SSP - JOURNAL OF CIVIL ENGINEERING Vol. 9, Issue 2, 2014
DOI: 10.2478/sspjce-2014-0015
Geometry of Prismatic Tensegrity Constructions Composed
of Three and Four-strut Cells
Tatiana Olejníková
Technical University of Košice
Civil Engineering Faculty, Institute of Construction Technology and Management
e-mail: [email protected]
Abstract
In the paper there is described geometry of double layer tensegrity constructions composed of prismatic cells
with rhombic configuration of three or four strut bases so-called prismatic Tensegrity constructions. There are
described bi-dimensional assemblies creating double layer grids of three or four-strut cells with a node-on-node
junction. The grids can be planar, of one or two curvature constructions.
Key words: tensegrityconstruction, equilibrium, compression, tension, grid, cell
1
Introduction
The term „tensegrity“ is a combination of words tension and integrity and represents the state
in which the shape of the object is created by a closed system of tension and compression.
The term tensegrity originally used in architecture is introduced as an naming of the principle
keeping shapes of living organisms at all levels, from single cell to a set of muscles, sinews
and bones. Tensegrity allows a balance between stability and mobility, which are two basic
requirements for the musculoskeletal system. Tensegrity constructions enable maximum
efficiency by minimum essential energy and used material. In nature, this principle is the
demonstration of the wisdom which the world is created. Tensegrity describes a structural
principle, where shape and structure are guaranteed by a closed system of cables and struts,
create the cells. Compressed elements can stabilize the construction, even if not touching, on
their positions. They are held only by tension of cables.
This paper describes the geometric relations of tensegrity constructions composed of
prismatic cells. Cells contain three types of components: horizontal cables, vertical cables and
struts connected in a nodes. Each node connects two horizontal cables lying in a horizontal
plane, one vertical cable and one strut. Vertical cables and struts connect nodes in different
horizontal planes (orbits). The elements in one orbit have the same length.
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2
Geometry of prismatic three-strut tensegrity grids
The geometry of the spatial constructions is completely defined by its structure described by
coordinates of nodes P (x(k ), y (k ), z (k )) , k = 1,..., n , where n is number of nodes in a cell. All
internal components of cell – struts have the same length “c”. Primary geometry of the cell
was defined as a triangular prism (Fig.1 - axonometry of triangular prism). This construction
was unstable and the new shape of the cell was created by the relative rotation of two
triangles in the parallel planes (Fig.2 - axonometry and top view of the modified prism).
6
4
h
3
5
1
r
1
2
Figure 1
3
6
5
π6
4
2
Figure 2
For a given value of the length of the strut are specified cable lengths to form a stabilized selfstressing cell, which will be referred to as “null self-stress equilibrium geometry”. The
geometric distance between nodes corresponds strictly to the length of manufactured
components. Two geometrical ranges can be identified. If the relationship between the lengths
of the struts and cables is ratio s c = 2 = 1,467 and the relative rotation between triangles is
π 6 . When the parameter r is specified, then the length of the struts is s = r 3 + 2 3 , the
length of the all cables is c = r 3 and the height of the equilibrium is h = r 1 + 3 (Fig.2).
2.1
Connecting cells into grid constructions
Described the general three-strut cells of the same dimensions can be joined together in
several ways: the connection node on node, node on cable, cable on cable.
In Fig.3 a) the junction is operated with a single vertical (bracing) cable: each of two ends of
the strut lies in the different horizontal planes and it will be used in a plane configuration
leading to a double layer tensegrity grids, in the types b), c) the junction is operated with
horizontal cables.
Connection of cells by type a) will be used to create a planar grid, and after the modification
of the cell will be used to create the spherical grid construction.
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a)
b)
Figure 3: Junction modes “node on cable”
c)
In Fig.5 is displayed the planar double-leyer tensegrity grid created by the three-strut cells
connected by nodes on cables of type a) (Fig.4).
