slump and yield

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CorrelationbetweenYieldStressandSlump:ComparisonbetweenNumericalSimulationsandConcreteRheometersResultsARTICLEinMATERIALSANDSTRUCTURESAUGUST2006ImpactFactor:1.39DOI:10.1617/s11527-005-9035-2

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Materials and Structures (2006) 39:501509DOI 10.1617/s11527-005-9035-2

Correlation between yield stress and slump: Comparisonbetween numerical simulations and concrete rheometersresultsN. Roussel

Received: 29 June 2005 / Accepted: 21 July 2005C RILEM 2006

Abstract Results of numerical flow simulations fortwo slump geometries, the ASTM Abrams cone and apaste cone, are presented. These results are comparedto experimental results in the case of a cone filled withcement pastes in order to validate the proposed nu-merical method and the chosen boundary conditions.The correlation between slump and yield stress ob-tained numerically for the ASTM Abrams cone is thencompared to the experimental correlations obtainedby testing concrete with different rheometers dur-ing comparative studies that were organized at LCPCNantes (France) in 2000 and MB Cleveland (USA) in2003.

Resume Des resultats de simulations numeriquesdecoulements sont presentes pour deux geometriesdessais, le cone dAbrams et le mini cone. La valida-tion de la methode numerique et des choix portant surles conditions aux limites de lecoulement est realiseepar comparaison avec des resultats experimentauxsur pates de ciment dans le cas du mini cone. Lacorrelation obtenue numeriquement dans le cas ducone dAbrams est ensuite comparee aux correlationsexperimentales obtenues pour differents rheome`tres a`betons lors des campagnes de comparaisons orga-nisees au LCPC Nantes (France) en 2000 et a` MBCleveland (USA) en 2003.

N. RousselLCPC Paris, France

1. Introduction

Fresh cementitious materials, as many materials in in-dustry or nature, behave as fluids with a yield stress,which is the minimum stress for irreversible deforma-tion and flow to occur. This yield stress is an uniquematerial property and may, in the case of cement pastes(i.e. fine particles), be measured using conventionalrheological tools. For example, Couette Viscometer [1]or parallel plates rheometer [2] are used in the labo-ratory to measure the yield stress value. In the caseof concretes containing coarse aggregates, large scalesrheometers had to be developed (the BTRHEOM [3],the BML [4] or the two-point test [5]). Even if, in situ,simpler and cheaper tests such as the slump test [6]are still often preferred, these apparatus represent a bigstep forward in the field of concrete science. However,there still exists a discrepancy between the various con-crete rheometers, [7,8]. These apparatus give the samerheological classification of materials but they do notgive the same absolute values of the rheological pa-rameters. On the other hand, the slump test does notgive any value of a physical parameter at all. Its resultcan not be expressed in physical rheological units butit has also proved through the years to be able to clas-sify different materials in terms of their ability to becast.

The aim of this paper is to propose a theoretical cor-relation between slump and yield stress and to compareit to the experimental correlations obtained in the tworheometers comparison campaign [7,8]. In the first part

502 Materials and Structures (2006) 39:501509

Fig. 1 Initial cone shape and cylindrical coordinates.

Table 1 Cone geometries

Cone ASTM Abrams cone paste cone

H0 (mm) 300 50Rmin(mm) 200 35Rmax (mm) 100 50

of the present work, results of numerical simulationsare presented for the ASTM Abrams cone [6] and apaste cone test [9]. The geometries of these molds aregiven in Figure 1 and Table 1. It should be noted thatwe are not using here what is called mini slump butthe ASTM C230 flow table conical mold This coneis used usually in conjunction with the flow table. Itis not the case here. The originality of the present ap-proach lies in using a rigorous three dimensional ex-pression of the behaviour law and plasticity criterion(read 2.2).

The numerical results are first compared to thepaste cone test results and standard Vane test mea-surements [10] on cement pastes. The good agree-ment between the obtained numerical results and ex-perimental values over a wide range of yield stressconfirms the validity of the numerical approach andallows us to use the numerical results obtained forASTM Abrams cone to predict slump in terms of yieldstress.

In the last part of this paper, the numerical correla-tion between slump and yield stress is compared to thecorrelation between slump and yield stress obtained forthree concrete rheometers: the BTRHEOM, the BMLand the two-point test.

