Elastic Buckling with Infinitely Many Local Modes

ABSTRACT

ABSTRACT In some cases, asymptotic methods present an appealing alternative to full nonlinear analyses. In other cases, the value of an asymptotic analysis may merely be that, in a qualitative way, it can characterize the behavior of a structure. Whether an asymptotic method is applied for one or the other purpose it is of interest to attempt an estimation of its range of validity. The present paper addresses this question for an asymptotic method to predict imperfection sensitivity of elastic structures with mode interaction. The particular structure that is investigated possesses an infinity of nearly simultaneous local buckling modes. It is found that very few of these modes need to be taken into account.     

Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction

Abstract

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely an overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect, as was done in a previous study by Byskov and Hutchinson, may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.     

Postbuckling Behavior of Thin-Walled Open Cross-Section Compression Members

ABSTRACT

The postbuckling behavior of simply supported columns with thin-walled open cross section is investigated by means of the general nonlinear theory of elastic stability. Fourth-order terms in the series expansion of the total potential energy are disregarded.

It is shown that interaction between linearly independent simultaneous buckling modes is responsible for neutral equilibrium at bifurcation if the column cross section has two axes of symmetry, and unstable if it has only one. If the buckling modes are not simultaneous, the equilibrium is neutral in both cases. Finally, the equilibrium at bifurcation is usually unstable if the cross section has no axis of symmetry.     

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.     

Mode Interaction in Thin-Walled Structural Members∗

Thin-walled structural members are susceptible to failure by interaction of local and overall modes of buckling. A comprehensive analytical approach that employs finite strips and the theory of mode interaction is outlined for study of the problem. Theoretical predictions of collapse loads agree well with the experimental results, despite the uncertainty caused by the lack of data on local imperfections of experimental specimens. Illustrative examples explore to some extent the following specific questions: optimality of design for simultaneous buckling, imperfection-sensitivity of compression members having unstiffened plate elements, and the nature of interaction of local and lateral-torsional buckling.     

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.     

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.     

Development mechanism of local plastic buckling in bars subjected to axial impact

In order to clarify the developmental mechanism of the local plastic buckling and the interaction between axial wave and buckling deformation in an axially impacted slender-bar, the non-linear dynamic equations in the incremental form are derived and solved by use of the finite difference method, with the axial wave front treated as a moving boundary. The initial local-buckling deflection given by the characteristic-value analysis is used as the initial condition of the solution of the equations, instead of the initial imperfection that is assumed in literatures. It is found that the initial buckling deflection with one half-wave, occurring near the impacted end, develops into the higher post-buckling mode with several half-waves, as the axial compression waves propagate forward. The numerical results show that no strain reversal occurs at the early stage of post-buckling process, and the solution corresponding to the tangent-modulus theory is valid for the dynamic plastic post-buckling response of the bar at this stage. The theoretical results are in good agreement with the experimental results reported in the literature.     

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.     

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。     

直杆碰撞刚性壁弹塑性动力后屈曲有限元分析

利用显式动力学有限元方法对直杆弹塑性动力后屈曲进行了分析,模拟了直杆轴向碰撞动力屈曲的变形及发展过程。分析中在直杆碰撞端局部临界屈曲长度范围内引入半正弦波形式的初始缺陷,计算结果与文献中的实验数据获得了很好的一致。分析结果表明,随着碰撞过程中所产生的应力波逐渐向前传播,后屈曲变形过程中所呈现的多个半波形式的高阶屈曲模态由初始具有单个半波形式的简单屈曲模态迅速演变而成。分析结果同时也揭示了直杆动力屈曲变形发展的机理,以及轴向应力波和屈曲变形的相互作用规律。     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

Dynamic buckling mode of an elastic bar

The conception of buckling relative initial imperfection is presented in this paper. According to Boulli-Euler beam equation, the dynamic buckling mode of an elastic bar under the homogeneous boundary conditions can be derived by applying the preferred mode analytical method. As an example, the dynamic buckling mode of an elastic bar clamped at both ends is discussed.     

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。     

Simplified prediction of the creep buckling of cylinders under external pressure. Part 2: Experimental verification

In this paper, the simplified method, proposed in (Combescure, 1998), for the prediction of creep buckling is compared to experimental results. The model is applied to predict the buckling time of two sets of experiments on cylinders subjected to uniform external pressure. It is shown that the proposed model is satisfactory for this type of prediction: in all cases, the times up to failure predicted by the model are generally lower than the experimental failure times. The model is rather conservative for thicker cylinders. However, it appears that a very detailed geometrical imperfection survey would be necessary if a highly accurate assessement of the creep failure time were sought. It has been observed experimentally that creep buckling is a very dangerous failure mode: nothing seems to happen during a very long “incubation” period but, when the initial imperfection reaches some critical value, buckling then suddenly occurs. For thin cylinders, the level of creep strain at which the instability starts to develop is much lower than the strain at which the tertiary creep initiates; the instability is thus clearly generated from the interaction between the material and the geometrical nonlinearity.     

