cable stayed bridges non linear effects

Cable Stayed BridgesCable Stayed BridgesNon Non –– Linear EffectsLinear Effects

Tony DempseyTony Dempsey

ROUGHAN & O’DONOVANConsulting Engineers

Presentation Layout

1. Introduction

2. Cable-Stayed Bridges - Steel

Theory & Examples

2

Theory & Examples

3. Cable-Stayed Bridges - Concrete

Theory & Examples

4. Cable-Stayed Bridges - Composite

Examples

1. Introduction

• Cable Stayed Bridges – Non Linearity

Geometric Non Linear (GNL) – Large Displacement

Material Non Linear (MNL) – Moment Curvature

3

Material Non Linear (MNL) – Moment Curvature

Non Linear Time Dependent Effects (TDE)

Non Linear Cable Elements (NLE)

Non – Linear Combinations (GNL / MNL / TDE / NLE)

Cable – Rupture & Plastic Analysis

• Cable Stayed Bridges – Static Linear Analysis

• BS 5400 Part 3: Clause 10

• First Principle Approach


Steel Pylon Design – Second Order Effects

4

• Perry Robertson Failure Criteria

• First Principle Approach

02

2

4

4

=+dx

yd

EI

P

dx

yd

Ey

EyEyσσ

σησσησσ −

++−

++=

2

2

)1(

2

)1(

0.80

1.00

1.20

y

Euler Failure Curve

Mean Axial Stress

Perry Robertson Failure Curve

BS 5400 Part 3 Curve A

BS 5400 Part 3 Curve B

BS 5400 Part 3 Curve C

BS 5400 Part 3 Curve D



5

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200

Slenderness Ratio

Ra

tio

σ σ σ σc / σσ σσ

y


BS 449

BS5950 Curve A

BS5950 Curve B

BS5950 Curve C


Samuel Beckett Bridge, Dublin, Ireland

6

Courtesy Santiago Calatrava



7


Strabane Footbridges, Northern Ireland

8



0.80

1.00

1.20

σσ σσy

Euler Failure Curve

Mean Axial Stress






9

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200

Slenderness Ratio

Ra

tio



BS 449

BS5950 Curve A

BS5950 Curve B

BS5950 Curve C



0.80

1.00

1.20

σσ σσy

Euler Failure Curve

Mean Axial Stress





10

0.00

0.20

0.40

0.60

0.80

0 50 100 150 200

Slenderness Ratio

Ra

tio




BS 449

BS5950 Curve A

BS5950 Curve B

BS5950 Curve C

Analysis A = ULS DL + SDL + Wind

Analysis B = ULS DL + SDL + Wind + Back-Stay Imbalance

Analysis C = ULS DL + SDL Wind + Construction Tolerance

Analysis D = ULS DL + SDL Wind + Back-Stay Imbalance + Constr. Tol.

2.5



11

0.0

0.5

1.0

1.5

2.0

-2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00

Transverse Displacement (m)

Lo

ad

Fa

cto

r

Pylon Tip - Analysis D

Pylon M12 - Analysis D

Pylon Tip - Analysis A

Pylon M12 - Analysis A

Pylon Tip - Analysis B

Pylon M12 - Analysis B

Pylon Tip - Analysis C

Pylon M12 - Analysis C

Analysis A = ULS DL + SDL + Wind

Analysis B = ULS DL + SDL + Wind + Back-Stay Imbalance

Analysis C = ULS DL + SDL Wind + Construction Tolerance

Analysis D = ULS DL + SDL Wind + Back-Stay Imbalance + Constr. Tol.

