Reliability Based Design:

Optimization of Suspension Bridges with

Emphasis on Aerodynamic Stability

Ibuki Kusano

M. Cid, A. Baldomir, F. Nieto, J.A. Jurado, S.

Hernandez

Outline

2

1. Motivation

2. Reliability Based Design Optimization

3. Reliability analysis of flutter

4. Application example

5. Summary

Structural Optimization

3

• Widely Used Technique (e.g., aerospace, automobile, defense)

• But Not So Common in Civil Engineering

• Many Uncertainties

• Reliability Based Design Optimization Considers

Uncertainty Parameters Explicitly

Benefits & Payoffs

4

Robust Optimum Design + Reduce Carbon Emission

What Is Optimization?

5

6

YesFinal

Design

Optimum

Design?

ModifiedDesign

Optimization

Algorithm

Initial

Design

No

General Optimization Flow Chart

Analysis

Uncertainty in Analysis?

7

Uncertainty in Parameters

8

Sampling Methods

• Monte Carlo Sampling

• Latin Hypercube Sampling

• Importance Sampling

• 1st Order Reliability Method

• 2nd Order Reliability Method

Moment Methods

Two Methods

9

( ) ( ) ( )G R S x x x R: resistance S: Load

First Order Reliability Method (FORM)

Eurocode for bridges (EN1990)β=5.2 for 1 year period β=4.3 for 50 years β=3.8 for 100 years (Pf =7E-5)

Limit State Function:

Probability of Failure:

random variables

β

Safe region

G(u)=0

Failure region

U*

min T U U

x1 and x2i

i

i x

i

x

xu

μxi: mean values

σxi: standard deviation

10

Adding Reliability

YesFinal

Design

Optimum

Design?

ModifiedDesign

Optimization

Algorithm

Initial

Design

No

Analysis

• Determine βT

β >βT?

β

Reliability

analysis

RIA method

What is Flutter?

• Aerodynamic instability of flexible structures

• Fluid structure interaction

• Coupling of modes

• Zero effective damping

Reliability Analysis of flutter

11

Scanlan’s Formulation

* * * * * *

1 5 2 4 6 3

* * * 2 2 * * *

5 1 2 6 4 3

* * 2 * * * 2 *

5 1 2 6 4 3

1 1· ·

2 2

a

a

a

D P P BP v P P BP v

L VKB H H BH w V K H H BH w

M BA BA B A BA BA B A

fa=

12

j j ji

j j

2 2

j

j

j j

αj=0 flutter instability

frequency

structural damping

Methods for Flutter Analysis

Full Bridge Model Test

13

+ Quasi Steady theory

Fully Computational Method

*Akashi bridge full model, PWRI

Hybrid Method

*Messina bridge sectional model, U. of Coruna

1. Definition of the Deck Baseline Geometry and Design Range

Flutter analysis for Different Deck Shapes

14

1. Definition of the Deck Baseline Geometry and Design Range

Flutter analysis for Different Deck Shapes

14ΔB (%)

ΔH

(%

)

G1

2. Sampling Plan of Computational

Fluid Dynamics (CFD) Models

3. CFD Simulations by OpenFoam (kω-SST turbulence model)

15

Flutter analysis for Different Deck Shapes

*Railings not included in the model

16

4. Wind Tunnel Test Validations

Flutter analysis for Different Deck Shapes

5. Kriging surrogate model construction

CL CD CM CL’ CD’ CM’

17

,0*

1

' oMC

AK

,0*

2

' oM

A

CA

K

,0*

3 2

' oMC

AK

,0 ,0*

1

' o oL DC C

HK

,0 ,0*

2

' o oL D

H

C CH

K

,0*

3 2

' oLC

HK

6. Quasi-steady formulation to define flutter derivatives

Flutter analysis for Different Deck Shapes

Vf

Define

Deck

ShapeModified FEM

Multi-Modal

Analysis

18

Flowchart: Flutter Analysis

QS

Φ and ω

RBDO Formulation: Shape & Size

19

Design Optimization

6 design variables:

Obj. Func.

Min: bridge deck volume

Constraints:

g1: probabilistic flutter

g2: side limits

g3: deck max. vertical displacement

under overload case

g4: max main cable stress

Reliability Analysis

1( ) exp( ) exp exp( )

w

w wx w

X Xf X

7 random variables:

( ) ( )f i wG V x x x

xi: rand. variables of force coeff.

xw: rand. variable of extreme

wind

Flowchart: RBDO

i

G

u

f

i

V

x

20

Application Example: Great Belt East Bridge

Scanlan’s G1 Section

21

Mode Shapes and Frequencies

Mode Type Frequency (Hz)

2 VS 0.098

5 VS 0.131

11 LS 0.186

12 LS 0.195

13 LA 0.213

14 LS 0.213

15 VS 0.216

18 VS 0.249

19 LA 0.275

20 VS 0.282

21 TS/LS 0.285

22 VS 0.285

23 VA 0.286

24 TS/LS 0.290

d1

d2

d3

d4

ΔH=0; ΔB=0

d=[12, 10, 10, 10] (mm)

Flutter Analysis: Initial Design

22

Vf=78.2 m/s

K=0.47

V: vert. L: lat. T: tors. S: symm. A: asymm.

Limit State Function:

Probability of Failure:

Random variables:

– Case A: Extreme Wind Velocity

– Case B: Force Coefficients, Derivatives, Extreme Wind Velocity

Messina example –reliability analysis

( )f wG V x x

( ) 0fP P G d x,

Random variables of force

coefficients

Case random var. CV β Pf Vf (MPP) V*(MPP)

A xw 0.07 12.01 1.57E-33 78.20 13.22

B xw and xi 0.2 7.58 1.73E-14 62.13 12.33

Reliability Analysis of GB Bridge

23

6 Design Variables 7 Random Variables

Initial Design =7.58

Target Reliabilities T=6 and 8

1( ) exp( ) exp exp( )

w

w wx w

X Xf X

RBDO Formulation

24

DS410

βT=6.0

βT=8.0

t1: top platet2: bottom platet3: upper sidet4: lower side

RBDO Results

25

Objective Function

26

RBDO Results

Objective Function

βT Vf ΔH ΔB d1 d2 d3 d4

% variation

obj. func.

initial 78.2 0 0 12.0 10.0 10.0 10.0 -

6 69.45 10.00 4.51 6.48 6.63 6.24 6.24 -26.07

8 82.10 10.00 -0.32 9.51 11.82 8.02 9.58 -4.76

26

RBDO Results

RBDO Results Summary

1. RBDO Provides Accurate & Competitive Optimum Design for Considering

Uncertainty Explicitly.

2. Fully Numerical Approach of Flutter Velocity Computation Permits the Shape

Optimization of Bridge Decks.

3. More Probabilistic Constraints in the Future Study (aerodynamic instabilities,

turbulence effects, traffic loads, temperature loads…)

Summary

27

Thank you!

Thank you for your attention.

I hope you enjoyed the presentation