Non-uniform Elastic Foundation Research Articles

Influence of non-uniform elastic foundations on free vibration behavior of nanocomposite plates interacting with a fluid environment based on a novel shear deformation theory

Influence of non-uniform elastic foundations on free vibration behavior of nanocomposite plates interacting with a fluid environment based on a novel shear deformation theory

Finite Element Analysis for the Free Vibration of a Rigid Pavement resting on a Non-uniform Elastic Foundation

This paper presents a finite element analysis of the free vibration behavior of rigid pavements resting on non-uniform foundations. The rigid pavement was modeled using the Mindlin plate theory, while the supporting soil medium was approximated by a Winkler model with non-uniform stiffness. A finite element formulation was established to govern the equation of free vibration for rigid pavements. Subsequently, a computer program was developed based on the proposed algorithm, enabling the determination of natural frequencies and mode shapes. The accuracy of the proposed method was verified by comparing numerical examples of free vibration with analytical results. These numerical examples also demonstrate the significant influence of the foundation stiffness on natural frequencies and mode shapes.

Analysis of Concrete Pavement Slab resting on Non-uniform Elastic Foundation using the Finite Element Method

During the rigid pavement design, the concrete slab working on the base/subbase and subgrade is usually modeled as the slab on the elastic foundation. However, the non-uniform distribution of materials in the base or subgrade layers naturally exists in real conditions and should be considered. In this paper, a finite element method for calculating concrete pavement slabs on an elastic foundation with non-uniform stiffness distribution is developed. This study applies 4-node finite elements and Mindlin plate theory to formulate the finite element equations. The results predicted by the proposed approach are verified analytically. Calculation examples are conducted with the practical settings to investigate the influence of slab and foundation stiffness parameters on the slab displacement.

Dynamic stability of slightly curved tensioned pipe conveying pressurized hot two phase fluid resting on non uniform foundation

The dynamic stability of slightly curved tensioned pipes conveying pressurized two phase fluid, under thermal loadings are investigated in this paper. Furthermore, these pipes are resting on nonuniform elastic foundations. The equation of motion of the system is derived using the extended Hamilton’s principle for open dynamic systems and considering the Euler–Bernoulli theory for slender beams. The free vibration and the effects of two phase fluid flow, geometric imperfections, thermal, tension and pressure on the pipe system resting on nonuniform foundations are investigated numerically. Three different boundary conditions (BC) are considered, namely simply-simply (SS) supported, simply-clamped (SC) supported and clamped-clamped (CC) supported. The numerical method of Finite Element was used on the partial differential equation. This method is used to accurately predict steady state natural frequency responses of a dynamical system through several numerical investigations. It was found that the stability of slightly curved pipes conveying two phase flow are significantly affected by the boundary condition. It is shown that the initial curvature term increases the critical velocities for all the nine modes considered, the increase is significant in the first and second critical velocities compared to the other modes. Initial curvature effect is more pronounced in simple supported ends. Further-more the influence of the fluid velocity, temperature, pressure and axial tension is analysed; the results show that the vibration frequency decreases when the fluid velocity, temperature, and pressure increase or the axial tension decreases, and the critical fluid velocity decreases when the temperature and pressure increase or the axial tension decreases. For the pipe on nonuniform foundation, an increased stiffness leads to an increased in the first critical velocity for all the BC. However, the pattern of the increment depends on the attachment parameter, initial curvature, and the vapour quality. Some of these results are completely absent in pipes conveying single flows.

Static and dynamic thermal post-buckling analysis of imperfect sigmoid FG cylindrical shells resting on a non-uniform elastic foundation

In this study, an approach combining semi-analytical and analytical methods is utilized to investigate the dynamic and static thermal post-buckling behavior of imperfect sigmoid functionally graded (SFG) cylindrical shells in a thermal environment. Two types of SFG cylindrical shells consisting of ceramic-metal-ceramic (CMC) layers and metal-ceramic-metal (MCM) layers are considered. The structure is considered to rest on a non-uniform elastic foundation, and the parameters of elastic foundation used in the research are defined by using Selvadurai's methodology. Material distributions of the shells are graded through the thickness direction following a sigmoid distribution in terms of volume fraction. The nonlinear governing equation of the cylindrical shells is derived by applying the von-Kármán equation and the Donnell shell theory. Based on Galerkin's method, a discretized nonlinear governing equation is obtained for analyzing the behavior of the shells. To systematically study the dynamic thermal post-buckling response of the shells, the fourth-order P-T method is utilized. The research results obtained are verified. Moreover, to further validate the dynamic thermal post-buckling, comparisons are made with the fourth-order Runge-Kutta method. It is noted that the fourth-order P-T method has illustrated advantages over that of the fourth-order Runge-Kutta method in terms of accuracy and reliability. Furthermore, the fourth-order P-T method consumes less CPU time than that of the fourth-order Runge-Kutta method.

