Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

Wang, Xiancheng; Li, Wei; Yao, Jianjun; Wan, Zhenshuai; Fu, Yu; Sheng, Tang

doi:https://doi.org/10.1155/2019/7527148

Mathematical Problems in Engineering

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Abstract Introduction Materials and Methods Discussion Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 7527148 | https://doi.org/10.1155/2019/7527148

Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

Xiancheng Wang,^1,2Wei Li,¹Jianjun Yao,³Zhenshuai Wan,³Yu Fu,³and Tang Sheng³

Academic Editor: Francesco Tornabene

Received09 Jan 2019

Accepted20 Feb 2019

Published13 Mar 2019

Abstract

In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.

1. Introduction

The functionally graded (FG) materials are widely used in aerospace, automobile, and civil engineering due to continuous variation of material properties along the thickness direction [1–7]. As is known to us, the FG shallow shells-structures, i.e., FG plate, FG circular cylindrical shallow shell, FG spherical shallow shell, and FG hyperbolic paraboloidal shallow shell, have attracted considerable attention for its high strength and stiffness [8–12]. It is noticed that the FG shallow shells are unavoidably suffered from dynamic loads, which can lead to fatigue wear and structural damage [13–15]. Therefore, it is essential to study the free vibration characteristics of FG shallow shell structures.

Recently, extensive research efforts focused on vibration of FG shallow shells have been made by various shell theories, such as classical shallow shell theory (CSST) [16], first-order shear deformation theory (FSDT) [17], and higher-order shear deformation theory (HSDT) [18]. Furthermore, numerous calculation methods, i.e., Ritz method, finite element method (FEM), generalized differential quadrature method, and wave propagation approach, have also been developed [19–27].

Most of the above researches are limited to the classical boundary restraints (clamped, free, simply supported, and shear-diaphragm supported), which requires constant modification of the solution procedure according to the variation of the boundary restraints [28–30]. However, in practical engineering applications, the boundary restraints are not always in certain classical case. There are many possible boundaries such as nonuniform boundaries, elastic edge boundaries, and point-supported boundaries [31–34]. Consequently, an efficient and accurate method for vibration analysis of FG shallow shells with general boundary restraints including classical and elastic edge boundaries is needed. On the other hand, among the FG shallow shells, the existing results of FG sandwich shallow shells (FGSSS) are too scarce for the comparative studies and engineering applications [35–37]. Talebitooti et al. [38] studied the sound transmission across FG laminated sandwich cylindrical shells by using three-dimensional elasticity theory. Hao et al. [39] investigated the nonlinear forced vibrations and natural frequency of FG doubly curved shallow shells with a rectangular base based on the third-order shear deformation theory. Sofiyev [40] presented a modified form of FSDT to solve the bucking problem of FG sandwich truncated conical shells. Trinh and Kim [41] employed the fourth-order Runge-Kutta method to obtain analytical closed-form solutions for thin FGSSS with double curvature testing on elastic bases.

Therefore, this paper presents an exact method for the free vibration of FGSSS on Winkler and Pasternak foundations based on first-order shear deformation shallow shell theory (FSDSST). Regardless of boundaries, the standard double Fourier cosine series and several closed-form supplementary functions are introduced to describe the displacement and rotation components of FGSSS so as to eliminate the possible jumps and boundary discontinuities. The current results are checked by comparing with those results published in other literature. Numerous new vibration results for FGSSS with different curvature types, geometrical parameters, and boundary restraints resting on Winkler and Pasternak foundations are presented. The results show that vibration frequencies of FGSSS are strongly influenced by the boundaries. Furthermore, the effect of inertia, shear deformation, and foundation coefficients on free vibration characteristic is comprehensively investigated by comparing FSDSST with CSST.

2. Materials and Methods

2.1. Model Description and Material Properties

The basic configuration of FGSSS in rectangular planform is depicted in Figure 1. Herein, the length, width, and total thickness are represented by a, b, and h, respectively. The FGSSS considered here are characterized by the middle surface, which can be obtained by [45]where and denote the radii of curvature in the directions of x and y, respectively. is the corresponding radius of twist. In this work, we set , , and as constants. The x and y coordinates are parallel to boundaries so that is infinity. Herein, three types of independent springs including translational, rotational, and torsional springs are used to realize the given boundaries of FGSSS by setting the stiffness of the springs (x=0, a, y=0, b) as various values, which are , , , , and , respectively. For instance, the free boundary can be easily generated by setting the spring stiffness values into zero, and the stiffness values are set to infinite (a larger number, 10¹⁵ N/m) to achieve a clamped boundary restraint. On the other hand, for the elastic foundations, and are denoted as linear Winkler foundation and Pasternak foundation coefficients, respectively.

