Stochastic Structural Dynamics. Lecture-32. Probabilistic methods in earthquake engineering-1

Transcription

1 Sochasic Srucural Dyamics Lecure-3 Probabilisic mehods i earhquake eieeri- Dr C S Maohar Deparme of Civil Eieeri Professor of Srucural Eieeri Idia Isiue of Sciece Baalore 56 Idia maohar@civil.iisc.ere.i

2 Recall We have developed mehods o sudy sysems overed by MX CX KX F X X G, where X & X are specified,ad G is a vecor radom process. The aalysis has icluded characerizaio of respose momes, pdf-s of sysem saes, ad reliabiliy measures. The equaios overi he behavior of srucures subjeced o earhquake suppor moios have form similar o he above equaio. Therefore, wha are he issues ha we eed o cosider afresh?

3 Focus: uceraiies i vibraory respose of srucures duri a earhquake Sochasic models for roud moios Respose specrum, PSD ad ime hisories Modal combiaio rules Seismic risk aalysis Performace based srucural desi (PBSD) Aim: To iroduce he basic ideas ad faciliae fuure self sudy 3

4 Uceraiies i earhquake eieeri problems Whe, where, ad how earhquakes occur Earhquakes The deails of roud moio The effec of roud moio o srucure Dyamic respose Damae ad loss 4

5 Aleraives for earhquake load specificaio Time hisories Moe Carlo simulaios Duhamel's ieral Specral esimaio Specrum compaible accelerorams Power specral desiy Ereme value heory Respose specra Specrum compaibe PSD usi ereme value heory No suied for reliabiliy aalysis 5

6 Refereces R W Clouh ad J P Pezie, 993, Dyamics of srucures, McGraw Hill, NY N C Niam ad S Narayaa, 994, Applicaios of radom vibraios, Narosa, New Delhi 6

7 Respose specra 7

8 Respose specrum Effec based model for earhquake roud acceleraio Equaio overi oal displaceme mv c( v v ) k( v v ) Relaive displaceme v ( ) v( ) v( ) mv cv kv mv v v vv () 8

9 v ( ) ep[ ( )]si[ d( )][ mv( )] d m d ep[ ( )]si[ d( )] v( ) d d. d v ( ) ep[ ( )]si[ ( )] v( ) d v () v ( )ep[ ( )]cos[ ( )] d v ( ) ep[ ( )]si[ ( )] d 9

10 c k v ( ) ( v v ) ( v v) v v m m () ( ) ( )ep[ ( )]si[ ( )] v v d v ( )ep[ ( )]cos[ ( )] d Remark Sysems wih he same ad respod ideically o he same roud moio.

11 Respose quaiies of ieres v (): Maimum relaive displaceme Force i he spri (colum) ad hece he sresses i he colum are proporioal o v ( ) v (): Of ieres i he sudy of secodary sysems Poudi of adjace buildis

12 Force i he spri f () k[ v () v ()] kv() s m v() A () ma() W f () base shear s A A v ( ): has uis of acceleraio (pseudo-acceleraio) seismic coefficie Weih of he buildi seismic coefficie= base shear Base shear heih of he buildi= base mome

13 U kv () m v () m m [ v ( )] V ( ) U srai eery sored v : has uis of velociy pseudo-velociy 3

14 Le a family of sdof sysems wih dampi {η} ad aural frequecies {ω } be subjeced o a ive roud acceleraio. Le us deermie he peak resposes over ime as a fucio of {η} ad {ω }. 4

15 Defiiios S S S S S d v a (, ) ma v( ) Specral relaive displaceme pa (, ) Sd (, ) Specral pseudo (, ) ma v ( ) Specral relaive velociy (, ) ma v ( ) Specral absolue acceleraio acceleraio (, ) S (, ) Specral pseudo velociy pv d 5/3/ 5

16 Defiiios S d (ω,η) S v (ω,η) S a (ω,η) S pa (ω,η) S pv (ω,η) as a fucio of ω wih η as a parameer is called he respose specrum for Relaive displaceme Relaive velociy Absolue acceleraio Pseudo acceleraio Pseudo velociy 5/3/ 6

