## Description

Description

FIBONACCI SPACE STRUCTURES

Technical Field

[1] Embodiments described herein generally relate to

Metal spatial structures

Shell structures

Structural design

Conceptual design

Bio-based structures

Building Physics

Background

[2] Efficient Use of material is the significance of space frame. Domes with a wide range of general arrangements have been built since Graham Bell introduced the Space frame. Arrangement of members in space frame is Structural general arrangement of the structure. NPL 1, NPL 2, NPL 3, NPL 5 and NPL 6 describe a wide range of general arrangements .

[3] Domes are nature’s perfect structure and provide a unique environment for every use.

Domes are the strongest portable structures known to man. These structures require a lot of time to construct and the material costs would be higher than other construction systems. Braced domes decrease time to construct and the material costs which would be higher in other dome construction systems.

[4] Buckminster Fuller invented the Geodesic dome which proved to be the most stiff structural general arrangement so far. Fuller selected an icosahedron as the starting point for optimum bracing. Icosahedron 30 edges, 20 faces( all equilateral triangles), 12 vertices. Edges of icosahedron are subdivided to provide additional vertices to keep the lengths within a limit and also so that all the members are of equal lengths.

[5] Richter analysed different bracing systems and assigned strength and stiffness efficiencies for each of them respectively:

[6] Nature inspired design:The foremost and basic context for any structure to be built is nature. Apart from the fact that nature affects architecture, it still exists as one of the main inspirations for design concepts. Generally this is done to make structures more functional and efficient. Symmetry of nature is well observed in Fibonacci Spiral arrangement.

[7] As quoted by Galileo Galilei ’The laws of nature are written by the hand ofGod in the language of mathematics.’

The series follow

F0= 0,

F 1= I ,

and for n> 1

Fn = Fn-l+Fn-2

Fn=Fn-l+Fn-2 where

Fn is the nth value of the sequence. 0, 1, I , 2, 3, 5, 8, 13, 21, 34, 55. . . The ratio of Fn/Fn- 1 is known as the Golden ratio.

[8] Formex Algebra: Formian is a software that generates forms and structures. The software can be used to save time for modelling general arrangement. Replication of line segments, conversion of intrinsic plot to extrinsic plot are some of the advantages of the software. NPL 4 and NPL 7 give insights into Formex algebra and generation tools

Summary of Invention

[9] Symmetry of nature is well observed in Fibonacci Spiral arrangement. An application of nature’s symmetry in space frames for generating new ways of arranging elements in space frames. Force transfer differs from one structural general arrangement to another. The force transfer in Fibonacci spiral geometry is applied to braced domes.

[10] Structures like buildings and towers constitute a framework of elements intersecting at nodes. Framework in which main structural elements are interconnected in a Fibonacci pattern of arrangements intersecting to form triangles imposing a Fibonacci spiral onto the differentiable surfaces of revolution, dividing the Fibonacci spiral into smaller segments generated the invention. [11 ] In some embodiments, the pattern comprises configuration of general arrangement of Fibonacci space structure and Fibonacci space structure N, a configuration of structural elements forming a truss in which interconnecting nodes lie substantially in a common differentiable surface and an intrinsic plot contained within the differentiable surface, wherein the nodes are points on Fibonacci intrinsic plots.

[12] In certain embodiments space structures comprises a configuration of structural elements forming a truss in which interconnecting nodes lie substantially in a common differentiable surface and a curve generated the differentiable surface by rotation along an axis wherein the projection of nodes on any plane perpendicular to the axis substantially lies on the great Fibonacci spiral. In one embodiment, elements are interconnected in a Fibonacci A and Fibonacci N patterns to form a three way grid defining substantially isosceles triangles, interconnecting nodes lie substantially in a common differentiable surface and innermost points of truss lie substantially in a common differentiable surface and coaxial with the differentiable surface thereof.

Advantageous Effects of Invention

[13] The general arrangements described could be used for a building structure, a skylight, or other enclosure such as a gazebo, a greenhouse, or an arbor. One of the major advantages is that, in a pre-assembly state, the entire structure can be packaged very compactly.

[14] The geometry developed will contribute to engineering as well as science.

