shield tunnel

STUDY ON MECHANICAL BEHAVIOR AND

DESIGN OF COMPOSITE SEGMENT

FOR SHIELD TUNNEL

合成セグメントの力学的挙動および設計

法に関する研究

October 2009

Civil and Environmental Engineering

Graduate School of Science and Engineering

WASEDA University

早稲田大学大学院理工学研究科建設工学専攻

WENJUN ZHANG

張穏軍

STUDY ON MECHANICAL BEHAVIOR AND

DESIGN OF COMPOSITE SEGMENT

FOR SHIELD TUNNEL

Dissertation submitted to the faculty of

Science and Engineering of Waseda University for the degree of

Doctor of Philosophy in Engineering

October 2009

WENJUN ZHANG

The dissertation of Wenjun ZHANG is by Prof. Hiroshi SEKI Prof. Teruhiko YODA Prof. Osamu KIYOMIYA Prof. Atsushi KOIZUMI, Committee Chair

WASEDA University, Tokyo, JAPAN

Copyright 2009

By WENJUN ZHANG

I

ABSTRACT

The use of underground space is not just to excavate deeply, but also to enlarge the

cross-section of tunnel and to replace the circular shape with rectangular, multi-circle or

other shape of cross-section in recent years. For these reasons, hydraulic pressure and

earth pressure acting on a tunnel lining are high and the occurred resultant forces are

very large. Therefore, tunnel lining must satisfy the required performances under severe

conditions. However, traditional tunnel linings (called segment in shield tunnel, e.g. steel

segment, concrete segment, and ductile cast iron segment) can not satisfy the required

performances, because which have disadvantages in economy, product, transportation,

and assembly. The need exists for developing new type segment as composite segment.

When designing composite segment, special considerations need to be taken for the

effects of the interface slip, the confinement effect of steel tube on the deflection of the

concrete infill, the number and arrangement of shear connectors, the contributions of

resultant forces in steel tube and concrete infill, and the deflection behavior of composite

section.

By laboratory tests of composite segments, the advantages of light weight, high

strength, superior ductility compared to traditional segments can be achieved. The slip

effect creates an additional bending moment, local bucking of steel plate, and concrete

cracking, which reduce the carrying capacity of composite segment.

The proposed FEM model and mechanical model are suitable for general analysis of

tunnel lining of composite segment under combined loads.

A comparison is made between the cross-section of RC segment and the cross-section

of Closed-composite segment. The reduction of segment thickness is obtained using

Closed-composite segment and this indicates that Closed-composite segment is suitable

for decreasing muck, construction period and the outside diameter of the shield machine.

In general, the costs of construction, risk, and maintenance will be decreased

Key Words: Composite segment; Shield tunnel; Confinement effect; Slip effect;

Nonlinear analysis; limit state design method

III

ACKNOWLEDGES

I would like to thank my supervisor Prof. Atsushi Koizumi for his support,

encouragement, guidance, and advice.

Many thanks are also due the staff at structural laboratory for their help with my tests

and measurements.

All the students at the Koizumi laboratory are appreciated for being such good

colleagues. Special thanks, to the master-students for laughter and highly informal

discussion. Thanks to Takuya Omura for your beautiful pictures. Thanks to Tetsutaro

Suzuki at Maeda Corporation for ordering the experimental data. Thanks to Takeshi

Suzuki for making the Moleman model.

Further, gratefully thank the financial support provided for this study by Japan Iron

and Steel Federation (JISF), and experimental data provided for DRC segment and

SSPC segment by Ministry of Land, Infrastructure, Transport and Tourism (MLIT),

Nippon Civic Consulting Engineers Co., Ltd (NCC), and Japan Steel Segment

Association (JSSA). Special thanks, to Masami Shirato at MLIT and Kunihiko Takimoto

at Kajima Corporation for help.

Finally, Mom, Wife, Son, and my brothers, I would like to thank you very much for

always supporting and being there for me.

V

CONTENTS

ABSTRACT ……………………………………………... V ACKNOWLEDGMENTS …………………………………………III CONTENTS …………………………………………..V LIST of FIGURES ………………………………………….. IX

LIST of TABLES ………………………………………….. XIII

NOTATION …………………………………………. XVII

Chapter 1. INTRODUCTION

1.1 Preface ……………………………...1

1.2 Purpose and scope of this study ……………………….......4

1.3 Layout of this dissertation …...………………………......6

1.4 References …………………………………..7

Chapter 2. MATERIAL PROPERTIES 2.1 Steel component …………………………...9

2.2 Concrete component …………………………............11

2.2.1 Concrete properties ……………………………12

2.2.2 Yield and failure criterion for concrete ………….………….........21

2.3 Shear connectors ……..……………………...22

2.3.1 Behavior of shear Stud ………….…………………24

2.3.2 Behavior of rib connector ………….…….……..........28

2.4 References ………………………......29

Chapter 3. TEST PROGRAM OF CLOSED-COMPOSITE SEGMENT

3.1 Introduction ……………………………35

3.2 Test program …….………………………35

3.2.1 Test specimens ...…………………………35

3.2.2 Test setup …………………………...37

VI

3.3 Test results and discussion .…...…………...…….……43

3.3.1 Load-deflection response ……………………………44

3.3.2 Strain distribution …….…….………….........49

3.3.3 Sensitivity study ….….…….………….........53

3.3.4 Failure modes ….….…….………….........54

3.4 Summary ..…………………………..55

3.5 References ……………………………..56

Chapter 4. FEM ANALYSIS OF CLOSED-COMPOSITE SEGMENT

4.1 Introduction …………………………....57

4.2 Finite element model ….……………………….57

4.2.1 Structure model ……………………………57

4.2.2 Material model ………………………….....59

4.3 Analysis method .………………………......62

4.3.1 Contact analysis …………………………….63

4.3.2 Buckling analysis …….…….…………...........67

4.4 Results of analysis and discussion ………………………......68

4.4.1 Load-deflection response ………………………….....69

4.4.2 Strain distribution …….…….…………..........75

4.4.3 Failure modes …….…….…………........76

4.4.4 Contact status …….…….…………........79

4.4.5 Stress distribution …….…….…………........80

4.5 Discussion of contact analysis ………………………......81

4.6 Summary …..……………………….82

4.7 References …………………………...82

Chapter 5. A MECHANICAL MODEL OF CLOSED COMPOSITE

SEGMENT 5.1 Introduction ……………………...83

5.2 Structural modeling .……………………..84

5.2.1 Basic assumptions ……………………….84

VII

5.2.2 Equilibrium and compatibility conditions ……………......85

5.3 Material model …………………...93

5.3.1 Concrete .…………………........93

5.3.2 Steel …………………......96

5.3.3 Reinforcement embedded in concrete ………..………........96

5.3.4 Shear stud connector ……………………........98

5.4 Cross sectional analysis …………………….........98

5.5 Results and discussion of the proposed method ……….........105

5.7 Summary ..……………………….108

5.8 References …………………….......109

Chapter 6. VERIFICATION OF THE PROPOSED FEM AND

MECHANICAL MODELS 6.1 Introduction ……………………..111

6.2 SSPC segment .…………………….111

6.2.1 Test specimens ………………………111

6.2.2 Test setup ………………….....113

6.2.3 Test results and discussion ………………….....114

6.2.4 FEM results and discussion ………………….....117

6.3 DRC segment .…………………….123

6.3.1 Test specimens ………………………123

6.3.2 Test setup ………………….....128

6.3.3 Test results and discussion ………………….....133

6.3.4 FEM results and discussion ………………….....138

6.4 Results and discussion of the mechanical model ……….........147

6.5 Summary ..……………………….151

6.6 References …………………….......152

Chapter 7. DESIGN OF COMPOSITE SEGMENT

7.1 Introduction ………………………153

7.2 Calculation of loads .………………………..155

VIII

7.2.1 Earth pressure and water pressure …..……………………158

7.2.2 Self weight ………..........................163

7.2.3 Surcharge ………..........................163

7.3 Structural calculation ………………...............163

7.3.1 Elastic equation method ………………………165

7.3.2 Calculation of shell-spring model ………………………......167

7.4 Check of safety of segmental lining .…………………….....171

7.4.1 Allowable stress design method ………………………..171

7.4.2 Limit state design method …………………………176

7.5 Design example …………………….....184

7.5.1 Dimensions of segment . ………………………..184

7.5.2 Ground Conditions ………………………...184

7.5.3 Load Conditions ………………………...185

7.5.4 Calculation of member forces ……………………….186

7.5.5 Designing segmental lining and checking safety ……………192

7.6 Summary ……………………199

7.7 References ……………….......202

Chapter 8. CONCLUSIONS Abstract in Japanese List of papers

IX

List of Figures

Fig.1.1 Schematic and photo of composite segments.……………….…………………...1

Fig.2.1 Stress-Strain curve of the structural steel..………………....……………….......9

Fig.2.2 Von Mises' yield criterion.………………......………………………………......11

Fig.2.3 Stress-strain curves for unconfined concrete………………..........…………….14

Fig.2.4 CFST member………………......………………........……….………………...18

Fig.2.5 Buyukozturk yield and failure envelopes for concrete…………………............22

Fig.2.6 Shear connectors…………………………….……………….…………………23

Fig.2.7 Load-slip curves for shear studs……………………………….………….........26

Fig.3.1 Manufacturing process of Closed-composite segment…..……………………...35

Fig.3.2 Detail of Closed-composite segment specimens…………….…………………36

Fig.3.3 Biaxial Structure Test Machine and measuring instruments…..….……..........38

Fig.3.4 Test setup for Closed-composite segment specimens………………………......39

Fig.3.5 The arrangement of displacement transducers…………………………………39

Fig.3.6 The strain gauges arrangement of Closed-composite segment specimens……42

Fig.3.7 The method for determining the yield load from load-deformation curves……43

Fig.3.8 Load-deflection curves at midspan………………...…………………………...48

Fig.3.9 Measured strain distribution on the surfaces of steel plates along the midspan

section……………………………………...…………………………...50

Fig.3.10 Measured strain distribution along the height of main girder and mortar

surfaces…………….……………………………………...…………………51

Fig.3.11 Load-relative slip strain on interface between top skin plate and mortar infill..52

Fig.3.12 Load-strain curves on skin plates and the edge of main girders……...….........52

Fig.3.13 Effects of the changing thicknesses of steel tube on ultimate carrying

capacity..……………………………………….....…………………………..53

Fig.3.14 Failure modes of the Closed-composite segment specimens………………..54

Fig.4.1 Finite element model of Closed-composite segment…….…..………………..58

Fig.4.2 Von Mises yield surface in the principal stress space.….……………………….59

Fig.4.3 Concrete failure surface in the principal stress space.….……………………….59

Fig.4.4 Stress-strain relation for unconfined concrete under uniaxial compression.....61

X

Fig.4.5 MSC.Marc ‘Continuous’ friction model through Eq. (4.10)..………………......64

Fig.4.6 Stick-Slip friction procedure in MSC.Marc……..…………………………….66

Fig.4.7 Load-deflection curves for Case15 Closed-segment specimen from different

analyses……….…..……………………….……..…………………………….69

Fig.4.8 Load-deflection curves for tested Closed-segment specimens at midspan…......72

Fig.4.9 Relative slip distribution for Closed-segment specimens without shear studs of

Case10………………….…..……………………….……..……………………74

Fig.4.10 Relative slip distribution for Closed-segment specimens with shear studs of

Case14……………….…..……………………….……..……………………75

Fig.4.11 Load-relative slip relationship for Closed-segment specimens with shear

studs……………….…..……………………….……..……………………75

Fig.4.12 Strain distribution for main girder of Closed-composite segment specimens...76

Fig.4.13 Local buckling of Closed-composite segment specimens without shear studs..77

Fig.4.14 Local buckling of Closed-composite segment specimens with shear studs…...77

Fig.4.15 Concrete infill failure of Closed-composite segment specimens…………….78

Fig.4.16 Contact status of Closed-composite segment specimens……………………...79

Fig.4.17 x-directional stress in skin plates of Closed-composite segment specimens...80

Fig.4.18 Effect of friction on the stiffness of Closed-composite segment specimens…..81

Fig.5.1 Calculation model for the composite segment………………………………….85

Fig.5.2 Kinematic of composite segment…….……………………………………......88

Fig.5.3 Strain distribution of composite segment…………………………………......88

Fig.5.4 Load definitions.………………………………………..…………………........91

Fig.5.5 Calculation model for additional moment………………..…………………......99

Fig.5.6 Discretization of composite section…………………….………..………........100

Fig.5.7 Strain and stress distribution of composite segment……………………........102

Fig.5.8 Bending moment-curvature curves of Closed-composite segments...…….......107

Fig.5.9 Load-deflection curves of Closed-composite segments at midspan.....….........108

Fig.6.1 Manufacturing process of SSPC segment……………………….…………….111

Fig.6.2 Detail of SSPC segment specimen…………………..…………………….......112

Fig.6.3 Test setup for SSPC segment specimens…………..…………………………..113

Fig.6.4 The arrangement of measuring instruments on SSPC segment specimens…....114

XI

Fig.6.5 Load-deflection curves of SSPC segment specimens at midspan.….…………115

Fig.6.6 Measured strain distribution along the height main girder…………..…….....116

Fig.6.7 Failure modes of SSPC segment specimens………………..…………….....117

Fig.6.8 Finite element model of an SSPC segment…………….……………….....118

Fig.6.9 Load-deflection curves for tested SSPC segment specimens at midspan……119

Fig.6.10 Load-relative slip relationship for SSPC segment specimens…..……………119

Fig.6.11 Strain distribution for main girder of SSPC segment specimens………….....120

Fig.6.12 Concrete infill failure of SSPC segment specimens………..…………….....121

Fig.6.13 Contact status of SSPC segment specimens………………………………...122

Fig.6.14 x-directional stress in steel tube of SSPC segment specimens….…….........123

Fig.6.15 DRC segment..……………………………………...……………………….123

Fig.6.16 Arrangement of the discharge channel tunnel and ground condition of 4th

section..……………………………………...……………………………..124

Fig.6.17 Segmental ring arrangement…………………………………………………125

Fig.6.18 Configuration and reinforcing bar layout of A1P~A6P segments…………..126

Fig.6.19 Configuration and reinforcing bar layout of DRC segment specimens...……127

Fig.6.20 Test setup for DRC segment specimens…………..………..........................129

Fig.6.21 Testing steps.………..................................................………..........................130

Fig.6.22 The arrangement of displacement transducers on DRC segment specimens...130

Fig.6.23 The arrangement of strain gauges on DRC segment specimens………..……132

Fig.6.24. Load-deflection curves of DRC segment specimens at midspan………......133

Fig.6.25. Measured strain distribution…………………………...………………….....135

Fig.6.26. Load-relative slip curves of DRC segment specimens……………………..137

Fig.6.27. Failure modes under combined positive bending and axial loads…………138

Fig.6.28 Finite element model of a DRC segment…………………..……….....139

Fig.6.29 Load-deflection curves for tested DRC segment specimens at midspan…..140

Fig.6.30 Load-relative slip relationship for DRC segment specimens………………...141

Fig.6.31 Strain distribution for the members of DRC segment specimens…………...143

Fig.6.32 Concrete infill failure of DRC segment specimens………..……………….144

Fig.6.33 Contact status of DRC segment specimens…………………….……………145

Fig.6.34 x-directional stress in skin plate of DRC segment specimens………………146

XII

Fig.6.35 Bending moment-curvature curves of composite segments….………………150

Fig.6.36 Load-deflection curves of composite segments at midspan……………….…151

Fig.7.1 Flow chart of shield tunnel lining design….………………….………………154

Fig.7.2 Flow chart of calculation of the loads…….………………….………………156

Fig.7.3 Calculation model of loosing earth pressure.………………….………………158

Fig.7.4 Structural models of the segmental lining………………….………………163

Fig.7.5 Concept of the additional rate of bending rigidity…………….………………164

Fig.7.6 Stress condition of joint member..……………….…………….………………167

Fig.7.7 Modeled joint member by springs..……………..…………….………………167

Fig.7.8 Bolt and nut……………………….……………..…………….………………168

Fig.7.9 Stress-strain curve………………...……………..…………….………………181

Fig.7.10 Ground condition………………..……………..…………….………………185

Fig.7.11 Structural model………………..……………..…………….………………187

Fig.7.12 Schematic of joint…...…………..……………..…………….………………187

Fig.7.13 Distributions of member forces of A-Ring in circumferential direction…190

Fig.7.14 Distributions of member forces of B-Ring in circumferential direction…191

Fig.7.15 Contour of member forces of segmental lining assembled in a staggered

pattern…...…………..……………..…………….……………………….…192

Fig.7.16 Transition of ultimate limit states for RC segment…….…………………193

Fig.7.17 Section of RC segment and arrangement of main reinforcements…………194

Fig.7.18 Axial force-moment interaction diagram of RC segment.…………………194

Fig.7.19 Transition of ultimate limit states for Closed-composite segment…………196

Fig.7.20 Resultant axial force in each element………………………….…………197

Fig.7.21 Distribution of connectors in segment with distributed loads…….…………197

Fig.7.22 Section of Closed-composite segment and arrangement of shear studs……199

Fig.7.23 Axial force-moment interaction diagram of Closed-composite segment……199

XIII

List of Tables

Table1.1 Conduits under national roads in the wards of Tokyo………………..………...1

Table2.1. The parameters of unconfined concrete……………………………………....13

Table2.2. Stress-Strain models for confined concrete based on Sargin et al…………....15

Table2.3. Stress-Strain models for confined concrete based on Kent and Park………...16

Table2.4. Stress-Strain models for confined concrete based on Kent and Park………...17

Table2.5 Experimental and analytical curve parameters of Candappa et al…………20

Table2.6 Experimental and analytical curve parameters of Attard-Setunge……………20

Table2.7 Experimental and analytical curve parameters of Imran-Pantazopoulou……..20

Table2.8 Coefficients for static stiffness of a shear stud per Equation 2.20..……...26

Table3.1 Details of Closed-segment specimens…………………….…………………36

Table3.2 Mechanical material properties for Closed-composite segment specimens....37

Table3.3 Experimental results of Closed-composite segment specimens.…………......44

Table5.6 Comparison of elastic bending capacities of composite segment specimens..105

Table5.7 Comparison of ultimate bending capacities of composite segment specimens

...…………………………...…………………………106

Table6.1 Details of SSPC segment specimens.……………...………………………...112

Table6.2 Mechanical material properties for SSPC segment specimens………….....112

Table6.3 Experimental results of SSPC segment specimens...………….....…….……115

Table6.4 Details of DRC segment specimens……….……….………………………128

Table6.5 Mechanical material properties for DRC segment specimens.………….......128

Table6.6 Comparison of elastic bending capacities of composite segment specimens..148

Table6.7 Comparison of ultimate bending capacities of composite segment specimens

...…………………………...…………………………149

Table7.1 Earth pressure acting on the lining by Terzaghi…………………………..157

Table7.2. Coefficient (λ ) of lateral earth pressure and coefficient ( k ) of ground reaction

...…………………………...…………………………157

Table7.3. Examples of notation used in the guidelines (Soil condition)...…………….160

Table7.4. Examples of notation used in the guidelines………….……………….......161

XIV

Table7.5 Equations of member forces for conventional model/modified conventional

model...……...…….....…………………………...…………………………165 Table7.6 Effective ratio of the bending rigidity of η and additional rate of ζ ….……166 Table7.7 Spring constant of soil reaction...……………………………………………170

Table7.8 Allowable stresses of concrete for segment (N/mm2)…………………….....171

Table7.9 Allowable stresses of cast-in-place reinforced concrete (N/mm2)….…….....172

Table7.10 Allowable stresses of cast-in-place plain concrete (N/mm2)…..….…….....172

Table7.11 Allowable stresses of reinforcement(N/mm2)..……….…………………….172

Table7.12 Allowable stresses of steel material and welds(N/mm2).…………………173

Table7.13 Allowable buckling stresses(N/mm2)………………...………………….....173

Table7.14 Allowable stresses for local buckling of steel segment(N/mm2)……….....174

Table7.15 Allowable stresses of spheroidal graphite cast iron (N/mm2)..……….....174

Table7.16 Allowable stresses for buckling of spheroidal graphite cast iron (N/mm2)...174

Table7.17 Allowable stresses for buckling of ductile cast iron segment (N/mm2)….....175

Table7.18 Allowable stresses of steel casting for welded structure (N/mm2)…..….....175

Table7.19 Allowable stresses of bolt (N/mm2)...…..............................................….....175

Table7.20 Characteristic values of the strength of concrete for segment (N/mm2)...176

Table7.21 Characteristic values of the strength of cast-in-place reinforced concrete

(N/mm2) ...…..............................................…..... .........................…...........177

Table7.22 Characteristic values of the strength of reinforcement (N/mm2)…………...177

Table7.23 Characteristic values of the strength of steel material and welds (N/mm2)...178

Table7.24 Characteristic values of steel buckling(N/mm2)…………..........................178

Table7.25 Characteristic values for local buckling of steel segment(N/mm2)..........179

Table7.26 Characteristic values of the strength of spheroidal graphite cast iron

(N/mm2)....…..............................................…...............................…...........179

Table7.27 Characteristic values of the strength for buckling (N/mm2)………..........179

Table7.28 Characteristic values for local buckling of ductile cast iron segment (N/mm2)

....…..............................................…...............................…...........180

Table7.29 Characteristic values of steel casting for welded structure(N/mm2)....180

Table7.30 Characteristic values of the strength of bolt(N/mm2)………………..180

Table7.31 Elastic modulus of concrete (segment) (N/mm2)……………………181

XV

Table7.32 Elastic modulus of steel, reinforcement, and spheroidal graphite cast iron

(N/mm2) ....…..............................................…...............................…..........181

Table7.33 Poisson’s ratio………………………………………………………………181

Table7.34 Nominal standard for material factor………………………………………182

Table7.35 Nominal standard for member factor (Concrete segment)…………………182

Table7.36 Nominal standard for member factor (Steel and cast iron segment)………183 Table7.37 Nominal standard for load factor fγ ……………………..…………………183

Table7.38 Nominal standard for structural analysis factor….…………………………183

Table7.39 Nominal standard of structure factor...…………………..…………………184

Table7.40 Combined load case……………….....…………………..…………………185

Table7.41 Spring constant of soil reaction……..…………………..…………………187

Table7.42 Member forces of segmental lining (A-Ring)…….……..…………………188

Table7.43 Comparison of structural type…………………….……..…………………200

XVII

NOTATION

The following notation is used in this book. Generally, only one meaning is assigned

to each symbol, but in cases where more than one meaning is possible, then the correct

one will be evident from the context in which it is used.

