Buckling of multi-lamellae compression flanges of welded I-beams: A unilateral elasto-plastic plate-stability problem

Buckling of compression flanges of welded I-beams consisting of more than one lamella is characterized by partial loss of contact between the lamella welded to the web and the neighbouring lamella. The boundaries of the contact regions of the buckling modes of the two lamellae are originally unknown. Thus, the present elasto-plastic plate-stability problem is coupled with a contact problem. It represents a unilateral problem because eigenforms with penetrations of lamellae are physically impossible. This problem is solved first for the elastic material domain. The Rayleigh-Ritz method is used for determining symmetric eigenforms. A finite-strip technique is employed for determining unsymmetric eigenforms. The solution of the unilateral elastic plate-buckling problems serves as the starting point for the iterative solution of the corresponding elasto-plastic plate-stability problem. The buckling pressures obtained are compared with corresponding buckling stresses resulting from a classical design procedure disregarding the interaction of the lamellae at buckling.     

FINITE ELEMENT MODELLING OF SPATIALLY CHAOTIC STRUCTURES

A simple non-linear mechanical system comprising a pin-jointed string of finite-length links, supported by elastic springs at the pins and compressed by an axial load, is viewed from two perspectives. When seen as an initial-value problem, equilibrium equations provide an iterative non-linear mapping. When seen as a boundary-value problem, it becomes a simple finite element model. At loads less than the critical buckling load, a preferred buckling configuration is found that is localized along the length. In the limit of infinite length this is described as a homoclinic connection in phase space, joining the flat equilibrium state to itself. The infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle, within which also appear periodic, quasi-periodic, and chaotic spatial solutions. Implications in the finite element setting are discussed. © 1997 John Wiley & Sons, Ltd.     

Nonlocal buckling of double-nanoplate-systems under biaxial compression

This paper reports an analytical study on the buckling of double-nanoplate-system (DNPS) subjected to biaxial compression using nonlocal elasticity theory. The two nanoplates of DNPS are bonded by an elastic medium. Nonlocal plate theory is utilized for deriving the governing equations. An analytical method is used for determining the buckling load of DNPS under biaxial compression. Difference between nonlocal uniaxial and biaxial buckling in DNPS is shown. Both synchronous and asynchronous buckling phenomenon of biaxially compressed DNPS is highlighted. Study shows that the small-scale effects in biaxially compressed DNPS increases with increasing values of nonlocal parameter for the case of synchronous modes of buckling than in the asynchronous modes of buckling. The buckling load decrease with increase of value of nonlocal parameter or scale coefficient. In biaxial compression higher buckling modes are subjected to higher nonlocal effects in DNPS. Further the study shows that the increase of stiffness parameter brings uniaxial and biaxial buckling phenomenon closer while increase of aspect ratio widen uniaxial and biaxial buckling phenomenon.     

Buckling of fibers in fiber-reinforced composites

Elastic stability of fibers in fiber-reinforced composite materials subject to compressive loading is studied. The transversal buckling mode is considered, and two limiting cases, the dilute and non-dilute composites are analyzed. In the case of a non-dilute composite, the cylindrical model and the lubrication approximation are applied. The original problem is reduced to a problem of stability of a rod on elastic foundation. Through the solution of this problem a simple formula for the buckling load is obtained. In the case of a dilute composite, the solution of a problem of stability of a compressed rod in elastic plane is used. On the basis of the obtained solutions in two limiting cases the interpolation formulae are derived. These formulae describe buckling of fiber in the fiber-reinforced composite for any value of the fiber volume fraction. Comparison with known numerical and experimental results is carried out, and the sufficient accuracy of the derived formulae is demonstrated.     

