Construction Stage Analysis of Cable-Stayed Bridges
by Marko Justus Grabow Thesis submitted to the Faculty of the Technical University of Hamburg Harburg in partial fulfilment of the requirements for the degree of Diplom - Ingenieur in Civil and Environmental Engineering
June 29th, 2004 Hamburg, Germany
Arbeitsbereich Baustatik und Stahlbau
Structures Research Group
Univ.-Prof. Dr.-Ing. Uwe Starossek
Dr. Park Chan Min
Picture on the front cover: Existing Jindo Bridge in the south of the Republic of Korea
Construction Stage Analysis of Cable-Stayed Bridges Marko Justus Grabow Abstract Different means and methods exist in the construction industry for the erection of bridges. In the planning and the execution of the complex construction operations, the effects of the chosen erection methods need to be taken into consideration to achieve a safe and economical process. For a defined span range, the cable-stayed structure is a bridge type which offers an aesthetic shape and also a cost-effective solution for crossing rivers and valleys. The Cantilevering Method is a widely used procedure for the construction of the superstructure. In the structural analysis of this process, changes in the geometry and boundary conditions as well as the material properties and other structural details must be considered. Temporary construction loads and boundary conditions act only during the construction, depending on the method and the sequence of the erection. However, these construction loads can produce considerable stresses in the unfinished structure. Due to its lack of resistance against failure, a detailed investigation prior to the construction is essential. Not only the influence of individual structural elements, such as the non-linear behaviour of the stay cable, but also the performance of the composed structure in the various stages must be taken into account. Furthermore, time dependent material properties such as creep and shrinkage play a major role, especially in the case of bridges where the main girder is fabricated of cast-insitu concrete segments or composite sections. The issues and considerations required to develop a save and economical construction sequence are expatiated in this thesis. An example of a construction stage analysis is provided in detailed for the Second Jindo Bridge. This bridge is a steel cable-stayed bridge with a main span of 344 metres, and is erected with the Cantilever Construction Method. The overall construction process is modelled and analysed.
Table of Contents
Table of Contents
Acknowledgement........................................................................................ V List of Figures ............................................................................................................................. VI List of Tables .............................................................................................................................XII List of Symbols and Units........................................................................................................ XVI
1
2
General task............................................................................................ 1 1.1
Introduction ................................................................................................................. 1
1.2
Overview ..................................................................................................................... 3
1.3
Thesis organisation ...................................................................................................... 4
Cable-Stayed Bridges ............................................................................. 6 2.1
History of cable-stayed bridges ................................................................................... 6
2.2
Stay-cables................................................................................................................. 13
2.3
Erection of cable-stayed bridges................................................................................ 15
2.3.1 2.3.2 2.3.3 2.3.4
3
Static arrangement of cable-stayed bridges .................................................... 15 Erection procedures ........................................................................................ 18 Construction of the pylon ............................................................................... 23 Erection of the main girder using the cantilever method ................................ 24
General description of a Construction Stage Analysis......................... 29 3.1
Designed Cable Forces .............................................................................................. 29
3.2
Construction Stage Analysis...................................................................................... 34
3.3
Construction Stage Analysis by MiDAS ................................................................... 37
3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.4 3.4.1 3.4.2
The analysis programme ................................................................................. 37 Structural Data ................................................................................................ 38 Unknown Load Factor .................................................................................... 39 Backward Analysis ......................................................................................... 41 Forward Analysis............................................................................................ 43 Forward Method = Backward Method............................................................ 45 Influence matrix......................................................................................................... 46 Calculation of influence matrices ................................................................... 46 Influence matrix calculated by MiDAS .......................................................... 47 I
Table of Contents 3.5 3.5.1 3.5.2 3.5.3
Time dependent effects ................................................................................... 49 Non-linearity effects ....................................................................................... 61 Temperature .................................................................................................... 74
3.6
Modelling and tuning of cables.................................................................................. 75
3.7
Construction control and monitoring ......................................................................... 79
3.7.1 3.7.2 3.7.3 3.7.4 3.7.5
4
General considerations and uncertainties................................................................... 48
Construction Control Systems......................................................................... 80 Adjustment instruction on site ........................................................................ 81 Methods of cable-stay adjustment................................................................... 82 Control of deck geometry................................................................................ 83 Computational Systems................................................................................... 84
Example of a Cable-Stayed Bridge including temporary supports......91 4.1
Model data ................................................................................................................. 91
4.2
Different restrictions for the Unknown Load Factor.................................................. 93
4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.5 4.5.1 4.5.2 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.8
Case I: Use of different connections girder-pylon, restricted displacement ... 93 Case II: Restricted moment distribution ......................................................... 95 Optimisation Method for ideal cable forces by influence matrix............................... 97 Adjustment of the girder elevation.................................................................. 98 Adjustment of cable forces............................................................................ 101 Summary of the adjustment calculation ........................................................ 102 Backward and forward analysis ............................................................................... 103 Backward analysis......................................................................................... 103 Forward analysis ........................................................................................... 107 Influence of the activation time of the Girder-Pylon connection.................. 109 Construction stage analysis considering creep and shrinkage ................................. 115 General conditions ........................................................................................ 115 Modelling creep and shrinkage ..................................................................... 116 Camber Control........................................................................................................ 120 General calculation method........................................................................... 120 Camber calculation for the Case I example .................................................. 125 Camber calculation for the Case II example ................................................. 129 Construction Errors.................................................................................................. 130 Error in cable force ....................................................................................... 131 Error in elevation of a segment ..................................................................... 135 Cable elements in construction stages ..................................................................... 142
II
Table of Contents
5
Model of the Second Jindo Bridge..................................................... 150 5.1
Location of the bridge.............................................................................................. 150
5.2
Structure................................................................................................................... 152
5.3
Erection Options ...................................................................................................... 152
5.3.1 5.3.2 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.5 5.5.1 5.5.2 5.5.3 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5
Modelling of the bridge with MiDAS...................................................................... 154 Nodes155 Elements ....................................................................................................... 157 Material Properties........................................................................................ 158 Section Properties ......................................................................................... 159 Boundary Conditions .................................................................................... 161 Loading ......................................................................................................... 161 Initial Cable Forces.................................................................................................. 165 Optimised Structural Moments ..................................................................... 166 Limited Cable Forces.................................................................................... 167 Limited cable forces and restricted bending moment ................................... 171 Back- and Forward Analyses of the Second Jindo Bridge ...................................... 173 Construction Stages ...................................................................................... 174 Modified Construction Stage Analysis ......................................................... 180 Linear Backward and Forward Analyses...................................................... 182 Differences in the application of cable elements .......................................... 189 Cable elements in forward analysis .............................................................. 194
5.7
Comparison with Hyundai and RM Results ............................................................ 202
5.8
Minimum and maximum allowable stresses............................................................ 205
5.8.1 5.8.2 5.8.3
6
Erection of the towers and side spans ........................................................... 153 Erection of the main span ............................................................................. 153
Tension forces in the cable stays .................................................................. 205 Maximal stresses in the girder segments ...................................................... 208 Maximum stresses in the pylon..................................................................... 211
5.9
Fabrication Camber of the Second Jindo Bridge ..................................................... 212
5.10
Unstressed cable length L0....................................................................................... 215
5.11
Final comments on the construction stage analysis ................................................. 219
Conclusion.......................................................................................... 221 6.1
Summary.................................................................................................................. 221
6.2
Contribution............................................................................................................. 228
6.3
Recommendation ..................................................................................................... 228
6.4
Final comments on the analysis programme MiDAS .............................................. 229
III
Table of Contents
Literature ...................................................................................................231 References............................................................................................................................. 231 Bibliography ......................................................................................................................... 232 Internet ................................................................................................................................. 235
Appendix ...................................................................................................237 Appendix A: Creep calculation CEB-FIP 1990 .................................................................... 237 Appendix B: Extracts from the correspondence with MiDAS support................................. 238 Appendix C: Model of the Second Jindo Bridge .................................................................. 240 Appendix D: Allowable and existing stresses....................................................................... 243 Appendix E: Unstressed cable length L0 ............................................................................... 249 Appendix F: Cross sections and plans of the Second Jindo Bridge ...................................... 252
IV
List of Figures
Acknowledgement
I would like to express my gratitude to my teacher Prof. Starossek for his courses and his interest and support of his students. As a main part of this thesis was developed in relation to an internship at the Korea Highway Corporation, I thank him deeply for organizing the unique opportunity to visit Korea and for giving me the time to find the confidence in my decision. I also appreciate his providing advice and support throughout my work on this thesis. Furthermore, I would like to express my sincere thanks to Dr. Park Chan-Min for inviting me to Korea and for his guidance during my internship. I greatly appreciated that I could learn much from his rich experience in the construction field. My life has been enriched by his delightful storytelling. I truly enjoyed the various visits and the trips that we undertook. I would also like to thank Mr. Choi Gyeong Hag and his family for their far-reaching help and profound hospitality during my stay. The diverse activities gave me great pleasure and an insight into the Korean tradition. I can only hope that they could learn a little about the German culture as well. I would like to express my sincere appreciation to the whole structural research group and all the people that I have come in contact with during my stay at the Korea Highway Corporation. Thanks for the kindness and open-heartedness. I will keep them in good remembrance. Finally, and most importantly, I am most grateful for the continuous love and support that my family has always given me, during my work on this thesis and all the time prior.
V
List of Figures
List of Figures
Figure 2-1: The Albert Bridge across the Thames in London....................................................... 6 Figure 2-2: The Strömsund Bridge................................................................................................ 7 Figure 2-3: The Köhlbrand Bridge................................................................................................ 8 Figure 2-4: The Maracaibo Bridge................................................................................................ 9 Figure 2-5: The Alex Fraser Bridge during its construction ....................................................... 10 Figure 2-6: The Hitsuishijima Bridge ......................................................................................... 11 Figure 2-7: The Sutong Yangtze River Bridge............................................................................ 12 Figure 2-8: New PWS cable........................................................................................................ 14 Figure 2-9: Fan systems .............................................................................................................. 16 Figure 2-10: Multi-cable harp systems with intermediate supports in the side span................... 17 Figure 2-11: Erection on temporary supports.............................................................................. 19 Figure 2-12: Erection by free cantilever method......................................................................... 21 Figure 3-1: Illustration of the pendulum rule .............................................................................. 30 Figure 3-2: Unit Load Case Method for determining the ideal state........................................... 33 Figure 3-3: Unit Load Case Method for construction stage analysis .......................................... 36 Figure 3-4: Cable-stayed example............................................................................................... 38 Figure 3-5: Flowchart for cable initial prestress calculation ....................................................... 39 Figure 3-6: Moment self-weight & unit pretension load [tonf]................................................... 40 Figure 3-7: Results of the Unknown Load Factor calculation .................................................... 40 Figure 3-8: Moment self -weight & initial pretension load [tonf]............................................... 40 Figure 3-9: Sequence for backward analysis............................................................................... 42 Figure 3-10: Applied cable forces [tonf] ..................................................................................... 44 Figure 3-11: Moment distribution; forward analysis before adding the support [tonfm]............ 44 Figure 3-12: Moment distribution in the last step of the forward analysis [tonfm] .................... 44 VI
List of Figures Figure 3-13: Discontinuity between two segments ..................................................................... 45 Figure 3-14: Influence Matrix of displacement [mm]................................................................. 47 Figure 3-15: Time dependent concrete deformation ................................................................... 50 Figure 3-16: Creep isochrones .................................................................................................... 55 Figure 3-17: Definition of the Creep-Function J......................................................................... 55 Figure 3-18: Verification model for creep & shrinkage.............................................................. 59 Figure 3-19: a) Creep coefficient b) Shrinkage strain................................................................. 60 Figure 3-20 Force/deflection curve............................................................................................. 64 Figure 3-21 Horizontal and inclined stay cable .......................................................................... 64 Figure 3-22 Deformed and uniformed cable element ................................................................. 67 Figure 3-23 Newton-Raphson Method........................................................................................ 69 Figure 3-24 Verification example of non-linear analysis............................................................ 70 Figure 3-25: Uniform variation of temperature, a) without supports or with supports but no friction on bearings, b) with supports and fixed bearings ........................................................... 74 Figure 3-26: Deflection produced by: a) temperature variation in the deck, b) temperature variation only in the side span..................................................................................................... 74 Figure 3-27: Deflections produced by an increase of temperature a) in symmetrical cablestayed cantilevers b) with side span on supports......................................................................... 75 Figure 3-28: Stay adjustment definition...................................................................................... 77 Figure 3-29: Deflections produced by construction with final cable forces a) in case of symmetrical cable-stayed cantilever b) in case of bridge with intermediate supports ................ 78 Figure 3-30 Theoretical and actual deck profile ......................................................................... 84 Figure 3-31: Example of camber error........................................................................................ 89 Figure 4-1: Structural system ...................................................................................................... 91 Figure 4-2: Moment Distribution under Self Weight [tonfm], Case I......................................... 93 Figure 4-3: Idealised moment distribution after restricted deformation [tonfm], Case I ............ 94 Figure 4-4: Deformation dz after restricted deformation [mm], Case I ...................................... 95 Figure 4-5: Moment distribution after restricted deformation [tonfm], Case II.......................... 96 Figure 4-6: Ideal moment distribution after moment restriction [tonfm], Case II ...................... 96 Figure 4-7: Displacement after moment restriction [mm], Case II ............................................. 97 VII
List of Figures Figure 4-8: Moment distribution backward analysis [tonfm], Case I........................................ 103 Figure 4-9: Deformation dz backward analysis [mm], Case I................................................... 104 Figure 4-10: Before removing the cable and activating the support [mm]................................ 105 Figure 4-11: Addition of support to the deformed (a) and the original structure (b) [mm] ...... 106 Figure 4-12: Deformation when first part of the side span is erected [mm] ............................. 107 Figure 4-13: Moment distribution forward analysis [tonfm], Case I ........................................ 108 Figure 4-14: Deformation dz forward analysis [mm], Case I.................................................... 108 Figure 4-15: Different normal forces back- and forward analysis [tonf], Case II ..................... 108 Figure 4-16: Moment distribution due to considered gap in normal forces in the girder of forward and backward analysis [tonfm]; Case II ...................................................................... 109 Figure 4-17: Moment distribution forward analysis with changed girder-pylon connection, neglecting normal forces in the key segment [tonfm], Case II.................................................. 110 Figure 4-18: Changed backward analysis a) normal force, b) horizontal displacement, Case II ................................................................................................................................................ 111 Figure 4-19: Changed forward analysis a) horizontal displacement, b) normal force, Case II ................................................................................................................................................ 111 Figure 4-20: Moment distribution changed forward analysis, applying a horizontal displacement [tonfm], Case II ................................................................................................... 112 Figure 4-21: Horizontal displacement changed forward analysis, applying a horizontal displacement [mm], Case II....................................................................................................... 112 Figure 4-22: Horizontal displacement changed forward analysis, applying a horizontal displacement, Case II ................................................................................................................ 113 Figure 4-23: Bending moment in the girder [tonfm] a) 1 day after applying additional load, b) 10 days after applying additional load, c) after 5000 days.................................................... 118 Figure 4-24: Vertical displacement of the main girder [mm].................................................... 119 Figure 4-25: Camber and deformation ...................................................................................... 120 Figure 4-26: Cantilever ............................................................................................................. 121 Figure 4-27: a) Current displacement b) Construction camber ................................................. 121 Figure 4-28: Erection of a cantilever......................................................................................... 122 Figure 4-29: Erection of a cantilever, current displacement value............................................ 123 Figure 4-30: Fabrication camber, real displacement [mm] ....................................................... 124 Figure 4-31: Construction camber graph, negative net displacement [mm] (Case I model)..... 128 VIII
List of Figures Figure 4-32: Fabrication camber [mm] (Case I model) ............................................................ 128 Figure 4-33: Fabrication camber [mm] (Case II model) ........................................................... 129 Figure 4-34: Construction camber [mm] (Case II model)......................................................... 130 Figure 4-35: Vertical displacement considering cable tension error......................................... 131 Figure 4-36: Final moment distribution due to changed pre-stressing in cable 4 [tonfm] ........ 132 Figure 4-37: Final moment distribution after restressing of cable 1 to 5 [tonfm]..................... 135 Figure 4-38: Final moment distribution after elevation adjustment [tonfm]............................. 138 Figure 4-39: Elastic link in order to model an error in the girder elevation.............................. 139 Figure 4-40: Vertical displacement original system and system including error in girder elevation .................................................................................................................................... 140 Figure 4-41: Fabrication camber [mm] ..................................................................................... 141 Figure 4-42: Structural system of harp type cable stayed bridge (dimensions in [m]) ............. 142 Figure 4-43: Non-linear analysis of a single cable (cable 6 in the model Figure 4-42) [m] and [kN] .................................................................................................................................... 144 Figure 4-44: Deflected shape of the girder due to non-linear analysis and different initial tension [m] ................................................................................................................................ 145 Figure 4-45: Comparison of deflected shapes,.......................................................................... 146 Figure 4-46: Cable installation in the linear truss model [kN].................................................. 148 Figure 4-47: Cable installation in the Ernst truss model [kN] .................................................. 148 Figure 5-1 Location of Second Jindo Bridge ............................................................................ 151 Figure 5-2: Girder elevation in the side and main span [m]...................................................... 156 Figure 5-3: Working points at the pylon ................................................................................... 157 Figure 5-4: Working points at the girder................................................................................... 157 Figure 5-5: Cable-girder connection and tied down condition using elastic links .................... 157 Figure 5-6 Element numbers..................................................................................................... 158 Figure 5-7: Material numbers.................................................................................................... 160 Figure 5-8: Traffic load Korean Standard ................................................................................. 165 Figure 5-9: Moment distribution restricted displacement [tonfm]............................................ 166 Figure 5-10: Moment distribution restricted cable forces [tonfm]............................................ 168 IX
List of Figures Figure 5-11: Displacement dz restricted cable forces [mm]...................................................... 169 Figure 5-12: Moment distribution restricted cable forces & bending moments in the girder [tonfm]....................................................................................................................................... 172 Figure 5-13: Construction stages 1-39 ...................................................................................... 179 Figure 5-14: Bending moment before opening the bridge [tonfm] (Case A) ............................ 181 Figure 5-15: Moment forward and backward analysis Case A [tonfm] .................................... 184 Figure 5-16: Moment forward analysis, considering the tension forces due to the SelfWeight function Case A [tonfm] ............................................................................................... 186 Figure 5-17: CS 4-Installation of cable 6, considering an effective stiffness in forward and backward analysis [tonf] ........................................................................................................... 190 Figure 5-18: CS 16 installation of cable 10, considering an effective stiffness in forward and backward analysis [tonf]..................................................................................................... 190 Figure 5-19: Final moment distribution using cable elements, considering the effect of ......... 191 Figure 5-20: CS 16 installation of cable 10, cables stressed in 5 steps [tonf] ........................... 192 Figure 5-21: Pylon and side span before the installation of the first cable[tonfm] ................... 193 Figure 5-22: Moment distribution in the main girder using cable elements, considering and neglecting the effect of the Self-Weight function [tonfm] ......................................................... 194 Figure 5-23: Vertical displacement neglecting the effect of the Self-Weight function [mm], Case B ....................................................................................................................................... 197 Figure 5-24: Vertical displacement at the tip of the cantilever for each construction step [mm] .......................................................................................................................................... 198 Figure 5-25: Maximum and minimum moments from forward analysis using cable elements and Case B values given in Table 5-15 [tonfm]......................................................... 201 Figure 5-26: Final moment [tonfm], Hyundai initial tension, same loading and construction sequence .................................................................................................................................... 203 Figure 5-27: Final moment [tonfm], RM initial tension, changed self weight, same construction sequence................................................................................................................ 203 Figure 5-28: Vertical displacement dz due to changed initial cable forces [mm] ..................... 204 Figure 5-29: Moment envelope due to traffic load [tonfm] (no dead weight considered) ........ 209 Figure 5-30: Load distribution for the maximum bending moment in the centre of the main span............................................................................................................................................ 210 Figure 5-31: Maximum moment at the top of the pylon during the erection of cable 1 [tonfm]....................................................................................................................................... 211 Figure 5-32: General manufacture camber [mm]...................................................................... 212 X
List of Figures Figure 5-33: Cable .................................................................................................................... 216 Figure A-1: Cable element with the length ds........................................................................... 249
XI
List of Tables
List of Tables
Table 2-1: The 18 longest cable-stayed bridges .......................................................................... 12 Table 3-1: Input data ................................................................................................................... 39 Table 3-2 Flowchart for backward analysis ................................................................................ 41 Table 3-3: Cable forces [tomf] .................................................................................................... 43 Table 3-4: Input data verification example creep & shrinkage ................................................... 59 Table 3-5: Creep and shrinkage data verification example ......................................................... 60 Table 3-6: Result table static verification example for creep...................................................... 61 Table 3-7: Structural classification and calculation procedure ................................................... 62 Table 3-8: Verification table non-linear analysis ........................................................................ 70 Table 3-9: Compared permanent loads for different bridge types, the bridges are about 20 metre wide ................................................................................................................................... 77 Table 4-1: Modified input data.................................................................................................... 92 Table 4-2: Ideal cable forces for different elastic link types ....................................................... 94 Table 4-3: Construction stage analysis data of the backward calculation ................................. 104 Table 4-4: Calculation of detension force [tonf] ....................................................................... 105 Table 4-5: Initial cable forces obtained from backward analysis (Case I model) [tonf] ........... 106 Table 4-6: New construction stage data for backward analysis ................................................ 110 Table 4-7: Cable forces obtained from backward analysis (Case II model), changed construction sequence [tonf] ..................................................................................................... 110 Table 4-8: Horizontal displacement after installing the first cable (Case II b model) [mm]..... 114 Table 4-9: Horizontal displacement after applying the additional load (Case II b model) [mm] .......................................................................................................................................... 114 Table 4-10: Input data CEB-FIP code ....................................................................................... 116 Table 4-11: Construction time schedule.................................................................................... 117 Table 4-12: Real displacement table [mm] ............................................................................... 123 Table 4-13: Total net displacement and construction camber data [mm] ................................. 124 XII
List of Tables Table 4-14: Calculation table of the current displacement (Case I model)............................... 126 Table 4-15: Calculation table for real and net displacement (Case I model) ............................ 126 Table 4-16: Calculation table for construction camber (Case I model) .................................... 127 Table 4-17: Construction camber table (Case I model) ............................................................ 127 Table 4-18: Fabrication camber table (Case I model)............................................................... 128 Table 4-19: Cable forces due to changed pre-stressing in cable 4 ............................................ 131 Table 4-20: Cable forces due to elevation adjustment .............................................................. 138 Table 4-21: Required rotational stiffness obtained from MiDAS............................................. 140 Table 4-22: Fabrication camber data [mm]............................................................................... 141 Table 4-23: Property table for harp system............................................................................... 142 Table 4-24: Initial pretension according to the sag to span ratio [kN]...................................... 145 Table 4-25: Tension forces in cable 3 & 4 due to adapted stiffness ......................................... 149 Table 5-1: Main Geometric Data Second Jindo Bridge ............................................................ 152 Table 5-2: Material property table ............................................................................................ 159 Table 5-3: Cross section table ................................................................................................... 160 Table 5-4: Boundary table......................................................................................................... 161 Table 5-5: Segment load table................................................................................................... 163 Table 5-6: Calculated distributed load ...................................................................................... 163 Table 5-7: Unknown Load Factor restrictions .......................................................................... 166 Table 5-8: Theoretical ideal cable forces .................................................................................. 167 Table 5-9: Allowable tension forces in [tonf] ........................................................................... 167 Table 5-10: Additional Unknown Load Factor restrictions ...................................................... 168 Table 5-11: Summary table of ideal cable forces...................................................................... 169 Table 5-12: Unknown Load Factor restrictions including limited moments in the main girder ......................................................................................................................................... 171 Table 5-13: Summary table of ideal cable forces including moment restriction ...................... 173 Table 5-14: Sequence of cable erection .................................................................................... 180 Table 5-15: Initial cable forces from backward analysis .......................................................... 182 XIII
List of Tables Table 5-16: Difference in cable tensions between forward - and backward analysis ............... 183 Table 5-17: Changed initial cable forces considering the tension due to the self-weight of the cables ................................................................................................................................... 185 Table 5-18: Results of forward - and backward analysis .......................................................... 188 Table 5-19: Initial cable forces from backward analysis [tonf]................................................. 189 Table 5-20: Difference in cable tension forward - and backward analysis considering an effective stiffness [tonf]............................................................................................................. 191 Table 5-21: Comparison of cable forces obtained from different calculations ......................... 195 Table 5-22: Final cable forces truss and cable elements (forward analysis) ............................. 196 Table 5-23: Vertical displacement at the tip of the cantilever [mm]......................................... 197 Table 5-24: Cable forces back- and forward analysis using truss elements and Case B values given in Table 5-15 ........................................................................................................ 199 Table 5-25: Comparison of cable forces using truss and cable elements in forward analysis for Case B values given in Table 5-15 ...................................................................................... 200 Table 5-26: Comparison of cable forces obtained from different calculations ......................... 202 Table 5-27: Control maximum cable forces during construction.............................................. 206 Table 5-28: Calculation of angle and cable force due to concentrated load.............................. 207 Table 5-29: Control maximum cable due to live load ............................................................... 207 Table 5-30: Allowable stresses for SM400-steel....................................................................... 208 Table 5-31: Load cases to consider the maximum load cases for traffic load........................... 210 Table 5-32: Camber data [mm] ................................................................................................. 212 Table 5-33: Control calculation for construction camber data .................................................. 213 Table 5-34: Real horizontal displacement final state [mm] ...................................................... 213 Table 5-35: Longitudinal deformation of each segment [mm].................................................. 213 Table 5-36: Construction camber data ...................................................................................... 214 Table A-1: Node coordinates .................................................................................................... 240 Table A-2: Element table .......................................................................................................... 241 Table A-3: Elastic link table...................................................................................................... 242 Table A-4: Control of allowable stresses of the girder segments during construction.............. 244 Table A-5: Control of allowable stresses of the girder segments under live load condition..... 246 XIV
List of Tables Table A-6: Control of allowable stresses of the pylon due to construction loads..................... 247 Table A-7: Control of allowable stresses of the pylon under live load condition..................... 248
XV
List of Symbols
List of Symbols and Units
Scala, Vectors and Matrices c
chord length of a cable
S
cable force
E
modulus of elasticity
q
distributed load
Er
allowable error range
t
time
F
general force
u
displacement
l
horizontal projected length of a
w
weight per unit length
cable
ε
strain
L
cable length
σ
stress
M
bending moment
α
error factor or angle
N
normal force
A
adjustment vector
M
vector of shim thickness at each
E
adjustment error vector
I
ideal state vector
δ
deflection or displacement vector
S
cable force vector
Rf
field measurements of member
Z
superposed error mode vector
B
relation between strain and nodal
K
stiffness matrix
displacement
ρ
weighting matrix
R
sum of internal and external
D
influence displacement representing
cable
matrix or
elastic
the
forces
of
generalized forces
matrix T
relationship
influence matrix for tension forces in the cables
between the stress and strain F
and displacements
error influence matrix
XVI
List of Symbols Indices A
target value
Mi
MiDAS result
c
concrete
N
net
C
cable
P
permanent load
Co
construction
Py
pylon
cr
creep
R
real
eff
effective
sag
sag
el
elastic
sec
secant
er
error
sh
shrinkage
fac
factor
T
temperature
fi
final state
tan
tangential (or T)
G
girder
tot
total
L
large displacement
0
initial condition
∂()
partial differentiation
T
transformation of a matrix or vector
Mathematic operations d( )
simple differentiation
d( )/dt differentiation after time
Units
1 N/mm² kN/mm² kN/m² MN/m² tonf/mm² tonf/m²
N/mm² 1 103 10-3 1 9.807*103 9.807*10-3
kN/mm² 10-3 1 10-6 10-3 9.807 9.807*10-6
kN/m² 103 106 1 103 9.807*106 9.807
XVII
MN/m² 1 103 10-3 1 9.907*103 9.807*10-3
tonf/mm² 1.02*10-4 1.02*10-1 1.02*10-7 1.02*10-4 1 10-6
tonf/m² 1.02*102 1.02*105 1.02*10-1 1.02*102 10-6 1
XVIII
Chapter 1: General task
1
1 General task
This first chapter gives the necessary information on the topics of the study to enable the reader to place these into the right context. On the other hand, the main subjects relating to the investigations performed in this study are describes. An overview of the general considerations in the analyses of construction stages will be given in this part as well. A brief summary of each chapter of this thesis is then provided as a reference for the reader.
1.1 Introduction The construction of bridge superstructures is a highly complex process due to the interrelationships between the applied erection methods and the manifold internal and external effects concerning loads and material behaviour, and also to the environmental influences. When planning to build a bridge, engineers are required to come up with the most feasible way of erecting the structure in a safe and economic manner. Finding the optimum solution is based on comparing alternative techniques of erecting the bridge, along with the consideration of the different means and methods that can be employed and the implications on schedule and budget. An analysis of these methods always has to consider the bridge itself, as well as the characteristics of the site at which it is to be erected. This study deals with the constructability and the modelling of the construction stages of cablestayed brides erected with the cantilevering method. Cable-stayed bridges are structural systems which are effectively composed of cables, the main girder and towers. This bridge form has a fine-looking appearance and fits in with most surrounding environments. The structural systems can be varied by changing the tower shapes and the cable arrangements. Up to a span length of 1000 metres, the cable stayed system is considered as an economical solution. In addition to the static analysis of dead and live load, the dynamic analysis and that of wind loads, a detailed investigation of the construction sequence is essential. The interrelationship
Chapter 1: General task
2
between the growing, yet unfinished structure, and the various kinds of loads that affect the construction is a major issue in the actual field operation. The main objective of this study is to compile and review related topics that are of concern in the analysis and the modelling of the construction process. The focus of interest is on the Cantilever Construction Method and the accompanying issues. This thesis is supposed to serve as an understandable introduction to the broad topic of how to analyse, plan and deal with the complex construction process. While the Cantilevering Construction is with certainty the main erection method preferred in the construction of cable-stayed bridges, other methods exist as well and may be in some cases, depending on the characteristics of the actual bridge project, even more feasible. However, this study only mentions certain constructability aspects of other erection methods in brief. Two major sources of information are used in the first part of this thesis. Literature on the history of bridge construction is utilised to outline the development and the different types of construction methods. Following sections on the construction stage analysis and the related concerns are based on professional literature on the state-of-the-art of cable-stayed bridge engineering. In the second part of this thesis, the concept and the problems relating to a construction stage analysis are illustrated by simple structural systems. Besides the complex erection process, the difficulties which occur when modelling these step-by-step conditions are also explained. The case study of the Second Jindo Bridge, that is located at the south coast of South Korea, is provided as a real-life construction example in order to complement the theoretical part of this study. This concrete example helps to gain a better understanding of the construction stage analysis of cable-stayed bridges. The computer programme MiDAS is used to model and analyse the examples. In order to give other users a guideline on the application, the programme and its features are described. Conclusions are drawn in the final part of this study.
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1.2 Overview Due to the high degree of indeterminacy of cable-stayed structures, an extensive degree of understanding for both design and construction is required. In comparison to other types of conventional bridges, cable-stayed bridges demand sophisticated structural analyses and design techniques. With an optimized adjustment of the cable forces, it is possible to achieve an “ideal state”, at which the girder and the pylon are compressed with little bending only. The “ideal state” of a cable-stayed bridge is associated with the minimized total bending energy accumulated along the girder. This results in a possible design of slender decks. The materials for the deck and the pylons can be efficiently utilized. Moreover, in case of concrete decks, it has dominant influence on the creeping behaviour. At the time of construction, the deck segments are connected with cables so that each cable (or a pair of cables in the case of two cable planes) approximately takes the weight of one segment, with the length corresponding with the longitudinal distance between two cables. In the final state, the effect of other dead loads, such as pavement, curbs, fence, etc., as well as the traffic loads must be taken into account. There are different methods of determining the cable forces. Two simple ones can be assumed: •
a simple supported beam
•
a continuous supported beam
Furthermore, simple formulas which consider the self-weight of the cable and the stiffness of the girder and the pylon are developed. Analytic programmes often use an optimisation method. In this method, to minimize the material used in the girder and the pylon, bending moments and the deflection of the deck and the pylon are limited to prescribed tolerances with the purpose of determining the required tension forces in the stay cables. For the determination of the cable prestress forces that are induced at the time of the cable installation, the initial equilibrium state for dead load at the final stage must be determined first. Then, using backward and forward analyses, the construction stage analysis can be performed according to the construction sequence.
Chapter 1: General task
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During the construction of cable-stayed bridges, there are mainly two kinds of errors that frequently occur: •
Tension force error in the cables
•
Geometric error in controlling the elevation of the deck
Discrepancies of parameter values between design and reality, such as the modules of elasticity, the mass density of the concrete or the weight of the girder segments, are unavoidable, but possible irregularities may influence the structural performance. Accumulations of these errors must be avoided to ensure a safe design. Therefore, during the construction period, the structure must be continuously monitored so that the most suitable adjustment can be obtained whenever corrections become necessary. In general, there are two possible adjustment procedures: •
Adjustment of the cable forces
•
Adjustment of the girder elevation
The first case may change both, the internal forces and the configuration of the structure. The latter adjustment only changes the length of the cable and does not induce any change in the internal forces of the structure. In the service stage of concrete bridges, the cable force may need to be adjusted to recover an optimal structural state because of concrete creep effects. This short introduction demonstrated the complexity of the erection of cable-stayed bridges. In the following chapters, the mentioned topics will be described and discussed in detail.
1.3 Thesis organisation Chapter 1 of this thesis contains introductory information. It provides the reader with an overview of construction stage analyses and a brief description of each chapter. Chapter2 covers the historical background of cable-stayed bridge constructions, outlining the developments of this type of bridge in the last decades and gives the salient examples for each era. Different erection procedures are also outlined.
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Chapter 3 deals with the general description of construction stage analyses using the cantilevering method for the erection of cable-stayed bridges. By analysing a simple structural system, the procedure using the analysis programme MiDAS is illustrated. The general purpose of using influence matrices is presented. The special functions offered by MiDAS are described and the matrix is evaluated for the given example. Specific considerations and uncertainties, which should be taken into account in the construction process, are clarified to contribute to the reader’s overall understanding. The modelling approaches to cable-stays and the philosophy of tuning sequences during the erection and in the final state of the bridge are also described. Finally, the construction control and the monitoring systems are mentioned. Chapter 4 concerns itself with the construction stage analysis of a more complex example including temporary supports. The important issues and the considerations necessary for a reliable construction stage analysis are presented in more detail. The optimisation method is used to determine the cable forces to achieve an ideal state. Using back- and forward analyses, the initial cable forces are evaluated for the time of erecting the stay-cables. As creep and shrinkage are important factors to be included in the analysis, the method of considering these effects is illustrated. To ensure a successful erection process, the camber control is a main issue in the construction stage analysis. Moreover, the camber calculation is demonstrated in this chapter and the functions offered by MiDAS are introduced and controlled. Various construction errors are assumed to be incorporated in the already built structure. The errors are modelled and possible solutions are given to adjust the discrepancies. Finally, the influence of non-linearity due to cable elements is investigated. The accuracy of the cable elements is then proved. Chapter 5 encompasses the case study, the Second Jindo Bridge in the south of the Republic of Korea. Different erection methods are discussed. The generation of the model for the construction stage analysis is illustrated in detail, including the change of boundary conditions and variations in loading. The ideal cable forces are established and a construction stage analysis is performed. In order to rate the modelled system and the obtained initial cable forces, the results are compared with other calculations. The minimum and maximum stresses are proved to be in the allowable limits. Chapter 6 recapitulates the contributions made in this thesis and calls attention to further related areas of research that may be worth exploring.
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6
2 Cable-Stayed Bridges
In this chapter, a general overview of cable-stayed bridges and their different erection options are given. The development in the field of cable-stayed bridges is shown by means of a historical outline first. Then, after introducing the importance of static arrangements, the erection methods are explained with the focus on the cantilevering method.
2.1 History of cable-stayed bridges The principle of supporting a bridge deck with inclined tension members leading to the towers on either side of the span has been known for centuries. Already in 1823, the French engineer Navier published the results of a study on bridges with the deck stiffened by wrought iron chains taking both, a fan shaped and a harp shaped system, into consideration. However, due to the imperfections during the fabrication and the erection in early stayed bridges, it was very difficult to arrive at an even distribution of the loads between all stays. Furthermore, without the reliable tensile strength of steel wires, cable-stayed bridges did not become an interesting option, whereas systems in which the suspension system was combined with the stayed system, was used in major bridges in the second half of the 19th century. The Albert Bridge from 1873 across the Thames in London or the Brooklyn Bridge designed by Roebling are examples of this period.
Figure 2-1: The Albert Bridge across the Thames in London [70]
The first modern cables-stayed bridge was the Strömsund Bridge in Sweden, designed by Dischinger. The bridge is of a three span range and has a main span of 182.6 m with two side
Chapter 2: Cable-Stayed Bridges
7
spans of 74.7 m. The stays are arranged according to a pure fan system with two pairs of stays radiating from each pylon top. The steel pylons are of the portal type, supporting the two vertical cable systems arranged on either side of the bridge deck.
Figure 2-2: The Strömsund Bridge [8]
In the following years, numerous innovative cable-stayed bridges were constructed in Germany. The Theodor Heuss Bridge across the Rhine was opened to traffic in 1957. With a main span of 260 m, the bridge introduced the harp-shaped cable system with parallel stays and a freestanding pylon. The Severins Bridge, erected in 1959, was the first application of an A-Shaped pylon combined with transversally inclined cable planes. It was also the first to be constructed as an asymmetrical two span bridge with a single pylon positioned at one side of the river banks. The first cable-stayed bridge with a central cable plane, with the pylon and the stay cables positioned in the centre of the motorway, was the Norderelbe Bridge in Hamburg. In the following years, this system became the preferred solution for the majority of cable-stayed bridges constructed in Germany, e.g. the Leverkusen Bridge and the Maxau Bridge across the Rhine. These bridges have the same centrally arranged cable plane but the cable system is of a harp configuration. The development of cable-stayed bridges also required improvements in the techniques of structural analysis, allowing the calculation of cable forces throughout the erection period. The efficient use of all cables in the final state, as well as a favourable distribution of dead load moments had to be ensured. The first cable-stayed bridges only had a limited number of cables, which were generally composed of several prefabricated strands. The first multi-cable bridges were designed by Homberg. The Friedrich Ebert Bridge contains a central cable plane with two pylons, each supporting 2x20 stays.
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In multi-cable systems, the girder is supported more continuously. The cable forces, that are to be transmitted at each anchor point, are reduced so that a local strengthening of the girder is not necessary. It also has important advantages during the erection. Shorter deck cantilevers are required to reach from one anchor point to the next. This leads to simpler construction processes and, as it should be realized later, to slender decks. In 1972, the first parallel-wire strands were used in the Mannheim-Ludwigshafen Bridge across the Rhine. Additionally, the bridge introduced a new design concept. In the main span, the deck girder is entirely made out of steel, while the side span is made out of concrete. With a maximum free side span of 65 m and a main span of 287 m, the higher dead load of the side span reduces the requirements for a vertical anchoring of the girder. The Köhlbrand Bridge (1974) in the port of Hamburg was the first application of the multi-cable system with double cable planes supported by A-shaped pylons. With a modified fan-system during the construction, no temporary supports or temporary stays were required.
Figure 2-3: The Köhlbrand Bridge [68]
The first twenty years in the evolution of cable-stayed bridges took place, to a large extent, in Germany. Under the large influence of German developments, cable-stayed bridges become more popular in other countries, too. In the UK, the Wye Bridge was completed in 1965. This bridge is quite unique by having only one set of stays leading from the pylons to the deck. Based on a similar design, the Erskine Bridge in Scotland was constructed in 1971 with a main span of 305 m. Because this bridge also has only one stay leading from each of the two pylons to the deck despite its length, the girder has to span more than 100 m without a support from the cable system. During the erection, it was necessary to use temporary stays to reduce the moment in the deck girder when cantilevering in the main span. In France, the Saint Nazaire Bridge (1975) across the Loire River was the first cable-stayed bridge to span more than 400 m.
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The beginning of modern cable-stayed bridges was to a large extent dominated by steel bridges with orthotropic decks together with plate or box girders and cellular pylons. In the Maracaibo Bridge (1962) in Venezuela, which was designed by Morandi, the pylon and the deck girder are made of concrete. However, because of the unusual design and abnormal proportions, this stays an exception and hardly a typical example of the bridge type described above. Nevertheless, it was the first multi-span cable-stayed bridge.
Figure 2-4: The Maracaibo Bridge [70]
Another early example for the use of multi-cable systems in a concrete cable-stayed bridge is the Pasco-Kennewick Bridge. The deck is supported by a double cable system in the fan configuration. The stays, which are made of a single parallel-wire strand, are inside a grouted polyethylene tube. The deck girder was erected by the segmental method using heavy prefabricated elements. After ship collisions with pylons, original bridges were replaced by cable-stayed bridges to increase the open width. This was a further proof of the superiority of cables-stayed systems. To replace the original arch bridge, the Tjörn Bridge was built with a span of 366 m, 86 m longer than the original one, which allowed both pylons to be located on land. The Tjörn Bridge belongs to the group of cable-stayed bridges with different structural materials in the side and the main spans. The side spans are designed as continuous concrete girders with intermediate supports at each cable anchor point, whereas the main span is made of a steel box with an orthotropic steel deck. Also after a ship collision accident, the new Sunshine Skyway Bridge, a single bridge having a 360 m long cable-stayed main span, was decided to replace the existing two parallel bridges. At its completion in 1986, the Sunshine Skyway Bridge was the longest cable-stayed bridge in the USA. Prior to its construction, two designs were considered, one based on a composite deck and two cable planes, and the other on a concrete box girder and a single central cable plane. In this
Chapter 2: Cable-Stayed Bridges
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case, the second option was chosen. However, the alternative of the composite girder was subsequently applied in the Alex Fraser Bridge at Vancouver, which became the longest cablestayed bridge in the time between 1986 and 1991.
Figure 2-5: The Alex Fraser Bridge during its construction [67]
The advantages of composite girders were used during the construction of the Alex Fraser Bridge. The cantilevering from one cable anchor point to the next was easily achieved by the relatively light steel girder. The stay cables had been added before the heavy concrete deck was erected by precast slabs. The concrete slab could be efficiently utilized to transfer the axial compression through the deck, which is induced into the girder by the horizontal components of the stay cable forces. In the following years, after the completion of the Alex Fraser Bridge, the system of composite girders was generally preferred for the majority of cable-stayed bridges in North America. Major developments of cable-stayed bridges can also be found in the Far East. In 1977, the first double deck cable-stayed bridge – the Rokko Bridge- was completed in Japan. In a much larger scale, the double deck concept was later used for the twin cable-stayed bridges, the Hitsuishijima and Iwagurojima Bridge. Each of the two neighbouring bridges has a span of 185 m – 420 m – 185 m. The traffic runs on a two level truss with a four-lane expressway on the upper deck and a double track railway on the lower deck. The cable systems are of the modified fan configuration with two vertical cable planes positioned directly above the deck trusses. As extensively used in Japan, parallel-wire strands are applied for the stays.
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11
Figure 2-6: The Hitsuishijima Bridge [69]
At present, the Tatara Bridge (1999, Japan) is the world’s longest cable-stayed bridge, measuring 1480 m in total length. With an 890 m centre span, it is 34 m longer than the one in the Normandy Bridge in France. The main tower of this bridge is 220 m high and designed in an inverted Y shape. It has a cross section with corners cut for a higher wind resistance stability. The bridge is very flexible due to not only its length but also to the low girder-depth. The girder-depth to span-length ratio is about 1/300. One side span is 270 m, while the other side span is 320 m. To prevent this large difference in the lengths of the centre and the side spans from causing dead load unbalance problems, PC girders are installed at each end of both side span sections. These function as counterweight girders to resist negative reactions. Steel girders are used in the remaining part of the side span and in the main span. The girder is designed as a slender box girder and contains three cells, each 2.7 m high. The box girders are attached to fairings in order to ensure wind stability. The cables are installed in two-plane multi-fan systems with a maximum cable length of about 460 m. The cables of the bridge have dimpled surfaces, similar to that of a golf ball, to resist vibration caused by both wind and rain. However, the evolution of cable-stayed bridges continues and in the near future, they will exceed the magical 1000 m length. At 1596 m in length, the Stonecutters Bridge is a part of Hong Kong's plan to develop its infrastructure. The main span will be 1018 m and the side span 2x289 m. The pylons are designed with a height of 289 m. The Sutong Bridge over the Yangtze River will span 1,088 metres making it 70 metres longer than the Stonecutter Bridge. The full length of this bridge is 7600 meters. The height of the central span of 62 meters will enable fourth and fifth generation container ships to pass through in virtually any weather. The bridge is designed on six-lane expressway standards with a maximum vehicle speed of 100 km. The construction has already commenced and is expected to be continued until 2008.
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Figure 2-7: The Sutong Yangtze River Bridge [66]
Table 2-1 shows the 18 longest cable-stayed spans. It is remarkable that twelve of the longest cable-stayed bridges already built are located in the Far East. No.
Name
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Sutong Bridge Stonecutters Bridge Tatara Bridge Normandie Bridge Nanjing Bridge Wuhan Baishazhou Qingzhou Minjiang Br. Yangpu Bridge Meiko Chuo Bridge Xupu Bridge Rion-Antirion Bridge Skarnsund Bridge Queshi Bridge Tsurumi Tsubasa Bridge Jingzhou Bridge Øresund Bridge Ikuchi Bridge Higashi-Kobe
Span [m] 1088 1018 890 856 628 618 605 602 590 590 560 530 518 510 500 490 490 485
Girder material main span Steel Steel Steel Steel/Conc Steel Steel Composite Composite Steel Composite Composite Concrete Composite Steel Steel Steel -
Traffic
Country
Year
Road Road Road Road Road Road Road Road Road Road Road Road Road Road Road Road & rail Road Road
China China Japan France China China China China Japan China Greece Norway China Japan China Denmark/Sweden Japan Japan
ca. 2008 ca. 2008 1999 1995 2001 2000 1998 1993 1997 1996 2004 1991 1999 1994 2002 2000 1991 1992
Table 2-1: The 18 longest cable-stayed bridges
In the last twenty years, cable-stayed bridges have developed to become dominating in bridge constructions with the span range from 200 m to 500 m. Under specific conditions, the cablestayed bridges may even be a competition to suspension bridges up to spans more than 1000 m. Table 2-1 also shows that the girder in the main span is dominantly fabricated by steel and, up to a span of 600 m, also by composite sections.
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2.2 Stay-cables The cable-stay is a basic element in all cable-stayed bridges and therefore a short introduction to the different existing cable types will be given. More details on the cable-stay technology with an emphasis on the corrosion protection are given by Ito [31]. Gimsing also describes the basic types and mechanical properties of structural cables in his text book [8], and the recommendations made by Setra [13] provide a fine guideline to all topics relating to cablestays, too. The cables require excellent mechanical properties, such as a high tensile strength, a high elastic modulus, a sectional compactness and also ease of handling during the installation. Furthermore, it is important that the cables have a high corrosion resistance and a satisfactory fatigue strength. The first cable-stayed bridges in Germany employed locked-coil cables. The locked coil rope (LCR) is composed of two types of twisted wires, normally of round wires in the core layers and of T- and Z-shaped wires in the outer layers. The LCR has a smooth surface and compact cross sections. Compared to other cable types, these are stiffer to handle. These cables are hardly used nowadays, mainly due to the rather complicated anchorage details and the difficulty of their replacement. From 1970 to 1985, most cable-stayed bridges applied parallel-wire cables. Some bridges were also built with parallel bars. Then, the Brotonne Bridge in France and later the Sunshine Skyway Bridge in the United States, were the first to use cable-stays made of 7-wire prestressing strands. Bar stay cables consist of round steel bars with a diameter of 26-36 mm and are covered by a steel pipe. To provide protection against corrosion, the inside is filled with cement grout. Since the lengths of the bars cannot be too long, coupling is normally necessary. However, this type of stay member has been scarcely used, particularly not for long cable-stayed bridges. Parallel wire stay cables (PWS) are composed of a bundle of pre-stressing wires with a diameter of 6-7 mm in a polyethylene or stainless steel pipe filled with cement grout as a corrosion protection. The parallel wires are kept in place by twisting a steel rope around the bundle. A bundle of wires forms a hexagonal cross section, and in some cases, numerous PWSs are formed into one large round cable on site. These parallel wire cables were widely used on both prestressed concrete (PC) and steel cable-stayed bridges. Later on, it became more common to utilize galvanized wires and wax as fillings in the pipes, as this is a non-cracking and ductile
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material. The shop-prefabricated parallel wire strands could be extensively found in Japanese bridges. In the 1980s, these cables were improved to the New PWS system. In the New PWS cable, the wire bundle is slightly twisted up to 3-4° so that the wire bundle is enabled to reel easily and the strands made self-compacting under axial tension without spoiling the mechanical properties. It is also characteristic of the New PWS cables to have the protecting polyethylene cover extruded directly onto the wire bundle so that no void volume exists between the wires and the surrounding pipe (Figure 2-8). Through the elimination of the spiral rope and the voids for cement grouting, the New PWS cables become more compact than traditional PWS cables.
Figure 2-8: New PWS cable [13]
New PWS cables are fabricated in sizes ranging from 7 No. 7 mm to 421 No. 7 mm wires. The longest stay cable of this type is 460 m long with the outer diameter of 165 mm and is used on the Tatara Bridge. In the 1980s, the technology of the parallel-strand cables (PSC) evolved towards higher protection of cables and improved fatigue performance of the anchorages by using wedges. However, a method was developed in the 1990s to protect the parallel strands individually by an extruded high density polyethylene sheath. Parallel-strand and parallel wire cables are similarly composed, with the sole exception of the individual 7 mm wires substituted by seven-wire strands. A parallel-strand cable can be fabricated as a complete unit, such as the parallel-wire cables. Yet, it has become more common to insert and stress the seven-wire strands one by one; a procedure called Isotension Method. Each strand is tensioned with a mono-strand jack and the application of appropriate devices finally ensures that all installed strands have the same tension. This reduces the size of the stressing equipment but increases somewhat the amount of work to be carried out on site.
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2.3 Erection of cable-stayed bridges In order to compare cable-stayed bridges with other bridge types, these must be categorised into medium and long span bridges. For the medium span length, cable-stayed bridges have to compete with conventional structures and they were often built only for aesthetic reasons when technical conditions as the span length did not require them. A serious analysis of construction methods has improved their economical efficiency. In the medium-span range cable-stayed bridges can be built on temporary supports or constructed by rotation. There is also a possible to install the deck by the incremental launching method. However, cable-stayed bridges have experienced great success during the last twenty years in the field of long span bridges. The cantilever method is an economical and practical solution for the construction of long cable-stayed bridges. In this method each new segment is built or installed and subsequently supported by a new cable (or a pair of cables) which balances its weight. In addition to the construction of the pylon, the different construction methods for the main structural parts, i.e. the cables and the girder, are mentioned in the following chapters to give a complete overview of the different possible construction procedures. The main focus of this thesis is the cantilever method which is therefore dealt with especial attention.
2.3.1 Static arrangement of cable-stayed bridges For cable-stayed bridges the choice of the structural system is an important factor in the design process. The common systems in cables-stayed bridges are the fan and the harp systems. The fan system is mostly applied in the form of a modified fan system in which the cable anchorage points are spread over a certain height at the pylon top. While studying the rigidity offered by the cable system itself and by assuming that the girder and the pylon only provide axial resistance, Gimsing defines the fan-shaped system a system which is stable of the first order [8]. This means that an equilibrium can be achieved without assuming any displacement. All stay cables must be fixed to the pylon top so that the horizontal components of the forces in any of the cables can be transferred to the anchor cable. Thus, the stability of the fan-shaped system depends to a large extent on the anchor cable. It is also required for this cable to be in tension for any loading case.
