1. Application of the Direct Optimized Probabilistic Calculation Martin Krejsa Department of Structural Mechanics Faculty of Civil Engineering VSB - Technical University Ostrava Czech Republic

2. Synopsis of Presentation Direct Optimized Probabilistic Calculation – DOProC method Background and introduction to the probabilistic method Program package ProbCalc Probabilistic calculation approach to the propagation of a fatigue crack using DOProC method Propagation of cracks from the outer edge Propagation of cracks from the surface Probabilistic calculation of fatigue cracks propagating using FCProbCalc program The choice of integration method Analysis of accuracy in relation to the time of calculation Conclusions Application of the Direct Optimized Probabilistic Calculation1 / 26

3. Essential of DOProC Characteristics Should be effectively used for the assessment of structural reliabilities and/or for other probabilistic calculations. Input random variables (load, geometry, material properties, imperfections) are expressed by the empirical or parametric distributions in histograms. Reliability function under analysis can be expressed analytically or using DLL library. Error of calculation is done only by discretization of input and output variables and numerical error. The number of intervals (classes) of each histogram is extremely important for the number of needed numerical operations and required computing time. The number of numerical operations can be reduced using optimizing techniques of the probabilistic calculation. Direct Optimized Probabilistic Calculation - DOProC Method2 / 26

4. Principle of Numerical Calculation B = f(A 1, A 2, …, A j, … A n ) Direct Optimized Probabilistic Calculation - DOProC Method3 / 26

5. Optimizing Techniques in DOProC Grouping of input random variables, which can be expressed by the common histogram. Interval optimizing - decreasing the number of intervals in input variable histograms. Zonal optimizing - each histogram is divided into areas (zones) depending on their share in the result (failure). Trend optimization – using correct or incorrect trend of input variable on the result. Grouping of partial calculations results. Parallelization of the calculation – calculation is proceeded on number of processors. Combination of the mentioned optimizing techniques. Direct Optimized Probabilistic Calculation - DOProC Method4 / 26

6. Program Utility ProbCalc 5 / 26 Analytical definition of calculation model Calculator Reliability function Command line Grouping of input random variables Table of input random variables Direct Optimized Probabilistic Calculation - DOProC Method

7. Program Utility ProbCalc Probability of failure p f = 1,28.10 -6 meets requirements of EN1990 for consequences class RC3/CC3 with design probability p d = 8,4.10 -6 Histogram of reliability function RF 6 / 26Direct Optimized Probabilistic Calculation - DOProC Method

8. Usage of ProbCalc Probabilistic assessment of load combinations, Probabilistic reliability assessment of cross-sections and systems of statically (in)definite load-bearing constructions, Probabilistic aproach to assessment of mass concrete and fibrous concrete mixtures, Reliability assessment of arch supports in underground and mining workings, Reliability assessment of load-bearing constructions under impact loads, Probabilistic calculation of fatigue crack progression in steel structures and bridges. Direct Optimized Probabilistic Calculation - DOProC Method7 / 26

9. Program FCProbCalc Probabilistic calculation of fatigue cracks propagating using FCProbCalc program8 / 26 FCProbCalc program desktop - entry of input quantities

10. Look to the reviewed road bridge Photo: J.Odrobiňák Bridge’s crosswise cut Detail of Solved Steel Bridge’s Flange Photo: J.Odrobiňák Reliability Assessment of Steel Bridge’s Flange in Tension Probabilistic calculation of fatigue cracks propagating using FCProbCalc program9 / 26

11. Some of input values - no way to obtain using measurement, can be approximate only. Quantity Type of parametric distribution Parameters Mean Value Standard Deviation Oscillation of stress peaks  [MPa] Normal303 Total number of oscillation of stress peaks per year N [-]Normal10 6 10 5 Initial size of the crack a 0 [mm]Lognormal0,20,05 Smallest measurable size of the crack a d [mm]Normal100,6 Yield stress of material f y [MPa]Lognormal28028 Nominal stress in flange  [MPa] Normal20020 List of random input variables QuantityMean Value Constant of material m3 Constant of material C2,2.10 -13 Flange width b f [mm]400 Flange thickness t f [mm]25 List of constant input variables Real values Approximate values Overview of Variable Input Quantities Probabilistic calculation of fatigue cracks propagating using FCProbCalc program10 / 26

