Cable Stayed Bridges Non Linear Effects
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Transcript of Cable Stayed Bridges Non Linear Effects
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Cable Stayed BridgesCable Stayed BridgesNon Non –– Linear EffectsLinear Effects
Tony DempseyTony Dempsey
ROUGHAN & O’DONOVANConsulting Engineers
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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
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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
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• BS 5400 Part 3: Clause 10
• First Principle Approach
2. Cable-Stayed Bridges - Steel
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(
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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
2. Cable-Stayed Bridges - Steel
Steel Pylon Design – Second Order Effects
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 5400 Part 3 Curve D
BS 449
BS5950 Curve A
BS5950 Curve B
BS5950 Curve C
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2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
6
Courtesy Santiago Calatrava
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2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
7
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2. Cable-Stayed Bridges - Steel
Strabane Footbridges, Northern Ireland
8
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2. Cable-Stayed Bridges - Steel
Steel Pylon Design – Second Order Effects
0.80
1.00
1.20
σσ σσy
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
9
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
BS 5400 Part 3 Curve D
BS 449
BS5950 Curve A
BS5950 Curve B
BS5950 Curve C
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2. Cable-Stayed Bridges - Steel
Steel Pylon Design – Second Order Effects
0.80
1.00
1.20
σσ σσy
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
10
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
BS 5400 Part 3 Curve C
BS 5400 Part 3 Curve D
BS 449
BS5950 Curve A
BS5950 Curve B
BS5950 Curve C
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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
2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
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
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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
2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
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
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2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
13
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2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
14
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2. Cable-Stayed Bridges - Steel
Samuel Beckett Bridge, Dublin, Ireland
15
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3. Cable-Stayed Bridges - Concrete
Boyne Bridge, Meath / Louth, Ireland
16
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3. Cable-Stayed Bridges - Concrete
Dublin Eastern Bypass, Ireland
17
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3. Cable-Stayed Bridges - Concrete
Dublin Eastern Bypass, Ireland
18
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3. Cable-Stayed Bridges - Concrete
Taney Bridge, Ireland
19
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3. Cable-Stayed Bridges - Concrete
Taney Bridge, Ireland
20
Tower
Design
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3. Cable-Stayed Bridges - Concrete
Pylon Design – Critical Loadcase & Location
21
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3. Cable-Stayed Bridges - Concrete
Second Order Effects – Bending Moments
22
First order First & second orderStructure
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• Implications for Taney Bridge
• Methods and Codes
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Investigation
23
• Simple Cantilever Strut
• Methods and Codes
• Cable-Stay Bridge Design - Example
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• Elastic theory – Closed Form Solution
• Numerical Geometric Non-Linear Analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
24
• BS5400 Part 4
• Eurocode 2
• FIP / CEB
• Curvature Estimation Methods
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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
3. Cable-Stayed Bridges - Concrete
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
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3. Cable-Stayed Bridges - Concrete
Taney Bridge – Free Standing Tower
26
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• Elastic theory – closed form solution
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
27
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EI
xM
dx
xwd )()(2
2
−=
Moment–curvature relationship:
3. Cable-Stayed Bridges - Concrete
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 =
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• Elastic theory – closed form solution
• Numerical geometric non-linear analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
29
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• Incremental load application
• Iterative techniques – equilibrium maintained
3. Cable-Stayed Bridges - Concrete
Numerical Geometric Non-Linear Analysis
30
• Stiffness revision
• Load – deformation path history
• Structural analysis packages
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• Elastic theory – closed form solution
• Numerical geometric non-linear analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
31
• BS5400 Part 4
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• Slenderness
moment
−
+=
y
e
y
ey
ixtxh
l
h
lNhMM
0035.01
1750
2
• Eccentricity at
collapse
10/2
ueadd le ψ=
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
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
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• Elastic theory – closed form solution
• Numerical geometric non-linear analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
33
• BS5400 Part 4
• Eurocode 2
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Three Methods
• Numerical Non-Linear Analysis
3. Cable-Stayed Bridges - Concrete
Eurocode 2
34
• Linear Second Order Analysis - Reduced Stiffness
• Curvature Estimation Methods
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Linear second order analysis with reduced stiffness
• Reduced stiffness
3. Cable-Stayed Bridges - Concrete
Eurocode 2
35
• Total bending moment
• Buckling load factor Ed
B
N
N=λ
• Simplified total bending
moment
−=
1λ
λOEdEd MM
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• Elastic theory – closed form solution
• Numerical geometric non-linear analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
36
• BS5400 Part 4
• Eurocode 2
• FIP / CEB
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• Step 1 – First Order Eccentricity at ULS ULS
ULS
ULSN
Me =
MaxRdMe
