Buckling and Vibration Performance of a Composite Laminated Plate with Elastic Boundaries Subjected to Local Thermal Loading

In this paper, a model is established for the calculation of the vibrations of a composite laminated plate with elastic boundary conditions subjected to local thermal loading..e model is based on first-order shear deformation theory using the finite element method. .e influence of boundary conditions, heating area, and heating location on buckling and vibrations of a composite laminated plate was investigated, and there were two stages in which the critical temperature increased sharply during the transition from free boundary to simply supported and rigid fixed boundaries. .e thermal buckling of locally heated laminated plates is generally not checked in practical applications unless the heated area exceeds approximately 10% of the total area of the plates. .e stronger the boundary constraint is, the greater the influence of the heated area is on the vibrational frequencies of the composite laminated plate.


Introduction
Because of their excellent strength, stiffness, light weight, and malleability, plates made of composite materials have received increasing consideration for structural applications in aerospace, shipbuilding, and other areas of industrial engineering. However, composite material plates are often compromised by exposure to thermal loads. For example, temperature changes cause stresses on ship structures that cannot be ignored when at sea. ere can be very large temperature changes in the work environment, and temperature stresses can be particularly large on main engines, auxiliary engine compartments, pipelines, LNG facilities, and so on [1].
Many paper results have been published about vibration, buckling, and static performance analyses of composite structures. Vangipuram and Ganesan [2] studied the buckling and vibration of a rectangular composite viscoelastic sandwich plate with different boundary conditions subjected to thermal loading. Jeyaraj et al. [3] investigated the vibration and acoustic radiation properties of composite laminated plates with different combinations of boundary conditions subjected to various thermal loads based on the finite-boundary element method. Shiau et al. [4] applied the finite element method to study the thermal buckling behaviour of composite laminated plates with different boundary condition combinations of simply supported and clamped edges, and the results indicated that the thermal buckling mode is mainly dependent on the E1/E2 ratio, fibre orientation, and aspect ratio of the plate. Biswal et al. [5] performed numerical and experimental investigations on the free vibration behaviour of woven fibreglass/epoxy laminated composite shells for C-C-C-C and S-S-S-S boundary conditions subjected to hygrothermal environments. e results showed that there was a reduction in the natural frequency of laminated composite shells with increasing temperature and moisture concentrations. Zhao et al. [6] utilised the first-order shear deformation theory to study the effects of temperature and humidity on vibration performance of orthogonal anisotropic laminated plates with four simply supported edges, and the influence of temperature on the vibration frequencies of single-layer plates was more obvious than that of humidity.
According to a review of the literature, most previous studies on composite laminated plates are confined to classical boundary conditions, such as free, simply supported, clamped, and their combinations. However, a variety of possible boundary supports encountered in practical engineering applications may not always be classical in nature, and they will always be elastic along the supports. Unfortunately, studies of the effects of elastic boundaries have mostly focused on the dynamic performance of structures, and to the best of the author's knowledge, to date, no paper has dealt with the buckling performance of structures with elastic boundaries. Moreover, the temperature distribution on the composite laminated plate is usually uneven. For example, as a hypersonic structure accelerates in the atmosphere, the plate boundary acts as a radiator when connected to the cooler substructure, resulting in the generation of aerodynamic heat on its surface [7]. Few studies have dealt with how local heating affects the vibration properties of composite structures. Over the last several years, rapid growth in the design of structures and industrial processes has required development of computational models that reflect practical engineering problems; therefore, it has great significance to study the buckling and vibration performance of composite laminated plates with arbitrary elastic edge supports subjected to local thermal loads. e motivation of this study is to solve these two problems.
It is commonly believed that the finite element method (FEM) is a powerful tool for solving complex problems because of its strong adaptability. Recently, Alimirzaei et al. [8] considered the nonlinear static, buckling, and vibration of viscoelastic microcomposite beams reinforced by various distributions of boron nitride nanotubes (BNNTs) with initial geometrical imperfection by modified strain gradient theory using FEM. Hirane et al. [9] employed a novel higherorder layer-wise finite element model for static and free vibration analyses of functionally graded material (FGM) sandwich plates. In this paper, the buckling and vibration behaviour of a composite laminated plate subjected to elastic boundaries was studied by the finite element method. e paper is arranged in four sections. Following this introduction, Section 2 provides the theoretical background of the composite laminated plate and elastic boundary, and the corresponding matrix representation is given. In Section 3, the effects of boundary conditions, heating position, and heated area on the buckling and vibration performance of composite laminated plates subjected to local thermal loading are discussed. e final section summarizes the main results of the paper. Figure 1, based on first-order shear deformation theory, the deformation field of the composite laminated plate can be expressed as follows [6]:

