Winkler-Pasternak Elastic Foundation Research Articles

Free vibration of FG-GPLRC conical panel on elastic foundation

Present research is aimed to investigate the free vibration behavior of functionally graded (FG) nanocomposite conical panel reinforced by graphene platelets (GPLs) on the elastic foundation. Winkler-Pasternak elastic foundation surrounds the mentioned shell. For each ply, graphaene platelets are randomly oriented and uniformly dispersed in an isotropic matrix. It is assumed that the Volume fraction of GPLs reainforcement could be different from layer to layer according to a functionally graded pattern. The effective elastic modulus of the conical panel is estimated according to the modified Halpin-Tsai rule in this manuscript. Cone is modeled based on the first order shear deformation theory (FSDT). Hamilton’s principle and generalized differential quadrature (GDQ) approach are also used to derive and discrete the equations of motion. Some evaluations are provided to compare the natural frequencies between current study and some experimental and theoretical investigations. After validation of the accuracy of the present formulation and method, natural frequencies and the corresponding mode shapes of FG-GPLRC conical panel are developed for different parameters such as boundary conditions, GPLs volume fraction, types of functionally graded and elastic foundation coefficients.

Size-dependent instability of organic solar cell resting on Winkler–Pasternak elastic foundation based on the modified strain gradient theory

The present study employs the modified strain gradient theory (MSGT) in conjunction with the refined shear deformation plate theory to explore the buckling behaviour of simply supported and clamped OSC. The Winkler-Pasternak elastic foundation is implemented to idealise the foundation. The size-dependent effect of the OSC is captured by the three length scale parameters within the MSGT. The Hamilton principle is used to derive the equations of motion and the boundary conditions, and the Galerkin procedure is subsequently implemented to obtain the critical buckling load. Subsequently, the framework is extended to the thermally induced buckling behaviour, and three types of temperature rise patterns, namely uniform, linear and nonlinear temperature variations, along the thickness of the OSC are considered. Several verification studies are conducted to illustrate the accuracy of the present method. Besides, size-dependent material properties are taken into consideration during the numerical experiments. Thorough studies are conducted to demonstrate the difference between critical buckling loads obtained from the MSGT, the modified couple stress theory (MCST), and the classical plate theory (CPT) models. Furthermore, the effects of length scale parameter (h/l), the aspect ratio (a/b), the length-to-thickness ratio (a/h) and the Winkler-Pasternak elastic foundation parameters on the buckling behaviour of the OSC are also revealed by the numerical results.

Analysis of functionally graded plates resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical loading

In the present study, a new fifth-order shear and normal deformation theory is developed and applied for the bending analysis of functionally graded (FG) plates resting on two-parameter Winkler–Pasternak elastic foundation subjected to non-linear hygro-thermo-mechanical loading. The theory involves the effects of transverse shear and normal deformations, i.e. thickness stretching. Navier’s solution technique is used to obtain analytical solutions for simply supported FG plates. The results are presented in non-dimensional form and are compared with previously published results in the literature. The present study has the following novelties. The present polynomial-type theory is computationally simpler than non-polynomial-type plate theories which are mathematically complicated, tedious and more cumbersome. For the accurate structural analysis of composite plates under hygro-thermal loading, consideration of thickness coordinate up to third-order polynomial is not sufficient. Therefore, in the present theory, thickness coordinate is expanded up to fifth-order polynomial to get the accurate displacements and stresses. Transverse normal stress/strain plays an important role in the modeling of thick plates which is neglected by many theories available in the literature.

Dynamic behavior of FGM viscoelastic plates resting on elastic foundations

The free vibration (FV) and dynamic stability (DS) analyzes are presented for functionally graded viscoelastic plates (FGVPs) under compressive load and resting on elastic foundations (EFs). Winkler elastic foundation models and Pasternak elastic foundation models are used as elastic foundations. The basic equations of FGVPs interacting with EFs are derived using the concepts of Boltzmann and Volterra. An analytical method for studying the DS and FV of FGVPs interacting with EFs is developed using the integro-differential equations. To solve the current problem, Galerkin and Laplace methods are used. A technique for the analysis of DS and FV of FGVPs on the EFs is developed. To confirm the proposed formulation, the results are compared with other available solutions. Finally, the influences of EFs, volume fractions and rheological constants on the critical times and frequencies depending on the geometrical characteristics and loading parameters are examined.

