Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Structural function from sequential, interacting elastic instabilities

E. D. Wheatcroft

E. D. Wheatcroft

Bristol Composites Institute, Queen’s Building, University Walk, Bristol BS8 1TR UK

[email protected]

Contribution: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Validation, Visualization, Writing – original draft, Writing – review & editing

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J. Shen

J. Shen

Bristol Composites Institute, Queen’s Building, University Walk, Bristol BS8 1TR UK

Contribution: Conceptualization, Funding acquisition, Methodology, Supervision, Writing – review & editing

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R. M. J. Groh

R. M. J. Groh

Bristol Composites Institute, Queen’s Building, University Walk, Bristol BS8 1TR UK

Contribution: Conceptualization, Funding acquisition, Methodology, Software, Supervision, Writing – review & editing

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A. Pirrera

A. Pirrera

Bristol Composites Institute, Queen’s Building, University Walk, Bristol BS8 1TR UK

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M. Schenk

M. Schenk

Bristol Composites Institute, Queen’s Building, University Walk, Bristol BS8 1TR UK

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    Abstract

    Elastic instabilities have traditionally been considered a failure mechanism; however, recent years have seen numerous studies exploiting instabilities as a means to achieve structural functionality. By contrast, interacting instabilities and compound buckling are still largely viewed as a failure mechanism. In this paper, we show that interacting instabilities can also be exploited to achieve bespoke functionality. We focus on ‘sequential instabilities’, whose associated critical points cannot both lie on a fundamental equilibrium path. We obtain sequential instabilities by combining canonical bifurcations, (e.g. limit point, pitchfork) as building-blocks. Initially, this concept is explored through simple bar-and-spring models that are found to have several properties not exhibited by the building blocks from which they are constructed. Further, the utility of the building block approach, and that of sequential, interacting instabilities, is demonstrated through the development of a morphing structure which must rapidly deploy after a critical input displacement is attained, and meet specific post-deployment stiffness requirements. Two design concepts are proposed, each comprising building blocks to mirror the fundamental working principles identified through the bar-and-spring models. Finite-element models of the design solutions are presented, demonstrating how the designs positively use sequential, interacting instabilities in order to meet the challenging requirements of the application. This work extends the contextual framework of instabilities that can be used to create structures with novel functionality.

    1. Introduction

    The loss of elastic stability has traditionally been regarded as a failure mechanism in engineering, and was explicitly avoided when designing structures [1]. With the exception of a few notable cases (e.g. aircraft fuselage panels, where a partial loss of stability is considered to be acceptable [2,3]) any structural response requiring traversal of unstable post-buckling paths has generally been deemed uncontrollable, often unpredictable, and therefore detrimental. Indeed, when noting in his seminal thesis that unstable post-buckling paths can ultimately re-stabilize in the advanced post-buckling regime, Koiter remarked that such equilibrium states can be reached only on passing through [an] unstable buckling state, making their practical usefulness, to say the least, doubtful [4]. However, this narrative has been challenged more recently, and instabilities have been exploited for novel functionalities. This has led to the creation of well-behaved nonlinear structures [5] such as shape-adaptive structures [69], deployable structures [1012], zero or negative stiffness structures [13,14], energy harvesters [15], as well as non-destructive testing techniques [1620].

    By contrast, interacting instabilities are generally still viewed as a failure mechanism [2124]. It is well understood that buckling modes with coincident or nearby critical points can ‘interact’ with one another [2528], a phenomenon known as mode interaction, or sometimes ‘modal contamination’ [16,24,27]. Mode interaction can cause the response at super-critical (stable in the post-buckling regime) bifurcations to become subcritical (i.e. destabilized), even if each buckling mode is super-critical when triggered individually [1,4,2128]. This destabilization can lead to sudden deflections and a buckling load which is highly sensitive to initial imperfections. These effects have been referred to as ‘the erosion of an optimum design’ [1,26,29], and can also occur when two critical points are exactly coincident, creating a degenerate critical point (also sometimes referred to as compound buckling [1,25,3032]). The ‘interaction’ described in the above examples is between buckling modes, each with an associated mode shape, and these mode shapes can provide a simple and accurate coordinate basis in which to describe the deformations of a structure around a given critical point. However, in this paper, we shall investigate the advanced post-buckling regime of structures, well away from any critical points. So the notion of ‘modes’ and their ‘interaction’ with one another becomes less helpful. Nonetheless, in all of the structures we investigate, we show that the stability of their advanced post-buckling paths is linked to the proximity of their associated critical points. Therefore, for the sake of consistency with existing literature, we shall refer to said critical points as interacting instabilities. Because of the detrimental effects of modal interaction, engineers are cautioned against optimizing a structure such that two buckling modes occur simultaneously—a so-called ‘naive optimization’ [1], and coincident buckling modes are usually avoided in structural design practice, particularly if the magnitude or nature of imperfections cannot be well controlled. Nonetheless, there are a handful of examples in published literature which contradict the traditional narrative and deliberately exploit the properties of interacting instabilities for functionality. For example, interacting instabilities have been used to design compliant mechanisms with a near-zero stiffness snap-through response, which is achieved by compressing a beam with two coincident buckling modes [3335]. Another example is found in meta-materials, where structures consisting of an array of snap-through arches have been used to create programmable solids [36], and for the purposes of energy dissipation [37].

    In this paper, we add to this line of research efforts on interacting instabilities by using their properties—such as rapid destabilization and re-stabilization, or the possibility for large dynamic post-buckling deflections—to achieve complex functionality for well-behaved nonlinear structures, which would not be possible if only a single instability were used. For the first time, to the best of our knowledge, we craft sequential instabilities by combining canonical instabilities in a building block approach, with the ultimate aim of generating beneficial interactive behaviour. We use this new design approach to fulfil the challenging actuation requirements of a highly nonlinear morphing composite structure.

