1. Chapter 4 System Reliability Analysis of Structures

2. Chapter 4: System Reliability Analysis of Structures 4.2 Reliability of Simple Systems 4.3 Reliability Bounds for Structural Systems 4.1 Elements and Systems Contents

3. 4.1 Elements and Systems Chapter 4 System Reliability Analysis of Structures

4. 4.1 Elements and Systems …1 – There are two extreme types of structural elements that are commonly considered in system reliability analyses. – These extreme types are brittle members and ductile members. – A member is classified as brittle if the member becomes completely ineffective after it fails. – A member is classified as ductile if the member is able to maintain its load-carrying capacity after it fails. Load Displacement Load Brittle memberDuctile member

5. 4.1 Elements and Systems …2 – In conducting system reliability analysis of structures, it is convenient to distinguish brittle and ductile members by using different symbols. Load Displacement Load Brittle memberDuctile member

6. 4.2 Reliability of Simple Systems Chapter 4 System Reliability Analysis of Structures

7. 4.2 Reliability of Simple Systems …1 4.2.1 Reliability of Series Systems  Consider a series system with n elements. Let represent the event “safety of the ith element”, represent the event “failure of the ith element”. S and F represent the events “safety” and “failure” of the system respectively. – Assume that all failure events are all statistically independent, we have – The probability of failure of the system is larger than that of either element.

8. 4.2 Reliability of Simple Systems …2 4.2.2 Reliability of Parallel Systems Consider a parallel system with n perfect ductile elements. Assume that all failure events are all statistically independent.  – The probability of failure of the system is smaller than the probability of failure of either component.

9. 4.2 Reliability of Simple Systems …3 4.2.2 Reliability of Parallel Systems … – The system strength is the sum of all the strengths of the elements, – Consider a special system that has n parallel, uncorrelated, identically distributed elements. – The coefficient of variation for a system of n parallel, uncorrelated, identically distributed elements is smaller than the coefficient of variation of each element.

10. 4.2 Reliability of Simple Systems …4 4.2.3 Reliability of Hybrid or Combined Systems         parallel series system series parallel system

11. Example 4.1 ~ 4.4 4.2 Reliability of Simple Systems …5 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. P. 257, Example 9.1 P. 258, Example 9.2 P. 261, Example 9.3 P. 253, Example 9.4

12. 4.3 Reliability Bounds for Structural Systems Chapter 4 System Reliability Analysis of Structures

13. 4.3 Reliability Bounds for Structural Systems …1 4.3.1 Series Systems with Positive Correlation – For a series system with positive correlation between pairs of elements, that is,, the probability of failure must satisfy – The lower bound is the probability of failure when all elements are fully correlated, that is,. In this case, the most unsafe element determines the reliability of the system. – The upper bound is the probability of failure when all elements are statistically independent,or uncorrelated, that is,. In this case, the system survives only if all the elements survive.

14. 4.3 Reliability Bounds for Structural Systems …2 4.3.2 Parallel Systems with Positive Correlation – For a parallel system with n ductile elements, the bounds on the probability of failure for the parallel system with positive correlation are as follows: – The upper bound is the probability of failure when all elements are perfectly correlated, that is,. In this case, the safest element determines the reliability of the system. – The lower bound is the probability of failure when all elements are statistically independent,. In this case, the system fails only if all the elements fail.

15. Example 4.5 ~ 4.6 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. P. 268, Example 9.5 P. 269, Example 9.6 4.3 Reliability Bounds for Structural Systems …3

16. Homework 4 4.1 Solve the problem 9.2 in text book on P.286. 4.2 Solve the problem 9.3 in text book on P.286. 4.3 Solve the problem 9.4 in text book on P.286. 4.4 Solve the problem 9.5 in text book on P.286. Chapter4: Homework 4

17. End of Chapter 4