## 2 Derivation of \(e-log\sigma^{\prime}{}_{v}\) Equations for Laboratory Test Results
### Equations for the Hysteresis
#### 2.2.2 Equations for the Asymmetric Hysteresis  
The general characteristics of asymmetric hysteresis are illustrated in Figure 4. While the unloading portion is still the same as that in symmetric hysteresis, the reloading curve distinctly slopes downward at the end compared to the symmetric hysteresis case. In this study, we introduce only one additional variable \(\Psi\) to the equation of symmetric hysteresis to create asymmetric hysteresis. \[e=\left[\frac{\left(log\sigma^{\prime}{}_{v}-h_{1}\right)^{2}}{4C_{1}}+k_{1} \right]\left[1-\Psi\left(log\left(\frac{\sigma^{\prime}{}_{v}}{\sigma^{\prime} {}_{\omega}}\right)^{6}\right)\right] \tag{18}\]  
Equation (18) shows the relationship used to represent the reloading curve for both symmetric (\(\Psi=0\)) and asymmetric (\(\Psi>0\)) hysteresis, which is valid only for stress that does not exceed the stress value on the AJOP line. Notably, the value of term \(\left[1-\Psi\left(log\left(\sigma^{\prime}{}_{v}/\sigma^{\prime}{}_{\omega} \right)^{6}\right)\right]\) in Equation (18) will be equal to 1 when the stress point is on the starting point of reloading. This results in the slope at this point remaining the same as \(x_{0}\) for the symmetric reloading curves. The change in void ratio \(\Delta e\) and the secant slope \(\overline{C_{r}}(ReA)\) between the starting point of reloading (\(\sigma^{\prime}{}_{\omega}\)) and any point on the asymmetrical reloading (\(\sigma^{\prime}{}_{v}\)) of Equation (18) can be calculated using Equations (19) and (20), respectively. \[\Delta e=e-e_{\omega}=\left[\frac{\left(log\sigma^{\prime}{}_{v}-h_{1}\right) ^{2}}{4C_{1}}+k_{1}\right]\left[1-\Psi\left(log\left(\frac{\sigma^{\prime}{}_ {v}}{\sigma^{\prime}{}_{\omega}}\right)^{6}\right)\right]-\left[\frac{\left( log\sigma^{\prime}{}_{\omega}-h_{1}\right)^{2}}{4C_{1}}+k_{1}\right] \tag{19}\]  
\[\overline{C_{r}}(ReA)=-\frac{\Delta e}{\Delta log\sigma^{\prime}{}_{v}}=\frac {\left[\frac{\left(log\sigma^{\prime}{}_{v}-h_{1}\right)^{2}}{4C_{1}}+k_{1} \right]\left[1-\Psi\left(log\left(\frac{\sigma^{\prime}{}_{v}}{\sigma^{\prime }{}_{\omega}}\right)^{6}\right)\right]-\left[\frac{\left(log\sigma^{\prime}{} _{\omega}-h_{1}\right)^{2}}{4C_{1}}+k_{1}\right]}{log\left(\frac{\sigma^{ \prime}{}_{\omega}}{\sigma^{\prime}{}_{\omega}}\right)} \tag{20}\]  
It is important to note that the loading behavior of overconsolidated clay from the initial stress \(\sigma^{\prime}{}_{\omega}\) to the final stress \(\sigma^{\prime}{}_{f}\) can be divided into two cases: (1) when \(\sigma^{\prime}{}_{f}<\sigma^{\prime}{}_{p}\), \(\sigma^{\prime}{}_{f}\)  
Figure 4: Effect of parameter \(\Psi\) on the asymmetric hysteresis. remains on the reloading curve as shown in Figure 5a, therefore settlement can be directly calculated using \(\Delta e\) in Equation (19); and (2) when \(\sigma^{\prime}{}_{f}>\sigma^{\prime}{}_{p}\), \(\sigma^{\prime}{}_{f}\) thus shifts to the VCL as shown in Figure 5b, and the settlement could be calculated using \(\Delta e\) in Equation (21). This distinction ensures accurate settlement calculations under different loading conditions. Using Equations (19) or (21), the error of \(\Delta e\) in the approximation by LF can be eliminated. \[\Delta e=\overline{Cr}(Un)\cdot log(OCR)+\alpha log\left(\frac{\sigma^{ \prime}{}_{f}}{\sigma^{\prime}{}_{p}}\right)+\sqrt{\theta+\left(\alpha log \left(\frac{\sigma^{\prime}{}_{f}}{R}\right)\right)^{2}}-\sqrt{\theta+\left( \alpha log\left(\frac{\sigma^{\prime}{}_{p}}{R}\right)\right)^{2}} \tag{21}\]  
Although \(C\), \(h\), \(k\), and \(\Psi\) are derived from the laboratory test results, these parameters can also be applied to the unloading at other stress values \(\sigma^{\prime}{}_{\theta}\) without graph refit.