The Use of One of the Potential Theory Methods to Study the Static Deformation of Composite Cylindrical Shells

In view of the problem complexity, numerical methods, mainly finite and boundary element methods, are fairly widely used in the calculation of the static deformation of composite shells. The paper presents a new approach to the calculation of the static deformation of composite cylindrical shells using potential theory methods, which made it possible to obtain a solution to the set problem in the closed analytical form. Longitudinally infinite cylindrical shells, which are sequentially rigidly joined together (by clamping the outer edges of the formed composite structure), are considered. Orthogonal curvilinear coordinates in the middle of each of the cylindrical shells have been chosen. The problem has been solved in displacements. The system of differential equations of the static deformation of each of the cylindrical shells being part of the compound body was written according to the Vlasov shallow shell theory. The composite structure was under the action of a normal surface load. The expressions for internal forces and moments are given in terms of the components of the vector of displacements and the corresponding derivatives of them. For the system of differential equations, an exact solution has been found. The problem was solved by the method of variation of arbitrary constants. To determine the integration constants, the rigid joint conditions of cylindrical shells (from physical considerations) and the generally known conditions of rigid clamping of the edges of a compound body have been formulated. A system of linear algebraic equations for the desired constants has been derived. As a result of solving the system and substituting the found constants into the expressions for displacement vectors, the final solution of the problem in the form of potential images has been obtained with the aid of constructed Green matrices for this problem. The calculation method is illustrated by some numerical results, which describe the stress state of the object under investigation, composed of two in-series connected sections, as a function of the wavelength of the cylindrical shells. Analysis of the stress state characteristics shows that both bending moment and transverse force increase with decreasing wavelength of the cylindrical sections. This can be accounted for by the way of joining the cylindrical shells in the composite structure. Deformation schemes of composite cylindrical shells (under unilateral normal load) are presented with a purely illustrative purpose as evidence of the capabilities of the proposed method.