Static bending and free vibration of organic solar cell resting on Winkler-Pasternak elastic foundation through the modified strain gradient theory

Organic solar cell (OSC), which is deemed to be the most promising third generation solar energy application, is developing vigorously. Based on the modified strain gradient theory (MSGT) and the refined shear deformation plate theory, static bending and free vibration of the size-dependent OSC are thoroughly investigated in this paper. A Winkler-Pasternak elastic foundation is considered for the OSC. A multiscale suitable plate analysis framework (i.e., both macro- and micro plates can be handled) is developed herein. Three length scale parameters are incorporated in the presented analysis to capture the size-dependency of the OSC. By setting two or three of them into zero, the presented model could degenerate into the modified couple stress theory (MCST) and the classical plate theory (CPT). The derivation of the governing equations and the corresponding boundary conditions are conducted by Hamilton principle. The Navier-type solution is employed for solving the governing equations of the simply supported OSC. The accuracy of the presented method is validated. Extensive numerical experiments have been conducted to investigate the differences between the adopted MSGT, the MCST and the CPT. Moreover, the impacts of the geometrical configuration as well as the elastic foundation parameters on the static bending and free vibration characteristics are illustrated in the numerical studies. This paper also explores the thickness of the active layer effect on the free vibration behaviour in combination with the power conversion efficiency (PCE) of the OSC.

Nonlinear Flutter Suppression and Thermal Buckling Elimination for Composite Lattice Sandwich Panels

A mechanism is proposed to suppress the nonlinear flutter and thermal buckling of composite lattice sandwich panels with tetrahedral truss core in supersonic airflow by means of the Winkler–Pasternak elastic foundation. The strain-displacement relations are evaluated according to the von Kármán large deflection and first-order shear deformation theories. The aerodynamic pressure acting on the sandwich panel is formulated by the supersonic piston theory. The nonlinear equations of motion of the structure are established by Hamilton’s principle in conjunction with the assumed mode method. By using the Winkler–Pasternak elastic foundation, an effective method for eliminating the nonlinear thermal buckling of the structure is presented. The natural frequencies of the sandwich panels are validated through comparing with the published results. The parametric investigations on the shearing layer and Winkler parameters are conducted. The influences of several parameters, including ply angle of laminated face sheets, and radius and material properties of the trusses on the nonlinear flutter, and thermal buckling behaviors are investigated. The numerical results attest that the nonlinear thermal buckling effect can be completely eliminated, and the nonlinear flutter of the composite lattice sandwich panels can be effectively suppressed by adjusting the elastic foundation parameters.

Free vibration analysis of metal foam core sandwich beams on elastic foundation using Chebyshev collocation method

In this paper, free vibration of a metal foam core sandwich (MFCS) beam embedded in Winkler–Pasternak elastic foundation is studied using the Chebyshev collocation method (CCM). This method can achieve high precision within the range allowed by the effective number of bits of computers. Three foam distribution types along the thickness direction are considered for the core. The Timoshenko beam theory is adopted and Hamilton’s principle is utilized to derive the boundary conditions and governing equations of the model. The numerical results show that natural frequencies of the sandwich beam initially increase and then decrease with the rise in thickness of metal foam core. By arranging the foam distribution in the core, natural frequencies of the sandwich beam can be significantly changed. Moreover, natural frequencies of the uniform foam distribution beam are insensitive to the foam coefficient. For the beam with non-uniform foam distribution, however, the natural frequencies increase or decrease with the foam coefficient, depending closely on the foam type. In addition, the present method is validated by comparing with the published ones for special cases.

