# Assessment of Concrete Strength Using the Combination of NDT--Review and Performance Analysis
## 5 Performance of the Evaluation Using Monte Carlo Simulations  
In real practices, after the in situ measurements and strength testing in laboratory, these results are used to identify/calibrate a conversion model between the NDT measurements and concrete strengths. This methodology can be affected by some factors, such as the number of cores and the uncertainty of NDT measurements. To take into account  
Figure 3: In situ case study of using the calibration methods on the SonReb model (**a**) uncalibrated model, (**b**) calibration by \(\Delta\)-method, (**c**) calibration by \(k\)-method, (**d**) the error of calibration of the two methods (shifting and multiplication) [7].  
these effects, recent works [24; 48] proposed to simulate the methodology using the Monte Carlo method.  
The model calibration is based on a random selected number of test results or number of cores (NC). The process of strength estimation goes first through the identification step and then the prediction step. The strength estimation in the identification step is based on the NC of the test results or cores chosen for the calibration. On the other hand, the strength estimation in the prediction step is based on a number NT-NC of test results, where NC is the number of test or cores used to identify the model and NT is the total number of test locations or tested samples for NDT measurements.  
After the random selection of NC from NT test locations, the regression analysis is used to identify three linear models corresponding to three cases: using pulse velocity method \(V\), using rebound hammer method \(R\), and using combined method \((V+R)\). Each identified model is used to estimate the local strength, which is used to calculate the estimated mean strength \(\overline{f}_{cest}\), the estimated strength standard deviation (strength variability) \(\mathsf{s}(f_{cest})\), and errors [42].  
The figure below shows three Fitting Error Curves (FEC) and Prediction Error Curves (PEC) corresponding to three cases (single technique \(V\), single technique \(R\), and combination of \(V+R\)).  
From Figure 4, the following can be noted:  
* The fitting error RMSEfit increases as NC increases. This is rational because the number of points to be fitted using a model having a fixed number of parameters is increased.
* The other interesting observation is about the points with RMSEfit = 0 when NC = 2 for the case of a single technique and NC = 3 for the case of combination of NDT techniques. This means that when NC is equal to the number of model parameters, the model parameters can be identified without any fitting error, but the prediction error is very large, i.e., the models have poor predictive ability. This discrepancy must be pointed out, since a very low number of cores is a common practice.
* The prediction error exhibits an adverse pattern to that of fitting error, since it decreases while NC increases by increasing NC.
* For each of the three cases (\(V\) and \(R\) measurements alone or in combination), the difference between the FEC and PEC decreases as NC increases.
* For all cases, the error of prediction starts stabilisation after six cores. After nine cores, there is no significant improvement in the strength prediction. Therefore, no core is needed after this number.  
Figure 4: Fitting and prediction errors (FEC and PEC) as a function of NC resulting from analyzing the in situ dataset S1 [40].