Roma - 12 marzo 2019

Transcription

1 Roma - 12 marzo 2019 Dipendenza dell affidabilità strutturale dalle assunzioni iniziali Analisi di sensitività Pietro Croce Dipartimento di Ingegneria Civile e Industriale Univ. di Pisa

2 Azioni γ F = γ f γ Sd γ f γ Sd Incertezza di Modello Azioni/sollecitazioni Aleatorietà dell azione γ M = γ m γ Rd γ m γ Rd Aleatorietà delle caratteristiche meccaniche Incertezza di modello

3 Relative frequency Density Plot (Shifted Lognormal) - [A1_792] Yield strength [MPa]

4 b 1/P f 2,3 93 2, , , , , Distribuzioni normali

5 2 1,8 1,6 1,4 1,2 1 b=3,8 P f =1/14000 Resistenza (COV=0,15) Sollecitazione 0,8 0,6 0,4 0,2 0 S m S k S d =R d R k 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 R m

6 4,5 4 3,5 3 2,5 2 1,5 1 b 2 =3,8 R 2m R 2k b 1 =3,8 P f =1/14000 Resistenza Mat. 1 (COV=0,15) Sollecitazione Resistenza Mat.2 (COV=0,06) m(1) =1,4 m(2) =1,0 M(1) =1,5 M(2) =1,05 0,5 S k 0 S m S d =R 1d =R 2d R 1k 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 R 1m

7 2 1,8 1,6 b 3 =3,8 Resistenza Mat. 1 (COV=0,15) Sollecitazione Resistenza Mat.3 (COV=0,25) 1,4 1,2 1 0,8 0,6 0,4 b 1 =3,8 P f =1/14000 m(1) =1,4 m(2) =1,85 M(1) =1,5 M(2) =2,0 0,2 0 S k S d =R 1d =R 3d R 3k R 3m S m R 1k R 1m

8 Wind speed Commonly adopted extreme value distributions

9 Wind speed

10 Wind speed Annual maxima of wind speed at Pisa airport weather station

11 Wind speed Pisa airport annual maxima elaboration

12 Wind actions Annual maxima of wind speed at Pisa airport weather station

13 Wind speed Pisa airport annual maxima elaboration Vk=27.45 m/s Vk=27.6 m/s Sk=0.68

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16 Steel S275 V=0.07 W k /(G k +Q k ) Cases considered for extreme wind velocity Gumbel distribution 3-parameters Weibull distribution GPD V=0.1 V=0.2

17 b GPD V=0.1 Weibull V=0,1 Gumbel V= Wk/(Gk+Qk) b-w k /(G k +Q k ) diagrams for various extreme maxima distributions for wind (V=0.1)

18 b GPD V=0.2 Weibull V=0,2 Gumbel V= Wk/(Gk+Qk) b-w k /(G k +Q k ) diagrams for various extreme maxima distributions for wind (V=0.2)

19 Reliability decreases when the wind action is very high Reliability depends on the distribution assumed for extreme maxima Wind pressure model is still an open question (each relevant coefficient needs to be discussion)

20 Fundamental combination (ULS) F d = γ G,i G k,i + γ Q,1 Q k,1 + γ Q,j ψ 0,j Q k,j + γ P P k i j>1 Eqn (formerly 6.10) F d = i γ G,i G k,i + γ Q,1 ψ 0,1 Q k,1 + γ Q,j ψ 0,j Q k,j + γ P P k j>1 ξ i γ G,i G k,i + γ Q,1 Q k,1 + γ Q,j ψ 0,j Q k,j + γ P P k i j>1 Eqns a+b (formerly 6.10 a+b)

21 Fundamental combination (ULS) γ G,i G k,i + γ P P k F d = i ξ i γ G,i G k,i + γ Q,1 Q k,1 + γ Q,j ψ 0,j Q k,j + γ P P k i j>1 Eqns a+b (formerly 6.10 a+b mod) F d = γ G1,i G 1k,i + γ G2,i G 2k,i + γ Q,1 Q k,1 + γ Q,j ψ 0,j Q k,j + γ P P k i i j>1 PT proposal (rejected)

