Stable Post-buckling Behaviour Research Articles

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections

In this paper, the stability of steel members with a complex initial geometrical imperfection pattern are analysed. This issue is extremely important in the case of slender structures, characterised by multiple close critical loads and modal interactions, which can lead to unstable post-critical paths and imperfection sensitivity. Despite the fact that the loss of stability, as a result of complex geometrical imperfections, is a very common mechanism for the destruction of slender steel structures, there is still no unambiguous and adequate research in the literature and in scientific research taking into account multimodal buckling. Therefore, in this study, special attention was focused on the analysis of the equilibrium path of the structure in the pre- and post-buckling range. This was studied by introducing a model of a structure composed of four rigid bars connected by elastic nodes. For this model, as well as for the structure with and without initial geometrical imperfections, a set of nonlinear algebraic equations of equilibrium was developed. A complex pattern of imperfections was taken into account using a linear superposition of buckling modes obtained from a linear eigenvalue problem. In order to investigate the nature of bifurcation points, the concept of minimum of potential energy was adopted. By means of numerous examples, the influence of imperfections on the structural behaviour was discussed. It was found that, for special imperfection patterns, an increase in the amplitude of initial geometrical imperfection can result in an increase in the value of the critical load defining the bifurcation point. In these cases, initial geometrical imperfections can play a positive role, resulting in stable post-buckling behaviour. This phenomenon corresponds with the so-called “modal nudging” which aims to improve the buckling response of slender elastic structures by introducing a small disturbance in the primary geometry of the structure, which results in equilibrium paths of greater load-carrying capacity. Among other observations, a snap-through phenomenon caused by transition from the local to the global minimum of potential energy was also noted. The observed snap-through was caused by the specific configuration of initial geometrical imperfections, which in this case played quite a dangerous role. It should be emphasised that the proposed model structure allows for a full description of the post-critical behaviour and a trace of the influence of complex imperfection configurations in a simple and clear manner.

Cyclic behavior of oval hollow section (OHS) beam-columns

This paper seeks to gain an in-depth understanding of the seismic behavior of oval hollow section (OHS) beam-columns through experimental and numerical investigations. Quasi-static tests on nine OHS beam-column specimens with varying tube thicknesses, axial load ratios and bending directions are conducted, where the failure mode, yield/ultimate strength, ductility, and energy dissipation-related quantities are discussed in detail. Following the experimental program, a numerical study on a total of 360 OHS beam-column models with broadened section and material parameters is further conducted, and design recommendations for OHS beam-columns subjected to seismic conditions are proposed. Among other findings, the study reveals that local buckling at the fixed base is the governing failure mode for the considered members, but the failure mechanism varies under different bending directions. The flat web plate of the OHS is a critical area which is prone to local buckling, although stable post-buckling behavior can be exhibited. Earlier local buckling and more severe damage to the column base are induced when the axial load increases, resulting in decreased ductility. The plastic moment resistance is better mobilized with the increase in the tube thickness and when under major axis bending. The parametric study finally leads to practical ductility-oriented and strength-oriented design curves, where both least-square regression and lower-bound solutions are provided.

Elastic buckling of thin-walled polyhedral pipe liners encased in a circular pipe under uniform external pressure

In this study, a thin-walled polyhedral polymer pipe liner is proposed for the internal rehabilitation of a deteriorated/cracked underground circular metal pipe. The pipe liner is externally confined and subjected to the hydrostatic pressure of water seeped through the cracked pipe. The critical buckling pressure of the pipe liner is derived analytically based on the principle of minimum potential energy and compared with that of a cylindrical pipe liner. A finite element model of the pipe liner is established and analyzed to understand pressure-deformation equilibrium paths and the stability of post-buckling behavior. The analytical buckling pressure is in excellent agreement with the numerical results. The buckling pressure of a polyhedral liner increases with the increase of thickness-to-radius ratio and the decrease of the number of sides in polygon base shape. In comparison with the cylindrical liner, the polyhedral liner can increase buckling pressure up to 10 times but result in a less stable post-buckling behavior.

On the interaction of uni-directional and bi-directional buckling of a plate supported by an elastic foundation

A thin flat rectangular plate supported on its edges and subjected to in-plane loading exhibits stable post-buckling behaviour. However, the introduction of a nonlinear (softening) elastic foundation may cause the response to become unstable. Here, the post-buckling of such a structure is investigated and several important phenomena are identified, including the transition of patterns from stripes to spots and back again. The interaction between these forms is of importance for understanding the possible post-buckling behaviours of this structural system. In addition, both periodic and some localized responses are found to exist as the dimensions of the plate are increased and this becomes relevant when the characteristic wavelengths of the buckle pattern are small compared with the size of the plate. Potential applications of the model range from macroscopic industrial manufacturing of structural elements to the understanding of micro- and nanoscale deformations in materials.

