139c-modelling of reinforced concrete structures-kamiŃski-2011

ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING

Vol. XI 2011 No. 3

Modelling of reinforced concrete structuresand composite structures with concrete strength degradationtaken into consideration

P. KMIECIK, M. KAMIŃSKIWrocław University of Technology, Wybrzeże Wyspiańskiego 25, 50-370 Wrocław, Poland.

Because of the properties of the material (concrete), computer simulations in the field of reinforcedconcrete 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 undercompound stress are described in this paper. The parameters are illustrated using the Concrete DamagedPlasticity 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 crushingunder compression. However, concrete strength determined in simple states of stress (uni-axial compression or tension) radically differs from the one determined in complex statesof stress. For example, the same concrete under biaxial compression reaches strength ofbetween ten and twenty per cent higher than in the uniaxial state while in the hydro-staticstate (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 widerange 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 islocated 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-dilatationalstrain energy and the boundary surface itself in the stress space assumes the shape of a

P. KMIECIK, M. KAMIŃSKI624

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 withthe actual behaviour of concrete 0.

Fig. 1. Drucker–Prager boundary surface 0: a) view, b) deviatoric cross section

The CDP (Concrete Damaged Plasticity) model used in the ABAQUS software isa modification of the Drucker–Prager strength hypothesis. In recent years the latter hasbeen 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 itis governed by parameter Kc.

Fig. 2. Deviatoric cross section of failure surface in CDP model 0

Modelling of reinforced concrete structures and composite structures... 625

Physically, parameter Kc is interpreted as a ratio of the distances between thehydrostatic 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 itassumes the value of 1, the deviatoric cross section of the failure surface becomesa circle (as in the classic Drucker–Prager strength hypothesis). Majewski reportsthat according to experimental results this value for mean normal stress equal tozero amounts to 0.6 and slowly increases with decreasing mean stress. The CDPmodel 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 andWarnke (1975). It is a theoretical-experimental criterion based on triaxial stresstest results.

Similarly, the shape of the plane’s meridians in the stress space changes. Experi-mental results indicate that the meridians are curves. In the CDP model the plasticpotential surface in the meridional plane assumes the form of a hyperbola. The shapeis adjusted through eccentricity (plastic potential eccentricity). It is a small positivevalue 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 thesegment between the vertex of the hyperbola and the intersection of the asymptotes ofthis hyperbola (the centre of the hyperbola). Parameter eccentricity can be calculatedas a ratio of tensile strength to compressive strength [4]. The CDP model recommendsto assume є = 0.1. When є = 0, the surface in the meridional plane becomes a straightline (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 theconcrete undergoes failure under biaxial compression. σb0/σc0 ( fb0 / fc0) is a ratio of thestrength in the biaxial state to the strength in the uniaxial state. The most reliable inthis regard are the experimental results reported by Kupler (1969). After their approxi-mation with the elliptic equation, uniform biaxial compression strength fcc is equal to1.16248 fc 0. The ABAQUS user’s manual specifies default σb0/σc0 = 1.16.

The last parameter characterizing the performance of concrete under compoundstress is dilation angle, i.e. the angle of inclination of the failure surface towards the


hydrostatic axis, measured in the meridional plane. Physically, dilation angle ψ is in-terpreted as a concrete internal friction angle. In simulations usually ψ = 36° 0,0 or ψ= 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 stressParameter name ValueDilatation angle 36

Eccentricity 0.1fbo /fco 1.16κ 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 ABAQUSuser’s manual. The other parameters describing the performance of concrete are deter-mined for uniaxial stress. Table 1 shows the model’s parameters characterizing its per-formance under compound stress.