2. Literature study

2.1. Relations between slump and yield stress

Several attempts can be found in the literature in or-der to relate slump to yield stress. Murata [11] andSchowalter and Christensen [12] wrote a relation be-tween slump and yield stress by assuming that the conecould be divided into two parts. In the upper part, theshear stress does not reach the yield stress and no flowoccurs. In the lower part of the cone, the shear stressinduced by the self-weight of the material is higherthan the yield stress and flow occurs. The height of theflowing lower part decreases until the shear stress inthis zone becomes equal to the yield stress, after whichthe flow stops. Schowalter and Christensen [12] wrotea relation between the final total height of the coneand the yield stress that did not depend on the mouldgeometry. This relation or similar ones were success-fully validated by Clayton and co-workers [13] or Saakand co-workers [14] in the case of cylindrical moulds.However, in the case of conical moulds, a discrepancybetween predicted and measured slumps was systemat-ically obtained. In the above studies, the experimentalresults suggested that these relations and the fact thatthey did not depend on the mould geometry are validfor high slumps (i.e. low yield stress) (Clayton and co-workers [12], Pashias and co-workers [15], Saak andco-workers [13]). In the case of the ASTM Abramscone, Hu and co-workers [16] gave a semi empiricalcorrelation between the yield stress 0 (Pa) measuredusing the BTRHEOM, the density and the slump(mm):

s = 300 347(0 212)

(1)

2.2. Yield criterion

It should be noted that all the above analytical ap-proaches involve a unidimensional expression of theyield criterion and behaviour law: flow occurs or stopswhen the shear stress becomes higher or lower than theyield stress. The other components of the stress ten-sor are not taken into account when writing a scalaryield criterion. This greatly simplifies the analysis ofthe flow but is valid only if the flow is dominated byshear stresses (i.e. the diagonal terms of the devia-toric stress tensor can be neglected compared to the

Materials and Structures (2006) 39:501509 503

shear stress). In fact, this assumption is true only inthe ideal two-dimensional case studied by Coussot andco-workers [17] (very high slumps). This simplifica-tion is similar to the use of the lubrication theory inthe squeezing flow literature [18]. The squeezing flowtest is a simple compression test carried out on cylin-drical samples with reduced slenderness. The appa-ratus consists in two coaxial circular parallel plates,without any rotation. By using the lubrication theory,only the shear stress is considered and the flow can beeasily studied but this theory generates what is calledthe squeezing flow paradox. On the plane of sym-metry between the two plates, the shear stress equalszero. The unidimensional yield criterion is of coursenot fulfilled and the material should flow as a solidbody (i.e. plug flow). But Lipscomb and Denn [19]have demonstrated that plug regions can not exist insqueezing flow of yield stress fluids. Clearly a solidbody can not move radially outward with a velocitythat increases with the radial coordinate as the con-servation equations demand. Wilson [20] has pointedout that this paradoxical yielded/unyielded region isdue to the neglect of the extensional stresses close tothe centre plane where they are of a higher order thanthe shear stress. A proper three-dimensional criterion isthereafter needed to avoid this paradox and Adams andco-workers [21] can be quoted a comprehensive yieldcriterion is one which is based upon a combination ofall the acting components of the stress. In this paper,a 3D yield criterion will be used as it is the only way toobtain correct quantitative results when the flow is nota purely shearing flow.

2.3. Numerical simulations

Several authors have also developed numerical simu-lations of this free surface stoppage flow. Two typesof method are available when trying to simulate theflow of a rough suspension such as concrete. The firstmethod consists in only considering an homogeneousfluid whereas the second one takes into account thepresence of the particles. The homogeneous approachis easier to implement but is valid only when the small-est characteristic dimension (thickness of the flow, sizeof the mould, spacing between bars) of the flow is highcompared to the size of the biggest particle (5 timeslarger for example in [22]) and when the material stayshomogeneous (which seems to be the case during aslump test on a standard concrete). As accepted in

literature, a Bingham model is, in this case, a possi-ble rheological model. On the other hand, when theabove conditions are not fulfilled, it is then needed totake into account the presence of the particles [23, 24,25]. This type of approach is a lot more complex butis the only suitable technique when confined flows orflows between steel bars are studied. In this paper, wewill only use an homogeneous approach. As a con-sequence, the validity of the presented results will bedoubtful when the smallest characteristic dimension ofthe flow becomes lower than 5 times the biggest parti-cles or, in other words, when the slump is higher than25 cm.

