Effect Of Elastic Foundation Research Articles

Free vibration of functionally graded beams resting on Winkler-Pasternak foundation

In the present study, the free vibration of functionally graded beams resting on two parameter elastic foundation was examined. The properties of the functionally graded materials were presumed to vary continuously along the thickness direction. The foundation medium was assumed to be linear, homogeneous, and isotropic, and it was modeled by the Winkler-Pasternak model with two parameters for describing the reaction of the elastic foundation on the beam. The functionally graded beam was modeled with classical beam theory. The governing equation including the effects of functionally graded material properties, Winkler-Pasternak elastic foundation was solved using separation of variables. The eigenvalues of yielding fundamental equation versus clamped-clamped, clamped-free, clamped-simply supported, and simply supported-simply supported boundary conditions were found. To corroborate the results, comparisons were carried out with available results for homogeneous and functionally graded beams. The effects of Winkler-Pasternak type elastic foundation and functionally graded material properties on the values of dimensionless frequency parameter of beams were discussed. Briefly, it was found that the dimensionless frequency parameters of beam change according to material properties, presence of elastic foundation, and boundary conditions; moreover, the separate effects of these quantities on each other are interesting.

On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment

This study provides an exact solution for the size dependent buckling and post-buckling behavior of functionally graded (FG) micro-beams with arbitrary boundary conditions which are subjected to combined thermo-mechanical loading. To this end, a theoretical formulation including the effects of size dependency, elastic foundation and uniform temperature distribution is first derived using the modified couple stress theory and through the principle of minimum total potential energy. Next, the nonlinear equations governing bending and stretching behavior of FG micro-beams are uncoupled to a fourth-order ordinary differential equation. Finally, the differential operator method is utilized to exactly solve the decoupled equation. Also, a Fourier series solution is presented for doubly-simply supported FG micro-beams to show the importance of exact solution. In the numerical results section, the effects of the geometric ratios, material distribution, temperature variation, and material length scale parameter on the post-buckling behavior are discussed in detail. Findings show that the Fourier series solution is not able to correctly predict the post-buckling behavior of FG micro-beams, since the effect of flexural-extensional coupling stiffness term appearing in the natural boundary condition is ignored. Also, it is seen that critical values of axial traction obtained from the post-buckling analysis are significantly varied with the transverse force per unit length and temperature variation, while the buckling analysis predicts that buckling values of beam remain constant. Therefore, it can be concluded that the buckling analysis is inadequate for analyzing FG micro-beams under thermo-mechanical loadings.

Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy's HSDT

This work presents an analytical approach to investigate buckling and post-buckling behavior of FGM plate with porosities resting on elastic foundations and subjected to mechanical, thermal and thermomechanical loads. The formulations are based on Reddy's higher-order shear deformation plate theory taking into consideration Von Karman nonlinearity, initial geometrical imperfections, and Pasternak type of elastic foundations. By applying Galerkin method, closed-form relations of buckling loads and post-buckling equilibrium paths for simply supported plates are determined. Numerical results are carried out to show the effects of porosity distribution characteristics (Porosity-I and Porosity-II), geometrical parameters, material properties and elastic foundations on the mechanical, thermal and thermomechanical buckling loads and post-buckling resistance capacity of the porous FGM plates.

Multi-objective optimization of functionally graded materials, thickness and aspect ratio in micro-beams embedded in an elastic medium

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.

Postbuckling of magneto-electro-elastic CNT-MT composite nanotubes resting on a nonlinear elastic medium in a non-uniform thermal environment

As a first endeavor, the effect of nonlinear elastic foundation on the postbuckling behavior of smart magneto-electro-elastic (MEE) composite nanotubes is investigated. The composite nanotube is affected by a non-uniform thermal environment. A typical MEE composite nanotube consists of microtubules (MTs) and carbon nanotubes (CNTs) with a MEE cylindrical nanoshell for smart control. It is assumed that the nanoscale layers of the system are coupled by a polymer matrix or filament network depending on the application. In addition to thermal loads, magneto-electro-mechanical loads are applied to the composite nanostructure. Length scale effects are taken into account using the nonlocal elasticity theory. The principle of virtual work and von Karman’s relations are used to derive the nonlinear governing differential equations of MEE CNT-MT nanotubes. Using Galerkin’s method, nonlinear critical buckling loads are determined. Various types of non-uniform temperature distribution in the radial direction are considered. Finally, the effects of various parameters such as the nonlinear constant of elastic medium, thermal loading factor and small scale coefficient on the postbuckling of MEE CNT-MT nanotubes are studied.

