Application Of Variational Inequality To Lower Limit Analysis In Geomechanics

In the study of geotechnical mechanics,the problem of stability and deformation is always the two core problem,especially the stability of the slope,which has always been a hot topic in the field of engineering.In the current study of slope stability,the main methods used include rigid body limit equilibrium method and finite element strength reduction method.The lower limit analysis method based on the lower limit theorem has a rigorous mechanical basis in theory,and can give a safety factor with practical engineering value.Therefore,this paper mainly studies the variational inequality model of the lower limit and its projection contraction algorithm.The main results and conclusions as follows:(1)This paper mainly discusses the lower limit principle of limit analysis theory,and introduces the basic idea of solving the load under the lower limit of rock and soil.(2)On the basis of the lower limit theorem,the static permissible stress field is established by the finite element method,and its set form is obtained,and the corresponding mathematical nonlinear programming model is constructed.The strict mathematical transformation relationship between the lower bound principle and variational inequalities is derived theoretically.It is proved that both of them are convex optimization problems in essence.(3)The projection analytic solution of the Mohr-Coulomb criterion is derived,which provides a basis for solving the problem by using the projection contraction algorithm of Prof.He Bingsheng of Nanjing University.(4)Finally,based on the MATLAB numerical analysis software,a variational inequality method program is worked out.The calculation results of the variational inequality method and the traditional method are compared with two simple solutions to the ultimate load of the soil slope.The variational inequality method and the correctness of the program are illustrated.This is the first variational inequality method applied to this field.