Winkler-type Elastic Foundation Research Articles

Buckling analysis of stiffened functionally graded multilayer graphene platelet reinforced composite plate with circular cutout embedded on elastic support subjected to in-plane normal and shear loads

Buckling analyses of orthogonally stiffened functionally graded graphene platelet reinforced composite plate with circular cutout supporting on Winkler elastic foundation is conducted according to the third order shear deformation plate theory and the finite element method to achieve the desired solutions. Material properties of the composite plate are evaluated using Rule of mixture and Halpin-Tsai approach. Graphene platelet reinforcements and Poly methyl methacrylate matrix are considered as the two components of the composite media. The material of the stiffeners is selected the same as that of the polymer matrix. In one efficient case, adding a set of stiffeners increased the critical buckling load by 41.8 % with only a 10.3 % increase in weight of the structure. Influence of broad range of factors such as GPLs nanofillers distribution pattern and weight fraction, Winkler-type elastic foundation, uniaxial and biaxial normal and shear loads, plate thickness and aspect ratio, width, height and number of longitudinal and transverse stiffeners are reported in several tabular and graphical data in detail.

Buckling response of functionally graded multilayer graphene platelet-reinforced composite plates with circular/elliptical cutouts supporting on an elastic foundation under normal and shear loads

The present article deals with the buckling response of functionally graded multilayer graphene platelet-reinforced composite (FG-GPL RC) rectangular plates with circular/elliptical cutouts resting on a Winkler-type elastic foundation under uniaxial and biaxial normal and shear loads. Rule of mixtures and the Halpin–Tsai approach are applied to obtain the effective Poisson’s ratio, mass density, and elastic modulus of the reinforced composite. The governing equations are developed by applying the third-order shear deformation plate theory. Then, the finite element procedure is used to solve the problem. Four different types of graphene platelet distributions, namely, UD, FG-X, FG-V, and FG-O, are considered. A broad range of factors such as plate aspect ratio, plate slenderness ratio, applying uniaxial and biaxial normal and shear loads to the plate, several Winkler elastic foundation stiffness parameters, different displacement boundary conditions, the effect of size of the circular cutout and orientation of the elliptical cutout, and the influence of GPL weight fraction are discussed in several tabular and graphical data in detail.

Investigating the Elastic Response of Smart Cylinders Under Asymmetric Loading

This paper investigates the hygrothermal-magneto-elastic response of functionally graded piezomagnetic (FGPM) cylinders under asymmetric loading. The cylinders are supported by a Winkler-type elastic foundation, and their properties vary with the radius according to a power-law function. By solving 2D equations of Fickian diffusion and Fourier relations, the distribution of asymmetric moisture concentration and temperature field is determined. Incorporating constitutive equations into mechanical and magnetic equilibrium equations yields three second-order partial differential equations. The equations are solved using the separation of variables and complex Fourier series. Simulation results demonstrate the influence of hygrothermal loading, magnetic field, elastic foundation, and material inhomogeneity on the cylinder's response.

Estimation of plate parameters from vertical displacement data using a family of plate models

We develop a method for estimation of parameters of an elastic plate resting on a Winkler-type elastic foundation solely from data on the vertical displacements of the plate. The method allows one to estimate components of the external body force density field, plate thickness, elastic foundation stiffness parameters, horizontal displacements of the plate, and stresses. The key idea of the method is that multiple plate models are used simultaneously, namely the proposed reduced three-dimensional (R3D) plate model, the Mindlin plate model, and the thin plate model. The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. The hierarchical relationship among the plate models allows one to incorporate prior information into the estimation technique. The applicability of the proposed estimation method is illustrated by a numerical example. For more information see https://ejde.math.txstate.edu/Volumes/2023/15/abstr.html

