Postcritical imperfection sensitivity of sandwich or homogenized orthotropic columns soft in shear and in transverse deformation

The previous energetic variational analysis of critical loads and of the choice of finite strain measure for structures very weak in shear, remaining in a state of small strain, is extended to the initial postcritical behavior. For this purpose, consideration of the transverse deformation is found to be essential. It is shown that imperfection sensitivity of such structures, particularly laminate-foam sandwich plates, can arise for a certain range of stiffness and geometric parameters, depending on the proper value of parameter m of the Doyle–Ericksen finite strain tensor, as determined in the previous analysis. The bifurcation is symmetric and Koiter’s 2/3-power law is followed. The analytical predictions of maximum load reductions due to imperfection sensitivity are verified by finite element simulations. The possibility of interaction between different instability modes, particularly lateral deflection and bulging, is also explored, with the conclusion that lateral deflection dominates in common practical situations.     

Influence of general imperfections in axially loaded cylindrical shells

The buckling of an axially loaded cylindrical shell is considered when imperfection components corresponding to all of the classical buckling modes are taken into consideration. The analysis represents an extension of Koiter's axisymmetric solution and in the asymptotic sense due to Koiter the imperfections considered are as general as possible. The results obtained reveal many interesting aspects of shell buckling which arize for various imperfection forms. The buckling behaviour which results is associated with both bifurcation and limit point critical states.     

板片空间结构初始缺陷的动力稳定性影响分析

对板片空间结构体系的研究大都局限于静力载荷工况，结构在动力载荷作用下的稳定性及失效特征研究仍相对缺乏．板片空间钢结构先天存在各样缺陷，而缺陷对结构稳定性的影响不可忽略．本文引入拉丁超立方法，结合Budiansky-Roth 准则，提出适用于求解含初始缺陷结构动力临界荷载的方法，对板片空间结构初始缺陷的动力稳定性开展研究．研究表明，相比于其它抽样方法，拉丁超立方法能显著减少样本数，有效提高计算效率．针对典型的板片空间结构，施加不同方向的阶跃荷载，分析了节点位置偏差缺陷、杆件截面尺寸缺陷以及板片厚度缺陷对板片空间结构临界承载力的影响．结果显示，节点位置偏差缺陷对该结构临界承载力影响最为显著，其它两类缺陷影响甚微．     

On a probabilistic stability theory for imperfection sensitive structures

The concept of almost sure sample stability and sample stability in probability are formulated for elastic systems. Using a Koiter type approach these concepts are used in the analysis of imperfection sensitive structures. The applied load and the initial geometric imperfections are introduced into the analysis as random quantities. A compressed beam of finite length on a nonlinear elastic foundation is used in an example calculation.     

基于模态缺陷的受压球壳屈曲特性研究

针对受压球壳非线性屈曲过程，对含初始缺陷受压球壳的稳定性进行研究。根据EN1993-1-6（2007）规范，给出不同制造等级壳体的等效缺陷值计算方法；基于线性特征值分析的模态构型给出初始缺陷的分布。利用非线性有限元弧长法对球壳受压失稳过程进行数值模拟，得到屈曲前后球壳变形情况及全过程载荷-位移曲线。计算一致缺陷模态法和N阶缺陷模态法对应的球壳屈曲临界载荷，结果表明，受压球壳对缺陷较敏感，承载能力随缺陷值增大而降低；一致缺陷模态法计算便捷，在工程应用上具有合理性，N阶缺陷模态法考虑高阶模态缺陷构型，结果更加全面，可以为工程中缺陷结构稳定性设计提供参考。     

Imperfection sensitivity of ultimate buckling strength of elastic-plastic square plates under compression

The mechanism of imperfection sensitivity of elastic-plastic plates under compression is complex as they undergo elastic and/or plastic buckling, dependent on their width-thickness ratio. For elastic buckling, the Koiter power law is an established means to describe the imperfection sensitivity. Yet, for plastic buckling, there is no such an established way to describe it. In this paper, the quadratic power law is advanced to describe imperfection-insensitive plastic buckling behavior. The Koiter power law is extended by implementing the quadratic law so as to describe the elastic and plastic buckling in a synthetic manner. The finite-displacement, elastic-plastic analysis was conducted on simply-supported square plates under compression by varying the plate thickness and the initial deflection of a sinusoidal form. In association with an increase of the plate slenderness parameter (decrease of plate thickness), the predominant buckling is shown to change from (1) plastic buckling to (2) unstable elastic-plastic buckling and to (3) elastic stable bifurcation followed by a maximum point of load. In accordance with the change of the mechanism of buckling, the power law is changed pertinently to describe the complex imperfection sensitivity of the compression plates in a synthetic manner. The extended imperfection sensitivity law is thus advanced as a simple and strong tool to describe the ultimate buckling strength of elastic-plastic plates.     

