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.     

Imperfection sensitivity of laminated conical shells

The sensitivity of laminated conical shells to imperfection is considered, via the initial post-buckling analysis, on the basis of three different shell theories: Donnell’s, Sanders’, and Timoshenko’s. Unlike isotropic conical shells or laminated cylindrical shells, in the case of laminated conical shells the thickness and the material properties vary with the shell coordinates, which complicates the problem considerably. The main objective of the study is to investigate the influence of the variation of the stiffness coefficients on the buckling behavior and on the imperfection sensitivity of laminated conical shells. It is felt that by finding the various parameters that influence the shell’s imperfection sensitivity, it is possible to improve the behavior of the whole structure.A special Level-1 computer code ISOLCS (Imperfection Sensitivity of Laminated Conical Shells) had been developed. ISOLCS calculates the classical buckling load and the imperfection sensitivity via Koiter’s theory of laminated conical shells with consideration to the variation of the material properties in the shell’s coordinates. The range of validity of the Level-1 predictions by ISOLCS is verified by the Level-3 code STAGS-A.     

Multimode Interaction in Axially Stiffened Cylindrical Shells

ABSTRACT

A number of studies show that nonlinear interaction between a shortwave local panel buckling mode and a longwave overall shell buckling mode usually causes strong imperfection sensitivity in stiffened shells. Interaction is particularly strong for simultaneous or nearly simultaneous modes. Mode interaction between two overall buckling modes, as well as interaction between two overall and one local mode, has received much less attention. The present study indicates that for certain imperfection combinations, such mode interactions are important. On the other hand, in most cases, interaction between one overall mode and one local mode governs.     

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.     

Buckling driven debonding in sandwich columns

A compression loaded sandwich column that contains a debond is analyzed using a geometrically non-linear finite element model. The model includes a cohesive zone along one face sheet/core interface whereby the debond can extend by interface crack growth. Two geometrical imperfections are introduced; a global imperfection of the sandwich column axis and a local imperfection of the debonded face sheet axis. The model predicts the sandwich column to be very sensitive to the initial debond length and the local face sheet imperfection. The study shows that the sensitivity to the face sheet imperfection results from two mechanisms: (a) interaction of local debond buckling and global buckling and (b) the development of a damaged zone at the debond crack tip. Based on the pronounced imperfection sensitivity, the author predicts that an experimental measurement of the strength of sandwich structures may exhibit a large scatter caused by geometrical variations between test specimens.     

Imperfection sensitivity of optimal symmetric braced frames against buckling

Imperfection sensitivity characteristics of the non-linear buckling load factors of non-optimal and optimal symmetric frames are investigated. The frames are subjected to symmetric proportional vertical loads. The optimization problem is formulated under constraints on linear buckling load factors. Although the buckling load factors corresponding to sway and non-sway modes coincide at the optimum design, the non-sway-type asymmetric bifurcation point disappears as a result of geometrically non-linear analysis. Therefore, the maximum allowable load factors of perfect and imperfect systems should be determined by assigning displacement constraints. It is shown that quantitative evaluation is possible for imperfection sensitivity and mode interaction based on the higher order differential coefficients of the total potential energy even for frames of which the critical points should be numerically obtained. Numerical examples are presented to show that the properties of the non-sway bifurcation point are similar to those of a symmetric bifurcation point, and the interaction between sway and non-sway modes does not always lead to enhancement of imperfection sensitivity.     

Post-buckling dynamic behavior of periodically supported imperfect shells

A circular cylindrical shell, periodically supported and subjected to step-loading in the form of lateral or hydrostatic pressure, is studied. Using the time-dependent von Kármán-Donnell equations, its imperfection sensitivity is examined and a simple asymptotic expression for the dynamic buckling load, valid for small imperfections, is obtained. There is a simple relation, independent of the imperfection, between the dynamic and static buckling loads.     

Numerical study of the effects of constitutive models on plastic buckling of plate elements

Compared with experiments, the J₂ deformation theory of plasticity is known to predict plastic buckling with better accuracy than the more accepted incremental J₂ flow theory. This paradox is commonly known as the ‘plastic buckling paradox’. In an attempt to analyse this discrepancy, the two mentioned constitutive models were implemented in a non-linear finite element code, along with a third non-associative J₂ flow theory. The latter model incorporates a vertex-type plastic flow rule. Using these three constitutive models, the buckling behaviour of plate outstand elements was investigated. Comparisons between the buckling strengths derived are presented. The non-linear static buckling simulations show that the instability introduced by the alternative flow rule of the non-associative model has substantial influence on the buckling behaviour. The acceptance of only small departures from normality was shown to reduce the predicted ultimate capacity of the plates. Furthermore, for plates with small plate slendernesses it was found that the imperfection sensitivity was significantly reduced when using the non-associative flow rule.     

Sensitivity analysis of an optimized bar-spring model with hill-top branching

Summary Characteristics of optimal solutions under nonlinear buckling constraints are investigated by using a bar-spring model. It is demonstrated that optimization under buckling constraints of a symmetric system often leads to a structure with hill-top branching, where a limit point and bifurcation points coincide. A general formulation is derived for imperfection sensitivity of the critical load factor corresponding to a hill-top branching point. It is shown that the critical load is not imperfection-sensitive even for the case where an asymmetric bifurcation point exists at the limit point.     

