Proceedings of the IASS Annual Symposium 2016 “Spatial Structures in the 21st Century” 26–30 September, 2016, Tokyo, Japan K. Kawaguchi, M. Ohsaki, T. Takeuchi (eds.)
Some Morphological Aspects of Configurations H. NOOSHINa and OA. SAMAVATI* * Consulting Engineer No. 2, Unit 2, Vanak St., Vanak Sq., Tehran, Iran
[email protected] a
Emeritus Professor, University of Surrey, UK
Abstract In the present paper, to begin with, the attention is focused on a number of basic characteristics of the configurations. These characteristics concern such morphological aspects as ‘curviance’ and ‘connectivity’. The term ‘curviance’ means the ‘way in which a configuration is curved’ and the term ‘connectivity’ means the ‘way in which the components of a configuration are interrelated’. The morphological characteristics discussed in the paper are then used in considering the constitution of a proposed classification for the lattice spatial structural forms. In this classification system, it turns out that the ‘curviance’ is the main factor that governs the constitution. Also, at the next level, for the configurations that have the same curviance, it is the pattern that governs the classification. Then comes such aspects as the ‘geometric proportions’ involved and the ‘frequency’ of the pattern that are considered as the distinguishing features for the classification. Keywords: Structural morphology, Classification of structural forms
1. Introduction Successful structural design demands close collaboration of a number of specialists. These include architects, structural engineers, geotechnical engineers, etc. The teamwork and the inclusion of essential specialisations become even more important when dealing with major spatial structures. Also, an aspect of the design which is always crucially important is the choice of the ‘structural form’. The time and energy spent initially in finding a good ‘general form’ for the structure will often prove to be well worth the effort in producing satisfactory final results. Therefore, the increasing attention, in the recent years, towards branches of knowledge related to structural shapes and forms is well justified. In particular, ‘structural morphology’ and ‘configuration processing’ are subjects that have attracted much research and development. ‘Structural morphology’ is the field of knowledge for the study of ‘structural forms’ and ‘configuration processing’ is the branch of knowledge providing conceptual tools for the solution of morphological problems, as well as the generation and manipulation of configurations.
2. Properties of configurations This section is concerned with the description of different properties of (structural) configurations. The main purpose is to help with the accuracy and clarity of communication of information and description of ideas related to ‘structural morphology’.
Copyright © 2016 by H. NOOSHIN, OA. SAMAVATI Published by the International Association for Shell and Spatial Structures (IASS) with permission.
Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
2.1. Connectivity A fundamental property of any configuration is its ‘connectivity’. Connectivity is the description of the manner in which the components (elements) constitute the configuration. Connectivity is not concerned with such aspects as length, angle, area and volume. As far as connectivity is concerned, all that matters is how the components of the configuration are related together. If a configuration is assumed to have been made from highly elastic rubber bands (and/or rubber sheets), then any amount of stretching and/or distortion of the configuration will not change the connectivity, as long as no component is added or removed and as long as no connection is added or removed. The term ‘compret’ or ‘topology’ may also be used instead of ‘connectivity’. For example, consider the configuration shown in Figure 1a. This is a lattice diamatic dome with a span of 30 m, a central rise of 9 m, six sectors and a triangulated pattern with frequency 4. To show the sectors clearly, the elements on the boundaries of sectors are shown in thicker line. For details regarding diamatic domes, see Nooshin and Disney [2], Section 2.5.
Figure 1. Three different diamatic domes with the same connectivity
Figure 1b represents another lattice diamatic dome which is identical to that of Figure 1a in every respect, except for its central rise which is 3 m, that is, a third of the rise of the dome of Figure 1a. Also, Figure 1c represents a dome obtained by ‘scalloping’ the dome of Figure 1b, see Nooshin et al. [5]. The domes in Figures 1a, 1b and 1c have, obviously, major differences. However, all the three domes have exactly the same connectivity (compret). To wit, the differences between these domes do not affect the number of elements and the manner in which the elements are connected together. Figures 2a, 2b and 2c are also diamatic domes similar to those of Figure 1, with the only difference being the frequency of the subdivisions. This frequency is 4 for the domes of Figure 1 and 6 for the domes of Figure 2. Again, all the three dome configurations in Figure 2 have exactly the same connectivity. However, the connectivity of the domes of Figure 2 is different from that of the domes of Figure 1. Seemingly, there are similarities between the forms of the domes of Figures 1 and 2. However, the connectivity of the domes of Figure 2 is different from that of the domes of Figure 1, since, in spite of the similarities, they have different numbers of elements.
