Abstract
The article is dedicated to the discussion on the exact dynamic stiffness matrix method applied to the problems of elastic stability of engineering structures. The detailed formulation of the member dynamic stiffness matrix for beams is presented along with the general guidelines on automatisation of the assembly of member dynamic stiffness matrices into the global matrix that corresponds to the whole structure. The advantage of the dynamic stiffness matrix in case of parametric studies is explained. The problem of computing the eigenvalues of transcendental matrix is addressed. The straightforward approach as well as a powerful Witrick-Williams algorithm are discussed in details. The general guidelines on programming the DS matrix method are given as well.
Highlights
Elastic stability problems faced by engineers these days feature extremely high levels of complexity
The member dynamic stiffness (DS) matrices as well as the assembled DS matrices are transcendental which means that their entries are transcendental functions of eigenvalues and define exact relations between nodal forces and nodal displacements of each member in a structure. In such a way the DS matrix method possesses considerable advantages compared to the finite element (FE) method in terms of computation efficiency and accuracy
The approach was discussed in details at all stages
Summary
Elastic stability problems faced by engineers these days feature extremely high levels of complexity. This article is dedicated to the discussion of the exact Dynamic Stiffness method and its application to the problems of elastic stability. The member DS matrices as well as the assembled DS matrices are transcendental which means that their entries are transcendental functions of eigenvalues and define exact relations between nodal forces and nodal displacements of each member in a structure In such a way the DS matrix method possesses considerable advantages compared to the finite element (FE) method in terms of computation efficiency and accuracy. We discuss the methods of computing the eigenvalues of the DS matrix on a simple illustrative example We study both the straightforward evaluation of the zeros of the determinant of the DS matrix and the Wittrick-Williams (W-W) algorithm. We believe that illustration of the application of the algorithm on a simple example is essential
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