Cable Stayed Bridges Non – Linear Effects Tony Dempsey
ROUGHAN & O’DONOVAN Consulting Engineers
Presentation Layout
1. Introduction 2. Cable-Stayed Bridges - Steel Theory & Examples
3. Cable-Stayed Bridges - Concrete Theory & Examples
4. Cable-Stayed Bridges - Composite Examples
2
1. Introduction
• Cable Stayed Bridges – Non Linearity Geometric Non Linear (GNL) – Large Displacement 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 3
2. Cable-Stayed Bridges - Steel Steel Pylon Design – Second Order Effects
• BS 5400 Part 3: Clause 10 • First Principle Approach • Perry Robertson Failure Criteria d4y P d2y + =0 4 2 EI dx dx
σ=
σ y + (1 + η )σ E 2
σ y + (1 + η )σ E
−
2
2
− σ yσ E 4
2. Cable-Stayed Bridges - Steel Steel Pylon Design – Second Order Effects 1.20
Euler Failure Curve Mean Axial Stress Perry Robertson Failure Curve
1.00
BS 5400 Part 3 Curve A
Ratio σc / σy
BS 5400 Part 3 Curve B BS 5400 Part 3 Curve C BS 5400 Part 3 Curve D
0.80
BS 449 BS5950 Curve A BS5950 Curve B
0.60
BS5950 Curve C
0.40
0.20
0.00 0
50
100
150
200
Slenderness Ratio
5
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland
Courtesy Santiago Calatrava 6
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland
7
2. Cable-Stayed Bridges - Steel Strabane Footbridges, Northern Ireland
8
2. Cable-Stayed Bridges - Steel Steel Pylon Design – Second Order Effects 1.20
Euler Failure Curve Mean Axial Stress Perry Robertson Failure Curve
1.00
BS 5400 Part 3 Curve A
Ratio σc / σy
BS 5400 Part 3 Curve B BS 5400 Part 3 Curve C BS 5400 Part 3 Curve D
0.80
BS 449 BS5950 Curve A BS5950 Curve B
0.60
BS5950 Curve C
0.40
0.20
0.00 0
50
100
150
200
Slenderness Ratio
9
2. Cable-Stayed Bridges - Steel Steel Pylon Design – Second Order Effects 1.20
Euler Failure Curve Mean Axial Stress Perry Robertson Failure Curve
1.00
Ratio σc / σy
BS 5400 Part 3 Curve A BS 5400 Part 3 Curve B
0.80
BS 5400 Part 3 Curve C BS 5400 Part 3 Curve D BS 449
0.60
BS5950 Curve A BS5950 Curve B BS5950 Curve C
0.40
0.20
0.00 0
50
100
150
200
Slenderness Ratio
10
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland 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
Load Factor
2.0
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
1.5
1.0
0.5
-2.00
-1.00
0.0 0.00
1.00
2.00
3.00
4.00
Transverse Displacement (m)
11
5.00
6.00
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland 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
Load Factor
2.0
1.5
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
-0.20
-0.15
-0.10
1.0
0.5
-0.05
0.0 0.00
0.05
0.10
0.15
Transverse Displacement (m)
12
0.20
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland
13
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland
14
2. Cable-Stayed Bridges - Steel Samuel Beckett Bridge, Dublin, Ireland
15
3. Cable-Stayed Bridges - Concrete Boyne Bridge, Meath / Louth, Ireland
16
3. Cable-Stayed Bridges - Concrete Dublin Eastern Bypass, Ireland
17
3. Cable-Stayed Bridges - Concrete Dublin Eastern Bypass, Ireland
18
3. Cable-Stayed Bridges - Concrete Taney Bridge, Ireland
19
3. Cable-Stayed Bridges - Concrete Taney Bridge, Ireland
Tower Design 20
3. Cable-Stayed Bridges - Concrete Pylon Design – Critical Loadcase & Location
21
3. Cable-Stayed Bridges - Concrete Second Order Effects – Bending Moments
Structure
First order
First & second order 22
3. Cable-Stayed Bridges - Concrete Second Order Effects – Investigation
• Implications for Taney Bridge • Methods and Codes • Simple Cantilever Strut • Cable-Stay Bridge Design - Example
23
