## 2 Derivation of \(e-log\sigma^{\prime}{}_{v}\) Equations for Laboratory Test Results
### Equation for the Virgin Compression Line (VCL)
The sensitivity of parameters \(\Gamma\), \(R\), \(\alpha\), and \(\theta\) is summarized as follows. Figure 1: Comparison of the consolidation test with the best-fit LF, CF, and AJOP equations. 1. Parameter \(\Gamma\) primarily controls the asymptotic upper bound of the void ratio. As such, it has a strong influence on the vertical position of the curve at low stress levels. A small change in a affects the curve globally, particularly in the initial loading phase. 2. Parameter \(R\) governs the horizontal shift of the curve along the logarithmic stress axis. While its effect is not as dominant as that of \(\alpha\), it moderately influences the stress range at which transition between recompression and virgin compression occurs. Improper selection of \(R\) may cause a misalignment with the test data, especially in the mid-stress range. 3. Parameter \(\alpha\) controls the slope of the curve in the virgin compression region. It exhibits high sensitivity in the high-stress domain, and an over- or underestimation of \(\alpha\) can result in significant errors in predicted settlements, particularly for deeper soil layers or higher loads. 4. Parameter \(\theta\) adjusts the degree of curvature of the curved portion of the virgin compression line. It is most sensitive in controlling the smoothness of the transition zone. A small change in \(\theta\) can significantly alter the curvature behavior. When \(\theta=0\), the model reduces to a piecewise function with a sharp corner; as \(\theta\) increases, the transition becomes smoother, better capturing gradual changes in stiffness. Due to the interdependency and nonlinear interaction among these parameters, a least-squares fitting procedure is recommended to optimize their values simultaneously. This enhances both the robustness and reproducibility of the model calibration process. The secant slope \(\overline{C_{c}}\) of the AJOP function can be determined using Equation (7). This equation provides an accurate calculation of \(\Delta\epsilon\) between two points on VCL for normally consolidated clay without the estimation. Figure 2 shows the AJOP graph for the effects of various fitting parameters. In Equation (7), \(\sigma^{\prime}{}_{v1}\) and \(\sigma^{\prime}{}_{v2}\) represent the vertical effective stresses at the initial point of stress interval (lower value) and the final point of interval (higher value), respectively, and \(\sigma^{\prime}{}_{v2}>\sigma^{\prime}{}_{v1}\). The changes in \(\Delta\epsilon\) between these two points can be calculated by multiplying \(\overline{C_{c}}\) in Equation (7) by \(log(\sigma^{\prime}{}_{v2}/\sigma^{\prime}{}_{v1})\). The application of the AJOP equation and its secant slope for calculating settlement in normally consolidated clays is straightforward. By substituting \(C_{c}\) in Equation (1) with \(\overline{C_{c}}\) in Equation (7), the errors in the calculation can be effectively eliminated across all soil depths. It is worth noting that the traditional method, which relies on linear approximation and a constant \(C_{c}\), often results in significant calculation errors, particularly for shallow soils near the ground surface, and these errors tend to decrease with depth. This discrepancy arises because the initial stress near the surface is relatively low, resulting in the initial stress point, located on the curved portion of the \(e-log\sigma^{\prime}{}_{v}\) rather than on the straight portion. In addition, the stress induced by the surface loads is comparatively higher when compared to deeper soil layers. Therefore, application of the LF causes a dramatic error of \(\Delta\epsilon\). When comparing the capabilities of the AJOP and CF equations, it becomes evident that CF can address the aforementioned issues similarly to AJOP. However, the use of CF introduces significant computational errors at greater depths due to the absence of a linear segment in its graph.