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.     

On the load distribution of thin-walled beams subjected to bending with respect to the cross-section distortion

The paper deals with the estimation of the load distribution under which distortion of the cross-sections of thin-walled beams subjected to bending cannot occur. It is assumed that beam walls are hinged along their longitudinal edges. Beams with closed and open rectangular cross-sections, with three or two cells, with two or one axis of symmetry are considered. It is shown that the problem can be treated by two equivalent beams, defined by the shear flow zero points of the beam with rigid cross-section. The beam load must be distributed in the plane of beam walls, proportionally to the cross-section moments of inertia of the equivalent beams. Some illustrative examples are given.     

Numerical and analytical study on deep biaxial bending collapse of thin-walled beams

The objective of the paper is to get a fundamental understanding of large rotation response of thin-walled prismatic beams subjected to biaxial bending. Square and rectangular section beams were subjected to a cantilever bending with hinge rotation up to 40°. A general purpose nonlinear FE code PAM-CRASH was used to generate results. Altogether 94 computer runs were made by changing aspect ratio, condition of the loading and bending plane with respect to principal inertia axes of the cross section. The main emphasis was placed on the determination of initial and subsequent yield loci. It was also shown that the direction of the vector of generalized strain rates obeys the normality rule. Calculated in the paper was also a critical angle defining a preferable direction of bending. The present finding can be used to better interpret results of FE calculation of automotive bodies and to develop simplified crash-oriented design tools.     

Vlasov torsion of elastic-ideally-plastic beams of thin-walled open cross-section

Prismatic beams of thin-walled open cross-section are studied. The sectional forces are a bimoment B and a Vlasov torsional moment M_w. The material is assumed to be elastic-ideally-plastic. Special consideration is given to the monosymmetric I-section. Exact and approximate solutions are derived for a cantilevered I-beam. These solutions are compared in a numerical example.     

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.     

Vlasov torsion of non-linearly elastic beams of thin-walled open cross-section

A beam lamina of thin-walled open cross-section is considered. The sectional forces are a bimoment (warping moment) B and a Vlasov torsional moment M_w. A formula for the sectorial co-ordinate containing three unknowns and being suitable for numerical calculations is given. The material is assumed to be non-linearly elastic with the constitutive equation = K|σ|ⁿ sgn σ in uniaxial tension. The relation M_w = B′ is shown to hold also for n ≠ 1. The location of the centre of twist as a function of n is determined for a monosymmetric I cross-section. The torsion-bending analogue for beams is found to be valid also for n ≠ 1.     

Lateral-torsional buckling analysis of thin-walled beams including shear and pre-buckling deformation effects

In this paper, lateral-torsional buckling behavior of open-section thin-walled beams is investigated based on a geometrically nonlinear formulation, which considers the effects of shear deformations. A finite element numerical solution along with an incremental-iterative solution procedure is adopted to trace the pre-buckling as well as the post-buckling equilibrium paths. Formulation is applicable to a general type of open-section and load position effects are also included. Numerical results are validated through comparisons with experimental results and those based on other formulations presented in the literature. Comparisons have also been made between the results based on fully nonlinear analysis and linearized buckling analysis in order to illustrate the effects of pre-buckling deformations as well as the shear deformations on the buckling load predictions. Examples illustrate the influence of beam slenderness and moment gradient on the effects of pre-buckling deformations in predicting bucking loads.     

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.     

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton’s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin’s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.     

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.     

The sectorial co-ordinate of elastic-ideally-plastic beams of three thin-walled open cross-sections

A beam lamina of thin-walled open cross-section is considered. The sectional forces are a bimoment B and a Vlasov torsional moment M_w. The material is assumed to be elastic-ideally-plastic. The change in sectorial co-ordinate with increasing bicurvature is studied for three frequently used cross-sections.     

A consistent approach to linear stability of thin-walled beams of open section

This paper presents a stability criterion for thin-walled beams of curvilinear section, based on the positive definiteness of the second variation of the potential energy. The internal stresses are derived according to a shell model which incorporates the kinematic restraints of thin-walled beams. In addition to the classical terms of the potential energy, the geometric effects of an applied torque, as well as flexural-extensional and torsional-extensional coupling terms, are accounted for.     

