Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform Was employed in the analysis of the semi-infinite elastic foundation with vertical load. And an integral representations for the displacements of the semi-infinite elastic foundation was presented. The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semiinfinite elastic foundation. Some computational results and the analysis on the influence of parameters were presented.

Bending and Free Vibration Analysis of Porous-Functionally-Graded(PFG)Beams Resting on Elastic Foundations

The bending and free vibration of porous functionally graded(PFG)beams resting on elastic foundations are analyzed.The material features of the PFG beam are assumed to vary continuously through the thickness according to the volume fraction of components.The foundation medium is also considered to be linear,homogeneous,and isotropic,and modeled using the Winkler-Pasternak law.The hyperbolic shear deformation theory is applied for the kinematic relations,and the equations of motion are obtained using the Hamilton’s principle.An analytical solution is presented accordingly,assuming that the PFG beam is simply supported.Comparisons with the open literature are implemented to verify the validity of such a formulation.The effects of the elastic foundations,porosity volume percentage and span-to-depth ratio are finally discussed in detail.

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded(FG)beams supported by a viscoelastic Winkler-Pasternak foundation is presented.The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation,which were not considered in most literature on this subject,and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven(ε-D)and stress-driven(σ-D)two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered,which can address both the stiffness softening and toughing effects due to scale reduction.The generalized differential quadrature method(GDQM)is used to solve the complex eigenvalue problem.After verifying the solution procedure,a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained.Subsequently,the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.

NONLINEAR RESPONSES OF A FLUID-CONVEYING PIPE EMBEDDED IN NONLINEAR ELASTIC FOUNDATIONS

The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial differential equation. For the analytical model, the effect of the nonlinear elastic foundation is modeled by a nonlinear restraining force. By using an iterative algorithm, a set of ordinary differential dynamical equations derived from the equation of motion of the system are solved numerically and then the bifurcations are analyzed. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits of the oscillations. The intermittency transition to chaos has been found to arise.

Buckling analysis of functionally graded material(FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material （FGM） coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable- coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

Buckling analysis of shear deformable composite conical shells reinforced by CNTs subjected to combined loading on the two-parameter elastic foundation

The main objective of this study is to investigate the buckling analysis of CCSs reinforced by CNTs subjected to combined loading of hydrostatic pressure and axial compression resting on the twoparameter elastic foundation(T-P-EF).It is one of the first attempts to derive the governing equations of the CCSs reinforced with CNTs,based on a generalized first-order shear deformation shell theory(FSDST)which includes shell-foundation interaction.By adopting the extended mixing rule,the effective material properties of CCSs reinforced by CNTs with linear distributions are approximated by introducing some efficiency parameters.Three carbon nanotube distribution in the matrix,i.e.uniform distribution(U)and V and X-types linear distribution are taken into account.The stability equations are solved by using the Galerkin procedure to determine the combined buckling loads(CBLs)of the structure selected here.The numerical illustrations cover CBLs characteristics of CCSs reinforced by CNTs in the presence of the T-P-EF.Finally,a parametric study is carried out to study the influences of the foundation parameters,the volume fraction of carbon nanotubes and the types of reinforcement on the CBLs.

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

Based on Reissner plate theory and Hamilton variational principle, the nonlinear equations of motion were derived for the moderate thickness rectangular plates with transverse surface penetrating crack on the two-parameter foundation. Under the condition of free boundary, a set of trial functions satisfying all boundary conditions and crack＇s continuous conditions were proposed. By employing the Galerkin method and the harmonic balance method, the nonlinear vibration equations were solved and the nonlinear vibration behaviors of the plate were analyzed. In numerical computation, the effects of the different location and depth of crack, the different structural parameters of plates and the different physical parameters of foundation on the nonlinear amplitude frequency response curves of the plate were discussed.

