Abstract
The vertices of two specific eigenvectors, obtained from a novel linear eigenvalue problem, describe two curves on the surface of an N-dimensional unit hypersphere. N denotes the number of degrees of freedom in the framework of structural analysis by the Finite Element Method. The radii of curvature of these two curves are 0 and 1. They correlate with pure stretching and pure bending, respectively, of structures. The two coefficient matrices of the eigenvalue problem are the tangent stiffness matrix at the load level considered and the one at the onset of loading. The goals of this paper are to report on the numerical verification of the aforesaid geometric-mechanical synergism and to summarize current attempts of its extension to combinations of stretching and bending of structures.
Highlights
It is preferable that the loads are mainly carried by membrane and axial forces instead of bending moments
The lower bound of this ratio is zero, and it refers to pure stretching
The basic idea of this work is to find a geometric quantity such that its lower and upper bound agrees with the bounds of (U − UM )/U. This is done with the help of a novel linear eigenvalue problem in the framework of the Finite Element Method (FEM)
Summary
It is preferable that the loads are mainly carried by membrane and axial forces instead of bending moments. A numerical challenge of the CLE was a sufficiently accurate finite-difference approximation of the derivative of the tangent stiffness matrix [4,9] Another forerunner of the present work was a paper by Mang [6], characterized by the replacement of ρ by k a , where a denotes the acceleration of a fictitious particle moving along a curve on the surface of an N -dimensional unit hypersphere, obtained by the CLE, and where k stands for a proportionality factor.
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