12. Resistance of the structure R : Cumulated effect of loads S : N is total number oscillation of stress peak  needed to increase the crack from a 0 to a d or a ac N 0 is number of stress range cycles in time of fatigue crack initialization C, m are material constants a 0 is initial crack size a d is detectable crack size a ac is acceptable crack size F (a) is calibration function, which corresponds to propagation behavior of the crack Probabilistic calculation approach to the propagation of a fatigue crack where Probabilistic calculation of fatigue cracks propagating using FCProbCalc program11 / 26

13. All of these three events creates full space of event, which can come in time t, can be applied: Probability of crack undetection in time t : where a d is minimal detectable crack size Probability of crack detection in time t, crack size a ( t ) is less than tolerable size a ac : Probability of crack detection in time t, crack size a ( t ) is equal or greater than tolerable (acceptable) size a ac : Probability of Defined Random Events Probabilistic calculation of fatigue cracks propagating using FCProbCalc program12 / 26

14. Crack’s propagations from the edge or from the surface are possible to monitor according to initial crack position. Weakness of the same flange increased from the edge is quicker then from the surface one. Places of Fatigue Damage Danger’s Concentration Probabilistic calculation of fatigue cracks propagating using FCProbCalc program13 / 26

15. The propagation of the fatigue crack from the edge can be expressed by means of a calibration function F (a) : where a is crack size, b is flange width (in this case 400 mm). Fatigue Crack’s Propagations from the Edge Probabilistic calculation of fatigue cracks propagating using FCProbCalc program14 / 26

16. where where  is geometrical parameter. For sizeable width b is valid, that f w =1. Coefficient S : Fatigue Crack’s Propagations from the Surface Calibration function F (a) in general definition: Probabilistic calculation of fatigue cracks propagating using FCProbCalc program15 / 26

17. Cumulated Effect of Loads S Total number oscillation of stress peak per 111 years Determined for each year of the bridge operation using time step equals 1 year. Probabilistic calculation of fatigue cracks propagating using FCProbCalc program16 / 26 Total number oscillation of stress peak per 49 years

18. Structure Resistance R ad Crack’s propagations from the edge Crack’s propagations from the esurface Yet possible to select 5 types of integration methods Probabilistic calculation of fatigue cracks propagating using FCProbCalc program17 / 26

19. Numerical integration Rectangular method Simpson method Romberg method Adaptive method Gauss quadrature Gaussian quadrature integration points Probabilistic calculation of fatigue cracks propagating using FCProbCalc program18 / 26

20. Probability of Defined Random Events The probability of the event U, D and F, depending on years of operation of the bridge The crack from the edge 0 to 80 years of operation The crack from the surface 0 to 150 years of operation Probabilistic calculation of fatigue cracks propagating using FCProbCalc program19 / 26

21. Construction Inspection Times Fatigue crack from the edge FCProbCalc program desktop Probabilistic calculation of fatigue cracks propagating using FCProbCalc program20 / 26

22. Construction Inspection Times Fatigue crack from the edge The resulting probabilities and times of construction inspections The dependence of the probability of failure P f and years of operation of the bridge Adaptive numerical integration method chosen with a parameter tol 0 = 1.10 -4 Probabilistic calculation of fatigue cracks propagating using FCProbCalc program21 / 26

23. Construction Inspection Times Fatigue crack from the surface FCProbCalc program desktop Probabilistic calculation of fatigue cracks propagating using FCProbCalc program22 / 26

24. Construction Inspection Times Fatigue crack from the surface The resulting probabilities and times of construction inspections The dependence of the probability of failure P f and years of operation of the bridge Probabilistic calculation of fatigue cracks propagating using FCProbCalc program23 / 26 Adaptive numerical integration method chosen with a parameter tol 0 = 1.10 -4

25. Comparison of the calculated 1 st time inspection Calculated times of the 1 st inspection of the bridge construction with a particular attention on used numerical integration method Fatigue crack from the edge Probabilistic calculation of fatigue cracks propagating using FCProbCalc program24 / 26

26. Time of the Calculation Calculation time for each type of numerical integration and the specified number of intervals of input random variables Fatigue crack from the edge Probabilistic calculation of fatigue cracks propagating using FCProbCalc program25 / 26

27. Conclusion Presentation approached the newly developed probabilistic method DOProC, applications and developed software tools (for details see eg http://www.fast.vsb.cz/popv/).http://www.fast.vsb.cz/popv/ DOProC method appears to be an effective means for obtaining solutions for a number of probability problems. 26 / 26 DOProC method can solve such propagation of fatigue cracks in steel bridges and structures and to determine the inspection times using conditional probability. Application of the Direct Optimized Probabilistic Calculation

28. Thank you for your attention ! Ostrava, Moravian-Silesian Region