A frequency dependent rectangular plate finite dynamic element for high precision transverse vibration analysis of simply supported plates

The need for adding to the many existing rectangular plate elements is justified and the inclusion of the twist parameter, in the degrees of freedom selected, is shown to be essential. Using a rational choice of parameters to form the deflection functions for a simply supported plate element, a constant term, sixteen degrees of freedom rectangular element is developed which is shown to predict exactly the first mode frequency and mode shape when used in whole plate modelling. The properties of this ‘dynamic’ element are combined with the properties of an existing ‘static’ element to obtain the frequency dependent properties of a ‘blended’ element. It is demonstrated that in order to obtain high accuracy in eigenvalue determination it is necessary to use three separate modellings of the plate using a square element and two rectangular elements which are oriented orthogonally. From 3 × 3 and 2 × 5 modelling, it is shown that the frequencies of the first 20 modes of a simply supported square plate, resulting from matrix eigenvalue determinations are obtainable within one per cent of the exact values. Modes 11 and 18 are determined exactly. Mode shape distortions associated with frequency errors are examined and shown to be minimal for the first 20 modes.     

Numerical solution of cracked thin plates subjected to bending, twisting and shear loads

A semi-analytical method namely fractal finite element method is presented for the determination of mode I and mode II moment intensity factors for thin plate with crack using Kirchhoff's theory. Using the concept of fractal geometry, infinite many of finite elements is generated virtually around the crack border. Based on the analytical global displacement function, numerous degrees of freedom (DOF) are transformed to a small set of generalised coordinates in an expeditious way. The stress intensity factors can be obtained directly from the generalized coordinates. No post-processing and special finite elements are required to develop for extracting the stress intensity factors. Examples of cracked plate subjected to bending, twisting and shear loads are given to illustrate the accuracy and efficiency of the present method. The influence of finite boundaries on the calculation of the moment intensity factors is studied in details. Very accuracy results when compare with the theoretical and numerical counterparts are found.     

Three-dimensional absolute nodal co-ordinate formulation: plate problem

In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no infinitesimal or finite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate coordinates. Using this interpretation of the plate coordinates the new method does not require the use of co-ordinate transformation to define the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The stiffness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental finite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain. © 1997 John Wiley & Sons, Ltd.     

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.     

A simple method for including shear deformations in thin plate elements

A method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution. It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications. This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.     

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C¹ approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.     

Dynamics of the bilinear Mindlin plate element

Plate and shell finite elements based upon Mindlin plate theory and evaluated by one-point quadrature are attractive because of their effectiveness for both thick and thin shells, at minimal computing cost. However, these elements are rank-deficient and must be stabilized to obtain reliable behaviour. Methods for achieving full rank of the stiffness and for stabilizing element behaviour in static analysis and in explicit dynamic calculations exist and are quite effective. This paper addresses the remaining issue of controlling spurious modes of response in vibration analysis and implicit dynamic solutions. Several alternatives for the element mass formulation are examined in detail. We show that non-physical dynamic modes present a potential problem with most mass matrix formulations, and that spurious modes other than the familiar hourglassing motion are possible. A combination of projection methods and reduced quadrature is suggested which eliminates these deficiencies and produces accurate numerical results. The remaining techniques investigated give rise to anomalous behaviour which make them unsuitable for general use.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

爆炸载荷作用下双向加筋方板的大挠度塑性动力响应

从分别列出加筋板面板以及加强筋的运动方程出发,分析了爆炸载荷作用下双向加筋固支方板的大挠度塑性动力响应。分析表明:取决于加强筋的相对刚度以及爆炸载荷峰值的大小,加筋板的运动将呈现3种不同的模式。该文限于讨论加筋板的总体变形模式,具体讨论了十字加筋以及双十字加筋固支方板在忽略弯矩影响下的薄膜解法。理论结果与已有的试验结果在多数情况下符合良好,表明该文提出的简化理论分析方法能对爆炸载荷下双向加筋方板的永久变形做出较为合理的预报。     

Deformation and failure of blast-loaded square plates

Numerical results for clamped, thin square steel plates subjected to blast loading are presented. The numerical analysis is based on a finite element formulation, which includes the nonlinear effects of geometry and material as well as strain rate sensitivity. A phenomenological interactive failure criterion comprising bending, tension and transverse shear is proposed to predict the various modes of failure. A node release algorithm is developed to simulate the progression of plate rupture from the boundary. The analysis is continued in the post-failure phase to account for the free flight deformation of the torn plate. The predicted failure modes for a blast-loaded plate are presented and compared with previously published experimental data.     

