Statistical Equations to Estimate the In-situ Concrete Compressive Strength from Non-destructive Tests

The aim of this study is to propose reliable equations to estimate the in-situ concrete compressive strength from the non-destructive test. Three equations were proposed: the first equation considers the number of rebound hummer only, the second equation consider the ultrasonic pulse velocity only, and the third equation combines the number of rebound hummer and the ultrasonic pulse velocity. The proposed equations were derived from non-linear regression analysis and they were calibrated with the test results of 372 concrete specimens compiled from the literature. The performance of the proposed equations was tested by comparing their strength estimations with those of related existing equations from literature. Comparisons revealed that the proposed ultrasonic pulse velocity and combined equations achieved better agreements with the test results than the related existing equations, whereas the proposed and the existing rebound hummer equations were inconsistent. 
 

‫المشيدة.‬‫المنشأت‬ ‫تقييم‬ ‫المدمجة,‬ 1. INTRODUCTION Some existing structures experience modifications instead of demolishing and reconstruction either to lengthen their life or to deal with a new assigned function.In such cases, structural assessment is required to determine the capacity of structures and to examine their ability in accommodating the new imposed loadings.The assessment of any structure requires information about of the geometry of structural members, the loadings criteria, and the mechanical properties of materials.The determination of compressive strength of in-situ concrete can be regarded as the key element in the assessment of existing reinforced concrete structures.Compression destructive test on concrete cores drilled out from existing structures is considered the most reliable method used to evaluate the compressive strength of in-situ concrete.In addition, cores can be used to determine density, water absorptions, tensile strength, and concrete expansion, The use of rebound hammer and ultrasonic pulse velocity in practice is disadvantaged by the lack of consistent strength equations.Therefore, there is a necessity to form general strength equations that is calibrated with wide range of datasets and correlate well with a large number of test specimens.
To that end, three statistical equations based on non-linear regression analysis are proposed to improve the compressive strength estimations of in-situ concrete.The first equation is a function of the numbers of rebound hummer, the second equation is a function of the ultrasonic pulse velocity, and the third equation is a function of both the number of rebound hammer and the ultrasonic pulse velocity.The proposed equations are calibrated with 372 test datasets compiled from previous experimental studies.The performance of the proposed equations is examined by comparing their strength estimations with those of related existing equations.

REBOUND HUMMER
The Schmidt rebound hammer is a simple inexpensive method that used to evaluate the hardness surface of concrete.As shown in The relationship between the rebound number and the concrete strength is basically derived from the wave propagation mechanism, Akashi and Amasaki, 1984.In which, the ratio of the compressive wave induced by the generated impact on the concrete surface and the reflected compression wave associated with the reaction force, which could be empirically related to the concrete compressive strength.

ULTRASONIC PULSE VELOCITY
The principle of this method is to measure the velocity of compression waves through a solid medium, refer to General strength equations, however, still can be established and the influence of the factors described above can be taken into account indirectly if the proposed equations were to be calibrated with a large number of specimens constructed from various mixtures and ingredients, cured in various conditions, and tested at various ages.Thus, a total of 372 datasets of concrete specimens tested by Schmidt rebound hammer, ultrasonic pulse velocity, and axial crushed compressive strength were compiled from literature.Specimens were constructed using different mixtures, from various types of cement and aggregates from different countries, subjected to different curing conditions, and tested at various ages.Details of concrete properties and test programs are not described here, and only the salient points are summarized in Table 1.Specimens considered were cubes, cylinder, beams, columns, and foundations from various test programs.The ultrasonic pulse velocity,, of the compiled specimens ranged between 2.2 km/s and 5.4 km/s, the rebound number, , from 20 to 53.5, and the crushed cylinder compressive strength,  , from 16.3 MPa to 48.7 MPa.Statistics of the datasets are presented in Table 1.
It is important to note that the crushed concrete strength is converted from a cube value,  , to a cylinder one using the following equation, Neville, 2011:   = 0.8   Eq. ( 2)

EXISTING EQUATIONS
A critical review about the selected existing equations revealed that these equations are generally varied in terms of their mathematical nature and the considered variables.Equations are either linear, non-linear, or exponential.Some equations are single variable consider either the numbers of rebound hummer or the ultrasonic pulse velocity, others consider the combined measurements of rebound hummer and ultrasonic pulse velocity.Table 2 lists the selected existing equations from literature.

