## 4 Measurement Uncertainty Evaluation
### Measurement Uncertainty for the Optical Method  
Due to the nonlinear models (Equations (9) and (10)) applied in the determination of the individual scale coefficients, the Monte Carlo method (MCM) [12] was used in the evaluation of the corresponding measurement uncertainties. Estimates of the reference targets image and world coordinates, as exemplified in Tables 3 and 4, were used in the performed simulations assuming uniform probability density functions (PDF) for the image coordinates with a semi-amplitude of 0.25 pixels, and a Gaussian PDF was used for the world coordinates with standard uncertainties equal to the obtained sample experimental standard deviations (see Tables 1 and 2). Table 6 exemplifies the results obtained for the intermediate and output quantities (in this case for the individual scale coefficient calculated with targets 1 and 2 of referential B), based on \(10^{5}\) trials. Although a triangular PDF was obtained for the \(d_{1,2}\) intermediate quantity, the scale coefficient PDF showed a geometrical shape close to a Gaussian PDF. Suitable computational accuracy levels were obtained in the performed numerical simulations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Quantity** & **Estimate** & **95\% Expanded Uncertainty** & **Computational Accuracy** & **Numerical PDF** \\ \hline \hline \(D_{1,2}\) & 90.52 mm & 0.097 mm & \(<\)0.002 mm & \\ \(d_{1,2}\) & 448.68 pixels & 0.78 pixel & \(<\)0.01 pixel & \\ \(K_{1,2}\) & 0.200 70 mm-pixel\({}^{-1}\) & mm-pixel\({}^{-1}\) & mm-pixel\({}^{-1}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the MCM simulation for the determination of the measurement uncertainty of scale coefficient \(K_{1,2}\) obtained with referential B. PDF: probability density functions. The application of the uncertainty propagation law to expressions (6,7) allows determining the standard measurement uncertainty related to the dimensional optical measurement, given by  
\[u(L)=u(X)=u(Y)=[l^{2}\cdot u^{2}(K_{\text{average}})+K^{2}_{\text{average}}\cdot u ^{2}(l)]^{0.5} \tag{17}\]  
where the standard uncertainty of the targets' image coordinates is majorated by a spatial resolution equal to 0.25 pixel, considering a uniform PDF, i.e.,  
\[u(l)=u(x)=u(y)=0.25\;\text{pixel}/\sqrt{3}=0.07\;\text{pixel}. \tag{18}\]  
The standard uncertainty of the average scale coefficient corresponds to  
\[u(K_{\text{average}})=[u^{2}_{\text{max}}(K_{i})+s^{2}(K_{i})/n]^{0.5} \tag{19}\]  
where \(s(K_{i})\) is the experimental standard deviation of the scale coefficient measurement sample with dimension \(n\) (equal to 28, see Tables 10 and 11 in Appendix B). This measurement uncertainty is majorated by the use of the maximum measurement uncertainty, \(u_{\text{max}}(K_{i})\), which was obtained in the complete set of individual scale coefficients. Table 7 shows the obtained estimates and standard uncertainties of the average scale coefficients for both referentials A and B. These results were used in the determination of the standard measurement uncertainty related to the dimensional optical measurement (by use of Equation (17)), being posteriorly used to calculate the strain standard uncertainty (Equation (16)) and, in a last stage, the modulus of elasticity and the Poisson ratio. Figure 14 shows the measurement estimates and standard uncertainties obtained for the tested masonry specimens, taking into account the specimen's cross-section area and maximum load force measurement uncertainties described in Appendix C.  
With respect to the modulus of elasticity, a standard uncertainty between 0.07 and 0.60 GPa was obtained considering a measurement sample with dimensions equal to four (two measurements on the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Referential** & **Estimate** & **Standard Uncertainty** \\  & **/mm-pixel\({}^{-1}\)** & **/mm-pixel\({}^{-1}\)** \\ \hline A & 0.196 50 & 0.000 97 \\ B & 0.205 59 & 0.000 63 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Cross-section area measurements in tested masonry specimens.