## Appendix A Case Study Project 1: Lengshuibe Auxiliary CSGRD Project
\[\beta=P_{tv}+FA_{tv}-CA_{sv}\times(\frac{CA_{sv}}{CA_{sv}})\] (A2)  
where \(\beta\) is the mortar margin; \(FA_{tv}\) is the total volume of fine aggregate per unit of volume; \(CA_{sv}\) is the total volume of coarse aggregate per unit of volume; \(CA_{sv}\) is the total void volume of the coarse aggregate per unit of volume; and the ratio \(\frac{CA_{sv}}{CA_{sv}}\) is the void ratio of the coarse aggregate. The total volume of the mix is given by the following expression:  
\[T_{v}=V_{agg}+V_{paste}\] (A3)  
where \(T_{v}\) is the total volume; \(V_{agg}\) (fine aggregate and coarse aggregate) is the volume of aggregates; and \(V_{paste}\) (cement, fly ash, water, and admixtures) is the volume of paste. The solid volume of aggregates and pastes is computed from the following equation:  
\[\Phi=\frac{P_{b\;m}}{SG_{m}\;\times\;P_{w}}\times F_{m}\;(\%)\] (A4)  
where \(\Phi\) is the packing density or volume of solid material; \(P_{b\;m}\) is the bulk density of material [30]; \(SG_{m}\) is the specific gravity of material; \(P_{w}\) is the density of water; and \(F_{m}\) is the fraction of material in the total volume of the mix [22]. The void volume can be computed as follows:  
\[V_{v}=T_{vs}\;-\phi\] (A5)  
where \(V_{v}\) is the void volume within the solid content, and \(T_{vs}\) is the total volume of the solid content. The void volume, or the void ratio, is influenced by the aggregate grading and the ability to consolidate and compact the fresh mixture [22]. * **Sample Calculation of the Determination of the Pearson Coefficient and Statistical Significance** Computations of the Pearson correlation and the statistical significance of the paste and mortar margins with the 14 th-day test age compressive strength of the Shaping I Hydropower CSGRD are provided as follows: (i) Data on the \(\alpha\), \(\beta\), and the 14 th-day compressive strength values were collected as indicated in Table 15. (ii) Verification of criteria for the usage of the Pearson correlation coefficient:The paste margin (0.068, 0.072, 0.070,..., 0.095) is continuous. The mortar margin (0.337, 0.242,..., 0.167) is continuous. The 14 th-day compressive strength (MPa) (1.8, 2.6, 5.2,..., 15.1) is continuous. This criterion is met, as the variables are continuous by observation. The values can change smoothly and are not confined to specific points, allowing for a wide range of possibilities. Checking the linearity of the relationship between the variables. The linearity of the paste and mortar margins with the 14 th-day compressive strength was assessed visually using a scatter plot or statistical linear regression in Figure A3 (only the paste margin and the same applies to the mortar margin). To check linearity, Y, the dependent variable (14 th-day compression strength), was plotted against X, the independent variable (paste or mortar margin), to see if the points roughly followed a straight line, and then the linear regression model was fitted, and the fit was inspected visually as shown in Figure A3. From the scatter plot in Figure A3, the parameters are linear. Checking how closely the data are normally distributed. Due to the complexity of the Shapiro-Wilk expression initially proposed, the \(p\)-values were computed using Python's scipy.stats.shapiro() function version v1.14.1, which automated the calculation. The Shapiro-Wilk test \(p\)-values were obtained using a computational approach built into the statistical software, which handled the complexities of the calculation. The \(p\)-values were derived for each dataset using the steps below: (i) Input data: The data for paste margin, mortar margin, and 14 th-day compression strength were entered into the function. (ii) Execution of Shapiro-Wilk test: The scipy.stats.shapiro() function was applied to each dataset. This function internally computed the test statistic W and then determined the \(p\)-value associated with that W based on the sample size and distribution.