DEVELOPMENT OF A SECOND-ORDER INELASTIC ANALYSIS METHOD
ACCOUNTED FOR CONSTRUCTION STAGE EFFECTS ON THE BEHAVIOUR OF
PRESTRESSED STEEL STRUCTURES
Thuy Thi My Nguyen MEng
Submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy
School of Civil Engineering and Built Environment
Faculty of Science and Engineering
Queensland University of Technology
2018
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures i
Dedication
To my parents, husband and two sons with love
ii Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
Keywords
Beam-column element; Build-Kill technique; Construction sequence; Constructional
displacement; Deformed geometry; Direct solving method; Equivalent load approach;
Finite element method; Higher-order element; Influence matrix method; Initial force;
Interdependent behaviour; Iterative solution approach; Load sequence; Nonlinear
geometry; Nonlinear material; Numerical solution procedure; Prestressed steel; Pre-
tension process analysis; Refined plastic hinge; Step by step technique; Updated
Lagrangian coordinate.
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures iii
Abstract
In prestressed steel structures, prestressed member forces are always difficult to
maintain during the construction phase, because the displacements of those members
incurred during construction can release their specific prestressed forces. This is
particularly true in the case of lacking temporary supports and/or stability precaution.
This phenomenon may be further exacerbated by nonlinearities owing to large
prestress load. It implies that the prestressed load level of a prestressed structure is
hardly preserved at its final stage when those constructional displacements are
inevitable. Unfortunately, existing analysis approaches of prestressed steel structures
have often neglected the effects of constructional displacements, in particular, the
constructional displacements that occurred between two constitutive stages subjected
to the deformed geometry of the previous stage, on which the newly structural parts
are built at the next construction sequence. These unaccounted effects, in turn, produce
the nonlinearities that impair the structural safety at the construction phase. To this
end, this study presents a second-order inelastic analysis to take the nonlinearities of a
prestressed steel structure at construction sequence into account, in which the
nonlinear effects from the constructional displacements of a structure on its prestress
loads are continuously evaluated at any sequence until its final stage.
These constructional displacements at a construction stage are commonly due to
gravity and constructional loads, which makes the original alignment at the next
construction stage hard to maintain. In order to preserve the alignment at the next stage
along with minimising the member’s length, the position technique for installation at
the next stage subjected to these constructional displacements is developed by virtue
of the nonlinear least-square approach. The construction sequence is simulated using
the step-by-step method together with the ‘Build and Kill’ technique for establishing
the global tangent stiffness matrix. Additionally, the higher-order element formulation
is employed to capture the nonlinear geometric effects. At the same time, the proposed
method reliant on the refined plastic-hinge approach can also evaluate the structural
safety at service condition, which is prone to material nonlinearities, such that the
structural performance of a prestressed steel structure sensitive to constructional
displacements can be predicted. Thanks to all these merits, the proposed approach can
properly capture the effects of constructional displacements that occurred in each
iv Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
construction stage and also in between two constitutive stages. As a result, any
instability, excessive deflections of structural members, or the possibility of structural
collapse during construction can be predicted and avoided.
Additionally, another important feature in the design of prestressed steel
structure is the interaction among prestressed members in the entire system. Under the
circumstances of the presence of many prestressed members in the system and the
limited capability of tensioning equipment make it impossible to prestress all members
simultaneously. When one member is prestressed to its target value for the optimal
capacity of the system, the target values in other tensioning members will immediately
change due to the interdependent behaviour of all tensioning members in the system.
As a result, batched and repeated tensioning schemes are unavoidable so that the
required tensioning control force and/or displacement of each tensioning member can
be computed to achieve the final target state. In order words, another important feature
in the design of prestressed steel structure is to predict properly the prestressing forces
required to achieve a target prestressed state which is significantly influenced by the
interdependent behaviour among all prestressed members in the entire system.
Therefore, a comprehensive linear elastic analysis of the pre-tension process is
presented based on the Influence matrix approach in which four different types of
Influence matrix are introduced and two different solving methods are brought forth.
The direct solving method solves for an accurate solution, whereas the iterative solving
method repeatedly amends trial values to achieve an approximate solution. Through a
series of numerical examples, the analysis result shows that various kinds of
complicated batched and/or repeated tensioning schemes can be analysed reliably,
effectively, and efficiently.
However, as Influence matrices are set up based on the principle of linear
superposition, the pre-tension process analysis based on the Influence matrix approach
is limited to the linear elastic range only. Further, as constructional displacements have
direct effects on structural nodal coordinates, they, in turn, influences the structural
behaviour of prestressed steel structures. In this case, the iterative solution approach is
obviously a more appropriate one. Unfortunately, existing iterative solution methods
for the pre-tension process analysis often cannot accurately capture all the
constructional displacements that may take place, especially the constructional
displacements occurred between two constitutive stages subjected to the deformed
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures v
geometry of the previous stage, on which the new structural parts are built in the next
construction sequence. To this end, this research further presents an iterative solution
approach for the pre-tension process analysis which searches for the prestressing
forces required in order to achieve a target prestressed state. By incorporating the
above nonlinear construction stage analysis, this iterative approach can account for the
effects of displacements incurred within a construction stage and in between two
constitutive construction stages as well as all the inelastic material behaviour which
may take place during the construction phase. By accounting for this particular effect
a target prestressed state can successfully be achieved and the required prestressing
forces predicted by the present iterative solution method are more accurate and the
errors between the measured member forces after finished tensioning and the desired
target values can be reduced. This, in turn, reduces the number of cyclic pre-tensions
on the construction site and also construction time and cost.
Finally, this research introduces a new analysis approach for the prestressed steel
structures that properly take into account all the construction stage effects on the
behaviour of prestressed steel structures during construction, in particular, the effects
of the deformed geometry of previous construction stage on the position of newly
installed members of the current construction stage. Consequently, the behaviour of
prestressed steel structures during construction can be accurately evaluated. This is
especially important in large-scale and/or complicated structures under construction
that lack temporary supports or stability precautions in addition to the presence of large
prestressing forces. With the present analysis approach, any instability and excessive
deflection of structural members or the possibility of structural collapse during
construction can be avoided. Overall, this research is a successful candidate to
integrate the structural engineering design into each sequence of the construction
phases of a building project, and further extend its realm to the architectural design as
the building information modelling.
vi Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
Publications
Refereed International Journal Papers
1. Nguyen, T. M. T., Iu, C. K., & Chan, T. H. T. (under review). Nonlinear analysis
of construction sequence accounting for constructional displacement at stages:
Part I. Algorithm. Steel and Composite Structures.
2. Nguyen, T. M. T., Iu, C. K., & Chan, T. H. T. (under review). Nonlinear analysis
of construction sequence accounting for constructional displacement at stages:
Part II. Application. Steel and Composite Structures.
3. Nguyen, T. M. T., Iu, C. K., & Chan, T. H. T. (to be submitted). Nonlinear
construction analysis of prestressed steel structure considering construction
effects. Engineering Structures.
4. Nguyen, T. M. T., & Iu, C. K. (to be submitted). Interdependent Behaviour for a
control system of prestressed steel structures.
5. Nguyen, T. M. T., Iu, C. K., & Chan, T. H. T. (to be submitted). Pre-tension
process analysis of prestressed steel structures accounting for constructional
displacements.
Refereed International Conference Paper
Nguyen, T.M.T. and Iu, C.K. (2015), "A Thorough Investigation of The
Interdependent Behaviour of Prestressed System." In The 2015 World Congress on
Advances in Structural Engineering and Mechanics, Incheon, Korea, August
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures vii
Table of Contents
Keywords ............................................................................................................................................... ii
Abstract ................................................................................................................................................. iii
Publications ............................................................................................................................................ vi
Table of Contents ................................................................................................................................. vii
List of Figures ......................................................................................................................................... x
List of Tables ....................................................................................................................................... xii
List of Abbreviations and Symbols ...................................................................................................... xiv
Statement of Original Authorship ..................................................................................................... xviii
Acknowledgements .............................................................................................................................. xix
CHAPTER 1: INTRODUCTION ....................................................................................................... 1
1.1 Background .................................................................................................................................. 1
1.2 Research problem ........................................................................................................................ 1
1.3 Aim and Scope ............................................................................................................................. 3
1.4 Objectives and methodology ........................................................................................................ 3
1.5 Outcome and significance ............................................................................................................ 5
1.6 Thesis Outline .............................................................................................................................. 6
CHAPTER 2: LITERATURE REVIEW ........................................................................................... 8
2.1 Overview ...................................................................................................................................... 8
2.2 Construction stage analyses ......................................................................................................... 8
2.3 Construction stage effects on the behaviour of prestressed steel structures ............................... 15
2.4 The interdependent behaviour among prestressed members in an entire prestressed steel structure ................................................................................................................................................. 17
2.4.1 Linear elastic analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure ............................................................. 17
2.4.2 Nonlinear inelastic analysis of the interdependent behaviour among prestressed members in a whole prestressed steel structure .............................................................. 20
2.5 The simulation of prestressing force .......................................................................................... 23 2.5.1 Equivalent load approach ............................................................................................... 23 2.5.2 Initial stress approach ..................................................................................................... 24 2.5.3 Initial deformation approach ........................................................................................... 24 2.5.4 Decreasing temperature approach ................................................................................... 24
2.6 Nonlinear Geometric formulation .............................................................................................. 27 2.6.1 Stability function approach ............................................................................................. 27 2.6.2 Higher-order element formulation approach .................................................................. 27
2.7 Inelastic Material formulation .................................................................................................... 28 2.7.1 Plastic zone approach ..................................................................................................... 29 2.7.2 Plastic hinge approach .................................................................................................... 30
2.8 Numerical solution method ........................................................................................................ 31
2.9 Summary and research problem ................................................................................................. 33
CHAPTER 3: SECOND-ORDER INELASTIC BEHAVIOUR OF STEEL STRUCTURES ACCOUNTED FOR CONSTRUCTION STAGE EFFECTS ......................................................... 36
viii Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
3.1 Introduction ............................................................................................................................... 36
3.2 Second-order inelastic analysis .................................................................................................. 37 3.2.1 Elastic higher-order element formulation ....................................................................... 37 3.2.2 Refined plastic hinge stiffness approach ........................................................................ 38
3.3 Positioning technique based on constructional displacements of previous construction stages 39 3.3.1 Technique to locate the coordinates of the newly built nodes at the current
construction stage ........................................................................................................... 39 3.3.2 Methodology of positioning the geometry for the whole construction sequence ........... 44
3.4 Nonlinear analysis of construction sequence ............................................................................. 49 3.4.1 ‘Build and Kill’ technique for assembling the global stiffness matrix ........................... 49 3.4.2 Nonlinear solution procedure of construction sequence ................................................. 51
3.5 Numerical verifications .............................................................................................................. 54 3.5.1 Load-deflection relation of a cantilever under multistage construction .......................... 55 3.5.2 Two-bay three-storey frame (second-order elastic behaviour) ....................................... 57 3.5.3 Three-storey building frame (second-order inelastic behaviour) .................................... 65 3.5.4 Slope truss (second-order elastic behaviour with initial force) ....................................... 77 3.5.5 Shallow hexagonal dome (second-order elastic behaviour) ........................................... 81 3.5.6 20-storey space steel building (second-order inelastic behaviour) ................................. 85
3.6 Conclusion ................................................................................................................................. 94
CHAPTER 4: SECOND-ORDER INELASTIC BEHAVIOUR OF PRESTRESSED STEEL STRUCTURES ACCOUNTED FOR CONSTRUCTION STAGE EFFECTS ............................. 97
4.1 Introduction ............................................................................................................................... 97
4.2 Nonlinear analysis of construction sequence of prestressed steel structures ............................. 98
4.3 Numerical verifications ............................................................................................................ 100 4.3.1 Arch bridge ................................................................................................................... 100 4.3.2 Frame column ............................................................................................................... 106 4.3.3 Shallow dome ............................................................................................................... 110
4.4 Conclusions ............................................................................................................................. 116
CHAPTER 5: LINEAR ANALYSIS OF THE INTERDEPENDENT BEHAVIOUR AMONG PRESTRESSED MEMBERS IN AN ENTIRE PRESTRESSED STEEL STRUCTURE .......... 119
5.1 Introduction ............................................................................................................................. 119
5.2 Influence matrix ....................................................................................................................... 120 5.2.1 Definition of IFM ......................................................................................................... 120 5.2.2 Different types of IFM .................................................................................................. 120
5.3 Effect of installation process vs tensioning process on IFM .................................................... 121
5.4 Effect of determinate vs indeterminate structure on IFM ........................................................ 122
5.5 Setup of IFM ............................................................................................................................ 122
5.6 Numerical solution procedures ................................................................................................ 124 5.6.1 Governing equation ...................................................................................................... 124 5.6.2 Direct solving method .................................................................................................. 125 5.6.3 Iterative solving method ............................................................................................... 127
5.7 Numerical verifications ............................................................................................................ 129 5.7.1 Frame column ............................................................................................................... 130 5.7.2 Arch Bridge .................................................................................................................. 134 5.7.3 Space grid structure ...................................................................................................... 136 5.7.4 Hybrid structure ............................................................................................................ 140
5.8 Discussion and conclusion ....................................................................................................... 142
CHAPTER 6: NONLINEAR ANALYSIS OF THE INTERDEPENDENT BEHAVIOUR AMONG PRESTRESSED MEMBERS IN AN ENTIRE PRESTRESSED STEEL STRUCTURE 144
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures ix
6.1 Introduction .............................................................................................................................. 144
6.2 Iterative solution procedure for prestressed target forces accounted for construction stage effects ................................................................................................................................................. 147
6.3 A simple illustration of the present iterative approach............................................................. 153
6.4 Numerical verifications ............................................................................................................ 154 6.4.1 Space grid structure ...................................................................................................... 154 6.4.2 Frame column structure ................................................................................................ 158 6.4.3 Arch bridge ................................................................................................................... 163
6.5 Discussion ................................................................................................................................ 168
6.6 Conclusion ............................................................................................................................... 169
CHAPTER 7: CONCLUSIONS AND FUTURE WORKS ........................................................... 172 7.1 Conclusions .............................................................................................................................. 172
7.1.1 Summary ....................................................................................................................... 172 7.1.2 Research contribution ................................................................................................... 174 7.1.3 Research significance ................................................................................................... 174 7.1.4 Research innovation ..................................................................................................... 178
7.2 Future work .............................................................................................................................. 180
REFERENCES .................................................................................................................................. 181
APPENDICES ................................................................................................................................... 192 Appendix A Higher-order element formulation ....................................................................... 192
A.1 TANGENT STIFFNESS MATRIX .......................................................................................... 192
A.2 SECANT STIFFNESS MATRIX .............................................................................................. 193 Appendix B Refined plastic hinge formulation ........................................................................ 194
x Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
List of Figures
Figure 1.1. The change of geometry during construction of a prestressed space gird structures with four prestressed members at the bottom chords. ............................................................ 2
Figure 1.2. Analysis process of prestressed steel structure accounted for the effects of construction sequence, geometric and material nonlinearities ............................................... 5
Figure 2.1. Deformed geometry of a three-storey frame under three construction stages ...................... 9
Figure 2.2. The geometry of the second construction stage, which unaccounted (a) and accounted (b) for the deformed geometry of previous construction stage ........................... 14
Figure 2.3. Discretization of frame in plastic zone method (S. L. Chan & Chui, 2000) ....................... 30
Figure 2.4. Newton-Raphson method (S. L. Chan & Chui, 2000) ........................................................ 32
Figure 2.5. Arc-length method (S. L. Chan & Chui, 2000) .................................................................. 32
Figure 2.6. Minimum residual displacement method (S. L. Chan & Chui, 2000) ................................ 32
Figure 3.1. Equilibrium conditions of higher-order beam-column element with element load effect .................................................................................................................................... 37
Figure 3.2. Principle to locate new nodes at current construction stage for 2D system ........................ 43
Figure 3.3. Mapping algorithm for the deformed geometry of all construction stages ........................ 46
Figure 3.4. Mapping methodology for the deformed geometry at all construction stages .................... 47
Figure 3.5. One-one mapping: one primary node to one secondary node ............................................ 47
Figure 3.6. Multi-to-one mapping: more than one primary nodes to one secondary node ................... 48
Figure 3.7. Repeated mapping: repetitive procedure of both one-one and multi-one mapping ............ 49
Figure 3.8. Procedure of the ‘Build’ and ‘Kill’ technique to formulate the system analysis ................ 51
Figure 3.9. The procedure of nonlinear analysis of construction stage analysis ................................... 54
Figure 3.10. Finite element models of 25m cantilever under construction ........................................... 55
Figure 3.11. Dimensions and section properties of two-bay three-storey steel frame .......................... 58
Figure 3.12. Comparison of deflected shapes between using deformed and undeformed coordinates ........................................................................................................................... 59
Figure 3.13. Vertical deflection at node 2 of the frame from 1st to the 3rd stage ................................. 62
Figure 3.14. Lateral deflection at each floor of the frame from 1st stage to the 3rd stage .................... 65
Figure 3.15. Geometry, applied loads, section, and material properties of a three-storey frame .......... 66
Figure 3.16. Original and deformed geometry of the three-storey frame ............................................. 67
Figure 3.17. Horizontal and vertical displacements at nodes 1, 2 & 3 for different stages – without bracing .................................................................................................................... 76
Figure 3.18. Geometry, section and material properties of slop truss ................................................... 78
Figure 3.19. Deflected shape of a slope truss under different stages .................................................... 78
Figure 3.20. Vertical deflection at nodes A & B of slope truss against the total construction load factor ............................................................................................................................ 81
Figure 3.21. Geometry and applied loads of shallow dome under different stages .............................. 82
Figure 3.22. Vertical displacements at nodes A & B for different stages ............................................. 84
Figure 3.23. Plan and elevation views of 20-storey space steel building.............................................. 86
Figure 3.24. Deflected shapes and locations of the plastic hinges of the 20-storey building ............... 87
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures xi
Figure 3.25. Elastic displacements at corners A & B during construction ........................................... 91
Figure 3.26. Inelastic displacements at corners A & B during construction ......................................... 93
Figure 4.1. Procedure of nonlinear analysis of construction sequence of prestressed steel structure ................................................................................................................................ 99
Figure 4.2. Original geometry, section properties and applied load of arch bridge structure ............. 101
Figure 4.3. Vertical deflections at node A during construction .......................................................... 104
Figure 4.4. Vertical deflections at node B during construction .......................................................... 104
Figure 4.5. Vertical deflections at node A by conventional analyses ................................................. 105
Figure 4.6. Vertical deflections at node B by conventional analyses.................................................. 106
Figure 4.7. Layout of frame column and its construction sequence ................................................... 107
Figure 4.8. Vertical displacement at top A during construction against ‘total load factor’ ................ 109
Figure 4.9. Geometry of the shallow prestressed dome and its construction sequence ...................... 111
Figure 4.10. Vertical displacements at nodes A & B against the total load factor .............................. 115
Figure 4.11. Vertical displacements at nodes A & B by the conventional analysis ............................ 116
Figure 5.1. Example of setup of IFM based on the batched tensioning process ................................. 123
Figure 5.2. Example of setup of IFM based on the repeated tensioning process ................................ 124
Figure 5.3. A particular case in which the IFMs are singular ............................................................. 127
Figure 5.4. Iterative solution procedure using one-criterion IFMs ..................................................... 129
Figure 5.5. Iterative solution procedure using two-criterion IFMs ..................................................... 130
Figure 5.6. The structural model of frame column and applied load .................................................. 131
Figure 5.7. Geometry and applied load ............................................................................................... 135
Figure 5.8. The perspective view of space grid structure (Dong & Yuan, 2007) ............................... 137
Figure 5.9. Convergent rate of the analysis using D and DF matrices in Scheme 3(a) ....................... 137
Figure 5.10. The perspective view of hybrid frame (Zhuo & Ishikawa, 2004) ................................... 140
Figure 5.11. The convergent rate of the analysis using F and DF matrices ........................................ 141
Figure 6.1. Effects of construction sequence on prestressing forces .................................................. 145
Figure 6.2. Iterative solution procedure for target prestressed forces accounted for construction stage effects ........................................................................................................................ 151
Figure 6.3. Iterative solution process for target prestressed forces ..................................................... 152
Figure 6.4. Illustration of the iterative solution process for a three-storey frame under three construction stages ............................................................................................................. 153
Figure 6.5. Original geometry of space grid structure and the arrangement of prestressed members in the study of Dong and Yuan (2007) ................................................................ 155
Figure 6.6. Original geometry, sectional properties, and applied load of frame column .................... 159
Figure 6.7. Vertical displacement at node A under different schemes ................................................ 162
Figure 6.8. Geometry, applied load, and pre-tension schemes of Arch Bridge .................................. 164
Figure 6.9. Vertical displacement at nodes A & B during construction ............................................. 168
xii Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
List of Tables
Table 2.1. Previous literature of the construction stage analysis – Project-oriented ........................... 10
Table 2.2. Previous literature of the construction stage analysis based on birth-death element technique .............................................................................................................................. 12
Table 2.3 Previous literature of the construction stage analysis based on step-by-step technique .............................................................................................................................. 13
Table 2.4. Previous literature of the interdependent behaviour of prestressed steel structures ........... 22
Table 2.5. Different modelling of prestressing forces of prestressed steel structures in previous studies. ................................................................................................................................. 26
Table 3.1. Nodal deformations separately at different stages according to conventional approach (m) ........................................................................................................................ 56
Table 3.2. Nodal deformations at different stages according to the present method (m) ..................... 57
Table 3.3. Bending moments at different stages (kNm) ......................................................................... 57
Table 3.4. Bending moment at section 1 at different stages (kNm) ....................................................... 61
Table 3.5. Vertical deflection at node 2 at different stages (mm) ......................................................... 61
Table 3.6. Bending moment at the section of node 1 at different stages (kNm) – without bracing ....... 67
Table 3.7. Horizontal displacement at node 1 at different stages (mm) – without bracing .................. 67
Table 3.8. Horizontal displacement at node 1 at different stages (mm) – with bracing ........................ 77
Table 3.9. Axial force of element 1 (kN) from various approaches ....................................................... 79
Table 3.10. Vertical displacements at A and B (mm) from various approaches ................................... 79
Table 3.11. Support reactions (kN) from various approaches .............................................................. 81
Table 3.12. Axial force of element 1 (kN) ............................................................................................. 83
Table 3.13. Vertical displacements at Nodes A & B (mm) .................................................................... 83
Table 3.14. Elastic horizontal displacement at nodes A & B in mm ..................................................... 88
Table 4.1. Construction sequences of Arch Bridge ............................................................................. 101
Table 4.2. Prestress forces of hangers at different stages by different approaches (kN) .................... 102
Table 4.3. Vertical displacements at nodes A & B by different approaches (mm) .............................. 102
Table 4.4. Construction sequences of frame column ........................................................................... 107
Table 4.5. Prestress forces during construction by different approaches (kN) ................................... 108
Table 4.6. Vertical displacement at node A during construction by different approaches (mm) ........ 108
Table 4.7. Construction sequences of shallow dome ........................................................................... 111
Table 4.8. Prestress member forces of 6 bottom chords at different stages (kN) ................................ 112
Table 4.9. Vertical displacement at nodes A & B at different stages (mm) ......................................... 112
Table 5.1. Construction sequences of frame column ........................................................................... 132
Table 5.2. Member forces in tensioning members of frame column (kN) ............................................ 133
Table 5.3. Nodal displacements in tensioning members of frame column (mm) ................................. 134
Table 5.4. Construction sequence of Arch bridge structure ................................................................ 135
Table 5.5. Nodal displacements of hangers (mm) ............................................................................... 136
Table 5.6. Member forces in hangers (kN) .......................................................................................... 136
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures xiii
Table 5.7. Member forces in tensioning bottom chords of space grid structure (kN) ......................... 138
Table 5.8. Nodal displacements in tensioning bottom chords of space grid structure (mm) ............... 139
Table 5.9. Construction sequence of Hybrid structure ........................................................................ 140
Table 5.10. Member forces of the prestressed members of Hybrid frame according to the present method (kN) ........................................................................................................... 141
Table 5.11. Member forces of the prestressed members of Hybrid frame according to Zhuo & Ishikawa (kN) ..................................................................................................................... 142
Table 6.1. Construction sequence of space grid structure .................................................................. 155
Table 6.2. The required prestressing forces and final prestressed member forces (kN) ..................... 156
Table 6.3. Vertical displacements at Node A (mm). ............................................................................ 156
Table 6.4. Prestressed member forces during construction of different approaches (kN). ................. 157
Table 6.5. Prestressed member forces during construction (kN). ....................................................... 157
Table 6.6. Displacements at Node A during construction (mm) .......................................................... 158
Table 6.7. Construction sequences of frame column ........................................................................... 159
Table 6.8. The prestressed member forces after finished tensioning and final prestressed member forces (kN) ............................................................................................................ 160
Table 6.9. Vertical displacement at top A (mm) .................................................................................. 160
Table 6.10. Variation of prestressed member forces during construction under different schemes (kN) ...................................................................................................................... 163
Table 6.11. Construction sequences of Arch Bridge ........................................................................... 164
Table 6.12. The prestressed member forces after finished tensioning and final prestressed member forces (kN) ............................................................................................................ 165
Table 6.13. Vertical displacement at node A (mm). ............................................................................ 165
Table 6.14. Prestressed member forces during construction under different schemes (kN) ............... 166
Table 6.15. Vertical displacement at Node A under different schemes (kN) ....................................... 166
xiv Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
List of Abbreviations and Symbols
This is the list of common abbreviations and symbols used in this thesis. All
abbreviations and symbols are defined in the text when they first appear.
Abbreviations
CA Conventional analysis
CS Construction stage
CSA Construction stage analysis
Def Deformed geometry
Hor. Horizontal
IFM Influence matrix
LA Linear analysis
PH Plastic hinge
Supp. Support
Und Undeformed geometry
Ver. Vertical
Symbols
A Cross-sectional area
pA Cross-sectional area of prestressed steel
A, akj IFM and its coefficient
D Initial member deformation or lack of fit
E Young modulus of elasticity
tE Tangent modulus
pE Elastic modulus of prestressed steel
EI/L Elastic flexural stiffness
EA/L Elastic axial stiffness
ue ∆= Axial shortening
F,F ∆ Force, incremental forces mft and nt the total nodal forces and the total number of nodal forces about all
degrees of freedom of the whole construction sequence mfc the cumulative force, including dead and constructional loads, up to
the load level at the current stage
fkj, dkj, dfkj & fdkj Coefficient of F, D, DF & FD IFMs
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures xv
g Uniform load
G Shear modulus of elasticity
I Section moment of area about major axis
zy II , Second moment of area about two principal axes
J St. Venant torsional constant Tr
mr
m J,J The Jacobian matrix, its transpose
JT1(j), JT2(j) The functions of first and second node of the jth element
L Length of an element or member
MM ∆, Moment, incremental moment
Nn , Ne Number of nodes and elements in each construction stage
Ncs The total number of construction stages
n The total number of tensioning members or the number of primary
nodes of the secondary node s
P Member axial force
crP Critical buckling load 'jk
p , 'jk
δ Member force and nodal displacement of member (batch) k
'jj
p , 'jj
δ Member force and nodal displacement of member (batch) j itself
mjkp , m
jkδ Member forces and nodal displacement of member k after member j
is prestressed in the mth tensioning round
0j0j δ,p Member force, nodal displacement before tension
q Stability parameter
RR ∆, Member resistance, incremental member resistance
( )βim r The residual or change of member length at the mth current
stage
S Spring stiffness of the plastic hinge at yielding
s Secondary node mjt , m
ju The tensioning control force, displacement of member j
U Internal strain energy
u,u ∆ Displacement, Incremental displacement
φ,w,v,u Axial deformation along x-axis, transverse displacement along y- and
z-axis, twist about x-axis respectively
xvi Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
smsmsm u,u,u 321 The coordinate of the secondary node s at the mth current stage
ooo u,u,u 321 Original coordinates of the secondary node s under designation
ioioio u,u,u 321 The original coordinate of primary node i
z,y,x Element centroid coordinate system
V External work done
pe ZZ , Elastic, plastic moduli
F, D, DF & FD Force based IFM, displacement-based IFM, displacement-force
based IFM; and force-displacement based IFM mmmmjjjj &,, fddfdf Column jth of F, D, DF and FD matrices
mf The total nodal force vector
mft The nodal force vector due to the loads imposed on the built
structure at the current mth stage mfin The initial force vector due to the change in member lengths at the
current mth stage
pm f The equivalent nodal forces due to prestress
fep The member prestressing force vector
ft Total nodal load vector for whole construction sequence
( )yx M,M,P=f Resultant stresses mKT Global stiffness matrix
ke Element stiffness matrix
ks Secant stiffness matrix
pt, δt The target force vector
mjp , m
jδ The vector of member forces, displacements
nnp δ, The member force, displacement vector after tensioning n prestressed
members
mR Total element resistance
T The transformation matrix
t, u The tensioning control force, displacement vector
mug The geometry of a structure at the current stage m-1u The deformed geometry at the previous stage
mup The change of geometry because of the positioning technique mu Total displacement at the current stage
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures xvii
[ ]sssT u,u,u 321 ∆∆∆β = The parameter vector of the restraint equations, i.e. the
change of nodal displacements in x-, y- and z-axes oβ Initial inputted value of the parameter vector
∆ Member elongation
φi(f) and φf(f) Initial and full yield surface of steel sections
bs MM ∆∆ , Incremental applied moment at node and element respectively
mjp∆ , m
jδ∆ The incremental force, displacement of member j
sb , θ∆θ∆ Incremental rotation at section and beam nodes respectively
imimim u,u,u 31
21
11 ∆∆∆ −−− The incremental displacement of primary node i at the (m-1)th
previous stage sss u,u,u 321 ∆∆∆ Incremental nodal displacements of the secondary node s that
satisfied the restraint equation ( )βim r
mλ The ‘total construction load factor’
µ Strain hardening parameters
Π Total potential energy
σ , ε Stress, strain
oo ,εσ Initial stress, initial strain
2121 zzyy ,,, θθθθ Rotation about y- and z-axes at the two end nodes respectively
ψ Unbalanced forces
L/x=ξ Relative coordinate in x-direction
m∆f Unbalanced force vector m∆Lin Vector of the change in member length at axial degree of freedom at
the current stage
δ∆∆ ,p The unbalanced control force, displacement vector
m∆u Incremental deformations mm
p , δεε The deviation vector of the member force, displacement after finish
tensioning with its target value
pε The deviation vector of the member force after finish tensioning with
its target value
xviii Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures
Statement of Original Authorship
The work contained in this thesis has not been previously submitted to meet
requirements for an award at this or any other higher education institution. To the best
of my knowledge and belief, the thesis contains no material previously published or
written by another person except where due reference is made.
Signature:
Date: April 2018
QUT Verified Signature
Development of a Second-Order Inelastic Analysis Method Accounted for Construction Stage Effects on the Behaviour of Prestressed Steel Structures xix
Acknowledgements
I would like to express my sincere gratitude to my Principal Supervisor, Professor
Tommy H.T. Chan for his valuable advice and ongoing support throughout my PhD
candidature. I also express my sincere thanks to my External Supervisor, Dr Chi Kin
Iu for his valuable suggestions, guidance, and motivation throughout my three-year
research journey. I also thank my Associate Supervisor, Assoc. Prof Bo Xia for his
advice and suggestions for my final thesis. And my thanks to Dr Andy Nguyen for His
advice and time in proofreading my thesis.
My grateful gratitude is given to the financial support of QUT to conduct this research.
My sincere thanks are given to all QUT staff members especially those of the IT unit,
the Document Delivery unit, the HDR support office for their assistance and
enthusiastic prompt responses to my numerous requests.
My thanks are also given to my colleagues at the School of Civil Engineering and Built
Environment for their comments and sharing knowledge.
Finally, I express my gratitude to my family for their love and encouragement, helping
me being able to complete this research.
Queensland University of Technology
Brisbane, Australia
April 2018
Thuy Nguyen
Chapter 1: Introduction 1
Chapter 1: Introduction
In this chapter, the research background is outlined in section 1.1 and the
research problem is identified in section 1.2. The aim and scope are presented in
section 1.3, while section 1.4 describes the objectives and methodology of this
research. Finally, section 1.5 includes an outline of the remaining chapters of the
thesis.
1.1 BACKGROUND
In recent decades, pre-tension has been widely applied in space steel structures
to increase structural load carrying capacity, improve structural rigidity, and reduce
structural deformation. Therefore, prestressed structures can cover a larger span with
a smaller structural weight, and hence become more aesthetic as being slender.
However, the most critical stage of prestressed steel structures is often the construction
phase, while part of large-scale and complicated structures under construction lack
temporary supports or stability precaution. Further, large pre-tension forces applied on
the most often unsupported structure with a small structural stiffness possibly triggers
nonlinear geometric behaviour and even inelastic deformation.
1.2 RESEARCH PROBLEM
During the construction stage, when prestressed members are installed and
prestressed, the structural geometry continuously change due to the mutual influence
between prestressing load and structural deformation as illustrated in Figure 1.1.
Subsequently, this constructional displacement incurs the change of structural
geometry and load redistribution correspondently to the change in the structural
stiffness (Z. Chen et al., 2015; Y. Liu & Chan, 2011; X. Wu et al., 2005).
2 Chapter 1: Introduction
Original geometry
Deformed geometry when the first
prestressed member P1 is assembled and
tensioned
Deformed geometry when the second
prestressed member P2 is assembled and
tensioned
Deformed geometry when the third
prestressed member P3 is assembled and
tensioned
Deformed geometry when the fourth
prestressed member P4 is assembled and
tensioned
Figure 1.1. The change of geometry during construction of a prestressed space gird structures with four prestressed members at the bottom chords.
Meanwhile, these constructional displacements also create significant errors in
the predicted prestressing forces, because only a small change in member length can
induce a large change in member force (Deng et al., 2011). As a result, the target force
P1 P2
P3 P4
2 Chapter 1: Introduction
and/or displacement to achieve a specific prestressed effect, which is important to
ensure the structural performance of a structure at its service condition, cannot be
achieved. In other words, the effects of sequential loading and construction process are
crucial in the analysis of prestressed steel structures in order to obtain a truer structural
response or to predict more accurate required prestressed forces and/or displacements
for a target prestressed state. Consequently, structural safety during construction can
be ensured and construction time and cost can be reduced (Y. Liu & Chan, 2011; X.
Wu et al., 2005). Unfortunately, previous research has often neglected the effects of
constructional displacements in the analysis of prestressed steel structures, which
produce the nonlinearities that impair the structural safety during the construction
phase.
Further, under the circumstances of the presence of many prestressed members
in the system, it is difficult to prestress all members simultaneously especially in large-
scale complicated structures or when the control forces/displacements of prestressed
members are not the same. When one member is prestressed to its target force and/or
displacement, the prestressing forces and/or displacements in other prestressed
members will immediately change due to the interaction among prestressed members
in the system as shown in Figure 1.1. Consequently, structural analysis of the entire
prestressed steel structure, to control the prestressing forces during construction, is
quite difficult but critical. However, there is limited research on the interdependent
behaviour of all prestressed members in an entire prestressed system. Moreover,
previous research has often been based on the theoretical original structural model and
has neglected the effects of constructional displacements in the pre-tension process
analysis, which produces errors in the predicted prestressing forces and in turn deviates
the final prestressed members’ forces from their target values. This requires
subsequent cyclic pre-tension onsite to compensate for those errors; hence,
construction time and costs are increased.
To this end, a nonlinear construction stage analysis, which can account for all
the construction stage effects properly, is needed and a comprehensive investigation
of the interdependent behaviour of all prestressed members within an entire prestressed
steel structure is demanded. Consequently, a better understanding of the behaviour of
this structural type is obtained; whereas the capability to obtain a more economical
design and to reduce construction time and cost can be achieved.
Chapter 1: Introduction 3
1.3 AIM AND SCOPE
The aim of this research is to investigate the construction stage effects on the
behaviour of prestressed steel structures, especially the effects of the deformed
geometry of previous construction stage on the position of newly installed members
of the current construction stage. At the same time, all the nonlinear geometric and
material effects (if any), that occurred within each construction stage and between two
constitutive stages can be captured properly.
This research focuses only on prestressed steel structures constructed of beam-
column elements because the higher-order element formulation established for the
beam-column element is employed. Therefore, the studies of prestressed cable
structures where prestressed cables may be slack, such as prestressed cable trusses,
suspen-domes, or suspension bridges, are beyond the scope of this research.
1.4 OBJECTIVES AND METHODOLOGY
To achieve the research aim, the two objectives are:
1. Investigate the construction stage effects on the behaviour of prestressed steel
structures during construction, in particular, the effects of the deformed geometry
of previous construction stage on the position of newly installed members of the
current construction stage.
2. As constructional displacements have direct effects on nodal coordinates, which
in turn affects the final prestressed member forces and/or displacements through
the interaction among prestressed members in the system, the second objective is
to investigate this particular effects by means of the pre-tension process analysis
of prestressed steel systems.
By accomplishing these two objectives, a thorough understanding of the
construction stage effects on the behaviour of prestressed steel structures during
construction can be achieved.
These two objectives are fulfilled through the following research tasks:
4 Chapter 1: Introduction
1. Propose an approach capable of accounting for the effects of the deformed
geometry of previous construction stage on the position of newly installed
members of the current construction stage. In order to minimise the change in
member length, the nodal positioning technique is presented based on the
nonlinear least-square approach from which the geometric mapping process is
built. At the same time, higher-order element formulation and refined plastic-
hinge approach are employed to capture all the nonlinear geometric and inelastic
material behaviours that may take place during construction. (Objective 1)
2. Employ the proposed approach from task 1 to investigate the construction stage
effects on the behaviour of prestressed steel structures during construction. The
structures of interest herein include plane and space frames, plane and space
trusses, which represent the major prestressed steel structural types apart from the
cable structures that are beyond the scope of this research. (Objective 1)
3. In order to investigate the interdependent effect on the target prestressed forces of
prestressed steel structures during construction, this research at first needs to
explore the advantages and disadvantages of different approaches for the pre-
tension process analysis that existed in literature to identify which approach is the
best to be adopted in this research. As later discussed in Section 2.4, there exist
the two main approaches for the pre-tension process analysis, which are the
Influence Matrix (IFM) and iterative solution. The iterative solution approach
searches for the tensioning control force/displacement by amending the trial
inputted value iteratively to achieve a target prestressed state at the end. On the
contrary, in the IFM approach, IFM that represents the mutual influences of
prestressed members in the structural system needs to be established first. Then
the tensioning control force/displacement will be obtained based on these IFMs.
Hence, as compared with the iterative solution approach, IFM has the advantage
of providing the analyst with a thorough understanding of the interdependent
behaviour among prestressed members in the system by means of its coefficients.
Therefore, the third task is to propose a linear analysis to investigate the
interdependent effect on the target prestressed forces of prestressed steel structures
during construction based on Influence matrix approach. (Objective 2)
Chapter 1: Introduction 5
4. It should be pointed out that this influence matrix approach is set up based on the
principle of linear superposition. Consequently, nonlinear geometric or inelastic
material behaviour, which may take place during construction, could not be
accounted for properly. Therefore, the fourth task is to propose a nonlinear
analysis to investigate the interdependent effect on the target prestressed forces of
prestressed steel structures during construction based on iterative solution
approach which is capable of capturing all the geometric and material nonlinearity
if any under the construction phase. (Objective 2)
1.5 OUTCOME AND SIGNIFICANCE
By accomplishing the four required tasks, a nonlinear analysis approach for the
prestressed steel structures that take into account properly all the construction stage
effects on the behaviour of prestressed steel structures during construction can be
achieved. The present analysis can be grossly generalised as in Figure 1.2.
Figure 1.2. Analysis process of prestressed steel structure accounted for the effects of construction sequence, geometric and material nonlinearities
The innovation of the present approach is it can properly and successfully
capture all the nonlinear geometric and material effects (if any) occurring within each
construction stage and in particular in between two constitutive stages which are the
main drawbacks of most of the existing analysis approaches. Consequently, the
behaviour of prestressed steel structures during construction can be accurately
evaluated. It is especially important in large-scale and/or complicated structures under
construction lack temporary supports or stability precautions in addition to the
6 Chapter 1: Introduction
presence of large prestressing forces. With the present analysis approach, any
instability and excessive deflection of structural members or the possibility of
structural collapse during construction can be avoided. A more-economic design can
be achieved, and construction time and cost can be reduced.
