# Assessment of Concrete Strength Using the Combination of NDT--Review and Performance Analysis
## Combining NDT Techniques Using the SonReb Method for Concrete-Strength Evaluation
### Bivaried Regression Model for the SonReb Method
#### 2.2.1 Types of Models  
For combining the pulse velocities and rebound numbers with cores, different model forms have been considered by researchers, such as bilinear, double power, exponential, polynomial, and other miscellaneous forms. In the study presented by [PERSON] gathered about 69 models from the literature. Table 1 shows several models derived by different researchers.  
A possibility promoted by some researchers, see for examples [24; 25], is to combine (in addition to NDT measurements) several concrete characteristics (water-to-cement ratio, aggregate-to-cement ratio, admixture content, concrete density, age, etc.) in the model. However, the main drawback of this type of model is the need to know the concrete characteristics as inputs in the model, while they usually remain unknown in old structures. Therefore, in this study, models with only NDT measurements as the inputs are considered.  
The power form is the most widely used one in the literature, and it is written as follows:  
\[f_{c}=a\times V^{b}\times R^{c} \tag{1}\]  
where \(a\), \(b\), and \(c\) are the model parameters to be identified. The least squares minimization [37] can be used to derive the values of the model parameters; however, it is necessary to first put the model in a linear form using the logarithmic transformation. Consequently, Equation (1) becomes:  
\[lnf_{c}=Ina+b\times lnV+c\times lnR \tag{2}\]  
As an example, the SonReb model provided by [23] is \(f_{c}=1.15\)\(\times\)\(10^{-10}\)\(V^{2.6}\)\(R^{1.3}\). The graphical representation of this model is shown in Figure 1. From this figure, the compressive strength corresponding to any values of \(V\) and \(R\) can be evaluated (for example, for \(V=3800\) m/s and \(R=40\), the compressive strength \(f_{c}=30\) MPa).  
\begin{table}
\begin{tabular}{c c} \hline
**Model (\(f_{cest}\) in MPa, \(V\) in km/s)** & **Title 2** \\ \hline \(f_{cest}=8.630V+1.416R\)\(-\) 51.581 & [26] \\ \(f_{cest}=7.695\)\(+\)\(10^{-11}\)\(V^{2.6}\)\(R^{1.4}\) & [17; 27] \\ \(f_{cest}=0^{0.446V\)\(+\)\(0.048R\)} & [28] \\ \(f_{cest}=0.67\)\(\mathrm{e}^{0.72V\)\(+\)\(0.04R\)} & [29] \\ \(f_{cest}=0.42\)\(R^{0.63}\)\(\mathrm{e}^{0.58V}\) & [30] \\ \(f_{cest}=-173.04\)\(+\)\(4.07V^{2}\)\(+\)\(57.96V\)\(+\)\(1.31R\) & [31] \\ \(f_{cest}=-21.1\)\(+\)\(1.24R\)\(+\)\(0.058V^{4}\) & [32] \\ \(f_{cest}=-76.30\)\(+\)\(0.17\)\(V^{0.4}\)\(R^{0.7}\) & [33] \\ \(f_{cest}=(R/(3.64\)\(+\)\(0.023R\)\(-\)\(0.56V))^{2}\) & [34] \\ \(f_{cest}=44.8V\)\(+\)\(0.77R\)\(-\)\(194\) & [35] \\ \(f_{cest}=1.974\)\(\mathrm{e}^{0.000542V\)\(+\)\(0.01605R\)} & [36] \\ \(f_{cest}=0.01174V\)\(+\)\(0.37R\)\(-\)\(28.44\) & [18] \\ \(f_{cest}=10^{-4.251}\)\(V^{1.281}\)\(R^{0.686}\) & [36] \\ \hline \end{tabular}
\end{table}
Table 1: Some models derived by different researchers in order to estimate strength using combined rebound hammer and pulse velocity techniques.  
It is necessary to note here that this model and each one of the models presented in Table 1 were identified from experimental work on a specific concrete, which explains the variability in the values of the model parameters from one model to another. Generally, the identification of the SonReb model for a specific concrete requires a dataset of NDT measurements (\(V\) and \(R\)) and compressive strengths \(f_{c}\) from a destructive test on samples. This dataset (\(V\), \(R\), and \(f_{c}\)) is used to identify the model parameters, and, consequently, the identified model can be used to assess the compressive strength within the concrete for which this model is derived. If this model is used to assess the compressive strength within another concrete, the calibration of this model is very important. The principles of calibration will be discussed later.