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.     

Plastic buckling of circular tubes under axial compression—part II: Analysis

In the second part of this study, the evolution of uniform axisymmetric wrinkling in axially compressed cylinders is modeled using the principle of virtual work. A version of this formulation also allows localization of wrinkling. The model domain is assigned an initial axisymmetric imperfection of a chosen amplitude and the wavelength yielded by the first bifurcation check. The solution correctly simulates the growth of wrinkles and results in a limit load instability. The limit strain is influenced by the amplitude of the imperfection. Beyond the limit load, wrinkling tends to localize, eventually leading to local folding.The possibility of bifurcation of the axisymmetric solution to non-axisymmetric buckling modes is examined by using a dedicated bifurcation check. The bifurcation check was found to yield such buckling modes correctly. The evolution of such buckling modes is simulated by a separate non-axisymmetric model assigned imperfections with axisymmetric and nonaxisymmetric components. The domain analyzed is one characteristic wavelength long (2λ_C). Initially, compression activates mainly axisymmetric deformation. In the neighborhood of the bifurcation point, non-axisymmetric deformation starts to develop, eventually leading to a limit load instability. Experimental responses were simulated with accuracy by assigning appropriate values to the two imperfection amplitudes. Prediction of the limit strains for the whole range of diameter-to-thickness ratios (D/t) considered in the experiments was achieved by making the amplitude of the non-axisymmetric imperfection proportional to (D/t)²/m³ (m is the circumferential wavenumber). Matching all aspects of the experiments required inclusion of the anisotropy measured in the tubes tested through Hill's yield criterion in all models.     

Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes

Three possible failure modes have been identified in tube hydroforming: buckling, wrinkling and bursting. A general theoretical framework is proposed for analyzing these failure modes as an elastoplastic bifurcation problem. This framework enables advanced yield criteria and various strain-hardening laws to be readily incorporated into the analysis. The effect of plastic deformation on the geometric instability in tube hydroforming, such as global buckling, axisymmetric wrinkling and asymmetric wrinkling, is precisely treated by using the exact plane stress moduli tensor. A mathematical formulation for predicting the localized condition for bursting failure is established herein. Furthermore, the critical conditions governing the onset of buckling, axisymmetric wrinkling and asymmetric wrinkling are derived in closed-form expressions for the critical axial compressive stresses. Closed-form solutions for the critical stress are developed based on Neale–Hutchinson's constitutive equation and an assumed deformation theory of plasticity. It is demonstrated that the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one under the context of tube hydroforming with applied internal pressure and hence the asymmetric wrinkling mode can be excluded in the analysis of tube hydroforming. Parametric studies show that buckling and axisymmetric wrinkling are strongly dependent on geometric parameters such as t₀/r₀ and r₀/ℓ₀, and that axisymmetric wrinkling is the predominant mode for short tubes while global buckling occurs for long slender tubes.     

Nonlinear in-plane elastic buckling of shallow circular arches under uniform radial and thermal loading

This paper presents a nonlinear in-plane elastic buckling analysis of circular shallow arches that are subjected both to a uniform temperature field and to a uniform radial load field. A virtual work method is used to establish nonlinear equilibrium equations and buckling equilibrium equations, and analytical solutions for the limit instability and bifurcation buckling loads are obtained. It is found that the temperature influences the limit instability, bifurcation buckling and postbuckling behaviour of shallow arches significantly. The limit instability and bifurcation buckling loads increase with an increase of the temperature. A maximum temperature is shown to exist for the occurrence of bifurcation buckling of shallow arches, and when the temperature is higher than this value, bifurcation buckling of an arch is not possible.An arch geometric parameter is introduced to define switches between the limit instability and bifurcation buckling modes, and between buckling and no buckling. Formulae and methods for the calculation of the limiting values of the arch geometric parameter are developed. It is also found that the limiting values of the arch geometric parameter decrease with an increase of the temperature.     

