Space structure configurations are elegant and impressive but, unless one is equipped with suitable conceptual tools, they are rather difficult to generate. The emerging branch of geometry which is called 'formex configuration processing' provides the conceptual tools that are needed for convenient handling of space structure configurations. The concepts of formex configuration processing allow innovative structural engineers and architects to make full use of their creative power in evolving imaginative as well as economical space structure forms.

In the context of formex configuration processing, the term 'configuration' is used to mean an 'arrangement of parts'. The assembly of all the elements of a structure, for instance, is a configuration. The most common usage of the term configuration is in reference to a geometric composition consisting of points and/or lines and/or surfaces.

A configuration may be described using a numerical model (that is, an arrangement of numbers). In particular, the internal computer representation of a configuration is bound to be in terms of a numerical model. The term 'configuration processing' is used to mean the 'creation and manipulation of numerical models that represent configurations'. In particular, the term 'formex configuration processing' is used to mean configuration processing with the aid of the concepts of 'formex algebra'. Formex algebra is a mathematical system that provides simple and elegant conceptual tools for the representation and processing of configurations. The basic ideas from which formex algebra has emerged were evolved in the early seventies [7] and the first textbook on the subject appeared in 1984 [8]. A concise description of the current state of the concepts of formex algebra is given in Ref. 9. A convenient medium for using the concepts of formex configuration processing is the programming language 'Formian' [9]. The use of this programming language is illustrated through an example.

Consider the dome configurations of Figs 7b to 7h. Domes of this kind are referred to as 'scallop domes' and are obtained by 'arching' the segments of a normal dome in various ways [10]. Now, suppose that one wants to produce a scallop dome from the basic diamatic pattern of Fig.7a. A convenient way of approaching the problem is to write a 'formex formulation' that can generate all the possible configurations of interest with parameters representing the varying features. Such a formex formulation is shown in Fig. 8. The formulation describes the disposition of the elements of the dome and the coordinates of its nodal points using the notation of formex algebra.

A reader who is not familiar with the concepts of formex algebra needs a time investment of a few days to learn about the ideas of formex algebra before the implications of the formulation can be followed in detail. However, the inclusion of the formulation of Fig. 8 at this point is only intended to give an indication of the 'general look' of a formex formulation. Indeed, the following discussion regarding the use of the formulation of Fig. 8 does not depend on the understanding of the details of the formulation. The only point that needs to be noticed, however, is that the first line of the formulation assigns values to four parameters a, g, m and p.

Figure 7: Examples of domes configurations represented by a generic formex formulation

In an actual design situation such as the case under consideration, one is not always completely sure about all the required details at the outset. It is, therefore, prudent to formulate the problem 'generically' (that is, in terms of parameters), so that the effects of variations in different features can be examined easily.

In the formulation of Fig. 8, parameter a (standing for 'amplitude') represents the 'rise' of the segmental arches. The value of a for the dome of Fig. 7b is chosen to be 1 unit of length (the radius of the dome is 10 units of length). A variant of the dome of Fig. 7b is shown in Fig. 7c, where the amplitude is increased from 1 to 2.

The number of segments into which the dome is to be divided for scalloping is governed by the value of parameter g (standing for 'gauge angle'). For the dome of Fig. 7c, the gauge angle is 60° and, therefore, the dome has 6 arched segments (to make a full circle). For the dome of Fig. 7d, the gauge angle is chosen to be 30° resulting in 12 arched segments.

The parameter m (standing for 'mode') determines the form of the arches. Mode 1 corresponds to a parabolic arch form as used for the domes of Figs 7b to 7d. Mode 2 corresponds to a sinusoidal arch form as shown in Fig. 7e. The parameter p (standing for 'prominence') controls the 'horizontal curving' of the segments. When prominence is equal to zero then there will be no horizontal curving. A positive prominence will create 'outward curving' of the segments, as in Fig. 7f, and a negative prominence will give rise to an 'inward curving', as in Fig. 7g. Fig. 7h shows a scallop dome which is identical in every respect to the dome of Fig. 7d except for the presence of prominence.

With the formulation of Fig. 8 working in Formian, it will be easy to examine the various possibilities and chose the combination of parameters that gives rise to the best solution. For viewing the configuration that corresponds to a choice of parameters, all that is required is to enter values for the parameters and the required configuration will appear on the Formian screen after a few seconds. The formulation of Fig. 8 may also be used to generate data for structural analysis. The data can be prepared by Formian in a suitable format for input to a structural analysis package such as ABAQUS, LUSAS or SAP. Also, information about the configuration may be sent, through Formian, to a graphics package such as AutoCAD or Corel for further processing.

The above example of formex configuration processing relates to a lattice space structure. However, a similar approach may be used for continuous space structures. In such a case, the configuration to be formulated will be a finite element mesh that represents the continuous structure.


References

• [7] Nooshin, H. Algebraic Representation and Processing of Structural Configurations, International Journal of Computers and Structures, June 1975
• [8] Nooshin, H. Formex Configuration Processing in Structural Engineering, Elsevier Applied Science Publishers, London, 1984 (Obtainable from Chapman & Hall Publishers)
• [9] Nooshin, H and Disney, P. Formian 2, Multi-Science Publishing Co Ltd, 1997
• [10] Nooshin, H, Tomatsuri, H and Fujimoto, M. Scallop Domes, Proceedings of the International Symposium on Shell and Spatial Structures: Design, Performance and Economics, Singapore, November 1997.