Interaction Of Buckling Modes Research Articles

Local–distortional buckling mode of steel cold-formed columns: Generalized direct strength design approach

Cold-Formed Steel (CFS) members are highly susceptible to several structural instability phenomena, involving either individual (Local — L, Distortional — D and Global — G) or interaction (LD, LG, DG, and LDG) buckling modes. The buckling mode interaction of CFS columns and beams may lead to strength erosion compared to isolated modes and must be considered in structural design. The North American, Australian/New Zealand and Brazilian standards incorporated the Direct Strength Method (DSM) that provides accurate design approach for columns and beams affected by L, D, G, and LG buckling modes. However, the coupled LD, DG, and LDG buckling modes are not currently considered in the DSM and many recent research projects have been focused on these topics. The procedure adopted by DSM estimates the CFS member strength through experimentally calibrated design strength equations and curves, based on the relationship between the member elastic stability and the steel yielding property, respectively the elastic critical buckling load and the plastic load, represented by the slenderness factor. The more accurate the buckling load, the more improved will be the result of the column strength by the DSM equations. In this context, the present paper presents the main results of the investigation, based on both experimental and finite element method numerical results, that conducted to calibrated strength curves for CFS unstiffened sections columns without holes covering LD interaction buckling mode. To take into account the effect of the LD buckling, the proposed procedure for design purposes incorporates the slenderness factor ratio RλDL=λD/λL as the main variable, and λL, λD, λG as the local, distortional and global slenderness factors, respectively.

CFS Columns Under Buckling Interaction: Direct Strength Method Generalised Approach

AbstractCold‐Formed Steel (CFS) members are highly susceptible to several structural instability phenomena, involving either individual (Local – L, Distortional – D and Global – G) or interaction (LD, LG, DG, and LDG) buckling modes. The buckling mode interaction of CFS columns and beams may lead to strength erosion compared to isolated modes and must be considered in structural design. The North American, Australian/New Zealand and Brazilian standards incorporated the Direct Strength Method (DSM) that provides accurate design approach for columns and beams affected by L, D, G, and LG buckling modes. Moreover, the coupled LD, DG, and LDG buckling modes are not currently considered in the DSM and many research projects have been focused on these topics. The procedure adopted by DSM estimates the CFS member strength through experimentally calibrated design strength curves, considering the relationship between the member elastic stability and the steel yielding property. In this context, the present paper presents the main results of the investigation, based on both experimental and finite element method numerical results, that conducted to calibrated Winter‐type strength curves for CFS columns covering LD, DG, and LDG buckling modes. To take into account the effect of the buckling modes interaction, the proposed procedure for design purposes incorporates the slenderness factors ratio RλDL = λD/λL and λG/ λmaxLD, with λmaxLD = max {λL, λD} and λL, λD, λG as the local, distortional and global slenderness factors, respectively.

Explicit approach for elastic local buckling analysis of thin-walled channels under combined bending and shear

For flexural thin-walled section beams, the ultimate strength is significantly affected by the interaction of different local buckling modes associated with combined bending and shear stresses. This paper provides a comprehensive analytical and numerical study of elastic buckling of thin-walled channels subjected to different combinations of bending and shear stresses. The commercial ABAQUS software package using Finite Element Method (FEM) has been used to perform the elastic buckling analyses of the channel sections. The results are benchmarked against those from the Semi-Analytical Finite Strip Method (SAFSM) for the cases of pure bending and shear only. Two typical types of channels with and without lips are selected for investigation. To study the effect of different combinations of bending and shear on the buckling stresses, different normal stress distributions and equilibrating shear stresses are applied at two end cross-sections of the channels to maintain static equilibrium. Different types of interactions which occur during the buckling analyses are studied by varying the ratios of bending moments resulting from the modelled longitudinal stresses. A new concept for the elastic local buckling interaction based on the Average Bending Moment (ABM) at buckling is provided. In addition, an explicit approach for local buckling loads of channels under different combinations of bending and shear is also introduced for the practical design use. The results from this explicit solution for elastic buckling can be used as direct inputs for an explicit Direct Strength Method (DSM) design of members under combined bending and shear.

