波形钢腹板组合槽形梁抗扭性能试验研究

为了研究新型结构预应力波形钢腹板组合槽形梁的抗扭性能,基于约束扭转理论,推导了波形钢腹板组合槽形梁的约束扭转翘曲应力表达式和扇形惯性矩;考虑了波形钢腹板的褶皱效应对纵向刚度的影响,计入波形钢腹板的剪切变形对约束扭转刚度的降低,得到了扇形惯性矩修正公式;最终建立了集中扭矩作用下的扭转角计算公式。完成了2片波形钢腹板组合槽形试验梁的偏载试验和3片相同梁的对称加载试验,试验表明:90 kN以内的偏载作用下,试验梁的荷载-位移曲线基本呈线性;试验梁两侧竖向位移的平均值与对称荷载作用下的竖向位移基本相同;试验梁的偏载系数位于1.2~1.3,比波形钢腹板组合箱梁增大10%左右。理论计算、试验测试和有限元分析表明:该文建立的扭转角计算公式采用修正过的扇形惯性矩进行计算,具有良好的精度。     

Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams

This paper presents a new finite element formulation, referred to as reference surface element (RSE) model, for numerical prediction of dynamic behaviour of delaminated composite beams and plates using the finite element method. The RSE formulation can be readily incorporated into all elements based on the Timoshenko beam theory and the Reissner–Mindlin plate theory taking into account the transverse shear deformations. The ‘free model' and ‘constrained model' for dynamic analysis of delaminated composite beams and/or plates have been unified in this RSE formulation. The RSE formulation has been applied to an existing 2-node Timoshenko beam element taking into account the transverse shear deformations and the bending–extension coupling. Frequencies and vibration mode shapes are determined through solving an eigenvalue problem. Numerical results show that the present RSE model is reliable and practical when used to predict frequencies and mode shapes of delaminated composite beams. The RSE formulation has also been used to investigate the effects of the number, size and interfacial loci of delaminations on frequencies and mode shapes of composite beams.     

Instability of laminated composite thin-walled open-section beams

Instability of thin-walled open-section laminated composite beams is studied using the finite element method. A two-noded, 8 df per node thin-walled open-section laminated composite beam finite element has been used. The displacements of the element reference axis are expressed in terms of one-dimensional first order Hermite interpolation polynomials, and line member assumptions are invoked in formulation of the elastic stiffness matrix and geometric stiffness matrix. The nonlinear expressions for the strains occurring in thin-walled open-section beams, when subjected to axial, flexural and torsional loads, are incorporated in a general instability analysis. Several problems for which continuum solutions (exact/approximate) are possible have been solved in order to evaluate the performance of finite element. Next its applicability is demonstrated by predicting the buckling loads for the following problems of laminated composites: (i) two layer (45°/−45°) composite Z section cantilever beam and (ii) three layer (0°/45°/0°) composite Z section cantilever beam.     

The restrained torsional response of open section carbon fibre composite beams

The analysis procedure outlined in this paper essentially makes use of the existing isotropic theories of torsion suitability modified to account for the non isotropic nature of typical carbon fibre composite material.

The warping and St Venant torsional stiffnesses of the beams are determined using the appropriate equivalent engineering elastic constants of the composite material which correspond to the membrane and bending modes of action respectively.

The differential equation governing the constrained torsional equilibrium of the open section beams is solved exactly in the paper for Z and channel sections with some emphasis being given to the influence of ply stacking sequence.

Theoretical results are presented in graphical form and these depict the variations in warping displacement, warping shear flow and longitudinal or axial constraint force intensity with applied torque for the cantilever beam condition with torque applied at the free end.

The paper also gives details of finite element studies of the composite beams and of an experimental programme of work pertaining solely to the behaviour of composite Z beams.

Comparisons between theory, finite element and experiment are presented and these are seen to give exceptionally close agreement.