A =Area; constant of material

bA =Area of bottom skin plate section

cA =Area of concrete section

cuA =Area of uncracked concrete section

mbA =Area of main girder section in tension

mtA =Area of main girder section in compression

tA =Area of top skin plate section

shA =Cross-sectional area of the shank of a shear stud

B =Width of segment; constant of material; width or span of opening

C =Constant of material; ratio

D =Diameter of an equivalent circular section

D0 =Diameter of segmental lining

hdd =Diameter of the head of a shear stud

shd =Diameter of the shank of a shear stud

cE =Elastic modulus of the concrete

iE =Initial tangent modulus

fE =Secant modulus of concrete measured at peak stress

sE =Elastic modulus of the steel

stE =Strain hardening modulus of steel

EA =Axial rigidity

EI =Bending rigidity

cf = Compressive strength of unconfined concrete

ccf = Compressive strength of confined concrete

tf ′ =Tensile strength of unconfined concrete

XVIII

lxf , lyf =Lateral confining pressures in the two orthogonal directions

ypf =Yield stress of steel plate

udf ′ =Post-buckling stress of steel plate

H =Height of channel connector

Ht =Height of opening

shh =Height of the shank of a shear stud

bI =Moment of inertia of the bottom skin plate section

cuI =Moment of inertia of the uncracked concrete section

mbI =Moment of inertia of main girder section in tension

mtI =Moment of inertia of main girder section in compression

tI =Moment of inertia of the top skin plate section

0K =Ratio between lateral earth pressure and vertical earth pressure

k = Coefficient of ground reaction; local buckling coefficient;

bk =Axial rigidity of bolt cak =Rigidity of axial spring csk =Rigidity of shear spring ckθ =Rigidity of rotational spring

dk =Shape factor

puk =Axial rigidity of joint plate in compression

plk =Axial rigidity of joint plate considering the initial tightening bolt induced compressive strain released

siK =Initial stiffness of a shear stud

L =Length of segment

rL =Length of channel shear connector

bM =Moment carried by the bottom skin plate

cM =Moment carried by the concrete infill

tM =Moment carried by the top skin plate

mbM =Moment carried by the main girders in tension

mtM =Moment carried by the main girders in compression

N-A =Position of neutral axis

n =Number of shear connectors in a group; modular ratio;

XIX

P =Applied load

P0 =Surcharge

uP =Ultimate tensile strength of a shear stud

uQ =Ultimate shear strength of a shear stud

nQ =Ultimate shear strength of a channel shear connector

q =Shear flow; shear flow force; longitudinal shear force per unit length

0R =Outer radius of the lining

cR =Radius of controid of the lining

r =Radius of the internal corner

ar =Radius of a bolt hole

wr =Radius of a washer

S =Shear force of segmental lining

ults =Slip at fracture

bT =Axial force carried by the bottom skin plate

cT =Axial force carried by the concrete infill

tT =Axial force carried by the top skin plate

mbT =Axial force carried by the main girders in tension

mtT =Axial force carried by the main girders in compression

t =Thickness of plate

tj =Thickness of joint plate

tm =Thickness of main girder

ts =Thickness of skin plate

wt =Thickness of a washer

V = ransverse shear load;

bV =Shear force carried by the bottom skin plate

cV =Shear force carried by the concrete infill

tV =Shear force carried by the top skin plate

mbV =Shear force carried by the main girders in tension

mtV =Shear force carried by the main girders in compression

W =Weight of lining per meter in longitudinal direction

XX

α =A function of uniaxial compressive strength of concrete cf ′

β =Coefficient of variability

Vβ =Reduction factor

tγ =The top relative slip between the top skin plate and concrete infill

bγ =The bottom relative slip between the bottom skin plate and concrete

infill

δ =Deflection ε =Strain

cε =Strain in concrete

,h rupε =Ultimate transverse strain in the steel jacket at rupture σ =Stress

1σ , 2σ , 3σ =Principal stresses

1 2σ ′ ′ =Shear stress

1I =First invariant of stress tensor

2J =Second invariant of stress tensor

φ =Curvature

The Mechanical Behavior and Design of Composite Segment for Shield Tunnel

1

Chapter 1. Introduction 1.1 PREFACE

The subsurface under roads in big cities is already crowded with underground

facilities such as railroads, and tunnels for electricity, gas, communications, sewage and

drainage, and conduits. For example, the subsurface of a national highway in Tokyo has

about 33 km of conduits per kilometer of road as shown in Table1.1. For this reason,

newly constructed infrastructure such as subways etc. must be constructed deeper year

by year in order to avoid those conduits [1].

Table 1.1 Conduits under National Roads in the Wards of Tokyo (161.2km of national road

under direct administration of National Government)

Total length (km) No. of kilometers (km) of underground facilities per one road km

Telephone lines 2684.1 16.7

Electric lines 1660.7 10.3

Gas 325.9 2.0

Water supply lines 364.6 2.3

Sewage lines 315.7 2.0

Total 5351.0 33.3

From MLIT

Note: 1. Statistical data of April, 2004

2. Total length refers to total length of under-road conduits

3. Not include conduits to each building.

The shield tunneling method has rapidly become popular in Japan as an effective

tunneling method in urban and for constructing infrastructures. Recently, it has often

become necessary to construct tunnels under severe conditions and to develop new

techniques because there are some restrictions on the alignment of tunnels and the width

of street where the tunnels are located.

Introduction

2

Transverse joint Skin plate

Main girderCircumferential joint

Sealing grooveJoint plate

Segment hanger

Stud

Filling concrete

Transverse joint

Circumferential joint

Ductile cast iron

Infilled concrete

Reinforcing bar

(a) Closed-composite segment (b) Ductile Cast iron and reinforced concrete segment (DRC)[3]

(c) Steel Segment with Pre-filled Concrete (SSPC)[4]

Fig.1.1. Schematic and photo of composite segments

The use of underground space is not just to excavate deeply, but also to enlarge the

cross-section of tunnel and to replace the circular shape with rectangular, multi-circle or

other shape of cross-section in recent years. For these reasons, hydraulic pressure and

earth pressure acting on a tunnel lining are high and the occurred resultant forces are

very large. Therefore, tunnel lining (called segment in shield tunnel, hereinafter, called

segment) must satisfy the required performances under severe conditions. However,

traditional tunnel linings (called segment in shield tunnel, e.g. steel segment, concrete

segment, and ductile cast iron segment) can not satisfy the required performances,

because which have disadvantages in economy, product, transportation, and assembly

[2]. Therefore, the composite segments combined thin steel tube and concrete were

developed as shown in Fig.1.1. These composite segments have the following


3

advantages: (a) reducing the producing periods of composite segment by using steel

tube as permanent formwork for the concrete; (b) obtaining high dimensional accuracy

because of minimizing the deflection of steel form in welding process; (c)

manufacturing an arbitrary section and an arbitrary shape because of excellent

weldability; (d) not be damaged easily like concrete segment in assembling stage

because of six or five sides covered with the steel tube, and compositing a uniform

tunnel section; (e) easily assembling the composite segments because of having high

stiffness, the few openings of the joints, and high dimensional accuracy; (f) decreasing

the construction cost with decreasing the outside diameter of the shield machine and

muck, because the reduction of segment thickness can be achieved; (g) resisting large

flexural moment because of having large carrying capacity; (h) resisting the particular

loads for neighboring construction and sharply curved construction by increasing

thickness of the skin plates; (i) a rational structure for seismic design because of having

superior ductility; (j) the segment width can be enlarged for having stronger main

girders of steel.

Segments can also be classified by the materials used for their production, e.g.

concrete, steel, cast iron or a combination of these. Each material has its own peculiarity

as follows.

Compressive and buckling failure due to thrust force of shield jacks or earth

pressure seldom occurs, because concrete segments have fairly large rigidity and high

resistibility. They have also durability and provide excellent water-tightness if they can

be properly handled and assembled. However, the corner edges of RC segment are

easily damaged because of the segment weight and weak tensile strength. Careful

attention must be paid when the concrete form is removed and the concrete segments

are transported and assembled.

Steel segments are easy to handle, to work, and to modify on site because of

relatively light weight and uniform material, and can ensure the strength, and excellent

weldability. However, steel segments are easy to deform in comparison with the

concrete segments and thus needs to be made consideration to buckling failure in the

case of excessive thrust force or excessive back grouting pressure.

Ductile cast iron segments have excellent strength, good dimensional accuracy as

Introduction

4

products, and also provide excellent waterproof performance. They, like steel segment,

needs to be made consideration to buckling failure or to apply proper corrosion

protection, when the secondary lining is not executed.

Composite segments are composed of steel tube and concrete or Ductile cast iron

tube and concrete. Although composite segments are more expensive than RC segments,

which take advantage of the speed of construction, light weight and high strength of

steel, and the high-mass, rigidity, damping properties, and economy of concrete. In

addition, composite segments can resist the large sectional force induced by

asymmetrical pressure, high hydraulic pressure and earth pressure, and decrease the

height of the used segment and muck of the excavated tunnel cross-section.

1.2 PURPOSE AND SCOPE OF THIS STUDY

The knowledge of this composite interactions as well as elemental behavior

involved in composite structure has developed rapidly during the past several decades.

Much effort has been put forth to better understand and model the behavior of the

composite structure. Research on the subject has been conducted worldwide (U.S.,

Japan, Canada, Europe and Australia).

However, a rational design method for the composite segments can not be

established, because the mechanical behavior of the composite segments is still not clear.

Especially, it is true problem that how to evaluate the effects of the interface slip and the

confinement effect of steel tube on the deflection of the concrete infill, to install the

number and arrangement of shear connectors, and to evaluate the contributions of

resultant forces in steel tube and concrete infill, and the deflection behavior of

composite section.

In general, many underground structures are designed by the allowable stress design

method. In this method various indeterminate factors such as variations in material

properties, acting loads, precision of estimated design loads, analysis model and

structural calculation method etc., are simply assumed based on factors of safety.

Shield tunnel linings (hereinafter called as segmental rings) consist of segments and

many connecting joints, and show complicated mechanical behavior under combined


5

loads. Meanwhile, it is difficult to precisely define the loads acting on segment rings

due to variation in construction conditions. Therefore, the allowable stress design

method as a simplified solution is still used in the design of segments for shield tunnel.

However, it is possible to predict the variations in acting loads and material

properties etc., because of the high technical developments of FEA and measuring

method for the last few years. Therefore, the allowable stress design method is not

rational in economy, because it is not able to directly assess the variations in material

strength, member size, and loads etc. On the contrary, the limit state design is able to

assess the variations in material strength, member size, and loads etc using factors of

safety based on the theories of probability, statistics, and reliability.

The limit state design requires the structure to satisfy two principal criteria: the

ultimate limit state (ULS) and the serviceability limit state (SLS). A limit state is a set of

performance criteria (e.g. vibration levels, deflection, strength, stability, buckling,

twisting, and collapse) that must be met when the structure is subject to loads.

To satisfy the ultimate limit state, the structure must not collapse when subjected to

the peak design load for which it was designed. A structure is deemed to satisfy the

ultimate limit state criteria if all factored bending, shear, and tensile or compressive

stresses are below the factored resistance calculated for the section under consideration.

The limit state criteria can also be set in terms of stress rather than load. Thus the

structural element being analyzed (e.g. a beam or a column or other load bearing

element, such as walls) is shown to be safe when the factored loads are less than their

factored resistance.

To satisfy the serviceability limit state criteria, a structure must remain functional

for its intended use subject to service loads, and as such the structure must not cause

occupant discomfort under design life.

It is true problem that the limit state design is not currently used in the segment

design for shield tunnel. Therefore, one of the purposes of developing a mechanical

model for composite segment is to provide tools suitable for limit state design. The

paper does not address safety coefficients as its purpose is to underscore the phenomena

involved in the issue rather than measuring structural safety.

The purpose of this study can be summarized as follows:

Introduction

6

(1) To evaluate the reliability of the existing provisions for the design of shear

connectors;

(2) To evaluate the reliability of the existing models of the unconfined concrete/

confined concrete;

(3) To study the mechanical behavior of composite segments using the experiments;

(4) To develop a FEM model to study the mechanical behavior of composite segments;

(5) To develop a mechanical model which can be used to analyze the nonlinear

behavior of composite segments with discrete partial shear connection under

combined loads.

(6) Applying the proposed mechanical model in the design method of composite

segment.

1.3 LAYOUT OF THIS DISSERTATION

In order to get an overview of this dissertation the following chapters are list below

with short description of the content.

Chapter 1 briefly introduces the research significance and the research tasks.

In Chapter 2, shear connectors are presented in general. The most common shear

connectors are studied but the focus is on the headed shear stud and rib shear connector.

This chapter also includes theories for mechanical properties of steel and unconfined

concrete/confined concrete materials.

Chapter 3 investigates the mechanical properties of Closed-composite segment by

using the experimental tests, which includes the deflection, load carrying capacity, and

the confinement effect of the composite segments applied on pure bending, when the

thickness of the plates, the dimensional size of shear connectors, the width and length of

the segments are changed.

Chapter 4 deals with structure simulation using the Finite Element Method. The

smeared cracked concrete model and contact analysis of interface between steel tube,

shear connectors, and concrete infill are considered. A comparison between the analyzed

and experimental results indicates that the proposed finite element model can simulate

the mechanical behavior of Closed-composite segments.


7

In Chapter 5, a nonlinear fiber element analysis method is developed for the

inelastic analysis and design of concrete infill steel tubular composite segments with

local buckling and slips effects. Sectional geometry, residual stresses and strain

hardening of steel tubes and confined concrete models were considered in the proposed

mechanical model. The local buckling, slip and effective strength formulas are

incorporated into the nonlinear analysis procedures to account for local buckling and

slip effects on the strength and ductility performance of composite segments under

combined loads. Comparisons are made between the experimental results of

Closed-composite segment and the mechanical predictions of behavior using the

proposed method. Good agreement is found and this indicates that the proposed method

is suitable for general analysis of Closed-composite segment.

Chapter 6 deals with structure simulation of SSPC segment and DRC segment using

FEM and the proposed method. Good agreement is found and this indicates that the

proposed finite element model and the proposed mechanical model is suitable for

general analysis of others type composite segments.

Chapter 7 deals with the cross-section design of composite segment of the fourth

section of the Tokyo Metropolitan Area Outer Underground Discharge Channel based

on the above proposed model.

Finally, Chapter 8 summarizes the outcomes of this research work, draws associated

conclusions.

1.4 REFERENCES 1) Ministry of Land, Infrastructure and Transport Government of Japan (MLIT), 2005.

Progress in the use of the Deep Underground.

2) Japan Society of Civil Engineers, 2006. Standard specifications for tunneling-2006,

Shield tunnels.

3) Masami, Shirato, et al.2003. Development of new composite segment and application

to the tunneling project. Journal of JSCE, No.728, 157-174. (In Japanese)

4) Japan Steel Segment Association (JSSA), 1995. The report of development of Steel

Segment with Pre-filled Concrete(SSPC) . (In Japanese)

The Mechanical Behavior and design of Composite Segment for Shield Tunnel

9

Chapter 2. Material Properties

2.1 STEEL COMPONENT

The primary purpose of the steel element in a composite beam is to carry tensile

stresses, while in composite columns the steel shares in the carrying of compressive

stresses with the concrete. It is the high strength of the steel, coupled with its ductility,

which makes it such a vital component of a composite member.

The steel section is usually made from so-called mild or structural steel. The real

stress-strain curve for a tensile coupon of SS400 steel is shown in Fig. 2.1. Tensile and

yield strengths of the structural steel were obtained by tensional testing according to

Japanese Structural Steel Specification. Initially, the stress-strain curve is linear with an

elastic modulus sE . For most mild steels, the modulus sE is close to 5 22.0 10 N/mm× , and

this is the value generally used in design. If the stress is removed in the elastic zone, the

0.000 0.003 0.006 0.009 0.012 0.0150

50

100

150

200

250

300

350

400

4.5mm

25mm

Strain hardeningPlastic flow

Plastic plateau

Elastic

stE1

1

5 22.0 10 N/mmsE = ×

Stre

ss (N

/mm

2 )

Strain

Real strain-stress(SS400)

Fig.2.1 Stress-Strain curve of the structural steel

Material Properties

10

steel recovers perfectly on unloading. The linear elastic behavior continues until the yield stress yf is reached, at a yield strain y sf / Eyε = . Further straining results in plastic

flow with little or no increase in stress until the strain hardening strain is reached. The

stress in the steel then increases until its ultimate tensile strength is attained. The

cross-section then begins to neck down, with large reductions in the cross-sectional area,

until the steel finally fractures.

Undoubtably, the most important strength property of the steel element is its yield strength yf . In most composite applications, this value is usually between about 250 and

350 N/mm2, although in some structures it may be higher, and it depends largely on the

chemical constituents of the steel, primarily carbon and manganese. The yield stress is

increased with increased amounts of these elements, as well as the amount of working

which takes place during the rolling process. Higher yield stresses are also observed

under higher strain rates /d dtε of loading. Generally speaking, the higher the yield

stress, the less is the plastic plateau in Fig. 2.1 and consequently the ductility is

decreased. Because of this, many structural steel standards place limits on the yield

stress of the steel that may be used, since ductility is a desired requirement in structural

design.

Under uniaxial compression, the stress-strain characteristics of the steel section are

roughly the same as those in tension up to the plastic range. The yield stress yf determined from a tensile strength test is generally accepted as being the same

for compression, along with the elastic modulus sE . However, the steel section under

compression is often subjected to buckling or instability effects.

Quite often, it is appropriate to treat the stress in the steel section as being uniaxial.

However, the general state of stress at a point in a thin-wall member is one of biaxial

tension and/or compression, and yielding under these conditions is not so simply

determined, which uses the notation of Trahair and Bradford [1,2]. The most accepted

theory of two-dimensional yielding under biaxial stresses is the von Mises' maximum

distortion energy theory, and the stresses at yield according to this theory satisfy the

condition

2 2 2 21 1 2 2 1 23 yfσ σ σ σ σ′ ′ ′ ′ ′ ′− + + = (2.1)


11

where 1σ ′ , 2σ ′ are the normal stress, and 1 2σ ′ ′ is the shear stress at the point. For the case

where1' and 2' are the principal stress directions 1 and 2, Eq.(2.1) takes the form of the

ellipse shown in Fig.2.2, while for the case of pure shear( 1 2 0σ σ′ ′= = ,so that

1 2 1 2σ σ σ ′ ′= − = ), Eq.(2.1) reduces to

1 2 3y

y

fσ τ′ ′ = = (2.2)

which defines the shear yield stress, so that the close approximation 0.6y yfτ = is

often used in design.

(a) Two-dimensional Stress Space

Yield surface

Prin

cipa

l stre

ss ra

tio σ

2/f y

Principal stress ratio σ2/f

y1.0

1.0

1.0

1.0

Uniaxial compression

Pure shear

Uniaxial tension

1σ

2σ 3σ

Yield surface

Elastic region

(b) π-Plane

Fig.2.2 Von Mises' yield criterion

2.2 CONCRETE COMPONENT

The second major component contributing to the strength and stiffness of a composite

member is the concrete. The concrete is produced by mixing cement powder with coarse

aggregate (gravel), fine aggregate (sand) and water. Quite often, fly ash and slag waste

from steel blast furnaces are added to increase the workability of the wet concrete mix,

and to reduce the cost of the cement which in terms of mass is the most expensive major

component of the concrete mix.

The strength of hardened concrete varies inversely with its water/cement ratio.

Because the workability of the concrete is reduced as this ratio decreases, it is not

uncommon to introduce various organic admixtures, apart from slag and fly ash, such as

lingo-sulphonate to the mix. A water/cement ratio (by weight) of at least 0.25 is required

Material Properties

12

to hydrate the cement properly, and water/cement ratios in the range 0.35 to 0.50 are

commonly used for normal strength concretes.

Limit states or load and resistance factor design necessitate that both strength and

stiffness requirements of the concrete are met. In achieving these requirements, it must

be noted that the properties of the concrete.

2.2.1 Concrete properties

Concrete is a variable material, and identical strength tests undertaken at a given time

after casting show significant variability. However, the mean strength cmf in uniaxial

compression increases with concrete age. The major shortfall of the concrete portion of

a composite member is its low tensile strength, so that strengths usually quoted for

concrete are in terms of the uniaxial compressive strength.

Concrete compressive strengths are determined by testing specimens of identical

shape, which have been cured under the same conditions, in a stiff hydraulic testing

machine. In Japan, the standard test specimen is a 50/100 mm diameter cylinder, which

is 100/150 mm high. North America and in Australia, the standard test specimen is a

150 mm diameter cylinder, which is 300 mm high with its top capped with sulphur. On

the other hand, British practice is to use a 150 mm sided cube. Because of the shape

effects, the cube strengths cuf are higher than the cylinder strengths cf . Generally

throughout this paper, reference will be made to cylinder strengths of Japanese standard

test specimen.

It is well known that confinement of concrete is effective in increasing its strength

and deformation capacity. It is generally agreed that the strength and stiffness of

confined concrete increases with the stiffness of the confining material as well as the

compressive strength of the unconfined concrete. Because of confinement effect, both

the unconfined and confined properties of concrete must be addressed. These properties

are studied in the following.

(a) Constitutive Models for Unconfined Concrete

In the concrete compressive stage, the stress-strain relation proposed by Carreira and

Chu [3] has been employed to model the elastic-plastic material characteristics with


13

strain softening:

1

cc

cc

c

c

f

α

εαε

σεαε

⎛ ⎞′ ⎜ ⎟⎜ ⎟′⎝ ⎠=

⎛ ⎞− + ⎜ ⎟⎜ ⎟′⎝ ⎠

(2.3)

where cσ is compressive stress in concrete( 2N/mm ); cε is strain in concrete; cf ′ is

uniaxial compressive strength of concrete( 2N/mm ); cε ′ is strain corresponding to cf ′ ;

and α as a function of uniaxial compressive strength of concrete cf ′ , can be estimated

by the following formula[4] 3

1.55cfαβ′⎡ ⎤

= +⎢ ⎥⎣ ⎦

(2.4)

The coefficient of variability β increases when increasing the compressive strength of

the concrete. Therefore, if 221.0N/mmcf ′ = , 22.0β = and if 280.0N/mmcf ′ = , 71.4β = ,

for intermediate stress gradients, β can be determined by linear interpolations. The

proposed relation for β was found using regression analysis based experimental values

shown in Table 2.1.

Table 2.1. The parameters of unconfined concrete

Compressive strength

2(N/mm )cf ′

Elastic Modulus

2(kN/mm )cE

Tensile strength 2(N/mm )tf ′

Peak strain

cε ′ Unit weight

(kN/m3) β

21 22 1.75 0.00205 23 22.0

24 25 1.91 0.00219 23 24.5

27 26 2.07 0.00232 23 27.1

30 28 2.22 0.00245 23 29.6

40 31 2.69 0.00283 23 37.3

50 33 3.12 0.00316 23 39.5

60 35 3.53 0.00346 23 46.5

70 37 3.91 0.00374 23 55.0

80 39 4.27 0.00400 23 71.4

6447.2 10c cfε −′ ′= × or ci2 /cf E′ [5]; ciE is the initial tangent modulus of concrete

Material Properties

14

Concrete in tension is considered as a linear-elastic material until the uniaxial tensile strength tf ′ , can be estimated by the following equation [5]

2 / 30.23( )t cf f′ ′= (2.5)

A comparison of the calculated results of the above proposed constitutive relation for

unconfined concrete and the test results is shown in Fig.2.3. It is shown that the above

proposed constitutive relation is consistent and agrees well with the test results.

