Shear deformable finite beam elements for composite box beams

The shear deformable thin-walled composite beams with closed cross-sections have been developed for coupled flexural, torsional, and buckling analyses. A theoretical model applicable to the thin-walled laminated composite box beams is presented by taking into account all the structural couplings coming from the material anisotropy and the shear deformation effects. The current composite beam includes the transverse shear and the restrained warping induced shear deformation by using the first-order shear deformation beam theory. Seven governing equations are derived for the coupled axial-flexural-torsional-shearing buckling based on the principle of minimum total potential energy. Based on the present analytical model, three different types of finite composite beam elements, namely, linear, quadratic and cubic elements are developed to analyze the flexural, torsional, and buckling problems. In order to demonstrate the accuracy and superiority of the beam theory and the finite beam elements developed by this study,numerical solutions are presented and compared with the results obtained by other researchers and the detailed threedimensional analysis results using the shell elements of ABAQUS. Especially, the influences of the modulus ratio and the simplified assumptions in stress–strain relations on the deflection, twisting angle, and critical buckling loads of composite box beams are investigated.     

On the cross-section distortion of thin-walled beams with multi-cell cross-sections subjected to bending

This paper deals with distortion of the cross-section contour of thin-walled beams with simple multi-cell closed rectangular cross-sections. The cross-section distortion is considered in the limit case. It is assumed that beam plates are hinged together along their longitudinal edges. Double symmetric three and two-cell closed cross-sections are considered. The stresses and displacements are obtained in the closed analytical form. The additional stresses and displacements due to distortion with respect to the stresses and displacements of the ordinary theory of bending are obtained. The boundary conditions are given in the general form. Some illustrative examples are given.     

Asymptotic theory for static behavior of elastic anisotropic I-beams

End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.     

A finite element model for linear viscoelastic behaviour of pultruded thin-walled beams under general loadings

A finite element model for orthotropic thin-walled beams subject to long-term loadings is presented. The hypothesis, rather usual for thin-walled beams, of cross-sections remaining undistorted in their own planes after deformation is introduced, so reducing the number of d.o.f.’s and, consequently, the computational effort of the analysis. The model is used to perform linear viscoelastic analysis of prismatic beams with general cross-sections, i.e., open, closed or multi-cell. As far as the constitutive viscoelastic law is concerned, a generalized linear Maxwell model is adopted. Making use of the exponential algorithm, differential equations are written in incremental form and integration is performed adopting time intervals of variable length. Numerical examples are finally presented, concerning glass-fibre pultruded shapes under long-term loadings. Displacement evolution with time and stress redistribution adopting different creep laws are presented. Convergence features of the proposed finite element and time integration procedure are also shown.     

梁在固有振动中的对偶关系

本文研究具有齐次边界的Euler-Bernoulli梁在固有振动中的对偶关系.将两种截面变化不同、但固有频率完全相同的梁定义为异截面对偶梁.通过位移描述和弯矩描述,指出具有齐次边界条件的变截面梁共有如下7类异截面对偶:一是自由-自由梁与固支-固支梁,二是滑支-自由梁与滑支-固支梁(及其镜像),三是铰支-自由梁与铰支-固支梁(及其镜像),四是铰支-滑支梁与铰支-滑支梁(及其镜像),五是滑支-滑支梁与滑支-滑支梁,六是铰支-铰支梁与铰支-铰支梁,七是固支-自由梁与自由-固支梁.在此基础上,将两种截面变化相同、固有频率也相同的梁定义为同截面对偶梁.研究表明,当且仅当梁的截面积函数和截面惯性矩函数具有特定指数函数形式时,前4类异截面对偶梁能成为同截面对偶梁.对于等截面梁,上述前3类同截面对偶仍可保持,而第4类同截面对偶退化为彼此镜像.此时,通过引入转角描述可发现等截面梁产生新对偶,即滑支-滑支梁与铰支-铰支梁对偶.上述等截面梁的对偶均具有如下特征,即对偶中的一种梁具有静定约束,另一种梁具有静不定约束.      

