Probability-Based Seismic Assessments: Implementing Wide-Range Nonlinear Dynamic Analysis Methods

Transcription

1 Probbility-Bsed Seismic Assessments: Implementing Nonliner Dynmic Anlysis Methods Ftim Jlyer Postdoctorl Scholr University of Rome L Spienz Cliforni Institute of Technology (CIT)

2 Outline A brief Introduction to Probbilistic Performnce-Bsed Erthquke Engineering Demnd nd Cpcity Fctor Design (DCFD) Formt Probbilistic Representtion for the Ground Motion Intensity Mesure (IM) Implementing Non-Liner Dynmic Anlysis Methods Probbilistic Representtion for Structurl Demnd Given IM Probbilistic Representtion for Structurl Limit Stte Cpcity A Cse-Study for Probbilistic Seismic Assessment

3 A Brief Introduction to Probbilistic Performnce-Bsed Erthquke Engineering

4 Deterministic Anlysis nd Probbilistic Anlysis (how necessry is probbilistic modeling?) Deterministic nlysis cn be employed for processing nd orgnizing the vilble informtion bout physicl phenomenon Probbilistic nlysis is needed for tking into considertion the missing informtion bout phenomenon Therefore, rther thn being lterntive pproches to nlyzing physicl phenomenon, probbilistic nd deterministic modeling re two sides of the sme coin.

5 Chllenges in Seismic Anlysis of Structures (from probbilistic nlysis point of view) One of the most significnt nd chllenging steps in the nlysis of system is to understnd the ttributes of future input to the system nd to distinguish those cusing extreme behvior in it. Being events with extreme consequences, erthqukes re mongst the most significnt input to structurl system. The smll frequency nd the lrge uncertinty ssocited with their occurrence, renders erthqukes lso one of the most difficult structurl input ctegories to predict.

6 Erthqukes: Extreme, Rre, nd Uncertin

7 A Seismic Fult Zone A building locted in seismic fult zone is subject to seismic thret from the surrounding ctives fults source i: Sn Andres Fult site: Vn Nuys (M,r) Fults of Los Angeles region

8 Probbilistic Performnce-Bsed Erthquke Engineering One of the min ttributes distinguishing performnce-bsed erthquke engineering from trditionl erthquke engineering is the definition of quntifible performnce objectives. Performnce objectives re quntified usully bsed on life-cycle cost considertions, which encompss vrious prmeters ffecting structurl performnce, such s, structurl, non-structurl or contents dmge, nd humn csulties. Probbilistic performnce-bsed engineering cn be distinguished by defining probbilistic performnce objectives.

9 Probbilistic Performnce Objective The performnce objective cn be stted in terms of the men nnul frequency of exceeding limit stte, e.g., collpse λls P o λ LS is the men nnul frequency of exceeding limit stte P 0 is the llowble probbility level

10 Probbilistic Performnce Objective in Structurl Terms The probbilistic performnce objective cn be stted in structurl terms s the men nnul frequency of exceeding demnd exceeding limit stte cpcity λ LS = P( D > CLS M m0 ) λ( M m0 ) Po m 0 C LS D is the moment mgnitude for n erthquke of interest is the structurl cpcity for limit stte LS is the structurl demnd

11 A Probbilistic Representtion for Structurl Demnd Men nnul rte of exceeding given vlue of demnd Also known s Hzrd Curve for Demnd λd( y) = P( D > y M m0) λ( M m0 ) 10 0 Annul Frequency of Exceeding Mximum Interstory Drift 10 1 λ θmx λ D (y) θ mx y

12 A Probbilistic Representtion for Structurl Limit Stte Cpcity ηˆc LS βˆ C LS P( C y) LS

13 Implicit Probbilistic Anlysis in Current Seismic Design nd Assessment Procedures Current seismic design procedures (FEMA 356, ATC-40) tke into ccount the uncertinty in the ground motion implicitly by defining design erthqukes with prescribed probbilities of being exceeded in given time period (e.g., P o =10% probbility in 50 yers). Men Annul Frequency of Exceeding PGA Also Known s PGA Hzrd Curve 10% in 50 yers PGA=0.40g

14 Alterntive Direct Probbilistic Representtions of Ground Motion Uncertinty A Probbilistic Representtion of Ground Motion using Intensity Mesures (IM-Bsed, FEMA-SAC Guidelines, PEER Methodology) B Complete Probbilistic Representtion of the Ground Motion Time History

