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.     

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.     

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.     

A small strain,moderately large deflection finite element beam model with cross-sectional warping effect

A finite element model that can be applied to helicopter rotor blades has been developed with a particular emphasis on the proper representation of out-of-plane warping of arbitrary cross-sections. The model can describe accurately coupled bending, torsion and extensional behavior of beams undergoing small strain, moderately large deflection. The model can also handle beams with arbitrary cross-sections, taper, pretwist and planform. A main feature of the present approach is to introduce small warping displacement superimposed over flat cross-sections of a shear-flexible beam in the direction of the reference axis in deformed configuration. The nonlinear equilibrium equation resulting from finite element approximation is solved by the Newton-Raphson method. Numerical tests involving simple isotropic beams demonstrate the validity of the present approach.     

复合材料空间薄壁梁的有限元分析模型

在剪切梁理论的基础上, 采用9 节点平面单元模拟梁任意截面形状; 采用27 节点体单元, 模拟截面出平面外的二次翘曲位移, 从而建立了空间复合材料任意截面薄壁梁考虑二次翘曲的有限元分析模型。根据本文中导出的复合材料有限元模型编制了相应的分析计算程序。算例表明: 本文中建立的复合材料薄壁梁模型正确, 可以用于考虑多种耦合影响因素作用下复杂结构空间薄壁复合材料梁的有限元分析计算。      

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.     

Warping torsion in thin-walled open section beams using the semiloof beam element

Many structural members including metallic, reinforced polymer and concrete members must be modelled, in the simplest approximation, as thin-walled beams whose cross-sections exhibit significant, out-of-plane warping owing to torsion. The elastic capabilities of the semiloof beam element are extended to include warping torsion of thin-walled open section beams. The performance of the extended element is compared with that of the original formulation.     

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.     

Investigation of Functionally Graded and Viscoelastic Layers Effects on the Dynamic Behavior of Functionally Graded Electro-Rheological Sandwich Beams

Vibration characteristics of functionally graded electro-rheological (FGER) sandwich beams are investigated. While a vast majority of studies have been reported about functionally graded material (FGM) or electrorheological fluids (ERF) composite beams, few, if any, works are conducted about FGER models. In order to validate the present finite element formulation of the FGER beam model, the results of the developed finite element (FE) model are compared with the results of an experimental test on a fabricated ERF composite beam. The effects of FGM volume fraction index, electric field, and thickness of the viscoelastic core are studied on the natural frequencies and modal loss factors of the FGER beam.     

大变形薄壁复合材料旋转梁静动态特性分析

该文研究了大变形薄壁复合材料旋转梁的静、动态特性以及复合材料旋转梁不同铺层角、转速对固有频率的影响变化规律。在梁截面上设立翘曲节点引入翘曲变形,假设应变沿薄壁厚度呈二次曲线形式变化;由Hamilton原理,推导了复合材料梁的大变形公式;建立了考虑剪切、翘曲薄壁复合材料大变形梁静、动态分析的有限元模型,并编写了相应的计算程序。分析了复合材料薄壁梁大变形状态下的静态变形行为;在旋转状态下,考虑哥氏力影响,不同转速、不同铺层角对复合材料梁动态特性的影响。计算结果表明:能够用该文建立的模型计算复合材料梁的静动态特性,与相关文献计算结果进行比较,说明了模型的优越性。     

Active damping of rotating composite thin-walled beams using MFC actuators and PVDF sensors

The object of this research is to enhance the damping performance for vibration suppression of rotating composite thin-walled beams using MFC actuators and PVDF sensors. The formulation is based on single cell composite beam including a warping function, centrifugal force, Coriolis acceleration and piezoelectric effect. Adaptive capability of the beam is acquired through the use of a negative velocity feedback control algorithm. Numerical analysis is performed using finite element method and Newmark time integration method is used to calculate the time response of the model. It is observed that the feedback control gain has an effect on damping performance. The paper continues with an investigation into influences of parameters such as the rotating speed and the fiber orientation in host structures. Also, it is confirmed that effective damping performance is achievable through the suitable arrangement and distributed size of sensor and actuator pair using case study.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

压电复合梁热机电耦合有限元模型

压电材料应用于航天结构形状或振动控制时,可能会受到热场、力场和电场的共同作用。为分析处于热场、力场和电场共同作用下的压电复合结构,文中基于高阶剪切变形理论、高阶电势模型和线性温度分布假设,利用虚功原理建立了压电复合梁结构的热-机-电耦合有限元模型。该模型可应用于热机电耦合压电复合结构的形状与振动控制研究。利用本文模型对压电双晶片梁、压电复合悬臂梁进行了数值仿真,仿真结果与文献给出的理论结果和实验值吻合良好,表明本文模型是正确有效的。     