Vibration Analysis of Functionally Graded Timoshenko Beams on Winkler–Pasternak Elastic Foundation

This paper aims to analyze dynamic response of axially functionally graded (FG) Timoshenko beams on Winkler–Pasternak elastic foundation. Spring constant and material inhomogeneity are used as research parameters in the dynamic analysis. In this study, ODEs in Laplace domain are calculated by using complementary functions method. The analysis is performed in the Laplace domain, and calculations are converted to the time domain by using Durbin’s algorithm. The effects of material gradient index, translational and rotational spring constants, slenderness ratio, different boundary conditions and impulsive loads on the dynamic response of FG Timoshenko beams are investigated.

On the statics of fullerene structures

In the present study, thorough research on bending analysis of the fullerene structures (such as C60, C720, C2160, and C4860) resting on the Winkler-Pasternak elastic foundation and under external loading is conducted. Small-scale effects on the results are taken into consideration via nonlocal elasticity theory. The geometry of fullerene structures is assumed as a sphere and doubly-curved structure. The static governing equations for fullerene structures are derived by implementing the principle of minimum potential energy on the basis of the nonlocal first-order shear deformation theory. Various types of boundary conditions including the free edges are considered. The elastic foundation is applied locally on the particular area of the structure, on the contrary of the most available researches that have considered it on the whole of problem geometry. For this aim, an innovative formulation is presented in which simulates the elastic foundation locally. The resulting equations are solved by using the semi-analytical polynomial method (SAPM). Since no similar research has been done before, the bending results of a spherical structure (whereas the fullerene structure can be approximated by a sphere) are compared and validated with the results of ABAQUS software. Also, the effects of several important parameters such as the type of fullerene structure, the nonlocal parameter, and the elastic foundation constants on the results are investigated in detail.

Nonlinear Hygro-Thermo-Mechanical Analysis of Functionally Graded Plates Using a Fifth-Order Plate Theory

In the present research, a new fifth-order plate theory is developed and applied to evaluate the static response of functionally graded (FG) plates resting on Winkler–Pasternak elastic foundation under nonlinear hygro-thermo-mechanical loading. This theory involves polynomial shape functions expanded up to fifth order in terms of the thickness coordinates. The effects of shear deformation and normal deformation are accounted; hence, it can be called as quasi-3D plate theory and abbreviated as FOSNDT. The theory involves nine unknowns. Zero transverse shear stress conditions at top and bottom surfaces are satisfied using constitutive relations. Analytical solutions are obtained using the double Fourier series technique suggested by Navier. The non-dimensional displacements and stresses are compared with available results.

Nonlinear vibrations, bifurcations and chaos of lattice sandwich composite panels on Winkler–Pasternak elastic foundations with thermal effects in supersonic airflow

This paper is devoted to investigate the nonlinear dynamic characteristics of lattice sandwich composite panels resting on Winkler–Pasternak elastic foundations under simultaneous aerodynamic and thermal loads in supersonic airflow. The first-order shear deformation and von Karman large deflection theories are applied in the structural modeling. The supersonic piston theory is used to model the aerodynamic pressure acting on the lattice sandwich composite panel. The equation of motion of the structure is established by Hamilton’s principle with the assumed mode method. The nonlinear vibration responses of the lattice sandwich composite panel under different aerodynamic pressures are computed. In addition, the influences of several significant parameters including the ply angle of laminated face sheets, elastic foundation parameters, aerodynamic pressure and temperature change on the nonlinear aerothermoelastic characteristics and the route to chaos for the lattice sandwich composite panel are investigated. Time histories, phase maps, Poincare plots and fast Fourier transform frequency spectra are presented to identify the periodic, quasi-periodic and chaotic motions.

Bending Behavior of Sandwich Plates with Aggregated CNT-Reinforced Face Sheets

The main aim of this paper is to investigate bending behavior in sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets with considering the effects of carbon nanotube (CNT) aggregation. The sandwich plates are assumed resting on Winkler-Pasternak elastic foundation and a mesh-free method based on first order shear deformation theory (FSDT) is developed to analyze the deflection of sandwich plates. In the face sheets, volume fraction of CNTs and their clusters are considered to be changed along the thickness. To estimate the material properties of the nanocomposite, Eshelby-Mori-Tanaka approach is applied. In the mesh-free analysis, moving least squares (MLS) shape functions are employed to approximate the displacement field and transformation method is used for imposition of essential boundary conditions. The effects of CNT volume fraction, distribution and degree of aggregation, and also boundary conditions and geometric dimensions are investigated on the bending behavior of the sandwich plates. It is observed that in the same value of cluster volume, FG distribution of clusters leads to less deflection in these structures.

