Performance based Engineering

Transcription

1 Probability based based Methods in Performance based Engineering Fatemeh Jalayer University of Naples Federico II A RELAXED WORKSHOP ON PERFORMANCE-BASED EARTHQUAKE ENGINEERING July

2 The characterization of structural modeling uncertainties in existing buildings. Real time post earthquake assessment of civil structures in the presence of after shocks Multi hazard approach for structural assessment under severe loading conditions

3 One of the main challenges in sesimic performance of existing buildings is the characterization of the modeling uncertainties. The European seismic codes already address the structural modeling uncertainties through the use of confidence factors. The confidence factors (CF) are classified based on different knowledge levels els (KL).

4 The Knowledge Levels: EC8 Part 3 KL Inspections of reinforcement details Testing of Materials (sample/floor) (% elements*) Limited 20 1 Extended 50 2 Comprehensive 80 3

5 An alternative to CF: The SAC FEMA Formulation The structure is verified based on the following: 1 k 2 1 k 2 β Y S a β UC b 2 b 2 b η e e 1 η P o = a Sa Po Y Y In the static case, the formulation reduces to: η Y η e 1 β 2 2 UC 1 η Y is the median for structural performance parameter. β UC represents the overall effect of modeling dli uncertainties titi in on the structural performance parameter.

6 Characterization of Uncertainties (Prior Distributions) The material properties The Defects The distance between shear reinforcement Beam anchorage Column superposition Positioning of bars (columns) Missing bar (beams) Error in diameter (columns) uniform Uniform Uniform Uniform Uniform Uniform Systematic per floor and typology Systematic per floor Systematic per floor Systematic per floor and typology Systematic per floor and typology Systematic per floor and typology f c LN f y LN Expert judgment grouping interval plausibility DATABASE

7 Updating the Probability bilit Distributions ib ti for Structural Modeling Parameters Using Bayesian Updating In the absence of test data, it is assumed that the inspected components all verify the original design values.

8 Calculating the Structural Performance Variable Y Non linear static analysis Capacity Spectrum Method Knowledge levels (KL0, KL1, KL2, KL3) Structural analysis (Pushover) Fragility Curves (h Y, b UC ) Non linear time history analysis The component shear demand to capcity ratio is calculated. A suite of ground motion records are selected Linear regression is performed on Y versus S a (a, b, b Y Sa ) Knowledge levels (KL0, KL1, KL2, KL3) Structural analysis (dynamic) Fragility site specific hazard integration

9 An Efficient Method for Estimation of Robust Reliability A small sample of structural models is generated using Monte Carlo Simulation and probability distribution for the structural t lfragility parameters is calculated. l

10 The structural fragility 20 simulations Non linear static η Y = 50% β UC 1 84% log 2 16%

11 Comaprison with standard Monte Carlo Simulation

12 Seismic risk curves 30 simulations non linear time history P O =0.002

13 Some Recommendations The suitable approach for assessment of existing buildings is the probabilistic one which account for all the uncertainties. The prior probability distributions for the structural modeling uncertainties can be classified and tabulated based on the a survey of expert opinion and experience. Different classes of existing buildings need to be classified. Based on the prior probability distributions the values for b UC need to be evaluated tabulated for different levels of knowledge and for different building classes. The Efficient method presented herein can be quite useful for the assessment of strategic buildings.

14 Other efforts related to this topic: A survey form is soon going to be available on the site of RELUIS. This would serve as a database for developing prior probability distribution for structural modeling parameters. A range of simplified methods for seismic assessment of existing buildings are studied. The results presented herein are going to be verified with existing in situ test results.

15 Introduction A strategic structure could be subject to more than one critical action during its service life. A sustainable design needs to consider all the possible critical actions which the structure could be subjected to. Given the uncertainty tit involved, it seems inevitable it to address the sustainable design based on a probabilistic framework.

16 Multi hazard Design/Assessment Considering i a particular case in which h the critical ii events are earthquake and blast, the design/assessment criterion can be written as: ν = PC ( EQ) ν + PC ( Blast) ν ν C EQ Blast dm

17 Blast Loading The induced overpressure normallyconsists of a positive decaying phase followed by a weaker negative phase P [Mp pa] p(t) p 0 p 0 p(t) t t t [s]

18 Characterization of Uncertainties The uncertain quantities of interest are the amount of explosive W and its position with respect to a fixed point within the structure R.

19 The Analysis Procedure 1. A local dynamic analysis is performed on the column elements affected by the blast. 2. A kinematic plastic analysis is performed on the damaged structure in order to evaluate whether the structure is able to carry the gravity loads in its post explosion state.

