Global Mode Interaction Research Articles

Variational modelling of local–global mode interaction in long rectangular hollow section struts with Ramberg–Osgood type material nonlinearity

A variational model describing the nonlinear mode interaction in thin-walled box-section struts under pure axial compression made from a nonlinear material obeying the Ramberg–Osgood law is presented. The formulation combines continuous displacement functions and generalized coordinates, leading to the derivation of a system of differential and integral equations that describe the static equilibrium response of the strut. Solving the system of equations using numerical continuation techniques reveals the strongly unstable post-buckling response arising from combined geometrical and material nonlinearities during the interactive buckling of the global and local buckling modes—the resulting behaviour being more unstable with decreasing material hardening. A finite element (FE) model is also devised and reveals very similar post-buckling behaviour as highlighted in the variational model. The results compare very well in terms of the mechanical destabilization and the post-buckling deformation, which verifies the analytical model.

Local–global mode interaction in thin-walled inelastic rectangular hollow section struts part 1: Nonlinear finite element analysis

Abstract Mass efficient thin-walled rectangular hollow section (RHS) struts have been shown to be susceptible to local–global mode interaction and exhibit sensitivity to imperfections. Material nonlinearity may increase the imperfection sensitivity of such members further and affect the final failure mode. Nonlinear finite element (FE) models for welded inelastic thin-walled RHS struts with pre-defined local and global geometric imperfections alongside residual stresses are developed within the commercial package Abaqus and validated against two independent experimental studies. Based on the validated FE model, the effects of material nonlinearity and residual stresses from welding on the ultimate load, mechanical behaviour and the imperfection sensitivity of struts are investigated. A simplified method to determine the initial geometric imperfection amplitude introduced in the FE model with residual stresses explicitly modelled within the ECCS framework is proposed for the first time. The experimental and numerical results in conjunction with existing experimental results from the literature are employed in the companion paper for the assessment of the Effective Width Method and Direct Strength Method, which also forms the basis for a new set of design recommendations.

Local–global mode interaction in thin-walled inelastic rectangular hollow section struts part 2: Assessment of existing design guidance and new recommendations

Abstract The second part of the study on the local–global mode interaction in thin-walled inelastic rectangular hollow section struts focuses on design guidance. Based on the validated finite element (FE) model from the companion paper, a framework for fully automating FE model generation, submission and post-processing for geometric and material nonlinear analysis with imperfections is first presented. The ultimate load data for specimens with different cross-section aspect ratios, cross-sectional slenderness, global slenderness and welding options are generated. The current design rules for thin-walled welded RHS struts are assessed using the numerical results and existing experimental results from the literature by means of structural reliability analysis in accordance with the methodology presented in Annex D of Eurocode EN1990. A modified Direct Strength Method (DSM) relationship is then proposed and it is demonstrated to provide superior ultimate load predictions than the current guidelines.

Slenderness effects in thin-walled I-section struts susceptible to local–global mode interaction

A variational model that describes the nonlinear interaction between the global and local buckling of a thin-walled I-section strut under pure compression is presented and subsequently exploited. A parametric study is conducted for two limiting cases, where the flange–web joint is assumed to be pinned or rigid. For a chosen set of geometries, the most undesirable parametric spaces are identified for both global and local slendernesses, in terms of the strut length and the flange width respectively, where highly unstable behavior is observed in the post-buckling range. Practical implications are discussed in terms of the idealized design strength relationship.

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.

Cellular buckling in stiffened plates

An analytical model based on variational principles for a thin-walled stiffened plate subjected to axial compression is presented. A system of nonlinear differential and integral equations is derived and solved using numerical continuation. The results show that the system is susceptible to highly unstable local–global mode interaction after an initial instability is triggered. Moreover, snap-backs in the response showing sequential destabilization and restabilization, known as cellular buckling or snaking, arise. The analytical model is compared with static finite element (FE) models for joint conditions between the stiffener and the main plate that have significant rotational restraint. However, it is known from previous studies that the behaviour, where the same joint is insignificantly restrained rotationally, is captured better by an analytical approach than by standard FE methods; the latter being unable to capture cellular buckling behaviour even though the phenomenon is clearly observed in laboratory experiments.

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Direct numerical simulations (DNS) of the wake of a circular disk placed normal to a uniform flow show that, as the Reynolds number is increased, the flow undergoes a sequence of successive bifurcations, each state being characterized by specific time and space symmetry breaking or recovering (Fabre, Auguste & Magnaudet, Phys. Fluids, vol. 20 (5), 2008, p. 1). To explain this bifurcation scenario, we investigate the stability of the axisymmetric steady wake in the framework of the global stability theory. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers Re and characteristics of the destabilizing modes agree with the study of Natarajan & Acrivos (J. Fluid Mech., vol. 254, 1993, p. 323): the first destabilization occurs for a stationary mode of azimuthal wavenumber m = 1 at RecA = 116.9, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers m ± 1 at RecB = 125.3. Since these critical Reynolds numbers are close to one another, we use a multiple time scale expansion to compute analytically the leading-order equations that describe the nonlinear interaction of these three leading eigenmodes. This set of equations is given by imposing, at third order in the expansion, a Fredholm alternative to avoid any secular term. It turns out to be identical to the normal form predicted by symmetry arguments. Though, all coefficients of the normal form are here analytically computed as the scalar product of an adjoint global mode with a resonant third-order forcing term, arising from the second-order base flow modification and harmonics generation. We show that all nonlinear interactions between modes take place in the recirculation bubble, as the contribution to the scalar product of regions located outside the recirculation bubble is zero. The normal form accurately predicts the sequence of bifurcations, the associated thresholds and symmetry properties observed in the DNS calculations.