Efficient CUF-based method for the vibrations of thin-walled open cross-section beams under compression

Abstract

This study proposes an efficient method for the evaluation of vibrations and buckling in thin-walled beams with complex geometries subjected to progressive compressive loads. A comprehensive study is conducted in order to investigate the effects of compressive loads on the natural frequencies of the thin-walled beams. Namely, a numerical simulation of the Vibration Correlation Technique is provided in this study. Finite Elements (FEs) are built in the framework of the Carrera Unified Formulation (CUF), and the displacements of complex geometric shapes of the thin-walled beams are evaluated using low- to higher-order Taylor and Lagrange polynomials. The results are compared with the experimental results of the available literature and the numerical results by the shell models. The cross-sectional deformations of the beam due to the vibration modes are also compared, and the importance of structural theories capable of accurate detection of complex cross-sectional deformations is highlighted. The obtained results are demonstrated to be promising and accurate and match reasonably well with the experiments and shell models, which are more expensive in terms of computational costs compared to the efficient CUF ones proposed here.

Introduction

Thin-walled beam structures with open cross-sections are widely used in different applications such as mechanical, aerospace, and civil engineering. Beams are structures with one dimension much larger than the other two and primarily subjected to lateral loads, resulting in the bending of their reference axes [1]. One-dimensional beam theories were developed due to the simplicity and lower computational costs. Beam theories were initially formulated by Euler [2], Bernoulli [3], Timoshenko [4], [5], and further by Saint-Venant [6], [7]. Dynamic behaviors of thin-walled beams by using experimental, numerical, and analytical approaches were investigated [8], [9]. In many cases, the beam, columns, and plate structures are subjected to axial compression [10], [11], [12]. Therefore, the accurate understanding of the vibration and buckling of these structures under compression plays a pivotal role in the safe and reliable engineering design. The Vibration Correlation Technique (VCT) [13] was introduced as a non-destructive method to evaluate buckling loads of structures under progressive compressive loads. According to the VCT, the natural frequencies of the structure are decreased by the compressive loads. By the assumption that the vibration modes are similar to the buckling ones, the critical buckling load can be extrapolated as the load which causes zero natural frequency [14]. A review paper by Abramovich [15] studied the use of the VCT for buckling load prediction of thin-walled structures, including the available literature for different structures such as columns, beams, plates, panels, and cylindrical shells.

Vibration and buckling of beam structures under compression were investigated by many researchers [16], [17], [18], [19]. Abramovich [20] studied the natural frequencies in the Timoshenko beams with different boundary conditions subjected to compressive axial loads. Prokic [21] worked on the flexure–torsion coupled vibrations of axially loaded thin-walled beams by exact solutions. Carpinteri et al. [22] studied the evolution of fundamental frequency in slender beams under axial displacements, and analyzed the effects of geometrical imperfections and constraint conditions. Some research studies are focused on the double-beam systems under compressive axial loads [23], [24]. Piana et al. [25] worked on the buckling and natural frequencies of non-symmetric thin-walled beams. They used a universal machine for the experimental compression tests that imposed a relative axial displacement in the ends of the aluminum thin-walled beams. The same authors worked on the thin-walled beams with symmetric cross-sections considering the warping effects [26]. Aquaro [27] focused on the torsional instabilities of thin-walled open cross-section beams with stiffeners placed in different positions. Zmuda [28] presented a numerical analysis on the axially-loaded steel beams and considered linearly-elastic and elastic-perfectly plastic material behaviors for the cold-formed lipped channel section beams. Elkaimbillah et al. [29] proposed a one-dimensional finite element (FE) model for the forced nonlinear vibration response of thin-walled composite beams with open variable cross-sections. Cabral et al. [14] used the VCT for the experimental and numerical analysis of pre-stressed laminated reinforced panels. The results of experimental tests were compared by advanced FE models using the Carrera Unified Formulation (CUF). By using the CUF, Pagani et al. [30] worked on the VCT for analysis of beam structures in the geometrical nonlinear framework.

The CUF was demonstrated as an efficient and accurate method to solve the vibration analysis problems of structures [31], [32], [33]. A hierarchical FE by using the CUF was presented by Carrera et al. [34]. Mass and stiffness matrices of the structure in terms of the independent fundamental nuclei were formulated. Petrolo et al. [35] investigated the free vibration response of compact and bridge-like cross-sections. The higher-order theories based on the CUF were used for the free vibration response of different beam cross-sections [36]. By opportunely modifying the fundamental nucleus of the mass matrix, Pagani et al. [37] evaluated the effect of nonstructural localized inertia on the free vibration response of thin-walled structures. Dan et al. [38] studied the free vibration of simply supported beams with solid and thin-walled cross-sections using the CUF. Xu et al. [39] used Lagrange polynomial expansion for the free vibration analysis of thin-walled beams, and presented an efficient FE method based on the CUF. Pagani et al. [40] investigated the frequency and mode change of beam structures subjected to geometrical nonlinearities in the large displacements and rotations. They focused on the large-deflection and post-buckling of different compact and thin-walled beams. Furthermore, the influence of large-displacements on the vibration of composite beams was evaluated [41].

In this paper, an efficient CUF-based method for evaluating vibrations and buckling in the thin-walled beams with complex open cross-sections subjected to progressive compressive loads is presented. A comprehensive study is conducted in order to investigate the effect of compressive loads on the natural frequencies of the beams subjected to progressive compressive loads. The CUF-based FEs with the Taylor and Lagrange expansions are implemented and compared with the experimental results of the available literature and the numerical results by the shell models. Moreover, the cross-sectional deformations of the beam due to the vibration modes are compared, and the importance of structural theories capable of accurate detection of these cross-sectional deformations is outlined. The obtained results by the CUF-based FE method correlate reasonably well with the experiments and shell models, which are considerably more expensive in terms of computational costs.