Генерация коррелированных случайных величин с нормальными наборами распределительных

64a83713-10a2-44ff-9708-11add58ba7da 85 Генерация коррелированных случайных величин с нормальными наборами распределительных Димитриос Торис Создает коррелированные случайные переменные из некоррелированных случайных величин с нормальным распределением Generation of correlated random variable sets with normal distribution Dimitrios Toris Generates correlated random variables from uncorrelated random variables with normal distribution 3 5 false false 0 7 decimal

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For this problem, the cohesion (c) and friction angle (ϕ) for the soil is given as follows (assume normal distribution):

- c (mean) = 200 psf (std. dev. = 80 psf)
- ϕ (mean) = 35 deg (std. dev. = 5 deg)

Assume that the correlation between cohesion (c) and friction angle (ϕ) can be taken as -0.5, and calculate:

- the ultimate bearing capacity of the footing for D = 4 feet, B = 8 feet and γ = 110 pounds per cubic feet
- the probability of failure and system reliability if the factor of safety for the footing load is 3.0.

Number of trials:

n.tr 10000 :

Normal statistics of the variables:

μ.ϕ 35 deg * :

σ.ϕ 5 deg * :

μ.c 200 psf * :

σ.c 80 psf * :

ρ 0.5 - :

ρ = Correlation coefficient

Footing dimensions, soil unit weight (γ), and desired factor of safety (FoS):

γ 110 lbf ft 3^/ * :

B 8 ft * :

D 4 ft * :

FoS 3.0 :

Generation of a set of random numbers with standard normal distribution:

xa n.tr Random :

xb n.tr Random :

i 1 n.tr range Coh i el xa i el 01 ICDF.Normal : Phi i el xb i el 01 ICDF.Normal : 21 line for

Generation of correlated random variables:

i 1 n.tr range c i el μ.c psf / σ.c psf / Coh i el * + (psf (* : ϕ i el μ.ϕ deg / σ.ϕ deg / Coh i el ρ * Phi i el 1 ρ 2^- (0.5^* + (* + (deg (* : 21 line for

Calculate ultimate bearing capacity (Q):

i 1 n.tr range N.q i el π ϕ i el tan * exp 45 deg * 12 / ϕ i el * + tan (2^* : N.γ i el 2 N.q i el 1 + (* ϕ i el tan * : N.c i el N.q i el 1 - (ϕ i el cot * : Q i el c i el B * N.c i el * γ D * B * N.q i el * + 12 / γ * B 2^* N.γ i el * + : 41 line for

N.q π ϕ tan * exp 45 deg * 12 / ϕ * + tan (2^* vectorize :

N.γ 2 N.q 1 + (* ϕ tan * vectorize :

N.c N.q 1 - (ϕ cot * vectorize :

Q c B * N.c * γ D * B * N.q * + 12 / γ * B 2^* N.γ * + vectorize :

Calculate the probability of failure (PF) and reliability (R):

q Q Mean FoS / : tonf ft / 80.2

Define "failure rate (FR)" as when the ultimate footing load exceeds the allowable footing load (q), and calculate probability of failure as the number of times failure occurs divided by the total number of trials:

i 1 n.tr range Q i el q < FR i el 1 : FR i el 0 : if 11 line for

PF FR sum n.tr / : 0.078

R 1 PF - : % 92.2