Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

Flexural–torsional behavior of thin-walled composite space frames

A general analytical model based on the first-order shear deformable beam theory applicable to thin-walled composite space frames with arbitrary lay-ups under external loads is presented. This model accounts for all the structural coupling coming from the material anisotropy. The seven governing equations are derived from the principle of the stationary value of total potential energy. A displacement-based one-dimensional 14 degree-of-freedom space beam model which includes the effects of shear deformation, warping is developed to solve the problem. Numerical results are obtained to investigate the effects of fiber orientation on flexural–torsional responses of thin-walled composite space frame under vertical load.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

The twisting of a transversely braced thin-walled beam with allowance for localized flange deformations

The torsional stiffness of a channel-section beam braced by evenly spaced transverse beams is studied. Previous work by Vlasov and the writer is extended theoretically to include localized deformations which take place in the flanges. A family of curves, resulting from a finite difference analysis of the flange deformations, is used in conjunction with a simple expression to predict a “reduced” second moment of area for the transverse beams from which a modified St. Venant torsional constant for the braced channel can be calculated. Experiments on a series of braced Perspex channels show good agreement with the theoretical predictions when the ratio of the length of the transverse beams to their depth is large but as the beams become “deep”, the engineer's theory of bending with the usual shear correction inadequately describes their behaviour. A correction factor, obtained from a graph based upon a finite element analysis, is used to obtain closer agreement for all length to depth ratios.     

Implications of warping restraint on statics and dynamics of elastically tailored thin-walled composite beams

Static response and free vibration of elastically tailored thin-walled beams accounting for the warping restraint effect are investigated via an exact solution methodology within the context of a refined beam model. Analytical results obtained from the restrained warping model are compared with those based on its Saint-Venant model counterpart. It is revealed that the beam slenderness and thickness ratio, as well as the elastic anisotropy, considered in conjunction with the warping restraint have profound effects on the static and dynamic response characteristics. It is also shown that even for anisotropic composite thin-walled beams with high slenderness ratios, warping restraint can still be significant, implying the inadequacy of merely considering the geometric aspects in the modeling of anisotropic composite thin-walled beams.     

Stiffness matrix of thin-walled curved beam for spatially coupled stability analysis

The exact stiffness matrix for the spatially coupled stability analysis of thin-walled curved beam with non-symmetric cross-section subjected to uniform compression is newly presented. For this, the elastic strain energy including the axially/flexurally/torsionally coupled terms and the potential energy due to the initial stress resultant are introduced. Then, equilibrium equations and force–deformation relationship are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Next, the exact element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the finite element solutions using the Hermitian curved beam elements. In addition, the coupling of symmetric and anti-symmetric modes at the buckling load crossover with change in curvature of beam is investigated.     

汽车前纵梁薄壁结构碰撞仿真及优化设计研究

针对前纵梁现有结构吸能特性的不足,对其结构进行了改进,使前纵梁在正面碰撞过程中吸收更多能量。根据前纵梁板厚、焊点的间距及焊点大小等设计参数,建立了前纵梁正面碰撞的有限元模型。采用HyperMesh和LS-DYNA软件,对前纵梁碰撞过程进行了模拟仿真。通过加诱导槽及加加强板的方式,对前纵梁结构进行了优化。研究结果表明,优化后的前纵梁单位质量吸收的能量提高了63.3%。通过该方法使前纵梁提高了吸能特性,降低了车辆碰撞对乘客造成的损害。     

An alternative approach to the analytical determination of the moment capacity of a thin-walled channel steel section beam

