Download Numerical methods

8. Numerical methods for reliability
computations
Objectives
• Learn how to approximate failure probability using Level
I, Level II and Level III methods
1
Level I, II, III methods
Crude,
economical
• Level I methods: use one parameter to account
for the uncertainty in each random variable
• Level II methods: use two parameters to account
for the uncertainty in each random variable
(usually the mean and the standard deviation)
• Level III methods: take into account for the
probability distributions of the random
variables.
More accurate,
But also more
expensive 2
First order, second moment methods
• Reduced random variables. Independent, standard normal
variables obtained from a transformation of random
variables.
• Safety index, =distance from straight line representing the
boundary between survival and failure and origin in space
of reduced variables. Failure probability=(- ).
• Of all combinations of the r.v. that entail failure the one
corresponding to the most probable failure point is the
most likely.
3
z2
Constant joint
probability density
O
OA=safety index
OB=safety index if X1 were
deterministic
OC=safety index if X2 were
deterministic
X2 more important than X1
C
z1
B
A, most probable failure
point
4
Lessons learnt (refer to the previous slide)
• If performance function is linear, it is easy to compute the
failure probability exactly.
• If we plot the limit state function in the space of the
random variables, then this function is represented by a
straight line.
• The safety index, , is equal to the distance of the straight
line representing the limit state function to the origin. The
probability of failure is (-).
• Of all combinations of values of the values of the random
variables, the one corresponding to the most probable
failure point, A, is the most likely
5
Finding the most probable point when the
performance function is linear and the random
variables are independent, standard normal
g(z1,z2 )=a0+a1z1+a2z2
z2

 z* 
1   g ( 0 ) g
 *
 2
 z2 
g
z*1, z*2
g<0
gradient of g
g>0
O
z1
6
Estimating failure probability using linear
approximation of the performance function about MPP:
First order methods (FORM) or First order, second moment
methods (FOSM)
• It is important to approximate g-function accurately in the
vicinity of MPP to estimate failure probability accurately.
Therefore, we should use linear Taylor expansion about the
MPP.
• Case A: standard normal, independent random variables
• Case B: correlated standard normal random variables
• Case C: independent variables with arbitrary probability
distributions
• Case D: dependent variables with arbitrary probability
distributions (beyond the scope of this class)
7
Case A: normal independent random
variables
Transform variables into standard normal
Find MPP:
Find z1*,…,zn*
To minimize (z1*2+…+zn*2)1/2
So that g(z1*,…,zn*)=0
Find safety index,  and P(F)
8
Performance function in the space of reduced random variables
Limit state
function,g(z)=0
Linear approximation of
limit state function, g(z)=0
z2
MPP: z1*, z2*
Failure region
Gradient of g
  z1*2  z2*2
O
Safe region
z1
P( F )   ...  f Z1,..., Z n ( z1,..., zn )dz1...dzn   (   )
g 0
9
Case B Correlated, standard
normal random variables
• Need to transform correlated variables into
uncorrelated
• Transformation: rotation of coordinate
system
• Y=TTX
• T: matrix whose columns are the
eigenvectors of the covariance matrix
10
Independent standard normal random variables
11
M
Positively correlated standard normal random
variables
12
M
Case C:Independent random variables
with arbitrary probability distribution
• Transform random variables to standard normal
using Rosemblatt transformation (the same
transformation is used for generating random
variables with arbitrary probability distributions)
( z )  FX ( x)
 (z )
z
FX (x)
x
x, z
13
Reliability computations: practical
considerations
• Available methods: FORM, Monte Carlo, direct integration
of PDF
• FORM: can be tricky but it can be the only feasible
approach for many complex problems
– Validate the results using Monte-Carlo for in a few cases
– Conduct parametric studies to understand what the important
variables are
– Understand the behavior of the performance function
– Remove the unimportant random variables and repeat the analysis
to see if the results change
– Repeat the procedure for finding MPP starting from different initial
points to see if there are multiple MPPs
14
Suggested reading
• Der Kiureghian, “First- and Second-Order
Reliability Methods,” Engineering Design
Reliability Handbook, CRC press, 2004, p.
14-1.
• Thoft-Christensen, “System Reliability,”
Engineering Design Reliability Handbook,
CRC press, 2004, p. 15-1.
15