Governing Equations. As shown in
where u 0 , v 0 , and w 0 are the displacements of a generic point at the middle surface in the x, y, and z directions, respectively. e bending rotations of a vector normal to the midsurface are denoted by ϕ x and ϕ y when measured in the x − z and y − z planes, respectively. Consequently, the strain-displacement equations can be expressed as follows [10]: where ε x and ε y represent bending strains and c xy , c yz , and c xz represent transverse shear strains. e composite laminated plate is designed to be composed of N layers. e thickness of each layer is t k so that t � N k�1 t k is the total thickness of the composite laminated plate. When the laminated plate is subjected to a constant temperature, the stress-strain relations of the k th layer are of the form as follows [11]: where σ x and σ y and τ xy , τ yz , and τ xz represent the stress components and Q ij are transformed reduced stiffnesses, which can be expressed in terms of the orientation angle and engineering constant of the material. For an in-depth discussion of the meaning of the laminate, see reference [12].
ΔT is the thermal load. α x and α y are coefficients of thermal y z x t Figure 1: Schematic diagram of a composite laminated plate. expansion along the x and y axes, respectively, and α xy is the apparent coefficient of thermal shear. e stress resultants are as follows [11]: and the moment resultants are as follows: where t is the thickness of the composite laminated plate. e total potential energy is the sum of the bending strain energy of the laminated plate and the potential energy of the laminated plate under the action of in-plane forces due to the thermal load, which can be expressed as follows [13]: where π is the bending strain energy of the laminated plate and V is the potential energy of the laminated plate. ey can be written as follows [13]: where [K s ] is the stiffness matrix and [K σ ] is the thermal stress stiffness matrix. e superscript e denotes that these variables are defined on the element. n is number of finite elements. q is the nodal displacement vector. Figure 2, the strain energy U of the elastic boundary in the plate element can be represented by the following [14]:

Elastic Boundary. As demonstrated in
where k tx0 , k tx1 , k ty0 , and k ty1 are the corresponding boundary support stiffnesses and K rx0 K rx1 , K ry0 , and K ry1 are the corresponding boundary rotational stiffnesses of the composite laminated plate. e stiffness matrix representation of the elastic boundary is obtained from the above equation as follows: where δ { } e is the displacement vector of the element node, and the elastic boundary stiffness matrix K b e can be expressed as follows: Figure 2: Locally heated laminated plate with elastic boundary subjected to local thermal load.
where [N w ] is the element shape function. e elastic boundary matrix of the composite laminated plate can be expressed as follows:

Vibration Eigenvalue Problem.
When the composite laminated plate is in the critical buckling state corresponding to the neutral equilibrium condition, the second derivative of the total potential energy is set to zero; that is, the following buckling eigenvalue problem is obtained [2]: where [K] is the global stiffness matrix, namely, is the structural stiffness matrix. For thermal buckling, the product of λ and the initial estimate ΔT are the critical buckling temperature. at which the plate buckles due to thermal stresses.
[U] represents the generalized displacement vector of the composite laminated plate. e vibration eigenvalue problem of the laminated plate subjected to local thermal loading and the thermal load can be expressed as follows [15]: where [M] is the mass matrix of the composite laminated plate and ω represents the vibration frequency.

Numerical Results and Discussion
A schematic diagram of the composite laminated plate subjected to local thermal loading and the coordinate system is shown in Figure 2. e composite laminated plate is made of carbon/epoxy with the following material and geometric properties (unless otherwise stated): the physical material parameters and structural dimensions of the laminated plate are as follows: density ρ � 1520 kg/m 3 , E 1 � 133.86 Gpa, e critical temperature of the laminated plate with rigid fixed boundary condition is shown in Table 3, and the numerical results are in good agreement with the results in the literature. e next step is to calculate the natural frequencies of the laminated plate. e geometrical size of the plate is a � 0.15 m and b � 0.15 m, and the boundary condition is CFFF; that is, one side is rigidly fixed and the other side is a free boundary. Material I is glass/epoxy resin, the density of the laminated plate is ρ � 1420.05 kg/m 3 Table 4. e slight difference between the numerical and experimental results is caused by the limitations of the ideal boundary conditions of the laminated plate. 4 Shock and Vibration

