Elastic Post-buckling Behaviour Research Articles

Post-buckling behaviour of thin-walled regular polygonal tubes subjected to bending

This work presents and discusses numerical results concerning the elastic post-buckling behaviour and imperfection sensitivity of regular convex polygonal cross-section (RCPS) tubular beams buckling in local, distortional and mixed local–distortional modes, a topic currently lacking research. This study is carried out in the framework of Generalised Beam Theory (GBT) geometrically non-linear analyses, enriched with a branch switching technique, and takes advantage of the GBT intrinsic modal nature to shed new light on the mechanics underlying the post-buckling behaviour of these members. Due to the small half-wavelength of all the buckling phenomena addressed, only simply supported members under uniform bending are investigated. In particular, this work investigates the post-buckling behaviour and imperfection sensitivity of RCPS beams (i) exhibiting several wall numbers (6, 8, 10, 14, 20, 30) with distinct combinations of circumradius-to-thickness ratios (ii) having distinct lengths, and (iii) containing critical-mode initial geometrical imperfections with several amplitudes. Relevant displacement profiles and modal participation diagrams are provided along trivial and non-trivial equilibrium paths, in order to draw meaningful conclusions concerning the post-buckling behaviour of RCPS tubes under bending. For comparison and validation purposes, ABAQUS shell finite element results are also presented.

In-plane stiffness of imperfect thin rectangular plates subjected to biaxial loads in elastic post-buckling region

The elastic post-buckling behavior of thin plates covers a relatively vast region in which geometry nonlinearity (large deflection) and material linearity (Hooke's low) are realized. In this region, a thin rectangular plate has constant stiffnesses in both orthogonal directions. Few simplified analysis guidelines have been analytically represented for in-plane stiffnesses of an elastic post-buckled thin plate subjected to biaxial loads. In this study, Marguerre's equations (the generalized form of von Karman equations), which describe the elastic post-buckling behavior of imperfect thin plates, are solved. Galerkin's method is used to solve these equations in a semi-analytical procedure. Simply supported imperfect thin rectangular plates are considered, and the stresses and displacements functions are obtained in two orthogonal directions to determine corresponding in-plane stiffnesses of the plate. Also, the maximum applicable load is obtained so that the material's linear behavior is maintained. The semi-analytical procedure has accuracy enough to predict the in-plane stiffness of post-buckled plates and can be easily used for practical purposes.

On the Lateral-Torsional Elastic Post-Buckling and Strength of Channel Steel Beams

This paper aims at shedding new light on the elastic post-buckling behavior and strength (elastoplastic collapse) of steel channel section beams undergoing lateral-torsional (LT) buckling. This new insight is acquired using a two-node geometrically exact beam finite element developed by the author, which can handle large displacements, including torsion-related warping, Wagner effects, plasticity, residual stresses and arbitrary initial configurations (namely geometrical imperfections). Several cross-sections and loading/support conditions are analyzed. The results obtained show that the LT buckling behavior of channels is asymmetric, meaning that the direction of the geometric imperfection can influence significantly the strength of the beams. Moreover, it is demonstrated that the loading and support conditions play a major role, leading to a significant scatter of results which hinder the definition of a single buckling curve for design purposes. Nevertheless, in all cases considered, the buckling strengths obtained fall significantly above those predicted by the current version of Eurocode 3.

Bilinear Load–Displacement Curve of Semi-supported Steel Shear Walls

Elastic stiffness and ultimate shear capacity are two main parameters of a structural system to obtain its ideal bilinear load–displacement. In the previous studies, the ultimate shear capacity of semi-supported steel shear walls (SSSW) which are a new lateral resisting system, has been determined. In this system, wall plates do not have any direct connection to the main columns of structure and they are connected to secondary columns which do not carry the gravity loads. The used thin plate in the SSSW elastically buckles at low levels of lateral loads and the wall plate stays on a fairly vast region with elastic post-buckling behavior (elastic stiffness). In this study, the Von-Karman plate equations are solved by the Galerkin method to find displacement field of the wall plate in the elastic post-buckling region as well as the maximum shear load after which the plasticity expand in the wall plate. Thus, the elastic stiffness of system is calculated. As the analytical procedure is complicated, the method is applied on 144 examples with different material and geometrical properties. Using linear regression technique, a concise formula is proposed to predict the elastic stiffness of system. The dimensions of wall plate are only the effective parameters in the suggested formula and the elastic stiffness is independent of the overturning moment, section of secondary columns and yield stress of material. Using the ultimate shear capacity and elastic stiffness, an ideal bilinear curve is obtained for the lateral load versus the horizontal displacement. The shear capacity at the end of elastic post-buckling region and out of plane displacement are acceptably validated with those of FE analysis for some examples.

