STRUCTURAL DESIGN OF A PRACTICAL SUSPENDOME

Transcription

1 Advanced Steel Constructon Vol. 4, No. 4, pp (2008) 323 STRUCTURAL DESIGN OF A PRACTICAL SUSPENDOME Zh-Hong Zhang 1,*, Qng-Shua Cao 1, Sh-Ln Dong 2 and Xue-Y Fu 1 1* Ph. D., Chna Constructon (Shenzhen) Desgn Internatonal, NO.138, Kangjan Road, Shangha, Chna, *(Correspondng author: Emal: zhangzh@zuaa.zju.edu.cn) 2 Professor, Space Structure Research Center, Zhejang Unversty, Hangzhou, Zhejang Provnce, Chna, Receved: 8 October 2007; Revsed: 7 December 2007; Accepted: 10 December 2007 ABSTRACT: Structural desgn of a practcal sphercal suspendome wth the dameter of 122m was carred out n Chna Constructon (Shenzhen) Desgn Internatonal (CCDI) n The suspendome structure s a new type of large-span spatal structure whch s wdely used n the sports buldngs. Several suspendome structures have been constructed as the structural roof n the sports arena n Chna n recent years. As s known to all, prestresses n the cable-strut system are crucal to the tensegrc system, and the determnaton of prestress level and dstrbuton s somewhat complcated. However, no provsons has been provded by current chnese desgn codes of practce for the suspendome structure. Ths paper gves out the detaled prestress desgn procedure for the practcal sports arena whch s to be bult n Jnan Cty, Chna. General purpose fnte element package ANSYS s utlzed for the analyses. The self-nternal-force mode and the prestress level rato among three rng cables are nvestgated to determne the prestress n the cable. Lnear statc analyses are then carred out to valdate the prestress effcency. It has been shown that prestress defned by ths way has much nfluence on the deformaton and the nternal force of the upper retculated shell structure. The lnear elastc bucklng and the geometrcally nonlnear stablty analyss are also presented. The snap-through phenomenon of the structure s nvestgated to determne the crtcal load carryng capacty of stablty. The nfulence of lve load dstrbuton patterns and mperfectons on suspendome s addressed n detals. The results from the studes can not only be refered for drect desgn use, and also for the desgn of smlar hybrd structures. Keywords: Space structure, suspendome, tensegrc system, prestress, sports buldng, bucklng, egenmode, geometrcally nonlnearty, mperfecton 1. INTRODUCTION Wth the development of the socety, more and more novel structures whch can cover larger space are needed. Large-span space structures have been put nto applcaton at a rapd rate for recent several decades. Based on the demands, several new types of space structure are developed (Lu [1] and Dong et al. [2]): the cable dome structure, the suspendome structure, the beam strng structure, etc. Suspendome s a new type of large-span pre-stressed space structure whch has only developed for more than one decade. It s a hybrd structure composed of the upper rgd sngle-layer lattced shell and the lower flexble cable-bar tensle system (Fgure 1). Suspendome structure conssts of three types of member: the rgd member n the upper snglelayer dome, the vertcal strut and the cable n the lower tensegrc system. Suspendome can be derved from two types of structure: 1) It s formed by replacng the upper cables of the cable dome by the rgd members, usually the sngle-layer lattced dome. Compared wth the cable dome, prestress n the suspendome can be observably reduced. As the upper rgd members can provde certan ntal stffness to the structure, form fndng analyses s no longer necessary and the constructon technques are smplfed. The upper rgd member can resst both the axal force and the bendng moments to ncrease the rgdty of the structure n one hand, and n the other hand the stress/force flow n suspendome may be closed, thus make the structure a self-equlbrum system, and weak bearng system s possble(zhang [3], Zhang [4]). 2) The upper sngle-layer lattced dome s strengthened by the lower tensegrc system. Ths method reasonably enhances the stffness of the lattced shell and mproves the bucklng capacty. As the bearng horzontal reacton nduced by the

