Free vibration and buckling response of hat-stiffened composite panels under general loading

The paper presents finite element free vibration and buckling analysis of laminated hat-stiffened shallow and deep shells using arbitrarily oriented stiffener formulation. Modified approach for modelling the curved stiffener is implemented using necessary transformations. A simplified stiffener formulation is presented to accommodate various shapes of stiffener shapes in developing the rigidity matrix for the finite element formulation. Investigation has been carried out on free vibration and buckling analyses of laminated composite stiffened shell structures with laminated open section (rectangular or ‘T’ shaped) and closed section (‘hat’ shaped) stiffeners. Parametric study on the hat-stiffened panels for the free vibration and buckling analyses confirms that the closed section stiffener being torsionally rigid is found to show better performance over open section stiffeners.     

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t^1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t^1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t^2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t^2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t^1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.     

Misalignment effects on the load capacity of a hydraulic cylinder

Analytical and experimental investigations of typical hydraulic cylinders have indicated that their load capacities are significantly different from those obtained from simple buckling analysis of idealized systems. In any case, an increase in the friction coefficient at the restrained ends changes the actuator's limit load, while an increase in the initial maximum deflection (initial misalignment) decreases the limit load. A common practice of most cylinder manufacturers is to use a safety factor (between 2.5 and 4) to determine the service load after the critical load (buckling) is obtained by simple analytical procedures treating the cylinder as a perfect stepped column. The intricate aspects of friction effects have been deliberately left aside in this present work. Nevertheless, friction and interaction between mechanism and actuator in the buckling characteristics will be presented in the ongoing paper, which will follow this work. Authors know that, in a real system, the cylinder tube-rod interface is not rigid. Due to the flexibility of guide rings and clearances between components, misalignment (an angular deflection which increases with increasing axial load) exits at the interface. When initial imperfection angle exists, there is no sudden buckling. Then, stresses and deflections increase with increasing load. After repetitive use, the tolerance between the parts will become larger, consequently increasing the initial deflection, which has been proved to considerably decrease the load capacities of the power cylinders. From this analysis, a theoretical and experimental work has been carried out in order to show the advantages and disadvantages of the current design methods, characterizing the critical factors that cause the collapse and proposing useful design criterions. The present work aims to describe the behaviour of actuators under load capacity with experimental validation.     

Post-buckling and delamination propagation behavior of composite laminates with embedded delamination

Delamination occurred due to poor manufacturing process or in-service actions significantly affects the mechanical and failure behavior of laminated composite structures. In this study, the buckling and post-buckling delamination behavior of laminated composite with an embedded initial delamination under in-plane compression was studied experimentally and numerically. First, compression tests for laminated composite specimens with embeded initial delamination were performed and the buckling and delamination responses were obtained. Then the experimental test was numerically simulated using finite element methods with the progressive failure accounted for by using cohesive zone modeling. The load-displacement curve, strain behavior and delamination shapes of experimental specimens obtained from load cells, strain gages installed at different locations, and C scan images, respectively, were compared with the FEM results, and good agreements were attained. The effect of the buckling modes, laminate stacking sequence and shape of initial delamination on the buckling load and propagation behavior was studied by considering different ply stacking and shapes of initial delaminations. It was found that the buckling mode determined the growth direction of the delamination propagation, and the stacking sequence influenced the extent of the propagation area, while the orientation of the delamination affected the buckling loads.     

大展弦比机翼屈曲及后屈曲分析

针对某大展弦比复合材料无人机机翼结构,利用有限元素法进行机翼结构的屈曲及后屈曲分析.首先采用MSC.Patran/Nastran软件进行结构屈曲分析,目的是确定机翼结构易发生屈曲的区域;然后对不同区域进行分析,确定需要进行后屈曲分析的细节模型.对细节模型进行后屈曲分析时采用MSC.Patran/Marc软件,分析方法采用弧长法.通过机翼结构的后屈曲分析,可以分析出结构的失稳路径.结果表明,该机翼存在屈曲失稳问题.该机翼的地面静力试验结果验证分析结果的可信性.     

