a simplified strategy for force finding analysis of suspendomes

8/12/2019 A Simplified Strategy for Force Finding Analysis of Suspendomes

1/13

Engineering Structures 32 (2010) 306318

Contents lists available atScienceDirect

Engineering Structures

journal homepage:www.elsevier.com/locate/engstruct

A simplified strategy for force finding analysis of suspendomes

Qing-shuai Cao a,, Zhi-hong Zhang b

a Huasen Architectural & Engineering Design Consultants Ltd., Hangzhou 310012, Chinab China Construction(Shenzhen) Design International, Shanghai 200235, China

a r t i c l e i n f o

Article history:

Received 17 December 2008Received in revised form

8 September 2009

Accepted 9 September 2009

Available online 8 October 2009

Keywords:

Space structure

Suspendome

Sports buildingTensegrity system

Self-internal-force mode

Prestress

a b s t r a c t

The suspendome has been widely used as the structural roof system of sports buildings in recent years.

It is a kind of hybrid space structure composed of an upper rigid single-layer latticed shell and a lowerflexible tensegrity (cablestrut) system. The prestress level in the lower cablestrut system is of great

significancefor thesuspendome structurebecause it has no initial geometric stiffness(for a ribring type)beforeprestress is introduced into the lower tensegritysystem. The traditional solution for calculatingtheself-internal-force mode and the prestress force level (forcefinding) is somewhat complicated; in general

it is based on the Equilibrium Matrix Theory. In the present paper, a simplified computational strategyfor the determination of the self-internal-force mode based on the nodal equilibrium is presented for the

tensegrity system in a suspendome which is grounded on a newly developed method: the Local AnalysisMethod. Two types of cablestrut arrangement, the Levy system and the Geiger system, are addressed,

andthe characteristic of each type is expounded. An analytical solution forthe self-internal-force mode ofthe lower cablestrut system is put forward together with the equivalent nodal force acting on the upper

single-layer dome for the two types of cablestrut arrangement. The determination of the prestress levelof the lower tensegrity system is then elucidated on the ground of the initial architectural configuration,

the counterbalance of the bearing reaction, the equivalent nodal force, and the wind-induced slackeningeffect. Anillustrativeexampleis appended inthe endto validate theefficiency of this simplified method. It

is shownthat force finding, at theviewpoint of structuraldesign, based on this methodis of great accuracyand efficiency. The prestress in the outermost ring generally has the highest level among the cablestrutsystem, and has the most influence on the structural performance of the suspendome. The results from

thestudies canbe referred to notonly fordirectdesign usein practicalengineering, butalso forthe designof similar hybrid space structures.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

A tensegrity system is nowadays a hot topic of large-spanspace structures. Cablestrut systems as the tensegrity system aremainly used in two kinds of space structure. The first kind consistsonly of tension-only cables and compression-only struts, in whichall elements are pin jointed and no bending moment exists. Asthe tensegrity structure has hardly any natural rigidity, it wouldbecome stable only if proper prestress is introduced to the system.The rigidity of the structure results from the self-equilibration offorces between the tensile cables and the compressive struts [ 1,2].The structural investigation involves firstly the morphologicalanalysis including the geometric configuration finding (formfinding) [3,4] and the determination of prestress level (forcefinding). Analytical and experimental researches on the structuralbehavior of tensegrity structures have been performed during

Corresponding author. Tel.: +86 0571 87397327.

E-mail address: [email protected](Q.S. Cao).

the past several decades [58]. The most successful applicationof the tensegrity system is the cable dome structure proposedby Geiger and Levy, which is exemplified by several famous

gymnasiums built for the Olympic Games [9]. The second kind canbe found in the so-called hybrid space structures, composed of therigid primary structure, tension-only cables and compression-onlystruts. In the hybrid structure, the cablestrut system is used tostiffen the primary structure, which may be a column [1014], abeam [15,16],an arch [17], a plane or space truss, or even a latticedshell [18]. As a result, various types of hybrid space structuresare explicitly proposed, such as the cable-stayed column, thebeamstring structure, the bowstring structure, the suspendome

structure, etc. Many investigations have been performed forhybrid space structures, in which structural characteristics suchas the buckling strength and the structural rigidity are provedto be increased greatly[19,20]. The cable-stiffened latticed shell:suspendome is a typical example of the hybrid space structure,which is the focus of this paper.

The suspendome is a newly developed large-span hybrid spacestructure which has been widely used for the structural roof

0141-0296/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.09.017
http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2009.09.017http://dx.doi.org/10.1016/j.engstruct.2009.09.017mailto:[email protected]://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstruct


2/13

Q.S. Cao, Z.H. Zhang / Engineering Structures 32 (2010) 306318 307

Fig. 1. Suspendome: Beijing University of Technologys Gymnasium (Badminton

and Eurythmics Arena for the 2008 Beijing 29th Olympic Games, China).

system of sports buildings in recent years (Figs. 13). As a newtype of hybrid space structure which organically integrates the

lower flexible tensegrity (cablestrut) system into the rigid uppersingle-layer latticed dome, it has been attracting engineers and

researchers throughout the world since it was put forward, and ithas caused vigorous vitality [21]. A suspendome is composed of

three kinds of member: rigid beam members connected usuallywith welded hollow balls in the upper single-layer latticed shell,

and compression-only vertical struts and the tension-only cablesin the lower tensegrity system [22,23]. A suspendome can be

derived from two types of structure. (1) It is formed by replacingthe upper cables of the cable dome by rigid members, usually a

single-layer latticed dome. Compared with the cable domes, theprestress level in the suspendome can be observably reduced.

As the upper rigid members can provide certain initial stiffnessto the structure, the construction techniques are also simplified.

The upper rigid member can resist both the axial force and thebending moments to increase the rigidity of the structure on

the one hand, and on the other hand the internal stress/forceflow in suspendome could be built in a closed loop, thus making

the structure a self-equilibrated system, and a weak boundarybearing system becomes possible [22,23]. (2) The upper single-

layer latticed dome is strengthened by the lower tensegrity system.The suspendome performs like a double-layer latticed dome when

prestress is introduced to the lower tensegrity system, thereforeenhancingthe stiffnessof the latticed shell andimproving the load-

carrying capacity of stability. These characteristics make it possiblefor the dome to have a larger span, satisfying extremely well the

requirements of architectural design for larger spaces. In addition,the horizontal bearing reaction induced by the service load is the

reverse of that induced by thetensegrity system,and thehorizontalbearing reaction can be reduced to zero if proper prestress in

the tensegrity system is introduced. Compared to the single-layerdome, the internal force distribution is improved substantially

for a suspendome under service load. As a result, the internalforces transfer from the upper shell to the lower high-strength

cablestrut system. The maximum value of internal force in theupper single-layerdome memberscan consequentlybe reduced by

the tensegrity system[2225]. The high tensile strength of cablesis fully employed in the rigidflexible composite structure, and the

self-weight of the upper single-layer dome (steel weight index)is reduced [23]. Thereby, the suspendome is known as a hybrid

space structure.

The suspendome was first brought forward as a new indepen-dent space structure in 1993 [2628]. Experimental research on

the structural static and dynamic behavior of suspendomes has

been well investigated through a full-size model, in which thelower tensegrity system showed great influence on the boundary

Fig. 2. Suspendome: Jinan Olympic Sports Centers Gymnasium (centerpiece for

the 2009 11th National Olympic Games, China).

Fig.3. Suspendome:Changzhou Olympic Sports Centers Gymnasium(centerpiece

for the 2010 Provincial Olympic Games, China).

reaction and the member internal force of the upper shell [28].

