On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.     

On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates

A novel superposition method based on the symplectic geometry approach is presented for exact bending analysis of rectangular cantilever thin plates. The basic equations for rectangular thin plate are first transferred into Hamilton canonical equations. By the symplectic geometry method, the analytic solutions to some problems for plates with slidingly supported edges are derived. Then the exact bending solutions of rectangular cantilever thin plates are obtained using the method of superposition. The symplectic superposition method developed in this paper is completely rational compared with the conventional analytical ones because the predetermination of deflection functions, which is indispensable in existing methods, is dispelled.     

NONLINEAR BENDING OF SIMPLY SUPPORTED RECTANGULAR SANDWICH PLATES

In this paper,fundamental equations and boundary conditions of the nonlinearbending theory for a rectangular sandwicl plate with a soft core are derived by meansof the method of calculus of variations.Then the nonlinear bending for a simplysupported rectangular sandwich plate under the uniform lateral load is investigated byuse of the perturbation method and a quite accurate analytic solution is obtained.     

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.     

Nonlinear bending of simply supported symmetric laminated cross-ply rectangular plates

Based on the von Kármán-type theory of plates,nonlinear bending problems of simplysupported symmetric laminated cross-ply rectangular plates under the combined action ofpressure and inplane load are investigated in this paper.The solution which satisfies thegoverning equations and boundary conditions is obtained by using the double Fourier seriesmethod.     

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

An exact solution for the bending of point-supported orthotropic rectangular thin plates

A closed series solution is proposed for the bending of point-supported orthotropic rectangular thin plates. The positions of support points and the distribution of transverse load are arbitrary. If the number of simply supported points gradually increases the solution can infinitely approach to Navier's solution. For the square plate simply supported on the middle of each edge and free at each corner, the results are very close to the numerical solutions in the past.     

Accurate variational approach for free vibration of simply supported anisotropic rectangular plates

The Ritz method is one of the most elegant and useful approximate methods for analyzing free vibration of laminated composite plates. It is simple to use and also straightforward to implement. However, the Ritz method has its own difficulty in determining the natural frequencies of simply supported laminated anisotropic plates. This is caused by the fact that the natural boundary conditions in the vibration of anisotropic plates can never be exactly satisfied by a solution in a variables separable form. As a result, the calculated natural frequencies would be expected to converge to solutions a little higher than true ones. To overcome this difficulty, this paper presents a simple variational formulation with Ritz procedure in which all the natural boundary conditions are implemented in an averaging manner. It is revealed that the proposed method can produce lower upper bound solutions compared with the conventional Ritz method where the geometric boundary conditions can only be satisfied by the assumed deflection functions.     

Ultimate strength of postbuckling for simply supported rectangular composite thin plates under compression

1.IntroductionAsiswellthnown,thepostbucklingmetalplatescanbesuccessivelysubjectedtotheload.Forsomemetalplates,theultimatestrengthcanreachthirtytimesofthebucklingstressll].Inordertostudythebearingcapacityofthecompositeplatesthroughtestsandtheoreticalanalysis,f'ttiluretestsofthe283specimensofGFRPthinrectangularplatesundercompres.sionarecarriedout.ItisprovedthattheGFRPplatesinpostbucklingstillhavehigherbearingcapacity,forthetestedspecimensshowedultimatestrengthwhichisfourtotwenty-livetiniesth…     

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method （SSDQM）. The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model （ALM） is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.     

THE BENDING OF THIN RECTANGULAR PLATES WITH MIXED SUPPORTED SEGMENTS OF STRAIGHT EDGES

ＴＨＥＢＥＮＤＩＮＧＯＦＴＨＩＮＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＳＷＩＴＨＭＩＸＥＤＳＵＰＰＯＲＴＥＤＳＥＧＭＥＮＴＳＯＦＳＴＲＡＩＧＨＴＥＤＧＥＳＣｈｅｎＬｉ－ｚｈｉ（陈立志）（ＱｉｎｈｕａｎｇｄａｏＰｏｒｔＡｕｉｈｏｒｉｔｙ,Ｑｉｎｈｕａｎｇｄａ...     

