2021-08 structural optimization of superstructure
TRANSCRIPT
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Geotechnical Engineering Thesis
2021-08
STRUCTURAL OPTIMIZATION OF
SUPERSTRUCTURE PARAMETER OF
EXTRADOSED CABLE STAYED
BRIDGE USING GENETIC
ALGORITHM, As A CASE STUDY ON
ABAY RIVER BRIDGE
ABEBA, LAMESGIN SIMEGN
http://ir.bdu.edu.et/handle/123456789/12573
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BAHIR DAR UNIVERSITY
BAHIR DAR INSTITUTE OF TECHNOLOGY
SCHOOL OF RESEARCH AND POSTGRADUATE STUDIES
FACULTY OF CIVIL AND WATER RESOURCE ENGINEERING
STRUCTURAL ENGINEERING
MSc. Thesis
On
STRUCTURAL OPTIMIZATION OF SUPERSTRUCTURE
PARAMETER OF EXTRADOSED CABLE STAYED BRIDGE
USING GENETIC ALGORITHM, As A CASE STUDY ON ABAY
RIVER BRIDGE
By
ABEBA LAMESGIN SIMEGN
Aug. 2021
Bahir Dar, Ethiopia
i
BAHIR DAR UNIVERSITY
BAHIR DAR INSTITUTE OF TECHNOLOGY
FACULTY OF CIVIL AND WATER RESOURCES ENGINEERING
Structural Optimization of Superstructure Parameter of Extradosed
Cable- Stayed Bridge Using Genetic Algorithm, As A Case Study On
Abay River Bridge
By
Abeba Lamesgin Simegn
A thesis submitted
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Structural Engineering
Advisor: Eng. Ghulam Rasool
Co-Advisor: Eng. Wubishet Jemanenh
Aug. 2021
Bahir Dar, Ethiopia
©2021 Abeba Lamesgin Simegn
iii
DECLARATION
This is to certify that the thesis entitled ―Structural Optimization of Superstructure
Parameter of Extradosed Cable-Stayed Bridge using Genetic Algorithm, as a Case
study on Abay River Bridge” submitted in partial fulfillment of the requirements for
the degree of Master of Science in Structural Engineering under Faculty of Civil and
Water Resources Engineering, Bahir Dar Institute of Technology , is a record of
original work carried out by me and has never been submitted to this or any other
institution to get any other degree or certificates. The assistance and help I received
during the course of this investigation have been duly acknowledged.
Abeba Lamesgin Simegn ______ 03/08/2021
Name of the Candidate Signature Date
iv
BAHIR DAR UNIVERSITY
BAHIR DAR INSTITUTE OF TECHNOLOGY-
SCHOOL OF RESEARCH AND GRADUATE STUDIES
FACULTY OF CIVIL AND WATER RESOURCE ENGINEERING
APPROVAL OF THESIS FOR DEFENSE
I hereby certify that I have supervised, read, and evaluated this thesis titled ―Structural
Optimization of Superstructure Parameter of Extradosed Cable-Stayed Bridge
using Genetic Algorithm, as a Case study on Abay River Bridge prepared by Abeba
Lamesgin under my guidance. I recommend the submission of the thesis for oral
defense.
Ghulam Rasool (Eng) 22/06/2021
Advisor’s name Signature Date
vii
ACKNOWLEDGEMENTS
First of all, I would like to thank ―The Almighty‖ for giving me the time, courage,
patience, initiation, and determination of doing the research.
I would like to express my thankfulness to structural engineering department head Mr.
Alemayehu Golla for his commitment, valuable guidance, and willingness to share his
knowledge from beginning of research up to end of the research work.
I would like to express my gratitude to my thesis supervisors Eng. Ghulam Rasool, Eng
Fkre silassie worku and Eng. Wubeshet Jemaneh for their proper guidance, invaluable
advice and support throughout this research period.
I would also like to express my appreciation to Botek Head of Structural Department of
Amir Reza Poorbakhshaie, and all structural department members for their willingness
to answer questions for any aspect of conventional design work of Abay bridge.
I would also like to express my thanks to Dr. Hanibal Lemma and Dr. Abrham Gebre
who have helped me through my thesis work.
Also, I would like to express my appreciation to ―China Communication Construction
Company‖ for their willingness to giving any aspect of conventional design output Abay
bridge.
A special thanks to my family, outstandingly to my uncle, my mother, and my father, for
all their unconditional support.
viii
ABSTRACT
Weight of post-tensioning prestressing extradosed cable-stay bridge superstructure part
formed from concrete, non-prestressing reinforcement, and prestressing reinforcement
and stay cable tendon weight. Main load-carrying members, which are, girder, stayed-
cable, and pylon are considered to apply structural optimization techniques in the
superstructure component of the extradosed cable-stayed bridge. This research paper
evaluated the optimum depth of the girder with pylon height, angle of stay cable, and
effect of concrete grade in the three main parameters of an extradosed cable-stay bridge.
Then identification of significant and insignificant design variables using sensitivity
analysis by considering cable stiffness, girder weight, and stay cable tension. Structural
optimization was carried out by taking the minimization of the total material weight of
girders, pylon, and angle of stayed-cable as an objective function and all requirements of
strength, stability, serviceability, and fatigue as constraint functions. As a case study
Abay‘s extradosed cable-stayed bridge first design by China Communication
Construction Company. It has two twin box girders with 24.7m width and a length of
380m. The width of the top of the box girder is 24.7m, and the width of the bottom plate
gradually changes from 9m to the end fulcrum. The main tower adopts a double-column
tower, the tower beam is consolidated. The tower root size is 4m x2m (lateral), and the
tower top dimension is 3mx2m as respectively. The bridge is subject to five main load
cases, dead, live, wind, settlement, and temperature loads. This paper gives an optimum
cross-section of the superstructure main component of the Abay bridge by using the
fixed load parameter that has already been defined by the designer company. Effects of
girder depth, angle of cable-stayed, and pylon height with the effect of concrete grade on
the optimum weight were investigated. The results of structural optimization indicate
that optimum girder depth is 5.129m at pier level and 2.62m at span. The optimum pylon
height was found to be 24.827m. And optimum stay cable length was reduced from
conventional design output length by 10% from total values. Optimum design of pylon
height reduced weight of conventional design by 20.03% and optimum design of box
girder reduced girder weight by 19.47%. Paper gives the more reduced weight of bridge
by 49.47% from conventional design output by using same material grade.
Keywords: Box girder depth, Height of Pylon, Stay cable angle, and Concrete
grade, Sensitivity analysis
ix
ABBREVIATIONS
AASHTO America Association of State Highway and Transportation Officials
ASTM American Society for Testing and Materials,
ERA Ethiopia Road Authority
FEM Finite Element Method
GA Genetic Algorithm
SLS Serviceability limit state
ULS Ultimate limit state
CCCC China Communication Construction Company
AWS American Welding Society,
IM Dynamic load allowance or impact factor
M Multiple presence factors
LRFD load resistance factor design
x
NOTATIONS
cp- Tensile strain in the concrete at the level of the tendon at decompression stage.
o -Compressive strain at the extreme top fiber at service load stage
oc- Compressive strain in the concrete at the level of the tendon
s - Tensile strain in the reinforcing steel at working loads
A - Cross-sectional area of concrete (mm2 )
a - Depth of equivalent rectangular stress block (mm)
a‘- Distance from the left support to the point of truckload for which deflection is to be
computed.
Ac - Area of concrete cross-section (mm2 )
Act - Area of the cracked transformed section under service limit state (mm2
)
At - Effective tension area of concrete surrounding one bar (mm2)
Ap- Area of prestressing steel (mm2 )
As- Area of non-prestressed steel tension reinforcement (mm2 )
As‘- Area of non-prestressed steel compression zone reinforcement (mm2 )
Av- Cross-sectional area of shear reinforcement within a distance S (mm2)
be -Width of compression face of the section of exterior girder (mm)
bi -Width of compression face of the section of interior girder (mm)
bw -Web width of the cross-section (mm)
C -Resultant compressive force in the compression zone of concrete (N)
c - Depth of the neutral axis (mm)
Cn - Compressive force in the compression zone of concrete used to reduce the resultant
Compressive force C when NA depth exceeds flange thickness (N)
Wp - Weight of prestressing Reinforcement steel (Kg/m3)
Ws - Weight of non-prestressing Reinforcement steel (Kg/m3)
d- Distance from extreme compression fiber to centroid of non-prestressed tension
xi
reinforcement (mm)
dc -Thickness of concrete cover measured from extreme tension fiber to centroid of the
closest bar (mm)
de - Depth from extreme compression fiber to centroid of tensile force (mm)
dp - Depth from extreme compression fiber to centroid of prestressing steel (mm)
ds‘- Distance from extreme compression fiber to centroid of non-prestressed compression
zone reinforcement (mm)
dv - Effective depth of shearing force (N)
dz-Depth from extreme compression fiber to centroid of resultant compression
force(mm)
dzn -Depth from extreme compression fiber to centroid of compression force Cn (mm)
e - Eccentricity of prestressing force from the centroid of the section (mm)
Ec- Modulus of elastic of concrete (N/mm2 )
Ep- Modulus of elastic of prestressing steel (N/mm2 )
Es Modulus of elastic of reinforcing steel (N/mm2 )
fbr Stress range at the extreme bottom fiber(N/mm2 )
fc‘ Specified cylindrical compressive strength of concrete (N/mm2 )
fcpe -Compressive stress in concrete due to effective pre-stress forces only (N/mm2 )
fct - Maximum allowable compressive stress in concrete at initial pre-stress (N/mm2 )
fcw -Maximum allowable compressive stress in concrete at service load (N/mm2 )
ffp - Stress range in prestressing steel due to fatigue load (N/mm2 )
ffs - Stress range in reinforcing steel due to fatigue load (N/mm2 )
finf - Stress at the extreme bottom fiber for a given eccentricity e (N/mm2 )
fmin- Minimum live load stress where there is stress reversal (N/mm2 )
fp - Total stress in prestressing tendons at the application of service loads (N/mm2 )
fpe- Effective stress in prestressing steel (N/mm2 )
xii
fps - Average stress in prestressing steel (N/mm2 )
fpu - Ultimate tensile strength of prestressing steel (N/mm2
)
fpy- Yield strength of prestressing steel (N/mm2 )
fr - Modulus of rupture (N/mm2 )
fs Stress in steel reinforcement at the application of service loads (N/mm2 )
ftr Stress range at the extreme top fiber (N/mm2 )
ftt -Maximum allowable tensile stress in concrete at initial prestress (N/mm2 )
ftw - Maximum allowable tensile stress in concrete at service load (N/mm2 )
Fx - Forces acting in the horizontal direction (N)
Fy - Yield strength of non-prestressed steel tension reinforcement (N/mm2 )
fy‘ -Yield strength of non-prestressed steel compression zone reinforcement (N/mm2 )
gs - Girder spacing (mm)
h - Height of the deformation (mm)
h - Prestress loss factor
h - Overall depth of the section (mm)
h1 - Distance from the centroid of tensile steel to NA depth (mm)
h2 - Depth from extreme compression fiber to the depth of NA (mm)
hf - Thickness of the flange (mm)
I - Second moment of area or moment of inertia of concrete cross-section (mm4 )
Ict - Moment of inertia of cracked transformed section under service limit state (mm4
)
Ie - Effective moment of inertia of the section (mm4 )
L- Span length of the girder (mm)
M3- Working moment at service limit state III (Nmm)
Mcr- Cracking moment (Nmm)
Md- Ultimate factored design moment due to all loads (Nmm)
Mf - Maximum fatigue load moment (Nmm)
xiii
Mg - Total un-factored dead load moment (Nmm)
Mmin - Minimum moment due to self-weight or during handling of the member (Nmm)
Mn -Nominal moment of resistance (Nmm)
Mr -Total factored moment of resistance of the section (Nmm)
Mw- Working moment at service limit state I (Nmm)
np- Modular ratio of prestressing steel
ns- Modular ratio of reinforcing steel P Prestressing force (N)
r -Base radius of the deformation (mm) and S Spacing of stirrups (mm)
Tp -Tension force in the prestressing steel at service limit state (N)
Ts -Tension force in the reinforcing steel at service limit state (N)
Vc -Shear resisting force due to tensile stress in the concrete (N)
Vn -Nominal shear resistance (N)
Vp - Component of prestressing force in the direction of shearing force (N)
Vs- Shear resisting force due to tensile stress in traverse reinforcement (N)
Vu- Factored design shearing forced distance from the face of support (N)
who - Width of overhang (mm)
Wstr- Weight of stirrups (g)
x -Distance from left support to a point at which maximum service load moment occurs.
y - NA depth of the cracked section under service limit state (mm)
yb- Depth from extreme bottom fiber to centroid of the section (mm)
yct- Depth from extreme compression fiber to centroid of cracked section (mm)
yt - Depth from extreme top fiber to centroid of the section (mm)
Zb- Section modulus of the extreme bottom fiber (mm3
)
Zc -Section modulus for the extreme fiber of the composite section where tensile stress is
caused by externally applied loads (mm3
)
Znc- Section modulus for the extreme fiber of monolithic or non-composite section
xiv
where tensile stress is caused by externally applied loads (mm3 ) that is Zb
Zt- Section modulus of the extreme top fiber (mm3).
Δall - Allowable deflection for the live load (mm)
Δd - Total long term deflection due to dead load (mm)
Δdi- Immediate deflection due to dead load (mm)
Δkl - Deflection due to truckload (mm)
ΔLL- Deflection due to living load (mm)
ΔLn - Deflection due to design lane load (mm)
Δp - Upward deflection due to prestressing force (mm)
Φ- Resistance factor
1 - Stress block factor s Density of reinforcement steel
κ - Correction factor for closely spaced anchorages
aeff -Lateral dimension of the effective bearing area measured parallel to the larger
dimension of the cross-section (mm)
beff- Lateral dimension of the effective bearing area measured parallel to the smaller
dimension of the cross-section (mm)
Pw - Width of bearing plate or pad (mm)
L- Length of bearing pad (mm)
de -Effective depth from extreme compression fiber to centroid of tensile force (mm)
t - Member thickness (mm)
s - Center-to-center spacing of anchorages (mm)
n - Number of anchorages in a row
ℓc -Longitudinal extent of confining reinforcement of the local zone but not more than
the larger of 1.15 aeff or 1.15 beff (mm)
Ag - Gross area of the bearing plate calculated following the requirements herein (mm2 )
xv
Ab - Effective net area of the bearing plate calculated as the area Ag, minus the area of
openings in the bearing plate (mm2 )
f ′ci - Nominal concrete strength at the time of application of tendon force (Mpa)
A - Maximum area of the portion of the supporting surface that is similar to the loaded
area and concentric with it and does not overlap similar areas for adjacent anchorage
devices (mm2 )
Tburst -Tensile force in the anchorage zone acting ahead of the anchorage device and
transverse to the tendon axis (N)
Pu -Factored tendon force (N)
dburst -Distance from anchorage device to the centroid of the bursting force, Tburst(mm)
a - Lateral dimension of the anchorage device or group of devices in the direction
considered (mm)
e -Eccentricity of the anchorage device or group of devices concerning the centroid of
the cross-section; always taken as positive (mm)
h- Lateral dimension of the cross-section in the direction considered (mm)
α - Angle of inclination of a tendon force concerning the centerline of the member;
positive for concentric tendons or if the anchor force points toward the centroid of the
section; negative if the anchor force points away from the centroid of the section.
1 - The basic partial coefficient for steel for the fatigue test of the stay cables
2 - The partial coefficient taking into account the effect of grouping (fatigue tests carried
out on separated wires or strands or the full size of the stay cable)
3 - The partial coefficient taking into account the conversion of the fatigue test values
into characteristic values
Fi -Required re-stressing or de-stressing force for the stay cable no. i
Fi-∞-0 -Change of the stay cable force between the time infinity and the time of the
construction completion
.Li - Shortening length of stay cable no. i
Mi - Change of the bending moment at the anchorage point of stay cable no. i or at the
xvi
intermediate support of a continuous beam
i - Change of vertical deflection at the anchorage point of stay cable no. i
11- Vertical deflections for construction method no. 1 in the first stage
41- Vertical deflections for construction method no. 4 in the first stage
12- Vertical deflections for construction method no. 1 in the second stage
42 - Vertical deflections for construction method no. 4 in the second stage
m1 -Vertical deflection at point m1 due to the combination of the dead load and the
stay cable forces
s1 -Vertical deflection at point s1 due to the combination of the dead load and stay
cable forces
per - Permissible stress variation
L -Stress variation due to living load
-Test Stress variation considered in the fatigue test
, el - Strain(mm) and Elastic strain(mm) respectively
C-cr - Strain of concrete due to the creep effect
C-sh - Strain of concrete due to the shrinkage effect
- Ratio between the side span and main span lengths
I - Angle of the stay cable no. i with the horizontal line
- Ratio between the uniform live and dead loads acting on the deck L Partial length of
the deck
- Allowable stress in the stay cable under SLS loads
w - Axial stress in the stay cable due to dead load
q - Axial stress in the stay cable due to living load
UTS - Ultimate tensile stress of the stay cable material
I - Creep coefficient of the beam elements in span no. i
xvii
TABLE OF CONTENTS
DECLARATION ......................................................................................................... iii
APPROVAL OF THESIS FOR DEFENSE ................................................................. iv
DEDICATION .............................................................................................................. vi
ACKNOWLEDGEMENTS ......................................................................................... vii
ABBREVIATIONS ...................................................................................................... ix
LIST OF FIGURES .................................................................................................... xxi
LIST OF TABLES .................................................................................................... xxiv
1. INTRODUCTION .................................................................................................. 1
Background ............................................................................................... 1 1.1
Problem Statement .................................................................................... 2 1.2
Objective of Study .................................................................................... 3 1.3
1.3.1 General Objective ..................................................................................... 3
1.3.2 Specific Objectives ................................................................................... 3
Significance of the Study .......................................................................... 3 1.4
Scope and limitation of the study.............................................................. 4 1.5
2. LITERATURE REVIEW ....................................................................................... 5
2.1 Extradosed cable-stayed bridge ................................................................ 5
2.1.1 Advantages of extradosed cable-stayed bridge ......................................... 5
2.1.2 The disadvantage of extradosed cable-stayed bridge ................................ 6
2.2 Structural Optimization Techniques ......................................................... 6
2.2.1 Linear programming ................................................................................. 7
2.2.2 Nonlinear Programming............................................................................ 7
2.3 Forms of Structural Optimization ............................................................. 8
2.3.1 Shape Optimization ................................................................................... 8
2.3.2 Size optimization. ..................................................................................... 8
2.3.3 Topology Optimization ............................................................................. 8
xviii
2.4 Genetic Algorithm .................................................................................... 9
2.4.1 The major advantage of Genetic Algorithm ............................................. 9
2.4.2 The major disadvantage of Genetic Algorithm ......................................... 9
2.4.3 Application of Genetic Algorithm .......................................................... 10
2.4.4 Sensitivity analysis for design variable................................................... 10
2.5 Optimization Problem Formulation ........................................................ 11
2.6 Objective Function .................................................................................. 11
2.7 Design Variables ..................................................................................... 11
2.8 Design Constraints .................................................................................. 11
2.8.1 Stayed cable force design constraint values............................................ 12
2.8.2 Cast-in-Place Post-Tensioned Concrete Box Girder............................... 31
2.8.3 Pylon design ............................................................................................ 60
2.8.4 Need of optimization............................................................................... 67
3. METHODOLOGY ............................................................................................... 68
3.1 General .................................................................................................... 68
3.2 Method of Structural Analysis ................................................................ 68
3.3 Method of Design Optimization ............................................................. 68
3.4 Materials Used ........................................................................................ 69
3.4.1 Prestressed Reinforcement ...................................................................... 70
3.4.2 Reinforcement ......................................................................................... 70
3.5 Optimization Procedure with GA in Matlab ........................................... 71
3.5.1 Running the model .................................................................................. 75
3.6 Study Variables ....................................................................................... 75
3.6.1 Independent variables ............................................................................. 75
3.6.2 Dependent variables ................................................................................ 76
3.7 Load Analysis ......................................................................................... 76
3.7.1 Load Cases and Load Combinations ....................................................... 76
xix
3.7.2 Design Truck Load ................................................................................. 77
3.7.3 Braking force .......................................................................................... 77
3.7.4 Wind Load .............................................................................................. 77
3.7.5 Load Combination and Load Factors ...................................................... 88
3.8 Design Philosophy .................................................................................. 89
3.9 Optimization Problem Formulation ........................................................ 89
3.10 Numerical Model .................................................................................... 90
3.10.1 Numerical modeling of prestressing box girder extradosed cable-
stayed Bridge ........................................................................................................ 90
3.10.2 Numerical modeling of stay cable extradosed stayed- Cable
Bridge..……. ........................................................................................................ 91
3.10.3 Numerical modeling of pylon parameter of extradosed cable-stayed
Bridge………… ................................................................................................... 92
3.11 Fixed Design Variables ........................................................................... 93
3.12 Design Variables ..................................................................................... 93
3.13 Objective Function .................................................................................. 94
3.14 Constraint Function ................................................................................. 95
3.15 Sensitivity Analysis of Optimum Section of Extradosed Cable Stay
Bridge………….. ..................................................................................................... 95
4. RESULTS AND DISCUSSIONS ........................................................................ 96
4.1 Effect of depth of box girder of on optimum weight of extradosed cable-
stayed bridge ............................................................................................................ 96
4.2 Effect of Concrete Grades of on optimum the weight bridge and the
depth of girder of extradosed cable-stayed bridge ................................................... 97
4.3 Effect of the unit cost of Concrete grade on the optimum weight of the
bridge…… .............................................................................................................. 100
4.4 The optimum height of the pylon.......................................................... 102
4.5 Effect of concrete grade on the optimum height of the pylon .............. 103
xx
4.6 Optimum angle of Cables Stay ............................................................. 104
4.7 Effect of concrete grade on the angle of the stayed cable ..................... 106
4.8 Comparison of the conventional and optimal design approach ............ 106
4.8.1 Comparison of conventional versus optimal design of box girder ....... 107
4.8.2 Comparison of conventional and optimal design of pylon ................... 109
4.8.3 Comparison of Conventional versus optimal design of cable .............. 110
4.9 Analysis of parametric sensitivity ......................................................... 112
4.9.1 Parametric sensitivity of girder weight. ................................................ 113
4.9.2 Parametric sensitivity of cable stiffness. ............................................... 118
4.9.3 Parametric sensitivity of cable tension ................................................. 121
5 CONCLUSION AND RECOMMENDATION ................................................. 124
5.1 Conclusion ............................................................................................ 124
5.2 Recommendation .................................................................................. 125
REFERENCE ............................................................................................................. 126
APPENDIX ................................................................................................................ 130
Annex 1 Extradosed Cable-Stayed Bridge structural modeling............................. 130
Annex 2 Design calculation process of the pylon .................................................. 130
Annex 3 Design Optimization Code using GA in Matlab for angle of stay cables 130
Annex 4 Design Optimization Code using GA in Matlab for box girder .............. 140
Annex 5 Design Optimization Code Using GA in Matlab for Pylon of Extradosed
Bridge ..................................................................................................................... 155
Appendix 6 Design Optimization Validation in Excel spreadsheet for stay cable of
extradosed cable-stayed bridge .............................................................................. 161
Appendix 7 Design Optimization Validation in Excel spreadsheet for box girder for
extradosed cable-stayed bridge .............................................................................. 163
xxi
LIST OF FIGURES
Figure 2.8-1 an inclined stay-cable layout and its component (Chen & Duan, 2000b)
...................................................................................................................................... 13
Figure 2.8-2 Determination of cable force corresponding to a specified distribution of
dead load moments adopted (Niels J.Gimsing, 2012) ................................................. 15
Figure 2.8-3 extradosed bridge with a deck of variable depth under the variable dead
load W(x) and the equivalent vertical stay cable forces wc (Dipl.-Ing. & Ägypten,
2013). ........................................................................................................................... 16
Figure 2.8-4Dead load and the moment of inertia of the deck cross-section change to
constant along the entire length of the deck (Dipl.-Ing. & Ägypten, 2013) ................ 16
Figure 2.8-5 dead load moment distribution(Dipl.-Ing. & Ägypten, 2013). ............... 16
Figure 2.8-6 Ms1(x) due to a unit vertical load acting at s1. ...................................... 17
Figure 2.8-7 Mm1(x) due to the vertical load acting at m1 ........................................... 18
Figure 2.8-8 Bending moment due to unit load on s1(Dipl.-Ing. & Ägypten, 2013) .. 19
Figure 2.8-9Bending moment for unit vertical load act on m1 .................................... 20
Figure 2.8-10 Prestressing configurations for extradosed bridges (Dipl.-Ing. &
Ägypten, 2013). ........................................................................................................... 21
Figure 2.8-11 Fp and Mf place due to bottom straight tendons at the middle of the
main span (Dipl.-Ing. & Ägypten, 2013) ..................................................................... 23
Figure 2.8-12 Place of Fp and Ms due to top straight tendons at the pylons (Dipl.-Ing.
& Ägypten, 2013). ....................................................................................................... 24
Figure 2.8-13Fp and Mo due to bottom straight tendons in the side spans (Dipl.-Ing.
& Ägypten, 2013). ....................................................................................................... 24
Figure 2.8-14 Weq, P and Fp due to draped tendons at the middle of the main
span(Dipl.-Ing. & Ägypten, 2013). .............................................................................. 24
Figure 2.8-15Forces of P, Weq, and Fp due to draped tendons in the side spans (Dipl.-
Ing. & Ägypten, 2013). ................................................................................................ 25
Figure 2.8-16 Basic systems with cable support (left) stay cable alone ( right) stay
cable +pylon(Niels J.Gimsing, 2012) .......................................................................... 26
Figure 2.8-17 Chord force T and chord length c of stay cable(Niels J.Gimsing, 2012)
...................................................................................................................................... 28
Figure 2.8-18 Two conditions for horizontal stay cable with chord forces T1 and T2,
respectively(Niels J.Gimsing, 2012). ........................................................................... 29
xxii
Figure 2.8-19 The vertical sag kv and relative sag kc of a horizontal stay cable(Niels
J.Gimsing, 2012) .......................................................................................................... 30
Figure 2.8-20 Cross section of trapezoidal box girder ............................................... 32
Figure 2.8-21A flange section at normal moment capacity state(Chen & Duan, 2000a)
...................................................................................................................................... 35
Figure 2.8-22 Cracked Transformed Section (Mast, 1998). ........................................ 44
Figure 2.8-23 Anchorage set loss model(F. Abebe, 2016). ......................................... 57
Figure 2.8-24 Limited longitudinal displacements of the pylon top in a system with a
fixed bearing under the deck (Niels J.Gimsing, 2012). ............................................... 65
Figure 2.8-25 linear variation of the cross-sectional area of the pylon above the
deck(Niels J.Gimsing, 2012)........................................................................................ 66
Figure 3.4-1Working flow genetic algorithm for optimization ................................... 75
Figure 3.6-1 Characteristics of the Design Truck adopted from AASHTO Bridge
Design Specification 2010 ........................................................................................... 77
Figure 3.7-2 Cross section of pylon and wind direction .............................................. 78
Figure 3.6-3Wind direction on bridge deck (Fig 8.2 EN 1991-1-4) ............................ 81
Figure 3.7-4 Positive Vertical Temperature Gradient(Load and Resistance Factor
Design., 2015) .............................................................................................................. 88
Figure 3.10-1 Cross-section of the bridge deck ........................................................... 91
Figure 3.10-2 Cables number and geometry of the Abay extradosed stay cable bridge
...................................................................................................................................... 91
Figure 4.2-1 Effect of grades of Concrete on Optimum weight of the bridge ............. 99
Figure 4.2-2 Concrete specified compressive strength of concrete versus with
optimum depth and optimum weight. ........................................................................ 100
Figure 4.3-1 Effect of the unit cost of concrete grade on the optimum weight of the
bridge ......................................................................................................................... 102
Figure 4.5-1Effect of concrete grade on height and weight of pylon ........................ 104
Figure 4.8-1Weight Comparison of Optimum and Conventional Design box section
.................................................................................................................................... 108
Figure 4.8-2Comparison cumulative conventional design output and optimum design
output of bridge girders section ................................................................................. 109
Figure 4.8-3Weight comparison of the two design outputs of the pylon................... 110
Figure 4.8-4Comparison of conventional and optimum stay cable length and the
reduced amount .......................................................................................................... 112
xxiii
Figure 4.9-1Variation of girder deflection due to dead load when the weight of girder
decreasing 3.5% ......................................................................................................... 115
Figure 4.9-2 Variation of girder deflection due to live load when the weight of girder
decreasing 3.5% ......................................................................................................... 116
Figure 4.9-3 Variation of cable force when the weight of girder decreasing 3.5% ... 117
Figure 4.9-4 Variation of girder stress when the weight of girder decrease 3.5% .... 118
Figure 4.9-5Variation of girder stress when Modulus change through the length of the
bridge ......................................................................................................................... 121
Figure 4.9-6 Stay cable force from mid-span to pier ................................................. 122
Figure 4.9-7Stay cable tension force from side-span to pier ..................................... 123
xxiv
LIST OF TABLES
Table 2.8-1values of k(AASHTO LRFD 2010 BridgeDesignSpecifications 5th
Ed..Pdf, 2010) .............................................................................................................. 36
Table 3.4-1reinforcement steel strength (China Communication Construction
Campany Limited, 2020) ............................................................................................. 70
Table 3.7-1 Braking force values ................................................................................. 77
Table 3.7-2 Pylon Wind Load Calculation .................................................................. 79
Table 3.6-3 Pylon Wind Load Calculation ................................................................. 80
Table 3.6-4Wind Load Calculation Table ................................................................... 81
Table 3.7-5Beam Vertical Wind Load Calculation ..................................................... 82
Table 3.7-6 Side span cable base frequency fn calculation .......................................... 83
Table 3.7-7 Middle span Cable base frequency fn calculation ..................................... 83
Table 3.7-8 for calculating critical wind speed of wake vibration of side span cable . 85
Table 3.6-9 Calculation table of critical wind speed of wake vibration of mid-span
cable ............................................................................................................................. 85
Table 3.7-10 If calculation table of the flutter stability index ...................................... 86
Table 3.7-11 Uf table for flutter critical wind speed .................................................... 86
Table 3.7-12 Flutter stability list ................................................................................. 86
Table 3.7-13 Calculation table of vortex-induced resonance amplitude YMAX of stay
cable ............................................................................................................................. 87
Table 3.7-14 Temperature ranges ................................................................................ 87
Table 3.7-15 Basis for temperature gradients .............................................................. 88
Table 3.7-16 Load combination ................................................................................... 89
Table 3.10-1 Coding of design related to box girder variables .................................... 91
Table 3.10-2 Designation of Design Variables ............................................................ 91
Table 3.10-3Coding of design related to cable-stay variables ..................................... 92
Table 3.10-4Designation of Design Variables ............................................................. 92
Table 3.10-5Coding of design related to pylon variables ............................................ 92
Table 3.10-6 Designation of Design Variables ........................................................... 92
Table 3.11-1 Material property .................................................................................... 93
Table 4.1-1effect of depth of box girder of on optimum weight of extradosed cable
stayed bridge ................................................................................................................ 97
Table 4.2-1 Effect of grades of concrete on the optimum weight ............................... 98
xxv
Table 4.2-2Cumulative optimum weight of box girder in grade concrete 35 up to
75(Mpa)........................................................................................................................ 99
Table 4.3-1Unit price cost of concrete grade(Votorantim Cimentos & St. Marys CBM,
2021)and (Peng et al., 2019) ...................................................................................... 101
Table 4.3-2 Unit cost effect of concrete grade on optimum weight box girder ......... 101
Table 4.5-1Effect of concrete grade on the optimum height and weight of pylon .... 103
Table 4.6-1Optimum angle of stay cable for fixed span length for side span ........... 105
Table 4.6-2Optimum angle of stay cable for fixed span length for Mid-span ........... 105
Table 4.7-1Effect of Concrete Grades on Optimum angle of stay cable ................... 106
Table 4.8-1Comparison of conventional versus optimal design of box girder .......... 107
Table 4.8-2 Cumulative mass of 90m bridge segment .............................................. 108
Table 4.8-3 Conventional weight of pylon and optimum weight of pylon with the
reduced amount of weight .......................................................................................... 109
Table 4.8-4Comparison of Conventional stays cable length with optimum length side
span stay cable ........................................................................................................... 111
Table 4.8-5Comparison of Conventional stays cable length with optimum length
middle span stay cable ............................................................................................... 111
Table 4.8-6Cumulative conventional stay cable length and optimum stay cable length
and reduced length ..................................................................................................... 112
Table 4.9-1Variation of girder deflection due to dead load when the weight of girder
decreasing 3.5% ......................................................................................................... 114
Table 4.9-2Variation of girder deflection due to live load though the length of the
bridge when the weight of girder decreasing 3.5% .................................................... 115
Table 4.9-3 Variation of cable force when the weight of girder decreasing 3.5%
before and after Shrinkage and creep ......................................................................... 117
Table 4.9-4 Variation of girder tension stress when Modulus changes through the
length of the bridge .................................................................................................... 120
Table 4.9-5Stay cable force from mid-span to pier ................................................... 121
Table 4.9-6 Stay cable tension force from side-span to pier ...................................... 122
1
1. INTRODUCTION
Background 1.1
The extradosed bridge is a relatively new type of structure that has been developed
since the 1990s. The first such structure was the Odawara Blue way bridge, which
was designed and constructed in Japan (Shirono, Y., Takuwa, I., Kasuga, A., and
Okamoto, 1993). The extradosed concept precursors are Ganter Bridge in Switzerland
and the bridge in Rzuchów in Poland, both built-in 1980. Nevertheless, Jacques
Mathivat is most commonly credited as an inventor of extradosed terminology and its
design concepts by publishing his ideas in 1988 (Miskiewicz & Pyrzowski, 2018a).
