Winkler-Pasternak Elastic Foundation Research Articles

A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation

By using high-order deformation theory, this paper presents a novel approach for analysing the mechanical behaviours of functionally graded porous beams resting on a Winkler-Pasternak elastic foundation. The governing equations are derived from the Lagrange principle. Legendre-Ritz polynomial functions, which possess the simplicity of algebraic polynomials and the efficiency of orthogonal polynomials, are developed to solve the problem. Three boundary conditions accompanied by two porosity types, including symmetric and asymmetric distributions of beams, are considered. The effects of porosity, boundary condition, foundation parameter, span-to-height ratio, and distribution type on the mechanical behaviours of porous beams are investigated. Some new results are presented for the first time as a benchmark for future studies.

Dynamic vibration analysis of double-layer auxetic FGP sandwich plates under blast loads using improved first-order shear plate theory

In this study, the characteristics of static bending and free and forced vibration of double-layer auxetic functionally graded porous (FGP) sandwich plates with shear connectors under blast load are explored comprehensively, where the structure of the plate comprises two layers joined by shear connectors. Each layer comprises auxetic FGM sandwich material, containing a core component made of auxetic honeycomb with a negative Poisson's ratio, while the upper and bottom two components are made from FGP materials. The whole plate is supported by a two-parameter Winkler-Pasternak elastic foundation. The key innovation of the proposed theory is that the transverse shear stresses are zero at the two free surfaces of each layer. In contrast to previous first-order shear deformation theories, no shear correction factor is required. To solve the plate issue when the boundary condition was entirely supported, Navier's exact solution was devised. On the other hand, in order to address the plate behavior in the event that the boundary condition was altered in any way, a plate element with nine nodes was used. Besides, to determine whether or not these results are accurate, a complete comparison approach that makes use of reliable claims has been used. The problem model's special case situations are used in order to accomplish this goal. When the object being developed is exposed to explosive loads, the findings that were derived from this research have the potential to be used to the construction of both military and civil works.

Nonlocal large deflection analysis of a cantilever nanobeam on a nonlinear Winkler-Pasternak elastic foundation and under uniformly distributed lateral load

Nonlocal large deflection analysis of a cantilever nanobeam on a nonlinear Winkler-Pasternak elastic foundation and under uniformly distributed lateral load

Study on dynamic stability of magneto-electro-thermo-elastic cylindrical nanoshells resting on Winkler–Pasternak elastic foundations using nonlocal strain gradient theory

Study on dynamic stability of magneto-electro-thermo-elastic cylindrical nanoshells resting on Winkler–Pasternak elastic foundations using nonlocal strain gradient theory

Dynamical and chaotic analyses of single-variable-edge cylindrical panels made of sandwich auxetic honeycomb core layer in thermal environment

This paper proposes a new structure of the cylindrical panel called single-variable-edge (SVE) panel resting on a Winkler–Pasternak elastic foundation to form a mechanical system. The panel is made of sandwich structures with a negative Poisson’s ratio auxetic honeycomb core layer. The classical shell theory and Von-Karman nonlinearity assumption are used to establish the motion equation, then Galerkin’s method is applied to obtain the characteristics of the system. The comparisons to the previous article and the finite element analysis are shown in the validation section to confirm the reliability and accuracy of the present calculating procedure. In the results section, the dynamical characteristics of thermo-mechanical systems are investigated through the impact of materials and geometrics. In addition, the dynamics and chaos phenomena of the SVE panel under the critical external load are presented in three phases: stability, the beginning of stabilization and completely chaos state. The new structure has great potential in practical engineering application and is expected to contribute knowledge to the plate/shell research field.

Nonlinear dynamic stability analysis of axial impact loaded structures via the nonlocal strain gradient theory

