development of a second rder inelastic analysis method ... · influence matrix are introduced and...

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 5.2. 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 6.8. 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


tensioned

Deformed geometry when the third


tensioned

Deformed geometry when the fourth


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


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.


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:


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)


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


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


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).


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


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.


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*


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.


(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


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.


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


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


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


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.


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


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.


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


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


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


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.


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


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


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


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.


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


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


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)


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.


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.


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)


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

qq

qq

qq

qq

qq

qq

q

Lq

qq

qq

qq

qq

qq

qq

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)


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


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


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


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


(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


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


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.


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.


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


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


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


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.


‘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.


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,


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.


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


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


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


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


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


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


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.


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.


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


δSAP2000=-0.155δCSA=-0.111

δANSYS=-0.04


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


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


(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


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


δCSA=-0.055

δSAP2000=−0.161

δANSYS=-0.003


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


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


Table 3.7. Horizontal displacement at node 1 at different stages (mm) – without bracing


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


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.


(a1) Elastic horizontal displacements at nodes 1, 2 & 3

(a2) Lateral displacement at node 1 from 1st to the 3rd stages

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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

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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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δSAP2000=1.27

δANSYS=1.11δCSA~ 0.04




0.0

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0.6

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1.0

1.2

0.0 5.0 10.0 15.0 20.0 25.0

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δSAP2000=-0.93δANSYS=2.52

δCSA=-0.6

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0.0 5.0 10.0 15.0 20.0 25.0 30.0

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δSAP2000=-9.36

δANSYS=3.22δCSA=-1.6


(b1) Elastic vertical displacements at nodes 1, 2 & 3

(b2) Elastic vertical displacements at node 1

0.0

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0.8

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-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

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δSAP2000 ~ 0

δANSYS=-0.02

δCSA~ 0




0.0

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-9.00 -8.00 -7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00

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δSAP200=1.67

δANSYS=-0.03

δCSA~ 0

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δSAP2000=5.03δANSYS=-0.03

δCSA~ 0


(c1) Inelastic horizontal displacements at nodes 1, 2 & 3

(c2) Inelastic horizontal displacement at node 1

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Inelastic horizontal displacement (mm)

δ2=-10.2δ1=-9.1

δ3=-11.4

Limit loadλCSA = 0.98

1st PHλCSA =0.76

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δSAP2000=-18.6

δANSYS=1.06

δCSA= -9.1


1st PHλCSA =0.76




0.0

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δSAP2000=-25.53



1st PH λCSA =0.76

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1.2

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δSAP2000=-36.61



1st PHλCSA =0.76


(d1) Inelastic vertical displacements at nodes 1, 2 & 3

(d2) Inelastic vertical displacement at node 1

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-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


1st PH λCSA =0.76

0.0

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-7.00 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00

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δCSA=-0.04Limit loadλCSA = 0.98

1st PHλCSA =0.76




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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1st PHλCSA =0.76

0.0

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1.2

-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00

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1st PHλCSA =0.76


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


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


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


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


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


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.


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

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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


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


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


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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-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

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Vertical displacement (mm)

δA=0.78


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).


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


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


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 &


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

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Elastic horizontal displacement in x-direction (mm)

δ2A=17.0 δ6Α=30.0 δ10A=49.0


Elastic horizontal displacement at nodes 2B, 6B & 10B

Elastic vertical displacement at nodes 2A, 6A & 10A

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Elastic horizontal displacement in x-direction (mm)

δ2B=18.4 δ6Β=68.0 δ10B=247.0

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-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


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

Tota

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δ2B=0.03δ6B=0.1δ10B=-1.1


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

0.4

0.6

0.8

1.0

1.2

0.0 500.0 1,000.0 1,500.0 2,000.0 2,500.0

Tota

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Inelastic horizontal displacement in x-direction (mm)

δ2A=173δ6Α=580

δ10A=580

10A. CSA

1st PHλCSA = 0.702


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

Tota

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Inelastic horizontal displacemt in x-direction (mm)

δ2B=25.5 δ6B=78.0 δ10B=283.0

1st PHλCSA = 0.702



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

1.0

1.2

-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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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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δ2B=-0.66δ6B=-1.30δ10B=-2.70


10B. CSA

1st PHλCSA = 0.702


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


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


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


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


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.


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


(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


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


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


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

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0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

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Vertical displacement in mm

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


CS2CS3

0.418

0.430

1st PHλANSYS=0.920

1st PHλCSA=0.863δANSYS=-10.2

δSAP2000=24.3

δCA=2.34

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

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

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1st PHλCA=0.803 1st PH

λSAP2000=1.033

1st PHλANSYS=0.917

δANSYS=-2.12

δSAP2000=-2.52


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

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

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1st PHλCA=0.803


1st PHλANSYS=0.917

δANSYS=-12.5

δSAP2000=14.4

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


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)


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


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

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

Tota

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CS2

CS10.01

0.899

δCA=-0.003δANSYS=-0.003δSAP2000=-0.003

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0

Tota

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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


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


(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)


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)


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


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.


(a) Vertical displacements at nodes A & B at service stage

(b) Vertical displacements at node A against load factor till limit load level

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-45.0 -40.0 -35.0 -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0

Tota

l con

stru

ctio

n lo

ad fa

ctor


CS2

0.720

δSAP2000=-2.966δANSYS=0.048δCA=-0.012

CS10.240

δSAP2000=0.063δANSYS=0.102

δCA=-0.227

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

Tota

l con

stru

ctio

n lo

ad fa

ctor


CS2

CS10.240

0.720

1st PHλCA=1.120


1st PHλCSA=1.121

λANSYS=1.106λCA.Lim=1.122

λCSA.Lim=1.122 λSAP2000=1.106

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.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-100.0 -80.0 -60.0 -40.0 -20.0 0.0

Tota

l con

stru

ctio

n lo

ad fa

ctor


CS2

0.72

1st PHλCA=1.120


1st PHλCSA=1.121

λANSYS=1.106

λCA.Lim=1.122

λCSA.Lim=1.122

λSAP2000=1.106


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

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-40.0 -35.0 -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0

Tota

l con

stru

ctio

n lo

ad fa

ctor


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


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


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


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


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


[ ]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


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),


( ) ( )[ ]( )

( ) ( )[ ]( )

( ) ( )[ ]( )( ) ( )[ ]( )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


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


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


Figure 5.4. Iterative solution procedure using one-criterion IFMs


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.


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


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.



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


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


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


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.


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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,


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


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


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


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



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


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



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


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


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


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


(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


Scheme Stage Description Target force Pt (kN)

1

1 Construct main frame and struts 1, 2, 3 & 4


2 Prestress struts 2 & 3 3.15

3 Apply vertical load -50kN

2

1 Construct main frame and struts 1 & 4


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


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


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)


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


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.


(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


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


0.444

0.889

δ1=~0.03

δ2=~0.03

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


(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


3 Assemble hangers 2, 3 & girders 2, 3


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


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


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).


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


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


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


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


(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


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


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.


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


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.


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


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.


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


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


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


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.


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)

development of a second rder inelastic analysis method ... · influence matrix are introduced and...

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