The extradosed bridge can be defined as the structure being between the girder bridge
and the cable-stayed bridge (Mermigas & A, 2008). The extradosed prestressed bridge
in essence provides a transition structure type between conventional prestressed girder
bridges and cable-stayed bridges(Stroh, 2012). The feature of the extradosed bridges
is the larger girder stiffness in comparison to that of the cable-stayed bridges. Stay-
cables in the extradosed bridges can be stressed to a relatively high level, similar to
use in prestressed girder structures since the stress variation under live loads in stay
cable is usually lower in comparison with the cable-stayed bridges (Jerzy Onysyk,
Wojciech Barcik, 2017).
Currently the world's longest extradosed bridge, according to structurae.net, is
ArrahChhapra Bridge in India with 1920 m of total main bridge length (16 spans, 120
m each), built in 2017. In turn, the longest span world record belongs to KisoGawa
Bridge in Japan, 275 m long. The European bridges reach lower achievements.
However, one of the records belongs to a Polish structure. It started in 2013 when the
extradosed bridge was completed in Kwidzyn over the Vistula River. This structure,
with its main span length of 204 m, became the record holder in this category in
Europe (Biliszczuk et al., 2017). Recently, at the end of 2017, the bridge MS-3
construction was finished along the road DK-16 near Ostróda. The longest European
span length achievement has (Miskiewicz & Pyrzowski, 2018b).
The first extradosed concrete box girder deck and pylons bridge in Ethiopia is
renaissance bridge has 4m Girder depth at pier point and main span 145m and 78m
each side up to abutment.
2
The overall bridge length is 303m. This bridge opened at 2008 E.C(Kaljima
Corporation, 2019). The second prestressing concrete extradosed cable-stayed bridge
is the Abay Bridge. It has 380m in length .main span is 180m and the side span is
100m.
Problem Statement 1.2
The target of the research is on weight minimization of superstructure components of
extradosed cable-stayed bridges. The use of the conventional design method leads to
oversize structural members. Because large iterations by the conventional method are
so tedious and time consumes. Therefore traditional design methods have mostly had
non-optimal structural members in terms of size, shape, and topology. Besides, the
practice of structural optimization has been overlooked in civil engineering. But in the
world, there is a Limited resource of construction material. In extradosed cable-stayed
bridge to apply optimization, consider the main parameters of the superstructure.
Because this type of bridge has hybrid nature; bridges have significant additional
complexity and oversizing of structure(Tejashree G. Chitari1 & 1ME, 2019). The
existing extradosed cable-stayed bridges have high self-weighted and the effect of
concrete grade is high affects the weight of the bridge.
Thesis fills the gap of optimum design parameters of superstructure components of
extradosed cable-stayed bridge. The parameters of the study were the depth of box
girder, the height of the pylon, angle of cables stay, and effect of concrete of grade on
those three main parameters as independent variables and as dependent variables,
dimension of side cantilever, bottom width of slab, and prestressing reinforcement,
and non-prestressing reinforcing concerning the weight of the bridge. Therefore the
study is about structural optimization of superstructure main components of the
extradosed cable-stayed bridge on Abay River. The paper answers the optimum value
of the main load-carrying component parameter in extradosed bridge by considering
the concrete grade and sensitivity analysis of parameter that concerned the structure.
The previous research work gap was no one was the study about optimum box girder
depth with the height of pylon and angle of cable-stay in extradosed cable-stay bridge
with fixed span length (Tejashree G. Chiari*1 & ME, 2019). Therefore this paper
answers the optimum value of the main load-carrying component parameter that has
not been done so far.
3
Objective of Study 1.3
1.3.1 General Objective
The main objective of the research is weight optimization of superstructure main
component of extradosed cable-stayed bridge by using genetic algorithm.
1.3.2 Specific Objectives
To analyze the effect of depth of box girder extradosed cable-stayed bridge on
optimum weight.
To determine concrete grade effect on the optimum weight of bridge and depth of
girder of extradosed cable-stayed bridge.
To determine unit cost concrete grade effect on the optimum weight of box girder
To investigate the optimum angle of stayed cable.
To examine concrete grade on the optimum angle of the stayed cable
To compute the optimum height of the pylon
To examine concrete grade on the optimum height of the pylon
To compare the conventional and optimal design of superstructure components of
extradosed cable-stayed bridge.
Significance of the Study 1.4
The research specified on weight minimization of extradosed cable-stay bridge
structures facilitates the use of structural optimization methods in structural design
practice. The purpose of this thesis is to contribute to the close gap between
conventional design and optimum design works by implementing weight optimization
in practical works. The significance of structural optimize of the superstructure of
extradosed cable-stayed bridge is important not only weight reduction and also
economically, but it also gives better economical section and aesthetic appearance. Its
uses for the structural engineers and for the student to create familiarity about
structural optimization with any civil structure design work with regarding the
reducing structural weight.
As a case study, any decision-maker of the Ethiopian government can use this paper
to build these extradosed cable-stay bridges in other places of the country. Because
paper gives optimum dimension and weight of extradosed cable-stay bridge.
4
The weight minimization use for the bridge to reduced settlement and self-weight and
for the owner of the bridge used by minimizes the cost of the material.
Scope and limitation of the study 1.5
Consider the superstructure part as a case study on the Abay River Bridge to minimize
the weight of the bridge. The literature review and the case study focus on structural
optimization of extradosed cable-stay bridges of superstructure main components. The
superstructure component of different parametric consider case by case and subjected
to routine iterations of optimization by genetic algorithm to find the optimum weight
of the bridge. For Abay bridge construction, materials use prestressing strands for
girder prestressing reinforcement will be uncoated, low-relaxation, seven-wire strand,
Type 1x7 (d15.2mm), complying with AASHTO M 203/M (ASTM A 416/A 416M),
Type 1860Mpa(Grade 270) with the tensile strength of 1860Mpa use. The
optimization process uses two codes AASHTO Bridge Design Specification 2010 and
ERA 2013 Bridge Design Manual. Any ultimate and serviceability check is based on
these two codes. This paper, considering live load as per AASHTO LRFD Load and
ERA Bridge Design Manual 2013. The models are analyses by applying dead load,
live load, wind load, temperature, and settlement according to ERA 2013.
5
2. LITERATURE REVIEW
2.1 Extradosed cable-stayed bridge
The concept of an extradosed bridge is based on a combination of post-tensioned
girder bridges and cable-stay bridges. In some situations, bridges with higher ratios of
span-to-depth arrangement, beyond the capacity of internal post-tensioning, are
required. To achieve this requirement, higher eccentricity/load balancing is needed
from the post-tensioning tendons(Özel et al., n.d.).
An extradosed bridge has the characteristics of a lower tower, a more rigid main
beam, and a more concentrated cable layout. Cable force layout and corresponding
tower height will have crucial impacts on the structural performance of extradosed
bridge(Chang-Huan Kou1, a Tsung-Ta Wu2, b, Pei-Yu Lin3, 2014). It has two
different structural systems, the cable suspension system and the stiff deck bending
system. By reducing the deck stiffness, the bridge has the behavior like a cable-stayed
bridge; on the other hand, by increasing the deck stiffness it will behave more like the
traditional box-girder bridge. It was estimated that an extradosed bridge has a stiffness
ratio of around 30%(Tejashree G. Chiari*1 & ME, 2019). The pre-stressing force will
improve the stress performance of the main beam, therefore, the cable force layout
and corresponding tower height will have crucial impacts on the structural
performance of extradosed bridge(Kou et al., 2014). Tower height affects how the
loads are shared between the cables and the girder. Mainly how live loads are carried,
and how much change in live load, or fatigue, the cable is subjected to(Mermigas &
A, 2008). The reduced cable inclination in an extradosed bridge leads to an increase in
the axial load in the deck and a decrease in the vertical component of force at the
cable anchorages. Thus, the function of the extradosed cables is also to prestress the
deck, not only to provide vertical support as in a cable-stayed bridge(Mermigas & A,
2008). The estimated sizes of the stay cables should then be checked for compliance
with the requirements of the fatigue limit state and the strength limit state. This
process should be repeated until the minimum size/weight of stay cables is reached.
2.1.1 Advantages of extradosed cable-stayed bridge
1. A shallow structural depth below the roadway is preferable, either to meet
clearance requirements.
6
2. Tall piers over a deep valley do not permit a cable-stayed tower to be
aesthetically pleasing when the portion of the tower above the deck is around
half of the height between the deck and the ground.
3. There are height restrictions imposed by a nearby airport that limit the height of
the towers overhead.
4. The cross-section of the approach spans on a long viaduct can be made to span
further with extradosed pre-stressing. Extradosed prestressing can be kept to a
minimum by using as many internal and external tendons in the girder of the
extradosed span as in the approach spans(Mermigas & A, 2008).
2.1.2 The disadvantage of extradosed cable-stayed bridge
I. It has many uncertainty behaviors. Due to their hybrid nature can lead to
significant additional complexity in their design, as the response of the bridge
to applied loads (Tejashree G. Chitari1 & 1ME, 2019).
II. In extradosed prestressed bridges, the prestressing tendons in the negative
moment region over supports are moved outside the box girder to increase
their eccentricity (Saad, 2000).
III. For extradosed bridges with concrete decks, a combination of the stay cable
forces and the Prestressing forces inside the deck section may be selected to
eliminate the bending moment due to dead load along the entire length of the
deck(Dipl.-Ing. & Ägypten, 2013).
2.2 Structural Optimization Techniques
Structural optimization is the subject of making an assemblage of materials that
sustains loads in the best way. To fix ideas, think of a situation where a load is to be
transmitted from a region in space to a fixed support. To find the structure that
performs this task in the best possible way(Klarbring &An, 2008). In using the
mathematical programming methods, the process of optimization begins with an
acceptable design point. A new point is selected suitably to minimize the objective
function. The search for another new point is continued from the previous point until
the optimum point is reached. There are several well-established techniques for
selecting a new point and proceeding towards the optimum point, depending upon the
nature of the problem, such as linear and nonlinear programming( J. Abebe wubishet,
2018).
7
2.2.1 Linear programming
In a linear programming problem, the objective function and constraints are linear
functions of the design variables, and the solution is based on the elementary
properties of systems of linear equations. The properties of systems of proportionality,
additively, divisibility, and deterministic features are utilized in the mathematical
formulation of the linear programming problem. A linear function in three-
dimensional spaces is a plane representing the locus of all design points. In n-
dimensional space, the surface so defined is a hyperplane. In these cases, the
intersections of the constraints give solutions which are the simultaneous solutions of
the constraint equations meeting at that point (Rechenberg, 1973).
2.2.2 Nonlinear Programming
In nonlinear programming problems, the objective function and the constraints are
nonlinear functions of the design variables. Several techniques have been developed
for the solution of nonlinear programming problems (Rechenberg, 1973). Some of the
prominent techniques are
1. The method of feasible direction can be grouped under the direct methods of
approach on general nonlinear Inequality constrained optimization problems.
From starting from an initial feasible point, the nearest boundary is reached and a
new feasible direction is found. An appropriate step is taken along this feasible
direction to get the new design point. The procedure is repeated until the optimum
design point is reached(Gladwell, 1991).
2. In the sequential unconstrained minimization technique, the constrained
minimization problem is converted into an unconstrained one by introducing an
interior or exterior penalty function(Gladwell, 1991).
3. In sequential linear programming, the nonlinear objective function and
constraints are linearized in the vicinity of the starting point and a new design
point is obtained by solving the linear programming problem. The sequence of
linearizing in the neighborhood and solving by linear programming is continued
from the new point till the optimum is reached(Gladwell, 1991).
4. Dynamic programming which is widely applied in operations research and
economics is a mathematical approach for multi-stage decision problems.
8
This approach is well suited to the optimal design of certain kinds of structures,
in general, those in which the interaction between different parts is rather simple.
The main limitation of dynamic programming is that it does not lend itself to the
construction of general-purpose computer programs suitable for a wide range of
distinct problems(Gladwell, 1991).
2.3 Forms of Structural Optimization
2.3.1 Shape Optimization
Shape optimization is performed similarly to topology optimization. The main
difference is in how the design variables are defined. Design variables are the
coordinates of the boundary. The process of shape optimization consists of three
modules. Geometrical representation, structural analysis, and optimization algorithms.
Select a geometrical representation is the first step in the shape optimization process,
The nodal coordinates are chosen as design variables (Prashant Kumar Srivastava1*,
Simant2 & 1Asst., 2017).
2.3.2 Size optimization.
Sizing optimization is the simplest form of structural optimization. The shape of the
structure is known and the objective is to optimize the structure by adjusting the sizes
of the components. Here the design variables are the sizes of the structural elements
(Prashant Kumar Srivastava1*, Simant2 & 1Asst., 2017).
2.3.3 Topology Optimization
Topology optimization is the most general structural optimization technique and it is
mainly considered in a conceptual design stage. By topology optimization, we
understand finding a structure without knowing its final form beforehand. Only the
environment, optimality criteria, and constraints are known. The major Civil
Engineering representatives serve as a decision tool in selecting an appropriate static
scheme of the desired structure. They are mostly applied to the pin-jointed structures,
where the nodal coordinates of joints are optimization variables. Based on the position
of supports and objective functions, several historically well-known schemes can be
discovered.
9
The typical example of this optimization form within the reinforced concrete area is
the placement of steel reinforcing bars into a concrete block. In other words, search
for the most suitable strut-and-tie model(Bendsøe, M. P. and Sigmund, 2003).
2.4 Genetic Algorithm
Genetic Algorithms are global optimization techniques developed by John Holland in
1975(S.N.Sivanandam, 2008). They belong to the family of evolutionary algorithms
that search for solutions to optimization problems by "evolving" better and better
solutions. Thus this search is based on Darwin‘s theory of survival of the fittest.
Genetic algorithms are ideally suited for unconstrained optimization problems. As the
present problem is a constrained optimization one, it is necessary to transform it into
an unconstrained problem to solve it using Genetic Algorithms. Transformation
methods achieve this by either using exterior or interior penalty functions. This
method is shown to be highly advantageous in practical structural design
problems(Kirsch, 1993). Hence, traditional transformations using penalty or barrier
functions are not appropriate for genetic algorithms. A formulation based on the
violations of normalized constraints is proposed in this paper.
2.4.1 The major advantage of Genetic Algorithm
It doesn‘t have many mathematical requirements for the optimization problem.
Due to its evolutionary nature, the genetic algorithm will search for the solution
without regard to the specific inner working.
Genetic Algorithm can handle any kind of objective function and any kind of
constraint( i.e. linear or nonlinear) defined on discrete, continuous
It provides us great flexibility to hybridize with domain-dependent heuristics to
make an efficient implementation for a specific problem (Parsaei, 1997).
2.4.2 The major disadvantage of Genetic Algorithm
A genetic algorithm is an unconstraint optimization method. We must provide
an external penalty function for structural optimization
GA requires less information about the problem, but designing an objective
function and getting the representation and operators right can be difficult.
GA implementation is still an art.
10
2.4.3 Application of Genetic Algorithm
Genetic Algorithms are the heuristic search and optimization techniques that
mimic the process of natural evolution(Bhattacharjya, 2015).
Structural design- Size, Shape, Topology optimization
Control Gas- pipeline, missile evasion
Design -Aircraft design, keyboard configuration, communication networks
Security- Encryption and Description(Parsaei, 1997).
2.4.4 Sensitivity analysis for design variable
Sensitivity analysis is conducted to evaluate the dependence of structural
performances on design or imperfection parameters. As stated in the Preface,
dependent on parameters to be employed, sensitivity analysis in structural stability
can be classified as follows. In the design sensitivity analysis, employed as
parameters are design variables, such as member stiffness‘s and geometrical variables.
The sensitivity (differential) coefficients of structural responses, such as
displacements, stresses, and buckling loads, concerning these parameters are obtained.
These coefficients, in turn, are put to use in gradient-based optimization
algorithms(Introduction to Design Sensitivity Analysis, 2015). The main purpose of
global sensitivity analysis is to identify the most significant model parameters
affecting a specific model response. This helps engineers to improve the model
understanding and provides valuable information to reduce computational effort in
structural optimization. Structural optimization is characterized by a set of design
parameters, constraints, and objective functions formulated on basis of model
responses. The computational effort of a structural optimization depends besides the
complexity of the computational model heavily on the number of design parameters.
However, in many cases, an objective function is dominated only by a few design
parameters. The result of global sensitivity analysis may be used to select the most
significant design parameters from several potential candidates and thereby reduce
optimization problems by insignificant ones(Reuter & Liebscher, 2009).
11
2.5 Optimization Problem Formulation
Optimization problem formulation is started by defining all elements that need for any
constraints function. The ingredients of a structural optimization computer code
include finite element analysis, sensitivity analysis, and optimization. . Each of these
is now avail- able, but is seldom contained in a single computer code. Notably,
sensitivity analysis must often be calculated as a post-processing operation to the
finite element analysis. These various aspects of structural optimization are dis-
cussed, with emphasis on sensitivity calculations. Examples are given to demonstrate
the present state of the art. It is argued that, while experts in the field can now create
this capability by combining existing software, this is still a major task.
2.6 Objective Function
A function is used to classify designs. For every possible design, f returns a number
that indicates the goodness of the design. Usually, we choose f such that a small value
is better than a large one (a minimization problem). Frequently f measures weight,
displacement in a given direction, effective stress, or even cost of
production(Klarbring & An, 2008).
2.7 Design Variables
A function or vector that represents the design, and which can be changed during
optimization. It may represent geometry or a choice of material. When it describes
geometry, it may relate to a sophisticated interpolation of shape or it may simply be
the area of a bar, or the thickness of a sheet(Klarbring & An, 2008).
2.8 Design Constraints
Design constraints are conditions that need to happen for a project to be
successful. Design constraints help narrow choices when creating a project. Design
constraints can feel like a negative thing, but they help shape the project to fit the
exact needs of the client. Any set of values for the design variables represents a design
of the structure. Some designs are useful solutions to the optimization problem, but
others might be inadequate in terms of function, behavior, or other considerations. If a
design meets all the requirements placed on it, it will be called a feasible design. The
restrictions that must be satisfied to produce a feasible design are called constraints
(Kirsch, 1993).
12
2.8.1 Stayed cable force design constraint values
1. Equivalent Modulus of Elasticity for Stay Cables
Stay cable carry the load of the girder and transfer it to the tower. The cables in an
extradosed cable-stayed bridge are all inclined shown in Figure 2.8-1.
Equivalent elastic modulus of inclined cables
( )
2.1
Where
Eeq -The equivalent elastic modulus of inclined cables
E - The cable effective elastic modulus ((205Gpa Modulus of elasticity)
L0 - The horizontal projected length of the cable;
γ - The weight per unit volume of cable (87 kN/m³ for strand)
f - The cable tensile stress (Mpa)
The actual stiffness of an inclined cable varies with the inclination angle
* ( )+
2.2
Where
G - Total cable weight
Cable tension force (N)
Aeff - Cross-sectional area of the cable (mm2)
E - Young‘s modulus single cable (N/mm2)
If the cable tension T changes from T1 to T2, the equivalent cable stiffness state in
equation 2.3(Chen & Duan, 2000b).
* ( ) (
)+
2.3
13
Figure 2.8-1 an inclined stay-cable layout and its component (Chen & Duan, 2000b)
2. Preliminary Design of Stay Cables at the Serviceability Limit States
(Setral, 2002) limit the allowable stress of a stay cable fa to between 0.46 and 0.60 of
the guaranteed ultimate tensile strength fpu, for a maximum axial stress range due to
living load at SLS ΔσL between 140 Mpa and 50 Mpa(Dipl.-Ing. & Ägypten, 2013).
(
)
2.4
{
(
)
}
2.5
Where
- Allowable stress of a stay cable
fpu - Ultimate tensile strength
- Stress variation due to SLS live loads
Ultimate stress at SLS for the stay cables
- Allowable axial stress at SLS for the stay cables
3. Verification of Stay Cables at the Fatigue Limit State
PTI, (2001) fatigue load consists of a single design truck, in a single lane, and the load
effect is then increased by a Dynamic Load Allowance of 15% and by a factor of 1.4
to account for longer spans of cable-stayed bridges.
( ) ( ) ( )
( ) ( )
( )
14
Where
γ - The load factor of 0.75.
(ΔF) - The stress range due to the passage of the fatigue load
(ΔF)TH - The constant amplitude fatigue threshold (taken as 110 Mpa for parallel
strands)
4. Verification of Stay Cables at the Ultimate Limit States
SETRA (2001), material resistance factor for extradosed cables is 0.75 if the cables
have been mechanically tested to ensure ultimate and fatigue strength, 0.67 if they
have not been tested.
5. The cable forces due to dead load
The cable forces, Ti, can be found from the dead load moments and dead load moment
distribution through the following equation(Niels J.Gimsing, 2012).
4
( )
5
2.6
Where
Ti - Cable forces
Dead load moments at the cable anchor points (and at the pylons)
Length of bridge between cable anchorage
- The intensity of the deck dead load between each anchor point
- The angle of stay cable inclination
Where
W -Total dead load of the deck
Wc - Equivalent vertical stay cable forces
L - Main span length of the bridge
- Ratio between side span to the main span
15
Figure 2.8-2 Determination of cable force corresponding to a specified distribution of dead
load moments adopted (Niels J.Gimsing, 2012)
6. Calculation of the desired equivalent vertical stay cable forces wc
Extradosed cable-stay Bridge has a deck of variable depth under the variable dead
load W(x) and the equivalent vertical stay cable forces wc is shown in Figure 2.8-3.
Vertical components of the stay cable forces wc are calculated by assuming that
deflection of the deck at the points s1, m1, m3 and s2 are equal to zero (i.e. deck
under dead load will behave as a continuous beam on 8-supports at the points s1, m1,
m3, and s2, in addition to the already existing rigid supports at points 1 to 4.
Accordingly, using the virtual work method, the following two equations may be
written for the structure Figure 2.8-3. Equal/symmetrical stay cable forces on both
sides of the pylon are needed to avoid bending or rotation of the pylon under dead
load.
∫ ( ) ( )
( )
( )
-∫
( ) ( )
( )
( )
0 2.7
∫ ( ) ( )
( )
( )
, ∫
( ) ( )
( )
( )
2.8
Zs1 - Vertical deflection of the deck at s1 due to the combination of w(x) and wc
Zm1 - Vertical deflection of the deck at m1 due to the combination of w(x) and wc
ms1(x) - The bending moment at any point, along the entire length of the deck, due to
a vertical unit load F=1 acting at s1
16
Mm1(x) - The bending moment at any point, along the entire length of the deck, due to
a vertical unit load F=1 acting at m1
I(x) - Moment of inertia of the deck cross-section at any point along the length
- The ratio between the side span and the main span length
Figure 2.8-3 extradosed bridge with a deck of variable depth under the variable dead load
W(x) and the equivalent vertical stay cable forces wc (Dipl.-Ing. & Ägypten, 2013).
Figure 2.8-4Dead load and the moment of inertia of the deck cross-section change to constant
along the entire length of the deck (Dipl.-Ing. & Ägypten, 2013)
Figure 2.8-5 dead load moment distribution(Dipl.-Ing. & Ägypten, 2013).
M(x) evaluate in 11-zones as follows(Dipl.-Ing. & Ägypten, 2013).
17
( )
{
( )
( )
( ) ( )
(
)
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( ) }
2.9
Figure 2.8-6 Ms1(x) due to a unit vertical load acting at s1.
Figure 2.8-6 represent may be written for 4-zones as follows
( )
{
( )
( ) ( )
( )
( ) ( ) }
2.10
18
Figure 2.8-7 Mm1(x) due to the vertical load acting at m1
In Figure 2.8-7 Mm1(x), calculated Mm1(x) may be written for 4- zones from the
bridge as
( )
{
( )
( )
( ) ( ) }
2.11
Where
R1 -The reaction force at the support (1) for the structure
M2 - The bending moment at the support (2)
f -An unknown factor depending only on the parameter
( )
( )
.
/
2.12
(
)
( )
2.13
2.14
Where
R2 - The reaction force at support (2) and calculated by using equation 2.13.
R3 - Reaction force at support (3) and it is equal to R2 due to structural symmetry.
Q2r - The shear force and it.
Q3r - The shear forces on the support (3)
19
( )
2.15
( )
( )
2.16
( )
2.17
( )
2.18
Where
Q2rs1 - The shears force on the right-hand side of the support (2)
Q3rs1 - The shear force on the right-hand side of the support (3) for the structure
shown in Figure 2.8-8
R1s1, M2s1, and M3s1 – The reaction force at the support (1) and bending moments at
supports (2) and (3) respectively.
R4s1, R3s1, R2s1- The reaction forces at the support (4), (3), and (2).
Figure 2.8-8 Bending moment due to unit load on s1(Dipl.-Ing. & Ägypten, 2013)
,
( )
( ) 2.19
( ) 2.20
( )
2.21
( )
2.22
20
2.23
Where
R1m1- The reaction force at the support (1).
M2m1 and M3m1 - The bending moments at the supports (2) and (3) respectively for the
structure.
R4m1, R3m1, and R2m1 - The reaction forces at the support (4), (3), and (2)
Q2rm1 - The shear force on the right-hand side of the support (2) for the structure
Q3rm1 - The shear forces on the right-hand side of the support (3) for the structure and
calculated by using equation 2.23
Figure 2.8-9Bending moment for unit vertical load act on m1
1. Desired stay cable force for each stay cable FDI under dead load
After calculating wc. Desired stay cable force for each stay cable FDi under dead load
was determined, by using equation 2.24
( )
2.24
wc- Equivalent cable-stayed force
n- Number of stay cable
Ɵ- Angle between stay cable and horizontal distance
Stay cables in extradosed bridges carry approximately 70% of the dead load of the
bridge in extradosed bridges (Stroh, 2012).
( )
( )
2.25
21
2. Stay cable force of the cable no. i under shortening/re-stressing of any stay
cable by the forces Fi.
Stay cable force of the cable no. i under the effect of the dead load before the
shortening/re-stressing of any stay cable by the forces Fi.
( )
( )
2.26
The force F1 is dependent on the ―e‖. By rearrangement of equation
( )
.
( )/
2.27
Where
L- Main span length of the bridge
W- A dead load of the deck
Ɵ- Angle between stay cable and horizontal decks
e- Eccentricity
3. Effect of prestressing cable on the stayed-cable force in extradosed bridges
Prestressing cable is applied to the girder. Bending moment distribution allows the
possible use of draped tendons within the side and main spans of extradosed bridges.
The value of stay cable forces that get from dead load decrease by prestressing force
effect(Dipl.-Ing. & Ägypten, 2013).
Figure 2.8-10 Prestressing configurations for extradosed bridges (Dipl.-Ing. & Ägypten,
2013).
22
The forces resulting from the bottom straight tendons at the middle of the main span
Figure 2.8-11(Dipl.-Ing. & Ägypten, 2013).
Mf = - Fp eb, , M2= -3 Mf
2.28
R1f = - R2f =
= -3
2.29
When the forces resulting from the top straight tendons at the pylons (Dipl.-Ing. &
Ägypten, 2013).
= = .
( ) / 2.30
= - = - .
( ) / 2.31
When the forces resulting from the bottom straight tendons in the side spans (Dipl.-
Ing. & Ägypten, 2013).
= Fpeb , M2 =
. /
( ) = = -R2o =
2.32
The forces resulting from draped tendons at the middle of the main span Figure
2.8-14(Dipl.-Ing. & Ägypten, 2013).
=
Fb , P =
. ( )
( )
/
.
/
2.33
=
, =
( )(
)
( )
2.34
R1p = - ( ), M2p = R1pƞL , M2 = M2w + M2p 2.35
The forces resulting from draped tendons in the side span. The forces due to the
equivalent uniform load alone can be determined by using the equations
= weq Lo
. /
, = weq Lo
. /
2.36
23
(
+
(r1 λ L – r2(ƞ L – λ L - ))
+
+
( – )
( )
= M2w
L.
/
2.37
= -
. /
( )
2.38
The final forces resulting from the equivalent uniform loads and the 2x2
vertical nodal loads P can be determined then by superposition (Dipl.-Ing. & Ägypten,
2013).
∑
( ( )( ) ( ) ( )
( )
( )
2.39
2.40
( ) , 2.41
Figure 2.8-11 Fp and Mf place due to bottom straight tendons at the middle of the main span
(Dipl.-Ing. & Ägypten, 2013)
24
Figure 2.8-12 Place of Fp and Ms due to top straight tendons at the pylons (Dipl.-Ing. &
Ägypten, 2013).
Figure 2.8-13Fp and Mo due to bottom straight tendons in the side spans (Dipl.-Ing. &
Ägypten, 2013).
Figure 2.8-14 Weq, P and Fp due to draped tendons at the middle of the main span(Dipl.-Ing.
& Ägypten, 2013).
25
Figure 2.8-15Forces of P, Weq, and Fp due to draped tendons in the side spans (Dipl.-
Ing. & Ägypten, 2013).
4. Stay cable force under prestressing force
[ ( .
/ )
( ( )
( )
]
2.42
5. Total stay cable force for each cable in two directions
2.43
6. Quantity of cable steel
Quantity of cable steel Qcb,1 for a single cable with length lcb and axial force Tcb is
defined by:
2.44
26
Where
fcbd - The design stress (allowable cable stress) and
The density of the cable material.
For a cable system composed of n cable elements, the theoretical quantity for the total
system is found by a summation.
∑
∑
2.45
Quantity of stay cable material is expressed by
4
5
2.46
hopt=λ i.e stay cable inclination corresponding to an angle 45o
in Figure 2.8-16
vertical deflection at the lower stay cable anchorage can be expressed:
4
5
2.47
Figure 2.8-16 Basic systems with cable support (left) stay cable alone ( right) stay cable
+pylon(Niels J.Gimsing, 2012)
The minimum tension in the anchor cable occurs for traffic load in the side spans
expressing equation 2.48 (Niels J.Gimsing, 2012).
∑ ∑ ( )
2.48
Where
n - The number of loading points in one main span half and
27
m - The number of loading points in the side span.
The minimum tension in the anchor cable
The maximum tension max Tac in the anchor cable occurs for traffic load in the main
state in Equation 2.49 (Niels J.Gimsing, 2012)
∑ ( ) ∑ (
2.49
The stress ratio kac = min ac/max ac =minTac/max Tac in the anchor cable is then
determined by(Niels J.Gimsing, 2012)
∑ ∑ ( )
∑ ( ) ∑ (
2.50
With a specified value of kac and a given main span length, this expression can be
used to determine the side span length. For a continuous system with the uniform load
as shown in the expression for kac in equation 2.51(Niels J.Gimsing, 2012).
( )
( )
2.51
For a bridge with kac=0.4 and p=0.25g, corresponding to a typical road bridge
situation, la/lm becomes 0.38 so that in this case the side span length could amount to
almost 40% of the main span length without(Niels J.Gimsing, 2012)
7. Stay cables resist only tensile forces
-The specified minimum tensile resistance=
D=Diameter of the stay cable
2.52
According to AASHTO LRFD [2007], the maximum deflection of bridge deck under
live load should satisfy the following constraint
28
2.53
Maximum deflection limit of bridge deck due to living load
The allowable deflection limit prescribed by AASHTO LRFD [2007]
L-The total length of the bridge
8. Axial stress in stay cable due to dead load
2.54
9. Axial stress in stay cable due to dead load
Stay cable resist (25- 30) % of live load in extradosed stay cable bridge(Tejashree G.
Chitari*1 & ME, 2019).
= 2.55
10. Stay cable under varying chord force
Cables are subjected to axial tension the most relevant deformational characteristic is
the relationship between the chord force T and the chord length c. With the cable dead
load distributed uniformly along the curve, the correct cable curve is a catenary. Two
conditions, characterized by the chord forces T1 and T2, are considered
Figure 2.8-17 Chord force T and chord length c of stay cable(Niels J.Gimsing, 2012)
29
Figure 2.8-18 Two conditions for horizontal stay cable with chord forces T1 and T2,
respectively(Niels J.Gimsing, 2012).
In Condition 1 the equation of the catenary is given by(Niels J.Gimsing, 2012)
6 4
(
)5 (
)7
2.56
Cable two
8 [
(
)] 4
( )
59
2.57
Where
Gcb - The cable dead load per unit length
d - The cable elongation
The cable lengths s1 and s2 are determined by(Niels J.Gimsing, 2012).
( ( )
)
( (
)
2.58
The total elongation from the unstressed condition are found from
[ (
)
]
[ (
)
] 2.59
If , s2 expressing
.
/ .
/
The equation of state for cable now leads to the following
expression for shown that equation 2.60
30
( )
0
.
/
.
/1
0 .
/ .
/1
.
/
.
/
2.60
Introducing T1=A, T2=A2 and gcb=Acb
( )
0
.
/
.
/1
0 .
/ .
/1
.
/
2.61
Where
s1 -The cable stress in Condition 1,
s2 - The cable stress in Condition 2, and
gcb -The density (weight per unit volume) of the cable material
Figure 2.8-19 The vertical sag kv and relative sag kc of a horizontal stay cable(Niels
J.Gimsing, 2012)
The vertical sag k of a horizontal stay cable is determined by:
2.62
It appears that the relative sag k/c is proportional to the cable length c. So that the
relative sag k/c remains below 1/100 for cable lengths up to 400 m. For an inclined
stay cable, the sag kc perpendicular to the chord is given by
31
2.63
And vertical sag kv
2.64
2.8.2 Cast-in-Place Post-Tensioned Concrete Box Girder
In unsymmetrical sections, the stresses due to M alone may be variable for a constant
value of y, because the x- and y-axes are not the principal axes.
