Abstract— The article deals with the problem of formation of a mathematical model and algorithm of numerical study of flexible elastic elements in the form of thin-walled shells. The proposed algorithm allows us to quickly solve the problem of numerical synthesis of the structure. The main problem for the synthesis of these elements is to provide the required deformation. The deformation of shell elements is essentially a nonlinear process. The main relations of the version of the theory of thin axisymmetric shells are known. However, the study of the elements under consideration requires taking into account some peculiarities. The paper presents the derivation of the equations describing the axisymmetric deformation of thin-walled shells for several practically important cases, as well as reflects the features that must be taken into account for the cases under consideration. The proposed algorithm allows in the process of modeling the element to carry out a kind of programming of the properties of the future design and as a result provide the required performance characteristics in the process of numerical design. The article presents the results of the application of this technique for the design of real products based on flexible elements with controlled elastic deformation.