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。     

Influence of stiffener location on the stability of stiffened plates under compression and in-plane bending

The elastic behaviour of stiffened plates under non-uniform edge compression is investigated. As a first stage, an energy formulation, in which the structure is modelled as assembled plate and beam elements, is presented. A Sequential Quadratic Programming (SQP) algorithm is then used to evaluate the buckling load and the associated buckling mode for given plate/stiffener geometric properties. Results are presented showing the influence of the stiffener location on the stability of the structure under combined compression and bending. Then, a strategy is presented for optimum location of the stiffener. The proposed methodology can be used to develop an improved design procedure for efficient design of stiffened plates under this type of loading.     

碟形封头压力容器在内压作用下的弹塑性屈曲及后屈曲行为研究

基于ABAQUS对受内压碟形封头压力容器的弹塑性屈曲及后屈曲行为进行分析.通过非对称网格剖分技术诱发结构的屈曲行为,采用Riks算法捕捉完整的屈曲及后屈曲路径,分析屈曲载荷和后屈曲形态.计算结果表明,在内压作用下碟形封头压力容器过渡区域出现波状的分叉屈曲形态,随着加载的继续,过渡段将逐渐生成肉眼可视的褶皱,进入后屈曲阶段.后屈曲路径可以分为第一阶段与第二阶段.对引入初始厚度缺陷的结果分析可知,初始缺陷厚度的布置将对后屈曲路径产生影响,并降低结构的分叉屈曲载荷.     

Elastic buckling of ring-stiffened cone–cylinder intersections under internal pressure

Cone–cylinder intersections are commonly found in pressure vessels and piping. Examples include conical end closures to cylindrical vessels and conical reducers between cylinders of different radii. In the case of a cone large end-to-cylinder intersection under internal pressure, the intersection is subject to a large circumferential compressive force. Both the cone and the cylinder may be thickened near the intersection to resist this compression, but it is often convenient and necessary to augment further the strength of the intersection using an annular plate ring stiffener. Under this large circumferential compression, the intersection may fail by elastic buckling, plastic buckling or plastic collapse. This paper describes an investigation of the elastic buckling strength of ring-stiffened cone–cylinder intersections. Two buckling modes are identified: a shell mode for thin intersections with a shallow cone (a cone with its apex half angle approaching 90°) and/or a relatively stocky ring stiffener, and a ring mode for other cases. An existing elastic buckling approximation for annular plate rings in steel silos is found to be applicable to the intersection when it buckles in the ring mode. New approximate design equations are also established for the shell mode. In addition, simple expressions are identified which relate the number of circumferential buckling waves to the geometric parameters of the intersection.     

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

Buckling phenomena related to rolling and levelling of sheet metal

The paper deals with analytical and numerical considerations of buckling phenomena in thin plates or strips under in-plane loads which typically appear during rolling and levelling, i.e. straightening by stretching, of sheet metal. Buckling due to self-equilibrating residual stresses, caused by the rolling process, in eventual conjunction with global tensile stresses (denoted as “rolling buckling”) as well as buckling during the levelling process (denoted as “stretching buckling” or “towel buckling”) are considered. Analytical estimates are derived and compared against results of numerical simulations and field observations. Mode jumping by varying the global strip tension is explained on the basis of the derived analytical solutions. It is shown how from the waves, i.e. height and length, observed on the strip sliding over or lying on a rigid plane one can provide information about the distribution of the differences in the plastic strains over the width of the strip which lead to the buckled configuration. And, vice versa, knowledge of the plastic strain distribution can be used for estimating the expected wave heights representing a measure for the geometrical quality of the rolled product. The influence of the dead weight of the strip on the post-buckling pattern is also discussed on the basis of non-linear analyses.     

Parametric studies on buckling loads and critical speeds of microdrill bits

Transverse bending vibrations of the spinning microdrill bit subjected to a compressive axial load are developed based on the Timoshenko beam theory. The system equations of motion are discretized into the form of time-dependent ordinary differential equations by the finite element method. Two types of eigenvalue problems are formulated and utilized to study the effects of the drill helix angle, flute length and diameter on the buckling load and critical speed of microdrill bits with different supported ends. Equivalent formulae similar to those of untwisted Euler beams are established to predict critical buckling loads and critical speeds for microdrills and provide results with sufficient accuracy. The effect of rotational speed on the buckling load, and the influence of thrust force on critical speed are also investigated. A Galef-type equation associated with critical speed, thrust force and buckling load is formulated.     