And the impact of the product-moment of inertia must be considered. For a vertical
load, producing M only, the stress at a point identified by the coordinates x and y may
be obtained from (Zhongguo(John) Ma, Maher K. Tadros, 2004).
1. Geometric constraint of trapezoidal box girder
4(
)5
2.65
(
) ,( )-
Where
tw - Thickness of web(mm)
ws - Clear web spacing
tb - Thickness of the bottom flange
tft - Thickness of the top flange
2. Stresses at Final Service Conditions
Stresses due to full load plus effective prestress can be calculated with the aid of the
unsymmetrical bending formula, equation 2.66, but with two sets of section
properties. The properties of the box section shown in Figure 2.8-20 should be used
with prestressing and girder weight.
32
Figure 2.8-20 Cross section of trapezoidal box girder
Top fiber subjected to tension at stress transfer stage:
+
2.66
2.67
Bottom fiber subjected to compression at stress transfer.
2.68
Top fiber subjected to compression at service loads:
(
)
(
)
2.69
Bottom fiber subjected to tension at service loads:
(
)
(
)
2.70
< . ( )
/ ( ) ( ) .
/
( ) .
/
( )
=
( ) ( ) ( ) .
/
33
[
( ) ( ) . ( )
/
( ) .
/ .
( ) ( )/
( ) .
/ .
( ) ( ) ( )
/
( ) ( ) ( ) .
/
]
( )
( ) (
)
(
)
( )
( )
4 ( )
5
4 ( ) ( )
5 ( ) ( )
( ( ))
( )
(
)
[
( )
( ) .
( )
/
4( ( )
5
( ) ( ( )
)
.
/ ( ( ) )
( ( )
) ( ( ) )
4 ( ) ( ( )
) 5
4( ) ( )
5
(
( ( )
( ))
( ( ) ) ( ( )
)
( ( ) ) (
( )
4 ( ) ( )
( )
(
)5
)
)
]
.
/
( ) .
/
+
( ) .
( ) ( )/
34
Where
y2 –height of opening of box girder
y3-width of opening of box girder
Stresss at point P1 due to self-weight moment Mdx=
=
2.71
The stress at bottom due to prestressing force at release is
,
(
)
2.72
Stress at Point P1 due to superimposed dead load moment (kN-m) and live load
moment (kN-m) is:
,
(
) 2.73
3. Live load
Live-load stresses are mostly determined by the evaluation of influence lines.
However, the stress at a given location in a cable-stayed bridge is usually a
combination of several force components. The stress, f, of a point at the bottom
flange(Chen & Duan, 2000b)
⁄
⁄ K
Where
A - The cross-sectional area,
I - The moment of inertia,
y - The distance from the neutral axis, and
35
c - A stress influence coefficient due to the cable force
K -Anchored in the vicinity.
P - The axial force and
M - The bending moment.
The above equation can be rewritten as
2.74
Where
a1, a2, and a3- Depend on the effective width, location of the point, and other global
and local geometric configurations.
f- Combined influence line obtained by adding up the three terms multiplied by the
corresponding constants a1, a2, and a3, respectively.
4. Flexural Strength
For members with bonded tendons, strain is linearly distributed across a section. Non-
prestressed reinforcement reaches the yield strength, and the corresponding stresses in
the prestressing tendons are compatible based on plane section assumptions. For a
member with a flanged section Figure 2.8-21 subjected to uniaxial bending, the
equations of equilibrium are used to give a nominal moment resistance(Chen & Duan,
2000a).
Figure 2.8-21A flange section at normal moment capacity state(Chen & Duan, 2000a)
NA axis depth for evaluation at the strength limit state is given by AASHTO LRFD
Equation 5.7.3.1-3. If
( )
………T
36
section else
……… rectangular section. For rectangular or
T sections where , average stress in Prestressing steel is given by
AASHTO LRFD Equation (
) (
)
(
) for
Table 2.8-1values of k(AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf, 2010)
Type of Tendon Value of k
Low relaxation strand 0.9 0.28
Stress-relieved strand and Type 1 high-strength bar 0.85 0.38
Type 2 high-strength bar 0.8 0.48
Effective depth from extreme compression fiber to centroid of tensile force, de is
given by AASHTO LRFD Equation 5.7.3.3.1-2
Depth of equivalent rectangular stress block, a=1.c, and the nominal moment of
resistance Mn is given by AASHTO LRFD Equation 5.7.3.2.2-1 stated as follows.
If
.
/ .
/
.
/
( )(
) .
/ .
/
.
/
2.75
Where
fpe -Compressive stress in concrete due to effective pre-stresses(N/mm2
)
fd -Stress due to un-factored self-weight(N/mm2
)
fpe and fd - Stress at extreme fiber where tensile stresses are produced by externally
applied loads. (N/mm2
)
Ap – Area of prestressing steel (mm2)
fpe– Effective stress in prestressing steel (N/mm2
)
37
fpu – Ultimate tensile strength of prestressing steel (N/mm2 )
fpy – Yield strength of prestressing steel (N/mm2
fps – Average stress in prestressing steel (N/mm2 )
dp –Distance from extreme compression fiber to centroid of prestressing tendons(mm)
de – Depth from extreme compression fiber to centroid of tensile force (mm)
As – Area of non-prestressed steel tension reinforcement (mm2)
fy – Yield strength of non-prestressed steel tension reinforcement (N/mm2 )
d – Distance from extreme compression fiber to centroid of non-prestressed tension
reinforcement (mm)
As‘ – Area of non-prestressed steel compression zone reinforcement (mm2)
fy‘ – Yield strength of steel compression zone reinforcement (N/mm2 )
ds‘ – Distance from extreme compression fiber to centroid of non-prestressed
compression zone reinforcement (mm)
fc‘ – Specified cylindrical compressive strength of concrete (N/mm2 )
b – Width of the cross-section in compression zone (mm)
bw – Web width of the cross-section (mm)
1 – Stress block factor, 1 = 0.85 for fc‘ = 28Mpa and reduced by 0.05 for each
7Mpa increment of fc‘ and 1 ≥ 0.65
hs – Depth of the deck slab or flange thickness (mm)
c – Depth of the neutral axis (mm)
a – Depth of equivalent rectangular stress block (mm)
Md – Ultimate factored design moment due to all loads (Nmm)
Mn – Nominal moment of resistance (Nmm)
Φ – Resistance factor
Aeff- Area of the concrete section
Ap - Area of prestressed tension reinforcement
b -Top flange width of box girder
38
d- Distance from extreme compression fiber to centroid of tension reinforcement
f -Concrete stress
f1 to f7- Concrete stress at critical checking Points 1 to 7
f’c- Specified compressive strength of concrete
f - Specified compressive strength of concrete at the transfer of prestress
f‘ - Ultimate strength of prestressing steel
fsu* -Stress in prestressing steel at ultimate load
Ix, Iy, Ixy-Second moments of inertia about x- and y axes of cross-section
5. Estimate Crack Angle θ
The LRFD method of shear design involves several cycles of iteration. The first step
is to estimate a value of ζ, the angle of inclination of diagonal compressive stress, ζ =
26.5 degrees. This simplifies the equation somewhat by setting coefficient 0.5 cotζ =
1.0.
6. Shear Strength
The shear resistance is contributed by the concrete, the transverse reinforcement, and
the vertical component of prestressing force. The modified compression field theory-
based shear design strength was adopted by the AASHTO-LRFD and has the
formula(AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf, 2010). Thus
nominal shear resistance,
4
5
:
⁄;
2-76
For 2-77
Where
de- Effective shear depth is given by AASHTO LTRFD Article 5.8.2.9
- The effective depth between the resultants of the tensile and compressive forces
due to flexure.
39
- The effective web width is determined by subtracting the diameters of un-grouted
ducts or one half the diameters of grouted ducts
- Nominal shear resistance
- The shear resistance is contributed by the concrete
- The shear resistance is contributed by the transverse reinforcement
- The shear resistance is contributed by the vertical component of prestressing force
5. Required Vertical Reinforcement, Vs
( )
2.78
( )
2.79
,
√
√
2.80
Then from this, we have the following
2.81
2.82
Where
bv -The effective web width determined by subtracting the diameters of un-grouted
ducts or one half the diameters of grouted ducts;
dv - The effective depth between the resultants of the tensile and compressive forces
due to flexure.
Av - The area of transverse reinforcement within distance s;
s - The spacing of stirrups;
α The angle of inclination of transverse reinforcement to the longitudinal axis;
40
-Factor indicating the ability of diagonally cracked concrete to transmit tension;
ζ - The angle of inclination of diagonal compressive stresses
Shear stress v and strain εx in the reinforcement on the flexural tension side of the
member is determined by
2.83
Where
Mu - Factored moment
Nu - Axial force (taken as positive if compressive)
fpo- Stress in prestressing steel when the stress in the surrounding concrete is zero
fpe -Effective stress after losses
When the value of εx calculated from the above equation is negative, its absolute value
shall be reduced by multiplying by the factor Fε, taken as
Where
Es, Ep, and Ec- Modulus of elasticity for reinforcement, prestressing steel, and
concrete, respectively
Ac-The area of concrete on the flexural tension side of the member Minimum
transverse reinforcement
√
Maximum spacing of transverse reinforcement
For 2
41
2.84
For 2
whereasfci=fc‘
2.85
7. Longitudinal reinforcement
At each section, the tensile capacity of the longitudinal reinforcement on the flexural
tension side of the member shall satisfy the following requirement.
+.⌊
⌋ /
take Ɵ=45 =1
=
⌊
⌋-0.5min0
1 2.86
Where,
Nd – Factored longitudinal tension force (N)
Md- Factored moment for Action (kN.m)
Longitudinal reinforcement on the flexural tension side should satisfy the following
conditions, AASHTO LRFD Article 5.8.3.5:
.
/ Assume =45, cot=1
2.87
Minimum spacing of traverse reinforcement, S
√
√
2.88
Maximum spacing of transverse reinforcement, S
if ⌈
⌉
2
2
2.89
42
Else, for 2
whereas fci=fc‘
2
2.90
Where;
Vu – Factored design shearing force d distance from the face of support (N)
Vn – Nominal shear resistance (N)
Vc - Shear resisting force due to tensile stress in the concrete (N)
Vs – Shear resisting force due to tensile stress in traverse reinforcement (N)
Vp – Component of prestressing force in the direction of shearing force (N)
S – Spacing of stirrups (mm)
Av- Cross-sectional area of shear reinforcement within a distance S (mm2)
dv - Effective depth of shearing force (N) in (mm)
8. Limits of reinforcement (AASHTO LRFD Article 5.7.3.3)
Maximum reinforcement limit
=
0 2.91
Minimum reinforcement limit
Where as
( )
Where
φ- Flexural resistance factor 1.0 for prestressed concrete and 0.9 for reinforced
concrete
Mcr -The cracking moment strength
Mn- Nominal flexural resistance
43
9. The minimum amount of reinforcement
The amount of prestressed and non-prestressed tensile reinforcement shall be
adequate to develop factored flexural resistance Mr which shall not be the lesser of 1.2
times cracking moment and 1.33 times factored design moment as equated below.
Cracking moment, ( ) (
)
For monolithic section substitute for then, ( )
, .
/ And
2.92
2.93
As per ACI-318 1989 minimum area of flexural reinforcement shall not be less than
0.4% of ( )
2.94
Where,
fcpe – Compressive stress in concrete due to effective prestress forces (N/mm2 )
Mg – Total un-factored dead load moment (Nmm)
Md – Total factored design moment (Nmm)
Mr – Total factored moment of resistance of the section (Nmm)
Mcr – Cracking moment (Nmm)
Zc – Section modulus for the extreme fiber of the composite section where tensile
stress is caused by externally applied loads (mm3)
Znc – Section modulus for the extreme fiber of monolithic or non-composite section
where tensile stress is caused by externally applied loads (mm3) that is Zb
fr – Modulus of rupture (N/mm2
)
44
10. Permissible stresses in the reinforcement steels
The stress and forces acting on a cracked post-tension prestressed concrete section
subjected to a moment a working or service load moment of Mw over the cracking
moment Mcr is shown in Figure 2.8-22 below(Mast, 1998).
Figure 2.8-22 Cracked Transformed Section (Mast, 1998).
From the cracked section analysis we have the following equations.
a. On application of Mw or at stage (1) stress in the prestressing tendons is
b. Consider a fictitious load stage (2) corresponding to complete decompression of
the concrete, at which there is zero concrete strain throughout the entire depth as
shown in Figure 2.8-22 compatibility of deformation of concrete and steel
requires that changes in strain in the tendon are the same as that in the concrete at
that level and the stress in tendon due to this stain is given by:
0
1From which 0
1
c. During the application of Mw, the concrete compressive strain in the bottom fiber
reduces to zero and then becomes tensile. With Mw acting the tensile strain in the
reinforcing steel is s and the strain in the concrete at the level of the tendon have
changed from a compression oc to a tension of cp( wubishet J. Abebe, 2018).
From the linearity of strain distribution, these strains can be defined in terms of
the neutral axis depth y and top fiber strain o. Let the extreme top fiber strain be
45
o and NA depth y, and then from strain compatibility one can drive the strains
and stresses in reinforcing steel and prestressing tendons at their respective depths.
Once the strains are evaluated the corresponding stresses can be obtained from the
stress-strain relationship as shown in the following steps. The strain in reinforcing
steel at a depth is d is
and the corresponding stress.
; Similarity the tensile strain in concrete at the level of
prestressing steel or a depth is =
; the Prestressing tendons undergo.
This strain and thus the stress in the tendons are
and
stress in concrete at extreme top fiber is . The Prestressing steel undergoes
the stress of [ ] during the application of Mw so that the total tensile stress in
the tendon is . Tensile force in Prestressing and reinforcing steel
respectively. . In the concrete compressive zone, the
resultant compressive force is; by which is acting at a depth
. This
equation is valid if the neutral axis lies in the flange that is and if , the
force C shall be reduced Cn given by; ( )( )
which can be
regarded as negative. Force and acting at a depth,
the incremental
strain, sought as loading passes from the stage (2) to stage (3) can be defined in
terms of neutral axis depth, y as;
( )
( ( ) .
/) 4 .
/ .
/5
This equation is for the flanged section so that substitute bw by b if y is less than or
equal to hf. From equilibrium of moments have
If 2.95
2.96
Since Mw is known to solve for the strain co and NA depth y and then the equilibrium
of x (horizontal of forces should be checked).
46
∑
2.97
Location of the centroid of the cracked transformed section from extreme top fiber, yct
is given
If
( )
( ) by else
The cross-sectional area of the cracked transformed section, Act will be; if
( )
The second moment of area or moment of inertia of the cracked transformed section,
Ict is; if
.
/
( )
( ) 4
5
( ) ( )
.
/
( ) ( )
If then 2.98
Where,
fp1 – Incremental stress in prestressing tendons before the application of service loads
or at stage (1) (N/mm2)
fp2 – Incremental stress in prestressing tendons as the section passes from before the
application of service loads stage (1) to decompression stage (2) (N/mm2)
fp3 – Incremental stress in prestressing tendons due to change of stress from
compression to tension in the concrete located at the level of the tendon (N/mm2)
fco – Stress in extreme top fiber during application of service load moment(N/mm2 )
fs – Stress in steel reinforcement at the application of service loads (N/mm2)
fp – Total stress in prestressing tendons at the application of service loads (N/mm2 )
47
oc– Compressive strain in the concrete at the level of the tendon(mm)
cp – Tensile strain in the concrete at the level of the tendon(mm)
o –compressive strain at the extreme top fiber (mm)
s – Tensile strain in the reinforcing steel at working loads
np – Modular ratio of prestressing steel
ns – Modular ratio of reinforcing steel
Ec –modulus of elasticity of concrete (N/mm2)
Es – Modulus of elastic of reinforcing steel (N/mm2)
Ep – Modulus of elastic of prestressing steel (N/mm2)
Fx – Forces acting in the horizontal direction (N)
Ts – Tension force in the reinforcing steel at service limit state (N)
Tp – Tension force in the prestressing steel at service limit state (N)
C – Resultant compressive force in the compression zone of concrete (N)
Cn – Compressive force in the compression zone of concrete used to reduce the
resultant compressive force C when NA depth exceeds flange thickness (N)
y – NA depth of the cracked section under service limit state (mm)
dz – Depth from extreme compression fiber to centroid of resultant compression force
C (mm)
dzn – Depth from extreme compression fiber to centroid of compression force Cn
(mm)
yct – Depth from extreme compression fiber to centroid of cracked section (mm)
Act – Area cracked transformed section under service limit state (mm2)
Ict – Moment of inertia of cracked transformed section under service limit state (mm4)
11. Deflection control (ASHTO LRFD Article 5.7.3.6)
Deflection and camber calculations shall consider dead load, live load, prestressing,
erection loads, concrete creep and shrinkage, and steel relaxation.
48
Immediate or instantaneous deflection is computed by taking the effective moment of
inertia, the effective moment of inertia used to calculate the instantaneous deflection
is given by
{.
/
( .
/
)
}
And =0.63
Where,
Ie- Effective moment of inertia (mm4)
Mck – Cracking moment (Nmm)
frk – Modulus of rupture of concrete (N/mm2)
12. Deflection due to dead loads and prestressing force
Instantaneous deflection for permanent loads calculation, i
Instantaneous deflection due to dead load, di=∬ ( )
Additional long-term deflection d1= . di λ 8
9
Thus total deflection to dead load, . For parabolic tendon profile with
central anchor upward deflection due to prestress is
2.99
13. Deflection due to living loads
When investigating the maximum absolute deflection, all of the design lanes should
be loaded. For live load deflection evaluation, design vehicular live load of AASHTO
HL-93 where the vehicle load includes the impact factor IM and the multiple presence
factor m. In general, the deflection at the point of maximum moment, x due to each
design truckload at a distance a, from the left support is given by live load deflection
due to design truckload will be
For x=a,
. For
( ) individual truckload
deflection. Thus total design truck deflection will be ∑ .
49
Deflection due to each design lane load
where
In the computation of live load deflection design truckload alone or design lane load
plus 25% of design truckload whichever is the greater as stated in AASHTO article
3.6.1.3.2. Thus live load deflection is, {
. Allowable live load
deflection,
. Thus limit live load deflection,
Hence
2.100
Where:
Ie – Effective moment of inertia of the section (mm4)
Δdi – Immediate deflection due to dead load (mm)
Δd – Total long term deflection due to dead load (mm)
Δp – Upward deflection due to prestressing force (mm)
Δkl – Deflection due to truckload (mm)
ΔLn – Deflection due to design lane load (mm)
ΔLL – Deflection due to living load (mm)
Δall – Allowable deflection for the live load (mm)
x – Distance from left support to a point at which maximum service load moment
occurs.
a – Distance from the left support to the point of truckload for which deflection is to
be computed.
b – Distance from the right support to the point of truckload for which deflection is to
be computed.
14. Limit of the crack width
For dry air or protective membrane (class I) exposure conditions, and assumed
allowable crack width is 0.41mm. The expressions that have figured prominently in
the development of the crack control provisions in the ACI code. These equations are
respectively.
50
( )
( ) 2
3 2.101
Where,
fs - Service load stress in non prestressed steel (Mpa)
h1 – Distance from the centroid of tensile steel to NA depth (mm)
h2 – Depth from extreme compression fiber to depth of NA (mm)
dc Thickness of concrete cover from extreme tension fiber to centroid (mm)
Act – Effective tension area of concrete surrounding one bar (mm2)
15. Fatigue limit state (AASHTO LRFD Article 5.5.3)
The stress range in reinforcing steel resulting from fatigue load is ( )
And stress range in prestressing steel resulting from fatigue load is
( )
. The allowable stress range in reinforcing steel is given by
.
/ by setting
Where,
r/h– Ratio of base radius-to-height of rolled-on transverse deformation (a value of 0.3
can be used in the absence of specific data). Allowable stress range in prestressing
tendons with the radius of curvature larger than 9000mm shall be less than 125Mpa.
Thus, the following constraints stated;
2.102
2.103
Where,
fffs – Stress range in reinforcing steel due to fatigue load (N/mm2 )
fffp – Stress range in prestressing steel due to fatigue load (N/mm2 )
Mf – Maximum fatigue load moment (Nmm)
51
fmin – Minimum live load stress where there is stress reversal (N/mm2 )
r – Base radius of the deformation (mm) and
h –The height of the deformation (mm)
16. Torsion
Calculation of torsion resistance by formula(5.8.2.1-4) in《AASHTO Bridge
Design Specification 2010》. . In which
4
5
2.104
Where:
Tu- Factored torsional moment (Nmm)
Tcr - Torsional cracking moment (Nmm)
Acp-Total area enclosed by the outside perimeter of concrete cross-section (mm2)
Pc -The length of the outside perimeter of the concrete section (mm)
17. Longitudinal Reinforcement due to torsion
The provisions of Article 5.8.3.5 shall apply as amended, herein, to include torsion.
The longitudinal reinforcement in solid sections shall be proportioned to satisfy Eq.
5.8.3.6.3-1:
+ √.⌈
⌉ /
.
/
In the box girder sections, longitudinal reinforcement for torsion, in addition to that
required for flexure, shall not be less than
+ √.⌈
⌉ /
.
/
( )
2.105
Where:
52
ph - Perimeter of the centerline of closed transverse torsion reinforcement (mm)
18. Minimum Area of Interface Shear Reinforcement
The cross-sectional area of the interface shear reinforcement, Avf, crossing the
interface area, Acv, shall satisfy the equation of 5.8.4.4 in AASHTO
. Amount needed to resist 1.33Vui /φ as determined using Eq.5.8.4.1-3
2.106
Where
Avf- The cross-sectional area of the interface shear reinforcement
Acv- Crossing the interface area
19. Torsional Reinforcement
Where a consideration of torsional effects is required by Article 5.8.6.3, torsion
reinforcement shall be provided.
The nominal torsional resistance from transverse reinforcement shall be based on a
truss model with 45-degree diagonals and shall be computed as:
.
The minimum additional longitudinal reinforcement for torsion, Aℓ , shall satisfy:
2.107
Where:
Av - Area of transverse shear reinforcement (mm2)
Aℓ - Total area of longitudinal torsion reinforcement in the exterior web of the box
girder (mm2)
Tu -Applied factored torsional moment (Nmm)
53
Ph - Perimeter of the polygon defined by the centroids of the longitudinal chords of
the space truss resisting torsion.
Ao- Area enclosed by shear flow path, including the area of holes, if any (mm2)
fy -Yield strength of additional longitudinal
20. Edge Tension Forces
The longitudinal edge tension force may be determined from an analysis of a section
located at one-half the depth of the section away from the loaded surface taken as a
beam subjected to combined flexure and axial load. The force may be taken as equal
to the longitudinal edge tension force but not less than that specified in (AASHTO
LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf, 2010) in Article 5.10.9.3.2)
21. Bursting Forces
This calculation is checked by (AASHTO LRFD 2010 BridgeDesignSpecifications
5th Ed..Pdf, 2010). Calculation of bursting forces by formula(5.10.9.6.3-1and(
5.10.9.6.3-2 ) in(AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf,
2010).
∑ .
/ 0∑ 1
The location of the bursting force, dburst, may be taken as:
( ) 2.108
Where:
Tburst -Tensile force in the anchorage zone acting ahead of the anchorage device and
transverse to the tendon axis (N)
Pu - Factored tendon force (N)
dburst -Distance from anchorage device to the centroid of the bursting force(m.)
a -Lateral dimension of anchorage device or group of devices in direction (mm)
e -Eccentricity of the anchorage device (mm)
h - Lateral dimension of the cross-section in the direction considered (mm)
α - Angle of inclination of a tendon force concerning the centerline of the member
54
22. Bursting Forces Results of End Beam Prestressed reinforcement
∑ .
/ ,∑ -The location of the bursting force, dburst,
may be taken as:
( ) 2.109
23. Bursting Forces Results of Tooth Block
∑ .
/ ,∑ -The location of the bursting force, dburst,
may be taken as:
( ) 2.110
24. Bursting Forces Results of Stay Cables
∑ .
/ ,∑ -The location of the bursting force, dburst,
may be taken as:
( ) 2.111
25. Compressive Stress of Concrete in Front of Anchor Head
Calculation of Compressive Stress of Concrete in Front of Anchor Head by formula(
5.10.9.6.2-1)and(5.10.9.6.2-2)in(AASHTO LRFD 2010
BridgeDesignSpecifications 5th Ed..Pdf, 2010).
In which if ; then (
) .
/
Ifs ; then k=1, fca< 0.7φf'ci
2.112
Where:
κ -Correction factor for closely spaced anchorages
aeff -Lateral dimension of the effective bearing area measured parallel to the larger
dimension of the cross-section (mm)
55
beff - Lateral dimension of the effective bearing area measured parallel to the smaller
dimension of the cross-section (mm)
Pu -Factored tendon force (N)
t- Member thickness (mm)
s - Center-to-center spacing of anchorages (mm)
n - Number of anchorages in a row
ℓc - Longitudinal extent of confining reinforcement of the local zone but not more
than the larger of 1.15 aeff or 1.15 beff (mm)
Ab- Effective bearing area (mm2)
The effective bearing area, Ab, in Eq. 5.10.9.6.2-1 shall be taken as the larger of the
anchor bearing plate area; Aplate; or the bearing area of the confined concrete in the
local zone,
Aconf, with the following limitations:
• If Aplate controls,
A plate shall not be taken larger than 4/πAconf.
• If Aconf controls, the maximum dimension of Aconf shall not be more than twice the
maximum dimension of Aplate or three times the minimum dimension of A plate. If any
of these limits are violated, the effective bearing area, Ab, shall be based on Aplate.
Deductions shall be made for the area of the duct in the determination of Ab. If a
group of anchorages is closely spaced in two directions, the product of the correction
factors, κ, for each direction shall be used, as specified in Eq. 5.10.9.6.2-1.
26. Punching Shear
=0.125√ ( ) 2.113
Where:
f ′c- Specified strength of concrete at 28 days (Mpa)
W- Width of bearing plate or pad as shown in Figure 5.13.2.5.4-1 (mm)
L- Length of bearing pad as shown in Figure 5.13.2.5.4-1 (mm)
de -Effective depth from extreme compression fiber to centroid of tensile force(mm)
56
27. Losses of Pre-stress
Pre-stress losses can be divided into two categories:
i. Instantaneous losses
Including losses due to anchorage set (fpA), friction between tendons and
surrounding materials (∆fpF), and elastic shortening of concrete (fpES) during the
construction stage. Time-dependent losses including losses due to shrinkage (fpSR),
creep (fpsR), and relaxation of the steel (fpR) during the service life(F. Abebe, 2016).
The total pre-stress loss (fpT) is dependent on the pre-stressing methods.
For post-tensioned members
ii. Anchorage Set Loss
The anchorage set loss changes linearly within the length (LpA), the effect of
anchorage set on the cable stress can be estimated by the following formula:
0
1,
√
( )
Where
∆L - The thickness of anchorage set;
E - The modulus of elasticity of anchorage set;
∆f - The change in stress due to anchor set;
LPa - The length influenced by anchor set;
LpF - The length to a point where loss is known; and
x - The horizontal distance from the jacking end to the point considered
57
Figure 2.8-23 Anchorage set loss model(F. Abebe, 2016).
iii. Friction Loss
For a post-tensioned member, friction losses are caused by the tendon profile
curvature effect and the local deviation in tendon profile wobble effects. AASHTO-
LRFD specifies the following formula:
( ( )
Where
fpj - Stress in the prestressing steel at jacking (Mpa)
K - Wobble friction coefficient (per ft of the tendon)
µ - Curvature friction coefficient
x - Length of a prestressing tendon from the jacking end (mm)
α - Sum of absolute values of angular change of prestressing steel path from jacking
end, (o)
e - Base of Napierian logarithms
iv. Elastic Shortening Loss ∆fpES
Equation for calculation of elastic shortening in the LRFD Commentary(AASHTO
LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf, 2010).
( ) (
)
( )
( )
58
v. Time-Dependent Losses
AASHTO-LRFD provides the approximate lump sum estimation of time-dependent
losses ∆fpTM resulting from shrinkage and creep of concrete and relaxation of
prestressing steel(AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed., 2010).
vi. Refined Estimation
d. Shrinkage loss. Shrinkage loss can be determined
e. Creep loss: creep loss can be predicted by
2.114
Where
fcgp -Concrete stress at the center of gravity of prestressing steel at transfer, and
∆fcdp - Concrete stress change at the center of gravity of prestressing steel due to
permanent loads, except the load acting at the time the prestressing force
vii. Relaxation Loss:
The total relaxation loss (∆fpR) includes two parts: relaxation at the time of transfer
∆fpR1 and after transfer ∆fpR2. For low relaxation strands, the relaxation in the pre-
stressing strands equals 30% of the equation shown below:
( )
{
6
7
6
7
For stress-relieved strands
[ ( ) for post-tensioning
Where
t - The time estimated in days from testing to transfer.
- Loss due to elastic shortening (Mpa)
- Loss due to creep of concrete (Mpa)
59
-Loss due to shrinkage (Mpa)
viii. Post-Tensioned Members
The loss due to elastic shortening in post-tensioned members, other than slab systems,
maybe 5.9.5.2.3b(AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed..Pdf,
2010).
Where
N - Number of identical prestressing tendons
fcgp - Sum of concrete stresses at the center of gravity of prestressing tendons due to
the prestressing force after jacking and the self-weight of the member (Mpa)
fcgp - Values may be calculated using steel stress reduced below the initial value by a
margin dependent on elastic shortening, relaxation, and friction effects.
The loss is due to elastic shortening in post-tensioned members, other than slab
systems determined by the following alternative equation 2.115.
( )
( )
2.115
Where:
Aps- Area of prestressing steel (mm2)
Ag -Gross area of section (mm2)
Eci- Modulus of elasticity of concrete at transfer (N/mm2)
Ep - Modulus of elasticity of prestressing tendons (N/mm2)
em - Average eccentricity at mid-span (mm)
fpbt - Stress in prestressing steel immediately before transfer as specified (Mpa)
Ig - Moment of inertia of the gross concrete section (mm4)
Mg - Mid-span moment due to member self-weights (Nmm.)
N - Number of identical prestressing tendons
fpj - Stress in the prestressing steel at jacking (Mpa)
60
ix. Creep Losses
The equation for creep follows:
2.116
Where
Δfcdp -Equals the change in concrete stress due to externally applied dead loads
excluding self-weight.
2.8.3 Pylon design
The primary function of the pylon is to transmit the forces arising from anchoring the
stays cable. The pylon should ideally carry these forces by axial compression where
possible such that any eccentricity of loading is minimized(Nippon Koei Co., 2005).
The tower height is a particularly important parameter because it influences how the
loads are shared between the stay-cables and the girder. Specifically how they live
loads are carried, and how much change in live load, or fatigue, the cable is subjected
to (Stroh, 2012). When investigating the buckling of the pylon simple support at the
top and a fixed base, provided the column is subjected to the axial load Npt, the
relevant angular change Df at the bottom, as well as the longitudinal wind load. The
resistance of the pylon top against horizontal displacements due to elongation of the
anchor cable depends on the ratio between the axial force Npt and the critical force
Ncr(Niels J.Gimsing, 2012).
The pylon is linearly tapered with cross-sectional shapes of k (=4)-sided regular
polygons with circum radii r measured from the centroid to a vertex at any coordinate
x.
( )
Where
E - The material modulus of elasticity
I - Moment inertia of the prismatic column,
γcr - The buckling load factor depends on the ratio of moment inertia and support
condition of the column(Dharma & Suryoatmono, 2019).
61
Fw = γA, 2.117
Where
γ - The weight density of the column material.
The cross-sectional area of the plane area at x
I - Second moment of the plane area at x
The pylon is subjected to an external compressive load P at the head end and its own
self-weight W (= γV). When P increases and reaches the buckling load Pcr, the
column with a buckling length l buckles and forms the buckled-mode shape
represented by the solid curve(Lee & Lee, 2021). Taper pylon function of r at x
expressing mathematically in the equation
Where
n -Taper ratio
rh - Head radius
rt - Toe radius is introduced.
Linear taper function, r is expressed in terms of x as follows. Where n1 = n − 1. The
variable functions of A and I for the 4-sided regular polygon at x can be gained as
equation 2.118(Lee & Lee, 2021).
2.118
Where C1 and C2 are
.
/ .
/
.
/ .
/ 0 .
/1
Where
k ( ) is integer side number and k= for the circular cross-section. The column
volume V is determined as
∫
∫
,
( ) 2.119
62
Where
l - The buckling length of the column subjected to an external buckling load B
W (= γV) - self-weight
rt- can be obtained in terms of V as √
A and I can be obtained in terms of V as
,
The self-weight intensity Fw at x caused by the γ value of the column material is given
by
The axial force N at x is obtained as
∫
.
/ ,
2.120
Where
γV – The equal to the total column weight W.
Pcr- The buckling load
Fw- The self-weight intensity
The bending moment M is given by the relationship between load and deformation
based on the small deflection theory as
2.121
Differentiating Equation 2.121 twice yields
2.122
After substituting equations the equation yields
[ (
)]
2.123
From the equation, the first and second derivatives of I are determined, respectively:
63
, 2.124
Substituting equation 2.131 equation 2.130
[ (
)]
2.125
Following the system, parameters are cast into non-dimensional forms equation 2.126
;,
;
;
2.126
0.
/1
+
2.127
√
6
(
)
7
2.128
Where
(ξ, ε)-Non-dimensional Cartesian coordinates,
β- The buckling load parameter, and
λ - The self-weight parameter.