2 324 Structural Desgn of a Practcal Suspendome servce load s reverse to that nduced by the tensegrc system, the bearng horzontal reacton can be reduced to zero f proper prestress n the cables are ntroduced. The maxmum value of axal force n the upper sngle-layer dome members can also be reduced by the tensegrc system (Zhang [3], Zhang [4], Kang et al. [5], Ktporncha [6]). So suspendome s also called hybrd space structure. + = Fgure 1. Suspendome: the Lattced Shell and the Cable-Strut System zero state ntal state prestress Fgure 2. Zero State and Intal State of Suspendome The suspendome desgn procedure s a lttle complcated. On the bass of the constructon procedure of the suspendome, three typcal reference confguratons are defned (shown n Fgure 2): 1) the zero state geometry, s the equlbrum state wthout prestress and wthout self-weght n the structure to determne the loftng durng constructon. 2) the ntal state geometry, s the equlbrum state wth prestress n the structure and under self weght after camberng, correspondng to the constructed confguraton of the structure. 3) the loadng state geometry, s the equlbrum state after loaded. The structural deformaton n ths paper s based on the ntal state as the reference confguraton, and the ntal stresses n the members of the upper sngle layer dome are also taken nto consderaton. Analytcal and expermental research on the structural behavor of suspendomes has been performed n varous approaches by the researchers (Zhang [3], Zhang [4,7], Kang et al. [5], Ktporncha [6]). Based on Equlbrum Matrx Theory (Calladne and Pellegrno [8,9,10,11]), a new method called local analyses method to determne the structural self-nternal-force mode s put forward (Zhang [3], Zhang [4,7]). The lower tensegrc system s detached from the upper sngle layer dome as an ndependent system. Boundary restrants are appled to the lower ndependent tensegrc system to determne the self-nternal-force modes. The ntal prestress dstrbuton of the tensegrc system can be obtaned by the superposton of self-nternal-force modes. The nteracton of the lower tensegrc system to the upper sngle layer dome s appled as the equvalent load to obtan the ntal stress dstrbuton of upper sngle-layer dome (the ntal state, showed n Fgure 2.). For detaled nformaton, please refer to Reference [3].

3 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 325 Fgure 3. Arscape of Jnan Olympc Sports Center-the Gymnasum (Chna, 2009) Several suspendome structures have been constructed n Chna n recent years (Lu [1] and Dong et al. [2]). However, no provsons has been provded by the current chnese desgn codes of practce for suspendome. The desgn method for the above mentoned constructed suspendome seems dscretonary and unorganzed. Ths paper gves out the detaled desgn procedure for the practcal sports arena whch s to be bult n Jnan Cty, Chna, at the prelmnary desgn stage (shown n Fgure 3). 2. THE DESIGN MODEL 2.1 General Informaton of the Arena Jnan Olympc Sports Center-the Gymnasum s a mult-purpose buldng and located n Jnan Cty, Chna. The faclty wll serve as the center-pece of the local Olympc Games held n the cty n The buldng s n the shape of the ellpse wth the longtudnal length 220 meters and the lattudnal length 168 meters (Fgure 3). Two adjacent accessoral gymnastc buldng for warmng up before the event are desgned n the two ends of the Gymnasum. The buldng has an approxmate archtectural area 60,000 m 2 whch s to accommodate more than 12,000 audences. The archtectural style of the roof exposts full-boded regonal characterstcs- the lotus flower and the wllow leaf (Fgure 3). Suspendome structure s chosen as the roof of the gymnasum. The span and the rse of the dome are 122.0m and 12.2m respectvely, makng the rse to span rato 1/10. The Sunflower-Kewtt patterned hybrd sngle-layer dome s used n the upper part of the roof structure. The roof s covered wth the rgd profled steel sheet. Three rngs of cable-strut are arranged n the suspendome (Fgure 4 and Fgure 8). The suspendome conssts of 1230 ppe members, 379 nodes, and 48 sphercal steel bearngs. The frame-shearwall concrete structure s used n the lower part of the buldng. The roof bearng jonts s supported by alternate columns and the rng beam.

4 326 Structural Desgn of a Practcal Suspendome 3rd rng 2nd rng a) Top vew 1st rng b) Sde vew 2.2 Fnte Element Model Fgure 4. The Suspendome Structure for the Desgn All analyses are carred out usng the commercal general purpose fnte element package ANSYS. The 3-D quadratc beam element BEAM188 whch has three dsplacement and three rotatonal degrees-of-freedom (DOFs) s used to dscretze the members n the upper dome and the spar element LINK10 wth the tenson-only (or compresson-only) opton whch has three dsplacement degrees-of-freedom (DOFs) s used to model the cables and the struts. The materal Q345B (GB [12]) s used n the members of upper sngle layer dome and the vertcal struts whch have propertes typcal of steel: an elastc modulus E of 2.06x10 5 N/mm 2, a Posson s rato of 0.3 and a desgned strength f of 310N/mm 2. The cables are made of hgh-strength seven-steel-wres whch have an elastc modulus E of 1.90x10 5 N/mm 2, a Posson s rato of 0.3 and the ultmate tensle capacty 1670 N/mm 2 (Shen et al. [13]). Both types of the materal have the mass densty of 7850kg/m 3, and the thermal expanson coeffcent of 1.2x10-5 / o C. Each fnte element node has sx degrees of freedom ncludng translatons n the x, y, and z drectons and rotatons about the x, y, and z drectons. Welded gapless steel ppes wth the dameter 351mm and the thckness from 12mm, 14mm to 16mm are used n the upper sngle-layer dome. The chord member and the bracng member are connected each other to form the ntersectng jont (GB [12]). As the bracng members are welded drectly wth the chord members, the connectons are modeled as the rgd jonts. The upper steel roof s supported by the lower concrete structure wth the sphercal steel bearngs. Regardng the boundary condtons to the roof model, the connectons are treated as hnged jonts,.e, all translatonal degrees of freedom (DOFs) are perpherally fxed at the bottom of the structure (Ux, Uy, Uz). The fnte element model s shown n Fgure 5.

