Periods of vibration of braced frames with outriggers

Transcription

1 Avalable onlne at Proceda Engneerng 0 (01 ) Steel Structures and Brdges 01 Perods of vbraton of braced frames wth outrggers M. Ncoreac a * and J.C.D. Hoenderkamp b a Techncal Unversty of Cluj-Napoca, C. Dacovcu 15, Cluj-Napoca 0000, Romana b Endhoven Unversty of Technology, P.O. Box 513, 5600 MB Endhoven, The Netherlands Abstract Ths paper presents smple equatons for the natural lateral and rotatonal perods of vbraton for hgh-rse steel structures comprsng braced frames wth outrgger trusses. The structural floor plan can be symmetrc or asymmetrc combnatons of dentcal bents. Each braced frame s modelled by a cantlever wth a bendng stffness and rackng shear stffness. The outrggers are represented by a rotatonal sprng. The stffness of the outrgger sprng s dependent on three specfc modes of behavour: bendng and rackng shear n the outrgger truss and axal lengthenng or shortenng of the exteror columns. Ths yelds a smple calculaton of the maxmum horzontal deflecton of the structure. For the rotatonal frequences, the bendng, rackng shear and sprng stffness parameters of the steel structure each are combned to yeld a torsonal stffness, a warpng stffness and a warpng sprng stffness whch allow an assessment of the rotaton at the top. The approxmate method of analyss relates the natural frequences of the structure to the top deflecton and rotaton when the self-weght s taken as a dstrbuted horzontal load. In the calculatons a braced frame s represented by a rgd stck model wth ts overall flexblty assgned to a rotatonal sprng at the base. Results obtaned by the smplfed method are compared to those from a fnte element analyss. The approxmate method gves acceptable conservatve results for prelmnary analyss. 01 Publshed by Elsever Ltd. Selecton and revew under responsblty of Unversty of Žlna, FCE, Slovaka. Open access under CC BY-NC-ND lcense. Keywords: braced frames; outrgger trusses; rotatonal sprng; warpng sprng. 1. Introducton In the early desgn stages of a tall buldng, there are some crtera that need to be satsfed n order to assure that the chosen structural system s an optmal one. The maxmum lateral dsplacements are controlled by employng the drft ndex whch s generally accepted to be 1/500. The natural perod of free vbraton s also an mportant factor and s used for obtanng the desgn base shear. Several emprcal estmates are gven by dfferent sesmc codes. The ASCE 7-10 [1], UBC-97 [], and Eurocode 8 [3] gve an equaton for estmatng * Tel fax: E-mal address: monca.ncoreac@mecon.utcluj.ro Publshed by Elsever Ltd. do: /j.proeng Open access under CC BY-NC-ND lcense.