Ho's theorem in global-local mode interaction of pin-jointed bar structures

This paper is focused on the interaction phenomena among a global critical mode and some local Eulerian critical modes in pin-jointed structures. These phenomena are framed within Koiter's theory of elastic instability, by an asymptotic reduction into cubic systems. The aim is to present an algorithm for the appraisal of the lowest critical load characterizing the structure under the effect of small imperfections. First of all, the Ho's theorem, concerning the definition of the most dangerous imperfection, is presented and discussed. Then, a FEM code aimed at the determination of the most dangerous shape for the imperfection, and at performing the related sensitivity analysis, is implemented, by superimposing a proper FE beam model (able to model Eulerian instability) to a non-linear FE model for spatial pin-jointed structures. Some numerical results having a practical interest are presented and discussed.     

Imperfection sensitivity of curved panels under combined compression and shear

The initial buckling load of curved panels under compressive loads is substantially reduced by the existence of imperfections, in particular geometric imperfections. It is therefore essential that these imperfections are considered in analysing components which incorporate such panels in order to accurately predict their buckling behaviour. Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections, however, to obtain accurate results information is required on the exact size and shape of the imperfection to be modelled. In most cases this data is not available. It is therefore generally recommended that the imperfections are modelled on the first eigenmode and have an amplitude selected according to the manufacturing procedure. This paper presents the effects of varying imperfection shape and amplitude on the buckling and postbuckling behaviour of one specific case, a curved panel under combined shear and compression, to test the accuracy of such recommendations.     

Flutter instability in imperfect structural systems

The periodic oscillations of discrete structural systems with follower-force loading are studied in the region of the critical point. The case in which small initial geometric imperfections exist in the structure is also examined. Flutter instability is found to be much less sensitive to initial imperfections than is static instability. Critical loads are destabilized (or stabilized) in either a linear or parabolic fashion with imperfection amplitude. The postcritical characteristics of the flutter exhibited cannot be altered by initial imperfections.     

Stress analysis of hyperbolic shells of revolution with non-axisymmetrical geometric imperfections

In analyzing hyperbolic shells of revolution with non-axisymmeteric imperfections, an approximate method based on simulating the effect of imperfections by the application of fictitious normal pressure loading on the perfect shell is investigated. In the analysis of a shell of revolution with a bulge-type imperfection under non-axisymmetric loads, an efficient algorithm of applying the method is developed: the effect of individual curvature errors on stress resultants and couples are separately considered, while the interactions among various curvature errors are properly treated in the analysis by an iterative procedure. This algorithm avoids repeated analyses for non-axisymmetric loads and may be implemented with a purely axisymmetric analysis capability.A hyperbolic cooling tower shell with a bulge-type imperfection is analyzed under dead load and wind load conditions by the equivalent load method. A direct analysis of the imperfect shell is also made by a specialized finite element program. Through numerical studies, the accuracy and applicability of the equivalent load method are examined.     

Nonlinear buckling optimization of composite structures considering “worst” shape imperfections

Nonlinear buckling optimization is introduced as a method for doing laminate optimization on generalized composite shell structures exhibiting nonlinear behaviour where the objective is to maximize the buckling load. The method is based on geometrically nonlinear analyses and uses gradient information of the nonlinear buckling load in combination with mathematical programming to solve the problem. Thin-walled optimal laminated structures may have risk of a relatively high sensitivity to geometric imperfections. This is investigated by the concepts of “worst” imperfections and an optimization method to determine the “worst” shape imperfections is presented where the objective is to minimize the buckling load subject to imperfection amplitude constraints. The ability of the nonlinear buckling optimization formulation to solve the laminate problem and determine the “worst” shape imperfections is illustrated by several numerical examples of composite laminated structures and the application of both formulations gives useful insight into the interaction between laminate design and geometric imperfections.     

Postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments

A postbuckling analysis is presented for a shear deformable functionally graded cylindrical shell of finite length subjected to combined axial and radial loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect cylindrical shells with two constituent materials subjected to combined axial and radial mechanical loads and under different sets of thermal environments. The results reveal that the temperature field and volume fraction distribution have a significant effect on the postbuckling behavior, but they have a small effect on the imperfection sensitivity of the functionally graded shell.     

几何缺陷浅拱的动力稳定性分析

研究了几何缺陷对粘弹性铰支浅拱动力稳定性能的影响。从达朗贝尔原理和欧拉-贝努利假定出发推导了粘弹性铰支浅拱在正弦分布突加荷载作用下的动力学控制方程,并采用Galerkin截断法得到了可用龙格-库塔法求解的无量纲化非线性微分方程组。同时引入能有效追踪结构动力后屈曲路径的广义位移控制法,对含几何缺陷浅拱的响应曲线进行几何、材料双重非线性有限元分析。用这两种方法分析了前三阶谐波缺陷对浅拱动力稳定性能的影响,其中动力临界荷载由B-R准则判定。主要结论有:材料粘弹性使浅拱动力临界荷载增大且结构响应曲线与弹性情况差别很大;二阶谐波缺陷影响显著,它使动力临界荷载明显下降且使得浅拱粘弹性动力临界荷载可能低于弹性动力临界荷载。     

铁磁梁式板磁弹性初始后屈曲及缺陷敏感性分析

从铁磁板的磁弹性广义变分原理出发，通过对铁磁板内外的磁场和板变形 场的摄动技术以及采用经典Koiter理论，对横向磁场中铁磁悬臂、简支、固支梁式薄 板磁弹性稳定性的初始后屈曲行为及缺陷敏感性进行了定性研究. 解析地给出了悬臂板对初 始缺陷敏感，简支和固支板对初始缺陷不敏感等结果，对悬臂梁式板理论预测的临界磁场值 往往大于实验观测结果的现象从定性上给予了合理解释.     

Imperfection sensitivity of coincident buckling systems

A two-degree of freedom structural system with both buckling loads equal or nearly so is examined, allowing for the presence of small imperfections specified by the values of two general imperfection parameters. A discrete generalized coordinates energy method is used, and the system considered is non-symmetric in the two modes of deflection. Conclusions regarding the types of behaviour possible are obtained by examination of all possible equilibrium path shapes. Comparison is made to behaviour of idealized systems. Change in buckling load value with imperfection size, or imperfection sensitivity, is established and imperfection sensitivity surface forms are sketched.     

Postbuckling and imperfection sensitivity of fixed-end and free-end struts on elastic foundation

Summary A study of the postbuckling and imperfection sensitivity of fixed-end and free-end struts on a Winkler elastic foundation is carried out. The configuration and stability of the postbuckling paths bifurcating from the critical points are analysed. For the most part of foundation stiffness, the corresponding postbuckling paths are shown to be falling with respect to load and be unstable. This indicates that, for almost all values of foundation stiffness, the buckling loads of the struts will be sensitive to imperfections. We also obtain imperfection sensitivity of the struts with respect to geometric imperfections having the shape of buckling modes. Received 30 October 1998, accepted for publication 30 March 1999     

Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings

A postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical and thermal loads. The temperature field considered is assumed to be of uniform distribution over the plate surface and through the plate thickness and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A two step perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, volume fraction distribution, applied voltage, the character of in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Stability of Geometrically Imperfect General-View Shells

An efficient numerical algorithm is created on the basis of the method of curvilinear meshes, the method of basis reduction, and the asymptotic Koiter theory. It enables us to study load curves to reveal singular (limit and bifurcation) points and to analyze their behavior at branch points. Such an approach allows analyzing the nonlinear deformation of shell structures of arbitrary form and their sensitivity to imperfections. The technique is illustrated by an example of a conic panel with imperfections in the shape     

The Shallow Arch—General Buckling,Postbuckling, and Imperfection Analysis∗

A general discussion of the behavior of the shallow circular arch is presented. It is shown that, irrespective of specific loading or boundary conditions, it is possible to arrive at general conclusions regarding buckling, postbuckling, and imperfection sensitivity. General methods of analysis are established which lead to the determination of points of bifurcation and of postbuckling paths under symmetric loads. Modifications accounting for antisymmetric load components are introduced, with special emphasis on their asymptotic and limit load effect.

A typical numerical example is carried through in detail. The solution is “exact” in the sense of shallow arch theory. Its asymptotic behavior conforms to the asymptotic theory of Koiter.