Search

Four-point bending tests have been a staple in many structural engineering experiments as a reliable way of assessing the bending resistance of circular hollow sections, tubes, and cylindrical shells, and they continue to be widely performed. However, relatively little attention appears to have been paid to quantifying the effects of different boundary conditions on the test outcome. In particular, the restraint or freedom given to the cross section at the ends of a specimen to ovalize can have a significant impact when the specimen is in an appropriate length range. Ovalization is an elastic geometrically nonlinear phenomenon that is known to reduce the elastic bending resistance by as much as half in long tubes or cylinders. This paper presents a short distillation of some recent advances in understanding the buckling of cylindrical shells under uniform bending, identifying the strong influence of cylinder length on cross-section ovalization. A sample set of three-dimensional load application arrangements used in existing four-point bending tests was simulated using finite elements, allowing an assessment of the differences caused by prebuckling ovalization and its effect on the tested bending resistance. The study is limited to elastic behavior to identify the effect of ovalization alone in reducing stiffness without material nonlinearity. The outcomes demonstrate that maintaining circularity at inner load application points by appropriate stiffening has a significant effect. With freedom to ovalize, a significant reduction in stiffness occurs, leading to much lower bending resistance at buckling than may be achievable in practical applications. [ABSTRACT FROM AUTHOR]

Efforts are ongoing to characterise a comprehensive resistance function for cylindrical shells under uniform bending, a ubiquitous structural system that finds application in load-bearing circular hollow sections, tubes, piles, pipelines, wind turbine support towers, chimneys and silos. A recent computational study by Rotter et al. demonstrated that nonlinear buckling of perfect elastic cylinders under bending is governed by four length-dependent domains –'short', 'medium', 'transitional' and 'long'– depending on the relative influence of end boundary conditions and cross-sectional ovalisation. The study additionally transformed its resistance predictions into compact algebraic relationships for use as design equations within the recently developed framework of reference resistance design. This article extends on the above to present a detailed computational investigation into the imperfection sensitivity of thin elastic cylindrical shells across the most important length domains, using automation to carry out the vast number of necessary finite element analyses. Geometric imperfections in three forms – the classical linear buckling eigenmode, an imposed cross-sectional ovalisation and a realistic manufacturing 'weld depression' defect – are applied to demonstrate that imperfection sensitivity is strongly length dependent but significantly less severe than for the closely related load case of cylinders under uniform axial compression. The axisymmetric weld depression almost always controls as the most deleterious imperfection. The data are processed computationally to offer an accurate yet conservative lower-bound algebraic design characterisation of imperfection sensitivity for use within the RRD framework. The outcomes are relevant to researchers and designers of large metal shells under bending and will appeal to computational enthusiasts who are encouraged to adopt the automation methodology described herein to explore other structural systems. [ABSTRACT FROM AUTHOR]

A recent computational study identified four distinct domains of stability behaviour at different lengths in thin elastic cylindrical shells under global bending. Cylinders of sufficient length suffer from fully-developed cross-sectional ovalisation and fail by local buckling at a moment very close to the Brazier prediction. Progressively shorter cylinders experience less ovalisation owing to the increasingly strong restraint provided by the boundary at the edges. Very short thin cylinders, however, restrain the formation of even a local buckle and fail through a limit point instability at moments and curvatures significantly in excess of the classical elastic prediction. This limit point behaviour is not caused by ovalisation but by the growth of a destabilising fold on the compressed meridian. The nonlinear behaviour of very short cylinders under global bending is investigated in detail herein, covering a wide range of lengths, radius to thickness ratios and boundary conditions with both restrained and unrestrained meridional rotations corresponding to ‘clamped’ and ‘simply-supported’ conditions respectively. Two types of imperfections are investigated, the critical buckling eigenmode and a realistic manufacturing-related ‘weld depression’. A complex insensitivity to these imperfections is revealed owing to a pre-buckling stress state dominated by local compatibility bending, and the cylinder length is confirmed as playing a crucial role in governing this behaviour. The study contributes to the characterisation of multi-segment shells with very short individual cylindrical segments, often found in the aerospace and marine industries as well as in specialised civil engineering applications such as LIPP® silos. [ABSTRACT FROM AUTHOR]

