Investigation on non-linear vibration in arched beam for bridges construction via AGM method

https://doi.org/10.1016/j.amc.2016.11.008Get rights and content

Abstract

Analyzing and modeling the vibrational behavior of arched bridges during earthquake in order to decrease the exerted damages to the structure is a very hard task to do. In the present paper, this item has been performed analytically for the first time. Due to the importance of arched bridges building as a great structure in the human being civilization and its specifications such as transferring vertical loads to its arcs and the lack of bending moments and shearing forces, this case study is devoted to this special issue. Therefore, the nonlinear vibration of arched bridges has been modeled and simulated by an arched beam with harmonic vertical loads and its behavior has been investigated by analyzing a nonlinear partial differential equation governing on the system. It is notable that the procedure has been done analytically by AGM (Akbari-Ganji's Method). It has found that angular frequency is increased significantly by escalating the amount of the center outlet. Furthermore, comparisons have been made between the obtained results by numerical Method (rkf-45) and AGM in order to assess the scientific validity. Obtained results indicate the precision of the achieved answers by AGM method.

Introduction

According to rapid progress of nonlinear sciences, a heightening interest amongst researchers has been emerged in the field of analytical asymptotic techniques, particularly for nonlinear problems in the field of vibrations, such as Mechanical, Earthquake and Civil Engineering [1], [2]. Indeed, equations governing on the vibrational systems are highly nonlinear and there are not precise analytical solutions. It is noteworthy that numerical solutions rarely give intuitive insights into the effects of various parameters associated with problems. Therefore, in most of the situations, recent studies have been centered on the use of effective analytical methods in order to investigate these kinds of problems precisely. Consequently, many new techniques have been presented in open literature, for instance Harmonic Balance Method (HBM) [3], [4], Homotopy Perturbation Method (HPM) [5], [6], the Elliptic Lindstedt–Poincare method (LP) [7], [8], Adomian Decomposition Method [9], [10] and the Krylov Bogolioubov Mitropopolsky Method (KBM) [11]. Another analytical method called Differential Transformation Method or DTM [12]. Also, Variational Iteration Method (VIM) which was first proposed by He [13], [14] can be considered as a suitable approximation method. Moreover, VIM has successfully been applied to solve various engineering applications. For instance, the classical Blasius equation was solved by using VIM and also this method was utilized to give approximate solutions for some well-known non-linear problems [15]. After that He's Amplitude Frequency Formulation method [16], [17], which was first presented by Ji-Huan He, has been attracted much attention in last decade. In addition, Exp-function method can be applied in the field of vibration successfully like the research done by Ganji et al. [18], [19]. The response of arched beam bridges to dynamic loads is one of the most determining factors in the useful life of bridges. Various forms of dynamic loads on arched bridges are exerted according to their characteristics. Also its modified designing can help us to increase the chance of reliability and economical point of view. Dynamic of bridges is one of the most attractive subjects amongst engineers, for instance the response of bridge elements to the traffic load. Another issue facing researchers is the influence of harmonic loads or moving loads on the arched bridges. However, it is important to solve the nonlinear differential equations governing on the arched bridges in order to analyze the physical behavior of this kind of structures in spite of its complexity. In this paper, the nonlinear differential equation of an arched beam with a harmonic load in different modes of vibration and force transmission to the foundation of the system are investigated. Furthermore, the differential equation governing on this system has been solved analytically by AGM (Akbari-Ganji's Method) [20], [21], and 22]. On the basis of the obtained results by the afore-mentioned methods in different fields of study particularly in mechanical and civil engineering, it is necessary to mention that the above methods do not have this ability to gain the solution of the presented problem in high precision and accuracy, so nonlinear partial differential equations such as the presented problem in this case study should be solved by utilizing new approaches like AGM [23], [24].

Section snippets

The analytical method (Akbari-Ganji's method (AGM))

In general, vibrational equations and their initial conditions are defined for different systems as follows: f(u¨,u˙,u,F0sin(ω0t))=0

Parameter (ω0) angular frequency of the harmonic force exerted on the system and (F0) the maximum amplitude of its. And initial conditions as follows: {u(t)=u0,u˙(t)=0,att=0}

Application

Consider an arched shaped beam with a hinged support and fixed end as a suitable model for a bridge with certain physical properties such as bending stiffness EI, cross sectional area A, density ρ, length L and a live harmonic load in the middle of the beam WL (center outlet e) in Fig. 1.

It is citable that a concentrated sinusoidal force f(x, t) is exerted on the center of the vibrational system. Based on the given information, the nonlinear partial differential equation governing on the

Result and discussion

Results of difference between AGM and numerical method (Figs. 14 and 15), depict the difference between AGM and numerical method is worthless which demonstrates the accuracy of this technique. According to these figures, it is clear that AGM is an acceptable procedure for solving nonlinear differential equations and the charts of AGM and Numerical solutions are overlapped which show the precision of the AGM at the solution of nonlinear equations. Therefore, using AGM technique and Considering

Conclusions

In the present paper, the physical behavior of arched bridges has been analyzed by utilizing a simplified model as an arched beam. In order to accomplish the above procedure, the nonlinear differential equation governing on the proposed model has been assessed and analyzed analytically for the first time by a combination of Akbari-Ganji's Method (AGM) and the separation method. Then the obtained solution has been compared with numerical solution and the results indicate the precision of the

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