Overall, this research is a successful candidate to integrate the structural
engineering design into each sequence of the construction phase of a building project,
and further extend its realm to the architectural design as the building information
modelling.
1.6 THESIS OUTLINE
This section presents the format of this thesis.
Chapter 1 establishes the necessity of an investigation of the behaviour of prestressed
steel structures during construction, introduces the proposed methodology and
highlights the research outcome.
Chapter 2 presents the research problem established by analysing previous studies of
the behaviour of prestressed steel structures during construction.
Chapter 3 investigates the construction stage effects on the structural behaviour of
steel structures. The methodology to account for the nonlinear geometric effects in
each construction stage and between two constitutive stages, namely the effects of the
change of geometry of previous construction stages on the position of newly installed
members of the current construction stage, is presented herein.
Chapter 4 investigates the construction stage effects on the behaviour of prestressed
steel structures, which are more prone to nonlinearity, during construction based on
the methodology proposed in chapter 3.
Chapter 5 presents a linear analysis to investigate the interdependent behaviour of all
prestressed members in the entire prestressed steel structure during construction based
on Influence matrix approach.
Chapter 6 presents a nonlinear analysis to investigate the interdependent behaviour of
all prestressed members in the entire prestressed steel structure during construction
Chapter 1: Introduction 7
based on iterative solution approach, which can account for the construction stage
effects.
Chapter 7 concludes and recommends future work.
Chapter 2: Literature review 8
Chapter 2: Literature review
2.1 OVERVIEW
The literature review focuses on previous studies of construction stage analysis
in section 2.2; in particular, the construction stage effects on the behaviour of
prestressed steel structure in section 2.3. Previous studies of the interdependent
behaviour among prestressed members in an entire structure are reviewed in section
2.4. At the same time, for a sophisticated investigation of the behaviour of prestressed
steel systems during construction, three other main aspects needed to be considered
are the simulation of prestressing forces, reviewed in section 2.5; numerical
approaches to capture the nonlinear geometric effects, reviewed in section 2.6;
inelastic material behaviours, reviewed in section 2.7; together with the nonlinear
numerical solution procedures, reviewed in section 2.8. Finally, section 2.9
summarises and identifies the research problem, as well as analysis approaches, will
be employed in this research.
2.2 CONSTRUCTION STAGE ANALYSES
Large-scale and complicated structures are often under phased construction. As
the internal force distribution and structural geometry are continuously changed during
construction (Z. Chen et al., 2015; Kuroedov et al., 2016; G. Wang, 2000), and further
the construction effects may even change the structural behaviour in later service and
limit states (Choi & Kim, 1985; Jayasinghe & Jayasena, 2004; H. Kim & Cho, 2005;
Marí et al., 2003; Moragaspitiya et al., 2013; Samarakkody et al., 2014; Subramanian
& Velayutham, 2015; Yip et al., 2011; J. Zhang et al., 2012; Z. W. Zhao et al., 2016),
the construction simulation analysis of this structural type is crucial. Figure 2.1
illustrates the difference in the behaviour of a three-storey frame under three
construction stages according to the analyses accounted and unaccounted for
construction sequence. Once the construction sequence is accounted in the analysis,
the deflected shape of the frame in Figure 2.1(a) illustrates the frame leans back to the
original position at the 3rd stage. On the contrary, the whole frame deflects to the right
side if the construction sequence is neglected as shown in Figure 2.1(b).
Chapter 2: Literature review 9
Accounted for the construction stage
effects
Unaccounted for the construction stage
effects
Figure 2.1. Deformed geometry of a three-storey frame under three construction stages
Many construction simulation analyses to evaluate the structural behaviour at
each construction stage have been carried out in the past decades. For example, the
studies of X. Wu et al. (2005) of the erection procedure of the spatial steel shell
structure of National Grand Theatre, China; Guo et al. (2007) to determine the pre-set
deformation for the inclined couple towers of the new CCTV headquarters; Fan et al.
(2007) to determine the erection scheme of the complex spatial steel structure at the
National Stadium, China; Hu et al. (2009) of the full construction process of Palms
together Dagoba of Famen temple; Xie et al. (2009) of the construction process of a
polyhedron space rigid steel structure of the National swimming centre, China; H.
Wang et al. (2011) of the construction process of the mega-grid suspen-dome of Dalian
gymnasium; Y. J. Liu et al. (2011) of the construction process for the extension of the
large spatial steel structure of Ordos airport terminal; Feng et al. (2012) of the
construction process of the steel roof of a terminal building of a civil airport; Tian et
al. (2012) of a single-layer folded-plane lattice shell of Universidad sports centre; J. G.
Zhang et al. (2012) of the hyperbolic paraboloid steel grid structure of the Ocean
university gymnasium, China using cantilever expansion technique; Yang et al. (2012)
of the construction scheme determination of the multistorey cantilever steel structure
of the National fitness centre, Mongolia; W. Zhang et al. (2012) of the complex spatial
steel structure of the Hefei international convention and exhibition centre, China. As
summarised in Table 2.1, these studies focused on a particular structure, so they have
not verified their approaches to a general structural type. In short, their studies are
mainly project-oriented instead of technology development.
10 Chapter 2: Literature review
Table 2.1. Previous literature of the construction stage analysis – Project-oriented
Author Project Structure
X. Wu et al. (2005) National grand
theatre, China
Large spatial steel shell structure
Fan et al. (2007) National stadium,
China
Complex spatial steel structure
Guo et al. (2007) CCTV headquarters Inclined couple towers
Hu et al. (2009) Famen temple Palms together Dagoba
Xie et al. (2009) National swimming
centre, China
Polyhedron space rigid steel
structure
Y. J. Liu et al.
(2011)
Ordos airport
terminal
Large span spatial steel structure
H. Wang et al.
(2011); H. J. Wang
et al. (2014)
Dalian gymnasium Inclined mega-grid suspen-dome
Zhang et al. (2011) Xinjiang exhibition
centre
Truss string structure
Feng et al. (2012) Civic Airport Terminal roof
Tian et al. (2012) Universidad Sports
Centre
Single-layer folded-plane lattice
shell roof
Wei and Zhang
(2012)
Cultural building,
China
Large span steel truss
Yang et al., (2012) National fitness
centre, Mongolia
Multistorey large cantilevered steel
structure
J. G. Zhang et al.
(2012)
Ocean University
Gymnasium, China
Hyperbolic paraboloid steel grid
structure
W. Zhang et al.
(2012)
Login hall, China Complex spatial steel structure
Zhu et al. (2014) Shiyan stadium Stay cables supported latticed shell
Chapter 2: Literature review 11
Among these studies, element birth-death technique and step-by-step technique
were most widely used. Based on the element birth-death technique, the stiffness
matrix of a complete structure with its undeformed geometry is constructed at the
beginning of the analysis. Then the death technique is used to disable members at the
construction stage when those members have not yet been installed accordingly; For
example, a trivial coefficient is used to multiply with the corresponding element
stiffness. Those coefficients in the stiffness matrix will be removed, once their
correspondent members are put in place at the current construction stage (Fan et al.,
2007; Guo et al., 2007; Hu et al., 2009; Lozano-Galant et al., 2012; H. J. Wang et al.,
2013; Yang et al., 2012; J. G. Zhang et al., 2012; W. Zhang et al., 2012). Previous
literature of construction stage analysis based on birth-death element technique are
summarised in Table 2.2.
However, the birth-death technique heralds that the stiffness matrix of a structure
at a construction stage being founded on its original undeformed geometry by
activating the coefficients at the corresponding rows and columns. This technique
sometimes results in the distortion of the structural stiffness matrix when based on the
original undeformed geometry instead of the deformed geometry at the previous
construction stage (Z. Chen et al., 2015; Guo & Liu, 2008). It is termed as the floating
problem in computational technique, which affects the convergence and the accuracy
of the analysis regarding properly capturing the structural behaviour when the structure
is critical to the geometric nonlinear effects. Hence, it may even make some particular
structure adversely result in the collapse during construction (Guo et al., 2007).
In contrast, the structure is simulated and analysed by reliant on the step-by-step
technique, which build up the element stiffness matrix of a structure according to the
deformed geometry at the last construction sequence (Z. Chen et al., 2015; Guo et al.,
2007; Hu et al., 2009; H. S. Kim & Shin, 2011; Y. Liu & Chan, 2011; Y. J. Liu et al.,
2011; Qu et al., 2009). Previous literature of construction stage analysis based on step-
by-step technique is summarised in Table 2.3.
12 Chapter 2: Literature review
Table 2.2. Previous literature of the construction stage analysis based on birth-death element
technique
Author Structure
Fan et al. (2007) Complex spatial steel structure
Guo et al. (2007) Inclined couple towers
Xie et al. (2009) Polyhedron space rigid steel structure
Ge et al. (2010) Prestressed cantilever truss*
Zhou et al. (2010b) Arch-supported prestressed grid*
Jiang, Shi, et al. (2011) Elliptic paraboloid radial beam string structure*
H. Wang et al. (2011) Inclined mega-grid suspen-dome*
Zhang et al. (2011) Truss string structure*
H. Liu et al. (2012) Suspen-dome*
Lozano-Galant et al. (2012) Cable-stayed bridges*
Tang and Zhou (2012) Plum blossom-shape steel roof*
Y. Wang et al. (2012) Suspen-dome*
Yang et al., (2012) Multistorey large cantilevered steel structure
J. G. Zhang et al. (2012) Hyperbolic paraboloid steel grid structure
W. Zhang et al. (2012) Complex spatial steel structure
Zhuo et al. (2012) Single-layer folded space grid structure*
Z. q. Li et al. (2012) Suspen-dome*
H. J. Wang et al. (2013) Super high-rise building
J. Li et al. (2014) Cable supported barrel shell*
Zhou et al. (2014) Suspen-dome*
Chapter 2: Literature review 13
Table 2.3 Previous literature of the construction stage analysis based on step-by-step technique
Author Structure
Zhuo and Ishikawa (2004) Prestressed hybrid structure*
Dong and Yuan (2007) Prestressed space grid*
Guo et al. (2007) Inclined couple towers
Zhuo et al. (2008) Tension structures*
Hu et al. (2009) Coupling cantilever structure
Qu et al. (2009) Pre-tensioned reticulated structures*
Y. M. Li et al. (2010) Suspen-dome system*
Pan and Wei (2010) Long span steel structures
H. S. Kim and Shin (2011) High-rise building
Liu & Chan, (2011) Frame structures
Y. J. Liu et al. (2011) Large span spatial steel structure
Z. Chen et al. (2015) Conch-shaped latticed roof*
However, these methods of analyses are indispensable to evaluate the behaviour,
such as the constructional displacements at the construction stage, which always
continuously change the geometry of a structure for the installation at the next stage.
Figure 2.2 illustrates the difference in the structural geometry of the second storey,
which is built upon the deformed geometry of the first storey, in two different
situations accounted and unaccounted for the deformed geometry of previous
construction stage. In Figure 2.2(a), the geometry of the second storey which is built
upon the deformed one of the first storey and tries to maintain the original coordinates
of the two nodes m & n. It infers that the displacements of the two nodes k & l in the
first construction stage, on which the upper structure is built, is neglected. On the
contrary, in Figure 2.2(b), the new coordinates of the two nodes m & n have been
adjusted in alignment with the deformed coordinated of the two nodes k & l in the first
construction stage.
Chapter 2: Literature review 14
(a) (b)
Figure 2.2. The geometry of the second construction stage, which unaccounted (a) and accounted (b) for the deformed geometry of previous construction stage
It is interesting to note that when the structure is built upon the deformed
geometry of the first storey and tries to maintain the original undeformed geometry of
the second storey as shown in Figure 2.2(a), the geometry of the newly built structure
is much distorted compared with those based on the deformed geometry as shown in
Figure 2.2(b). It heralds that significant initial forces can be built up in the two columns
of the second storey if those members are already prefabricated.
Subsequently, the constructional displacement imposed on a structure at the
construction phase incurs the change of its geometry and load redistribution according
to the change in the stiffness of a structure (Z. Chen et al., 2015; Y. Liu & Chan, 2011;
X. Wu et al., 2005). Meanwhile, the member lengths may also be changed due to these
constructional displacements that may, in turn, lead to the premature material
nonlinear behaviour during construction. It, therefore, drew an amount of research
interest and attention for the effects of sequential loading and construction process in
the structural analysis in order to predict the accurate behaviour of a structure and
ensure its structural safety during construction (X. Wu et al., 2005). According to the
conventional design approach, the strength and stability of a structure are often reliant
on a final structure with the original undeformed alignment. As a result, many
members may be under-designed such that the instability and excessive deflections of
those members are unavoidable (Y. Liu & Chan, 2011).
Unfortunately, limited literature (Z. Chen et al., 2015; Guo & Liu, 2008)
explained comprehensively how does the effect of the construction sequence or the
Chapter 2: Literature review 15
deformed geometry due to constructional displacements at previous stages influence
the overall behaviour of a structure at its final stage (Z. Chen et al., 2015; Guo & Liu,
2008). Guo et al. (2007) and Y. J. Liu et al. (2011) presented a method to account for
the geometric nonlinearities due to constructional displacements, which imposes the
structural pre-deformation to the stiffness formulation of a structure at the
corresponding construction stages. However, the geometric nonlinear effect due to the
deformed structural geometry remained unsolved. While Pan and Wei (2010)
presented an approach to adjust the constraint condition by setting a pre-angle between
the pre-erected and post-erected sub-structures, but this approach is inconvenient
because of the pre-setting of the angle between active and inactive members. H. S.
Kim and Shin (2011) proposed a lumped construction sequences approach to account
for the elastic shortening of columns in high-rise building while the change of nodal
coordinates in other directions are not mentioned.
To this end, nonlinear effects due to the change of structural geometry and
material yielding that may take place during the construction process are focused in
this study. A nonlinear construction stage analysis accounted for construction stage
effects is proposed, validated, and employed to investigate the behaviour of different
steel structural types with the details given in chapter 3.
2.3 CONSTRUCTION STAGE EFFECTS ON THE BEHAVIOUR OF PRESTRESSED STEEL STRUCTURES
The above section focuses on previous studies of construction stage analysis of
general structures, whereas this section focuses on the effects of the construction stages
on prestressed steel structures in particular.
In the recent decades, tensioning technique has been widely applied in spatial
steel structures to increase structural load carrying capacity, improve structural
rigidity, and reduce structural deformation. Therefore, prestressed structures can cover
a larger span with a smaller structural weight, and hence become more aesthetic as
being slender (R Levy & Hanaor, 1992). However, the most critical stage of
prestressed systems is often at the construction phase, while part of the large-scale and
complicated structures under construction lack temporary supports or stability
precautions (Z. Zhao et al., 2015). Further, due to large prestressing forces applied on
the most often unsupported structure with a small structural stiffness that possibly
triggers nonlinear geometric behaviour and even inelastic deformation.
16 Chapter 2: Literature review
In this regard, a construction simulation analysis is crucial to evaluate the
structural behaviour at each construction stage. The element birth-death technique and
step-by-step technique are hence most widely used for the simulation of construction
sequence. Previous studies based on the element birth-death technique are the studies
of Ge et al. (2010) of prestressed cantilevered truss; of Zhou et al. (2010b) of an arch-
supported prestressed grid structure; He et al. (2011) of prestressed space reticulated
structures; Jiang, Shi, et al. (2011) of beam string structures with elliptic-parabolic
radial shape; H. Wang et al. (2011) of Dalian gym; H. Liu et al. (2012) of the
construction process of suspen-dome; Tang and Zhou (2012) of a plum blossom-shape
steel roof; Z. Zhou et al. (2012) of a single-layer folded space grid structure accounted
for the change of temperature; J. Li et al. (2014) of cable-supported barrel shell
structure; H. B. Liu et al. (2014) of the construction process of suspen-dome; and Zhou
et al. (2014) of suspen-dome structure. However, the birth-death technique heralds that
the stiffness matrices of a structure at the new construction stage being founded on the
original undeformed geometry (Z. Chen et al., 2015). Previous literature of
construction stage analysis of prestressed steel structures (indicated with *) based on
birth-death element technique are summarised in Table 2.2.
In contrast, previous studies relied on the step-by-step technique are those of
Zhuo and Ishikawa (2004) of a hybrid structure; Dong and Yuan (2007) of space grid
structure; Zhuo et al. (2008) for tension structures. However, these analysis methods
are indispensable to evaluate the behaviour, in particular, constructional displacements
at each stage, which continuously change the geometry of a structure during
construction under the circumstance of inadequate temporary supports and lateral
bracings. Previous literature of construction stage analysis of prestressed steel
structures (indicate with *) based on step-by-step technique are summarised in Table
2.3.
Under the construction phase, when the prestressed members are installed and
tensioned, the geometry of a structure will immediately change due to the interaction
between prestressing load and deformation in the system. Subsequently, this
constructional displacement incurs the change of structural geometry and load
redistribution correspondently to the change of the structural stiffness (Z. Chen et al.,
2015; Y. Liu & Chan, 2011; X. Wu et al., 2005; Zhang & Sun, 2011; X. Z. Zhao et al.,
2007). Meanwhile, these constructional displacements also create significant errors in
Chapter 2: Literature review 17
the predicted prestressing forces, because a small change in member length can induce
a large change in member force (Deng et al., 2011). As a result, the target force and/or
displacement for a specific prestressed effect, which was required to ensure the
structural performance of a structure at service condition, could not be achieved at the
end of the construction phase. In other words, the effects of sequential loading and
construction process are crucial in the analysis of a prestressed steel system in order to
predict a more accurate structural response in order to ensure structural safety during
construction and also to reduce construction time and cost (Y. Liu & Chan, 2011; X.
Wu et al., 2005).
Unfortunately, the prestressing forces required and/or displacements or the
tensioning control forces and/or displacements are often determined during the design
stage based on the theoretical structural model. Therefore, the first part of this research
introduces a new analysis method that is able to capture properly all the change of
geometry occurred within each construction stage and in between two constitutive
stages. The proposed approach is then employed to investigate these nonlinear effects
on the behaviour of slender prestressed steel structures. Consequently, a better
understanding of the behaviour of this structural type is obtained and, a more
economical design can be achieved. The details of this study are given in chapter 4.
2.4 THE INTERDEPENDENT BEHAVIOUR AMONG PRESTRESSED MEMBERS IN AN ENTIRE PRESTRESSED STEEL STRUCTURE
The above section focuses on previous studies of the effects of the construction
stages on the behaviour of prestressed steel structures; whereas this section focuses on
the construction stage effects on the interdependent behaviour among prestressed
members of the whole prestressed steel structure to achieve a target prestressed state.
2.4.1 Linear elastic analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
In recent decades, the application of pre-tension in spatial steel structures has
been more common due to many advantages of this structural type. Unfortunately,
under the circumstances of the presence of many prestressed members in the system,
it is difficult to prestress all members simultaneously especially in complicated
structures or the control forces/displacements of prestressed members are not the same.
As a result, the batched and repeated tensioning schemes are unavoidable. When one
member is prestressed to its target value for the optimal capacity of the system, the
18 Chapter 2: Literature review
target values in other tensioning members will immediately change due to the
interdependent behaviour among tensioning members in the system. The pre-tension
process analysis is therefore important such that the required tensioning control force
and/or displacement of each tensioning member can be computed to achieve the final
target state. In order words, once the predicted value is applied on each tensioning
member according to the predetermined construction scheme, the forces and/or
displacements in tensioning members at the target state must reach their target values
after tensioning instead of blindly and endlessly supplemental tension. This problem
needs to be solved beforehand in the design stage and usually based on the theoretical
structural model.
Aim at solving this problem, Zhuo and Ishikawa (2004) proposed a ‘tensile force
compensation analysis method’ to search for the control prestressing force by iteration.
The study is applied to hybrid structures, composed of struts and cables, in which cable
is assembled and prestressed one by one. The control force is solved by a number of
compensation iterative cycles. Another approach to search for the tensile control force
is the study of Dong and Yuan (2007). In this study, the ‘initial internal force method’
was introduced to analyse the pre-tension process of prestressed space grid structures.
This approach needs to establish a set of equations of ‘initial internal forces’ in order
to solve for the required control forces by the recursive approach. This approach is
applicable for many pre-tension schemes such as one by one, i.e. one prestressed
member at a time; batch by batch, i.e. some prestressed members at a time; as well as
simultaneous, i.e. all prestressed members at once. However, the application is limited
to the linear range of behaviour only and the repeated tensioning scheme in which
prestressed members are prestressed multiple times was not addressed. While, J. Li et
al. (2014) proposed a nonlinear simulation analysis also using cyclic iteration method
for cable-supported barrel shell structures to search for its nodal coordinate and also
pre-tension forces of cables, analysed by ANSYS software. This method in basic is
similar to the ‘tensile force compensation analysis method’ in the study of Zhuo and
Ishikawa (2004).
While the above studies directly searched for the tensile control forces of
prestressed steel structure, other studies tried to determine the equivalent member
initial deformation or initial strain to achieve a target prestressed state. Zhou et al.
(2010b) presented an analysis of the pre-tension process of arch supported prestressed
Chapter 2: Literature review 19
grid structures, based on member initial deformation or lack of fit. The mixed influence
matrix approach and iterative approximation approach were introduced to solve for the
required initial deformations or lack of fits of prestressed cables. It should be pointed
out that the mixed influence matrix, in general, are asymmetric which results in
difficulty to inverse and the approach is applicable for the linear range of behaviour
only. On the other hand, the iterative approximation approach could handle the
nonlinear geometric effects. However, due to the approach based on the iteration of
member deformation, as only a small change of member deformation may result in a
large change in member force, extremely slow convergence or even no convergence
was noticed in some cases (Nguyen & Iu, 2015a). It makes this approach inefficient.
Later on, Zhou et al. (2014) combined an iterative method for form finding and the
sequential analysis method for a pre-tension process simulation of suspen-dome
structures. In the meantime, He et al. (2011) provided a method to calculate the initial
strains of cables to meet the design requirements also by iteration. This method in basic
is similar to the ‘tensile force compensation analysis method’ in the study of Zhuo and
Ishikawa (2004) and is applied for prestressed space reticulated structures.
Overall, the two main approaches for the pre-tension process analysis are the
Influence Matrix (IFM) and iterative solution. The iterative solution approach searches
for the tensioning control force/displacement by amending the trial inputted value
iteratively to achieve a target prestressed state at the end. On the contrary, in the IFM
approach, IFM that represents the mutual influences of prestressed members in the
structural system needs to be established first. Then the tensioning control
force/displacement will be obtained based on these IFMs. Hence, as compared with
the iterative solution approach, IFM has the advantage of providing the analyst with a
thorough understanding of the interdependent behaviour among prestressed members
in the system by means of its coefficients.
Unfortunately, previous studies of the interdependent behaviour among
prestressed members within the entire structures are still limited. Aiming to achieve
more practical engineering applications, the second part of this research presents a
comprehensive investigation of the linear elastic interdependent behaviour of
prestressed steel structure based on influence matrix (IFM) in a reliable, effective, and
efficient manner. Detail of this study is given in Chapter 5.
20 Chapter 2: Literature review
2.4.2 Nonlinear inelastic analysis of the interdependent behaviour among prestressed members in a whole prestressed steel structure
Another important feature in the pre-tension process analysis is that there are
often existed influence factors that make the actual structural state in construction
somewhat different with the theoretical structural model on which is based in the
design stage to predict the tensioning control forces. Those factors can be such as
simplified assumptions of the theoretical structural model, fabricated errors,
temperature loads, or friction of structural components. That makes the design
prestressed state could not be achieved even though the pre-tension scheme and the
tensile control forces and/or displacements analysed beforehand has already been
followed. As a result, the actual tensioning control forces and/or displacements need
to be re-analysed during construction.
Among these influenced factors, construction stage effect is an important one.
During construction, when prestressed members are installed and tensioned, the
geometry of a structure will immediately change due to the mutual influence between
the prestressing load and deformation in the system. Subsequently, this constructional
displacement incurs the change of the structural geometry and load redistribution
correspondently to the change of the structural stiffness (Z. Chen et al., 2015; Y. Liu
& Chan, 2011; X. Wu et al., 2005). Meanwhile, these constructional displacements
also create significant errors in the predicted prestressing forces, because of the change
of member orientation and of the change in member length. At the same time, as the
member length has changed, this initial member deformation or the so-called lack of
fit, in turn, induces a requirement of a constructional initial force in order to pre-tension
or pre-compress the corresponding member to resume its original length in order to be
able to fit into its designed position. As a result, the target forces and/or displacements
for the designed prestressed state could not be achieved and in turn, the structural
performance of a structure at service condition could not be ensured (Y. Liu & Chan,
2011; J. Wu et al., 2015; X. Wu et al., 2005). Moreover, numerous cyclic pre-tension
on site could not be avoided in order to finally obtain the design requirements.
Once again, previous researchers have tried to reduce the errors in the predicted
required prestressing values. Zhuo et al. (2008) extended the ‘tensile force
compensation analysis method’ in their previous study (Zhuo & Ishikawa, 2004) to
adjust the tensile controlling force based on the measured values of cable forces on
Chapter 2: Literature review 21
site. However, this new method also based on the analysis of the same theoretical
structural model, i.e. the nonlinear effect due to the deformed structural geometry
during construction was not accounted for. Hence, the solution may not be reliable in
case there is a large difference between the theoretical and the actual structural model.
Later Zhou et al. (2010c) introduced the ‘pre-tension scheme decision analysis
method’ using an iterative calculation approach based on the recursive of cable forces
based on influence matrix to adjust the controlling force iteratively (Zhou et al.,
2010a). However, influence matrix approach limits the application of this study to the
linear behaviour range only. Further, if high accuracy is required, a large number of
iteration is unavoidable, especially when there are many prestressed members in the
structure and also extremely slow convergence or even no convergence was noticed in
some particular cases (discussion is given in chapter 5). Moreover, the change of
geometry of the previous stage on the position of newly installed nodes of the current
construction stage was not addressed either. After that, Feng et al. (2013) proposed a
probabilistic finite element analysis based on the nonlinear mapping function of the
neural network to predict the control forces of next construction stages. The study
accounted for construction errors due to geometric and material parameters to achieve
a designed prestressed state of space grid structures. Previous literature of the
interdependent behaviour of prestressed steel structures is summarised in Table 2.4.
22 Chapter 2: Literature review
Table 2.4. Previous literature of the interdependent behaviour of prestressed steel structures
Author Structures Method Analysis type
Zhuo and Ishikawa
(2004) Hybrid frame
Tensile force
compensation analysis
method
Linear analysis
Zhuo et al. (2008) Hybrid frame
Tensile force
correction calculation
method
Linear analysis
Dong and Yuan
(2007) Space grid structure
Initial internal force
method Linear analysis
Zhou et al. (2010a) Arch supported space
grid structure Pre-tension process Linear analysis
Zhou et al. (2010b)
Arch supported space
grids
Mixed influence
matrix approach Linear analysis
Iterative
approximation
approach
Nonlinear
elastic analysis
Zhou et al. (2010c) Arch supported space
grid structure
Influence matrix
approach Linear analysis
Zhou et al. (2014) Suspen-dome
Iterative solution for
form finding and
sequential analysis
method
Nonlinear
elastic analysis
Feng et al. (2013) Space grid structure Tensioning process
feedback control
Probabilistic
analysis
Li et al. (2014) Cable supported barrel
shell Pre-tension process
Nonlinear
elastic analysis
However, most of the above approaches had to carry out after the (first) pre-
tension phase, as they needed to rely on the real values of prestressing forces and/or
displacements measured on site to adjust the tensioning values in order to re-meet the
design requirements. Hence, they are considered as somewhat passive approaches as
they do not actively solve the core of the problems that affect the precision of the
prestress. Further, previous research often neglected constructional displacements
based on the deformed geometry of a prestressed steel structure, which produces the
nonlinearities that impair the structural safety during construction. Therefore, the
second part of this research proposes an iterative solution approach to search for the
Chapter 2: Literature review 23
required prestressing forces and/or displacements to reach a target prestressed state
which accounted for the construction stage effects. Consequently, a more accurate
required prestressing forces and/or displacements can be predicted; the number of
cyclic pre-tension on site can be reduced and construction time and cost can be saved.
Details of this study are given in chapter 6.
2.5 THE SIMULATION OF PRESTRESSING FORCE
An important feature in the numerical analysis of prestressed steel structures is
the simulation of prestressing force. Many approaches that have been used are the
equivalent load, initial stress, initial deformation, or change in temperature approaches.
2.5.1 Equivalent load approach
A large amount of previous researchers simulated prestress as equivalent nodal
loads applied on the structures. Based on this approach, a variety of prestressed
structural types such as beams, stayed columns, trusses and frames have been studied.
For example, the analytical study of the linear elastic behaviour of prestressed steel
beam with straight tendon profile of Hoadley (1961); the elastic buckling load induced
by eccentric straight tendons in prestressed steel beams of Bradford (1991); a new
formulae for strength and stability check for concentric tendon case prestressed steel
beams proposed by Tocháček and Ferjenčík (1992), belonging to the former
Czechoslovak national standard; a static linear elastic analysis for prestressed steel
continuous-span girders with uniform cross-section of Troitsky et al. (1989); an
analysis for simply supported box beam, considered the relations between the change
of prestressing force due to the deformation of the steel beam under applied loading
proposed by Jia and Liang (2011); a formula to estimate deflection considering the
combined effects of prestress and external load of prestressed steel beams proposed by
Ponnada and Vipparthy (2013); the study of the relation between initial prestress in
the stays and the buckling load of the perfectly straight column by Hafez et al. (1979);
the elastic buckling of a stayed column perfectly straight and the lateral deflection
stability of a stayed column with initial out of straightness by Smith (1985); a stability
analysis and parametric study of the relation between initial prestress in the stays,
initial imperfection and strength of stayed columns proposed by S. L. Chan et al.
(2002); a series of studies focus on the optimal pre-tension force, geometric
configuration, geometric imperfection and stability of prestressed stay columns by
24 Chapter 2: Literature review
Saito & Wadee, from 2008 to 2010. Overall, it can be seen that the equivalent load
approach provides a clear picture of the prestressing forces applied to the prestressed
structures.
2.5.2 Initial stress approach
Prestress can also be simulated as an initial stress of members. For example in
the study of the relation between prestress and member force and the derivation of the
stiffness of the tendon with different profiles of post-tensioned plane trusses by Ayyub
et al. (1990). Prestress was first modelled as applied post-tensioned stress, and post-
tensioned force of the tendon was then computed; a nonlinear analysis of beam string
structure based on the principle of virtual work by Jiang, Xu, et al. (2011), prestressing
force was inputted as initial stress at first and then the prestressing force was computed.
It can be seen that the initial stress is later transferred to equivalent applied load hence
this approach is quite similar to the equivalent load approach.
2.5.3 Initial deformation approach
In a study of optimal prestress for a minimum weight design of statically
indeterminate structures, the concept ‘prestressing by lack of fit’ was introduced by L.
P. Felton and Hofmeister (1970) in which prestress was simulated as an initial member
deformation or lack of fit of the prefabricated member. Based on this approach, a
number of studies were presented later on. For example, a study of space truss in which
prestress was firstly simulated as an initial deformation or lack of fit, and then
employed to compute the equivalent initial prestress load by Lewis P Felton and Dobbs
(1977); the formulated optimal criteria for simple truss design of Spillers and Levy
(1984); the proposal of an optimal design for space truss of R Levy and Hanaor (1992);
the study to enhance the design of space truss of Hanaor and Levy (1985); the pre-
tension process analysis of arch supported prestressed grid structures by Zhou et al.
(2010b). Overall, it can be seen that using initial deformation approach, prestress is
later transferred into equivalent applied loads hence this approach is also similar to the
equivalent load approach.
2.5.4 Decreasing temperature approach
Another approach, in which initial prestress was modelled as a temperature
change that manipulates the temperature around the tendon elements, usually cooling
down the tendons to achieve the desired prestressing force, was also used by previous
Chapter 2: Literature review 25
researchers. For example, the study of limit analysis of cable tensioned structures by
J. Y. R. Liew et al. (2001); the study of the collapse behaviour of bowstring column,
another form of stayed column by J.Y.R. Liew and Li (2006); an analysis of assembled
truss reinforced by cables was proposed, based on the inelastic displacement by the
principle of virtual work of Y. Z. Zhou et al. (2012). It is noticed that the decreasing
temperature approach often needs to be used together with a temperature analysis.
Prestress modelling of some of the previous researches can be summarised in Table
2.5.
Overall, in simulating prestressing force, the approach that modelled
prestressing force as equivalent nodal loads were predominantly adopted by most
previous researchers, as summarised in Table 2.5, because the equivalent load
approach provides a clear picture of the prestressing forces applied on the prestressed
structures. Other approach modelled prestressing force as initial stress or initial
deformation imposed on the structure, which then induced equivalent nodal forces.
Therefore, these approaches, in general, are similar to the equivalent load approach.
Hence, the equivalent nodal load approach is employed in this study to simulate
prestressing force.
26 Chapter 2: Literature review
Table 2.5. Different modelling of prestressing forces of prestressed steel structures in previous
studies.
Author Prestress simulation Studied structure
Hoadley (1961) Equivalent load Simply supported I beam
Bradford (1991) Equivalent load Simply supported plate girder
Tocháček and Ferjenčík (1992) Equivalent load Centrally compressed strut
Ponnada and Vipparthy (2013) Equivalent load Simply supported I beam
Jia (2009) Jia and Liang (2011)
Equivalent load Simply supported box beam
Troitsky et al. (1989) Equivalent load Continuous girder
Ronghe and Gupta (2002) Equivalent load Simply supported plate girder
Belletti and Gasperi (2010) Equivalent load Simply supported I beam
Hafez et al. (1979) Equivalent load Single cross-arm stayed column
Temple et al. (1984) Equivalent load Single cross-arm stayed column
Smith (1985) Equivalent load Single cross-arm stayed column
S. L. Chan et al. (2002) Equivalent load Multi cross-arm stayed column
Saito and Wadee (2008) Saito and Wadee (2009a)
Equivalent load
Single cross-arm stayed column
Saito and Wadee (2009b) Equivalent load Single cross-arm stayed column
Saito and Wadee (2010) Equivalent load Single cross-arm stayed column
Osofero et al. (2012) Single cross-arm stayed column
Osofero et al. (2013) Equivalent load Single cross-arm stayed column
Ronghe and Gupta (2002) Equivalent load One storey frame
Zhuo and Ishikawa (2004) Equivalent load Hybrid structure
Dong and Yuan (2007) Equivalent load Space truss
Ayyub et al. (1990) Initial stress Plane truss
Jiang, Xu, et al. (2011) Initial stress Beam string
Lewis P Felton and Dobbs (1977) Initial deformation Space truss
Spillers and Levy (1984) Initial deformation Space truss
R Levy and Hanaor (1992) Initial deformation Space truss
Zhou et al. (2010b) Initial deformation Arch supported grid
J. Y. R. Liew et al. (2001) Decreasing temperature
Bowstring column
J.Y.R. Liew and Li (2006) Decreasing temperature
Bowstring column Bowstring frame
Y. Z. Zhou et al. (2012) Decreasing temperature
Assembled truss reinforced by cable
Chapter 2: Literature review 27
2.6 NONLINEAR GEOMETRIC FORMULATION
An important feature of the nonlinear analysis is to capture properly any
geometric nonlinearity that may take place; two main approaches often used by
previous researchers were the stability function and higher-order element formulation.
This section analyses the advantages and disadvantages of these two approaches in
order to choose an appropriate one for this research.
2.6.1 Stability function approach
By solving the equilibrium equation of a beam-column element, the stability
function approach develops the element stiffness matrix. Based on this approach, many
researchers have successfully modified different aspects of geometric nonlinear
behaviour of structures such as the studies of Oran (1973a, 1973b) of plane and space
frames, W. F. Chen et al. (2001) of three-dimensional steel frame, King et al. (1992)
for steel frame design, S. E. Kim and Chen (1996) of braced steel frame, J. Y. R. Liew
et al. (2002) for fire analysis of steel structures, J. Y. R. Liew et al. (2000) for the
nonlinear analysis of space frames, J. Y. R. Liew and Hong (2004) of explosion and
fire analysis of steel frames.
However, because there are different solutions for axial loads, whether it is
tensile or compressive so the approach loses its generality. Besides, when the axial
force is small, the solution may encounter a numerical problem. Moreover, if the
section properties change along the member length, the element matrix needs to be
derived according to S. L. Chan and Zhou (1994). At the same time, stability functions
have different expression under different element loads that discourage practical
applications (Iu, 2015). Hence, another suitable approach is the higher-order element
formulation approach.
2.6.2 Higher-order element formulation approach
To model geometric nonlinearity due to large displacement that may occur in
general steel structures, previous research works often used the cubic Hermite finite
element formulation which assumed a linear or quadratic variation for axial
deformation and a cubic polynomial for transverse deformation (Jennings, 1968).
From this formulation, the geometric stiffness matrix is derived. However, due to the
assumption of linear interpolation function for curvature, the result based on this
formulation involves some approximation of the behaviour of the structure under large
28 Chapter 2: Literature review
axial load. According to S. L. Chan (2001), unless the axial force in an element is less
than 40% of the Euler buckling force, one cubic Hermite element can still be used to
model a structural member. Outside of this range, especially in the case of prestressed
structures, the corresponding member must be divided into two or more elements in
order to reach the required accuracy. This will result in more computational efforts.
The higher-order element formulation assuming quartic polynomials (Iu &
Bradford, 2012a; Izzuddin, 1990; So & Chan, 1991) or quintic polynomials (S. L. Chan
& Zhou, 1994) for transverse deformation is quite straightforward to overcome the
demerit of the cubic formulation. Using this approach, a variety of issues related to the
change of geometry of steel structures have been solved more accurately and
efficiently, such as the studies of S. L. Chan and Zhou (1995) accounted for initial
member imperfection of steel frames; S. W. Liu et al. (2012) for hybrid-steel concrete
frames; Zhou and Chan (2004) which proposed a fifth-order element formulation
capable of simulating one element per member that can capture the nonlinear
geometric behaviour of steel frame.
Belonging to these studies, Iu and Bradford (2012a) derived a fourth-order
element based on the force equilibrium equation at mid-span to capture the geometric
nonlinear behaviour of steel frames. Based on this formulation, an advanced analysis
for steel structure was presented by Iu and Bradford (2010). The proposed method can
model nonlinear geometry of large steel frame structures, included large displacement,
member bowing and buckling effects together with its notable advantage that is the
ability to use only a single element to model a structural member but the required
accuracy is still reached. This obviously reduces a large computational work. Further,
the generalised element load method based on higher-order element formulation was
also introduced to ensure the accuracy for the whole element length under arbitrary
transverse element loading patterns (Iu, 2016a; Iu & Bradford, 2015).
With all of these advantages, the refined higher-order element formulation of Iu
and Bradford (2015) is employed in this study to capture the nonlinear geometric
behaviour of prestressed steel structures.
2.7 INELASTIC MATERIAL FORMULATION
Another important feature of the nonlinear analysis is to capture properly any
material nonlinearity that may take place, of which there are two main analysis
Chapter 2: Literature review 29
methods. The first one is referred to as the Distributed plasticity or Plastic zone
method, while the other one is recognised as the Lump plasticity or Plastic hinge
method. This section analyses the advantages and disadvantages of these two methods
and an appropriate one is then chosen to employ in this research.
2.7.1 Plastic zone approach
The plastic zone method discretises structural members both along their length
and through their cross sections into finite numbers layers, each of which is assumed
to be in a uniaxial stress. Then, stress-strain can be captured for all fibres as illustrated
in Figure 2.3. In case of plasticity, especially with prestressed steel structures, the
spread of plasticity due to increasing load is traced by the sequential yielding of the
elements. Hence, the method can account for the gradual spread of plasticity within
the whole volume of the structures (S. L. Chan & Chui, 2000). In addition, the stress-
strain relationship is explicit and is used to compute moments and forces directly.
