Buckling of vibrations of shear-flexible orthotropic plates subjected to mixed boundary conditions

An approximate technique is presented for the analysis of buckling and vibrations of free-form, orthotropic, shear-flexible (‘Mindlin’) plates subject to mixed boundary conditions. The method falls into the category of Rayleigh/ Ritz-techniques; however, by using Lagrangian multipliers to ‘relax’ the geometric boundary conditions, the selection of appropriate trial functions is made considerably simpler. Accuracy and reliability of the proposed technique is demonstrated on the basisof several sample problems.     

Short vs. long column behavior of pultruded glass-fiber reinforced polymer composites

Analytical and experimental studies were conducted for the purpose of establishing a distinguishing criterion between short and long FRP composite column behaviors. The results of investigating 24 full-scale GFRP composite columns are presented. The experimental studies utilized specimens with commonly used ‘Universal’ and ‘Box’ cross-sections which were manufactured by the pultrusion process using E-glass fibers in various forms as reinforcement and polyester and vinylester as binding matrices. The effective slenderness ratios (L/r) of the investigated columns were 3.79, 32.7, 47.9, 63.1 and 75.4 for the ‘Universal’ section and 9.38, 53.9 and 78.9 for the ‘Box’ section. The specimen lengths were 18 inches (0.46 m), 8 foot (2.44 m), 12 foot (3.66 m), 16 foot (4.88 m) and 19.25 foot (5.87 m). All columns were tested in a vertical position and under compressive axial static loading and the fundamental pinned–pinned end-conditions. The columns’ compressive strains, buckling and crippling loads, lateral displacements, initial curvatures, and modes of failure were documented during the course of this investigation. Orthotropic mechanical properties of the composites were experimentally obtained utilizing 44 coupons cut from the column specimens. Analytically, Euler's formula was employed to obtain critical loads for the slender columns. For short columns, the classical plate theory was used to predict columns’ buckling loads. Based on experimental evaluations and analytical results, a slenderness ratio based criterion was established for distinguishing between short and long composite column behaviors. Further conclusions and design recommendations were made.     

Probabilistic approach to elastic-plastic coupled instability

Interaction of elastic and plastic instability modes is considered. Buckling resistance of columns is treated as the lesser value of plastic resistance R_pl of a critical cross-section and the Euler critical force R_cr. The random variables R_pl and R_cr have log-normal probability distributions. Their coefficients of variation are supposed equal and independent from the slenderness ratio λ in the first model of random instability. A simple formula for the buckling factor is derived thanks to the introduction of equivalent Weibull distribution functions. Strict formulae are derived in the second model of coupled instability for the columns of equal median values R_pl = R_cr when the relative slenderness ratthe columns λ( = 1. The design values R_d of the elastic-plastic resistance are reduced by an analysis factor C which depends on the slenderness λ( and the imperfection class, a, b, c or d, defined for European multiple buckling curves. Optimal values of resistance factor γ_R and analysis factor γ_C are determined so that the risks of exceeding the design values R_d and C_d are equal.     

Some comments on the numerical analysis of plates and thin-walled structures

The paper briefly reviews the theoretical analysis of plates structures that might exhibit multiple ‘loading paths’ and highlights the need for engineers using non-linear numerical modelling to be aware of the multi-mode phenomenon and to ensure that the modelling is set up in such a manner that the various ‘loading paths’ and possible changes of path would be incorporated in the modelling response. The paper presents a simple example of numerical analysis of thin-plate buckling that involves ‘coupled buckling modes’ and provides comments on suitable methods for defining in a simple and straightforward way the numerical modelling that could ensure that results from computer analysis describe the physically correct relationship between applied loadings and deformations of thin-walled structural components.     

Understanding imperfection-sensitivity in the buckling of thin-walled shells

The buckling of thin-walled shell structures under load is still imperfectly understood, in spite of much research over the past 50 years. In this paper the author traces the history of the ideas which have been deployed in order to shed light on what is often referred to as ‘imperfection-sensitive’ buckling behaviour of shells. The ideas, which recur in various combinations, involve interaction of competing buckling modes, nonlinear behaviour, the growth of initial geometric imperfections under load and the alteration of the distribution of membrane stress as imperfections grow. The author claims that there are strong grounds for supposing that ‘locked in’ initial stresses on account of imperfect initial geometry and the static indeterminacy of boundary conditions of real shells have a pronounced effect on the buckling performance. This effect has been ignored in the past, and is the subject of a current experimental study.     

GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections

This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and (ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip or shell finite element analyses.     

Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shells. Part A: Experimental investigation and effective width evaluation

A box-like stringer-stiffened thin-walled CFRP structure was subjected to load cases well beyond the limits of local buckling. The development of the deflection pattern was recorded via optical means and analysed numerically. In addition, the structure was modelled and analysed using the MARC FE program in the nonlinear deflection range.

