Partial Elastic Foundation Research Articles

A study on static bending behavior of partially elastically supported functionally graded plate with porous voids and prediction of deformation through deep learning

This research explores the static bending behaviour of functionally graded rectangular plates with porous voids. It addresses maximum deformation and static bending factors under uniform-pressure, considering variables such as porous-void distributions, full and partial elastic foundations, and edge constraints. The Rayleigh-Ritz method combined with algebraic polynomials is employed to obtain the numerical solutions. The convergence test shows computing efficiency, while the validation tests against public data and ANSYS findings verify the accuracy of the present numerical model. Additionally, this research presents a deep learning-based Artificial-Neural-Network model for deformation prediction to enhance the depth of the analysis without extensive numerical simulations.

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

Vibration response of FG-CNTRC plate resting on non-homogeneous elastic foundation via a closed-form dynamic stiffness formulation

In this work, a closed-form dynamic stiffness matrix has been formulated to study the influence of various configurations of non-homogeneous elastic foundation on the free vibration characteristics of carbon nanotube-reinforced functionally graded (FG-CNTRC) plate. The plate kinematic variables are defined based on the first order shear deformation theory. Through the application of Hamilton’s principle, frequency dependent dynamic stiffness (DS) matrix is formulated for a Levy type plate element supported on the Winkler-Pasternak foundation. Appropriate assembly of these DS matrices leads to the formation of global dynamic stiffness matrix, for the non-homogeneously supported plate, which is finally solved using the well-established Wittrick–Williams algorithm to compute the mode shapes and natural frequencies to a desired accuracy. Thus, the present work exploits the applicability of the dynamic stiffness method (DSM) for the free vibration study of the FG-CNTRC plate supported by partial and non-homogeneous elastic foundations. A comparative study is also carried out to ascertain the exactness and the computational efficiency of the dynamic stiffness method. Results from the parametric studies show that the non-homogeneous supports greatly influence the frequency as well as the modeshape of the plate. It is observed that the Pasternak parameter has a profound effect on the response of the plates as compared to that of the Winkler parameter. Effect of different CNT distributions on the frequency behavior are also discussed in detail. It can be emphasized that the new set of DSM computed results reported in the paper for diverse plate configurations may serve as reference results for the future research studies.

Free Vibration and Buckling Characteristics of Uniform Beam: A Modified Segmented Rod Method

This paper proposes a modified segmented rod method (MSRM) in order to facilitate the introduction of boundary conditions in the analysis of free vibrations and buckling of Euler–Bernoulli beams. First, the basic idea of proposed modifications is presented in the case of elementary length beam, and then the idea is generalized to the full-length beam. In addition to facilitating the introduction of boundary conditions, by comparing the results of MSRM with segmented rod method (SRM) and Hencky bar-chain method (HBM), a faster convergence of MSRM results to the exact ones was observed. Although the case of uniform beam is considered in this paper, proposed MSRM may contribute to greater use of segmented beam approach in solving of various static, free vibration or buckling problems of non-uniform beams. Also, it was shown that MSRM can be easily adapted for the analysis of beams that resting on partial elastic foundation.

Buckling and Free Vibration Analysis of Fiber Metal-laminated Plates Resting on Partial Elastic Foundation

This research presents, buckling and free vibration analysis of fiber metal-laminated (FML) plates on a total and partial elastic foundation using the generalized differential quadrature method (GDQM). The partial foundation consists of multi-section Winkler and Pasternak type elastic foundation. Taking into consideration the first-order shear deformation theory (FSDT), FML plate is modeled and its equations of motion and boundary conditions are derived using Hamilton's principle. The formulations include Heaviside function effects due to the nonhomogeneous foundation. The novelty of this study is considering the effects of partial foundation and in-plane loading, in addition to considering the various boundary conditions of FML plate. A computer program is written using the present formulation for calculating the natural frequencies and buckling loadings of composite plates without contacting with elastic foundation and composite plates resting on partial foundations. The validation is done by comparison of continuous element model with available results in the literature. The results show that the constant of total or partial spring, elastic foundation parameter, thickness ratio, frequency mode number and boundary conditions play an important role on the critical buckling load and natural frequency of the FML plate resting on partial foundation under in-plane force.

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.

