BEAM STIFFNESS MATRIX BASED ON THE ELASTICITY EQUATIONS

A new general beam stiffness matrix which accounts for bending, torsion and shear deformation is derived from an elasticity solution of the beam. The influence of shear and torsion is considered using a 3×3 matrix of deformation coefficients. Numerical examples are presented to demonstrate the behaviour of the deformation coefficients and of the beam stiffness matrix. © 1997 by John Wiley & Sons, Ltd.     

MIXED ELEMENTS IN THE SENSITIVITY ANALYSIS OF STRUCTURES

In this paper is developed the sensitivity analysis of structures based on mixed elements. It examines the sensitivity of static, dynamic and stability behaviour of structures. The results are compared with analytical and displacement finite element solutions. The advantages and disadvantages of mixed elements in sensitivity analysis of structures are discussed with reference to applications.     

Viscoelastic Timoshenko beam theory

The concept of elastic Timoshenko shear coefficients is used as a guide for linear viscoelastic Euler-Bernoulli beams subjected to simultaneous bending and twisting. It is shown that the corresponding Timoshenko viscoelastic functions now depend not only on material properties and geometry as they do in elasticity, but also additionally on stresses and their time histories. Possible viscoelastic definitions are formulated and evaluated. In general, the viscoelastic relations are sufficiently complicated so that the elastic-viscoelastic correspondence principle (analogy) cannot be applied. This is particularly true for, but not limited to, elastic shear coefficients which are Poisson ratio dependent. Expressions for equivalent viscoelastic Timoshenko shear functions must, therefore, be derived de novo on a case by case basis, taking in to account specific relaxation moduli, stresses, temperatures and their time histories. Thus the elastic simplicity and generality is lost and hence rendering the use of viscoelastic Timoshenko shear functions as highly impractical. Consequently, it is necessary to directly solve the coupled viscoelastic beam governing relations for bending and twisting deflections by using appropriate solution protocols as discussed herein.     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

FREE VIBRATION ANALYSIS OF KIRCHHOFF PLATES RESTING ON ELASTIC FOUNDATION BY MIXED FINITE ELEMENT FORMULATION BASED ON GÂTEAUX DIFFERENTIAL

The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger–Reissner and Hu–Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C⁰ class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.     

On the dynamic behaviour of the Timoshenko beam finite elements

First, various finite element models of the Timoshenko beam theory for static analysis are reviewed, and a novel derivation of the 4 × 4 stiffness matrix (for the pure bending case) of the superconvergent finite element model for static problems is presented using two alternative approaches: (1) assumed-strain finite element model of the conventional Timoshenko beam theory, and (2) assumed-displacement finite element model of a modified Timoshenko beam theory. Next, dynamic versions of various finite element models are discussed. Numerical results for natural frequencies of simply supported beams are presented to evaluate various Timoshenko beam finite elements. It is found that the reduced integration element predicts the natural frequencies accurately, provided a sufficient number of elements is used. The research reported herein is supported by theOscar S. Wyatt Endowed Chair.     

Timoshenko梁能量守恒逐步积分算法

该文提出了Timoshenko梁非线性动力分析的能量守恒逐步积分算法。采用共旋技术考虑结构的几何非线性,空间离散采用相关插值形式,避免了剪切锁定现象。在时间离散时利用多参数修正方法对等效的节点动力平衡方程进行修正,实现了离散系统在保守荷载作用下的能量守恒。算法具备二阶局部精度,与已有的平均加速度方法和隐式中点方法相比,具有更好的数值稳定性。在二维情形下与Simo方法对比,指出了Simo方法在受保守外弯矩作用时系统能量不守恒。最后,通过三个数值模拟算例验证了算法的性能和能量守恒特性。     

广义协调元方法的收敛性

本文从弹性力学平面问题理论及分区广义势能原理入手,说明广义协凋元方法的基本理论,讨论了广义协调元方法的特点;并证明了按边协调和周协调条件构造的广义协调元的解的收敛性与唯一性。     

Timoshenko梁厚薄通用降阶单元的构造及其自适应求解

该文以含有弹性地基的Timoshenko梁和Euler梁为模型,提出厚薄梁通用降阶单元,采用泛函变分法构造了内置的厚薄梁通用的误差估计器,并构建了自适应有限元计算方法。该降阶单元继承了降阶单元在其他应用领域中的优点,对于厚薄梁,均可以给出按最大模度量的、满足用户指定的误差限的降阶单元解答。文中给出一系列典型算例以验证本法的可行性、有效性和可靠性。     

Reliability issues in hybrid optoelectronic interconnect systems

New technologies such as optical interconnects have been developed to realize very high speeds and massive data throughput. However, key reliability problems must be solved before commercialization occurs. Addressing the reliability issue, this paper presents the performance degradation of hybrid optoelectronic interconnect systems due to coefficient of thermal expansion mismatch between the optoelectronic component and the microlens and low cycle thermal fatigue of flip-chip solder bond between the photodiode array and the receiver electronic circuitry.