3. Stress-strain curve for uniaxial compression

The stress-strain relation for a given concrete can be most accurately described onthe basis of uniaxial compression tests carried out on it. Having obtained a graph fromlaboratory tests one should transform the variables. Inelastic strains in

cε~ 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 εεε −= (1)

.0

0 Ecel

cσε = (2)

Fig. 5. Definition of inelastic strains 0

When transforming strains, one should consider from what moment the materialshould 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 toconcrete strength from this formula:

.80

exp1lim ⎟⎠⎞

⎜⎝⎛ −−= cfe (3)


This ceiling can be simply arbitrarily assumed as 0.4 fcm. Eurocode 2 specifies themodulus of elasticity for concrete to be secant in a range of 0–0.4 fcm. Since the basicdefinition of the material already covers the shear modulus and the longitudinalmodulus of concrete, at this stage it is good to assume such an inelastic phase thresh-old that the initial value of Young’s modulus and the secant value determined accord-ing to the standard will be convergent. In most numerical analyses it is rather not theinitial behaviour of the material, but the stage in which it reaches its yield strengthwhich is investigated. Thanks to the level of 0.4·fcm there 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 toone for the total loss of load-bearing capacity. These values can also be obtained fromuniaxial compression tests, by calculating the ratio of the stress for the declining partof 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

σεε−

−= (4)

where E0 stands 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 cancalculate stress σc for uniaxial compression and its effective stress cσ .

( ) ( ),~1 0pl

cccc Ed εεσ −−= (5)

( ) ( ).~1 0

plcc

c

cc E

dεεσσ −=

−= (6)

3.1. Plotting stress-strain curve without detailed laboratory test results

On the basis of uniaxial compression test results one can accurately determine theway 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 orderto 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)


where: fcm [MPa],Ecm [GPa].Other values defining the location of characteristic points on the graph are strain εc1

at average compressive strength and ultimate strain εcu 0:

( ) ,7.0 31.01 cmc f=ε (8)

εcu = 3.5 ‰. (9)

The formulas (8–9) 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 −−−−=ε (10)

( )[ ].0215.0exp10011.0004.0 cmcu f−−−=ε (11)

Knowing the values of the above one can determine the points which the graph shouldintersect.

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 mostpopular formulas are presented in Table 2, but the original symbols have been re-placed with the uniform denotations used in Eurocode 2.


Choosing a proper formula form to describe relation σc – εc one should notewhether the longitudinal modulus of elasticity represents initial value Ec (at stressσc = 0) or that of secant modulus Ecm. Most of the formulas use initial modulus Ecwhich 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 (Comité Euro-Internationaldu Béton), this function is not flexible enough to correctly describe the perform-ance of concrete.

Table 2. Stress-strain relation for nonlinear behaviour of structureFormula name/

source Formula form Variables

Madrid parabola⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1211

c

cccc E

εεεσ ( )1, ccc Ef εσ =

Desay& Krishnan

formula

2

1

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

c

c

ccc

E

εε

εσ( )1, ccc Ef εσ =

EN 1992-1-1( )η

ηησ21

2

−+−

=k

kfcmc

cm

ccm f

Ek 105.1 ε= ,

1c

c

εεη =

( )1,, ccmcmc fEf εσ =

Majewskiformula

ccc E εσ = if cmc felim≤σ

( )( )

( )( ) ( )

cmc

cmc

ccm

c

ccmc

feif

eef

eef

eef

lim

lim

2lim

1lim

2lim

2

1lim

2lim

14122

142

>

⎪⎪

⎭

⎪⎪

⎬

⎫

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛

−−

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=

σ

εε

εεσ

( )lim2 efEc

cmc −=

ε,

elim in formula (3)

( )1,, ccmcc fEf εσ =

Wang & Hsuformula

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

2

112

c

c

c

ccmc f

ζεε

ζεεζσ if 1

1

≤c

c

ζεε

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=2

1

1/21/1

ζζεεζσ cc

cmc f if 11

>c

c

ζεε

( )1, ccmc ff εσ =

Sáenz formula 32ccc

cc DCBA εεε

εσ+++

=

symbols in formula (12)⎟⎟⎠

⎞⎜⎜⎝

⎛=

11,,,,

cuc

cucmcc

ffEf

εεσ


Fig. 7. Property of 2nd order parabola

The 2nd order parabola has this property that the tangent of the angle of a tangentpassing 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 thesecant passing through the same point and the extremum of the parabola, relative tothe same horizontal axis.