Modal analysis of the post-buckling behaviour of cylindrical steel panels under compression: Imperfection sensitivity and local2 interaction

Abstract This paper presents and discusses numerical results on the (i) elastic post-buckling behaviour and (ii) imperfection sensitivity of simply supported cylindrical steel panels under uniform compression. The results presented are obtained by means of a geometrically non-linear Generalised Beam Theory (GBT) formulation previously reported by the authors, which can include arbitrary member initial geometrical imperfections. The modal decomposition features of GBT enables extending the knowledge on the mechanics underlying these structural elements, which cannot be obtained with standard shell finite element analysis. The work begins by describing the GBT buckling analysis of four curved panels with distinct curvatures. Then, post-buckling results are presented and discussed for each of the four panels geometries, by considering (i) two distinct critical-mode initial geometrical imperfections shapes and (ii) five distinct amplitudes to assess the imperfection sensitivity – if the “sign” of the initial geometrical imperfection is relevant, ten amplitudes are considered. These results provide the evolution, along the equilibrium paths, of relevant modal displacement profiles, modal participation diagrams and deformed configurations. For comparison and validation purposes, shell finite element results are also reported.

Post-buckling behaviour of thin-walled regular polygonal tubular columns undergoing local-distortional interaction

This work presents and discusses numerical results of an ongoing investigation concerning the elastic post-buckling behaviour and imperfection sensitivity of regular convex polygonal cross-section (RCPS) tubular columns affected by local-distortional (L-D) interaction. This investigation is conducted by means of geometrically non-linear Generalized Beam Theory (GBT) analyses, extending the knowledge on the “pure” local and distortional post-buckling behaviours of RCPS tubes recently reported by Martins et al. [1]. Due to their inherent modal nature, the GBT analyses make it possible to acquire in-depth knowledge on the structural behaviour of RCPS tubes and also to include, in a straightforward fashion, arbitrary member initial geometrical imperfections. Columns with distortional-to-local critical buckling load ratios (i) close to 1 (“true L-D interaction” – the most trivial L-D interaction type) and (ii) significantly above 1 (“secondary-distortional bifurcation L-D interaction” – the most relevant L-D interaction type) are investigated. Local/distortional critical-mode initial geometrical imperfections and (linear) combinations of both are considered in the former columns, while only local initial imperfections are dealt with in the latter ones – several amplitudes are dealt with to assess the imperfection sensitivity. For comparison and validation purposes, ABAQUS shell finite element analysis results are also reported.

Distortional-global interaction in lipped channel and zed-section beams: Strength, relevance and DSM design

This work aims at presenting and discussing numerical results concerning the post-buckling behaviour, strength and design of cold-formed steel simply supported lipped channel and zed-section beams under uniform bending and undergoing distortional-global (D-G) interaction – two different support conditions are considered in the lipped channel beams. The relevance of the interaction effects is assessed by identifying the beams whose ultimate strength and/or failure mode are visibly affected by them. Distinct (i) global-to-distortional critical buckling moment ratios (RGD) and (ii) yield-to-non-critical buckling (distortional or global) moment ratios (Ry) are considered, which are expected to lead to different failure mode natures (global, distortional or interactive). For each beam type, combinations of 41 geometries and 11 yield stresses are considered, in order (i) to characterise the beams experiencing “true D-G interaction” (RGD ≈ 1.0) and “secondary global-bifurcation D-G interaction” (RGD > 1.0 and high Ry) and (ii) to investigate the possible occurrence of “secondary distortional-bifurcation D-G interaction” (RGD < 1.0 and high Ry). Moreover, an in-depth investigation on the elastic post-buckling behaviour is carried out, to assess how the initial geometrical imperfection shape/configuration influences the behaviour and strength of beams prone to D-G interaction. The results presented consist of (i) relevant non-linear equilibrium paths, (ii) deformed configuration evolutions along those paths and (iii) figures providing the failure mode characterisation. Then, the numerical failure moments obtained are compared with their predictions by means of (i) the currently available Direct Strength Method (DSM) beam distortional and global strength curves, and also (if necessary) (ii) proposed DSM-based design approaches, specifically developed to handle beam D-G interactive failures.

