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.

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.

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.

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]

Modified Muravskii model for elastic foundations

A frequency equation for the vibration of an engine seating and an equation for pressure under the bottom of the engine are obtained.The present approach extends the so called Muravskii model possessing high practical accuracy of the ground modeling with its simultaneous simplicity.

Buckling Analysis of FG Porous Truncated Conical Shells Resting on Elastic Foundations in the Framework of the Shear Deformation Theory

In this article,an analytical method is proposed to analyze of the linear buckling behavior of the FG porous truncated conical shells subjected to a uniform axial compressive load and resting on the Pasternak elastic foundation.The material properties including Young’s modulus,shear modulus and density are assumed to vary in the thickness direction.Three types of FG porous distributions including symmetric porosity distribution,non-symmetric porosity and uniform porosity distribution are considered.The governing equations of the FG porous truncated conical shells are obtained by using the first-order shear deformation theory(FSDT).With the help of the Galerkin method,the expressions for critical buckling loads are obtained in closed forms.The reliability of the obtained results is verified by comparing the present solutions with the published solutions.Finally,the numerical results show the effects of shell characteristics,porosity distribution,porosity coefficient,and elastic foundation on the critical buckling load.

Thermodynamical Bending of FGM Sandwich Plates Resting on Pasternak’s Elastic Foundations

The analysis of thermoelastic deformations of a simply supported functionally graded material(FGM)sandwich plates subjected to a time harmonic sinusoidal temperature field on the top surface and varying through-the-thickness is illustrated in this paper.The FGM sandwich plates are assumed to be made of three layers and resting on Pasternak’s elastic foundations.The volume fractions of the constituents of the upper and lower layers and,hence,the effective material properties of them are assumed to vary in the thickness direction only whereas the core layer is still homogeneous.When in-plane sinusoidal variations of the displacements and the temperature that identically satisfy the boundary conditions at the edges,the governing equations of motion are solved analytically by using various shear deformation theories as well as the classical one.The influences of the time parameter,power law index,temperature exponent,top-to-bottom surface temperature ratio,side-to-thickness ratio and the foundation parameters on the dynamic bending are investigated.

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper,the static analysis of functionally graded(FG)circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach.The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson’s ratio remains constant.The solution is obtained by employing the state space method(SSM)to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method(DQM)to approximate the radial variations of the parameters.The effects of different parameters(e.g.,material property gradient index,elastic foundation coefficients,the surfaces conditions(hard or soft surface of the plate on foundation),plate geometric parameters and edges condition)on the deformation and stress distributions of the FG circular plates are investigated.

A THREE-DIMENSIONAL ELASTICITY SOLUTION FOR TWO-DIRECTIONAL FGM ANNULAR PLATES WITH NON-UNIFORM ELASTIC FOUNDATIONS SUBJECTED TO NORMAL AND SHEAR TRACTIONS

In the present paper, bending and stress analyses of two-directional functionally graded （FG） annular plates resting on non-uniform two-parameter Winkler-Pasternak founda- tions and subjected to normal and in-plane-shear tractions is investigated using the exact three- dimensional theory of elasticity. Neither the in-plane shear loading nor the influence of the two- directional material heterogeneity has been investigated by the researchers before. The solution is obtained by employing the state space and differential quadrature methods. The material proper- ties are assumed to vary in both transverse and radial directions. Three different types of variations of the stiffness of the foundation are considered in the radial direction： linear, parabolic, and sinu- soidal. The convergence analysis and the comparative studies demonstrate the high accuracy and high convergence rate of the present approach. A parametric study consisting of evaluating effects of different parameters （e.g., exponents of the material properties laws, the thickness to radius ratio, trends of variations of the foundation stiffness, and different edge conditions） is carried out. The results are reported for the first time and are discussed in detail.

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.

Dynamic response of honeycomb-FGS shells subjected to the dynamic loading using non-polynomial higher-order IGA

The main goal of this study is to use higher-order isogeometric analysis(IGA)to study the dynamic response of sandwich shells with an auxetic honeycomb core and two different functionally graded materials(FGM)skin layers(namely honeycomb-FGS shells)subjected to dynamic loading.Touratier's non-polynomial higher-order shear deformation theory(HSDT)is used due to its simplicity and performance.The governing equation is derived from Hamilton's principle.After verifying the present approach,the effect of input parameters on the dynamic response of honeycomb-FGS shells is carried out in detail.

Free vibration and buckling analysis of polymeric composite beams reinforced by functionally graded bamboo fbers

Natural fibers have been extensively researched as reinforcement materials in polymers on account of their environmental and economic advantages in comparison with synthetic fibers in the recent years.Bamboo fibers are renowned for their good mechanical properties,abundance,and short cycle growth.As beams are one of the fundamental structural components and are susceptible to mechanical loads in engineering applications,this paper performs a study on the free vibration and buckling responses of bamboo fiber reinforced composite(BFRC)beams on the elastic foundation.Three different functionally graded(FG)layouts and a uniform one are the considered distributions for unidirectional long bamboo fibers across the thickness.The elastic properties of the composite are determined with the law of mixture.Employing Hamilton’s principle,the governing equations of motion are obtained.The generalized differential quadrature method(GDQM)is then applied to the equations to obtain the results.The achieved outcomes exhibit that the natural frequency and buckling load values vary as the fiber volume fractions and distributions,elastic foundation stiffness values,and boundary conditions(BCs)and slenderness ratio of the beam change.Furthermore,a comparative study is conducted between the derived analysis outcomes for BFRC and homogenous polymer beams to examine the effectiveness of bamboo fibers as reinforcement materials,demonstrating the significant enhancements in both vibration and buckling responses,with the exception of natural frequencies for cantilever beams on the Pasternak foundation with the FG-◇fiber distribution.Eventually,the obtained analysis results of BFRC beams are also compared with those for carbon nanotube reinforced composite(CNTRC)beams found in the literature,indicating that the buckling loads and natural frequencies of BFRC beams are lower than those of CNTRC beams.

A method to Estimate Dynamic Responses of VLFS Based on Multi-Floating-Module Model Connected by Elastic Hinges

Based on the elastic foundation beam theory and the multi-floating-module hydrodynamic theory,a novel method is proposed to estimate the dynamic responses of VLFS(Very Large Floating Structure).In still water,a VLFS can be simplified as an elastic foundation beam model or a multi-floating-module model connected by elastic hinges.According to equivalent displacement of the two models in static analysis,the problem of rotation stiffness of elastic hinges can be solved.Then,based on the potential flow theory,the dynamic responding analysis of multi-floatingmodule model under wave loads can be computed in ANSYS-AQWA software.By assembling the time domain analysis results of each module,the dynamic responses of the VLFS can be obtained.Validation of the method is conducted through a series of comparison calculations,which mainly includes a continuous structure and a three-part structure connected by hinges in regular waves.The results of this paper method show a satisfactory agreement with the experiment and calculation data given in relative references.

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.

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 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.

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.

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.

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.