Elsevier

Applied Mathematical Modelling

Volume 92, April 2021, Pages 380-409
Applied Mathematical Modelling

On the use of reproducing kernel particle finite strip method in the static, stability and free vibration analysis of FG plates with different boundary conditions and diverse internal supports

https://doi.org/10.1016/j.apm.2020.11.009Get rights and content

Highlights

  • Static, stability and free vibration analysis of FG plate are carried out.

  • Free vibration of plates with inner point, line and area supports is studied.

  • The finite strip method with longitudinal RKPM functions based on FSDT is used.

  • A decent way is used to impose boundary conditions considering meshfree functions.

  • Elimination of shear-locking, as an undesirable phenomenon, is elaborated.

Abstract

This paper is focused on the application of a novel reproducing kernel particle finite strip method (RKP-FSM) for the static, mechanical and thermal buckling, and free vibration analyses of rectangular functionally graded (FG) plates with different boundary conditions and internal supports. Conventional reproducing kernel particle method (RKPM) is incorporated into the finite strip method (FSM), and the governing equations are derived based on the first-order shear deformation theory (FSDT). As a convenient approach, boundary conditions are imposed using a simple method based on the RKPM correction function and the essence of the FSDT displacement field. Several examples are studied using this method, and the results are compared to those obtained from other numerical methods. The proposed method is also extended to consider different support geometries and complex inner conditions, and the results are validated using COMSOL finite element software. The shear-locking error, which occurs when considering C0 plate theories, is noticeably reduced by considering the RKPM parameters in the finite strip formulation.

Introduction

After introduction of finite element method (FEM) [1], development of novel numerical methods to solve partial differential equations (PDEs) started to attract interest from researchers. The finite strip method (FSM), introduced by Cheung [2], could be considered as an extension of FEM. The FSM enabled a more efficient analysis of plates, especially those with rectangular and parallelogram shapes. However, conventional FSM cannot consider complicated boundary conditions or internal supports. These issues led to development of new types of FSM with more sophisticated longitudinal and transversal shape functions.

Azhari et al. [3] augmented the FSM with bubble functions in the transverse direction to reduce the minimum number of strips required for convergence. Liew et al. [4] incorporated the reproducing kernel particle method (RKPM) shape functions along the strips’ transversal direction into the FSM.

The FSM has also been improved by modifying the shape functions along the strips’ longitudinal direction. Azhari et al. [5] proposed a new kind of harmonic shape functions in the longitudinal direction of strips for plates’ buckling analysis. Cheung and Fan [6] introduced a new FSM that could consider various boundary conditions using new functions instead of harmonic ones in the strips’ longitudinal direction. Khezri et al. [7] proposed the RKP-FSM in which generalized RKPM shape functions were used in the longitudinal direction of the strips. This method was applied to perform the buckling and free vibration analysis of thin plates with abrupt thickness change and continuous inner line support across the plate [7]. The RKP-FSM was also used to assess the behavior of composite laminated plates [8,9].

Moreover, Khezri et al. [10] used both mesh-free and finite strip methods to analyze plates with perforations and cracks. Furthermore, a unified approach was proposed for the mesh-free analysis of thin to moderately thick plates based on a shear-locking free Mindlin theory [11]. Ming et al. [12] used the reproducing kernel method to model cracked curved shells. Also, another mesh-free evaluation of cracked folded structures can be found in [13]. Sadamoto et al. [14] used the RKPM to perform the buckling analysis of cylindrical shells with and without cutouts.

Wang et al. [15] proposed a locking-free stabilized conforming nodal integration for mesh-free Mindlin-Reissner plate formulation. On this subject, another study has been conducted using Galerkin mesh-free method in Ref. [16]. Mousavi et al. [17] presented coupled Galerkin mesh-free in conjunction with FSM for static and buckling analysis of skew plates and plates with an inner hole. Naghsh et al. [18] carried out a thermal buckling analysis of point-supported laminated composite plates using the Galerkin mesh-free method. The boundary element method and FSM were combined by Najarzadeh et al. [19] to perform free vibration and buckling analysis of thin plates subjected to high gradients stresses.

Development of the RKPM was originated from the smoothed particle hydrodynamics (SPH) introduced by Gingold and Monaghan [20] and Lucy [21] for astrophysics applications. Lack of the partition of unity in the SPH led to the absence of rigid body motion capability and deficiency in dealing with boundary conditions. To overcome these shortcomings, Liu et al. [22] introduced the RKPM.

