Reliability-based assessment of existing structures

Transcription

1 Reliability-based assessment of existing structures Sven Thelandersson Structural Engineering - Lund University 1

2 General background The value of existing built infrastructure in the developed world is tremendous It is becoming increasingly older since much were built the decades after World war 2. Accelerating development in society requires changes in use and reconstruction High economic incentive if existing facilities can be used longer and be utilised for new needs. Structural Engineering - Lund University 2

3 Example: Bridges in the European railway network Age of bridges < 20 years 11% years 22 % years 32 % >100 years 35 % Type of bridges Concrete 23 % Metallic 21 % Arches (mostly masonry) 41 % Composite (steel/concrete) 14 % Source: EU-project Sustainable bridges Structural Engineering - Lund University 3

4 Why do we need to assess reliability of an existing structure? Deviations (e.g. in loads, usage) from the original project description Indications from periodic investigation of its state Doubts about safety due to evidence of damage Unusual incidents during use (e.g. vehicle impact, fire, earthquakes) Inadequate serviceability Discovery of design or construction errors Planned change of use of the structure Expiry of residual service life based on earlier assessment of the structure Structural Engineering - Lund University 4

5 Residual service life - definition b Safe b target Unsafe t 0 t now Time Residual service life Structural Engineering - Lund University 5

6 General features in assessment of existing structures The safety is usually first verified with standard methods based on deterministic code rules. If this indicates that safety is not sufficient, there are three alternative decisions 1. Perform more detailed analysis and investigation of the structure 2. Strengthen the structure 3. Demolish and replace the structure Usually the cheapest alternative if sufficient safety can be verified Structural Engineering - Lund University 6

7 Limit states Ultimate limit state Is the structure safe enough and for how long? Problematic since we usually have to assess response against extreme loading which may occur in the future Serviceability limit state Easier since the structure can be regarded as tested in reality. Either adequate or defects are known Structural Engineering - Lund University 7

8 Bases for assessment of safety of an existing structure Documentation of design (drawings and other documentation) Field experience from operation, monitoring or inspections Requalification analysis (often leads to inadequate formal safety when new methods and codes are used) Economical analysis Structural Engineering - Lund University 8

9 Relative cost to increase safety For a new structure the additional cost to increase safety is low (e.g add some reinforcement) For an existing structure this cost is usually very high One could argue that the target reliability could be set lower in that case, but this is usually not possible for political reasons Structural Engineering - Lund University 9

10 JCSS Target Reliability Indices (per year) Relative safety cost Low consequences of failure moderate consequences of failure high consequences of failure high medium small b 3.1 b 3.3 b 3.7 P F P F P F 510 P F 10 b 3.7 b 4.2 b 4.4 P F P F 10 P F 510 b 4.2 b 4.4 b P F 510 P F 10

11 Assessment procedure in three phases Reliability-based assessment is often employed in the third phase Faber (2001) Structural Engineering - Lund University 11

12 Advantages and possibilities by use of probabilistic methods for assessment of safety for existing structures 1. Makes it possible to successively introduce specific information about the particular structure* into the analysis. 2. Gives a measure of safety for the structure, which can be used as a basis for decision about the future of the structure. 3. Sensitivity analyses shows the importance of different factors. 4. Usually less conservative than generic code rules *Information in structural codes is intended to be valid for all possible structures on the market Structural Engineering - Lund University 12

13 Bayesian probabilistic reassessment of structures Structural Engineering - Lund University 13

14 Risk-based decision analysis C=0*y+1*8=8 Mil. Sfr C=0.0018*6.55+0,9982 *2=2,01 Mil. Sfr C=0,0036*4,55+0,9964 *0=0,016 Mil. Sfr Faber (2001) Structural Engineering - Lund University 14

15 Information which may be used to update the reliability of a structure The fact that the structure has survived so far Tests of the structural materials in the structure Geometrical measures Damage and deterioration Proof loading Static and dynamic response to controlled loading Information of actual loading on the specific structure Structural Engineering - Lund University 15

16 Probabilistic updating based on proof loading Based on conditional probability P( R r R P( R U r R l) P( R l) l) Structural Engineering - Lund University 16

17 Simple example: a steel bar R s s Suppose that we need to increase the capacity of the reinforcement with 10 % due to changed loading conditions! Structural Engineering - Lund University 17

18 Prior decision analysis The first step is to evaluate the capacity based on available first hand (a priori) information (e.g. generic information about the actual material quality). An a priori probabilistic model for the yield strength of the steel can be formulated on this basis. If this is not sufficient further investigations may be performed to update the information. This first step is called prior decision analysis Structural Engineering - Lund University 18

19 Posterior decision analysis A number of tests of the material is performed and the probabilistic model describing the yield strength is updated. The analysis based on the new information may show that the reliability is sufficient or not. This type of analys is referred to as a posterior decision analysis since it is performed after new information has been obtained Structural Engineering - Lund University 19

20 Pre-posterior decision analysis As a tool for planning the investigations to update the information a so called pre-posterior analysis can be made. The idea is to perform a posterior analysis before the tests are performed, assuming that the results are in line with the prior information. In this analysis all costs for strengthening, tests, etc. are considered, given that the requirement for reliability is fulfilled. The result is used as a basis to decide about investigation plans and other measures to be taken. Structural Engineering - Lund University 20

