Reliability-based Design Optimization in...

Reliability-based Design Optimization

in OpenSees

Terje Haukaas, UBC, Vancouver

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


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


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)


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


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


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 …


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


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


Objects

designVariablePositioner

designVariable costFunction

constraintFunction

objectiveFunction

ReliabilityDomain

evaluateFun evaluateGradFun

NonlinSingleIneqOpt

PolakHeNonlinSingleIneqOpt

NonlinMultiIneqOpt

PolakHeNonlinMultiIneqOpt

ReliabilityAnalysis

DSA-MOOAAnalysis DSA-SAnalysis

evaluateFun evaluateGradFunLinMultiIneqOpt

LSSOLLinMultiIneqOpt


Response Gradients

� Direct differentiation method (DDM)

u

v

v

d

du ∂

∂

∂

∂

∂

∂=

∂

∂ gg

fixed

intn

extnn

n nvvvd

PPdK |

∂

∂−

∂

∂=

∂

∂


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


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


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


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 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


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


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


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


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


Acknowledgements

� Hong Liang, former graduate student at UBC, Vancouver

� Johannes O. Royset, Naval Postgraduate School, Monterey


Thank you for your attention!

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