变截面压电层合梁自由振动分析

考虑压电材料的质量效应和刚度效应,将表面粘贴或埋入式压电悬臂梁看作变截面梁,研究压电材料对智能结构固有特性的影响。基于一阶剪切变形理论导出压电层合梁的抗弯刚度和横向剪切刚度,计及梁的剪切变形和转动惯量,采用Timoshenko理论推导变截面压电层合梁的频率方程。给出了T300/970压电层合梁和硬铝压电层合梁的前3阶固有频率,并和有限元结果、等截面梁的计算结果进行比较。计算表明,压电材料对压电结构固有频率和固有振型的影响显著,在以振动控制为目标的压电结构动力学建模过程中,有必要考虑压电材料的质量和刚度。     

Vibration analysis of rotating composite beams using a finite element model with warping degrees of freedom

A finite element beam formulation that properly takes into account the warping of the cross-sections has been extended to the free vibration analysis of rotating and nonrotating composite beams. The formulation allows transverse shear deformation and the warping effects are incorporated by superimposing warping displacements that are parallel to the beam axis in the deformed configuration. For modeling of thin and moderately thick walled sections, the strain is assumed to be linear through the wall thickness. Numerical tests were conducted to calculate the natural frequencies of cantilever composite beams with various ply layups. Correlations of the calculated natural frequencies with experimentally measured values demonstrate the validity of the present approach. Although only rectangular solid and box beams were considered for numerical tests, the formulation allows modeling of beams with complicated cross-sections, tapers, pretwists and arbitrary planforms.     

Dynamic behavior of cross-ply laminated beams with piezoelectric layers

The dynamic behavior of cross-ply non-symmetric composite beams, having uniform piezoelectric layers is analysed. A first-order Timoshenko type analysis is applied to obtain the equations of motion, which include shear deformation, rotary inertia, bending-stretching coupling terms and induced axial strains caused by the piezoelectric material. Using the principle of virtual work, the coupled equations of motion and the relevant boundary conditions are obtained. For a laminated beam having uniform piezoelectric layers the induced strains appear only in the boundary conditions yielding time dependent ones. Therefore, a special procedure involving orthogonality of the coupled Timoshenko type natural vibrational modes of the beam is applied to help understanding of the dynamic behavior of the non-symmetric laminated beam and to investigate the influence of the induced strains (by the piezoelectric layers) on the dynamic behavior while keeping an ‘open-loop’ control. Typical types of laminates and piezoelectric materials are used to calculate natural frequencies and mode shapes. Numerical results for various parameters of laminated beams are presented to stress the better applicability and suitability of the present approach to the analysis of dynamic behavior of laminated composite beams with piezoelectric layers.     

Reddy高阶组合梁的非线性有限元分析

该文基于Reddy高阶梁理论,提出了小变形双层组合梁的隐式运动学假定;应用拉格朗日乘子法,将该隐式关系引入到组合梁的最小势能原理,得到了考虑各子梁和粘结滑移层非线性材料特性的高阶组合梁非线性位移法有限单元,且该单元可以容易地转化为非线性Timoshenko和Euler-Bernoulli组合梁有限单元。随后,该研究分别应用提出的Reddy、Timoshenko和Euler-Bernoulli组合梁有限单元对双跨连续钢-混凝土组合梁进行了准静力分析,考察剪切效应对组合梁构件的挠度、粘结层滑移和截面应力的影响,且参数分析了组合梁的跨高比对剪切效应的影响。参数分析表明:短粗组合梁结构往往表现出显著的剪切效应,Newmark假定不再适用。     

C beam elements based on the Refined Zigzag Theory for multilayered composite and sandwich laminates