    (a) Sequential instabilities

    As a structure is loaded from its undeformed configuration it initially follows a fundamental equilibrium path, which may pass through multiple critical points. These can be either limit points or symmetry-breaking branching points (provided said symmetries have not already been broken by initial imperfections). Our interest is in exploiting the existence of these multiple instabilities for functionality, specifically so-called ‘sequential instabilities’. We define a pair of instabilities as non-sequential when both of their corresponding critical points lie on a structure’s fundamental equilibrium path. This is tantamount to stating that the modes can be physically realized in any order—buckling into one mode does not require the prior development of the other—hence they need not occur in any particular sequence. An example is the buckling of a slender column subjected to axial compression (the well-known Euler column). This may buckle into one of an infinite number of modes, each with an associated critical point, but all critical points lie on the fundamental path [1]. This is illustrated by the schematic equilibrium paths in figure 1a, with the pair of critical points C1 and C2. By contrast, a pair of instabilities is considered sequential when their associated critical points do not both lie on the fundamental path, unless they are exactly coincident. In general, the second critical point of the pair lies on a secondary path, which bifurcates from the fundamental path at the first critical point, as illustrated in figure 1b. In other words, there is only one physically meaningful sequence in which these modes can occur, and hence, they are sequential.1

    Figure 1.

    Figure 1. (a) A schematic diagram of a non-sequential pair of critical points; both C1 and C2 lie on the fundamental path. Λ is the load parameter, which may be enforced as a load or a displacement, and q1 and q2 are the generalized coordinates associated with C1 and C2. (b) A schematic diagram of a pair of sequential critical points; C2 can only be accessed via a secondary path, which bifurcates from the fundamental path at C1.

    An example of sequential instabilities might be the global-local buckling of a thin-walled rectangular hollow section (RHS) strut subjected to axial compression. This is investigated in detail by Shen & Wadee [24], who note the existence of a sinusoidal global mode as well as two local modes (which they refer to as local modes I and II). Provided such a strut is sufficiently slender, it initially buckles into the global mode. However, a second critical point is soon encountered, from which a branch associated with local mode II emanates. This is the only way that local mode II can be realized; hence, it is sequential with the global mode.

    Two instabilities can interact with one other regardless whether they form a sequential or non-sequential [26,38] pair. We focus on the interaction of sequential instabilities as they may be easily constructed by combining multiple sub-structures (referred to hereafter as ‘building blocks’) whose individual post-buckling responses are known a priori. This approach also helps us to understand the underlying working mechanism of sequential instabilities, meaning simple models can be used to synthesize more complex mechanical designs with the same working principle. However, it is perfectly possible that a non-sequential, interacting pair of instabilities could also be used to achieve functionality. Nonetheless, the fascinating peculiarity of sequential instabilities is that their interaction leads to properties that the individual instabilities do not exhibit on their own. This property suggests that sequential, interacting instabilities open up new design paradigms for achieving functionality through nonlinearity and buckling.

    The remainder of this paper is structured as follows: in §2, we analyse bar-and-spring structures that use building blocks to create sequential, interacting instabilities, and use energy-based arguments to provide an insight into their working mechanism. Section 3 discusses a particular application where the unique properties of sequential instabilities are desirable; namely, the design of a passive, morphing aerodynamic spoiler for the wing of a commercial airliner, or other large-scale aircraft. We then present finite-element models of structures which meet the needs of this application using the same sequential instabilities discussed in §2. Finally, conclusions are drawn in §4.

    2. Analysis of sequential instabilities

    In this section, we explore the behaviour of two different types of sequential instability using simple, discrete degree of freedom (DOF), bar-and-spring models. The two models investigated consist of (i) two branching point bifurcation structures placed in sequence, and (ii) a limit point bifurcation placed in sequence with a branching point bifurcation structure. The equilibrium manifolds of these structures are explored using analytical methods, as well as a nonlinear finite-element (FE) analysis in conjunction with generalized path-following techniques [5]. We study the constituents of the structures’ strain energy during buckling to provide insight into how the sequential instabilities in these structures interact. Using simple models constructed from building blocks makes it easier to understand the underlying physical principles which govern the behaviour of these sequential instabilities, and thus allows us to explore the range of possible responses. Furthermore, the building-block approach means these models can then be used to synthesize mechanical designs which use sequential instabilities to achieve the same functionality.

    (a) Double branching point structure

    Figure 2 shows a simple bar-and-spring example of a structure containing sequential instabilities, specifically sequential branching point bifurcations. The structure consists of two bar-and-spring building blocks, each designed to reflect the behaviour of an Euler strut. Each strut consists of two axial springs and a rotary spring, which represent the axial and bending stiffness of the continuum equivalent, respectively. The horizontal strut is loaded by an enforced compressive input displacement, uin, leading to the reaction loads P. Its centre is freely pinned to one end of the vertical strut. The transverse deflection of the horizontal strut (ucouple) serves as the input to the vertical strut. Since the structure contains two elements, which each exhibit a branching point bifurcation, it is referred to hereafter as the ‘double branching point structure’.

    Figure 2.

    Figure 2. A double branching point structure; the bar-and-spring model is constructed from two building blocks connected in series to produce a structure containing sequential instabilities. Each building block consists of two axial springs (stiffness k1 and k2, rest length L01 and L02) attached at a pin joint supported by a rotary spring (torsional stiffness R1 and R2). The structure is loaded via an imposed axial displacement uin (with reaction force P) and it is assumed that the symmetry of each building block is preserved during loading. The transverse displacement of the centre of the horizontal strut (ucouple) serves as the input to the second building block, with functional output deflection uout. This simple structure contains two separate branching point bifurcations which are sequential with one another.

    We now present an analysis of the double branching point structure. This is conducted analytically, taking advantage of the model’s simplicity. Assuming that the symmetry of both struts is always preserved, the structure has three geometric degrees of freedom. We use parameters uin, θ1 and θ2 to simplify the mathematical expressions involved. These can be readily converted into the more geometrically intuitive DOF space of uin, ucouple and uout once the analysis is complete. The total potential energy can then be expressed as

    Π(θ1,θ2,uin,P)=UW=k1x12+k2x22+2R1θ12+2R2θ22Puin,2.1
    where U is the internal strain energy, W the external work, P the input load, with x1 and x2 being geometric parameters defined as
    x1=2L01uin2L01cosθ12cosθ1andx2=4L02(1cosθ2)(2L01uin)tanθ14cosθ2.}2.2
    Note this energy statement assumes the structure is free from imperfections in its undeformed state. This assumption allows the ideal equilibrium paths of the structure to be found, so any branching points can be fully investigated. In a real structure, imperfections have the effect of breaking the branching points, forcing the structure to follow a single equilibrium path [1]. Dividing equation (2.1) by k1L012 non-dimensionalizes the total potential energy (hereinafter the overbar symbol denotes a dimensionless quantity)
    Π¯(θ1,θ2,u¯in,P¯)=x¯12+αx¯22+2ηθ12+2βηθ22P¯u¯in,2.3
    where
    x¯1(θ1,u¯in)=x1L01,u¯in=uinL01andx¯2(θ1,θ2,u¯in,γ)=x2L01,P¯=Pk1L01,}2.4
    and structural parameters are defined by
    α=k2k1,β=R2R1,γ=L02L01,η=R1k1L012.2.5
    In combination with the three non-dimensional DOFs, θ1, θ2 and u¯in, and the reaction force, P¯, these non-dimensional groups fully define the state of the structure.

