Computational Modeling of Reinforced Concrete Structures

Hartl H., Beer G. : "Computational Modeling of Reinforced Concrete Structures", Freytag B.,

Stebernjak B. (Hsg.), Festschrift zum 60. Geburtstag von Lutz Sparowitz, TU-Graz, pp. 105-

114 (2000)

2 Computational Modeling of Reinforced Concrete Structures

COMPUTATIONAL MODELING OF REINFORCED CONCRETESTRUCTURES

Helmut HartlTU-Graz, Institute for Structural Concrete

Gernot BeerTU-Graz, Institute for Structural Analysis

ABSTRACT: Some issues of implementation of non linear finite element modeling of reinforced concretestructures are presented. Firstly, the elasto-plastic and the elasto-visco-plastic return algorithms are compared.A method is introduced which allows to simulate the elasto-plastic solution within an iterative procedurewhere no derivative of the yield function is needed. Secondly, the update of stresses within the supplementaryslip model is discussed. The steel stress is initially overestimated due to the perfect bond assumption in theglobal stiffness matrix. The supplementary slip model relaxes this steel stress, thus even within amonotonically increasing external load a reverse loading situation is obtained numerically within theiterations. Two stress update techniques will be discussed, where this overestimation of the steel stress neverresults in plastic strain at the end of the iteration. Finally, the number of sample points employed for the stressrecovery is discussed. Due to high stress gradients within a concrete structure, it seems to be necessary toemploy sample points close to the boundary of the elements. This is studied numerically on a beam for thespecific situation of reinforced concrete.

1 VISION

About two decades ago, on a visit to the Universityof Applied Sciences in Vienna, the second authorwas shown some visionary ideas by Lutz Sparowitzon an integrated approach to the design of reinforcedconcrete structures. The vision was of a completelyautomatic method of calculation and design. Startingfrom the general idea, for example the plan ofconstructing a bridge as part of a highway, theconcept involved the automatic generation andtransfer of all data until construction (reinforcement)plans were automatically generated.A powerful data base is necessary to achieve thisvision i.e. one which maintains all the informationrelated to the object and passes information from theinitial design to the analysis model and finally to theconstruction drawings. A major issue is thesimplification of the actual structure for numericalanalysis.One option of numerical analysis is based on

continuum mechanics. If the FE-method is appliedfor a comprehensive assessment of reinforcedorprestressed concrete structures, the program mustbe able to account for the nonlinearity of theconcrete and for the presence of the rebars. In such amodel we may define every single rebar in the inputphase but this information should be provided indigital form from the data base so no “hand” inputshould be necessary. The merit of such a scheme isthat the amount of idealization is reducedconsiderably and that the structure response can becomputed more accurately.With this contribution, to mark the 60th birthday ofLutz Sparowitz, we hope to make a smallcontribution towards the realization of his vision.

2 INTRODUCTION

When the stress state is beyond the yield surface aniterative procedure has to be employed in order toreturn the stress state to the yield surface (Owen[1]). Several methods are available therefore which

Helmut Hartl & Gernot Beer

can be split into two families. The elasto-plastic andthe elasto-visco-plastic approaches. For somegeneral constitutive models the former might bedifficult to implement since the needed derivative ofthe yield function can be involved and thecomputation may be expensive. The latter does notneed the derivative of the yield function. The crux isto find the right time step length in order to avoidunnecessary computation time on the one hand andto find convergence on the other hand. Here, thesetwo algorithms are compared. A procedure whichcomputes the elasto-plastic solution employing theelasto-visco-plastic theory is shown in section 3.The rebars are incorporated according to theembedded approach (Elwi [2]). Within thisapproach, rigid bond is assumed at the level of theglobal stiffness matrix. Hence, the steel stress isoverestimated due to the perfect bond assumption inthe global stiffness matrix. The supplementary slipmodel is able to account for bond slip situationsbetween rebars (or tendons) and concrete. Interfaceelements (Beer [3]) are introduced upon the knowndisplacement field of the parent elements at thematerial level of the reinforcing steel (Hartl [4,5])and the steel stress will be relaxed. Due to thisscheme, a monotonically increasing external loadwill result in a reverse loading situation for the steelwithin the iterations, as shown in section 4. Twostress update techniques (Crisfield [6], Nyssen [7])will be discussed. In both methods theoverestimation of the steel stress at the beginning ofthe iteration does not result in a plastic strain at theend of the iteration.The effect of the employed integration scheme onthe numerical results is discussed in Zienkiewicz[8]. Concrete structures have in general a very highstress gradient and stresses have the maximum valueat the boundaries of the domain. Thus, it seems to benecessary to employ sample points for the stressrecovery and the subsequent integration of residualstresses to residual forces close to the boundary ofthe domain. The effect of the number and location ofthe sample points dependent on the physicalnonlinearity of reinforced concrete is discussed on anumerical example in section 5.The work presented here is implemented into aBoundary Element / Finite Element program(BEFE [9]). It provides a useful pre- and post-processing environment. Some tools formanipulating the reinforcement were added and aninterface to a CAD program is in preparation [10].

