Coupled Instabilities in Metal Structures

The following sections are included:

• Preface

• List of Contributors

• Contents

In this paper a brief summary of the different types of instability is presented on the basis of [1, 2]. Using the results of the elementary catastrophe theory the possible types of singularities are shown and a simple model for almost every case.

A direct one-dimensional beam model accounting for warping is introduced. The bi-shear and bimoment naturally result as quantities spending power on the warping and its first derivative. Balance of power provides local balance equations. Standard inner constraints are imposed. By means of suitable non linear hyperelastic constitutive equations, two coupled bifurcations for axially loaded beams are examined. Interaction shows that in both cases the beam is sensitive to initial imperfections.

Catastrophe theory applied to imperfection-sensitivity analyses is presented. By considering the problem as an imperfect version of a higher order catastrophe, a topologically good approximate solution can be obtained. Moreover, the solution seems to be valid not only in the neighborhood of the critical point, it can be considered as a quasi-global solution. The solution can be controlled by comparing it with correct sections of the imperfection-sensitivity surface, obtained in graphical way. By means of the parametric description of classes of catastrophes, the functions of imperfection-sensitivity can be obtained, moreover, the concerning multivalued imperfection-sensitivity surfaces can be illustrated as well.

Experimental studies on the axially-loaded sandwich panel are analysed against the recent developments in the theoretical modelling of interactive localized buckling. Improvements to current models are proposed such that the models may be extended to more complicated geometries and loading cases.

Full nonlinear analysis of large structural models is a very time-consuming problem. Therefore, approximation methods of critical load estimation are often used. The classical approximation method is the Southwell one. The paper deals with analogous problems. There is presented an analytical method. It can be used to estimate critical loads on the basis of experimental and numerical data as well. The results are obtained according to the distribution of the nonlinear problem solution in the basis of eigenvectors. There are presented several versions of the method. Total and nonlinear parts of displacements are considered in these versions. The comparison of results obtained by the incremental and non-incremental versions is presented too.

By using a continuum formulation, a variational approach is developed in order to perform the design sensitivity analysis of the nonlinear structural stability problem. The formulation is addressed in a unified manner by considering the general stability problem of nonconservative structures. Conservative structures are considered as particular cases of such problem. The variational principle of dynamics and the total lagrangean description of the motion, as well as the concepts of adjacent and adjoint displacements, are combined to formulate the stability equations and define critical and bifurcation points. By another hand, the design sensitivity equations are derived with basis on the concepts of adjoint structure and design control or reference volume.

The constitutive operator has a crucial role in the tangent stiffness, consequently, in the stability of structures. The development of the constitutive operator has a long history. The literature and research on this field is very large, extended and ramified. Recently the strain softening cases, damage and localization are in the focus of research. However, the effect of constitutive operator is modified if the nonlinear properties of other state variables are also considered. In this paper a brief summary of the history and development of the constitutive operator is presented. Then, the effect of this operator embedded into the structural tangent stiffness matrix is analyzed, under the influence of other structural nonlinearities. The paper is closed by numerical applications.

The currently available numerical modeling and computational techniques are reviewed. They fall under the following broad categories: Simplified models which combine elementary mechanics of beams suitably modified to account for local buckling, asymptotic procedures applicable to any structure undergoing elastic buckling , finite element analyses with elements carrying local buckling information , and general purpose computational algorithm s and tools. The scope and limitations of each of these approaches are discussed in this review.

The instability of a thin-walled beam with distortional flattening of a cross-section under bending is studied. An extended beam theory is proposed, and the coupled elastic buckling mode is investigated based on the finite deformation theory of distortion. Critical buckling loads are obtained analytically for a beam with a closed cross-section, and a new buckling mode is observed. To verify the result, numerical calculations are also carried out using FEM.

Practical numerical stability analyses with finite elements are generally carried out by studying the load-deformation behaviour of imperfect structures. Apart from this conventional approach an extensive theoretical basis for extended analyses exists. Main aspects are described in short form. The central issue, however, is the application of the methods on a problem of interest in practice – the tapered column section with thin-walled cross section parts. Here the methods and algorithms given in literature are used successfully to solve the specific problem of coupling of local and global instabilities. This enables to obtain further information, e. g. regarding the quality of critical points or imperfection sensitivity.

The interaction of local and overall buckling has been analyzed using novel finite elements in which local buckling deformation has been embedded. Amplitude modulation, a key feature of the interactive buckling has been incorporated in the element formulation. This work on interactive buckling of stiffened plates and shells has been extended in two important respects :

(i) The inclusion of a key secondary local mode in cases where the cross-section has complete or approximate double symmetry, and (ii) The introduction of a simple approach for capturing localization of the local mode. In order to model localization effectively, maximum possible freedom for the accentuation or alleviation of local buckling deformation in each of the member elements/plates is required. It is shown that this achieved easily by incorporating a single local buckling mode in the analysis, but selecting independent amplitude modulation functions for the elements.

A correct modelling of thin-walled members behaviour requires a due account for global lateral buckling, but also for local buckling of the profile walls. In this perspective, a special beam element is developed. The displacement field of this element allows to model any in plane global behaviour (tension, compression, in plane bending, in plane column buckling.) as well as the local buckling of the profile walls. The element is developed for bifurcation analysis (computation of critical loads) as well as for non linear analysis, including coupling between global and local phenomena.