The sectorial co-ordinate of non-linearly elastic beams of three thin-walled open cross-sections

A beam lamina of thin-walled open cross-section is considered. The sectional forces are a bimoment B and a Vlasov torsional moment M_w. The material is assumed to be non-linearly elastic with the constitutive equation = K|σ|ⁿ sgn σ in uniaxial tension. A slight redefinition of the generalized sectorial moment of inertia I⁽ⁿ⁾_ω is given. The sectorial co-ordinate as a function of n is determined for three commonly used cross-sections.     

基于圆轴扭转理论的安全带限力杆设计

当车辆发生碰撞时,安全带对乘员安全具有重要的保护作用,但是,普通安全带对乘员胸部的冲击力较大,加大了普通安全带对乘员的伤害。限力型汽车安全带能够有效减小汽车碰撞对乘员胸部伤害,通过研究限力型卷收器的限力杆设计,推导计算过程,结合试验验证,为限力杆的设计提供理论依据和计算方法,提高安全带的本土化设计能力。     

薄壁碰撞吸能部件的可适应响应面法轻量优化分析

薄壁部件在碰撞过程中发生塑性变形,从而吸能以保护其它部件的安全,在此基础上轻量化也是一个值得考虑的研究方向。采用有限元分析方法进行结构分析,并通过试验设计空间选择少量样本获得响应面模型,响应面法是数学方法和统计方法结合的产物,用这种方法可以寻找考虑了输入变量值和目标函数之后的响应最佳值。最后调用DYNA求解器进行批处理,取得优化的满足多约束条件的结果数据,碰撞时间得到延长,吸能大小变化很小,优化的结果是满足碰撞安全的,同时达到了轻量化的目的。     

Analysis of structures with thin-walled open sections

The problems of incorporating the effects of cross-sectional warping, offset shear centre and arbitrary orientation of an element in an assembled structure are examined. The kinematics of a variety of joint intersections and stiffening schemes are incorporated and transformation matrices are developed which enable solutions to be obtained comparing with a number of experimental and analytical studies.     

Torsional crushing of foam-filled thin-walled square columns

Torsional crushing behavior of foam-filled thin-walled square columns were investigated analytically, numerically and experimentally. The lower and upper bounds on the torsional resistance of foam-filled columns were established analytically. Numerical simulations were carried out and showed that the presence of the filler changes the torsional collapse mechanism and gives rise to higher order sectional collapse modes, which results in a higher torsional resistance. Torsional experiments were performed and results were compared to the analytical and numerical solutions with reasonably good agreement. It was found that bonding of the foam to the walls changes the deformation mode by spreading deformation over the whole length. The corresponding torsional resistance is also larger for the first 40° of rotation. It is concluded that fitting prismatic members with the aluminum foam of a density ranging from 0.14 to 0.28 g/cm³ can double the energy absorption of a given member.     

Energy absorption of pressurized thin-walled circular tubes under axial crushing

By pressurizing cellular materials, honeycombs, or thin-walled structures, their energy absorption can be greatly enhanced, and this enhancement can be controlled by the applied pressure. This concept shines light on the possibility of achieving adaptive energy absorption. To investigate the effect of internal pressure on energy absorption of thin-walled structures, this paper presents a study of axial crushing of pressurized thin-walled circular tubes. In the experiments, three groups of circular tubes with radius/thickness ratio R/t=120-200 were axially compressed under different pressurizing conditions. The results show that with an increase of internal pressure, the deformation mode switches from diamond mode with sharp corners to that with round corners, and eventually to ring mode. In diamond mode, the mean force of the tubes increases linearly with internal pressure. The enhancement comes from two mechanisms: direct effect of pressure and indirect effect due to interaction between pressure and tube wall. After the deformation switches to ring mode, the enhancement resulting from the second mechanism becomes weaker. Based on experimental observations, the deformation mode, energy dissipation mechanisms as well as interaction between internal pressure and tube wall are analyzed theoretically and the theoretical results are in good agreement with the experimental ones.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams

An improved third order shear deformation theory is employed to investigate thermal buckling and vibration of the functionally graded beams. A power law distribution is used to describe the variation of volume fraction of material compositions. The functionally graded material properties are assumed to vary smoothly and continuously across the thickness of the beams. The Ritz method is adopted to solve the eigenvalue problems that are associated with thermal buckling and vibration in various types of immovable boundary conditions. The parametric study covered in this paper includes the effects of material composition, temperature-dependent material properties, and slenderness ratio.