Nonlinear dynamic response and vibration of imperfect eccentrically stiffened sandwich third-order shear deformable FGM cylindrical panels in thermal environments

Abstract

This study follows an analytical approach to investigate the nonlinear dynamic response and vibration of eccentrically stiffened sandwich functionally graded material (FGM) cylindrical panels with metal–ceramic layers on elastic foundations in thermal environments. It is assumed that the FGM cylindrical panel is reinforced by the eccentrically longitudinal and transversal stiffeners and subjected to mechanical and thermal loads. The material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution. Based on the Reddy’s third-order shear deformation shell theory, the motion and compatibility equations are derived taking into account geometrical nonlinearity and Pasternak-type elastic foundations. The outstanding feature of this study is that both FGM cylindrical panel and stiffeners are assumed to be deformed in the presence of temperature. Explicit relation of deflection–time curves and frequencies of FGM cylindrical panel are determined by applying stress function, Galerkin method and fourth-order Runge-Kutta method. The influences of material and geometrical parameters, elastic foundations and stiffeners on the nonlinear dynamic and vibration of the sandwich FGM panels are discussed in detail. The obtained results are validated by comparing with other results in the literature.

Keywords

Nonlinear dynamic response and vibration eccentrically stiffened sandwich FGM cylindrical panels the Reddy’s third-order shear deformation shell theory elastic foundations thermal environment

Introduction

Functionally graded materials (FGMs) are microscopically inhomogeneous made from a mixture of metal and ceramic, and their mechanical properties vary smoothly and continuously from one surface to the other. Functionally graded structures, such as cylindrical panels and cylindrical shells in recent years, play an important part in the modern industries. They are used as primary load-carrying parts in spacecraft and missile structures which are usually exposed to very severe loading conditions. Therefore, globally, researches on static and dynamic stability of these structures have received special attention. Song et al. [1] studied the active vibration control of carbon nanotube-reinforced functionally graded composite cylindrical shell using piezoelectric materials. Mehralian et al. [2] investigated buckling of anisotropic piezoelectric cylindrical shells subjected to axial compression and lateral pressure based on the new modified couple stress theory and using the shear deformation theory with the von Kármán geometrical nonlinearity. Mirzaei and Kiani [3] researched free vibration characteristics of composite plates reinforced with single-walled carbon nanotubes. Sofiyev et al. [4] dealt with the nonlinear vibration of orthotropic cylindrical shells on the nonlinear elastic foundations. Hosseini-Hashemi et al. [5] concentrated on presenting a reliable and accurate method for free vibration of functionally graded viscoelastic cylindrical panel under various boundary conditions. Based on 3D elasticity theory, Norouzi and Alibeigloo [6] carried out thermoviscoelastic analysis of FGM cylindrical panel under thermal and/or mechanical load, while Ahmed [7] investigated how the corrugation parameters and the Winkler foundation affect the buckling behavior of isotropic and orthotropic thin-elliptic cylindrical shells with cosine-shaped meridian subjected to radial loads. Quan et al. [8] focused on the nonlinear dynamic analysis and vibration of shear deformable imperfect eccentrically stiffened functionally graded thick cylindrical panels, taking into account the damping when subjected to mechanical loads. Further, Chen et al. [9] studied the structural–acoustic radiation problem of cylindrical shell structures with complex acoustic boundary conditions. Shen [10] presented modeling and analysis for the postbuckling of carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations subjected to lateral pressure in thermal environments. Recently, Duc et al. [11] investigated the nonlinear buckling and postbuckling of eccentrically stiffened functionally graded thin-elliptical cylindrical shells surrounded on elastic foundations in thermal environment.

A thick structure is defined as a structure with a thickness which is large compared to its other dimensions, but in which deformations are still not large compared to thickness. When studying nonlinear analysis of thick structures, first-order shear deformation theory and higher order shear deformation theory are usually used, in which the effects of transverse shear and normal stresses in structures are taken into account. Despite their complexity, the mechanical behaviors of thick structures, such as bending, vibration, stability, buckling, etc., have attracted the attention of many researchers. Zhou and Zhu [12] utilized the third-order shear deformation plate theory to analyze the vibration and bending of the simply supported magneto-electro-elastic rectangular plates. Sayyaadi et al. [13] presented an analytical solution for power output from a piezoelectric shallow shell energy harvester using higher order shear deformation theory, and Selim et al. [14] studied the free vibration behavior of carbon nanotube-reinforced functionally graded composite plates in a thermal environment based on Reddy’s higher order shear deformation theory. Wattanasakulpong et al. [15] employed an improved third-order shear deformation theory to investigate free and forced vibration responses of FGM plates. Yang and Shen [16] and Huang and Shen [17] dealt with the nonlinear vibration and dynamic response of functionally graded material plates in thermal environments. Amirpour et al. [18] introduced the deformation solution of FGM plates with variation of material stiffness through their length using higher order shear deformation theory including stretching effects. Groh and Weaver [19] discussed the static inconsistencies that arise when modelling the flexural behavior of beams, plates, and shells with clamped boundary conditions using a certain class of axiomatic, higher order shear deformation theory. He et al. [20] presented a finite element formulation based on the classical laminated plate theory for the shape and vibration control of the FGM plates with integrated piezoelectric sensors and actuators. Quan and Duc [21] investigated the nonlinear static and dynamic stability of imperfect eccentrically stiffened FGM higher order shear deformable double-curved shallow shell on elastic foundations in thermal environments. Shao et al. [22] presented a unified formulation which is based on a general refined shear deformation beam theory to conduct free vibration analysis of composite laminated beams subjected to general boundary conditions.

Stiffeners are secondary plates or sections which are attached to structures to stiffen them against out-of-plane deformations. Stiffeners can be critical to the performance of the structures since a properly placed stiffener can increase the capacity of the member it supports. Therefore, in recent years, many investigations have been carried out on the mechanical behaviors of structures which are reinforced by stiffeners. Yang et al. [23] tested eight stiffened square concrete-filled steel tubular stub columns with slender sections of encasing steel and two non-stiffened counterparts subjected to axial compressive load. Xu et al. [24] developed a new effective smeared stiffener method to compute the global buckling load of grid-stiffened composite panels. Turner and Vizzini [25] conducted a study to determine the effect of integral stiffeners on the damage growth and ultimate strength of sandwich panels with impact damage. Tong and Guo [26] investigated the elastic buckling behavior of steel trapezoidal corrugated shear walls with vertical stiffeners. Feng et al. [27] studied the effect of impact damage positions on the buckling and postbuckling behaviors of stiffened composite panels under axial compression. Duc et al. [28] presented an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads. Farahani et al. [29] dealt with an analytical approach of the buckling behavior of an FGM circular cylindrical shell under axial pressure with external axial and circumferential stiffeners. Shen et al. [30] developed a theoretical model to predict sound transmission loss across periodically and orthogonally stiffened composite laminate sandwich structures using the first-order shear deformation theory. Recently, Duc [31] studied the nonlinear dynamic response of higher order shear deformable sandwich functionally graded circular cylindrical shells with outer surface-bonded piezoelectric actuator on elastic foundations subjected to thermo-electro-mechanical and damping loads.

New contribution of the paper is that this is the first investigation that successfully established the modeling and analytical formulations for the nonlinear vibration and dynamic response of eccentrically stiffened sandwich FGM thick cylindrical panels with metal–ceramic layers subjected to mechanical loads in thermal environments using the Reddy’s third-order shear deformation shell theory. The outstanding feature of this study is that both FGM cylindrical panel and stiffeners are assumed to be deformed in the presence of temperature. The nonlinear equations are solved by the Galerkin method and fourth-order Runge-Kutta method.