Buckling and Post-Buckling Behaviors of a Variable Stiffness Composite Laminated Wing Box Structure

The buckling and post-buckling behaviors of variable stiffness composite laminates (VSCL) with curvilinear fibers were investigated and compared with constant stiffness composite laminates (CSCL) with straight fibers. A VSCL box structure was evaluated under a pure bending moment. The results of the comparative test showed that the critical buckling load of the VSCL box was approximately 3% higher than that of the CSCL box. However, the post-buckling load-bearing capacity was similar due to the layup angle and the immature status of the material processing technology. The properties of the VSCL and CSCL boxes under a pure bending moment were simulated using the Hashin criterion and cohesive interface elements. The simulation results are consistent with the experimental results in stiffness, critical buckling load and failure modes but not in post-buckling load capacity. The results of the experiment, the simulation and laminated plate theory show that VSCL greatly improves the critical buckling load but has little influence on the post-buckling load-bearing capacity.     

On the dynamic stress intensity factors in finite cracked bodies under harmonic loading

A method of computing stress intensity factors under harmonic loading is suggested. The method is based on the representation of the stress intensity factors in form of superposition of the modal stress intensity factors corresponding to the normalized free vibration modes with some weight coefficients. The motion equations are built with the help of the finite element method. The singularity of stresses in the crack tip is taken into account using special finite elements. To demonstrate the accuracy and possibilities of the method, the dependence of the stress intensity factors of the first and second kind on frequency in square plate with inclined central crack under harmonic extension-compression was calculated.     

Exact analytical solutions for bending of rectangular plates with a partial internal line support

This paper studies the behavior of uniformly loaded rectangular thin plates with a partial internal line support. The highlight of the problem is that the analytical formulation explicitly considers the moment singularities that occur at the tips of partial internal line supports. The proper finite Hankel transform is used to transform a pair of dual-series equations obtained from the mixed conditions along the partial internal line support to a single Fredholm integral equation. Numerical results concerning deflection, bending moment, resultant forces, and bending-stress intensity factors are given for a square plate. Some results are also compared with the case of a square plate without partial internal line support.     

Coupled use of reduced integration and non-conforming modes in quadratic Mindlin plate element

The applications of the reduced integration technique, and the addition of non-conforming modes and their coupling to Mindlin plate elements to improve their basic behaviour are reviewed and the establishment of a series of new plate elements by combined use of these schemes is presented in this paper. The element formulation is based upon the quadratic Mindlin plate concept. The results obtained by new elements converge to the exact solutions very rapidly as the mesh is refined and show reliable solutions even for severely distorted meshes. The new elements have the requisite numbers of zero eigenvalues associated with rigid body modes to avoid spurious zero energy modes. These elements are shown to solve the shear locking problem completely so that they are applicable to a wide range of plate problems, giving a high accuracy for both thick and thin plates.     

大型刚体惯性参数识别的三线扭摆系统实验方法改进研究

精确获取汽车动力总成刚体惯性参数是发动机悬置系统设计的重要前提之一。利用三线扭摆法测量刚体单轴转动惯量精度较高的特点,基于表面固定点确定刚体方位的三点定位方法和测量6个~9个不同方位的多次测量原理,发展了一套适合于大型复杂刚体的惯性参数识别方法。关键技术有:(1)选取刚体表面三个定位点定义一个刚体随动坐标系以描述刚体方位;(2)通过测量刚体定位点至托盘表面参考点(定义一个整体坐标系)的距离,计算出定位点在整体坐标系下的坐标和两个坐标系之间的转换关系;(3)求出各组实验中在动坐标系下的刚体转轴方位和转动惯量;(4)运用最小二乘原理,求解多个转轴的最优交点得到动坐标系下的刚体质心坐标,求解由转动惯量转轴定理导出的线性方程组得到刚体惯性矩阵。实验方法中容易引起误差的环节较多,但是可以根据最小二乘原理进行逐级误差估计和控制。通过误差分析、长方体质量块实验验证和大量的汽车动力总成惯量参数识别实验,证明了该方法的实用性和可靠性。     