Model tests on a small-scale suspendome with a diameter of 3.0 mand a rise of 0.45 m have also been carried out [26], in which theload-carrying capacity of stability and the rigidity were provedto be increased greatly compared with the single-layer dome.Finite element (FE) analysis results also showed good agree-ment with results from experiments. Analytical researches on sus-pendomes have also been carried out in various approaches byresearchers[22,24,25]. One of the most challenging issues in sus-pendome design involves the determination of the initial prestressdistribution and prestress level in the lower cablestrut system. Asis known to all, the structural integral stiffness Kconsists mainlyofthe integral geometricalstiffness KG andthe integral linearstiffnessKL(K =KG + KL). For theflexible cables in the lower tensegrity sys-tem, the initial geometrical stiffness KGis usually so small as to beomitted before prestress is introduced,and the integral linear stiff-nessKL usually accounts for the structural integral stiffness. Thus,the determination of rational initial prestress in the lower tenseg-rity system is of great importance for a suspendome. An analyti-cal solution, such as the Equilibrium Matrix Theory [2931], hasbeen put forward for the determination of the self-internal-forcemode in the tensegrity system. The self-internal-force mode andthe mechanical displacement mode can be obtained on the basis ofthe decomposition of the equilibrium matrix through a numericalmethod: singular value decomposition or Gauss elimination [31].For better elucidation of the tensegrity system, the concept ofintegral feasible prestress involving a specific equilibrium statewas put forward based on the independent self-equilibrium stressmode of mechanics [9]; in this case, the cables are tension-onlyelements and the struts are compression-only elements. The Equi-

librium Matrix Theory was shown to be advantageous for tenseg-rity systems such as the cable dome, but was a little farfetched


3/13

308 Q.S. Cao, Z.H. Zhang / Engineering Structures 32 (2010) 306318

for hybrid (rigidflexible) space structures, such as a suspendomeor beam string structure. It mainly results from the complexity ofbeam elements in the upper shell which have a great number ofdegrees of freedom (DOFs), consequently resulting in a great num-

ber of self-internal-force modes and difficulty in calculation. Toexpand the theory to be used for the hybrid structure, the LocalAnalysis Method[22]was then developed based on the subma-trix of the equilibrium matrix, in which the lower cables and strutsare detached from the suspendome and analyzed separately. Therestrains of the upper single-layer dome to the lower cablestrutsystem are applied as boundary conditions for the analyses. Theinternal prestress distribution of the upper latticed shell can alsobe determined by the equivalent reactions of the lower tensegritysystem.

The suspendome has become popular in recent years due toits attractive structural performance, which is exemplified by thesports buildings built worldwide in recent decades, such as theHikarigaoka Dome with a span of 35 m in Japan in 1994, andthe Fureai Dome with a span of 46 m in Japan in 1997 [21]. The

suspendome has also been developed rapidly in China, such as theKiewitt suspendome with a span of 34.5 m and a rise of 4.6 m(rise to span ratio 1/7.5) constructed in Tianjin, in 2001 [21,24].

The most attractive are three gymnasiums presently constructedin China: (1) Beijing University of Technologys Gymnasium (Fig. 1,constructed in 2006) with a span of 93 m and a rise of 10 m(rise to span ratio 1/9.3),[32]. It was used as the Badminton andEurythmics Arena forthe 29thBeijing Olympic Games held in 2008.(2) Jinan Olympic Sports Centers Gymnasium(Fig. 2,constructedin2008)with a spanof 122 m and a riseof 12.2m (rise tospan ratio1/10). It served as the centerpiece of the 11th National OlympicGames held in the cityin 2009 [23]. (3) Changzhou Olympic SportsCenters Gymnasium (Fig. 3, constructed in 2007) whose roof takesthe shape of an ellipse with a long span of 120 m, a short span of

80 m and a rise of 21.45 m. It will serve as the centerpiece of theProvincial Olympic Games to be held in the city in 2010.

Oneof the most significant problems forsuspendome design in-

volvesinitiallythe determinationof the prestress level in the lowertensegrity system. Whilst the force finding analysis has been ex-tensively explored by numerical approaches [23,24,33], few stud-ies have investigated the rational prestress level in suspendomes,and the optimized prestress level based on the structural perfor-mance of suspendomes has received even less attention. This pa-per focuses mainly on the simplified calculation method for thedetermination of self-internal-force mode and the rational pre-stress force level (force finding) in the lower tensegrity systembased on the Local Analysis Method [22], for the purpose of directdesign use in practical engineering of suspendomes.

2. Levy and Geiger types of the lower cablestrut system

A single-layer spherical latticed shell is generally used for theupper constituent part of suspendomes together with the lowertensegritysystem. On thebasis of theform of lattice arrangement,aspherical single-layer shellis usually categorized into the followingtypes: the Keiwitt Pattern, the Sunflower Pattern, the SchwedlerPattern, the Geodesic Pattern, etc., or a hybrid pattern combiningtwoor more of the above types [21]. The category of a suspendomecan be specified on the basis of the type of the upper single-layerlatticed shell[34].

Generally, for the lower tensegrity system in a suspendome,there are two types of cablestrut arrangements, called the Levysystem and the Geiger system, as shown inFig. 4.The main differ-ence between these two types is the arrangement of the diagonalcables. For the Levy type of cablestrut arrangementFig. 4(1), the

vertical strut and the diagonal cables are not in the same verticalplane. The positions of the vertical struts are staggered by 180/m

in the circumferential direction compared with the neighboring

rings; herein,m is the number of equally divided segments of the

hoop cable. The vertical strut in the outer ring is connected at the

top end with two adjacent diagonal cables in the neighboring inner

ring (the connection joints in a practical suspendome are as shown

inFig. 5), thus making the cablestrut system a geometrically sta-

ble system in both radial and circumferential directions. For the

Geiger type of cablestrut arrangementFig. 4(2), the vertical strut

and the diagonal cable are in the same vertical plane and are ar-

ranged radially and circumferentially (ribring type). The vertical

strut in the outer ring is only radially connected at the top endwith

one diagonal cable in the neighboring inner ring (the connection

joints in a practical suspendome are as shown inFig. 6), thus mak-

ing the cablestrut system a geometrically stable system only in

the radial direction. Structural measures, such as adding of the di-

agonal cables or rods in the hoop plane, are used to strengthen the

circumferential stiffness. With respect to their different character-

istics, the Levytype usually matches a sunflower patterned latticed

dome, and the Geiger type usually matches a Schwedler latticed

dome. As for the sunflower patterned latticed dome, the lattices

become denser and denser from the boundary to the center, re-

sulting in the uneven length of members and difficulty during the

fabrication and construction. For the improvement of the member

arrangement of the sunflower patterned latticed dome, a hybrid

lattice arrangement is preferred as a rule, in which several central

rings of members are rearranged in another pattern, such as the

Keiwitt pattern. Such engineering examples could be established

by Beijing University of Technologys Gymnasium (with five rings

of cables and struts of Levy type) Fig. 7(a), Jinan Olympic Sports

Centers Gymnasium(with three rings of cablesand strutsof Geiger

type)Fig. 7(b), and Changzhou Olympic Sports Centers Gymna-

sium (with six rings of cables and struts of Levy type) Fig. 7(c). The

inner Keiwitt pattern and outer sunflower hybrid pattern are used

in these three single-layer shells.

3. Determination of the self-internal-force mode

On the basis of the construction procedure of the suspendome,

three typical reference configurations are defined explicitly in the

Chinese Technical Specification for Prestressed Steel Structure[34]:

(1) The zero state geometry, which refers to an equilibrium state

without prestress and without self-weight in the structure to

determine the lofting during construction. (2) The initial state

geometry, which is the equilibrium state with prestress in the

structure and with self-weight and partial or all service loads (for

example 1.0 times the dead load plus 0.5 times the live load)

after cambering, corresponding to the constructed configuration

of the structure. (3) The loading state geometry, which is the

equilibrium state after loading. The initial configuration (i.e. the

architectural configuration when constructed) is usually presentedby thearchitect as the referenceconfiguration forstructural design.