15 Choice of IM The spectrl ccelertion t the smll-mplitude fundmentl period of the structure denoted by ( T 1 ) or simply, S is dopted s the intensity mesure (IM). S c u(t) k m= 1 M,r T 1 = period of the oscilltor u(t) t ξ = dmping S 4π (T 1,ξ) = 2 T 2 coefficien t mx(bs(u(t)) )

16 Choice of Structurl Response Prmeter D = θ mx We hve chosen the mximum inter-story drift ngle, θ mx, displcementbsed structurl response, s the structurl response prmeter δ = θ h h 105 M, R θ h (l l )/ θ = mx( θ (t )) mx

17 Structurl Limit Stte: Globl Dynmic Instbility C = θ LS cp The onset of globl dynmic instbility

18 The IM-Bsed Approch for Clculting the Annul Frequency of Exceeding Limit Stte Cpcity λ LS = P( D CLS M m0 ) λ( M m0 ) Po λls = P(θ > θcp θ ) p( θ S ) dλs (S ) θ mx S mx mx mx λ LS is lso known s the limit stte probbility or probbility of filure

19 Demnd nd Cpcity Fctored Design (DCFD)

20 Demnd nd Cpcity Fctor Design (DCFD) A probbilistic ssessment criterion: λls P o After lgebric mnipultions nd mking set of simplifying ssumptions, n LRFD-like probbilistic design criterion for given llowble probbility level, P o, cn be derived: Fctored Demnd (Po) Fctored Cpcity

21 Min Assumptions Leding to Closed-form Expression for (DCFD) Spectrl ccelertion hzrd is power-lw function of the spectrl ccelertion: λ S ( x) = Demnd (given spectrl ccelertion) cn be described by lognorml distribution with constnt stndrd devition. k o x k β θ mx ( S S ) = const. Medin demnd is power-lw function of the spectrl ccelertion: η θmx S Medin cpcity is described by lognorml distribution with constnt medin nd stndrd devition (with respect to spectrl ccelertion). η θ, β = const. cp θ cp = x = x b

22 A Closed-Form Anlyticl Solution the Annul Frequency of Exceeding Limit Stte Cpcity λ LS = λ S ( S η θcp ) e 1 k 2 b 2 2 β 2 θmx S e 1 k 2 b 2 2 β 2 θcp S η C LS C = η LS 1 b is the spectrl ccelertion corresponding to medin cpcity.

23 Closed-Form Presenttion of DCFD Formt λls P o After lgebric mnipultions nd mking set of simplifying ssumptions, n LRFD-like probbilistic design criterion for given llowble probbility level, Po, cn be derived: η θ P0 mx S e 1 k β 2 b 1 k 2 b 2 2 θmx S β θ cp η θ cp e Fctored Demnd (Po) Fctored Cpcity

24 A Closed-Form Anlyticl Solution the Annul Frequency of Exceeding Structurl Demnd y y 1 b λ (Also Known s Drift Hzrd) θ mx ( y) = λ S ) e 2 2 β 2 θ mx S Where S = is the spectrl ccelertion corresponding to medin demnd y. ( S y 1 k 2 b Drift hzrd curve - closed form

25 A Grphic Presenttion of DCFD formt: Drift hzrd curve - closed form P 0 λ LS F.D. Fctored Demnd (Po) F.C. Fctored Cpcity

26 Probbilistic Representtion for the Ground Motion Intensity Mesure (IM)

27 The IM-Bsed Approch for Clculting the Annul Frequency of Exceeding Limit Stte Cpcity λ LS = P( D CLS M m0 ) λ( M m0 ) Po λls = P(θ > θcp θ ) p( θ S ) dλs (S ) θ mx S mx mx mx λ LS is lso known s the limit stte probbility or probbility of filure

28 Probbilistic Representtion for IM for given M nd r Ground motion uncertinty cn be represented by dopting prmeters known s the intensity mesures (IM). A probbilistic representtion of the (dopted) intensity mesure cn be constructed for given mgnitude nd distnce using empiricl ttenution models tht re developed from dtbse of ground motion records. The reltion between IM nd ground motion prmeters, such s mgnitude nd distnce, cn be expressed in the following generic form: ln uncertin vrible for missing informtion IM = f ( M, r) + ε σ ln IM M, r known first moment known second moment

