# Experimental and Empirical Studies to Evaluate Liquefaction Resistance of Partially Saturated Sands
## 5 Proposed Empirical Functions
### Excess Pore-Pressure Ratios  
The excess pore-pressure ratio (\(r_{u}\)), which accommodates the effects of excess pore pressure and the initial effective confining stress, is an important design parameter. This study assumed that a reasonable assessment of excess pore-pressure ratios can be achieved by establishing an inverse correlation between the liquefaction-resistance ratio (\(LRR\)) and the normalized excess pore-pressure ratio (\(\frac{r_{u-part}}{r_{u-sat}}\)) as described in Equation (11). A similar approach was undertaken in previous research [50; 51].  
\[\frac{r_{u-part}}{r_{u-sat}}=\frac{1}{LRR} \tag{11}\]  
Figure 15: Variation in liquefaction-resistance ratio with the degree of saturation under different field conditions.  
A group of experimental data was gathered from the literature through a comprehensive review of the elemental (e.g., undrained cyclic triaxial or simple shear) tests conducted on different types of sand with various degrees of saturation, effective confining stresses, and back pressures. The left window in Figure 16 shows the relationship between the liquefaction-resistance ratio and degree of saturation; this was obtained via an analysis of the test data from the present study and previous research. The right window in Figure 16 shows the normalized excess pore-pressure ratios (\(\frac{r_{p-err}}{r_{q-sat}}\)) calculated using Equation (11) as a function of the degree of saturation. Figure 16 also compares the predictions of Equations (10) and (11) with the experimental data.  
It is clear in Figure 16 that the proposed method successfully captured the decreasing trend in normalized excess pore-pressure ratios with decreasing degrees of saturation. A relative density of 40%; a back pressure of 100 kPa; and effective confining stresses of 20 kPa (shallower depth), 50 kPa (middle depth), and 100 kPa (deeper depth) provided an upper, an average, and lower boundaries for the normalized excess pore-pressure ratios, respectively. There were some discrepancies between the experimental and empirical results, particularly at an \(S_{r}\) of 95%. From a design perspective, this variation was, however, understandable when considering the experimental uncertainties and complexities inherent in assessing the cyclic behavior of partially saturated sands.  
The predictions of the empirical method were also compared with the physical modeling (e.g., centrifuge and shaking table) test data, which provided further validation of the results with different test and loading conditions. The excess pore-pressure data gathered from centrifuge tests [65] and 1 g shaking table tests [66] were processed and presented in the form of a relationship between the normalized excess pore-pressure ratio and degree of saturation (Figure 17). It appeared that the proposed methodology successfully captured the decreasing trend in the experimentally determined excess pore-pressure ratios with decreasing degrees of saturation. There were some differences in the measured and predicted values that can be mainly ascribed to a limited number of test data and experimental uncertainties such as the non-uniform distribution of the air/gas bubbles observed in physical model tests [65]. From an overall response perspective, this difference is expected to be acceptable in design practice when considering the complex nature of the problem under consideration.  
Figure 16: Validation of the predictions against elemental testing data [17; 34; 35; 36; 37; 38; 40].  
The above results may suggest that when following the proposed methodology, the liquefaction resistance of partially saturated sand can be estimated if the liquefaction resistance of the nearly/fully saturated soils is known; the same applies to excess pore-pressure ratios.