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

40

236.6(N/mm )cf ′ =0.00239cε ′ =

concrete

β=35.2

150m

m

100mm

Stre

ss(N

/mm

2 )

Strain(μ)

Test stress-strain curve Eq.(2.3) stress-strain curve

0 500 1000 1500 2000 2500 30000

6

12

18

24

30

36

42

0.002346cε ′ =

241.7(N/mm )cf ′ =

concrete

β=39.5 150m

m

100mm

Stre

ss(N

/mm

2 )

Strain(μ)


(a) (b)

0 1000 2000 3000 4000 50000

10

20

30

40

50

60

70

80

90

0.004707cε ′ =

284.4(N/mm )cf ′ =

mortar

β=71.4

100m

m

50mm

Stre

ss(N

/mm

2 )

Strain(μ)


0 1000 2000 3000 4000 5000 60000

10

20

30

40

50

60

70

80

90

cf ′

cf ′

cf ′cf ′

cf ′

cf ′

cf ′

cf ′cf ′

Stre

ss(N

/mm

2 )

Strain(μ)

=21N/mm2

=24N/mm2

=27N/mm2

=30N/mm2 =40N/mm2

=50N/mm2

=60N/mm2

=70N/mm2

=80N/mm2

(c) (d) Fig.2.3 Stress-strain curves for unconfined concrete

(b) Constitutive Models for Confined Concrete

It is known that the increase in strength of confined concrete is a result of the

combination of lateral pressure and axial compression, which put the concrete in a


15

triaxial stress state. The lateral pressure is provided by lateral steel reinforcement and a

steel jacket. Based on the test results, various stress-strain models for confined concrete

have been proposed, such as Sheikh and Uzumeri[6], Mander et al.[7], and Cusson and

Paultre[8]models. Existing stress-strain models for confined, unconfined normal, as

well as high strength concrete can be divided into three broad categories. One group of

researchers used a form of equation proposed by Sargin et al.[9](Table 2.2). The second

group of researchers proposed second order parabola for the ascending branch and a

straight line for the descending branch and their studies were based on equations

proposed by Kent and Park [10]( Table2.3). The third group developed stress-strain

relations based on equations suggested by Popovics[11] (Table 2.4).

In these stress-strain models, 1 1( , )σ ε are the coordinates of any point in the

stress-strain curve, coε is the peak axial strain of unconfined concrete strength cf , ccε

is the peak axial strain of confined concrete strength ccf , lf is the confining pressure,

cE is the elastic modulus of concrete, iE is the initial tangent modulus, and fE is the

secant modulus of concrete measured at peak stress.

Table 2.2. Stress-Strain Models for Confined Concrete Based on Sargin et al. [9]

2 2( 1) / 1 ( 2)Y AX D X A X DX⎡ ⎤ ⎡ ⎤= + − + − +⎣ ⎦ ⎣ ⎦ ; 1 cc/Y fσ= ; 1 cc/X ε ε=

Researcher A D

Sargin et al. [9] /c co cE kfε 30.65 7.25 10cf−− ×

Wang et al. [12] Different parameters for ascending and descending branches Ahmad and Shah [13] /i fE E

oct1.111 0.876 4.0883( / )cA fτ+ − ; ( )oct23 c lf fτ = −

El-Dash and Ahmad[14] /c fE E 0.033

sp(16.5 ) /( / )c lf f s d⎡ ⎤⎣ ⎦

Attard and Setunge [15] cc cc/iE fε ( ) ( ) ( ) ( )2 2 2

cc cc cc1 / 1 / (1 ) / / 1 /l l lA f f A f f f fα α α⎡ ⎤ ⎡ ⎤− − + − −⎣ ⎦ ⎣ ⎦ Assa et al. [16]

cc cc/cE fε ( ) ( )( ) ( )2 2

80 cc 80 cc 80 cc/ 0.2 1.6 / 0.8 / /Aε ε ε ε ε ε⎡ ⎤− + +⎣ ⎦

80ε is the strain corresponding to 80% of the peak stress, cc0.80 f

Material Properties

16

Table 2.3. Stress-Strain Models for Confined Concrete Based on Kent and Park [10]

Researcher Ascending branch ( )1σ Descending branch ( )1σ

Kent and Park[10] ( ) ( )2

cc 1 12 / 0.002 / 0.002f ε ε⎡ ⎤−⎣ ⎦ ( )cc m 11 0.002f Z ε⎡ ⎤− −⎣ ⎦

Sheikh and

Uzumeri [6] ( ) ( )2c 1 cc 1 cc2 / /Kf ε ε ε ε⎡ ⎤−⎣ ⎦ ( )cc m 1 cc1f Z ε ε⎡ ⎤− −⎣ ⎦

Park et al. [17] ( ) ( )2

c 1 12 / 0.002 / 0.002Kf K Kε ε⎡ ⎤−⎣ ⎦

c

1.0 s yhfK

fρ

= + ;

sρ = volumetric ratio of transverse

steel in concrete core;

yhf = yield strength of transverse steel

( )c m 11 0.002Kf Z Kε⎡ ⎤− −⎣ ⎦

Scott et al. [18] Same as Park et al. (1982)

Samra [19] Same as Kent and Park (1971)

Saatcioglu and

Razvi [20] ( ) ( )1/(1 2 )2

cc 1 cc 1 cc2 / /K

f ε ε ε ε+

⎡ ⎤−⎣ ⎦ ( )cc m 1 cc1f Z ε ε⎡ ⎤− −⎣ ⎦

Saatcioglu et al.

[21] Same as Saatcioglu and Razvi (1992)

Razvi and

Saatcioglu [22] cc /( 1 )rf xr r x− − ( )cc m 1 cc1f Z ε ε⎡ ⎤− −⎣ ⎦

Mendis et al. [23]

( ) ( )2c 1 cc 1 cc2 / /Kf ε ε ε ε⎡ ⎤−⎣ ⎦ ( )c m 1 cc1Kf Z ε ε⎡ ⎤− −⎣ ⎦

Shah et al. [24]

( )cc 1 cc1 / Af ε ε⎡ ⎤−⎣ ⎦ ( )1.151 cc

cckf e ε ε− −


17

Table 2.4. Stress-Strain Models for Confined Concrete Based on Kent and Park [10]

Researcher Ascending branch ( )1σ Descending branch ( )1σ

Popovics[11] ( ) ( )c 1 co 1 co/ /( 1) / nf n nε ε ε ε⎡ ⎤− +⎣ ⎦ ; c0.058 1n f= +

Carreira and Chu[3] cc /( 1 )f xββ β× − − Mander et al.[25] cc /( 1 )rf r r x× − −

Hsu and Hsu [26] ( ) ( ){ }cc 1 cc 1 cc/ / 1 / kf k kε ε ε ε⎡ ⎤− −⎣ ⎦ 0.8( )0.5

cc0.3 dx xf e− −

Cusson and Paultre [8] cc /( 1 )f xββ β× − − 1 1 cc 2( )

cck kf e ε ε− −

Wee et al. [27]

cc /( 1 )f xββ β× − −

( )c co

11 / if E

βε

=−

21 cc /( 1 )kk f x ββ β× − −

3.0

1c

50kf

⎛ ⎞= ⎜ ⎟⎝ ⎠

;1.3

2c

50kf

⎛ ⎞= ⎜ ⎟⎝ ⎠

Hoshikuma et al.

[28]

( ) 1c 1 1 cc1 (1/ ) / nE nε ε ε −⎡ ⎤−⎣ ⎦

c cc

c cc cc

EnE f

εε

=−

( )cc des cccf E ε ε− −

desE is deterioration rate,

and can be calculated by

regression analysis of test

data in the range of ccε to

cuε

In general, we assume that the strength of confined concrete is related to the

contribution of the confinement pressure. Therefore, the strength of confined concrete

can be expressed as the sum of the strength of unconfined concrete and the strength

increase due to the confining stress. For the concrete filled steel tubular(CFST) as

shown in Fig.2.4, the core concrete and the steel tube are in a complex three

dimensional stress state, because the lateral deformation of concrete is confined by the

steel tube when CFST structures are compressed in the axial direction. Therefore, the lateral confining pressures lxf , lyf must be evaluated in the two orthogonal directions,

respectively. For the lateral confining pressure lf as shown in Fig.2.4(c) is calculated as

Material Properties

18

Steel jacket

Concrete

2t

1t

b

h H

B

(a) CFST member details and dimensions

Confined core

Unconfined

H

B (b) Effectively confined region by steel jacket

lf

s lk f

EquivalentAverage

Actual

(c) Variation of confinement pressure

Fig.2.4 CFST member

follows[29]:

,2 s

l lx ly h ruptEf f fD

ε= = = (2.6)

where sE andt are the elastic modulus and the thickness of steel jacket respectively, t is


19

taken as the smaller of 1t and 2t ; ,h rupε is the ultimate transverse strain in the steel jacket

at rupture; and D is the diameter of an equivalent circular section given as 2 2D h b= + ,where h is the short side and b is the long side of the cross-section.

As a function of ratio of the effectively confined area eA to the cross sectional area cA ;

shape factor can be expressed as [30]

( ) ( )2 22 213 (1 )

es

c s

A b r h rkA bh ρ

− + −= = −

− (2.7)

where r is the radius of the internal corner; and sρ is the ratio of the longitudinal steel

jacket in the cross-section. The effective lateral pressure lf ′ acting on the concrete due to

the steel jacket can be calculated as follows:

l s lf k f′ = (2.8)

where sk is the effectiveness coefficient representing the ratio of the effectively

confined area to the total cross section area. lf can be considered as the possible

maximum confining pressure that exerted by steel jacket on the concrete core whereas

lf ′ can be accepted as minimum confining pressure assumed uniformly distributed over the surface of the concrete core as given in Eq. (2.6).

Attard and Setunge [31] proposed the equations for axial strain corresponding to peak axial stress ccε

( )ccc

c

1 17 0.06 l

c

fff

εε

⎛ ⎞′= + − ⎜ ⎟′′ ⎝ ⎠ (2.9)

where lf is confining pressure and cε ′ is axial strain corresponding to peak uniaxial compressive strength. Peak axial stress for confined concrete ccf is defined as

cc

c t

1lf ff f

α⎛ ⎞= +⎜ ⎟′ ′⎝ ⎠

(2.10)

where constantα = given by

( ) 0.21

c

t

1.25 1 0.062 lf ff

α−⎛ ⎞ ′= +⎜ ⎟′⎝ ⎠

(2.11)

and the tensile strength tf ′ is given by Eq.(2.5). The experimental data from the triaxial tests conducted by Ansari and Li[32], Attard

Material Properties

20

and Setunge[31], and Candappa et al.[33] were used to verify the calculated

ccf and ccε by the above Eqs.(2.9) and (2.10).

Table 2.5 Experimental and Analytical Curve Parameters of Candappa et al.

Analytical Experimental

cf

(MPa)

lf

(MPa)

coε

(10-3)

ccε

(10-3)

ccf

(MPa)

ccε

(10-3)

ccf

(MPa)

cc

cc

(an)(exp)

ff

cc

cc

(an)(exp)

εε

100 4 2.80 4.03 132.6 3.91 132.4 1.002 1.031

100 8 2.80 5.26 158.2 6.09 156.4 1.012 0.864

200m

m

98mm

100 12 2.80 6.50 180.2 7.11 170.7 1.056 0.914

Table 2.6 Experimental and Analytical Curve Parameters of Attard-Setunge


cf

(MPa)

lf

(MPa)

coε

(10-3)

ccε

(10-3)

ccf

(MPa)

ccε

(10-3)

ccf

(MPa)

cc

cc

(an)(exp)

ff

cc

cc

(an)(exp)

εε

110 5 2.90 4.27 150.2 3.80 150.0 1.001 1.124

110 10 2.90 5.64 181.0 4.7 171.3 1.057 1.200

200m

m

100mm

110 15 2.90 7.01 207.0 5.35 192.0 1.078 1.310

Table 2.7 Experimental and Analytical Curve Parameters of Imran-Pantazopoulou [34]


cf

(MPa)

lf

(MPa)

coε

(10-3)

ccε

(10-3)

ccf

(MPa)

ccε

(10-3)

ccf

(MPa)

cc

cc

(an)(exp)

ff

cc

cc

(an)(exp)

εε

73.4 3.20 3.25 5.03 93.9 4.95 96.1 0.977 1.016

73.4 6.40 3.25 6.82 111.0 6.50 108.7 1.021 1.049

73.4 12.80 3.25 10.39 139.5 10.45 125.6 1.111 0.994

73.4 25.60 3.25 17.53 185.7 20.25 168.6 1.101 0.866

73.4 38.40 3.25 24.67 225.0 31.05 204.0 1.103 0.795

110m

m

55mm

73.4 51.20 3.25 31.81 261.0 40.90 240.5 1.085 0.778


21

The predictions of the confinement models agree well with the experimental data, and

are shown in Tables. 2.5, 2.6 and 2.7.

The stress-strain curve by Montoya et al.[35] was adopted to model the compressive behavior of confined concrete. The stress cσ is related to the strain cε using the

following formula:

2

1.0

ccc

c c

cc cc

f

A B Cf f

σε ε

=⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2.12)

where dA k= ;sec

2 ABE

= ; 2sec

ACE

= ; and seccc

cc

fEε

= . The shape factor dk is given by the

following formula:

2

80

14

ccd

cc c

fkε ε⎛ ⎞= ⎜ ⎟−⎝ ⎠

(2.13)

where 80cε is the strain corresponding to 80% of the peak stress, cc0.80 f ,and given by the

following formula:

( )80 1.5 89.5 0.6 lc c c

c

fff

ε ε ⎡ ⎤′ ′= + −⎢ ⎥′⎣ ⎦ (2.14)

2.2.2 Yield and failure criterion for concrete

The Buyukozturk yield criterion [36] (adopted in MSC.MARC Code [37]) is used in

nonlinear analysis to identify the yielding condition of concrete. This criterion of

isotropic hardening and associated flow rule is developed to account for the two major

sources of nonlinearity: the progressive cracking of concrete in tension, and the

nonlinear response of concrete under multi-axial compression. Using this criterion,

incremental stress-strain relationships are established in suitable form for the nonlinear

finite element analysis.

Material Properties

22

tf ′

2σ

1σ

Com

pres

sion

Failure surface

tf

cf−

cf ′−

bcf ′−

bcf−

First yield surface

Compression-Compression zone

Compression-tension zone

Compression

Tension-tension zone

Fig.2.5 Buyukozturk yield and failure envelopes for concrete

The proposed failure criterion for concrete by Buyukozturk shown in Fig.2.5 is given

by the following formula:

2 22 0 1 1 03 3J I Iβσ α σ+ + = (2.15)

where 1I is the first invariant of stress tensor; 2J is the second invariant of stress tensor;

and β ,α and 0σ are material constants. These constants are to be determined from

the test data. For the concrete strength range used by Liu et al [38], and Kupfer et al [39], β ,α and 0σ constants were determined by a numerical trial procedure. The best fit

was found by

3β = , 1/ 5α = and 0 / 3Pσ = (2.16)

where P is uniaxial compressive strength of concrete.

2.3 Shear connectors

The bond which must be achieved between the steel element and the concrete

element in a composite member is crucial to the composite action. When the two

elements only touch at an interface such as shown in Fig. 2.6, then they are often tied

together using mechanical forms of shear connection, examples of which are given in


23

Head

ShankWeld collar H

eigh

t tw

(a) Stud (b) Bolt (c) Channel (d) Rib

Concrete element

tf

tw tw

tf

Fig.2.6 Shear connectors

Fig. 2.6. In all cases, the bond must be designed to resist the longitudinal shear forces at

the steel-concrete interface. However, the bond must also be designed to prevent

separation between the steel and concrete elements in order to ensure that the curvature

in the steel and concrete elements is the same. Hence the interface bond must be able to

resist both tensile forces normal to the steel-concrete interface, and shear forces parallel

to the steel-concrete interface.

Shear stud connectors, as shown in Fig. 2.6. (a), are probably the most common type

of mechanical shear connector used, and consist of a bolt that is electrically welded to

the steel member using an automatic welding procedure. The shank and the weld-collar

adjacent to the steel plate are designed to resist the longitudinal shear load, whereas the

head is designed to resist the tensile loads that are normal to the steel-concrete interface.

Bolts can also be attached directly to the plate, prior to casting the concrete, through

friction welding by spinning the bolt whilst in contact with the flange, or by bolting as

shown in Fig. 2.6. (b). in hand welded channels (c), the longitudinal shear load is

resisted mainly by the bottom flange of the channel whilst the top flange resists the

tensile loads normal to the steel-concrete interface. (d) Rib shear connectors rely on

friction and on an aggregate interlock effect.

There is an enormous variety of mechanical shear connectors varying in shape, size,

and methods of attachment. However, they all have the following important similarities.

They are steel dowels embedded in a concrete medium, they have a component that is

designed to transmit longitudinal shear forces, they have a component that is designed

to resist normal tensile forces and hence prevent separation at the steel-concrete

Material Properties

24

interface, and they all impart highly concentrated loads onto the concrete element.

In steel and concrete composite structures, the shear connectors significantly

affecting deformation and maximum carrying capacity [2], is mostly realized by means

of deformable studs [40,41], whereby steel plate to concrete infill shear occurs with

relative slip causing partial interaction [42]. This connection features limited slip

capacity and requires checking to ensure composite beam bending capacity with no

early connection failure [43]. Therefore, currently, the mechanical behavior of shear

connectors used in composite segments for shield tunnel-grouped headed stud and rib

connector will be studied in the following subsections:

2.3.1 Behavior of Shear Stud Although many researchers have investigated the static strength of the shear stud

since the 1950s, perhaps the most extensive research on the static behavior of the

headed stud was performed by Ollgaard, Slutter, and Fisher [44]. They looked at the

effect of the compressive and tensile strength, density, aggregate type, and modulus of

elasticity of the concrete, the diameter of the stud, and the number of connectors per

slab in a standard push-out test. Johnson and Molenstra [45] investigated the effect of

the strength and modulus of elasticity of stud material on the static capacity of the shear

connector and found it to be influential.

(a) Strength of Shear Stud

Many design equations have been developed to estimate the ultimate static strength

of the stud shear connectors. Different researchers have found different variables to be

influential on the static strength. Ollgaard, Slutter, and Fisher [44] proposed an equation

based on concrete properties and on the ultimate tensile strength of shear stud. Oehlers

et al [2] modified the proposed equation of shear strength by Ollgaard, Slutter, and

Fisher. They assumed concrete failure based on a 45 degrees cone, and gave the shear

strength of a shear stud in composite beam: 0.40

0.65 0.35sh

1.35.3 ( ) cu su c

s

EQ A f fEn

⎛ ⎞⎛ ⎞ ′= − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

(2.17)

where uQ is ultimate shear strength of a shear stud(N); n is the number of shear studs in


25

a group; shA is the cross-sectional area of the shank of a shear stud(mm2); cf ′ is uniaxial

compressive strength of the concrete( 2N/mm ); suf is the ultimate tensile strength of a

shear stud( 2N/mm ); cE is elastic modulus of the concrete( 2N/mm ); sE is elastic modulus

of the steel( 2N/mm ).

The ultimate tensile strength of a shear stud is given by the following equation [2]:

sh sh hdsh 2

sh

( )0.642.5 cu

f h h dP A

dn

′ +⎛ ⎞= −⎜ ⎟⎝ ⎠

(2.18)

where uP is ultimate tensile strength of a shear stud(N); shh is the height of the shank of

a shear stud(mm); hdd is the diameter of the head of a shear stud(mm); shd is the diameter

of the shank of a shear stud(mm) .

(b) Load-Slip Curve

Ollgaard, Slutter, and Fisher [44] also derived an empirical expression on the

load-slip relationship for shear stud:

( )0.40.70871uQ Q e δ−= − (2.19)

where Q is the applied load(N); δ is the slip of shear stud(mm).

This equation has a vertical slope at zero load. This was observed by Ollgaard, Slutter,

and Fisher [44] in the load-slip curves due to the bond between the concrete slab and the

steel girder. However, bond at the steel-concrete interface may be lost after being

subjected to service loads for some period of time. Therefore, it is believed that Eq.

(2.19) overestimates the initial stiffness of the shear stud. The stiffness at 0.5 uQ was

proposed as initial stiffness of a shear stud by Oehlers and Bradford [2]. It can be seen

in Eq. (2.20) that the initial tangent stiffness siK increases with the cylinder strength cf ′ .

( )sh 0.16 0.0017u

si

c

QKd f

=′−

(2.20)

where siK is initial stiffness of a shear stud(N/mm)

Oehlers and Coughlan[41] derived the stiffness of the stud shear connector under

static and dynamic loads from 116 push-out test results. From the results of 42 push-out

specimens with 19 mm and 22 mm diameter shear studs, a static load-slip curve was

Material Properties

26

derived from linear regression analyses. Eq. (2.21) shows the load-slip relationship as

the ratio of the slip to the shear stud diameter.

( ) shcA B f dδ ′= + ⋅ (2.20)

The coefficients A and B are listed in Table 2.8[46]. Fig.2.7 shows load-slip

relationships for a shear stud under static loading according to Eqs.(2.19) and (2.20).

Maximum strength of the stud shear connector is assumed to be 95.9kN.

Table 2.8 Coefficients for static stiffness of a shear stud per Equation 2.20 [46]

/ uQ Q A(10-3) B(10-2) / uQ Q A(10-3) B(10-2)

0.1 22 20 0.85 138 72

0.2 40 37 0.9 156 70

0.3 52 48 0.95 223 119

0.4 63 55 0.99 319 170

0.5 80 73 1.0 371 208

0.6 102 96 1.0 406 251

0.7 120 102 0.99* 475 356

0.8 143 108 0.99* 453 178

*: Reducing loads

0.0 2.5 5.0 7.5 10.00

20

40

60

80

100

Load

(kN

)

Slip(mm)

Ollgaard et al. (1971) Oehlers and Coughlan (1986)

Fig.2.7 Load-slip curves for shear studs


27

(c) Ultimate Slip Capacity

Oehlers and Bradford [43] gave the descending branch of the load-slip curve when

fracture of a shear stud. If fracture of a shear stud is assumed to occur when the load has reduced by 1 % from its peak, then the mean value of ults , is given by

( )ult sh0.48 0.0042 cs f d′= − (2.21)

where again the units are in N and mm. The lower 95% characteristic ultimate slip is

given by substituting 0.42 for 0.48 in Eq. (2.21). It can be seen that the slip at fracture

ults reduces as the cylinder strength cf ′ increases, and hence connectors encased in

strong concrete are less ductile than those in weak concrete, and so are more prone to

fracture. The stiffness siK and the slip ults were derived from experimental tests in

which the compressive cylinder strengths cf ′varied from 23N/mm2 to 82N/mm2.

(d) Effect of spacing on shear capacity

Shear studs are often arranged longitudinally and transversally with smaller spacing

between studs (this is referred to hereafter as the grouped arrangement). If shear studs

are grouped very closely together in the connection, the required performance may not

be satisfied. Investigations on grouped arrangement of shear studs indicated that the

strength of shear stud is different in the different grouped arrangement because of the interaction with the next shear studs. The reduction factorη proposed by Okada et al

[47] is given by

2

2

2

2

2

0.023 0.70 3 13, 25(N/mm )

0.021 0.73 3 13, 30(N/mm )

0.016 0.80 3 13, 40(N/mm )

0.013 0.84 3 13, 50(N/mm )

1 13, 25(N/mm )

l l c

l l c

l l c

l l c

l c

a a f

a a f

a a f

a a f

a f

η

⎧ ′+ ≤ < =⎪⎪ ′+ ≤ < =⎪⎪ ′= + ≤ < =⎨⎪⎪ ′+ ≤ < =⎪⎪ ′≥ ≥⎩

(2.22)

where 1 sh( / )la S d= is the longitudinal spacing factor; 1S is the longitudinal spacing of

shear studs(mm).