The initial prestress distribution in the tensegrity system (force

finding) and structural deformation are always based on the initial

state geometry of suspendome as the reference configuration in

this paper, except if it is specifically stated. It is also proved

that the discrepancy of geometric configurations between the

zero state and the initial state is rather slight, and this is even

more obvious for a suspendome with a comparatively rigid upper

latticed shell. Different from the tensegrity structure, the form

finding analysis for suspendome, from the viewpoint of structural

design, is usually dispensable for the sake of the simplification of

design,whilea force findinganalysis is essential. In addition, a form

finding analysis is also necessary for cases where the geometric

configuration under zero state is expected to be accuratelydetermined.


4/13


(a) Plan view

(b) Front view

(c) Perspective view (a) Plan view

(b) Front view

(c) Perspective view

(1) Levy type of cablestrut arrangement. (2) Geiger type of cablestrut arrangement.

Fig. 4. Two types of cablestrut arrangement in the lower tensegrity system. (Note: DC, diagonal cable; HC, hoop cable; VB, vertical strut.)

Fig. 5. Real Levy type of cablestrut connection joint used in Beijing University of Technologys Gymnasium, China.

Fig. 6. Real Geiger type of cablestrut connection joint used in Jinan Olympic Sports Centers Gymnasium, China.

The previous method (Equilibrium Matrix Theory) for thedetermination of the self-internal-force mode in the lowertensegrity system seems very complicated and lengthy, and itis usually incomprehensive to the practicing engineers. In thissection, a simplified strategy for calculation of the self-internal-force mode is presented based on the Local Analysis Method[22].

3.1. Simplified calculation method for initial prestress distribution of

the Levy type arrangement

For the lower tensegrity system with n rings of cables andstruts, all of the rings are independent of each other, as every

ring is connected only with the upper single-layer dome[22,23];here,n (n = 1, 2, . . . i, . . ., from the outermost to the innermostring) is the number of rings in the lower tensegrity system.Nodal equilibrium can be obtained separately from each ring ofthe cablestrut system. Typical nodal force equilibrium of thecablestrut arrangement of Levy type in the lower tensegritysystem is shown inFig. 8.

The suspendome is usually designed as a centrosymmetricspherical structure with a certain topology rule of member con-nection. So the structural deformation and the internal force are

induced regularly under the symmetrical load (such as the full-span dead or live load). As the upper single-layer dome is con-


5/13


(a) Beijing University of Technologys

Gymnasium.

(b) Jinan Olympic Sports Centers

Gymnasium.

(c) Changzhou Olympic Sports Centers

Gymnasium.

Fig. 7. Hybrid lattice in the upper single-layer shell.

Fig. 8. 3-D axial force equilibrium in typical nodeifor the Levy type arrangement.

nected with the lower tensegrity system through m nodes (m isthe number of equally divided segments of the hoop cable) at

the top of the vertical strut in each ring, these m nodes havethe same displacement for symmetry under full-span uniformlydistributed load (which is usually the case) and the equivalentnodal force induced by the ith ring of cables and struts. The mnodes at the upper single-layer dome are specified as one group,and the number of groups is equal to that of the rings in thelower tensegrity system. So the group number is also specified asGroup 1, . . . , Groupi, . . . , Groupnfrom the outer to the inner.

3.1.1. Determination of the self-internal-force mode

In the vertical symmetrical plane formed by the intersectionof the Symmetry line and the vertical strut and the horizontalprojected plane, the nodal force equilibrium equations in thevertical and the horizontal directions at node i can be expressed

as follows:2 cos(i)Ndi+ Nvi = 0

2 sin(i) cos(i/2)Ndi 2 cos(i/2)Nhi = 0 (1)

where

Nhi denotes the axial force of the hoop cable in the ith ring;Ndi denotes the axial force of the diagonal cable in the ith ring;Nvi denotes the axial force of the vertical strut in the ith ring;i denotes the inner angle between the neighboring hoop cables

in theith ring;i denotes the angle between the diagonal cable and the vertical

strut in theith ring;i denotes the inner angle between the projections in the

horizontal projected plane of the two neighboring diagonalcables in theith ring.

The force equilibrium in the third direction is satisfied automat-ically for symmetry.

Solving the above equations, the axial force Nvi, Ndi can beobtained expressed byNhi:

Nvi = 2cot(i)

cos(i/2)

cos(i/2) Nhi = kviNhi (2)

Ndi = csc(i)cos(i/2)

cos(i/2)Nhi = kdiNhi (3)

wherekviand kdiare given by

kvi = 2cot(i)cos(i/2)

cos(i/2)

kdi = csc(i)cos(i/2)

cos(i/2).

AssumingNhi = 1, and that a unit constant khi = 1 is specified,the self-internal-force mode in the ith ring of the tensegrity systemcan be expressed as follows:

[Nhi, Ndi, Nvi]T =[khi, kdi, kvi]T . (4)

Herein, superscript T denotes the transpose of the vector, and kvi =Nvi/Nhiis the coefficient of prestress efficiency.

It should be pointed out that the prestress force of theoutermost ring has the most influence on the suspendome [23,24].In other words, the most effective way to improve the rigidity ofsuspendome is to increase the angle 1, as the coefficient kv1 isproportional to the hyper nonlinear function cot(), provided theangle 1, 1 remains unchanged (which is usually the case). Forinstance, if1 is initially 80

, and is optimized to 75, then thecoefficient kv1increases by 54.3% fora small increment of the angle1(cot(80

) = 0.1736, cot(75) = 0.2679), although the anglesi, i, iin different rings are actually very close.

3.1.2. Determination of the equivalent nodal force

For a suspendome with n rings of cables and struts, theequivalent nodal forces in each group induced by the n rings ofcables and struts which act on the connection joints of the uppersingle-layer dome with the vertical strut are investigated andpresented in detail in this section.

For thenth (innermost) group of nodes (upper end node of thevertical strut), with which the upper single-layer latticed shell isonly stiffened by a vertical strut, the equivalent nodal force is thecounterforce of the axial force of the vertical strut; thus,

FEhi = 0 (i= n) (5)

FEvi = Nvi = kviNhi (i = n). (6)

For the ith (i = 1, 2, . . . , n 1) group of nodes, the uppersingle-layer latticed dome is stiffened by the vertical struts of the


6/13


ith ring and meanwhile pulled radially downwards by the diagonalcables of the (i + 1)th ring and the upward supporting force tothe upper single-layer dome is usually counterbalanced partiallyby the diagonal cable; thus the equivalent nodal forces inducedby the lower cables and struts in the horizontal (radial) and thevertical directions are expressed as

FEhi = 2cos(i+1/2 /mi+1)

cos(i+1/2)

cos(i+1/2) Nh,i+1

= kEhiNh,i+1 (i = 1, 2, . . . , n1) (7)

FEvi = Nvi Nd,i+1cos(i+1) = Nvi+ Nv,i+1= kviNhi kv,i+1Nh,i+1 (i = 1, 2, . . . , n1) (8)

in whichkEhiis given by

kEhi = 2cos(i+1/2 /mi+1)cos(i+1/2)

cos(i+1/2), and

FEhi denotes the horizontal equivalent nodal force acting on thenodes of theith group;

FEvi denotes the vertical equivalent nodal force acting on the nodesof theith group;

mi+1denotes the number of equally divided segments of the hoop

cable in the (i+1)th ring; denotes the circumference ratio

.=3.14159.