Modeling of eccentrically stiffened FGM cylindrical panels on elastic foundations

Consider an eccentrically stiffened FGM cylindrical panel with the radii of curvature, thickness, axial length and arc length of the panel are R, h, a and b, respectively. The FGM cylindrical panel is assumed to be rested on elastic foundations (Figure 1). A coordinate system $(x, y, z)$ is established in which $(x, y)$ plane on the middle surface of the panel and z on thickness direction $(- h / 2 \leq z \leq h / 2)$ as shown in Figure 1. The panel is reinforced by eccentrically longitudinal and transversal stiffeners (Figure 2). The width and thickness of longitudinal and transversal stiffeners are denoted by $d_{x}, h_{x}$ and $d_{y}, h_{y}$ , respectively; $s_{x}, s_{y}$ are the spacing of the longitudinal and transversal stiffeners. The quantities $A_{x}, A_{y}$ are the cross-sectional areas of stiffeners, and $I_{x}, I_{y}, z_{x}, z_{y}$ are the second moments of cross-sectional areas and the eccentricities of stiffeners with respect to the middle surface of panel, respectively. E₀ is Young’s modulus of stiffeners. In order to provide continuity between the panel and stiffeners, the stiffeners are made of full ceramic $(E_{0} = E_{c}) .$

By applying the power law distribution, the volume fractions of ceramic and metal of the FGM cylindrical panels are assumed as [21,28]

\begin{matrix} V_{m} (z) = (\frac{2 z + h}{2 h})^{N}, \\ V_{c} (z) = 1 - V_{m} (z) \end{matrix}

(1)

where N is volume fraction index (

0 \leq N < \infty

), subscripts m and c stand for the metal and ceramic constituents, respectively.

It is assumed that the effective properties $\Pr_{eff} (z, T)$ of FGM cylindrical panel, such as the elastic modulus $E (z, T)$ , the mass density $ρ (z, T)$ and the thermal expansion coefficient $α (z, T)$ , vary in the thickness direction z and are determined by linear rule of mixture as

\Pr_{e ff} (z, T) = \Pr_{c} (T) V_{c} (z) + \Pr_{m} (T) V_{m} (z)

(2)

A material property $\Pr$ in equation (2) can be expressed as a nonlinear function of temperature [10,28,32]

\Pr (T) = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T + P_{2} T^{2} + P_{3} T^{3})

(3)

where

T = T_{0} + Δ T,

Δ T

is the temperature increment in the environment containing the cylindrical panel and

T_{0} = 300 K

(room temperature),

P_{0}, P_{- 1}, P_{1}, P_{2}

and P₃ are coefficients characterizing the constituent materials and determined by experiments.

The effective properties in equation (2) of the FGM cylindrical panel are obtained by substituting equation (1) into equation (2) as

\begin{matrix} [E (z, T), ρ (z, T), α (z, T)] = [E_{m} (T), ρ_{m} (T), α_{m} (T)] \\ + [E_{cm} (T), ρ_{cm} (T), α_{cm} (T)] (\frac{2 z + h}{2 h})^{N} \end{matrix}

(4)

where

\begin{matrix} E_{cm} (z, T) = E_{c} (z, T) - E_{m} (z, T), ρ_{cm} (z, T) = ρ_{c} (z, T) - ρ_{m} (z, T), \\ α_{cm} (z, T) = α_{c} (z, T) - α_{m} (z, T) \end{matrix}

(5)

and the Poisson’s ratio is assumed to be constant

ν (z) = v = const

[33,34].

The cylindrical panel–foundation interaction of Pasternak model is given by

q_{e} = k_{1} w - k_{2} \nabla^{2} w

(6)

where

\nabla^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

, w is the deflection of the cylindrical panel, k₁ is Winkler foundation modulus and k₂ is the shear layer foundation stiffness of Pasternak model.

Basic equations

In this study, the Reddy’s third order-shear deformation shell theory is used to derive basic equations to investigate the nonlinear dynamic response and vibration of eccentrically stiffened FGM cylindrical panels on elastic foundations in thermal environments.

The normal strains $ɛ_{x}, ɛ_{y}$ , in-plane shear strain $γ_{xy}$ and transverse shear deformations $γ_{xz}, γ_{yz}$ at the distance z from the mid-plane are given as

\begin{matrix} [\begin{matrix} ɛ_{x} \\ ɛ_{y} \\ γ_{xy} \end{matrix}] = [\begin{matrix} ɛ_{x}^{0} \\ ɛ_{y}^{0} \\ γ_{xy}^{0} \end{matrix}] + z [\begin{matrix} k_{x}^{1} \\ k_{y}^{1} \\ k_{xy}^{1} \end{matrix}] + z^{3} [\begin{matrix} k_{x}^{3} \\ k_{y}^{3} \\ k_{xy}^{3} \end{matrix}], \\ [\begin{matrix} γ_{xz} \\ γ_{yz} \end{matrix}] = [\begin{matrix} γ_{xz}^{0} \\ γ_{yz}^{0} \end{matrix}] + z^{2} [\begin{matrix} k_{xz}^{2} \\ k_{yz}^{2} \end{matrix}] \end{matrix}

(7)

where

\begin{matrix} [\begin{matrix} ɛ_{x}^{0} \\ ɛ_{y}^{0} \\ γ_{xy}^{0} \end{matrix}] = [\begin{matrix} \frac{\partial u}{\partial x} + \frac{1}{2} (\frac{\partial w}{\partial x})^{2} \\ \frac{\partial v}{\partial y} - \frac{w}{R} + \frac{1}{2} (\frac{\partial w}{\partial y})^{2} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \end{matrix}], [\begin{matrix} γ_{xz}^{0} \\ γ_{yz}^{0} \end{matrix}] = - 3 c_{1} [\begin{matrix} \frac{\partial w}{\partial x} + φ_{x} \\ \frac{\partial w}{\partial y} + φ_{y} \end{matrix}], \\ [\begin{matrix} k_{x}^{1} \\ k_{y}^{1} \\ k_{xy}^{1} \end{matrix}] = [\begin{matrix} \frac{\partial φ_{x}}{\partial x} \\ \frac{\partial φ_{y}}{\partial y} \\ \frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial y} \end{matrix}], [\begin{matrix} k_{x}^{3} \\ k_{y}^{3} \\ k_{xy}^{3} \end{matrix}] = - c_{1} [\begin{matrix} \frac{\partial φ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial x^{2}} \\ \frac{\partial φ_{y}}{\partial y} + \frac{\partial^{2} w}{\partial y^{2}} \\ \frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial y} + 2 \frac{\partial^{2} w}{\partial x \partial y} \end{matrix}], \\ [\begin{matrix} k_{xz}^{2} \\ k_{yz}^{2} \end{matrix}] = - 3 c_{1} [\begin{matrix} \frac{\partial w}{\partial x} + φ_{x} \\ \frac{\partial w}{\partial y} + φ_{y} \end{matrix}] \end{matrix}

(8)

in which

c_{1} = 4 / (3 h^{2})

and geometrical nonlinearity is incorporated. Also,

u, v

are the displacement components along the

x, y

directions, respectively, and

φ_{x}, φ_{y}

are the rotations of normals to the mid-surface with respect to y and x axes, respectively.