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

Refined Series Methodology for the Fully Coupled Thin-Walled Laminated Beams Considering Foundation Effects

The refined power series solutions are presented for the coupled static analysis of thin-walled laminated beams resting on elastic foundation. For this purpose, the elastic strain energy considering the material and structural coupling effects and the energy including the foundation effects are constructed. The equilibrium equations and the force-displacement relationships are derived from the extended Hamilton's principle, and the explicit expressions for displacement parameters are presented based on power series expansions of displacement components. Finally, the member stiffness matrix is determined by using the force-displacement relationships. For comparison, the finite element model based on the Hermite cubic interpolation polynomial is presented. In order to verify the accuracy and the superiority of the laminated beam element developed by this study, the numerical solutions are presented and compared with results obtained from the regular finite beam elements and the ABAQUS's shell elements. The influences of the fiber angle change and the boundary conditions on the coupled behavior of laminated beams with mono-symmetric I-sections are investigated.     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

基于塑性铰理论的车身概念设计正碰分析与优化方法

在概念设计阶段,车身碰撞安全性能评价是一个难点问题,需要详细的结构模型,本文基于塑性铰理论提出采用梁单元简化模型对框架车身进行概念设计阶段的耐撞性评估和优化设计方法。首先,介绍了关于箱型截面薄壁梁弯曲特性研究的理论模型与计算过程,接着赋予梁单元塑性铰的特性,模拟薄壁梁变形,再对框架车身进行了碰撞仿真。将仿真结果与详细模型对比,以分析简化模型的精度及可靠性。最后,以此为基础对框架车身进行耐撞性优化。结果表明,该简化模型易于创建,且有较高的精度,可用于概念设计阶段梁结构的设计工作。     

Free vibration of axially loaded thin-walled composite Timoshenko beams

Based on shear-deformable beam theory, free vibration of thin-walled composite Timoshenko beams with arbitrary layups under a constant axial force is presented. This model accounts for all the structural coupling coming from material anisotropy. Governing equations for flexural-torsional-shearing coupled vibrations are derived from Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibrations. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite beams to investigate the effects of shear deformation, axial force, fiber angle, modulus ratio on the natural frequencies, corresponding vibration mode shapes and load–frequency interaction curves.     

Response of flexure–torsion coupled composite thin-walled beams with closed cross-sections to random loads

A study of the flexure–torsion coupled random response of the composite beams with solid or thin-walled closed-sections subjected to various types of concentrated and distributed random excitations is dealt with in this paper. The effects of flexure–torsion coupling, shear deformation and rotary inertia are included in the present formulations. The random excitations are assumed to be stationary, ergodic and Gaussian. Analytical expressions for the displacement response of the composite beams are obtained by using normal mode superposition method combined with frequency response function method. The present method can produce the effective solutions for the composite Timoshenko beams with circumferentially antisymmetric (CAS) configuration and more general beam assemblages of connected beams. The influences of flexure–torsion coupling, shear deformation and rotary inertia on the random response of an appropriately chosen composite beam from the literature are demonstrated and discussed.     

A generalized Adadorov theory for anisotropic thin-walled beams

This paper presents a generalized Adadorov theory for anisotropic thin—walled beams. The theory takes account of the shear strain of the middle surface, which exerts a significant influence on the anisotropic thin-walled beams. A new approach is established to solve the governing equations, which have the same form for both open and closed section beams. The numerical examples show that the effects of the shear strain cannot be neglected for this class of beams.This work was part of research project supported by the National Natural Science Foundation of China     

薄壁截面梁的歪曲屈曲

本文在梁的整体屈曲分析中,废除了传统的刚性截面假定(刚周边假定),允许梁截面自由地歪曲,进而研究了薄壁截面梁的歪曲屈曲性能。分析中采用了样条有限条法,考虑了各种不同的荷载形式、支承条件和边界约束。这一方法与有限元法相比较,计算工作量大大地减少。数值计算结果表明,在梁长细比的较大范围内,歪曲屈曲模型对梁的设计起控制作用。     

