Two-parameter Elastic Foundation Research Articles

A comprehensive study on the size-dependent hygrothermal analysis of exponentially graded microplates on elastic foundations

In this paper, the effects of hygrothermal conditions on various behaviors, such as bending, free vibration, mechanical and thermal buckling, of exponentially graded microplates lying on two-parameter elastic foundations are investigated. The trigonometric four-variable plate theory incorporated to the modified couple stress theory (MCST) is employed to derive the equations of motion. The present MCST contains an internal material length scale parameter, thus it can capture the size effect. The microplate is assumed to be subjected to a temperature rise and moisture concentration which are varied linearly through the thickness of the plate. Based on an exponential law, the material properties of the microplate are graded only in z direction. The equations of motion are solved analytically to obtain the displacements, stresses, eigenfrequencies and critical buckling load and temperature of the microplates. The present results are validated by comparing them with those previously published. The numerical examples reveal that considering the size effect and/or the elastic foundations leads to an increment in plate stiffness and thereby leads to a decrement in the deflection and an increment in eigenfrequency and buckling loads. It is also shown that the size effect is negligible for the thicker plate.

Nonlinear Static and Dynamic Thermal Buckling Analysis of Spiral Stiffened Functionally Graded Cylindrical Shells with Elastic Foundation

In this paper, an analytical method is used to study the nonlinear static and dynamic thermal buckling analysis of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells. The SSFG cylindrical shell is surrounded by a linear and nonlinear elastic foundation. The proposed linear model is based on the two-parameter elastic foundation (Winkler and Pasternak). A three-parameter elastic foundation with hardening/softening cubic nonlinearity is used for nonlinear model. The material properties are temperature dependent and assumed to be continuously graded in the thickness direction. Also, for thermal buckling analysis, the uniform and linear temperature distribution in thickness direction is considered. The SSFG cylindrical shells are considered with various angles for spiral stiffeners. The strain–displacement relations are obtained based on the von Kármán nonlinear equations and the classical plate theory of shells. The smeared stiffener technique and the Galerkin method are applied to solve the nonlinear problem. In order to find the nonlinear dynamic thermal buckling responses, the fourth-order Runge–Kutta method is used. To validate the results, comparisons are made with those available in literature and good agreements are shown. The effects of various geometrical and material parameters are investigated on the nonlinear static and dynamic thermal buckling response of SSFG cylindrical shells.

Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT

This paper presents an analytical study of wave propagation in simply supported graduated functional plates resting on a two-parameter elastic foundation (Pasternak model) using a new theory of high order shear strain. Unlike other higher order theories, the number of unknowns and governing equations of the present theory is only four unknown displacement functions, which is even lower than the theory of first order shear deformation (FSDT). Unlike other elements, the present work includes a new field of motion, which introduces indeterminate integral variables. The properties of the materials are assumed to be ordered in the thickness direction according to the two power law distributions in terms of volume fractions of the constituents. The wave propagation equations in FG plates are derived using the principle of virtual displacements. The analytical dispersion relation of the FG plate is obtained by solving an eigenvalue problem. Numerical examples selected from the literature are illustrated. A good agreement is obtained between the numerical results of the current theory and those of reference. A parametric study is presented to examine the effect of material gradation, thickness ratio and elastic foundation on the free vibration and phase velocity of the FG plate.

Statics, stability and vibration of non-prismatic linear beam-columns with semirigid connections on elastic foundation

A method that evaluates the static, stability and vibration behavior of non-prismatic Euler-Bernoulli straight beam-columns resting on a two-parameter elastic foundation with generalized end conditions is presented. The static stiffness matrix and shape functions of the element are derived including the combined effects of semi-rigid connections and lateral springs at both ends using the conjugate beam principle. The fixity factors of a non-prismatic element are defined and used in the definition of its characteristic matrices according to the consistent method. It is shown that static, stability and vibration analysis of non-prismatic beam-columns under any transverse loading and axial end loads with generalized end conditions can be carried out once the main flexibility coefficients of the member are determined. Three comprehensive examples are presented that show the simplicity and accuracy of the proposed method. The calculated results using the proposed approach are compared to those available in the technical literature.