Hooke’s law for an FGM cylindrical panel with temperature-dependent properties is defined as [8,11]

\begin{matrix} (σ_{x}, σ_{y}) = \frac{E (z, T)}{1 - ν^{2}} [(ɛ_{x}, ɛ_{y}) + ν (ɛ_{y}, ɛ_{x}) - (1 + ν) α (z, T) Δ T (1, 1)] \\ (σ_{xy}, σ_{xz}, σ_{yz}) = \frac{E (z, T)}{2 (1 + ν)} (γ_{xy}, γ_{xz}, γ_{yz}) \end{matrix}

(9)

The stress–strain relations of the stiffeners can be given as follows

(σ_{x}^{s}, σ_{y}^{s}) = E_{0} (z, T) (ɛ_{x}, ɛ_{y}) - \frac{E_{0} (z, T)}{(1 - 2 ν_{0})} α_{0} (z, T) Δ T (1, 1)

(10)

In this paper, we assumed that the force and moment resultants of an eccentrically stiffened FGM cylindrical panel under temperature can be constructed by the combination of the integrations of the stresses over the panel thickness and those of corresponding stiffener contributions (including the thermal stress in the stiffeners) as follows

\begin{matrix} (N_{i}, M_{i}, P_{i}) = \int_{- h / 2}^{h / 2} σ_{i} (1, z, z^{3}) d z + \int_{h / 2}^{h / 2 + h_{i}} σ_{i}^{s} (1, z, z^{3}) \frac{d_{i}^{T}}{s_{i}^{T}} d z, i = x, y \\ (N_{xy}, M_{xy}, P_{xy}) = \int_{- h / 2}^{h / 2} σ_{xy} (1, z, z^{3}) d z, (Q_{i}, R_{i}) = \int_{- h / 2}^{h / 2} σ_{iz} (1, z^{2}) d z, i = x, y \end{matrix}

(11)

It is assumed that both material properties and geometrical shapes of stiffeners depend on temperature, and they are deformed in the presence of temperature. After the thermal deformation process, the geometrical parameters of stiffeners can be determined as [8,11,32]

\begin{matrix} d_{x}^{T} = d_{x} (1 + α_{0} Δ T), d_{y}^{T} = d_{y} (1 + α_{0} Δ T), h_{x}^{T} = h_{x} (1 + α_{0} Δ T), h_{y}^{T} = h_{y} (1 + α_{0} Δ T) \\ s_{x}^{T} = s_{x} (1 + α_{0} Δ T), s_{y}^{T} = s_{y} (1 + α_{0} Δ T), z_{x}^{T} = z_{x} (1 + α_{0} Δ T), z_{y}^{T} = z_{y} (1 + α_{0} Δ T) \end{matrix}

(12)

Substitution of equation (7) into equations (9) and (10) and the result into equation (11) yields the constitutive relations as

\begin{matrix} N_{x} = B_{11} ɛ_{x}^{0} + B_{12} ɛ_{y}^{0} + B_{13} k_{x}^{1} + B_{14} k_{y}^{1} + B_{15} k_{x}^{3} + B_{16} k_{y}^{3} - B_{17} Φ_{1} - B_{18} Φ_{1 x}^{s}, \\ N_{y} = B_{12} ɛ_{x}^{0} + B_{22} ɛ_{y}^{0} + B_{24} k_{y}^{1} + B_{14} k_{x}^{1} + B_{16} k_{x}^{3} + B_{26} k_{y}^{3} - B_{17} Φ_{1} - B_{28} Φ_{1 y}^{s}, \\ N_{xy} = B_{31} γ_{xy}^{0} + B_{32} k_{xy}^{1} + B_{33} k_{xy}^{3}, \\ M_{x} = B_{13} ɛ_{x}^{0} + B_{14} ɛ_{y}^{0} + B_{43} k_{x}^{1} + B_{44} k_{y}^{1} + B_{45} k_{x}^{3} + B_{46} k_{y}^{3} - B_{17} Φ_{2} - B_{18} Φ_{2 x}^{s}, \\ M_{y} = B_{14} ɛ_{x}^{0} + B_{24} ɛ_{y}^{0} + B_{44} k_{x}^{1} + B_{54} k_{y}^{1} + B_{46} k_{x}^{3} + B_{56} k_{y}^{3} - B_{17} Φ_{2} - B_{28} Φ_{2 y}^{s}, \\ M_{xy} = B_{32} γ_{xy}^{0} + B_{62} k_{xy}^{1} + B_{63} k_{xy}^{3}, \\ P_{x} = B_{71} ɛ_{x}^{0} + B_{16} ɛ_{y}^{0} + B_{73} k_{x}^{1} + B_{46} k_{y}^{1} + B_{75} k_{x}^{3} + B_{76} k_{y}^{3} - B_{17} Φ_{4} - B_{18} Φ_{4 x}^{s}, \\ P_{y} = B_{16} ɛ_{x}^{0} + B_{82} ɛ_{y}^{0} + B_{46} k_{x}^{1} + B_{84} k_{y}^{1} + B_{76} k_{x}^{3} + B_{86} k_{y}^{3} - B_{17} Φ_{4} - B_{28} Φ_{4 y}^{s}, \\ P_{xy} = B_{33} γ_{xy}^{0} + B_{63} k_{xy}^{1} + B_{93} k_{xy}^{3}, \\ Q_{x} = B_{31} γ_{xz}^{0} + B_{62} k_{xz}^{2}, Q_{y} = B_{31} γ_{yz}^{0} + B_{62} k_{yz}^{2}, \\ R_{x} = B_{62} γ_{xz}^{0} + B_{63} k_{xz}^{2}, R_{y} = B_{62} γ_{yz}^{0} + B_{63} k_{yz}^{2} \end{matrix}

(13)

with the detail of coefficients

B_{1 i} (i = \bar{1, 8}), B_{2 j} (j = 2, 4, 6, 8), B_{3 k} (k = \bar{1, 3}), B_{4 l} (l = \bar{3, 6}),

B_{54}, B_{56,} B_{62}, B_{63}, B_{7 m} (m = 1, 3, 5, 6), B_{8 n} (n = 2, 4, 6), B_{93}

are given in Appendix 1.

The nonlinear motion equations of FGM cylindrical panel are [32]

\frac{\partial N_{x}}{\partial x} + \frac{\partial N_{xy}}{\partial y} = \bar{I_{1}} \frac{\partial^{2} u}{\partial t^{2}} + \bar{I_{2}} \frac{\partial^{2} φ_{x}}{\partial t^{2}} - \bar{I_{3}} \frac{\partial^{3} w}{\partial t^{2} \partial x}

(14a)

\frac{\partial N_{xy}}{\partial x} + \frac{\partial N_{y}}{\partial y} = \bar{I_{1}^{*}} \frac{\partial^{2} v}{\partial t^{2}} + \bar{I_{2}^{*}} \frac{\partial^{2} φ_{y}}{\partial t^{2}} - \bar{I_{3}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y}

(14b)

\begin{matrix} \frac{\partial Q_{x}}{\partial x} + \frac{\partial Q_{y}}{\partial y} - 3 c_{1} (\frac{\partial R_{x}}{\partial x} + \frac{\partial R_{y}}{\partial y}) + c_{1} (\frac{\partial^{2} P_{x}}{\partial x^{2}} + 2 \frac{\partial^{2} P_{y}}{\partial x \partial y} + \frac{\partial^{2} P_{y}}{\partial y^{2}}) \\ + \frac{N_{y}}{R} + q + N_{x} \frac{\partial^{2} w}{\partial x^{2}} + 2 N_{xy} \frac{\partial^{2} w}{\partial x \partial y} + N_{y} \frac{\partial^{2} w}{\partial y^{2}} - k_{1} w + k_{2} \nabla^{2} w = I_{1} \frac{\partial^{2} w}{\partial t^{2}} \\ + 2 ɛ I_{1} \frac{\partial w}{\partial t} + \bar{I_{3}} \frac{\partial^{3} u}{\partial t^{2} \partial x} + \bar{I_{5}} \frac{\partial^{3} φ_{x}}{\partial t^{2} \partial x} + \bar{I_{3}^{*}} \frac{\partial^{3} v}{\partial t^{2} \partial y} + \bar{I_{5}^{*}} \frac{\partial^{3} φ_{y}}{\partial t^{2} \partial y} - c_{1}^{2} I_{7} (\frac{\partial^{4} w}{\partial t^{2} \partial x^{2}} + \frac{\partial^{4} w}{\partial t^{2} \partial y^{2}}) \end{matrix}

(14c)

\begin{matrix} \frac{\partial M_{x}}{\partial x} & + \frac{\partial M_{xy}}{\partial y} - Q_{x} + 3 c_{1} R_{x} - c_{1} (\frac{\partial P_{x}}{\partial x} + \frac{\partial P_{xy}}{\partial y}) = \bar{I_{2}} \frac{\partial^{2} u}{\partial t^{2}} + \bar{I_{4}} \frac{\partial^{2} φ_{x}}{\partial t^{2}} - \bar{I_{5}} \frac{\partial^{3} w}{\partial t^{2} \partial x} \end{matrix}