Modal nudging in nonlinear elasticity: Tailoring the elastic post-buckling behaviour of engineering structures

The buckling and post-buckling behaviour of slender structures is increasingly being harnessed for smart functionalities. Equally, the post-buckling regime of many traditional engineering structures is not being used for design and may therefore harbour latent load-bearing capacity for further structural efficiency. Both applications can benefit from a robust means of modifying and controlling the post-buckling behaviour for a specific purpose. To this end, we introduce a structural design paradigm termed modal nudging, which can be used to tailor the post-buckling response of slender engineering structures without any significant increase in mass. Modal nudging uses deformation modes of stable post-buckled equilibria to perturb the undeformed baseline geometry of the structure imperceptibly, thereby favouring the seeded post-buckling response over potential alternatives. The benefits of this technique are enhanced control over the post-buckling behaviour, such as modal differentiation for smart structures that use snap-buckling for shape adaptation, or alternatively, increased load-carrying capacity, increased compliance or a shift from imperfection sensitivity to imperfection insensitivity. Although these concepts are, in theory, of general applicability, we concentrate here on planar frame structures analysed using the nonlinear finite element method and numerical continuation procedures. Using these computational techniques, we show that planar frame structures may exhibit isolated regions of stable equilibria in otherwise unstable post-buckling regimes, or indeed stable equilibria entirely disconnected from the natural structural response. In both cases, the load-carrying capacity of these isolated stable equilibria is greater than the natural structural response of the frames. Using the concept of modal nudging it is possible to “nudge” the frames onto these equilibrium paths of greater load-carrying capacity. Due to the scale invariance of modal nudging, these findings may impact the design of structures from the micro- to the macro-scale.

On the mechanics of distortional-global interaction in fixed-ended columns

This work reports the results of a numerical investigation concerning the elastic post-buckling behaviour of fixed-ended thin-walled lipped channel and zed-section columns affected by distortional-global (D-G) interaction – “unexpected” features associated with the global buckling nature are unveiled in both cases. The columns analysed (i) exhibit cross-section dimensions and lengths ensuring the simultaneous occurrence of distortional and global critical buckling, and (ii) contain critical-mode initial geometrical imperfections (linear combinations of the critical distortional and global buckling mode shapes). For comparison and clarification purposes, the post-buckling behaviour of lipped channel columns buckling in an “isolated” distortional mode and containing “pure” distortional and global (flexural-torsional) initial geometrical imperfections are also analysed. The results presented and discussed are obtained through geometrically non-linear Generalised Beam Theory (GBT) analyses and provide the evolution, along given equilibrium paths, of the column deformed configuration (expressed in GBT modal form), relevant displacement profiles and modal participation diagrams, making it possible to acquire in-depth insight on the mechanics underlying D-G interaction in columns. Finally, particular attention is paid to interpreting the differences exhibited by the various column post-buckling behaviours investigated.

On the mechanics of local-distortional interaction in thin-walled lipped channel beams

This paper investigates the elastic post-buckling behaviour of simply supported thin-walled lipped channel beams undergoing local-distortional (L-D) interaction. The beams are uniformly bent about the major-axis and experience flange-triggered local buckling (most common situation in practice). Three beam geometries are considered, each exhibiting a different type of L-D interaction, namely (i) “true L-D interaction” (close local and distortional critical buckling moments – two critical-mode initial geometrical imperfections are used, akin to the competing critical buckling modes), (ii) “secondary local-bifurcation L-D interaction” or (iii) “secondary distortional-bifurcation L-D interaction” (critical distortional-to-local buckling moment ratio well below or above 1.0, respectively, with “high enough” yield stresses). The results presented and discussed are obtained through elastic geometrically non-linear Generalised Beam Theory (GBT) analyses and provide the evolution, along given equilibrium paths, of the beam deformed configuration (expressed in GBT modal form), relevant displacement profiles and modal participation diagrams, making it possible to acquire in-depth knowledge on the beam L-D interaction mechanics. Particular attention is devoted to interpreting the quantitative and qualitative differences exhibited by the beam post-buckling behaviours associated with the three aforementioned L-D interaction types.

