Plastic buckling of circular tubes under axial compression—part I: Experiments

Elastic buckling of cylindrical shells due to axial compression results in sudden and catastrophic failure. By contrast, for thicker shells that buckle in the plastic range, failure is preceded by a cascade of events, where the first instability and failure can be separated by strains of 1–5%. The first instability is uniform axisymmetric wrinkling that is typically treated as a plastic bifurcation. The wrinkle amplitude gradually grows and, in the process, reduces the axial rigidity of the shell. This eventually leads to a limit load instability, beyond which the cylinder fails by localized collapse. For some combinations of geometric and material characteristics, this limit load can be preceded by a second bifurcation that involves a non-axisymmetric mode of deformation. Again, this buckling mode localizes resulting in failure.The problem is revisited using a combination of experiments and analysis. In Part I, we present the results of an experimental study involving stainless steel specimens with diameter-to-thickness ratios between 23 and 52. Fifteen specimens were designed and machined to achieve uniform loading conditions in the test section. They were subsequently compressed to failure under displacement control. Along the way, the evolution of wrinkles was monitored using a special surface-scanning device. Bifurcation buckling based on the J₂ deformation theory of plasticity was used to establish the onset of wrinkling. Comparison of measured and calculated results revealed that the wrinkle wavelength was significantly overpredicted. The cause of the discrepancy is shown to be anisotropy present in the tubes used. Modeling of the postbuckling response and the prediction of the limit load instability follows in Part II.     

Plastic buckling of tubes under axial compression and internal pressure

The plastic buckling and collapse of long cylinders under combined internal pressure and axial compression was investigated through a combination of experiments and analysis. Stainless-steel cylinders with diameter-to-thickness values of 28.3 and 39.8 were compressed to failure at fixed values of internal pressure up to values 75% of the yield pressure. The first effect of internal pressure is a lowering of the axial stress–strain response. In addition, at some plastic strain level, the cylinder develops uniform axisymmetric wrinkling. Under continued compression, the wrinkles grow stably, gradually reducing the axial rigidity of the structure and eventually lead to a limit load instability. All pressurized cylinders remained axisymmetric until the end of the test past the limit load.The critical stress and wavelength were established using classical plastic bifurcation theory based on the deformation theory of plasticity. The evolution of wrinkling, and the resultant limit state, were established by modeling a periodic domain that is one half of the critical wavelength long. The domain was assigned an initial imperfection corresponding to the axisymmetric buckling mode calculated through the bifurcation check. The inelastic material behavior was modeled through the flow theory of plasticity with isotropic hardening. The variations of the axial response and of the limit strain with pressure observed in the experiments were reproduced well by the model. Inclusion of Hill-type anisotropic yielding in all constitutive models was required for good agreement between predictions and experiments.     

Quasi-static axial compression of thin-walled circular aluminium tubes

This paper presents further experimental investigations into axial compression of thin-walled circular tubes, a classical problem studied for several decades. A total of 70 quasi-static tests were conducted on circular 6060 aluminium tubes in the T5, as-received condition. The range of D/t considered was expanded over previous studies to D/t=10–450. Collapse modes were observed for L/D10 and a mode classification chart developed. The average crush force, F_AV, was non-dimensionalised and an empirical formula established as F_AV/M_P=72.3(D/t)^0.32. It was found that test results for both axi-symmetric and non-symmetric modes lie on a single curve. Comprehensive comparisons have been made between existing theories and our test results for F_AV. This has revealed some shortcomings, suggesting that further theoretical work may be required. It was found that the ratio of F_MAX/F_AV increased substantially with an increase in the D/t ratio. The effect of filling aluminium tubes with different density polyurethane foam was also briefly examined.     