(14d)

\begin{matrix} \frac{\partial M_{xy}}{\partial x} & + \frac{\partial M_{y}}{\partial y} - Q_{y} + 3 c_{1} R_{y} - c_{1} (\frac{\partial P_{xy}}{\partial x} + \frac{\partial P_{y}}{\partial y}) = \bar{I_{2}^{*}} \frac{\partial^{2} v}{\partial t^{2}} + \bar{I_{4}^{*}} \frac{\partial^{2} φ_{y}}{\partial t^{2}} - \bar{I_{5}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y} \end{matrix}

(14e)

in which q is an external pressure uniformly distributed on the surface of the FGM cylindrical panel, ɛ is the viscous damping coefficient and

\begin{matrix} \bar{I_{1}} = I_{1}, \bar{I_{1}^{*}} = I_{1} + \frac{2 I_{2}}{R}, \bar{I_{2}} = I_{2} - c_{1} I_{4}, \bar{I_{2}^{*}} = I_{2} + \frac{I_{3}}{R} - c_{1} I_{4} - \frac{c_{1} I_{5}}{R}, \\ \bar{I_{3}} = c_{1} I_{4}, \bar{I_{3}^{*}} = c_{1} I_{4} + \frac{c_{1} I_{5}}{R}, \bar{I_{4}} = \bar{I_{4}^{*}} = I_{3} - 2 c_{1} I_{5} + c_{1}^{2} I_{7}, \bar{I_{5}} = \bar{I_{5}^{*}} = c_{1} I_{5} - c_{1}^{2} I_{7}, \\ (I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{7}) = \int_{- h / 2}^{h / 2} ρ (z) (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z \end{matrix}

(15)

The geometrical compatibility equation for an FGM cylindrical panel is written as [32]

\frac{\partial^{2} ɛ_{x}^{0}}{\partial y^{2}} + \frac{\partial^{2} ɛ_{y}^{0}}{\partial x^{2}} - \frac{\partial^{2} γ_{xy}^{0}}{\partial x \partial y} = {\frac{\partial^{2} w}{\partial x \partial y}}^{2} - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} - \frac{1}{R} \frac{\partial^{2} w}{\partial x^{2}}

(16)

From equation (13), one can write

\begin{matrix} ɛ_{x}^{0} = I_{11}^{*} \frac{\partial^{2} f}{\partial y^{2}} - I_{12}^{*} \frac{\partial^{2} f}{\partial x^{2}} + I_{13}^{*} \frac{\partial φ_{x}}{\partial x} + I_{14}^{*} \frac{\partial φ_{y}}{\partial y} - c_{1} I_{15}^{*} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial φ_{x}}{\partial x}) \\ - c_{1} I_{16}^{*} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial φ_{y}}{\partial y}) + I_{17}^{*} Φ_{1} + I_{18}^{*} Φ_{1 x}^{s} + I_{19}^{*} Φ_{1 y}^{s}, \\ ɛ_{y}^{0} = I_{21}^{*} \frac{\partial^{2} f}{\partial x^{2}} - I_{12}^{*} \frac{\partial^{2} f}{\partial y^{2}} + I_{23}^{*} \frac{\partial φ_{x}}{\partial x} + I_{24}^{*} \frac{\partial φ_{y}}{\partial y} - c_{1} I_{25}^{*} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial φ_{x}}{\partial x}) \\ - c_{1} I_{26}^{*} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial φ_{y}}{\partial y}) + I_{27}^{*} Φ_{1} + I_{29}^{*} Φ_{1 x}^{s} + I_{29}^{*} Φ_{1 y}^{s}, \\ γ_{xy}^{0} = - I_{31}^{*} \frac{\partial^{2} f}{\partial x \partial y} + I_{32}^{*} (\frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial x}) - c_{1} I_{33}^{*} (2 \frac{\partial^{2} w}{\partial x \partial y} + \frac{\partial φ_{x}}{\partial y} + \frac{\partial φ_{y}}{\partial x}) \end{matrix}

(17)

in which

\begin{matrix} Δ = I_{11} I_{22} - I_{12}^{2}, I_{11}^{*} = \frac{I_{22}}{Δ}, I_{12}^{*} = \frac{I_{12}}{Δ}, I_{13}^{*} = \frac{I_{14} I_{12} - I_{13} I_{22}}{Δ}, I_{14}^{*} = \frac{I_{24} I_{12} - I_{14} I_{22}}{Δ}, \\ I_{15}^{*} = \frac{I_{16} I_{12} - I_{15} I_{22}}{Δ}, I_{16}^{*} = \frac{I_{26} I_{12} - I_{16} I_{22}}{Δ}, I_{17}^{*} = \frac{I_{22} I_{17} - I_{12} I_{17}}{Δ}, I_{18}^{*} = \frac{I_{18} I_{22}}{Δ}, \\ I_{19}^{*} = - \frac{I_{28} I_{12}}{Δ}, I_{21}^{*} = \frac{I_{11}}{Δ}, I_{23}^{*} = \frac{I_{13} I_{12} - I_{11} I_{14}}{Δ}, I_{24}^{*} = \frac{I_{14} I_{12} - I_{11} I_{24}}{Δ}, \\ I_{25}^{*} = \frac{I_{15} I_{12} - I_{11} I_{16}}{Δ}, I_{26}^{*} = \frac{I_{16} I_{12} - I_{11} I_{26}}{Δ}, I_{27}^{*} = \frac{I_{11} I_{17} - I_{12} I_{17}}{Δ}, \\ I_{28}^{*} = - \frac{I_{18} I_{12}}{Δ}, I_{29}^{*} = \frac{I_{11} I_{28}}{Δ}, I_{31}^{*} = \frac{1}{I_{31}}, I_{32}^{*} = - \frac{I_{32}}{I_{31}}, I_{33}^{*} = - \frac{I_{33}}{I_{31}} \end{matrix}

(18)

and the stress function

f (x, y, t)

is defined as

N_{x} = \frac{\partial^{2} f}{\partial y^{2}}, N_{y} = \frac{\partial^{2} f}{\partial x^{2}}, N_{xy} = - \frac{\partial^{2} f}{\partial x \partial y}

(19)

Imposing equation (19) into equations (14a) and (14b) yields

\frac{\partial^{2} u}{\partial t^{2}} = - \frac{\bar{I_{2}}}{\bar{I_{1}}} \frac{\partial^{2} φ_{x}}{\partial t^{2}} + \frac{\bar{I_{3}}}{\bar{I_{1}}} \frac{\partial^{3} w}{\partial t^{2} \partial x}

(20a)

\frac{\partial^{2} v}{\partial t^{2}} = - \frac{\bar{I_{2}^{*}}}{\bar{I_{1}^{*}}} \frac{\partial^{2} φ_{y}}{\partial t^{2}} + \frac{\bar{I_{3}^{*}}}{\bar{I_{1}^{*}}} \frac{\partial^{3} w}{\partial t^{2} \partial y}

(20b)

Substituting equations (8), (13), (20a) and (20b) into equations (7c) to (7e) leads to

\begin{matrix} L_{11} (w) & + L_{12} (φ_{x}) + L_{13} (φ_{y}) + S (w, f) + q = I_{1} \frac{\partial^{2} w}{\partial t^{2}} + 2 ɛ I_{1} \frac{\partial w}{\partial t} + \bar{\bar{I_{5}}} \frac{\partial^{3} φ_{x}}{\partial t^{2} \partial x} \\ + \bar{\bar{I_{5}^{*}}} \frac{\partial^{3} φ_{y}}{\partial t^{2} \partial y} + \bar{\bar{I_{7}}} \frac{\partial^{4} w}{\partial t^{2} \partial x^{2}} + \bar{\bar{I_{7}^{*}}} \frac{\partial^{4} w}{\partial t^{2} \partial y^{2}} \end{matrix}