17 Specrum Cooes "frequecy" o -ais. "Frequecy" ofe refers o he frequecy parameer used i defii Fourier rasform. I his coe we alk of ime ad frequecy domai represeaios of sials. I he coe of respose specrum "frequecy" is o he Fourier frequecy bu he aural frequecies of a family of sdof sysems. Ofe period is ploed o he -ais. 7

18 Limii behavior of he specra as v v vv ( ) lim vv( ) lim ma v ( ) ma v( ) lim S (, ) ma v ( ) pa ma v ( ) ZPAor PGA ZPA: zero period acceleraio PGA: peak roud acceleraio 8

19 Limii behavior of respose specra as v () ep[ ( )]si[ d( )] v( ) d si[ d ( )] lim v ( ) lim v ( ) d ( ) v ( ) d v ( ) lim v ( ) v( ) d lim ma v ( ) ma v ( ) d lim S (, ) ma v ( ) d 9

20 Spv (, ) ma v ( )si ( ) d Sv(, ) ma v( )cos ( ) d S (, ) S (, ) ecep for very small. pv v Sa(, ) ma v( )si ( ) d S pa (, ) ma v ( )si ( ) d S (, ) S (, ) S (, ) a pa pv &. S (, ) S (, ) a pv

21 Triparie plos S (, ), S (, )& S (, ) d pv pa S (, ) S (, ) pv d S S (, ) S (, ) d pv (, ) S (, ) a pv lo S (, ) lo S (, ) lo d pv lo S (, ) lo S (, ) lo a pv

22 lo S T, pa lo S T, pv lo S T, d Naural period s

23 Why so may respose specra? Each respose specrum provides a physically meaiful quaiy S S S d (, ), ) pv (, ) pa ( Peak deformaio Peak srai eery Peak force i he spri Base shear Base mome 5/3/ 3

24 The shape of he respose specrum ca be approimaed more readily for desi purposes wih he aid of all hree specral quaiies ha ay oe of hem ake aloe. Helps i udersadi characerisics of respose specra. Helps i cosruci respose specra. Helps i relai srucural dyamics coceps o buildi codes. 4

25 S pa T, Smooh desi respose specra Soil codiio Dampi PGA ZPA. Naural period s PGA : arrived a based o seismic hazard aalysis 5

26 Facors ha ifluece respose specrum a a ive sie Source mechaism Epiceral disace Focal deph Geoloical codiios Richer s maiude Soil codiio Dampi ad siffess of he sysem 6

27 Aleraives for earhquake load specificaio Time hisories Moe Carlo simulaios Duhamel's ieral Specral esimaio Specrum compaible accelerorams Power specral desiy Ereme value heory Respose specra Specrum compaibe PSD usi ereme value heory No suied for reliabiliy aalysis 7

28 How o eerae a respose specrum compaible wih a ive PSD? X ~, ma m m N S X P T ep zero mea, saioary, Gaussia radom process; wih H S d& H S d T ep 8

29 For a ive probabiliy p, he correspodi is ive by pep ep l l T Le R, be he ive pseudo-acceleraio respose specrum. T p We ierpre R, as he p- h perceile poi. R, l l p (ypically p=84%) T 9

30 How o eerae a PSD compaible wih a ive respose specrum? ~, zero mea, saioary, Gaussia radom process; N S To a firs approimaio we assume H S H S S & d 3

31 , l l To a firs approimaio we hus e,. l l S R p T R S p T

32 Seps () Se ieraio N= () Sar wih he iiial uess o he PSD ive by S N R, T l (3)Evaluae ad usi N H S d& N H S d l p 3

33 N (4)Evaluae R, l l p. T (5)Obai a improved esimae of PSD usi N N R( ) S ( ) S ( ) N R ( ) (6)Sop ieraios if he PSD fucio has covered; if o o o sep 3. 33

34 6 4 Tare Derived Acceleraio m/s 8 6 Pseudo acceleraio respose specrum Frequecy rad/s 34