基于并行计算的三自由度并联机床动力学解析模型

基于运动学分析、凯恩动力学方程和数字-符号方法,建立了三自由度并联机床的动力学解析模型.将广义坐标、构件的质量和转动惯量处理为符号量.将动力学模型矩阵的推导问题转化为特定条件下运用运动学和动力学计算公式求解驱动力的问题,由计算机自动生成动力学模型矩阵中的各元素的实时代码.文中引入了一种新的标量矩阵与矢量矩阵的乘法运算,研究了广义坐标和构件的质量对驱动力的影响规律.构造了动力学解析模型的并行算法,节省了计算时间.给出了动力学模型矩阵元素的实时代码生成和驱动力矩与参数关系的具体数值实例.     

Plate end debonding in the constant bending moment zone of plated beams

Reinforced concrete (RC) beams strengthened in flexure by externally bonding fibre reinforced polymer (FRP) or steel plate on their tension face are susceptible to premature plate end debonding failures. Safe design of such a strengthened RC beam demands a reliable and predictive debonding strength model. There are two special cases of plate end debonding failures: flexural debonding for cases when the plate terminates within a constant bending moment region (CMR), and shear debonding for the case when the plate terminates where the shear force is large but the bending moment is minimal. A general plate end debonding case is usually considered as an interaction between these two special cases. This paper is concerned with flexural debonding. A brief review of existing models is presented before the plate end interfacial stresses are examined. Three new models with different levels of accuracy are then developed: a closed-form theoretical model based on a simplified interfacial fracture mechanics analysis; a semi-empirical model; and a wholly empirical model. These three models together with two existing models are assessed against a carefully constructed test database containing 67 test data from an extensive literature survey.     

Plate end debonding in the constant bending moment zone of plated beams

Reinforced concrete (RC) beams strengthened in flexure by externally bonding fibre reinforced polymer (FRP) or steel plate on their tension face are susceptible to premature plate end debonding failures. Safe design of such a strengthened RC beam demands a reliable and predictive debonding strength model. There are two special cases of plate end debonding failures: flexural debonding for cases when the plate terminates within a constant bending moment region (CMR), and shear debonding for the case when the plate terminates where the shear force is large but the bending moment is minimal. A general plate end debonding case is usually considered as an interaction between these two special cases. This paper is concerned with flexural debonding. A brief review of existing models is presented before the plate end interfacial stresses are examined. Three new models with different levels of accuracy are then developed: a closed-form theoretical model based on a simplified interfacial fracture mechanics analysis; a semi-empirical model; and a wholly empirical model. These three models together with two existing models are assessed against a carefully constructed test database containing 67 test data from an extensive literature survey.     

Pure modes for elastic waves in crystals: mathematical modeling and search

A universal search method of pure longitudinal and pure transverse modes for elastic wave propagation in crystals, in general piezoelectrics, is presented. A mathematical model of pure modes for elastic waves based on adiabatic state equations for an arbitrary anisotropic piezoelectric medium and its equation of motion under elastic deformations in the rotated Cartesian coordinates is constructed. The condition for longitudinal normals is that all non-diagonal matrix elements of the effective elastic stiffness coefficients in the corresponding wave equation are equal to zero. Equating to zero non-diagonal elements only in one row of this matrix, one can obtain the condition for transverse normals. A computer program is prepared which allows to find the pure modes for elastic waves in crystals and to calculate their characteristics when symmetry class, elastic, piezoelectric, dielectric constants, and crystal density are known.     

Acceleration sensitivity of crystal resonators affected by the mass and location of electrodes

Predominant thickness-shear frequencies and modes of a crystal plate with electrodes of arbitrary shape and mass distribution are obtained by a finite-element method, based on Mindlin's first-order equations with platings. These frequencies and modes are used in a perturbation method for computing the acceleration sensitivity of crystal resonators with electrodes. Computations are made for a square AT-cut quartz plate that is supported by a four-point mount and coated with identical square and uniform electrodes on the upper and lower faces of the plate. To study the effect of uneven distribution of electrode mass, acceleration sensitivities are calculated when a small mass is added at various locations near the edges of the square electrodes. It is found that the percent increase of the acceleration sensitivity of the resonator with a small added mass to that of the resonator without added mass ranges from 3.8% to 541.7%, depending on the location of the small mass placed at the edges of the electrodes.