Thermal postbuckling of shear deformable CNT-reinforced composite plates with tangentially restrained edges and temperature-dependent properties

Buckling and postbuckling behaviors of moderately thick composite plates reinforced by single-walled carbon nanotubes (SWCNTs), rested on elastic foundations and subjected to two types of thermal loading are investigated in this article. Carbon nanotubes (CNTs) are reinforced into isotropic polymer matrix according to functional rules in which volume fractions of constituents are graded in the thickness direction. Material properties of constituents are assumed to be temperature-dependent and effective properties of nanocomposite are estimated by extended rule of mixture. Formulations are based on first-order shear deformation theory taking von Karman nonlinearity, initial geometrical imperfection, tangential constraints of edges, and two-parameter elastic foundation into consideration. Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive nonlinear temperature–deflection relations from which buckling temperatures and postbuckling equilibrium paths are determined by an iteration algorithm. Novel findings of the present study are that deteriorative influences of temperature-dependent properties on the postbuckling behavior become more serious as plate edges are partially movable, CNT volume fraction is higher, elastic foundations are stiffer, plates are thicker, and/or temperature linearly changed across the thickness.

Effects of nonlinear hygrothermomechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory

In the present study, a simple four-unknown exponential shear deformation theory is developed for the bending of functionally graded material (FGM) rectangular plates resting on two-parameter elastic foundation and subjected to nonlinear hygrothermomechanical loading. The elastic properties, coefficient of thermal expansion, and coefficient of moisture expansion of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. Unlike first-order and other higher-order plate theories, the present theory has four independent unknowns. The in-plane displacement field of the present theory uses exponential functions in terms of thickness co-ordinate for calculating out-of-plane shearing strains. The transverse displacement includes bending and shear components. The principle of virtual displacement is employed to derive the governing equations and associated boundary conditions. A Navier solution technique is employed to obtained an analytical solutions. The elastic foundation is modelled as two-parameter Winkler–Pasternak foundation. The numerical results obtained are compared with previously published results wherever possible to prove the efficacy and accuracy of the present theory. The effects of stiffness and gradient index of the foundation on the hygrothermomechanical responses of the plates are discussed.

Lateral vibrations of embedded hetero-junction carbon nanotubes based on the nonlocal strain gradient theory: Analytical and differential quadrature element (DQE) methods

In the current paper, lateral vibrations of embedded hetero-junction carbon nanotubes (HJCNTs) are investigated. HJCNT and its surrounding medium are idealized as two different connected nonlocal strain gradient Rayleigh beams and a two-parameter (Pasternak) elastic foundation, respectively. The governing differential equation is derived based on the Hamilton's principle, and the classical and higher order boundary conditions are obtained using the weighted residual approach. The sixth-order differential equation is then solved analytically and numerically. The differential quadrature element method based on the Hermit interpolation functions is employed and developed to the stepped nanobeams to establish the reliability of the analytical method. The effects of the nonlocal parameter, the strain gradient parameter, the type of boundary conditions, the type of adopted higher order boundary conditions and the elastic foundation parameters on the frequency and mode shapes are studied and discussed in details. This paper provides an appropriate source not only for the vibrational study of HJCNTs, but also for comparing the vibration results of embedded uniform or stepped nanobeams obtained based on different theories that consider the size-dependent effects.

Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation

This paper presents an investigation on free vibration of functionally graded (FG) porous nanocomposite plates reinforced with a small amount of graphene platelets (GPLs) and supported by the two-parameter elastic foundations with different boundary conditions. Four different porosity distributions of the parent metal were reinforced by four various GPLs dispersion patterns (evenly or unevenly) along the thickness direction. To obtain graded distribution in both porosity and GPLs, the Young’s modulus, shear modulus and density of nanocomposites are assumed to vary through the thickness direction due to variation of internal pores and the volume fraction of GPLs. By employing Halpin-Tsai micromechanics model, effective elastic modulus of the nanocomposites is obtained according to the assumption of closed-cell cellular solids under Gaussian Random Field scheme. The governing equations of FG GPLs-reinforced metal foams resting on elastic foundations are derived from Hamilton’s principle by the means of classic plate theory with the consideration of von Kármán strain-displacement relation. Applying the differential quadrature method, the dimensionless natural frequencies of porous nanocomposite plates with different boundary conditions are obtained and present method is thoroughly validated with the results in open literature. The comprehensive parametric studies on different porosity coefficients, GPL dispersion patterns, weight fraction of GPL, aspect ratios, thickness ratios and parameters of elastic foundation on free vibration of FG GPLs-reinforced porous plates with various boundary conditions. An interesting finding is that the increase of porosity coefficients would lead to the linearly decrease of both mass density and stiffness of the plates, but the increase of porosity coefficients does not always induce the decrease of natural frequencies. Moreover, both porosity distribution and GPL dispersion pattern have a distinct effect on dynamic characteristics, the former plays a more important role than the latter in analysing mechanical properties of FG GPLs-reinforced plates.

Eigenfrequencies of advanced composite plates using an efficient hybrid quasi-3D shear deformation theory

This research investigates the free vibration analysis of advanced composite plates such as functionally graded plates (FGPs) resting on a two-parameter elastic foundations using a hybrid quasi-3D (trigonometric as well as polynomial) higher-order shear deformation theory (HSDT). This present theory, which does not require shear correction factor, accounts for shear deformation and thickness stretching effects by a sinusoidal and parabolic variation of all displacements across the thickness. Governing equations of motion for FGM plates are derived from Hamilton\' s principle. The closed form solutions are obtained by using Navier technique, and natural frequencies are found, for simply supported plates, by solving the results of eigenvalue problems. The accuracy of the present method is verified by comparing the obtained results with First-order shear deformation theory, and other predicted by quasi-3D higher-order shear deformation theories. It can be concluded that the proposed theory is efficient and simple in predicting the natural frequencies of functionally graded plates on elastic foundations.

Quasi 3-D trigonometric plate theory for bending analysis of EG plates resting on Pasternak foundations

Abstract This paper deals with the bending analysis of exponentially graded material (EGM) plates resting on two-parameter elastic foundations according to a trigonometric shear deformation plate theory (TPT) using Navier’s technique. The normal and shear deformations are both includes. The present TPT does not need a shear correction factors. The material properties of plate like, Lamé’s coefficients convert exponentially in a given constant orientation. The equilibrium equations according to the EG plate resting on Pasternak foundations are presented. Numerical results for the EG thick plate on elastic foundations are presented. A good comparison of results with those being in the literature. The influences played by Winkler and Pasternak parameters, side-to-thickness ratio, inhomogeneity parameter and aspect ratio on the bending responses of EG plates are debated.

Nonlinear vibration of functionally graded sandwich shallow spherical caps resting on elastic foundations by using first-order shear deformation theory in thermal environment

A semi-analytical approach to investigate the nonlinear vibration axisymmetric analysis of functionally graded sandwich shallow spherical caps under external pressure resting on elastic foundation in thermal environment is presented. The governing equations are derived by using the first-order shear deformation theory taking into account von Karman geometrical nonlinearity and Pasternak’s two-parameter elastic foundation. The motion equations are determined by Galerkin method and the obtained equation is numerically solved by using Runge–Kutta method. Results of nonlinear dynamic responses show the effects of foundation, material, geometric parameters, and temperature change on the nonlinear vibration of shells.

Nonlinear stability of sandwich beams with carbon nanotube reinforced faces on elastic foundation under thermal loading

In this research, post-buckling response of sandwich beams with carbon nanotube reinforced face sheets subjected to uniform temperature rise loading and resting on a two-parameter elastic foundation is investigated. A single-layer theory formulation based on the first-order shear deformation beam theory is used. Material properties of the media are obtained according to a refined rule of mixtures approach which contains efficiency parameters. Suitable for the large deformations, von-Kármán strains are taken into consideration. The elastic foundation is modelled as the Pasternak model which takes into account the shear interaction of the springs. Material properties of the face sheets are considered to be position and temperature dependent. The governing equations of the system are obtained using the Ritz method for various combinations of clamped, simply supported and sliding supported edges. Post-buckling equilibrium path of the beam is obtained according to an iterative displacement control strategy. Numerical results of the present study are compared with the available data in the open literature. Then, the numerical results are provided to explore the effect of side-to-thickness ratio, volume fraction of carbon nanotube, distribution pattern of carbon nanotube, the ratio of face thickness-to-host thickness, boundary conditions and elastic foundation.

Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment

Abstract In the present work, thermo-elastic vibrational behavior of thick microbeams embedded in a two-parameter elastic foundation is studied. A Winkler-Pasternak type elastic foundation model is employed to simulate the interactions between microbeam and elastic medium. Size-dependent constitutive equations and associated boundary conditions are obtained by applying dynamic version of virtual work's principle based on modified couple stress and various beam theories. Several numerical examples are presented to examine the sensibility of various parameters associated with slenderness ratio, temperature rise, length scale, Winkler and shear layer parameters on the natural frequencies and critical temperature point of embedded microbeams.

A study on the vibrational properties of weight-efficient plates made of material with functionally graded porosity

Functionally graded (FG) porous structures are a novel class of weight-efficient engineering materials and characterized by introducing graded non-uniform porosities. Due to the wide range of applications of porous materials in lightweight structures, biomedical systems and separation processes, in this paper, three-dimensional (3D) free vibration analyses of light-weight plates made of uniform and non-uniform graded porous materials resting on two-parameter elastic foundations are investigated. The mathematical formulation is based on linear and small strain elasticity assumptions. The plate with FG porosity is simply supported at the edges and is assumed to have three variations of porosities through the thickness. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. Validation and convergence studies are also done to demonstrate the accuracy of the results. The natural frequencies of the simply-supported plate made of material with functionally graded porosity are calculated and presented in both tabular and graphical forms. The effects of different porosity distributions, porosity parameter, elastic coefficients of foundation and geometrical parameters on the results are investigated. The results of the present analysis indicate the advantages of using a plate with graded porosity over a traditional plate with uniform porosity.

Thermodynamic effect on the bending response of elastic foundation FG plate by using a novel four variable refined plate theory

ABSTRACTThis article deals with the study of the thermodynamic behavior of functionally graded material plates resting on two-parameter elastic foundation. An analytical solution based on a new shear refined deformation theory is presented. The displacement field used in the present refined theory contains undetermined integral forms and involves only four unknowns to derive. The plate is assumed simply supported and subjected to two different temperatures fields across its thickness. The mechanical characteristics of the plate are assumed to be varied across the thickness according to a simple exponential law distribution. The governing equations and boundary conditions are derived using the principle of virtual displacements and Navier solution technique is adopted to derive analytical solutions. A detailed numerical study of the present new refined theory is carried out to examine the influence of the time’s parameter, foundation’s parameters and deflection on the bending response of the FG plate.

Nonlinear bending analysis of anisotropic laminated tubular beams based on higher-order theory subjected to different kinds of distributed loads

Pipelines, and more generally long tubular structures, are major oil and gas industry tools used in exploration, drilling, production, and transmission. Technical challenges of the field have spawned significant research and development efforts in the mechanical behavior and the modes of failure of long tubes and pipes under various loads. Nonlinear bending analysis of anisotropic laminated composite tubular beams with different kinds of distributed loads resting on a two-parameter elastic foundation is investigated. Based on Hamilton's principle and the displacement field by Laurent series expansion form with a von Kármán-type of kinematic nonlinearity, the governing differential equations are obtained. Composite tubular beams with clamped-clamped, clamped-hinged, and hinged-hinged boundary conditions including two kinds of end conditions, namely movable and immovable are considered. A numerical solution of the transverse deflection of tubular beams by using Galerkin's method is employed to determine the relations between distributed loads and deflections of a composite beam with or without initial axial loads. The results of the present formulation concern the large-deflection bending behavior of laminated beams with different geometric and material parameters, distributed loads, end conditions and effect on elastic foundation, the obtained numerical results are also compared with the results of nonlinear finite element method. Some new results for anisotropic laminated tubular beams, such as effects of the end conditions, load characteristics and foundation stiffness are obtained for future references in the design of the civil, mechanical, aeronautical and aerospace industries.