    The equilibrium condition requires the first derivatives of Π¯ with respect to the DOFs to be zero:

    Π¯θ1=Π¯θ2=Π¯u¯in=0.2.6
    In turn, the stability of an equilibrium configuration is determined using the tangent stiffness matrix of the structure. The non-dimensional form of the tangent stiffness matrix K¯T is the Hessian of the non-dimensional potential energy Π¯. When the system is loaded by controlling the displacement uin, K¯T is equal to
    K¯T=[2Π¯θ122Π¯θ1θ22Π¯θ2θ12Π¯θ22].2.7
    If both eigenvalues of K¯T are positive for a given equilibrium configuration, then that equilibrium is stable; if either is negative then the equilibrium is unstable [1,39]. The symbolic toolbox in Matlab [40] was used to solve equation (2.6). Where closed-form solutions were not found, a numerical approach was adopted. In all cases, a numerical approach was adopted to calculate the eigenvalues of K¯T. If a change in stability was detected between adjacent solutions along the equilibrium path, then a Newton–Raphson process was used to pin-point the critical point.

    The equilibrium paths of the perfect structure with non-dimensional parameters α=1, β=0.5, γ=0.7 and η=0.05 are shown in figure 3a. Note that the paths are shown in the more intuitive parameter space of dimensionless input displacement u¯in, coupling parameter u¯couple=ucouple/L01 and output deflection u¯out=uout/L01. This structure has a fundamental path consisting only of pure axial compression in the horizontal strut. Once the critical point B1 is reached, the horizontal strut buckles, causing the structure to leave the fundamental branch. This leads to either positive or negative ucouple deflections. If ucouple<0, then the vertical strut is placed into tension and no further critical points are encountered. However, if ucouple>0, then the vertical strut is placed into compression, ultimately causing it to buckle at B2 and leading to uout deflections. These two instabilities are sequential, as B2 can only be reached via a secondary branch which bifurcates from the fundamental at B1. Figure 3a also shows that creating a pair of sequential instabilities from two super-critical branching point building blocks has resulted in interactive buckling: The tertiary path bifurcating from the critical point B2 is subcritical, and, as shown later, this only occurs when the two branching points B1 and B2 are in close proximity on the equilibrium path.

    Figure 3.

    Figure 3. (a) The equilibrium paths of the double branching point structure with the non-dimensional structural parameters α=1, β=0.5, γ=0.7 and η=0.05 and assuming no initial imperfections. The second bifurcation point, B2, lies on a secondary branch bifurcating from the fundamental path at B1, making this pair of instabilities sequential. Furthermore, these critical points interact to destabilize the post-buckling path between B2 and the limit point L1. In practice, this causes the structure to ‘snap’ into region A when B2 is surpassed. (b) The normalized internal energies, U¯, of the double branching point structure with respect to normalized arc-length, s¯, along the equilibrium path. The bold line bounding the shaded area from the top shows the total non-dimensionalized internal energy, U¯Tot, coloured blue when equilibria are stable and red when unstable. The proximity of B2 to B1 causes the two critical points to interact, leading to a reduction in the energy of the horizontal strut (U¯k1+U¯R1= the sum of the two purple regions) beyond B2 which the vertical strut (U¯k2+U¯R2= the sum of the two green regions) is unable to counteract, thus leading to instability.

    In order to gain insight into the structural response and the interaction of instabilities, we consider the non-dimensionalized internal strain energy U¯ of the structure and its constituents along the equilibrium paths. Figure 3b shows the structure’s non-dimensionalized internal strain energy U¯ as a function of normalized arc-length s¯=s/L01 along the equilibrium paths shown in figure 3a. The path shown is the lowest energy route through the parameter space that leads to buckling of both struts (i.e. from the origin to B1, to B2, to L1, to A etc.), so neither the unstable fundamental path beyond B1, the unstable primary bifurcated branch beyond B2, nor the primary path for u¯couple<0 are shown. The purple areas on these plots correspond to the energy of the horizontal strut, and the green areas to that of the vertical strut. These areas are ‘stacked’ on top of one another to represent the total non-dimensionalized internal energy, U¯Tot, at a particular s¯. This visualizes how the subcomponents of the structure exchange energy with one another as the equilibrium manifold is traversed. If one part of the structure releases energy faster than another part can store energy, then U¯Tot will decrease leading to an unstable equilibrium path. Critical points occur when U¯Tot is stationary in terms of s¯. This is because a critical point is, by definition, a point of neutral stability, where small2 perturbations away from the equilibrium configuration result in no change to the energy of the structure.

    Figure 3b shows that the fundamental path is stable until B1 as expected from figure 3a. At B1, the horizontal strut buckles and springs k1 begin to release strain energy. If only the second-order changes in s¯ are considered, then this decrease in U¯k1 is matched exactly by an increase in U¯R1 and U¯k2. Indeed, this is why B1 occurs at this particular point on the fundamental path; all the energy released by the linear springs k1 during a second-order perturbation away from the fundamental state is stored by springs R1 and k2. An important consequence of this is that B1 occurs at a higher u¯in than it would without the support of the vertical strut, because both springs R1 and k2 are available to store energy during such a perturbation. Higher-order effects come into consideration as the structure moves towards B2 along the primary bifurcated branch. In this case, the springs k2 and R1 are able to store energy at a faster rate than the k1 springs release energy, so B1 constitutes a local energy minimum and the path is super-critical. When B2 is reached, the energy landscape is again flat to second order. However, the buckling of the vertical strut means springs k2 cannot continue to store the energy released from springs k1 (as can be seen from the approximately constant height of the dark green band beyond B2 in figure 3b). This results in a lack of support for the horizontal strut’s axial springs, which must therefore be unloaded in order to maintain equilibrium, releasing energy rapidly (as indicated by the sudden drop in the light purple areas). The increases in U¯R1 and U¯R2 are unable to counter this decrease, so U¯Tot decreases, leading to a local energy maximum and an unstable symmetric branching point at B2. The path only re-stabilizes at L1, when the increases in energy from the two rotary springs are able to absorb the release of energy from the k1 springs. In practice, the structure does not follow this path quasi-statically, and ‘snaps’ dynamically from B2 to region A (cf. figure 3a). In a physical prototype of the model with mass and friction, the released energy would be converted to kinetic energy and then eventually dissipated by viscous effects.