This paper concerns the post critical behaviour of I and Channel beams. For this, the critical loads are first determined using analytical solutions. The eccentricities effects of loads from the shear center are studied in the post critical state. The flexion–torsion coupling are observed. The nonlinear analysis gives that the behaviour is characterised by a stable bifurcation point only when the loads line are passing through the shear center. After the critical loads have been reached, the beams exhibit an increase in displacements and stresses. Numerical simulations are conducted on Finite Element code. 3D beams with warping are used.

A model for the investigation of structures with plate buckling risk is presented by applying finite element analysis with code-related modified algorithms. The advanced finite element analysis is used for the determination of the lowest buckling loads for the complete structure and loading situations as well as the calculation and localisation of the equivalent stresses of the buckling structure. A good approach for the following buckling design calculation is the application of the Huber-Mises-Hencky yield criterion. Great attention has to be paid for structures with low or no post-buckling strength reserves. Here the used buckling capacity curve suggested by Winter has to be reduced. Some methods for the estimation of the post-buckling carrying capacity and the interaction between plate- and column-like behaviour of such structures are presented.

As a result of the presence of unavoidable initial imperfections, all stability phenomena in compressed thin-walled bars (i.e. the buckling of the bar as a whole and that of its plates progress in a permanent interaction. Geometrical imperfections (i.e. the imperfections of the cross-sectional shape and the out-of-straightness of the column) are to a high degree uncertain. In the authors study of the impact of such imperfections on the load-carrying capacity of compressed thin-walled bars, the relevant geometrical quantities are treated as random variables. In so doing, the "shape" of the imperfections is assumed to be affine to the first eigenvalue modes of the corresponding global and local buckling problems. In the paper, the problem is studied, on columns with typical slenderness ratios and having a closed (box) cross-section, using the programme system ANSYS, able to take account of both large deflections and plastic deformations. The randomness of the problem is dealt with by adopting the simulation technique Latin Hypercube Sampling. The statistical parameters of the ultimate loads of the columns are assessed with due account to the statistical character of the influence of initial imperfections. This is expressed by means of sensitivity analysis.

The General Report focus to problems: (1) the plastic-elastic interactive buckling of thin-walled members, and (2), the characterisation and codification of imperfections for advanced numerical analysis of these members. A review of last research achievements on this topic are presented and analysed.

The aim of this work is to set up the methodologies of experimentation on industrial shelves in order to study their mechanical behaviors. This derives from the necessity to support the planning with realistic data, that furnish more certain and suitable safety coefficients. We have concentrated the experimentation on a particular typology of shelf: the Door-Pallet The examined modular elements are constituted by different typologies of columns and different beams (shape and dimensions) but not for technologies of realization: elements obtained by different thickness metallic plates subjected to cold workmanships like cutting, perforation, folding and welding. The column/crossbeam connection has been made by means a particular system of hookup.

The paper sets out the governing equations for the overall buckling of locally buckled columns compressed between fixed ends. The governing equations incorporate the rigidities of the locally buckled section against overall minor and major axis flexure as well as warping torsion. Doubly, singly and point-symmetric cross-sections are considered. Buckling curves are obtained for an I-section, U-section (or channel section) and Z-section, representing doubly, singly and point-symmetric sections respectively. The buckling curves show that local buckling can change the critical overall mode from a flexural-torsional mode to a purely flexural mode in the case of U-sections. For Z-sections, the flexural modes are uncoupled for a non-locally buckled section but coupled for a locally buckled section.

The paper summarises the differential equations for the overall bifurcation of locally buckled point-symmetric columns. It is shown that flexural buckling about the minor and major principal axes are coupled in a locally buckled point-symmetric column whereas the buckling modes are uncoupled in a non-locally buckled column. The governing equations are solved for columns fixed at the ends and applied to Z-sections. Overall bifurcation curves are obtained for five Z-sections with increasingly slender flanges. It is shown that local buckling reduces the elastic torsional buckling load more so than the flexural buckling load for Z-sections, and that local buckling can cause a mode switch from the flexural to the torsional mode in Z-sections with very slender flanges.

The paper deals with the analysis of buckling, post-buckling behaviour and load carrying capacity of prismatic pole structures. The asymptotic expansion established by Byskov-Hutchinson is used in the second order approximation. The thin-walled tubular columns are simply supported at the ends and subject to the uniform and eccentric compression. Several types of cross-sections with and without intermediate stiffeners are considered. The present work is the continuation of a previous paper of the authors (5) where the modal interaction of thin-walled composite beam-columns was investigated.