21 Bayesian updating of material properties 1,00 0,80 0,60 Ffy 0,40 Prior Post 0,20 0, fy [MPA] Statistical methods are available to update probability distribution functions based on new test information Can be found in e.g. JCSS-publication: Probabilistic assessment of existing structures. Ed. by Diamantides D. Can be purchased through Structural Engineering - Lund University 21

22 Example: Important steel bar in tension. Is the yield strength sufficient? Decision tree Risk (1 1, ) 0 1, $ 1, , Risk (1 1, ) , $ Structural Engineering - Lund University 22

23 Decision based on a priori information Alternative a 1 (strengthening) is most favourable Now make some tests of the material and update the probabilities of failure Structural Engineering - Lund University 23

24 Example: Important steel bar in tension. Is the yield strength sufficient? Decision tree with updated failure probabilities Risk (1 0, ) 0 0, $ 0, , Risk (1 0, ) , $ Structural Engineering - Lund University 24

25 Decision based on posterior information Alternative a o (do nothing) Most favourable Structural Engineering - Lund University 25

26 Updating based on indirect information Utilization of information not originating from the structure itself to upgrade the reliability. This is possible if the information is correlated to the structure through common loading, correlated materials or degradation process. Structural Engineering - Lund University 26

27 Example: steel bar We know that a similar steel bar in another structure has been subjected to a known proof load and survived. This bar comes from the same manufacturer but probably not from the same batch. The correlation between the two bars is estimated to 0,8 The figure shows the updated probability of failure as a function of the proof load Structural Engineering - Lund University 27

28 Updating based on inspections example fatigue of steel with inspection of crack propagation Updating at each inspection conditional on no crack found Structural Engineering - Lund University 28

29 Example: Öland bridge (from Fredrik Carlsson) Completed1969 B=180 kn Length: 6 km Four lanes (two in each direction) Kantbal Mittlinj Kantbal Critical section: Moment capacity of cantilever slabs Structural Engineering - Lund University 29

30 Material properties c300 Ks40 12c300 Ks60 16c300 Ks60 12c300 Ks60 12c300 Ks60 From drawings: Rebars: Ks40 and Ks60 Concrete: K450 10c300 Ks40 10c300 Ks Information is needed about properties of old material qualities Structural Engineering - Lund University 30

31 A conservative value for the dynamic amplification of the load A/B is specified in the code A/B Load model Kritiskt snitt A= axle pressure B=boogie pressure According to the deterministic assessment code the concentrated load A/B should be placed as far out as possible Structural Engineering - Lund University 31

32 Structural Engineering - Lund University 32 Limit state function T M B M G M cc f s st f s st f M C M C M C bf C A f C d A f C Z T B G st st st 2 2, ,66 3 2,6 0,22 0, a I I a B r a B b a B M eff T

33 Updating of random variables In this case load conditions concerning sideways positions were measured on the bridge and expressed as a random variable The dynamic amplification of the traffic load was described as a random variable based on test results in literature The verification was made by reliability analysis to achieve the target reliability specified by the road administration Structural Engineering - Lund University 33

34 Basic random variables Variabel Symbol Fördelning Medelvärde Standardav. Tryckhållfasthet betong: f cc Lognormal 47,5 MPa 7,6 MPa Drag hållfasthet f st Lognormal 667,6 MPa 30 MPa armering: Armeringsarea: A s Konstant 1,80*10-3 m 2 /m - Effektiv höjd: d Lognormal 0,238 m 0,01 m Bredden: b Konstant 1,0 m - Moment av M G Normal 29,2 knm/m 1,5 knm/m egentyngd: Moment av M B Normal 12,8 knm/m 1,3 knm/m beläggning: Boogietryck: B Normal Beräknas 0,05B Dynamisk Normal 1,07 0,07 förstoringsfaktor: Hävarm: a Normal* 2,085 0,24 m Tröghetsmoment: I 1 Konstant 6,92*10-3 m 4 - Tröghetsmoment: I 2 Konstant 2,25*10-3 m 4 - Modell osäkerhet C fst Lognormal 1,0 0,05 armering: Modell osäkerhet C fcc Lognormal 1,0 0,05 betong: Modell osäkerhet C MG Normal 0 1,5 egentyngd: Modell osäkerhet C MB Normal 0 0,65 beläggning: Modell osäkerhet trafiklast: C MT Normal 1,0 0,05 Structural Engineering - Lund University 34

35 Results COMREL was used to calculate reliability index and sensitivities and the allowable boogie pressure corresponding to target reliability = 4,75 was determined to B= 603 kn This can be compared with the allowable boogie pressure determined from standard code-based classification, which was B code = 408 kn Structural Engineering - Lund University 35

36 Sensitivities 0,5 0,4 0,3 0,2 -värden 0,1 0-0,1 Cfst d fst fcc Cfcc CMB MB CMG MG a CMT B e -0,2-0,3-0,4-0,5 Grundvariabler Structural Engineering - Lund University 36

37 Possible improvements of the results through additional information More accurate control of rebar positions Improved knowledge about yield strength of rebars Transversal positions of heavy vehicles Improved knowledge of vehicle widths Improved knowledge about dynamic amplification factor (Proof loading) Structural Engineering - Lund University 37

38 A number of critical sections for the Öland bridge were investigated in a similar way. This verified axle and boogie pressures for the bridge to the satisfaction of the bridge owner Trafikverket. This has saved a lot of money which can be spent on traffic safety elsewhere. Structural Engineering - Lund University 38