    As alluded to above, the destabilization of the post-buckling path becomes less pronounced as their associated critical points become spaced further apart within the parameter space of the structure. Ultimately, if the two critical points are sufficiently well spaced then B2 becomes a super-critical bifurcation point and the post-buckling response of the structure is fully stable. This is the case in figure 4, which shows the equilibrium paths and energy contributions of the structure for the case where α=1, β=1, γ=0.7 and η=0.05, i.e. the same structural parameters as in figure 3b, only with a doubling of β. Note that this change does not affect the shape of the branch bifurcating from B1. However, it does affect its stability, and B2 is shifted further up this branch and away from B1 [41].

    Figure 4.

    Figure 4. (a) Equilibrium paths of the double branching point structure with α=1, β=1, γ=0.7 and η=0.05. The post-buckling path is fully super-critical so L1 has disappeared (cf. figure 3a). (b) Non-dimensionalized internal energy versus arc-length of the structure with the same parameters. Colour coding, etc. is as per figure 3b. The increased spacing of B1 and B2 has led to a smaller deficit in the horizontal strut’s energy (sum of purple regions) beyond B2, and therefore a stable equilibrium path.

    The energy breakdown depicted in figure 4b shows that the primary branch is stable as before, but the secondary branch bifurcating from B2 is now also stable, leading to a fully super-critical response. The increased separation of the two critical points means the horizontal strut has advanced further into its post-buckling regime when B2 is reached, so more of the strain energy stored in the horizontal strut has been transferred from compression in its axial springs (U¯k1) to bending in the torsional spring R1 and compression in the axial springs k2 (U¯R1 and U¯k2, respectively). This increased support for the k1 springs leads to a less drastic energy release when the vertical strut buckles. This can be seen by comparing the ‘dip’ in the purple areas beyond B2 in figures 3b and 4b. In the latter case, the vertical strut is able to make up for the energy deficit left by the horizontal strut, and B2 becomes a local energy minimum with a fully stable post-buckling path.

    (b) Branching point plus limit point structure

    In this section, we apply similar analysis techniques to a different pair of sequential instabilities; namely a limit point bifurcation in sequence with a branching point bifurcation. Again, we use a bar-and-spring model to explore the behaviour of this pair of instabilities. This model is shown in figure 5. The inspiration for this structure comes from von Kármán’s early work into cylinder buckling, where experiments were carried out on an Euler column supported at its mid-point by an arch [42]. For this reason, we refer to this bar-and-spring model as the von Kármán strut. The system consists of a bar-and-spring Euler strut whose centre node is attached to a modified spring-loaded von Mises truss [1]. Compared with the classic von Mises truss, this truss has an additional rotary spring, R2, at the top node. This additional rotational spring allows a more general snap-through response to be investigated—in its absence the von Mises truss building block always exhibits a bistable snap-through response, however the addition of spring R2 allows a monostable snap-through to be achieved.

    Figure 5.

    Figure 5. A bar-and-spring model featuring two building blocks: a branching point in sequence with a limit point. We refer to this as the von Kármán strut. The branching point bifurcation is given by a bar-and-spring Euler strut made up of the axial springs with stiffness k1 and a rotary spring with stiffness R1 as before. The centre of this strut undergoes post-buckling deflections ucouple in response to the enforced deflection uin. Buckling of this strut in turn loads the second building block: a von Mises truss made up of the axial springs of stiffness k2 and k3 and rotary spring with stiffness R1. This second building block gives the limit point bifurcation Output deflections, uout, are measured at the apex of the truss.

    The structure shares the same geometric DOFs as the double branching point structure (input displacement u¯in, coupling deflection u¯couple and output displacement u¯out), and its elastic properties can be fully described using α, γ, η, and the three additional dimensionless groups

    ζ=R2k1L012,μ=k3k1andθ02,2.8
    with parameters as indicated in figure 5 and where θ02 is the value of θ2 when the structure is unstressed. Due to the increased complexity of the problem, an in-house nonlinear FE code with a generalized path following solver [5] was adopted to explore the equilibrium paths of this structure, rather than the analytical approach used previously. This nonlinear FE solver offers a number of advanced capabilities not available in commercial FE solvers. Specifically, it can pinpoint critical points; switch onto any bifurcated branches emanating from critical points and trace the loci of critical points as a particular structural parameter is varied.

    Figure 6 shows the equilibrium paths of the von Kármán strut for two different sets of structural parameters as well as the corresponding energy versus arc-length plots. The addition of the von Mises truss to the bar-and-spring Euler strut clearly introduces a top-down asymmetry to the structure, and this is reflected in the equilibrium paths shown in figure 6a,c; the structure now undergoes asymmetric transcritical branching at point T1, as opposed to the symmetric branching at point B1 in the previous structure. Recall that the presence of the vertical strut in the double branching point structure has the effect of increasing the u¯in at which B1 occurs by providing linear support until B2, whereupon the vertical strut itself buckles. In the current structure, the von Mises truss also provides additional support to the horizontal strut, but its stiffness diminishes with increasing u¯couple. Therefore, the transverse support for the horizontal strut diminishes as soon as it buckles in the positive u¯couple direction, leading to an asymmetric (i.e. transcritical) branching point and an unstable post-buckling path for u¯couple>0. If the u¯couple<0 branch is followed from T1, the von Mises truss stiffens, leading to a stable post-buckling response. The destabilized branch can be seen in the internal energy versus arc length plot (figure 6b), in which the horizontal strut (purple areas) rapidly releases energy beyond point T1.3 The energy released by the horizontal strut is partly absorbed by the von Mises truss. However, the maximum energy storage capacity of this sub-structure is soon reached, and it snaps through at point ST. Only once this has occurred can the system re-stabilize at the limit point L1. Note that ST is the point at which the von Mises truss building block alone would become unstable if subjected to an enforced downward load at the ucouple node.

    Figure 6.

