Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity

Abstract The paper considers the possibility of making more exact the theory of plates based on Kirchhoff's hypothesis. The prob;em of the bending of a plate is formulated as a three-dimensional problem of the theory of elasticity which can be solved by an iteration process; it is assumed that one of the extensions of the region under consideration is small compared with the other two. The required state of stress of the plate is presented as the sum of a slowly damped state of stress, derived by means of a basic iteration process, and states of stress which are rapidly damped with increase in distance from the edge, and which are derived by means of auxiliary iteration processes. Such an approach is often used in the symptotic integration of differential equations (see [1]) and corresponds to the physical nature of the problem. The basic iteration process enables us to find the state of stress which is given as a first approximation by the classical theory. The auxiliary iteration process allows us to take into account the stress distribution at the edges which were discussed in attempts to make the classical theory more exact by repalcing Kirchhoff's hypothesis by alternative assumptions (see, for example, [2–6]).