22 Rand. Var. R X G S Variable distr COV F x k MU Concrete MU Steel MU Timber LogN MU Masonry Input Jäger MU Soil / / Concrete compressive strength Structural steel yielding strength Re-bar yield strength Solid timber bending strength Glulam timber bending strength Masonry compression Input Jäger Masonry shear Input Jäger LogN Soil internal friction Soil drained cohesion Soil undrained shear strength Timber bending MOE Masonry MOE Input Jäger Steel bending MOE Concrete compression MOE 1.00 To be completed Concrete Steel Timber Norm Masonry Input Jäger Soil F X

23 Rand. Var. P * P Variable distr COV F x k G Permanent load Norm G Permanent load (large COV) Norm Q Q (1yr) MU Wind LogN To be completed MU Snow 1 LogN for Cpe, Mean values for others Mu+sigma for Cr a, mean values for others MU Snow mean MU Imposed LogN Mean value Wind pressure Gumbel Snow on ground Gumbel Imposed Gumbel (LogN) 1.00 To be completed 0.98 Model uncertainty - LogNormal Two cases: Mean value 0.35 and 0.51 for wind 0.20 and 0.28 for snow

24 α G = g sk g sk + g Pk α G = 1 ; 0.6; 0.8; α Q = q k g sk + g Pk + q k α Q = 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8 Design equation according to: Eq (6.10) G Q Eq a+b (6.10 a+b) Eq prop Eq a+b prop G Q x y 0,1 y 0, G1 G2 Qw Qs G1 G2 Qw Qs x y 0,1 y 0,

25 Probability of failure is evaluated for two cases: 1 year and 50 years Steel Concrete Glulam

26 Steel Snow Eq. 6.10

27 Steel Snow Eq a+b

28 Steel Snow MU COV=0.20

29 Steel Snow MU COV 0.20

30 Steel Snow MU COV=0.51

31 Steel Snow MU COV 0.51

32 Concrete Snow Eq. 6.10

33 Concrete Snow MU COV=0.28

34 Concrete Snow MU COV 0.28

35 Concrete Wind Eq a+b prop

36 Concrete Wind MU COV=0.51

37 Concrete Wind MU COV=0.51

38 Glulam Snow MU COV=0.28

39 Concrete Snow MU COV 0.28

40 Glulam Wind MU COV=0.51

41 Glulam Wind MU COV=0.51

42 Snow Resulting pdfs

43 Snow CDFs

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45 Wind Resulting CDFs

46 Design equation (eq of pren1990:2019) p r k γ M = 1 α Q γ G g k + α Q γ Q q k (1) γ M = 1.00 (steel); γ G = 1.35 γ Q = 1.50 p is a suitable parameter granting that (1) is satisfied α Q is a parameter expressing the relative weigth of variable and permanent actions

47 Probability of failure P f = P p θ R r 1 α Q g + α Q θ Q q < 0 Hypotheses

48 Considered cases 1 year reference period considering permanent actions and wind actions including variable load model uncertainty θ Q ; 1 year reference period considering permanent actions and wind actions excluding θ Q ; 50 years reference period considering permanent actions and wind actions including variable load model uncertainty θ Q ; 50 years reference period considering permanent actions and wind actions excluding θ Q. P f = P p θ R r 1 α Q g + α Q θ Q q < 0

49 Reference values of β t and P ft Reference value of β t and P ft have been determined referring to the following conditions: min w i β i β t 2 3 min w i,j β i,j β tj 2 min w i P fi P ft 2 5

50 Different distribution of weights have been considered Case 1: (basic case) α Q = 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80 subcase 1.1 like case 1 (most refined coverage) α Q = 0.20, 0.25, 0.30, 0.35, , 075, 0.80 Case 2: (basic case shifted by -0.1) α Q = 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70 Case 3: (higher influence of permanent loads) α Q = 0.05, 0.15, 0.25, 0.35, 0.45, 0.55 subcase 3.1 like case 3 (most refined coverage) α Q = 0.05, 0.10, 0.15,..., 0.50, 0.55 Case 4: (basic case shifted by +0.05) α Q = 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85 Case 5: like case 3, α Q = 0.05, 0.15, 0.25, 0.35, 0.45, 0.55, with linearly decreasing weighs (relative weights are 3 for α Q = 0.05; 2.5 for α Q = 0.15; 2.0 for α Q = 0.25; 1.5 for α Q = 0.35; 1.0 for α Q = 0.45; 0.5 for α Q = 0.55).