    Figure 6. (a) The equilibrium paths of the von Kármán strut for the case where α=1, γ=0.637, η=0.05, ζ=25×106, μ=1 and θ02=35. The branching and limit points interact strongly to create a single unstable portion of the path, between the trans-critical branching point T1 and the limit point L1. (b) The corresponding internal energy profile with respect to arc-length. The energy released by the horizontal strut (U¯k1+U¯R1= the sum of the two purple regions) when T1 is surpassed is greater than the amount of energy the von Mises truss can store before it snaps through at point ST (U¯k3+U¯k2+U¯R2= the sum of the green regions. Note that for this set of parameters U¯R2 is too small to resolve on the plot). As with the double branching-point energy plots, the unstable portion of the fundamental path beyond T1 is not shown, and neither is the post-buckling branch for u¯couple<0. (c) The same equilibrium paths when μ=0.5. The snap-through of the von Mises truss is well separated from the branching of the horizontal strut, creating two additional limit points, L1a and L1b. The loci of the three limit points as μ is increased from 0.5 are shown in black. The green triangle is the point where L1a and L1b meet and disappear. This occurs when μ=0.68 (d) The corresponding internal energy profile, showing that the energy released at T1 alone can no longer invert the von Mises truss. The total energy alone for the case when μ=0.68 is shown in fine red and blue lines for comparison. The two limit points L1a and L1b have coalesced at a cusp. If μ were increased even slightly further then the critical points would disappear entirely.

    It is possible to spread these sequential critical points further apart in a similar way to the double branching point structure. However, unlike in the case of the double branching point structure, the interaction between the critical points can never be fully removed, only reduced. This is the situation shown in figure 6c,d. The only change in the structural properties from the equilibrium paths shown in figures 6a is that μ has been reduced from 1 to 0.5. The path in figure 6c exhibits the same trans-critical branching point as in figure 6a. The only way to fully transform this to a symmetric branching point would be to set θ02=0, but this would also remove the limit point from the structure entirely. For the case shown in figure 6d, where μ=0.5, the energy released during the initial post-buckling of the strut is no longer sufficient to overload the von Mises truss. Here, the green segments create an energy surplus which can overcome the deficit caused by the purple segments, leading to a re-stabilization at L1a. Further loading is then needed to completely fill the energy reserves of the von Mises truss. When this happens, the path again destabilizes at L1b before finally re-stabilizing at L1 as it did before. Note that L1b is not the equivalent of point ST from figure 6a,b. The reason for this is that the horizontal strut does not impose a pure load control boundary condition on the von Mises truss. Instead it imposes a more complex load–displacement profile, which is dependent on its own configuration and component stiffnesses. This means that the location of the point at which the overall system destabilizes, i.e. L1b, is dependent on all elements of the structure, not just the von Mises truss. The loci of the limit points as μ is increased from 0.5 are shown in black on figure 6c. Note that L1a and L1b move closer together before meeting and disappearing at a cusp point marked by the green triangle, leaving a path which is unstable between T1 and L1 as for the path shown in figure 6a.

    3. Function from sequential, interacting instabilities

    In the preceding sections, sequential instabilities were defined and investigated using simple bar-and-spring models. In this section, we intentionally harness the features of sequential, interacting instabilities to achieve new functionality, demonstrating that interacting instabilities need not be thought of purely as a failure mode. We introduce key functional requirements for a passively actuated morphing structure and present engineering designs which meet these requirements using the sequential, interacting instabilities discussed in §2. The designs are investigated using nonlinear FE models.

    (a) A passively-actuated morphing structure

    A driving motivation for our work is the design of a passively actuated spoiler which can be mounted on the top wing skin of a commercial airliner, or other large-scale aircraft. The device should deploy into the airflow during extreme gust loading, dumping lift and thus alleviating the peak aerodynamic loading on the wing. This load reduction would allow the mass of the wing structure to be reduced, in turn reducing lift-induced drag and therefore fuel burn. The device must then stow itself once the gust has passed in order to avoid adding unwanted drag to the standard cruise configuration of the aircraft. Uniquely, the only input actuation to this device is the enforced compression from the wing, i.e. no other sensors, electronics or actuators are required to operate it, hence it is described as ‘passive’. In order to achieve this, the device will incorporate structural nonlinearity into its design and will exploit structural instabilities for deployment and stowage.

    Clearly, there are many practical considerations in the design and manufacture of such a device; however, at its core, it has three structural requirements. These are visualized in figure 7, which shows a schematic of the desired equilibrium path of the structure in terms of output displacement (uout, or how far the device protrudes into the airflow above the wing) versus input actuation (uin, or the amount of enforced compression from the wing). The three requirements are:

    1. stable stowed configuration: output deflection u{out} is insensitive to uin below a triggering threshold (uin, trig).

    2. deployment phase: the spoiler should ‘pop up’ with a large, rapid increase in u{out}.

    3. stable deployed configuration: output displacement u{out} again shows low sensitivity to uin to maintain deployed state.

    Figure 7.

    Figure 7. A schematic equilibrium diagram for a passively actuated ‘pop-up’ spoiler. In the stowed configuration, the structure exhibits almost no output deflection (in order to remain flush with the wing) until the input displacement uin reaches a critical value, uin,trig. Next, the spoiler deploys, exhibiting a sudden and large increase in output deflection from uout,trig to uout,dep. As uin is subsequently decreased below uin,stow the reverse occurs, and the spoiler rapidly stows.

    These requirements effectively demand a binary device, which is either fully stowed or fully deployed depending on the magnitude of the input actuation u{in}. This response is not only highly nonlinear, but requires a sudden change in configuration when u{in} reaches a critical level. However, a single structural instability may not be able to achieve the bespoke functionality required. For instance, a limit point structure can suddenly deploy as it snaps through, but there may exist significant and undesired output deflections before that occurs. Similarly, a super-critical branching point structure undergoes minimal deflections prior to buckling, but its post-buckling path is stable, implying that deployment deformations would be limited in magnitude and speed. By contrast, the sequential, interacting instabilities explored in §2 offer a much richer nonlinear response, including the sudden destabilization and ultimate re-stabilization demanded by figure 7 (cf. figures 3a, 6a and c). Therefore, sequential, interacting structural instabilities are explored as a means of meeting the design requirements. A further benefit of these structures is the building-block approach used in their construction, which provides a design methodology where two individual structural components with known behaviour can be combined to achieve a target output response. This methodology was used to synthesize the design concepts presented here, as well as understand and tailor their respective responses.

