reliability-based design optimization in...
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Reliability-based Design Optimization
in OpenSees
Terje Haukaas, UBC, Vancouver
Developer Symposium, Richmond Field Station, August 24, 2005
Developer Symposium, Richmond Field Station, August 24, 2005
Vision
Rational decision making under uncertainty (in performance-based engineering)
� “Is the structural performance OK?”
� Prediction of structural performance
� Can only be done in a probabilistic sense
� � Reliability analysis
� “Is the structural performance OK?”
� Prediction of structural performance
� Can only be done in a probabilistic sense
� � Reliability analysis
� “If not, which parameter to change?”
� Response sensitivities
� Importance measures
� “If not, which parameter to change?”
� Response sensitivities
� Importance measures
� Ultimate question: “Have we balanced cost and safety?”
� Reliability-based design optimization (RBDO)
� Rational decision making
� Ultimate question: “Have we balanced cost and safety?”
� Reliability-based design optimization (RBDO)
� Rational decision making
Developer Symposium, Richmond Field Station, August 24, 2005
Decision Making
� Scientists: Can research forever without making a decision
� Engineers: Must make decisions under uncertainty
� The decision can be to perform further studies, or to make a final decision
� What is an acceptable design? / How to determine target safety level?
� Scientists: Can research forever without making a decision
� Engineers: Must make decisions under uncertainty
� The decision can be to perform further studies, or to make a final decision
� What is an acceptable design? / How to determine target safety level?
1) Calibration towards the current “accepted” design practice
2) Calibration towards the background risk in society
3) Determination of social consensus level
4) Reliability-based optimal structural design; balance cost and safety
1) Calibration towards the current “accepted” design practice
2) Calibration towards the background risk in society
3) Determination of social consensus level
4) Reliability-based optimal structural design; balance cost and safety
Developer Symposium, Richmond Field Station, August 24, 2005
Reliability-based Optimal Design
Design variable
Expected cost of failure
Initial cost
Total expected cost
Optimum
� Minimize total expected cost, subject to constraints
No reliability constraints?
{ } | min 0 0f ≤+ ff pcc
c0 = initial cost
cf = cost of failure
pf = probability of failure
f = constraints on design variables
Reliability-based
design optimization
(RBDO)
Developer Symposium, Richmond Field Station, August 24, 2005
Determination of Costs and Probabilities
cf Cost of failure
Damage cost
Downtime cost
Value of life, cost of injury
Present value of future events
pf Probability of failure
Finite element reliability analysis
Issues: multiple failure modes / dependence / system reliability analysis
c0 Cost of construction
New structures
Retrofit of old structures
Developer Symposium, Richmond Field Station, August 24, 2005
Objective
� Implement RBDO analysis capabilities in main-stream engineering analysis software
� Open-source
� Object-oriented
� Already extended with reliability and response sensitivity capabilities
� Framework of analysis tools: ReliabilityAnalysis
FindDesignPoint SearchDirection StepSizeRule
GFunEvaluatorRandomNumberGenerator GradGEvaluator
ProbabilityTransformation
ConvergenceCheck
Developer Symposium, Richmond Field Station, August 24, 2005
Problems and Solutions
� Problem: { } pp , | mintargetff0 ≤≤+ 0fff pcc
� Challenges
� Computational cost
� Potentially non-smooth reliability
� Solution techniques
� Nested bi-level approach
� Mono-level approach
� Genetic algorithms
� Response surface methods and neural networks
� … and …
Developer Symposium, Richmond Field Station, August 24, 2005
The Decoupled Sequential Approach(Ref: Royset, Polak, Der Kiureghian)
� “Semi-infinite optimization problem”
� Method of Outer Approximations (Polak 1997)
� Solution
{ } 0 , , | mintarget0 fff paapacc ≤≤=≤⋅+ 0f
� Replace the constraint pf = a by the requirement that the limit-state function
be zero at a distance –Φ-1(pf) from the origin
� Do not allow the limit-state function to be negative within a ball in the space
of random variables
� Reformulation
� Replace the failure probability by an auxiliary variable
Developer Symposium, Richmond Field Station, August 24, 2005
Implementations in OpenSees
1) Solve the semi-infinite optimization for a given ball radius
2) Run reliability analysis with method-of-choice
3) Update the radius of the ball and repeat
1) Solve the semi-infinite optimization for a given ball radius
2) Run reliability analysis with method-of-choice
3) Update the radius of the ball and repeat
FFffgg jj ∇∇∇ ,,,,,
RBDO Algorithm
x, u
Finite element reliability module
Developer Symposium, Richmond Field Station, August 24, 2005
Objects
designVariablePositioner
designVariable costFunction
constraintFunction
objectiveFunction
ReliabilityDomain
evaluateFun evaluateGradFun
NonlinSingleIneqOpt
PolakHeNonlinSingleIneqOpt
NonlinMultiIneqOpt
PolakHeNonlinMultiIneqOpt
ReliabilityAnalysis
DSA-MOOAAnalysis DSA-SAnalysis
evaluateFun evaluateGradFunLinMultiIneqOpt
LSSOLLinMultiIneqOpt
Developer Symposium, Richmond Field Station, August 24, 2005
Response Gradients
� Direct differentiation method (DDM)
u
v
v
d
du ∂
∂
∂
∂
∂
∂=
∂
∂ gg
fixed
intn
extnn
n nvvvd
PPdK |
∂
∂−
∂
∂=
∂
∂
Developer Symposium, Richmond Field Station, August 24, 2005
Gradient Discontinuities stress
strain
Steel01
yF Eα
yFγ
yF−yFγ−
smoothing
E
confined concrete
reinforced steel layer
20 fibers
unconfined concrete
20 fibers
Column Fiber Section Beam Fiber Section
unconfined concrete
2 fibers
Developer Symposium, Richmond Field Station, August 24, 2005
Example� Six-story reinforced concrete frame in Vancouver
� Load case: 1.0 × dead load + 0.5 × live load + 1.0 × earthquake load
� Original element dimensions:
width (b) x depth(h)
Interior columns 500 x 500 mm
Exterior columns 450 x 450 mm
First 3 storys beams 400 x 600 mm
Top 3 storys beams 400 x 550 mm
� Drift ratio control:
Roof displacement is 297mm by
static pushover analysis.