The paper analyses the influence of local-sectional geometrical imperfection shapes on the local-distortional buckling modes of stub columns and on the interactive buckling of cold-formed steel members. The erosion of theoretical interactive buckling load, due both to the imperfections and interactive buckling effect is numerically evaluated

The load carrying capacity of thin-walled cross-sections is usually governed by local plate buckling. The non-linear stress distribution in the postcritical range can cause a shift of the effective centroid. This leads to an eccentricity of the applied force in pin-ended singly symmetric compression members and hence induces overall bending. A perfect, centrally loaded I-section has a double symmetric postcritical stress distribution. Therefore, no centroid shift occurs. But in a real construction, a geometrical imperfection has to be considered. For that reason, the case of an axial force plus an additional bending moment always exists. Consequently, a centrally loaded doubly symmetric member also shows a shift of the effective centroid. In international codes there are differences in opinion whether this unintentional eccentricity must be taken into account or not. An extensive parametric study was carried out in order to assess the effects concerning the effective global slenderness and the ultimate load. The study leads to the conclusion that the influence of the centroid shift can be covered by an additional equivalent geometrical imperfection. For practical application welded thin-walled I-sections are classified with regard to the European buckling curves.

This paper considers the design of the stiffened flanges of cassette sections in the context of Eurocode 3 under both axial compressive load and bending which places the wide flange in compression. The primary methodology used in this research is "Generalised Beam Theory" which allows the individual buckling modes to be considered individually and in predetermined combinations.

The paper describes a combined analysis of both coupled instability (interactive buckling) and post-failure behaviour of thin-walled channel-section beam-column under bending or/and compression. The local and global buckling analysis is carried out on the basis of non-linear stability equations, using the asymptotic Koiter method in the second order approximation and applying compatibility conditions for adjacent beam walls. As the result of the interactive buckling and post-buckling analysis some different modes of failure are predicted and are used as a base for the determination of theoretical models of plastic mechanisms of failure. The analysis of post-failure behaviour is performed using the energy method based on the assumptions of the rigid-plastic theory with additional assumptions which enable to incorporate the strain-hardening into consideration. Solution of the post-failure problem is based on the Principle of Virtual Works. The post-buckling path obtained from the interactive buckling analysis is combined with the failure path gained from the plastic mechanism analysis. A point in the loading path corresponding to a stress state in the boundary layer in the flange under compression which fulfils a yield criterion is aknowledged as a lower bound estimation of the ultimate load. The upper bound estimation of the load-carrying capacity is carried out on the basis of the plastic mechanism analysis. This approach allows the authors to evaluate the structure's behaviour in the whole range of loading: up to and beyond the ultimate load. Numerical calculations for a channel-section beam-column made from structural steel are carried out.

The objective of this paper is to present a study, both numerical and experimental, of the local post-buckling behaviour of cold-formed steel rack columns, with particular attention paid to the influence of the column local boundary conditions. First, spline finite strip analyses are employed to investigate (i) the influence of fixed and longitudinally rigid end conditions on the column linear stability and post-buckling (local plate and distortional) behaviours and (ii) the occurrence of local mode interaction. Finally, a test program, presently under way is briefly reported. The test set-up, procedure and data acquisition system are described and a few experimental results (DM behaviour) are presented and compared with numerical ones.

The paper summarises the principles underpinning good experimentation of thin-walled structural members. The key features include the selection of appropriate gain control strategy and control loop variable, as well as careful end bearing design and techniques for measuring buckling displacements at advanced stages of loading. The paper focuses on the testing of thin-walled columns as this type of test presents one of the most demanding problems in terms of securing a stable control loop for capturing post-ultimate response. The principles are described with reference to the test rigs developed at the University of Sydney over the past 15 years.

The paper gives test results on the compression strength of aluminium columns with small closed and open cross-sections made of extruded profiles. Experimental results are compared with computed strength values based on the finite element method and elastic-plastic, Ramberg-Osgood and experimental multi-linear stress-strain curves. The experimental results are also compared with the values given by the expressions in the European prestandard EC9.

The extrusion manufacturing process of aluminium allows profiles with a large variety of cross-sections. If these are thin-walled, cross-sectional instability might occur, which is caused by interaction of local buckling modes of connecting plate elements. To investigate this phenomenon, an extensive set of experiments is performed on uniformly compressed aluminium RHS-sections at Eindhoven University of Technology. In-plane and out-of-plane plate deformations are visualised using a so-called ESPI-laser system. The tests are used to validate a finite element model, which results are presented as well. In a subsequent research this model will be used to analyse the failure modes of more complex sections.

The objective of the paper is to report about the main results and conclusions of a new stage of experimental research conducted in Prague on the coupled problem of a many times repeated loss of stability and cumulative damage in the slender webs of steel plate girders subjected to many times repeated loading. In so doing, the paper analyses the experimental data obtained, particularly from the point of view of the initiation and propagation of fatigue cracks in breathing webs, their failure mechanism and relation to the fatigue limit state of the whole girder.

Two sets of tests have been carried out on hot-rolled UPE-channel sections with parallel flanges. In the first set, the sections were loaded in bending about the major axis, with load applied at the shear centre, to determine the lateral torsional buckling load. In the second set the sections were loaded in the plane of the web which caused failure through bending and torsion. Much effort has been put into avoiding unwanted influences of load application. A test rig was designed such to minimise these effects.

In the first set the members failed through sudden lateral torsional buckling. The predicted buckling resistance load according to Eurocode 3 is conservative when compared to the experimental failure load. In the second set the mebers failed gradually through bending and torsion. The failure loads were, as was expected, lower than in the first set of experiments.