Material Properties

28

(e) Effect of plate thickness on shear capacity

Tests at Southampton University examined the effect of varying plate thickness and

distance between plates [48-51]. The reduction factor Vβ induced by the plate thickness

is given by

yp(0.024 0.76) and 1355V V

ftβ β= + ≤ (2.23)

where ypf is yield stress of steel plate ( 2yp 355N/mmf ≤ ), and t is thickness of steel

plate ( 10mmt ≤ ).

2.3.2 Behavior of Rib connector

The rib shear connectors (shown in Fig.2.6 (d)) can be considered as the deformed

types of the channel shear connector (shown in Fig.2.6 (c)). Therefore, the strength

calculation formula for rib shear connector can use the strength equation of channel

shear connector. The current American Standard (AISC 1993[52]) provides the

following equation for calculating the strength of a channel shear connector embedded

in a solid concrete slab:

f w r0.3( 0.5 )n c cQ t t L f E′= + (2.24)

where nQ is in Newton; ft is the flange thickness of channel shear connector(mm); wt

is the web thickness of channel shear connector(mm); and rL is the length of channel

shear connector(mm).

However, the study of Amit Pashan[53] reported that the web thickness, the area of

the channel in contact with the concrete should be considered in the calculation formula

for the strength of channel shear connector. Therefore, Amit Pashan proposed the

following equation for calculating the strength of a channel shear connector embedded

in a solid concrete slab:

2f w r w r wn c c c cQ At t f BL H f Ct L f Dt f′ ′ ′ ′= + + + (2.25)

where A, B, C, D are constants to be determined; H is Height of channel connector(mm).

The experimental results of the 18 push-out specimens with solid concrete slabs


29

were used for the regression analysis, and yielded A, B, C, D as

A=597.058; B=5.378; C=-19.066; D=-23.672.

If the above yielded coefficients A, B, C and D are rounded to 597, 5.4, -19 and

-23.7 respectively, and substituted these values into Eq.(2.25). The final form of the

proposed equation is:

2f w r w r w597 5.4 19 23.7n c c c cQ t t f L H f t L f t f′ ′ ′ ′= + − − (2.26)

where nQ is in Newton; ft is the flange thickness of channel shear connector(mm); wt

is the web thickness of channel shear connector(mm); and rL is the length of channel

shear connector(mm); H is Height of channel connector(mm); cf ′ is uniaxial

compressive strength of concrete( 2N/mm ).

2.4. REFERENCES 1) Trahair, N.S. and Bradford, M.A., 1991. The Behaviour and Design of Steel

Structures, revised 2nd edn., Chapman and Hall, London.

2) Deric J Oehlers, Mark A Bradford, 1995: Composite steel and concrete structural

members, UK: Pergamon, Elsevier Science ,1-28.

3) Carreira, D. J., K.H. Chu,1985. Stress-strain relationship for plain concrete in

compression. J., Proc., ACI, 82(11), 797-804.

4) Wenjun, Zhang, Atsushi,Koizumi,2007. Relative slip of interface between concrete

and steel of closed-composite segment. Modern tunnelling technology. pp40-45.

5) Japan concrete committee, 2005. Standard specifications for concrete structures-2002

(structural performance verification).

6) Sheikh, S. A., and Uzumeri, S. M.,1982 . Analytical model for concrete confinement

in tied columns. J. Struct. Div. ASCE, 108 12 ,2703-2722.

7) Mander, J. B., Priestley, M. J. N., and Park, R. ,1988. Theoretical stress-strain model

for confined concrete.” J. Struct. Eng., 114(8), 1804-1826.

8) Cusson, D., and Paultre, P. ,1995. Stress–strain model for confined high strength

concrete.” J. Struct. Eng., 121(3), 468–477.

9) Sargin, M., Ghosh, S. K., and Handa, V. K. ,1971. Effects of lateral reinforcement

Material Properties

30

upon the strength and deformation properties of concrete. Mag. Concrete Res.,

23(75-76), 99-110.

10) Kent, D. C., and Park, R.,1971. Flexural members with confined concrete.J. Struct.

Div. ASCE, 97(7), 1969-1990.

11) Popovics, S., 1973. A numerical approach to the complete stress-strain curve of

concrete. Cem. Concr. Res., 3(5), 583-599.

12) Wang, P. T., Shah, S. P., and Naaman, A. E. ,1978. Stress-strain curves of normal

and lightweight concrete in compression. ACI J., 75(11),603-611.

13) Ahmad, S. H., and Shah, S. P. ,1982. Stress-strain curves of concrete confined by

spiral reinforcement.” ACI J., 79(6), 484-490.

14) El-Dash, K. M., and Ahmad, S. H. ,1995. A model for stress-strain relationship of

spirally confined normal and high-strength concrete columns. Mag. Concrete Res.,

47(171), 177-184.

15) Attard, M. M., and Setunge, S. ,1996. Stress-strain relationship of confined and

unconfined concrete. ACI Mater. J., 93(5), 432-442.

16) Assa, B., Nishiyama, M., and Watanabe, F. ,2001. New approach for modeling

confined concrete. I: Circular columns. J. Struct. Eng.,127(7), 743-750.

17) Park, R., Priestley, M. J. N., and Gill, W. D. ,1982. Ductility of square confined

concrete columns. J. Struct. Div. ASCE, 108(4), 929-950.

18) Scott, B. D., Park, R., and Priestley, N. ,1982. Stress-strain behavior of concrete

confined by overlapping hoops at low and high strain rates. ACI J., 79(1), 13-27.

19) Samra, R. M., 1990. Ductility analysis of confined columns. J. Struct. Eng., 116(11),

3148-3161.

20) Saatcioglu, M., and Razvi, S. R.,1992. Strength and ductility of confined concrete.”

J. Struct. Eng., 118(6), 1590-1607.

21) Saatcioglu, M., Salamat, A. H., and Razvi, S. R. ,1995. “Confined columns under

eccentric loading.” J. Struct. Eng., 121(11), 1547-1556.

22) Razvi, S., and Saatcioglu, M.,1999. Confinement model for high strength concrete.”

J. Struct. Eng., 125(3), 281-289.

23) Mendis, P., Pendyala, R., and Setunge, S.,2000. Stress-strain model to predict the

full-range moment curvature behavior of high-strength concrete sections.” Mag.


31

Concrete Res., 52(4), 227-234.

24) Shah, S. P., A., and Arnold, R.,1983. Cyclic loading of spirally reinforced concrete.”

J. Struct. Eng., 109(7), 1695-1710.

25) Mander, J. B., Priestley, M. J. N., and Park, R.,1988. Observed stress-strain

behavior of confined concrete.” J. Struct. Eng., 114(8),1827-1849.

26) Hsu, L. S., and Hsu, C. T. T.,1994. Complete stress-strain behavior of high-strength

concrete under compression. Mag. Concrete Res., 46(169), 301-312.

27) Wee, T. H., Chin, M. S., and Mansur, M. A.,1996. Stress-strain relationship of high

strength concrete in compression. J. Mater. Civ.Eng., 8(2), 70-76.

28) Hoshikuma, J., Kazuhiko, K., Kazuhiko, N., and Taylor, A. W.,1996. A model for

confinement effect on stress-strain relation of reinforced concrete columns for

seismic design.” Proc., 11th World Conf. on Earthquake Engineering, Elsevier

Science, London.

29) B. Doran a, H.O. Koksal, and T. Turgay,2009. Nonlinear finite element modeling of

rectangular/square concrete columns confined with FRP. Material and design.

30) ACI Committee 440,2002. Design and construction of externally bonded FRP

systems for strengthening concrete structures (ACI 440.2R-02), Farmington Hills

(MI):American Concrete Institute.

31) Attard, M. M., and Setunge, S., 1996. Stress-strain relationship of confined and

unconfined concrete.” ACI Mater. J., 93(5), 432-442.

32) Ansari, F., and Li, Q. ,1998. High-strength concrete subjected to triaxial

compression.” ACI Mater. J., 95(6), 747-755.

33) Candappa, D. ,2000. The constitutive behavior of high strength concrete under

lateral confinement. PhD thesis, Monash Univ., Clayton,VIC, Australia.

34) Imran, I., and Pantazopoulou, S. J.,1996. Experimental study of plain concrete

under triaxial stress.” ACI Mater. J., 93(6), 589-601.

35) Montoya, E.,2003. Behavior and analysis of confined concrete. Ph.D. thesis, Univ.

of Toronto, Toronto.

36) Oral,Buyukozturk, 1977. Nonlinear analysis of reinforced concrete structure.

Computers & Structures, Vol.7, 149-156.

37) MSC.Marc Documentation, 2005. Volume A Theory and User Information. MSC

Material Properties

32

Software.

38) T. C. Y. Liu, A. H. Nilson and F. O. Slate, 1972. Stress-strain response and fracture

of concrete in uniaxial and biaxial compression. ACZ J., Proc. 69(5) 291-295.

39) H. Kupfer, H. K. Hilsdorf and H. Rusch, 1969. Behavior of concrete under biaxial

stresses. ACZ .Z. Pmt. 66(8), 656-666.

40) Ollgaard JG, Slutter RG, Fisher JW,1971. Shear strength of connectors in

lightweight and normaldensity concrete. AISC Engng, 55–64.

41) Oehlers DJ, Coughlan CG., 1986. The shear stiffness of stud shear connectors in

composite beams. J Construct Steel Res,73-84.

42) Oehlers DJ, Nguyen NT, Ahmed M, Bradford MA., 1997. Partial interaction in

composite steel and concrete beams with full shear connection. J Construct Steel

Res,41(2/3),235-48.

43) Oehlers DJ, Sved G.,1995. Composite beams with limited-slip-capacity shear

connectors. J Struct Eng.,32-38.

44) Ollgaard, J.G., Slutter, R.G., Fisher, J.W.,1971. Shear Strength of Stud Shear

Connectors in Lightweight and Normal-Weight Concrete. AISC Engineering

Journal, 8, 55-64.

45) Johnson, R.P., and Molenstra, I.N., 1991. Partial Shear Connection in Composite

Beams for Buildings. Proc. Instn Civ. Engrs, Part 2, 91, 679-704.

46) Gunup Kwon et al, 2007. Strengthening Existing Non-Composite Steel Bridge

Girders Using Post-Installed Shear Connectors. CTR Technical Report: 0-4124-1.

47) Jun,Okada,Teruhiko,Yoda, and Jean-paul, Lebet,2006. A study of the grouped

arrangements of stud connectors on shear strength behavior. Structural

Eng./Earthquake Eng.,JSCE,23(1),75-89.

48) Moy SSJ, Xiao RY, Lilistone D, 1998. Tests for British steel on the shear strength of

the studs used in the Bi-steel system. Department of Civil and Environmental

Engineering, University of Southampton.

49) Clubley SK,Moy SSJ, Xiao RY,2003. Shear strength of steel-concrete-steel

composite panels, Part I-testing and numerical modelling. Journal of

Constructional Steel Research, 59(6),781-94.

50) M. Xie, N. Foundoukosb, J.C. Chapmanb,2004. Experimental and numerical


33

investigation on the shear behaviour of friction-welded bar–plate connections

embedded in concrete. Journal of Constructional Steel Research,61, 625-649.

51) Bowerman HG, Gough MS, King CM, 1999. Bi-steel design and construction guide.

Scunthorpe (London), British Steel Ltd.

52) AISC. 1993. Load and resistance factor design specifications for structural steel

buildings. American Institute of Steel Construction, Chicago,Illinois.

53) Amit Pashan,2006. Behaviour of channel shear connectors: Push-out tests. Master

thesis, Saskatchewan Univ., Saskatoon, Saskatchewan, Canada.


35

Chapter 3. Test Program of Closed-Composite Segment

3.1 INTRODUCTION

The test program involved the testing of 10 Closed-composite segments without shear

studs, and 6 Closed-composite segments with shear studs. For Closed-composite

segments, this paper is to investigate the deflection behavior, load carrying capacity,

relative slip, and the confinement effect of steel tube, when changing the thickness of

the steel plates, dimension of headed shear studs, and the width and length of composite

segments.

3.2 TEST PROGRAM

3.2.1 Test Specimens

Sixteen steel-mortar filled composite segment models are designed, manufactured,

and air-cured. The constructing process of the Closed-composite segments is shown in

Fig.3.1.

(a) Welding shear studs (b) Installing strain gauges (c) Welding steel tube

(d) Mixing material infill (e) Casting mortar

Fig.3.1 Manufacturing process of Closed-composite segment

Test Program of Closed-Composite Segment

36

Table 3.1. Details of Closed-segment specimens

Dimension Thickness Stud arrangement

Skin plate

Main girder

Joint plate spacing Specimen Width

B(mm) Length L(mm)

Height H(mm)

ts(mm) tm(mm) tj(mm)

Welded position

Shankdiameterd(mm) S1(mm) S2(mm)

Case1 200 900 100 4.5 4.5 4.5 No stud

Case2 200 900 100 4.5 4.5 3.2 No stud

Case3 200 900 100 4.5 4.5 6.0 No stud

Case4 200 900 100 4.5 3.2 4.5 No stud

Case5 200 900 100 4.5 6.0 4.5 No stud

Case6 200 900 100 3.2 4.5 4.5 No stud

Case7 200 900 100 6.0 4.5 4.5 No stud

Case8 200 900 100 4.0 4.5 -- No stud

Case9 300 900 100 3.2 6.0 6.0 No stud

Case10 500 900 100 4.5 4.5 4.5 No stud

Case11 200 900 100 4.5 4.5 4.5 Skin plate 4.0 65 37.5

Case12 300 900 100 3.2 6.0 6.0 Skin plate 8.0 60 60

Case13 300 900 100 3.2 6.0 -- Skin plate 8.0 60 60

Case14 500 900 100 4.5 4.5 4.5 Skin plate 4.0 65 37.5

Case15 750 2100 150 4.5 9.0 9.0 Skin plate 13.0 150 100

Case16 1000 2100 150 4.5 9.0 9.0 Skin plate 13.0 150 150

B

t s tj

s2

B

H

L

Skin plate

L

H

tj

s1

AA

B

B

A-A section

B-B section

Joint plate

Joint plate

Main girder

Skin plate Stud

Main girderMortar T

dD

Details of studWelded Case d D T

Case11,14 Case12,13 Case15,16

4.08.0

13.0

11.016.0 22.0

10.510.5 10.5

30.043.0

50.0

Fig.3.2 Detail of Closed-composite segment specimens [1]

The details of the Closed-composite segment specimens are shown in Table3.1 and

Fig.3.2. The compressive strength of the mortar at 28 days was determined by testing

standard 100×50 mm mortar cylinders according to Japanese Concrete Specification.

Tensile and yield strengths of the structural steel were obtained by tensional testing


37

according to Japanese Structural Steel Specification. Mechanical properties of structural

steel (SS400) and mortar are shown in Table 3.2. The mix proportions of the mortar are

as follows: Cement: 666.7 kg/m3; Water: 266.7 kg/m3; Sand: 1333.3 kg/m3 (size: 5mm);

Cement/Water ratio: 40%; Water-reducing admixture: 666.7 cc/m3; Polymer dispersion:

2.22 kg/m3.

Table 3.2. Mechanical material properties for Closed-composite segment specimens

Structural steel (SS400) Mortar

Specimen Yield strength

2(N/mm )yf

Tensile strength

2(N/mm )sf ′

Young modulus

2(N/mm )sE


2(N/mm )cf ′

Young modulus

2(N/mm )cE

Case1 325 448 2.0×105 52.3 2.41×104

Case2 325 448 2.0×105 52.3 2.41×104

Case3 325 448 2.0×105 52.3 2.41×104

Case4 325 448 2.0×105 52.3 2.41×104

Case5 325 448 2.0×105 52.3 2.41×104

Case6 325 448 2.0×105 52.3 2.41×104

Case7 325 448 2.0×105 52.3 2.41×104

Case8 325 448 2.0×105 52.3 2.41×104

Case9 333 423 2.05×105 86.8 2.85×104

Case10 303 430 2.25×105 73.5 2.36×104

Case11 325 448 2.0×105 52.3 2.41×104

Case12 333 423 2.05×105 86.8 2.85×104

Case13 333 423 2.05×105 86.8 2.85×104

Case14 303 430 2.25×105 73.5 2.36×104

Case15 328 457 1.95×105 77.6 2.57×104

Case16 328 457 1.95×105 77.6 2.57×104

3.2.2 Test Setup

The simply supported Closed-composite segment specimens were loaded

symmetrically at four points within the span using a spreader beam. In these test

arrangements, such loading led, in theory, pure bending between applied forces of the

composite segment specimens can be obtained between the two loading points without

the presence of shear and axial forces. The Biaxial Structure Test Machine shown in Fig.

3.3 of 5000kN capability was used to apply monotonic load. The test specimens of


38

Case1 to Case8 were designed to study the effects of the thickness of the steel plates

changing on load carrying capacity; The test specimens of Case9 to Case14 were

designed to study the behavior of composite segments with/without shear studs; Case15,

Case16 were designed to study the interface relative slip between the skin plates and the

mortar infill shown in Fig.3.4. The applied load was measured and recorded using a

load cell. The loads were increased gradually with an increment of approximately 5kN

until the ultimate carrying capacities of Case1 to Case8, Case10, Case11, and Case13.

However, for Case9, Case12 to Case16, initial loading control was based on readings of

the load applied through the load cell, until the load-deflection response became

non-linear. Thereafter, loading was shifted to displacement control based on increments

of midspan deflection, until failure of the specimen was observed or until very large

deflections occurred. Resistance strain gauges were fixed to the inside/outside of the

steel tube and mortar infill. Finally, all the data records were made on data logger (data

acquisition equipment). Displacement transducers and Strain gauges located on key

points shown in Fig.3.5 and Fig.3.6 were used to measure strains and deflections

respectively.

(a) Biaxial Structure Test Machine (c) Data logger

Fig. 3.3 Biaxial Structure Test Machine and measuring instruments

(b) Switch box


39

The numbers 1,2,3 etc. denote the serial numbers of displacement transducer　　:displacement transducer Unit : mm

Case15,16

204 2 6

14

1016

13

91917

7

11158

12

3 1 518

200 250 250 200 1

3

2

6 4

450 600

2100

8

7

600

95

450

Case1 to Case14

900

Fig.3.5 The arrangement of displacement transducers

100 200 300 200 100

900

CL

Composite segment

Abutment

Spreader beam

Load cell

Teflon sheetRubber plate

P

Unit：mmTest setup for Case1~Case14

Roller support(hinged and movable)

(a)Test setup for Case1~Case14

Composite segment

2100

700

Unit：mm

LC

120

150

150

900

190

150

P

450250 250450

AbutmentRubber plate

Teflon sheetRubber plate

Spreader beam

Test setup for Case15,Case16

(b)Test setup for Case15, Case16

Fig. 3.4 Test setup for Closed-composite segment specimens


40

4@25=100

4@33

=122

4@18

=72

3434

18.5

4@50

=200

18.5

200

900

35

1

13151719 30

242628

42

6

252321

2729

1216141820

39

41

40

383752

810

11 22

79

33

35 313236

34

200 250 250 200

Top skin plate

Bottom skin plate

Main girder

Main girder

The outside steel tube of Case1~Case8,Case11

Joint plate

(a) The strain gauges arrangement of Case1~Case8, Case11

1112

1314

3

56

4

12

7

8

910

7776

7574

6463

6665

67

68

69

7071

7372

15161718 26

252423

30292827

22212019

90919293

82838485

81807978

89888786

105106

107108

9495

969798

99100

104103

10210131

32

33

34

3536

39

40

37

385960

6162

5556

5758

44

4546

42

43

41

The inside of top skin plateA

D C

B D

A B

C

A B

C D

C D

A B

C'

B'A'

D' B'

C'D'

A'48 52

54505349

5147

900

180 270 270 180

300

100

900

900

180 270 270 180

300

1545

60

45

6060

888

A' B'

A' B'

888 888

C' D'

C' D'

900

180 270 270 180

300

900

180 270 270 180

300

1545

6060

6030

1010

2737

Case9,Case12,Case13

The outside of top skin plate

The inside of main girder The outside of main girder

The inside of main girder The outside of main girder

100

100

100

The inside of bottom skin plate The outside of bottom skin plate

　：Strain gauges for steel : Strain gauge for mortar Unit: mm

(b) The strain gauges arrangement of Case9, Case12, and Case13


41

106105

103

102

101

012

3

4

5

6

143

142

141

140

139

138

137

124123

122

121

120

119

118 18

19

20

21

22

23

24

37

38

39

40

41

4243

63 64 65 66 67

109

107

111110

108

129

127

125

128

126

148

146

144145

147

165 164 163167 166

7

9

11

8

10

25

27

29

26

28

44

46

48

45

47

68

66

64

62

60

58

56

40

38

36

34

32

30

28

12

13

14

15

16

17

100

117

116

115

114

113112

30

31

32

33

34

35

36

49

50

51

52

53

54

55

156

155

158

159

160

161

162

6162

6059

5857

56

4@18

=72 18

.518

.5

12@

40=4

80

6@75

=450

2525

4@18=7218.5 18.5

180 250

500

900

104

10

20 20

The outside of steel tube of Case10,Case14

Main girder


The outside of bottom skin plate

Main girder

Join

t pla

te

Join

t pla

te18

.518

.5

20

4@18

=72

　：Strain gauges for steel Unit: mm

(c) The strain gauges arrangement of Case10, Case14


42

A B

C D

C' D'

A' B'

50@

18=9

00

45.5

50@

9=45

045

.5

50@

9=45

0

45.5

50@

12=6

00

37.5

@4=

150

37.5

@4=

150

18@2=36

18@2=36

25

３０

３１

３２

３３

３４

３５

３６

３７

３８

３９

450 600 600 4502100

D'

B'

146(24)

147(25)

148(26)

D

B

195.

515

0@2=

300

134(12)

135(13)

136(14)

131(9)

132(10)

133(11)

75

120.

5 C'140(18)

141(19)

142(20)

143(21)

144(22)

145(23)

120.

515

0@5=

750

149(27)

150(28)

151(29)

A'

137(15)

138(16)

139(17)

123(1)

124(2)

125(3)

126(4)

127(5)

128(6)

129(7)

130(8)

195.

515

0@4=

600

195.

5

150@

2=30

0

C

A

195.

5

2100

AA'

C'C

15@6=90

20@3=60 BB'

D'D

The numbers 1,2,3 etc. denote the serial numbers of strain gauges

1000

150

1000

150

45.5

20

21

2223

24

25

26

27

28

29

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

115117119121

116118120122

99101103105

100102104106

8081

838485

86

87

88

89

90

78 79

82

4243

464748

49

50

51

52

53

40

45

5455

56

57

58

5960616264

65

6667

68

69

70

717273747577

44

76

9593

9291

949696 98

111109

108107

110112113 114


The inside of top skin plate

The outside of main girder


The inside of bottom skin plate


Strain gauges Grouting hole Shear studs

The arragement of Case16

D'

B'

B

149(24)

150(25)

151(26)

D

100@

2=20

0

139(14)

138(13)

137（12）

142（17）

141(16)

140(15)

170.

5

152(27)

153(28)

154(29)

143(18)

144(19)

145(20)

146(21)

147(22)

148(23)

100@

5=50

0

C'

A'

A

130（5）

129（4）

128(3)

127(2)

126(1)

100@

4=40

0

133（8）

132(7)

131(6)

170.