(21a)

L_{21} (w) + L_{22} (φ_{x}) + L_{23} (φ_{y}) + L_{24} (f) = \bar{\bar{I_{3}}} \frac{\partial^{2} φ_{x}}{\partial t^{2}} - \bar{\bar{I_{5}}} \frac{\partial^{3} w}{\partial t^{2} \partial x}

(21b)

L_{31} (w) + L_{32} (φ_{x}) + L_{33} (φ_{y}) + L_{34} (f) = \bar{\bar{I_{3}^{*}}} \frac{\partial^{2} φ_{y}}{\partial t^{2}} - \bar{\bar{I_{5}^{*}}} \frac{\partial^{3} w}{\partial t^{2} \partial y}

(21c)

in which

\begin{matrix} L_{11} (w) = D_{11} \frac{\partial^{2} w}{\partial x^{2}} + D_{11} \frac{\partial^{2} w}{\partial y^{2}} + D_{12} \frac{\partial^{4} w}{\partial x^{4}} + D_{13} \frac{\partial^{4} w}{\partial x^{2} \partial y^{2}} + D_{14} \frac{\partial^{4} w}{\partial y^{4}} \\ - k_{1} w + k_{2} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}), \\ L_{12} (φ_{x}) = D_{11} \frac{\partial φ_{x}}{\partial x} + D_{15} \frac{\partial^{3} φ_{x}}{\partial x^{3}} + D_{16} \frac{\partial^{3} φ_{x}}{\partial x \partial y^{2}}, \\ L_{13} (φ_{y}) = D_{12} \frac{\partial φ_{y}}{\partial y} + D_{17} \frac{\partial^{3} φ_{y}}{\partial y^{3}} + D_{18} \frac{\partial^{3} φ_{x}}{\partial x^{2} \partial y}, \\ S (w, f) = \frac{\partial^{2} f}{\partial y^{2}} \frac{\partial^{2} w}{\partial x^{2}} - 2 \frac{\partial^{2} f}{\partial x \partial y} \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} f}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} + \frac{1}{R} \frac{\partial^{2} f}{\partial y^{2}}, \\ L_{21} (w) = D_{21} \frac{\partial w}{\partial x} + D_{22} \frac{\partial^{3} w}{\partial x^{3}} + D_{23} \frac{\partial^{3} w}{\partial x \partial y^{2}}, \\ L_{22} (φ_{x}) = D_{21} φ_{x} + D_{24} \frac{\partial^{2} φ_{x}}{\partial x^{2}} + D_{25} \frac{\partial^{2} φ_{x}}{\partial y^{2}}, \\ L_{23} (φ_{y}) = D_{26} \frac{\partial^{2} φ_{y}}{\partial x \partial y}, L_{24} (f) = D_{27} \frac{\partial^{3} f}{\partial x^{3}} + D_{28} \frac{\partial^{3} f}{\partial x \partial y^{2}}, \\ L_{31} (w) = D_{21} \frac{\partial w}{\partial y} + D_{31} \frac{\partial^{3} w}{\partial x^{2} \partial y} + D_{32} \frac{\partial^{3} w}{\partial x^{3}}, \\ L_{32} (φ_{x}) = D_{33} \frac{\partial^{2} φ_{x}}{\partial x \partial y}, \\ L_{33} (φ_{y}) = D_{21} φ_{y} + D_{34} \frac{\partial^{2} φ_{y}}{\partial x^{2}} + D_{35} \frac{\partial^{2} φ_{y}}{\partial y^{2}}, \\ L_{34} (f) = D_{36} \frac{\partial^{3} f}{\partial x^{2} \partial y} + B_{37} \frac{\partial^{3} f}{\partial y^{3}} \end{matrix}

(22)

and the detail of coefficients

D_{1 i} (i = \bar{1, 8}), D_{2 j} (j = \bar{1, 8}), D_{3 k} (k = \bar{1, 7})

may be found in Appendix 2.

Introducing equation (17) into equation (16) gives the compatibility equation of the eccentrically stiffened FGM cylindrical panel as

\begin{matrix} I_{21}^{*} \frac{\partial^{4} f}{\partial x^{4}} + I_{11}^{*} \frac{\partial^{4} f}{\partial y^{4}} + J_{1} \frac{\partial^{4} f}{\partial x^{2} \partial y^{2}} + J_{2} \frac{\partial^{3} φ_{x}}{\partial x^{3}} + J_{3} \frac{\partial^{3} φ_{x}}{\partial x \partial y^{2}} + J_{4} \frac{\partial^{3} φ_{y}}{\partial y^{3}} \\ + J_{5} \frac{\partial^{3} φ_{y}}{\partial y \partial x^{2}} - c_{1} I_{25}^{*} \frac{\partial^{4} w}{\partial x^{4}} - c_{1} I_{16}^{*} \frac{\partial^{4} w}{\partial y^{4}} + J_{6} \frac{\partial^{4} w}{\partial x^{2} \partial y^{2}} \\ - ({\frac{\partial^{2} w}{\partial x \partial y}}^{2} - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} - \frac{1}{R} \frac{\partial^{2} w}{\partial y^{2}}) = 0 \end{matrix}

(23)

where

\begin{matrix} J_{1} = I_{31}^{*} - 2 I_{12}^{*}, J_{2} = I_{23}^{*} - c_{1} I_{25}^{*}, J_{3} = I_{13}^{*} - c_{1} I_{15}^{*} - I_{32}^{*} + c_{1} I_{33}^{*}, \\ J_{4} = I_{14}^{*} - c_{1} I_{16}^{*}, J_{5} = I_{24}^{*} - c_{1} I_{26}^{*} - I_{32}^{*} + c_{1} I_{33}^{*}, J_{6} = - c_{1} I_{15}^{*} - c_{1} I_{26}^{*} + 2 c_{1} I_{33}^{*} \end{matrix}

(24)

Equations (21) and (23) are nonlinear equations in terms of variables $w, φ_{x}, φ_{y}$ and f and used to investigate the nonlinear vibration and dynamic response of FGM thick eccentrically stiffened cylindrical panels in thermal environments using the Reddy’s third-order shear deformation shell theory.

Nonlinear vibration analysis

In the present study, four edges of the FGM cylindrical panel are assumed to be simply supported and freely movable. The associated boundary conditions are

\begin{matrix} w = N_{xy} = φ_{y} = M_{x} = P_{x} = 0, N_{x} = N_{x 0} atx = 0, a \\ w = N_{xy} = φ_{x} = M_{y} = P_{y} = 0, N_{y} = N_{y 0} aty = 0, b \end{matrix}

(25)

where

N_{x 0}, N_{y 0}

are in-plane compressive loads at movable edges.

The mentioned condition (25) can be satisfied identically if the approximate solutions are represented by [8,21,32]

\begin{matrix} [\begin{matrix} w (x, y, t) \\ φ_{x} (x, y, t) \\ φ_{y} (x, y, t) \end{matrix}] = [\begin{matrix} W (t) sin λ_{m} x sin δ_{n} y \\ Φ_{x} (t) cos λ_{m} x sin δ_{n} y \\ Φ_{y} (t) sin λ_{m} xc os δ_{n} y \end{matrix}] \end{matrix}

(26)

where

λ_{m} = m π / a

δ_{n} = n π / b;

m, n

are odd natural numbers representing the number of half waves in the x and y directions, respectively, and

W (t), Φ_{x}, Φ_{y}

are the time-dependent amplitudes.

