## Calculating the probability of failure by Zhou & Novak

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benissimo
Posts: 3
Joined: Mon May 30, 2016 10:39 am
Location: HSLU

### Calculating the probability of failure by Zhou & Novak

Hi, I am new to Opensees and I would be happy if someone could help me.

I would like to calculate the probability of failure of a reinforced concrete frame under horizontal loads. I do not want to carry a FORM analysis, but with a method according to Zhou & Novak "integration formulas to evaluate functions of random variables" in 1988.

This works as follows:
I have n random variables, which may be correlated.
With the method of Zhou & Novak I have 2 * n ^ 2 + 1 points, to which I have to run a FEM calculation.
So for example an elastic frame in which only the Young's modulus is a lognormal distribution with mu = 30000N/mm^2, sd = 1000N/mm ^2
2 * 1 ^ 2 + 1 = 3
Now I have 3 FEM calculations with the following points as Young's modulus:

x1 = 29983 N / mm ^ 2
x2 = 31765 N / mm ^ 2
x3 = 28301 N / mm ^ 2

this 3 deformations "G" at the top of the frame should get saved in the same output-file.

G1 (x1), G2 (x2), G3 (x3)

At the end, the individual calculations are weighted.

w1 = 0.666
w2 = 0.166
w3 = 0.166

The first two moments be then:

E [G] = w1 * G1 + w2 * G2 + w3 * G3
sd ^ 2 = E [G ^ 2] -E [G] ^ 2
beta = mu / sd
pf = NORMINV (- beta)

My real question is: can you read a list of parameters in Opensees, for example, with 10 random variables.
This list contains 2 * 10 ^ 2 + 1 = 201 lines and 10 columns for 201 calculations and 10 random variables. The calculation starts with the parameters in line 1, for example, save the horizontal deformation d1 in the output-file "calculation.out".
Thereafter, a further calculation for the line 2 is performed with the parameters 1-10 and save d2 in the same output-file "calculation.out" and so on. This, until one arrives at line "201".
At the end, the individual results are weighted again.

I hope that I could make my question comprehensible.
(Sorry for my english)

#Exemple:

# Create nodes
# tag X Y
node 1 0.0 0.0
node 2 \$width 0.0
node 3 0.0 \$height
node 4 \$width \$height

# Fix supports at base of columns
# tag DX DY RZ
fix 1 1 1 1
fix 2 1 1 1

# Define materials for nonlinear columns
# ------------------------------------------
# CONCRETE tag f'c ec0 f'cu ecu
# Core concrete (confined)
uniaxialMaterial Concrete01 1 -6.0 -0.004 -5.0 -0.014 #Here, parameters should be taken from a list

# Cover concrete (unconfined)
uniaxialMaterial Concrete01 2 -5.0 -0.002 0.0 -0.006 #Here, parameters should be taken from a list

# STEEL
# Reinforcing steel
set fy 60.0; # Yield stress #Here, parameters should be taken from a list
set E 30000.0; # Young's modulus #Here, parameters should be taken from a list
# tag fy E0 b
uniaxialMaterial Steel01 3 \$fy \$E 0.01