5

136(11)

135(10)

134(9)

C

A

750

C'

750

A'

C

100/

3@10

=333

20@2＝4025@4＝100

100/

3@12

=400

20@2＝4025@4＝100

D'

B'

B

100/

3@20

=666

100/

3@10

=333

D2223242526272829303132

2100

123456789101112131415161718192021

37.5

37.5

37.5

34353637383940414243

33

75 75

A'A

B'B

150 118

120122124

119121123125

102104106108

103105107109

2100AA' 15@6=90

20@3=60 BB'

150 98

9695

94

9799100 101

114112

111110

113115116 117

69 70

73

75

77

798081

76

78

7172

74

4445

48

50

52

545556

51

53

4647

49

5758

63

65

6768

6466

59

626160

868584

89

91

9394

9092

85

888786

30.5

30.5


The inside of bottom skin plate



The inside of top skin plate


The numbers 1,2,3 etc. denote the serial numbers of strain gauges

Strain gauges Grouting hole Shear studs

The arragement of Case15

(d) The strain gauges arrangement of Case15, Case16

Fig.3.6 The strain gauges arrangement of Closed-composite segment specimens


43

3.3. TEST RESULTS AND DISCUSSION

Table3.3 gives the values of the ultimate load, the midspan deflections at failure, and

the midspan yield deflections for each of the composite segment specimens. The

midspan yield deflections were taken as the average deflections (measured from the

displacement transducers located at the midspan) corresponding to yield loads equal to

about 80% of the ultimate loads. Table3.3 shows that the deflections of the composite

segments with shear studs are equal to about 70% to 80% of the ultimate deflections

after yielding. The yield load is defined as the load at the intersection of the two lines,

representing the “kink” in the curve where an abrupt change in stiffness occurs. This

bilinear approximation of the yield load shown in Fig.3.7 has been used since the 1970s.

Deflection(mm)

Pmax

Tang

ent l

ine1

Tangent line2

Pyield

Load

(kN

)

Fig.3.7. The method for determining the yield load from load-deformation curves


44

Table 3.3. Experimental results of Closed-composite segment specimens

Specimen Yield load(kN)

Ultimate load (kN)

Yield load /Ultimate load

(%)

Yield deflection

yδ (mm)

Deflection at failure uδ (mm)

( ) /u y uδ δ δ−

(%)

Case1 380 470 80.9 3.35 8.19 59.1

Case2 375 450 83.3 3.50 7.39 52.6

Case3 380 470 80.9 3.35 9.62 65.2

Case4 315 425 74.1 3.0 8.74 65.7

Case5 415 500 83.0 3.49 9.31 62.5

Case6 298 370 80.5 3.16 7.78 59.3

Case7 410 530 77.4 3.28 10.74 69.5

Case8 260 345 75.4 2.87 10.87 73.6

Case9 250 380 65.7 2.88 16.6 82.7

Case10 570 705 80.9 3.9 13.0 70.0

Case11 400 505 79.2 3.01 11.3 73.4

Case12 352 427 82.4 3.89 27.9 86.1

Case13 304 425 71.6 3.53 20.4 82.7

Case14 620 790 78.5 3.52 20.2 82.6

Case15 1000 1222 81.8 10.48 35.7 70.6

Case16 1244 1533 81.1 12.06 39.1 69.2

3.3.1 Load-Deflection Response

Load and midspan deflection relationships for all tested specimens can be described

by the load-midspan deflection curves shown in Fig.3.8. It can be observed that the

load-midspan deflection curves are linear up to a load corresponding to the yield point.

After yielding, composite segments with shear studs represents very large deflection or

excellent ductility, and maintain load carrying capacity to achieve the ultimate

deflection. Fig.3.8 (a) shows the effect of thickness of steel tube on load carrying


45

capacity. It can be observed that the thickness of steel tube markedly affect the load

carrying capacities of the composite segments. Fig.3.8 (b), (c), and (d) show the effect

of shear studs on load carrying capacity. It can be observed that the load carrying

capacities of the composite segments with shear studs are larger than those of the

composite segments without shear studs, and composite structural members show

differently the degree of shear connection. From Fig.8 (e), it can be observed that the

joint plates as shear connectors markedly affect the load carrying capacities of the

composite segments without shear studs. However, the joint plates only affect the

deflection of the range from yield load to ultimate load, and not affect the load carrying

capacities of the composite segments with shear studs shown in Fig.8 (f). It can be

concluded that joint plate should be considered as shear connectors, and resist the shear

force occurring in the interface between steel tube and concrete infill with shear studs in

ultimate limit analysis.

0 2 4 6 8 10 120

50

100

150

200

250

300

350

400

450

500

550

Case3

Case2

Case1

Case6

Case4

Case5Case7

Load

(kN

)

Deflection at midspan(mm)

Yield pointCase1 (3.35,380)Case2 (3.50,375)Case3 (3.35,380)Case4 (3.00,315)Case5 (3.49,415)Case6 (3.16,298)Case7 (3.28,410)

(a) Load-deflection curves for Case1~Case7 without shear studs


46

0 2 4 6 8 10 120

50

100

150

200

250

300

350

400

450

500

550

900 200 100mmL B H× × = × ×

Case1 without shear studs

Case11 with shear studs

Load

(kN

)


Yield pointCase1 (3.35,380)Case11 (3.01,400)

(b) Load-deflection curves for Case1 without shear studs and Case11 with shear studs

0 2 4 6 8 10 12 14 16 18 20 220

100

200

300

400

500

600

700

800

900 500 100mmL B H× × = × ×



Load

(kN

)



(c) Load-deflection curves for Case10 without shear studs and Case14 with shear studs


47

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

900 300 100L B H× × = × ×


Case12 with shear studsLo

ad(k

N)



(d) Load-deflection curves for Case9 without shear studs and Case12 with shear studs

0 2 4 6 8 10 120

50

100

150

200

250

300

350

400

450

500

550

900 200 100mmL B H× × = × ×Note: Case1 and Case8 without shear studs

Case1 with joint plates

Case8 without joint plates

Load

(kN

)



(e) Load-deflection curves for Case1 with joint plates and Case8 without joint plates


48

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

Note: Case12 and Case13 with shear studs

900 300 100L B H× × = × ×

Case13 without joint plates

Case12 with joint platesLo

ad(k

N)



(f) Load-deflection curves for Case12 with joint plates and Case13 without joint plates

0 6 12 18 24 30 36 420

200

400

600

800

1000

1200

1400

1600

Case16: 2100 1000 150L B H× × = × ×

Case15: 2100 750 150L B H× × = × ×

Case15

Case16

Load

(kN

)



(g) Load-deflection curves for Case15 and Case16 with shear studs

Fig.3.8. Load-deflection curves at midspan


49

3.3.2 Strain Distribution

Fig.3.9 shows the measured strain distribution along the composite segment width

(with the origin at the center of the top skin plates) on the surfaces of the skin plates and

main girders. Such curves are displayed under different loading levels for the midspan

sections of the composite segment specimens. In general, the strain values on the

surfaces of the skin plates of the composite segments without shear studs are larger than

those of the composite segments with shear studs (Fig.3.9(a), (b)), and the top skin plate

of the composite segment without shear studs appears the local buckling(the strain

values on the surfaces of the top skin plate shift from negative to positive), it can be

concluded that no shear connection is provided between the mortar infill and the steel

plates, and the two components work independently resisting the moment. It is observed

from Fig.3.9(c) that the strain values on the skin plates increase and decrease repeatedly,

and the strain distribution shows markedly wavy change with increasing load. The strain

response is attributed to the shear lag effect of shear studs embedded in the skin plates.

Some values of the strains in the skin plates are larger than the values of the strains on

the edges of the main girders. The confinement effect of the main girders

on the deflection of the skin plates is greater. Based on testing investigation, it can be

-420 -350 -280 -210 -140 -70 0 70 140 210 280 350 420

-2000

-1500

-1000

-500

0

500

1000

1500

2000

Stud position

200kN

150kN

100kN

Bottom skin plateGirderBottom skin plate Girder

(mm)Distance from the middle of the width of the top skin plate

Top skin plate

Mic

rostr

ain

Case9 Case12 Case9 Case12 Case9 Case12

(a) Measured strain distribution on the surfaces of steel plates along the midspan section for

Case9 without shear studs and Case12 with shear studs


50

-630 -540 -450 -360 -270 -180 -90 0 90 180 270 360 450 540 630

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

Stud position

400kN

200kN

100kN

Bottom skin plateGirderBottom skin plate Girder


Top skin plate

Mic

rostr

ain

Case10 Case14 Case10 Case14 Case10 Case14

(b) Measured strain distribution on the surfaces of steel plates along the midspan section for

Case10 without shear studs and Case14 with shear studs

-900 -600 -300 0 300 600 900

-1500

-1000

-500

0

500

1000

1500

Bottom skin plateGirderBottom skin plate GirderStud position


Top skin plate

Mic

rost

rain

Case15 40kN 200kN 400kN 600kN

(c) Measured strain distribution on the surfaces of steel plates along the midspan section for

Case15 with shear studs Fig.3.9. Measured strain distribution on the surfaces of steel plates along the midspan section

observed that the shear studs and the main girders can affect not only the overall

response (e.g., shape of the load-deflection curve), but also local results (e.g., strain

distributions along the skin plates).


51

-1600 -1200 -800 -400 0 400 800 1200 1600 2000 2400

100

80

60

40

20

0 Strain of mortar surface for typical applied load

Strain of mortar surface

Case2 without shear studs900 200 100mmL B H× × = × ×

100kN 150kN 250kN 350kN

Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

-1600 -1200 -800 -400 0 400 800 1200 1600 2000 2400

100

80

60

40

20

0 Strain of mortar surface for typical applied load


Case11 with shear studs900 200 100mmL B H× × = × ×

100kN 150kN 250kN 350kN

Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

-2000-1500-1000 -500 0 500 1000 1500 2000 2500 3000 3500

100

80

60

40

20

0

100kN 300kN 400kN 500kN 600kN


Strain of mortar surface for typical applied load

900 500 100mmL B H× × = × ×


Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

-2000-1500-1000 -500 0 500 1000 1500 2000 2500 3000 3500

100

80

60

40

20

0



900 500 100mmL B H× × = × ×

100kN 300kN 500kN 600kN 700kN


Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

-1800 -1200 -600 0 600 1200 1800 2400 3000160

140

120

100

80

60

40

20

0



Case15 with shear studs2100 750 150mmL B H× × = × ×

Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

200kN 400kN 600kN 800kN 1000kN

-3000-2400-1800-1200 -600 0 600 1200 1800 2400 3000160

140

120

100

80

60

40

20

0



2100 1000 150mmL B H× × = × ×

400kN 700kN 1100kN 1400kN


Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

Fig.3.10. Measured strain distribution along the height of main girder and mortar surfaces


52

-5000 -4000 -3000 -2000 -1000 0 1000 20000

250

500

750

1000

1250

2100 750 150mmL B H× × = × ×

Case15Lo

ad(k

N)

Microstrain

Strain on the top skin plate contacting mortar infill Strain on mortar contacting the top skin plate Relative slip strain on the interface of the top

skin plate and mortar infill

Fig.3.11. Load-Relative slip strain on interface between top skin plate and mortar infill

-3000 -2500 -2000 -1500 -1000 -500 00

250

500

750

1000

1250

Compressive edge

2100 750 150mmL B H× × = × ×

Case15

Microstrain

Load

(kN

)

The average value of the all strain gauages on top skin plate The strain value of the compressive edge of main girder

0 3000 6000 9000 12000 15000 180000

250

500

750

1000

1250

Tensional edge

2100 750 150mmL B H× × = × ×

Case15

The average value of all strain gauages on bottom skin plate The strain value of the tensional edge of main girder

Microstrain

Load

(kN

)

Fig.3.12. Load-strain curves on skin plates and the edge of main girders [2]

Fig.3.10 shows the typical applied load-strain curves of the test data for main girders

and mortar surfaces. The strain values for the main girders were taken as the average

strain values (from the measured strain based on the symmetrically-installed strain

gauges on the midspan main girders). It can be observed that the plane sections remain

plane after loading, because the fitted curves of the strain distribution along the height

main girder is linear in elastic and plastic ranges, and relative strain occurs on interface

between the skin plates and mortar infill. Fig.3.11 shows the applied load-relative strain

curves of the test data for interface between the top skin plate and mortar infill. It can be

observed that the relative strain occurs on interface between the skin plate and mortar

infill, and increases as the applied load increases. The strain values for the skin plates

were taken as the average strain values (from the measured strain based on the installed


53

strain gauges on the inside and outside of skin plates along the width of the specimen)

shown in Fig.3.12. The strain values of the edges of main girders are about equal to the

average values of the skin plate. Therefore, it can be concluded that the shear studs

installed in compressive and tensional sides resist the compressive resultant force

occurring on the compressive main girder and top skin plate, and on the tensional

resultant force occurring on the tensional main girder and bottom skin plate,

respectively.

3.3.3 Sensitivity Study

An investigation was performed to evaluate the sensitivity of the overall response of

Closed-composite segments (represented by ultimate loads) to likely variations in the

thicknesses of the steel plates. Composite segment specimens of Case1 to Case7 without

shear studs described in Table3.1 were used for the sensitivity study. The dimension of

the compared standard case (Case1) is L×W×H = 900×200×100 mm; all the thicknesses

of skin plates, main girders, and joint plates are 4.5 mm. If the ratios of ultimate

carrying capacity are defined as the ratios of the ultimate carrying capacities of Case2,

Case3, etc to the ultimate carrying capacity of Case1, the relationship of the ratio of

ultimate carrying capacity and percentage change of the thicknesses of the steel plates is

shown in Fig.3.13. It can be observed that the thicknesses of the skin plates obviously

affect the ultimate carrying capacities of the Closed-composite segments.

70 80 90 100 110 120 130 1400.6

0.8

1.0

1.2

The ultimate carrying capacity of Case2~Case7ui :PThe ultimate carrying capacity of standard case(Case1)us :P

(%) :ρ

ui

us

(%) ( 2,3,4,5,6,7)P iP

ρ = =

Ratio of the ultimate carrying capacity(%)

Rat

io o

f ulti

mat

e ca

rryi

ng c

apac

ity

Percentage change of the thicknesses of steel plates(%)

Thicknesses of skin plates Thicknesses of main girders Thicknesses of joint plates

Fig.3.13. Effects of the changing thicknesses of steel tube on ultimate carrying capacity


54

3.3.4 Failure Modes

The failure modes of the composite segments with/without shear studs are shown in

Fig.3.14. The local buckling failure mode of the composite segments with shear studs

(Fig.3.14 (a)) behaves differently from that of the composite segments without shear

studs(Fig. 3.14 (b)), where the top skin plate buckled between two adjacent rows of

shear studs. Local buckling failure causes the top skin plate to separate from the mortar

infill. Fig. 3.14 (c) shows the mortar infill crushing failure after the buckled top skin

plate has been removed from the surface to reveal the mortar infill. It can be observed

from Fig. 3.14 (c) that the failure of mortar infill in compression is crushing when

reaching its ultimate strain.

(a) Local buckling of specimen with studs (b) Local buckling of specimen without studs

(c) Mortar infill crushing failure

Fig.3.14. Failure modes of the Closed-composite segment specimens


55

3.4. SUMMARY

Experimental studies on the mechanical behavior of Closed-composite segments

with/without shear studs or ribs were carried out under pure bending load. The main

conclusions are obtained as follows:

(1) The failure modes of the Closed-composite segments with shear studs are the top

skin plate buckled between two adjacent rows of shear studs, and concrete infill

crushing after the buckled top skin plate.

(2) For the Closed-composite segments, local buckling of skin plate in compression

before yielding can be achieved by limiting the ratio of shear stud spacing/plate

thickness to constant.

(3) The relative slip occurs at interface between concrete infill and the steel tube for

Closed-composite segment.

(4) Considering the confinement effect by the adjacent segment ring on longitudinal

direction in actual tunnel, and the small deflection on transversal direction, the effect

of this deflection can be neglected in actual design of segment.

(5) The joint plates only affect the deflection of the range from yield load to ultimate

load, and not affect the load carrying capacities of the composite segments with

shear studs. It can be concluded that joint plate should be considered as shear

connectors, and resist the shear force occurring in the interface between steel tube

and concrete infill with shear studs in ultimate limit analysis.

(6) The strain distribution in the skin plates is non-uniformly distributed along the width.

The strain response is attributed to the shear lag effect. Therefore, the effective

width of the skin plates should be defined based on stress gradient.

(7) The plane sections remain plane after loading, because the strain distribution along

the height main girder is linear.


56

3.5. REFERENCES

1) Wenjun, Zhang, Atsushi, Koizumi, 2007.Relative slip of interface between concrete

and steel of closed-composite segment. Modern tunnelling technology. pp40-45. (In

Japanese)

2) Wenjun, Zhang, Atsushi, Koizumi, 2009. Flexural behavior of composite segment.

Journal of JSCE. Vol.65,No2,246-263. (In Japanese)


57

Chapter 4. FEM Analysis of Closed-Composite Segment

4.1 INTRODUCTION

Using the finite element method to involve complex structures or interactions among

structural members has been one of the motivations for applying this method in this

study. Comparing with simple mechanical models, finite element models may offer

more accurate analyses because of the ability to model the material and interaction of

each member of the composite segment in more detail. Further, the response history of

virtually any part of the model can be obtained. In this method, element and material

model types play an important role for the entire analysis. Selection of element and

material model types for the analysis is based on the structural system and any specific

need or emphasis of the study.

In this study, the MSC.Marc software package is used to simulate the mechanical

behavior of the Closed-composite segments, and the material properties of each

composite structural member described in Chapter 3 are used in the following finite

element analysis.

4.2 FINITE ELEMENT MODEL

4.2.1 Structure Model Only one quarter of each the Closed-composite segment specimens are modeled

taking advantage of symmetry in two mutually perpendicular vertical planes shown in

Fig4.1. Symmetric boundary conditions are applied at the two vertical planes of

symmetry. 8-Node 3D solid elements with tri-linear interpolation and nonlinear

interpolation are used to model steel plates and concrete, respectively. Discrete shear

studs are modeled using 3D beam elements with elastic-plastic behavior, which can

assume a slip bond between the shear studs and the surrounding mortar, because it is

observed that shear studs separate from the surrounding concrete in these

Closed-composite segment specimens. The cross-sectional area of the beam element

FEM Analysis of Closed-Composite Segment

58

was modified to make it equivalent in both strength and stiffness to the actual shear

studs in composite segment. The vertical and horizontal constraints are applied at the

bottom of the support along its centre line and mid-span section respectively. Vertical

load is distributed to all the nodes on top of the loading plate, with nodes at symmetry

plane getting half of the load at rest of the nodes.

Skin plateveiw-1

veiw-2

view-1 view-2

Skin plate

x

yz

y

x

y

z

Full scale model

Main girder

Joint plate

Main girder MortarJoint plateMortarSkin plate

One quarter model considering symmetry of load and structure Bottom skin plate(Soild element)

Shear stud(3D Beam element)

Top skin plate(Soild element)

Mortar (Soild element)

Interface

Interface

(a) Closed-composite segment (b) FEM model

(c) Boundary condition (f) Modeled shear studs

Fig.4.1 Finite element model of Closed-composite segment


59

4.2.2 Material Model

The material properties were obtained from elemental tests as presented in Chapter 2,

so material model used in finite element analysis are briefed in the following sections.

Incremental plastic flow theory is applied for the steel, shear connector and concrete

materials whereas nonlinear elasticity theory is applied for shear bond interaction. J2-

plasticity (von Mises) with associated flow rule is used for the steel material of the steel

tube and shear connector. In this case, the yield surface is independent of the hydrostatic

component of the stress vector as shown in Fig.4.2. The top skin plate buckling at the

maximum positive moment region was observed in some specimens, therefore, buckling

is assumed in the model. On the other hand, the concrete material is pressure dependent.

The general shape of failure surface for concrete material is illustrated in Fig.4.3.

MSC.Marc uses the Buyukozturk failure surface, a two-parameter model, for concrete

material (Buyukozturk 1975). This model is valid for the nonlinear response of concrete

under multi-axial compression.

Recent developments in the application of fracture mechanics to concrete in tension

enabled a fracture mechanics model to be used for the tensile portion of the concrete.

Concrete behavior in tension is assumed as linearly elastic. Concrete cracks, and softens

linearly until a strain of 0.005 where it completely loses its tensile load carrying capacity, when principal stress in tension reaches tensile strength tf ′ . Tensile softening

Failure surface

π plane

2σ

3σ

1σ

Hydrostatic axis

Yield surface

Failure surfaceπ plane2σ

3σ

1σ

Hydrostatic axis

Fig.4.2 Von Mises yield surface Fig.4.3 Concrete failure surface

in the principal stress space in the principal stress space


60

modulus is calculated based on this assumption. Tensile strength is adopted based on

Japanese Concrete Specification:

2 / 30.23( )t cf f′ ′= (4.1)

where, cf ′ is uniaxial compressive strength of concrete( 2N/mm ). The reduction in shear modulus due to mortar cracking was defined as a function of

direct strain across the crack in the shear retention model. The shear modulus of cracked mortar is defined as eG Gϕ= , where eG is elastic shear modulus of uncracked mortar;

ϕ is reduction factor, which is given by the following equation (Thevendran 1999[1])

maxmax

max

(1 )

0

cc

c

for

for

ε ε εεϕ

ε ε

⎧ − <⎪= ⎨⎪ ≥⎩

(4.2)

where cε is direct strain across the crack. The shear retention model states that the shear

stiffness of open cracks reduces linearly to zero as the crack opening increases.

It is well known that confinement of concrete is effective in increasing its strength

and deformation capacity. It is generally agreed that the strength and stiffness of

confined concrete increases with the stiffness of the confining material as well as the

compressive strength of the unconfined concrete. Because of confinement effect, both

the unconfined and confined properties of concrete must be addressed. These properties

are briefed in the following.

(a) Constitutive models for unconfined concrete

The uniaxial stress-strain relation for concrete in compression is modeled using the

proposed equation by Carreira and Chu. This equation is given by:

1

cc

cc

c

c

f

α

εαε

σεαε

⎛ ⎞′ ⎜ ⎟⎜ ⎟′⎝ ⎠=

⎛ ⎞− + ⎜ ⎟⎜ ⎟′⎝ ⎠

(4.3)



61

3

1.55cfαβ′⎡ ⎤

= +⎢ ⎥⎣ ⎦

1

cc

cc

c

c

f

α

εαε

σεαε

⎛ ⎞′ ⎜ ⎟⎜ ⎟′⎝ ⎠=

⎛ ⎞− + ⎜ ⎟⎜ ⎟′⎝ ⎠

cε ′

cuε

tuε

tf ′

cf ′

ε

σ

Fig.4.4 Stress-strain relation for unconfined concrete under uniaxial compression

uniaxial compressive strength of concrete( 2N/mm ); cε ′ is strain corresponding to cf ′ ;

and α as a function of uniaxial compressive strength of concrete cf ′ . Fig.4.4 shows the

stress-strain relation for concrete.