Figure 4: Connection of three cells
Figure 5: Planar double-leyer tensegrity
Nodes P(k ) of one cell in the planar grid are determined by the coordinates
P ( x(k ), y (k ), z (k )) :
x(k ) = k 0 r cos ϕ(k ), y (k ) = k 0 r sin ϕ(k ), z (k ) = h1
for k = 1,2,3
ϕ(k ) = −
for k = 4,5,6 ϕ(k ) =
π
2π
+ (k - 1) , h1 = 0
12
3
(1)
π
2π
+ (k - 4) , h1 = h = r 1 + 3
3
12
where parameter k 0 = 1 , r is the distance between vertex and center of the triangle, which is
created by three horizontal cables of one cell, h is the height of the cell (Fig.2).
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Coordinates x1 (i, j, k ), y1 (i, j, k ), z1 (i, j, k ) of the nodes P(i, j, k ) of this cell in the planar grid
displayed in Fig.5 are described by equations (2), (3), (4):
x1 (i, j, k ) = x(k ) + d x ( j), y1 (i, j, k ) = y (k ) + d y ( j), z1 (i, j, k ) = z (k ) ,
d x (j) = vx (i) cos(
jπ
jπ
jπ
jπ
) - v y (i) sin( ) , d y (j) = vx (i) sin( ) + v y (i) cos( ) ,
3
3
3
3
(2)
(3)
where i = 1,...,6, j = 1,...,6, k = 1,...,6 , coordinates x(k ), y (k ), z (k ) are expressed in (1),
v1 =
3r
3r
, v2 =
, vx (1) = vy (1) = 0, vx (2 ) = v1 , vy (2) = v2 ,
2
4
vx (3) =
5
9
v1 , vy (3) = v2 , vx (4) = 2v1 , v y (4) = 2v2 , vx (5) = 3v1 ,
4
2
vy (5) = 3v2 , vx (6 ) =
(4)
9
11
3
v1 , vy (6) = v2 , vx (7 ) = v1 , v y (7 ) = 8v2 .
4
2
2
Adhering of the principle of elementary cell stabilized by self-stressing state can modify this
cell to connection of the grid creates a spherical surface with radius R. The elementary cell
can be modified by using parameter k0 in the equations (1) for nodes of the upper triangle of
cell, where k = 4,5,6 , i = 1, ...,6, j = 1, ...,6 .
Transformation equations of coordinates of the nodes in the lower layer from the planar grid
to the spherical grid with radius R and the centre S(0,0,-R) for k = 1,2,3 , i = 1,...,6, j = 1,...,6
and the parameter k 0 = 1 in equations (1) are:
P( x1 (i, j, k ), y1 (i, j, k ), z1 (i, j, k )) → P′( x′ (i, j, k ), y′ (i, j, k ), z′ (i, j, k ))
x′ (i, j, k ) = R cos u (i, j, k ) cos v (i, j, k )
y′ (i, j, k ) = R cos u (i, j, k ) sin v (i, j, k )
(5)
z′ (i, j, k ) = R sin u (i, j, k ) − R
where
u (i, j, k ) =
π d (i, j, k )
−
, d (i, j, k ) = x12 (i, j, k ) + y12 (i, j, k ),
R
2
x (i, j, k )
v (i, j, k ) = sgn y1 (i, j, k ) arccos 1
d (i, j, k )
(6)
Coordinates x1 (i, j, k ), y1 (i, j, k ) are expressed in the equations (2).