It is obvious from Eq. (8) that any improvement of the rigidity ofthe upper single-layer dome would be irrealizable if the axial forceof the vertical strut in the ith ring is less than that in the (i+ 1)thring. In order to strengthen the vertical rigidity of the upper single-layer dome, the upward resultant force FEvi (i = 1, 2, . . . , n 1)should be not less than zero. That is to say, the axial force of thevertical strut in the outer ring should be greater than that in theinner ring; thus,

FEvi > 0, i.e.

Nvi > Nv,i+1, or

kvi

Nhi

>kv,i+1

Nh,i+1

. (9)

For the boundary joints, the equivalent nodal forces acting onthe boundary which are induced by the diagonal cable of theoutermost (i = 1) ring of cables and struts are expressed in thehorizontal (radial) and vertical directions as follows:

FEh1 = 2cos(1/2 /m1)cos(1/2)

cos(1/2)Nh1 = kEh1Nh1 (10)

FEv1 = 2cos(1)Nd1 =Nv1 = kv1Nh1 (11)

in whichkEh1is given by

kEh1 = 2cos(1/2 /m1)cos(1/2)

cos(1/2), and

FEh1 denotesthe bearing horizontal equivalent nodal forceinducedby the first ring;FEv1 denotes thebearing vertical equivalent nodal force induced by

the first ring;m1 denotes the number of equally divided segments of the hoop

cable in the first ring.

Generally, a suspendome is supported at the bottom ring beam.The bearing joints are restrained in the circumferential, radialand vertical directions (Ux, Uy, Uz). Large reactions are inducedwhen the upper single-layer dome is under the service loador prestressed by the lower tensegrity system separately. Fromfinite element analysis, it is found out that the inner ( i > 1)rings of cables and struts have little influence on the boundarybearing reactions and the boundary reactions induced by the low

tensegrity system are mainly originated from the outermost ( i =1) ring of cables and struts. Herein, the analytical expression

Fig.9. 3-D axial forceequilibriumin typicalnode i forthe Geiger typearrangement.

for the joint reaction induced by the lower tensegrity system ischiefly addressed. For an upper single-layer dome prestressed bythe outermost ring of cables and struts, the force equilibriumin the vertical direction is automatically satisfied; consequentlythe bearing joint reaction in the vertical direction is zero. The

bearing joint reactions in the horizontal (radial) direction resultfrom the horizontal component of axial forces of the diagonal cablein the outermost ring and the horizontal pull force induced by thearch effect of the dome when supported by the vertical strutin the outermost ring. The bearing joint horizontal (radial) andvertical reactionsRh and Rv induced by the tensegrity system canbe expressed as

Rh = 2cos(1/2 /m1)cos(1/2)

cos(1/2)Nh1 tan(1)Nv1

= kEh1Nh1+ Rha = kEh1Nh1+ kRa1Nh1 = kRhNh1 (12)

Rv =0 (13)

in whichkRa1andkRhare given by

kRa1 =

2cos(1/2)

cos(1/2) ;

kRh = kEh1+ kRa1 = 2 [cos(1/2 /m1)+1]cos(1/2)

cos(1/2), and

Rha =kRa1Nh1 denotes the horizontal pull force induced by thearch effect of the dome when supported by thevertical strut in the outermost ring; it is an approx-imate value obtained by the numerical fitting;

Rh =kRhNh1 denotes the bearing joint horizontal reaction in-duced by the outermost ring of cables and struts;

Rv denotes the bearing vertical reaction induced by theoutermost ring of cables and struts.

3.2. Simplified calculation method for initial prestress distribution of

the Geiger type arrangement

Based on the same method, the self-internal-force mode for acablestrut arrangement of Geigertype is presented as follows. Thetypical nodal force equilibrium state for Geiger type in the lowertensegrity system is illustrated inFig. 9.

3.2.1. Determination of the self-internal-force mode

In the vertical symmetrical plane formed by the intersection ofthe Sym. line and the vertical strut and the horizontal projectedplane, the force equilibrium equations in the vertical and thehorizontal directions in nodeican be expressed as

Nvi+ Ndicos (i) = 0Ndisin (i)2Nhicos (i/2) = 0

(14)

where


7/13


Nhi denotes the axial force of the hoop cable in the ith ring;Ndi denotes the axial force of the diagonal cable in the ith ring;Nvi denotes the axial force of the vertical strut in the ith ring;i denotes the inner angle between the neighboring hoop cables

in theith ring;i denotes the angle between the diagonal cable and the vertical

strut in theith ring.

The force equilibrium in the third direction is also satisfiedautomatically for symmetry.Combining the above equations, the axial force Nvi, Ndi can beobtained expressed byNhi:

Nvi = 2cos (i/2)

tan (i)Nhi = kviNhi (15)

Ndi =2cos (i/2)

sin (i)Nhi =kdiNhi (16)

wherekviand kdiare given by

kvi =2cos (i/2)

tan (i)

kdi =2cos (i/2)

sin (i)

.

AssumingNhi = 1, and that a unit constant khi = 1 is specified,the self-internal-force mode in the ith ring of the tensegrity systemcan be expressed as follows:

[Nhi, Ndi, Nvi]T =[khi, kdi, kvi]

T . (17)

3.2.2. Determination of the equivalent nodal force

For the nth (innermost) group of nodes, the equivalent nodalforce is expressed as

FEhi = 0 (i= n) (18)

FEvi = Nvi = kviNhi (i = n). (19)

For theith (i = 1, 2, . . . n 1) group of nodes, the equivalentnodal forces in the horizontal (radial) and the vertical directionsare expressed as

FEhi = 2cos(i+1/2)Nh,i+1= kEhiNh,i+1 (i = 1, 2, . . . , n1) (20)

FEvi = Nvi Nd,i+1cos(i+1) = Nvi+ Nv,i+1= kviNhi kv,i+1Nh,i+1 (i = 1, 2, . . . , n1) (21)

in whichkEhiis given by

kEhi = 2cos(i+1/2), and

FEhi denotes the horizontal equivalent nodal force acting at thenodes of theith group;

Fvhi denotes the vertical equivalent nodal force acting at the nodesof theith group.

In order to strengthen the vertical rigidity of the upper single-layer dome, the upward resultant force FEvi (i = 1, 2, . . . , n 1)should be also not less than zero; thus

FEvi >0, i.e.

Nvi > Nv,i+1, or

kviNhi > kv,i+1Nh,i+1. (22)

For the boundary joints, the equivalent nodal forces acting onthe boundary which are induced by the diagonal cable of theoutermost (i = 1) ring of cables and struts are expressed in thehorizontal (radial) and vertical directions as follows:

FEh1 =2 cos(1/2)Nh1 =kEh1Nh1 (23)

FEv1 =2 cos(1)Nd1 = Nv1 = kv1Nh1 (24)

in whichkEh1is given bykEh1 = 2 cos(1/2), and

FEh1 denotes the bearing horizontal equivalent nodalforce inducedby the first ring;

FEv1 denotes the bearing vertical equivalent nodal force induced bythe first ring.

On the basis of the same viewpoint, the bearing joint horizontal(radial) and vertical reactionsRh and Rv induced by the tensegritysystem can be expressed as

Rh = 2cos(1/2)Nh1 tan(1)Nv1= kEh1Nh1+ Rha = kEh1Nh1+ kRa1Nh1 =kRhNh1 (25)

Rv =0 (26)

in whichkRa1and kRhare given by

kRa1 =2 cos(1/2)

kRh = kEh1+ kRa1 = 4 cos(1/2), and

Rha = kRa1Nh1 denotes the horizontal pull force induced by thearch effect of the dome when supported by thevertical strut in the outermost ring; it is an approx-imate value obtained by the numerical fitting;

Rh = kRhNh1 denotes the bearing joint horizontal reaction in-

duced by the outermost ring of cables and struts;Rv denotes the bearing vertical reaction induced by the

outermost ring of cables and struts.