2.5



12

0.0

0.5

1.0

1.5

2.0

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20

Transverse Displacement (m)

Lo

ad

Fa

cto

r

Pylon Tip - Analysis D

Pylon M12 - Analysis D

Pylon Tip - Analysis A

Pylon M12 - Analysis A

Pylon Tip - Analysis B

Pylon M12 - Analysis B

Pylon Tip - Analysis C

Pylon M12 - Analysis C



13



14



15


Boyne Bridge, Meath / Louth, Ireland

16


Dublin Eastern Bypass, Ireland

17


Dublin Eastern Bypass, Ireland

18


Taney Bridge, Ireland

19


Taney Bridge, Ireland

20

Tower

Design


Pylon Design – Critical Loadcase & Location

21


Second Order Effects – Bending Moments

22

First order First & second orderStructure

• Implications for Taney Bridge

• Methods and Codes


Second Order Effects – Investigation

23

• Simple Cantilever Strut

• Methods and Codes

• Cable-Stay Bridge Design - Example

• Elastic theory – Closed Form Solution

• Numerical Geometric Non-Linear Analysis


Second Order Effects – Methods & Codes

24

• BS5400 Part 4

• Eurocode 2

• FIP / CEB

• Curvature Estimation Methods

50

60

BS

54

00

Sle

nd

ern

es

s D

efi

nit

ion

150

200

FIP

/ E

C2

Sle

nd

ern

es

s D

efi

nit

ion

BS5400

FIP / EC2

BS 5400 Slenderness Upper Limit Upper Slenderness Limit

FIP Upper Slenderness Limit

Taney Pylon - No Cables


Slenderness Definition – BS 5400 / EC 2 / FIP

25

0

10

20

30

40

0 20 40 60 80 100 120 140

Effective Length (m)

BS

54

00

Sle

nd

ern

es

s D

efi

nit

ion

0

50

100

FIP

/ E

C2

Sle

nd

ern

es

s D

efi

nit

ion

BS 5400 Slenderness Limit

FIP / EC2 Slenderness Limit

BS 5400 Slenderness Upper Limit Upper Slenderness Limit

for FIP Equilibrium Method

Taney Pylon


Taney Bridge – Free Standing Tower

26

• Elastic theory – closed form solution



27

EI

xM

dx

xwd )()(2

2

−=

Moment–curvature relationship:


Elastic Theory – Closed Form Solution

28

)()()( 21 xMxMxM +=Total Moment:

EI

xq

dx

xwd

EI

xQ

dx

xwd )()()()(2

2

4

4

=+Deflection Equation:

Solution: Solve for w(x)

Second Order Moment:

)()()(2 xQxwxM =


• Numerical geometric non-linear analysis



29

• Incremental load application

• Iterative techniques – equilibrium maintained


Numerical Geometric Non-Linear Analysis

30

• Stiffness revision

• Load – deformation path history

• Structural analysis packages





31

• BS5400 Part 4

• Slenderness

moment

−

+=

y

e

y

ey

ixtxh

l

h

lNhMM

0035.01

1750

2

• Eccentricity at

collapse

10/2

ueadd le ψ=



32

collapse

• Curvature

(material failure)

+=

d

Ef smyu

u

γφεψ

• Curvature reduction

(stability failure)

• Eccentricity

−

=

y

e

y

ey

addh

l

h

lhe

0035.01

1750

2

250000hle





33

• BS5400 Part 4

• Eurocode 2

Three Methods

• Numerical Non-Linear Analysis


Eurocode 2

34

• Linear Second Order Analysis - Reduced Stiffness


Linear second order analysis with reduced stiffness

• Reduced stiffness


Eurocode 2

35

• Total bending moment

• Buckling load factor Ed

B

N

N=λ

• Simplified total bending

moment

−=

1λ

λOEdEd MM





36

• BS5400 Part 4

• Eurocode 2

• FIP / CEB

• Step 1 – First Order Eccentricity at ULS ULS

ULS

ULSN

Me =

MaxRdMe

,=


FIP / CEB

37

• Step 2 – Eccentricity WatershedRd

MaxRd

RdN

Me

,=

• Step 3 – Reduced Flexural Stiffness

tyEccentriciSmallee

tyEccentricieargLee

RdULS

RdULS

⇒<

⇒≥

MaxRdd

Rdd

dMEItyEccentriciSmall

dMEIyEccentrciteargL

,180

180

=⇒

=⇒

(a)