The paper deals with the development and computational assessment of three- and two-node beam finite elements based on the Refined Zigzag Theory (RZT) for the analysis of multilayered composite and sandwich beams. RZT is a recently proposed structural theory that accounts for the stretching, bending, and transverse shear deformations, and which provides substantial improvements over previously developed zigzag and higher-order theories. This new theory is analytically rigorous, variationally consistent, and computationally attractive. The theory is not affected by anomalies of most previous zigzag and higher-order theories, such as the vanishing of transverse shear stress and force at clamped boundaries. In contrast to Timoshenko theory, RZT does not employ shear correction factors to yield accurate results. From the computational mechanics perspective RZT requires C⁰-continuous shape functions and thus enables the development of efficient displacement-type finite elements. The focus of this paper is to explore several low-order beam finite elements that offer the best compromise between computational efficiency and accuracy. The initial attention is on the choice of shape functions that do not admit shear locking effects in slender beams. For this purpose, anisoparametric (aka interdependent) interpolations are adapted to approximate the four independent kinematic variables that are necessary to model the planar beam deformations. To achieve simple two-node elements, several types of constraint conditions are examined and corresponding deflection shape-functions are derived. It is recognized that the constraint condition requiring a constant variation of the transverse shear force gives rise to a remarkably accurate two-node beam element. The proposed elements and their predictive capabilities are assessed using several elastostatic example problems, where simply supported and cantilevered beams are analyzed over a range of lamination sequences, heterogeneous material properties, and slenderness ratios.     

《薄壁Timoshenko梁弯扭耦合振动的动态有限元法》

通过直接求解单对称均匀薄壁Timoshenko梁单元弯扭耦合振动的运动微分方程,推导了其精确的动态刚度矩阵。在本文研究中考虑了弯扭耦合、翘曲刚度、转动惯量和剪切变形的影响。针对某弯扭耦合的薄壁梁算例,应用本文推导的动态刚度矩阵,采用自动Muller法和结合频率扫描法的二分法求解频率特征方程,计算了该薄壁梁的固有特性,并讨论了翘曲刚度、剪切变形和转动惯量对该弯扭耦合薄壁梁的固有频率和模态形状的影响。数值结果验证了本文方法的精确性和有效性,并指出随着模态阶次的增加,剪切变形、转动惯量和翘曲刚度对薄壁梁的固有特性的影响更加显著。     

考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁解析解

通过在Newmark 模型中引入(1)描述横向非均匀分布的纵向位移的翘曲形函数和(2)描述钢梁腹板剪切变形的Timoshenko 梁假定,建立了一个能考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁模型,并推导了均布荷载作用下的解析解。最后通过4 个算例验证了模型和解析解的正确性和适用性,并显示了考虑组合梁剪切变形的必要性。另外,算例还表明,在组合梁的三维有限元建模中采用Timoshenko 梁单元来考虑钢梁的剪切变形会导致显著的误差。     

Deformation of composite beams under transverse impact loading

Results of experiments on determining the contact force with transverse impact on carbon-filled plastic beams are presented. The problem of transverse impact on beams is stated and solved based on the finite-element method using the Timoshenko shear theory. The Runge-Kutta-Felberg method is used for integrating the main equation of the finite-element method. Analysis of the results of experiments and calculations shows that the form, magnitude, and duration of the contact force of impact depend on elastic and strength characteristics of the contact surface and are determined by the spectrum of natural frequencies of vibrations. Maximum nonsteady deformations in bending and shear are realized for beams, the first natural frequencies of which are comparable with the impact pulse duration. The existence of time-intermediate peaks of nonsteady bending and shear deformations in various sections of the beam, as well as span-intermediate maxima of bending and shear deformations, makes it possible to state and solve the problem of optimizing the structure and form of composite beams in relation to the external nonsteady action.Translated from Problemy Prochnosti, No. 2, pp. 114–119, February, 1990.     

An accurate laminated element for piezoelectric smart beams including peel stress

This paper presents a laminated element for piezoelectric (PZT) smart beams in taking into account peel stresses. In the finite element analysis (FEA) formulation, a coupled electrical and mechanical beam element is used to model PZT patches, and a conventional structural element is used to model a host beam. A continuous adhesive element with shear and peel stiffness is derived to form a PZT laminated element. For a smart beam with a partially bonded PZT patch or distributed PZTs, the laminated element is applied to an area of the host beam with PZTs and the conventional element is used in the host beam where no PZT is bonded. A novel PZT laminated element is firstly derived based on the Timoshenko beam theory, in which the FEA formulation based on the Euler-Bernoulli beam theory can be considered as its special case. FEA numerical results of static and dynamic analyses based on the Euler-Bernoulli beam theory are compared with the exact static and dynamic solutions to validate the present FEA formulation. The present FEA framework based on the Timoshenko beam theory is then used to investigate the effects of PZT debondings on static behaviors and dynamic responses, and an original and effective procedure for detecting debondings in PZT actuators or sensors is proposed.The authors are grateful to the support of the Australian Research Council through a Large Grant Scheme (Grant No. A10009074).     

