The objective of this paper is to investigate the member length regularity of lattice domes. Also, a recently developed configuration for lattice domes generated by Surface Sphere Packing technique is compared to five other lattice dome configurations

BEHNEJAD, S Alireza

NOOSHIN, Hoshyar

PARKE, Gerard

University of Surrey

    ISIS Congress-Festival Symmetry: Art and Science Special Theme: Labyrinth and Symmetry Session/ Symposium: ………………………………………………………………… Bali, Crete, 9-15 September 2013 MEMBER LENGTH REGULARITY OF LATTICE DOMES BEHNEJAD, PARKE AND NOOSHIN Name: S. Alireza, BEHNEJAD. Specialty: Architecture, Structural Engineering Affiliation: Research Student, Space Structures Research Centre, University of Surrey Address: Civil and Environmental Engineering Department, University of Surrey, Guildford, UK E-mail: a.behnejad@surrey.ac.uk Fields of interest: Architectural Technology, Structural Systems, Spatial Structures and Education (Also, Art, Geometry and Literature). Publications: Length Equalization of Elements in Single Layer Lattice Spatial Structures, Sphere Packing for Regularisation of Lattice Domes.  Name: Gerard, PARKE. Specialty: Structural Engineering Affiliation: Professor of Structural Engineering and Head of Department of Civil and Environmental Engineering, University of Surrey Address: Civil and Environmental Engineering Department, University of Surrey, Guildford, UK E-mail: g.parke@surrey.ac.uk Fields of interest: Steel Spatial Structures, Steel Bridge Structures, Glass Structures. Publications: - Parke, G. A. R. (1985) Comparison of Structural Behaviour of Various Types of Braced Barrel Vaults, In: Makowski, Z. S. ed., Analysis, Design and Construction of Braced Barrel Vaults, London: Elsevier Applied Science Publishers, 90-120. - Overend, M. and Parke, G. A. R. (2002) Glass Space Structures, In: Parke, G. A. R. and Disney, P. eds., Space Structures 5, [Conference organized by the Space Structures Research Centre, University of Surrey, Guildford, UK, August 19-21, 2002], Vol. 1, London: Thomas Telford, 529-540.  Name: Hoshyar, NOOSHIN. Specialty: Spatial Structures Affiliation: Emeritus Professor, Space Structures Research Centre, University of Surrey Address: Civil and Environmental Engineering Department, University of Surrey, Guildford, UK E-mail: h.nooshin@surrey.ac.uk Fields of interest: Spatial Structures and Configuration Processing. Publications: - Nooshin, H. (1984) Formex Configuration Processing in Structural Engineering, London: Elsevier Applied Science Publishers, xi + 273 pp. - Nooshin, H. (1997) Computer-Aided Processing of Polyhedric Configuration, In: Gabriel, J. F. ed., Beyond the Cube, New York: John Wiley & Sons Inc., 343-384. - Nooshin, H. and Disney P. (2009) Formex Configuration Processing, In Motro, R. ed., Structural Morphology and Configuration Processing of Space Structures, Brentwood: Multi-Science Publishing Co Ltd, 249-549.       Abstract: The objective of this paper is to investigate the member length regularity of lattice domes. Also, a recently developed configuration for lattice domes generated by Surface Sphere Packing technique is compared to five other lattice dome configurations. INTRODUCTION A lattice dome consists of a number of members, usually, with different lengths. However, having too many different member lengths is undesirable from the point of view of manufacture, sorting and assembly. Although the construction of lattice domes is simplified by computerised high-tech facilities, the problem with excessive member length variation still exist. 1 LATTICE DOME CONFIGURATIONS Some examples of lattice domes are shown in Fig. 1. Fig. 1a shows a view of the dome configuration derived using Surface Sphere Packing (Behnejad, 2012, pp. 7-12). The ultimate aim of the Surface Sphere Packing technique is to generate a new type of pattern on spherical cap with a high proportion of equal length members. The sphere packing technique is used by a number of researchers including Tόth, Tarnai and Teshima. Some other lattice dome configurations (on spherical caps) are also shown in Figs 1b to 1e. These configurations are commonly used for construction of lattice domes around the world. Also, a mallow dome configuration which has a non-spherical surface is shown in Fig. 1f. 2 REGULARITY OF LATTICE DOMES There is no all-embracing definition for the term “regularity” of a structural configuration. However, there is normally a reasonable degree of agreement between people regarding what is and what is not regular. The degree of regularity of a lattice dome can be investigated by looking at the variation of the member lengths, shape of faces, form of vertices, … (Nooshin et al., 2011). In the present work, the term regularity is mainly used in the sense of a high proportion of the members being of the same length. Having a high degree of equal length members in a pattern of a lattice dome will have some benefits, specially from the construction point of view. The benefits include decreasing the time and cost of manufacturing of members, simplifying the sorting and labelling of members and increasing the speed of assembly with lower probability of mistakes in the position of members.       Fig. 1 Six different patterns for lattice domes together with the length profile chart and the regularity indicators of each pattern. The indicators consist of the number of members (n), number of member length types (b), length ratio (r) and length deviation (d). These indicators are discussed in Section 3.     3 REGULARITY INDICATORS Consider the bar charts in the middle area of Fig. 1. The top chart in the right side of Fig. 1a, for instance, represents information about the member lengths of the dome configuration shown in Fig. 1a. The numbers on the horizontal axis represent proportions of the ratio of the member length and the average member length. The vertical axis in the chart represents the number of members of the same length as a percentage of the total number of the members. This chart is referred to as a “length profile chart” or a “LEP chart” (Nooshin et al., 1997, pp. 170-172). A LEP chart gives a visual representation of the degree of scatter of the member lengths in a configuration. Also, values for four items n, b, r and d are given at the right side of each LEP chart in Figs 1a to 1f. The definition of the items are as follows:  n represents the “total number of members”  b represents the “number of member lengths” which, in a regular configuration, should be relatively small.  r represents the “length ratio”, being the ratio of maximum member length and minimum member length. The more regular the configuration, the closer to one the value of r will be.  d represents the “deviation” being the ratio of standard deviation of member length / average member length. The more regular the configuration, the closer to zero the value of d will be. The regularity indicators help to evaluate the member length regularity in a configuration. Also, comparing regularity in different configurations can be done using the regularity indicators. However, other indicators, with different point of view, can be defined for evaluation of regularity in configuration. 4 A COMPARISON The number of members in each of the configurations of Figs 1a to 1f varies from 500 in the diamatic configuration, Fig. 1d, and 575 in Sphere Packing configuration, Fig. 1a. The LEP charts in Fig. 1 show that the Sphere Packing configuration has the maximum proportion of members with the same length (more than 45%). Also, the Sphere Packing configuration has 13 member lengths which is close to that of the geodesic configuration, while the other configurations in Fig. 1 have more scatter with 18 member lengths. REFERENCES Behnejad, S. A. (2012) Member Length Equalisation in Lattice Domes, [MSc. Dissertation], Guildford: University of Surrey, 58 pp. Nooshin, H., Ishikawa, K., Disney, P. L. and Butterworth, J. W. (1997) The Traviation Process, Journal of the International Association for Shell and Spatial Structures, 38, No. 125, 165-175. Nooshin, H., Mohammadi, N. and Parke, G. (2011) Regularity of Geodesic Domes, [Abstract], In: Nethecot, D. A. and Pellegrino, S., eds., Taller, Longer, Lighter, [Conference organized by the International Association for Bridge and Structural Engineering and the International Association for Shell and Spatial Structures, London, UK, September 20-23, 2011], London: Hemming Group, 406. 

Member Length Regularity of Lattice Domes

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