Mathematical model for predicting the ultrasonic pulse velocity of concrete

Abstract This paper proposes a mathematical model approach to predict the Ultrasonic Pulse Velocity (UPV) of concrete. This paper presents the formulation, calibration, evaluation, and validation of the proposed UPV model. An experimental program was developed to calibrate and evaluate the proposed model. Furthermore, the model was validated using a separate group of measurements in the experimental program, to test its accuracy and generalization capability. Analysis of the results reveals that the proposed model provides a good fit to the experimental data and does not contain outliers or discerning pattern. The corresponding standard error and determination coefficient are 45 m/s and 0.92, respectively. Model validation demonstrates high predictability and generalization capability. The corresponding standard error and determination coefficient are 74 m/s and 0.91, respectively.


Introduction
Nondestructive testings, such as radiographic and ultrasonic techniques, have been routinely used for metals.Those methods have been extended to concrete and appeared in standards, such as the American Society for Testing and Materials (ASTM), the Canadian Standards Association (CSA), the International Standards Organization (ISO), and the British Standards Institute (BSI) (Malhotra & Carino, 2004).Ultrasonic Pulse Velocity (UPV) has been investigated by many researchers to find relationships between certain properties of concrete and UPV measurements.Lee and Lee (2020) developed a model to predict setting time and compressive strength of concrete utilizing UPV testing at early age of concrete.They have also quoted other investigators who reported that UPV had been applied to investigate mechanical properties and integrity of concrete structures.Adamati et al. (2017) tested UPV for analysis of existing structures and confirmed previous studies that the technique is useful as an inspection tool and provides a measure of homogeneity and quality of structures and can further inform investigation strategies.Some researchers, such as Mendes et al. (2019) attempted to use UPV as a performance parameter for concrete mix design, as a means of evaluating concrete durability.Their research, however, only assessed the effect of different binders on UPV and compressive strength measurement after 28 and 91 days.Besides, Mendes et al. have not provided a model correlating mix design parameters with UPV but rather presented their findings in dosage diagrams.They concluded that the concretes they tested showed correlations between UPV and w/b ratio only at 28 days.The current articles do not provide mathematical models linking mix design characteristics with UPV measurement.
This paper proposes a model to predict UPV values utilizing mix design parameters describing cement hydration, porosity of concrete, and the packing density of aggregates.The importance of this study is that it provides a basis of predicting UPV based on mix design parameters, which in turn has been linked by several researchers to durability and compressive strength of concrete (Güçlüer, 2019;Hedjazi & Castillo, 2020).The British Standards in particular (BS EN 12504-4, 1991), stipulated that UPV measurement can be used for the determination of the presence of cracks or voids in concrete as well as changes in its properties with time.Hence, the current study with its UPV predictive model enhances the ability of engineers to predict performance characteristics of concrete at the stage of mix design, leading to savings later on by avoiding lower-quality concrete and allowing to design more durable concrete, hence improving sustainability and life span of structures.
In order to formulate the UPV model, the development of at least three sub-models was needed to describe four parameters that describe characteristics of concrete mixtures.They are cement hydration, water to cementing material ratio, air voids, and the packing density of aggregates.The hydration variable was sub-modeled utilizing the degree of cement hydration, which defines the ratio between the quantity of hydrated cement to the original quantity.The critical degree of hydration was employed in the development of the hydration sub-model based on the work of Rasmussen et al. (2002), who have shown, using experimental data, that cement starts hydrating when degree of hydration α reaches a threshold value referred to as critical degree of hydration.
Water to cementing material ratio and air voids fraction were used as variables in the second sub-model representing porosity of concrete.In developing the sub-model describing the packing density of aggregate, the ratio of ϕ / ϕ max , where ϕ denotes the packing density of aggregates and ϕ max denoting the maximum packing density of aggregates, was employed.In order to calibrate and validate the model, an experimental program was designed, where concrete mixtures were prepared using proportions and ranges of the design variables of mixtures: w/c, water content, bulk volume of coarse aggregate (CA), maximum size of CA, and air entrainment.Those values and ranges of those variables were developed following the statistical fractional factorial design method (Montgomery & Runger, 2003) and the range recommended by the ACI 211 (1991) for designing and proportioning normal concrete mixtures.UPV measures were taken on concrete cylinders prepared from those mixtures after 28 days.
The following sections provide details on the development and formulation of the proposed UPV model function of concrete mixture variables, design of the experimental program, and calibration, evaluation, and validation of the proposed model.