This paper shows studies on essentially experimental basis of columns strength in thin-walled steel plate (cold-formed profile), submitted to centred compression. Special attention is given to instability phenomena, including local-global buckling interaction for the case of cold-formed "S" profiles. The research was based on a large series of tests that were performed for stub, intermediate and slender columns and incorporated the analysis of some buckling curves propositions. Reliability prediction of the proposed buckling curve was verified for different groups of experimental results, including local-flexural, distorsional-flexural and local-torsional flexural buckling interaction.

An experimental study on new type steel structural members forming triangular hollow section with pipes at each apex is described. Uniformly compressed and eccentrically loaded test are conducted. The test results indicate that the strength of pipes is far greater than the expected and the pipes work as fixed end against the local buckling of adjacent plates. In addition, it is found that each plate buckles independently due to the existence of pipes between plates and the final collapse of the member is caused by the local failure of pipes.

A brief state-of-the-art in the field of current approaches to the design of steel plates and plated structures subjected to quasi-constant loading is presented in the first part of the general report, attention being paid to both ultimate and service conditions. However, as very significant progress was promoted in this domain over the last three decades, great number of researchers worldwide having contributed to this, more attention is paid - in the second part of the general report - to the behaviour of plated structures under many times repeated loading, i.e. to the so-called breathing phenomenon in them.

The design of longitudinally stiffened plates is on the agenda in connection with the conversion of Eurocode 3 to EN. There are several open questions of which the effective area of a longitudinally stiffened plate under uniform compression is studied in this paper. The method is performing a parametric FE study in order to overcome the lack of systematic experimental data. A calibration of the FE model against experimental date is done to check its predicting power. The result is that the model is correct within 10%and the parametric study shows an unexpected variation of the resistance with the length of the plate.

It is essential to know how the compression members change the stiffness under growing load. This can be determined by the analysis of the load – displacement equilibrium paths established in our case for columns built up from steel strips and steel stiffened shells. The degree of sensitivity of the stiffened plates on the individual types of imperfections in the postbuckling range has been determined for various proportions of transverse and longitudinal dimensions. Numeral solutions have been obtained using LUSAS program. In the plate – strips of columns having both box and rectangular hollow cross-sections, the load – displacement equilibrium paths have been determined up to the load – bearing capacity point using the numeral incremental method of integration of the equilibrium equations. This method has been used for assumed initial crookedness of column axes. All these theoretical analyses have been supported by own experimental investigations carried out on full-scale models. The theoretical and experimental results of Scientific Centres have been considered as well. A big deal of attention has been paid to the longitudinal stiffness of plates during the procedure of optimisation.

Buckling of webs of crane runway girders is caused by local stresses from transverse patch loads and beam stresses from bending moment and shear force. Buckling design with respect to the related German or European standard requires the evaluation of buckling slendernesses for each of the three single stress components. Within this procedure the resistances of the plate and the cross section respectively are at first calculated independent for transverse force, bending moment and shear force. Finally the interaction of these components is considered via interaction relations. The procedure suggested in the study is based on simultaneous treatment of the problem under combined loading.

This paper deals with the behaviour of thin-walled closed cross-section. Through a simple example, the different relevant effects (bending, torsion, distortion, local force) are examined. First, the elastic distribution of longitudinal stresses is determined by means of analytical, numerical methods, and with tests as well. Then, the ultimate load-bearing capacity is predicted in different ways, namely by a method which combines the critical stress with yielding condition and by Eurocode 3 regulations. The estimated capacities are compared to test results. Finally, on the base of comparison some conclusions are summed up.

Crane runway girders are loaded by wheel loads creating global axial bending stresses and local transverse stresses. Within the design process webs of these girders have to checked against local buckling due to both stress components. If webs are unstiffened and the distance between the wheels is short compared to beam span interaction in buckling under the wheels may occur. It is the object of the present study to numerically investigate possible interaction in case of two wheels acting on crane girders. Critical elastic buckling loads are given for two wheel loads and compared to the case of a single load. Using these results buckling design for this problem will be safe and economic.

The results of theoretical and experimental studies of the elastic buckling and ultimate resistance of slender plate girder web panels to bending, shear and patch loading, and combinations of such loading, are presented. Theoretical studies of buckling resistance are based on numerical solution of the governing energy equation, in which the pre-buckling membrane stresses due to patch loading are approximated by a statically admissible stress field, reduced in magnitude to provide close correlation with more exact results. Experimental and theoretical studies of ultimate resistance are based on numerous test results, approximate elasto-plastic mechanism solutions, and semi-empirical interaction formulae validated by test results.

This study is aimed to check the effectiveness of U-shaped vertical stiffeners of the steel box girder support diaphragms. The results of the author's previous studies on the steel box girder support diaphragms indicate that such a diaphragm is collapsed by the local deformation with small waves at its bottom part near a shoe, and the torsional local buckling-like behaviour is observed at the bottom of the load bearing stiffeners. To avoid the torsional buckling, the stiffener should have more torsional rigidity such as U-shaped ribs. In this study, numerical analysis is made on the support diaphragms with the load bearing stiffener having U-shaped section to confirm the contribution of the torsional rigidity of the stiffener to the strength and behaviour of the diaphragms, and check the effectiveness of the U-shaped stiffeners. Dimensions and thickness of the U-ribs are adopted as parameters. Through the analysis, it is found that use of a U-rib type stiffener as a load bearing stiffener of a support diaphragm is effective to improve the strength of the diaphragm.