Drift ratio is
297/23100 = 1.3% < limit of 2%
� Reliability analysis results:
Reliability index β = 3.25
Failure probability pf = 0.0577%
H6
H5
H4
H3
H2
H1
Developer Symposium, Richmond Field Station, August 24, 2005
Example
confined concrete
reinforced steel layer
20 fibers
unconfined concrete
20 fibers
Column Fiber Section Beam Fiber Section
unconfined concrete
2 fibers
Nonlinear material models
Developer Symposium, Richmond Field Station, August 24, 2005
modulus of elasticity of steellognormal0.70.05200000
MPa
steel bar strengthlognormal0.70.15400 MPa
modulus of elasticity of
unconfined concretelognormal0.70.1015000 MPa
unconfined concrete strengthlognormal0.70.1530 MPa
modulus of elasticity of
confined concretelognormal0.70.109750 MPa
confined concrete strengthlognormal0.70.1539 MPa
lateral load on rooflognormal0.15131890 kNH6
lateral load on floor 5lognormal0.15109780 kNH5
lateral load on floor 4lognormal0.1589100 kNH4
lateral load on floor 3lognormal0.1570070 kNH3
lateral load on floor 2lognormal0.1548950 kNH2
lateral load on floor 1lognormal
0.7
0.1528490 kNH1
DescriptionTypec.c.c.o.v
.MeanVariable
78 random variables
81 cccc EE L
14
'
1
'
cc ff L
141 cc EE L
141 yy ff L
141 EE L
8
'
1
'
cccc ff L
Developer Symposium, Richmond Field Station, August 24, 2005
18 design variables
area of reinforced bars of top three stories’ beams0.0024m2A6
width and depth of exterior columns of top three stories0.40×0.55mb4× h
4
area of reinforced bars of first three stories’ beams0.0024m2A5
width and depth of exterior columns of first three stories0.40×0.60mb5× h
5
half of the area of reinforced bars of interior columns of
top three stories0.003m2A
4
width and depth of interior columns of top three stories0.50×0.50mb4× h
4
half of the area of reinforced bars of interior columns of
first three stories0.003m2A
3
width and depth of interior columns of first three stories0.50×0.50mb3× h
3
half of the area of reinforced bars of exterior columns of
top three stories0.003m2A
2
width and depth of exterior columns of top three stories0.45×0.45mb2× h
2
half of the area of reinforced bars of exterior columns of
first three stories0.003m2A
1
width and depth of exterior columns of first three stories0.45×0.45mb1× h
1
DescriptionInitial ValueVariable
Developer Symposium, Richmond Field Station, August 24, 2005
Problem Definition
� Limit-state function: roof%2m1.23 dg −×=
� Cost functions � Objective functions:
⋅⋅++⋅⋅⋅+⋅⋅+
⋅⋅++⋅⋅⋅+=
∑∑
∑∑
==
==
)100()2100( 5)(
)100()2100(
6
5
4
1
6
5
4
1
iiiii
iiiiif
iiiii
iiiii
LAhbLAhbp
LAhbLAhbF
x
Cost of failure is 5 times initial cost Cost of steel is 100 times that
of concrete (per volume)
� Structural constraints:
� 0 ≤ bi, hi
� 0.5 ≤ bi/hi ≤ 2
�iiiii hbAhb ⋅≤≤⋅ 02.001.0
Developer Symposium, Richmond Field Station, August 24, 2005
Reliability-based Design Optimization
� Initial design:
� Reliability index: 3.12
� Total expected cost: 130.7
� Optimized design:
� Reliability index: 3.0
� Total expected cost: 85.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.08 0.16 0.24 0.32 0.40 0.48
Roof Drift (m)
Late
ral
Load
Fact
or
(1) Original mean point
(2) Original MPP
(3) Optimal mean point
(4) Optimal MPP
Developer Symposium, Richmond Field Station, August 24, 2005
Concluding Remarks
� Sophisticated structural model
� Implementation is software that is increasingly
employed in the earthquake engineering community
� Response gradients obtained by the DDM
� Valuable tool for performance-based engineering:
guide for design improvement
� Technical issues
� Expansion of reliability constraints
� Computational cost
� Handling of nonlinearities
Developer Symposium, Richmond Field Station, August 24, 2005
Acknowledgements
� Hong Liang, former graduate student at UBC, Vancouver
� Johannes O. Royset, Naval Postgraduate School, Monterey
Developer Symposium, Richmond Field Station, August 24, 2005
Thank you for your attention!