Large deflection analysis of rectangular plates by combined perturbation and finite strip method

The perturbation method and finite strip method are combined to solve the largedeflection bending problems of rectangular plates.Perturbation method is used to reducethe nonlinear differential equations into a series of linear differential equations.The finitestrip method is then employed to tackle these linear equations.Some calculation examplesare compared with those got by other methods.     

Karman型正交异性矩形薄板非线性弯曲的统一求解方法

本文对Karman型四边支承正交异性薄板在5种不同边界条件下的几何非线性弯曲进行了统一分析。所设的位移函数均为梁振动函数。它们精确地满足边界条件，利用Galerkin方法和位移函数的正交属性，转换控制方程为非线性代数方程。用“稳定化双共轭梯度法”求解稀疏矩阵线性方程组以及“可调节参数的修正迭代法”求解非线性代数方程组，最后给出了相应的数值结果。     

Nonlinear bendings of symmetrically layered anisotropic rectangular plates

This paper has studied the nonlinear bending of symmetrically layered anisotropic rectangular plates under various supports. The uniformlys valid N-order asymptotic solutions of the deflection and stress function are derived by the singular perturbation methods offered in [1]. The analysis and calculations are given for simply and clamped supported, rectangular plates subjected to combined edge tensions and lateral loading in conjunction with the modified Galerkin procedure (a method of weighted residuals).     

Validation of the Rayleigh–Ritz method for the postbuckling analysis of rectangular plates with application to delamination growth

Postbuckling solutions of laminated rectangular plates are obtained by the Rayleigh–Ritz method using von Karman’s nonlinear strain displacement relations and high-order polynomial expansions of the displacements. The potential energy function and the nonlinear algebraic equations governing the undetermined coefficients are obtained by Mathematica. Reasonably accurate solutions for the membrane forces, the bending and twisting moments and the pointwise energy-release rates generally require more undeterminated coefficients. Such refined postbuckling solutions show significant non-uniformity of the in-plane forces and strains and certain boundary effects characterized by concentration of the curvatures and the bending moment.     

Nonlinear bending behavior of orthotropic Mindlin plate resting on orthotropic Pasternak foundation using GDQM

Rectangular plates resting on elastic foundations are operational activities of large transportation aircraft on runways, footings, foundation of spillway dam, civil building in cold regions, and bridge structures. Hence, in the present work, nonlinear bending analysis of embedded rectangular plates is investigated based on orthotropic Mindlin plate theory. The elastic medium is simulated by orthotropic Pasternak foundation. Adopting the nonlinear strain–displacement relation, the governing equations are derived based on energy method and Hamilton’s principle. The generalized differential quadrature method is performed for the case when all four ends are clamped supported. The influences of the plate thickness, shear-locking, elastic medium constants, and applied force on the nonlinear bending of the rectangular plate are studied. Results indicate that increasing the plate thickness decreases the deflection of the plate. It is also observed that increasing the applied force increases the deflection of the plate. Furthermore, considering elastic medium decreases deflection of the plate, and the effect of the Pasternak-type is higher than the Winkler-type on the maximum deflection of the plate. Also, it is found that the present results have good agreement with previous researches.     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

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.     

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads in thermal environments

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads is investigated. A new micro-macro-mechanical model of unit cells is suggested. In this model, a 3D braided composite may be considered as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate. The material properties of the epoxy are expressed as a linear function of temperature. Based on Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, analytical solutions for nonlinear bending behavior of simply supported 3D braided rectangular plates are obtained using mixed Galerkin-perturbation method. The numerical examples concern effects of geometric parameters, of fiber volume fraction, braiding angle and load boundary condition.     

Vibration and stability of hybrid plate based on elasticity theory

The governing equations of elasticity theory for natural vibration and buck- ling of anisotropic plate are derived from Hellinger-Reissner＇s variational principle with nonlinear strain-displacement relations. Simply supported rectangular hybrid plates are studied with a precise integration method. This method, in contrast to the traditional finite difference approximation, gives highly precise numerical results that approach the full computer precision. So the results for natural vibration and stability of hybrid plates presented in the paper can be riewed as approximate analytical solutions. Furthermore, several types of coupling effects such as coupling between bending and twisting, and coupling between extension and bending, when the layer stacking sequence is asymmetric, are considered by only one set of governing equations.     