    Two preliminary conceptual design embodiments are developed for the morphing pop-up spoiler. Each is designed to be mounted to the upper surface of an aircraft wing, and uses the compression of the upper wing skin during gust loading as the input displacement for deployment. The two designs presented each use one of the sequential, interacting instabilities discussed in §2 to achieve the novel functionality required. As previously mentioned, this is achieved using a building block approach to synthesize the designs from two explicit structural components whose responses are known a priori.

    (b) Double branching point embodiment

    Figure 8a shows a candidate design for a morphing pop-up spoiler that uses two branching point instabilities in sequence to achieve the required functionality. The device is shown in plan view on the left, and it is envisaged that air would flow over it from the bottom to the top of the figure. The design is intended to be symmetric about the marked plane and consists of two building blocks, mirroring the double branching point bar-and-spring model. The first building block is referred to as the ‘spine’. This is simply a beam, which in the intended application would be attached to the top skin of the underlying wing by pinned joints at its ends. The compression of the wing skin would enforce a compressive displacement uin on the spine as shown in the figure. The application of this compression ultimately causes the spine to buckle, leading to ucouple deflections and the first branching point bifurcation. The spine is rigidly attached to the ‘fin’, a thin planar shell of tapered planform which is the second building block. When the spine buckles it enforces a rotation about the z-axis on the fin’s lower edge, which ultimately causes the fin to buckle out of the plane with deflections uout. The buckling of the fin is the second branching point bifurcation, and its post-buckled shape is intended to disrupt the impinging airflow, causing the flow to separate from the underlying aerofoil. In turn, this would lead to a reduction in the lift generated by the wing and a load-alleviating effect. It is shown below that the two branching point bifurcations in this structure are sequential, so this design functions in a conceptually identical manner to the double branching point bar-and-spring model.

    Figure 8.

    Figure 8. (a) A finite-element model of a candidate morphing structure which functions using two coupled building blocks (the spine and the fin) to create a double branching point sequential instability, mirroring the functionality of the bar-and-spring model in figure 2. The model is shown in its deformed configuration under applied input displacement uin from the wing. The intended flow direction is from bottom to top. The colour map pertains to magnitude of deflection from the undeformed state. (b) Equilibrium paths of the double branching point FE model for two different fin thicknesses, t. Imperfections have been seeded into the undeformed mesh, so the branching point B2 has become the limit point L2 (cf. figure 3a). The inset shows the same paths in uinuout space. The path for t=1.2mm exhibits all of the required features from figure 7, demonstrating that sequential, interacting instabilities can be used for functionality. (c) The structure’s internal strain energy versus arc-length plot when t=1.2mm. The sequential branching point instabilities interact strongly, and the energy increase from the buckling of the fin is unable to counteract the decrease caused by the buckling of the spine. This leads to a subcritical post-buckling response and rapid deployment. (d) The strain energy versus arc-length plot when t=2mm. The branching points have separated to the extent that they no longer interact; the strain energy increases monotonically and deployment is gradual and stable.

    The design concept was developed and explored by producing an FE model of it in the commercial FE package Abaqus [43]. This model is shown in figure 8a and, as mentioned above, represents only half of the full design which is intended to be symmetric about the labelled plane. This reduces the computational effort, but has the disadvantage of implicitly restraining any symmetry breaking deflection modes. The spine and the fin are discretized using 131 two-noded B31 beam elements and 1342 four-noded, reduced-integration s4r shell elements, respectively. Both parts of the model have the same isotropic material properties of E=70GPa and ν=0.3, which were chosen so as to reflect the properties of aluminium. The spine has two parts, one with a cross-section of 6×6mm, and the other with a cross-section of 10mm×10mm. The node where uin is applied is restrained from translating in the y-direction, and the whole spine is constrained to move only in the xy plane. The only unconstrained degree of freedom at the node where ucouple is measured is translation in the y-direction. Nodes on the edge of the fin at the symmetry plane are restrained from translating along the x-axis, as well as from rotating about the y-axis. A mesh sensitivity study was carried out and mesh density was found to have a negligible effect on the converged equilibria.

    Since there are no facilities in Abaqus to pin-point bifurcation points and switch branches, geometric imperfections were seeded into the undeformed mesh so as to trace the nonlinear equilibrium path of this structure. The imperfection field was generated by running a separate analysis in which a small y-axis deflection was applied to the mesh at the ucouple node, and a small z-axis deflection was applied at the uout node. The resultant deformations throughout the mesh were then seeded into the main model. If these imperfections were not included, then the model would continue to deform along its unstable symmetry-preserving fundamental path. Seeding such imperfections has the effect of ‘breaking’ any bifurcations, meaning the structure follows a single smooth path, rather than branching when it reaches a bifurcation point. The smaller the imperfections, the closer the broken path is to that of the imperfection-free structure [1]. For this reason, the magnitude of the imperfections was chosen to be as small as possible while still causing the structure to buckle, and to ensure that ucouple>0 and uout>0. The chosen imperfection field is by no means optimal, and other fields may yield a more desirable response. It was also found that, as expected for a subcritical branching point, the response was highly sensitive to the magnitude of initial imperfections, though no formal study on this was conducted. The equilibrium manifolds of this FE model are shown in figure 8b. The stability of the equilibrium states was determined by interrogating the *.MSG file generated by Abaqus for warnings of negative eigenvalues in the tangent stiffness matrix. Note that because imperfections have broken the bifurcations, the boundary between stability and instability at B2 in figures 3a and 4a has become a limit point, labelled as L2 in figure 8. The two curves in figure 8b each correspond to a model with different fin thickness. Varying the fin thickness, t, increases the bending stiffness of the fin, EI, which is found to have a similar effect on the behaviour of the structure as varying β has on the double branching point bar-and-spring structure4 [41]. Indeed, it can be seen from figure 8b that increasing the fin thickness reduces the degree of interaction between the branching points to the stage where the response is fully stable, as was the case in figure 4a. The same parallels can be observed by comparing the strain energy versus arc-length plots for the FE model (figure 8c,d) to those for the bar-and-spring model (figures 3b and 4b): When the fin thickness is small, the critical points interact strongly, and the energy contribution of the fin (green areas in figure 8) when it buckles is unable to overcome the deficit left by the spine (purple areas). The thicker post-buckled fin is able to store much more energy and provide much more support to the spine. This not only means that the fin contributes more to the overall energy, but also slows down the rate at which the spine releases energy, ultimately leading to a fully stable response.

