Keywords: structural optimisation, truss, dimensionless numbers, scale, self weight, buckling.

This paper introduces the concept of scale into the morphological indicators. Morphological indicators were first touched in references [1] and [2], but it was in reference [3] that their full potential was first established. Morphological indicators are dimensionless numbers that represent a physical property of structures. The two most important morphological indicators are the indicator of volume and the indicator of displacement. In this paper we will work with the indicator of volume W. It is the volume of an isomorphic structure with unit span, with at least one section of each element dimensioned on its unit allowable stress, subjected to a load system with a unit resultant. This number decouples the efficiency of a structure from its scale. Indeed, in its simplest formulation, the indicator of volume only depends on the slenderness of the rectangle in which the structure is inscribed. If phenomena like buckling, self weight and deck weight (for trusses) are taken into account [4], the indicator of volume will no longer be independant of the scale of the problem. However, the explicit study of the influence of the scale on the optimal design has not been performed yet.

In this paper, three phenomena that are influenced by the scale are studied: buckling, self weight and deck weight. Buckling sensitivity is proportional to the square root of the span L. Self weight contributions are proportional to the span L and deck weight is also proportional to the span L, from the moment the height of the deck reaches its maximum height. Their effect is illustrated with two examples: a beam and a two mesh warren truss. The value of their indicator of volume increases as expected with span, until a limit span is reached. One way to design better warren trusses is to let the mesh number vary. This will generate better results, because buckling lengths will decrease as well as deck spans. An alternative lay-out is the so-called multiwarren, where the deck beams of the master warren is a warren truss. This process can be repeated until deck contributions are minimised. This yields better results than simply varying the mesh numbers of a warren truss.

The conclusion of the examples is that scale effects have an important impact on the optimal design. Future work should incorporate other important phenomena, like dynamics [5] and the weight of connections. In stead of solving buckling globally (through the choice of a topology), it can also be solved locally (by using cable stayed columns [5]).

1

H. Cox, "The Design of Structures of Least Weight", Pergamon Press, Oxford, 1965.

2

W.S. Hemp, "Optimum Structures", Clarendon, 1973.

3

Ph. Samyn, "Étude Comparée du Volume et du Déplacement de Structures Isostatiques Bidimensionnelles sous Charges Verticales entre Deux Appuis", Ph.D Thesis, Université de Liège, Belgium, 1999.

4

P. Latteur, "Optimisation et prédimensionnement des treillis, arcs, poutres et câbles sur base d'indicateurs morphologiques", Ph.D Thesis, Vrije Universiteit Brussel, Belgium, 2000.

5

J. Van Steirteghem, "A Contribution to the theory of Morphological Indicators: (In)stabilities and Dynamics", Ph.D Thesis, Vrije Universiteit Brussel, Belgium, 2006.

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