Two-parameter Foundation Research Articles

Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment

The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic Pasternak foundation is investigated. The motion equations are derived applying the Hamilton principle. We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction factor is required. A 16-noded Mindlin plate element of the Lagrange family which is free from shear locking due to small thickness of the plate used. The material properties are assumed to be temperature-dependent and expressed as a nonlinear function of temperature. Because the FGM plates are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which differs from the geometric mid-plane. In the pre-buckling range natural frequencies decrease ultimately reaching zero for critical stress in the bifurcation point.

Axisymmetric loading of an elastic-plastic plate on a general two-parameter foundation

Axisymmetric loading of an elastic-plastic plate on a general two-parameter foundation

Static analysis of a thick ring on a unilateral elastic foundation

A thick ring on a unilateral elastic foundation can be used to model important applications such as non-pneumatic tires or bushing bearings. This paper presents a reduced-order compensation scheme for computing the static deformation response of a thick ring supported by a unilateral elastic foundation to an arbitrary applied force. The ring considered is an orthotropic and extensible ring that can be treated as a Timoshenko beam. The elastic foundation is a two-parameter foundation with a linear torsional stiffness but a unilateral radial stiffness whose value vanishes when compressed or tensioned. The paper first derives the deformation response for the linear foundation case for which Fourier expansion techniques can be applied to obtain an analytical solution. Then, the nonlinear unilateral foundation problem is solved via an iterative compensation scheme that identifies regions with vanishing radial stiffness and applies a compensation force to the linear foundation model to counteract the excessive foundation forces that would not be there with the unilateral foundation. This scheme avoids the need for solving the complex set of nonlinear differential equations and gives a computationally efficient tool for rapidly analyzing and designing such systems. Representative results are compared with Finite Element Analysis (FEA) results to illustrate the validity of the proposed approach.

Recursive differentiation method: application to the analysis of beams on two parameter foundations

The recursive differentiation method (RDM) is introduced and employed to obtain analytical solutions for static and dynamic stability parameters of beams resting on two-parameter foundations in various different end conditions. The present analysis reflects the reliability, efficiency and simplicity of the proposed RDM in tackling boundary value problems. In fact, it is widely common that the critical load accompanied with the first buckling mode is the smallest critical load, and then it is the dominant factor in the static stability analysis. In contrast, the present analysis indicates that such a conclusion is correct only for the case of beams without foundations or in the case of a weak foundation relative to the beam. It is proved that critical loads accompanied with higher buckling modes may be smaller than those accompanied with the lower modes and then it may control the stability analysis. The same phenomenon exists for natural frequencies in the presence of an axial load. Several illustrations are introduced to highlight the effects of both the foundation stiffness and beam slenderness on the critical loads and natural frequencies.

General Analysis of Timoshenko Beams on Elastic Foundation

General analytical solutions for stability, free and forced vibration of an axially loaded Timoshenko beam resting on a two-parameter foundation subjected to nonuniform lateral excitation are obtained using recursive differentiation method (RDM). Elastic restraints for rotation and translation are assumed at the beam ends to investigate the effect of support weakening on the beam behavior. However, the effects of rotational inertia and shear stress induced from the axial load are considered. The obtained solutions are verified first and then used to investigate the significance of different parameters on the beam behavior. In addition, solutions of forced vibration are analyzed to highlight the effects of excitation nonhomogeneity on the beam behavior.

