BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

Two fundamental solutions for bending problem of Reissner's plates on twoparameter foundation are derived by means of Fouier integral transformation of generalized function in this paper.On the basis of virtual work principles, three boundary integral equations which fit for arbitrary shapes, loads and boundary conditions of thick plates are presented according to Hu Haichang's theory about Reissner's plates. It provides the fundamental theories for the application of BEM. A numerical example is given for clamped, simply supported and free boundary conditions. The results obtained are satisfactory as compared with the analytical methods.

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.

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]

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.

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.

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.

Calculation method of composite foundation sedimentation of grouting pile with cover plate under embankment load

Grouting pile is a new soft soil foundation treatment method with characteristics such as no vibration, no noise, no soil compaction, light construction machines and quick construction velocity and so on. At present, study on reinforcement mechanism and design calculation method of composite foundation of grouting pile is initially started without design specifications, so it is usually required to draw on design specifications of stump pile when designing composite foundation of grouting pile while grouting pile has its characteristics and difference although reinforcement mechanisms and construction processes of two types of piles are similar. Sedimentation formula of composite foundation of grouting pile with cover plate is educed and a suitable deformation mode is proposed by aiming to deformation characteristics of composite foundation of grouting pile with cover plate under embankment load on basis of relevant sedimentation theories of composite foundation by combination of characteristics of composite foundation of grouting pile. The sedimentation calculation formula of grouting pile with cover plate under embankment load is educed according to balance relation of force and displacement coordination conditions by elastic theory and sedimentation calculation model established is validated by sedimentation monitoring documents of one expressway in China.

Tensionless contact of a finite circular plate

A general formulation is developed for the contact behavior of a finite circular plate with a tensionless elastic foundation. The gap distance between the plate and elastic foundation is incorporated as an important parameter. Unlike the previous models with zero gap distance and large/infinite plate radius, which assumes the lift-off/separation of a flexural plate from its supporting elastic foundation, this study shows that lift-off may not occur. The results show how the contact area varies with the plate radius, boundary conditions and gap distance. When the plate radius becomes large enough and the gap distance is reduced to zero, the converged contact radius close to the previous ones is obtained.

Synergistic instability of coal pillar and roof system and filling method based on plate model

The security challenges from room and pillar gobs include land subsidence, spontaneous combustion of coal pillars and mine flood caused by gob water. To explore the instability mechanism of room and pillar gob, we established a mechanical model of elastic plate on elastic foundation in which pillars and hard roofs were considered as continuous Winkler foundations and elastic plates, respectively. The synergetic instability of pillar and roof system was analyzed based on plate bending theory and catastrophe theory. In addition, mechanical conditions and math criterion of roof failure and overall instability of coal pillar and roof system were given. Through analyzing both advantages and disadvantages of some technologies such as induced caving, filling, gob sealing and isolation, we presented a new filling method named box-filling, in view of box foundation theory, to control the disasters of ground collapse, water inrush and mine fire. In a gob's treatment project in Ordos, safety assessment and filling design of a room and pillar gob have been done by the mechanical model. The results show that the gob will collapse when the pillars' average yield band is wider than 0.93 m, and box-filling can control land collapse, mine flood and mine fire economically and efficiently. So it is worth to study further and popularize.

部分弹性地基Mindlin板自由振动的Hamilton方法

采用Hamilton方法研究Pasternak型部分弹性地基上Mindlin板自由振动问题。引入合适的状态向量,将Mindlin板的控制微分方程转化为相应的Hamilton形式,进一步求解Hamilton算子的本征值和本征函数系,并给出具体的数值算例,数值模拟了部分弹性地基上板的固有频率。

Elastic Modulus Limit of Support Layer in Twin-Block Ballastless Track Based on Train Load Effect

The support layer is an important component of twin-block ballastless track. The modulus of the support layer is an important design parameter and must be carefully solved. We studied the bending stress and deformation of track slab and support layer due to train load using the beam-plate finite element model on elastic foundation. The results show that support layer type has great impact on both support layer deformation and the stress on subgrade, but has little impact on the bending stress of either track slab or support layer. The continuous support layer type, and articulated support layer type with shear transfer device at their ends, are recommended. In order to keep the stress in the support layer less than that in track slab, the modulus of the continuous, unit, and articulated types of support layer （ in unit twin-block ballastless track）, and the support layer in continuous twin-block ballastless track, should not be larger than 15, 22, 20.5 and 5 GPa, respectively. In addition, the modulus of the unit-type support layer should not be more than 20 GPa, to ensure the step in support layer remains less than 1 mm.