On internal stability problems in structured materials

Summary Internal instability of structured materials and elements is occasionally observed and has been verified by experiments. The specific problems in this contribution treated analytically and numerically consist of internal instability phenomena of prestressed elastic materials and the so-called membrane buckling of thin elastic plates and shells. A stability theory taking into consideration independent local rotations is outlined; this theory is then used to treat the membrane buckling of cylindrical shells and the in-plane buckling of rectangular plates. It is shown that under certain circumstances, the in-plane buckling mode may precede the out-of-plane buckling deformation. To simulate the internal stability phenomena numerically, a number of discrete models of structured materials are considered; based on these models numerical Finite Element (FE) buckling analyses are carried out, including linear analyses for the membrane buckling of a circular cylindrical shell model and the in-plane buckling analysis of a flat plate. The FE simulations effectively afford buckling loads and buckling modes.     

An energy conservative symplectic methodology for buckling of cylindrical shells under axial compression

This study concerns a theoretical analysis on the buckling of cylindrical shells under axial compression in a new energy conservative symplectic system. By introducing four pairs of dual variables and employing the Legendre transformation, the governing equations that are expressed in stress function and radial displacement are re-arranged into Hamiltonian’s canonical equations. The critical loads and buckling modes are reduced to solving for symplectic eigenvalues and eigensolutions, respectively. The obtained results conclude that buckling solutions are mainly grouped into two types according to their nature of different buckling modes: non-uniform buckling with deflection localized at the vicinity of the ends and uniform buckling with deformation waves distributed uniformly along the axial direction, and the complete solving space only consists of the basic eigensolutions. The influence of geometric parameters and boundary conditions on the critical loads and buckling modes is discussed in detail, and some insights into this problem are analyzed.     

On chessboard buckling modes in compressed materials

The article is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard at a certain level of initial strain. The critical strain is established to be uniquely determined from the system of equations of the equilibrium bifurcation; however, there are many modes of buckling corresponding to this strain. A solution of the system of the bifurcation equations is considered in the form of double periodic functions of the kind sin r ₁ x ₁ sin r ₂ x ₂. The uncertainty in the buckling mode implies that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers, we examine the initial supercritical behavior of the material. Only two types of buckling modes (the shear type and the volume type) are possible. It is established that the buckling mode of the volume type is a chessboard-like one, and the mode of the shear type is not chessboard like. The stability of the supercritical equilibrium state is discussed.     

Quasistatic critical states of strings for deep drilling

We pose the problem of buckling of an elongated twisted rotating rod subjected to tension-compression and containing an internal flow of homogeneous fluid. We derive the resolving equations capable of modeling the stability of strings for deep drilling, propose a procedure for their solution, and consider typical examples. The critical values of the parameters of the system specifying its elastic equilibrium are determined and the stability loss modes of are established. __________ Translated from Problemy Prochnosti, No. 5, pp. 109–119, September–October, 2006.     

Buckling analysis of rotationally periodic structures using shell element with relative degrees of freedom

By considering the characteristics of deformation of rotationally periodic structures subjected to rotationally periodic loads, the periodic structure is divided into several identical substructures in this paper. If the structure is really periodic but not axisymmetric, the number of the substructures can be defined accordingly. If the structure is axisymmetric (special in the case of the periodic), the structure can be divided into any number of substructures. It means, in this case, the number of substructures is independent of the number of buckling waves. The degrees of freedom (DOFs) of joint nodes between the neighbouring substructures are classified as master and slave ones. The stress and strain conditions of the whole structure are obtained by solving the elastic static equations for only one substructure by introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get the bifurcation buckling load and mode for the whole rotationally periodic structure by solving the eigenvalue problem for only one substructure without introducing any additional approximation. Finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is then derived. Different measures of tackling internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the high efficiency and validity of this method. Copyright © 2001 John Wiley & Sons, Ltd.     

