The method of local approximations, whose basic concepts are given in [1, 2], is based on the representation of the strains of shells in small regions (elements) by Taylor series expansions. If we confine ourselves to a small element and refer its geometry to a certain simple geometrical object, say a plane, the strain relations can be simplified. Depending on the degree of local approximation, various nonlinear models of the strains of shells are obtained. Here we investigate the most effective finite-element models based on the first terms of local approximations.1. In the neighborhood of a point O the centroidal surface of the shell is defined by the equationwhere r is the radius vector of the surface, rP is the radius vector of the plane tangent to the shell surface at the point O, ~" = ~'(~1, ~2) is a function describing shape of the surface in the neighborhood of the point under consideration, ~z and ~2 are orthogonal coordinates on the tangent plane, and a subscript after a con#ma denotes differentiation with respect to the corresponding coordinate. Summation over repeated indexes is used everywhere. Neglecting small terms ~-,2 < < I, we obtain an expression for the normal vector n to the shell surface

We consider an elastic rod divided into a number of finite elements. We represent the displacement of each element in the form of a sum of the element displacement as a solid body which we later call the solid displacements, and elastic displacements associated with strains. Then for the nodal displacements q of the element we can write q = qt +q~, (1)where the subscripts t and e refer to the solid and elastic displacements, respectively.With respect to the elastic displacements we assume that they are sufficiently small as compared with the element dimensions. Therefore, in a nonlinear analysis the solid displacements will be the principal part of the total displacements.The solid displacements are defined by three components in the plane case and six in the three-dimensional case.The approach proposed is sufficiently general and can be applied to an analysis of any structures. In this paper it is applied to an analysis of the large displacements of plane elastic rods.In an example of arbitrary bending of an initially rectilinear rod, we show that the proposed method of separation of variables permits the application of linear relationships between the elastic displacements and strains in the element. (3)Here ut and vt are the element displacements as a solid body (determined from the conditions that e and ~ equal zero), and ue and Ve are the elastic displacements associated with deformation of the element.Projections of the elastic displacements on the axes of the global x, y and local gi*, ~2" coordinate systems are connected by the following relationships: where @t is the angle of rotation of the element as a solid, and the asterisk indicates that the displacement projection belongs to the ~*, ~=* coordinate system.The $I*, $2* axes are oriented along the tangent and the normal to the elastic line, respectively.Since the solid body displacements are defined by three constants, then they can always be selected such that the following three equalities:would be satisfied at a certain point p of the element; for instance, at the origin of the coordinates ~l*, ~2", i.e., the actual displacements equal the solid body displacements at this point. Substituting (3) into (2) with the relationships (4) taken into account, andNovosibirsk.

No abstract

No abstract

No abstract