Nonlinear dynamic response of acoustically excited and thermally loaded composite plates resting on elastic foundations

In order to ensure integrity of thermal protection system (TPS) subjected to a combination of thermal and acoustic loadings, a thin composite plate resting on a two-parameter elastic foundation is used to characterize the behavior of the thin top facesheet of TPS. The nonlinear dynamic response of a thermal loaded, acoustic excited plate is investigated. A theoretical model is developed based on Kirchhoff thin plate assumptions and von Kármán-type equation. General static condensation and Galerkin's method are used to derive a set of ordinary differential equations with cubic nonlinearity related to nonlinear coupling between mid-plane stretching and transverse deflection. The reduced-order model has been validated by comparison of postbuckled displacements with those obtained from full-order FEM analysis. Variations of transverse displacement and in-plane strain statistics with acoustic loading level and temperature rising are presented. It is proposed that the in-plane strain located on the plate surface is dominated by the competition of the linear and quadratic nonlinear modal amplitude terms, thus the characteristic of the strain histogram can be used to identify oscillation transition from no snap-through to persistent dynamic snap-through for the thermally buckled plate. The skewness of the strain histogram can be used to evaluate the degree of dynamic geometrical nonlinearity quantitatively for the postbuckled plate with symmetric snap-through motion.

Effect of thermal gradient load on thermo-elastic vibrational behavior of sandwich plates reinforced by carbon nanotube agglomerations

In this study, we investigated the effect of thermal gradient load on natural frequencies of sandwich plates with polymer-based nanocomposite face sheets reinforced by functionally graded (FG) single-walled carbon nanotubes (SWCNTs) agglomerations. Volume fractions and agglomerations of CNTs change across the thickness of each nanocomposite face sheet and temperature-dependent properties of polymer/CNT composite can be estimated with the Eshelby-Mori-Tanaka method. First-order shear deformation theory and a moving least square (MLS) shape function based mesh-free method have also been developed for free vibration and steady state thermal analysis on sandwich plates on two-parameter elastic foundations. We investigated the effects of orientation, aggregation, volume fraction and distribution of CNT. We also studied the effects of essential boundary conditions, sandwich dimensions, and elastic foundation coefficients on frequency behavior of sandwich plates.

An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations

In this study, a four-node quadrilateral partial mixed plate element with low degrees of freedom (dofs) is developed for static and free vibration analysis of functionally graded material (FGM) plates rested on Winkler-Pasternak elastic foundations. The formulation of the presented finite element model is based on a parametrized mixed variational principle which is developed recently by the first author. The presented finite element model considers the effects of shear deformations and normal flexibility of the FGM plates without using any shear correction factor. It also fulfills the boundary conditions of the transverse shear and normal stresses on the top and bottom surfaces of the plate. Beside these capabilities, the number of unknown field variables of the plate is only six. The presented partial mixed finite element model has been validated through comparison with the results of the three-dimensional (3D) theory of elasticity and the results obtained from the classical and high-order plate theories available in the open literature.

Buckling analysis of tapered BDFGM nano-beam under variable axial compression resting on elastic medium

The current study presents a new technique in the framework of the nonlocal elasticity theory for a comprehensive buckling analysis of Euler-Bernoulli nano-beams made up of bidirectional functionally graded material (BDFGM). The mechanical properties are considered by exponential and arbitrary variations for axial and transverse directions, respectively. The various circumstances including tapering, resting on two-parameter elastic foundation, step-wise or continuous variations of axial loading, various shapes of sections with various distribution laws of mechanical properties and various boundary conditions like the multi-span beams are taken into account. As far as we know, for the first time in the current work, the buckling analyses of BDFGM nano-beams are carried out under mentioned circumstances. The critical buckling loads and mode shapes are calculated by using energy method and a new technique based on calculus of variations and collocation method. Fast convergence and excellent agreement with the known data in literature, wherever possible, presents the efficiency of proposed technique. The effects of boundary conditions, material and taper constants, foundation moduli, variable axial compression and small-scale of nano-beam on the buckling loads and mode shapes are investigated. Moreover the analytical solutions, for the simpler cases are provided in appendices.