Classical shell finite elements usually employ low-order polynomial shape functions to interpolate between nodal displacement and rotational degrees of freedom. Consequently, carefully-designed fine meshes are often required to accurately capture regions of high local curvature, such as at the ‘boundary layer’ of bending that occurs in cylindrical shells near a boundary or discontinuity. This significantly increases the computational cost of any analysis. This paper is a ‘proof of concept’ illustration of a novel cylindrical axisymmetric shell element that is enriched with rigorously-derived transcendental shape functions to exactly capture the bending boundary layer. When complemented with simple polynomials to express the membrane displacements, a single boundary layer shell element is able to support very complex displacement and stress fields that are exact for distributed element loads of up to second order. A single element is usually sufficient per shell segment in a multi-strake shell. The predictions of the novel element are compared against analytical solutions, a classical axisymmetric shell element with polynomial shape functions and the ABAQUS S4R shell element in three problems of increasing complexity and practical relevance. The element displays excellent numerical results with only a fraction of the total degrees of freedom and involves virtually no mesh design. The shell theory employed at present is kept deliberately simple for illustration purposes, though the formulation will be extended in future work. [ABSTRACT FROM AUTHOR]

• Statistical reconstruction of lost dataset of axially compressed cylindrical shells. • Existing test data is unrepresentative of full-scale civil engineering metal shells. • Destructive tests of full-scale civil engineering shells will never be performed. • Part of legacy dataset then reproduced computationally or 'synthetically'. • Case made to fully embrace simulation in future reliability studies. In the structural engineering limit state design philosophy of the Eurocodes, a target reliability is achieved by using partial safety factors on both the loading and the resistance that are, in principle, to be calibrated based on test data. The partial factor that is currently used for the buckling limit state of metal shells has been adopted from the knowledge base on other structural elements and has been retained for reasons of historical continuity and to maintain a close relationship between all Eurocode steel design standards. However, the mechanics of the behaviour of thin-walled metal shells gives strong reasons to believe that the partial factors for buckling should be dependent on the shell geometrical form, slenderness, load case and quality of fabrication for the target reliability to be consistently achieved. None of these are currently considered in defining the partial factor for resistance. The most ubiquitous thin-walled metal shell structures are imperfection-sensitive cylinders under uniform axial compression. A dataset of many hundreds of these test results has thus been accumulated over many decades, though it is of variable quality and sparsely documented. This paper shows that this dataset is an entirely inappropriate basis on which to calibrate the safety level of full-scale metal civil engineering shells. Indeed, the professional community should face the uncomfortable reckoning that an experimental test dataset suitable for the reliable calibration of the safety level of design relationships for full-scale metal civil engineering shells may likely never come into existance, and that the bolder approach of extensive computational simulation must instead be embraced. [ABSTRACT FROM AUTHOR]

Computational problems in structural engineering are growing ever larger and solutions must increasingly be based on correspondingly large datasets obtained from detailed parametric sweeps. However, the acquisition of computational datasets of useful size is also becoming increasingly unfeasible without extensive use of automation. In computational shell buckling studies, particularly those of thin-walled shells under complex loading conditions, an important qualitative piece of information is the class of buckling mode which reveals the dominant destabilising membrane stress components. Unfortunately, the diversity of geometries that can be encountered in computational shell buckling studies is truly vast, and there is currently no way to rapidly assess the buckling mode without laborious direct human observation of the model output. This paper presents an automated classification tool for linear bifurcation buckling eigenmodes in cylindrical shells such as those found as wind turbine support towers, chimneys, silos, tanks, piles and pipelines. It is based on a convolutional neural network implemented using the PyTorch machine learning framework. The adopted network architecture is based on those widely adopted for image classification and recognition tasks, chosen based on a stratified five-fold cross-validation exercise. The network is trained on a purposefully generated basic dataset of 13,392 linear bifurcation buckling eigenmodes modes encoded as chromatic signatures in .jpg images (enhanced to 25,726 by transformations). An example parametric sweep of a cylindrical shell under unsymmetrical wind loading illustrates the performance of the classifier. A GitHub repository offers Python scripts and instructions on how to download the dataset and trained network. • Buckle classifier developed based on a convolutional neural network architecture. • Eigenmodes classified into eleven classes based on dominant destabilising stresses. • Automated generation of prototypical dataset for cross-validation and training. • Enhancement of dataset for release training via probabilistic transformations. • Classifier used in ongoing development of the RRD method for EN 1993-1-6 [ABSTRACT FROM AUTHOR]

Abstract: Many thin-walled cylindrical shells are used in structural applications in which the dominant loading condition is global bending. Key examples include chimneys, wind turbine support towers, pipelines, horizontal tanks, tubular piles and silos. The buckling behaviour of these structures in bending is complex due to the coupling between cross-section ovalisation and local bifurcation buckling. Analytical treatments of this problem have a history going back almost a century and still constitute an active and challenging research area. This paper investigates in detail the effect of cylinder length on the nonlinear elastic buckling behaviour of clamped cylindrical tubes under global bending, covering a very wide range of lengths. It is found that the behaviour may be classified into four distinct length-dependent domains with clearly-defined boundaries which have here been assigned the names ‘short’, ‘medium’, ‘transitional’ and ‘long’. Algebraic characterisations of the computed nonlinear moment–length relationships are proposed for design purposes. [Copyright &y& Elsevier]

