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.

Application of strength reduction method to dynamic anti-sliding stability analysis of high gravity dam with complex dam foundation

Considering that there are some limitations in analyzing the anti-sliding seismic stability of dam-foundation systems with the traditional pseudo-static method and response spectrum method, the dynamic strength reduction method was used to study the deep anti-sliding stability of a high gravity dam with a complex dam foundation in response to strong earthquake-induced ground action. Based on static anti-sliding stability analysis of the dam foundation undertaken by decreasing the shear strength parameters of the rock mass in equal proportion, the seismic time history analysis was carried out. The proposed instability criterion for the dynamic strength reduction method was that the peak values of dynamic displacements and plastic strain energy change suddenly with the increase of the strength reduction factor. The elasto-plastic behavior of the dam foundation was idealized using the Drucker-Prager yield criterion based on the associated flow rule assumption. The result of elasto-plastic time history analysis of an overflow dam monolith based on the dynamic strength reduction method was compared with that of the dynamic linear elastic analysis, and the reliability of elasto-plastic time history analysis was confirmed. The results also show that the safety factors of the dam-foundation system in the static and dynamic cases are 3.25 and 3.0, respectively, and that the F2 fault has a significant influence on the anti-sliding stability of the high gravity dam. It is also concluded that the proposed instability criterion for the dynamic strength reduction method is feasible.

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.

Calculation of Hydro-Dynamic Stability of the Soil Inside Bucket in the Process of Bucket Foundation Penetration

The offshore platform with bucket foundations is a;new type of offshore platform that distinguishes from traditional template platforms by replacing driven piles with bucket foundations. The suction penentration of bucket foundation is a complicated hydro-dynamic process. The key of this process is the seepage field caused by the difference of pressure applied on purpose inside and outside the bucket. The appearance and developement of seepage field has a decisive influence on the suction penetration process. In this study, the finite element analysis method is applied to the dynamic simulation of the seepage field of suction penetration of bucket foundation. A criterion is suggested to distinguish the hydro-dynamic stability of the soil inside the bucket according to the critical hydraulic gradient method. The reliability of the model and its applicability to engineering practice have been proved through comparison between the results of model test and finite element calculation.

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.

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.

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.

Parametric Study of Dynamic Wrinkling in a Thin Sheet on Elastic Foundation

This work presents an approximate analytical study of the problem of dynamic wrinkling of a thin metal sheet under a specified time varying tension. The problem is investigated in the framework of the dynamic stability of a nonlinear plate model on elastic foundation which namely takes into account the nonlinear mechanics of mid-plane stretching and the dependence of the membrane force on this mechanics. The plate is assumed to be a wide rectangular slab, hinged at two opposite ends and free at the long ends, which can be deformed in a cylindrical shape so that the governing in-plane bending equation of motion takes the same form as that of a beam (e.g. lateral strip) element. An approximate analytical analysis of the beam wrinkling behavior under sinusoidal parametric excitation is carried out by using the assumed single mode wrinkling motion to reduce the beam field nonlinear partial differential equation to that of a single degree of freedom non-linear oscillator. A first order stability analysis of an approximate analytical solution obtained using the Multi-Time-Scales (MMS) method is used to derive a criterion defining critical driving frequency in terms of system parameters for the initiation of wrinkling motion in the thin metal sheet. Results obtained using this criterion is presented for selected values of system parameters.

In-Plane Elastic Stability of Arches under a Radial Concentrated Load

This paper is concerned with the in-plane elastic stability of arches subjected to a radial concentrated load. The equilibrium equation for pin-ended circular arches is established by using energy method, and it is proved that the axial force is nearly a constant along the circumference of the circular arches. Based on force method, the equation for the primary eigen function is derived and solved, and the approximate analytical solution of critical instability load is obtained. Numerical examples are given and discussed.

Analysis of vibration stability of fluid conveying pipe on the two-parameter foundation with elastic support boundary conditions

In this study,the vibration stability of fluid conveying pipe resting on two-parameter foundation is in-vestigated under four different elastic support boundary conditions.The harmonic differential quadrature(HDQ)method is applied to solve the governing vibration equation derived based on Euler–Bernoulli beam theory subject to the elastic foundation and boundary conditions.As a result,a general set of second-order ordinary differential equations emerges,and by appropriately setting the stiffness of the end springs,one can easily study the dynamics of various systems with classical or non-classical bound-ary conditions.The numerical simulations are conducted to study the pipe instability performance with respect to various boundary conditions,elastic support parameters,elastic foundation parameters and fluid mass ratios.The numerical model is validated by comparison with published data.It is found that the elastic support boundary conditions have significant effects on the stability of pipe resting on elas-tic foundation.The pipe stability performance is very sensitive to the two elastic foundation parameters.Larger fluid mass ratio enhances the pipe flutter stability performance but has no effects on the diver-gence.

