Keywords: conceptual design, volume optimisation, morphological indicators, genetic algorithms, trusses, stiffness, strength, static displacements.

Within the framework of sustainable development we strive for constructions with a minimum volume of material. When we only consider the criteria of resistance and buckling, [1,2] and prove that even at the stage of conceptual design a clear hierarchy among the different typologies can be established by using morphological indicators (MI). Up to now, stiffness constraints have never been considered in the use of MI. Only the resistance criterion was met with volume as the objective function. This strategy is called design for strength and often results in lightweight structures with a problematic lack of stiffness, which implicates a non-negligible volume increase to meet the (imposed) stiffness criteria. In this paper we investigate the influence of the stiffness constraints on the optimal design by combining MI and genetic algorithms (GAs). Only the upper limit on the static displacement is considered.

An optimisation process that involves this stiffness constraint at the stage of the conceptual design is developed in [3] and leads to an optimum with a reduced need to change the structure drastically afterwards. Nevertheless, the commonly used classic structural typologies (arc, beam and truss) still present a lack of stiffness. Based on those conclusions, one can decide to enlarge the search space for new, less compliant solutions. In this paper we limit ourselves to trusses.

Using MI a truss optimisation method is developed in which each construction is coupled with a dimensionless number. The calculations for classic trusses is undertaken analytically. The designer can choose from a list of already known solutions, (e.g. Warren, Howe, Pratt). For more complex loads and not a priori known truss topologies this method requires much calculation time. Since we want to enlarge the possible truss topologies, a logic link between GA and MI is established: the mean criterion for an individual to survive is to have a small volume indicator. After a number of generations an optimum arises under the given load condition. When considering the stiffness constraint(s) the fitness of an individual strongly depends on (not) violating those constraints. The optimisation process is implemented in the design strategy considering the upper limit on the static displacements and the search space is gradually enlarged.

Some essential, time saving guidelines at the conceptual design stage arise from this study: in general one can say that the optimal number of panels and the corresponding slenderness must increase for higher buckling sensitivity of the compression bars, in order to reduce their buckling length. The buckling sensitivity of a truss (with given material and form factor of the bars) can be assessed by means of the structural index . The smaller this index, the larger the buckling sensitivity of the truss [4]. This index clearly demonstrates the scale sensitivity of a design problem in relation to buckling.

On the other hand, the upper limit on static displacements is a dimensionless and a scaleless constraint (expressed as a maximum value of the displacement indicator). The results are independent of the length of the problem, although we observe that extra material against buckling also provides extra stiffness against displacements and vice versa. However, this influence is limited; and as a consequence the scale effect is quasi-inexistent. Moreover, the stiffest trusses show a small number of panels and smaller optimal slenderness. It can be stated that a design problem with strict displacements constraints presents an optimum solution which is different from the optimum found against buckling.

The comparison of a large amount of truss types within a manageable calculation time (thanks to a GA-MI combination) in a search to stiffer solutions, confirms the superiority of classic Warren truss. Only for small slendernesses a volume gain is obtained by lowering the truss height close to the supports (i.e. a slightly parabolic shaped upper chord). More 'exotic' alternatives do not provide structurally more efficient trusses. This is certainly confirmed from the point of view of assembly and production costs.

1

P. Samyn, "Etude Comparée du Volume et du Déplacement de Structures Isostatiques Bidimensionelles sous Charges Verticales entre Deux Appuis , Vers outil d'évaluation de prédimensionnement des structures" (Tome 1 à 4), PhD thesis, Université de Liège, Belgium, 1999.

2

P. Latteur, "Optimisation et Prz'edimensionnement des Treillis, Arcs, Poutres et Câbles sur Base d'Indicateurs Morphologiques" (Tome 1 à 4), PhD Thesis, Vrije Universiteit Brussel, Belgium, 2000.

3

T. Vandenbergh, "Influence of stiffness constraints on optimal design of trusses using morphological indicators", Proceedings for The Third International Conference on High Performance Structures and Materials, Ostend, 2006. doi:10.2495/HPSM06004

4

J. Van Steirteghem, "A Contribution to the Optimisation of Structures Using Morphological Indicators: (In)Stability and Dynamics", PhD Thesis, Vrije Universiteit Brussel, Belgium, 2006.

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