In engineering applications, there has been an overwhelming tendency towards portability, miniaturization, and integration in recent years. To link the intrinsic size dependency feature of the small-scale structure with its structural stability, the nonlocal strain gradient theory, which captures the size effect in a more general size-dependent continuum-based model, is introduced to explore the nonlinear dynamic stability behaviour of nanoplates. Four types of axial impact loading configurations, namely, sinusoidal, exponential, rectangular, and damping, are considered. Some practical factors, such as Winkler-Pasternak elastic foundation and damping, are taken into account in the analysis. The equations of motion for the size-dependent initially imperfect plate are derived in the framework of the first-order shear deformation plate theory in conjunction with the Von Kármán nonlinear terms. Then the Airy stress function corresponding to simply supported nanoplate is introduced; then, by applying the Galerkin method, the obtained differential equations are addressed by the fourth-order Runge-Kutta algorithm. Subsequently, the specific value of the critical dynamic buckling load is determined by the Volmir criterion. Organic solar cells (OSCs), a type of emerging solar-to-electrical energy conversion nanodevice, are used as an illustrative example within the existing framework. The effects of size dependency in conjunction with the pulse load configuration, the initial imperfection, the elastic foundation, as well as the damping ratio on the nonlinear dynamic buckling behaviour of the OSC are thoroughly investigated.

Buckling of porous FG sandwich plates subjected to various non-uniform compressions and resting on Winkler–Pasternak elastic foundation using a finite element model based on the high-order shear deformation theory

Buckling of porous FG sandwich plates subjected to various non-uniform compressions and resting on Winkler–Pasternak elastic foundation using a finite element model based on the high-order shear deformation theory

Bending and Free Vibration Analysis of Functionally Graded Sandwich Pile on Elastic Foundation Using Rayleigh-Ritz Method

The present study aims to carry out a parametric investigation on a pile made of different materials, and subjected to static and dynamic loading resulting from the weight of the structure and the shear force acting on the head of the pile. It is worth specifying that the pile is anchored in the ground on a Winkler-Pasternak elastic foundation that is made up of five different layers. In this study, the equilibrium equations were treated using the high-order theory with a new shear deformation shape function of a cylindrical pile made of an isotropic material and a sandwich functionally graded material.The results obtained with a numerical study based on the minimization of energies by the Rayleigh-Ritz method was carried out in order to highlight the influence of the geometric ratio, the volume fraction of the functionally graded material, and the type of loading on the vibration frequencies and the admissible stresses in order to determine the most appropriate material for the pile so that extreme stresses can be absorbed. The precision and the applicability of the method with new solution of the transverse shear deformation are demonstrated through this study with modeling of the layered property of the soil, were then compared with those reported in previous work available in the literature. It turned out that these results are in good agreement with each other, for the different materials of pile and for all boundary conditions considered. The results of pile with sandwich functionally graded material, it is revealed that the role of this material has an influence on the type of loading, especially, in the case statique of a pile subjected to a compound bending weight and the shear effect, in case dynamique the pile with vibration flexible or rigid .

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

Wavelet-based techniques for Hygro-Magneto-Thermo vibration of nonlocal strain gradient nanobeam resting on Winkler-Pasternak elastic foundation

The primary objective of this article is to investigate the vibration characteristics of a nanobeam embedded in a Winkler-Pasternak elastic foundation adopting wavelet-based two relatively recent techniques, namely the Haar Wavelet Method (HWM) and the Higher Order Haar Wavelet Method (HOHWM). While subjected to a longitudinal magnetic field, the nanobeam is also exposed to hygroscopic and thermal environments. Hamilton's principle is being used to model the nanobeam within the frameworks of Euler-Bernoulli beam theory and nonlocal strain gradient theory. The effects of several parameters such as, the small scale, length scale, magnetic field intensity, hygroscopic, thermal, Winkler modulus and shear modulus on frequency parameters, are thoroughly investigated for PP (Pined-Pined) and CC (Clamped-Clamped) boundary conditions. Richardson's formula is used to determine the rate of convergence of both the Haar wavelet and higher order Haar wavelet methods, and the results obtained by the Haar wavelet technique are also extrapolated, revealing that the rate of convergence of extrapolated HWM is comparable to the higher order Haar wavelet method. The effectiveness of the higher order Haar wavelet method over the Haar wavelet method is demonstrated using PP (Pined-Pined) boundary condition as a test case. Furthermore, the results of this study have been cross-validated with previously published literature in specific circumstances, exhibiting exceptional agreement.