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Transformation equations of coordinates of the nodes in the upper layer from planar grid to
the spherical grid with the radius R+h and the centre S(0,0,-R) for k = 4,5,6 and the parameter
R+h
in equations (1) are (Figs.6,7):
k0 =
R
x′ (i, j, k ) = ( R + h) cos u (i, j, k ) cos v (i, j, k )
y′ (i, j, k ) = ( R + h) cos u (i, j, k ) sin v (i, j, k )
(7)
z ′ (i, j, k ) = ( R + h) sin u (i, j, k ) − R
where
d (i, j, k ) = x12 (i, j, k ) + y12 (i, j, k ), u (i, j, k ) =
π d (i, j, k )
−
,
2
R+h
(8)
x (i, j, k )
v (i, j, k ) = sgn y1 (i, j, k ) arccos 1
d (i, j, k )
z
P (k )
x
0
P (i, j, k )
d(i,j,k)
P (i, j, k )
P ′ (i, j, k )
u(i,j,k
y
P′ (i, j, k )
v(i,j,k)
S
Figure 6
Figure 7
In Fig.8 there are displayed the nodes and cables of the triangles of the bottom layer on the
spherical surface. Fig.9 displays the triangles of the bottom layer of the tensegrity grid located
on the tangent plane of the spherical surface and in Fig.10 are these triangles placed on the
spherical surface together with a spherical surface. Fig.11 shows the grid composed of
modified three-strut cells, creating a spherical tensegrity construction.
Figure 8
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Figure 9
Figure 10
Figure 11: Spherical tensegrity construction
3
Geometry of prismatic four-strut tensegrity grids
Geometry of four-strut cell is dependent on a single parameter a (Fig.12). The strut length is
expressed by s = a 20 3 , the length of the cables in the lower horizontal plane is c1 = a 2 ,
in the upper horizontal plane is c2 = 2a and the height of the cell is h = a 5 3 .
7
8
5
7
4
6
8
6
4
1
3
2
3
1
a
5
h
2
Figure 12: Four-strut cell
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The planar tensegrity grid is created by connection of four-strut cells of type node on node.
Coordinates of nodes P (k ) = ( x(k ), y (k ), z (k )) for k = 1,2,3,4 and k = 5,6,7,8 of one cell in
the planar grid are expressed in equations (9)
π π
for k = 1,2,3,4 and ϕ (k ) = (k - 1) + , z (k ) = 0,
2 4
P (k ) = ( x(k ), y (k ), z (k )) =
(
2a cos ϕ (k ) + a,
π
for k = 5,6,7,8 and ϕ (k ) = (k - 4 ) , z (k ) = a
2
)
2a sin ϕ(k ) + a, z (k )
1
+ 2
2
(9)
P (k ) = ( x(k ), y (k ), z (k )) = ( a cos ϕ (k ) + a, a sin ϕ(k ) + a, z (k ))
where parameters a, h are illustrated in Fig.10. Planar tensegrity construction is created by
connecting of 3×3 four-strut cells. Coordinates of nodes P(i, j, k ) in the planar grid for
i = 1,...,3, j = 1,...,3, k = 1,...,8 are expressed in the equations (10)
x (i, j, k ) = x (k ) + 2a (i − 1),
y (i, j, k ) = y (k ) + 2a ( j − 1),
(10)
z (i, j, k ) = z (k ),
where x (k ), y (k ), z (k ) are coordinates of nodes P(k ) in one cell expressed in equations (9).
Figure13: Planar four-strut tensegrity grid
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The elementary four-strut cell can be modified such that the grid created by these cells is
surface with one curvature (cylindrical surface in Fig.14). Then the coordinates of nodes in
this grid can be described by equations (11), where the nodes lying in the plane can be
transformed to the nodes lying on the cylindrical surfaces wih radii R and R+h,
P (k ) → P (i, j, k ) , i = 1,...,7, j = 1,...,7
x (i, j, k ) = x (k )cos α i − ( z (k ) + R ) sin α i ,
y (i, j, k ) = y (k ) + 2a ( j − 1),
(12)
z (i, j, k ) = x (k ) sin α i + ( z (k ) + R )cos α i − R,
where the parameter a is modified to the parameter b = a (1 + h R ) for nodes
P(k ), for k = 6,8 , the parameter α i = −6α + (i − 1) 2α , where α = arctan(a R ) , R is the
radius of the created cylindrical surface.
