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Metal wind turbine support towers are very tall and slender shell structures designed to exhibit a stepwise varying distribution of optimised wall thicknesses, with strakes in the upper regions of the tower usually being much thinner than those in the lower regions. Each strake is an individual shell and potentially a critical location for failure, and as the failure location is rarely obvious in advance each strake in theory requires careful meshing in a finite element analysis. It is not unusual for over twenty individual strakes to be present in a design, and the computational cost involved in modelling such a structure with finite elements, particularly in nonlinear analyses, can quickly become prohibitive for execution on a personal workstation. Compromises in mesh resolution must often be made, usually to the detriment of the quality of the global solution. This paper explores a simple hybrid beam-shell modelling technique that permits an efficient and insightful analysis of multi-strake wind turbine support towers. It consists of modelling all but a handful of the strakes with beam elements or rigid bodies which have a negligible computational cost compared to shell elements, and to focus the deployment of expensive shell elements only on strakes of interest as part of a resistance assessment. As only strakes meshed with shell elements participate in a failure mechanism, the technique allows the realistic exploration of the relative criticality of all tower strakes. The technique is illustrated on a real design of a 1.5 MW 25-strake wind turbine tower. • A technique using combinations of beam and shell elements for multi-strake towers. • Full shell element models are expensive and provide little auxiliary information. • Beam-shell models permit the exploration of criticality in all tower strakes. • Potentials for optimisation of these complex structures under complex load cases. • LBAs, MNAs and GMNIAs, including pitfalls relating to eigenmode imperfections. [ABSTRACT FROM AUTHOR]

Multi-strake cylindrical and conical shells of revolution are complex but commonplace industrial structures which are composed of multiple segments of varying wall thickness. They find application as tanks, silos, circular hollow sections, aerospace structures and wind turbine support towers, amongst others. The modelling of such structures with classical finite elements interpolated using low order polynomial shape functions presents a particular challenge, because many elements must be sacrificed solely in order to accurately represent the regions of local compatibility bending, so-called ‘boundary layers’, near shell boundaries, changes of wall thickness and at other discontinuities. Partitioning schemes must be applied to localise mesh refinement within the boundary layers and avoid excessive model runtimes, a particular concern in incremental nonlinear analyses of large models where matrix systems are handled repeatedly. In a previous paper, the authors introduced a novel axisymmetric cylindrical shell finite element that was enriched with transcendental shape functions to capture the bending boundary layer exactly, permitting significant economies in the element and degrees of freedom count, mesh design and model generation effort. One element is sufficient per wall strake. This paper extends this work to conical geometries, where axisymmetric elements enriched with Bessel functions accurately capture the bending boundary layer for both ‘shallow’ and ‘steep’ conical strakes, which are characterised by interacting and independent boundary layers, respectively. The bending shape functions are integrated numerically, with several integration schemes investigated for accuracy and efficiency. The potential of the element is illustrated through a stress analysis of a real 22-strake metal wind turbine support tower under self-weight. The work is part of a wider project to design a general three-dimensional ‘boundary layer’ element. [ABSTRACT FROM AUTHOR]

The new Reference Resistance Design (RRD) method, recently developed by Rotter [1] , for the manual dimensioning of metal shell structures effectively permits an analyst working with only a calculator or spreadsheet to take full advantage of the realism and accuracy of an advanced nonlinear finite element (FE) calculation. The method achieves this by reformulating the outcomes of a vast programme of parametric FE calculations in terms of six algebraic parameters and two resistances, each representing a physical aspect of the shell's behaviour. The formidable challenge now is to establish these parameters and resistances for the most important shell geometries and load cases. The systems that have received by far the most research attention for RRD are that of a cylindrical shell under uniform axial compression and uniform bending. Their partial algebraic characterisations required thousands of finite element calculations to be performed across a four-dimensional parameter hyperspace (i.e. length, radius to thickness ratio, imperfection amplitude, linear strain hardening modulus). Handling so many nonlinear finite element models is time-consuming and the quantities of data generated can be overwhelming. This paper illustrates a computational strategy to deal with both issues that may help researchers establish sets of RRD parameters for other important shell systems with greater confidence and accuracy. The methodology involves full automation of model generation, submission, termination and processing with object-oriented scripting, illustrated using code and pseudocode fragments. [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]