Dynamic bending of an ice wedge resting on a winkler-type elastic foundation

For most ice – sloping structure interactions, the incoming ice floes' failures are determined by the formation of circumferential cracks. This physical process can be simplified as analyzing the bending failure of an ice wedges resting on a fluid foundation. In history, closed-form analytical/empirical solutions have been developed for the static bending problem; and numerical solutions have been attempted for the dynamic scenarios. This paper revisits this classic problem and conducts extensive Finite Element Method (FEM) – based simulations on the dynamic bending of an ice wedge resting on a Winkler-type elastic foundation. The simulations are based on inputs (i.e., ice wedge geometry, loading radius and loading rate) within ranges that are typical for engineering applications. Based on the simulation, a database of ‘ice breaking load’ and ‘ice breaking length’ is constructed. Then we applied the Artificial Neural Network (ANN) method to establish the general relationship between the varying inputs (i.e., ice wedge angle, loading radius and rate) with the target outputs (i.e., breaking load and length). Such relationship is expressed in simple closed-form (i.e., Eq. (13)) allowing for easy, efficient and wide engineering applications. In the process of developing the ANN model, based on extensive FEM-simulations, we managed to extend Nevel's (1972) analytical solution. We also quantitatively demonstrated many well-known dynamic effects in this classic problem, e.g., a faster loading rate leads to a larger ice breaking load and a shorter ice breaking length. In addition, we also uncovered the failure pattern transition of an ice wedge, i.e., when an ice wedge's angle is below 100°, the circumferential crack will develop first; however, when the ice wedge gets wider than around 100°, depending on the loading radius and interaction velocity, the radial crack is more prone to develop first.

On the dynamic stability of delaminated composite beams under free vibration

This work deals with the analysis of the free vibration problem of elastic delaminated composite beams. The work mainly consists of a model development and improvement stage based on the first-order shear deformable beam theory. A general model is developed taking the bending-extensional coupling into account. The specified problem is a built-in beam with free end, and one of the novelties of this work is the consideration of the fact that a built-in beam cannot be fixed rigidly in reality. Thus, a Winkler-type elastic foundation is applied along the built-in length. The total potential energy and the governing equations of the delaminated and intact parts of the beam are also captured. The problem is solved in two ways: analytically and numerically by using the finite element method, respectively. Applying the developed models the natural frequencies, mode shapes as well as the stress resultants are determined. The comparison of natural frequencies to those measured experimentally shows that the built-in length resting on Winkler-type elastic foundation influences significantly the agreement between model and experiment. In the final stage, the parametric excitation phenomenon taking place in the delaminated part is analyzed using a local model and the harmonic balance method. The dynamic buckling is characterized by some stability diagrams, and it is shown that the applied model is very sensitive to the frequency leading to somewhat controversial critical amplitudes compared to measurements.

Effects of time delay on the dynamical behavior of nonlinear beam on elastic foundation under periodic loadings: Chaotic detection and it control

In this paper, we analyze the effects of the time-delay feedback position on the dynamical behavior of the nonlinear beam on elastic foundation under periodic external force and the chaos control. Firstly, by using the formulation of Lagrange as well as geometric analysis, the nonlinear equation of the system with time-delay is established. In this case, the system rests on a Winkler-type elastic foundation soil that acts on its bottom interface (soil-beam). Secondly, the equilibrium points are found, their stability studied and the condition of Hopf bifurcation established. Thirdly, by using the Melnikov process, the analytical constraints necessary to have chaos or not as a behavior of the system are sought, which is confirmed by numerical investigations, with plotting of the time series, Lyapunov exponent and the bifurcation diagram, which are used to study the effects of time-delay on dynamical behavior of the system. This time-delay being used as a control parameter. The results show that when the value of time delay is small, the ergodic tori and resonance cycles with different rotation numbers on the torus can be observed. When the value of time-delay increases the chaos behaviors take place in the system. Taking into account the fact that chaotic situations are considered disturbance and are harmful for operation, the control method based on the time-delay parameter is used to quench and also to prevent chaotic behaviors.