An analytical method of combined plastic mechanism and elastic approaches has been developed to predict the moment capacity of a thin-walled channel steel section beam. The plastic mechanism approach is performed by analyzing an idealized model of bending-plastic-hinge-collapse mechanism of the beam bent about its minor neutral axis. This approach adopts a concept of the equilibrium between external energy and the one dissipating in the hinge mechanism. Another analytical approach has been done by analyzing the investigated beam according to an elastic-bending theory. In the elastic analysis, the application of an effective width concept has also been considered to account for the effect of local buckling on the bending element of the beam. These both analytical approaches are then used in the method of cut-off strength to estimate a theoretical moment capacity of the beam bent about its minor neutral axis. An attempt has also been carried out to correlate the estimated moment capacity obtained to another one about a major neutral axis. In order to assess the accuracy of the analytical method developed, its predicted results are verified using the data obtained from experiments and a design code specially used for cold-formed steel structural members. The data of moment capacity about the minor neutral axis is compared to the one measured in a number of flexural-bending tests to failure on the similar channel steel section beams. Meanwhile, the data of moment capacity about the major neutral axis is compared to the one calculated using the design code. A statistical analysis of verified data populations indicates that the mean value of deviated data from experimental one is 1.025 with the standard deviation of 0.087 and from the design code is 1.004 with the standard deviation of 0.111. These statistical measures clearly mean that the analytical model presented herein, on average, tends to estimate conservatively the moment capacity of the investigated beam about its both unsymmetrical and symmetrical-neutral axes by less than 5% and this is certainly still within acceptable limits of ±20%.     

Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms

Several approaches exist for calculating the torsional compliance of rectangular cross-section beams, but most depend on the relative magnitude of the cross-section thickness and width, which might be changing during the design phase (especially for design optimization) or is variant for variable cross-section beams such as circular flexure hinges and tapered bars. After summarizing current equations and analyzing their computational accuracy, two new equations are proposed, which are thickness-to-width ratio independent, and suitable for variable cross-section beams and optimization design of torsional elements in compliant mechanisms. The closed-form equations for the torsional compliance of elliptical and circular flexure hinges are derived by using the new equations.     

类椭圆截面吊臂的约束扭转特性研究

综合分析常见椭圆截面和大圆角矩形截面大吨位起重机吊臂结构力学性能,提出了一种类椭圆截面起重机吊臂结构。根据吊臂旋转平面受力状态,可将基本臂简化为闭口曲边截面悬臂薄壁梁力学模型。然后基于乌曼斯基理论,推导了该力学模型的约束扭转特性计算公式。并基于MSC.PATRAN/NASTRAN有限元数值分析验证了理论计算公式的正确性。通过算例分析,获取了一类椭圆截面吊臂的约束扭转翘曲正应力和翘曲位移等重要分析参数。研究表明,推导的理论公式可为类椭圆起重机吊臂的约束扭转分析提供一种简便可行的计算方法。     

基于有限元法的矩形薄壁梁轴向耐撞性能优化

矩形薄壁空心直梁是汽车纵梁经常采用的结构形式,并且是承受正面碰撞的主要部件,其结构的耐撞性及碰撞吸能优化是现代汽车研究的重要内容.以矩形薄壁梁为研究对象,基于动态有限元分析模型,利用LS-DYNA软件中的OPT模块技术,研究矩形薄壁梁横截面的边长比变化对其耐撞性能的影响,对薄壁直梁在轴向冲击载荷下的耐撞性能进行了优化仿真.优化结果表明,当矩形两个边长的比值为0.6时,结构的碰撞力峰值最小,吸收的塑性变形能最大,具有最好的耐撞性能.     

Geometrically nonlinear theory of thin-walled composite box beams using shear-deformable beam theory

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings is presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The nonlinear governing equations are derived and solved by means of an incremental Newton-Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beams under vertical load to investigate the effects of shear deformation, geometric nonlinearity and fiber orientation on axial-flexural-torsional response.     

一种清洁型薄壁异型件夹具设计

对薄壁异型件夹具进行了设计。充分考虑到薄壁零件刚度低,加工工艺差的特点,针对零件的特殊要求,提出了相应的定位和夹紧方案。通过Pro/ENGINEER软件环境下对零件和夹具进行了三维模型设计,使得该夹具的设计直观化。并通过对其定位误差和夹紧力的理论计算分析,证明该设计能够满足此薄壁零件的加工工艺要求。     

Damping of the torsional vibrations of a transmission

A method is proposed for damping the torsional vibrations of an automobile transmission. In this method, the rigidity of the damper is reduced to a value at which vibration is impossible.     