Buckling and vibration analysis of non-homogeneous rectangular Kirchhoff plates resting on two-parameter foundation

Effect of two parameter foundation has been analyzed on the transverse vibrations of non-homogeneous rectangular plates of uniform thickness when the two opposite edges are simply supported and these are subjected to linearly varying in-plane forces on the basis of Kirchhoff plate theory. The non-homogeneity of the plate material is assumed to arise due to the exponential variation in young’s modulus and density of the plate material along one direction. Using Levy approach, the partial differential equation governing the motion of such plates has been reduced to an ordinary differential equation. Differential quadrature method has been used to obtain the frequency equations for two different combinations of clamped and simply supported boundary conditions at the other two edges. These frequency equations have been solved numerically using MATLAB. The lowest three roots of these equations have been reported as first three natural frequencies corresponding to the first three modes of vibration. The effect of various parameters such as foundation parameters, non-homogeneity parameter, density parameter, aspect ratio, in-plane force parameter and loading parameter has been investigated on the frequencies. By allowing the frequency to approach zero, the critical buckling loads for various value of different parameters have been computed. Three dimensional mode shapes for a specified plate for both the boundary conditions have been plotted. A comparison of results with those available in literature has been presented.

Improved nonlinear displacement-based beam element on a two-parameter foundation

This paper presents a nonlinear beam element on a two-parameter foundation. A set of governing differential equations of the problem (strong form) is first derived. The displacement-based beam-foundation element with improved displacement shape functions (weak form) is then formulated based on virtual displacement principle. The improved functions are analytically derived based on homogeneous solution to the governing differential equilibrium equation of the problem and are employed to enhance the model accuracy. Tonti’s diagrams are used to conveniently represent the equations that govern both the strong and weak forms of the problem. An averaging technique previously proposed by the authors is employed to determine system parameters needed in evaluating the displacement shape functions. Finally, two numerical simulations are used to verify the accuracy and the efficiency of the proposed beam model. The first simulation is used to perform convergence studies of the proposed model and to show its accuracy in representing both global and local responses. The second simulation is used to address effects of the two-parameter foundation model on system responses when compared to the Winkler foundation model.

A nth-Order Shear Deformation Theory for the Free Vibration Analysis of the Isotropic Plates on Elastic Foundations

A nth-order shear deformation theory for free vibration of the isotropic plates resting on a two-parameter Pasternak foundations is developed. The present theory does not require shear correction factor, and satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Governing equations are derived from the principle of virtual displacements. Meshless global collocation method based on the thin plate spline radial basis function is used to solve the governing differential equations. The accuracy of the present theory is demonstrated by comparing the present results with available published results.

The Mode and Nature Frequency of Pipes Conveying Fluid with Distributed Follower Force on Two-Parameter Foundation

The vibration differential equation of pipes conveying fluid with distributed follower force on two-parameter foundation is derived, the mode and nature frequency of clamped-clamped pipes with distributed follower force are calculated by complex modal. The relations between the nature frequency and the velocity of foundation liner stiffness and shear stiffness and mass ratio are studied. The numerical calculation shows that these parameters have a great influence to the nature frequency of the system and have certain effects to the characters vibrations of pipes.

Hamiltonian structural analysis of curved beams with or without generalized two-parameter foundation

The solution of curved Timoshenko beams with or without generalized two-parameter elastic foundation is presented. Beam can be subjected to any kind of loads and imposed external actions, distributed or concentrated along the beam. It can have external and internal restraints and any kind of internal kinematical or mechanical discontinuity. Moreover, the beam may have any spatial curved geometry, by dividing the entire structure into segments of constant curvature and constant elastic properties, each segment resting or not on elastic foundation. The foundation has six parameters like a generalized Winkler soil with the addition of other two parameters involving the link between settlements in transverse direction, as occurs for Pasternak soil model. The solution is obtained by the Hamiltonian structural analysis method, based on an energetic approach, solving a mixed canonical Hamiltonian system of twelve differential equations, leading to the fundamental matrix. The solution, numerically expressed or in closed form, is fast and simple to be implemented on a software, being the solving matrix always of order 12 for any kind of geometry and loads, without increasing in computational complexity. Numerical examples are given, comparing the results of the proposed procedure with the literature data, for rectilinear and curved beams with different soil properties.