New design methodologies are being developed to allow stocky steel members to attain and exceed the full plastic condition. For theoretical validation, such methods require a characterisation of the uniaxial stress-strain behaviour of structural steel beyond an idealised elastic-plastic representation. However, the strain hardening properties of carbon steels are not currently guaranteed by the standards or by any steel manufacturer. Assumptions must thus be made on what values of these properties are appropriate, often based on limited information in the form of individual stress-strain curves. There is very little consistency in the choices made. This paper first illustrates, using an example elastic-plastic finite element calculation, that a stocky tubular structure can attain the full plastic condition at slendernesses comparable with those defined in current standards and supported by experiment when using only a very modest level of strain hardening, initiated at first yield. It is then hypothesised that the yield plateau in the stress-strain curve for structural carbon steels, classically treated as flat and with zero tangent modulus, actually has a small but statistically significant positive finite gradient. Finally, a robust set of linear regression analyses of yield plateau gradients extracted from 225 tensile tests appears to support this hypothesis, finding that the plateau gradient is of the order of 0.3% of the initial elastic modulus, consistent with what the finite element example suggests is sufficient to reproduce the full plastic condition at experimentally-supported slendernesses. [ABSTRACT FROM AUTHOR]

• Fully reproducible worked solution to computational WTST benchmark. • Illustration of automatable, geometrically scalable meshing protocol. • Illustration of prEN 1993-1-6-compliant GMNIA imperfection amplitude calibration. • Eigenmode and weld depression imperfections require different calibrations. • Benchmark makes start of public repository of GMNIA verification examples. An assessment of the elastic–plastic buckling limit state for multi-strake wind turbine support towers poses a particular challenge for the modern finite element analyst, who must competently navigate numerous modelling choices related to the tug-of-war between meshing and computational cost, the use of solvers that are robust to highly nonlinear behaviour, the potential for multiple near-simultaneously critical failure locations, the complex issue of imperfection sensitivity and finally the interpretation of the data into a safe and economic design. This paper presents a detailed reference solution to the computational buckling analysis of a standardised benchmark problem of an 8-MW multi-strake wind turbine support tower. Both linear and nonlinear analyses are performed, including advanced GMNIA with several different models of geometric imperfections. The crucial issue of interpreting the imperfection amplitude in a way that is compliant with the new prEN 1993-1-6 is discussed in detail. The solution presented herein is intended for use by analysts in both industry and academia for training, verification and calibration of finite element models and is intended to initiate a public repository of such computational solutions for metal civil engineering shell structures. This paper is the second of a pair. The first paper [37] presents a synthesis of 29 submissions to an international round-robin exercise performed on the same benchmark problem. [ABSTRACT FROM AUTHOR]

Abstract: Many problems in structural mechanics involve complex principal stress fields that are not orthogonal to the geometric axis of the structure. Such problems are often analysed with finite elements, but the quality of a finite element solution may be sensitive to the orientation of the mesh with respect to the principal axes of stress. This paper presents the outline of a procedure to generate well-structured inclined quadrilateral finite element meshes for the analysis of thin plate and shell structures. The procedure was developed using the commercial FE pre-processor ABAQUS CAE and the Python script language, though it may readily be applied in any pre-processor which supports an external scripting functionality. A set of mesh convergence studies using linear buckling analyses are presented on four benchmark problems with known analytical solutions to illustrate the effect of inclined meshes on the accuracy of the computed solution. These illustrations are intended to raise an awareness of the subtle but important relationship between mesh and stress field orientation and are presented for the benefit of practising finite element analysts in structural engineering. [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]

Abstract: The unsymmetrical discharge of a granular solid from a thin-walled cylindrical metal silo is well known to be a potential prelude to catastrophic buckling failure. The mechanics of this structure have only been very slowly unraveled in recent years with the help of powerful nonlinear finite element analyses. The associated buckling collapse is now known to be caused by localized axial membrane compression and occurs under predominantly elastic conditions. However, such high compressive stress concentrations are also known to produce significantly less imperfection sensitivity than uniform compression. For this reason, the search for an appropriately detrimental imperfection form under eccentric discharge has been quite a long one. This study presents and explores a novel form of long-wave imperfection using a superellipse to parametrise the entire shell geometry. This imperfection, termed ‘superelliptical flattening’, is judged to be potentially present in practical silo construction, and is shown to cause significant decreases in the nonlinear buckling strength of an imperfect slender silo under eccentric discharge. [Copyright &y& Elsevier]

Abstract: The axisymmetric linear bending theory of shells is treated for thin-walled orthotropic cylindrical shells under any smooth axial distribution of normal and shear pressures. The equations are developed, solved and explored in this paper. The derivation is presented in terms of a generalised Hooke’s Law with coupling between the axial membrane stress resultant and axial bending moment. This formulation permits the shell to be alternatively treated as a composite isotropic cylinder with axial stiffeners, rendering it useful for many practical problems. A linear kinematic relationship is assumed between the generalised strains and displacements. Expressions for the linear axial bending half-wavelength are presented for special cases of the stiffness matrix. The equations developed here are simple enough to be applied to the analysis of anisotropic thin-walled cylindrical shells using basic spreadsheet tools, removing the need to perform an onerous finite element analysis. Engineering applications potentially include corrugated metal, axially-stiffened or reinforced concrete silos under granular solid pressures, tanks under hydrostatic pressures, tubular piles under earth pressures, gas-filled cisterns and chimneys. [Copyright &y& Elsevier]

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]