NONLINEAR RESPONSE OF GEOMETRICALLY IMPERFECT STIFFENED FLAT PANELS UNDER THERMOMECHANICAL LOADING

A study of the nonlinear response of orthogonally stiffened isotropic flat panels exposed to a temperature rise and a lateral pressure field is presented. A theoretical analysis and the numerical results emphasize the effects played by the bi/uniaxial stiffeners and unavoidable initial geometric imperfections on the thermomechanical load-carrying capacity of stiffened panels, and pertinent conclusions are outlined.     

PUBLICATIONS ON THERMAL STRESSES

The study of the nonlinear response of sandwich flat panels exposed to thermomechanical loading systems is the topic of this article. The sandwich structure considered in this article consists of a thick core layer bonded by the face layers, which are assumed to be symmetrically located with respect to the midplane of the overall structure. The loads involved in this analysis consist of biaxial compressive edge loads, a lateral pressure, and a nonuniform temperature field. The effects of the unavoidable initial geometric imperfections and the character of tangential boundary conditions are incorporated, and their implications upon the structural response are explored. In short, the results of this study are intended to provide pertinent information on the thermomechanical load-carrying capacity of flat sandwich structures.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.     

Nonlinear free vibration of spherical shell panel using higher order shear deformation theory – A finite element approach

A nonlinear finite element model for geometrically large amplitude free vibration analysis of doubly curved composite spherical shell panel is presented using higher order shear deformation theory (HSDT). The nonlinearity is introduced in the Green–Lagrange sense. The governing equations of the vibrated shell panel are derived using the Variational approach. Frequency ratios (nonlinear frequency to linear frequency) of the spherical panels are determined as a function of shell amplitude ratio. The results are computed for different orthotropicity ratios, stacking sequences, thickness ratios, amplitude ratios and boundary conditions and also compared with those available in literature.     

Thermomechanical nonlinear analysis of axially compressed carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations with tangentially restrained edges

Buckling, postbuckling, and nonlinear responses of composite cylindrical panels reinforced by single-walled carbon nanotubes (CNTs), supported by an elastic foundation, exposed to elevated temperature and axially compressed by uniform load are investigated in this article. Distribution of CNTs is uniform or graded in the thickness direction and the effective properties of CNT-reinforced composite are assumed to be temperature dependent, and are estimated by extended rule of mixture through a micromechanical model. Governing equations are established based on thin shell theory taking von Kármán–Donnell nonlinearity, initial geometrical imperfection, Pasternak-type elastic foundation and tangential elastic constraints of boundary edges into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive explicit expressions of load–deflection relation from which critical buckling loads can be obtained. Unlike works in the literature, the present study accounts for elasticity of tangential restraint of two unloaded straight edges in model of cylindrical panel. The study also gives conditions for which bifurcation type buckling response can occur and novel findings in numerical examples.     

Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments

In this article, nonlinear vibration and dynamic response of imperfect functionally graded materials (FGM) thick double-curved shallow shells resting on elastic foundations are investigated using Reddy's third-order shear deformation shell theory in thermal environments. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The FGM shells are subjected to mechanical, damping, and thermal loads. The Galerkin method and fourth-order \hboxRunge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation, and dynamic response of the shells. In numerical results, the effects of geometrical parameters, material properties, imperfections, shear deformation, the elastic foundations, mechanical, thermal and damping loads on the nonlinear dynamic response, and nonlinear vibration of FGM double-curved shallow shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.     

Postbuckling and Vibration Analysis of Antisymmetric Angle-Ply Composite Plates

ABSTRACT

Postbuckling and postbuckled vibration analysis of antisymmetric angle-ply composite plates subjected to a thermo-mechanical loading is investigated in this article. The nonlinear partial differential equations of equilibrium are derived based on higher-order shear deformation theory, incorporating von Kármán nonlinear strain displacement relations and initial geometric imperfections. The solutions to the governing nonlinear partial differential equations are sought using the multi-term Galerkin technique. The nonlinear equilibrium paths are traced using the Newton-Raphson method in conjunction with the Riks approach. The free vibration frequencies of postbuckled plate about the static equilibrium state are reported by solving the associated linear eigenvalue problem. Results are presented for a simply supported antisymmetric angle-ply laminated plate, clearly bringing out the effects of uniform through-thickness temperature distribution and mechanical edge load on postbuckling and vibration behavior of the plate.     

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.     

Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM

In this article, the nonlinear vibration frequencies of functionally graded carbon nanotube-reinforced composite doubly curved shell panels under elevated thermal environment are numerically investigated using finite element method. The doubly curved carbon nanotube-reinforced shell panel has been modeled mathematically using higher-order kinematics theory and Green–Lagrange geometrical nonlinear strains. The properties of the individual constituents of the graded composite are assumed to be temperature dependent. In addition, the properties of the media are obtained based on the modified rule of mixture. The carbon nanotubes are dispersed nonuniformly through the thickness direction. The large deformation kinematic effects on the structural responses are counted by including all the nonlinear higher-order terms in the formulation. The desired nonlinear responses are computed numerically using our in-house computer code in conjunction with the direct iterative scheme. The convergence and the accuracy of the present numerical model have been checked by solving various numerical examples. Nonlinear mechanical responses were affected by several other design parameters and explored numerically for the thickness ratios, volume fractions, temperature loading, type of geometries, and type of grading under the uniform thermal environment.     

AEROTHERMOELASTIC PHENOMENA OF AEROSPACE AND COMPOSITE STRUCTURES

The thermal postbuckling and aerodynamic-thermal load analysis of cylindrical laminated panels has been performed using the finite element method. To consider large deflections due to thermomechanical loads, the von Karman nonlinear displacement-strain relationships based on layerwise theory are applied. The cylindrical arc-length method is used to take account of the snapping phenomena. The panel flutter analysis of cylindrical panels subject to thermal stresses is carried out using Hans Krumhaar's supersonic piston theory. For the enhancement of the postbuckling and panel flutter behavior subjected to thermal load, the shape memory alloy hybrid composite (SMAHC) panel is investigated.     

Nonlinear thermoelastic deflection of temperature-dependent FGM curved shallow shell under nonlinear thermal loading

The geometrically nonlinear thermomechanical transverse deflection responses of the functionally graded curved structure under the influence of nonlinear thermal field are reported in this article. For the numerical analysis, a nonlinear mathematical model is derived using the higher-order shear deformation theory and Green–Lagrange nonlinear strains. The current model includes all of the nonlinear higher-order terms to achieve the true flexure of the structure under the combined action of loads. It is assumed that the panel structure is exposed to nonuniform temperature field combined with the transversely distributed mechanical load. Additionally, the properties of material constituents are assumed to vary with the nonuniform temperature load and corresponding properties are evaluated considering dependency and independence of temperature. Furthermore, the panel material grading has been obtained mathematically with the help of Voigt’s micromechanical rule together with the power-law distribution. The system of equations is obtained using the variational principle and solved numerically using the finite element steps in association with the direct iterative method. The stability of the present numerical model has been established through the convergence test and compared with the benchmark results to show the validity. Finally, numerical experimentations have been carried out for different parameters and discussed in detail.     

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels

Based on the three-dimensional elasticity theory, free vibration analysis of functionally graded (FG) curved thick panels under various boundary conditions is studied. Panel with two opposite edges simply supported and arbitrary boundary conditions at the other edges are considered. Two different models of material properties variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential distribution of the material properties through the thickness are considered. Differential quadrature method in conjunction with the trigonometric functions is used to discretize the governing equations. With a continuous material properties variation assumption through the thickness of the curved panel, differential quadrature method is efficiently used to discretize the governing equations and to implement the related boundary conditions at the top and bottom surfaces of the curved panel and in strong form. The convergence of the method is demonstrated and to validate the results, comparisons are made with the solutions for isotropic and FG curved panels. By examining the results of thick FG curved panels for various geometrical and material parameters and subjected to different boundary conditions, the influence of these parameters and in particular, those due to functionally graded material parameters are studied.     

Buckling of Functionally Graded Cylindrical Shells Under Combined Thermal and Compressive Loads

This article focuses on analytical solutions for bifurcation buckling of FGM cylindrical shells under thermal and compressive loads. A new solution methodology is established based on Hamilton's principle. The fundamental problem is subsequently transformed into the solutions of symplectic eigenvalues and eigenvectors, respectively. Then, by applying a unidirectional Galerkin method, imperfection sensitivity of an imperfect FGM cylindrical shell is discussed in detail. The solutions reveal that boundary conditions, volume fraction exponent, FGM properties, and temperature rise distribution significantly influence the buckling behavior. Critical stresses are reduced greatly due to the existence of initial geometric imperfections.     

Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells

This article presents analytical solutions for the nonlinear static and dynamic stability of imperfect eccentrically stiffened functionally graded material (FGM) higher order shear deformable double curved shallow shell on elastic foundations in thermal environments. It is assumed that the shell’s properties depend on temperature and change according to the power functions of the shell thickness. The shell is reinforced by the eccentrically longitudinal and transversal stiffeners made of full metal. Equilibrium, motion, and compatibility equations are derived using Reddy’s higher order shear deformation shell theory and taking into account the effects of initial geometric imperfection and the thermal stress in both the shells and stiffeners. The Galerkin method is applied to determine load–deflection and deflection–time curves. For the dynamical response, motion equations are numerically solved using Runge–Kutta method. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. The influences of inhomogeneous parameters, dimensional parameters, stiffeners, elastic foundations, initial imperfection, and temperature increment on the nonlinear static and dynamic stability of thick FGM double curved shallow shells are discussed in detail. Results for various problems are included to verify the accuracy and e?ciency of the approach.     