Separation of traveling and standing waves in a finite dispersive string with partial or continuous viscoelastic foundation

The free and forced vibrations of a linear string with a local spring-damper on a partial elastic foundation, as well as a linear string on a viscoelastic foundation conceptualized as a continuous distribution of springs and dampers, are studied in this paper. Exact, analytical results are obtained for the free and forced response to a harmonic excitation applied at one end of the string. Relations between mode complexity and energy confinement with the dispersion in the string system are examined for the steady-state forced vibration, and numerical methods are applied to simulate the transient evolution of energy propagation. Eigenvalue loci veering and normal mode localization are observed for weakly coupled subsystems, when the foundation stiffness is sufficiently large, for both the spatially symmetric and asymmetric systems. The forced vibration results show that nonproportional damping-induced mode complexity, for which there are co-existing regions of purely traveling waves and standing waves, is attainable for the dispersive string system. However, this wave transition phenomenon depends strongly on the location of the attached discrete spring-damper relative to the foundation and whether the excitation frequency Ω is above or below the cutoff frequency ωc. When Ω<ωc, the wave transition cannot be attained for a string on an elastic foundation, but is possible if the string is on a viscoelastic foundation. Although this study is primarily formulated for a harmonic boundary excitation at one end of the string, generalization of the mode complexity can be deduced for the steady-state forced response of the string-foundation system to synchronous end excitations and is confirmed numerically. This work represents a novel study to understand the wave transitions in a dispersive structural system and lays the groundwork for potentially effective passive vibration control strategies.

Asymmetric thermal buckling of annular plates on a partial elastic foundation

ABSTRACTIn this investigation, the asymmetrical buckling behavior of isotropic homogeneous annular plates resting on a partial Winkler-type elastic foundation under uniform temperature elevation is investigated. First-order shear deformation plate theory is used to obtain the governing equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained under the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearized 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 generalized differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Benchmark results are given in tabular and graphical presentations for combinations of simply supported and clamped types of boundary conditions. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and the boundary conditions.

Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation

This paper is concerned with the development of the Hencky bar-chain model (HBM) for the buckling and vibration analyses of non-uniform beams resting on partial variable elastic foundation. The HBM allows analysts to obtain beam buckling and vibration solutions by solving a set of algebraic equations instead of a differential equation. To get the HBM, we resort to the first order central finite difference model (FDM) because both discrete models have phenomenological similarities. Based on bending moment, shear force and deflection equivalence between the two discrete beam models, the expressions for the internal spring stiffness, the end spring stiffness and the elastic foundation stiffness for the HBM are obtained. Some example problems are considered to demonstrate the simplicity of the HBM for buckling and vibration analyses of non-uniform beams on partial variable elastic foundation by taking the number of segments of the HBM to infinity. The effects of the foundation length and stiffness on the buckling load and natural frequencies are also discussed. Naturally, the HBM can be used to obtain solutions for articulated beams or beam-like structures with repetitive cells as well.

Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells

The free vibration characteristics of fluid-filled functionally graded cylindrical shells buried partially in elastic foundations are investigated by an analytical method. The elastic foundation of partial axial and angular dimensions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear deformation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The governing equation is obtained using the Rayleigh–Ritz method and a variation approach. The fluid is described by the classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method. The free vibration of fluid-filled functionally graded cylindrical shells buried partially in elastic foundations is investigated by an analytical method. The elastic foundation of partial axial and angular dimensions is represented by the Pasternak model. Shell motion is represented by first-order shear deformation theory. The governing equation is obtained using the Rayleigh–Ritz method. The fluid is described by classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method.

Modal response of short pipeline spans on partial elastic foundations

Very short pipeline free spans in flooded areas, river crossings, straits and inlets have been given a lot of attention due to recent failures caused by vortex-induced vibrations (VIV). Performing VIV analyses of free spanning pipelines requires accurate prediction of modal response quantities. Currently, no simple analytical solutions are available for short free spans supported by soils with a finite stiffness. Based on the Buckingham Pi theorem, the solution space for short free spans is spanned in the present study. Closed form analytical solutions are determined based on surface fitting to the developed solution space. The novel analytical solutions are accurate, easily implementable and thoroughly verified, and represent a significant improvement to current approaches found in design standards. An accurate expression for the buckling load of short spans is also derived, and asymptotic properties of the buckling load and frequency equations as the span length approaches zero are explored.

Three dimensional biaxial buckling analysis of functionally graded annular sector plate fully or partially supported on Winkler elastic foundation

In this paper, buckling analysis of functionally graded annular sector plates subjected to uniform in-plane compressive loads based on three dimensional theory of elasticity is investigated. Moreover, influence of full or partial Winkler-type elastic foundations is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. The material properties vary continuously through the thickness of plate according to a power law distribution while Poisson's ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on finite element orthogonal integral equations. Buckling loads are obtained based on a generalized geometric stiffness concept. In this regard, effects of both pre- and post-buckling modes are considered. The effects of material gradient exponent, different sector angles, aspect ratio, thickness ratio, loading condition, different shapes of partial elastic foundation and foundation stiffness on the buckling loads and mode shapes of movable simply supported FGM annular sector plates have been investigated to draw practical conclusions.