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In a harp-shaped cable system an equilibrium condition cannot be achieved by the cable system itself. But this does not imply that the total structural system of the bridge is unstable. The bending stiffness of the stiffening girder and the pylons add the stability that lacks in the cable system. A complete stabilization of the harp system can be obtained when intermediate supports are placed in the side spans under the cable anchor points. This can be seen in several large bridges with a harp-shaped system, such as the Oberkasseler Bridge in Düsseldorf. Even small changes in the supporting condition or the addition of a few structural elements often change the structure to a significant extent. Although the cable system forms only one part in the total structural system of the bridge, it highly influences the overall stiffness. In bridges with one main span several different solutions exist for arranging a fan-shaped cable system, e.g. a single pylon with only one fan system in the main span and a single anchor cable in the side span, or two pylons with fan system on both sides. The adopted solution often depends on the local conditions at the bridge site. In early cable-stayed bridges only few concentrated stay cables were used, as it is the case in the Strömsund Bridge (Figure 2-2). In bridges with few stay cables the fans could often be arranged symmetrically about the pylon as the large bending stiffness of the girder prevented a too large unloading of the anchor cables under traffic load in the side span (a traffic load only in the side span decreases the tension in the anchor cable). In modern cable-stayed bridges with a multi-cable system and a slender stiffening girder, a symmetrical fan is not possible with the top cable in the side span forming the anchor cable. The required minimum tension in the anchor cable cannot be achieved for all loading cases. Consequently, skew fans, - in which the side span is shorter than the main span - , are generally in use. In such a system the anchor cables must have a considerably lager crosssection than the normal cables do. As mentioned before, in modern cable-stayed bridges with more cables, there is no need for cross sections with large inertia. Walther [14] demonstrates in his work that the longitudinal bending moments increase with the larger bending stiffness without reducing the stresses in the pylon or in the stay cables. In these cases the stiffness of the deck has only a minor contribution to the overall structural stiffness. Thus, there is no need for a high bending stiffness of the girder.
a) Fan system with shorter side span than main span
b) Modified fan system with an anchorage cable composed of several stay cables
Figure 2-9: Fan systems [30]
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The side-to-main span ratio has a very strong influence on the stress ratio of the anchor cable and the deformation of the system. Gimsing [8] shows the effects of different ratios in a parametric study and suggests short side spans to increase the structural stiffness using ratio values from 0.25 to 0.5. The anchor cable is often composed of a number of individual strands, as in the Köhlbrand Bridge where the anchor cable is formed by three individual cables. In this case the chosen side-to-main span ratio is 0.3. A solution to the application of a symmetrical fan structure in a multi cable-system is the displacement of the end pier from the side span towards the pylon. By this arrangement, it is possible to reduce the side span length to less than half of the main span length. In this case, the side span cables near the support are activated to form the anchor cable. It should be noticed that this can induce a local bending in the stiffening girder near the intermediate pier, which does not appear in systems with a concentrated anchor cable connected to the girder above the end pier. As for the fan system, the first cable-stayed bridges built with a harp system only had few symmetrically arranged stays and a very stiffening girder. Here, the pylon was usually slender. As mentioned before, with intermediate supports in the side span, the global stability of the structural system can be achieved without a bending stiffness of the girder or the pylon. Thus, the girder and the pylon can be designed to be more slender. But in case of few concentrated stays, the local bending in the stiffening girder must be considered. In modern multi-cable harp bridges the local bending of the girder can be reduced and a very slender girder can therefore be applied if the global stiffness is achieved without the bending stiffness of the girder. A heavy pylon with a considerable stiffness may be advantageous in this case. The harp system may be chosen for aesthetic reasons, but in general, they are less economical.
Figure 2-10: Multi-cable harp systems with intermediate supports in the side span
In contrast to harp systems with all cable stays parallel, the inner stay cables in a modified harp system do not have the same angle at the girder and the pylon. In the Rhine Bridge at Flehe, a bridge with intermediate supports in the side span, the stay cables are arranged in a true harp shape with parallel cables in the sides, whereas the main span cables form a modified harp.
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The construction method is also considerably influenced by the longitudinal configuration of the bridge. To simplify matters, only the bridge type with one main cable-stayed span suspended from two pylons is considered in the following description of construction methods. Bridges with long side spans are divided into two types: a) Typical three span bridges which can be extended on both sides by a series of non-cablestayed access spans b) Bridges where the main span is balanced by access spans on the intermediate supports. As mentioned previously, the existence of intermediate supports has a favourable influence on the structural behaviour, which can be summarized as below: •
The anchor cables are distributed over the side spans and not concentrated on the abutments
•
The deformation and deflection of the pylons are reduced when the main span is loaded.
•
The bending moment in the pylons is reduced when the main span is loaded
•
Bending moments and deflections are small in case of loaded side spans
The existence of intermediate supports has a dominant influence on the chosen erection method. For example, intermediate piers allow the use of the incremental lunching method whenever this is reasonable. When the main span is built by the cantilever method and connected with the side span, the temporary stability of the cantilever increases during the construction.
2.3.2 Erection procedures The erection method which is applied in the construction of a cable-stayed bridge clearly depends on the size of the structure, the structural system and the conditions found at the intended location. In general, there are four different construction methods possible: •
Construction on temporary supports
•
Construction by rotation
•
Construction by incremental launching
•
Construction by the cantilever method
The different solutions are explained comprehensively in the next chapter.
Chapter 2: Cable-Stayed Bridges
2.3.2.1
19
Construction on temporary supports
A straightforward solution is to erect the entire girder on temporary supports before adding the cables. In the four stages illustrated in Figure 2-11, the following main operations are performed:
Figure 2-11: Erection on temporary supports [8]
Stage 1:
Erection of the stiffening girder on permanent and temporary supports. Any of the procedures used for the construction of girder bridges can be applied in this stage.
Stage 2:
Erection of the pylons from the deck of the completed girder.
Stage 3:
Installation of the stay cables. In this stage the cables only need to be tensioned moderately as the final tensioning takes place in the following stage.
Stage 4:
After the installation of all cables the temporary supports can be removed. During this process, the load is transferred to the cable system. Since the girder deflects downwards, it is necessary to erect the girder in an elevated position to reach the desired final position.
This erection procedure offers the advantage that the girder can be erected continuously from one end to the other. The procedure leads to an efficient control of the geometry and the cable tension. The disadvantage is related to the temporary supports that are applied. In many cases, the erection of temporary supports, - often over a large water depth in the main span - , is not economical so that the procedure itself is not feasible.
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2.3.2.2
Construction by rotation
If the construction of temporary supports, for example in a river, is undesirable from the financial viewpoint or for any other reason, it is possible to build one or two cable-stayed cantilevers on the shore parallel to the bank and to rotate it around its pylon. In this case, the conditions are similar to the erection on temporary supports as described in the chapter before, with a final closure after the rotation. The cables are tensioned in a single operation which disengages the cantilever from the temporary supports. This technique was applied for several bridges, such as the Illhof pedestrian bridge and the Ben Ahin cable-stayed bridge. The design of the foundation and the pylon depends enormously on the erection technique. Due to the rotation process, excessive outcentered forces producing high bending moments in the footing must be avoided. Placidi proposed a construction by rotation for the Gilly bridge, where the deck rotates around one of the two pylon legs and the other pylon leg turns on a temporary concrete support.
2.3.2.3
Construction by incremental launching
The incremental launching technique has been used on many bridges since its introduction in 1961 by Leonardt. In this method, the superstructure or a part of it is cast-in-situ at a stationary location behind one of the abutments. The completed or partially completed structure is then jacked horizontally into place. Subsequent segments can then be cast onto the already completed section and, in turn, pushed onto the piers. The procedure has the advantage that it, like the balanced cantilever technique, does not require falsework to cast the girder. Moreover, heavy erection equipment is not necessary. The only required special equipment is a light steel truss, which works for a launching nose to reduce the cantilever moment during the launching. The launching method can also be applied for steel or composite decks as these are often used in the main span of a cable-stayed bridge. The method is applied for the construction of small and medium sized cable-stayed bridges. It is favourable for intermediate piers in the side span, but requires temporary supports in the main span. After the structure is built, all stays can be tensioned in a single operation. Launching the structure through an already built pylon can become a problem when the deck is suspended with cables anchored laterally to the deck, because the anchor blocks on both sides widen the deck. If the pylon is made of two vertical columns outside the width, generated by the anchor blocks, the cables will induce transversal forces in the columns. In this case, a bracing between the columns is necessary. There is also a
Chapter 2: Cable-Stayed Bridges
21
possibility to build the higher part of the pylon after the launching. Since this is not a practical concept for the construction, Kretz [54] proposed to launch the steel structure with the already built steel pylons, which are rigidly connected to the deck, for the erection of the Chateauneuf Bridge. The matter is simpler with pylons which have the shape of an inverted V or Y, as these can be designed to leave out the necessary place for the deck. There are alternative erection techniques for the main span, such as the cantilever method, which is described in the next chapter.
2.3.2.4
Construction by cantilever method
In a classical three-span bridge temporary supports can be completely avoided if the superstructure is erected by the free-cantilever method. This method is illustrated in Figure 2-12. The procedure comprises the following stages:
Figure 2-12: Erection by free cantilever method [8]
Stage 1:
The pylon and the girder units above the main piers are erected.
Stage 2:
A balanced free cantilever is initiated by using derrick cranes which operate on the deck to lift up the girder segments. These are transported to the site on barges.
Stage 3:
As the cantilevers grow, the stay cables are installed and tensioned to their initial forces to carry the weight of the newly erected segment.
Stage 4:
The bridge is closed at midspan and the additional loading is applied.
22
Chapter 2: Cable-Stayed Bridges
The cable anchor points should be chosen so that a free cantilevering of the stiffening girder from one cable anchor point to the next is allowed without a temporary support being required. Temporary stability can become a problem during the erection. The structure must be able to withstand unsymmetrical construction forces. These loads occur already in the simple case of installing a new segment that extends one of the two cantilever arms. Dynamic forces may be produced by accidental situations, such as the fall of a mobile carriage. Wind effects can also become very important due to the static temporary configuration. Furthermore, in some cases, seismic effects are considered for the construction period. The nature of the problems varies with the conditions of the connection between the deck and the pylon. If there is a ridged connection between the deck and the pylon, construction problems can be limited. But because of thermal effects, only few cable-stayed bridges are built with a fixed connection in the final state. It is more frequent that the deck is simply supported by the pylon or not supported at all. In this case, it is necessary to install a temporary connection between the girder and a support. This can be the pylon, or if the deck is very low over the ground, it can be supported by a temporary structure independent from the pylon. Producing an elastic connection by using, for example, neoprene bearings, the deck can be supported by the pier to transverse bending moments and still allow longitudinal deformations of the deck. The temporary stability becomes a greater problem with the increase of the span length. In a number of erections, the procedures in Figure 2-11 and Figure 2-12 were combined so that the side spans were erected on temporary supports and the main span by free cantilevering. By this erection procedure, the application of temporary supports is limited to the side spans, where water depths or other reasons seldom exclude such supports. To take advantage of the intermediate supports in the side spans for the erection of the bridge, the side spans are built in the first step and after that, the long cantilever in the main span, which is then stabilised by the already existing side spans. In general, three methods are adopted for constructing the main girder in the side spans: the construction of balanced cantilevers on each of the intermediate supports, the construction by incremental launching or the construction of the access spans on scaffoldings. The principals of one-side cantilevering into the main span will be especially advantageous if the side spans can be erected without requiring temporary supports in the side spans. For the Normandie Bridge [53] over the river Seine, the concrete access spans were built on multiple intermediate supports by incremental launching up to the last pier. Balanced cantilevers were then built in concrete from each of the two pylons to close the side spans and to create long concrete cantilevers in the main bay. The remaining part of the central span was then built
Chapter 2: Cable-Stayed Bridges
23
by the cantilever method with a steel box girder. The free cantilever length was 428 m (half main span), which corresponds to a cantilever-to-width ratio of approximately 20. In order to avoid uplift reactions on the intermediate supports, a difference in weight between the access spans and the main span was created by the use of prestressed concrete on one side and steel on the other side. All the erection stages were investigated both analytically and experimentally by full model tests of the erection phase, in which the cantilever had reached its midspan. The erection equipment, such as the purposed crane, was carefully modelled as well. The investigations revealed that the girder could oscillate laterally. A large tuned-mass damper was therefore installed on the bridge deck to suppress such an oscillation.
2.3.3 Construction of the pylon The material for the pylon can be of steel or concrete. Depending on the material and the height of the structure, there are different erection options. For small steel pylons, as they can be found in cable-stayed bridges of moderate size, the erection can be carried out using mobile cranes or floating cranes. For larger pylons, too, can land or see based cranes be used; at least for the erection of the lower part. Due to the evolution in the size of floating cranes since the 1970s, it has become possible to erect the entire pylon in one or two pieces. For very high pylons it is necessary to erect the upper part by a climbing crane, which follows the pylon as it grows. The crane applied to hoist the prefabricated pylon parts into place generally consists of a derrick crane on a latticed strut attached to both pylon legs. It is also common to use tower cranes, which can be stabilized against buckling by struts leading to the permanent pylon legs. For concrete pylons, either slipforming or climbing scaffolding can be applied when casting the pylon legs. The slipforming will lead to short construction periods because of the continuous casting, but may cause problems in the continuous delivery of ready mixed concrete. Therefore, slipforming can be a preferred solution if the pylon is built on land where the concrete can be reached easily. The case of the pylon cast far from the coast and all the concrete obliged to being brought in by sea is the main reason for using climbing scaffoldings. Special procedures have to be applied when the horizontal struts between the two legs are cast. It is necessary to fasten a transverse girder to the pylon legs to support the formwork until the concrete is hardened.
24
Chapter 2: Cable-Stayed Bridges
In cable-stayed bridges with A- or diamond-shaped pylons, the legs are more leaning and produce considerable bending moments. For this reason, it is necessary to install additional temporary struts between the two legs during the construction. Traditionally, the pylons are flexible in the longitudinal direction to follow the displacement of the cable system. During the construction of the pylon, the cable-stays are not in place and the pylon must have the necessary resistance against horizontal wind loads by itself. The most critical phase for the pylon during the construction occurs when the pylon is at its full height with no cables installed. In this construction stage, wind induced oscillations of the pylon in the longitudinal direction may be experienced. Owing to the steel structures’ smaller mass, larger flexibility and lower dumping, the problems of the wind-induced oscillation are more significant for steel than for concrete pylons. For large bridges, it has become universal to perform wind tunnel tests to check the stability of the free-standing pylon and to determine whether it is necessary to take special measures. In concrete pylons, a relatively modest amount of vertical reinforcement is required in the completed structure due to the effective pre-stressing by the vertical compressive force from the cable system. Thus, when only designed for the final stage, the pylon can have an insufficient bending strength during the construction. The construction stage control is therefore a matter including all structural parts.
2.3.4 Erection of the main girder using the cantilever method The cantilevering method for the erection of the main deck is investigated in more detail in this part of the paper. The material used for the fabrication of the dick influences the erection procedure. Depending on the material, such as steel or concrete, different construction steps are necessary. To highlight the main issues, this part is divided into cast-in-place, precaste and steel- and composite decks. Each part is described thoroughly in the following.
Chapter 2: Cable-Stayed Bridges
2.3.4.1
25
Cast-in-place concrete decks
During the erection of cast-in-place decks, a mobile carriage supports the weight of the fresh concrete of the new segment by means of longitudinal beams or frames extending out into the cantilever from the last segment. The use of a classical mobile carriage in the erection can produce high unfavourable temporary bending moments in the deck. For the construction of one of the first concrete cable-stayed bridges, the Coatzacoalos Bridge in Mexico [54] for example, long prestressed tendons had to be placed in the box girder, which were of no use for the complete structure. The cables were anchored in the deck axes every second segment. Before tensioning the cable of the last two finished segments, it was necessary to concrete a new one and to move the mobile derrick further. High negative bending moments were produced in this situation and had to be consequently balanced. The situation can become critical if the deck has limited inertia, as it has become more and more frequent with the development in the design of cable-stayed bridges. The deck bending capacity becomes very limited. The situation of this case can be improved by reducing the distance between the cables. Furthermore, referring to the example given before, it is an advantage to improve the mobile carriage so that the cables can be tensioned before moving the carriage. Another option for the erection of the main concrete girder is the use of cable-stayed mobile carriage. There are two possible techniques: The first is to use temporary cables to stabilize the mobile carriage in each of its successive positions. The second is to employ final cables to stay the mobile carriage in the erection position; these are later anchored to the concreted segment. The advantage is that rather long segments can be concreted corresponding to the distance between two successive cables. The construction of flexible decks is possible without creating high bending moments. It is then possible to build the successive segments with no pre-stressing tendon during the construction. Tendons, which may be required for the final condition, for example, to reduce cracks in the concrete due to life load, can be installed when it is practicable and easy. The horizontal component of the stay force is carried either by a mobile carriage or by a precast member which becomes part of the future segment. The permanent stay can also be anchored in the final deck if the stay anchor structure is extended ahead of the whole section. In the case of the Isere Bridge, the center section part, where the stays are anchored, was cast in the first phase and the remaining structure in the second. The sequence of the construction and the applied tension force to the stay cables during the installation have dominant influence on the creep effect, which is explained in detail in Chapter 3.5.1.1.
Chapter 2: Cable-Stayed Bridges
26
2.3.4.2
Precaste segments
Precast segmental bridges become economical for relatively large bridges where the associated cost for setting up a casting yard can be compensated by the speed of casting segments and the speed of the erection process. The pre-fabrication may be more interesting if the method is used for both, the main and the side spans. For the Sunshine Skyway Bridge in Florida, a similar cross section was used throughout the bridge. The 120-ton segments were precast in a yard close to the site and delivered by barges. The segments were lifted into place and mounted on the previously completed portion of the deck. For the James River Bride in Virginia, twin parallel precast box girders were employed. A single plan of stays was designed for the main span and the two boxes were connected by a transverse frame at each stay anchor point. According to Virlogeux [54], cast-in-situ constructions have advantages for cable-stayed bridges because it allows some limited tensile stresses during the erection. With an ideal state for the final construction, the bridge is then in good conditions to experience limited live load which produce no tensile stress in the concrete elements at all. Extreme live loads can then be balanced by partially prestressed concrete members. Partial pre-stressing is not possible with precast segments and they may be preferred in bridges which have a relatively important inertia.
2.3.4.3
Steel and composite decks
Derrick cranes or floating cranes are usually used for the construction of steel and composite decks. Generally, the lifting capacity of standard derrick cranes is smaller than the capacity of lifting struts or floating cranes. Thus, the maximum lifting capacity of derrick cranes used for the erection hardly exceeds 200 tons [8]. It is therefore necessary to use smaller erection units and accept a larger number of erection joints. To keep the weight of each segment below the lifting capacity of the derrick cranes, the girder has to be sometimes split not only transversally but also longitudinally. As an example, the girder of the Parana Bridges is split transversally into three parts, with two outer parts (the edge girders) and the remaining inner part. As described before, high bending moments are produced in the structure during the lifting operation and must be considered in the choice of the segment size. The classical method of constructing steel decks consists in the lifting of the successive steel segments with a derrick, which is installed on the already built part of the bridge. After lifting a new segment with the derrick, it is welded to the previous one, while still suspended to the
Chapter 2: Cable-Stayed Bridges
27
derrick. After this, the cables are installed and tensioned to allow the derrick to move forward on the newly placed segment, ready to lift another one. Composite cable-stayed bridges have a steel girder and a concrete deck which participate together in providing stiffness and resistance to the applied bending and the axial forces on the bridge. The usage of steel permits the structure to be lighter than concrete decks. As for a pure steel deck, the steel elements are prefabricated with a high quality control and a dimensional accuracy. The concrete deck forms the roadway, usually with traffic wearing surface made from asphalt or a concrete overlay, and also carries most of the axial load of the cables. The construction of a composite deck is a little bit more complicated. The steel structure is first built by the cantilever method, segment after segment, and suspended to the successive cables. The concrete slab is then cast or installed in case of precaste elements, segment after segment again. It is also possible to cast the slab just after placing the steel structure that it covers. The disadvantage is that such a solution may create a congestion in the construction operation and slows down the erection process. It can be better to have a distance of two or three segments that allow an independent progression of the two cycles. A greater distance is unfavourable because the steel has to suffer greater forces during the construction and can become unnecessarily critical. Attention to the details of the connections of the concrete deck has to be paid if the strength, the durability and the constructability are to be maintained. An example for the erection process is given by the Uddevalla Bridge [23]: The cross section in the main span of 414 m is a composite structure of an open steel grid and prefabricated concrete slab elements. The steel girder consists of two longitudinal edge-beams, three cross beams and two cable anchorages, which have a weight of approximately 70 tonnes. It is lifted by a derrick and temporary fixed to the previous steel section. After controlling the local geometry in the elevation and in the plan, the welding of the main beam is completed. Since the steel girder is very flexible, supporting bracings are used to ensure correct geometry during the lifting and the installation operation. The bracings also provide a lateral support of the slender cross beams when loaded with the concrete elements. Then, the cables are installed and stressed to a first stressing stage. Afterwards, the derrick lifts the concrete elements in place. The geometry and the cable forces are checked and possible adjustments performed. The joints between the concrete elements and the edge beams are cast. As the last step, the cables are stressed to their final length and the derrick is moved in position for lifting the next segment. The cable forces, which are applied in the first stressing operation, have to be carefully determined so that no deflections or stresses are built into the deck section when the composite structure is established. Thus, it is an important control stage which regards the geometry and the cable forces.
28
Chapter 2: Cable-Stayed Bridges
Additionally, a final tuning of the back stay is performed parallel to the application of the secondary dead loads as the pavement of asphalt layer. This example illustrates the complexity of erection procedures and the importance of construction stage analyses. In some cable-stayed bridges, temporary erection stays are used because of the considerable distance between the cable supported points of the stiffening girder. The distance is adapted to the final situation of the deck and the bridge size, but may not be consider the construction condition. Temporary cables can then be progressively installed, tensioned and removed during the construction between two successive final cables. In these bridges an overstressing may occur either in the girder section or in the last erected stay cable during the free-cantilevering from one cable anchor point to the next. A temporary erection stay reduces both the bending moment in the stiffening girder and the tension in the permanent stay cable. Alternatively, the cantilevered part of the stiffening girder may be supported by a secondary cable system which comprises a temporary pylon with a set of temporary erection stays. In this case, the tension in the permanent stay cables remains practically unchanged. Thus, this arrangement is only practicable if the problem of overstressing is confined to the stiffening girder. However, it must be mentioned that temporary measures in form of temporary supports, stays or pylons, which are employed only during the erection, are costly as they have to be fabricated, erected and removed before completing the structure. Therefore, in very early design steps, the structural system should be chosen to fulfil the requirements in both the final operational phase and in the construction phase.
Chapter 3: General description of a Construction Stage Analysis
29
3 General description of a Construction Stage Analysis
This chapter gives the reader a general overview of a construction stage analysis of cable-stayed bridges and the important issues to be taken into consideration to allow the development of a save construction sequence. In the following, the role of designed cable forces is illustrated and the process of a construction stage analysis outlined by a simple model. The computer programme MiDAS is then introduced and the main features for the analysis are explained. Time dependent effects of the material will be clarified, as these effects must be considered in the analysis of concrete or composite section bridges in order to develop a construction process which finally fits the defined structural shape. Structural non-linearities are further an important issue. An explanation of the significance of these effects, that must be properly included in the analysis, is given. Additionally, the tuning sequence is a main part to be defined by the engineers for the erection of the bridge. Depending on the structural system, it might be more or less reasonable to choose retension steps directly from the design. The influence of the stressing sequence will be clarified. Finally, various monitoring and control procedures as well as computational systems are explained.
3.1 Designed Cable Forces The permanent state of stress in a cable-stayed bridge subject to its dead load is determined by the tension forces in the cable stays. They are introduced to reduce the bending moments in the main girder and to support the reactions in the bridge structure. The cable tension should be chosen in a way that bending moments in the girders and the pylons are eliminated or at least reduced as much as possible. Hence, the deck and pylon would be mainly under compression under the dead load. In case of a concert deck, this reduces creep-induced deflections and the corresponding uncertainties in concrete members. Furthermore, second-order effects decrease [14]. The designer first selects the final condition and then evaluates the initial cable forces. The initial tension forces, which must be applied to the cable-stays at the time of installation, are determined by a precise construction stage analysis.
Chapter 3: General description of a Construction Stage Analysis
30
To estimate the cable forces, the simplest method is to assume a bridge segment to be a simple beam supported by cables. The method of the simple supported beam can be used in the preliminary design stage to estimate the cable area. Another method is to assume that, under the dead load, the main girder behaves like a continuous beam and the inclined stay cables provide rigid supports for the girder. The vertical component of the forces in stay cables are equal to the support reactions calculated on this basis. Virlogeux [56] describes the first procedure by the pendulum method: The cable tensions are evaluated from the key section to the pylon by balancing the load to produce no bending moment, e.g. at the cable anchorage. Distributing the weight of each segment at the two corresponding anchorages, it follows that the cable force Si is (Figure 3-1):
S i * sin α i = 12 * ( Pi + Pi +1 )
(3-1)
and from the equilibrium of horizontal forces, the normal force Ni:
N i * cos β i = N i +1 * cos β i +1 + S i * cos α i
(3-2)
The cable force Si and the normal force Ni result in:
Si =
1 2*sin α i
Ni =
1 cos β i
* ( Pi + Pi +1 )
(3-3)
Gi + Gi +1 N i +1 * cos β i +1 + 2 * tan α i
(3-4)
In order to achieve minimum bending in the pylon, the horizontal projection of cable tension must be balanced:
S ' i * cos α ' i = S i * cos α i
Figure 3-1: Illustration of the pendulum rule [13]
(3-5)
Chapter 3: General description of a Construction Stage Analysis
31
Herzog [9]and Gimsing [8] developed easy hand-formulae to predict the cable forces by taking the stiffness of the girder and pylon into consideration. Since the cable stayed structure is a highly undetermined system, there is no unique solution for calculating the initial cable forces directly. Usually it is an iterative process to find an economical solution. As mentioned before, the moment and displacement distribution along the girder and the pylon can reach the ideal state by adjusting the cable stresses. Using vector and matrix calculations, the moment or the displacement of an ideal state I can be written as:
I = [i1
... in ]
T
i2
(3-6)
n is the total number of the targets that need to be satisfied and T stands for the transformation of a matrix or a vector. The approach to the ideal state is to make Equation 3-6 as close to a designated value as possible. The result cable stresses S can be written as
S = [s1
s2
... s m ] , T
(3-7)
in which m is the number of cables to be adjusted. By analysing the response of the unit prestress applied to each tuning cable, the influence values of all the targets can be obtained. When m rounds of the analysis are done, the influence matrix T can be written as:
t11 t T = 21 ... t n1
t12 ... ... t n2
... t1m ... ... , ... ... ... t nm
(3-8)
where tnm is the response at the target n by pre-stressing the unit stress at cable m. Thus, their relation can be set down as
T *S = I
(3-9)
If the number of cables that are to be tuned is the same as the number of targets, the setting I to the designated target values, the cable stresses S can be obtained accordingly by solving the linear Equation 3-9. In this case, engineering experience is required to select the proper target values. In this method, m cannot be greater than n. If, as in most cases, m is less than n, the
Chapter 3: General description of a Construction Stage Analysis
32
cable stresses can be optimized so that the error of the target value and the designated state is kept to a minimum. The method of square error minimization is an effective way to obtain the optimal I. A is the adjustment value which has the same form as I. E describes the error between A and I, and can be written as:
E = A− I
(3-10)
The optimization of the cable stress is to minimize Ω, the square of E. As a definition, Ω can be written as:
Ω = (A − I ) * (A − I ) T
(3-11)
The principal condition to minimize Ω is:
∂Ω = 0 , i = 1,2,3..., m ∂S i
(3-12)
Using the matrix differential and considering Equation 3-9 and 3-11, the following equation can be obtained:
T TT * S = T T A
(3-13)
After calculating S from the linear equation group in Equation 3-13, the optimized target value can be calculated by Equation 3-9. An example is presented in a later chapter. A very similar method can be applied for adjustments of errors during the erection process and is explained in detail in Chapter 3.7. Many analysis programmes use an optimization method for determining the cables forces. Under permanent loads, a criterion (objective function) is chosen in a way that the internal forces, mainly the bending moments, are evenly distributed and small. The deflection of the structure can be limited to prescribed tolerances, too.
Chapter 3: General description of a Construction Stage Analysis
33
Bruer and Pircher [17] favour a numerical approach to reduce the calculation effort, the Unit Load Method. For the final stage structure including its total dead load, unit load cases as well as the ideal moment diagram must be defined. Commonly, the selected unit forces are: •
A unit shortening of the cable or a unit tensioning causing an axial cable shortening
•
A unit translation of a rigid support or an element. A transverse or longitudinal movement changes the moments in the deck by changing the cable forces which act on the deck.
The same number of unit loading cases must be defined as the number of “Fixed Moment” points, chosen on the structural model to represent the ideal moment diagram. Figure 3-2 illustrates this procedure.
Figure 3-2: Unit Load Case Method for determining the ideal state 0
The ideal dead load bending moment diagram is defined for the deck girder. As shown in the figure, the bending moments for nine points along the girder are described from position A to I. Nine unit load cases are selected for setting up the simultaneous equations – eight unknown stay cable forces and one unit translation at the end support. In this case, the linear equation system follows to be:
M A = M P + M T 1=1 * x1 + ... + M T 8=1 * x8 + M TJ * x9
(3-14)
:
M I = M P + M T 1=1 * x1 + ... + M T 8=1 * x8 + M TJ * x9 where MA to MI is the final stage moment at positions A to I including tensioning and movement at the end supports, MP the permanent load moment at the current position without cable
34
Chapter 3: General description of a Construction Stage Analysis
tensioning and support movement, MT1=1 to MT 8 =1 the bending moments due to each unit tensioning at position A to I, and MJ the bending moment due to unit jacking of the end support. The solution of the equations, the unknown xi, is the factor by which the unit loads must be multiplied to achieve the defined moment distribution. This basic solution defines the cable forces and the jacking force for the final stage, but does not include the effect of any chosen construction sequence, creep, the 2nd Order Theory and the non linearity of the cables due to the sagging effects. However, in the analysis programme RM2000, developed by Pircher, a method which allows the consideration of non-linear problems in the optimisation process is introduced. This approach is called the AddCon Method [43] [45] (Additional Constrain Method) and is an extension of the Unit Load Method. It is possible to include time effects (creep and shrinkage) and non-linear structural behaviours (non-linearity of cable elements and P-delta effects) in the calculation. An iterative process is used to appropriately factorise the user-defined unit loading cases so that the defined constrains are achieved. User-defined constrains can be a set of forces/moments, stresses or displacements or a combination of both, which must be fulfilled under the applied loading. The analysis programme MiDAS also provides the Unknown Load Factor function, which is based on an optimization technique. Similar to the described method above, this can be used to calculate the load factors that satisfy specific boundary conditions (constrains) defined for a system. An example of this procedure is given in Chapter 3.3.3. There, it is explained in detail how to apply this method in the analysis of a structural system.
3.2 Construction Stage Analysis In order to complete the design of a cable-stayed bridge and to analyse the structure in full, a construction stage analysis must be included. The construction stages should be modelled to check the stability of each step separately. Depending on the erection method, the structural system can significantly change. Sometimes, a change of a system during the construction can result in a more critical condition for the structure compared to that of the final state. Therefore, an accurate construction stage analysis should be carried out to check and to review the stresses in the cables, the girder and the pylon. In no stage is a compression in the cables possible. This would be a theoretical case using truss elements for the cables in the analysis of the structure.
Chapter 3: General description of a Construction Stage Analysis
35
Furthermore, the geometric profile of the girder is also very important during the construction. To avoid serious problems, it must be ensured that both cantilever ends meet smoothly together in the final construction stages. It should be noted that the internal forces of the structure and the elevation of the girder vary in the construction process. This usually happens because of the bridge segments that are built by a few components, the heavy lifting operations during the erection in case of prefabricated segments and the erection equipment, which is often placed at different positions during the construction. Additionally, because errors such as the weight of the segment and tension forces in the cables, wrong values for material properties, etc. may occur, monitoring and adjustment during erection are absolutely needed. The simulation of the construction process must be adjusted to the field measurements during the construction. In a later chapter, these procedures are illustrated in detail. The general objectives of the simulation are: •
Determination of the required tension forces in the cable stays at each construction stage
•
Specification of the fabrication girder geometry
•
Setting the elevation of the girder segment
•
Calculation of the structural deformation at each construction stage
•
Controlling of the stresses in the girder and pylon sections
Construction state analyses for a cable-stayed bridge can be classified into the forward and the backward analysis. The forward analysis reflects the real construction sequence, whereas the backward analysis is performed by regarding the state of the final structure. The elements and loads are eliminated in the reverse sequence to the real construction sequence. It is assumed that all of the creep and shrinkage deformations of concrete are completed, i.e., a state 5 years or 1500 days after the completion of the bridge construction [5].The cable prestresses, which are induced during the construction, can be calculated by using the backward analysis. These results are the guidance for the forward erection and stressing. Using the forward analysis, time dependent effects can be considered and implied in the design. To perform a construction stage analysis, construction stages must be defined. Main girders, cables, boundary conditions, loads, etc. are activated or deactivated to consider the effect and the change on the structure. The changing of the structural system by modifying the boundary condition or removing certain elements might cause gaps between the forward and backward analysis. Activation errors can
36
Chapter 3: General description of a Construction Stage Analysis
occur and remain in the system or even increase as the construction stage analysis continues. In the simple case of a steel structure, where no time-dependent effect must be considered, the last stage in the forward analysis should theoretically correspond to the first stage in the backward analysis. This will hardly be the case in an analysis considering time dependent effects, as the difference between the forward and backward analysis increases. However, these two methods may be alternatively applied until a convergence is reached within a tolerable range. For the construction stage analysis, Bruer [17] proposes an extended method of the before explained Unit Load Method. The unit load cases are applied to the different structural systems, which exist at different construction stages, as illustrated in the figure below.
Figure 3-3: Unit Load Case Method for construction stage analysis [17]
The loading cases for each construction stage are combined to form the set of simultaneous equations, which are solved to find the required multiplication factors for the unit loading. In this procedure, the influence of the different stages is considered and by applying an iterative process, the calculation results in the defined ideal system in the final construction stage. In the analysis programme RM2000, time dependent effects, as well as non linearity can be included in this procedure using the AddCon function as it was briefly explained before. However, this programme has not been used in the construction stage analysis presented in this thesis and therefore it is difficult to comment this procedure or to judge about the reliability of this function. The method of modelling a construction stage analysis by the computer programme MiDAS is explained in more detail in the following chapter. To give the reader an overview of the methods and the different phases of the calculation, a simple cable-stayed bridge is being analysed.
Chapter 3: General description of a Construction Stage Analysis
37
3.3 Construction Stage Analysis by MiDAS The following example explains techniques for modelling a simple cable-stayed bridge using the structural analysis programme MiDAS/Civil. A short introduction to the programme is given first. Then, the calculation of the ideal cable forces using the Unknown Load Factor function is demonstrated. Furthermore, a backward and a forward analysis are performed in Chapter 3.3.4 and 3.3.5. respectively.
3.3.1 The analysis programme MiDAS/Civil is an analysis programme for modelling and analysing structural systems. It offers special features for analysing civil structures, such as box girder or composite bridges. The programme provides inter alia the Construction Stage Analysis feature, which allows the creation of construction stages by changing the structural system. This function offers the addition or the removal of elements. The boundary condition can be changed and loadings can be activated or deactivated in order to simulate the real condition for each construction stage. It is capable of carrying out analysis reflecting the erection and the dismantlement (backward analysis) of structures. In the construction stage input, time steps can be defined to represent the loading and unloading time for load changes within a given construction stage without changes in the structural system. For modelling structural systems, the programme offers various element types. The main elements used for the analysis of cable stayed bridges are General Beam elements and Truss elements or Tension-Truss Cable elements (catenary cable element). However, numerous other structural elements exist, too, such as the plate and shell elements. Initial pre-stressing forces can be calculated through optimizing the equilibrium state. The calculation of the ideal cable prestressing forces by the optimization is restricted to the linear analysis as the different loadings are superposed. The initial cable pre-stressing forces are obtained by the Unknown Load Factor function and the initial equilibrium state analysis of a completed cable-stayed bridge.
Chapter 3: General description of a Construction Stage Analysis
38
For static systems, linear and non-linear analyses can be performed by using truss and cable elements. Geometric non-linearities can be considered by including P-delta effects in the calculation or by performing a large displacement analysis. The programme also offers to consider structural non-linear behaviour in the construction stage analysis. In the Construction Stage Analysis Data, it can be defined to perform a large displacement analysis. It was tried to use these functions and to perform a non-linear construction stage analysis. However, when contacting the MiDAS support-team, it turned out that these functions were not available at the time of working out this thesis. The main e-mail contact with Mr. Lee from MiDAS is listed in the appendix. These details are reported in more detail in later chapters. Furthermore, material behaviors as creep and shrinkage can be modelled. These features are generally explained and also tested in the construction stage analysis for a simple cable-stayed system in Chapter 4.5. More information on this programme is given in the following chapters and can also be found in detail in the analysis reference [60] or in the on-line manuals [61].
3.3.2 Structural Data The following structural system is used for the calculation in the example to illustrate a simplified modelling of a construction stage analysis. Table 3-1 gives a list of the input data which are assumed in the model.
7
1
3 2
1
2
3
4
Figure 3-4: Cable-stayed example
5
Chapter 3: General description of a Construction Stage Analysis Data Area A Deck Stiffens I Deck Poisson Ratio υ Deck Modulus of Elasticity ECable /EDeck
Value 4.38 m2 0.92 m4 0.3 5.25
Data Area A Cable 1 Area A Cable 2 Area A Cable 3
39 Value 0.0115 m2 0.0062 m2 0.0208 m2
Table 3-1: Input data
3.3.3 Unknown Load Factor The first step of analysing a cable-stayed bridge, in order to perform a construction stage analysis, is to evaluate the ideal cable forces for the final structure under its self-weight. The general procedure of calculating these ideal cable forces by the Unknown Load Factor function in MiDAS is outlined in Figure 3-5. The function allows the determination of superposition factors for previously calculated load cases to obtain a prescribed state in the structure by combining those load cases. As far as a solution for realizing the user defined conditions exists, the factors will be calculated. To determine the unknown load factors for each cable stay to achieve an ideal state, a unit pretension load is applied for each cable. Performing a linear analysis, the programme computes the influence on the structure due to each unit tension load. In the Unknown Load Data the unit load cases are then defined as an Unknown Load. Furthermore, the structural restrictions for e.g. moment or displacement values, which are to be realized through the load factors in the combined load case, must be defined. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:
Create a model of the structural system ↓ Generate the load conditions for the dead load and a unit pretension load for each cable ↓ Assign the dead loads and the unit loads to the elements ↓ Define the load combination for the dead loads and the unit loads after the performance of a linear analysis Define boundary conditions which have to be fulfilled ↓ Calculate Unknown Load Factors for each cable Figure 3-5: Flowchart for cable initial prestress calculation
In this example, constraints are specified to restrict the displacement of the main girder and the pylon. The displacements of Node 2, 3 and 7 are limited. In the same way, it is also possible to
Chapter 3: General description of a Construction Stage Analysis
40
define constraints, for example, for moment values or tension forces in the cables. Since the unknown load factors are calculated on the basis of the superposition of different load cases, truss elements must be defined for the cables; it is not possible to use cable elements. Figure 3-6 shows the moment distribution under the structural self-weight and a unit pretension load in each of the cables. The figure clearly demonstrates that the structural performance can be improved by a higher pretension in the first cable.
Figure 3-6: Moment self-weight & unit pretension load [tonf]
In order to fulfil the defined restrictions, the load factors are calculated. Figure 3-7 illustrates the result table given by MiDAS.
Figure 3-7: Results of the Unknown Load Factor calculation
For this example, the factors for Cable 1, 2 and 3 are 5.829, 3.105 and 10.724 respectively. The values can be found in Figure 3-7 in the first column. Figure 3-8 shows the resulting moment distribution including the factors for the tension forces in the cable stays 1 to 3.
Figure 3-8: Moment self -weight & initial pretension load [tonf]
Chapter 3: General description of a Construction Stage Analysis
41
The distribution of the bending moments is changed into the direction of a continuous beam condition. Thus, the moment distribution is more equal and the maximum moment is reduced. For the application of the Unknown Load function, the displacement has been restricted, whereas the main target in the structural design is an equal distribution of the bending moments. However, the example shows the correlation of both parameters. As the bridge turns into a more complex structure containing a higher number of cables, it becomes more difficult to define the essential restrictions, which are needed in order to calculate the most efficient ideal cable forces. Experience is helpful in this case. Care must be taken in the selection of sensible and unrelated restrictions. If a specified condition is in conflict with another defined requirement, a singularity will result and there will be no solution.
3.3.4 Backward Analysis After the determination of the ideal cable forces, the principles of a backward analysis by MiDAS are explained. In the previous calculation, only one element has been used per segment. In the backward and forward calculation the model is refined using five elements per girder. The following table shows the steps carried out to perform a backward analysis for the described example. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:
Define the initial cable forces (internal forces) ↓ Define each construction stage and the names ↓ Define the elements by group which are added or deleted ↓ Define the boundary condition by group which are added or deleted Define the loading conditions by group which are added or deleted ↓ Define the elements, the boundary and the loading belonging to each stage Table 3-2 Flowchart for backward analysis
The construction stages which are defined for this simple example are shown in Figure 3-9. The numbering for the cables and the girder elements starts from the left to the right. In the backward analysis the complete structure is dismantled step by step. The initial cable forces,
42
Chapter 3: General description of a Construction Stage Analysis
which must be applied at the time of installing the cables in the forward analysis, can be obtained from the backward analysis before removing the corresponding cable. Construction stage 0
Construction stage 1
-/-
- remove support
Construction stage 2
Construction stage 3
- remove girder 1
- remove cable 1
Construction stage 4
Construction stage 5
- remove girder 2
- remove cable 2
Construction stage 6
Construction stage 7
- remove cable 3
- remove girder 3 Figure 3-9: Sequence for backward analysis
Chapter 3: General description of a Construction Stage Analysis
43
The calculated cable forces for each construction stage are used as external prestress loads for the forward analysis. The cable forces are shown in Table 3-3. CS 0 [tonf] 562.19 288.94 1000.06
CS 1 [tonf] 835.04 292.20 1348.44
CS 2 [tonf] 302.82 243.06 625.97
CS 3 [tonf] -/577.98 577.38
CS 4 [tonf] -/217.57 217.57
CS 5 [tonf] -/-/118.78
CS 6 [tonf] -/-/-/-
CS 7 [tonf] -/-/-/-
Table 3-3: Cable forces [tomf]
A simple way to check whether or not the loads are correctly considered in the analysis is to control the reaction forces in the final state. The applied loads are: ton Ftot := 11 ⋅ 80m m
Ftot = 880ton
The reaction forces in the MiDAS programme are: F1 := 86.43⋅ ton
F2 := 1414.28ton ⋅
F3 := −620.71ton
Control: Ftot_Mi := F1 + F2 + F3
Ftot_Mi = 880ton
The calculation is correct as far as the reaction forces are identical. The correct consideration of the influence of the construction sequence is not proved here, but will be shown later on.
3.3.5 Forward Analysis In the forward analysis, the calculated cable forces from the backward analysis are applied as external initial tension forces. That means that the forces are treated as external loads to support the structure at the construction stage of installing and pretensioning the cables. In the case of internal forces, as used in the backward analysis, the tension values reduce, depending on the loads and the support of the structure based on its stiffness. The applied cable forces are shown in the next figure.
Chapter 3: General description of a Construction Stage Analysis
44
Figure 3-10: Applied cable forces [tonf]
The construction sequence in the forward analysis is the same as in the backward analysis. The next figure shows the moment distribution before adding the support on the left side of the bridge.
Figure 3-11: Moment distribution; forward analysis before adding the support [tonfm]
Even if the support is being added in the next step, there still is a vertical deformation in the system. It is possible to add a boundary condition to the deformed or the undeformed structure. In both cases, the results are different from the target values as shown in Construction Stage 0 in the backward analysis. The problem can be solved by applying the reaction force F1 instead of the support. The resulting moment distribution for adding the support and using a force are illustrated in the figure below. a) Construction stage 0
b) Construction stage 0
- adding support to the original structure
- adding reaction force F1
Figure 3-12: Moment distribution in the last step of the forward analysis [tonfm]
In addition to the moment distribution, the reaction forces and deformations of both analytical methods (backward and forward analyses using the reaction force F1) are compared. The values are identical within a very limited range.
Chapter 3: General description of a Construction Stage Analysis
45
This simple example explains the purpose of performing a backward and a forward analysis to control the construction stages. It also demonstrates the importance of taking remaining deformations into consideration when structural groups are activated or deactivated and the boundary conditions are changed. Otherwise, it is not possible to perform a back- and forward analysis with satisfying results. A more complex example is given in Chapter 4, illustrating the analysis of a cable-stayed bridge in detail.
3.3.6 Forward Method = Backward Method As far as no creep or shrinkage is considered in the calculation, the obtained results from the forward and backward analyses are identical. However, in addition to the already mentioned problems and considerations which are required to perform a construction stage analysis, a few more remarks have to be given. The backward method requires a stress-free situation that has to be achieved for the elements before their deactivation. In case of a support, it requires some movement or balance loads until the reaction becomes zero before its removal. A segment may be removed by applying its weight in an upward direction first to create a zero stress condition. In MiDAS, the results of all prior construction stages are accumulated and applied to the current stage. Once activated, elements, boundary conditions and loads remain active until they are deactivated. When an element is being removed, the internal forces are internally imposed to the contiguous remaining elements in the opposite directions. Therefore, in MIDAS, it is not necessary to apply the loads in the opposite direction before removing the loads or the elements.
Figure 3-13: Discontinuity between two segments
46
Chapter 3: General description of a Construction Stage Analysis
Forward and backward analyses give different results in case both are applied straight forward. The forward method automatically applies some discontinuities at every joint between two segments as it can be seen in Figure 3-13. The backward method assumes that the new segments are added to the previous stage in the tangential direction. Therefore, when comparing both results, there are differences in the deformed shape. The moment distribution stays the same as before, since the internal forces are not influenced by this circumstance. Using the Initial Tangent Displacement for Erected Structures option in MiDAS, the real displacement as well as the rotational angle is calculated for the elements installed in the following stage. This option allows to install new segments tangentially and to avoid any discontinuities. In programmes which do not offer this function, it may be a solution to install all segments of the main girder in the fist stage and to apply only the self-weight in the corresponding construction stages. In this way, the segments are always considered tangentially to the already existing structure.
3.4 Influence matrix The MiDAS programme offers the function of calculating the influence matrix, which can be used to calculate the cable forces in order to achieve the ideal state or to determine the adjustment procedures to reduce errors during the construction process. Using this function, the influence matrix for displacement and tension forces can be calculated. First, it is demonstrated how to calculate the influence matrix. Later, for a better understanding of the performed calculation, the influence matrix calculated by MiDAS is used to control the Unknown Load Factor calculation.
3.4.1 Calculation of influence matrices The ideal stay stressing value and the cable adjustment values can be determined by using the influence matrix analysis. It may be required to calculate different kinds of influence matrices, such as for displacements or cable forces. Generally, the influence matrix is assembled by applying unit loads to the cables which are being stressed in separate load cases and by determining the changes in the structure. For the influence matrix of cable forces, the change in
Chapter 3: General description of a Construction Stage Analysis
47
the tension force in each cable due to a unit load is evaluated by a computer analysis. The values can be stored in a vector of stay forces, which represents a single column in the influence matrix. The same procedure can be used to calculate the influence matrix for displacements.
3.4.2 Influence matrix calculated by MiDAS For the example introduced in Chapter 3.3.2, Figure 3-14 shows the influence matrix for the displacement obtained from the Unknown Load function, which has been previously explained. In order to illustrate the utilisation of the influence matrix, the load factor which has been obtained in the previous chapter is recalculated in the following.
Figure 3-14: Influence Matrix of displacement [mm]
The initial vertical displacement at Node 2 and 3, and the horizontal movement of the top at the pylon (Node 7), including the distributed load and the unit pretension in all cables, are (in [mm]):
160.165 δ0 := 116.484 −52.005 The deformation caused by the distributed load is:
−191.257 δP := −143.006 −52.477 Because there should be no deformation at the defined nodes
0.000 δfi := 0.000 0.000 ,
Chapter 3: General description of a Construction Stage Analysis
48
the target value of adjustment is: δA := δfi − δP
191.257 δA = 143.006 52.477
Using the transposed displacement matrix
11.2905 11.4092 8.3947 D = 3.9936 15.9948 6.5341 −6.0153 −2.3586 8.8463 T
and solving the equation DT*x = δA, the load factor x (Sfac) is (in [tonf]):
5.829 Sfac = 3.105 10.724 . The calculation is controlled by
191.257 D ⋅ Sfac = 143.006 52.477 , T
which is exactly the value δA. The values of Sfac are identical to the load factor which has been calculated before. A similar calculation is performed using the cable force influence matrix. Compared to the previously described procedure, the calculation seems to be unstable and very sensitive to small deviations.
3.5 General considerations and uncertainties In a construction stage analysis of a cable-stayed bridge, the calculation is firstly made in a linear analysis which considering the effects of building the structure in different stages, together with the exact loading condition for each step. Later, in a more detailed analysis, a nonlinear calculation is made taking the non-linear effects of the cable sag and P-delta effects into consideration. Additionally, creep and shrinkage must be included in the analysis because both time dependent effects and non-linearity affect the structural performance. Moreover, the temperature at the construction site may also influence the structural parts. These topics are investigated in more detail in the following chapters.
Chapter 3: General description of a Construction Stage Analysis
49
3.5.1 Time dependent effects The stress and the strain in a reinforced or prestressed concrete structure are subject to change for a long period of time during which the creep and the shrinkage of concrete and the relaxation of the steel used for pre-stressing gradually develop. With the age of concrete, its modulus of elasticity increases. A stress applied on concrete instantaneously produces strain. If the stress is sustained, the strain progressively increases with the time due to creep. Thus, the magnitude of the instantaneous strain and the creep depends upon the age of concrete at loading and the length of the period after loading. Besides concrete, also steel exhibits some creep if it is subject to stresses higher than 50% of its strength. In practice, the steel used for pre-stressing may be subjected in the service condition to a stress of 0.5 to 0.8 of its strength. If a tendon is stretched between two fixed points, constant strain is sustained but the stress progressively decreases due to creep. This relaxation in tension is of concern in deformations of prestressed concrete members.
3.5.1.1
Creep and shrinkage
In concrete or composite cable-stayed bridges, time dependent effects of creep and shrinkage have a significant effect on the geometry so that the final stress state in the completed bridge must be included in the analysis. In order to perform a reliable construction stage analysis, the load history and the casting sequence must be considered. Each individual construction stage must be modelled in the erection analysis to obtain accurate predictions of the final stress and the final profile. The weight of the deck segments and the equipment must be modelled correctly. The weight of a deck segment should be devided into components and the weight incrementally added according to the intended erection cycle. Figure 3-15 shows the total deformation in a structure. The true elastic strain in the figure represents the reduction of elastic strain as a result of concrete strength gaining in relative to time.
Chapter 3: General description of a Construction Stage Analysis
50
Figure 3-15: Time dependent concrete deformation
Creep deformations in a member are a function of sustained stresses. A high-strength concrete yields less creep deformations in comparison to a lower strength concrete under an identical stress. The magnitudes of creep deformations can be 1.5 ~ 3.0 times more than those of elastic deformations. About 50% of the total creep deformations take place within the first few months and the majority of creep deformations occur in about 5 years [5]. The effect of creep in concrete members depends on the following factors: •
Creep increases with increasing water-cement ratio
•
Creep decreases with increasing age and strength of the concrete when it is stressed
•
Creep deformations increase with the increase in ambient temperature and the decrease in humidity
•
It also depends on many other factors relating to the quality of concrete and the conditions of exposure, such as the type, the amount and the maximum size of the aggregate, the type of cement, etc.
