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This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents. [ABSTRACT FROM AUTHOR]

Existing numerical analysis methods often incur high computational costs when directly solving multi-degree-of-freedom systems due to computer capacity limitations. To address this issue, this paper took the composite laminated concentrically stiffened rectangular plate as the research object. The modal synthesis method was employed to reduce the degrees of freedom of the model, enabling the solution of complex dynamic characteristics. In the research process, the domain decomposition theory was used to divide the research object into plate and rib substructures. Based on the spectral Chebyshev method, the substructure dynamic model of the composite laminated rectangular plate was constructed. Subsequently, the function expressions of the complex load excitation on each substructure were derived, and the fixed modal synthesis method was introduced to construct the spectral Chebyshev reduced model of the substructures. According to the artificial spring theory equivalent, the interface force, and the whole reduction model of the composite laminated concentrically stiffened rectangular plate were obtained by assembling the substructure models. The time–domain and frequency–domain dynamic behaviors of composite laminated concentrically stiffened rectangular plates are yielded by solving the whole reduced model. The novelty of this work lies in combining the spectral Chebyshev method with the modal synthesis method. The established reduced model no longer depended on mesh quality and required fewer matrix dimensions and computation time compared to the whole model, while still achieving sufficiently accurate results. Additionally, based on the reduced model, an effective parametric study scheme was proposed to analyze the impact of dimensions and material properties of rib substructures on the dynamic behaviors of the composite laminated concentrically stiffened rectangular plate. These findings can be used to guide the optimal design of laminated concentrically stiffened rectangular plates in engineering practice. [ABSTRACT FROM AUTHOR]

Orthogonally stiffened cylindrical shell coupled with internal rectangular plate is widely applied in engineering industries. The significant effects of the internal plate on vibration signatures of the shell have been demonstrated. Little attention has been paid to vibration analysis in the wavenumber–frequency domain. This paper aims to conduct the circumferential modal coupling analysis of the coupled model using the semi-analytical wavenumber–frequency method. The coupled model is disassembled into the stiffened cylindrical shell and rectangular plate. The interface potential between the shell and rectangular plate is derived based on a unified variational modeling procedure, in which the transformation of the coordinate system must be conducted. The displacement fields of the shell are expanded with the Fourier series along the circumferential direction. The vibratory field in wavenumber–frequency space is analytically obtained. The numerical method is utilized to illustrate the convergence and accuracy of the proposed method. The results demonstrate that the internal plate affects modal characteristics and then results in the alterations of the number and the shifts of resonance peaks. The phenomenon can be demonstrated by the circumferential modal contribution of the vibration response. Besides, it should be noted that the dominant circumferential modes for the input energy and vibrational energy are inconsistent owing to the strong shell–plate coupling effect. [ABSTRACT FROM AUTHOR]

In this article, the nonlinear vibration analysis of a rectangular hyperelastic membrane embedded within the nonlinear elastic foundation exposed to uniformly distributed hydrostatic pressure is investigated. The aim is to study the nonlinear vibration behaviors of a rectangular hyperelastic membrane when both primary and 1:1 internal resonances happen in the system. Also, by considering the system embedded within the nonlinear elastic foundation, the model of the structure has been developed close to the real situation. The structure is made of an isotropic, homogeneous, incompressible, and hyperelastic material. To model the system, the neo-Hookean hyperelastic constitutive law is used. The nonlinear elastic foundations include two linear terms Winkler and Pasternak, and a nonlinear cubic term. The governing equations are obtained with Hamilton's principle of thin hyperelastic membrane and the assumption of finite deformations. Then, using Galerkin's method, the equation of motion in the transverse direction, which is considered in two modes, is discretized. The resonant case includes the 1:1 internal resonance between two modes and primary resonance, in which the excitation frequency is close to the natural frequency of the first mode. In this analysis, the responses of nonlinear vibration behavior, including primary and 1:1 internal resonances, are obtained with analytical and numerical methods. The analytical and numerical methods are, respectively, the multiple scales method and the fourth-order Runge–Kutta method. Finally, the effects of the elastic foundation parameters and geometrical characteristics on the nonlinear vibration behavior are examined. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear forced vibration responses of axially moving functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular thin plates are investigated. In the frame of classical plate theory and von Kármán geometric nonlinear theory, the nonlinear partial differential equations of motion for axially moving FG-CNTRC plates are established by Hamilton's principle and discretized through the Galerkin method. The validity of the present model is verified by comparing the natural frequencies and the critical divergence velocity of the axially moving plate with the existing results in the literature. The frequency response curves (FRCs) of axially moving FG-CNTRC plates subjected to transverse harmonic concentrate excitation are calculated by using the incremental harmonic balance (IHB) method. The stability and bifurcation characteristics of period solutions obtained through the IHB method are analyzed by Floquet theory. The isolated detached resonance curves (DRCs) are observed inside the main continuous FRCs of axially moving FG-CNTRC plates. Saddle-node bifurcation, Hopf bifurcation, quasi-periodic motion and the internal resonance caused by the modal coupling effect are analyzed. The numerical results illustrate the effects of the CNT distribution, volume fraction, axially moving velocity, aspect ratio and width-to-thickness ratio on the nonlinear dynamic responses of axially moving FG-CNTRC plates. [ABSTRACT FROM AUTHOR]

