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.     

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.     

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.     

T-ring stiffened cone cylinder intersection under internal pressure

Cone–cylinder junctions are vastly used in the industries such as oil refineries and aeronautics. They can be seen in pressure vessels and piping such as tanks' roofs and pipes' reducers. When cone–cylinder junctions are subjected to the internal pressure, compression stresses are established near the joining point of the cone to cylinder and make the junction susceptible to non-symmetric buckling failure or axisymmetric failure. As it is practical to increase the shell wall thickness locally near the point of intersection, sometimes it is more convenient to attach a ring-beam exactly to the point of intersection. Only limited work has been done on the T-ring stiffened cone–cylinder junctions under internal pressure. In this study, experimental behavior along with numerical analysis of T-ring stiffened cone–cylinder intersection under internal pressure has been dealt and experimental results such as buckling mode and load are presented here and compared with numerical results. It can be seen that by wise consideration and manipulated use of material properties and geometric imperfections in nonlinear analysis, buckling mode and load resulted from non-linear analysis are compatible with that of experimental results. Two classes of non-linear analyses have been carried out and compared with each other, then it was inferred that even though pattern of geometrical imperfection is effective in determination of buckling modes, but in these kinds of structures it is not necessarily used for the analysis of buckling loads. Finally experimental results were compared with design proposals. It is shown that these proposals can conservatively predict the failure loads.     

Torsional buckling behaviour of stiffened cylinders under combined loading

In this study cylindrical boundary conditions for finite element analysis are formulated that allow torsional displacement and buckling of a sector of a cylinder of half axial height, and of a circumferential arc angle that will divide into 360°. Finite element tests are carried out on un-stiffened elastic cylinders to verify the method of analysis against classical elastic torsional buckling theory.Elastic–plastic limit point finite element tests are carried out on ring and stringer stiffened and stringer stiffened cylinders to investigate the effects of stiffeners on post-buckling behaviour in torsion.A stringer stiffened cylinder is subjected to many combinations of axial force and surface pressure in the elastic range of response and then tested to failure in torsion to investigate the effects of axial and surface pressure loads on the resistance to plastic collapse in torsion.     

A semi-analytical model for global buckling and postbuckling analysis of stiffened panels

A computational model for global buckling and postbuckling analysis of stiffened panels is derived. The loads considered are biaxial in-plane compression or tension, shear, and lateral pressure. Deflections are assumed in the form of trigonometric function series, and the principle of stationary potential energy is used for deriving the equilibrium equations. Lateral pressure is accounted for by taking the deflection as a combination of a clamped and a simply supported deflection mode. The global buckling model is based on Marguerre’s nonlinear plate theory, by deriving a set of anisotropic stiffness coefficients to account for the plate stiffening. Local buckling is treated in a separate local model developed previously. The anisotropic stiffness coefficients used in the global model are derived from the local analysis. Together, the two models provide a tool for buckling assessment of stiffened panels. Implemented in the computer code PULS, developed at Det Norske Veritas, local and global stresses are combined in an incremental procedure. Ultimate limit state estimates for design are obtained by calculating the stresses at certain critical points, and using the onset of yielding due to membrane stress as the limiting criterion.     

Some thoughtd on mode interaction in thin-walled columns under uniform compression

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with thin-walled trapezoidal columns under uniform compression. The asymptotic expansion established by Byskov and Hutchinson is also used here. The present paper is devoted to an improved study of the equilibrium path in the initial post-buckling behaviour of imperfect structures. The results obtained include the effect of interaction of the ‘primary’ local mode and a ‘secondary’ local mode having the same wavelength as the primary. In this paper the analysis of a few buckling mode interactions is presented.     

Fabrication of small models of large cylinders with extensive welding for buckling experiments

Cylindrical shells in large steel silos and tanks are commonly constructed from a large number of curved panels joined by many circumferential and meridional welds (referred to as the panel method hereafter). The extensive use of welding in these shells is a unique feature not previously studied in laboratory buckling experiments due to the great difficulty in fabricating realistic small-scale model shells. This paper presents an innovative technique for the fabrication of small models of such large steel cylindrical shells constructed from many welded panels. The experimental set-ups to implement this technique in the laboratory are also described. The new technique consists of two stages: (a) production of a high quality model by rolling two sheets (or a single sheet) and welding along the meridional seams; and (b) ‘welding’ in the form of controlled heat input in a required pattern of circumferential and meridional ‘welds’ on the central portion of the shell surface. The imperfections in an example specimen are also examined to show that they have a realistic pattern. The observed buckling behavior of this specimen is presented and discussed. The specimen buckled at a very low load, confirming that the welding-induced imperfections in such shells are severely detrimental to the buckling strength.     

