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.     

An analytical model for the collapsing deformation of thin-walled rectangular tube in rotary draw bending

The collapsing deformation of an outer flange is the key factor affecting the forming quality of a thin-walled rectangular tube during the rotary draw bending process. Therefore, the collapsing deformation is a problem that needs an urgent solution. Firstly, based on the simplified model for loads and deformation of the outer flange, the force acted by the core die and the bending moment acted by the clamp die are obtained analytically. Then, the analytical formula of collapsing deformation is deduced based on the theory of plate and shell, and finally, the analytical model is validated by comparison with simulated and experimental results. The study is of great significance to elevate the forming quality of a thin-walled rectangular tube during the rotary draw bending process.     

Springback after the biaxial elastic-plastic pure bending of a rectangular plate—I

The elastic springback undergone by a rectangular plate after submitting it to biaxial elastic-plastic pure bending is determined. The analysis is first based on non-proportional loading using the Tresca criterion and then on proportional loading for both the Mises and Tresca criteria. Some numerical results and examples are given.     

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.     

Effect of clearance on wrinkling of thin-walled rectangular tube in rotary draw bending process

The clearances between tube and various dies have a significant and complicated influence on the onset of wrinkling during the rotary draw bending process. To study the effect of clearance on wrinkling, a 3D finite element (FE) model of the process for thin-walled rectangular aluminum alloy tube has been built using the explicit code ABAQUS/Explicit and validated by comparing the experiment. Then, simulation and analysis of the process have been carried out based on the model. The influence laws of clearances between tube and various dies on wrinkling have been studied and the reasonable combination of clearances obtained using the combination method of 3D FE simulation and orthogonal experimental design. The results show that with the increase of tube mandrel clearance, Δc _m, and tube bending die clearance, Δc _b, the wrinkling wave number decreases rapidly, whereas the wrinkling wave height increases sharply. The effects of tube wiper die clearance, Δc _w, and tube pressure die clearance, Δc _p, on wrinkling are not significant. The reasonable combination of clearances is Δc _m?=?0.15 mm, Δc _b?=?0 mm, Δc _w?=?0 mm, and Δc _p?=?0 mm. These achievements are helpful to the design and optimization of the process.     

Effect of an ultralight metal filler on the bending collapse behavior of thin-walled prismatic columns

The effect of low-density metal filler, such as aluminum foam or honeycomb, is studied on the bending collapse resistance of thin–walled prismatic columns. A combination of analytical and numerical results is used to predict the initial and post collapse response of empty and filled columns. Closed-formed solutions for the bending-rotation characteristics are constructed in terms of the geometrical parameters and the filler strength. The low-density metal core retards sectional collapse of the thin-wall column, and increases bending resistance for the same rotation angle. Numerical simulations show that, in terms of achieving the highest energy absorption to weight ratio, columns with aluminum honeycomb or foam core are preferable to thickening the column wall. Moreover, the presence of adhesive improved the specific energy absorption significantly.     

Modeling and bending vibration control of nonuniform thin-walled rotating beams incorporating adaptive capabilities

This paper addresses the problems of modeling and bending vibration control of tapered rotating blades modeled as nonuniform thin-walled beams and incorporating adaptive capabilities. The blade model incorporates non-classical features such as transverse shear, secondary warping and includes the centrifugal and Coriolis force fields. For the non-adaptive system, an assessment of a number of non-classical features including the taper characteristics is accomplished. The adaptive capabilities are provided by a system of piezoactuators bonded to the structure surface and spread over the entire span of the beam. Based on the converse piezoelectric effect and on the out-of-phase actuation, the piezoactuators produce a localized strain field in response to the applied voltage, and consequently, a change of the dynamic response characteristics is induced. A combined feedback control law relating the piezoelectrically induced transversal bending moment at the beam tip with the kinematical response quantities appropriately selected is used, and the beneficial effects upon the closed-loop dynamic characteristics of the blade are highlighted.     

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.     

Inelastic behaviour of rectangular beams subjected to steady axial load and cyclic bending

Analytical and numerical solutions are given for the inelastic behaviour of a rectangular cross-section beam, subjected to a steady axial load and cyclic, fully reversed load-controlled bending. Elastic-perfectly plastic, linear isotropic and kinematic materials are examined and solutions for the strain accumulation and cyclic reverse plastic strains obtained. Also the effect of introducing a creep dwell period into the cycle is considered.     

