M. Ahmer Wadee

Cellular buckling from mode interaction in I-beams under uniform bending

Beams made from thin-walled elements, while very efficient in terms of the structural strength and stiffness to weight ratios, can be susceptible to highly complex instability phenomena. A nonlinear analytical formulation based on variational principles for the ubiquitous I-beam with thin flanges under uniform bending is presented. The resulting system of differential and integral equations are solved using numerical continuation techniques such that the response far into the post-buckling range can be portrayed. The interaction between global lateral-torsional buckling of the beam and local buckling of the flange plate is found to oblige the buckling deformation to localize initially at the beam midspan with subsequent cellular buckling (snaking) being predicted theoretically for the first time. Solutions from the model compare very favourably with a series of classic experiments and some newly conducted tests which also exhibit the predicted sequence of localized followed by cellular buckling.

Longitudinally inhomogeneous deformation patterns in isotropic tubes under pure bending

A variational model is formulated that accounts for the localization of deformation due to buckling under pure bending of thin-walled elastic tubes with circular cross-sections. Previous studies have successfully modelled the gradual process of ovalization of the cross-section with an accompanying progressive reduction in stiffness but these theories have had insufficient freedom to incorporate any longitudinal variation in the tube. Here, energy methods and small-strain nonlinear elastic theory are used to model the combined effects of cross-section deformation and localized longitudinal buckling. Results are compared with a number of case studies, including a nanotube, and it is found that the model gives rise to behaviours that correlate well with some published physical experiments and numerical studies.

Cellular buckling in I-section struts

An analytical model that describes the interactive buckling of a thin-walled I-section strut under pure compression based on variational principles is presented. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weakly stable global buckling mode and the strongly stable local buckling mode. The resulting behaviour is highly unstable and when the model is extended to include geometric imperfections it compares excellently with some recently published experiments.

Geometric modelling of kink banding in laminated structures

An analytical model founded on geometric and potential energy principles for kink band deformation in laminated composite struts is presented. It is adapted from an earlier successful study on confined layered structures that was formulated to model kink band formation in the folding of geological layers. This studys principal aim was to explore the underlying mechanisms governing the kinking response of flat, laminated components comprising unidirectional composite laminae. A pilot parametric study indicates that the key features of the mechanical response are captured well and that quantitative comparisons with experiments presented in the literature are highly encouraging.

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.

Delamination from localized instabilities in compression sandwich panels

Abstract Compressed sandwich structures, comprising two stiff face plates separated by a softer core material, while designed principally as efficient integral structures, can lose this quality when modes of overall (Euler) buckling and local buckling interact to produce buckle localization. Moreover, differing responses occur depending on the profile of the face plate buckling pattern relative to the core. An earlier variational formulation, leading to a system of nonlinear differential equations subject to boundary and integral constraints, has been adapted to account for an asymmetric response which manifests itself in delamination — the face plate coming apart from the core material. These are solved in a numerical continuation package, such that the response far into the unstable range can be portrayed.

Localized buckling in sandwich struts with pre-existing delaminations and geometrical imperfections

Abstract A recent model for the nonlinear structural response of compression sandwich struts is developed further to account for pre-existing face–core delaminations and initial imperfections in the strut geometry. Whilst the pre-existing delaminations only take effect after the critical bifurcation for overall (Euler) buckling, it is found that the secondary instability associated with localized buckling occurs earlier than for the initially perfectly bonded strut. More severe instabilities can also be promoted by superimposing geometric imperfections. In combination with delamination, the practical structural response can be highly unstable.

Lateral Stability of Imperfect Discretely Braced Steel Beams

The lateral stability of imperfect discretely braced steel beams is analyzed using Rayleigh-Ritz approximations for the lateral deflection and the angle of twist. Initially, it is assumed that these degrees of freedom can be represented by functions comprising only single harmonics; this is then compared with the more accurate representation of the displacement functions by full Fourier series. It is confirmed by linear eigenvalue analysis that the beam can realistically buckle into two separate classes of modes: a finite number of node-displacing modes, equal to the number of restraints provided, and an infinite number of single harmonic buckling modes, where the restraint nodes remain undeflected. Closed-form analytical relations are derived for the elastic critical moment of the beam, the forces induced in the restraints, and the minimum stiffness required to enforce the first internodal buckling mode. The position of the restraint above or below the shear center is shown to influence the overall buckling behavior of the beam. The analytical results for the critical moment of the beam are validated by the finite-element program LTBeam, whereas the results for the deflected shape of the beam are validated by the numerical continuation software AUTO-07p, with very close agreement between the analytical and the numerical results.

Numerical Studies on the Buckling Resistance of Prestressed Stayed Columns

The structural behaviour of prestressed stayed columns is investigated through nonlinear finite element modelling. The models were developed using the commercial software ABAQUS and validated against a series of recently conducted experiments. The sensitivity of the load-carrying capacity to the geometry of the stayed column, the initially applied prestress level within the stays and the initial global imperfection is investigated through parametric studies. It is found that there is a substantial increase in load-carrying capacity with increasing cross-arm length, provided the critical buckling mode remains symmetric. Once the critical buckling mode becomes antisymmetric, mode interaction becomes significant and the load-carrying capacity reaches a plateau and the component generally becomes more sensitive to imperfections. It is also found that the relative level of initial prestress required to maximize the load-carrying capacity of a given stayed column tends to reduce with increasing cross-arm length.

Modeling of Interactive Buckling in Sandwich Struts with Functionally Graded Cores

AbstractAn analytical pilot model for interactive buckling in sandwich struts with cores made from a functionally graded material based on total potential energy principles is presented. Using a Timoshenko beam approach, a system of nonlinear differential and integral equations is derived that predicts critical and secondary instabilities. These are validated against numerical simulations performed within the commercial finite-element package Abaqus. Good agreement is found, and this offers encouragement for more elaborate models to be devised that can account for face-core delamination—a feature where functionally graded materials are known to offer distinct advantages.