Two-parameter Elastic Foundation Research Articles

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

Nonlocal strain gradient-based nonlinear vibration analysis of nonlinear FG three-phase HJCNT/MWCNT/epoxy composite microbeam resonators using a modified micromechanical model

This paper aims to study nonlinear dynamic behavior of functionally graded (FG) three-phase composite microbeam resonators made of an epoxy matrix and two reinforcements namely multi-walled carbon nanotubes (MWCNTs) and hetero-junction carbon nanotubes (HJCNTs). The effective mechanical properties of the composite microbeam are obtained using the modified Halpin-Tsai micromechanical model. The microbeam surrounding medium is simulated using a two-parameter elastic foundation. The von-Karman’s geometric nonlinearity relations are incorporated and the equations of motion are derived based on the nonlocal strain gradient Euler–Bernoulli beam model. A new closed-form analytical solution is obtained using the homotopy perturbation method. The effects of vibration amplitude, nanofiber volume fraction, nanofiber distribution pattern, small-scale parameters and the foundation parameters on the nonlinear frequency and deflection of the FG three-phase composite microbeams are studied in detail. The findings of the paper are valuable for researchers in the field of microbeam resonators.

A New Higher-Order Finite Element Model for Free Vibration and Buckling of Functionally Graded Sandwich Beams with Porous Core Resting on a Two-Parameter Elastic Foundation Using Quasi-3D Theory

A New Higher-Order Finite Element Model for Free Vibration and Buckling of Functionally Graded Sandwich Beams with Porous Core Resting on a Two-Parameter Elastic Foundation Using Quasi-3D Theory

Natural frequencies and modal shapes of folded sandwich plates made of porous core and FG-CNTRC coating layers resting on two parameters elastic foundation

Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Cutout effects on the dynamic analysis of laminated composite plates resting on two-parameter elastic foundation

This paper investigates the free vibration analysis of cutout laminated composite plates resting on an elastic foundation of the Winkler and Pasternak type. An efficient isoparametric 9-noded Lagrangean C0 finite element method is developed based on five degrees of freedom (d.o.f) first-order shear deformation theory (FSDT), which has the benefit of the requirement of only one variable along the thickness to model the plate equations to obtain the numerical results from the formulation. The equation of motion of the plate is derived using the virtual work principle. The accuracy and efficiency of the current formulation are validated with established results concerning free vibration analysis of laminated composite plates on elastic foundations and plates containing cutouts. It is found that the results obtained using the present formulation have variation within ±1.5% of the established results. After sufficient validation of the present formulation, new parametric results are given for cutout laminated composite plates resting on elastic foundations with varying elastic foundation stiffness parameters, boundary conditions, cutouts, side-to-thickness ratios, and the aspect ratio of plate and fiber angle orientations. The impact of cutouts on the dynamic behavior of laminated composite plates is investigated, and interestingly, it is observed that cutouts alter the frequency response of plates when introduced to plates without elastic foundations or plates resting on elastic foundations having low stiffness parameters; however, when cutouts are introduced to plates with high elastic foundation stiffness, the cutouts have very negligible impact on the dynamic behavior of plates. It is observed that the layout and the pattern of the cutout in the laminated composite plate affect the dynamic behavior of plates at all elastic foundation stiffnesses.

Vibrational characteristics of attached hemispherical-cylindrical ocean shell-structure reinforced by CNT nanocomposite with graded pores on two-parameter elastic foundations applicable in offshore components

One of the essential simulations of offshore components of ocean and marine structures can be counted as a modeling of the interaction between the systems and soil's behavior on the dynamic of the offshore systems. Based on this, for the first time, the fundamental frequencies of the porous nanocomposite attached hemisphere-cylinder shell system (AHCSS), counted as ocean and marine offshore structures, reinforced by carbon nanotube (CNT) rested on the Winkler-Pasternak elastic foundations, added up as soil's behavior simulation, are investigated in this paper. In detail, to expand the effectiveness of the CNT along the matrix, eight distinct functionally graded (FG) forms besides uniform distributions are prepared. Also, two forms of porosity which are FG distributed along the FG-CNT nanocomposite, are realized to add the effect of the porosity on the nanocomposite. Addedly, the first shear deformation theory and Donnell's scheme are coupled to arrange the affiliations of the AHCSS. In addition, Hamilton's principle gets the central motion equations (CMEs) of the AHCSS. Then, the half-analytical procedure, named the generalized differential quadrature method, discretizes the CMEs. Finally, the eigenvalue method finds the frequencies of the porous nanocomposite AHCSS. Remarkably, various numerical samples are prepared and solved to indicate the effects of the Winkler-Pasternak foundations, boundary cases, porosity forms, FG-CNT models, and physical values on the frequencies of the porous nanocomposite AHCSS.