Local buckling and ultimate strength of slender elliptical hollow sections in compression

The local buckling behaviour and ultimate cross-sectional strength of tubular elliptical profiles in compression is examined in this study through numerical modelling. The numerical models were first validated against previous experimental data with good agreement observed, enabling an extensive parametric study to be performed. A total of 270 elliptical sections were simulated in order to examine the influence of cross-section aspect ratio, geometric imperfections and local slendernesses. The obtained ultimate capacities, load–deformation responses and failure modes are discussed. It was found that for lower cross-section aspect ratios the behaviour of the elliptical hollow sections (EHS) was similar to that of cylindrical shells across a number of metrics; however, as the aspect ratio increased, more plate-like stable postbuckling behaviour was observed. Imperfection sensitivity was found to decrease with increasing slenderness and aspect ratio. The influence of the shape of the initial imperfection on the strengths of the EHS columns was also assessed and was found to be generally limited. Finally, a design method has been proposed for Class 4 EHS members that reflects the reduction in capacity due to local buckling with increasing slenderness, but also recognises the improved postbuckling stability with increasing aspect ratio; the proposals were shown to provide safe and accurate predictions for the strengths of the EHS columns with nondimensional local slendernesses up to 2.5 and aspect ratios from 1.1 to 5.0.

Buckling prediction of panels using the vibration correlation technique

The Vibration Correlation Technique (VCT) for experimentally nondestructive determination of buckling loads of thin-walled structures is applied to stringer stiffened curved panels manufactured both from aluminum and laminated composite material. The modal behavior of the panels is investigated by exciting the structures using the modal hammer method. Natural frequencies of the panels are recorded as a function of the applied axial compression load. Unlike shell structures which present a non-stable post-buckling behavior, the stringer stiffened panels show a stable post-buckling behavior, enabling the measurement of the natural frequencies up to the actual experimental buckling load. The modal behavior of compressed panels is compared for reference to shells, yielding areas of applicability for VCT to predict efficiently the buckling loads of thin-walled structures. Guidelines are then formulated for the application of the VCT.

Elastic buckling of Cassini ovaloidal shells under external pressure - theoretical study

The paper is devoted to shells of revolution with positive and negative Gaussian curvature. The meridian of shells is a plane curve in the Cassini oval form. Geometrical properties of the middle surface of the shell of revolution based on this curve are presented. The membrane state of stress of a family of shells with constant capacity and mass under uniform pressure is described analytically and numerically with the use of the FEM (the ANSYS system). The critical pressure, buckling modes and equilibrium paths of analysed shells are calculated numerically. The advantages of a pressure vessel made in the form of Cassini ovaloidal shell, such as the lack of edge effect and a stable post-buckling behaviour, are pointed out. The results of the analytical and numerical investigations are compared and presented in tables and figures.

Buckling behavior of floating-roof steel tanks under measured differential settlement

Large vertical cylindrical steel tanks constructed on soft foundations may bring forth various types of settlement, which is well known as the uniform settlement, the planar tilt and the differential settlement. Measured settlements of steel tanks in several practical projects are first summarized according to the in-site surveying data, and two kinds of settlement patterns are clearly defined, which are referred to as the global differential settlement and the localized differential settlement. Two practical floating-roof steel tanks are taken as the illustrative examples of large steel tanks, which are marked TK-2020 with the height-to-radius ratio of 0.35 as the representatives of large volume tanks and TK-2090 with the height-to-radius ratio of 0.7 as representatives of smaller volume tanks in this paper. Buckling behaviors of the two illustrative tanks under the global and localized differential settlement are investigated using the general-purpose finite element computer package ABAQUS by means of the geometrical nonlinearity algorithm. It is shown that for tanks under global differential settlement, local buckling occurs first at the eave wind girder, followed by a stable post-buckling behavior, so that the local buckling of the wind girder can be taken as the serviceability limit state and the post-buckling strength can be utilized in structural design of tanks. Moreover, the behavior of tanks under localized differential settlement is related to the degree of localization. The buckling behavior in the case of a large circumferential central angle for localized settlement is similar to that under global differential settlement, while the effect of the localized settlement with a relatively small central angle is more obvious than that of the global differential settlement, and the snap-through buckling induced by localized settlement would govern the design.

Stability Analysis of Floating-Roof Tanks under Differential Settlement

Vertical cylindrical steel tanks are sensitive to differential settlement beneath the tank wall. Most previous studies were based on idealized harmonic settlement, however, for thin shell structures of high nonlinear behavior, it is obviously inappropriate to obtain the results under real settlement by simple summation of harmonic solutions. Real settlement of steel tanks can be grouped into two types - global differential settlement and local differential settlement. This paper examines the nonlinear response and stability behavior of floating-roof steel tanks under both types of settlement. It is shown that, for tanks under global differential settlement, local buckling occurs at the eave ring, followed by a stable post-buckling behavior, and final failure is by overall buckling of the tank shell; while for tanks under local differential settlement, the structural response is related to the degree of localization. At highly localized settlement, local snap-through buckling occurs at the tank wall, but it does not lead to a serious consequence, and the post-buckling behavior can also be utilized in design.