Post-buckling behaviour in variable stiffness cylindrical panels under compression loading with modal interaction effects

Variable Angle Tow (VAT) composite panels, whereby fibre angle distributions vary continuously in-plane, have more scope for tuning structural properties than traditional straight fibre laminated composite materials. Currently, the geometrically non-linear response of such structures is typically modelled using path-following methods implemented in commercial finite element analysis. However the high computational cost given from both the incremental-iterative procedure and fine mesh required for accurate modelling of the variable trajectory of fibre direction, can be excessively time consuming. Driven by these shortcomings, we use a fast multi-modal Koiter asymptotic method for investigating the nonlinear buckling behaviour of VAT cylindrical panels in compression. In doing so, this paper provides new insight into the multi-modal description and nonlinear buckling mode interactions which are responsible for the highly nonlinear behaviour of cylindrical panels in compression with a particular emphasis on the nonlinear components in the mathematical asymptotic description. As such, through an extensive parametric virtual testing programme we construct, for each panel under investigation, an asymptotic solution projecting the equilibrium equations in the subspace of its first 30 buckling modes. Then, at several points of the resulting equilibrium path, the percentage contribution of each of these modes to the solution is measured. On the basis of the percentage obtained, those modes which give the largest contribution to the solution are identified. The main novelty of this work is that these results are used as a priori information for re-projecting the equilibrium equations into a new modal subspace to which only the buckling modes with the largest participation belong. Moreover, for the first time, we observe that in a multi-modal Koiter-inspired description the modal shapes giving the largest contribution to the asymptotic solution have the same degree of symmetry. In this way, for a generic panel under consideration, we show that it is possible to obtain a more computationally-efficient asymptotic description than current approaches. Finally, the new computed equilibrium paths are compared to benchmark results using the commercial finite element software ABAQUS and the computational advantages given by using a priori information in the asymptotic expansion are commented upon.

Experimental investigation and numerical analysis of pin-ended extruded aluminium alloy unequal angle columns

The structural performance of aluminium alloy unequal angle columns was investigated in this study. Fourteen columns made of two extruded aluminium alloy unequal angle cross-sections were tested. Complementary material property tests and geometric imperfection measurements were also carried out. The test setup, procedure and results were fully reported. After the tests, numerical models were developed to replicate the tested pin-ended extruded aluminium alloy unequal angle column behaviour. The validated finite element models were then adopted to analyse the effect of the configuration of initial geometric imperfections, which proved to be pivotal in mode interaction of the column buckling performance. Parametric studies were carried out with specific initial imperfection configuration to generate further numerical data. The ultimate resistances derived from the experiments and finite element analyses were employed to evaluate the applicability of the current codified standards. The result comparison shows that current design provisions yield unsafe predictions for pin-ended unequal angle columns since the buckling mode interaction is not considered. The development of applicable design proposals and further investigation into buckling modal participation for pin-ended extruded aluminium alloy unequal angle columns are required.

Inelastic buckling mode interactions and resonance instabilities in thin-walled columns due to the fatigue damage