Therefore, this approach is considered as more accurate and is used to establish
benchmark solutions as verification studies for other approaches. Many researchers
have studied nonlinear behaviour of structures by this method such as the studies of S.
L. Chan (1989) for the analysis of tubular beam-column and frames, Teh and Clarke
(1999) for the analysis of space steel frames. However, this method is only suitable for
simple structure design, as it requires a great amount of calculation work and its
convergent rate is inefficient. Hence, another more efficient approach was proposed is
the plastic hinge approach.
30 Chapter 2: Literature review
Figure 2.3. Discretization of frame in plastic zone method (S. L. Chan & Chui, 2000)
2.7.2 Plastic hinge approach
The plastic hinge method assumes that post-elastic deformations are
concentrated at the zero-length plastic hinges at the ends of the elastic element. An
equivalent force-deformation relationship is used to control the plasticization of the
cross-section at the ends of the element. This method was later refined and improved
by many researchers such as the studies of Yau and Chan (1994) by introducing the
spring-in-series model, S.L. Chan and Chui (1997) of the design-based analysis of steel
frames; S. L. Chan and Chui (2000) for the analysis under static and cyclic loadings,
S. L. Chan et al. (2005) of portal frames accounting for imperfection and semi-rigid
connections; Liew et al. (1993; 1993) for semi-rigid frame design and recently in the
study of system reliability of Thai et al. (2016).
Further, the moment and force interactive equation from different design code
can be incorporated into the analysis procedure to fulfil the design code requirement
directly. These merits make the refined plastic hinge method more efficient and more
preferable for engineering design practice (S. L. Chan & Chui, 2000). It makes the
application of this method is not just limited to steel structures. Using this approach,
Iu (2008) studied the nonlinear behaviour of composite beams with arbitrary sections.
In Iu’s study, the gradual yielding and full plasticity of the composite section were
modelled by the spring stiffness of a plastic hinge formerly proposed by Iu and Chan
(2004). Later Iu et al. (2009) modified this method by using both axial and bending
plastic hinges for the analysis of composite frame structures and for steel structures in
the study of Iu and Bradford (2012b). This refined plastic hinge can capture material
Chapter 2: Literature review 31
yielding for an entire frame structure effectively (Iu, 2016b, 2016c; Iu & Bradford,
2012b).
Besides, using this approach, one structural member can be modelled by only
one or two elements. It infers that there is no requirement to discretize a structural
member into many elements along its length as in plastic zone method, but the required
accuracy is still ensured (S. E. Kim & Chen, 1998). With all of these merits, the refined
plastic hinge method is more efficient than the plastic zone method.
Therefore, refined plastic-hinge approach (Iu & Bradford, 2012b) is employed
to simulate the inelastic material behaviour of prestressed steel structures in this
research.
2.8 NUMERICAL SOLUTION METHOD
While the nonlinear geometric and material nonlinearities are captured by
higher-order formulation and refined plastic-hinge approach, numerical solution
method needs to be employed to trace the nonlinear equilibrium path. Usually,
incremental iterative methods were performed to trace the load versus displacement
relationship. Belonging to the incremental iterative methods, the Newton-Raphson
method, which keeps the load constant within a load cycle as illustrated in Figure 2.4,
is well-known for its computational efficiency and accuracy according to S. L. Chan
and Chui (2000); Clarke and Hancock (1990); Rezaiee-Pajand et al. (2013a).
Therefore, it has been widely used by previous researchers (Belletti & Gasperi, 2010;
Fedczuk & Skowroński, 2002).
However, the Newton-Raphson method diverges within the vicinity of limit
points or when the structures experience softening behaviour (S. L. Chan & Chui,
2000; Rezaiee-Pajand et al., 2013a). A number of numerical techniques have been
developed to overcome this problem. Some notable and widely used methods that need
to be mentioned are the constant Arc-length method and the Minimum residual
displacement method. The Arc-length method, which controls constant work done
within a load cycle (Crisfield, 1981b, 1983; Ramm, 1981) as illustrated in Figure 2.5,
is capable of tracing the equilibrium path through the limit points. This method has
been proved to be efficient when the structures exhibit softening behaviour (snap
through problems) so it has been used by many researchers according to Clarke and
Hancock (1990); Rezaiee-Pajand et al. (2013a, 2013b). It is also important to mention
32 Chapter 2: Literature review
the Minimum residual displacement method, proposed by S. L. Chan (1988) as shown
in Figure 2.6. This method aims at searching the direction leading to the minimum
displacement error that is the true aim of a numerical solution procedure. Therefore
this method follows the shortest path to achieve a solution point and it was proven to
be capable of passing limit points in most cases without failure proofs (S. L. Chan,
1988; Clarke & Hancock, 1990) and there is no requirement to solve a quadratic
equation in case of a negative root.
Figure 2.4. Newton-Raphson method (S. L. Chan & Chui, 2000)
Figure 2.5. Arc-length method (S. L. Chan & Chui, 2000)
Figure 2.6. Minimum residual displacement method (S. L. Chan & Chui, 2000)
Chapter 2: Literature review 33
Overall, the Newton-Raphson method proves to be very efficient to trace the
equilibrium path up to the vicinity of the limit points, whereas the Minimum residual
displacement method or the Arc-length method is appropriate for the vicinity of limit
points and post-buckling path. Therefore, these methods will be used in this research.
2.9 SUMMARY AND RESEARCH PROBLEM
This literature review establishes the research problem and the numerical
approaches, which are employed in this research.
1. As previous researchers often neglected constructional displacements based on
the deformed geometry of a prestressed steel structure, which produces the
nonlinearities that impair the structural safety during construction, the first part of
this research investigates the construction stage effects on the behaviour of
prestressed steel structures. A better understanding of the behaviour of this
structural type can be obtained and consequently, a more economical design can
be achieved. First, the methodology to investigate the effects of the constructional
displacements is set up and validated in chapter 3. Second, the proposed method
is employed to study the construction stage effects on the behaviour of prestressed
steel structure as presented in chapter 4.
2. There is limited literature that studied the mutual influence among prestressed and
non-prestressed members within an entire prestressed steel structure, especially
the analysis of the whole pre-tension process. It is noticed that to achieve an
economic design, smaller and slender member are essential. On the contrary,
slender members are prone to buckling, so these effects need to be considered in
the analysis of the entire structural system. Aiming to achieve more practical
engineering applications, the second part of this research presents a
comprehensive investigation of the interdependent behaviour of prestressed steel
structure based on influence matrix (IFM) in a reliable, effective, and efficient
manner. Once the IFM is established, a complete analysis for the whole tensioning
process can be obtained in which the required tensioning control forces and/or
displacements needed to apply upon each prestressed member in order to finally
meet the requirements for a specific design (target) stage instead of tensioning by
trial and error. Details of the study are presented in Chapter 5.
34 Chapter 2: Literature review
3. However, as IFM approach based on the superposition prerequisite, the
application of this approach is limited to the linear elastic behaviour range only.
Therefore, an iterative solution approach for the pre-tension process analysis of
prestressed steel structures that is capable to properly account for all the nonlinear
construction stage effects is demanded. By accounting for the effects of the
construction stage effects, the errors between the measured members’ forces after
finished tensioning with the designed target values can be reduced which in turn
reduces the number of cyclic pre-tension on site as well as construction time and
cost. The details of this study are presented in Chapter 6.
4. To simulate prestressing forces, the approach that modelled prestressing forces as
equivalent nodal loads applied on the structure were predominantly adopted by
most previous researchers because this approach provides a clear picture of the
prestressing forces applied to the prestressed structures. Therefore, equivalent
nodal load approach is employed in this study.
5. Higher-order element formulation has been proven to be capable of capturing the
geometry of structures accurately and efficiently through previous literature. In
addition, the plastic hinge approach has been proven being able to simulate
material nonlinear behaviour more effectively while the analysis resultant
accuracy is similar to the plastic zone approach. Moreover, the higher order finite
element formulation together with refined plastic-hinge approach allows using a
single element to model a structural member. Hence, these methods can benefit
the analysis of an entire large prestressed steel system. Therefore, higher-order
element formulation and refined plastic-hinge approach are employed in this
research to capture the nonlinear geometric and inelastic material behaviour of
prestressed steel systems.
Chapter 2: Literature review 35
6. In order to trace the nonlinear equilibrium path from the onset of loading up to the
limit point, Newton-Raphson method is used because this method gives the exact
structural response for an input load level and it follows the shortest path to
achieve the solution point. However, Newton-Raphson method diverges within
the vicinity of limit points or when the structures experience softening behaviour.
In these cases, arc-length or minimum residual displacement method is employed
instead due to these two methods have proven to be quite effective in tracing the
equilibrium path in case the structure exhibits snapping, softening or stiffening
behaviour.
7. As this research is based on the higher-order element formulation, which is
applicable for the beam-column element, the application of this research covers a
variety of structural types such as frames, arches, and trusses, apart from cable
structures that are beyond the scope of this research.
36 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
3.1 INTRODUCTION
The conventional design of a structure is based on the structural geometry at its
final stage of construction. Unfortunately, the displacement of a structure during the
construction phase is sometimes unavoidable, especially when bracings and temporary
supports are not always available for the sake of speeding the construction sequence.
It leads to nonlinear effects, which are not accountable by the conventional design
approach since the superposition principle is invalid to account the nonlinearities at
each construction stage by considering the nonlinearities only at the final stage.
Unfortunately, limited literature explained comprehensively how does the effect
of the construction sequence or the deformed geometry due to constructional
displacements at previous stages influence the overall behaviour of a structure at its
final stage as discussed in section 2.2. To this end, this chapter presents the algorithm
of a second-order inelastic analysis to take into account the nonlinearities during the
whole construction sequence. The nonlinear effect due to constructional displacement
or the deformed structural geometry is continuously evaluated until the final stage.
These constructional displacements at a construction stage are commonly due to
gravity and constructional loads, which makes the original alignment at the next
construction stage hard to maintain.
For the sake of minimising the change in member lengths, the newly installed
nodal positions at the next construction stage accounted for these constructional
displacements are determined by virtue of the nonlinear least-square approach with
details in section 3.3.1. While the methodology to locate the new geometry of an entire
newly built structure at each construction stage is required which is demonstrated in
section 3.3.2. Further, the higher-order element formulation is resorted to capture the
nonlinear geometric effects (including, P-δ & P-∆ effect, large deformation and
buckling), whereas the material nonlinearities (including, gradual yielding, full
plasticity and strain-hardening effect due to interaction) is reliant on the refined plastic-
hinge approach, which is given in section 3.2.1 & 3.2.2, respectively. The nonlinear
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 37
solution procedure of construction sequence accounted for the geometric and material
nonlinearities as well as the construction sequence, in which the re-assembling
stiffness matrix of a structure resorts to the step-by-step technique. Their details are
discussed in section 3.4. The present algorithm is then employed to analyse a series of
numerical verifications in section 3.5. The results are independently compared with
those obtained from other researchers and other approaches. Finally, some conclusions
are drawn in section 3.6. Finally, this nonlinear construction stage analysis is a
successful candidate to integrate the structural engineering design into the each
sequence of the construction phases of a building project, and further extend its realm
to the architectural design as the building information modelling. This chapter
accomplished task 1 and partially fulfilled objective 1 of this research.
3.2 SECOND-ORDER INELASTIC ANALYSIS
3.2.1 Elastic higher-order element formulation
The geometric nonlinearity is taken into account by the higher-order element
formulation, developed by Iu and Bradford (2015). The higher-order polynomial
transverse displacement function of an element as expressed in Eq. (3.1), which
satisfies not only the compatibility condition in Eqs. (3.2) & (3.3), but also the force
equilibrium equations with element load effects as in Eqs. (3.4) & (3.5), is used to
derive a higher-order element, as originally proposed by S. L. Chan and Zhou (1994).
Further, the elastic material law follows in the higher-order element function.
y
xv
PP
Equilibrium of beam-column element about z-axis
W ωMz2
Mz1
Mo & So
Figure 3.1. Equilibrium conditions of higher-order beam-column element with element load effect
∑=p
ii xc)x(v
1 (3.1)
Unknown coefficients are determined based on boundary conditions. The
compatibility conditions of an element at the nodes are given in Eqs. (3.2) & (3.3),
0=v and 1zx
v θ=∂∂ at 0=ξ , (3.2)
38 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
0=v and 2zxv θ=
∂∂ at 1=ξ , (3.3)
where L/x=ξ and L is the element length.
The bending and shear force equilibrium equations with element load effects at
mid-span, 21 /=ξ , written in Eqs. (3.4) & (3.5),
ozz MMMPv
xvEI +
−+=
∂∂
221
2
2
, (3.4)
ozz S
LMM
xuP
xvEI +
++
∂∂
=∂∂ 212
2
2
. (3.5)
The higher-order displacement function with element effects is then derived as,
( )( )
( )( )
( )
( )( )
( )( )
( )
( ) [ ] ( ) [ ]54322
432
2
5432432
1
5432432
25480
248
8040
8010
80980
8027240
482
484
482548
8040
8010
80980
8027240
482
484
482548
ξξξξξξξ
θξξξξξξξ
θξξξξξξξξ
−+−+
++−+
−
+
++
−+
++
++
−+
+
++
−+
++
+
++
−+
++
++
−+
+
−+
++
+−=
qLS
qLM
Lq
q
Lq
qv
oo
z
z
(3.6) in which q is an axial load or stability parameter and EI the flexural rigidity about the
z-axis, ( )zEI/PLq 2= ; L/x=ξ ; the equivalent mid-span moment 0M and shear force
0S under different types of element load can be found in the studies of Iu and Bradford
(2015) and Iu (2015).
The transverse displacement in the z-direction can be formulated in the same
way. The comprehensive illustration of the higher-order elastic element stiffness
formulation, as well as its efficacious and reliable convergence, can be found in the
study of Iu and Bradford (2010) whereas the profound implication of the element load
effects is discussed in the study of Iu (2015).
3.2.2 Refined plastic hinge stiffness approach
The material nonlinearity is taken into account by the refined plastic-hinge
approach (Iu & Bradford, 2012b), that simulates the gradual yielding and full plasticity
of section at node and strain hardening by the axial and bending spring stiffness of the
plastic hinge as follows
( )( ) µ
φφ
+−
−=
11
ff
i
f
LEIS (3.7)
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 39
in which S is the spring stiffness of plastic hinges at yielding; EI/L is the elastic flexural
stiffness; EA/L is the elastic axial stiffness; µ is the strain hardening parameters first
introduced by Iu and Chan (2004); φi(f) and φf(f) are initial and full yield surface of
steel sections as written in Eqs. (3.8) & (3.9), respectively,
( ) 1251251
80=++=
py
y
px
x
yi M
M.M
M.P.
Pfφ . (3.8)
( )( )[ ] ( )[ ] 1
11 3
2
31=
−+
−=
α
φypy
y
.ypx
xf
PPM
M
PPM
Mf (3.9)
in which ( )yx M,M,P=f are resultant stresses. The plastic hinge spring is activated
whenever φi(f) > 1. This plastic hinge stiffness in Eq. (3.7) is incorporated directly into
the tangent and secant stiffness matrix of the elastic stiffness formulation.
Comprehensive formulation and its details are discussed in Iu and Bradford (2012b).
3.3 POSITIONING TECHNIQUE BASED ON CONSTRUCTIONAL DISPLACEMENTS OF PREVIOUS CONSTRUCTION STAGES
3.3.1 Technique to locate the coordinates of the newly built nodes at the current construction stage
As aforementioned, most of the nonlinear analyses of construction sequence in
the literature are to formulate a structure at the current mth construction stage in
conformity with the original undeformed geometry. In contrast, when being in
conformity with the deformed geometry of the previous (m-1)th construction stage due
to the constructional displacements, the positioning technique is necessary to locate
the change of nodal coordinates of the members being built at the mth current
construction stage. In order words, the new position of the nodes being built in the
current mth construction stage, named as secondary nodes, are located based on the
deformed coordinates of the nodes built in previous construction stages, named as
primary nodes.
In fact, if the newly built structure at the current mth construction stage is built
upon the deformed structural parts, previously built at the (m-1)th stage, and at the same
time try to maintain its original undeformed geometry of those newly built nodes, large
deformations of the newly built members at the current mth stage can be incurred. To
this end, there are two different principles that can be adopted to locate the position of
newly installed nodes. The first principle is to keep minimum change in length of
40 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
newly built members in case these members have been pre-fabricated, while the
second principle is to keep minimum change in shape of the newly built structural parts
in case of prefabrication of these structural modules. The restraint equations of these
two principles can be formulated as in Eqs. (3.10) & (3.11) respectively.
( ) ( ) ( ) 03
1
23
1
21 ≈−−−−+= ∑∑==
−
j
ij
osj
o
j
ij
mij
osj
sj
oi
m uuuuuur ∆∆β (3.10)
( )( ) ( )
( )
( ) ( )
( )0
2
2
1 3
1
2
3
1
211 3
1
2
1 3
1
21
3
1
211111 3
1
21
→
−
−−−
−
−−+
−−+−−−+
=
∏∑
∑∑∑
∏∑
∑∑∑
+
= =
=
++
= =
+
= =
−
=
−+−++
= =
−
i
ik j
kj
onj
o
j
ij
oij
oi
ik j
kj
onj
o
i
ik j
kj
mkj
onj
nj
o
j
ij
mij
oij
mij
oi
ik j
kj
mkj
onj
nj
o
im
uu
uuuuarccos
uuuu
uuuuuuuuarccosr
∆∆
∆∆∆∆β
, (3.11)
in which ( )βim r is the residual or change of member length at the mth current stage,
which is the nth set of restraint equations with respect to each member depending
on the parameter vector [ ]sssT u,u,u 321 ∆∆∆β = . Under this circumstance, the
parameter vector β is the change of nodal displacements in x-, y- and z-axes,
so j means the dimension, for instance, j = 1 implies the displacement u1 is in
x-axis; ioioio u,u,u 321 , are the original coordinate of primary node i (with 1 ≤ i ≤ n); imimim u,u,u 3
12
11
1 ∆∆∆ −−− is the incremental displacement of primary node i at the (m-1)th
previous stage. In short, this positioning technique was applied to search the nodal
coordinates of a newly built structure at the current mth construction stage.
It is interesting to note that although the principle of minimum change in length
of the newly built members is followed, the initial force due to the constructional
displacement at the (m-1)th stage may also incur; especially when the newly built
structure is highly redundant. Therefore, this initial force is anticipated to be significant
when the newly built structure is based on the original undeformed geometry but the
previously built structural part has already been deformed.
Pre-fabrication of members are more common compared with pre-fabrication of
structural modules. At the same time, it can be foreseen that the analysis based on the
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 41
principle to minimise the change in members’ lengths only account for the change of
nodal coordinates. As a result, the new geometry of the later construction stage is
defined with accounting for only the change in nodal coordinates and initial forces
induced from the member length changes if any. On the contrary, the analysis based
on the principle to minimise the change in modular shapes, all the changes of nodal
rotations as well as initial moments (if any) are also accounted for. In that case, the
analysis is more accurate and obviously more complicated. Therefore, this study firstly
adopted a principle of minimum change in length of the newly built members in order
to develop a positioning technique to locate the nodal coordinates of a newly built
structure at the current mth construction stage subjected to the change of geometry at
the previous (m-1)th construction stage. This study will then be a foundation for the
analysis approach based on the principle to minimise the change in modular shapes to
be built on.
The positioning technique is to locate the position of a new node s (a particular
node number) of the deformed geometry of newly built structure also known as the
secondary node at the mth current stage that is connected to other nodes i (with 1 ≤ i ≤
n < s) of the previously built structure at the (m-1)th previous termed as the primary
nodes. In this regard, the principle of minimum change in newly built member’s
lengths can be formulated based on the algorithm of the nonlinear least-square
technique to solve the restraint equations with respect to each member, which is
identical to the number of the primary node n under this circumstance for simplicity
and clarity. In summary, this positioning technique is proposed to determine the nodal
coordinates of a secondary node s at the mth current stage subjected to the restraint
condition of Eq. (3.10) dependent on a set of primary nodes i (with 1 ≤ i ≤ n).
Therefore, there are three conditions in the positioning technique for the three-
dimensional problems; they are under-determinate, determinate, and over-determinate
conditions.
In case of the under-determinate condition for a three-dimensional problem,
there is only one restraint equation (n = 1), which implies a secondary node connects
to one primary node as conceptually shown in the two dimension of Figure 3.2(a). The
obvious solution of incremental nodal displacements of the secondary node is equal to ij
nj uu ∆∆β == for j = 1, 2, 3 in x-, y- and z-axes. When there are two primary nodes
or two members are connected to a secondary node s (n = 2), the minimum distance
42 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
of parameter vector β from the original undeformed coordinate of the secondary node
s is chosen from possible solutions from Eq. (3.12).
In case n = 3 of the determinate condition as shown in the two dimension of
Figure 3.2(b), the exact solution of incremental nodal displacements of a secondary
node can be obtained by using the relation of,
( )
( ) ( ) ( ) ( )( )∑
∑
=
−−−−−−
=
−−
−+−−−
=−−+
3
1
1111321213211221
3
1
3111212
5050j
jm
jm
jo
jm
jm
jo
jm
jm
jjj
mj
oj
mj
o
uuuuuuu.u.
uuuuu
∆∆∆∆
∆∆∆ (3.12)
In case n ≥ 4 of the over-determinate condition for a spatial structure as shown
in the two dimension of Figure 3.2(c), Eq. (3.10) becomes an over-determined system
of the nonlinear equations, where the best approximate solution is a least-square
solution as
( ) ( ) ( ) ( )∑ ∑∑∑−
= ==
−−
=
−−−−+==
1
1
23
1
23
1
211
1
2n
i j
ij
onj
o
j
ij
mij
onj
nj
on
ii
mm uuuuuurS ∆∆ββ (3.13)
(a) Under-determined system (b) Determined system
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 43
(c) Over-determined system
Figure 3.2. Principle to locate new nodes at current construction stage for 2D system
Since the system of the nonlinear equations is over-determined, the solution of
the incremental nodal displacements of a secondary node s can be reached iteratively
by an initial value of the parameter vector oβ . It means the coordinate of a secondary
node is based on the original undeformed geometry as the initial condition. Hence,
using the Gauss-Newton algorithm (Gratton et al., 2007), the best approximate solution
of parameter vector 1+kβ at (k+1)th iteration can be given as long as the difference of
the parameter β is significantly unobvious,
( ) ( )kmTr
mr
mTr
mkk r.J.J.J βββ 11 −+ −= , (3.14)
in which Tr
mr
m J,J is the Jacobian matrix, its transpose and its entries are given in Eq.
(3.15).
( ) ( ) ( ) ( )∑=
−− −−+−−+=∂
∂=
3
1
211
j
ij
mij
osj
sj
oij
mij
osj
sj
o
j
ki
m
ijrm uuuu/uuuurJ ∆∆∆∆
ββ
, (3.15)
Once the incremental nodal displacements of a secondary node is known, the
new nodal coordinate of a secondary node in respective axes is given as,
ssosmssosmssosm uuu,uuu,uuu 333222111 ∆∆∆ +=+=+= , (3.16)
in which smsmsm u,u,u 321 is the coordinate of the secondary node s at the mth current
stage; ooo u,u,u 321 is its original coordinates under design; sss u,u,u 321 ∆∆∆ is its incremental
nodal displacements that satisfy the restraint of Eq. (3.10). In summary, this
44 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
positioning technique is to locate the nodal coordinate of a secondary node, which is
associated with newly built members at the current construction stage, based on one
or more primary nodes with their deformed coordinates at the previous construction
stage. This positioning technique is repeated for the whole construction sequence by
virtue of the mapping methodology as discussed in the following section.
3.3.2 Methodology of positioning the geometry for the whole construction sequence
The positioning technique in section 3.3.1 is targeted for the location of a
secondary node (i.e. nodal coordinates of newly built structural part) at the current
stage with recourse to the deformed geometry of the primary node(s) (i.e. nodal
coordinates of previously built structural part, on which the newly built part will be
built). It means this positioning technique is just implemented for the current
construction stage and hence not enough for the whole construction sequence.
Therefore, the mapping methodology is indispensable to regulate the positioning
technique in order to determine the nodal coordinates of a structure from stage to stage.
First is the mapping procedure. There are three kinds of mapping procedure (i.e.
one to one mapping; multi to one mapping and repeated mapping). The principle of
the basic mapping methodology is that the secondary nodes at the mth construction
stage can be directly related to the primary node(s) at the (m-1)th stage according to the
member connectivity (i.e. JT1(j) & JT2(j)), which are the functions of first and second
node of the jth element. First, when the ith secondary node is selected, primary node is
searched complying with the member connectivity. Once it is matched according to
the connectivity, a primary node with respect to the ith current secondary is found and
the number of primary nodes (i.e. n) is increased accordingly as illustrated in Figure
3.3. The procedure is repeated until all primary nodes are searched and during
searching, element number j of the member connectivity is firstly incremented (e.g. j
≤ Ne), and subsequently is the node number i (e.g. i ≤ Nn). Thus, the number of n is
counted in the course of this searching procedure.
When it is one to one mapping (i.e. n = 1) that one primary node links with a
secondary node whose coordinate of the deformed geometry is obtained by Eq. (3.10)
with ij
nj uu ∆∆β == . When it is multi to one mapping (i.e. n > 1) that more than one
primary nodes connect with a secondary node whose deformed coordinate can be
determined by using Eqs. (3.12) or (3.13). Moreover, if a secondary node of the newly
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 45
built structural part at the mth current stage may not be connected to any primary nodes.
However, the secondary node under concern is connected to other secondary nodes at
the mth current stage whose coordinates have just been determined. Under this
circumstance, the mapping and positioning procedures are repeatedly implemented
similarly to the one illustrated in Figure 3.4 until the deformed nodal coordinates of all
members of the newly built structural part of the current stage are all determined. This
is named as the repeated mapping procedure. This mapping procedure is necessary
since the positioning procedure as previously discussed in section 3.3.1 for these two
situations are different.
Second is the positioning procedure. The nodal coordinates of each secondary
node are computed according to the deformed geometry of its primary node(s). As a
result, the nodal coordinates of the secondary node are more likely different from their
original undeformed coordinates, which is useful to measure the change in geometry
due to construction.
Finally, these procedures at a particular stage, including mapping procedure,
positioning procedure, and repeated mapping procedure, are repeated to execute from
stage to stage until the nodal coordinates of the whole structure subjected to the
constructional displacements are defined.
46 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Figure 3.3. Mapping algorithm for the deformed geometry of all construction stages
The mapping procedures, including one-one, multi-one and repeated mapping,
are illustrated in the following examples as Figure 3.5, Figure 3.6 & Figure 3.7
respectively. Further, these examples also demonstrate the difference between the
deformed geometry due to constructional displacements adopted in this study and the
original undeformed geometry mostly used in the literature.
The first example is one-one mapping. It implies one primary node connecting
with a secondary node, for example, secondary node m & n connects with k & l,
respectively, as shown in Figure 3.5(a). The nodal coordinates of the secondary nodes
with deformed geometry are predicted by the positioning technique as denoted by m’
& n’ in Figure 3.5 (a), whereas the original undeformed geometry used mostly (Z.
Chen et al., 2015) is shown in Figure 3.5(b). Once the deformed coordinates of
secondary nodes m’ and n’ are obtained as Figure 3.5(a), the deformed geometry of all
newly built members are defined in a construction stage.
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 47
Figure 3.4. Mapping methodology for the deformed geometry at all construction stages
(a) New geometry based on the deformed coordinate
(b) New geometry based on the original coordinate
Figure 3.5. One-one mapping: one primary node to one secondary node
The second example is multi-one mapping. It implies more than one primary
nodes link with one secondary node, for instance, a secondary node r connects with
four primary nodes l, k, o & p as well as a secondary node q associates with two
primary nodes k & o as illustrated in Figure 3.6. Similarly, the deformed geometry of
secondary nodes r’ & q’ are computed by the present positioning technique as shown
in Figure 3.6(a). Once the deformed coordinates of secondary nodes r’ and q’ are
48 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
obtained as Figure 3.6(a), the deformed geometry of all newly built members in this
slope truss is defined at a construction stage.
New geometry based on deformed coordinate
New geometry based on original coordinate
Figure 3.6. Multi-to-one mapping: more than one primary nodes to one secondary node
The third example is repeated mapping. This mapping combines the one-one and
multi-one mapping such that the deformed coordinates of all secondary nodes at
current stage are determined, especially when a multiplicity of members are defined
in one single construction stage, which can help speed up the process of construction
sequence. In this example as Figure 3.7, there are the secondary nodes m, n, q & r, and
the secondary node r does not relate to the primary nodes l, k, o & p. Under this
circumstance, the deformed coordinates of the secondary node r’ are obtained based
on the deformed geometry of the secondary nodes m’, n’ & q’ by virtue of the use of
both one-one and multi-one mapping in the multiple times as depicted in Figure 3.7(a).
It is interesting to note that when the structure based on the original undeformed
geometry as given in Figure 3.5(b), Figure 3.6(b) & Figure 3.7(b), the geometry of the
newly built structure is much distorted compared with those based on the deformed
geometry as shown in Figure 3.5(a), Figure 3.6(a) & Figure 3.7(a). It heralds that
significant initial forces can be built up in the members if those members are already
prefabricated. This initial force can cause premature material yielding. In addition,
initial forces can also be induced when using the deformed geometry, such as Figure
3.6(a) & Figure 3.7(a), since these structures are indeterminate. In other sense, there is
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 49
no initial force induced in the determinate structure, such as the newly built structure
under one-one mapping.
(a) New geometry based on deformed coordinate
(b) New geometry based on original coordinate
Figure 3.7. Repeated mapping: repetitive procedure of both one-one and multi-one mapping
3.4 NONLINEAR ANALYSIS OF CONSTRUCTION SEQUENCE
3.4.1 ‘Build and Kill’ technique for assembling the global stiffness matrix
The previous section is the positioning and mapping procedure at a particular
construction stage, whereas the present section is to demonstrate the stiffness
formulation at each construction stage and nonlinear solution procedure for all stages
to capture the behaviour of a structure during the whole construction sequence.
Since the behaviour of a structure due to the construction loads and its gravity
changes at different construction stages, for example, new members installed and
temporary supports or bracing removed during construction, the structure at different
stage illustrates different behaviour, which should be truly represented by the stiffness
formulation of a structure. Therefore, this study utilises the step-by-step technique to
formulate the ‘Build and Kill’ technique, as the step-by-step technique can evaluate
the behaviour based on the current structural form, such as the deformed geometry.
In regard to the ‘Build’ technique, the global stiffness matrix mKT of a structure
is re-assembled at each mth current stage as Figure 3.8(a), when the new members are
50 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
installed. The dead and construction loads imposed on the previously built structural
parts (i.e. 1st to (m-1)th stage) accumulated in the deformation vectors, while the force
vector mf only included the load entries of the corresponding degrees of freedom of the
newly built members at the current mth construction stage. After the global tangent
stiffness of a structure including the new members is re-assembled at the current stage,
the incremental deformations m∆u of the structure can then be determined as shown in
the equation in Figure 3.8(a). The resistance of all members mR associated with these
deformations is evaluated through the secant stiffness formulation as comprehensively
discussed in section 3.4.2.
In regard to the ‘‘Kill’’ technique, the global stiffness matrix mKT of a structure
is re-assembled by deleting the stiffness coefficients of the corresponding elements
being removed as illustrated in Figure 3.8(b), such as temporary supports and bracings,
at the current mth stage. Under this circumstance, the corresponding redundant degrees
of freedom are restrained; whereas the effects of the dead and construction loads
imposed on those members of the previous construction stages will be redistributed
among the remained members of the system at the mth current stage. It should be noted
that if there is no other dead and construction loads act on the remained members at
the mth current stage, the force vector mf is a null vector. Under this circumstance, a
dummy load should be applied at the selected critical degree of freedom in order to
preserve the equilibrium equation given in Figure 3.8(b) effective. In addition, the
internal forces and deformations of the structure at the previous and current stages are
maintained. The global tangent stiffness of a structure excluding the removal members
is re-assembled at the current stage. The incremental deformations m∆u of the structure
are generated mainly because the internal forces and deformations redistributed based
on the new equilibrium path in line with the new tangent stiffness formulation as
demonstrated in Figure 3.8(b). The resistance of all members mR is obtained by the
secant stiffness formulation.
In summary, the ‘Build’ and ‘Kill’ technique basically change the equilibrium
path by means of re-assembling the stiffness matrix of a structure with its deformed
geometry at the current stage, which is incorporated into the nonlinear solution
procedure of construction sequence as comprehensively discussed in section 3.4.2.
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 51
‘Build’ technique to reassemble global tangent stiffness from the previously built structure (left)
to the newly built structure (right) at the current stage
‘‘Kill’’ technique to reassemble global tangent stiffness from the previously built structure (left)
to the latest structure (right) after removal of some structural components at the current stage
Figure 3.8. Procedure of the ‘Build’ and ‘Kill’ technique to formulate the system analysis
3.4.2 Nonlinear solution procedure of construction sequence
This section provides an overall insight of the nonlinear solution procedure of
construction sequence. In this study, the Newton-Raphson method is used to trace the
nonlinear equilibrium path mostly for its reliable convergence (S. L. Chan & Chui,
2000; Rezaiee-Pajand et al., 2013a, 2013b) so this section only refers to the Newton-
Raphson method. However, the arc-length method (Crisfield, 1981a) and the minimum
residual displacement method (S. L. Chan, 1988) are also employed to trace the
nonlinear equilibrium path with load decrement; for example the softening behaviour
due to the material yielding or the equilibrium path of the removal construction
process.
52 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Further, one of the emphases of this study is to investigate the influence on the
structural behaviour when considering the deformed geometry due to the
constructional displacements from stage to stage, on which the principle of the
minimum member length is based. Therefore, the geometry of a structure mug at the
current stage composes of the deformed geometry m-1u at the previous stage due to the
constructional displacements and the change of geometry of a structure mup because of
the positioning at the current stage according to section 3.3.1. Consequently, the
geometry of a structure mu at current stage is written as,
pmm
gm uuu += −1 , (3.17)
It should be remarked that the change of geometry of a structure mup due to
positioning technique mostly help to alleviate the deformations of a newly built
member, because the change of geometry mup depends on the deformed geometry u1−m ,
which means the relative deformations of the new built member is less critical;
especially when the newly built structure is determinate.
When the geometry of a structure at construction phase complies with the
deformed geometry due to constructional displacements and the original member
length is preserved, the geometry of a structure at the current stage may stress up the
member as commonly termed as the initial force; in particular, when the newly built
structure is indeterminate. Thus, the initial force mfin (if any) on the newly built
member in the global coordinate is given as,
inm
eT
inm k LTf ∆= , (3.18)
in which m∆Lin is a vector of the change in member length at the axial degree of
freedom at the current stage; T is the transformation matrix; ke is the element stiffness
matrix. Thus, the nodal force vector mf keep accumulating at the current stage is
obtained as,
tm
inmm fff += , (3.19)
in which mft is the nodal force vector due to the loads imposed on the built structure at
the current mth stage. The global tangent stiffness mKT of the built structure at the
current stage is then re-assembled based on the ‘Build’ and ‘Kill’ technique as
discussed in section 3.4.1. The incremental displacement m∆u and element resistance
vector m∆R at the current stage are respectively written as,
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 53
fKu mT
mm ⋅=∆ −1 (3.20)
uTR ∆∆ ms
m k ⋅⋅= (3.21)
in which ks is the secant stiffness formulation (as discussed in details in Iu (2016a)).
And the refined plastic-hinge formulation is also incorporated into the secant stiffness
formulation ks, which is comprehensively mentioned in Iu and Bradford (2012b). The
total element resistance mR and total displacement mu at the current stage can then be
obtained respectively,
uuuuuu ∆∆ mg
mmp
mmm +=++= −1 (3.22)
RRR ∆mmm += −1 , (3.23) Therefore, the unbalanced force of a structure at the current stage is obtained as,
Rff mmm - =∆ (3.24) If the nodal displacement m∆u and the unbalanced force m∆f are satisfied the
convergent criteria at the mth current stage, the above procedure from Eqs. (3.17) ~
(3.24) is repeated for the next (m+1)th construction stage, at which the load level
accumulating at (m+1)th stage are written as,
tmmm fff ⋅+=+ λ1 , (3.25)
in which ft is the total nodal loads for whole construction sequence in order to trace
the whole equilibrium path; mλ is the ‘total construction load factor’, which is
commensurate to the load level at the mth stage given as,
tmm nS=λ , (3.26)
in which ∑=
=tn
it
mc
mm ffS1
; mfc is the cumulative force, including dead and constructional
loads, up to the load level at the current stage; mft and nt are respectively the total nodal
forces and the total number of nodal forces about all degrees of freedom of the whole
construction sequence. As a result, the equilibrium path for the whole construction
sequence can be traced in the reference of the ‘total construction load factor’ until the
total construction stage Ncs is reached. The present nonlinear solution procedure of
construction sequence is also summarised in the flowchart of Figure 3.9.
54 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Figure 3.9. The procedure of nonlinear analysis of construction stage analysis
It should be noticed that, as the proposed construction stage analysis is based on
the step-by-step method, the numerical incremental solution is employed within each
construction stage. The results of nodal displacements and member forces are carried
over as the initial conditions of the next construction stage. It inferred that the
numerical incremental iterative solution is performed within a constant load level.
3.5 NUMERICAL VERIFICATIONS
The present nonlinear analysis of construction sequence based on the present
method is to evaluate the behaviour of a steel structure under construction phase. Thus,
the present method based on the deformed geometry due to constructional
displacements was validated through a number of independent research reports.
Because of its new realm of research, other approaches including SAP2000 using the
step-by-step technique based on original undeformed coordinates (SAP, 2010) and
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 55
ANSYS using the element birth-death technique based on original undeformed
geometry (ANSYS, 2009), were also exploited for comparison. Moreover, all studies
in the examples also included a comparison with the conventional approach, which
implemented the analysis of the complete structure at one, processed by Iu and
Bradford (2012a); (2012b). At first, a cantilever under five construction stages is
studied to verify the tangent stiffness matrix for the step-by-step technique complying
with its deformed geometry at the previous construction stages. Then a number of
structures are under-investigated included plane frames under vertical construction, a
slope truss under horizontal construction, and three-dimensional structures with a
space dome and a 20-storey steel space building. It should be noticed that in order to
clearly capture any small change in the structural behaviour that might take place
during construction, most of the structures under investigation are simple and small-
scale structures. A high-rise building was after all studied to quantify the construction
stage effects that might happen in case of large-scale structures.
3.5.1 Load-deflection relation of a cantilever under multistage construction
A 25m cantilever is subjected to its self-weight, whose nodal displacements and
axial forces are of concern under different construction stages. This structure is divided
into 5 present higher-order elements under 5 construction stages and its section and
material properties and the construction stages are given in Figure 3.10.
Figure 3.10. Finite element models of 25m cantilever under construction
56 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
This example aims at verifying the tangent stiffness matrix for the step-by-step
technique complying with its deformed geometry at the previous construction stages.
Firstly, the conventional method of Iu and Bradford (2012a) was employed to analyse
the behaviour of the structure based on its deformed geometry at each stage
individually. For example, the cantilever is undeformed at first stage as shown in
Figure 3.10. At the second and subsequent stages, its geometry is based on the
deformed coordinates of the previous stage due to its gravity as illustrated by the solid
and dotted lines in Figure 3.10. Its deflection at node 2 and axial force of element 1
are respectively tabulated in Table 3.1 & Table 3.2 for brevity, which was analysed at
each construction stage separately by virtue of the conventional approach of Iu and
Bradford (2012a). Secondly, the present method was employed to analyse the
cantilever continuously from stage to stage. The behaviour of the cantilever, including
deflection at node 2 and axial force of element 1, using the present method, are given
in Table 3.2 & Table 3.3.