3 CONVERGENCE IN NONLINEARSITUATIONS

A material may behave elastically as long as alimiting stress state described by the yield surface isnot exceeded. The displacements (u) are obtained ina FE-analysis from the equation: uFK =!"1 ,where K is the stiffness matrix and F are the appliedforces. Strains (! ) can be calculated fromdisplacements by compatibility: uB! != , where Bis the strain displacement matrix. These strains arerelated to stresses by employing a constitutive law.As long as this relation is linear and the stiffnesscorresponds to K, equilibrium will be gained bydefault. Beyond the elastic limit, an iterativeprocedure is needed in order to fulfill theconstitutive relation and equilibrium.

3.1 Review of the Elasto-Plastic Return

The general form of this procedure is

"D""

##!!$"= Q

el % (1)

where" corrected stress on the yield surface

el" elastically predicted stress by the stiffnessmatrix

%$ plastic strain rate multiplier (scalar)D constitutive matrixQ plastic potential function====& &&&&el

&&&&T &&&&

&

F (yield surface)

Figure 1 Return to the Yield Surface within theelasto-plastic approach

Eq. (1) must be evaluated at a distinct point withinthe applied increment. It can be computed at thetransition point ( T" ) of the elastically predictedincrement with the yield surface (forward Eulerprocedure) or at the final point (" ) where the stressis returned to the yield surface (backward Eulerprocedure). Evaluation of Eq. (1) at the end of theelastically predicted increment ( el" ) is very

"D

##!!$ Q%


appealing, since this point is obtainedstraightforwardly. It can be viewed as a form of thebackward Euler algorithm. A detailed discussion ofthese elasto-plastic algorithm for associativeplasticity is presented in Crisfield [6].The plastic strain rate multiplier %$ is a scalarwhich is obtained by two basic principles of elasto-plasticity. First, the plastic strain increment ( pd! ) isperpendicular to the plastic potential function ( Q );second, the stress state must never exceed the yieldsurface ( F ). The derivation for associative plasticityis given in Owen [1], for non associative plasticity

%$ is given in Potts [11]

"D

" ##

##=$ QF

F%(2)

where F is the yield function

Eq (2) requires the derivative of the yield functionand the derivative of the plastic potential function inorder to obtain a stress state which is on (or moreprecisely: close to) the yield surface. If theincrements are either not very small and/or the yieldsurface has a strong curvature, the computed stressstate may have a considerably offset to the yieldsurface, no matter at which reference stress state

%$ is computed. In general, the computedprediction of the corrected stress state lies outsidethe yield surface as shown in Figure 2.

Figure 2 Offset of the numerical solution

In order to correct for this offset Crisfield [12] andOritz [13] recommended to repeat the procedure(Eq. (1) and (2)) until the new stress state issufficient close to the yield function (“operatorsplitting”).

