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re-334Accepted 19 December 2011Keywords:Concrete strengthNondestructive testing (NDT)On site assessmentRebound hammerndconcan analysis of experimental data gathered by many authors in laboratory studies as well as on site, (c)quality of strength estimate are identied. Two NDT techniques (UPV and rebound) are prioritized andnon-decess to material properties while remaining rapid and of moderatecost. The quality of estimation is a key issue since it can impact thedecisions regarding maintenance [106]. However, NDT techniquesare sensitive rst to physical properties and provide only an indi-rect way towards material mechanical performances [32]. Quanti-fying a mechanical property like strength is the highest level ofrequirement for assessment (lower ones being detecting or rank-spect to given problems [20,114,15] or to dene the most promis-culty to correlate the values of physical NDT measurements withthe mechanical properties has been pointed out for a long time.NDT is currently used in combination with destructive tests(cores) or semi-destructive tests which provide more direct infor-mation [53]. Rebound measurement and ultrasonic pulse velocity(UPV) are among the most widely used NDT methods regardingconcrete strength assessment [74] and a recent European standardprovides a formal solution on how concrete strength can be esti-mated from in situ testing [36]. However this standard requiresat least 15 cores from the site to be used in order to establish aConstruction and Building Materials 33 (2012) 139163Contents lists available atBevE-mail address: denis.breysse@u-bordeaux1.frtexts: (a) when some damage has developed through time, (b)when new requirements have to be addressed, because of changesin regulations or in the loads to be supported, (c) when the mate-rial condition must be checked because of some suspicion e.g.when the concrete in control cast cylinders may differ from theconcrete in the building itself! In any case, nondestructive testing(NDT) techniques offer an interesting approach, since they give ac-ing paths for developments [87]. It is usually agreed that thequality of assessment is limited due to sources of uncertaintiesarising at various levels and caused: by the testing method, by sys-tematic interferences with the environment, by random interfer-ences (due to material intrinsic variability), by human factorinuence and by data interpretation, including errors in the modelbetween what is measured and what is looked for [49,10]. The dif-structural engineers who need to feed structural computationswith material data. Such assessment is required in various con-Strength assessment of existing buildings is a key challenge for of data processing for a better assessment of building materials.Some authors have tried to synthesize the abilities of NDT with re-SonRebUltrasonic pulse velocity1. Introduction the challenge ofassessment0950-0618/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.conbuildmat.2011.12.103many empirical strength-NDT models are analyzed. It is shown that the measurement error has a muchlarger inuence on the quality of estimate than the model error. The key issue of calibration is addressedand a proposal is made in the case of the SonReb combined approach. It is based on the use of a priordouble power law model, with only one parameter to identify. The analysis of real datasets from labora-tory studies and from real size buildings show that one can reach a root mean square error (RMSE) onstrength of about 4 MPa. Synthetic simulations are developed in order to better understand the roleplayed by the strength range and the measurement error. They show that the number of calibration corescan be signicantly reduced without deteriorating the quality of assessment. It is also shown that theoptimal calibration approach depends on the number of cores. 2012 Elsevier Ltd. All rights reserved.structive strength ing), since values are expected, even with some range of uncertain-ties [11].Much research has been devoted to the development of NDT orAvailable online 29 March 2012the development and analysis of synthetic simulations designed in order to reproduce the main patternsexhibited with real data while better controlling inuencing parameters. The key factors inuencing theNondestructive evaluation of concrete stperspective by combining NDT methodsD. BreysseUniversity Bordeaux, I2M, UMR 5295, F-33400 Talence, France CNRS, I2M, UMR 5295, Fa r t i c l e i n f oArticle history:Received 26 July 2011Received in revised form 19 December 2011a b s t r a c tThis paper analyzes why aassess on site strength ofConstruction andjournal homepage: www.elsll rights reserved.ngth: An historical review and a new00 Talence, France Arts et Metiers ParisTech, I2M, UMR 5295, F-33400 Talence, Francehow nondestructive testing (NDT) measurements can be used in order torete. It is based on (a) an in-depth critical review of existing models, (b)SciVerse ScienceDirectuilding Materialsier .com/locate /conbui ldmatildincalibration curve. This requirement increases the cost of NDTinvestigation and limits its practical use. The development and val-idation of a methodology that would lead with an acceptable levelof condence to a reliable strength assessment remains a key issue.A main point is that of calibration, i.e. that of building and using areliable relationship between NDT values and strength.The difculties encountered with calibration lead to the devel-opment of combined approaches since a second measurementcan enable correction of the inuence of some uncontrolled fac-tor(s) on a rst measurement. Combined methods were developedrst by RILEM (Technical Committees 7 and 43) based on seminalworks from Facaoaru [41]. Among the large number of possiblecombinations the SonReb combination which combines UPV mea-surements and rebound hammer measurements is the most widelyknown and used. Since its rst developments (thanks to RILEM) inRumania and Eastern Europe, it has spread in many other countries[22,27,103] and was standardized in China in 2005 [24]. The corre-lation between NDT measurements and strength is established ona standard concrete mixture. Five inuence coefcients are intro-duced to account for the effect of inuencing factors. This limitsthe practical use of the method. The main advantages of SonRebmethod remain its easiness and low cost. It can be used on anytype of structure and concrete, measurements do not require ahigh level of expertise and large area can be investigated at a rela-tive high speed.A common statement is that while neither UPV nor rebound are,when used individually, appropriate to predict an accurate estima-tion for concrete strength, the use of combined methods producesmore trustworthy results that are closer to the true values whencompared to the use of the above methods individually. Such astatement is however somehow optimistic since the combined ap-proach leads to contrasted results. It was even said that they haveonly provided marginal improvements [91]. A large number ofrelationships have been proposed in order to estimate the strengthfrom a couple of (UPV, rebound) values. It appears that there is nota unique relationship and that calibration remains a key issue, as itis the case for individual methods [14].However research remains very active, aiming at developingand validating combined approaches. This can be explained by anincreasing need for evaluating the condition of existing buildings.It is the case for instance of the seismic retrotting of public build-ings in Italy, where standards were recently modied [17]. Accord-ing to the Ordinanza P.C.M. n. 3431 [89], a suitable assessment ofconcrete compressive strength can be obtained by integrating re-sults from destructive tests with those from non-destructive testshaving proved suitability.This approach is conrmed by the recent D.M. 14/01/2008Norme Tecniche per le Costruzioni which requires that the char-acterization of mechanical properties of materials in existing struc-tures be obtained from material testing, in addition to availabledocumentation and in situ inspections. Estimate of concrete com-pressive strength through non-destructive testing (rebound, ultra-sonic, combined Sonreb methods) relies upon suitability ofcorrelation formulas. However current formulas have usually beencalibrated based on concrete samples that were realized ad hoc,thus representative of new concretes and new buildings, i.e. thesecalibrations are usually not accounting for peculiarities of existingbuildings [30]. It is also stated that they are not valid for concreteof poor quality [17].The general panorama remains complex and confusing. It isdifcult:(a) to draw conclusions on the practical added-value of NDT140 D. Breysse / Construction and Bumethods (alone and/or in combination for quantitative on-site strength assessment,(b) to explain why NDT assessment appears to be effective insome situations and noneffective in other situations,(c) to understand what could be rules of good practice forachieving acceptable results regarding strength assessment.The purpose of this synthetic paper is to answer these questions.It is based on an extensive literature review in the eld of nonde-structive strength assessment, concerning both laboratory studiesand on site investigations. After an analysis of the involved physicalphenomena and of the commonly accepted results, we will try tounderstand why this problem is complex and why apparently con-tradictory conclusions have been reached. Advantages and weak-nesses of the methods will be analyzed, always keeping in mindthat the real aim is on-site assessment and that some patterns, likecarbonation, which have no inuence in laboratory studies at earlyage may play a very important role on site, when investigating a30-year old building. Most of the paper will be focused on UPV andreboundmeasurements (used individually or in combination Son-Reb),when other techniques and combinations could have been dis-cussed (in the following, V and R respectively denotes themeasuredvaluesofUPVand rebound). This choicehasbeenmade for three rea-sons: (a) focusing on the most commonly used NDT methods whileavoiding a too lengthy paper, (b) the fundamental issues addressedwould have been similar for other NDT methods, (c) the main con-clusions and proposal would have been similar for other NDTmeth-ods. Some other NDT methods will be briey discussed in order toshow the general character of some statements.Synthetic simulationswill be used in order to reproduce the con-text of NDT and its main patterns. They will highlight the respectiveweights of the quality ofmeasurements, and of themodel error. Twopossible approaches for calibrationwill be compared anda syntheticproposal onhow to calibrateNDT resultswill be undertaken. To con-clude, the issueof assessment at thebuilding scalewill be shortly ad-dressed, mainly based on recent Italian works [76].This paper is restricted to the estimation of average proper-ties, keeping apart some other issues like that of characteristicstrength assessment which deserves further considerations, com-bining statistics and analysis of variability. The reader interested inthis specic issue is invited to report to recent papers or books[46,80,108].2. A review of uncertainty and variability in NDT measurementsWe will rst review why UPV and rebound measurement canprovide an efcient means to estimate concrete strength. The qual-ity of estimation may be affected by some errors and uncertainties.Two main causes of uncertainties will be discussed: (a) factorsother than strength that may inuence the NDT measurementand thus the NDT-strength relationship, (b) the NDT measurementvariability and its roots.Variability can be analyzed at several scales [12]: at a very local scale, whenmeasurements are repeatedwhile keep-ing sensors at a given location themeasurement process is not fullyrepeatable, because of some randomness in the measurementdevice or data processing and because of the uctuation of someexternal inuencing factors (e.g. temperature or air humidity), at a local scale, when measurements are repeated while movingsensors in a small area where the material is assumed to keepthe same properties, some additional variability is induced by:(a) the lack of repeatability of the measurement, e.g. the exactdistance between emitter and receiver may uctuate, (b) theshort range material variability, due to its heterogeneity, e.g.g Materials 33 (2012) 139163the fact that the sensor may be near an aggregate in one caseand on cement paste in another case, at a larger scale, additional variability has several causes includ-ing intrinsic long range material variability, e.g. due to batch tobatch variability for a given mix, or uctuations in the castingand curing process. At this scale some variability may also beinduced by deterioration or damage development.When one aims at assessing material condition and at identify-ing contrast in material properties, the last degree of variability i.e.that induced by damage, is usually the target of the investigation. Itcan be called signal. All other sources of variability, whatevertheir source, may be considered as noise. The challenge for theinvestigator is to differentiate between this noise (either due tomaterial intrinsic variability or to the measurement process) and UPV is also dependent on other parameters like cement per-centage, cement type, addition of y ash, etc., even if their inu-ence is usually smaller than that of aggregates; another main inuencing factor is relative humidity in the con-crete. It is reported that the pulse velocity may increase up to 5%between the air-dry and saturated condition [21]. If the effect ofmoisture is not considered, erroneous conclusions may bedrawn about in-place strength, especially in mature concrete; in reinforced concrete, rebar are known to have a signicantinuence, since they can offer an easier path for wave propaga-tion at higher velocity. It is recommended to localize rst therebar, then to measure UPV at locations where the inuenceof rebar can be neglected; inuence of temperature can be neglected between 0 C and30 C. Some inuence has been noted for hot or cold weather, a last series of inuencing factors are defects of all types, like voidsD. Breysse / Construction and Building Materials 33 (2012) 139163 141the signal. It is therefore important to know what levels of noisevariability can be encountered in practice at each scale. In the fol-lowing, the three respective levels of variability will be called cv1(very local), cv2 (local), cv3 (large scale), where cv stands for coef-cient of variation which is the standard deviation divided by themean value. Some considerations on the appropriate number ofnondestructive tests will conclude this section.2.1. Ultrasonic pulse velocity measurement (V) with regard to strengthassessmentThe basic reason for using UPV for strength assessment is thatthe velocity of ultrasonic waves in concrete shows a large increasein the rst days after casting, when setting then hardening develop.On one hand, theoretical considerations support a direct link be-tween Youngs modulus and wave velocity in an elastic media,on the other hand, empirical relations have been long recognizedbetween modulus and strength. Thus it seems logical to think thatUPV can provide a direct means to assess strength.A rst