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",Philippe Samyn, Étude de la morphologie des structures à l'aide des indicateurs de volume et de déplacement, Académie royale de Belgique, Bruxelles, 2004, 482 p; www.samynandpartners.com (for online e-book), (). since the reference thesis,Philippe Samyn, Étude comparée du volume et du déplacement de structures bidimensionnelles, sous charges verticales entre deux appuis vers un outil d'évaluation et de prédimensionnement des structures, Tome I: Mémoire, 175 p.; Tome II: Annexes, 184 p.; Tome III: Figures, 197 p. (4th of July 1999); Tome IV: Epilogue, 33 p. + 14 figures (1st of December 1999). PhD Thesis in Applied Sciences, University of Liège. 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).

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}).

In material mechanics and for the structural elements under a specific loading case, the factor {Ge} corresponds to the "form factor" for elements of continuous section out of a solid material (without voids).

The constituting material might however present a microstructure with voids. This cellular structure enhances than the form factor, whatever the loading case.

The factor {Ma} characterizes a material which efficiency might be compared to another for a given loading case, and the independently of the form factor {Ge}.

The indicators W = σV/FL and Δ = δE/σL just defined, characterize the macrostructures, while the same notations and symbols in small letters, w = σv/fl and Δ = δE/σl, refer to the structural element.

The figure 1 gives the values of W and Δ for the structural element subject to traction, compression, bending and shear. The left column relates to the limitation of stress and the right column to the limitation of deformation. It shows the direct relation of W to {Ge}{Ma} as:

, thus

and

or for given dimensions and loading case.

Then, as W and Δ depend only on :

and:

which for a given loading case, is the specific weight of a macrostructure per unit of force and length, depending only from the geometry through L/H, and the materials though σ/ρ.

Wρ/σ includes thus, the material factor {Ma} (ρ/σ and ρ/E for tension and compression without buckling, ρ/E1/2 for compression limited by buckling, ρ/σ2/3 and ρ/E1/2 for pure bending, ρ and ρ/G for pure shear) and the form factor {Ge}.

All other factor being equal, a cluster of tubes with a diameter H and a wall thickness e, compared to a solid bar of equal volume in a material characterized by ρ, σ, E et G, presents an apparent density ρa = 4k(1 − k)ρ with k = e/H, allowable stress σa = 4k(1 − k)σ,

The Young's modulus is and the shear modulus is .

Thus

and

This explains the better performances of lighter materials for structural elements subject to compression or bending.

This indicator allows to compare the efficiency of macrostructures including geometry and material.

It echos the work of M.F. Ashby: "Materials Selection in Mechanical Design" (1992).M.F. Ashby, Materials Selection in Mechanical Design, 311 pages, 1997. Butterworth-Heinemann edition, division of Reed Educational and Professional Publishing Ltd. Oxford (first edition in 1992 by Pergamon Press Ltd), UK. He analyses {Ge} and {Ma} separately as, for his studies, {Ma} relates to a large amount of the materials physical properties.

Different and complementary, it can also be placed alongside work carried out since 1969 by the Institut Für Leichte Flächentragwerke in Stuttgart under the direction of Frei Otto and now Werner SobekK, which refers to indices named Tra and Bic.Il Publikationen: Institut Für Leichte Flächentragwerke, Universität Stuttgart, plaffenwaldring, 14, 70569 Stuttgart; (Current director: Prof. Dr. Ir. Werner Sobek). tel 00.49.711.685.35.99 – fax 00.49.711.685.37.89 The Tra is defined as the product of the length of the trajectory of the force Fr, (causing the collapse of the structure) onto the supports by the intensity of this force, and the Bic is the relationship of the mass of the structure with Tra.

Since ρ* is the density of the material (in kg/m3), and α is, like W, a constant depending on the type of structure and the loading case:

therefore, with stress reached under

and as

Unlike W, which is dimensionless, Bic is expressed in kg/Nm. Therefore, depending on the material, an independent comparison of different morphologies is not possible. Despite the abundance of their works, none of them mention W and its relationship with L/H.

It appears that only V. Quintas RipollValentin Quintas Ripoll, Pro. Titular Dep. Estructuras de Edificación E.T.S Arquitectura. Universidad Politécnica de Madrid, Sobre el teorema de Maxwell y la optimización de arcos de cubierta, Informes de la construcción, Vol 40, n°400, marzo/abril 1989, pages 57 à 70, Madrid.,Valentin Quintas Ripoll, Sobre las formas de minimo volumen de las celosías de sección constante, Informes de la Construcción, Vol 43, n°418, marzo/abril 1992, pages 61 à 77, Madrid. and W. Zalewski and St. KusW. Zalewski, St. Kus, Shaping structures for least Weight, proceedings of the I.A.S.S International Congress on Shells and Spatial Structures, Stuttgart, 1992, pages 376 à 383. mentioned the volume indicator W without examining it in depth.

Validity limits of W and Δ

In general, the second order effects have very little influence on W, but they can have a significant impact on Δ. W and Δ therefore also depend on E/σ. The shearing force T may be crucial in the case of short and continuous elements, subject to bending so that W does not fall below a given value, regardless of the reduction of the slenderness L/H. However, this limitation is very theoretical because it is always possible to remove it by transferring material from the flanges to the web of the section, close to the supports. The stress σ to which the structure can be subjected depends on the nature, the internal geometry, the production method and the implementation of the materials, as well as several other factors, including the dimensional accuracy of the actual construction, the nature of the connections of the components or their fire resistance, but also the skill with which the geometry of the structure is designed to cope with elastic instability. Pierre Latteur,P. Latteur, Optimisation des treillis, arcs, poutres et câbles sur base d'indicateurs morphologiques – Application aux structures soumises en partie ou en totalité au flambement, Tome I: Thesis, 328 p.; Tome II: Thesis appendix, 12 p.; Tome III: Appendix to chapter 2, 432 p. (May 2000). Ph.D. Thesis in Applied Sciences, Vrije Universiteit Brussel. who discovered the buckling indicator, studied the influence of elastic instability on W and Δ.

In this regard, the existence of anchoring points of an element in traction may reduce the apparent permissible stress to the same level as the reduction necessary to take into account a moderate level of elastic instability. The influence on W of the buckling of the compressed parts on one side and of the anchoring points at the extremities of an element in traction on the other side is analysed on pages 30 to 58 in the « reference book ».