Vibration and buckling characteristics of weld-bonded rectangular plates using the flexibility function approach

Structures with a combination of spot welds and adhesive bonding, often referred to as weld-bonded structures, are likely to see increasing usage in automotive and other engineering structures. The present study considers a representative weld-bonded rectangular plate having simple supports on two opposite edges and weld-bonded support conditions with periodic spot welds along the other two edges. The study shows that the flexibility function approach for modeling free edges with point supports [Bapat AV, Venkatramani N, Suryanarayan S. Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates. Journal of Sound and Vibration 1988;120(1):127–40; Bapat AV, Venkatramani N, Suryanarayan S. A new approach for the representation of a point support in the analysis of plates. Journal of Sound and Vibration 1988;120(1):107–25; Bapat AV, Venkatramani N, Suryanarayan S. The use of flexibility functions with negative domains in the vibration analysis of asymmetrically point-supported rectangular plates. Journal of Sound and Vibration 1988;124(3):555–76; Bapat AV, Suryanarayan S. Free vibrations of periodically point-supported rectangular plates. Journal of Sound and Vibration 1989;132(3):491–509; Bapat AV, Suryanarayan S. The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges. Journal of Sound and Vibration 1989;128(2):203–33; Bapat AV, Suryanarayan S. Free vibrations of rectangular plates with interior point supports. Journal of Sound and Vibration 1989;134(2):291–313; Bapat AV, Suryanarayan S. Importance of satisfaction of point-support compatibility conditions in the simulation of point supports by the flexibility function approach. Journal of Sound and Vibration 1990;137(2):191–207; Bapat AV, Suryanarayan S. A fictitious foundation approach to vibration analysis of plates with interior point. Journal of Sound and Vibration 1992;155(2):325–41; Bapat AV, Suryanarayan S. A theoretical basis for the experimental realization of boundary conditions in the vibration analysis of plates. Journal of Sound and Vibration 1993;163(3):463–78], used in the direct series solution of the governing differential equations, can be employed very effectively to study the vibration and buckling characteristics of the weld-bonded rectangular plates. This is done by using a flexibility function constructed in terms of Fourier components to model the weld-bonded edge that represents the finite uniform flexibility of the adhesively bonded segment of the weld-bonded edge along with zero flexibility at the spot welds modeled as discrete point supports. A detailed convergence study shows that by a proper choice of the number of terms used to represent the flexibility function and the number of terms in the Levy sine series for the solution of the plate displacement, accurate results can be obtained for vibration and buckling characteristics. This paper also includes a parametric study undertaken to show the effect of plate geometry, number of spot welds and adhesive joint parameters. The paper also discusses how such parametric studies can be of use to the designer in arriving at an optimal joint configuration of weld-bonded rectangular plates from linear elastic buckling and free vibration considerations.     

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.     

正置正交加筋薄壁柱壳优化设计参数化有限元方法研究

利用有限元软件建立轴压正置正交加筋薄壁圆柱壳的参数化有限元分析模型,研究结构参数对薄壁加筋圆柱壳结构的临界载荷和屈曲模式的影响。随着蒙皮厚度的增加,结构的屈曲模式由局部屈曲逐步变化到总体屈曲,屈曲载荷上升;随着加筋厚度或宽度的增加,由总体屈曲变化到局部屈曲,屈曲载荷上升。通过等体积时的参数变化对屈曲载荷和屈曲模式的影响研究,表明在对应某体积的设计中,只有一种设计使结构屈曲载荷达到最大,而当此最大的屈曲载荷等于设计载荷时,是最轻重量的设计。在此基础上发展一种基于APDL(Ansys parametric design language)语言的薄壁加筋圆柱壳结构优化设计方法,利用该方法给出设计算例的优化结果。     