PurposeThe problems of inelastic instability (buckling) and dynamic instability (resonance) have been the subject of extensive investigation and have received wide attention from the structural mechanics community. This paper aims to tackle these problems in thin-walled structures, taking into account geometrical and/or material non-linearity.Design/methodology/approachThe inelastic buckling mode interactions and resonance instabilities of prismatic thin-walled columns are analysed by implementing the semi-analytical finite strip method (FSM). A scalar damage parameter is implemented in conjunction with a material modelling named rheological-dynamical analogy to address stiffness reduction induced by the fatigue damage.FindingsInelastic buckling stresses lag behind the elastic buckling stresses across all modes, which is a consequence of the viscoelastic behaviour of materials. Because of the lag, the same column length does not always correspond to the same mode at the elastic and inelastic critical stress.Originality/valueThis paper presents the influence of mode interactions on the effective stresses and resonance instabilities in thin-walled columns due to the fatigue damage. These mode interactions have a great influence on damage variables because of the fatigue and effective stresses around mode transitions. In its usual semi-analytical form, the FSM cannot be used to solve the mode interaction problem explained in this paper, because this technique ignores the important influence of interaction of the buckling modes when applied only for undamaged state of structure

Flexible filaments buckle into helicoidal shapes in strong compressional flows

The occurrence of coiled or helical morphologies is common in nature, from plant roots to DNA packaging into viral capsids, as well as in applications such as oil drilling processes. In many examples, chiral structures result from the buckling of a straight fiber either with intrinsic twist or to which end moments have been applied in addition to compression forces. Here, we elucidate a generic way to form regular helicoidal shapes from achiral straight filaments transported in viscous flows with free ends. Through a combination of experiments using fluorescently labeled actin filaments in microfluidic divergent flows and of two distinct sets of numerical simulations, we demonstrate the robustness of helix formation. A nonlinear stability analysis is performed and explains the emergence of such chiral structures from the nonlinear interaction of perpendicular planar buckling modes, an effect that solely requires a strong compressional flow, independent of the exact nature of the fiber or type of flow field. The fundamental mechanism for the uncovered morphological transition and characterization of the emerging conformations advance our understanding of several biological and industrial processes and can also be exploited for the controlled microfabrication of chiral objects.

Advances in the Direct Strength Method of cold-formed steel design

Twenty years ago, the Direct Strength Method (DSM) was first proposed as an efficient alternative to traditional design methods for thin-walled cold-formed steel (CFS) structural members. DSM may be regarded as one implementation of a class of generalized slenderness methods that are used in structural design. The objective of this paper is to provide additional context on how DSM-based design fits within past design knowledge and summarize the current state of development and future potential of the approach. Although DSM is regarded as a relatively new member design approach, its fundamental principles have long been used to explore relationships between stability, material yielding, and strength. The first ten years of DSM development (1998–2008) focused largely on simple prismatic CFS members under single actions and was conducted by a small group of researchers. In the last ten years of DSM development both the scope of DSM-based solutions and the researchers who are contributing to this development have broadly expanded. Since 2008 significant advances in DSM have occurred in strength prediction in shear, torsion, and combined loading; DSM efforts have also addressed buckling mode interactions, built-up members, elevated temperatures, member optimization, stiffness and ductility predictions, system-level implementations and more. DSM’s ability to directly integrate elastic buckling simulation and provide efficient structural predictions for complex folded cross-sections remains one of its central advantages. Looking to the future DSM is well positioned to provide useful and efficient structural predictions and current efforts will continue to expand the scope and applicability of DSM.

Lipped channel cold-formed steel columns under local-distortional buckling mode interaction

The present research is aimed at verifying design rules for cold-formed steel (CFS) members under axial compression when local-distortional buckling modes (LD) is achieved. Sets of experimental results previously available were applied to calibrate a shell finite element model addressed to survey the structural behavior of CFS lipped channels under LD bucking interaction. A large range of distortional-local slenderness ratio was considered in order to obtain numerical results of LD interaction including strong and weak interaction conditions. Based on both experimental and numerical results, the nature of LD interaction was assessed and the column strength variation was observed according with the representative slenderness variables, λL, λD and RλDL, respectively local and distortional slenderness factors and distortional-local slenderness ratio λD/λL. The column strength variation was established based on the Winter-type equation, conceived to cover the complete range of LD interaction. The proposed solution follows the direct strength method concept, is easy to apply and accurate enough for the structural design of lipped CFS columns. Additional research is under way to include other CFS types in the proposed design approach.