ANALYSIS OF COUPLED-MODE FLUTTER OF PIPES CONVEYING FLUID ON THE ELASTIC FOUNDATION

The governing equation of solid-liquid couple vibration of pipe conveying fluid on the elastic foundation was derived. The critical velocity and complex frequency of pipe conveying fluid on Winkler elastic foundation and two-parameter foundation were calculated by po,ver series method. Compared,with pipe without considering elastic foundation, the numerical results show that elastic foundation can increase the critical flow velocity of static instability and dynamic instability of pipe. And the increase of foundation parameters may increase the critical flow velocity of static instability and dynamic instability of pipe, thereby delays the occurrence of divergence and flutter instability of pipe. For higher mass ratio beta, in the combination of certain foundation parameters, pipe behaves the phenomenon of restabilization and redivergence after the occurrence of static instability, and then coupled-mode flutter takes place.

ANALYSIS OF BENDING, VIBRATION AND STABILITY FOR THIN PLATE ON ELASTIC FOUNDATION BY THE MULTIVARIABLE SPLINE ELEMENT METHOD

In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable spline element equations are derived, based on the mixed variational principle. The analysis and calculations of bending, vibration and stability of the plates on elastic foundation are presented in the paper. Because the field functions of plate on elastic foundation are assumed independently, the precision of the field variables of bending moments and displacement is high.

Influence of elastic foundations and carbon nanotube reinforcement on the hydrostatic buckling pressure of truncated conical shells

In this study,the effects of elastic foundations(EFs)and carbon nanotube(CNT)reinforcement on the hydrostatic buckling pressure(HBP)of truncated conical shells(TCSs)are investigated.The first order shear deformation theory(FOSDT)is generalized to the buckling problem of TCSs reinforced with CNTs resting on the EFs for the first time.The material properties of composite TCSs reinforced with CNTs are graded linearly according to the thickness coordinate.The Winkler elastic foundation(W-EF)and Pasternak elastic foundation(P-EF)are considered as the EF.The basic relations and equations of TCSs reinforced with CNTs on the EFs are obtained in the framework of the FOSDT and solved using the Galerkin method.One of the innovations in this study is to obtain a closed-form solution for the HBP of TCSs reinforced with CNTs on the EFs.Finally,the effects of the EFs and various types CNT reinforcements on the HBP are investigated simultaneously.The obtained results are compared with the results in the literature,and the accuracy of results is confirmed.

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 elas- tic foundations are investigated by an analytical method. The elastic foundation of partial axial and angular dimen- sions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear defor- mation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material prop- erties 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.

REMARKS OF SOME PROBLEMS FOR RECTANGULAR THIN PLATES WITH FREE EDGES ON ELASTIC FOUNDATIONS

For the bending, stability and vibrations of rectangular thin plates with free edges on elastic foundations, in this paper we give a flexural function which exactly satisfies not only all the boundary conditions on free edges but also the conditions at free corner points. Applying energy variation principle, we give equations defining parameters in flexural function, stability equation, frequency equation, and general formulae of minimum critical force and minimum eigenfrequency as well.

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foundation of finite height bounded to the surface of a saturated half-space. The foundation is subjected to time-harmonic vertical loadings. First, the transform solutions for the governing equations of the saturated media are obtained. Then, based on the assumption that the contact between the foundation and the half-space is fully relaxed and the halfspace is completely pervious or impervious, this dynamic mixed boundary-value problem can lead to dual integral equations, which can be further reduced to the Predhohn integral equations of the second kind and solved by numerical procedures. In the numerical extortples, the dynamic colnpliances, displacements and pore pressure are developed for a wide range of frequencies and material/geometrical properties of the saturated soil-foundation system. In most of the cases, the dynamic behavior of an elastic foundation resting on the saturated media significantly differs from that of a rigid disc on the saturated half-space. The solutions obtained can be used to study a variety of wave propagation problems and dynamic soil-structure interactions.