Figure 14: Cylindrical four-strut tensegrity grid
Coordinates of nodes P ′ (i, j, k ) in the grid creates the surface with two curvatures R1 and R2
can be expressed by equations
x ′ (i, j, k ) = x (k )cos α i − ( z (k ) + R1 ) sin α i ,
y ′ (i, j, k ) = y (k ),
(12)
z ′ (i, j, k ) = x (k ) sin α i + ( z (k ) + R1 )cos α − R1 ,
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x(i, j, k ) = x′(i, j, k )
y (i, j, k ) = y′(i, j, k )cos β j − ( z ′(i, j, k ) + R2 ) sin β j ,
(13)
z (i, j, k ) = y′(i, j, k )sin β j + (z ′(i, j, k ) + R2 ) cos β j − R2 ,
where the parameter a for nodes P(i, j, k ), if k = 6,8 in equations (10) is modified to
b1 = a (1 + h R1 ) and for nodes P(i, j, k ), if k = 5,7 to b2 = a (1 + h R2 ) , α = arctan(a r1 ) ,
β = arctan(a r2 ) , α i = −6α + (i − 1) 2α , βi = −6β + ( j − 1) 2β where r1 = sgn 1 R1 and
r2 = sgn 2 R2 are radii of curvature of the cylindrical grid.
In Fig.15 is displayed tensegrity construction, that is created by 7×7 two times modified fourstrut cells, where radii of curvature of created surface are the same and have the same
orientation ( r1 = r2 ). This construction creates a part of the spherical surface.
Figure 15: Spherical four-strut tensegrity grid
4
Conclusion
This paper aimed to provide some examples of a planar or a non planar prismatic tensegrity
grid constructios. Some are only geometrical studies without equilibrium considerations. It is
not easy to define tensegrity constructions. It could be claimed that everything in the universe
is tensegrity with properties related to the continuum of tensioned components. Tensegrity
structures are the most recent addition to the array of systems available to designers. The
concept itself is about eighty years old and it came not from within the construction industry
but from the world of arts. Although its basic building blocks are very simple – a compression
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component and a tension component – the manner in which they are assembled in a complete,
the stable system is by no means obvious.
The idea was adopted into architecture in 1960 when Maciej Gintowt and Maciej Krasiński,
architects of Spodek, a venue in Katowice in Poland, designed it as one of the first major
structures to employ the principle of tensegrity. The roof uses an inclined surface held in
check by a system of cables holding up its circumference. Another example of a practical
implementation of the tensegrity system is Seoul Olympic Gymnastic Arena designed by
David Geiger in the 1980 for the 1988 Summer Olympics. The Georgia Dome which was
built for the 1996 Summer Olympics is a large tensegrity structure of similar design to the
aforementioned Gymnastics Hall.
Acknowledgements
This work was supported by VEGA 1/0321/12 „Theoretical and experimental analysis of adaptive
cable and tensegrity systems under static and dynamic stress considering the effect of wind and
seismic“.
References
[1]
Kmeť S., Platko P., Mojdis M. (2010). Design and analysis of tensegrity systems. The
international Journal “Transport & Logistics”. Volume 18. pp. 42-49
[2]
BURKHARDT R.W. (2008). A Practical Guide to Tensegrity. Cambridge, USA. MA 021420021
[3]
Motro R. (2003). Tensegrity. Structural Systems for the Future. Kogan Page Limited, London
and Sterling.
http://www.google.com/books?hl=sk&lr=&id=0n0K6zOB0sC&oi=fnd&pg=PR7&dq=Tensegrity:+Structural+Systems+for+the+Future&ots=85AsV
SIo_l&sig=amBKpKIADyZVPlpc0vVQ03g0OUI#v=onepage&q&f=false
[4]
Olejníková T. (2010). Geometry of tensegrity systems. In: Proceedings of scientific works
„Innovative approach to modeling of intelligent construction components in building 2010“.
Košice 2010. pp98-104
[5]
Olejníková T. (2011). Prismatic Tensegrity systems. SSP-Journal of Civil Engineering, Selected
Scientific Papers. Volume 6 Issue 2. pp. 39-46
[6]
Olejníková T. (2012). Double Layer Tensegrity Grids. Acta Polytechnica Hungarica, Journal of
Applied Sciences. Volume 9 Issue 5. pp. 95-106
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