# Research on Mechanical Properties of Core-Filled Desulfurized Gypsum Masonry
## 4 Elastic Modulus Calculation Formula  
Currently, there is no unified constitutive model for the compressive stress-strain relationship of masonry structures, and research on the constitutive relationship of gypsum masonry by domestic and foreign scholars is relatively limited. In contrast, the study of the constitutive relationship of concrete is more extensive and well established. Based on the constitutive relationship models in the form of a parabola proposed by [PERSON] [26], [PERSON], and [PERSON] [27], all stress-strain data for grouted masonry and reinforced masonry were normalized, as illustrated in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Specimen** & **Specimen** & **Height-to** & **Stability** & **Vertical** & **Cross-Sectional** & **Test Value of** & **Calculated** & \\
**Group** & **Supplement** & **Thickness** & **Factor of** & **Reinforced** & **Reinforcement** & **Area of Vertical** & **Strength of** & **Value of** & **Compressive** \\  & & **Ratio** \(\beta\) & **Rainford** & **Cross-Sectional** & **Reinforcement** & **Reinforced** & **Strength \(N_{c,m}\)** & **Strength \(N_{c,m}\)** & \\  & & & **Masonry**\(\varphi\) & **Area**\(A\) (mm\({}^{2}\)) & \(A_{\text{s}}^{*}\) (mm\({}^{2}\)) & **Masonry**\(f_{c}\) & **Masonry**\(f_{c}\) & **(MPa)** & **(MPa)** & \\ \hline \hline \multirow{4}{*}{C20} & C20-2-1 & \multirow{4}{*}{3.85} & C20-2-2 & \multirow{4}{*}{3.85} & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 6.90 & \multirow{4}{*}{7.13} & 0.97 \\  & C20-2-3 & & & & & & 6.87 & & 0.96 \\  & C25-2-1 & \multirow{4}{*}{3.85} & C25-2-1 & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 8.02 & \multirow{4}{*}{7.13} & 1.03 \\  & C25-2-2 & & & & & 7.98 & & 1.03 \\  & C25-2-3 & \multirow{4}{*}{3.85} & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 7.87 & \multirow{4}{*}{7.13} & 1.01 \\  & C30-2-1 & & & & & 8.67 & & 1.02 \\  & C30-2-3 & & & & & 8.49 & & 1.00 \\  & C35-2-1 & \multirow{4}{*}{3.85} & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 8.55 & \multirow{4}{*}{7.13} & 1.01 \\  & C35-2-2 & & & & & 8.49 & & 1.00 \\  & C35-2-1 & & & & & 9.12 & & 0.99 \\  & C35-2-2 & \multirow{4}{*}{3.85} & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 9.18 & \multirow{4}{*}{7.13} & 1.00 \\  & C40-2-1 & & & & & 8.95 & & 0.97 \\ C40 & C40-2-2 & & & & & 9.77 & & 1.00 \\ C40 & C40-2-2 & \multirow{4}{*}{3.85} & \multirow{4}{*}{0.985} & \multirow{4}{*}{120,000} & \multirow{4}{*}{1231.5} & 9.73 & \multirow{4}{*}{7.13} & 0.99 \\  & C40-2-3 & & & & & 9.67 & & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of test values and fitted formula values for compressive strength of reinforced masonry. Figure 8: Stress–strain points after normalization. The normalization process for grouted masonry involves setting the horizontal coordinate  
\[x=\frac{\varepsilon}{\varepsilon_{c,i}} \tag{12}\]  
and the vertical coordinate  
\[y=\frac{\sigma}{f_{c,i}} \tag{13}\]  
based on test data. Similarly, for reinforced masonry, the normalization process involves setting the horizontal coordinate  
\[x=\frac{\varepsilon}{\varepsilon_{c,\text{m}}} \tag{14}\]  
and the vertical coordinate  
\[y=\frac{\sigma}{f_{c,\text{m}}} \tag{15}\]  
based on test data. All stress-strain data for grouted masonry and reinforced masonry were normalized, and referencing the constitutive relationship model for concrete proposed by [PERSON], a regression analysis of the stress-strain relationship for grouted masonry and reinforced masonry was conducted using the least squares method, as depicted in Figure 9.  
In summary, the fitting formula for grouted masonry yields an R-squared value of 0.99 for the first segment and 0.98 for the second segment. For reinforced masonry, the fitting formula achieves an R-squared (COD) value of 0.976 for the first segment and 0.977 for the second segment, indicating that the fitted formulas align well with the statistical test data.