(b) Constitutive models for confined concrete

It is known that the increase in strength of confined concrete is a result of the

combination of lateral pressure and axial compression, which put the concrete in a

triaxial stress state. Based on the test results, various stress-strain models for confined

concrete have been proposed, such as Sheikh and Uzumeri(1982), Mander et al. (1988),

and Cusson and Paultre(1995)models, etc. Montoya et al. proposed a concrete

confinement model for steel tube confined concrete is found good to predict the shape of stress-strain curve. The stress cσ is related to the strain cε using the following

formula:

2

1.0

ccc

c c

cc cc

f

A B Cf f

σε ε

=⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4.4)

where dA k= ;sec

2 ABE

= ; 2sec

ACE

= ; and seccc

cc

fEε


following formula:


62

2

80

14

ccd

cc c

fkε ε

⎛ ⎞= ⎜ ⎟−⎝ ⎠ (4.5)

where 80cε is the strain corresponding to 80% of the peak stress, cc0.80 f ,and given by the

following formula:

( )80 co 1.5 89.5 0.6 lc c

c

fff

ε ε ⎡ ⎤= + −⎢ ⎥⎣ ⎦ (4.6)

Peak axial stress for confined concrete ccf is defined as

cc

c t

1lf ff f

α⎛ ⎞′

= +⎜ ⎟⎜ ⎟⎝ ⎠

(4.7)


( ) 0.21

c

t

1.25 1 0.062 lf ff

α−⎛ ⎞ ′= +⎜ ⎟′⎝ ⎠

(4.8)

and the tensile strength tf is given by Eq.(4.1).

The effective lateral pressure lf ′acting on the concrete due to the steel jacket can be

calculated as follows:

l s lf k f′ = (4.9)


confined area to the total cross section area and lf is the possible maximum confining

pressure that exerted by steel jacket on the concrete core. sk and lf can be calculated by

Eqs.(2.8) and (2.6) respectively.

4.3 ANALYSIS METHOD

Among the three sources of nonlinearity: material, geometrical and boundary

(contact), the three are applicable to composite segment problems in this study. FEA is

an approximate technique, and there exist many methods to solve the basic equations. In

nonlinear FEA, two popular incremental equilibrium equations are: full Newton


63

-Raphson and modified Newton-Raphson. The full Newton-Raphson (N-R) method

assembles and solves the stiffness matrix for every iteration process. It has quadratic

convergence properties, which means in subsequent iterations the relative error

decreases quadratically. It gives good results for most nonlinear. Therefore, the full

Newton-Raphson (N-R) method is used in this study. The integration scheme used to

trace the equilibrium path is the arc-length method. The loading achieved by applying a

fixed displacement is distributed to all the nodes on top of the loading plate, and

convergence tolerance on displacement is defined as 0.1. Boundary (contact) and

geometrical nonlinearities are presented as the following sections.

4.3.1 Contact Analysis (Boundary Nonlinearity) Analysis is performed for slip condition to investigate the interface between

concrete and steel tube. A contact analysis possesses a provision for slip, and can be

used to simulate the slip between concrete and steel tube. Separate nodes are assigned

for concrete and steel tube at the same location. Frication is also considered in the

contact analysis for studying the interface between concrete and steel tube. In

MSC.Marc, the two available friction models are referred to in the documentation [2] as

the ‘Coulomb’ model and the ‘Stick-Slip’ model. The names are unfortunate since both

are approximate implementations of the theoretical Coulomb model. The following

sections briefly describe the implementation of these two friction models in MSC.Marc.

Full details can be found in [2]. Contact is a nonlinear boundary value problem. During

contact, mechanical loads and perhaps heat are transmitted across the area of contact. If

friction is present, shear forces are also transmitted. In MSC.Marc, areas of potential

contact do not need to be known prior to the analysis.

(a) Continuous Friction Model

The Coulomb friction law is expressed in the MSC.Marc documentation [1] as:

fr n tσ μσ≤ − ⋅ (4.10)

where nσ is the magnitude of the normal stress, frσ is the tangential (friction) stress


64

vector, μ is the friction coefficient, t is the tangential unit vector in the direction of the relative velocity ( )r rt v / v= and rv is the relative sliding velocity. Notations in bold face in Eq.(4.10) indicate vectors while a dot is used to indicate the scalar product of two vectors. Clearly, this equation results in a discontinuity in frσ when the relative velocity rv changes sign. This will cause numerical convergence difficulties, so in the ‘Continuous’ friction model, the discontinuity in Eq. (4.10) is removed by the use of a smoothing arctangent function, resulting in:

rfr n

V

2 varctan tR

σ μσπ

⎛ ⎞≤ − ⎜ ⎟⎝ ⎠

(4.10)

where VR is the value of the relative sliding velocity when sliding occurs. As noted by

Winistőrfer and Mottram [3], this parameter is a nonphysical quantity introduced as a

part of the approximate ‘Continuous’ implementation. Nevertheless, it is an important

parameter in determining how closely the model represents the discontinuity in the Coulomb model. As can be seen in Fig.4.5, for large values of VR the model fails to

represent the step function behavior accurately and results in a reduced effective friction

force, while a very small value results in a very good representation, but like the step

function, can result in nonconvergence. Note that no distinction is made between static

and kinetic friction coefficients in this model.

-10 -8 -6 -4 -2 0 2 4 6 8 10

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0Equation(4.10)

2n 5(N/mm )σ =

0.3μ =

rv

frσ

Rv=0.01Rv=0.1Rv=1Rv=10

Fig.4.5 MSC.Marc ‘Continuous’ friction model through Eq. (4.10).


65

(b) Stick-Slip Friction Model

Similar to the procedures outlined in [3], the Stick-Slip model attempts to capture the

discontinuity in the idealized Coulomb model through a series of conditional statements.

The model is based on a force implementation of the Coulomb law:

t nf f tμ≤ − ⋅ (4.11)

where tf is the tangential or friction force and nf is the magnitude of the normal

force. For the friction coefficient, the user supplies a kinetic coefficient kμ and an

‘overshoot parameter’ α (typically >1); a static coefficient sμ is then deduced from:

s

k

μαμ

= (4.11)

The program flow as provided in [1] is shown schematically in Fig.4.6, in which tuΔ

represents the tangential relative displacement between two contacted components. A

node is considered sticking until the tangential (friction) force reaches a critical value

nfμ . It then slips in the same direction until tuΔ either changes sign or becomes very

small (signifying that sliding has either changed direction or ceased). It can be seen that

before a node is allowed to change the direction of its slip, it must pass through a sticking phase, to test again if the tangential force exceeds the critical value nfμ . The

procedure also requires testing to determine if the tangential force value has converged:

tp

t

f1 1f

e e− ≤ ≤ + (4.12)

where ptf is the tangential force in the previous iteration and e is a convergence

tolerance which can be user-defined. To apply the Stick-Slip model, the following

parameters need to be supplied:

α , the parameter relating the static and kinetic friction coefficients (default

value=1.05; can be user-defined) β , the relative change in displacement tolerance above which a node is considered

slipping (default value=10-6; can be user-defined) ε , the ‘small’ number set so that βε is ‘very small’, (value=10-6; cannot be

user-defined); βε is the relative change in displacement tolerance below which

motion is assumed to have ceased forcing a change from slipping to sticking mode

e, convergence tolerance on the tangential force (default=0.05; can be user-defined).


66

t s nf fμ≤

t s nf fμ>tu βεΔ ≈or if

tu βΔ >andt tf 0u⋅Δ >

tu βΔ >andt tf 0u⋅Δ <

t 0uΔ ≈

Remain in sticking mode if:

Change to slipping mode if:

Remain in slipping mode if:

Change to sticking mode if:

No Yes

Determine soulutionof next iteration

Assume sticking mode

Assume slipping mode

Initial contact

Fig.4.6 Stick-Slip friction procedure in MSC.Marc (from [1])

(c) Bond-Slip Model for Steel Reinforcement

In general, a reinforced concrete member is subjected to axial tensile force. When the

stress attains the tensile strength of concrete, the first crack appears and the relative slip

occurs between steel reinforcement and surrounding concrete. The following constitutive relation between the bond stress vτ and the slip v is used in Hanswille’s

theory [4]:

( ) ( )Nv cux Af v xτ = (4.13)

where A and N are constants and cuf is the cubic compressive strength of concrete, and

x is the longitudinal coordinate of the member. Note that N is not a non-dimensional

parameter but dimensional one. If the unit of length is cm, then Hanswille[3] reported

that A=0.58 and N=0.3 are standard values for a deformed bar.

This feature of contact analysis allows for the automated solution of problems where

contact occurs between deformable and deformable/rigid bodies. It does not require

special elements to be placed at the points of contact. This contact algorithm

automatically detects nodes entering contact and generates the appropriate constraints to

insure no penetration occurs and maintains compatibility of displacements across

touching surfaces.


67

4.3.2 Buckling Analysis (Geometrical Nonlinearity) In MSC.Marc[1], Buckling analysis allows the user to determine at what load the

structure will collapse. It can detect the buckling of a structure when the structure’s

stiffness matrix approaches a singular value. It can extract the eigenvalue in a linear

analysis to obtain the linear buckling load. It can also perform eigenvalue analysis for

buckling load in a nonlinear problem based on the incremental stiffness matrices.

The buckling option estimates the maximum load that can be applied to a

geometrically nonlinear structure before instability sets in. To activate the buckling

option in the program, use the parameter BUCKLE. If a nonlinear buckling analysis is

performed, also use the parameter LARGE DISP.

Use the history definition option BUCKLE to input control tolerances for buckling

load estimation (eigenvalue extraction by a power sweep or Lanczos method). It can

estimate the buckling load after every load increment. The BUCKLE INCREMENT

option can be used if a collapse load calculation is required at multiple increments.

The linear buckling load analysis is correct when take a very small load step in

increment zero, or make sure the solution has converged before buckling load analysis

(if multiple increments are taken). Linear buckling (after increment zero) can be done

without using the LARGE DISP parameter, in which case the restriction on the load step

size no longer applies. This value should be used with caution, as it is not conservative

in predicting the actual collapse of structures.

In a buckling problem that involves material nonlinearity (for example, plasticity),

the nonlinear problem must be solved incrementally. During the analysis, a failure to

converge in the iteration process or nonpositive definite stiffness signals the plastic

collapse.

For extremely nonlinear problems, the BUCKLE option cannot produce accurate

results. In that case, the history definition option AUTO INCREMENT allows automatic

load stepping in a quasi-static fashion for both geometric large displacement and

material (elastic-plastic) nonlinear problems. The option can handle elastic-plastic

snap-through phenomena. Therefore, the post-buckling behavior of structures can be

analyzed. Full details can be found in [1].


68

The buckling option solves the following eigenvalue problem by the inverse power

sweep method:

( ), , 0GK K u uλ σ φ+ Δ Δ Δ =⎡ ⎤⎣ ⎦ (4.14)

where GKΔ is assumed to be a linear function of the load increment to PΔ cause

buckling.

The geometric stiffness used for the buckling load calculation is based on the stress

and displacement state change at the start of the last increment. However, the stress and

strain states are not updated during the buckling analysis. The buckling load is therefore

estimated by:

(beginning)P Pλ+ Δ (4.15)

where for increments greater than 1, (beginning)P is the load applied at the beginning

of the increment prior to the buckling analyses, and λ is the value obtained by the

power sweep or Lanczos method.

The control tolerances for the inverse power sweep method are the maximum number

of iterations in the power sweep and the convergence tolerance. The power sweep

terminates when the difference between the eigenvalues in two consecutive sweeps

divided by the eigenvalue is less than the tolerance. The Lanczos method concludes

when the normalized difference between all eigenvalues satisfies the tolerance. The

maximum number of iterations and the tolerance are specified through the BUCKLE

history definition option.

4.4 RESULTS OF ANALYSIS AND DISCUSSION

All composite segment specimens were analyzed using 3D finite element analysis of

MSC.Marc software package. Parameters of Closed-segment specimens are described in

Chapter 3.2.1. Initially, a Closed-composite segment specimen was analyzed with an

assumption of perfect bond between steel tube and concrete infill. It can be seen that the

stiffness of analysis result is larger than that of the experimental result shown in Fig.4.7.


69

For instance, the stiffness from the analysis is about 150% of the experimental stiffness.

Therefore, contact analysis was used to study the interface interaction. Various

conditions of the interface were assumed. Analysis was performed for a range of friction

coefficient from 0(frictionless) to 0.3. Contact analysis with a friction coefficient of 0.1

is found to produce favorably comparable result shown in Fig.4.7. The analyzed results

of the composite segment specimens are compared with the experimental results in the

following sections.

0 5 10 15 20 25 30 35 40 450

200

400

600

800

1000

1200

1400

2100 750 150mmL B H× × = × ×

Load

(kN

)


Case15Exprimental result Perfect bond analysis Contact analysis (friction Coeff.0.2)Contact analysis (friction Coeff.0.1)

Fig.4.7 Load-deflection curves for Case15 Closed-segment specimen from different analyses

4.4.1 Load-Deflection Response

Among the analysis results, load vs. midspan deflection response histories of

Closed-segment specimens (with shear studs, without shear studs) are shown in Fig. 4.8.

In general, the analytical results agree with the experimental values. The analytical

model successfully described the interface slip behavior, because contact analysis (steel

tube and concrete infill) is used.


70

0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)


Case1Exprimental result FEM result

0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)



(a) Case1 without shear studs (b) Case2 without shear studs

0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)



0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)



(c) Case3 without shear studs (d) Case4 without shear studs

0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)



0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)



(e) Case5 without shear studs (f) Case6 without shear studs


71

0 2 4 6 8 10 120

50100150200250300350400450500550600

900 200 100L B H× × = × ×

Load

(kN

)



0 2 4 6 8 10 120

50

100

150

200

250

300

350

400

450

900 200 100L B H× × = × ×

Load

(kN

)



(g) Case7 without shear studs (h) Case8 without shear studs

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

900 300 100L B H× × = × ×

Load

(kN

)



0 3 6 9 12 150

100

200

300

400

500

600

700

800

900 500 100mmL B H× × = × ×

Load

(kN

)



(i) Case9 without shear studs (j) Case10 without shear studs

0 5 10 15 20 250

50

100

150

200

250

300

350

400

450

900 200 100L B H× × = × ×

Load

(kN

)



0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

900 300 100L B H× × = × ×

Load

(kN

)



(k) Case11 with shear studs (l) Case12 with shear studs


72

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

Note: Case13 with shear studs

900 300 100L B H× × = × ×

Load

(kN

)


Case13 without joint platesExprimental result FEM result

0 3 6 9 12 15

0

100

200

300

400

500

600

700

800

900 500 100mmL B H× × = × ×

Load

(kN

)



(m) Case13 with shear studs (n) Case14 with shear studs

0 5 10 15 20 25 30 35 40 450

200

400

600

800

1000

1200

1400

2100 750 150mmL B H× × = × ×

Load

(kN

)



0 5 10 15 20 25 30 35 40 450

200

400

600

800

1000

1200

1400

1600

2100 1000 150mmL B H× × = × ×

Load

(kN

)



(o) Case15 with shear studs (p) Case16 with shear studs

Fig.4.8 Load-deflection curves for tested Closed-segment specimens at midspan

Comparing the analyzed results with the experimental results, it can be seen that the

load-deflection curves from numerical analysis agree with the experimental load-

load-deflection curves shown in Fig. 4.8.

Figs.4.9 and 4.10 show the relative slip along the length of Closed-segment

specimens and load-relative slip relationship, respectively. It can be seen that the top

maximum relative slip between the top skin plate and concrete infill occurs in shear

region near the supports, and the bottom maximum relative slip occurs in concrete

cracking region. Therefore, the maximum slip depends on overall segment behavior and


73

that a significant contribution to slip increase comes from segment length, plasticization

of the most stressed zones and concrete infill cracks. The calculated maximum slip by

simplified assumption that shear forces along the connection is constant and

disregarding the portion close to the zero slip point where shear force decreases.

Meanwhile, the interface slip-load curves of composite segment are linear up to the

ultimate load shown in Fig.11. Therefore, it can be assumed that the shear stress at the

interface is proportional to the interface slip in analytical solutions.

Non-deform Deformed

(a) The deformation contour of the top interface

Non-deform Deformed (b) The deformation contour of the bottom interface


74

0 50 100 150 200 250 300 350 400 4500.0

0.4

0.8

1.2

1.6

2.0

Case10 without shear stud relative slip between top skin plate and concrete infill relative slip between bottom skin plate and concrete infill

900 500 100mmL B H× × = × ×

Hor

izon

tal r

elat

ive

slip

(mm

)

Half length of closed-composite segment(mm) Fig.4.9 Relative slip distribution for Closed-segment specimens without shear studs of Case10

Non-deform Deformed

(a) The deformation contour of the top interface

Non-deform Deformed

(b) The deformation contour of the bottom interface


75

0 50 100 150 200 250 300 350 400 4500.0

0.4

0.8

1.2

1.6

2.0 Case14 with shear stud relative slip between top skin plate and concrete infill relative slip between bottom skin plate and concrete infill

900 500 100mmL B H× × = × ×H

oriz

onta

l rel

ativ

e sli

p (m

m)

Half length of closed-composite segment(mm)

Fig.4.10 Relative slip distribution for Closed-segment specimens with shear studs of Case14

0.00 0.02 0.04 0.06 0.08 0.10 0.120

200

400

600

800

1000

1200

300mm distance from midspan

Load

(kN

)

Relative slip between the top skin plate and mortar infill (mm)

Nonlinear analysis(MSC.Marc) Experiment(Case15 with shear studs)

Fig.4.11 Load-relative slip relationship for Closed-segment specimens with shear studs

4.4.2 Strain Distribution

Fig.4.12 shows the strain distributions at the measured locations. The strain

distributions of nonlinear analysis show good agreement with the experimental results

in main girders. The tendency of the strain distribution of nonlinear analysis shows

close agreement with the experimental value in the skin plates.


76

-3000 -2000 -1000 0 1000 2000 3000 4000100

80

60

40

20

0

Line PlotFEM result Experimental result

900 500 100mmL B H× × = × ×

100kN 300kN 400kN 500kN 600kN


Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

-3000 -2000 -1000 0 1000 2000 3000 4000 5000100

80

60

40

20

0


100kN 300kN 500kN 600kN 700kN

900 500 100mmL B H× × = × ×Case14 with shear studs

Hei

ght o

f mai

n gi

rder

(mm

)

Compressive edge

Microstrain

Fig.4.12 Strain distribution for main girder of Closed-composite segment specimens

4.4.3 Failure Modes

The local buckling failure of the Closed-composite segments without/with shear studs

is shown in Figs.4.13 and 4.14, respectively. The local buckling failure mode of the

composite segments with shear studs (Fig.4.14) behaves differently from that of the

composite segments without shear studs (Fig.4.13), where the top skin plate buckled

between two adjacent rows of shear studs. Local buckling failure causes the top skin

plate to separate from the concrete infill. The failure of concrete infill is shown in

Fig.15.


77

Fig.4.13 Local buckling of Closed-composite segment specimens without shear studs

Fig.4.14 Local buckling failure of Closed-composite segment specimens with shear studs


78

(a) Equivalent of cracking strain in tension (b) Longitudinal cracks in tension

(c) Plastic strain in compression (d) Concrete crushing in compression

Fig.4.15 Concrete infill failure of Closed-composite segment specimens

Figs.4.13 and 4.14 show images of the finite element model and the experiments for

Closed-composite segment specimens, with those deformed geometry. Figs.4.13, 4.14,

and 4.15 compare the failure pattern from composite segment specimens with the local

buckling and cracking results from the finite element model. It can be seen that there is

agreement between the local buckling distribution in the simulation and the local

buckling which appeared during the test. Even though the discrete nature of the concrete

cracks is not able to be captured by the smeared crack model, the crack pattern predicted

using the finite element model is similar to the observed pattern.


79

4.4.4 Contact Status

Contact status, contact force were carefully analyzed throughout the incremental

solution and the model incorporating a non-positive definite solution seemed to be of

satisfactory quality because no anomalies existed in the stresses or strains and contact

penetration was not evident. Fig.4.16 shows the images of contact status and contact

force of Closed-composite segment specimens.

(a) 600kN contact status (without shear studs)

(b) 600kN contact status (with shear studs)

Fig.4.16 Contact status of Closed-composite segment specimens


80

Fig.4.16 compares the contact status of Closed-composite segment specimens with

shear studs and without shear studs. It can be seen that contact area of Closed-composite

segment specimens with shear studs is larger than that of Closed-composite segment

specimens without shear studs. Therefore, shear studs prevent the steel tube to separate

from the concrete infill.

4.4.5 Stress Distribution

Fig.4.17 shows the distributions of the longitudinal stress (x-direction) in the skin

plates of Closed-composite segment specimens. It can be seen that apparent uniform

stress can be found at the skin plates, if neglecting stress concentration around shear

studs. Stress of top skin plate still increases as the applied load increases, when the top

skin plate buckled.

Fig.4.17 x-directional stress in skin plates of Closed-composite segment specimens


81

It can be seen from Fig.4.17 that the local buckling of the skin plate in compression

should be adopted in analysis and design of composite segment, according to different

degree of shear connection. The skin plate in tension can be assumed as the member of

full effective cross-section.

4.5 DISCUSSION OF CONTACT ANALYSIS

All Closed-composite segment specimens were analyzed with an assumption of

perfect bond between concrete infill and steel tube. Closed-composite segment

specimens were found too stiff from the analysis as compared to the experimental

results shown in Fig.4.18.Later, contact analysis was performed to study the interface

interaction, and Coulomb friction model was used in this study. Various conditions of

the interface were assumed. Analysis was performed for a range of friction coefficient

from 0 (frictionless) to 0.3. Contact analysis with a friction coefficient of 0.1, and the relative sliding velocity ( VR ) of default value 0.1 are found to produce favorably

comparable result shown in Figs.4.8 and 4.18.

0 5 10 15 20 25 30 35 40 450

200

400

600

800

1000

1200

1400

2100 750 150mmL B H× × = × ×

Load

(kN

)


Case15Exprimental result Contact analysis (friction Coeff.0.2)Contact analysis (friction Coeff.0.1)Contact analysis (frictionless)

Fig.4.18 Effect of friction on the stiffness of Closed-composite segment specimens


82

4.6. SUMMARY

The proposed finite element model using MSC.Marc have been examined for their

ability to simulate the mechanical behavior of Closed-composite segments. The main

conclusions are obtained as follows:

(1) Contact analysis was found to be able to simulate the slip at the interface between

steel tube and concrete infill. For the composite segments with no axial load,

frictionless condition could simulate the load-deflection curves from the

experimental results. For the composite segments, a friction coefficient of 0.1

produced convincing results.

(2) Discrete shear studs modeled using 3D beam elements is able to simulate the

mechanical behavior of shear connectors.

(3) The effective width of the skin plate in compression should be adopted in analysis

and design of composite segment, according to different degree of shear connection.

The skin plate in tension can be assumed as the member of full effective

cross-section.

4.7. REFERENCES

1) Thevendran, V., et al,1999. Nonlinear analysis of steel-concrete composite beams

curved in plan. Finite Elem. Anal. Design, 32(3), 125-139.

2) MSC.Marc Documentation, 2005. Volume A Theory and User Information. MSC

Software.

3) Winistőrfer, A. and Mottram, J.T, 2001. Finite Element Analysis of Non-Laminated

Composite Pin-Loaded Straps for Civil Engineering, Journal of Composites Materials,

35(7):577–601.

4) Hanswille, G,1986. Zur Rißbreitenbeschränkung bei Verbundträgern, Technisch-

Weissenschaftliche Mitteilungen, Institut für Konstruktiven Ingenieurbau

Ruhr-Universität Bochum, Mittelilung Nr. 86-1.


83

Chapter 5. A Mechanical Model of Closed-Composite Segment

5.1 INTRODUCTION

In general, many underground structures are designed by the allowable stress design

method. In this method various indeterminate factors such as variations in material

properties, acting loads, precision of estimated design loads, analysis model and

structural calculation method etc., are simply assumed based on factors of safety.