The total deck creep is made of the creep due to post-tensioning loads, axial compression in the deck from the cables and the bending moments in the deck. Post-tensioning is often provided in those sections of the bridge deck which have low axial compression (e.g. the segments near the centre of the bridge) in order to prevent cracking that can be caused by high bending moments due to live load conditions. In case of a prestressed girder segments, the pre-stressing is generally applied in one of the two ways: pre-tensioning or post-tensioning. When pre-tensioning is applied, a tendon is stretched into the form in which the concrete member is cast. After the concrete has attained sufficient strength, the tendon is cut. Because of the bond with the concrete, the tendon cannot regain its original length and thus a compressive
Chapter 3: General description of a Construction Stage Analysis
51
force is transferred to the concrete. This causes a shortening of the concrete member and is accompanied by an instantaneous loss of a part of the prestress in the tendon. A slip usually occurs at the extremities of the member. When post-tensioning is employed, the tendon passes through a duct which is placed in the concrete before casting. After attaining a specified strength, tension is applied on the tendon, which is anchored to the concrete at the two ends. Later, the duct is grouted with cement mortar. During tensioning the tendon, before its anchorage, the strain in steel and concrete are not compatible. The concrete shortens without causing instantaneous loss of the prestress forces. After the transfer, a perfect bond is assumed between the tendon, the grout, the duct and the concrete. This assumption is not justified when the tendon is left unbonded, but, in most practical calculations, the incompatibility in strain between the strain in an unbonded tendon and the concrete may be ignored [7]. At each cable anchorage, axial load is induced to the girder and spreads laterally into the deck. Over the time, the deck creeps under this axial loads. Because of the creep effects, it is important to select convenient cable tensions which produce no bending moments during the construction. If the designer wants to reduce creep effects due to bending of the deck, it is efficient to erect the bridge with cable tensions adjusted to the construction loads and not to the final ones. The strain that occurs during the application of the stress is referred to as the instantaneous strain and is expressed as follows:
ε el (t , t 0 ) =
σ c (t 0 ) E c (t 0 )
,
(3-15)
where σ c (t 0 ) is the concrete stress and E c (t 0 ) the modulus of elasticity of the concrete at the age t0, the time of application of the stress. Under sustained stress, the strain increases with time due to creep and the total strain is:
ε (t , t 0 ) =
σ c (t 0 ) E c (t 0 )
* [1 + φ (t , t 0 )] , (shrinkage is neglected)
(3-16)
where φ (t ,t 0 ) is a dimensionless coefficient, which is a function of the age at loading t0 and the age t for which the strain is calculated. The coefficient represents the ratio of creep to the
52
Chapter 3: General description of a Construction Stage Analysis
instantaneous strain. The value increases over the length of the period (t-t0) during which the stress is sustained. For codes used in practice, different formulas are developed for the prediction of creep, based on a product or summation approach. For example, in the ACI 1982 code, the product function considers the age of concrete at loading and the loading time:
ε cr (t ) ≈ f 1 (t 0 ) * f 2 (t − t 0 )
(3-17)
In the summation approach of the CEB-FIP 1978 and DIN 4227 1979, the creep is divided into a delayed elastic part (f3) and a yield part (f4 and f5).
ε cr (t ) ≈ f 3 (t − t 0 ) + f 4 (t ) − f 5 (t 0 ) .
(3-18)
Newer models, such as the CEB-FIP 1990 and the EC2 respectively, are based on a product approach, which considers latest researches, and include a certain non-linearity of creep in case of high stresses. Drying of concrete in air results in shrinkage. If the change in volume by shrinkage is restrained, stresses develop. In reinforced concrete structures, the restraint may be caused by the reinforced steel, by the supports or by the differences in the change of volume. Stresses caused by shrinkage are generally reduced by the effect of creep of the concrete. To minimize shrinkage, precast concrete forms are usually aged for 6 months or more before utilized in the construction of the bridge [5]. However, the infill concrete, which is caste-in place, exhibits a normal shrinkage. The axial shortening and the bending deformation of the girder must be taken into account in order to predict the correct deck geometry. In case of a composite section, the change in axial and bending stiffness, which occurs when the girder is made composite with the deck, must be considered. In the subsequent analysis, the girder resists additional loading or unloading as a composite member. Felber [25] reveals the importance to ensure that the cast-in axial and bending deformations and stresses are properly modelled in order to obtain accurate predictions of the deformations and stresses for all subsequent construction stages. For axial stresses and deformations, this can be achieved by tracking the forces and deformations to which the section is subjected at the time it is made composite. This information is then used to determine additional strains which are applied in the subsequent stages to achieve the correct length of the composite member and thus, the correct overall deformation and stresses.
Chapter 3: General description of a Construction Stage Analysis
53
Shrinkage is a function of time, which is independent from stress in the concrete member. Therefore, the shrinkage can be generally expressed as:
ε sh (t , t 0 ) = ε sh 0 * f (t , t 0 ) ,
(3-19)
where εsh0 represents the shrinkage coefficient at the final time, f(t,t0) is a function of time and t stands for the time of observation and t0 for the initial time of the shrinkage. Concrete towers are currently a preferred solution for long cable-stayed bridges. Steel tower segments are considerably simpler to construct because only elastic shortenings needs to be considered. A concrete tower is usually built too tall by a calculated amount to account for creep and shrinkage, as well as the elastic shortening due to the applied dead load. This height adjustment should be included in an analysis model because it effects the predicted deck profile for the initial stages of construction. The calculated height for the compensation is usually applied just below the cable connections. The self-weight of the tower makes up a considerable portion of the total dead load at the base of the tower. In large structures the tower self-weight is applied to a “young” concrete, while the tower concrete might be already a year old before much of the load from the self-weight of the deck is applied. Therefore, a large portion of the total tower creep originates from its self-weight. As it is also the case for shrinkage, all the shortenings of the tower due to creep, that occurs prior to the completion of the tower, is compensated by setting the formwork to the desired elevation during the construction.
3.5.1.2
Relaxation of prestress steel
As already mentioned, steel is also subject to creep. A short introduction is given in the following. The effect of creep on prestressed steel is commonly evaluated by a relaxation test, in which a tendon is stretched and maintained at a constant length and temperature. The loss in the tension is measured over a long period. The relaxation under constant strain, as in a constant length test, is referred to as intrinsic relaxation, ∆σ pr . The equation widely used for the intrinsic relaxation at any time t of stress-relieved wires or strands is:
∆σ pr
σ p0
=−
log(t − t 0 ) σ p 0 − 0.55 , f 10 py
(3-20)
Chapter 3: General description of a Construction Stage Analysis
54
where fpy is the yielded stress and defined as the stress at a stain of 0.01. The ratio fpy to the characteristic tensile stress fptk varies between 0.8 and 0.9, with the lower value for pre-stressing bars and the higher value for low-relaxation strands ((t-t0) is the period in hours for which the tendon is stretched). In the absence of a reliable relaxation test, MC-90 code suggests the intrinsic relaxation values. The Eurocode 2 allows the use of relaxation slightly different values from MC-90. Details can be found in the CEB-FIP Model Code 1990 [4].
3.5.1.3
Modelling of creep
In order to obtain the strain due to creep εcr, different methods can be applied. A range of various options can be found in the literature. Bažant [2] gives a detailed overview. As it has already mentioned before, creep is a phenomenon in which deformations occur under sustained loads with time and without necessarily additional loads. As such, a time history of stresses and the time itself become important for determining the creep. Creep does not only increase deformations, it also affects the pre-stressing in tendons and thereby the structural behaviour. In order to accurately account for time dependent variables, a time history of stresses in a member and creep coefficients for numerous loading ages are required. For this reason, controlling construction stages only by a backward analysis is not sufficient in concrete or composite bridges. Creep is a non-mechanical deformation, which means that only deformation can occur without accompanying stresses unless constrains are imposed. The material behaviour of concrete is usually represented by a viscous-elastic-plastic model, because of its properties with time. For the service ability loading, a linear viscoelasticity is an acceptable assumption for most cases. Besides a significantly simpler formulation of the model, superposition can be used for different stress and strain histories. Assuming a small strain behaviour, the total strain ε(t) of a uniaxially loaded concrete specimen at a time t after the casting of the concrete can be separated as [2]:
ε (t ) = ε el (t ) + ε cr (t ) + ε sh (t ) + ε T (t ) = ε σ (t ) + ε 0 (t ) ,
(3-21)
εel is the elastic strain, εcr the creep strain, εsh the shrinkage strain and εT the thermal dilatation. The elastic and the creep strains are stress produced strains εσ, whereas ε0 is the stressindependent inelastic strain consisting of εsh and εT. The dependence of creep on stress can be
Chapter 3: General description of a Construction Stage Analysis
55
shown graphically by so-called creep isochrones. These are lines that connect the values of strain produced by various constant stresses during the same time. ε (t)
C(t,t0)
J(t,t0) curve
1/E(t0) time
Figure 3-16: Creep isochrones [2]
Figure 3-17: Definition of the Creep-Function J and the Specific Creep C
Figure 3-16 indicates that the creep is approximately linear for stress below a certain range. Within this limits, which are within the usual service range, the creep curves can be described as:
ε σ (t ) = ε el (t ) + ε cr (t ) = σ * J (t , t 0 )
(3-22)
where J(t,t0) represents the total strain under a unit stress and is defined as Creep-Function (also called Compliance Function). For different time steps with ∆t → 0, Equation 3-22 follows to t
ε σ (t ) = ∫ J (t , t 0 ) * dσ (t 0 )
(3-23)
t0
A typical shape of the function is sketched in Figure 3-17, which can also be expressed as:
J (t , t 0 ) =
1 + C (t , t 0 ) , E (t 0 )
(3-24)
where E(t0) represents the modulus of elasticity at the time of the load application, and C(t,t0) the resulting creep deformation at the age t, which is referred to as the specific creep (or the creep compliance). The creep function can be also expressed in terms of a ratio relative to the elastic deformation.
J (t , t 0 ) =
1 + φ (t , t 0 ) , E (t 0 )
(3-25)
Chapter 3: General description of a Construction Stage Analysis
56
where φ (t , t 0 ) is defined as the creep coefficient, representing the ratio of creep to the elastic deformation. Analytical solutions for Equation 3-23 may be found by using Laplace-Transformation. In this case, the initial value problem is transformed into a usual linear elastic problem. However, this method requires the storing of the complete history of stresses and strains for each time step at all integration points. For a more efficient numerical expression, the total creep, starting from a particular time t0 and lasting till a final time t, can be expressed as an integration of creep due to the stress resulting from each stage by transforming ε cr (t ) to Equation 3-26: t
ε cr (t ) = ∫ C (t 0 , t − t 0 ) t0
∂σ (t 0 ) dt 0 ∂t 0
(3-26)
Assuming that the stress at each stage is constant, the total creep strain can be simplified as a function of the sum of the strain at each stage as the following: n −1
ε cr ,n = ∑ ∆σ j C (t j , t n − j )
(3-27)
j =1
Using the above expression, the incremental creep strain ∆εcr,n between the stages tn and tn-1 can be expressed as n −1
n−2
j =1
j =1
∆ε cr , n = ε cr , n − ε cr , n −1 = ∑ ∆σ j C (t j , t n − j ) −∑ ∆σ j C (t j , t n − j )
(3-28)
If the specific creep C in Equation 3-28 is expressed in degenerate kernel, the incremental creep strain can be calculated without saving the entire stress time history [2]. By this method, the integral-type creep law is converted to a rate-type creep law. Among others, Carol substitutes the specific creep by a sum of negative exponential functions (Dirichlet functional summation) as: m
[
]
C (t 0 , t − t 0 ) = ∑ a i (t 0 ) * 1 − e −( t −t0 ) / Γi , i =1
(3-29)
where ai is the coefficient related to the initial shape of the specific creep curve at the loading application time t0. Γi represents values related to the shapes of the creep curves over a period of
Chapter 3: General description of a Construction Stage Analysis
57
time. With this method, the incremental strain for each element at each stage can be obtained from the resulting stress from the immediately preceding stage and the modified stress accumulated to the previous stage. The employed programme MiDAS uses a similar approach in order to compute the creep strain, which is explained in detail in the analysis references [60]. The method is greatly affected by the analysed time interval. In the general cases, time intervals of construction stages are relatively short and thus, do not present any problems. However, if a long construction stage interval is specified in MiDAS, it may be necessary to define internal sub-times. It has already been mentioned that the relationship between the current elastic strain and the plastic strain due to creep is basically linear. Nevertheless, considering creep and shrinkage makes the calculation non-linear since the change of the internal loads changes the creep and shrinkage effects as well. To complete the list of possible methods for finding a solution for Equation 3-23, it should be mentioned that the integral-creep low can also be converted into a system of differential equations. It is then possible to transform the system into a linear form. The differential equations may be interpreted in terms of rheologic models consistent of springs and dampers, for example the Kelvin- and Maxwell-Model. However, not every J(t,t0)-function can be represented. According to Bažant [2], in case of aging, the equivalence of various spring and damper rheologic models and their equivalence to a general linear integral-type creep law is not true Pircher [42] presents a finite-difference-scheme in the time domain to predict time dependent internal forces and displacement due to creep and shrinkage. The derivation of the creep effects is founded on ε el * φ = ε cr , which has already been given in Equation 3-16. It is shown through a series of mathematical equations, that the effect of creep can be treated in a linear manner. The essence of the method is that all the creep influences on the final distribution of internal forces and displacements are related in a linear way to the elastic strain. The latter itself initially causes the creep. Therefore the creep effects can be considered by multiplying the elasticity matrix [Kel] by a factor
1 to get the solution for a single 1 + φ * 0.5
time step by using exactly the same finite element formulation as used for usual static analysis. The factor 0.5 corresponds to a time stepping strategy (0.5 for Crank-Nicolson 0).
Chapter 3: General description of a Construction Stage Analysis
58
The principals of the linear superposition can be applied and the total creep which occurs during a single time step can be formulated as below, considering for example one of the prescribed ideal moment positions as shown in Figure 3-2 given in Equation 3-14.
M cr = M P + M ct =1 * x1 + M ct = 2 * x 2 + M ct =3 * x3 + ...
(3-30)
Mcr therefore consists of one part, which is related to the permanent load, and the other parts are related to the unit loads as described above, which are linearly coupled. In Chapter 3.1 the Unit Load concept has been introduced. The same basic concept can be applied here. The effect of creep for the permanent loads and for unit loads are decomposed into separate contributions from each time interval and then summed up. Based on the example given in Chapter 3.1, the system of equations for defining x1 to x9 remains linear.
3.5.1.4
Verification example for creep
As already mentioned, MiDAS offers the option to consider creep and shrinkage in construction stage analyses. This chapter gives a short introduction into the application of these functions in the calculation and a simple verification analysis is also presented. The creep and shrinkage parameters must be defined by using the Time Dependent Material Function or one of the already prepared creep & shrinkage models. In the first case, the user directly defines the time dependent material properties of concrete. In the second case, an implemented design code can be chosen. MiDAS provides the following codes: •
CEB-FIP 1990 (adopted by the Comité Européen du Béton)
•
ACI 209 Code (American Concrete Institute)
•
PCA
•
Combined ACI & PCA
Creep can also be considered by modifying the modulus of elasticity. But this only provides rather approximate calculations. Different parameters must be specified for the models, describing environmental data, material properties and/or cross section properties. The CEB-FIP model is widely used. Many new design codes are based on these rules. For example, the Eurocode specification and the DIN code follow the same concept and only a few details are different. The CEB-FIP document gives a detailed description of the formulas used [4].
Chapter 3: General description of a Construction Stage Analysis
59
When the time dependent material properties are defined, they must be related to the material data by using Time Dependent Material Link. In the construction stage analysis, a specific duration for each construction stage can be defined, representing the actual time of the erection of the current stage. Within a construction stage, in which the model and boundary condition remain unchanged, alterations in load application timing or in additional loads can be included through additional steps. Considering the behaviour of creep, additional steps within one construction stage may be created in a logarithmic scale by defining the step number and using the Auto Generation option. Then the extra steps are automatically generated. The user can also assign the number of days for each additional step so that the load changes can be revealed within a defined period. To reflect the effects of creep and shrinkage on the elements, the age of activation must be defined in the Construction Stage Analysis Data. The age of an element group represents the time that elapses between the concrete casting prior the start of the current construction stage being defined. The age of an element characterises the time span between the concrete casting and the removal of falsework/formwork (the elements are loaded) for horizontal members such as segments. The following figure presents the structural system, which is analysed in the static verification example.
P = 100tonf
L = 10m Figure 3-18: Verification model for creep & shrinkage
Data Area A Stiffens I Poisson Ratio υ Modulus of Elasticity E
Value 1.00 m2 0.08333 m4 0.18 3.63*106 tonf/m²
Table 3-4: Input data verification example creep & shrinkage
60
Chapter 3: General description of a Construction Stage Analysis
Creep and shrinkage are modelled with the following input data: Creep and shrinkage data Code Compressive strength of concrete at the age of 28 days Relative Humidity Notational size of member Age of concrete at the beginning of shrinkage Age of concrete at the beginning of loading
CEB-FIP 4000 tonf/m² 80% 0.4 3 days 7 days
Table 3-5: Creep and shrinkage data verification example
Figure 3-19 shows the time dependent material function for creep (a) and shrinkage (b) calculated by MiDAS in accordance with the defined input data.
Figure 3-19: a) Creep coefficient b) Shrinkage strain
The cantilever is modelled with one element which is activated in the first construction stage. The first loading P1 is applied at an age of 7 days. After 60 days, a second load P2 is activated in the second construction stage and in the Construction Stage 3, a third load P3 is applied after 180 days. The total calculation simulates 460 days. Each load Pi is 100 tonf. The elastic shortening of the cantilever due to one load is ∆Le = 0.2755 mm. In order to control the creep calculation, a simplified hand calculation is performed. The strain is calculated as ε = ε el + ε cr = (1 + φ (t , t 0 )) * ε el , - the shrinkage is neglected. The creep coefficients are obtained by using the time function graphs determined by Ghali [7]. In a second calculation, the formulas from the CEB-FIP Model Code 1990 are used to calculate the total creep. The formulas and the performed calculation are given in the appendix. The calculated axial creep deformation for each construction stage can be found in Table 3-6. The last two columns show the total deformation by elastic and creep effects.
Chapter 3: General description of a Construction Stage Analysis
CS
Day
φ1
1 2 3 4
7 60 180 460
0.75 0.90 1.20
Simplified calculation φ 2 φ 3 ∆LcrP1 ∆LcrP2 [mm] [mm] 0.2066 0.75 0.2480 0.2066 0.85 0.70 0.3306 0.2342
61 Results ∆Lges [mm]
∆LcrP3 [mm]
0.1929
Simplified
Theoretical
Midas
0.276 0.758 1.281 1.584
0.276 0.764 1.301 1.620
0.276 0.768 1.301 1.603
Table 3-6: Result table static verification example for creep
Considering that for the simplified calculation the creep coefficients have been taken from a graph without detailed values, the discrepancies between the values are acceptable. The theoretical values obtained from the code formulas show more exact result. For the creep value after 460 day, the difference between the theoretical and the MiDAS value is 1.05%, which is tolerable.
3.5.2 Non-linearity effects The geometric non-linear effects of cable-stayed bridges include three different parts, which can be included in an analysis: •
Sag effects of long cables
•
P-Delta effects or initial stress (2nd Order Theory)
•
Large deformations (3rd Order Theory)
Since the cable tensions are determined to minimize the bending moments in the deck and to set off only centred forces, the permanent loads produce almost only normal forces. In modern cable-stayed bridges with a limited bending moment in the deck, the deflections are extremely small, and therefore, second-order effects are limited. Even if the deck has a desired camber, the cable tensions are adjusted so that normal forces follow the deck line and second order forces are not produced [56]. Thus, Virlogeux states that the distribution of permanent loads can be evaluated by a first order computation because of the desired geometry and the distribution of forces, and the structure must only be checked by a final control with a second order computation for live loads. However, this is not the case for all construction situations. With a mobile derrick lifting a new segment, the erection loads may produce important negative bending moments.
Chapter 3: General description of a Construction Stage Analysis
62
If the last erected cables then suffer high tensile stresses, the cables behind support low tensile stresses owing to the bending forces and the possible upward movement of the deck. The apparent modulus of elasticity of cables can change considerably in such situations and it is necessary to consider the non-linearity of cables. In general, for longer cables, the sag effect increases and in analysis models, it is practice to use the effective Young’s modulus [22] or to compute the cable stiffness by an iteration process according to the cable stress in any particular stage. As the main span length increases, the geometric non-linear effects can no longer be ignored. The initial stress effects (2nd Order or P-Delta effects) should be taken into consideration, not only for the construction stage analysis, but also for the final stage, which might influence the ideal state. According to Roik [12], the deformations in cable-stayed bridges are usually small compared to the overall structural dimension. Therefore, the 2nd Order Theory may be sufficient to analyse the main structural system. For the analysis of spans over 600 meters, large deformations should be taken into account [65]. Based on the large deformation but small strain assumption, the global equilibrium equation can be established together with the consideration of the initial stress effects. In addition to the linear stiffness, the tangential stiffness of a geometric non-linear structure includes the geometric stiffness and the large deformation stiffness. This is explained in more detail in Chapter 3.5.2.2. In general, there is no rule or specific suggestion when or under which circumstances a nonlinear calculation should be made. The ratio of the cable element length and the girder stiffness often indicates a tendency. As mentioned before, smaller structures can be understood to be structures with relatively high girder stiffness, where effects such as P-Delta, the cable sagging etc. do not have a significant effect on the structural behaviour. The following table is suggested by TDV. The influence of each function on a large cable-stayed structure is shown as a percentage of the results derived without consideration of the function. Classification Small structure Final system Small structure Construction Stage Large structure Final system Large structure Construction Stage
P-Delta
Non-linear cable
Large displacement
X X
X
X 10-20%
X 10-20%
X
X
Table 3-7: Structural classification and calculation procedure [45]
X 10%
Chapter 3: General description of a Construction Stage Analysis
63
Each cable-stayed bridge is unique and it is possible to consider all or none of the special effects for each structure. The decision on whether or not to consider non-linear effects and that on the appropriate choice of the various combinations for the needs will remain the engineers’ responsibility. Therefore, at least the final results should be checked by a non-linear analysis. In MiDAS, a large displacement analysis can be applied to both the general static and the construction stage analysis. The consideration of P-Delta effects alone, which may be sufficient, is not possible for construction stage calculations. This function is only available for general static and dynamic analyses. Finally, stability becomes a critical factor for building a long-span cable-stayed bridge. Both the lateral and vertical stability analyses in the maximum cantilever stage should be conducted. As for the deck, a stability analysis should also be performed for the pylon. After the determination of the initial stresses, the critical load can be obtained from solving the eigenvalue problem. However, the value only gives the upper limit of the stability since a perfect stability problem rarely occurs in actual engineering situations.
3.5.2.1
Cable elements
The axial stiffness of cables depends on two factors, namely the sag of the cable and the deformation of the cable steel. The sag has a softening effect on the cable stiffness so that this results in a non-linear force-displacement relationship. For large values of sag, the cable has a relatively low stiffness. As the sag decreases, the cable stiffness increases and the behaviour of the cable comes close to a truss bar tension element. The basic formulation of the static behaviour of cables is formulated by Peterson [11]. An extensive study on cables and cable systems is also provided by Gimsing [8]. Further details of the behaviour of cable elements in cable-stay bridges may be also found in the recommendations on cable stays [13]. Here, a detailed overview of static and dynamic analysis is given. In structural application it is convenient to link the cable force to the elongation from a given condition, e.g. its dead load condition, rather than to the total cable length. Figure 3-20 illustrates the force deflection curve of a straight cable with its origin at its dead load condition.
Chapter 3: General description of a Construction Stage Analysis
64
Figure 3-20 Force/deflection curve
For stay-cables, it is usually assumed that the cable has a parabolic profile. The parable-based solution is easier to use than the more exact catenary-based solution. This assumption is valid if the sag to span ratio of the cable is of the order of about 1:12 or less. It is further assumed that the sag and the mechanical strain effect can be uncoupled. Considering a straight cable of a uniform cross section A, which is subject to an end displacement δ in the direction of its chord, the new end force σ 2*A is then given by an implicit equation,
δ c
=
(σ 1 − σ 2 ) γ 2 c 2 + E 24
1 2 − 12 σ 1 σ2
(3-31)
where c is the chord length, E the modulus of elasticity, γ the weight per unit volume and
σ 1*A the element axial force in the initial condition.
Figure 3-21 Horizontal and inclined stay cable [11]
The deformational characteristics of the inclined cable are the same as those of a horizontal cable with the same chord length c, but subjected to a vertical dead load g*cos α . By substituting γ by γ *cos α and c by l/ cos α , Equation 3-31 becomes:
δ c
=
(σ 1 − σ 2 ) γ 2 l 2 + E 24
1 2 − 12 σ σ2 1
(3-32)
Chapter 3: General description of a Construction Stage Analysis
65
In this equation, the first term expresses the elastic elongation of the cable and the second the effect of the sag variations. The secant modulus of the cable can be given by the following equations:
1 1 γ 2 l 2 σ 1 + σ 2 = + 24 σ 1 2 * σ 2 2 Esec E
(3-33)
And, respectively, the tangent stiffness can be found by setting σ 1 = σ 2 as:
1 1 γ 2l 2 = + E tan E 12σ 13
(3-34)
The non-linear behaviour of the sagging cable will complicate the structural calculation if the used analysis programme is based on an elastic behaviour on each member. In such a case, the non-linear behaviour of the cable should be considered by specifying the tangent or secant modulus as given in the formulas above. The secant modulus, based on the parabolic instead of the correct catenary solution, gives a good approximation as the error remains below 1% for cables up to 300 m and below 2% for cable lengths up to 750 m. For the tangent modulus, only the cable stress in the initial condition has to be known, which makes it easier to use. On the other hand, the use of the tangent modulus may give rather erroneous results for large cable lengths and large traffic-to-dead load ratios [8]. In the analysis of the construction process of a cable-stay bridge, the high stress variations in the cables due to the derrick movements and the lifting operations of new segments will make it difficult to calculate a proper E-modulus. However, in modern analysis computer programmes, cable elements can be specified, and the cable stiffness can be calculated by an iterative process. Care must be taken not to get the false impression that a cable stay becomes stiffer as its stress ratio increases. In fact, the axial stiffness of a stay cable is governed by the product of Eeq*A rather than by the tangent modulus alone. As the cable area A increases with decreasing stress level, the product Eeq*A may very well increase despite the decrease of Eeq. In case of applying non-linear cable elements in a linear analysis MiDAS calculates an effective stiffness. In this programme, cable elements must be defined as Tension only/Hook/Cable elements. To consider the sag effect of cable elements in a linear analysis in this system, an equivalent truss element is used for the cable element. The stiffness of the equivalent truss element is composed of the elastic stiffness and the stiffness resulting from the sag.
Chapter 3: General description of a Construction Stage Analysis
66
In the MiADS analysis reference [60], the following formula is given for the calculation of the effective stiffness of a cable:
K eff =
1 EA = 1 / K sag + 1 / K el w 2 L2 EA L1 + 12 S 3
K sag =
12 S 3 EA , K el = 2 3 L w L
(3-35)
(3-36), (3-37)
where w is the weight density per unit length ( w = γ * A ) of the cable and T= σ *A represents the tension force in the cable. The parameter L is not defined. However, to be consistent with the formulas given before, L should be substituted by the length of the horizontally projection l in the term Ksag, and, respectively, by the chord length c in the term Kel. The analysis procedure for using the non-linear element is explained in the following five steps: 1) The stiffness of non-linear elements at the linear state are used to formulate the global stiffness matrix and the load vector. 2) The global stiffness matrix and the load vectors are used to perform a static analysis to obtain displacement and member forces. 3) The global stiffness matrix and the load vector is reformulated. 4) If the Method 1 is used, in which the analysis is performed without changing the stiffness term while varying the loading term, the non-linear stiffness is computed by using the displacements and member forces obtained in Step 2, which is then used to reformulate the loading term. If the Method 2 is used, where the analysis is performed with changing the stiffness term, the stiffness of non-linear elements is computed first by using the resulting displacement and member forces, which is then used to determine the global stiffness matrix. 5) Step 2 and 3 are repeated until the convergence requirements are satisfied. In this procedure the element stiffness changes with changing displacements and member forces due to applied loadings. Therefore, a linear superposition of the results from individual loading cases is prohibited. In case of a geometric non-linear analysis in MiDAS, a cable element is automatically transformed into an elastic catenary cable element. This considers the tangential stiffness which
Chapter 3: General description of a Construction Stage Analysis
67
is briefly explained below. The figure shows a two dimensional cable element in its initial and deformed state.
i
S01
S02
∆1, ∆2
S2
S03
j
i’ S1
S04
x ∆3, ∆4
S4
j’ S3
y Figure 3-22 Deformed and uniformed cable element
The equilibriums of the nodal forces and the displacement can be expressed as: S1 = -S3
lx = lx0 – ∆1 + ∆3 = f(S1, S2)
S2 = - S4 -∆S
ly = ly0 – ∆2 + ∆4 = g(S1, S2)
The differential equation for each directional length is given below. By rearranging the loaddisplacement relation, a flexibility matrix F can be obtained. The tangential stiffness K is then obtained by inverting the flexibility matrix.
dl x =
∂f ∂f dS1 + dS 2 ∂S1 ∂S 2
(3-38)
dl y =
∂g ∂g dS1 + dS 2 ∂S1 ∂S 2
(3-39)
In matrix form, the above given equations can be transformed into
dl x dS1 dl = [F ]* , dS y 2
(3-40)
By considering K=F-1, this can be written as
dl x dS1 dS = [K ]* dl 2 y
(3-41)
Since the stiffness of the cable cannot be obtained immediately, an iterative analysis is carried out until it reaches an equilibrium state.
Chapter 3: General description of a Construction Stage Analysis
68
3.5.2.2
Large displacement analysis (3rd Order Theory)
For long cable-stayed bridges, it may be required to consider large deformations in the analysis of construction stages as well as in that for the final state [65]. In order to perform a construction stage analysis with MiDAS, it is not possible to consider only 2nd Order effects. To include geometrical non-linearities, a large deformation analysis must be performed. For that reason, the theory of the method is explained in more detail. The equilibrium condition between internal and external forces has to be satisfied as:
R(u ) = ∫ B T σ * dV − f = 0
(3-42)
where R is the sum of internal and external generalized forces, u the displacement, f the external forces acting on the structure, σ the stress. B defines the relationship between the strain and the nodal displacements as:
ε = Bu
(3-43)
The bar suffix represents the relationship that is no longer linear since the large displacement is considered. The strain is now depending on the displacement as:
B = B0 + BL (u )
(3-44)
The assumption that the strains are reasonable small follows to the general elastic relation:
σ = D(ε − ε 0 ) + σ 0 ,
(3-45)
where D is the elastic matrix representing the relationship between the stress and the strain, ε0 the initial strain and σ0 the initial stress. The Newton-Raphson method is a widely used method to calculate the displacement u in equilibrium with the given external load f, as shown in Figure 3-23. The relation between du and dR is:
dR = ( K 0 + K σ + K L )du = K T du
(3-46)
in which KT is the tangential stiffness matrix. K0 represents the usual small displacement
∫
stiffness matrix K 0 = B0T DB0 dV , Kσ the initial stress stiffness or geometric matrix, which V
Chapter 3: General description of a Construction Stage Analysis
69
can be written as K σ du = dB LT σdV , and KL the large displacement matrix given by
∫
V
K L = ∫ ( B0T DB L + B LT DB L + B LT DB0 )dV .
(3-47)
V
The procedure of the Newton-Raphson iteration method to solve this non-linear problem comprises the following steps: 1)
Obtain the first approximation of the displacement as u0, based on the elastic linear solution.
2)
R0, the unbalance force, is found using Equation 3-42 with appropriate definition of B and stress as given by Equation 3-45.
3)
K T 0 is established
4)
the correction of the displacement is calculated by solving the equation
K T 0 ∆u 0 = R0 Step 2 to 4 is repeated until Rm becomes sufficiently small.
Figure 3-23 Newton-Raphson Method [65]
Chapter 3: General description of a Construction Stage Analysis
70
3.5.2.3
Verification example for non-linear analysis using MiDAS
In order to consider non-linear behaviour in the construction stage analysis, a general static example is first analysed using MiDAS. Figure 3-24 shows a fixed beam, which is loaded with a horizontal and a vertical force. MiDAS provides the function to consider the P-Delta effect or to perform a non-linear analysis. V = 50kN H = 100kN h = 10m
Figure 3-24 Verification example of non-linear analysis
Analysis Method Elastic
P-Delta
Non-linear analysis
Load Definition Two load cases: 1) H 2) V One load case: 1) H Two load cases: 1) H 2) V One load case: 1) H; V Load sequence not defined Load sequence: 1) V 2) H Load sequence: 1) H 2) V
M1 [kNm]
M2 [kNm]
LC [kNm]
Displacement ∆x [cm]
-1000
0
-1000
49.42
-1000
-/-
-1000
49.42
-1025.46
0
-1025.46
50.93
-1025.46
-/-
-1025.46
50.93
-998.56
0
-998.56
49.32
0
-1023.86
-1023.86
50.80
-998.56
-1023.86
-2022.42
50.80
Table 3-8: Verification table non-linear analysis
Different calculations are preformed, as it can be seen in the table above. In the first calculation, only elastic behaviour is considered. The moment at the support is equal to the horizontal force times the length.
Chapter 3: General description of a Construction Stage Analysis
71
Considering the P-Delta effect, the increase of the moment is about 25kN, which is approximately the vertical force times the deflection of 0.50m. Compared to the elastic calculation, not only the moment, but also the value of displacement is higher as a result of the 2nd Order deformation. The load cases, which should be considered in the analysis, must be defined in the P-Delta Analysis Control. The vertical and horizontal loads are stored in one and in another analysis, in two different load cases to control the calculation. Both results are the same, which means that the programme automatically considers the effect of the previous load cases. Performing a non-linear analysis, it is important to define the load sequence at the Non-linear Analysis Data. The order of the load cases is not important; it only influences the output data. As it can be seen in the table, applying first the vertical load and afterward the horizontal load, the moment M1 is zero, but the additional stresses from the vertical load are considered in the moment M2. If the horizontal load is defined first and then the vertical load, M1 is the moment value due to the first load. M2 gives the total moment value including the increase because of the additional normal stresses in the beam. The values obtained from the non-linear analysis are slightly smaller than the P-Delta values, which may be related to the iteration process. The load case LC is always the sum of both.
3.5.2.4
Further comments on the non-linear construction stage analysis in MiDAS
In Chapter 3.5.2.2 it has already been mentioned that, in the analysis of construction stages in MiDAS, it is not possible to include P-delta effects alone, even if this may be sufficient for the analysis. To consider geometrical non-linear effect in the analysis, in the Construction Stage Analysis Data the function Include Non-linear Analysis must be activated. With the definition of the construction stages, the load sequence is defined as well and it is not required to define it in the Non-linear Analysis Data, as it is the case for usual static analysis. The number of load steps, the maximum numbers of iteration per step and the convergence criteria are also defined in the Construction Stage Analysis Data. In a non-linear analysis, the cable effects can be considered as well and it is possible to define a pretension value or a length to unstrained length ratio. In a linear analysis these data is not considered. For the cable elements the tangential stiffness is calculated as it was explained before.
72
Chapter 3: General description of a Construction Stage Analysis
When performing a non-linear analysis, it is not possible to include any time dependent effects. Creep and shrinkage must be analysis in a separate calculation. As there are a number of possible combinations for considering non-linear effects, the user must beware of randomly applying all combinations together. It can be easier to first study the effect of the cables individually and combine it later with other geometric non-linearity. The calculation time increases analogue to each additional non-linear effect considered. However, at the time of working out this document and performing a construction stage analysis for the Second Jindo Bridge, the offered MiDAS functions did not work. The options were tested by a simple model, but the calculated results were not reasonable and therefore, it was not possible to illustrate these functions in more detail and to study the effect on the different structural systems during the construction process. Unfortunately the MiDAS support could not help in these problems immediately. The construction stage analysis functions are new features in the MiDAS programme and not all functions are working properly until now. The support stated that they are trying to solve the related problems and they might be able to present a version providing these functions in a couple of month. Some parts of the correspondence with Mr. Lee from MiDAS are given in the appendix.
3.5.2.5
Buckling analysis of the main girder
The buckling failure of a cable-stayed bridge girder is the result of the interaction between axial and bending forces. In the final state of construction, it typically occurs either at the point of maximum bending moment, i.e. the third point of the side span, or close to the point of maximum axial load, i.e. the tower. In the first cable-stayed bridges, the cables were widely spaced along the girder and a stiff girder used to span in between. Due to small spans and relatively large girders, buckling was rarely a governing design criterion. When later cable-stayed bridges with continuously spaced cables were introduced, the approximate buckling analysis was based on a beam on an elastic foundation. At that time, cable-stayed spans were in the 300 m range. Tang presents a paper containing analyses of a number of realistic cable-stayed bridge configurations. For buckling, all of these bridges
Chapter 3: General description of a Construction Stage Analysis
73
showed load factors between 4.4 and 6 times the dead load. The accuracy of this method depends upon the proximity of the assumed buckled shape to the actual buckling mode. Due to the reduced bending demands in the cable-stayed bridge with closely spaced cables, the size of the deck girder was reduced. Walter [14] introduced cable-stayed bridges with simple concrete slabs. For a bridge with a main span length of 200 m and thickness of the deck of 50 cm, a model test was carried out to confirm the calculated buckling strength. It could be experimentally proved that, for cable-stayed bridges with thin concrete decks and continuously spaced cables, no particular buckling instability exists. The main dimensions of the girder depend on the longitudinal and transversal bending moments as well as on normal forces induced at the cable anchorage points. Usually, the maximum normal force is close to the pylon, and it can be necessary to increase the girder thickness in this area. In the design of long cable-stays bridges nowadays, there are a number of trends which are tending to reduce the load factor against the girder buckling. Longer spans cause an increase in both, the girder axial load and the girder lateral bending moments under wind loads. As mentioned before, reduced girder depths and the use of slab decks reduce the buckling stiffness of the system. Due to the difficulty of anchoring steep cables in the tower, sometimes cable supports in the region of the tower are omitted, which has a negative effect on the buckling stiffnes. Furthermore, the buckling load of the system is reduced because of design criteria specifying a hypothetical one or two cable out situation. During the construction, the cable-stayed bridge may experience a lack of buckling stiffness due to some missing boundary restrains afforded to the permanent bridge. Geometrical construction errors in the deck and a large variation of cable tension during the cable tuning can also produce critical conditions. Taylor [51] examined the effects of some of these factors by a series of parameter studies. It is concluded that, at the current upper limit of cable-stayed bridge, design buckling failures of the girder must be a significant design consideration. The cable distribution on the Tatara Bridge is interrupted over a region on each side of the tower. It was calculated that, with the first two cables in the main span removed, the buckling load of the Tatara Bridge drops from 3.9 times the dead load to 1.9. For the Evripos Bridge in Greece with a main span of 215m and a slab thickness of 0.45m, an elastic buckling load of 2.4 times the dead load was calculated. When inelastic behaviour in the slab is taken into account, the buckling factor is only about 1.5. Therefore, it is not sufficient to analyse the elastic buckling capacity; material non-linearity should be considered. The calculation of the buckling load
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during the construction was not performed. But it can be concluded that the mentioned effects influence the structural performance and should be examined for the construction stages, too.
3.5.3 Temperature Thermal effects are important for the geometrical control of all types of bridges. When the sun heats the deck, thermal gradients produce a downward camber of the cantilever. However, there are also other thermal effects producing deflections during the construction in a cable-stayed bridge, and not only thermal gradient. In case of a uniform variation of the structure temperature, the displacement is limited to small deflections and bending moments when the fixed point of the cantilever is at the pylon. Even with intermediate supports in the side spans, this is also be the case if there is no friction on bearings, Figure 3-25 a). The situation is different when bearings are fixed on intermediate supports. Then, the deck moves towards the main span when temperature increases, which is illustrated in Figure 3-25 b).
Figure 3-25: Uniform variation of temperature, a) without supports or with supports but no friction on bearings, b) with supports and fixed bearings [56]
Due to the different thermal inertia, steel members in the deck heat quicker than concrete ones. The average temperature in a steel member can be 5° C higher as a in concrete member. The difference in temperature between concrete and steel members produces deflections and forces which must be considered during the construction. Figure 3-26 shows the deformations produced by temperature variation in the deck.
Figure 3-26: Deflection produced by: a) temperature variation in the deck, b) temperature variation only in the side span [56]
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75
Furthermore, the difference in temperature between cables and structure can be responsible for important deflections and bending moments, illustrated in Figure 3-27.
Figure 3-27: Deflections produced by an increase of temperature a) in symmetrical cable-stayed cantilevers b) with side span on supports [56]
Even if measurements are made early in the morning, geometry control cannot be conducted from the absolute geometry of the cantilever. Each new segment must be geometrically characterized by reference to the previous one, in relative geometry, to ensure a correct installation. In the case of the Pont de Normandie [16], 8 thermocouples were installed in different parts of the segments and 4 thermocouples were inserted in the strands of a reference cable for the monitoring during the construction process. Furthermore, 8 thermocouples are distributed over the surface of the full height of the pylons. Regular measurements were taken early in the morning to determine the average temperature and the temperature gradient of the deck and the pylon. In practice, it was found that the appearance of temperature gradients in the pylon is a too complex phenomenon and appraising these gradients could not be envisaged. Due to an excessive scatter of the measurements only a vague correlation with the pylon movements could be established posterior. From the structural point of view, these effects have no significant influence during the construction period when the structure is still a cantilever, but should be considered before the joints are closed to avoid that the temperature gradient is “locked into” the pylon.On the other hand, the average temperature measurements in the deck and in the cables were far more interpretable and were used in the monitoring and adjustment procedures.
3.6 Modelling and tuning of cables As already mentioned before, stay adjustment is a major topic in cable-stay bridges. This issue directly controls the stress distribution in the structure as well as the final geometry. It concerns both, analyses during detailed design and tensioning procedures during the erection on site. The sequence of tensioning operations must fulfil several targets:
Chapter 3: General description of a Construction Stage Analysis
76 •
Practicability of stay installation and construction simplicity. From this point of view, the ideal solution would be to stress each cable in a single step, at the time when it is installed. Stressing of the stay cables is an expensive procedure and could be minimized.
•
Structural design objectives. Acceptable stresses in the structure and the stay cables during the erection phases and after the completion must be maintained. Ideally, readjusting stays at different stages would make it possible to reduce bending stresses in the structure almost to zero as erection proceeds.
In many cases, the stays are tensioned in two steps: Firstly at installation, then just prior to finishing. In some projects, a global re-tensioning operation is required after a few years of bridge operation in order to compensate for creep effects, as in case of the Brotonne Bridge in France. Seven years after the completion of the bridge, the cable-stays had to be re-adjusted to compensate the effect of creep [20]. As described in detail in Chapter 3.1, the search for an adjustment solution usually starts from the final erection stage. Once the set of tensions at bridge completion has been selected, the tensioning sequence must be defined. Using stage by stage structural software as described before, the bridge construction is simulated. The installation of a cable-stay is idealised by activating a new bar or cable finite element. For adjusting the stay at this stage (initial tension at the time of erection), there are different options for imposing an appropriate initial condition to the related element. The initial condition may be expressed by using the value S of the tension (external or internal loading), that must exist in the element after assembly. However, the use of the stressing force S as a parameter suffers a severe drawback: The stressing force is not an intrinsic characterization of the cable-stay pre-loading. Stressing a stay with the same force S in presence or in absence of temporary loads, such as a mobile crane, does not yield the same tension in the stay at the bridge completion. For example, any modification in carriage weight during the design requires resuming the computation of all the stressing forces to reach the same state at the end of construction. This is a very time consuming and exhausting process. As another option, stay adjustment can be achieved by assembling a cable element with a proper initial length L0. In the literature, L0 is referred as the cable unstressed length (or cable neutral length). The unstressed length of a cable stay, which can be seen as the length of the cable when it is laid on a flat support, is an intrinsic parameter. Because the cable length L0 has a valuable meaning only when it is compared with the distance between the anchorage nodes in the model theoretical geometry, the dimensionless parameter ε is often preferred: ε =
∆L L − L0 = . In L L
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77
most cable-stayed bridges, the value of this parameter relating to a completed bridge under permanent loading ranges between 2.5*10-3 and 3.5*10-3[13]. If the mobile carriage weight changes during the design, these values, that describe the adopted stay adjustment, are not impacted. Of course, it must be checked if the new weight values still produce allowable stresses.
Figure 3-28: Stay adjustment definition [37]
If the bridge is directly built with the final length of the cables, their tensions will not balance the structural self-weight during construction. The lack of balance will be much greater for a steel bridge than for a concrete one. This is illustrated in Table 3-9, which compares the selfweight of different bridge types with the construction loads. Bridge Type Concrete Composite Steel
Steel Concrete
Self-weight [t/m] 24 4 12 8
Equipment [t/m] 4
Total [t/m] 28
% of equipment 14 %
4
20
20 %
4
12
33 %
Table 3-9: Compared permanent loads for different bridge types, the bridges are about 20 metre wide
In case of a composite bridge, in which the concrete slab is built some segments behind the steel structure, it is impossible to place the cable on the steel structure with its final length because it is extremely light in this stage. In this situation, a first tension is given to the cables which are later adjusted with the increased load. Despite the specific problems already mentioned, it is possible to erect bridges by giving directly the cables their final length (or final tension). In this case, the cantilever is cambered upward, so that a corresponding moment is produced. In modern cable-stayed bridges with flexible decks, the girder can generally suffer the camber much better than bridges with stiffer decks. However, in addition to some operations which are needed for the closure to produce the
78
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necessary angular continuity, such a procedure will produce more creep effects than with cable forces adopted to the construction loads and a later adjustment. In case of erecting the cables according to their final loading, if intermediate supports in the side-span exist, it may turn out that the pylon bents backward because of a lack of load in the main span. Therefore, such a tensioning procedure cannot be adopted if the pylon is not able to resist these deformations. To avoid cable adjustments as much as possible, a practicable solution can be to adapt the cable lengths of the side span to the construction loads so that the pylon is left free of bending moments, and to install the cables in the main span with their final lengths. The resulting upward camber in the main span can be tempered by the weight of the mobile derrick used for the lifting operations. By reducing the length of the backstays, it is possible to produce a limited backward camber in the pylon in order to minimize deflections and bending moments while lifting a new segment.
Figure 3-29: Deflections produced by construction with final cable forces a) in case of symmetrical cable-stayed cantilever b) in case of bridge with intermediate supports [56]
A similar tensioning concept was used in the construction of the Normandy Bridge. Two analyses were carried out corresponding to the final situation of the long cantilever in the main span, just before the closure [55]. One of the analyses considered the final cable lengths which are selected to balance all permanent loads. The other one considers the cable lengths adapted to self-weight only. The calculation revealed that the pylon could not resist the bending moment in case of the first situation. Finally, the cable lengths of the backstays were adapted to the construction loads and the main span was built with the final cable length. In some bridge constructions, it is preferred to build the whole structure with cable lengths or pre deformations adapted to the construction loads (self-weight, erection equipment, mobile carriages or derrick). This may lead to better designs with limited creep effects and simpler closure procedures. However, the disadvantage is that all cable lengths have to be adjusted after the completion of the bridge after the final equipments are installed.
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3.7 Construction control and monitoring During the construction of a cable-stayed bridge, discrepancies may occur between the actual state and the state of design expectation. The main error factors can be divided into three categories as follows: •
•
•
Structural Analysis Errors: -
Error in material properties (Young’ modules, etc.)
-
Incorrect dead load estimation
-
Incorrect stiffness estimation (moment of inertia, section area, etc.)
-
Improper boundary condition
-
Numerical error in computation
Construction Errors: -
Defective installation (improper location)
-
Improper field joint by bolt
-
Improper field joint by welding
Fabrication Errors: -
Mistake in size (cable, girder, tower)
-
Defects in joint detail
On the other hand, a system error may occur in the measuring of the deflection in the girder or the pylon, because of defect load cells, etc. Since it is impossible to eliminate all errors, there are two basic requirements for the completed structure: •
The geometric profile must match the designed shape within a limited range
•
The internal forces, especially the bending moments in the girder and the pylon, must be within the designed envelope values
If the error of the girder elevation deviating from the design value is small, the error can be reduced by adjusting the elevation of the segment by inducing an extra angle between two adjacent deck segments. In this case, only the geometrical position of the girder is changed, producing a modification in the length of the cable without a change in the internal forces. If the error is not small, it is necessary to adjust the cable forces. In this case, the geometric position changes and changes occur in the internal forces in the structure. Both methods are illustrated in Chapter 4.7 by the example calculations.
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80
The limits of errors in the cable tension and the bridge configuration should be carefully determined. Too small limits can possibly result in a failure in satisfying the requirements, although smaller those are wanted for an accurate construction. Depending on the various conditions, the requirements for the cable adjustment change. In case that little margin rests in the cable tension, an adjustment to reduce the tension error is required. On the other hand, in the cantilever erection before the closure of the girder, the correction of girder configuration is the dominant target. Whatsoever, cable force adjustments are not preferable because it takes time and increases the construction cost. The general procedure at each stage is to find out the correct length of the cables and to appropriately set the elevation of the segment. In the ideal case, cable tensioning is performed for the new stay only, but as explained in the previous chapter, this is not always possible. A comprehensive adjustment is often applied before connecting the two cantilever ends. For example, during the construction of the SeoHae Bridge [40] in Korea with the main span of 470 m, several adjustments have been performed. In the early stage of the construction, the error in deck level has been adjusted because a small error in the beginning generates a larger error later on. In such a situation the superstructure is short and stiff, and a great change in cable forces is required to adjust the geometry. Because the level of the girder becomes very important in constructing the main span closure, another adjustment has been performed just before constructing the main span closure. In this stage, the error in the global geometry is easier to correct due to the flexible deck. However, the error in the local geometry could not be fully corrected.
3.7.1 Construction Control Systems To guarantee structural safety and to achieve the design aim, a computer system is installed at the erection site to monitor and control the structural performance. A construction controlling system consists of four subsystems, which can be divided into: •
Measuring system: The following data are measured at each erection cycle. -
deflection of the girder
-
cable forces
-
horizontal displacement of the pylon
-
stresses of sections in the girder and the pylon
Chapter 3: General description of a Construction Stage Analysis -
modulus of electricity and mass density of concrete
-
creep and shrinkage of concrete
-
temperature gradient in the structure
81
The measurements should be done at 6.00 – 7.00 am to eliminate thermal effects as much as possible. Furthermore, the measurements should be done at a step when forces are almost centred [56]. •
Error and sensitivity analysis system: The temperature effects are first determined and removed. Then the sensitivity of structural parameters, such as the self-weight, stiffness of the girder segments, etc., is analysed. Through the analysis, the causes of errors should be detected and explained. It is very important to understand the reasons for geometrical differences in order to select the adapted amendments.
•
Control/prediction system: The measurement values are compared to those of the design expectation. If the differences are lower than the prescribed limits, then the construction can be continued with the next stage. Otherwise the calculations mentioned above are used to eliminate or reduce those errors through proper methods.
•
New design system: Since the structural parameters for the already completed part have deviated from the designed values, the design expectations must be updated with the changes made to adjust the structure. The sequential construction now follows new design values to achieve the final state as intended.