In addition, the geometrical stability of the independent ringof cables and struts in the two kinds of tensegrity system is alsoinvestigated by the Equilibrium Matrix Theory [2931]. The resultsfrom singular value decomposition of the equilibrium matrixindicate that the number of independent self-internal-force modess = 1 and the number of independent inextensional displacementmodes m = 0 for the Levy type arrangement, ands = 1,m = 1for the Geiger type arrangement. In most cases, some diagonalcables are usually added in the vertically hoop plane for theGeiger type arrangement; the system thus becomes kinematicallydeterminate. As no internal inextensional displacement exists,

both types of tensegrity system studied here are geometricallystable.

4. Determination of prestressforce level in the lower tensegrity

system

The introduction of prestress is of great significance for asuspendome because it has no initial geometric stiffness beforebeing prestressed in the lower tensegrity system. No criteriaof recommendations have been provided so far for a rationalprestress level in the lower tensegrity system of a suspendome,although some recommendations are available from Refs.[3537]on cable-tensioned structures and bowstring structures. Theprestress ratio among the hoop cables of each ring is specified

discretionarily only to satisfy inequality(9)or(22)in the previousliteratures; for instance, the prestress ratio from the outermostto the innermost hoop cables is defined empirically as 5:4:3:2:1for a suspendome with five rings of cables and struts [24,25,33].The prestress level has great influence on the load-carryingcapacity and the initial configuration of the suspendome[22,23].The proper prestress level is supposed to not only assure theconsistency with the architectural configuration, but also tosimplify the constructionof the suspendome. This is because a verylarge prestress level would undoubtedly add to the difficulty ofelemental fabrication and structural construction. Accordingly, areasonable prestress ratio should be specified on the ground of itseffects.

It is also addressed in Chinese technical specification [34] that

the determination of the prestress level in the low tensegritysystem should be grounded on the principle that the equivalent


8/13


nodal force of the service load on the upper single-layer dome

is counteracted by the tensegrity system to a minimum and thebearing reaction is reduced to a minimum. This section presents

the design method for obtaining a reasonable prestress level inthe lower tensegrity system based on the initial architecturalconfiguration, the bearing counterbalance, the equivalent nodal

force and the wind-induced slackening effect.

4.1. Design method based on the initial architectural configuration

On the basis of the definition of the initial state of a sus-

pendome [34], the upper single-layer dome identical to the ar-chitectural configuration is supposed to keep undeformed when

it performs under the service load combined with the equivalentforce of the lower tensegritysystem. Fora suspendome with n ringsof cables and struts, the consistent equation of the nodal vertical

deflection in thengroups is expressed as follows:

d11 . . . d1n

... dij...

dn1 . . . dnn

nn

Nh1...

Nhj

...Nhn

n1

+

dL1...

dLi

...dLn

n1

=

0...

0

...

0

n1

(27a)

which is rewritten for short as follows:

[D] {N}+{dL}={0} (27b)

in which

[D] =

d11 . . . d1n... dij

...dn1 . . . dnn

nn

is the flexibility matrix;

{N} =

Nh1, . . . , Nhj, . . . , NhnT

is the force vector of the hoop

cables;

{dL} =[dL1, . . . , dLi, . . . , dLn]T is the vertical displacementvector of the nodes in the ngroups under service load;dijis the vertical displacement of the nodes in the ith groupresulting from the equivalent force of the jth ring (usuallyassuming the unit axial force in the hoop cable, i.e. Nhj = 1);

dLiis the vertical displacement of the nodes in the ith groupinduced by the full-span uniformly distributed service load(e.g. 1.0 dead load+1.0 live load).

Solving the above equation, the axial force in the hoop cables ofthe lower tensegrity system can be obtained by

{N}= [D]1 {dL} (28)

where [D]1 is the inverse matrix of[D], and the prestress level

of the hoop cables in each ring can be obtained. Subsequently,the prestress level of the whole tensegrity system is also obtainedaccording to the internal force mode defined by Eq.(4)or Eq.(17).

The coefficients d ij and dLi are very convenient to identify by

carrying out finite element (FE) analyses, in which only the FEmodel of the upper single-layerdome is needed.In the definition of

dLi, the uniformlydistributed loadp is generallyrequired to convertto the equivalent nodal force for simplification of calculation.The equivalent nodal force of the service load p is derived from

the product of the service load p with the nodal tributary areawhich is shown in Fig. 10. The nodal tributary area Ai in the

horizontal projective plane is accurately defined; it is formed bythe enclosure of the centroids of the relevant areas (the areas

related in the common node i) for the inner node and the centroids

of the relevant area and midpoint of the boundary element forthe boundary node(Fig. 10). Assuming that the enclosed area is

Fig. 10. Nodal tributary area of the inner node and the boundary node in the

latticed shell.

Ai about nodei, then the equivalent nodal force can be expressedas

FEqvi =Aip. (29)

FEqvi is the vertical equivalent nodal force induced by the serviceload;

Ai is theith node tributary Area;p is the uniformly distributed service load, the nominal

combination of the load effect may be chosen as the serviceloadp.

As thevertical deflections of nodesin the n groups are designedto be zero in the initial state, the designed configuration is identicalfor these nodes and approximate to the architecturalconfigurationforother nodes. Accordingly,the designed configuration could bestsatisfy the architectural requirements.

4.2. Design method based on the counterbalance of bearing reaction

For an integral prestressed structure, such as a cable dome ora membrane structure, a relatively rigid boundary, such as a rigidring beam, is absolutely essential as a constituent of the structure.Compared to an integral prestressed structure, a suspendome is

usually satisfied with a relatively flexible boundary as it is a self-equilibrium system. Theoretically speaking, if proper prestress isintroduced, the horizontal restraint at the boundary is no longerneeded when the horizontal reaction induced by the prestressforce in the tensegrity system could counterbalance that inducedby the service load. On the grounds of the above standpoint, thesecond design criterion could be obtained:

Rh + RL = 0, or (30)

kRhNh1 + RL = 0 (31)

in which

Rh is the bearing joint horizontal reaction induced by theoutermostring of cablesand struts as definedin Eq. (12) or (25);

RL is the bearing joint horizontal reaction induced by the full-spanuniformly distributed service load, as can be determined by the

finite element analysis of a single-layer dome.

Solving the above equations,Nh1can be obtained:

Nh1 = RL/kRh. (32)

Subsequently, the prestress force in other hoops Nhi(i = 2,. . . , i, . . . , n) canbe established assuming the same prestress forceratio among the hoop cables as defined in Eq.(28).The prestresslevel of the whole tensegritysystemcan be also obtained accordingto the internal force mode defined by Eq.(4)or Eq.(17).

4.3. Design method based on the counterbalance of equivalent nodal

force

One of principles of the determination of prestress force level

is the reduction of the equivalent nodal force of the upper single-layer dome induced by the service load, which is also specified


9/13


(a) Uniform wind suction. (b) Uniform self-weighted pressure.

Fig. 11. Schematic pressure distribution of wind action and roof weight. (Note: This is not a real but rather an approximate wind pressure distribution on a shallow dome

shape.)

in the Chinese Technical Specification [34]. Namely, the vertical

equivalent nodal forces induced by the lower cables and struts andthat induced by the service load are counteracted by each other atthe same nodes of the ngroups in the vertical direction; thus

FEvi+ FEqvi = 0 (i = 1, 2, . . . , n) (33)

in which

FEvi is the vertical equivalent nodal force induced by the lowercables and struts which acts at the nodes of the ith group, as

defined in Eqs.(6)and(8)or(19)and(21);FEqvi is the vertical equivalent nodal force induced by the service

load as defined in Eq.(29).