2 M. Ncoreac and J.C.D. Hoenderkamp / Proceda Engneerng 0 ( 01 ) the natural vbraton perod of a braced frame: T = C t H 0.75 where C t = and H s the heght of the buldng n meters. Other codes, lke ATC 3-06 [], gve a dfferent formula: T = 0.09H/D -0.5 where D s the dmenson of the buldng, parallel to the appled force. Ells [5] consdered the frequences of 163 buldngs and recommended the followng equaton for the translatonal frequency: T= H/6 and for the fundamental torsonal mode T φ = H/7. Varatons on these frequences can be found n the lterature, for steel structures: T = H/5, T φ = H/78 or T φ =H/108(for truss bracng) [6]. The dynamc behavor of outrgger structures was frst studed by Rutenberg [7] who presented graphcal solutons for the frst three natural perods of vbraton for hgh-rse shear walls wth nfntely rgd outrggers. The free vbraton of outrgger braced shear wall structures was also studed by Moudarres and Coull [8] wth the help of a transfer matrx technque. The suggested solutons do not allow a rapd assessment of the vbraton perods. The am of ths study s to present smple approxmate equatons for the fundamental lateral and rotatonal perods of vbraton for tall buldngs wth a symmetrc or an asymmetrc structural floor plan comprsng dentcal steel braced frames wth trussed outrggers.. Modelng of outrgger structure The method of analyss s based on an earler presented method for the calculaton of the optmum locaton of the outrgger up the heght of a braced frame structure [9] where the structure as shown n Fgure 1(a) s represented by seven stffness parameters. The central braced frame has a unform bendng stffness EI b and a rackng shear stffness GA b. The outrgger structure conssts of a truss on both sdes of the frame wth a bendng stffness EI o and a rackng shear stffness GA o. The axal stffness of the exteror columns below the outrgger level s combned to form a global bendng stffness EI c. The outrggers and exteror columns are replaced by a sngle rotatonal sprng at outrgger level, as shown n Fgure 1, wth stffness: K b ( EI 1 o ) 1 ( hgao zo EI c o ) (1) where H s the total heght of the structure; z o s the locaton of the outrgger measured from the base of the structure; α = /b s a non-dmensonal parameter where b s the length of the outrgger and s the dstance between an exteror column and the centrod of the braced frame and h s the heght of the outrgger. When the structure s subjected to a horzontal unform load the restranng moment n the sprng and the lateral deflecton at the top of the structure then are: M r 3 w ( zoh zo H zo 3) EI b w (1 zo GAb ) 1 Ko zo EI b 1 ( hgab ) () max wh EIb wh GAb Mr zo(1 zo / ) EIb 1 8 GA (3) For the three dmensonal model of the structure the salent planar propertes EI b, GA b and K o wll be used to obtan the rotatonal stffness parameters of the structure. The warpng stffness EI, torsonal stffness GJ and a warpng sprng stffness parameter K can qute easly be obtaned as follows, where c s the dstance between an outrgger bent and the elastc neutral axs of the structure. EI EI c ; GJ GA b; c ; K K c () ω b; ω o; b

3 300 M. Ncoreac and J.C.D. Hoenderkamp / Proceda Engneerng 0 ( 01 ) Fg. 1 (a) braced frame wth outrgger trusses; (b) smplfed model for planar dynamc analyss; (c) floor plan of asymmetrc buldng wth braced frames and outrgger trusses; (d) smplfed model for torsonal dynamc analyss It can qute easly be shown that the restranng b-moment and the maxmum rotaton at the top are: B 3 m t ( zoh zoh zo 3) EIω mt (1 zo) GJ 1 Kω zo EIω hgj ω 1 8 ω t ω o o ω m H EI m H GJ B z (1 z / ) EI 1 GJ (6) top t where m t s the horzontal load w multpled by the eccentrcty of the loadng. The above equatons allow a rapd assessment of the maxmum lateral dsplacement of the elastc axs and rotaton about ths axs. 3. Natural perods of vbraton For slender lateral load resstng structures where the frst mode shape can be taken to be smlar to that of a flexural cantlever beam wth a unformly dstrbuted stffness and mass, the frst natural frequency ω can be approxmated by the generally accepted equaton [10] gven n expresson (7a), where s the mass per unt heght of the buldng n kg/m and EI s the flexural stffness of the structure. Rewrtng equaton (7a) by applyng the mass of the buldng as a horzontal load, the horzontal perod of vbraton can be expressed as a functon of the top deflecton of the flexural cantlever δ max (the gravtatonal acceleraton s 9.81 m/s ): EI H T 1.61 max (7a), (7b) The horzontal deflecton confguraton of a typcal braced frame wth a sngle outrgger devates sgnfcantly from that of a pure flexural cantlever as shown n Fgure. The bendng deflecton dsplays a sngle curvature over the heght of the structure, whle a braced frame typcally shows a double curvature deflected shape. The horzontal dsplacement pattern of a braced frame wth an outrgger structure has a trple curvature. In order to select a smplfed deflected confguraton, closer to the trple curvature, t s suggested to adopt a straght lne whch can be represented by a rgd cantlever. The flexblty of the structure can then be completely represented by a sngle rotatonal sprng at the base of the structure. For ths system the equaton of moton s: (5) H 3 3 d K dt θ 0 (8)

