1. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING MODELING ERRORS AND UNCERTAINTY IN ENGINEERING COMPUTATIONS Reliability of Structural Reliability Estimation Isaac Elishakoff and Roberta Santoro Florida Atlantic University,Florida,USA University of Palermo,ITALY February 22-24,2006-Georgia Tech Savannah NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro REC-2006

2. Governing Equation NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro First order central difference method w: Transversal displacement EI: Bending stiffness P: Axial compressive load L: Lenght of the column N: Number of segments h=L/Nh=L/N

3. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro Expression of General Solution

4. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro General Solution Where: Integration constants

5. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro Case of SIMPLY SUPPORTED COLUMN Boundary conditions: CRITICAL LOAD

6. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro PROBABILISTIC ANALYSIS MODULUS OF EASTICITY E:CONTINOUS RANDOM VARIABLE (assigned probability distribution function F E (e) with e>0) REQUIREMENT TO AVOID BUCKLING PHENOMENA: RELIABILITY R: or

7. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro PROBABILISTIC DESIGN OF THE COLUMN Column performs acceptably if: codified reliability valuetollerable level of unreliability L lenght of the column: design parameter (depending on N and r 0 ) or

8. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro Exponentially Distributed Elasticity Modulus (continue) Mathematical expectation Variance APPROXIMATE RELIABILITY

9. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro APPROXIMATE LENGHT By demanding that: EXACT RELIABILITY (continue)

10. ACTUAL RELIABILITY NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro or Evaluating R actual for increasing number of N we obtain values that are always greater than r 0 or equal r 0. (continue)

11. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro PERCENTAGE ERRORS BETWEEN R actual AND r 0 FOR INCREASING VALUE OF N

12. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro PERCENTAGE ERRORS BETWEEN P f,actual AND p 0 FOR INCREASING VALUE OF N

13. NSF WORKSHOP ON RELIABLE ENGINEERING COMPUTING “Reliability of Structural Reliability Estimation” Isaac Elishakoff and Roberta Santoro CONCLUSIONS In the example that is presented in this investigation the actual probability of failure turns out to be smaller than the tolerable level. This appears to be a good news for the reliability of the reliability calculation of the finite difference method. Is actual probability of failure greater than, equal to, or less than the tolerable probability of failure that is pertinent to the ideal, error-free situation?