Table 3.1. Nodal deformations separately at different stages according to conventional approach (m)
Node
number
Stage Original coordinate The deformed coordinate
x y z x y z
2 1 5 0 0 5.000000 -0.00041 0
2 2 5.000000 -0.00041 0 4.999999 -0.00234 0
2 3 4.999999 -0.00234 0 4.999996 -0.00593 0
2 4 4.999996 -0.00593 0 4.999988 -0.01118 0
2 5 4.999988 -0.01118 0 4.999967 -0.01808 0
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 57
Table 3.2. Nodal deformations at different stages according to the present method (m)
Node number
Stage Original coordinate The deformed coordinate
x y z x y z
2 1 5 0 0 4.99999998 -0.00041 0
2 4.99999946 -0.00240 0
3 4.99999651 -0.00594 0
4 4.99998760 -0.01120 0
5 4.99996740 -0.01810 0
Table 3.3. Bending moments at different stages (kNm)
Element number
Stage Initial force
Member force due to self-weight
Total member force
Conventional approach
Present approach
1 1 0.0 24.3 24.3 24.3 2 24.3 87.5 111.8 112.0 3 111.8 146.0 257.8 257.0 4 257.8 204.0 461.8 462.0
5 461.8 262.0 723.8 724.0
The deformations between the conventional approach and present method are in
a very good agreement as evidenced in Table 3.1 & Table 3.2. Further, the comparison
of the bending moment between the two approaches is also very consistent as
demonstrated in Table 3.3. In summary, there is a very good agreement between the
present method and the conventional approach at a particular stage, when the
maximum error of both nodal displacements and bending moment is less than 0.5%.
Therefore, the present step-by-step technique for assembling the tangent stiffness is
effective to simulate the behaviour of a structure from stage to stage.
3.5.2 Two-bay three-storey frame (second-order elastic behaviour)
For the sake of investigating the structural behaviour during construction
sequence, the two-bay three-storey steel frame under its vertical gravity was first
studied by Y. Liu and Chan (2011), whose formulation referred to the step-by-step
technique with recourse to the original undeformed geometry of a structure, was re-
investigated by the present method. The structural geometry and member sections were
given in Figure 3.11. All members are bending about their major axes. This structure
is built by three construction stages, and each storey is constructed at each stage. The
major characteristic of this frame is that there are two transfer beams at the first floor
58 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
of each bay, such as node 2 as indicated in Figure 3.11, which makes the vertical
deflection at these locations unbounded and critical to the construction stage effect.
Figure 3.11. Dimensions and section properties of two-bay three-storey steel frame
Firstly, the deflected shapes of the frame for each construction stage were
compared among the present algorithm as Figure 3.12(a) (i.e. based on the deformed
coordinates at the previous stage), approaches commonly adopted including Y. Liu
and Chan (2011) and SAP2000 as Figure 3.12(b) (i.e. based on the undeformed
coordinate at the previous stage), and the conventional approach without taking the
constructional displacements into account as Figure 3.12(c). In summary, NIDA (Y.
Liu & Chan, 2011), SAP2000 (SAP, 2010) and the present method are all resorted to
the step-by-step technique, and the major difference is put on the deformed geometry
allowing for the constructional displacement according to the present study.
i) Positioning using deformed coordinates at 1st stage ii) Deflected shape using deformed coordinates at 2nd stage
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 59
iii) Positioning using deformed coordinates at 2nd stage iv) Deflected shape using deformed coordinates at the 3rd stage
Fig. 3-12(a) Positioning technique and deflected shapes based on the deformed coordinates
i) Positioning using undeformed coordinates at 1st stage ii) Deflected shape using undeformed coordinates at 2nd stage
iii) Positioning using undeformed coordinates at 2nd stage iv) Deflected shape using undeformed coordinates at the 3rd stage
Fig. 3-12(b) Positioning technique and deflected shapes based on the undeformed coordinates
Fig. 3-12(c) Deflected shapes based on the conventional approach
Figure 3.12. Comparison of deflected shapes between using deformed and undeformed coordinates
60 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
According to the present method, the positioning technique was applied to locate
the deformed geometry of a structure at 2nd and the 3rd stages as indicated by the blue
dashed lines in Figure 3.12(a; i & iii), respectively. Similarly, Figure 3.12(b; i & iii)
illustrates the positioning technique based on the original undeformed geometry of a
structure at 2nd and the 3rd stages by the grey dotted lines. The deflected shapes of all
approaches are indicated by the solid brown lines in Figure 3.12. The deflected shapes
between the two approaches in Figure 3.12(a) & (b) are very similar and cannot
distinguish visually. The difference of the deflected shapes in percentage at 2nd and the
3rd stages between two approaches using deformed and undeformed geometry is
illustrated in Figure 3.12(b; ii & iv). The maximum difference was always sought at
the top of the frame at the current stage, and negative percentage means the deflected
shape from Figure 3.12(b) is less than those of Figure 3.12(a) and vice versa.
It is interesting to point out that the deflected shapes of both approaches shown
in Figure 3.12(a & b) are similar but different from those of the conventional approach
in Figure 3.12 (c) that the frame leans back to the original position at the 3rd stage,
which is also consistent with the observation from Y. Liu and Chan (2011). This is
because when the frame displaces laterally due to the P-∆ effect, the floor inclines back
owing to the rigid joint at each beam-column connection, and then the newly built
structure at the current stage is built on this inclining back floor. This inclining back
behaviour becomes obvious at the 3rd stage when it is gradually built up on each floor
at each stage; especially the rectangular frame of which the redundancy of newly built
structure at the current stage is less that allows significant lateral displacements. In this
sense, if the bracing is installed to the rectangular frame structure, the newly built
structure becomes less redundant and restraints the lateral displacements from the
inclining back, which results in less lateral displacements. Therefore, this inclining
back behaviour can help to restore the original shape. This inclining back behaviour
can also be seen in the rectangular building frame of sections 3.5.3 & 3.5.6.
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 61
Table 3.4. Bending moment at section 1 at different stages (kNm)
CS Present method NIDA SAP2000 ANSYS
1 -15.0 -19.6 -19.5 -19.3
2 -33.1
-33.0 -33.2 -25.8
3 -47.6
-42.4 -42.8 -29.5
Conventional approach -30.4 -30.3 -29.7
Table 3.5. Vertical deflection at node 2 at different stages (mm)
CS Present method SAP2000 ANSYS
1 -0.126 -0.142 -0.148
2 -0.259
-0.287 -0.225
3 -0.354
-0.394 -0.279
Conventional approach -0.239 -0.273
Moreover, the bending moment at the section indicated by 1 and vertical
displacement at node 2 in Figure 3.12 from the present method are listed in Table 3.4
& Table 3.5 respectively, in which the corresponding values from the conventional
approach, Y. Liu and Chan (2011), SAP2000 and ANSYS are also compared. The
vertical deflection at node 2 is also graphically presented in Figure 3.13. Further, the
lateral deflection at nodes 3, 4 & 5 shown in Figure 3.11 is of much concern, which is
illustrated in Figure 3.14. The comparison among the conventional approach, NIDA
(Y. Liu & Chan, 2011) and SAP2000 are all consistent as generally shown in Table
3.4 & Table 3.5. However, ANSYS always exhibits a bit stiffer that leads to less
deflection and loading distribution, which may attribute to the birth-death element
technique in which the stiffness of inactive members of all the following stages are
still somewhat available by merely using the trivial coefficients to deactivate their
contribution.
62 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Comparison between the present method and the conventional approach
Comparison between different approaches
Figure 3.13. Vertical deflection at node 2 of the frame from 1st to the 3rd stage
In regard to the load-deformation relation, the vertical deflection at node 2 is
given in Figure 3.13(a), in which CS1 stands for the 1st construction stage and others
are similar. At the same time, CSA and CA are denoted as the construction sequence
analysis and conventional analysis respectively. The discrepancy of vertical
deflections between the present method and the conventional approach become larger
at the following stages because of the unbounded characteristic in verticality of the
frame that the inclining back behaviour as seen in Figure 3.12 exacerbates the vertical
deflection at node 2. The difference of the vertical deflections between these two
approaches is around 30% because the stiffness of a structure between the two
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00
Tota
l con
stru
ctio
nal l
oad
fact
or
Vertical deflection (mm)
δCS3=-0.111
δCS2=-0.099
δCS1=-0.046
CS1
CS2
CS3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00
Tota
l con
stru
ctio
nal l
oad
fact
or
Vertical deflection (mm)
δSAP2000=-0.155δCSA=-0.111
δANSYS=-0.04
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 63
approaches is different. Especially, the conventional approach formulates the stiffness
of a structure based on the undeformed original geometry as dashed lines in Figure
3.13(a) without stages, whereas the stiffness matrix of the present method is assembled
when accounting for the deformed geometry as the solid lines in Figure 3.13(a) at each
corresponding stage. However, the difference of δcs3 = 0.111mm at the 3rd stage is not
critical for the building construction. In addition, the load-deflection curve of vertical
displacement at node 2 from the present method is consistent with those from the
construction stage analyses of SAP2000 and ANSYS as shown in Figure 3.13(b).
From Figure 3.14(a), (b) & (c), the lateral displacements at nodes 3 & 4 from the
construction stage analysis of SAP2000 and ANSYS are very consistent except that at
node 5, but the pattern of lateral displacement is still within an acceptable level. Further,
the lateral displacements at each floor have a similar pattern that the lateral
displacement at each floor from the present method approaches toward to and even
exceeds the one of the conventional approach. This phenomenon can be attributed to
the deficiency of superposition principle for the construction sequence that the total
nonlinear effect from the conventional approach is not identical to the accumulation
of the nonlinear effect from each construction stage. Specifically, the load vector mft
accumulates the construction load imposed on the erected structure until current mth
stage only and the equilibrium equation for the current stage is then formulated, of
which the stiffness formulation KT always exhibits not stiffening as the one
considering the stiffness from all members of a whole structure. The nonlinear effect
from this equilibrium path of each stage is cumulative to the total effect from all
construction stages. In contrast, the load vector ft from the conventional approach
includes all construction loads overall structure simultaneously. In addition, because
of this, the load-deformation curves accounting for the construction sequence must be
approaching to those of the conventional approach stage by stage, but cannot be
exactly coincident with the one at its final stage, because of the deficiency of the
superposition principle in the nonlinear range. Sometimes the load-deformation curves
can even intersect those of the conventional approach at later construction stages. For
example, the lateral displacements at nodes 3 & 4 are greater than those according to
the conventional approach at the 3rd stage as shown in Figure 3.14(a) & (b). The
incremental lateral displacements at the first (i.e. node 3) and second floor (i.e. node
4) from the present method are less than those of the conventional approach owing to
64 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
the softening stiffness formulation at the first and second stage when compared to the
whole structural stiffness at 1st and 2nd stages. However, the total lateral displacements
at first and second floor at final stage are greater compared with those of the
conventional approach. This is attributed to the deficiency of superposition principle.
In summary, the larger incremental displacement at first and second floor from the
softening erecting structure is eventually cumulative (i.e. tm tm ff =∑ ) to the greater
displacements than those from the conventional approach at its final stage. Therefore,
the intersection in lateral displacements between the analyses of construction sequence
and the conventional approach normally emerges at the later stage. It can also be
observed from the lateral displacement at nodes 3, 4 & 5 in Figure 3.14 (a), (b) & (c)
that the frame shifts from left to right at the second stage when the P-∆ effect takes
place.
(a) Lateral displacement at node 3 from 1st to the 3rd stages
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Tota
l con
stru
ctio
n lo
ad fa
ctor
Horizontal displacement (mm)
δCSA=0.031
δSAP2000=0.036δANSYS=-0.003
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 65
(b) Lateral displacement at node 4 from 2nd to the 3rd stages
(c) Lateral displacement at node 5 at the 3rd stage
Figure 3.14. Lateral deflection at each floor of the frame from 1st stage to the 3rd stage
3.5.3 Three-storey building frame (second-order inelastic behaviour)
A single-bay three-storey building frame with and without cross-bracings under
three construction stages was of concern as shown in Figure 3.15, with each storey was
built at each stage. The geometry, section of members, material properties for all
members and applied loads during the construction are displayed in Figure 3.15. The
large vertical loads P are acting on the building frame in order to simulate the
behaviour of a tall building, and the horizontal loads at each floor, H is due to the wind
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Tota
l con
stru
ctio
n lo
ad fa
ctor
Horizontal displacement (mm)
δCSA=0.024
δSAP2000=0.063
δANSYS=-0.005
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Tota
l con
stru
ctio
n lo
ad fa
ctor
Horizontal displacement (mm)
δCSA=-0.055
δSAP2000=−0.161
δANSYS=-0.003
66 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
effect. Therefore, this building frame was created to study the P-∆ effect as well as the
inelasticity on the behaviour at different stages; especially very vulnerable to the case
without cross-bracings. Therefore, the emphasis of this study was mainly put on the
case without cross-bracings. This building frame was studied by the present method
and compared with both SAP2000 and ANSYS. The conventional approach was also
carried out by the Iu and Bradford (2012a); (2012b) for comparison study. The
inelastic analysis was by virtue of minimum residual displacement method to trace the
potential softening effect.
3-storey frame without bracing (b) 3-storey frame with bracing
Figure 3.15. Geometry, applied loads, section, and material properties of a three-storey frame
Firstly, the second-order inelastic deflected shapes of the building frame without
bracings at various stages are displayed in Figure 3.16. The inclining back behaviour
is again observed at the 2nd and the 3rd stages, and the horizontal displacement at first
floor at first stage is obviously larger when subjected to horizontal loads. It heralds
that the inclining back behaviour is mainly contributed from the deformed geometry
at its previous stage, where the newly built structure at the current stage is built on in
an attempt to restore its original compatibility condition, which is greater than the
lateral load component.
According to Table 3.6 & Table 3.7, the elastic result of bending moment at the
end section of node 1 and horizontal displacement at node 1 shown in Figure 3.15(a)
at the first stage from three different approaches (i.e. the present method, SAP2000
and ANSYS) agrees very well. Elastic and inelastic results from Table 3.6 & Table 3.7
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 67
at final stage are consistent with those from the conventional approach. However, the
difference in the elastic bending moment and horizontal elastic displacement at node
1 becomes apparent at second and third stages, because the cumulative difference
between nodal coordinates between deformed geometry (i.e. present method) and
original undeformed geometry (i.e. SAP2000 and ANSYS) is up to 15% in moment
and 25% in displacement. It can be seen that the positioning technique reliant on the
deformed and undeformed geometry is sensitive to the deformation of a structure;
especially those at the later stage because of the cumulative effect.
(a) Positioning using deformed
coordinates at 1st stage (b) Deflected shape at 2nd stage (c) Positioning using deformed
coordinates at 2nd stage (d) Deflected shape at the 3rd
stage
Figure 3.16. Original and deformed geometry of the three-storey frame
Table 3.6. Bending moment at the section of node 1 at different stages (kNm) – without bracing
CS Present analysis Present analysis SAP2000 ANSYS
Inelastic Elastic Elastic Elastic
1 -17.6 -17.6 -17.4 -17.5
2 -54.8 -54.8 -53.5 -53.4
3 -109.0 -102.0 -100.1 -101.9
Conventional approach -120.0 -100.0
Table 3.7. Horizontal displacement at node 1 at different stages (mm) – without bracing
CS Present analysis Present analysis SAP2000 ANSYS
Inelastic Elastic Elastic Elastic 1 2.14 2.14 2.39 2.35 2 5.15 5.15 5.80 5.68 3 14.70 8.85 10.08 9.92
Conventional approach 23.80 8.81
Furthermore, in the case of a frame without bracing, the present method was
compared with the conventional approach in terms of the horizontal displacement at
nodes 1, 2 & 3 indicated in Figure 3.15(a) for the corresponding stages as depicted in
Figure 3.17(a1), (b1), (c1) & (d1), respectively. It can be seen in Figure 3.17(a1) that
68 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
the difference in elastic horizontal displacements at nodes 1, 2 & 3 between the present
method and the conventional approach increases at first and second stages, but seems
closing the gap at the third stage. Further, concerning the inelastic horizontal
displacements in Figure 3.17(c1), the approaching phenomenon is still emerging. The
first plastic hinge (PH) occurs at the bottom of the right column at the load factor of λ
= 0.757 according to the present method, whereas it is formed at the load factor of λ =
0.761 when reliant on the conventional approach. However, the inelastic horizontal
displacements at each floor increase drastically at the limit load of a structure λ = 0.98,
because of the ductility of the material when sufficient plastic hinges are formed.
As for the vertical displacement from Figure 3.17(b1) & (d1), the elastic and
inelastic vertical displacements can reach to each other exactly and very closely at the
final stage respectively. It implies that the superposition principle is effective in this
regard, since the P-∆ effect and material yielding by bending, such as the plastic hinge
approach, cannot significantly contribute to the nonlinearities. The difference of
vertical displacement at nodes 2 and 3 are mainly due to the deformed geometry of mup
from the positioning at 2nd and the 3rd stage respectively as shown in Figure 3.17(b1).
Since the superposition principle is effective in vertical displacement and mup is
additional nodal displacement due to positioning, which does not change the
equilibrium path of a structure (e.g. tangent stiffness formulation KT), the equilibrium
point in vertical displacement of the present method and conventional approach at final
stage if the same; especially elastic vertical displacement. It is interesting to note in
Figure 3.17(b1) & (d1) that the vertical displacements at node 1 are very consistent
between the present method and the conventional approach because the geometry of a
structure at first stage is based on its undeformed geometry, where mup is absent.
Therefore, the construction stage effect on vertical displacement of a continuous
column is insignificant. On the other hand, when the vertical displacement is
unbounded like section 3.5.2, the difference in vertical displacement between the
nonlinear analysis of construction sequence and the conventional approach is
pronounced.
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 69
(a1) Elastic horizontal displacements at nodes 1, 2 & 3
(a2) Lateral displacement at node 1 from 1st to the 3rd stages
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0
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Elastic horizontal displacement (mm)
δ2=-0.60δ1=0.04 δ3=-1.60
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 2.0 4.0 6.0 8.0 10.0 12.0
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Elastic horizontal displacement (mm)
δSAP2000=1.27
δANSYS=1.11δCSA~ 0.04
70 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
(a3) Lateral displacement at node 2 from 1st to the 3rd stages
(a4) Lateral displacement at node 3 from 1st to the 3rd stages
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0
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Elastic horizontal displacement (mm)
δSAP2000=-0.93δANSYS=2.52
δCSA=-0.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0 30.0
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Elastic horizontal displacement (mm)
δSAP2000=-9.36
δANSYS=3.22δCSA=-1.6
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 71
(b1) Elastic vertical displacements at nodes 1, 2 & 3
(b2) Elastic vertical displacements at node 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0
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Elastic vertical deflection (mm)
δ2=0 δ1=0δ3=0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
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Elastic vertical deflection (mm)
δSAP2000 ~ 0
δANSYS=-0.02
δCSA~ 0
72 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
(b3) Elastic vertical displacements at node 2
(b4) Elastic vertical displacements at node 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-9.00 -8.00 -7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
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Elastic vertical deflection (mm)
δSAP200=1.67
δANSYS=-0.03
δCSA~ 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
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Elastic vertical deflection (mm)
δSAP2000=5.03δANSYS=-0.03
δCSA~ 0
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 73
(c1) Inelastic horizontal displacements at nodes 1, 2 & 3
(c2) Inelastic horizontal displacement at node 1
0.0
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0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0
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Inelastic horizontal displacement (mm)
δ2=-10.2δ1=-9.1
δ3=-11.4
Limit loadλCSA = 0.98
1st PHλCSA =0.76
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0 30.0
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Inelastic horizontal displacement (mm)
δSAP2000=-18.6
δANSYS=1.06
δCSA= -9.1
Limit loadλCSA = 0.98
1st PHλCSA =0.76
74 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
(c3) Inelastic horizontal displacement at node 2
(c4) Inelastic horizontal displacement at node 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
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Inelastic horizontal displacement (mm)
δSAP2000=-25.53
δANSYS=3.9δCSA=-10.2
Limit loadλCSA = 0.98
1st PH λCSA =0.76
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 10.0 20.0 30.0 40.0 50.0
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Inelastic horizontal displacement (mm)
δSAP2000=-36.61
δANSYS=5.67δCSA=-11.4
Limit loadλCSA = 0.98
1st PHλCSA =0.76
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 75
(d1) Inelastic vertical displacements at nodes 1, 2 & 3
(d2) Inelastic vertical displacement at node 1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0
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Inelastic vertical deflection (mm)
δ2=-0.05 δ1=-0.04δ3=-0.01
Limit loadλCSA = 0.98
1st PH λCSA =0.76
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
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Inelastic vertical deflection (mm)
δSAP2000=1.15δANSYS=-1.38
δCSA=-0.04Limit loadλCSA = 0.98
1st PHλCSA =0.76
76 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
(d3) Inelastic vertical displacement at node 2
(d4) Inelastic vertical displacement at node 3
Figure 3.17. Horizontal and vertical displacements at nodes 1, 2 & 3 for different stages – without bracing
It can be seen from Figure 3.17 that the displacements at nodes 1, 2 & 3 are in
very good agreement with those from SAP2000 and ANSYS generally. However, the
displacements at node 3 from the present method are not very consistent with those
from SAP2000 and ANSYS, because the present method adopts the deformed
0.0
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0.6
0.8
1.0
1.2
-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
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Inelastic vertical deflection (mm)
δSAP2000=3.68δANSYS=-1.54
δCSA=-0.05Limit loadλCSA = 0.98
1st PHλCSA =0.76
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
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Inelastic vertical deflection (mm)
δSAP2000=7.93δANSYS=-1.62
δCSA=-0.01Limit loadλCSA = 0.98
1st PHλCSA =0.76
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 77
geometry that accumulates when compared to those of undeformed geometry from
SAP2000 and ANSYS. Despite this, this inconsistency is still insignificant thanks to
the negligible magnitude of difference when compared with the dimension of a
structure.
The rectangular frame was only focused in this paper so far. Thus, the 3-storey
frame with cross-bracings shown in Figure 3.15(b) was under investigation. The
inclining back behaviour is still observed but much less severe, while the elastic and
inelastic horizontal displacements are obviously insignificant as tabulated in Table 3.8.
The elastic/inelastic horizontal/vertical displacements at the corresponding locations
are similar with those without bracing as given in Figure 3.17, but with less magnitude.
The reason is thanks to its high redundancy and no large inelastic horizontal
displacement due to ductility as Figure 3.17(c) when the first plastic hinge is only
formed at a higher load level of λ = 0.909 at the base of the right column. Therefore,
the second-order effect and inelastic displacements are not severe for the building
framed structure with bracing, and the structural design for the construction phase is
not necessary in this sense.
Table 3.8. Horizontal displacement at node 1 at different stages (mm) – with bracing
CS Present analysis Present analysis SAP2000 ANSYS
Inelastic Elastic Elastic Elastic
1 0.432 0.432 0.435 0.439
2 0.950 0.950 0.961 0.959
3 1.517 1.515 1.538 1.537
CA 1.519 1.516
3.5.4 Slope truss (second-order elastic behaviour with initial force)
A slope truss subjected to its self-weight was of interest for investigating the
transverse bending behaviour of a structure under the horizontal construction, such as
the cantilever construction method for the bridge structure, and the ‘‘Kill’ technique’
in the step-by-step technique could be verified. Further, the initial force was under
study, when this slope truss is highly redundant. The geometry, section of all members,
material properties, and support conditions of the truss at each construction stage is
illustrated in Figure 3.18. This slope truss is under three construction stages as
demonstrated in Figure 3.19. It is reminded that the gravity only applied on the erecting
structure at the 1st and 2nd stage, and there is no additional load available at the 3rd
78 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
stage except those at 1st and 2nd stage, which is the removal of the temporary supports
as seen in Figure 3.19(d).
Figure 3.18. Geometry, section and material properties of slop truss
(a) Positioning technique using original undeformed geometry at 1st stage
(b) Deformed shape at 1st stage and positioning technique for 2nd stage
(c) Deformed shape at 2nd stage and no load imposed at the 3rd stage
(d) Deformed shape at the 3rd stage after removal of temporary supports
Figure 3.19. Deflected shape of a slope truss under different stages
The present analysis, SAP2000 and ANSYS were employed to study the
structural behaviour under construction sequence, whereas Iu and Bradford (2012a);
(2012b) was implemented as the conventional approach. It should be noted that the
initial force was excluded from SAP2000 and ANSYS. The axial force in element 1
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 79
and vertical deflections at nodes A & B indicated in Figure 3.18 by various approaches
at different stages are tabulated in Table 3.9 & Table 3.10. The reactions at the left end
and temporary supports as shown in Figure 3.18 at each stage are listed in Table 3.11.
The load-deflection curves at nodes A & B are plotted in Figure 3.20, in which δini2 &
δini3 means deflection including the initial force in CS2 & CS3 respectively.
Table 3.9. Axial force of element 1 (kN) from various approaches
Construction stage
Present method SAP2000 ANSYS Initial
force No initial force Deviation (%)
2 -1.83 -1.11 39.3 -1.11 -2.23 3 -4.59 -2.53 44.9 -1.87 -3.00
Conventional approach -3.00
Table 3.10. Vertical displacements at A and B (mm) from various approaches
Construction stage
Present method SAP2000 ANSYS
Initial force No initial force Deviation (%)
A B A B A B A B A B
1 -0.12 -0.12 0.0 -0.12 -0.12 2 -0.13 -0.12 -0.11 -0.11 -19.6 -9.1 -0.10 -0.11 -0.13 -0.15 3 -1.30 -0.57 -1.08 -0.54 -20.4 -5.9 -1.05 -0.57 -1.52 -0.82
Conventional approach -1.04 -0.56
It was found that the initial force is insignificant for the rectangular building
framed structure as demonstrated in the sections 3.5.2 & 3.5.3 when they are less
redundant; specifically the newly built structure at the current stage. In this regard, the
initial force is of interest in this example, because this slope truss is highly redundant
when including the bracing, which can generate the member force in the newly built
members due to being restrained by other connected members. Table 3.9 indicates that
the initial force can increase the axial force significantly, which may incur the
premature material yielding.
Further, the initial force can also lead to the increase in the vertical displacements,
such as the vertical deflection at nodes A and B as shown in Figure 3.18 according to
Table 3.10. The vertical displacements at nodes A and B against the total construction
load factor were also plotted in Figure 3.20. There is a large difference in vertical
displacements at nodes A and B. It can be pointed out that the conventional approach
is futile in analysing the structural behaviour before and after temporary supports
80 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
removal when the conventional approach is not targeted for any construction sequence.
Hence, the difference in vertical deflection at nodes A & B at the final stage between
the present method and the conventional approach is obvious according to Table 3.10
and Figure 3.20. Especially from the second stage to the final stage, according to the
present method, the tangent stiffness was reformulated by removal of the two elements
of temporary supports that could not be implemented by the conventional approach.
Because of the large difference in vertical displacement with the effect of construction
sequence, the cantilever construction method from both ends may not meet at the final
stage. Hence, the present sophisticated design method of the construction sequence is
highly recommended for the horizontal construction technique.
Support reaction at left end and temporary supports are tabulated in Table 3.11
in order to verify the present ‘‘Kill’ technique’. Because of the symmetric property,
the reaction at one left end and temporary support are given in Table 3.11. The applied
loads of gravity on erecting structure at 1st, 2nd and the 3rd stages are respectively
4.8kN, 3.2kN and 0kN. Hence the total reactions at all supports are 4.79kN, 8kN and
8kN at respective 1st, 2nd and the 3rd stages as indicated in Table 3.11, which can match
with the applied load at three stages. It is interesting to remark that the internal load
redistribution after removal of the temporary supports at the 3rd stage made use of the
‘Kill’ technique. The reaction at a temporary support is 3.16kN, which shares the
majority of gravity loads at 2nd stage, while the reaction at end support is only 0.84kN.
After removal of the temporary supports, the tangent stiffness of a whole system was
reformulated on the basis of the ‘Kill’ technique as in section 3.4.1 and the new
equilibrium solution point was searching by means of the procedure stated in Section
3.4.2, in which the internal stresses in two elements of the temporary supports were
withdrawn from the system. For the sake of the balance of the equilibrium of the
system at the 3rd stage, the deformations of the structure increase pronouncedly as
illustrated in Figure 3.19(d) that provoke the increase in internal stresses of the
elements. Eventually, in order to balance the larger internal stresses of the elements at
the end supports, the reactions at both end supports increase to 8kN as the total applied
loads at the 3rd stage. In addition, the horizontal reactions at both end-supports
increase dramatically at the 3rd stage as shown in Table 3.11, since the deformation of
the slope truss is remarkable at the 3rd stage as seen in Figure 3.19(d). It also implies
the present method can successfully capture the change of geometry of a structure.
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 81
Figure 3.20. Vertical deflection at nodes A & B of slope truss against the total construction load factor
Table 3.11. Support reactions (kN) from various approaches
Construction stage
Present analysis SAP2000 ANSYS
End support Temp. Sup. End support Temp.
Sup. End support Temp. Sup.
Hor. Ver. Ver. Hor. Ver. Ver. Hor. Ver. Ver.
1 0.00 0.84 1.56 0.00 0.84 1.56 0.00 0.84 1.56 2 -0.09 0.84 3.16 0.50 0.83 3.17 1.10 0.83 3.17 3 14.40 4.00 14.30 4.00 14.55 4.00
Conventional approach 14.50 4.00
3.5.5 Shallow hexagonal dome (second-order elastic behaviour)
This example extends the previous plane truss (i.e. horizontal long span structure)
to the study of a space dome in order to investigate the three-dimensional behaviour
of a space structure; in particularly the geometric nonlinear effects subjected to the
deformed geometry of the previous stage, on which the newly built structure is built at
current stage. Therefore, a shallow hexagonal dome was of concern as shown in Figure
3.21. The geometry and dimension of the space dome are illustrated in Figure 3.21 (a).
The section of GB-SSP68x3mm is used for all members and elastic modulus is
200kN/mm2. The dome is under self-weight and constructed by three stages as
demonstrated in Figure 3.21 (b - d).
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-1.40 -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00
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Vertical deflection (mm)
δiniA3=-0.22 δiniA2=0.021δiniB3=-0.032
δiniB2=-0.01
Node A.CSA with Ini.
Node B.CSA with Ini.
Node A. CSA Node B. CSAδB3=-0.014δA3=-0.26
82 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
The axial force of element 1 and vertical deflections at nodes A & B as shown
in Figure 3.21 by various approaches at different stages are listed in Table 3.12 &
Table 3.13, respectively.
3D-view of shallow hexagonal dome
1st stage (c) 2nd stage
the 3rd stage
Figure 3.21. Geometry and applied loads of shallow dome under different stages
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 83
Table 3.12. Axial force of element 1 (kN)
Construction stage Present method SAP2000 ANSYS
3 -125.0 -99.1 -100.7
Conventional approach -101.0
Table 3.13. Vertical displacements at Nodes A & B (mm)
Construction stage Present method SAP2000 ANSYS
A B A B A B
1 -3.73 -3.96 -4.08
2 -4.46 -5.93 -4.08
3 -5.80 -83.40 -6.59 -60.61 -4.65 -61.14
Conventional approach
-4.76 -61.4
Based on the deformed geometry from the present analysis, it results in more
than 10% difference when compared with those from SAP2000 and ANSYS based on
the undeformed geometry. It is noted that this change in nodal displacement of the top
of the crown (i.e. node B) due to the displacement mup of 7.8mm from the positioning
technique in the vertical direction at the 3rd stage, which contributes up to 65% in the
total difference in vertical displacement between the present method and SAP2000. It
can be seen from Figure 3.22 that the vertical displacements at nodes A and B from
the present analysis exhibit lower stiffness of an erecting structure when compared
with the one from the conventional approach. In addition, both vertical displacements
approach to those from the conventional approach and then exceed to form an
intersection; in particular, for the vertical displacement at B. It means the vertical
displacements accounting for the effect of construction sequence are more critical to
governing the service condition of the shallow hexagonal dome. Further, the vertical
displacements at B of the present method and the conventional approach both illustrate
the second-order effect at a similar load level of about λ = 0.35. Unfortunately, the
vertical displacement at B from the present method only reaches the maximum load
level of λ = 0.92 because of the buckling of the crown. It can be attributed to the present
method based on the deformed geometry of the structure at each stage that makes the
members of the crown as shown in Figure 3.21(a) & (d) slightly longer that triggers
the pre-buckling phenomenon, which is always inevitable in the construction process.
In addition, according to Figure 3.22(b), the vertical displacement at A bounces back
84 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
at the load level of about λ = 0.8, at which the vertical displacement at B begins to
intersect those from the conventional approach as seen in Figure 3.22(a). It is because
the large deflection of the crown at B causes the flipping back of the deflection at A,
which is similar to the snap-through buckling behaviour.
According to the above observations, the deformed geometry of an erecting
structure due to the constructional displacements at each stage can cause the pre-
buckling before the ‘total construction load level’ that cannot be captured by the
conventional approach. Therefore, the nonlinear analysis of the construction sequence
seems to be indispensable to continuously monitor the structural response of an
erecting structure at each stage for the structural safety at the construction phase.
Vertical displacements at nodes A & B
Vertical displacements at nodes A
Figure 3.22. Vertical displacements at nodes A & B for different stages
0.0
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0.4
0.6
0.8
1.0
1.2
-100.00 -80.00 -60.00 -40.00 -20.00 0.00 20.00
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Vertical displacement (mm)
δA=0.78δB=-21.73
Nod
eA.
CA
0.0
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0.4
0.6
0.8
1.0
1.2
-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00
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Vertical displacement (mm)
δA=0.78
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 85
3.5.6 20-storey space steel building (second-order inelastic behaviour)
A 20-storey space steel building carries vertical load 4.8kN/m2 and lateral wind
load 0.96kN/m2 in the x-direction, which was also studied by a number of researchers
(e.g. Iu (2016b); J.Y.R. Liew et al. (2001)). The layout, section of members and
material properties are stated in Figure 3.23. This 20-storey building is constructed
under 10 stages, and every two storeys are built in each construction stage. There is no
bracing and temporary support to resist the lateral wind load and vertical load during
construction. This study comprises both second-order elastic and second-order
inelastic cases, which are solved by the Newton-Raphson and Minimum residual
displacement method, respectively. The horizontal displacements at nodes A & B from
these approaches at the corresponding stages are tabulated in Table 3.14. The locations
of nodes A & B are indicated in the plan view of Figure 3.23(a) and those locations at
different stages are shown in the elevation view of Figure 3.23(b).
The present nonlinear analysis of construction sequence can evaluate the
behaviour of the 20-storey steel building structure in the course of construction, of
which the deflected shape is given in Figure 3.24(a) and compare with those from the
conventional approach. It can be seen from Figure 3.24(a) that the inclining back
behaviour in both x- and z-directions are sought when accounting for the construction
sequence, which is similar to the findings from the rectangular framed structures in the
previous examples. The elastic horizontal displacements at each floor are tabulated in
Table 3.14 compared with those from SAP2000 and ANSYS, in which all
displacements at different stages are generally consistent with each other. And the
locations of the plastic hinges, as illustrated in Figure 3.24(b), are obtained from the
inelastic analysis of construction sequence, which is similar to the observation from
other studies including J.Y.R. Liew et al. (2001) and Iu (2016b).
86 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Plan view of space steel building
Elevation view of space steel building
Figure 3.23. Plan and elevation views of 20-storey space steel building
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 87
Deflected shapes of construction sequence and conventional analyses
Locations of the plastic hinges
Figure 3.24. Deflected shapes and locations of the plastic hinges of the 20-storey building
88 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Table 3.14. Elastic horizontal displacement at nodes A & B in mm
Construction stage Present method SAP2000 ANSYS 1A 1B 1A 1B 1A 1B
1 2.9 0.5 3.0 1.6 3.0 1.6 2 9.2 4.3 8.6 4.9 8.3 4.8
3 15.3 8.0 14.9 8.4 14.5 8.4
4 21.7 11.4 21.5 12.2 21.2 12.3
5 28.2 15.1 28.3 16.2 28.1 16.3
6 34.9 19.0 35.4 20.2 35.4 20.4
7 41.9 22.9 42.8 24.5 44.5 24.6
8 49.1 27.3 50.6 28.8 51.8 29.0
9 56.9 31.2 58.7 33.3 55.3 14.0
10 64.9 36.4 65.2 36.9 63.3 30.0
Conventional analysis 61.3 33.6
The locations of horizontal and vertical displacements at the selected levels at
particular stages were chosen for comprehensive second-order elastic and inelastic
studies in Figure 3.25 & Figure 3.26. The levels of floors (Figure 3.23) are at the lower
(i.e. 2A & 2B), middle (i.e. 6A & 6B) and top floors (i.e. 10A & 10B) such that the
deformation and torsional behaviour along the elevation can be studied. First, for the
second-order elastic analysis, the elastic horizontal displacements at the grip lines of
A & B at particular levels are displayed in Figure 3.25(a) & (b).
The similar pattern of forming an intersection in the elastic horizontal
displacements was found at different levels at both grip lines of A & B, which is
explained in the section 3.5.2 in detail. The intersection in Figure 3.25 means the
deformations accounting for the construction sequence increase faster compare with
those from the conventional approach amid the construction progress. In this regard,
the intersection implies the effect of construction sequence may dictate the design,
while the maximum lateral drift is usually a critical requirement for the tall building.
The lateral load can lead to the intersection being formed at an early stage when
compared to the elastic horizontal displacements of the section 3.5.2 as shown in
Figure 3.25. In addition, the height of the floor can also contribute to the increase in
the horizontal displacement, such as δ10 > δ6 > δ2 generally, which embodies
cumulative effect. According to Figure 3.25(a) & (b), the horizontal displacements
along a grip line A are more critical than those in grip line B because of the asymmetric
geometry in the plan. And the difference in the horizontal displacements between A &
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 89
B increases at a different rate along with the increase in floor level, which means the
twisted behaviour at the top floor is very critical; especially the horizontal
displacement δ10 are much larger than those from the conventional approach. It should
be highlighted that the δ10B is about 200mm.
For the elastic vertical displacements from Figure 3.25(c) & (d), the vertical
displacements at the selected floors at final stage mostly converge to those from the
conventional approach as explained in the section 3.5.3 comprehensively. Even when
the vertical displacements are greater, their magnitudes are very insignificant when
compared to the horizontal displacements. It is noteworthy that the convex load-
deflection curve of δ2A in Figure 3.25(c) means stiffer, which opposes to other
observation. It can be attributed to the inclining back behaviour as demonstrated in
Figure 3.24(a) that push down the column A to reduce the vertical displacement against
load factor and this effect becomes dominant at the floors at the lower levels. In general,
the vertical displacement along a continuous column accounting for the construction
sequence effect is not critical.
Elastic horizontal displacement at nodes 2A, 6A & 10A
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0
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Elastic horizontal displacement in x-direction (mm)
δ2A=17.0 δ6Α=30.0 δ10A=49.0
90 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
Elastic horizontal displacement at nodes 2B, 6B & 10B
Elastic vertical displacement at nodes 2A, 6A & 10A
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
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Elastic horizontal displacement in x-direction (mm)
δ2B=18.4 δ6Β=68.0 δ10B=247.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-10.00 -8.00 -6.00 -4.00 -2.00 0.00
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Elastic vertical delfection (mm)
δ2A=0.07δ6Α=0.41δ10A=0.62
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 91
Elastic vertical displacement at nodes 2B, 6B & 10B
Figure 3.25. Elastic displacements at corners A & B during construction
According to the present second-order inelastic analysis, the first plastic hinge is
formed on the second floor of the second frame at λ = 0.702 as shown in Figure 3.24(b),
whereas according to the conventional approach of J.Y.R. Liew et al. (2001) it was
obtained on the third floor of the first frame at λ = 0.784. It means the formation of the
plastic hinge is early when accounting for the effect of construction sequence because
the stiffness of the built structure is lower than the one of the final complete structure.
At 8th stage and onward, the number of plastic hinge increases substantially when the
total construction load level approaches to its limit load.
In regard to the inelastic horizontal displacements at A & B-grip lines at different
floors, it can be found in Figure 3.26(a) & (b) that the horizontal displacements are
less than those from the conventional approach, but can eventually exceed those and
form intersections at some stages, which is similar to the behaviour of elastic
horizontal displacements. However, the distinct feature of inelasticity from elastic
behaviour is the large deflection behaviour near the limit load thanks to the ductility.