THICK RECTANGULAR PLATES WITH FREE EDGES ON ELASTIC FOUNDATIONS

In this paper an accurate solution for the thick rectangular plate with free edges laid on elastic foundation is presented. The superposition method of trigonometric series is used. The method can solve this kind of plates directly and simply. Its results completely satisfy the boundary conditions of the four free edges and nicely agree with the solutions by Wang Ke-lin and Huang Yi[2]

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES ON AN ELASTIC FOUNDATION

On the basis of von Karnmnequations, the axisymmetric buckling and post-buckling of annular plates on anelastic foundation is so wematically discussed byusing shooting

GENERAL SOLUTION FOR DYNAMICAL PROBLEM OF INFINITE BEAM ON AN ELASTIC FOUNDATION

Based on the theory of Eider-Bernoulli beam and Winkler assumption for elastic foundation, a mathematical model is presented. By using Fourier transformation for space variable, Laplace transformation for time variable and convolution theorem for their inverse transformations, a general solution for dynamical problem of infinite beam on an elastic foundation is obtained. Finally, the cases of free vibration,impulsive response and moving load are also discussed.

SOUND RADIATION FROM PLATES ON ELASTIC FOUNDATION UNDER ACTION OF MOVING LINE FORCES

An analysis is made of the problem of sound radiation from infinite one-dimensional plateson elastic foundation, when the plates are subjected to the action of harmonic line forces movingat subsonic speeds (M 【 1). The expressions of nondimensional sound power are formulated andthe asymptotic forms of sound power in the low frequency regions are derived. The radiatedsound power is shown as a function of the stiffness of elastic foundation, in terms of stiffness fac-torψ, the moving speed of line force, in terms of Math number M, and the frequency, in termsof wavenumber ratio γ . The effects of the parameter ψ in conjunction with the parameters Mand γ on the radiated sound power level and the phenomenon of coincidence radiation are alsoinvestigated in detail.

A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION

In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential eguation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous eguations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.

A Unified Dimensionless Parameter for Finite Element Mesh for Beams Resting on Elastic Foundation

Discretising a structure into elements is a key step in finite element(FE)analysis.The discretised geometry used to formulate an FE model can greatly affect accuracy and validity.This paper presents a unified dimensionless parameter to generate a mesh of cubic FEs for the analysis of very long beams resting on an elastic foundation.A uniform beam resting on elastic foundation with various values of flexural stiffness and elastic supporting coefficients subject to static load and moving load is used to illustrate the application of the proposed parameter.The numerical results show that(a)Even if the values of the flexural stiffness of the beam and elastic supporting coefficient of the elastic foundation are different,the same proposed parameter“s”can ensure the same accuracy of the FE solution,but the accuracy may differ for use of the same element length;(b)The proposed dimensionless parameter“s”can indeed be used as a unified index to generate the mesh for a beam resting on elastic foundation,whereas the use of the same element length as a criterion may be misleading;(c)The errors between the FE and analytical solutions for the maximum vertical displacement,shear force and bending moment of the beam increase with the dimensionless parameter“s”;and(d)For the given allowable errors for the vertical displacement,shear force and bending moment of the beam under static load and moving load,the corresponding values of the proposed parameter are provided to guide the mesh generation.

Prediction Model of Abutment Pressure Affected by Far-Field Hard Stratum Based on Elastic Foundation Theory

In view of the three-dimensional dynamic abutment pressure,the influence of the far-field hard stratum(FHS)in deep,thick coal seams is indeterminant.Based on elastic foundation theory,a three-dimensional dynamic prediction model of the abutment pressure was established.Using this model,the dynamic change in the coal seam abutment pressure caused by the movement of the FHS was studied,and a method for determining the dynamic change range of the abutment pressure was developed.The results of the new prediction model of the abutment pressure are slightly higher than the measured values,with an error of 0.51%,which avoids the shortcomings of the results because the Winkler foundation model results are lower than the measured values and have an error of 9.98%.As time progresses,the abutment pressure and its distribution range are affected by the FHS movement,which has the characteristics of gradually increasing dynamic change until the FHS fractures.The peak value of the abutment pressure increases linearly with time,and the influence range increases with time following a power function with an exponent of less than 1.The influence range of the FHS movement on the abutment pressure ahead of the working face,behind the working face,and along the working face is 10 times,25 times,and 17 times the mining thickness,respectively.According to the actual geological parameters,the dynamic change range of the coal seam abutment pressure was determined by drawing an additional stress curve and by determining the threshold value.These research results are of great significance to the partition optimization of the roadway support design of deep,thick coal seams.