Other authors have previously developed homoge-neous numerical simulations of the slump flow. Tani-gawa and Mori [26] developed an innovative visco-plastic finite element analysis introducing a frictionalinterface law at the base of the slumping cone. Theycalculated the slump in terms of the yield stress but,as they did not have any experimental way to measurethe rheological parameters of concrete, they did notcompare their results to experimental measurements.Later, Schowalter and Christensen [12] compared theiranalytical prediction to Tanigawa and Mori numeri-cal results and found a good agreement. It should benoted that both predictions were based on a unidimen-sional plasticity criterion. Hu [27] assumed that theshape of the deposit stayed conical and calculated thestate of stress using an elastoplastic finite element anal-ysis. Once again, a unidimensional yield criterion wasconsidered.

Recently, Chamberlain and co-workers [28] calcu-lated rigorously stresses in a purely plastic cylindricalsample using either von Mises or Tresca plasticity crite-rion in order to determine the height of incipient failure,which is the height of material required to just initiateflow for a given cylinder radius. This is equivalent tocalculating the critical yield stress for which flow doesstart or does not for a given cylindrical geometry. Theyfound a discrepancy between the mono dimensionalapproximation written by Schowalter and Chistensen[12] and their three dimensional approach. They alsostudied the dependency of the critical yield stress onthe cylinder radius.

2.4. Behaviour at the interface

Most analysis in the literature are carried out assum-ing sticky flow at the base of the deposit. If we go into


further details, the following distinction could be made.In the case of fluid concretes (i.e. low yield stress, highslumps), this assumption is probably valid as these con-cretes behaves just like suspensions but, in the case ofhigh yield stresses, this assumption should be ques-tioned. A concrete with a high yield stress may be ob-tained by two different trends in the mix proportioning.On one hand, the amount of cement or fine particlesmay be high. The colloidal force network that can bebuilt between these fine particles increases the yieldstress of the mixture. The concrete is similar to a densefine suspension and, in this case, the assumption of asticking flow is also valid as experimentally obtained byPashias [15]. On the other hand, the amount of coarseparticles may be high. The behaviour of the obtainedconcrete becomes closer to the behaviour of a cohesivegranular material. In this case, as for a granular ma-terial, the behaviour at the interface may be frictional.Because of the uncertainty in the behaviour at the inter-face, the results obtained with a sticky flow assumptionshould be considered with care in the case of high yieldstress concretes as the validity of the assumption of asticking flow depends on the tested material aspect andmix proportioning.

More recently, Chamberlain and co-workers [28]studied the influence of the plate roughness on the crit-ical yield stress assuming a coulomb type friction lawat the interface involving a friction coefficient equalto zero for a perfect slip case. They showed that, abovea critical value c depending on the cylinder radius,there was no influence of the friction parameter on theheight of incipient failure (or critical yield stress) andthat the interface could then be considered as perfectlyrough. They also showed that the difference betweenheight of incipient failure predicted for the perfect slip( = 0) and perfectly rough (c = 0) cases increasedfrom zero for small radii to 18% for a radius equal to20/g. In the case of typical concrete, this referenceradius becomes 0.16 m (yield stress of the order of mag-nitude of 2000 Pa and density around 2500 kg/m3). Theradius of the ASTM Abrams cone being equal to 0.1m, the error made while neglecting the friction at theinterface in the case of concrete should be lower than18% for low slumps and high yield stress.

2.5. Effect of the inertia

In all the above analytical studies, inertia (or dynamic)effects are neglected. Because of this simplifying