Multi-resonance analysis of a viscoelastically supported orthotropic plate with an inclined crack

Vibration behavior of an orthotropic cracked plate embedded in a nonlinear Winkler–Pasternak elastic foundation is studied in this paper. The plate’s imperfection is due to an inclined crack started on the surface and continued in the depth of the plate. Crack is located at the center of the plate and has an arbitrary angle of orientation with respect to its side edge. The harmonically excited plate is modeled based on the Kirchhoff’s plate theory. Galerkin’s procedure is applied to develop the time-dependent ordinary differential equations of the system, and multiple timescales method is used to obtain the solution. Two different phenomena, namely first primary resonance and the primary resonance with the 1:3 internal resonance, are investigated from the frequency responses of the system. The first four natural modes of vibration for the case of external forced vibration are studied, and for the combined primary and 1:3 internal resonances, the four-to-one internal resonances are considered. The effects of various parameters on these phenomena have been examined, and the results are discussed in detail.

Large Displacement Static Analysis of Composite Elliptic Panels of Revolution having Variable Thickness and Resting on Winkler-Pasternak Elastic Foundation

Large Displacement Static Analysis of Composite Elliptic Panels of Revolution having Variable Thickness and Resting on Winkler-Pasternak Elastic Foundation

New hybrid approach for free vibration and stability analyses of axially functionally graded Euler-Bernoulli beams with variable cross-section resting on uniform Winkler-Pasternak foundation

Abstract In this paper, a new hybrid approach is presented based on the combination of the power series expansions and the Rayleigh-Ritz method for stability and free vibration analyses of axially functionally graded non-uniform beams resting on constant Winkler-Pasternak elastic foundation. In the proposed novel technique, the power series approximation is first adopted to solve the motion equation. Regarding this numerical methodology, the transverse displacement and all mechanical properties are expanded in terms of power series of a known degree. By solving the eigenvalue problem, one can acquire the fundamental natural frequencies. According to aforementioned method, the expression of vibrational mode shape is also determined. Based on the similarities existing between the vibrational and buckling deformation shapes, Rayleigh-Ritz method is finally employed to construct eigenvalue problem for obtaining the critical loads. In order to illustrate the correctness and convergence of the method, several numerical examples of axially non-homogeneous and homogeneous beams are conducted. The obtained outcomes are compared to the results of Finite Element Analysis in terms of ANSYS software and those of other available numerical and analytical solutions. The accuracy of the method is then remarked.

New Shape Function for the Bending Analysis of Functionally Graded Plate

The bending analysis of thick and moderately thick functionally graded square and rectangular plates as well as plates on Winkler–Pasternak elastic foundation subjected to sinusoidal transverse load is presented in this paper. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. This paper presents the methodology of the application of the high order shear deformation theory based on the shape functions. A new shape function has been developed and the obtained results are compared to the results obtained with 13 different shape functions presented in the literature. Also, the validity and accuracy of the developed theory was verified by comparing those results with the results obtained using the third order shear deformation theory and 3D theories. In order to determine the procedure for the analysis and the prediction of behavior of functionally graded plates, the new program code in the software package MATLAB has been developed based on the theories studied in this paper. The effects of transversal shear deformation, side-to-thickness ratio, and volume fraction distributions are studied and appropriate conclusions are given.