Shield tunnel linings (hereinafter called as segment rings) consist of segments and

many connecting joints, and show complicated mechanical behavior under combined

loads. Meanwhile, it is difficult to precisely define the loads acting on segment rings due

to variation in construction conditions. Therefore, the allowable stress design method as

a simplified solution is still used in the design of segments for shield tunnel.

However, it is possible to predict the variations in acting loads and material properties

etc., because of the high technical developments of FEA and measuring method for the

last few years. Therefore, the allowable stress design method is not rational in economy,

because it is not able to directly evaluate the variations in material strength, member size,

and loads etc. On the contrary, the limit state design is able to evaluate the variations in

material strength, member size, and loads etc using factors of safety based on the

theories of probability, statistics, and reliability.

The limit state design requires the structure to satisfy two principal criteria: the

ultimate limit state (ULS) and the serviceability limit state (SLS). A limit state is a set of

performance criteria (e.g. vibration levels, deflection, strength, stability, buckling,

twisting, collapse) that must be met when the structure is subject to loads.

To satisfy the ultimate limit state, the structure must not collapse when subjected to

the peak design load for which it was designed. A structure is deemed to satisfy the

ultimate limit state criteria if all factored bending, shear, and tensile or compressive

stresses are below the factored resistance calculated for the section under consideration.

The limit state criteria can also be set in terms of stress rather than load. Thus the

structural element being analyzed (e.g. a beam or a column or other load carrying

element, such as walls) is shown to be safe when the factored loads are less than their

A Mechanical Model of Closed-Composite Segment

84

factored resistance.

To satisfy the serviceability limit state criteria, a structure must remain functional for

its intended use subject to service loads, and as such the structure must not cause

occupant discomfort under design life.

It is true problem that the limit state design is not currently used in the segment design

for shield tunnel. Therefore, one of the purposes of developing a mechanical model for

composite segment is to provide tools suitable for limit state design. The paper does not

address safety coefficients as its purpose is to underscore the phenomena involved in the

issue rather than measuring structural safety.

The proposed mechanical model analyzes the nonlinear behavior of a composite

segment with discrete partial shear connection under combined loads. This model is

based on the two fields mixed force-displacement equation with nonlinear constitutive

relationships for the components. The tension stiffening effect is taken into account by

using the relationship proposed in the CEB-FIB Model Code 90 [1]. The model

considers the concept of partial interaction allowing for the occurrence of slip at the

interface between steel tube and concrete infill, and local buckling of steel plate in

compression.

5.2 STRUCTURAL MODELING

The main ingredients of the proposed formulas consists of (a) fiber element formulas

for the composite segment; (b) nonlinear constitutive relationships for the component

materials; (c) model for steel embedded in concrete.

5.2.1 Basic Assumptions Following the analytical method for the composite segment, it is assumed that:

(1) Throughout the depth of the cross-section, the strain distribution is linear, but a

discontinuity exists at the interface between steel tube and concrete infill due to

slip;

(2) No uplift or vertical separation occurs between steel tube and concrete infill;

therefore two parts of the composite section have the same rotation and the same


85

curvature;

(3) The post local buckling stress of the skin plate in compression is equal to the local

buckling strength;

(4) The shear connectors are considered to be discrete elements with uniform spacing.

The shear stress at the interface is proportional to the slip;

(5) The joint plates are considered as shear connectors;

(6) The member cross-section is subdivided into concrete and steel layers (fiber

element); (7) When the reinforcement is in tension, all layers in the effective area c,effA are

replaced by a single layer of steel embedded in the concrete.

5.2.2 Equilibrium and Compatibility Conditions The following presents the three conditions of equilibrium which were necessary to

be satisfied at all cross-sections of composite segment at any load stage. The description

of the conditions is shown in Fig. 5.1.

(b) Shear connector element(a) Forces on a composite segment

tc

b bsν b bsν

t tsνt tsν

zx

cV

cTcM

tttb

bdxν

tdxν

b mbV V+

b mbM M+

b mbT T+

t mtV V+ t mtT T+

t mtM M+

bdxτ

tdxτ

( )b m b b m bdV V V V+ + +

c cdV V+

t mt t mtd( )T T T T+ + +t mt t mt+d( )M M M M+ +

c c+dM M

c cdT T+

b mb b mb+d( )M M M M+ +

b mb b mb+d( )T T T T+ +

( )t m t t m t+ dV V V V+ +

b m bV V+

cV

t m tV V+

b mbM M+

b mbT T+

cT

cM

t mtT T+

t mtM M+

dx

Fig.5.1 Calculation model for the composite segment


86

(a) Interaction Force Equilibrium

Due to the discrete nature of the shear connectors, the internal forces (bending, axial

force, and shear force) distributions in the concrete infill and in the steel tube are now

discontinuous with jumps at each connector. To derive the equilibrium conditions for a

composite segment with discrete shear connectors, it needs to consider firstly the

equilibrium of an infinitesimal segment without shear connector and the equilibrium at

the cross-section containing a shear connector shown in Fig. 5.1 (b). The first set of

equilibrium equations, which apply between two consecutive connectors, is readily

obtained by expressing the equilibrium of a small element of the composite segment

with a finite length dx in the horizontal direction, and subjected to internal forces (Fig.

5.1(a)). The equilibrium conditions result in the following set of equations:

t mt( )t

d T Tdx

ν+= − (5.1)

b mb( )b

d T Tdx

ν+= − (5.2)

( )t mt t t tt mt

( )2 2

d M M t dxV Vdx

ν τ++ + = − (5.3)

( )b mb b b bb mb

( )2 2

d M M t dxV Vdx

ν τ++ + = − (5.4)

( )t bc cu cuc t b c2 2 2

dxdM t tV tdx

τ τν ν

+⎛ ⎞+ = + − +⎜ ⎟⎝ ⎠

(5.5)

where tT is axial force carried by the top skin plate; mtT is axial force carried by the

main girders in compression; bT is axial force carried by the bottom skin plate; mbT is

axial force carried by the main girders in tension; cT is axial force carried by the

concrete infill; tM is moment carried by the top skin plate; bM is moment carried by the

bottom skin plate; cM is moment carried the concrete infill; mtM is moment carried by

the main girders in compression; mbM is moment carried by the main girders in tension;

tV is shear force carried by the top skin plate; cV is shear force carried by the concrete

infill; mtV is shear force carried by the main girders in compression; bV is shear force

carried by the bottom skin plate; mbV is shear force carried by the main girders in


87

tension;ν is shear stress at interface between steel tube and concrete infill; τ is normal stress at interface between steel tube and concrete infill; tt is thickness of the top skin

plate; bt is thickness of the bottom skin plate; ct is depth of the concrete infill; and cut is

depth of the uncracked concrete infill.

Combining Eqs. (5.3), (5.4) and (5.5) yields

( ) ( )t t cu b c b cutube ctube c

22 2

t t t t tdM dM V Vdx dx

ν ν+ + −+ + + = + (5.6)

where tube t mt mb bM M M M M= + + + ; and tube t mt mb bV V V V V= + + + .

(b) Displacement Equilibrium

The curvature and the axial deformation at any section are related to the composite

segment displacements through kinematic relations. Under small displacements and

neglecting the relative transverse displacement between the steel tube and concrete infill,

these relationships are as follows (Fig.5.2):

( ) ( )tddt

u xxx

ε = (5.7)

( ) ( )cddc

u xxx

ε = (5.8)

( ) ( )bddb

u xxx

ε = (5.9)

( ) ( )2dd

xxx

υφ = (5.10)

( ) ( ) ( ) ( ) ( )t cc t

d2 dt

t t xx u x u xx

υγ+

= − + (5.11)

( ) ( ) ( ) ( ) ( )b cb c

d2 db

t t xx u x u xx

υγ+

= − + (5.12)

where u is the longitudinal displacement; υ is the transversal displacement, tε is the

strain at the section centroid of the top skin plate; cε is the strain at the section centroid

of the concrete; bε is the strain at the section centroid of the bottom skin plate; and φ is

the curvature.


88

υ

υ

υθ

θ

θ

cu

bu

tu

γ b

γ t

z

y

x

Fig.5.2 Kinematic of composite segment

tcuht

hb

ε

εt2

εct

tttb

tc

Cracked concrete

εt1εc1

εb2εb1

εc2

Uncracked concrete

Fig.5.3 Strain distribution of composite segment

Slip occurs at the interface between steel tube and concrete infill based on the

analysis of experimental data and FEM results in Chapter 3 and Chapter 4 respectively.

Therefore, the rate of change of slip equals the strain difference at the interface and if

tensile strains are assumed positive, the following equations may be derived shown in

Fig.5.3.

The assumption (5) gives


89

t t

t tt

sn Kνγ = (5.13)

b b

b bb

sn Kνγ = (5.14)

where tγ is the top relative slip between the top skin plate and concrete infill; bγ is the

bottom relative slip between the top skin plate and concrete infill; ts is the longitudinal

spacing of the top shear studs; bs is the longitudinal spacing of the bottom shear studs;

tn is number of the top shear studs in a group; bn is number of the top shear studs in a

group; tK is stiffness of a top shear stud; and bK is stiffness of a bottom shear stud.

Taking the derivative with respect to x in Eqs. (5.13) and (5.14) and then using Eqs.

(5.1) and (5.2) gives the differential equations. The relative slip strains at the interface

are calculated as 2

t t mtt2 c12

t t

d d ( )d d

tst

s T Tx n K xγε ε ε+

= = − = − (5.15)

2b b mb

c2 b12b b

d d ( )d d

bsb

s T Tx n K xγε ε ε+

= = − = − (5.16)

Strains at the top skin plate ( t2ε ), the top of the concrete infill ( c1ε ), the bottom of

the concrete infill ( c2ε ), and the bottom skin plate ( b1ε ) are calculated from the bending

and axial force as

t mt t mt tt2

t t m mt t t m mt 2T T M T t

E A E A E I E Iε

⎛ ⎞ ⎛ ⎞+ += − +⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

(5.17)

c c cuc1

c cu c cu 2T M t

E A E Iε = − − (5.18)

c c cuc2 c

c cu c cu 2T M tt

E A E Iε ⎛ ⎞= − − −⎜ ⎟

⎝ ⎠ (5.19)

b mb b mb bb1

b b m mb b b m mb 2T T M M t

E A E A E I E Iε

⎛ ⎞ ⎛ ⎞+ += −⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

(5.20)

where tA is area of the top skin plate section; mtA is area of main girder section in

compression; cuA is the uncracked concrete section; bA is the bottom skin plate section;

mbA is main girder section in tension; tE is elastic modulus of the top skin plate; mE is

elastic modulus of main girders; cE is elastic modulus of the concrete infill; bE is


90

elastic modulus of the bottom skin plate; tI is moment of inertia of the top skin plate

section; mtI is moment of inertia of main girder section in compression; cuI is moment of

inertia of the uncracked concrete section; mbI is moment of inertia of main girder section

in tension; and bI is moment of inertia of the bottom skin plate section.

The assumption (3) gives

t mt b mb c

t t m mt b b m mb c cu

M T M M ME I E I E I E I E I

φ + += = =

+ + (5.21)

Substituting Eqs. (5.17), (5.18), (5.19), (5.20), and (5.21) into Eqs.(5.15) and (5.16)

gives:

c t mtt

c cu t t m mt

dd

t T T Thx E A E A E Aγ φ

⎛ ⎞+= + − ⎜ ⎟+⎝ ⎠

(5.22)

b mb cb

b b m mb c cu

dd

b T T Thx E A E A E Aγ φ

⎛ ⎞+= − −⎜ ⎟+⎝ ⎠

(5.23)

where ( )t t cu / 2h t t= + ; and ( )b c b cu2 / 2h t t t= + − .

Taking the derivative with respect to x in Eqs. (5.22) and (5.23) and then giving the

following differential equations:

1 1 1t t b Vγ α γ β γ η′′ = − + + (5.24)

2 2 2b b t Vγ α γ β γ η′′ = − + + (5.25)

where 2

t t1

t t t m mt c cu

1 1tn K hs EI E A E A E A

α⎛ ⎞

= + +⎜ ⎟+⎝ ⎠; b t b

1b c cu

1bn K h hs EI E A

β⎛ ⎞

= −⎜ ⎟⎝ ⎠

;

2b b

2b b b m mb c cu

1 1bn K hs EI E A E A E A

α⎛ ⎞

= + +⎜ ⎟+⎝ ⎠; t b t

2t c cu

1tn K h hs EI E A

β⎛ ⎞

= −⎜ ⎟⎝ ⎠

;

t1

hEI

η = ; b2

hEI

η = ;

t t m mt b b m mb c cuEI E I E I E I E I E I= + + + + ; and tube cV V V= + .

The fourth order differential equations for the relative slips, tγ and bγ , are

( ) ( ) ( )(4)1 2 1 2 1 2 2 1 1 2t t t Vγ α α γ α α β β γ α η βη′′+ + − − = + (5.26)


91

( ) ( ) ( )(4)1 2 1 2 1 2 1 2 2 1b b b Vγ α α γ α α β β γ αη β η′′+ + − − = + (5.27)

Considering the boundary conditions of 0 0

0t bx xγ γ

= == = and

/ 2 / 20t bx L x L

γ γ= =

′ ′= = ,

The particular solutions of Eqs.(5.26) and (5.27) are as follows:

x

y

bb

L/2L/2

L/2L/2

L/2L/2

q

P/2 P/2

P

(c) Uniformly distributed load

(b) Symmetrical two point loads

(a) Concentrate load at midspan

Fig.5.4 Load definitions

* 1 2 1 2

1 2 1 2t Vηα βηγ

α α β β+

=−

; * 1 2 1 2

1 2 1 2b Vη β αηγ

α α β β+

=−

(5.28)

Defining

t b b

1 2 1 2 t t c cu t t m mt

t b t1 2 1 2 b b

c cu b b m mb

//

t

b

h h hn K s E A E A E AC h h hn K s

E A E A E A

η β αηηα βη

+ ++ +

= =++ +

+

,then b tCγ γ= , Eq.(5.24)

can be rewritten as:

21 0t t Vγ λ γ η′′ − − = (5.29)

where 21 1Cλ β α= − .


92

For the load cases shown in Fig.5.4, solving Eq. (5.29) and using the boundary conditions that

0 00t bx x

γ γ= == = ,

/ 2 / 20t bx L x L

γ γ= =

′ ′= = and / 2V P= gives the relative

slip solutions

( )( )

1 1

2 1

L x L x

t L

P e e e

e

λ λ λ λ

λ

ηγ

− − −

−

+ − −=

+ (5.30)

( )( )

12

1

2 1

L x L x

b L

P e e eC

e

λ λ λ λ

λ

ηγ

λ

− − −

−

+ − −=

+ (5.31)

Correspondingly, the relative slip strain solutions are

( )( )

1

2 1

x x L

st L

P e e

e

λ λ λ

λ

η λε

− −

−

−=

+ (5.32)

( )( )

1

2 1

x x L

sb L

P e eC

e

λ λ λ

λ

η λε

− −

−

−=

+ (5.33)

The additional curvature due to slip is calculated as

st sb

hε εφ +

Δ ≈ (5.34)

where t b ch h h h= + +

(c) Effect of Slip on Deflection

Considering the boundary conditions of ( )δ /2 0LΔ = and ( )δ /2 0L′Δ = , the

additional deflection at midspan of composite segments can be calculated by integrating

the additional curvature along the length of the composite segment.

For the case of simply supported composite segments with a concentrate load shown

in Fig.5.4 (a), the additional deflections due to slip is derived as

( )( )1 1

1δ (1 )

4 2 1

L

L

eLC Ph h e

λ

λη

λ

−⎛ ⎞−⎜ ⎟Δ = + +⎜ ⎟+⎝ ⎠

(5.35)

Similarly, for symmetrical two point loads (total load=P) and uniformly distributed


93

load q shown in Figs. Fig.5.4 (b) and (c), the additional deflections due to slip are

derived, respectively, as

( )2 12δ (1 )

4 2 1

b L b

L

L b e eC Ph h e

λ λ λ

λη

λ

−⎛ ⎞− −Δ = + +⎜ ⎟⎜ ⎟+⎝ ⎠

(5.36)

( )2 ( ) / 2

3 1 2

2 1δ (1 )8 1

L L

L

L e eC qh h e

λ λ

λη

λ⎛ ⎞− −

Δ = + +⎜ ⎟⎜ ⎟+⎝ ⎠ (5.37)

where 1δΔ is additional deflection due to slip for concentrate load; 2δΔ is additional

deflection due to slip for symmetrical two point loads; 3δΔ is additional deflection due to

slip for uniformly distributed load.

The total deflection calculated from elastic deformation and slip-induced deflection

is

( )( )

3

1 1

1δ (1 )

48 4 2 1

L

L

ePL LC PEI h h e

λ

λη

λ

−⎛ ⎞−⎜ ⎟= + + +⎜ ⎟+⎝ ⎠

(5.38)

( )

( )

2

1

δ 2 312 2 2

2(1 )4 2 1

b L b

L

P L Lb b b L bEI

L b e eC Ph h e

λ λ λ

λη

λ

−

⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎛ ⎞− −

+ + +⎜ ⎟⎜ ⎟+⎝ ⎠

(5.39)

( )4 2 ( ) / 2

3 1 2

5 2 1δ (1 )384 8 1

L L

L

qL L e eC qEI h h e

λ λ

λη

λ⎛ ⎞− −

= + + +⎜ ⎟⎜ ⎟+⎝ ⎠ (5.40)

5.3 MATERIAL MODELS

The material behavior is described using explicit relationships between the total

stress and the total strain with appropriate loading/unloading conditions.

5.3.1 Concrete (a) Unconfined Concrete

The stress-strain relationship suggested by Carreira and Chu [1] is adopted in


94

unconfined concrete for compression region. The initial portion of the ascending branch

is linearly elastic, but at about 30% of the ultimate strength, the presence of micro-cracks

leads to a nonlinear behavior, with a reduction in tangent modulus.

1

cc

cc

c

c

f

α

εαε

σεαε

⎛ ⎞′ ⎜ ⎟⎜ ⎟′⎝ ⎠=

⎛ ⎞− + ⎜ ⎟⎜ ⎟′⎝ ⎠

(5.41)


uniaxial compressive strength of concrete( 2N/mm ); ci2 /c cf Eε ′ ′= ; ciE the initial tangent

modulus of the concrete; and α as a function of uniaxial compressive strength of

concrete cf ′ , can be estimated by the following formula

3

1.55cfαβ′⎡ ⎤

= +⎢ ⎥⎣ ⎦

(5.42)

The coefficient of variability β increases when increasing the compressive strength of

the concrete. Therefore, if 221.0N/mmcf ′ = , 22.0β = and if 280.0N/mmcf ′ = , 71.4β = ,

for intermediate stress gradients, β can be determined by linear interpolations. In tension region, the stress-strain relationship is described by the CEB-FIB Model

Code 90 [2].

ci ci

ci

ci

0 0.9 /

0.1 (0.00015 ) 0.9 / 0.000150.00015 0.9 /

0 0.00015

c c t

tc t c t c

t

c

E f E

ff f Ef E

ε ε

σ ε ε

ε

⎧ ′< ≤⎪

′⎪⎪ ′ ′= − − < ≤⎨ ′−⎪⎪ >⎪⎩

(5.43)

(b) Confined Concrete

In compression region, Montoya et al.[3] proposed a concrete confinement model for steel tube confined concrete is adopted for confined concrete. The stress cσ is related to


95

the strain cε using the following formula:

2

1.0

ccc

c c

cc cc

f

A B Cf f

σε ε

=⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(5.44)

where dA k= ;sec

2 ABE

= ; 2sec

ACE

= ; and seccc

cc

fEε


following formula:

2

80

14

ccd

cc c

fkε ε

⎛ ⎞= ⎜ ⎟−⎝ ⎠ (5.45)

where cc cc

(17 0.06 ) lc c

fff

ε ε ε ⎛ ⎞′ ′ ′= + − ⎜ ⎟′⎝ ⎠; 80cε is the strain corresponding to 80% of the

peak stress, cc0.80 f ,and given by the following formula:

( )80 1.5 89.5 0.6 lc c c

c

fff

ε ε ⎡ ⎤′ ′= + −⎢ ⎥′⎣ ⎦ (5.46)

Peak axial stress for confined concrete ccf is defined as

cc

c t

1lf ff f

α⎛ ⎞′

= +⎜ ⎟⎜ ⎟′ ′⎝ ⎠ (5.47)


( ) 0.21

c

t

1.25 1 0.062 lf ff

α−⎛ ⎞ ′= +⎜ ⎟′⎝ ⎠

(5.48)

and the tensile strength 2 / 3t 0.23( )cf f′ ′= .

The effective lateral pressure lf ′acting on the concrete due to the steel jacket can be

calculated as follows:

l s lf k f′ = (5.49)


96


confined area to the total cross section area and lf is the possible maximum confining

pressure that exerted by steel jacket on the concrete core. sk and lf can be calculated by

Eqs.(2.8) and (2.6) expressed in Chapter 2 respectively.

In tension region, the above stress-strain relationship proposed by the CEB-FIB

Model Code 90 [2] is adopted.

5.3.2 Steel

In the present study, the steel is modeled as an elastic-plastic material incorporating

strain hardening. Specifically, the relationship is linearly elastic up to yielding, plastic

between the elastic limit and the commencement of strain hardening, linear hardening

occurs up to the ultimate tensile stress shown in Fig.2.1.

However, local buckling occurs on the thin steel plates of concrete infill steel tube

members. In general, local buckling of thin steel plates depends on the plate aspect ratio,

width-to-thickness ratio, applied edge stress gradients, boundary conditions, geometric

imperfections, and residual stresses. According to Guideline of buckling design of JSCE code [5], the post-buckling stress udf ′ is calculated as:

( )/ud s syf t s E f′ = (5.50)

where t is the thickness of the steel plate; s is the longitudinal spacing of shear connectors; sE is the elastic modulus of the steel plate; and syf is the yield strength of

the steel plate.

5.3.3 Reinforcement embedded in concrete

When uncracked concrete is in tension, the tensile force is distributed between the

reinforcement and the concrete in proportion to their respective stiffness, and cracking

occurs when the stress reaches a value corresponding to the tensile strength of the

concrete. In a cracked cross-section all tensile forces are balanced by the steel encased in

the concrete only. However, between adjacent cracks, tensile forces are transmitted from

the steel to the surrounding concrete by bond forces. The contribution of the concrete

may be considered to increase the stiffness of the tensile reinforcement. This effect is


97

called tension-stiffening. To describe this effect, a number of models have been proposed.

The majority of the models are based on the mean axial stress and the mean axial strain

of the concrete member in the reinforced concrete, [5-9].

To take the tension stiffening effect into account, the stress average strain

relationship of steel embedded in concrete proposed by the CEB-FIB model [2] is

considered to describe the behavior of the reinforced concrete members in tension.