assumption, the influence of the lifting speed of themould (that depends on the operator) or of the plasticviscosity (if a Bingham model is chosen to describethe material behaviour) on the measured slump havenot been studied: the velocity of the flow and its ki-netic energy are not taken into account and the finalshape is calculated as a quasi static state assuming it isreached slowly enough. In other words, what happensbefore stoppage of the flow does not influence the shapeat stoppage. Tatersall and Banfill [29] experimentallyconcluded that the slump of fresh concrete is indeedhighly correlated with yield stress but is not signifi-cantly affected by the plastic viscosity. This conclusionwas also reached by Murata [11]. Let us check here thatneglecting inertia effects is a correct assumption on atheoretical point of view: let us roughly compare thetypical inertia stress (I = V 2) to the material yieldstress. For a small slump the flow duration is of the or-der of magnitude of 1s for a slump of the order of 10 cm.We thus have I 20 Pa, a value much smaller than thematerial yield stress in that case (typically larger thanseveral hundreds of Pa). For a large slump the flow du-ration is of the order of magnitude of 23 s for a slumpof the order of 20 cm. We thus have again I 20 Pa, avalue once again much smaller than the material yieldstress in that case (typically larger than several tenthsof Pascals). This confirms the experimental deductionsof [29] and [11].

As the final shape only depends on the yield stressand as the final shape is the only point of interest forthe engineer, we will only study this aspect of the flowin his paper.

3. Numerical simulations

The cylindrical frame of reference (O, r, , z) is shownon Figure 1. p is the pressure, is the stress tensorand (d) is the deviatoric stress tensor. s is the slump.s is the dimensionless slump and 0 the dimensionlessyield stress as defined by Showalter and Christensen[12].

s = s/H0 0 = t0/gH0

It can already be noted here that, although this scal-ing was suitable for Schowalter results, it does not apply


so well to the present results as they appear to dependon the cone geometry. We chose to use this scalinghowever as it allows us to plot the predicted results forthe two studied geometries on the same figure.

As already stated, in this work, we wanted to im-plement a proper three dimensional yield criterion (see2.2). A 3D Bingham model was thus used to describethe tested fluid behaviour. Moreover, in order to avoidthe undetermination of the strain state when the yieldcriterion is not fulfilled, the material was assumed to be-have as an incompressible elastic solid up to the yieldstress, beyond which it behaves as a Bingham fluid.Other methods exist to avoid this undetermination suchas the biviscosity model introduced by ODonovan andTanner [30] or the exponential model proposed by Pa-panastasiou [31] but, as whether or not flow starts isconcerned, it was of particular interest to use a type ofmodel where a situation of no flow at all (just deforma-tion) can exist. This was the case here. The computa-tional fluid mechanics code Flow3DR [32] was chosento solve the fluid mechanics equation. Flow3DR is ageneral purpose computer program with many capa-bilities. Using input data, the user can select differentphysical options to represent a wide variety of fluidflow phenomena. The program can be operated in sev-eral modes corresponding to different limiting casesof the general fluid equations. In the opinion of thepresent author, FLOW3D is very user friendly whendealing with otherwise complex free surface transientflow of non Newtonian fluids.

The invariant generalization of a Bingham fluid usedhere is the one proposed by Oldroyd [33] based on thethree dimensional von Mises yield criterion:

(d) =[

0

/

(

12

d : d)1/2

+ ]

d,12(d) : (d) 20

where d is the strain rate tensor and the plastic vis-cosity. When the flow is dominated by shear stress andshear rate, the previous relation simplifies to the famousBingham scalar model:

= 0 + , 0

The generated grid in the case of the paste cone isshown in Figure 2. The one used for the ASTM Abramscone is similar. The cell size is smaller in the zoneswhere the shearing is the highest.

Fig. 2 Calculation grid and initial hydrostatic pressure in thecement paste before lifting of the paste cone mould. Note thatthe horizontal to vertical scale ratio is not equal to 1.

The initial pressure is hydrostatic and the speed atwhich the mould is numerically lifted is infinite asthe mould simply disappeared at t = 0 s. This bru-tal lifting could have generated non negligible iner-tia effects taken into account by the code. The finalpredicted shape could have been affected by the highmean kinetic energy of the flow that is not, as in real-ity, dissipated by the lifting of the mould. To solve thisproblem, we chose to study numerical fluids with highplastic viscosity. This indeed generated slow flows, inwhich inertia effects were negligible no matter the lift-ing speed. The calculations were thus carried out witha plastic viscosity equal to 300 Pa.s for concrete in theASTM Abrams cone and 10 Pa.s for cement paste in thepaste cone (far above the traditional plastic viscositiesof these materials). It may be reminded here that, asthe final shape only depends on the yield stress wheninertia effects are negligible which is the case of thereal ASTM Abrams cone (read 2.5), the values chosenfor the plastic viscosity do not matter. The obtained nu-merical results are plotted on Figure 3. Examples oftwo dimensional predicted shapes are shown on Figure4 for the ASTM Abrams cone. The presence of an un-yielded zone (usual in this type of simulation) can benoted.