-The self-weight buckling length for which the column buckles
In particular, the self-weight buckling length L and self-weight buckling stress σ
caused only by the self-weight W with P = 0 are obtained using Equations 2.129
respectively.
√
.
/
2.129
Cables that are passed through saddles must be dimensioned to account for the
bending stress due to curvature fc determined as follows (Mermigas& A, 2008).
64
2.130
Therefore, the total stress in the cable is:
2.131
Where
E - The elastic modulus of the wire or strand;
r - The radius of the wire, strand, or bundle
R - The radius of the saddle bend.
As a case study, Abay extradosed cable-stayed bridge has a cross-sectional area of the
pylon that varies linearly from 3*2 at the top to the value 2*4 at deck level. The force
Npt acting at the pylon top will be the resultant of all vertical components of the cable
forces at the supporting point. Horizontal equilibrium between the two-chord forces
TA and TC, the axial force RT acting on the pylon will be the resultant of these two
forces(Niels J.Gimsing, 2012).
2.132
As the pylon is fixed at the base and longitudinally supported at the top by the anchor
cable, the effective column length will be approximately 0.7hpl.
2.133
Conversely, the horizontal restraint offered by the anchor cable implies that the force
Npt from the cable system does not have to stay vertical but might turn into an
inclined direction. Under longitudinal wind load, the anchor cable renders support to
the pylon. When investigating buckling of the pylon it can generally be assumed that
the anchor cable constitutes fixed support in the longitudinal direction. Thus, the
analysis should be based on investigating a column fixed at the base and simply
supported at the top, and subjected to an axial force Npt, a top displacement dh, and a
longitudinal wind load(Niels J.Gimsing, 2012).
65
Figure 2.8-24 Limited longitudinal displacements of the pylon top in a system with a fixed
bearing under the deck (Niels J.Gimsing, 2012).
2.134
If a linear analysis is performed, then the distinction between forces directed towards
a fixed point or remaining vertical becomes unimportant and the forces from each
cable system can then be represented by a single force equal to the resultant of all
tangential cable forces. Fixing of column-type pylons to piers, longitudinal restraint of
the pylon top can be achieved by fixing the pylon to the substructure, but this will
require a substantial increase of the cross-sectional dimensions to achieve equally
small displacements as those found in a system with an anchor cable. For the
horizontal displacement of the pylon tops, the following expressions can be derived:
2.135
2.136
Which
ac - The tensile stress in the anchor cable;
- The modulus of the elasticity for the cable;
pb - The bending stress in the lower part of the pylon; and
- The modulus of elasticity for the material used in the pylon
-The height of the pylon above the deck
66
-the height of the pylon below the deck
The quantity Qpd of the pylon above deck then becomes(Niels J.Gimsing, 2012).
Figure 2.8-25 linear variation of the cross-sectional area of the pylon above the deck(Niels
J.Gimsing, 2012).
=
2.137
Vertical projection of forces acting on the pylon of asymmetrical with
= 2(gp) + 2 + = = 2
2.138
= 2 ( )
( )
Where
The quantity of the pylon
Effect of nonlinearity
Nonlinearity effects including cable due to self-weight of stay cable and p-delta
effects due to interaction of deck and tower are also considered in the analysis of both
bridge types. Reduced or equivalent modulus of elasticity of stay cables is determined
by
( )
Known as Ernst‘s formula in which Eeq is the equivalent modulus of elasticity,
Where
E - The effective material modulus of elasticity.
67
A - The cross-sectional area of stay cable,
w - Cable weight per unit length,
L -Horizontal projected length and
T - The tensile force in stay cable
2.8.4 Need of optimization
The major need for structural optimization is to review and enhance the current
state of structural design techniques and to show possibilities of these methods in
structural work( wubishet J. Abebe, 2018).
structural optimization needs to determine the most suitable combination of design
variables, so to achieve the satisfactory performance of the structure subject to the
behavioral and geometric constraints impose, with the goal of optimality being
defined by the objective function for specified loading or environmental
conditions(Klarbring & An, 2008).
It allows a better manipulation of material, thus decreasing structure self-weight
and saving material costs(AN, n.d.).
Also, give the structure a higher aesthetic value. Therefore structural optimization
is important for any design to get the most economical output.
68
3. METHODOLOGY
3.1 General
Structural optimization is the subject of making an assemblage of materials that
sustains loads in the best way. Term best to make the structure as light as possible, to
make the structure as stiff as possible, and to make it as insensitive to buckling or
instability as possible. Maximizations or minimizations cannot be performed without
any constraints. Quantities that are usually constrained in structural optimization
problems are stresses, displacements, and/or geometry (Klarbring & An, 2008).
Nonlinear constraint value is used for structural optimization of extradosed cable-stay
bridge. Because any civil design has its own restriction or limit values. So extradosed
cable-stay bridge has its restriction for each element design of the bridge for the
pylon, stay cable, and for post-tensioning box girder. Each limit values takes as
constraint for optimization.
3.2 Method of Structural Analysis
The method of structural analysis with the use of simplified load distribution factors
for distributing the loads among between box girders, stay cable. Abay bridge was
built in an earthquake-free zone in Bahir Dar(EBCS EN- 8 Part 1, 2014). But wind
load and vehicle load are investigated as dynamic load. MIDAS CIVIL 2019 use for
Structural analysis and model of extradosed cable-stay bridge. Worst load effects
were investigated under applicable load combinations was given in design. All load
effect is considering in structural analysis of the bridge. After analysis, take the
moment and shear value of the section then finding the optimum section of the bridge.
3.3 Method of Design Optimization
In this research design, optimization problems were handled with the use of an
evolutionary or genetic algorithm (GA) after it has been tested under simple manually
solved optimization problems. Recent advances in the field of computational
intelligence led to several promising optimization algorithms. These algorithms have
the potential to find optimal or nearly optimal solutions to complex problems within a
reasonable time frame. Structural optimization is a research field where algorithms are
applied to optimally design structures. It is essentially the combination of two
research fields, structural mechanics and computational intelligence.
69
In these thesis objectives, the concrete grade effect on the weight of girder, pylon, and
angle of stay cable have been considered case by case to compare the weight of
parameter and recommend for practical use.
After testing by simple optimization problem, the present problem had connected to
the optimizer, the optimization of superstructure parameter the such as depth of
girder, the height of the pylon, angle of stay cable, concrete grade, bottom slab width,
and side cantilever dimension with constant span 380m with constant carriage width
of 24.7m and Fe-420 grade of steel were carried out. Worst load effects are
investigated under applicable load combinations and then the results are input into GA
optimization code prepared within the built-in MATLAB R2016a software. After that,
the code was run to generate the outputs and the validity of the result was verified by
exporting it into an excel spreadsheet.
GA based optimization depends on three important aspects:
Coding of design variables.
Evaluation of fitness of each solution string.
Application of genetic parameters (selection, cross-over, and mutation) to
generate the next generation of solution strings.
Geometry to fulfill all the requisites from the ULS and SLS as well as to provide
minimized the total weight of the structure subject to constraints on deflection and
stresses in the structure under a given load.
3.4 Materials Used
The cable passes through the pylon through the embedded steel conduit and is
tensioned on the main beam. By standard, a stay cable consists of a prestressed strand
with high strength and low relaxation. The stay cables are made in a factory. C1~C3
is composed of 43- 15.2 (1x7) strands and C4~C9 is composed of 55-15.2 (1x7)
strands. The steel strand is galvanized (China Communication Construction Company
Limited, 2020).
70
3.4.1 Prestressed Reinforcement
For post-tension prestressing strand:
I. The strand should be low-relaxation, seven-wire, Type 1x7 (d15.2mm), with the
tensile strength of 1860Mpa (Grade 270), complying with AASHTO M 203/M
(ASTM A 416/A 416M).
II. The properties of the anchorages should be according to ERA 2013 standard.
III. The prestressing strand should be tensioned with 4296.6KN (75% specified
strength) +friction of the anchorage and jack for M15-22, 3710.7KN (75%
specified strength) +friction of the anchorage, and jack for M15-19, 3320.1KN
(75% specified strength) +friction of the anchorage and jack for M15-17.
IV. The strength of the grouting materials should be no less than 55Mpa and comply
with the corresponding specifications
V. The properties of the duct should comply with corresponding specifications, and
connecting of the ducts should be done with one size bigger and minimum length
should be no less than 30cm. The reinforcement welding procedures and
requirements should comply with AWS D1.4/D1.4M:2018 structural welding
code—steel reinforcing bars, during welding, the minimum overlap length should
be 20cm, the welding length should be minimum 5d or 10cm(take the bigger
value) for double side welding, minimum 10d for single-side welding, and in one
location, the welding joint percentage should not more than 50%, and the stagger
distance should be minimum 35d or 500mm((take the bigger value) (China
Communication Construction Company Limited, 2020).
3.4.2 Reinforcement
The steel bars used in the design of the beam are of high yield strength as shown in
Table 3.4-1
Table 3.4-1reinforcement steel strength (China Communication Construction Campany
Limited, 2020)
Grade of Steel Min. yield strength, fy [Mpa] Modulus of Elasticity, Es [Mpa] Dia.
Grade 420 420 2x105
>=D12
(Max.D32)
71
3.5 Optimization Procedure with GA in Matlab
Procedures involved in design optimization by GA in Matlab were given in the
following steps.
Step 1: Define fitness function. (i.e. define weight function as objective). Open
Matlab and HOME menu click the New Script button. The new script edit field is
displayed under the EDITOR menu. Use % symbol to write a comment for readers in
which Matlab could not read. It is used for defining code for users.
If % appears before any statement. The new script starts to type the fitness function
and give it its original name, in this case, let it be Extramaincompfun(x) to denote
the objective function of extradosed box girder parameter of the main load-carrying
component.
Move in other weight parameters before stating the objective function as follows and
list the weight function. The objective function must write as end letter fun.
Now define the weight function as specified below
72
And save it suppressing ―CTRL+S‖ to file folder, it must save by the same name of
objective function ‗Extramaincompfun‘ which is the name of objective function
defined earlier.
Step 2. Define the constraint function.
Then add a new script and type the constraint functions as follows. Next enter
parameters of constrained functions. Note C and Ceq stands for nonlinear inequality
and equality constrained functions respectively.
After defining all parameters in terms of design variables, next state the constraint
functions as follows.
73
Step 3. Define the main function and develop GA.
Add a new script and define the boundaries of design variables in the main file
74
Next call GA with the following syntax to solve the problem. Save this file with a
name, let it be ‗maincodeBG.m’ for this case.
Step 4: Run the code. Generate optimization results. After the end of development,
GA runs it. The program prompts to change the folder so that click change folder.
Unless the current folder is active or being opened by the program, Matlab couldn‘t
understand the newly developed code and solve it.
If all constraints are fulfilled print the outputs the optimum values of the design
variable and if not adjust lower and upper bounds and repeat it
75
3.5.1 Running the model
Workflow of structural optimization by MATLAB by genetic algorithm(Chessa,
2020)
3.6 Study Variables
Study variables have two parts that are independent variables and dependent
variables.
3.6.1 Independent variables
Depth of girder.
Define constant geometry value
Generate initial number of samples in the population
(NPOP);
Decoding of design variables
Evaluate constraints, objective function and
penalized objective function
Apply genetic operators
(reproduction, crossover and
mutation)
optimum
solution
No
Yes
Check
convergenc
Figure 3.5-1Working flow genetic algorithm for optimization
76
Height of pylon
Stay cable angle.
Concrete grade
Sensitivity analysis
3.6.2 Dependent variables
Weight optimization of prestressed post-tension superstructure part of extradosed
cable-stay Bridge. The weight component of the bridge contains the optimum weight
of non-prestressing reinforcement, torsional reinforcement, concrete and prestressing
reinforcement.
3.7 Load Analysis
The structural model of Abay Bridge was done by MIDAS CIVIL 2019. Linear static
structural analysis was made using distribution factors for shear and moment given
under AASHTO Article 4.4.2.2. In the analysis of loads, a dynamic load allowance or
impact factor (IM) of 15% for fatigue and fracture limit state and 33% for all other
limit states was considered as stated in AASTO Article 3.6.2.1. These factor accounts
for hammering when riding surface discontinuities exist, and long undulations when
the settlement or resonant excitation occurs. Depending on the number of lanes loaded
multiple presence factors (m) specified under AASHTO Article 3.6.1.1.2 were used to
modify the vehicular live loads for the probability that vehicular live loads occur
together in a fully loaded state.
3.7.1 Load Cases and Load Combinations
Live loads are forces that act momentarily on structures due to moving loads. The
following Live loads are considered in the design of Abay bridge structures 《
AASHTO Bridge Design Specification 2010》3.6.1.1.6 Pedestrian Loads Pedestrian
and Bicycle of 3.6 m/10-3
. These are Traffic Loads due to trucks and Lorries. For the
new bridge design, the vehicular live load considered was《AASHTO Bridge Design
Specification 2010》 . This comprises the Design Truck or Design Tandem and
Design Lane Load.
77
3.7.2 Design Truck Load
The total design truckload is 325kN distributed between 3 axles the front axle load
being 35 KN and two rear axle loadings being145 KN each. The relative spacing of
the three axles is illustrated in 《AASHTO Bridge Design Specification 2010》.
Figure 3.7-1 Characteristics of the Design Truck adopted from AASHTO Bridge Design
Specification 2010
3.7.3 Braking force
The longitudinal displacement of the car when it is broken on the bridge and the beam
body causes the bridge to be stressed. According to the 《AASHTO Bridge Design
Specification 2010》, 3.6.4 adopts 5% of the designed truck plus lane load or 5% of
the designed series plus lane load, selects the maximum braking force of the truck,
and arranges 3 lanes. The uniform braking force on the 380m bridge length.
Table 3.7-1 Braking force values
Braking force Axle
weight(kN)
Bridge
length(
m)
Uniform
force(N/mm)
Single
lane
coefficient
Three lane
reduction
factor
Uniform
force
(kN/m)
Truck 320.3 0.05
1.29
Design tandem 222.4 0.05 1.26
Lane 380 9.3 3534 0.05
3.7.4 Wind Load
In the case of footbridges, the height of the moving load is to be taken as 2 meters
throughout the length of the span.
78
The wind pressure effect is considered as a horizontal force acting in such a direction
that resultant stresses in the member under consideration are the maximum. The
nature of the wind load is dynamic. This means that its magnitude varies concerning
time and space.
1. Horizontal Wind Pressure
The design wind speed is based on the formula in the ERA Road Design Manual
(3.8.1.1-1).
[
] [
]
3.1
Where
V0 (m/s) - The average maximum wind speed for 10 minutes in the condition of flat
open ground, a terrain clearance of 10m and a return period of 10, 50 or 100 years
The V0 friction coefficient is 17.6km/hr. according to the urban and suburban
coefficients;
V10=105km/hr. according to C3.8.1.1;
VB=160km/hr.;
Z -The height from ground or horizontal plane, value is=16m;
Z0=1m.
VDZ=22.2m/s.
The calculation process of horizontal wind load of the pylon. The calculation process
is as follows.
Figure 3.7-2 Cross section of pylon and wind direction
79
Table 3.7-2 Pylon Wind Load Calculation
Node
No V10
(km/h)
V0
(km/hr)
Vb
(km/hr
)
Z
(M)
Vdz
(km/hr
)
Vdz
(m/s)
PB
(Mpa)
PD(Mpa
)
Width of
Pylon
(m)
1 105 17.6 160 12.8 73.6 20.4 0.0024 0.0005 2.00
2 105 17.6 160 13.3 74.7 20.8 0.0024 0.0005 2.00
3 105 17.6 160 13.8 75.8 21.1 0.0024 0.0005 2.00
4 105 17.6 160 14.3 76.8 21.3 0.0024 0.0006 2.00
5 105 17.6 160 14.8 77.8 21.6 0.0024 0.0006 2.00
6 105 17.6 160 15.3 78.8 21.9 0.0024 0.0006 2.00
7 105 17.6 160 15.8 79.7 22.1 0.0024 0.0006 2.00
8 105 17.6 160 16.3 80.6 22.4 0.0024 0.0006 2.00
9 105 17.6 160 16.8 81.5 22.6 0.0024 0.0006 2.00
10 105 17.6 160 17.3 82.3 22.9 0.0024 0.0006 2.00
11 105 17.6 160 17.8 83.1 23.1 0.0024 0.0006 2.00
12 105 17.6 160 18.3 83.9 23.3 0.0024 0.0007 2.00
13 105 17.6 160 18.8 84.7 23.5 0.0024 0.0007 2.00
14 105 17.6 160 19.3 85.5 23.7 0.0024 0.0007 2.00
15 105 17.6 160 19.8 86.2 23.9 0.0024 0.0007 2.00
16 105 17.6 160 20.3 86.9 24.1 0.0024 0.0007 2.00
17 105 17.6 160 20.8 87.6 24.3 0.0024 0.0007 2.00
18 105 17.6 160 21.3 88.3 24.5 0.0024 0.0007 2.00
19 105 17.6 160 21.8 89.0 24.7 0.0024 0.0007 2.00
20 105 17.6 160 22.3 89.6 24.9 0.0024 0.0008 2.00
21 105 17.6 160 22.8 90.3 25.1 0.0024 0.0008 2.00
22 105 17.6 160 23.3 90.9 25.3 0.0024 0.0008 2.00
23 105 17.6 160 23.8 91.5 25.4 0.0024 0.0008 2.00
24 105 17.6 160 24.3 92.1 25.6 0.0024 0.0008 2.00
25 105 17.6 160 24.8 92.7 25.8 0.0024 0.0008 2.00
26 105 17.6 160 25.3 93.3 25.9 0.0024 0.0008 2.00
27 105 17.6 160 25.8 93.9 26.1 0.0024 0.0008 2.00
28 105 17.6 160 26.3 94.4 26.2 0.0024 0.0008 2.00
29 105 17.6 160 26.8 95.0 26.4 0.0024 0.0008 2.00
30 105 17.6 160 27.3 95.5 26.5 0.0024 0.0009 2.00
31 105 17.6 160 27.9 96.1 26.7 0.0024 0.0009 2.00
32 105 17.6 160 28.5 96.7 26.9 0.0024 0.0009 2.00
33 105 17.6 160 29.1 97.3 27.0 0.0024 0.0009 2.00
34 105 17.6 160 29.7 97.9 27.2 0.0024 0.0009 2.00
35 105 17.6 160 30.3 98.5 27.4 0.0024 0.0009 2.00
80
Table 3.7-3 Pylon Wind Load Calculation
Node
No
V10
(km/h)
V0
(km/hr) Vb(km/hr) Z (M) Vdz(km/hr) Vdz(m/s) PB(Mpa) PD(Mpa)
Width of
Pylon(m)
36 105 17.6 160 30.9 99.1 27.5 0.0024 0.0009 2
37 105 17.6 160 31.5 99.6 27.7 0.0024 0.0009 2
38 105 17.6 160 32.1 100.2 27.8 0.0024 0.0009 2
39 105 17.6 160 32.7 100.7 28 0.0024 0.001 2
35 105 17.6 160 33.3 101.2 28.1 0.0024 0.001 2
36 105 17.6 160 33.9 101.7 28.3 0.0024 0.001 2
38 105 17.6 160 34.5 102.2 28.4 0.0024 0.001 2
39 105 17.6 160 35.1 102.7 28.5 0.0024 0.001 2
40 105 17.6 160 35.7 103.2 28.7 0.0024 0.001 2
41 105 17.6 160 36.3 103.7 28.8 0.0024 0.001 2
42 105 17.6 160 36.9 104.2 28.9 0.0024 0.001 2
43 105 17.6 160 37.5 104.7 29.1 0.0024 0.001 2
44 105 17.6 160 38.1 105.1 29.2 0.0024 0.001 2
45 105 17.6 160 38.7 105.6 29.3 0.0024 0.001 2
46 105 17.6 160 29.7 97.9 27.2 0.0024 0.0009 2
47 105 17.6 160 30.3 98.5 27.4 0.0024 0.0009 2
48 105 17.6 160 30.9 99.1 27.5 0.0024 0.0009 2
49 105 17.6 160 31.5 99.6 27.7 0.0024 0.09 2.0
2. Wind Pressure on Structures
[
]
3.2
Where
PB =0.0024MPa;
- The design wind speed when calculating at height z
PD-The wind pressure on structure values of PD= 0.0006Mpa,
Vb- The design benchmark wind speed when calculating height is z
Zo=1 (m),
Wl=basic wind pressure, value=1.46 N/mm
Structural wind = beam height + guardrail height * PD
Wind load on the pier is calculated by, Wind Load= Width of Pier* Pd*1000(kN/m).
81
Table 3.7-4Wind Load Calculation Table
Nod
e
No
V10(km/h
r)
V0(km/h
r)
Vb(km/h
r)
Z(M
)
Vdz(km/h
r)
Vdz(m/
s)
Pd(Mpa
)
Width
of
Pier(
M)
Wind
Load(kN/
m)
1 105 17.6 160 28 96.2 26.7 0.00087 9.3 8.03
3 105 17.6 160 13 74.1 20.6 0.00051 9.1 4.69
5 105 17.6 160 12 71.8 19.9 0.00048 9 4.35
7 105 17.6 160 11 69.2 19.2 0.00045 8.9 3.99
9 105 17.6 160 10 66.5 18.5 0.00041 8.8 3.63
11 105 17.6 160 9 63.4 17.6 0.00038 8.6 3.26
12 105 17.6 160 8 60 16.7 0.00034 8.5 2.88
13 105 17.6 160 7 56.2 15.6 0.0003 2.49
3. Vertical Wind Load on the beam
In EN 1991-1-4, wind load on bridge decks are considered to be coming from the
longitudinal (y-direction) or transverse (x-direction) axes and these actions generate
stresses in the x, y, x directions of the bridge deck. During analysis, the wind must be
considering only one direction (either x or y direction) for each load combination. In
the case of the Abay bridge During the construction process, the main beam on one
side of the pylon is considered to bear 1 time of vertical wind load, and the main beam
on the other side bears 0.5 times vertical wind load.
Figure 3.7-3Wind direction on bridge deck (Fig 8.2 EN 1991-1-4)
82
According to《ERA 2013 Bridge Design Manual》,PD=0.00096Mpa. The calculation
process is as follows.
Table 3.7-5Beam Vertical Wind Load Calculation
PD(Mpa) Width of Beam(M) Wind Load (kN/m)
0.000960 24.70 23.71
4. Wind Pressure on Vehicles
According to 3.8.1.3 of 《AASHTO Bridge Design Specification 2010》, the moving
force is 1.46kN/m, and this force is 1.8m on the bridge. The vehicle wind is
2.63kN/m.
5. Symmetric vertical curved fundamental frequency fb.
For Extradosed Bridge without auxiliary piers, fb value is 0.611Hz according to the
empirical formula. According to the calculation results of the overall bridge model, fb
is 0.937Hz.
6. Symmetric vertical curved fundamental frequency fb
The symmetric vertical bending fundamental frequency fb of this bridge is calculated
as 0.937Hz according to the overall model. AASHTO Bridge Design Specification
2010.
7. Symmetric torsional fundamental ft
The value of ft is 1.043Hz according to the empirical formula
8. Cable base frequency fn
The calculation table of cable base frequency fn is as follows: m1=63 kg/m from (C1-
C4)
(
)
3.3
Where
m2=mass per length of each stay cable, value 63 Kg/m from (C1-C3) and 78Kg/m
from (C4-C9)
n=1 for all cable l -length
83
Table 3.7-6 Side span cable base frequency fn calculation
Side-span
Cable Number Fn (Hz) M (kg/m) F(N) L (m) N
C1 5.16 63 5059000 27.480 1
C2 4.23 63 5215000 33.980 1
C3 3.58 63 5352000 40.685 1
C4 2.86 78 5704000 47.358 1
C5 2.52 78 5820000 54.264 1
C6 2.25 78 5902000 61.217 1
C7 2.02 78 5947000 68.204 1
C8 1.84 78 5955000 75.215 1
C9 1.67 78 5909000 82.245 1
Table 3.7-7 Middle span Cable base frequency fn calculation
Middle-span
Cable Number Fn (Hz) M (kg/m) F (N) L (M) N
C1 5.17 63 5041000 27.356 1
C2 4.24 63 5179000 33.837 1
C3 3.59 63 5320000 40.527 1
C4 2.86 78 5675000 47.188 1
C5 2.52 78 5800000 54.080 1
C6 2.25 78 5894000 61.021 1
C7 2.03 78 5955000 67.996 1
C8 1.85 78 5982000 74.995 1
C9 1.69 78 5962000 82.012 1
9. Galloping stability
Galloping is an aeroelastic self-excited phenomenon, characterized by low
frequencies and high amplitudes. It was observed for the first time on ice-covered
power lines subject to strong wind. Galloping is a typical instability induced by the
coupling of aerodynamic forces, which affect the structure along with its vibration.
The vibration of the structure periodically changes the angle of attack of wind.
Changes in the angle of attack induce change in aerodynamic forces affecting the
structure, changing the structure‘s response(Jafari, 2019). The first simplified criterion
(assuming the model of a single-degree-of-freedom system) concerning the instability
connected with galloping was introduced by Glauert-Den Hartog and is as follows.
84
(
) 3.4
Where
α - The angle of attack,
CL and CD-Aerodynamic coefficients of the lift force and drag force, respectively.
A necessary condition for galloping to take place (as part of quasi-steady theory) is
the occurrence of negative aero-elastic damping in the system. Cable with circular
cross-section cannot gallop, because of its geometrical symmetry (
⁄ )
unless the cross-section is changed. Ice accumulation on a cable causes changes in its
cross-section and, as a result, it leads to the cable‘s aerodynamic instability.
10. Wake galloping for groups of cables
Wake galloping is an elliptical motion caused by the variability of forces that affects
cables that are in the wake of other structural elements (e.g. pylons or other cables).
This phenomenon occurs at high wind velocities and is associated with high vibration
amplitudes. Cooper has proposed an approximate global stability criterion in the form
of minimum critical wind velocity above which wake galloping can be expected.
Minimum critical velocity is proportional to a square root of the Scruton number,
frequency of cable‘s vibration, and diameter. At small distances between cables (from
2 to 6 x cable diameters), leeward cables move along an almost circular orbit; for
larger distances, the orbit becomes elliptical, with the main axis of the ellipsis situated
approximately in the same direction as the wind direction. These types of vibration
have been observed on overhead power lines, with amplitudes around 20 x cable
diameters and with cables arranged with a span between them of 10 to 20 x diameters
(Golebiowska & Peszynski, 2018). The main beam and cable tower of the bridge are
made of concrete structure with large damping and no galloping effect.
11. Wake galloping
The minimum longitudinal spacing of the stay cables of this bridge is about 3m,
which appears near the main tower end of C1 ~ C4. The diameter of the stay cables is
0.24m, and the wake gallop vibration test should be carried out for the stay cables.
85
:
⁄
⁄ ;
3.5
Where
δs=0.001,
ρ-The constant related to the density of air, value =1.25 kg/m3,
Dc=0.24
Table 3.7-8 for calculating critical wind speed of wake vibration of side span cable
Side-span
Cable
Number
Uwg (m/s) Cwg Fn (Hz) M (kg/m) δs Ρ (kg/m3) Dc (m)
C1 92.6 80 5.16 63 0.001 1.25 0.24
C2 76.0 80 4.23 63 0.001 1.25 0.24
C3 64.3 80 3.58 63 0.001 1.25 0.24
C4 57.1 80 2.86 78 0.001 1.25 0.24
C5 56.6 90 2.52 78 0.001 1.25 0.24
C6 50.5 90 2.25 78 0.001 1.25 0.24
C7 50.6 100 2.02 78 0.001 1.25 0.24
C8 45.9 100 1.84 78 0.001 1.25 0.24
C9 41.8 100 1.67 78 0.001 1.25 0.24
Table 3.7-9 Calculation table of critical wind speed of wake vibration of mid-span cable
Middle-span
Cable
Number Uwg(m/s) Cwg Fn(Hz) M(kg/m) δs Ρ(kg/m
3) Dc(M)
C1 92.9 80 5.17 63 0.001 1.25 0.24
C2 76.1 80 4.24 63 0.001 1.25 0.24
C3 64.4 80 3.59 63 0.001 1.25 0.24
C4 57.1 80 2.86 78 0.001 1.25 0.24
C5 56.7 90 2.52 78 0.001 1.25 0.24
C6 50.6 90 2.25 78 0.001 1.25 0.24
C7 50.8 100 2.03 78 0.001 1.25 0.24
C8 46.1 100 1.85 78 0.001 1.25 0.24
C9 42.1 100 1.69 78 0.001 1.25 0.24
86
It can be seen from the above table that the minimum critical wind speed of the wake
gallop of the stay cable is 41.8m, the design reference wind speed of the stay cable is
26.7m/s, and the partial coefficient of the wake gallop stability is 1.2,
41.8>26.7 * 1.2 = 32.1
12. Flutter stability
Table 3.7-10 If calculation table of the flutter stability index
If Ks Μ Ud(m/s) ft(Hz) B(M) M(kg/m) Ρ(kg/m3)
1.79 12 114.8 20.6 1.043 12.35 68714 1.25
As can be seen from the above table, the flutter stability index If =0.9<2.Therefore,
the Uf table of the critical flutter wind speed of this bridge is as follows:
Table 3.7-11 Uf table for flutter critical wind speed
Uf(m/s) εs Ηα Uco(m/s) Μ ft(Hz) B(m) b(m) R(M)
160.5 0.9 0.7 254.7 114.8 1.043 24.7 12.35 1.68
Table 3.7-12 Flutter stability list
Uf (m/s) εf εt εα Ud(m/s) εf εtεαUd (m/s)
160.5 1.4 1.33 1 20.6 38.4
The critical wind velocity Uf is much higher than the f t Ud, and the bridge flutter
meets the design requirements.
Uf> Ud 3.6
13. Structural vortex-induced resonance
Both the main beam and cable tower of this bridge adopt concrete structure, with
large damping. Aerodynamic negative damping is not enough to overcome structural
damping, and no vorticity resonance effect is generated.
87
14. Stay cable vortex-induced resonance
Table 3.7-13 Calculation table of vortex-induced resonance amplitude YMAX of stay cable
ymax (M) St Sc Dc(M) σCl M(kg/m) δs Ρ(kg/m3)
0.02 0.2 1.08 0.24 0.45 78 0.001 1.25
The vortex-induced amplitude of the stay cable meets the design requirements.
15. Temperature force
1. Temperature Load
Temperature loads are forces due to temperature change. The following temperature
loads are considered in the design of structures.
2. Uniform Temperature
According to the ERA Bridge Design Manual provision 3.12.2, the ranges of
temperature shall be as specified in ERA Bridge Design Manual Table 3.12.2-1. The
difference between the extended lower or upper boundary and the base construction
temperature assumed in the design shall be used to calculate thermal deformation
effects.
Table 3.7-14 Temperature ranges
Steel or Aluminium Concrete Wood
-5° to 50°C 5° to 35°C 0° to 35°C
-5° to 50°C 5° to 35°C 0° to 35°C
16. Temperature Gradient
According to the ERA Bridge Design Manual provision 3.12.3, positive temperature
values shall be taken as specified for various deck surface conditions in ERA Bridge
Design Manual Table 3.12.3-1. Negative temperature values shall be obtained by
multiplying the values specified in ERA Bridge Design Manual Table 3.12.3-1 by -
0.30 for plain concrete decks and -0.20 for decks with an asphalt overlay.
88
Table 3.7-15 Basis for temperature gradients
Zone T1 (°C) T2 (°C) T3 (°C)
1 (Lowland, below +1 500 m level) 30 8 0
2 (Highland, above +1 500 m level) 35 10 5
Positive Vertical Temperature Gradient in Concrete and Steel Superstructures
Figure 3.7-4 Positive Vertical Temperature Gradient(Load and Resistance Factor Design.,
2015)
17. Settlement
The settlement position of the support is selected according to the position of the main
beam support to simulate the position of the main beam center point. The settlement
of the main pier support is calculated by -0.5cm.
3.7.5 Load Combination and Load Factors
Structural components shall be proportioned to satisfy the requirements at all
appropriate service and strength limit states. The load factors and load combination
were specified in 《AASHTO Bridge Design Specification 2010》.
89
Table 3.7-16 Load combination
Load Combination Limit State Participation in portfolio projects and coefficients
DC DW LL
Strength I 1.25 1.5 1.75
Strength V 1.25 1.5 1.35
Service I 1 1 1
Service III 1 1 0.8
3.8 Design Philosophy
The design philosophy used in this thesis is the AASHTO load and resistance factor
design (LRFD) approach stated in Article 1.3.2 is given below.
∑
3.7
Where,
i – Load modifier as per ODOT recommendation is a value of 1.05 used.
j – Load factor, statically based multipliers applied to force effects.
Φ – Resistance factor, statically based multipliers applied to nominal resistance; a
value of 0.90 is used for both shear and flexure as given in AASHTO LRFD Article
5.5.4.
Qi – Force effects.
Rn – Nominal resistance.
3.9 Optimization Problem Formulation
In this thesis, problem formulation was based on linear elastic analysis and ultimate
strength method of design with the consideration of serviceability constraints as per
AASHTO LRFD 2010 code was used. Optimization problem algorithm develops by
GA. In GA, the population is a collection of candidate solutions. An important feature
of a population, especially in the early generation of its evolution, is its genetic
diversity. The too-small population size may lead to a scarcity of genetic diversity.
90
It may result in a population dominated by almost equal chromosomes and then, after
decoding the genes and evaluating the objective function it converges quickly but
leads to a local optimum. On the other hand, if there is too large a population, the
overabundance of genetic diversity can lead to the clustering of individuals around
different local optimal. But the mating of individuals belonging to different clusters
can produce children lacking the good genetic part of either of the parents. In
addition, the manipulation of large populations is excessively expensive in terms of
computer time(S.N.Sivanandam, 2008). Thus the proper selection of population size
is extremely important.
