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NEW SOLUTION SYSTEM FOR PLATE BENDING AND ITS APPLICatION
板弯曲求解新体系及其应用

Zhong Wanxie,Yao Wei-an,
钟万勰
,姚伟岸

力学学报 , 1999,
Abstract: The governing equation for plate bending is biharmonic equation, the traditionalsolution methodology is the semi-inverse one that causes limitations. The Airy stress function isusually applied traditionally for plane elasticity problems, it also satisfies biharmonic equation.The analogy between plate bending and plane elasticity problems had been noticed long before,but their solution systems are different each other, and the analogy relationship had not been usedsystematically. In this paper, the analogy is set up and perfect further.The deflection w for plate bending correspond to the Airy stress function in plane elasticity,conversely, the displacements in plane elasticity correspond to two bending moment functionsop. I gb. in plate bending. Based on the analogy between plate bending and plane elasticity problems,Hamiltonian system can also be applied to plate bending problem, that is, the problem can be solvedin symplectic space that consists of curvatures and bending moment functions. So correspondingto the principle of minimum potential energy and the Hellinger-Reissuer variational principle forplane elasticity, the new variational principles of minimum complementary energy and the Pro-H-R in terms of bending moment for plate bending can be proposed respectively. Carrying outthe variations, a set of new governing equations and solution for plate bending clajssical theory ispresented.The new methodology presents the analytical solutions in plate strip via the methods ofseparation of variables and eigenfunction-vector expansion, it breaks through the limit of traditionalsemi-inverse solution. The new one for plate simply support on both sides is equivalent to the Levysingle trigonometrical series expansion method, but it is not the same as the classical semi-inverseone, is derived rationally and analytical. Therefore it can easily be applied to other lateral boundaryconditions, which is very difficult for semi-inverse method, such as the both sides free plate givenin this paper. The results show that the new methodology will have vast application vista.
STATISTICAL ANALYSIS OF BENDING STRESS INTENSITY FACTORS FOR CRACKED PLATE WITH UNCERTAIN PARAMETERS
具有随机参数的含裂纹板弯曲应力强度因子的统计分析