Vibration analysis of cracked plates resting on elastic foundation via moving least squares differential quadrature method

In this work, moving least squares differential quadrature scheme is employed to formulate and solve the free vibration analysis of cracked plates resting on the Winkler–Pasternak elastic foundation. Two types of cracks as a part-through surface and internal crack are studied in a rectangular plate. Formulation of the problem is dependent on transverse shear deformation theory and line spring model. Also, continuous weight functions around the crack are produced by diffraction method. Boundary conditions are augmented in the equation of motion and the problem is converted to a discrete eigenvalue problem. The accuracy and efficiency of the present results is validated by comparing with exact, experimental and numerical results. Moreover, the effects of different boundary conditions, crack length, crack depth and foundation parameters on the vibration frequencies and mode shapes are examined in detail. The obtained results may be used for structural health monitoring.

Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation

Free vibration analysis is carried out on axially inhomogeneous beams resting on Winkler-Pasternak elastic foundation. The material properties of the beam like Young’s modulus, modulus of rigidity and material density are considered to be varying along the length direction following constant, linear and exponential material models. The beam is subjected to different combinations of clamped and simply supported boundary conditions. The formulation is based on Timoshenko beam theory and energy method along with Hamilton’s principle is used to derive the governing equations. The effect of material gradation and the 2 parameters of elastic foundation on the natural frequencies are studied in detail. The present results are validated by comparing them with established ones and satisfactory matching is observed.
 HIGHLIGHTS
 
 Free vibration behavior of axially functionally graded beams in investigated
 Three different material models and three boundary conditions are considered
 Effect of two parameter elastic foundation is considered
 Timoshenko beam theory and Rayleigh Ritz method are employed
 It is found that the stiffness of elastic foundation significantly affects the natural frequency of the beam
 
 GRAPHICAL ABSTRACT

Sigmoid functionally graded plates embedded on Winkler-Pasternak foundation: Free vibration analysis by dynamic stiffness method

In this present paper, the dynamic stiffness method has been formulated to calculate the natural frequency of sigmoid functionally graded material (S-FGM) plate embedded on the Winkler-Pasternak elastic foundation. The material properties of S-FGM continuously vary along the transverse direction of the plate by using two power-law variations in terms of volume fraction of the constituent's material. Hamilton's principle is implemented to derive the governing partial differential equation of motion based on the classical plate theory considering the physical neutral surf ace of the FGM rectangular plate. The Wittrick -Williams algorithm is applied as a solution technique to solve the transcendental nature of the dynamic stiffness matrix and extract the natural frequencies of the FGM plate with the desired accuracy. The S-FGM plate parameters’ variation of natural frequencies with the change of parametric numerical values (aspect ratio, sigmoid volume fraction index, boundary conditions and elastic foundation parameters, density ratio, and modulus ratio) are also highlighted. The DSM results are compared and validated with the available published literature. A new set of natural frequency results for the S-FGM plate embedded on the Winkler-Pasternak elastic foundation are generated.

Free vibration of functionally graded beam embedded in Winkler-Pasternak elastic foundation with geometrical uncertainties using symmetric Gaussian fuzzy number

Free vibration of functionally graded beam embedded in Winkler-Pasternak elastic foundation with geometrical uncertainties using symmetric Gaussian fuzzy number

Nonlinear Free Vibration Analysis of In-plane Bi-directional Functionally Graded Plate with Porosities Resting on Elastic Foundations

This paper deals with the nonlinear free vibration analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with porosities which are resting on Winkler–Pasternak elastic foundations. The material properties of the IBFG plate are assumed to be graded along the length and width of the plate according to the power-law distribution, as well as, even and uneven types are taken into account for porosity distributions. Equations of motion are developed by means of Hamilton’s principle and von Karman nonlinearity strain–displacement relations based on classical plate theory (CPT). Afterward, the time-dependent nonlinear equations are derived by applying the Galerkin procedure. The nonlinear frequency is determined by using modified Poincare–Lindstedt method (MPLM). Numerical results are obtained in tabular and graphical form to examine the effects of some system key parameters such as porosity coefficients, distribution patterns, gradient indices, elastic foundation coefficients, aspect ratio and vibration amplitude on the nonlinear frequency of the porous IBFG plate. To validate the analysis, the results of this paper have been compared to the published data and good agreements have been found.