A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates

A new 4-node quadrilateral finite element is developed for the analysis of laminated composite plates containing distributed piezoelectric layers (surface bonded or embedded). The mechanical part of the element formulation is based on the first-order shear deformation theory. The formulation is established by generalizing that of the high performance Mindlin plate element ARS-Q12, which was derived based on the DKQ element formulation and Timoshenko’s beam theory. The layerwise linear theory is applied to deal with electric potential. Therefore, the number of electrical DOF is a variable depending on the number of plate sub-layers. Thus, there is no need to make any special assumptions with regards to the through-thickness variation of the electric potential, which is the true situation. Furthermore, a new “partial hybrid”-enhanced procedure is presented to improve the stresses solutions, especially for the calculation of transverse shear stresses. The proposed element, denoted as CTMQE, is free of shear locking and it exhibits excellent capability in the analysis of thin to moderately thick piezoelectric laminated composite plates.     

Enhanced magnetoelectric effect in multiferroic composite beams due to flexoelectricity and transverse deformations

This paper is concerned with the exploration of the role of transverse normal and shear deformations on enhancing the magnetoelectric (ME) coefficient of multiferroic bilayer composite beams composed of a piezoelectric layer and a piezomagnetic layer. Analytical models have been derived based on the displacement field which accounts for both the transverse normal and shear deformations, Timoshenko beam theory and Euler Bernoulli beam theory. The induced flexoelectricity in the piezoelectric layer due to axial strain gradient and transverse shear strain gradient has also been taken into consideration for estimating the ME coefficient. It has been found that the contribution of transverse normal strain in the piezoelectric layer for enhancing the ME coefficient is significantly larger than that due to axial strain, transverse shear strain and flexoelectricity. For the particular values of the thicknesses of the piezoelectric layer and the piezomagnetic layer, the ME coefficient remains invariant for both thick and thin multiferroic composite beams.     

Finite element analysis and parametric study of steel-concrete composite beams

A finite element model is presented to predict the stresses and deformations in steel-concrete composite beams. The model takes into account the effect of cracking and tension-stiffening in the tensed concrete, and of longitudinal slip between the steel beam and the concrete slab due to the ‘partial interaction’ of theconnectors. Some comparisons with experimental data available in literature are reported to validate the efficiency of the proposed model. Finally, a parametric study was done to investigate the effects of the geometric and mechanical variables as boundary conditions and the slip modulus of the connectors.     

Free vibrations of a multi-span Timoshenko beam carrying multiple spring-mass systems

Structural elements supporting motors or engines are frequently seen in technological applications. The operation of a machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass system and Bernoulli-Euler multi-span beams carrying multiple spring-mass systems are plenty, but on Timoshenko multi-span beams carrying multiple spring-mass systems is fewer. This paper aims at determining the natural frequencies and mode shapes of a Timoshenko multi-span beam. The model allows to analyse the influence of the shear effect and spring-mass systems on the dynamic behaviour of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Timoshenko multi-span beam calculated by using secant method for non-trivial solution are compared with the natural frequencies of multi-span beam calculated by using Bernoulli-Euler Beam Theory (EBT) in literature; the mode shapes are presented in graphs.     

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.     