The crown pillar in the stope structure belongs to the category of thick plate, and its thickness determines the stability of the engineering structure. The key to determining the safe thickness of the crown pillar lies in solving its deflection function. Previous researchers often used the Galerkin method and other methods except the Ritz method to solve the deflection function of the crown pillar under simple boundary conditions. However, these methods are difficult to solve for the deflection function of the crown pillar under complex boundary conditions. Therefore, this paper builds upon the Reissner plate theory and introduces the Ritz method while considering the influence of strain components εz, γyz, and γzx on the flexural deformation of the crown pillar. Thus, the Reissner–Ritz method for solving the deflection function of the crown pillar in the stope is developed. Taking the failure of the crown pillar in stope 27 in the + 280 m section of the Daxin Manganese Mine as an example, firstly, the maximum tensile stress of the crown pillar is compared with its tensile strength to determine the safe thickness of the crown pillar in stope 27. The correctness of the chosen safe thickness for the crown pillar in stope 27 is verified using the Reissner–Ritz method. Then, FLAC3D is used to model and analyze the 8 m thick crown pillar in stope 27, and the accuracy of the Reissner–Ritz method in determining the safe thickness of the crown pillar is verified through finite element analysis. Finally, a comparison is made between the Reissner–Ritz method and the Galerkin method in solving the deflection function of the crown pillar under the uniformly distributed load, considering both simple and complex boundary conditions. The research results show that the Reissner–Ritz method has significant advantages over the Galerkin method in solving the deflection function of the crown pillar under the uniformly distributed load, even under complex boundary conditions. The findings are of great significance for solving the deflection function of thick plates under both simple and complex boundary conditions under the uniformly distributed load and determining the safe thickness of thick plates. Highlights: Based on the Reissner thick plate theory, the Ritz method is introduced, taking into account the influence of strain components εz, γyz, and γzx on the bending deformation of the crown pillar, forming the Reissner–Ritz method for solving the deflection function of the crown pillar in the stope. The range of safe thickness values for the crown pillar is determined based on the thickness-span ratio and safety factor, incorporating engineering experience to finalize the safe thickness of the crown pillar in the stope. The accuracy of the Reissner–Ritz method in determining the safe thickness of the four-sided fixed support crown pillar under the uniformly distributed load is verified through finite element analysis. Comparing and analyzing the calculation results of the Reissner Ritz method with the Galerkin method. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the flutter characteristics of nanocomposite cantilever trapezoidal plates with non-uniform thickness enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs) which are distributed functionally graded (FG) in the axial direction. It is assumed that the thickness of the plate and the volume fraction of the nanofillers vary in one direction from the wider clamped edge of the plate to the outer narrower free one. The modeling of the plate is done using the first-order shear deformation theory (FSDT) and the aerodynamic pressure generated by the aerodynamic pressure is modeled using the linear approximation of the piston theory. The material properties of the plate are calculated using the mixing rule (ROM) and the Halpin–Tsai model. The governing equations and boundary conditions at the clamped and free edges of the plate are derived via Hamilton's principle. An approximate solution is applied using the differential quadrature method (DQM) to calculate the natural frequencies and the damping ratios of the plate. Numerical examples show that it is possible to find an optimal thickness variation profile that provides the greatest aeroelastic stability. It is concluded that by considering the same value for the mass fractions of the nanofillers, the highest aeroelastic stability can be attained by utilizing the GNPs as the reinforcers. It is found that to attain further improvement in aeroelastic stability, most nanofillers should be distributed near the clamped edge and away from the outer free edge. [ABSTRACT FROM AUTHOR]

Thin rectangular plates are considered basic structures in various sectors like aerospace, civil, and mechanical engineering. Moreover, isotropic and laminated composite plates subjected to transverse normal loading and undergoing small and large deflections have been extensively studied and published in the literature. Yet, it seems that the particular case of long thin plates having a high aspect ratio appears to be almost ignored by various scholars despite its engineering importance. The present study tries to fill this gap, yielding novel findings regarding the structural behavior of long thin plates in the small- and large-deflection regimes. In contrast to what is normally assumed in the literature, namely that a long plate with a high aspect ratio can be considered an infinitely long plate, the present results clearly show that the structural effects of the ends continue to exist near the remote ends of the long plate. An innovative finding is that long plates would (only on movable boundary conditions for the large-deflection regime) exhibit a larger mid-width displacement in comparison with deflections of infinitely long plates. This innovative higher deflection appears for both small and large-deflection regimes for both all-around simply supported and all-around clamped boundary conditions. This new finding was shown to be valid for both isotropic and orthotropic materials and presents a novel engineering approach for the old assumption well quoted in the literature that a relatively long plate on any boundary condition can be considered an infinite plate. Based on the present research, it is recommended that this assumption should be used carefully as the largest plate mid-deflection might occur at finite aspect ratios. [ABSTRACT FROM AUTHOR]

A theoretical model of the normal sound insulation of a rectangular locally resonant (LR) plate with simply supported boundary conditions is established. The accuracy of the model is verified through comparison with finite element simulation results. Considering a 1-m2 aluminum plate as the research object, a non-dominated genetic algorithm is introduced. The algorithm optimizes the lightweight sound insulation of an LR finite plate with a center frequency band of 20–800 Hz as the optimization objective of maximum average sound insulation and minimum mass. The optimization parameters are substrate thickness and local resonance parameters (including resonator target frequency (f obj), number of resonators, additional mass ratio, and resonator damping). Optimization results indicate that the average sound insulation fluctuates at approximately 2 dB due to changes in f obj. The results also provide an optimal solution to the algorithm. The optimal sound insulation value curve of LR plates corresponding to different surface densities of intermediate improved cases is fitted and compared with the sound insulation value of an equal-mass bare plate. Under lightweight optimization, the resonators are deemed capable of improving the local frequency band sound insulation. In contrast, the wideband average sound insulation can approximate but not surpass the insulation provided by the equal-mass bare plate. In practical applications, an LR plate that can improve the local frequency band sound insulation and ensure the overall lightweight sound insulation performance of the wide frequency band can be obtained. This can be achieved by analyzing the frequency spectrum of optimal points in the lightweight sound insulation optimization process. Relevant research can provide a basis for the evaluation and design of the low-frequency sound insulation of plates for transportation equipment, such as airplanes, high-speed trains, and cars. [ABSTRACT FROM AUTHOR]