The calculated values of the slump confirm the factthat slump depends of course on yield stress and den-sity but also on the tested volume and initial height.Indeed, the predicted dimensionless slump is differ-ent for the two cone geometries. As already stated, thescaling suggested by Schowalter and Christensen [12],although suitable for his own experimental results, doesnot apply here.

4. Comparison with experimental results

Measurements were carried out using the paste conegeometry given in Table 1 while the yield stress


Fig. 3 Dimensionless slump in terms of dimensionless yieldstress for the ASTM Abrams cone and paste cone. Both numericalpredictions and experimental results are plotted for the paste conetest.

was measured using the Vane procedure described byNGuyen and Boger [10]. It can be noted that two dif-ferent Vane test geometries were used on the HAAKEViscoTesterR VT550 to measure the yield stresses ofthe tested mixtures. Indeed, as the yield stresses of thestudied mixtures varied from 0.6 Pa to 300 Pa, it wasnecessary to change the geometry of the rotating toolin order to measure an acceptable torque with a suffi-cient precision. Several types of cements and mix pro-portioning were tested but the mixing procedure wasalways the same: the dry ingredients are first mixed for2 min at the lowest mixer rpm setting (260 rpm), thenthe fluids are added and all are mixed for 2 min. Themixer is then stopped to scrape its edges. A higher ro-tation speed (700 rpm) is applied for 15 min followedby a 15 min final mixing phase at the lowest mixer rpmsetting. This chosen mixing procedure had two advan-tages: it ensured good particle dispersions in the fluidphase and gave any chemical binders enough time toact.

The cone experimental results were obtained by var-ious users over a year period. The plate surface was thesame for all the tests. The maximum particle size ofthese materials was 100 m. It was largely smallerthan the characteristic size of the mould. It was alsosmaller than the minimum height measured after col-lapsing of the sample (4 mm). There was no waiting

Fig. 4 Examples of obtained shapes for the ASTM Abramscone (a) yield stress = 2600 Pa (b) yield stress = 2000 Pa. Den-sity = 2500 kg/m3 for both simulations.

time between the filling of the mould and its lifting toprevent any thixotropic effect from increasing the yieldstress. The mould was slowly lifted in order to elimi-nate any inertial effects that could take place. But, asdifferent operators realized these tests, this slow lift-ing speed was probably not constant. The spread andheight measurements were done after a 2 min waitingtime. For each test, two perpendicular diameters andthe maximum thickness of the collapsed sample weremeasured. The obtained results are plotted on Figure 3.The agreement between the numerical simulations andthe experimental results is very good. The experimentalcritical dimensionless yield stress seems to be around


0.58. It is however difficult to precisely check the valueof this critical yield stress as it is impossible not to de-form the cone while lifting the mould. The good agree-ment between numerical and experimental results onthe entire studied range of yield stress confirm the factthat no sliding occurs at the base of the deposit and thatthe use of a proper three-dimensional plastic criterionallows a correct quantitative numerical prediction ofthe final slump.

5. Astm abrams cone and concrete rheometers

The validation of the proposed numerical method ob-tained in the previous section allowed us to use theproposed numerical method to predict the ASTM coneslump in terms of the tested concrete yield stress. Apartfrom the possible sliding at the interface that may occurin the case of high yield stress concretes (see 2.4), theflow in the case of the ASTM Abrams cone is identi-cal. The numerical correlation between slump in mmand the ratio yield stress/density is plotted on Figure 5along with the experimental results obtained in the tworheometers comparison campaign. MBT-LCPC com-parison carried out at LCPC (Nantes, France) in 2000[7] and at MB (Cleveland, USA) in 2003 [8]. It has to

Fig. 5 Yield stress-slump correlations. Experimental results forvarious rheometers and numerical correlations.

be emphasized that we do not have at the moment anyway to measure the real value of the yield stress. It hasbeen proven in [7] and [8] that the rheometers did notgive the same results. This does not mean that one iscorrect and the others are wrong, this could mean thatthey are all wrong or at least that none of them is cor-rect on the entire range of yield stress tested. Thus, theyet unanswered question is What is the real correla-tion between yield stress and slump? In the frame ofthis paper, this means that we do not have any way toconclude on the validity on the simulations in the caseof concrete unless by reminding that they proved to bevery efficient to simulate the flow of cement pastes andthat all the theoretical assumptions to carry valid simu-lations in the case of concrete are fulfilled in the rangeof 5 to 25 cm slump.

