Two-parameter Elastic Foundation Research Articles

Vibration behavior of functionally graded sandwich beam with porous core and nanocomposite layers

This paper presents the influence of carbon nanotubes (CNTs) waviness, aspect ratio, internal pores and graphene platelets (GPLs) on the vibrational behavior of functionally graded nanocomposite sandwich beams resting on two-parameter elastic foundations. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness of upper and bottom layers of the sandwich beam and their mechanical properties are estimated by an extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes\' effective aspect ratio as the filler content increases. The core of structure is porous and the internal pores and graphene platelets (GPLs) are distributed in the matrix of core either uniformly or non-uniformly according to three different patterns. The elastic properties of the nanocomposite are obtained by employing Halpin-Tsai micromechanics model. The equations of motion are derived based on Timoshenko beam theory and employing Hamilton\'s principle. The problem is modeled using a semi-analytical approach composed of generalized differential quadrature method (GDQM) and series solution adopted to solve the equations of motion. Detailed parametric studies are carried out to investigate carbon nanotubes (CNTs) waviness, CNT aspect ratio, porosity coefficient, porosity distribution, graphene platelets (GPLs) distribution, Winkler foundation modulus, shear elastic foundation modulus and geometrical conditions on the vibrational behavior of the sandwich structure.

A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin\'s approach

In this work, the buckling analysis of material sandwich plates based on a two-parameter elastic foundation under various boundary conditions is investigated on the basis of a new theory of refined trigonometric shear deformation. This theory includes indeterminate integral variables and contains only four unknowns in which any shear correction factor not used, with even less than the conventional theory of first shear strain (FSDT). Applying the principle of virtual displacements, the governing equations and boundary conditions are obtained. To solve the buckling problem for different boundary conditions, Galerkin\'s approach is utilized for symmetric EGM sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of plate aspect ratio, elastic foundation coefficients, ratio, side-to-thickness ratio and boundary conditions on the buckling response of FGM sandwich plates. A good agreement between the results obtained and the available solutions of existing shear deformation theories that have a greater number of unknowns proves to demonstrate the precision of the proposed theory.

An analytical framework for rectangular FGM tapered plate resting on the elastic foundation

This article presented the free vibration analysis of tapered rectangular functionally graded material (FGM) plates resting on the two-parameter (Winkler and Pasternak) elastic foundation model. Higher-order shear deformation theory (HSDT) to satisfy the traction free boundary conditions on the top and bottom surface of the sigmoid FGM (S-FGM) plate. The material properties are varied according to the two simple power-law distributions of volume fractions of the constituents. The thickness of the plate linearly varies in the y–direction to both sides of the middle plane. The variational Hamilton principle establishes the equation of motions, and Galerkin-Vlasov’s method is used to solve the equation of motion. The free vibration is evaluated with the inclusion of an elastic foundation with different taper ratios. A parametric study has been investigated for the different span ratio, aspect ratio, volume exponent index, taper ratio, and combination of elastic foundation.

Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment

In this paper, free vibrations of Porous Functionally Graded Beams (P-FGBs), resting on two-parameter elastic foundations, and exposed to three forms of thermal field, uniform, linear, and sinusoidal, are studied using a Refined Higher-order shear Deformation Theory. The present theory accounts for shear deformation by considering a constant transverse displacement and a higher-order variation of the axial displacement through the thickness of the beam. The stress-free boundary conditions are satisfied on the upper and lower surfaces of the beam without using any shear correction factor. The material properties are temperature-dependent and vary continuously through the depth direction of the beam, based on a modified power-law rule, in which two kinds of porosity distributions, uniform, and nonuniform, through the cross-section area of the beam, are considered. Hamilton’s principle is applied to obtain governing equations of motion, which are solved using a Navier-type analytical solution for simply supported P-FGB. Numerical examples are proposed and discussed in detail, to prove the effect of the thermal environment, the porosity distribution, and the influence of several parameters such as the power-law index, porosity volume fraction, slenderness ratio, and elastic foundation parameters on the critical buckling temperatures and the natural frequencies of the P-FGB.

Dynamic Analysis of Circular Plates in Contact with Fluid and Resting on Two-Parameter Foundations

The dynamic behaviour of a circular plate in contact with fluid and resting on two-parameter elastic foundations is of interest in the field of geotechnics, structure, highway, railway, oil and gas and mechanical engineering. In this work, the dynamic behaviour of circular plate in contact with fluid and resting on Winkler and Pasternak foundations is investigated. The coupled differential equation of the system is analysed using differential transformation method. Good agreements are established when the results of the analytical solutions are compared to the results of the experimental investigation as reported in literature. The analytical solutions obtained are used to investigate the effects of elastic foundation parameters on natural frequency, combine foundation parameters on natural frequency, plate in contact with fluid and that of radial and circumferential stress on mode shapes. From the results, it is observed that, increases in elastic foundation parameter increases the natural frequency in all cases. Presence of fluid reduces the natural frequency of the plate. Also, it is established that mode shapes are not altered by the presence of fluid. However, mode shape displacement occurs due to presence of radial and circumferential stresses. Since the study provides a physical insight into the vibration mode of the structure, it is expected that the study will enhance better understanding on the dynamic behaviour of a circular plate in contact with fluid and resting on two-parameters elastic foundation.

