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.     

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.     

Horizontal cylinder-in-cylinder buckling under compression and torsion: Review and application to composite drill pipe

Available analytical results and experiments on the structural behavior of constrained horizontal cylinders subjected to axial compression, torsion, and gravitational loads are reviewed. Such configurations are of interest to the oil-drilling field and provide static design expressions for steel tubulars. The buckling problem is similar to restrained railroad tracks and submerged/underwater pipelines under thermal expansion. Due to outer cylinder constraint and gravitational loads, analysis has shown that long cylinders initiate buckling at loads significantly higher than classical Euler buckling loads. For these constrained long cylinders, buckling initiates in a sinusoidal mode that snakes along the lower surface of the constraining cylinder. Classic analytical expressions hold that as the axial load increases, the cylinder achieves an overall helically buckled state in which the buckled cylinder forms a helix spiraling around the inner surface of the constraining cylinder. Torsion is shown to have little effect on either buckling load but controls the sense/direction of the helical buckling. Little experimental data exist on constrained cylinder buckling, and it is unclear how the initiating sinusoidal mode transitions to the helical mode. Implications of the buckling progression for composite cylinder applications are described including the finding that composites perform poorly relative to steel on the metric of buckling due to lower density and axial stiffness; composites perform well on the metric of lock-up length when friction is considered. Based on this review and findings for composite cylinders, recommendations are made for further work.     

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.     

On the stability of elastically supported cantilever with continuous lateral restraint

A solution for the buckling load of an elastically supported cantilever with continuous lateral restraint under uniform distributed axial load is developed. The restraining media are distributed along the centroidal axis and the shear forces are assumed to be proportional to the respective derivative. Boundary conditions are developed for three cases: (a) fixed supports, (b) the cantilever on an elastic support while the restraining media is fixed and (c) both cantilevers and restraining media on an elastic support.     

Collapse mode transitions of thin tubes with wall thickness, end condition and shape eccentricity

Transition of deformation mode shapes of round aluminum tubes from axisymmetric concertina to non-axisymmetric diamond mode have been studied with varying tube wall thickness, boundary conditions and tube shape eccentricities. Quasi-static axial compression experiments were carried out on as received aluminum tubes and tubes with wall thickness eccentricity, incorporated by off center machining. Tubes were of D/t=29 and L/D=1.4. The numerical simulation of the collapse phenomenon has been undertaken using a static non-linear finite element analysis in ANSYS with a fine mesh discretization of the tube domain and small incremental displacements as load steps. Convergence studies for the finite element model with respect to load step size and mesh density have also been established. The numerical results are found to compare well with the experimental load compression and energy absorption responses both for the axisymmetric concertina and non-axisymmetric diamond collapse modes. Having validated the numerical model with experiments, it has been used to undertake a systematic study of the load–deformation characteristics, energy absorption response and collapse mode transition of the tubes in varying configurations of wall thickness, shape and inplane boundary condition eccentricities. Dependence of tube collapse characteristics and collapse mode transitions on such eccentricities have been discussed.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

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.     

大挠度后屈曲倾斜梁结构的非线性力学特性

基于弹性梁的几何非线性大挠度屈曲理论,建立两端固定对称倾斜支撑梁结构的大挠度后屈曲控制微分方程,采用几何非线性隐式变形协调关系来表达强非线性超静定边值问题,得到描述倾斜梁大挠度后屈曲行为的精确解析解.采用数值方法求解含有第一、二类椭圆积分的强非线性微分方程,给出不同倾角梁结构从初始屈曲到后屈曲并发生两态跳转过程中的位形曲线及非线性刚度.根据最小能量原理和挠曲线拐点个数,分析对称屈曲模态与非对称屈曲模态之间相互跳转的内在联系及其对结构非线性刚度突变的影响,得到了屈曲模态之间的转换条件.跳转过程的数值仿真表明,倾斜支撑梁结构发生大挠度后屈曲时具有明显的双稳态特性且只出现低阶(1、2阶)屈曲模态,仿真计算结果与试验结果相一致.     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

Misalignment effects on the load capacity of a hydraulic cylinder

Analytical and experimental investigations of typical hydraulic cylinders have indicated that their load capacities are significantly different from those obtained from simple buckling analysis of idealized systems. In any case, an increase in the friction coefficient at the restrained ends changes the actuator's limit load, while an increase in the initial maximum deflection (initial misalignment) decreases the limit load. A common practice of most cylinder manufacturers is to use a safety factor (between 2.5 and 4) to determine the service load after the critical load (buckling) is obtained by simple analytical procedures treating the cylinder as a perfect stepped column. The intricate aspects of friction effects have been deliberately left aside in this present work. Nevertheless, friction and interaction between mechanism and actuator in the buckling characteristics will be presented in the ongoing paper, which will follow this work. Authors know that, in a real system, the cylinder tube-rod interface is not rigid. Due to the flexibility of guide rings and clearances between components, misalignment (an angular deflection which increases with increasing axial load) exits at the interface. When initial imperfection angle exists, there is no sudden buckling. Then, stresses and deflections increase with increasing load. After repetitive use, the tolerance between the parts will become larger, consequently increasing the initial deflection, which has been proved to considerably decrease the load capacities of the power cylinders. From this analysis, a theoretical and experimental work has been carried out in order to show the advantages and disadvantages of the current design methods, characterizing the critical factors that cause the collapse and proposing useful design criterions. The present work aims to describe the behaviour of actuators under load capacity with experimental validation.     