Geometrically nonlinear dynamic analysis of organic solar cell resting on Winkler-Pasternak elastic foundation under thermal environment

The nonlinear dynamic responses of a nanocomposite organic solar cell (NCOSC) are developed through the classical plate theory. The investigated NCOSC consists of five layers which are including Al, P3HT: PCBM, PEDOT: PSS, IOT and glass. A uniformly distributed external excitation is exerted on the simply supported NCOSC. The impacts of the Winkler-Pasternak elastic foundation, thermal environment and damping on the nonlinear dynamic responses of the NCOSC are investigated. The equations of motion and geometric compatibility of the NCOSC with the consideration of the von Kármán nonlinearity are derived. The governing equation of the dynamic system is formulated by employing the Galerkin and the fourth-order Runge-Kutta methods. Several numerical experiments are thoroughly presented to report the effects of damping ratio, temperature variations, and elastic foundation parameters on the frequency–amplitude curves and nonlinear dynamic response of the NCOSC. The numerical studies indicate that the existence of the Winkler-Pasternak elastic foundation effectively reduces the dynamic response of the NCOSC. In addition, the damping and thermal variation depress the vibration of the NCOSC but with relatively less efficiency compared with the Winkler- Pasternak elastic foundation.

A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates

This paper presents a model which is effective and simple based upon Second Order Shear Deformation Theory (SSDT) to comprehensive study of Functionally Graded Carbon Nanotube Reinforced Composite (FG-CNTRC) plates including size effects for the first time. Also it is the first time that the size-dependent static, stability and dynamic response of FG-CNTRC plates considering Winkler–Pasternak elastic foundation effects are investigated. It is assumed that the material properties of FG-CNTRC are varied through the thickness direction using four different distributions of carbon nanotubes (CNTs). In order to include the size effects in our modeling, the nonlocal elasticity theory presented by Eringen is utilized. The influences of different parameters such as volume fraction of carbon nanotubes, nonlocality and elastic foundation effects on the response of FG-CNTRC plates are studied. Numerical results prove high accuracy and reliability of the present method in comparison with other available numerical or analytical methods.

Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler–Pasternak elastic foundation

The nonlinear vibration and the dynamic buckling of a graphene platelet reinforced sandwich functionally graded porous (GPL-SFGP) plate are thoroughly investigated in this paper. The investigated GPL-SFGP plate consists of two metal face layers and a functionally graded porous core with graphene platelet reinforcement. The effects of the Winkler–Pasternak elastic foundation, thermal environment and damping are incorporated. The open-cell metal foam model is implemented to model the mechanical properties of the porous core. Axial compressive stress is applied on the GPL-SFGP plate by exerting various compressive loading speeds at one edge of the plate. Grounded on the classical plate theory, both motion and geometric compatibility equations of the plate are deduced by introducing the Von Kármán strain-displacement relationship and stress function. Both the Galerkin and the fourth-order Runge–Kutta methods are implemented to solve the governing equation of the dynamic system. Meticulously designed numerical experiments are conducted to identify the critical influential factors of the dynamic stability of the GPL-SFGP plate. The influences of loading speed, damping ratio, temperature variation, initial imperfection, elastic foundation parameters, porosity, GPL weight fraction and the dimensions of the GPL on the overall dynamic stability of the GPL-SFGP plate are evidently demonstrated.

Vibration of FG-CNTRC annular sector plates resting on the Winkler-Pasternak elastic foundation under a periodic radial compressive load

The vibration and stability of functionally graded carbon nanotube-reinforced (FG-CNTRC) composite annular sector plates resting on Winkler-Pasternak elastic foundation subjected to a periodic radial compressive load are investigated for various boundary conditions. To this end, a shear deformable plate model is established according to a parabolic theory which can interpret the shear deformation and rotary inertia effects without using any shear correction factor. The modified micromechanical technique is employed to compute the effective material properties of the FG-CNTRCs. The discretized form of higher-order governing equations is directly accessed by employing Hamilton’s principle and the variational differential quadrature (VDQ) method. In addition, by considering the applied radial compressive force as a periodic function, the discretized equations are expressed as the Mathieu–Hill equations, and then the instability regions are determined via the Bolotin’s scheme. In numerical results, the effects of the CNT distribution pattern and volume fraction, geometry parameters, sector angle, elastic foundation and static load factor on the stability of FG-CNTRC annular sector plates subject to arbitrary edge conditions are thoroughly discussed.