Topology optimization of thin‐walled box beam structures based on the higher‐order beam theory

The investigation aims to formulate ground‐structure based topology optimization approach by using a higher‐order beam theory suitable for thin‐walled box beam structures. While earlier studies use the Timoshenko or Euler beams to form a ground‐structure, they are not suitable for a structure consisting of thin‐walled closed beams. The higher‐order beam theory takes into an additional account sectional deformations of a thin‐walled box beam such as warping and distortion. Therefore, a method to connect ground beams at a joint and a technique to represent different joint connectivity states should be investigated for streamlined topology optimization. Several numerical case studies involving different loading and boundary conditions are considered to show the effectiveness of employing a higher‐order beam theory for the ground‐structure based topology optimization of thin‐walled box beam structures. Through the numerical results, this work shows significant difference between optimized beam layouts based on the Timoshenko beam theory and those based on a more accurate higher‐order beam theory for a structure consisting of thin‐walled box beams. Copyright © 2015 John Wiley & Sons, Ltd.     

解析型Winkler弹性地基梁单元构造

该文采用Winkler弹性地基梁理论确定了弹性地基梁的挠度方程解析通解; 根据最小势能原理建立了解析型Winkler弹性地基欧拉梁及铁摩辛柯梁的单元刚度及等效节点荷载; 得到了解析型弹性地基欧拉梁单元AWFB-E及铁摩辛柯梁单元AWFB-T。同时,论文还采用传统里兹法求得了相应的Winkler弹性地基欧拉梁及铁摩辛柯梁单元刚度矩阵,得到了里兹法弹性地基欧拉梁单元RWFB-E及铁摩辛柯梁单元RWFB-T。对该文构建的两类单元与一般梁-基体系有限元分析结果及理论解析解进行了对比。对比结果表明,传统里兹法由于其多项式形函数无法精确模拟弹性地基梁变形,因此其结果与理论解析解有误差,但随着单元数量增多其误差减小; 采用解析型单元进行计算时,无论单元数量多少,得到的均为“真实”解,说明解析试函数法求得的位移形函数比一般的多项式形函数精确,得到的弹性地基梁单元具备解析型、精确性的特点,可应用于解决实际工程问题。     

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.     

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.     

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.     

Timoshenko梁功率流主动控制研究

为了研究扰动影响下梁式结构的动力学响应与主动控制,首先基于Timoshenko梁理论,采用行波方法建立了悬臂梁结构的动力学模型并获得了其在扰动下的精确动力学响应,进一步得到结构中传播的功率流,并以此为目标函数,优化得到了最优控制力的大小与相位,然后对结构施加最优控制力,实现了Timoshenko梁结构的功率流主动控制。对Timoshenko梁结构动力学响应与功率流主动控制方法进行了数值计算,并与Euler-Bernoulli梁理论计算结果进行了对比分析。结果表明：采用行波方法计算梁结构的动力学响应准确可靠;Timoshenko梁模型较Euler-Bernoulli梁模型在中、高频段更为精确,且更接近工程实际;通过数值计算与分析验证了基于行波方法功率流主动控制的正确性与有效性,并且功率流主动控制可以明显降低梁式结构全频域内的抖动。     

Behaviour of a two-layered beam under impact loading

A dynamic model for multilayered beams under impact loading is proposed. The model is based on Timoshenko beam theory but accommodates interlayer slip. Different impact loads as well as different interfacial situations are investigated. The transient behaviour of a two-layered beam is determined by the characteristic method solution of the dynamic governing equations. The results are found to be in close agreement with the finite element analysis code ABAQUS.     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

Free vibration analysis of beams by using a third-order shear deformation theory

In this study, free vibration of beams with different boundary conditions is analysed within the framework of the third-order shear deformation theory. The boundary conditions of beams are satisfied using Lagrange multipliers. To apply the Lagrange’s equations, trial functions denoting the deflections and the rotations of the cross-section of the beam are expressed in polynomial form. Using Lagrange’s equations, the problem is reduced to the solution of a system of algebraic equations. The first six eigenvalues of the considered beams are calculated for different thickness-to-length ratios. The results are compared with the previous results based on Timoshenko and Euler-Bernoulli beam theories.     