It is clearly indicated that fibre orientation significantly influences the restrained torsional behaviour of thin-walled open-section composite beams. It is also clear that the use of the appropriate equivalent engineering elastic material constants in the theory is able to closely predict actual behaviour.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

The behaviour of open and closed section carbon fibre composite beams subjected to constrained torsion

A simple engineering theoretical approach is presented in this paper which is able to predict the initial constrained torsional response of a specific class of thin-walled open-section and single-cell closed-section carbon fibre composite beams. The flat walls of the composite beams are symmetrically laminated about their own mid-planes and possess membrane orthotropy. The laminated flats are assembled in such a way that the stiffness distribution round the section is of a symmetrically disposed nature and thus the flanges of a composite box-section, for example, can have a different lay-up configuration to that of the section webs. Beams of this type are essentially uncoupled in their overall stiffnesses and thus it is possible to apply axial load or bending to the sections without inducing torsional behaviour.

The analysis procedures for such beams will, of course, be considerably less complex in nature than those associated with beams of a more general lay-up configuration. Indeed, the analysis approach adopted in this paper simply makes use of the existing theories of torsion appropriate to isotropic construction and these are then suitably modified to account for the non-isotropic nature of the composite material. The torsional and warping rigidities for use in the analysis of the composite beams are thus duly determined through the use of the appropriate equivalent engineering elastic constants of the individual thin composite walls and the concept of effective thickness is employed to account for the different stiffnesses in the walls.

In the paper some detailed attention is paid to the effects of primary and secondary warping restraint on the torsional response of open section beams and the distinct differences between sections whose behaviour is governed predominantly by primary effects and those whose response is associated solely with secondary effects are discussed. The stress systems set up in open-section and single-cell closed-section carbon fibre composite beams when subjected to torsion with variable twist are examined in the paper and in particular it is shown that although the shear flow due to primary warping restraint in open-section beams serves in part to equilibrate the applied torque, that in closed box section beams is completely self equilibrating. Comparisons are given in the paper between theory and experiment and between theory and finite element solutions and these are shown to give good agreement for the Z, angle and box section beams considered.     

Theory of thin-walled composite beams with single and double-cell sections

In the present work, a mixed beam approach that combines both the stiffness and the flexibility formulation in a unified manner has been performed to analyze the elastically-coupled composite beams with closed cross-sections. The analysis model includes the effects of elastic couplings, shell wall thickness, torsion warping, and constrained warping. The Reissner’s semi-complementary energy functional is used to derive the beam force–displacement relations. The theory is validated against detailed finite element analysis results for coupled composite beams with single and double-celled box sections. Various layup cases of box beams with bending–torsion or extension–torsion couplings are considered to evaluate whether all the significant non-classical structural effects of composites are captured in the beam theory. Good correlation of the present theory with the finite element analysis is obtained over the different cases. Numerical results showing the accuracy of the approach are demonstrated in the framework of the analysis.     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

Simple Efficient Smart Finite Elements for the Analysis of Smart Composite Beams

This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams. The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements. The overall smart composite beam is composed of a laminated substrate conventional composite beam, and a piezoelectric layer attached at the top surface of the substrate beam. The piezoelectric layer acts as the actuator layer of the smart beam. Alternate finite element models of the beams, based on an “equivalent single layer high order shear deformation theory”, and a “layer-wise high order shear deformation theory”, are also derived for the purpose of investigating the required number of elements across the thickness of the overall smart composite beams. Several cross-ply substrate beams are considered for presenting the results. The responses computed by the present new “smart finite element model” excellently match with those obtained by the exact solutions. The new smart finite elements developed here reveal that the development of finite element models of smart composite beams does not require the use of conventional first order or high order or layer-wise shear deformation theories of beams. Instead, the use of the presently developed locking-free 4-node elements based on conventional linear piezo-elasticity is sufficient.     

