Two-parameter Elastic Foundation Research Articles

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.

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.

Exact analytical solution for static deflection of Timoshenko composite beams on two-parameter elastic foundations

New exact analytical solutions are presented for the static deflection of coupled Timoshenko composite beams resting on two-parameter elastic foundations subject to arbitrary boundary and loading conditions. Governing differential equations are obtained from the principle of virtual work in which four degrees of freedom, namely axial elongation, twist, bending and transverse shear are coupled. In order to obtain the analytical solutions describing the static deflection of coupled Timoshenko composite beams resting on elastic foundations, the governing equations are first rewritten in matrix form to enable decoupling of axial displacement and twist from bending and transverse shear. Applying a separation of variables technique, axial elongation and twist are eliminated and the matrix differential equation for bending and transverse shear is transformed into a system of first-order equations which is solved using a fundamental matrix approach. The results obtained from the analytical solutions are firstly verified by comparison with solutions from the literature for isotropic homogeneous beams on elastic foundations and then with the numerical results from the Chebyshev collocation method obtained for composite beams on elastic foundations, and found to be in excellent agreement. The influence of elastic foundation coefficients, coupling terms and length-to-thickness ratio on the static deflection of Timoshenko composite beams resting on elastic foundations are investigated and discussed.

Natural frequency analysis of imperfect FG sandwich plates resting on Winkler-Pasternak foundation

The present research analyzes the natural frequencies of the imperfect functionally graded sandwich plate (FGSP) comprised of porous face sheets made of functionally graded materials (FGMs) and an isotropic homogeneous core resting on the elastic foundation. To accomplish this, the material characteristics of the FGSP are taken to be changed incessantly along the thickness direction based on the volume fraction of constituents expressed by the modified rule of the mixture, which includes porosity volume fraction with three diverse kinds of porosity distribution models. Furthermore, to describe the two-parameter elastic foundation's response on the imperfect FGSP, the medium is supposed to be linear, homogenous, and isotropic, and it has been modelled using the Winkler-Pasternak model. Moreover, in the kinematic relationship of the imperfect FGSP resting on the Winkler-Pasternak foundation, third-order shear deformation theory (TSDT) is used, and the motion equations are set up employing Hamilton's principle. For natural frequency analysis of imperfect FGSPs resting on the Winkler-Pasternak foundation with simply supported edges, an analytical solution is obtained. To verify the current formulation, comprehensive comparisons are performed with the available data. The impacts of the two-parameter elastic foundation, porosity volume fraction, porosity distribution types, lay-up scheme, and side to thickness ratio, on the values of the non-dimensional fundamental natural frequency (NDFNF) of the imperfect FGSPs, are investigated, thoroughly.

Finite Element Modeling for Static Bending Behaviors of Rotating FGM Porous Beams with Geometrical Imperfections Resting on Elastic Foundation and Subjected to Axial Compression

The static bending analysis of the FG porous beam resting on the two-parameter elastic foundation is initially carried out using a combination of Reddy’s high-order shear deformation theory and the finite element technique, where the initial geometrical imperfection and rotation movement in one fixed axis are calculated. Through the power-law distribution function with porosities, material characteristics vary constantly from one surface to the next in the direction of thickness, and the beam is concurrently impacted by an acting force perpendicular to the beam axis and an axial compressive force. The stiffness matrix of the beam element changes as a result, and the static bending response of this beam is significantly different from that of ordinary beams. Comparison cases with published findings are used to verify the computational theory. The calculations clearly reveal many innovations for rotating beams that are influenced by many different kinds of loads, which may be used to the designing, manufacturing, and usage of these structures in reality.

Free vibration of piezoelectric FGM sandwich beam with porous core embedded in thermal environment and elastic foundation

This paper aims to present thermo-electrical free vibration characteristics of functionally graded material (FGM) sandwich beam placed on the two-parameter elastic foundation. The beam is constructed of a foam core, two middle FGM layers, and two outer piezoelectric layers. It is assumed that the beam is subjected to a constant voltage and a uniform/linear temperature distribution. Physical properties of the core and two middle layers vary smoothly through the thickness according to the cosine and power-law forms, respectively. Lagrange equations in conjunction with the Reddy third-order beam theory is employed to derive the governing equations of motion. A simple polynomial trial function-based Ritz method is adopted for the approximation of the displacement field to obtain the vibration response. The correctness of the study is verified by comparisons with other authors’ published results. Influences of geometry parameters, material property distribution, applied voltage, elastic foundation, temperature distribution, temperature change, porosity coefficient, span-to-height ratio, and boundary conditions are investigated through parametric studies.