3.7.2 Adjustment instruction on site The adjustment instructions supplied by the designer to the site engineer are often expressed in terms of stressing forces to be applied to the cables at their anchorage points at the given construction stage. Provided that the actual site conditions are very similar to those taken into account in the design, the theoretical force value can be used straightforward. As mentioned before, such an ideal situation is seldom met in practice. A current practice consists in determining the tensioning force value by updating the computational model with values measured on site shortly before the stressing operation. As many cable-stayed bridges designed nowadays have a very slender deck, and result in an extreme flexibility, a slight error ∆S in the tensioning force S produces a significant vertical
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82
displacement ∆w of the deck. In such a situation stays are adjusted by controlling the altitude of the deck, rather than the stressing forces. An alternative to monitor stressing forces or vertical displacements is to use the initial length L0 as introduced in Chapter 3.6. In this case, the cable stays must be prepared with their exact unstressed length L0. For the installation of the cables, they must be extended to attach their two ends to the structure. To base the initial stay adjustment operations on the unstressed length has the following advantages: •
Actual loading conditions, such as temperature or construction loads, do not influence the adjustment values.
•
The procedure is applicable for rigid and flexible decks
This method is often used for prefabricated stays, which are in one stage only. The same procedure can also be applied to stays erected by threading strands one by one, where each strand is cut at the precise length. Besides the accurate cable length, the anchorage location must be controlled with a precision of some millimetres. Otherwise, the unstressed cable length does not constitute a suitable adjustment parameter for practical purpose. For concrete decks cast-insitu, the anchorage position is generally set by the mobile carriage. Thus, the anchorages are located with a tolerance reaching several centimetres, so that this results in a poor quality using cable length L0. However, the unstressed length L0 provides a valuable parameter to crosscheck the adjustment data supplied by other methods.
3.7.3 Methods of cable-stay adjustment In practical terms, there are two methods for adjusting cable stay tensions, which are used for initial adjustments and re-adjustments of the structure during the construction. •
Displacement of the anchorage relative to the structure: The cable length is adjusted by inserting some plates (shim plates) in front of the cable anchor sockets. This is usually called “shim adjustment”. Another technical option is to hold the anchorage head in position by screwing a concentric nut in or out, or relocate nuts on threaded transfer rods securing the anchorage to the structure. For the adjustment, the tension of the cable stay must be temporarily removed from the bearing using a high-capacity jack, which takes the full force of the cable.
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83
Tensioning and adjusting by re-jacking: In general, anchorages using split-cone wedges are reversible, and therefore allow strands to be re-jacked. It must be ensured that a sufficient extra length is left in the anchorage head and that a displacement of the strands is not prevented by a rigid fill. This type of anchorage makes it possible to adjust the force and the length of the cable stay. In contrast to the adjustment method described above, the anchorage does not move relative to the structure. Wedges leave jaw marks on the strands, from which fatigue cracks can be initiated if they were left on the taut length of the strand. Therefore, the method of adjustment by re-jacking must be used only for retensioning and never for detensioning.
3.7.4 Control of deck geometry The position of each new segment is usually given with reference to the previous one, by the relative geometry. The absolute geometry is too sensitive to a great number of parameters to be a reliable base for a segmental construction. The positioning in relative terms makes it possible to obtain a more accurate profile even with various interfering effects. Modern designs tend to slender decks, which increase some problems due to the larger flexibility. Characterizing each new segment by its relative geometry with reference to the previous segment cannot be done with the same precautions in slender decks compared to classical box- girders. Local deformations between the anchorages of the cables must be considered. They can seriously alter the definition of the reference line used as a basis for the relative geometry of the new segment. Therefore, to reduce local uncertainties, the reference line should have a clear definition, for example the line between the two anchorage points of the previously installed segment. In case of a cast-in-situ deck, the positioning should not be based on the actual concrete surface. Due to construction tolerances, there are variations in the scale of centimetres. Consequently, an ideal theoretical profile line attached to the deck must be considered.
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Chapter 3: General description of a Construction Stage Analysis
Figure 3-30 Theoretical and actual deck profile [37]
The actual surface w can be calculated as:
w = z ref − z c − η ,
(3-48)
where η is the vertical offset between the theoretical profile line attached to the deck and a given benchmark, zc the divergence to the reference line and zref is the elevation of the reference line. When a new segment is welded to a previous one, in steel constructions shrinkage develops in the welds. This can induce deflection by producing a downward or an upward angle in the joints. These weld-shortenings should be measured to control the erection precisely.
3.7.5 Computational Systems As stated in earlier chapters, a proper tension adjustment is one of the most important aspects in the construction of cable-stayed bridges. Since cables are installed in turn in the cantilever erection, the adjustment can be made in two ways. Method 1: adjust all cables in the final stage of construction Method 2: adjust limited cables (usually cables installed in the current stage) in the arbitrary intermediate stages. Although Method 1 is apparently superior to Method 2 from the point of view of the final residual errors, Method 2 is frequently employed mainly because of the economical reasons [28]. The following calculation procedure can be applied for both methods.
Chapter 3: General description of a Construction Stage Analysis •
85
Trial and error method: When cable tensions are chosen as a control item, the influence matrix T, which describes the influence value of the cable tension due to each unit shim thickness, is given as follows:
t11 t T = 21 ... t n1
t12 ... ... t n2
... t1m ... ... , ... ... ... t nm
(3-49)
where tnm is the tension increment at cable n when a unit shim thickness is insert to the mth cable, n the total number of cables. If M is the shim thickness of each cable and ∆S is the effect on each cable, the following relation results:
T * M = ∆S
(3-50)
If the cable tension is different from the required value by ∆S, the required shim thickness H gives –∆S,
T * M = − ∆S .
(3-51)
When the camber is selected as a control item, the relations are the same as above. The influence matrix D, which expresses the influence value of displacement due to each unit shim thickness, is given as:
d11 d D = 21 ... d n1
d12 ... ... d n2
... d1m ... ... , ... ... ... d nm
(3-52)
where dnm is the displacement increment at section n when a unit shim thickness is inserted to the mth cable and n the total number of cables. As before, the following equation can be obtained:
D * M = ∆δ ,
(3-53)
where ∆δ is the effect on the deflection. If the deflection of the girder is different from the required value by ∆δ, the required shim thickness M gives – ∆δ,
D * M = − ∆δ .
(3-54)
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86
If there is a tension error ∆S or a deflection error ∆δ, the trial shim thickness can be calculated. In case of cable-stayed bridges with many stays, this method takes many trials and sometimes it is difficult to obtain the suitable shim thickness. •
Optimization method: To improve the trail and error method, it was proposed to determine the required shim thickness, which eliminates the tension error ∆S or camber error ∆δ, using Equation 3-55 or 3-56 [39].
M = −T −1 * ∆S
(3-55)
M = − D −1 * ∆δ
(3-56)
The shim thickness, which is required from this calculation, becomes too large and impractical because errors are eliminated which should not or cannot be eliminated. Therefore, shim thicknesses are calculated by the optimisation method taking an allowable error range ErS and Erδ for ∆S and ∆δ into consideration,
M = −T −1 * (∆S ± ErS )
(3-57)
M = − D −1 * (∆δ ± Erδ )
(3-58)
In order to reduce the error uniformly, Fujisawa [27] proposes the so-called least square method (LSM) to find an optimum adjustment. •
Last square optimisation method: Using the already given relation
T * S = ∆S ,
(3-59)
where T is the influence matrix of cable forces, S the cable adjustment vector in terms of cable forces, and ∆S the change in the cable forces. Since the target adjustment A, which is required to bring the existing cable forces S0 to a theoretical value SA, is not always obtainable in the actual project, there is always some difference between SA and S0. The remaining adjustment E can be treated as the error in the optimisation process, which is to be minimised. The target adjustment can be calculated from
A = S A − S0
(3-60)
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87
The optimisation error is determined as
E = A − ∆S = A − T * S
(3-61)
In the least square method, the remaining adjustment is minimized as following:
ω = Σe n 2 ω = Σ(a n − Σt nm s m )2
(
ω = Σ a n 2 − 2a n Σt nm s m + (Σt nm s m )2
)
(3-62)
When taking the partial differential of the object function ω with respect to the change in the cable force and let it be zero, it becomes
∂ω / ∂s k = Σ(− 2a n t nk + 2t nk Σt nm s m ) ∂ω / ∂s k = 2Σt nk Σt nm s m − 2Σt nk a n = 0
(3-63)
If this is expressed in the form of matrices, the Equation 3-63 follows to
T TT * S − T T A = 0
(3-64)
and solved for S to
S = (T T T ) −1 T T * A
(3-65)
This is the change in cable force to minimise the remaining adjustment E. If a weighting function W, which is an n by n diagonal matrix, is introduced, cables which may have more adjustment than others can be controlled [28].
S = (T T W T WT ) −1 T T W T W * A
(3-66)
This procedure is especially useful in the case when multiple cables are adjusted, e.g. major cable adjustment during the construction and the final cable adjustment. The method is an unconstrained optimisation. The object function does not have any restrain on the stresses. It just minimises the variation in the structural geometry without any constrains. The theoretical optimum cable forces may not be used because these forces can cause excessive stresses. Therefore, a limited adjustment error should still be allowed.
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88
The same procedure, as just described can be applied when the girder elevation is selected as a control item. In this case, the influence matrix must be adapted to the changed method. Example calculations are presented in Chapter 4.7. It should be mentioned again, that, when the error factors are due to cable length only, the error for both the tension and the camber can be eliminated simultaneously by the same shim thickness. However, when the errors for control items are caused by the various kinds of factors, for example a higher flexibility or a different weight of the girder, they cannot be eliminated at the same time. Tanaka presents a method of estimating cable tension adjustments through the application of system identification (SI) [49]. In this method, error factors are estimated quantitatively. This permits the prediction of the final construction state for the bridge, which is used for the cable tension adjustment, reducing the camber error or member force error. The procedure of cable adjustment using the SI method can be briefly outlined as: (1) Calculate deviations between the field measured values and the design aim (2) Apply SI method to quantify the error factors α (see error factor analysis below) (3) Predict the final stage by performing a forward analysis using revised input data (a model with errors) (4) If the deviation of the configuration or the member force is larger than the allowable tolerance, a cable tension adjustment is necessary. •
Error factor analysis: The error vector Z consists of camber errors and member errors, the deviations between filed measured values and the design values of member forces and displacements. Z is assumed to be a superposition of error modes, given in Equation 3-67. For example, the measured camber error mode may be caused by the sum of the increase in dead load of span (2) and the decrease in stiffness of span (1).
Chapter 3: General description of a Construction Stage Analysis
89
Figure 3-31: Example of camber error [33]
The error factor analysis determines what kind of error factors contribute quantitatively towards member forces or configuration errors. N
Z = ∑ α i * Fi ,
or in matrix form written as: Z = F * α
(3-67)
i =1
f 11 f F = 21 ... f n1
f 12 ...
... ...
...
...
f n2
...
f 1m α 1 α ... 2 and α = : ... f nm α n
(3-68), (3-69)
where Z is the error vector, F the error influence matrix, n the number of field measurement items, m the number of error factors and α the error contribution rate vector. Assuming Rf to be field measurements of member forces and displacements, and Rf0 to be corresponding calculated values obtained without an error model, R can then be defined as:
Rf ≈ Rf 0 + Z .
(3-70)
In reality, Equation 3-70 is an approximation and can be transformed into the following optimization problem:
φ = ( Rf 0 + Z − Rf ) 2 → Minimize .
(3-71)
After introducing Rf0-Rf = rf and differentiating Equation 3-71 with respect to α, it follows:
α = −[F t * F ] * F t * rf . −1
(3-72)
Chapter 3: General description of a Construction Stage Analysis
90
If a weighting matrix [ρ] is introduced to account for dimensional adjustments and field measurement uncertainties, then follows
α = −[F t * ρ * F ] * F t * ρ * rf −1
(3-73)
where ρ is a diagonal matrix. Tanaka compares the method of determining the optimum adjustment for each construction step with the SI method. The first is the case of making cable tension adjustments without predicting the final state. However, the adjustment in the SI method is based on the predicted final step and thus, it is not always an optimum solution for the current construction step. For the error in the final step, the comparison of both procedures demonstrates the advantage of the SI method. In practical application of the SI method, careful attention must be paid to determine the unknown error factors α. Satisfying estimations can be done by repeated application of the procedure to each construction step. Fujusawa [28] states that the objective of an economic cable adjustment is not to know the accurate structural error but to predict the response of the structure in future stages. Thus, even if the identified errors are wrong, any arbitrary error factors may be used as an equivalent error to the actual one for the prediction of the final state, if it can be assumed that the change in the structural characteristics up to the final stage does not significantly affect the response to the error identified in the current stage. However, if the choice of error factors or assumptions of those magnitudes do not fulfil this condition or if field measured data contain measurement errors, the estimation of the error factors becomes unreliable. In order to overcome these problems, a system identification method has been developed, in which measured field data are assumed to be fuzzy data and a fuzzy regression analysis is applied to the process of the system identification [33]. The Fuzzy System Identification method (FSI) can be solved by using a linear programming algorithm. A similar method has been formulated, also developed by the application of the fuzzy regression, to calculate the thickness of the shim plates, the Fuzzy Shim Adjustment Method (FSA) [32]. Because of limited shim plates prepared at site, Furuta [29] introducs the thickness of the shim plates as a discrete variable. The problem is formulated as an optimization problem with an objective function of reducing the number of cables to be adjusted in addition to the errors of cable tension and deflection.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
91
4 Example of a Cable-Stayed Bridge including temporary supports
The structural system introduced in Chapter 4.1 is now analysed in order to perform a construction stage analysis of a more complex symmetrical cable-stayed bridge including temporary supports in the side span. The tension forces are adjusted to achieve an idealised moment distribution. Different methods of using the Unknown Load Factor function are demonstrated in Chapters 4.2.1 and 4.2.2. Furthermore, the influence matrix is used to recalculate the ideal cable forces and to explain how cable adjustments may be calculated. Moreover, the construction stage analysis is presented and the basic camber calculations are described. During the construction, various errors may occur so that these must be included in the model to obtain reliable adjustment options. Different errors are assumed and it is demonstrated how these errors can be handled in modelling the structural system. Finally, the influence of cable elements is investigated in detail and the MiDAS` problems relating to a nonlinear analysis in combination with a construction stage analysis are commented.
4.1 Model data The following model is used for the example calculations presented in the subsequent chapters. A simple structure is chosen in order to clarify main considerations in modelling the construction stage analysis.
5 1 2
G1
3
G2
G3
4
G4
Figure 4-1: Structural system
G5
Key-G
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The total height of the pylon is 30m; the distance from the girder to the top 20m. The distances between the cables are 12m, 9*16m and 12m. Because of the symmetry of the bridge, it suffices to model only one side. The cable numbering starts from the right to the left, as indicated in the figure. The girder Segments are defined as G1 to G5, plus the segment in the centre Key-G. For the modelled construction stages, Segment G2 is defined from Node 4 to 9, Segment G3 from 9 to 12. The other segments have the length in-between two cables. Similar to the previous example, the following input data is used for the Case I model: Data Stiffness I Deck Stiffness I Pylon =3*I Deck Modulus of Elasticity Ecable /Edeck Modulus of Elasticity EPylon/EDeck Poisson Ratio υ Deck
Value 0.92 m4 2.76 m4 5.25 1.00 0.3
Data Area A Deck Area A Pylon Area A Cable 1 Area A Cable 2,3 Area A Cable4,5 Weight Cable ρCable
Value 4.38 m2 1.00 m2 0.0208 m2 0.0062 m2 2 * 0.0062 m2 (7.85 tonf/m³)
Table 4-1: Modified input data
A self-weight of the girder of 11ton/m is assumed and furthermore an additional load of 11 ton/m is applied. In this system, the cables are modelled with truss elements. This simplifies the analysis because the non-linear behaviour of the cables is neglected. In a second model (Case II) the girder property is changed. A Eurocode standard concrete C30/37 is used. The other input data for the model stay the same as in Table 4-1. MiDAS offers to consider the self-weight of a structure by a special self-weight load case. The self-weight function is used for the Case II model. Therefore, with the concrete density of 2.4 tonf/m³, the girder weight is 10.51 tonf/m, instead of 11 tonf/m which is used for the Case I model. According to the results of some prior calculations, it was investigated that, in the case of external loads, the initially applied tension forces are superposed with the self-weight of the cable, as it will be shown later in detail. The initial tension values, as obtained from the backward analysis, must be applied as external loads in the forward analysis. Since the applied values of initial cable forces change depending on its self-weight, it is difficult to compare and control the back- and forward analyses. To apply the same tension values, the change can be already considered in the initial value. A simpler method is to neglect the self-weight of the cables in the back- and forward analyses. In this case, the applied external tensioning stays the same. The latter method is applied in the analysis of this model. The influence of cable elements is studied in detail in Chapter 4.8.
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4.2 Different restrictions for the Unknown Load Factor The Unknown Load Factor function is, as already described before, used to determine the ideal cable forces for the final state of the structure. The possibility of different structural restrictions in order to evaluate the ideal cable forces is investigated. Prior to this, the different options of modelling the girder-pylon connections are examined.
4.2.1 Case I: Use of different connections girder-pylon, restricted displacement The girder and the pylon are connected during the erection period and sometimes, these connections remain after the completion of the bridge. For a general case, this condition can be modelled using various functions. When using MiDAS, there are three different possibilities which may be used, i.e. the elastic links type Gen and Ridget and the Ridged Link. The elastic link type Gen can be specified for all directions and rotations, whereas the Ridget type is fixed. The Ridged Link has a master and one or more slave nodes for which the direction or the rotation can be specified. The different influences of these options on the calculated initial cable forces and the structural behaviour are compared. A value of 1+e8 tonf/m is used for the elastic link type Gen. As already described in the previous chapter, the initial cable forces can be calculated using the Unknown Load Factor function provided by MiDAS. Figure 4-2 illustrates the moment distribution under its self-weight and when 11tonf/m additional load is included.
Figure 4-2: Moment Distribution under Self Weight [tonfm], Case I
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To calculate the initial cable forces, Nodes 4, 12, 16 and 20 are constrained within a vertical movement of 0.001m. The horizontal movement of Node 106 is restricted to the same value. The determined load factors for the applied unit pretension load are:
Elastic Link GEN Elastic Link RIGID Ridged Link
Cable 1 [tonf]
Cable 2 [tonf]
Cable 3 [tonf]
Cable 4 [tonf]
Cable 5 [tonf]
1888.307
624.076
580.407
688.431
925.523
1944.022
603.92
602.94
681.92
928.20
1888.318
624.049
580.381
688.446
925.519
Table 4-2: Ideal cable forces for different elastic link types
The calculated results are slightly different. Because of the possibility to define the elastic link type Gen for different conditions, the function seems to be the most suitable for modelling the interaction between the girder and the pylon during and after the construction process. The elastic link type Ridget is fixed in all directions. It is only possible to model limited conditions between the girder and the pylon. The Ridget Link shows very similar results as the elastic link type Gen and would be an alternative solution to model the connection. However, in the following calculation, the connection is modelled by using the elastic link type Gen as specific values can be defined for all directions. Applying the tension loads obtained with the elastic link Gen, the final moment distribution and the vertical displacement are given in Figure 4-3 and Figure 4-4.
Figure 4-3: Idealised moment distribution after restricted deformation [tonfm], Case I
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Figure 4-4: Deformation dz after restricted deformation [mm], Case I
The moment in the main girder shows a distribution similar to a continuously supported beam. There is nearly no bending moment in the pylon. The deformed shape shows very limited deflection at the restricted nodes. Whatsoever, there are some possible improvements as it will be explained in the next chapter.
4.2.2 Case II: Restricted moment distribution In the previous calculation, the ideal cable forces have been determined by restricting the displacement for the Unknown Load Factor calculation. This procedure results in a moment distribution which is not smooth, as it can be seen in Figure 4-3. The moment values at the connection point of the cable-girder are much higher than the values between the cables. The displacement is limited, but, as this is not the target value, it is also possible to restrict the moment distribution in the girder. For the Case II model, the ideal cable forces are recalculated using moment restrictions. The analysed system is the same as introduced in Chapter 4.1, with the difference that the material properties for the girder are changed to concrete (=> Case II model). Even so, compared to the Case I model, the structural changes for the example have only a minor influence. In Chapter 4.5, the model is used for the investigation of the creep and shrinkage effects. In order to get a first value for the moment restriction, it is possible to use the formulas presented in Chapter 3.1. In the calculation offered in this chapter, the displacements are restricted first, and possible moment restrictions are then obtained from this calculation. Figure 4-5 presents the moment distribution in case of a restricted displacement.
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Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Figure 4-5: Moment distribution after restricted deformation [tonfm], Case II
The moment restriction is chosen to balance the moments between the anchorage point and the field values. Therefore, the anchorage moments in the main span are restricted to a value of -350 tonfm. The value in the side span is restricted to -400 tonfm. Additionally, it is important to restrict the displacement at the top of the pylon; otherwise, there will be a high moment at the bearing point and the pylon will lean inward the main span.
Figure 4-6: Ideal moment distribution after moment restriction [tonfm], Case II
To illustrate the smooth balanced moment distribution in the girder, the whole system is shown in Figure 4-6. The figure also indicates a higher moment in the girder at the pylon. Usually, a stiffer cross section is used in this area.
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Figure 4-7: Displacement after moment restriction [mm], Case II
The deformed structure (Figure 4-7) shows a higher deformation than in the previous method. However, these deformations are of minor importance as the deformation is neutralised by the camber control. The basic camber calculation will be presented in Chapter 4.6.
4.3 Optimisation Method for ideal cable forces by influence matrix In Chapter 3.4, the basics of influence matrix has been introduced. For a simple example, the initial cable forces have been found by solving a linear equation system. As the structure becomes more complex and a higher degree of restrictions are made, it is no longer a linear system. In this case, a solution can be calculated by using the optimisation method. In this chapter, this method is used to estimate the initial tension forces for the Case I solution. Because of unavoidable construction errors, retensioning of the cable forces is sometimes inevitable. The influence matrix for the cable forces can be used similerly to calculate the amount of additional tension force, as it will be shown later.
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4.3.1 Adjustment of the girder elevation The displacement influence matrix for the Case I system described above is as follows (in [mm]):
−0.092215 −0.044746 DGP := 0.006457 0.006512 0.020938
0.068260 0.137747 0.180271 −0.025830
0.134238 0.143768 0.112069 −0.026464 0.100998 0.277691 0.343818 −0.011257 0.047678 0.228162 0.447222 −0.000180 0.003647 0.045054 0.073228 0.059141
The first to the fifth column describe the horizontal displacement at Node 104 and the vertical displacement at Node 12, 16, 20 and 4 due to a unit internal pretension load in Cable 1 to 5. To facilitate the useage of the influence matrices as offered by MiDAS, the transposed matrix must be formed, which is:
−0.0922 0.0683 T DGP = 0.1377 0.1803 −0.0258
−0.0447 0.0065
0.0065
0.0036
0.1342
0.101
0.0451
0.1438
0.2777
0.0732
0.1121
0.3438
0.0591 −0.0265 −0.0113
0.0477 0.2282 0.4472 −0.0002 0.0209
In order to calculate the initial cable forces, an objective function must be defined; this can be given for example as: fδ( x) := −0.092215x − 0.044746x ⋅ + 0.006457x ⋅ + 0.006512x ⋅ + 0.020938x ⋅ 0 1 2 3 4
The restricted boundaries for the Case I model have been previously explained and are also used for this calculation. Under the self-weight and the additional load, the horizontal deformation at node number 106 and the vertical displacement at Nodes 12, 16, 20 and 4 can be derived from the MiDAS calculation. In order to fit the specified restrictions in full, the vector
−175.4449 323.7417 δP := 773.0095 1102.7613 −36.1424 describes the displacements (in [mm]) that should be adjusted by the initial cable forces.
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The upper and lower limit and the calculation are defined as:
1 −175.4449 1 323.7417 δfi_up := 1 + 773.0095 1 1102.7613 1 −36.1424
−1 −175.4449 −1 323.7417 δfi_low := −1 + 773.0095 −1 1102.7613 −1 −36.1424
The limit is the same as the ones used in the Unknown Load Factor calculation. In Mathcad, an optimisation problem can be programmed as: x := 0 4
Given T
DGP ⋅ x ≤ δfi_up T
DGP ⋅ x ≥ δfi_low x≥ 0
The calculated internal stressing value Sfac in comparison to the MiDAS result Sfac_Mi is:
( )
Sfac := Maximize fδ , x
1888.298 624.078 Sfac = 580.412 688.428 925.527
1888.307 624.076 Sfac_Mi := 580.407 688.431 925.523
It should be noted that the result has already been given in Table 4-2 which has been obtained from the MiDAS calculation. The adjusted displacement is: T
T
δfi := DGP ⋅ Sfac
δfi_Mi := DGP ⋅ Sfac_Mi
−174.445 322.742 δfi = 774.009 1101.761 −35.142
−174.446 322.742 δfi_Mi = 774.01 1101.762 −35.143
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The calculated adjusted displacement is close to the required value δP and identical to the MiDAS results. Furthermore, the increase of the tension in Cable 1 to 5 can be calculated with the influence matrix for the cable forces, which is (in [tonf]):
0.274701 −0.365979 T := 0.037337 0.047120 0.147971
−0.146558 0.014952 0.012807 0.029186
0.087018 0.576166 −0.215164 −0.094019 0.047711 −0.317009 0.603796 −0.259832 0.045654 −0.190883 −0.358048 0.640314 0.706830 0.087018 0.032383 0.022487
In this matrix the columns represent the change in Cable 1 to 5 respectively, due to a unit internal pretension load in Cable 1 to 5. With the transposed matrix
0.2747 −0.1466 T T = 0.015 0.0128 0.0292
−0.366 0.0373 0.0471 0.7068 0.087 0.087
0.0477
0.5762 −0.317
0.0324 −0.2152 0.6038 0.0225 −0.094 −0.2598
0.0457 −0.1909 −0.358 0.6403 0.148
the increase of the tension is (in [tonf])
T
∆S := T ⋅ Sfac
481.379 289.978 ∆S = 22.049 3.796 428.328 .
The next vector gives the tension forces before tensioning the cables.
1209.23 240.48 S0 := 488.72 647.97 477.58 Adding the additional forces, the final cable forces are:
Sfi := S0 + ∆S
1690.61 530.46 Sfi = 510.77 651.77 905.91
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The result obtained from MiDAS is:
1690.61 530.45 Sfi_Mi := 510.77 651.77 905.91 This result is identical to the calculated one, which proves the exactness of the presented method.
4.3.2 Adjustment of cable forces In the same way as the displacement is adjusted, it is possible to correct the cable forces. As already mentioned, it may be necessary to retension the cables to a specified value during the construction. For this example calculation, the additional implied tension force shall be (in [tonf]:
481.379 289.978 ∆SA := 22.049 3.796 428.328 An objective function can be defined from the influence matrix for the cable forces for example. to: fT( x) := 0.274701x − 0.365979x ⋅ + 0.037337x ⋅ + 0.04712x ⋅ + 0.147971x ⋅ 0 1 2 3 4
In order to achieve results close to the target value, the upper and lower limit is set to:
0.1 481.379 0.1 289.978 ∆Sfi_up := 0.1 + 22.049 0.1 3.796 0.1 428.328
−0.1 481.379 −0.1 289.978 ∆Sfi_.low := −0.1 + 22.049 −0.1 3.796 −0.1 428.328
The results from this calculation Sfac and from MiDAS Sfac-Mi are:
(
)
Sfac := Maximize fT , x
1890.576 624.752 Sfac = 578.953 687.025 924.46
1888.307 624.076 Sfac_Mi := 580.407 688.431 925.523
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In order to control the results, the vectors are multiplied with the matrix TT, which yields slightly different results. T
T
∆Sfi := T ⋅ Sfac
∆Sfi_Mi := T ⋅ Sfac_Mi
481.479 289.878 ∆Sfi = 21.949 3.696 428.228
481.381 289.975 ∆Sfi_Mi = 22.045 3.801 428.325
Nevertheless, comparing these, it is justified to state that both values are close to the target value ∆SA.
4.3.3 Summary of the adjustment calculation In the case presented before, both adjusted values - the one for the displacement and that for the force – match and by eliminating one error, the other is adjusted as well. On the real construction site, this is very rarely the case, as it has been explained previously. With the cable forces and the geometry retaining in a limited range, it is the designers decision how to adjust and to balance both of them. As the forces and the geometry may have a gap from the theoretical target, a satisfying condition for both should be achieved. As explained in detail in Chapter 3.5, the final condition can be achieved by using length control of the cables. This method assumes that the unstressed length for each cable is correctly calculated and the cable length is then exactly fabricated. The structural performance can also be improved by force control. It is important to ensure that the right cable forces are applied to the cables while they are retensioned. On the site, the cables are adjusted one after another. This means that if one cable is retensioned, it will influence the forces in those next to it. They will get looser so that this circumstance must be considered in the tension force applied first. Since this procedure is difficult and has some uncertainties in applying the exact forces to the cables, the required length ∆L, which must be pulled out by a hydraulic jack, is used very often. The occurring elongation and the place where to cut down the cable can be determined from the actual length and the retension force. If the length ∆L is pulled out, the load is applied, too. This method enables a reliable procedure to stress the cables to the adjusted tension force.
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If there are errors in the cable forces or in the geometry, and some adjustments become necessary, the theoretical unstressed length L0, which can be compared with the actual unstressed length, and the theoretical and the actual cable forces, provide guidance for an appropriate adjustment. It should be mentioned that the internal cable forces are calculated in this example. They can be used in the theoretical way of modelling. In contrast to this, the tension force on the site acts as an external load. As there is no separate function to model cable shortening, the process of pulling out the cable by a hydraulic jack may be modelled by applying a temperature load.
4.4 Backward and forward analysis The following paragraph describes the backward and forward analyses for the example introduced at the beginning of this chapter. The results obtained in Chapter 4.2.1 are used as input values to describe the backward procedure for the model. Differences between the backand forward analyses are investigated in detail.
4.4.1 Backward analysis In order to estimate the initial cable forces using the backward analysis, the results obtained from the Unknown Load Factors must be stored in one load case and applied as internal pretension loads as already described in Chapter 3.3. As the loading is added in one step on the structure, there are fine differences in the moment distribution and deformed shape between the following figures below and the ones given in Figure 4-3 and Figure 4-4.
Figure 4-8: Moment distribution backward analysis [tonfm], Case I
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Figure 4-9: Deformation dz backward analysis [mm], Case I
The difference should be mentioned, but since they are so minimal, they can be neglected. The table below describes the construction sequence as it is defined for the backward analysis. For the forward analysis, the same sequence is used in reverse order. In this case, the cables are stressed to a defined pretension load at the time of their activation. When the key segment is installed in the forward analysis, the moment at the tip of the key-segment will be zero. To consider this situation, a detension load is used in the backward analysis to bring the moment to zero. For the forward analysis, this means that a retension of the cable is needed. Const. Stage
Structure Activation Deactivation
-All Elements
CS0
Boundary *) Activation Deactivation -dz side span -Pylon bottom -dx & ry centre of main span -Elastic link dz Girder/Pylon
Load Activation -Self-Weight -Pretension Load -Additional Load
CS1
-Additional Load -Elastic link dx Girder/Pylon
CS2 CS3
-De-Tension
CS4
-Key Segment
CS5 CS6 CS7 CS8 CS9 CS10
-Cable 5 -Segment 5 -Cable 4 -Cable 1 -Segment 4 -Cable 3
CS11
-Cable 2
CS12
-Segment 3
*)
Deactivation
-dx & ry centre of main span
-Temp. Support dz (Node 4)
dz is in vertical - and dx in horizontal direction
Table 4-3: Construction stage analysis data of the backward calculation
The value for the detension load is determined by employing a few calculations and iterate until the target value of zero moment has been reached. The next table shows the trial and error method which is used.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports De Tension Value [tonf] -600 -615 -613 -612 -612.3
Percent of final cable force (925.523 tonf) ~ 64.8% ~ 66.4% ~ 66.2% ~ 66.1% ~ 66.2%
105
Moment at tip [tonfm] -37.018 8.206 2.176 -0.839 0.066
Table 4-4: Calculation of detension force
For the construction stage analysis, MiDAS offers the option of activating supports to the original or deformed structure. This is very important because the displacement and the moments in the system should be the same in both back- and forward analyses and changing boundary conditions can influence the condition severely. The addition of supports to the deformed structure in the backward analysis fixes the deformations induced on the system according to the construction stage. This results in wrong reaction forces and, as the calculation proceeds, the moment distribution and the displacements become different from the according stage in the forward analysis. The backward- and forward analyses then give different results. Applying supports to the original structure sets the deformation to zero. In the analysis of this example, in construction stage CS11, a temporary vertical fixation is activated to support the girder in the side span. The influence of the activation on the deformed and the original system is shown in Figure 4-11.
Figure 4-10: Before removing the cable and activating the support [mm]
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Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
a)
b) Figure 4-11: Addition of support to the deformed (a) and the original structure (b) [mm]
The reaction force at Node 4 is 177.66tonf in case a) and 55.76tonf in case b). In the forward analysis the reaction force in this state is 55.76tonf and therefore the same as for the case when the support is being activated to the original structure in the backward analysis. If there is a gap between the reaction forces, it is also possible to use the reaction force value from the forward analysis and apply it as a nodal load in the backward analysis. This is similar to the procedure shown in the general description of a construction stage analysis in Chapter 3.3.4. Likewise, this procedure may be utilized if the employed programme lacks the above function. Then, in the next construction step the nodal load can be removed and the support can be activated. Because the nodal load has been used before, allowing some small deformations of the system, there will be no deformation left at the concerning node when the boundary condition is changed. From the backward analysis, the following cable forces are calculated for the installation steps of the stay-cables in the forward analysis: Cable Number (in order of construction) Cable 2 Cable 3 Cable 1 Cable 4 Cable 5 Re tension Cable 5
Cable Force at Stage of Installation [tonf] 229.396 286.989 477.460 267.922 132.337 391.018
Table 4-5: Initial cable forces obtained from backward analysis (Case I model)
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4.4.2 Forward analysis In the forward analysis, the modelling of the erection starts with the pylon. In the next stage Node 4 to 9 and the corresponding elements are activated, supported by the temporary vertical (dz) fixation at Node 4. Then, the remaining part of the side span is installed before the erection sequence continues in reverse order as given in Table 4-3. In the forward analysis, the new activated girder elements are installed tangentially. The function Initial Tangent Displacement for Erected Structures must be activated in the Construction Stage Analysis Control Data. Either all elements or only a special defined element group can be added tangentially. In the case of applying all elements tangentially, there will be a small error in the deformed shape. After the installation of the first part of the side span, a deformation occurs as it can be seen in Figure 4-12. If the succeeding part of the side span is installed tangentially, the segment will have an upward movement at the bearing at node number 1 – even though the boundary condition is activated at the original position. This circumstance can be avoided by defining a separate group containing only the main girder elements. In this case, the upward movement at the bearing in the side span is zero. As described in the previous chapter, the initial pre-stressing forces are applied as internal forces in the backward analysis. The cable forces in the forward analysis are applied as external loads, either by using the add or the replace function. In this calculation, in the Construction Stage Analysis Control Data, the external force, given in Table 4-5, are added.
Figure 4-12: Deformation when first part of the side span is erected [mm]
Figure 4-13 shows the moment distribution and Figure 4-14 the deformed shape after the final construction stage. The results are very similar to that of the construction stage CS0 in the backward analysis, which are presented in Figure 4-8 and Figure 4-9. Thus, the calculation can be regarded as a basically successfully performed back- and forward analysis.
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Figure 4-13: Moment distribution forward analysis [tonfm], Case I
Figure 4-14: Deformation dz forward analysis [mm], Case I
The back- and forward analyses are also performed for the Case II model so that some further important facts about the procedure can be clarified. Comparing the results of the backward and forward methods, the moment distribution and the cable forces are exactly the same up to the construction stage 4 (in the backward and the corresponding stage in the forward analysis). In the following stages, there are slight differences in both models. It has been explained before that the moment in the centre of the main span is reduced to a zero value before removing the support in the centre of the main span, which represents the centreline of the model. In this case, the moment is removed, but there is still a normal force in the system as it can be seen in Figure 4-15 a. In the forward analysis, the middle segment is activated in a zero force condition with no normal forces (Figure 4-15 b). For that reason, there are small differences in the subsequent stages. a) Backward analysis
b) Forward analysis
Figure 4-15: Different normal forces back- and forward analysis [tonf], Case II
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By applying a nodal load in the analysis, the same moment distribution can be achieved for the final construction stage in the forward as calculated in the backward analysis. The gap in the normal forces must be applied before the centre fixation is activated, which is for this example 47.67 tonf on the Node 20. There are still some differences in mm range in the deformed shape, which may be related to the numerical calculation and is negligible. a) Forward analysis
a) Analysis without normal force; max moment pylon: 13 tonfm
b) Forward analysis corrected *2)
b) Analysis including normal force; max moment pylon: -26 tonfm
Figure 4-16: Moment distribution due to considered gap in normal forces in the girder of forward and backward analysis [tonfm]; Case II
In the unknown load factor calculation for the Case II model, the restrictions have been chosen to obtain an equal moment distribution in the girder (Figure 4-6). This condition can also be found in Figure 4-16 b). The differences to Figure a) are minor, however, the influence of neglecting the normal force increases if the construction sequence is changed, as it can be found in the next chapter.
4.4.3 Influence of the activation time of the Girder-Pylon connection In Chapter 2.3.2.4 the importance of the connection between the girder and the pylon has been clarified. In the previous construction stage analysis, the elastic link girder-pylon is activated in longitudinal direction before the detensioning of the cable and the deactivation of the connection of the key-segments. In the forward analysis, this is equal to the fixation between the girder and the pylon until the closure of the bridge and the retensioning of the cables. On the real construction site, the fixation may be released before connecting both cantilevers to adjust the girder in longitudinal direction first. Before the key-segment is inserted, a set back of the girder is often performed. For the Case II model, the construction sequence is changed to illustrate the differences. Table 4-6 shows the new sequence; construction stages CS0 and CS5 to CS12 are the same as before.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
110 Const. Stage CS2
Structure Activation Deactivation
CS3
Activation
Boundary Deactivation
Activation -De-Tension
Load Deactivation
-dx & ry centre of main span
-Key Segment -Elastic link dx Girder/Pylon
CS4
Table 4-6: New construction stage data for backward analysis
Table 4-7 compares the initial cable forces for the case that the construction sequence is changed with the original sequence, which has been given in Table 4-3. Cable force at stage of installation [tonf] Girder/Pylon fixed until retensioning Girder/Pylon fixation removed before (sequence as seen in Table 4-3) activating the key-seg. elements 177.836 86.831 226.958 246.882 519.801 416.619 297.589 298.989 129.794 129.898 406.114 405.930
Cable Number*) Cable 2 Cable 3 Cable 1 Cable 4 Cable 5 Re tension cable 5 *)
in order of construction
Table 4-7: Cable forces obtained from backward analysis (Case II model), changed construction sequence
It can be concluded that the correct modelling is very important as the initial cable forces change significantly. Especially Cable 2 varies extremely in both calculations. If the normal force in the girder is being neglected, as explained in the previous chapter, the discrepancy will influence the final moment distribution enormously in case of a changed construction sequence, as it can be seen in Figure 4-17. There is a high moment value at the bottom of the pylon.
Figure 4-17: Moment distribution forward analysis with changed girder-pylon connection, neglecting normal forces in the key segment [tonfm], Case II
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In this case, the neglected normal force value in the key-segment is -89.549 tonf (Figure 418 a). After the deactivation of the centre support in the backward analysis, the horizontal displacement is 112.45 mm. To consider the normal force in the key-segment before removing the support in the forward analysis, a displacement can also be used instead of a nodal force.
a) Normal force [tonf]
Backward analysis b) Horizontal displacement [mm]
Figure 4-18: Changed backward analysis a) normal force, b) horizontal displacement, Case II
A nodal displacement is applied to node number 22, which produces a normal force shown in Figure 4-19 b). Forward analysis a) Horizontal displacement [mm]
b) Normal force [tonf]
Figure 4-19: Changed forward analysis a) horizontal displacement, b) normal force, Case II
The final moment distribution (Figure 4-20) represents an idealised moment distribution with no bending in the pylon. This example illustrates the importance of the correct consideration of the activation time because it influences the required initial cable forces and the final state of the structure. Furthermore, taking the remaining forces in the system into account is supremely important as it is not possible to control forward- and backward analyses with close results.
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Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Figure 4-20: Moment distribution changed forward analysis, applying a horizontal displacement [tonfm], Case II
In the deformed shape, there are still discrepancies between both analysis methods. The final state in the forward analysis shows a horizontal displacement in the main girder, as it can be seen in Figure 4-21. The centre of the span shows a displacement of -111mm. This horizontal movement must be considered in the construction. Otherwise, the two cantilevers may not meet in the middle of the main span and cannot be easily connected.
Figure 4-21: Horizontal displacement changed forward analysis, applying a horizontal displacement [mm], Case II
In the final state in the backward analysis, there is no horizontal displacement. The last stage in the backward analysis after the removal of last cable can be seen in Figure 4-22 a. In this case, there is a horizontal displacement, whereas the corresponding stage in the forward analysis starts from a zero condition until the first cable has been installed.
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113
Horizontal displacement [mm] before installing the first cable a) Backward analysis
b) Forward analysis
Figure 4-22: Horizontal displacement changed forward analysis, applying a horizontal displacement, Case II
It would be easy to assume that the problem can be solved by applying an appropriate displacement in the forward analysis in order to start with the same condition. But this would only be a theoretical solution for the analysis, since the structure on the real construction site will not be erected including a horizontal movement before the key-segment is installed to close the bridge. However, in case of applying a horizontal displacement of 114.37 mm (value from the backward analysis) in the forward analysis, the condition is equal for the horizontal deformation in the current construction stage. At the time of activating the nodal displacement and moving the whole girder, the temporary connection between the pylon and the girder is released to avoid a displacement of the pylon. Afterward the connection is again installed. In the vertical displacement, a small variation in the order of 2mm remains in the structure. Nevertheless, in the next stage when the first cable is installed, there is again a gap between the forward and the backward analysis, as it is given in Table 4-8. The horizontal displacement at the top of the pylon and at the girder-pylon connection is different. The cable forces are identical.
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114
Horizontal displacement [mm] after installing the first cable Location Backward analysis Forward analysis Top of pylon -52*) -31*) Node 4 108 114 Connection girder pylon *) Tower leans into the side span
Table 4-8: Horizontal displacement after installing the first cable (Case II b model)
Because there is already a little difference, the gap in the subsequent stages increases more and more. Especially the cable force of the anchor cable (Cable 1) changes. This results in a high bending moment in the pylon in the final state. In this case, the horizontal displacement at the top of the pylon is 112mm. Even if the horizontal displacement in the girder is reduced from 111mm (Figure 4-21) to -1mm, with the high bending moment in the pylon, this condition is definitely not the same as the backward condition. Horizontal displacement [mm] after applying the additional load (final stage) Location Backward analysis Forward analysis Top of pylon -1*) 112*) Node 4 4 114 Connection girder-pylon Node 22 0 -1 Centre of the main span *)
Negative value: Tower leans into the main span
Table 4-9: Horizontal displacement after applying the additional load (Case II b model)
Theoretically, the back- and forward analyses should be identical. However, there are some differences in the numerical process and it seems to be difficult to equalise both methods in full. Besides the distribution of the internal forces, the deformations in vertical and horizontal direction must be properly evaluated, too, to consider these in the fabrication of the individual segments. Bearing this in mind, it may be more reliable to follow the result from the forward analysis, as it represents the real construction sequence. The backward analysis can be used to determine the initial cable forces and as a control option for the forward analysis. Furthermore, for concrete and composite cable-stayed bridges, time dependent effects can only be taken into account in the forward analysis. The influence of creep and shrinkage is demonstrated in the next chapter.
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115
4.5 Construction stage analysis considering creep and shrinkage The general concerns of creep and shrinkage effects are explained in the following part of this thesis. An example calculation will illustrate these effects in a construction stage analysis of the simple cable-stayed bridge introduced in Chapter 4.1.
4.5.1 General conditions In a cable-stayed bridge, as well as in all other types of bridges, a concrete girder is often composed of three types of material: concrete, prestressed steel and non-prestressed steel. Concrete of more than one type is often employed in one cross section, for example where the lower part is precast, and installed by a derrick, and the upper part is cast-in-situ. A similar case is the composite section, where the steel structure is lifted first and the deck concreted on site. Concrete exhibits the properties of creep and shrinkage and prestressed steel loses part of its tension due to relaxation. This means that the components which form one section tend to have different strains. However, because of the bond, the difference in strain is restrained. Thus, the stresses in concrete and the two types of reinforcement change with time as creep, shrinkage and relaxation develop. In determined structures, elongations or end rotations are not restrained by supports. Creep, shrinkage and relaxation of steel change the distribution of stress and strain in the section, but change neither the reaction nor the induced stress resultants (values of axial force or bending moment acting on the section). The time effects usually produce a reduction of tension in the prestressed steel, a reduction of compression in the concrete and an increase of compression in the non-prestressed steel. When taking the effect of creep on a statically indeterminate structure made of concrete as a homogeneous material into consideration and neglecting the presence of reinforcement, a larger displacement occurs, but has no effect on the reaction forces. Nevertheless, when a concrete structure is constructed and loaded in stages, or when the member cross section contains reinforcement, creep cannot occur freely. Similar to temperature expansions, restrained creep induces stresses. Because of the different ages of structural parts when constructed in stages, the concrete has different creep coefficients. Precast parts are often
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made continuous with other members by casting joints and pre-stressing, so that the boundary condition for the members changes during the construction. In these cases creep influences the reactions and the internal forces, too. Over its time, creep will have the tendency to change the internal forces into the direction of the “continuous beam condition”. Additionally, internal forces are also produced when changes in the length of members due to shrinkage are restricted. Because shrinkage is always accompanied by creep, the internal values due to shrinkage are well below the values that would develop if shrinkage were to occur alone. An analysis of the time-dependent stresses and the deformations in uncracked and cracked concrete structures is thoroughly done by Ghali [7].
4.5.2 Modelling creep and shrinkage For the example used in this chapter, a construction stage analysis is performed to study creep and shrinkage effects. The used functions offered by MiDAS are explained. The creep and shrinkage is modelled by the CEB-FIP code. The input data given in Table 4-10 are specified. Creep and shrinkage data Compressive strength of concrete at the age of 28 days: 3800tonf/ m2 Relative Humidity of ambient environment 70% National size of member: 0.5m Type of cement Normal or rapid hardening cement (N,R) Age of concrete at the beginning of shrinkage: 3 days Table 4-10: Input data CEB-FIP code
In the construction stage analysis data, the construction periods must be defined in order to consider load histories in the creep and shrinkage calculation.
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For this example, the following construction time is defined: Constr. Stage CS0 CS0a CSb CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12a CS12
Construction Pylon Girder at Pylon (G2) Girder left side span (G1) 1st Girder main span (G3) Cable 2 Cable 3 2nd Girder main span (G4) Cable 1 Cable 4 3rd Girder main span (G5) Cable 5 Girder middle segment (CG) Connection of middle segments Retension cable 5 Remove elastic link dx Pylon/Girder Applying the additional load -/-
Day(s) 34 12 12 12 1 1 12 1 1 12 1 12 1 1 1+9 5000
Table 4-11: Construction time schedule
In each of the steps, the corresponding segment loading is also activated. The boundary conditions are identical as already described in Table 4-3. For the girder elements an age of 2 days is defined, which considers the time of loading or rather the time of removing the formwork after casting the concrete. Figure 4-23 shows the moment distribution in the girder after 1, 10 and 5000 days after applying the additional load. The upper curve in Figure 4-23 shows the moment distribution including creep, the lower curve only considers the construction sequence without any creep or shrinkage. The shrinkage values are low and therefore the curve considering creep and shrinkage is nearly identical with the upper one (only creep, no shrinkage). Because of the assumption that the formwork is being removed after 2 days, the creep curve and the moment curve as constructed (no creep or shrinkage) are close in the centre of the main girder. At the pylon, a high increase in the moment value is visible. 10 days after applying the additional loading (Figure 4-23b), it can be seen that the high moment value at the pylon is reduced and near the key-segment, the influence of creep is increased. Shrinkage still has a minor effect, which changes in the course of a longer period. After 5000 days (Figure 4-23c) the values of creep and shrinkage are of the same magnitude. The upper curve in Figure 4-23c indicates the moment distribution including the creep. In this example calculation, due to shrinkage, the creep effect is balanced and thus, the moment curve including creep and shrinkage becomes close to the moment curve which considers only the construction sequence.
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-1200 Case II model 1 day after applying additional load
-1000 -800 -600 -400
0
20
40
60
80
-200 0 200 400 -1200
Case II model 10 days after applying additional load
-1000 -800 -600 -400
0
20
40
60
80
-200 0 200 400 -1200 -1000
Case II model after 5000 days
-800 -600 -400
0
20
40
60
80
-200 0 200
Pylon
400
Moment as constructed (no creep or shrinkage) Moment including creep (no shrinkage) Moment including creep & shrinkage
Figure 4-23: Bending moment in the girder [tonfm] a) 1 day after applying additional load, b) 10 days after applying additional load, c) after 5000 days
The calculated results are greatly influenced by the defined material properties, the loading time and the erection sequence. However, the main behaviour can be clarified. Besides the changes on the moment distribution in indeterminate structures, time dependent effects have a significant influence on the overall deformation. As already mentioned before, the vertical deflection of the main girder due to dead load, post-tensioning (if applied) as well as long term effects of creep and shrinkage should be predicted during the design process. In Chapter 3 it has been clarified that the deflection depends on a large extent on the method of construction of the structure. The age of a newly installed segment has a dominant role when it is post-tensioned and/or when the formwork is removed in order to construct the next segment.
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119
Therefore it can be expected, that the actual deflections of the structure will be different from that of the predicted during the design due to changed assumptions. As a result, the deflections should be recalculated on the basis of the actual construction sequence by the contractor’s engineer. The permanent deflection of the structure after all creep deflections have occurred, normally 10 to 15 years [5], may be objectionable from the perspective of the riding comfort for the users or for the confidence of the general public. Although there is no structural problem with a span with noticeable sag, it will not inspire public confidence. For these reasons, a camber is normally cast into the structure so that the permanent deflection of the bridge is nearly zero.
0
10 20 30 40 50 60 70 80 Figure 4-24 illustrates the vertical deflection in the main girder, which has been obtained from 0 the example calculation. -50 0
20
40
60
80
0 -50
-100 -150
-100 -150
-200
-200
-250
-250
-300
-300 -350
As conctructed (no creep & shrinkage)
After 5000 days
1 day after applying additional load
After 5000 days, built on scaffoldings
-350
10 days after applying additional load
Figure 4-24: Vertical displacement of the main girder [mm]
Based on the input data for the CEB-FIP code given before, the graphic clearly shows that most of the time, dependent deformation occurs in a period after loading. The difference between 10 days and 5000 days of applying the additional load is only a minor part, compared to the deflection that has already proceeded. With the defined input parameters, the calculated ratio of total to elastic deformation is nearly 5, which is too high for a real construction, but this is of no importance here. A second analysis is performed, assuming the bridge is erected on scaffoldings in the same period as used before. The deformation after 5000 days is plotted in Figure 4-24. The result indicates a much lower vertical displacement, which is caused by the different construction method producing lower stresses in the sections during the construction.
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4.6 Camber Control The calculation method for the camber data is presented in the following chapter. Later on, the camber data is analysed for the Case I and II model.
4.6.1 General calculation method Right from the design or planning state, one of the problems of construction engineering to be resolved is the cambering of the individual bridge components. The design planes show the geometric outline of the bridge, i.e. its shape under the designated load condition. This is commonly the full dead load including the additional dead load due to pavement etc. at normal temperature. The contractor, however, fabricates the bridge segments under the no-load condition, and at “shop temperature”. The difference between the shape of a member under full dead load and normal temperature, and its shape at the no-load condition and shop temperature, is the segment camber. Figure 4-25 illustrates the camber control by a camber and the deformation sketch.
Figure 4-25: Camber and deformation
The general procedure of calculating the camber data is demonstrated in the following. The figure below shows the cantilever, which is assumed to be installed in three steps. In this case, not only the actual deformation under its direct loading, but also the additional deflection due to the weight of following segments must be considered in the camber calculation. Usually, the subsequent segments are connected with a prefabricated angle to balance the deformations.