Bending analysis of P-FGM plates resting on nonuniform elastic foundations and subjected to thermo-mechanical loading

Bending analysis of P-FGM plates resting on nonuniform elastic foundations and subjected to thermo-mechanical loading

Effects of non-uniform elastic foundation on the nonlinear vibration of nanocomposite plates in thermal environment using Selvadurai methodology

This paper focuses on effects of non-uniform elastic foundation, that impact on non-linear thermal dynamics of the simply supported plate reinforced by functionally graded (FG) graphene nanoplatelets (GNPs). The Halpin-Tsai micromechanical model and the rule of mixtures are employed to calculate the material properties. By adjusting the volume fraction of matrix/GNPs in the thickness direction, the various distribution patterns of the plates are considered as the uniform distribution (UD) and functionally graded (FG) reinforcements. The governing equations is expressed by using the classical theory, Von Karman-Donnell geometrical nonlinearity assumption, and combining the temperature change. Then the dynamical behaviors of the FG-GNPs reinforced plates are obtained by applying the Airy's stress function and the Galerkin's method. The obtained results are described by the column charts, the 2D and 3D graphs and written by codes of Wolfram-Mathematica. In addition, the present study scrutinizes the effects of the thermal environment, GNPs weight fraction, GNPs distribution patterns, damping coefficient and geometric parameters. More importantly, the influences of the longitudinal variable thickness of Pasternak’s subgrade model on the dynamical characteristics are analyzed to optimize structures. Altogether these findings have significant applications in industries engineering and lead to breakthroughs in the microstructures design.

Free Vibration Analysis of Bidirectional Functionally Graded Conical/Cylindrical Shells and Annular Plates on Nonlinear Elastic Foundations, Based on a Unified Differential Transform Analytical Formulation

In the present research, a unified formulation for free vibration analysis of the bidirectional functionally graded conical and cylindrical shells and annular plates on elastic foundations is developed. To cover more individual cases and optimally tailored material properties, the material properties are assumed to vary in both the meridian/radial and transverse directions. The shell/plate is assumed to be supported by a non-uniform Winkler-type elastic foundation in addition to the edge constraints. Therefore, the considered problem contains some complexities that have not been considered together in the available researches. The proposed unified formulation is derived based on the principle of minimum total potential energy and solved using a differential transform analytical method whose center is located at the outer edge of the shell or plate; so that the resulting semi-analytical solution can be employed not only for truncated conical shells and annular plates, but also for complete conical shells and circular plates. Accuracy of results of the proposed unified formulation is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS finite element analysis code. A variety of the edge condition combinations are considered in the results section. A comprehensive parametric study including assessment of influences of the material properties indices, thickness to radius ratio, stiffness distribution of the elastic foundation, and various boundary conditions, is accomplished. Results reveal that influence of the meridian variations of the material properties on the natural frequencies is more remarkable than that of the transverse gradation.

Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

For arbitrary two-point boundary condition, which makes the corresponding linear uniform problem well-posed, we obtain an existence and uniqueness result for the boundary value problem of finite beam deflection resting on arbitrary nonlinear non-uniform elastic foundation. The difference between the desired solution and the corresponding linear uniform one in L^{infty} sense is bounded explicitly in terms of given inputs of the problem. Our results seamlessly unify linear uniform and nonlinear non-uniform problems and lead to an iteration algorithm for uniformly approximating the desired deflection.

Flexural vibration and power flow analyses of axially loaded beams with general boundary and non-uniform elastic foundations

In this article, flexural vibration and its power flow of an axially loaded beam with arbitrary boundary and non-uniform elastic foundation are analyzed via energy principle in conjunction with Rayleigh–Ritz procedure. A general solution is assumed as a modified version of Fourier series supplemented with boundary-smoothing auxiliary terms. Effects of axial load and elastic foundation of the beam are taken into account in terms of potential energy in system Lagrangian. Critical load of this axially loaded beam is obtained under elastic boundary condition and non-uniform foundation. Numerical examples are then presented to demonstrate the reliability and effectiveness of the established model by comparing results with those available in literature or calculated using finite element method. Results show that the current model can make an accurate dynamic analysis for the elastically restrained beam with axial load and non-uniform elastic foundation. Influence of some important factors, including boundary restraint, axial load, and non-uniform foundation, on the vibration characteristics and power flow transmission are studied and addressed. It is found that the supporting condition has a significant influence on power flow distribution across the whole beam, in which the shear force component plays a dominant role for the power flow transmission.