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – Closed Form Solution • Numerical Geometric Non-Linear Analysis • BS5400 Part 4 • Eurocode 2 • FIP / CEB • Curvature Estimation Methods 24
3. Cable-Stayed Bridges - Concrete Slenderness Definition – BS 5400 / EC 2 / FIP
200
Taney Pylon - No Cables
50
Upper Slenderness Limit for FIP Equilibrium Method
BS 5400 Slenderness Upper Limit
40
30
150
100
Taney Pylon 20
BS 5400 Slenderness Limit
50
FIP / EC2 Slenderness Limit
10
0 0
20
40
60
80
100
120
Effective Length (m)
25
0 140
FIP / EC2 Slenderness Definition
BS5400 Slenderness Definition
FIP Upper Slenderness Limit
BS5400 FIP / EC2
60
3. Cable-Stayed Bridges - Concrete Taney Bridge – Free Standing Tower
26
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution
27
3. Cable-Stayed Bridges - Concrete Elastic Theory – Closed Form Solution Moment–curvature relationship: d 2 w( x ) M ( x) = − dx 2 EI
Total Moment:
M ( x) = M 1 ( x) + M 2 ( x)
Deflection Equation:
d 4 w( x ) Q ( x ) d 2 w( x ) q( x ) + = dx 4 EI EI dx 2
Solution:
Solve for w(x)
Second Order Moment:
M 2 ( x ) = w( x )Q ( x ) 28
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution • Numerical geometric non-linear analysis
29
3. Cable-Stayed Bridges - Concrete Numerical Geometric Non-Linear Analysis • Incremental load application • Iterative techniques – equilibrium maintained • Stiffness revision • Load – deformation path history • Structural analysis packages
30
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution • Numerical geometric non-linear analysis • BS5400 Part 4
31
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Slenderness moment
Nhy le M tx = M ix + 1750 h y
2
0.0035le 1 − h y
• Eccentricity at collapse
eadd = le2ψ u / 10
• Curvature (material failure)
φε u + f y γ m E s ψ u = d
• Curvature reduction (stability failure)
le 50000h 2
• Eccentricity
eadd
h y le = 1750 h y
2
0.0035l e 1 − hy 32
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution • Numerical geometric non-linear analysis • BS5400 Part 4 • Eurocode 2
33
3. Cable-Stayed Bridges - Concrete Eurocode 2 Three Methods • Numerical Non-Linear Analysis • Linear Second Order Analysis - Reduced Stiffness • Curvature Estimation Methods
34
3. Cable-Stayed Bridges - Concrete Eurocode 2 Linear second order analysis with reduced stiffness • Reduced stiffness • Total bending moment NB N Ed
• Buckling load factor
λ=
• Simplified total bending moment
λ M Ed = M OEd λ − 1
35
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution • Numerical geometric non-linear analysis • BS5400 Part 4 • Eurocode 2 • FIP / CEB
36
3. Cable-Stayed Bridges - Concrete FIP / CEB
• Step 1 – First Order Eccentricity at ULS
eULS =
• Step 2 – Eccentricity Watershed
eRd =
• Step 3 – Reduced Flexural Stiffness eULS ≥ eRd
(a) e < e ULS Rd (b) Large Small
⇒ Large
Eccentricity
⇒ Small
Eccentricity
Eccentrcit y ⇒ EI d = 180dM Rd Eccentrici ty ⇒ EI d = 180dM Rd ,Max 37
M ULS N ULS
M Rd ,Max N Rd
3. Cable-Stayed Bridges - Concrete FIP / CEB
• Step 4 – Second Order Moment λ M Sd 2 = N Sd w1 ( x) λ − 1
• Step 5 – Final Eccentricity Check eULS +
M Sd 2 compared with eRd NULS
38
3. Cable-Stayed Bridges - Concrete Second Order Effects – Methods & Codes • Elastic theory – closed form solution • Numerical geometric non-linear analysis • BS5400 Part 4 • Eurocode 2 • FIP / CEB • Curvature Estimation Methods 39
3. Cable-Stayed Bridges - Concrete Curvature Estimation Methods 25000
Bending Moment (kN-m)
20000
Slenderness = 40 15000
10000
5000
0
40
9 00 0.
