Generalised path-following for well-behaved nonlinear structures

Recent years have seen a research revival in structural stability analysis. This renewed interest stems from a paradigm shift regarding the role of buckling instabilities in engineering design—from detrimental sources of catastrophic failure to novel opportunities for functionality. Novel nonlinear structures take the form of optimised thin-walled structures that operate safely in the post-buckling regime; shape-morphing structures that exploit multi-stability to snap and pop between different configurations; and meta-materials that derive novel material properties from a cascade of choreographed instabilities. Hence, elastic instabilities are no longer considered as structural failures but rather exploited for repeatable well-behaved adaptations. In this article we focus on shape-morphing—a bio-inspired design strategy that intends to conform structures to different operating conditions. Computational tools that integrate easily with established methods used in industry, and that are capable of capturing the full phase diagram of compound instabilities and entangled post-buckling paths typical of these structures, are limited. Such a capability is crucial, however, as confidence in predictive tools can be key in enabling non-conventional designs. One potential candidate in this regard is generalised path-following, which combines the computational robustness of numerical continuation algorithms with the geometric versatility of the finite element method. In this paper we collate an array of successful computational tools introduced by other researchers, and introduce our own developments, to present a modelling framework fit for analysing and designing with well-behaved nonlinear structures in industry and academia. Particularly, we show that the full complexity of multi-snap events of morphing composite laminates is robustly captured by generalised path-following algorithms, and that the ability to determine loci of singular points with respect to a set of parameters is especially useful for tracing the boundaries of bistability in parameter space. Furthermore, we shed new insight into the mechanics of multi-stable laminates, showing that the multi-stability and snapping behaviour of these structures is much richer than previously assumed, featuring many unstable post-buckling branches and localised regions of stability

Exploring the design space of nonlinear shallow arches with generalised path-following

The classic snap-through problem of shallow arches is revisited using the so-called generalised path-following technique. Classical buckling theory is a popular tool for designing structures prone to instabilities, albeit with limited applicability as it assumes a linear pre-buckling state. While incremental-iterative nonlinear finite element methods are more accurate, they are considered to be complex and costly for parametric studies. In this regard, a powerful approach for exploring the entire design space of nonlinear structures is the generalised path-following technique. Within this framework, a nonlinear finite element model is coupled with a numerical continuation solver to provide an accurate and robust way of evaluating multi-parametric structural problems. The capabilities of this technique are exemplified here by studying the effects of four different parameters on the structural behaviour of shallow arches, namely, mid span transverse loading, arch rise height, distribution of cross-sectional area along the span, and total volume of the arch. In particular, the distribution of area has a pronounced effect on the nonlinear load-displacement response and can therefore be used effectively for elastic tailoring. Most importantly, we illustrate the risks entailed in optimising the shape of arches using linear assumptions, which arise because the design drivers influencing linear and nonlinear designs are in fact topologically opposed

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

A 2D equivalent single-layer fomulation for the effect of transverse shear on laminated plates with curvilinear fibres

The Hellinger–Reissner mixed variational principle in conjunction with Lagrange multipliers is used to model transverse shear stresses, in bending of variable stiffness laminated plates, using a reduced two-dimensional formulation. The effect of transverse shear stresses on the bending deflection of variable angle tow (VAT) laminates is assessed. The novel formulation features multiple shear correction factors that are functions of the bending rigidity terms Dij, their first and second derivatives and the Timoshenko shear factor χ. The new set of governing equations are solved, in their strong form, using the Differential Quadrature Method (DQM) and the accuracy and robustness of the solution technique verified using a 2D “thick” shell and 3D high-fidelity finite element model. The derived theory is superior to the “thick” 2D shell model in capturing transverse shear effects and shows good accuracy compared to the full 3D solution for thicknesses within the range of practical engineering laminates. The derived equations degenerate to Classical Laminate Analysis for very thin configurations but discrepancies as large as 43% are observed for span-to-thickness ratios of 10:1. Finally, the specific VAT panels under investigation are affected more by transverse shear deformation than a corresponding homogeneous quasi-isotropic laminate