Design of beneficial geometric imperfections for elastic collapse of thin-walled box columns

This paper deals with the design of beneficial geometric imperfections for elastic collapse of thin-walled box columns of square cross-section under axial compression. From the point of view of high elastic post-collapse stiffness, it is desirable to force the column to collapse with a larger number of axial half-waves than with the preferred wave number which is approximately equal to the aspect ratio. By introducing a beneficial geometric imperfection with such a larger wave number and if the magnitude of such imperfection exceeds the transitional value, it is found that the column will collapse in the beneficial mode in the initial postbuckling (assume to be elastic) finite-deflection regime. Equilibrium paths of typical box columns are plotted and analyzed. The two-mode potential energy is found to fall into the category of a double-cusp catastrophe.     

Interaction between local and overall buckling

Numerical solutions of the equations governing the interaction of local and overall modes on box columns are presented. In confirmation of previous results these solutions show that elastic buckling is highly sensitive to changes in the magnitude of both local and overall small imperfections, and that the strong optimum indicated for coincident mode design is effectively eliminated when this imperfection sensitivity as well as effects of plasticity are included. It is also clearly demonstrated that the reduced modulus approach provides a conservative estimate of the elastic load-carrying capacity.The inclusion of plasticity into the analysis shows that the effects of elastic interaction may also be highly significant in controlling the plastic collapse load, and for most practical cases this plastic interactive buckling will control design. An extension of the Perry-Robertson procedure is shown to provide a convenient means of presenting the design information. Finally, the usefulness of the van der Neut model as the starting point of the analysis is verified.     

壳体屈曲分析中关于能力减弱系数的剖析

工程中通常采用有限元中的线性屈曲方法和非线性屈曲方法来求解壳体的屈曲载荷。对于复杂结构(如钢安全壳),若直接采用非线性屈曲方法求解,除计算工作量大外,还不容易得到符合实际要求的临界载荷。因此,工程中常通过线弹性方法来获得符合实际要求的临界载荷。介绍了壳体屈曲的线弹性理论解,然后利用ANSYS软件求解圆柱壳受轴向和侧向外压的屈曲载荷,并与理论解进行了对比;最后通过对比前人的试验结果与线弹性理论得到的上临界值来剖析"能力减弱系数"的含义和合理性。结果表明:采用线弹性方法来求解壳体屈曲问题是可行的,但必须进一步考虑"能力减弱系数"、"安全系数"和"塑性折减系数"后才能得到工程上所需的临界屈曲载荷值。     

Linear and nonlinear buckling analysis of a locally stretched plate

Uniformly stretched thin plates do not buckle unless they are in special boundary conditions. However, buckling commonly occurs around discontinuities, such as cracks, cuts, narrow slits, holes, and different openings, of such plates. This study aims to show that buckling can also occur in thin plates that contain no defect or singularity when the stretching is local. This specific stability problem is analyzed with the finite element method. A brief literature review on stretched plates is presented. Linear and nonlinear buckling stress analyses are conducted for a partially stretched rectangular plate, and various load cases are considered to investigate the influence of the partial loading expanse on the critical tensile buckling load. Results are summarized in iso-stress areas, tables and graphs. Local stretching on one end of the plate induces buckling in the thin plate even without geometrical imperfection.     

Minimum weight design of beams against failure under uncertain loading by convex analysis

Under operational conditions, some loads acting on a beam are known (deterministic loads), but there usually exist other loads the magnitude and distribution of which are unpredictable (uncertain loads). If the uncertainty in the loading is not taken into account in the design, the likelihood of failure increases. In the present study beams are designed for minimum weight subject to maximum stress and buckling load criteria and under deterministic and uncertain transverse loads. The uncertain load, which is subject to a constraint on its L ₂ norm, is determined to maximize the normal stress using a convex analysis. The location of the maximum stress is determined under the combination of deterministic and worst-case uncertain loads. The minimum weight design is obtained by determining the minimum cross-sectional area subject to stress and buckling load constraints. Results are given for a number of problem parameters including the axial load, elastic foundation modulus and uncertainty levels.     

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.     

Effect of Porosity and Normal Load on Dry Sliding Friction of Polymer Foam Blocks

In this study, the effect of porosity on the dry sliding fiction of ethylene-vinyl acetate (EVA) foams was investigated under different normal load conditions. EVA foam blocks with varying porosities were slid against a smooth stainless steel plate under dry conditions. The friction coefficient increased with increasing porosity under all of the normal load conditions. In addition, the contact area was estimated using a contact model considering elastic buckling of the cell walls (elastic collapse). The elastic collapse area in the anterior portion of the EVA foam block increased with increasing normal load and porosity, which resulted in an increased contact area. Furthermore, the friction coefficient was positively correlated with the estimated contact area divided by the normal load, indicating that adhesion friction increases with increasing porosity of polymer foams. These results may contribute to the design of high-friction, lightweight shoe sole tread blocks prepared using polymer foam blocks.     

Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods

In this paper, the buckling analysis of laminated composite plates reinforced by single-walled carbon nanotubes (SWCNTs) is carried out using an analytical approach as well as the finite element method. The developed model is based on the classical laminated plate theory (CLPT) and the third-order shear deformation theory for moderately thick laminated plates. The critical buckling loads for the symmetrical layup are determined for different support edges. The Mori-Tanaka method is employed to calculate the effective elastic modulus of composites having aligned oriented straight nanotubes. The effect of the agglomeration of the randomly oriented straight nanotubes on the critical buckling load is also analyzed. The results of analytical solution are compared and verified with the FEM calculations The critical buckling loads obtained by the finite element and the analytical methods for different layup and boundary conditions are in good agreement with each other. In this article, the effects of the carbon nanotubes (CNTs) orientation angle, the edge conditions, and the aspect ratio on the critical buckling load are also demonstrated using both the analytical and finite element methods.     

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.