Magnetoelastic bending and stability of soft ferromagnetic rectangular plates

I.IntroductionSofar,twokindsofphenomenaofmagnetoelasticinteractiontoferromagneticplatesinappliedmagneticfieldshavebeenfound.Oneisthemagnetoelasticbuchlingorillstabllitytoacantileveredbeam-plateinatransversemagneticfieldt'],thatisthephen1Onlenollot'negiltivemagneticstiffness,whiletheotheristheincreaseofnaturalfrequencyoftheplatewhichisfreelyviratinginanin-planemagneticfieldl=],thatisthephenomenonofpositivemagneticstiffness.Inordertosimulatethesetwokindsofexperimentalphenomena,sometheoreticalmo…     

弹性地基上四边自由矩形厚板非线性大挠度弯曲

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｌｉｎ一阶剪切变形板理论,采用摄动－Ｇａｌｅｒｋｉｎ混合法,给出双参数弹性地基上四边自由矩形中厚板在对称分布局部荷载作用下的大挠度弯曲渐近解,满足全部自由边界条件和控制方程,同时讨论弹性地基刚度系数对自由矩形厚板大挠度弯曲的影响。     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

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.     

Bending problems of rectangular Reissner plate with free edges laid on tensionless Winkler foundations

In this paper,an analytical method for solving the bending problems of rectangularReissner plate with free edges under arbitrary loads laid on tensionless Winkler foundationsis proposed.By assuming proper form of Fourier series with supplementary terms,whichmeet derivable conditions,for deflection and shear force functions,the basic differentialequations with given boundary conditions can be transformed into a set of simple infinitealgebraic equations.For common Winkler foundations,this set of equations can be solveddirectly and for tensionless Winkler foundations,it is a set of weak nonlinear algebraicequations,the solution of which can be obtained easily by using iterative procedures.     

Reciprocal theorem method for solving the problems of bending of thick rectangular plates

ＲＥＣＩＰＲＯＣＡＬＴＨＥＯＲＥＭＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＰＲＯＢＬＥＭＳＯＦＢＥＮＤＩＮＧＯＦＴＨＩＣＫＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＳＦｕＢａｏ－ｌｉａｎ（付宝连）Ｔａｎｗｅｎ－ｆｅｎｇ（谭文锋）（ＹａｎｓｈａｎＵｎｉｒｅｒｓｉｔ...     

Effects of transverse shear on the nonlinear bending of rectangular plates laminated of bimodular composite materials

This paper investigates the application of Dynamic-Relaxation (DR) method to the problems of nonlinear bending of rectangular plates laminated of bimodular composite materials. The classical lamination theory and a shear deformation theory of layered composite plates, taking account of large rotations (in the von Karman sense) are employed separately to analyze the subject. It has been found here that the estimation of the fictitious densities which control the convergence and numerical stability of nonlinear DR solution considering transverse shear effect still needs to be further investigated. In this paper, a procedure to calculate fictitious densities has been presented; hence the numerical stability of this topic has been ensured. In this paper the main steps of solving the nonlinear bending of bimodular composite laminates by means of DR method are outlined. The numerical results are given for simply supported, two-layer cross-ply rectangular plates made of mildly bimodular material (Boron-Epoxy (B-E)) and highly bimodular materials (Aramid-Rubber (A-R) and Polyester-Rubber (P-R)) under sinusoidally distributed and uniformly distributed transverse loads. The results obtained have been compared with linear results and those obtained for laminates fabricated from conventional composite materials, the elastic moduli of which are identical with the tensile moduli of the bimodular materials. In addition, the effect of transverse shear deformation on the nondimensionalized center deflection has been studied.The main contents of this paper were presented at the International Symposium of Composite Materials and Structures (June 1986, Beijing).The authors thank Prof. Zhou Li for his guidance.     

Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series

The bending of a cantilever rectangular plate is a very complicated problem in thetheory of plates.For a long time,there have been only approximate solutions for thisproblem by energy methods and numerical methods.since 1979,Prof.F.V.Chang of Tsing Hua University obtained,by the method ofsuperposition,a series of analytic solutions for cantilever rectangular plates under uniformload and concentrated load.In this paper,the two-direction trigonometric series is used to obtain the solution forthe bending of cantilever rectangular plates under uniform load.The obtained results arecompared with the results by the method of superposition.The comparison shows that theresults of these two methods are in good agreement,hence they are mutually confirmed to becorrect.     

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.     

基于Lenosky原子作用势单层石墨烯片的力学模型

基于对Lenosky 碳-碳共价键作用势连续化得到的单层石墨烯的势能和Hamilton 原理,导出了单层石墨烯的动力学方程. 使用该数学模型及Galerkin 方法,研究了矩形单层石墨烯片的静力挠曲问题. 结果显示,石墨烯片的几何尺寸较小时,弯曲刚度对结构的受力影响较大,可用板理论来描述;随着结构尺寸的增大,弯曲刚度的影响迅速降低;当矩形石墨烯片的短边尺寸大于10 nm 时,可以忽略弯曲刚度对结构的影响,使用薄膜理论来描述单层石墨烯的力学性质.