Non-linear buckling of an FGM truncated conical shell surrounded by an elastic medium

In this paper, the non-linear buckling of the truncated conical shell made of functionally graded materials (FGMs) surrounded by an elastic medium has been studied using the large deformation theory with von Karman–Donnell-type of kinematic non-linearity. A two-parameter foundation model (Pasternak-type) is used to describe the shell–foundation interaction. The FGM properties are assumed to vary continuously through the thickness direction. The fundamental relations, the modified Donnell type non-linear stability and compatibility equations of the FGM truncated conical shell resting on the Pasternak-type elastic foundation are derived. By using the Superposition and Galerkin methods, the non-linear stability equations for the FGM truncated conical shell is solved. Finally, influences of variations of Winkler foundation stiffness and shear subgrade modulus of the foundation, compositional profiles and shell characteristics on the dimensionless critical non-linear axial load are investigated. The present results are compared with the available data for a special case.

Flexural Vibration Band Gaps Characteristics in Phononic Crystal Euler Beams on Two-Parameter Foundation

The vibration band gaps in the phononic crystal (PC) Euler beam on two-parameter foundation are investigated theoretically by the combination of the transfer matrix (TM) method and Bloch theorem. Compared with the Winkler foundation with only the foundation reaction modulus k, the shear modulus [Formula: see text] is taken into account simultaneously to consider the continuity of deformation of the foundation. The band structures of a lead-copper PC Euler beam either on two-parameter foundation or on Winkler (one-parameter) foundation show that the introduction of the shear modulus [Formula: see text] has significant impact on the distribution of band gaps. The first band gap has not any evolution with the increase of [Formula: see text], while the other band gaps trend to higher frequency and their ranges become narrower. The analysis shows that the parameters of foundation and the length ratio of segments can be conditionally selected to obtain adjustable band gaps.

Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium

In this study, the torsional vibration and buckling analysis of cylindrical shell with functionally graded (FG) coatings surrounded by an elastic medium is investigated. The material properties of FG coatings are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. A two-parameter foundation model or Pasternak foundation model is used to describe the shell–foundation interaction. The basic equations are derived and solved by using Galerkin method and the expressions for dimensionless torsional buckling load and dimensionless torsional frequency parameter of the cylindrical shell with FG coatings resting on the Pasternak elastic foundation are obtained. Finally, the influences of geometrical parameters, volume fraction distribution of FG coatings and foundation stiffnesses on the dimensionless torsional buckling load and dimensionless torsional frequency parameter are discussed. The present results are compared with the available data for a special case.

Flexural transient response of elastically supported elliptical plates under in-plane loads using Mathieu functions

The elaborated method of eigenfunction expansion in elliptic coordinates is employed to obtain an exact time-domain series solution, involving products of angular and radial Mathieu functions, for the forced flexural vibrations of a thin elastic plate of elliptical planform. The plate is supported by a constant moduli two-parameter foundation, while elastically restrained against translation and rotation at its edge, and subjected to the combined action of uniform in-plane static edge forces and general arbitrary time-dependent transverse loads with arbitrary initial conditions. Numerical calculations are carried out for the displacement response of clamped or simply supported elliptical plates of selected aspect ratios in various practical loading configurations (i.e., an impulsive point load, a point force in circular motion, a uniformly distributed harmonic load, and a blast load), with or without an elastic foundation, while taking the effects of initial tension or compression below the buckling load into consideration. Limiting cases are considered and good agreements with available results as well as with the computations made by using a commercial finite element package are obtained.

Analysis of partially embedded beams in two-parameter foundation

In this study, Pasternak foundation model, which is a two parameter foundation model, is used to analyze the behavior of laterally loaded beams embedded in semi-infinite media. Total potential energy variation of the system is written to formulate the problem that yielded the required field equations and the boundary conditions. Shear force discontinuities are exposed within the boundary conditions by variational method and are validated by photo elastic experiments. Exact solution of the deflection of the beam is obtained. Both foundation parameters are obtained by self calibration for this particular problem and loading type in this study. It is shown that, like the first parameter k, the second foundation parameter G also depends not only on the material type but also on the geometry and the loading type of the system. On the other hand, surface deflection of the semi infinite media under singular loading is obtained and another method is proposed to determine the foundation parameters using the solution of this problem.