3.2 Review of the Elasto-Visco-Plastic Return

"

!##QFtp

t $=$

d'''' d''''vp F

d&&&&vp Q

Figure 3 Illustration of the elasto-visco-plasticapproach

A different procedure for the return to the yieldsurface is the elasto-visco-plastic approach. Perzynaproposed a theory [14,15] which is able to accountsimultaneously for two material behaviors: plasticityand rheology. It is assumed that plastic strains needtime to develop. Hence, stress states beyond theyield surface (F > 0) are allowed for t ( ) withinthis method. The viscosity parameter (*) is set tounity when rheology is not of interest. Then, thetime ( t ) has no physical meaning any more, it willbe reduced to a convergence parameter ( t ) only.The stress at ( tt $+ ) within the return procedure tothe yield surface is (Beer [16])

"D""

##QFtttt $!"=$+ (3)

where

"tt $+ stress at tt $+

"##QFt$ visco-plastic strain increment within

increment t$ step function:

! F = 0 if F ! 0! F = f(F) if F > 0

In the subsequent section only the product t$ ·!F isof interest any more. Thus, the value 1.0 is assignedto !F if F > 0, this is f(F) = F

0.

The strain rate "## /Q gets multiplied by the scalar

t$ ·!F . A small value of t$ should assureconvergence but can be time consuming. On theother hand, a large time step can deteriorate theaccuracy significantly. For some classical models,analytical time step lengths have been found byCormeau [17], but not for the general case.


Zienkiewicz and Cormeau [18] have reported that atime step above a certain magnitude might result inan unstable iteration indicated by oscillation ofresidual forces. They gave an empirical rule for thetime step and they limit this rule to associatedplasticity with classical models. Applying this ruleto general models with a non associative flow ruleachieved indeed only limited success (Hartl [19]).

3.3 Proposed strain rate multiplier (time step)

Eq (1) and Eq (3) are similar. Both $% and t$ ·! Fcan be referred to a scaling value for the strain rate.From the comparison of the two approaches followsthat any $% which produces a stress state along thepath from the elastic prediction ( el" ) to the returnpoint on the yield surface (" ) is acceptable withinthe iterative scheme. This multiplier, which resultsin the limiting stress state on the yield surface, is ofinterest for a fast computation.

0=!"#

$

##$!""

D" QF % (4)

The condition given in Eq (4) can be fulfillednumerically by employing an iterative scheme.Here, the Newton root finding algorithm isemployed. The derivative of nF is approximatedwith the finite differences between two trial points.

0

1

11 "

""

D""##!

""!"=

"

"+

QFF

F

nn

nn

nnn (5)

F

n0

n1 n2

"

Figure 4 Iterative scheme for obtaining a stress stateon the yield surface

The initial trial stress is elo "" = . Then, a smallvalue is assigned to $% and 1" is computed byevaluation of Eq (1). For all subsequent steps Eq (5)applies and a fast convergence is obtained. The finalstress is always on the yield surface.There is no need within this method to compute the

strain rate multiplier $% or t$ explicitly, since thefinal stress state " , which is on the yield surface, isknown at the end of the iteration process of Eq (5).A merit of this approach is that the derivative of theyield function (F ) is not needed in order to compute" . This is useful when the yield function is notsmooth throughout the stress space, since only asmooth plastic potential function is necessary for theprocedure presented here. This procedure isappealing as well when the derivative of the yieldfunction is very involved and costly to compute.An offset of the obtained stress state to the yieldsurface as indicated in Figure 2 cannot occur withinthe proposed method, the obtained stress state isalways on the yield surface. Nevertheless, thedirection of plastic flow (given by "##Q ) changesusually its orientation within the iteration process tothe final stress state on the yield surface (" ). Hence,the solution can still be erroneous. Improvementscan be made in different manners. One way is tosplit the elastic increment in sub increments asillustrated in Figure 5a. Another approach is to splitthe returning stress increment into sub increments asillustrated in Figure 5b.

a) split of elastic increment b) split of exceeding incrementFigure 5 splitting of the stress increment

A third method is similar to this one illustrated inFigure 5b. The corrective increment is scaled by afixed factor. Thus, only a certain ratio of theexceeding stresses are mapped as residual nodalforces and get reapplied on the domain. This followsthe concept of visco-plasticity, where a stress statebeyond the yield surface is allowed for t ( ).An automatic scaling procedure for the excessincrements is proposed. The user defines themaximum ratio of the excess increment, which maybe mapped as residual forces within one step. Thisratio is called time scale. A value of 1.0 for the timescale gives the elasto-plastic solution. If the iterativeprocedure follows a convergent path, the residualforces will decrease with each iteration step.Divergence is indicated by increasing residualforces. For each divergent iteration the time scale is


reduced by 30%. When a convergent path isachieved again, the time scale will be increased by10% for each stable iteration until the maximumtime scale is reached again.