Figure 2. Higher frequency versions of the domes of Figure 1
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Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
2.2. Metrics Another fundamental property of a configuration relates to the information about the magnitudes associated with various aspects of the configuration. These are referred to as ‘metric properties’ and include coordinates, distances, areas, angles, etc. The collection of all the metric particulars of a configuration is referred to as the ‘metrics’ of the configuration. The terms ‘normic’ and ‘normics’ can also be used instead of ‘metric’ and ‘metrics’. Naturally, any change in a configuration will alter some of its metric details. Therefore, the metrics of a configuration are unique to that configuration and different configurations cannot have the same metrics. 2.3. Pattern The ‘pattern’ of a configuration is a ‘visual effect’ that is created by a combination of factors including the number, forms and proportions of the elements of the configuration. Alternatively, a pattern may be defined as being a (visually) recognisable ‘formation’ of the elements of a configuration. A pattern is not, usually, considered to change if it is repeated or scaled or is (moderately) deformed, as long as it remains (visually) recognisable as the ‘same pattern’. A pattern on a rubber balloon will deform in various ways when the balloon is inflated. However, this will not, necessarily, change the pattern, unless, the deformations are such that the pattern becomes ‘unrecognisable’.
Figure 3. Different diamatic domes with sectoral lamella pattern
For example, the dome of Figure 1a has 6 sectors which are divided into triangles. It may be said that the dome of Figure 1a has a ‘sectoral ternate (triangulated) pattern’. It is also seen that the other two domes in Figure 1 have the same ‘pattern’. Actually, all six domes in Figures 1 and 2 have the same pattern. In fact, for these domes, any frequency of triangulation and/or any number of sectors would result in the same pattern. On the other hand, the domes in Figure 3 have the same span, rise and scalloping particulars as the domes in Figures 1 and 2. However, the ‘pattern’ of the domes of Figure 3 is different. Although, just as for Figures 1 and 2, all three domes of Figure 3 have the same pattern. The domes of Figure 3 have a ‘sectoral lamella pattern’. 2.4. Circumsurface The term ‘circumsurface’ refers to a surface that contains all the nodal points of a single layer configuration. For instance, consider the dome of Figure 1a. Here, all the nodal points of the dome lie on a ‘spherical surface’ and this surface will be the circumsurface of the dome. The same applies to the dome configurations of Figures 1b, 2a, 2b, 3a and 3b. In processing a configuration, if it is subjected to any transformation, then its circumsurface will also be subjected to the same transformation. For example, when the dome of Figure 1b is transformed into the scallop dome of Figure 1c, then the spherical circumsurface of Figure 1b will be transformed into a surface that may be described as a ‘scalloped sphere’. This way of describing a circumsurface by reference to a ‘basic surface’ (like sphere) and the ‘manner in which it is transformed’ (like scalloped), will be, particularly, useful for describing circumsurfaces of ‘freeforms’. A single layer configuration may have all of its nodal points not on one but on a combination of two or more surfaces. In this case, the configuration is said to have a ‘compound circumsurface’. 3
Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
For example, consider the configuration shown in Figure 4. This is an ovate (egg-shaped) diamatic dome half of which is spherical and the other half is ellipsoidal, see Nooshin and Disney [2], Section 2.5. In other words, all the nodal points of the left part of the dome lie on a spherical surface and all the nodal points of the right part lie on an ellipsoidal surface. Thus, the dome has a compound circumsurface consisting of a combination of spherical and ellipsoidal surfaces. Note that, the circumsurface of a configuration contains all of its nodal points. However, this does not necessarily imply that the elements of the configuration also lie in the circumsurface. The elements may or may not lie in the circumsurface depending on the particulars of the configuration and its circumsurface.
Figure 4. An ovate diamatic dome with a compound circumsurface
In the case of simple circumsurfaces like sphere, cylinder, torus, etc, one can use the terms ‘circumsphere’, ‘circumcylinder’, ‘circumtorus’, etc, as alternatives for circumsurface. Also, when a circumsurface has a radius then one can use the term ‘circumradius’ to mean ‘radius of circumsurface’. The concept of a ‘circumsurface’ is also applicable to multi-layer configurations, with each layer having its own circumsurface. For a multi-layer configuration, the circumsurfaces of different layers may or may not be of the same form. 2.5. Curviance The term ‘curviance’ means the ‘manner in which a configuration is curved’. To exemplify the usage of the term, one can, for instance, say:
‘Curviance’ of a configuration is concerned with its ‘form’, irrespective of its connectivity and pattern. The three dome configurations in Figure 1 have the same ‘connectivity’ and ‘pattern’ but different ‘curviances’. Innovative architects create interesting spatial structures with exciting freeform ‘curviances’.