problem is that the range of variation of velocity withtime is much lower when concrete is older than 28 days: sensitiv-ity of UPV to the rate of concrete strength is extremely high in therst few days but considerably lower after about 57 days,depending on curing conditions [104]. Other problems arise frommany other factors which may affect to a different degree strengthand UPV. The inuence of these factors has been pointed out andquantied by many authors [42,109,99,97,71,81,72,112,54]. A syn-thetic view is given in Table 1. The factors with most inuence arethe following ones: in the concrete mix, the main inuence is that of aggregate:aggregate percentage, aggregate maximum size and aggregatetype. UPV can increase by several hundreds of m/s for higherpercentage or larger aggregates. The type and density of aggre-gate has also a large inuence: while in undamaged rocks UPVcan vary from less than 4000 m/s to more than 6000 m/s, achange of 400 m/s is common in concrete [81,84,2];Table 1Inuencing factors on UPV measurements.Concrete mix Aggregate PercentageaMaximum sizebType (density)aCement paste PercentagecTypecOthers Fly ash contentbWater/cement ratioaHumiditybOthers AgebRebarbVoids, cracksaa High inuence.b Average inuence.c Moderate inuence.and cracks. If concrete is cracked, the wave propagation process isaffected and measurements must be analyzed carefully; the inuence of age must be noted, the kinetics of developmentwith time being quicker for UPV than for strength (UPV is moreor less stabilized after few weeks); several congurations can be used for UPV measurements:transmission or direct, i.e. through concrete, semi-direct, i.e.with sensors on two orthogonal faces, or indirect, i.e. withtwo sensors on the same face. Indirect measurements are moresensitive to near surface condition and can be affected by car-bonation for instance.Measurement variability has been assessed in various studies,both in laboratory and on site. Table 2 synthesizes this information.In Table 2, the SENSO reference corresponds to a French collabo-rative research project in which a specic attention was paid to theevaluation of NDT measurements at various scales [5,6]. The givenvariability corresponds to the measurement of UPV through trans-mission, at 50 kHz frequency [47].On site, large scale variability (for instance between all beams atthe same level of a building) as identied by Masi and Vona [76]can be much larger, up to 30% or 40%. This reveals the inuenceof damage and cracking since open cracks can disturb or preventwave propagation. In this case, the UPV cannot be considered asa material property. An analysis of the specic inuence of de-fects is then required.Except in the case of highly cracked structures, the magnitudeof cv is about 12.5%, meaning that UPV provides a highly repeat-able measurement in the case of an homogeneous concrete. Thiscorresponds to a standard deviation of 40100 m/s if the averagevelocity value is 4000 m/s.Table 2Magnitude of UPV variability at various scales, in the laboratory and on site.Laboratory cv1 1% [70]cv2 0.52% [73]0.52% [71] quoting various sources1.52.5% [2,29]1.3% (SENSO)4% [117]cv3 1.55% [73]2% (SENSO)34% ([43], on damaged columns)On site cv1cv2 1% (SENSO)2.5% [20]cv3 1.56% [75]2% (SENSO)36% [94]410% [76]2.2. Rebound hammer measurement (R) with regard to strengthassessmentRebound measurement consists in a direct mechanical solicita-tion of the structure. The rebound value is correlated with thehardness of the near-surface concrete. Knowing that the hardnesslogically increases when the porosity decreases and that stiffnessand hardness are empirically correlated with strength, rebound isexpected to provide a logical means for strength evaluation.Due to its simplicity and low cost the rebound hammer is themost widely used nondestructive test for concrete. It was initiallydesigned in Switzerland by Ernst Schmidt and a correlation wasdeveloped between the compressive strength of standard cubesand rebound number. However, it appeared soon that the correla-tion was not unique and had to be modied for different devices ordifferent conditions of testing.It is usually assumed that the rebound concerns mostly the rst30 mm of concrete and is thus sensitive to near-surface properties.Rebound hammer is logically blind to more internal changes in142 D. Breysse / Construction and Buildinconcrete properties. Existing differences between internal and sur-face properties of concrete have to be carefully examined sincethey may inuence the quality of material assessment. The bestknown inuencing factor is carbonation whose progressive devel-opment creates a harder layer near surface. If the bias due to car-bonation is forgotten the error on strength evaluation may reach50%. Using data from 1 year to 40 years old buildings [4] haveshown that, for a same concrete strength, the rebound value is 4points higher in the structure than on a laboratory cube. Correctionprocedures have been proposed [59] and even standardized in Chi-na [57].As is the case for UPV, many other factors may affect to a differ-ent degree internal strength of concrete and rebound number.Their inuence has been pointed out and quantied by manyauthors [101,104,99,55,91,97,71,110] and a synthetic view is givenin Table 3.The factors with most inuence are (Table 3): age, after 2 months, because of carbonation; the type of aggregate, with a difference of 6 points between cal-careous and siliceous aggregates which are harder. A specialcalibration is also needed for lightweight aggregates; humidity with an decrease of about 20% between dry-air con-crete and saturated concrete. This difference may be moreimportant when the concrete strength is not too high;Table 3Inuencing factors on average rebound measurements.Concrete mix Aggregate Maximum sizebType (hardness/density)aCementpastePercentagecType of cementcHumiditybFactors inuencingsurface and near-surfaceconditionAge/carbonationaSurface regularity andrugositybType of formwork andcuringbOthers Rigidity of the component(thickness)bTemperaturecVoidsaa High inuence.b Average inuence.c Moderate inuence. all factors that can inuence the surface regularity and rough-ness. The main effect of an irregular surface is the increase ofvariability between successive measurements; all factors that can inuence near-surface properties, like thetype of formwork and curing conditions: because drying of con-crete is modied, there is a change in near-surface porosity. Thepresence of air bubbles or voids can also affect themeasurement; rebound number is lower if the rigidity of the tested componentis smaller. This has a direct effect since the rebound numbermeasured on a core is lower than that directly provided onthe structure (even without carbonation effect); the rebound number is device specic and depends on thedirection of testing (horizontal, upwards or downwards),because of gravity, but this item can be easily corrected.Most published data on rebound measurements deal with cor-relations with standard strength tests, rather than repeatability.Conclusions about repeatability are often in conict because of dif-ferences in test designs or in data analysis. For instance, is the var-iability calculated on individual rebound values (within testvariability) or on the average taken on a set of N (N being usuallybetween 10 and 12 points)? The results are obviously different. Ta-ble 4 synthesizes information from various sources on reboundvariability (calculated on individual rebound values, except whenin italics).On site, large scale variability as identied by Masi and Vona[76] can be much larger, up to 23%. This variability includes all ef-fects cited above, e.g. the fact that, because of rigidity inuence, therebound value may be different on a given component if it is mea-sured in the middle section or at a free end. However, this revealsthe inuence of damage and cracking. Except in the case of highlycracked structures, the magnitude of cv is about 510%, meaningthat the rebound measurement is not as repeatable as UPV mea-surement. This corresponds to a standard deviation of 24 if theaverage rebound value if 40.2.3. How many measurements?In Sections 3 and 4, it will be explained how concrete strengthcan be estimated from the NDT measurements (either V or R) usinga conversion curve. The challenge will be to get a value which canbe considered as representative of the investigated volume.The number of replications of NDT measurements that are re-quired in order to obtain a representative value depends on (a)the within test variability, (b) the allowable error between thesample average and the true average, (c) the condence level thatthe allowable error is not exceeded. In fact the number is oftenbased on customary practice. The number of in place tests shouldbe chosen so as to have an uncertainty for onsite results identicalto that obtained for direct standard compression tests. This comesto 10 replications for rebound and 2 replications for UPV at eachtest location [91].Once a conversion curve fc(NDT) is assumed, the accuracy of thestrength estimate also depends on the quality of this conversionmodel and on its shape (it is better if the NDT measurement ismore sensitive to the strength fc). This issue has been treated byBungey et al. [20] who considers that if the number of tests N ismore than 10, the strength mean value is estimated with a 15/pN% with a condence level of 95%. Leshchinsky et al. [66] has pro-vided an expression of strength variability as a function of NDTvariability, of the partial derivative dfc/dNDT and of the determina-tion coefcient of the conversion model fc(NDT).g Materials 33 (2012) 139163If the measurements with a given technique i are assumed tofollow a Gaussian distribution of known standard deviation s(NDTi)the true mean value E(NDTi) of NDTi belongs to the intervalildin[ka2 s(NDTi)/pN] around the empirical average value at the levelof condence (1 a. For a condence level 1 a = 95%, ka2 = 1.96and the interval is 1.96 s(NDTi)/pN.Thus one can calculate the minimal number of measurementswhich is required in order to estimate the NDT measurement ata given accuracy level x and with a 95% condence level:N P 1:96cvNDTi=x2 1where cv(NDTi) = s(NDTi)/E(NDTi)With the variability identied in Tables 2 and 4 it comes: for UPV, N = 1 measurement is enough if cv = 1% to get the rep-resentative velocity with an accuracy x of 2% while N = 2 mea-surements are required if cv = 2.5% for an accuracy x of 4%, for rebound, with cv = 10%, and considering that the measure-ment is the average rebound number on 1012 individual val-ues, N = 1 measurement leads to estimate the representativerebound value with an accuracy x of 20%, while the accuracyis respectively equal to 10% and 6% for N = 4 and 10.Table 4Magnitude of average rebound number variability at various scales, in the laboratoryand on site.Laboratory cv1 5% [78]cv2 4% [117]5.5% [78]5.57% [29]69% [66]610% [2]411% [119]About 10%, from analysis of several sources [22]4% [23]acv3 7% [78]3.55% [73]a N = 21.55% [119]aAbout 10% ([43], on damaged columns)aOn site cv3 56% for Czech standards [18]12% for EN 12504-2 standard on rebound [19]37% in homogeneous parts of a structure [76]a713% [94]aa Means that cv was calculated between averaged rebound values calculated on Nindividual measurements.D. Breysse / Construction and Bu3. Monovariate conversion curves: establishing and calibratingrelationshipsAn extensive review of available literature reveals that a hugenumber of relationships have been proposed in order to derivethe concrete strength from NDT measured values. This huge num-ber asks the question of what is the best model (if it exists).Many apparent inconsistencies between results can also bepointed out. Our purpose is to explain the reasons why so manydifferent models exist. It will be shown that a profound consistencyexists between results which justies a different approach.We will analyze how it is possible, by using a model and addinga calibration step, to correctly estimate the strength. Calibrationis a key point, which deserves specic attention. Several optionswill be compared and some proposals will be drawn. These itemsare developed here for two specic NDT techniques (UPV and re-bound), but most analyses and comments would remain valid foralternative NDT techniques.3.1. Building a conversion curve the weight of uncontrolledparametersMany possible inuencing factors have been identied in Tables1 and 3. Their role is twofold: (a) on one hand, it is assumed thattheir inuence on strength and on NDT is not identical, (b) onthe other hand and this is a consequence of the rst point ifa conversion model fc(NDT) is built, this model is not stable whenthe value of the inuencing factor changes.For instance the age of concrete has been identied as inuenc-ing in Table 1. This means that the age at which the conversionmodel between strength and UPV is built has some inuence.The reason is that strength goes on increasing after seven dayswhile the UPV is more or less stable after this time. Another exam-ple is that of the aggregate type: the conversion model (fc(V) orfc(R)) changes with the aggregate type since heavier or harderaggregates tend to increase the UPV and rebound value more thanconcrete strength.This leads to the calibration process. In fact, if one calls X theinuencing factor (like aggregate type or age in the precedingexamples) two options exist for calibration: The rst option is calibration on inputs. Different levels of X areconsidered and different subsets of data pairs {NDT, fc} are iden-tied, one for each level of X. A conversion curve is built for eachX level. When the curve will be used for estimation, the userwill only have to identify the accurate X value and to choosethe relevant curve. The second option is calibration on outputs. Here all {NDT, fc}pairs, whatever X level is, are considered together and a uniquecurve is built (this model is less explicit since it is inuenced byan uncontrolled parameter). All pairs corresponding to a xed Xlevel can thus be analyzed regarding the residuals between realand predicted value. These residuals correspond to the inuenceof X (by a comparison with a reference X value, that of the pop-ulation on which the general model has been built). A correctingfactor for calibration is nally identied for each X value.These two options had been presented in a slightly differentway by [67]. They enable the user to account for the inuence ofa (known) inuencing parameter. They however require: (a) thatthis parameter has been identied, (b) that it is measurable andcontrollable, (c) that one has enough experimental data to proceedto calibration. In real cases, the number of inuencing (most ofthem uncontrolled) parameters can be very large, as seen in Tables1 and 3 for two specic NDT methods. Each inuencing parameterwould require a correction/calibration stage, notwithstanding withpossible interactions between parameters. This process quicklycomes to be unpracticable.Before analyzing existing conversion models into greater de-tails, it is useful to have a further look on the inuence of uncon-trolled factors on the quality of models. Figs. 1 and 2 are builtfrom synthetic simulations. They illustrate the model identicationfc(V) with two types of aggregates (G1 and G2). The simulationsconsisted in: assuming a prior (theoretical) law fc(V) as an exponential func-tion: fc = a exp (b V), accounting for the inuence of aggregate type, with G1 and G2being respectively D(V) m/s slower/quicker than the generalmodel for the same strength, building a set of 20 theoretical pairs {V, fc}, adding error measurements on both V and fc, identifying the models from the measured data, with two pos-sibilities: (a) a general model from the full set of pairs, (b) onemodel for each aggregate type.The values taken for the synthetic models have been chosen ing Materials 33 (2012) 139163 143order to reproduce experimental patterns observed by [42,33,81]who studied the calibration of strength/UPV models with varioustypes of aggregates. Figs. 1 and 2 correspond to two magnitudes144 D. Breysse / Construction and Buildinof the measurement error, assumed to be Gaussian and of zeromean. The numerical data for simulations are: a = 0.06, b = 1.44 s/km, D(UV) = 0.15 km/s. Fig. 1 corresponds to small measurementerror (cv(V) = 1%, cv(fc) = 2%), while Fig. 2 corresponds to largerones (cv(V) = 2%, cv(fc) = 4%). These values are in agreement withthose synthesized in Table 2 for UPV and those identied by [7]for compressive strength.Figs.1 and 2 show that: the identied model is very dependent on the data set. Forinstance, if the model is identied separately on subsets G1and G2, two models can be built with r2 above 0.94. They leadto a difference of about 300 m/s for the same strength. If G1and G2 are taken together the identied model is much lessinformant (r2 = 0.79), because of the uncontrolled inuencingparameter (aggregate type), the same can be said when measurement errors are larger (hereboth on UPV and strength), the only difference being that thequality of the models decreases, both for separate and for globalidentication, the identied values for a and b coefcients are different fromthe true ones (known in the synthetic model). The differencecan be very large when measurement errors increase.This point is also a mark of statistical uncertainty: since only 10data pairs are taken for each aggregate type, the identied model issensitive to the limited sample size and the identied parametersFig. 