The failure of torispherical ends of pressure vessels due to instability and plastic deformation—An experimental investigation

Twelve model pressure vessels with torispherical ends have been tested under internal pressure to investigate failure by instability and plastic deformation. The models covered combinations of three head heights and four thicknesses. All the thicker specimens with internal diameter-thickness ratios of 53, 106 and 212 failed by plastic deformation. The three thin specimens with an internal diameter-thickness ratio of 530 failed by buckling of the torus due to the circumferential compressive stresses. The experimental results for limit pressure and instability failure are compared with theoretical values. The effect of change of geometry is significant particularly for the larger head heights. For these specimens the experimental limit pressure is higher relative to theoretical predictions than is the case for the smaller head heights. A simple approximate theory is presented for predicting the pressure at which buckling occurs in the torus. The correlation of the new predictions with the experimental values is not good, but the new predictions are lower than those previously published.     

On the analysis of sheet metal wrinkling

An analysis for the prediction of wrinkling in curved sheets during metal forming is presented. Using a local approach, similar to that employed for conventional forming limit diagram representations, we construct “wrinkling limit curves” (WLCs) which represent the combinations of the critical principal stresses for wrinkling in curved sheet elements. Wrinkling limit curves are first determined using a bifurcation analysis for plastic buckling in short-wavelength shallow modes. A study of the effects of material properties and sheet geometry on the critical conditions for wrinkling is carried out. We then analyse the effects of geometric imperfections on wrinkling. This analysis is based on the implementation of a finite element scheme. The influence of nonproportional loading is also investigated. In our analysis the material is assumed to be isotropic, elastic-plastic with the plastic part modelled using both J₂ deformation theory and J₂ flow theory of plasticity.     

Buckling and collapse of cylinders with one end open and one end simply supported with varying axial restraint

This paper examines the buckling and collapse of cylindrical shells under axial load with one end radially and tangentially fixed, with varying axial fixity, and the other end free. The bifurcation loads are found for elastic cylinders, while collapse loads are found for both elastic and elastic-perfectly plastic cylinders. The varying axial restraint is applied in the form of linear springs. The eigenvalue buckling loads are calculated with conditions matching those of a classical analysis. Bifurcation loads are shown to be a function of the axial restraint; as the axial restraint is increased, the bifurcation load increases dramatically, until it reaches that of a semi-infinite, open ended cylinder. A non-dimensional form of the axial spring stiffness is proposed, and shown to be applicable across a range of geometries.The collapse load and imperfection sensitivity of cylinders with the boundary conditions examined here is also found to be a function of the axial restraint. Cylinders with low axial restraint are shown to be imperfection insensitive, with collapse loads above, or close to, the bifurcation load. As the amount of axial restraint increases, the collapse behaviour displays a degree of imperfection sensitivity associated with more usual boundary conditions.     

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.     

Deformation and failure mechanisms of lattice cylindrical shells under axial loading

The lattice cylindrical shells wound from the planar lattice plates, which have significant applications in aerospace engineering, exhibit different deformation modes with their planar counterparts because of the curvature of the cell wall. In this paper, deformation mechanisms are systematically investigated and failure analyses are conducted for the lattice cylindrical shells with various core topologies. Analytical models are proposed to predict the axial stiffness, critical elastic buckling load or effective yield strength of these shells. Finite element simulations are carried out to identify the validity of the models. The models can be employed for the optimal design. As an example, we construct the failure map for the Kagome lattice cylindrical shell made from an elastic ideally-plastic material. Various failure mechanisms, including yielding, global elastic buckling and local elastic buckling are taken into account. Moreover, optimizations are performed to minimize the weight for a given stiffness or load-carrying capacity for three types of lattice cylindrical shells. It is found that the Kagome and triangular lattice cylindrical shells have almost equivalent load-bearing capacity and both significantly outperform the hexagonal one under axial compression.     

薄板起皱失稳数值模拟计算方法

汽车、航天、航空等领域的零件制造轻量化趋势致使管板材成形起皱失稳缺陷的预测成为行业的重点关注问题。基于目前常规数值模拟算法未包含起皱失稳判据从而无法解决起皱数值预测的现状,利用平板对角拉伸试验作为对比验证对象,利用大型平台软件ABAQUS证实了用于模拟薄板变形的常规壳单元动态显式算法（^*DYNAMIC）以及用于结构件屈曲计算的弧长法（^*RIKS）无法用于计算薄板成形起皱问题,尝试并比较了“用于微缺陷引入的特征值分析（^*BUCKLE）+^*DYNAMIC分析”相结合分析方法、“^*BUCKLE分析+^*RIKS分析”相结合的分析方法以及“三维实体单元动态显式算法”三种方式对薄板起皱失稳问题的运算结果,围绕计算精度、适用范围、算法特征等方面对这三种算法进行研究,确立了从多方面考察综合性能最佳的薄板成形起皱失稳数值模拟方法。     