PHYSICS-INFORMED NEURAL NETWORKS FOR ELASTIC PLATE PROBLEMS WITH BENDING AND WINKLER-TYPE CONTACT EFFECTS

Kirchhoff plate bending and Winkler-type contact problems with different boundary conditions are solved with the use of physics-informed neural networks (PINN). The PINN is built on the base of mechanics laws and deep learning. The idea of the technique includes fitting the governing partial differential equations at collocation points and then training the neural network with the use of optimization techniques. Training of the neural network is performed by numerical optimization using Adam’s method and the L-BFGS (Limited- Broyden–Fletcher–Goldfarb–Shanno) algorithm. The error loss function and the computational error of the approximate solution (output of the neural network) of the bending problem and contact problem with Winkler type elastic foundation are shown on examples. The predictions of the NN are investigated for different values of the foundation’s constants. The effectiveness of the proposed framework is demonstrated through numerical experiments with different numbers of epochs, hidden layers, neurons and numbers of collocation points. The Tensorflow deep learning and scientific computing package of Python is used through a Jupyter Notebook.

Study of non-homogeneity effects on natural frequencies and mode shapes of polar orthotropic annular plate of exponentially varying thickness rest on elastic foundation

To analyze natural frequencies of vibrations of an orthotropic annular plate of exponentially varying thickness resting on Winkler type elastic foundation, a mathematical model is constructed using Hamilton’s energy principle. Numerical simulation is carried out using quintic spline method for three different combinations of edge conditions. Effect of non-homogeneity variation, together with elastic foundation and orthotropy, on the natural frequencies of vibration is illustrated for variety of thickness variation for the first three modes. To compare parametric effect on a specific plate, transverse displacements are presented in normalized form. Accuracy of the results and validity of numerical method are demonstrated by comparing the existing results in the literature.

Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al2O3) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e0a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

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.

Modeling and Simulation of Vibrations of Non-Homogeneous Annular Plate of Quadratic Thickness Resting on Elastic Foundation

Mathematical modeling is presented to analyze natural frequencies of vibrations of an isotropic annular plate of quadratic varying thickness resting on Winkler type elastic foundation where numerical simulation is carried out using quintic spline technique for three different combinations of edge conditions. Effect of elastic foundation, together with nonhomogeneity variation, on the natural frequencies of vibration is illustrated for variety of thickness variation for the first three modes. To compare parametric effect on a specific plate, transverse displacements are presented in normalized form. Accuracy of the results and validity of numerical method is demonstrated by comparing the existing results in the literature.

Two-way Hertzian spring in numerical analysis of coupled train-track system vibrations

The paper presents and compares two methods for simulation the train-track system vibrations, differing in the wheel-rail contact model used. In the first method, two-way Hertzian spring is used, in the second – a non-deformable constraint. In both methods, a flat computational model is assumed, consisting of an Euler-Bernoulli beam resting on a Winkler-type elastic foundation with damping and a set of rail vehicles modeled by dynamic systems with ten degrees of freedom. The results of numerical analysis are presented, in order to determine an influence of the contact constraints’ deformability on the vibration simulations. It is found that the replacement of non-deformable contact constraints by two-way Hertzian springs has no significant effect on track vibration simulations and has a little effect on vibration simulations of vehicle body and bogie. The developed simulation method can be used for numerical studies of the phenomenon of instantaneous detachments of wheels from rails, after minor modifications directed to introduce one-way Hertzian springs (i.e. not carrying tensile forces) being a more accurate contact model.

FROM WINKLER’S FOUNDATION TO POPOV’S FOUNDATION

In recent years, the method of dimensionality reduction (MDR) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. The MDR modeling framework introduces an equivalent punch profile and a one-dimensional Winkler-type elastic foundation, called henceforth Popov’s foundation. While the former mainly accounts for the geometry of contact configuration, the Popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). The discussion is illustrated with an example of the Kendall-type adhesive contact for an isotropic elastic half-space.