35 .35.3 Derived Used.5 PSD (m /s 4 )/(rad/s) Frequecy rad/s 35

36 5 Acceleraio m/s Time s 36

37 . Velociy m/s Time s 37

38 Displaceme m Time s 38

39 Variace(σ ) =.893 m /s 4 Sd. deviaio (σ ) =.437 m/s 3σ = 3.3 m/s Time hisory o. Maimum ( m/s ) Miimum ( m/s ) Zero Peak Acceleraio (ZPA) =3.53 m/s 39

40 .9 Probabiliy Normal Simulaio PDF of he simulaed samples of roud acceleraio Acceleraio m/s 4

41 .35.3 Tare Simulaio PSD (m /s 4 )/(rad/s) PSD fucio from samples of roud acceleraio Frequecy rad/s 4

42 Simulaio Gumbel Probabliy Acceleraio m/s 4

43 Respose specrum mehod provides maimum respose of sdof sysems as a fucio of aural frequecies ad dampi raios. How o deal wih mdof sysems? Hope : MDOF sysems ca be decomposed io a se of ucoupled sdof oscillaors. How ca his be ake advaae of? 43

44 Buildi frame uder earhquake suppor moio 44

45 45 Equaio of moio for oal displaceme ) ( ) ( c k k k k k k k k k k c c c c c c c c c m m m Relaive displaceme y y y 3 3 Equaio of moio for relaive displaceme m m m y y y k k k k k k k k k y y y c c c c c c c c c y y y m m m T M KY CY Y M ) (

46 MY CY KY M () Y Z T T T T T T T T,,, T M I K Dia C Dia M Z C Z K Z M z z z M z () ep( )[ A cos B si ] d d ( ) h ( ) d 46

47 Displaceme: y ( ) z ( ) N N k k {ep( )[ A cos B si ] ( ) h ( ) d} k d d Elasic forces: Fs KY KZ MZ Base shear: V N F s Overuri mome: M N F o s 47

48 MV CV KV M{} v ( ) V Z MM KK M z C z K z M{} v ( ) v ( ) M{} z z z v ( ) M modal paricipaio facor or modal eciaio facor 48

49 v v vv T ( ) v ( ) ep[ ( )]si[ ( )] v( ) d z() v() M V Z V () z () V () v () p p k k k k M Pu ( ) v ( ) p k k Vk k M ma v ( ) S (, ) ma V ( )? k k D T k 49

50 Spri force F () KV() KZ s Recall T M F () [ ] MZ s F () [ ] M{ v ()} M s F () M{ a ()} s M ma F ( )? s K 5

51 Base shear N V F () F () B si s i VB M a( ) M Recall M{} M a () M a() VB a() N M M N an () M N N VB a(); ma VB? M T 5

52 Noe : The quaiy has uis of mass ad is called M he effecive modal mass. Prove ha N M oal mass. MT M Toal mass: Z M MZ M M Z M Z MZ Z MT M M M M M M T N M N M QED 5

53 Modal combiaio rules : wha is he basic problem? iv EIy my cy y(,) ; y(,) ; EIy( L,) ; EIy( L,) y, z, iv EIz mz cz m z(,) ; z(,) ; EIz( L,) ; EIz( L,) 53

54 Eiefucio epasio, z a wih ;,,, Wha we kow based o respose specrum based aalysis? We kow ma a ;,,,. T a a a Wha we wish o kow? T ma z, ma a T 54

55 6 4 Tare Derived Acceleraio m/s Frequecy rad/s 55

56 Difficuly ma ma a a T T Remarks The erema of a for =,,, are likely o occur a differe imes ad hey may have differe sis. Respose specra do o coai iformaio o imes a which erema occur or do hey sore he sis of he erema. T ma a ca occur a a ime isa * a which oe of a ; =,,, eed o aai heir respecive eremum values. 56

57 Problem of modal combiaio rules How bes o obai ma a i erms of ma T a sysem? T, ad he modal characerisics of he vibrai Modal characerisics: aural frequecies, mode shapes, modal dampi raios, ad paricipaio facors. These rules are formulaed based o mehods of radom vibraio aalysis. 57