This work deals with the analysis of elasto-plastic post-buckling state of rectangular plates subject to combined loads such as: uniform compression, in-plane bending and shear. The isotropic plates are initially deflected. The analysis is carried out on the basis of nonlinear theory of plates involving plasticity. The solution is obtained in the analytical-numerical way using Prandtl-Reuss equations. As a result of numerical calculations the load versus displacement curves are drawn.

Investigation of girders with thin-walled undulating webs under local loading is presented. Results of 27 tests, which cover various transverse eccentricities of loading and various support conditions, are described. Statistical evaluation of all test results in accordance with Annex Z of Eurocode 3 resulted into formulas of characteristic and design local load resistance of the undulating webs. Large-deflection elasto-plastic analysis (ANSYS software) was calibrated against test results. Parametrical study covering the full range of beams with undulating webs produced in Poland and used in Central European countries ("WT girders") was evaluated. Former experimental formulas were slightly rearranged to fit all the theoretical results and recommended for design.

This paper presents the results of combined experimental and analytical/numerical studies of the response and instability of tapered cantilevers with a T-shaped cross-section and the tip of the web in compression. Although the design of a T-shaped cross-section with the lower part in compression is not best, it takes advantages in fabrication, corrosion protection and because of the slenderness in aesthetic way. Probably for those reasons, nowadays, tapered or nontapered cantilevers with a T-shaped cross-section can be found in building and bridge design more often. Local and Lateral-Torsional buckling may occur and might be coupled (Distortional Buckling). First, critical loads of a widly range of different cantilever geometries, loading cases and boundary conditions were determined. Next the real load-carrying capacities of those cantilevers were examined. Finally a function of reduction is presented. The load-carrying capacity can be derived easily and, furthermore, now it is possible to do design calculations and optimization of tapered cantilevers with T-shaped cross-section.

The buckling behaviour of shell structures is widely controlled by coupled instability phenomena. Three practically important metal shell cases with coupled (or interactive, respectively) instability behaviour are reviewed herein: Unstiffened cylindrical steel shells under combined stressing by axial, circumferential and shearing membrane forces, in which coupling of single load instability modes may occur; ring-stiffened cylindrical steel shells under external pressure and stringer-stiffened steel shells under axial compression, in which coupling of local and global instability modes may occur.

The paper presents a theoretical and numerical studies of the buckling of steel shells subjected simultaneously to axial and pressure loading. The type of combined non-uniform loading used for this analysis (horizontal pressure and wall frictional pressure which occurs in the silos) are given in the Eurocode 1 part 4. Abaqus program was used in a study of the influence of the parameters affecting the strength of the shell. The results of this parametric studie were used to develop a new semi-analytical method for the prediction of the ultimate stress resistance of the stiffened and isotropic shells. The corresponding formulae are compared, in different examples of geometrical shells, with the results obtained by finite element simulation and by the formulae for buckling design given in the recommendation EC3.

Non-linear behaviour and coupled instabilities of steel cylindrical tank shells subjected to meridional edge deformation of harmonic pattern are investigated. Harmonic number n is shown to have a strong influence on the shear and meridional buckling interaction. Practical applications are related to the design and maintenance of storage tanks and silos. The shear buckling mode is observed at small harmonic number n (<5) in shells of uniform thickness; with increasing n, the mode is dominated by axial buckling. As the shell deforms in the shear buckling mode, it continues to carry higher load until local buckling occurs near the base. In tapered shells, shear buckling occurs in the upper region of the shell at high n.

Guyed masts with tubular shaft and one level of guys are taken into consideration. New equations are derived for global analysis. They are geometrically nonlinear and coupled. Conservation of mass of the members and overall stability of the mast with respect to overturning are the principles. The shaft is subject to wind action and second order forces due to lateral sway of the mast. Sag of guys is evaluated by cubic polynomials which satisfy conditions at the ends of guys. The forces introduced from the guys to the shaft are nonconservative because their directions change during deformation of the system. A scalar effective moment is defined for local instability of the tubular shaft because of interacting bending moment, axial force and shear force. Resistance of the cylindrical shell is assessed for the joint probability of elastic and plastic unstable behaviour. Resistance of the shaft is checked at the critical cross-section where the effective moment reaches its maximum. Deflection of the shaft at the critical point and components of internal forces amplify the effective moment.

Aggressive environments can provoke corrosion of thin cylindrical shells widely used in different engineering applications. This work treats the influence of thickness imperfections on buckling of cylindrical shells under hydrostatic pressure. An experimental setup permitting buckling analysis under pressure up to 16 MPa was built up. The specimens were pressurised in a test chamber by oil pressure under quasi-displacement controlled conditions. Parameters that were surveyed include shell geometry and thickness distribution. Localised thickness imperfections were induced by electroerosion. A numerical analysis methodology, together with the formulation of one finite element permitting to analyse circumferential and axial distribution of geometrical and thickness imperfections was performed. Two parametric studies treating modal and localised thickness imperfections as well as their coupling with geometrical ones were realised. In the case of modal imperfections two imperfection modes appeared to be the most detrimental; the mode I imperfection and the mode 2n imperfection, where n is the linear bifurcation mode of the perfect shell with nominal thickness. Thickness variation in these cases was taken to be symmetrical to the shell's mid surface. With each of the analyses an analytical numerical study for the infinitely long cylinder under external pressure is done as well. Localised thickness imperfections at the external surface of the shell are also presented. For each of these two parametrical analyses an estimate of the buckling load reduction is established as a function of the BATDORF Z parameter. The domain of the imperfection parameters variation for validating the proposed estimate using experimental analysis is given. The presentation of experimental methodology is given. Interpretations and numerical validations of the performed tests are given as well. Finally the buckling load estimate is compared with the test results.