By introducing equation (26) into the compatibility equation (23), we define the stress function as

\begin{matrix} f = A_{1} (t) cos 2 λ_{m} x + A_{2} (t) cos 2 δ_{n} y + A_{3} (t) sin λ_{m} x sin δ_{n} y \\ + \frac{1}{2} N_{x 0} y^{2} + \frac{1}{2} N_{y 0} x^{2} \end{matrix}

(27)

with

\begin{matrix} A_{1} = \frac{δ_{n}^{2}}{32 I_{21}^{*} λ_{m}^{2}} W (W + 2 μ h), A_{2} = \frac{λ_{m}^{2}}{32 I_{11}^{*} δ_{n}^{2}} W (W + 2 μ h), \\ A_{3} = Q_{1} W + Q_{2} Φ_{x} + Q_{3} Φ_{y} \end{matrix}

(28)

and

\begin{matrix} Q_{1} = \frac{\frac{λ_{m}^{2}}{R} + c_{1} I_{25}^{*} λ_{m}^{4} + c_{1} I_{16}^{*} δ_{n}^{4} - J_{6} λ_{m}^{2} δ_{n}^{2}}{I_{21}^{*} λ_{m}^{4} + J_{1} λ_{m}^{2} δ_{n}^{2} + I_{11}^{*} δ_{n}^{4}}, Q_{2} = \frac{- (J_{2} λ_{m}^{3} + J_{3} λ_{m} δ_{n}^{2})}{I_{21}^{*} λ_{m}^{4} + J_{1} λ_{m}^{2} δ_{n}^{2} + I_{11}^{*} δ_{n}^{4}}, \\ Q_{3} = \frac{- (J_{4} δ_{n}^{3} + J_{5} λ_{m}^{2} δ_{n})}{I_{21}^{*} λ_{m}^{4} + J_{1} λ_{m}^{2} δ_{n}^{2} + I_{11}^{*} δ_{n}^{4}} \end{matrix}

(29)

Replacing equations (26) and (27) into equation (21) and then applying Galerkin method to the resulting equations yields

\begin{matrix} l_{11} W + l_{12} Φ_{x} + l_{13} Φ_{y} + l_{14} (W + μ h) Φ_{x} + l_{15} (W + μ h) Φ_{y} \\ + [n_{1} - (N_{x 0} λ_{m}^{2} + N_{y 0} δ_{n}^{2})] (W + μ h) + n_{2} W (W + μ h) \\ + n_{3} W (W + 2 μ h) + n_{4} W (W + μ h) (W + 2 μ h) \\ + n_{5} q + n_{5} \frac{N_{y 0}}{R} = I_{0} \frac{\partial^{2} W}{\partial t^{2}} + 2 ɛ I_{1} \frac{\partial W}{\partial t} - λ_{m} \bar{\bar{I_{5}}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - δ_{n} \bar{\bar{I_{5}^{*}}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} \end{matrix}

(30a)

\begin{matrix} l_{21} W + l_{22} Φ_{x} + l_{23} Φ_{y} + n_{6} (W + μ h) + n_{7} W (W + 2 μ h) = \bar{\bar{I_{3}}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - λ_{m} \bar{\bar{I_{5}}} \frac{\partial^{2} W}{\partial t^{2}} \end{matrix}

(30b)

\begin{matrix} l_{31} W + l_{32} Φ_{x} + l_{33} Φ_{y} + n_{8} (W + μ h) + n_{9} W (W + 2 μ h) = \bar{\bar{I_{3}}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} - δ_{n} \bar{\bar{I_{5}}} \frac{\partial^{2} W}{\partial t^{2}} \end{matrix}

(30c)

in which the detail of coefficients

l_{1 i} (i = \bar{1, 3}), l_{jk} (j = \bar{2, 3}, k = \bar{1, 2}), n_{m} (m = \bar{1, 9})

may be found in Appendix 3.

This is the basic equation to determine the nonlinear vibration of an eccentrically stiffened thick FGM cylindrical panel on elastic foundations in thermal environments.

Nonlinear dynamic response

Consider an eccentrically stiffened FGM cylindrical panel acted on by an uniformly distributed transverse load $q = Q sin Ω t$ (Q is the amplitude of uniformly excited load, Ω is the frequency of the load). The system equation (30) has the form as

\begin{matrix} l_{11} W + l_{12} Φ_{x} + l_{13} Φ_{y} + l_{14} (W + μ h) Φ_{x} + l_{15} (W + μ h) Φ_{y} \\ + [n_{1} - (N_{x 0} λ_{m}^{2} + N_{y 0} δ_{n}^{2})] (W + μ h) + n_{2} W (W + μ h) \\ + n_{3} W (W + 2 μ h) + n_{4} W (W + μ h) (W + 2 μ h) \\ + n_{5} Q sin Ω t + n_{5} \frac{N_{y 0}}{R} = I_{0} \frac{\partial^{2} W}{\partial t^{2}} + 2 ɛ I_{1} \frac{\partial W}{\partial t} - λ_{m} \bar{\bar{I_{5}}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - δ_{n} \bar{\bar{I_{5}^{*}}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} \end{matrix}

(31a)

\begin{matrix} l_{21} W + l_{22} Φ_{x} + l_{23} Φ_{y} + n_{6} (W + μ h) + n_{7} W (W + 2 μ h) = \bar{\bar{I_{3}}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - λ_{m} \bar{\bar{I_{5}}} \frac{\partial^{2} W}{\partial t^{2}} \end{matrix}

(31b)

\begin{matrix} l_{31} W + l_{32} Φ_{x} + l_{33} Φ_{y} + n_{8} (W + μ h) + n_{9} W (W + 2 μ h) = \bar{\bar{I_{3}}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} - δ_{n} \bar{\bar{I_{5}}} \frac{\partial^{2} W}{\partial t^{2}} \end{matrix}

(31c)

The nonlinear dynamic response of eccentrically stiffened FGM cylindrical panel can be obtained by solving this equation combined with initial conditions to be assumed as $W (0) = 0, \frac{d W}{dt} (0) = 0$ by using the fourth-order Runge–Kutta method.

Natural frequencies

In the case of $q = 0$ , the natural frequencies of the eccentrically stiffened FGM cylindrical panel can be determined by solving the following equation

\begin{matrix} | \begin{matrix} l_{11} + n_{1} + I_{0} ω^{2} & l_{12} - λ_{m} \bar{\bar{I_{5}}} ω^{2} & l_{13} - δ_{n} \bar{\bar{I_{5}^{*}}} ω^{2} \\ l_{21} + n_{6} - λ_{m} \bar{\bar{I_{5}}} ω^{2} & l_{22} + \bar{\bar{I_{3}}} ω^{2} & l_{23} \\ l_{31} + n_{8} - δ_{n} \bar{\bar{I_{5}}} ω^{2} & l_{32} & l_{33} + + \bar{\bar{I_{3}}} ω^{2} \end{matrix} | = 0 \end{matrix}

(32)

Three angular frequencies of the FGM cylindrical panel in the axial, circumferential and radial directions are determined by solving equation (30) and the smallest one is being considered.

Numerical results and discussion

Validation

Firstly, Table 1 shows the comparison of the natural frequencies

ω (Hz)

for simply supported Al₂O₃/Ti-6Al-4 V square FGM plate

(a = b = 0.4 m, h = 0.005 m)

for the two special cases of isotropy in this paper with the results presented by Yang and Shen [16], Huang and Shen [17] based on Reddy’s higher order shear deformation plate theory and He et al. [20] using the classical laminated plate theory. The material properties are given as:

E_{m} = 105.7 GPa, ν_{m} = 0.2981, ρ_{m} = 4429 {kg / m}^{3}

for Ti-6Al-4 V and

E_{c} = 320.24 GPa, ν_{c} = 0.26, ρ_{c} = 3750 {kg / m}^{3}

for Al₂O₃ _. From Table 1, good agreements are observed for the present solution and the numerical results of three publications.

Table 1.

Comparison of natural frequencies for isotropic square FGM plates ( $a = b = 0.4 m, h = 0.005 m$ ).