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Figure 4-26: Cantilever
Figure 4-27 a) describes the current deflection of installing a new segment and Figure 4-27 b) the fabrication camber to realize a zero displacement condition in the last stage of the construction. 2
1
3 δ 31
δ 11
δ 21
δ 32
δ 31 δ 11 δ 12 δ 13
δ 12
δ 13
δ 22
δ 21 δ 22 δ23
δ 33
δ 32
δ 23 δ 33 δ 31 δ 32 δ 33
Figure 4-27: a) Current displacement b) Construction camber
The current displacement is only that of the new installed segment without considering any other existing displacements of previous installations. The real displacement of each segment can be calculated on the basis of the current displacement.
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After all segments are put in place, the real displacement can be calculated for each node as: Node 1 = δ11 + δ12 + δ13 Node 2 = δ 21 + δ 22 + δ 23
⇒
Real displ., fabrication camber
Node 3 = δ 31 + δ 32 + δ 33 where δ expresses the deflection, the first indices describes the location of the deformation (node number) and the second indices describes due to which segment the deformation occurs. The real displacement has the meaning of the general fabrication camber. The total net displacement, which is the meaning of the construction camber, can be determined as: Node 1 = δ11 + δ12 + δ13 Node 2 = δ 22 + δ 23
⇒
Total net displ., construction camber
Node 3 = δ 33 In case of the given example, the first segment must be installed with a specific angle, which produces an upward deformation of δ11 + δ12 + δ13 at the tip of the segment. Due to its selfweight, the segment will be subject to the downward movement of δ 11 . The next segment must be installed with an angle again, producing a deformation at the tip of δ 22 + δ 23 , which will be pulled downward by - δ 22 because of its self-weight. The remaining value of δ 23 will be eliminated with the installation of the last segment, which has to be fixed to the previous segment with an angle generating a deformation of δ 33 . The total structural upward displacement produced by the angle-connections, will be balanced by the structural self-weight. The described method is illustrated in Figure 4-27 b). A simple example is calculated to illustrate the procedure as explained before. Figure 4-28 shows a cantilever which is erected in three stages. A distributed load of 25 tonf/m, a stiffness I of 0.92 m4, an elasticity E = 2.1*105 N/mm and cross section area A = 4.38 m2 is assumed in the calculation. CS 1 CS 2
10 tonf/m 10 tonf/m 10 tonf/m
CS 3 Figure 4-28: Erection of a cantilever
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123
The figure below shows the current displacement for each erection step. All elements are activated in the first step in order to prove the tangential activation of the elements in the MiDAS programme. The loads are applied sequentially. Current step displacement [mm] CS 1 CS 2 CS 3 Figure 4-29: Erection of a cantilever, current displacement value
The current displacement for construction stage CS 1 is controlled by a simple calculation. With the displacement u1 (δ11) at Node 1 q := 25
tonf m
5
E := 2.1⋅ 10
N
4
I := 0.92⋅ m
2
L := 10⋅ m
mm
4
u 1 :=
q⋅ L
8⋅ E⋅ I
u 1 = 1.586mm
and the angle ψ, the displacement for Node 2 (δ12) and 3 (δ13) can be calculated as: 3
ψ :=
q⋅ L
−4
ψ = 2.115 × 10
6E⋅ I
∆u 2ψ := ψ ⋅ L
u 2 := ∆u 2ψ + u 1
∆u 3ψ := ψ ⋅ 2L
u 3 := ∆u 3ψ + u 1
∆u 2ψ = 2.115mm
u 2 = 3.701mm
∆u 3ψ = 4.23mm
u 3 = 5.816mm
From the current displacement, the real and the net displacement can be calculated as already explained. The real displacement values are calculated in Table 4-12. The real displacements are also calculated with MiDAS by using a tangential erection for the new activated elements. The computed results, given in the table, are identical with the calculated values. Real displacement [mm] CS 1
uR1 = δ11 = 1.586
CS 2
uR1 = δ11+δ12 = 1.586+7.402 = 8.988 uR2 = δ21+δ22 = 3.701+21.678 = 25.379
CS 3
uR1 = δ11+δ12+δ13 = 1.586+7.402+13.747 = 22.735 uR2 = δ21+δ22+δ23 = 3.701+21.678+46.529 = 71.908 uR3 = δ31+δ32+δ33 = 5.816+36.483+86.185 = 128.484
Real displacement MiDAS [mm]
Table 4-12: Real displacement table
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124
Table 4-15 illustrates the calculation of the net displacement and the construction camber data. The results obtained form the MiDAS calculation is given as well. Both results are identical. Total net displacement [mm] CS 1
uN1 = δ11 = 1.586
CS 2
uN1 = δ11+δ12 = 8.988 uN2 = δ22 = 21.678
CS 3
uN1 = δ11+δ12+δ13 = 22.735 uN2 = δ22+δ23 = 68.207 uN3 = δ33 = 86.185
Construction camber [mm] uCo1 = δ11+δ12+δ13 = 22.735 uCo1 = δ12+δ13 = 21.149 uCo2 = δ22+δ23 = 68.207 uCo1 = δ13 = 13.747 uCo2 = δ23 = 46.529 uCo3 = δ33 = 86.185 uCo1 = 0 uCo2 = 0 uCo3 = 0
Construction camber MiDAS [mm]
Table 4-13: Total net displacement and construction camber data
The required angle, which must be considered between two segments to achieve a final structure with no deformation, can be evaluated from the real displacement data. Figure 4-30 indicates the angles which have to be calculated for the fabrication of the individual segments.
Ψ3
Ψ2
Ψ1
Figure 4-30: Fabrication camber, real displacement [mm]
With the real displacement and the current displacement u R1 := 22.735mm ⋅ u R2 := 71.908mm ⋅
δ12 := 3.701⋅ mm δ13 := 5.816⋅ mm
u R3 := 128.484mm ⋅
δ23 := 36.483mm ⋅
the angles can be calculated as the following:
uR1 lSeg
ψ 1 := atan
ψ 1 = 0.13deg
( )
∆u Seg1 := lSeg ⋅ sin ψ 1
∆u Seg1 = 22.735mm
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports ∆u Seg2 := u R2 − 2⋅ ∆u Seg1
125
∆u Seg2 = 26.438mm
∆u Seg2 lSeg
ψ 2 := atan
ψ 2 = 0.151deg
∆u Seg3 := u R3 − 3⋅ u R1 − 2⋅ ∆u Seg2
∆u Seg3 = 7.403mm
∆u Seg3 lSeg
ψ 3 := atan
ψ 3 = 0.042deg
To prove that the required girder elevation at the time of installing the segment is achieved by the calculated angles, the flowing calculation shall control the condition:
( )
u CS1 := lSeg ⋅ tan ψ 1
u CS1 = 22.735mm
( )
u CS2 := 2⋅ ∆u Seg1 − δ12 + lSeg ⋅ tan ψ 2
(
( )
( ))
u CS2 = 68.207mm
( )
u CS3 := 3⋅ tan ψ 1 + 2⋅ tan ψ 2 ⋅ lSeg − δ13 − δ23 + lSeg ⋅ tan ψ 3
u CS3 = 86.185mm
The girder elevation is the same as given in the construction camber data in Table 4-13.
4.6.2 Camber calculation for the Case I example For the bridge-example Case I, introduced in Chapter 4.2.1, the camber control is calculated and compared with the values given by MiDAS. The calculated displacement values of the structure can be seen in the deformed shape view. These values are given in the table below for Node 12, 16 and 20, as obtained from the MiDAS computation. The location of the nodes can be seen in Figure 4-1. The current displacement can be calculated by subtracting the displacement value of the previous step from the actual step.
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Numerical values of displacement in dz direction [mm] 1) Node 12 Node 16 Node 20 -71.78 -105.1 163.31 16.47 -389.51 127.64 -156.91 225.68 195.76 146.74 -96.12 -608.12 160.15 -13.86 -395.39 129.1 -129.64 -665.86 129.1 -129.64 -665.86 158.29 10.79 -389.31 -3.47 -375.2 -940.35
Constr. Stage CS0 CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12 1)
Calculated current displacement [mm] Node 12 -71.78 -33.32 2) 268.41 -146.84 111.17 98.04 -78.94 13.41 -31.05 0 29.19 -161.76
Node 16 -389.51 232.6 352.67 -291.88 82.26 -115.78 0 140.43 -385.99
Node 20 -608.12 212.73 -270.47 0 276.55 -551.04
Values obtained from MiDAS; 2) e.g. = -105.1 - (-71.78)
Table 4-14: Calculation table of the current displacement (Case I model)
As described briefly above, by using the current displacement, the real and the net displacement can be calculated; for Node 12, 16 and 20, these values are given in the next table. Constr. Stage CS0 CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12 1)
Calculated real displacement [mm] Node 12 Node 16 Node 20 0 -68.97 1) -102.29 2) 166.12 19.28 -13.19 1) 130.45 219.41 228.49 572.08 149.55 280.2 331.04 1) 162.96 362.46 543.77 131.91 246.68 273.3 131.91 246.68 273.3 161.1 387.11 549.85 -0.62 1.12 -1.19
Total net displacement [mm] Node 12 Node 16 Node 20 -71.78 -105.10 3) 163.31 16.47 -389.51 127.64 -156.91 225.68 195.76 146.74 -96.12 -608.12 160.15 -13.86 -395.39 129.10 -129.64 -665.86 129.10 -129.64 -665.86 158.29 10.79 -389.31 -3.43 -375.20 -940.35
obtained from MiDAS, 2) e.g. = -68.97 + (- 33,32) Table 4-14; 3) e.g. = -71.78 + (-33.32) Table 4-14
Table 4-15: Calculation table for real and net displacement (Case I model)
The actual real displacement is the current displacement plus the accumulated real displacement from the previous steps. The change in each construction stage can be predicted from the net displacement and the current displacement. The target displacements for Node 12, 16 and 20 are presented in Table 4-16. At each node, the negative upper value represents the camber to be incorporated at the time of installing the segment. The lower values represent the change of displacement at the subsequent construction stages and can be calculated from the actual displacement minus the current displacement. Due to the construction sequence and the structural dead load, the total amount of displacement at the end of the construction should be compensated as the final structure is assumed to be with no deformation.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports Constr. Stage CS0 CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12 1)
127
General construction camber [mm] Node 12 Node 16 Node 20 -3.43 1) 68.352) 101.67 -166.74 -375.20 1) -19.90 14.31 -131.07 -218.29 -229.11 -570.96 -940.35 1) -150.17 -279.08 -332.23 -163.58 -361.34 -544.96 -132.53 -245.56 -274.49 -132.53 -245.56 -274.49 -161.72 -385.99 -551.04 0.00 0.00 0.00
from Table 4-15; 2) e.g. = -3.43-(-71.78) Table 4-14
Table 4-16: Calculation table for construction camber (Case I model)
For a construction stage analysis, MiDAS offers the option to calculate a construction camber table with the General Camber function. If the initial tangent displacement for the erected structure in the analysis option is switched on, new segments are added tangentially and the real displacement for each construction step is calculated. In order to produce the camber output, a structural group must be specified to which the main girder, the key segment and the supports are assigned. In the General Camber Control option, the structural group and the constructed direction must be defined. The results for the general construction camber obtained by the MiDAS function are given in the next table.
Table 4-17: Construction camber table [mm] (Case I model)
The values found in Table 4-17 are very close to the ones calculated in Table 4-16 for Node 12, 16 and 20, which proves the calculation performed by the programme. MiDAS also offers to plot the results on a diagram; this is given in Figure 4-31.
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Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Figure 4-31: Construction camber graph, negative net displacement [mm] (Case I model)
The X-axis of the graph represents the length (or nodes) of the girder and the Y-axis the corresponding camber. Two values are produced for each node. The upper value represents again the camber to be incorporated at the time of installing the segment, and the lower values represent the displacements at the subsequent construction stages. Furthermore, the general fabrication camber is given in a table and as a graph below. The values in the table represent the real displacement as it is also found in the deformed plot of the structure after finishing the erection, given in Figure 4-14.
Table 4-18: Fabrication camber table [mm] (Case I model)
Figure 4-32: Fabrication camber [mm] (Case I model)
The values found in the graph are the same deformation values with opposite sign as illustrated in the deformed shape plot and are also given in the table above. In case of a steel deck, the angles between the different segments, which are required for an erection of the structure with no deformation, can be calculated from this data, as it has been illustrated in the previous chapter. The angle is then considered in the fabrication of the individual segments.
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129
The initial cable forces for the Case I model have been found by restricting the vertical displacement. For that reason, the fabrication camber has this up- and downward shape, which may not be understandable right away. In the next chapter, the fabrication camber is also illustrated for the Case II model, which shows more reasonable deformations.
4.6.3 Camber calculation for the Case II example In the Case II model, the ideal cable forces are determined by restricting the moment distribution in the girder. As already illustrated before (Figure 4-7), in this case, a much higher deformation occurs in the main girder so that it must be balanced by the camber erection. For this second analysis creep and shrinkage are also considered. These additionally increase the structural deformation. Figure 4-33 shows the fabrication camber including and excluding creep and shrinkage effects. The upper graph includes a calculation of 5000 days as explained earlier. The lower graph only considers the elastic deformation.
Camber [mm]
Fabrication Camber 350 300 250 200 150 100 50 0 -50 1
4
7
9
12
16
20
22
Node Number Including creep & shrinkage
No creep & shrinkage
Figure 4-33: Fabrication camber [mm] (Case II model)
The construction camber (Figure 4-34) also shows an upper and a lower graph. The upper graph represents again the creep and shrinkage effects. Two values are given for each node. The first value is the camber required at the time of installing the formwork. The second includes the current step displacement due to its self-weight and the time dependent effects of the actual step.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
130
Camber [mm]
Construction Camber
1000 800 600 400 200 0 -200 1
22 Node Number Including creep & shrinkage
No creep & shrinkage
Figure 4-34: Construction camber [mm] (Case II model)
In order to calculate the values of creep and shrinkage effects properly, the actual duration of each construction stage must be considered. In the construction stage analysis with MiDAS, the number of days must be defined for each step. To calculate the values for the camber data of the girder, the creep and shrinkage deformations within each step must be considered. These data are not accessible if no additional step results are saved. The summation of the different loadings as results from dead load, creep, shrinkage and possible erection loads will only show the summation of the actual additional step, but not the result from the total current construction stage. Therefore, in case of applying additional steps within a construction step, it is important to save the results for the additional steps as well.
4.7 Construction Errors As described in Chapter 3.7, during the construction process, there may be various factors responsible for discrepancies between the theoretical design values and the actual measured data on site. Some potential errors are investigated in more detail in the following. Different errors from the theoretical data are assumed and their influence is investigated. Furthermore, some suggestions are made for proceeding in case of revealed construction errors.
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4.7.1 Error in cable force For the simple example of the already introduced cable-stayed bridge (Case II model), it is assumed that an error in the tensioning has occurred. During the construction, a cable is installed with a lower pretension value than it has initially been designed. For the example calculation, Cable 4 is installed with only 90% of its designed magnitude. In MiDAS the defined pretension value is changed to 267.83 tonf. At the time of installing Cable 4, the tension values in the other cables also change due to the lower pretension as given in Table 4-19. Cable Nr. 1 2 3 4 5
Time of installing cable 4 [tonf] Orig. system Wrong CF % 549.22 546.28 -0.5 170.00 165.86 -2.4 193.48 225.21 +16.4 297.59 267.83 -10.0 -/-/-
Last step [tonf] Orig. system Sfi_A Wrong CF Sfi_0 1698.55 1695.55 398.61 394.45 407.56 439.29 649.47 619.71 895.09 895.09
% -0.2 -1.0 +7.8 -4.6 0.0
Table 4-19: Cable forces due to changed pre-stressing in cable 4
Because Cable 4 is installed with a lower pre-stressing load, the tension in the neighbouring cables is influenced as well. The tension in Cable 3 increases about 16% at the time of installing Cable 4. The change in elevation of the already constructed girder is illustrated in Figure 4-35. Vertical displacement [mm] Original
Error calculation
Figure 4-35: Vertical displacement considering cable tension error
The upward movement of the deck at the time of installing Cable 4 is lower than in the original system. As a consequence, the tension forces in the cables change in the subsequent stages. Table 4-19 also illustrates the cable forces in the final state in comparison to the original system. The vertical displacement in the centre of the bridges increases from -70 mm to -148 mm.
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Figure 4-36: Final moment distribution due to changed pre-stressing in cable 4 [tonfm]
The simplest condition to adjust the cable force is when the error is already detected at the time the cable is being installed. In this situation, the cable can be retensioned without having further influence on the following stages. If the adjustment procedure is applied in a later stage, the restressing will have considerable influence on the overall structure. It is not possible to just retension Cable 4 to the designed value. It also changes the cable forces in the neighbouring cables but it is unlikely that they will have the target values. It is assumed that the cable forces are adjusted in the final state after the additional loading is applied. In Chapter 4.3, the influence matrix for internal forces has been introduced for the Case I model. For the Case II model (different girder properties and changed loading compared to Case I model), the influence matrix for internal forces is calculated using the Unknown Load Factor function. This matrix can then be used to calculate the influence matrix for external loads, which follows to (in [tonf]): 1 −0.514263 T := 0.080443 0.098118 0.242717
−0.52461 0.06762
0.057567 0.110399
0.143179 1 −0.35814 −0.166614 0.085797 −0.513188 1 −0.41332 0.079739 −0.307917 −0.533143 1 . 1
0.117982 0.049345 0.035553
This matrix is derived from a factor multiplication of the influence vectors of each cable force. Assuming a linear condition, the retension values can be obtained with the influence matrix for the external forces. It must be noted that the influence matrix obtained from MiDAS is actually the transposed matrix compared to the general definitions used in Chapter 3.7. This must be considered for the application of the given formulas.
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The required additional tension ∆Sfi can be evaluated from the final existing cable forces Sfi_0 and the design forces Sfi_A. The rows of the vector represent Cable 1 to 5 from up to down.
1695.55 394.45 Sfi_0 := 439.29 619.71 895.09
1698.55 398.61 Sfi_A := 407.56 649.47 895.09
3.00 4.16 ∆Sfi = −31.73 29.76 0.00
The optimisation method has been explained in Chapter 4.3. The same method is used to calculate the retension values for two different conditions. In the first one, it is assumed that there is no possibility to detension the cables. In this case, a solution can only be found with an increase of the upper restrain value ∆Sfi_up to (in [tonf])
4.15 3 4.15 4.16 ∆Sfi_up := 4.15 + −31.73 4.15 29.76 4.15 0
−0.1 3 −0.1 4.16 ∆Sfi_low := −0.1 + −31.73 −0.1 29.76 −0.1 0 .
The determined values for Cable 2, 4 and 5 and the final cable forces are given as x := 0 4
Given T
T ⋅ x ≤ ∆Sfi_up T
T ⋅ x ≥ ∆Sfi_low x≥ 0
(
)
∆Sfi_star := Minimize fT , x
0.00 3.12 ∆Sfi_star = 0.00 43.50 18.27
1702.65 402.76 Sfi = 411.71 653.62 895.49
Comparing the design forces Sfi_A and the final cable force after the adjustment Sfi, there are obviously some differences.
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Better results can be achieved if it is assumed that the cable system allows changes in the effective length of the cable, for example by inserting shim plates at the anchorage points. In this case the new calculated values are
−0.09 2.46 ∆Sfi_star = −13.12 30.39 10.40
1698.65 398.51 Sfi = 407.66 649.37 895.19 ,
with a restriction of the upper and lower values to ±0.10. When the last square method is directly applied as given in Equation 3-65, which does not include an upper or lower tolerance, the calculated adjustment vector ∆Sfi_star_LSM is:
(
T
∆Sfi_star_LSM := T⋅ T
)− 1⋅ T⋅ ∆Sfi
0.006 2.669 ∆Sfi_star_LSM = −13.375 30.269 10.187
1698.55 398.61 Sfi_LSM = 407.56 649.47 895.09
By setting the upper and the lower limits to ±0.00, the identical value is calculated with the previously applied procedure. To model the retensioning with MiDAS, the calculated values as given in the vector ∆Sfi_star are applied in the construction stage analysis after the final load step. It should be noted, that the adjustment procedure on site is usually performed by restressing one cabal after another. This is considered by the definition of five additional construction steps for each stressing process (definition in the construction stage analysis control: add external forces). It is also important that the stressing sequence effects the influence matrix and the tension forces. In case of stressing two or more cables simultaneously, the condition must be regarded when the influence matrix is formed. Therefore, it causes wrong results if the stressing is applied in only one step. Considering this situation, the final cable forces determined with MiDAS after the last tensioning are close to the calculated value
1698.67 398.51 Sfi_Mi := 407.69 649.39 895.16
.
The moment distribution after the cable adjustment is distributed more equally as it can be seen in Figure 4-37. However, as the construction sequence is different from the original purposed
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sequence with a limited retensioning in the stages after the closure of the bridge, the overall distribution is different. Here, the negative moments increase and the positive moments decrease. In the girder, the total moment distribution moves upward. The moment distribution for the girder in the original system has been given in Figure 4-16 a.
Figure 4-37: Final moment distribution after restressing of cable 1 to 5 [tonfm]
Due to the restressing of the cables, the girder elevation changes. After the last step, the vertical displacement in the centre of the girder is reduced to -126 mm. Assuming the initially calculated displacement of -70 mm is incorporated in the prefabricated camber deformation, there sill is a gap of 56 mm which must be adjusted. The example shows that, in this case, the designed cable forces may be achieved by the adjustment, but the girder elevation remains with errors. Therefore, in most of the cases the error of cable forces and girder elevation must be balanced in order for both of them to be within an allowable range.
4.7.2 Error in elevation of a segment In this chapter an error in the elevation of the girder is assumed. The error can be adjusted either by changing the cable forces with retention operations or, in case of smaller discrepancies from the target condition, by changing the camber in the remaining segments. Both methods shall be demonstrated in the following.
136
4.7.2.1
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Adjustment by cable forces
If the error in the girder elevation is large, it is necessary to retension the cables, which will change both the internal forces of the structure and the girder level. As in the example given before, it is assumed that Cable 4 is erected with wrong initial stressing. Due to wrong tensioning of the cable, the girder elevation must be adjusted. As it is the same case for the tension forces, if a large error occurs during the construction, the adjustment of the girder elevation must proceed immediately before continuing the erection. An overall adjustment may be performed before connecting both cantilevers to close the bridge. However, to exemplify the general influence and to show how to calculate the additional tension forces, the girder elevation shall be also corrected in the last stage as it was assumed in the chapter before. Figure 4-35 showed the difference in the girder level after Cable 4 has been installed. The final displacement in the last stage of the forward analysis is given in the vector δfi_0. The displacement, as it has been obtained from the original system without changing the initial tension in Cable 4, is given in δfi_A. In these vectors the vertical displacements at Node 4, 12, 16, 20 and 22 and the horizontal displacement at the top of the pylon at Node 106 are represented by the 1st to 6th value respectively. The location of the nodes is given in Figure 4-1. It is assumed that the vertical displacement δfi_A is considered in the camber construction of the girder. Therefore, the additional displacement ∆δfi_A must be adjusted by retensioning operations. The values below are given in [mm].
2 −43 −89 δfi_0 := −129 −148 0
1 −32 −53 δfi_A := −65 −70 0
−1 11 36 ∆δfi_A = 64 78 0
The influence matrix for the displacement can be found by using the Unknown Load Function offered by MiDAS again. The displacement of the according nodes is restricted and the influence matrix is calculated. As it has also been the case in the previous chapter, this influence matrix is based on internal forces and must be factorized and rearranged for the purposed operations. The calculation can be done in a spreadsheet to find the required matrix from.
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The influence matrix for the external forces follows to (in [mm])
−0.083147 0.070955 DGP := −0.041707 −0.018946 −0.001992
0.230012 0.465652 0.609062 0.628109 −0.318524
0.209085 0.234892 0.194025 0.187227 0.013359 0.160815 0.431504 0.538937 0.548178 0.013575 0.076382 0.334835 0.636946 0.684844 0.034020 0.006504 0.060452 0.097387 0.102365 −0.062885
It should be noted that the given influence matrix is not exactly the matrix for the system to be adjusted. Because the influence matrix is evaluated with the Unknown Load Function, the system which is employed has a different deformation and therefore a different angle between the girder and the cables. The change in the angle influences the calculated vertical girder elevation. However, this difference is small enough to be neglected. The upper and lower restriction values must be increased again, otherwise it is not possible to find a solution. The following vectors state the lowest possible limits.
1 1 1 −11 1 −36 δfi_up := − 4 −64 1 −78 1 0
−1 1 −1 −11 −1 −36 δfi_low := − −1 −64 −4.9 −78 −1 0
The utilisation of the optimisation method gives the required additional cable forces ∆Sfi_star and the new calculated displacement δfi.
10.273 −12.424 ∆Sfi_star = 4.788 −7.497 103.721
0 −33 −52 δfi = −61 −75 1
Applying the calculated values in separate stressing steps, the displacement values δfi_Mi, calculated by MiDAS are the same as determined before. However, there is still a discrepancy between the target value δfi_A and the final calculation of ∆δer. The highest divergence is in the centre of the girder at node number 22 of -5 mm. But as mentioned earlier, with the specified adjustment requirements for the girder nodes and the top of the pylon, it has not been possible to fine-tune the girder elevation closer to the target values.
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0 −33 −52 δfi_Mi := −61 −75 1
1 −32 −53 δfi_A = −65 −70 0
−1 −1 1 ∆δer = 4 −5 1
Because of the focus on the girder elevation in this calculation, the change in cable forces has been neglected. Table 4-20 gives the cable forces after the girder elevation has been adjusted in comparison to the tension forces with originally designed cable tuning. Cable Nr. 1 2 3 4 5
Last step [tonf] Orig. system Wrong CF 1698.55 1737.17 398.61 384.88 407.56 415.60 649.47 554.77 895.09 1001.95
% 2.2 -3.4 2.0 -14.6 11.9
Table 4-20: Cable forces due to elevation adjustment
Not only are the cables forces influenced by the restressing operations, the overall internal forces are redistributed in fact, too. Figure 4-38 describes the final moment distribution after the adjustments.
Figure 4-38: Final moment distribution after elevation adjustment [tonfm]
The moment distribution indicates a high moment at the anchorage point of Cable 5. In the centre of the bridge, a high discrepancy between the actual vertical displacement and the target displacement has been assumed, which must to be adjusted. Due to this large difference, the restressing operation causes this high moment at the anchorage point of Cable 5.
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Nevertheless, this and the previous example show that, if the correction procedure mainly focuses on the cable forces, there will be a gap in the girder elevation and vice versa, so that correcting the girder elevation results in a wrong cable force. It is the designers purpose to find an appropriate balance between both. Cable forces and girder elevations must be within the allowable range in the final state to ensure reliability and serviceability of the structure.
4.7.2.2
Adjustment by camber
In this chapter, it is assumed that an error in the girder elevation has occurred due to a wrong installed segment. This may happen in case of a prefabrication inaccuracy of the individual segment. It can also be the case that at site operation, for example during the welding, an additional angle is induced to the newly installed segment. For the example it is assumed that girder G4 is erected with an error producing an extra vertical displacement of -50 mm after the installation. In order to model this condition, the model must be slightly modified. At the start of the 4th segment (at Node 12, see Figure 4-1) a further node is created very close to the existing one. Then segment 4 is defined to start with Node 200 and both girder parts are connected by an elastic link, as it can be seen in the figure below.
Figure 4-39: Elastic link in order to model an error in the girder elevation
For the elastic link, the type Gen is chosen, which allows to define six stiffness values, - three directions and three rotations. High values for all degrees of freedom are defined to generate a rigid connection. In the construction stage analysis, the elastic link and Node 200 is activated at the same time of installing the previous segment (Segment 3). This is important because otherwise the tangential activation of the segments and the recalculation of the following cambers will result in errors. At the tip of the girder, the real vertical displacement is -110.57 mm after activating Segment 4. In the corresponding step of the original analysis without errors, the value is -110.76 mm. The small discrepancy is neglected so that it proves the reliability of the proposed method. In the next construction step, the elastic link can be replaced by another elastic link with a changed rotational stiffness, allowing a further displacement in order to
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140
achieve the error value as it has been measured on the construction site. The additional displacement for this example is -50.19 mm, which includes the small discrepancy as mentioned. The rotational stiffness value is found as given in the table below. Stiffness [tonf*m/rad] *104 50 41.653 42.815
Vertical Displacement [mm] -43.01 -51.63 -50.23
Target [mm] -50.19
∆ Stiffness [tonf*m/rad] *104 -8.347 +1.162 -/-
New Stiffness [tonf*m/rad] *104 41.653 42.815 -/-
Table 4-21: Required rotational stiffness obtained from MiDAS
In the stage of activating the elastic link with the calculated rotational stiffness, the vertical real displacement is -160.79 mm. In the next construction stage, the active elastic link must be replaced again by a rigid connection. There is also the possibility to model the error displacement with only one additional step. However, it is easier to control the different changes by using different construction stages. In this example, it has been assumed that the segment is installed with an error in the girder elevation which goes downward. Due to the temporary installation of an elastic link and the self-weight of the girder, the required error can be modelled. This procedure is not possible in case of an upward error. In this case, an elastic link can be used but additionally an external force, which produces the required deformation, must be applied. The load must be removed in the next step and at the same time, a fixed connection between both girder parts must be activated. Since the internal forces do not change, the moment distribution and the cable forces of both systems are identical. At the final state, the vertical deformation for the original system (as already given in Figure 4-34) and the system including the erection error are given below. Vertical displacement [mm] of the girder (Case II model) Original system
System including construction error
Figure 4-40: Vertical displacement original system and system including error in girder elevation
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Due to the error in connecting girder G4 with the already installed girder G3 and because of the tangential erection of the following segments, the girder elevation changes in the subsequent construction stages and results in the final condition as seen in the deformed plot. The error can be compensated by changing the camber of the following segments, compared to the initial design with no additional errors. By modelling the error as described before, the General Camber function can be used to calculate the changes in the remaining segments. For the node at the tip of the segments, Table 4-22 gives the initially designed camber data and the required changes due to the error.
Orig. system Error system
N1 0.00 0.00
N4 -1.00 -0.98
N7 6.03 6.04
N9 14.52 14.51
N 12 32.38 32.35
N 16 52.85 103.02
N 20 64.98 165.39
N 22 69.56 195.08
Table 4-22: Fabrication camber data [mm]
Node 16 defines the end of Girder 4, which has been assumed to be installed inaccurately and therefore the camber of this segment is neglected or already included in the structure. Only the two remaining segments, which are not installed at this time of construction, can be changed and can adjust the error so the final structure has no deformation.
Figure 4-41: Fabrication camber [mm]
The fabrication camber data given in Table 4-20 is transformed into a graphical form including the error for the system.
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4.8 Cable elements in construction stages In this chapter, the influence of the element type for the cable stays, as well as the analysis method shall be investigated. During the construction, the cables may be installed with low initial tension values or, as the construction proceeds, this situation can occur in some other stages. A low tension value produces a higher deflection of the cable, which directly influences its stiffness. The effect of various sags to span ratio, according to the calculation method, is examined. For the following calculations a different structural system than in the examples given before is used. The new system is a harp type system as it can be seen below. The bridge is analysed under the distributed service load of 40 kN/m and various initial tension forces.
Figure 4-42: Structural system of harp type cable stayed bridge (dimensions in [m])
Table 4-23 gives the properties of the structural elements used for the model. Element Girder Tower
Cables
Girder Top Middle Bottom Top Middle Bottom
Area (cm2) 3200 2140 2360 2580 677 271 239
Moment of Inertia (m4) 1.130 0.248 0.321 0.395
Unit Weight (kN/m3) 24
Modulus of Elasticity (Mpa) 200000
24
200000
-/-
78
165500
Table 4-23: Property table for harp system
The cables shall be installed with an initial sag to span ration of 1/60, 1/80, 1/100 and 1/120 respectively. This can be achieved by applying a pretension force on the cable elements. The required pre-stressing values are given in Table 4-24. The calculation for the top cable with a sag to span ratio of 1/120 is given in the following. The value is controlled by a single cable model.
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Calculation for the top cable, sad to span ratio R of 1/120: From the cable area ACtop and the cable density γ the weight per metre cable length L is calculated. 2
A Ctop := 677cm
γ := 78
kN 3
m
kN wtop = 5.281 m
wtop := A Ctop ⋅ γ
The projected length ltop of the cable is: ltop := 3⋅ 45.7m
ltop = 137.1m
With the angle cosα t.m.b. between cable and girder, the weight qtop is determined (load per metre based on the projected length). 3⋅ 45.7
cos αt.m.b. :=
2
61 + ( 3⋅ 45.7)
q top :=
2
cos αt.m.b. = 0.914
wtop q top = 5.78
cos αt.m.b.
kN m
For a sag to span ratio R120, the maximal sag is given as 1 R120 := 120
ftop120 := ltop ⋅ R120
ftop120 = 1.143m
The required initial pre-stressing value Ttop120 is calculated from the horizontal component Htop120 of the cable force S(x). R=1/120:
q top ⋅ ltop
2
Htop120 := 8⋅ ftop120 Ttop120 :=
Htop120 cos αt.m.b.
Htop120 = 11885.9kN
Ttop120 = 13009.3kN
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Control of a single cable In order to control this calculation and the MiDAS programme, the top cable is modelled separately. Fixed boundaries are applied to the upper node and a nodal load of Htop120 of 11885.9 kN is applied in horizontal direction at the lower node. In the Element Table for each element, a pretension load Ttop120 of 13009.3 kN is entered. Figure 4-43 shows the generated model and the calculated maximal displacement perpendicular to the undeformed cable.
Figure 4-43: Non-linear analysis of a single cable (cable 6 in the model Figure 4-42) [m] and [kN]
The maximal sag, which is calculated as follows, shows only a small divergence to the intended value ftop120. fMiDAS := 1.0467m
fmax :=
fMiDAS cos αt.m.b.
fmax = 1.146m
Furthermore the cable force at the lower end shall be compared. With the vertical distance of the two anchorage points htop, the cable force is calculated as h top := 61m
h top wtop ⋅ ltop S0 := Htop120⋅ 1 + − ltop 2⋅ Htop120 The value obtained from MiDAS is S0_MiDAS := 12854.44kN ⋅
2
S0 = 12866.45kN
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The formulas used for the calculation assume a parable shape of the deformed cable and a maximal sag at the mid span, whereas the computer calculates on the basis of a more exact catenary model. The small discrepancies in the results obtained from the formulas and the computation may be influenced by these differences. After the calculation is controlled with satisfying results, the pretension values given in Table 4-24 are applied for the different cases of sag to span ratio. R=1/60 6504.7 1735.9 765.4
Top Cable Middle Cable Bottom Cable
1/80 8672.9 2314.5 1020.6
1/100 10841.1 2893.1 1275.7
1/120 13009.3 3471.7 1530.9
Table 4-24: Initial pretension according to the sag to span ratio [kN]
For a non-linear analysis, the effect of the initial cable sag, which is a function of the initial pretension of the cable, is shown in Figure 4-44. The differences between the deflection values for R=1/60 and R=1/80 are higher than those for R=1/100 and R=1/120. On the other hand, the differences between the initial cable forces are the same. For example, for the top cable, the increase in the initial tension is constantly 2168 kN. This behaviour shows the non-linearity of cables. 0,4 Tower
0,3
CL
Deflection [m]
0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 -0,5
R=1/60
R=1/80
R=1/100
R=1/120
Figure 4-44: Deflected shape of the girder due to non-linear analysis and different initial tension [m]
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Figure 4-45 shows the results from a linear analysis (truss elements), a linear cable analysis (Ernst truss) and those from a non-linear cable analysis (elastic catenary cable analysis). In the case of a linear analysis, especially the displacement values obtained for high and low sag to span ratio show a large discrepancy to the values determined by the elastic catenary cable analysis. 0,4 Tower
0,3
CL
Deflection [m]
0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 -0,5
R=1/60
R=1/80
R=1/100
R=1/120
Figure 4-45: Comparison of deflected shapes, Elastic centenary cable (point + joint line), Ernst truss (point), elastic truss (line) [m]
In practice, an initial sag ratio R less than 1/100 is usually used [13]. For these ratios, the results illustrated for the equivalent cable stiffness proposed by Ernst show close results to the catenary cable elements. For this reason, the method has been adopted by many investigators. Nevertheless, judging from the assumptions used in the theory, the method holds in limited cases for cables with low sag to span ratio. Therefore, particularly in order to perform more reliable and reasonable analyses for each stage of the construction, a more exact method should be used instead of using the Ernst truss elements. However, at the time of working out this document and performing a construction stage analysis for the Second Jindo Bridge, MiDAS did not offer the option of using catenary cable elements, which would imply the performance of non-linear analyses included in the construction stage analysis. A non-linear analysis is not possible in combination with a construction stage analysis. The MiDAS support stated that they are trying to solve the related problems and they may be able to present a version providing these functions in a couple of month (see appendix). For that reason, for a final structural analysis, the decision is whether to choose a linear analysis with truss elements or cable elements in a linear analysis which considers an equivalent stiffness. From the results given above, it seems to be reasonable to use the Ernst truss formulation. However, it must be mentioned that, in order to obtain the above given reasonable results, all loads have to be stored in one load case. In the construction stage analysis various loads are
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removed and activated during the different construction stages. It must be proved wheter the loads, which are activated and deactivated in one construction step, are treated separately or if they are automatically stored in one load case. The last condition is a requirement in order to use the Ernst truss elements in a construction stage analysis; otherwise, the results will not be reliable. In order to prove the approach used by MiDAS to consider loads in a construction stage analysis, the cable system of the harp type is investigated in more detail. A simplified construction stage analysis is modelled, consisting of 11 steps. The cable stays are modelled by Tens–Truss Cable elements (Ernst truss). In the first stage, the pylon, the elements of the left side of the bridge and the first segment of the main span are activated. In the following stages, the remaining segments and cables are added separately step by step. For each construction stage, the self-weight of the structural member plus a construction load of 40 kN/m is applied. The following three different calculations are performed: 1. The self-weight and the construction load are stored in one load case 2. The self-weight is stored in one load case. The construction load is splitted in two extra load cases, all loads are activated the same day 3. The different loads are modelled as given in 2. Additional time steps are generated in the construction stages and for each stage, the loads are activated on different days for the same structural system. The examined results show the same values for the first two cases. It signifies that the load activation for each construction stage is treated as one load case. Consequently, the equivalent stiffness is calculated on the basis of the load changes in the system, which means that Ernst truss elements can also be used in a construction stage analysis. However, the results produced by the analysis of the third case are highly different. Therefore, in order to use the Ernst formulation, the loads must be activated on the same day. Load activations on different days are treated as different load cases and the results are superposed, which causes wrong results. During the step by step construction process, the stresses in the already erected cable stay change due to the different loading conditions. Thus, it must be finally proved if the effective stiffness of the cables is adapted to the various tension forces throughout the analysis. To investigate this condition, the generated construction stage model of the harp type is used for further calculations.
148
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Cable 4 is erected with a pretension load of 800 kN before Cable 3 is activated in the next stage with an pretension force of 1500 kN (bottom cables). The tension forces in these tow cables are compared using truss and cable elements (Ernst truss) for the stay cables. In the linear truss model, the maximal tension load in Cable 4 is 818.95 kN after installing the cable. Due to its self-weight, the load is higher than the initially applied 800 kN. However, the load increases to 1488.49 kN after the erection of Cable 3. Erection of cable 4
Erection of cable 3
Figure 4-46: Cable installation in the linear truss model [kN]
In the Ernst truss model the same initial pretension loads are used, but the tension value in Cable 4 is only 1390.16 kN after the activation of Cable 3. Because of a low initial pretension, this great difference is caused by the reduction of the cable stiffness due to the sag effect. Erection of cable 4
Erection of cable 3
Figure 4-47: Cable installation in the Ernst truss model [kN]
With the tension force obtained from the Ernst truss model, the effective stiffness Keff can be recalculated. In order to achieve the same stiffness in the elastic truss model as in the Ernst model, the modulus of elasticity can be changed in the input data. In the first analysis (Truss E1), the new modulus of elasticity Enew, which is applied in the truss model, is recalculated form the initial cable force of 818.95 kN.
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149
With the tension force Sb (b stands for bottom cable) and the cable area ACb, the modulus of elasticity Eb, the weight per metre cable length wb and the projected length of the cable lb, Sb := 818.95kN ⋅ kN wb = 1.86 m
2
A Cb = 239cm Eb := 165500
N
lb := 45.70⋅ m
2
mm
the effective elasticity can be calculated using Formula 3-34 2 2 γ ⋅ lb 1 Etanb := + Eb 3 12σ1
−1 4 N
Etanb = 3.0902× 10
2
mm
The value Enew is entered in the truss model. Because of the changed elasticity, the cable force is reduced from 1488.49 kN to 1184.36 kN. Compared with the tension forces in the Ernst model, the cable force is too low, which means that the stiffness is too low. For the second analysis (Truss E2), a new equivalent E-modulus is calculated with the tension force of Cable 4 after the erection of Cable 3. The new calculated modulus of elasticity is given in the result table below. Because of the higher tension force, the value Enew increases as well. Table 4-25 gives tension forces obtained from the different calculations and the applied modulus of elasticity. Ernst model Erection Cable 3 Cable 4 stage [kN] [kN]
Cable 3 [kN]
Cable 4 [kN]
Modulus of Elasticity [N/mm²]
Cable 3 [kN]
Cable 4 [kN]
Modulus of Elasticity [N/mm²]
Cable 4 Cable 3
-/1518.95
818.95 1184.36
3.0902*104
-/1518.95
818.95 1391.13
8.7543*104
-/1518.95
818.95 1390.16
Truss E1
Truss E2
Table 4-25: Tension forces in cable 3 & 4 due to adapted stiffness
For the truss and the Ernst model, the result of the second analysis shows very similar tension forces for both construction steps. Since the effective stiffness and the tension load in the cable affect each other, the erection of Cable 3 will influence both of it in the already erected Cable 4. Application of the same effective stiffness for Cable 4 in the linear truss model, as calculated from the tension force in the Ernst model after the installation of the second cable, results in the same tension. Therefore, it can be concluded that the effective stiffness of the Ernst truss elements is recalculated for each new construction stage considering the load changes in the stay cable. As a result, it seems to be possible to use the Ernst truss formulation as implemented in MiDAS in construction stage analysis.
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Chapter 5: Model of the Second Jindo Bridge
5 Model of the Second Jindo Bridge
The following chapter describes the modelling and the construction stage analysis of the Second Jindo Bridge. After the description of the considerations made in order to generate an appropriate model for the analysis, the boundary conditions and the assumed loading during the construction and for the final state are given in detail. The unknown load function, as described above, is applied for attaining the ideal cable forces. Using various structural restrictions, different solutions of possible cable forces are determined. A backward and a forward analysis is performed and the results are compared. The influence and problems of the double activation of elements and the boundary activation is discussed. Furthermore, for the final analysis, the effect of the non-linearity of the stay cables is considered. The obtained results are compared with other independent construction stage analyses. The camber data is provided to achieve a construction with no deformation under permanent loads after the completion of the erection. Additionally, in order to ensure a safe erection process, the occurring maximum and minimum stresses are controlled. The calculation of the unstressed cable length is exemplified. Finally, a closing conclusion on modelling the erection process of the Second Jindo Bridge is drawn.
5.1 Location of the bridge The Second Jindo Bridge is located in the southern part of the Republic of Korea. The construction site is marked on the map shown in Figure 5-1. It is under construction now and will be finished by the end of the year 2004. The bridge is built to connect the main land from the Mokpo side with the Jindo Island. A cable-stayed bridge was constructed in 1984 at the narrowest points of the straits. It enables the bridge to have the shortest possible span without the use of marine foundations. The new bridge will be the second crossing to ensure a reliable transportation connection.
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151
Figure 5-1 Location of Second Jindo Bridge
At the beginning of the project, site surveys were carried out to determine possible route alignments for the bridge. Eight alternative crossing locations had been investigated. The associated risks and the work-scope for each option were discussed. After considering the bridge length, the distance from the existing bridge, the aerodynamic complicity and several additional location factors, it was concluded that the most economical solution is to duplicate the existing bridge. The new bridge is being constructed as closely as possible to the existing one to enable the same short span. In the area of the existing north tower foundation, a major rock fault dips steeply toward the east. If the new structure were to be constructed to the east of the existing bridge, a very deep and expensive excavation would have been required for the construction of the foundation of the north tower. Out of this cognition, it was decided to locate the new bridge to the west side of the existing structure. The dimensions of the main tower legs determine the minimum distance of the existing and the new structure. The new tower leg has a similar width as the existing one and is positioned adjacently. The centre line separating the existing from the new superstructure is around 20 metres, which can be seen on the provided plans.
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5.2 Structure The Second Jindo Bridge is a cable-stay structure with the total length of 484 m. The main span is 344 m long and both side spans consists of 70 m. As already mentioned, the new bridge has a similar main span as the existing one. For aesthetic reasons, the same side span arrangement is retained. To realise a cable stay spacing of 17 meters, as already used for the first bridge, the main girder is of a steel box girder structure, which is based on the deck structure of the same external dimensions as the existing structure. The bridge will accommodate two lanes of traffic on the 12.55 m wide deck. The superstructure is made out of hollow steel sections, whereas the tower substructure is made of reinforced concrete. The side to main span ratio ls/lm has a low value of 70.00/344.00 = 0.20. Therefore, the box girder in the side span is filled with concrete to act as a counterweight. Detailed plans on the girder sections and the cable diameter employed in the bridge are given in the appendix. The arrangements and dimensions of the cross sections of the main tower can be found in the plans in the appendix as well. The main dimensions of the bridge are shown in the plan provided in the appendix. A summary of the geometric data can be seen in the table below. Geometric Data Second Jindo Bridge Length main span 344.00 m Length side span 70.00 m Height pylon above girder 65.40 m Total height pylon 88.90 m Number of cables main span 9 Number of cables side span 6 Cable spacing main span 17.00 m Segment length 17.00 m Length key-seg 12.00 m Table 5-1: Main Geometric Data Second Jindo Bridge
5.3 Erection Options This chapter describes the erection of the Second Jindo Bridge. The erection of the towers and side spans are straightforward operations on land and are therefore described only briefly. The erection of the main span deck steelwork is significantly more complex. In the beginning, a
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153
number of options have been considered. These options are briefly described and the merits of the finally chosen one are explained.
5.3.1 Erection of the towers and side spans The tops of the towers are within the reach of a large crawler or a mobile crane so that these have been selected for the erection. This method is simple and economical. The leg members are divided into four segments and the tower head is split into three segments so that they are suitable for the transport and the lifting. The different prefabricated parts can be seen in the appendix. Because of the possibility of aerodynamically induced vibrations of the towers in the longitudinal direction caused by vortex shedding, a damper must be provided during the construction until the first pair of cables is installed. The erection of the side span box girder will also be achieved by using a crawler or a mobile crane. Four temporary supports will be positioned in the side span until its cable stays are installed. Additionally, Segment 6 (first segment in the main span) will be supported until Cable 7 (first cable in the main span) is being stressed.
5.3.2 Erection of the main span The three most practical erection options that could have been applied for the construction are listed below. Option1: In this option, the complete box girder segments are delivered to their erection position by a barge and lifted directly to their final position by a derrick crane. With a dynamic positioning control system linked to a number of steerable propellers, the delivery barge must be in an accurate position beneath the erection point until the deck segment has been lifted from the barge. To lift the segments, the crane is positioned at the end of the previously erected section. The drawings in the appendix illustrate this operation. Option2: In this option, the complete box girder segments are delivered by a barge to the shoreline under the already erected deck. A gantry crane supported on rails at the edges below the deck will lift the segments and move them out to its erection position. A lifting device
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positioned on the end of the previously erected deck will then lift up the segment to connect it to its final position. Option3: Another option is to divide the box segments along their longitudinal centre line into two parts. The half boxes are delivered to the side by a barge, lifted up to the deck level, moved through the gap between the first cable stay and the tower, and placed on a trailer. Each half box will then be brought out on this trailer to the end of the cantilever from where it will be lifted into its final position using a derrick crane. In option 1, it is most difficult to maintain the delivery barge in its position with sufficient accuracy over the period of time required to lift the deck segment clear. As an advantage, it can be mentioned that the number of lift and transfer operations is reduced to a minimum. The only lifting operation is that from the delivery barge to the final segment position. Consequently, taking these aspects into account, this is the most favourable option. In option 2, the complexity of segment movements and lifting operations is of considerable increase and is thus not suitable for the construction process in comparison with the other options. In option 3, the problem is that the half box segments must be produced to a high level of accuracy to ensure a trouble-free fit on the site. The production of half box segments with consistently accurate cross sectional geometry is more difficult and expensive than that of complete boxes. Furthermore, the additional site splicing work would demand more time for each erection cycle in the main span and might be unacceptable for the overall programme.
5.4 Modelling of the bridge with MiDAS The main purpose of the model is to analyse the structural members, as well as the behaviour of the bridge during its construction. Since there are no dynamic effects considered in the analysis, a 2-D model is sufficient for the investigation. In addition to it, the symmetric shape of the bridge can be used for further simplification. Only one half of the structure is generated and the boundary conditions are adjusted. The provided planes and cross section tables of the bridge are given in the appendix. Unfortunately, in some cases the available data sowed different values or lacks in information. In these cases, reasonable assumptions have been made.
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5.4.1 Nodes The drawings of the bridge outline the final geometry of the bridge. In geometric modelling, the nodal location, or rather the way of applying the section properties to the generated elements must be considered. If no other eccentric connection is defined, the nodes lay in the centreline of the sections. In this model, the nodal coordinates of the girder define the upper part of the deck. In MiDAS, an eccentric connection is considered using the Offset function for the cross section properties. In the girder, a linear gradient of 5% can be found in the side span up to Segment 7, which is the first segment in the main span. With the girder elevation given as 20.00 metres at the abutment and 23.50 metres at the pylon, the girder elevation is defined by the function
y1 ( x) = 0.05 x + 20 .
(5-1)
The girder shape in the main span can be expressed as a following type:
g − g1 2 y ( x) = 2 x + g 1 * x + G.EL , 2L
(5-2)
where g2 and g1 describe the girder gradient at the start and end point of the function and G.EL the girder elevation at the point x = 0, located at the start point of the function. With Equation 51, the last value becomes to y1 (86.30) = 0.05 * 86.30 + 20 = 24.315 . The remaining girder length L between the first segments in the main span is 311.4 meter. In order to achieve a girder elevation of 28.35 metres in the centre of the bridge, as this value is given in the plans, the girder gradient is increased to 5.183%. With the calculated values, Equation 5-2 can be given as:
0.05183 − (−0.05183) 2 y 2 (x) = x + 0.05183 * x + 24.315 2 * 311.4
(5-3)
For the final modelling, the x-coordinates are located in the centre of the main span. Figure 5-2 illustrates the determined girder shape. The detailed node coordinates are given in the appendix.
Chapter 5: Model of the Second Jindo Bridge
156 30 28 26 24 22 20 -250
-200
-150
-100
-50 0 side span
50 main span
100
150
200
Figure 5-2: Girder elevation in the side and main span [m]
The pylon nodes are generated in the centreline of the pylon. Figure 5-3 illustrates the location of the cable-pylon connection as constructed on the site and the distance to the centreline as modelled for the analysis. Because of the dependence of the forces in the cables on the angle between the pylon and the cable, attention should be paid for modelling these details accurately. There are two possible ways of taking this situation into consideration: One is to use an eccentric connection between elements representing the cables and the pylon nodes, or to define extra nodes located in the working point (W.P.C) and to use ridged links to fix them with the pylon. The other option is to extend the cable elements and model the working point B (W.P.B.) to properly generate the angle of the cables. This second option is used for modelling the bridge in MiDAS. The calculation sheet for the working point B, as well as a list of the coordinates of nodal location, is presented in the appendix. The same situation is found in the girder–cable connections, but in this case, additional nodes are generated. For Cable 1 to 4, the extra nodes are located –1.71 metres in dz and -1.84 metres in dx direction; for Cable 5 to 15, the nodes are generated -0.54 metres in dz direction relative to the girder nodes. Figure 5-5 shows the nodes and ridged links used to connect the nodes with the girder nodes. By this method, the angle between the pylon-cable and the girder-cable can be modelled properly, so that the determined cable forces will consider the real existing angle.