1. Inuence of aggregate type on the UPV-strength relationship (smallmeasurement errors).change at each simulation.This simple example highlights the inuence of uncontrolledfactors on the quality of the identied model. One main questionis always: can we consider the data set as homogeneous? If itFig. 2. Inuence of aggregate type on the UPV-strength relationship (largemeasurement errors).appears to be so, is it relevant to identify a single model. If it isnot, one has two possibilities: either splitting the data set between more homogeneous sub-sets (e.g. aggregate types G1 and G2) and identify one specicmodel for each subset, or identifying a more general and thus less predictive model forthe whole set. In this case it may be possible, in a second step, toanalyze the residuals between measured and predicted valuesand to check if they can be correlated with some inuencingfactor.The advantage of specic models is that they better t data. Forpractical use, they however require to know the value of the inu-encing parameter. In practice one often has to make a correctionwhose magnitude depends on the value of the inuencing param-eter. A general model (universal law) is less accurate and it has ahigher uncertainty but it can be used without any furtherinformation.One must also keep in mind that the number of possible inu-encing factors can be very large. For instance [31] showed that itwas possible to identify a single power law for the UPV-strengthrelationship for concrete less than 90 days old, but that the qualityof t is improved if a different relationship was established foreach age.In addition some interactions can exist between these factors.One must always wonder how far it is reasonable to go whenone tries to identify homogeneous subsets and specic models,since the domain of application of too specic models can be verylimited.3.2. UPV-strength relationships: a review3.2.1. Model typesMany authors have studied how UPV can be correlated withconcrete strength. An extensive review of their contributions hasbeen undertaken. The purpose is to get a general overview and tohighlight what may appear to be as common rules. In fact apparentinconsistencies are common: many different types of models areused and, for each type of model, the values of coefcients canchange within a large range. The variety of types of models wasnoted in an analysis of national standards by [63]. . . It seems thatthere is no general law. Table 5 synthesizes some informationabout 70 models which have been gathered from literature. It doesnot pretend to be exhaustive, but it is probably representative.Three types of models are more commonly found: exponentiallaw, power law and linear law. The two rst types of models, i.e.exponential and power laws, will now be analyzed in more detail.3.2.2. Analysis of exponential modelsTwenty-six models are considered for analysis. They are all ofthe type fc = a exp (b V) and have been published between 1957and 2010. They cover a wide variety of concrete mixes (with meanstrength varying between 14 and 45 MPa and individual valuesranging from 5 MPa to more than 80 MPa), combining laboratorystudies and measurements on existing structures. a and b coef-cients vary in a very large range, with the following extreme values(all models have been harmonized by considering that V is ex-pressed in m/s and b in s/m): a = 0.0012, b = 0.00227 [35]. a = 2.901, b = 0.0006 [30].g Materials 33 (2012) 139163As a general case, b decreases when a increases. To the best ofour knowledge, nobody has ever tried to nd some consistency be-tween these results. An interesting feature however appears when that Aticis model has been identied on concretes with UPVranging between 4600 and 5700 m/s. Thus the dotted line onFig. 5 mostly corresponds to extrapolation of the model towardslower velocities. That Changs model has been identied on concretes with UPVranging between 1900 and 3800 m/s. Thus the dotted line onD. Breysse / Construction and Building Materials 33 (2012) 139163 145b is plotted against a since all (a, b) pairs are more or less distrib-uted along a straight line on a semi-log diagram (Fig. 3).From Fig. 3, it can be said that all models, despite their apparentdiversity, follow the same rule, with a linear relationship betweenthe two coefcients:b a ln a b 2with a = 0.00021359, b = 0.00082086, and a determination coef-cient r2 = 0.965.Thus, instead of considering that the models have two indepen-dent coefcients a and b, it is possible to write:fc a expbaV 3where b(a) is given by Eq. (2). There is only one parameter (a) in Eq.(3) and all possible models cross at a common point (Vref, fcref). Aftersome modications, it appears that Vref = 1/a. It comes:V ref 1=0:00021359 4681 m=s and f cref 46:6 MPaThus, Eq. (3) can be rewritten in a slightly different form:fc fcref expbV V ref 4where fcref = 46.6 MPa and Vref = 4681 m/s. Fig. 4 plots a series ofcurves corresponding to seven different exponential models the ex-treme cases being (a = 0.000615, b = 0.0024) and (a = 2.812,b = 0.0006). This range has been chosen in agreement with thatfound in existing models. The latter pair, with a very small b valuecorresponds to a quite linear model.All curves cross at the same point, which is obviously not thecase with original models. For the sake of comparison, six originalmodels are plotted at Fig. 5, their characteristics being given inFig. 3. Relation between the 2 coefcients a and b of exponential functions for UPV-strength models.Table 6. These models have been selected either becausetheir correspondence with the reference point (Vref = 4681 m/s,fcref = 46.6 MPa) is maximum this is the case of the rst four mod-els or because they show the largest difference this is the caseof the two last.In fact, if plotted on Fig. 3, the (a, b) pairs of the rst four modelswould be very close to the regression line while the pairs for Aticiand Chang models would be at a signicant distance (below theline for Aticis and above for Changs). For a better understanding,one must add:Table 5Information about UPV-strength models from literature review.Type of model Expression Number of parameteExponential a exp (bV) 2Power a Vb 2Linear a V + b 2Polynomial (2) a V2 + b V + c 3Miscelleanous 2 or 3Fig. 5 mostly corresponds to extrapolation of the model towardshigher velocities.In both cases, the original model was tted on a limited domain(regarding UPV as well as strength) and it cannot be extrapolatedto a wider domain without care.To summarize these rst considerations, we are clearly not sug-gesting that there may be a universal relation between a and bcoefcients, as expressed in Eq. (1). However we think that allmodels previously established by various authors show a generalconsistency, clearly visible on Fig. 3. Nevertheless even with thehigh coefcient of determination on Fig. 3, one cannot concludethat all models must belong to the family of models drawn onFig. 4. Such an assumption would currently lead to high errors.We will now make a similar analysis of power law models.3.2.3. Analysis of power law modelsSeventeenmodels are considered for analysis. They are all of thetype fc = a Vb and have been published between 1988 and 2010. aand b coefcients vary in a very large range, with the following ex-treme values (all models have been harmonized by consideringthat V is expressed in km/s): a = 2.09 107, b = 12.809 [39]. a = 2.057, b = 1.7447 [58].As a general tendency, a decreases when b increases. Most b val-ues are between 2 and 6, with a mean value equal to 4.9. To thebest of our knowledge, nobody has ever tried to nd some consis-tency between these results. An interesting feature however ap-pears when b is plotted against a since, as was the case forexponential models, all (a, b) pairs are distributed along a straightline on a semi-log diagram which corresponds to a linear relation-ship between the two coefcients:b a ln a b 5with a = 0.684, b = 2.3934, and a determination coefcientr2 = 0.9942.Thus it is possible write Eq. (6), with only one independentcoefcient:fc aVba 6As for exponential models (see Section 3.2.2) this induces a directlink between a and b and all possible models cross at a commonpoint (Vref, fcref). After some calculations, Eq. (6) can be rewrittenin a slightly different form:fc fcrefxV=V refb 7where fcref = 33.1 MPa and Vref = 4314 m/s. Obviously, original mod-els do not cross at the same point. For the sake of comparison, sevenrs Number of references Original references26 [60,42]17 [92]18 [73]6 [61]3power law fc = a Vb is looked for, b must be very large to capturethe experimental pattern.Some authors have tried to justify a type of model that would bethe good one. One can cite [77] who have taken a fc = f(V4) func-tion, arguing that, in linear elasticity, V varies as the square root ofYoungs modulus and that experimental data on concrete agreewith a fc = k E2 relationship. On another way [68] have unconvinc-ingly tried to prove through models that the UPV-strength rela-tionship must be exponential. [113] tried to nd an unreachableuniversal law. He synthesized many laws found in the literatureand built a median law, which writes fc = 0.0872 exp (1.29 V). Itmust be clear that such a law cannot describe the effect of all inu-encing parameters and would not be of any use in practice.The most common models (exponential and power law) havetwo parameters which are identied independently, but a strikinglaw has emerged from their comparative analysis. For these twotypes of models, one single parameter seems to be enough to de-scribe the intrinsic physics underlying the UPV-strength relation-ilding Materials 33 (2012) 139163Fig. 4. Consistency between all exponential models and crossing point.(Vref = 4681 m/s, fcref = 46.6 MPa).146 D. Breysse / Construction and Buoriginal models are plotted in Fig. 6, their characteristics being gi-ven in Table 7. These models have been selected either because oftheir correspondence with the reference point (Vref = 4314 m/s,fcref = 33.1 MPa) is maximum this is the case of the rst four mod-els or because they show the largest difference this is the case ofthe three last.As seen with exponential models, some consistency between allpower law models has been found, leading to consider a relation-ship with only one unknown parameter instead of two.3.2.4. Some nal considerations on UPV-strength relationshipsFrom an in-depth review of 70 existing empirical models, it hasbeen shown in Section 3.2.1 that many types of empirical relation-ships can be used, the most common being the exponential model.As noted by [91], over a wide range of maturity, the relationshipbetween compressive strength and UPV is highly nonlinear... Thus,the sensitivity of the pulse velocity as an indicator of change inconcrete strength decreases with increasing maturity andstrength.... This explains why the identied coefcients in thelaws can vary a lot. When concrete is mature, strength can increasesignicantly while the change in UPV remains small. Thus if aFig. 5. Illustration of the variety of exponential models (sources for modelsprovided in references).Table 6Characteristics of some exponential models.Author [35] [60] [100] [30] [3] [25]a 0.0012 0.0141 0.06 2.901 0.0316 0.15833b 0.00227 0.0017 0.00144 0.0006 0.0013 0.0014ship. The second parameter, a, can be expressed as a function ofthe rst one, b.Some comments can also be made about the quality and repre-sentativeness of correlations: The UPV value does not change drastically between a 25 MPa-strength concrete and a 40-MPa one. Among all existing models,two thirds of them predict less than 400 m/s of differencebetween these two concretes. If there is only a limited contrastbetween stronger and weaker concretes, the contrast inUPV may be very small. In such a case, the signal to noiseratio, i.e. the expected deterministic difference on UPV dividedby the magnitude of error measurement on UPV may be verylow. The result may be a poor correlation and a low level of sig-nicance of the identied parameters. Existing correlations are mainly identied in the laboratory,from concrete specimens whose age is limited (typically lessthan few months). Two main differences must be noted whenthese correlations are used to predict on site properties: (a)the variability of measurements is larger on site than in thelab (see Table 2), (b) carbonation effects are limited in the labwhile they become signicant when concrete is older, whichis the case with on-site existing structures.Even if a correlation has been carefully established in laboratorystudies, these reasons explain why a calibration process will benecessary for on-site concrete strength assessment.Fig. 6. Illustration of the variety of fc(V) power law models (sources for modelsprovided in references). [111] have analyzed more into details the inuence of carbon-ation which has a severe impact on rebound measurement ,ildin3.3. Rebound number-strength relationships: a review3.3.1. Model typesMany authors have studied how rebound measurement R canbe correlated with concrete strength. An extensive review of theircontributions has been undertaken. The purpose is to get a generaloverview and to highlight what may appear as common rules. Infact apparent inconsistencies are common: many different typesof models are used and, for each