Based on the three-dimensional elasticity theory, the free vibration of functionally graded porous (FGP) sandwich rectangular plates is studied, and a unified solution for free vibration of the plates is proposed in this study. The arbitrary boundary conditions of FGP sandwich rectangular plates are simulated by using the Rayleigh–Ritz method combined with artificial spring theory. The calculation performances of the unified solution for FGP sandwich rectangular plates such as convergence speed and computational efficiency are compared extensively under different displacement functions. In addition, three kinds of elastic foundation (Winkler/Pasternak/Kerr foundations) and three porosity distributions are considered. Some benchmark results and accurate values for the free vibration of FGP sandwich rectangular plates resting on elastic foundations are given. Finally, the effects of diverse structural parameters, elastic foundations with different parameters, and boundary conditions on the free vibration of the FGP sandwich rectangular plates are analyzed. [ABSTRACT FROM AUTHOR]

This article aims to analyze rectangular functionally graded material plates with rectangular cutouts of diverse sizes, numbers, and positions for free vibration using Mindlin's first-order shear deformation theory (FSDT). Isoparametric plate elements of nine nodes and five degrees of freedom at each node were used for the present finite element formulation. The Poisson ratio is assumed to be constant throughout the plate. At first, results piled up from the present finite element formulation were compared with existing results collected from various previous journals for different isotropic plates and functionally graded material plates. After verification of finite element formulation, some new results (nondimensional fundamental frequency) were obtained considering various shapes, sizes, numbers, positions of cutouts, thickness ratio, aspect ratio, FGM power law index, and different edge conditions. Dynamic analysis of FGM plate with interior support along rectangular cutout edges is an integral part of this study. [ABSTRACT FROM AUTHOR]

Screw piles have a greater bearing capacity than straight piles due to their larger helix. However, an excessively large helix can cause bending and reduce the soil bearing capacity. This study investigates the failure pattern and mechanical performance of screw pile helices through full-scale load tests and numerical analyses. The results revealed that the helix buckled at its connection to the shaft. Additionally, the geological characteristics of the soil in which the pile was located had a negligible effect on the mechanical properties of the helix. Furthermore, the shape of the anchor plate (flat or helical) had a negligible effect on the load-bearing properties of the pile or the mechanical properties of the anchor plate itself. To simplify the analysis, the screw pile helix was assumed to be a flat circular plate. For a uniformly loaded flat circular plate with fixed inner edges, the result of Roark's formula satisfactorily agreed with the measured maximum radial normal stress in the helix. Moreover, the value given by Roark's formula for a flat circular plate with simply supported inner edges agreed well with the measured helix deformation. [ABSTRACT FROM AUTHOR]

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge. [ABSTRACT FROM AUTHOR]

A new method for evaluating the effective flange width of simply supported rectangular box-girders in bending is illustrated. It aims to overcome the lack in the literature of a general method able to supply the effective flange width for a general distribution of loads along the beam axis, not restricted, i.e., to the few cases reported in standard codes. The method consists of finding an analytical expression for the effective width under sinusoidal loads of arbitrary wavelength and combining the results in the framework of a Fourier analysis. The model allows investigating the shear-lag effects on displacements and stresses. The analytical formulation proposed is validated against numerical results supplied by refined Finite Element analyses. [ABSTRACT FROM AUTHOR]

Purpose: Due to the remarkable physical properties of the graphene, the graphene-reinforced composite structures can be applied generally in the aerospace and other engineering fields. However, for the vibration amplitude attenuation of the graphene-reinforced structures, this is a challenging problem for researchers. The integrated piezoelectric sensors and actuators have many advantages as active intelligent materials in many mechatronic and vibration control systems. Therefore, the integrated piezoelectric actuator and sensor are designed by sticking piezoelectric patches on the surface of the FG–GRLCC rectangular plate to study the vibration amplitude attenuation. Methods: Based on the extended Halpin–Tsai micromechanical model, the elastic coefficients of the graphene-reinforced plate are formulated. Considering the classical laminated plate theory, the Hamilton principle, and the piezoelectric coupling effect, the dynamic equations of the piezoelectric FG–GRLCC rectangular plate are derived. The ordinary differential control equations of motion for the plate are obtained by applying the Galerkin method. To reduce vibration amplitude, the parameters of the positive position feedback (PPF) controller are designed using the Routh–Hurwitz stability criterion and the H ∞ optimal strategy. Results: The vibration characteristics for the piezoelectric FG–GRLCC rectangular plate with four different graphene distributions are studied. Numerical simulations are given to study the vibration amplitude attenuation via the proposed PPF control algorithm. In addition, the PPF control algorithm and the velocity feedback controller are compared. Conclusion: The designed PPF control algorithm can reduce the vibration amplitude for the four different graphene distribution for the FG–GRLCC rectangular plate subjected to the transverse excitation. In addition, the key of the vibration suppression is that the frequency of the PPF controller is same with the frequency of the structure. [ABSTRACT FROM AUTHOR]

The combined influences of initial geometrical imperfection, elasticity of in-plane constraints of edges, elastic foundations and elevated temperature on the nonlinear vibration and dynamical response of carbon nanotube-reinforced composite rectangular plates are investigated in this paper. Carbon nanotubes (CNTs) are reinforced into matrix according to functionally graded distributions. The properties of CNTs and matrix are assumed to be temperature dependent and effective properties of nanocomposite are determined using an extended rule of mixture. Governing equations in terms of deflection and stress function are established within the framework of thin plate theory including von Kármán nonlinearity, geometric imperfection and interactive pressure from elastic foundation. Analytical solutions are assumed for simply supported plates and Galerkin method is applied to result in nonlinear time differential equation. This nonlinear equation is solved using fourth-order Runge–Kutta scheme. Parametric studies are executed to examine numerous influences on the natural frequency, nonlinear to linear frequency ratio and nonlinear dynamical response of CNT-reinforced composite plates. The results reveal that increase in imperfection size results in increase and decrease in natural frequencies and the amplitude of forced vibration, respectively. In contrast, it is found that the elevated temperature reduces the natural frequencies and enhances the amplitude of forced vibration. [ABSTRACT FROM AUTHOR]