Relations between the stress components and cross-sectional distortion of thin-walled rectangular waveguide tube in rotary draw bending process

The significant cross-sectional distortion is one of the major problems in the bending of thin-walled rectangular waveguide tube. The cross-sectional distortion, which contains the flange distortion and the web distortion, depends on the stress components distribution. In this paper, the cross-sectional distortion characteristics are investigated using a three-dimensional finite element (FE) model. Results show that: the maximum flange distortion locates at the symmetric line; meanwhile, the maximum web distortion locates at the extrados ridge of the tube. The deformation zone of the tube can be divided into three sub-zones considering the loads and deformation, viz., the clamp die affect zone, the middle zone, and the mandrel/cores affect zone. Then the underlying relations between the cross-sectional distortion and the stress components are obtained. It is found that the flange distortion has a close relation with the circumferential stress. At the same time, the web distortion is relevant to both the tangential and the circumferential stress. The above relations are verified by FE models with different cores number. Moreover, some guidelines are introduced to help reduce the cross-sectional distortion.     

一种薄壁梁加强结构及其动态抗弯性能分析

针对大客车车身骨架侧墙立柱薄壁梁结构侧翻耐撞性能薄弱的特点,提出了一种等强度贴板加强立柱梁结构.采用LS-DYNA软件对单根悬臂方管在侧向冲击载荷下的动力响应进行了数值模拟,比较分析了不同加强方式下立柱的抗弯性能.结果表明,与薄壁方管、贴板薄壁方管和泡沫铝填充方管相比,等强度贴板方管立柱的吸能性能(或比吸能)显著提高,整体结构在指定冲击位移内充分参与变形,没有出现因局部塌陷而产生的塑性铰,说明这种加强立柱结构是客车侧翻耐撞性立柱的良好型式.     

Non-linear non-uniform torsion of thin-walled beams

A theoretical and experimental investigation is made to study the non-linear behaviour of thin-walled elastic beams of open sections subjected to non-uniform torsion. The differential equations describing the behaviour and the associated boundary conditions are presented. A perturbation solution is carried out to obtain the torque-rotation characteristics and also the axial strain distribution. The accuracy of the perturbation solution is checked against a numerical solution of the governing equation.

A set of experimental tests on two I-sections is carried out to verify the theoretical calculation. It is shown that the torque-rotation behaviour exhibits a hardening non-linear characteristic. The agreement between theory and test results is reasonable.     

Elastoplastic buckling of rectangular plates in biaxial compression/tension

This paper examines the elastoplastic buckling of a rectangular plate, with various boundary conditions, under uniform compression combined with uniform tension (or compression) in the perpendicular direction. The analysis is based on the standard linear buckling equations and material behaviour is modelled by the small strain J₂ flow and deformation theories of plasticity. A detailed parametric study has been made for Al 7075 T6 over a range of plate geometries (a/b=0.25–4,a/h≈20–100) and with three sets of boundary conditions (four simply supported boundaries and the symmetric combinations of clamped/simply supported sides). For sufficiently thin plates we recover with both theories the classical elastic results. However, for thicker plates there is a remarkable difference in the buckling loads predicted by these two theories. Apart from the expected observation that deformation theory gives lower critical stresses than those obtained from the flow theory, we report on the existence of an optimal loading path for the deformation theory model. Buckling loads attained along the optimal path—specified by particular compression/tension ratios—are the highest possible over the entire space of loading histories. By contrast, no similar optimum has been found with the flow theory. This discrepancy in the buckling behaviour, obtained from the two competing plastic theories, provides a possibly new illustration of the plastic buckling paradox.     

An investigation on collapse behavior of shear localization in elasto-thermo-viscoplastic materials

The stress collapse in the formation of shear bands in elasto-thermo-viscoplatic materials is systematically studied within the framework of one-dimensional formulation via analytical and numerical methods. The elastic energy released in a domain is found to play an important role in the collapse behavior of shear localization. A non-dimensional parameter named the stability indicator is introduced to characterize the collapse behavior, with approximate forms of the incremental governing equations. The stability indicator offers useful information regarding the degree of severity of an abrupt change of deformations during the stress collapse. Numerical experiments are carried out to verify the stability indicator by varying material properties.     

