1. 7. Reliability based design Objectives Learn formulation of reliability design problem. Understand difference between reliability-based design and deterministic design Learn how to quickly approximate reliability of component using mean value, first order, second moment method Be exposed to three example problems involving analysis and design

2. Deterministic vs. Reliability Design Find design variables To minimize cost (or weight) so that safety factor  minimum acceptable value Find design variables To minimize cost (or weight) so that reliability  minimum acceptable value

3. Approximate calculation of reliability Mean value, first order, second moment method (MVFOSM) method Assumptions: –Random variables are Gaussian –Linear Taylor approximation of performance function, g(x 1,…,x n ), is accurate (performance function; positive implies survival, negative implies failure)

4. Approximate calculation of reliability (continued) Linear Taylor approximation of g about mean values of random variables:

5. Approximate calculation of reliability (continued) The derivatives are calculated at the mean values of r.v.s Mean value of g Standard deviation of g

6. Approximate calculation of reliability (continued) Failure probability =  (-  ) Reliability=1-failure probability

7. Generic procedure for solving reliability design problem Develop equation for performance function Develop method for calculating the reliability as a function of the design variable(s). Since many iterations are required in design we need an approximate method (e.g. MVFOSM) for calculating reliability at this stage Find the optimum values of the design variable(s) so that the requirement for acceptable reliability as well as other requirements are satisfied Validate the final design, i.e. compute the reliability of the final design using an accurate method (for example Monte- Carlo simulation) Redesign the system if needed.

8. Example 1 Tension element Given probability distribution of axial load, ultimate strength and coefficient of variation of diameter, find mean diameter of element that has reliability 0.9999. P P d

9. Example 2: Design of I-beam Find web height of I-beam to meet given reliability target

10. Example 3: Design-oriented analysis of torsion bar of a truck

11. Suggested reading Frangopol, D, M., and Maute, K., “Reliability- based Optimization of Civil and Aerospace Structural Systems,” Engineering Design Reliability Handbook, CRC press, 2004, p. 24-1. Mourelatos, Z., P., et al., “Probabilistic Analysis and Design in Automotive Industry,” Engineering Design Reliability Handbook, CRC press, 2004, p. 38-1.