Volume and displacement indicators for an architectural structure

The volume (W) and displacement (Δ) indicators have been discovered by Philippe Samyn in 1997 to help the search for the optimal geometry of architectural structures.

A resistant structure is anything amorphous or living, which supports the forces to which it is normally subjected without breaking.

Characterised by its shape and its constituent materials, and three-dimensional by nature, the amorphous structure usually has a two-dimensional geometry, which is given a thickness, or a three-dimensional geometry (the three-dimensional structure). The latter is formed of a pair of two-dimensional structures on non-parallel planes or three-dimensional curved volumes as in the case for any living things, including a specific group: the shells (a three-dimensional surface with a thickness). An example of these is the structure of cars, boats or planes, or even human skulls, sea shells or the stem of a dandelion.

The geometry of most "architectural" structures (such as buildings or bridges) is twodimensional and it is essential to study this aspect, whether for aesthetic, commodity-related or economic reasons. Several criteria are therefore taken into account in its definition.

The study is limited to the quest of the geometry giving the structure of minimum volume.

The cost of a structure depends on the nature and the quantity of the materials used as well as the tools and human resoures required for its production.

Although technological progress has reduced the cost of tools and the amount of human resources required, and despite the fact that computerised calculation tools can now be used to determine the dimension of a structure so that the load it bears at every point is within the admissible limits allowed by its constituent materials, it is also necessary for its geometry to be optimal. It is far from simple to find this optimal point because the choice available is so vast.

Furthermore, the resistance of the structure is not the only criterion to take into account. In many cases, it is also important to ensure that it will not undergo excessive deformation under static loads or that it does not vibrate to inconvenient or dangerous levels when subjected to dynamic loads.

Volume and displacement indocators, W and Δ, discovered by Philippe Samyn in August 1997, are useful tools in this regard. This approach does not take into account phenomena of elastic instability. Indeed, it can be shown that it is always possible to design a structure so that this effect becomes negligible.

The objective is to ascertain the optimal morphology for a two-dimensional structure with constant thickness, which:

Each form chosen corresponds to a volume of material V (in m³) and a maximum deformation δ (in m). Their calculation depends on the factors L, H, E, σ and F. These calculations are long and tedious, they cloud the objective of finding the optimal form.

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