4 M. Ncoreac and J.C.D. Hoenderkamp / Proceda Engneerng 0 ( 01 ) Fg.. (a) Cantlever beam model wth rotatonal sprng; (b) deflecton confguratons of flexural cantlever, (c) braced frame, (d) outrgger braced structure and (e) rgd cantlever. In equaton (8) K s the rotatonal stffness representng the structure and H 3 /3 s ts mass moment of nerta. The angular frequency s expressed as a functon of the maxmum dsplacement max of the structure when the mass of the buldng s horzontally appled to the structure (see eqn. (9a)). Rewrtng Equaton (9a) yelds the approxmate smple expresson (9b) for the natural perod of vbraton for a braced frame wth an outrgger truss. Ths gves a 1.5% longer perod than the more famlar equaton. 3 3Kθ H 1. 5g max max T (9a), (9b) For the rotatonal natural perod of vbraton of a structure comprsng dentcal braced frames wth outrgger trusses the angular frequency can analogously be approxmated by: ω EI r H where r a b 1 (10) where EI s the warpng stffness along the heght and r s the radus of gyraton of the buldng.n whch a and b are the wdth and depth of the buldng. The perod of torsonal vbraton then becomes: T 1.61 r (11) max where φ max s the maxmum angle of twst of the structure subjected to a unformly dstrbuted torque of magntude g r. Analogously to the analyss of planar structures the same holds true for the rotatonal analyss of the structure,.e. the mproved perod of torsonal vbraton becomes: T r (1) max Equatons (13) and (16) can be used for a rapd assessment of the natural lateral and rotatonal perods of vbraton of the buldng.

5 30 M. Ncoreac and J.C.D. Hoenderkamp / Proceda Engneerng 0 ( 01 ) Accuracy of the smplfed method In order to determne the accuracy of the suggested equatons, a lmted error analyss was performed on sx typcal hgh-rse structures: one symmetrc and one asymmetrc floor plan as shown n resp. n Fg. 3 (a) and Fg. 3 (b); three buldng heghts of 100 m, 160 m and 00 m; outrgger locatons at the top (z o =H), at mdheght (z o =H/) and at three quarters up the heght of the structure (z o =3H/); two sets of sectonal propertes for the structural elements whch are gven n Table 1. The two 100 m hgh buldngs have dentcal one-bay X- braced frames, whle the other structures have dentcal two-bays X-braced frames, resp. Fg 3(c) and (d). Each outrgger truss s 1 m long and has a sngle storey heght of m wth four X-braced frame segments for the 100 m hgh buldngs, whle for the 160 m and 00 m hgh buldngs the outrgger trusses are two storeys hgh (8 m) wth 8 X-braced frame segments. The elastc modulus of steel s taken: E = kn/m. The structural parameters for the braced frames wth outrggers are computed usng expressons gven earler [9]. The mass per unt volume for the dynamc analyss s 00 kg/m 3. Fg. 3 Structural systems: (a) floor plan A1, (b) floor plan A, (load eccentrctes accordng to EN , Secton 7.1. [11]), (c) outrgger braced frame H=100m, (d) outrgger braced frames H=160m and H=00m Table 1 Cross sectonal areas of the structural elements Buldng Braced Frames Exteror Outrgger beams Outrgger dagonals Heght m Columns m Dagonals m Columns m Max. m Mn. m Max. m Mn. m H A1 A A1 A A1 A A1 A A1 A A1 A A1 A The natural lateral and torsonal perods of vbraton of the structures, gven by equatons (13) and (16), are presented n resp. Table and Table 3. The results from fnte element analyses done wth the SAP000 F.E. package are gven n brackets. 5. Conclusons The proposals for the natural perods of vbraton gven by equatons (13) and (16) gve results that are wthn % of those obtaned from fnte element analyses. Also, the percentage absolute maxmum and average errors for the rgd stck model (eqns. 13 and 16) are smaller than those obtaned for the flexural cantlever beam model (eqns.10 and 15).