Mechanical performance of innovative GFRP-bamboo-wood sandwich beams: Experimental and modelling investigation

Innovative GFRP-bamboo-wood sandwich beams were developed and investigated experimentally and by modeling. The effects of the thickness of the GFRP and bamboo layers on the overall structural performance in bending were clarified. It was shown that an increase of thickness of the bamboo and GFRP layers could significantly increase the flexural stiffness and ultimate load of the sandwich beams. ANSYS was used to parametrically analyze the material efficiency and to obtain optimal solutions for the thickness of the GFRP, wood, and bamboo layers. The total depth of 60 mm and the thickness of 6 mm for bamboo and of 4.5 mm for GFRP presented the best material efficiency in terms of stiffness enhancement. A simplified model based on Timoshenko beam theory was proposed to predict the load-deflection behavior of the sandwich beams, where the section transformation method was used to calculate the stress distribution along the depth of the sandwich beams. The calculated results showed good correlation with the experimental and numerical results. Design optimization in terms of self-weight and cost of the proposed sandwich beam was conducted using MATLAB and ANSYS, and the optimized thicknes was obtained with minimized self-weight, cost, and acceptable mechanical performance.     

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.     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method

A modeling method for flapwise and chordwise bending vibration analysis of rotating pre-twisted Timoshenko beams is introduced. In the present modeling method, the shear and the rotary inertia effects on the modal characteristics are correctly included based on the Timoshenko beam theory. The kinetic and potential energy expressions of this model are derived from the Rayleigh–Ritz method, using a set of hybrid deformation variables. The equations of motion of the rotating beam are derived from the kinetic and potential energy expressions introduced in the present study. The equations thus derived are transmitted into dimensionless forms in which main dimensionless parameters are identified. The effects of dimensionless parameters such as the hub radius ratio, slenderness ration, etc. on the natural frequencies and modal characteristics of rotating pre-twisted beams are successfully examined through numerical studies. Finally the resonance frequency of the rotating beam is evaluated.     

Generalized Timoshenko modelling of composite beam structures: sensitivity analysis and optimal design

This article describes a new approach to design the cross-section layer orientations of composite laminated beam structures. The beams are modelled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the variational asymptotic beam section analysis) methodology is used to compute the cross-sectional model for a generalized Timoshenko model, which was embedded in the finite element solver FEAP. Optimal design is performed with respect to the layers’ orientation. The design sensitivity analysis is analytically formulated and implemented. The direct differentiation method is used to evaluate the response sensitivities with respect to the design variables. Thus, the design sensitivities of the Timoshenko stiffness computed by VABS methodology are imbedded into the modified VABS program and linked to the beam finite element solver. The modified method of feasible directions and sequential quadratic programming algorithms are used to seek the optimal continuous solution of a set of numerical examples. The buckling load associated with the twist–bend instability of cantilever composite beams, which may have several cross-section geometries, is improved in the optimization procedure.     

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。     

Shear coefficients for thin-walled composite beams

The shear coefficient in Timoshenko beam theory is obtained for thin-walled beams constructed of laminated panels of composite material using a variation of the method due to Cowper. Formulae are presented for a class of such composite beams. Comparisons are made with Cowper's original formulae for the case of an isotropic beam. The effect of shear deformation under static loading of typical composite beams is investigated. A procedure is outlined for the distribution of plies in the laminated panels to achieve optimal response under static or dynamic loading.     

不同梁理论之间简支梁特征值的解析关系

利用Euler-Bernoulli梁理论(EBT)、Timoshenko梁理论(一阶理论,TBT)和Reddy三阶梁理论(RBT)之间,梁的特征值问题在数学上的相似性,研究了不同梁理论之间特征值的关系。将特征值问题的求解转化为一个代数方程的求解,并导出了不同梁理论之间梁的特征值之间的精确解析关系。因此,只要已知梁的经典结果(临界载荷和固有频率),便很容易从这些关系中获得一阶和三阶梁理论下的相应结果。另外,从这些关系中获得的含有剪切变形影响的结果,可以用于检验一阶和三阶梁理论下梁数值结果的有效性、收敛性以及精确性等问题。     

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.     

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。     

A nonlinear Timoshenko beam formulation based on the modified couple stress theory

This paper presents a nonlinear size-dependent Timoshenko beam model based on the modified couple stress theory, a non-classical continuum theory capable of capturing the size effects. The nonlinear behavior of the new model is due to the present of induced mid-plane stretching, a prevalent phenomenon in beams with two immovable supports. The Hamilton principle is employed to determine the governing partial differential equations as well as the boundary conditions. A hinged–hinged beam is chosen as an example to delineate the nonlinear size-dependent static and free-vibration behaviors of the derived formulation. The solution for the static bending is obtained numerically. The solution for the free-vibration is presented analytically utilizing the method of multiple scales, one of the perturbation techniques.