The solution of the nonlinear (NL) vibration problem of the interaction of laminated plates made of exponentially graded orthotropic layers (EGOLs) with elastic foundations within the Kirchhoff–Love theory (KLT) is developed using the modified Lindstedt–Poincaré method for the first time. Young's modulus and the material density of the orthotropic layers of laminated plates are assumed to vary exponentially in the direction of thickness, and Poisson's ratio is assumed to be constant. The governing equations are derived as equations of motion and compatibility using the stress–strain relationship within the framework of KLT and von Karman-type nonlinear theory. NL partial differential equations are reduced to NL ordinary differential equations by the Galerkin method and solved by using the modified Lindstedt–Poincaré method to obtain unique amplitude-dependent expressions for the NL frequency. The proposed solution is validated by comparing the results for laminated plates consisting of exponentially graded orthotropic layers with the results for laminated homogeneous orthotropic plates. Finally, a series of examples are presented to illustrate numerical results on the nonlinear frequency of rectangular plates composed of homogeneous and exponentially graded layers. The effects of the exponential change in the material gradient in the layers, the arrangement and number of the layers, the elastic foundations, the plate aspect ratio and the nonlinearity of the frequency are investigated. [ABSTRACT FROM AUTHOR]

In this work, a firefly algorithm was implemented and used to optimise composite structures in the postbuckling regime. In the first case studied, the goal was to maximise the post-buckling load of a rectangular plate subjected, independently, to shear load and uniaxial compression. The orientations of the layers served as design variables, while a maximal transverse displacement was considered as a constraint. Next, a reinforced flat panel was studied, with the goal of maximising the shear load in the post-buckling regime while constrained by the Tsai-Wu criterion. The design variables were the positions of the stiffeners and the orientations of the layers of the laminate. The degree of improvement in the maximum post-buckling load depended on the specific problem and ranged from 2.5 to 36 % compared to baseline designs. The selection of the structures chosen for the analyses ensured that the firefly algorithm was tested with progressively more challenging optimisation problems. The results suggested that the firefly algorithm could be used in the design of laminated composite structures. [ABSTRACT FROM AUTHOR]

Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz's method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied. [ABSTRACT FROM AUTHOR]

The buckling and postbuckling behaviors of functionally graded graphene platelets-reinforced composite (FG-GPLRC) circular plates are studied based on the classical nonlinear von Karman plate theory. The effective Young's modulus of the composite is estimated using the modified Halpin–Tsai micromechanical model, and the effective Poisson's ratio is estimated by the rule of mixtures. Governing equations of the problem are derived based on the Hamilton principle and the numerical solutions of critical loads and postbuckling deflection–load relationships are calculated using the shooting method. Different from the existing linear buckling analysis based on the Terriftz criterion, the study with considering the global deformation of the plates, we analyze the influencing factors of the critical buckling loads and postbuckling paths of the FG-GPLRC circular plates subjected to uniformly distributed radial pressure. The results show that the content, geometric parameters and distribution pattern of GPL have great influences on the critical buckling loads and the post-buckling bearing capacities of the circular FG-GPLRC plates. [ABSTRACT FROM AUTHOR]

In the present study, the effect of non-linear thickness variation and that of Winkler's foundation is investigated on radially symmetric vibrations of bi-directional FGM circular plates subjected to uniform in-plane peripheral loading. It is considered that the elastic modulus, as well as density of the FGM, varies in two mutually perpendicular directions (i.e., radial and transverse). This variation follows the power law. However, the variation in thickness of FGM plate varies along the radial direction and follows quadratic variation. Hamilton's principle is used to obtain the coupled differential equations governing the motion. The approximate solutions of these equations for simply supported and clamped edge conditions are obtained by an elegant numerical technique: the Harmonic differential quadrature method. The effect of density parameter, gradient index, heterogeneity parameter, taper parameters, and thickness parameter of the plate is analyzed on the vibration characteristics for the first two modes of vibration for different values of foundation parameter together with in-plane peripheral loading parameter. Furthermore, critical buckling loads in compression by taking the frequencies zero are evaluated for both plates. The validity of the current approach is established by comparing the frequency parameter with true values and with the results of other authors. [ABSTRACT FROM AUTHOR]

The present paper deals with the modal analysis of sigmoid functionally graded (S-FGM) rectangular plate resting on elastic foundation by using the dynamic stiffness method (DSM). The DSM is formulated based on the exact solutions of the governing differential equations, and thereby it results in the very accurate computation of the natural frequencies. To obtain the DSM results for thicker plates, the study incorporates first-order shear deformation theory (FSDT) which includes the important effects of transverse shear deformation and rotatory inertia. The governing equations and the associated natural boundary conditions are derived using Hamilton's principle, and the solution is sought in the Levy form where two opposite edges of the plate are simply supported. The present study also contributes by highlighting mistakes in the classical plate theory (CPT)-based DSM formulation published in a recent work and presents a correct CPT-based mathematical formulation. For both these cases, the frequency-dependent dynamic stiffness matrix of the S-FGM plate gives rise to the transcendental eigenvalue problem, which is solved by using the Wittrick and Williams algorithm. Comparison with the available literature establishes the accuracy of the method. In addition, a parametric study is presented for various geometric and stiffness parameters of the elastically supported S-FGM plates using both CPT- and FSDT-based formulations, and accurate frequency results are reported. [ABSTRACT FROM AUTHOR]

This paper investigates the size-dependent vibration and buckling behavior of nanoplates described by the nonlocal Kirchhoff model analytically. The nonlocal and local constitutive relations, governing equations and boundary conditions are comprehensively discussed. A shifted governing equation in terms of local stress resultants with associated local boundary conditions is chosen in the current work. The direct separation of variables method is formulated based on the Hamiltonian dual form of the governing differential equation to address the eigenvalue problems of nonlocal rectangular plates with each edge either simply supported or clamped. The closed-form eigen-solutions are acquired for the first time for the cases where two adjacent clamped edges exist. The validity and accuracy of the present approach are verified by comparison with published analytical and numerical results. Parametric studies are performed to investigate the influences of nonlocal parameter and boundary conditions on the size-dependency of natural frequencies and critical buckling loads of nanoplates. [ABSTRACT FROM AUTHOR]