Cylindrical Shells Made of Stainless Steel – Investigation of Postbuckling

Abstract The relevant load case of open thin-walled shells is often wind loading during construction. Because of the missing stabilization effect of the roof they show a very high sensitivity to buckling which results into higher wall thicknesses. As part of the European RFCS research project BiogaSS the Institute for Metal and Lightweight Structures of the University of Duisburg-Essen carried out investigations on open thin-walled tanks made of austenitic and duplex stainless steels under wind load to study a possible economic advantage which might be gained from the consideration of the elastic postbuckling behaviour. This contribution presents not only experimental and numerical results but also first recommendations regarding the range of possible buckling reduction factors which might be incorporated in future revisions of EN 1993-1-6 and EN 1993-4-2.

Harnessing Seeded Geometric Imperfection to Design Cylindrical Shells With Tunable Elastic Postbuckling Behavior

Geometric imperfection, known as a detrimental effect on the buckling load of cylindrical shells, has a new role under the emerging trend of using buckling for smart purposes. Eigenshape-based geometries were designed on the shell surface with the aim of tailoring the postbuckling response. Fourteen seeded geometric imperfection (SGI) cylinders were fabricated using polymer-based 3D printing, and their postbuckling responses were numerically simulated with a general-purpose finite element program. Results on the prototyped SGI cylinders showed a tunable elastic postbuckling response in terms of initial and final stiffness, the maximum load drop from mode switching, and the number of snap-buckling events. A response contour and discrete map is presented to show how the number of waves in the axial and circumferential directions in the seeded eigenshape imperfection can control the elastic postbuckling response. SGI cylinders provide diverse design opportunities for controllable unstable response and are good candidates for use in smart and adaptive materials/structures.

Elastic postbuckling response of axially-loaded cylindrical shells with seeded geometric imperfection design

Elastic instabilities (such as buckling) have been recognized as a promising phenomenon to design smart materials and mechanical systems. Thin-walled cylindrical shells under axial compression can attain multiple bifurcation points in their postbuckling regime due to the natural transverse deformation restraint provided by their geometry; but harnessing such behavior for smart purposes is lacking extensive study due to its notoriously high imperfection sensitivity. In this paper, the concept of seeded geometric imperfection (SGI) design is proposed to modify and control the elastic postbuckling behavior of cylindrical shells. Eigenvalue-based mode shapes were used as basic geometric forms to generate a seeded imperfection. Prototyped SGI cylindrical shells were fabricated through 3D printing and tested under loading–unloading cycles. Numerical and experimental results suggest that the SGI cylindrical shells are less sensitive to initial imperfections and load variation than uniform ones. Cylindrical shells with seeded geometry can be potentially used in the design of smart devices and mechanical systems such as energy harvesters and self-powered sensors.

Tailoring the elastic postbuckling response of cylindrical shells: A route for exploiting instabilities in materials and mechanical systems

Postbuckling response, long considered mainly as a failure limit state is gaining increased interest for smart applications, such as energy harvesting, frequency tuning, sensing, actuation, etc. This letter explores the potential of cylindrical shells under axial compression, for which localized buckling events along with high-rate local deformations can be attained in their elastic postbuckling regime under smaller axial system deformation input, as a prototype for harnessing elastic instabilities. Three avenues are presented to tailor and control the elastic postbuckling response of axially-compressed shells: (1) by introducing seeded geometric imperfections (SGI); (2) by introducing non-uniform stiffness distributions (NSD); and (3) by providing lateral constraints and interactions (LCI). Prototyped cylindrical shells were fabricated through 3D printing and tested under loading–unloading cycles. Experimental results show that, with appropriate selection of geometry, material, and stiffness distribution, these three concepts offer significant advantages over uniform cylindrical shells for use of their notoriously unreliable elastic postbuckling response. This work provides new knowledge on the possibilities and means to design the cylindrical shells with controlled elastic postbuckling behavior and opens new avenues for using this structural form for applications in smart materials and structures.