Dispersion and Energy Characteristics of Bending Waves in a Plate Lying on a Two-Parameter Elastic Foundation

The propagation of bending waves in a plate resting on a two-parameter elastic foundation is considered. In contrast to the classical Kirchhoff model, the mathematical model used here takes into account not only the kinetic and potential energies of bending vibrations, but also the kinetic energy due to the inertia of rotation of the plate elements during bending. The dispersion equation, phase velocity, energy transfer velocity, and energy characteristics of waves propagating in the plate are analyzed depending on the ratio of the coefficients determining the shear and compression stiffness of the elastic foundation. Conditions are found under which waves with phase and group velocities having opposite directions (frequently called “backward” waves), can exist in the plate. It is demonstrated that such waves significantly change the behavior of the energy flux. In addition, relations are found that relate the kinematic and average energy characteristics of the waves.

Dispersion and Energy Characteristics of Bending Waves in a Plate Lying on a Two-Parameter Elastic Foundation

The propagation of bending waves in a plate resting on a two-parameter elastic foundation is considered. In contrast to the classical Kirchhoff model, the mathematical model used here takes into account not only the kinetic and potential energies of bending vibrations, but also the kinetic energy due to the inertia of rotation of the plate elements during bending. The dispersion equation, phase velocity, energy transfer velocity, and energy characteristics of waves propagating in the plate are analyzed depending on the ratio of the coefficients determining the shear and compression stiffness of the elastic foundation. Conditions are found under which waves with phase and group velocities having opposite directions (frequently called “backward” waves), can exist in the plate. It is demonstrated that such waves significantly change the behavior of the energy flux. In addition, relations are found that relate the kinematic and average energy characteristics of the waves.

Porosity and size effects on electro-hygrothermal bending of FG sandwich piezoelectric cylindrical shells with porous core via a four-variable shell theory

The sinusoidal four-variable shear deformation shell theory is employed, for the first time, to investigate the size dependent electro-hygrothermal bending of functionally graded (FG) sandwich piezoelectric cylindrical shells integrated with porous core. This model is assumed to be rested on two-parameter elastic foundations and subjected to elevated temperature, moisture concentration and external electric voltage as well as transverse mechanical loads. In order to take into account the size effect, the modified couple stress theory is employed containing only one length scale parameter. The core layer is assumed to be functionally graded porous material. Whereas, the face sheets are made of a functionally graded piezoelectric material which having gradient change along the thickness direction. Based on the sinusoidal four-variable shell theory, four governing equations are obtained involving external forces and foundation interaction, in addition to an equation associated to the electrical potential. An analytical solution for the obtained equations is presented to get the displacements and stresses of the FG sandwich piezoelectric cylindrical shells. Influences of the geometric parameters, material length scale parameter, temperature rise, moisture concentration, electric voltage, porosity factor, core thickness and foundation coefficients on the bending of the FG sandwich piezoelectric cylindrical shells resting on elastic foundations are discussed.

Mathematical modeling and solution of nonlinear vibration problem of laminated plates with CNT originating layers interacting with two-parameter elastic foundation

Generalizing the first-order shear deformation plate theory (FOPT) proposed by Ambartsumyan (Theory of anisotropic plates, Nauka, Moscow, 1967 (in Russian)) to the heterogeneous laminated nanocomposite plates and the nonlinear vibration problem is analytically solved taking into account an elastic medium in this study for the first time. The Pasternak-type elastic foundation model (PT-EF) is used as the elastic medium model. After creating the mathematical models of laminated rectangular plates with CNT originating layers on the PT-EF, the large amplitude stress–strain relationships and motion equations are derived in the form of nonlinear partial differential equations (PDEs) within FOPT. Then, by applying Galerkin's method to the derived equations, it is reduced to a nonlinear ordinary differential equation (NL-ODE) containing the second- and third-order nonlinear terms of the deflection function for laminated rectangular plates composed of nanocomposite layers. The NL-ODE is solved by the semi-inverse method, and the nonlinear frequency–amplitude relationship for the laminated plates consisting of CNT originating layers resting on the PT-EF is established within FOPT for the first time. From these relations, similar relations can be obtained particularly for the unconstrained laminated and monolayer CNT patterns plates. After comparing the accuracy of the obtained formulas with the reliable results in the literature, comprehensive numerical analyses are performed.