5 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 327 Fgure 5. Fnte Element Model of the Suspendome 3. DETERMINATION OF THE SELF-INTERNAL-FORCE MODE Ths secton determnes the self-nternal-force mode of the tensegrc system based on the local analyss method (Zhang [3]). In the lower tensegrc system, the cables and the vertcal struts are hnged n the jonts. The cables are tenson-only elements and the vertcal struts are compresson-only elements. Nodal force equlbrum at each jont can be expressed n the x, y, and z drectons. The equlbrum equaton of the lower cable-bar system s shown n Eq. 1 as follows. A t = p (1) [ ]{ } { } n whch, [ A ] s the equlbrum matrx of the system-the functon of nodal coordnates, { t } s the nternal force vector of the elements, { p } s the equvalent force vector at the nodes. Typcal unt of the lower tensegrc system s shown n Fgure 6. By Sngular Value Decomposton of the equlbrum matrx [ A ] (Calladne and Pellegrno [8,10,11]), the number of the self-nternal-force mode s can be obtaned as s=3. Fgure 7 shows the axal force equlbrum n node, where, Nhc s the axal force n the horzontal (hoop) cable; N dc s the axal force n the dagonal (radal) cable; N vb s the axal force n the vertcal strut, α s the angle between the horzontal hoop cables and β s the angle between the dagonal cable and the vertcal strut. DC3 VB3 DC1 VB1 DC2 HC2 VB2 HC3 HC1 Fgure 6. Typcal Unt of the LowerTensegrc System Note: DC, the dagonal cable; HC, the hoop cable; VB, the vertcal strut. 1,2,3 refers to the frst (outmost), second (mddle) and thrd (nnermost) rng of the tensegrc system respectvely.

6 328 Structural Desgn of a Practcal Suspendome N vb N hc N dc β α N hc Fgure 7. Axal Force Equlbrum n Typcal Node. Accordng to the force equlbrum n the vertcal and the hoop drecton, the equlbrum equaton n node can be expressed as: N = N cos( β ) (2) vb dc ( β ) 2 cos( α 2) Ndcsn = Nhc (3) Combnng Eq. 2 and Eq. 3, the followng equatons can be obtaned: 2cos( α 2) Ndc = Nhc sn( β ) (4) 2cos( α 2) Nvb = Nhc tan ( β ) (5) Gvng N hc = 1, the self-nternal-force mode n the th rng of the tensegrc system s: 2cos( α 2) 2cos( α 2) [ Nhc, Ndc, N vb ] = 1,, sn ( β) tan ( β) (6) Three hoop rngs altogether are desgned n the structure (shown n Fgure 4, 5, 6 and 8.). The angle α (=1,2,3) n ths model s 165 o, and β s 80 o, 75 o, 72 o respectvely for =1,2,3. The self-nternal-force modes of the lower cable-strut system for the outer, mddle and nner rng are as follows: T t = t, t, t (7) [ ] [ 1 2 3] where, { t 1} = [ 1.0, , ] T { t 2} = [ 1.0, , ] T { t 3} = [ 1.0, , ] T