Efficient Quadrature Solution for Composite Plates with Variable Thickness Resting on Non-Uniform and Nonlinear Elastic Foundation

Efficient Quadrature Solution for Composite Plates with Variable Thickness Resting on Non-Uniform and Nonlinear Elastic Foundation

Analysis of isotropic rectangular plate resting on non-uniform elastic foundation using Ritz approach

Abstract An analysis of bending of plate on elastic foundation dealing with the Ritz method is presented. The elastic foundation is modeled as Winkler foundation with linear variation stiffness. The solution is obtained by applying Ritz method for simply supported rectangular plate. The displacements, moments and stresses for plate predicted by analytical method are then compared to those of finite element method which provided as a verification.

Nonlinear analysis for extreme large bending deflection of a rectangular plate on non-uniform elastic foundations

An analysis is presented for different bearing loadings of a plate resting on various linear and nonlinear foundations subjected to different boundary conditions such as the circled clamped, simply supported, and mixed boundary conditions. The highly accurate solutions of the plate deflection and the stress under different work conditions are obtained by the novel wavelet-homotopy technique, which are in full agreement with previous ones in literature. Different from previous studies, our solutions are also valid for the extra large plate bending cases, which are rarely considered before. Particularly, we consider the important case that the connection coefficients of various foundations are variational, which seems to be overlooked in previous studies owing to their extreme difficulties in mathematical treatments and programming. Besides, to overcome the limitation of existing wavelet technique of poor capability on handling complex boundary conditions, we reconstruct the boundary wavelet by the Coiflets so that it can be used to handle the governing partial differential equations subjected to nonhomogeneous boundary conditions. Moreover, we introduce the homotopy iteration technique so that the computational efficiency improves to a large extent as compared with the traditional Homotopy Analysis Method (HAM) technique. It is expected the proposed wavelet-homotopy method can be as a new generation of analytical tool for solving strong nonlinear problems subjected to complicated boundary conditions, especially for those with variable coefficients.

Bending of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment using an accurate theory

This article presents the bending analysis of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment. Theoretical formulations are based on a recently developed refined shear deformation theory. The displacement field of the present theory is chosen based on nonlinear variations in the in-plane displacements through the thickness of the plate. The present theory satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional trigonometric shear deformation theory, the present refined shear deformation theory contains only four unknowns as against five in case of other shear deformation theories. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a simple power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as non-uniform foundation. The results of the shear deformation theories are compared together. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio and elastic foundation parameters on the thermo-mechanical behavior of functionally graded plates. Numerical results show that the present theory can archive accuracy comparable to the existing higher order shear deformation theories that contain more number of unknowns.

The Method of Matrix Differential Equation to Calculate the Natural Frequencies of Variable Cross-section Beams on the Non-uniform Elastic Foundations

The tradition methods to calculate the natural frequencies of beam on the elastic foundation were based on high order differential equation by defection. This paper derived the general equation based on the one order matrix differential equation represented by the mixed variable (a vector formed by deflection, angle of rotation, bending moment and shearing force) to calculate the natural frequencies of the variable cross-section beams on the non-uniform elastic foundations. To calculate the natural frequencies of this kind of beams, we divide the beams into several sections, calculate the transfer matrices of each section, multiply the matrices in proper order and obtain the overall transfer matrix of the mixed variable from one end to the other end of the variable cross-section beam on non-uniform elastic foundations. According to the conditions of boundary at both ends of the beam, we can calculate out the natural frequencies in MATLAB easily.

Buckling of Nonprismatic Column on Varying Elastic Foundation with Arbitrary Boundary Conditions

Buckling of nonprismatic single columns with arbitrary boundary conditions resting on a nonuniform elastic foundation may be considered as the most generalized treatment of the subject. The buckling differential equation for such columns is extremely difficult to solve analytically. Thus, the authors propose a numerical approach by discretizing the column into a finite number of segments. Each segment has constantsE(modulus of elasticity),I(moment of inertia), andβ(subgrade stiffness). Next, an exact analytical solution is derived for each prismatic segment resting on uniform elastic foundation. These segments are then assembled in a matrix from which the critical buckling load is obtained. The derived formulation accounts for different end boundary conditions. Validation is performed by benchmarking the present results against analytical solutions found in the literature, showing excellent agreement. After validation, more examples are solved to illustrate the power and flexibility of the proposed method. Overall, the proposed method provides reasonable results, and the examples solved demonstrate the versatility of the developed approach and some of its many possible applications.