Mode sequence	$N = 0$				$N = 2000$
Mode sequence	Yang and Shen [16]	Huang and Shen [17]	He et al. [20]	Present	Yang and Shen [16]	Huang and Shen [17]	He et al. [20]	Present
1	143.96	144.94	144.66	144.82	261.46	271.03	268.92	269.54
2	360.07	362.04	360.53	360.68	653.14	677.04	669.40	672.63
3	360.07	362.04	360.53	360.68	653.14	677.04	669.40	672.63
4	568.88	578.78	569.89	572.44	1044.31	1082.38	1052.49	1075.84
5	718.22	723.06	720.57	722.51	1304.79	1352.24	1338.52	1348.96
6	718.22	723.06	720.57	722.51	1304.79	1352.24	1338.52	1348.96
7	916.40	939.19	919.74	925.17	1694.98	1756.49	1695.23	1783.17
8	916.40	939.19	919.74	925.17	1694.98	1756.49	1695.23	1783.17
9	1207.09	1226.19	1225.72	1225.81	2214.34	2294.47	2280.95	2286.93
10	1207.09	1226.19	1225.72	1225.81	2214.34	2294.47	2280.95	2286.93

Table 2.

Material properties of the constituent materials of the considered FGM panel.

Material	Property	P₀	P₁	P₂	P₃
Si₃N₄ (Ceramic)	E (Pa)	348.43e9	−3.70e-4	2.160e-7	−8.946e-11
	ρ(kg/m³)	2370	0	0	0
	$α (K^{- 1})$	5.8723e-6	9.095e-4	0	0
	$k (W / mK)$	13.723	0	0	0
SUS304 (Metal)	E (Pa)	201.04e9	3.079e-4	−6.534e-7	0
	ρ(kg/m³)	8166	0	0	0
	$α (K^{- 1})$	12.330e-6	8.086e-4	0	0
	$k (W / mK)$	15.379	0	0	0

Secondly, Figure 3 compares the dynamic response of simply supported eccentrically stiffened S-FGM cylindrical panel with metal–ceramic–metal layers under uniform external pressure using the results of Quan et al. [8] in the case of temperature-independent properties. Again, this comparison study also shows that the present result agrees very well the existing result.

Next, we will investigate the effects of the volume fraction index, the geometrical dimensions, elastic foundations, imperfections and stiffeners on the nonlinear response of the eccentrically stiffened FGM cylindrical panel.

The effective material properties with dependent temperature in equation (5) are listed in Table 2, and the Poisson’s ratio is $v = 0.3$ [21,28,32 –34].

The parameters for the stiffeners are [29]

\begin{matrix} s_{1} = s_{2} = 0.4 m, z_{1} = z_{2} = 0.0225 m, \\ h_{1} = h_{2} = 0.003 m, d_{1} = d_{2} = 0.004 m \end{matrix}

(33)

Natural frequencies

Table 3 shows the influences of elastic foundations with two coefficients k₁ and k₂, geometrical parameter

b / h

, volume fraction index N and stiffeners on the natural frequencies of simply supported FGM cylindrical panel in thermal environments. Obviously, the value of the natural oscillation frequency increases when the values k₁ and k₂ increase. Furthermore, the Pasternak elastic foundation influences on the natural oscillation frequency larger than the Winkler foundation. The results of Table 3 also shows that decrease of volume fraction index N and ratio

b / h

leads to increase of the natural oscillation frequencies of FGM cylindrical panel. Moreover, the natural frequency of eccentrically stiffened FGM cylindrical panel is higher than one of FGM cylindrical panels without stiffeners with the same geometrical and material parameters.

Table 3.

Effects of elastic foundations, ratio $b / h$ , volume fraction index N and stiffeners on natural frequencies $(s^{- 1})$ of FGM cylindrical panel in thermal environments.

$\begin{matrix} (k_{1} (GPa / m), \\ k_{2} (GPa . m)) \end{matrix}$	$b / h$	$N = 0$		$N = 1$		$N = 5$
	$b / h$	Stiffened	Un-stiffened	Stiffened	Un-stiffened	Stiffened	Un-stiffened
(0,0)	20	3.079e3	2.971e3	1.930e3	1.722e3	1.725e3	1.465e3
	30	2.237e3	2.001e3	1.581e3	1.161e3	1.489e3	986.9
	40	1.920e3	1.528e3	1.527e3	891.7	1.484e3	753.9
(0.1,0)	20	3.163e3	3.058e3	2.084e3	1.892e3	1.916e3	1.685e3
	30	2.407e3	2.190e3	1.850e3	1.507e3	1.804e3	1.419e3
	40	2.177e3	1.841e3	1.817e3	1.423e3	1.794e3	1.398e3
(0.1,0.02)	20	3.476e3	3.381e3	3.010e3	2.624e3	2.901e3	2.726e3
	30	2.985e3	2.812e3	2.658e3	2.452e3	2.713e3	2.474e3
	40	2.982e3	2.747e3	2.602e3	2.432e3	2.531e3	2.362e3
(0,0.04)	20	3.693e3	3.603e3	3.471e3	3.242e3	3.626e3	3.394e3
	30	3.464e3	3.263e3	3.129e3	2.939e3	3.229e3	3.031e3
	40	3.353e3	3.199e3	2.930e3	2.797e3	2.907e3	2.761e3

Effects of volume fraction index

Figure 4 considers the influences of volume fraction index N on the nonlinear dynamic response of eccentrically stiffened FGM cylindrical panels with movable edges in thermal environments. It is clear that the fluctuation amplitude of the eccentrically stiffened FGM cylindrical panel increases when the volume fraction index N decreases. This is reasonable because from equation (1), it is easy to see that when the volume fraction index N decreases, the volume fraction of ceramic decreases (the volume fraction of metal increases). Furthermore, the elastic module of ceramic is higher than metal ( $E_{c} > E_{m}$ ). Therefore, a decrease of volume fraction index N leads to a decrease of the elastic module of FGM, which is the reason for an increase of the fluctuation amplitude of the eccentrically stiffened FGM cylindrical panel.

The ceramic volume fraction is decreased; moreover, elastic module of ceramic is higher than metal ( $E_{c} > E_{m}$ ).

Effects of temperature increment

Effect of temperature increment, $Δ T$ , on the nonlinear dynamic response of the eccentrically stiffened FGM cylindrical panels in thermal environments is shown in Figure 5. The result from this figure shows that the fluctuation amplitude of eccentrically stiffened FGM cylindrical panel increases when temperature increment $Δ T$ increases.

Effects of elastic foundations

Figures 6 and 7 indicate the effect of elastic foundations on the nonlinear dynamic response of eccentrically stiffened FGM cylindrical panel in thermal environments with $a / b = 1, a / h = 20, N = 1, P_{x} = 0, P_{y} = 0$ . As can be seen, elastic foundations have beneficial effects on the nonlinear dynamic response of eccentrically stiffened FGM cylindrical panels. Especially, the panel fluctuation amplitude becomes considerably lower due to the support of elastic foundations. In addition, the beneficial effect of the Pasternak foundation with the module k₂ on the dynamic response of the eccentrically stiffened FGM cylindrical panels is better than the Winkler one with the parameter k₁.

Effects of stiffeners

Figure 8 illustrates the effects of stiffeners on the nonlinear dynamic response of the FGM cylindrical panels with movable edges in thermal environments. Obviously, the fluctuation amplitude of the eccentrically stiffened FGM cylindrical panel is lower than the FGM cylindrical panel without stiffeners. In other words, the stiffeners strongly decrease the amplitude of the FGM cylindrical panels.

Effects of force amplitude

Figure 9 indicates the effect of exciting force amplitude Q on nonlinear dynamic response of FGM cylindrical panels in thermal environments. Three values $Q = 25, 500 {N / m}^{2}, Q = 27, 500 {N / m}^{2}$ and $Q = 30, 500 {N / m}^{2}$ are used. It is easy to see that the nonlinear dynamic response amplitude of the FGM cylindrical panel increases when the value of the exciting force amplitude Q increases.

