2. Problem 4.6 of your textbook (Haldar& Mahadevan): The magnitude of a load act

Question

2. Problem 4.6 of your textbook (Haldar& Mahadevan): The magnitude of a load acting on a structure can be modeled by a normal distribution with a mean of 100 kip and a Lognormal standard deviation of 20 kip. (a) If the design load is considered to be the 90h percentile value, determine the design (b) If the design load is considered to be the mean 2 standard deviation value, what (c) A load of magnitude less than zero is physically illogical; calculate its probability. load is the probability that it will be exceeded? Is a normal distribution appropriate to model the load?

Explanation / Answer

mean value = 100 kip

Standard deviation = 20 kip

(a) Here the design load is K, we say

Pr(X < K ; 100 kip ; 20 kip) = 0.90

so.

Z = 1.282

Here ln = ln (100) - 1/2 ln (202/1002 + 1) = 4.6052 - 0.0196 = 4.5856

in = sqrt [ln ((202/1002 + 1) ] = 0.198

Z = (ln X - ln)/in

1.282 = (ln X - 4.5856)/0.198

ln X = 4.8395

X=126.405 kip

(b) If the design load is mean + 2 standard deviation that means the design value is 120 kip,

so we have to calculate,

Pr(X > 140 kip ; 100 kip ; 20 kip) = 1 - Pr(X <= 140 kip; 100 kip ; 20 kip)

Z =(ln 140 - 4.5856)/0.198

Z = 1.7982

Pr(X > 140 kip ; 100 kip ; 20 kip) = 1 - Pr(X <= 140 kip; 100 kip ; 20 kip) = 1- Pr(Z < 1.7982) = 1 - 0.9639 = 0.0361

(c) Pr(x < 0 ; 100 kip ; 20 kip) = 0

as ln (0) =- infnity so lognormal distribution is appropriate for this model as it will show zero probability for less than zero values.

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