At the CIMS'96 Conference in Liege, the author presented a General Report with the same title. This Report expands on some of the subject areas that were considered in the earlier Report. Particular attention is given to elastic buckling modes for portal frames, the influence of semi-rigid joints and elastic-plastic instability.

This paper presents a coupled second-order analysis of frameworks by an improved continuous medium technique. Buckling mode superposition and simplified second-order amplification formulas are taken into account. The method developed is applicable to a complete class of frameworks including the ideal shear frame panel, in which the beams are assumed to be rigid, and the associated columns system, for which the rigidity of beams are negligible. The main results can be extended to other types of lateral stiffening elements as built-up columns.

Frame corner joints built up from cold-formed C-profiles and mechanical fasteners (metric bolts and self-drilling screws) are investigated by experimental program. 26 specimens with different profile and connection parameters are studied. The behaviour is analysed by the measured moment–rotation relationship of the joints. The experienced behaviour is combined phenomena of different local instabilities of the profiles and the connection performance. On the bases of the governing phenomenon three behaviour modes of the joints are determined and explained in the paper.

A convenient solution for medium span truss bridges is a half-through design. Converting of the through-truss bridges to this type, prevents possible vehicle collisions. Stability of the compressed chords of the main trusses of the half-through bridges can be ensured by additional strengthening.

The objective of this paper is to present the results of a study concerning coupled global instabilities in pitched-roof frames. The nature (stability and symmetry) of the post-critical equilibrium paths associated to the anti-symmetric (ASM), symmetric (SM) and coupled (CM) modes is investigated and the corresponding frame imperfection-sensitivities are established. The behaviour of 2 d.o.f. mechanical model is first (analytically) studied, after which both analytical and numerical (FEM) results are obtained for real pitched-roof frames. The analytical results presented deal with the post-buckling behaviour of perfect frames associated to bifurcation in the ASM, SM and CM. The imperfect frames are analysed by means of the FEM and the results used to establish and discuss the frame imperfection-sensitivity under mode interaction. In particular, the most detrimental imperfection configuration is identified.

This general report continues the reports of the same author concerning the theory of static and dynamic coupled instabilities presented during the 1992 and 1996 CIMS Conferences [1], [3], in order to extend the state-of-the-art of the problem with new aspects. The correlation between equilibrium and motion instabilities, classification of dynamic loading, arrest of progressive collapse and impact loading are discussed.

Von Kármán theory has been used for the description of the post-buckling behaviour of a thin-walled panel with geometrical imperfections and residual stresses. By including inertia forces the problem is extended into dynamics Using Hamilton's principle in incremental form the problem of the free vibration has been established. Illustrative examples of the buckling of the column and a slender web loaded in compression emphasizing different types of support in the direction of the column axis or the neutral surface of the slender web are presented. The influence of the mode of the geometrical imperfections is shown. In the case of the of a slender web loaded by shear the buckling mode is different as the mode of the free vibration of an unloaded web. The comparison of the theoretical and experimental investigation is done.

The theory and the experiments have proved the sensitivity of the circular frequency of free vibration on the level of the load and different types of initial imperfections. This knowledge can be used as an inverse idea. Measuring of the natural frequencies can give us a picture of the stresses and imperfections in a thin-walled structure.

In this paper we characterize the behaviour of a plate-like latticed modular structure ("refined" structure) using a a two-dimensional continuum endowed with affine micro-structure ("coarse" model) which provides a compendious description of the displacements field. Such a coarse model is obtained by the identification of the constitutive parameters for internal and inertial bulk actions via the "refined" one. The identification of the constitutive parameters results from assuming that the internal power density and the inertial bulk actions power, in the coarse model, are equal to suitable averages of the internal power and of the inertial bulk actions power, defined for any module of the latticed structure, respectively.

The paper studies the appearance of the local buckling phenomenon due to a geometrical imperfection or a non-uniform loading scheme, under static loads, for single layer reticulated shells. The local buckling phenomenon in study is a node instability, in the shape of a "snap-through". In order that the node instability to remain a local phenomenon, the equilibrium position of the structure after the static jump, due to the "snap-through", has to be stable. In this paper this position was disturbed and it was studied the response of the structure. It was possible to demonstrate that if it is taken into account the dynamic effect of the loads, in general the local buckling phenomenon propagates into a general instability, so that finally the analysed domes have changed their curvatures. The software used for all this numerical simulations was ANSYS 5.3. [6]

Due to the fact that a local node instability can spread into a general instability for single-layer reticulated shells, if the dynamic effect of the loads is considered, this paper deals with the possible ways of stopping this spreading, by modifying the geometrical characteristics of the cross-sections of the members. There were taken into account geometrical imperfections or non-uniform loads applied on the analysed shells There were found out the members that needed to be modified, and also the new values for the geometrical characteristics of the cross-sections of these members — both of them for two types of single-layer reticulated structures. The software used for all these numerical simulations was ANSYS 5.3.