UPV model formulation
The proposed model for predicting UPV in concrete is based on the hypothesis that UPV is a function that describes different aspects of the concrete mixture.The model comprises four parameters, namely: cement hydration that also accounts for the chemical composition of cement, water to cementing material ratio together of the fraction of air voids and the packing density of aggregates.Each of those variables is a function of other parameters describing the concrete mixture, and a sub-model will be developed to describe each of those variables.The following general form of the UPV model is proposed: Where: UPV = Ultrasonic Pulse Velocity (m/s or km/s) α = degree of cement hydration w/c = water to cementing materials ratio V a = Volume fraction of air entrapped and/or entrained relative to total volume of the mix ϕ = packing density of aggregates

Cement hydration sub-model
When Portland cement is mixed with water it hydrates, forming two major products, calcium silicate hydrates and calcium hydroxide.The degree of cement hydration (α) represents the ratio between the quantity of hydrated cement to the original quantity.Mehta and Monteiro (2006) pointed out that the degree of cement hydration reflects concrete evolution.Beek et al. (1999) have suggested a bi-linear relationship between the compressive strength of concrete (f' c ) and α depending on the w/c.This goes Schindler and Folliard (2005) mentioned cement composition and fineness, water-to-cement ratio (w/c), mineral admixtures, concrete age, temperature, and moisture content, as factors that contribute to the degree of cement hydration, and hence evolution of concrete characteristics.Rasmussen et al. (2002), have shown, using experimental data, that cement starts hydrating when α reaches a threshold value referred to as critical degree of hydration, α cr .For Portland cement, α cr was estimated as the product of w/c and a constant "k" equal to 0.43.Accordingly, in this study, α will be related to UPV using the following relationship: Where: The cement hydration prediction model developed by Schindler and Folliard (2015) was adopted in this study owing to its comprehensiveness and good predictabilities.

Porosity sub-model
Density, and consequently strength, of concrete is affected by the presence of capillary pores.The lower the density of concrete is, the lower its strength.It was established that capillary pores are increased with the increase of w/c (de Larrard, 1999).Popovics and Ujhelyi (2008) suggested a model that describes the combined effect of w/c as well as entrained air pores on the density and strength of concrete.Chidiac et al. (2012) confirmed that Popovic and Uihelvi's model had shown a high degree of predictability of strength of concrete.Therefore, the same model is adopted in this paper to correlate the two variables of w/c and entrained air, with UPV: Where: (w+V a )/c = the weight fractions of water and air relative to the cementing materials in the concrete mixture C = a calibration constant which depends on the specimen shape and test conditions.

Packing density sub-model
Research by de Larrard (1999), revealed that an increase in the packing density of aggregates yields an increase in the strength and durability of hardened concrete by allowing for a reduction in the water to cement ratio for the same workability.Optimum packing density of aggregates yields optimum hardened properties by reducing concrete porosity (Johansen & Andersen, 1996;Tasi et al., 2006;Wong & Kwan, 2005).
In this study, UPV is related to the packing density ratio of aggregates as follows: Where: ϕ = packing density of aggregates ϕ max = maximum packing density of aggregates D = a calibration constant which depends on the shapes and sizes of the aggregate particles.
The above sub-model follows the relationship suggested by de Larrard (1999), which showed that strength of concrete is a function of the ratio of packing density of aggregates (volume fraction) in a concrete mixture to its maximum packing density, ϕ/ϕ max.For a given mixture, packing density, which provides an indirect measurement of porosity, accounts for mixture parameters of aggregate gradation and proportions of aggregates and cement.The maximum packing density of aggregates, ϕ max , is function of the size, shape, surface texture, and volume fraction of solids in addition to the method of compaction (de Larrard, 1999; Wong & Kwan, 2005).Particle packing models can be used to calculate ϕ max .In this paper, the Modified Toufar Model (MTM) (Toufar et al., 1976) was used because of its generic form, its high level of predictability and its simplicity relative to other models.Other particle packing models that have been employed to calculate ϕ max include the Compressible Packing Model (CPM) and Theory of Particle Mixtures Model (TPM) (Moutassem, 2016).

UPV model
The above sub-models, combined together, yield the following proposed model for predicting the UPV of concrete at a certain age: Where A, B, C, and D are calibration constants.Constant A and UPV are in m/s or km/s.All other ratios and constant B are dimensionless terms.

Experimental program
An experimental program was developed to evaluate the accuracy and precision of the model in predicting the UPV measure on concrete cylinders after 28 days.Design variables traditionally employed for proportioning concrete mixtures, namely w/c, water content, bulk volume of coarse aggregate (CA), maximum size of CA, and air entrainment, were adopted for the design of the program.

