Thermal Buckling Analysis of Perforated Functionally Graded Plates

In this research, the buckling behavior of functionally graded (FG) plates under thermal loading is investigated based on finite element analysis. It is assumed the plate is subjected to a uniform temperature rise across plate thickness. First-order shear deformation theory (FSDT) is utilized for developing the solution method. By using an appropriately designed mesh structure for a perforated plate, the critical thermal buckling temperature is obtained by numerical solution of the problem based on finite element method (FEM). The FG plate is perforated by multiple cutouts. The number of cutouts is assumed one, two, four, or six. Also different geometrical shapes of cutouts including triangle, square, rhombus, pentagon, hexagon, and circle are considered. The influence of the number of cutouts and their geometrical shapes on thermal buckling response is investigated. The effects of the number of sides of cutouts from three (triangle) to infinity (circle) are discussed. Two different boundary conditions are taken into account. Also the influences of the distance between the cutouts and the orientation of cutouts on critical buckling temperature are studied. In addition, the effects of the orientation of ellipse cutouts are studied. Some remarkable conclusions are gained that can be useful in practical applications.     

Thermal Buckling of Multi-Walled Carbon Nanotubes Based on a Rigorous van der Waals Interaction

This paper reports the result of an investigation on the axially compressed buckling of multi-walled carbon nanotubes under thermal load, based on a rigorous van der Waals interaction which is dependent on the change of interlayer spacing and the radii of tubes. From the point of view of continuum modeling, each of the concentric tubes of multi-walled carbon nanotubes is considered as an individual elastic shell and coupled with any two tubes through a rigorous van der Waals interaction force. Based on this model, some example calculations are carried out to describe the effect of temperature changes and van der Waals interaction models on the axially critical load of multi-walled carbon nanotubes. Some results obtained show that the axial buckling stress of multi-walled carbon nanotubes under thermal environment is dependent on the wave number of axially buckling modes, and the wave numbers corresponding to the minimum axial stress are not unique for the multi-walled carbon nanotubes under thermal environments. On the other hand, a rigorous van der Waals interaction force can make the axially critical load of multi-walled nanotubes under thermal loading increase. The effect of thermal environments on the axially critical stress of multi-walled nanotubes gradually increases as the axial half wavenumber (m) of buckling modes increases.     

THERMAL AND MECHANICAL INSTABILITY OF AN IMPERFECT SHALLOW SPHERICAL CAP

In this article, the thermal and mechanical buckling loads of a cap of a shallow spherical shell of isotropic material and geometrically imperfect shell are considered. The equilibrium and stability equations are based on Donnell-Mushtari-Velasov (DMV) theory and are derived using the variational method. The Sander's nonlinear strain-displacement relations are used. The shell is under external pressure for mechanical loading and uniform temperature rise and radial temperature difference for thermal loadings. A simply supported boundary condition is assumed. The solutions for thermal and mechanical buckling loads are obtained using the stability equations and the Galerkin method. One-term approximation for the middle-plane shell displacement is considered. The expressions for the thermal and mechanical buckling loads are obtained analytically and are given by closed-form solutions.     

Nonlocal Flugge Shell Model for Thermal Buckling of Multi-Walled Carbon Nanotubes with Layerwise Boundary Conditions

Presented herein is the thermal buckling analysis of multi-walled carbon nanotubes on the basis of nonlocal Flugge shell model capturing small scale effects. Based upon the continuum mechanics, a multiple-shell model is adopted in which the nested tubes are coupled with each other through the van der Waals interlayer interaction. The utilized van der Waals model incorporating the interlayer interactions between any two layers, whether adjacent or non-adjacent is curvature dependent. To analytically solve the problem, the Rayleigh–Ritz method was implemented to the variational form equivalent to the Flugge type equations. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. It is shown that the shell-like thermal buckling is significantly sensitive to the nonlocal parameter variation, whereas the column-like thermal buckling remains unaffected when the nonlocal parameter is varied.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

REFINED THEORY FOR THERMOELASTIC STABILITY OF FUNCTIONALLY GRADED CIRCULAR PLATES

Axisymmetric thermal and mechanical buckling of functionally graded circular plates is considered. Equilibrium and stability equations under thermal and mechanical loads are derived based on first-order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental ordinary differential equations is established. Buckling analysis of a functionally graded plate under uniform temperature rise, linear and nonlinear gradient through the thickness, and uniform radial compression are considered, and the critical buckling loads are derived for clamped edge plates. The results are compared with the buckling loads obtained for a functionally graded plate based on the classical plate theory given in the literature.     

Effects of Thermal Loading on the Buckling and Vibration of Ring-Stiffened Functionally Graded Shell

A theoretical method is developed to investigate the effects of thermal load and ring stiffeners on buckling and vibration characteristics of the functionally graded cylindrical shells, based on the first-order shear deformation theory (FSDT) considering rotary inertia. Heat conduction equation across the shell thickness is used to determine the temperature distribution. Material properties are assumed to be graded across the shell wall thickness of according to a power-law, in terms of the volume fractions of the constituents. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. The effects of stiffener's number and size on natural frequency of functionally graded cylindrical shells are investigated. Moreover, the influences of material composition, thermal loading and shell geometry parameters on buckling and vibration are studied. The obtained results have been compared with the analytical results of other researchers, which showed good agreement. The new features of thermal vibration and buckling of ring-stiffened functionally graded cylindrical shells and some meaningful and interesting results obtained in this article are helpful for the application and the design of functionally graded structures under thermal and mechanical loads.     

