Reduction of dimensions in random, elastic soil medium

The paper deals with an application of the plane strain analysis in a stochastic three!dimensional soil medium[ In a framework of random elasticity theory\ the geostatical state of stresses and the problem of a unit force acting in a statistically homogeneous half!space are considered[ Only the modulus of elasticity is considered to be random and is modelled as a three!dimensional "2!D# homogeneous random _eld[ As the result of imposed constrains due to the plane strain assumption the additional body and surface forces are induced[ In order to determine them\ additional equations must be introduced[ The equations in a form of constrain relations are proposed in this paper[ These equations are also valid for a case of uniformly distributed external loading[ First\ the two!dimensional "1!D# problem and its reduction to the uni!axial strain state\ for the gravity forces and uniform\ unlimited surface loading is considered[ Then\ it is generalised into a 1!D schematization of the 2!D state[ Next\ the problem of a unit force acting in a statistically homogeneous half!space is considered[ For a 2!D state of stress and strain the resulting stresses are compared with those for a 1!D state[ These stresses for the multidimensional state of strain and stress are presented as a sum of two components[ The _rst one re~ects plane strain state stresses and is given in a form of a 2!D random _eld[ This term allows for incorporating a spatial\ 2!D soil variability into a two!dimensional analysis[ The second component can be treated as a correction term and it represents the longitudinal in~uence of a 2!D analysis[ Some numerical results are presented in this paper[ The proposed method can be regarded as a framework for further research aiming at application to a variety of geotechnical problems\ for which the plane strain state is assumed