# Influence of Degree of Saturation on Soil-Pile Interactions for Piles in Expansive Soils
## 4 Evaluation of Pile Uplift Force and Parameters  
### Calculation of Pile Uplift Force  
The uplift force vs. time was measured throughout the pile uplift tests. The change in uplift force with time was analyzed by fitting the observed data to a hyperbolic model, as described by Equation (13).  
\[U=\frac{t}{a+bt} \tag{13}\]  
where \(U\) is the uplift force, and \"\(a\)\" and \"\(b\)\" are fitting curve parameters. \"\(t\)\" represents the time since inundation started.  
By taking the limit of Equation (13) as \"\(t\)\" approaches infinity, we obtain:  
\[\frac{1}{U}=\frac{a}{t}+b \tag{14}\]  
To determine the fitting curve parameters \"\(a\)\" and \"\(b\)\", a linear graph is conducted using the relationship between \(1/U\) and \(1/t\). The slope of this linear graph corresponds to the parameter \"\(a\)\" and the intercept represents the parameter \"\(b\)\". The regression analysis is used to calculate these parameters. The observed data and hyperbolic fitting curve for the initial DOS of 70% are depicted in Figure 8. It is observed in Figure 8 that the majority of the uplift force developed during the early stage of the heaving process.  
The results illustrated in Figure 8 are comparable to the findings reported by [PERSON] (2007) [40]. Specifically, Figure 9 presents a similar trend, depicting pile heave data observed at the TRACON site in Denver, Colorado, USA, underlain by expansive soils. In that study, the heave data observed between 2001 and 2006 exhibited a hyperbolic pattern. The response pattern shown in Figure 9 exhibits a strong resemblance to the outcomes of the present study, as illustrated in Figure 8. It should be noted that Figures 8 and 9  
Figure 7: Schematic view of the pile uplift model test for determining the interface shear stress.  
represent different physical quantities--uplift force in our experimental model and pile heave measured at the TRACON site. However, the comparison was made to emphasize the similarity in the overall response trend rather than to directly equate the magnitudes or units of the two curves. Both figures show a hyperbolic pattern during the wetting-induced heaving process, where a significant portion of the total response (either uplift force or displacement) develops in the early stages. This trend is critical for understanding the time-dependent behavior of expansive soils and their effect on pile performance.  
It is also noteworthy that an increase in swelling pressure, as indicated by Equation (10), results in a corresponding increase in uplift force. As the degree of saturation progresses over time, the swelling pressure intensifies, thereby inducing greater strain or heave, as illustrated in Figure 1. Consequently, although Figure 8 depicts the variation in uplift force with time and Figure 9 represents the heave response, the two remain comparable. This is because the uplift force, driven by the increasing swelling pressure, directly contributes to heave development.  
Figure 10 illustrates the relationships between the measured ultimate uplift stress and the initial degree of saturation, as well as the initial matric suction. Figure 10 shows that the ultimate uplift shear stress decreases nonlinearly with increasing initial degree of saturation, but decreases with increasing initial matric suction. It is believed that the  
Figure 8: Comparison between the observed data and the hyperbolic fitting curve of the soil sample at the initial DOS of 70%.  
Figure 9: Pile heave pattern at the TRACON site, Denver, Colorado, USA, reproduced with permission from [PERSON], Design Principles for Foundations on Expansive Soils, Ph.D. Dissertation, Colorado State University, 2007 [40].  
observed phenomena are primarily due to the change in the swelling pressures of the specimens at various initial stress state conditions, as presented in Table 4.