## 4 Measurement Uncertainty Evaluation  
### Calculation of the Output Measurement Uncertainties  
As mentioned before, three main output quantities are usually defined in the context of the determination of the compressive mechanical testing of masonry specimens: the compressive strength, the modulus of elasticity, and the Poisson ratio defined, respectively, by Equations (1)-(3). The application of the uncertainty propagation law [11] to these mathematical models allows obtaining expressions for the corresponding measurement uncertainties  
\[u(\alpha)=[(1/A^{2})\cdot u^{2}(F_{\mathrm{max}})+(F_{\mathrm{max}}{}^{2}/A^{ 4})\cdot u^{2}(A)]^{0.5} \tag{11}\]  
\[u(E)=[[1/(9\cdot{\varepsilon_{\mathrm{v}}}^{2}\cdot A^{2})]\cdot u^{2}(F_{ \mathrm{max}})+[F_{\mathrm{max}}{}^{2}/(9\cdot{\varepsilon_{\mathrm{v}}}^{4} \cdot A^{2})]\cdot u^{2}(\varepsilon_{\mathrm{v}})+[F_{\mathrm{max}}{}^{2}/(9 \cdot{\varepsilon_{\mathrm{v}}}^{2}\cdot A^{4})]\cdot u^{2}(A)]^{0.5} \tag{12}\]  
\[u(\
u)=\{(1/({\varepsilon_{\mathrm{v}}}^{2})\cdot u^{2}(\varepsilon_{\mathrm{h} })+({\varepsilon_{\mathrm{h}}}^{2}/{\varepsilon_{\mathrm{v}}}^{4})\cdot u^{2} (\varepsilon_{\mathrm{v}}))\}^{0.5} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multirow{2}{*}{**Specimen**} & \multirow{2}{*}{**Compressive**} & \multicolumn{2}{c}{**Contact Measurement Chain**} & \multicolumn{2}{c}{**Optical Measurement Chain**} \\ \cline{3-6}  & & & **Modulus of** & & **Modulus of** & \\
**Identification** & & **Type** & **/MPa** & **Elasticity/GPa** & **Poisson Ratio/GPa** & **Poisson Ratio** \\ \hline M0F3 & & 2.053 & 1.32 & 0.31 & 1.47 & 0.32 \\ M0F7 & Hollow & 2.272 & 0.96 & 0.06 & 0.73 & 0.07 \\ M0F9 & & 2.194 & 1.99 & 0.05 & 0.52 & 0.08 \\ Average value & 2.173 & 1.42 & 0.14 & 0.91 & 0.16 \\ Average std. deviation & 0.064 & 0.30 & 0.09 & 0.29 & 0.08 \\ M0M9 & & 5.338 & 2.12 & 0.10 & 1.71 & 0.17 \\ M0M10 & Solid & 7.515 & 1.87 & 0.06 & 1.50 & 0.36 \\ M0M12 & & 4.194 & 3.20 & 0.17 & 1.75 & 0.07 \\ Average value & 5.682 & 2.40 & 0.11 & 1.65 & 0.20 \\ Average std. deviation & 0.974 & 0.41 & 0.03 & 0.08 & 0.09 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mechanical properties estimates of some tested masonry specimens.  
where \(u(A)\) is given by  
\[u^{2}(A)=w^{2}\cdot u^{2}(l)+l^{2}\cdot u^{2}(w). \tag{14}\]  
The measurement uncertainty of the vertical or horizontal strain, \(\varepsilon\), is given by  
\[u(\varepsilon)=[(1/{L_{0}}^{2})\cdot u^{2}(\Delta L)+(\Delta L^{2}/{L_{0}}^{4}) \cdot u^{2}(L_{0})]^{0.5}. \tag{15}\]  
This expression can be simplified based on the \(\Delta L\) definition given by Expression (5) and on the assumption that \(u(L)=u(L_{0})\), originating the following expression for the strain measurement uncertainty  
\[u(\varepsilon)=[(2+\Delta L^{2}/{L_{0}}^{2})\cdot(1/{L_{0}}^{2})\cdot u^{2}(L )]^{0.5}, \tag{16}\]  
which is related to each dimensional method (optical or contact).  
Appendix C lists the measurement uncertainty calculations related to the specimen's cross-section area and maximum load force, while the following Sections 4.2 and 4.3 describe, in detail, the determination of the strain measurement uncertainty, respectively, for the optical and the contact method.