,=
3. Cable-Stayed Bridges - Concrete
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)
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• Step 4 – Second Order Moment
= )(
λxwNM
3. Cable-Stayed Bridges - Concrete
FIP / CEB
38
• Step 5 – Final Eccentricity Check
−=
1)(12
λ
λxwNM SdSd
Rd
ULS
SdULS e withcompared
N
Me 2+
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• Elastic theory – closed form solution
• Numerical geometric non-linear analysis
3. Cable-Stayed Bridges - Concrete
Second Order Effects – Methods & Codes
39
• BS5400 Part 4
• Eurocode 2
• FIP / CEB
• Curvature Estimation Methods
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15000
20000
25000
Bendin
g M
om
ent (k
N-m
)
Slenderness = 40
3. Cable-Stayed Bridges - Concrete
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
)
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3. Cable-Stayed Bridges - Concrete
Simple Example – Cantilever Strut
41
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• Variation of slenderness ratio
3. Cable-Stayed Bridges - Concrete
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
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8
10
12
BS5400 Part 4
Moment 1st Order
Elastic Theory
FIP
Numerical NL Analysis
Eurocode 2
Curvature Method
3. Cable-Stayed Bridges - Concrete
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
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25
30
35
40
Colu
mn
Heig
ht (m
)
BS5400 Part 4
Moment 1st Order
Elastic Theory
FIP
Numerical NL Analysis
Eurocode 2
3. Cable-Stayed Bridges - Concrete
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
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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
3. Cable-Stayed Bridges - Concrete
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
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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
3. Cable-Stayed Bridges - Concrete
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
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25000
30000
35000
40000
45000
Be
nd
ing
Mo
me
nt
(kN
-m)
3. Cable-Stayed Bridges - Concrete
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
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25000
30000
35000
40000
45000
Be
nd
ing
Mo
me
nt
(kN
-m)
3. Cable-Stayed Bridges - Concrete
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
BS5400 (MaterialFailure)BS5400 (Reduced)
FIP & Elastic Theory
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30000
35000
40000
45000
Be
nd
ing
Mo
me
nt
(kN
-m)
3. Cable-Stayed Bridges - Concrete
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
BS5400 (MaterialFailure)BS5400 (Reduced)
FIP & Elastic Theory
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• Determine Buckling Factor, λ
• Determine First Order Eccentricity
3. Cable-Stayed Bridges - Concrete
Taney Bridge Pylon Design
50
• Determine First Order Eccentricity
• Application of Codes and Methods
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3. Cable-Stayed Bridges - Concrete
Taney Bridge Pylon Design
51
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Buckling Magnification
Comment Factor Factor
E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1
Flexural Stiffness (EI)
Deck Pylon
3. Cable-Stayed Bridges - Concrete
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
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Buckling Magnification
Comment Factor Factor
E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1
Flexural Stiffness (EI)
Deck Pylon
3. Cable-Stayed Bridges - Concrete
Taney Bridge Buckling Factor
53
Gross properties EST IG EST IG 13.5 1.08
FIP (pylon) EST IG 5.5 1.220.22ESTIG
EST = Youngs modulus – short term
IG = Uncracked second moment of area
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Buckling Magnification
Comment Factor Factor
E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1
Flexural Stiffness (EI)
Deck Pylon
3. Cable-Stayed Bridges - Concrete
Taney Bridge Buckling Factor
54
Gross properties EST IG EST IG 13.5 1.08
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
EST = Youngs modulus – short term
IG = Uncracked second moment of area
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Buckling Magnification
Comment Factor Factor
E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1
Flexural Stiffness (EI)
Deck Pylon
3. Cable-Stayed Bridges - Concrete
Taney Bridge Buckling Factor
55
Gross properties EST IG EST IG 13.5 1.08
FIP (pylon) EST IG 5.5 1.22
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
EST = Youngs modulus – short term
IG = Uncracked second moment of area
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Buckling Magnification
Comment Factor Factor
E I E I λλλλ λ / λ−1λ / λ−1λ / λ−1λ / λ−1
Flexural Stiffness (EI)
Deck Pylon
3. Cable-Stayed Bridges - Concrete
Taney Bridge Buckling Factor
56
Gross properties EST IG EST IG 13.5 1.08
FIP (pylon) EST IG 5.5 1.22
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
EST = Youngs modulus – short term
IG = Uncracked second moment of area
EIS = Secant Flexural Stiffness
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Cable-Stayed Span
Buckling Mode Shape 1
Anchor Span
LUSAS Modeller 13.3 January 15, 2003
3. Cable-Stayed Bridges - Concrete
Taney Bridge Buckling Factor
57
Anchor Span
T IT LE:
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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
Numerical NL Analysis
Section Capacity
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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)
3. Cable-Stayed Bridges - Concrete
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
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• 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
3. Cable-Stayed Bridges - Concrete
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
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3. Cable-Stayed Bridges - Concrete
Second Order Effects – Extradosed Bridges
61
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4. Cable-Stayed Bridges- Composite
Monastery Road Bridge, Dublin, Ireland
62
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4. Cable-Stayed Bridges- Composite
Waterford Footbridge, Ireland
63
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4. Cable-Stayed Bridges- Composite
Waterford Footbridge, Ireland
64
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4. Cable-Stayed Bridges- Composite
Narrow Water Bridge, Ireland / Northern Ireland
65
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4. Cable-Stayed Bridges- Composite
Narrow Water Bridge, Ireland / Northern Ireland
66
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4. Cable-Stayed Bridges- Composite
Narrow Water Bridge, Ireland / Northern Ireland
67
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4. Cable-Stayed Bridges- Composite
New Wear Bridge, Sunderland
68
Courtesy TECHNIKER / SPENCE
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Thank You
69
Thank You