Analytical post-buckling model of corrugated board panels using digital image correlation measurements

The optimisation of board packages often rely on their load bearing capacity. Then it seems attractive to measure how such thin-walled structures deform using for instance kinematic field measurement techniques, and to incorporate, at least partially, the gained kinematic information within mechanical models. Digital Image Correlation (DIC) can provide a vivid description of the buckling of box panels, e.g. during box compression tests. Therefore, we propose an analytical plate model to predict the elastic post-buckling behaviour of corrugated board box panels where the kinematic boundary conditions emanate from DIC measurements. Comparing experimental and calculated strain fields on the outer liner of board panels as well as box compression force lend some confidence to the model. Further results reveal the heterogeneity of in-plane forces, bending and twisting moments the box panels have to withstand as well as strain fields in usually inaccessible regions of the panels such as the inner liner. Thereby an improvement of the structure of box panels can be envisaged.

Non-linear GBT formulation for open-section thin-walled members with arbitrary support conditions

This paper presents the development and illustrates the application of a beam finite element based on Generalised Beam Theory (GBT) and intended to analyse the elastic post-buckling behaviour of thin-walled steel members exhibiting arbitrary open cross-section and with non-standard support conditions ( e.g., in-span bracing systems). After briefly reviewing the main concepts and procedures required to obtain the GBT system of non-linear equilibrium equations, the paper describes the steps involved in the numerical implementation (incremental–iterative strategy) of a non-linear beam finite element that incorporates the influence of the non-standard support conditions. Finally, the application and capabilities of the proposed GBT-based beam finite element are illustrated by means of the presentation and discussion of numerical results concerning the post-buckling behaviour of lipped I-section columns and lipped channel columns and beams with and without localised displacement restraints. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the commercial code A nsys.

Prediction of the elasto-plastic post-buckling strength of uniformly compressed plates from the fictitious elastic strain at failure

This paper discusses the use of the theory of elasticity to determine the post-buckling strength of uniformly loaded square simply supported thin (steel) plates with sinusoidal-shaped initial imperfections and longitudinal edges free to wave in plane. Based on the findings from a FEM parameter study two main types of failure are distinguished: edge failure and center failure. The parameters determining which failure mode occurs are explained using a simple two-strip model. To determine the elastic post-buckling behavior of plates a modified far post-buckling solution is proposed, which is simpler to use than the existing far post-buckling solutions reported in the literature, giving results with engineering accuracy for loads up to about three times the buckling load. It is shown that determining the post-buckling strength as the elastic load corresponding to first membrane yield (as is done in the effective width method) paradoxically gives reasonable results for plates failing by center failure, but very conservative results for plates failing by edge failure. For plates failing by edge failure a more accurate strength prediction is obtained by deriving empirical expressions for a fictitious elastic strain at failure.

On the post-buckling behaviour of plates under stress gradient

In this paper the elastic post-buckling behaviour of plates under non-uniform compressive edge stress is investigated. The compatibility differential equations is first solved analytically and then an approximate solution of the equilibrium equation is obtained using the Galerkin method. Explicit expressions are derived for the load-deflection, ultimate strength and membrane stress distributions. Analytical effective width formulations, based on the characteristics of the stress field of the buckled plate, are proposed for this general loading condition. The predicted load-deflection expression is compared with independent test results. Results are also presented detailing the load-deflection behaviour and stress distribution for various aspect ratios.

Three-dimensional nonlinear analysis of elastic arches

This paper presents a curved finite element model for the three-dimensional nonlinear analysis of elastic arches. The model includes higher-order curvatures which make the order of the bending strains consistent with that of the membrane strains, and the same low-order polynomials are used for all the displacements. No approximations are made for the twist rotations, and the nonlinear strains and the incremental stiffness matrix include the exact coupling due to the twist rotations. All significant terms of strains are retained. Without these terms, superimposed rigid body motions may lead to overstiff solutions. The model also includes warping of the cross-section and the Wagner effect. Application of the model to several numerical examples and comparison with the existing experimental and analytical results demonstrate that the model is much more effective and efficient than other methods in terms of accuracy, the number of elements needed for convergence, and the ability to pass maximum load points. The model is also used to investigate the elastic flexural-torsional buckling and the elastic postbuckling behaviour of arches.

Axisymmetric elastic postbuckling of circular plates

This paper deals with the elastic postbuckling behaviour of axisymmetric thin plates. The analysis is based on an alternative boundary integral equation formulation which has been applied successfully in connection with the large deflection analysis of plates and shells. Numerical results are obtained and compared with corresponding ones available in the literature. Further applications of the proposed formulation to more advanced problems are also discussed.