Purpose: Due to the significance of the effect size on microstructures and also due to the fact that classical continuum theory cannot predict their behavior, higher order continuum theories should be used for modeling and obtaining the governing partial differential equations of microplates. For considering the size effect, the strain gradient theory is used. Vibration control of a microplate is studied using a linear boundary control law, and by utilizing a Lyapunov functional, the boundary stabilization of the system will be proved. The subject of vibration control of a strain gradient microplate utilizing the PDE control theory. The boundary action, in which the actuator is placed on the free sides of the plate, is used to stabilize mechanical vibrations. Method: The governing equation and boundary condition of strain gradient microplate using Hamilton principle are derived. The boundary control problem of a CFFC (Clamped-Free-Free-Clamped) non-classical microplate using Lyapunov stability theorem and the design of a proper controller for the system are stablisged. A Lyapunov-based functional is used to demonstrate closed-loop system boundary stabilization. The verification of the theoretical results is accomplished using the finite-element method. New non-classical microplate element stiffness and mass matrices are developed based on the strain gradient theory. To derive strain gradient plate element stiffness and mass matrices, a four-node rectangular element is utilized. The verification of the theoretical results is accomplished using the finite-element method. Numerical method (Finite Element Method) is used for microplate simulation. Results and Conclusion: Boundary stabilization of a vibrating strain gradient plate has been achieved using the Lyapunov stability theorem and boundary control. The boundary action is used to stabilize mechanical vibrations. The boundary PDE control technique used in this study provides various advantages. This simulations shows good agreement with the theoretical results. The deflection, slope, velocity and angular velocity are comprised in open loop and closed loop. Appling control force {F} at the free sides of the plate, causes to stabilize the vibrating system. [ABSTRACT FROM AUTHOR]

In this article, a new mathematical model to develop porous skew partially functionally graded (SPFG) magneto-electro-elastic (MEE) plate from a stepped functionally grade (SFG) plate is proposed. The combination of Barium Titanate (BaTiO3) and Cobalt Ferrite (CoFe2O4) is graded stepwise to achieve SFG plates. Such, SFG plate is approximated using modified power law for functional variation through its thickness. Modified first-order shear deformation theory (FSDT) is deployed in the current study to develop mathematical models. The geometrical modification from rectangular to skew plate is achieved using transformation matrix. Porosity in the partial functionally graded MEE plate is taken as local density. Free vibration study is carried out to reflect the influence of partial gradation, porosity distribution, porosity volume, and skew angle on the natural frequency of porous SPFG MEE plate. The influence of various types of porosity distribution, partial gradation, and skew angle on the primary quantities such as displacements, potentials, and secondary quantities such as stresses, electric displacement, and magnetic induction on SPFG MEE plate is studied. The stability characteristics of the SPFG MEE plate are also investigated to study the buckling behavior under various porosity distributions, partial gradation, and skew angles. The effect of change in aspect ratio, and thickness ratio on the free vibration and static behavior is also investigated. [ABSTRACT FROM AUTHOR]

This study investigates the vibration control of a plate subjected to moving masses. The plate model reflects the dynamic behavior more accurately than the beam model which is usually used in the existing literature. However, the vibration control of plates subjected to moving masses is also more challenging as more degrees of freedom need to be taken into account. In addition, the measurement of system status and uncertainty problems in practical application is another nodus. In this paper, combining an adaptive sliding‐mode control (ASMC) strategy and a modal coordinate reconstruction (MCR) method, the MCR‐ASMC method is proposed. The MCR method is employed to capture the plate dynamic behavior for feedback to the ASMC. The ASMC strategy realizes vibration control through the adaptive updating of parameters. Numerical results demonstrate the effectiveness of the proposed method and excellent control performance in vibration suppression of moving mass problems. [ABSTRACT FROM AUTHOR]

An analytical formulation based on a quasi-3D shear deformation theory is proposed to examine the bending response of functionally porous/perfect plates supported on a Kerr/Pasternak/Winkler elastic foundation in hygro-thermo-mechanical conditions. Two kinds of plates, perfect and porous, were studied. For the perfect plates, Mori–Tanaka, LRVE, Voigt, and Reuss models were employed to calculate their mechanical properties changing across the thickness. For the porous plates, different porosity patterns were considered to estimate their mechanical properties. After establishing the governing equations, the Navier solution for simply supported plates was used to calculate their displacements and stresses. The results obtained were compared with data published in the literature, and a good agreement was revealed. Some numerical results were used to investigate the effects of the porosity function, porosity percentage, micromechanical models, and the elastic foundation on the displacements and stresses of a simply supported porous/perfect plate under hygro-thermo-mechanical loadings. The results obtained reveal that these variables have a great influence on the response of the plate in hygro-thermo-mechanical environments. [ABSTRACT FROM AUTHOR]

Curved structures are used in many engineering applications. Temperature affects the performance of structural vibration and acoustics and can damage the equipment when its effects are severe. A theoretical model of the sound transmission loss (STL) of a simply supported curved plate considering the effect of temperature was established. Moreover, a numerical solution was obtained using the modal superposition method. The finite element results were compared to validate the theoretical model. The effects of temperature and acoustic loading on the modal frequency and STL were considered, and the variation laws of the radius of curvature, opening angle, and loss factor on the STL were analyzed. The results showed that the modal frequency increased with decreasing temperature, the frequency ratio of the same-order modal frequency varied approximately linearly with temperature, and the trough of the STL shifted toward high frequencies. The pitch angle substantially influenced the STL, whereas the azimuth angle was negligible. A larger radius increased the effect of temperature. In addition, an increase in the opening angle formed an aggregation of STL at (1, 1)-order modal frequencies. The loss factor affects only the amplitude of the trough position of the STL. Therefore, the proposed theoretical model can aid in the vibroacoustic design of curved plates in a thermal environment. [ABSTRACT FROM AUTHOR]