Plastic mechanism analysis of circular tubes under pure bending

There are a number of solutions available to predict the response of a circular steel tube under pure bending. However, most of these solutions are based on an elasto-plastic treatment, which is complex and difficult to use in any routine design. This paper describes a theoretical treatment to predict the moment-rotation response of circular hollow steel tubes of varying D/t ratios under pure bending. The Mamalis et al. (J. Mech. Sci. 1989;203:411–7) kinematics model for a circular tube under a controlled moment gradient was modified to include the effect of ovalisation along the length of the tube. Inextensional deformation and rigid plastic material behaviour were assumed in the derivation of the deformation energy. The plasticity observed in the tests was assumed to spread linearly along the length of the tube. Two local plastic mechanisms (Star and Diamond shapes) were studied to model the behaviour observed in the tests especially during the unloading stage. The theoretical predictions are compared with the experimental results recently obtained by Elchalakani et al. (Quartral. J. Struct. Eng. 2000;3(3):1–16). Good agreement was found between the theoretical predictions and experimental moment-rotation responses, particularly for the Star shape mechanism. A closed-form solution is presented suitable for spreadsheet programming commonly used in routine design.     

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.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

轴心受压高强钢圆管稳定承载性能研究

对770MPa级高强钢管件的初始几何缺陷、残余应力和本构关系进行了测试,完成了66根770MPa级不同规格高强钢管件的轴压极限承载力试验,结合仿真分析,验证了添加初始缺陷的一致模态法分析高强钢管件极限承载力的准确性。     

On bend-stretch forming of aluminum extruded tubes — II: analysis

Part II describes a two-dimensional model of the bend-stretch-pressure forming process, which assumes that the tube shape and all applied loads are uniform along the length. This simplification results in a significant improvement in computational time over corresponding three-dimensional models. The model is validated through comparisons with experimental results discussed in Part I and its application to the design process is illustrated. The model is then used to study the current manufacturing process and to evaluate some possible alternative loading histories. The effect of tension and pressure on the cross-sectional distortion, springback and net elongation are discussed in light of the predictions. Additionally, the merits of changing the tension and pressure after forming are discussed. The simulations and corresponding experiments are used as the basis for a design methodology for selecting the order and magnitude of the loads applied during the forming process.     

铝合金E形截面轴压杆件屈曲分析方法的探讨

分别采用弹性理论公式法、ANSYS软件中的特征值屈曲分析法和非线性屈曲分析法计算两端简支的铝合金E形截面薄壁杆件在轴心受压时的弯扭屈曲临界载荷。计算结果表明,采用梁单元建模并用非线性法对E形截面薄壁杆件进行轴压屈曲分析,是一种既能满足一般工程设计要求又简单易行的计算方法。     

Buckling of elastic—plastic square tubes under bending

This paper addresses the response, buckling and collapse of long, thin-walled, seamless steel square tubes under pure bending using a combined experimental and analytical approach. All tubes tested had nominal cross-sections with height equal to 1 in. (25.4 mm) and ranged in height-to-thickness ratios (h/t) from 15.4 to 28.6. The experiments were conducted under curvature control. It was found that the deformation of the cross-section that accompanied bending was uniform along the tubes for low values of curvature. At higher values, periodic ripples with wavelengths approximately equal to the width of the cross-section appeared on the compression flange. These ripples increased in amplitude with further bending. For tubes with higher h∼ the increase was more pronounced. Tubes with lower h/t showed more moderate increases in ripple amplitude but developed regions spanning several ripples in which the cross-section deformation was more pronounced. In all cases, collapse occurred when a kink formed on the compression flange of the tube.Rayleigh—Ritz type formulations based on the principle of virtual work were developed to predict the response and buckling of the tubes. Results include predictions of the response considering the effect of uniform cross-section deformation and predictions of the critical curvature at which the ripples appear. The numerical results are in good agreement with experimental observations.     