(b)

• Step 4 – Second Order Moment

= )(

λxwNM


FIP / CEB

38

• Step 5 – Final Eccentricity Check

−=

1)(12

λ

λxwNM SdSd

Rd

ULS

SdULS e withcompared

N

Me 2+





39

• BS5400 Part 4

• Eurocode 2

• FIP / CEB


15000

20000

25000

Bendin

g M

om

ent (k

N-m

)

Slenderness = 40


Curvature Estimation Methods

40

0

5000

10000

15000

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

Curvature (m-1

)

Bendin

g M

om

ent (k

N-m

)


Simple Example – Cantilever Strut

41

• Variation of slenderness ratio


Simple Example – Cantilever Strut

42

• Low first order moment (slenderness = 26)

- varying axial load 5000kN – 35000kN

• High first order moment (slenderness = 26)

- varying axial load 5000kN – 35000kN

8

10

12

BS5400 Part 4

Moment 1st Order

Elastic Theory

FIP

Numerical NL Analysis

Eurocode 2

Curvature Method


Slenderness = 12

43

0

2

4

6

8

0 1000 2000 3000 4000 5000 6000 7000 8000

Moment (kN-m)

Colu

mn

Heig

ht

(m)

Curvature Method

25

30

35

40

Colu

mn

Heig

ht (m

)

BS5400 Part 4

Moment 1st Order

Elastic Theory

FIP


Eurocode 2


Slenderness = 40

44

0

5

10

15

20

25

0 5000 10000 15000 20000 25000

Moment (kN-m)

Colu

mn

Heig

ht (m

)

Curvature Method

Section Capacity

5

6

7

8

9

Rati

o S

ec

on

d O

rder

Mo

men

t /

Fir

st

Ord

er

Mo

men

t

FIP / Elastic Theory / Numerical NL Analysis

EC2

Curvature Method

BS5400


Summary Low First Order Moment

45

0

1

2

3

4

5

1 2 3 4 5 6 7 8 9

Buckling Factor

Rati

o S

ec

on

d O

rder

Mo

men

t /

Fir

st

Ord

er

Mo

men

t

4

5

6

Ra

tio

Se

co

nd

Ord

er

Mo

me

nt

/

Fir

st

Ord

er

Mo

men

t

FIP / Elastic Theory / Numerical NL Analysis

EC2

Curvature Method

BS5400


Summary High First Order Moment

46

0

1

2

3

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Buckling Factor

Ra

tio

Se

co

nd

Ord

er

Mo

me

nt

/

Fir

st

Ord

er

Mo

men

t

25000

30000

35000

40000

45000

Be

nd

ing

Mo

me

nt

(kN

-m)


Method Comparison

47

0

5000

10000

15000

20000

25000

0.000

0.001

0.002

0.003

0.004

0.005

Curvature (m-1

)

Be

nd

ing

Mo

me

nt

(kN

-m)

First Order Moment

BS5400 (MaterialFailure)BS5400 (Reduced)

FIP & Elastic Theory

25000

30000

35000

40000

45000

Be

nd

ing

Mo

me

nt

(kN

-m)


Method Comparison

48

0

5000

10000

15000

20000

25000

0.000

0.001

0.002

0.003

0.004

0.005

Curvature (m-1

)

Be

nd

ing

Mo

me

nt

(kN

-m)

First Order Moment



30000

35000

40000

45000

Be

nd

ing

Mo

me

nt

(kN

-m)


Method Comparison

49

0

5000

10000

15000

20000

25000

0.000

0.001

0.002

0.003

0.004

0.005

Curvature (m-1

)

Be

nd

ing

Mo

me

nt

(kN

-m)