Free Vibration Analysis for Elastic Plate Resting on Two-Parameter Foundation by Coupling FEM and BEM

Based on two-parameter foundation model and Mindlin plate theory, the FEM equation for free vibration analysis of elastic plates resting on elastic foundation is derived by Hamilton variation principle. The effect of foundation beneath the plate is combined in the stiffness matrix of the plate element, and the effect of the foundation outside the plate domain is taken into account by boundary element method. By coupling FEM and BEM, numerical analyses for the free vibration of foundation plates are carried out. Calculated frequencies are in good agreement with measured results, which proves the accuracy and efficiency of the present approach.

Natural frequency of laminated orthotropic shells with different boundary conditions and resting on the Pasternak type elastic foundation

This paper deals with the vibration analysis of laminated orthotropic shells with different boundary conditions and resting on elastic foundations. Boundary conditions are clamped–clamped ( C– C) and sliding–sliding ( SL– SL). A two-parameter foundation model (Pasternak type) is used to describe the shell–foundation interaction, from which Winkler foundation model can be easily obtained as a limiting case. The modified Donnell type dynamic stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells resting on elastic foundations. Applying Galerkin methods, the natural frequency of the laminated orthotropic truncated conical shell for different boundary conditions, are obtained. The appropriate formulas for laminated cylindrical and complete conical shells are found as special cases. Finally, influences of the elastic foundations, the boundary conditions, the number and stacking sequence, and the variations of shell characteristics on the natural frequency are investigated. The results are compared with their counterparts in the literature.

Response of a completely free beam on a tensionless Pasternak foundation subjected to dynamic load

Static and dynamic responses of a completely free elastic beam resting on a two-parameter tensionless Pasternak foundation are investigated by assuming that the beam is symmetrically subjected to a uniformly distributed load and concentrated load at its middle. Governing equations of the problem are obtained and solved by paying attention on the boundary conditions of the problem including the concentrated edge foundation reaction in the case of complete contact and lift-off condition of the beam ina two-parameter foundation. The nonlinear governing equation of the problem is evaluated numerically by adopting an iterative procedure. Numerical results are presented in figures to demonstrate the non-linear behavior of the beam-foundation system for various values of the parameters of the problem comparatively by considering the static and dynamic loading cases.

Elastic Waves in Pipe Resting on Two-Parameter Foundation

A method based on Hamiltonian system in complex field is presented in curvilinear coordinates to study elastic waves in pipes of various shapes on two-parameter foundation. The method and its computer program are verified and applied to analyze the axial wave propagation problem of elliptical pipe embedded in foundation. Numerical results show the dispersion changes of varying degree in the presence of foundation and reveal significant influences of the second foundation parameter especially in the low frequency range. The promising and effective way of controlling propagating waves by adjusting the shear ability of foundation is also indicated in the results.

Elasto-Plasticity Solution on Bearing Characteristic of Cast-in-Place Concrete Pipe Pile with Large Diameter in Layered Soil under Lateral Load

Aimed at problems lying in the exiting methods for calculation of laterally loaded piles installed in layered soils, the elasto-plasticity solution of cast-in-place concrete pipe pile with large diameter (PCC pile) in layered soil under lateral load based two-parameter foundation model and unity limiting force profile, deduced the finite difference solution based unity limiting force profile(LFP), established the calculation program of the displacement and internal force of pile. It was shown that the calculated pile moment and pile displacement coincided very well by computation and comparison of model test and FEM, verified the method’s correction. Meanwhile, the method was more precise and efficient than the existing conversion methods of subgrade modulus. Therefore, it could be useful for the design and analysis of PCC pile under lateral load.