The stiffening load-deflection curve at B-grip line gets stiffer observed in Figure
3.26(b), which can be also captured by the conventional approach of Iu (2016b)
because the 4th frame as shown in Figure 3.23(a) is stiffer and cause the torsional
behaviour. For the inelastic vertical displacements as shown in Figure 3.26(c) & (d),
the vertical displacement of the continuous column is always less critical when
compared to the horizontal displacement. Similarly, the vertical displacements are less
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-20.0 -18.0 -16.0 -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0
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Elastic vertical deflection (mm)
δ2B=0.03δ6B=0.1δ10B=-1.1
92 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
than those from the conventional approach, until the sufficient plastic hinges are
formed to cause the large deflection behaviour.
Inelastic horizontal displacement at nodes 2A, 6A & 10A
Inelastic horizontal displacement at nodes 2B, 6B & 10B
0.0
0.2
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0.8
1.0
1.2
0.0 500.0 1,000.0 1,500.0 2,000.0 2,500.0
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Inelastic horizontal displacement in x-direction (mm)
δ2A=173δ6Α=580
δ10A=580
10A. CSA
1st PHλCSA = 0.702
Limit loadλCSA = 0.98
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
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Inelastic horizontal displacemt in x-direction (mm)
δ2B=25.5 δ6B=78.0 δ10B=283.0
1st PHλCSA = 0.702
Limit loadλCSA = 0.98
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 93
Inelastic vertical displacement at nodes 2A, 6A & 10A
Inelastic vertical displacement at nodes 2B, 6B & 10B
Figure 3.26. Inelastic displacements at corners A & B during construction
In summary, the maximum lateral drift is normally a critical design requirement.
Unfortunately, the lateral loads together with the effect of construction sequence
because of lower stiffness at each construction stage can create considerable lateral
drift to govern the design of a structure at construction phase; especially the high-rise
building.
0.0
0.2
0.4
0.6
0.8
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-50.0 -45.0 -40.0 -35.0 -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0
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Inelastic vertical deflection (mm)
δ2A=-6.22δ6Α=-22.9
δ10A=-22.4
2A. CSA
10A. CSA
6A. CSA 10A. CALimit loadλ CSA= 0.98
1st PHλCSA = 0.702
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-25.0 -20.0 -15.0 -10.0 -5.0 0.0
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Inelastic vertical deflection (mm)
δ2B=-0.66δ6B=-1.30δ10B=-2.70
Limit loadλCSA = 0.98
10B. CSA
1st PHλCSA = 0.702
94 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
3.6 CONCLUSION
In conclusion, when the analysis considering the construction sequence effects,
the major numerical phenomenon is that the stiffness KT of the erecting structure until
the current stage is lower than those of a whole structure at its final stage, and also the
load mft imposed on the erecting structure is only considered until that stage. Therefore,
the cumulative behaviour of a structure under the construction sequence effect at final
stage may not be same with those from the conventional approach in most cases,
mainly because of the deficiency of superposition principle in the nonlinear range,
which depends on the type of a structure under particular circumstance as discussed in
the following.
The peculiar inclining back behaviour when considering the construction sequence
effect is commonly observed in the building framed structure because the P-∆ effect
causes the sway of the structure. In the meantime, the floor inclines slightly owing
to the rigid beam-column connection, and subsequently, it makes the newly built
structure leans back against the sway; especially the less redundant rectangular
building framed structures. The inclining back behaviour of a high-rise building at
final stage becomes severe when the inclining back effect is cumulative. The
inclining back behaviour is less severe when bracings are applied, but it may incur
large initial forces due to high redundancy, which can subsequently cause the
premature material yielding.
The cumulative effect of the structural behaviour can be built up against the
construction stages. However, in order to assure no numerical drift-off error
embedded at final equilibrium point at the final stage. The tolerance level of
convergent criteria is recommended to set tight, such as the incremental
displacements ∆u and unbalanced forces ∆f is 0.1% or less of the total
displacements u and load vector f, for the sake of the reliable equilibrium solution
at the final stage. In particular when the number of construction stage is enormous.
Mostly, the elastic horizontal displacements of a building structure accounting for
the construction sequence effect are less than, but approaching to those from the
conventional analysis as aforementioned at early construction stages. While
eventually, the horizontal displacements may exceed those of conventional
approach at later stages, such as the intersection of the load-lateral displacement
curve, which depends on a number of factors, including the lateral loads, bracing
Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects 95
for lateral stability, the number of floors. One certain thing is that the horizontal
displacements of a building structure at final construction stage are hardly identical
with those from the conventional approach exactly because of the deficiency of the
superposition principle.
Under the circumstances of the horizontal displacement greater than that of the
conventional approach, it means the effect of the construction sequence can dictate
the design of a structure, while the maximum lateral drift is usually a critical design
requirement of a high-rise building structure.
The vertical displacement of the continuous column is less critical. However, when
the vertical displacement is unbounded, for example transverse-bending structures,
the vertical displacement becomes larger against the construction stages unlike the
pattern of the horizontal displacement. Therefore, the nonlinear analysis of
construction sequence become indispensable to monitor the deformation behaviour
at different stages in order to prevent excessive deflection; especially when using
the cantilever construction method from both ends and both sides of the structures
are asymmetric, the vertical deformation at final stage seems very critical under this
circumstance to ensure its connection from both ends.
In contrast to the horizontal displacement, the vertical displacement of a continuous
column of a building structure with the construction sequence effect most likely to
reach the vertical displacement from the conventional approach at its final stage
generally. Since the P-∆ effect and material yielding by bending when reliant on
the plastic hinge approach cannot contribute any effect in the vertical direction
directly and hence the deficiency of the superposition principle absents under this
condition. And because of this, when the deformed geometry mup due to positioning
technique vanishes, such as at the 1st construction stage, the vertical displacement
from the present method is very similar or same with those of the conventional
approach.
Under the case of the unbounded vertical displacement, such as the long span
structure, the deformed geometry of an erecting structure due to the constructional
displacements at stage may trigger the pre-buckling or snap-through buckling prior
to the total load level that cannot be captured by the conventional approach.
Therefore, the nonlinear analysis of the construction sequence to continuously
96 Chapter 3: Second-order inelastic behaviour of steel structures accounted for construction stage effects
monitor the structural response of an erecting structure at each construction stage
seems to be indispensable for the structural safety during the construction phase.
Inelasticity always exacerbates the deficiency of superposition principle between
the nonlinear analysis of construction sequence at the final stage and the
conventional approach, which compared to the elastic condition, which is mainly
attributed to the large ductility of the material. In this sense, the advanced
computational technique such as the present nonlinear analysis of construction
sequence is indispensable for the reliable and sophisticated design approach.
In summary, the sophisticated and advanced design technique of a structure
using the nonlinear analysis of construction sequence is desirable to ensure the
construction performance at each construction stage. Further, this nonlinear analysis
can bridge the gap between the structural engineering design and the construction
sequence such that the holistic design of a building project including the architectural
design, structural engineering design, and construction sequence can be materialised.
This nonlinear construction stage analysis is then employed to investigate the
construction stage effects on the behaviour of prestressed steel structures in particular
with details presented in the following chapter.
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 97
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
4.1 INTRODUCTION
In the recent decades, tensioning technique has been widely applied in spatial
steel structures to increase structural load carrying capacity, improve structural
rigidity, and reduce structural deformation. Therefore, prestressed structures can cover
a larger span with a smaller structural weight, and hence become more aesthetic as
being slender. However, the most critical stage of prestressed systems is often at the
construction phase, while part of the large-scale and complicated structures under
construction lack temporary supports or stability precautions. Further, the prestressed
member forces of prestressed steel structures under construction phase are always hard
to maintain, because the displacements of those members incurred in the construction
sequence can release their specific prestressed forces. Those constructional
displacements may be further exacerbated by the nonlinearities owing to large
prestress loads applied. It implies that the performance of a prestressed structure is
hard to predict at construction phase when the specific prestress force is preserved at
its final stage and those constructional displacements are inevitable.
Unfortunately, limited literature investigated comprehensively those effects on
the behaviour of prestressed steel structures as discussed in section 2.3. To this end,
this chapter presents a second-order inelastic analysis to take the nonlinearities of
prestressed steel structures at the construction phase into account, in which the
nonlinear effects due to constructional displacements on prestress loads are
continuously evaluated at any sequence until the final stage. In order to preserve the
alignment at the next construction stage with minimising the member lengths’ change,
the position technique for installation at the next stage subjected to these constructional
displacements is newly developed by virtue of the nonlinear least-square approach to
account for these nonlinearities. At the same, the higher-order element formulation (Iu
& Bradford, 2012a) was therefore resorted to capturing the nonlinear geometric
effects, of which the accurate second-order element load solutions with the efficacious
and reliable convergence could be attained (i.e. Iu and Bradford (2010) and Iu (2015)).
98 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
Meanwhile, the proposed method reliant on the refined plastic-hinge approach (Iu &
Bradford, 2012b) could also evaluate the structural safety at service condition, which
is prone to the material nonlinearities, such that the structural performance of a
prestressed steel structure sensitive to the constructional displacements could be
predicted. This chapter accomplished task 2 and together with chapter 3 fulfilled
objective 1 of this research.
For the sake of construction simulation, there are two major factors to account
for constructional displacement at sequence according to the proposed method. First
is the newly proposed positioning technique to allow for the geometry of structure
updated depending on the constructional displacements at the next construction stage.
This technique was developed to locate a new coordinate of the deformed geometry at
the next construction stage as detailed in section 3.3. Second is the step-by-step
technique to simulate the constructional displacements at the current stage in the
sequence, which is discussed in section 4.2. The proposed method was then employed
to analyse different types of prestressed steel structure whose results were compared
with other approaches with details in section 4.3.
4.2 NONLINEAR ANALYSIS OF CONSTRUCTION SEQUENCE OF PRESTRESSED STEEL STRUCTURES
This section is to demonstrate the nonlinear solution procedures for all
construction stages to capture the behaviour of prestressed steel structures during the
whole construction sequence. In other words, this section presents the numerical
formulation of the construction sequence analysis of prestressed steel structures
accounted for constructional displacements. The details of this nonlinear construction
stage analysis for general steel structures are presented in chapter 3.
About the simulation of the prestress effect, the prestress effect on a prestressed
steel structure is usually to pre-tension a member before installing it into the structural
system, such that the member restores its original geometry, during which the prestress
effect of the member, is generated. This prestress effect in the structural system can
enhance the load carrying capacity and reduce the deflections of the structure. The
prestressing force of a member, including tensioning or compression, is simulated by
the equivalent load approach, which was widespread in the numerical methods of the
prestressed structure and has been comprehensively discussed in section 2.5. In this
sense, the prestressed member force of a member can be considered by a couple of
99
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 99
nodal forces at its both ends to capture the initial stress or prestress of the member.
Therefore, a set of nodal load vector pm f for the mth construction stage in the global
system is established equivalent to the prestress effect of the member fep but opposite
in sense. Thus, the nodal force vector mf at the current stage is obtained as,
pm
tm
inmm ffff ++= , (4.1)
in which mft is the total nodal force vector due to the loads imposed on the newly built
elements at the mth current stage; other symbols are already defined in section 3.4.
For clarity purpose, the present nonlinear solution procedure of construction
sequence of prestressed structures is summarised in the flowchart of Figure 4.1.
Figure 4.1. Procedure of nonlinear analysis of construction sequence of prestressed steel structure
100 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
It should be noticed that, as the proposed construction stage analysis is based on
the step-by-step method, the numerical incremental solution is employed within each
construction stage. The results of nodal displacements and member forces are carried
over as the initial conditions of the next construction stage. It inferred that the
numerical incremental iterative solution is performed within a constant load level.
4.3 NUMERICAL VERIFICATIONS
In this numerical verification, the present method resorted to the step-by-step
technique based on the deformed geometry (Def) and other approaches of the
construction analysis were employed for verification. The latter included SAP2000
using the step-by-step technique based on the undeformed geometry (Und) (SAP,
2010); ANSYS using the birth and death technique based on the undeformed geometry
(Und) (ANSYS, 2009); and the conventional approach of Iu and Bradford (2012a);
(2012b) which analyses the complete structure without taking the construction
sequence into account. Because SAP2000 and ANSYS base on the original
undeformed coordinates of a complete structure at the final stage, it implies no
positioning technique being applied in SAP2000 and ANSYS for these examples.
4.3.1 Arch bridge
The transverse bending behaviour of a singly-symmetric arch bridge subjected
to its self-weight at construction phase and loading at its service stage was investigated.
The hangers of this structure were prestressed at construction phase to enhance its
transverse load carrying capacity. The effect of the construction sequence on the
prestressed load of the hangers was the focus. The geometry, section properties, and
applied load are shown in Figure 4.2, and the construction and pre-tension scheme are
listed in Table 4.1, of which all four hangers 1 to 4 are simultaneously tensioned as
shown in Figure 4.2, in order to highlight the effects of construction stage on the
structural behaviour and prestressing forces. The vertical nodal load applied on the
girder at the final stage is 100kN to account for dead and imposed loads of concrete
slab at the service condition.
101
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 101
(a) Original geometry, section properties
(b) Dead load of arch bridge structure
(c) Nodal prestressing force yielding to target member force
Figure 4.2. Original geometry, section properties and applied load of arch bridge structure
It is reminded that due to the highly nonlinear behaviour, the variable load
method (S. L. Chan, 1988; Crisfield, 1981a) could be employed to trace the
equilibrium path.
Table 4.1. Construction sequences of Arch Bridge
Step Construction sequence Applied load or Prestressing force (kN)
1 Construct upper arch
2 Construct girders 1, 2, 3 & 4
Assemble hangers 1, 2, 3 & 4
Prestress hangers 1 & 4 7.5 Prestress hangers 2 & 3 3.75
3 Construct middle girder
4 Impose load from deck -100
The prestressed member forces of the four hangers and vertical deflections at
nodes A & B during construction as indicated in Figure 4.2 by various approaches are
respectively listed in Table 4.2 & Table 4.3. Very good agreement of prestressed
102 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
member forces among the present method, SAP2000 and ANSYS with deviation is
less than 1.5%. In contrast, deviation of vertical deflections between the present
method, based on the deformed geometry, with SAP2000 and ANSYS, based on the
undeformed geometry (ANSYS, 2009), is larger among 15% in average. It should be
highlighted that the positioning technique of the present method can locate the
coordinates of the bridge girder/deck based on both criteria of the original hanger
length and the deformed geometry of upper arch at the first construction stage as shown
in Table 4.3. Those common practical construction factors always affect the alignment
of the bridge deck, and subsequently the structural behaviour at the construction phase.
It also interestingly indicates from Table 4.3 that the vertical displacements at A & B
are larger compared with those based on the undeformed geometry without increasing
member forces because of the non-mechanical positioning displacements up for bridge
deck alignment. Therefore, the present method can accurately account for the
constructional factors and calibrate the prestress member forces such that the target
forces can accommodate the practical construction factors, such as deformed geometry
or constructional displacements during the construction phase, once they are identified.
Table 4.2. Prestress forces of hangers at different stages by different approaches (kN)
Member CS Present method SAP2000 ANSYS
Def Und Und
Hangers
1.4 2 7.50 7.50 7.50
3 7.51 7.50 7.50
4 110.9 109.4 111.6
Hangers
2.3 2 3.75 3.75 3.75
3 7.49 7.50 7.49
4 104.4 105.8 103.7
Table 4.3. Vertical displacements at nodes A & B by different approaches (mm)
Node Stage Present method SAP2000 ANSYS
Def Und Und Und
A 2 0.412 0.015 0.031 0.902 3 1.0375 0.642 0.683 1.582
4 12.871 12.899 9.988 11.117
103
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 103
According to the present inelastic analysis accounted for construction stage
effects or constructional displacements, the 1st plastic hinge (PH) is formed at the load
level of λ = 0.863 as shown in Figure 4.3. Further, the limit load capacity of the arch
bridge is λ = 1.055 and λ = 1.139 according to the analyses based on the deformed and
undeformed geometry, respectively. Therefore, the practical construction effects can
undermine the limit load capacity that should be taken into consideration.
The vertical displacements at nodes A & B against the total load factor λ
analysed by the present method, SAP2000 & ANSYS are plotted in Figure 4.3 &
Figure 4.4 respectively, in which the results without construction effects by the
conventional approach (Iu & Bradford, 2012a, 2012b) are also shown. It can be seen
that the vertical displacements at nodes A & B from the present analysis exhibit lower
stiffness in the early stage when compared with the one from the conventional
approach (CA). While the results shown in Figure 4.3 & Figure 4.4 of the present
method and SAP2000 are very consistent. The vertical displacements at nodes A & B
approach to the one of the conventional approach (CA) and then intersect in Figure 4.3
& Figure 4.4 respectively. Moreover, the vertical displacements at B from both present
method and conventional approach illustrate large inelastic deformations at a similar
load level of about λ = 0.8. Since the first plastic hinge is formed at the load level of λ
= 0.863 by virtue of the present method accounted for the construction sequence,
whereas the first plastic hinge at about λ = 0.8, 0.92, 1.045 according to the
conventional approach (Iu & Bradford, 2012a, 2012b), ANSYS and SAP2000,
respectively. Further, the limit load capacity of the arch bridge is also consistent, as λ
= 1.055, 1.016, 1.063 according to the present method, ANSYS and SAP2000,
respectively.
B 2 -10.227 -2.334 -2.403 -11.096
3 -13.927 -6.042 -6.207 -15.122
4 -130.21 -124.95 -108.16 -142.58
104 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
Figure 4.3. Vertical deflections at node A during construction
Figure 4.4. Vertical deflections at node B during construction
The vertical displacements at nodes A & B against the load factor according to
the conventional approaches (i.e. neglect the construction sequence), including (Iu &
Bradford, 2012a, 2012b), SAP2000 and ANSYS, are plotted in Figure 4.5 & Figure
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0
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1st PHλCA=0.803
1st PHλSAP2000=1.045
CS2 CS30.418 0.430
1st PHλANSYS=0.920
1st PHλCSA=0.863
δANSYS=-2.06
δSAP2000=-3.19
δCA=0.07
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-250.0 -200.0 -150.0 -100.0 -50.0 0.0
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1st PHλCA=0.803
1st PHλSAP2000=1.045
CS2CS3
0.418
0.430
1st PHλANSYS=0.920
1st PHλCSA=0.863δANSYS=-10.2
δSAP2000=24.3
δCA=2.34
105
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 105
4.6. In which, the simultaneous pre-tension scheme for all vertical hangers 1 to 4 are
implemented as mentioned in Table 4.1 in order to verify the equivalent prestressed
load approach employed in this study. It can be seen that the results from three
conventional approaches are in good agreement. The load-deflection curve according
to ANSYS generally exhibits a bit higher stiffness compared with the other two
approaches and the difference in vertical deflections at nodes A & B increase gradually
at higher load level; especially after the formation of the first plastic hinge. From
Figure 4.5 & Figure 4.6, the load-deflection curves from SAP2000 and ANSYS seem
linear and even after the formation of the first plastic hinge.
Figure 4.5. Vertical deflections at node A by conventional analyses
0.0
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1.0
1.2
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
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Vertical displacement in mm
1st PHλCA=0.803 1st PH
λSAP2000=1.033
1st PHλANSYS=0.917
δANSYS=-2.12
δSAP2000=-2.52
106 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
Figure 4.6. Vertical deflections at node B by conventional analyses
4.3.2 Frame column
A frame column is designed to increase its axial compression capacity by
prestressing the four struts in between the main frames, whose geometry and section
properties are shown in Figure 4.7(a), as the prestress forces in the struts can offset a
part of the external axial compression. The construction and simultaneous pre-tension
scheme are summarised in Table 4.4, which is also shown graphically in Figure 4.7(b)-
(d), and the vertical nodal load applied on the top of the column is 100kN, which
simulates the dead and service loads at service condition. Further, the vertical nodal
load applied on the top of the column increases gradually until the limit load capacity
of the column. The construction sequence is that the main frame is built at first stage
as in Figure 4.7(b) and the four struts are simultaneously prestressed as in Figure
4.7(c). As a result, the nodal positioning technique is not effective in this example.
0.0
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-160.0 -140.0 -120.0 -100.0 -80.0 -60.0 -40.0 -20.0 0.0
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Vertical displacement in mm
1st PHλCA=0.803
1st PHλSAP2000=1.033
1st PHλANSYS=0.917
δANSYS=-12.5
δSAP2000=14.4
107
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 107
Table 4.4. Construction sequences of frame column
Stage Construction sequence Applied load or Prestressing force (kN)
1 Construct the whole frame
2 Prestress struts 1 & 2 1.24
Prestress struts 3 & 4 0.64
3 Apply vertical load -100
(a) Layout of frame Col. (b) Frame at 1st stage (c) Prestress at 2nd stage (d) Loading at the 3rd
stage
Figure 4.7. Layout of frame column and its construction sequence
In regard to the service stage, the prestressed member forces and vertical
deflection at top A indicated in Figure 4.7(a) during construction by various
approaches are listed in Table 4.5 & Table 4.6, in which the target force aimed at the
2nd stage of both the present method and SAP2000 can be held effective. The results
from the present analysis are consistent with those from SAP2000 based on the step-
by-step technique to simulate the construction stage effects. On the other hand, the
deviation of target force and vertical displacement between the present method and
ANSYS based on the birth and death technique is a bit greater but still less than 1.5%
in general. The load-deflection curve of node A at service stage is plotted in Figure
4.8(a), in which large difference in vertical displacement at top A from the
conventional approach is found at the 2nd prestress stage according to the present
method. The prestress at the 2nd stage changed the behaviour of the frame column
dramatically (i.e. from downward to upward displacement) that cannot be predicted by
108 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
the conventional approach. In addition, Figure 4.8(a) also shows that the difference of
the vertical displacement at A at service stage among SAP2000, ANSYS and the
present method compared to the conventional approach (Iu & Bradford, 2012a, 2012b)
are -0.003mm. It is proved that the construction stage techniques, i.e. the step-by-step
and the ‘birth and death’ technique, are adequate to capture the effect of construction
sequence at the final or service stage of a complete structure in linear elastic range.
Table 4.5. Prestress forces during construction by different approaches (kN)
Member Stage Present method SAP2000 ANSYS
Def Und Und
Strut 1.2 1 1.23 1.24 1.22 2 1.24 1.24 1.23
3 13.60 13.60 13.43
Strut 3.4 1 0.63 0.63 0.64 2 0.64 0.64 0.64
3 6.93 6.93 6.99
Table 4.6. Vertical displacement at node A during construction by different approaches (mm)
Node Stage Present method SAP2000 ANSYS
Def Und Und
A 1 -0.1346 -0.1350 -0.1349 2 -0.0810 -0.0810 -0.0814
3 -1.4273 -1.4300 -1.4304
Concerning the inelastic analysis, the first plastic hinge (PH) from the present
method is found at the load level of λ = 1.732, whereas the first PH from SAP2000
and ANSYS are respectively at λ = 1.739 and 1.759 as displayed in Figure 4.8(b).
Moreover, the limit load capacities of the frame column predicted by different
approaches are also given in Figure 4.8(b). According to the present method,
maximum carrying load level is at the load level of λ = 1.778, whereas they are λ =
1.750 and λ = 1.795 according to SAP2000 & ANSYS respectively, which are very
consistent. It implies that there is a minor strength reserve at the inelastic range of the
frame column. In contrast, the limit load capacity by virtue of the conventional
approach (Iu & Bradford, 2012a, 2012b) is at λ = 1.778, which is same as the one from
the present analysis. Therefore, the numerical techniques, such as step-by-step
technique used in the construction analysis, insignificantly contribute to the limit load
of the structure except for the inelastic analysis, such as the plastic hinge approach. In
109
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 109
other words, the effect of construction sequence has no remarkable contribution to the
limit load capacity of a linear elastic structure.
(a) Load-deflection curve at node A by different approaches at the service load level
(b) Load-deflection curve at node A by different approaches till the limit load level
Figure 4.8. Vertical displacement at top A during construction against ‘total load factor’
It can be seen that the results from three approaches of construction analysis are
consistent. Moreover, the behaviour of the structure (e.g. load capacity; deformations)
at the service load condition evaluated by the construction analysis, including the
present method, and the conventional approach is similar. However, the behaviour of
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-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
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Vertical displacement in mm
CS2
CS10.01
0.899
δCA=-0.003δANSYS=-0.003δSAP2000=-0.003
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1.0
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1.6
1.8
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-14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0
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Vertical displacement in mm
CS2
CS10.01
0.899
1st PHλSAP2000=1.739 1st PH
λCSA=1.732
1st PHλANSYS=1.759
λCA=1.778λANSYS=1.795
λSAP2000=1.750
λCSA=1.778
110 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
prestressed structures is quite different during the construction phase. As a result, the
construction analysis is indispensable to capture the behaviour of the prestressed
structure for its structural safety during construction.
4.3.3 Shallow dome
This example investigated the construction stage effects on the structural
behaviour of a prestressed symmetric spatial dome prone to second-order effects; in
particular, the influence on target prestress forces of the six bottom chords, which
could, in turn, change the limit load capacity. The geometry and dimension of the
shallow dome with the six prestressed bottom chords are illustrated in Figure 4.9(a) &
(b). All members are made of CHS88.6x3.9mm with elastic modulus 205kN/mm2 and
yield stress 275N/mm2. The dome is under its own weight and constructed by three
stages as demonstrated in Figure 4.9(c), (d) & (e). Prestressed member forces of 25kN
are applied for all six prestressed members at the 2nd stage, and the equivalent nodal
loads are shown in Figure 4.9(d). The vertical concentrated loads of the dead and
service load are imposed on the crown at the 3rd stage in Figure 4.9(e), such that the
limit loading capacity of the shallow dome can be traced after the service stage. The
construction sequence of three stages of the shallow dome is summarised in Table 4.7.
111
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 111
(a) Original geometry
(b) Arrangement of 6 prestressed bottom chords (c) 1st stage: frame construction
(d) 2nd stage: prestress (e) the 3rd stage: crown construction
Figure 4.9. Geometry of the shallow prestressed dome and its construction sequence
Table 4.7. Construction sequences of shallow dome
Stage Construction sequence Applied load or Prestressing forces (kN)
1 Construct the main frame
2 Prestress 6 bottom chords 23.2
3 Construct the crown and apply load as shown in Figure 4.9(e)
112 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
Table 4.8. Prestress member forces of 6 bottom chords at different stages (kN)
Member Stage Present method SAP2000 ANSYS
Def Und Und
3 1 23.05 23.04 23.02
2 23.23 23.23 23.28
3 61.66 60.61 62.32
11 1 22.99 23.00 22.95
2 23.20 23.21 23.25
3 56.95 56.96 56.71
19 1 22.89 22.88 22.73
2 23.16 23.18 23.18
3 54.65 54.65 53.47
27 1 23.00 22.99 22.91
2 23.21 23.21 23.24
3 58.69 58.66 58.75
33 1 22.87 22.87 22.68
2 23.16 23.17 23.16
3 55.53 55.54 54.11
39 1 23.03 23.03 23.04
2 23.21 23.22 23.27
3 57.12 57.10 57.30
Table 4.9. Vertical displacement at nodes A & B at different stages (mm)
Node Stage Present method SAP2000 ANSYS
Def Und Und
A 1 -5.465 -5.465 -5.470 2 -3.638 -3.640 -3.638
3 -9.784 -9.721 -9.682
B 3 -36.420 -39.386 -36.372
The member forces of six prestressed bottom chords and vertical deflections at
nodes A & B indicated in Figure 4.9(b) by various approaches during construction are
listed in Table 4.8 & Table 4.9, respectively.
According to Table 4.8 & Table 4.9, the prestressed member forces and vertical
deflections at nodes A & B from the present method are consistent with those from
SAP2000 and ANSYS. It should be noted that because the present method based on
the deformed geometry, which leads to the coordinates of the main frame being slightly
different from its original layout. For example, the change of nodal displacement at
113
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 113
node B due to the positioning technique up, aiming to maintain the minimum change
in member length of the six members of the crown is 2mm upward. It, in turn, affected
the target prestressed member forces of six bottom chords at the 3nd stage. As a result,
there exists a deviation in the final prestressed member forces among the present
method, SAP2000 and ANSYS as in Table 4.8. In Table 4.9, the vertical displacement
at node A from the present method is less than 1.5% difference with other approaches.
The vertical displacements at node A against the total load factor from the
present method, SAP2000 & ANSYS are also plotted in Figure 4.10. It is similar to
the example in section 4.3.2 that the zip-zap load-deflection curve of vertical
displacement at node A can be found in Figure 4.10(a) & (b) because the prestressing
forces of the bottom chords change this displacement from downward to upward
dramatically at the 2nd stage. It can be seen that the load-deflection curves of the three
approaches (i.e. SAP2000 (δSAP); ANSYS (δANSYS); (Iu & Bradford, 2012a, 2012b)
(δCA)) are in good agreement at the service stage of λ = 1 when compared with the
conventional approach.
Concerning the inelastic analysis, the first plastic hinge (PH) according to the
present method is formed at the load level of λ = 1.121, whereas it is formed a bit
earlier at λ = 1.101 according to SAP2000. It is noted that structure still remains elastic
being evaluated by ANSYS. In Figure 4.10, regarding the limit load capacity, the
present method evaluates it at λ = 1.122 right after the first plastic hinge formation,
whereas it reached a slightly lower load level at λ = 1.106 according to SAP2000 &
ANSYS.
114 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
(a) Vertical displacements at nodes A & B at service stage
(b) Vertical displacements at node A against load factor till limit load level
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Vertical displacement in mm
CS2
0.720
δSAP2000=-2.966δANSYS=0.048δCA=-0.012
CS10.240
δSAP2000=0.063δANSYS=0.102
δCA=-0.227
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Vertical displacement in mm
CS2
CS10.240
0.720
1st PHλCA=1.120
1st PHλSAP2000=1.101
1st PHλCSA=1.121
λANSYS=1.106λCA.Lim=1.122
λCSA.Lim=1.122 λSAP2000=1.106
115
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 115
(c) Vertical displacements at node B against load factor till limit load level
Figure 4.10. Vertical displacements at nodes A & B against the total load factor
The buckling of the crown governs the limit load capacity of the shallow dome
as observed from Figure 4.10(b) & (c). Based on the deformed geometry of the
structure according to construction sequence from the present method, node B is
positioned upward and then it lengthens the members of the crown due to the
constructional displacements, which makes the shape of the dome (i.e. the crown)
critical to second-order effect. On the other hand, when compared with the
conventional approach as shown in Figure 4.10, the same limit load level λ = 1.122 is
reached.
Furthermore, the load-deflection curve at node A neglecting construction
sequence is also plot in Figure 4.10. Compared with the one from the present analysis,
the load-deflection curve at node A by the conventional approach (Iu & Bradford,
2012a, 2012b) similarly exhibits higher stiffness. While the present method and the
conventional approach can both capture a similar second-order effect that the vertical
displacement at A bounces back due to large deflection of the crown at B. This
bouncing back phenomenon appears at the load level of about λ = 1.12 according to
the present method. Similarly, the load-deflection curves at A & B neglecting the
construction sequence are summarised in Figure 4.11 by different approaches (i.e. Iu
and Bradford (2012a); (2012b), SAP2000 & ANSYS), at which the similar second-
order effect is also predicted by those methods.
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-100.0 -80.0 -60.0 -40.0 -20.0 0.0
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Vertical displacement in mm
CS2
0.72
1st PHλCA=1.120
1st PHλSAP2000=1.101
1st PHλCSA=1.121
λANSYS=1.106
λCA.Lim=1.122
λCSA.Lim=1.122
λSAP2000=1.106
116 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
Through the above studies, the construction stage effects can influence the
behaviour of prestressed steel structure; in particular, the structural behaviour during
construction and its prestressed member forces, which can, in turn, affect its optimal
performance. Therefore, the present construction analysis of a structure with its
construction sequence is highly recommended for a sophisticated evaluation of the
behaviour of the prestressed steel structures at the construction phase.
Figure 4.11. Vertical displacements at nodes A & B by the conventional analysis
4.4 CONCLUSIONS
This paper presents the nonlinear analysis of the prestressed steel structure with
the construction sequence, which can accommodate the constructional displacements
that incur at the construction phase. In particular, the positioning technique and the
mapping methodology are proposed herein. The positioning technique, based on the
principle of minimum change in the newly erected member’s lengths, is necessary to
locate the change of nodal coordinates of the newly built members of the current
construction stage because of the deformed geometry of the structure of the previous
construction stage. While the mapping methodology is essential to regulate the
positioning technique in order to determine the nodal coordinates of the deformed
geometry of a structure from stage to stage. Through the numerical study, there are a
few conclusive remarks drawn for the sake of the optimal performance of prestressed
steel structures at the service stage.
This study found that the behaviour of steel structures at the construction phase
is quite different from the final complete structure at service stage. Through the
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117
Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects 117
numerical analysis in section 4.3, the deviation of final displacements between
the present approach accounted for construction sequence and the conventional
approach unaccounted this effect is up to 8%. However, concerning the
structural dimension, its rigidity, and prestressed magnitude of those case
studies, it is considered that the deviation might be enlarged in other situations.
In particular, when there is no temporary supports and stability measure at
construction stage, which makes prestressed steel structure critical to nonlinear
effects. Thus, the structural safety of slender steel structures, which are prone
to the second-order effect, should be monitored during construction.
The positioning technique adjusts the nodal coordinates of newly built members
or structure stage by stage in order to accommodate the practical constructional
displacements. It is important to note that when the structure constructed based
on its original undeformed geometry; the geometry of the newly erected
structural part is much distorted compared with those constructed based on the
deformed geometry as the present study. It heralds that the significant initial
forces can be built up in the members if they are already prefabricated. This
initial force can cause the premature material yielding as discussed in section
3.5.4. Thus, construction simulation analysis is necessary to reflect a true
structural behaviour at construction phase such that the optimal performance of
a structure can be preserved as those specified at the design stage.
Constructional displacements directly change nodal coordinates, which induces
initial member deformations or the so-called ‘lack of fit’ of prestressed
members and in turn affects prestressing forces. Consequently, the final
prestressed member forces may deviate from the target design values, which
consequently affect the optimal performance of a structure. The general
deviation found through the numerical studies in section 4.3 could be up to 10%.
Hence, this particular effect should be accounted in the construction analysis of
prestressed slender steel structures.
The positioning technique adjusts the nodal coordinates of newly built members
or structural parts stage by stage in order to accommodate the practical
constructional displacements. Thus, construction simulation analysis is
necessary to reflect a true structural behaviour.
Even though the loading sequence (i.e. prestress load and applied load
sequences) does not change the structural response at full load level before the
118 Chapter 4: Second-order inelastic behaviour of prestressed steel structures accounted for construction stage effects
formation of plastic hinges, it affects the inputted target prestressed member
forces of a structure for the sake of minimum pre-tension process. Therefore,
the loading sequence should be monitored by using the construction simulation
analysis for an efficient prestress construction process.
In this sense, the advanced computational technique such as the present
nonlinear analysis of construction sequence is indispensable for a reliable and
sophisticated design approach of prestressed steel structures at the construction stage.
Unfortunately, the common nonlinear construction analyses (e.g. SAP2000 and
ANSYS) accounted for constructional displacements at construction phase are not yet
mature and adequate. As those analyses are able to capture the change of geometry
within each construction stage, whereas those change occurred between two
constitutive stages are not yet properly accounted for. Therefore, practitioners should
properly evaluate the behaviour of prestressed steel structures, in particular due to all
kinds of constructional displacements, at any construction sequence as well as monitor
the target forces being attained at a particular stage, by means of construction
simulation analysis in order to ensure structural safety during construction and
efficient pre-tension construction process.
As the construction sequence has direct effects on the behaviour of prestressed
steel structures, the following chapters present solution approaches to search for the
required prestressing forces to achieve a target prestressed stage which takes into
account the construction stage effects.
Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 119
Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
5.1 INTRODUCTION
In recent decades, the pre-tension has been widely applied in space steel
structures to increase its load carrying capacity, improve its structural rigidity, and
reduce the structural deformation. Therefore, prestressed steel structures can cover a
larger span with a smaller structural weight, and hence become more aesthetic as being
slender (R. Levy et al., 1994).
Unfortunately, under the circumstances of the presence of many prestressed
members in the system and the limited capability of tensioning equipment, it is
impossible to prestress all members simultaneously. When one member is prestressed
to its target value for the optimal capacity of the system, the target values in other
tensioning members will immediately change due to the interdependent behaviour of
all tensioning members in the system. As a result, the batched and repeated tensioning
schemes are unavoidable such that the required tensioning control force and/or
displacement of each tensioning member can be computed to achieve the final target
state. In order words, once the predicted value is applied on each tensioning member
according to the predetermined construction scheme, the forces and/or displacements
in tensioning members at the target state must reach the target values after tensioning.
Due to the lack of more practical engineering applications as discussed in section
2.4.1, this chapter presents a comprehensive investigation of the interdependent
behaviour of prestressed steel structures based on influence matrix (IFM) in a reliable,
effective, and efficient manner. It should be noticed that IFM approach was chosen in
this study as compared with the iterative solution approach (Zhou et al., 2010b), IFM
has the advantage of providing the analyst with a thorough understanding of the
interdependent behaviour among prestressed members in the system by means of its
coefficients. As the coefficients of IFM represent the mutual influences of prestressed
members in the structural system, once the IFM is established, a complete analysis for
the entire tensioning process can be obtained. It inferred that analysts could also choose
the optimal tensioning scheme, in which the required tensioning control forces and/or
120 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
displacements needed to apply upon prestressed members in order to finally meet the
requirements for a specific design (target) stage instead of tensioning by trial and error.
The reliability, effectiveness, and efficiency of this approach are founded on a series
of numerical verification. This chapter accomplished task 3 and partially fulfilled
Objective 2 of this research.
5.2 INFLUENCE MATRIX
5.2.1 Definition of IFM
=
nnnn
n
n
a..aa........
a..aaa..aa
A
21
22221
11211
, (5.1)
Influence Matrix (IFM) represents the mutual influences of prestressed members
in the structural system, whose quantities are the system state parameters, as written
in Eq. (5.1). In which, matrix coefficient akj is the variation of a specific quantity
(member force or displacement) of member k once a specific quantity of member j
increases by one unit; n is the total number of tensioning members. In short, column j
of IFM represents the variation of a specific quantity of other prestressed members
once a specific quantity of member j increases by one unit.
5.2.2 Different types of IFM
There are four types of IFM that represent the interdependent response of
member forces and/or displacements during the tensioning process (Nguyen & Iu,
2015b); they are force based (F matrix); displacement based (D matrix); displacement-
force based (DF matrix); and force-displacement based (FD matrix). They can be sub-
divided into two groups; one-criterion IFM, including F and D matrices (i.e. single
control value) and two-criterion IFM, including DF and FD matrices (i.e. dual control
values).
One-criterion IFM
Concerning the structural characteristics on the choice of IFM, force is likely the
control criterion for cable-type members, whereas displacement is likely the control
value for hanger-type members. The coefficient fkj of F matrix is the force variation of
member k when the force of member j increases by one unit, while the coefficient dkj
121
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 121
of D matrix is the displacement variation of member k when the displacement of
member j increases by one unit.
Two-criterion IFM
In case, there are two criteria under control, DF matrix or FD matrix can be used.
The coefficient dfkj of DF matrix is the displacement variation of member k when the
force of member j increases by one unit, while the coefficient fdkj of FD matrix is the
force variation of member k when the displacement of member j increases by one unit.
The coefficients of IFMs can be obtained from the finite element analysis by
virtue of Iu and Bradford (2012a), (2012b). The choice of IFM depends on the
structural characteristics and the construction conditions, especially the monitoring
unit and the detective equipment available on site, that a suitable type of IFM will be
used for a specific situation. Another important consideration is the determinant of the
IFM that also influences which type of IFM is suitable for a particular situation. The
detailed discussion is given in sections 5.3 & 5.4.
5.3 EFFECT OF INSTALLATION PROCESS VS TENSIONING PROCESS ON IFM
In this study, the installation process refers to the tensioning member being
assembled in the system, whereas the tensioning process refers to the tensioning
member, already installed in the system, being prestressed. These processes can adjust
the coefficients of IFM of all kinds. In short, the installation process means the system
characteristic (i.e. stiffness/flexibility) of IFM changes step-by-step according to the
construction sequence of the physical members. On the other hand, the tensioning
process means the system characteristic of IFM changes step-by-step according to the
quantities (i.e. displacements/ forces) imposed on the tensioning members that have
been already assembled in the system. The former process varies the
stiffness/flexibility of the system that produces the quantities of displacements/forces
on the tensioning members. The latter process adjusts the quantities of
displacements/forces on the tensioning members directly. Therefore, IFM can be
expressed in different forms based on the different construction process.