STATIC BENDING ANALYSIS OF NANOPLATES ON DISCONTINUOUS ELASTIC FOUNDATION WITH FLEXOELECTRIC EFFECT

In this study, rectangular plates subjected to static loads and supported on a discontinuous two-parameter elastic foundation are analyzed. The formulae for the computations are developed from improved shear deformation theory, where the displacement w is divided into bending and shear strain components. The results obtained from this article and other publications are evaluated to prove the accuracy of the proposed theory and the calculation program. Then, a parameter study is carried out to capture the effect of some material and geometrical parameters on the static response of structures. Especially the influence of the flexoelectric effect and discontinuous foundation is investigated. As two elastic foundation parameters and flexoelectric coefficient are increased, the non-dimensional maximum deflection of the plate decreases, indicating that these factors improve the plate's stiffness.

Influence Surface Coefficients of Plates Resting on Pasternak Foundation

In this study, internal force influence surface coefficients of plates resting on Pasternak foundation are obtained using the finite element method applying the classical Müller-Breslau Principle. The two-parameter elastic foundation is represented by the inclusion of the elastic bedding and shear parameter matrix terms of a 4-noded soil finite element to the corresponding stiffness matrix terms of the plate finite element used in the implementation. The loading matrices used to determine the influence surfaces directly are derived from the governing equations and the element matrices which are obtained using the Betti’s law. Internal force influence surface coefficients of plates with and without elastic foundation are given comparatively through numerical examples, confirming the values with another approach and the literature.

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory

Effect of simultaneous compressive and inertia loads on the bifurcation stability of shear deformable functionally graded annular fabrications reinforced with graphenes

In the current study, the stability of a functionally graded annular plate reinforced with a limited content of graphene platelets (FG-GPLRC) resting on a two-parameter elastic foundation is investigated. The structure is under the coincident effect of radial compressive forces and inertial load due to angular velocity. It is considered that the GPL nanofillers are randomly oriented, and their weight fraction varies with a layer-wise pattern through the thickness direction. The equivalent Young’s modulus and other material properties of the annular FG-GPLRC plate are evaluated by utilizing the Halpin–Tsai micromechanics rule. It is considered that the plate is resting on a two-parameter elastic foundation. The governing equations are extracted based on the first-order shear deformation theory (FSDT) and the von-Kármán type of geometric nonlinearity. Applying the virtual work principle and the adjacent equilibrium standard, the plate’s equations of stability are established. The derived governing equations are solved using a combined analytical–numerical (TE-GDQ) method, including trigonometric expansion in the circumferential direction and the generalized differential quadratures method through the radial direction. Comprehensive novel numerical results is derived to evaluate the effect of weight fraction of nanoparticles, the type of FGM, geometric dimensions, various boundary conditions, and surrounding elastic foundation on the buckling behavior of the plate.

Buckling of Timoshenko beam under two-parameter elastic foundations

This paper exposes buckling solutions of a plane, quasi-static Timoshenko beam with small transformation subjected to a longitudinal force and surrounded by an elastic wall modeled by two-parameter elastic foundations. A non-dimensional analysis of associated Haringx and Engesser model is performed and buckling stress and shape are exposed analytically. Relations for rigidity of the wall and buckling solutions were made for different regimes and for both models using asymptotic approach. Introducing the yield limit gives a simple criterion in terms of stiffness foundation and slenderness ratio for which buckling or irreversible transformation occur.