Effects of geometrical parameters

Figures 10 to 12 show the effects of geometrical parameters on nonlinear dynamic response of eccentrically stiffened FGM cylindrical panel in thermal environments with $N = 1, k_{1} = 0.3 GPa / m, k_{2} = 0.01 GPa . m, Δ T = 300 K, ɛ = 0.1$ . Specifically, Figures 10 and 11 show the influences of ratios $b / a$ and $b / h$ on the nonlinear dynamic response of eccentrically stiffened FGM panels, which indicates that an increase of ratio $b / a$ or $b / h$ leads to an increase of the panel fluctuation amplitude. The nonlinear dynamic response of the eccentrically stiffened FGM panels with various values of ratio $R / h$ is illustrated in Figure 12. As can be observed, the amplitude of the FGM shell increases when decreasing the ratio $R / h$ .

Conclusions

This work studied the nonlinear vibration and dynamic response of eccentrically stiffened sandwich FGM thick cylindrical panels with metal–ceramic layers on Pasternalk elastic foundations in thermal environments. The sandwich FGM panel is reinforced by the eccentrically longitudinal and transversal stiffeners and subjected to mechanical load and temperature. The sandwich panel’s properties with metal–ceramic layers depend on temperature and change according to the nonlinear function of the thickness. Governing equations are derived using the Reddy’s third-order shear deformation shell theory and solved by the Galerkin method and fourth-order Runge-Kutta method. The present numerical results are compared with other results obtained from the literature. Some conclusions can be obtained from the numerical analysis in this study:

The stiffener system strongly reduces the amplitude of the eccentrically stiffened sandwich FGM cylindrical panels.

Temperature increment has a significant negative influence on the nonlinear vibration of the eccentrically stiffened sandwich FGM cylindrical panels.

Elastic foundations have beneficial effect on the nonlinear dynamic response and natural frequencies of eccentrically stiffened sandwich FGM cylindrical panels. Moreover, the influence of nonlinear Pasternalk foundation is stronger than the linear Winkler foundation.

The excitation force strongly influences the dynamic response of the eccentrically stiffened sandwich FGM cylindrical panels.

The geometrical parameters have strong influences on the nonlinear vibration of the eccentrically stiffened sandwich FGM cylindrical panels.

Footnotes

Author's Note

Nguyen D Duc is also affiliated with National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, Korea.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Grant in Mechanics code 107.02-2015.03 of the National Foundation for Science and Technology Development of Vietnam – NAFOSTED. The authors are grateful for this support.

Appendix 1

References

Song

Zhang

Liew

. Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches. Compos Struct 2016; 158: 92–100.

Mehralian

Tadi Beni

Ansari

. On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure. Int J Mech Sci 2016; 119: 155–169.

Mirzaei

Kiani

. Free vibration of functionally graded carbon nanotube reinforced composite cylindrical panels. Compos Struct 2016; 142: 45–56.

Sofiyev

Karaca

Zerin

. Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory. Compos Struct 2017; 159: 53–62.

Hosseini-Hashemi

Abaei

Ilkhani

. Free vibrations of functionally graded viscoelastic cylindrical panel under various boundary conditions. Compos Struct 2015; 126: 1–15.

Norouzi

Alibeigloo

. Three dimensional thermoviscoelastic analysis of a simply supported FGM cylindrical panel. Compos Struct 2016; 148: 181–190.

Ahmed

. Buckling behavior of a radially loaded corrugated orthotropic thin-elliptic cylindrical shell on an elastic foundation. Thin-Walled Struct 2016; 107: 90–100.

Quan

Tran

Tuan

, et al. Nonlinear dynamic analysis and vibration of shear deformable eccentrically stiffened S-FGM cylindrical panels with metal-ceramic-metal layers resting on elastic foundations. Compos Struct 2015; 126: 16–33.

Chen

Liang

. Vibro-acoustic characteristics of cylindrical shells with complex acoustic boundary conditions. Ocean Eng 2016; 126: 12–21.

10.

Shen

. Postbuckling of nanotube-reinforced composite cylindrical panels resting on elastic foundations subjected to lateral pressure in thermal environments. Eng Struct 2016; 122: 174–183.

11.

Duc

Tuan

Tran

, et al. Nonlinear stability of eccentrically stiffened S-FGM elliptical cylindrical shells in thermal environment. Thin-Walled Struct 2016; 108: 280–290.

12.

Zhou

Zhu

. Vibration and bending analysis of multiferroic rectangular plates using third-order shear deformation theory. Compos Struct 2016; 153: 712–723.

13.

Sayyaadi

Rahnama

Askari Farsangi

. Energy harvesting via shallow cylindrical and spherical piezoelectric panels using higher order shear deformation theory. Compos Struct 2016; 147: 155–167.

14.

Selim

Zhang

Liew

. Vibration analysis of CNT reinforced functionally graded composite plates in a thermal environment based on Reddy’s higher-order shear deformation theory. Compos Struct 2015; 156: 276–290.

15.

Wattanasakulpong

Prusty

Kelly

. Free and forced vibration analysis using improved third-order shear deformation theory for functionally graded plates under high temperature loading. J Sandw Struct Mater 2013; 15: 583–606.

16.

Yang

Shen

. Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J Sound Vibr 2002; 255: 579–602.

17.

Huang

Shen

. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int J Solids Struct 2004; 41: 2403–2427.

18.

Amirpour

Das

Saavedra Flores

. Analytical solutions for elastic deformation of functionally graded thick plates with in-plane stiffness variation using higher order shear deformation theory. Compos Part B: Eng 2016; 94: 109–121.

19.

Groh

RMJ

Weaver

. Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells. Compos Struct 2015; 120: 231–245.

20.

Sivashanker

, et al. Active control of FGM plates with integrated piezoelectric sensors and actuators. Int J Solids Struct 2001; 38: 1641–1655.

21.

Quan

Duc

. Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells. J Therm Stress 2016; 5739: 1–26.

22.

Shao

Wang

, et al. Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions. Compos Part B: Eng 2017; 108: 75–90.

23.

Yang

Wang

. Effect of reinforcement stiffeners on square concrete-filled steel tubular columns subjected to axial compressive load. Thin-Walled Struct 2014; 82: 132–144.

24.

Tong

Liu

, et al. A new effective smeared stiffener method for global buckling analysis of grid stiffened composite panels. Compos Struct 2016; 158: 83–91.

25.

Turner

Vizzini

. Response of impacted sandwich panels with integral stiffeners. J Sandw Struct Mater 2004; 6: 313–326.

26.

Tong

Guo

. Elastic buckling behavior of steel trapezoidal corrugated shear walls with vertical stiffeners. Thin-Walled Struct 2015; 95: 31–39.

27.

Feng

Zhang

Tan

, et al. Effect of impact damage positions on the buckling and post-buckling behaviors of stiffened composite panel. Compos Struct 2016; 155: 184–196.

28.

Duc

Tuan

Quan

, et al. Nonlinear mechanical, thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations. Thin-Walled Struct 2015; 96: 155–168.

29.

Farahani

Azarafza

Barati

. Mechanical buckling of a functionally graded cylindrical shell with axial and circumferential stiffeners using the third-order shear deformation theory. Comptes Rendus – Mecanique 2014; 342: 501–512.

30.

Shen

Xin

. Sound transmission across composite laminate sandwiches: influence of orthogonal stiffeners and laminate layup. Compos Struct 2016; 143: 310–316.

31.

Duc

. Nonlinear thermo-electro-mechanical dynamic response of shear deformable piezoelectric sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations. J Sandw Struct Mater 2016. doi:10.1177/1099636216653266.

32.

Duc

. Nonlinear static and dynamic stability of functionally graded plates and shells, Hanoi, Vietnam: Vietnam National University Press, 2014.

33.

Duc

Quan

. Nonlinear dynamic analysis of imperfect FGM double curved thin shallow shells with temperature-dependent properties on elastic foundation. J Vibr Control 2015; 21: 1340–1362.

34.

Duc

Quan

. Transient responses of functionally graded double curved shallow shells with temperature-dependent material properties in thermal environment. Eur J Mech – A: Solids 2014; 47: 101–123.