The three-dimensional bending behavior of a viscoelastic functionally graded material (FGM) layer embedded between piezoelectric layers and subjected to an electric field as well as a uniform transverse pressure is studied. An analytical solution is computed for a simply supported viscoelastic smart FGM plate using the state space technique along the thickness direction and a Fourier expansion along the in-plane coordinates. The governing differential equations in the time domain are transformed to the Laplace domain, solved in the Laplace domain, and transformed back to the time domain using the inverse Laplace transform. In the present study, the relaxation modulus of the FGM layer is assumed in the form of a Prony series, and it varies according to the power law in the thickness direction. The validity of the proposed approach is assessed by comparison of the numerical results of the approach with those of published works in the literature. The effects of geometric dimensions, electromechanical loads, and relaxation time constant on the behavior of the smart plate are investigated. [ABSTRACT FROM AUTHOR]

In this paper, thermal buckling and free vibration of a rectangular plate reinforced with carbon nanotubes (CNT) are investigated within the framework of third-order shear deformation theory (TSDT). CNT distribution along the thickness direction of the plate is uniformly or functionally graded. The equivalent properties of the reinforced composite plate are calculated based on the extended rule of mixture. Governing equations of motion are derived using the Hamilton principle. Obtained governing differential equations are analyzed by utilizing Fourier series expansion along the longitudinal and latitudinal direction for simply supported edges boundary conditions whereas for other edges boundary conditions we used the differential quadrature method (DQM) to solve numerically. Validation of the present formulation is assessed by comparing the results with those reported in the open literature. The effect of CNT volume fraction, the different patterns of CNT distribution, temperature difference, aspect ratio, and thickness-to-length ratio on buckling and vibration behavior of carbon nanotube-reinforced composite (CNTRC) plate is studied. The numerical illustration reveals that in the FG-X pattern of CNT distribution natural frequency and thermal buckling load is more significant compared with the other patterns of CNT distribution. [ABSTRACT FROM AUTHOR]

The paper elaborates the principally new variant of finite element method in application to plate problem. It differs from classical FEM approach by, at least, three points. First, it uses the strong differential formulation rather than the weak one and suppose the approximate analytical solution of all differential equations. Second, it explicitly uses all geometrical and physical parameters in the procedure of solution, rather than some chosen ones, for example, displacement and angles of rotation as usually done in FEM formulation. Third, the conjugation between adjacent elements occurs between the adjacent sections rather than in polygon vertexes. These conditions require the continuity of displacements, angles, moments and forces. Each side of rectangular elements is characterized by 6 main parameters, so, at whole there are 24 parameters for each rectangular element. The right and upper sides’ parameters are considered as output ones, and they are related with lower and left sides ones by matrix equations, which allows to apply transfer matrix method for the compilation of the resulting system of equations for the whole plate. The numerical examples for the thin-walled and Mindlin plates show the high efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

New accurate buckling analysis for rectangular orthotropic thin plates with complicated non-classical boundary restraints are conducted through adopting the finite Fourier integral transform approach. Non-classical boundaries such as rotationally restrained edges increase the mathematical difficulty in processing problems of plates, which leads to rare analytical results for benchmark use. The proposed approach is implemented in the framework of integral transform theory, in which trial function for the deflection is not necessary, and offers uniform solution procedures for problems of plates with various boundaries via adopting different integral kernels. The main merits of the approach employed is to enable one to change the complicated title problem into dealing with linear algebraic equations easily solved. Via altering the rotational spring factors introduced, buckling behaviors of plates with Levy-type boundaries and non-Levy-type boundaries can also be studied. Finally, all the given results including critical load factor and mode shape match the FEM analysis exactly, which illustrate the accuracy and validity of the method. [ABSTRACT FROM AUTHOR]

The complex nature of the anisotropic plate bending problems undergoing large deflection necessitates the need for an efficient approach. The generalized matrix-form formulation proposed herein is based on the Ritz method to handle bending of rectangular anisotropic thin plates undergoing large deflection. This matrix formulation is then encoded in a Mathematica program that provides an automated solution for the problem. The derived matrix-form formulation and the developed code are tested by two numerical examples of composite laminated plates involving different degrees of anisotropy. In both examples, the accuracy of the proposed method is demonstrated by comparing the present results with FEM and the available results from the literature. The results of the investigated plates showed the effect of extension–shear and bending–twisting couplings on the symmetry of the deflection, bending moments, and in-plane forces. [ABSTRACT FROM AUTHOR]

In this paper, new tangent shape function-based higher-order transverse shear deformation theory (NTHSDT) is proposed to compute the buckling behavior of the elastically supported functionally graded material (FGM) sandwich plates under porous medium. The proposed theory is found to be variationally consistent and fulfills the zero traction boundary conditions on the bottom and top layer without a shear correction factor. The material properties are presumed to be graded in the thickness direction as characterized by a modified power law distribution in terms of volume fraction of constituents. The governing equations are derived using Hamilton's Principle. A strong form of solution discretizes the governing equations by employing a thin plate spline radial basis function-based collocation (TSRBFC) method. The proposed theory is efficient, reliable, and is in close agreement with the results in the literature. Comparison studies show that the NTHSDT is more accurate than other plate theories and is simple in analyzing buckling behavior. A parametric study is done to examine the effects of grading index, porosity index, sandwich schemes, aspect ratio, side-to-length thickness ratio and foundation stiffness. [ABSTRACT FROM AUTHOR]

We prove the existence of an optimal domain for minimizing the buckling load among all, possibly unbounded, open subsets of R n ( n ≥ 2 ) with given measure. Our approach is based on the extension of a two-dimensional existence result of Ashbaugh and Bucur and on the idea of Alt and Caffarelli to focus on the eigenfunction. [ABSTRACT FROM AUTHOR]