Postbuckling of truss-lattice shear panels using exact theory

A new solution is developed to model the stable postbuckling behaviour of a truss-lattice shear panel. The mode shapes are derived through load equilibrium conditions and are based on the load in the structural members. The load in each member of a single-cell panel can be calculated exactly, without the need for an iterative postbuckling path, and the method produces excellent results in initial and advanced postbuckling. Comparisons are drawn with an alternative analytical method and the commonplace finite element approach. A Rayleigh‐Ritz method based on a Fourier approximation to the mode shape provides the useful progression from an unbuckled to a buckled structure giving excellent results in initial postbuckling, although is limited for advanced postbuckling. The standard finite element method for this problem produces accurate results but with limited detail around the buckling load owing to the presence of an imperfection in the shape of the initial mode, which is required to initiate the postbuckling analysis.

Introducing a discrete modelling technique for buckling of panels under combined loading

The paper introduces a discrete model to describe the buckling of a stiffened panel beam under a complex loading environment. The study begins by examining the existing load interaction equation for a continuous panel. Experimental and finite element investigations establish the validity of considering the critical panel of a more complex structure in isolation. The paper then devises a discrete model for this critical panel, which was validated for a range of boundary conditions using anti-optimisation. The numerical results show that the discrete model exhibits the buckling behaviour of a continuous panel under combined loading. Recent studies established that the truss-lattice configuration has stable post-buckling behaviour and derived fast analysis technique for such a structure. It is therefore concluded that the truss-lattice model introduced in the present paper can offer a fast analysis formulation for buckling (and potentially post-buckling) of multiple-panel beams suitable for optimisation.

Problems of structural optimization for post-buckling behaviour

A proposal of a new approach to the optimal design of structures under stability constraints is presented. It is shown that the standard problem of maximization of the instability load may be modified so as to obtain a specified post-critical behaviour of the designed structure. The modified optimal structure represents stable post-buckling behaviour either locally, that is, in the vicinity of the critical point, or for a specified range of generalized displacements. First, some rigid–elastic finite-degree-of-freedom models are optimized, giving an insight into the modified design problems. Then a classification of the new optimization problems is presented. Various forms of instability are taken into account and a broad selection of objective as well as constraint functions is proposed. Based on the presented classification and following the proposed optimization concept, detailed formulations of nonlinear problems of design for post-buckling behaviour are given.

Stiffened steel plates under uniaxial compression

The stability of steel plates stiffened with tee-shape sections under uniaxial compression and combined uniaxial compression and bending was investigated using a finite element model. The emphasis of the work presented in this paper was to find the parameters that uniquely describe the strength and behaviour of stiffened steel plates. A finite element model, validated using the results of tests on full-size stiffened plate panels, was used to investigate the scale effect for five dimensionless parameters. The parameters investigated were: the transverse slenderness of the plate, the slenderness of the web and flange of the stiffener, the ratio of torsional slenderness of the stiffener to the transverse slenderness of the plate, and the stiffener-to-plate area ratio. Average magnitude residual stresses and initial imperfections were assumed for this study. A parametric study covering a wide range of dimensionless parameters indicated that stiffened steel plates do not fail by stiffener tripping unless a bending moment is applied to create flexural compressive stresses in the stiffener. Although plate buckling and overall buckling were found to lead to a very stable post-buckling behaviour, the interaction between these two buckling modes was found to give rise to a sudden loss of capacity following initial plate buckling. The plate transverse slenderness, the stiffener slenderness-to-plate slenderness ratio, and the stiffener-to-plate area ratio were found to have a significant effect on this behaviour. A comparison of the numerical analysis results with API and DnV design guidelines indicates that the guidelines predict stiffened steel plate capacity with various degrees of success, depending on the governing mode of failure. Neither guidelines address the potential interaction-buckling phenomenon.

Effects of Welding-Induced Imperfections on Behavior of Space Shuttle Superlightweight Tank

Results of linear-bifurcation buckling and nonlinear analyses of the Space Shuttle superlightweight external liquid-oxygen tank are presented for an important prelaunch loading condition. These results show the effects of actual, measured welding-induced initial geometric imperfections on an important response mode for thin-walled shellsthataresubjectedtocombinedmechanicalandthermalloads.Thistypeofinitialgeometricimperfectionmay be encountered in the design of other liquid-fuel launch vehicles. Results are presented that show that the liquidoxygen tank will buckle in the barrel section, but at load levels nearly four times the magnitude of the operational load level, and will exhibit stable postbuckling behavior. The actual measured imperfections are located in this sectionofthetank. Resultsofimperfection sensitivity analysesarepresented thatshowthat thelargestdegradation in the apparent membrane stiffnesses of the liquid-oxygen tank barrel section is caused by an imperfection shape that is in the form of the linear-bifurcation buckling mode with a relatively small amplitude. These results also show that the effect of the relatively large-amplitude measured imperfection is benign.