Factors at Influence Critical Temperatures and Vibration Frequencies.
It is necessary to investigate the determining factor for the buckling failure of composite laminated plates with elastic boundaries subjected to local thermal loads. e numerical results of nonuniform in-plane resultant forces due to thermal load were presented. e following values were taken: a � 0.5 m, b � 0.5 m, t � 0.00625 m, a 0 � 0.2 m, and b 0 � 0.2 m. Figure 3 shows typical cases of the in-plane resultant force distribution for a composite laminated plate at the buckling temperature. e compressive forces in both the x and y directions over the heated region dominate the occurrence of thermal buckling. In addition, comparatively small values of in-plane shear resultant forces N xy can be found at four corners of the heated region.

Boundary Conditions.
e size of the composite laminated plate is a � 1 m, b � 1 m, and t � a/50, the layer of composite laminated plate is [0/90/90/0], and the surrounding boundary adopts the same constraint. e curve showing the change in critical temperature of the composite laminated plate with boundary support stiffness and boundary rotation stiffness varies (free boundary-simply supported boundary-rigid fixed boundary) and is shown in Figure 4. ere are two first-order critical temperatures in Figure 4. e figure shows that there are two stages in which the critical temperature increases sharply, and the influence of the boundary support stiffness is greater for the secondorder critical temperature than the first-order critical temperature, i.e., the boundary support stiffness changes from k t � 1e5 Pa to k t � 1e7 Pa. In the second stage, i.e., the Table 1: Natural frequencies of the heated orthotropic plate (Hz; T � 303.15 K − 1 ).     Shock and Vibration 5 boundary rotational stiffness changes from k r � 1e4 N/rad to k r � 1e8 N/rad, the two first-order critical temperatures of the composite laminated plate increase significantly, and the increased temperature range of the second stage is larger than that of the first stage, which indicates that the boundary rotational stiffness ratio has a greater influence on the critical temperature of the composite laminated plate.

Heating Position.
e geometric size and physical parameters of the composite laminated plate are taken as a � 1 m, b � 1 m, t � a/100, a 0 � 0.4 m, and b 0 � 0.4 m. e distance d 2 � 0.5b remains unchanged, i.e., only d 1 changes from 0.2a to 0.5a when the heating position is changed. e curve of critical temperature vs. heating position varies as shown in Figure 5. e critical temperature of the composite laminated plate increases gradually when the boundary support stiffness is small (5e4 Pa). e critical temperature of the composite laminated plate decreases gradually mainly because the boundary stiffness of composite laminated plates is smaller than the stiffness of the laminated plates, and the boundary of the composite laminated plates is basically unrestricted. However, the composite laminated plate is not prone to buckling when there is thermal load near the boundary, and the stiffness of the boundary support is greater than 2e5 Pa, as shown in Figure 5. e variation trend is basically unchanged when the stiffness of the boundary support reaches a certain value. As the stiffness of the boundary support is infinite and the rotational stiffness of the boundary increases gradually, the difference in the critical temperature of the composite laminated plate increases when the heating position is at the boundary (d 1 � 0.2a) and the heating position is in the middle of the composite laminated plate (d 1 � 0.5a), as shown in Figure 6. e boundary stiffness has an important influence on the position of the minimum temperature required for buckling locally heated composite laminates; that is, the position of the minimum critical temperature is closer to the boundary when the boundary stiffness is small, while the position of the minimum critical temperature is at the midpoint, the boundary stiffness is large. In other words, the influence of boundary stiffness on the minimum critical temperature of laminates mainly depends on the relative stiffness of the  boundary and the plates. For example, when the boundary support stiffness is small, that is, the relative stiffness between the boundary and the laminate is small, the laminate is more prone to buckling when the local thermal load is near the boundary.

Heated Area.
e curve of critical temperature vs. heating area of the laminated plate structure is shown in Figure 7 where d 1 � 0.5a, d 2 � 0.5b, k t � 1e7 Pa, and K r � 1e6 N/rad. e critical temperature of the composite laminated plate is large when the heated area is small, and it decreases rapidly when the critical area gradually increases. e critical temperature decreases slowly when the heated area exceeds approximately 10% of the total area of the composite laminated plate. erefore, the thermal buckling of locally heated composite laminated plates is generally not checked in practical applications, and thermal buckling calculations are necessary only when the heated area is large and the thermal load is high.

