The Solution Asymptotics of Classical and Nonclassical, Static and Dynamic Boundary-Value Problems for Thin Bodies

An asymptotic method for solution of classical and nonclassical boundary-value problems of the theory of elasticity for thin bodies (beams, rods, plates, and shells) is expounded. Studies on the asymptotic theory of thin bodies are reviewed. Asymptotic results are compared with those obtained by other applied theories. The asymptotic approach has been found out to be related to Saint Venant's principle. The correctness of this principle is mathematically proved for one class of problems. A fundamentally new asymptotics in the components of the stress tensor and the displacement vector is revealed in considering new classes of problems. On their basis, the applicability domains are outlined for various models of understructures. Solutions are obtained to certain classes of dynamic problems for thin bodies, particularly, those simulating seismic effects. The resonance conditions are established and ways of preventing them are pointed out.