3.3.1 Application to Concrete

In this program (BEFE) concrete is modeled withOttosen yield criterion [20]. The Ottosen criterionseems to have corners as shown in Figure 6b.However, a detailed investigation shows that it issmooth throughout the entire stress space. Eventough, within the classical elasto-plastic algorithmand employed normality rule for the flow, a smallincrement would be necessary at stress states inthese regions where the yield surface has a strongcurvature. In addition, the derivative of the yieldfunction is very involved symbolically, and thenumerical evaluation would be time consuming aswell.In order to account for mode I cracks in a simple butrobust fashion the yield criterion is combined withRankine tension cut off, which is not smooth.Additional edges are obtained at the intersection ofthe two surfaces. The edges could be roundednumerically if associated flow has to be considered.But the direction of plastic flow is not well knownfor concrete materials and neither of importance inmany cases. Thus, the van Mises cylinder isemployed as plastic potential surface.

yield surface:Rankine tension cut offOttosen criterion

plastic potential surf.:van Mises cylinder

a) &1/&2 plane (plane stress) b) octahedral plane

Figure 6 Constitutive model for concrete

4 SUPPLEMENTARY SLIP MODEL -STRESS UPDATE

When reinforcement is modeled utilizing theembedded approach (Elwi [2]) and Hartl [5]), thesame displacement field is assigned to the parentelement (2D/3D) and to the one dimensional rebarelement. Perfect bond is obtained. A characteristicstress distribution along the rebar is shown in Figure7b. Bond slip/tendon sliding can be accounted for atthe material level by introducing interface elementssupplementary between rebar and concrete after thedisplacement field has been computed (Hartl [4, 5]).Due to bond slip the steel stress can relax asindicated in Figure 7c.

a) beam subjected to pure moment loading

b) rebar stress by assuming rigid bond

c) rebar stress after employing the supplementary slip model

Figure 7 Stress distribution along the rebar

The basic idea of the supplementary slip algorithmis illustrated in Figure 8 on a truss analogy. Thetruss members are the rebars and the supports arethe concrete. Since we account for slip, the supportsare linked to the rebars by bond springs (which areimplemented as continuous interface elements (Beer[3]). The strains of the parent element along therebar path get integrated and are referred toprescribed displacements of the supports. Thesesupport displacements get transferred to the rebarsby bond springs. The arising relative displacementof a rebar support node and the adjacent rebar nodeis referred to as bond slip. The differences of theforces in the rebar elements now, compared to forcesin these elements with rigid bond, are forces due tobond slip. These forces are mapped back as residualnodal forces of the parent element. The interfaceforces need not to be considered within the residualforces. These forces act on both sides on the rebarand on the concrete but with opposite sign, thusthese interface forces cancel out.


node points rebar points integration point

rebar (truss element)

support (concrete) bond spring (continuous interface element) prescribed displacement

Figure 8 Illustration of the supplementary slipalgorithm on a truss analogy

Within the standard loop over all parent elements,the strains of the rebars along the portion within therespective parent element are computed. After thisloop is finished, the strains along each rebar areknown and the supplementary slip algorithm may beinvoked.When stiffness terms for the interface elements areintegrated, the stiffness modulus is obtained byrelating the relative displacement between the rebarand the according concrete to the constitutiverelation of the interface. The interface stiffness willdecrease with increasing slip as indicated in Figure9. Thus, an iterative scheme is needed.

Figure 9 Generic form of the bond slip relation

Within this iterative scheme the slip will increasemonotonically. The steel stress will beoverestimated in the first iteration followed by acontinuous unloading in all subsequent iterations.This causes no problems as long as the steel doesnot exceed the elastic limit as shown in Figure 10a.The numerical simulation of the load path might gofrom A over B in the first iteration to C and D in thesubsequent iterations. However in nature the loadpath goes directly from A to D, since the slip arisessimultaneously with the applied load.

