A Semianalytical Approach for Nonlinear Dynamic System of Shallow Arches Using Higher Order Multistep Taylor Method

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9567619, 17 pages https://doi.org/10.1155/2018/9567619 Research Article A Semianalytical Approach for Nonlinear Dynamic System of Shallow Arches Using Higher Order Multistep Taylor Method Sudeok Shon ,1 Soohong Ahn,2 Seungjae Lee ,3 and Junhong Ha4 1 Department of Architectural Engineering, Korea University of Technology and Education, Cheonan 31253, Republic of Korea Acrovision Co. Ltd., B-1411, 70 Dusan-ro, Geumcheon-gu, Seoul 08584, Republic of Korea 3 Interdisciplinary Program in Creative Engineering, Korea University of Technology and Education, Cheonan 31253, Republic of Korea 4 School of Liberal Arts, Korea University of Technology and Education, Cheonan 31252, Republic of Korea 2 Correspondence should be addressed to Seungjae Lee; Received 21 November 2017; Revised 21 April 2018; Accepted 24 April 2018; Published 19 August 2018 Academic Editor: Stefano Lenci Copyright © 2018 Sudeok Shon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study aimed at obtaining a semianalytical solution for nonlinear dynamic system of shallow arches. Taylor method was applied to find the analytical solution, and an investigation of their dynamic characteristic was carried out to verify the applicability of this methodology for the shallow arches under step or periodic excitation. A polynomial solution can be obtained from this multistep approach with respect to time, and direct buckling as well as indirect buckling of the shallow arches can be observed, also. The results indicated that the dynamic buckling load level was higher with higher shape factor. Additionally, a change of attractor in phase space was investigated. Coupling in symmetric mode as well as asymmetric mode was observed in case of indirect buckling, and a sensitive response was also manifested during sinusoidal and beating excitation. These results of applying multistep Taylor series for the investigation of displacement response and attractor change revealed that this analytical approach was valid in explaining the dynamic buckling behavior of shallow arches under direct and indirect snapping. 1. Introduction Large span roof structures are generally designed as arches and spherical shells. In other words, the structural performance of the arch depends on its shape. Thus, lighter, thinner, and larger space can be made with less materials compared to traditional flat structures. Shallow arches have received considerable attention in the field of architecture because of their beautiful shape and economic efficiency. However, the response to transverse loading of the shallow arch is quite different from that of the flat beam. The structural design also requires dynamic behavior analysis under various loads. Since a large roof can be treated separately from the substructures, the arch’s boundary is usually designed by hinge, and with shallow arch roofs the primary interest is in the instability of vertical direction behavior. The dynamic unstable behavior of shallow arches is generally complex depending on the initial condition, e.g., geometrically imperfection, and shape of the arches. In particular, direct snapping in symmetric mode and indirect snapping due to coupling in asymmetric mode manifest very sensitive behavior and progress to chaotic behavior, attracting deep interest from many researchers. Investigation of dynamic instability of shallow arches has started with the fundamental study of Hoff and Bruce [1] in the mid-1950s and was followed by many researches afterwards in the beginning (Humphreys [2]; Lock [3]; Hsu [4]; Ariaratnam and Sankar [5]; Sundararajan and Kumani [6]; Donaldson and Plaut [7]; and Kounadis et al. [8]). These early researches on shallow arches resulted in rebounding studies of chaotic motion or global dynamic behavior (Blair et al. [9]; Levitas et al. [10]), accurate solution of free-vibration (Tseng et al. [11]; De Rosa and Franciosi [12]), internal resonance (Bi and Dai [13]; Lacarbonara and Rega [14]), and buckling under boundary condition and moving load (Kong et al. [15]; Chen and Lin [16, 17]). Recently, many investigations for design and construction of shallow arch roofs have been conducted, including the identification and stability for shallow arches (Ha et al. [18]), and sensitivity of shallow arch (Virgin et al. [19]). On the other hand, the initially 2 curved or buckled microbeams in mechanics (Farokhi and Ghayesh [20]) have been studied in terms of nonlinearity, geometrical imperfection, resonance, and chaotic behaviors. Various numerical researches have been conducted on the nonlinear behaviors of the microbeams and their application. However, this buckled configuration of the microbeams is due to some external effects such as temperature increase or compressive forces, which are not associated with the arch roof. Even though in the study of arch roofs the range of arch shapes and sizes differs from that of microbeams, these investigations are still useful for all engineering fields. The evaluation of dynamic buckling of shallow arch roof generally uses energy method or phase space of system approach or direct approach (Lin and Chen [21]) of investigating significant growth of displacement response (Budiansky and Roth [22]) by obtaining dynamic response of the governing equations in a wide range of parameters. This process of dealing with dynamic loading requires numerical method (Belytschko [23]; Argyris et al. [24]) or analytical techniques for solving nonlinear differential equations, and the process of formulating the equations differs by the methodology. One of the direct purposes of dynamic analysis is finding out the transient response to external force and ultimately attaining the solution to motion equations. However, it is difficult to obtain an analytical solution for dynamic system of shallow arches under the various external forces. The techniques to obtain an analytical solution to nonlinear differential equations have been considerably investigated in the past (Sadighi et al. [25]). Analytical or semianalytical methods include the traditional Taylor’s power series method (Barrio [26]; Barrio et al. [27]), Adomian decomposition method (Adomian and Rach [28]; Adomian and Rach [29]), homotopy analysis method (Liao [30]), and homotopy perturbation method (He [31]; He [32]; and Chowdhury et al. [33]). Unlike numerical ones, these methods are limited in their applicability and are heavily dependent on the influence of their parameters. The Taylor series method, which has the longest history, provides an accurate analytical solution with excellent stability. Recently, Barrio et al. [27] explained that the Taylor series method provides advantages including easy formation with the method of using variable order (...truncated)