Simple buckling and vibration analyses of beam or spring connected structures

The problem considered is the buckling or vibration of a system of n columns, or n general structures, which are suitably related to each other and are connected together by sets of springs, or inextensible beams, which are also suitably related to each other. It is shown that all the critical loads, natural frequencies and modes which are required in buckling and vibration problems can be found from n substitute systems which each consist of one column or structure and a single set of springs. The derivation of these substitute systems involves the solution of a very simple linear eigenvalue problem of order n, which has closed form solutions for several of the special cases considered. The reduction of the original system to n substitute systems has been adapted to permit the use of design procedures which avoid a complete analysis of each trial design.