It should be emphasized that, as the behaviour at thebase of the deposit is unknown for high yield stresses(possible sliding, read Section 2.4.), the validity of theobtained numerical results should be limited to slumpshigher than 5 cm. Moreover, for slumps higher than25 cm, the thickness of the flowing layer of concretebecomes of the same order as the size of the biggestaggregate and the homogeneous fluid mechanics ap-proach proposed here is not valid any more (read Sec-tion 2.3.).

However, from the numerical predicted results, asimple linear approximation may be written for slumpsbetween 5 cm and 25 cm.

s = 25.5 17.60

(2)

It is not the aim of the present work to conclude aboutthe efficiency of any concrete rheometer compared toanother. However, it seems that, in the 525 cm slumprange, the BTRheom correlation between measuredyield stress and measured slump is the closest to thenumerical correlation. For the lowest yield stresses andthe highest slumps, it seems that the BTRheom correla-tion overestimates the yield stress. This may be linkedto the fact that the sliding assumption at the peripheralinterface needed in the analysis of the BTRheom data iscorrect in the traditional concrete range (see 2.4) but,when the yield stress becomes lower, the behaviourat the interface becomes sticky and the shear stressat the interface may not be neglected any more com-pared to the measured yield stress. The spontaneousformation of a limit layer (made up with water and fine


elements) limiting the friction may not occur for lowyield stress concrete that behave just like traditionalsuspensions and the flow pattern may differ from thetheoretical one used to calculate the yield stress fromthe torque measurements. During the comparison cam-paign at MB (Cleveland, USA) [8], this specificity ofthe BTRHEOM was already spotted when comparingthe various rheometers by measuring purely viscousoils. Concerning the two other apparatus, their mainpotential drawbacks may be reminded here:

- The ratio between the gap between the BML cylin-ders and the maximum particle size is small. In thegeometry used in [8], the gap was only 45 mm, thusonly allowing, on a theoretical point of view, the test-ing of concretes with particles smaller than 9 mm[22]. When it is not the case, the consequence, ifthere is one, is very difficult to predict. However, inthe case of fluid concretes such as Self CompactingConcretes (SCC) often prepared with smaller parti-cles, the BML could prove the most suitable of thethree apparatus in the opinion of the present author.It can be noted that the BML correlation betweenyield stress and slumps gets closer to the numericalpredictions obtained here when the yield stress is ofthe order of several hundreds Pa.

- Because of the complexity of the flow in the two-points test, the stress (or velocity) field can not be rig-orously calculated. The rheological parameters aresimply extrapolated from the global measurementswithout taking into account the fact that some nonflowing zones, the sizes of which depend on the ro-tation speed and of the tested material itself, mightappear in the sample closed to the rotating impelleror to the external walls. However, in the case offluid concretes or SCC, this phenomenon may beneglectible. As for the BML, the two points test cor-relation between yield stress and slumps gets closerto the numerical predictions obtained here when theyield stress decreases below the order of several hun-dreds Pa.

6. Conclusion

In the first part of the present work, results of numer-ical simulations using a three dimensional expressionof the behaviour law and plasticity criterion for theASTM Abrams cone and a paste cone test have been

presented. The obtained numerical results have beenfirst compared to paste cone test results and standardVane test measurements on cement pastes. The goodagreement between the obtained numerical results andexperimental values over a wide range of yield stresshave confirmed the validity of the proposed numericalapproach and its associated boundary conditions andallowed us to use the numerical results obtained forASTM cone to predict slump in terms of yield stress. Asimple numerical correlation between slump and yieldstress has then been proposed.

In the last part of this paper, this numerical correla-tion has been compared to the experimental correlationbetween slump and yield stress obtained for three con-crete rheometers: the BTRHEOM, the BML and thetwo-point test during comparison campaigns held in2000 and 2003. It seems that, in the range of traditionalconcretes (slump between 50 mm and 250 mm), thereis a good agreement between the BTRHEOM measure-ment and the theoretical prediction.

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