Behaviour of axially loaded lipped channel

Thin plates have a stable post buckling behaviour so that its failure load is materially higher than its buckling load. This characteristics makes thin plates very suitable structural elements in lighweight structures. For a thin-walled channel which consists of a system of connected thin plates, the structure will develop local, torsional, flexural buckling or a combination of these modes under axially compressive loads. Either buckle of an individual plate component or buckle as a system of connected plates may cause a reduction in the stiffness of the section. Hence, a lower load carrying capacity is expected. In this paper, the finite strip method is employed to investigate the buckling behaviour of axially loaded lipped channels. In the formulation, by modifying the displacement function, the buckling solution of other end support conditions besides the simply supported case is obtained. Furthermore, utilizing the eigenvectors from the elastic buckling solution, the lower bound method is used to evaluate the initial post buckling stiffness of an adequately edged stiffened thin-walled channel.

Compressive strength of edge-loaded corrugated board panels

Postbuckling strength of simply supported corrugated board panels subjected to edge compressive loading has been studied experimentally using a specially developed test fixture. Although the load versus out-of-plane displacement response was highly sensitive to the presence of initial imperfections in the panels, the collapse loads did not vary much, which is attributed to the stable postbuckling behavior of the plates. Thin plates collapsed at nearly twice the buckling load, while thick panels collapsed at loads below the elastic critical buckling load. Local buckling of the facing on the concave side of the buckled plate was observed at load levels close to the collapse load. The plate collapse was triggered by compressive failure of the facings that initiated at the unloaded edges. A simplified design analysis was derived based on approximate postbuckling analysis and compared with an existing design formula for corrugated board panels and boxes.

Application of a modified donnell technique for the determination of critical loads of imperfect plates

Methods existing for the determination of critical loads of plates which exhibit a stable postbuckling behavior, like the Southwell method, inflection point, stiffness reversal, Donnell extrapolation method and others, yield inconsistent results. This is mainly due to the fact that the behavior of a plate depends strongly on its initial geometric imperfection and boundary conditions. A numerical study was carried out to examine the behavior of imperfect plates subjected to in-plane loading. The investigation was aimed at determining the effects of initial geometric imperfection magnitude, boundary conditions and gravity on the behavior of the plate, thus developing a consistent method for the determination of the critical load corresponding to a plate once its behavior is provided either by tests or analytically. The present approach is based on application of a modified version of the Donnell method which yields the critical load as well as the initial geometric imperfection of the investigated plate. The numerical studies involved intensive use of the ADINA code for application of the proposed approach to experimental and analytical data and comparison with experimental results.

Buckling and Postbuckling Analyses of Struts with Discrete Supports

Previous studies on the buckling of an elastic strut with discrete flexible transverse supports indicate that a sufficient support stiffness may be found such that the flexible supports would act as rigid supports for a hinged‐end strut. In the present work, a comprehensive study of the buckling and postbuckling behavior of both hinged‐end and fixed‐end struts is carried out. It is revealed that the fixed‐end struts behaves differently from the hinged‐end struts and a clear‐cut suffcient support stiffness cannot be defined from the result of a buckling analysis. A postbuckling analysis indicates that there is a region of support stiffness, in which the postbuckling behavior of the strut is unstable. Therefbre, a sufficient support stiffness may be defined as one that would lead to a stable postbuckling behavior. With this new concept a single support stiffness value is suggested for the design of struts with fixed‐end, hinged‐end, or partially constrained end‐conditions.

Further Results on the Stability of Viscoelastic Columns

Recent results published by the authors on the stability behaviour of columns made of time-dependent materials are extended in a number of ways. Firstly, the closed-form expression obtained for the safe load limit of a simply supported column made of a linear three-element model material, is generalized for an arbitrary linearly viscoelastic constitutive law. The result, obtained by means of the static stability approach, is confirmed by an asymptotic solution of the dynamic stability equations. The same solution technique is used to generalize this expression for columns with arbitrary boundary conditions. Even though columns as structural members exhibit stable post-buckling behaviour, there are structural configurations, involving compression members, the overall load deflection behaviour of which indicate unstable post-buckling characteristics. A simple example is used to alert the designer to the possibility of encountering such configurations and the danger associated with such post-buckling behaviour in the case of structures made of time-dependent materials.