In this paper a low dimensional model, which retains the essential non-linear terms, is used to study the parametric instability and escape from the pre-buckling potential well (dynamic buckling) of a thin-walled cylindrical shells subjected to time-dependent axial loads. For this, Donnell's shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. In this study, special attention is given to the determination of the stability boundaries in control space and to the identification of the bifurcation phenomena associated with these boundaries

This paper presents an analytical technique based on the finite element method for determining the dynamic load-bearing behavior of linear thin-walled bars when subjected to impact loads from any direction. The theoretical basis for the method is the general theory of bending and torsion of thin-walled bars under consideration of shear deformation. The method accounts both for geometric non-linearities in the structure and for physical non-linearities in the material. The behavior at the contact point is idealized by means of a specially developed elasto-plastic contact spring. The numerical algorithm is described by a step-by-step integration technique in the time domain.

Current report is addressed to provide a review on the influence of material non-linearity on the buckling behaviour of metal structures by considering both singly and coupled instability modes. It also explores the possibility to use a unified approach allowing the material behaviour to be properly accounted for. Obviously, being the subject an open research field, this paper cannot be a comprehensive "state-of-the-art". Instead, it represents an attempt to provide a general overview on this topic, giving some indications on previous studies, current trends and further research developments.

The H-shaped beam with large depth-thickness ratio has a tendency to be collapsed by the web buckling. And the web bears the shear stress. So this shear stress influences the coupled instability of flange plate and web plate composed H-shaped beam. In this study, the influence of shear stress in cross section to the plate coupled instability is investigated, and the estimation method of the plastic deformation capacity of H-shaped steel beams is discussed. The behavior of H-shaped steel beams is investigated by numerical analysis used by finite element method. Analysis models are cantilever beams. Parameters of analytical models are the width and the thickness of plate elements consisted H shaped beams, and the shear span ratio to depth.

In the paper the effects of horizontal-stiffener-end-gap. stiffener size and steel material are studied on ultimate behaviour of plate girders. The analysis is completed by experimentally verified finite element models. Five behaviour modes are determined. The main focus is on the application of new structural steel material; the improved behaviour by advanced, low-yield-ratio steel is emphasised.

The present work shows the relevance of the numerical simulation in the case of plastic buckling of thick cylindrical shells under axial compression. It is well known that the cylindrical shells can exhibit several critical configurations. Furthermore, the difficulty to corroborate the test and the simulation resides in the presence of different modes of bifurcation for the near loads. Herein, this difficulty is examined, based on an analysis of possible plastic bifurcation along the incremental path, this leads to the possible estimation of the real behavior.

The exhaustion of the load-bearing capacity of the slender flanges of the bars representing the class 4 according to Eurocode 3 and being under combined bending and compression is reached by the formation of the plastic hinge mechanism which should not be confused with the simple plastic hinge known from the familiar theory of plastic analysis. The exhaustion of the load capacity runs suddenly for loads which are lesser from the critical load of the overall stability. In the analysis of the load capacity of thin-walled bars the initial geometrical imperfections have to be considered. The random critical stresses of the thin-walled bars initiating the formation of a plastic hinge mechanism are analyzed in this paper. Proposed way of the load capacity assessment combines the analysis of the overall buckling and local exhaustion of the load capacity of the thin-walled bars under compression.

The evaluation of the complete load-deformation history pointed out that thin-walled members have a different post-critical curve for different cross section types. The aim of the paper is to establish the theoretical rigid-plastic curve for thin-walled beams with double symmetrical cross section formed of two channel or lipped channel profiles. The study of plastic mechanisms for these members is governed by the local plastic mechanisms for thin-walled steel plates of flanges subjected to in-plane compression. The type of plastic mechanism determines the rigid-plastic curve shape. The influence of geometrical imperfections is also included. A program named DUCTROT-TWM was elaborated for drawing the post-critical curve for thin-walled beams in function of cross section types and beam characteristics.

This paper presents and discusses non-linear equivalent elastic models that reproduce the behaviour of steel joints loaded in bending. In these models, each substitute sub-system for an elastic-plastic spring corresponds to a critical load of the global system, each arising at different load (moment) levels along the fundamental equilibrium path. It is shown that these modes present a highly interactive behaviour, resulting in a cascade of secondary bifurcations that yield a broken line with reducing stiffness, effectively reproducing the original response of the non-linear elastic-plastic joint model.

Appearance of the new European standards gave the opportunity to make comparison between the results of the formerly carried out experimental program and the BS. 8118. Wide ranged parametrical calculations were made to find relation between the optimum value of the ultimate load and the proportion of very thin walled aluminium sections. For case of asymmetrical section rough estimations are given.

A brief review of the research on the optimum design of structures with stability criteria is presented. The stability criteria are derived in a variational fashion. The optimal design problem is formulated and several optimal design criteria are presented for structures with stability constraints.