A considerable number of studies have been accomplished to impose boundary conditions in RKPM. Gosz et al. [23] proposed a method based on the correction function of RKPM, which easily provides Kronecker delta property for the shape functions at both ends of the domain (assuming 1-D domain). Moreover, it was noted that using particles with a variable kernel support radius may provide smoother shape functions [23]. Chen et al. [24] presented a transformation technique that provides the Kronecker delta at particles. They used this method for large deformation analysis of rubber. The transformation technique is considered an easy way to impose support conditions even inside the domain. However, this method may cause extra computational costs due to full range shape functions. Sadamoto et al. [25] performed a buckling analysis of stiffened plate structures by an improved mesh-free flat shell formulation, in which the transformation technique by Chen and Wang were used to impose the essential boundary conditions.

Considering the difficulties of enforcing derivative-type boundary conditions using the conventional RKPM, Shodja and Hashemian [26] incorporated the first derivative of function into the RKPM formula named the gradient RKPM. Behzadan et al. [27] developed a generalized form of RKPM called the GRKPM in which any order of function derivatives could participate in the formulation. A generalized form of collocation method was also introduced in Ref. [24] to impose EBCs. The radial basis RKPM has recently been used to analyze geometrically nonlinear problems made of functionally graded materials [28].

Mesh-free methods are widely used for the analysis of plate and shell structures. A finite rotation mesh-free formulation has been applied for analysis of geometrically nonlinear flat, curved, and folded shells in Ref. [29]. Isogeometric analysis (IGA), as a robust mesh-free technique, has been considerably employed for the examination of plates with complex geometries. Some studies which used FSDT to assess buckling and free vibration behavior of composite plates can be found in [30,31].

Furthermore, the discrete singular convolution method (DSC) has been turned into account to analyze plates. This method discretizes the spatial derivatives and reduces the given partial differential equations into a standard eigenvalue problem based on singular convolution formula. Here some examples of using the DSC for free vibration and buckling analysis of isotropic and orthotropic plates can be found [32], [33]–34].

Meanwhile, there was progress in manufacturing and analyzing plates made of newly developed materials named functionally graded materials (FGMs) introduced by the Japanese in the mid-80s [35]. Several advancements have been seen in the mechanical and thermal behaviors of structural members by utilizing FGMs. The general concept of FG plates is a gradual change of materials, especially metal and ceramic, along the plate's thickness.

Plenty of researches have been carried out investigating FGMs. Some of the most relevant are outlined here. Reddy [36] used a finite element model and the Navier solution for analyzing thick FG plates using the third-order shear deformation theory (TSDT) and compared the results with those obtained from FSDT. Also, a three-dimensional elasticity solution for the bending of simply supported FG rectangular plates was proposed by Kashtalyan [37]. Qian et al. [38] carried out static and dynamic studies of rectangular FG plates adopting the meshless local Petrov-Galerkin (MLPG) method and higher-order shear and normal deformable plate theory (HOSNDPT).

Static response analysis of simply supported FG rectangular plates has been accomplished by Zenkour [39], using a generalized shear deformation theory. Nguyen et al. [40] studied FG plates’ static response using the FSDT and determined the transverse shear correction factors for regular FG plates and sandwich panels. Using higher-order shear deformation theory (HSDT), Talha and Singh [41] studied free vibration and static responses of FG plates using continuous isoparametric lagrangian finite element model and 117 degrees of freedom per element. On this subject, Tang et al. [42] performed a nonlinear analysis of FGM plates based on the FSDT and isogeometric analysis (IGA) method.

A thermal buckling study with four types of thermal loading and using Reddy HSDT were accomplished by Javaheri and Eslami [43] to observe material distribution effect on the buckling temperature change of nonhomogeneous plates. Moreover, there are a couple of papers focusing on the analysis of FG plates using the FSM. Sarrami-Foroushani et al. [44] examined the buckling of FG stiffened and unstiffened plates using harmonic shape functions in longitudinal direction. Foroughi and Azhari [45] conducted mechanical and free vibration analysis of thick rectangular FG plates resting on elastic foundation using a B-spline FSM based on TSDT.