Basically, curviance is a ‘qualitative’ concept providing a general idea of the form of a configuration. However, the description of curviance may also include metric information. In describing the ‘curviance’ of a configuration, the required level of details depends on the context. For instance, to describe the curviance of the dome of Figure 1a, one may say that:
‘Curviance has the form of a cap of sphere’ or ‘Curviance has the form of a cap of sphere with medium rise’ or ‘Curviance has the form of a cap of sphere with a span of 30 m and a rise of 10 m.
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Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
The curviance of a configuration will have the form of the region of its ‘circumsurface’ in which it is situated. To elaborate, consider Figure 5 showing a toroidal surface, together with three types of toroidal barrel vaults, namely, ‘outbent’, ‘inbent’ and ‘sidebent’ toroidal barrel vaults. A toroidal barrel vault assumes an ‘outbent shape’ if it is situated in the region of ‘outer ridge’ of the toroidal surface. Also, a toroidal barrel vault assumes an ‘inbent shape’ or a ‘sidebent shape’ if it is situated in the region of ‘inner ridge’ or ‘side ridge’ of the toroidal surface. The toroidal surface of Figure 5 contains all the nodal points of all the three barrel vaults. Therefore, it is the ‘circumsurface’ of all the three outbent, inbent and sidebent barrel vaults. However, the ‘curviances’ of these three barrel vaults are significantly different, dictated by the forms of the regions of the circumsurface in which they are situated. Even in the case of a simple spherical circumsurface, depending on the proportions of span and rise, the curviance can assume different shapes.
Figure 5. A toroidal surface together with three types of toroidal barrel vaults
Finally, it should be mentioned that an effective way of describing a ‘curviance’ is to liken it to a ‘known shape’ and using such descriptive terms as: doughnut-shaped, ovate (egg-shaped), onionshaped, funnel-shaped, leaf-shaped, umbrella-shaped, etc..
3. Classification of Lattice Spatial Structures A major objective of the present paper is to consider the classification of lattice spatial structures. The constitution of a suitable classification is shown in Table 1. In the Classification Table (Table 1), the lattice spatial forms are divided into eleven families as follows: Plane grids, barrel vaults, domes, levic forms, hypar forms, pyramidal forms, towers, polyhedric forms, foldable forms, freeforms and paragenic
3.1. Plane Grids Each of the first three families of forms in the Table is further subdivided into a number of families. To be specific, the family of ‘plane grids’ is subdivided into the family of ‘single layer grids’ and the family of ‘multi-layer grids’. A single layer grid, or simply a ‘grid’, consists of an arrangement of interconnected ‘bar elements (line elements)’ lying in a plane. There are many possible patterns for a grid of which two examples are shown in the Classification Table. A multi-layer grid consists of two or more, usually parallel, plane grids that are joined together through bars interconnecting the layers. These interconnecting members are referred to as ‘web’ elements. A multi-layer grid, in most practical cases, has only two layers, that is, it is a ‘double layer grid’. Two examples of double layer grids are shown in the Table.
