An analytical solution for in-plane free vibration and stability of loaded elliptic arches

In-plane free vibration and stability analyses of elliptic arches subjected to a uniformly distributed vertical static loading are performed here. A variational principle is applied to derive the governing equations for free vibration and stability of preloaded arches, considering the effect of the extensibility of the arch centerline but neglecting the effect of shear deformation. Particular attention is given to present a general procedure for combining series solutions with stiffness matrixes to construct an analytical solution for free vibration and stability of loaded arches with varying curvature. The correctness of the proposed solution is verified through a convergence study on the vibration frequencies of a loaded circular arch and by comparing the results with published data. The solution is further applied to investigate the behaviors of clamped or fixed-free elliptic arches.     

矩形中厚板自由振动问题的辛本征展开定理

研究了矩形中厚板自由振动问题导出的一个Hamilton算子的本征值问题.在广义位移与内力构成的混合边界条件下,首先求解了相应算子的本征函数.接着,证明了本征函数系的完备性,这为使用分离变量法求解相应问题提供了可行性.最后,根据文中证明的展开定理获得了问题的一般解,并给出了具体的数值算例.     

矩形中厚板自由振动问题的哈密顿体系与辛几何解法

以矩形中厚板的胡海昌方程为基础,将中厚板自由振动问题导入哈密顿体系,然后利用辛几何中的分离变量和本征函数展开的方法求出了对边简支板自由振动的精确解．文中采用的辛方法不必事先人为地引入试函数,而是通过完全理性的推导,从而突破了传统半逆解法的限制,使得问题的求解更加合理,易于推广．计算实例证明了本文推导结果的正确性．     

Axisymmetric vibration of bimodulus thick circular and annular plates

The fundamental natural frequencies of axisymmetric circular and annular plates subjected to a combination of a pure bending stress and extensional stress in the plane of the plate are investigated. The thick plate ring element model which includes the effect of transverse shear deformation is created for axisymmetric free vibration problems. The obtained results of non-dimensional natural frequency coefficient compared with the closed form solutions for ordinary plates are shown to be very good. The effects of various parameters on the natural frequency and neutral surface locations are studied. The bimodulus properties are shown to have significant influence on the natural frequency.     

Post-buckling of linearly tapered,moderately thick circular plates by finite element method

The post-buckling behaviour of linearly tapered, moderately thick circular plates subjected to an external compressive load is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and central thickness to radius ratio. Both simply supported and clamped boundary conditions are considered in the present study.     

An element for static, vibration and buckling analysis of thick laminated plates

A conforming finite element formulation of the equations governing composite multilayered plates using Reddy's higher-order theory is presented. The element has eight degrees of freedom, u₀, v₀, w, ∂w/∂x, ∂w/∂y, ∂²w/∂x∂y, γ_x, γ_y, per node. The transverse displacement of the present element is described by a modified bicubic displacement function while the in-plane displacements and shear-rotations are interpolated quadraticly. The element is evaluated for its accuracy in the analysis of static, vibration, and buckling of anisotropic rectangular plates with different lamination schemes and boundary conditions. The conforming finite element described here for the higher-order theory gives fairly accurate results for displacements, stresses, buckling loads, and natural frequencies.     

Post-buckling behaviour of moderately thick circular plates with edges elastically restrained against bending rotation

A finite element formulation is presented to study the post-buckling behaviour of circular plates with the effects of shear deformation and elastic restraints against bending rotation at the edges of the plates considered. A set of numerical results are included which clearly bring out variation of post-buckling characteristics with the varying values of thickness of the plate and the stiffness of the spring supports.     

Asymmetric vibration analysis of thick composite circular cylindrical shells with variable thickness

Thick isotropic and composite shells of revolution are analyzed for vibration characteristics. The theory used is that proposed by Naghdi, which includes the thickness normal strain and shear deformation. A curved-semianalytical finite element is used to solve the problem. Parametric studies are conducted to study the effect of various parameters on the vibration behavior of shells. The effect of thickness variation along the axial direction with a constraint on mass is also studied.     

An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures

In the present work, an efficient facet shell element for the geometrically nonlinear analysis of laminated composite structures using the corotational approach is developed. The facet element is developed by combining the discrete Kirchhoff-Mindlin triangular bending element (DKMT), and the optimal membrane triangular element (OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix is formulated. Using corotational formulation and the proposed facet element, some example laminated composite structures with geometric nonlinearity are analyzed, and the results are compared with those found using other facet elements.     

Effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates

The effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates is studied in this paper. Finite element formulation is employed to obtain the non-linear to linear period ratios for some rectangular plates. A conforming finite element of rectangular shape wherein the effects of shear deformation and rotatory inertia are included, is developed and used for the analysis. Results are presented for both simply supported and clamped boundary conditions.     

粘弹性夹层圆板自由振动的理论解

研究了粘弹性夹层圆板的自由振动特性.基于经典弹性薄板理论和Kelvin-Voigt粘弹性本构方程,建立了粘弹性夹层圆板振动控制方程.采用分离变量法导出了粘弹性夹层圆板的自然频率及振型解析表达式,计算了固支和简支粘弹性夹层圆板的自然频率,并与有限元计算结果进行比较;讨论了粘弹性夹层圆板的夹心层比率对自然频率及衰减系数的影响.研究表明：（1）随着夹心层厚度的增大,系统频率先增大后减小,高阶时该趋势表现更为明显;（2）随着夹心层厚度的增大,衰减系数一直增大,高阶时该趋势表现更为明显.     

A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges

On the basis of three-dimensional elasticity, this paper presents a free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries. The analysis is based on a recursive solution suitable for three-dimensional vibration analysis of simply supported plates. Clamped boundary conditions are imposed by suppressing the edge displacements of a number of planes which are parallel to the mid-plane. This is achieved by coupling a number of different vibration modes of the corresponding simply supported plate using a Lagrange multiplier method. A satisfactory solution can be obtained by choosing suitably larger numbers of the coupled vibration modes and the fixed planes across the thickness of the plate. Numerical results are presented to show the convergence of the solution. Results are also obtained for either isotropic or cross-ply laminated plates having different combinations of simply supported and clamped boundaries.     

HLXVIB-A computer code for free vibration analysis of circular helicoidal bars

The development and details of the computer code HLXVIB are presented as a viable means to assess the dynamic structural response of prismatic circular helicoidal bars. To this end, the development of the dynamic transport (transfer) matrix T^RL_RL is first outlined for a typical helicoidal segment. Then, the implementation of the resulting matrix in the program HLXVIB is discussed to emphasize particular attractive features of the program. Several parametric studies are presented to illustrate the application of the program to studying the vibration characteristics of circular helicoidal bars as one type of the more general class of three-dimensional bars.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

多跨下承式拱桥面内全局动力学理论与自由振动研究

为弥补高维工程结构有限元分析对结构参数优化设计等的局限,本文基于欧拉伯努利梁理论与传递矩阵法,研究大跨度拱桥面内自由振动问题.首先,基于多跨拱桥的刚度分析,建立系统的全局动力学模型.其次,基于传递矩阵法建立系统的全局动力学理论,最后,以一座四跨下承式拱桥的平面力学模型,求解其面内自由振动时的固有频率与振型,并将所得结果与用同样参数建模的有限元分析结果对比,证明本文所建理论对求解该类问题的有效性与精确性.此外,通过整桥各跨矢跨比、吊杆截面面积、拱肋惯性矩等的多组参数组合分析了该系统面内自振频率的分布规律.结果表明：随着拱桥矢高增大,系统固有频率减小,因拱桥质量的快速增加,对整桥面内刚度影响显著;增大吊杆截面面积可在一定程度内增大拱桥的面内刚度,导致系统频率在一定范围内增大,且观察到Veering现象.     

圆弧拱的面内非线性动力学分析

对圆弧拱面内自由振动的频率和非线性内共振现象进行了研究.通过假设满足位移边界条件的振型函数对圆弧拱的面内非线性方程进行Galerkin一阶模态截断,先对派生系统的线性频率进行求解,就不同参数对频率的影响进行分析;然后利用多尺度法对离散模型进行求解.结果表明:截面高度对径向振动频率有显著影响,但是对切向振动频率影响很小;另外,就本文方程而言,可能存在三种内共振形式,但内共振现象的产生与否还决定于圆弧拱的边界条件.     

Post-buckling analysis of moderately thick elastic annular plates

Post-buckling behaviour of moderately thick elastic annular plates is studied in this paper using a simplified finite element formulation. Radial load ratios for various radii ratios and thickness parameters are presented in the form of tables bringing out clearly the effect of radii ratios and thickness parameters.     

Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.     

Post-buckling behaviour of moderately thick cylindrically orthotropic annular plates

A finite element formulation including the effects of shear deformation and cylindrically orthotropic material properties is described for studying the post-buckling behaviour of annular plates. Numerical results for the buckling load parameter and ratios of nonlinear load parameter to buckling load parameter for various values of orthotropic properties, thicknesses and radii ratios of the plates are presented.