The formulation of the optimization problem had been made by utilizing the interior
penalty function method as an optimization method to minimize the objective
function representing the weight of the superstructure component.
3.10 Numerical Model
Modeling is a way to create a virtual representation of a real-world system that
includes software and hardware. If the software components of this model are driven
by mathematical relationships, it can simulate this virtual representation under a wide
range of conditions to see how it behaves. Numeric models are the
basic numeric representation of linear systems or components of linear systems. In
this paper extradosed stay cable bridge superstructure component numerical model in
Matlab.
3.10.1 Numerical modeling of prestressing box girder extradosed cable-stayed
Bridge
In numerical modeling of prestressing post-tensioning box girder extradosed stayed-
Cable Bridge. Abay extradosed stay cable has an Unsymmetrical post-tensioning
prestressing girder. Design variables prestressing box girder include overall depth,
Area of non Prestressing reinforcement, Area of Prestressing reinforcement, the width
of bottom deck, spacing of traverse reinforcement (m), and NA depth of the cracked
transformed section (m). These variables are listed in Table 3.10-1.
91
Figure 3.10-1 Cross-section of the bridge deck
Table 3.10-1 Coding of design related to box girder variables
No Description Notation
1 Overall depth (D) D
2 Width of side cantilever(m) W1
3 Bottom slab width(m) W2
4 Fulcrum Section area(mm2) Acp
5 Area of non Prestressing reinforcement (mm2) As
6 Area of Prestressing reinforcement (mm2) Ap
7 NA depth of the cracked transformed section(m) Y
8 spacing of traverse reinforcement(m) S
Table 3.10-2 Designation of Design Variables
Variable Designation D W2 W1 At As Ap Y S
Matlab code Designation x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8)
3.10.2 Numerical modeling of stay cable extradosed stayed- Cable Bridge
Figure 3.10-2 Cables number and geometry of the Abay extradosed stay cable bridge
92
Table 3.10-3Coding of design related to cable-stay variables
No Descriptions Notations
1 The angle of stay cable A
2 The ratio between equivalent stay cable force and dead load wc/w
3 Length of stay cable Lcb
4 Number of stay cable N
5 Horizontal length of between pier and first anchorage point HL
6 cable stayed spacing on pylon Sv
7 cable Stayed spacing on girder Sh
Table 3.10-4Designation of Design Variables
Variable Designation A R Lcb N HL Sv Sh
Matlab code
Designation x(1) x(2) x(3) x(4) x(5) x(6) x(7)
3.10.3 Numerical modeling of pylon parameter of extradosed cable-stayed
Bridge
The main parameter related to pylon height is the numerical model. This parameter
must be an independent variable. If one variable expressing by other parameters the
variables must be removed from the model. Pylon is one of the main loading
components it transferred cable force into the pier.
Table 3.10-5Coding of design related to pylon variables
No Description Notations
1 Height of pylon above deck H
2 Area of non prestressing reinforcement
bars(mm2)
As
Table 3.10-6 Designation of Design Variables
Variable Designation H As
Matlab code Designation x(1) x(2)
93
3.11 Fixed Design Variables
In the case study of structural optimization of Abay extradosed stayed cable bridge,
the width of the deck including the distance between barriers, the thickness of deck,
length of span, number of lanes, tower height below deck, cross-section, and shape of
the tower are kept constant as they are governed by the traffic requirement. The cross-
section of the tower is assumed to be a solid reinforced concrete box. Weight of
anchorage also fixed load in the bridge.
Also, reinforcement steel grade, modulus of elasticity, and unit weight of concrete and
reinforcement, the magnitude of dead and live loads were assumed to be fixed or pre-
assigned parameters. Consequently, the total weight of the structure was calculated
using fixed parameters of the density of the unit volume of concrete and the unit
weight of reinforcement. Values of fixed parameters and the defined materials
property are given in Table 3.11-1.
Table 3.11-1 Material property
Items Properties Values
Modulus of Elasticity of Concrete Ec 27660 N/mm2
Modulus of Elasticity of prestressing steel Ep 195000 N/mm2
Modulus of Elasticity of non-prestressing steel Es 200000 N/mm2
Specified compressive strength of concrete girder Fc 55 N/mm2
Specified compressive strength of concrete pylon
50N/mm2
Yield strength of reinforcing bars Fy 420 N/mm2
Ultimate tensile strength of prestressing steel Fpu 1860 N/mm2
The density of reinforcement steel 7.850x109
3.12 Design Variables
The formulations of an optimization problem begin with noticing the underlying
geometric design variables.
94
These variables should be independent of each other. If one of the design variables
can be expressed in terms of the other variable then that variable can be removed from
the numerical model.
3.13 Objective Function
In structural design, the dominant objective was to minimize the structural weight (i.e.
it minimized cost). There can be multi-objective functions such as minimizing cost,
maximize performance and maximize reliability. But generally, it is avoided by
choosing the most important objective as the objective function and the other
objective functions were included as constraints by restricting their values within a
certain range. In this research minimization of the initial weight of bridge girders,
pylon, and stayed cable was carried out. So the total weight is the weight of concrete
and the weight of non Prestressing reinforcement and prestressing steel and the weight
of stay cable. The function below defines the total weight of extradosed superstructure
components in terms of weight of concrete, stay cable, Prestressing reinforcement,
and non Prestressing reinforcement.
( )
(( )
)
( )
)
( )
3.8
Where
c- Concrete density (g/mm3)
S-Density non-prestressing reinforcement bar (g/mm3)
p -Density of Prestressing reinforcement bar (g/mm3)
Cab-Stay cable bar density (g/mm3)
Aeff – Area of concrete cross-section (mm2)
As – Area of longitudinal reinforcement (mm2)
Ap – Area of prestressing tendons (mm2)
L – Span length of the girder (mm)
Wstr– Weight of stirrups (g)
95
vp- Total volume of pylon(mm3)
vs- Volume of non Prestressing reinforcement (mm3)
3.14 Constraint Function
The constraints reflect design requirements in the optimization problem. In other
words, they limit the range of acceptable designs in the problem. In this research, the
constraints relevant to the design of Prestressing post-tensioning box girder, stay cable
angle and pylon height are applied using a penalty function. Generally, structural
design is required to conform to the number of inequality constraints related to
stresses, deflection, dimensional relationships, and other code requirements. The
constraint functions formulated were constrained nonlinear programming problems
for numerical solutions of post-tension box girders and stay cable angle and pylon.
These formulations were coded in the script for constraint function definition in GA
packages of Matlab software.
3.15 Sensitivity Analysis of Optimum Section of Extradosed Cable Stay
Bridge
Sensitivity analysis is conducted to evaluate the dependence of structural
performances on design or imperfection parameters. As stated in the Preface,
dependent on parameters to be employed, sensitivity analysis in structural stability.
Sensitivity analysis is a promising way to provide a tuned analytical model and assess
the actual dynamic characteristics of superstructures such as extradosed cable-stayed
bridges. Sensitivity analysis is a technique to determine the influence of different
properties, such as deflection of bridge, stress of bottom and top girder due to
variation of elastic modulus, stress after and before loss of shrinkage and creep loss
and geometrical parameters and stay cable tension. For sensitivity analysis of
optimization output done by Microsoft excel and JMP Pro 14 used.
96
4. RESULTS AND DISCUSSIONS
The main load-carrying component of extradosed cable-stay bridge was considering
in the structural optimization process. Structural optimization applies to the parameter
of the box girder, the parameter of pylon height, and the parameter of the angle of stay
cable on the optimum weight of extradosed cable-stayed bridge using a genetic
algorithm. The formulated optimization problem was coded into Matlab R2016b
software to run the optimization genetic algorithm. The various parameters such as
effect of concrete grades on optimum weight, optimum girder depth, and optimum
height of the pylon, and optimum angle of stay cable and comparisons of conventional
design and optimum output have been studied by using GA and evaluate sensitivity
analysis of girder weight, cable stiffness and stay cable tension. All result of
parameters was discussed as follows.
4.1 Effect of depth of box girder of on optimum weight of extradosed
cable-stayed bridge
By considering all relevant loads of the bridge the optimum design output of bridge
girder depth was done. Optimization for girder depth of Abay extradosed cable-stayed
bridge did form point of pier up to the mid-span of the bridge. It has 90m in length.
Because this span represents the whole girder of the bridge. The nature of the bridge
depth girder is parabolic shape because of that the depth was optimized in every 10m
section of the bridge by using maximum moment and shear at that point. Maximum
moment and shear of the bridge at different sections obtained from analysis finite
element MIDAS CIVIL 2019 software. The optimum depth of extradosed cable-
stayed bridge was done by considering all parameters that were affected by the depth
of girders such as prestressing reinforcement, non- prestressing reinforcement,
torsional reinforcement, spacing of transverse reinforcement, and neutral axis depth
were optimized other‘s parameters are fixed.
97
Table 4.1-1effect of depth of box girder of on optimum weight of extradosed cable stayed
bridge
Span
Length
h opt
(mm) At(mm
2) As(mm
2) Ap(mm
2) Y(mm) S(mm)
Wop
use (T)
0 5129 5.64E+03 1.68E+08 3.90E+04 2166.6 100 5.47E+03
10 4971 7.15E+03 1.62E+08 3.76E+04 2106.6 120 5.11E+03
20 4571 2.58E+03 7.99E+07 2.38E+03 2056.6 150 4.92E+03
30 4418 7.31E+03 8.19E+07 2.40E+03 1866.6 200 3.97E+03
40 3818 4.92E+03 8.47E+07 4.99E+03 1766.6 220 3.63E+03
50 3516 7.04E+03 1.11E+08 9.99E+03 1666.6 230 3.32E+03
60 3511 3.78E+03 1.28E+08 1.65E+04 1566.6 250 2.98E+03
70 3506 4.36E+03 1.20E+08 1.03E+04 1466.6 260 2.69E+03
80 3131 9.97E+03 6.51E+03 7.97E+04 1366.6 270 2.47E+03
90 2620 3.75E+03 1.05E+09 7.02E+04 1166.6 275 2.31E+03
The result shows that at the point of the pier the girder depth of is 5.129 and at the
point of mid-span and abutment 2.62m. This result closed to the recommended depth
of box girder of extradosed cable-stay bridge that is the depth of girder L/34 at point
of the pier and point of mid-span and point of abutment depth of girder must greater
L/70 (Mermigas, 2008) and (fib Guidance for good bridge design (2000)). The result
obtained from this finding was fall in the range recommended by (Bujnak et al., 2013)
4.2 Effect of Concrete Grades of on optimum the weight bridge and the
depth of girder of extradosed cable-stayed bridge
Effect of concrete grade on the optimum weight of extradosed cable-stay bridge was
given below in Table 4.2-1. Recommended value of concrete grade for long-span
bridge 35 up to 70 Mpa in extradosed cable-stayed bridge. It can be noted from this
value of the table. The relation between the concrete grade and the weight of the
bridge is reversed, with the concrete grade increasing with decreasing weight of the
bridge. Due to the use of high strength concrete effect of concrete grade, not the same
relation when moves from C-35 up to C-75 concrete grade.
98
Table 4.2-1 Effect of grades of concrete on the optimum weight
Section
of
bridge
Optimum weight of bridge in (T)
C-35 C-40 C-45 C-50 C-55 C-60 C-65 C-70 C-75
0 5500 5497 5490 5480 5470 5460 5450 5435 5430
10 5210 5199 5190 5110 5110 5010 4980 4964 4960
20 4930 4927 4920 4920 4910 4900 4880 4860 4860
30 4000 3997 3989 3983 3970 3960 3950 3940 3940
40 3370 3368 3360 3360 3320 3310 3300 3280 3280
50 3659 3650 3650 3630 3630 3620 3600 3560 3560
60 2990 2989 2980 2980 2980 2960 2950 2930 2930
70 2690 2691 2690 2690 2690 2670 2650 2630 2630
80 2500 2490 2480 2480 2470 2450 2430 2410 2410
90 2380 2369 2360 2360 2310 2280 2260 2240 2240
Cumulative optimum weight showed that at lower concrete grade mass large. For high
strength, the concrete optimum weight of the bridge decreased. Because the high
quality of concrete reduced the size of the box girder.
99
Table 4.2-2Cumulative optimum weight of box girder in grade concrete 35 up to 75(Mpa)
Specified Compressive strength of
concrete ,fc, Mpa
Cumulative Mass box girder of extradosed cable-
stay bridge(Ton)
C-35 37229
C-40 37187
C-45 37098
C-50 36993
C-55 36860
C-60 36620
C-65 36450
C-70 36249
C-75 36140
The optimum mass versus the specified compressive strength of concrete draws
graphically as shown in Figure 4.2-1. The graph shows that the effect of concrete
grades on the optimum weight of the bridge is like decreasing graph. It shows
maximum weight at concrete grade C-35 up to C-45. But at concrete strength
increased weight of the bridge is decreasing.
Figure 4.2-1 Effect of grades of Concrete on Optimum weight of the bridge
Effect of concrete grade on optimum weight and optimum depth of box girder at point
of the pier is shown in Figure 4.2-2.
35600
35800
36000
36200
36400
36600
36800
37000
37200
37400
C-40 C-45 C-50 C-55 C-60 C-65 C-70 C-75
Op
tim
um
wei
gh
t(T
)
Specified Compressive Strength of Concrete, fc, Mpa
Optimum
weight(T)
100
At the point of the pier, the load is maximum, there is no support cable. Load resist
and transfer to pier only by box girder. Because of that maximum cross-section of the
box girder appeared at the pier. A graph represents optimum weight and depth have
directly related.
Figure 4.2-2 Concrete specified compressive strength of concrete versus with optimum depth
and optimum weight.
4.3 Effect of the unit cost of Concrete grade on the optimum weight of the
bridge
The weight of the bridge is decreasing when the specified compressive strength of
concrete is increasing from 35Mpa to 75Mpa. But due to using of high-quality
ingredients for higher strength concrete grade cost is highly increase, so it is
impossible to select the optimum concrete grade for Abay Bridge by only considering
the weight of the bridge. Therefore investigating unit cost effect on the bridge is very
essential for selecting optimum concrete grade for Abay Bridge. To determine the unit
costs of each concrete grade first determine the mixing ratio of concrete grade. The
mix design ratio of concrete grade C-35 is up to the C-75 grade list in Table 4.3-1.
There is no fixed mixing ratio like standard concrete grade. The mixing ratio depends
on the quality of cement and types of admixtures.
4900
5000
5100
5200
5300
5400
5500
5600
C-35 C-40 C-45 C-50 C-55 C-60 C-65 C-70 C-75
Opti
mum
wei
ght(
T)
Specified Compressive Strength of Concrete, fc, Mpa
optimum
depth(m)
Optimum
weight(T)
101
Table 4.3-1Unit price cost of concrete grade(Votorantim Cimentos & St. Marys CBM,
2021)and (Peng et al., 2019)
Strengt
h of
concret
e grade
Cemen
t(kg)
Sand
(kg)
Stone
(kg)
Water(
kg)
Chemical
Admixt
ure(Kg)
Mixing ratio of
concrete Birr/m3
C-35 424 581 1179 195 0 1:1.37:2.781 :0.5 9526.85
C-40 488 528 1176 196 0 1:1.1:2.41:0. 4 10176.41
C45 410 610 1170 180 120 1:1.488:2.85 4:0.44:0.3 11475.52
C-50 410 618 1169 171 125 1:1.51:2.851 :0.417:0.305 12341.6
C-55 410 618 1168.9 159.8 130 1:1.51:2.851 :0.39:0.317 12774.64
C-60 410 628 1166.3 152.6 145 1:1.532:2.84 45:0.372:0.3 54 13424.2
C-65 410 625.5 1161.6 150 150 1:1.526:2.83 :0.366:0.366 14073.75
C-70 410 620 1151.4 145.6 145 1:1.512:2.81 :0.355:0.354 14723.31
C-75 420 622.5 1156.1 144.1 155 1:1.482:2.75 3:0.343:0.36 9 15372.87
Based on the unit cost of concrete grade, related it effect of optimum weight on it unit
cost.
Table 4.3-2 Unit cost effect of concrete grade on optimum weight box girder
Grade of Concrete
grade
The unit cost of
concrete(birr/m3)
Cost Ratio Optimum weight, opt(T)* Cost
Ratio
35 9526.85 1 3.72E+04
40 10176.41 1.07 3.97E+04
45 11475.52 1.2 4.47E+04
50 12341.6 1.3 4.79E+04
55 12774.64 1.34 4.94E+04
60 13424.2 1.41 5.16E+04
65 14073.75 1.48 5.38E+04
70 14713.31 1.54 5.60E+04
75 15262.87 1.60 5.81E+04
A graphical representation of the effect of the unit cost of concrete on the optimum
weight of the bridge was shown below.
102
Figure 4.3-1 Effect of the unit cost of concrete grade on the optimum weight of the bridge
The unit cost of concrete grade ratio time‘s optimum weight gives the effect of cost of
concrete on box girder depth. The graph represents the unit cost of concrete grade
effect on the optimum weight of the bridge. As the graph show effect of the unit cost
of concrete grade changes the graph direction from o right. Because high strength
concrete achieved its strength by using admixture and jelly in addition to other
concrete gradients. These lend cost effect is higher than the weight effect when we
move from C-35 to C-75 concrete grade. Therefore based on the value of the table
and graph concrete grade from C-45 up to C-55 is the optimal concrete grade for the
box girder of Abay Bridge. The graph in this range gives intermediate point less
weight and less cost. Using concrete grade from C-45 up to C-55 is not having much
effect. So for Abay extradosed cable-stayed bridge using C-45 concrete grade is more
economical than C-55 concrete grade.
4.4 The optimum height of the pylon
A pylon is a tapered column that is connected to all cables and transmits cable forces
to the pier. The result of optimum pylon height is 24.43m for a fixed span length. As
the grade of concrete increases pylon height and weight are decreased. By optimizing
the height of the column, it will also change the inclination of the cable. Also, stay
cable force depend on the height of the pylon. As pylon height is more optimized
vertical stay cable force reduces. It acts likes prestressing force on decks.
0.00E+00
1.00E+04
2.00E+04
3.00E+04
4.00E+04
5.00E+04
6.00E+04
7.00E+04
35 40 45 50 55 60 65 70 75
Op
tim
um
Wei
gh
t,(T
)
Specified Compressive Strength of Concrete, fc', Mpa
unit cost of
concrete
grade(birr/m3)
103
The value of pylon height getting from optimization is in the range that
recommended values of tower height ratios from a H/L range of 1/7 to 1/13 (Stroh,
2012). The population of existing extradosed cable-stay bridges has not followed
Mathivat‘s original suggestion that a 1/15 height/span ratio would be the optimal
value. Based on existing bridges tower heights that have been used are slightly taller
than recommended by Mathivat‘s(Stroh, 2012).
4.5 Effect of concrete grade on the optimum height of the pylon
Pylon is a vertical member and it acts as a compressive member. Concrete grade
affects the height of the pylon and the optimum weight of the pylon. Effect of
concrete grade on pylon is like same that box girder effect. The optimization of pylon
height above the deck is done by taking the fixed force of stayed cable. And pylon is
fixed connected to piers. Effect of concrete grade on the optimum height of pylon and
weight are the list Table 4.5-1.
Table 4.5-1Effect of concrete grade on the optimum height and weight of pylon
Grades of Concrete, Mpa Optimum height pylon(m) Optimum pylon weight of bridge(T)
C-75 23.02 436.47
C-70 23.12 437.47
C-65 23.65 438.47
C-60 23.88 439.47
C-55 24.43 440.48
C-50 25.37 441.5
C-45 26.98 442.51
C-40 27.5 443.89
C-35 27.9 445.9
104
Figure 4.5-1Effect of concrete grade on height and weight of pylon
4.6 Optimum angle of Cables Stay
The angle of stay cable is one of the most critical factors in the design of extradosed
cable-stayed bridges. Using a small angle of stay cables leads to large prestressing
cable forces in the deck, and lends a small vertical cable force component. It also
affects the length of stay cable tendon. As the angle of stay cable increase length of
stay cable increases and weight also increase. Excessive cable forces also increase the
stress concentrations at the anchorage points in both the pylon and the deck. A large
angle of stay cable increases the vertical force of stay cables this lends increasing
pylon height. Optimization of the angle of stay cable of extradosed cable-stay bridge
done by considering 60%- 85% dead load and 30% of the live load (approximately
10% dead load) resisted by supporting stay cable. In the optimization process
minimum, dead load resisted by stay cables is 79%. Effect of stay cable force will
improve the stress performance of the main beam; therefore, the cable force layout
and corresponding tower height have critical impacts on the structural performance of
extradosed bridge.
430
435
440
445
450
0
5
10
15
20
25
30
C-75 C-70 C-65 C-60 C-55 C-50 C-45 C-40 C-35
Op
tim
um
p
ylo
n w
eigh
t o
f b
rid
ge(
T)
Opti
mum
hei
ght
pylo
n (
m)
specificied compressive strength of concrete, fc', Mpa
Optimum height pylon(m) Optimum pylon weight of bridge(T)
105
Table 4.6-1Optimum angle of stay cable for fixed span length for side span
Stay cable number optimum Angle of stay
cable(o)
optimum Length of stay
cable (m)
Side-span
C1 30.99 26.37
C2
26.46 32.27
C3
24.01 38.27
C4
22.24 44.31
C5
20.89 50.38
C6
19.83 56.48
C7
18.97 62.59
C8
18.27 68.72
C9 17.69 74.85
Table 4.6-2Optimum angle of stay cable for fixed span length for Mid-span
Stay cable number optimum Angle of stay
cable(o)
optimum Length of stay
cable (m)
Mid-span
C1 30.98 26.37
C2
26.46 32.27
C3
24.01 38.27
C4
22.24 44.31
C5
20.89 50.38
C6
19.83 56.48
C7
18.97 62.59
C8
18.27 68.72
C9 17.69 74.85
106
Values of stay cable angle getting in the range that recommended by (Stroh, 2012)
angle of extradosed cable-stay bridge must greater than 15o.
4.7 Effect of concrete grade on the angle of the stayed cable
When the grade of concrete decreased the amount of dead load is increased. The
angle of supported cable also increases as the dead load is increased because stay
cable in extradosed cable bridge carries at least 70% of total dead load and 30% of
live load. As the angle of the stay cable increases the amount of vertical reaction in
the stay cable is increasing. As used minimum stay cable angle most of the load is
resisted by girder. Effect of concrete grade with stay cable angle in reverse but the
effect on each cable angle is minimum. Effect of Concrete grade on the angle of stay
cable was list in tabular form Table 4.7-1.
Table 4.7-1Effect of Concrete Grades on Optimum angle of stay cable
Stay cable
number
stay cable angle for different concrete grade
C-45 C-50 C-55 C-60 C-65 C-70 C-75
a. Side span cables
C1 32.13 31.01 30.99 30.02 30.01 29.99 29.11
C2 28.12 27.21 26.46 26.01 26.36 26 25.86
C3 27.91 25.98 24.01 23.51 23.01 22.98 21.01
C4 23.04 22.89 22.24 21.98 21.44 21.24 21.21
C5 21.86 21.02 20.89 20.13 20.09 19.89 19.19
C6 19.83 19.83 19.83 19.63 19.13 18.83 19.13
C7 19.97 19.37 18.97 18.67 18.32 18.07 17.97
C8 19.52 18.97 18.27 18.21 18.1 17.89 17.27
C9 18.99 18.61 17.69 17.64 17.02 16.99 16.69
b. Middle span cables
C1 32.13 31.01 30.98 30.02 30.01 39.99 29.11
C2 28.12 27.21 26.46 26.01 26.36 26 25.86
C3 27.91 25.98 24.01 23.51 23.01 22.98 21.01
C4 23.04 22.89 22.24 21.98 21.44 21.24 21.21
C5 21.86 21.02 20.89 20.13 20.09 19.89 19.19
C6 19.83 19.83 30.99 19.63 19.13 18.83 19.13
C7 19.97 19.37 26.46 18.67 18.32 18.97 17.97
C8 19.52 18.97 24.01 18.21 18.1
17.89 17.27
C9 18.99 18.61 22.24 17.64 17.02
16.99 16.69
4.8 Comparison of the conventional and optimal design approach
Conventional design of Abay extradosed cable stays bridge. It models by software
finite element software MIDAS CIVIL 2019.
107
Conventional design output gives more weight than optimal design. Because during
conventional design not done many iterations like optimization design in Matlab. It
tests 5000 population size in a single iteration. The optimal design of the Abay
extradosed cable-stay bridge gives more reducing weight than conventional ones.
Summary of comparison of the weight of optimum design and conventional design
output of the three main superstructures parameters were discussed.
4.8.1 Comparison of conventional versus optimal design of box girder
The weight at the point of the pier is maximum. Because there is a maximum moment
and shear at the point of the pier. It needs more cross-sections. In the two designs
output weight is maximum at pier point. But by comparing conventional weight with
the optimum weight of the bridge deck optimum design given more reduced weight. It
reduced weight by 19.47% of the conventional design output of box girder weight.
This reduced weight was important for the bridge by reducing settlement and for the
owner of the bridge saving material usage. The amount of weight in each section of
the bridge was discussed as follows in Figure 4.8-1.
Table 4.8-1Comparison of conventional versus optimal design of box girder
Section
Bridge(m)
Conventional weight of
bridge(T)
Optimum Weight of
Bridge(T) Reduced Weight(T)
0 6800 5470 1420
10 5790 5110 680
20 5690 4910 777
30 5040 3970 1070
40 4810 3320 1490
50 4440 3310 1130
60 4200 2980 1220
70 3970 2690 1280
80 3690 2470 1220
90 3500 2310 1190
Total weight 38500 31100 7400
Weight Comparison of Optimum and Conventional Design output of box girder
graphed below in Figure 4.8-1.
108
Figure 4.8-1Weight Comparison of Optimum and Conventional Design box section
Bar graph representation of conventional design with optimum design output and
reduced weight of box girder was represented by graph using the cumulative mass of
bridge segment.
Table 4.8-2 Cumulative mass of 90m bridge segment
Conventional bridge weight(T) optimum bridge weight(T) Reduced weight(T)
38500 31100 7400
Bar graphs simply show the decreasing amount of weight by comparing the optimum
design with the conventional one.
0.00E+00
1.00E+03
2.00E+03
3.00E+03
4.00E+03
5.00E+03
6.00E+03
7.00E+03
8.00E+03
0 10 20 30 40 50 60 70 80 90
Conventional weight
of bridge(T)
Optimum Weight of
Bridge(T)
109
Figure 4.8-2Comparison cumulative conventional design output and optimum design output
of bridge girders section
4.8.2 Comparison of conventional and optimal design of pylon
By comparing the conventional weight of pylon with an optimum weight of pylon,
optimum design output gives more reduced weight. It reduces the amount of material
by 20.03% from the conventional weight of pylon.
Table 4.8-3 Conventional weight of pylon and optimum weight of pylon with the reduced
amount of weight
Conventional pylon weight (T) Optimum pylon weight (T) Reducing pylon of weight t(T)
552.04 441.48 110.56
0.00E+00
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
3.00E+04
3.50E+04
4.00E+04
4.50E+04
Conventional
bridge weight(T)
optimum bridge
weight(T)
Reduced
weight(T)
Conventional
bridge weight(T)
optimum bridge
weight(T)
Reduced
weight(T)
110
Weight reduction represents by a bar graph as shown in Figure 4.8-3
Figure 4.8-3Weight comparison of the two design outputs of the pylon
4.8.3 Comparison of Conventional versus optimal design of cable
Abay extradosed cable-stayed bridge has 9 cables in the single pylon. Bridge support
twin pylon. So that at point of pier 18 cable is support girder. Consider half-bridge
width for a single pylon. Cables carry 70% dead load and 30% of live load. Stay cable
design takes 10% of a dead load of deck equal to live load. By considering load effect
for stay cable-stay optimum length and conventional length are shown.
0
100
200
300
400
500
600
Conv. pylon wt.
(Ton)
Opt. pylon wt.
(Ton)
Redu. pylon of
wt. (Ton)
Conv. pylon wt.
(Ton)
Opt. pylon wt. (Ton)
Redu. pylon of wt.
(Ton)
111
Table 4.8-4Comparison of Conventional stays cable length with optimum length side span
stay cable
Cable position Actual length(m) The optimum length of
stay cable (m) Reduced length, (%)
Side span cables
C1 29.13 26.37 9.49
C2 35.63 32.27 9.42
C3 42.34 38.27 9.62
C4 49.01 44.31 9.59
C5 55.91 50.38 9.88
C6 62.87 57.48 8.57
C7 69.85 63.59 8.96
C8 76.87 69.72 9.3
C9 83.89 75.85 9.58
Table 4.8-5Comparison of Conventional stays cable length with optimum length middle span
stay cable
Cable position Actual
length(m)
The optimum length of stay
cable (m) Reduced length, (%)
Middle span cables
C1 29.01 26.87 7.39
C2 35.49 32.27 9.06
C3 42.18 38.27 9.28
C4 48.84 44.31 9.28
C5 55.73 50.38 9.59
C6 62.67 56.48 9.87
C7 69.65 63.59 8.69
C8 76.64 69.72 9.03
C9 83.66 75.85 9.33
To compare conventional stay cable length and optimum stay cable length, optimum
length is more economical it reduced the length by 10% of conventional design output
of cable length.
112
Table 4.8-6Cumulative conventional stay cable length and optimum stay cable length and
reduced length
Convectional Length of Stay
Cable(m)
Optimum Stay Cable
Length(m) Reduced Stay Cable Length (%)
1009.37 908.49 10
Figure 4.8-4Comparison of conventional and optimum stay cable length and the reduced
amount
4.9 Analysis of parametric sensitivity
In this paper, three design parameters are selected: the main girder weight, cable
stiffness, and cable tension. Each parameter is modified separately to different scales
while the other two parameters are unchanged, then the structural analysis is
conducted to obtain the variation on girder deflection, cable force, and stress at the
control section of the main girders. Therefore, primary sensitive parameters and
secondary sensitive parameters are identified according to how they rate in terms of
their effect quantity.
1009.37(m)
908.5(m)
100.87
0
200
400
600
800
1000
1200
1
Convetional Length of Stay Cable Optimum Stay Cable Length
Reduced Stay Cable Length
113
4.9.1 Parametric sensitivity of girder weight.
The Abay River Highway Bridge is a 380m span prestressing post-tension extradosed
cable-stayed bridge. Bridge divide 1050 node and 153 elements. For construction
used cantilever casting method, its effect on the girder deflection due to dead load and
live load, cable force, and stress at the control section of the main girders is analyzed
on the completed bridge state and the results are presented in Figure 4.9-1, Figure
4.9-2 and Figure 4.9-3. All data of in each node and element getting from finite
element MIDAS CIVIL 2019.
114
Table 4.9-1Variation of girder deflection due to dead load when the weight of girder
decreasing 3.5%
Main beam Node Deflection(mm) Main beam Node Deflection(mm)
5 0 500 -12
10 0 505 -12
15 0 510 -12
20 0 515 -12
25 0 520 -12
30 0 525 -12
35 0 530 -12
40 0 535 -12
45 0 540 -12
50 0 545 -12
55 0 550 -12
60 0 555 -12
65 0 1000 12
70 0 1005 12
75 0 1010 12
80 0 1015 12
85 0 1020 12
90 0 1025 12
95 0 1030 12
100 0 1035 12
105 0 1040 12
110 0 1045 12
115 0 1050 12
120 0 1055 12
125 0 500 -12
130 0 505 -12
135 0 510 -12
140 0 515 -12
145 0 520 -12
150 0 525 -12
115
Figure 4.9-1Variation of girder deflection due to dead load when the weight of girder
decreasing 3.5%
Limit of deflection due to live to load (L/800)=225.00 mm for construction of bridge.
Table 4.9-2Variation of girder deflection due to live load though the length of the bridge
when the weight of girder decreasing 3.5%
Node DZ (mm) Node DZ (mm)
1 1.79 80 69.19
5 17.65 85 66.39
10 42.41 90 59.28
15 58.55 95 51.31
20 63.10 100 41.11
25 60.44 105 26.95
30 48.61 110 10.57
35 34.21 115 9.81
40 3.98 120 38.58
45 14.79 125 52.51
50 30.81 130 60.79
55 41.70 135 62.59
60 53.69 140 55.45
65 61.49 145 41.46
70 66.61 150 11.26
75 69.45 153 1.80
-15
-10
-5
0
5
10
15
0 200 400 600 800 1000 1200
Dis
pla
cem
ent(
mm
)
Main beam Node
116
Figure 4.9-2 Variation of girder deflection due to live load when the weight of girder
decreasing 3.5%
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
0 50 100 150 200
Def
lect
ion o
f dec
k d
eck f
or
live
load
(m
m)
Node
117
Table 4.9-3 Variation of cable force when the weight of girder decreasing 3.5% before and
after Shrinkage and creep
Shrinkage and creep(before) Shrinkage and creep(after) All losses of cables
Elem Force-I (kN) Force-J (kN) Force-I (kN) Force-J (kN) Force-I (kN) Force-J (kN)
500 5908 5924 5324 5340 584 584
505 5704 5717 5190 5203 514 514
520 5185 5194 4807 4815 379 379
525 5954 5969 5401 5415 553 553
530 5954 5968 5402 5416 552 552
535 5185 5193 4807 4815 378 378
550 5703 5715 5189 5202 514 514
555 5904 5920 5320 5336 584 584
1000 5908 5924 5324 5340 584 584
1005 5704 5717 5190 5203 514 514
1020 5185 5194 4807 4815 379 379
1025 5954 5969 5401 5415 553 553
1030 5954 5968 5402 5416 552 552
1035 5185 5193 4807 4815 378 378
1050 5703 5715 5189 5202 514 514
1055 5904 5920 5320 5336 584 584
Figure 4.9-3 Variation of cable force when the weight of girder decreasing 3.5%
4400
4600
4800
5000
5200
5400
5600
5800
6000
6200
500
503
506
519
522
525
528
531
534
547
550
553
100
0
100
3
100
6
101
9
102
2
102
5
102
8
103
1
103
4
104
7
105
0
105
3
Ten
sion(k
N)
Element
118
Figure 4.9-4 Variation of girder stress when the weight of girder decrease 3.5%
The above figures show that when the concrete box girder weight decrease by 3%-5%
from the pier to mid-span, the effect on the main beam elevation and the cable force
of the bridge completed state is greater. While the stress dispersion of negative
moment is generated at the control section of the beam, in the light of the above
passage, can get that the effect on the bridge completed state is great when the beam
weight is increased, so should control the cubic amount of concrete strictly in the
construction.