type of model, the values of coef-cients can change within a large range. . . It seems that there is nogeneral law. A comprehensive literature review was recently per-formed by Szylagyi and Borosnyoi[110] who identied 60 modelsand analyzed 40 of them. Table 8 synthesizes some informationabout 89 models which have gathered from literature. It does notpretend to be exhaustive but it is probably representative. Threetypes of models are more commonly found: power law, linearlaw and second order polynomial law. Exponential model can beencountered but it is much less common than for UPV-strengthcorrelation.The rst type of model, i.e. power law model, will now beanalyzed.3.3.2. Analysis of power law modelsThirty-one models are considered for analysis. They are all ofthe type fc = a Rb and have been published between 1953 and2010. a and b coefcients vary in a very large range, with the fol-lowing extreme values: a = 3.54 105, b = 3.81 [39], a = 1.017, b = 0.968 [58].As a general tendency, a decreases when b increases. b valuesrange between 1 and 4, with a mean value equal to 2.1. Theseexponents are about the half of those for the UPV-strength powerlaw models (Section 3.2.3). This can be explained by a relativelylarger range of variation for R than for V. To the best of our knowl-edge, nobody has ever tried to nd some consistency betweenthese results. An interesting feature appears when b is plottedagainst a since, as it was the case for exponential and power lawmodels for UPV, all (a, b) pairs are more or less distributed alonga straight line on a semi-log diagram which corresponds to a linearrelationship between the two coefcients:b a ln a b 8with a = 0.267, b = 1.000, and a determination coefcientr2 = 0.979.Table 7Characteristics of some power law models.Author [44] [9] [72] [90] [92] [118] [58]a (V inkm/s)1.304 0.171 0.036 0.000241 0.00834 0.00220 1.745b 2.222 3.593 4.696 8.1272 6.074 6.289 2.057D. Breysse / Construction and BuThus it is possible to write a law with only one independentcoefcient, as it has been done at Section 3.2.3:fc aRba 9Eq. (9) induces a direct link between a and b and all possible modelscross at a common point (Rref, fcref). After some modications, onecan write:fc fcrefxR=Rrefb 10where fcref = 42.4 MPa and Rref = 42.4.thus on the rebound-strength correlation. They proposed theSBZ model, from the name of the authors, which combines aseries of ve specic models respectively describing: (a) theway material strength depends on water to cement ratio, (b)how strength develops with time, (c) how rebound and strengthare correlated, (d) how carbonation develops with time, accord-ing to water to cement ratio, (e) how carbonation inuencesrebound. Such a model could make it possible to correct therebound measurements in order to compensate the inuenceof carbonation on the outer layers of old age concrete, most models have been identied from laboratory studies, withyoung age concrete (typically up to 28 days, and in some casesup to few months) while the main challenge is to assess con-crete in existing carbonated structures. In addition, these lab-originating laws may be not representative of on-site contextwhere the material variability may be larger (including damageand cracking), the measurement error higher (see Table 4) andwhere the environment may have some inuence [30].As seen previously with UPV-strength models, a high level ofconsistency between all power law models for rebound-strengthcorrelation has been found, leading to consider a model with onlyObviously, original models do not cross at a common point. Forthe sake of comparison, seven original models are plotted at Fig. 7,their characteristics being given in Table 9. These models havebeen selected either because their correspondence with the refer-ence point (Rref = 42.4, fcref = 42.4 MPa) is maximum this is thecase of the rst three models or because they show the largestdifference this is the case for the last three.One must add, for a better understanding: the three crossing curves, despite their similarity, correspond toa very different b value, between 1.47 and 2.40, that Bellanders model was very soon noted as overestimatingstrength [46], while De Almeidas model is based on data fromhigh performance concrete, with strength ranging between 40and 120 MPa. Thus the dotted line on Fig. 7 mostly correspondsto extrapolation of the model towards lower strengths, that Nuceras model has been identied from measurements onstructures, where carbonation had probably developed. Therange of strength was very low (526 MPa) and the statisticalvalidity of the model is questionable (determination coefcientof the regression r2 being equal to only 0.24).3.3.3. Some nal considerations on rebound-strength relationshipsFrom an in-depth review of 89 existing empirical models, it hasbeen shown that many types of empirical relationships can beused, the most usual being the power law model. Existing modelshad yet been compared by [46] and recently by (Szylagyi and Boro-snyoi, 2009). The latter study conrmed the large variability inexisting models and that each model is only valid on the domainwhere it has been built. We are even less optimistic since somemodels from the literature may be not valid even on this restricteddomain, because of a lack of care in analyzing statistical results(like signicance of regression parameters). In any case, it is impos-sible to use any model for another concrete or a different context.Two other reasons can be advanced for restraining the use andextension of existing models:g Materials 33 (2012) 139163 147one unknown parameter instead of two. A calibration process willbe necessary in order to use such a model to predict on-site con-crete strength.eteildinTable 8Information about R-strength models from literature review.Function Expression Number of paramPower a Rb 2Linear a R + b 2Polynomial (2) a R2 + b R + c 3Exponential a exp (bR) 2Miscelleanous 24Fig. 7. Illustration of the variety of fc(R) power law models (sources for modelsprovided in references).148 D. Breysse / Construction and Bu3.4. How single NDT methods can be used for estimating strength: thenecessity of calibrationWe will identify in the following the main errors that must beavoided when one develops a model for estimating concretestrength from NDT measurements. Since previous sections have fo-cused on UPV and rebound measurements, the following com-ments and recommendations are addressed to these two types ofNDT measurements. They however remain valid for all types oftechniques.3.4.1. Synthesis of errors to avoid3.4.1.1. Do not look for an universal NDT-strength relationship thatsimply does not exist. Many reasons prevent the establishment of aunique NDT-strength law that would cover any situation. This evi-dence has been widely shared by experts, but it does not preventsome authors to pursue such an unreachable goal.One can read for instance: The aim of the work was to establisha general and direct relationship between the compressivestrength, the ultra-sonic pulse velocity and the rebound number,regardless of differences in mix proportions and age of concretes.[31] or This research aims to nd unied relationship [. . .] byusing statistical methods in the analyzing process depending onlaboratory tests [. . .] and nding correlation curves to predict thestrength of concrete much better. [85].When speaking about universality one must differentiate be-tween: (a) the type of model, (b) the value of themodel coefcients.Regarding the rst point, our literature review has conrmed thatthere is no universal model. According to ACI [91] the use of non-destructive tests in the eld should be preceded by the developmentTable 9Characteristics of some fc(V) power law models.Author [82] [28] [34] [8] [31] [86]a (V in km/s) 0.0051 0.024 0.167 0.008 1.0407 0.007b 2.3956 1.9898 1.4664 2.466 1.155 2.012are more or less equivalent. All are able to describe linear or quasilinear variations as well as highly nonlinear variations. Because ofthe measurement errors, the NDT-strength relationship may be lin-ear if the range of variation of strength (thus that of NDT measure-ments) remains small, when a nonlinear tendency appears when;(a) the range of variation is larger, (b) measurements are accurate.The preference for one given type of model is mainly subjective.Regarding the model parameters (coefcients) we have shownthat a striking consistency exists between most available models.This is probably a general rule whose practical consequence is thatincreasing the number of parameters is useless. We will later inthis paper (Sections 5.3 and 5.4) propose a strategy for concreteassessment in which the number of unknown parameters will bedrastically reduced.3.4.1.2. Avoid meaningless comparisons and check the statisticalvalidity of the model. A common practice in research papers is toconsider various earlier models from the literature and, by workingon a dataset obtained from a specic experimental program, tocompare their merits to those of a new model specically builtby the authors. In fact such a work is meaningless since it comesto compare a yet existing model with given xed parameters, toa new model whose unknown parameters are tted to describeat best the specic dataset. In fact calibration is always necessary.Another point is that the statistical validity of the model is notreally analyzed in many cases. Common practice is to give only thevalue of the model parameters and the determination coefcientr2. This may be not sufcient to draw conclusions. Precisely, thestandard error on these parameters must be known since theyof correlation curves, from laboratory tests done on standard con-crete specimens (cylindrical or cubic)madewith the samematerialsused in the structures concrete that is under evaluation.An interesting work has been undertaken by [39] who devel-oped a database from various sources. As these authors said thecorrelations presented [. . .] are intended as a reference to pointout the trends of experimental data, and not necessarily a new pro-posal of additional formulae. The data analysis enabled to esti-mate condence intervals for strength from the NDTmeasurement value. For instance, if R = 30, the average estimatedstrength is 20 MPa and the 90% condence interval is [16 MPa,30 MPa]. The prediction however remains very uncertain, whichmay make it useless in many cases.A careful attention paid to datasets shows that all model typesrs Number of references Original references31 [26]28 [115]15 [62]8 [100]7g Materials 33 (2012) 139163indicate the level of signicance of the regression (if the standarderror/mean value ratio is large, this means that the identied valueis not reliable). In any case, the value of the uncertainty/error onthe estimated strength value deserves more attention than the sin-gle determination coefcient. Regarding standards ACI [91] pro-vides guidance about the number of levels of the strength scaleto be tested, the number of repetitions of tests and on how to ana-lyze the test results.3.4.1.3. Be careful with any extension of the domain of validity (i.e.extrapolation). Applying the identied model to a wider domain isusually difcult for two reasons:out [14]. Kheder, pointing out the limits of the calibration processSonic and Rebound. Some works have shown that it is also possibleildin From a statistical point of view, extrapolation is possible, butthe quality of estimation decreases as the distance from the ini-tial domain increases. It is only when the average relationshipbetween NDT and strength and the standard errors on the val-ues of identied coefcients are known that the extrapolationcan be tested, and the level of condence of the estimationcalculated. Extending to a wider domain usually means trying to cover theinuence of additional factors. For instance it is possible to cal-ibrate NDT-strength relationships for high performance con-cretes but [18] has shown that if the standard nationalrebound-strength law (identied on normal concrete) is extrap-olated to try to estimate HPC strength, it tends to underestimateit.3.4.1.4. Think how the model can be used. As illustrated on Figs. 1and 2, the two main reasons explaining why there is no universallaw are that: in any experimental program, all measurements have someuncertainty/error, and the quality of the identication processdepends on these uncertainties, one has many inuencing factors, most of them beinguncontrolled.The rst problem can be solved by improving the quality ofmeasurements and by increasing their number. The second prob-lem is more complex. A natural strategy could be, once an inuenc-ing factor is identied, to build a series of models, each modelcorresponding to a given value of the inuencing factor. Once thesemodels are built it is possible, if the inuencing factor can also bemeasured on site, to use the model corresponding to this value inorder to estimate the strength. This way has been followed bymany authors.For instance [42] who initiated many works in that eld used anexponential model for the UPV-strength relation. He analyzed theinuenceof factors like cement content, typeof aggregate, aggregatesize distribution, curing temperature. Corrections were identiedfor each of them. Kheder quantied the inuence ofwater to cementratio, aggregate to cement ratio and specic weight. [3] comparedidentiedmodels by accounting for variations in age, cement quan-tity, y ash addition. The inuence of aggregate typehas been the to-pic of many works, among which those from [71,81]. The latteridentied seven parameters that must be identied and worked to-gether in order to get a quality factor for the aggregate which canlater be introduced as a correcting factor in the model.This last example shows what can be the limits of such correla-tions. The value of the inuencing factor(s) must be known for theconcrete to be tested. This is a hard task when one works with anold building, where parameters like the cement content or theaggregate size distribution cannot easily be estimated. Anotherlimit has been pointed more than thirty years ago [101] by a tech-nical committee devoted to strength assessment with reboundmeasurements: Some experts are of the opinion that it is possibleto apply correction factors for most of the important inuences butit is probable that if a large number of such factors were required,the estimate of strength is likely to be of low accuracy. It is possiblethat a simultaneous change of two inuences leading to the needto apply correction factors could result in an interaction factor(i.e. the principle of superposition may not apply) which will inu-ence the estimate of strength in a different way from that predictedfrom the separate action of these inuences. This statement re-D. Breysse / Construction and Bumains fully valid and it is the main drawback of a correction/cali-bration process that would concern input data, i.e. based on theidentication of impact and value of inuencing factors.to combine: two techniques like UPV and ultrasonic pulse attenuation (UPA)[68,98], or combining a ND technique with a semi-destructiveone like penetration test [118,72] or pull-out [14]; and rebound with a third technique like penetration resistance,leading to SonRebWin, where Win comes from the name ofthe Windsor probe [98,72,37].on inputs (i.e. that described at Section 3.4.1.d) recommended atwo-stage calibration process, in which on-site results