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Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
3.2. Barrel Vaults The family of barrel vaults is subdivided into fourteen families, each of which is represented by two examples in Table 1. The common characteristic of all these fourteen families is that their ‘curviance’ resembles a form of ‘barrel vault’. The simplest shape of a barrel vault has a cylindrical form with a constant cross-section along its axis. This cross-section may be circular, elliptic, parabolic, etc. However, a barrel vault may have many other shapes. The cross-section of a barrel vault need not necessarily be constant along its axis. Thus, the curviance of a barrel vault may have the form of a paraboloid, hyperboloid, hypar, etc. Also, the axis of a barrel vault may be inclined giving rise to a ‘sloping’ barrel vault. Furthermore, the axis of a barrel vault need not be straight. Thus, a barrel vault may have the form of a part of a toroid, see Figure 5. This would give rise to ‘inbent’, ‘outbent’ and ‘sidebent; ‘toroidal barrel vaults’. Most of the examples of barrel vaults shown are ‘single layer’, however, any variety of barrel vault may also be multi-layered, with two or more layers. The names of some families of barrel vaults in the Classification Table require explanation. These names are ‘scallop’, ‘levic’, ‘palm’ and ‘barrel-dome’. Brief descriptions of these terms are as follows:
A scallop barrel vault is a barrel vault which is divided into segments each of which is ‘raised’ or ‘lowered’. The concept represents the extension of the idea of a ‘scallop dome’ applied to barrel vaults, see Nooshin et al. [5]. To describe a ‘levic form’, consider a ‘plane pattern (grid)’ and let (the nodal points of) this pattern be subjected to (one or more) transformations that have ‘bulging’ effects on the pattern in the ‘perpendicular’ direction to the plane of the pattern. The resulting configuration, is referred to as a ‘levic form’. If the pattern for creation of a levic form is multi-layer, then the different layers may be turned into levic forms in similar ways or in different ways. The term ‘levic’ is of Latin origin and implies ‘raised’. A levic form may be a dome, barrel vault, hypar, etc. A levic form may also be a freeform. A palm barrel vault is a ‘dome-like’ structure whose central part is turned into support (column). So, the structure is effectively a ‘roundish corridor’. The use of the term ‘palm’ is due to the imagined similarity of the form with a ‘palm tree’, with the central support representing the ‘stem’ of the tree. A ‘barrel-dome’ is a barrel vault with two semi-domes forming its gable ends, see Nooshin and Disney [3], Section 3.5.
It is to be noted that ‘curviance’ is the only feature necessary to consider for the classification of the barrel vaults, both at the initial level of deciding whether a form is a barrel vault and at the level of deciding on the specific ‘family’. On the other hand, when the curviance of a number of forms are the same, then to differentiate between them, it is necessary to consider other features. The feature that is normally considered after ‘curviance’ is the ‘pattern’. For instance, in considering the classification of plane grids, since the curviance of all of them is the same (that is, planar), then the distinguishing feature among the plane grids, is the pattern of the grids.
3.3. Domes The family of ‘domes’ in the Classification Table is subdivided into fourteen families, each of which is represented by two examples. The principal common characteristic of the configurations in these families is that the overall form of the ‘curviance’ is a ‘roundish enclosure’, which is, obviously, necessary for a structure to be considered a ‘dome’. When it comes to the distinguishing features of the families, in most cases it is based on the curviance but in some cases it is based on the pattern. The distinguishing feature among the first four families of domes in the Classification Table is related to ‘pattern’. Indeed, the words ‘ribbed’, ‘Schwedler’, ‘lamella’ and ‘diamatic’ are names of patterns, see Nooshin and Disney [1], Sections 1.9 and [2], Section 2.5. However, in the case of the remaining ten families of domes, the deciding feature is the ‘curviance’. The reason for the pattern being the 6
Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
deciding feature for the first four families is that the curviance of each of these four families is either a ‘spherical cap’ or a ‘cap’ of another surface of revolution. Thus, the curviance here cannot be the distinguishing feature. Table 1: Classification Table Plane Grids: Multilayer grids
Single layer grids
Barrel Vaults:
Cylindrical barrel vaults
Paraboloidal barrel vaults
Hperboloidal barrel vaults
Hypar barrel vaults
Ellipsoidal barrel vaults
Toroidal barrel vaults
Palm barrel vaults
Scallop barrel vaults
Undulated barrel vaults
Levic barrel vaults
Compound barrel vaults
Freeform barrel vaults
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Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
Domes:
Table 1: Classification Table (continued)
Ribbed domes
Schwedler domes
Lamella domes
Diamatic domes
Ovate domes
Scallop domes
Onion domes
Mallow domes
Geodesic domes
Conical domes
Faceted domes
Levic domes
Compound domes
Freeform domes
Barrel-domes
Hypar forms
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Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
Table 1: Classification Table (continued)
Pyramidal forms
Polyhedra
Towers
Foldable forms
Paragenic forms
Freeforms
The other families of domes in the Classification Table include: ‘ovate domes’ (egg shaped), see Nooshin and Disney [1], Section 1.9, ‘scallop domes’ (with raised or lowered sectors), see Nooshin et al. [5], ‘onion domes’, see Nooshin and Disney [3], Section 3.4.2, ‘mallow domes’ (formed from hypar sectors), see Nooshin and Disney [3], Section 3.2.1, ‘geodesic domes’, see Champion [4], ‘conical domes’, see Nooshin and Disney [3], Section 3.3.1, ‘faceted domes’ (having faceted outside surface), ‘levic domes’ (as explained for ‘levic barrel vaults’), ‘compound domes’, see Nooshin and Disney [3], Section 3.5 and ‘freeform domes’. As in the case of the barrel vaults, any variety of dome may be also ‘multi-layer’.