7 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu DETERMINATION OF CABLE PRESTRESS RATIO AMONG THREE HOOPS OF THE CABLE-STRUT SYSTEM The number of rngs n the tensegrc system and the prestress level rato among the rngs have much nfulence on the structural behavor, theren, the outmost rng wth prestress s the most effectve (Zhang [3], Kang et al. [5]). As s mentoned above, steel ppes wth dameter 351mm and thchness from 12, 14 to 16mm are used n the upper sngle-layer dome. The member arrangements of the upper sngle-layer dome are shown n Fgure 8. Characterstc combned load between unformly dstrbuted dead load 0.8kN/m 2, and unformly dstrbuted lve load 0.3kN/m 2 s appled to the roof as the servce load (GB [14]). Each nodal external concentrated force s determned by the product of the combned load and ts trbutary area. The nodal trbutary area s calculated by self-developed program. Lnear statc analyss s then carred out. As the three rngs of cable-strut of the tensegrc system are assumed to be ndependent of each other. The actons of each rng are appled separately to the upper sngle layer dome as the equvalent load. The prestressed force N hc n each rng s assumed provsonally to be T 0 = 1000kN, when the dome s only stffened by the sngle rng. The dsplacements of the upper sngle layer dome under dead and lve load and each rng equvalent loads are lsted n Table 1 together wth the horzontal radal reacton of the bearng jont for the typcal nodes defned n Fgure 8. strut Φ351x12 Φ351x14 Φ351x16 Fgure 8 Member Arrangements for Upper Sngle Layer Dome and Typcal Node and Element Numbers Table 1. Vertcal Dsplacements and Horzontal Radal Reactons for Three Rngs of EL Node Vertcal Dsplacement (mm) Jont Horzontal Node 9 Node 6 Node 4 Radal Reacton n Node 13 (kn) Dead +Lve Load st Rng EL nd Rng EL rd Rng EL Note: EL, the equvalent load.

8 330 Structural Desgn of a Practcal Suspendome Relatonshp between the servce load and three-rng equvalent loads s establshed to determne the prestressed force rato among three rngs of cable-strut system. The vertcal dsplacement of the typcal Node 4, 6 and 9 (the nodes at the top of the vertcal struts) s consdered to be zero under the dead and lve load and the three rngs of equvalent loads. So the relaton can be expressed as: B ϕ = b (8) [ ]{ } { } where, [ B] = {} b = [ , , ] T Solvng the above equaton, { ϕ } can be obtaned: { ϕ } = [ , , ] T (9) From the above results, the prestress force rato for the outmost rng has the largest value and consequently s the most effectve. The prestress force (unt: kn) n the tensegrc system can then be noted as: HC DC VB st = = 2nd rd [ T ] T dag { ϕ} [] t n whch, [ t] [ t, t, t ] T =, refers to Eq. 7, dag { ϕ} = Rng Rng Rng The jont horzontal radal reacton under three-rng equvalent load s R e = kN, t s opposte to and much greater than the reacton R s =1030.3kN nduced by the servce load. If full-scale prestress force s ntroduced to the cables, the reacton of bearng jont would be as large as 1800kN or so. Ths also adds dffculty for the desgn of the bearngs and the prestress needs a second order optmzaton. For the wnd load case, the reacton nduced n the same bearng jont may be reversed for the case of randomcty of the wnd. The jont horzontal radal reacton under wnd load s R w =± (10) kn. Assume that the prestress rato among three rngs keeps unchanged, a reducton factor γ s proposed to reduce the prestress level n the cable-strut system. The followng equaton s used to determne the reducton factorγ : γ Re + Rs = 0.5 abs( Rw) (11) where, Re, Rs, R w s the jont horzontal radal reacton under three-rng equvalent load, the servce load and the wnd load respectvely, and abs(r w ) s the absolute value of the bearng reacton under wnd load. Substtutng the value to Eq. 11, γ can be obtaned: γ = The optmzed prestress force (unt: kn) for the tensegrc system s determned as follows: HC DC VB st Rng [ T ] = nd Rng (12) rd Rng

9 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 331 The cables and struts are desgned by the prestress forces obtaned by Eq. 12. The geometrc parameters and the desgned prestress forces of the lower cable-strut system s shown n Table 2. As the compressve elements, the lateral stablty of the vertcal struts should be verfed. The Euler crtcal bucklng capacty of the struts n the three rngs s -341kN, -204kN and -93.1kN 2 respectvely by the equaton N ( ) 2 cr = π EI μl (GB [12]), where, N cr s the Euler crtcal bucklng force, E s the materal elastc modulus, I = π d 4 64 s the nerta moment of the secton, d s the strut dameter, μ s the coeffcent of effectve length and μ =1, l s the geometrc length of the vertcal strut. Hence, the struts bucklng capacty s satsfed. Table 2. Prestresses and Geometres of the Lower Tensegrc System 1st Rng 2nd Rng 3rd Rng HC1 DC1 VB1 HC2 DC2 VB2 HC3 DC3 VB3 Secton 2Φ7x85 Φ7x55 Φ203x6 Φ7x31 Φ7x13 Φ152x4.5 Φ7x7 Φ7x7 Φ95x3.5 Area (mm 2 ) Inerta Moment(m 4 ) e e e-6 Length (m) Intal stran (με) Prestress Force (kn) STATIC ANALYSES Ths secton presents the results of lnear statc response of suspendome and sngle-layer dome (SLD) under servce load to valdate the effcency of the desgned prestresses (prestress stffenng effect s consdered). Based on the lnear superposton theory, the structural dsplacement at the top of the strut s zero under combned dead and lve load wth the full-scale three-rng prestress. The deformaton for non-optmal prestress suspendome under servce loads s shown n Fgure 9. The maxmum vertcal dsplacement occurs between the bearng jont and the outmost rng (Node 11, Fgure 8), and the value s only 34.23mm. The deformaton for the optmzed prestress suspendome under servce loads s shown n Fgure 10. The maxmum vertcal dsplacement occurs between the outmost rng and the mddle rng (Node 7, Fgure 8), and the value s only 26.42mm. Fgure 9. Deformaton of Suspendome under the Servce Load ( γ = 1.0,scaled by a factor of 100)