29 Probbilistic Representtion for IM for given M nd r Empiricl Attenution Model Spectrl ccelertion is 1 Probbility of Exceeding Spectrl Accelertion for Scenrio Erthquke, M=7, r=20km, Deep Soil described by log-norml 0.9 distribution. The prmeters of this distribution, nmely, men nd stndrd devition, re predicted by the ttenution reltion. Probbility of Exceeding S (T 1 ) P[ S > ln x x M, r] = 1 Φ( σ ln S f ( M, r) ) M, r 0.1 Abrhmson nd Silv (1997) Attenution Reltion SS, Atkinson nd Silv (2000) Stochstic Model S (T 1 =0.80 sec) [g]

30 A Rnge of Possible Erthquke Events A scenrio Erthquke Event is represented by moment mgnitude M nd source-site-distnce r. source i: Sn Andres Fult site: Vn Nuys (M,R) Fults of Los Angeles region

31 λ S ( x) Probbilistic Representtion of IM men nnul rte of exceeding given spectrl ccelertion vlue, lso known s spectrl ccelertion hzrd = N N λi ( S > x) = λi ( M m0 ) I( S > x M, r, ε ) p( M, r, ε ) dmdrdε i= 1 i= 1 ll M,r nd ε probbility model for missing informti S = x Spectrl ccelertion hzrd curve for: T=0.85sec - Vn Nuys, CA Attenution lw: Abrhmson nd Silv, horizontl motion on soil

32 Approximting the Hzrd Curve with Line in the Region of Interest P 0 =0.03 P 0 = k=2.7 S =0.40g S =0.70g λ S ( S ) = k 0 S k

33 Probbilistic Representtion for Structurl Demnd given IM Implementing Non-Liner Dynmic Anlysis Methods

34 The IM-Bsed Approch for Clculting the Annul Frequency of Exceeding Limit Stte Cpcity λ LS = P( D CLS M m0 ) λ( M m0 ) Po λls = P(θ > θcp θ ) p( θ S ) dλs (S ) θ mx S mx mx mx λ LS is lso known s the limit stte probbility or probbility of filure

35 Ground Motion Record Selection A suite of 30 rel ground motion recordings on soil (soil types C nd D by the Geo-mtrix definition) re selected from Pcific Erthquke Engineering Reserch Center (PEER) dtbse. The records re chosen from bin of ground motions records with moment mgnitude between 6.5 nd 7.0 nd closest distnce to the site between 15 to 32 kilometers. 10 Histogrm: Record Selection in Tble II M r 30 km M i, r i S (T 1 )

36 Structurl Model: An Existing RC Frme Structure in Los Angeles Are M, R,θ Bem-column model with stiffness nd strength degrdtion in sher nd flexure using DRAIN2D-UW by J. Pincheir et l.

37 Multiple-Stripe Anlysis Multiple Stripe Anlysis Existing RC Frme, Los Angeles S (T 1 ) Ground Motion Recordings, PEER dtbse 6.5 M 7.0, 15 r 30 km θ mx Multiple-stripe nlysis (MSA) is non-liner dynmic nlysis method in which, stripes of structurl response vlues re obtined by subjecting structurl model to suite of ground motion records scled to multiple levels of spectrl ccelertion.

38 Multiple-Stripe Anlysis A Log-Norml Probbilistic Representtion for the Stripe Response ˆ β θ mx S = 0.70 = 0.49 ˆ η θ S = mx = Sttistics of the stripe of responses cn be used to estimte the medin nd frctionl stndrd devition t ech spectrl ccelertion level

39 Multiple-Stripe Anlysis A Log-Norml Probbilistic Representtion for the Stripe Response ˆ β θ mx S = 0.70 = 0.49 ˆ η θ mx S = = Sttistics of the stripe of responses cn be used to estimte the medin nd frctionl stndrd devition t ech spectrl ccelertion level

40 Closed-Form Presenttion of DCFD Formt λls P o After lgebric mnipultions nd mking set of simplifying ssumptions, n LRFD-like probbilistic design criterion for given llowble probbility level, Po, cn be derived: η θ P0 mx S e 1 k β 2 b 1 k 2 b 2 2 θmx S β θ cp η θ cp e Fctored Demnd (Po) Fctored Cpcity

41 A Closed-Form Anlyticl Solution the Annul Frequency of Exceeding Structurl Demnd (Also Known s Drift Hzrd) y y 1 b λ θ mx ( y) = λ S ) e 2 2 β 2 θ mx S Where S = is the spectrl ccelertion corresponding to medin demnd y. ( S y 1 k 2 b Drift hzrd curve - closed form