Concrete mixture design composition
Table 1 shows the values of the CA size, w/c, water content, and bulk volume of CA in the concrete mixtures designed for the purpose of this research.Those values were developed following the statistical fractional factorial design method (Montgomery & Runger, 2003) and the range recommended by the (1991) for designing and proportioning normal concrete mixtures.The range selected for w/c was from 0.4 to 0.7.The water content values ranged from 175 to 205 kg/m 3 , covering the full range of slump for air-entrained concrete.The cement content ranged from 250 to 513 kg/m 3 .The bulk volume of CA per unit volume of concrete ranged from 0.45 to 0.69.Two CA maximum sizes were used: 20 mm and 14 mm.An air entraining agent was used.Table 2 shows a total of 20 concrete mixtures designed utilizing those ranges.

Materials and properties
All concrete mixtures were prepared using crushed limestone, siliceous sand, OPC, airentraining admixture (AEA), and water.Hydraulic cement GU-type 10 obtained from Lafarge North America.The chemical and physical properties of OPC are summarized in Table 3.
Crushed limestone CA with 20 mm and 14 mm nominal maximum aggregate sizes were utilized.These aggregates were obtained from Lafarge, North America's Dundas quarry.The specific gravities, absorption values, and bulk density for the 20 mm CA are 2.75, 0.92%, and 1636 kg/m 3 , respectively, and for the 14 mm CA are 2.74, 0.88%, and 1576 kg/m 3 , respectively.The sand was also obtained from Lafarge North America.The specific gravities, absorption values, bulk densities, and fineness modulus for the sand are 2.71, 1.58%, 1812 kg/m 3 , and 2.72, respectively.The bulk density, specific gravity, and absorption for CA and sand were  determined following ASTM C127-04 ASTM C127 (2015) and ASTM C128-04 ASTM C128 (2015c), respectively.The particle size distribution was evaluated and found to conform to the ASTM requirements.Micro Air, produced by BASF, which meets the requirements of ASTM C260/C260M (2010), was used to entrain air.
The modified Toufar et al. (1976) was used to compute the maximum packing density of aggregates, ϕ max , for each concrete mixture.After determining the volume fractions of the fine aggregate and CA particles from the mixture, their characteristic diameters and their measured maximum oven-dried packing densities with rodding as the method of compaction, the MTM was employed to calculate ϕ max .Experimental measurement of the maximum packing densities was carried out using three specimens and taking the average values.The measured maximum packing densities of sand, 14 mm maximum CA size, and 20 mm maximum CA size are 0.669, 0.575, and 0.595, respectively, and the corresponding standard deviations are 0.016, 0.011, and 0.012.The average characteristic diameters that corresponds to 63.2% passing, as recommended by Goltermann et al. (1997), were measured for sand, 14 mm maximum aggregate size and 20 mm maximum aggregate size.The corresponding values are 1.1 mm, 10.4 mm, and 14.3 mm, respectively, with the corresponding standard deviations of 0.05 mm, 1.48 mm, and 1.22 mm.

Experimental procedure
The experimental procedure, consistent for all concrete mixtures, included mixing, placing, consolidating, curing, and testing.Mixing and placing of concrete is in accordance with ASTM C192/ 192M (2016a).The amount of slump and air content for each concrete mixture were determined in accordance ASTM C143/C143M (2015d) and ASTM C231/C231M (2014a), respectively.Following casting and placing of concrete, the specimens were then sealed and cured in a standard curing room at a temperature of 23°C and a relative humidity exceeding 95%.For each concrete mixture, three 100 × 200 mm standard cylinders were cast, consolidated by rodding and finished in accordance with ASTM C192/C192M (2016a).The compressive strength was evaluated at 28 days  following ASTM C39/C39M (2016c).Testing for UPV was conducted following ASTM C597/ C597M (2016d).