The Effect of Multiple Buckling Modes on the Postbuckling Behavior of Plane Elastic Frames. Part II. Asymmetric Frames

The postbuckling behavior of an asymmetric one-bay, two-storey frame with clamped edges is analyzed. Columns have different bending stiffnesses and are pairwise of the same length. By assuming columns to be inextensible and shear undeformable, and beams to be rigid, two buckling modes are possible which are described by the sidesway of the lower floor with a rigid horizontal displacement of the upper floor and a sidesway of the upper floor, the lower floor undergoing no displacement. By properly selecting the ratios EI/h² (EI being the bending stiffness of a column and h its length) the two buckling modes may occur simultaneously. A third buckling mode is also possible which is characterized by no displacement of the horizontal beams and local deflection of one or more columns in the shape of a beam with fixed edges. This third case will not be considered in this paper. The Koiter general nonlinear theory of elastic stability recast in a form convenient for the development of finite elment models along lines similar to the recent presentation by Budiansky has been employed in the analysis. Nonlinear constraints on the field variable φ (φ being the cross-section rotation) are accounted for by means of Lagrangian multipliers. Results show that the postbuckling behavior of a single buckling mode is always asymmetric unstable and depends both on the degree of asymmetry of the structure and on the ratio h/l, l being the frame span. The occurrence of simultaneous buckling modes exacerbates the imperfection sensitivity of the structure.     

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     

含轴对称初始缺陷的具有正交各向异性表层的夹层圆柱壳的非线性屈曲

夹层圆柱壳具有很高的结构效能。在许多工程结构中被广泛采用。本文研究分析了含有轴对称初始缺陷的夹层圆柱壳在轴压下的非线性屈曲问题。该夹层壳具有正交各向异性表层和各向同性可承剪的夹心．利用Stein的前屈曲一致理论得出了前屈曲挠度随轴向载荷及缺陷参数的变化情况，运用Galerkin法导出了屈曲控制方程，并进行了数值计算，得到了屈曲载荷、缺陷幅值、缺陷波数、夹心模量等参量之间的关系．结果表明与壳体实际屈曲模态相同的初始缺陷具有很大的危害性，可以通过增加壳体表层的轴向弹性模量或优化夹心的有关参数等途径来提高屈曲载荷，改善壳体屈曲性能。     

薄壁圆筒壳初始几何缺陷不确定性量化的极大熵方法

针对薄壁圆筒壳结构轴压屈曲载荷的缺陷敏感性以及真实几何缺陷的不确定性,提出一种基于实测缺陷数据和极大熵原理的初始缺陷建模与屈曲载荷预测方法。首先,将初始几何缺陷视为二维随机场,并利用实测缺陷数据和Karhunen-Loève展开法将初始缺陷的随机场建模转化为随机向量的建模;其次,利用极大熵方法确定随机向量的概率分布;最后,基于所构建的初始缺陷随机模型,利用MCMC抽样方法和确定性屈曲分析方法,进行随机屈曲分析并给出基于可靠度的屈曲载荷折减因子。数值算例表明,与直接假设随机场相关结构的方法相比,本文方法的结果是对薄壁圆筒壳屈曲载荷的一个更无偏估计。     

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.     

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.     

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.     

基于弧长法的受压球壳稳定性分析

对受压球壳进行特征值屈曲分析,得到了前6阶屈曲模态及线性屈曲临界载荷;采用弧长法进行非线性有限元分析,对理想球壳施加初始扰动,通过2次扰动值折半的方法求得引起结构屈曲的最小扰动值,追踪到了屈曲分支点和全过程载荷-位移路线。基于前6阶屈曲模态位移,在受压球壳中分别引入2.5mm和1mm两种缺陷值,分析缺陷对球壳屈曲特性的影响。结果表明:取壳厚的0.5%即0.05mm时,得最小扰动值,近似模型与完善结构极值载荷的差值为0.93%;球壳是缺陷敏感性结构,缺陷的幅值和分布都对其极限载荷有影响,缺陷幅值与厚度比为0.1时,缺陷球壳承载力相对理想结构下降了约11%,缺陷幅值与厚度比为0.25时,承载力相对下降了约30%,说明提高球壳稳定性需要提高球壳加工精度。     

Optimum design of hierarchical stiffened shells for low imperfection sensitivity

A concept of hierarchical stiffened shell is proposed in this study, aiming at reducing the imperfection sen- sitivity without adding additional weight. Hierarchical stiffened shell is composed of major stiffeners and minor stiff- eners, and the minor stiffeners are generally distributed between adjacent major stiffeners. For various types of geo- metric imperfections, e.g., eigenmode-shape imperfections, hierarchical stiffened shell shows significantly low imper- fection sensitivity compared to traditional stiffened shell. Furthermore, a surrogate-based optimization framework is proposed to search for the hierarchical optimum design. Then, two optimum designs based on two different opti- mization objectives （including the critical buckling load and the weighted sum of collapse loads of geometrically imperfect shells with small- and large-amplitude imperfections） are compared and discussed in detail. The illustrative example demonstrates the inherent superiority of hierarchical stiffened shells in resisting imperfections and the effectiveness of the proposed framework. Moreover, the decrease of imperfection sensitivity can finally be converted into a decrease of structural weight, which is particularly important in the development of large-diameter launch vehicles.     

Effect of multiple localized geometric imperfections on stability of thin axisymmetric cylindrical shells under axial compression

Stability of imperfect elastic cylindrical shells which are subjected to uniform axial compression is analyzed by using the finite element method. Multiple interacting localized axisymmetric initial geometric imperfections, having either triangular or wavelet shapes, were considered. The effect of a single localized geometric imperfection was analyzed in order to assess the most adverse configuration in terms of shell aspect ratios. Then two or three geometric imperfections of a given shape and which were uniformly distributed along the shell length were introduced to quantify their global effect on the shell buckling strength. It was shown that with two or three interacting geometric imperfections further reduction of the buckling load is obtained. In the ranges of parameters that were investigated, the imperfection wavelength was found to be the major factor influencing shell stability; it is followed by the imperfection amplitude, then by the interval distance separating the localized imperfections. In a wide range of parameters this last factor was recognized to have almost no effect on buckling stresses.