Modelling of Reinforced Concrete Structures

8/12/2019 Modelling of Reinforced Concrete Structures

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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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P. KMIECIK, M. KAMISKI624

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

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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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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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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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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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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[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.

Modelling of Reinforced Concrete Structures

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