Studies of nonlinear theories for thin shells

In order to formulate the equations for the study here, the Fourier expansions upon the system of orthonormal polynomials are used. It may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions. Based on the principle of virtual work the equilibrium equations of various orders are formulated. In particular, the system of thirdorder is given in detail, thus providing the reference for accuracy analysis of lower-order equations. A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis. Therefore the functions used are specified and the analysis rendered is no longer a formal one. The analysis will show that the Kirchhoff-Love's theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentially of the first order, too.