Vibration Frequencies of a Composite Laminated Plate
Subjected to Local ermal Loading. Assuming size of the laminated plate is a � 1 m, b � 1 m, and t � a/100 m, the thermal load is set as the critical temperature when the heating area is the whole laminated plate, so the fundamental frequency of the composite laminated plate is equal to 0 Hz when the heated area is equal to the laminated plate area. e curve of the fundamental frequency of the composite laminated plate vs. heating area for boundary conditions is shown in Figure 8. e fundamental frequency of the laminated plate decreases with increase in heated area. e effect of the heating area on the fundamental frequency of the composite laminated plate becomes greater when the boundary constraint is strengthened. For example, when k t � 1e5 Pa and K r � 0 N/rad and the heating area increases from 10% to 25%, the fundamental frequency of the composite laminated plate decreases by 8%. When k t � 1e12 Pa and K r � 1e12 N/rad and the heating area increases from 10% to 25%, the fundamental frequency of the composite laminated plate decreases by 15%. In addition, the two stages of the sharp increase in fundamental frequency have nothing to do with the heated area but are related to the boundary conditions.
Assuming size of the laminated plate is a � 0.5m, b � 0.5m and t � a/80m. When a 0 � 0.4a and b 0 � 0.4b, the heating area is 16% of the entire area of the composite laminated plate, and the thermal load is the minimum  Shock and Vibration 7 critical temperature of the laminated plate structure when the heating position changes. e curves of the frequency change of the composite laminated plate at different heating positions are shown in Figure 9. As shown in the figure, the first-order frequency of the composite laminated plate is large when the boundary supports stiffness k t � 1e5 Pa and K r � 0 N/rad, and the heating position is close to the centre of the laminated plates. e first-order frequency of the composite laminated plate is the smallest when the heating position is the centre of the composite laminated plate, and it is the largest when the stiffness of the boundary support increases to k � 4e5 Pa and K r � 0 N/rad, and the heating position is approximately d 1 � 0.3a. e thermal stress of the composite laminated plate caused by the thermal load is larger due to the enhancement of the boundary stiffness when the boundary support stiffness continues to increase to k � 1e15 Pa and K r � 0 N/rad; in this instance, the firstorder frequency of the composite laminated plate is the smallest when the heating position is at the centre of the laminated plate. When the boundary support stiffness continues to increase to k � 1e15 Pa and K r � 0 N/rad, the boundary condition is equivalent to rigid fixation, and the difference between the first-order frequencies of the composite laminated plate increases when the heating area is at the boundary and central position because the boundary constraint is strengthened and the thermal stress difference of the composite laminated plate increases.

Conclusions
ere are two stages in which the critical temperature increases sharply during the transition from a free boundary to a simply supported boundary and a rigid fixed boundary. e effect of the boundary rotational stiffness on the critical temperature of the composite laminated plate is greater than that of the boundary support stiffness. When the stiffness of the boundary support is small, the composite laminated plate is prone to buckling if the local temperature load is near the boundary. However, buckling easily occurs when the local temperature load is located at the centre of the laminate if the boundary support stiffness reaches a certain value. e thermal buckling of locally heated laminated plates is  Figure 9: Vibration frequencies of composite laminated plates with different heating positions: (a) k t � 1e5 Pa and K r � 0 N/rad; (b) k t � 4e5 Pa and K r � 0 N/rad; (c) k t � 1e15 Pa and K r � 0 N/rad; (d) k t � 1e15 Pa and K r � 1e15 N/rad. 8 Shock and Vibration generally not checked in practical applications. It is necessary to carry out thermal buckling calculations only when the heated area exceeds approximately 10% of the total laminate area. When the stiffness of the boundary support is small, the frequency of the composite laminated plate increases as the heating position approaches the centre of the laminated plate. When the stiffness of the boundary support reaches a certain value, the fundamental frequency of the laminated plate decreases as the heating position approaches the centre of the composite laminated plate. e stronger the boundary constraint is, the greater the influence of the heating area is on the vibration frequencies of the composite laminated plate.
Data Availability e data generated or analyzed during this study are included in this article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.