In this paper, we mainly aimed to design a reduced-order extended state observer-based active vibration controller for a structural vibration control system with total disturbances, i.e., model uncertainties, higher harmonics, and external excitations. A reduced-order extended state observer (RESO)-based nonsingular terminal sliding mode vibration control (RESO–NTSMVC) method is proposed for the vibration suppression of an all-clamped plate structure with an inertial actuator. First, a second-order state space model of the thin plate, with an inertial actuator, was established by solving the dynamic partial differential equation and analyzing the physical model. Second, the total disturbances, i.e., model uncertainties, higher harmonics, and external excitations, were estimated and compensated for by using a RESO via a feedforward part. Third, a NTSMVC based on an estimated value was designed to obtain a fast-tracking rate and effective vibration suppression performance. In addition, the stability of the closed-loop system was proven by using a Lyapunov stability criterion. Finally, a semi-physical experimental instrument was built based on the MATLAB/Simulink real-time environment and the NI-PCIE6343 acquisition card to verify strong anti-disturbance performance and effective vibration control performance of the designed method. The experimental comparison results showed that the vibration amplitudes of the proposed method could be reduced by 11.7 dB, when the traditional extended state observer-based nonsingular terminal sliding mode vibration control (ESO–NTSMVC) method achieved a control effect of only 6.5 dB. The comparative experimental results showed that the proposed method possessed better vibration suppression performance and anti-disturbance performance. [ABSTRACT FROM AUTHOR]

This study proposes a force plate with a planar spring and an eddy current displacement sensor to measure the ground reaction force (GRF) of a small insect and reveal its motion characteristics. The proposed force plate comprises a circular aluminum plate, four aluminum springs symmetrically connected to the plate, and an eddy current displacement sensor under the plate. The diameter and thickness of the fabricated plate were 8 and 0.1 mm, respectively. The spring width was 0.4 mm. When a force is applied to the plate, the plate moves vertically downward. Then, an eddy current displacement sensor detects the plate displacement without contact. The applied force can be measured using Hooke's law. The proposed force plate has the advantages of ease of fabrication and cost-effectiveness. The central displacement variation and resonant frequency of the designed springs were evaluated by simulation. Then, we calibrated the fabricated force plate to obtain the sensitivity variation and resonant frequency. The experimental results suggest that the proposed force plate can effectively measure the GRF of a small insect. [ABSTRACT FROM AUTHOR]

• Free vibration of elastically line-hinged orthotropic plates is studied. • Rotationally restrained edges are incorporated in the analysis. • Symplectic analytical free vibration solutions are obtained. • Comprehensive results under various boundary and hinge connection cases are presented. The elastically line-hinged orthotropic rectangular plates with rotationally restrained edges are commonly used in engineering applications such as deployable structures. However, it is intractable to obtain the analytical solutions for the free vibration problems of such structures owing to the challenges in processing the high-order partial differential equations. Here, we make a first attempt to deal with such issues within the Hamilton system-based symplectic space. The problem of a plate is transformed into the symplectic space from the Euclidean space, and the subplates that may be analyzed analytically by the symplectic superposition method are then obtained with the division of the entire plate. The complex boundary and connection forms are achieved by enforcing mechanical quantities with undetermined expansion coefficients on the edges of the subplates. By integrating the solutions of the subplates, the final solution of an elastically line-hinged orthotropic rectangular plate with rotationally restrained edges is accessible. The proposed solution scheme is performed with rational derivations, with no requirement for pre-defined solution forms. Comprehensive results with validations under various boundary and hinge connection cases are presented. Moreover, detailed parametric investigations are conducted, which could facilitate the engineering design of deployable structures. [ABSTRACT FROM AUTHOR]

Although the thermal buckling behavior of thin plates has received great attention, only a limited number of studies have reported the thermal buckling analysis of damaged rectangular plates. A long-term loading can induce material damage on such plates, resulting in the premature failure. The present work addresses the thermal buckling of thin rectangular plates with damage. The continuum damage mechanics approach is used to represent material damage, which is accordance with the stiffness reduction. Stability equilibrium equations of rectangular plates with damage under thermal loads are derived based on classical plate theory. Take a concrete rectangular plate with damage on four edges simply supported under pressure as an example, the analytical solutions of temperature variation function and the effects of geometry size on the buckling temperature are obtained. The numerical results indicate that considering material damage in thermal buckling analysis is more in line with engineering practice and improves the design requirements of reliability. The correctness of the present numerical results through a comparison of the present results and the existing results. [ABSTRACT FROM AUTHOR]

Considering the geometric nonlinearity and geometric imperfection of stiffened cylinders, the generalized similitude conditions and scaling laws for axial compression post-buckling are established by applying the similitude transformation to the total energy of the structure. The post-buckling similarity of stiffened cylinders is numerically analyzed, and scale laws, through innovative application of the stiffness formulas and deformation scale factor. Based on three type geometrical imperfections, the effects of radius to effective thickness ratios, stiffened cross-sections, boundary conditions and imperfection coefficients are investigated by post-buckling partial similarity simulation of stiffened cylinders under axial compression. The results show that the partial similarity can be well realized by changing other parameters or selecting a model with a similar Poisson's ratio for the prototypical material on the premise of invariant stiffness scale factors. Partial similarity simulation of axial compression post-buckling of stiffened cylinders is not affected by radius to effective thickness ratios (23.76–268.95), classical simply supported and fixed support boundary conditions, rectangular or T-type stiffened cross-sections and three type of geometric imperfections. [ABSTRACT FROM AUTHOR]

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations. [ABSTRACT FROM AUTHOR]

