nawaciwoci materiau, jakim jest beton. W przeciwiestwie do stali, jest to materia, ktry
podczas ciskania wykazuje nieliniowojuod samego pocztku swojej pracy. Ponadto pod-czas rozcigania ulega znacznie szybszej degradacji, co skutkuje problemami natury nume-rycznej. W niniejszej pracy opisano parametry niezbdne do prawidowego zamodelowania
betonu w zoonym stanie naprenia. Parametry te przedstawiono na przykadzie modeluConcrete Damaged Plasticity zawartego w programie ABAQUS.