8 00 0.
7 00 0.
6 00 0.
5 00 0.
4 00 0.
3 00 0.
2 00 0.
1 00 0.
0 00 0.
-1
Curv ature (m )
3. Cable-Stayed Bridges - Concrete Simple Example – Cantilever Strut
41
3. Cable-Stayed Bridges - Concrete Simple Example – Cantilever Strut
• Variation of slenderness ratio • Low first order moment (slenderness = 26) - varying axial load 5000kN – 35000kN • High first order moment (slenderness = 26) - varying axial load 5000kN – 35000kN
42
3. Cable-Stayed Bridges - Concrete Slenderness = 12 12
BS5400 Part 4 Moment 1st Order Elastic Theory FIP Numerical NL Analysis Eurocode 2 Curvature Method
10
Column Height (m)
8
6
4
2
0 0
1000
2000
3000
4000
5000
M ome nt (kN-m)
6000
7000 43
8000
3. Cable-Stayed Bridges - Concrete Slenderness = 40 40
BS5400 Part 4 Moment 1st Order
35
Elastic Theory FIP
30
Column Height (m)
Numerical NL Analysis Eurocode 2
25
Curvature Method Section Capacity
20
15
10
5
0 0
5000
10000
15000
20000
25000
M ome nt (kN-m)
44
3. Cable-Stayed Bridges - Concrete Summary Low First Order Moment Ratio Second Order Moment / First Order Moment
9 FIP / Elastic Theory / Numerical NL Analysis
8
EC2 Curvature Method
7
BS5400 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
Buckling Factor
45
9
3. Cable-Stayed Bridges - Concrete Summary High First Order Moment Ratio Second Order Moment / First Order Moment
6 FIP / Elastic Theory / Numerical NL Analysis EC2 Curvature Method BS5400
5 4 3 2 1 0 1
1.5
2
2.5
3
3.5
4
4.5
5
Buckling Factor
46
5.5
3. Cable-Stayed Bridges - Concrete Method Comparison 45000
Bending Moment (kN-m)
40000 35000 30000 25000 20000
First Order Moment
15000
BS5400 (Material Failure) BS5400 (Reduced)
10000 5000
FIP & Elastic Theory
0
5
4
3
2
1
0
47
00 0.
00 0.
00 0.
00 0.
00 0.
00 0.
-1
Curvature (m )
3. Cable-Stayed Bridges - Concrete Method Comparison 45000
Bending Moment (kN-m)
40000 35000 30000 25000 20000
First Order Moment
15000
BS5400 (Material Failure) BS5400 (Reduced)
10000 5000
FIP & Elastic Theory
0
5
4
3
2
1
0
48
00 0.
00 0.
00 0.
00 0.
00 0.
00 0.
-1
Curvature (m )
3. Cable-Stayed Bridges - Concrete Method Comparison 45000
Bending Moment (kN-m)
40000 35000 30000 25000 20000
First Order Moment
15000
BS5400 (Material Failure) BS5400 (Reduced)
10000 5000
FIP & Elastic Theory
0
49
5 00 0.
4 00 0.
3 00 0.
2 00 0.
1 00 0.
0 00 0.