Vibration analysis of thin-walled curved beams using DQM

The differential quadrature method (DQM) is applied to computation of the eigenvalues of small amplitude free vibration for thin-walled curved beams including a warping contribution. Natural frequencies are calculated for singlespan, curved, wide-flange uniform beams having a range of nondimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curvature, included angle of the curve, polar mass moment of inertia, and various end conditions. Results are compared with existing exact and numerical solutions by other methods for cases in which they are available. It is found that the DQM gives good accuracy even when only a limited number of grid points is used. In addition, results are given for a cantilever beam not previously considered for this problem. Finally, parametric results are presented in dimensionless form.     

Identification of multiple cracks in beams under bending

The identification of a transverse crack on a beam is the subject of many investigators. Identifying the crack means to find its position and depth. In many cases there are more than one cracks on a beam. Then the solutions, or the combinations of parameters characterising the cracks are more and the problem becomes more complicated particularly when the crack must be identified using one more parameter, the relative each other angular position.In the present paper the dynamic behaviour of a cracked beam with two transverse surface cracks is studied. Each crack is characterised by its depth, position and relative angle. Both cracks are considered to lie in arbitrary angular positions with respect to the longitudinal axis of the beam and at any distance from the left end. A local compliance matrix of two degrees of freedom, bending in the horizontal and the vertical planes is used to model the rotating transverse crack in the shaft and is calculated based on the available expressions of the stress intensity factors and the associated expressions for the strain energy release rates. The compliance matrix is calculated for the first time at any angle of rotation. Thus, the compliance is given as a function of both the crack depth and the angular location. These expressions are usable, due to the stress intensity function limitations, only for limited regions around the zero angular position of the crack and not for every crack angle. For these cases, B-spline curves are used to interpolate the known points and a function in analytical form is given for every crack depth and angle. It is well known that when a crack exists in a structure, such as a beam, then the natural frequency of vibration decreases. This reduction is studied here for six independent parameters namely the depth, the location, and the rotational angle of each crack. By keeping these six parameters constant, the first three flexural eigenmodes can be computed and plotted.Due to its sensitivity in slope or displacement changes the theory of wavelets is used here to identify the locations of the cracks reducing thus the number of independent parameters. As it is well known the existence of a crack on a beam in bending, creates in the elastic line of the beam a slope discontinuity analog generally to the crack depth and additionally here to the angular position. The wavelet transformation of a vibration mode or of the vibration response of the structure under some circumstances could be used to locate the cracks. If the positions are known, then the depths and the respective angles can be determined. Here the diagrams of the first three eigenvalues versus both the crack depth and the rotational angle, are used to identify the remaining unknown parameters for both cracks.     

An experimental investigation into the performance of externally pressurized rectangular air bearings

An experimental programme which was designed to investigate the performance characteristics of externally pressurized rectangular air bearings considering the effects of pad geometry and inlet gas conditions is presented. Three forms of rectangular bearings were considered: one arrangement consisted of continuous plane surfaces, the second had a recessed pad and a plane bed and the third had a plane bed and a recessed pad with a step in the bearing area following the recess.

The results revealed that the recess length has a dominant effect on the pressure distribution in the air film. The effect of pressure depression is obvious at extremely small clearances and supply pressures. Such an effect is exaggerated when a step is introduced in the pad surface. The introduction of the step improves the load-carrying capacity of the bearing as long as the bearing is free from shock-wave formation.     

Non-linear bending of beams of variable cross-section

For the non-linear bending of cantilever beams of variable cross-section, the effect of large deformations, but with linear elasticity, is considered. The governing integral equation is solved by a numerical iterative procedure. Results for some typical cases are obtained and compared with some of those available in the literature.     

Bending collapse of rectangular and square section tubes

The bending collapse behaviour of rectangular and square section tubes is studied theoretically and experimentally. A limit analysis technique was employed and a set of formulae relating the hinge moment and associated angle of rotation was derived. The theoretical predictions were verified by comparison with 56 quasi-static bending tests on 27 different sections having aspect ratios ( ) varying between 3.0 and 0.33 and with width to wall thickness ratios ( ) between 128 and 9.14. Very good agreement was found between the theoretical predictions and experimental results for the whole range of sections.