were gath-ered on 510 points.More generally, the principle is that the (NDTi, fci) pairs, where icorresponds to the ith point of coring and where the NDT can becarried out either on the structure or directly on the core, serveas a reference for calibration. US standards [91] as well as Euro-pean ones [36] thus combine a prior model and a procedure for cal-ibration. Fig. 8 synthesizes how EN-13791 standards [52]recommend to identify strength from on-site NDT measurements.It is specied for rebound but the same gure can be drawn forUPV measurements. Information obtained on cores can be used:(a) either for establishing a specic model (correlation curve) Ap-proach A, (b) or for correcting, by shifting, the rst estimates givenby a prior model Approach B. Prior models are provided for UPVand rebound measurements. At least 18 cores which is a high num-ber in current practice are required for Approach 1, and at least 9(NDTi, fci) pairs are required for Approach 2. These large numbersrestrain the practical use of NDT, whose interest would be to re-duce the number of destructive tests to an acceptable minimum.We will come back later (Section 6.5) to the inuence of this num-ber of cores/pairs on the quality of the estimate.4. Combination of nondestructive techniques for strengthassessmentThe idea of combining two or several nondestructive techniquesto better assess material properties is a very general idea[11,79,13]. A RILEM Technical Committee (TC 207-INR) chairedby the author has recently been devoted to this issue. We will re-strain here to the case of combining UPV and rebound measure-ments. This combination has received the name of SonReb, forHowever, some (partial) corrections remain possible and maybe of interest. [104] have developed several regression laws byconsidering only two inuencing factors: the age of concrete (lessthan 7 days, between 7 days and 3 months, older than 3 months)and the aggregate type. They argued that these two parameterscan be easily controlled on site. t could be of interest to analyzeseparately the inuence of humidity on UPV, as it was done by[102] or of carbonation on rebound, as suggested by [111] whodeveloped a very original and convincing approach.3.4.2. Calibration process and how it is dened in European standardsEN-13791An alternative approach consists in calibrating the NDT-strength model by using additional information directly gatheredon the concrete to be tested. The idea is, at few well chosen pointson the structure, to get information on concrete strength and NDTmeasurements. This needs some destructive tests which are usu-ally compressive tests performed on cores. Variants are possiblewith reduced-size cores [16] or semi-destructive tests like pull-g Materials 33 (2012) 139163 149Increasing the number of techniques obviously improves thequality of t for a given set of experimental data [69] but may benot optimal because of (a) the additional cost, (b) the practicalproblems due to statistical relevancy and calibration. Since mostresearch efforts have been devoted to sonic and rebound combina-tion, we will concentrate on it.4.1. SonReb combination: an historical perspectiveRebound and ultrasonic pulse velocity measurements can becarried out quickly and easily. The idea of their combinationemerged in the sixties, with the seminal ideas of Facaoaru whohad worked earlier on ultrasonics [42] and other works in EasternEurope [107]. The underlying concept is that if the two methodsare inuenced in different ways by the same factor, their combineduse can cancel the effect of this factor and improve the accuracy ofthe estimated strength. It is the case, for example, with humidity: ahigher humidity increases pulse velocity but decreases the re-bound number. A careful attention to Tables 1 and 3 shows thatit is possible that other inuencing factors have an adverse effecton UPV and rebound and can have less inuence when the twomethods are used in combination. [61] also consider that UPV en-ables to investigate the concrete properties at a greater depth thanspecimens or cores and when the composition is known, and 1520% when only the composition is known. In fact, the nomogramis no more than a conversion curve like those analyzed at Section 3,but here the strength is estimated from the values of two nonde-structive parameters instead of a single one. It results a series ofcurves like on Figs. 9 and 10, instead of a single one. Two maindrawbacks have been pointed: since many factors are possibly inuencing either UPV orrebound (cf Tables 1 and 3), many correcting coefcients mustbe identied, making the procedure as difcult as for monovar-iate regressions, when some correcting coefcients have been identied, theirvalues as well as the basic correlations were tted on a seriesof concretes and cannot be extrapolated to the general case,To add a last point, the benets of combination appear to becontroversial. [61] stated that: It appears that by using two non-destructive test methods, it is not possible to gain an increase ofprecision in results, if one of the methods used is substantially lessprecise (in this case the rebound hammer method) than the other150 D. Breysse / Construction and Building Materials 33 (2012) 139163rebound, thus providing additional information on the material.[96] argues that the combination is informative since V and R donot develop at the same rate with time.[40] applied the combination by taking the mean of three UPVmeasurements and the mean of six readings for rebound numbersR, and then determined the compressive strength by using three-dimensional surfaces (called nomograms) where the concretestrength is expressed as a function of two variables: fc = f (V, R).These ideas were applied in Italy [27] and by [45].Facaoaru later chaired the RILEM Technical Committee 43 whopublished its recommendations in the 90s, after having workedduring many years on this topic [41]. The TC43 guidelines explainhow it is rst necessary to build a reference curve, from results ob-tained on 75200 specimens, with at least 30 specimens for eachrange of 10 MPa, and to use correcting factors if the concrete toanalyze differs from the reference by one or several parameters:type of cement, aggregate type, aggregate size, etc. Figs. 9 and 10show two examples of nomograms: the rst one gives iso-strengthcurves while the second one gives iso-rebound curves.The accuracy of the estimated strength (the range comprising90% of all the results) is considered to be 1014% when the corre-lation relationship is developed with known strength values of castFig. 8. Schematic view of two possible appr(in this case the ultrasonic pulse method). This statement clearlypoints at a fundamental issue that had been addressed for mono-variate correlations: that of the required accuracy of the measure-ments. It adds that if a rst technique has been used, adding asecond technique which would not be accurate enough is wastingenergy. This issue will be addressed more in details on real data atSections 5.2 and 5.4 and on synthetic data at Section 6.4.2.4.2. Bivariate relationships: a review4.2.1. Model typesSeveral methods can be used to build the model. For instance: [41] or [27] suggest to split the whole data set into subsets ofequal strength and to build the (R, V) curves for each strengthclass, [99] considers subsets of equal rebound values and builds fc(V)curves for each rebound class.The more practical way is probably to directly identify the bilin-ear regression by minimizing an error function in a spreadsheet,oaches for calibration, from EN 13791.since this method directly provides the magnitude of residuals(model errors). The use of nonexplicit models has developed re-cently. It is for instance possible to estimate strength by using aneural network [37,112] but extrapolation is therefore impossible.An extensive review of existing models has been undertaken,with the purpose to get a general overview, and to highlight whatmay appear to be common rules. As for monovariate models, dif-ferent types of models have been used and, for each type of model,the values of coefcients can change within a large range. A uni-versal law, which could be used for any concrete has beensearched for by many authors. Some have tried to combine severalsets of data [113,99,38,85] to nd such a law. These efforts havebeen in vain.Table 10 synthesizes some information about 59 models whichhave been gathered in literature. This information does not pretendto be exhaustive, but it is probably representative. Two types of many uncontrolled factors still exist, like concrete properties,water content, the way of coring, the core size and shape, etc. Using two techniques in combination instead of a single onechanges nothing about calibration which remains a mandatorystep. The methodology described in Fig. 8 remains valid and twoapproaches can be used: Approach A consists in looking for a multivariate regressionfunction between the (R, V) values measured on the concretespecimens (either directly on the structure, on cores taken fromthe structure or on cubes cast with the same concrete) and fcvalues measured on cores or cubes. To be consistent with themost common models, we will consider a double power lawmodel and we will identify the three coefcients of the bilinearmultivariate regression:ln fc ln a b lnV c lnR 11which corresponds to fc = a Vb Rc Approach B consists in using a prior model, e.g. a (fcest = a Vb Rc)model, where b and c are given, and to calibrate the a value. Cal-ibration can be done by calculating the mean value of estimatedstrength fcest,mean and the mean value of experimental strengthfcexp,mean:k fcest;mean=fcexp;mean 12Thus the calibrated model can be written as:D. Breysse / Construction and Buildin the measurement error of NDT technique is not usuallyaccounted for to establish correlations and the level of con-dence of the identied model is not often provided.In two cases the b or c exponent have even been found negative,which clearly contradicts the laws of physics and is surely the re-models are more commonly found: double power law and bilinearlaw. Both need three parameters.The merits of different models have often been compared [56].For instance, in recent Italian works on seismic retrotting basedon SonReb methodology, the concrete strength can be assessedby three models and their results compared [17].4.2.2. Analysis of double power law modelsWe will focus on the analysis of double power law models fc = aVb Rc which are the most common ones. The values of b and c coef-cients are varying in a wide range, as shown in Fig. 11 where thecurve fc = 30 MPa has been drawn according to the equations givenby fourteen models.Even if some consistency is visible between most of the curves,regarding their slope (i.e. the relative inuence of change in V and Ron the strength), there is a high variability. Some curves are moreor less parallel but with some shift. This corresponds to a differentvalue of the a coefcient, whose consequence is a high uncertaintyon the strength value. This variability has several causes: some relationships have been identied on laboratory speci-mens and others on cores,Fig. 9. Nomogram from RILEM TC43.sult of a careless statistical analysis on measurements with a highnoise level. One must be very cautious when considering modelsfrom the literature, especially when the authors have not providedthe level of signicance of the identied coefcients.If one limits consideration to the set of twelve models for whichc/b was ranging between 1 and 3, one nds that b varies between1.6 and 4.8 and c varies between 0.9 and 1.8. The respective aver-age values are b = 2.6 and c = 1.3. These values are very close fromthose given by RILEM (b = 2.6, c = 1.4, [51]. Fig. 12 is the nomogramcorresponding to this last model in which the iso-strength curvesbetween 20 and 50 MPa have been drawn.5. Quality of the estimation with SonReb after calibration5.1. Calibration methodologyFig. 10. Other nomogram (after [55].g Materials 33 (2012) 139163 151fcest cal a=kVbRc 13Table 10Information about UPV-R-strength models from literature review.Type of model Expression Number ofPower-power a Vb Rc 3Bilinear a + bV + cR 3Double exponential a exp (bV) exp (cR) 3Polynomial (24) 3 or 4Miscelleanous 2548005000 Schickert PascaleLenzi IdrissouGiacchetti (RILEM) Cianfrone-FacaoaruGalan-1 Tanigawa et alV (m/s)152 D. Breysse / Construction and BuildinIn the following, the prior model will befcest aV2:60R1:30 14with a = 1.15 1010. The choice of this value, somehow arbitrary,has been done in agreement with experimental data (see Sec-tion 5.2). In any case, this value will be modied after calibration.In Approach A, three degrees of freedom (a, b and c) must beidentied while in Approach B, b and c are assumed to be known(and constant in all cases) and there is only one unknown param-eter. The efciency of these two approaches will be compared on a30003200340036003800400042004400460020 25 30 35 40 45 50Pucinotti R. Di Leo A., Pascale GArioglu and Koyluoglu GasparikGalan-2 KhederRFig. 11. Iso (fc = 30 MPa) curves according to 14 existing double power law models.variety of data taken from the literature, both from laboratorystudies and from real structures, in order to cover a wide eld ofsituations.5.2. Laboratory studies: experimental data and model identicationMany authors have undertaken research works on the use ofNDT techniques for a better assessment of concrete strength. Mostof these works are laboratory studies. Fifteen studies are consid-ered here, corresponding to a total of 645 (fc, R, V) data sets. Ta-ble 11 summarizes the concrete properties for these studies.One must note that most of these studies are limited to 28 day-old concretes, the range of properties being covered either by mea-suring the properties since the earliest ages or by considering aFig. 12. Nomogram of iso-strength curves for the Giacchetti model:fc = 7.695 1011 V2.6 R1.4.wide range of water/cement ratio. Two studies concern high per-formance concretes.For each of the 15 data series, one can identify the best power lawregressionmodels: (a)with a single variable, V or R, (b)with V and R.In the last case, this comes to t the best regressionmodel accordingto Eq. (11) (Approach A). Table 12 summarizes the values of theexponents identied in these three cases. Values in italics corre-spond to cases for which the coefcient is considered as not beingstatistically signicant (Students test at 95% level of condence).These cases usually correspond to low values of the exponents thatcan appear when the effect of one variable is overshadowed by thedominant effect of the second variable. This conrms that it is man-datory,whena regressionmodel (like a doublepower law) is built, tocheck the statistical relevancy of its parameters.The values of the exponents for single regressions vary from astudy to another but are statistically signicant in all cases. Ifone excludes two studies (Muniandys and Lees), for which thevery high values of V exponent are explained by a very limitedrange of variation for V (see Table 12), and if one keeps only statis-tically signicant results, the respective average values of theexponents are: respectively 4.8 and 2.2 for V and R in single regression models.These values are very close to those identied in the literaturestudy (4.9 for V cf Sections 3.2.3 and 2.1 for R cf Section 3.3.2), 4.0 for V and 1.5 for R in