Local buckling of wide-flange thin-walled anisotropic composite beams

The present contribution deals with the onset of local buckling of compressively loaded thin-walled beams with open I, C, Z, T and L-cross-sections made of laminated composite materials. The method employs a discrete plate analysis approach in the course of which each structural subelement of interest—which presently is the flange—of the thin-walled cross-section is considered as a separate composite plate with elastic rotational restraints at those edges where an adjacent substructural element is located. While in many investigations the lamination schemes of webs and flanges are considered to be purely orthotropic, in the present paper the laminate layups are allowed to be of an arbitrary non-orthotropic nature, which also allows for the analysis of laminates with inherent bending–torsion coupling. The analysis of the buckling loads of the flanges of thin-walled composite beams is performed using the Ritz-method for which some especially adjusted displacement shape functions are employed. For the case of pure orthotropy, a novel closed-form solution is described. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy, which makes it very attractive for actual practical applications whenever the local stability behaviour of wide-flange thin-walled composite beams is to be considered.     

Buckling of thin-walled columns accounting for initial geometrical imperfections

The paper is devoted to the effect of some geometrical imperfections on the critical buckling load of axially compressed thin-walled I-columns. The analytical formulas for the critical torsional and flexural buckling loads accounting for the initial curvature of the column axis or the twist angle respectively are derived. The classical assumptions of theory of thin-walled beams with non-deformable cross-sections are adopted. The non-linear differential equations are derived and the critical buckling loads are approximated by means of the Galerkin’s method. Comparison of analytical results to numerical analysis of simply supported I-columns by means of finite element method (FEM) is provided. Moreover the analytical formulas is adapted to I-columns with lipped flanges and satisfactory agreement of analytical and numerical results of stability analysis is observed.     

Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications

This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by three-dimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701–734.] is analyzed and the results are compared.     

Theory of open thin-walled composite and prestressed beams

Summary Composite and prestressed beams having open thin-walled cross-sections involving concrete as an aging linear viscoelastic material, prestressing steel with relaxation property, and two other steel members as elastic materials are considered. The expressions for stresses and generalized displacements referring to any given open thin-walled composite cross-section and to any compliance function of concrete are developed without mathematical neglects. In the framework of the theory of composite and prestressed open thin-walled beams there exists only one approximate solution based on the approximate algebraic stress-strain relation for concrete where the prestressing steel is introduced as an elastic material. The exact theory is significant because on the basis of it is possible to create approximate calculating methods and because we can verify whether approximate methods are sufficiently accurate for the engineering practice.

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Theorie dünnwandiger, offner und vorgespannter Verbundträger

Übersicht Behandelt werden vorgespannte Verbundträger mit beliebigem dünnwandigen, offenen Profil. Der Verbund besteht aus alterndem, linear-viskoelastischen Beton, relaxierendem Saannstahl und zwei weiteren elastischen Stahlstegen. Die Ausdrücke für die Spannungen und verallgemeinerten Verschiebungen werden ohne Einschränkung der Nachgiebigkeitsfunktion des Betons hergeleitet. Im Rahmen der Theorie existiert bisher nur eine Näherungslösung; sie beruht auf einer Näherungsbeziehung zwischen den Spannungen und Verzerrungen des Betons, wobei der Spannstahl als elastisch eingeführt wird. Die exakte Theorie ist deshalb bedeutsam, weil man mit ihr näherungsweise Berechnungsmethoden entwickeln kann und prüfen kann, ob Näherungsmethoden hinreichend genau sind für ingenieurmäßige Zwecke.

     

复合材料薄壁旋转梁的研究进展

本文综述了近年来复合材料薄壁旋转梁的研究进展,着重介绍了围绕梁平面外翘曲问题,各国学者在非线性梁理论、用解析法和有限元法构造复合材料薄壁梁模型方面做的工作,讨论了此领域需进一步研究的问题．