Biaxial buckling analysis of double-orthotropic microplate-systems including in-plane magnetic field based on strain gradient theory

Background/purposeThis paper deals with analysis of biaxial buckling behavior of double-orthotropic microplate system including in-plane magnetic field, using strain gradient theory.MethodsTwo Kirchhoff microplates are coupled by an internal elastic medium and also are limited to the external Pasternak elastic foundation. Utilizing the principle of total potential energy, the equilibrium equations of motion for three cases (out-of-phase buckling, in-phase buckling and buckling with a plate) are acquired. In this study, we assumed boundary conditions of all the edges are simply supported. In order to get exact solution for buckling load of system, Navier approach which satisfies the simply supported boundary conditions is applied.ResultsVariations of the buckling load of double-microplate system subjected to biaxial compression corresponding to various values of the thickness, length scale parameter, magnetic field, stiffness of internal and external elastic medium, aspect ratio, shear stiffness of the Pasternak foundation and biaxial compression ratio are investigated. Furthermore, influence of higher modes on buckling load is shown. By comparing the numerical results, it is found that dimensionless buckling load ratio for in-phase mode is more than those of out of phase and one microplate fixed. Also it is shown that the value of buckling load ratio reduces, when non-dimensional length scale parameter increases.ConclusionHowever, we found when properties of plate are orthotropic the buckling load ratio is more than isotropic state. Also, by considering the effect of magnetic field, non-dimensional buckling load ratio reduces.     

用初函数法研究平板的稳定性问题

本文从三维数学弹性稳定理论的基本方程出发，以位移和横向应力为自变量，运用初函数法分析了面内双向均匀受压和均匀受剪的平板弹性稳定性问题，导出了矩形平板对称失稳和反对称失稳的控制方程。文中以四边简支双向均匀受压的矩形板为例，得到二种失稳模式，并证明在弹性范围内不可能产生对称失稳。计算结果表明，临界载荷低于经典理论给出的结果。     

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed-form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross-section column, a 7-story frame, and a Kiewitt single-layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.     

Finite Element modeling of Nomex^® honeycomb cores : Failure and effective elastic properties

The purpose of the present study is to determine the components of the effective elasticity tensor and the failure properties of Nomex^® honeycomb cores. In order to carry out this study, the NidaCore software, a program dedicated to Nomex®Cores predictions, has been developed using the Finite Element tool Cast3M-CEA. This software is based on periodic homogenization techniques and on the modelling of structural instability phenomena. The homogenization of the periodic microstructure is realized thanks to a strain energy approach. It assumes the mechanical equivalence between the microstructures of a RVE and a similar homogeneous macroscopic volume. The key point of the modelling is that by determining the RVE buckling modes, the ultimate stress values of the homogenized core can be deduced. The numerical analysis shows an adequation between the first critical buckling modes computed and the ultimate stress values experimentally observed in standardized tests. This approach, strengthened by experimental data, makes it possible to devise a failure criterion. This criterion relies on the understanding of the mechanical effects caused by a local damage. In order to improve the reliability of the software, predicted mechanical properties are compared with data given by the Euro-Composite company and with results coming from crushing tests performed on sandwich specimens. This study brings out a powerful tool for the determination of the mechanical properties of most Nomex^® honeycomb core used in nautical construction. The design of oceanic sailing race boats is one of its most outstanding practical applications.     

纵向变厚度(LP)矩形钢板弹性屈曲系数的理论分析

纵向变厚度(LP)钢板因适应现代化建筑超高层及大跨度的发展使用需求,将广泛应用于工程结构中。因LP钢板几何形状的特殊性,其局部屈曲变形更容易发生在靠近薄端的位置,并且钢板局部稳定性能与相同受力状态下的等厚度钢板差异较大,需要进行深入研究。基于能量原理,采用Galerkin和Rayleigh-Ritz法(GRM),对单向均匀受压荷载作用下四种不同边界条件下的矩形LP钢板的弹性屈曲系数的计算公式进行了理论推导,并采用ANSYS有限元软件验证了公式的正确性。最后得到了不同厚度放大系数下,四种边界条件下的矩形钢板的弹性屈曲系数与钢板长宽比的关系曲线图,进一步为LP钢板的工程应用提供理论指导和设计依据。     