An elastoplastic analysis of flange wrinkling in deep drawing process

The onset of flange wrinkling of a deep drawing cup is analyzed as an elastoplastic bifurcation problem. The flange is modeled as an elastoplastic annular plate subject to axisymmetric radial tension along its inner edge. As observed in the laboratory as well as practical industrial applications, aluminum alloy sheets usually wrinkle in the plastic range. Therefore, the critical condition governing the onset of elastoplastic wrinkling is formulated within the context of the general bifurcation theory. A closed-form solution for the critical drawing stress is developed based on an assumed nonlinear plastic stress field and the deformation theory of plasticity. The theory properly accounts for the plastic anisotropy of the aluminum sheets and the critical drawing stress at the onset of wrinkling is also compared against the one employing the flow theory of plasticity. The predicted critical bifurcation stress and the wave numbers are compared to those obtained by Senior's one-dimensional theory. It is demonstrated that there is a strong dependency of the critical bifurcated stress at the onset of wrinkling on the shear stress induced on the flange. The effects of flange width, drawing ratios, material properties, strain hardening on the onset of wrinkling are investigated. The differences between the present theoretical approach and Senior's theory are emphasized.     

Inelastic wrinkling and collapse of tubes under combined bending and internal pressure

The problem of inelastic bending and collapse of tubes in the presence of internal pressure is investigated using experiments and analyses. The experiments involve 1.5-inch diameter, D/t=52 stainless steel tubes bent to failure at fixed values of pressure. The moment-curvature response is governed by the inelastic characteristics of the material. Bending induces some ovalization to the tube cross section while, simultaneously, the internal pressure causes the circumference to grow. Following some inelastic deformation, small amplitude axial wrinkles appear on the compressed side of the tube, and their amplitude grows stably as bending progresses. Eventually, wrinkling localizes, causing catastrophic failure usually in the form of an outward bulge. Internal pressure stabilizes the structure, it increases the wavelength of the wrinkles and can increase significantly the curvature at collapse. The onset of wrinkling is established by a custom bifurcation buckling formulation. The evolution of wrinkling and its eventual localization are simulated successfully using a FE shell model. The material is represented as an anisotropic elastic-plastic solid using the flow theory, while the models are assigned initial geometric imperfections with the wavelength of the wrinkling bifurcation mode. It is demonstrated that successful prediction of collapse requires very accurate representation of the material inelastic properties including yield anisotropies, and that as expected, the collapse curvature is sensitive to the imperfection amplitude and wavelength imposed.     

Coupled buckling and plastic instability for tube hydroforming

In this paper, the hydroforming limit of isotropic tubes subjected to internal hydraulic pressure and independent axial load is discussed.Swift's criterion is often used in this case for the prediction of diffuse plastic instability. Here, we first highlight the existence of two different Swift's criteria (for sheets and for tubes).Then, we recall that these types of approaches do not take into account buckling induced by axial loading. In fact, buckling may obviously occur before plastic instability; consequently, Swift's criteria must not be used alone to predict instability in the case of tube hydroforming.Numerical simulation was used to confirm these points and to analyse both the buckling and striction phenomena together. The two types of instability must be treated together in a reasonable approach to the hydroforming process.In this paper, the material verifies a “J2-flow” constitutive rate constitutive law. Jaumann's derivative was chosen and the Prandtl–Reuss equations with von Mises’ yield criterion and the associated flow rule were used. Isotropic hardening was taken into account.     

Plastic instability in dual-pressure tube-hydroforming process

The tube-hydroforming process has become an indispensable manufacturing technique in recent years. Successful tube hydroforming requires bulging to take place without causing any type of instability such as bursting, wrinkling or buckling. The dual-pressure tube-hydroforming process was introduced to achieve a favorable tri-axial stress state in the deformation process. In this paper, the effect of applying counter pressure on plastic instability of thin-walled tubes is analyzed. It is concluded that in dual-pressure tube hydroforming, the onset of plastic instability is delayed and the ductility of the metal is increased.     