The purposes of this paper are to study bending, buckling, and vibration by considering micro-scale effects using the Kirchhoff thin-plate theory and to consider small deflections, neglecting higher-order nonlinear terms. The governing equations for the bending, buckling, and vibration of the system are obtained using the equilibrium method coupled with the Kirchhoff thin-plate theory and a modified couple stress theory (MCST). The concept of the equivalent bending stiffness (EBS) of micro-thin plates is proposed to describe the scale effect. The Navier method is used to obtain analytical solutions for the bending, buckling, and free vibration of thin plates under simply supported boundary conditions with scale effects. The numerical results are presented to investigate the influence of scale effects on deflection, critical buckling load, buckling topography, and thin-plate natural frequency. The results show that the scale effect increases the equivalent stiffness of the thin plate, which leads to a decrease in deflection, a larger critical buckling load, and an increase in natural frequency, but does not affect the buckling topography. The MSCT is invalid when the thickness is greater than 10 times the scale effect parameter, thus defining the scope of application of the scale effect. This research study may contribute to the design of micro-scale devices such as MEMSs/NEMSs. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a micro-nano particle on the micro-thin plate driven by an acoustic signal was investigated, including the particle kinematics mode, kinematics equation, and trajectory control. According to Newton's kinematic theorem, analyzing the forces acting on the particle, the kinematic modes of the driven particle are distinguished with specific mathematical conditions, which are classified as slide, bounce, and stable modes strictly planned on a thin plate area. Based on the theory of kinematic modal analysis, the simulation results reveal the distribution rules of particle motion modes against the driving signal or plate geometry. The particle kinematics equation governing the sliding movement on the thin plate was then derived in light of the interaction between the particle and driving signal, based on which, the particle trajectory was drawn and analyzed in detail. For the purpose of controlling the particle trajectory, the control problem was designed in accordance with a linear active disturbance rejection controller (LADRC). Further, a guidance law was proposed, and the corresponding controller was designed to realize the linear trajectory following. [ABSTRACT FROM AUTHOR]

Considering the problems of model uncertainties, higher harmonics, uncertain boundary conditions, external excitations, and system time delay in practical vibration control system, a novel active vibration control method is proposed to suppress the vibration of a thin plate structure with acceleration sensor and piezoelectric bimorph actuator in this paper. First, a nonlinear extended state observer (NESO)-based controller is designed to ensure the anti-disturbance performance of the structural vibration control system. Then, an enhanced differentiator-based time delay compensation method is introduced to improve the vibration suppression performance of the NESO-based controller. A real time hardware-in-the-loop benchmark for an all-clamped piezoelectric thin plate is designed to verify and compare the performance of the developed controller against conventional ESO-based methods (linear ESO with/without time delay compensation, NESO without time compensation). The best vibration suppression and disturbance rejection performance of the proposed NESO-based controller with an enhanced time delay compensator is verified in the comparative experimental results. This work is able to provide practitioners with vital guidance in designing active vibration control system in the presence of disturbances and time delay. [ABSTRACT FROM AUTHOR]

This paper presents some new analytic thermal buckling solutions of temperature-dependent moderately thick functionally graded (FG) rectangular plates with non-Lévy-type constraints within the symplectic solution framework in the Hamiltonian system. An original problem is reduced to the superposition of two constructed subproblems that are analytically solved via the rigorous symplectic elasticity approach with mathematical techniques such as variable separation in the symplectic space and symplectic eigen expansion. The physical neutral surface is employed to remove the stretching-bending coupling in constitutive equations. The main characteristic of the present symplectic superposition strategy is on the rational derivation without assumption on solution forms, which, however, is hard to achieve by conventional semi-inverse methods. Comprehensive benchmark results are presented, including critical buckling temperatures as well as mode shapes of typical non-Lévy-type FG plates in uniform and nonlinear temperature fields, and are verified by available numerical results. The effects of boundary constraint, aspect ratio, volume fraction exponent, and thickness-to-width ratio on buckling temperatures are quantitatively investigated. The occurrence of thermal buckling is further studied by presenting a failure mode map covering a wide range of geometric parameters. [ABSTRACT FROM AUTHOR]

A mode-counteraction-based self-decoupling method is proposed for the circularly polarized (CP) multi-input multi-output (MIMO) microstrip patch antenna (MPA) array. The array simply consists of two identical diagonal-fed rectangular MPAs, without any extra decoupling structure. It is found that by properly adjusting the patch dimensions and feeding positions, the orthogonal TM1, 0 and TM0, 1 modes can be simultaneously excited in the active MPA to generate the CP field. Meanwhile, the coupled $E$ -fields of the two modes in the adjacent passive MPA can counteract each other, generating a null-field region. When the feeding probe of the passive MPA is placed within the null-field region, no signal can be received at the port, and consequently, an extremely low mutual coupling of −59.4 dB is achieved, without scarifying any other radiation performance of the MPA. In addition, this self-decoupling method is also applicable to other types of MPA arrays and four-element MPA arrays, exhibiting great versatility and extensibility. [ABSTRACT FROM AUTHOR]

Size-dependent functionally graded material thin plate buckling and post-buckling problems are considered using the framework of the MCST (Modified Couple Stress Theory). Based on modified couple stress theory and power law, the post-buckling deflection and critical buckling load of simply supported functionally graded material thin plate are derived using Hamilton's minimum potential energy principle. The analysis compares the simulation results of linear buckling and nonlinear buckling. Innovatively, a power-law distribution with scale effects is considered. The influences of scale effect parameters l and power-law index parameters k on buckling displacement, load, and strain energy of plates have been investigated. In this article, it is found that the critical buckling displacement, critical buckling load, and buckling strain energy increase with increases in the power-law index parameters k. The membrane energy decreases as the power-law index parameter increases. If the upper and lower layers are swapped, the opposite result is obtained. In comparison, the scale effect parameter is more influential than the power-law exponent. The critical buckling displacement in the x-direction is not affected by scale effects. The critical buckling load, the membrane energy, and buckling strain energy increase as the scale effect parameter increases. Scale effects increase material stiffness compared with traditional theory, and the power-law index parameters affect FGM properties such as elastic modulus, Poisson's ratio, density, etc. Both scale effects parameters and power-law index parameters have important effects on the mechanical behavior of materials. [ABSTRACT FROM AUTHOR]