Resonance analyses of a pipeline-riser system conveying gas-liquid two-phase flow with flow-pattern evolution

The pipeline-riser systems are applied extensively in engineering systems. The flow patterns of gas-liquid two-phase flow evolve abundantly in pipeline-riser system, and it might induce the combination and parametric resonance, which threatens the security and stability of the piping system. In order to evaluate the resonance effect caused by internal two-phase flow, a dynamic model capable of describing axial, bending vibrations and fluid-structure interaction was formulated using FEM (Finite Element Method), where the void fractions of the two-phase flow were predicted by empirical correlations. To understand the flow pattern evolution and the pressure fluctuation characteristics, the experiments of observations and pressure testing were carried out, and the flow pattern map has been drawn. The effects of void fraction and elastic foundation on structural natural characteristics were investigated in detail. Then, the resonance analysis of the experimental facility and the verification experiments were performed to study the vibration characteristics. The results presented that the flow patterns in the riser were generalized into five types, and the fluctuating frequencies were mainly concentrated in low frequencies. The supporting foundation could increase the structural natural frequency significantly, and enhancing elastic foundation coefficient was effective to eliminate the resonance caused by internal two-phase flow.

Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method

Due to the capability of Eringen's nonlocal elasticity theory to capture the small length scale effect, it is widely used to study the mechanical behaviors of nanostructures. Previous studies have indicated that in some cases, the differential form of this theory cannot correctly predict the behavior of structure, and the integral form should be employed to avoid obtaining inconsistent results. The present study deals with the bending analysis of nanoplates resting on elastic foundation based on the integral formulation of Eringen's nonlocal theory. Since the formulation is presented in a general form, arbitrary kernel functions can be used. The first order shear deformation plate theory is considered to model the nanoplates, and the governing equations for both integral and differential forms are presented. Finally, the finite element method is applied to solve the problem. Selected results are given to investigate the effects of elastic foundation and to compare the predictions of integral nonlocal model with those of its differential nonlocal and local counterparts. It is found that by the use of proposed integral formulation of Eringen's nonlocal model, the paradox observed for the cantilever nanoplate is resolved.

Effect of elastic foundations on stochastic thermally induced post buckling response of functionally graded materials plates with uncertain material properties

Effect of elastic foundations on stochastic thermally induced post buckling response of functionally graded materials plates with uncertain material properties

Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

A unified formulation for modeling of inhomogeneous nonlocal beams

In this article, buckling and free vibration of functionally graded (FG) nanobeams resting on elastic foundation are investigated by developing various higher order beam theories which capture shear deformation influences through the thickness of the beam without the need for shear correction factors. The elastic foundation is modeled as linear Winkler springs as well as Pasternak shear layer. The material properties of FG nanobeam are supposed to change gradually along the thickness through the Mori–Tanaka model. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. From Hamilton\'s principle, the nonlocal governing equations of motion are derived and then solved applying analytical solution. To verify the validity of the developed theories, the results of the present work are compared with those available in literature. The effects of shear deformation, elastic foundation, gradient index, nonlocal parameter and slenderness ratio on the buckling and free vibration behavior of FG nanobeams are studied.

Dynamic analysis of stepped composite cylindrical shells surrounded by Pasternak elastic foundations based on the continuous element method

This research presents a continuous element model for solving vibration problems of stepped composite cylindrical shells surrounded by Pasternak foundations with various boundary conditions. Based on the First Order Shear Deformation Theory (FSDT), the equations of motion of the circular cylindrical shell are introduced and the dynamic stiffness matrix is obtained for each segment of the uniform shell. The interesting assembly procedure of continuous element method (CEM) is adopted to analyze the dynamic behavior of the stepped composite cylindrical shell surrounded by an elastic foundation. Free vibrations and harmonic responses of different configurations of stepped composite cylindrical shells on elastic foundations are examined. Effects of structural parameters, stepped thickness and elastic foundations on the free vibration responses of stepped composite cylindrical shells are also presented. Comparisons with previously published results and finite element (FE) analyses show that the proposed technique saves data storage volume and calculating time, and is accurate and efficient for widening the studied frequency range.

Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.

Effects of Elastic Foundations on Flexural Response of Shear Deformable Laminated Plates Subjected to Transverse Uniform Lateral Pressure Uncertain System Environment and Hygrothermomechanical Loading

Effects of Elastic Foundations on Flexural Response of Shear Deformable Laminated Plates Subjected to Transverse Uniform Lateral Pressure Uncertain System Environment and Hygrothermomechanical Loading

Divergence instability of pipes conveying fluid with uncertain flow velocity

This article deals with investigation of probabilistic stability of pipes conveying fluid with stochastic flow velocity in time domain. As a matter of fact, this study has focused on the randomness effects of flow velocity on stability of pipes conveying fluid while most of research efforts have only focused on the influences of deterministic parameters on the system stability. The Euler–Bernoulli beam and plug flow theory are employed to model pipe structure and internal flow, respectively. In addition, flow velocity is considered as a stationary random process with Gaussian distribution. Afterwards, the stochastic averaging method and Routh’s stability criterion are used so as to investigate the stability conditions of system. Consequently, the effects of boundary conditions, viscoelastic damping, mass ratio, and elastic foundation on the stability regions are discussed. Results delineate that the critical mean flow velocity decreases by increasing power spectral density (PSD) of the random velocity. Moreover, by increasing PSD from zero, the type effects of boundary condition and presence of elastic foundation are diminished, while the influences of viscoelastic damping and mass ratio could increase. Finally, to have a more applicable study, regression analysis is utilized to develop design equations and facilitate further analyses for design purposes.

Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory

ABSTRACTIn this research, vibration characteristics of axially functionally graded nanobeams resting on variable elastic foundation are investigated based on nonlocal strain gradient theory. This nonclassical nanobeam model contains a length scale parameter to explore the influence of strain gradients and also a nonlocal parameter to study the long-range interactions between the particles. The present model can degenerate into the classical models if the material length scale parameter and the nonlocal stress field parameter are both taken to be zero. Elastic foundation consists of two layers: a Winkler layer with variable stiffness and a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. Material properties are graded axially via a power-law distribution scheme. Hamilton's principle is employed to derive the governing equations that are solved applying a Galerkin-based solution for different boundary edges. Comparison study is also performed to verify the present formulation with those of previous papers. Results are presented to investigate the influences of the nonlocal and length scale parameters, various material compositions, elastic foundation parameters, type of foundation and various boundary conditions on the vibration frequencies of AFG nanobeams in detail.

Elastic foundation effects on arch dams

Earthquake response of an arch dam should be calculated under ground motion effects. This study presents three-dimensional linear earthquake response of an arch dam. Thereby, we considered different ground motion effects and also foundation conditions in the finite element analyses. For this purpose, the Type 3 double curvature arch dam was selected for application. All numerical analyses are carried out by SAP2000 program for empty reservoir cases. In the scope of this study, linear modal time-history analyses are performed using three dimensional finite element model of the arch dam and arch dam-foundation interaction systems. According to numerical analyses, maximum horizontal displacements and maximum normal stresses are presented by dam height in the largest section. These results are evaluated for rigid and various elastic foundation conditions. Furthermore, near-fault and far-field ground motion effects on the selected arch dam are taken into account by different accelerograms obtained from the Loma Prieta earthquake at various distances.

An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model

ABSTRACTEmploying the variational differential quadrature (VDQ) method, the effects of initial thermal loading on the vibrational behavior of embedded single-walled carbon nanotubes (SWCNTs) based on the nonlocal shell model are studied. According to the first-order shear deformation theory and considering Eringen's nonlocal elasticity theory, the energy functionality of the system is presented and discretized using the VDQ method. The effects of thermal loading and elastic foundation are simultaneously taken into account. The use of the numerical discretization technique in the context of variational formulation reduces the order of differentiation in the governing equations and consequently improves the convergence rate. The accuracy of the present model is first checked by comparison with molecular dynamics simulation results and those of other methods. The effects of involved parameters are then investigated on the fundamental frequencies of thermally preloaded embedded SWCNTs. The results imply that the thermal loading has a significant effect on the vibration analysis of embedded SWCNTs.

Nonlinear stability of functionally graded material (FGM) sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell’s shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material (FGM) sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.

Postbuckling of piezolaminated cylindrical shells with eccentrically/concentrically stiffeners surrounded by nonlinear elastic foundations

A fully analytical approach on thermal and mechanical buckling and postbuckling of cylindrical shell surrounded on nonlinear elastic foundation is presented. The shell is composed of composite material, piezoelectric actuator, and eccentrically/concentrically isotropic stringers and rings. The equilibrium and compatibility equations of shell are derived based on Kirchhoff assumptions taking into account von Karman nonlinearity. Two types of simply-supported boundary conditions are considered as freely movable and immovable edges. The equations are solved by definitions of stress function and applying Galerkin method. Numerical examples are well verified with available data in the literature. Several parametric investigations are conducted to examine the effects of voltage, different stiffeners, lay-up configuration, and nonlinear elastic foundations on equilibrium paths.

Postbuckling and deflection response of imperfect piezo-composite plates resting on elastic foundations under in-plane and lateral compression and electro-thermal loading

ABSTRACTThis article presents closed-form solution for the buckling, postbuckling, and nonlinear deflection of laminated composite plate with two piezoelectric actuators on elastic foundation and under transverse pressure, in-plane compression, thermal, and electrical loads for the first time in the literature. Material properties are assumed to be temperature-dependent. Two cases of boundary conditions and initial imperfection of the plate are considered. The formulations are based on the classical laminated plate theory (CLPT). The Galerkin method is employed to obtain load–deflection relations. The effects of elastic foundation, lateral pressure, in-plane compression, temperature dependency of material properties, electrical and thermal loading, imperfection, and aspect ratio are studied.