bivariate models. These values appearto be larger than those identied previously. One must also notethat these coefcients are much less stable than those for singleregression models.Approach B has also been used for each of the 15 data series,with the prior model of Eq. (14). The last column in Table 12 sum-marizes the k values. It shows that, with a same prior model in allcases (without any tting), the error on the average strength canvary from 27% (De Almeida) to +39% (Oktar). This error is obvi-ously reduced to zero after calibration (cf Eq. (13)).5.3. Laboratory studies: efciency of calibrationThe quality of the strength estimation will now be compared forsix options for calculating fcest: fcest1: Prior model, Gasparik model: fcest = 8.06 108 V1.851.246parameters Number of references References33 [27]15 [73]7 [116]6 [8]8g Materials 33 (2012) 139163R , fcest2: Prior model, authors model: fcest = 1.15 1010 V2.60 R1.30, fcest3: Approach A, single regression model with V only, fcest4: Approach A, single regression model, with R only, fcest5: Approach A, double regression model with V and R, fcest6: Approach B, authors model after calibration with k.The Gasparik model [48] has been chosen because it has beenwidely used in recent years in the context of seismic retrotting[17]. All estimations will be compared regarding three criteria:the r2 coefcient of determination, the root mean square error(RMSE) and the average value of the absolute relative error e. Itis expected that:e 15.2.1.7.5.1.0.2.1.7.9.1.8.0.2.6ildin uncalibrated models (i.e. fcest1 and fcest2) have a larger error, the bivariate regression fcest5 is more accurate than singleregression models fcest3 and fcest4, the estimation with a unique prior model fcest6 is less accuratethan a specic calibration, since the latter has three degrees offreedom while the former has only one.Tables 1315 synthesize (respectively for r2, RMSE and e) theperformances of the 6 options for the 15 data series. The twolast columns in bold characters correspond to the best resultsthat can be reached with the two approaches. It must be empha-sized that it is assumed here that the calibration (whatever theApproach) is done by using information obtained on all speci-mens, and not on a limited sample. Figs. 13 and 14 illustratethe quality of estimation with and without calibration (sevenpoints with strength above 130 MPa remain out of scale).Fig. 15 show the cumulated distribution of the error for the lasttwo models.A rst analysis of these three tables shows that: uncalibrated models have obviously larger errors with RMSE ofabout 10 MPa against 47 MPa for calibrated models. Similarconclusions can be made for r2 or e, with Approach A the comparison between monovariate andTable 11Mean values and range of variations of strength (MPa), rebound and UPV (m/s) for thMeannb fc RCianfrone [27] 1979 110 45.6 34Knaze Beno [61] 1984 63 27.2 37De Almeida [31] 1993 60 79.2 42Oktar [88] 1996 60 23.6 30Arioglu [1] 1998 12 129.6 60Qasrawi [99] 2000 24 25.7 32Cummings [29] 2004 12 50.1 45Domingo [34] 2005 12 28.8 33Idrissou [56] 2006 36 41.0 31Wu [117] 2008 90 49.0 43Muniandy [83] 2009 36 30.1 25Machado [71] 2009 60 36.9 33Lee [64] 2009 20 31.9 32Sbartai [105] 2010 26 39.0 33Lenzi [65] 2010 24 43.5 43D. Breysse / Construction and Bubivariate models conrms that using techniques in combinationusually reduces the error, with average RMSE decreasing from67 MPa to 4.4 MPa, calibration with Approach B is slightly less efcient than cali-bration with Approach A, but it must be reminded that it isbased on a single universal curve and a simple multiplication.Figs. 13 and 14 illustrate how calibration improves the qualityof the estimate. The difference between the real strength and itsestimation results a bandwith of about 10 MPa. Fig. 15 shows thatthe error is between 5 MPa in 80% of the cases for fcest5 and in 70%of the cases for fcest6.The efciency of combination has been widely discussed inthe literature, without consensus. Positive results have oftenbeen published, but [95] wrote: Unfortunately, analysis of suchstrength estimates suggests that the use of the rebound test, log-ical as it may have seemed several decades ago, contributes lit-tle, if any, to the increase of the ultrasonic strength estimation.His explanation was that a same inuencing factor (e.g. aggre-gate type) can have a similar inuence on the two measure-ments (a denser aggregate increases both velocity and reboundnumber). In such a situation, carrying out two measurements in-stead of one does not increase the quality of estimation, as dis-cussed at Section 4.1.Other explanations lay on measurement uncertainties. Tables1315 also reveal contradictory results: combination may be efcient is some cases (Lenzi, Cianfrone,Oktar, Lee Jian, and Qasrawi), which correspond to cases whereeach technique gives alone a correct but not perfect result, combination may be useless in other cases, either because a sin-gle technique gives very good results (Domingo) or, more often,because one technique has a much poorer quality than theother. It is the case here for instance for poor rebound measure-ments (KnazeBeno, Cummings, Munyandy) or poor velocitymeasurements (De Almeida, Machado, and Wu). In such case,adding a very noisy measurement to the rst one, which is ofacceptable quality, cannot provide added value.To summarize these results, the efciency of the strength esti-mation depends on three elements:(a) The sensitivity of the property to be estimated (e.g. strength)to the NDT measurement.(b) The range of variation of the measured values (either V or R):data series (nb is the number of measurements, age in days).Range AgeV fc R V4911 67 26 1620 7284206 18 14 566 284771 78 31 770 33654461 35 28 1870 285177 105 17 774 1564229 35 22 1040 4199 54 16 768 283654324 36 26 952 1l44825 36 10 800 7284714 43 36 1350 4505 34 11 370 7284292 46 22 990 3284401 31 13 330 7284708 45 20 590 284344 35 10 605 28g Materials 33 (2012) 139163 153for a given sensitivity, the larger the range, the better thestatistical regression model. For instance, measuring V isuseful only if some contrast exists on V. As seen at Sec-tion 2.1, this contrast can be high at early ages but is muchlower when concrete is mature and undamaged.(c) The magnitude of the measurement error: for a given sensi-tivity and a given range of variation, the smaller the mea-surement error, the more accurate the statistical regressionmodel.Two other issues must be considered: the combination is efcient only if the sensitivity of the twomeasurements to the uncontrolled inuencing factors isdifferent, the prior model must be relevant. Here a double power lawmodel was considered for Approach A and for Approach B (Eq.(14)). In both cases, model error can have some inuence onthe quality of the nal result. However, rst analyses, notdetailed here, have shown that this inuence is much lowerthan that due to other causes.licsildinTable 12Coefcients identied for the 15 data series and Approaches A and B numbers in ita154 D. Breysse / Construction and Bu5.4. On site investigation: experimental data, model identication andefciency of calibrationThe number of on site investigation studies in which originaldata have been published is far less smaller than for laboratoryApproach A, single regressionV exponent R exponentCianfrone 5.00 2.09Knaze-Beno 4.65 1.80De Almeida 5.54 1.21Oktar 2.02 0.94Arioglu 6.08 3.00Qasrawi 6.58 1.92Cummings 6.25 2.36Domingo 5.63 1.47Idrissou 5.28 2.76Wu 0.54 1.06Muniandy 12.26 2.17Machado 4.81 2.15Lee 13.53 2.19Sbartai 6.27 1.73Lenzi 4.09 6.27a Corresponds to a level of condence to 90% according to Students test.Table 13Performance of approaches with and without calibration: determination coefcients r2.parameter (est6).Without calibration Calfcest1 fcest2 fcesCianfrone [26] 0.912 0.914 0.7Knaze Beno [60] 0.782 0.790 0.8De Almeida [30] 0.827 0.827 0.6Oktar [87] 0.815 0.815 0.7Arioglu [1] 0.918 0.917 0.9Qasrawi [98] 0.890 0.898 0.8Cummings [28] 0.770 0.783 0.9Domingo [33] 0.974 0.974 0.9Idrissou [55] 0.803 0.801 0.6Wu [118] 0.930 0.914 0.5Muniandy [82] 0.751 0.754 0.8Machado [70] 0.892 0.889 0.6LeeJian [63] 0.866 0.868 0.8Sbartai [104] 0.707 0.710 0.5Lenzi [64] 0.895 0.895 0.7average 0.849 0.850 0.7Table 14Performance of approaches with and without calibration: RMSE (MPa).Without calibration Calfcest1 fcest2 fcesCianfrone [26] 5.2 5.5 8.3Knaze Beno [60] 5.6 6.8 1.8De Almeida [30] 24.5 23.4 11.Oktar [87] 10.7 10.1 4.2Arioglu [1] 24.0 25.3 10.Qasrawi [98] 4.9 5.6 4.7Cummings [28] 12.4 11.8 3.9Domingo [33] 5.7 6.5 1.7Idrissou [55] 6.2 5.3 6.4Wu [118] 12.5 15.2 8.6Muniandy [82] 9.4 8.0 3.9Machado [70] 7.2 6.0 6.5LeeJian [63] 14.2 13.9 6.4Sbartai [104] 6.7 6.6 8.5Lenzi [64] 4.7 5.0 9.7average 10.3 10.3 6.4when the level of condence is below 95%.Approach A, double power law Approach Bg Materials 33 (2012) 139163studies. Four studies, all originating from Italy, are analyzed here,corresponding to a total of 64 data sets. Table 16 summarizes theconcrete properties for these studies.These studies concern old concretes from buildings or bridges.The mechanical properties as well as the NDT values are muchb c k2.17 1.41 1.044.37 0.27 1.241.04 1.02 0.733.12 1.74 1.395.87 0.11 0.845.17 0.70 1.156.40 0.13 0.871.63 1.06 1.151.22 2.30 0.963.36 1.17 1.3411.30 0.23 0.810.51 2.00 0.877.50 1.18 0.982.03a 1.45 1.013.27 2.22 1.04Bold characters for SonReb Approach A (est5) and calibrated model with only oneibration - Approach A Cal. Approach Bt3 fcest4 fcest5 fcest649 0.863 0.915 0.91458 0.429 0.863 0.79086 0.854 0.846 0.82708 0.815 0.815 0.81523 0.877 0.925 0.91764 0.697 0.914 0.89849 0.307 0.946 0.78371 0.960 0.971 0.97465 0.813 0.824 0.80171 0.930 0.944 0.91440 0.670 0.845 0.75445 0.924 0.928 0.88982 0.948 0.963 0.86808 0.673 0.707 0.71055 0.780 0.887 0.89572 0.759 0.886 0.850ibration - Approach A Cal. Approach Bt3 fcest4 fcest5 fcest66.4 4.9 5.03.6 1.8 2.24 7.8 8.0 12.14.6 3.3 3.82 12.9 10.1 12.16.1 4.0 3.514.6 4.0 9.22.1 1.7 3.24.8 4.6 5.03.3 3.1 8.35.9 3.8 5.03.1 3.0 3.74.3 3.6 5.36.9 6.6 6.519.6 3.9 4.97.1 4.4 6.0Table 15Performance of approaches with and without calibration: average absolute relative errorWithout calibration Calfcest1 fcest2 fcesCianfrone [26] 0.10 0.11 0.1Knaze Beno [60] 0.21 0.25 0.06De Almeida [30] 0.31 0.29 0.1Oktar [87] 0.42 0.34 0.1Arioglu [1] 0.16 0.16 0.06Qasrawi [98] 0.22 0.25 0.1Cummings [28] 0.16 0.16 0.07Domingo [33] 0.15 0.16 0.06Idrissou [55] 0.10 0.10 0.1Wu [118] 0.19 0.22 0.1Muniandy [82] 0.23 0.20 0.1Machado [70] 0.17 0.13 0.1LeeJian [63] 0.20 0.20 0.1Sbartai [104] 0.15 0.15 0.2Lenzi [64] 0.10 0.12 0.1average 0.17 0.19 0.1Fig. 13. Comparison between measured and estimated values, without calibration,model 2.Fig. 14. Comparison between measured and estimated values, with calibration,model 6.Fig. 15. Cumulated distribution of the error between estimated and experimentalstrength, for 645 data points.D. Breysse / Construction and Buildine.ibration - Approach A Cal. Approach Bt3 fcest4 fcest5 fcest67 0.13 0.10 0.100.12 0.06 0.071 0.08 0.08 0.135 0.16 0.12 0.150.08 0.06 0.085 0.29 0.13 0.170.24 0.07 0.170.06 0.05 0.112 0.10 0.09 0.104 0.06 0.05 0.173 0.19 0.12 0.166 0.07 0.06 0.095 0.12 0.09 0.142 0.15 0.15 0.156 0.27 0.07 0.103 0.14 0.09 0.13Table 16Mean values and range of variations of strength (MPa), rebound and UPV (m/s) for thefour data series.Mean Rangenb fc R V fc R VBrognoli [17] 2007 10 15.8 30.0 2669 13 16 1566Masi [75] 2008 18 18.1 26.3 2765 29 25 2780Gennaro Santori [50] 2010 13 36.3 45.0 4490 39 10 872Nucera [86] 2010 23 14.0 41.4 3769 21 19 1140g Materials 33 (2012) 139163 155lower than those in Table 12, because concrete may be damaged ormay had very poor characteristics at origin. For instance, the dataset from Masi contains some strength values of about 5 MPa, re-bound numbers of about 20 and velocities less than 2000 m/s. An-other difference with laboratory studies is the higher level ofvariability of measurements, which can be due either to harderconditions or to a larger material variability, including in somecases damage and cracking.As it was done at Section 5.2 for laboratory studies, for each ofthe 4 data series, the best power law regression models can beidentied: (a) with a single variable, V or R, (b) with V and R. Inthe last case, this comes to t the best regression model accordingto Eq. (11) (Approach A). Table 17 summarizes the values of theexponents identied in these three cases. Values in italics corre-spond to cases for which the coefcient is considered as not beingstatistically signicant (Students test at 95% level of condence).The values of the exponents for single regressions vary from astudy to another but are statistically signicant in most cases,which is not the case for bi-variate models, where the effect ofone variable may be overshadowed by the dominant effect of thesecond variable.Approach B has also been used for each of the 4 data series, withthe prior model of Eq. (14). The last column in Table 17 summa-rizes the k values. It shows that, with a same prior model in allcases (without any tting), the error on the average strength is dra-matically larger than in laboratory studies, with possible over (orunder) estimation of about 100%.Tables 12 and 17 cannot be compared on a rm statistical basis,because they are too few datasets. However it seems that, whenconsidering only statistically sounded regressions, the c values(i.e. R exponents) are somehow similar when the b values (V expo-nents) appear to be lower on site. This may be due to a higher mea-surement error on site and will have to be conrmed by furtherstudies.Tables 1820 show some similarities with the laboratory stud-ies but also some major differences: uncalibrated estimates remain very far from real values, andcannot be used, if calibration is done with Approach A (i.e.) with a set-specicmodel, identied on each data set, the estimation is highlyimproved, with an average RMSE of 3.6 MPa, comparable to thatobtained for laboratory studies. The interest of combining thetechniques is conrmed in some cases (Brognoli and Gennaro-Santori) where it reduces the RMSE but not in others. The rea-sons are known: combination is useless if the quality of onetechnique is much lower than that of the other technique: thisis the case for UPV measurements for Masi and rebound mea-surements for Nucera, the efciency of Approach B is not systematic and the resultingerror on the estimated strength is too high in two cases (Brogn-oli and Masi). The reason is probably that for these two data setsthe prior model does not describe accurately the apparentdependency of strength to UPV or rebound, as seen from VApproachA seems tobe themost efcient but the results havebeenobtained by using all data for calibration (e.g. data on between 10 and23 cores). If one looks at data from Masi; due to a large variability onmeasurements, the b exponent in the double power lawmodel is neg-ative. This is clearly in contradictionwithphysics. Fromasimplemath-ematical point of view, this model is however the best t fordescribing the correlation between strength and NDT values. Never-theless, the model is not well adapted for predicting strength on an-other data set. This point deserves a more detailed analysis whichmust focus on the inuence of the sample size for calibration.These rst results show that, if the methodology of combinationenables a real improvement of the strength