计算E形截面受弯构件整体稳定临界弯距单元类型选择的探讨

本文分别采用弹性理论公式、ANSYS软件的梁单元和壳单元计算单跨简支受弯铝合金E形截面薄壁构件在不同载荷形式、不同载荷作用点位置以及不同跨距时的弹性整体失稳临界弯矩。对计算结果的分析表明,使用壳单元计算E形截面薄壁构件受弯整体稳定临界弯矩时,能达到较好的精度。     

Elastoplastic dynamic instability of long circular cylindrical shells under pure bending

This paper presents a numerical study which is concerned with the prediction of the response and instabilities in long circular cylindrical shells under dynamic pure bending. Of particular interest is the response of such shells, bent into the plastic range of the material, and the various instability characteristics of the shells under dynamic bending (sudden step load). It was found that the major deformation characteristic of the shells is essentially similar to that observed in the static bending when the applied moment is much smaller than the critical dynamic moment. However, when the applied moment is close to the critical dynamic moment, the ovalization of the shell cross-section was found to be localized over a length of several shell diameters in the central region, even though the response of the shell curvature was shown to be still stable in this case. When the applied moment reaches the critical dynamic moment, the response of the shell curvature was shown significantly increasing with time and the shell buckled catastrophically. For thicker shells, it was found that the development of localized ovalization of the shell cross-section is the major factor that causes shell dynamic instability. For thinner shells, however, besides the localized ovalization, the bifurcation induced by short wavelength ripples on the compressed side of the shell was also observed in the initial buckling patterns. After the bifurcation, the initial buckling pattern was replaced by the final postbuckling mode characterized by a localized sharp cupping in the centre of the shell.     

Thermoelastic lateral-torsional buckling of fixed slender beams under linear temperature gradient

The thermal expansions and rotations that result from a linear in-plane temperature gradient field are fully restrained at the ends of a fixed beam. These restrained expansions and rotations will produce internal bending and compressive actions in the beam, and these actions increase with an increase of the temperature differential and average temperature of the linear temperature gradient field. When these actions reach critical values, the fixed beam may bifurcate from its primary equilibrium state to a buckled equilibrium configuration. This paper presents a systematic treatment of classical buckling analysis for thermoelastic lateral-torsional buckling and for in-plane thermoelastic flexural buckling of a fixed beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is shown that the effective centroid and shear centre, rather than the geometric centroid and shear centre, should be used in formulating the thermoelastic prebuckling and buckling analysis and that the effects of temperature on the buckling resistance need to be considered. The thermoelastic lateral-torsional buckling of a fixed beam under a linear temperature gradient field is more complicated than its mechanical counterpart for uniform bending or for uniform compression, and iterative methods are needed to obtain accurate solutions.     

蜂窝夹芯胞元壳的屈曲特性研究

以正六边形蜂窝夹芯的胞元壳为例,将Euler细长杆的稳定理论与板壳力学中的稳定理论相结合,提出了薄壁胞元壳轴向受压临界载荷的计算公式和屈曲模态的三重级数表达式。根据胞元壳不同的失效形式,将其分成了短胞元壳、长胞元壳和特长胞元壳,确定了不同类型胞元壳的临界载荷控制方程和其对应的一阶屈曲模态,在此基础上提出了胞元壳临界载荷总图。用3D有限元数值模拟技术,通过对铝质正六边形蜂窝胞元壳进行模拟,将其结果与公式的计算结果相比,两者非常接近,证实了理论分析的有效性。其结论对蜂窝夹层结构和薄壁棱柱壳的理论研究有重要的指导意义。     

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

Free vibration and buckling response of hat-stiffened composite panels under general loading

The paper presents finite element free vibration and buckling analysis of laminated hat-stiffened shallow and deep shells using arbitrarily oriented stiffener formulation. Modified approach for modelling the curved stiffener is implemented using necessary transformations. A simplified stiffener formulation is presented to accommodate various shapes of stiffener shapes in developing the rigidity matrix for the finite element formulation. Investigation has been carried out on free vibration and buckling analyses of laminated composite stiffened shell structures with laminated open section (rectangular or ‘T’ shaped) and closed section (‘hat’ shaped) stiffeners. Parametric study on the hat-stiffened panels for the free vibration and buckling analyses confirms that the closed section stiffener being torsionally rigid is found to show better performance over open section stiffeners.     

