DETERMINISTIC AND PROBABILISTIC ANALYSIS OF SHALLOW FOUNDATIONS ON LAYERED SOILS

Abstract

Soil is generally considered as homogeneous and isotropic. However it is the outcome of various geological processes. Soil properties are determined by various field and experimental methods. Therefore, it is important to understand the behaviour of structures built on it. Analysis and design of shallow foundations has been one of the important aspects of Geotechnical engineering. While designing any foundation, the foundation is checked for two criteria, namely, shear failure and settlement criterion. The settlement of the foundation should be within a permissible value to have satisfactory service of the foundation for its lifetime. Different empirical, analytical and numerical methods have been proposed by various research workers for the analysis and design of foundations considering linear as well as nonlinear stress strain behaviour. The soil parameters which are obtained from the results of different in-situ and laboratory tests highlight the fact that there is a fair amount of variability in the soil parameters. Therefore this suggests that soil parameters should be modelled as random rather than deterministic. However, the design engineers usually address this issue by adopting a mean value of various test results in theoretical prediction of the settlement and stresses in soil domain. In recent years, a variety of approaches like probabilistic and stochastic methods have been developed to predict the behaviour of soil foundation system. Critical review of literature suggests that the conventional methods available are considering the linear stress strain relationship. Further the arithmetic mean of multiple tests was adopted for the soil properties resulting in large factors of safety in the design. However no attempt has so far been made for considering nonlinear stress strain behaviour of soil layers in probabilistic analysis. Keeping the above facts in view, there is a need to investigate behaviour of shallow foundations considering the aspects of linear stochastic analysis, nonlinear analysis and nonlinear probabilistic analysis: The soil foundation system consists of a strip surface footing over a three layered soil with different properties have been considered. These layers have been overlying a rigid base and the soil parameters considered in the analysis are modulus of elasticity and Poisson’s ratio of different layers. The footing has been subjected to a uniformly distributed load. ii For the linear stochastic analysis of strip footings, the elastic modulii of soil layers have been treated as a stationary stochastic field defined by mean, variance and the autocorrelation distance (ACD). Realizations of elastic modulii, generated by solving a stochastic differential equation, have been fed to a deterministic finite difference elastic model to generate realizations of dependent stochastic fields comprising, settlement of footing and interfacial vertical stresses. Subsequently these realizations have been analyzed to evolve probability distribution functions, variance and autocorrelation function (ACF) of the dependent stochastic fields. The linear stochastic algorithm which is used to simulate the spatial variations of soil properties combined with the deterministic finite difference model is coded using C- Language. It has been revealed that ACF of these fields are independent of the ACF of elastic modulii of soil layers. The variance of settlement has been found to increase as the ACD of elastic modulii increases, implying requirement of a larger factor of safety (FOS) when random soils display low frequency variations. On the other hand the variance of vertical stress is larger at smaller ACD of elastic modulii, indicating that for vertical stress, larger FOS is required when the random soils display high frequency variations. Strip footings have been analyzed considering the non-linear stress-strain behaviour of soils which has been modelled by Kondner’s hyperbolic stress-strain constitutive relationship. The mixed method that combines advantages of both the incremental and iterative procedures using finite element method has been used to solve the nonlinear response. The nonlinear finite element method has been coded in FORTRAN 90. The layered soil system has been discretized using eight noded two-dimensional elements assuming plane strain condition. Six noded zero thickness elements have been used to simulate the interface between soil layers. A parametric study has been conducted to understand the influence of various parameters on stresses and displacements. It has been observed that these parameters have significant influence on displacements and stresses in the entire soil system. Finally, the nonlinear finite element analysis of strip footings has been combined with a probabilistic approach to consider the inherent variability in soil layers. This helped in studying the reliability of predicting the uncertain parameters of soil and their influence on settlements and stresses on the layered soil system. Initial elastic modulii of all the soil layers have been considered to be the uncertain parameters that follow lognormal distribution. Estimated FOS for settlement and the normal interface stresses have been linked to risk iii factor and the coefficient of variation (COV) of random variables. A thorough investigation on effect of key parameters implied the fact that, for a given risk level, the FOS is mainly function of COV of soil modulus while the other parameters have negligible influence on response. This study facilitated to develop mathematical expression linking FOS, risk of failure and the COV of elastic moduli. With the help of these expressions, FOS for a given risk level and COV can be readily calculated.