A series solution for spatially coupled deflection analysis of thin-walled Timoshenko curved beam with and without elastic foundation

The exact solutions for the spatially coupled deflection and the normal stress at an arbitrary location of a crosssection of the thin-walled Timoshenko curved beam with symmetric and non-symmetric cross-sections with and without two types of elastic foundations are newly presented using series solutions for the displacement parameters. The equilibrium equations and the force-deformation relations are derived from the elastic strain energy including the effects of shear deformation and the axial-flexural-torsional coupling, and the strain energy considering the foundation effects. The explicit expressions for displacement parameters are derived by applying the power series expansions of displacement components to the simultaneous ordinary differential equations. Next, the element stiffness matrix is determined by using the force-deformation relationships. The normal stress at any arbitrary location of the cross-section for a curved beam is evaluated from the stiffness matrix. To verify the validity and the accuracy of this study, the displacements and the normal stresses of curved beams are presented and compared with the analytical solutions, the finite element results using the isoparametric curved beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of SAP2000. This paper was recommended for publication in revised form by Associate Editor Maenghyo Cho Nam-Il Kim received his B.S. degree in Civil and Environmental Engineering from Sungkyunkwan University, Korea, in 1996. He then received his M.S. and Ph.D. degrees from Sungkyunkwan University in 1998 and 2004, respectively. Dr. Kim is currently a research professor at Civil and Environmental Engineering at Myongji University in Korea. Dr. Kim’s research interests include stability and vibration of steel and composite structures. Dong Ku Shin received his B.S. and M.S. degrees in Civil Engineering from Seoul National University, Korea, in 1983 and 1985, respectively. He then received his Ph.D. degree from Virginia Tech. at Blacksburg, VA, USA, in 1990. Dr. Shin is currently a professor of Civil and Environmental Engineering Department at Myongji University in Korea. Prof. Shin’s research interests include LRFD design of steel bridges and stability of composite structures.     

Deep drawing of a thin-walled hemisphere

A model of deep drawing of a thin-walled hemisphere with a flat bottom from a plane blank by a rigid punch, considering the work hardening and wall thickness variation is developed. Elastic bending and von Mises membrane rigid-plastic strain with different friction coefficients at the punch and die contact boundaries are considered. The computational model determines the distribution of the wall thickness and the material work hardening along the shell generatrix, the drawing force versus punch displacement relation, and the critical parameters of the process in which some defects are probable.     

一种新颖实用的薄壁深孔零件加工方法

在分析深孔薄壁零件的工艺特性和加工难点的基础上,提出了一套实用有效的加工工艺方案,并设计制造了一套具有较高刚性的专用刀具和专用支撑夹具.借助于专用刀具和支撑夹具,上述工艺方案得以顺利实施,并很好的保证了零件的加工质量,达到了各项技术要求.实际使用证明,应用此套加工工艺不仅提高了生产效率,同时还有效降低了加工成本.     

Control reversal and torsional divergence analysis for a high-aspect-ratio wing

Modern aircraft tends to adopt more flexible wing structures, such as high-aspect-ratio wings. They also feature advancements in performance, such as increased flight speeds. Thus, their aeroelastic characteristics and related safety factors became more important. By analyzing the static aeroelasticity of aircraft accurately, such as the control reversal and torsional divergence speed, it is expected that safe flights can be guaranteed within a prescribed flight envelope. In this paper, the control surface effectiveness and torsional divergence were analyzed using the following engineering tools: nonlinear multi-body dynamic analysis, DYMORE, cross-sectional analysis, two-cell analysis or VABS, and two-dimensional aerodynamic coefficient analysis for the subsonic isolated airfoil XFOIL. The results were verified through comparisons with those based on a typical section model with thin strip theory. Aluminum and composite single-cell box beam wings were analyzed in this paper. It is found that the control reversal and torsional divergence flight speeds as measured here were in good agreement with those predicted by a typical section model.     

Nonlinear torsional vibrations of a stretched rod

A thin elastic stretched rod is considered. Nonlinear torsional vibrations of the rod are investigated. The Reissner variational principle is used to determine the vibration amplitude. It is shown that, in contrast to the linear case, the amplitude is a limited quantity.     

Investigations of the torsional vibrations of a shaft with disks

Investigations of the direct and inverse spectral problems of torsional vibrations of a shaft with disks were performed. The Tolle method was used to solve the direct problem. The influence of the characteristics of a shaft with loads on the frequencies of its torsional vibrations was studied using the solution of the direct problem. It was shown that an increase in the moments of inertia of disk masses leads to a decrease in vibration frequencies and that an increase in the torsional rigidity of segments of a shaft leads to an increase in the system’s vibration frequencies. Examples, tables, and plots of the corresponding dependences are presented.