Sunday, 18 December 2022

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory

 

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory


Abstract

In this study, the spectral element method to assess free vibration of axially-loaded functionally-graded beams according to the first-order shear deformation theory is developed. In this approach, the underlying equations of motion and related boundary conditions were determined by using Hamilton's principle. Analytical solutions are presented for simply-simply, clamped-clamped, clamped-free, and clamped-simply supported FG beams. The general solutions corresponding to governing equations determined for axially loaded FG beams are presented in the spectral-element matrix. A comparison is performed to validate the proposed formulation and solution between the obtained results with the existing solutions provided in previous studies. Finally, the parametric study is developed to investigate the impacts of slenderness ratio, end supports, gradient parameter, and axial load value on the free vibration characteristics in the axially-loaded FG beams. A comparison of the results of the developed theory and the results of the existing solutions shows that the proposed solution developed based on the spectral element method is the appropriate accuracy and efficiency in determining the dimensionless natural frequencies parameter. Based on the findings, all four parameters are effective in the free vibration of axially-loaded FG beams.

Introduction

The mechanical characteristics of functionally-graded materials (FGMs) change gradually along with the given directions (Gorji Azandariani et al., 2022; Rajasekaran et al., 2022; Yang et al., 2021). FGMs have begun to find their way into beams due to technological developments and achievements. Considering the broad potential of FG beams in engineering applications, it is of proper great importance to better understand their mechanical behavior to develop these kinds of structures. Consequently, buckling and vibration problems of FG beams have been investigated using various numerical and analytical approaches according to different beam theories (Dangi et al., 2021; Darban et al., 2021; Gorji Azandariani et al., 2021). Also, stability and free vibration problems of FG nanobeam/nanoplate have been investigated using various numerical and analytical approaches according to different beam theories (Ji et al., 2020; Luo et al., 2021; Shen et al., 2020; Van Vinh and Tounsi, 2021, 2022; Vinh, 2022; Vinh et al., 2022).

Aydogdu and Taskin (2007) explored the free vibration of a simply-supported (S–S) FG beam based on classical, parabolic, and exponential shear deformation theories. Sina et al. (2009) provided a new beam theory to study the free vibration of FG beams. These authors set the normal lateral stress of the beam to zero. In a study done on the FG beams, the bending and free vibration problems were investigated using different higher-order shear deformation theories. These developed theories describe the higher-order variations that occurred in the transverse shear strain through the depth of the beam (Thai and Vo, 2012). In another work performed by some scholars, a new first-order shear deformation beam theory (FSDT) was developed for solving the free vibration problem in axially-loaded rectangular FG beams. This study utilized the in-plate stress equilibrium equations to extract the transverse shear stiffness (Nguyen et al., 2013).

Furthermore, the Lagrange multiplier strategy was also employed for evaluating the free vibration problem in the FG beams according to different higher-order beam theories (Şimşek, 2010a, 2010b, 2010c; Şimşek and Kocatürk, 2009). The vibration and buckling behavior of FG sandwich beams have also been analytically investigated by employing the higher-order beam theory developed by Nguyen et al. (2015). A hyperbolic distribution is assumed for transverse shear stress. In two other studies done by the same research group, an efficient and simple analytical method was developed to analyze the dynamic/static behaviors in the FG beams using the theory of elasticity (Li, 2008; Li et al., 2010; Nguyen et al., 2021). The thermoelastic behavior of FG beams was studied using an FSDT-based beam element proposed by Chakraborty et al. (2003). The vibration of axially-loaded FG Timoshenko non-uniform beams was investigated using the novel technique proposed by Huang et al. (2013). In addition to the approaches mentioned above, a new two-node six-degree-of-freedom beam element was also developed to assess the free vibration problem in FG beams, along with a finite element model, constructed to dynamically analyze layered FG beams according to the third-order zigzag theory (Alshorbagy et al., 2011; Kapuria et al., 2008). In two different studies, the classic Euler-Bernoulli theory and Timoshenko beam theory were applied to develop a dynamic stiffness matrix (DSM) for investigating the free vibration problem in the FG beams (Phi et al., 2021; Su et al., 2013; Su and Banerjee, 2015). There have been many studies on the FG beams beyond this article's scope (Barretta et al., 2016; Barretta and Luciano, 2015; Reddy et al., 2020; Vo et al., 2015; Wang, 2021).