In this regard, the tensioning process can be sub-divided into two schemes; they
are batched tensioning (i.e. batch-by-batch), which refers to tensioning member(s) by
member(s), and repeated tensioning (i.e. time-by-time), which stands for tensioning
122 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
the same member(s) time after time. Therefore, both schemes can determine the way
to set up IFM.
5.4 EFFECT OF DETERMINATE VS INDETERMINATE STRUCTURE ON IFM
If a member of an external or internal determinate system is prestressed, this
tensioning member causes no change to other tensioning members, as there is no
restraining force or redundancy to influence other tensioning members. It implies that
there is no mutual influence among these structural members, and thereby the
determinants of IFMs, in this case, are zero. On the contrary, when a member of an
external or internal indeterminate system is prestressed, this tensioning member
induces the displacements and forces on other tensioning members because of the
redundancy. As a result, the determinants of IFMs of the indeterminate system are not
zero. Despite this, the null determinant of IFMs can be also found in a particular
indeterminate system, such as the members are prestressed symmetrically in a
structurally symmetric system. Therefore, the determinant of IFMs is one important
criterion that influences which type of IFM is suitable for a particular situation.
5.5 SETUP OF IFM
The quantities of the control values of IFMs, i.e. prestressed member force or
nodal displacement, can be expressed as the lack of fit of tensioning member, which
was first introduced by L. P. Felton and Hofmeister (1970) and its application was later
improved by Hanaor and Levy (1985). Both control values can be determined
according to two major tensioning processes, e.g. the batched and repeated tensioning
process (Nguyen & Iu, 2015b).
Concerning the batched tensioning process, IFM is constructed by imposing -1 unit
lack of fit (it induces a tension and vice versa) upon tensioning member(s) j. The
IFM coefficients of F, D, DF, and FD matrices are given in Eq. (5.2).
'
'
kj'
'
kj'
'
kj'
'
kj
jj
jk
jj
jk
jj
jk
jj
jkp
fd,p
df,d,pp
fδ
δ
δ
δ==== , (5.2)
123
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 123
Figure 5.1. Example of setup of IFM based on the batched tensioning process
in which 'jk
p and 'jk
δ are member force and nodal displacement of member (batch) k;
'jj
p and 'jj
δ are member force and nodal displacement of member (batch) j itself.
Figure 5.1 illustrates an example of setting up F matrix based on the batched
tensioning process. In which -1 unit lack of fit is imposed on member 1 of the complete
structure, to construct the first column [ ]T'p'p
'p'p
1 11
12
11
11 ,=f of F; and then -1 unit lack of fit is
imposed on member 2 of the same complete structure, to construct the second column
[ ]T'p'p
'p'p
2 22
22
22
21 ,=f of F. It is noted that since member forces or nodal displacements are
computed based on the final structure, they are the total in quantity.
In regard to the repeated tensioning process, the setup of IFM is based on the
member forces and/or displacements at different stages of the construction
sequence as Eq. (5.3),
mj,j
mjj
mk,j
mjkm
kjmj,j
mjj
mk,j
mjkm
kjmj,j
mjj
mk,j
mjkm
kjmj,j
mjj
mk,j
mjkm
kj
ppfd,
ppdf;d,
pppp
f1
1
1
1
1
1
1
1
−
−
−
−
−
−
−
−
−
−=
−
−=
−
−=
−
−=
δδδδ
δδδδ
, (5.3)
in which, for the mth tensioning round, mjkp and m
jkδ are member forces and nodal
displacement of member k after member j is prestressed; the nominator and
denominator of the IFM coefficient respectively represents the incremental force or
displacement of member k and j respectively from stage (j-1) when member (j-1) is
prestressed, to stage j when member j is prestressed. In order words, m denotes the
repeated tension, whereas j stands for the batched tension. Figure 5.2 demonstrates an
example of setting up F matrix based on the repeated tensioning process, in which the
control forces/displacements are imposed on the correspondent members step by step.
Hence, for the first round tension, in the first step, the first column of F matrix can be
constructed, as [ ]Tpppp
pppp1
101
111
02112
01111
01111 ,
−−
−−=f with p0j is the member force before tension; whereas
in the second step, the second column of F matrix can be established as
124 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
[ ]Tpppp
pppp1
2 112
122
112
122
112
122
111
121 ,
−−
−−=f . It is noted that since the member forces and nodal displacements
are obtained step-by-step, they are incremental in quantity.
Figure 5.2. Example of setup of IFM based on the repeated tensioning process
It is interestingly remarked that the coefficients of F & FD matrices represent
the mutual influence of member forces in local coordinate, whereas the coefficients of
D & DF matrices represent the mutual influence of nodal displacements in the global
coordinate. It is generally because the prestressed member forces can be measured on
site, whereas the nodal displacement can be monitored as its vertical and/or horizontal
components on site.
As far as the IFM according to a specific tensioning process has been established,
the relation among tensioning members is mandatorily formulated by means of
solution procedure, which is comprehensively discussed in the following section.
5.6 NUMERICAL SOLUTION PROCEDURES
5.6.1 Governing equation
With the definition of the IFM, the mutual relation between member forces
and/or displacements of tensioning members according to a particular batched and/or
repeated tensioning process can be formulated in the matrix form as Eq. (5.4), when
member j is prestressed for the mth round,
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
mj
;p
;p
11
11
−−
−−
+=+=
+=+=
pfdpdf
dpfp
δ∆∆
δ∆∆
δδ
δδ, (5.4)
in which Tmjn
mj
mj
mj ]p..p,p[ 21=p , Tm
jnmj
mj
mj ]..,[ δδδ 21=δ are the vector of member
forces/displacements respectively; mj,j
mj
mj ptp 1−−=∆ , m
j,1jmj
mj uδ −−=∆ δ are respectively
the incremental force/displacement of member j, which can be regarded as the
unbalanced effects from the tensioning process; mjt , m
ju are the tensioning control
force/displacement of member j respectively; Tmnj
mj2
mj1
m ]f..f,f[j =f ,
125
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 125
Tmnj
mj2
mj1
m ]d..d,d[j =d , Tmnj
mj2
mj1
m ]df..df,df[j =df , Tmnj
mj2
mj1
m ]fd..fd,fd[j =fd are the column
jth of F, D, DF and FD matrices, respectively.
Hence, the first terms on the right-hand side of Eq. (5.4) is the incremental
force/displacement vector induced when member j is prestressed. The physical
interpretation of Eq. (5.4) is that the member force/displacement at current jth
tensioning stage ( mjp or m
jδ ) is equal to the sum of tensioning effect at (j-1)th step ( m1j −p
or m1j−δ ) and the unbalanced effect from the previous step ( m
jp∆ or mjδ∆ ). This
unbalanced effect is distributed in the structure complying with the corresponding
column of IFM (i.e. mj
mj
mj
mj ,,, fddfdf ). Equation (5.4) is the fundamental governing
equation of a prestressed steel structure.
5.6.2 Direct solving method
Once the IFM has been established, the required tensioning control forces and/or
displacements can be determined in order to achieve the final target state by setting up
a set of n equations for n tensioning members, in which n unknown tensioning control
forces and/or displacements of n tensioning members can be computed as follows,
After tensioning n prestressed members, the target prestressed state needs to be
achieved. Hence, the set of Eq. (5.4) becomes Eq. (5.5) for the mth tensioning round,
mt
mn
mn
mn
mn
mt
mn
mn
mn
mn
mt
mn
mn
mn
mn
mt
mn
mn
mn
mn
;p;p
ppfdpdfdppfp
=+==+=
=+==+=
−−
−−
11
11
δ∆∆
δ∆∆
δδδ
δδδ. (5.5)
By solving directly the set of Eq. (5.5), the required control prestressing forces/
displacements can be determined.
A simple example of applying the direct solving method is illustrated when there
are only two tensioning members in the system, prestressed one by one for one round,
the whole set (n = 2) of Eq. (5.5) based on F matrix can be expressed as Eq. (5.6),
[ ] ( )[ ] [ ][ ] ( )[ ] [ ] [ ]T
ttTTT
TTT
ppppffptpp
ppffptpp
211211221212222212
0201211101112111
=+−==
+−==
p
p , (5.6)
By back substitute p1 into p2 to achieve the target state, the required control
forces, [ ]T21 tt=t can be obtained by directly solving the set of Eq. (5.6). To extend to
4 IFMs with n = 2, the required unbalanced control values of member 1 are given in
Eq. (5.7) and those of member 2 to achieve the target state are written in Eq. (5.8),
126 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
( ) ( )[ ]( )
( ) ( )[ ]( )
( ) ( )[ ]( )( ) ( )[ ]( )12212211
02212011220111
12212211
02212011220111
12212211
02212011220111
12212211
02212011220111
fdfdfdfdppfdppfd
u
dfdfdfdfdfdf
ptp
dddddd
u
ffffppfppf
ptp
tt
tt
tt
tt
−−−−
=−=
−−−−
=−=
−−−−
−=−=
−−−−
=−=
δδ∆
δδδδ∆
δδδδδδ∆
∆
, (5.7)
in which 0j0j δ,p is member force, nodal displacement before tension. It is interesting
to remark that Eq. (5-7) is the incremental control force/displacement of prestressed
member 1 (i.e. ∆p1). Its effect carries over on prestressed member 2 (i.e. ∆p2) through
the mutual influence of the whole structure, termed as the stage effect as the second
term of the right-hand side of Eq. (5.8).
( ) ( )[ ]( ) ( )
( ) ( )[ ]( ) ( )
( ) ( )[ ]( ) ( )
( ) ( )[ ]( ) ( )01121
12212211
02211011210222
0112112212211
02211011210222
0112112212211
02211011210222
0112112212211
02211011210222
ptfdfdfdfdfd
ppfdppfdu
udfdfdfdfdf
dfdfptp
uddddd
ddu
ptfffff
ppfppfptp
tt
tt
tt
tt
−+−
−−−−=−=
−+−
−−−−=−=
−+−
−−−−=−=
−+−
−−−−=−=
δδ∆
δδδδδ
∆
δδδδδ
δδ∆
∆
, (5.8)
Similarly, for a general situation of n tensioning members in the system at the
mth round tension, by back substitute mjp into m
j 1+p , mjδ into m
j 1+δ , for n times and
rearrange (Zhou et al., 2010a), the set of n equations based on F’, D’, DF’, and FD’
can be expressed in matrix form as Eq. (5.9),
( ) ( )( ) ( ) 1111
1111
−−−−
−−−−
−=−−=−
−=−−=−mn
mt
mn
mmn
mt
mn
m
mn
mt
mn
mmn
mt
mn
m
;..;
ppptFD'.uDF'uD'ppptF'.
δδδ
δδδ, (5.9)
whose coefficients are expressed in Eq. (5.10), in which mjlf → and m
jld → stand for the
algebraic sum of IFM coefficients that consist of all paths from l to j; the odd term is
negative and the even term is positive, i.e. mm ff 3223 =→ , mmmm f.fff 21323113 −=→ .
∑∑
∑∑
+=→
+=→
+=→
+=→
−=−=
−=−=
n
jljl
mkl
mkj
m'kj
n
jljl
mkl
mkj
m'kj
n
jljl
mkl
mkj
m'kj
n
jljl
mkl
mkj
m'kj
dfdfdfd;fdfdfdf
dddd;ffff
11
11, (5.10)
127
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 127
Therefore, n unknown control values of the particular mth tensioning round to
achieve the target stage, i.e. tm or um, can be obtained by directly solving Eq. (5.9).
Unfortunately, this approach is futile when the determinant of IFM is equal to zero,
e.g. one particular case is when all the IFM’s coefficients are equal to 1, in a symmetric
structure which all members are symmetricaly prestressed as illustrated in Figure 5.3.
Figure 5.3. A particular case in which the IFMs are singular
Hence, when member 1 is prestressed, the change in member force/displacement
due to tensioning member 2 is always equal to the one of member 1. Specifically, the
denominators of Eqs. (5.7) & (5.8) which are the determinants of the corresponding
IFMs are equal to zero. Another example of this particular case can be found in section
5.7.3.
Apart from no reliable solution available for those particular cases, another
drawback of the direct solving method is to solve a set of n equations of n unknowns,
i.e. a large amount of calculation work is indispensable, and including n-time back-
substitution and factorization, and a set of complicated equations is resulted in. In this
sense, the iterative solving method as discussed in the following section is therefore
preferable.
5.6.3 Iterative solving method
The iterative solving method reaches the solution step-by-step reliably; even can
yield the solutions for the futile particular cases mentioned in section 5.6.2. Further,
this method can be implemented numerically, which can make full use of the
computational technology. Similarly, there are four types of IFM, which can be
subdivided into two categories: one-criterion and two-criterion. The iterative solving
procedure can be formulated in conjunction with the governing equation of Eq. (5.4)
for a specific mth tensioning round as follows:
Step1. For the first iteration, i = 1, the control values (member forces/nodal
displacements) are assumed equal to the target values, the unbalance between the
128 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
controls and target values can be eliminated through the subsequent iterations.
tmj
it
mj
i , δ== upt , (5.11)
in which mj
i t and mj
i u are the tensioning control force and displacement vector
respectively.
Step2. The member forces and displacements induced after tensioning each
member/batch, mj
i p and mj
i δ , are then computed for all n members/batches.
Step3. The difference between the member forces and/or displacements after
prestressing n tensioning members, ni p and n
i δ , are checked using Eq. (5.12). Only
one criterion needs to be checked when using one-criterion IFMs, whereas both criteria
are controlled when using two-criterion IFMs.
mt
mn
im
mt
mn
imp ,
δδ
εε −=−= 11 δpp
(5.12)
Step4. If the deviation is larger than a required tolerance, the tensioning control
values are adjusted by summing up with the deviated values as in Eq. (5.13) and repeat
Step2 for the (i+1)th iteration.
( ) ( )mt
mn
imimimt
mn
imimi , δδ −+=−+= ++ uupptt 11 (5.13)
The procedure is repeated until convergence is detected. The numerical solution
procedures for both one-criterion and two-criterion IFMs are shown in Figure 5.4 &
Figure 5.5, respectively.
Further, the procedure of the mth tensioning round from Step1 to Step4 is
repeated until the total number of tensioning round r is reached. The determinant of
IFM is an important parameter that controls the convergent rate of the analysis, and
which types of IFM should be used for a particular situation. The discussion and
methodology to obtain the optimal tensioning scheme are given in section 5.7.
129
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 129
Figure 5.4. Iterative solution procedure using one-criterion IFMs
5.7 NUMERICAL VERIFICATIONS
This section demonstrates the applications of different types of IFM under the
various installation sequence and tensioning schemes (i.e. batched and repeated
tensioning process) to reach a target prestressed state so that the optimal tensioning
scheme, i.e. minimum inputted pre-tension and minimal construction cost, can be then
chosen. The singly and doubly symmetric structural systems include frame column,
arch bridge, space grid and hybrid structure, in which the members are prestressed
symmetrically or asymmetrically. It notes that the bold and underlined numbers in all
tables represent the control values and the target values, respectively.
130 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Figure 5.5. Iterative solution procedure using two-criterion IFMs
5.7.1 Frame column
In this example, application of F and FD matrices is demonstrated for a singly-
symmetric frame column. Installation sequences including batch by batch and all
simultaneously as well as the batched and repeated tensioning schemes with the load
application stage at the end or in the middle of the tensioning stage are studied.
A frame column under compression load in Figure 5.6 was investigated under
four tensioning schemes as stated in Table 5.1. The member forces (from F & FD
matrices) and nodal displacements (from FD matrix) according to different schemes
are tabulated in Table 5.2 & Table 5.3respectively. Only the final analysis results of
the prestressed members are given for simplicity.
131
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 131
Figure 5.6. The structural model of frame column and applied load
Scheme 1(a) shows that accurate control values are achieved right at the first
iteration when employing FD matrix as compared with at least 4 iterations required
when employing DF matrix. In Scheme 1(d), as the determinant of F matrix is small,
slow convergence is noticed with up to 61 iterations required to reach the same
accuracy as compared with only 2 iterations required by employing FD matrix. The
application of FD matrix is therefore found to be more effective and efficient, which
unfortunately is not covered in the previous literature.
Further, the required tensioning control forces and displacements in schemes
1(b) & 1(c), symmetric pre-tensions, are found to be the smallest among the 4 schemes
as justified in Table 5.2 & Table 5.3.
132 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 5.1. Construction sequences of frame column
Construction sequence
Step Assemblage member jth Tensioning member jth Target force pt
Scheme 1(a)
1 11 11 20
2 12 12 10
3 13 13 20
4 14 14 10
5 Apply dead load
Scheme 1(b)
1 11.14 11.14 20
2 12.13 12.13 10
3 Apply dead load
Scheme 1(c) 1 11.12.13.14 11.12.13.14 20
2 Apply dead load
Scheme 1(d) 1 (m =1) 11 11 10 2 12 12 5 3 13 13 5 4 14 14 10
5
Apply dead load 6 (m =2)
11 20
7 12 10 8 13 10 9 14 20
133
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 133
Table 5.2. Member forces in tensioning members of frame column (kN)
Scheme 1(a) Member 11 12 13 14
Assemble and tensioning member:
11 19.78
12 3.62 11.07
13 8.16 1.78 8.54
14 7.65 3.70 3.70 7.65
Apply dead load 20.01 10.00 10.00 20.00
Scheme 1(b) Batch 1 2
Assemble and tension batch: 1 (11,14) 14.45
2 (12,13) 7.65 3.70
Apply dead load 20.00 10.00
Scheme 1(c) Member 11 12 13 14
Assemble and tension member
7.64 3.71 3.71 7.64
Apply dead load 20.00 10.00 10.00 20.00
Scheme 1(d) Member 11 12 13 14
Assemble and first-time tension member:
11 (m=1) 9.89
12 1.81 5.54
13 4.08 0.89 4.27
14 3.82 1.85 1.85 3.82
Apply dead load 16.18 8.15 8.15 16.18
Second time tensioning member:
11 (m=2) 47.54 -11.71 16.00 14.10
12 5.42 41.06 -22.91 30.76
13 22.08 2.15 29.86 -11.36
14 20.00 10.00 10.00 20.00
134 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 5.3. Nodal displacements in tensioning members of frame column (mm)
Scheme 1(a) Node 10 9 8 7
Assemble and tension member: 11 -45.63 -66.86 -48.33 -16.89
12 -45.76 -97.00 -81.22 -30.13
13 -45.72 -97.10 -97.02 -40.20
14 -45.72 -97.08 -97.08 -45.72
Apply dead load -45.62 -97.01 -97.01 -45.62
Scheme 1(b) Node 10, 7 9, 8
Assemble and tension batch: 1 -45.67 -84.16
2 -45.73 -97.10
Apply dead load -45.63 -97.03
Scheme 1(c) Node 10 9 8 7
Assemble and tension member
-45.67 -97.10 -97.10 -45.67
Apply dead load -45.63 -97.03 -97.03 -45.63
Scheme 1(d) Node 10 9 8 7
First time tensioning member: 11 (m =1) -22.82 -33.43 -24.17 -8.45
12 -22.88 -48.51 -40.62 -15.07
13 -22.86 -48.56 -48.53 -20.10
14 -22.86 -48.55 -48.55 -22.86
Apply dead load -22.76 -48.48 -48.48 -22.76
Second time tensioning member: 11 (m =2) -45.31 -48.63 -48.33 -22.68
12 -45.65 -96.63 -48.75 -22.55
13 -45.51 -97.05 -96.75 -22.89
14 -45.53 -96.97 -96.97 -45.53
5.7.2 Arch Bridge
This example demonstrates the application of D and DF matrices for a singly-
symmetric structure; Installation sequence including batch by batch and all
simultaneously, as well as batched and repeated tensioning schemes, at the end of
which load application stage is implemented, are studied.
An arch bridge structure in Figure 5.7 under two symmetric tensioning schemes
as summarised in Table 5.4 was studied.
135
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 135
Figure 5.7. Geometry and applied load
Table 5.4. Construction sequence of Arch bridge structure
Construction sequence
Step Assemble batch jth Tension batch jth δt (mm)
Scheme 2(a)
1 8,11 8,11 2 9,10 9,10 3 Apply dead load 0
Scheme 2(d)
1 (m =1) 8,11 8,11 2.8
2 9,10 9,10 4.5
3 (m =2) 8,11 0
4 9,10 0
5 Apply dead load
The nodal displacements (from D & DF matrices) and member forces (from DF
matrix) are given in Table 5.5 & Table 5.6. The combination of installation and
tensioning batch by batch of Scheme 2(a) is found to be optimal, as it requires smaller
tensioning control force. It should be noted that the change in member force is more
obvious than the change in nodal displacement, while Table 5.5 only presents the
vertical nodal displacement.
136 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 5.5. Nodal displacements of hangers (mm)
Scheme 2(a) Node 8.11 9.10
Assemble and tension batch: 8,11 5.79 9.80 9,10 5.60 8.99
Apply dead load 0.00 0.00
Scheme 2(b) Node 8.11 9.10
First time tension batch: 8,11 (m =1) 2.89 4.90 9,10 2.80 4.50
Second time tension batch: 8,11 (m =2) 4.76 5.32 9,10 5.60 8.99
Apply dead load 0.00 0.00
Table 5.6. Member forces in hangers (kN)
Scheme 2(a) Batch jth 8.11 9.10
Assemble and tension batch: 8,11 10.34
9,10 12.88 -1.43
Apply dead load 31.80 17.73
Scheme 2(b) Batch jth 8.11 9.10
First time tension batch: 8,11 (m =1) 5.17
9,10 6.44 -0.71
Second time tension batch: 8,11 (m =2) 24.44 -7.94 9,10 12.88 -1.43
Apply dead load 31.80 17.73
5.7.3 Space grid structure
This example demonstrates the accuracy of this approach and the choice of a
particular type of IFM for a particular situation. The space grid structure previously
studied by Dong and Yuan (2007) as shown in Figure 5.8, was re-investigated. The
peculiar feature of this system is doubly structural symmetry, and possible tensioning
symmetry with 4 bottom chords are prestressed to enhance its load carrying capacity.
The structure is subjected to a uniform vertical load 1.0kN/m2.
The tensioning control forces of the six schemes obtained in Table 5.7 are in
good agreement with the solutions of Dong and Yuan (2007), from which the
maximum discrepancy found in Scheme 3(b) is only 3.4%. The tensioning control
displacements of the six schemes are also obtained in Table 5.8.
137
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 137
Figure 5.8. The perspective view of space grid structure (Dong & Yuan, 2007)
The determinant of the D matrix of Scheme 3(a) is close to zero because of its
structural symmetry. Hence, slow convergence was noticed. Thus, a comparison of the
convergent rate between DF and D matrices of the same scheme 3(a) is present in
terms of the displacement control values as shown in Figure 5.9. It is remarked that
although the average error of the analysis employing D matrix is not too large after 30
iterations, the deviation from displacement control values of each member still remains
high (more than 30%). On contrary, employing DF matrix can yield accurate
displacement control values within a few iterations as demonstrated in Figure 5.9.
Similarly, the determinants of D matrices of schemes 3(b) & 3(e) are zero exactly,
because of both structural as well as tensioning symmetries. Nodal displacements and
tensioning control displacements can only be obtained by DF matrix for schemes 3(b)
& 3(e) respectively. Therefore, the two-criterion IFMs are considered as more
effective than the one-criterion IFMs.
Figure 5.9. Convergent rate of the analysis using D and DF matrices in Scheme 3(a)
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50
Aver
age
eror
in %
Number of iterations
D matrix
DF matrix
138 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 5.7. Member forces in tensioning bottom chords of space grid structure (kN)
First scheme Member 1 2 3 4
Assemble and tension member: 1 56.7 2 54.1 54.1 3 52.3 51.6 52.3 4 50.0 50.0 50.0 50.0
Second scheme Batch 1.3 2.4
Assemble and tension batch: 1.3 54.7
2.4 50.0 50.0
Third scheme Member 1 2 3 4
Assemble and tension member simultaneously 50.0 50.0 50.0 50.0
Four scheme Member 1 2 3 4
Apply 1/2 first vertical load 22.7 22.7 22.7 22.7
Tension member: 1 27.9 22.7 22.7 22.7
2 27.6 27.6 22.7 22.7
3 27.5 27.4 27.5 22.7
4 27.3 27.3 27.3 27.3
Apply second 1/2 vertical load 50.0 50.0 50.0 50.0
Fifth scheme Batch 1.3 2.4
Apply 1/2 first vertical load 22.7 22.7 Tension batch: 1.3 27.5 27.3
2.4 22.5 27.3
Apply second 1/2 vertical load 50.0 50.0
Sixth scheme Member 1 2 3 4
Apply 1/2 first vertical load 22.7 22.7 22.7 22.7
Tension member simultaneously 27.3 27.3 27.3 27.3
Apply second 1/2 vertical load 50.0 50.0 50.0 50.0
139
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 139
Table 5.8. Nodal displacements in tensioning bottom chords of space grid structure (mm)
Scheme 3(a) Node 1 2 3 4
Assemble and tensioning member: 1 -0.31 -0.31 0.27 0.27
2 -0.04 -0.60 -0.04 0.52
3 0.22 -0.33 -0.33 0.22
4 -0.08 -0.08 -0.08 -0.08
Scheme 3(b) Node 1.3 2.4
Assemble and tension batch: 1,3 -0.04 -0.04
2,4 -0.08 -0.08
Scheme 3(c) Node 1 2 3 4
Assemble and tensioning member simultaneously -0.08 -0.08 -0.08 -0.08
Scheme 3(d) Node 1 2 3 4
Apply first 1/2 vertical load 1.86 1.86 1.86 1.86
Tensioning member: 1 1.83 1.83 1.88 1.88
2 1.86 1.80 1.86 1.91
3 1.88 1.83 1.83 1.88
4 1.85 1.85 1.85 1.85
Apply second 1/2 vertical load 3.71 3.71 3.71 3.71
Scheme 3(e) Node 1.3 2.4
Apply first 1/2 vertical load 1.86 1.86
Tension batch: 1,3 1.86 1.86
2,4 1.85 1.85
Apply second 1/2 vertical load 3.71 3.71
Scheme 3(f) Node 1 2 3 4
Apply first 1/2 vertical load 1,2,3,4 1.86 1.86 1.86 1.86
Tensioning member simultaneously 1.85 1.85 1.85 1.85
Apply second 1/2 vertical load 1,2,3,4 (m=2) 3.71 3.71 3.71 3.71
140 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
5.7.4 Hybrid structure
The hybrid frame, first studied by Zhuo and Ishikawa (2004) in Figure 5.10, was
re-investigated in order to demonstrate the reliability and efficiency of this approach.
The main frame is constructed of tube 300x300x12mm; whereas struts and braces of
pipe φ114.3x5.6mm. The elastic modulus of steel tube and pipe are 210kN/mm2. The
section of prestressed members is φ30 with elastic modulus 160kN/mm2. There are
three construction stages as summarised in Table 5.9. The results according to the
present approach are shown in Table 5.10 that the tensioning control forces are in good
agreement with those from Zhuo and Ishikawa (2004) listed in Table 5.11, whose
average deviation is less than 4%. Besides, the iterative solving method is very
efficient as shown in Figure 5.11, of which high accuracy can be achieved only after
two or three iterations. As a result, a large amount of calculation work can be reduced
as compared with the study of Zhuo and Ishikawa (2004). The approach based on
iterative solving method is therefore efficient and reliable. Further, the force-based
approach always compromises with others with 4% difference in terms of accuracy.
Figure 5.10. The perspective view of hybrid frame (Zhuo & Ishikawa, 2004)
Table 5.9. Construction sequence of Hybrid structure
Construction stage
Step Description Applied load/Control forces
or Target force
1 Construct the main frame and its braces 1.05kN/m
1 ~ 8 Assemble and tension C1 ~ C8 in turn 100kN
2 Erect roof 0.4kN/m2
3 1 ~ 8 Re-tension C1 ~ C8 in turn 200kN
141
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 141
Figure 5.11. The convergent rate of the analysis using F and DF matrices
Table 5.10. Member forces of the prestressed members of Hybrid frame according to the present
method (kN)
Stage Member 1 2 3 4 5 6 7 8
Stage1. Assemble and first-time tension member:
1 100.0
2 86.9 100.0
3 79.7 86.5 100.0
4 81.6 78.2 86.4 100.0
5 82.4 79.4 78.1 86.5 100.0
6 82.9 80.1 79.3 78.2 86.4 100.0
7 82.9 80.6 80.0 79.5 78.0 86.4 100.0
8 82.9 80.6 80.5 80.3 80.0 78.0 86.6 100.0
Stage 2. Erect the roof 107.8 110.6 110.9 111.1 110.8 108.3 116.6 124.8
Stage3. Second time tension member:
1 221.8 95.3 101.2 113.4 111.8 108.9 116.6 124.9
2 206.3 226.0 85.1 102.2 113.2 109.7 117.2 124.9
3 195.9 208.8 226.4 84.9 101.5 111.1 118.1 125.5
4 198.3 197.0 209.0 226.8 84.3 99.4 119.6 126.5
5 199.3 198.6 197.3 209.6 225.8 82.1 107.8 129.0
6 200.0 199.4 198.7 197.9 208.5 223.2 90.7 118.6
7 200.0 200.0 199.5 199.2 198.1 208.1 212.8 104.1
8 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0
-16
-14
-12
-10
-8
-6
-4
-2
00 5 10 15
Aver
age
eror
in %
Number of iterations
F matrix DF matrix
142 Chapter 5: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 5.11. Member forces of the prestressed members of Hybrid frame according to Zhuo &
Ishikawa (kN)
Stage Member 1 2 3 4 5 6 7 8
Stage1. Assemble and first-time tension member:
1 100.0
2 84.7 100.0
3 78.4 81.2 100.0
4 83.8 71.8 81.4 100.0
5 84.7 76.4 72.2 81.5 100.0
6 85.2 77.4 76.7 72.2 81.1 100.0
7 84.9 78.1 77.8 76.7 71.4 81.3 100.0
8 84.9 77.8 78.4 77.7 77.5 72.8 84.0 100.0
Stage 2. Erect the roof 114.1 113.0 113.2 116.3 113.1 107.6 119.2 129.2
Stage3. Second time tension member:
1 220.8 95.6 104.1 119.8 114.1 108.4 118.8 129.2
2 201.4 232.7 80.3 105.5 120.0 109.6 119.9 128.7
3 190.5 207.0 232.9 79.5 105.0 115.9 121.2 129.6
4 198.3 191.5 206.8 234.7 79.2 100.9 127.6 130.8
5 199.5 197.9 191.8 208.9 234.4 74.8 112.2 138.6
6 200.4 199.3 198.2 193.6 207.9 230.1 86.0 127.5
7 200.0 200.3 199.4 199.2 194.5 207.7 214.8 109.4
8 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0
5.8 DISCUSSION AND CONCLUSION
This study presents comprehensively the linear elastic analysis to investigate the
interdependent behaviour among prestressed members in the entire prestressed steel
structures. In summary, the mutual influences of all prestressed members in the system
under various batched and repeated tensioning schemes and various installation
sequences are studied based on four types of IFM, by either the direct or iterative
solving method in order to meet the design requirement for a specific target prestressed
state.
From the above examples, the two-criterion IFMs, which control both force and
displacement, is considered as more effective than the one-criterion IFMs; especially
the force-displacement based (FD matrix), which is firstly introduced in this study. It
is particularly true when the determinant of a single criterion IFM is zero or close to
zero because the coefficient of FD matrix is significant without rounding error. Under
this circumstance, IMF is singular, and thus the direct solving method is completely
invalid. Fortunately, the iterative solving method can lead to the final solutions but
143
Chapter 4: Linear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 143
with slow convergence. In this regard, the iterative solving method is more reliable.
Further, this study remarks that, for a particular situation as indicated by example 5.7.3,
a symmetric structural system with symmetric tensioning process leads to the null
determinant of any IFMs, which cannot be solved by the direct solving method, but it
does not mean deficiency. Instead, this particular situation implies that no mutual
influence whatsoever from other tensioning members, and therefore any target values
imposed to the symmetric system is absolutely valid without disturbing other
tensioning members. Hence, the determinant of IFMs is an important criterion in the
analysis.
For the sake of lowering construction costs by reducing the tensioning control
values, the combination of installation and tensioning batch by batch is necessary for
an optimal tensioning scheme, especially keeping the batch of tensioning members
symmetric (i.e. determinant of IFMs is small) can mitigate the large differential
deformations in a symmetric structure. In summary, the superiority of IFMs can be
generally denoted by FD > DF > F > D matrices, and structural and tensioning
symmetry often provoke to the optimal scheme. Therefore, the analyst needs to
understand the characteristics of the prestressed system, the construction conditions,
and the tensioning process to decide which type of IFM is the most suitable for a
particular situation.
Overall, the interdependent behaviour among prestressed members in the system
under the effects of construction sequence or tensioning sequence can be investigated
reliably, effectively and efficiently by IFM method presented in this chapter. It should
be pointed out that this influence matrix approach is set up based on the principle of
linear superposition. Consequently, nonlinear geometric or inelastic material
behaviour, which may take place during construction, could not be accounted for
properly. Therefore, the following chapter presents an iterative solution approach for
the pre-tension process analysis, which is capable of capturing all the geometric and
material nonlinearity if any under the construction phase.
Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 144
Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
6.1 INTRODUCTION
The application of pre-tension in spatial steel structures are more and more
widespread in recent decades thanks to many advantages such as increasing its load
carrying capacity, improving the structural rigidity, and reducing the structural
deformation. Unfortunately, under the circumstances of the presence of many
prestressed members in the system, it is difficult to prestress all members
simultaneously especially in complicated structures or when the control
forces/displacements are not the same, it makes the batched and repeated tensioning
schemes are unavoidable. As a result, when one member is tensioned to its target force
and/or displacement, the already achieved target values in other prestressed members
will immediately change due to the interdependent behaviour of all members in the
system. Therefore, the key problem is to determine the prestressing forces required
and/or displacements of all prestressed members in the system such that by tensioning
all prestressed members to their control values, their final forces and/or displacements
in the system will reach the target values successfully instead of tensioning by trial and
error. In order words, once the predicted prestressing force and/or displacement is
applied on each tensioning member according to the predetermined pre-tension
scheme, its final force and/or displacement at the target state must reach its target value
once the pre-tension process is accomplished.
Past research proposed analysis approaches to determine the prestressing forces
required and/or displacements in order to achieve a target prestressed state should be
mentioned are the studies of Dong and Yuan (2007) of space grid structures; Zhou et
al. (2010b) of arch supported prestressed grids structures; Zhuo and Ishikawa (2004)
of hybrid structure. In these studies, the required prestressing forces and/or
displacements and the tensioning control forces and/or displacements are often
determined during the design stage based on the theoretical structural model.
145
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 145
Unfortunately, due to constructional displacements, there often exists a deviation
between a member’s original geometry ij and its deformed one i’j’ as shown in Figure
6.1(c). It makes the prestressed member could not be installed into the structure
normally due to the member length has changed from its original length ijL into 'j'iL
and its orientation also deviated from its original one. In case, the prestressed member
needs to reach a target force after finish tensioning, the required prestressing force
needed 'piF is obviously different with the one needed based on its original geometry
piF as shown in Figure 6.1(c). Further, as the member length has changed an amount
ij'j'i LLL −=∆ , this ‘initial member deformation’ or the so-called ‘lack of fit’ in turn
induces a requirement of a ‘constructional initial force’ mfin in order to pre-tension or
pre-compress the corresponding member to resume its original length to be able to fit
into its designed position. This constructional initial force may, in turn, trigger the
premature material yielding during construction. A detailed discussion of this
particular effect is given in section 3.5.4. Consequently, the target forces and/or
displacements could not be achieved and numerous cyclic pre-tension on site could
not be avoided in order to finally obtain the design requirements. It obviously increases
the construction time and cost.
(a) Original undeformed geometry (b) The deformed geometry
(c) The change of equivalent prestressing forces applied on the system due to the change of
geometry
Figure 6.1. Effects of construction sequence on prestressing forces
Due to constructional displacements has direct effects on structural nodal
coordinates, it, in turn, influences prestressed member forces. As only a small change
146 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
in nodal coordinate can induce a large change in member forces, especially when
lacking temporary supports and stability precautions. In order to search for the
prestressing forces required to achieve a desired prestressed state, this study presents
an iterative solution approach, which takes into account the nonlinearities of
prestressed steel structures at the construction phase by embedding the nonlinear
construction stage analysis proposed in chapter 3. Consequently, this iterative
approach can account for the effects of displacements incurred within a construction
stage and in between two constitutive stages. The numerical verifications show that a
target prestressed state can successfully be achieved and that constructional
displacements directly affected the required prestressing forces and alter structural
behaviour. By accounting for this particular effect, the required prestressing forces
predicted by the present method are more accurate and the errors between the
measured member forces after finished tensioning and the desired target values can be
reduced, which in turn reduces the number of cyclic pre-tension on the construction
site. As a result, construction time and cost can also be reduced. This chapter
accomplished task 4 and together with Chapter 5 fulfilled Objective 2 of this research.
Once again, previous researchers have tried to reduce the errors of the predicted
required prestressing values (Zhou et al., 2010a; Zhou et al., 2010c; Zhuo et al., 2008).
However, these approaches had to carry out after the pre-tension phase, as they most
often needed to rely on the measured values of prestressing forces and/or
displacements on site to adjust the tensioning values. Hence, they are considered as
somewhat passive approaches as they do not actively solve the core of the problems
that affect the precision of the prestress. Besides, e.g. in the study of Zhou et al.
(2010c), the tensioning control forces were re-calculated using the influence matrix
that was set up based on the measured values of cable forces. It makes Zhou’s approach
somewhat limited to the linear elastic range only.
Overall, previous approaches often neglected constructional displacements and
based on the original undeformed geometry in the pre-tension process analysis. As a
result, all the change in nodal coordinates which induces the change in member length
during construction is unaccounted for which create significant errors in the predicted
required prestressing values and the tensioning control values. Consequently, the
target forces/displacements could not be achieved and numerous cyclic pre-tension on
site could not be avoided. It obviously increases the construction time and cost.
147
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 147
To this end, effects of construction sequence on the required prestressing forces
were focused in this study. The higher-order element formulation developed by Iu and
Bradford (2015) was resorted to capturing the nonlinear geometric effects (including,
P-δ & P-∆ effect, large deformation and buckling). The comprehensive illustration of
the higher-order elastic element stiffness formulation, as well as its efficacious and
reliable convergence, can be found in the study of Iu and Bradford (2010) and
summarised in Section 3.2.1; whereas the profound implication of the element load
effects was discussed in the study of Iu (2015). At the same time, the material
nonlinearities (including, gradual yielding, full plasticity, and strain-hardening effect
due to interaction) were reliant on the refined plastic-hinge approach with the
comprehensive formulation and its details were discussed in Iu and Bradford (2012b)
and summarised in Section 3.2.2. Further, since the geometry of newly built structure
keeps changing depending on the constructional displacements at previous stages, the
positioning technique to locate new coordinate of a member at the next construction
stage, as well as the methodology required to locate the new geometry of a newly
erected structure introduced in section 3.3, were employed. An iterative solution
approach to search the prestressing forces required to achieve a target prestressed state
accounted for constructional displacements is then presented in section 6.2. Finally,
the present method was employed to analyse the pre-tension process of different types
of prestressed steel structures in section 6.4. The results were validated by SAP2000
or general finite element method, which supports construction stage analysis.
6.2 ITERATIVE SOLUTION PROCEDURE FOR PRESTRESSED TARGET FORCES ACCOUNTED FOR CONSTRUCTION STAGE EFFECTS
This section is to demonstrate the iterative solution procedure to search for the
prestressing forces required in order to achieve a target prestressed stage. Iterative
solution approach is used in this study as this approach can reach the solution at the
target state step by step reliably (Nguyen & Iu, 2015a; Zhou et al., 2010b; Zhuo &
Ishikawa, 2004). Moreover, this approach can be implemented numerically, which can
make full use of the computational technology.