경사진 부분 탄성 지지부를 갖는 원통셸의 자유진동

The free vibration characteristics of cylindrical shells on inclined partial elastic foundations are investigated by an analytical method. The cylindrical shell is partially surrounded by the elastic foundations, these are represented by the Winkler or Pasternak model. The area of elastic foundation is not uniform and varies along the axial direction of the shell. The motion of shell is represented by first-order shear deformation theory(FSDT) to account for rotary inertia and transverse shear strains. The governing equation is obtained using the Rayleigh-Ritz method and a variation approach. To validate the present method, the numerical example is presented and compared with the present FEA results. The numerical results reveal that the elastic foundation has significant effect on vibration characteristics.

Jumping instabilities in the post-buckling of a beam on a partial nonlinear foundation

Mode jumping is an instability phenomenon in the post-buckling region, which causes a sudden change in the equilibrium configuration and is thus harmful to structure. The configuration of a partial elastic foundation can directly induce mode coupling from the buckling stage and through the whole post-buckling region. The mode coupling effect due to the configuration of partial foundation on mode jumping is investigated and demonstrated to be an important factor of determining mode jumping. By properly choosing the partial elastic foundation configuration, mode jumping can be avoided.

Free vibrations of free spanning offshore pipelines

Fast and accurate methods for determining pipeline eigenfrequencies and associated bending stresses are essential to free span design, and hence of great interest to the pipeline industry. The Rayleigh–Ritz approach has been applied in combination with a displacement field taken as a Fourier sine series to obtain a novel, semi-analytical solution to an initially curved Euler–Bernoulli beam on a partial elastic foundation subject to axial force. The proposed methodology for calculation of free span eigenfrequencies determines static and harmonic response with excellent accuracy, and includes non-linear geometric effects and the pipe stiffening effect caused by static deformation due to gravity.

Instability of heated circular FGM plates on a partial Winkler-type foundation

Thermal buckling analysis of a transversely graded circular plate attached to a centric partial elastic foundation is studied, analytically. Thermomechanical properties of the circular plate are distributed across the thickness based on a power law function. The governing equations of the plate are obtained by means of the classical plate theory. A conventional Winkler-type foundation is assumed to be in contact with the plate which acts in compression as well as in tension. Proper boundary conditions are chosen after pre-buckling analysis of the plate, and stability equations are established via the adjacent equilibrium criterion. To analyze the thermal stability problem, the plate is divided into two sections, a foundation-less domain and an in-contact region. An exact procedure is presented to accurately predict the critical buckling temperature as well as the buckled configuration of the plate. Analysis of various involved parameters including the Winkler parameter, foundation radius, power law index, and loading type is presented. It is concluded that while the loading is symmetric, in many cases, the buckled configuration of the plate is asymmetric.

부분 탄성지지된 유체 저장 원통셸의 자유진동

The free vibration characteristics of fluid-filled cylindrical shells on partial elastic foundations are investigated by an analytical method. The cylindrical shell is fully or partially surrounded by the elastic foundations, these are represented by the Winkler or Pasternak model. The motion of shell is represented by the first order shear deformation theory to account for rotary inertia and transverse shear strains. The steady flow of fluid is described by the classical potential flow theory. The fluid-structure interaction is considered in the analysis. The effect of internal fluid can be considered by imposing a relation between the fluid pressure and the radial displacement of the structure at the interface. To validate the present method, the numerical example is presented and compared with the available existing results.

On the free vibration response of rectangular plates, partially supported on elastic foundation

Rectangular plates on distributed elastic foundations are widely employed in footings and raft foundations of variety of structures. In particular, mounted columns and single footings may partially occupy the rectangular plate of any kind.This study deals with free vibration problem of thin rectangular plates on Winkler and Pasternak elastic foundation model which is distributed over a particular arbitrary area of the plate. Closed form solutions are developed through solving the governing differential equations of plates. Moreover, a novel mathematical approach is proposed to find the exact analytical solution of free vibration of plates with mixed or fully-clamped boundary conditions. Results are discussed in detail thorough parametric and verification studies. It was concluded that proposed mathematical method can successfully obtain the exact solution to free vibration problems of plates on partial elastic foundations with clamped and/or simply-supported boundary conditions.

Exact solution to free vibration of beams partially supported by an elastic foundation

This study pertains to the free vibration problem of beams on an elastic foundation of the Winkler type, which is distributed over a particular length of the beam. Closed form solutions are developed by solving the governing differential equations of beams. Moreover, an innovative mathematical approach is proposed to find the precise analytical solution of the free vibration of beams with mixed boundary conditions. Results are discussed in detail through verification studies. Ultimately, it was concluded that the proposed mathematical method could successfully obtain the exact solution to the free vibration problem of beams on partial elastic foundations under mixed boundary conditions.

Elastic stability of columns on partial elastic foundations under subtangential loading

Divergence and flutter instability of a cantilevered beam on a partial elastic foundation subjected to a subtangential load at the free end are investigated. The dependence of divergence and flutter critical loads on the tangency coefficient is studied parametrically to illustrate the effects of the extent, position, and modulus of the partial foundation. Particular attention is paid to the stabilizing and destabilizing effects of the various parameters.