The arrangement when applied to other branches of engineering will increase efficiency of the system, product etc…

[15] Typical usage of the Fibonacci structural general arrangement is where a roof is needed over a very large area, such as a sports stadium, swimming pools or where a circular structure is desired for aesthetic reasons or required for functional usage. [16] Nature inspired conceptual designs could provide alternative arrangements. Studies for finding better structural general arrangements will be improved from the invention.

[17] The general arrangements described could be used for lattice towers such as transmission towers .mobile towers, broadcasting towers, hyperboloid structures, paraboloid structures.

Brief Description of Drawings

Fig.1

[18] [Fig.1 ] illustrates Fibonacci spiral

Fig.2

[19] [Fig.2] is a diagrammatic representation of Projection of fibonacci spiral onto hemisphere

Fig.3

[20] [Fig.3] illustrates Fibonacci sequence observed on snail shells and spiral galaxies

Fig.4

[21] [Fig.4] illustrates Fibonacci spiral intrinsic plot, 0°to 90°

Fig.5

[22] [Fig.5] is a diagrammatic representation of Structural general arrangement of Fibonacci A dome

Fig.6

[23] [Fig.6] illustrates Fibonacci spiral intrinsic plot, 90°to 180

Fig.7

[24] [Fig.7] is a diagrammatic representation of Structural general arrangement of Fibonacci N dome

Fig.8

[25] [Fig.8] illustrates congruence of General arrangement of sunflower seeds and top view of Fibonacci A structural framework

Fig.9

[26] [Fig.9] illustrates top view of Fibonacci N structural framework

Fig.10

[27] [Fig.10] illustrates Fibonacci N and Fibonacci A Structural general arrangements

Fig.11

[28] [Fig.11 ] is an elevational view of general arrangements of Fibonacci space structures.

Description of Embodiments

[29] Fibonacci space structures are generally disclosed. Certain embodiments comprise applying (e.g., spirally) a Fibonacci sequence to a spatial structure(e.g., braced dome) such that the Fibonacci spiral is projected onto the structure. In some embodiments, the structures described here in are configured of structural elements forming a truss in which interconnecting nodes lie substantially in a common differentiable surface; and an intrinsic plot contained within the differentiable surface, wherein the nodes are points on Fibonacci intrinsic plots.

[30] The term helix used herein describes the imposed spiral in [Fig 2], Imposition may be done onto any differentiable spiral. Differentiable surfaces may be surfaces of revolution. Differentiable surfaces can be generated by rotating curves such as parabola, hyperbola, ellipse, quarter circle etc. The Pappus’ conical spiral, Three-dimensional Fermat’s spiral, Three-dimensional hyperbolic spiral, Three-dimensional lituus spiral, Three-dimensional logarithmic spiral, Three-dimensional hyperboloidal spiral, Three-dimensional catenoidal spiral are other examples of helix which may generated by imposition of spiral onto differentiable surfaces as described in NPL 10.

[31] In one embodiment, the Fibonacci spiral can be represented by equations [Math. 1], [Math. 2], [Math. 3] and [Math. 4] is depicted in [Fig. 1], The radius vector rotating about the centre of the hemisphere passes through arcs of increasing radius in Fibonacci sequence. The radius vector makes an angle [3 with CA. The magnitude of the radius vector changes as the radius vector rotates. Boundary conditions are obtained after completion of each Fibonacci quarter circle. The angle between the radius vector and the side of the unit length is 360°-p. Using the cosine rule for the triangle ABC from the [Fig. 1] we get [Math. 2], The rate of change of slopes at the intersection of the curves [Math. 1], [Math. 2], [Math. 3] and [Math. 4] is equal.