The geometrical imperfections of the test structure were recorded by mechanical scanning and optical methods and introduced into the mathematical model. For the perfect (‘ideal’) and the geometrically imperfect (‘real’) structural model, the results of the FE analyses were compared and used to judge the effect of geometric imperfections on the postbuckling behaviour of the structure.

The effective axial stiffness for the various postbuckling states was evaluated and related to analytical estimates of effective width values for orthotropic sheet-like panels.     

考虑SD效应的地基承载力广义强度安全系数

现有的地基承载力安全系数其缺点在于不能体现土体的三维受力状态,广义强度安全系数采用π平面上广义剪应力与强度包络线的比值可以更好的考虑土体的三维受力情形,因而能更好的体现地基承载力的安全特性。为了研究SD效应即土在三轴压缩与三轴拉伸强度不等对地基承载力安全系数的影响,本文首先基于广义非线性强度准则研究了表征SD效应大小R及准则参数α的关系,将R划分成三种情况。然后运用变换应力空间方法描述了基于广义非线性强度准则下的广义强度安全系数的表达式。算例结果表明,SD效应参数R对于地基承载力安全系数影响较大。     

A semi-analytical model for local post-buckling analysis of stringer- and frame-stiffened cylindrical panels

A fast semi-analytical model for the post-buckling analysis of stiffened cylindrical panels is presented. The panel is comprised of a skin (shell) and stiffeners in both longitudinal (stringers) and circumferential direction (frames). Local buckling modes are considered where the skin may buckle within a bay and may induce rotation of the stiffeners. Stringers and frames are considered as structural elements and are thus not ‘smeared’ onto the skin. Large out-of-plane deflections and thus non-linear strain–displacement relations of skin and stiffeners are taken into account. The displacements of skin and stiffeners are approximated by trigonometric functions (Fourier series). First, a linear buckling eigenvalue analysis is carried out and some combination of buckling eigenmodes is chosen as imperfection. Then the load history is started and the Fourier coefficients are determined by minimizing the stiffened panel's energy at each load level. A curve-tracing algorithm, the Riks method, is used to solve the equations. The present model can be used to assess the post-buckling behavior of stiffened panels, for example, aircraft fuselage sections.     

Tangible virtuality—perceptions of computer-aided and physical modelling

Designing—giving form to new objects or environments—is largely a question of anticipating the workings of spatial and material environments, which can become ‘reality’ only by being built. Until ‘realized’, a design is essentially a figment of the designer's imagination, although his or her ideas may be laid down and conveyed to others via specialized design media. In this way, impressions of the design may be shared with clients, colleagues or other ‘actors’ in the design process.

Such products of the designer's imaging process can be relatively abstract or begin to approach future reality. Form & Media research can be ‘revealing’, stimulating insights concerning preferences, working processes and the effects of products of the designer's imagination. In the past 10 years, we have gained considerable practical experience with both virtual and tangible (scale) models. We have compared different techniques in conference workshops, within educational settings and in our Form & Media research laboratory. The research projects ranged from the development of practical techniques and working methods to protocol analyses of designing architects.

This contribution draws comparisons between different computer-aided modelling techniques, with an indication of their perspectives, making use of the experience gained from various experiments in an educational context, and will highlight the potentials for different combinations of digital and physical modelling techniques.     

Towards a rationally based elastic-plastic shell buckling design methodology

The ‘reduced stiffness method’ for the analysis of shell buckling was developed to overcome a trend towards increasingly sophisticated analysis that has become divorced from its basically simple underlying physics. This paper outlines the developments of the reduced stiffness method from its origins in the late 1960s, through its experimental confirmation, generalisation and elaboration over the past 20 years, to its more recent consolidation using carefully controlled non-linear numerical experiments. It is suggested that the method has now reached a stage where it could profitably be adopted as a basis for an improved shell buckling design methodology.     

不同自由端约束条件下双轴对称工字形截面悬臂梁的稳定性

对自由端侧向位移和扭转简支约束、自由端仅扭转约束和自由端仅简支扭转弹性约束条件下,双轴对称工字形截面悬臂梁的稳定性进行研究,并提出临界弯矩的计算公式。这一计算公式能够考虑悬臂梁长细比的变化,以及横向荷载作用点沿截面高度方向变化对其临界弯矩的影响。通过与有限元结果的比较可以发现,所提出的计算公式可以获得精度良好的临界弯矩值。     

Flexural capacity of thin wall fibre reinforced cement

A method of calculating the flexural strength (cracking) of fibre reinforced cement floor and wall systems is presented. The method for flexural strength is based on non-linear stress-strain relationship at failure tension and corresponding stress distribution in section at cracking and subsequent failure. The method is verified in tests of full-scale ‘V’ and ‘U’ shaped panels made of asbestos-cement which were tested as simply supported beams designed for the construction of low cost housing and wall facades. Additional tests were performed on small sized samples to verify the mechanical properties of material used.     