复合材料梁的几何精确非线性建模及非经典效应

基于Hodges的广义Timoshenko梁理论对具有任意剖面形状、任意材料分布及大变形的复合材料梁进行几何精确非线性建模,采用旋转张量分解法计算梁内任意一点的应变,采用变分渐近法确定梁剖面的任意翘曲,采用平衡方程由二次渐近精确的应变能导出广义Timoshenko应变能,采用广义Hamilton原理建立梁的几何精确非线性运动方程。将所建模型用于复合材料梁的静动力分析,通过与实验数据的对比,验证了建模方法的准确性,并进一步研究了剖面翘曲及横向剪切变形非经典效应对复合材料梁的影响。研究表明,剖面翘曲对复合材料梁的静变形和固有频率有显著影响,横向剪切变形对复合材料梁的静变形和固有频率的影响与梁的长度/剖面高度比有关。     

A higher order plate theory for dynamic stability analysis of delaminated composite plates

A higher order shear deformation theory is used to investigate the instability associated with delaminated composite plates subject to dynamic loads. Both transverse shear and rotary inertia effects are taken into account. The procedure is implemented using the finite element method. Delamination is modeled using the penalty parameter approach. The natural frequencies are computed and compared with NASTRAN 3D results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions, plate thickness and boundary conditions. As expected the natural frequencies and the critical buckling load of the delaminated plate are lower than those of the nondelaminated plate. They decrease with increase in delamination length. Increase in delamination length causes instability regions to be shifted to lower parametric resonance frequencies and the normalized width of the instability regions to increase.     

Optimum Design of Laminated Composite Plates Undergoing Large Amplitude Oscillations

The optimum design of composite laminated plates under going large amplitude free vibration is discussed. Von Karman's nonlinear strain displacement relations are considered to account for large amplitude. A higher order shear deformation theory with parabolic variation of transverse shear stresses through thickness is used in the finite element formulation. A nine-noded isoparametric element with 7 dof per node is adopted. Ritz formulation for nonlinear finite element analysis is implemented and the direct iteration method is used to solve the governing nonlinear equation. Optimization is carried out using genetic algorithm (GA) with tournament selection scheme.     

含分层损伤复合材料加筋层合板的动承载能力

采用有限元方法研究了含穿透分层损伤复合材料加筋层合板的动力响应和承载能力。根据复合材料层合板一阶剪切理论, 推导了复合材料层合板单元的刚度阵和质量阵列式;同时采用Adams 应变能法与Rayleigh阻尼模型相结合的方法, 构造了相应的阻尼阵列式;为了防止在低阶模态中分层处出现的上、下子板不合理的嵌入现象, 建立了含分层损伤复合材料加筋层合板动力分析中分层分析模型和虚拟界面联接模型。并采用Tsai提出的刚度退化准则和动力响应分析的精细积分法, 对在动荷载作用下含分层损伤复合材料加筋层合板结构进行了破坏和承载能力分析。通过典型算例分析, 分别讨论了外载频率、分层深度、筋的位置以及破坏过程中刚度退化对含损伤复合材料加筋层合板动力响应特征和承载能力的影响, 得到了一些具有理论和工程价值的结论。     

Shear deformation effect in nonlinear analysis of spatial composite beams subjected to variable axial loading by bem

Summary In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account the shear deformation effect. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

Coupled deflection analysis of thin-walled Timoshenko laminated composite beams

For the deflection analyses of thin-walled Timoshenko laminated composite beams with the mono- symmetric I-, channel-, and L-shaped sections, the stiffness matrices are derived based on the solutions of the simultaneous ordinary differential equations. A general thin-walled composite beam theory considering shear deformation effect is developed by introducing Vlasov’s assumptions. The shear stiffnesses of thin-walled composite beams are explicitly derived from the energy equivalence. The equilibrium equations and force-deformation relations are derived from energy principles. By introducing 14 displacement parameters, a generalized eigenvalue problem that has complex eigenvalues and multiple zero eigenvalues is formulated. Polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to the equilibrium equations. Then the displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to nonzero and zero eigenvalues, respectively. Finally, the stiffness matrices are evaluated by applying the member force-displacement relations to the displacement functions. In addition, the finite beam element formulation based on the classical Lagrangian interpolation polynomial is presented. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the finite element results using the isoparametric beam elements and the detailed three-dimensional analysis results using the shell elements of ABAQUS. Particularly the effects of shear deformations on the deflection of thin-walled composite beams with the mono-symmetric I-, channel-, and L-shaped sections with various lamination schemes are investigated.