GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.     

Coupled refined layerwise theory for dynamic free and forced response of piezoelectric laminated composite and sandwich beams

This work extends a previously presented coupled refined layerwise theory to dynamic analysis of piezoelectric laminated composite and sandwich beams. Contrary to most of the available theories, all the kinematic and stress boundary conditions are satisfied at the interfaces of the piezoelectric layers with the non-zero longitudinal electric field. Moreover, both electrical transverse normal strains and transverse flexibility are taken into account for the first time in the present theory. In the presented formulation a high-order polynomial, an exponential expression and a layerwise term containing the electric field are included in the describing expression of the in-plane displacement of the beam. For the transverse displacement, the coupled refined model uses a combination of continuous piecewise fourth-order polynomials with a layerwise representation of electrical unknowns. The electric field is also approximated as linear across the thickness direction of piezoelectric layers. One of advantages of the present theory is that the mechanical number of the unknown parameters is very small and is independent of the number of the layers. For validation of the proposed model, various free and forced vibration tests for thin and thick laminated/sandwich piezoelectric beams are carried out. For various electrical and mechanical boundary conditions, excellent correlation has been found between the results obtained from the proposed formulation with those resulted from the three-dimensional theory of piezoelasticity.     

An admissible form of an inelastic constitutive equation—I. General theory

From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.     

A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.     

Moderately large forced oblique vibrations of elastic- viscoplastic deteriorating slightly curved beams

Summary An integral equation formulation for the dynamic biaxial response of slightly curved elastic-viscoplastic beams is presented in the context of a multiple field analysis, which takes into account the geometrically nonlinear influence of moderately large deflections. Materials are considered in the regime of rate-dependent plasticity and are subjected to accumulated ductile damage. The latter is modeled by the growth of voids in the plastic zones of an initially porous elastic material. Inelastic defects of the material are considered in the linear elastic background beam by a second imposed strain field (eigenstrains). Geometrically nonlinear effects of large deflections under conditions of immovable supports are approximately taken into account. By inspection, they render another “strain field” to be imposed on the linear background beam. Superposition applies in the linear elastic background in an incremental formulation. Linear methods, as those based on Green's functions and Duhamel's integral, are used to account for the given loads as well as for the resultants of the imposed strain fields. The intensity and the distribution of the imposed strain fields are calculated incrementally in a time-stepping procedure. They are determined by the constitutive law and by application of the nonlinear geometric relations. The numerical procedure resulting from the multiple fields in the elastic background is illustrated for two cases: (1) a preloaded viscoplastic beam of rectangular cross section is subjected to oblique flexural vibrations when forced by a sinusoidal load, and (2) an I-beam with a prescribed initial curvature is severely impacted and thus driven into the plastic regime. Accepted for publication 22 November 1996     

考虑弹性地基切向抗力的变截面圆弧壳梁分析

本文根据弹性薄壳理论,考虑了弹性地基径向和切向抗力的影响,解决了变截面圆弧壳地基梁的难题,作者所提出的近似解析法和加权余量法、传递矩阵法较有限单元法简便.     

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.     

A coupled refined high-order global-local theory and finite element model for static electromechanical response of smart multilayered/sandwich beams

In the present study, a coupled refined high-order global-local theory is developed for predicting fully coupled behavior of smart multilayered/sandwich beams under electromechanical conditions. The proposed theory considers effects of transverse normal stress and transverse flexibility which is important for beams including soft cores or beams with drastic material properties changes through depth. Effects of induced transverse normal strains through the piezoelectric layers are also included in this study. In the presence of non-zero in-plane electric field component, all the kinematic and stress continuity conditions are satisfied at layer interfaces. In addition, for the first time, conditions of non-zero shear and normal tractions are satisfied even while the bottom or the top layer of the beam is piezoelectric. A combination of polynomial and exponential expressions with a layerwise term containing first order differentiation of electrical unknowns is used to introduce the in-plane displacement field. Also, the transverse displacement field is formulated utilizing a combination of continuous piecewise fourth-order polynomial with a layerwise representation of electrical unknowns. Finally, a quadratic electric potential is used across the thickness of each piezoelectric layer. It is worthy to note that in the proposed shear locking-free finite element formulation, the number of mechanical unknowns is independent of the number of layers. Excellent correlation has been found between the results obtained from the proposed formulation for thin and thick piezoelectric beams with those resulted from the three-dimensional theory of piezoelasticity. Moreover, the proposed finite element model is computationally economic.     

层合结构压电器件的机电耦合效应

压电传感器和致动器都可以看成是一种复合材料层合板结构，由压电材料层和非压电(弹性)材料层交替铺设而成。对于这类任意铺设的层合板悬臂梁结构，我们推导出了表示力学变形与外加电场之间耦合效应的解析表达式。进而，又推导出了两类(一类为单层压电-弹性层，另一类为双层压电-弹性层)层合型悬臂梁结构机电耦合性能的解析公式。在该机电耦合模型中，包括了两个压电常数d211和d222。此外，还建立了含压电材料的有限元算式，进行了实验测量。最后，通过比较解析解(包括考虑了d222参数的理论值和没有考虑d222参数的理论值)，实验值以及有限元计算结果，发现它们吻合得很好，而且考虑d222是十分必要的。     

An Exact Theory for Circular, End-Loaded, Anisotropic Beams of Narrow Rectangular Cross Section

The exact theory of linearly elastic beams developed by Ladevèze and Ladevèze and Simmonds is limited to prismaticbeams. Herein, the theory is extended to circular beams of narrow rectangular cross section, using the equations of plane stress for an anisotropic elastic body. Explicit formulas are given for the cross-sectional material operators that appear in the overall beamlike stress–strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized strain). The extension to circular beams is non-trivial, especially for full anisotropy, because the analogues of the Saint-Venant solutions, that are key in the exact theory of straight beams, are more complicated.     