20 The Period of vibation and the first mode shape for fixed end beam 1 T L = 2π m EI v(0.5, t) = Y( t) x/l

21 Analysis of Impact loading for fixed end beam 4 2 vxt (, ) vxt (, ) t EI + m = p (1 ) t t 4 2 x t t EI plus plus vxt (,) vxt (,) + m = 0 t > t 4 2 x t plus p(t) p0 p(t) p0 t t

22 Closed form Solution for the Local Dynamic Analysis The maximum response will most likely be in the freevibrationresponse response phase: Yt ( plus ) Yt () = Yt ( plus )cos ωt+ sinωt ω Yt ( 2 plus ) ρ = Yt ( plus ) + ω M max = 1.26 L 2 2 EIρ

23 Kinematic plastic analysis on damaged structure Minimize u s = j= 1 M θ subject to both θ = tθ and e= te m pj j j i ij i i i i= 1 m λ C u = = e s j = 1 m M i= 1 i pj te θ i j

24 Blast Fragility Using simulation based reliability methods for risk assessment: PC ( Blast) N sim i= 1 I CBlast N sim ( θ ) I ( θ ) = 0 if λ > λ I ( θ) = 1 if λ λ CBlast C Cth, CBlast C Cth, i

25 Numerical lexample Type A Type A Type A Type A Type A T yp e A T yp e B B* Type A/Type B Type A/Type B* T yp e B T yp e A Type A Type A Type A Type A Type A T yp e A T yp e B T yp e B T yp e B T yp e B T yp e A y Type A Type A Type A Type A x

26 Characterization ti of Uncertainties ti Blast scenario 30% 70% 1 st floor: Car bomb W=200 kg 500 kg Explosion takes place inside the structure 20% 20% 20% 20% 20% 2 nd floor: Backpac 3 rd floor: Backpac 4 th floor: Backpac kb bomb b 5 th floor: Backpa k bomb k bomb W=15 ck W=15 W=15 kg 35 bomb kg 35 kg 35 kg W=15 kg kg kg 35 kg Explosion takes place outside a 10 m stand-off distance from the structure t Truck bomb W=15,000 kg - 25,000kg 10% 90% Car bomb W=200 kg - 500kg

27 Blast Fragility P(λ <= λ C B Blast) λ C

28 Blast Scenarios Leading to Collapse storey #

29 Calculating Seismic Fragility 2000 Equivalent SDOF System F y 1400 Force [k kn] d y d max Displacement [meters]

30 Seismic Fragility Curve

31 Multi hazard Ananlysis ν C 5 = ν Blast ν Blast non-strategic struture strategic struture

32 Conclusions The methodology exploits the particular characteristics of the blast action and its effect on the structure in order to achieve maximum efficiency in the calculations. The simulation i results highlight hli h the location of critical ii blast scenarios on the structural geometry and the risk prone areas. A simple and effective prevention strategy would be to limit or to deny the access to critical zones within the structure.

33 Other efforts related to this topic: The algorithm for plastic limit analysis is modified for application to concentrically braced frames. The multi hazard approach presented herein is used to compare viable retrofit options for an existing building. A methodology for estimation of expected life cycle cost is developed that incorporates the multi hazard approach presented herein. This methodologyis used to compare various retrofit strategies based on life cycle cost criteria. The methodologypresented for life cycle cost assessment can be applied to seismic assessment of structures in an after shock prone area.

34 Jalayer F, Iervolino I., Manfredi G. Structural Modeling Uncertainties and their Influence on Seismic Assessment of Existing RC Structures. Under review, Structural Safety, Asprone D, Jalayer F, Prota A, Manfredi G. Proposal of a Probabilistic Model for multi hazard assessment of structures in seismic zones subjected to blast for the limit state of collapse. Structural Safety, In press, 2009.

35 L Aquila 2009 after shock sequence

36 The after shock sequence parameters Omori law Gutenberg Richter law A non homogeneous Poisson process

37 The after shock sequence parameters L Aquila 2009 sequence a= 1.85; p=0.80; b=1.05 Generic California sequence a= 1.67; p=1.08; b=0.91

38 The after shock hazard

39 Probability of exceeding a limit state LS in time T max Which is the fatal event? Non homogeneous Poisson process Already damaged or intact (eventually repaired)? How many times it has been damaged?

40 Estimation of limit state probabilities Equivalent SDOF system Already damaged by the main shock ground motion residual displacement

41 Estimation of limit state probabilities Given a selection of ground motions, each one is applied k times on the structure maximum displacement Y The maximum displacement Y is related to the spectral acceleration of the ground motion at the fundamental period

42 Estimation of limit state probabilities The limit state probabilities, given S a, are evaluated The limit state probabilities are then obtained by integrating over the hazard

43 The limit i state probabilities bili i in a 24h interval Differentiating the limit state probabilities, the daily probabilities bili i of exceeding the limit i state are obtained ΔTT = 24 h = 1 day

44 The case study T 1 =0.58s; d y =0.034m034m

45

46 Conclusions The procedure allows the quantification of the time variant probabilities of exceeding various discrete limit states for a structure in an after shock prone environment The limit state probabilities strongly depend on the residual damage induced in the structure by the main shock; if this value is small, the structure immediately verify the life safety requirements The presented procedure can represent the core of a DSS (Decision Support System) tool, to be used in postearthquake emergency management