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.     

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.     

Timoshenko beam-based stability and natural frequency analysis for heavy load mechanical spindles

The heavy load mechanical spindle is an important functional component in a 5-axis computer numerical control (CNC) machine tool, which is used to process large and complex free-form surfaces. It is necessary to obtain the natural frequency and analyze the spindle stability for improving the machining precision. In this paper, Timoshenko beam theory is introduced to model the mechanical spindle shaft, where the centrifugal force and gyroscopic effects are considered. Stability of the heavy load mechanical spindle shaft is analyzed, and the buckling load of the spindle shaft is obtained under different rotational speeds. The natural frequency of spindle is investigated in a freedom and restraint state, respectively. Comparing the proposed method with the simplified hollow cylinder and shaft prototype in the freedom state, the results show that they are highly correlated with experimental results. For the restraint state, the axial load, rotational speed, gyroscopic effect, and centrifugal force are discussed, and all of these parameters affect the natural frequency. The proposed modeling approach can be used for spindle design and optimization in a given machining process and can be easily extended to other spindle design.     

Exact static analysis of partially composite beams and beam-columns

The ordinary differential equations and general solutions for the deflection and internal actions and, especially, the pertaining consistent boundary conditions for partially composite Euler–Bernoulli beams and beam-columns are presented. Static loading conditions, including transverse and axial loading and first- and second-order analyses are considered. The theoretical procedure is applicable to general loading and boundary conditions for uniform composite beams and beam-columns with interlayer slip. Further, the exact closed form characteristic equations and their associated exact buckling length coefficients for composite columns with interlayer slip are derived for the four Euler boundary conditions. It is shown that these coefficients are the same as those for ordinary fully composite (solid) columns, except for the Euler clamped-pinned case. For the clamped-pinned case, the difference between the exact buckling length coefficient and the corresponding value for solid columns is less than 1.8% depending on the so-called composite action parameter and relative bending stiffness parameter. Correspondingly, the maximum deviation between the exact and approximate buckling load is at most 2.5%. These small differences can in most practical cases be neglected. Also, the maximum theoretical range for the relative bending stiffness for partially composite beams and beam-columns is derived. An effective bending stiffness, valuable in the determination of the critical buckling load for partially composite members, is derived. This effective bending stiffness is also suitable for analysing approximate deflections and internal actions or stresses in composite beams with flexible shear connection. The beam-column analysis is applied to a specific case. The difference in the approaches to the first- and second-order analysis is illustrated and the results clearly show the magnification in the actions and displacements due to the second-order effect. The magnification of the internal axial forces is different from magnifications obtained for the other internal actions, since only that portion of an internal axial force that is induced by bending is magnified by the second-order effect.     

Investigating elastic stability of cylindrical shell with an elastic core under axial compression by energy method

In this paper, the elastic axisymmetric buckling of a thin, isotropic and simply supported cylindrical shell with an elastic core under axial compression has been analyzed using energy method. The nonlinear strain-displacement relations in general cylindrical coordinates are simplified using Sanders kinematic relations (Sanders, 1963) for axial compression. Equilibrium equations are obtained by using minimum potential energy together with Euler equations applied for potential energy function in cylindrical shell. To acquire stability equation of cylindrical shell with an elastic core, minimum potential energy theory and Trefftz criteria are implemented. Stability and compatibility equations for an imperfect cylindrical shell with an elastic core are also obtained by the energy method, and the buckling analysis of shell is carried out using Galerkin method. Critical load curves versus the aspect ratio are obtained and analyzed for a cylindrical shell with an elastic core. It is concluded that the application of an elastic core increases elastic stability and significantly reduces the weight of cylindrical shells.     

Theoretical study of dynamic elastic buckling of columns subjected to intermediate velocity impact loads

Dynamic elastic buckling of simply supported columns subjected to intermediate velocity impact is theoretically studied in this paper. The dynamic buckling equation is set up. Theoretical solution of dynamic responses of columns subjected to a half-sine shape intermediate velocity impact load is derived. Based on the characteristics of the theoretical solution, a dynamic buckling criterion is proposed to determine the critical buckling condition and to estimate the dynamic buckling critical load. Theoretical results obtained in this study are compared with experimental data. They are also compared with numerical results obtained by other authors. Good agreements between them are observed.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.     

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。     

Destabilizing effect of additional restraint on elastic bar structures

This paper revises the well-known theorem; “The addition of restraint to a structure cannot decrease its elastic critical load”. It is pointed out by theoretical arguments and simple examples that this theorem is valid only if the displacement components in the buckling direction are additionally restrained. But additional restraints on pre-buckling displacements, which are orthogonal to the direction of buckling, can decrease the elastic critical load factor of the structure. In the case of additional restraints on prebuckling displacements, by the Dunkerley theorem, a bracketing formula is presented which gives lower and upper bounds to the minimum positive critical load factor of the modified structure with the help of minimum positive critical load factors of different loading patterns on the original structure.     

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。     

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.     