estimate in the contextof laboratory studies, it appears to be less efcient with on-sitedata. The main reasons are: an usually smaller size of the data set which, combined to the lar-ger measurement uncertainty, increases the uncertainty on theregression model. Among the four data sets, only one (Gennari-Santori) corresponds to a regression model which is statisticallysignicant for both b and c exponents. Increasing the databaseby collecting a larger amount of data must be a priority,Table 17Coefcients identied for the 4 data series and Approaches A and B.Approach A, single regression Approach A, double power law Approach BV exponent R exponent b c kBrognoli 1.06 1.64 0.35 1.27 0.55Masi 1.11 1.70 0.21 1.95 0.59Gennaro-Santori 3.41 2.62 2.52 1.91 1.37Nucera 3.86 1.28 3.77 0.14 2.06n.ibrat33178365199ibrat3156 D. Breysse / Construction and Building Materials 33 (2012) 139163exponent values and b values in Table 17.Table 18Performance of approaches with and without calibration: coefcients of determinatioWithout calibration Calfcestl fcest2 fcesBrognoli 0.823 0.801 0.7Masi 0.695 0.682 0.5Gennaro Santori 0.694 0.695 0.5Nucera 0.444 0.484 0.5Average 0.664 0.666 0.5Table 19Performance of approaches with and without calibration: RMSE (MPa).Without calibration Calfcestl fcest2 fcesBrognoli 4.9 7.5 8.2Masi 8.6 10.1 5.5Gennaro Santori 17.5 15.7 7.0Nucera 21.7 16.7 3.1Average 13.2 12.5 6.0Table 20Performance of approaches with and without calibration: average absolute relative errorWithout calibration Calibrafcestl fcest2 fcest3Brognoli 0.30 0.47 0.53Masi 0.40 0.53 0.37GennaroSantori 0.51 0.45 0.15Nucera 1.66 1.22 0.15Average 0.72 0.67 0.30tion approach A Calibration approach Bfcest4 fcest5 fcest60.827 0.889 0.8010.767 0.765 0.6820.426 0.704 0.6950.140 0.553 0.4840.540 0.728 0.666tion approach A Calibration approach Bfcest4 fcest5 fcest63.8 1.4 (8.0)4.3 4.4 (14.1)7.8 5.6 5.84.4 3.1 3.45.0 3.6 (7.8)e.tion approach A Calibration approach Bfcest4 fcest5 fcest60.23 0.08 (0.40)0.29 0.29 (0.58)0.16 0.12 0.140.28 0.15 0.180.24 0.16 (0.32)case of Approach B, the prior model is that dened at Eq. (14).ildin a larger variability ofmaterial properties for on sitemeasurements,with a range of variation of strength andNDTmeasurementswhichis different, and a larger magnitude of measurement errors, coring can induce some spurious effects, like additional damage.In some cases [86] the correlation between the velocity mea-sured directly on the structure and the velocity measured onthe cores taken at the same points may be very poor, the prior law has been identied from laboratory studies, wherethere is no (or little) carbonation. In old structures, carbonationhas developed with a signicant effect on rebound and a lowereffect on UPV. It would be useful to get a larger data set, in orderto check if the prior model is adapted to this different situation, there is another difference in the physics underlying the corre-lations: while in the laboratory the low V or R value correspondto the earliest ages of concrete, in old structures, they probablycorrespond to damaged material. The physical mechanisms aredifferent, but the same prior regression model was consideredin all cases. Further studies could be dedicated to the inuenceof damage (namely distributed cracking) on the NDTmeasurements.6. Using virtual NDT assessment to better understand and to gofurther6.1. Virtual NDT assessment: objectives and principlesThe results discussed above have been obtained on the basis ofavailable experimental data from the literature. The effects of somefundamental items have been discussed: range of variation of theproperties, measurement error, model uncertainty in relation withthe number of cores. . . In any case, the possibilities offered by realdata sets remain limited, since many parameters remain unknownor uncontrolled, and since it is not possible to perform a parametricstudy. To deepen the analysis, an alternative is offered by syntheticsimulations. The idea is to reproduce, within the computer, thephysics involved and the main patterns of the assessment method-ology and to analyze, by the means of simulation, what are themost determinant factors governing the quality of NDT strengthassessment.The synthetic world created within the computer must mimicas closely as possible the real world. It must reproduce: (a) the dis-tribution of material properties (e.g. strength) and inuencing fac-tors, (b) the physics involved between these parameters and thoseinvolved in the NDT measurements (e.g. rebound and UPV), (c) theerrors and uncertainties induced at various stages: measurement,modeling and calibration. . .Synthetic simulations will be used to clarify issues like: How the quality of the techniques can inuence the efciency oftheir combination? Is calibration with Approach A more efcient than withApproach B? What is the inuence of the number of cores on the quality ofthe estimation?6.2. Framework and development of the synthetic simulationsThe general principles of the synthetic approach works in foursteps: Step 1 generation. A data set of material properties is gener-ated, the focus being given on material strength fc and on a spe-cic inuencing factor, concrete humidity (described throughD. Breysse / Construction and Busaturation rate X) which is known to be inuencing, cf Tables1 and 3). These two parameters are randomly simulated accord-ing to a prior statistical distribution representing the materialThe number n of cores used for calibration is a parameter whichcan vary from 1 to any value. Monovariate models are also sys-tematically identied for comparison. Step 4 validation. Because of measurement errors, model errorand statistical uncertainty (limited size of the core set used forcalibration), the estimated strength differs from the truestrength (which, in this synthetic approach, is available andhad been generated at Step 1).Internal consistency between the four sensitivity exponents canbe checked since it is possible to combine Eqs. (15) and (16) and, byeliminating X, to express the strength as a function of R and V. Itthen becomes:bx=cx kvariability. It is assumed that material strength has a Gaussiandistribution N(fcm, s(fc)) and that saturation has a truncatedGaussian distribution N (Xm, s(X)), with X 6 100%. Step 2 measurement. Nondestructive measurements are sim-ulated and two values V (for UPV velocity) and R (for rebound)are generated. These values are calculated in two steps: (a) atheoretical value is generated, according to relationships V (fc,X) and R (fc, V) and then, (b) a random error is added, corre-sponding for each technique to the measurement error. Thiserror is assumed to have a Gaussian distribution N(0, s(R or V)).The relationships for V and R are the following:V V ref fc=fcref1=bf X=Xref1=bx 15R Rreffc=fcref1=cfX=Xref1=cx 16where the reference values (ref) are arbitrary values introduced inorder to normalize the equations, and have no inuence on the re-sults. The exponents quantify the sensitivity of V and R to strengthvariations and humidity variations. The reference values areRref = 40, Vref = 4000 m/s, Xref = 85% and fcref = 40 MPa.The exponent values have been carefully chosen, in order toaccurately describe what is observed in practice. The strength sen-sitivity exponents bf and cf have respectively been taken equal to4.90 and 2.10, in agreement with what has been identied in Sec-tions 3.2.3 and 3.3.2. The humidity sensitivity exponents bx and cxhave respectively been taken equal to 0.13 and 0.28. They respec-tively correspond, if one compares the value at saturation(X = 100%) with the value for a dry-air specimen (assumed atX = 65%): to a 12% decrease for rebound, when [101] or Czech StandardCSN 731373 after [19] suggest a 20% decrease and [58] suggestsa 34 points decrease, i.e. less than 10%, to a 6% increase for V, i.e. 270 m/s for V = 4500 m/s. This is inagreement with experimental data: the increase in velocitybetween dry cured specimens and wet cured specimens rangedbetween 300 and 400 m/s for Ref. [77]; [58] suggests anincrease between 200 and 400 m/s; experimental results inthe SENSO program have shown a velocity increase from4365 m s1 for X = 85% to 4630 m s1 for the same concrete sat-urated [70]. Step 3 calibration. The calibration stage consists of using thenondestructive measurement values and the concrete strengthvalues obtained at the same points. The two approaches (Aand B) can be used in order to build the fcest = f (R, V) model.In case of Approach A, the best double power law is tted. Ing Materials 33 (2012) 139163 157fc=fcref R=RrefV=V ref 17where k = bf cf cx/(cf cxbf bx).Eq. (17) is a double power law.When it is rewritten as fc = a Vb Rc,one nds b = 2.35 and c = 1.09. These value are strikingly close to theaveragevalues identied in the literature review(Section4.2.2). Thisconvincingly conrms the consistency of the simulated physics.Regarding the error measurements, based on the magnitude ofvariability identied from the literature review (cf Tables 2 and 4),second technique varies. These two situations are considered intion with UPV measurement of varying quality. Fig. 18 shows whatpoor measurements). It can be seen that:ponents of the regression models (Approach A).Theoretical value Identied value (average sd)2.10 1.65 0.334.90 3.48 0.52Table 22Quality of strength assessment (RMSE, in MPa), for four approaches.Approach A Approach BMonovariate, R Monovariate, V Bivariate R + V Bivariate R + V3.99 0.66 3.81 0.29 2.94 0.32 3.66 0.18158 D. Breysse / Construction and Building Materials 33 (2012) 139163three situations have been considered: High quality measurements : s(R) = 1, s(V) = 50 m/s. Average quality measurements : s(R) = 2, s(V) = 100 m/s. Low quality measurements : s(R) = 4, s(V) = 200 m/s.Since the simulation process is stochastic, it must be repeated acertain number of times in order to achieve signicant results,either regarding average values or their variability. In practice, 10simulations seem to be enough for stabilizing the results.6.3. Comparison between synthetic world and real worldThe rst simulations are designed in order to reproduce pat-terns similar to those observed on real data and discussed in Sec-tions 5.3 and 5.4. These simulations are done with: strengthdistribution N(35, 7), saturation rate distribution N(0.85, 0.05),average quality measurements with s(R) = 2 and s(V) = 100 m/s.The calibration is carried out on 13 cores, either with Approach Aor with Approach B (with the model dened at Eq. (14)). The sim-ulation is repeated ten times in order to derive average values andstandard deviation for each result. Tables 21 and 22 summarize theresults. Table 21 compares the theoretical and identied values forR and V exponents both for monovariate and bivariate power laws.All identied average values of the coefcients are slightlysmaller than real values (known in this synthetic world, which isnot the case in real world). The difference is larger for monovariatelaw exponents, because of the effect of the uncontrolled parameter.For the bivariate law, the difference is limited and smaller than thestandard variation. This demonstrates the ability of Approach A tocorrectly identify the exponents of the bivariate regression model.Regarding the quality of assessment, the various RMSEs are ofabout 34 MPa, and two facts are conrmed by simulations: (a)the combination of NDT measurements improves the quality andreduces RMSE, (b) Approach A is slightly more efcient than Ap-proach B, since the prior model used in Approach B does not ex-actly t the real strength-NDT measurements relationship.However both approaches lead to interesting results. The RMSEsvalues are slightly smaller than those identied on real measure-ments (Tables 13 and 18), but the difference is small. It is explainedby the choice made on measurement error values, probably largerin reality than in the simulations.In any case, all these comparisons conrm the global ability ofour synthetic world to mimic the main patterns of the real one.6.4. Inuence of the range of variation of material properties and of themeasurement errorThe synthetic world having been validated, one can now usesimulations in order to better identify the role played by main fac-Table 21Comparison between theoretical and identied values from the simulations for the exMonovariate regression R exponentV exponentBivariate regression R exponentV exponentRMSE on strength assessment results in each case.The ve curves respectively correspond to the use of a singletechnique, and to the use of techniques in combination, for varyingmagnitudes of the error on UPV (from perfect measurements tothe following.The simulations are carried out with the same parameters thanin Section 6.3, with Approach A and bivariate calibration on 13cores. The range of variation of strength is N(35, 7).Inarststep,weconsider thethreesituationsdenedatSection6.2:high quality measurements (s(R) = 1, s(V) = 50 m/s); average qualitymeasurements (s(R) = 2, s(V) = 100m/s); low quality measurements(s(R) = 4, s(V) = 200 m/s). Simulations lead toRMSEsbeing respectivelyequal to (average s.d.): 1.54 0.11, 2.94 0.32, 5.27 0.57. This con-rms that a high quality of the techniques is required in order toachieveahighefciencyof the strengthestimate. Theeffectof theerrormeasurement is quite linear, since the RMSE is respectively multi-plied by two and four in the second and third cases.In a second step, we consider that rebound measurements ofvarying quality are used either as a single technique or in combina-tors. The quality of assessment mainly depends on two items: therange of variation of strength (i.e. contrast) and the magnitude ofthe NDT measurement error (i.e. noise).6.4.1. Inuence of the strength rangeThe simulations are carried out with the same parameters thanin Section 6.3, with Approach A and bivariate calibration on 13cores. The only difference is the range of variation of strength, forwhich three possibilities are analyzed: strength distributionN(35, 3.5), N(35, 7), N(35, 10.5), respectively corresponding to acoefcient of variation of 10%, 20% and 30%. The statistical distribu-tion of the uncontrolled factor remains unmodied. Figs. 16 and 17show the correlation between estimated and true strength for therst and third cases. It is easy to see that the quality of correlationincreases when the signal to noise ratio increases, i.e. here whenthe strength range increases.The determination coefcient is respectively equal to (aver-age s.d.): 0.61 0.03, 0.85 0.02 and 0.93 0.01. The RMSE is in allcases below 3MPa and seems to be slightly smaller in the rst case.6.4.2. Inuence of the measurement errorTwo issues can be addressed when two NDT techniques arecombined. One case is to consider that their quality varies at thesame rate (the two techniques are together of good/average orpoor quality). The second case is when the quality of a rst tech-nique is taken as constant (i.e. average) while the quality of the1.09 0.98 0.312.35 1.85 0.59 the RMSE increases in all cases with error on R, combining a second technique improves the assessment: RMSEis lower for combination, with the exception of poor qualityUPV measurements, that do not improve the results when com-pared to the use of a single technique. This conrms the con-tradictory literature results discussed at Section 5.3, wherethe combination may appear useless in some situations, when combination is used, the higher the quality of the secondtechnique, the more efcient the combination.All results obtained in this section have been obtained after calibra-tion on13 cores. They are somehow optimum regarding the possibleefciency of NDT strength assessment. If the number of cores is re-duced, the quality of assessment decreases because of additionaluncertainties, resulting inmodel error. In the case of ApproachA,mod-el error appears because the regressionmodel is established from a setof limited size. In the caseofApproachB, thek factor for calibration (seeTable 16) is identiedwith a higher uncertainty. Reducing the numberof cores has however a direct effect on the cost of investigation and it isalways a possible option. The effect of reducing the number of coreswill be analyzed in the following.6.5. Inuence of the number of cores n used for calibrationVery limited information is available in the existing literatureabout the effect of core number reduction on the quality of NDTassessment. It only recently that [16,93] have focused on this issue,on the basis of real on-site data. [16] have compared what can ob-tained with a full set of 14 cores, to what can be obtained when thisregarding average RMSE when n decreases, and the standard devi-ation on RMSE increases only slightly. For a number of cores n,there is some calibration error due to the variability of the esti-mated concrete strength for a limited set of (fcexp, R, V). The coef-cient of variation of the estimated strength for an innite numberof cores can be calculated according to the following equation:cvfcest2 c cvR2 b cvV2 18where b and c are the two exponents of the prior model (Eq. (14))and cv(R) and cv(V) the respective coefcients of variation corre-sponding to the two measurement errors. When n decreases, theuncertainty on fcest (for a given level of condence) increases as1/pn. This tendency can be seen on error bars in Fig. 20.Fig. 19 shows that the performance of Approach A is less stable,since the uncertainty increases more rapidly when n decreases.D. Breysse / Construction and BuildinFig. 