THE LIMIT ANALYSIS IN SOLID MECHANICS AND A SUGGESTED GENERALIZED VARIATIONAL PRINCIPLE
TSIEN LING-HI TSOON WAN-SHIA(Dairen Institute of Technology)
The first part of this paper deals with a brief survey and discussion of the limit analysis in solid mechanics. Since the complete establishment of the well-known theorems of the upper and lower bounds, considerable advances have been made in the limit analysis as a branch of applied plasticity. Now, exact calculation of the plastic limit load is feasible with no apperciable difficulty for rigid framed structures, consisting of systems of members subjected mainly to bending action. In the field of two and three dimensional structures, especially in the plate and shell problems, although many results have also been found for a wide variety of practical problems, but further progress seems to be very difficult in encountering with more complicate problems. Progress is restricted by the fact that, the limit theorems cannot in these cases give results sufficiently approached to the upper and lower bounds. Moreover, the application of lower bound theorem is especially difficult.The second part of this paper suggests a generalized variational principle, in which both stress distribution σij and velocity field vi are introduced and vary independently. This variational principle is equivalent mathematically to the whole set of equations, which must be satisfied by the limit analysis: equalibrium, mechanism, yield condition and flow law. It is proved that with independently assumed kinematically admissible velocity field and statically admissible stress distribution, the generalized variational principle gives the approximate plastic limit load, lying between the upper and lower bounds obtained from bound theorems. Moreover, numerical examples for circular plates are carried out to show that, the generalized variational principle gives rather stable answers for different combinations of assumed stress distribution and velocity field. It is remarked, furthermore, that the generalized variational principle can be applied to the limit analysis dealing with non-homogeneous as well as anisotropic perfectly plastic materials.