4.9.2 Parametric sensitivity of cable stiffness.
In the case of the same length of structural members, structural stiffness depends on
the elastic modulus and the geometric properties of A and I, and the support
conditions of the elastic state of the material. For the extradosed cable-stayed bridge,
on the premise that the support condition simulation accuracy, the stiffness depends
on the E, A, and I of the beam, tower, and cable. As a result of limited space, only the
cable stiffness sensitivity analysis was operated, the stiffness error of cable mainly
comes from modulus E and area A of steel wire, which under normal circumstances,
the production of steel have high accuracy, so the error of its area A is limited, while
the elastic modulus of high strength steel is generally more stable.
-2
-1
0
1
2
3
4
5
6
7
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
Stre
sss(
N/m
m2
)
Element
Top
Stress(N/mm^2)
Bottom
stress(N/mm^2)
119
So these two errors are relatively small, but because the span of the extradosed cable-
stayed bridge is generally relatively large, the cable sag effect will cause the axial
stiffness of cable down, usually, through the conversion of the elastic modulus to
modify, this paper analyzes the impact on the bridge state when cable elastic modulus
E increase by 3 %. Girder tension stress in variation of elastic modulus in top stress
tension (FTL)and bottom stress tension( FBL) of box girder in each element of bridge
length is state in Table 4.9-4.
120
Table 4.9-4 Variation of girder tension stress when Modulus changes through the length of
the bridge
Elem FTL
(N/mm^2)
FBL
(N/mm^2)
1 -1.06 -0.29
2 0.79 8.95
3 1.16 6.88
4 1.30 6.34
5 1.52 5.40
6 1.68 4.62
7 1.75 3.94
8 2.09 2.69
9 2.14 1.91
10 1.25 4.50
11 3.19 1.32
12 3.07 1.41
13 2.83 1.57
14 2.59 1.69
15 2.51 1.70
16 2.48 1.69
17 2.48 1.68
18 2.47 1.67
19 2.48 1.66
20 2.47 1.65
21 2.46 1.67
22 2.42 1.72
23 2.44 1.75
24 2.52 1.74
25 2.74 1.62
26 0.62 6.57
27 1.31 4.11
28 1.97 2.49
29 1.97 2.82
30 1.71 4.13
31 1.61 4.71
32 1.47 5.43
33 1.26 6.33
34 1.23 6.99
121
Figure 4.9-5Variation of girder stress when Modulus change through the length of the bridge
4.9.3 Parametric sensitivity of cable tension
All the cables of Abay Bridge take repeatedly stretching method. In this paper; the
last tension of every stay-cable is increased by 2% when doing tension sensitivity
analysis. Its effect on the girder deflection, cable force, and stress at the control
section of the main girders on the completed bridge state is analyzed.
Table 4.9-5Stay cable force from mid-span to pier
Side-span
Cable Number F (N)
C1 5059000
C2 5215000
C3 5352000
C4 5704000
C5 5820000
C6 5902000
C7 5947000
C8 5955000
C9 5909000
-2
0
2
4
6
8
10
FTL(N/mm^2)
FBL(N/mm^2)
122
Figure 4.9-6 Stay cable force from mid-span to pier
Table 4.9-6 Stay cable tension force from side-span to pier
Middle-span
Cable Number F (N)
C1 5041000
C2 5179000
C3 5320000
C4 5675000
C5 5800000
C6 5894000
C7 5955000
C8 5982000
C9 5962000
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
0 2 4 6 8 10
Ten
sio
n F
orc
e(K
N)
Cable No
Cable Tension(KN)
Cable Tension(KN)
123
Figure 4.9-7Stay cable tension force from side-span to pier
4800
5000
5200
5400
5600
5800
6000
6200
0 2 4 6 8 10 12
Ten
sio
n f
orc
e(K
N)
Cable No
Cable Tension(KN)
Cable Tension(KN)
124
5 CONCLUSION AND RECOMMENDATION
5.1 Conclusion
The goal of this research was to optimize the superstructure weight of prestressed
post-tensioning extradosed cable-stayed bridge by considering the depth of girder
with the height of pylon and angle stay cable. Those three parameters are the main
loading carrying components of the bridge. This paper state that saves material
amount by considering these three main superstructure parameters. The amount of
material described in terms of prestressing reinforcement, concrete, and reinforcement
bar by comparing conventional design out of Abay extradosed cable-stayed bridge
that builds in Abay River. The following conclusions were drawn from the present
work.
1. The result indicates that the bridge has unwanted mass. As comparing between
conventional depth and optimum depth, optimum depth given more reducing
weight. This gives the minimum weight of concrete, prestressing reinforcement,
and non-prestressing reinforcement for each section of the bridge. The result
shows that at the point of pier depth of girder is 5.129m and at point of mid-span
and abutment 2.62m.
2. Concrete grade and weight of bridge have an inverse relation
3. As a study of the effect unit cost of the concrete grade on optimal weight was
given that the optimal grade of concrete for prestressing posts tensioning
extradosed cable-stay bridge is form C-45 up to C-55 concrete grade is optimal.
4. The optimum height of the pylon above the deck is 24.43m for a fixed span length
of 180m.
5. As result shows that amount of stay cable parameter depends on the weight of the
girder and the amount of prestressing force applies on decks. As the amount of
stay cable force increase with increasing dead load amount. For stay cable angle
optimization considering prestressing force.
6. The effect of concrete grade on the angle of stay cable is less because it more
depends on prestressing force on deck. Because of using fixed prestressing force
in deck change angle less.
7. To compare the conventional and optimal design of box girder, pylon, and stayed
cable length using by weight of the bridge, the optimal design gives less weight.
125
The total reducing amount is 10% from stay cable length and 20.03% from pylon
weight and 19.47% % from box girder weight. Generally, 49.47% of the weight is
reduced from conventional design output. So that for the design of extradosed
cable-stayed bridge design use optimization techniques gives the best result.
8. Based on the parameters sensitivity analysis of the weight of girder, the stiffness
and tension of cable the influences of the weight of girder and the tension of cable
are noticeable and need amended as the main design parameters when the bridge
model are being calculated. Through the parameters sensitivity analysis of
structure, the bridge model can be optimized. Combining the optimization with
the technology of model amended can guarantee the accuracy of bridge
construction control.
5.2 Recommendation
In this study, the effect of box girder on optimum weight, effect of concrete grades on
the weight of three main parameters (box girder pylon and angle of stay cable), pylon
height, and angle of stay cable were investigated.
It is recommended to use 5.129m girder depth at the pier and 2.62m at mid-span
for a 180m main span length.
It is recommended to use C-45 concrete grade for box girder of extradosed bridge.
Use the height of pylon 24.43m for a 180m span length.
Using the optimum angle of stay cable 32.13o for using optimum grade of
concrete C-45 in girder.
For extradosed cable-stayed bridge, optimum superstructure value is 5.129m
depth of girder at point of the pier and 2.62m at point of mid-span and optimum
height of pylon is 24.43m and optimum first stay cable angle is 32.13o for using
optimum concrete grade C-45.
For using the optimum output of superstructure parameter of Abay Bridge, must
consider optimization on substructure component of bridge.
The parameters sensitivity analysis of structure is useful for parameter
recognition and correct for error adjustment in the construction process.
126
REFERENCE
Abebe, wubishet J. (2018). Optimization of PC Girders for Bridge Design_by
WubishetJ.pdf. Jimma University.
Abebe, F. (2016). Department of Civil and Environmental Engineering Design of
Precast Pre-Stressed Concrete Girder Bridges As an Economical Solution for
Congested Urban Areas Design of Precast Pre-Stressed Concrete Girder
Bridges As an Economical Solution For Congested Urb. Addis Ababa Institute
Of Technology.
An, Y. F. (N.D.). An Optimization Index To Identifyidentify The Optimal Design
Solution Of Bridges. University Of Trento Department.
Introduction to Design Sensitivity Analysis, (2015).
Bendsøe, M. P. and Sigmund, O. (2003). Topology Optimization: Theory, Methods
and Applications. Springer-Verlag.
Bhattacharjya, R. K. (2015). Genetic Algorithms.pdf.
Biliszczuk, J., Onysyk, J., Barcik, W., Toczkiewicz, R., & Tukendorf, A. (2017).
Extradosed Bridges in Poland—Design and Construction. Frontiers in Built
Environment, 2(January), 1–9. https://doi.org/10.3389/fbuil.2016.00037
Bujnak, J., Odrobinak, J., & Vican, J. (2013). Extradosed bridge - Theoretical and
experimental verification. Procedia Engineering, 65, 327–334.
https://doi.org/10.1016/j.proeng.2013.09.050
Chang-Huan Kou1, a, Tsung-Ta Wu2, b, Pei-Yu Lin3, C. (2014). Optimization of
Cable Force of Extradosed Bridges. Vols 587-5.
Chen, E. W., & Duan, L. (2000a). Bridge Engineering Handbook.
Chen, E. W., & Duan, L. (2000b). Cable-Stayed Bridges Man-Chung. In Bridge
Engineering.
Chessa, J. (2002). matlab_fem.pdf.
China Communications Construction Company Limited. (2020). Bahirdarabay River
Bridge And Approach Roaddesign-Build Project: Lot 1 - Abay River Bridge And
127
Adjacent Concrete Structures.
Dharma, A. P., & Suryoatmono, B. (2019). Non-Linear Buckling Analysis of Axially
Loaded Column with Non-Prismatic. 5(3).
Dipl.-Ing., vorgelegt von, & Ägypten, E. A. E. S. aus. (2013). Form Finding For
Cable-Stayed And Extradosed Bridges. der Technischen Universität Berlin zur.
EBCS EN- 8 Part. (2014). EBCS- 8 Part 1 Design of Structures R for (Issue
December).
Gladwell, G. M. . (1991).
f i i i h f f f.
Golebiowska, I., & Peszynski, K. (2018). Cable vibration caused by wind. EPJ Web
of Conferences, 180, 1–6. https://doi.org/10.1051/epjconf/201818002031
Load and Resistance Factor Design., 18 Engineering Journal 74 (2015).
Jafari, M. (2019). Aerodynamic-load identification and prediction of wind-induced
vibration of structural / power-line cables and traffic signal structures.
Jerzy Onysyk,Wojciech Barcik, R. T. and A. T. (2017). Extradosed bridges in
poland-design and construction. 1–11.
Kaljima Corporation. (2019). Ethiopia‘s First Extradosed Bridge to Span the Blue
Nile Gorge. 35(May 2006), 30. https://doi.org/10.36621/0397-004-998-003
Kirsch, P. U. (1993).
Structural_Optimization_Fundamentals_and_Applications_by_Prof_Uri.pdf.
Klarbring, P. W. C. · A., & An. (2008).
An_Introduction_to_Structural_Optimization_by_Peter_W_Christensen.pdf.
Kou, C., Wu, T., & Lin, P. (2014). Optimization of Cable Force of Extradosed
Bridges. 589, 1577–1580. https://doi.org/10.4028/www.scientific.net/AMM.587-
589.1577
Lee, B. K., & Lee, J. K. (2021). Buckling of Tapered Heavy Columns with Constant
Volume. 1–14.
128
Mast, R. F. (1998). Analysis of cracked prestressed concrete sections: A practical
approach. PCI Journal, 43(4), 80–90.
https://doi.org/10.15554/pcij.07011998.80.91
Mermigas, by K. K., & A. (2008). Behaviour and Design of Extradosed Bridges.
University of Toronto ©.
Miskiewicz, M., & Pyrzowski, L. (2018a). Load test of new European record holder
in span length among extradosed type bridges. E3S Web of Conferences, 63, 1–2.
https://doi.org/10.1051/e3sconf/20186300006
Miskiewicz, M., & Pyrzowski, L. (2018b). Load test of new European record holder
in span length among extradosed type bridges. E3S Web of Conferences, 63, 1–6.
https://doi.org/10.1051/e3sconf/20186300006
Niels J.Gimsing, C. T. G. (2012). Cable Supported Bridges- Concept and Design .pdf.
Technical University of Denmark.
Nippon Koei Co., L. (2005). The Feasibility Study of Padma Bridge Final Report.
AASHTO LRFD 2010 BridgeDesignSpecifications 5th Ed..pdf, (2010).
Özel, C. N., Özkul, Ö., Erdoğan, E., Ergüner, K., & Görgülü, G. (n.d.). The First
x B i g i : D ig f h A Ç ı B i g . 1–8.
PARSAEI, H. R. (1997).
Genetic_Algorithms_and_Engineering_Design_by_Mitsuo_Gen,_Runwei.pdf.
Peng, S., Rong, C., Cheng, H., Wang, X., Li, M., Tang, B., & Li, X. (2019).
Mechanical Properties of High-Strength High-Performance Reinforced Concrete
Shaft Lining Structures in Deep Freezing Wells. Advances in Civil Engineering,
2019. https://doi.org/10.1155/2019/2430652
Prashant Kumar Srivastava1*, Simant2, S. S., & 1Asst. (2017). Structural
Optimization Methods: A General Review. International Journal of Innovative
Research in Science, 6(9,), 3.
PTI. (2001). 3. Recommendations for stay cable design, testing and installation.pdf.
Rechenberg, I. (1973). . Evolution strategy: Optimization of technical systems by
129
means of biological evolution. Fromman-Holzboog, Stuttgart.
Reuter, U., & Liebscher, M. (2009). Global Sensitivity Analysis in Structural
Optimization. 7th European LS-DYNA Conference, Salzburg.
http://www.dynamore.de/dynamore/documents/papers/euro2009/f-i-03.pdf
S.N.Sivanandam. (2008).
Introduction_to_Genetic_Algorithms_by_S_N_Sivanandam,_S_N_Deepa.pdf.
Saad, F. (2000). Structural Optimization of Extradosed Bridges.
Setral. (2002). 2. Cable stays.pdf.
Shirono, Y., Takuwa, I., Kasuga, A., and Okamoto, H. (1993). The design of an
extradosed prestressed concrete bridge – the Odawara Port Bridge.
Stroh, S. L. (2012). On The Development Of The Extradosed Bridge Concept (Issue
January). University of South Florida.
Tejashree G. Chitari*1, T. N. N., & ME. (2019). The Study of Pylon Height Variation
for Cost Optimization of Extradosed Bridge.
Tejashree G. Chitari1, T. N. N., & 1ME. (2019). Cost Optimization of Extradosed
Bridge by varying Cable Position Tejashree.
Votorantim Cimentos, & St. Marys CBM. (2021). 2021 General Contractors Price
List. http://www.canadabuildingmaterials.com/en-ca/Pages/Products and
Services/GTA-Pricelist.aspx
Zhongguo(John) Ma, Maher K. Tadros, C. S. P. (2004). Prestressed Concrete Box
Girders Made from Precast Concrete Unsymmetrical Sections.pdf.
130
APPENDIX
Annex 1 Extradosed Cable-Stayed Bridge structural modeling
Table 1 Maximum moment and shear value of Abay extradosed cable-stayed bridge(China
Communications Construction Company Limited, 2020)
Item Strength(KN.m) Action (KN.m)
Bending
moment
Mid span 566323 333176
Support -1901784 -886238
Item Strength(kN) Action(kN)
Max 73587 40500
Min 73587 -40500
Annex 2 Design calculation process of the pylon
Table 2 Value Axial force and bending moment of pylon under different load combination
Loads ULS
Strength Ⅰ Strength Ⅱ Strength Ⅲ
DC 1.25 1.25 1.25
DW 1.5 1.5 1.5
LL 1.75 1.35 1.5
WS 1.4
FR 1 1 1
Axial Force(kN) 1357.3 1053.1 823
Moment(kN.m) 16683.5 13179.8 1387.1
Force(kN) Shear 58847.8 62836.9 59607.8
Annex 3 Design Optimization Code using GA in Matlab for angle of stay
cables
1) Stay cable design
Fitness Function for Stay Cable in Extradosed Cable Stay Bridge
Function Y = staycableobjfun(x)
% weight parameter for stay cable
%pc = 0.0848; % unit weight (N/mm3)
yca=0.0023; % density of the cable material low relaxation strand
dca=15.2; % diameter of stay cable
131
n=46; % number of cable in single tendon
Aca=pi*dca^2/4; % area of one bar in cable tendon
av =n*Aca*x(3); % area of f16mm for shear reinforcement within a
distance S (mm2)
% weight function of stay cable
Y= x (4)*yca*av; % weight function of stay cable of extradosed
cable-stay bridge component
Nonlinear Constraint Functions Definition for Stay Cable for
Extradosed Cable-Stay Bridge
Function [C,ceq] = Extracableconst(x)
% Material properties
fc = 45; % cylindrical compr.strength for box girder (N/mm2)
fpu = 1860; % specified tensile strength of prestressing
reinforcement (Mpa)(fpu = 270,000 psi (low-relaxation strand;)
Eca= 190e3; % typical modulus of elasticity of stay cable (N/mm^2)
Qult=1.6e3; %Ultimate tensile stress of the stay cable material
% stress limits in concrete
fcbd =0.6*fpu; % for stays cable are designed to a maximum
allowable tensile strength(fcbd is the design stress (allowable
cable stress)
Qa=0.45*Qult; % Allowable stress in the stay cable under SLS loads
ftw = 0.5.*sqrt(fc); % Allowable tensile stress at working loads
Ec=27660; % Young’s modulus of concrete (N/mm2)
Mw= 40218.79e6*0.3; % Service limit state-I bending moment (Nmm)
w=3.3727E+11+(0.1*3.3727E+11); % total load that apply on girder
(dead load + live load)
%% section properties of box girder
Re=24700; % total width of roadway (mm)
Angle=45; % angle of external web inclination (o)
dduth=32; % diameter of duct(mm_)
Cov=50; % concrete cover of box girder (mm)
z1=90000; % half bridge length of bridge (mm)
w1=0.03404*z1+2.2e3;
tb=min([1/20*w1, 200]); % thickness of bottom flange
w2=2.315e3; % optimum side width of cantilever
w3=0.0645*z1+9e3; % optimum bottom width of box girder
132
tw=max([1/36*w1+cov + dduth, 200+dduth]); % thickness of web(mm)
%% equations for effective depth of reinforcing steel
db = 32; % assumed diam. of bar assume it (mm).
Agg = 25; % maximum aggregate size (mm)
Sh = max ([1.5*db, 1.5*Agg, 38]); % clear spacing of parallel bars
(horizontal) (mm)
Sv = max ([25, db]); % clear spacing between layers of bars
(vertically) (mm)
tft=max([200, 150+dductho]); % thickness of top flange
as = pi*db^2/4; % area of a single reinf. Bar (mm2)
nb =As/as; % Number of bars
npr = min([(tw+Sh-124)/(Sh+db), nb]); % Number of bars per a row
nr = nb/npr; % Number of reinforcement rows
hr = nr*db+ Sv*(nr-1); % Height of reinforcement rows
dst = 62+hr/2; % depth from extreme tension fiber to centroid of
reinf. Steel (mm)
d = w1-dst; % effective depth of reinforcement steel (mm)
% effective depth of prestressing steel
dsrd = 15.2; % assumed diam. of prestressing low relaxation strand
(mm)
Nspt = 31; % number of strands per tendon
ap = 0.77.*pi.*dsrd.^2./4; % steel area of a single strand
(mm2)(using a reduction factor of 77 % of nominal area of the
strand)
dduct = 125; % diameter of duct, (mm)
Sduct = 38; % clear vertical and horizontal spacing of ducts (mm)
nst = w4/ap ; % number of strands required
nt = nst/Nspt; % Number of tendons
ntr = min([(tw+Sduct-200)/(dduct+Sduct),nt]); % Number of tendons
per a row
nrt = nt/ntr; % Number of rows of prestressing tendons
hrt = dduct.*nrt+Sduct*(nrt-1); % height of rows of prestressing
tendons
dpt = 50+12+Sduct+25+hr+hrt./2; % Depth from extreme tension fiber
to centroid of prestressing tendons (mm)
133
dp = w1-dpt; % Depth from extreme top fiber to centroid of
prestressing steel (mm)
d1 =5129; % optimum depth of box girder (mm)
W1= 2135; % optimum side cantilever width (mm)
W2 = 8650; % optimum bottom slab width (mm)
n=3; % number of opening cell
Gs=(x (3)-(n+1)*tw)/(n+1); % spacing between web or internal girder
Aeff=Re*tft+x (3)*tb+ (d1-tb-tft)*tw*(n-1)+(d1-tb-tft)/sin…
(43.9)*tw*2; % total area of concrete
yt=(W2)*tb^2/2+Re*tft*(d 1-tft/2)+(d1-tb-tft)*tw*(n-1)*(d1-tb…
-tft)/2+ (d1-tb-tft)/sin (43.9)*tw*1/3*(d1-tb-tft)/sin…
(43.9)*2)/Aeff; % depth from c.g of section to extreme bottom
fiber(mm)
yb = d1-yt; % depth from c.g of section to extreme top fiber (mm)
Iy=W2*tft^3/12+W2*tft*(yt-tft/2)^2+tw*(x(1)-tb-tft)^3/12*(n-
1)+tw*(x(1)-tb-tft)*((x(1)-tb-tft)/2-yt)^2+tw*((x(1)-tb-
tft)/sin(43.9))^3/12+tw*(x(1)-tb-tft)/sin(43.9)*(((x(1)-tb-
tft)/sin(43.9))*1/3-yt)+Re*tft^3/12+Re*tft*(tft/2-yt)^2; % Gross
moment of inertia of concrete in y axis
Zb=Iy/yb; % section modulus of the extreme bottom fiber (mm3)
Zt=Iy/yt; % section modulus of the extreme top fiber (mm3)
xb=((Re.*(Re/2).*tft)+W2*tb*(W1+(d1-tb-tft)/tan(43.9)+W2/2)+(d1-tb-
tft)/sin(43.9)*tw*(W1+(d1-tb-tft)/tan(43.9)*2/3)+(d1-tb-
tft)/sin(43.9)*tw*(W1+(d1-tb-tft)/tan(43.9)+W2+(d1-tb-
tft)/tan(43.9)*1/3)+(d1-tb-tft)*tw*(W1+(d1-tb-
tft)/tan(43.9)+Gs)+(d1-tb-tft)*tw*( W1+(d1-tb-tft)/tan(43.9)+(n-
1)*Gs))/Aeff; % depth from c.g of section to extreme bottom fiber
in x- axis(mm)
Ixeff=tft*Re^3/12+tft*Re*(Re/2-xb)^2+tb*W2^3/12+tb*W2*(W1+ (d1+tb….
-tft)/tan (43.9) +W2/2-xb) ^2+ ((d1-tb-tft)/tan (43.9))*tw^3/12…
+ ((d1-tb-tft)/tan (43.9))*tw*(x(2)+(d1-tb-tft)/tan (43.9)*2/3…
-xb)^2+(d1-tb-tft)/sin (43.9)*tw^3/12+ (d1-tb-tft)/sin(43.9)*tw…
*(W1+W2+(d1-tb-tft)/tan (43.9)+(d1-tb-tft)/tan(43.9)*1/3-xb)^2…
+(d1-tb-tft)*tw^3/12+ (d1-tb-tft)*tw*(W1+ (d1-tb-tft)/tan (43.9)....
+Gs-xb) ^2+ (d1-tb-tft)*tw^3/12+(d1-tb-tft)*tw*(W1+((d1-tb…
-tft)/tan (43.9) +2*Gs-xb) ^2);
134
Ixy=xb.*yb.*Aeff;
finf = ftw/0.85+Mw/(0.85*Zb); % extreme bottom fiber stress, finf
developed at a given eccentricity e (N/mm2)
e = yb-dpt; % possible maximum eccentricity of prestressing force
from c.g.c (mm)
P = Aeff*finf*Zb/(Zb+Aeff*e); % minimum prestressing force at a
known eccentricity, e(N)
%% Geometric properties
r=0.555; % Ratio of main span and side span
L=180e3; % Length main span length of bridge (mm)
% total stress in the cable
dca=15.2; % diameter of wire
wc =x(2)*w; % desired equivalent vertical stay cable forces
% initial tension loading in stay cable
E=205e3; % modulus of elasticity for stay cable (Gpa)
yca=0.0023; % density of the cable material
f=0.6*fpu; % cable tensile stress
Eeq=E/(1+(x(3)*yca)^2/12*f^3); % equivalent modulus of elasticity
fn=0.00346884; %unknown factor depending only on the parameter 0.65
M2= (-w*L^2*(1+0.65^3)+wc*L^2*(62/125+24*fn)/(8*0.65+12)); %bending
moment at the support (2) for the structure
M3=M2; %bending moment at the support (2) for the structure
R1= (M2-w*(0.6*L)^2/2-6*wc*L^2/100)/r*L; % reaction force at the
support (1) for the structure
R2= (w*(L+2*0.65*L)-0.8*L*wc-2*R1)/2=R3; % reaction force at the
support (2) for the structure
Q2r=-R1+w*r*L-0.2*L*wc-R2; % shear force on the right hand side of
the support (2) for the structure
Q3r=Q2r+w.*L-0.4*L.*wc-R3; % shear force on the right hand side of
the support (3) for the structure
M3s1=-0.01633; % bending moments at the supports (3) (kN.m)
M2s1=-2*(0.65+1)*M3s1; % bending moments at the supports (3) (kN.m)
Rlsl= (M2s1-0.3*L)/(r*L);
R4s1=M3s1/0.65*L;
R3s1= (M2s1-R4s1*((r+1)*L)^2/L);
R2s1= (M3s1-(Rlsl*(r+1)*L-1.3*L)/L);
135
Q2rs1=-Rlsl-R2s1-1; % the shears force on the right-hand side of the
support (2)
Q3rs1=-Rlsl-R2s1-1-R3s1; % the shear force on the right-hand side of
the support (3) for the structure
M2m1= (0.714*(1+r)-0.273)/(4*(1+r)^2-1)*L; %bending moment at
support (2)
R1ml= (M2m1)/(r*L);
M3m1=0.357*L-2*(1.65)*M2m1;
R4m1=M3m1/0.65*L;
R2m1= (M3m1-R1ml*((1+r)*L-0.7*L)/L);
R3m1= (M2m1-R4m1*((1+r)*L-0.3*L)/L); %
Q2m1=-R1ml-R2m1; % shear force on the right hand side of the support
(2) for the structure
Q3m1=-R1ml-R2m1-1-R3m1; % shear force on the right hand side of the
support (3) for the structure
% M(x) calculation in 11 zones
z=38e3; % section distance on bridge (moment determine every z
section)
if (-r*L<=z<=0.4*L) ; % first section distance of bridge
M(z) =R1*(r*L+z)-w*(r*L+z)^2/2; % bending moment in first section
elseif(-0.4*L<=z<=0.2*L)
M(z) =R1*(r*L+z)-w*((r*L+z)^2)/2+wc*((0.4*L+z))^2/2; % bending
moment in second section
elseif(-0.2*L<z<=0)
M(z) =R1*(r*L+z)-w(r*L+z)^2/2+0.2*L*wc* (0.3*L+z); %bending moment
in third section
elseif (0<z<=0.2*L)
M(z)=M2-(w*((z)^2)/2)-(Q2r*(z)); %bending moment in fourth section
else if(0.2*L<z<=0.4*L)
M (z) =M2-w*((z)^2)/2-Q2r*(z)-wc*(z-0.2*L)^2/2; %bending moment in
fifth section
elseif(0.4*L<z<0.6*L)
M (z) =M2-w*(z^2)/2-Q2r*(z) +0.2*L*wc*(z-0.3*L); %bending moment in
sixth section
elseif 0.6*L<z<=0.8*L
136
M(z)=M2-w*(z^2)/2-Q2r*(z)+0.2*L*wc*(z-0.3*L)+wc*(z-0.6*L)^2/2;
%bending moment in seventh section
elseif(0.8*L<=z)&&(z<=L)
M(z)=M2-w*(z^2)/2-Q2r*(z)+0.4*L*wc*(z-0.5*L); %bending moment in
eight section
elseif L<z<=1.2*L
M(z)= M3-w*((z-L)^2/2)-Q3r(z-L); % bending moment in ninth section
elseif 1.2*L<z<=1.4*L
M(z) = M3-w*((z-L)^2)/2-Q3r*(z-L)+wc*(z-1.2*L)^2/2; %bending moment
in tenth section
elseif 1.4*L<z<=(1+r)
M(z) = M3-w*((w-L)^2)/2-Q3r*(z)+0.2*L*wc*(z-1.3*L); %bending moment
in eleventh section
% the bending moment at any point, the entire length of the deck,
due to a vertical unit load F=1 acting at s1
if(-r*L<=z<0.3*L)
Ms1 (z) = Rlsl*(r*L+z);
elseif -3*L<z<=0
Ms1 (z) = Rlsl*((r*L+z) + (0.3*L+z));
elseif 0<z<=0.3*L
Ms1(z) = M2s1-Q2rs1*z;
elseif (L<z<=(r+1)*L)
Ms1(z) = M3s1-Q3rs1*(z-L);
% the bending moment at any point, along the entire length of the
deck, due to a vertical unit load F=1 acting at m1
if(-r*L<z<=0)
Mm1 (z) =R1ml*(r*L+z);
Elseif (0<z<=0.3*L)
Mm1 (z) =M2m1-Q2m1*z;
elseif(0.3*L<z<=L)
Mm1 (z) =M2m1-Q2m1*z+ (z-0.3*L);
elseif (L<z<=(1+r)*L)
Mm1 (z) =M3m1-Q3m1*(z-L);
H=x (3) ^2-22e3^2;
FDi= (0.2*wc*L/(2*x (4)*sin(x(1)))); % desired stay cable force for
each stay cable FDi under dead load may be determined,
137
%the forces resulting from the bottom straight tendons at the middle
of the main span
xb=24.7e2/2;
eb=5.129e3./2-xb-cov;
Mf=-Fp.*eb; % moment of prestressing force in decks
M21=-3*Mf*Lf/L* (1/(3+2*r)); % bending moment at support two
R1f=M21/(r*L); % reaction at support 1 and 2
R2f=-(-0.3*Mf*Lf/(r*L^2)*1/(3+2*r)); %the forces resulting from the
top straight tendons at the pylons
et=w1/2-Zt-cov;
Ls=100e3; % from center to center of cable stayed
Ms=Fp*et;
M22=Ms*(1-(24*Ls*r*L-3*Ls^2)/(12*r*L^2*(1+r)-4*r^2*L^2));
R1s=-Ms*(1-(24*Ls*r*L-3*Ls^2)/(12*r^2*L^3*(1+r)-4*r^3*L^3));
R2s=-R1s;
%the forces resulting from the bottom straight tendons in the side
spans
Mo=-Fp*eb;
Lamda=0.03;
Lo=63e3;
M23=-6*Mo*Lo*(lamda*L+Lo/2)/(3*r*L^2*(r+1)-r^2*L^2);
R1o=M23/(r*L);
M23=R1o*r*L;
R2o=-M23/(r*L);
R1o=-R2o;
%the forces resulting from draped tendons at the middle of the main
span
Weq=8.*eb*Fp./L.^2;
Tc1=-4*eb/Lf*Fp;
M2w=Weq*Lf/48*(3*(L-Lf) ^2+6.*Lf.*(L-Lf) +2*Lf.^2/L.* (r/3+1/2));
R1w=M2w/r*L;
R2w1=-Weq*Lf/2-R1w;
R2p1=3*P*r*L*(2*r*L+L+Lf)*(r*L+ (L-Lf)/2)-2*P*r^3*L^3/
(6*r^2*L^3*(r+1)-2*r^3*L^2);
R1p=-(R2p1+Tc1);
M2p=R1p*r*L;
138
M24=M2w+M2p;
% the forces resulting from draped tendons in the side spans
%the forces due to the equivalent uniform load Weq alone, can be
determined by using the equations
Weq= (8*eb/Lo^2)*Fp;
r1= (Lo*(r*L-lamda*L-Lo/2)/ (0.55*L))*Weq;
r2=Weq*Lo*(lamda*L+Lo/2)/r*L; % where r1 and r2 are supplementary
variables and will be used below to determine M2w
%1/3*lamda^3/r*L^2*r1+Lo*r2*(r*L-lamda*L- Lo)*((lamda*L+Lo/2)/r*L)…
+ 1/2*Lo*(r1*lamd*L-r2*(r*L-lamda*L-Lo))*(lamda*L+Lo/3)/(r*L)…
+ 1/12*Weq*Lo^3*(y*L+Lo/2)/(r*L)+1/2*r2*(r*L-y*L-Lo)^2*(r*L…
-2/3*(r*L-y*L-Lo)/(r*L))=M2w*L*(1/2+r/3);
M2w=1/3*lamda^3/r*L^2*r1+Lo*r2*(r*L-lamda*L-
Lo)*((lamda*L+Lo/2)/r*L)+1/2*Lo.*(r1.*lamda*L-r2*(r*L-lamda*L-
Lo))*(lamda*L+Lo/3)/(r*L)+1/12*Weq*Lo^3*(lamda*L+Lo/2)/(r*L)+1/2*r2*
(r*L-lamda*L-Lo)^2*(r*L-2/3*(r*L-lamda*L-Lo)/(r*L))/(L*(1/2+r/3)); %
R1w1=-(Weq*Lo*(r*L-lamda*L-Lo/2)-M2w)/(r.*L);
R2w2=-(Weq*Lo+R1w1);
%The final forces resulting from the equivalent uniform loads Weq
and the 2x2 vertical nodal loads P, can be determined then by
superposition.