Micro-buckling in the nanocomposite structure of biological materials

Nanocomposite structure, consisting of hard mineral and soft protein, is the elementary building block of biological materials, where the mineral crystals are arranged in a staggered manner in protein matrix. This special alignment of mineral is supposed to be crucial to the structural stability of the biological materials under compressive load, but the underlying mechanism is not yet clear. In this study, we performed analytical analysis on the buckling strength of the nanocomposite structure by explicitly considering the staggered alignment of the mineral crystals, as well as the coordination among the minerals during the buckling deformation. Two local buckling modes of the nanostructure were identified, i.e., the symmetric mode and anti-symmetric mode. We showed that the symmetric mode often happens at large aspect ratio and large volume fraction of mineral, while the anti-symmetric happens at small aspect ratio and small volume fraction. In addition, we showed that because of the coordination of minerals with the help of their staggered alignment, the buckling strength of these two modes approached to that of the ideally continuous fiber reinforced composites at large aspect ratio given by Rosen's model, insensitive to the existing “gap”-like flaws between mineral tips. Furthermore, we identified a mechanism of buckling mode transition from local to global buckling with increase of aspect ratio, which was attributed to the biphasic dependence of the buckling strength on the aspect ratio. That is, for small aspect ratio, the local buckling strength is smaller than that of global buckling so that it dominates the buckling behavior of the nanocomposite; for comparatively larger aspect ratio, the local buckling strength is higher than that of global buckling so that the global buckling dominates the buckling behavior. We also found that the hierarchical structure can effectively enhance the buckling strength, particularly, this structural design enables biological nanocomposites to avoid local buckling so as to achieve global buckling at macroscopic scales through hierarchical design. These features are remarkably important for the mechanical functions of biological materials, such as bone, teeth and nacre, which often sustain large compressive load.     

Secondary buckling and tertiary states of a beam on a non-linear elastic foundation

An axially compressed beam resting on a non-linear foundation undergoes a loss of stability (buckling) via a supercritical pitchfork bifurcation. In the post-buckled regime, it has been shown that under certain circumstances the system may experience a secondary bifurcation. This second bifurcation destablizes the primary buckling mode and the system “jumps” to a higher mode; for this reason, this phenomenon is often referred to as mode jumping. This work investigates two new aspects related to the problem of mode jumping. First, a three mode analysis is conducted. This analysis shows the usual primary and secondary buckling events. But it also shows stable solutions involving the third mode. However, for the cases studied here, there is no natural loading path that leads to this solution branch, i.e. only a contrived loading history would result in this solution. Second, the effect of an initial geometric imperfection is considered. This breaks the symmetry of the system and significantly complicates the bifurcation diagram.     

Postbuckling analysis of elastic shells of revolution considering mode switching and interaction

The postbuckling response of shells is known to exhibit complex phenomena including mode switching and interaction, particularly in the advanced postbuckling range. The existing literature contains many initial postbuckling analyses as well as advanced postbuckling analyses for a single buckling mode, but little work is available on the advanced postbuckling analysis of shells of revolution considering mode switching and interaction. In this paper, a numerical method for the advanced postbuckling analysis of thin shells of revolution subject to torsionless axisymmetric loads is presented, in which such mode switching and interaction are properly captured. Numerical results obtained using the present method for several typical problems not only demonstrate the capability of the method, but also lead to significant observations concerning the postbuckling behavior of thin shells of revolution. In particular, the results show that strong interaction between different harmonic modes may exist and the transition of deformation mode from one to another is gradual. Consequently, the conventional approach of finding the postbuckling path of a shell as the lower festoon curve of postbuckling paths of individual harmonic modes is not valid and is at best a convenient approximation.     

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.     

On the Buckling of Axially Compressed Thin Cylindrical Shells

ABSTRACT

A simple general method for the evaluation of the effect of shape imperfections on the buckling of thin shells is briefly presented. This method is applied to the axially compressed thin cylindrical shell resulting in an efficient numerical procedure for the computation of its buckling strength. The procedure is applicable to any sufficiently smooth imperfection pattern and has given results in good agreement with the available experimental data.