任永坚,朱位秋,丁浩江,胡海昌
力学学报 , 1991,
Abstract: This paper applies stochastic finite element method based on local averages of random fields to analyse statistical features of bending stress intensity factors for cracked plates with uncertain parameters. The stiffness matrix of a hybrid crack-tip singular element is first derived, then by use of first-order Taylor expansion the mean and variance of stress intensity factors are formulated. As an example the effects of uncertainties of Young's modulus. Possion's ratio and plate thickness on plate bending s...
Bending and Free Vibration Analysis of Isotropic Plate Using Refined Plate Theory  [PDF]
I.I. Sayyad
Bonfring International Journal of Industrial Engineering and Management Science , 2013, DOI: 10.9756/bijiems.4390
Abstract: In this paper, Trigonometric shear deformation theory is applied for bending and free vibration analysis of thick plate. In this theory in plane displacement field uses sinusoidal function in terms of thickness coordinate. It accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free surface conditions at the top and bottom surfaces of the plate. The theory obviates the need of shear correction factor like other higher order or equivalent shear deformation theories. Simply supported thick isotropic plate is considered for detail numerical study. Navier?s solution technique is employed for the analytical solution. The results are obtained for displacements, stresses and natural bending mode frequencies and compared with those of other refined theories and exact solution from theory of elasticity.
Analytical solution for multilayer plates using general layerwise plate theory  [PDF]
Vuksanovi? ?or?e M.,?etkovi? Marina
Facta Universitatis Series : Architecture and Civil Engineering , 2005, DOI: 10.2298/fuace0502121v
Abstract: This paper deals with closed-form solution for static analysis of simply supported composite plate, based on generalized laminate plate theory (GLPT). The mathematical model assumes piece-wise linear variation of in-plane displacement components and a constant transverse displacement through the thickness. It also include discrete transverse shear effect into the assumed displacement field, thus providing accurate prediction of transverse shear stresses. Namely, transverse stresses satisfy Hook's law, 3D equilibrium equations and traction free boundary conditions. With assumed displacement field, linear strain-displacement relation, and constitutive equations of the lamina, equilibrium equations are derived using principle of virtual displacements. Navier-type closed form solution of GLPT, is derived for simply supported plate, made of orthotropic laminae, loaded by harmonic and uniform distribution of transverse pressure. Results are compared with 3D elasticity solutions and excellent agreement is found.
Analytical solution for free vibrations of simply supported transversally inextensible homogeneous rectangular plate  [PDF]
Milan Batista
Physics , 2010,
Abstract: In article, the exact solution of sinusoidal loaded simply supported elastic transversally inextensible rectangular plate is given. The expressions for displacement and stress components are derived and asymptotic expansion with respect to plate thickness are present. The frequency factors for plate thickness to width ratio 0.01, 0.1, 0.2 and 0.4 and various ratios of plate length to width are given.
Semi-Analytical Solution Based on Strip Method for Buckling and Vibration of Isotropic Plate  [PDF]
Mohamed A. El-Sayad,Ahmed M. Farag
Journal of Applied Mathematics , 2013, DOI: 10.1155/2013/796274
Abstract: The present paper achieves a semianalytical solution for the buckling and vibration of isotropic rectangular plates. Two opposite edges of plate are simply supported and others are either free, simply supported, or clamped restrained against rotation. The general Levy type solution and strip technique are employed with transition matrix method to develop a semianalytical approach for analyzing the buckling and vibration of rectangular plates. The present analytical approach depends on reducing the strips number of the decomposed domain of plate without escaping the results accuracy. For this target, the transition matrix is expressed analytically as a series with sufficient truncation numbers. The effect of the uni-axial and bi-axial in-plane forces on the natural frequency parameters and mode shapes of restrained plate is studied. The critical buckling of rectangular plate under compressive in-plane forces is also examined. Analytical results of buckling loads and vibration frequencies are obtained for various types of boundary conditions. The influences of the aspect ratios, buckling forces, and coefficients of restraint on the buckling and vibration behavior of rectangular plates are investigated. The presented analytical results may serve as benchmark solutions for such plates. The convergence and efficiency of the present technique are demonstrated by several numerical examples compared with those available in the published literature. The results show fast convergence and stability in good agreement with compressions. 1. Introduction The buckling problem of a thin rectangular elastic plate subjected to in-plane compressive forces is important in the aircraft and automotive industries. Kumar Panda and Ramachandra [1] offered a brief historical review on this subject. Due to the additional complexity of achieving the analytical solution of the plate problems under nonclassical boundary conditions, analysis for the effect of the in-plane force and buckling of the plate becomes difficult. Several methods, such as Rayleigh-Ritz, finite element, finite difference, and Fourier series method, are available. Singh and Dey [2] discussed the transverse vibration of rectangular plates subjected to in-plane forces by a difference based on variational approach. Finite strip transition matrix method (FSTM) was used by Farag and Ashour [3, 4] as a numerical technique depending on Runge-Kutta method to solve the vibration of rectangular plate as an initial value problem. Consequently, the method has been improved and applied successfully for several problems of
An approximate analytical solution of free convection problem for vertical isothermal plate via transverse coordinate Taylor expansion  [PDF]
Sergey Leble,Witold M. Lewandowski
Physics , 2013,
Abstract: The model under consideration is based on approximate analytical solution of two dimensional stationary Navier-Stokes and Fourier-Kirchhoff equations. Approximations are based on the typical for natural convection assumptions: the fluid noncompressibility and Bousinesq approximation. We also assume that ortogonal to the plate component (x) of velocity is neglectible small. The solution of the boundary problem is represented as a Taylor Series in $x$ coordinate for velocity and temperature which introduces functions of vertical coordinate (y), as coefficients of the expansion. The correspondent boundary problem formulation depends on parameters specific for the problem: Grashoff number, the plate height (L) and gravity constant. The main result of the paper is the set of equations for the coefficient functions for example choice of expansion terms number. The nonzero velocity at the starting point of a flow appears in such approach as a development of convecntional boundary layer theory formulation.
Matched asymptotic solution to a class of singularly perturbed thin plate bending problem
一类奇摄动薄板弯曲问题的匹配渐近解