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。     

Theory and numerics of three‐dimensional beams with elastoplastic material behaviour

A theory of space curved beams with arbitrary cross‐sections and an associated finite element formulation is presented. Within the present beam theory the reference point, the centroid, the centre of shear and the loading point are arbitrary points of the cross‐section. The beam strains are based on a kinematic assumption where torsion‐warping deformation is included. Each node of the derived finite element possesses seven degrees of freedom. The update of the rotational parameters at the finite element nodes is achieved in an additive way. Applying the isoparametric concept the kinematic quantities are approximated using Lagrangian interpolation functions. Since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells. Due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load‐carrying capacity of beam structures. Copyright © 2000 John Wiley & Sons, Ltd.     

A C1 finite element including transverse shear and torsion warping for rectangular sandwich beams

A new three‐noded C¹ beam finite element is derived for the analysis of sandwich beams. The formulation includes transverse shear and warping due to torsion. It also accounts for the interlaminar continuity conditions at the interfaces between the layers, and the boundary conditions at the upper and lower surfaces of the beam. The transverse shear deformation is represented by a cosine function of a higher order. This allows us to avoid using shear correction factors. A warping function obtained from a three‐dimensional elasticity solution is used in the present model. Since the field consistency approach is accounted for interpolating the transverse strain and torsional strain, an exact integration scheme is employed in evaluating the strain energy terms. Performance of the element is tested by comparing the present results with exact three‐dimensional solu‐tions available for laminates under bending, and the elasticity three‐dimensional solution deduced from the de Saint‐Venant solution including both torsion with warping and bending. In addition, three‐dimensional solid finite elements using 27 noded‐brick elements have been used to bring out a reference solution not available for sandwich structures having high shear modular ratio between skins and core. A detailed parametric study is carried out to show the effects of various parameters such as length‐to‐thickness ratio, shear modular ratio, boundary conditions, free (de Saint‐Venant) and constrained torsion. Copyright © 1999 John Wiley & Sons, Ltd.     

Prediction of material coupling effect on structural damping of composite beams and blades

The present paper investigates the effect of material coupling on static and modal characteristics of composite structures. Incorporation of stiffness and damping coupling terms into a beam formulation yields equivalent section stiffness and damping properties. Building upon the damping mechanics, an extended beam finite element is developed capable of providing the stiffness and damping matrices of the structure. Validation cases on beams and blades demonstrate the importance of all stiffness and damping terms. Numerical results validate the predicted effect of material coupling on static characteristics of composite box-section beams. The effect of the full coupling damping matrices on modal frequencies and structural modal damping of composite beams is investigated. Box-section beams and small blade models with various ply angle laminations at the girder segments are considered. Finally, the developed finite element is applied to the prediction of the modal characteristics of a 19 m realistic wind-turbine model blade.     

Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections

 In this paper shear correction factors for arbitrary shaped beam cross-sections are calculated. Based on the equations of linear elasticity and further assumptions for the stress field the boundary value problem and a variational formulation are developed. The shear stresses are obtained from derivatives of the warping function. The developed element formulation can easily be implemented in a standard finite element program. Continuity conditions which occur for multiple connected domains are automatically fulfilled.     

复合材料薄壁梁力学特性分析

综述了人们在建立研究复合材料薄壁梁力学特性的非线性梁理论及分析结构剖面特性,确立结构算子参数等方面所做的工作以及这些研究工作的特点。同时,结合作者的工作,介绍了近年来各国学者在复合材料薄壁梁力学特性研究上的进展。      

Investigation of composite box beam dynamics using a higher-order theory

A higher-order composite box beam theory is developed to model beams with arbitrary wall thicknesses. The theory, which is based on a refined displacement field, approximates the three-dimensional elasticity solution so that the beam cross-sectional properties are not reduced to one-dimensional beam parameters. Both inplane and out-of-plane warping are included automatically in the formulation. The model can accurately capture the tranverse shear stresses through the thickness of each wall while satisfying stress-free boundary conditions on the inner and outer surfaces of the beam. Numerical results are presented for beams with varying wall thicknesses and aspect ratios. The static results are correlated with available experimental data and show excellent agreement. Dynamic results presented show the importance of including inplane and out-of-plane warping deformations in the formulation.