[32] [Math. 1]

For 0°<x<180°-arctan(2),

√5 (2) (sin (y))(1 + 2 √l5)(cos (x + arctan (2))) =(1 + 2 √l5)2(sin (y))2+(5-82)

[33] [Math. 2]

For 180°-arctan(2)< x <270°-arctan(4),

√2 (2) (sin (y))(1 + 2 √l5)(cos (x-arctan (1 ))) =(1 + 2 √l5)2(sin (y))2+(2-52)

[34] [Math. 3]

For 270°-arctan(4)< x <360°-arctan(6),

√2 (2) (sin (y))(1 + 2 √l5)(cos (x+ 5 arctan (1 ))) =(1 + 2 √l5)2(sin (y))2+(2-32)

[35] [Math. 4]

For 360°-arctan(6)< x <360°,

√1 (2) (sin (y))(1 + 2 √l5)(cos (x-180)) =(1 + 2 √l5)2+(sin (y))2+(1 -22)

[36] The centre of the spiral described in [Fig. 1] coincides with the pole of the hemisphere in [Fig. 2] .The spiral is imposed on the hemisphere such that the projection of the spiral on the equatorial plane follows the Fibonacci series. Equations [Math. 1], [Math. 2], [Math. 3] and [Math. 4] results plots shown in [Fig.4] and [Fig. 6], Two intrinsic plots can be obtained for different limits of y.

[Fig. 4] depicts an intrinsic plot for limits 0°<y<90°and 0°<x<360° which is concave up. [Fig. 6] depicts an intrinsic plot for limits 90°<y<180°and 0°<x<360°.The intrinsic plot in figure 4 is concave down. The intrinsic plots are replicated and reflected to form a 2-D array followed by connecting lateral connections to form an arrangement of triangles.[37] Equations [Math. 1], [Math. 2], [Math. 3] and [Math. 4] can be further generalised equations of [Math, n] corresponding to the nth quarter circle in the fibonacci approximated spiral. Plots [Fig. 4] and [Fig. 6] are fragmented dividing horizontal axis into equal units. [Fig. 4] and [Fig. 6] correspond to the intrinsic plot of the single spiral of [Fig. 5] and [Fig. 7],

[38] The points in [Fig. 5] and [Fig. 7] correspond to nodes which are generally called connectors of elements. Node connectors are generally made of steel. Node connectors behave like hinged connections.

[39] The elements are generally made of steel, timber and aluminum. The elements refer to the line segments in [Fig. 5] and [Fig. 7], The elements have the function of stretching. The linear elements are axially loaded in tension or compression. Elements are described as members in the non patent literature. The individual elements may be rolled, extruded or fabricated steel sections. The bottom elements of the dome’s general arrangement can be replaced with a Tension ring connecting bottom nodes. Tension ring transfers load from super-structure to the sub-structure. If a poured concrete foundation is used, the lowermost struts and fastenings , or the ends of such struts, may be embedded in the foundation, in which case the concrete or portions thereof is poured after erection of space structure has been completed. [40] Erection with precision is involved. Elements to be fabricated accurately as effect falls in all three directions. Cantilever method or lift method or sub assembly methods of erection sequence to be used to construct.

[41] The points in [Fig. 8] and [Fig. 9] or nodes of truss connect members and the cladding which transfers load of cladding to the substructure. The substructure transfers load to earth. Most preferred node connectors in spatial structural systems are the Mero system, Triodetic system, Unibat system, Space deck system, Nodus system and Unistrut system.

[42] Those skilled in the art given the guidance and teaching of this specification would be capable of determining suitable methods for tuning the mathematical properties of Fibonacci spiral components into space frames.

[43] The Fibonacci space structures may be covered with plastic skins, membranes inside or outside or both, or with other materials. Openings for access, light, sun and air are provided as desired.

[44] Multilayered structures may be designed enlarging the pattern of nodes and elements formed coaxially and an enlarged layer can be mapped to the first said layer defining a topology between layers.

[45] Fibonacci N structures generally deviate maximum from their free form surfaces or surfaces of revolution which makes them suitable to resist cyclic loading. Fibonacci A structures generally deviate minimum from their free form surfaces or surfaces of revolution which makes them highly efficient.

Industrial Applicability

[46] Commercial and industrial structures like

[47] Auditoriums

[48] The general arrangements described could be used for a building structure, a skylight, or other enclosure such as a gazebo, a greenhouse, or an arbor. One of the major advantages is that, in a pre-assembly state, the entire structure can be packaged very compactly. [49] The geometry developed will contribute to engineering as well as science. The arrangement when applied to other branches of engineering will increase efficiency of the system, product etc…

[50] Typical usage of the Fibonacci structural general arrangement is where a roof is needed over a very large area, such as a sports stadium, swimming pools or where a circular structure is desired for aesthetic reasons or required for functional usage.