CFS lipped channel columns affected by L-D-G interaction. Part I: Experimental investigation

This work deals with the structural behaviour and ultimate strength of fixed-ended cold-formed steel (CFS) lipped channel columns experiencing various levels of local-distortional-global buckling mode interaction. After briefly addressing the column specimen geometry selection (carefully chosen to ensure more or less closeness between the critical buckling stresses associated with these three buckling mode types), the paper reports the results of an experimental investigation involving a set of 17 columns with several cross-section dimensions, lengths and yield stresses, which is aimed at (i) providing experimental evidence of the occurrence of the triple mode interaction under consideration and (ii) quantifying its effect on the column deformed configuration evolution and failure load erosion (drop). Various aspects concerning the test set-up and procedure are described in some detail before presenting and discussing the obtained column experimental results, which basically consist of (i) measured cross-section dimensions, lengths and initial displacements (geometrical imperfections), (ii) stress-strain curves and yield stresses determined from tensile coupon tests, (iii) recorded non-linear equilibrium paths (applied load vs. various relevant displacements) and ultimate strength values, and (iv) observed deformed configurations and collapse mechanisms. The test results are used, in a companion paper (Part II), (i) to calibrate and validate numerical simulations and (ii) to support the proposal and to assess the merits of design approaches based on the Direct Strength Method (DSM) and intended to handle L-D-G interactive failures.

CFS lipped channel columns affected by L-D-G interaction. Part II: Numerical simulations and design considerations

This work deals with the structural behaviour, ultimate strength and design of fixed-ended cold-formed steel (CFS) lipped channel columns experiencing various levels of local-distortional-global buckling mode interaction (more or less close critical local, distortional and global loads). First, a comparison between experimental results, obtained from three tests carried out at The University of Hong Kong (UHK), and the corresponding Abaqus shell numerical simulations are presented and discussed – the UHK test campaign is reported in Part I of this two-part paper. The comparison (i) involves equilibrium paths, deformed configurations, failure loads and collapse mechanisms, and (ii) provides enough evidence to ensure that the numerical model developed may be deemed validated. Then, this numerical model is employed to perform a parametric study, aimed at obtaining numerical failure load data concerning lipped channel columns experiencing different levels of local-distortional-global interaction – 368 fixed-ended columns, exhibiting different geometries (cross-section dimensions and lengths) and various yield stresses, thus covering a wide slenderness range. Finally, the present (Part II) paper also assesses the quality of the failure load estimates provided by various design approaches, based on the Direct Strength Method (DSM), for CFS columns affected by the triple coupling phenomenon under consideration. This assessment is based on the comparison with both (i) experimental failure loads, namely those reported in Part I and others collected from the literature, and (ii) numerical failure loads, obtained from Abaqus shell finite element analyses (SFEA) and either reported in this work or gathered from previous publications by the authors.

Compressive strength of composite laminates with delamination-induced interaction of panel and sublaminate buckling modes

Compression After Impact (CAI) strength is critical to the safety and weight of carbon fibre aircraft. In this paper, the standard aerospace industry practice of using separate analyses and tests for panel buckling and CAI strength is challenged. Composite panels with a range of stacking sequences were artificially delaminated and subject to compression testing in a fixture that allowed local sublaminate and global panel buckling modes to interact. Compared to panels without delamination, interaction of buckling modes reduced panel buckling strains by up to 29%. Similarly, compared to delaminated panels restrained against panel buckling, interaction reduced delamination propagation strains by up to 49%. These results are the first to indicate that restriction of interaction during CAI testing is unconservative and therefore potentially unsafe. A novel integration of an analytical Strip model, for sublaminate buckling driven delamination propagation, and a Shanley model, for determining increased local strain due to sublaminate-buckling-induced panel curvature, is used to calculate the reduction in strength due to buckling mode interaction. Assuming a typical sublaminate post to pre-buckling stiffness ratio of 0.65, the difference in integrated model and experimental results is<11% – a level of accuracy that will allow the integrated model to drive initial design studies.