For the nth (innermost)group of nodes, based on Eq. (6) or (19),the Eq.(33)is rewritten as

Nhi = FEqvi/kvi (i = n). (34)

For theith (i = 1, 2, . . . n1) group of nodes, based on Eq.(8)or(21),Eq.(33)is rewritten in detail as

FEqvi = kviNhi kv,i+1Nh,i+1, or (35)

Nhi =

kv,i+1Nh,i+1 FEqvi

/kvi (i = 1, 2, . . . , n1) (36)

and the prestress level of the hoop cables in each ring can then beobtained. Subsequently, the prestress level of the whole tensegritysystem can be also obtained according to the internal force mode

defined by Eq. (4) or Eq. (17). With the above two methods,finite element analyses are not required for the determination ofthe prestress force level, and the design based on the method isthe simplest. Since the structural configuration is not taken into

consideration during design, there is usually a great discrepancybetween the designed and the anticipated configurations.

4.4. Design method based on the wind-induced slackening effect

It is noteworthy that wind action is supposed to introducean upward pressure (suction) to the whole structural roof for a

shallow dome, such as with the rise to span ratio f/L being lessthan 0.25 [3840]. For a shallow dome withf/L = 0.1, the shapecoefficient s ranges only from 0.85 to 1.0 for areas all over

the dome, and this is considered as a constant s = 1.0 inthis paper. The wind suction on the roof may be determined bythe expression w = zzsw0 [38]. For simplification in thepreliminary design, the dynamic factor zis assumed as 1.61.8,

and z = 0.5(zb +zt), inwhich zb, ztarethe exposure factorsfor wind pressure in height level above ground at the domesbottom and top, respectively.The values arerough approximations

and are expected to be conservative [41]. The typical windpressure distribution is assumed to be approximately uniform;

this is illustrated in Fig. 11(a). Light cladding materials such asa membrane or a rigid profiled steel sheet are normally used in

the roof for a large span structure. As a result, the uniform self-weighted load is usually less than 1.0 kN/m2, and the approximatedistribution of self-weighted load is illustrated in Fig. 11(b). Itis presumed that the weight of the roof is easily counteractedand surpassed by the wind load. In such circumstance, the wind-induced effect would dominate the structural design, inducing theunloading of members in the lower tensegrity system. It should

be guaranteed to avoid the slackening of cables for the designedprestress level, or the whole tensegrity system would not work.

Based on the membrane theoryof shells [42,43], the continuousdistributed reactionqw , qgin the boundary can be obtained for aspherical shell under two cases of pressure distribution, as showninFig. 11.

The meridional membrane resulting force qwper unit length inthe continuous shell for uniform wind suction (Fig. 11(a)) can becalculated by the following expression [42]:

qw =1

2wr.

The horizontal radial drag force in each bearing joint causedby wind pressure can thus be determined by integral of the aboveexpression and expressed as

Rwh =1

2wrL cos ( ) sin

m1

. (37)

The meridional membrane resulting forceqgper unit length inthe continuousshell foruniform self-weightedload (Fig. 11(b))canbe calculated by the following expression [42]:

qg = 1

1+cos ( )gr.

The horizontal radial push force in each bearing joint caused byself-weighted load is thus expressed as

Rgh = grLcos ( )

1+cos ()sin

m1

. (38)

As is described in many reports [40,41], the bearing reaction

induced by the wind suction is normally opposite to that inducedby the service load. The cables in the low tensegrity system,especially cables in the outermost ring, would slacken only if thereaction induced by the wind is greater than that induced by theself-weight. This is especially obvious for diagonal cables in theoutermost ring directly connected to the bearing joint. The bearing

junction used in a practical engineering case, Jinan Olympic SportsCenters Gymnasium, is shown inFig. 12.To avoid the slackeningof cables in the lower tensegrity system, especially cables in theoutermost ring, the following inequality should be satisfied:

Rh >Rwh + Rgh, i.e.

(Nh1)min =1

kRh

Rwh + Rgh

=

rL cos ( )

kRhsin

m1

12

w 11+cos ()

g

(39)


10/13


Fig.12. Bearing junctionin a practicalsuspendome(Note:DC, diagonalcable;CRB,

concrete ring beam; LSM, latticed shell member).

(a) Plan view

(b) Front view

(c) Perspective view

Fig. 13. Sunflower patterned suspendome for illustration.

in which

w is the uniform wind suction;w0 is the reference wind pressure;

g is the uniform self-weighted pressure;L is the span of the dome;

f is the rise of the dome;r is the radius of the spherical dome;

qw is the meridional reaction per unit length underw;qg is the meridional reaction per unit length underg.

If(Nh1)min determined by Eq.(39)is less than zero, it suggests

that the wind suction is so small as not to induce slackening ofthe cables. If a positive(Nh1)min is expected, the prestress level ofthe whole tensegrity system can also be determined, as stated inSection4.2.

As mentioned before, the prestress level is of great significancefor a suspendome structure from both technical and economicaspects. Too high or too low a prestress force level is equallydisadvantageous for a suspendome. Generally, for a low prestresslevel, the following shortcoming would be expected: (1) it wouldcontribute little to the improvement of the limiting load-carryingcapacity of stability, (2) it would make little improvement tothe boundary condition, and (3) large differences would beexistent between the designed configuration and the architecturalconfiguration. In spite of the above disadvantages, a suspendome

at a low prestress level is favorable for technical operation andfor cost of fabrication and construction. The proper prestress

Fig. 14. Numbers of typical nodes and elements in the upper latticed shell.

Fig. 15. Vertical displacement in typical nodes (under 1.0 dead+1.0 live).

level should be grounded on the comparison of the cost and thetechnique, and a design method on the combination of the abovecriteria is usually anticipated. The prestress level for a practicalsuspendome design (Jinan Olympic Sports Centers Gymnasium)was successfully determined. For detailed information, pleaserefer to Ref. [23]. In addition, the designed prestress force levelis supposed to satisfy the structural requirements, such as thedeflection and the internal force allowance, the load-carryingcapacity of stability, etc., under any load case. Otherwise, theprestress force level should be redesigned until all requirementsare satisfied.

5. An illustrative example

A sunflower patterned suspendome is chosen as an example;it has a span of 60 m and a height of 6 m, making the riseto span ratio 1/10 (Fig. 13). Steel pipes with diameter 180 mmand thickness 10 mm are used for all members in the uppersingle-layer dome and the vertical struts. Two rings of cablesand struts of Geiger type are arranged in the lower tensegritysystem, in which the angles are 1 = 150

, 1 = 70, 2 =

150, 2 = 75. The cables are made of high-strength seven-

steel wires (seven strands with nineteen wires per strand). Thematerial Q345B [44]is used in the members of upper single-layerdome and the vertical struts which have properties typical of steel:an elastic modulus of 2.06 105 N/mm2, a Poissons ratio of 0.3and a designed strength of 310 N/mm2. The cables have an elasticmodulus of 1.90 105 N/mm2, a Poissons ratio of 0.3 and the

ultimate tensile capacity 1670 N/mm2

[45]. Both types of materialhave a mass density of 7850 kg/m3. The uniformly distributed


11/13


(a) Radial element. (b) Hoop element.