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Chapter 5: Model of the Second Jindo Bridge
Figure 5-3: Working points at the pylon
157
Figure 5-4: Working points at the girder
Figure 5-5: Cable-girder connection and tied down condition using elastic links
5.4.2 Elements Beam elements are used to model the girder and the pylon. Generally, the element has 6 degrees of freedom per node, reflecting axial, shear, bending and torsional stiffness. Since a twodimensional analysis is performed, in the Structural Type option the model is defined to be in the x-z plane, so that the degrees of freedom in dy-direction are ignored. The beam element is formulated on the basis of the Timoshenko Beam Theory reflecting shear deformations. Concentrated loads, distributed loads, temperature gradient loads and prestress loads can be applied to beam elements. The cables are modelled with a truss element, which means that sagging effects are neglected. Firstly, it is assumed that these are negligible, but they are included in the final construction stage analysis. The set back is a common procedure in the construction of cable-stayed bridges. This is usually done before the closure and the addition of the key segment. This is a highly non-linear process, and using truss elements in a finite element analysis may cause significant errors. The set back
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158
is not modelled in this analysis but it should be mentioned to consider this problem in case of any other analyses in which it is included. In a final analysis, the sagging effects of the cables are considered and the truss elements are changed to Tension-Truss Cable elements, which consider the effective stiffness in a linear analysis by applying the Ernst-formula. Figure 5-6 shows the element location and its numbering. An element table is also given in the appendix.
Figure 5-6 Element numbers
5.4.3 Material Properties Table 5-2 shows the materials defined for several structural parts. For the girder, the different steel definitions have no meaning as the density is set to zero and the self-weight is applied as an external distributed loading. For the cables and the pylon, the weight is considered in the Self-Weight function. The extra loading from the coating is integrated in the increased density.
Chapter 5: Model of the Second Jindo Bridge ID
Name
Typ
1 2 3 4 5 6 7 8 9 10
Steel_g=10.60;0 Concrete Cable Steel_g=14.52;0 Steel_g=10.99 Steel_g=9.42 Cb(1-6)*1.0437 Cb(7-9)*1.0472 Cb(10-13)*1.0537 Cb(14-15)*1.0453
User Defined User Defined User Defined User Defined User Defined User Defined User Defined User Defined User Defined User Defined
159 Elsaticity [tonf/m^2] 2.10e+007 2.50e+006 2.00e+007 2.10e+007 2.10e+007 2.10e+007 2.00e+007 2.00e+007 2.00e+007 2.00e+007
Poisson 0.296 0.167 0.235 0.296 0.296 0.296 0.235 0.235 0.235 0.235
Density [tonf/m^3] 0.0000 2.5000 7.8500 0.0000 10.9900 9.4200 8.1930 8.2205 8.2715 8.2056
Table 5-2: Material property table
5.4.4 Section Properties The main girder is composed of four section types. Type 1 is used for the middle span. At the pylon, where higher moments occur, Type 3 is installed. As mentioned before, the sections in the side span are filled with concrete to act as a counterweight. In this area, section Type 2 is used. At the end of the side span, where the four backstay cables are connected to the girder, a very stiff and heavy steel girder section is constructed to resist the high forces; here, Type 4 is utilized. The change from section Type 4 to 2 is represented by a cross section using the average value of both sections. Above, it is explained that the offset function is applied to consider that the generated nodes are located on the upper deck level of the final geometry. The values Cyp and Cym, Czp and Czm define the centroid of the segments. For the concert part of the pylon, cross section 6 to 8 is employed. Since there are no exact dimensions available in the plans given in the appendix, assumptions are made in order to calculate the cross section properties. However, as this part is very stiff, it does not have much influence on the overall deformation of the structure. Four different diameters are used for the cables. These can be found in Table 5-3 in the cross section number 15 to 18. Because of the two dimensional modelling, the section area of each cable is duplicated.
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Figure 5-7: Material numbers
For all cross sections, the values are entered as listed in the table below. The values for the cross sections have been provided by the bridge engineering company COWI KOREA and can be found in the appendix. It should be noted that the local axes of the given sheets and the MiDAS input data are inverted. ID
Name
1 Section1 2 Section2 3 Section3 4 Section4 5 SectionV 6 Pylon1 7 Pylon2 8 Pylon3 9 Section5 10 Section5-7 11 Section7 12 Section9 13 Section11 14 Section12 15 Cable(2*151) 16 Cable(2*139) 17 Cable(2*109) 18 Cabel(2*73)
Area [m^2]
Asy [m^2]
Asz [m^2]
Ixx [m^4]
Iyy [m^4]
Izz [m^4]
Cyp [m]
Cym [m]
Czp [m]
Czm [m]
0.4624 0.4859 0.7593 0.9360 0.7109 59.1200 33.6200 98.1437 0.7872 0.6812 0.5751 0.6675 1.0159 0.6070 0.0116 0.0107 0.0084 0.0056
0.2627 0.3285 0.4869 0.5031 0.4158 100 100 100 0.2105 0.181 0.1515 0.1786 0.3403 0.1795 0.0105 0.0096 0.0076 0.0051
0.0225 0.0217 0.047 0.0662 0.044 100 100 100 0.2887 0.2465 0.2044 0.2549 0.3621 0.3351 0.0105 0.0096 0.0076 0.0051
1.314 1.284 2.390 3.589 2.437 923 289 100 1.000 1.000 1.000 1.000 1.000 1.000 0 0 0 0
0.565 0.5696 0.9738 1.421 0.9952 116 102 100 0.7642 0.6629 0.5616 0.645 1.080 0.5074 0 0 0 0
6.283 6.777 10.376 16.888 11.328 697 492 100 0.510 0.441 0.372 0.442 3.583 0.680 0 0 0 0
6.095 6.093 6.090 6.090 6.090 0 0 0 0 0 0 0 0 0 0 0 0 0
6.095 6.093 6.090 6.090 6.090 0 0 0 0 0 0 0 0 0 0 0 0 0
1.005 1.003 1.407 1.622 1.622 0 0 0 0 0 0 0 0 0 0 0 0 0
1.745 1.747 1.354 1.164 1.164 0 0 0 0 0 0 0 0 0 0 0 0 0
Table 5-3: Cross section table
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161
5.4.5 Boundary Conditions At the final state of the construction, the side spans are supported in the vertical direction at the anchorage block. To model the tied down condition from the girder, Node 3 and Node 1001 are fixed using an elastic link defined in the local dx direction. Node 1001 is fixed in all degrees of freedom. Figure 5-5 shows this supporting condition. At the pylon, the girder is held in place by a rubbery support, which is vertically very stiff and horizontally loose. During the construction, the side span is supported by four temporary bents. Segment 6 is supported by a truss structure connected to the pylon, which is removed after the installation of Cable 7. This state is modelled with an additional temporary support in the vertical direction at Node 17. Table 5-4 gives the node numbers and supporting conditions during and after the construction. The elastic link table is provided in the appendix. Node
dx
dz
ry
Group
8 10 12 13 17 36 101 1001
0 0 0 0 0 1 1 1
1 1 1 1 1 0 1 1
0 0 0 0 0 1 1 0
Bent8 Bent10 Bent12 Bent13 Bent17 Middle Pylon Left Sup
Table 5-4: Boundary table
5.4.6 Loading For the construction stage analysis, the dead load and the construction load are considered. Additional dead loads representing the pavement, the installations on the bridge, etc. are applied after the completion of the bridge. To calculate the maximum tension forces in the cables and the maximum bending moments in the girder, the traffic load is applied on the structure to simulate the open condition of the bridge. The different loads are explained below. The sensitivity of cable-stayed bridges to wind loads is often greater during their critical construction phases, which are before the closure of the bridge, than at other times during the service of the structure. High bending moments may occur from buffeting forces of the wind and an increased probability of vortex-shedding induced oscillation due to the lower weight of
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162
the deck and less structural damping. In cases of long span cable-stayed bridges, wind tunnel tests are often performed to study the behaviour of the aeroelastic model during the erection. However, this is a completely different field of analysis and should only be briefly mentioned. The possibility of static wind loads is neglected in the performed analysis of the Second Jindo Bridge.
5.4.6.1
Permanent Load
Table 5-5 below shows the given values of the dead load for the segments. Unfortunately, there were uncertainties because the calculated distributed loads (Line 5) do not match the given values of total segment loads (Line 3) and segment length (Line4) when divided by one another. For the MiDAS model, the values are recalculated as shown in Table 5-6. Line 7 is the dead load for the concrete weight, filled in the segments in the side span. The filling is done in two steps. Half of the concrete is filled after the erection of the side span. The rest is placed in the boxes in the final state after removing the derrick crane. Line 8 gives the load for the additional dead load applied after the last construction step.
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Table 5-5: Segment load table
Calculated distributed Dead Loads for the Segments Load [ton/m]
Seg 1
Seg 2
Seg 3
Seg 4
Seg 5
Seg 6
Seg 7
Seg 8
Seg 9
Seg 10
Seg 11
Seg 12
Seg 13
Seg 14
Seg 15
Key Seg
19.345
4.019
4.539
4.641
11.879
4.99
4.825
4.755
4.761
4.768
4.777
4.786
4.796
4.814
4.826
4.604
Table 5-6: Calculated distributed load
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164
5.4.6.2
Construction Load
A heavy lifting device is required for installing new segments. The self-weight of the crane will have severe influence on the maximum bending moments in the previously erected segments. In order to perform a reliable construction stage analysis, it is important to have exact information on the self-weight of the crane and also to assume reasonable values for the additional construction load. In some bridge constructions, the influence of these loadings was underestimated. A self-weight of 50% less than the actual weight of the crane resulted in serious errors in the calculations. For this calculation, a total derrick weight of 85tonf is assumed according to the data provided by COWI KOREA. Furthermore, a distributed construction load of 1.00 tonf/m is applied on the segments. In the given construction process, it is assumed that this load is being removed after closing the bridge. The derrick crane is removed in the same step. Compared to the self-weight of the segments, the 1.00 tonf/m value seems to be very high. It may be reasonable to consider removing this load in parts as the construction of the new segments continues. In a later discussion, the hired construction company stated that they actually presumed this change in their construction analysis. After installing a new segment the construction load of 1.00 tonf/m on the segment before the last one is removed. During the time of cable installation, a working platform is moved to the tip of the last segment. After positioning the cable, the working platform is moved back until it is needed for the next cable setting. A weight of 10 tonf is estimated for the working platform. An additional positive effect of this platform is that, during the stressing operation of the new installed cable, the upward movement and the high bending moments can be reduced by the weight of the platform at the tip of the cantilever.
Chapter 5: Model of the Second Jindo Bridge
5.4.6.3
165
Live Load
Figure 5-8 shows an abstract from the Korean Standard for traffic loads on bridges.
Figure 5-8: Traffic load Korean Standard
Since the Second Jindo Bridge has two lanes, a distributed loading of 2,54tonf/m is assigned. After the last construction step, the load is applied in order to calculate the approximate maximum cable forces and the bending moments. Additionally, two lorries (Pm for max. moment) of a weight of 10.8 tonf are considered. Service pipes will be installed under the main girder. A load of 0.283 tonf/m is assumed for the pipes. A detailed analysis is usually performed in the design process and therefore, for the construction stage analysis, no exact study is carried out. To consider the traffic load in the minimal and maximal stresses in the structure, a moving load analysis is performed.
5.5 Initial Cable Forces In order to calculate the initial cable forces when installing a new cable, the Unknown Load Factor function is used again, as already explained in detail before. Different restrictions are made to optimise the structure. An optimised case representing a very small deformation, smooth bending moments in the girder and no moment in the pylon is presented. In this case, the tension forces in the first four cables, which can be seen as one backstay cable, are widespread and exceed the allowable forces. Since this solution is not realistic, a more practical solution is introduced.
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5.5.1 Optimised Structural Moments An extensive study is carried out using various restrictions to calculate an optimal moment distribution in the structure, including low moments in the girder and pylon top and bottom. The best solution is found applying the following restrictions:
Side span dz [m] Main Span dz [m] Top of Pylon dx [m] Beam Moments [tonfm] Truss Force [tonf]
Node/Element 2, 4,5, 6, 9 and 11 18, 20, 22, 22, 24, 26, 28, 30, 32 and 34 153 101 12 504
Upper Bound 0.01 0.005 0.005 300 800 750
Lower Bound -0.01 -0.005 -0.005 -300 0 0
Table 5-7: Unknown Load Factor restrictions
The moment in Element 12 is restricted so as to balance the moments in the girder on the left and right hand side of Cable 6. To reduce the moment in the pylon, Element 101 (at the bottom of the pylon) is limited to 300 tonfm. Figure 5-9 shows the resulting moment distribution in the bridge. The maximum moment at the top of the pylon is -179 tonfm and at the bottom 300 tonfm. The maximum moment at the anchorage of the backstays is -2008 tonfm. At the pylon, the moment in the girder is -1425tonfm.
Figure 5-9: Moment distribution restricted displacement [tonfm]
Table 5-8 shows the cable forces. Even if the moment distribution above has a minimal range, it is remarkable that the forces in cable 1 to 4, which have the same diameter and their distance is only 1.60m, lay in a range of 260 tonf and 750 tonf. However, this solution may not be practical if the cable forces exceed the allowable values when live load is applied on the structure. If this should be the case, theoretically, the cable diameters must be changed to tension the cable to the calculated forces. Nevertheless, this is not an optimal solution due to the unbalanced cable forces in the backstay and the induced high bending moment at their fixed point. Furthermore,
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in the case of the Second Jindo Bridge, the cables are already fabricated and diameters can no longer be changed. Because of these reasons, the allowable tension forces are considered in the determination of the ideal state. Cable
Element Number
Internal Cable Force [tonf]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
350.43 580.56 802.64 852.06 493.04 607.54 130.34 156.17 176.60 213.47 248.57 290.31 322.65 360.43 489.12
External Cable Force [tonf] 260.47 486.38 703.35 750.00 408.98 541.66 128.04 159.86 182.17 221.08 254.51 294.68 325.77 362.96 490.80
Table 5-8: Theoretical ideal cable forces
5.5.2 Limited Cable Forces In contrast to the previously presented case, the maximum cable forces are now restricted. The allowable values are calculated in Table 5-9.
Cable
1-6 7-9 10-13 14-15
Section Area fpu ID [cm^2] [kgf/cm^2] Cable (2*151) Cable (2*73) Cable (2*109) Cable 2*139)
Allow. Tension Construction (0.56*) double per Cable Cable
Allow. Tension Structure in service (0.45*) double per Cable Cable
Restricted Value per double Cable Cable
116.2
18000
585.75
1171.50
470.69
941.38
300
600
56.2
18000
283.15
566.29
227.53
455.06
145
290
83.9
18000
422.86
845.71
339.80
679.59
215
430
107.0
18000
539.18
1078.36
433.27
866.54
275
550
Table 5-9: Allowable tension forces in [tonf]
During the construction phase, cable stays are subject to erection loads, which can be greater than those expected in the service lifetime of the structure. Since the duration of these
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168
construction-phase situations is short, the acceptable stress ratio can be increased in cables stays for this period. The tension of a cable stay should prove: FSLS < 0.56 Fck during construction and FSLS < 0.45 Fck for the serviceability limit state. These values FSLS, as well as the allowable tensile stress fpu, were given according to the Korean Standard and the cables applied in the structure. Since the bridge is not analysed in full detail in this report and the analysis is focused on the construction stage analysis, there has been no live load applied until now. Nevertheless, these loads must be considered and therefore the cable forces are limited to the “Restricted Values” as it can be seen in Table 5-9. Later, the cable forces and the limited values must be checked against each other. In addition to the restrictions of the cable forces, the following conditions were defined for the analysis of the ideal state. In order to find an equilibrium state, the limit range of the deformation must be set higher in comparison to the previous calculation. Using the same restrictions for the deformation as applied before, it is not possible to fulfil the specified conditions. Using the newly calculated cable forces, Figure 5-10 and Figure 5-11 respectively show the moment distribution and the deformed shape.
Side Span dz [m] Main Span dz [m] Top of Pylon dx [m]
Node/Element 2, 4,5, 6, 9 and 11 18, 20, 22, 22, 24, 26, 28, 30, 32 and 34 136 and 153
Upper Bound 0.03 0.008 0.008
Table 5-10: Additional Unknown Load Factor restrictions
Figure 5-10: Moment distribution restricted cable forces [tonfm]
Lower Bound -0.03 -0.008 -0.008
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169
Figure 5-11: Displacement dz restricted cable forces [mm]
The maximum moment at the top of the pylon is now -388 tonfm and 590 tonfm at the bottom. The maximum moment in the girder is -1481 tonfm at the anchorage of the backstays and -1436 tonfm at the pylon. The horizontal movement at the top of the pylon is -8mm, which is still within an acceptable range. Table 5-11 presents the newly calculated cable forces as compared to the results calculated in Chapter 5.5.1. Cable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Element Number 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
Cable Force previous Calculation External[tonf] 260.47 486.38 703.35 750.00 408.98 541.66 128.04 159.86 182.17 221.08 254.51 294.68 325.77 362.96 490.80
Restricted Cable Force External [tonf] 452.74 512.69 568.20 600.00 493.04 511.79 129.03 157.19 184.33 220.76 257.03 283.47 328.60 397.95 460.74
Restricted Cable Force Internal [tonf] 547.61 610.16 669.07 702.48 589.13 577.88 131.12 152.85 178.12 211.86 249.52 277.30 323.89 392.87 454.90
Table 5-11: Summary table of ideal cable forces
In the optimised moment analysis (third column), the sum of cable 1 to 4 is 2200.20tonf. A change of 3.12% resulting in 2133.63tonf can be found in the second analysis. The difference in the forces is taken by a higher moment in the pylon.
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Due to the moment in the top of the pylon a simple stress check can be made as follows: M Py := 388.13tonf ⋅ ⋅m 4
NPy := −3629tonf 2
Iy := 1.0803m
A Py := 1.0159m
z := 1.430⋅ m KSCE - ASD96 fy := 2400⋅
kgf 2
cm fd := 2000⋅
kgf 2
cm σPy :=
NPy A Py
M Py
−
Iy
⋅z
σPy = −408.60
kgf 2
< fd
cm
EC 2:
fyk := 2400
kgf
γ m := 1.1
2
cm
γ f := 1.35 fyd :=
fyk
fyd = 2181.82
γm
A Py
2
M d = 523.98tonf ⋅ m
Nd := γ f ⋅ NPy Nd
kgf cm
M d := γ f ⋅ M Py
σd :=
self-weight
−
Nd = −4899.15tonf Md Iy
⋅z
σd = −551.60
kgf 2
< fyd
cm
The allowable stress is below the actual stress at the top of the pylon. In summary, it can be stated that, for a complex structure, the unknown load function is a very useful tool. However, on the other hand, it is also a highly sensitive approach to calculate the initial cable forces. In order to build a stable system, there are many possible solutions. The results previously explained exemplify only two reasonable ones after all. The results greatly depend on the restrictions imposed. For example, without the restriction for Node 136, the upper part of the pylon shows a double curved deformation, which may not be a satisfying outcome. When the internal cable forces are applied, the structural performance must be analysed carefully and changes for the restricted conditions will be often necessary. The calculated values act as a guideline, but must be still analysed and interpreted. As the applied method is a
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constrained optimisation, a limited range of the defined restrictions increase the internal stresses, which must be considered when choosing an upper and lower value for the restriction.
5.5.3 Limited cable forces and restricted bending moment In order to find reasonable ideal cable forces, a further calculation is performed, including a restriction of the cable forces and the bending moments in the main girder. The restriction of vertical displacements, as it has been used in the previous calculation, is removed. To prevent high bending moments in the pylon, the horizontal displacement is still constrained at Node 136 and 153. The final analysis should be performed by taking the influence of sag effects of the cables into consideration and hence, cable elements are to be used. Due to the sag effect of the cable, the effective stiffness will change. In order to find reasonable tension forces for the final state, two alternative solutions are investigated separately and compared afterwards. The restricted values of the cable forces are used as given in Table 5-9. Table 5-12 shows the additional conditions applied for both cases. Case A
J-nodes at element
Side Span My [tonfm]
Main Span My [tonfm]
Top of Pylon dx [m]
Case B
8 10 17 19 21, 23, 25, 27 29 31 33
Upper Bound -640 -730 -150 -170 -170 -170 -190 -225
Lower Bound -/-/-160 -180 -180 -180 -200 -235
Upper Bound -640 -730 -175 -190 -195 -215 -215 -225
Lower Bound -/-/-185 -200 -205 -225 -225 -235
136 and 153
0.008
-0.008
0.008
-0.008
Table 5-12: Unknown Load Factor restrictions including limited moments in the main girder
The value for the moments restrictions are obtained using the result from the previous calculation. The moment distribution obtained from the previous calculation, which is given in Figure 5-10, shows much higher moment values at the anchorage point of the cable than found in the girder between two cables. Therefore, in order to increase the field moments and to reduce the moment at the anchorage point, in Case I, a value of -150 to -225 tonfm is chosen. In Case II, the negative moments at the anchorage points are increased from -175 to -225 tonfm to consider some effects of the cables. Furthermore, the moments in the side span are restricted. The final moment distribution is given in the figures below.
172
Chapter 5: Model of the Second Jindo Bridge
a) Case A
b) Case B Figure 5-12: Moment distribution restricted cable forces & bending moments in the girder [tonfm]
In the Case A model, the maximum moment at the top of the pylon is now -404 tonfm and 572 tonfm at the bottom. The maximum moment in the girder is -1445 tonfm at the anchorage of the backstays and -1464 tonfm at the pylon. In the Case B model, the maximum moment at the top of the pylon is now -401 tonfm and at the bottom 588 tonfm. The maximum moment in the girder is -1458 tonfm at the anchorage of the backstays and -1276 tonfm at the pylon. The differences of both cases are small, but Case B has higher moments at the anchorage points in the main girder. The horizontal movement at the top of the pylon is in both cases -8mm (tower leans into the side span). Table 5-13 shows the calculated internal and external ideal cable forces in case of including the moment restriction for both analyses. The results show some differences when compared to the results obtained from the previous analysis (also given in the table); but in general, the values are not far apart.
Chapter 5: Model of the Second Jindo Bridge
Cable
Element number
Restricted cable force external [tonf]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
452.74 512.69 568.20 600.00 493.04 511.79 129.03 157.19 184.33 220.76 257.03 283.47 328.60 397.95 460.74
173 Restricted cable force & bending moments external forces [tonf] Case A Case B 469.11 465.35 513.50 512.46 552.75 554.34 595.70 599.92 481.74 480.90 532.31 540.50 126.60 135.87 157.90 157.19 184.24 184.65 220.30 220.31 254.37 254.38 291.39 288.97 329.00 334.44 377.01 377.82 476.51 472.40
Restricted cable force & bending moments internal forces [tonf]] Case A Case B 561.99 560.04 609.20 610.02 652.06 655.55 696.92 703.10 583.56 584.36 617.43 626.18 116.09 136.85 137.68 151.69 161.91 177.27 189.14 209.67 227.17 245.25 268.14 281.33 309.21 328.03 355.27 370.33 457.03 464.77
Table 5-13: Summary table of ideal cable forces including moment restriction
As already described in Chapter 4.2.2, in case of restricting the bending moments without any limitation on the vertical displacement, the deformation of the structure will increase. In this calculation, the vertical displacement extends to -40 mm in Case A and to -12 mm in Case B in the centre of the bridge. However, at this stage of the analysis, the deformed structure has no important meaning. Since the structural deformation changes due to some smaller variations in the cable forces and the following proceedings (back- and forward analyses), the camber calculation should not base on the data of the ideal state calculation.
5.6 Back- and Forward Analyses of the Second Jindo Bridge The performed backward and forward analyses are presented in the next chapter. The cable forces evaluated from the backward analysis are applied as initial cable forces in the forward analysis. Both results are compared. The construction stages are investigated and some possible changes are suggested. As already mentioned, the non-linear behaviour of the cables is considered in the final analysis by changing the element type.
174
Chapter 5: Model of the Second Jindo Bridge
5.6.1 Construction Stages The construction sequence, as it has been given initially, is presented in the appendix. After several calculations, this sequence is modified in some stages. In particular, the order of the cable installation is changed, which is described in more detail later on. The following tables give an overview of the construction stage analyses as modelled in MiDAS. The nodal loads, which are in the graphics, represent the derrick crane, the working platform and the lifting of a newly installed segment. In order to keep the graphics clear, the distributed loads are not shown. The permanent and temporary boundary conditions are also shown in the plots.
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175
Stage 0
Stage 1 – Add Crane
Stage 2 – L-seg7
Stage 3 – Seg7
Stage 4 – Cable6
Stage 5 – Cable7
Stage 6 – L-seg8
Stage 7- seg8
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176
Stage 8 – Cable5
Stage 9- Cable8
Stage 10 – L-seg9
Stage 11- seg9
Stage 12 – Cable9
Stage 13- Cable3
Stage 14 – L-seg10
Stage 15- seg10
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177
Stage 16- Cable 10
Stage 17 – L-seg11
Stage 18- seg11
Stage 29 – Cable2
Stage 20- Cable11
Stage 21 – L-seg12
Stage 22- seg12
Stage 23 – Cable 1
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178
Stage 24- Cable13
Stage 25 – L-seg14
Stage 26- seg14
Stage 27 – Cable14
Stage 28- L-seg15
Stage 29 – seg15
Stage 30- Cable4
Stage 31 – Cable15
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179
Stage 32- L-key-seg
Stage 33 – key-seg
Stage 34- Move Derrick
Stage 38- Remove Horizontal Fix Pylon
Stage 35 – Closing the bridge
Stage 36- Re-Tension
Stage 37 – Remove Derrick
Stage 39 – Remove Constr. Load Apply Additional Load
Seg: Segment; L-seg: Lift segment; load values in [tonf]
Figure 5-13: Construction stages 1-39
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180
5.6.2 Modified Construction Stage Analysis Double cable activation: The original construction stage data recommends that Cable 6 and 7 (Element 506 and 507) should be installed simultaneously in Step 8. From the point of modelling this condition, it is found that the order of activation and stressing the cables have severe influence on the resulting forces of these cables. There is a great difference in the initial tension force for the cables whether both are activated and stressed in one step or only activated at the same time but tensioned one after another in different stages. In order to maintain reliable initial tensioning values for Cable 6 and 7, construction step 8 is modified. Keeping in mind that, at the construction site, it may cause problems to concurrently ensure cable installation and tension, it is suggested to install and tension Cable 6 first and then to erect Cable 7 in the next step. This order is also illustrated in the sequence above. Removal of temporary supports: The temporary bents in the side span are removed after the cable installation. This can be seen in Step 8 and 10 of the original data. Since there is still compression in the bents at Node 12, 13 and 17, the supports remain until Cable 7 is installed as it can be seen in Step 4 and 5 on the previous pages. The temporary bents at Node 8 and 10 are removed when Cable 5 is activated. Order of cable installation: The order of cable installation, which is presented in the illustrated construction stage analysis, corresponds with the order in the original data. In this case, cable 4 is the last cable installed in the side span. A reasonable change in the order of cable installation is to install cable 4 after the installation of cable 9 (elements 504 and 509). In the side span, the sequence of the installation can then be continued from right to left as the construction continues. Table 5-14 summarises the possible sequences of the cable erection.
A) Original data B) Order as seen above C) Possible change
6&7 6 6
-/7 7
5 5 5
8 8 8
9 9 9
Order of cable erection 3 10 2 11 12 3 10 2 11 12 4 10 3 11 12
1 1 2
13 13 13
14 14 14
4 4 1
Table 5-14: Sequence of cable erection
Retension of Cable 15: In the performed construction stage analysis, in the backward analysis, Cable 15 is detensioned before removing the symmetrical boundary condition in the centre of the bridge in order to ensure a zero moment condition. This procedure is solely used for modelling this condition. As it can be seen by the example of the Case A calculation in Figure 5-15, the moment has a small value of -25tonfm before opening the bridge.
15 15 15
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181
Figure 5-14: Bending moment before opening the bridge [tonfm] (Case A)
It is possible to move the derrick crane a little closer to the middle of the bridge to achieve the same situation of a zero moment in the key segment. This can be realised in the model by refining the mesh and the creation of nodes at the intended positions. Another method is to apply the derrick loads, as concentrated forces, directly on the beam elements instead of using nodal loads. In this way, it is theoretically possible to construct the cable-stayed bridge without the need to retension any cables. However, on the site, as there are discrepancies between the target and real geometry, it may be unavoidable to use retension to adjust some construction errors. In the forward analysis, Cable 15 is retensioned after the closure of the key segment. Closing the key segment: Shortly before finishing the construction, the closing of the cablestayed bridge is a difficult phase as it will influence its overall structural behaviour. The following list shall explain the procedure in more detail: 1. Release the temporary longitudinal fixation of the main girder at the pylon 2. Survey the actual formation of the bridge 3. Determine the amount of set-back 4. Perform the set back 5. Lift the key segment with the crane on the Mokpo side and welding it with the Jido side (construction stage 32-33). For this period, the crane on the Jido side is moved back 6. Move the derrick crane on the Jindo side closer to the middle of the bridge to balance its the level. 7. Adjust displacements and connect the two sides 8. Cast the concrete in the side span, remove the derrick crane and the erection equipment, add the additional load Connecting the key segment in this order, it is assumed that it will be fabricated in one segment with a length of 12metres. Following this sequence, it is very complicated to connect the key
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182
segment with the Jindo side as it is lifted from the Mokpo side. In order to balance the level after the connection, it is not possible to use the crane on the Jindo side for this lifting operation. To avoid problems related to this construction process, there is a possibility to fabricate the key segment in two 6 metre parts. Thus, each segment can be lifted and connected with the derrick on the same side, which is far easier to control. For the final closure, the two segments must be closed in the centreline of the bridge.
5.6.3 Linear Backward and Forward Analyses The internal cable forces obtained in the calculation described in Chapter 5.5.3 (limited cable forces and restricted bending moments) are used in the backward analysis. For the first analysis, the order of cable installation is performed as illustrated in the figures of the construction stages above, which corresponds to the cable installation sequence B given in Table 5-14. For the forward analysis, the initial cable forces are taken from the backward analysis in the corresponding stage. Table 5-15 shows the pretension forces, which are used for the forward analysis. Cable Nr.
El Nr.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15 (retension)
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 515
Applied tension force [tonf] Case A Case B 253.45 250.29 178.24 176.74 78.54 79.19 475.27 479.55 79.91 81.53 132.60 133.08 170.17 177.57 151.66 150.64 160.11 159.88 179.58 178.85 209.73 209.03 242.70 240.83 276.11 279.00 307.51 305.61 371.39 367.27 434.87 430.77
Table 5-15: Initial cable forces from backward analysis
MiDAS calculates two values for the cable forces in the truss elements, one at the top and one at the bottom of the cable. The weight of the cable reduces the pre-stressing from the top of the pylon to the anchorage point at the girder. Table 5-15 shows the lower stressing values. The
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183
cable forces are applied as external loads, using the Replace function as described previously. The results of the back- and forward analyses are discussed in the following chapters.
5.6.3.1
Results Backward and Forward Analysis
After completing the construction and adding the additional self-weight in the final stage, the difference in the maximal cable forces between the performed back- and forward analyses is shown in Table 5-16.
Cable Nr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
∆ S final [tonf] ∆ S final [%] Case A Case B Case A Case B 3.13 3.15 -0.67 -0.68 1.83 1.85 -0.36 -0.36 -0.25 -0.23 0.04 0.04 4.00 4.01 -0.68 -0.67 -1.57 -1.56 0.33 0.32 -0.47 -0.43 0.09 0.08 -0.04 0.00 0.03 0.00 -0.25 -0.24 0.16 0.15 -0.43 -0.42 0.23 0.23 -0.28 -0.27 0.13 0.12 -0.50 -0.49 0.19 0.19 -0.68 -0.67 0.23 0.23 -0.62 -0.61 0.19 0.18 0.66 0.68 -0.18 -0.18 4.29 4.29 -0.91 -0.92
Table 5-16: Difference in cable tensions between forward - and backward analysis
As evident in the table, the differences between the back- and forward analyses are less than 1% in the final state, - which is a quite satisfactory result. Nevertheless, when investigating the moment distribution in detail, larger discrepancies can be evaluated. For Case A, Figure 5-15 shows the final moment distribution obtained from the forward and the backward analysis. Comparing the final moment distribution from the backward analysis (first step) with the one from the Unknown Load Factor calculation given in Figure 5-12 a), the backward method shows a slightly smaller moment in the main girder. However, comparing the backward and forward analyses, the moment distribution in the main girder shows much higher differences as it can be seen in the figures below.
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184
a) First step in backward analysis
b) Last step in forward analysis Figure 5-15: Moment forward and backward analysis Case A [tonfm]
Even if the cable forces have only small changes, following the real construction process, the negative bending moments at the anchorage points have up to -60 tonfm higher magnitudes in the forward analysis. In order to ensure that there are no differences in the models, which might be responsible for these gaps, both models have been controlled – however, both models are identical. Similar differences can be found for the Case B cable forces. Therefore, the differences can only be related to small variations in the initial tension forces at the time of activating the elements. In each step of the forward analysis, the values are below the ones in the backward analysis. Exactly the same stressing values as obtained from the backward analysis are applied at the time of installing the cable, but it is not possible to achieve the same tension force in the actual forward calculation. As it has been mentioned earlier, MiDAS calculates the cable forces for the end and the start node of the truss or cable element. In this analysis, the lower values at the girder anchorage point are always used in the forward analysis. In case of applying the higher values from the backward analysis in the forward method, the tension forces in the forward analysis are always higher than the ones in the backward analysis, - which is the opposite case as it is given here. It is assumed that, in the
Chapter 5: Model of the Second Jindo Bridge
185
numerical method, the influence of the self-weight of the cable is always added to the defined stressing value. This explains that the values are never exactly the same as the input values. It should be mentioned again that this is also the case when the Replace function is activated in the Cable-Pretension Force Control function. In case of modelling a structure without the cable self-weight, the tension values are exactly the same as defined in the input data. In order to prove whether or not these small variations in the initial tension forces are responsible for the large discrepancies in the final moment distribution, the tension values due to the self-weight are considered in the pre-stressing loads defined in the input data. The following table gives the tension loads due to the self-weight of the cables and the new initial tension loads, which are defined in the forward analysis for Case A and B.
Cable Nr.
El Nr.
Initial tension due to self-weigth [tonf]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15 (retension)
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 515
-3.36 -3.31 -3.26 -3.21 -3.01 -2.89 -1.34 -1.33 -1.33 -2.01 -2.02 -2.03 -2.05 -2.63 -2.66 -2.66
New applied tension force [tonf] Case A Case B 256.81 253.65 181.55 180.05 81.79 82.45 478.48 482.76 82.92 84.54 135.49 135.97 171.51 178.91 152.99 151.97 161.43 161.21 181.59 180.86 211.75 211.05 244.72 242.86 278.16 281.05 310.14 308.23 437.53 433.43 374.05 369.93
Change in % Case A 1.33 1.86 4.15 0.67 3.77 2.18 0.79 0.88 0.83 1.12 0.96 0.84 0.74 0.85 0.61 0.72
Case B 1.34 1.87 4.12 0.67 3.69 2.17 0.75 0.88 0.83 1.12 0.97 0.84 0.74 0.86 0.62 0.72
Table 5-17: Changed initial cable forces considering the tension due to the self-weight of the cables
The values of the initial cable tension due to the self-weight of the stay cables given in Table 5-17 are calculated using MiDAS (NB: truss elements are still applied in the model). The value is controlled for cable 6. With the cable area A=116.2 cm², a self-weight of ρ = 8.193 tonf/m³ and the total length of the element L = 65.81 m, the tension force can be calculated. Considering an angle α = 67.20° between the truss element and the girder, a tension value of 2.89 tonf is calculated, the same as given in the table obtained from MiDAS. Figure 5-16 shows the final moment distribution for Case A, which considers the new calculated initial tension forces. Compared to the moment distribution in the backward analysis (Figure 5-15), both results are very close now. Analysing the stage by stage construction sequence, the
Chapter 5: Model of the Second Jindo Bridge
186
cable forces and the moment distributions are similar until the closure of the bridge. In the following stages, there are some differences because of the existence of a normal force in the key segment in the backward analysis which does not occur in the forward analysis. In Chapter 4.4, the influence of the normal forces has already been illustrated. In this analysis, a gap of -81 tonf exists, which is responsible for the small variation in the moment distribution close to the centre of the bridge. In the centreline of the bridge, the vertical displacement is -44 mm in the first step in the backward analysis. In the forward analysis, which uses the initial tension forces as they are directly obtained from the backward analysis, the vertical displacement is-141 mm. The vertical displacement is only -41 mm in the last calculation, which is very close to the backward result. Comparing the other stages, the differences are within 1 or 2 mm. These calculations demonstrate the importance of accurate modelling, the exact load information and the high influence of the initial cable forces. Minor differences in the initial tension load influence the internal forces, as well as the overall deformations and can easily produce problems during the construction process.
Figure 5-16: Moment forward analysis, considering the tension forces due to the Self-Weight function Case A [tonfm]
The horizontal displacement at the top of the pylon is -3 mm. The moment at the bottom of the pylon is reduced to 199 tonfm. The maximum moment at the top of the pylon remains at -405 tonfm. The maximum moment in the girder is -1440 tonfm at the anchorage of the backstays and -1478 tonfm at the pylon.
Chapter 5: Model of the Second Jindo Bridge
5.6.3.2
187
Different moment distribution at the time of temporary supports
Some differences still exist in the first stages until the temporary supports are being removed between the forward and backward analyses. These variations are small, but they should be mentioned here. The following figures show the moment distribution considering the corrected Case A tension values as the initial pre-stressing in the input data. The values for the vertical reaction forces are also given for each support. The bold numbers indicate the differences in the calculated results. As the construction continues, the gaps become smaller. The initial discrepancy may be relating to activation errors and deformations that remain in the system in the backward analysis. Results from Forward Analysis:
Results from Backward Analysis:
Stage 0
Stage 0
Vertical reaction forces [tonf]: Left: 150, Bent8: 124, Bent10:114, Bent12:87, Bent13:103,Pylon: 3985, Bent17: 46
Vertical reaction forces [tonf]: Left: 150, Bent8: 124, Bent10:114, Bent12:84, Bent13:117, Pylon: 3969, Bent17: 51
Stage 4 – Cable6
Stage 4 – Cable6
Vertical reaction forces [tonf]: Left: 129, Bent8: 216, Bent10:228, Bent12:86, Bent13:167, Pylon: 4185, Bent17: 164
Vertical reaction forces [tonf]: Left: 129, Bent8: 216, Bent10:229, Bent12:87, Bent13:181, Pylon: 4166, Bent17: 169
Chapter 5: Model of the Second Jindo Bridge
188 Stage 5 – Cable7
Stage 5 – Cable7
Vertical reaction forces [tonf]: Left: 130, Bent8:194, Bent10:295, Pylon:4570
Vertical reaction forces [tonf]: Left: 130, Bent8:193, Bent10:298, Pylon:4568
Stage 7- seg8
Stage 7- seg8
Vertical reaction forces [tonf]: Left: 129, Bent8:232, Bent10:145, Pylon:4781
Vertical reaction forces [tonf]: Left: 129, Bent8:231, Bent10:148, Pylon:4779
Table 5-18: Results of forward - and backward analysis (moment values in [tonfm])
It can be seen that the conditions during the forward and backward analyses are not identical. The moment distribution has some small differences. There are also variations in the reaction force of the temporary bents. In the forward analysis, the moments in the pylon are slightly larger. In construction stage 8, in which all temporary supports are removed, the reaction forces become close to each other. Since the discrepancies are reduced until the bents are finally removed, the gaps are not investigated in more detail. However, both analysis methods should be compared carefully to control each other. Attention is required especially when boundary conditions are changed. In this analysis, the differences are small, but this must not always be the case in other calculations. The situation is similar if the analysis is performed using the Case B values.
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189
5.6.4 Differences in the application of cable elements In order to consider the influence of the sagging effects of the cables, the truss elements are replaced by Tens-Truss Cable elements in the following analysis. As described in detail in chapter 4.8, in a linear analysis, these elements calculate an effective stiffness for the elements representing the cables. In the analysis carried out before, these effects are entirely neglected. Applying the Case B tension values in the model, a back- and forward analysis is performed. The initial tension forces to be applied at the time of the cable erection change. In comparison to the values calculated previously, the values obtained from the backward analysis are given in the following table.
Cable Nr.
El Nr.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15 (retension)
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 515
Tension force from backward analysis [tonf] Truss elements neglecting Considering an cable effects Case B effective stiffness 250.29 255.64 176.74 186.52 79.19 108.26 479.55 479.40 81.53 101.56 133.08 144.70 177.57 176.24 150.64 152.49 159.88 161.31 178.85 185.78 209.03 214.04 240.83 244.52 279.00 281.07 305.61 313.99 367.27 367.54 430.77 428.77
Applied tension forces in forward analysis [tonf] 259.00 189.83 111.52 482.61 104.57 147.59 177.58 153.82 162.64 187.79 216.06 246.55 283.12 316.62 370.20 431.43
Table 5-19: Initial cable forces from backward analysis
There are obviously differences in the cable forces when the different element types are applied. These are related to the effective stiffness which is now considered. The last column in the table shows the corrected tension values which consider the changes in the tension forces due to the self-weight of the elements (see Table 5-17). By neglecting an effective stiffness, the backward and forward analyses can be performed with identical results. However, when the cable forces given in the table above in the forward analysis are applied, the results are not identical with the results from the backward analysis any more. Theoretically, this should be the case, but as it can be seen in the following, there are considerable gaps.
Chapter 5: Model of the Second Jindo Bridge
190
Figure 5-17 shows the installation of the first cable (Cable 6). At the time of installing the cables, the forces in the back- and forward analyses are identical. Forward analysis
Backward analysis
Figure 5-17: CS 4-Installation of cable 6, considering an effective stiffness in forward and backward analysis [tonf]
Nevertheless, as the construction continues, the tension forces in the already installed cables change and a gap between both methods develop. Forward analysis
Backward analysis
Figure 5-18: CS 16 installation of cable 10, considering an effective stiffness in forward and backward analysis [tonf]
Table 5-20 presents the final cable forces for the back-and forward analyses. The differences between both values are given as well. The greatest gap between both calculations can be found in the tension value of Cable 9, which is 5.3%. Furthermore, Cable 5, 6, 13 and 14 show variations of more than 2%. These values seem to be small but as it has been proved before, minor variation in the tension forces already have an influence on the final moment distribution. The moment distribution of the finished structure is illustrated in Figure 5-19.
Chapter 5: Model of the Second Jindo Bridge Cable Nr 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Backward analysis [tonf] 458.98 506.25 548.54 593.84 476.68 529.50 131.42 154.19 182.03 216.39 250.57 285.16 330.57 372.66 467.14
191 Forward analysis [tonf] 456.88 503.05 555.04 589.80 462.75 518.22 132.29 151.16 172.45 214.59 246.92 281.19 323.36 385.97 461.84
∆ S final [tonf] 2.10 3.21 -6.50 4.04 13.93 11.27 -0.86 3.04 9.59 1.81 3.65 3.96 7.21 -13.31 5.29
[%] -0.46 -0.63 1.18 -0.68 -2.92 -2.13 0.66 -1.97 -5.27 -0.83 -1.46 -1.39 -2.18 3.57 -1.13
Table 5-20: Difference in cable tension forward - and backward analysis considering an effective stiffness
Figure 5-19: Final moment distribution using cable elements, considering the effect of the self-weight of the cables in the initial tension values[tonfm]
To find a reason for the discrepancies in both methods, in the forward analysis, the tension forces are applied in five separate construction stages. Errors can occur due to a linearization of non-linear effects. In case of changing the structural system in large steps, a linearization, which may be sufficient in smaller calculation steps, can result in errors. Considering an effective stiffness of the cables, the introduced forces at the time of installing the cable will also influence the stiffness of the neighbouring cables. By splitting the tension procedure of the new cable into five steps, it should be investigated if the error reduces because of a stepwise adaptation of the cable stiffness due to smaller changes of the internal forces in the structural system. However, the result of the changed system shows nearly the same tension values in the cables as the calculation before. This becomes apparent when Figure 5-20 is compared with Figure 5-18; the maximal variation (in Cable 5) is below 0.5%. Therefore, the calculation process of the effective
Chapter 5: Model of the Second Jindo Bridge
192
stiffness appears not to be the reason for the gap in the tension values. The method of dividing the cable stressing is applied in the backward analysis, too. In this case, the cables are gradually detensioned before the elements are removed. However, this procedure does not influence the tension values which have to be applied to the stay cables as initial forces, either. Forward analysis cable stressing in 5 steps
Figure 5-20: CS 16 installation of cable 10, cables stressed in 5 steps [tonf]
In Figure 5-17, the installation of the first cable (Cable 6) has been illustrated. It is proved that in the forward and backward analysis, the initial applied tension forces are identical. However, at the time of installing the first cable, the tension forces are identical but there already exists a gap in the deformed shape. The horizontal deformation in the top of the pylon is -320 mm in the forward analysis and -245 mm in the backward analysis (pylon leans into the side span) Usually, the final state of the backward analysis is the first state in the forward analysis. For the linear backward analysis, which does not consider an effective stiffness in the cable elements, Table 5-18 shows that there is no deformation at the top of the pylon before the installation of the first cable. With the installation of the first cable, forces which bend the pylon and introduce deformations are applied. This condition is the same in the forward analysis. For the backward analysis, which considers the effective stiffness of the cable elements, the calculation still shows bending moments and deformations in the pylon after removing the last cable as it is illustrated in the next figure.
Chapter 5: Model of the Second Jindo Bridge Forward analysis
193 Backward analysis
Figure 5-21: Pylon and side span before the installation of the first cable [tonfm]
The maximum moment in the pylon is 376 tonfm and the horizontal displacement is 38 mm into the main span. There is no vertical deformation in the corresponding step in the forward analysis. In the next step, when Cable 6 is removed or installed in the forward analysis, the moment at the bottom of the pylon is -3192 tonfm in the backward- and -3560 tonfm in the forward analysis. If the already existing moment in the backward analysis is considered, the gap in the moment value is only -7 tonfm. It seems to be possible that the difference in the cable forces in both methods is related to a different initial condition in the forward analysis and, respectively, the remaining forces in the backward analysis. The figures of the modelled construction stages show a construction load of 25 tonf at the top of the pylon. By a sideward leaning of the pylon, the existing moment may be produced by this construction load and the self-weight of the pylon. The construction load is compared to the pylon weight light and therefore, it can only have a minor influence. This is proved by an analysis without the construction load at the top of the pylon. In this case, the moments and the deformations remain identical. However, if the same model is analysed with truss elements, which do not calculate an effective stiffness, the moment in the pylon and its vertical displacement vanishes. Since there should be no influence of the different calculation procedures at the time when all cables are removed and also other non-linearity are not considered, it is not possible to explain the existing moment in the backward analysis. Nevertheless, this gap may be responsible for the variations in the forward analysis when compared to the backward method.
194
Chapter 5: Model of the Second Jindo Bridge
5.6.5 Cable elements in Forward Analysis In the previous chapter, a gap between back- and forward analysis has been found, which may be relating to a different structural condition at the time the first cable is installed in the forward analysis. Nevertheless, the sagging effects of the cables are considered by an effective stiffness in the forward analysis. Due to the variations in the forward and backward methods, it appears to be more reliable to control the final set of initial cable forces by using the forward analysis. The analysis is performed for the values of Case A and B, which has been found from the backward analysis which neglects an effective stiffness. Besides the corrected initial cable forces, as they are given in Table 5-17, the cable tension forces, which neglect the change due to the self-weight of the cables (values Table 5-15), are also included in the comparison of the different results. The results using the initial cable forces which have been obtained form a backward analysis neglecting an effective stiffness seem to be more satisfying than the results when an effective stiffness in their determination is considered. The following figures show the final moment distribution obtained from the forward analysis.
a) Considering the tension forces due to the Self-Weight function Case A
b) Considering the tension forces due to the Self-Weight function Case B
c) Neglecting the tension forces due to the Self-Weight function Case A
d) Neglecting the tension forces due to the Self-Weight function Case B Figure 5-22: Moment distribution in the main girder using cable elements, considering and neglecting the effect of the Self-Weight function [tonfm]
Chapter 5: Model of the Second Jindo Bridge
195
To compare the differences between truss and cable elements for Case A, Figure 5-22 c) can be compared with the bending moments given in Figure 5-15 b), and for the corrected initial tension forces, Figure 5-22 a) with Figure 5-16. Due to the influence of the cable elements, the effective stiffness decreases so that it leads to changes in the moment distribution. The diagrams show that the negative moment at the anchorage point is reduced and the moments between the cables increased. The main girder is less supported by the cables. Not only should the moment distribution of the main girder be considered to choose the most proper initial cable forces, the overall structural behaviour should be taken into account in this decision, too. For the final sate, the table below gives further key values, which should be focused on.
Min. moment girder side span [tonfm] Max. moment girder side span [tonfm] Min. moment girder at the pylon [tonfm] Min. moment top of the pylon [tonfm] Max. moment bottom of the pylon [tonfm] Horiz. displacement top of the pylon [mm]
Consider changes due to the Self-Weight function Case A Case B -1402 -1409 778 848 -1498 -1341 -359 -357 188 242 2 7
Neglecting changes due to the Self-Weight function Case A Case B -1404 -1418 719 813 -1562 -1392 -357 -353 382 351 15 13
Table 5-21: Comparison of cable forces obtained from different calculations
Comparing the corrected and not corrected initial cable forces, the final moment distributions show some minor differences. The results from Case A and B, which consider the effect of the Self-Weight function (Figure a and b), show higher maximal values between the anchorage points of two cables and lower minimum moments at the connection cable-girder. The minimum girder moments in the side span are equal for all cases. Case A may show a preferred moment distribution in the side span because the maximum moment in the girder between cable six and the pylon decreases about 80 tonfm, but on the other hand, the minimum moment at the pylon increases. An advantage of the corrected values is a minor horizontal displacement at the top of the pylon. However, in the Case B, which neglects the correction due to the changes in the initial cable tensioning as obtained from the backward analysis (Figure d), the moment in the main span is distributed more equally. The negative values at the anchorage point of the cables are higher and therefore the moments between two cables reduced. For this reason, the pretension values obtained from this case appear to give the most reasonable results. The displacement of 13 mm at the top of the pylon is still tolerable. Therefore, in the following steps of the construction stage analysis, the stressing values are used as proposed for Case B in Table 5-15.