双轴受压下混凝土动态力学特性试验研究

为了解双轴受压下混凝土动态力学特性,采用真三轴材料试验机,对边长为300 mm的立方体混凝土试件在应变速率为10-5、10-4、10-3s-1下进行双轴动态受压试验,研究了双轴受压下混凝土的抗压强度、弹性模量、峰值应变等力学特性,并与单轴压缩试验进行了比较。结果表明,双轴应力状态下的力学参数与单轴应力状态时相比均有所提高,且双轴受压时混凝土的各项力学参数受侧压力和加载速率的共同影响,并呈现出不同的变化规律。     

Thermal Buckling of Imperfect Functionally Graded Cylindrical Shells Based on the Wan–Donnell Model

ABSTRACT

A thermal buckling analysis of an imperfect functionally graded cylindrical shell is considered using the Wan–Donnell model for initial geometrical imperfections. Derivation of the equations is based on the first-order classical shell theory using the Sanders nonlinear kinematic relations. Results for the buckling loads are obtained in the closed form. The effects of shell geometry and volume fraction exponent of functionally graded material on the buckling load are investigated. The results are validated with known data in the literature.     

Nonlinear dynamic response and vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shear deformable plates with temperature-dependent material properties and surrounded on elastic foundations

Based on Reddy’s third-order shear deformation plate theory, the nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates on elastic foundations subjected to dynamic loads and temperature are presented. The plates are reinforced by single-walled carbon nanotubes which vary according to the linear functions of the plate thickness. The plate’s effective material properties are assumed to depend on temperature and estimated through the rule of mixture. By applying the Airy stress function, Galerkin method and fourth-order Runge–Kutta method, nonlinear dynamic response and natural frequency for imperfect FG-CNTRC plates are determined. In numerical results, the influences of geometrical parameters, elastic foundations, initial imperfection, dynamic loads, temperature increment, and nanotube volume fraction on the nonlinear vibration of FG-CNTRC plates are investigated. The obtained results are validated by comparing with those of other authors.     

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.     

Investigation of Aerothermoelastic Behaviors of Functionally Graded Panels in Supersonic Flows

Considering the potentials of Functionally Graded Panels (FGPs) in aerospace field, it is necessary to study the aerothermoelastic behaviors of FGPs in supersonic flows. In this study, Piston Theory Aerodynamics (PTA) and Eckert reference enthalpy method are used to model aerodynamic force and heating, respectively. The 2-D heat conduction equation is solved and the impact of elevated temperature on the mechanical properties of FGPs is considered to build an aerothermoelastic two-way coupling model of FGPs, and Finite Element Method (FEM) is used to approach the solution. As the results, it is found that there exist three different regions in the bifurcation diagram, namely, thermal buckling region, critical region and flutter region. Due to the inhomogeneous distribution of thermal expansion coefficient, the panel buckles up first and then buckles down via vibration, as thermal buckling happens. Also, irregular vibrations are observed in the critical region of bifurcation diagram. In the flutter region, the dynamic behavior of FGPs is discontinuous and very sensitive to initial conditions. With the impact of aerothermoelastic two-way coupling, different FGPs behaviors lead to the differences in temperature distribution. In particular, the final buckling position and vibration center move to lower positions, and lower temperature region near leading edge is left in the FGPs, because of thermal moment. Also, regular vibrations, rather than irregular vibrations, are easy to extract more principal and regular POD (Proper Orthogonal Decomposition) modes. The results presented could be applied to the analysis and design of Functionally Graded Panels in supersonic flows.     

Analysis and study of two‐dimensional parameter bifurcation of wind power farms and composite loads

This study focuses on the stability of power system based on codimension‐two bifurcation theory. In this paper, we investigate the impact of load modeling on permissible wind power generation margins in distribution networks. The study considers codimension‐two bifurcations of equilibria and limit cycles in wind power systems depending on varying two parameters simultaneously. The principle parameter is the wind power generation, and the other parameter depends on the different types of loads. The types of loads are ZIP, exponential recovery, dynamic induction loads, and composite load models. To study the effects of the induction motor loads, the proportion of the static component in the motor load is changed and assessed with respect to their mechanical loads. Wind generation margin boundaries are traced, and saddle‐node, Hopf, and limit‐induced bifurcation branches are obtained, delimiting the stable and unstable operating regions in the parameter space. The analysis presented in this paper can pave the way for determining methods for improving and monitoring these margins with consideration to the system parameters and load composition.