10 332 Structural Desgn of a Practcal Suspendome The vertcal dsplacement for the typcal numbered nodes (as defned n Fgure 8) of the sngle-layer dome and the suspendome under servce load s shown n Fgure 11. The vertcal dsplacement of the sngle-layer dome s larger than that of the suspendome. The sngle layer dome has the largest downward dsplacement and occurs at Node 7. The non-optmzed prestress suspendome has the smallest downward dsplacement and occurs at Node 9. The deformaton seems uneven and fluctuates greatly up and down n the vertcal drecton for the unreasonable tensegrc prestress dstrbuton and even upward dsplacement n Node 5 s nduced. Ths reveals the fact that the outmost rng of the tensegrc system has the most effects on the suspendome structure, and the other two rngs has relatvely low effects on the structure. After prestress optmzaton, the deformaton becomes unform, and the largest downward dsplacement s 26.42mm, correspondng to the rato of the deflecton to the span 1/4618. The dsplacement of the optmzed tensegrc system greatly satsfes the structural allowable deflecton specfcatons L/400 (L s the dome span) (GB JGJ [15])and the archtectural desgn. Fgure 10. Deformaton of Suspendome under the Servce Load ( γ = , scaled by a factor of 100) Vertcal deflecton(mm SLD γ=1.0 γ= Node number Fgure 11. Vertcal Dsplacement of Typcal Numbered Nodes under Servce Load

11 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 333 Normally, tensle axal forces n the members of the upper suspendome are nduced by the lower tensegrty system. It s ncompatble wth the compressve axal force nduced by the servce load. So proper prestress ntroduced n lower tensegrc system wll greatly counteract the effects caused by the servce load. The horzontal radal reactons are 320.8kN n the optmzed suspendome and 963.5kN n the sngle layer dome under servce load. The maxmum axal forces are 469.3kN, 496.6kN, 545.8kN n the optmzed suspendome and 498.2kN, 483.5kN, 645kN n the sngle-layer dome under the same servce load for ppe secton Φ351x12, Φ351x14, Φ351x16 respectvely. It shows that not only the bearng reactons but also the ppe member axal force of the suspendome can decrease greatly compared to the sngle-layer dome under the same servce load. The axal force for the typcal numbered radal members (as defned n Fgure 8) of the snglelayer dome and the suspendome under servce load s shown n Fgure 12. The axal force of the sngle-layer dome s larger than that of the suspendome. The axal force n the non-optmzed prestress suspendome s greatly uneven n the radal drecton for the unreasonable tensegrc prestress dstrbuton. The axal force n Member 9 (connected wth the frst rng, as stated n Fgure 8) s near to zero. Ths suggests agan that the outmost rng of the tensegrc system has the most effects on the statc property of the suspendome structure. The axal forces n the optmzed suspendome are very even, and t s favorable for the member desgn. Axal force (kn) SLD γ=1.0 γ= Element number Fgure 12. Radal Axal Force of Typcal Numbered Elements under Servce Load Fgure 13 shows the axal force for the typcal numbered hoop members (as defned n Fgure 8) of the sngle layer dome and the suspendome under servce load. The characterstcs of the axal forces dstrbuton are very smlar to that shown n Fgure 12. The axal forces n the optmzed suspendome are very even, and s also favorable for the member desgn. It s noteworthy that the stress rato (the rato of the peak sectonal stress of the members to the materal desgn strength f ) of the elements n the dome under servce load s usually comparatvely small rangng from 0.1 to 0.5, thereby the materal strength s unlkely to govern the structural desgn. From the above analyses, t can be concluded that the optmzed prestress defned by Equaton 12 s harmonous wth both the axal force and the deformaton of the structure. The optmzed prestress level s used for all analyses below.