The nonlinear dynamic buckling behaviour of imperfect solar cells subjected to impact load

Solar cells are becoming a strong competitor in the new energy market due to their superior ability to generate electricity in an environmentally friendly and sustainable manner. The present study is devoted to presenting a theoretical framework for nonlinear dynamic buckling behaviours of imperfect multilayer solar cells subjected to impact loading resting on the elastic foundation. Two types of solar cell models, namely, organic solar cell (OSC) and perovskite solar cell (PSC), with simply supported and clamped boundary conditions are investigated. Sinusoidal, exponential, rectangular, and damping pulse functions are considered Based on the first-order shear deformation plate theory, the nonlinearity are introduced with the aid of von Kármán theory. The equations of the dynamic system of the plate with the consideration of large-deflection are derived by the Galerkin method and then solved by the fourth-order Runge–Kutta methods. After validation, some parametric experiments are performed to explore the influences of the pulse duration, pulse function pulse amplitude, initial imperfection, boundary conditions, Winkler–Pasternak​ elastic foundation coefficients, and damping ratios on the dynamic stability of the structures.

Bending deflection and stress analyses in a thin functionally graded material skew plate with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined in-plane and uniform lateral loads using the extended Kantorovich method

In this study, bending deflection and stress analyses have been conducted for a thin skew plate made of functionally graded material (FGM) with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined loads including uniform transverse load, normal and shear in-plane forces, and planar body forces. The Cartesian partial differential equation governing the bending deflection of the skew plate has been converted into a partial differential equation in oblique coordinates using the conversion relations. Then, by employing the variational principle and residual weighted Galerkin method and using the Extended Kantorovich Method (EKM), the equation has been converted to a set of linear differential equations in terms of two functions in the longitudinal and transverse directions of the oblique plate, and afterward, the equation has been solved using the iterative solution method. Different boundary conditions in a combined form of simply and clamped supports have been investigated and their effects on bending deflection and generated in-plane normal and shear stresses are discussed.

Nonlinear Vibration of Multi-Walled Carbon Nanotubes with Initial Curvature Resting on Elastic Foundations in a Nonlinear Thermo-Magnetic Environment

The present work focuses on nonlinear dynamics models of multi-walled carbon nanotubes with initial curvature resting on Winkler-Pasternak elastic foundations in a nonlinear thermomagnetic environment using nonlocal elasticity theory. The derived systems of nonlinear vibration models are solved with the aid of the Galerkin decomposition and the homotopy perturbation method. Effects of temperature, magnetic field, multi-layer, and other thermomechanical parameters on the dynamic responses of the slightly curved multi-walled carbon nanotubes are investigated and discussed. As the temperature increases, the frequency ratio decreases as the linear natural frequency of the system increases. The results reveal that the frequency ratios decrease as the number of nanotube walls, temperature, spring constants, magnetic field strength, and the ratio of the radius of curvature to the length of the slightly curved nanotubes increase. These trends are the same for all the boundary conditions considered. However, clamped-simple and clamped-clamped supported multi-walled nanotube have the highest and lowest frequency ratio, respectively. Also, from the parametric studies to control nonlinear vibration of the carbon nanotubes, it is shown that quadruple-walled carbon nanotubes can be taken as pure linear vibration even at any value of linear Winkler and Pasternak constants. Therefore, this can be used for the restraining of the chaos vibration in the objective structure. These research findings will assist the designers and manufacturers in developing multi-walled carbon nanotubes for various structural, electrical, mechanical, and biological applications, especially in the areas of designing nanoelectronics, nanodevices, nanomechanical systems, nanobiological devices, and nanocomposites, and particularly when they are subjected to thermal loads, magnetic fields and elastic foundations.

A Quasi 3D Modified Strain Gradient Formulation for Static Bending of Functionally Graded Micro beams Resting on Winkler-Pasternak Elastic Foundation

This paper presents the bending analysis of simply supported functionally graded (FG) size dependent beams based on modified strain gradient theory. The shear and normal deformations are considered in displacement field according to hyperbolic shear deformation theory. Governing equations and corresponding boundary conditions for FG micro beam are derived utilizing principle of minimum total potential energy. Mori–Tanaka homogenization scheme and the classical rule of mixture are used for prediction of material properties through the thickness. Effects of Winkler-Pasternak elastic foundation parameters are studied for different side to thickness ratios. Effects of different aspect ratios, elastic foundation parameters, power law gradient indexes and different loading conditions are investigated. The efficiency and accuracy of present model is demonstrated by comparing to the existing results in especial cases.