Surface effects on the mechanical properties of nanoporous materials

Using the theory of surface elasticity, we investigate the mechanical properties of nanoporous materials. The classical theory of porous materials is modified to account for surface effects, which become increasingly important as the characteristic sizes of microstructures shrink to nanometers. First, a refined Timoshenko beam model is presented to predict the effective elastic modulus of nanoporous materials. Then the surface effects on the elastic microstructural buckling behavior of nanoporous materials are examined. In particular, nanoporous gold is taken as an example to illustrate the application of the proposed model. The results reveal that both the elastic modulus and the critical buckling behavior of nanoporous materials exhibit a distinct dependence on the characteristic sizes of microstructures, e.g. the average ligament width.     

两种边界条件直杆的撞击屈曲及其应力波效应

在对Bernoulli-Euler梁运动方程定性分析的基础上,利用非零解的条件,直接展开含双参数的特征行列式得到了两端固定和一端铰支一端固定直杆动力屈曲临界载荷和相应的屈曲模态;证明了b<-α4/4时,模态方程的解不代表杆的动力屈曲。进一步,讨论了杆-杆类直杆动力屈曲过程中的应力波效应,其理论预测结果和相应的撞击实验结果显示出相当好的一致性。     

弹性地基上加热弹性圆板的热过屈曲及临界屈曲模态跃迁

基于von Kármán薄板理论建立了Winkler弹性基础上弹性圆板在均匀升温下的轴对称热过屈曲控制方程。这是一组以中面位移为基本未知量的非线性常微分方程,其中包含了温度载荷和弹性地基刚度两个参数。采用打靶法数值求解相应的非线性两点边值问题,获得了周边不可移简支圆板的热屈曲及热过屈曲响应。绘出了前三阶屈曲模态对应的临界温度载荷随地基参数连续变化的特性曲线,获得了反映临界热屈曲模态跃迁特性的地基参数值。给出了弹性圆板按一阶模态失稳后的热过屈曲平衡路径和平衡构形,分析了地基刚度参数对临界屈曲温度载荷以及过屈曲平衡构形的影响。     

Post-buckling of elastic annular plates at multiple eigenvalues

The buckling and post-buckling problem of elastic annular plates is treated at multiple eigenvalues making strong use of symmetry properties of the problem in deriving bifurcation equations (amplitude equations of the critical modes) from the von Karman plate equations by means of the Ljapunov-Schmidt method. Further, a restricted generic unfolding of the bifurcation equations is made and some practically interesting results concerning the post-buckling behaviour are presented.     

Post-buckling of elastic annular plates at multiple eigenvalues

The buckling and post-buckling problem of elastic annular plates is treated at multiple eigenvalues making strong use of symmetry properties of the problem in deriving bifurcation equations (amplitude equations of the critical modes) from the von Karman plate equations by means of the Ljapunov-Schmidt method. Further, a restricted generic unfolding of the bifurcation equations is made and some practically interesting results concerning the post-buckling behaviour are presented.     

多跨弹性支承连续矩形薄板屈曲分析

针对钢箱梁和混凝土薄壁箱梁受压翼缘的稳定问题,基于状态-空间向量法,提出了一种用于弹性支承连续矩形薄板弹性屈曲分析的计算方法。与有限条法结果对比,验证了该方法的可靠性。分析了跨间弹性支承刚度和布置以及荷载参数对屈曲的影响,结果表明:跨间弹性支承对连续矩形薄板屈曲影响明显,屈曲系数随着弹性支承刚度的增大呈非线性增长;等间距、等刚度布置弹性支承有利于板的稳定性,弹性支承的刚度或间距差别越大,对板的稳定性越不利;不同荷载工况下,弹性支承刚度-屈曲系数关系曲线的变化规律基本相同,弹性支承刚度较小时,荷载参数对屈曲系数影响显著,单向受压的屈曲系数可达双向等值受压的两倍。