Post-buckling behavior of carbon fiber epoxy composite plates

In this study, the pre-buckling and post-buckling behaviors of layered composite plates which were made of woven carbon fiber fabric with a circular hole in the middle were investigated experimentally and numerically. Firstly, load-displacement graphs of composite plates with different hole diameters were experimentally obtained under compressive load. Then the numerical load-displacement graphs of the plates were found with the ANSYS package program which used the finite element method. As a result, after linear buckling experimental and numerical results were found to be compatible with each other. In addition, damage behavior of plates after buckling with the aid of Tsai-Wu damage criterion was obtained similar to experimental results. The increase in hole diameter did not change the load-displacement behavior characteristics of the plates after buckling. However, it has reduced maximum damage load and maximum failure displacement. The stress at the perimeter of the hole increased significantly with the increase of the vertical displacement with immediately after the buckling but later was not significantly affected by this increase.

     

多墙翼面结构后屈曲强度试验和数值研究

采用静态试验和有限元分析相结合的方法研究了复合材料中厚蒙皮多墙翼面结构的后屈曲性态、强度和破坏过程。有限元分析取得的载荷-位移曲线和试验结果的一致性说明了有限元模型的正确性。在此模型基础上,采用非线性弧长法追踪外载-位移路径,研究了盒段在压缩和集中力两种不同载荷下的后屈曲行为。给出了屈曲临界载荷和屈曲变形性态,分析了后屈曲强度和破坏过程。     

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.     

轴向冲击载荷下圆柱壳动力屈曲的计算机仿真

圆柱壳是工程中最常用的结构元件之一,对圆柱壳在各种基本载荷以及在不同类型载荷的联合作用下的屈曲问题研究一直是应用力学界和结构工程界十分关注的课题,与静力屈曲相比,圆柱壳的动力屈曲问题的研究还不尽完善.文章通过有限元软件ANSYS研究圆柱壳轴向冲击下的动力屈曲问题.     

直角切口柔性铰链串联支链的屈曲分析

利用材料力学弯曲变形理论的挠曲线近似微分方程,建立计算直角切口柔性铰链串联支链屈曲临界力的数学模型,在建模过程中定义平均屈曲临界挠度系数并给出其确定方法.采用商用有限元软件ANSYS8.0对大量不同参数的串联支链模型进行有限元屈曲分析,分析结果与根据数学模型计算得到的屈曲临界力十分接近,说明了所建数学模型的正确性.研制测试柔性铰链串联支链屈曲临界力的专用试验装置,针对加载载荷尺寸对柔性支链屈曲的影响,提出载荷等效长度的概念.试验结果表明:屈曲临界力的试验值与理论计算值相吻合,从而验证所建数学模型具有较高的参考价值,可以作为柔性铰链串联支链屈曲优化设计的指导理论.     

复合材料中厚层合板在轴压位移和剪切作用下的稳定性研究

采用线性屈曲分析方法和弧长法对单向压缩和剪切两种载荷作用下复合材料中厚板的稳定性问题进行研究,给出两种载荷作用下的一阶失稳模态以及失稳平衡路径,并得出在单向压缩载荷作用下,过分叉点后载荷的微小增加将引起板的横向大挠度变形,而不能继续承受更大的载荷,而板在剪切载荷的作用下不发生失稳的结论。     

水平井中钻柱屈曲的非线性有限元分析

建立了直井中钻柱屈曲的平衡方程及对应的泛函表达式,使用有限元法对不同约束下水平井中钻柱从稳定到非线性屈曲的整个过程进行了分析。力学模型中考虑了重力和扭矩对屈曲的影响。分析结果表明:钻柱的屈曲是一个从局部屈曲到总体屈曲的过程,屈曲段的井壁约束力、钻柱弯矩和屈曲位移呈周期性变化;重力在水平井中对屈曲有较强的抑制作用;边界约束对屈曲的影响不可忽略;扭矩对屈曲的影响很小,可以忽略。