Fig. 16. Axial force in typical elements (under 1.0 dead+1.0 live).

service load in the gravity projected direction is applied to the

structure: dead 1.0 kN/m2

(structural gravity load included), live0.3 kN/m2 [44]. The uniform wind suction on dome externalsurface is assumed to be 2.0 kN/m2. The analysis is carried outusing the commercial general purpose finite element package

ANSYS. The isoparametric beam element BEAM188 is used todiscretize the members in the upper latticed shell and the sparelement LINK8 is used to model the struts. As high tensile stressesare usually introduced to the cables, the sagging effect due to self-weight can be left out, and the spar element LINK10 is used tomodel the cables. All translational degrees of freedom (DOFs) areperipherally fixed at the bottom of the structure (Ux, Uy, Uz).

The equivalent nodal forces of service load (1.0 dead plus 1.0live) based on Eq.(29)are 42.536, 15.207, 33.410, 49.394,64.523,78.535,47.885 kN from Node 1 to Node 7, as showninFig. 14.The vertical displacement under service load from the

finite element analysis is as follows:{dL}= [71.524, 43.144]

T mm.

The self-internal-force mode is obtained by Eq.(4)as follows:

[kh1, kd1, kv1]T = [1, 0.5509, 0.1884]T;

[kh2, kd2, kv2]T = [1, 0.5359, 0.1387]T.

The prestress level is determined by the method stated inSections 4.14.4, and listed in Table 1. The prestress is alsopresented based on the method stated in Ref. [24]. Due to differentboundary conditions from Ref. [24], the vertical displacement atNodes 1, 3, 5 (Fig. 14) is chosen as the critical displacement inthis example. The table shows that the prestresses obtained fromthe different methods differ from each other: the prestress in thefirst hoop cable based on Ref. [24]-5 is almost 2.05 times thatbased on the method in Section 4.3. The structural behavior ofthe suspendome with the various prestress levels in Table 1 isinvestigated under service load (1.0 dead plus 1.0 live). The verticaldisplacement in typical nodes (Fig. 14) is shown in Fig. 15.It isshown that prestress Level 1 is most rational from the viewpoint ofstructural configuration. Upward displacement of the suspendomeunder prestress Level 5 and Level 7 even occurs in a major portionof the structure due to the unreasonable prestress ratio. Theaxial force in typical elements is shown in Fig. 16. The radialelements are usually compressive for Level 1Level 4, but a tensileforce for Level 5 and Level 7 is also observed. The hoop elementnear the structural center is tensile and that near the boundaryis compressive; the discrepancy between the compressive andtensile force tends to increase for Level 5 and Level 7. Thissuggests that prestress Level 5 and Level 7 are unfavorable for the

redistribution of internal forces and thus unpractical due to theunreasonable prestress ratio.

6. Nonlinearity during self-prestressing

As is known, one of the notable features of a tensegrity systemis the self-adaptiveness, which refers to the capacity of the systemto adjust and redistribute increasingly the integral stiffness duringprestressing. Self-adaptiveness also indicates high nonlinearitywith large displacement; herein, the geometrical stiffness KG ofthe system increases with the increasing of the prestresses. Insuch cases, the effects of prestressing on the integral stiffness Kof tensegrity system should not be negligible, and the geometricalstiffness KG might so much as account for the structural integralstiffness. Prestressing of a tensegrity system is closely correlatedwith the topology, the morphology, the prestress distribution andprestress level. The process of self-prestressing thus completes theconstruction of a tensegrity system.

Whilst the effects of geometrical and material nonlinearity onrigid spacestructures under external actions have been extensivelyexplored [46], studies on the nonlinearity of a flexible tensegritysystem during prestressing and under external actions have notamply been carried out. Kebiche et al. [47]proposed a calculationmethod for a tensegrity system taking geometrical nonlinearityinto account with the Total Lagrangian formulation. A four-strut tensegrity system under external actions and various self-stress levels was investigated. The procedure for the nonlinearelastoplastic analysis of tensegrity systems under static loads wasalso presented [48]; it considered both geometric and materialnonlinearities, using an updated Lagrangian formulation and amodified NewtonRaphson iterative scheme with incrementalloading. Yuan et al. [9] performed a geometrically nonlinearanalysis of the static behavior of cable domes on the basis

of the available integral feasible prestress state. A nonlinearanalysis of hybrid structures composed of rigid elements and acablestrut system waspresentedby Liew et al.[3537],in which acable-tensioning technique was used to investigate the nonlinearbehavior of hybrid structures during self-prestressing and underexternal actions. A cable truss, dome with a cablestrut system,barrel vault stiffened by cables, and square-grid and hexagonal-grid space trusses were considered as illustrative examples intheir work [35,36]. An advanced analysis method considering bothgeometrical and material nonlinearities has been developed topredict the collapse behavior of pre-tensioned steel structures intheir recent work[37], in which bowstring structures were usedas examples. Their contributions have provided a good insight intothe behavior of hybrid space structures by nonlinear analysis.

In this section, the nonlinearity of a suspendome during self-prestressing is investigated using the same model as that shown in


12/13


Table 1

Prestress level in the l ower cablestrut system (unit: kN).

Method Level HC1 DC1 VB1 HC2 DC2 VB2

Section 4.1 1 840.7 463.1 158.4 327.4 175.5 45.4

Section 4.2 2 677.2 373.1 127.6 263.7 141.4 36.6

Section 4.3 3 519.8 286.4 97.9 240.9 129. 1 33.4

Section 4.4 4 522.8 288.0 98.5 203.6 109.1 28.2

Ref.[24]-1 5 1040.6 573.3 196.0 520.3 278.8 72.2

Ref.[24]-3 6 641.4 353.3 120.8 320.7 171.9 44.5Ref.[24]-5 7 1064.1 586.2 200.5 532.1 285.2 73.8

Fig. 17. Prestressing displacement curves for various prestress levels.

Fig. 13.The arc-length solution technique by iterative equilibriumis used to follow the structural full-course responses. The behaviorof a suspendome during self-prestressing (considering 1.0 deadplus1.0 live) is shown inFig. 17by relating the vertical deflectionof Node 1 (Fig. 14) and the various prestress levels determinedinTable 1. As different zero state geometry would be producedfor various prestress level, the zero state configuration is usedas the reference configuration for the definition of structuraldeformation during self-prestressing. It is shown in Fig. 17 thatthe self-prestressing process for large prestress level (e.g. Level 5and Level 7) is highly nonlinear; nevertheless it is almost linear forrelatively small prestress levels (e.g. Levels 14). It is also provedby the nonlinear FE results that the material nonlinearity startswell beyond the initiation of geometric nonlinearity; usually thepeak elastic Von Mises equivalent strain never exceeds 0.2%. So,an analysis taking only geometrical nonlinearity into account isfeasible for a suspendome during self-prestressing.

7. Conclusions

The prestress force level in the lower cablestrut system of

a suspendome is of great significance to the structural behavior.This paper is mainly concerned with the simplified computationalmethod for the determination of the self-internal-force modeand the prestress force level in the lower tensegrity system ofa suspendome. An analytical solution for the self-internal-forcemode of two types of low cablestrut arrangement, called theLevy system and the Geiger system, is put forward together withthe equivalent nodal force acting on the upper single-layer domeand the reaction on the boundary ring beam. The determinationof the prestress force level of the lower tensegrity system isalso elucidated on the grounds of the initial configuration, thecounterbalance of the bearing reaction, the equivalent nodal forceand the wind-induced slackening effect. The results from thestudies can be referred to not only for direct design use for asuspendome in practical engineering, but also for the design ofsimilar hybrid space structures, such as beam string structures, etc.

Acknowledgments

The authors gratefullyacknowledge the support of the Commit-tee of National Science Foundation of China (Grant 50638050#).The authors also wish to thank the anonymous reviewers for theirconstructive comments and suggestions.