First Order Moment



• Determine Buckling Factor, λ

• Determine First Order Eccentricity


Taney Bridge Pylon Design

50

• Determine First Order Eccentricity

• Application of Codes and Methods


Taney Bridge Pylon Design

51

Buckling Magnification

Comment Factor Factor

E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1

Flexural Stiffness (EI)

Deck Pylon


Taney Bridge Buckling Factor

52

Gross properties EST IG EST IG 13.5 1.08

EST = Youngs modulus – short term

IG = Uncracked second moment of area





Deck Pylon



53


FIP (pylon) EST IG 5.5 1.220.22ESTIG







Deck Pylon



54


FIP (pylon) EST IG 5.5 1.22

Deck Creep (φ = 2) φ = 2) φ = 2) φ = 2) 0.5EST IG 5.2 1.24

0.22ESTIG

0.22ESTIG







Deck Pylon



55



Deck Creep (φ = 2) 0.5EST IG 5.2 1.24

FIP (deck)+Creep (φ = 2)φ = 2)φ = 2)φ = 2) 3.3 1.44

0.22ESTIG

0.22ESTIG

0.06ESTIG 0.22ESTIG







Deck Pylon



56



Deck Creep (φ = 2) 0.5EST IG 5.2 1.24

FIP (deck)+Creep (φ = 2) 3.3 1.44

EIS (deck)+Creep (φ = 1.72)φ = 1.72)φ = 1.72)φ = 1.72) 5.0 1.250.48ESTIG 0.22ESTIG

0.22ESTIG

0.22ESTIG

0.06ESTIG 0.22ESTIG



EIS = Secant Flexural Stiffness

Cable-Stayed Span

Buckling Mode Shape 1

Anchor Span

LUSAS Modeller 13.3 January 15, 2003



57

Anchor Span

T IT LE:

30

35

40

45

Py

lon

He

igh

t (m

)3. Cable-Stayed Bridges - Concrete

Taney Bridge First & Second Order Moments

58

0

5

10

15

20

25

-80000

-60000

-40000

-20000

0 20000

40000

60000

80000

Bending Moment (kN-m)

Py

lon

He

igh

t (m

)

First Order Moment

BS5400

FIP

Eurocode 2


Section Capacity

2.0

2.5

3.0

Ra

tio

Se

co

nd

Ord

er

Mo

me

nt

/

Fir

st

Ord

er

Mo

me

nt

FIP / Elastic Theory / Numerical NL AnalysisEC2Curvature MethodBS5400FIP (Taney)Numerical NL Analysis (Taney)EC2 (Taney)BS5400 (Taney)


First & Second Order Moments

59

0.0

0.5

1.0

1.5

1 2 3 4 5 6 7 8

Buckling Factor

Ra

tio

Se

co

nd

Ord

er

Mo

me

nt

/

Fir

st

Ord

er

Mo

me

nt

• Low first order moment & buckling factor λ > 3

Recommendation: Use EC2 or FIPRecommendation: Use EC2 or FIP

• High first order moment & buckling factor λ > 3


First & Second Order Moments

60

• High first order moment & buckling factor λ > 3

Recommendation: Use EC2 / FIP / BS 5400Recommendation: Use EC2 / FIP / BS 5400

• Buckling factor λ < 3

Recommendation: Curvature Methods Recommendation: Curvature Methods //

Geometric & Material NonGeometric & Material Non--Linear AnalysisLinear Analysis


Second Order Effects – Extradosed Bridges

61

4. Cable-Stayed Bridges- Composite

Monastery Road Bridge, Dublin, Ireland

62


Waterford Footbridge, Ireland

63


Waterford Footbridge, Ireland

64


Narrow Water Bridge, Ireland / Northern Ireland

65



66



67


New Wear Bridge, Sunderland

68

Courtesy TECHNIKER / SPENCE

Thank You

69

Thank You

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