Plastic buckling of conical shells under axial compression

Analysis and experimental results are presented on the plastic axisymmetric buckling of steep, truncated conical shells under axial compression. Specimens of 6061-T6 aluminum and type 416 stainless steel were tested; in spite of the considerable difference in the stress-strain curves for the two materials, the buckling modes observed in the experiments for geometrically identical cones were the same. Perturbation analysis, which takes account of the continuous change in direction of the plastic strain rate vector during buckling, is found to describe the essential features of the observed buckling deformation.     

Limit loading of cylindrical shells subjected to local circumferential bending moments

Experiments have been carried out on twelve mild steel cylindrical shells each of which had a rigid radial square bar attached at its mid length. Strain and deformation patterns in the shell were measured when a hoop bending moment was applied to it through the attachment. Though some elastic stress distributions were obtained, the observations were mainly concerned with plastic deformations, the determination of plastic limit moments and the onset of instability. Measured deformation patterns were used to suggest a collapse mechanism from which a theoretical upper bound for the plastic limit moment was determined. Theoretical and experimental limit moments were compared for all specimens which had ratios varying between 13·75 and 67·5, and attachment half width/cylinder radius ratios varying between and .     

Bifurcation analysis of wrinkling formation for anisotropic sheet

Surface distortions in the form of localized buckles and wrinkles are often observed in the sheet metal forming process. In many cases the presence of wrinkles in the final praduct is unacceptable for the purposes of assembly. Because of the trend in recent years towards thinner gauges and higher strength, wrinkling is increasingly becoming a more common and troublesome mode of failure in sheet metal forming. In this study, a numerical analysis for evaluating a wrinkling limit diagram (WLD) for an anisotropic sheet subjected to biaxial plane stress is presented. Here the scheme of plastic bifurcation theory for thin shells based on the Donnell-Mushtari-Viasov shell theory is used. The effects of the various material parameters (yield stress, strain hardening coefficient and normal anisotropy parameter) and geometric parameters on WLD are investigated numerically and compared with Kawai's and Havranek's experiment(1975).     

Contrast between elastic and plastic buckling of axially compressed conical shells with edge constraint

Analysis of plastic buckling of axially compressed truncated conical shells taking account of edge constraint due to friction is presented and related to earlier experimental results. It is found that the time-dependence of the external load in the equilibrium equations is essential in the mathematical formulation. Localization of buckling deformation near the boundary appears to be a distinctive feature of plastic buckling, in contrast to elastic buckling treated on the basis of the same equilibrium equations, boundary conditions, and geometric relations. In the plastic analysis, rigid-plastic material behaviour based on simple J₂ flow theory is used.     

Deformation and stability of an elastica under a point force and constrained by a flat surface

This paper studies the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. The elastica is constrained by a rigid plane wall containing the two ends. Three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. A vibration method is adopted to determine the stability of the calculated deformations. In order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an Eulerian version of the governing equations is adopted. It is found that all the point-contact deformations are unstable. On the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. In the secondary buckling, the length of the line-contact segment and the axial force satisfy the Euler buckling criteria for a pinned-clamped column. On the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the Euler buckling load. The theoretical predictions are confirmed by experimental observations.     

初始缺陷和比例加载路径对圆柱壳弹塑性稳定性的影响

应用非线性弹塑性稳定性理论研究圆柱壳在轴向力和内压联合作用下的弹塑性稳定性问题,得到了不同比例加载路径、初始缺陷和塑性变形发展程度对弹塑性失稳的影响规律曲线。结果表明,在内压的作用下,柱壳的临界轴向压应力在初始阶段有所提高,但从某一内压值开始将会有所下降。对于同一比例加载路径,随着缺陷因子的增加,临界屈曲轴向压应力随着相应地递减。而对于同样的缺陷因子,比例加载参数小于1.0时,临界屈曲轴向压应力随着比例加载参数的增大而增大;比例加载参数大于1.0时,临界屈曲轴向压应力随着比例加载参数的增大而减小;比例加载参数等于1.0为理想加载比例参数值。该研究丰富了内高压成形工艺的理论基础。