Effect of variable elastic foundations on static behavior of functionally graded plates using sinusoidal shear deformation

In the present research, static analysis of functionally graded (FG) plates resting on variables elastic foundation and subjected to mechanical loads is examined using four variable shear deformation theories The present theory utilized an undefined integral to reduce the number of the unknown from five unknown, as such higher theory and first shear theory, to only four unknown, without needing a shear corrector coefficient. The elastic medium is assumed as two-parameter elastic foundation (Pasternak-Winkler) with considering a variation in the Winkler layer along of a side of the functionally graded plate; however, the shear layer of Pasternak is supposed to be constant. The material proprieties of FG plates are supposed to vary smoothly through-the-thickness direction according to the power-law distribution. The equilibrium equations of functionally graded plates resting on the elastic foundation are derived using the principle of the virtual work. The solution of simply supported FG plates resting on two-parameter elastic foundation is obtained by using technique of Navier. Numerical results and validation for functionally plates are presented, the accuracy of the present method to predict static behavior of functionally graded plates and the influence of variable elastic foundation on stress and displacement of functionally graded plates are shown.

Nonlinear free vibration of cracked functionally graded graphene platelet-reinforced nanocomposite beams in thermal environments

The present paper studies the nonlinear free vibration of edge-cracked graphene nanoplatelet (GPL)-reinforced composite laminated beams resting on a two-parameter elastic foundation in thermal environments. GPL nanofillers are assumed to be randomly oriented and are functionally graded distributed in a layer-wise pattern along the beam thickness. The temperature field considered is assumed to be a uniformly distributed in the domain of the beam. Effective material properties of the GPL-reinforced nanocomposite (GPLRC) are estimated by micromechanical models. Stress intensity factors are calculated based on the finite element methods. In the framework of the first order shear deformation beam theory, the equations of motion of cracked GPLRC beams are established including the von Kármán-type geometric nonlinearity. The bending stiffness of the cracked section is evaluated by the massless rotational spring model. The differential quadrature method is used to calculate the linear and nonlinear natural frequencies of the cracked beams. Numerical results illustrate the effects of GPL distribution pattern and concentration, GPL geometry and size, crack length, foundation stiffnesses and temperature variation on the linear and nonlinear free vibration characteristics of the cracked GPLRC beams.

A rigorous analytical solution of interfacial stresses and overall stiffness of beam structures bonded with partially covered plates

Beam structures with partially covered plates have been widely used in macro members and micro devices, such as concrete beams with fiber-reinforced plastic (FRP) plates, aircraft structures with FRP patches and smart beams with actuators/sensors. Based on the two-parameter elastic foundation model and the classical Euler–Bernoulli beam model, a rigorous analytical solution of interfacial stresses and overall stiffness of plated beams is obtained, and influences of the interface and plate properties are presented. The limitation of the assumption of the same curvature of host beam and plate, which was widely adopted in previous studies, has been clarified. And the present solution is applicable to a wider range of interface and plate properties, especially for large values of plate thickness, plate modulus and interfacial stiffness, as well as small values of plate length. The saturation phenomenon of overall stiffness is observed in some cases, and critical values of the interface and plate parameters corresponding to the saturation limit of overall stiffness can be referred for design. The results can provide guidance for the optimal design of widely-used plated beams to prevent premature interfacial debonding and acquire better overall performance, such as higher stiffness with FRP plates or shape control with actuators, by manipulating the interface and plate properties.

Discrete-Continual Finite Element Method for Semianalytical Analysis of Plates on Two-Parameter Elastic Foundation. Part 1: Continual Formulations of the Problem and Approximations

This paper is proposed discrete-continual finite element method (DCFEM) for semianalytical analysis of plates on two-parameter elastic foundation. First of all, the operational and discrete-continual formulations of the problem are described, then the approximation of unknown functions and its partial derivatives are obtained. Later the internal forces, local stiffness matrices and local load vectors are evaluated, then these local vectors and matrices are assembled to obtain resultant multipoint boundary problem and in the final step the problem is solved by the exact analytical method. This method allowed to obtain exact analytical solutions of boundary problems along the regular direction, and these solutions remain exact for arbitrary influences such as any force or moment, soil parameters, intermediate constraints and connections. Using this method, the problem remains continual in the one direction (basic), while the discrete (finite element) approximation is carried out with respect to the non-basis direction. The structural discontinuities in the analytical direction can be taken into account, by addition of new appropriate boundary conditions at the relevant section. The DCFEM increases the accuracy of the solution and significantly reduces the computational efforts, especially within analysis of extended plates such as strip foundations.