[51] The general arrangements described could be used for lattice towers such as transmission towers .mobile towers, hyperboloid towers, paraboloid towers.

[52] Skylights

[53] Canopies

[54] Toll booths

[55] Exhibition halls

[56] Sports stadiums

[57] hyperboloid towers

Patent Literature

[58] PTL 1 : Patent US20060213145A1

[59] PTL 2: Patent US2371421

[60] PTL 3: Patent US3062340

[61] PTL 4: Patent US3364633

[62] PTL 5: Patent US3645104

[63] PTL 6: Patent W02004083564A2

[64] PTL 7: Patent US4587777

[65] PTL 8: Patent US1773851 A

[66] PTL 9: Patent US2351419 [67] PTL 10: Patent US2682235

[68] PTL 11 : Patent US2905113

[69] PTL 12: Patent US2914074

[70] PTL 13: Patent US2986241

[71] PTL 14: Patent US20180366833

[72] PTL 15: Patent US3063521

[73] PTL 16: Patent US3139957

[74] PTL 17: Patent US3197927

[75] PTL 18: Patent US3203144

[76] PTL 19: Patent US3354591

[77] PTL 20: Patent US7143550

[78] PTL 21 : Patent US7900405

[79] PTL 22: Patent US20210128460A1

Non Patent Literature

[80] NPL 1 : Ramaswamy G.S. and M. Eekhout, ’’Preliminary design”, inAnalysis, Design and Construction of Steel Space Frame, Telford Publication, U.K,

1999. https://doi.Org/10.1680/adacossf.30145.0003

[81] NPL 2: Ramaswamy G.S. and M. Eekhout, ’’Introduction to space frames”, inAnalysis, Design and Construction of Steel Space Frame, Telford Publication, U. K, 1999. https://doi.Org/10.1680/adacossf.30145.0001

[82] NPL 3: Ramaswamy G.S. and M. Eekhout, ’’Structural design of space frame com-ponents”, inAnalysis, Design and Construction of Steel Space Frame, TelfordPublication, U.K, 1999. https://doi.org/10.1680/adacossf.30145.0002

[83] NPL 4: Ramaswamy G.S. and M. Eekhout, ’’Time- and labour- saving aids for pre- and post-processing tasks”, inAnalysis, Design and Construction of SteelSpace Frame, Telford Publication, U.K,

1999. https://doi.Org/10.1680/adacossf.30145.0005

[84] NPL 5: Ramaswamy G.S. and M. Eekhout, ’’Space trusses for long spans”, inAnal-ysis, Design and Construction of Steel Space Frame, Telford Publication, U. K, 1999. https://doi.Org/10.1680/adacossf.30145.0006

[85] NPL 6: Ramaswamy G.S. and M. Eekhout, ’’Braced domes”, inAnalysis, Design and Construction of Steel Space Frame, Telford Publication, U.K,

1999. https://d0i.0rg/l 0.1680/adacossf.30145.0008

[86] NPL 7: Hoshyar Nooshin and Peter Disney Space Structures Research Centre, Department of Civil Engineering, University of Surrey, Formex Configurationprocessing I Guildford, Surrey GU2 7XH, United

Kingdom, 2000. https://d0i.0rg/l 0.1260/2F0266351001494955

[87] NPL 8: J. D. RENTON, University of Oxford, GENERAL PROPERTIES OF SPACEGRIDS, Department of Engineering

Science, 1970.https://doi.org/10.1016/0020-7403(70)90055-X

[88] NPL 9: Ar. Anjali Prashant Kshirsagar Asst. Professor, Deccan Institute ofTechnology. Ar. Seema Santosh Malani Asst. Professor, Deccan Instituteof Technology, Kolhapur Er. Vikramsinh S. Tiware Asst. Professor, BharatiVidyapeeth, Kolhapur Shivaji University, Biomimicry – Nature Inspired Building Structures

[89] NPL 10: Spirals on surfaces of revolution Cristian Lazureanu Department of Mathematics, Politehnica University of Timisoara Piata Victoriei nr. 2, Timisoara, 300006, RonTania

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