Effect of Initial Imperfections on Thermal Buckling of Functionally Graded Plates

ABSTRACT

Thermal buckling analysis of rectangular functionally graded plates with initial geometrical imperfections is presented in this article. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the first-order shear deformation plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through the thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading, namely: uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect functionally graded plate. The influence of transverse shear on thermal buckling load is discussed.     

OPTIMIZATION CONTROL OF COMPOSITE LAMINATES FOR MAXIMUM THERMAL BUCKLING AND MINIMUM VIBRATIONAL RESPONSE

The problem of maximizing the thermal buckling and minimizing the vibrational response of composite laminates is solved using optimal design and active control procedures. The problem is formulated based on a first-order shear deformation laminate theory with various cases of boundary conditions. The design objective is to maximize thermal buckling using ply thickness and the fiber orientation angle as design variables. The active control objective is to minimize the laminate vibrational response with the minimum possible expenditure of control energy. The vibrational response is expressed in terms of the total elastic energy of the laminate and a penalty functional of closed-loop control force. Liapunov-Bellman theory is used to obtain solutions for controlled deflections and optimal control force. Comparative examples are given for angle-ply antisymmetric laminates subjected to a uniform temperature distribution. A general representation for the design variables is presented such that the ply thickness is a function of the number of layers. Some of the obtained numerical results are compared with their counterparts in the literature.     

THERMAL BUCKLING ANALYSIS OF INHOMOGENEOUS RECTANGULAR PLATE DUE TO UNIFORM HEAT SUPPLY

This paper is concerned with the thermal buckling analysis of an isotropic inhomogeneous rectangular plate subjected to the arbitrary thermal loads. The fundamental equations system is derived by introducing the technique of the newly defined position of the reference plane, which allows us to analyze the problem using an elementary plate theory. It is assumed that the material properties such as the coefficient of linear thermal expansion α, the thermal conductivity λ, and Young's modulus of elasticity, E, are changed in the thickness direction with the power law of the coordinate variable, whereas Poisson's ratio ν is assumed to be constant. As an illustrative example, we consider the thermal buckling problem of a simply supported inhomogeneous rectangular plate due to uniform heat supply. Numerical calculations are carried out for several cases taking into account the variations of the inhomogeneous material properties, aspect ratio, and width-to-thickness ratio.     

Thermal Buckling of Piezolaminated Plates Subjected to Different Loading Conditions

A thermal buckling analysis is presented for simply supported rectangular laminated composite plates that are covered with top and bottom piezoelectric actuators, and subjected to the combined action of thermal load and constant applied actuator voltage. The thermomechanical properties of composite and piezoelectric materials are assumed to be linear functions of the temperature. The formulations of the equations are based on the higher-order laminated plate theory of Reddy and using the Sanders nonlinear kinematic relations. The closed-form solutions for the buckling temperature are obtained through the Galerkin procedure and solving the resultant eigenvalue problem, which are convenient to be used in engineering design applications. Numerical examples are presented to verify the proposed method. The effects of the plate geometry, fiber orientation in composite layers, lay-up configuration, different utilized piezoelectric materials, temperature dependency of material properties, thermal conductivity, and energy generation on the buckling load are investigated.     

STUDY ON LOCAL DELAMINATED THERMAL BUCKLING OF COMPOSITE LAMINATED PLATES

Based on the Whitcomb delaminated buckling model, a Rayleigh-Ritz analysis is presented to study the local thermal buckling problem of elliptical, rectangular, triangular, and lemniscate delaminations in a symmetric composite laminated plate. The critical temperatures of the laminated plate with various shaped local delaminations and different stacking patterns are obtained by utilizing the energy principle. The geometrical axis of various shapes of local delamination is arbitrary. The stacking sequence of base laminated plates is symmetric, but the stacking sequence of the sublayer is asymmetric. Finally, our experimental study on the mechanism of delaminated buckling failure in a laminated plate with a single elliptical and rectangular delamination near the surface of the laminated plate under thermal load was accomplished. Analytical predictions for the critical temperature yielding the local delamination buckling are shown to correlate well with experimental results for a number of different delamination shapes.     

Mechanical and thermal buckling of exponentially graded sandwich plates

This article presents an analytical solution for the mechanical and thermal buckling of exponentially graded material (EGM) sandwich plates. The solution is obtained using a four-variable refined plate theory. Two types of sandwich plates are investigated: one with EGM face sheets and homogeneous core; the other with EGM core and homogeneous face sheets. The governing equations are derived based on the principle of virtual work and then solved through Navier method. The results on critical buckling load and temperature change of simply supported EGM sandwich plates are obtained. The influences of several parameters on buckling behaviors are discussed.     

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

In this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.     