Xu Hui,Chen Li-Hua,Mo Jia-Qi,
徐惠
,陈丽华,莫嘉琪

物理学报 , 2011,
Abstract: The thin plate-bending problem is studied. Introducing the stretched variables, the internal layer solutions near the boundary are constructed for the fourth order singularly perturbed boundary problem. Then matching the solutions with outer solution and using the theory of the composite expansion, the asymptotic solution is obtained finally.
An Improved Model for Bending of Thin Viscoelastic Plate on Elastic Foundation  [PDF]
Zhi-Da Li, Ting-Qing Yang, Wen-Bo Luo
Natural Science (NS) , 2009, DOI: 10.4236/ns.2009.12014
Abstract: An improved model for bending of thin viscoe-lastic plate resting on Winkler foundation is presented. The thin plate is linear viscoelastic and subjected to normal distributed loading, the effect of normal stress along the plate thickness on the deflection and internal forces is taken into account. The basic equations for internal forces and stress distribution are derived based on the general viscoelastic theory under small deformation condition. The reduced equations for elastic case are given as well. It is shown that the proposed model reveals a larger flex-ural rigidity compared to that in classic models, in which the normal stress along the plate thickness is neglected.
Scattering of flexural waves in mindlin''s plates of soft ferromagnetic materials with a cutout
含孔软铁磁材料Mindlin板中弹性波散射问题

Hu Chao Han Gang Huang Wenhu,
胡超
,韩刚,黄文虎

力学学报 , 2004,
Abstract: The problem of elastic waveguide and dynamic stress concentrations in plates with a cutout is the important subject in solid mechanics. The cutout in structures has influence directly on the loading capacity and the lifetime of structures, therefore, some researchers have devoted to theoretical analysis and experimental research in the world. Considered dynamic stress concentration or intensity factors, the classical theory of thin plate has disadvantage. Thick plate theory proposed by Mindlin made up for the shortage classical theory of thin plate including the effect of transverse shear deformation and rotator inertia. The satisfying result is gained in engineering. In the 1960's, with wave function expansion method, Pao Yih-Hsing first studied the problem of the flexural wave scattering and dynamic stress concentrations in Mindlin's thick plates with circular cavity and gave an analytical solution and numerical results. With the development of modern science and technology, the ferromagnetic materials have been applied to superconduct nuclear power station and magnetic levitation trains. It has better physical and mechanical property. The stress on the contour of a cavity or crack in ferromagnetic materials may be increase in a uniform magnetic field. It has a influence on the carrying capacity and the lifetime of structures. According to the many references, the dynamical behavior of ferromagnetic elastic structures can be significantly affected by the presence of a uniform magnetic field. Based on the theory of magneto-elastic interaction, Japanese researchers analyzed scattering of flexural wave and the dynamic bending moment intensity factors in cracked Mindlin plates of ferromagnetic materials and gave numerical results. They used Fourier transforms to reduce the mixed boundary value problem to a Fredholm integral equation that can be solved numerically. In this paper, based on the equation of wave motion in Mindlin's plate of magneto-elastic interaction, using wave function expansion method, the scattering of flexural wave and dynamic stress concentrations in a plate of ferromagnetic materials with a cutout are investigated. According to analysis and numerical results, the magnetic induction intensity has great influence on the dynamic stress concentration factors at low frequency.
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