Question

4.29 A structure is designed with a capacity to withstand a wind velocity of 150 kph; i.e., any wind velocity up to 150 kph will not cause damage to the structure. In a hurricane-prone area, the maximum wind velocity during a hurricane may be modeled as an extreme Type / asymptotic distribution with a mean velocity of 100 kph and a c.o.v. of 0.45. (a) For a structure with the above wind-resistant capacity, what would be the probability of wind damage during a hurricane? (b) If the permissible damage probability were to be reduced to 1/10 of the original design, as described in (a) above, what should be the wind-resistant capacity of the revised design (in kph)? (c) If the occurrence of hurricanes in the area is modeled as a Poisson process with a return period of 200 years, what is the probability of damage to the original structure over a life of 100 years? Also, what would be the corresponding damage proba- bility to the revised structure? Assume that damages between hurricanes are statistically independent. (d) Suppose that three structures with the original design were built in the area, each with the same wind damage probability. What is the probability that at least one of the structures will be damaged over the life of 100 years?

Answer 1

ANSWERS Given that A structure is designed with a capacity to with stand a wind velocity of 150 keph. a) find the probability of wind damage during a hurricane. p(y > 150) = 1- Fyly) - - Here, probability is p, cumulative distribution function of asymptotic from for the Type I distribution of The largest value of his fy (9) - Express The calmulative distribution Scanned with CamScanner

function of asymptotic form for the Type I distribution of the dargest value of no Fyl99 - exp (-(4-n)) – @ » Here, sample site is n, and on parameters are an and un & find the distribution parameter (&p) 6 - (MySy) e * Here, distribution parameter is ap me any mean velocity is My and & coefficient of variation of relocity is by * Rearrange the above ocuation to find an In "/37 (lly sy). Scanned with CamScanner

o substitute 100 for My and ous for Sy, to find and wher e ano Jo (100 x ous) n a 2n = 0.0295 -> kind the distribution parameter (un) Un = My - Van Iteve , distribution parameter is Un, mean velocity is My & Euler's number is r and distribution is an. & Euler's number value is 0.5772 Un = 100 0.5772 00985 . Un = 79.75 Scanned with CamScanner

» find The cumulative distribution function of asymptotic form for The Type 1 distribution of The largest value of n using equation @ x y - substitute oars for sp, 150 79.75 for un în ein Fy (y) a exp (--00285 (150-79.75)) *Cap 6-6-2.0093) a exep (-0.135) damage during hind The probability of wind a hurricane equation ④ Scanned with CamScanner

- substitute 0.874 fox Ey(y) in equation ply > 150) - | -0,874 0.136 - thence, the probability of wind damage during a hurricane is" 0.1261 b) permissible damage probability is reduced to 1/10th of the original design Pa 01191/10 1.po 0.01267 y find the revised design wind resistance capacity of the structure. Ies Scanned with Cam Scanner

- po 1 - esp le ap (7 - Un) ] prolata in erp (-8-00989 (y-7973) erp (-20.015 (4-24-, 1–0-0126 = 0.987u - Take natural logarithm, to eliminate exponential -0.0955 14-7976) lo 0.9874 - - 0.01268 1:00:0235 14-14.39 0.01268) -> Take natural logarithm, to eliminate exponential -0.0155 y-79.75) - Un 0.01268 y = 79.75 = 153.25 Scanned with CamScanner

> Hence, The revised design wind resistance capacity of the structure is "233 kph." e find the mean probability rate of original structure damage due to hurricane for one year p=1/200) Co.126) p = 0.00063 find the probability of original structure damage due to hurricane over 100 years. Pl damage over 100 yrs) : 1-P (no damage owy 100 you) al-exp (-0.00063 X 100) ol-0.9389 2 0.061 Scanned with CamScanner

Hence. The probability of original structure damage due to hurricane over 100 years is" 0.001 - find the mean probability rate of revised structure damage due to hurricane for one year. po laco) tesom (0.0128) p. 0.000063 o find the probability of revised structure damage due to hurricane over 100 years. >> pl damage over 100 yas ) - 1-P (no damage over loo yas) -la exp(-0.000063 X 1002 -1-0.9937 - 0.00628 cs Scanned with CamScanner

Hence, the probability of revised structure damage due to hurricane over 100 years is 0.00098 " d) find the probability at least one structure Corginal structure out of 3 will be damaged over 100 years. plat least 1 out of 3 structure damage) = a l plnone of the 3 structure damaged] -1-(1-0.061) - 1-0.8279 = 0.172 Scanned with CamScanner

*. Hence. The probability at least one structure (orginal structure ) out of 3 will be damaged over 100 years is "0.172" Scanned with Canes