Discrete-Continual Finite Element Method for Semianalytical Analysis of Plates on Two-Parameter Elastic Foundation. Part 2: Discrete-Continual Formulation of the Problem, Numerical Samples

This paper is devoted to application of discrete-continual finite element method (DCFEM) for semianalytical solution of problems of analysis of plates on two-parameter elastic foundation. Corresponding resultant multipoint boundary problem for system of ordinary differential equations of the first order with piecewise-constant coefficients is presented. Numerous samples of problems of structural mechanics have been considered. These samples allow demonstration of some advantages of application of the suggested approach to semianalytical analysis of plates on elastic foundation.

A beam formulation with 3D capabilities for the free vibration analysis of thin-walled metallic and composite structures

Abstract The present article proposes an advanced Ritz formulation with 3D capabilities for the analysis of thin-walled metallic and composite beam structures. Various set of admissible functions such as: Legendre, Chebyshev and algebraic polynomials, as well as hybrid functions are employed. The investigation is carried out by using the method of power series expansion of displacement components based on 2D-Taylor polynomials. The governing equations (GEs) are derived in their weak form via Hamilton's Principle and are solved by using the Ritz method. Convergence and stability of both cross-section and admissible functions have been thoroughly analysed. The high level of accuracy of the proposed formulation has been comprehensively examined in three selected case studies: (i) slender beams with arbitrary boundary conditions, and both solid and thin-walled cross-sections; (ii) a fully-clamped three-layer non-homogenous circular cylindrical shell; (iii) a fully-clamped functionally graded material (FGM) box resting on two-parameter elastic foundation.

Analysis of Rectangular Plates on Resting Winkler and Two-Parameter Elastic Foundation Models by Finite Difference Method

Analysis of Rectangular Plates on Resting Winkler and Two-Parameter Elastic Foundation Models by Finite Difference Method

Analysis of Rectangular Plates on Resting Winkler and Two-Parameter Elastic Foundation Models by Finite Difference Method

Analysis of rectangular plates on a rising elastic foundation is common when designing the foundation of civil, traffic, and irrigation works. The current research presents the results of the analysis of rectangular plates on rising Winkler elastic and two-parameter foundation using the finite difference method. The computational programming was done on Matlab. The results of the research verify the accuracy of the FEM and are in agreement with findings in the literature. In addition, this research also compares the calculation results of two foundation models and investigates the change of plate height ratio to plate size.

Rotary Inertia and Higher Modes Effect on the Dynamic Response of Timoshenko Beams on Two-Parameter Elastic Foundation

This study investigates the dynamic response of the axially loaded Timoshenko beams on two-parameter foundation with generalized elastic end conditions. A simplified modal analysis procedure is presented for the vibration analysis by using the conventional separation of variables method. The effects of higher modes and rotary inertia on the dynamic response of the forced Timoshenko beam are highlighted. A numerical example is presented to compare the modal responses of bending moment and shear force of the beams on Winkler and Vlasov type foundations with and without the rotary inertia effect.

Critical buckling temperature and force in porous sandwich plates with CNT-reinforced nanocomposite layers

This paper presents the thermal and mechanical buckling behaviors of lightweight polymeric nanocomposite sandwich plates containing uniformly dispersed (UD) pores resting on two-parameter elastic foundations. The outer layers of the proposed sandwich plates were assumed to be made of functionally graded (FG) carbon nanotube (CNT)-reinforced polymeric nanocomposite. For nanocomposite layers, the considerable effect of cluster formation of randomly-oriented CNTs was considered and material properties were evaluated through Eshelby-Mori-Tanaka (EMT)'s approach with the definition of cluster state. Furthermore, the first and third order shear deformation theories (FSDT and TSDT) of plates were employed to define the total energy function of sandwich plates. The governed buckling equations were facilitated using a mesh-free method. The effects of porosity, nanofiller characterizations, elastic foundation coefficients, sandwich plate dimensions and boundary conditions on buckling behavior were explored. The obtained results showed that porosity considerably improved thermal buckling behavior; however, it reduced the critical mechanical loads of sandwich plates. Moreover, adding CNTs into outer layers simultaneously improved the mechanical and thermal buckling responses of sandwich plates.