Application of Trefftz theory in thin-plate buckling with in-plane pre-buckling deformations

The paper presents the application of Trefftz initial stress theory in determining the elastic buckling load of rectangular thin plates under in-plane stress resultants, with allowance for in-plane pre-buckling deformations. Exact buckling solutions for simply supported plates are obtained and compared with earlier results furnished by Ziegler (1983, Ing.-Arch.53, 61 [1]). Although somewhat different, due to the use of different theories, both sets of results show that the buckling loads may be reduced when pre-buckling deformations are accounted for. This observation is contrary to that for column buckling where it has been shown that the allowance for pre-buckling shortening results in a higher buckling load. This reduction in plate-buckling load is significant especially for plates with large thickness-to-span ratios and subject to a combination of tensile and compressive in-plane stress resultants that increase pre-buckling deformations.     

Process parameter analysis of inertia friction welding nickel-based superalloy

A two-dimensional axisymmetric model for the inertia friction welding (IFW) of a nickel-based superalloy was developed. The influences from the axial pressure, initial rotational speed, and moment of inertia of the flywheel on the interface temperature and axial shortening were systemically examined. The analysis shows that the mechanical energy mainly depends on the initial rotational speed, and a relatively high axial pressure will increase conversion efficiency from mechanical energy to effective welding heat. The axial shortening is found to be approximately proportional to the square of initial rotational speed while logarithmical to the axial pressure. Based on this work, the weldability criteria for IFW nickel-based superalloy was established. Additionally, the approach for welding parameter optimization was performed considering the evolution of temperature profiles from various parameters. The results show that the axial pressure has a more obvious effect on the width of high-temperature zone than the rotational speed during the quick shortening stage.     

The dynamics of symmetric post-buckling

This work examines the relationship between natural frequency and compressive load for a conservative elastic mechanical system characterised by symmetric buckling. Application of some general theory shows that for a perfect system, exhibiting a stable-symmetric point of bifurcation, the initial post-buckling curve of compressive load against the square of the natural frequency is linear and has a slope of that of the pre-buckling curve. Simple link models and discretised continuous systems are analysed to illustrate the results of the general theory. These models are also used to show that the initial post-buckling curve is independent of axial inertia. The influence of initial imperfections is then considered and compared with experimental evidence.     

Buckling of braced monosymmetric cantilevers

The paper examines the elastic lateral buckling of monosymmetric cantilevers with a discrete intermediate brace. On the basis of the Timoshenko energy approach, the buckling capacities are determined by direct minimization of the ‘generalized Rayleigh quotient’. Solutions are expressed in terms of the easily calculated beam parameter, , and the degree of beam monosymmetry parameter, , which indicate readily the form of the monosymmetric section. The influences of lateral brace at different levels, of rotational brace and of full brace on the buckling capacities for varying brace locations, height of load application above the shear centre and the degrees of beam monosymmetry are investigated.     

线性缓冲材料对质量块撞击下的弹性杆动态屈曲影响研究

针对刚性质量块撞击情况,对弹性直杆在线性缓冲材料防护下的动态屈曲进行分析,得到撞击条件下杆体屈曲发生的临界弹簧刚度。以杆体整个长度发生轴向应变的最小弹性屈曲载荷为判据,分析了临界弹簧刚度与载荷作用时间、载荷幅值、缓冲材料及杆体长度的关系,以及缓冲材料位置对于杆体屈曲防护的影响等。结果表明：撞击初速和刚性块质量是引起临界弹簧刚度改变的两个因素,缓冲材料的安放位置在一定条件下影响着弹性杆屈曲防护。所得结论对于在刚性质量块作用下弹性直杆的动态屈曲防护问题提供了一定的理论依据。     

管轴压成形失稳起皱力学过程的研究

管轴压成形的先进性、优势及成形过程的成功实施受到变形区开裂和传力区起皱等失稳因素的制约,为此采用理论解析与试验相结合的方法,对失稳起皱的力学过程、影响因素及影响规律进行了研究,并给出了失稳判据,以实现稳定成形时变形参数的取值范围的确定。研究表明:成形过程的失稳起皱受到几何参数、材料参数及摩擦边界条件的共同作用。其中,模具圆角曲率半径是决定性因素,并且存在一个最大和最小圆角半径区域使成形过程顺利进行而不发生失稳起皱;增大管坯材料硬化指数,减小相对厚度及接触面的摩擦,有利于减少失稳的发生。对牌号为5052,原始管平均直径d_0=31.0mm,41.0mm铝管的应用表明所提出的判据是可靠实用的。     

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。     

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.     

Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory

The small-scale effect on the torsional buckling of a double-walled carbon nanotube (DWCNT) embedded on Winkler and Pasternak foundations is investigated in this study using the theory of nonlocal elasticity. The effects of the surrounding elastic medium, such as the spring constant of the Winkler type and the shear constant of the Pasternak type, as well as the van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. Finally, based on the theory of nonlocal elasticity and by employing the continuum models, an elastic double-shell model is presented for the nonlocal torsional buckling load of a DWCNT. It is seen from the results that the shear constant of the Pasternak type increases the nonlocal critical torsional buckling load, while the difference between the presence and the absence of the shear constant of the Pasternak type becomes large. It is shown that the nonlocal critical buckling load is lower than the local critical buckling load. Moreover, a simplified analysis is carried out to estimate the nonlocal critical torque for the torsional buckling of a DWCNT.