Bending problem of rectangular plates of unidirectional nonhomogeneity and variable thickness with two opposite edges simply supported under arbitrary distributed loads

Using the step reduction method^[1,2] suggested by the first author of this paper, we investigate the problem indicated in the title and obtain the stepped approximate solutions. As an example, the case of a square plate of linearly varying thickness with four edges simply supported under linearly distributed loads is calculated. The obtained results agree well with those given in [3] and thus the exactness of the new method is verified.Projects Suppotred by the Science and Technic Fund of the National Education Committee of the People's Republic of China.Although this paper has published in Journal of Lanzhou University, Special Number of Mechanics, No. 1, (1979) (in Chinese), it republishes here in order to correct some mistakes due to calculation.     

The simply supported rectangular plate: An exact,three dimensional,linear elasticity solution

An exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness is presented within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a pointwise manner and is in the form of a double Fourier sine series. The classical Navier solution for the problem is shown to be the limit of the present solution as the plate thickness aspect ratio approaches zero. It is noted that the solution presented provides a benchmark against which approximate theories of transversely loaded plates may be measured. The new elasticity solution also provides a heuristic basis for a novel theory of thick plates of arbitrary planform and edge support recently given by the author.     

三边简支一边自由混凝土矩形薄板的热弯曲

基于薄板的小挠度理论和叠加原理,考虑横向变温情况,将温度作用下的三边简支一边自由矩形薄板看作是面内温差作用下的四边简支矩形薄板和自由边上挠度作用下的三边简支一边自由矩形薄板的叠加,得到了温度作用下三边简支一边自由混凝土矩形薄板的挠度和弯矩解析解.首先通过在自由边界上试设具有待定参数的挠度函数,采用李维解法推导出三边简支一边自由矩形薄板在自由边界挠度作用下的挠度方程;其次利用横向变温作用下四边简支矩形薄板的求解得到待定参数;再采用叠加原理得出横向变温作用下三边简支一边自由矩形薄板的挠度和弯矩解析解;最后利用MATLAB编制程序得到了横向变温作用下三边简支一边自由矩形薄板的计算系数用表,为工程结构中三边简支一边自由混凝土矩形薄板在热环境下的设计计算提供了理论依据.     

On the frequency response of moderately thick simply supported rectangular plates with arbitrary lamination

A here-to-fore unavailable analytical solution to the boundary-value problem of free vibration response of arbitrarily laminated moderately thick rectangular plates, subjected to an admissible boundary condition is presented. A novel boundary continuous solution approach is employed to solve the highly coupled governing partial differential equations that arise from the implementation of the Yang-Norris-Stavsky (1966) theory incorporating first-order shear deformation, and rotary and in-plane inertias. The numerical results presented for various parametric effects to study first lowest seven eigenvalues, and corresponding eigenvectors, can be capitalized as bench-marks for a future comparison. An eight-node isoparametrics shear-flexible element is considered to validate its performance with respect to the present solutions.     

Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading

This paper presents the analytical solutions for the bending of orthotropic rectangular thin plates by the double finite integral transform, which, as an effective tool in solving plate problems, should have received attention. As a representative and difficult problem in the theory of plates, free plates’ bending is successfully solved to demonstrate the accuracy of the method by comparing the present analytical solutions with those from the literature as well as those by the finite element method. With the proper integral transform kernels, the proposed solution procedure is applicable to the bending of orthotropic rectangular plates with all combinations of simply supported, clamped and free boundary conditions, which serves as an elegant approach to analytical solutions of plate bending problems.     

多层正交异性矩形薄板弯曲振动稳定解析解

构造了带有补充项的双重正弦傅里叶级数通解来求解各种边界条件的多层正交各向异性矩形薄板的弯曲、振动和稳定问题.将坐标轴取在中性面上,求出用挠度表示的应力表达式,然后由横截面上每单位宽度的应力合成板的内力;再将层合板的内力代入板的平衡方程中得到板的控制方程,将多层板的物理参数折算为等价的单层板物理参数;最后联立控制方程与边界条件,求得未知量的系数并代入本文的通解中.本文的通解不需要叠加即可求解各种边界条件的板的弯曲、振动和稳定问题;现有的对于单层板的研究都可以用本文的方法拓展到多层板领域;对于复杂边界条件的板,也可以使用该通解分析.