Reliability and Sensitivity
Analysis with OpenSees
Armen Der Kiureghian
UC Berkeley
OpenSees Workshop
UC Berkeley
September 3, 2010
Outline

Formulation of structural reliability problem

Solution methods

Uncertainty propagation

Response sensitivity analysis

Sensitivity/importance measures

Methods implemented in OpenSees

Example – probabilistic pushover analysis

Stochastic nonlinear dynamic analysis

Example – fragility analysis of hysteretic system

Summary and conclusions
Formulation of structural reliability
 X1 
 
X  
X 
 n
Vector of random
variables
pf
f X ( x)
Distribution of X
x
Failure domain
pf 
f
x
X
x2
( x ) dx
fX(x)
Failure probability
X
x1
Formulation of structural reliability
Component reliability problem
x  g (x)  0
System reliability problem




 x    g i (x)  0


 k iCk

g (x), gi (x)
limit-state functions (must be continuously differentiable)
e.g., gi (x)  cr  i (x) failure due to excessive ith story drift
Solution by First-Order Reliability Method (FORM)
u  T(x), g (x)  G (u)
x2
u*  arg min  u G (u)  0
u
  α  u*
α
X
fX(x)
reliabilit y index
G (u)
G (u)
u u*
p f     
x1
u2
u*
β
Der Kiureghian, A. (2005). First- and second-order reliability
methods. Chapter 14 in Engineering design reliability handbook,
E. Nikolaidis, D. M. Ghiocel and S. Singhal, Edts., CRC Press,
Boca Raton, FL.
α
U
u1
FORM
approx
Solution by Importance Sampling (IS)
u2
1
pf 
N
(u i )
I G (u i )  0

h(u i )
i 1
N
u i simulated according to h(u i )
u1
Uncertainty propagation
  ( X) response quantity of interest
First-order approximations:
   (M X )
 2   X  Σ XX X  T
MX
mean vector
 XX
covariance matrix
 X  gradient row-vector evaluated at the mean
Response sensitivity analysis
(x)
For both FORM and FOSM, we need
xi
Available methods in OpenSees:

Finite difference

Direct Differentiation Method (DDM)
Differentiate equations of motion and solve for response derivative
equations as adjoint to the equations of motion. Equations for the
response derivative are linear, even for nonlinear response.
DDM is more stable, accurate and efficient than finite difference.

Zhang, Y., and A. Der Kiureghian (1993). Dynamic response sensitivity of inelastic structures. Comp.
Methods Appl. Mech. Engrg., 108(1), 23-36.

Haukaas, T., and A. Der Kiureghian (2005). Parameter sensitivity and importance measures in nonlinear
finite element reliability analysis. J. Engineering Mechanics, ASCE, 131(10): 1013-1026.
Sensitivity/Importance measures
FOSM:
(x)
i
xi
FORM:
α  relative importance of u variables
γ  relative importance of X variables
  