弹性圆柱壳的稳定性优化设计

研究任意轴对称边界条件下和受均布法向载荷作用圆柱壳的稳定性优化设计问题，即极大化屈曲临界载荷。利用能量原理分析轴对称变厚度圆柱壳的分支点屈曲，将求解屈曲临界载荷变成求解广义特征值方程，使圆柱壳稳定性优化设计成为极大化最小特征值问题。实际算例验证了本方法的有效性。研究结果可用于圆柱壳的加肋优化设计。     

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.     

Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions

Buckling analysis of nanobeams is investigated using nonlocal continuum beam models of the different classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Levinson beam theory (LBT). To this end, Eringen’s equations of nonlocal elasticity are incorporated into the classical beam theories for buckling of nanobeams with rectangular cross-section. In contrast to the classical theories, the nonlocal elastic beam models developed here have the capability to predict critical buckling loads that allowing for the inclusion of size effects. The values of critical buckling loads corresponding to four commonly used boundary conditions are obtained using state-space method. The results are presented for different geometric parameters, boundary conditions, and values of nonlocal parameter to show the effects of each of them in detail. Then the results are fitted with those of molecular dynamics simulations through a nonlinear least square fitting procedure to find the appropriate values of nonlocal parameter for the buckling analysis of nanobeams relevant to each type of nonlocal beam model and boundary conditions.analysis.     

Analytical buckling solutions for mindlin plates involving free edges

This paper investigates the buckling behaviour of rectangular Mindlin plates having two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The proper boundary conditions at free edges subjected to in-plane loads have been examined. The buckling analysis is performed by applying the concept of state space to the Levy-type solution method to obtain the closed-form critical loads from the governing differential equations. The results, where possible, are compared with existing solutions to verify the validity of the solution method. The differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are reported. Several design charts representing the essential features of the critical load characteristics of rectangular plates with two opposite edges simply supported at least one free edge are obtained. The critical loads can be determined from the design charts without difficulty.     

胶层厚度对胶接波纹板三点弯曲力学性能影响研究

三明治波纹板采用轻质化和薄壁化的结构设计方法，可以作为承载和吸能结构应用于汽车底盘前纵梁、车身Y字梁等，而现有文献缺少针对胶层厚度对波纹板性能影响的研究。通过制备胶接波纹板并开展三点弯曲工况准静态加载失效试验，分析了不同胶层厚度对三明治波纹板整体承载和吸能特性的影响规律。试验发现，不同胶层厚度下铝合金胶接波纹板结构承载能力和破坏样貌存在差异，相比于0.5 mm和1.0 mm,0.2 mm胶层厚度的三明治结构的各项力学性能和稳定性较好。随后建立胶接波纹板三点弯曲加载有限元模型，实现了对波纹板整体加载变形过程和胶层损伤的模拟，通过与试验结果对比验证了有限元建模方法的有效性，为轻质金属胶接波纹板结构设计和性能评价提供可靠方法。     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method

The buckling of thin rectangular plates with nonlinearly distributed loadings along two opposite plate edges is analyzed by using the differential quadrature (DQ) method. The problem is considerably more complicated since it requires that first the plane elasticity problem be solved to obtain the distribution of in-plane stresses, and then the buckling problem be solved. Thus, very few analytical solutions (the only one available in the literature is for rectangular plates with all edges simply supported) have been available in the literature thus far. Detailed formulations and solution procedures are given herein. Nine combinations of boundary conditions and various aspect ratios are considered. Comparisons are made with a few existing analytical and/or finite element data. It has been found that a fast convergent rate can be achieved by the DQ method with non-uniform grids and very accurate results are obtained for the first time. It has also been found that the DQ results, verified by the finite element method with NASTRAN, are not quite close to the newly reported analytical solution. A possible reason is given to explain the difference.     