PROPOSED EQUATIONS
The following equations are derived from non-linear regression analysis conducted using SPSS Statistics 22, 2016, to estimate the in-situ concrete compressive strength from non-destructive tests.The proposed equations are calibrated with the test results of 372 specimens compiled from literature.The first equation is a single variable that considers the numbers of rebound hummer, , in the estimation of concrete compressive strength:    = 16.9 *  0.13 + 3.6 * 10 −9 *  5.6 Eq. ( 3) The second equation is also a single variable which considers the ultrasonic pulse velocity, , in the estimation of concrete compressive strength:    = 15.9 *  0.26 + 9 * 10 −5 *  7.25 Eq. ( 4) The third equation combines the number of rebound hummer, , and the ultrasonic pulse velocity, , as expressed as follows:    = 4.41 *  0.46 + 21 * 10 −5 *  6.8 Eq. ( 5) It is important to note that the estimated concrete compressive strength,    , represents a cylinder concrete strength in MPa.The ultrasonic pulse velocity, , of Eq. ( 4) and ( 5) in /.

VALIDATION OF THE PROPOSED EQUATIONS
Figures 3 to 5 show the proposed equations (solid line), their correlation coefficient (R), and the test results of 372 specimens from literature.As can be inferred from Figure 3, the proposed rebound hummer equation, Eq. ( 3), provided inconsistent strength estimations with a correlation coefficient (R) of 0.39.In contrast, the proposed ultrasonic pulse velocity equation and the combined equation provided consistent strength estimations (refer to Figures 4 and 5), with correlation coefficient (R) of 0.78 and 0.81, respectively.The higher correlation coefficient of Eq. ( 5) as compared with that of Eq. ( 4) clearly shows the improvement of concrete strength estimations when the combined methods used together.However, the correlation coefficient (R) alone is not always a satisfactory proof of the accuracy of a proposed equation, Montgomery and Runger, 1998.Hence, further comparisons are carried out between the estimated strength by the proposed equations and available experimental results, see Figure 6 to 8. In these figures, the line of perfect equality (diagonal solid line) and the 90% prediction interval (the two dashed lines) are included to examine the accuracy of the proposed equations.Figure 6 proves again that Eq. ( 3) is inconsistent and widely scattered, as the majority of the estimated strength to test ratios are falling out of the upper and the lower bounds of the 90% prediction interval.Unlike Eq. ( 3), the proposed equations, Eq. ( 4) and Eq. ( 5), have correlated well with the test results.As can be seen from Figures 7 and 8 that, the majority of strength estimated strength to test ratios fall within the upper and the lower bounds of the 90% prediction interval.The lower standard deviation of Eq. ( 5), as compared with that of Eq. ( 4), clearly shows the improvement of concrete strength estimations when the combined rebound hummer and ultrasonic pulse velocity measurements used together.Further comparisons between the strength estimations provided by the rebound hummer equations (Table 3) and those provided by the ultrasonic pulse velocity equations (Table 4) showed that the ultrasonic pulse velocity method is more efficient in the strength estimations than the rebound hummer method, Pucinotti, 2015; Qasrawi, 2000.C. Combined rebound hummer and ultrasonic pulse velocity equations Table 5 compares the mean, the standard deviation, the correlation coefficient, and the root mean square error of the proposed equation, Eq. ( 5), and the existing combined equations.It appears that the proposed equation provided better strength estimations than the existing equations.Again, this is because of the wide range dataset used in the calibrations with the proposed equation.4, 5, and 6, the correlations with the test results are in general improved using the combined equations.This would be expected because when rebound hummer and ultrasonic pulse velocity methods are combined together, their shortcomings are reduced to minimum, as the rebound hummer method provides information of the concrete surface and the ultrasonic pulse velocity method provides insight information about the concrete properties, Huang et al., 2011; Nobile, 2014; Amini et al., 2016.