12 334 Structural Desgn of a Practcal Suspendome Axal force (kn) -800 SLD -700 γ= γ= Element number Fgure 13. Hoop Axal Force of Typcal Numbered Elements under Servce Load 6. LINEAR ELASTIC BUCKLING ANALYSES Bucklng analyss, n general, may be dvded nto lnear elastc bucklng (bfurcaton) and geometrcally nonlnear bucklng (snap-through) analyss. Geometrcally nonlnear analyses are usually used to determne the lmt load of the structure whch are usually started wth a lnear bucklng analyss. Lnear bucklng analyss predcts the theoretcal bucklng strength (the bfurcaton pont) of an deal lnear elastc structure. The results are bucklng modes and load factors (egenvalue). Load factors are estmated for an upper lmt of the ultmate load. The bucklng modes are related to a structure that mantans ts shape up to bucklng. If the load appled to the structure s P, the crtcal bucklng load s λ P. The equaton of bfurcaton bucklng s expressed as an egenvalue problem: K + λ S ψ = (13) ([ ] [ ]){ } 0 where: [ K ] s the stffness matrx of the system, [ S ] s the stress stffness matrx, λ s a load factor, the th egenvalue determnng bucklng load, { ψ } s the th egenvector determnng the bucklng mode. Ths secton presents the lnear bucklng of suspendome usng a lnearzed model of elastc stablty. Two load cases are consdered for ths secton and the below: Load Case 1, the dead load combng the full-span lve load (LC1); Load Case 2, the dead load combng the half-span lve load (LC2). Half-span lve load such as the snow load and the constructon load s proved to decrease the structural load-carryng capacty n most crcumstances and often neglected by the structural desgner. The egenvalues are lsted n Table 3, whch shows that the bucklng capacty of the suspendome s very hgh and the capacty for LC1 s hgher than that for LC2 for the nfluence of the unsymmetrcal dstrbuton of the lve load. The lowest bfurcaton egenvalue amounts to for LC1 and to for LC2 respectvely. The bucklng capacty decreases by 10.47% for the nfluence of the unsymmetrcal lve load..

13 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 335 Table 3. Egenvalues of Suspendome for LC1 and LC2 n LC LC a) 1 st mode b) 2 nd mode c) 3 rd mode d) 4 th mode e) 5 th mode f) 6 th mode Fgure 14. The Anteror Sx Egenbucklng Modes of Suspendome for LC1 Fgure 14. and Fgure 15. show the anteror sx egenbucklng modes of suspendome LC1 and LC2. The egenvalues are very close and the bucklng modes are very dense for the two cases for the nfluence of the cable-strut prestresses. The bucklng modes are usually n symmetrcal shape for structure for LC1 and unsymmetrcal shape for LC2. It can also be observed from the fgures that the bucklng regon of the suspendome for LC1 s more localzed n the center and bucklng usually begn wth great deformaton n the central area. Ths phenomenon can be establshed by the fact that the central area of the dome wth a dameter about 30 meters s sngle-layered wthn the thrd rng of cable-strut system, thus makes actually a secondary sngle-layer lattced dome whch s unstffened by the lower tensegrc system. Snce the suspendome s manly stffened by the outmost rng of the tensegrc system, the local weaker stffness detected n the structural center s reasonable and expected. For LC2, the most localzed deformaton s usually n the area adjacent to the strcutual boundary for the nfluence of the unsymmetrcal lve load. As the ncrease of the bucklng number n, the bucklng regon becomes extended to the structural boundary. The bucklng deformaton nduces more fluctuatons on the surface, and consequently shfts to the global bucklng mode.

