Parameter Variance of the Duncan Formula for Nonlinear Shear Strength of Coarse-Grained Soil

The reliability analysis of slope stability is significantly influenced by the variance of the soil’s shear strength. Currently, the shear strength of coarse-grained soil is commonly determined using the Duncan formula, which establishes a relationship between the shear strength and confining pressure. Specifically, the parameters of the Duncan formula are estimated based on available test data using least squares regression, which mitigates the limitations associated with small sample sizes and significant errors in the classic grouped data method. However, hypothesis testing in mathematical statistics reveals that the residuals from the classic least squares estimation exhibit heteroscedasticity and correlation, violating the fundamental assumptions required for least squares regression. Consequently, parameter estimates obtained through classic least squares regression have large variances, leading to unreliable statistical inferences. To address this issue, we propose a generalized least squares method that eliminates the heteroscedasticity and correlation of the residuals. Triaxial test data for five different coarse-grained soils are analyzed using the proposed method. The results show that the mean values of the estimated parameters of the Duncan formula are close to those obtained using the classic method, while their variances are significantly reduced, demonstrating the effectiveness of the proposed approach. The reliability analysis of the anti-sliding stability of an infinitely long slope shows that the strength parameters estimated by the classical least squares method tend to underestimate the stability of the slope due to the large variance. Therefore, the Duncan nonlinear shear strength parameters of coarse-grained soils should be estimated using the generalized least squares method that eliminates the heteroscedasticity and correlation of the regression residuals.