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed, Some complex dynamic performances such as perioddoubling motion and quasi-period motion are discussed.

Nonlinear thermo-mechanical stability of eccentrically stiffened functionally graded material sandwich doubly curved shallow shells with general sigmoid law and power law according to third-order shear deformation theory

In this paper, the nonlinear analysis of stability of functionally graded ma- terial （FGM） sandwich doubly curved shallow shells is studied under thermo-mechanical loads with material properties obeying the general sigmoid law and power law of four ma- terial models. Shells are reinforced by the FGM stiffeners and rest on elastic foundations. Theoretical formulations are derived by the third-order shear deformation theory （TSDT） with the von Karman-type nonlinearity taking into account the initial geometrical im- perfection and smeared stiffener technique. The explicit expressions for determining the critical buckling load and the post-buckling mechanical and thermal load-deflection curves are obtained by the Galerkin method. Two iterative algorithms are presented. The effects of the stiffeners, the thermal element, the distribution law of material, the initial imper- fection, the foundation, and the geometrical parameters on buckling and post-buckling of shells are investigated.

Stability and Nonlinear Response Analysis of Parametric Vibration for Elastically Constrained Pipes Conveying Pulsating Fluid

Usually,the stability analysis of pipes with pulsating flow velocities is for rigidly constrained pipes or cantilevered pipes.In this paper,the effects of elastic constraints on pipe stability and nonlinear responses under pulsating velocities are investigated.A mechanical model of a fluid-conveying pipe under the constraints of elastic clamps is established.A partial differentialintegral nonlinear equation governing the lateral vibration of the pipe is derived.The natural frequencies and mode functions of the pipe are obtained.Moreover,the stable boundary and nonlinear steady-state responses of the parametric vibration for the pipe are established approximately.Furthermore,the analytical solutions are verified numerically.The results of this work reveal some interesting conclusions.It is found that the elastic constraint stiffness in the direction perpendicular to the axis of the pipe does not affect the critical flow velocity of the pipe.However,the constraint stiffness has a significant effect on the instability boundary of the pipe with pulsating flow velocities.Interestingly,an increase in the stiffness of the constraint increases the instable region of the pipe under parametric excitation.However,when the constraint stiffness is increased,the steady-state response amplitude of the nonlinear vibration for the pipe is significantly reduced.Therefore,the effects of the constraint stiffness on the instable region and vibration responses of the fluid-conveying pipe are different when the flow velocity is pulsating.

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement.Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement.The mixed method is locking-free,i.e.,the convergence does not deteriorate in the nearly incompressible or incompressible case.The optimal convergence order is shown in the L^(2)-norm for stress and in the broken H1-norm and L2-norm for displacement,respectively.Finally,some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.

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.

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bemoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method （DTM） has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.

STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES

Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes.Two kinds of penalty terms are introduced in the stabilized mixed formulation,which are the jump penalty term for the displacement and the divergence penalty term for the stress.We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress,where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation.The stabilized mixed method is locking-free.The optimal convergence order is derived in the L^(2)-norm for stress and in the broken H^(1)-norm and L^(2)-norm for displacement.A numerical test is carried out to verify the optimal convergence of the stabilized method.

平行不对中约束下大范围转动弹性梁非线性动力学建模与振动机制

为了有效抑制或消除不对中约束响应,研究其非线性动力学特性和振动机制。针对柔性大范围转动弹性梁机构,综合应用Galerkin法和Hamilton原理建立转动梁的动力学模型。借助转子动力学理论和拉格朗日法推导联轴器平行不对中运动学约束方程。基于运动学约束关系,利用行波叠加原理构建弹性梁在平行不对中约束下的非线性动力学模型。应用高阶Runge-Kutta法分析弹性梁非线性动力学行为。研究结果表明:转速为10 rad/s时,联轴器不对中量对弹性梁的动力学特性影响较小。当转速为150 rad/s、不对中量从0增加到0.50 mm时,梁的横向振幅则由0.18 mm激增到10.06 mm,提高了2个数量级,而纵向振幅也由0.13 mm增至7.30 mm,表明高速下不对中量与系统稳定性存在强耦合,其微小变化会引起显著的突发性、非线性激振。转速和不对中量的增加会增大联轴器离心力且呈倍频、径向交替变化,引起转动梁的外激励突变导致系统快速失稳,这也是大型旋转机械运行速度不高的重要原因。