14 336 Structural Desgn of a Practcal Suspendome a) 1 st mode b) 2 nd mode c) 3 rd mode d) 4 th mode e) 5 th mode f) 6 th mode Fgure 15. The Anteror Sx Egenbucklng Modes of Suspendome for LC2 7. GEOMETRICALLY NONLINEAR STABILITY ANALYSIS Egenvalue bucklng analyss predcts the theoretcal bucklng strength (the bfurcaton pont) of an deal lnear elastc structure. However, mperfectons and nonlneartes prevent most real-world structures from achevng ther theoretcal elastc bucklng strength. Thus, egenvalue bucklng analyss often yelds unconservatve results, and should generally not be used for the desgn of actual structures. The nonlnear bucklng consders a load-dependent prebucklng deformaton durng loadng up to the structural nstablty. In ths case, the nstablty s connected wth relatvely large dsplacement ampltudes wthout a sgnfcant change of the equlbrum path. For the slender tensegrc system as suspendome structure, large dsplacement may be nduced n the structure when loaded. So, t s proper to carry out the geometrcally nonlnear analyses for suspendome structure. The geometrcally nonlnear analyses account for the ntal geometrc mperfecton, whch could be obtaned by the lnear superposton of lnear bucklng egenmodes. The ntal geometrc mperfecton, hereby, can be expressed n the followng expresson: n { δ} a { ψ } = (14) = 1 where, { } δ s the mperfecton vector, a s the scale coeffcent assocated wth the th egenvector, { ψ } s the th egenvector determned by Eq. 13. Usually the lowest egenvector s only used to determne the mperfecton dstrbuton. The maxmum value of mperfecton s chosen as L/300 (L s the dome span) (GB JGJ [15] ) n the desgn,.e. δ max s equal to mm. For the normalzed egenmode shape, a 1 s equal to δ max and a s equal to 0 (>1) n Eq. 14.

15 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 337 Ths secton presents the geometrcal nonlnearty of the suspendome. The Arc-length soluton technque combned wth the Newton-Raphson method s adopted for the analyses. The bucklng deformaton and crtcal bucklng strength are manly addressed for the geometrcally perfect and mperfect structure for LC1 and LC2. The load-deflecton curves approves the typcal nstablty mode of the snap-through phenomenon for geometrcally perfect and mperfect models. It s shown that the equlbrum path s hghly nonlnear, especally for the postbucklng stage. In the prebucklng stage, the equlbrum path for the perfect model nclnes wth a sheer ascend, and the slope turns to be gentle when close to the crtcal pont. After the maxmum load factor (crtcal pont) s reached, the path usually descends wth a rather steep load decay, even the curve backtracks to an unstable equlbrum path. The load factor correspondng to the crtcal pont s used as the bucklng strength for the perfect model. In addton, the equlbrum path for the mperfect model ascends more gentlely before the crtcal pont s reached. Large deflecton s nduced when the structure buckles whch s usually more than ten tmes of that for Fgure 16. Bucklng Mode of Suspendome for LC1 Fgure 17. Bucklng Mode of Suspendome for LC2

16 338 Structural Desgn of a Practcal Suspendome Fgure 18. Bucklng Mode of Suspendome for LC1 wth L/300 Imperfecton the perfect model, and s unlkely to satsfy the deflecton lmtatons. In the equlbrum path, no maxmum load factor s reached n the range of deflecton specfcatons whch should not be more L/400 (GB JGJ [15] ). The load factor correspondng to the vertcal deflecton wth the magntude of L/400 s specfed as the bucklng strength for the mperfect model. After the structure reachs ts crtcal pont, the equlbrum paths usually tend to ncrease n a plateau and the structure possess good postbucklng strength retenton capablty. The global stffness of the structure tend to degenerate to naught. Ths seems to be a clear sgn of nstablty of the structure and the falure may be a sudden phenomenon. The postbucklng s so complcated that the equlbrum path s dffcult to trace down. The maxmum load factors are 9.91 and 8.89 for the geometrcally perfect model, and 5.22 and 4.59 for the geometrcally mperfect model for LC1 and LC2 respectvely. The structrual lmt load decrease by 10.29% of perfect model and 12.07% of mperfect model for the nfluence of unsymmetrcal dstrbuton of the lve load. The nfluence of mperfecton s even more destructve to the structure. The lmt load decrease by 47.33% for LC1 and 48.37% for LC2. The bucklng modes are shown n Fgure for the geometrcally perfect and mperfect structure for LC1 and LC2 respectvely. The deformatons for perfect models for LC1 are symmetrcal global bucklng modes and unsymmetrcal global modes for LC2. The most vertcal dsplacement for LC1 occurs n the structrual center (Node 1, Fgure 8). The deformaton for LC2 s more localzed (Node 11, Fgure 8) n the area to whch the half-span lve load s appled. The structural center (Node 1, Fgure 8) only deflects a small vertcal dsplacement. The most vertcal dsplacement occurs n the areas between the outmost rng and the boundary for the two cases, and ths show lttle agreement wth the results from the bfurcaton analyses. The deformatons for mperfect models for LC1 and LC2 are both unsymmetrcal local bucklng modes. As t s well known, large span structures are most senstve aganst mperfectons. From the bfurcaton analyses, the egen modes are loalzed n the center for LC1 and adjacent to the boundary for LC2. As the ntroducton of the egen-mperfecton, the revsed structural confguraton consequently has the most mperfecton n the cente for LC1 and the boundary area for LC2. The bucklng modes (Fgure 18 and 19) are localzed n the center for LC1 and adjacent to the boundary for LC2, and ths show good agreements wth the lnear bucklng modes.