& ====& B

B

C C D D

A A

' =' a) b)

Figure 10 Loading path for the steel within theiterative process

Other, when the first prediction of the steel stressexceeds the elastic limit as shown in Figure 10b. Aplastic strain increment applies to the steel. Furtheriterations unload the steel only elastically and resultin an erroneous solution, since the plastic strainincrement occurs only due to the iterative scheme,but it does not happen along loading from A to D.The strains get updated with the previous iteration asreference. This method is referred as “iterative strainupdate” in Chrisfield [6].Updating the strains always from the beginning ofthe iterative scheme (point A) overcomes this lack.This is referred as “incremental strain update” inChrisfield [6].A remedy within the “iterative strain update” wasproposed by Nyssen [7]. There, a plastic unloadingback to the last converged reference state (point A)is allowed.From an engineering point of view slip occurspreferable only at the beginning and at the end of therebars. This is in regions where the utilization of therebar is low and the rebar is far below the yieldlimit. Thus, situations according to Figure 10a aremore likely to happen. When unbonded tendons areconsidered, slip occurs along the entire tendonbetween the two fixed ends. Stresses in thesetendons are more or less constant along the entiretendon and nearly independent of the appliedexternal load on the domain. A local concentrationof strains in the concrete due to loading causes sliponly instead of a localization of high tendon forces.Thus, yielding is not a criterion for unbondedtendons and situations shown in Figure 10b do notapply as long as the increments are not too big.However, if the yield limit is exceeded in anunlucky situation and an “iterative strain update” isimplemented more deformations will be obtained. Inmost cases more deformations are considered as anerror on the save side.

+b

u


The performance of the supplementary slip modelwill be investigated in this regard by several casestudies and the scheme that seems to be mostappropriate will be employed.

5 STRESS RECOVERY ANDINTEGRATION OF RESIDUALSTRESSES

Concrete is only able to sustain a very small amountof tensile stresses. When the tensile capacity isexceeded cracks will develop. Additionaldeformation will follow until the steel strain is highenough that the additional steel force is equal to thereleased force by concrete cracking. On thecompression side of the cross section theseadditional deformations cause a concentration of theconcrete stress at the boundary of the cross section.Within the FE method the stresses close to theboundary of the elements are in general not relatedto the constitutive law since the stress recoverypoints are located within the element. (As long asthe Gauss-Legendre quadrature is employed, whichis standard in a FE analysis). A way to overcomethis lack is to employ more sample points. Whenplates are analyzed it is common to employ moresample points over the depth (z-axis) of the platecompared to the number of sample points along thex- and y-axis of the plate. Within a 3D analysisstresses may have the steepest gradient in anydirection, thus more sample points would benecessary for each direction in order to reproducethe technique employed in the 2D analysis. This iscomputational expensive, since the number ofsample points increase by the power of three.In Zienkiewicz [8] the influence of the integrationorder in situations of distorted elements is presented.A higher integration scheme is necessary in order tointegrate the element stiffness more accurate.However, in the subsequent computation othererrors occur. Some of these errors cancel out andsatisfactory results can be achieved by employing astandard integration scheme (2x2x2 or 3x3x3). Theexample presented below concentrates on thephysical nonlinearity of reinforced concrete. Theeffect of the stress recovery technique and theintegration scheme for the residual forces on theresults is studied and compared to analyticalsolutions.

5.1 Investigated Configuration

A single span beam as presented in Figure 11 isinvestigated. It is modeled with 48 elements. Theelements are three dimensional and parabolic shapefunctions are employed. The beam is 600cm long,the cross section is 20/40cm. The system axis of thereinforcement has a distance of 5cm from thebottom end of the cross section. The concrete (fc =30.0MPa, ft = 3.0MPa) is modeled with Ottosenfailure criterion [20] according to Modelcode 90[21]. The material is assumed to be linear elasticwithin the yield envelope (Ec = 30.000MPa). Mode Icracks are simulated by combining the Ottosenmodel with the Rankine tension cut off. The steelbehavior is modeled linear elastic/perfectly plastic.(fs = 550.0MPa, Es = 200.000MPa). On both ends amoment is applied as load. For all cases studied insection 5.2 the magnitude of this moment is0.016MNm. In section 5.3 the ultimate moment willbe computed.