42 Multiple-Stripe Anlysis Obtining Locl Prmeter Estimtes for Clcultion of Fctored Demnd P 0 S =0.70 P 0 S =0.40 b=2.70 b=1.60 ˆ β θ mx S = 0.70 = 0.49 ˆ β θ mx S = 0.40 = 0.35 ˆ η θ S = mx = ˆ η θ S = mx = Fctored demnd t multiple levels of llowble probbility cn be clculted using locl prmeter estimtes bsed on multiple-stripe Anlysis (MSA)

43 IM-Bsed Probbilistic Representtion for Structurl Demnd The uncertinty in the prediction of structurl response cn be described by the men nnul rte of exceeding given vlue of mximum inter-story drift ngle, y, which cn be evluted numericlly using the Totl Probbility Theorem: λ θ mx ( y) = P( θmx > y M m0) λ( M m0) P( θ mx > y M m0 ) = P( θmx > y IM ) p( IM M m0 0 probbilistic representtion of structurl response given IM ) dim probbilistic representtion of IM given n event of interest, M > m 0 λ θ mx ( y) = P( θ mx > y M m0) λ( M m0 ) = P( θmx > y IM ) 0 dλ IM men nnul rte of occurrence of n event of interest, M > m 0

44 Multiple-Stripe Anlysis A Non-Prmetric Probbilistic Representtion for the Stripe Response P( θ 30 1 > = = mx y S x) Iθ > ( 30 ) mx y x i= 1

45 The uncertinty in the prediction of structurl response cn be described by the probbility of exceeding given vlue of mximum inter-story drift ngle, y, which cn be evluted numericlly using the Totl Probbility Theorem: λ θ ( y) = P( θmx > y S = x) dλs ( x) dx mx x= Spectrl Accelertion Hzrd, RC Frme, Los Angeles 2 Multiple Stripe Anlysis Existing RC Frme, Los Angeles λ S (T 1 ) S (T 1 ) IM bsed: PSHA Direct: Stochstic Ground Motion Model S (T 1 =0.85) Ground Motion Recordings, PEER dtbse 6.5 M 7.0, 15 r 30 km θ mx

46 Multiple-Stripe Anlysis Men nnul rte of exceeding given vlue of mximum inter-story drift λ θ mx ( y) 33 yers return period P o = yers return period P o = D(0.03)= D(0.0084)= 0.02 Structurl Response, Mximum Inter-story Drift Rtio y

47 Probbilistic Representtion for Limit Stte Cpcity Implementing Non-Liner Dynmic Anlysis Methods

48 The IM-Bsed Approch for Clculting the Annul Frequency of Exceeding Limit Stte Cpcity λ LS = P( D CLS M m0 ) λ( M m0 ) Po λls = P(θ > θcp θ ) p( θ S ) dλs (S ) θ mx S mx mx mx λ LS is lso known s the limit stte probbility or probbility of filure

49 Incrementl Dynmic Anlysis (IDA) The IDA curve provides unique informtion bout the nture of the structurl response of n MDOF system to ground motion record.

50 Structurl Limit Stte: Globl Dynmic Instbility C = θ LS cp The onset of globl dynmic instbility

51 Incrementl Dynmic Anlysis (IDA) IDA curves for the suite of 30 ground motion records. Ech curve corresponds to ground motion record nd is obtined by connecting the individul response points for tht record. ) plot in rithmetic scle, b) sme plot in logrithmic scle

52 Incrementl Dynmic Anlysis (IDA) Determining the Onset of Globl Dynmic Instbility

53 Incrementl Dynmic Anlysis (IDA) A Log-Norml Probbilistic Representtion for Limit Stte Cpcity ˆ = 0.38 β θco ˆ = 0.39 β θco ˆ = η θco ˆ = 0.41 β θco The sttistics (medin nd frctionl stndrd devition) of the spectrl ccelertion nd the mximum inter-story drift rtio t the onset of globl dynmic instbility re estimted using the IDA curves.

54 Closed-Form Presenttion of DCFD Formt λls P o After lgebric mnipultions nd mking set of simplifying ssumptions, n LRFD-like probbilistic design criterion for given llowble probbility level, Po, cn be derived: η θ P0 mx S e 1 k β 2 b 1 k 2 b 2 2 θmx S β θ cp η θ cp e Fctored Demnd (Po) Fctored Cpcity

55 A Cse-Study for Probbilistic Seismic Assessment

56 Structurl Model: An Existing RC Frme Structure in Los Angeles Are M, R,θ Bem-column model with stiffness nd strength degrdtion in sher nd flexure using DRAIN2D-UW by J. Pincheir et l.