Chapter 5: Model of the Second Jindo Bridge
196
The following table summarises the final tension load due to the permanent loads, including the additional load. Cable Nr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 *)
Truss elements [tonf] 455.23 504.04 548.55 589.72 478.20 530.09 131.35 154.36 182.37 216.72 251.05 285.78 331.09 371.98 462.77
Cable elements [tonf] 449.26 494.59 533.72 587.17 474.04 535.43 130.32 152.27 177.35 214.90 247.45 282.35 325.87 377.11 461.30
∆%*) -1.31 -1.87 -2.70 -0.43 -0.87 1.01 -0.79 -1.35 -2.76 -0.84 -1.43 -1.20 -1.58 1.38 -0.32
100% = value truss
Table 5-22: Final cable forces truss and cable elements (forward analysis)
The differences in the cable forces are small, as it can also be seen at the %-value of the stress changes. However, for the moment distribution in the main girder, it has already been shown that these variations influence the structure. Furthermore, it will also affect the displacement, which is important for the camber calculation. Figure 5-23 shows the vertical displacement using truss and cable elements. The vertical displacement of the girder in the main span is apparently different.
a) Truss elements
Chapter 5: Model of the Second Jindo Bridge
197
b) Cable elements Figure 5-23: Vertical displacement neglecting the effect of the Self-Weight function [mm], Case B
Using cable elements, Figure b) indicates smaller displacements in the main girder with an upward movement to the centre of the bridge. In this case, the centre is 110 mm higher in the final state of the construction than in the model with truss elements. Table 5-23 describes the vertical displacement at the tip of the segment for each construction stage. The table also shows the discrepancy between the two models (∆ truss/cable-values). Const. stage Real displ. truss Real displ. cable ∆ truss/cable
Seg 7
Cb 6
Cb 7
L-seg 8
Seg 8
Cb 5
Cb 8
L-seg 9
Seg 9
Cb 9
Cb 3
L-seg 10
Seg 10
Cb 10
-9
-9
101
-20
-106
86
302
98
17
249
335
134
39
323
-9
-9
101
-20
-104
84
298
95
14
247
334
127
29
314
0
0
0
0
-2
2
4
3
3
2
1
7
10
9
Const. stage Real displ. truss Real displ. cable ∆ truss/cable
L-seg 11
Seg 11
Cb 2
Cb 11
L-seg 12
Seg 12
Cb 12
L-seg 13
Seg 13
Cb 1
Cb 13
L-seg 14
Seg 14
78
-58
52
359
86
-78
300
-58
-287
-155
331
-87
-351
69
-69
45
360
84
-79
310
-48
-273
-141
358
-60
-316
9
11
7
-1
2
1
-10
-10
-14
-14
-27
-27
-35
Const. stage Real displ. truss Real displ. cable ∆ truss/cable
Cb 14
L-seg 15
Seg 15
Cb 4
Cb 15
L-keyseg
keyseg
D-mid
Fix Pylon
closing
reten
remD
Add Load
219
-263
-574
-386
296
-113
244
80
80
80
83
334
-113
267
-213
-515
-330
386
-24
346
180
182
182
183
443
-3
-48
-50
-59
-56
-90
-89
-102
-100
-100
-100
-100
-109
-110
Table 5-23: Vertical displacement at the tip of the cantilever [mm]
As the cable length increases and thus the sagging effects do, too, the gap between both analyses extends. At the time of installing Cable 15, the difference between the two models has increased from 56 mm to 90 mm. It has been explained before that the installation of a cable
Chapter 5: Model of the Second Jindo Bridge
198
influences the tension force in the neighbouring cables. When Cable 15 is erected, the tension force in Cable 14 is reduced from 505 tonf to 287 tonf using truss elements, and from 502 tonf to 293 tonf using cable elements, and is therefore 2.09 % higher in the case of cable elements (Table 5-25). This circumstance can be responsible for the change in the discrepancy between the two models at the time of installing Cable 15 and the upward movement of the main girder close to the centre in the final state. Table 5-23 is given in a graphic form in the figure below. It shows the increase of the cable effect as the discrepancy between the linear analyses and the analyses which considers the cables by an effective stiffness increase with the activation of longer cables. 400 200 0 -200 -400 -600 0
5
10
15
20
Truss element
25
30
35
40
Ernst truss
Figure 5-24: Vertical displacement at the tip of the cantilever for each construction step [mm]
In Chapter 4.8, the reliability of using cable elements has been proved. It has been demonstrated that, during the construction, differences in the girder elevation can occur due to some variations in the cable forces and the sagging effects of the stays. Each new installed segment is activated tangentially, which increases the difference in the elevation level at the tip of the cantilever between cable and truss elements. According to the results obtained from the comparison of both element types in different analyses performed in Chapter 4.8, it seems to be reasonable to trust the determined results of the moment distribution and the deformations calculated by the application of the cable elements. Using the cable element calculation, the values of the girder elevation are controlled in more detail for the camber calculation in Chapter 5.9. Table 5-24 gives a detailed overview of the cable forces obtained from the back- and forward analyses at the time of installing the cables using truss elements. The values are given for the anchorage point cable-girder. The values at the pylon are, due to the weight of the cable, higher. Table 5-25 shows the difference in the cable forces using truss and cable elements.
199 Cable Forces Forward Analysis Case B [tonf] (values for J-Node), the discrepance between forward- and backward analysis is related to the influence of the self-weight of the elements representing the cables Cable add derick L-seg7 seg7 Cb6 Cb7 L-seg8 seg8 Cb5 Cb8 L-seg9 seg9 Cb9 Cb3 L-seg10 seg10 Cb10 L-seg11 seg11 1 2 3 75.93 205.39 4 5 78.52 105.91 246.45 174.89 233.99 6 130.19 278.86 328.92 371.54 392.41 380.55 388.91 7 176.23 294.46 158.40 156.53 156.97 148.26 8 149.31 157.96 158.23 157.18 9 158.54 158.94 172.10 10 176.84 11 12 13 14 15 Cable Forces Backward Analysis Case B [tonf] (values for J-Node) 1 2 3 4 5 6 133.08 282.77 7 177.57 8 9 10 11 12 13 14 15 Cable Forces Forward Analysis Case B [tonf] (values for J-Node) Cable L-seg14 seg14 Cb14 L-seg15 seg15 1 361.28 2 412.02 3 459.19 4 5 348.77 6 400.87 7 123.40 8 124.49 9 142.72 10 157.05 11 196.08 12 235.72 13 281.87 14 302.98 15
Cb4 347.43 394.61 437.45 476.34 314.74 382.78 114.79 108.05 122.30 133.43 195.58 276.12 378.47 505.32
Cb15 357.66 403.73 445.40 483.26 316.13 379.66 123.34 123.30 139.21 147.67 182.77 220.09 267.09 287.43 364.62
Cable Forces Backward Analysis Case B [tonf] (values for J-Node) 1 364.61 2 413.97 3 459.00 4 5 347.47 6 401.37 7 123.85 8 124.23 9 142.07 10 156.46 11 195.52 12 235.41 13 282.19 14 305.61 15
350.00 395.81 436.49 479.55 312.94 383.12 115.24 107.79 121.65 132.83 195.02 275.84 378.85 508.06
360.25 404.93 444.45 486.47 314.32 379.97 123.86 123.17 138.71 147.20 182.13 219.38 266.59 288.41 367.27
81.53 329.71 296.33
109.10 372.73 158.95 150.64
L-key-seg
key-seg
D-mid
fixPyl.
Cb2
Cb11
173.43 229.63
180.53 236.17
L-seg12
seg12
292.58 343.87
Cb12
L-seg13
seg13
Cb1 246.93 302.30 354.54
Cb13 254.40 309.09 360.61
246.55 394.37 131.82 159.50 223.25 348.44
247.62 390.84 139.00 147.00 168.84 188.19 207.01
295.11 395.68 131.98 137.18 159.43 183.10 214.57 238.79
305.26 402.62 112.64 113.06 145.75 206.13 306.65 410.90
305.42 397.58 127.10 129.89 150.20 169.79 207.21 240.10 276.94
79.19
208.70
176.74 230.74
183.85 237.28
295.98 345.05
250.29 282.17 507.24
257.77 311.09 360.47
249.68 393.71 156.62 158.08 159.88
175.18 381.36 157.08 158.37 160.29
234.30 389.74 148.24 156.61 171.90 178.85
245.86 395.07 131.81 158.94 223.07 350.53
246.91 391.49 139.08 146.33 168.11 188.57 209.03
294.41 396.29 132.19 136.61 158.59 182.77 215.18 240.83
231.25 408.59 101.41 104.76 127.83 178.65 271.62 374.67
304.16 398.12 127.42 129.45 149.40 169.21 207.11 240.85 279.00
closing
re-ten
remD
Add Load/Final J-Node I-Node 455.23 461.90 504.04 510.65 548.55 555.10 589.72 596.22 478.20 484.29 530.09 535.91 131.35 134.01 154.36 157.01 182.37 185.02 216.72 220.71 251.05 255.07 285.78 289.82 331.09 335.17 371.98 377.22 462.77 468.10
458.37 505.88 548.32 593.74 476.64 529.66 131.35 154.12 181.95 216.45 250.56 285.10 330.47 372.66 467.06
465.10 512.50 554.84 600.15 482.66 535.44 134.02 156.78 184.61 220.46 254.60 289.17 334.58 377.91 472.38
Table 5-24: Cable forces back- and forward analysis using truss elements and Case B values given in Table 5-15
Minimum and Maximum Cable Forces Case B [tonf] El Nr. Forw. *) Backw. *) ∆F 501 max 477.11 480.50 -3.39 501 min 253.65 257.01 -3.36 502 max 523.82 525.83 -2.01 502 min 180.05 183.36 -3.31 503 max 566.60 566.46 0.14 503 min 82.45 85.71 -3.26 504 max 596.14 600.15 -4.01 504 min 457.61 461.54 -3.93 505 max 484.22 482.66 1.56 505 min 84.54 87.56 -3.01 506 max 535.87 535.44 0.43 506 min 135.97 138.86 -2.89 507 max 297.14 299.01 -1.87 507 min 114.36 114.68 -0.33 508 max 292.50 293.81 -1.31 508 min 109.55 109.29 0.26 509 max 303.71 305.10 -1.39 509 min 124.48 123.82 0.66 510 max 352.46 354.54 -2.09 510 min 132.91 132.45 0.47 511 max 379.34 381.45 -2.11 511 min 162.48 161.95 0.54 512 max 414.97 417.10 -2.14 512 min 193.07 192.34 0.73 513 max 456.60 458.55 -1.95 513 min 240.07 239.40 0.67 514 max 510.70 513.44 -2.74 514 min 265.86 266.47 -0.61 515 max 491.59 494.25 -2.66 515 min 369.93 372.59 -2.66 *) values for I-Nodes
200
Cable Forces Forward Analysis Case B Truss Elements [tonf] (values for J-Node) Cable add derick L-seg7 seg7 Cb6 Cb7 L-seg8 Seg8 1 2 3 4 5 6 130.19 278.86 7 176.23 8 9 10 11 12 13 14 15 Cable Forces Forward Analysis Case B Cable Elements [tonf] (values for J-Node) 1 2 3 4 5 6 130.19 278.21 7 176.23 8 9 10 11 12 13 14 15 Cable Forces Forward Analysis Case B Truss Elements [tonf] (values for J-Node) Cable L-seg14 seg14 Cb14 L-seg15 seg15 Cb4 Cb15 1 361.28 347.43 357.66 2 412.02 394.61 403.73 3 459.19 437.45 445.40 4 476.34 483.26 5 348.77 314.74 316.13 6 400.87 382.78 379.66 7 123.40 114.79 123.34 8 124.49 108.05 123.30 9 142.72 122.30 139.21 10 157.05 133.43 147.67 11 196.08 195.58 182.77 12 235.72 276.12 220.09 13 281.87 378.47 267.09 14 302.98 505.32 287.43 15 364.62 Cable Forces Forward Analysis Case B Cable Elements [tonf] (values for J-Node) 1 358.86 344.11 355.19 2 406.64 387.86 397.84 3 448.57 425.26 434.06 4 476.34 484.06 5 346.37 311.82 313.45 6 406.42 388.22 384.86 7 121.89 112.94 122.17 8 122.18 105.44 121.39 9 138.16 117.68 134.73 10 155.73 134.37 146.38 11 195.47 195.00 180.23 12 236.71 276.33 218.46 13 282.11 377.99 264.32 14 302.98 502.32 293.43 15 364.62
Cb5
78.52 328.92 294.46
Cb8
L-seg9
seg9
105.91 371.54 158.40 149.31
78.52 329.77 294.27
103.32 374.82 157.92 149.31
L-key-seg
key-seg
Cb9
75.93
205.39
173.43 229.63
180.53 236.17
292.58 343.87
Cb1 246.93 302.30 354.54
174.89 380.55 156.97 158.23 158.94
233.99 388.91 148.26 157.18 172.10 176.84
246.55 394.37 131.82 159.50 223.25 348.44
247.62 390.84 139.00 147.00 168.84 188.19 207.01
295.11 395.68 131.98 137.18 159.43 183.10 214.57 238.79
305.26 402.62 112.64 113.06 145.75 206.13 306.65 410.90
305.42 397.58 127.10 129.89 150.20 169.79 207.21 240.10 276.94
75.93
197.36
173.43 222.08
180.35 228.94
289.17 335.39
246.93 282.17 507.24
254.69 305.49 351.26
238.66 397.25 155.34 157.69 158.54
170.44 384.01 156.02 158.20 159.15
233.65 394.11 146.83 156.54 172.31 176.84
244.40 399.19 130.53 159.06 223.57 347.45
245.82 395.78 137.36 145.39 167.29 190.22 207.01
293.71 401.12 130.27 134.94 156.27 184.19 215.76 238.79
231.25 408.59 101.41 104.76 127.83 178.65 271.62 374.67
303.30 402.91 125.51 127.55 146.11 169.54 207.99 241.66 276.94
closing
re-ten.
246.45 392.41 156.53 157.96 158.54
D-mid
fixPyl.
Cb3
L-seg10
remD
seg10
Cb10
Add Load/Final J-Node I-Node 455.23 461.90 504.04 510.65 548.55 555.10 589.72 596.22 478.20 484.29 530.09 535.91 131.35 134.01 154.36 157.01 182.37 185.02 216.72 220.71 251.05 255.07 285.78 289.82 331.09 335.17 371.98 377.22 462.77 468.10
449.26 494.59 533.72 587.17 474.04 535.43 130.32 152.27 177.35 214.90 247.45 282.35 325.87 377.11 461.30
455.97 501.20 540.25 593.62 480.10 541.23 132.99 154.92 179.99 218.88 251.47 286.41 329.95 382.39 466.62
L-seg11
seg11
Cb2
Cb11
L-seg12
seg12
Cb12
Minimum and Maximum Cable Forces Case B [tonf] Cable *) ∆F El Nr. Truss *). 501 max 477.11 470.89 6.22 501 min 253.65 253.65 0.00 502 max 523.82 515.06 8.76 502 min 180.05 180.05 0.00 503 max 566.60 552.91 13.69 503 min 82.45 82.45 0.00 504 max 596.14 593.58 2.56 504 min 457.61 458.49 -0.89 505 max 484.22 480.06 4.16 505 min 84.54 84.54 0.00 506 max 535.87 541.20 -5.33 506 min 135.97 135.97 0.00 507 max 297.14 296.95 0.19 507 min 114.36 112.58 1.78 508 max 292.50 291.85 0.65 508 min 109.55 106.90 2.65 509 max 303.71 303.29 0.42 509 min 124.48 119.74 4.73 510 max 352.46 351.46 1.00 510 min 132.91 133.23 -0.32 511 max 379.34 377.97 1.37 511 min 162.48 160.24 2.24 512 max 414.97 413.76 1.21 512 min 193.07 191.08 1.99 513 max 456.60 454.52 2.07 513 min 240.07 236.00 4.07 514 max 510.70 507.57 3.13 514 min 265.86 274.58 -8.71 515 max 491.59 489.47 2.12 515 min 369.93 369.93 0.00 *) values for I-Nodes
Table 5-25: Comparison of cable forces using truss and cable elements in forward analysis for Case B values given in Table 5-15
L-seg13
seg13
Cb13 254.40 309.09 360.61
Chapter 5: Model of the Second Jindo Bridge
201
Figure 5-25 a) and b) show the maximum and minimum moments during the construction of the bridge. These values must be checked to see if there are any unallowable stresses on structural parts during the erection.
a) Maximum moment
b) Minimum moment Figure 5-25: Maximum and minimum moments from forward analysis using cable elements and Case B values given in Table 5-15 [tonfm]
The stresses are controlled in Chapter 5.8.
Chapter 5: Model of the Second Jindo Bridge
202
5.7 Comparison with Hyundai and RM Results Besides the presented analysis in this report, there was also an independent construction stage analysis carried out by the Hyundai Institute of Construction Technology. Furthermore, the initial cable forces used in a RM – model of the Second Jindo Bridge are available. Table 5-26 shows the final cable forces after the construction and the applied tension forces at the cable installation. There are obviously some different cable forces chosen. Final State
*)
Construction State
Cable
Forward *) MiDAS Final
per Cable MiDAS
Final *) Hyundai
per Cable Hyundai
Initial Cable *) Forces MiDAS
Initial Cable *) Forces Hyundai
Initial Cable *) Forces RM
1 2 3 4 5 6
461.90 510.65 555.10 596.22 484.29 535.91
230.95 255.32 277.55 298.11 242.15 267.96
532.70 535.52 539.24 542.64 582.8 520.74
266.35 267.76 269.62 271.32 291.4 260.37
250.29 176.74 79.19 479.55 81.53 133.08
306.88 197.89 83.23 439.61 124.85 139.18
276.12 177.04 65.98 376.22 122.42 140.20
7
134.01
67.01
125.46
62.73
177.57
192.16
108.60
8 9 10 11 12 13 14 15
157.01 185.02 220.71 255.07 289.82 335.17 377.22 468.10 Sum Cable Forces 1-4
78.50 92.51 110.36 127.53 144.91 167.59 188.61 234.05
152.64 195.16 237.18 268.78 314.62 360.12 399.80 448.12
76.32 97.58 118.59 134.39 157.31 180.06 199.90 224.06
150.64 159.88 178.85 209.03 240.83 279.00 305.61 367.27
156.56 173.58 193.10 219.75 255.55 289.08 309.19 348.43
164.68 173.42 191.30 214.06 246.98 279.96 149.30 348.00
2123.86
2150.10
values for two cables
Table 5-26: Comparison of cable forces obtained from different calculations [tonf]
The summations of the first four cables, which are very close to each other and can be seen as one thick cable, are calculated for the MiDAS and Hyundai results. The total value is similar but the distribution of the cable forces in the side and main span is different. In the table, the initial cable forces are compared with each other in the last three columns. A large gap in the initial tension can be found for the force of Cable 7 compared to the other values of the other calculations. This can be relating to the difference in the performed construction stage analysis, in which e.g. in the RM-model two cables are installed at the same time. Another reason for the variations in the initial forces is the uncertainties in the loading condition as there were different values found for the self-weight and the construction loading. More important to remember is
Chapter 5: Model of the Second Jindo Bridge
203
that the system is highly indeterminate and that many possible solutions can be found representing a reasonable structural performance. In order to get an idea of the quality of the generated MiDAS model and the results from the other calculations, the values are used as input data in the construction stage analysis of the truss model. Applying the initial cable forces from the Hyundai calculation in the generated MiDAS model, Figure 5-26 shows the final moment distribution. In this calculation, the same loading and the construction stages as in the previous chapters are used. The girder and the pylon show higher moments for this calculation compared to Figure 5-22d) and Table 5-26.
Figure 5-26: Final moment [tonfm], Hyundai initial tension, same loading and construction sequence
The RM-model uses the same self-weight as calculated in the original self-weight table found in Table 5-5. Therefore, the model is simply changed to these loadings. Figure 5-27 illustrates a very reasonable moment distribution of the girder and the pylon. In this calculation, the loading is changed, but the construction stages are still the same as assumed for the MiDAS model.
Figure 5-27: Final moment [tonfm], RM initial tension, changed self weight, same construction sequence
204
Chapter 5: Model of the Second Jindo Bridge
Figure 5-28 a) and b) show the vertical displacement using the Hyundai and, respectively, the RM data. The Hyundai results show a high upward movement, whereas the results from the RM data identify a deflection of -322mm. For the construction, these deformations must be balanced with the camber geometry of each segment and a more accurate analysis should be performed as the cable effects are neglected in this calculation. However, both deformed shapes can be compared with Figure 5-23 a, which clarifies the vertical displacement when using the initial cable forces of the Case B values in a truss model.
a) Hyundai initial tension forces, same loading and construction sequence
b) RM initial tension forces, changed self weight, same construction sequence Figure 5-28: Vertical displacement dz due to changed initial cable forces [mm]
Chapter 5: Model of the Second Jindo Bridge
205
5.8 Minimum and maximum allowable stresses The maximum and minimum stresses are proved for the structural parts of the bridge in the following chapters. During the construction of the bridge the stresses must stay below the allowable values. This condition must be proved for the individual parts for all construction stages. Furthermore, it must be controlled whether or not the final structure can also fulfil the service ability limit state under life loads. To consider the effect of live loads in the design process, a detailed analysis is required to determine the maximum forces in the pylon, the girder and the cables. The influence lines must be calculated to localise maximum and minimum conditions. The structural non-linearity should be taken into account in order to prove the resistance of the structural parts. For the construction stage analysis, a simplified calculation is performed. A moving load analysis is carried out to calculate the maximum and minimum forces under the influence of live loads. In order to prove the service ability limit state of the cables, the results are superposed with the results obtained from the construction stage analysis, which considers the effect of the cables in the construction stages under the structural self-weight. For the live load condition, the stresses in the main girder are evaluated by modelling the maximum and minimum load conditions obtained from the moving load analysis. Since the segment properties are different in the side and the main span, a number of load cases are modelled to determine the required values for both the side and the main span. By this method, the non-linearity of the cables is taken into account. The bridge class and the loads to be considered are given in Chapter 5.4.6.3.
5.8.1 Tension forces in the cable stays During the construction of the bridge, it is not allowed for the cable forces to exceed the allowable values as they have been calculated in Table 5-9. The following table proves the maximum cable forces against the allowable tension forces. The values do not exceed the limited stresses during the erection process.
Chapter 5: Model of the Second Jindo Bridge
206
Cable 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
max max max max max max max max max max max max max max max
Truss Element 477.11 523.82 566.60 596.14 484.22 535.87 297.14 292.50 303.71 352.46 379.34 414.97 456.60 510.70 491.59
Cable Element 470.89 515.06 552.91 593.58 480.06 541.20 296.95 291.85 303.29 351.46 377.97 413.76 454.52 507.57 489.47
Allowable Cable Forces
1171.50
566.29
845.71
1078.36
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Table 5-27: Control maximum cable forces during construction [tonf]
As already mentioned, the maximum cable forces for the live load condition are determined by adding the values from the final construction stage and the result obtained from the moving load analysis. The maximum tension forces are given in Table 5-29. The moving load analysis is roughly controlled by the following calculation. In the first load case, the distributed traffic load and service load is equally applied on the side span; in the second case, there is no traffic load on the side span and only the main span is loaded and finally, the traffic load is applied on the whole deck. The concentrated load of 21.6tonf is directly applied to the cables, taking the angle between the cable and the girder into account. This method is very conservative as the case of full load on the side or the main span, with no load on the other side, rarely occurs. If it is assumed that the whole concentrated load is taken by one pair of cable, the girder stiffness is entirely neglected. In reality, the girder distributes the load to more than one cable and the actual tension force is lower. However, this calculation is only for a rough control of the cables and it can give at least some magnitude for the loading. Truss elements are used in the model for this simplified calculation. In Table 5-28, the angle for each cable is calculated. In the last column, the estimated cable load due to the concentrated load is given. The x1-value indicates the location of the cable node in horizontal direction, x2 value represents the second node of the cable, which is at the pylon, and ∆x is the absolute horizontal distance between both nodes. The z-values represent the same for the vertical direction.
Chapter 5: Model of the Second Jindo Bridge Cable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
x1 [m] -244.61 -243.02 -241.25 -239.84 -218.75 -197.50 -146.50 -129.50 -112.50 -95.50 -78.50 -61.50 -44.50 -27.50 -10.50
x2 [m] -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00 -172.00
∆x [m] 72.61 71.02 69.25 67.84 46.75 25.50 25.50 42.50 59.50 76.50 93.50 110.50 127.50 144.50 161.50
z1 [m] 18.25 18.33 18.41 18.49 20.62 21.68 24.23 25.01 25.70 26.28 26.78 27.17 27.47 27.68 27.79
207 z2 [m] 88.86 87.86 86.87 85.87 83.86 82.35 82.17 82.65 83.33 84.12 84.94 85.78 86.63 87.49 88.36
∆z [m] 70.61 69.53 68.46 67.38 63.24 60.67 57.94 57.64 57.64 57.84 58.16 58.61 59.16 59.81 60.57
α [rad] 0.77 0.77 0.78 0.78 0.93 1.17 1.16 0.94 0.77 0.65 0.56 0.49 0.43 0.39 0.36
S [tonf] 30.98 30.88 30.72 30.65 26.86 23.43 23.60 26.84 31.05 35.82 40.89 46.10 51.32 56.48 61.51
Table 5-28: Calculation of angle and cable force due to concentrated load
Table 5-29 shows a summary of the maximum cable forces for the live load conditions. The approximate calculation values are above the summation of the moving load analysis and the final cable forces which consider the cable effects. The magnitudes of the tension forces determined by both calculations are of the same dimension and it can be assumed that the moving load analysis is correctly applied.
Cable 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
max max max max max max max max max max max max max max max
Control calculation [tonf] Distributed Concentrated ∑ traffic & traffic load traffic load dead load 209.29 30.98 702.17 201.70 30.88 743.22 193.18 30.72 779.00 184.96 30.65 811.83 107.55 26.86 618.70 53.51 23.43 612.85 35.01 23.60 192.63 51.72 26.84 235.57 59.72 31.05 275.79 95.53 35.82 352.06 101.37 40.89 397.33 104.21 46.10 440.13 101.52 51.32 488.01 117.78 56.48 551.47 98.79 61.51 628.39
Maximal tension forces [tonf] Moving load ∑ traffic & analysis dead load 212.38 668.352 204.49 705.693 195.68 735.929 187.16 780.785 109.29 589.392 64.497 605.723 49.24 182.230 64.244 219.167 67.435 247.422 100.9 319.784 105.29 356.758 107.89 394.293 107.18 437.127 131.83 514.229 118.04 584.658
Allowable Cable Forces
941.4
455.1
679.6
866.5
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Table 5-29: Control maximum cable due to live load
As demonstrated, the forces are within an allowable range. Therefore, it can be concluded that the chosen initial cable forces and the resulting final tension loads in the final construction stage do not lead to unallowable condition in the service ability state.
Chapter 5: Model of the Second Jindo Bridge
208
Beside the maximum cable forces, the minimum cable forces must be controlled, too. For the construction process, the minimum cable forces have been given in Table 5-25 for the case of the application of cable and truss elements. Furthermore, the minimum values are controlled for the live load condition. At no time is there compression in the tension force.
5.8.2 Maximal stresses in the girder segments For the fabrication of the girder segment SM400-steel is employed. The yield strength fy and the allowable stresse fd are given in the table below according to the Korean Standard. In designing of steel structures, the Korean Standard uses the working stress method and therefore, no load factors are employed.In order to take buckling and non-linear effect into consideration the allowable stress value is reduced to the value fcal (all values have been given in Korea). fy [kgf/cm²] 2400
fd [kgf/cm²] 2000
Major axial compression fcal (0.645*fd) [kgf/cm²] 1290
Table 5-30: Allowable stresses for SM400-steel
Since the girder is mainly loaded by its normal force, an allowable stress fcal of 1290 kgf/cm² is applied. The maximum and minimum stresses in the main girder are calculated by
σ=
N M + *z. A I
The maximum and minimum stresses are proved for each segment for the construction process. The detailed values are given in the Table A-4 in the appendix. The stresses are within the limits. As for the cables, but also for the girder segments, the traffic load should be considered in an approximate calculation. In order to consider the final state of the internal forces as they occur after the final construction stage, the relevant load cases are generated in the construction stage analysis in additional steps. The non-linearity of the cables is taken into account for the traffic loads with this method. Figure 5-29 shows the maximal and minimal moment envelope calculated by the moving load analysis. The figures do not include any self-weight as they only include the traffic load.
Chapter 5: Model of the Second Jindo Bridge
209
a) Minimal moment
b) Maximal moment Figure 5-29: Moment envelope due to traffic load [tonfm] (no dead weight considered)
From the distribution of the bending moments, the elements of maximum and minimum values due to traffic load can be located for the side and the main span. Because the dead load moment distribution is equally distributed in the main span, it does not affect the location of the maximum and minimum values. Therefore, the loading condition which causes a maximum moment in the centre and a minimum value at two-thirds of the girder is modelled for the main span. The load distribution can easily be found by the Moving Load Tracer function offered by MiDAS, which locates the position satisfying the maximum or minimum condition of internal forces for a defined element. As an example, the load distribution for the maximum bending moment in the centre of the bridge is given in the following figure.
Chapter 5: Model of the Second Jindo Bridge
210
Figure 5-30: Load distribution for the maximum bending moment in the centre of the main span
Under the traffic load, the maximum in the side span occurs between Cable 5 and 6 (Element 10). Since the moment due to dead load (Figure 5-22) reveals a higher moment in the segment between Cable 6 and 7, and as it can be possible that this condition is the relevant one for the case of traffic load, too, the load distribution for both cases are modelled.
Min N element 9 Min N element 16 Min M element 9 Min M element 15 Min M element 24 Max M element 10 Max M element 13 Max M element 36
Side span 1-2 -/3-9 -/1-2 -/-/-/1-2 -/3-14 P: Node 10 1-2, 11-14 P: Node 13 1-2 -/-
Main span 15-36 P: Node 36 15-36 P: Node 36 15-35 P: Node 36 3-27 P: Node 18 15-20, 27-35 P: Node 36 -/-/-/-/15-20, 29-35 P: Node 36
Table 5-31: Load cases to consider the maximum load cases for traffic load
The table above gives the extra load cases modelled additionally after the final erection step in the construction stage analysis. The complete data proving the allowable stresses is given in the appendix. The stresses are within the limits.
Chapter 5: Model of the Second Jindo Bridge
211
5.8.3 Maximum stresses in the pylon During the erection of the individual cable stays and their stressing procedures, the pylon can experience high bending moments. Furthermore, excessive bending forces may occur during the lifting operation of a new deck segment with the derrick at the tip of the cantilever. At this stage, the cables are under high tension and can produce maximum moments, especially at the top of the pylon. Because the stay cables are not directly balanced by a cable on the opposite side of the pylon, and as there are small distances in the levels of the anchorages between the side and the main span, extreme bending values occur very locally. In the construction sequence as shown in this chapter, Cable 1 is installed before the erection of Cable 4. The maximum bending moments at the top of the pylon occurs in the segment between Cable 3 and 5 during the installation of Cable 1 (Figure 5-31). The order of cable installation influences the moments at the top of the pylon. For example, by installing the cables in their order, which means installing Cable 4 before Cable 3, the moment can be reduced.
Figure 5-31: Maximum moment at the top of the pylon during the erection of cable 1 [tonfm]
The table given in the appendix shows a detailed list of maximum and minimum moments. Again, the values are proved for the construction loads and for the live load condition. As it has been obtained from the moving load analysis, the minimum moment at the top of the pylon occurs with the live load at the beginning of the side span (Element 1-2) and a loading of 2/3 of the main span (Element 15-31), the concentrated load P is located at Node 23 in the main span. This condition is considered in the analysis. The actual stresses during the construction and under service conditions are within the allowable range.
Chapter 5: Model of the Second Jindo Bridge
212
5.9 Fabrication Camber of the Second Jindo Bridge In order to fabricate the different segments, the camber must be considered as it has been explained in detail in Chapter 4.6. The differences in the final vertical deformations by using truss elements in the model and considering cable effects has been clarified before. The reliability of the Ernst-formulation is proved and therefore, the results for the camber data are used including the sag effects of the cables. For the vertical deformation, the manufacture camber data is given in Table 5-32.
1 1.05
3
10
0.00 5.45
13
15
17
12.70
2.76
19
Node Number 21 23
25
27
29
10.37 13.70 17.09 18.78 18.17 15.82 12.22
31
33
35
36
8.15
4.78
3.04
3.20
Table 5-32: Camber data [mm]
Figure 5-32 presents the data in a graphical form.
Figure 5-32: General manufacture camber [m]
The general construction camber data is given in Table 5-36. For Node 36, the values listed in the data generated by the MiDAS General Camber function are controlled.
Chapter 5: Model of the Second Jindo Bridge
213
The following table lists the current displacements for each step after the activation of Node 36. Current vertical displacement at Node 36 [mm]
Stage key-seg D-Mid2/3 D-Mid Remove Fix-Pylon Closing ReTension RemDerick AddLoad Sum: MiDAS Value (Table 5-36)
136.61 259.23 -166.54 0.06 0 3.37 259.49 -445.89 46.33 46.33
Calculation value for stage D-Mid: MiDAS Value (Table 5-36)
-182.97 -182.97
Table 5-33: Control calculation for construction camber data
With the current displacement values, the construction camber data can be calculated. To check the camber data from MiDAS, two values are calculated as it can be seen in the table above. The values are equal to the ones determined by MiDAS, which can be found in Table 5-36. The same data presented for the vertical direction must be calculated for the longitudinal one in order to consider the horizontal deformation in the fabrication of the segments. MiDAS offers a feature to calculate these data, but there is a programme error in the Camber Control function. For the longitudinal displacement, the programme calculates the same values as given for the vertical direction, which is definitely not the case. Nevertheless, the data can be evaluated by using the graphical deformation plots. Table 5-34 gives the longitudinal displacement of the segments. 1
8
10
12
13
16
17
Node Number 19 21 23 25
6
6
3
0
-2
-5
-7
-9
-13
-15
-19
27
29
31
33
35
36
-21
-24
-25
-27
-27
-27
Table 5-34: Real horizontal displacement final state [mm] Seg. Nr. Node Nr. ∆ dx[mm]
1 18 0
2 810 3
3 1012 3
4 1213 2
5 1316 3
6 1617 2
7 1719 2
8 1921 4
9 2123 2
10 2325 4
11 2527 2
thick line indicates the location of the pylon
Table 5-35: Longitudinal deformation of each segment
12 2729 3
13 2931 1
14 3133 2
15 3335 0
key 3536 0
Chapter 5: Model of the Second Jindo Bridge
214
Node number Stage
1
3
10
13
15
17
19
-1.05
0.01
-5.45
-12.70
-2.76
-10.37
CS0
-0.77
0.01
-5.45
-12.70
-2.11
-10.37
addcrane
-0.90
0.01
-5.45
-12.70
-2.09
-10.37
L-seg7
-0.90
0.01
-5.45
-12.70
-2.07
-10.37
-16.19
seg7
-0.90
0.01
-5.45
-12.70
-2.09
-10.37
-5.11
Cable6
-0.90
0.01
-5.45
-12.70
-2.07
-10.37
-5.12
21
Cable7
-0.90
0.01
-5.45
7.48
-1.98
-55.41
-114.64
L-seg8
-0.91
0.01
-5.45
-9.79
-1.94
2.38
5.82
11.26
seg8
-0.91
0.01
-5.45
-12.46
-1.93
8.82
34.17
86.97
Cable5
-20.90
0.01
106.59
32.70
-1.90
-56.56
-85.77
-101.17
Cable8
-21.22
0.01
110.01
41.96
-1.90
-93.73
-187.20
-315.21
L-seg9
-12.18
0.01
53.91
16.51
-1.87
-49.69
-84.41
-112.49
23
25
27
29
31
33
35
36
-140.73
seg9
-10.21
0.01
41.66
11.00
-1.86
-40.43
-63.35
-59.84
-33.01
Cable9
-9.78
0.01
39.55
12.88
-1.86
-50.50
-103.47
-179.51
-266.21
Cb3
-15.04
0.01
73.36
27.21
-1.86
-71.95
-143.26
-242.46
-352.39
L-seg10
-10.26
0.01
45.75
16.17
-1.82
-56.82
-102.67
-135.61
-145.55
-155.92
Seg10
-9.24
0.01
39.85
13.84
-1.81
-53.74
-94.23
-113.77
-92.09
-47.44
Cable10
-8.84
0.01
37.67
13.35
-1.81
-53.13
-106.35
-174.70
-249.04
-332.43
L-seg11
-3.57
0.01
7.63
2.24
-1.77
-39.99
-77.88
-109.41
-109.91
-87.20
Seg11
-2.42
0.01
1.14
-0.13
-1.76
-37.28
-71.99
-95.65
-81.04
-25.30
52.85
Cb2
-7.62
0.01
32.05
12.44
-1.76
-55.78
-106.12
-149.55
-154.84
-119.07
-60.87
Cable11
-7.20
0.01
29.51
10.72
-1.75
-50.44
-101.92
-166.14
-228.63
-296.77
-375.38
-65.53
L-seg12
-3.39
0.01
8.35
3.34
-1.71
-42.80
-87.23
-134.49
-156.33
-141.06
-100.05
-60.99
Seg12
-2.57
0.01
3.82
1.79
-1.70
-41.26
-84.31
-128.08
-141.23
-108.70
-31.65
66.84
Cable12
-2.06
0.01
0.76
-0.44
-1.69
-34.47
-73.46
-122.81
-168.03
-210.76
-260.81
-322.33
L-seg13
2.26
0.01
-23.22
-9.08
-1.64
-24.29
-55.74
-92.96
-112.65
-99.54
-45.23
35.96
114.18
Seg13
3.18
0.01
-28.35
-10.91
-1.63
-22.17
-52.11
-86.90
-101.27
-76.12
0.05
122.39
265.11
Cable1
-2.42
0.01
0.13
0.11
-1.63
-38.02
-81.18
-132.67
-164.00
-156.05
-97.21
7.76
133.12
Cable13
-2.04
0.01
-2.37
-1.89
-1.62
-31.79
-69.17
-118.15
-164.26
-206.14
-248.52
-300.70
-366.18
L-seg14
1.14
0.00
-20.89
-8.67
-1.57
-23.41
-55.33
-97.85
-131.13
-140.96
-114.76
-45.03
51.49
143.76
Seg14
1.81
0.00
-24.82
-10.10
-1.55
-21.66
-52.50
-93.81
-124.54
-127.65
-86.66
8.74
150.65
311.68
Cable14
2.26
0.00
-27.63
-12.00
-1.55
-16.36
-41.20
-75.74
-108.75
-137.60
-163.27
-189.07
-224.04
-272.17
L-seg15
5.77
0.00
-48.13
-19.82
-1.49
-5.54
-22.83
-49.82
-73.92
-84.33
-69.11
-15.01
83.65
208.48
328.03
Seg15
6.51
0.00
-52.47
-21.48
-1.47
-3.26
-19.00
-44.51
-66.93
-73.64
-49.84
21.47
148.30
320.86
511.57
Cable4
5.38
0.00
-32.48
-10.94
-1.47
-20.22
-51.21
-96.13
-137.98
-163.96
-159.22
-106.79
1.29
155.13
327.13
Cable15
5.84
0.00
-35.09
-12.51
-1.46
-16.15
-41.78
-78.38
-115.24
-150.56
-185.47
-220.95
-261.42
-316.66
-389.26
L-key-seg
8.45
-0.00
-49.04
-17.67
-1.41
-8.76
-29.33
-61.61
-95.41
-125.30
-145.19
-144.95
-115.61
-55.31
20.71
46.33
Key-seg
7.53
0.00
-44.09
-15.84
-1.43
-11.39
-33.88
-68.16
-103.97
-137.08
-163.47
-175.81
-167.98
-140.68
-103.78
-90.28
D-Mid 2/3
6.83
0.00
-40.39
-14.52
-1.44
-13.22
-36.06
-68.12
-98.38
-124.49
-148.32
-174.63
-214.08
-270.77
-329.36
-349.51
D-Mid
7.32
0.00
-42.99
-15.44
-1.43
-11.97
-34.63
-68.39
-102.71
-133.75
-159.04
-174.52
-178.86
-178.06
-182.40
-182.97
Remove FixPylon
7.32
0.00
-42.93
-14.99
-0.89
-11.50
-34.32
-68.22
-102.61
-133.70
-159.01
-174.51
-178.87
-178.09
-182.45
-183.03
Closing
7.32
0.00
-42.93
-14.99
-0.89
-11.50
-34.32
-68.22
-102.61
-133.70
-159.01
-174.51
-178.87
-178.09
-182.45
-183.03
ReTension
7.32
0.00
-42.96
-15.01
-0.89
-11.46
-34.24
-68.10
-102.51
-133.74
-159.39
-175.49
-180.68
-180.80
-185.77
-186.40
RemDerrick
4.02
0.00
-25.27
-8.36
-0.92
-20.88
-51.64
-96.75
-146.57
-201.08
-261.02
-322.47
-378.51
-421.34
-443.91
-445.89
AddLoad
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Table 5-36: Construction camber data [mm]
Chapter 5: Model of the Second Jindo Bridge
215
5.10 Unstressed cable length L0 In the designed construction process, the cables are stressed in only one step and, theoretically, no second restressing is required. Restressing operations are limited to possible adjustment procedures due to construction errors but the time and the situation when they are required will be decided on the site, based on real time measurements. In Chapter 3.7, the importance of the unstressed cable length L0 has been clarified. For the prefabrication of the stay cables and the control of the bridge construction, the values of the unstressed cable length are essential. For the first cable (Element 501), the calculation is illustrated in more detail in the following. The derivation of the used formulas and parameters is provided in the appendix. The different working points have been illustrated in Figure 5-3 and Figure 5-4 in Chapter 5.4.1. For the first cable, Node 3002 (xa, ya) describes the working points A (W.P.A) at the anchorage point cable-girder and Node152 (xb,yb) represents the working point B (W.P.B) at the top of the pylon. xa := −244.610⋅ m
xb := −172.000⋅ m
y a := 18.252m ⋅
y b := 88.860m ⋅
The coordinate system is located in the axis of WPB and therefore the new coordinates follow to xA := xa − xb
xA = −72.61m
xB := xb − xb
xB = 0 m
y A := y a
y A = 18.252m
y B := y b
y B = 88.86m
The final cable forces, as they are obtained from the MiDAS calculation, are for two cables., They must be now calculated for one cable. Furthermore, the cross section of only one cable must be considered in the determination of the unstressed cable length. The cable force SA is the tension force at the working point A. S2A := 449.26tonf ⋅
SA := 2
A 2C := 0.011622m ⋅
A C :=
S2A 2 A 2C 2
SA = 224.63tonf 2
A C = 0.005811m
Chapter 5: Model of the Second Jindo Bridge
216
Figure 5-33: Cable
The following data is required for the analysis, too: γ C := 8.193
tonf
wC :=
3
A 2C⋅ γ C 2
m
tonf wC = 0.0476 m
7 tonf
EC := 2⋅ 10 ⋅
2
m
The angle α as shown in the Figure 5-33 is calculated as
yB − yA xB − xA
α := atan
α = 0.771
α = 44.199deg
With the angle α and the weight of the cable wC, the distributed load q can be determined. The horizontal part of the cable force H can be calculated as H=S(x)*cosθ. However, at this stage of the analysis, the angle is unknown and therefore, the angle α is used for the determination. q :=
wC
cos ( α )
q = 0.0664
tonf m
HC := SA ⋅ cos ( α )
HC = 161.04tonf
The developed equation for the cable geometry can be used to calculate the actual angle θ at the point WPA and to determine a new horizontal force H. The cable-equation can be defined by the parameters K1, K2 and K3. The values given below show the results after 3 iteration steps (=>indices 3). −4 1
q K23 := 2⋅ HC3
K23 = 2.0465× 10
(
K13 := tan ( α ) − K22⋅ xB + xA
)
K03 := y A − tan ( α ) ⋅ xA + K23⋅ xA ⋅ xB
K13 = 0.9873 K03 = 88.8600m
m
Chapter 5: Model of the Second Jindo Bridge
217
Assuming a parable shaped cable, the function f(x) defines the cable geometry. To control the analysis, the function is used to recalculate the values yA and yB. The results are exactly the same as the coordinates of the WPA and WPB. 2
f3( x) := K23⋅ x + K13⋅ x + K03
( ) f3( xB) = 88.86m
f3 xA = 18.252m
y A = 18.252m y B = 88.860m
The gradient of the cables is given by the first derivative of f(x), leading to g(x). g 3( x) :=
d f ( x) dx
g ( x) := 2K23⋅ x + K13
The angle θ at WPA and WPB can be calculated as the following:
( )
θ A3 := atan g xA
( )
θ B3 := atan g xB
g 3 xA = 0.9575 g 3 xB = 0.9874
( ( ))
θ A3 = 0.7637
θ A3 = 43.758deg
( ( ))
θ B3 = 0.7790
θ B3 = 44.633deg
Since the horizontal force H is constant, the values at point A and B must be identical. The cable force per cable at the pylon top is given as SB. It is shown below that the horizontal forces HC4 (at WPA) and HC4B (at WPB), calculated with the local angle θ, are equal.
(
HC4 := SA ⋅ cos θ A3 HC4 = 162.24tonf
)
SB = 227.99tonf
( )
HC4B := SB⋅ cos θ B3 HC4B = 162.24tonf
After the determination of the equation for the cable geometry, the actual required cable length must be calculated. It must be considered that, in the real structure, the cables are not spanned from working point A to B as the true cable length is from working point C to E. In the total cable length, the length of the guide tube of the anchor system must be included. As this length is not exactly known, an approximate location of the working point C and E is obtained from the given data. The given initial calculation sheet of the working points shows a displacement of ±0.75 cm for all cables in the dx direction (horizontal) to the working point C. For the working point E of cable 1, a displacement of -1.247 m in the dx direction is calculated from the given values.
Chapter 5: Model of the Second Jindo Bridge
218
The location in y direction is calculated by the developed equation. xC := −0.75m
( )
y C := f3 xC
y C = 88.120m
xE := xA − 1.247m
xE = −73.857m
( )
y E = 17.058m
y E := f3 xE
∆y := y E − y A
∆y = 1.194m
For the point WPC, the given data shows an yC value of 88.112 m and a distance ∆y between the points WPA and WPE of 1.200 m. The values only have minor differences which are related to a different cable force and the unknown data of the anchorage system. In order to determine the cable length L, the parameters a, b and c, which describe the function B(x) are calculated. 2
a := 1 + K13
b := 4⋅ K23⋅ K13
a = 1.975
b = 8.082 × 10
2
c := 4⋅ K23
−4 1
m
−7 1
c = 1.675 × 10
2
m
B(x) describes the integral function of the cable length as it is explained in more detail in the appendix. With the values of a, b, and c, B(x) and γ follow to:
( ) B( xE) = 1.916
2
B( x) := a + b ⋅ x + c⋅ x
B xC = 1.974
γ := 4⋅ a⋅ c − b
2
−7 1
γ = 6.701 × 10
2
m
The parameters I1(x), I2(x) and I3(x), which represent the solution for the integration of the cable length, are calculated as: I1( x) := I2( x) :=
b + 2⋅ c⋅ x 4⋅ c
⋅ B( x)
γ
I1 xE = 1618.330m
( )
I2 xE = 1221.618m
I2 xC = 1221.618m
3
8⋅ c
( )
I1 xC = 1694.073m
( ) ( )
2
I3( x) := ln( b + 2⋅ c⋅ x + 2⋅ c⋅ B( x) ) ⋅ m
( )
I3 xC = −6.236
The cable length follows to: LC( x) := I1( x) + I2( x) ⋅ I3( x)
( )
( )
LCable1 := LC xE − LC xC
( )
LC xC = −5923.688m
( )
LC xE = −6025.642m LCable1 = 101.954m
( )
I3 xE = −6.257
Chapter 5: Model of the Second Jindo Bridge
219
The elongation of the cable must be considered in the determination of the unstressed length. With the parameters ∆S1, ∆S2 and ∆S3 ∆S1 := 1 + K1
⋅ ( xC − xA )
2
∆S2 := 2⋅ K2⋅ K1⋅ xC − xA 2
∆S1 = 141.920m
2
∆S2 = −2.146m
4 2 3 3 ∆S3 := ⋅ K2 ⋅ xC − xA 3
∆S3 = 0.022m
the elongation of the cable can be determined as ∆S :=
HC3 A C⋅ EC
(
)
⋅ ∆S1 + ∆S2 + ∆S3
∆S = 0.195m
Finally, the unstressed length of cable 1 is L0Cable1 := LCable1 − ∆S
L0Cable1 = 101.759m
Furthermore, the thickness of the bearing plates, the shim plates and some extra length must be considered for the top and the bottom anchorages. The value given above does not include this additional length.
5.11 Final comments on the construction stage analysis In the preceding sections of this chapter the construction stage analysis of the Second Jindo Bridge has been introduced. An ideal state is determined under various restrictions, which forms the basis for the backward analysis. Particular attention is paid to the correct modelling of different construction stages. Effects of construction loads are taken into consideration in the calculation of stresses and deformations for each construction stage. Unfortunately, diverging values can be found for some data in the provided plans and tables so that assumptions had to be made in these cases. Besides some smaller differences, it is shown that, with a careful modelling, it is possible to perform an identical forward analysis with the results from the backward calculation. The comparison of the results from both methods is a very useful control option for the complete analysis. In the numerical process, gaps between both methods can easily occur and it is the user’s decision at which point to tolerate these and when to investigate the reasons for these in detail.
220
Chapter 5: Model of the Second Jindo Bridge
When applying cable elements in the construction stage analysis, some changes in the internal forces and the structural deformation take place which are related to the sagging effect of the cables. In the performed analysis, different initial cable forces are investigated. The results show only smaller variations, but after a subtle comparison of the results, one option is finally chosen. In the performed calculation, the construction sequence mainly follows the initially scheduled construction sequence. There is also the possibility to investigate another erection sequence of the stay cables which leads to different initial cable forces and changed maximal stresses during the erection. However, with the assumed construction process and the determined initial cable forces, the Second Jindo Bridge can be erected as it has been proved that the stresses are within the allowable range at any time.
Chapter 6: Conclusion
221
6 Conclusion
The following chapter summarizes the most important issues raised in this study and points out related areas for further research that may be of value for construction engineers. Final comments are given on the analysis programme MiDAS.