According to the studies conducted on the free vibration in the metallic beams, the axial force significantly affects the mode shapes and natural frequencies (Cheng and Tseng, 1973; Howson and Williams, 1973; Lai et al., 2012; Niknam et al., 2014); thus, it needs to be considered in beam vibration analysis. However, the data reported in the literature on the axially-loaded FG beams is still very limited. Kang and Li (2009) presented the Euler-Bernoulli beam model for free vibration of axially-loaded FG beams under S–S boundary conditions. Moreover, Trinh et al. (2016) developed an analytical solution for vibration and buckling of FG beams with various end supports subjected to mechanical loads. Sui et al. (2015) investigated the transverse free vibration of an axial moving beam made of FGM using Timoshenko's beam theory. They studied and evaluated the natural frequencies, vibration modes, and critical velocities of axial moving beams made of FGM. They also used the Hamilton principle to derive the governing equation and used a sophisticated fashion approach to achieve transverse dynamic behaviors such as vibration modes and natural frequencies. In this study, the critical velocity is determined numerically, and its changes are plotted based on the power law, the initial axial stress, and the length-to-thickness ratio. Yao et al. (2020) studied the transverse free vibration and wave propagation of FG microbeams with an axial motion based on a nonlocal theory and the Timoshenko beam model. Yao et al. (2020) hypothesized the properties of FG microbeam performs vary in thickness. Also, the studied parameters were the effects of gradient index, nonlocal parameter, and axial velocity on natural frequencies. In addition, the wave propagation properties of the Timoshenko microbeam were analyzed by FG and studied the significant effects of wave number and other variables on wave propagation frequencies and wave velocities. Zhu et al. (2021) presented vibrational analyzes on axial moving FG nanoplates exposed to hydrothermal environments and developed the governing equation of motion based on the Hamilton principle using nonlocal strain gradient theory. Their studies show that with increasing nonlocal parameters, gradient index, temperature change, humidity concentration, and axial velocity, vibration frequencies have decreased. Also, with increasing the characteristic parameter scale of materials and the aspect ratio, the frequencies have also increased.

According to the literature review, it is clear that several numerical and analytical studies have been performed to analyze the free vibrations of FG structures, including beams, plates, and shells. Although a large number of studies have been performed on the free vibration analysis of FG beams, on the other hand, the development of the spectral element method for free vibration of axial load grading functional beams using first-order shear deformation theory has not been investigated in studies. The spectral element method (SEM) exactly reflects the dynamic behavior of the structural elements since it is developed using the exact frequency-dependent dynamic shape functions satisfying the underlying equations of motion (Gopalakrishnan et al., 1992; Lee and Jang, 2010). Also, SEM, in contrast to the traditional finite element method, allows one to represent the entire uniform structural member as a single element, whatever its length, with no need for dividing the structural members into several fine elements for enhancing the accuracy of the solution. Consequently, the overall number of degree-of-freedoms applied in the dynamic analysis and the costs of the computations are reduced significantly. To the best of the authors' knowledge, no publication that uses the spectral element method to evaluate the free vibration of axial load-bearing functional calibration beams using first-order shear deformation theory is available. Hence, this issue has not been well studied in the literature and needs further study.

The authors know that the vibration problem in the FG beams has not been investigated using the SEM to date. The current study develops the SEM to evaluate the free vibration problems in the axially loaded FG beams based on the first-order shear deformation theory (FSDT). The underlying equations of motion and the corresponding boundary conditions are first determined by employing Hamilton's principle. The general solutions of the underlying equations of motion are then used to formulate the spectral-element matrix. Next, the proposed SEM's accurate performance is evaluated by comparing the captured vibration frequencies with the vibration frequencies reported in previous studies. Finally, the effect of various parameter variations, including material property gradient parameter, slender ratio, and boundary conditions, on the vibration frequencies of FG beams is explored in detail.

Section snippets

Underlying differential equations

For this section, a rectangular FG beam with a length of L, a width of b, and a height of h are considered (Fig. 1). The upper surface of the beam is made of ceramic, while its lower surface is made of metal. Moreover, to simplify the calculations, Poisson's ratio ν is considered constant. Furthermore, the following equations define the Mass density ρ and Young's modulus E (Su and Banerjee, 2015):E(z)=(Ec−Em)(zh+12)k+Emρ(z)=(ρc−ρm)(zh+12)k+ρmwhere c is the ceramic constituent, m is the metallic 

Numerical method and verification

In this section, the proposed method is verified using the obtained results with the existing solutions provided in previous studies. Two comparative studies were carried out as a means to evaluate the reliability of the presented approach. In Table 1, the dimensionless frequencies of the S–S FG beam are compared with the results of Nguyen et al. (2013), which were obtained using an analytical method according to the FSDT. In Table 1, the effects of the axial force on the natural frequencies

Parametric results

This section deals with a parametric study using the presented method of different parameters involved in the FG beams and boundary conditions. In this section, the effects of end support, slender ratios L/h, axial force N0, and gradient index k on the vibration frequencies of the FG beams will be evaluated. The materials used in the upper and lower surfaces of the FG beams and the cross-sectional dimensions are the same as in Ref. (Su and Banerjee, 2015). Eq. (34) is used to non-dimensionalize 

Conclusions

The current paper develops the spectral-element method SEM to evaluate free vibration problems in axially-loaded FG beams. The underlying partial differential equations and the corresponding boundary conditions were determined by employing Hamilton's principle. The general solutions acquired for the underlying equations of axially-loaded FG beams form the basis for developing the spectral element matrix. The natural frequencies were determined using the W–W algorithm as a solution approach. To

Authors contribution statements

Mojtaba Gorji Azandariani: Conceptualization, Investigation, Methodology, Project administration, Writing – original draft, Writing – review & editing. Mohammad Gholami: Data curation, Formal analysis, Methodology, Writing – original draft, Visualization. Elnaz Zare: Data curation, Formal analysis, Visualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (48)

There are more references available in the full text version of this article.

Thanks to 
a
Centre for Infrastructure Engineering, Western SydneyUniversity, Penrith, Australia
b
Department of Civil Engineering, Yasouj University, Yasouj, Iran

Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer

http://civilbaselife.blogspot.com


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