    Viewing these paths in the uinuout plane (see inset in figure 8b) reveals that the path for t=1.2mm exhibits the same key features demanded by the desired path in figure 7. This demonstrates how sequential, interacting instabilities can be used to achieve the bespoke functionality required for a morphing pop-up spoiler. The difference in behaviour between the two cases also highlights the utility of ‘tuning’ the structure by controlling the separation of the critical points. One drawback of the design in its current form is that a number of stress concentrations in excess of 1000 MPa formed near the tip of the fin. This was not unexpected given the sharp corners in this region of the model, which are of course generally avoided in structural design. Nonetheless, reducing these stresses would be a key priority in the further development of this design, possibly through the addition of cutouts or fillets.

    (c) von Kármán strut embodiment

    It is also possible to use the philosophy of the von Kármán strut, i.e. a branching point in sequence with a limit point, in the design of a pop-up spoiler. Such a design is shown in figure 9, and as before consists of two discrete structures which each emulate one of the building blocks in the corresponding bar-and-spring model. The branching point bifurcation is again formed by an isotropic spine which is pinned to the wing at either end (points P). The spine is coupled at its mid-point (point M) to a curved shell, which is intended to follow the curved profile of the wing’s leading edge in its undeformed state. The shell is rigidly clamped along its own leading edge as labelled in figure 9a. When input deflections are enforced on the spine by compression of the top wing skin it ultimately buckles upward with ucouple deflections. The shell is constrained to move only in the xz plane at points C, so ucouple deflections cause the direction of its principle curvature to rotate, leading to uout deflections as shown in figure 9b. The flow direction is from the bottom-right to the top-left of the figure, so this deployed configuration is intended to disrupt the flow, leading to separation and a load alleviating effect. The shell is not isotropic, but is instead constructed from a bistable composite laminate based on [44]. This gives it a snap-through response when ucouple is applied, mirroring the functionality of the von Mises truss in the bar-and-spring model. Such a snap-through response would not be exhibited to the same extent by an isotropic curved shell. This limit point bifurcation is sequential with the branching point provided by the spine, so the design functions in a conceptually identical manner to the von Kármán strut.

    Figure 9.

    Figure 9. (a) An FE model of a pop-up spoiler using a branching point structure—the spine—in sequence with a limit point structure—a curved bistable shell. The shell is constructed from a bistable laminate according to [44]. The spine and shell are connected at the spine’s mid-point (point M), and the shell is fully clamped along its leading edge. (b) The same model in its deformed configuration, under the application of the enforced compression uin. The spine has buckled (ucouple deflections) causing the shell to snap-through, measured as uout. The colour map pertains to the magnitude of deflection from the deformed configuration.

    As before, the design was developed by modelling it in Abaqus, and the spine was again discretized using 150 isotropic B31 elements with the properties of aluminium. It has a rectangular cross-section of 5×4mm, with the weak axis aligned with the x-axis. The shell encloses an angle of 43 when unstressed, and was discretized using 3800 s4r elements, whose properties were defined by a composite laminate. This was constructed of plies with the material properties in table 1, and with anti-symmetric ply orientations [+45/45/0/+45/45]. These are the same properties as the laminate described in [44], although some additional properties required by Abaqus have had to be assumed based on those given in the cited paper. The boundary conditions described above were applied: the leading edge nodes are fully restrained; points C are constrained to the xz plane and points P restrained from translating in either the x- or y-directions. The rotational DOF about the z- and x-axes at the centre node of the spine at point M is also restrained. Note that the latter restraint is intended to prevent the spine from buckling into its second Euler mode, which does not cause snap-through of the shell. This restraint is therefore necessary for the model to function, but would perhaps be difficult to implement in a physical prototype. The spine and the shell are coupled using a tie constraint between their respective nodes at point M. Only translational DOFs were tied, so the shell and the beam are free to rotate relative to one another. A mesh sensitivity study was carried out and mesh density was found to have a negligible effect on the converged solutions. von Mises stresses were checked and were found to be below 300 MPa for realistic configurations of the structure.

    Table 1. Material properties of the plies which make up the shell composite laminate. These are based on the properties given in [44].

    E11 E22 ν12 G12 G13 G23 thickness
    26.6 GPa 2.97 GPa 0.4 1.39 GPa 1.39 GPa 1.39 GPa 0.3 mm

    The equilibrium paths and energy plots of this FE model are shown in figure 10a,b. Note that the paths shown are for the imperfection-free structure, unlike in figure 8b. This was achieved by first using dummy forces to perturb the model away from its perfect geometry while applying a large uin. The perturbing dummy forces were removed once the model settled into its post-buckling regime, leaving the model on its imperfection-free post-buckling branch. This could then be followed back toward T1 using a Riks step. This approach was taken so that the ideal behaviour around the trans-critical branching point T1 could be fully investigated, even though this is not the path an imperfect structure would follow in reality. Stability of the converged equilibria was again taken from the *.MSG file.

    Figure 10.

    Figure 10. (a) The imperfection-free equilibrium paths of the FE model shown in figure 9a, viewed in uinuout space (full three-dimensional view inset). As with the von Kármán strut, the structure buckles at a trans-critical bifurcation point T1, and the post-buckling path re-stabilizes at L1. The paths exhibit the key parameters required by a pop-up spoiler: minimal deflections before a triggering load, beyond which large, unstable deflections are encountered before the path ultimately re-stabilizes. (b) The corresponding internal energy versus arc-length plot. The reduction in the energy of the spine (Uspine) when it buckles exceeds the energy storage capacity of the shell (Ushell), causing it to snap through. (c) The paths of the same structure when a compliant connector is introduced between the spine and the shell (see inset), showing how the sequential instabilities have become separated. In this configuration, the snap-through of the shell occurs at L1b, well away from the buckling of the spine at T1. (d) The corresponding internal energy versus arc-length plot, showing that the spine releases almost no energy at T1, and this is easily absorbed by the connector (Uconnector) and the shell.