Geometric factors affecting I-section struts experiencing local and strong-axis global buckling mode interaction

A recent analytical model describing the post-buckling behaviour of an I-section strut experiencing strong axis global–local buckling interaction is extended to investigate the effects of modifying the strut geometry. Using a combination of analytical and finite element (FE) methods, the global and local slendernesses are varied parametrically, in turn, to determine the geometries leading to regions of interactive behaviour. The effect of stress relieved initial global imperfections are also investigated. It is observed that the strut can exhibit one of five distinct post-buckling behaviours, the geometries for which are identified. The strut can exhibit global buckling only, local buckling only, global–local buckling interaction with either the global or local mode being triggered first or the most severe case where both global and local buckling modes are triggered simultaneously. The strut is found to be highly sensitive to initial imperfections in the interactive region; the implications for imperfection sensitivity on the design and the practical use of such components are discussed.

Local-distortional buckling interaction on cold-formed steel lipped channel beams

This paper deals with the ultimate strength, post-buckling behaviour and design of cold-formed steel lipped channel beams affected by local-distortional buckling mode interaction, subjected to uniform bending of the major axis. The cross-sectional dimensions and length of the beams were chosen to have almost equal local and distortional critical buckling stresses by using the GBTUL-2.0β and CUFSM software. The beams are analysed under pinned with warping free end conditions. 12 sections were selected for this study. The selected sections satisfy the geometric limitations for pre-qualified sections in AISI-S100:2007. Elastic buckling and non-linear finite element analyses were carried out using ABAQUS. The results were compared with corresponding results from experiments available in the literature. The numerical parametric study of 60 analyses of different geometries and yield stress values was undertaken. Based on the comparison of ultimate strengths obtained from the finite element analysis and Direct Strength Method (DSM), a design equation is proposed for lipped channel sections which have their elastic local and distortional buckling moments nearly equal.

Interactively Induced Localization in Thin-walled I-section Struts Buckling About the Strong Axis

A variational model describing the behaviour of a thin-walled I-section strut suffering from local–global buckling mode interaction is presented where global (Euler) buckling about the strong axis is the critical mode. A system of differential and integral equations is derived that describe the equilibrium states from variational principles and are solved numerically using the continuation and bifurcation software Auto-07p for the perfect case. Initially stress relieved out-of-straightness imperfections are subsequently introduced and the nonlinear response is modelled. The modelled interaction is between the critical global buckling mode about the strong axis and local buckling in the flange and web simultaneously, where the flange–web joint is assumed to be free to rotate as a rigid body. The initial eigenmode is shown to be destabilized at a secondary bifurcation where interactive buckling is triggered. A progressive change in the buckling mode is then observed, initially with local buckling localizing at the mid-span of the compression flange, which also triggers sympathetic local buckling in the web. The results from the analytical model have been validated using the commercial finite element (FE) software Abaqus with good comparisons presented for the initial post-buckling behaviour. The strut also exhibits sensitivity to initial out-of-straightness imperfections, with a notable decrease in the ultimate load as the imperfection size increases. The ultimate loads for a range of imperfection amplitudes are found using both analytical models and FE analysis, with very good correlation observed.

Experimental study on cold-formed steel web stiffened lipped channel columns undergoing distortional–global interaction

This paper describes the design and testing of cold-formed steel web stiffened lipped channel sections experiencing distortional–global buckling mode interaction under axial compression. The relevant literature was reviewed, sections were selected with nearly coincident distortional and global buckling loads to have strong interaction effects. Selected sections satisfy the geometric limitations for pre qualified sections in order to apply direct strength method (DSM) in Appendix 1 of the North American Specifications (AISI-S100:2007). A total of 15 web stiffened lipped channel compression members were tested under pin with warping restrained end condition. The experimental results were compared with the current design rules in AISI-DSM and the quality of DSM based design procedures based on distortional–global mode interaction was reviewed. In addition, a reliability analysis was performed to assess the quality of current DSM and D–G interaction design procedure to predict the ultimate capacity of the sections.