The effect of prestress is simulated by the equivalent load approach because this
approach provides a clear picture of the prestressing forces applied to the prestressed
structures and it has been widely used by previous researchers. The detailed discussion
148 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
is given in Section 2.5. At the same time, in order to be capable of accounting for the
effects of constructional displacements incurred within each construction stage and in
between two constitutive stages, the nonlinear construction stage analysis accounted
for constructional displacements at previous construction stages proposed in chapter 3
is employed. To trace the nonlinear equilibrium path, the Newton-Raphson method is
used mostly for its reliable convergence, so this section only refers to the Newton-
Raphson method.
As the effect of prestress is simulated as equivalent nodal prestressing forces, the
main feature of this iterative solution procedure is to search for the prestressing forces
required that can ensure the correspondent prestressed members can reach their
designed target forces tp at the end of the construction phase.
Step 1. For the first prestressed iteration, s = 1, prestressing forces mp
s f are
assigned initial values dependent on the designer’s experience; normally they can be
set equal to the correspondent target forces as in Eq. (6.1); the deviation between the
initial and desired prestressing forces will be eliminated through iteration.
tTm
ps pTf = (6.1)
in which, T and TT are the transformation matrix and its transpose; m is the current
construction stage.
Step 2. The construction stage analysis accounted for constructional
displacements is then implemented. One of the emphases of this study is to investigate
the influence on the prestressing forces required to achieve a target prestressed state
considering the deformed geometry due to the constructional displacements from stage
to stage. Hence, the geometry of a structure mg
s u at the current stage composes of the
deformed geometry 1-ms u at the previous stage due to the constructional displacements
and the change of geometry of a structure mp
s u because of the positioning at the current
stage based on the principle of the minimum member length presented in section 3.3.1.
Consequently, the geometry of a structure mg
s u at current stage for the s prestressed
iteration is written as,
mp
s1-msmg
s uuu += , (6.2)
When the geometry of a structure at construction phase complies with the
deformed geometry due to constructional displacements and the original member
149
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 149
length is preserved, the geometry of a structure at the current stage may stress up the
member as commonly termed as the constructional initial force; in particular, when
the newly built structure is indeterminate. Thus, the constructional initial force min
s f (if
any) on the newly built member in the global coordinate is given as,
min
se
Tmin
s LkTf ∆= , (6.3)
in which min
s L∆ is a vector of the change in member length at the axial degree of
freedom at the current stage; sek is the element stiffness matrix. Thus, the nodal force
vector ms f keep accumulating at the current stage is obtained as,
mt
mp
smin
sms ffff ++= , (6.4)
in which mft is the nodal force vector due to the loads imposed on the built structure at
the current mth stage. It should be noticed that this load vector is unchanged throughout
the entire prestressed iteration. The global tangent stiffness mKT of the built structure
at the current stage is then re-assembled based on the ‘Build’ and ‘Kill’ technique as
discussed in section 3.4.1. The incremental displacement mu∆s and element resistance
vector mR∆s at the current stage are respectively written as,
msT
mms fKu ⋅=∆ −1 (6.5)
mss
ms . uTkR ∆⋅=∆ (6.6) in which ks is the secant stiffness formulation (as discussed in details in Iu (2016a)).
And the refined plastic-hinge formulation is also incorporated into the secant stiffness
formulation ks, which is comprehensively mentioned in Iu and Bradford (2012b). The
total element resistance mRs and total displacement mus at the current stage can then
be obtained respectively,
msmg
smsmp
smsms uuuuuu ∆+=∆++= −1
(6.7)
( )epmms-1msms f++= RRR ∆ , (6.8)
Therefore, the unbalanced force of a structure at the current stage is obtained as,
msmsms Rff −=∆ (6.9)
If the nodal displacement mu∆s and the unbalanced force mf∆s are satisfied the
convergent criteria at the mth current stage, the above procedure from Eqs. (6.2) ~ (6.9)
is repeated for the next (m+1)th construction stage, at which the load level
accumulating at (m+1)th stage are written as,
150 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
tsmsms1ms . fff λ+=+ , (6.10)
in which tfs is the total nodal loads for whole construction sequence in order to trace
the whole equilibrium path; mλs is the total construction load factor, which is
commensurate to the load level at the mth stage given as,
tsmsms nS=λ , (6.11)
in which ∑=
=tn
i
mt
smc
sms ffS1
; mt
s f is the cumulative force, including dead and
constructional loads, up to the load level at the current stage; mt
s f and ts n are
respectively the total nodal forces and the total number of nodal forces about all
degrees of freedom of the whole construction sequence. As a result, the equilibrium
path for the whole construction sequence can be traced in the reference of the total
construction load factor until the total construction stage Ncs is reached. Further, due
to the prestressing forces are adjusted for each prestressed iteration, and they can be
different for each prestressed iteration. It implies that the total construction load level
is also adjusted by the iterative procedure.
Step 3. At the end of this construction stage analysis, prestressed member forces
are checked against their correspondent target values,
t
eps
p pf
−= 1ε (6.12)
If the deviation pε is larger than a set tolerance, the equivalent nodal prestressing
forces for the next prestressed iteration need to be amended by summing up with the
deviated values in order to ensure convergence as,
( )eps
tTm
psm
p1s fpTff −+=+ (6.13)
Return to Step 2, the construction stage analysis is then re-implemented under
these amended prestressing forces and the correspondent prestressed member forces
are checked again in step 3. The whole procedure is repeated until convergence is
detected. Once the convergence is reached, the prestressing forces required to reach
the target forces are determined. The entire iterative solution procedure is summarised
in the flowchart of Figure 6.2.
151
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 151
Figure 6.2. Iterative solution procedure for target prestressed forces accounted for construction stage effects
It should be noticed that in the present approach, there are 2 different iterative
procedures.
First is the ‘prestress cycle’ to search for the required prestressing force, named
as the outer cycle as it encompassed other cycles, i.e. the construction cycle and the
numerical analysis cycle illustrated in Figure 6.3. The outer cycle is a direct iterative
procedure in which the prestressing forces are assumed, Eq. (6.1), and used to calculate
the prestressed member forces and nodal displacements, which are then used to re-
calculate the new prestressing forces, Eq. (6.13), and re-compute the new member
152 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
forces and nodal displacements. This process is repeated until the member forces and
nodal displacements are very close to the target values.
Second is the ‘numerical solution procedure’ to search for the nonlinear
equilibrium path, named as the inner cycle. The inner cycle is an incremental iterative
procedure implemented within each construction stage under the correspondent
loadings applied which are divided into a series of small load increments within that
correspondent stage in order to capture properly the nonlinear structural behaviour.
Figure 6.3. Iterative solution process for target prestressed forces
In other words, the structural analysis by the incremental iterative approach (i.e.
the numerical solution cycle) accounts for the sharing load phenomenon of all the
members in the system within each construction stage. On the contrary, the mutual
influence among prestressed members in different construction stages is accounted by
153
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 153
the direct iterative approach (i.e. the prestress cycle) when the unbalance of the final
prestressed member forces and their target values are calculated and employed to
adjust the new inputted prestressing forces for the next prestressed iteration.
6.3 A SIMPLE ILLUSTRATION OF THE PRESENT ITERATIVE APPROACH
A simple illustration of this iterative solution approach for a three-storey frame
is shown in Figure 6.4. The frame is constructed in three stages with each storey
included one prestressed girder in each stage. The target forces of the girders at the
end of the whole construction phase are 321 ttt p&p,p respectively.
Figure 6.4. Illustration of the iterative solution process for a three-storey frame under three construction stages
154 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
6.4 NUMERICAL VERIFICATIONS
In this numerical verification, the present iterative solution method is employed
to determine the required prestressing forces for different structural types included
space grid, frame and arch bridge structure under different tensioning schemes
included one by one, batch-by-batch and simultaneous schemes. For validation, the
required prestressing forces obtained from the present method were re-applied on the
structures according to the construction sequence and analysed by general finite
element method for final member forces and nodal displacements. Those member
forces/nodal displacements were then checked against their correspondent target ones.
In this study, SAP2000, also resorted to the step-by-step technique and based on the
undeformed geometry (Und), was employed, i.e. no positioning technique being
applied in SAP2000 for these examples.
6.4.1 Space grid structure
The space grid structure in Figure 6.5, previously studied the linear elastic
behaviour by Dong and Yuan (2007) and by IFM approach as presented in section
5.7.3, was re-investigated accounted for the construction stage effects on the required
prestressing forces, namely the interdependent behaviour of space grid structure. In
this study, the one-by-one (Scheme 1) and batch-by-batch installation and tensioning
(schemes 2 & 2A) schemes as summarised in Table 6.1 were studied and compared
with the simultaneous scheme (SM) with the whole structure constructed at once and
under full loadings to investigate the construction stage effects on the structural
behaviour. As the main frame is constructed in the first construction stage together
with all its structural nodes, the nodal positioning technique is not applicable in this
case.
155
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 155
Figure 6.5. Original geometry of space grid structure and the arrangement of prestressed members in
the study of Dong and Yuan (2007)
Table 6.1. Construction sequence of space grid structure
Scheme Stage Description Target force Pt (kN)
Scheme 1
1 Construct steel grid
Assemble and tension C1 50
2 Assemble and tension C2 50
3 Assemble and tension C3 50
4 Assemble and tension C4 50
Scheme 2 1 Construct steel grid
Assemble and tension C1 & C3 50
2 Assemble and tension C2 & C4 50
Scheme 2A 1 Construct the whole structure Tension C1 & C3 50
2 Tension C2 & C4 50
The prestressing forces required to achieve the final target forces were analysed
by the iterative solution procedure presented in section 6.2, which accounted for
constructional displacements are shown in Table 6.2. For validation, the required
prestressing forces obtained from the present method were re-applied on the structures
according to the construction sequence and analysed by SAP2000 for member forces
and nodal displacements, which are listed in Table 6.2 & Table 6.3 respectively. The
results show that the target forces of all prestressed members were successfully
achieved as in Table 6.2 with the tolerance of target force set as %.%t
t 50=−
=p
pff∆ .
Vertical displacement at node A, indicated in Figure 6.5, from the present method was
156 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
also checked with the results from SAP2000 shown in Table 6.3 with very good
agreement can be seen. The reliability of the present study is confirmed. It should be
mentioned that in Scheme 2A, the whole structure was constructed first and tensioned
later. The prestressing forces of C1 & C3 in step 1 created -5.14kN in C2 & C4. When
C2 & C4 were tensioned later in step 2, the final prestressed member forces are 50kN
as in Table 6.5. Hence, the prestressed member forces due to the prestress of C2 & C4
alone are 55.14kN. While in Schemes 1 & 2, pre-tension one-by-one, batch-by-batch,
and there is no other applied load, the prestressed member forces after tensioning is
created by prestress alone.
Table 6.2. The required prestressing forces and final prestressed member forces (kN)
Scheme Member Prestressed member forces after tensioning
Final prestressed member forces
Present method FEM check Present method FEM check
1 C1 56.6 56.6 50.0 49.9 C2 54.0 54.0 50.0 49.9 C3 52.2 52.1 50.0 49.8 C4 50.0 49.9 50.0 49.9
2 C1.C3 54.6 54.6 50.0 50.0
C2.C4 50.0 50.0 50.0 50.0
2A C1.C3 55.1 55.1 50.0 49.9
C2.C4 50.0 49.9 50.0 49.9
SM C1. C2. C3. C4 50.0 49.9 50.0 49.9
Table 6.3. Vertical displacements at Node A (mm).
Scheme Stage Present method FEM check
1 1 0.50 0.93
2 1.78 1.78
3 2.56 2.57
4 3.27 3.29
2 1 0.97 1.80
2 3.29 3.30
2A 1 0.89 1.60
2 3.29 3.30
SM 1 1.78 3.29
Further, the analysed results by the present method and the linear analysis results
from Dong and Yuan (2007) were also compared in Table 6.4. Very good agreement
can be seen which highlight the reliability of this approach. Insignificant differences
between the two approaches appeared in both pre-tension schemes infers that the
157
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 157
effects of the change of geometry during construction on nodal displacements and
member forces are not remarkable when the structural behaviour remains elastic.
Table 6.4. Prestressed member forces during construction of different approaches (kN).
Scheme Analysis type Member Construction stage
2 3 4 5
1 Und C1 56.59 53.97 52.19 50.0
C2 53.98 51.58 50.0
C3 52.21 50.0
C4 50.0
Dong & Yuan C1 56.68 54.07 52.26 50.0
C2 54.06 51.62 50.0
C3 52.26 50.0
C4 50.0
2 Und C1. C3 54.64 50.0
C2. C4 50.0
Dong & Yuan C1. C3 54.67 50.0
C2. C4 50.0
SM Und C1. C2. C3. C4 50.0
Dong & Yuan C1. C2. C3. C4 50.0
Further, a comparison between schemes 2 & 2A was also conducted to highlight
the effects of construction sequence on prestressing forces. The difference between
scheme 2, assemble and tension batch by batch, and scheme 2A, all members are
assembled first and then tension batch by batch later, is the change of structural
stiffness during construction. As in scheme 2, the structural stiffness in CS2 is smaller
compared with the one of Scheme 2A, a bit smaller prestressing force is required as in
Table 6.5. On the contrary, as in Scheme 2A, the whole structures is constructed first,
hence vertical displacement at node A is a bit smaller compared with the one of scheme
2 in Table 6.6. However, the deviation is insignificant in this case when the structural
behaviour still remains elastic.
Table 6.5. Prestressed member forces during construction (kN).
Scheme Member Construction stage
1 2 2 C1. C3 54.64 50.0 C2. C4 50.0
2A C1. C3 55.10 50.0 C2. C4 -5.14 50.0
158 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Table 6.6. Displacements at Node A during construction (mm)
Scheme Construction stage Present method
2 1 0.971 2 3.293
2A 1 0.889 2 3.291
6.4.2 Frame column structure
A frame column with its original geometry and section properties shown in
Figure 6.6 carrying vertical load from roof and glass façade was analysed and
accounted for the construction effects on the required prestressing forces under two
tensioning schemes as stated in Table 6.7, shown in Figure 6.6(b) & Figure 6.6(c) and
also the simultaneous scheme (SM). As the main frame was constructed in the first
construction stage together with all its structural nodes, the nodal positioning technique
was not applicable in this case.
(a) Original geometry, sectional property and vertical load
159
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 159
(b1) CS1 (b2) CS2 (c1) CS1 (c2) CS2
(b) Pre-tension scheme 1 (c) Pre-tension scheme 2
Figure 6.6. Original geometry, sectional properties, and applied load of frame column
Table 6.7. Construction sequences of frame column
Scheme Stage Description Target force Pt (kN)
1
1 Construct main frame and struts 1, 2, 3 & 4
Prestress struts 1 & 4 6.18
2 Prestress struts 2 & 3 3.15
3 Apply vertical load -50kN
2
1 Construct main frame and struts 1 & 4
Prestress struts 1 & 4 6.18
2 Assemble and prestress struts 2 & 3 3.15
3 Apply vertical load -50kN
The prestressing forces required to achieve the target forces were searched by
the iterative solution process presented as listed in Table 6.8. The correspondent
member forces and nodal displacements counter checked by SAP2000 under the
required prestressing forces obtained by the present iterative method are tabulated in
Table 6.8 & Table 6.9. The target forces of all prestressed members are successfully
achieved with the tolerance of target force set as %.%t
t 50=−
=p
pff∆ . It should be
noticed that the tolerance of target force could be increased if necessary. In that case,
the number of iterations may be increased correspondently. Moreover, vertical
displacement at node A, indicated in Figure 6.6(a), according to the present analysis is
also in good agreement with the results checked by SAP2000. It should be mentioned
that in scheme 1, the whole structure was constructed first and tensioned later. The
prestressing forces of Struts 1 & 4 in step 1 created in Struts 2 & 3 an insignificant
160 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
members force -0.03kN shown in Table 6.10. Hence, the prestressed member forces
due to the pre-tension of Struts 2 & 3 alone are 0.1kN. While in schemes 2 & SM, pre-
tension batch by batch and all at once with no other applied load, the value of
prestressed member forces after tensioning are created by prestress alone.
Table 6.8. The prestressed member forces after finished tensioning and final prestressed member
forces (kN)
Scheme Struts Prestressed member forces after tensioning Final prestressed member
forces
Present method FEM check Present method FEM check
1 1.4 0.10 0.09 6.18 6.18 2.3 -0.02 -0.02 3.14 3.14
2 1.4 0.06 0.06 6.18 6.19
2.3 -0.01 -0.01 3.16 3.16
SM 1.4 6.18 6.18 6.18 6.18
2.3 3.14 3.14 3.14 3.14
Table 6.9. Vertical displacement at top A (mm)
Scheme Stage Present method FEM check
1 1 0.009 0.009
2 0.012 0.012
3 -0.398 -0.398
2 1 0.043 0.004
2 0.041 0.041
3 -0.369 -0.369
SM -0.398 -0.398
Further, vertical displacement at top A under different pre-tension schemes are
also plotted in Figure 6.7 with δsi stands for the deviation of vertical displacement of
scheme i compared with the simultaneous scheme. As in scheme 1, the complete frame
is already constructed before pre-tension, the difference between scheme 1 and the
simultaneous scheme is only the pre-tension sequence and loading sequence. The
deviation between these two schemes during construction can be seen in Figure 6.7(a)
indicated as δs1.1 & δs1.2 at the end of CS1 & CS2 respectively. However, the two load-
deflection curves can reach the same point at full load, δs1 ~ 0, it infers that the effect
of loading sequence on vertical displacement diminishes at the end of the construction.
On the other hand, in pre-tension scheme 2, installation, and pre-tension batch
by batch, the structural stiffness changes continuously during construction. Hence, the
161
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 161
difference between scheme 1 & 2 is the change in structural stiffness. Deviations
between these two schemes indicate as δ1, δ2 & δS2 at the end of CS1, CS2 & CS3
respectively in Figure 6.7(b). It is interesting that this effect of the difference in
structural stiffness of the two struts 2 & 3, indicated in Figure 6.6(b1) & (c1), remain
throughout the whole construction as δ1 ~ δ2 ~ δS2= 0.03mm. In order words, the
difference in the installation process affects the structural behaviour. In Scheme 1, the
frame is constructed first and prestressed later, so its stiffness in CS1 is larger than the
one in Scheme 2. However, due to the pre-tension of struts 1, 4 of Scheme 2 in stage
1 is the largest among the three schemes (15.575kN in Scheme 2 versus 6.18kN in
schemes 1 & SM), it makes top A moving upward the highest, hence the final
displacement at top A, in Scheme 2 is the smallest. Moreover, this deviation remains
throughout the construction phase.
Variation of prestressed member forces during construction and prestressed
member forces after finished tensioning (the underline values) in Schemes 1 & 2 can
be seen in Table 6.10. It should be highlighted that even though the prestressed
member forces are not large, the difference in the initial internal prestressing forces
between schemes 1 & 2 is significant. The initial internal prestressing forces are 6.18
& 3.15kN for struts 1, 4 & 2, 3 in Scheme 1; and 15.575 & -1.922kN for struts 1, 4 &
2, 3 in Scheme 2. The reason is due to the interdependent behaviour of struts 1, 4 & 2,
3. In Scheme 1, the pre-tension of struts 1, 4 in stage 1 creates a pre-compression -
0.03kN in struts 2, 3 that is later reduced to -0.02kN when struts 2, 3 are tensioned in
stage 2; and the pre-tension of struts 2.3, in turn, reduced the tensile in struts 1.4 to
0.07kN. On the other hand, in scheme 2, the pre-tension of struts 1, 4 has no effects on
struts 2.3, which are not yet installed in stage 1. When struts 2, 3 are later compressed
in stage 2; the tensile in struts 1.4 increased further to 0.08kN.
162 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
(a) Vertical displacement at top A during construction
(b) A detail
Figure 6.7. Vertical displacement at node A under different schemes
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10
Tota
l con
stru
ctio
nal l
oad
fact
or
Vertical displacement in mm
δS2=~0.03
Sche
me
1
0.444
0.889
δS1~ 0
δS1.1= 0.186
δS1.2= 0.405
A
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
Tota
l con
stru
ctio
nal l
oad
fact
or
Vertical displacement in mm
0.444
0.889
δ1=~0.03
δ2=~0.03
163
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 163
Table 6.10. Variation of prestressed member forces during construction under different schemes (kN)
Scheme Member Construction stage
1 2 3
1 1.4 0.10 0.07 6.18
2.3 -0.03 -0.02 3.14
2 1.4 0.06 0.08 6.18
2.3 -0.01 3.16
SM 1.4 6.18
2.3 3.14
6.4.3 Arch bridge
A singly symmetric structure subjected to its self-weight was of interest to
investigate the transverse bending behaviour under the horizontal construction. This
structure with its original geometry, sectional properties, and applied load shown in
Figure 6.8 was investigated under two pre-tension schemes listed in Table 6.11 and
shown in Figure 6.8(c) & Figure 6.8(d). The vertical nodal load applied on the girder
in CS4 of 100kN is to account for concrete slab load afterwards. Besides, the
simultaneous scheme (SM) with the complete structure constructed at the same time
and under full loadings was also studied to compare with the two other schemes to
highlight the construction stage effects.
(a) Original geometry and sectional property
(b) Gravity load
164 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
(c1) CS1 (c2) CS2
(c) Pre-tension scheme 1
(d1) CS2 (d2) CS3
(d) Pre-tension scheme 2
Figure 6.8. Geometry, applied load, and pre-tension schemes of Arch Bridge
Table 6.11. Construction sequences of Arch Bridge
Scheme Step Description Target force (kN)
Scheme 1
1 Construct all members
Prestress hangers 1 & 4 111
2 Prestress members 2 & 3 104
3 Apply slab load -100kN
Scheme 2
1 Construct upper arch
2 Assemble hangers 1, 4 & girders 1, 4
Prestress hangers 1 & 4 111
3 Assemble hangers 2, 3 & girders 2, 3
Prestress hangers 2 & 3 104
4 Assemble middle girder
5 Apply slab load -100kN
The prestressing forces required were searched by the present iterative approach.
These predicted prestressing forces were then re-applied on the structures according
to the construction sequence and analysed by SAP2000 or general finite element
method that supported construction stage analysis for final member forces and nodal
displacements listed in Table 6.12 & Table 6.13. The results showed that the target
forces of all prestressed members were successfully achieved with the deviation
between the present method and SAP2000 is around 1% in general. At the same time,
vertical displacement at node A from the present analysis was consistent with the
results checked by SAP2000 as in Table 6.13 with maximum deviation 18%. It should
be noticed that plastic hinge occurs in CS3 according to the present method whereas
the structure remains elastic according to SAP2000, which contributed to the
deviation. In other words, there is some deviation in nodal displacements and member
forces between the present method and SAP2000 counter checked in this study;
165
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 165
whereas better agreements were found in the other two studies in section 6.4.1 & 6.4.2
when the structures still remained elastic for both approaches.
It should be mentioned that in scheme 1, the whole structure was constructed
first and tensioned later. It is observed that the gravity of the upper arch alone created
in the four hangers a tensile of 7.5kN. Hence, in stage 1, the pre-tension of hangers 1
& 4 induced 0.01kN in tensile, whereas the pre-tension of hangers 2 & 3 in stage 2
induced 0.06kN in tensile. While in schemes 2, pre-tension batch by batch, the value
of prestressed member force after tensioning is created by prestress alone.
Table 6.12. The prestressed member forces after finished tensioning and final prestressed member
forces (kN)
Scheme Member Prestressed member forces after tensioning Final prestressed member forces
Present method FEM check Present method FEM check
Def Und Def Und
1 1.4 7.51 7.50 110.9 109.8
2.3 7.55 7.50 104.4 105.4
2 1.4 3.75 3.75 110.6 111.5
2.3 3.75 3.75 105.0 104.1
SM 1.4 110.9 109.8 110.9 109.8
2.3 104.4 105.4 104.4 105.4
Table 6.13. Vertical displacement at node A (mm).
Scheme Stage Present method FEM check
Def Und
1 1 1.707 1.790
2 1.724 1.790
3 13.409 10.990
2 2 15.625 15.660
3 15.797 15.970
4 16.199 16.620
5 23.894 25.470
SM 13.506 10.990
Further, variations of prestressed member forces during construction under
different pre-tension schemes are listed in Table 6.14. The difference between the
prestressed member forces after finished tensioning (the underline values) of Scheme
1 (the whole structures are constructed first and prestressed later) versus Scheme 2
(only the upper arch is constructed first and prestressed members are assembled and
166 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
prestressed batch by batch) is nearly 50%. It highlights the effect of construction
sequence on prestress. It should be highlighted that even though the prestressed
member forces are not large, the difference in the initial internal prestressing forces
between schemes 1 & 2 is significant. The initial internal prestressing forces are 110.9
& 104.4kN for hangers 1, 4 & 2, 3 in Scheme 1; and 5467 & -5317kN respectively in
Scheme 2. The reason is due to the interdependent behaviour of hangers 1, 4 & 2, 3.
In Scheme 1, the pre-tension of hangers 1, 4 in stage 1 creates a pre-compression -
0.01kN in hangers 2, 3 that is later increased to 7.55kN when hangers 2, 3 are tensioned
in stage 2; and the pre-tension of hangers 2.3, in turn, reduced the tensile in hangers
1.4 to 7.42kN. On the other hand, in scheme 2, the pre-tension of hangers 1, 4 has no
effects on hangers 2.3, which are not yet installed in stage 2. When hangers 2, 3 are
later compressed in stage 2, the tensile in hangers 1.4 increased further to 7.45kN.
Table 6.14. Prestressed member forces during construction under different schemes (kN)
Scheme Strut Construction stage
1 2 3 4 5
1 1.4 7.51 7.42 110.9
2.3 7.49 7.55 104.4
2 1.4 3.75 7.45 7.65 110.6
2.3 3.75 7.35 105.0
SM 1.4 110.9
2.3 104.4
Table 6.15. Vertical displacement at Node A under different schemes (kN)
Scheme Construction stage
1 2 3 4 5
1 1.707 1.724 13.409
2 15.625 15.797 16.199 23.894
SM 13.506
Vertical displacement at nodes A & B indicated in Figure 6.8(a) under different
pre-tension schemes were listed in Table 6.15 and plotted in Figure 6.9 with δsi stands
for deviation of vertical displacement of scheme i compared with the simultaneous
scheme. As in scheme 1, the complete frame was already constructed and prestressed
later, so the difference between scheme 1 and the simultaneous scheme is only the pre-
tension sequence or loading sequence. The remarkable deviation between these two
167
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 167
schemes during construction can be seen in Figure 6.9(a) & (b), e.g. δs1.1 at the end of
CS1. Similar to the observation in section 0, the two load-deflection curves can reach
the same point at full load, δs1 ~ 0, it implies that the effect of loading sequence or
tensioning sequence on vertical displacement diminishes at full loading.
On the other hand, in Scheme 2, installation, and pre-tension batch by batch, the
structural stiffness changes continuously during construction. Hence, the difference
between schemes 1 & 2 includes the change in structural stiffness. Significant
deviation between these two schemes can be seen in Figure 6.9(a) & (b), with δS2 =
21.9 & -24.2mm at nodes A & B at the end of the construction sequence. The reason
is mainly due to the dead load of the upper arch creates a portion of vertical
displacements of nodes A & B in Scheme 1 in CS1 & in the simultaneous scheme;
whereas there is no displacement created due to this loading at nodes A & B in Scheme
2. Overall, Scheme 2 has the smallest nodal displacements at A & B. In regard to
inelastic behaviour, the formation of the 1st PH is at the load level of λ = 0.829 and ~
1 in Schemes 1 & 2 respectively, a bit later compared with the simultaneous scheme
without construction stage effects at λ = 0.8. Hence, in order to achieve a true structural
response, effects of construction sequence and loading sequence should be accounted
for in the pre-tension analysis of prestressed steel structures to ensure there are not any
excessive member forces or nodal displacements may take place during the
construction phase.
(a) Node A
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0
Tota
l con
stru
ctio
n lo
ad fa
ctor
Vertical displacement in mm
Scheme 1CS2
CS1
0.992
0.497
δS1=0.174
δS2=21.867
CS20.502
CS30.999
CS41.0
CS1~ 0
1st PHλSM=0.8001st PH
λS1=0.829
1st PHλS2~ 1
A
δS1.1=-8.566
168 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
(b) Node B
Figure 6.9. Vertical displacement at nodes A & B during construction
6.5 DISCUSSION
This chapter presents an iterative solution procedure to search for the
prestressing forces required in order to achieve a target prestressed state. There are
some important notices of the present method as follows,
Due to the present iterative solution approach incorporates the nonlinear
construction stage analysis proposed in chapter 3, all constructional
displacements incurred within each construction stage and in between two
constitutive construction stages are successfully accounted in the predicted
required prestressing forces for a specific prestressed state.
Because prestress can be applied at any construction stage and because of the
interdependent behaviour among prestressed members in the structure, pre-
tension in the following construction stages can change the prestressed member
forces of those members already pre-tensioned in the previous construction
stages. Hence, the prestressed member forces are only checked against their
target values at the end of the whole construction stage analysis in order to
account for the interdependent behaviour of all prestressed members in the
system. Consequently, prestressed iteration needs to encompass the
construction stage analysis as shown in Figure 6.2. Otherwise, the interaction
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-160.0 -140.0 -120.0 -100.0 -80.0 -60.0 -40.0 -20.0 0.0 20.0
Tota
l con
stru
ctio
n lo
ad fa
ctor
Vertical displacement in mm
CS2
CS1
0.992
0.497
δS1=0.22
δS2=24.2
CS30.999
CS20.5021st PH
λSM=0.800
1st PHλS1=0.829
1st PHλS2 ~ 1
δS1.1=39.95
169
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 169
between prestressed members of different construction stages could hardly be
accounted.
The structural analysis by the incremental iterative approach, i.e. the inner
cycle of the solution process in Figure 6.3, accounts for the sharing load
phenomenon of all the members in the system within each construction stage.
On the contrary, the mutual influence among prestressed members in different
construction stages is accounted by the iterative prestressed cycle, i.e. the outer
cycle of the process when the unbalance of the final prestressed member forces
and their target values are calculated and employed to adjust the new inputted
prestressing force for the next prestressed iteration.
The inputted prestressed forces are unchanged for each prestressed iteration.
Hence, these prestressing forces remain unchanged for the whole construction
stage analysis. On the contrary, due to the prestressing forces are adjusted for
each prestressed iteration, and they are likely different for each prestressed
iteration as well as the total construction load level is likely different for each
prestressed iteration.
It should be noted that in the present iterative approach, prestressing forces
could be applied in different construction stages while the designed target state
is achieved at the end of the whole construction sequence.
In case nodal displacement is the control criteria, the iterative solving method
can be formulated similarly.
6.6 CONCLUSION
Through the above numerical analyses, some conclusions can be drawn as
follows,
Due to constructional displacements has direct effects on the nodal coordinates,
it, in turn, influences prestressed member forces. As in prestressed structures,
only a small change in nodal coordinate can induce the change in member
length, which in turn induces a large change in member forces in case of pre-
fabrication, which can be up to 10% difference. It is recommended to account
for this particular effect in the pre-tension process analysis of prestressed steel
structures.
170 Chapter 6: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure
Even though the effects of sequential construction and loading sequence
diminished at the end of the construction phase, they can change the response
of a prestressed steel structure during construction. This particular effect
should be accounted for in the pre-tension process analysis to ensure there are
not any excessive member forces or nodal displacements may take place during
the construction phase.
It was noticed that required number of prestressed iteration of the numerical
examples are less than five in general with the convergent criteria set as 0.1%.
It should be noticed that the convergent rate is an important criterion for the
iterative solution approach when the solution could not always be achieved by
some other existing methods. In particular, when the number of prestressed
members is large, or target member forces are not the same, e.g. the convergent
rate of the target iterative option of SAP2000 software is not high.
By accounting for the effects of the change of geometry during construction,
the errors between the measured member forces after finished tensioning with
the required prestressing forces predicted by the present method and the
designed target values can be reduced which in turn reduces the number of
cyclic pre-tension on the construction site and also reduces construction time
and cost. It was observed that the deviation between the analysis results
accounted and unaccounted for the change of geometry during construction can
be up to 18%. Moreover, the structural dimension, its rigidity, and prestress
magnitude also affect the tensioning control forces/displacements. It is
considered that the deviation might be enlarged in other situations. On the
contrary, when the structural behaviour limits within the linear elastic range,
there is an insignificant deviation between the two analysis approaches.
In summary, a sophisticated iterative solution method to search for the
prestressing forces required in order to achieve a target prestressed state of prestressed
steel structures, which is capable of accounting for construction stage effects, was
presented in this study. By accounting for this particular effect, the required
prestressing forces predicted by the present method are more accurate and the errors
between the measured member forces after finished tensioning and the desired target
values can be reduced, which in turn reduces the number of cyclic pre-tension on the
construction site. As a result, construction time and cost can also be reduced.
171
Chapter 7: Nonlinear analysis of the interdependent behaviour among prestressed members in an entire prestressed steel structure 171
172 Chapter 7: Conclusions and future works
Chapter 7: Conclusions and future works
7.1 CONCLUSIONS
7.1.1 Summary
In prestressed steel structures, prestressed member forces are always hard to
maintain under construction phase, because the displacements of those members
incurred during construction can release their specific prestressed forces. It is
particularly true in the case of lacking temporary supports and/or stability precautions.
This phenomenon may be further exacerbated by the nonlinearities owing to large
prestressing load. It implies that the prestressed load level of a prestressed structure is
hardly preserved at its final stage when those constructional displacements are
inevitable. To this end, this research presented a second-order inelastic analysis to take
the nonlinearities of a prestressed steel structure at construction sequence into account.
The nonlinear effects from the constructional displacements of a structure on its
prestress loads are monitored at any sequence until its final stage. These constructional
displacements at a construction stage are commonly due to its gravity, constructional
and prestressing loads, which makes the original alignment at the next construction
stage hard to maintain. In order to preserve the alignment at the next construction stage
with minimising the member length, the position technique for installation at the next
stage subjected to these constructional displacements was developed by virtue of the
nonlinear least-square approach. The construction sequence was simulated by the step-
by-step technique together with the ‘Build’ and ‘Kill’ technique for establishing the
structural stiffness matrix. While, the present nonlinear analysis of construction
sequence of prestressed steel structure can capture the geometric and material
nonlinearities with recourse to the higher-order element and the refined plastic-hinge
approach, respectively. Therefore, the present nonlinear analysis of construction
sequence can properly capture the construction stage effects on the behaviour of
prestressed steel structures during construction, in particular, the effects of the
deformed geometry of previous construction stage on the position of newly installed
members of the current construction stage. Consequently, the prestressed loads of a
steel structure at the construction sequence such that the optimal structural
173
Chapter 7: Conclusions and future works 173
performance and ultimate capacity under a specific prestress load level can be
evaluated. Hence, the first objective of this research was successfully achieved.
On the other hand, under the circumstances of the presence of many prestressed
members in the system, it is difficult to prestress all members simultaneously
especially in complicated structures or when the control forces/displacements are not
the same, it makes the batched and repeated tensioning schemes are unavoidable. As
a result, when one member is tensioned to its target force and/or displacement, the
already achieved target values in other prestressed members will immediately change
due to the interdependent behaviour of all members in the system. Therefore, the key
point in the design of prestressed steel structure is to predict properly the prestressing
forces required to achieve a target prestressed state which is significantly influenced
by the interdependent behaviour among prestressed members in the entire system.
Hence, a comprehensive analysis of the pre-tension process was presented based on
Influence matrix approach in which four different types of Influence matrix (IFM)
were introduced and two different solving methods were brought forth. The direct
solving method solves for the accurate solution, whereas the iterative solving method
repeatedly amends trial values to achieve an approximate solution. Therefore, based
on IFM approach, various kinds of complicated batched and/or repeated tensioning
schemes can be analysed reliably, effectively, and efficiently.
However, as IFMs are set up based on the principle of linear superposition, the
pre-tension process analysis based on IFM approach is limited to the linear elastic
range only. Further, as constructional displacement has direct effects on structural
nodal coordinates, it, in turn, influences the structural behaviour of prestressed steel
structures. This research further presented an iterative solution approach for the pre-
tension process analysis which searches for the prestressing forces required in order to
achieve a target prestressed state. By incorporating the proposed nonlinear
construction stage analysis, this iterative solution approach can accommodate not only
displacements incurred within a construction stage and in between two constitutive
stages and also any inelastic material effects that may take place in the construction
phase. Consequently, the second objective of this research was also achieved and the
aim of this research was successfully obtained.
174 Chapter 7: Conclusions and future works
7.1.2 Research contribution
The main contributions of this research are as follows,
1. The positioning technique to locate the nodal coordinates of a newly built structure
at the current mth construction stage subjected to the change of geometry at the
previous (m-1)th construction stage based on the principle of minimum change in
length of the newly built members was proposed.
2. The mapping algorithm to regulate the positioning technique in order to determine
the nodal coordinates of a structure from stage to stage was introduced.
3. The new construction stage analysis that can properly capture not only all the
change of geometry in each construction stage and between two constitutive
stages and any inelastic material behaviour that may take place during
construction was presented.
4. The new iterative solution approach for the pre-tension process analysis that can
properly account for all the change of geometry, in each construction stage and
between two constitutive stages, and any inelastic material behaviour that may
take place during construction was developed.
7.1.3 Research significance
Based on the outcomes of a series of numerical studies, the main findings of this
research are as follows,
1. In the analysis considering the construction sequence effects, the major numerical
phenomenon is that the stiffness KT of the erecting structure until the current stage
is lower than that of a whole structure at its final stage, and also the load mft
imposed on the erecting structure is only considered until that stage. Therefore,
the cumulative behaviour of a structure under the construction sequence effect at
final stage may not be same with those from the conventional approach in most
cases, mainly because of the deficiency of superposition principle in the nonlinear
range. Through the numerical studies, it was observed that through the numerical
studies in Section 3.5 included a twenty-storey building, the deviation in member
force and nodal displacement could be as much as 30% and 60% respectively if
construction stage effects were unaccounted.
175
Chapter 7: Conclusions and future works 175
2. It was found that the behaviour of prestressed steel structure at construction phase
is quite different from the complete structure at service stage. It is particularly
important when there is no temporary supports and stability measure at
construction stage, which makes prestressed steel structure critical to nonlinear
effects. It, therefore, alters the target prestressed member forces and as a result,
affects the optimal performance of the structure at service stage. Thus, the
structural safety of a slender steel structure, which is prone to the second-order
effect, is crucial to be monitored at its construction sequence.
3. The positioning technique adjusts the nodal coordinates of newly built members
or structure stage by stage in order to accommodate the practical constructional
displacements. It is important to note that when the structure constructed based on
its original undeformed geometry; the geometry of the newly erected structural
part is much distorted compared with those constructed based on the deformed
geometry as the present study. It heralds that the significant initial forces can be
built up in the members if they are already prefabricated. This initial force can
cause the premature material yielding. Thus, construction simulation analysis is
necessary to reflect a true structural behaviour at construction phase such that the
optimal performance of a structure can be preserved as those specified at the
design stage.
176 Chapter 7: Conclusions and future works
4. Constructional displacements directly change nodal coordinates, which induce
initial member deformations or the so-called ‘lack of fits’ of prestressed members
and alter final prestressed member forces. Hence, it affects the structural
performance and in some particular cases may adversely affect the structural
safety during construction. The general deviation found through the numerical
studies in section 4.3 could be up to 10%. However, concerning the structural
dimension, its rigidity, and prestressed magnitude of those case studies, it is
considered that the deviation might be enlarged in other situations. On the other
hand, if a specific target design force needed to be achieved, the predicted required
prestressing forces by the present method will be more accurate. Hence, the errors
between the measured member forces after finished tensioning with the designed
target values can be reduced. It, in turn, reduces the number of cyclic pre-tension
on the construction site and hence reduces construction time and cost. Therefore,
this particular effect is recommended to be accounted in the construction analysis
of prestressed steel structures.
