Geometric characteristics of the deformation state of the shells with orthogonal coordinate system of the middle surfaces

Abstract

The aim of this work is to receive the geometrical equations of strains of shells at the common orthogonal not conjugated coordinate system. At the most articles, textbooks and monographs on the theory and analysis of the thin shell there are considered the shells the coordinate system of which is given at the lines of main curvatures. Derivation of the geometric equations of the deformed state of the thin shells in the lines of main curvatures is given, specifically, at monographs of the theory of the thin shells of V.V. Novozhilov, K.F. Chernih, A.P. Filin and other Russian and foreign scientists. The standard methods of mathematic analyses, vector analysis and differential geometry are used to receive them. The method of tensor analysis is used for receiving the common equations of deformation of non orthogonal coordinate system of the middle shell surface of thin shell. The equations of deformation of the shells in common orthogonal coordinate system (not in the lines of main curvatures) are received on the base of this equation. Derivation of the geometric equations of deformations of thin shells in orthogonal not conjugated coordinate system on the base of differential geometry and vector analysis (without using of tensor analysis) is given at the article. This access may be used at textbooks as far as at most technical institutes the base of tensor analysis is not given.

Bionics in architecture and geometric modelling of thin shell surfaces

“…Taking into account structural requirements, the geometry of shells [7,10,12,13] is considered in detail, depending on different ways of their support on: walls (supporting rings); columns (columns); directly on foundations (solid or separate); supports, which are an organic transition between the shell and the foundation; combined (mixed) support on walls and cables; console support; "floating" foundations. The influence of overlapped span size on the choice of shell shape was studied.…”

“…In research works, a number of authors consider the problem of the stress-strain state of thin geometric shells under a vertical load arising under given rigid displacements of a number of cross sections [5][6][7][8]. Scientific papers [9][10][11][12] are devoted to the problems of nonlinear deformation of the stress-strain state of thin-walled building structures of complex geometric shape.…”

“…In the present study, an attempt is made to analyze different predetermined requirements to the designed shells and their geometrical interpretation is given. For example, based on the calculation requirements, the dependence of the calculation complexity on the geometry of the shell is shown, and the static features of shells with parabolic, hyperbolic and elliptical points are given [10][11][12]. Studies of the shells delineated by the transfer surface have been carried out.…”

Finite element modeling of the processes of elastic-plastic deformation of reclamation objects of the agro-industrial complex

“…(5) Subscripts 1, 2, 3 after the decimal point indicate that differentiation is performed along curvilinear coordinates 1 θ , 2 θ , and t . Applying the scalar product operation to (5), one can obtain the covariant components of the metric tensors at points…”

“…To reduce the payback period of such systems, the optimal design of objects and systems for reclamation purposes becomes very important, which is unthinkable without the use of modern numerical methods of calculation [1,2,3,4,5], which, in particular, includes the finite element method (FEM) [6,7,8,9]. Moreover, the application of this method at the present stage is based on the use as discretization elements, mainly, three-dimensional finite elements [10,11,12,13,14], as the most versatile, adapted for modeling both thin-walled fragments of reclamation structures (pipelines, tanks, tanks) and thick-walled massive bodies (dams, retaining walls, joints and others).…”