In this analysis, different curvature types of FGSSS models, namely, plate (), circular cylindrical shell (, ), spherical shell (, ), and hyperbolic paraboloidal shell (, ), are shown in Figure 2.

(a)

(b)

(c)

(d)

Typically, FG layers of shallow shell are made from a mixture of metal and ceramic materials in different proportions. The effective material properties are assumed to vary continuously through the thickness and can be obtained: where E, ρ, and μ represent Young’s modulus, mass density, and Poisson ratio, respectively, and the subscripts c and m represent the ceramic and metal phase, respectively. is the volume fraction of ceramic constituent. As shown in Figure 3, the representative FGSSS are considered in this work: Type 1 is composed of FG face sheets and homogeneous middle layer (Figure 3(a)); Type 2 has homogeneous face sheets and FG middle layer (Figure 3(b)). of FG sandwich shallow shell in the thickness direction is expressed aswhere the subscripts , , and are the gradient index used to determine the FG materials and only take nonnegative values. Typical values for metal and ceramic used in the FG layer of shallow shells are listed in Table 1. Type and Type FGSSS are composed of FG face sheets and homogeneous middle layer, while Type and Type FGSSS are composed of homogeneous face sheets and FG middle layer. It is found that the bilayered FGSSS can be acquired assigning appropriate ratios of thickness for each layer. And the thickness of each layer from bottom to top is expressed by the combination of three numbers; for instance, “” denotes that h₁: h₂: h₃=2:3:2. To describe the behavior of (3a) and (3b), the volume fractions in the direction of thickness for the FGSSS with various gradient index p are shown in Figure 4.

(a)

(b)

(c)

(d)

2.2. Stress-Strain Relations and Stress Resultants

To describe the shell clearly, u, v, and represent the displacements in the x, y, and z directions, and t is the time variable, respectively. The assumed displacement field for the FGSSS based on the FSDSST can be given bywhere and represent the rotations of the reference surface to the y and x axes, respectively. For the FGSSS the strains can be defined aswhere the normal and shear strains in the directions of x, y, and z are and . and are transverse shear strains. According to generalized Hooke’s law [46], the stress-strain relations of the shallow shells can be written aswhere the material elastic stiffness coefficients are defined in terms of the materials properties as

It should be noted that . By carrying the integration of stresses over the plate thickness, the force and moment resultants are obtained aswhere , , are the normal and shear forces, , , and are the bending and twisting moments. , are the transverse shear forces. Performing the integration in (9a), (9b), and (9c) yields where denotes the shear correction coefficient and is usually taken as 5/6. The stiffness coefficients , , and can be obtained as

2.3. Energy Functions and Governing Equations

A modified Fourier version based on Rayleigh-Ritz method is performed. The Rayleigh-Ritz method is a powerful tool in the field of vibration analysis, in which the undetermined coefficients in the displacement can be obtained by minimizing the Lagrangian energy function expression or making them equal to zero [47]. Then, a series of equations can be summed up in matrix form as a standard characteristic equation, which can be easily solved to obtain the desired frequencies and modes. The Lagrangian energy function of the FGSSS can be written as

The strain energy for the FGSSS during vibration is given in an integral form by

Substituting (5), (6), and (10)-(12) into (15), it can be written in terms of displacements and rotations components as

The kinetic energy T, external work , and deformation strain energy U_sp of the FGSSS can be seen in [48]. Additionally, the strain energy based on the Winkler and Pasternak foundations is

The governing equations for the FGSSS can be written in matrix form by combining (5), (6), and (10)-(12) based on Hamilton’s principle:

The coefficients of the linear operator are given in Appendix A.

