HW#5_f08_sol

# HW#5_f08_sol - CE 3102 Fall 2008 Homework #5 Due: 9:45 AM...

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CE 3102 Fall 2008 Homework #5 Due: 9:45 AM October 28, 2008 This is a team assignment. You may work together in teams of up to, but not exceeding, three people. Only one written submission is required from each team. However, every team member is responsible for knowing all of the material. 1. Problem 4.15, p. 194 A&T. (a) P=(1/2)CRV 2 Ln(P)=ln(1/2)+ln(C)+ln(R)+2ln(V) Since C, R, and V are lognormal random variables, ln(C), ln(R), ln(V) are normal random variables, ln(P) is a linear combination of normal random variables, so ln(P) is also normal, which means P is a lognormal random variable. The pdf for P then has the form = σ μ π ) ln( ) 2 / 1 ( exp 2 1 ) ( p p p f where )] [ln( 4 )] [ln( )] [ln( )] [ln( )] [ln( 2 )] [ln( )] [ln( ) 2 / 1 ln( )] [ln( 2 V VAR R VAR C VAR p VAR V E R E C E P E + + = = + + + = = Let )] [ln( )] [ln( 2 C VAR s C E m C C = = Since C is a lognormal random variable, we (ought to) know that 1 ) exp( 20 . 0 2 exp ] [ 80 . 1 2 2 = = + = = C C C C s s m C E δ or 568 . 0 2 0392 . ) 8 . 1 ln( 2 ]) [ ln( 0392 . 0 ) 1 2 ln(. ) 1 ln( 2 2 2 2 = = + = + = C C C C s C E m s

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Similarly 1844 . 0 695 . 4 00995 . 0 08 . 6 2 2 V V R R s m s m so finally 787 . 0 ) 1844 (. 4 00995 . 0392 . 185 . 3 ) 695 . 4 ( 2 08 . 6 568 . ) 2 / 1 ln( 2 = + + = + + σ μ (b) 404 . 0 596 . 1 ) 244 . 0 ( 1 ) 244 . 0 ( 787 . 185 . 3 ) 30 ln( 787 . 185 . 3 ) ln( )) 30 ln( ) (ln( ) 30 ( = = > = ⎛− > = > = > Z P Z P P P P P P P (c) Let S denote the strength of the tower. Failure occurs if the load exceeds the strength, so ) 0 ) ln( ) (ln( 1 ) ( ) ( > = > = > = S P P S P P S P P failure P S is lognormal so ln(S) is normal, and ln(P)-ln(S) is also normal. 489 . 4 ) 2 / 0225 (. ) 90 ln( )] [ln( 0225 . ) 1 ) 15 ln((. )] [ln( 2 = + = S E S VAR so 0735 . 9265 . 1 ) 45 . 1 ( 1 ) 45 . 1 ( ) 45 . 1 ( 0225 . 787 . ) 489 . 4 185 . 3 ( 0 0225 . 787 . ) 489 . 4 185 . 3 ( ) ln( ) ln( ) ( = = = > > = + > + = Z P Z P Z p S P P failure P (d) Suppose Let N denote the number of windstorms in 25 years, and let X be the number of those that are strong enough to cause failure. The probability a windstorm causes failure
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HW#5_f08_sol - CE 3102 Fall 2008 Homework #5 Due: 9:45 AM...

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