Fig. 8. Relation σc–εc for Madrid parabola depending on longitudinal modulus of elasticity

The consequence of this property of the parabola is either the exceedance of theconcrete’s strength for a correct initial modulus value or the necessity to lower thevalue in order to reach a specific stress value in the extreme. Figure 8 shows relationσc – εc for 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-tremum fcm = 26.81 MPa;

• Ec/Ecu = 2 – the doubled tangent of the angle of the secant passing through point(εc1, fcm), amounting to Ec = 25602 MPa, calculated extremum fcm = 24.00 MPa (correct);


• 0.4 fcm – the value of initial modulus Ec = 31808 MPa matched so that the curveintersects point (εc, 0.4·fcm), calculated extremum fcm = 29.81 MPa;

• Ec = Ecm – a straight line describing the elastic behaviour of the concrete up to(εc, 0.4·fcm).

As one can see, when initial modulus Ec is assumed to amount to Ecm, the strength ofthe concrete is much overrated despite the fact that the initial modulus is still underrated(numerically Ecm is not the highest value). In the case of parabolic relations one shouldartificially lower modulus Ec in order for the graph to intersect the correct value fcm.

A sufficiently detailed description of relation σc – εc has been proposed by Sáenz. Thefunction with a 3rd order polynomial in the denominator (Table 2) depends on the vari-ables:

( )( )

.

11

1,

,

1,12

2,1

12

1

234

13

21

1

13

4

13

4

3

43

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

−−−

==

==

−=

−−=

−+==

PPPPP

fEP

ffPP

fPPD

fPPC

fPPPB

EA

cm

cc

cu

cm

c

cu

ccmccm

cmc

εεε

εε(12)

Fig. 9. Comparison of curves σc-εc based on table 2 relations for grade C16/20 concrete


The above notation allows one to shape the function graph so that it intersectspoints: (εc1, fcm) and (εcu, fcu). The relation proposed by Wang and Hsu is an interestingnotation. These are two functions describing the curve’s ascending and descendingpart. They also include coefficient ζ representing the reduction in compressive stressof concrete resulting from locating reinforcing bars in the compressed zone. In figure9 ζ = 1.0 (no reinforcement taken into account). It is worth noticing that Wang andHsu relation, the Majewski relation and the Madrid parabola almost coincide. Thesame applies to the Desay and Krishanan relation and the Sáenz relation, but in thelatter case the same point (εcu, fcu) which followed from the Desay and Krishanan for-mula was assumed since a lower value of function fcu would result in an impropershape 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 ofthe 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 strain cktε

~ is used in CDP model numerical analyses. The aim isto take into account the phenomenon called tension stiffening. Concrete under tensionis not regarded as a brittle-elastic body and such phenomena as aggregate interlocking


in a crack and concrete-to-steel adhesion between cracks are taken into account. Thisassumption is valid when the pattern of cracks is fuzzy. Then stress in the tensionedzone does not decrease sharply but gradually. The strain after cracking is defined as thedifference between the total strain and the elastic strain for the undamaged material:

,~0eltt

ckt εεε −= (14)

.0c

telt E

εε = (15)

Plastic strain pltε

~ is calculated similarly as in the case of compression after defin-ing degradation parameter dt.

