7. Reliability based design Objectives вЂў Learn formulation of reliability design problem. Understand difference between reliability-based design and deterministic design вЂў Learn how to quickly approximate reliability of component using mean value, first order, second moment method вЂў Be exposed to three example problems involving analysis and design Deterministic vs. Reliability Design вЂў Find design variables вЂў To minimize cost (or weight) вЂў so that safety factor п‚і minimum acceptable value вЂў Find design variables вЂў To minimize cost (or weight) вЂў so that reliability п‚і minimum acceptable value Approximate calculation of reliability вЂў Mean value, first order, second moment method (MVFOSM) method вЂў Assumptions: вЂ“ Random variables are Gaussian вЂ“ Linear Taylor approximation of performance function, g(x1,вЂ¦,xn), is accurate (performance function; positive implies survival, negative implies failure) Approximate calculation of reliability (continued) Linear Taylor approximation of g about mean values of random variables: g ( x1 ,..., x n ) п‚» g ( E ( X 1 ),..., E ( X n )) пЂ« п‚¶gi ( x i пЂ E ( X i )) пѓҐ i пЂЅ 1,..., n п‚¶ x i Approximate calculation of reliability (continued) Mean value of g E ( g ( x1 ,..., x n )) п‚» g ( E ( X 1 ),..., E ( X n )) Standard deviation of g 2 пЃіg п‚» п‚¶g 2 2 п‚¶g п‚¶g ) пЃі пЂ« 2 пЃіX X пѓҐ пѓҐ пѓҐ i j X i i пЂЅ 1,..., n п‚¶ x i i пЂЅ 1,..., n j пЂЅ i пЂ« 1,..., n п‚¶ x i п‚¶ x j ( The derivatives are calculated at the mean values of r.v.s Approximate calculation of reliability (continued) Failure probability = пЃ†(-пЃў) Reliability=1-failure probability пЃў пЂЅ safety index пЂЅ E(g) пЃіg Generic procedure for solving reliability design problem вЂў Develop equation for performance function вЂў 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 вЂў Find the optimum values of the design variable(s) so that the requirement for acceptable reliability as well as other requirements are satisfied вЂў Validate the final design, i.e. compute the reliability of the final design using an accurate method (for example MonteCarlo simulation) вЂў Redesign the system if needed. Example 1 вЂў Tension element вЂў 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 Example 2: Design of I-beam вЂў Find web height of I-beam to meet given reliability target Example 3: Design-oriented analysis of torsion bar of a truck Suggested reading вЂў Frangopol, D, M., and Maute, K., вЂњReliabilitybased Optimization of Civil and Aerospace Structural Systems,вЂќ Engineering Design Reliability Handbook, CRC press, 2004, p. 24-1. вЂў Mourelatos, Z., P., et al., вЂњProbabilistic Analysis and Design in Automotive Industry,вЂќ Engineering Design Reliability Handbook, CRC press, 2004, p. 38-1.