钢-混凝土组合梁收缩徐变分析的有限元方法

基于按龄期调整的有效模量法,提出了部分剪力连接钢-混凝土组合梁在长期荷载作用下收缩徐变分析的简化有限元模型,并通过建立特殊的剪力连接件单元刚度矩阵和利用Newton-Raphson迭代方法考虑滑移效应,同时考虑了负弯矩区混凝土板开裂对组合梁刚度和强度的影响。利用该模型计算了连续组合梁在长期荷载作用下的挠度、应力、滑移量,计算结果与已有的理论计算结果和实验结果吻合,证明本模型用于分析钢-混凝土组合梁收缩徐变是可靠的。     

A new approach to the finite element modelling of beams with warping effect

As a first step toward developing a finite element formulation that can model coupling among extensional, bending and torsional behaviour of beams, a new method is proposed to properly represent the warping of arbitrary cross-sections. The basic approach is to introduce a small warping displacement superimposed over flat cross-sections of a shear-flexible beam in a deformed configuration. Numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach. The present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation.     

Dynamic Finite Element Analysis of Extensional-Torsional Coupled Vibration in Nonuniform Composite Beams

The application of a Dynamic Finite Element (DFE) technique to the extensional-torsional free vibration analysis of nonuniform composite beams, in the absence of flexural coupling, is presented. The proposed method is a fusion of the Galerkin weighted residual formulation and the Dynamic Stiffness Matrix (DSM) method, where the basis functions of approximation space are assumed to be the closed form solutions of the differential equations governing uncoupled extensional and torsional vibrations of the beam. The use of resulting dynamic trigonometric interpolation (shape) functions leads to a frequency dependent stiffness matrix, representing both mass and stiffness properties of the beam element. Assembly of the element matrices and the application of the boundary conditions then leads to a frequency dependent nonlinear eigenproblem, which is solved to evaluate the system natural frequencies and modes. Two illustrative examples of uniform and tapered cantilevered, Circumferentially Uniform Stiffness (CUS), hollow, composite beams are presented. The influence of ply fibre-angle on the natural frequencies is also studied. The correctness of the theory and the superiority of the proposed DFE over the contrasting DSM and conventional FEM methods are confirmed by the published results and numerical checks. The discussion of results is followed by some concluding remarks.     

Combined tension and bending testing of tapered composite laminates

A simple beam element used at Bell Helicopter was incorporated in the Computational Mechanics Testbed (COMET) finite element code at the Langley Research Center (LaRC) to analyze the responce of tappered laminates typical of flexbeams in composite rotor hubs. This beam element incorporated the influence of membrane loads on the flexural response of the tapered laminate configurations modeled and tested in a combined axial tension and bending (ATB) hydraulic load frame designed and built at LaRC. The moments generated from the finite element model were used in a tapered laminated plate theory analysis to estimate axial stresses on the surface of the tapered laminates due to combined bending and tension loads. Surfaces strains were calculated and compared to surface strains measured using strain gages mounted along the laminate length. The strain distributions correlated reasonably well with the analysis. The analysis was then used to examine the surface strain distribution in a non-linear tapered laminate where a similarly good correlation was obtained. Results indicate that simple finite element beam models may be used to identify tapered laminate configurations best suited for simulating the response of a composite flexbeam in a full scale rotor hub.The U.S. Government right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged.     

Variable twist torsion tests on carbon fibre composite beams

A detailed experimental programme of work is presented which examines the constrained torsional response of carbon fibre composite beams for the cantilevered configuration with torque applied at the free end. The behaviour of open-section beams and that of single-cell closed-section box beams is reported and discussed and the experimental findings are shown to corroborate simple engineering theoretical approaches. Tests have been carried out on zed and angle-section beams and thus the effects of primary and secondary warping restraint respectively on the torsional response of open-section beams are examined. The behaviour of zed-section beams is governed, in the main, by primary effects and that of angle-section beams is associated solely with secondary effects. The stress distributions along the beams and around the cross-sections are determined from the measured strains during test obtained from surface bonded strain gauges and comparison of these with theory is shown to be in good agreement. The essential differences in response to constrained torsion between open and closed-section beams are detailed in the paper and the importance of experimental tests in the analysis of composite structures is highlighted.     