对称双圆弧柱壳的非线性屈曲

对于潜艇艇体耐压壳结构,屈曲特性在设计中被广泛关注。针对一种新型潜艇耐压艇体结构-对称双圆弧环肋柱壳,推导了相应的弹塑性失稳系数。采用非线性大挠度理论,给出了静水压力作用下含初始缺陷的对称双圆弧环肋柱壳大挠度弹塑性屈曲临界压力计算式。讨论了开口角、周向相当波数和初始几何缺陷对临界压力的影响。计算结果表明,开口角对结构弹塑性屈曲的临界压力影响很小,而周向相当波数是影响临界压力的主要因素。     

Quantifying the accuracy of numerical collapse predictions for the design of submarine pressure hulls

An overview of current design practices for submarine pressure hulls is presented, along with the results of a survey of the literature that was conducted to determine standard nonlinear numerical modelling practices for those structures. The accuracies of the conventional submarine design formulae (SDF) and nonlinear numerical analyses for predicting pressure hull collapse are estimated by comparing predicted and experimental collapse loads from the literature. The conventional SDF are found to be accurate within approximately 20%, with 95% confidence, for intact pressure hulls. The accuracy of a wide range of nonlinear numerical methods, including axisymmetric finite difference and general shell finite element (FE) models, is found to be within approximately 16% with 95% confidence. The accuracy is found to be within 9% when only higher fidelity general shell FE models are considered. It is shown how the observations taken from the survey could serve as a starting point for establishing modelling guidelines, quantifying the accuracy of nonlinear FE analysis in pressure hull collapse calculations, and introducing this method into a design procedure by way of a partial safety factor.     

Static buckling analysis of thin cylindrical shell with centrally located dent under uniform lateral pressure

One of the common failure modes of thin cylindrical shell subjected to external pressure is buckling. The buckling pressure of these shell structures are dominantly affected by the geometrical imperfections present in the cylindrical shell which are very difficult to alleviate during manufacturing process. Dent is one of the common geometrical imperfections present in thin shell structures which may be formed due to mechanical damage caused by accidental loading or impact. In this work, influence of various dent parameters (dent length, dent width, dent depth and angle of orientation of the dent) on the critical buckling pressure of thin cylindrical shells with a centrally located dent is studied using non-linear static finite-element analysis of ANSYS under external pressure with simply supported boundary conditions at the top and bottom edges of the thin cylindrical shell.     

Buckling behavior of cold-formed zed-purlins partially restrained by steel sheeting

This paper presents a study on the buckling behaviour of purlin-sheeting systems under wind uplift loading. The restraint provided by the sheeting to the purlin is modeled by using two springs representing the translational and rotational restraints. The analysis is performed using finite strip methods in which the pre-buckling stress is calculated based on the same model used for the buckling analysis rather than taken as the ‘pure bending’ stress. The results obtained from this study show that, for both local and distortional buckling, the restraints have significant influence on the critical loads through their influence on the pre-buckling stress rather than directly on the buckling modes. While for lateral-torsional buckling, the influence of the restraints on the critical loads is mainly due to their influence on the buckling modes rather than the pre-buckling stress.     