P1=-(4*eb)/Lf*Fp;
Ls1=(r*L-lamda*L-Lo);
Ls2= (r.*L-lamda.*L);
Lsi=Ls1:Ls2;
R2p2=symsum((-P1*(3*r*L*(r*L+L+Lsi)*(r*L-Lsi)-(r*L…
-Lsi)^2*(r*L+2*Lsi)-3*Lsi^2*(r*L-Lsi))/(3*r^2*L^3*(r+1)-r.^3…
*L.^3)), i, 1, 2);
R1p1=-2.*P1-R2p2;
M2p=R1p*r*L+P1*(Ls1+Ls2);
M25=M2-(M24+M22+M2p);
FPTi= (R2f+R2s+R2o+R2w1+R2p1+R2w2+R2p2)/(2*(x(4)*sin(x (1))))…
-(R1f+R1s+R1o+R1w+R1p+R1w1+R1p1)/(2*(x(4)*sin(x(1))));
Fi=3*w*L/(x(4)*sin(x(1))*1/(1+3/2*e/L*cot(x (1)))); % stay cable
force of the cable no. i under the effect of the dead load prior to
the shortening/re-stressing of any stay cable by the forces Fi
139
yca=0.0023; % density of the cable material
% x (2); % horizontal length of stay cable position
FDI= (FDi-Fi-FPTi+Pcb);
H=x (5)*tan(x(1));
H1=H+x (6);
y=x(5)+x(7);
d= (fcbd/Eeq*(H1+y^2/H));
%H= (24.6e3-8*1.1e3):1.1e3:24.6e3; %vertical height of stay cable
position
%Qsptc=(yca/fcbd*(FDI)*x(3),1, 9); % total cable steel quantity
Qcb=x(3)*yca/fcbd*(H+x(5)^2/H)*(FDI); % quantity of cable material
d= (fcbd/Eeq)*(H+(x(5)^2/H)); % vertical deflection d at the lower
stay cable anchorage
FDI=Qcb*fcbd/yca/((H+(22e3^2/H)));
Tc=pi*dca^2/4*Qult;
Aca=53*pi*15.262/2;
Qw=FDI/Qcb; % Axial stress in the stay cable due to dead load
fb=0.937; %(Hz)
m= (Aca*yca*x(3)/x(3)); % mass per length of stay cable (kg/m)
ft=1.043; % Symmetric torsional fundamental(HZ)
fn=n/2* (I*((FDI/m)^2)); % Cable base frequency
%Nonlinear inequality constraints [c] written of the form gi(xi)<= 0
% stay cable force limit
g1=0.7-4*wc*56e3/w* (2*r*L+L);
g2=Zs1+ (M (z)*Ms1 (z))/ ((Ec)*Ixeff); % the vertical deflection of
deck at s1 due to the combination of w(x) and wc
g3=Zm1+M (z)*Mm1 (z)/ (Ec*Ixeff); % the vertical deflection of deck
at m1 due to the combination of W(x) and WC
g4= (fn-ft)*I5;
g5= (fn-fb)*I5;
g6=FDI-0.55*Tc; %Stay cables tensile forces limit
g7= (800*d/L-1)*I5; % allowable stay cable deflections
g8= (4*22e3*8*6e3/ (w*2*r*L+L))*I5;
g9=Qw-Qa;
C=[g1;g2;g3;g4;g5;g6;g7;g8;g9;g10;g11;g12];
Main Code for Running the Genetic Algorithm
140
% Problem parameters
% x(1) = a; x(2)=n ; x(3)=Lcb; x(4)=N x(5)=HL, x(6)=Sv, x(7)=Sh
objfcn=@staycableobjfun;
% set boundary values of variables
lb =[15 0.9 1e3 14 13e3 1e3 5e3]
ub=[35 1.9 90e3 18 24.9e3 1.4e3 7e3]
consfcn= @Extracableconst;
%% set ga options
Options = optimoptions(@ga,...
'PopulationSize', 5000...
'MaxGenerations', 100,...
'EliteCount',1,...
'Maxstallgeneration',100 ,...
'FitnessScalingFcn',@fitscalingprop,...
'NonlinearConstraintAlgorithm','auglag',...
'InitialPenalty',10,...
'PenaltyFactor',100,...
'FunctionTolerance', 1e-10, ...
'ConstraintTolerance', 1e-10,...
'PlotFcn',@gaplotbestf);
%Call |ga| to solve the Problem
% we can now call |ga| to solve the problem.
rng(1,'twister') % random number generator for reproducibility
[xbest,fbest,exitFlag] =
ga(@staycableobjfun,nvars,Aineq,bineq,Aeq,beq,lb,ub,@Extracableconst
, 1:3,options);
% _Analyze the Results
Display (xbest)
%% return optimal value
fprintf ('\n Y function returned by ga = %g\n', fbest)
Annex 4 Design Optimization Code using GA in Matlab for box girder
% this is fitness function for Extradosed main loading carrying
component
function z = Extramaincompfun(x)
% weight parameters box girder
141
pc = 0.0024; % density of concrete(g/mm3)
ps= 0.00785; % density of steel_prestressing strands and
reinforcing bars (g/mm3)
pp= 0.0023; % density of steel_prestressing strands and reinforcing
bars (g/mm3)
tb=min([1/36*x(1), 200]); %thickness of bottom flange
L=5000%length of half span of main span bridge (mm)
dduth=32; % diameter of duct
dductho=dduth+10;% diameter of duct hole
Asb = 0.004*tfb*((NG-1)*x(5)+x(2))*L+0.005*tfb*((NG-1)*x(5)+tw); %
total area of bottom slab reinf.
NL = 4; % number of legs of vertical stirrups
dsh = 16; % diam. of shear rebar (mm)
av = NL*pi*dsh^2/4; % area of f12mm for shear reinforcement within a
distance S (mm2)
tft =max([200, 150+dductho]); %thickness of top flange
Re=24700; % width of roadway (mm)
A1= Re*tft; % concrete cross sectional area of the top slab of box
girder (mm^2)
Ag= ((Re-2*x (2))/2-(x(3)/2))*((x(1)-tb))+(x(3)*(x(1)-tb)); % gross
cross sectional area of concrete (mm2)
ddut=32; % diameter of duct
dduth=ddut+10;% diameter of duct hole
cov=50; % concrete cover of box girder
n=3; number of opening cell
tw=max([1/36*x(1)+(cov+dduth),(200+dduth)]); % thickness of web(mm)
y2=x(1)-tb-30;% height of opening
y3=x (3)/3-(3-1)*tw; % width of opening
Ag= (1/2)*((Re-(2*x(2)))+x(3))*(x(1)-tb); % gross cross sectional
area of concrete (mm2)
y2=x (1)-tb-300; % height of opening
y3=x (3)/3-((3-1)*tw); % width of opening
Aho=3*(y3*y2)+(y3*y2); % area of opening cell (mm2)
Aeff = A1+Ag-Aho; % effective cross sectional area of concrete (mm2)
VAsb = 0.004*tb*((NG-1)*x(3)+tw)*L+0.005*tb.*((NG-1)*x (3) +tw) ^2;
% volume of bottom slab reinf (mm3)
142
Wstr =ps*av*(L/x(8)+1)*2*tw/2+2*(x(1)-300); % weight of stirrups (g)
% total weight component
z= pc*((Aeff-x(5)-x(6)-x(4)).*L-Wstr./ps)+ps*x(5)*L + ps*VAsb/NG +
Wstr+ pp.*x(6)*L-x(4)*ps*(x(3)+x(2)+(x(1)-tb)/sin(45));% weight
function extradosed superstructure component
% Non Linear Constraint Functions Definition for Pc Box Girder for
Extradosed-Girder Bridge
function [C,ceq] = Extramaincompconst(x)
% Problem parameters
% D = x(1), W1=x(2),W2= x(3),At=x(4),As= x(5),Ap =x(6) y=x(7),
%s=x(8)
% Material properties
fc =55; % cylindrical compr.strength for box girder (N/mm2)
fy = 420; % yield strength of reinforcing steel (N/mm2)>=D12
(Max.D32)
fpu = 1860; %specified tensile strength of prestressing
reinforcement (Mpa)(fpu = 270,000 psi (low-relaxation strand;)
fcbd=0.6.*fpu; % for stays cable are designed to a maximum
allowable tensile strength(fcbd is the design stress (allowable
cable stress)
fci= 0.8*fc; % fci=fc' Specified compressive strength of concrete at
transfer of prestress
fcii=fci*0.9; % fcii=fci'
fck=0.8*fc;
Ec=1820*sqrt (fc);
Eci=1820*sqrt(fci);
ftt1=0.63*sqrt(fci);
Fs=400; %Prestressed Reinforcement grade, used Edge Tension Forces
(Mpa)
pc = 0.0024; % density of concrete(g/mm3)
Ec = 27760; % Young's modulus of concrete (N/mm2)
Es = 2*10^5; % Young's modulus of reinforcing steel (N/mm2)
Ep = 1.995*10^5; % Young's modulus of prestressing strands (N/mm2)
ns = Es/Ec; % Modular ratio of reinforcing steel
np=Ep/Ec; % Modular ratio of prestressed strands
Fcuk=40; % cube strength of prestressed concrete (N/mm2)
143
% stress limits in concrete
fci = 0.8.*fc; % Specified compressive strength of concrete at
transfer of prestress
fct = 0.6*fci; % Allowable compressive stress at transfer of
prestress
% Compressive Stress Check of the Main Beam at SLS
PHicwatsupport=0.93; % compressive stress check factor for mid-span
PHiwamidspan=0.98; % compressive stress check factor for support
ftt2=0.6*pHiwamidspan*fci; %compressive strength of girder at top
fiber
fcw = 0.45*fc; % Allowable compressive stress at working loads
ftw = 0.5.*sqrt(fc); % Allowable tensile stress at working loads
% loading box girder
Md =-886238e6; % design moment at support due to Action (kN.m)
%Md=333176e6; % design moment at mid-span due to Action (kN.m)
Vd =-73587e3; % max shear due to Action (N)
Tu = 131.8; % maximum torsion due to action (N)
Pa=1980.3e3; % axial force (N)
Mw = 1880218.79e6; % Service limit state-I bending moment (Nmm)
M3 = 1696825.33e6; Service limit state-III bending moment (Nmm) for
tension control
Mf=5888.69e6; % fatigue load design bending moment (Nmm)
Mg=-1901784e6; %moment due to dead load at support point
%% Geometric properties
tb=min([1/36.*x(1), 200]); % thickness of bottom flange
%% section properties of box girder
Gs=(x(3)-(n+1)*tw)/(n+1);% spacing of interior girder tb=max([140,(gs-x(2))/30]);
be = min([L/4,12*x(3)+x(2),gs]);% effec.width for int. girder
bb=be;
Re=24700; % total width of roadway (mm)
wsup = 4000; % width of support pier(mm)
Angle=43.9; % angle of external web inclination ()
dduth =32; % diameter of duct
cov=50;% concrete cover of girder mm
Ao=201; % area enclosed by shear path, including the area of holes
144
dductho=dduth+10; % diameter of duct hole
Acp=x(3)+2*x(2)+(x(1)-tb)/sin (43.39); %Fulcrum Section area for
torsion (mm2)
tw=max([1/36.*x(1)+cov + dduth, 200+dduth]); % thickness of web(mm)
Ao=201; % area enclosed by shear path, including area of holes
dductho=dduth+10;% diameter of duct hole
tft=max([200, 150+dductho]); %thickness of top flange
tw=max([1/36.*x(1)+cov + dduth, 200+dduth]); % thickness of web(mm)
Gs=(x (3)-(n+1)*tw)/(n+1); % number of cell of box girder
Aeff=Re*tft+x (3)*tb+(x (1)-tb-tft)*tw*(n-1)…
+(x (1)-tb-tft)/sin (43.9)*tw*2;
yt=(x(3)*tb^2/2+Re*tft*(x(1)-tft/2)+(x(1)-tb-tft)*tw*(n-1)*(x(1)…
-tb-tft)/2+(x(1)-tb-tft)/sin(43.9)*tw*1/3*(x(1)-tb…
-tft)/sin(43.9)*2)/Aeff;
yb = x(1)- yt; % depth from c.g of section to extreme top fiber (mm)
Iy=x (3)*tft^3/12+x (3)*tft*(yt-tft/2) ^2+tw*(x (1)-tb-tft)
^3/12*(n-1)+tw*(x (1)-tb-tft)*((x (1)-tb-tft)/2-yt) ^2+tw*((x(1)…
-tb-tft)/sin (43.9)) ^3/12+tw*(x (1)-tb-tft)/sin (43.9)…
*(((x (1)-tb-tft)/sin (43.9))*1/3-yt)…
+Re*tft^3/12+Re*tft*(tft/2-yt)^2;
Zb=Iy/yb;
Zt=Iy/yt;
xb=((Re.*(Re/2).*tft)+x(3)*tb*(x(2)+(x(1)-tb-
tft)/tan(43.9)+x(3)/2)+(x(1)-tb… tft)/sin(43.9)*tw*(x(2)+(x(1)-tb-
tft)/tan(43.9)*2/3)+(x(1)-tb-tft)/sin(43.9)*tw*(x(2)+(x(1)…
-tb-tft)/tan(43.9)+x(3)+(x(1)-tb-tft)/tan(43.9)*1/3)+(x(1)-tb…
-tft)*tw*(x(2)+(x(1)-tb-tft)/tan(43.9)+Gs)+(x(1)-tb…
-tft)*tw*(x(2)+(x(1)-tb-tft)/tan(43.9)+(n-1)*Gs))/Aeff;
Ix=tft*Re^3/12+tft*Re*(Re/2-xb)^2+tb*x(3)^3/12+tb*x(3)*(x(2)…
+(x(1)+tb-tft)/tan(43.9)+x(3)/2-xb)^2+((x(1)-tb…
-tft)/tan(43.9))*tw^3/12+((x(1)-tb-tft)/tan (43.9))*tw*(x(2)…
+(x(1)-tb-tft)/tan (43.9)*2/3-xb)^2+(x (1)-tb…
-tft)/sin(43.9)*tw^3/12+(x(1)-tb-tft)/sin(43.9)*tw*(x(2)+x(3)+(x(1)…
-tb-tft)/tan(43.9)+(x(1)-tb-tft)/tan (43.9)*1/3-xb) ^2+(x (1)-tb…
-tft)*tw^3/12+(x (1)-tb-tft)*tw*(x(2) +(x(1)-tb-tft)/tan (43.9)…
+Gs-xb) ^2+(x (1)-tb-tft)*tw^3/12+(x (1)-tb-tft)*tw*(x(2)+((x(1)…
145
-tb-tft)/tan (43.9) +2*Gs-xb) ^2);
Ixy=xb*yb*Aeff;
%% equations for effective depth of reinforcing steel
db = 32; % assumed diam. of bar assume it (mm).
Agg = 25; % maxim aggregate size (mm)
Sh = max ([1.5*db, 1.5*Agg, 38]); % clear spacing of parallel bars
(horizontal) (mm)
Sv = max ([25, db]); % clear spacing between layers of bars
(vertically) (mm)
tft=max([200, 150+dductho]);%thickness of top flange
as = pi*db^2/4 ; % area of a single reinforcement bar (mm2)
nb = x(5)/as; % Number of bars
npr = min([(tw+Sh-124)/(Sh+db),nb]);% Number of bars per a row
nr = nb/npr; % Number of reinforcement rows
hr = nr*db+ Sv*(nr-1); % Height of reinforcement rows
dst =62+hr./2;%depth from extreme tension fiber to centroid of
reinforcement steel (mm)
d = x(1)-dst; % effective depth of reinforcement steel (mm)
% effective depth of prestressing reinforcement steel
dsrd =15.2;% assumed diameter of prestressing low relaxation strand
(mm)
Nspt = 31; % number of strands per tendon
ap = 0.77.*pi.*dsrd.^2./4; % steel area of a single strand
(mm2)(using a reduction factor of 77 % of nominal area of the
strand)
dduct = 125; %diameter of duct, (mm)
Sduct = 38; %clear vertical and horizontal spacing of ducts (mm)
nst = x(6)/ap;% number of strands required
nt = nst/Nspt; % Number of tendons
ntr= min([(tw+Sduct-200)/(dduct+Sduct),nt]); %Number of tendons per
a row
nrt = nt/ntr; % Number of rows of prestressing tendons
hrt = dduct*nrt+Sduct.*(nrt-1); %height of rows of prestressing
tendons
dpt = 50+12+Sduct+25+hr+hrt./2;%Depth from extreme tension fiber to
centroid of prestressing tendons (mm)
146
dp = x(1)-dpt;% Depth from extreme top fiber to centroid of
prestressing steel (mm)
% shear reinforcement steel
NL = 4; % No. of legs of vertical stirrups
dsh = 16; % diam. of bar for shear reinforcement (mm)
av = NL*pi*dsh^2/4;%area of shear reinforcement within a distance S
(mm2)
% NA depth c from equivalent stress block analyses
co = ((x(6)*fpu+x(5)*fy-0.85^2*fc*(be-tw)*tft)/(0.85^2*fc*tw)…
+(0.28*fpu)*x(6)./dp);
if (co > tft)
c = co; % NA depth for T section (mm)
c = (x (6)*fpu+x (5)*fy)/(0.85^2*fc*be+0.28*x(6)*fpu/dp); % NA depth
for rectangular section (mm)
fps = fpu*(1-0.28*c/dp); % Average stress in prestressing steel
(N/mm2)
de = (x(6)*fps*dp+x(5)*fy*d)/(x(6)*fps+(x(5)*fy)); % effective depth
from extreme compression fiber to centroid of tension force (mm)
a = 0.85*c; % depth of equivalent stress block (mm)
% Nominal flexural resistance, Mn
if(c>tft)
Mn = x (6)*fps*(dp-a/2) +x (5)*fy*(d-a/2) +0.85^2*fc*tft*(be-
tw)*(a/2-tft/2); % Mn for T section (mm)
Mn = x (6)*fps*(dp-a/2) +x (5)*fy*(d-a/2)-(x (5)*fy)*(ds'-a/2); % Mn
for rectangular section (mm)
%% extreme fiber stresses for computing Prestressing force
% total prestressing force at release
%% fsup = ftt-Mg/Zt; % extreme bottom fiber stress, finf developed
at a given eccentricity e (N/mm2)
finf = ftw/0.85+Mw/(0.85*Zb); % extreme bottom fiber stress, finf
developed at a given eccentricity e (N/mm2)
e = yb-dpt; % possible maximum eccentricity of prestressing force
from c.g (mm)
P = Aeff*finf*Zb/(Zb+Aeff*e); % x(5)*fpt; minimum prestressing force
at a known eccentricity, e (N)
% shearing force parameters
147
dv = max([0.9.*de,0.72*x(1),de-a/2]); % effective shear depth
Vu = Vd*(L/2-wsup/2-d)/L/2; % ultimate design shear force at a
distance d from face of support (N)
Vc = 0.083*2*sqrt (fc)*tw*dv; % shear capacity of concrete
% Shear Force Due To Required Vertical Reinforcement, Vs
teta=45; %Crack Angle
Vs = av*fy*dv*cot(teta)/(x(8)); % shear capacity of non prestressing
reinforcement
Vp = 0.85*P*(4*e/L); % shear capacity of non prestressing
reinforcement
bv=(Vu/0.9-Vp)/(0.25*dv*fc);
Vn = min ([(Vc+Vs+Vp),(0.25*fc*bv*dv+Vp)]); % nominal shear
resistance
% limits of longitudinal reinforcement
fcpe = 0.85*P*(1/Aeff+e/Zb); % compressive stress in concrete due to
effective prestress forces only (N/mm2)
fr = 0.97*sqrt(fc); % modulus of rupture (N/mm2)
Mcr = (fcpe+fr)*Ixeff/yt; % cracking moment (Nmm)
% limits of maximum reinforcement
% a). Using reinforcement index omega-om
Asn = 0;
rhp = x(5)/(be*d);
rhn = Asn/(be*d);
rhpr = x(6)/(be*dp);
Omp = rhp*fy/fc;
Omn = rhn*fy/fc;
Ompr = rhpr*fps/fc;
% b). Using imperic.
c/de <= 0.42
% cracked section analysis
fp1 = 0.85*P/Aeff; % stress in the prestressing tendons prior to the
application of Mw (N/mm2)
fp2 = 0.85*np*P*(e.^2/Ixeff+1/Aeff); % stress in prestressing
tendons due to decompression (N/mm2)
% incremental strain during the appl. of Mw
% let NA depth of cracked section be y = x (7)
148
if(x(7)> tft)
eo = (x(6)*(fp1+fp2))/(0.5*Ec*(tw*x(7)+(be-tw)*tft*(1+(x (7)-tft)/x
(7)))-(Es*x (5)*(d-x (7))/x (7) +Ep*x (6)*(dp.*x (7))/x (7)));
eo = (x(6))*(fp1+fp2)/(0.5*Ec*be*(x(7))-(Es*(x(5))*(d.)…
*(x(7)))/(x(7)+Ep.*(x(6))*(dp-x(7)/x (7)))));
fco = eo*Ec; % stress in concrete at service limit state (N/mm2)
fs = Es*eo*(d-x(7))/x(7); % tensile stress in reinforcing steel at
service stage (N/mm2)
fp3 = Ep*eo*(dp-x(7)/x(7); % tensile stress in prestressing steel at
service stage (N/mm2)
fpk = fp1+fp2+fp3; % total tensile stress in prestressing steel at
service stage (N/mm2)
Ts = x (5)*fs; % tension force in reinforcing steel at service limit
state (N)
Tp = x (6)*fpk; % tension force in prestressing steel at service
limit state (N)
Ca = 0.5*fco*be*x (7); % total compression force in concrete (N)
Cn = 0.5*fco*(be-tw)*(x (7)-tft)^2/x(7); % a force used to reduce
if x(7)>tft
dz = x(7)/3; % location of centroid of comp. force C from top (mm)
dzn = tft+(x(7)-tft)/3; % location of centroid of comp. force Cn
from top (mm)
% section properties of cracked transformed section
% --------------moment of inertia of cracked section---------------%
if(x(7)>tft)
Ict = tw.*x(7).^3./3+((be-tw).*tft.^3/12)+(be-tw).*tft*(x(7)…
-tft./2).^2+np.*x(6)*(dp-x(7))^2+ns.*x(7)*(d-x(7))^2; % 2nd moment
of area of cracked transformed section (mm4)
Ict = be*x(7)^3/3+np*x(6)*(dp-x(7))^2+ns*x(5)*(d-x(7))^2; % 2nd
moment of area of cracked transformed section (mm4)
% ---------------------deflection parameters----------------------%
frk = 0.63*sqrt(fc); % modulus of rupture for Ie computation (N/mm2)
Mck = frk*Ixeff/yb;%cracking moment for deflection computation (Nmm)
Ie = min([(Mck./Md).^2*(Mck./Md)*Ixeff…
+(1-(Mck./Md)^2*(Mck./Md).*(Ict),Ixeff]); %effective moment of
inertia for deflection calculation (mm4)
149
defD = (1.708E+19)./(Ec.*Ie); % total dead load deflection including
long term effects (mm)
defLL = (5.45E+17)/(Ec*Ie); % maximum live load deflection (mm)
defP = (0.85*5*L^2*e*P/(48*Ec*Ie)); % total effec. Prestressing
load deflection (mm)
% maximum crack width
cw1 = (fs - 40)*1e-3; % CEB-FIP-1970, crack width eq. (mm)
h1 = d-x (7)-dst; % depth from steel centroid to NA (mm)
h2 = d-x (7); % depth from NA ~ tension face (mm)
dc = 62+db./2; % concrete cover to closest bar layer (mm)
Atc = tw*2*dst/nb; % effective tension area of concrete per bar
(mm2)
cw2 = 0.076*(h2/h1)*fs*(dc*Atc)^(1/3)*1e-3*0.1451; % Gergely Lut2-
1968 crack equation (mm)
cw = max([cw1, cw2]); % maximum of the crack width given by the
above eqns.
cwa = 0.41; % allowable crack width for moderate exposure condition
%% fatigue stress ranges
ffs = ns* Mf.*(d-x(7))/Ict; % fatigue stress range in reinforcing
steel(N/mm2)
ffp = np* Mf.*(dp-x(7))/Ict; % fatigue stress range in prestressing
steel (N/mm2)
%% torsion resistance
Pc= (x (3) +2*(x (1)-tb)/tan (angle) +2*x (2)); % the length of the
outside perimeter of the concrete section
fpc=0.25*sqrt(fc);
Tcr=(0.125*sqrt(fc')*Acp^2)./Pc.*sqrt(1+fpc/(0.125*sqrt(fci)));
%% Axial force and moment capacity check
% Prmax=Nsp;
% Shear force capacity check
%% factored flexural resistance
Mr=0.9*Mn;
Mu=Vu*dv; % factored shear force
%% Stress Limits for Prestressing Tendons
fpj=0.8*fpu; % jacking stress for Low Relaxation Strand
150
fpt = 0.7*fpu; % Post-tensioning at anchorages and couplers
immediately after anchor set
fpy = 0.9*fpu; % Yield strength of Prestressing Strand
fpe = 0.8*fpy; % Allowable stress in tendons at working loads(stress
limitation)
%% Temporary Concrete Stress Limits at Jacking State before Losses
due to Creep and Shrinkage
% Fully Prestressed Components
fccj =0.55*(fci); % compressive stress limit of post tension
fctj=0.58*sqrt(fci); % tension stress limit of post tension---Area
with bonded reinforcement which is sufficient to resist 120% of the
tension force in the cracked concrete computed on the basis of
uncracked section
fsa=0.8*fr; %
dzn= tft+(x(7)-tft)/3;
% Bursting Forces Results of End Beam Prestressed reinforcement
As=201;
a1=309.64; %( mm)
alphab1=17; % angle of inclination of a tendon force with respect to
the centerline of the member;
rnb1=40; % Root number
Pub1=1395*19*139*1.2; % top and bottom--factored tendon force (N)
T=rnb1*As*Fs*phib;
Tburst=0.25*sum (Pub1.*(1-a1/x (2) +0.5*sum (Pub1*sin (alphab1))));
%Compression zone assumed in the strength limit state to the depth
of the actual compression zone stress block.
%Nonlinear inequality constraints [c] written of the form gi(xi)<= 0
g1 = ftt2-P*(1/Aeff+e./Zt)-Mg/Zt; % Top fiber subjected to tension
at stress transfer stage
g2 = ftt1-(1/Aeff)*P+Mx*(yb*Iyeff-xb*Ixy)/(Ixeff*Iyeff-
(Ixy)^2)+My*(xb*Ixeff-yb*Ixy)/(Ixeff*Iyeff-(Ixy)^2); % Top fiber
subjected to tension at stress transfer stage
g3= P*(1/Aeff+e./Zb)-Mg/Zb-fct; % Bottom fiber subjected to
compression at stress transfer:
g4=fct-P/Aeff-Mx.*(yb.*Iyeff-xb.*Ixy)./(Ixeff.*Iyeff…
151
-(Ixy).^2)+My.*(xb.*Ixeff-yb.*Ixy)/(Ixeff*Iyeff-(Ixy)^2);% Bottom
fiber subjected to compression at stress transfer
g5=ftt2-P/Aeff-Mx*(yb*Iyeff-xb*Ixy)/(Iyeff*Ixeff+Ixy^2)…
+ My*(xb*Ixeff-yb*Ixy/ (Iyeff*Ixeff+Ixy^2)); % Top fiber subjected
to tension at stress transfer stage
g6= 0.85*P.*(1/Aeff-e/Zt) +Mw/Zt-fcw;
g7= ftw-0.85.*P.*(1./Aeff+e./Zb) +M3 /Zb;
g8=fct-Pa/Aeff+Md*yb/2*Ixeff+c*k; % live load stress limit
g9 = (Md-0.9.*Mn); % flexural strength required
g10= Vu-0.9.*Vn; % shear strength required
g11= Vu/0.9-(0.25*fc*dv)-Vp; % web requirement for shear
% limits of flexural reinf.
g12= abs(Md)/(0.9*dv)+abs(Vu/0.9-Vp)-0.5*min([Vu/0.9,Vs])-x(5)*fy-
x(6)*fps; % longitudinal reinf.
g13 = Vu/0.9-(0.5*Vs)-Vp-(x (5)*fy)-x (6)*fps; % min. longitudinal
reinf.
g14= (Md/dv+0.5*Pa+0.5*Vu*cot (angle)-x (6)*fpu)/ (Es*x (5) +Ep*x
(6)); %strain in reinforcement on flexural tension side
g15 = 1.2*Mcr; % minimum flexural reinf. Requirement
g16 = 0.004*tw*yb-x (5)-x (6); % minimum flexural reinforcement
required
g17 = (Omp+Ompr-Omn-0.3);
g18 = (c/de-0.42); % maximum flexural reinforcement required
bv=tw-dduct;
g19=Vu-0.1*fci*bv*dv*x (8); % maximum spacing of transverse
reinforcement
% limits of traverse reinforcement
g20= (x (8)-fy*av/ (0.083*tw*sqrt (fc))); % shear reinforcement
if (abs(Vu-0.9.*Vp)/(0.9*dv*tw) < 0.125*fc)
g21 = (x (8)-0.8.*dv); % spacing of shear reinf.
g21 = x (8)-min ([0.4*dv, 300]); % spacing of shear reinf.
% service load stress limit
g22 = P - x (6)*fpt; % stress limit in tendons at transfer
g23 = fs- min ([206, 0.6*fy]); % stress limit in reinforcing steel
at service limit state
% deflection limit
152
radd = 0;
tol = 1e-6;
confcnvald = defD-defP-radd;
g24 = confcnvald-tol; % camber due to prestressing shall counter
balanced by dead load deflection
g25= -confcnvald-tol;
g26= defLL-L/10000; % limit of vehicular live load deflection
% Crack width
g27= (cw-cwa); % spacing of longitudinal bars for crack control
% fatigue stress limit
g28 = (ffs-161.5); % limit on fatigue stress limit in reinforcing
steel
g29= (ffp-125); % limit on fatigue stress limit in prestressing
steel
% service load degree of prestress
g26=1-Mdec/Mw;
% check equilibrium conditions
% summations of internal couple must equal to working moment
if x(8) >tb
rad = Mw;
dz=x(1)./2-cov;
tol = 1e-6;
confcnvalm =Ts*d+Tp*dp+Cn-Ca*(dz)-rad;
g30 = confcnvalm-tol;%sum of service load moments when NA depth y>hf
g31 = -confcnvalm-tol;
rad = Mw;
tol = 1e-6;
confcnvalm = Ts*d+Tp*dp-Ca.*(dz)-rad;
g32 = confcnvalm-tol;%sum of service load moments when NA depth y<hf
g33= -confcnvalm-tol;
if x(8) > tft
radf = 0;
tol = 1e-6;
confcnvalf = Ts+Tp+Cn-Ca-radf;
g34= confcnvalf-tol; %sum of service load moments when NA depth y>hf
g35 = -confcnvalf-tol;
153
radf = 0;
tol = 1e-6;
confcnvalf = Ts+Tp-Ca-radf;
g36= confcnvalf-tol; %sum of service load moments when NA depth y<hf
g37 = -confcnvalf-tol;
O=0.9; % torsional resistance factor
g38=0.25*O.*Tcr-(Tu); % torsional resistance
g39 = (0.20*x (1)-x (7));
g40 = (x (8) - 0.75*x (1));
g41= (L/800); % deflection limit of bridge
% Compressive Stress Check of the Main Beam at SLS
Factor =0.93; % phi*watsupport
g42= (0.6*factor*fck); % Under the SLS combination I, the
compressive stress limit of the main beam
%Tensile Stress Check of the Main Beam at SLS
g43= (0.25*sqrt (fc)); %for segmental- constructed bridge, the
tensile stress limit of the joint is 0.25*fc under the SLS
combination III,
% Cable Stress limitation Check
g44=0.55*fpk; % cable stress meet the specifications requirements
g45=Tu*Pc/(2*0.9*Ao*fy)-x (4); % torsional reinforcement limit
g46= (fccj-0.6*phiw*fci); % Compressive Stress Limits in Prestressed
Concrete at Service Limit State after Losses, Fully Prestressed
Components
g47= (fctj-3*sqrt(fci)); % tension Stress Limits in Prestressed
Concrete at Service Limit State after Losses, Fully Prestressed
Components
g48=Vd-Vn; % constraint of shear force
g49=Tburst-O*T; % bursting force checks
g50=Vu/0.9-0.5*Vs-Vp-x (5)*fs-x (6)*fpu;
g51=x (8)-(av*fy)/ (0.083*bv*sqrt (fc));
%Minimum amount of reinforcement
g52=1.2*Mcr-0.9*Mn;
g53=1.33*Md-0.9*Mn;
g54= Mu/(dv*0.9)+(Vu/0.9-Vp)-0.5*min([Vu/0.9,(x(5)*fy*dv)/x(8)])-
((x(5)*fy+x(6)*fpu));% limit of longitudinal reinforcement
154
g55=Md/O*dv+0.5*Pa/O+ cot (angle*sqrt ((Vu/O-Vp)-0.5*Vs) ^2+
(0.45*Pc*Tu))-(x (6)*fps+x (5)*fy); % longitudinal reinforcement due
to torsion
% nonlinear equality const. functions defn.
