Transcript Reliability analysis

Approximate methods for calculating
probability of failure
• Monte Carlo simulation
• First-order second-moment method (FOSM)
• Working with normal distributions is
appealing
– The reliability index
• Most probable point
• First order reliability method (FORM)
• The Rosenblatt transformation
SOURCE:
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Monte Carlo Simulation
• Given a random variable X and
a function h(X):
sample X: [x1,x2,…,xn];
Calculate [h(x1),h(x2),…,h(xn)]; use to approximate statistics of
h.
• Example: X is U[0,1]. Use MCS to find mean of X2
x=rand(10); y=x.^2; %generates 10x10 random matrix
sumy=sum(y)/10
sumy =0.4017 0.5279 0.1367 0.3501 0.3072 0.3362 0.3855
0.3646 0.5033 0.2666
sum(sumy)/10 ans =0.3580
• What is the true mean
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Evaluating probabilities of failure
• Failure is defined in terms of a limit state function
where failure occurs when g(r)>0, where r is a vector
of random variables.
Pf  m / N
• Probability of failure is estimated as the ratio of
number of positive g’s, m, to total MC sample size, N
• The accuracy of the estimate is poor unless N is
much larger than 1/Pf
Pf (1  Pf )
• For small Pf
  Pf  
  Pf 
Pf
N

1
m
Example
• Estimate the probability that x=N(0,1)>1
x=randn(1,1000); x1=0.5*(sign(x-1)+1); pf=sum(x1)/1000.; pf =0.1550
• Repeating the process obtained: 0.136, 0.159, 0.160, 0.172, 0.154,
0.166.
• Exact value 0.1587.
  Pf  
Pf (1  Pf )
N

0.1587  0.8413
 0.0116
1, 000
With pf from MC =0.136   Pf   0.108
  Pf 
Pf

1
1

 0.086
m
136
• In general, for 10% accuracy of probability you need 100 failed
samples.
The normal distribution is attractive
• It has the nice property that linear functions of
normal variables are normally distributed.
• Also, the sum of many random variables tends to be
normally distributed.
• Probability of failure varies over many orders of
magnitude.
• Reliability index, which is the number of standard
deviations away from the mean solves this problem.
   1 ( Pf ) Pf   (  )
 is the normal CDF
Approximation about mean
• Predecessor of FORM called first-order
second-moment method (FOSM)
g ( X )  g (  X )  g (  X )T  X   X 
Then, easy to show
1/2
 n  g 

2
 g  g (  X )  g       xi 
 i 1  xi 

(How can you use that to get rid of unimportant variables?)
The reliability index and probability of failure are
2
g

Pf   (  )
g
 is the normal CDF
Beam under central load example
• Probability of exceeding plastic moment
capacity g (P, L,W , T )  WT  PL / 4
P
L
W (plastic section
modulus)
T (yield stress)
mean
10kN
8m
0.0001m^3
600,000 kN/m^2
Standard deviation
2kN
0.1m
0.00002m^3
100,000kN/m^2
Reliability index for example
• Using the linear approximation get
 g  g (  X )  0.0001 600, 000  10  8 / 4  40kNm (safety factor of 3!)
g
 W  0.0001
T
g
  L / 4  2
P
g
 T  600, 000
W
g
  P / 4  2.5
L
1/2
 n  g 2 
 g       x2i 
 i 1  xi 


(0.0001100, 000) 2  (600, 000  0.000002) 2   2  2   ( 2.5  0.1) 2  16.13kNm
2

g
 2.48, Pf   (2.48)  0.0066
g
• Example 4.2 of CGC shows that if we change to g=T0.25PL/W we get 3.48 (0.00025, exact is 2.46 or Pf=0.0069)
Most probable point (MPP)
• The error due to the linear approximation is
exacerbated due to the fact that the expansion
may be about a point that is far from the failure
region (due to the safety margin).
• Hasofer and Lind suggested remedying this
problem by finding the most probable point and
linearizing about it.
• The joint distribution of all the random variables
assigns a probability density to every point in the
random space. The point with the highest density
on the line g=0 is the MPP.
Response minus capacity illustration
r=randn(1000,1)*1.25+10;
c=randn(1000,1)*1.5+13;
f=@(x) x;
fplot(f,[5,20])
hold on
plot(r,c,'ro')
xlabel('r')
ylabel('c')
20
c
15
10
5
5
10
15
r
20
Recipe for finding MPP with
independent normal variables
• Transform into standard normal variables
(zero mean and unity standard deviation)
ui 
xi   xi
x
i
• Find the point on g=0 of minimum distance to
origin. The point will be the MPP and the
distance to the origin will be the reliability
index based on linear approximation there.
  min U TU 
1/2
U g (U )  0
Visual
Response minus capacity
• Original and transformed variables
r  N (10,1.252 ),
c  N (13,1.52 )
ur  (r  10) /1.25 or r  10  1.25ur
uc  (c  13) /1.5 or c  13  1.5uc
• Distance in standard normal space
g  r  c  3  1.25ur  1.5uc  0  ur  2.4  1.2uc
d 2  ur2  uc2  uc2   2.4  1.2uc 
2
20
• MPP and reliability index
c
d 2
 0  uc  ur  1.091  r  c  11.36
uc
15
10
d min  1.543    Pf  0.0614
5
5
10
15
r
20
First order reliability method (FORM)
• Limit state g(X). Failure when g<0.
• Linear approximation of limit state together
with assumption that random variables are
normal.
• Approximate around most probable point.
• Then limit state is also normal variable.
• Reliability index is the distance of the mean of
g from zero measured in standard deviations.
General case
• If random variables are normal but correlated, a
linear transformation will transform them to
independent variables.
• If random variables are not normal, can be
transformed to normal with similar probability of
failure. See Section 4.1.5 of CGC (It is called the
Rosenblatt transformation)
• Murray Rosenblatt, Remarks on a Multivariate
Transformation, Ann. Math. Statist. Volume 23,
Number 3 (1952), 470-472.