5. Even though the loading sequence (i.e. prestress load and applied load sequences)
does not change the structural response at full load level before the formation of
plastic hinges, it affects the inputted target prestressed member forces of a
structure for the sake of minimum pre-tension process. Therefore, the loading
sequence should be continually evaluated with recourse to the construction
simulation analysis for an efficient prestressed construction process.
6. The cumulative effect of the structural behaviour can be built up against the
construction stages. However, in order to assure no numerical drift-off error
embedded at final equilibrium point at final stage, the tolerance level of
convergent criteria is recommended to set tight, such as the incremental
displacements ∆u and unbalanced forces ∆f is 0.1% or less of the total
displacements u and load vector f, for the sake of the reliable equilibrium solution
at final stage. In particular when the number of construction stage is enormous.
177
Chapter 7: Conclusions and future works 177
7. In regards to the pre-tension process analysis based on IFM approach, the two-
criterion IFMs, which control both force and displacement, is considered as more
effective than the one-criterion IFMs; especially the force-displacement based
(FD matrix), which is firstly introduced in this study. It is particularly true when
the determinant of a single criterion IFM is zero or close to zero, because the
coefficient of FD matrix is significant without rounding error. Under this
circumstance, IMF is singular, and thus the direct solving method is completely
invalid. Fortunately, the iterative solving method can lead to the final solutions
but with slow convergence. In this regard, the iterative solving method is more
reliable. Further, this research remarks that, for a particular situation as indicated
by example 5.7.3, a symmetric structural system with symmetric tensioning
process leads to the null determinant of any IFMs, which cannot be solved by the
direct solving method, but it does not mean deficiency. Instead, this particular
situation implies that no mutual influence whatsoever from other tensioning
members, and therefore any target values imposed to the symmetric system is
absolutely valid without disturbing another tensioning member. Hence, the
determinant of IFMs is an important criterion in the analysis.
8. For the sake of lowering construction costs by reducing the tensioning control
values, the combination of installation and tensioning batch by batch is necessary
for an optimal tensioning scheme, especially keeping the batch of tensioning
members symmetric (i.e. determinant of IFMs is small) can mitigate the large
differential deformations in a symmetric structure. In summary, the superiority of
IFMs can be generally denoted by FD > DF > F > D matrices, and structural and
tensioning symmetry often provoke to the optimal scheme. Therefore, the analyst
needs to understand the characteristics of the prestressed system, the construction
conditions, and the tensioning process to decide which type of IFM is the most
suitable for a particular situation.
9. It was noticed that the convergent rate of the present iterative solution procedure
for the pre-tension process analysis was high with the convergent criteria set as
0.1% and the number of prestressed iteration was less than five in general.
In this sense, the advanced computational techniques such as the present
nonlinear analysis approach is indispensable for a reliable and sophisticated design
approach of prestressed steel structures at the construction phase. Unfortunately, the
178 Chapter 7: Conclusions and future works
common nonlinear construction analyses (e.g. via SAP2000 and ANSYS) accounted
for constructional displacements at construction phase are not yet mature and adequate
as indicated in Section 6.6. Therefore, designers are recommended to account for all
the constructional displacements that may occur during construction within each
construction stage and also in between two constitutive stage in the construction stage
analysis of prestressed steel structures by means of the present construction analysis
approach in order to ensure that any instability, excessive deflections of structural
members, or the possibility of structural collapse during construction can be predicted
and avoided.
Further, by means of the present construction analysis approach, any change in
member forces or nodal displacements of prestressed steel structures can be evaluated
properly at any sequence during construction. At the same time, this research also
founds that different construction schemes induced different structural responses at the
service stage. It is recommended that constructionists should take the advantage of the
present construction analysis approach to choose the optimal and efficient prestress
construction process for a particular structure. Besides, as all the variation of member
forces and displacements throughout the construction phase can be properly predicted
by the proposed method, these data can serve as a monitoring unit on site to ensure
structural safety during construction.
7.1.4 Research innovation
The innovation of this research is the developed analysis approach for the
prestressed steel structures that takes into account properly all the construction stage
effects on the behaviour of prestressed steel structures during construction, in
particular the effects of the deformed geometry of previous construction stage on the
position of newly installed members of the current construction stage, which to the
best knowledge of the author has not been presented in previous literature.
Consequently, a thorough understanding of the construction stage effects on the
behaviour of an entire prestressed steel structure is now able to obtain. The behaviour
of prestressed steel structures during construction can be accurately evaluated,
especially large-scale and/or complicated structures under construction lack temporary
supports or stability precautions and under large prestressing forces applied so that
instability and excessive deflection of structural members or structural collapse during
construction can be avoided. Overall, this research is a successful candidate to
179
Chapter 7: Conclusions and future works 179
integrate the structural engineering design into each sequence of the construction
phases of a building project, and further extend its realm to the architectural design as
the building information modelling.
180 Chapter 7: Conclusions and future works
7.2 FUTURE WORK
1. As the present method of nonlinear construction stage analysis was established
based on the positioning technique presented in Section 3.3, which applies the
principle that minimises the change in all member lengths. As a result, the new
geometry of the later construction stage is defined with accounting for only the
change in nodal coordinates and initial forces. It infers that all the changes of nodal
rotations as well as initial moments (if any) are neglected. Therefore, the further
work is to propose a nonlinear construction stage analysis based on the positioning
technique, which applies the principle that minimises the change in member,
shapes, i.e. all the changes of nodal rotations as well as initial moments (if any)
are accounted for. In that case, the analysis is more accurate and obviously more
complicated.
2. As the controlled criteria of prestressed structures can be either member forces or
nodal displacements. Future work is to extend the application of the iterative
solution method proposed in Chapter 6 to achieve the design target displacements.
It should be noticed that the concept of the two approaches is similar. However,
the key point is to improve the numerical convergence in case displacement is the
control value. The reason is when iteration based on displacement or deformation,
the deformation is adjusted during iteration by an incremental deformation.
Unfortunately, only a small change of displacement may induce a large change in
equivalent force that can make the required number of iterations increase or even
worse, in some situations, under a large equivalent force applied, bulking may
happen during iteration which results in numerical divergence.
3. As no kind of prestressed loss is accounted in this research, further step needs to
be made for real engineering practice.
4. As the scope of this research is limited to prestressed steel structures constructed
by beam-column element, future work may extend the application for prestressed
with cable elements which may be slack during the construction phase.
References 181
References
ANSYS, I. (2009). Theory Reference for the Mechanical APDL and Mechanical Applications.
Ayyub, B. M., Ibrahim, A., & Schelling, D. (1990). Posttensioned trusses: Analysis and design. Journal of Structural Engineering, 116(6), 1491-1506.
Belletti, B., & Gasperi, A. (2010). Behavior of Prestressed Steel Beams. Journal of Structural Engineering, 136(9), 1131-1139. doi:doi:10.1061/(ASCE)ST.1943-541X.0000208
Bradford, M. A. (1991). Buckling of prestressed steel girders. Engineering journal, 28(3), 98-101.
Chan, S. L. (1988). Geometric and material non-linear analysis of beam-columns and frames using the minimum residual displacement method. International journal for numerical methods in engineering, 26(12), 2657-2669.
Chan, S. L. (1989). Inelastic post-buckling analysis of tubular beam-columns and frames. Engineering Structures, 11(1), 23-30.
Chan, S. L. (2001). Non-linear behavior and design of steel structures. Journal of Constructional Steel Research, 57(12), 1217-1231. doi:10.1016/S0143-974X(01)00050-5
Chan, S. L., & Chui, P. P. T. (1997). A generalized design-based elastoplastic analysis of steel frames by section assemblage concept. Engineering Structures, 19(8), 628-636. doi:http://dx.doi.org/10.1016/S0141-0296(96)00138-1
Chan, S. L., & Chui, P. P. T. (2000). Non-linear static and cyclic analysis of steel frames with semi-rigid connections: Amsterdam: Elsevier.
Chan, S. L., Huang, H. Y., & Fang, L. X. (2005). Advanced analysis of imperfect portal frames with semirigid base connections. Journal of Engineering Mechanics, 131(6), 633-640. doi:10.1061/(ASCE)0733-9399(2005)131:6(633)
Chan, S. L., Shu, G. P., & Lü, Z. T. (2002). Stability analysis and parametric study of pre-stressed stayed columns. Engineering Structures, 24(1), 115-124. doi:10.1016/S0141-0296(01)00026-8
Chan, S. L., & Zhou, Z. (1994). Pointwise Equilibrating Polynomial Element for Nonlinear Analysis of Frames. Journal of Structural Engineering, 120(6), 1703-1717. doi:doi:10.1061/(ASCE)0733-9445(1994)120:6(1703)
Chan, S. L., & Zhou, Z. (1995). Second-Order Elastic Analysis of Frames Using Single Imperfect Element per Member. Journal of Structural Engineering, 121(6), 939-945. doi:doi:10.1061/(ASCE)0733-9445(1995)121:6(939)
Chen, W. F., Kim, S. O., & Choi, S. H. (2001). Practical Second-Order Inelastic Analysis for Three- Dimensional Steel Frames. International Journal of Steel Structures, 1(3), 213.
182 References
Chen, Z., Zhao, Z., Zhu, H., Wang, X., & Yan, X. (2015). The step-by-step model technology considering nonlinear effect used for construction simulation analysis. International Journal of Steel Structures, 15(2), 271-284. doi:10.1007/s13296-015-6002-9
Choi, C. K., & Kim, E. D. (1985). Multistory frames under sequential gravity loads. Journal of Structural Engineering, 111(11), 2373-2384.
Clarke, M. J., & Hancock, G. J. (1990). A study of incremental-iterative strategies for non-linear analyses. International journal for numerical methods in engineering, 29(7), 1365-1391. doi:10.1002/nme.1620290702
Crisfield, M. A. (1981a). A fast incremental/iterative solution procedure that handles "snap-through". Computers and Structures, 13(1-3), 55-62. doi:10.1016/0045-7949(81)90108-5
Crisfield, M. A. (1981b). A fast incremental/iterative solution procedure that handles “snap-through”. Computers & structures, 13(1), 55-62.
Crisfield, M. A. (1983). An arc-length method including line searches and accelerations. International journal for numerical methods in engineering, 19(9), 1269-1289. doi:10.1002/nme.1620190902
Deng, H., Cheng, J., Jiang, B. W., & Lou, D. A. (2011). Member length error effect on cable-strut tensile structure. Zhejiang Daxue Xuebao (Gongxue Ban)/Journal of Zhejiang University (Engineering Science), 45(1), 68-74+86. doi:10.3785/j.issn.1008-973X.2011.01.011
Dong, S., & Yuan, X. (2007). Pretension process analysis of prestressed space grid structures. Journal of Constructional Steel Research, 63(3), 406-411.
Fan, Z., Liu, X., Hu, T., Fan, X., & Zhao, L. (2007). Simulation analysis on steel structure erection procedure of the National Stadium. Jianzhu Jiegou Xuebao/Journal of Building Structures, 28(2), 134-143.
Fedczuk, P., & Skowroński, W. (2002). NON-LINEAR ANALYSIS OF PLANE STEEL PRESTRESSED TRUSS IN FIRE. Journal of Civil Engineering and Management, 8(3), 177-183. doi:10.1080/13923730.2002.10531274
Felton, L. P., & Dobbs, M. (1977). On optimized prestressed trusses. AIAA journal, 15(7), 1037-1039.
Felton, L. P., & Hofmeister, L. D. (1970). Prestressing in structural synthesis. AIAA journal, 8(2), 363-364. doi:10.2514/3.5672
Feng, Y., Li, H., He, C., & Feng, C. (2012) Construction technology of steel structure roof in terminal building of a civil airport. Vol. 368-373. Advanced Materials Research (pp. 169-172).
Feng, Y., Zhou, Z., Meng, S., Wang, Y., & Wu, J. (2013). Research on tensioning process feedback control for prestressed space grid structures. Jianzhu Jiegou Xuebao/Journal of Building Structures, 34(10), 93-100.
183
References 183
Ge, J. Q., Zhou, S. H., Gu, P., Huang, J. Y., Zhang, G. J., & Zhang, Q. M. (2010). Simulation analysis on prestressing construction of the main stadium in Guiyang Olympic Sports Center. Building Structure, 12, 009.
Gratton, S., Lawless, A. S., & Nichols, N. K. (2007). Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM Journal on Optimization, 18(1), 106-132.
Guo, Y. L., & Liu, X. W. (2008). State nonlinear finite element method for construction mechanics analysis of steel structures. Gongcheng Lixue/Engineering Mechanics, 25(10), 19-24+37.
Guo, Y. L., Liu, X. W., Liu, L. Y., & Zhang, Q. L. (2007). Analysis method of pre-set construction deformation values for the new CCTV headquarters [J]. Industrial Construction, 9, 002.
Hafez, H. H., Temple, M. C., & Ellis, J. S. (1979). PRETENSIONING OF SINGLE-CROSSARM STAYED COLUMNS. ASCE J Struct Div, 105(2), 359-375.
Hanaor, A., & Levy, R. (1985). Imposed lack of fit as a means of enhancing space truss design. Space Struct, 1, 147-154.
He, Y., Zhou, X., & Zhou, J. (2011). Calculation method of the initial strains of cables in pretensioning construction of prestressed space reticulated structures. Paper presented at the 1st International Conference on Civil Engineering, Architecture and Building Materials, CEABM 2011, Haikou. http://www.scopus.com/inward/record.url?eid=2-s2.0-79957994577&partnerID=40&md5=e43c26c6605543e3e550560e12f59d82
Hoadley, P. G. (1961). An analytical study of the behaviour of prestressed steel beams. ProQuest Dissertations and Theses.
Hu, C. M., Zeng, F. K., Li, Y. H., & Yan, X. (2009). Construction process simulation and actual analysis of palms together dagoba in famen temple. Engineering Mechanics, 26, 153-157.
Iu, C. K. (2008). Inelastic finite element analysis of composite beams on the basis of the plastic hinge approach. Engineering Structures, 30(10), 2912-2922.
Iu, C. K. (2015). Generalised element load method for first- and second-order element solutions with element load effect. Engineering Structures, 92, 101-111. doi:http://dx.doi.org/10.1016/j.engstruct.2015.03.016
Iu, C. K. (2016a). Generalised element load method with whole domain accuracy for reliable structural design. Adv. Steel Constr, 12(4).
Iu, C. K. (2016b). Nonlinear analysis for the pre-and post-yield behaviour of a composite structure with the refined plastic hinge approach. Journal of Constructional Steel Research, 119, 1-16.
Iu, C. K. (2016c). Nonlinear analysis of the RC structure by higher-order element with the refined plastic hinge. Computers and Concrete, 17(5), 579-596. doi:10.12989/cac.2016.17.5.579
184 References
Iu, C. K., & Bradford, M. A. (2010). Second-order elastic finite element analysis of steel structures using a single element per member. Engineering Structures, 32(9), 2606-2616. doi:10.1016/j.engstruct.2010.04.033
Iu, C. K., & Bradford, M. A. (2012a). Higher-order non-linear analysis of steel structures part I: Elastic second-order formulation. Advanced Steel Construction, 8(2), 168-182.
Iu, C. K., & Bradford, M. A. (2012b). Higher-order non-linear analysis of steel structures part II: Refined plastic hinge formulation. Advanced Steel Construction, 8(2), 183-198.
Iu, C. K., & Bradford, M. A. (2015). Novel non-linear elastic structural analysis with generalised transverse element loads using a refined finite element. Advanced Steel Construction, 11(2), 223-249.
Iu, C. K., Bradford, M. A., & Chen, W. F. (2009). Second-order inelastic analysis of composite framed structures based on the refined plastic hinge method. Engineering Structures, 31(3), 799-813. doi:http://dx.doi.org/10.1016/j.engstruct.2008.12.007
Iu, C. K., & Chan, S. L. (2004). A simulation-based large deflection and inelastic analysis of steel frames under fire. Journal of Constructional Steel Research, 60(10), 1495-1524. doi:10.1016/j.jcsr.2004.03.002
Izzuddin, B. A. (1990). Nonlinear dynamic analysis of framed structures. Imperial College London (University of London).
Jayasinghe, M. T. R., & Jayasena, W. M. V. P. K. (2004). Effects of axial shortening of columns on design and construction of tall reinforced concrete buildings. Practice Periodical on Structural Design and Construction, 9(2), 70-78. doi:10.1061/(ASCE)1084-0680(2004)9:2(70)
Jennings, A. (1968). Frame analysis including change of geometry. J. Struct. Div., ASCE, 94(ST3), 627-644.
Jia, Y. (2009). Nonlinear analysis of prestressed steel box beams, Harbin.
Jia, Y., & Liang, D. (2011) Numerical analysis of prestressed steel box beams. 2011 International Conference on Structures and Building Materials, ICSBM 2011: Vol. 163-167 (pp. 862-865). Guangzhou.
Jiang, Z., Shi, K., Xu, M., & Cai, J. (2011). Analysis of nonlinear buckling and construction simulation for an elliptic paraboloid radial beam string structure. Tumu Gongcheng Xuebao/China Civil Engineering Journal, 44(12), 1-8.
Jiang, Z., Xu, M., Duan, W., Shi, K., Cai, J., & Wang, S. (2011) Nonlinear finite element analysis of beam string structure. 2011 International Conference on Structures and Building Materials, ICSBM 2011: Vol. 163-167 (pp. 2124-2130). Guangzhou.
Kim, H., & Cho, S. (2005). Column shortening of concrete cores and composite columns in a tall building. The structural design of tall and special buildings, 14(2), 175-190.
185
References 185
Kim, H. S., & Shin, S. H. (2011). Column Shortening Analysis with Lumped Construction Sequences. Procedia Engineering, 14, 1791-1798. doi:http://dx.doi.org/10.1016/j.proeng.2011.07.225
Kim, S. E., & Chen, W. F. (1996). Practical advanced analysis for braced steel frame design. Journal of Structural Engineering, 122(11), 1266-1274.
Kim, S. E., & Chen, W. F. (1998). A sensitivity study on number of elements in refined plastic-hinge analysis. Computers and Structures, 66(5), 665-673.
King, W. S., White, D. W., & Chen, W. F. (1992). Second-order inelastic analysis methods for steel-frame design. Journal of structural engineering New York, N.Y., 118(2), 408-428.
Kuroedov, V., Akimov, L., Frolov, A., Savchenko, A., Kuznetsov, A., & Kostenko, A. (2016). The Determination of Stress State of Structures Considering Sequence of Construction and Load Application. Paper presented at the MATEC Web of Conferences.
Levy, R., & Hanaor, A. (1992). Optimal design of prestressed trusses. Computers & Structures, 43(4), 741-744.
Levy, R., Hanaor, A., & Rizzuto, N. (1994). Experimental investigation of prestressing in double-layer grids. International Journal of Space Structures, 9(1), 21-26.
Li, J., Yu, J. H., Yan, X. Y., & Liu, H. B. (2014). Construction numerical simulation of cable-supported barrel shell structure considering construction nonlinearity. Paper presented at the 3rd International Conference on Manufacturing Engineering and Process, ICMEP 2014, Seoul.
Li, Y. M., Shi, Y. S., Zhang, Y. G., Zhang, L., Liu, J., & Wu, J. Z. (2010). Test research and computer simulation analysis of construction state during form finding for suspen-dome system. Journal of Harbin Institute of Technology (New Series), 17(5), 690-696.
Li, Z. q., Zhang, Z. h., Dong, S. l., & Fu, X. y. (2012). Construction sequence simulation of a practical suspen-dome in Jinan Olympic Center. Advanced Steel Construction, 8(1), 38-53.
Liew, J. Y. R., Chen, H., & Shanmugam, N. E. (2001). Inelastic analysis of steel frames with composite beams. Journal of Structural Engineering, 127(2), 194-202.
Liew, J. Y. R., Chen, H., Shanmugam, N. E., & Chen, W. F. (2000). Improved nonlinear plastic hinge analysis of space frame structures. Engineering Structures, 22(10), 1324-1338. doi:http://dx.doi.org/10.1016/S0141-0296(99)00085-1
Liew, J. Y. R., & Hong, C. (2004). Explosion and Fire Analysis of Steel Frames Using Fiber Element Approach. Journal of Structural Engineering, 130(7), 991-1000. doi:10.1061/(ASCE)0733-9445(2004)130:7(991)
Liew, J. Y. R., & Li, J. J. (2006). Advanced analysis of pre-tensioned bowstring structures. Int J Steel Struct, 6(2), 153-162.
186 References
Liew, J. Y. R., Punniyakotty, N. M., & Shanmugam, N. E. (2001). Limit-state Analysis and Design of Cable-tensioned Structures. International Journal of Space Structures, 16(2), 95-110. doi:10.1260/0266351011495205
Liew, J. Y. R., Tang, L. K., & Choo, Y. S. (2002). Advanced Analysis for Performance-based Design of Steel Structures Exposed to Fires. Journal of Structural Engineering, 128(12), 1584.
Liew, J. Y. R., White, D. W., & Chen, W. F. (1993). Limit states design of semi-rigid frames using advanced analysis: Part 2: Analysis and design. Journal of Constructional Steel Research, 26(1), 29-57. doi:http://dx.doi.org/10.1016/0143-974X(93)90066-2
Liew, J. Y. R., White, D. W., & Chen, W. F. (1993). Second-order refined plastic-hinge analysis for frame design. Part II. Journal of structural engineering New York, N.Y., 119(11), 3217-3237.
Liu, H., Chen, Z., & Zhou, T. (2012). Research on the process of pre-stressing construction of suspen-dome considering temperature effect. Advances in Structural Engineering, 15(3), 489-493. doi:10.1260/1369-4332.15.3.489
Liu, H. B., Han, Q. H., Chen, Z. H., Wang, X. D., Yan, R., & Zhao, B. (2014). Precision control method for pre-stressing construction of suspen-dome structures. Advanced Steel Construction, 10(4), 404-425.
Liu, S. W., Liu, Y. P., & Chan, S. L. (2012). Advanced analysis of hybrid steel and concrete frames: Part 1: Cross-section analysis technique and second-order analysis. Journal of Constructional Steel Research, 70, 326-336. doi:10.1016/j.jcsr.2011.09.003
Liu, Y., & Chan, S. L. (2011). Second-Order and Advanced Analysis of Structures Allowing for Load and Construction Sequences. Advances in Structural Engineering, 14(4), 635-646. doi:doi:10.1260/1369-4332.14.4.635
Liu, Y. J., Chen, Z. H., & Zhang, Y. L. (2011). Construction technique and simulation analysis of large-span spatial steel structure. Paper presented at the 2011 International Conference on Remote Sensing, Environment and Transportation Engineering, RSETE 2011 - Proceedings.
Lozano-Galant, J. A., Payá-Zaforteza, I., Xu, D., & Turmo, J. (2012). Analysis of the construction process of cable-stayed bridges built on temporary supports. Engineering Structures, 40, 95-106. doi:http://dx.doi.org/10.1016/j.engstruct.2012.02.005
Marí, A., Mirambell, E., & Estrada, I. (2003). Effects of construction process and slab prestressing on the serviceability behaviour of composite bridges. Journal of Constructional Steel Research, 59(2), 135-163. doi:http://dx.doi.org/10.1016/S0143-974X(02)00029-9
Moragaspitiya, H. N. P., Thambiratnam, D. P., Perera, N. J., & Chan, T. H. T. (2013). Development of a vibration based method to update axial shortening of vertical load bearing elements in reinforced concrete buildings. Engineering Structures, 46, 49-61. doi:10.1016/j.engstruct.2012.07.010
187
References 187
Nguyen, T. M. T., & Iu, C. K. (2015a, August). A Thorough Investigation of The Interdependent Behavior of Prestressed System. Paper presented at the The 2015 World Congress on Advances in Structural Engineering and Mechanics (ASEM15), Incheon, Korea.
Oran, C. (1973a). Tangent stiffness in plane frames. ASCE J Struct Div, 99(ST6), 983-985.
Oran, C. (1973b). Tangent stiffness in space frames. Journal of the Structural Division, 99(6), 987-1001.
Osofero, A. I., Wadee, M. A., & Gardner, L. (2012). Experimental study of critical and post-buckling behaviour of prestressed stayed columns. Journal of Constructional Steel Research, 79, 226-241. doi:10.1016/j.jcsr.2012.07.013
Osofero, A. I., Wadee, M. A., & Gardner, L. (2013). Numerical studies on the buckling resistance of prestressed stayed columns. Advances in Structural Engineering, 16(3), 487-498. doi:10.1260/1369-4332.16.3.487
Pan, D. Z., & Wei, D. M. (2010). Simulation Method of Step-by-Step Construction of Long-Span Steel Structures. Journal of South China University of Technology (Natural Science Edition), 9, 025.
Ponnada, M. R., & Vipparthy, R. (2013). Improved method of estimating deflection in prestressed steel I-beams. Asian Journal of Civil Engineering, 14(5), 766-772.
Qu, X. N., Luo, Y. Z., Zheng, J. H., & Zhang, Y. (2009). Method of cable force control during accumulative sliding construction for pretensioned reticulated structures. Gongcheng Lixue/Engineering Mechanics, 26(5), 178-182.
Ramm, E. (1981). Strategies for Tracing the Nonlinear Response Near Limit Points. In W. Wunderlich, E. Stein, & K. J. Bathe (Eds.), Nonlinear Finite Element Analysis in Structural Mechanics: Proceedings of the Europe-U.S. Workshop Ruhr-Universität Bochum, Germany, July 28–31, 1980 (pp. 63-89). Berlin, Heidelberg: Springer Berlin Heidelberg.
Rezaiee-Pajand, M., Ghalishooyan, M., & Salehi-Ahmadabad, M. (2013a). Comprehensive evaluation of structural geometrical nonlinear solution techniques Part I: Formulation and characteristics of the methods. Structural Engineering and Mechanics, 48(6), 849-878. doi:10.12989/sem.2013.48.6.849
Rezaiee-Pajand, M., Ghalishooyan, M., & Salehi-Ahmadabad, M. (2013b). Comprehensive evaluation of structural geometrical nonlinear solution techniques Part II: Comparing efficiencies of the methods. Structural Engineering and Mechanics, 48(6), 879-914. doi:10.12989/sem.2013.48.6.879
Ronghe, G. N., & Gupta, L. M. (2002). Parametric study of tendon profiles in prestressed steel plate girder. Advances in Structural Engineering, 5(2), 75-85.
Saito, D., & Wadee, M. A. (2008). Post-buckling behaviour of prestressed steel stayed columns. Engineering Structures, 30(5), 1224-1239. doi:10.1016/j.engstruct.2007.07.012
188 References
Saito, D., & Wadee, M. A. (2009a). Buckling behaviour of prestressed steel stayed columns with imperfections and stress limitation. Engineering Structures, 31(1), 1-15. doi:10.1016/j.engstruct.2008.07.006
Saito, D., & Wadee, M. A. (2009b). Numerical studies of interactive buckling in prestressed steel stayed columns. Engineering Structures, 31(2), 432-443. doi:10.1016/j.engstruct.2008.09.008
Saito, D., & Wadee, M. A. (2010). Optimal prestressing and configuration of stayed columns. Proceedings of the Institution of Civil Engineers: Structures and Buildings, 163(5), 343-355. doi:10.1680/stbu.2010.163.5.343
Samarakkody, D., Moragaspitiya, P., & Thambiratnam, D. (2014). Quantifying differential axial shortening in high rise buildings with Concrete Filled Steel Tube columns. Paper presented at the Proceedings of the 10th fib International PhD Symposium in Civil Engineering.
SAP, C. (2010). Analysis Reference Manual. Computer and Structures, Berkeley.
Smith, E. A. (1985). Behavior of columns with pretensioned stays. Journal of Structural Engineering, 111(5), 961-972.
So, A. K. W., & Chan, S. L. (1991). Buckling and geometrically nonlinear analysis of frames using one element/member. Journal of Constructional Steel Research, 20(4), 271-289. doi:http://dx.doi.org/10.1016/0143-974X(91)90078-F
Spillers, W. R., & Levy, R. (1984). Truss design: two loading conditions with prestress. Journal of Structural Engineering, 110(4), 677-687.
Subramanian, K., & Velayutham, M. (2015). Construction sequence analysis of multi story structures. International Journal of Earth Sciences and Engineering, 8(4), 1727-1735.
Tang, Q., & Zhou, J. (2012) Study on the calculation method of plum blossom-shaped steel roof pre-set deformation value in construction. Vol. 446-449. Advanced Materials Research (pp. 1993-1996).
Teh, L., & Clarke, M. (1999). Plastic-Zone Analysis of 3D Steel Frames Using Beam Elements. Journal of Structural Engineering, 125(11), 1328-1337. doi:doi:10.1061/(ASCE)0733-9445(1999)125:11(1328)
Temple, M., Prakash, M., & Ellis, J. (1984). Failure Criteria for Stayed Columns. Journal of Structural Engineering, 110(11), 2677-2689. doi:doi:10.1061/(ASCE)0733-9445(1984)110:11(2677)
Thai, H. T., Uy, B., Kang, W. H., & Hicks, S. (2016). System reliability evaluation of steel frames with semi-rigid connections. Journal of Constructional Steel Research, 121, 29-39. doi:10.1016/j.jcsr.2016.01.009
Tian, L., Hao, J., Wang, Y., & Zheng, J. (2012). Research on several key technologies for structural construction of main stadium for the Universidad Sports Centre. Jianzhu Jiegou Xuebao/Journal of Building Structures, 33(5), 9-15+70.
189
References 189
Tocháček, M., & Ferjenčík, T. l. P. (1992). Further stability problems of prestressed steel structures. Journal of Constructional Steel Research, 22(2), 79-86.
Troitsky, M. S., Zielinski, Z., & Rabbani, N. (1989). Prestressed‐Steel Continuous‐Span Girders. Journal of Structural Engineering, 115(6), 1357-1370. doi:doi:10.1061/(ASCE)0733-9445(1989)115:6(1357)
Wang, G. (2000). On mechanics of time-varying structures. China Civil Engineering Journal, 33(6), 105-108.
Wang, H., Fan, F., Qian, H., Zhi, X., & Zhu, E. (2011) Full-process analysis of pretensioning construction of Dalian gym. Vol. 163-167. 2011 International Conference on Structures and Building Materials, ICSBM 2011 (pp. 200-204). Guangzhou.
Wang, H. J., Fan, F., Zhi, X. D., Hang, G., Zhu, E. C., & Wang, H. (2013). Research of vertical deformation and predeformation of super high-rise buildings during construction. Engineering Mechanics, 2, 043.
Wang, H. J., Qian, H. L., & Fan, F. (2014) Experimentalanalysis of prestressing construction schemes of Dalian gym. Vol. 638-640. Applied Mechanics and Materials (pp. 1568-1574).
Wang, Y., Guo, Z., & Luo, B. (2012). Research on the mechanic analysis method of prestress construction process of large-span suspendome. Paper presented at the 2nd International Conference on Civil Engineering and Building Materials, CEBM 2012, Hong Kong.
Wei, S. H., & Zhang, J. C. (2012). Simulation Analysis on Construction Process for Large-Span Steel Truss. Grain Distribution Technology, 5, 005.
Wu, J., Frangopol, D. M., & Soliman, M. (2015). Geometry control simulation for long-span steel cable-stayed bridges based on geometrically nonlinear analysis. Engineering Structures, 90, 71-82. doi:10.1016/j.engstruct.2015.02.007
Wu, X., Gao, Z., & Li, Z. (2005). Analysis of whole erection process for steel shell of National Grand Theatre. Jianzhu Jiegou Xuebao/Journal of Building Structures, 26(5), 40-45.
Xie, G., Fu, X., Wu, L., Chen, D., & Gu, L. (2009). Construction simulation analysis of steel structure for National Swimming Center. Jianzhu Jiegou Xuebao/Journal of Building Structures, 30(6), 142-147.
Yang, W. G., Hong, G. S., Wang, M. Z., Yang, G. L., Zhang, G. J., Wang, S., . . . Zhou, W. S. (2012). Simulation analysis of construction process of multistory large cantilevered steel structure. Journal of Building Structures, 4, 013.
Yau, C., & Chan, S. L. (1994). Inelastic and Stability Analysis of Flexibly Connected Steel Frames by Springs‐in‐Series Model. Journal of Structural Engineering, 120(10), 2803-2819. doi:doi:10.1061/(ASCE)0733-9445(1994)120:10(2803)
Yip, H. L., Au, F. T. K., & Smith, S. T. (2011). Serviceability performance of prestressed concrete buildings taking into account long-term behaviour and construction sequence. Paper presented at the Procedia Engineering.
190 References
Zhang, J., Luo, X., Cai, J., Feng, J., & Yang, X. (2011) Influence of segmented construction methods on the prestressed state of truss string structure roof of Xinjiang Exhibition Center. Vol. 163-167. Advanced Materials Research (pp. 85-89).
Zhang, J., Luo, X. C., Cai, J. G., Yang, X. J., & Feng, J. (2012). Influence of segmented construction methods on the design state of the large-span truss string structure. Shenzhen Daxue Xuebao (Ligong Ban)/Journal of Shenzhen University Science and Engineering, 29(2), 142-147. doi:10.3724/SP.J.1249.2012.02142
Zhang, J., & Sun, K. (2011) Construction process simulation of cable dome. Vol. 94-96. Applied Mechanics and Materials (pp. 750-754).
Zhang, J. G., Gao, K. Y., & Zhang, T. B. (2012). Research on cantilever expansion construction technology for one comprehensive steel grid structure. Gongcheng Lixue/Engineering Mechanics, 29(SUPPL.1), 57-62. doi:10.6052/j.issn.1000-4750.2011.11.S009
Zhang, W., Wu, Z., & Chen, B. (2012) Simulation study on construction process of complex spatial steel structure based on the construction mechanics. Vol. 226-228. Applied Mechanics and Materials (pp. 1209-1213).
Zhao, X. Z., Chen, Y. Y., & Chen, J. X. (2007). Experimental and numerical investigations of beam-string structures during prestressing construction. Paper presented at the Proceedings of the 3rd International Conference on Steel and Composite Structures, ICSCS07 - Steel and Composite Structures.
Zhao, Z., Zhu, H., Chen, Z., & Du, Y. (2015). Optimizing the construction procedures of large-span structures based on a real-coded genetic algorithm. International Journal of Steel Structures, 15(3), 761-776. doi:10.1007/s13296-015-9020-8
Zhao, Z. W., Chen, Z. H., Liu, H. B., & Zhao, B. Z. (2016). Investigations on influence of erection process on buckling of large span structures by a novel numerical method. International Journal of Steel Structures, 16(3), 789-798. doi:10.1007/s13296-015-0151-8
Zhou, Y. Z., Jiang, K. B., Ding, Y., & Yang, J. K. (2012). Theoretical and FEM Research on Inelastic Displacement of Assembled Truss Bridge with Cable Reinforcement. Applied Mechanics and Materials, 178, 2038-2042.
Zhou, Z., & Chan, S. L. (2004). Elastoplastic and Large Deflection Analysis of Steel Frames by One Element per Member. I: One Hinge along Member. Journal of Structural Engineering, 130(4), 538-544. doi:doi:10.1061/(ASCE)0733-9445(2004)130:4(538)
Zhou, Z., Feng, Y. L., Meng, S. P., & Wu, J. (2014). A novel form analysis method considering pretension process for suspen-dome structures. KSCE Journal of Civil Engineering, 18(5), 1411-1420. doi:10.1007/s12205-014-0111-4
Zhou, Z., Meng, S., & Wu, J. (2010a). Pretension scheme decision based on observation values of cable force and displacement for prestressed space structures. Jianzhu Jiegou Xuebao/Journal of Building Structures, 31(3), 18-23.
191
References 191
Zhou, Z., Meng, S. P., & Wu, J. (2010b). Pretension Process Analysis of Arch-supported Prestressed Grid Structures Based on Member Initial Deformation. Advances in Structural Engineering, 13(4), 641-650.
Zhou, Z., Meng, S. P., & Wu, J. (2010c). Pretension process control based on cable force observation values for prestressed space grid structures. Structural Engineering and Mechanics, 34(6), 739-753.
Zhou, Z., Wu, J., Meng, S. P., & Yu, Q. (2012). Construction process analysis for a single-layer folded space grid structure in considering time-dependent effect. International Journal of Steel Structures, 12(2), 205-217. doi:10.1007/s13296-012-2005-y
Zhu, L. M., Zhu, Z. H., & Kang, C. J. (2014) Behavior and construction of stay cables in Shiyan stadium. Vol. 577. Applied Mechanics and Materials (pp. 1093-1096).
Zhuo, X., & Ishikawa, K. (2004). Tensile force compensation analysis method and application in construction of hybrid structures. International Journal of Space Structures, 19(1), 39-46.
Zhuo, X., Zhang, G. F., Ishikawa, K., & Lou, D. A. (2008). Tensile force correction calculation method for prestressed construction of tension structures. Journal of Zhejiang University: Science A, 9(9), 1201-1207. doi:10.1631/jzus.A0720090
192 Appendices
Appendices
Appendix A Higher-order element formulation
A.1 Tangent stiffness matrix
( )( )
( )( )
( )
( ) ( ){ } 1212211
213
322
213
322
1
22
48420
1135
1325
4828484
482035
2525
48365
4812
iiiii
ii
ii
i
GLEIbb
LEI
q
qqq
q
qqq
LEI
qM
=θ−θ+θ+θ=
θ−θ
+
++++
θ+θ
+
+++
=∂
∂
(A. 1)
( )( )
( )( )
( )
( ) ( ){ } 2212211
213
322
213
322
2
22
48420
1135
1325
4828484
482035
2525
48365
4812
iiiii
ii
ii
i
GL
EIbbLEI
q
qqq
q
qqq
LEI
qM
=θ−θ−θ+θ=
θ−θ
+
+++−
θ+θ
+
+++
=∂
∂
(A. 2)
( ) ( )
( )( )
( ) LH'bAL
IL
q
q
ALI
Leq
z,yiii
z,yiii
11
4835
48645
4816
1
22122
2214
2
2
=θ−θ−
=
θ−θ
+
−−−
=∂∂
∑∑ =
=
(A. 3)
( ) ( )( ) LH
G
'bAL
Ibbq i
z,yiii
iiii
i
1
22122
212211
1
22=
θ−θ−
θ−θ+θ+θ=
θ∂∂
∑=
, ( ) ( )( ) LH
G
bAL
Ibbq i
zyiii
iiii
i
2
,
22122
212211
2 '
22=
−−
−−+=
∂∂
∑=
θθ
θθθθθ
(A. 4)
193
Appendices 193
A.2 Secant stiffness matrix
( )2
32
1 481055
685
34569216
q
qqq
C+
+++= , ( )2
32
2 4842
25
5764608
q
qqq
C+
+++= (A. 5)
( )3
322
1 484035
12854818
5486
q
qqqxx
b+
+++= , ( )3
322
2 48480
1135
6654814482
q
qqqxxb
+
+++= (A. 6)
194 Appendices
Appendix B Refined plastic hinge formulation
The incremental stiffness relationship of the beam-column element is:
=
2
2
1
1
s
b
b
s
MMMM
∆∆∆∆
−−+
+−−
22
222221
121111
11
ss
ss
ss
ss
SSSSkk
kSkSSS
2
2
1
1
s
b
b
s
θ∆θ∆θ∆θ∆
, (B. 1)
in which subscripts 1 and 2 stand for first and second note respectively; subscripts s and b
stand for the spring rotation at joint and element rotation respectively; Ss1 and Ss2 are plastic
hinge stiffness; k represents the stiffness coefficient in the tangent stiffness matrix [ ]TK
(details can be found in (Iu & Bradford, 2012a)).
Condense the internal degree of freedom of the element and decompose (B.1).
Hence, the member resistance can be calculated,
( )( )
+−
−+
=
2
1
1112211
1222221
2221
1211
2
1 1s
s
sss
sss
b
b
SkSkSkSSkS
kkkk
MM
θ∆θ∆
β∆∆
(B. 2)
in which ( )( ) 2112222111 kkSkSk ss −++=β
The moment-rotation relationship with respect to local coordinates becomes
( )( )( )( )
+−+−
=
2
1
1112
222121
12212222
11
2
1
s
s
sssss
sssss
s
s
/SkSS/kSS/kSS/SkSS
MM
θ∆θ∆
ββββ
∆∆
(B. 3)