Ultimate strength of submarine pressure hulls with failure governed by inelastic buckling

The analysis of submarine pressure hull assumes great importance among structural engineers due to the complexity involved in the collapse mechanism of stiffened shell structures. In most of the cases, the failure of stiffened shell structures occurs due to elastic buckling. But for some combinations of shell-stiffener geometry and material characteristics, the structure can fail by inelastic buckling, for which the methods of analysis are meagre. In this paper, the analysis of submarine pressure hull structure in which the failure gets governed by inelastic buckling is demonstrated. Three different approaches have been employed to investigate the ultimate strength of the ring stiffened submarine pressure hull structure with inelastic buckling modes of failure. The methods used are ‘Johnson–Ostenfeld inelastic correction’, ‘imperfection method’ and ‘finite element approach’. A typical submarine shell structure has been analysed for the inelastic buckling failure using these three approaches and the results are discussed.     

Stability of variable thickness shear deformable plates—first order and high order analyses

This work presents analysis of the buckling loads for thick elastic rectangular plates with variable thickness and various combinations of boundary conditions. Both the first order and high order shear deformation plate theories have been applied to the plate's analysis. The effects of higher order non-linear strain terms (curvature terms) are considered as well. The governing equations and the boundary conditions are derived using the principle of minimum of potential energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the stability analysis of compressed members with variable cross-section, which provides for the derivation of the exact stiffness matrix of tapered strips including the effect of in-plane forces. The results from the two shear deformation theories are compared with those obtained by the classical thin plate's theory and with published results.     

The computational post buckling analysis of fuselage stiffened panels loaded in compression

Fuselage panels are commonly fabricated as skin–stringer constructions, which are permitted to locally buckle under normal flight loads. The current analysis methodologies used to determine the post buckling response behaviour of stiffened panels relies on applying simplifying assumptions with semi-empirical/empirical data. Using the finite element method and employing non-linear material and geometric analysis procedures, it is possible to model the post buckling behaviour of stiffened panels without having to place the same emphases on simplifying assumptions or empirical data. Investigation of element, mesh, idealisation, imperfection and solution procedure selection has been undertaken, with results validated against mechanical tests. The research undertaken has demonstrated that using a commercial implicit code, the finite element method can be used successfully to model the post buckling behaviour of flat riveted panels. The work has generated a series of guidelines for the non-linear computational analysis of flat riveted panels subjected to uniform axial compression.     

Shear buckling of thin-walled channel sections

The elastic buckling stresses of channel sections with and without lips and subject to shear forces parallel with the web are determined, where computational modelling of the thin-walled steel sections is implemented by means of a spline finite strip analysis. Both unlipped and lipped channels are studied, where the main variables are flange width, different boundary conditions and shear flow distribution. The channel sections are also analysed at different lengths, to investigate the effect of length/width ratio on the critical shear buckling stresses. Comparisons between cases and with classical solutions are included in this paper.     

Design procedure for the buckling analysis of reinforced concrete shells

The paper presents a simple procedure to establish the buckling load of shell structures. It is essentially based on the assumption that the various factors influencing buckling can be assessed as multipliers of the ‘classical’ critical load.

These factors are: imperfections, creep, plasticity, cracks and the steel reinforcement. The paper gives the values of these factors, thus establishing a method which can be used in practical design.     

叉筒网壳子结构圆柱面交叉立体桁架系巨型网格结构的稳定性研究

本文针对子结构为单层叉筒网壳的圆柱面交叉立体桁架系巨型网格结构,分析了结构的构成、形体参数、支承方式等;建立了几何非线性力学模型,编制了相应的稳定分析程序;针对本结构的特点,着重研究了结构的稳定性能、失稳形式(局部失稳与整体失稳),以参数分析的形式研究了结构局部失稳与整体失稳的关系,找出了不同跨度结构在整体与局部失稳临界状态某些参数的取值规律,并给出了整体与局部失稳状态的分界曲线,可为该结构形式的工程应用提供参考。     

Recent research advances on ECBL approach.: Part I: Plastic–elastic interactive buckling of cold-formed steel sections

The objective of this two parts paper is to present some recent developments and applications of erosion of critical bifurcation load (ECBL) approach for the interactive buckling. Two different types of problems are analysed: (1) plastic–elastic interactive buckling which implements into the Ayrton–Perry interaction formula the plastic strength of the stub columns evaluated by means of local plastic mechanism analysis, and (2) elastic–elastic interactive buckling for members with perforations.The first part of the paper analyses the occurrence of local plastic mechanisms in cold-formed steel sections in compression, and how they can be implemented in the ultimate limit state analysis of the members. Actually, the failure of thin-walled cold-formed members in compression always occurs with a local plastic mechanism. Starting from this observation, the authors suggest to use in the interactive local-overall buckling analysis the sectional plastic mechanism strength instead of traditional ‘effective section’. The ECBL approach is used to implement the proposed interactive buckling model. Results are compared with those of other two recent methods, namely the direct strength method and plastic effective width approach. Relevant tests are used to evaluate the three methods. Comparisons with European and American design codes are also presented in the paper.