References

[1] Quirant J, Kazi-Aoual MN, Motro R. Designing tensegrity systems: The case ofa double layer grid. Eng Struct 2003;25:112130.

[2] Feng Fu. Structural behavior and design methods of tensegrity domes. J

Construct Steel Res 2005;61:2335.[3] Linkwitz Klaus. About formfinding of double-curved structures. Eng struct1999;21:70918.

[4] Tibert AG, Pellegrino S. Review of form-finding methods for tensegritystructures. Internat J Space Struct 2003;18(4):20923.

[5] Adriaenssens SML, Barnes MR. Tensegrity spline beam and grid shellstructures. Eng Struct 2001;23:2936.

[6] Dube JeanFrancois, Angellier Nicolas, Crosnier Bernard. Comparison betweenexperimental tests and numerical simulations carried out on a tensegrityminigrid. Eng struct 2008;30:190512.

[7] Sultan Cornel, Corless Martin, Skelton RobertE. Linear dynamics of tensegritystructures. Eng Struct 2002;24:67185.

[8] Kawaguchi Mamoru, Tatemichi Ikuo, Chen Pei Shan. Optimum shapes of acable dome structure. Eng struct 1999;21:71925.

[9] Yuan XF,DongSL. Nonlinear analysisandoptimumdesign ofcabledomes. EngStruct 2002;24:96577.

[10] ChanSiu-Lai,Shu Gan-ping, L Zhi-tao.Stabilityanalysis andparametric studyof pre-stressed stayed columns. Eng Struct 2002;24:11524.

[11] Saito Daisuke, Ahmer Wadee M. Post-buckling behaviour of prestressed steelstayed columns. Eng Struct 2008;30:122439.

[12] Saito Daisuke, Ahmer Wadee M. Numerical studies of interactive buckling inprestressed steel stayed columns. Eng Struct 2009;31:43243.

[13] Saito D, Wadee MA. Buckling behaviour of prestressed steel stayed columnswith imperfections and stress limitation. Eng Struct 2009;31:115.

[14] de Araujo RR, et al. Experimental and numerical assessment of stayed steelcolumns. J Construct Steel Res 2008;64:10209.

[15] Xue Weichen, Liu Sheng. Design optimization and experimental study onbeam string structures. J Construct Steel Res 2009;65:7080.

[16] Wu Minger. Analytical method for the lateral buckling of the struts in beamstring structures. Eng struct 2008;30:230110.

[17] Wu Minger,SasakiMutsuro. Structural behaviors of anarch stiffenedby cables.Eng struct 2007;29:52941.

[18] Hangai Yasuhiko, Wu Minger. Analytical method of structural behaviours of ahybrid structure consisting of cables and rigid structures. Eng struct 1999;21:72636.

[19] Saitoh Masao, Okada Akira. The role of string in hybrid string structure. Engstruct 1999;21:75669.

[20] Hosozawa Osamu, et al. The role of cables in large span spatial structures. Engstruct 1999;21:795804.

[21] DongShi-ling, Luo Yao-zhi, ZhaoYang, et al. Analysis, design and constructionof new space structure. China Communication Press; 2007 [in Chinese].

[22] Zhang Zhi-hong. Theoretical research on large-span tensile spatial structurescomposed of cables, bars and beams. Ph.D. thesis. Space Structure ResearchCenter, Depart of Civil Engineering, Zhe Jiang University; 2003 [in Chinese].

[23] Zhang Zhi-hong, Cao Qing-shuai, Dong Shi-lin. Structural design of a practicalsuspendome. Internat J Adv Steel Construct(IJASC) 2008;4(4):32138.

[24] Kang Wen-jiang, Chen Zhi-hua, Lam Heung-Fai, Zuo Chen-ran. Analysis anddesign of the general and outmost-ring stiffened suspendome structures. EngStruct 2003;25:168595.

[25] Kitipornchai S, KangWen-jiang,Lam Heung-Fai, Albermani F. Factorsaffectingthe design and construction of Lamella suspendome systems.J Construct SteelRes 2005;61:76485.

[26] Kawaguchi M, Abe M, Hatato T. et al. Structural tests on the suspendomesystem. In: Proceedings of the IASS symposium. 1994. p. 38492.

[27] Kawaguchi M, AbeM, Hatato T. et al.Structural testson a full-sizesuspendomestructure. In: Proceedings of the IASS symposium. 1997. p. 4318.

[28] Kawaguchi M, Abe M, Tatemichi I. Design, test and realization of suspen-dome system. J IASS 1999;40(131):17992.

[29] Pellegrino S, Calladine CR. Matrix analysis of statically and kinematicallyindeterminate frameworks. Int J Solids Struct 1986;22(4):40928.


13/13


[30] Calladine CR, Pellegrino S. First-order infinitesimal mechanisms. Int J SolidsStruct 1991;27(4):50515.

[31] Pellegrino S. Structural computations with the singular value decompositionof the equilibrium matrix. Int J Solids Struct 1993;30(21):302536.

[32] Zhang Ai-lin, et al. Optimization and determination of prestressed suspend-dome system of the badminton gymnasium for 2008 Olympic Games. JBuilding Struct 2007;28(6):19 [in Chinese].

[33] Lu Ci-lin, Yin Si-ming,Liu Xi-liang.Modern prestressed steel structures. ChinaCommunication Press; 2007 [in Chinese].

[34] CECS 212. Technical specification for prestressed steel structure. Beijing:

Chinese planning press; 2006 [in Chinese].[35] PunniyakottyNM, Richard LiewJY, Shanmugam NE. Nonlinear analysisof self-

erecting framework by cable-tensioning technique. J Struct Eng, ASCE 2000;126(3):36170.

[36] Richard Liew JY, Punniyakotty NM, Shanmugam NE. Limit-state analysis anddesign of cable-tensioned structures. Internat J Space Struct 2001;16(2):95110.

[37] Richard Liew JY, Li Jin-Jun. Advanced analysis of pre-tensioned bowstringstructures. Steel Struct 2006;6(2):15362.

[38] GB50009. Load code for the design of building structures. Beijing: ChinaArchitecture & Building Press; 2001 [in Chinese].

[39] Uematsu Yasushi, et al. Model of wind pressure field on circular flat roofs andits application to load estimation. J Wind Eng Ind Aerodyn 2008;96:100314.

[40] Fu JY, et al. Prediction of wind loads on a large flat roof using fuzzy neuralnetworks. Eng Struct 2006;28:15361.

[41] Eurocode 1. Actions on structures general actions Part 14: Wind actions.European Standard EN 1991-2-4; 1995.

[42] Eurocode 3. Design of steel structuresPart 16, strengthand stabilityof shellstructures, European Standard EN 1993-1-6; 2007.

[43] Timoshenko S, Woinowsky-KriegerS. Theory of platesand shells.International

edition McGraw-Hill Book Company; 1959.[44] GB50017. Code for design of steel structures. Beijing: Chinese Planning Press;2003 [in Chinese].

[45] Shen Shi-zhao, et al. Design of cable structures. 2nd edition Beijing: ChinaArchitecture & Building Press; 2006 [in Chinese].

[46] Richard Liew JY, Tang LK. Advanced plastic hinge analysis for the design oftubular space. Eng struct 2000;22:76983.

[47] Kebiche K, Kazi-Aoual MN, Motro R. Geometrical non-linear analysis oftensegrity systems. Eng Struct 1999;21:12742.

[48] Kahla NabilBen, Kebiche K. Nonlinear elastoplastic analysis of tensegritysystems. Eng Struct 2000;22:155266.

a simplified strategy for force finding analysis of suspendomes

Documents