Highlights: [•] Cylindrical shells with helical features are not widely studied computationally. [•] The study proposes a robust and powerful method of parametric helical meshing. [•] Method is illustrated on the post-buckling behaviour of axially-compressed cylinder. [•] Imperfection sensitivity of spirally welded cylinder is investigated. [Copyright &y& Elsevier]

Abstract: This paper explores the use of solid continuum finite elements and shell finite elements in the modelling of the nonlinear plastic buckling behaviour of cylindrical metal tubes and shells under global bending. The assumptions of shell analysis become increasingly uncertain as the ratio of the radius of curvature to the thickness becomes smaller, but the classical literature does not draw a clear line to define when a shell treatment is inappropriate and a continuum model becomes essential. This is a particularly important question for the bending of tubular members, pipelines, chimneys, piles, towers and similar structures. This study is therefore concerned solely with the uniform bending of thin tubes or thick shells which fail by plastic buckling well into the strain-hardening range. The analyses employ finite element formulations available in the commercial software ABAQUS because this is the most widely used tool for parametric research studies in this domain with an extensive and diverse element library. The results are of general validity and are applicable to other finite element implementations. This paper thus seeks to determine the adequacy of a thin or thick shell approximation, taking into account geometric nonlinearity, complex equilibrium paths, limit points and bifurcation buckling, extensive material ductility and linear strain hardening. It aims to establish when it is viable to employ shell elements and when this decision will lead to outcomes that lack sufficient precision for engineering design purposes. The results show that both thin and thick (shear-flexible) shell elements may give a reasonably accurate prediction of the buckling moment under global uniform bending for cylindrical tubes as thick as R/t=10. A finite strain and thick shell formulation is additionally shown to model the ductility of such thick tubes well, even when ovalisation of the cross-section and strain hardening are included. The use of solid continuum elements to model tubes in bending is found to become increasingly uneconomical as the R/t ratio rises above 25 with reduced advantages over shell elements, both in terms of the accuracy of the solution and the computation time. [Copyright &y& Elsevier]

Eccentric discharge in slender metal silos is known to be one of the most critical load conditions, responsible for many silo buckling disasters in the past. The high failure rate may be significantly attributed to difficulties in devising a suitable wall pressure representation for this condition. Where the flow of stored solids is eccentric and has partial contact with the wall in a slender silo, the solid exerts much lower pressures than the adjacent stationary solid. This pressure drop leads to very high local axial compression and causes buckling failure. Experimentally measured pressures indicate that a significant rise in pressure may occur just outside the flow channel, but its form and magnitude are not yet well understood because very detailed and expensive instrumentation is needed to obtain data that can define it. This paper explores the nonlinear structural behavior and buckling of a slender metal silo with and without specific inclusion of an adjacent rise in pressure, to determine whether it is a necessary part of any design pressure representation. To assist this investigation, the mechanics of the nonlinear behavior of a cylindrical silo under this load condition is explored using the analogy of a propped cantilever slice beam. Advantage is taken of a particular load condition that leads, by chance, to buckling at the same load factor for both linear and geometrically nonlinear analyses. This special case permits the detrimental effect of wall flattening and the beneficial effect of the changing prebuckling stress pattern to be explored to give a deeper insight into the behavior. The slice-beam analogy may also be generalized to describe the nonlinear behavior of any thin-walled cylindrical shell under meridional strip-like loads acting on part of the circumference. [ABSTRACT FROM AUTHOR]

Abstract: Modern procedures for the design of shell structures against buckling have their basis in analytical studies of axisymmetric shell geometries under the very simple load cases of uniform compression, external pressure and torsion. Studies of more complex but realistic stress states were based on prebuckling analyses using either membrane theory or linear bending theory because even these involved considerable mathematical complexity. As a result, only limited conclusions for practical design could be drawn and the effects of geometric nonlinearity could not be assessed. With recent advances in computing power and nonlinear finite element programs, it is now possible to undertake nonlinear analyses of complex load patterns that would have been very difficult to do only a decade or so ago. A number of practical load cases lead to a strip of pressure down one meridian, of which the best known ones are probably wind on tanks, eccentric discharge in silos, local thermal differentials, and partial fluid filling of a cylinder. This paper explores some of the rather unexpected stress patterns and modes of buckling that are predicted to develop in thin-walled cylindrical shells under such unsymmetrical strips of normal pressure. The results of a parametric study are presented to show the influence of the circumferential spread of the pressure strip on the structural behaviour. It is shown that the structural response to such loads may be very different, depending on whether the load acts inward or outward, and whether geometric nonlinearity and geometric imperfections are also included in the assessment. [Copyright &y& Elsevier]

