Natural vibrations and instability of plane frames: Exact analytical solutions using power series

•Exact calculation of natural frequencies of vibration and buckling loads of complex planar frames using power series.•Employement of the theory of second order in frames members with simplicity and high reduction of computational cost.•Development of iterative algorithm of power series strongly diminishing the number of unknowns. The objective of this article is to introduce a practical procedure for determining analytical solutions (or at least with arbitrary precision) to free vibration and instability problems related to plane frames, by means of an extended power series method. This procedure leads to an important reduction in the number of unknowns to be handled. In the problem of eigenvalue calculation of frames (in dynamics to extract natural frequencies or in statics to extract buckling loads), the solution corresponds to the nullity of a determinant whose order is substantially smaller compared to the one found by other methods. In order to attain higher precision, other procedures require an increase in the quantity of unknowns, however in the case of the present procedure, only the degree of power is increased without enlarging the number of unknowns. A number of examples are presented in order to show the advantages of the present procedure applied to dynamics and instability of frames. Moreover comparisons with other computational approaches are included as well.