6 M. Ncoreac and J.C.D. Hoenderkamp / Proceda Engneerng 0 ( 01 ) Table Natural lateral perods of vbraton (from equaton (13) and from the F.E.M. analyss, n brackets) Buldng 0.05H 0.75 system H/ 3H/ H [1,, 3] Mn. Max. Mn. Max. Mn. Max A1 H=100m 3.83(3.73).6(.) 3.76(3.8).3(.35) 3.98(.11).1(.7) A H=100m 3.8(3.78).7(.69) 3.79(3.85).60(.60).09(.1).67(.71) A1 H=160m 3.53(3.7) 5.5(5.) 3.65(3.80) 5.3(5.30).19(.37) 5.37(5.38) A H=160m 3.0(3.7) 5.9(5.3) 3.5(3.63) 5.9(5.0).1(.3) 5.3(5.8) A1 H=00m 3.63(3.5) 5.65(5.51) 3.73(3.85) 5.(5.36).6(.1) 5.7(5.) A H=00m 3.91(3.80) 6.01(5.87).03(.15) 5.79(5.7).61(.75) 5.85(5.81) Table 3 Natural torsonal perods of vbraton (from equaton (16) and from the F.E.M. analyss, n brackets) Buldng H/7 system H/ 3H/ H [5] Mn. Max. Mn. Max. Mn. Max A1 H=100m 6.33(6.5) 7.37(7.30) 6.(6.3) 7.17(7.19) 6.60(6.80) 7.9(7.38) A H=100m 3.9(3.3).06(.01) 3.(3.9) 3.9(3.93) 3.50(3.60).00(.03) A1 H=160m 5.8(5.7) 9.13(8.98) 6.03(6.7) 8.80(8.75) 6.9(7.) 8.88(8.90) A H=160m.91(.80).70(.57) 3.03(3.10).53(.5) 3.5(3.6).57(.5) A1 H=00m 6.00(5.81) 9.35(9.10) 6.16(6.36) 8.97(8.87) 7.0(7.8) 9.05(9.00) A H=00m 3.35(3.6) 5.1(5.03) 3.5(3.55).95(.90) 3.9(.06) 5.00(.98) 0.09H -0.5 [] H/6 [5] H/108 [6] The very smple expressons gven by the varous buldng codes yeld much shorter perods of vbraton wth maxmum errors larger than 50%. It can be concluded that the suggested smplfed equatons for the calculaton of the frst lateral and torsonal frequences of hgh-rse buldngs wth symmetrc or asymmetrc structural floor plans comprsng combnatons of dentcal braced frames wth outrgger trusses yeld good results for the prelmnary stages of the structural desgn. Acknowledgements Ths paper was supported by the project "Doctoral studes n engneerng scences for developng the knowledge based socety-sidoc contract no. POSDRU/88/1.5/S/60078, project co-funded by the European Socal Fund through Sectoral Operatonal Program Human Resources References [1] Amercan Socety of Cvl Engneers: Mnmum Desgn Loads for Buldngs and Other Structures, ASCE 7-10, Amercan Socety of Cvl Engneers, 010. [] Internatonal Conference of Buldng Offcals (ICBO): Unform buldng code, Internatonal Conference of Buldng Offcals Whtter, [3] European Commttee for Standardzaton: Eurocode 8: Desgn of structures for earthquake resstance - Part 1: General rules, sesmc actons and rules for buldngs, 00. [] Appled Technology Councl: Tentatve provsons for the development of sesmc regulatons for buldngs: a cooperatve effort wth the desgn professons, buldng code nterests, and the research communty, U.S. Dept. of Commerce, Natonal Bureau of Standards, [5] Ells, B.: An Assessment of the Accuracy of Predctng the Fundamental Natural Frequences of Buldngs and the Implcatons Concernng the Dynamc Analyss of Structures, ICE Proceedngs 69, 1980, p [6] Lagomarsno, S.: Forecast models for dampng and vbraton perods of buldngs, Journal of Wnd Engneerng and Industral Aerodynamcs 8, 1993, p [7] Rutenberg, A.: Earthquake analyss of belted hgh-rse buldng structures, Engneerng Structures 1, 1979, p [8] Moudarres, F., Coull, A.: Free Vbratons of Outrgger-Braced Structures, ICE Proceedng 79, 1985, p [9] Hoenderkamp, J.C.D., Bakker, M.C.M. : Analyss of hgh-rse braced frames wth outrggers, The Structural Desgn of Tall and Specal Buldngs 1, 003, p [10] Dym, C.L., Shames, I.H.: "Sold Mechancs: A Varatonal Approach", McGraw-Hll Inc.,US, [11] European Commttee for Standardzaton, Eurocode 1: Actons on structures - General actons - Part 1-: Wnd actons, 00.