2.4. Admissible Displacement Functions

In this subsection, the free vibration of FGSSS on the Winkler and Pasternak foundation with general boundary restraints are considered. The displacement and rotation components of FGSSS consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to remove the potential jumps and boundary discontinuities, which can be expressed aswhere , . M and N are truncation numbers with respect to variables x and y. , , , , and represent the Fourier expansion coefficients of the cosine Fourier series, respectively. , , , , , , , , , and are the corresponding supplement coefficients. The auxiliary polynomial functions and are introduced to remove all the potential discontinuities associated with the first-order derivatives at the boundaries, which can be expressed as follows:

It can be verified that

Alternately, all the expansion coefficients in (19)-(23) are treated equally and independently as the orthogonal coordinates and solved from the Rayleigh-Ritz method. The vibration characteristic equation can be summed up:where the coefficient eigenvector is the unknown expansion coefficient that appears in the series expansions and can be determined by

K and M represent the stiffness and mass matrix of the FGSSS, respectively. Both of them are matrices and can be expressed as [48]

The stiffness matrix K and mass matrix M are given in Appendix B.

3. Numerous Results and Discussion

In this section, a comprehensive free vibration analysis for FGSSS on Winkler and Pasternak foundations with general boundary restraints is presented. Firstly, the current results are checked by comparing with those results published in other literature. Then, the vibration behaviors for FGSSS on Winkler and Pasternak foundations with various curvature types, distribution types, geometrical parameters, and boundary restraints are studied. Finally, the influence of vibration parameters including inertia, shear deformation, and foundation coefficient on the free vibration is illustrated.

3.1. FGSSS with General Boundary Restraints

In the engineering applications, the letters F, C, S, and SD represent the completely free, completely clamped, simply supported, and shear-diaphragm supported boundary restraints, respectively. Besides the aforementioned classical boundary restraints, three elastic boundary restraints, denoted by E¹, E², and E³, are considered in this paper. Taking edge at =0, for example, the seven boundaries and corresponding restraint parameters are shown in Table 2.

The reference bending stiffness . The accuracy and reliability of the current results are displayed in Tables 3 and 4. Table 3 compares the nondimensional frequency parameters of simply supported square FG sandwich plates with FG face sheets and homogeneous middle layer (1-0-1, 1-1-1, 1-2-1, 2-1-2, 2-2-1, and 1-8-1). The thickness-length ratios are taken to be h/a=0.01, 0.1, and 0.2. The gradient indexes are given as p=0.5, 5, and 10. Young’s modulus, mass density, and Poisson ratio are obtained in Table 1. The solutions given by Li et al. [42], by the 3D solutions and those of Hadji et al. [43], and by the RPT are provided for the comparisons. The differences are less than 1.530% and 2.015% for the worst case, respectively.

Meanwhile, The frequency parameters (, ) for Type and Type FGSSS with completely free boundary restraints are presented in Table 4. All layers are assumed to be of equal thickness. The geometrical parameters are given as a=b and h/b=0.1. The length-radii ratios used for this analysis are a/R=0.2 and 0.5, and the truncation number is M=N=12, 13, and 14. The frequency results are compared with the solutions given by Jin et al. [44]. A well agreement can be obtained.

Numerous new results for fundamental frequencies f (Hz) are presented in Table 5 for Type FGSSS with a variety of general boundary restraints. The gradient index and geometrical parameters are given as p=5, a/b=1, h/a=0.1, R_x/a=2 and 5. And the thickness for each layer is set to be equal. Young’s modulus, mass density, and Poisson ratio are showed in Table 1. The boundary restraints including SSSS, CCCC, CFCF, SCSC, CSDCSD, CE¹E¹E¹, SE²SE², and E³E³E³E³ are considered. It is obvious that the values of fundamental frequencies corresponding to different boundaries are quite sensitive to the change of geometrical parameters. The frequencies of the FGSSS increase with the increase of radius ratio (R_x/R_y) and the decrease of radius-length ratio (R_x/a). The first four mode shapes for the plate, circular cylindrical shallow shell, and spherical shallow shell with CCCC, SSSS, and SE²SE² boundaries are depicted in Figures 5, 6, and 7, respectively. The radius-length ratio R_x/a is set as 2.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