On bend-stretch forming of aluminum extruded tubes — I: experiments

Tubular aluminum frame parts for automotive applications are best produced by extrusion. The tubes are then cold formed to the required shape by prestretching, pressurizing and bending them over rigid dies. Tension prevents buckling of the compressed side and significantly reduces the springback on unloading. An unwanted byproduct of the process is distortion of the cross section. It has been found that modest levels of pressure can reduce this distortion. The selection of the level of tension and pressure for optimum forming is presently empirical. The study discussed herein seeks to develop a scientific basis for optimizing forming processes such that buckling is avoided and distortion and springback are minimized. Part I describes a custom bend-stretch-pressure forming facility developed for the study. The facility is operated by one pneumatic and two servohydraulic closed-loop systems. This allows computer control of the process, and affords selectable loading histories. The planar forming process was modeled by approximating the tube as a nonlinear elastic–plastic beam which can undergo large rotations. The model was shown capable of reproducing accurately the loading history experienced by different sections along the length of the part during forming. Representative results from forming experiments involving rectangular aluminum are presented. The results are used to discuss the effect of friction, tension and pressure on the cross-sectional distortion, springback and net elongation of the part. Part II presents a model for establishing the cross-sectional distortion induced during forming. The model is used in conjunction with experimental results to establish ways of optimizing the process.     

Large deformations of thin-walled circular tubes under transverse loading—II: Experimental study of the crushing of circular tubes by centrally applied opposed wedge-shaped indenters

The behaviour of aluminium and mild steel tubes of 2 in. dia., 0.064 in. thickness and of lengths ranging from 2–24 in. loaded centrally by opposed wedge-shaped indenters is examined. Three modes of deformation are identified and analysed on the basis of plastic work considerations. Reference is also made to the behaviour of a very thin-walled tube of diameter to thickness ratio equal to 190.     

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part II: Approximate analytical solution

In this paper the method of harmonic balance is applied to study the nonlinear dynamic response in forced oscillations of a third-order nonlinear partial differential system obtained in Part I of this study. By using improved mode expansion examined in Part I, attention is concerned on the dynamic response of a rotating circular cylindrical shell with respect to the effect of precession of vibrating shape in the spectral neighborhood of one of the lowest natural frequencies. It can be found that the results obtained with the method developed in this study agree with numerical simulation in Part I very well, which indicate that this method has a good accuracy and is efficient for the dynamic analysis of rotating circular cylindrical shells. The stability of the period solutions is also examined in detail.     

Large deformations of thin-walled circular tubes under transverse loading—III: Further experiments on the bending of simply supported tubes

Further experimental results are presented concerning the transverse loading of simply supported tubes supplementing those given in ref. (1). Surface stresses are examined by means of the brittle laquer technique and strain gauges. The influence of the geometrical parameters on the modes of deformation of the tubes is discussed in general terms.     

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part I: Numerical solution

The nonlinear dynamic response of a cantilever rotating circular cylindrical shell subjected to a harmonic excitation about one of the lowest natural frequency, corresponding to mode (m=1, n=6),where m indicates the number of axial half-waves and n indicates the number of circumferential waves, is investigated by using numerical method in this paper. The factor of precession of vibrating shape ? is obtained, with damping accounted for. The equation of motion is derived by using the Donnell’s nonlinear shallow-shell theory, and is general in the sense that it includes damping, Coriolis force and large-amplitude shell motion effects. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Three different mode expansions are studied for finding the proper one which is more contracted and accurate to investigate the principal mode (i.e., m=1, n=6) response. From the present investigation, it can be found that for principal mode resonant response, there are two traveling waves with different linear frequencies due to the effect of precession of vibrating shape of rotating circular cylindrical shells; the effects of additional modes n and k (multiples of frequency) on the principal mode resonant response are insignificant compared with an additional mode m, showing that it is better to adopt two neighboring axial modes to study the principal resonant response of the system.     

Large deformations of thin-walled circular tubes under transverse loading—I: An experimental survey of the bending of simply supported tubes under a central load

The behaviour of simply supported circular tubes under the action of the transverse loading of a wedge-shaped indenter up to the point of maximum load is examined. Three phases in the deformation are identified and an attempt is made to relate the various characteristic loads to the parameters of the situation.     