16. Estimated and true strength values (MPa), strength distribution N(35, 3.5).number is reduced to 10, 7 or 3 cores. Their idea was to build rst areference model with the full set and to quantify in a second stepwhat error is generated when the model is identied with a limitedFig. 17. Estimated and true strength values (MPa), strength distribution N(35,10.5).set, randomly generated from the full data set. Synthetic world of-fers the possibility of multiplying the simulations in order to betterunderstand what happens.In the following simulations, the parameters (strength and sat-uration rate distribution, NDT measurement errors) are the samethan at Section 6.3. As in previous simulations, perfect strengthmeasurements are assumed. The main variable is the number nof cores used for calibration, which is varied from 1 to 13, andthe two calibration approaches (A and B) are compared. Since thestatistical stability is lower when n decreases, each simulation isrepeated 20 times. For each n value, the average RMSE and its var-iability are analyzed. There is a slight difference between the twoapproaches: if one core is enough in order to apply Approach B(k is calculated from one pair of measured and estimatedstrengths), at least three cores are needed for Approach A, sincethe model to t (double power law) has three parameters.Figs. 19 and 20 compare the results obtained for the two ap-proaches. For a better understanding, an approach is highlightedon each gure, and the error bars (1 s.d.) for this approach areadded.In most cases, even with a small number of cores, the error re-mains acceptable with RMSE being less than 4 MPa. These valuescan be positively compared with those obtained with real data (Ta-ble 16), which conrms the ability of synthetic simulations toreproduce the real world. As noted in Tables 20 and 22, when alarge number of cores is available, the error is lower for ApproachA than for Approach B, a specic model giving a best t than a priormodel with only one degree of freedom. However, this does not re-main true when the number of cores is smaller. The two curvescross between n = 4 and n = 5. This means that if n < 5, it is betterto use Approach B for calibration.Fig. 20 shows that the performance of Approach B is very stable,Fig. 18. RMSE as a measure of strength assessment quality as a function of thequality of NDT measurements.g Materials 33 (2012) 139163 159When n is too small, because of the measurement errors on Rand V, the statistical t of the regression model is not reliable.There is thus a high probability of nding exponents of the powerildinFig. 19. RMSE as a function of core number, for two calibration approaches (errorbars for Approach A).160 D. Breysse / Construction and Bulaw which are very different from true values. With n = 3 negativeexponents have been found in 50% of cases (one must note thatsuch negative value are sometimes used by authors working onreal data [76]).Regarding the best adapted approach for calibration, it can beconcluded that: If n > 5 or 6, Approach A is more efcient. If n = 5 the two Approaches seem to have the same efciency. If n = 3 or 4, Approach B is more efcient. If n = 1 or 2, it is only possible to use Approach B, but uncer-tainty can be high.The same conclusions would have been reached if other valuesof measurement errors had been considered. Further studies areneeded to check how the range of variation in strength of investi-gated concrete may inuence the critical n value (here n = 5) forabove which Approach A appears to be more efcient.The interest also appears, for on-site investigation, to try toidentify in a rst stage those areas where the concrete propertiesprobably have extreme values (either very high or very low), in or-der to enable an intelligent coring. This way was suggested by[58] and is now followed in many studies by Italian researcherswanting to assess the concrete strength in old buildings[17,16,76,93]. The procedure could be: To perform NDT measurements in a large number of points,with a double interest: that of identifying more contrastedareas, and that of getting a statistical distribution of values. To select n points for coring and destructive tests. Calibrate the strength-NDT model (concentrating on one oranother Approach depending on n).experimental data.The efciency of combining two or several NDT measurementsFig. 20. RMSE as a function of core number, for two calibration approaches (errorbars for Approach B).has also been controversial since its origins. We have explained thereasons why no agreement has been found between researchersand why they could reach opposite conclusions. By analyzing realdata and synthetic data, we have explained that the combinationcan improve the assessment in some cases and that it can be with-out interest in others. Combination is efcient only when the qual-ity of the two techniques is of the same order (adding a poorquality technique to a high quality technique only leads to disap-A last step could be to use the whole population of NDT mea-surements in order to get an idea of the statistical distribution ofstrength, enabling to derive characteristic strength values, whichcan be used for structural assessment. The analysis of NDT valuedistribution can also indicate that material properties may nothave the same distribution in all the structure and to identify sep-arately several statistical distributions, e.g. one for each level or adifferent one for beams or columns [75,93].7. ConclusionsWe have tried in this paper to understand why and how NDTmeasurements can be used in order to assess on site strength ofconcrete. This work was based on (a) an in-depth critical reviewof existing models, (b) an analysis of experimental data gatheredby many authors in laboratory studies as well as on site, (c) thedevelopment and analysis of synthetic simulations designed in or-der to reproduce the main patterns exhibited with real data whilebetter controlling inuencing parameters. It is time now to drawsome conclusions, since the efciency of NDT measurements forconcrete assessment has remained controversial for a long time.It must be pointed out rst that the efciency and quality of thestrength estimation depend on three elements:(a) the sensitivity of the property being estimated (herestrength) to the NDT measurement,(b) the range of variation of the measured NDT values, in directrelation with range of variation of strength for the investi-gated concrete,(c) the magnitude of the measurement error: for a given sensi-tivity and a given range of variation, the smaller the mea-surement error, the more accurate the statistical regressionmodel.Two NDT techniques (UPV and rebound) have been highlightedin this study, because they have been the most widely studied andbecause many experimental datasets are available for independentanalysis.Regarding the empirical models, it has been conrmed that try-ing to identify a universal strength-NDT value relationship issimply wasted energy. Many types of models are available in theliterature, and they all show a high variability in the values of theircoefcients. A striking consistency has however been shown, forthree types of models (power law function and exponential forUPV, power law function for rebound). A rst conclusion is thatthe model error is much smaller than that due to the measurementuncertainties. The priority must be thus to reduce as much as pos-sible the NDT measurement error. For a given set of experimentaldata, the choice of a given model type has no important conse-quences, since all models lead more or less to the same qualityof assessment. The key issue is that of calibration, i.e. that of adapt-ing the coefcients of the given model in order to better t to theg Materials 33 (2012) 139163pointing results). The analysis of real datasets from laboratorystudies has shown that combining rebound and UPV can lead toa RMSE on strength of about 4 MPa. The same value has beenanalyzed. Synthetic simulations have shown how the quality ofNDT measurements directly impacts the quality of assessmentildin(with RMSE varying between 1 and 5 MPa) both for NDT used aloneand in combination.The issue of calibration has deserved specic attention, appliedto the case of SonReb combination. Two calibration approacheshave been compared, similar to those suggested by EN standardsfor measurements with a single nondestructive technique. Theyrespectively correspond to the identication of a specic correla-tion model from measurements on cores (Approach A) and to thecalibration of a prior model (Approach B). In the latter case, a dou-ble power law model (Eq. (14)) was identied after a careful anal-ysis of relevant literature. The advantage of Approach B is that itrequires the calibration of a unique parameter, making its use verysimple, while giving very good results regarding RMSE on esti-mated strength.In the last part of the paper, synthetic simulations have beendeveloped. They were carefully built, in order to mimic the mainpatterns exhibited in real investigations. They enabled to betterunderstand the role played by the strength range and the measure-ment error. They have also enabled to analyze the way the numberof cores inuences the quality of assessment. An interesting resultis that the quality of assessment remains correct even if the num-ber of cores is reduced well below what is suggested by standards.Calibration Approach A is the most efcient when one has morethan 6 cores while Approach B has better (and acceptable) resultsif one has only 3 or 4 cores.These conclusions have been reached with two specic NDTtechniques (namely UPV and rebound) but similar conclusionswould be reached with other NDT methods. The study was alsolimited to the combination of two NDT methods but it could beinteresting to analyze the interest of: adding a third NDT technique, like penetration test . However,this choice increases the investigation cost while the added-value of a third technique remains to be conrmed, replacing one NDT technique with a semi-destructive test, likepull-out. In this case, some studies have shown that the semi-destructive test may also provide an interesting alternative tosome cores [14].Another possible development of synthetic simulations wouldbe to consider the role of other inuencing factors, like carbonationof the aggregate type. Such simulations would only require exper-imental data, in order to be able to correctly reproduce the physicsinvolved.The future of concrete strength assessment by using NDT tech-niques, alone or in combination, remains to be written. One can besure that, on the basis of a correct understanding of the phenom-ena involved, very positive results will certainly be obtained. Onenext challenge will be to dene and validate optimal proceduresfor assessing real-size structures. 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Breysse / Construction and Building Materials 33 (2012) 139163 163Nondestructive evaluation of concrete strength: An historical review and a new perspective by combining NDT methods1 Introduction the challenge of non-destructive strength assessment2 A review of uncertainty and variability in NDT measurements2.1 Ultrasonic pulse velocity measurement (V) with regard to strength assessment2.2 Rebound hammer measurement (R) with regard to strength assessment2.3 How many measurements?3 Monovariate conversion curves: establishing and calibrating relationships3.1 Building a conversion curve the weight of uncontrolled parameters3.2 UPV-strength relationships: a review3.2.1 Model types3.2.2 Analysis of exponential models3.2.3 Analysis of power law models3.2.4 Some final considerations on UPV-strength relationships3.3 Rebound number-strength relationships: a review3.3.1 Model types3.3.2 Analysis of power law models3.3.3 Some final considerations on rebound-strength relationships3.4 How single NDT methods can be used for estimating strength: the necessity of calibration3.4.1 Synthesis of errors to avoid3.4.1.1 Do not look for an universal NDT-strength relationship that simply does not exist3.4.1.2 Avoid meaningless comparisons and check the statistical validity of the model3.4.1.3 Be careful with any extension of the domain of validity (i.e. extrapolation)3.4.1.4 Think how the model can be used3.4.2 Calibration process and how it is defined in European standards EN-137914 Combination of nondestructive techniques for strength assessment4.1 SonReb combination: an historical perspective4.2 Bivariate relationships: a review4.2.1 Model types4.2.2 Analysis of double power law models5 Quality of the estimation with SonReb after calibration5.1 Calibration methodology5.2 Laboratory studies: experimental data and model identification5.3 Laboratory studies: efficiency of calibration5.4 On site investigation: experimental data, model identification and efficiency of calibration6 Using virtual NDT assessment to better understand and to go further6.1 Virtual NDT assessment: objectives and principles6.2 Framework and development of the synthetic simulations6.3 Comparison between synthetic world and real world6.4 Influence of the range of variation of material properties and of the measurement error6.4.1 Influence of the strength range6.4.2 Influence of the measurement error6.5 Influence of the number of cores n used for calibration7 ConclusionsReferences