A new implicit enumeration method is proposed for discrete minimal weight design of shallow trusses with stability constraints. Structural instability constraints and member buckling are considered as well. In this approach the usual critical load level computation is replaced by an equilibrium path computation from the zero load level to the applied load level. The discrete optimal design method is formulated as an exterior point tree search problem that maintains an upper bound, the lowest weight of any feasible solution found so far. Any node generated can be immediately discarded if its weight is greater than or equal to the current upper bound value. To illustrate the efficiency of the algorithm, a shallow dome structure is tested.

Profiled steel sheeting is widely used for cladding, roofing and flooring. Its development is coupled with the rising interest in cold-formed light-weight steel structures. Due to the variety of different profiles available on the market, the question of optimum shapes arises. This article introduces a new technique by which these can be obtained on the ground of the Eurocode specifications. The developed algorithm uses finite strip method, and a novel technique of extreme value search, genetic algorithm. Demonstrative examples of simply supported and two span arrangements are shown and results are discussed.

This work deals with the optimal design of nonlinear structures where both geometric and path-dependent material nonlinearities are considered. Postcritical behavior is allowed. Critical and postcritical constraints are considered. Local and global stability have been introduced also as constraints. The classical critical load constraint is given with numerical advantage by a new method using displacement constraint. The Total Lagrangean description and a continuum variational formulation are used for the response and design sensitivity analysis. A continuation algorithm is used to implement the postcritical path. Optimal design is performed with respect to cross-section dimensions and configuration.

This paper is intend to present an overview of design procedures included in recent code editions, related with stability problems, non linear interaction between buckling modes and reliability aspects. This goal is addressed to cold-formed steel members, because of the large application of these structural elements in metal construction. Four different codes are considered – USA, Australia, Eurocode and Brazil, representing different views of the problem. Distortional buckling is specially focused, since it has been recently considered in code prescriptions. Computational tools for the design of thin-walled steel members were also underlined, considering its high importance for safe structural analysis and design.

The paper presents a model based on the Ayrton-Perry approach, concerning the design of monosymmetrical beam-columns. The interaction between flexural buckling within the plane of symmetry and flexural-torsional buckling is taken into account, as well as the influence of imperfections and that of partial plasticity. The imperfection sensitivity of beam-columns concerning flexural buckling is smaller than in the case of axially loaded members (columns), because of the presence of primary bending moments. The proposed model gives the possibility to analyse the separate influence of all relevant parameters. Simple empirical formulas are preferable for current cases, but their applicability cannot be arbitrarily extended to more complex cases, where a more detailed investigation is necessary.

Based on an analytical method, a parametric study is carried out on the bending response of steel beams with box cross-sections. Slenderness requirements for the classification of these beams are proposed, taking into account the interactive effects of flange and web local buckling. These serve as limits for application of different methods for analysis of steel framed structures. The present study is a continuation of a previous research of the authors on I-beams, where the detrimental influence of lateral-torsional buckling, coupled with the combined effects of the local buckling modes, has been clearly demonstrated. It has been shown that the usual cross-section classification deserves a generalization in terms of member classification.

This paper deals with the behaviour of Class 4 cross-sections under axial force and bi-axial bending. According to Eurocode 3 Part 1.1 Chapter 5.4.8.3 [1] two different, but equivalent criterion can be use to calculate the resistance of Class 4 cross-section. Through two simple examples, the different criterions are compared. In the first example a thin walled hollow-tube section is evaluated, and in the second example a thin walled HP European cross-section is evaluated in case of multicomponent load. Finally, on the base of comparison some conclusions are summed up.

In order to improve Eurocode 3 beam-column formulae, a research was led by a so-called "ECCS TC8 – ad-hoc working group", and, as a result, a new proposal dealing with plane and spatial behaviour has been suggested. It was derived according to the following requirements: theoretical background, strong physical meaning, consistency with the other formulae of the code, and accuracy. It should also cover all types of continuities: between the cross-section classes, from plasticity to elasticity as slenderness and axial force increase, and continuity between individual stability checks and cross-section verifications. In this paper, the theoretical basis are first established and then the proposal is detailed for in-plane behaviour. Extension to spatial behaviour is next tackled, without accounting for lateral torsional buckling. Finally, in order to check the accuracy of the proposal, the results of comparisons based on a large number of numerical simulations are presented. It shows that the proposal gives satisfactory results.

In European standards the method of effective widths is established to calculate the load-carrying capacity of thin-walled cross-sections in bending and compression. Eurocode 3 [1] employs the method of effective widths because, in comparison to the method of reduced stresses which is prefered in german standard DIN 18800, part 3, [2] the calculated load-carrying capacity of compound sections is usually higher. The following paper deals with examinations concerning coupled instabilities determined with the help of the method of effective widths. First, different parameters are analysed which influence the cross-sectional load-carrying capacity of class 4-members in bending and compression. The investigations lead to new effective widths for three-sided supported steel plates considering realistic support conditions of the longitudinal edges. Serviceability can be ensured. Based on these studies coupled instabilities, local buckling of plates and global flexural buckling of members, are analysed. The researches lead to modifications of the design rules of EC3. The proposed method is verified with the help of experimental tests of compressed thin-walled U-sections and numerical investigations of the test members. A comparison to the calculations of EC3 shows the improvements.