Natural Vibrations of Plane Frames under Compression Through a Power Series Solution

Abstract

The use of plane frames is very common within Structural Engineering. Standard loads derive in bending, shear and axial internal forces being the compression ones of special interest. Thus, a buckling study is mandatory in this structural type. A previous work of the research group addressed the frames buckling loads determination by means of a power series technique (in practice, polynomials are used in the applications). Separately, the natural vibration of frames was also solved. Now, an extension to the natural vibration problem of plane frames with members under compression is presented. The frames are open and no branches are considered. The governing linear differential equations are first stated together with the boundary and continuity conditions at each node linking consecutive members with arbitrary slope. The equations take into account the second order effect of the axial load. The power series algorithm is then introduced and a systematization is proposed. Various frame configurations are studied and the natural frequencies are found. The frequency decreasing effect due to the variation of the external load as it approaches to the buckling limit, is shown. Also, the convergence of the solution is assessed for increasing number of terms in the polynomials. This technique is useful as an alternative to other popular methods, such as finite elements. Here, each member of the frame is not divided into elements. Instead, the mode shape of each member is represented by the power series with arbitrary accuracy. Finally, it should be mentioned that the resulting matrix of the eigenvalue problem is always of dimension 6 by 6 disregarding the number of members of the open frame, contrary to other methods. Numerical examples and comparisons illustrate the proposed technique.