6.1 Summary After introductory remarks on the background, the intention and the organization of this study in Chapter 1, the history of cable-stayed bridges is illustrated in Chapter 2. This later section is supposed to elucidate the remarkable achievements that have been made from the time of the first cable-stayed bridges to the current days. Within the last 60 years, cable-stayed bridges have undergone major innovations both in their design and their construction. A high tensile strength of the stay cables allowed cable-stayed bridges to become an interesting option. With improved techniques of structural analyses, the design and the calculation of the cable forces during the construction has become more reliable. The cable-stayed bridge has established itself as a very economic and aesthetically satisfying bridge type in the last decades. The introduction of multi-cable systems made the construction of more slender decks possible. Analogue to the growing span lengths, the weight of concrete superstructures of the main span increased extensively and steel or composite girders became more economical. At present, parallel wires or parallel-strand cables are employed. A high protection against environmental influences allows a long lifetime. It can be asserted with certainty that technological advantages will continue to influence the ways in which cable-stayed bridges are designed and constructed. There is no denying that new structural concepts in connection with improved or newly engineered materials offer a wide range of possibilities for the future. The second chapter also provides an overview of different modern erection methods. By utilizing temporary supports the advantage of a continuous erection of the girder from one end to the other can be obtained, but, on the other hand, its limits emerge as valleys or rivers become deeper and non-accessible. In some of these cases, a construction by rotation will be more
222
Chapter 6: Conclusion
feasible for smaller bridges, where the main structure is erected on temporary supports and then rotated around its pylon. Incremental launching was developed in the early 1960s. It is characterized by the stationary construction of all superstructure segments. The segments are launched over the valley in small increments with hydraulic jacks. The method is commonly used for caste in situ segments, but can also be applied for steel or composite decks. Cantilevering can be carried out in two different manners. In case the cantilever system consists of two arms on both sides of the pier, it is called Balanced Cantilever Construction. The cantilever arms balance each other with their respective weight. In the second type, only one cantilever arm grows from its pier or abutment. The superstructure is usually supported by the connection with the side spans. Cantilevering has the important advantage of being an erection method with which the valley to be crossed is widely left unobstructed by the erection process; temporary supports are not required. The repetitive nature of the segmental construction, either with cast-in place or prefabricated steel or concrete segments, can be used very advantageously in the cantilevering. Once the cantilevering has been finished, the closure segment is placed between the cantilever arms to form a continuous superstructure. The erection of a cable-stayed bridge is based on the construction stage analysis, which is introduced in Chapter 3. The discussed analysis takes the complex behaviour of the material, the structural elements and a step by step changing system into consideration. Before analysing the actual erection process, the “ideal state” of the cable-stayed bridge, subject to its permanent loads, must be determined. The bending moments in the main girder and the pylon are minimized by the chosen cable forces. For a preliminary design, various hand formulas can be used. Structural analysis programmes apply optimisation methods to minimize the internal forces in the calculation of the ideal cable forces. The calculation considers user defined restrictions for forces or moments, stresses and displacements; the proper usage requires some experience to obtain reasonable results. In the analysis of the erection process, the construction stages are modelled to control the stability of each step separately, considering the permanent loads and the exact construction loads. The construction stage analysis can be classified into the forward and the backward analysis, whereas the first method follows the real erection sequence in contrast to the second method which starts from the final state and dismantles the structure step-by-step. Without any time effects, both methods are theoretically identical and may be used to control each other. However, there are some differences the user must be aware of. Activating elements
Chapter 6: Conclusion
223
straightforward expose differences in the deformed shape; activation errors can require additional corrections between both methods. The principal procedure of the construction stage analysis is outlined by a simple model. Influence matrices for cable forces or deformations, which are used for the determination of the ideal cable forces or for adjustment operations, can be evaluated by the analysis of unit load cases and by combining the influences separately in a matrix form. In concrete or composite cable-stayed bridges, time dependent effects have a significant effect on the geometry and the final stress state in the complete bridge must be included in the analyses. Major points in the time of structural analyses are the end of the construction and the state of “infinity”. Cantilevering the bridge superstructure subsequently with cast-in-place segments requires consideration of different segment ages and time dependent material properties. Usually, newly added segments are loaded when they have reached only a specific portion of the 28-day compressive strength of concrete. Young loaded concrete is susceptible to increased time dependent effects that depend on ambient conditions, i.e. a concrete shrinkage. The interaction between these issues makes cast-in-place constructions a challenging task. In order to minimise creep effects, it is important to select a convenient cable tension sequence, which produces only limited bending moments during the construction. Precast segments may be aged for 6 months or longer to reduce time dependent effects. The sag effects of the cables, the P-delta effects and the large-deformations are geometric nonlinear problems. In general, there is no rule when to consider these effects. In small structures, it can be sufficient to control the design process with a second order computation for the live loads, but this may not be sufficient in the analysis of the construction stages. The cable length and the stiffness of the girder indicate a tendency towards which effects to include. In a large cable-stayed structure, each of these effects can have an influence of 10-20 % of the results derived without the consideration of non-linearity. In the construction stage analysis with the analysis programme MiDAS, taking the P-delta effects alone into account is not possible, even if this may be sufficient. In this case, a large displacement analysis must be performed (3rd Order). Further uncertainties in the erection process are the thermal effects which must be considered in the control of the geometry. A main issue of the construction stage analysis is to evaluate the best possible sequence of the stressing operations of the stay cables under the consideration of the structural design objectivities, the simplicity in the construction and the cost effectiveness. The most simple and economic solution would be a single stressing step at the time of the installation of the cable. However, this solution may cause problems in the acceptability of the maximal stresses induced
224
Chapter 6: Conclusion
into the structure and the cables. Potential high bending moments increase the time dependent effects, which must be further considered in case of concrete structures. During the construction process, discrepancies between the actual state and the design values may occur. These are caused by a large scale of possible errors in the analysis, the construction or the prefabrication process of the structure or the structural elements. Therefore, continuous measurements and control systems at the construction site are essential. Describing the initial condition of the cable at its erection by the tension force suffers severely as the stressing force is not an intrinsic characterization of the pre-loading. Another option of initial stay adjustment can be achieved by fabricating the cables with their unstressed length. This procedure is also applicable to stays erected by the isotension method, where each strand is cut at its precise length. Readjustments can be performed either by shim adjustment or by rejacking operations. In the latter case, the process can be expressed in term of a force ∆S or a cable elongation ∆L. In order to determine the required adjustment values, different methods are employed. Similar to the calculation of the ideal cable forces, applying an optimisation method can evaluate proper adjustments. Calculating the error factors and predicting the final construction state including the errors can be more efficient than calculating an optimum adjustment for each construction stage. Chapter 4 provides an extensive study of the main issues which are considered in the modelling of a construction stage analysis of cable-stayed bridges. In most of the performed calculations, a symmetric fan type system including temporary supports in the side span is examined using the structural analysis programme MiDAS. To develop the “ideal state” system by an appropriate cable pre-stressing, different restrictions are applied in the analysis. With the restriction of the vertical displacement and at the top of the pylon, a continuous beam condition for the main girder can be achieved, including its possible disadvantage that the distribution of the internal loads show very high bending moments at the anchorage points and only limited values inbetween. As the actual target is a limited value of the bending moments in the girder and the pylon, a more reasonable method is the direct restriction of this parameter. If the designer aims at a more equal distribution of the bending moments, the first method can be a good guideline for the restriction values of the moments in the second method. A recalculation of the determined initial cable forces by matrix-operations clarifies the procedure. As an important fact, it is to be mentioned that the programme operates with internal forces in this part of the construction stage analysis. The examination of the more complex model shows the importance of the correct activation of changing boundary conditions. Not only does the activation to a deformed or unreformed
Chapter 6: Conclusion
225
structure influence the deformation shape, it also has influence on the distribution of the internal forces. In this context, it must be furthermore stated that the cable activation sequences is highly important. This is actually investigated in relation to the analysis of the Second Jindo Bridge in Chapter 5. There is a great difference in the initial cable forces in case of a double activation including stressing (two cables are activated and stressed in one construction stage), double activation and then stressing one cable after the other in different stages, or as a third method, activating the cables separately in different stages. The last case appears to be the most reliable as it is assumed that the actual installation process at the erection site is to erect and stress one cable after the other. From the theoretical point of view, backward and forward analyses only have equal results if the modelled conditions are the same as well. The backward analysis starts from an ideal system with some moments in the centre of the bridge, whereas in the forward analysis, there is a zero moment condition when the key segment is activated and the bridge is closed. The condition can be equalised by modelling de- and restressing operations in the back- and, respectively, the forward analysis (or in a more complex analysis, by moving the derrick crane into a proper position). However, there will be some normal forces in the key segment in the backward analysis, which have a small influence on the results. This influence also depends on the construction sequence, such as on the time of activating a temporary pylon-girder connection. The time of activating this temporary support at the pylon also influences the determined initial cable forces and therefore, a correct modelling of the construction sequence is very important. Finally, there are always some variations in the deformations, at least in the horizontal direction. The forward analysis starts with no horizontal movement of the deck before installing the first cable, whereas in the backward analysis, a horizontal movement remains in the structure after removing the last cable. Creep and shrinkage influence the internal forces as well as the final displacement. Over its time, creep has the tendency to change the internal moments in the direction of the “continuous beam condition” reduced by the effect of shrinkage. The cantilever construction method is compared with the erection on scaffoldings. Due to lower stresses in the section during the construction, the second erection method shows a much lower displacement. The structural deformation can be balanced by a prefabrication camber. The programme offers a function to determine this data which is checked by a control calculation. The construction camber represents the girder elevation to be incorporated at the time of installing a new segment. Under the permanent loads the structure finally ends in a condition with no deformation. Because of time dependent effect, an additional deformation must be considered in the erection of concrete
226
Chapter 6: Conclusion
segments when positioning the form traveller. This can also be calculated with the programme by including material non-linearity in the analysis. Two typical errors which occur during the erection of a cable-stayed bridge are wrong cable forces and errors in the girder elevation. To simulate tension errors, a cable is installed with only 90% of its initially designed value and a restressing is preformed in the final state. The required adjustment value is calculated by the influence matrix. Compared to the original system, an adjustment in the final state changes both, the internal forces and the girder elevation. An error in the girder elevation can be corrected by either retension operations or by changing the angle of the remaining segments. By retensioning all cables, a higher accuracy can be achieved. However, this method is not very economical. An adjustment by angles does not influence the internal forces. The calculations demonstrate that the focus on only one parameter may increases the error in the other. In practical condition, both deviations must be adjusted and limited within a specified tolerance. The effect of cables can be treated in three different ways: the effect can be neglected by using truss elements, considered by an effective stiffness or included in a non-linear analysis. The latter case cannot be applied in a construction stage analysis with the actual programme version. In a static analysis, the three options are compared under the effect of various initial stresses, and it is proved that, for sag to span ratios less then 1/80, the effective stiffness is very close to the results obtained from the non-linear analysis; the linear calculation, which neglects the sagging effects, shows large discrepancies. It is proved in a construction stage analysis that the effective stiffness considers the various load conditions and is therefore applicable in the analysis. Chapter 5 comprises the case study of this work, which is the Second Jindo Bridge in the south of the Republic of Korea. Background information on the location and objective of the project is provided first. The bridge is a cable-stay structure with a total length of 484 m, built with the cantilever method. The main span is 344 m long and both side spans consists of 70 m. The superstructure of the bridge consists of a single box girder which will accommodate two lanes of traffic on the 12.55 m wide deck. In order to balance the weight of the long main span, the box girder is partially filled with concrete in the side span. The pylons are of hollow steel sections and will be composed at the construction site. For the erection of the main span, the prefabricated steel segments are lifted from a delivery barge into position. A detailed construction stage analysis is performed using the analysis programme MiDAS. The main issues relating to the modelling of the structure are outlined. Specific attention is given on the location of the nods representing the anchorage points of the cables because a wrong angle
Chapter 6: Conclusion
227
between the deck and the cables will lead to a wrong estimation of the cable forces. The structural self-weight, as well as the constructions loads are precisely modelled. The ideal cable forces are determined to achieve an optimal structural performance due to its permanent loads. For the construction stage analysis, 40 different stages are modelled so that they allow to include each erection step. Following the erection sequence by the backward and forward methods, the different stages are analysed. The backward analysis reveals the initial tension forces to be applied at the time of installing the cable. These data is applied in the forward analysis which considers the real erection process. In the final analysis, the influence of the sagging effect of the cables is taken into account. The application of cable elements in the calculation does not only influence the distribution of the internal forces, but also the structural displacement during the erection process and in the final state. The deformations are balanced by the camber deformation and the reliability of this data is therefore of major importance. A comparison of the determined cable forces with the results from independent construction stage analyses shows some differences in the chosen cable forces. For the final state, a summation of the first four cables in the side span, which can be assumed to form the backstay cable, shows only a difference of 1.24%, whereas the cables in the main span reveals a gap in the cable forces of up to 7.44%. Most of the calculated tension forces are below the final forces of the compared data. However, there have been uncertainties about the structural self-weight which must be considered when comparing these results. No detailed information on the applied weight exists for the other analyses. In the performed analysis, the initial stressing in the cables is applied in terms of pretension load which is not an intrinsic parameter. Therefore, the construction loads must be precisely known to obtain the accurate initial cable forces. For all construction stages, the minimal and the maximal stresses are controlled to be within an allowable range. Under service loads, the stresses in the structural members increase and in order to consider this condition, a simplified moving load analysis is carried out. The results show that the occurring stresses are in the allowable scale. The construction stage analysis feature is a new function in the MiDAS programme. The temporary version does not allow to include any other geometric non-linearity in the construction stage analysis. A final calculation should be done to take these effects into account as they may increase the internal forces. Using the general camber function, the camber data is calculated and presented. The reliability of this function has been previously proved. A discussion of constructability issues concludes the chapter.
228
Chapter 6: Conclusion
6.2 Contribution The main objective of this study is to provide a comprehensible discussion on the concept of construction stage analyses of cable-stayed bridges. The main issues that are of concern in the analysis and the modelling of the construction process have been compiled and reviewed on the basis of literature. A clear scheme has been developed and presented to explain the interrelationships of the material, the structural elements, the loads and the step by step growing structure. The main problems and salient issues, for which the construction engineer has to sine special attention to while modelling the construction process, have been studied in-depth and clarified. It has been demonstrated that a precise analysis of all construction stages including the exact permanent and construction loads is the main key to develop a safe and economical construction process. The provision of the analysis of the real construction example has enhanced the concept outlined in previous parts of this study. This study is intended to serve as an understandable introduction into the topic of how to analyse the complex construction process of cable-stayed bridges constructed by the cantilever method. Furthermore, it is meant to give a guideline to other potential MiDAS users.
6.3 Recommendation In this study, the focus of attention lies on the cantilever construction which is the major method for erecting the superstructure of long cable-stayed bridges. Besides the permanent and live load conditions, the major topic that needs to be considered in the design of cable-stayed bridges is the effect of dynamic loads. Seismics and aerodynamics are the two major dynamic loads that have to be taken account of in the design of cable-stayed bridges. Both are very interesting themes which still offer a wide field of further research. In the concept and design of future cable-stayed bridges with an increasing span length, their influence on the overall structural layout becomes increasingly important and will with a great probability be the dominant factor in the design process. Further research provides the introduction of improved or new materials offering a range of possibilities for future bridges. According to the longer span, the self-weight of the structure is a main issue in the overall design process, which makes the utilisation of lighter material more interesting but may also increase the structural costs.
Chapter 6: Conclusion
229
In the field of different bridge types, the construction stage analysis and the different erection options with respect to constructability issues could also be a focus of a further research.
6.4 Final comments on the analysis programme MiDAS The analysis programme MiDAS has proved itself to be a powerful tool for modelling and analysing structural systems. The provision of the Construction Stage Analysis feature allows the generation of construction stages by the option to change the structural system and the loading conditions. In addition to the possibility to perform back- and forward analyses, material non-linear behaviour as creep and shrinkage can be modelled and also taken into consideration in the construction stage analysis. Various standard codes are offered to be applied in the analysis. The Unknown Load Factor function enables the user to calculate the ideal pre-stressing forces through an optimization of the static system by applying user defined restrictions. Unfortunately, the consideration of the structural non-linear behaviour in the construction stage analysis is not yet possible. However, the programme should be capable of solving these problems with a new version in the near furure. In a construction stage analysis with the actual version, the effect of the non-linearity of cables can only be considered by an effective stiffness. A disadvantage is that the different stages and stressing operations can only be defined in terms of forces. With some changes in the loading, for example a different self-weight of the derrick crane, the initial tension values must be adapted to these changes as well. This can only be done by a trial and error method, or a new analysis of the different stages. During the time of working out this study, five (!) new versions had been released. The study started with the MiDAS/Civil 5.9 version. Due to important improvements in the construction stage analyses, the new 6.0 version has been tested, which has then been replaced by two other versions because of analysis errors. Finally, the 6.1 version was released, by which the main part of this thesis has been developed. It was not possible to perform a linear back- and forward analyses with identical results for simple models with the first 6.0 versions and it also required quite an effort to analyze and clarify the calculated results. The consideration of the cable effects by an effective stiffness still
230
Chapter 6: Conclusion
reveals a gap between both methods when using the programme version 6.1. The reason for the differences in the back- and forward analyses could not be identified in detail. Owing to the changes in the coding data, it was not possible to open and run the previous input file with the newer version. A further difficulty appeared when it came to light that the help menu was very basic and the theoretical background for elements or calculation functions was only explained in brief. At the beginning of using the MiDAS programme, a help menu for the new features did not exist at all and some data were only available in Korean language. Therefore, it often required to contact the MiDAS support team, which again was not always easy because of lingual problems. It should be noted that for working out this thesis and thus, for overcoming the difficulties mentioned above, a considerable amount of energy and time had been investigated. However, most of the MiDAS problems could be solved by now and I hope that his thesis has contributed at least a little to the facilitation of the calculation and analysis of the complex construction process of cable-stayed bridges with MiDAS/Civil for future users.
Literature
231
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[10]
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Kaneyoshi, M., H. Tanaka, M. Kamei, and H. Furuta, New system identification technique using fuzzy regression analysis, Proceeding of ISUMA ’90 – The first international symposium on uncertainty modelling and analysis, 528-533, Maryland, USA, 1990
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Marchetti, M., C. Bertocchi, D. Regallet and G. Hochet, Elorn Bridge - stay and geometry adjustment, International Conference A.I.P.C. – F.I.P. – Cable-stayed and suspension bridges, 363-370, Volume 2, Deauville, 1994
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[40]
Paik, J., Y. Song, S. Cho and H. Kang, Geometry control of cable-stayed bridge in SeoHae Bridge, IABSE Conference on cable-supported Bridges, Seoul, Korea, 2001
[41]
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[45]
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[48]
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[54]
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[61]
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[63]
Satou,
Y.,
N.
Hirahara,
T.
Murata
and
M.
Asano,
Erection
of
Tatara
Bridge,
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[66]
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[67]
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[68]
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[69]
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[70]
Figure 2-1 and Figure 2-4 : http://www.structurae.net/de/index.cfm
Appendix
237
Appendix
Appendix A: Creep calculation CEB-FIP 1990 Code: CEB-FIP 1990 Compressive strength of concrete at the age of 28 days: f := 39.2266 [N/mm²] ck Relative Humidity:
RH := 80%
Notational size member:
d eff := 400
Age of concrete at the beginning of loading:
t := 460
Cement konstant:
α := 0
RH
1−
100% 3
0.46⋅
t0 := t0T⋅
[mm]
t0T := 7
Age of concrete:
φRH := 1 +
fcm := fck + 8
φRH = 1.274
β fcm :=
5.3
d eff
fcm
100
10
9
2 + t0T
1.2
+ 1
φ0 := φRH⋅ β fcm⋅ β t0
α
t0 = 7
β t0 :=
β fcm = 2.439
1 0.20
0.1 + t0
β t0 = 0.635
φ0 = 1.972
18 RH d eff β H := 150⋅ 1 + 1.2⋅ + 250 ⋅ 100% 100
β H = 1137.762
< 1500
( t − t0) β c := β H + ( t − t0)
φ := φ0⋅ β c
φ = 1.353
0.3
β c = 0.686
Appendix
238 6 tonf
E := 3.63⋅ 10
2
2
A Q := 1m
l := 10m
Pi := 100tonf
m σ :=
Pi
σ = 100
AQ
tonf 2
m
∆l0 := l⋅ εel
εel :=
σ E
−5
εel = 2.755 × 10
∆l0 = 0.275mm
φ11 := 0
φ21 := 0.0
φ31 := 0.0
∆l1 := 1⋅ ∆l0
∆l1 = 0.275mm
φ12 := 0.775
φ22 := 0.0
φ32 := 0.0
∆l2 := 2⋅ ∆l0 + φ12⋅ εel⋅ l
∆l2 = 0.764mm
φ13 := 1.074
φ23 := 0.648
φ33 := 0.0
φ14 := 1.353
φ24 := 0.876
φ34 := 0.653
( ) ∆l4 := 3⋅ ∆l0 + ( φ14 + φ24 + φ34) ⋅ εel⋅ l ∆l3 := 3⋅ ∆l0 + φ13 + φ23 ⋅ εel⋅ l
∆l3 = 1.301mm ∆l4 = 1.620mm
Appendix B: Extracts from the correspondence with MiDAS support E-mail 8th of March 2004: Dear Mr Lee, Maybe you remember me because you have already helped me with some problems a couple of weeks ago. Until last January, I made an internship at Korea Highway Corporation where I modelled a construction stage analysis for a cable-stayed bridge. Now I am back in Germany, but still work for Dr. Park Chan-Min on the project. Until I am finished, I have got a Hardlock for the MiDAS 6.1 version. I downloaded the latest version from the MiDAS Korea homepage, which runs on my computer now. I modelled a simple example for a forward analysis but had some problems. I wanted to consider large deformations (non-linear analysis) in my calculation, but I can only apply the load in one step. Compared to an elastic analysis, the results are not reasonable. For example, when I activate the first cable (truss element; construction stage: CS2), the bending moment in the supported girder increases and the pretension load, which I applied to the cable using external force and replace or add function, is not considered. If I want to apply the loads in more than one step, the calculation does not run. Activating elements tangentially is not possible either. Are these problems related to some input mistakes? I have sent you the input file from my model. Could you please help me with my problem? How is it possible to perform a non linear analysis; do I have to define a loading sequence at the non-linear analysis data? This work will also be my master thesis, which I have to hand over in a few weeks. Your support would be a great help for me. With best regards Marko
Appendix
239
E-mail 10th of March 2004: Dear Marko We are sorry the answer is late. We are trying to test your model. We'll give you an answer as soon as possible. Feel free to contact us anytime. With best regards Mr.Lee e-mail 23rd of March 2004: Dear Mr. Lee, I am still working on a construction stage analysis for cable-stayed bridges. I have sent you a model on 8th March, when I wanted to consider large displacement in the analysis. Have you already been able to solve my problem? Are there any input mistakes? Since I have to submit my report for my university in two weeks, it would be helpful if I could solve this problem soon. Are there other possibilities to consider p-delta or large displacement effects in a construction stage analysis? I also tried to perform a simple calculation considering creep and shrinkage. I tried to recalculate the verification example from the download page. However, I got different results than the ones written in the paper. Are there any additional help files?
Thanks for your support With best regards Marko Grabow e-mail 25th of March 2004: Dear Marko We're so sorry the answer is late. We tried to ask a development team to solve the problem. We understand that non-linear analysis such as p-delta or large displacements effects can't be considered consequently in a construction stage analysis until now. But we're trying to solve the problem. In several months, it will be possible to consider a non-linear analysis with a construction stage analysis. And the verification example from the download page shoud have been changed. We also upgraded MIDAS/Civil as Ver.6.1.1. And the verification example was changed. We attach the verification example. p.s: We'll send MIDAS/Civil Ver.6.1.1 by another mail.
With best regards Mr. Lee
Appendix
240
Appendix C: Model of the Second Jindo Bridge Nodes Table: N. N.
x [m]
z[m]
N. N.
x [m]
z[m]
N. N.
x [m]
z[m]
N. N.
x [m]
z[m]
1
-245.787
19.811
21
-125.100
25.742
105
-172.000
13.370
152
-172.000
88.860
2
-242.770
19.962
22
-112.500
26.241
106
-172.000
20.090
153
-172.000
88.920
3
-242.000
20.000
23
-108.100
26.402
132
-172.000
29.920
1001
-242.000
18.290
4
-241.179
20.041
24
-95.500
26.830
134
-172.000
46.920
3002
-244.610
18.252
5
-239.589
20.121
25
-91.100
26.967
136
-172.000
66.920
3004
-243.019
18.331
6
-237.999
20.200
26
-78.500
27.323
137
-172.000
76.420
3005
-241.429
18.411
7
-235.750
20.313
27
-74.100
27.435
138
-172.000
82.170
3006
-239.839
18.490
8
-234.750
20.363
28
-61.500
27.720
139
-172.000
82.350
3009
-218.750
20.617
9
-218.750
21.163
29
-57.100
27.807
140
-172.000
82.650
3011
-197.500
21.679
10
-214.350
21.383
30
-44.500
28.020
141
-172.000
83.330
3018
-146.500
24.227
11
-197.500
22.225
31
-40.100
28.082
142
-172.000
83.860
3020
-129.500
25.009
12
-193.100
22.445
32
-27.500
28.224
143
-172.000
84.120
3022
-112.500
25.695
13
-182.900
22.955
33
-23.100
28.261
144
-172.000
84.940
3024
-95.500
26.284
14
-173.400
23.430
34
-10.500
28.332
145
-172.000
85.780
3026
-78.500
26.777
15
-172.000
23.500
35
-6.100
28.344
146
-172.000
85.870
3028
-61.500
27.174
16
-168.450
23.678
36
0.000
28.350
147
-172.000
86.630
3030
-44.500
27.474
17
-155.700
24.315
101
-172.000
-4.500
148
-172.000
86.870
3032
-27.500
27.678
18
-146.500
24.773
102
-172.000
-2.500
149
-172.000
87.490
3034
-10.500
27.786
19
-142.100
24.985
103
-172.000
0.000
150
-172.000
87.860
20
-129.500
25.555
104
-172.000
12.370
151
-172.000
88.360
Table A-1: Node coordinates
Element Table: Element Typ
Material
Property
Node 1
Node 2
Element Typ
Material
Property
Node 1
Node 2
BEAM
4
4
1
2
BEAM
2
6
104
105
BEAM
4
4
2
3
BEAM
2
8
105
106
BEAM
4
4
3
4
BEAM
6
9
106
132
BEAM
4
4
4
5
BEAM
6
10
132
134
BEAM
4
4
5
6
BEAM
6
11
134
136
BEAM
4
4
6
7
BEAM
6
12
136
137
BEAM
1
5
7
8
BEAM
5
13
137
138
BEAM
1
2
8
9
BEAM
5
13
138
139
BEAM
1
2
9
10
BEAM
5
13
139
140
BEAM
1
2
10
11
BEAM
5
13
140
141
BEAM
1
2
11
12
BEAM
5
13
141
142
BEAM
1
2
12
13
BEAM
5
13
142
143
BEAM
1
3
13
14
BEAM
5
13
143
144
BEAM
1
3
14
15
BEAM
5
13
144
145
Appendix
241
Element Typ
Material
Property
Node 1
Node 2
Element Typ
Material
Property
Node 1
Node 2
BEAM
1
3
15
16
BEAM
5
13
145
146
BEAM
1
1
16
17
BEAM
5
14
146
147
BEAM
1
1
17
18
BEAM
5
14
147
148
BEAM
1
1
18
19
BEAM
5
14
148
149
BEAM
1
1
19
20
BEAM
5
14
149
150
BEAM
1
1
20
21
BEAM
5
14
150
151
BEAM
1
1
21
22
BEAM
5
14
151
152
BEAM
1
1
22
23
BEAM
5
14
152
153
BEAM
1
1
23
24
TRUSS
7
15
152
3002
BEAM
1
1
24
25
TRUSS
7
15
150
3004
BEAM
1
1
25
26
TRUSS
7
15
148
3005
BEAM
1
1
26
27
TRUSS
7
15
146
3006
BEAM
1
1
27
28
TRUSS
7
15
142
3009
BEAM
1
1
28
29
TRUSS
7
15
139
3011
BEAM
1
1
29
30
TRUSS
8
18
138
3018
BEAM
1
1
30
31
TRUSS
8
18
140
3020
BEAM
1
1
31
32
TRUSS
8
18
141
3022
BEAM
1
1
32
33
TRUSS
9
17
143
3024
BEAM
1
1
33
34
TRUSS
9
17
144
3026
BEAM
1
1
34
35
TRUSS
9
17
145
3028
BEAM
1
1
35
36
TRUSS
9
17
147
3030
BEAM
2
6
101
102
TRUSS
10
16
149
3032
BEAM
2
7
102
103
TRUSS
10
16
151
3034
BEAM
2
7
103
104
Table A-2: Element table
Appendix
242 Elastic Link Table: No
Node1
Node2
Typ
Sdx
Sdz
Sry
Group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
3002 3004 106 3 3005 3006 3009 3011 3018 3020 3022 3024 3026 3028 3030 3032 3034 106
2 4 15 1001 5 6 9 11 18 20 22 24 26 28 30 32 34 15
GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN GEN
100.000.000 100.000.000 418.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000
100.000.000 100.000.000 721 0.0000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000
100.000.000 100.000.000 0.00 0.00 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 100.000.000 0.00
Cable1 Cable2 ElaPylon Left Cable3 Cable4 Cable5 Cable6 Cable7 Cable8 Cable9 Cable10 Cable11 Cable12 Cable13 Cable14 Cable15 TempPylon
Table A-3: Elastic link table
Appendix
243
Appendix D: Allowable and existing stresses Maximum and minimum stresses in the girder segments due to construction loads: Segment No. Nmax
1
Mmax Mmin Nmax
2
Mmax Mmin Nmax
3
Mmax Mmin Nmax
4
Mmax Mmin Nmax
5
Mmax Mmin Nmax
6
Mmax Mmin Nmax
7
Mmax Mmin
8
Nmax Mmax
N [tonf] -1497.85 -1497.85 -9.44 -9.44 -1360.04 -1360.04 -1780.84 -1780.84 -47.95 -47.95 -1337.03 -1337.03 -1993.89 -1993.89 -47.95 -47.95 -1506.03 -1506.03 -1989.36 -1989.36 -207.74 -207.74 -1503.41 -1503.41 -1993.22 -1993.22 -207.74 -207.74 -1993.22 -1993.22 -1990.49 -1990.49 -71.67 -71.67 -122.08 -122.08 -1985.18 -1985.18 -153.48 -153.48 -5.95 -5.95 -1932.82 -1932.82 -88.78 -88.78
My [tonf*m] -1418.30 -1418.30 1427.78 1427.78 -1633.85 -1633.85 -622.18 -622.18 2786.29 2786.29 -1639.23 -1639.23 -770.36 -770.36 2786.29 2786.29 -1510.33 -1510.33 155.91 155.91 1885.26 1885.26 -523.90 -523.90 -1391.92 -1391.92 1885.26 1885.26 -1391.92 -1391.92 -859.85 -859.85 1356.06 1356.06 -910.83 -910.83 32.64 32.64 1007.89 1007.89 -1319.76 -1319.76 59.44 59.44 599.39 599.39
Property No.
4
2
1
1
3
1
1
1
fc
fcz
f
fcal
-160.03 -160.03 -1.01 -1.01 -145.30 -145.30 -366.50 -366.50 -9.87 -9.87 -275.17 -275.17 -431.20 -431.20 -10.37 -10.37 -325.70 -325.70 -430.22 -430.22 -44.93 -44.93 -325.13 -325.13 -262.51 -262.51 -27.36 -27.36 -262.51 -262.51 -430.47 -430.47 -15.50 -15.50 -26.40 -26.40 -429.32 -429.32 -33.19 -33.19 -1.29 -1.29 -418.00 -418.00 -19.20 -19.20
161.89 -116.18 -162.97 116.96 186.50 -133.84 109.56 -190.83 -490.63 854.57 288.65 -502.76 137.03 -237.93 -495.61 860.54 268.65 -466.46 -27.73 48.15 -335.34 582.26 93.19 -161.81 201.11 -193.54 -272.39 262.13 201.11 -193.54 152.95 -265.56 -241.21 418.82 162.01 -281.31 -5.81 10.08 -179.28 311.29 234.75 -407.61 -10.57 18.36 -106.62 185.12
1.87 -276.21 -163.98 115.95 41.19 -279.14 -256.94 -557.33 -500.50 844.70 13.48 -777.93 -294.18 -669.13 -505.98 850.17 -57.05 -792.16 -457.96 -382.07 -380.27 537.34 -231.94 -486.94 -61.40 -456.04 -299.75 234.77 -61.40 -456.04 -277.52 -696.03 -256.71 403.32 135.61 -307.71 -435.13 -419.24 -212.47 278.09 233.47 -408.89 -428.57 -399.64 -125.82 165.92
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
0.00 0.21 0.13 0.09 0.03 0.22 0.20 0.43 0.39 0.65 0.01 0.60 0.23 0.52 0.39 0.66 0.04 0.61 0.36 0.30 0.29 0.42 0.18 0.38 0.05 0.35 0.23 0.18 0.05 0.35 0.22 0.54 0.20 0.31 0.11 0.24 0.34 0.32 0.16 0.22 0.18 0.32 0.33 0.31 0.10 0.13
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
244 Segment No. Mmin Nmax
9
Mmax Mmin Nmax
10
Mmax Mmin Nmax
11
Mmax Mmin Nmax
12
Mmax Mmin Nmax
13
Mmax Mmin Nmax
14
Mmax Mmin Nmax
15
Mmax Mmin Nmax
key
Mmax Mmin
N [tonf] -219.80 -219.80 -1842.04 -1842.04 -115.27 -115.27 -537.27 -537.27 -1713.67 -1713.67 -177.46 -177.46 -625.19 -625.19 -1540.76 -1540.76 -212.68 -212.68 -715.20 -715.20 -1329.17 -1329.17 -890.15 -890.15 -811.74 -811.74 -1078.12 -1078.12 -638.04 -638.04 -467.60 -467.60 -795.41 -795.41 -343.77 -343.77 -467.04 -467.04 -455.95 -455.95 3.67 3.67 -2.57 -2.57 -2.44 -2.44 3.81 3.81 -2.44 -2.44
My [tonf*m] -1716.10 -1716.10 89.71 89.71 479.37 479.37 -1780.46 -1780.46 113.07 113.07 445.16 445.16 -2056.48 -2056.48 95.22 95.22 465.46 465.46 -2238.86 -2238.86 77.10 77.10 528.83 528.83 -2342.86 -2342.86 47.72 47.72 626.48 626.48 -2171.55 -2171.55 -1189.80 -1189.80 586.31 586.31 -1798.00 -1798.00 -696.90 -696.90 56.69 56.69 -1201.17 -1201.17 -994.39 -994.39 210.06 210.06 -994.39 -994.39
Property No.
1
1
1
1
1
1
1
1
fc
fcz
f
fcal
-47.53 -47.53 -398.37 -398.37 -24.93 -24.93 -116.19 -116.19 -370.60 -370.60 -38.38 -38.38 -135.21 -135.21 -333.21 -333.21 -45.99 -45.99 -154.67 -154.67 -287.45 -287.45 -192.51 -192.51 -175.55 -175.55 -233.16 -233.16 -137.98 -137.98 -101.12 -101.12 -172.02 -172.02 -74.34 -74.34 -101.00 -101.00 -98.61 -98.61 0.79 0.79 -0.56 -0.56 -0.53 -0.53 0.82 0.82 -0.53 -0.53
305.25 -530.02 -15.96 27.71 -85.27 148.05 316.70 -549.89 -20.11 34.92 -79.18 137.49 365.80 -635.14 -16.94 29.41 -82.79 143.76 398.24 -691.47 -13.71 23.81 -94.07 163.33 416.74 -723.59 -8.49 14.74 -111.44 193.49 386.27 -670.68 211.64 -367.47 -104.29 181.08 319.82 -555.31 123.96 -215.24 -10.08 17.51 213.66 -370.98 176.88 -307.12 -37.36 64.88 176.88 -307.12
257.72 -577.55 -414.32 -370.66 -110.20 123.12 200.51 -666.09 -390.72 -335.68 -117.56 99.11 230.59 -770.35 -350.15 -303.80 -128.79 97.76 243.57 -846.14 -301.16 -263.64 -286.57 -29.18 241.19 -899.14 -241.65 -218.42 -249.42 55.50 285.14 -771.81 39.62 -539.49 -178.64 106.74 218.82 -656.31 25.36 -313.84 -9.29 18.30 213.10 -371.54 176.35 -307.64 -36.54 65.70 176.35 -307.64
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
Table A-4: Control of allowable stresses of the girder segments during construction
0.20 0.45 0.32 0.29 0.09 0.10 0.16 0.52 0.30 0.26 0.09 0.08 0.18 0.60 0.27 0.24 0.10 0.08 0.19 0.66 0.23 0.20 0.22 0.02 0.19 0.70 0.19 0.17 0.19 0.04 0.22 0.60 0.03 0.42 0.14 0.08 0.17 0.51 0.02 0.24 0.01 0.01 0.17 0.29 0.14 0.24 0.03 0.05 0.14 0.24
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
245
Maximum and minimum stresses in the girder segments due to construction and traffic loads: Segment No. Nmax
1
Mmax Mmin Nmax
2
Mmax Mmin Nmax
3
Mmax Mmin Nmax
4
Mmax Mmin Nmax
5
Mmax Mmin Nmax
6
Mmax Mmin Nmax
7
Mmax Mmin Nmax
8
Mmax Mmin
N [tonf] -2083.78 -2083.78 -9.44 -9.44 -2083.78 -2083.78 -2415.04 -2415.04 -47.95 -47.95 -2415.04 -2415.04 -2623.11 -2623.11 -47.95 -47.95 -2622.18 -2622.18 -2618.57 -2618.57 -207.74 -207.74 -2618.57 -2618.57 -2633.97 -2633.97 -207.74 -207.74 -2246.77 -2246.77 -2630.73 -2630.73 -71.67 -71.67 -2243.52 -2243.52 -2623.62 -2623.62 -153.48 -153.48 -5.95 -5.95 -2558.17 -2558.17 -88.78 -88.78 -219.80 -219.80
My [tonf*m] -2397.29 -2397.29 1427.78 1427.78 -2397.29 -2397.29 -2192.06 -2192.06 2786.29 2786.29 -2192.06 -2192.06 -1761.36 -1761.36 2786.29 2786.29 -1773.43 -1773.43 -802.26 -802.26 1885.26 1885.26 -802.26 -802.26 -2192.67 -2192.67 1885.26 1885.26 -2676.85 -2676.85 -1462.51 -1462.51 1356.06 1356.06 -1861.61 -1861.61 -150.51 -150.51 1007.89 1007.89 -1319.76 -1319.76 -43.31 -43.31 599.39 599.39 -1716.10 -1716.10
Property No.
4
2
1
1
3
1
1
1
fc
fcz
f
fcal
-222.63 -222.63 -1.01 -1.01 -222.63 -222.63 -497.02 -497.02 -9.87 -9.87 -497.02 -497.02 -567.28 -567.28 -10.37 -10.37 -567.08 -567.08 -566.30 -566.30 -44.93 -44.93 -566.30 -566.30 -346.89 -346.89 -27.36 -27.36 -295.90 -295.90 -568.93 -568.93 -15.50 -15.50 -485.19 -485.19 -567.39 -567.39 -33.19 -33.19 -1.29 -1.29 -553.24 -553.24 -19.20 -19.20 -47.53 -47.53
273.64 -196.37 -162.97 116.96 273.64 -196.37 386.00 -672.32 -490.63 854.57 386.00 -672.32 313.30 -544.00 -495.61 860.54 315.45 -547.72 142.70 -247.78 -335.34 582.26 142.70 -247.78 316.81 -304.88 -272.39 262.13 386.77 -372.20 260.15 -451.70 -241.21 418.82 331.14 -574.96 26.77 -46.48 -179.28 311.29 234.75 -407.61 7.70 -13.38 -106.62 185.12 305.25 -530.02
51.01 -419.00 -163.98 115.95 51.01 -419.00 -111.03 -1169.34 -500.50 844.70 -111.03 -1169.34 -253.98 -1111.28 -505.98 850.17 -251.63 -1114.80 -423.60 -814.08 -380.27 537.34 -423.60 -814.08 -30.09 -651.77 -299.75 234.77 90.87 -668.10 -308.78 -1020.63 -256.71 403.32 -154.05 -1060.15 -540.62 -613.88 -212.47 278.09 233.47 -408.89 -545.53 -566.61 -125.82 165.92 257.72 -577.55
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
0.04 0.32 0.13 0.09 0.04 0.32 0.09 0.91 0.39 0.65 0.09 0.91 0.20 0.86 0.39 0.66 0.20 0.86 0.33 0.63 0.29 0.42 0.33 0.63 0.02 0.51 0.23 0.18 0.07 0.52 0.24 0.79 0.20 0.31 0.12 0.82 0.42 0.48 0.16 0.22 0.18 0.32 0.42 0.44 0.10 0.13 0.20 0.45
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
246 Segment No. Nmax
9
Mmax Mmin Nmax
10
Mmax Mmin Nmax
11
Mmax Mmin Nmax
12
Mmax Mmin Nmax
13
Mmax Mmin Nmax
14
Mmax Mmin Nmax
15
Mmax Mmin Nmax
key
Mmax Mmin
N [tonf] -2438.24 -2438.24 -115.27 -115.27 -537.27 -537.27 -2268.45 -2268.45 -1637.27 -1637.27 -625.19 -625.19 -2022.21 -2022.21 -1377.23 -1377.23 -715.20 -715.20 -1725.84 -1725.84 -1194.92 -1194.92 -811.74 -811.74 -1380.99 -1380.99 -638.04 -638.04 -467.60 -467.60 -987.18 -987.18 -521.98 -521.98 -467.04 -467.04 -521.76 -521.76 13.90 13.90 -2.57 -2.57 -2.61 -2.61 14.12 14.12 -2.44 -2.44
My [tonf*m] -68.42 -68.42 479.37 479.37 -1780.46 -1780.46 -153.88 -153.88 527.16 527.16 -2056.48 -2056.48 -223.37 -223.37 534.31 534.31 -2238.86 -2238.86 -91.86 -91.86 587.19 587.19 -2342.86 -2342.86 201.13 201.13 626.48 626.48 -2171.55 -2171.55 689.26 689.26 1197.37 1197.37 -1798.00 -1798.00 1197.37 1197.37 1846.41 1846.41 -1201.17 -1201.17 -24.35 -24.35 2184.07 2184.07 -994.39 -994.39
Property No.
1
1
1
1
1
1
1
1
fc
fcz
f
fcal
-527.30 -527.30 -24.93 -24.93 -116.19 -116.19 -490.58 -490.58 -354.08 -354.08 -135.21 -135.21 -437.33 -437.33 -297.84 -297.84 -154.67 -154.67 -373.24 -373.24 -258.42 -258.42 -175.55 -175.55 -298.66 -298.66 -137.98 -137.98 -101.12 -101.12 -213.49 -213.49 -112.88 -112.88 -101.00 -101.00 -112.84 -112.84 3.01 3.01 -0.56 -0.56 -0.56 -0.56 3.05 3.05 -0.53 -0.53
12.17 -21.13 -85.27 148.05 316.70 -549.89 27.37 -47.53 -93.77 162.81 365.80 -635.14 39.73 -68.99 -95.04 165.02 398.24 -691.47 16.34 -28.37 -104.45 181.35 416.74 -723.59 -35.78 62.12 -111.44 193.49 386.27 -670.68 -122.60 212.88 -212.98 369.81 319.82 -555.31 -212.98 369.81 -328.43 570.26 213.66 -370.98 4.33 -7.52 -388.49 674.55 176.88 -307.12
-515.13 -548.43 -110.20 123.12 200.51 -666.09 -463.21 -538.11 -447.85 -191.27 230.59 -770.35 -397.60 -506.32 -392.88 -132.82 243.57 -846.14 -356.90 -401.61 -362.86 -77.06 241.19 -899.14 -334.43 -236.54 -249.42 55.50 285.14 -771.81 -336.09 -0.61 -325.87 256.92 218.82 -656.31 -325.82 256.97 -325.43 573.27 213.10 -371.54 3.77 -8.08 -385.44 677.60 176.35 -307.64
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
0.40 0.43 0.09 0.10 0.16 0.52 0.36 0.42 0.35 0.15 0.18 0.60 0.31 0.39 0.30 0.10 0.19 0.66 0.28 0.31 0.28 0.06 0.19 0.70 0.26 0.18 0.19 0.04 0.22 0.60 0.26 0.00 0.25 0.20 0.17 0.51 0.25 0.20 0.25 0.44 0.17 0.29 0.00 0.01 0.30 0.53 0.14 0.24
Table A-5: Control of allowable stresses of the girder segments under live load condition
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
247
Maximum and minimum stresses in the pylon due to construction loads: Segment No. Nmax
5
Mmax Mmin Nmax
5-7
Mmax Mmin Nmax
7
Mmax Mmin Nmax
9
Mmax Mmin Nmax
11
Mmax Mmin Nmax
12
Mmax Mmin
N [tonf] -4089.5 -4089.5 -3261.6 -3261.6 -1152 -1152 -4016.6 -4016.6 -3188.7 -3188.7 -1079.1 -1079.1 -3907.6 -3907.6 -3907.6 -3907.6 -970.03 -970.03 -3799.2 -3799.2 -210.59 -210.59 -2280.1 -2280.1 -3739.5 -3739.5 -150.85 -150.85 -1074 -1074 -1543 -1543 -45.35 -45.35 -1091 -1091
My [tonf*m] 257.61 257.61 751.33 751.33 -3665.77 -3665.77 188.42 188.42 488.47 488.47 -3101.95 -3101.95 68.77 68.77 68.77 68.77 -2126.88 -2126.88 -71.99 -71.99 0 0 -1117.71 -1117.71 -138.85 -138.85 0 0 -1521.23 -1521.23 -189.38 -189.38 0 0 -1327.27 -1327.27
Property No.
9
10
11
12
13
14
fc
fcz
f
fcal
-519.50 -519.50 -414.32 -414.32 -146.34 -146.34 -589.64 -589.64 -468.10 -468.10 -158.41 -158.41 -679.46 -679.46 -679.46 -679.46 -168.67 -168.67 -569.17 -569.17 -31.55 -31.55 -341.59 -341.59 -368.09 -368.09 -14.85 -14.85 -105.72 -105.72 -254.20 -254.20 -7.47 -7.47 -179.73 -179.73
-47.53 47.53 -138.63 138.63 676.36 -676.36 -40.08 40.08 -103.90 103.90 659.79 -659.79 -17.30 17.30 -17.30 17.30 535.13 -535.13 15.79 -15.79 0.00 0.00 245.20 -245.20 18.38 -18.38 0.00 0.00 201.37 -201.37 51.51 -51.51 0.00 0.00 360.98 -360.98
-567.04 -471.97 -552.95 -275.70 530.02 -822.70 -629.72 -549.56 -571.99 -364.20 501.38 -818.20 -696.76 -662.15 -696.76 -662.15 366.46 -703.80 -553.38 -584.96 -31.55 -31.55 -96.38 -586.79 -349.71 -386.47 -14.85 -14.85 95.65 -307.08 -202.69 -305.71 -7.47 -7.47 181.26 -540.71
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
Table A-6: Control of allowable stresses of the pylon due to construction loads
0.44 0.37 0.43 0.21 0.41 0.64 0.49 0.43 0.44 0.28 0.39 0.63 0.54 0.51 0.54 0.51 0.28 0.55 0.43 0.45 0.02 0.02 0.07 0.45 0.27 0.30 0.01 0.01 0.07 0.24 0.16 0.24 0.01 0.01 0.14 0.42
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
248
Maximum and minimum stresses in the pylon due to construction and traffic loads: Segment No. Nmax
5
Mmax Mmin Nmax
5-7
Mmax Mmin Nmax
7
Mmax Mmin Nmax
9
Mmax Mmin Nmax
11
Mmax Mmin Nmax
12
Mmax Mmin
N [tonf] -5142.3 -5142.3 -5123.1 -5123.1 -1152 -1152 -5069.4 -5069.4 -5050.2 -5050.2 -1079.1 -1079.1 -4960.3 -4960.3 -4941.1 -4941.1 -970.03 -970.03 -4851.9 -4851.9 -3822.2 -3822.2 -2280.1 -2280.1 -4792.2 -4792.2 -3762.4 -3762.4 -1074 -1074 -2120.1 -2120.1 -45.35 -45.35 -1091 -1091
My [tonf*m] 1532.3 1532.3 1742.4 1742.4 -3665.77 -3665.77 1189.9 1189.9 1335.02 1335.02 -3101.95 -3101.95 597.76 597.76 630.5 630.5 -2126.88 -2126.88 -98.88 -98.88 44.15 44.15 -1117.71 -1117.71 -429.78 -429.78 50.43 50.43 -1521.23 -1521.23 -572.23 -572.23 0 0 -1327.27 -1327.27
Property No.
9
10
11
12
13
14
fc
fcz
f
fcal
-653.24 -653.24 -650.80 -650.80 -146.34 -146.34 -744.18 -744.18 -741.37 -741.37 -158.41 -158.41 -862.51 -862.51 -859.17 -859.17 -168.67 -168.67 -726.88 -726.88 -572.61 -572.61 -341.59 -341.59 -471.72 -471.72 -370.35 -370.35 -105.72 -105.72 -349.28 -349.28 -7.47 -7.47 -179.73 -179.73
-282.72 282.72 -321.48 321.48 676.36 -676.36 -253.09 253.09 -283.96 283.96 659.79 -659.79 -150.40 150.40 -158.64 158.64 535.13 -535.13 21.69 -21.69 -9.69 9.69 245.20 -245.20 56.89 -56.89 -6.68 6.68 201.37 -201.37 155.63 -155.63 0.00 0.00 360.98 -360.98
-935.96 -370.52 -972.28 -329.31 530.02 -822.70 -997.28 -491.09 -1025.33 -457.40 501.38 -818.20 -1012.91 -712.11 -1017.81 -700.54 366.46 -703.80 -705.19 -748.57 -582.30 -562.92 -96.38 -586.79 -414.83 -528.61 -377.03 -363.68 95.65 -307.08 -193.65 -504.91 -7.47 -7.47 181.26 -540.71
1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290 1290
Table A-7: Control of allowable stresses of the pylon under live load condition
0.73 0.29 0.75 0.26 0.41 0.64 0.77 0.38 0.79 0.35 0.39 0.63 0.79 0.55 0.79 0.54 0.28 0.55 0.55 0.58 0.45 0.44 0.07 0.45 0.32 0.41 0.29 0.28 0.07 0.24 0.15 0.39 0.01 0.01 0.14 0.42
OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK
Appendix
249
Appendix E: Unstressed cable length L0
Figure A-1: Cable element with the length ds
Forming the equilibrium condition for the cable element shown in Figure A-1. a general differential equation can be derived as
H
d2y = q ( x) . dx 2
(A-1)
Assuming the constancy of q and a parable cable-equation. Peterson [11] developed the following function for a cable with two suspension-points of different height:
[
]
y ( x) = x 2 − ( x A − x B )x + x A x B *
y − yA y x − yB xA q + B x+ A B . 2H x B − x A xB − x A
(A-2)
Transforming the equation as given above and introducing the parameters
K2 =
q . 2H
K 1 = tan α −
(A-3)
q (x A + x B ) and 2H
(A-4)
q x A xB 2H
(A-5)
K 0 = y A − x A tan α +
the cable function can be written as
y ( x) = K 2 x 2 + K 1 x + K 0 .
(A-6)
The first derivative of y(x) is the gradient of the cable written as:
y ( x) ′ =
dy = 2K 2 x + K1 dx
(A-7)
Appendix
250
In accordance to Figure A-1 and Equation A-7. the cable length ds can be formulated as
dy ds = dx 2 + dy 2 = dx 1 + dx
2
(A-8)
ds = 1 + (2 K 2 x + K 1 ) 2 dx
(A-9)
and the final cable length follows to the integration of
L = ∫ 1 + K 12 + 4 K 2 K 1 x + 4 K 22 x 2 dx .
(A-10)
In order to solve the integral of the cable length L. the parameters a. b. and c are introduced as
a = 1 + K 12
(A-11)
b = 4K 2 K1
(A-12)
c = 4K 22
(A-13)
and the integral can be given as a type of
B = a + bx + cx 2 .
(A-14)
This integral is of a standard type now and the solution can be given as (e.g. Bartsch [1])
∫
B dx =
b + 2cx γ dx B+ ∫ 4c 8c B
(A-15)
with
γ = 4ac − b 2
(A-16)
and
∫
dx B
=
1 c
(
)
ln b + 2cx + 2 cB .
(A-17)
Finally. the cable length L can be expressed as
L = ∫ B dx =
(
)
b + 2cx γ 1 B+ ln b + 2cx + 2 cB . 4c 8c c
(A-18)
Introducing the parameters I1. I2 and I3 as
I1 = I2 =
b + 2cx B 4c
γ 8c 3 2
(A-19)
(A-20)
Appendix
251
(
I 3 = ln b + 2cx + 2 cB
)
(A-21)
the solution of the final cable length can be written as
L = [I1 + I 2 * I 3 ]x= xBA . x= x
(A-22)
The elongation of the cable due to its tension force S must be considered in the determination of the initial cable length. Under the consideration of
S ( x) =
H . cos Θ
(A-23)
the elongation of a cable element ds can be written as
d (∆L ) =
Sds H ds . = * AE AE cos Θ
(A-24)
which can also be expressed as 2
H ds H ds H ds d (∆L ) = * = * ds = * dx . AE dx AE dx AE dx ds
(A-25)
By integration. the elongation over the total cable length follows to 2
x
H B ds ∆L = dx . AE x∫A dx
(A-26)
Using Equation A-8 and A-9 respectively. formula A-26 can be written as
H ∆L = AE
∫ (1 + K
xB
2 1
)
+ 4 K 2 K 1 x + 4 K 22 x 2 * dx .
(A-27)
xA
With the introduction of the parameters L1. L2 and L3 as
(
)
L1 = 1 + K 12 * ( x A − x B )
(
L2 = 2 K 2 K 1 * x A − x B L3 =
(
2
2
(A-28)
)
)
4 2 3 3 K 2 * x A − xB . 3
(A-29) (A-30)
Equation A-27 follows to
∆L =
H (L1 + L2 + L3 ) . AE
(A-31)
Finally. the unstressed cable length can be calculated from Equation A-32 as
L0 = L − ∆L
(A-32)
252
Appendix F: Cross sections and plans of the Second Jindo Bridge
Appendix
Appendix
253
254
Appendix
Appendix
255
256
Appendix
Appendix
257
258
Appendix
Appendix
259
260
Appendix
Appendix
261
262
Appendix
Appendix
263
264
Appendix
Appendix
265
266
Appendix
Appendix
267
