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Figure 3: Λ2 continuation; at the most efficient structure of Λ3 = 0.99993, the total volume of the

structure varies from 200 (Λ2 = 1) to 1000 (Λ2 = 5);

(a)

(b)

Figure 2: Λ3 continuation (a) Equilibrium surface of beam problem with fixed average area Λ2 = 1 in multi-parameter space. The black line represents the most efficient structure; (b) Top-down view of Figure (a);

Blue, stable; Red, one eigenvalue unstable; Green, two eigenvalues unstable

Embracing nonlinearities in structural designBradley Cox1, Rainer Groh1, Daniele Avitabile2 and Alberto Pirrera1

Introduction

Nonlinear phenomena are prevalent instructural and continuum mechanics, albeit notcommonly utilised. This general reluctance isdue to the lack of availability of sufficientlyrobust computational tools to solve thesecomplex problems. There appears to be littlequestion that the so-called incremental iterativemethods represent by far the most popularprocedures for the solution of nonlinearstructural mechanics. Of these, numericalcontinuation provides many advantages. Thenoteworthy, yet often overlooked, quality is itsapplication in optimising engineeringstructures.

Numerical continuation is coupled to anonlinear finite element analysis framework,where, for a benchmark nonlinear problem,arched beams are analysed using structuralbeam elements based on Timoshenkokinematics.

Future research

Numerical continuation

Results and discussion

Motivation

Model definition

Within the realm of nonlinear structuralanalysis it is generally understood that moststructures are somewhat overdesigned. This issimply to ensure that they respond linearly tothe externally applied loads. The motivationbehind the current research is to enableengineers to design new structures thatembrace well-behaved nonlinear deformationsthus potentially leading to significantly moreefficient structures, by removing redundant andunnecessary stiffness, and in turn mass.

Numerical continuation is a numerical methodused for computing solutions of a system ofparameterised nonlinear equations. Themethod is independent from the source ofnonlinearity - in terms of structural analysis - itcan be geometric, inertial, material, or it canarise from other physical phenomena such ascontact, damage, delamination, buckling drivendelamination, etc.

The numerical method offers a number ofadvantages. Most significantly, it allows for asystematic exploration of the design space, andit makes typically cumbersome and expensivenonlinear calculations, often requiring oneroususer intervention, relatively straightforward.

In essence, numerical continuation is builtaround the idea of the addition of someaugmenting conditions to the equilibriumequations. These conditions, specifying subsetsof equilibrium states with certain properties,will further allow the addition of the extracontrol parameters required [1] in any analysis.

Problem formulation

1 ACCIS, Queen’s Building, University of Bristol,

Bristol, BS8 1TR, UK2

School of Mathematical Sciences, University of

Nottingham, NG7 2RD, UK

The finite element equilibrium equations areformally expressed as

where 𝒇, the internal forces, are a function ofboth the current displacements, 𝒅 , andpossibly some other properties of the model,𝚲. This additional property parameter could intheory be anything from a change in geometry(area, length, height, etc.), to a change inmaterial properties (E, G, etc.) or auxiliaryload cases.

Model #1: Arched beam, Λ1

Model #2 & #3: Arched beam, Λ2 & Λ3

In order to illustrate the capabilities of numericalcontinuation a simple arched beam problem hasbeen evaluated.

Here, the vector 𝚲 = {Λ1, Λ2, Λ3, Λ4} comprises ofonly four parameters. The parameters are definedas follows:• Λ1 Externally applied load to centre-point of

arch.• Λ2 Average cross-sectional area (used to

evaluate total volume).• Λ3 Quadratic distribution of volume along

beam length (symmetric).• Λ4 Arch height.

This model delivers classic snap-through curvesas illustrated in Figure 1a.

The models presented herein are based on onefundamental load-displacement curve, where theload-factor 𝜆 = Λ1 is the first parameter evaluated– all other parameters remain fixed. This isnecessary due to the process involved incontinuing the parameters along an equilibriumpath. The solutions at each time-step on thisfundamental load-displacement path act as aninitial guess i.e. starting positions for theevaluation of other parameters.

Parameters Λ2 and Λ3 are once again solvedindependently of one another, but both arerelated to the change in cross-sectional areaalong the beam length, as illustrated in thefollowing expression:

Λ3 is physically limited to 0.99995 ≤ Λ3 ≤ 1.0002 toavoid elements containing zero or negativevolume (see Figure 1). Λ3 = 1 corresponds to aconstant cross-sectional area along the beamlength. Λ2 relates directly to the average area ofthe beam: when divided by the overall length it isused to evaluate the overall volume of the beamand hence mass.

References

To expand on our understanding of more complexnonlinear structures and to undertakecomprehensive imperfection sensitivity analyses.

Figure 1: Beam problem schematics; (a) Free-body diagram; (b) Parameter Λ3 with a maximum beam thickness at the centre; (c) Parameter Λ3 with the

maximum thickness at either end.

[1] C. Pacoste, A. Eriksson, A. Zdunek, Parameter dependence in the critical behaviour of shell structures: a numerical approach, IASS-IACM 2000

Figure 2a illustrates an equilibrium surface inwhich each load-displacement curve is solvedvarying the distribution of volume along thelength of the beam. Each curve corresponds to afixed load.

The most efficient design (Figure 2a: blackcurve) is found by evaluating each point on thesurface to find the best ratio of load-to-displacement. This design is found at Λ3 = 0.99993;which incidentally is the solution at which the first

bifurcation point and the first limit pointcoincide; this is illustrated graphically in Figure2b as the point where all three colours coincide.

The design of the arch is further evaluatedwith the continuation along parameter Λ2 whichessentially alters the solution into a 4th

dimension (see Figure 3). In this instance theequilibrium surface, and hence design space, isexpanded. Figure 3 can be used to designminimum weight structures for any given load orrequired displacement. The most importantfeature is the blue region. These solutionscorrespond to stable, and hence real, solutionsthat are physically observed.