Summary and conclusions
Three equations were derived from non-linear regression analysis to estimate the in-situ concrete compressive strength from non-destructive tests.The proposed equations were calibrated with the test results of 372 concrete specimens from literature.The first equation, Eq. ( 3), considers the numbers of rebound hummer only, the second equation, Eq. ( 4), considers the ultrasonic pulse velocity only, and the third equation, Eq. ( 5), combines the numbers of rebound hummer and the ultrasonic pulse velocity.The following conclusions are drawn from the present study: 1.The use of the proposed combined equation, Eq. ( 5), significantly improved the concrete strength estimations in comparisons with the proposed rebound hummer equation, Eq. ( 3), and the proposed ultrasonic pulse velocity equation, Eq. ( 4).This clearly shows that the use of combined equations provide a more consistent and reliable strength estimations than the single variable equations.
2. Despite of being less accurate than the combined methods in terms of concrete strength estimation, comparisons with the test results revealed that the use of ultrasonic pulse velocity method alone is more efficient than the use of rebound hummer method alone in terms of concrete strength estimations.3. Comparisons with the test results indicated that, the use of rebound method alone is not appropriate due to the high variability of concrete, which would consequently lead to inconsistent strength estimations.4. Statistical comparisons between the estimated strengths provided by the proposed equations and those provided by the related existing equations in terms of mean, standard deviation, correlation coefficient, and root mean square error showed that the proposed equations achieved better agreements with the test results than the existing ones. 5.The proposed combined equation is recommended to be used in daily practice.It achieved a mean of 1.00, the highest correlation coefficient of 0.81, and the least root mean square error of 4.46 MPa.
Bungey et al., 2006.The guideline of cores sampling and testing have been included in various standards, 214.4R-03, 2013; ASTM C42/C 42M, 2008.Despite of these benefits, this method suffers from certain drawbacks.Along with being relatively expensive and laborious, it might not be suitable to inspect all structural members either due to inconvenient accessibility or to avoid jeopardizing the integrity and durability of the structural members, especially for those in service, Vasanelli et al., 2017; Alwash et al., 2016.As a results, the number of cores would be limited and they might not cover the entire structure leading to inconclusive results, Alwash et al., 2015.To overcome the deficiencies associated with the usage of drilled cores, non-destructive tests are employed to assess the compressive strength of in-situ concrete.Rebound hammer and ultrasonic pulse velocity are considered the most popular methods used in practice, Malhotra and Carino, 2002.The assessment of concrete strength, using these methods, is derived from predictive equations furnished with devices usually calibrated with limited database, Vasanelli et al., 2017; Breysse and Martínez-Fernández, 2013; Proceq 2006.The use of such equations is therefore questionable when they applied to estimate the compressive strength of various types of concrete made from different mixtures and cured under various conditions, Breysse and Martínez-Fernández, 2013.The American standard, ACI 228.1R-13, 2013, and European standard, EN 13791, 2007, require that non-destructive tests need to be used in company with destructive core tests.The procedure recommended by these standards involves establishing a specific regression equation that combines the number of rebound hummer and the ultrasonic pulse velocity.This equation has to be validated by the test results of cores taken from the same location, where non-destructive tests conducted.To establish such equation, the American standard and the European standard (Alternative 1 approach) require the results of no less than twelve cores and eighteen cores, respectively, to be calibrated with the results of non-destructive tests.European standard (Alternative approach 2) allows to use the results of a smaller numbers of cores (nine cores) to modify an existing regression equation.It is believed that the number of cores imposed by the aforementioned standards is somewhat impractical and considerably high, Vasanelli et al., 2017; Alwash et al., 2016; Breysse and Martínez-Fernández, 2013.Practitioners are therefore have a tendency to ignore such limitations and carry on the in-situ concrete assessment using either predictive equations suggested by devices' producers or existing strength equation calibrated with a smaller number of cores than that required by such standards, Ali-Benyahia et al, 2017; Alwash et al., 2015; Breysse, 2013.Although various equations have been derived to estimate the in-situ concrete strength, these equations are usually calibrated with limited range of datasets at the time of derivation.They are generally inconsistent and scattered when they applied to estimate the concrete strength constructed from different materials due to concrete variability, Vasanelli et al., 2017.

Fig 1 ,
the device consists of a plunger, a hummer mass, a spring, and a sliding indicator, Malhotra and Carino, 2002.When the hummer is pushed against the concrete surface, the plunger generates an impact on the surface.The rebound number is simply the measured distance by the sliding indicator Malhotra and Carino, 2002.

