2C09 Design for seismic and climate changes

Transcription

1 2C09 Desg for sesmc ad clmate chages Lecture 08: Sesmc aalyss of elastc MDOF systems Aurel Strata, Poltehca Uversty of Tmsoara 06/04/2017 Europea Erasmus Mudus Master Course Sustaable Costructos uder atural Hazards ad Catastrophc Evets CZ-ERA MUDUS-EMMC

2 8.1. Effectve modal mass. Lecture outle 8.2. Modal respose spectrum aalyss The lateral force method. 8.4 Accdetal torso. Accoutg for torsoal effects structural aalyss. 8.5 Combato of the effects of the compoets of the sesmc acto. 2

3 Effectve modal mass Modal aalyss - equvalet statc forces -th mode -th mode cotrbutos r (t) to the respose quatty r(t): f t s A t s m s m L M T j j j 1 L m m j j j1 T 2 M m m j j j1

4 Effectve modal mass Respose quatty r (t) ca be expressed by: st r t r A t r st - modal statc respose, by applyg "forces" {s} Multstorey structures: base shear force V b s m s m j j j st * b j j j j1 j1 2 * 2 jj jj j1 j1 V s m L M M L m m

5 Effectve modal mass Base shear force -th mode: st b b substtutg V t V A t V st b * * M V t M A t b A SDOF system wth mass m, atural crcular frequecy ad crtcal dampg rato b V t ma t (2) Comparg eq. (1) ad (2) M * - effectve modal mass MDOF: oly the porto M * of the total mass of the structure s effectve producg the base shear force The sum of effectve modal masses over all modes s equal to the total mass of the structure * M m 1 j1 (1) j

6 Effectve modal mass

7 Spectral aalyss Modal aalyss: tme-hstory respose st r t r A t r t r t 1 r0 max t r t Desg - peak values of forces ad dsplacemets Spectral aalyss: drect determato of peak values of forces ad dsplacemets Peak respose r o of the cotrbuto r (t) the -th mode to the total respose r(t) st r r A 0 A - spectral pseudo-accelerato

8 Modal cotrb. ad total tme-hstory respose

9 Methods for combato of peak modal respose Absolute sum r r sutable for structures wth closely spaced atural modes of vbrato Square Root of Sum of Squares (SRSS): 2 0 r 0 1 r sutable for structures wth dstct modes of vbrato

10 Methods for combato of peak modal respose Complete quadratc combato (CQC): r r r r r r r

11 Spectral aalyss: summary Defe structural propertes - mass [m] ad stffess [k] matrces - crtcal dampg rato Determe (T =2/ ) ad {} Respose -th mode: - T ad pseudo-accelerato A from the respose spectrum - equvalet statc forces f s A - compute respose quatty r from forces {f}, for each respose quatty Combe modal cotrbutos r to obta total respose usg SRSS or CQC combato methods ote: geerally t s OT ecessary to cosder ALL modes of vbrato

12 Spectral aalyss: summary Defe propertes of the structure: - mass matrx [m] ad stffess matrx [k] - crtcal dampg rato [m] [k] Fd out atural crcular frequeces (wth the correspodg perods T = 2/ ) ad atural modes of vbrato {} {} 1, T 1 {} 2, T 2 {} 3, T 3

13 A For each mode of vbrato fd out: Pseudo acceleratos A from the respose spectrum correspodg to perods of vbrato T A 3 A 2 A 1 T 3 T 2 T 1 T f 31 f 32 f 33 f 21 f 22 f 23 Equvalet statc forces {f} f 11 f 12 f 13 {} 1, T 1 {} 2, T 2 {} 3, T 3 Respose r due to forces {f}, for each requred respose quatty (forces, dsplacemets, etc. M A1 M A2 M A3 r 1 r 2 r

14 Compute the total respose r by combg modal cotrbutos r (e.g. usg the SRSS method) M A =M A1 2 +M A2 2 +M A3 2 r

15 Modal respose spectrum aalyss Modal respose spectrum aalyss a.k.a. spectral aalyss Spectral aalyss: s the default aalyss method E ca be used always (also cases whe lateral force method caot be appled) umber of modes that eed to be cosdered aalyss: the sum of effectve modal masses for the cosdered modes should amout to at least 90% of the total mass of the structure, all modes wth effectve modal mass larger tha 5% of the total mass of the structure were cosdered aalyss Combato of modal respose: Sum of absolute values (ABS) Square root of sum of squares (SRSS) respose two modes k ad k+1 ca be cosdered depedet f T k ad T k+1 check the followg relatoshp: T k 1 0.9T k Complete quadratc combato (CQC) Results are geerally coservatve, but the correlato betwee tme ad sg of peak values of dfferet respose quattes s ot kow

16 Modal respose spectrum aalyss Results are geerally coservatve, but the correlato betwee tme ad sg of peak values of dfferet respose quattes s ot kow M

17 Lateral force method Ca be used for structures whose sesmc respose s ot flueced sgfcatly by hgher modes of vbrato E crtera for fulfllg the requremet above: structure wth T sec ad T 1 4T C structure regular elevato A smplfed spectral aalyss, that cosders the cotrbuto of the fudametal mode oly V M A * b F S T m b d 1 (V b1 F b ; A 1 S d (T 1 ); M 1* m )

