  We consider the problem of satisfaction of boundary conditions when the
generalized stress vector is given on the surfaces for elastic plates and
shells. This problem was open also both for refined theories in the wide sense
and hierarchical type models. This one for hierarchical models was formulated
by Vekua. In nonlinear cases the bending and compression-extension processes
did not split and for this aim we cited von Karman type system without variety
of ad hoc assumptions since, in the classical form of this system of DEs one of
them represents the condition of compatibility but it is not an equilibrium
equation. Thus, we created the mathematical theory of refined theories both in
linear and nonlinear cases for anisotropic nonhomogeneous elastic plates and
shells, approximately satisfying the corresponding system of partial
differential equations and boundary conditions on the surfaces. The optimal and
convenient refined theory might be chosen easily by selection of arbitrary
parameters; preliminarily a few necessary experimental measurements have been
made without using any simplifying hypotheses. The same problem is solved for
hierarchical models too.