Optimally reinforced cutouts in laminated circular cylindrical shells

The problem of designing a cutout in a load bearing structural member in the form of a shell, such that the cut structure maintains its stress state with a minimal departure from the stress state of the uncut structure is addressed herein. Symmetrically laminated composite circular cylindrical shells under hydrostatic compression and axial pressure are considered. Shallow thin shell (Donnell shell theory) lamination theory is utilized. The original (uncut) stiffness of the shell structures is recovered considerably by appropriately designing an edge reinforcement around the cutout. The buckling load of the designed shells are analyzed via the finite element method. An experimental investigation has been carried out to verify some of the results obtained from the finite element analysis. In the work presented, the reinforcement is modeled as a one-dimensional rod/beam type structural element.     

帽形截面耐撞薄壁结构的棱边强化与优化设计

为了满足车身轻量化要求,乘用车耐撞车身的关键承载结构越来越多采用超高强度薄钢板制作。但是为减小板厚而提高材料强度可能引发平板屈曲,导致薄壁箱型结构受压侧的极限承载力的降低。为此,根据平板和圆柱壳棱边的屈曲理论,讨论帽形截面的薄壁箱型结构受压侧的有棱边强化平板的后屈曲极限承载力。通过CAE数值模拟,求得单向均匀压缩带棱边强化平板的横截面应力分布和极限承载力。从而明确用棱边强化钢板部件替代热成形钢板部件在乘用车耐撞结构中应用的可行性。最后,梳理了棱边强化设计应用的前景和后续待研究的课题。     

Robust design of spot welds in automotive structures: A decision-making methodology

Automotive structures include thousands of spot welds whose design must allow the assembled vehicle to satisfy a wide variety of performance constraints including static, dynamic and crash criteria. The objective of a standard optimization strategy is to reduce the number of spot welds as much as possible while satisfying all the design objectives. However, a classical optimization of the spot weld distribution using an exhaustive search approach is simply not feasible due to the very high order of the design space and the subsequently prohibitive calculation costs. Moreover, even if this calculation could be done, the result would not necessarily be very informative with respect to the design robustness to manufacturing uncertainties (location of welds and defective welds) and to the degradation of spot welds due to fatigue effects over the lifetime of the vehicle. In this paper, a decision-making methodology is presented which allows some aspects of the robustness issues to be integrated into the spot weld design process. The starting point is a given distribution of spot welds on the structure, which is based on both engineering know-how and preliminary critical numerical results, in particular criteria such as crash behavior. An over-populated spot weld distribution is then built in order to satisfy the remaining design criteria, such as static torsion angle and modal behavior. Then, an efficient optimization procedure based on energy considerations is used to eliminate redundant spot welds while preserving as far as possible the nominal structural behavior. The resulting sub-optimal solution is then used to provide a decision indicator for defining effective quality control procedures (e.g. visual post-assembly inspection of a small number of critical spot welds) as well as designing redundancy into critical zones. The final part of the paper is related to comparing the robustness of competing designs. Some decision-making indicators are presented to help the analyst to plan robust resistance spot welds designs along with quality controls in order to insure a specified level of structural performance. All examples are presented on a full body-in-white structure (one million dofs and thousands spot welds).     

Large deflection analysis of elasto-plastic beams and frames

A general method for analysis of elasto-plastic beams and frames with large displacements is described in this paper. A hybrid-type beam element with 3df at each end is used in the analysis. The element stiffness matrix is obtained by inversion of a flexibility matrix, which is computed from an assumed distribution of internal forces along the element axis. This approach with approximations of the stress fields better imitates varying stiffness along the beam than the traditional approach with assumed displacement fields.Large displacement effects are taken into account by updating the geometry (both node co-ordinates and cross-section shapes) and accurately computing the elongations of each element. Partial yielding is considered across the height of the beam as well as along the element axis.