δ
i   reliabilit y importance of mean value s
  i 
  
η
i   reliabilit y importance of stdev values
 i 
Haukaas, T., and A. Der Kiureghian (2005). Parameter sensitivity and importance measures in
nonlinear finite element reliability analysis. J. Engineering Mechanics, ASCE, 131(10): 1013-1026.
Methods implemented in OpenSees
Propagation of uncertainty:
Estimate second moments
of response
Reliability analysis:
Estimate probability of
events defined in terms of
limit-state functions
Response sensitivity analysis:
Determine derivative of
response with respect to input
or structural properties
First-Order SecondMoment (FOSM)
• Mean
Monte Carlo sampling
(MCS)
• Correlation
First-Order Reliability
Method (FORM)
• “Design point”
Importance Sampling (IS)
• Parameter importance/
sensitivity measures
Second-Order Reliability
Method (SORM)
• Standard deviation
• Parameter importance
at mean point
• Probability of failure
 response
 parameter
The Direct Differentiation
Method (DDM)
•
Finite Difference scheme
(FD)
(Used in FOSM and
FORM analysis)
The I-880 Testbed Bridge
32 - 57 mm bars
1.7 m
1.7 m
9.3 m
12 - 35 mm bars
(6 each side)
8 - 11 mm bars
Section 1
20 - 44 mm bars
Section 1
2440 mm
15.6 m
Hoop Transverse Reinforcement
25 mm bars at 102 mm on center
Section 2
R 1170 mm
2590 mm
51 mm
36 - 43 mm bars
8 - 16 mm bars
127 mm
Section 2
320 random variables
g1 = uo - u(lo)
g2 = lo - l(uo)
g3 = u(20% tangent) - uo
g4 = l(20% tangent) - l o
First-Order Second-Moment Analysis
Second-moment
response statistics
for u(lo)
Second-moment response
statistics for l(uo)
FORM Analysis, g1, lo=0.20
Parameter Importance
g1 = 0.35 - u(l=0.2)
141
151
161
171
172
162
152
142
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
-0.603
-0.538
-0.280
0.240
0.232
-0.188
-0.177
0.135
-0.122
-0.100
0.091
0.083
-0.073
-0.058
-0.056
-0.048
0.046
0.040
-0.040
0.040
-0.032
0.031
0.029
-0.027
0.026
0.026
-0.023
-0.022
-0.022
0.021
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Node
Element
Node
Element
Element
Element
Element
141
142
151
142
142
152
1502
142
1602
161
141
152
141
142
162
142
142
152
1502
152
141
162
151
14002
141
14005
1602
142
151
162
y
y
y
f'c
cu
y
E
f'c
E
y
f'c
f'c
b
c
y
b
c
cu
E
f'c
E
f'c
f'c
y-crd.
c
y-crd.
E
E
b
c
FORM Analysis, g2, uo=0.30
Continuity of response derivative
FORM Analysis, g3 and g4
Haukaas, T., and A. Der Kiureghian (2007).
Methods and object-oriented software for FE
reliability and sensitivity analysis with application
to a bridge structure. Journal of Computing in Civil
Engineering, ASCE, 21(3):151-163.
Stochastic nonlinear dynamic analysis
Representation of stochastic ground motion
Stochasticity
A(t , u)  q(t )i 1 si (t )ui  q(t ) s(t )u
n
Temporal nonstationarity
q(ti)ui
ti-1
t i ti 1
18
Spectral nonstationarity
si(t) normalized response of a
time-dependent filter
t
Stochastic nonlinear dynamic analysis
Representation of stochastic ground motion
A(t , u)  q(t )i 1 si (t )ui  q(t ) s(t )u
n
10.0
acc (m /s2 )
5.0
0.0
-5.0
-10.0
0
10
20
30
40
tim e ( s )
Rezaeian, S. and A. Der Kiureghian (2009). Simulation of synthetic ground motions for specified
earthquake and site characteristics. Earthquake Engineering & Structural Dynamics, 39:1155-1180.
19
Stochastic nonlinear dynamic analysis
Prx  X (t , u)
tail probability for threshold x at time t
Solution of the above problem by
FORM leads to identification of a TailEquivalent Linear System (TELS).
u2
u*
TELS is solved by linear random
vibration methods to obtain response
statistics of interest, e.g., distribution of
extreme peak response, fragility curve.
X (t , u)  x
β( x, t )
a( x, t )
Fujimura, K., and A. Der Kiureghian (2007). Tail-equivalent linearization method for nonlinear
random vibration. Probabilistic Engineering Mechanics, 22:63-76.
20
u1
Application to MDOF hysteretic system
Node 6
m=3.0104 kg for
all nodes
300
Force (kN)
k0 =2.010 kN/m
4
Node 5
k0 =4.010 kN/m
4
Node 4
Node 3
Node 2
0.0
0.04
Disp. (m)
6th story
k0 =5.5104 kN/m
1000
k0 =6.5104 kN/m
k0 =7.0104 kN/m
Node 1
k0 =7.510 kN/m
4
21
-300
0.04
Force (kN)
1000-0.04
0.0
Disp. (m)
0.04
1st story
Smooth bilinear hysteresis model
Fragility curves for story drifts
First story
Sixth story
1.0E+0
Pr[x<maxt(0,20)|X(t)|]
1.0E+0
1% drift
1.0E-1
1.0E-1
2%
1.0E-2
3%
1.0E-4
1.0E-5
1.0E-5
1
1.5
3%
1.0E-3
1.0E-4
0.5
2%
1.0E-2
1.0E-3
0
1%
2
Sa(T = 0.576s,  = 0.05) in g’s
2.5
0
0.5
1
1.5
2
Sa(T = 0.576s,  = 0.05) in g’s
Der Kiureghian, A., and K. Fujimura (2009). Nonlinear stochastic dynamic analysis for performancebased earthquake engineering. Earthquake Engineering and Structural Dynamics, 38:719-738.
22
Summary and conclusions

OpenSees is a general-purpose structural analysis platform with
unique capabilities for sensitivity and reliability analysis.

Reliability analysis requires specification of uncertain quantities, their
distributions, and definition of performance via limit-state function(s).
It provides probability of exceeding specified performance limit(s).

Uncertainty propagation provides first-order approximation of mean,
variance and correlations of response quantities.

Sensitivity and importance measures provide insight into the relative
importance of variables and parameters.

Stochastic dynamic analysis is performed via tail-equivalent
linearization. TELM can be used to generate fragility functions (e.g., in
lieu of performing IDAs).
23