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.     

Nonlinear free vibration analysis of piezoelastic laminated plates with interface damage

This paper presents a nonlinear model for piezoelastic laminated plates with damage effect of the intra-layers and inter-laminar interfaces. Discontinuity of displacement and electric potential on the interfaces are depicted by three shape functions. By using the Hamilton variation principle, the three-dimensional nonlinear dynamic equations of piezoelastic laminated plates with damage effect are derived. Then, by using the Galerkin method, a mathematical solution is presented. In the numerical studies, effects of various factors on the natural frequencies and nonlinear amplitude-frequency response of the simply-supported peizoelastic laminated plates with interfacial imperfections are discussed. These factors include different damage models, thickness of the piezoelectric layer, side-to-thickness ratio, and length-to-width ratio.     

Static flexural analysis of elliptic Reissner–Mindlin plates on a Pasternak foundation by the triangular differential quadrature method

In this paper, static flexural analysis of elliptic Reissner–Mindlin plates resting on a Pasternak foundation is conducted using the triangular differential quadrature method (TDQM). A triangular serendipity transformation is proposed to map a curvilinear domain onto a unit isosceles right triangle. The applicability of the TDQM to problems on domains with curvilinear boundaries is improved significantly since no inner points are needed in the transformation. In the case of thin circular plates, the results of the present method are in excellent agreement with those of the available exact solution, demonstrating the effectiveness of the present method. Elliptic plates with various aspect ratios and thickness-to-width ratios under uniformly distributed load are studied.     

Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation

Nonlinear free vibration of symmetrically laminated magneto-electro-elastic rectangular plate resting on an elastic foundation is studied analytically. The plate is considered to be simply supported on all edges. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To model the motion of the plate, the first order shear deformation theory along with the Gauss's equations for electrostatics and magnetostatics are used. Then equations of motion are reduced to a single nonlinear ordinary differential equation which is solved analytically by multiple scales method. The results are compared with the published results and good agreement is found. Some numerical examples are presented to investigate the effects of several parameters on the linear and nonlinear behavior of these plates.     

基于局部移动Kriging无网格方法的层合板自由振动分析

利用基于局部移动Kriging插值无网格法对层合板自由振动进行了数值分析,基于一阶剪切层合理论导出了层合板振动的控制方程和边界条件,进一步得到了自由振动的离散化特征方程。由于Kriging插值函数具有Kronecker delta函数性质,可以直接施加本质边界条件。通过本文给出的方法,对不同边界条件、不同跨厚比、不同材料参数和铺设角度的层合板的振动频率进行了计算,均得到满意结果。最后用该方法对层合板的铺设角度进行优化设计,得到了与已有文献完全一致的优化结果。数值结果充分表明了无网格Kriging方法分析层合板自由振动问题的有效性和高精确度。     

NONLINEAR THREE-DIMENSIONAL ANALYSIS OF COMPOSITELAMINATED PLATES

ＮＯＮＬＩＮＥＡＲＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＡＬＹＳＩＳＯＦＣＯＭＰＯＳＩＴＥＬＡＭＩＮＡＴＥＤＰＬＡＴＥＳ￥(江晓禹，张相周，陈百屏)ＪｉａｎｇＸｉａｏｙｕ；（ＳｏｕｔｈｗｅｓｔｅｒｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ６...     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。     

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

I.Intr0ducti0nThenon1inearvibrationprob1emsofshellsofrevolutionarealwaysofgreatdifficultyandofgreatvaluetostudyfortheircomplexityinmathematicsandmechanicsaswellasinwideapplications.ManyinvestigatorshavemaderesearchontheseinoneWayoranother,butfewinvolvedth…     

新修正偶应力层合板理论中的几个基本假设对自由振动计算的影响

基于新修正偶应力理论,在对微细观尺度的复合材料层合梁/板进行力学响应计算时,往往采用一系列假设来简化模型。现有文献都全部或部分应用了这些假设,但对这些假设是否会对计算结果造成影响尚未进行充分讨论分析。本文建立了未经简化的新修正偶应力Reddy层合板模型,并对其自由振动进行了分析。通过数值算例的对比,讨论了常用的几个简化假设对微细观复合材料四边简支方板自振频率的影响以及适用范围。算例结果表明,常用的几个简化假设对于微尺度层合薄板自由振动的影响很小,对于厚板的低阶频率影响也很小,但对厚板的高阶频率影响显著。     

复合材料层合板壳的非线性热动态响应分析

本文基于高阶剪切变形理论，考虑同天应变和横向剪切应变的影响，对受热复合材料层合板壳的非线性热动态响应进行分析，计及了转动惯量的影响，给出了通用性较好的Ｃ＾０类有限元公式，文中数值算同现有文献和三维有限元计算结果进行了比较，证明了本文方法的精确、有效性，文中还就层合板的边界条件、纵厚比及铺设角度对非线性热动态响应的影响进行了分析。     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

对称角铺设复合材料层合梯形板振动分析的一种有效方法

本文给出了对称角铺设复合材料层合梯形板振动特性分析的一种有效方法，可方便地处理具有内部点支承及各种边界支承时矩形、平行四边形、三角形、等腰梯形与非等腰梯形层合板的横向弯曲振动问题．     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam

The quadrature method (OM) has been used in structural analysis only in recent years. In this study, OM is applied to flexural vibration analysis of a geometrically nonlinear beam. The numerical results by OM agree with the results by the finite element method. It is believed that this is the first attempt to solve a nonlinear dynamic problem by the quadrature method.     

基于逆复合二次径向基函数的对称复合材料层合板自由振动分析

利用逆复合二次径向基函数无网格配点法对Reddy的高阶剪切变形理论进行离散,预测了对称复合材料层合板的自由振动特性.将不同材料参数、几何尺寸和边界条件的层合板固有频率计算结果与相关文献中的结果进行对比,结果表明:逆复合二次径向基函数在对称复合材料层合板自由振动分析方面具有收敛性好及精度高等一系列优点.     

Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method

The generalized differential quadrature method (GDQM) is employed to consider the free vibration and critical speed of moderately thick rotating laminated composite conical shells with different boundary conditions developed from the first-order shear deformation theory (FSDT). The equations of motion are obtained applying Hamilton’s concept, which contain the influence of the centrifugal force, the Coriolis acceleration, and the preliminary hoop stress. In addition, the axial load is applied to the conical shell as a ratio of the global critical buckling load. The governing partial differential equations are given in the expressions of five components of displacement related to the points lying on the reference surface of the shell. Afterward, the governing differential equations are converted into a group of algebraic equations by using the GDQM. The outcomes are achieved considering the effects of stacking sequences, thickness of the shell, rotating velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes indicate that the rate of the convergence of frequencies is swift, and the numerical technique is superior stable. Three comparisons between the selected outcomes and those of other research are accomplished, and excellent agreement is achieved.     

变截面欧拉梁自由振动分析的重采样微分求积法

采用重采样微分求积法求解了变截面欧拉梁的自由振动问题。推导了变截面梁的控制方程离散格式,采用重采样矩阵方法对边界条件进行处理,给出了变截面梁自由振动算法。采用本文方法对不同类型截面形式和不同边界条件的变截面梁进行自由振动分析,并和其他解法进行比较。计算结果表明,本文方法可以适用于不同变截面类型和不同边界条件,计算精度与解析解吻合良好,具有良好的收敛性能。在同等精度条件下网格点数少于现有计算方法。重采样转换矩阵边界处理方法相比于传统边界处理方法具有更快的收敛性能。     

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.