40cm

5

20

Concrete: Steel:Ec = 30.000MPa Es = 200.000MPafc = 30,0MPa fs = 550,0MPaft = 3,0MPa

600cm

Figure 11 3D image, cross section and structuralsystem of the investigated single span beamsubjected to end moments


5.2 Beam subjected to service load

Figure 12 shows the stress distribution for plain(unreinforced) concrete before cracking. In this casethe material behaves linear elastic. The numericalresults with 2 sample points employed for the stressrecovery coincide with the analytical solution asindicated in Figure 12.

Figure 12 stress distribution of uncracked concrete,plain (unreinforced) situation,M = 0.016MNm

The distribution of stresses for the reinforceduncracked beam is shown in Figure 13. The beam isreinforced with 5 bars ,30mm, that is a very highreinforcement ratio. The numerical solution isidentical to the analytical solution for the concretestress and for the steel stress.

Figure 13 stress distribution of uncracked concrete,reinforcement: 5,30mm,M = 0.016MNm

The distribution of stresses for the sameconfiguration as discussed in Figure 13, butassuming cracked conditions for the concrete isshown in Figure 14. The constitutive relation for theconcrete in compression is still linear. In this case,

the analytical solution can be simulated very closelywith 2 stress recovery points. However, a lowerreinforcement ratio results into raising the neutralaxis of the beam and an offset of the numericalsolution will follow, since the integration willassume a wedge of tensile concrete stresses at thelower end of the upper element. These effects willbe studied later on in Figure 15.

Figure 14 stress distribution of cracked concrete,reinforcement: 5,30mm,M = 0.016MNm

For the example studied in Figure 15 thereinforcement is changed to 2,12mm, that isapproximately the minimum reinforcement requiredby design codes. The neutral axis of the beam rises.The stress distribution cannot be simulated well withtwo sample points in each direction. The concretestress is underestimated and the steel stress isoverestimated. However, equilibrium of externaland internal forces is still fulfilled. Predicted strainsare still in a close agreement to the analyticalsolution as shown in Figure 15b. Thus, more samplepoints are only required if a more accurateprediction of the concrete stresses is subject ofinterest. The analytical strains were computed basedon the assumption of a linear elastic / no tensionmaterial.

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Figure 15 stress distribution of cracked concrete,reinforcement: 2,12mm, M = 0.016MNm

5.3 Beam subjected to failure load

The load is now monotonically raised for the lowreinforced (2,12mm) cross section. At a certain loadlevel the steel stress reaches the yield limit. Thisload level is very close the ultimate load, since nofurther redistribution of tensile forces is possible.However, a small increase of the load is stillpossible since the neutral axis raises a bit due to theyielding of the steel. Thus, the cantilever of internalforces increases and the failure load does alsoincrease therefore. The final distribution of concretestresses is presented in Figure 16.

a) concrete stress (steel stress is for all three cases 550MPa)

Figure 16 failure load analysis;reinforcement: 2,12mm(- minimum reinforcement)

The analytical solution shows a concentration ofcompression forces in the upper region of the crosssection only. This stress concentration cannot beaccounted for by employing two sample points ineach direction, the concrete stresses are underpredicted. The failure load is underestimated sincethe cantilever of the inner forces is shorter than theanalytical one (the ultimate load is the yield force ofthe steel times this cantilever of the internal forces).However, although the concrete stresses are severelyunder predicted, the error of the failure moment issmall in magnitude and on the save side. If 6x6x6sample points are employed, the concrete stressesare predicted better but still poor in this specific caseof low reinforcement ratio. The computational effortincreases tremendously since 6x6x6 = 216 samplepoints have to be considered over the whole elementinstead of 8 for the 2x2x2 case. The ultimatemoment can be computed close to the analyticalsolution. However, the failure moment computedwith 2x2x2 sample points is not poor either.In Figure 17 is the reinforcement increased to2,30mm. This is approximately the maximumreinforcement allowed by design codes for thisbeam. Beyond this limit, the beam would not be ableto show a ductile failure, since failure would beinitiated by a sudden crushing of the concrete andnot by a ductile yielding of the steel. Due to the highreinforcement ratio, the capacity of the concrete isutilized highly. When 2x2x2 sample points areemployed, the computed concrete stresses aresatisfactory in the integration points, but theintegration assumes a extrapolation of stresses.Thus, the stresses above the upper integration pointswill extrapolated beyond the limit stress. This isindicated by the gray shaded triangle in Figure 17.The concrete stresses are truly overestimated.However, the computed failure load is notoverestimated. It is very close to the analytical onebut still a bit lower. The error is on the save sideagain. The distribution of the concrete stressescomputed with the 6x6x6 sample points is in goodagreement with the analytical data. The concretestress at the upper boundary of the cross section isnegligible over predicted. The prediction of theultimate moment is good as well, but thecomputational expense is high when six points areemployed in each direction.

