Deformation and Stress Analysis of Catenary Shell of Revolution

The catenary shells of revolution are widely used in constructions due to their unique mechanics' feature. However, no publications on this type of shells can be found in the literature. To have a better understanding of the deformation and stress of the catenary shells of revolution, we formulate the principal radii for two kinds of catenary shells of revolution and their displacement type governing equations. Numerical simulations are carried out based on both Reissner-Meissner mixed formulations and displacement formulations. Our investigations show that both deformation and stress response of elastic catenary shells of revolution are sensitive to its geometric parameter $c$, and reveal that the mechanics of the catenary shells of revolution does much better than the spherical shells. Two complete codes in Maple are provided.

Stress-strain state of shell of revolution analysis by using various formulations of three-dimensional finite elements

The aim of the work is to perform a comparative analysis of the results of analyzing arbitrarily loaded shells of revolution using finite element method in various formulations, namely, in the formulation of the displacement method and in the mixed formulation. Methods. To obtain the stiffness matrix of a finite element a functional based on the equality of the actual work of external and internal forces was applied. To obtain the deformation matrix in the mixed formulation the functional obtained from the previous one by replacing the actual work of internal forces in it with the difference of the total and additional work was used. Results. In the formulation of the displacement method for an eight-node hexahedral solid finite element, displacements and their first derivatives are taken as the nodal unknowns. Approximation of the displacements of the inner point of the finite element was carried out through the nodal unknowns on the basis of the Hermite polynomials of the third degree. For a finite element in the mixed formulation, displacements and stresses were taken as nodal unknowns. Approximation of the target finite element values through their nodal values in the mixed formulation was carried out on the basis of trilinear functions. It is shown on a test example that a finite element in the mixed formulation improves the accuracy of the strength parameters of the shell of revolution stress-strain state.

Optimal shells of revolution and main optimizations

Introduction. Optimization is a criterion, on the ground of which, comparative estimation of possible alternatives and selection of the best decisions is carried out. Cost of a shell, its minimum weight, absence of bending moments and tensile normal stresses, given stress state for acting external load, given bearing capacity when optimal shallowness, maximum external load, minimum weight under limitation on the value of natural frequencies of vibration and maximum displacements, absence of bending moments with taking into account internal pressure, dead weight, and centrifugal forces; maximum of critical force and something else can be criterion of selection of optimal shape of shell of revolution. Methods. The main criteria of optimality for shells of revolution and information sources for the 1970-2019 periods are presented in a paper. It will help to study previous results devoted to using optimizations and to set about further investigation. But there is no single approach to the definition of optimal shell of revolution and obviously will not be, because own optimizations are necessary for every concrete case of loading, or distribution of stresses along the thickness, or under the demands to the ratio of the volume and area of considered shell, or with due regard for different kind of expenses, and other demands. Results. For the first time, 24 criteria of optimality only for shells of revolution were discovered. The names of scientists offered presented criteria of optimality and the 45 references dealing with this question are pointed out. It is shown that principles put in the basis of optimal design and criteria of optimality must be given with the help of language quite naturally for computer. Having used optimi- For citation Krivoshapko S.N. (2019). Optimal shells of revolution and main optimizations. Structural Mechanics of Engineering Constructions and Buildings, 15 (3), 201-209. http://dx.doi.org/ 10.22363/1815-5235-2019-15-3-201-209 zations presented in the paper, designers can choose the criterion for their own design of optimal shell shape. Study of the prerequisites of the structural solutions in building and machine-building, the history of the development and perfecting of technologies of erection of shells of revolution will permit to generalize the experience accumulated by designers and to develop new fundamental solutions. Otherwise, architects, structural engineers, and designers will be repeating the achieved solutions in building, architecture, and machine-building.