-1
Curvature (m )
3. Cable-Stayed Bridges - Concrete Taney Bridge Pylon Design • Determine Buckling Factor, λ • Determine First Order Eccentricity • Application of Codes and Methods
50
3. Cable-Stayed Bridges - Concrete Taney Bridge Pylon Design
51
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor
Comment
Gross properties
Flexural Stiffness (EI) Deck Pylon E I E EST
IG
EST
I
Buckling Magnification Factor Factor λ λ / λ−1
IG
13.5
EST
= Youngs modulus – short term
IG
= Uncracked second moment of area
52
1.08
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor
Flexural Stiffness (EI) Deck Pylon E I E
I
Gross properties
EST
IG
IG
FIP (pylon)
EST
IG
Comment
EST
Buckling Magnification Factor Factor λ λ / λ−1
0.22ESTIG
13.5
1.08
5.5
1.22
EST
= Youngs modulus – short term
IG
= Uncracked second moment of area
53
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor
Flexural Stiffness (EI) Deck Pylon E I E
I
Gross properties
EST
IG
IG
FIP (pylon) φ = 2) Deck Creep (φ
EST
IG
0.5EST
IG
Comment
EST
Buckling Magnification Factor Factor λ λ / λ−1 13.5
1.08
0.22ESTIG
5.5
1.22
0.22ESTIG
5.2
1.24
EST
= Youngs modulus – short term
IG
= Uncracked second moment of area
54
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor
Flexural Stiffness (EI) Deck Pylon E I E
I
Gross properties
EST
IG
IG
FIP (pylon) Deck Creep (φ = 2)
EST
IG
0.5EST
IG
Comment
φ = 2) FIP (deck)+Creep (φ
0.06ESTIG
EST
Buckling Magnification Factor Factor λ λ / λ −1 13.5
1.08
0.22ESTIG
5.5
1.22
0.22ESTIG
5.2
1.24
0.22ESTIG
3.3
1.44
EST
= Youngs modulus – short term
IG
= Uncracked second moment of area
55
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor
Flexural Stiffness (EI) Deck Pylon E I E
I
Gross properties
EST
IG
IG
FIP (pylon) Deck Creep (φ = 2)
EST
IG
0.5EST
IG
Comment
EST
Buckling Magnification Factor Factor λ λ / λ−1 13.5
1.08
0.22ESTIG
5.5
1.22
0.22ESTIG
5.2
1.24
FIP (deck)+Creep (φ = 2)
0.06ESTIG
0.22ESTIG
3.3
1.44
φ = 1.72) EIS (deck)+Creep (φ
0.48ESTIG
0.22ESTIG
5.0
1.25
EST
= Youngs modulus – short term
IG
= Uncracked second moment of area
EIS
= Secant Flexural Stiffness 56
3. Cable-Stayed Bridges - Concrete Taney Bridge Buckling Factor LU SAS Modeller 13.3
January 15, 2003
Buckling Mode Shape 1
Cable-Stayed Span
Anchor Span
T IT LE:
57
3. Cable-Stayed Bridges - Concrete Taney Bridge First & Second Order Moments 45 40 35
Pylon Height (m)
30 25 20 15
First Order Moment BS5400
10
FIP Eurocode 2 Numerical NL Analysis
5
Section Capacity
0
58
0 00 80
0 00 60
0 00 40
0 00 20
0
00 00 -2
00 00 -4
00 00 -6
00 00 -8
Bending Moment (kN-m)
3. Cable-Stayed Bridges - Concrete First & Second Order Moments FIP / Elastic Theory / Numerical NL Analysis EC2 Curvature Method BS5400 FIP (Taney) Numerical NL Analysis (Taney) EC2 (Taney) BS5400 (Taney)
Ratio Second Order Moment / First Order Moment
3.0
2.5
2.0
1.5
1.0
0.5
0.0 1
2
3
4
5
6
7
Buckling Factor
59
8
3. Cable-Stayed Bridges - Concrete First & Second Order Moments • Low first order moment & buckling factor λ > 3 Recommendation: Use EC2 or FIP • High first order moment & buckling factor λ > 3 Recommendation: Use EC2 / FIP / BS 5400 • Buckling factor λ < 3 Recommendation: Curvature Methods / Geometric & Material NonNon-Linear Analysis
60
3. Cable-Stayed Bridges - Concrete Second Order Effects – Extradosed Bridges
61
4. Cable-Stayed Bridges- Composite Monastery Road Bridge, Dublin, Ireland
62
4. Cable-Stayed Bridges- Composite Waterford Footbridge, Ireland
63
4. Cable-Stayed Bridges- Composite Waterford Footbridge, Ireland
64
4. Cable-Stayed Bridges- Composite Narrow Water Bridge, Ireland / Northern Ireland
65
4. Cable-Stayed Bridges- Composite Narrow Water Bridge, Ireland / Northern Ireland
66
4. Cable-Stayed Bridges- Composite Narrow Water Bridge, Ireland / Northern Ireland
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4. Cable-Stayed Bridges- Composite New Wear Bridge, Sunderland
Courtesy TECHNIKER / SPENCE 68
Thank You
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