51 Case 1-1 year reference period - Permanent load + wind without model uncertainty Weight factors w α Q Target values β t P ft 1.077E-06 γ G = 1.03 γ Q = 1.67

52 Case 1-1 year reference period - Permanent load + wind without model uncertainty Weight factors w α Q Target values β t P ft 1.077E-06 γ G = 1.05 γ Q = D Plot of Φ 1 w i P fi P ft 2

53 Case 1-1 year reference period - Permanent load + wind with model uncertainty Target values β t P ft 2.638E-04 γ G = 0.88 γ Q = 1.81

54 Case 1-1 year reference period - Permanent load + wind with model uncertainty Target values β t P ft 2.638E-04 γ G = 0.88 γ Q = 1.70

55 Case 1-50 year reference period - Permanent load + wind without model uncertainty Weight factors w α Q Target values β t P ft 5.464E-04 γ G = 0.91 γ Q = 1.80

56 Case 1-50 year reference period - Permanent load + wind without model uncertainty γ G = 0.90 γ Q = D Plot of Φ 1 w i P fi P 2 ft Weight factors w α Q Target values β t P ft 5.464E-04

57 Case 1-50 year reference period - Permanent load + wind with model uncertainty Target values β t P ft 1.071E-02 γ G = 0.87 γ Q = 1.83

58 Case 1-50 year reference period - Permanent load + wind with model uncertainty Target values β t P ft 1.071E-02 γ G = 0.87 γ Q = 1.72

59 Summary of the sensitivity study (50 years reference)

60 Summary of the sensitivity study (1 year reference)

61 Conclusions Target reliability and then the solution is strongly influenced on the considered cases and on the model uncertainty; shifting the considered Q window results vary (even taking into account different number of subintervals can have some effect) for the same case, partial factors γ Q calibrated referring to β and partial factors calibrated referring to P f can differ up to , while γ G factors look very close due to small COV and no model uncertainty ; In general, γ Q values calibrated referring to P f are smaller and less sensitive to model uncertainty than those calibrated with respect β The Φ 1 w i P fi P ft 2 surfaces are characterized by extensive plateau indicating that the region of optimal solutions in terms of probability of failure is much more wide than crudely indicated by the maximum;

62 Conclusions Considering one variable action characterized by high COV and high model uncertainty and one permanent action characterized by small COV could lead to capricious results: in effect, surprisingly, increasing the relevance (case 3) or the relative weights (case 5) of permanent loads results in a moderate increase of γ G and in a much more evident increase of γ Q ; Since the calibration tends to uniform the calculated reliability, it increases the reliability in the region where α Q is high and it reduces the reliability in the region where α Q is low. But, comparing the figures before and after the calibration, it is evident that the decrease of the reliability in region where permanent actions dominates could be unacceptable. In addition, it should be considered that permanent actions are always present and that structures sensitive to them cannot rely on hidden safety.

63 Conclusions Hidden safety resources are often invoked to justify the apparent reduced reliability of structures whose design is dominated by variable actions (characterized by high COV). Disregarding hidden safety could affect effective reliability in high α Q, but, at present, hidden safety cannot be quantified. From the engineering point of view, tackling the calibration of partial factors as a pure mathematical challenge could lead to manifestly bizarre results. As already discussed, in many cases it results γ G < 1, especially when model uncertainty dominates; clearly, such finding cannot be accepted. The procedure, if not accompanied by sound engineering judgement, could lead in a very wrong direction: paradoxically, the effect of an increase of the uncertainty is not only, as expected, an increase of γ Q, but also a parallel, even relatively more pronounced, decrease of γ G.

64 Grazie per l attenzione