Many other methods and theories have been considered for different analysis of plates. Some of them included plates rested on an elastic foundation [46]. Also, a new analytic solution is now used widely for the analysis of plates [47]. Further assessment of plate behaviors from the aspect of theory and geometry can be found in Refs. [48,49].

In the present paper, the conventional form of RKPM, which has less complexity than GRKPM, is incorporated into the FSM. Comprehensive static, free vibration, and stability analyses of rectangular FG plates are conducted based on this method.

As an important application of the proposed method, free vibration analysis of plates with complex internal supports, i.e., different point supports and discontinuous line supports, is carried out. An analytical examination for point support is conducted in Ref. [50].

Furthermore, a fully constrained area has been presumed as interior support, and the results of the employed method in the case of internal supports are compared to the FEM solution. The results demonstrate the high accuracy of the proposed model for considering plates with diverse internal supports, along with a low computational cost. Moreover, additional examples (e.g. an L-shaped plate) with nonuniform RKPM nodes and strips distribution are particularly studied to evaluate the method's capability. Improvement is observed by logical scattering of nodes and strips.

Section 2 describes the theory and formulation of RKPM and RKP-FSM in conjunction with the FSDT. The governing equations for the static, free vibration, and stability analysis of FG plates are developed. Several case studies with different boundary conditions and plate materials are conducted in the results and discussion section. Also, the shear-locking error is evaluated, and a technique is proposed to reduce this error.

Section snippets

Theory

In this section, the mesh-free RKPM is introduced. Also, the governing equations of the FSDT for FG plates based on the power-law distribution of constituents are introduced.

Results and discussion

In this section, the assessment of results based on static, mechanical buckling, thermal buckling, and free vibration of FG plates is carried out using the RKP-FSM. The plate is assumed to have a rectangular shape and composed of ceramic at the top and aluminum at the bottom, and the ratio of the constituents varies along the thickness from pure ceramic to pure aluminum, based on the power-law distribution (see Eq. (18)).

The physical properties of aluminum and ceramic are considered asEm=70GPa,E

Conclusions

  • In this study, the FSM and RKPM 1-D functions were combined to carry out various types of analysis on FG thin and thick plates. Diverse static, stability, and free vibration analysis of homogeneous and nonhomogeneous plates were carried out. A close match was observed between the results obtained by the present method and other numerical methods. Also, a novel technique for simulating internal supports was proposed and applied in the free vibration analysis of plates with different intermediate

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      Nguyen-Thanh et al. [13] presented a coupling method of reproducing kernel particle method (RKPM) and isogeometric analysis for the static, dynamic and buckling analyses of 3D FGM plates and shells. Esfahani et al. [14] introduced a reproducing kernel particle finite strip method (RKP-FSM) for the static, dynamical and thermal buckling analyses of rectangular functionally graded plates with different boundary conditions and internal supports. Chu et al. [15,16] proposed a Hermite radial basis collocation method(HRBCM) for the vibration and buckling analysis of the FGM plates.

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      Since then, lots of numerical methods have been proposed in the literature to account for buckling and vibration of both thin and moderately thick plates. Representative numerical methods include the finite element method [5,6], differential quadrature method [7,8], boundary element method [9], meshless method [10–12], finite strip method [13,14]. Analytical models have also been proposed in the literature.

    • A simple finite element procedure for free vibration of rectangular thin and thick plates

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      The Rayleigh-Ritz method with different types of trial functions such as the Timoshenko beam functions [2], orthogonal polynomials [3–5], polynomial beam functions [6], static beam functions [7,8], B-spline functions [9], Chebyshev polynomials [10], crack functions [11], trigonometric functions [12,13], simple algebraic polynomials [14,15], Legendre polynomials [16], Jacobi polynomials [17] and special corner functions [18] has also been successfully applied to solve various plate problems. Other analytical and/or numerical methods applied in this area include, but are not limited to, the superposition method [19], the differential transformation method [20], different versions of the finite strip method [21–23], the discrete singular convolution method [24,25], the meshfree method [26,27], the spectral stiffness method [28–30], the spectral element method [31,32], the Fourier series method [33–35], the differential quadrature method (DQM) [36–38], the finite integral transform method [39,40], and the finite element method [41–43]. In general, as we mentioned above, the computation of the free vibration responses of the plate can be performed analytically and/or numerically (or approximately).

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