Quasi-3D integral model for thermomechanical buckling and vibration of FG porous nanoplates embedded in an elastic medium

Mechanical buckling, thermal buckling and free vibration behaviors of functionally graded (FG) porous nanoplates embedded in an elastic medium are investigated via a nonlocal strain gradient theory. Two dimensional (2D) and quasi three dimensional (quasi-3D) sinusoidal shear deformation theories are used in this study. The proposed theories have a displacement field with integral terms which include the effects of both transverse shear and normal deformations. Porosity-dependent material properties of the porous nanoplate are defined via a modified power-law function. Equations of motion are derived based on Hamilton’s principle which includes the effect of the two-parameter elastic foundations. Numerical results are presented to verify the accuracy of the present 2D and quasi-3D shear deformation theories by comparing them with the solutions given in literature with different 2D, quasi-3D and 3D solutions. The effects of many parameters like porosity factor, nonlocal, length scale parameters, plate aspect ratio, side-to-thickness ratio and gradient index on the buckling, thermal buckling and vibration of FG porous nanoplates are all discussed.

Free vibration analysis Silicon nanowires surrounded by elastic matrix by nonlocal finite element method

Higher-order theories are very important to investigate the mechanical properties and behaviors of nanoscale structures. In this study, a free vibration behavior of SiNW resting on elastic foundation is investigated via Eringen\'s nonlocal elasticity theory. Silicon Nanowire (SiNW) is modeled as simply supported both ends and clamped-free Euler-Bernoulli beam. Pasternak two-parameter elastic foundation model is used as foundation. Finite element formulation is obtained nonlocal Euler-Bernoulli beam theory. First, shape function of the Euler-Bernoulli beam is gained and then Galerkin weighted residual method is applied to the governing equations to obtain the stiffness and mass matrices including the foundation parameters and small scale parameter. Frequency values of SiNW is examined according to foundation and small scale parameters and the results are given by tables and graphs. The effects of small scale parameter, boundary conditions, foundation parameters on frequencies are investigated.

DYNAMIC ANALYSIS OF THE LAMINATED COMPOSITE PLATE RESTING ON TWO-PARAMETER ELASTIC FOUNDATION SUBJECTED TO MOVING MASS USING FINITE ELEMENT METHOD

This paper presents a model and an algorithm for solving the dynamic problem of the laminated composite plate resting on two-parameter elastic foundation subjected to moving mass. By using the finite element method and the homogenization theory for the laminated composite plate, the motion equations of composite plates resting on two-parameter elastic foundation subjected to moving mass are established, in which the elastic deformation foundation was modeled as a Pasternak’s elastic foundation. Through the proposed theory and the computed program in MATLAB environment, the numerical results are verified by comparing with other exact publications, and influences of geometrical and material properties such as the fiber angles of the structure, modulus ratio E1/E2 and the elastic foundation stiffnesses c1 and c2, etc., on the dynamic behaviours of the structure are investigated.

Static interaction analysis between beam and layered soil using a two-parameter elastic foundation

The work presents a finite element modeling of beam resting on a two-parameter layered soil. The behavior of the soil continuum and the beam are assumed to be linear and homogeneous isotropic. Using the strain energy expressions, the shear strain of the beam element and the soil foundation are taken together in the analysis. In this approach, the stiffness matrix of each component is elaborated and integrated in the finite element analysis. First, various examples are elaborated to show the effectiveness of the proposed approach and the ability of the numerical program developed for this concern. Second, the analysis is widened to study the influence of the soil properties on the interface continuum and on the beam responses. Third, a parametric study is carried out to highlight the effect of the position of springs at the interface continuum, the properties of soil, the deepest of the soil foundation and the ballast layer on the response of the interface and the beam itself. Moreover, shear deformations are presented to show the crucial influence on the beam, on the structure and on the interface behaviors. Obtained results show pertinent results corresponding to the interface continuum and the beam responses.

Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass

In this study, a modified finite element method (FEM) that can be used to analyse the transverse vibrations of a Timoshenko beam, made of functionally graded materials (FGMs), on a two-parameter foundation and subjected to a variable-velocity moving mass is presented. First, the motion equations of the FGM beam exposed to a moving mass are obtained by combining the mass interaction equations with the beam equations obtained using the first-order shear deformation theory (FSDT). Secondly, to obtain the equation of motion of the beam, mass and foundations, the rigidity of the two-parameter foundations, the Winkler and shear, then combined with the motion equations of the beam and mass using the Hamilton principle. The response of the system is investigated considering the effects of inertia and variable velocity of the moving mass, parameters of the foundations and material constituents of the FGM beam. The changes on the dynamic behaviour of the FGM Timoshenko beam, which are affected by the frequency variation of the considered system are highlighted.