Thermo-magneto-elasticity analysis of variable thickness annular FGM plates with asymmetric shear and normal loads and non-uniform elastic foundations

In the present research, a semi-analytical three-dimensional solution is developed for transversely graded annular/circular plates resting on non-uniform elastic foundations and subjected to magnetic and thermal fluxes and non-uniform and asymmetric mechanical loads. Thickness of the plate may vary exponentially along the radial direction and the inner and outer edges are restrained. All the thermal, magnetic, and elastic properties are assumed to vary in the transverse direction. Transverse distribution of the temperature is determined analytically for both the specified transverse temperature gradient and specified heat flux thermal conditions. Dependencies of the solution on the radial and transverse coordinates are determined through solving the resulting governing equations by using the differential quadrature and state space approaches in the radial and transverse directions, respectively. Effects of the material heterogeneity, thermal and mechanical boundary conditions, thickness variability, coefficients of the elastic foundation, shear to normal loads ratio, and magnetic field intensity are investigated comprehensively, through a parametric study. Results reveal that complex interactions may be observed among the elastic, thermal, and magnetic responses.

An analytical approach for bending and stress analysis of cross/angle-ply laminated composite plates under arbitrary non-uniform loads and elastic foundations

In the present study, a new analytical approach is developed for bending and stress analysis of angle-ply laminated composite and sandwich plates subjected to arbitrary non-uniform loads. Governing equations are derived based on the Mindlin–Reissner plate theory. For the first time, a system of three second order coupled partial differential equations is analytically solved by using power-series solution. The transverse shear stress is determined based on the three-dimensional theory of elasticity. Most of the available analytical solutions of the rectangular plates were presented for specific load and edge conditions. The proposed method can be applied for plates with arbitrary boundary conditions and various distributed loads. To demonstrate the generality and accuracy of the present method, comprehensive numerical examples are presented and compared with other available published results. The results show that the proposed solution can be applied for analysis of cross/angle-ply laminated composite and sandwich plates with various combinations of the edge conditions, orientation angle of the laminated composite plate, arbitrary non-uniform transversely distributed loads and non-uniform elastic foundation. The proposed method predicts excellent results for laminated composite plates. Also even for moderately thick sandwich plates, accuracy of the present results are in a good agreement with the exact solutions.

Small scale effect on the vibration of non-uniform nanoplates

Free vibration of non-uniform embedded nanoplates based on classical (Kirchhoff\'s) plate theory in conjunction with nonlocal elasticity theory has been studied. The nanoplate is assumed to be rested on two-parameter Winkler-Pasternak elastic foundation. Non-uniform material properties of nanoplates have been considered by taking linear as well as quadratic variations of Young\'s modulus and density along the space coordinates. Detailed analysis has been reported for all possible cases of such variations. Trial functions denoting transverse deflection of the plate are expressed in simple algebraic polynomial forms. Application of the present method converts the problem into generalised eigen value problem. The study aims to investigate the effects of non-uniform parameter, elastic foundation, nonlocal parameter, boundary condition, aspect ratio and length of nanoplates on the frequency parameters. Three- dimensional mode shapes for some of the boundary conditions have also been illustrated. One may note that present method is easier to handle any sets of boundary conditions at the edges.

Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations

In the present paper, a complicated and general problem of a functionally graded porous variable thickness circular plate subjected to non-axisymmetric and non-uniform shear and normal tractions and a magnetic actuation, and supported by a non-uniform Kerr elastic foundation is analyzed. Although such a combination of complexities may rarely be observed in the practical applications, the developed solution may identically be employed for many practical problems, as special cases. The governing equations are derived based on the three-dimensional theory of elasticity instead of using the approximate plate theories. The solution is proposed based on the differential quadrature and state space vector techniques in the radial and thickness directions, respectively. A comprehensive parametric study is presented to evaluate effects of the material, loading, boundary, and elastic foundation specifications on the resulting displacement, stress, Lorentz force, electromagnetic stress, and magnetic perturbation quantities.