Thermal vibration analysis of functionally graded porous plates with variable thickness resting on elastic foundations using finite element method

Finite element modeling was applied for the first time in this work to analyze the free vibration responses of functionally graded plates porous with varying thickness resting on two-parameter elastic foundations in temperature conditions. The mechanical qualities of materials are supposed to alter along the thickness direction of plates using the Voigt model, temperature conditions fluctuate according to a nonlinear law, and porosity is suggested according to even and un-even distributed principles. The thickness of the plate is a function of the two directions, x and y, and it is supported by an elastic foundation Pasternak’s with two stiffness factors. A quadrilateral element model was created and employed to handle the difficulties of FG plates with diverse boundary conditions based on the revised higher-order shear deformation theory. The correctness of the current theory and mechanical model is validated by comparing the numerical data of this study with those of trustworthy sources. A great variety of factors have been examined to capture the influence of the vibration response of the plate on geometric and material features. The article's results are scientifically noteworthy, contributing to computational mechanics' understanding of the mechanical behavior of this intricate structure.

Modeling of partially delaminated composite plates resting on two-parameter elastic foundation with improved layerwise theory

In this work, we aim to build the mathematical model for delaminated composite plates resting on elastic foundation by means of improved layerwise theory and finite element implementation for free vibration analysis. The proposed model takes into account the transverse normal and shear deformation of the elastic foundation so as to investigate their effect on the dynamic characteristics of delaminated plates. The four-node plate element is used for the finite element formulations with Langrage interpolation functions for in-plane structural unknowns and Hermite cubic interpolation functions for out-of-plane structural unknowns. The delamination is simulated by means of Heaviside unit step functions allowing for the discontinuity of displacement fields. The effects of both longitudinal locations and interfacial locations of delamination are investigated. The results show that when the delamination located closer to the clamped end and at the middle plane, it has much more severe influence on its natural frequencies. The foundation parameters also have significant effect on the natural frequencies and frequency shift due to delamination. Different layup stacking sequences are also considered in this study.

Dynamic response of porous E-FGM thick microplate resting on elastic foundation subjected to moving load with acceleration

For the first time, dynamic analysis of exponentially functionally graded material (E-FGM) including porosities is investigated. As a first endeavor, forced vibration response of thick microplates made of porous E-FGM when subjected to moving loads with an acceleration in speed is studied in this research study. The role of a two-parameter elastic foundation is also included. A powerful mathematical formulation supported by the assumptions of the general third-order shear deformation theory (GTSDT) as well as the modified couple stress theory (MCST) is developed to find an accurate model for thick size-dependent microplates. The effective material properties of imperfect FGM, which change exponentially along the z-direction, are established via developing a modified rule of mixture which includes the porosity imperfection. The governing equations for simply-supported microplate is found using a virtual work of Hamilton principle, and afterwards they are solved by developing a state space method (SSM) in conjunction with a set of mathematical series. The numerical results attempt to indicate the influence of variation in volume fraction index, porosity index, elastic foundations, and small-scale parameter, highlighted by different load speeds, acceleration ratios, and time histories of moving load under upper surface of E-FGM thick microplate. The parametrical studies can be used in better designing of micro/nanostructures made of porous FGMs under accelerated moving loads.

Natural Vibrations of a Cylindrical Shell with Fluid Partly Resting on a Two-Parameter Elastic Foundation

This paper presents the results of studies on natural vibrations of circular cylindrical shells containing liquid and resting on an elastic foundation, which is described by the Pasternak two-parameter model. In the meridional direction, the elastic medium is nonuniform and represents an alternation of sections in which the foundation is present or absent. The behavior of the elastic structure and the compressible fluid is described in the framework of classical shell theory based on the Kirchhoff–Love hypothesis and the Euler equations. The equations of motion of the shell are reduced to a system of ordinary differential equations with respect to new unknowns. The wave equation written for pressure in the fluid also reduces to a system of ordinary differential equations using the straight line method. The solution of the formulated boundary value problem is found by the Godunov orthogonal sweep method. The validity of the results obtained is confirmed by comparison with the known numerical-analytical solutions. The dependences of the minimum vibration frequencies on the characteristics of elastic medium with variable nonuniformity along the length of the structure have been obtained for cylindrical shells with different boundary conditions. It has been found that the violation of smoothness of the derived dependences is caused by a change of the vibration mode with minimum frequency and is determined both by the ratio of the size of the elastic foundation to the entire length of the shell and its stiffness, and also by a combination of boundary conditions set at the edges of the thin-walled structure.