3.2. FGSSS Resting on Winkler and Pasternak Foundations

In this subsection, the free vibration of Type (1-1-1) FGSSS on Winkler and Pasternak foundations with general boundary restraints is investigated. The geometric parameters and gradient index of the models are a=b=1 m, h/a=0.1, R_x/a=2, p=0.5 and 5. Winkler coefficient and Pasternak coefficient K_S are taken to be 10 and 100. Young’s modulus, mass density, and Poisson ratio of the structures are obtained in Table 1. As known in Table 6, numerous new solutions of fundamental frequencies f (Hz) for the FG sandwich plates, circular cylindrical, spherical, and hyperbolic paraboloidal shallow shells on Winkler and Pasternak foundations with various boundaries (SSSS, CCCC, CFCF, SCSC, CSDCSD, CE¹E¹E¹, SE²SE², and E³E³E³E³) are presented. These results can be used as the benchmark solutions for future research in this field. From the results, it is clear that the variation of the Winkler and Pasternak coefficients has the significant effect on frequencies of the shallow shells. In order to further explore the influence of foundation coefficients and K_S on the frequency parameter of the FGSSS, the detailed parametric study will be illustrated in the next subsection.

3.3. Studies on Free Vibration Parameters

In this subsection, the influence of vibration parameters including inertia, shear deformation, and foundation coefficient on the free vibration is illustrated. For CSST, the inertia and shear deformation are not considered in solving the vibration characteristic of FG shallow shells. Consequently, the modified FSDSST is addressed to tackle above problems.

The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported Type FGSSS with various thickness ratios and curvature types is shown in Figure 8. The gradient index and geometric parameters used are p=2, a/b=1, and R_x/a=2. It is clear that the difference of frequency parameters grows with the increase of thickness ratios for all curvature types. Mode shape of (2, 2) displayed larger errors than that of (1, 1), (2, 1), and (1, 2). Hence, CSST is not suitable for the FGSSS due to the important role of inertia and shear deformation in frequency calculation of high-order mode. Furthermore, the difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FG spherical shallow shells with different thickness ratios and anisotropic degrees are plotted in Figure 9. It can be seen that the difference of frequency parameters increases with the increase of anisotropic degrees, which suggests that the effect of inertia and shear deformation is magnified when anisotropic degrees are taken to be larger values. However, due to the influence of mode shape and anisotropic degrees, the difference of frequency parameters for some FG spherical shallow shells may show a decreasing trend after reaching a certain thickness ratio. Furthermore, the variations of frequency parameters Ω versus the foundation coefficients for simply supported Type (1-1-1) FGSSS are presented in Figure 10. It can be easily obtained that the change in frequencies parameters is very small when the Winkler and Pasternak foundation coefficients are less than 10⁵, while when the values of Winkler and Pasternak foundation coefficients are between 10⁵ and 10⁹, the frequencies parameters increase rapidly, which is a sensitive range. In the case of greater than 10⁹, the effect of Winkler and Pasternak foundation coefficients on the frequencies parameters Ω can be neglected.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

4. Conclusions

This paper presents a modified Fourier method for free vibration of FGSSS on Winkler and Pasternak foundations based on FSDSST. Vibration characteristics of the shallow shells have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components are described as a combining form of standard double Fourier cosine series and several closed-form supplementary functions in order to eliminate the potential jumps and boundary discontinuities. The present method displays better reliability and accuracy. Then, numerous new results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are presented, which may be used for benchmark solutions in the future research.

In addition, a comprehensive investigation concentrated on the free vibration characteristics of FGSSS is performed. The results show that the inertia, shear deformation, and foundation coefficients are verified to affect significantly the vibration frequencies of FGSSS. The difference of frequency parameters increases with the increase of anisotropic degrees, which proves that the effect of inertia and shear deformation is magnified when anisotropic degrees are taken to be larger values. And the change in frequencies parameters is small when the Winkler and Pasternak foundation coefficients are less than 10⁵, while when the values of Winkler and Pasternak foundation coefficients are between 10⁵ and 10⁹, the frequencies parameters increase rapidly. In the case of greater than 10⁹, the effect of foundation coefficients on the frequencies parameters can be neglected.

Appendix

A. Linear Differential Operator of the Governing Equation

The coefficients of the linear operator are as follows:

B. Stiffness and Mass Matrix

Submatrices in the stiffness matrix K and mass matrix M are listed as follows.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This project is supported by the Natural Science Foundation of Heilongjiang Province of China (Grant no. E2018019) and the Fundamental Research Funds for the Central Universities (Grant nos. HEUCFP201733 and HEUCFP201814).

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Copyright © 2019 Xiancheng Wang 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.

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