    The energy versus arc-length plot for this structure shows a strong qualitative resemblance to the von Kármán strut (figure 6b). As expected, the branching elements of the structure (purple areas) rapidly release energy beyond T1 because they are supported by a limit point structure, whose stiffness diminishes with increasing arc-length. The energy stored by the shell (green areas) is unable to compensate for this loss until L1 is reached, whereupon the structure re-stabilizes. Figure 10a shows that the equilibrium paths of this structure meet the requirements of our application (albeit with a shallower curve beyond L1), illustrating again that sequential, interacting instabilities can be used for functionality.

    As with the von Kármán strut when μ=1, there is a single region of instability between T1 and L1. In the bar-and-spring model, it was possible to introduce a re-stabilized region to the post-buckling path (cf. figure 6c) by reducing the stiffness of the spring k3, which connected the branching and limit point building blocks of the structure. In order to achieve this in the FE model, a short compliant member was placed between the mid-point of the spine and the connection point with the shell. The axial stiffness of this connector was around 10 000 times less than that of the spine. The spine had to be raised above the shell slightly in order to accommodate this member, which of course would not be practical if this device were fitted to an aircraft wing, so this modification was made purely to illustrate the tunability of the design in an intuitive way. If such compliance were needed in reality, then some sort of planar leaf spring would be more appropriate, or the spine could be placed inside the airofoil below the shell rather than above it. The modified structure, resulting equilibrium paths and energy arc-length plots are shown in figure 10c,d. These show that the branching point and limit point have indeed been separated by a stable region, leaving the portion of the post-buckling path between L1b and L1 unstable.

    One difference from the bar-and-spring model is that the trans-critical branching point appears to have been completely converted to a symmetric super-critical branching point by the introduction of compliance between the two building blocks. This is due to the large amount of compliance introduced, which makes the shell effectively rigid by comparison to the connector, so the latter provides near symmetric, linear support during the initial buckling of the spine. This in turn means the branching point at T1 is itself almost symmetric, to the point where its asymmetry cannot be seen at the scale shown in figure 10c,d.

    4. Conclusion

    Building on the burgeoning field of buckliphilia [45] or well-behaved nonlinear structures [5], we have demonstrated that interacting instabilities can be exploited for functionality, where traditionally they have been explicitly avoided in structural design due to subcritical post-buckling behaviour and high imperfection sensitivity. In particular, we have focused on a particular class of interacting instabilities which we refer to as sequential instabilities. We have defined these as a pair of instabilities where the secondary critical point is only to be found on a branch bifurcating from the primary critical point. It has been shown through two simple bar-and-spring models, analysed using energy methods, that the interaction of sequential instabilities can create structures with a much richer and more varied response than any individual instability could alone. Furthermore, the degree of interaction between these sequential instabilities could be ‘tuned’ by controlling their separation within the parameter space of the structure. The models were constructed using discrete ‘building blocks’ containing canonical limit point or pitchfork bifurcations. The response of these building blocks was known a priori, however their interactive mechanics led to behaviour that neither building block displays on its own. This building block approach meant the bar-and-spring models could be used to synthesize two mechanical designs for a particular application; namely a passive, morphing aerodynamic spoiler for installation on the wing of a commercial airliner. This device must ‘pop-up’ in response to excessive wing straining. This is but one application where the wider design space offered by interacting instabilities might be implemented. These two spoiler designs were intended to function using the same sequential, interacting instabilities as the bar-and-spring models on which they were based. FE models of the designs were used to verify that this was the case, and to demonstrate that their response could be tuned in the same way as the bar-and-spring models. However, the effect of imperfections was not explored in detail, though it was noted that the converged equilibria were highly sensitive to the magnitude of initial imperfections. The main design goal of these structures is their shape shifting capability, rather than their load carrying capacity as would be the case in the design of conventional structures. This places less emphasis on a precise prediction of buckling loads, however a formal quantitative characterization of this sensitivity should clearly be included in future studies of these structures. Furthermore, this sensitivity suggests that imperfections would need to be tightly controlled in any physical prototype if these designs were to function as intended. Similarly, the designs would have to respond robustly to any imperfections which could not be controlled during manufacture. A further consideration is the effect of aerodynamic loads on the deployment of these structures, and how effectively they can disrupt an impinging airflow to achieve a load alleviating effect. This will be the subject of future work. Furthermore, the concept of discrete building blocks was useful in conceiving these designs. However, a more continuous or blended structure may prove more desirable now that their principle of operation is understood.

    Data accessibility

    Supporting data are provided within this paper.

    Authors' contributions

    E.D.W.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, validation, visualization, writing—original draft, writing—review and editing; J.S. and R.M.J.G.: conceptualization, funding acquisition, methodology, supervision, writing—review and editing; R.M.J.G.: conceptualization, funding acquisition, methodology, software, supervision, writing—review and editing; A.P.: conceptualization, funding acquisition, methodology, software, supervision, writing—review and editing; M.S.: conceptualization, funding acquisition, project administration, supervision, writing—review and editing.

    All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

    Conflict of interest declaration

    We declare we have no competing interests.

    Funding

    E.D.W. is funded by an EPSRC DTP studentship (EP/T517872/1). R.M.J.G. is funded by the Royal Academy of Engineering under the Research Fellowship scheme (RF/201718/17178). J.S. is funded by The Leverhulme Trust through a Philip Leverhulme Prize awarded to R.M.J.G. The support of all funders is gratefully acknowledged.

    Acknowledgements

    The authors wish to thank Dr Enzo Cosentino of Airbus UK, who has supported this project through many fruitful discussions. The contributions of two anonymous reviewers were also gratefully received.

    Footnotes

    1 In some non-sequential instabilities, it may be the case that a critical point with an identical eigenvector to C2 (figure 1) may be reached by following the post-buckling path from C1 (the well-known Augusti column [1,25,27] is one such case). In these cases, the fact that C2 exists on the fundamental path in addition to the post-buckling path still means the pair of instabilities is non-sequential.

    2 Here, ‘small’ means that only first and second-order changes in s¯ are considered, as in fact both dU¯/ds¯ and d2U¯/ds¯2 are zero at B1, B2 and L1.

    3 The critical eigenvector is not tangential to the post-buckling path at a trans-critical branching point. This means that the energy versus arc-length plot is not stationary at T1, as it was for B1, B2 and L1 in figures 3b and 4b.

    4 Increasing t will also increase the in-plane stiffness of the fin, EA, which is analogous to increasing α in the bar and spring model. However, the effect of this variation in EA on the response is much less noticeable, because It3, whereas At. So, small changes in t have a much greater effect on I than on A.

    Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

    References