3.4. Other Families Finally, the other families of lattice spatial structures in the Classification Table include: ‘hypar forms’, see Nooshin and Disney [3], Section 3.2, ‘pyramidal forms’, see Nooshin and Disney [2], Section 2.2, ‘towers’, see Nooshin and Disney [2], Section 2.3 and [3], Section 3.3, ‘polyhedra, see Champion [4], ‘foldable forms’, see Nooshin and Disney [2], Section 2.4, ‘freeforms’ and ‘paragenic forms’, see Nooshin and Disney [3], Section 3.6. Each one of these families is represented by two examples in the Classification Table. The distinguishing feature for all of these families is the curviance. The Classification Table presented in the paper is not meant to be ‘exhaustive’ or ‘fixed’ or ‘final’. It is meant to be a general guide representing lattice spatial forms. However, other families of forms may be added to the Table and existing families may be removed or modified. Also, families may be subdivided into ‘subfamilies’. For example, the family of toroidal barrel vaults (see Figure 5) may be subdivided into three families of ‘inbent’, ‘outbent’ and ‘sidebent’ toroidal barrel vaults. 9
Proceedings of the IASS Annual Symposium 2016 Spatial Structures in the 21st Century
Every one of the families in the Table covers an unlimited number of possibilities. For example, consider the family of ‘cylindrical barrel vaults’. The following variations are all within the scope of this family:
Different shapes of cross-section for the cylinder, like circular, elliptic, parabolic, etc, as well as many irregular forms of cross-section, Number of layers of the cylinder, that is, single layer or multi-layer, Different patterns with different frequencies, Sloping cylindrical barrel vaults, with different degrees of inclination, Different rise-to-span ratios and length-to-span ratios.
An effective route for creation of new and innovative forms is to modify the basic cases of the lattice spatial structural forms. Such modified forms may be named using the established terminology in the Classification Table. For example, one can create an interesting form by ‘twisting’ an inbent toroidal barrel vault and this form could be referred to as a ‘twisted inbent toroidal barrel vault’.
4. Background The basic idea, of grouping various spatial structural forms into families, has been inspired by the book entitled ‘Steel Space Structures’ by Z S Makowski, published in 1965 [6]. Subsequently, an expanded version of Makowski’s classification was produced by the Authors of the present paper in an Appendix for the ‘Iranian Code of Practice for Spatial Structures’ [7]. This Appendix contains some 200 examples of lattice spatial structures, and has a parametric formex formulation for every one of these examples. An enhanced version of this Appendix is available in the new version of the configuration processing computer language Formian. This new version is called Formian-K.
5. Conclusion The paper represents an attempt to provide clear definitions for a number of important morphological terms relating to various aspects of configurations. Also, the paper provides a classification system for a major family of structural forms, namely, the family of lattice spatial structural forms. These are of value because clear definitions and well constituted classifications are helpful aids for the brain. They are helpful in effective thinking. They are helpful in allowing verbal and/or written explanations to be understood without ambiguity.
References [1] Nooshin H. and Disney P., Formex configuration processing I(*). International Journal of Space Structures, 2000; 15; 1-52. [2] Nooshin H. and Disney P., Formex configuration processing II(*). International Journal of Space Structures, 2001; 16; 1-56. [3] Nooshin H. and Disney P., Formex configuration processing III(*). International Journal of Space Structures, 2002; 17; 1-50. [4] Champion O.C., Polyhedric Configurations(*). PhD Thesis, University of Surrey, UK, 1997. [5] Nooshin H., Tomatsuri H. and Fujimoto M., Scallop Domes(*), in IASS 1997. International Symposium on Shell and Spatial Structures, Chiew S.P. (ed.), 1997, 651-660. [6] Makowski Z.S., Steel Space Structures. Published by Michael Joseph, 1965. (This book has also French and German editions.) [7] Iranian Code of Practice for Spatial Structures (in Persian), Publication No 400, 2011. (*): Also available from Formexia.com
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