The intrinsic theory of thin shells and plates. I. General theory

The development of the theory of thin shells and plates by many authors [l, 2, 3, 4] can be summarized under the following three main heads: (1) All theories are based upon certain simplifying unproved assumptions.For example: (a) the thickness remains unchanged during the deformation, (b) the middle surface in the unstrained state deforms into the middle surface in the strained state, (c) the normals of the unstrained middle surface deform into the normals of the strained middle surface.(2) All theories involve the use of displacement to describe the state of deformation.This plan works well in the theory of small deflection, but presents considerable difficulty in the case of large deflection.(3) The various approximations used in the theory of thin shells and plates are confusing.If one attempts to give a complete picture of the theory, one must be able to introduce a systematic method of approximation, which not only clears away the confusion of various approximations, but also leads to a complete classification of all thin shell and plate problems.The purpose of this paper is to give a systematic treatment of the general problem of the thin shell, which includes the problem of the thin plate as a special case.The work is based on the usual equations of elasticity for a finite body, supposed to be homogeneous and isotropic.The final equations of Part I are the three equations of *