18 Lateral force method Base shear force (E ): F S T m b d 1 S d (T 1 ) - ordate of the desg respose spectrum correspodg to fudametal perod T 1 m - total mass of the structure - correcto factor (cotrbuto of the fudametal mode of vbrato usg the cocept of effectve modal mass): = 0.85 f T 1 T C ad the structure s hgher tha two storeys, ad = 1.0 all other cases

19 Lateral force method Equvalet statc force at storey mode : where usg the expresso f m A m m 2 1 * 2 1 m M m * b A V M b b m m m f m A m V V m m m

20 Equvalet statc forces f Lateral force method Lateral force at storey (E ): F V b F b m 1 1 m ms ms F b - base shear force the fudametal mode of vbrato s - dsplacemet of the mass the fudametal mode shape - umber of storeys the structure m - storey mass

21 Lateral force method Fudametal mode shape ca be approxmated by a horzotal dsplacemets creasg learly wth heght F F b 1 mz mz F m z For structures wth heght <40m T 3 1 Ct H 4 C t = momet-resstg steel frames, C t = momet resstg reforced cocrete frames or steel eccetrcally braced frames, C t = 0.05 all other structures. F b

22 Accdetal torsoal effects Ucertates assocated wth dstrbuto of storey masses ad/or spatal varato of groud moto Accdetal eccetrcty e 1 = 0.05 L (E ) Spatal structural model: M 1 e1 F ±e 1x Y CM Fx ±e L CM y 1y X Fy Lx

23 Accdetal eccetrcty: lateral force method If the lateral stffess ad mass are symmetrcally dstrbuted pla ad uless the accdetal eccetrcty s take to accout by a more exact method, the accdetal torsoal effects may be accouted for by multplyg the acto effects the dvdual load resstg elemets resultg from the applcato of lateral forces by a factor For spatal models (3D): 10.6 x L e For plaar models (2D): 11.2 x L e 23

24 Accdetal eccetrcty: lateral force method x s the dstace of the elemet uder cosderato from the cetre of mass of the buldg pla, measured perpedcularly to the drecto of the sesmc acto cosdered; L e s the dstace betwee the two outermost lateral load resstg elemets, measured perpedcularly to the drecto of the sesmc acto cosdered. 24

25 Accdetal eccetrcty: spectral aalyss For spatal models (3D): the accdetal torsoal effects may be determed as the evelope of the effects resultg from the applcato of statc loadgs, cosstg of sets of torsoal momets M a about the vertcal axs of each storey : M e F a a M 3 M 2 M 1 M a s the torsoal momet appled at storey about ts vertcal axs; e a s the accdetal eccetrcty of storey mass for all relevat drectos; F s the horzotal force actg o storey, as derved usg the lateral force method. 25

26 Accdetal eccetrcty: spectral aalyss For plaar models (2D): the accdetal torsoal effects may be accouted for by multplyg the acto effects the dvdual load resstg elemets resultg from aalyss by a factor 11.2 x L e x s the dstace of the elemet uder cosderato from the cetre of mass of the buldg pla, measured perpedcularly to the drecto of the sesmc acto cosdered; L e s the dstace betwee the two outermost lateral load resstg elemets, measured perpedcularly to the drecto of the sesmc acto cosdered. 26

27 Compoets of the sesmc acto Sesmc acto has compoets alog three orthogoal axes: 2 horzotal compoets 1 vertcal compoets Peak values of a g for horzotal moto are OT recorded at the same tme stat Peak values of respose are OT recorded at the same tme stat accelerate, m/s 2 accelerate, m/s Vracea, , tmp, sicerc (B), EW Vracea, , ICERC (B), S tmp, s

28 Compoets of the sesmc acto Smultaeous acto of two orthogoal horzotal compoets (lateral force or spectral aalyss): Sesmc respose s evaluated separately for each drecto of sesmc acto Peak value of respose from the smultaeous acto of two horzotal compoets s obtaed by the SRSS combato of drectoal respose: E E E 2 2 Ed Edx Edy Alteratve method for combato of compoets of sesmc actos

29 Compoets of the sesmc acto Whe vertcal compoet s cosdered as well: E E E E (0.1) Ed Edx Edy Edz 0.3E 0.3E E (0.2) Edx Edy Edz E 0.3E 0.3E (0.3) Edx Edy Edz 0.3E E 0.3E (0.4) Edx Edy Edz

30 Vertcal compoet Vertcal compoet of sesmc acto shall be cosdered whe vertcal peak groud accelerato a gv 0.25g, ad the structure has oe of the followg characterstcs: has horzotal elemets spag over 20 m has catlever elemets wth a legth over 5 m has prestressed horzotal elemets has colums supported o beams s base-solated

31 Refereces / addtoal readg Al Chopra, "Dyamcs of Structures: Theory ad Applcatos to Earthquake Egeerg", Pretce-Hall, Upper Saddle Rver, ew Jersey, E :2004. "Eurocode 8: Desg of structures for earthquake resstace - Part 1: Geeral rules, sesmc actos ad rules for buldgs". 31

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