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Figure 17 failure load analysis,reinforcement: 2,30mm(- maximum reinforcement)

Following conclusions can be drawn from thisexample: A standard scheme (2x2x2 or 3x3x3sample points) is able to compute the deformationswell. Concrete stresses can be predicted well withthe standard sample points only in suchconfigurations where the stresses are notconcentrated at the outer boundary of the element.This is of minor importance for computing thefailure load since failure must be indicated by ayielding of the steel due to a design coderequirement in order to obtain always a ductilefailure mechanism. A accurate prediction ofconcrete stresses can be achieved in two ways. Thefirst way is to refine the mesh in the regions of highconcrete stresses. The second way is to employ moresample points for recovering the stress distribution.For computational reasons it is disregarded toemploy too much sample points for every run. Thusit is recommended to use a mesh which is able toapproximate the expected deformations in asatisfying way for the parameter studies. When thecomputation shows once promising results for agiven set of parameters, a final run where computing

time plays a minor role can be done with moresample points in order to compute more accurateconcrete stresses as well.

6 REFERENCES

[1] Owen D. R. J., Hinton E., “Finite Elements inPlasticity - Theory and Practice”, 1980,Pineridge Press, Swansea

[2] Elwi, A.E., Hrudey, T.M., “Finite ElementModel for Curved Embedded Reinforcement”,Journal of Engineering Mechanics, ASCE, ,1989, 115(4), pp. 740-754

[3] Beer, G., “An isoparametric joint/interfaceelement for finite element analysis”, Int. J. f.Num. Meth. Engng., 1985, 21, pp. 585-600

[4] Hartl, H., Elgamal A., “Nicht linearekontinuumsmechanische Modellierungvorgespannter Konstruktionen (Non LinearModeling of Prestressed Structures based on aContinuum Mechanics Approach)” (inEnglish), Heft 45 / September 2000,Österreichische Vereinigung für Beton- undBautechnik, Vienna, pp. 87-96

[5] Hartl H., Sparowitz L, Elgamal A., “The 3Dcomputational Modeling of Reinforced andPrestressed Concrete Structures”, BergmeisterK. (ed.), Proceedings of the 3rd InternationalPhD Symposium in Civil Engineering, Vienna,2000, vol. 2, pp 69-79

[6] Crisfield, M. A., “Non-linear finite elementanalysis of solids and structures”, 1997, Wiley

[7] Nyssen, C., “An efficient and accurate iterativemethod allowing large incremental steps tosolve elasto-plastic problems”,Computers&Structures, 1981, 13, pp. 63-71

[8] Zienkiewicz, O. C., Taylor, R. L., “The finiteelement method”, 4th ed, 1989, vol. 1,McGraw-Hill

[9] Beer, G., “BEFE user’s and reference manual”,CSS, Graz, 1999

[10] Hartl, H, “3D Computational Modeling ofReinforced Concrete Structures withApplication to Soil Structure Interaction”,Doct. Thesis, TU-Graz, in prep.

[11] Potts, D., Zdravkovi" L., “Finite elements ingeotechnical engineering”, Thomas Telford,1999

[12] Crisfield, M. A., “Consistent schemes forplasticity computation with Newton-Raphson

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Computational Modeling of Reinforced Concrete Structures