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.
Objective 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 resources 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 indicators, 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. It can indeed be shown that it is always possible to design a structure so that this effect becomes negligible.
The indicators The objective is to ascertain the optimal morphology for a two-dimensional structure with constant thickness, which:
fits in a rectangle of pre-determined dimensions, longitudinal L and horizontal H, expressed in metres (m); is made of one (or several) material(s) with a modulus of elasticity E, expressed in Pascals (Pa), and bearing a load at all points within its allowable stress(es) σ, expressed in Pascals (Pa); is resistant to the maximum loads to which it is subjected, in the form of a "resultant" F, expressed in Newtons (N). 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. It is, nevertheless, possible to overcome this problem by setting each factor to unity: while all other characteristics remain the same. Length L is therefore set to 1m, H to H/L, E and σ to 1Pa, and F to 1N. This "reduced" structure has a volume of material W= σV/ FL (the volume indicator) and a maximum deformation Δ = Eδ / σL (the displacement indicator). Their main characteristic is that they are numbers without physical dimensions (dimensionless) and their value, for every morphology considered, depends only on the ratio L/H, i.e. the geometric slenderness ratio of the form. This method can easily be applied to three-dimensional structures as illustrated in the following examples. The theory related to the indicators has been taught since 2000, and among other institutions, at the department of Civil Engineering and of Architecture at the Vrije Universiteit Brussel (VUB; section "material mechanics and constructions") leading to research and publications under the direction of Prof. Dr. Ir. Philippe Samyn (from 2000 to 2006); Prof. Dr. Ir. Willy Patrick De Wilde (from 2000 to 2011) and now Prof. Dr. Ir. Lincy Pyl. The "reference book", since the reference thesis, reports the developments of the theory at Samyn and Partners as well as the VUB, up to 2004. The theory is open to everyone who wants to contribute, W and Δ being to be calculated for any resistant structure as defined in paragraph 1 here above. Progresses in material sciences, robotics and three dimensional printing, lead to the creation of new structural forms lighter than the lightest known today. The geometry of minimal surfaces of constant thickness in a homogeneous material is, for example, substantially modified when thickness and/or local allowable stress are varying.
Macrostructure, structural element, microstructure and material The macrostructures considered here may be composed of "structural elements" which material presents a "microstructure". Whether searching to limit the stress or the deformation, macrostructure, structural element and microstructure have each, a weight Vρ, when ρ is the volumic weight of materials, in N/m3, function of the solicitations {F0} (for "force" in général) applied to them, of their size {L0} (for length or "size" in general), of their shape {Ge} (for geometry or "shape" in general), and of their constituting material {Ma} (for "material" in general).
V ρ ÷ { F 0 } { L 0 } { G e } { M a } {\displaystyle V\rho \div \{F_{0}\}\{L_{0}\}\{G_{e}\}\{M_{a}\}}
It can also be expressed as shape and material ({Ge}{Ma}) defining the weight (Vρ) for the structure of a given size under given force ({F0}{L0}).
V ρ { F 0 } { L 0 } ÷ { G e } { M a } {\displaystyle {\frac {V\rho }{\{F_{0}\}\{L_{0}\}}}\div \{G_{e}\}\{M_{a}\}}
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