The condition of eccentric discharge is known to be one of the most critical for the design of thin-walled cylindrical metal silos. Significant progress has been made in recent years in devising a relatively realistic set of representative pressures for this load case. However, the consequences these may have on the predicted structural behavior of a silo are not yet fully understood. This paper presents a detailed parametric study into the behavior of a custom-designed slender silo under a set of unsymmetrical pressures describing the action of an eccentric parallel-sided pipe flow channel of varying cross-sectional areas. The results are compared with the reference axisymmetric case of concentric discharge. It is found that the predicted behavior is very complex indeed, and that geometric nonlinearity is of much greater significance for cylindrical shells under unsymmetrical load patterns than under symmetrical patterns. Further, it is found that eigenmodeaffine imperfections, which are very deleterious under axisymmetric loading patterns, are instead beneficial to the buckling strength of a silo under eccentric discharge, thus making them unsuitable for use in design for this load condition. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive computational investigation of the elastic nonlinear buckling response of near-perfect and highly imperfect uniform-thickness thin cylindrical shells of varying length under combined uniform compression and bending. In particular, the elastic ovalization phenomenon in cylindrical shells of sufficient length under combined compression and bending was systematically investigated with finite elements for the first time. The study considered a representative range of practical lengths up to very long cylinders in which ovalization is fully developed under uniform bending and Euler column buckling controls under uniform axial compression. The imperfection sensitivity of the system was studied by introducing a single idealized axisymmetric weld depression imperfection at the midspan of the cylinder. The predictions permit an exploration of the nonlinear mechanics of the generally unfavorable interaction between bending and axial compression at the elastic nonlinear buckling limit state in thin long cylinders. The interaction is at its most unfavorable in cylinders where Euler column buckling is about to become critical, and is qualitatively very different from the favorable moment-force interaction at the reference plastic limit state of circular tubes. A simple closed-form algebraic characterization of the interaction is proposed considering both imperfections and ovalization. [ABSTRACT FROM AUTHOR]

Cylindrical metal shells with elliptical cross-sections are gaining increasing popularity as hollow sections due to their unique aesthetic appearance and different geometric properties about their two principal axes, with one axis exhibiting properties that are significantly more favourable than the other under flexure. However, in comparison with other hollow geometries, elliptical cross-sections have only recently begun receiving significant research attention. This is partly because even simple analytical treatments inevitably encounter cumbersome elliptical integrals that have no closed-form solutions, a problem now attenuated by powerful modern computing capabilities. A recent computational study investigated the nonlinear buckling resistance of perfect elastic circular cylindrical shells under uniform bending, establishing four distinct length-dependent domains of behaviour and characterising these in compact form using specially chosen dimensionless parameters. The present study extends this work to cylinders with elliptical cross-sections under bending about both principal axes. The same qualitative domains of length-dependent nonlinear elastic behaviour are found as for circular cylinders, but requiring a different algebraic characterisation that takes account of the varying elliptical radii. On the basis of computational results, a reference equation for the moment governing the ‘Brazier’ ovalisation cross-sectional failure mode for long elliptical thin-walled cylinders is deduced and presented for publication for the first time. [ABSTRACT FROM AUTHOR]

The resistance of cylindrical shells and tubes under uniform bending has received significant research attention in recent times, with a number of major projects aiming to characterise their strength through both experimental and numerical studies. However, the investigated cross-section slenderness ranges have mostly addressed low radius to thickness ratios where buckling occurs after significant plasticity and the influence of geometric imperfections is relatively minor. The behaviour under uniform bending of thinner imperfection-sensitive cylinders that fail by elastic buckling was largely omitted, as was the influence of finite length effects. The value of such resistance models that are only useful for thicker cylinders is therefore somewhat limited. This paper offers the most comprehensive known characterisation of the buckling and collapse resistance of isotropic cylindrical shells and tubes under uniform bending. Expressed within the modern framework of Reference Resistance Design (RRD), it holistically incorporates the effects of material plasticity, geometric nonlinearity and sensitivity to realistic and damaging weld depression imperfections. The characterisation was made possible by the authors' recently-developed novel methodology for mass automation of nonlinear shell buckling finite element analyses. A modification of the RRD formulation is proposed which facilitates its application to systems of low slenderness, and offers a compact algebraic characterisation of all potential imperfection amplitudes for this common shell structural condition. A reliability analysis is also performed. • Comprehensive computation exploration of cylindrical shells under uniform bending • Length-dependent plasticity, buckling and imperfection sensitivity relationship explored • Closed-form algebraic characterisation offered for entire range of behaviours. • Enhancement to Reference Resistance Design to account for imperfect thick shell behaviour • EN 1990 Annex D reliability analysis performed, with critique of underlying tests database. [ABSTRACT FROM AUTHOR]