Chapter 8 Fundamental Frequencies of Slender Beams Subject to Imposed Axial End Displacements G. Piana, A. Manuello, R. Malvano, and A. Carpinteri Abstract The dependence of the fundamental frequency on the axial load in slender beams subject to imposed axial end displacements was experimentally investigated. The considered specimens presented different geometrical imperfections (initial curvatures), and were tested in two different constraint conditions (hinged–hinged and hinged-clamped). The natural frequencies were extracted in both conditions of forced and free vibration, using an electromagnet and a laser displacement transducer. In addition, the responses observed during the experiments, for the hinged–hinged case, were reproduced by numerical simulations, obtaining a good agreement between numerical and experimental results. Keywords Experimental test • Numerical simulation • Fundamental frequency • Imposed displacement • Buckling 8.1 Introduction As is well known, the influence of applied compressive axial loads on the fundamental vibration frequency is strictly connected with the stability analysis of elastic slender beams [1]. For this reason, the correct evaluation of the fundamental frequency is of primary importance in designing new structures and components, as well as in monitoring existing ones. At the same time, if an internal axial load arises in a slender element as the consequence of an imposed (static) axial end displacement, then a different dynamic structural response is encountered respect to the case in which a beam end is free to slide, during transverse vibration, and a (constant) axial load is applied externally. This difference is due to the change in the axial boundary condition. Moreover, the presence of an initial curvature of the beam axis may significantly affect the aforesaid response. The latter situation—less infrequent than we might think—may appear when slender structures are subjected to external moving constraints. For example, for bridges and large structures the reduction in bearing capacity and the sudden collapse of compressed elements can be the result of constraint settlements or structural subsidence due to seismic events. Also, a situation analogous to that of a prescribed axial displacement in a slender beam is obtained when, for example, a uniform thermal expansion is restrained. In case of prescribed axial displacement, the internal axial load varies (oscillates) during the beam vibration, due to the change in axial strain; consequently, the variation of the frequency with the axial load will differ from that of the case in which a beam end is free to slide and a (constant) axial load is applied from outside [2, 3]. Furthermore, when both ends are axially restrained, the presence of an initial curvature contributes to the stiffness of the beam which behaves like a pinned shallow arch [4–7]. In the present contribution, we deal with an experimental investigation on the fundamental frequency evolution in slender steel beams subjected to axial end displacements [8]. Different geometrical imperfections, in terms of initial curvature, were considered in order to understand their influence on the evolution of the first natural frequency. In addition, each specimen G. Piana (*) • A. Manuello • A. Carpinteri Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: gianfranco.piana@polito.it R.Malvano Department of Mechanical and Aerospace Engineering, Fluid Dynamics Unit C/o Politecnico di Torino, National Institute of Metrological Research - INRIM, Corso Duca degli Abruzzi 24, 10129 Torino, Italy N. Sottos et al. (eds.), Experimental and Applied Mechanics, Volume 6: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06989-0_8, #The Society for Experimental Mechanics, Inc. 2015 59
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