Local/Distortional/Global buckling mode interaction on thin walled lipped channel columns

The buckling behaviour of pin ended cold-formed steel lipped channel columns affected by local / distortional / global buckling mode interaction under axial loading is simulated by using finite element software ABAQUS, and the results are compared with the test results available in the literature. The comparisons show that the proper analysis model can simulate the buckling behaviour and ultimate capacity of cold-formed steel columns. Then a parametric analysis is carried out for 15 column geometries of 3 yield stress values of different dimension, thickness and lengths. The cross sectional dimensions and length of the specimen have been chosen such that to have almost equal local, distortional and global critical buckling stresses by using the CUFSM software. The selected sections also satisfied the limitations given for pre-qualified sections in Direct Strength Method (DSM). After comparing the FEM column ultimate loads with the estimates predicted by the current Direct Strength Method (DSM) design curves against local, distortional and global failures, which clearly shows that they lead to inaccurate and often very unsafe ultimate load estimates. At the end, a design recommendation is made for current design practice of evaluating the ultimate strength of the lipped channel columns.

New Insight into Interaction of Buckling Modes with Stable Post-buckling Response

Buckling mode interaction of elastic systems in the presence of initial imperfections is well known to have a detrimental effect on the response of a wide range of structural systems. This has been demonstrated mostly analytically for simple models, assuming small displacements, thus obtaining results with questionable validity in the post-buckling range. In order to acquire additional insight into this issue, in the first part of the present paper, two different versions of the well-known 2-DOF Augusti model, whose independent buckling modes are both stable, are studied analytically without any simplifying assumptions with respect to the magnitude of deformation, in order to accurately demonstrate the coupling phenomena in the presence of imperfections in the pre- and post-buckling range. Depending on the nature of the structure’s rotational springs, its post-buckling equilibrium path may be either stable or unstable. Afterward, the elastic response of two examples of laced built-up columns is illustrated numerically, one characterized by interaction between in-plane global and local buckling and the other by in- and out-of-plane global buckling, featuring similar response to that of the two 2-DOF models, respectively, thus demonstrating occurrence of such behavior in actual structural systems.

Quasi-static axial compression of concentric expanded metal tubes

Previous studies have demonstrated that the failure mechanism and energy absorption capacity of expanded metal tubes strongly depends on the orientation of the cells. This paper presents an experimental investigation on the collapse of concentric expanded metal tubes subjected to quasi-static axial compression. Square tubes with two different cell orientations are tested to failure, and the energy absorption characteristics are calculated. The results show that the combination of cell geometries lead to a complex buckling mode interaction, which enhances the energy absorption capacity of expanded metal tubes.

Buckling mode interaction in prestressed stayed columns

Prestressed stayed columns offer an innovative and aesthetically attractive solution where designs demand long and slender elements under compression. The addition of cross-arms and pretensioned cables to the columns provides significant lateral and rotational support that can result in a marked increase of the buckling capacity. The current work presents a simplified model, comprising discrete rigid links and springs with a mathematically exact formulation including the destabilising effect of cable slackening, which mimics the response of stayed columns and reveals the governing physical parameters that drive the behaviour. A recent finite-element study had indicated that these structural components are likely to exhibit interactive buckling phenomena, particularly if the critical buckling mode is antisymmetric, leading to highly unstable responses. However, by its very nature, that study could not reveal the bifurcational structure of the equilibrium response. The results from the present model reveal similar behaviour to that observed in the earlier finite-element study. A sequence of bifurcations is found that switch the post-buckling response from one mode to another by way of a path of interactive buckling.