Fig 2 ,
Malhotra and Carino, 2002.As expressed by the following equation, the velocity of compression waves, , is a function of elastic properties of the medium, is the dynamic modulus of elasticity of the medium,  = (1 − ) (1 + )(1 − 2) ⁄ ,  is the dynamic Poisson's ratio, and  is the mass density of the medium.Following this law, a relationship could be established between the velocity measurement and the concrete compressive strength, in association with the existing relationship between the concrete compressive strength and the concrete modulus of elasticity, Pascale and Di Leo, 1984; Pascale et al., 2003.4. DATASETS The measurements of rebound hummer and ultrasonic pulse velocity were observed to be affected by the mechanical properties of concrete.The numbers of the rebound hummer was observed to be affected by the smoothness of concrete surface, concrete moisture state, and concrete carbonations, Breysse, 2012; Nobile, 2014; Qasrawi, 2000.And, the measurement of ultrasonic pulse velocity was observed to be affected by other mechanical properties such as the type of cement and aggregates, the proportions of concrete mixture, the existence of micro-cracks and steel reinforcements, and the age of concrete, Breysse, 2012; Nobile 2014; Qasrawi, 2000.The majority of previously developed equations, however, do not consider the mechanical properties of concrete such as the type of material, the mix design, and the age.They rely only upon the measurements of non-destructive tests.This is because of the high variability of concrete and the absence of accurate information about the materials properties especially for existing structures, Qasrawi, 2000; Nobile 2014; Amini et al., 2016.
This is hardly surprising due to the high variations in rebound numbers that reflect the hardness of concrete surface only, which would consequently lead to inconsistent strength estimations, Pucinotti, 2015; Qasrawi, 2000; Malhotra and Carino, 2002.B.Ultrasonic pulse velocity equationsTable4compares the mean, the standard deviation, the correlation coefficient, and the root mean square error of the proposed equation, Eq. (4), and the existing ultrasonic pulse velocity equations, Raouf, 1986; Qasrawi, 2000; Nash't et al., 2005; Hobbs et al., 2007.Comparisons revealed that the proposed equation achieved better agreement with the test results than existing equations.This is attributed to the wide range dataset employed in the calibration of the proposed equation as compared with the existing equations.The equations of Raouf, 1986 and Nash't et al., 2005 underestimated the concrete strength and those of Qasrawi, 2000 and Hobbs et al., 2007 overestimated the concrete strength.
The equation of Raouf, 1986 provided consistent strength estimations, while those of Nash't et al., 2005 and Amini et al., 2016 underestimated the concrete compressive strength and those of Giacchetti and Lacquaniti, 1980; Gasparik, 1992; Di Leo and Pascale, 1994; and Hobbs et al., 2007, overestimated the concrete compressive strength.By comparing Tables

Figure 3 .
Figure 3. Correlations of the estimated concrete compressive strength using Eq.(3) and the numbers of rebound hummer.

Figure 4 .
Figure 4. Correlations of the estimated concrete compressive strength using Eq.(4) and the ultrasonic pulse velocity.

Figure 5 .
Figure 5. Correlations of the estimated concrete compressive strength using Eq.(5), the ultrasonic pulse velocity, and the numbers of rebound hummer.

Table 3
compares the mean, the standard deviation, the correlation coefficient, and the root mean square error of the proposed equation, Eq. (3), and the existing rebound hummer equations,

Raouf, 1986; Qasrawi, 2000; Nash't et al., 2005; Hobbs et al., 2007.
As can be inferred from this table that all equations provided inconsistent strength estimations.The equations of Qasrawi

, 2000 and Nash't et al., 2005 underestimated
the concrete strength and those of Raouf, 1986 and Hobbs et al., 2007 overestimated the concrete strength.

Table 2 .
Selected equations from literature .

Table 3 .
Statistical results of the proposed and existing rebound hummer equations.

Table 4 .
Statistical results of the proposed and existing ultrasonic pulse velocity equations.

Table 5 .
Statistical results of the proposed and existing combined equations.Mean (   /   ) Standard deviation (   /   )