Finite plastic deformation of a circular membrane under hydrostatic pressure—II : Strain-rate effects

The large strain behaviour of a circular membrane under uniform hydrostatic pressure is examined for materials with transversely isotropic plastic properties. Time-independent material response is assumed, and both flow theory and deformation theory of plasticity are considered.A variational principle is first applied to derive the governing equations for the pressurized membrane. It is shown that a special solution previously given by Hill[7] for isotropic materials can readily be extended to certain transversely isotropic membranes.A finite element solution based on Hill's extremum principle[24] is then developed to study the influence of anisotropy on the finite deformation behaviour of the membrane. Conditions where localized-necking bifurcations become possible are also discussed.     

The plane-strain general yielding of notched members due to combined axial force and bending—II: Deep circular notches

The slip-line fields proposed by Green for the plane-strain general yielding of notched plates in pure bending are generalized to provide solutions for notched plates subjected to combined axial force and bending. Deep symmetrical circular notches and single circular notches are considered. For notched plates subjected to arbitrary combinations of axial force and bending, two constraint factors are obtained, T₁ due to the axial force and M₁ due to the bending moment. These constraint factors are shown to have characteristic relationships on the T₁−M₁ plane for the single notched plate and the symmetrically notched plate. The derivations of these general yielding loci are described in detail and the statical and kinematical limitations of the slip-line fields are discussed. Finally it is demonstrated that the corresponding strain-rate vectors are normal to these loci and that, therefore, these loci can be regarded as plastic potentials.     

Cross-sectional deformations of rectangular hollow sections in bending: Part I — experiments

This investigation deals with deformations of individual cross-sectional members as flanges and webs in bending of rectangular hollow sections. Part I describes the experimental work, while analytical models developed to predict pre- and post-buckling deformations are presented in a paper to follow (Part II). The experimental program involved rectangular single- and double-chamber aluminium alloy AA6060 extrusions with three different wall thicknesses. The profiles were given two distinct heat treatments to obtain different hardening characteristics. Multiaxial tests were performed to determine the mechanical properties of the materials. The profiles were then bent into a number of different bend radii. Measurements of strains, curvatures, deflections and bending forces were taken. The results show that cross-sectional distortions take place from the very beginning of bending; at first in the form of a uniform sagging-like deformation along the entire length of both sides of the bend until the inner (compressive) flange buckles into several half-waves, superimposing the pre-deformation modes. The instant at which buckling occurs is found to be mainly associated with the width-to-thickness ratio of the flange and the strain hardening characteristic of the material. The magnitude of pre- and post-deformations, however, appears to be more directly related to the actual width of the flange than to its slenderness. The material stress–strain curve is shown to have an increasingly effect on the distortions of members directly sustained to buckling as bending proceeds beyond the onset of buckling, leading to severely concentrated deformations for sections made of low hardening materials. The material has less impact on sagging of the outer (tensile) flange.     

Cross-sectional deformations of rectangular hollow sections in bending: Part II — analytical models

In this part, analytical models to predict the deflection of cross-sectional members such as flanges and webs are developed. The models are based on the deformation theory of plasticity along with the energy method, using appropriate shape functions capable of including the restraining effect of adjacent members. The present method provides explicit solutions of cross-sectional deformations prior to buckling, onset of buckling, as well as post-buckling deformations at different stages of bending. The predictions show that the suck-in of the tensile flange is closely related to geometry parameters, particularly the flange width. Plastic anisotropy appears to be the most significant material parameter. The width-to-thickness ratio tends to be the governing parameter with respect to buckling of the inner (compressive) flange. Also, the strain hardening of the material has a major effect on onset of buckling as well as post buckling deformations. Upon continued bending after buckling, the wavy deformation of the inner flange develops more rapidly than the more uniform deformation of the outer (tensile) flange. For relatively compact sections, however, the deformation mode of the compressive flange resembles that of the tensile flange without any typical buckling waves. There are also obvious interactions between deformations of different members. Comparing the theoretical predictions with the experimental results presented in Part I, a reasonably good agreement was found.