17 Zh-Hong Zhang, Qng-Shua Cao, Sh-Ln Dong and Xue-Y Fu 339 Fgure 19. Bucklng Mode of Suspendome for LC2 wth L/300 Imperfecton From the above analyses, the perfect and mperfect structure under LC1 and LC2 has large bucklng strength. At the same tme, local stffness n the structural center s comparatvely weak, and the secton of ppe elements n the center area (showed n Fgure 8.) are change from Φ351x12 to Φ351x14 n the end to strengthen the structural center. 8. CONCLUTIONS Prestress nduced n the cable and the strut have much nfluence on the deformaton and the nternal force of the suspendome structure. Proper prestresses should be defned accordng to the structural desgn requrements. The deflecton and the stress of the structure under servce load s usually easy to satsfy the desgn lmtaton specfed n the relevant codes. The mperfectons and lve load dstrbuton patterns have much nfluence on the bucklng strength and the modes. For large-span structure such as the suspendome, t s obvous that the materal strength and structural vertcal deflecton are most unlkely to govern the desgn, and desgn lmtatons should be based on the bucklng strength of the structure. The analyses above are only parts of the desgn procedure n the prelmnary desgn stage, further studes are carred out to nvestgate dynamc behavors under the sesmc effect (response spectrum and elastc-plastc dynamc tme-hstory analyses) and the wnd load (wnd-nduced vbraton), the constructon smulaton analyss and the progressve collapse (cable breakage) analyss s also carred out. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Commttee of Natonal Scence Foundaton of Chna (Grant #).

18 340 Structural Desgn of a Practcal Suspendome REFERENCES [1] Lu, X.L., Modern Spatal Structures, Tanjn Unversty Press, 2003 (n Chnese). [2] Dong, S.L., Luo, Y.Z. and Zhao, Y., Analyss, Desgn and Constructon of New Spatal Structure, Chna Communcaton Press, 2007 (n Chnese). [3] Zhang, Z.H., Theoretcal Research on Large-Span Tensle Spatal Structures Composed of Cables, Bars and Beams, PhD Thess, Space Structure Research Center, Depart of Cvl Engneerng, Zhe Jang Unversty, 2003 (n Chnese). [4] Zhang, M.S., Theoretcal Research on Suspendome, PhD Thess, Space Structure Research Center, Depart of Cvl Engneerng, Zhe Jang Unversty, 2004 (n Chnese). [5] Kang, W.J., Chen, Z.H., Lam, H.F. and Zuo, C.R., Analyss and Desgn of the General and Outmost-rng Stffened Suspendome Structures, Engneerng Structures, 2003, Vol. 25, No. 13, pp [6] Ktporncha, S., Kang, W.J., Lam, H.F. and Alberman, F., Factors Affectng the Desgn and Constructon of Lamella Suspendome Systems, Journal of Constructonal Steel Research, 2005, Vol. 61, pp [7] Zhang, M.S., Dong, S.L. and Zhang, Z.H., Dstrbuton of Intal Prestress and Stablty Analyss of Suspendome, Spatal Structures, Vol. 10, No. 2, pp (n Chnese). [8] Pellegrno, S. and Calladne, C.R., Matrx Analyss of Statcally and Knematcally Indetermnate Frameworks, Internatonal Journal of Solds Structures, 1986, Vol. 22, No. 40, pp [9] Calladne, C.R. and Pellegrno, S., Frst-order Infntesmal Mechansms, Internatonal Journal of Solds Structures, 1991, Vol. 27, No. 4, pp [10] Pellegrno, S., Analyss of Prestressed Mechansms [J]. Int. J. Solds Structures, 1990, Vol. 26, No. 12, pp [11] Pellegrno, S., Structural Computatons wth the Sngular Value Decomposton of the Equlbrum Matrx, Internatonal Journal of Solds Structures, 1993, Vol. 30, No. 2, pp [12] GB , Code for Desgn of Steel Structures, Chnese Plannng Press, Bejng (n Chnese). [13] Shen, S.Z., Desgn of Cable Structures, Chna Archtecture and Buldng Press, Bejng, 2 nd Edton, 2006 (n Chnese). [14] GB , Load Code for the Desgn of Buldng Structures, Chnese Plannng Press, Bejng (n Chnese). [15] GB JGJ , Techncal Specfcaton for Lattced Shells, Chnese Plannng Press, Bejng (n Chnese).