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。     

组合梁界面滑移的计算分析

钢-混凝土组合梁(以下简称组合梁)的界面滑移总是存在的,滑移的存在会降低组合梁的组合作用和刚度,增大挠度,要计算组合梁界面的滑移及挠度,对于简支梁在简单荷载情况下,还可得到解析解,但对于连续梁要得到解析解是十分困难的,另外简支梁的解析解十分冗长,实际运用十分不便。用有限元法计算组合梁的滑移和挠度将是很有效的,不受荷载及支撑条件限制,而有限元法的关键是单元刚度矩阵,该文用虚功原理推导了组合梁的单元刚度矩阵,并用自编的有限元程序对组合梁的滑移和挠度进行了计算,在简支情况下与解析解进行了对比和验证,误差很小,在1%以内。该文推导的单元刚度矩阵可用于小型的自编有限元软件,为快速经济地解决相关的大量实际工程问题奠定了基础。     

A C1 finite element for flexural and torsional analysis of rectangular piezoelectric laminated/sandwich composite beams

This work deals with the development of a new C¹ finite element for analysing the bending and torsional behaviour of rectangular piezoelectric laminated/sandwich composite beams. The formulation includes transverse shear, warping due to torsion, and elastic–electric coupling effects. It also accounts for the inter-layer continuity condition at the interfaces between layers, and the boundary conditions at the upper and lower surfaces of the beam. The shear strain is represented by a cosine function of a higher order in nature and thus avoiding shear correction factors. The warping function obtained from a three-dimensional elasticity solution is incorporated in the present model. An exact integration is employed in evaluating various energy terms due to the application of field consistency approach while interpolating the transverse shear and torsional strains. The variation of the electric potential through the thickness is taken care of in the formulation based on the observation of three-dimensional solution. The performance of the laminated piezoelectric element is tested comparing with analytical results as well as with the reference solutions evaluated using three-dimensional finite element procedure. A detailed study is conducted to highlight the influence of length-to-thickness ratio on the displacements, stresses and electric potential field of piezoelectric laminated beam structures subjected to flexural and torsional loadings. Copyright © 2004 John Wiley & Sons, Ltd.     

Active constrained layer damping of geometrically nonlinear vibration of rotating composite beams using 1-3 piezoelectric composite

In this paper, an analysis for active constrained layer damping (ACLD) of rotating composite beams undergoing geometrically non linear vibrations has been carried out. Commercially available vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) material has been used as the material of the constraining layer of the ACLD treatment. A finite element (FE) model has been derived to carry out the analysis. The substrate beam is considered thin and hence, first order shear deformation theory (FSDT) and von-Karman type nonlinear strain–displacement relations are used to derive the coupled electromechanical nonlinear FE model. The rotary effect has been suitably modelled by incorporating extensional strain energy due to centrifugal force. The Golla–Hughes–McTavish method has been employed to model the constrained viscoelastic layer of the ACLD treatment in the time domain. The numerical responses revealed that the ACLD treatment with 1-3 PZC constraining layer efficiently performs the task of active damping of geometrically nonlinear vibrations of the rotating composite beams. The effects of the fibre orientation angles of the angle-ply substrate beams and the 1-3 PZC constraining layer on the ACLD of the geometrically nonlinear vibrations have been investigated. Also, the effect of the thickness variations of the 1-3 PZC layer and the viscoelastic constrained layer on the damping characteristics of the overall rotating composite beams has been studied.     

Finite prism analysis of plates and shells

The finite prism technique introduced by Zienkiewicz and Too⁴ is extended to include 12-node prism elements and, more importantly, a novel offset beam element. The use of 12-node prism elements enables parabolic strain distributions to be simulated, this being useful for structures which have strongly tapered cross-sections. The offset beam element is used to simulate flexure and torsion of a beam whose centroid is offset from the main structure. The element is specified completely at the nodes of adjacent prism elements and so is not really an ‘element’ in the usual sense. The analysis is applied to thin and thick plates and to shells, with and without edge beams. It is shown to be more versatile than the finite strip method and it requires smaller computer resources than the finite element method. Experimental verification of the analysis is obtained by comparison with measurements for a double-T bridge deck tested by Loo¹⁴.