Methods of calculating the belt foundation on the elastic ground base of winkler type

Strip foundations on an elastic soil foundation and on a Winkler-type elastic foundation were used in the construction of buildings and structures from the most common types of shallow foundations. The article presents the results of studies related to the analysis of the features of the formation of structural solutions of strip foundations. It was found that various aspects associated with the development of multifactor mathematical models that determine the design parameters and modern regulatory requirements, provide conditions for not exceeding the limiting conditions of structures and soil base. It improves the functional quality indicators of strip foundations. Based on the results of the study, the need to study the possibilities of using strip foundations (beams on an elastic base) was identified, and the main provisions of the model of a Winkler-type soil base (with one, two and three bed coefficients) were considered. An algorithm is developed for determining the initial parameters and numerical solution of boundary value problems. The prospects of a shallow foundation model with a variable cross section or a beam of finite length with variable stiffness, which is accepted as one of the options for modifying a typical structural solution of a strip foundation, are shown.

Measurement of Vibrations of a Plate on Elastic Foundation

Vibrations of a plate on elastic foundation of Winkler type are investigated. A two-dimensional element having six nodal degrees of freedom (three displacements of the lower surface and three displacements of the upper surface) is used. Elastic foundation of Winkler type on the lower surface is used in the numerical model. Eigenmodes are calculated. Experimental investigations of vibrations were performed using a special experimental setup and typical experimental results are presented. Using the presented results places for recommended locations of Braille elements are determined. It is recommended to locate them at the places were the amplitude of vibrations is smallest. The stiffness parameters of the elastic foundation must be chosen in such a way that enables to shift the eigenfrequencies of the elastic structure from the excitation frequencies of the investigated mechanical device. DOI: http://dx.doi.org/10.5755/j01.mech.24.4.20742

Analytical analysis for the forced vibration of CNT surrounding elastic medium including thermal effect using nonlocal Euler-Bernoulli theory

This article studies the free and forced vibrations of the carbon nanotubes CNTs embedded in an elastic medium including thermal and dynamic load effects based on nonlocal Euler- Bernoulli beam. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The non-local Euler-Bernoulli beam model predicts lower resonance frequencies. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the temperature change on the non-dimensional natural frequency.

Vibration Analysis of Nano Beam Using Differential Transform Method Including Thermal Effect

In this paper, based on nonlocal Euler- Bernoulli beam model with arbitrary boundary conditions a simulation method called the Differential Transform Method (DTM) is employed to predict and to analysis the vibration of a single-walled carbon nanotube embedded in an elastic medium under thermal effect. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Ferstly, the Differential Transform Method is introduced. The research work reveals the significance of the small-scale coefficient, the vibrational mode number and the elastic medium on the non-dimensional natural frequency.

On Post-Critical Behavior of a Beam on an Elastic Foundation

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.

Dynamical response of an embedded nanobeam by using nonlocal integral stress models

Abstract An effective tool for investigating the mechanical response of nanostructures is considered the nonlocal continuum theory, capable of explaining the size effect phenomena. The key point of the nonlocal theory is the integral constitutive equation. A transformation of the integral form into the differential one was suggested by Eringen. Applying the nonlocal differential form to structural models, it gives rise to paradoxes and inconsistencies. Previous studies imply this transformation is not a vice-versa process in a finite domain. Recent research suggests that the nonlocal integral constitutive equation, used for the structural models' development, does not give rise to paradoxes and inconsistencies. This work focuses on employing the integral constitutive equation to explore the dynamical response of a nanobeam, embedded in an elastic medium and simulated as a Winkler type elastic foundation, for the first time. In our research endeavor, two models are used, i.e., the two phase nonlocal integral (TPNI) stress model and the modified kernel's model. In particular, the modified kernel, normalized in a finite domain, is employed to dynamical problems for the first time. What is more, both analytical and numerical methods are applied. Based on the results deduced, the dynamical response of an embedded nanobeam, through the use of integral models, presents a softening behavior compared to that of the classic model for all engineering benchmark problems that investigated. Our overall conclusions are encouraging of triggering the study of embedded type problems in applications for nanomaterials, nanocomposites and biomaterials.