Reliability – Based Design

1. b  Design mR = mS + bsS2 + sR2 Reliability – Based Design Observe for Case 1 in which R and S are both Normal If b PS Reliability  b = Reliability index = F-1(PS)

2. What makes a structure safer? • Normally we cannot control loading Building as ex, we can’t control wind • We can only control on resistance side We can build a larger column • Two ways: 1) higher μR higher mean capacity 2) lower σR  better quality control to reduce variation of strength of a column How about correlation between failure modes, or loading?

3. Case 3: If R is discrete, S is continuous P(R = r) 0.6 0.3 0.1 r 5 6 7 Example on Case 3 S = N(5, 1)

4. P(R = r) 1 r 5 6 7 P(R = r) 1 .8 .6 .4 .2 r 6 5 7 Challenge 1 • What if R is not discrete? Example on Case 4 S = N(5, 1) Cut into strips trapezium

5. Challenge 1 • If we use 5 strips,

6. Challenge 1

7. Central Limit Theorem S will approach a normal distribution regardless the individual probability distribution of Xiif N is large enough More importantly, S will have a mean of And Variance of Or a Variance of

8. fE(e) (a) E(E) = 0 1/2 -1 0 +1 e P4.27

9. (b) P(E < 0.01% of length of segment) (area under rectangular curve) (c) T = total mean = E1 + E2 +…+ E124 E(T) = 124  0 = 0 Var(T) = Var(E1) + Var(E2) +…+ Var(E124) = 124Var(E) = 124  1/3 = 41.33 sT = 6.43 Distribution of T  Normal (C.L.T) \ P(T < 0.01% of 2 miles) Assume s.i. if not S.I., P ?  By making more measurements, the % error is decreasing  P(within given tolerance limit) 

10. … X g(X) g’(mX) (X – mX) mx - mx Var(X) g(mX) mX X Taylor Series Approximation First order approximation:

11. If mK = 400 ; sK = 200 If M = 100  Example

12. K M M S.I. What if M is also random?  mM = 100; sM = 20 then What if not S.I.?

13. Challenge 2 Product of many normal is log-normal? Transform Take log on both sides, and let S = ln P, X = ln Y with N (λ,ξ) And find μx, σx (This is “moments” of r.v. but we are NOT transforming to normal) By CLT, S is Normal with Then P can be calculated using the transformation formulae

14. 4.17

15. 4.17