In order to plot curve σt – εt one should define the form of the weakening function.According to the ABAQUS user’s manual, stress can be linearly reduced to zero, start-ing from the moment of reaching the tensile strength for the total strain ten timeshigher than at the moment of reaching fctm. 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

if4.0

⎪⎭

⎪⎬

⎫

>⎟⎟⎠

⎞⎜⎜⎝

⎛=

≤=

crtt

crcmt

crttct

f

E

εεεεσ

εεεσ

(16)


where εcr stands for strain at concrete cracking. Since tension stiffening may consid-erably affect the results of the analysis and the relation needs calibrating for a givensimulation, it is proposed to use the modified Wang & Hsu formula for the weakeningfunction:

,if crt

n

t

crcmt f εε

εεσ >⎟⎟

⎠

⎞⎜⎜⎝

⎛= (17)

where n represents 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 themaximum number of steps when solving the problem by means of the Newton-Raphsonapproach, may prove to be insufficient. Therefore the CDP model uses viscosity parameterμ 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). Viscoplasticadjustment consists in choosing such μ > 0 that the ratio of the problem time step to μ ap-proaches infinity. This means that it is necessary to try to match the value of μ a few timesin order to find out how big an influence it has on the problem solution result inABAQUS/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 ismainly intended for the analysis of reinforced concrete structures and concrete-con-crete and steel-concrete composite structures. However, it is recommended that beforean analysis of the structure one should test the behaviour of the material itself, e.g. bycarrying out a numerical analysis of cylindrical samples under compression, in orderto compare it with the given stress-strain relation. Because of the character of concretefailure, some quantities can be rather assumed than determined in laboratory tests.Therefore the assumptions should be verified by comparing other parameters, e.g. thedeflection 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.

Wrocław 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 Systèmes.[2] Eurocode 2: Design of concrete structures. Part 1-1: general rules and rules for buildings,

Brussels, 2004.


[3] Godycki-Ćwirko T.: The mechanics of concrete, Arkady, Warsaw, 1982.[4] Jankowiak I., Kąkol W., Madaj A.: Identification of a continuous composite beam numeri-

cal model, based on experimental tests, 7th Conference on Composite Structures, ZielonaGóra, 2005, pp. 163–178.

[5] Jankowiak I., Madaj A.: Numerical modelling of the composite concrete-steel beam inter-layer bond, 8th Conference on Composite Structures, Zielona Góra, 2008, pp. 131–148.

[6] Kmita A., Kubiak J.: Investigation of concrete structures. Guide to laboratory classes,Wrocław University of Technology Publishing House, Wrocław, 1993.

[7] Lee J., Fenves G.L.: Plastic-damage model for cyclic loading of concrete structures, Journalof Engineering Mechanics, Vol. 124, No. 8, 1998, pp. 892–900.

[8] Lubliner J., Oliver J., Oller S, Oñate E.: A plastic-damage model for concrete, Interna-tional Journal of Solids and Structures, Vol. 25, 1989, pp. 299–329.

[9] Majewski S.: The mechanics of structural concrete in terms of elasto-plasticity, SilesianPolytechnic Publishing House, Gliwice, 2003.

[10] Szumigała A.: Composite steel-concrete beam and frame structures under momentaryload, Dissertation No. 408, Poznań Polytechnic Publishing House, Poznań, 2007.

[11] Wang T., Hsu T.T.C.: Nonlinear finite element analysis of concrete structures using newconstitutive models, Computers and Structures, Vol. 79, Iss. 32, 2001, pp. 2781–2791.

Modelowanie konstrukcji żelbetowych oraz zespolonychz uwzględnieniem degradacji wytrzymałościowej betonu

Symulacje komputerowe w dziedzinie konstrukcji żelbetowych są wyzwaniem z uwagi nawłaściwości materiału, jakim jest beton. W przeciwieństwie do stali, jest to materiał, którypodczas ściskania wykazuje nieliniowość już od samego początku swojej pracy. Ponadto pod-czas rozciągania ulega znacznie szybszej degradacji, co skutkuje problemami natury nume-rycznej. W niniejszej pracy opisano parametry niezbędne do prawidłowego zamodelowaniabetonu w złożonym stanie naprężenia. Parametry te przedstawiono na przykładzie modelu„Concrete Damaged Plasticity” zawartego w programie ABAQUS.

139c-modelling of reinforced concrete structures-kamiŃski-2011

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