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.

弹性地基上矩形薄板问题的Hamilton正则方程及解析解

利用辛算法求出弹性地基上矩形薄板问题的解析解,将弹性地基视为双参数弹性地基,直接从弹性矩形薄板的控制方程推导出了问题的Ham ilton正则方程,为求出任意边界条件下问题的理论解奠定了基础,并且通过算例验证了文中所采用方法的正确性.

弹性矩形板问题的Hamilton正则方程

为了采用辛算法求出弹性矩形板问题的解析解,文中直接从弹性矩形板的控制方程出发推导了弹性矩形板,其中包括弹性矩形薄板和厚板问题以及弹性地基上矩形薄板和厚板问题的Hamilton正则方程,为利用辛几何方法求出任意边界条件下这类问题的理论解奠定了基础.

双参数弹性地基上矩形薄板自由振动问题的Hamilton方法

运用矩阵多元多项式的带余除法把双参数弹性地基上矩形薄板的振动方程转化为Hamilton系统,利用分离变量法给出对应的Hamilton算子.通过计算得到对边简支问题所对应Hamilton算子的本征值和本征函数系,并证明了该本征函数系的辛正交性和在Cauchy主值意义下的完备性.根据本征函数系的完备性,得到对应Hamilton系统的通解,进而给出双参数弹性地基上对边简支矩形薄板问题振型函数的通解.此外,通过两个例子说明此方法可以计算出自由振动问题的频率和振型函数.

Influence of backfilling rate on the stability of the"backfilling bodyimmediate roof"cooperative bearing structure

To reduce the cost of backfilling coal mining and utilize the underground space of coal mines,a new backfilling mining method with low backfilling rate called constructional backfilling coal mining(CBCM)is proposed.The "backfilling body-immediate roof" cooperative bearing structure of CBCM is analyzed by establishing the model of the medium thick plate on an elastic foundation.The influence of the backfilling rate on the stability of overlying strata is analyzed by the numerical simulation experiment.The control effect of CBCM is verified by a physic similar simulation test.The economic benefit of CBCM is analyzed.The conclusions are:the deformation characteristics of the immediate roof and critical backfilling spacing in CBCM can be analyzed based on the Hu Haichang’s theory.Exerting the bearing capacity of the immediate roof is beneficial to the stability of the overlying strata.The CBCM has a good control effect on the overburden in Xinyang Mine when the backfilling rate is lower than 25%.The backfilling cost of per ton coal is 37.39 yuan/t when the backfilling rate is 13.7%,with a decrease rate of 56.63%than the full-filling.The research results can provide theoretical support for the application of CBCM in coal mining.

SEMI-ANALYTICAL SOLUTION FOR FREE VIBRATION OF THICK FUNCTIONALLY GRADED PLATES RESTED ON ELASTIC FOUNDATION WITH ELASTICALLY RESTRAINED EDGE

This paper deals with free vibration analysis of functionally graded thick circular plates resting on the Pasternak elastic foundation with edges elastically restrained against translation and rotation.Governing equations are obtained based on the first order shear deformation theory（FSDT） with the assumption that the mechanical properties of plate materials vary continuously in the thickness direction.A semi-analytical approach named differential transform method is adopted to transform the differential governing equations into algebraic recurrence equations.And eigenvalue equation for free vibration analysis is solved for arbitrary boundary conditions.Comparison between the obtained results and the results from analytical method confirms an excellent accuracy of the present approach.Afterwards,comprehensive studies on the FG plates rested on elastic foundation are presented.The effects of parameters,such as thickness-to-radius,material distribution,foundation stiffness parameters,different combinations of constraints at edges on the frequency,mode shape and modal stress are also investigated.