The paper discusses modelling of cyclic stress-strain behaviour of soil, in particular a simple model that can produce a desired stiffness and hysteretic damping for a given strain level as observed in laboratory testing is formulated. The unloading-reloading relationship is developed for total stress seismic site response analysis with appropriate damping at large strain. The constitutive model employs a hyperbolic equation as the backbone curve, and uses a modification of the extended Masing unloading-reloading relationship leading to correct measured modulus reduction and damping curves simultaneously. A quasi-static cyclic loading of increasing amplitude is used to demonstrate the model’s performance and its capability to allow improved modelling of the magnitude of energy dissipation based on an experimental program on native sandy soils from Christchurch, New Zealand.
Deformational properties of soil, in terms of modulus and damping, exert a great influence on seismic response of soil sites. However, these properties for sands containing some portion of fines particles have not been systematically addressed. In addition, simultaneous modelling of the modulus and damping behaviour of soils during cyclic loading is desirable. This study presents an experimental and computational investigation into the deformational properties of sands containing fines content in the context of site response analysis. The experimental investigation is carried on sandy soils sourced from Christchurch, New Zealand using a dynamic triaxial apparatus while the computational aspect is based on the framework of total-stress one-dimensional (1D) cyclic behaviour of soil. The experimental investigation focused on a systematic study on the deformational behaviour of sand with different amounts of fines content (particle diameter ≤ 75µm) under drained conditions. The silty sands were prepared by mixing clean sand with three different percentages of fines content. A series of bender element tests at small-strain range and stress-controlled dynamic triaxial tests at medium to high-strain ranges were conducted on samples of clean sand and silty sand. This allowed measurements of linear and nonlinear deformational properties of the same specimen for a wide strain range. The testing program was designed to quantify the effects of void ratio and fines content on the low-strain stiffness of the silty sand as well as on the nonlinear stress-strain relationship and corresponding shear modulus and damping properties as a function of cyclic shear strains. Shear wave velocity, Vs, and maximum shear modulus, Gmax, of silty sand was shown to be significantly smaller than the respective values for clean sands measured at the same void ratio, e, or same relative density, Dr. However, the test results showed that the difference in the level of nonlinearity between clean sand and silty sands was small. For loose samples prepared at an identical relative density, the behaviour of clean sand was slightly less nonlinear as compared to sandy soils with higher fines content. This difference in the nonlinear behaviour of clean sand and sandy soils was negligible for dense soils. Furthermore, no systematic influence of fines content on the material damping curve was observed for sands with fines content FC = 0 to 30%. In order to normalize the effects of fines on moduli of sands, equivalent granular void ratio, e*, was employed. This was done through quantifying the participation of fines content in the force transfer chain of the sand matrix. As such, a unified framework for modelling of the variability of shear wave velocity, Vs, (or shear modulus, Gmax) with void ratio was achieved for clean sands and sands with fines, irrespective of their fines content. Furthermore, modelling of the cyclic stress-strain behaviour based on this experimental program was investigated. The modelling effort focused on developing a simple constitutive model which simultaneously models the soil modulus and damping relationships with shear strains observed in laboratory tests. The backbone curve of the cyclic model was adopted based on a modified version of Kondner and Zelasko (MKZ) hyperbolic function, with a curvature coefficient, a. In order to simulate the hysteretic cycles, the conventional Masing rules (Pyke 1979) were revised. The parameter n, in the Masing’s criteria was assumed to be a function of material damping, h, measured in the laboratory. As such the modulus and damping produced by the numerical model could match the stress-strain behaviour observed in the laboratory over the course of this study. It was shown that the Masing parameter n, is strain-dependent and generally takes values of n ≤ 2. The model was then verified through element test simulations under different cyclic loadings. It was shown that the model could accurately simulate the modulus and the damping simultaneously. The model was then incorporated within the OpenSees computational platform and was used to scrutinize the effects of damping on one-dimensional seismic site response analysis. For this purpose, several strong motion stations which recorded the Canterbury earthquake sequence were selected. The soil profiles were modelled as semi-infinite horizontally layered deposits overlying a uniform half-space subjected to vertically propagating shear waves. The advantages and limitations of the nonlinear model in terms of simulating soil nonlinearity and associated material damping were further scrutinized. It was shown that generally, the conventional Masing criteria unconservatively may underestimate some response parameters such as spectral accelerations. This was shown to be due to larger hysteretic damping modelled by using conventional Masing criteria. In addition, maximum shear strains within the soil profiles were also computed smaller in comparison to the values calculated by the proposed model. Further analyses were performed to study the simulation of backbone curve beyond the strain ranges addressed in the experimental phase of this study. A key issue that was identified was that relying only on the modulus reduction curves to simulate the stress-strain behaviour of soil may not capture the actual soil strength at larger strains. Hence, strength properties of the soil layer should also be incorporated to accurately simulate the backbone curve.