Transverse Vibrations of Laminated Beams in Three-Dimensional Formulation

A new asymptotic method is proposed to describe the free and forced transverse vibrations of elastic laminated beams of arbitrary cross section using the three-dimensional elastic equations without additional hypotheses and constraints. For beams with layers of equal Poisson's ratio, the zero-order natural frequencies are equal to the natural frequencies predicted by classical beam theory based on the Bernoulli hypothesis. The method makes it possible to calculate the frequencies of free vibrations and amplitudes of forced vibrations with prescribed accuracy for the first natural modes __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 56–71, June 2005.     

Numerical analysis and elastic–plastic deformation behavior of anisotropically damaged solids

The present paper is concerned with the numerical modelling of the large elastic–plastic deformation behavior and localization prediction of ductile metals which are sensitive to hydrostatic stress and anisotropically damaged. The model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. The formulation relies on a multiplicative decomposition of the metric transformation tensor into elastic and damaged-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged configurations via associated metric transformations which allow for the interpretation as damage tensors. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach the constitutive framework is completed by different stress tensors, a yield criterion and a separate damage condition as well as corresponding potential functions. The evolution laws for plastic and damage strain rates are discussed in some detail. Estimates of the stress and strain histories are obtained via an explicit integration procedure which employs an inelastic (damage-plastic) predictor followed by an elastic corrector step. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. A variety of large strain elastic–plastic-damage problems including severe localization is presented, and the influence of different model parameters on the deformation and localization prediction of ductile metals is discussed.     

求解弹性梁的普遍化方法

介绍了一种求解弹性梁的新方法．该方法利用奇异函数与拉普拉斯变换相结合的方法导出弹性梁弯曲变形的普遍表达式，并利用边界条件确定约束力，对具有任意支承形式、受力状况和阶梯形状的静定或超静定梁具有普适性．     

基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。     

作大范围运动弹性梁刚—柔耦合动力学建模

利用弹性梁的变形理论和 Hamilton力学原理对作大范围运动弹性梁的刚 -柔耦合动力学建模理论进行了研究。分析了大范围运动对弹性梁的横向振动和纵向振动的影响 ,得到了大范围运动与弹性梁的中线耦合变形之间的耦合作用对该系统动力学性质有显著的影响 ,从而提出了作大范围运动弹性梁的刚柔耦合动力学模型     

Variational principles and optimal solutions of the inverse problems of creep bending of plates

It is shown that inverse problems of steady-state creep bending of plates in both the geometrically linear and nonlinear formulations can be represented in a variational formulation. Steady-state values of the obtained functionals corresponding to the solutions of the problems of inelastic deformation and elastic unloading are determined by applying a finite element procedure to the functionals. Optimal laws of creep deformation are formulated using the criterion of minimizing damage in the functionals of the inverse problems. The formulated problems are reduced to the problems solved by the finite element method using MSC.Marc software.     

Un modèle de stratifiés

A multi-particle model called M4-5n is proposed for the study of multi-layers having elastic and inelastic strains in the layers and non-continuous displacements at the interfaces between layers. We use Hellinger–Reissner's variational formulation to set up the model. Inelastic fields are supposed to be known in the modelling.     

A Numerical Method to Material and Geometric Nonlinear Analysis of Cable Structures

In this article, a numerical method is presented for computing the stiffness matrix and compatibility relations of cables under uniform distributed loads on any direction. Both the geometrical and material nonlinearities, including softening and yielding of material, are taken into account and the catenary cable element with gauss integration scheme is employed. The proposed formulation includes the effect of uniform distributed loads on any direction and 3D nodal forces to assess the geometric and possible material nonlinearity of cables. The derived equations are then applied to the analysis of cable structures. The accuracy of the responses obtained by the proposed method is evaluated by several benchmark solutions available in the literature. Results of numerical examples indicate the capability of the proposed stiffness matrix in prediction of the elastic and inelastic responses of cables. The proposed formulation is therefore recommended for cable elements to be used in the analysis of cable structures.     

Shakedown limits for reinforced beam structures under fluctuating loads

Shakedown of planar beam structures, with possible different reinforcements in the upper and lower layers of the beam members, subjected to quasi-statically as well as dynamically fluctuating loads is studied. With an assumption of elastic perfectly-plastic behaviour of the beams in bending, a reduced kinematic formulation for the safety factor determining shakedown limits is constructed, which is proved to be equivalent to (but simpler than) the original one, and ready for use in solving practical problems.     

层合结构压电器件的机电耦合响应

压电传感嚣和致动器都可以看成是由压电材料层和非压电(弹性)材料层交替铺设而成。对于这类任意铺设的层合板悬臂梁结构,推导出了表示力学变形与外加电场之间耦合效应的解析表达式。进而,又推导出了两类(一类为单层压电-弹性层。另一类为双层压电-弹性层)层合型悬臂梁结构机电耦合性能的解析公式。在该机电耦合模型中,包括了两个压电常数d211和d222。最后。通过比较解析解、实验值以及有限元计算结果,发现它们吻合得很好。