L. Schueremans, D. Van Gemert luc.schueremans@bwk.kuleuven.ac.be,

1. Using Monte Carlo and Directional Sampling combined with an Adaptive Response Surface for system reliability evaluation L. Schueremans, D. Van Gemert luc.schueremans@bwk.kuleuven.ac.be, dionys.VanGemert@bwk.kuleuven.ac.be Department of Civil Engineering KULeuven, Belgium Praha Euro-Sibram, June 24 tot 26, 2002, Czech Republic

2. Introduction Framework: • Ph. D. “Probabilistic evaluation of structural unreinforced masonry”, • Ongoing Research: “Use of Splines and Neural Networks in structural reliability - new issues in the applicability of probabilistic techniques for construction technology”. Target: • obtain an accurate value for the global pf, accounting for the exact PDF of the random variables; • minimize the number of LSFE, which is of increased importance for complex structures; • remain workable for a large number of random variables (n). In practice, the number of LSFE should remain proportional with the number of random variables (n).

3. Introduction Level II and Level III methods:

4. Introduction - reliability methods #LSFE~9n #LSFE~3/pf VI #LSFE~cte.n

5. Methods for System Reliability using an Adaptive Response Surface Real structure:high degree of mechanical complexity, numerical algorithms, non-linear FEM Response Surface:low order polynomial, Splines, Neural Network,... Reliability analysis Optimal scheme: DARS or MCARS+VI DARS:Matlab 6.1[Schueremans, 2001], Diana 7.1[Waarts,2000] MCARS+VI:Matlab 6.1[Schueremans, 2001]

6. DARS-Directional Adaptive Response surface Sampling

7. DARS -Directional Adaptive Response surface Sampling

8. DARS -Directional Adaptive Response surface Sampling

9. DARS -Directional Adaptive Response surface Sampling

10. DARS -Directional Adaptive Response surface Sampling

11. MCARS+VIMonte Carlo Adaptive Response surface Sampling+Variance Increase g LSF RS ladd Dg,add Dg,add gRS,i u lRS eg,i gLSF,i li Step 3: Monte Carlo Variance Increase on the Response Surface (vi). Sampling function: h=n-0.4 IF|gRS(v i)|<|g,add| calculate gLSF(vi) update RS update g,add Else .

12. DARS and MCARS+VI The number of direct LSFE remains proportional to the number of random variables (n), There is no preference for a certain failure mode. All contributing failure modes are accounted for, resulting in a safety value that includes the system behavior, thus on level III.

13. Safety of masonry Arch

14. Safety of masonry Arch To evaluate the stability of the arch, the thrust line method is used (Heyman, 1982), which is a Limit Analysis. Following assumptions are made: • blocs are infinitely resistant, • joints resist infinitely to compression • joints do not resist to traction • joints resist infinitely to shear An external program Calipous is used for the Limit State Function Evaluations [Smars, 2000]

15. Safety of masonry Arch Failure modes - limit states - limit analysis based on thrust lines

16. Safety of masonry Arch

17. Safety of masonry Arch Figure: DARS-outcome - reliability vernus number of samples N

18. Conclusions • Focus was on the use of combined reliability methods to obtain an accurate estimate of the global failre probability of a complete structure, within an minimum number of LSFE. • A level III method is presented and illustrated: (DARS/MCARS+VI • Ongoing research: Splines and Neural Network instead of low order polynomial for Adaptive Response Surface (ARS). • Acknowlegment: IWT-VL (Institute for the encouragement of Innovation by Science and Technology in Flanders).