In recent years, there has been a growing recognition of the importance of vertical ground motions in the seismic design of engineering structures. A comprehensive understanding of the small-strain constrained modulus M 0 , which is a key input soil parameter, is essential for conducting a reliable analysis of vertical site response. Natural soils in engineering scenarios are often subjected to various anisotropic stress states, and the role of such loading on M0 0 is a critical concern that remains incompletely understood. This paper presents a systematic experimental program aimed at addressing this issue. Using a triaxial apparatus, sand specimens initially isotropically consolidated were subjected to various anisotropic stress states, including triaxial compression and triaxial extension. The evolutions of M0 0 at different stress states were captured by exciting elastic compression waves through embedded bender-extender elements. The specimens were tested under a wide range of states in terms of void ratio, axial stress, and radial stress. The study demonstrates that the impact of stress anisotropy is complex, depending on the magnitude of the stress ratio, the loading mode, and the initial state of the specimen. A practical model is suggested for the improved characterization of M0 0 under the anisotropic stress states. This model considers two primary mechanisms that are associated with the effects of stress anisotropy.

The empirical expressions for predicting the small-strain shear modulus (G0) of granular soils in current engineering practice are established mainly on experimental data under isotropic stress conditions. In most geotechnical applications, however, soils are subjected to anisotropic stress conditions. The impact of stress anisotropy on G0 is a critical concern but is not yet fully understood. In this paper, we present a specifically designed experimental study to address the question. Various principal stress ratios were applied to isotropically consolidated sand specimens in a triaxial apparatus, and the elastic shear waves were generated by the bender elements installed in the apparatus such that the variations of G0 from isotropic stress states to anisotropic stress states were determined. Three quartz sands with different particle shapes were tested under a range of states in terms of void ratio, axial stress, and radial stress. The study shows that the impact of stress anisotropy is much more complicated than commonly thought. It depends on the magnitude of the stress ratio and the loading mode. A simple model that accounts for two primary mechanisms associated with the impact of stress anisotropy is proposed, and its performance is evaluated using various sources of data in the literature.