Determination of effective breadth and effective width of stiffened plates by finite strip analyses

A finite strip (FS) method is presented for the numerical investigation of two design parameters — effective breadth and effective width — of stiffened plates. For the effective breadth, stiffened plates under bending are studied. Due to the transverse bending loads there is shear transmission through the plate from the stiffener which leads to a non-uniform longitudinal stress distribution across the plate width. This phenomenon, termed as shear lag, can be represented by the ‘effective breadth concept’, and has been extensively studied by analytical methods. A linear FS method is presented which utilizes the advantages of decoupling of Fourier terms on the one hand and, on the other hand, allows the treatment of both webs and flanges using a plate model. A definitely different situation exists for estimating the effectiveness of the plate breadth (or width) of plates in the postbuckling range. The ‘concept of effect width’ is based on the fact that plates with supported longitudinal edges and/or stiffeners can accept additional load after buckling under longitudinal compression, and enables the designer to evaluate the postbuckling strength of plate structures simply by using the design parameter ‘effective width’. Several formulae (most of them empirically derived) exist for an approximative calculation of the load dependent value of the effective width. A nonlinear FS method is developed and applied to the investigation of the postcritical strength of locally buckled structures. An incremental successive iterative procedure is introduced for an effective numerical analysis.     

First order vs. second order reliability analysis of series structures

The paper presents an extension of the first order reliability method for computation of failure probabilities. The improvement is achieved by a better approximation to the limit state surface. The analysis is demonstrated for individual failure modes and for series structures. The results obtained are very close to ‘exact’ results obtained by simulation. The method is seen as a supplement to first order reliability methods for special cases. It is not meant to completely replace these methods.     

Postbuckling analysis of stiffened cylindrical shells under combined external pressure and axial compression

A new approach is extended to investigate the buckling and postbuckling behaviour of perfect and imperfect, stringer and ring stiffened cylindrical shells of finite length subject to combined loading of external pressure and axial compression. The formulations are based on a boundary layer theory which includes the edge effect in the postbuckling analysis of a thin shell. The analysis uses a singular perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Some interaction curves for perfect and imperfect stiffened cylindrical shells are given and compared well with experimental data. The effects of initial imperfection on the interactive buckling load and postbuckling behaviour of stiffened cylindrical shells have also been discussed.     

Experimental and Numerical Study on Buckling Behavior of a Rigidly Stiffened Plate with Tee Ribs

In the bridge structures, stiffened plates are usually designed as rigidly stiffened when the orthotropic steel box girder is used as the main load-bearing structure. Therefore, the buckling mode of stiffened plates is plate buckling which occurs in subpanel supported by stiffeners. The orthotropic steel box girder is used as the main girder for Egongyan Rail Special Bridge, which is a self-anchored suspension bridge. Plates of the steel girder are rigidly stiffened with unequal spacing open ribs, and the most slender stiffened plate is the mid web stiffened with Tee ribs. In order to ensure the safety of the bridge, the buckling behavior of the web and orthotropic steel box girder under axial compression, including ultimate strength, post-buckling behavior and failure modes, should be clearly investigated by experimental and numerical methods. The design, loading and testing methods of the 1:4 scale model of the orthotropic steel box girder are introduced in detail firstly. The orthotropic steel box girder and the stiffened web finite element (FE) models are validated by the test results, and the effects of residual stress and the magnitude of geometric imperfections are discussed roughly. Based on the validated web FE model, a detailed parametric study is performed to systematically investigate the effects of residual stress and geometric imperfections on buckling behavior of the web. The effect of shapes of geometric imperfections discussed is highlighted. Through tracing stress states, the failure modes of stiffened plate are in agreement with the experimental phenomenon to some extent. Results show that shapes of geometric imperfections have significantly influenced post-buckling behavior and failure modes of the web, but slightly affected the ultimate strength. It is advised that residual stress and geometric imperfections should be controlled to make full use of excellent performance of steel materials.     

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.     

Pulse pressure loading of aircraft structural panels

The main aim of the work reported in this paper was to assess the reponse of panels representative of aircraft structures, to pulse pressure loading by means of experimental and finite element (FE) techniques. A series of experiments were conducted to compare the responses and failure modes of stiffened, aluminium alloy panels, joined using conventional riveting techniques and laser welding. The failure pressures of the riveted panels ranged from 29 to 36 kPa and a number of principal modes of failure were identified, namely (1) rivet shear/tensile failure, (2) frame buckling, (3) stringer buckling and (4) frame rupture. In the case of the laser-welded panels, failure pressures were between 28 and 33 kPa and failure was dominated by significant tearing along the weld heat affected zone at the frame–skin interface. The FE models correctly modelled most of the principal modes of failure observed experimentally and central panel displacement was also predicted well up until the time at which failure initiated.