THERMAL BUCKLING OF FUNCTIONALLY GRADED PLATES BASED ON HIGHER ORDER THEORY

Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates. A buckling analysis of a functionally graded plate under four types of thermal loads is carried out and results in closed-form solutions. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates. The results are compared with the critical buckling temperatures obtained for functionally graded plates based on classical plate theory given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of functionally graded plates, whereas the classical plate theory overestimates buckling temperatures.     

Thermal buckling analysis of laminated composite beams using hyperbolic refined shear deformation theory

In this article, thermal buckling of laminated composite beams, based on hyperbolic refined shear deformation theory, presented for the first time, is formulated using the principle of minimum total potential energy. Navier’s analytical solution is derived to analytically solve the differential equations and the thermal critical buckling is presented in closed-form solution. The effects of temperature distribution, length to thickness ratio, modulus ratio, and thermal expansion coefficient ratio on thermal buckling of isotropic, orthotropic and laminated composite beams are investigated. The accuracy of the numerical model is verified by comparison with the available results in the literature.     

Thermal post-buckling of temperature dependent sandwich plates with FG-CNTRC face sheets

Present research investigates the thermal postbuckling of sandwich plates containing a stiff core and two thin carbon nanotube reinforced composite (CNTRC) face sheets. Properties of the core, carbon nanotubes (CNTs) and polymeric matrix of the faces are assumed to be temperature-dependent. It is assumed that CNTs as reinforcements may be distributed according to a functionally graded pattern. Plate is formulated based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity. The governing equations are obtained by the energy method with the aid of the Conventional Ritz method. Shape functions of the Ritz method are estimated according to the Chebyshev polynomials. A set of nonlinear eigenvalue equations is achieved. The obtained equations are homogeneous, coupled, and nonlinear in terms of both displacements and temperature. A successive displacement control strategy is implemented to trace the thermal postbuckling equilibrium path of the plate. It is shown that, with increasing the volume fraction of CNT, critical buckling temperature of sandwich plate increases and postbuckling deflection decreases. Furthermore, through a functionally graded distribution of volume fraction of CNTs across the thickness, critical buckling temperature of the sandwich plate may be enhanced and thermal postbuckling deflection may be alleviated.     

不同海况下近海超大型风力机动力学响应及结构损伤分析

以超大型DTU 10 MW单桩式近海风力机为研究对象,通过p-y曲线和非线性弹簧建立桩-土耦合模型,选取Kaimal风谱模型建立湍流风场,基于P-M谱定义不同频率波浪分布,并利用辐射/绕射理论计算波浪载荷,采用有限元方法对不同海况下单桩式风力机进行动力学响应、疲劳及屈曲分析。结果表明：不同海况波浪载荷作用下塔顶位移响应及等效应力峰值远小于风及风浪联合作用,其中风浪联合作用下风力机塔顶位移响应及等效应力略小于风载荷;波浪载荷对风载荷引起的单桩式风力机动力学响应具有一定抑制作用,此外相较于波浪载荷,风载荷为控制载荷;风载荷与风浪联合作用下风力机等效应力峰值位于塔顶与机舱连接处,波浪载荷风力机等效应力峰值位于支撑结构与桩基连接处;仅以风载荷预估风力机塔架疲劳寿命将导致预估不足;随着波浪载荷的增大,风力机失稳风险加大,波浪载荷不可忽略;不同海况下,风浪联合作用局部屈曲区域位于塔架中下端,在风力机抗风浪设计时,应重点关注此处;变桨效应可大幅降低风力机动力学响应、疲劳损伤及发生屈曲的风险。     

Finite Element Analysis of Thermal Buckling of Rectangular Laminated Composite Plates with Circular Cut-Out

In this work, thermal buckling analysis of symmetric and anti-symmetric laminated composite plates with a cut-out is presented. The plate is assumed to be subjected to a uniform temperature rise for different boundary conditions. The thermal buckling analysis is performed using the code developed in MATLAB software. The stiffness matrices and thermal force vector are derived according to first-order shear deformation theory (FSDT). To have more control on the mesh pattern around the cut-out, convenient meshes are manually constructed using a mesh generation algorithm in which mesh density around the hole can be controlled. The results of FEM code is compared with ABAQUS's solutions and with those available in the literature. After that, the effects of cut-out size, boundary conditions, plate aspect ratio, and stacking sequence on critical thermal buckling temperature are investigated for symmetric and anti-symmetric plates. Also, plates with cut-out located at positions other than the center of plate are investigated and useful conclusions are derived from the numerical results.     

Postbuckling analysis of shear deformable FG shallow spherical shell panel under nonuniform thermal environment

In this article, the buckling and postbuckling behavior of functionally graded spherical shell panel is examined under nonuniform thermal environment. The effective material properties of the graded structure are evaluated using the Voigt's micromechanical model through the power-law distribution. For the analysis purpose, a general nonlinear higher order mathematical model is developed in conjunction with Green–Lagrange geometrical nonlinearity. The governing equation is derived using variational principle and solved through the direct iterative method. The effect of different geometrical and material parameters on the buckling and postbuckling responses of the functionally graded shell panels is examined and discussed in detail.