Experimental results, analysis, and discussion
Calibration of UPV model constants was performed by minimizing the standard error (σ) between the measured experimental values and the model prediction values.The standard error, a global assessment of model predictions, was calculated using the following equation (Montgomery & Runger, 2003): ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi Where UPV model(i) and UPV exp(i) are the model and experimental UPV values corresponding to mix i, respectively.Parameters n and p are the number of test points and number of model constants, respectively.This global measure assesses the soundness and goodness of fit of the proposed model.The determination coefficient (R 2 ), which measures how well the regression predictions approximate the real data, was also calculated.Utilizing least square errors analysis, the calibration constants A, B, C, and D were found equal to 6813, 0.113, 0.719, and 0.766, respectively.The corresponding standard error (σ) and determination coefficient (R 2 ) were 45.2 m/s and 0.92, respectively.
Table 4 presents the experimental results and model predictions for each mix.The corresponding calculated values for the model parameters, α(t)-α cr , (w+V a )/c, and ϕ/ϕ max , are also given.Table 4 also presents the corresponding model error, considered as the ratio of experimental to predicted result for each mix.Accordingly, the mean, standard deviation and coefficient of variation for the model error are 1.00, 0.00952, and 0.000091, respectively.The percentage difference between predicted and measured compressive strength values range from 0.0% to 1.8% only.These results confirm the high degree of predictability of the proposed model.Utilizing the data in Table 4, Figure 1 was developed by plotting the values predicted by the model (UPV model ) against UPV measurements obtained from experiments on the concrete cylinders (UPV exp ) after 28 days.The figure shows the goodness of fit of the proposed model in comparison to the measured experimental data.The corresponding standard error and determination coefficient were 45.2 m/s and 0.92, respectively, which provides further evidence of the model ability to predict the UPV of concrete at 28 days.Bias and precision were evaluated using the limits of agreement plot as shown in Figure 2. Accordingly, it was revealed that the average difference between the experimental and model predictions is 0.0004 only, and the 95% confidence interval for the average difference is [−81.3, 81.3] with all data points falling within these lower and upper confidence limits.The soundness of the model was also evaluated by comparing the predicted UPV values to the    residual UPV values.As shown in Figure 3, the residuals are randomly scattered around zero, indicating no visible patterns or outliers, and thus confirming the good fitness of the model.
The significance of the model input parameters and their effect on the model predictions was evaluated using two methods.The first method involved plotting the input parameters individually against the outcome to investigate if a correlation exists and determine the corresponding determination coefficient for the best-fit relationship.Accordingly, it was revealed that the order of highest to lowest degree of correlation between model parameters and outcome corresponded to wþVa c , α t ð Þ À α cr , and ; ;max , respectively.The combined R 2 for the full model was 0.92.The second method involved removing each input parameter from the model, recalibrating the model using the remaining parameters and then determining the corresponding model standard error.This method was utilized to identify the most influential parameters on the model prediction.Results revealed that the most influential model parameters in order of importance were wþVa c , ; ;max , and α t ð Þ À α cr , respectively.The combined standard error for the full model was 45.2 m/s.
Moreover, the model was assessed by studying the trend and systematic deviation from the exact location.Table 5 and Figure 4 demonstrate that the model follows a similar trend to the experimental data and reveal no systematic deviation.Measure of the systematic deviation of the model from exact location was carried out by determining the regression coefficients in the linear regression model.As shown in Table 5, the 95% confidence interval for the intercept includes zero and the 95% confidence interval for the slope includes one.Furthermore, statistical evaluation of the model as a whole revealed that the proposed model is able to predict the outcome significantly, where the p-value was determined to be less than 0.00001.
In order to validate the model, the model was calibrated using only four mixes corresponding to the number of calibration constants, while the remaining 16 data points were used to assess the accuracy of model predictions.Figure 5 reveals a high degree of determination between the model results and the experimental data.The corresponding standard error and coefficient of determination are 74 m/s and 0.91, respectively.These results demonstrate the high degree of predictability of the proposed model.

Conclusions
In this study, a microstructure model for predicting the Ultrasonic Pulse Velocity (UPV) of concrete of 28 day age has been formulated, calibrated, evaluated, and validated.The model formulation comprises the following features: • The effect of cement hydration and concrete age on UPV were accounted for through incorporating the cement hydration sub-model to account for the chemical composition and properties of cement.
• The combined effect of capillary pores and air pores was accounted for by adopting a sub-model developed by Popovics and Ujhelhi (2008).
• The effect of other mixture parameters such as aggregate gradations and proportions of aggregates and cement, which affects the amount of porosity in the mix, was accounted for through incorporating a packing density of aggregates sub-model.
Model evaluation and validation revealed the following conclusions: • The proposed UPV model provides a good fit to the experimental data and does not contain outliers or discerning pattern.The corresponding standard error (σ) and determination coefficient (R 2 ) were 45.2 m/s and 0.92, respectively.The mean, standard deviation and coefficient of variation for the model error are 1.000, 0.00952, and 0.000091, respectively.
• The most influential input parameters on the model prediction, in order of importance, are revealed to be wþVa c , ; ;max , and α t ð Þ À α cr , respectively.
• Measure of the systematic deviation of the model from exact location was assessed and the results demonstrate that the model follows a similar trend to the experimental data and reveals no systematic deviation.Statistical evaluation of the model revealed that the proposed model can predict the outcome significantly, where the p-value was determined to be less than 0.00001.
• Model validation demonstrates a high degree of predictability where the corresponding standard error and determination coefficient are 74 m/s and 0.91, respectively.
The current study with its UPV predictive model enhances the ability of engineers to predict performance characteristics of concrete at the stage of mix design, leading to savings later on by avoiding lower-quality concrete and allowing to design more durable concrete, hence improving sustainability and life span of structures.

Figure
Figure 4. Data label versus experimental and model UPV values.