C=[g1;g2;g3;g4;g5;g6;g7;g8;g9;g10;g11;g12;g13;g14;g15;g16;g17;g18;g1
9;g20;g21;g22;g23;g24;g25;g26;g27;g28;g29;g30;g31;g32;g33;g34;g35;g3
6;g37;g38;g39;g40;g41;g42;g43;g44;g45;g46;g47;g48;g49;g50;g51;g52;g5
3;g54;g55;g56,g57,g58;]; % nonlinear inequality const. functions
defn.
%% Main Code for Running the GA Algorithm
% Problem parameters
% D= x(1), W1= x(2), W2 = x(3),M= x(4), As = x(5),% Ap = x6, y=x7,
s=x(8)
objfcn=@Extramaincompfun; % calling fitness function
consfcn= @Extramaincompconst; % calling constraint Function
% set boundary values of variables
lb =[4551 1350 7800 100 1*10^3 1*10^3 900 100];
ub=[7818*10^3 3350 9500 10000 1*10^9 1*10^8 2000 400];
%% Develop Genetic algorithm for optimization
options = optimoptions(@ga,...
'PopulationSize',5000, ...
'MaxGenerations',1000, ...
'EliteCount',10,...
'Maxstallgeneration',1000,...
'FitnessScalingFcn',@fitscalingprop, ...
'NonlinearConstraintAlgorithm','auglag', ...
'InitialPenalty',1.5,...
'PenaltyFactor',1, ...
'FunctionTolerance', 1e-10, ...
'ConstraintTolerance', 1e-10,...
'PlotFcn',@gaplotbestf)
%Call |ga| to solve the Problem
rng(1,'twister') % random number generator for reproducibility
[xbest, fbest, exitFlag] = ga (@Extramaincompfun, nvars, Aineq,
bineq, Aeq,…. Beq, lb, ub,@Extramaincompconst, 1:3, options);
% _Analyze the Results
155
display (xbest)
%% return optimal value
fprintf ('\n z function returned by ga = %g\n', fbest)
Annex 5 Design Optimization Code Using GA in Matlab for Pylon of
Extradosed Bridge
% this is fitness function for Pylon of Extradosed Bridge
Function Q = pylonobjfun(x)
% weight parameter for pylon
pc = 0.0024; % density of concrete(g/mm3)
ps= 0.00785; % density of steel_prestressing strands and
reinforcing bars (g/mm3)
NL = 4; % number of legs of vertical stirrups
dsh = 16; % diam. of shear rebar (mm)
av = NL*pi*dsh^2/4; % area of f16mm for shear reinforcement within a
distance S (mm2)
Wstr = pc*av*(x(1)/x(3)+1); % weight of stirrups (mm3)
%Wstr= pc*av*(x (1)/x(2)+1)*2*(x(1)/2+2*(x(1)-280));
%pp= 0.0023; % density of steel_prestressing strands and reinforcing
bars (g/mm3)
Ap1=2000*3000; % area at top of pylon (mm^2)
Ap2= 2000*4000; % area at bottom of pylon (m^2)
vp=(Ap1+Ap2)*x(1); % concrete volume of pylon (mm^3)
% weight function of pylon for extradosed bridges
Q= pc*((vp-x (3)*x (1))-Wstr/ps) + ps*x (3)*x (1) + Wstr; % weight
function extradosed superstructure component
% Non Linear Constraint Functions Definition for Pylon for
Extradosed Cable- Stay Bridge
Function [C,ceq] = pylonconst(x)
% material properties
fyk = 420; % standard strength non prestressed reinforcement
fyd=fyk/1.15; % yield strength of steel reinforcement
fc = 50; % cylindrical compr.strength for pylon (N/mm2)
fcd=0.8*0.85.*fc/1.5;
fci=0.8*fc;
pc = 0.0024; % density of concrete(g/mm3)
156
Ec = 0.043*pc*sqrt (fc); % Young's modulus of concrete (N/mm2) ----
pc density of concrete
Es = 2*10^5; % Young's modulus of reinforcing steel (N/mm2)
ns = Es/Ec; % Modular ratio of reinforcing steel
%% pylon section properties
z=1.195e3;
rh=1.5e3; % head radius
rt=2e3; % toe radius
n=rh/rt; % taper ratio
n1=n-1;
h=25e3;
F1=n1*h/x(1);
r=rt*F1; % To express the taper function of r at x mathematically
k=4; % number of side of pylon cross section
c1=k*sin(pi/k)*cos(pi/k);
c2=k/12*sin (pi/k)*(cos^3*(pi/k)*(3+tan^2*(pi/k)));
c3=1/3*(n^2+n+1);
I=c2*r^4; %second moment of the plane area at x
A=c1*r^2; %cross-sectional area
V=c1*c3*rt ^2*x (1); %The column volume V
rt =sqrt((V/(c1*c3*x(1))));
A= (V/(c3*x (1)))*F1^2;
I= (c2*V^2)/ (c1^2*c3^2*x (1) ^2)* F1^4; % moment of inertial of
tapered column
Fw=pc*A; % self-weight of pylon
F2=(n1^2)/3*(z^3/x(1)^3) +n1*z^2/x(1)^2 +z/x(1);
%N=Pcr+pc*V-integral (Fw, 0, x)*dx
Pcr=N-pc*V(1-F2/c3);
Q=diff (M,x)-N *diff(y,x);
%diff(Q,x)=diff(M,x,2)-diff(N,x)*diff(y,x)-N*(diff(y,x,2));
%diff(y,x,4)=-2/I*diff(I,x)*(diff(y,x,3)…
-1/I*(diff(I,x,2)*I*diff(y,x,2)-1./Ec*I*(B+pc*V(1-F2/c3
))*(diff(y,x,2)+pc*V/(c3*x(1))*(F1^2)/Ec*I*diff(y,x))));
Fw=pc*V/ (c3*x (1))*F1^2;
%M=-EI (diff(y, x, 2));
157
%diff(M,x,2)=-Ec*(diff(I,x,2)-(diff(y,x,2)-
2*Ec*diff(I,x)*(diff(y,x,3)-Ec*I(diff(y,x,4)))));
%diff(I,x)=(4*n1*c2*V^2)/(c1^2*c3^2*x(1)^3)* F1^3;
za=z/x(1);% non-dimensional Cartesian coordinates
f1=n1*za+1;
%diff(y,x,4)=-(8*n1/(x(1)*f1))*diff(y,x,3)-(12*n1^2/(x(1)^2*F1^2))*
diff(y,x,2)-(c1^2*c3^2)/(c2)*((x(1)^2)/(E*V^2)* (Pcr+pc*V(1-F2/c3 )*
1/(F1^4 )*(diff(y,x,2)
+(c1^2*c3*pc*x(1))/(c2*E*V)*1/(F1^2)*diff(y,x))));
beta=(Pcr*x(1)^4)/(Ec*V^2'); % buckling load parameter,
Pcr= (beta/x (1))^4*Ec*V^2; % buckling load
ba=y/x(1)'; % non-dimensional Cartesian coordinates
lamda=(pc.*x(1)^4)./Ec.*V';%self-weight parameter.
f2= (n1^2)/3*za^3+n1*za^2+za;
%diff(ba,za,4)=-(8*n1/F1*(diff(ba,za,3)-
(12*n1^2/f1.^2*(diff(ba,za,2))-(c1^2*c3^2)/c2*(beta+lamda.*(1-
f2/c3))*1/(f1^4)*(diff(ba,za,2)
+(c1^2*c3)/c2)*lamda/(f1^2)*diff(ba,za'))));
%diff (ba, za, 3) + (c1^2*c3^2)/ (n^4*c2)*diff (ba, za)*beta=0;
J= (pc*x (1)^4)/(E.*V'); %buckling self-weight parameter for P = 0
%diff(ba,za,4)=(-(8*n1/f1*diff(ba,za,3)-
(12*n1^2/(f1^2)*diff(ba,za,2)-(c1^2*c3^2)/c2*J*(1-
f2/c3)*1/(f1^4)*(diff(ba,za,2)+(c1^2*c3)/c2*J/(f1^2)*diff(ba,za'))))
;% L is the self-weight buckling length for which the column buckles
under self-weight alone
x (1)=nthroot((E*V^2)/Pcr,4)*beta;
L=nthroot (EV/pc, 4)*J;
Stress= (c3*Ec*V)/x(1)^3*(1-f2/c3)*J/(f1^2); %self-weight buckling
stress
E=1.995.*10 ^5; % elastic modulus of wire or strand
r= 99; % radius of strand
R= 150; %radius of saddle bend
f=E*r/R+P/A;
g11=stress-f;
hpl=24.5e3;
g12=L-0.7*hpl;
158
% loading
Tc=[5600e3;5600e3;5600e3;5800e3;5800e3;5800e3;5800e3;5800e3;5800e3];
% initial tension force for stay cable
Nsd=1357.3e18; % maximum axial force of pylon in strength I(N)
angle=[32.13; 28.12; 27.91; 23.04; 21.86; 19.83; 19.97;19.52;18.99];
Npt=2*Tc(1)*cos(angle(1))+2*Tc(2)*cos(angle(2))+2*Tc(3)…
*cos(angle(3))+2*Tc(4)*cos(angle(4))+2*Tc(5)*cos(angle(5))+2*Tc(6)*c
os(angle(6))+2*Tc(7)*cos(angle(7))+2*Tc(8)*cos(angle(8))+2*Tc(9)*cos
(angle(9));
Vu=62836.93e18;% maximum sheer force of pylon in strength I
% Axial ultimate load resistance (squash load) of a column (NRd)
Nrd= (Apt-x (3))*fcd+ x(3)*fyd; %Design Axial Load(N)
Vn=Nrd/((L1*L2)*fcd);
%critical load for buckling
Le=x (2)*1;
EIe=max (0.2*Ec*Ic+Es*Is, 0.4*Ec*Ic);
Ncr=pi*EIe/Le ^2;
c=pi^2/8*(x (1)*Pcr); % resistance of pylon against horizontal
displacement
% axial loads
pu=Tc.*2+Wpylon;
Mcy=324000; Mcz=0; Mc=x (3)*fyd;
% axial load and moment’s capacity check
Pr=1.9e8; %N) concentric max axial load pr-max
%checks whether the column is slender or not:
k=0.77; Lu=x (1);
r=120;
ph=240442;% axial load
vuy =7653.98;% design shear value in y direction
vuz =7653.98;% design shear value in z direction
%Nonlinear inequality constraints [c] written of the form gi(xi)<= 0
g1=Nsd/Ncr-0.1;
g2=pu/pr-1;
g3=Mcy/Mc-1;
g4=Mcz/Mc-1;
%shear force capacity check
159
g5=ph/vuy-1;
g6=ph/vuz-1;
g7=Vu./ph./Vn-1;
g8=pu-Nrd;
g9=34-k.*lu./r;
g10=Npt-Pcr; % limit of deflection state of pylon
C =[g1;g2;g3;g4;g5;g6;g7;g8;g9,g10;g11;g12];% nonlinear inequality
const. functions defn.
ceq = [];
%% MAIN CODE FOR RUNNING THE GA ALGORITHIM FOR PYLON DESIGN
% Problem parameters
% H= x (2), As = x (7)
% set boundary values of variables
lb = [24e3 200 1e2 ];
ub = [30e3 3000 2e5];
objfcn=@pylonobjfun; % call objective of pylon
consfcn=@pylonconst;% call constraint function of pylon
%% set ga options
opts = optimoptions(@ga, ...
'PopulationSize',5000, ...
'CreationFcn', @gacreationlinearfeasible, ...
'MaxGenerations',1000, ...
'EliteCount',10,...
'Maxstallgeneration', 1000, ...
'FitnessScalingFcn',@fitscalingprop, ...
'NonlinearConstraintAlgorithm','auglag', ...
'InitialPenalty',10,...
'PenaltyFactor',1000, ...
'FunctionTolerance', 1e-10, ...
'Constraint Tolerance', 1e-10,...
'PlotFcn', @gaplotbestf)
% _Call |ga| to solve the Problem_
rng (1,'twister') % random number generator for reproducibility
[xbest, fbest,
exitFlag]=ga(@pylonobjfun,nvars,[],[],[],[],lb,ub,@pylonconst,1:3,
opts);
160
% _Analyze the Results
disp(xbest);
%% return optimal value
fprintf('\n weight function returned by ga = %g\n', xbest);
Figure 1.Structural model optimum cross-section of extradosed cable-stay bridge by
SAP 2000v23
161
Appendix 6 Design Optimization Validation in Excel spreadsheet for stay
cable of extradosed cable-stayed bridge
gca 0.00 angle ratio length(m) number Lh(m) sv(m) sh(m)
Eca 190000.00 x(1) x(2) x(3) x(4) x(5) x(5) x(6)
da 15.20 30 1 26.37 18 19.282 1.132 5.3
Aca 181.37
av 4782.63 lu 15 0.6 14 1E+01 1 4
fpu 1860.00 ub 40 1 20 8E+01 1.4 7
Qult 1600.00
fc 55.00
fcbd=0.6*fpu 1116.00
173690400
QL 140.00 4570.8
Qa=0.45*Qult 720.00
Ec=4500*sqrt(fc) 4.34565763
L 180000.00 4781.92077
Mw 421879.00 135904574
ftw=0.5sqrt(fc) 3.71 0.48113347
et 4781.92
Ls 100000.00 -12154067.6
Lf 22000.00 135904574
r 0.55 6.4988E+11
lamda=0.03; 0.03 6.4988E+11
Lo=63e3; 63000.00 6.4988E+11
Mo=-Fp.*eb; 135904573.93
M2w 2000.00 -25945418.7
Weq=8*eb*Fp./L.^2 0.03 -262.074936
Tc1=-4.*eb./Lf.*Fp 0.00 -262.074936
Ls1=(r.*L-lamda.*L-Lo); 30600.00
w 45708.00 1.4726E+12
e 60.00 -12154067.6
H= x(5).*tan(x(1)) 11.13 587068.568
E 205000.00 -122.768359
n 1.00 14874933.3
I 1.00 -14875302.4
fpk= 1302.00 159089297
(ΔF)TH 110.00 -159089297
R2w1=-Weq.*Lf/2-R1w;
R2p1=3*P*r*L*(2*r*L+L+Lf)*(r*L+(L-Lf)/2)-2*P*r^3*L^3/(6*r^2*L^3*(r+1)-2*r^3*L^2);
R1p=-(R2p1+Tc1)
Check up of validity of Optim. Outputs for stay cables
%The forces resulting from the bottom straight tendons in the side spans
R1o=M23./(r*L);
R2o=-M23/(r*L), R1o=-R2o;
%The forces resulting from draped tendons at the middle of the main span
R1w=M2w/r.*L;
M23=-6*Mo*Lo*(lamda*L+Lo/2)/(3*r*L.^2*(r+1)-r.^2*L.^2);
M2w=Weq.*Lf/48*(3*(L-Lf)^2+6.*Lf.*(L-Lf)+2.*Lf.^2/L.*(r./3+1./2))
R2f=-(-3.*Mf.*Lf/(r.*L.^2)*1/(3+2.*r));
R1f=M21/(r*L)
extreme fiber stresses for computing Prestressing force
ql=0.1qd
M21=-3.*Mf.*Lf./L.*(1/(3+2.*r))
e = yb - dpt
M21=-3.*Mf.*Lf./L.*(1/(3+2.*r))
Ms=Fp.*et;
M22=Ms*(1-(24.*Ls.*r.*L-3*Ls^2)/(12*r*L^2*(1+r)-4*r^2*L^2))
R1s=-Ms*(1-(24*Ls*r*L-3*Ls^2)/(12*r^2*L^3*(1+r)-4*r^3*L^3))=-R2s
Possible use of concordant cable profiles for the extradosed bridges
Eeq=E./(1+(x(3).*yca)^2./12.*f̂ 3);
Fp=P = Aeff.*finf.*Zb./(Zb+Aeff.*e)
Mf=-Fp.*eb
Material Properties optim. Output variable
Desired stay cable force for each stay cable FDi under dead load
w=Aeff*ycon;
FDi=0.2*x(2).*w*L./(9.*sin(x(1)))
stress limits in concrete(Mpa)
stress limit in stay cable(Mpa)
finf = ftw/0.85+Mw/(0.85.*Zb)
162
14873607.2
-14875721.3
-24709.9225
-88596169
-183212979
9.00376606
2903578412
-1364944011
-3.79E+06
-2477429.55
103288.047
-1.6334E+10
2.90E+05
1270.51724
-1.45E-04
9.37E+18
2.92218965
1.0909E+17
-1.2047619 OK
-511 OK
-23067 OK
-9.2609E+18 OK
-9.2609E+18 OK
-1.37E+09 OK
458.057986 OK
0.46672156 OK
-7.20E+02 OK
-4.2555E+10 OK
-1.7699E+10 OK
-0.2 OK
-511 OK
d=(fcbd/Eeq).*(H+(x(5)^2./H))
FDI1=Qcb*fcbd/yca/((H+(x(5)^2/H)))
Tc=pi*dca^2/4.*Qult
%The forces due to the equivalent uniform load Weq alone, can be determined by using the equations
R1w1=-(Weq.*Lo.*(r*L-lamda.*L-Lo/2)-M2w)./(r.*L);
%% To calculate the wind resistance
fb=0.937
m=(Aca*yca)
fn=n./2.*(I.*((FDI./m).^2))
Aca=53*pi*15.262/2
FPTi=(R2f+R2s+R2o+R2w1+R2p1+R2w2+R2p2)./(9*sin(x(1)))-(R1f+R1s+R1o+R1w+R1p+R1w1+R1p1)./(9*sin(x(1)));
Fi=3.*w.*L./(8.*sin(x(1)).*1./(1+3./2.*e./L.*cot(x(1))))
FDI=(FDi-Fi-FPTi)./2
Ai=FDI/Qa*sin(x(1))
Qcb=x(3)*yca./fcbd.*(H+x(5)^2./H).*(FDI)
R2w2=-(Weq*Lo+R1w1);
%The final forces resulting from the equivalent uniform loads Weq and the 2x2 vertical nodal loads
P1=-(4.*eb)./Lf.*Fp;
R2p2=symsum((-P1.*(3.*r.*L.*(r.*L+L+Lsi).*(r.*L-Lsi)-(r.*L-Lsi).^2.*(r.*L+2.*Lsi)-
3.*Lsi.^2.*(r.*L-Lsi))/(3.*r.^2.*L.^3.*(r+1)-r.^3.*L.^3)),i,1,2)
R1p1=-2.*P1-R2p2;
g12=ql/qd-0.3
g13=QL-0.55*fpk
g9=Qw-Qa;
g10=18*FDI/tan(x(1));
g11=FDI-FDI1;
g4=(fn-ft)
g5=(fn-fb)
g6=FDI-0.55.*Tc
g7=(800*d/L-1)
g8=(4.*22e3*8.*6e3./(w.*2.*r.*L+L))*I5
%% nonlinear constraint function
g1=0.7-4.*wc.*56e3/w*(2.*r*L+L)
g2=QL-0.55*fpk
g3=Zm1+M(z)*Mm1(z)./(Ec.*Ixeff);
Qw=FDI./(Qcb*Ai);
163
Appendix 7 Design Optimization Validation in Excel spreadsheet for box
girder for extradosed cable-stayed bridge
D W1 W2 At As Ap y s
5129 2350 9000 5.64E+03 1.68E+08 39000 3900 350
fc= 55 x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8)
Ec = 27660
Es= 200000 lb 4.50E+03 2350 9000 1.00E+02 1*10^3 2*10^3 1500 100
Ep= 199500 ub 7000 2350 9000 1.00E+04 2*10^9 8*10^4 4500 400
fc= 55
ns= 7.23065799
np= 0.9975 5000 3500 1402 2 20689 15678 3578
fy 420 4 b
L= 10000 16 av 803.84 bex
N 3 bb
tw 200 tb
tft 250
vd -40500000
Md 8.86E+11
Mw 4021879 diam. Of bar,db 32 dia.strand dsrd 15.24
M3 -3.96E+10 area of single baras 803.84 140
Mf 1.39E+09 max. agg size,Agg 25
Tu 3.12E+02
Mg 1.90E+10 clc spacing of //bars sh 48 duct diam,DD 125
Ec.Ic.defDL dr spacing of bars layerssv 32 38
Ec.Ic.defLL no bars nb 25.7377 68.68
nr bars per row,npr 5.29 35
fci=0.8*fc 44 no of row of barnr 4.86535 1.9622857
fct=0.6*fci 26.4 Height of reint rowshr 279.382 2.07
ftt=0.63sqrt(fci) 4.178947236 d' dst 172.62 0.9479641
fcw=0.45*fc 24.75 d 4827.38 116.51815
ftw=0.5sqrt(fc) 3.708099244 dpt 466.64157
dp 4533.3584
fpu 1860 Pa 1.98E+06
fpy=0.9*fpu 1674 Ao 201
fpt=0.74*fpu 1376.4 o 0.9
fpe=0.8*fpy 216 y2 2951
fpk 1302 y3 700
Mx=xb.*P; 4.75712E+14
My=P.*yb; 5337426.572 Aho 8262800
A1 49400000 Ag 47139500
be 3.00E+03 300 tb
check up of validity of optim, outputs for box girder
Input fixed variab.
Concrete stress limit(Mpa)
Optimum output variable box girder
prestressing steel stress limits(Mpa)
optimum output variable box girder
number of leg of vertical strirups
dia. Of strirup
effec. depth of reinf. Steel
area of single strand,ap
dr [email protected] spcg,SD
No. strands,nsrd
no strand per tendon
No.of Tdn,nT
No,Tdn/row,ntr
effective depth of prestressing steel
no of. Rows, nrt
ht of rows of Tdn,hrt
164
88276700
2.4859781
5248.514
1.6E+15
2.046E+13
1.579E+15
30087.968
6.352E+14
4.678E+11
4.747E+13
2.405E+11
4.723E+13
2.167E+23
161.62209
4781.8725
1016.9405
7.06E+10
5107054.5
-8.11E+09
4.83E+03
4340996.3
-1.53E+17
16290.491
250
3692.88
-385722
909252.75
31968.296
1653.3792
942874.43
2.079E+14
6.4520927
-0.000838
yb=x1 - ytIgx=(2.*((x(2).*(tb.^3))./12+x(2)*tb.*((yb-tb./2)).^2+ (Re.*(tb).^3)/12+Re.*tb.*((yb-
tb/2)).^2+(x1-tb)./tan(43.9).*((x1-tb)^3).*1/18+((x1-tb)/tan(45)).*(x1-tb).*(2/3.*(x1-tb)-
yt).^2+(x3.*(x1-tb)^2/12
+x3.*(x1-tb).*(x1-tb-yt).^2)))
Iho=(y3.*((y2.^3)./12+y3*y2*(y2-yt)*(y2-yt)+(y3.*y2.^3)/18+(2/3.*(y2-yt).^2)))
Ixeff=Igx-Iho
Aeff=Re*tb+(1/2).*((Re-(2.*x2))+x3).*(x1-tb)-(x4.*(y3.*y2)+(y3.*y2))
yt= (Re.*tb.*(x1-tb./2)+x3.*(x1-tb).*((x1-tb)./2)+((x1-tb).*((x1-tb)./tan(43.9)).*2/3.*(x1-
tb)))./(Re.*tb)+(x3.*(x1-tb)+(x1-tb)+(x1-tb).*((x1-tb)./tan(angle)))
Section Properties
de = (x(6)*fps*dp+x(4)*fy*d)/(x(5)*fps+x(4)*fy) =
e = yb - dpt
P = Aeff.*finf.*Zb./(Zb+Aeff.*e)
co = ((x6.*fpu+x5.*fy-(x5'.*fy')-0.85^2.*fc.*(be-tw).*tft)./(0.85^2.*fc*tw)+
(0.28.*fpu).*x6./dp)
c =(if(co > tft,co,c = (x6.*fpu+x5.*fy)./(0.85.^2.*fc.*be+0.28.*x6.*fpu./dp))
fps = fpu.*(1-0.28.*c./dp)
Flexural reisitance
Minimum flexural reinf
a = 0.85*c =
wsup
Shear force resistance
Mn =if(c>hf), x(5)*fps*(dp-a/2)+x(4)*fy*(d-a/2)+0.85^2*fc*x(3)*(be-x(2))*(a/2-x(3)/2),
else, Mn = x(5)*fps*(dp-a/2)+x(4)*fy*(d-a/2)
dv = max([0.9*de,0.72*x(1),de-a/2])
Vu = Vd*(L/2-wsup/2-d)/(L/2) =
Vc = 0.083*2*sqrt(fc)*tw*dv =
Vs = fy*dv*av/(x(6)) =
Vp = 0.85*P*(4*e/L)
Vn = min([(Vc+Vs+Vp),(0.25*fc*tw*dv+Vp)])
fcpe = 0.85*P*(1/Ac+e/Zb)
fr = 0.97*sqrt(fc) =
Mcr = (fcpe+fr)*I/yb
xb=(Re.*(Re/2).*tb)+x(3).*(x(1)-tb).*(x(2)+(x(1)-tb)./tan(angle)+x(3)./2)+1/2.*(x(1)-
tb).*((x(1)-tb)./tan(angle).*(1/3.*(x(1)-tb)+x(2))+1/2.*(x(1).*tb).*((x(1)-
Zt = Ixeff./yt
Zb = Ixeff./yb
finf = ftw/0.85+Mw/(0.85.*Zb)
extreme fiber stresses for computing Prestressing force
Ixy=xb*yb*Aeff
Iyeff=Igy-Ihy
Ihy=(y2.*(y3.^3)./12+(y2*y3)*(xb-y3/2)^2+(y2.*(y3^3)/18+1/2*(y2*y3)*(xb-(3.*y3-
x2).^2+(y2.*(y3.^3)./18+1./2.*(y2.*y3).*(xb-y3./3-x2-x3.^2)))))
Igy=(x2.^3.*tb)./12+x2.*tb.*((xb-x2)./2).^2+(((Re-2.*x2^3.*tb)./12)+(Re-2.*x2).*tb.*(xb-
x2)-(Re-2.*x2/2).^2+(((x1-tb)./tan(RADIANS(angle))).^3.*(x1-tb ))./18+1/2.*((x1-
Pc= (x(3)+2*(x(1)-tb)/tan(angle)+2*x(2))
165
1058.0012
0
174.00743
0
0.0430145
1328.784
0
1.4546711
1330.2387
9.792E-06
9.767E-06
-5.01E+11
-2.38E+16
-2.38E+16
-2.38E+16
-1.38E+16
-1.34E+20
72540000
-5.4E+21
-6.62E+22
1300
7300
1.10E+15
4.672205
9.828E+11
2.167E+23
2.002E-12
7.182E-15
4.387E-13
-2.38E+13
754.76
927.38
78
2682.7563
-2.25E+16
0.0084343
0.0007946
15713121
ffs = ns* Mf*(d-y)/Ict
ffp = np* Mf*(dp-y)/Ict =
Prestressing indices
Mdec = x(5)*(fp1+fp2)*e =
h2 = d-x(7)
dc = 62+db/2
Atc = tw*2*dst/nb =
cw2 = 0.076*(h2/h1)*fs*(dc*Atc)^(1/3)*1e-3*0.1451 = 0
j). Fatigue stress limit
defP = 0.85*5*P*e*L^2/(48*Ec*Ie)
defLL = 2.63e15/(Ec*Ie) =
cw1 = (fs - 40)*1e-3 =
h1 = d-x(7) - dst =
h). Deflection limit
frk = 0.63*sqrt(fc)
Mck = frk*I/yb
Ie = min([(Mck/Mw)^3*I+(1-(Mck/Mw)^3)*Ict, I]) =
defD = 2.26e16/(Ec*Ie) =
C = 0.5*fco*be*x(7)
Cn = 0.5*fco*(be-x(2))*(x(7)-x(3))^2/x(7
dz = x(7)/3
dzn = x(3)+(x(7)-x(3))/3 =
Ict =if(y>hf)= tw*y^3/3+(be-tw)*tft^3/12+ (be-tw)*tft*(y-tft/2)^2+np*x(6)*(dp-y)^2+
ns*x(6)*(d-y)^2, else, Ict = be*y^3/3+np*x(6)*(dp-y)^2+ ns*x(5)*(d-y)^2
fp3 = Ep*eo*(dp-x(7))/x(7)
fp = fp1+fp2+fp3
fco = eo*Ec
Ts = x(5)*fs
Tp = x(6)*fp
Maximum limits of reinf.
a. c/de
Asn
rhp = x(4)/(be*d) =
rhn = Asn/(be*d)
rhpr = Ap/(be*dp)
fp1 = 0.85*P/Ac =
fp2 = 0.85*np*P*(e^2/I+1/Ac) = 1
eo =if(y>hf) =eo = (x(6)*(fp1+fp2))/(0.5.*Ec*(tw*x(7)+(be-tw)*tft* (1+(x(7)-tft)/x(7)))-
(Es*x(5)*(d-x(7))/x(7)+Ep*x(6)*(dp.*x(7))/x(7)));,else,
eo = (x(6)).*(fp1+fp2)./(0.5.*Ec.*be.*(x(7))-(Es.*(x(5)).*(d.*(x(7)))./(x(7)+Ep.*(x(6)).*(dp-
x(7)./x(7)))))
fs = Es*eo*(d-x(7))/x(7)
Omp = rhp*fy/fc =
Omn = rhn*fy/fc
Ompr = rhpr*fps/fc =
Omp+Ompr - Omn =
Cracked section analysis
166
-4.18E+00 OK
-1022.564 OK
-1026.742 OK
-2.69E+03 OK
-1.35E+04 OK
-1.32E+06 OK
-7.35E+02 OK
-1.38E+17 OK
-1234309 OK
-481010.5 OK
-7.04E+10 OK
-7.05E+10 OK
-4.43E-03 OK
-0.001006 OK
-1.68E+08 OK
-1330.209 OK
-1.12E+18 OK
75 OK
OK
-435016.5 OK
-296285.5 OK
-3.25E+03 OK
OK
-2.25E+16 OK
OK
-157.3211 OK
OK
-7.03E+10 OK
-2874.2 OK
-100.62 OK
-5.61E+03 OK
4.00E+24 OK
-4067274 OK
-1.38E+17 OK
-1.38E+17 OK
-60890 OK
-41442874 OK
-2390 OK
g5 = 0.85.*P.*(1./Aeff-e./Zt)+Mw./Zt-fcw;
g6 = ftw-0.85.*P.*(1./Aeff+e./Zb)+M3./Zb;
g7=fct-Pa/Aeff-Md*yb/2*Ixeff+c*k
g8= (Md-0.9.*Mn);
Non linear inequality constraints [c] written of the form gi(xi)<= 0
g1 = ftt-P.*(1./Aeff+e./Zt)-Mg./Zt
g2=ftt-P/Aeff-Mx*(yb*Iyeff-xb*Ixy)/(Iyeff*Ixeff+Ixy^2)+My*(xb*Ixeff-yb*Ixy/(Iyeff*Ixeff+Ixy^2))
g3=P/Aeff-Mx.*(yb.*Iyeff-xb.*Ixy)./(Ixeff.*Iyeff-(Ixy).^2)+My.*(xb.*Ixeff-yb.*Ixy)./(Ixeff.*Iyeff-(Ixy).^2)-fct
g4 = P.*(1./Aeff+e./Zb)-Mg./Zb-fct;
% Crack width
g21 = (cw-cwa);
g9= Vu-0.9.*Vn;
g10 = Vu./0.9-(0.25*fc*dv)-Vp;
g11 = abs(Md)/(0.9*dv)+abs(Vu/0.9-Vp)-0.5*min([Vu/0.9,Vs])-x(5)*fy-x(6)*fps;
g9 = Vu./0.9-(0.5.*Vs)-Vp-(x(5).*fy)-x(6).*fps;
g12=(Md/dv+0.5*Pa+0.5*Vu*cot(angle)-x(6)*fpu)/(Es*x(5)+Ep*x(6));
g13 = 1.2.*Mcr;
g14 = 0.004.*tw.*yb-x(5)-x(6
g15 = (Omp+Ompr-Omn-0.3);
%Minimum amount of reinforcement
g30=1.2*Mcr-0.9*Mn;
g31=1.33*Md-0.9*Mn;
g32= Mu/(dv*0.9)+(Vu/0.9-Vp)-0.5*min([Vu/0.9,(x(5)*fy*dv)/x(8)])-((x(5)*fy+x(6)*fpu));
g39=Vd-Vn
g40=Tburst-O*T
% fatigue stress limit
g22 = (ffs-161.5);
g23 = (ffp-125);
g24=Md/O*dv+0.5*Pa/O+cot(angle*sqrt((Vu/O-Vp)-0.5*Vs)^2+(0.45*Pc*Tu))-(x(6)*fps+x(5)*fy);
g25 = (0.20.*x(1)-x(7));
g26 = (x(7)- 0.75*x(1));
g27=Tu*Pc/(2*0.9*Ao*fy)-x(4);
g28=Vu/0.9-0.5*Vs-Vp-x(5)*fs-x(6)*fpu;
g29=x(8)-(av*fy)/(0.083*bv*sqrt(fc));
g16 =(c./de-0.42);
bv=tw-dduct;
% limits of traverse reinforcement
g17 = (x(7)-fy*av/(0.083*tw*sqrt(fc)))
g18= P - x(6).*fpt
g20 = fs- min([206,0.6.*fy])