57 Estimting the fctored demnd for the tolerble probbility, P o =0.0084: F. D.( P 0 ) = η θ mx F. D.(0.0084) = S ( P 0 s ) e e 1 k ( P 2 b( P 0 0 ) β ) (0.49) θmx S 2 ( P0 s ) = = 0.020

58 It cn be demonstrted tht Fctored demnd for the tolerble probbility P o, is equl to the demnd vlue tht is exceeded with frequency P o Annul Frequency of Exceeding Mximum Interstory Drift 10 1 λ θmx 10 2 P 0 = θ mx F.D.=0.020

59 Fctored cpcity estimtion for the limit stte of globl dynmic instbility: Getting help from the IDA's 2 Incrementl Dynmic Anlysis Holidy Inn, Vn Nuys 2 Incrementl Dynmic Anlysis Holidy Inn, Vn Nuys Spectrl Accelertion of "first" mode, S [g] s Spectrl Accelertion of "first" mode, S [g] s ˆ = 0.38 β θco ˆ = 0.39 β θco ˆ = η θco 8 ˆ = 0.41 β θco Mximum Interstory Drift Angle, θ mx Mximum Interstory Drift Angle, θ mx F. C. = η C LS e 1 k β 2 b 2 CLS = e (0.41) = = 0.026

60 Finlly the checking moment:? Fctored Cpcity Fctored Demnd (0.0084)

61 Assessment criterion for tolerble probbility, = : P o λ LS P o? = λ LS = θ mx S P(θmx > θ θmx ) p( θmx S ) dλ ( S ) = cp S 0.007

62 Summry nd Conclusions Probbilistic nlysis is n inseprble prt of the nlysis of ny physicl phenomenon. It is nothing but logicl method for quntifying the uncertinty or the missing informtion bout the physicl phenomenon. This provides rigorous method for nlysis which is quite generl with respect to the complexity of both the physicl model, which is bsed on vilble informtion, nd tht of the probbilistic model, which is bsed on missing informtion. A brief discussion on probbilistic nlysis nd its necessity in performnce-bsed erthquke engineering is presented.

63 Summry nd Conclusions (continued) A direct probbilistic representtion of ground motion nd (thereby) structurl response cn be constructed using prmeters known s intensity mesures. (IM-Bsed, FEMA- SAC Guidelines, PEER Methodology) A closed-form expression for the limit stte probbility is derived bsed on simplifying ssumptions. This leds to the derivtion of DCFD formt, closed-form design nd ssessment formt similr to the LRFD procedures. Probbility-bsed seismic performnce ssessment of structure cn be bsed on the results from lterntive non-liner dynmic nlysis procedures. These methods cn be used both to obtin prmeter estimtes for specific probbilistic ssessment criteri such s Demnd nd Cpcity Fctored Design (DCFD) nd lso to mke direct performnce ssessments using numericl integrtion methods. Alterntive (wide-rnge) non-liner dynmic procedures hve been used to ssess the design of n existing RC structure with degrding behvior in sher nd flexure.

64 Summry nd Conclusions (continued) The lterntive non-liner dynmic nlysis methods presented in this pper re referred to s wide-rnge methods since they cn mp out the structurl response to wide rnge of ground motion intensity nd structurl response levels. Multiple-stripe nlysis nd incrementl dynmic nlysis re the exmples of wide-rnge non-liner dynmic nlysis methods discussed here. It should be noted tht minimizing the number of structurl nlyses hs not been our primry concern in crrying out the non-liner dynmic procedures presented here.

65 This Presenttion is Prepred Bsed on the Following References: Jlyer, F., nd C. A. Cornell Alterntive non-liner demnd estimtion methods for probbility-bsed seismic ssessments - Prt I: wide rnge methods. submitted to Erthquke Spectr. Jlyer F. nd J. L. Beck Effects of Alterntive Representtions of Ground Motion Uncertinty on Seismic Risk Assessment of Structures. Submitted to Erthquke Engineering nd Structurl Dynmics (lso submitted s PEER report). Jlyer, F., nd C. A. Cornell A technicl frmework for probbility-bsed demnd nd cpcity fctor design (DCFD) seismic formts. Report PEER 2003/08. Berkeley, Clif.: University of Cliforni.

66 Thnk You