According to the CEB-FIB Model Code 90 [2] the mean stress-strain relationship of

embedded steel may be expressed as

( )

( )

( )

, s,m sr1

srn sr1sr1 , sr1 sr1 , srn

srn sr1

srnsrn , srn srn , sry

sry srn

sry , sr,sh

, sry sr,sh , srusru sry

, sru

11 s s m

s m s m

sys m s ms

sy s m

su sysy s m s m

su s m

En

f

ff f

f

f

ε ε ερσ σσ ε ε ε ε εε ε

σσ ε ε ε ε εσ ε ε

ε ε ε

ε ε ε ε εε ε

ε ε

⎧⎛ ⎞+ ≤⎪⎜ ⎟⎝ ⎠⎪

−+ − < ≤

−−

+ − < ≤= ⎨ −

< ≤−

+ − < ≤−

>

⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎩

(5.51)

where n and ρ are the modular ratio and the geometric ratios of reinforcing steel,

respectively; sr1σ is the steel stress in the crack, when the first crack has formed; srnσ is

the steel stress in the crack, when the last crack has formed; sr1ε and sr2ε are the steel

strains at the point of zero slip and at the crack when the cracking forces reach ctf ;

( )srn srn sr2 sr1/ 0.4sEε σ ε ε= − − ;

( )sry sy sr2 sr10.4ε ε ε ε= − − ;

( )sr,sh sh sr2 sr10.4ε ε ε ε= − − ;

( ) ( )sru sr,sh sr10.8 1 / /sy su sy stf f f Eε ε σ= + − − ;

0.01st sE E= .

In the present study, the steel is modeled as an elastic- plastic material incorporating

strain hardening. Specifically, the relationship is linearly elastic up to yielding, plastic

between the elastic limit and the commencement of strain hardening, linear hardening


98

occurs up to the ultimate tensile stress shown in Fig.2.1.

5.3.4 Shear stud connector

The constitutive relationship proposed by Ollgaard et al. [10] is considered for the shear stud connector. The analytical relationship between the shear force Q and the slip

δ of a shear stud is given by

( )0.40.70871uQ Q e δ−= − (5.52)

where Q is the applied load(N); δ is the slip of shear stud(mm).

The stiffness at 0.5 uQ was proposed as initial stiffness of a shear stud is given by

( )sh 0.16 0.0017u

si

c

QKd f

=′−

(5.53)

where siK is initial stiffness of a shear stud(N/mm); uQ is shear capacity of a shear stud

accounting for the effects of shear stud spacing and plate thickness, and given by

0.400.65 0.35

sh1.35.3 ( ) c

u V su cs

EQ A f fEn

ηβ⎛ ⎞⎛ ⎞ ′= − ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ (5.54)

whereη and Vβ are reduction factors of the shear capacity and described in Chapter 2.

5.4 CROSS SECTIONAL ANALYSIS

The procedure followed to predict the flexural behavior of the composite segment

sections is an incremental iterative method with secant stiffness formulation. For the

present numerical procedure, the cross section is divided into a finite number of discrete

layers (fiber model). Because the slip effect causes an equivalent additional moment to

the section, the elastic flexural capacity (corresponding to the first yielding of the

extreme fiber of the section) of the composite segment is reduced. The calculation model

for the additional moment is shown in Fig. 5.5.


99

hthb

M

+

-

= + +

εsb

εst

ΔNstΔMt

ΔNct

ΔNcb

ΔMbΔNsb

ε

εt2

εct

εt1εc1

εb2εb1

εc2

uncracked concrete

cracked concrete

Fig.5.5 Calculation model for additional moment

The slip-induced strains of the skin plates at the interfaces are

( )t

t cst st

tt t

ε ε′ =+

( )

b

b csb sb

tt t

ε ε′ =+

(5.55)

where stε ′ and sbε ′ are the slip-induced strains at the top and bottom interfaces, respectively.

By using stε ′ for the top skin plate sbε ′ for the bottom skin plate, approximately

0.5 stε ′ for the main girders in compression, and 0.5 sbε ′ for the main girders in tension, the

axial force variations in the skin plate sections due to slip are

( ) ( )tst t mt

t c

0.5s sttN E A A

t tεΔ = +

+ (5.56)

( ) ( )b

sb b mbb c

0.5s sbtN E A A

t tεΔ = +

+ (5.57)

where tA is area of the top skin plate; bA is area of the bottom skin plate; mtA is area of

the main girders in compression; and mbA is area of the main girders in tension.

The additional moment in the section shown in Fig.5.5 is

t b st c t sb c b1 1 1 13 3 3 3

M M M N t t N t t⎛ ⎞ ⎛ ⎞Δ = Δ + Δ = Δ + + Δ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(5.58)

Substituting Eqs. (5.56), (5.57) into Eq.(5.57), Eq.(5.58) can be rewritten as:


100

( ) ( )t bt mt b mb2 2

3 3s st s sbt tM E A A E A Aε εΔ = + + + (5.59)

Therefore, when considering the slip effect, the elastic moment in the section causing

the same stress state as M for a case without slip is

pM M M= − Δ (5.60)

The fiber element analysis is used to determine the bending moment M of the

member cross sections for a case without slip. In the fiber element method, a composite

section is discretized into many small regions (fiber elements) shown in Fig.5.6. Each

element represents a fiber of material running longitudinally along the member.

Constitutive models are based on the above described uniaxial stress-strain relationships

of materials. Stress resultants are obtained by numerical integration of stresses through

the composite cross section. In the present fiber analysis program, the origin of the

coordinate system is chosen to pass through the centroid of the composite section as

shown in Fig.5.6. The cross-sectional area of each fiber is automatically calculated based

on the discretization of fibers and the geometry of the composite section being

considered. Concrete fibers are grouped together as well as the steel fibers.

Steel fiber

Concrete fiber

y

x

H

Fig.5.6 Discretization of composite section


101

Member forces are determined as the stress resultants in a composite section by the

following equations:

, , , ,1 1

ns nc

s i s i c j c ji j

P A Aσ σ= =

= +∑ ∑ (5.61)

, , , ,1 1

ns nc

s i s i i c j c j ji j

M A y A yσ σ= =

= +∑ ∑ (5.62)

where P is the axial force; M is bending moment about the major principal axis; ,s iσ is

the longitudinal stress at centroid of steel fiber i ; ,s iA is is the area of steel fiber

i ; ,c jσ is the longitudinal stress at the centroid of concrete fiber j ; ,c jA is the area of

concrete fiber j ; iy is the coordinates of steel fiber i ; jy is the coordinates of concrete

fiber j ; ns is the total number of steel fiber elements, and nc is the total number of

concrete fiber elements. Compressive stresses are taken to be positive. Stresses in fibers

are calculated using fiber strains and material stress-strain relationships.

In the present method, the proposed secant algorithm by Liang [11] is used in the fiber element analysis program to iterate the depth and orientation of the neutral axis until equilibrium conditions are satisfied. The neutral axis depth ( nd ) is adjusted by using the following equation:

( ), 1 ,, 2 , 1

, 1 ,

n j n jn j n j

p j p j

d dd d

f f+

+ ++

−= −

− (5.63)

where the subscript j is the iteration number; and p nf P P= − , nP is the given axial

load. The convergence criterion for the neutral axis depth nd is defined by

, 1 ,n j n jd d e+ − ≤ (5.64)

where e is the convergence tolerance.

To simplify the calculation of ultimate carrying capacity, it is reasonably assumed

that the effect of transverse reinforcement can be ignored in carrying capacity

calculation. According to the position of the plastic neutral axis in the concrete infill, the

calculation of ultimate carrying capacity is discussed in the following section.


102

The strain and stress distributions over the section depth for composite segment are shown in Fig.5.7. The strain in the tension steel, ,s mε at failure is given by:

t b c,

ts m cc

t t t d cd t

ε ε+ + − −=

− (5.65)

where d is the depth of neutral axis; c is the depth of tension steel. For a segment section

with a constant width the parabolic portion of concrete stress distribution can be replaced by an equivalent rectangular block introducing the stress block factors 1α and

1β [12]. The resultant forces of the equivalent rectangular block have the same

magnitude and location as those of the actual parabolic stress distribution.

2cp cp

1 1c c

1 /3

ε εα β

ε ε

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟′ ′⎢ ⎥⎝ ⎠⎣ ⎦

(5.66)

cp c1

cp

4 /6 2 / c

ε εβ

ε ε′−

=′−

(5.67)

where cpε is the concrete strain at the end of the parabolic portion of the concrete stress

distribution. At failure, cpε is the ultimate compressive strain of an unconfined concrete

for unconfined segment while for steel tube-confined segment cp cε ε ′= .

tttb

tc

tm

,t1sf

,b1sfAs

c

Main reinforcement

s,mε

b

φ

cc st( )ε ε+

Equivalent Stressdistribution

Concrete

Mai

n gi

rder

Mai

n gi

rder 3d

tf ′tf ′

1 cfα ′

1 1dβcf ′

ccf2d

1d

dccf

cf ′

N.A

ctε

ccε

cε ′

Actual Stress distribution

Steel tubeConcrete

Strain distribution

Skin plate

Concrete infill

Skin plate

Composite section

Fig.5.7 Strain and stress distribution of composite segment


103

At failure, equilibrium conditions are imposed in terms of axial force, uP , and

bending moment, uM .

1 1 1 2 ,t1 t ,t1 t m

3 ,b1 b ,b1 t c m

u

( )2

( )2

cc cc s s

cts s s st

f ff d b d b f t b f d t t

f d b f t b f t t d t A f

P

α β′+′ + + + −

− − − + − −

=

(5.68)

( ) ( )

( )

( )

1 1 1 1 1 1 2 1 2

2,t1 t 1 2 t ,t1 t m

23 ,b1 b t c b

2,b1 t c m t c b

u

0.5 0.52

20.5 ( )3

( 0.5 )3

2 ( )3

cc cc

s s

cts

s s st

f ff d b d d d b d d

f t b d d t f d t t

f d b f t t t t d b

f t t d t A t t t d c f

M

α β β′+′ − + + +

+ + + −

− − + + −

− + − − + + − −

=

(5.69)

where

1

unconfined segment

confined segmentc

cc

dd

d εε

⎧⎪= ′⎛ ⎞⎨

⎜ ⎟⎪ ⎝ ⎠⎩

; 2

0 unconfined segment

1 confined segmentc

cc

dd ε

ε

⎧⎪= ′⎛ ⎞⎨ −⎜ ⎟⎪ ⎝ ⎠⎩

stf is the stress in tension steel; sA is the area of the main reinforcements.

For simplification, uM can be calculated as:

( ) s-t s-bu-tube full-Rd u-tube

s-t s-b1uk kM M M M

k k+

= + −+

(5.70)

where u-tubeM is the ultimate bending moment of steel tube; full-RdM is the ultimate bending

moment of a full composite segment have the same section; and s-tk , s-bk are degrees of

composite action and given by:

{ }d-t

s-tc c-tube

1min ,

FkN N

= ≤ (5.71)

{ }d-b

s-bc b-tube

1min ,

FkN N

= ≤ (5.72)


104

where d-t t uF nV ′= ; d-b b uF n V ′= ; cN is the compression force of concrete; c-tubeN is the

compression force of steel tube; b-tubeN is the tension force of steel tube; and uV ′ is the

shear capacity of a shear stud accounting for the effect of the joint plate, and given by:

u u m nnQV Q′ = +⋅

(5.73)

where uQ is the shear capacity of a shear stud accounting for the effects of shear stud

spacing and plate thickness; nQ (Eq.(2.26)) is the shear capacity of a joint plate; m is

the rows of a grouped shear studs; and n is columns of shear studs between the joint plate

and the analysis cross section.

In the performance-based analysis of composite segments under axial load and

bending, moment-curvature responses are obtained by incrementally increasing the

curvature and solving for the corresponding moment value for the given constant axial load ( nP ) and vertical load ( VP ). A set of moment-curvature curves for the composite

section can be generated by incrementally increasing the vertical load ( VP ) from zero to

the maximum value maxP .

The analysis procedure used to predict the composite segment strength can be

summarized as follows:

(1) Discretize the composite section into fine fiber elements.

(2) Set axial fiber strainsε ε= Δ .

(3). the neutral axis depth ( ),1 ,2,n nd d are set to H and H/2, and φ φ= Δ (4) Calculate fiber stresses using stress-strain relationships.

(5) Check local buckling and update steel fiber stresses accordingly. (6) Calculate the vertical force ( VP ) and bending moment ( nM ) that satisfy equilibrium

requirements.

(7). Adjust the depth of the neutral axis ( d ) using Eq. (5.64).

(8) Repeat Steps (3)-(7) until , 1 ,n j n jd d e+ − < .

(9) Using the Eq.(5.69) to calculate maxφ .

(11) if maxφ φ< , increase the vertical load by φ φ φ= + Δ , where max /100φ φΔ =

(12) Repeat Steps (3)-(11) until maxM

(13) Calculate the vertical force ( maxP ) and bending moment ( maxM ).


105

5.5 RESULTS AND DISCUSSION OF THE PROPOSED METHOD

Accuracy of the proposed mechanical model is verified by comparing the results

with those obtained experimentally. Comparison of the bending capacity and bending

moment-curvature, load-deflection responses obtained from the proposed mechanical

analyses with those obtained experimentally provides a means to verify the accuracy of

the proposed mechanical model.

Comparison of the predicted results derived from the analysis procedure and the experimental results in chapter 3 is shown in Table 5.6. ytM is measured yield bending

capacity. It shows that the elastic bending capacity without considering the slip is rather

larger than the experimental results. Taking into account the slip effect, the analysis

results are much closer to the measured values with a mean value of 1.058.

Table 5.6. Comparison of elastic bending capacities of composite segment specimens

Specimens ytM (kNm)

M (kNm)

MΔ (kNm)

pM (kNm)

p yt/M M

Case1 38.0 52.56 12.76 39.80 1.047

Case2 37.5 51.75 13.61 38.14 1.017

Case3 38.0 51.49 11.91 39.58 1.042

Case4 31.5 48.87 16.03 32.84 1.043

Case5 41.5 55.50 13.04 42.46 1.023

Case6 25.0 37.78 10.54 27.24 1.090

Case7 41.0 67.32 23.71 43.62 1.064

Case12 35.2 47.1 7.53 39.53 1.123

Case15 375.0 448.3 52.47 395.8 1.056

Case16 468.7 547.5 44.46 503.1 1.073

Mean value 1.058

Standard deviation 0.009

Table 5.7 shows the comparison between the experimental results and calculation values from the present study, where utM is measured ultimate bending capacity. It

shows that the simplified calculation for the ultimate bending moment is applicable in

practical design.


106

Table 5.7. Comparison of ultimate bending capacities of composite segment specimens

Specimens s-tk s-bk utM (kNm)

uM (kNm) ut u/M M

Case1 0.50 0.31 47 49.1 0.958

Case2 0.36 0.22 46 49.6 0.928

Case3 0.67 0.41 47 44.1 1.066

Case4 0.50 0.31 43 43.4 0.992

Case5 0.50 0.31 50.2 52.6 0.955

Case6 0.70 0.43 33 38.6 0.855

Case7 0.38 0.23 53.2 51.5 1.034

Case12 0.78 0.70 42.7 45.6 0.936

Case15 0.68 0.75 456 474.7 0.961

Case16 0.57 0.62 581 584.2 0.995

Mean value 0.968


The moment-curvature response is obtained by subdividing the cross-section into a

large number of horizontal layers. The ultimate strain which is assumed as 0.0038 as

recommended by the compressive tests of the unconfined concrete. For each step or

increment of strain, the depth to the neutral axis is determined by the strain distribution

in main girders，when the sum of all the forces acting on the section becomes zero (i.e.,

equilibrium of forces is satisfied). The moment of all the forces (acting on the section

layers) about the neutral axis is calculated and the curvature is determined by dividing

the concrete strain by the depth of the neutral axis. The entire curve is plotted by

repeating the above procedure until the ultimate strain in the confined concrete was

reached. The last point in the moment-curvature curve was the moment (flexural)

capacity of the section. The moment-curvature curves obtained using mechanical

analysis of the proposed model are compared with the curves obtained by the

experiments as shown in Fig.5.8. It can be observed that the curvatures predicted by the

proposed model are closer to the experimental curvatures.


107

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160

10

20

30

40

50

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)

Closed-composite segment (Case1)

Experimental result Proposed model

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350

10

20

30

40

50

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)



0.00 0.02 0.04 0.06 0.080

10

20

30

40

50

60

70

80

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)



0.00 0.02 0.04 0.06 0.08 0.10 0.120

50

100150

200

250

300

350400

450500

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)



Fig.5.8 Bending moment-curvature curves of Closed-composite segments

Fig.5.9 shows the load-midspan deflection curves of the experiments and mechanical

analysis of the proposed model. It can be observed that the load-midspan deflection

response predicted using the proposed model is similar to the corresponding

experimental plots. Therefore, it is evident from the comparisons between the results

calculated with the formula and the mechanical model that the calculated results with

reasonable accuracy. The difference between the experimental results and the calculated

results is considered that local buckling or yielding due to residual stress reduces the

rigidity of the composite segment section as well as cracks propagation.


108

0 2 4 6 8 10 120

50100150200250300350400450500550

900 200 100L B H× × = × ×

Load

(kN

)


Case1Exprimental result Proposed model

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

900 300 100L B H× × = × ×

Load

(kN

)Deflection at midspan(mm)


0 3 6 9 12 15 18 210

100

200

300

400

500

600

700

800

900 500 100mmL B H× × = × ×

Load

(kN

)



0 5 10 15 20 25 30 35 40 45

0

200

400

600

800

1000

1200

1400

2100 750 150mmL B H× × = × ×

Load

(kN

)



Fig.5.9 Load-deflection curves of Closed-composite segments at midspan

5.7. SUMMARY

A nonlinear fiber element analysis method has been presented in this paper for the

inelastic analysis and design of concrete infill steel tubular composite segments with

local buckling and slips effects. Sectional geometry, residual stresses and strain

hardening of steel tubes and confined concrete models were considered in the proposed

mechanical model. The local buckling, slip and effective strength formulas were

incorporated into the nonlinear analysis procedures to account for local buckling and slip

effects on the strength and ductility performance of composite segments under combined


109

loads.

This paper has presented the formulation of differential equations suitable for the

solution of partially interactive steel tube composite segment elements. Closed form

solutions for single, double and uniform load arrangements on simple span segments

have been presented. A step-wise linearization method has been described which allows

the non-linear affects of concrete cracking and shear stud stiffness to be incorporated.

The analysis involves non-dimensional constants which affect the steel plate

interaction. The effects are either direct when the constant relates to the plate for which

it is formulated or crossed when it affects the opposite plate for which it is formulated.

More attention should be paid on the shear capacity of shear studs in composite

segments. In current codes, only some coefficients are suggested for considering the

influences of the concrete strength, steel strength, and shear stud dimension. It was

found that the shear stud spacing and the thickness of the welded steel plate also have the effects on the shear capacity, so the coefficients η and Vβ are recommended.

Stopping criteria are required to terminate the iterative analysis process once

satisfactory solutions are obtained. In calculating the ultimate axial load of a composite

segment, the analysis may be stopped when axial strain in concrete exceeds the strain

corresponding to the peak stress of unconfined concrete or confined concrete. The

moment-curvature calculations may be stopped when the strain in steel exceeds the

specified ultimate strain or the strain in concrete exceeds the strain corresponding to the

peak stress of unconfined concrete or confined concrete.

Comparisons have been made between experimental results and the mechanical

predictions of behavior using the proposed method. Good agreement has been found and

this indicates that the method is suitable for general analysis of tunnel lining of

composite segment.

5.8. REFERENCES

1) Carreira, D. J., K.H. Chu,1985. Stress-strain relationship for plain concrete in

compression. J., Proc., ACI, 82(11), 797-804.

2) CEB-FIP model code 1990,1991. Lausanne (Switzerland): Comité Euro-International


110

du Béton.

3) Montoya, E.,2003. Behavior and analysis of confined concrete. Ph.D. thesis, Univ. of

Toronto, Toronto.

4) Scanlon A, Murray DW, 1974. Time dependent deflections of reinforced concrete slab

deflections. Journal of the Structural Division,100(9):1911-1924.

5) JSCE, 2005. Buckling design guideline.

6) Lin CS, Scordelis AC,1975. Nonlinear analysis of RC shells of general form. Journal

of the Structural Division;101(3):523-538.

7) Vebo A, Ghali A,1977. Moment curvature relation of reinforced concrete slabs.

Journal of the Structural Division,103(3),515-531.

8) Gilbert R, Warner R,1978. Tension stiffening in reinforced concrete slabs. Journal of

the Structural Division,104(12):1885-1900.

9) Q.H. Nguyen et al.,2009. Analysis of composite beams in the hogging moment

regions using a mixed finite element formulation. Journal of Constructional Steel

Research 65, 737-748.

10) Oehlers DJ, Sved G.,1995. Composite beams with limited-slip-capacity shear

connectors. J Struct Eng.,32-38.

11) Qing Quan Liang, 2008. Nonlinear analysis of short concrete-filled steel tubular

beam-columns under axial load and biaxial bending. Journal of Constructional Steel

Research 64, 295-304.

12) Collins MP, Mitchell D. Prestressed concrete basics; 1987. Ottawa,ON, Canada:

Canadian Prestressed Concrete Institute (CPCI).


111

Chapter 6. Verification of the Proposed FEM and Mechanical Models

6.1 INTRODUCTION

To verify the applicability and the accuracy of the results solved from the proposed

FEM and mechanical models in Chapters 4 and 5, SSPC segment and DRC segment are

simulated in this chapter. The associated theories, e.g., material properties, analysis

method, which are used in the analysis of SSPC segment and DRC segment and

described in Chapters 4 and 5.

6.2 SSPC SEGMENT

6.2.1 Test Specimens 2 Steel Segments with Pre-filled Concrete (SSPC segment) were manufactured. The

constructing process of SSPC segment is shown in Fig.6.1 [1].

The details of SSPC segment specimens are shown in Fig.6.2 and the dimensions of

SSPC segments are summarized in Table6.1. The SSPC segment specimens are 73.5°

central angle, 1981 mm inside diameter, 2106 mm outside diameter, 125 mm thickness.

Mechanical properties of steel and concrete are shown in Table6.2.

(a) Steel Segment (b) Casting concrete (c) Completed SSPC segment

Fig.6.1 Manufacturing process of SSPC segment

Verification of the Proposed FEM and Mechanical Models

112

Joint plate

Skin plate

Rib

8.0

938.0

3.0125

73.5o

Fig.6.2 Detail of SSPC segment specimen

Table 6.1. Details of SSPC segment specimens

Dimension Thickness Rib

Skin plate

Main girder

Joint plate Specimen Width

(mm)

inside diameter

(mm)

outside diameter

(mm)

central angle

(degree) ts(mm) tm(mm) tj(mm)

Height (mm)

Thickness(mm)

SSPC(Positive bending ) 1000 1981 2106 73.5 3.0 8.0 8.0 93.0 8.0

SSPC(Negative bending ) 1000 1981 2106 73.5 3.0 8.0 8.0 93.0 8.0

Note: Positive bending is the load applied on the outside (ground side) of SSPC; Negative bending is the load applied on the inside (tunnel side) of SSPC.

Table 6.2. Mechanical material properties for SSPC segment specimens

Structural steel Concrete

Specimen Yield strength

2(N/mm )yf

Tensile strength

2(N/mm )sf ′

Young modulus

2(N/mm )sE


2(N/mm )cf ′

Young modulus

2(N/mm )cE

SSPC 363 570 2.0×105 27.9 2.3×104


113

6.2.2 Test Setup The simply supported SSPC segment specimens were loaded symmetrically at four

points within the span using a spreader beam shown in Fig.6.3. In these test

arrangements, such loading led, in theory, positive and negative pure bending between

applied forces of the composite segment specimens can be obtained between the two

loading points without the presence of shear and axial forces. The applied load was

measured and recorded using a load cell. The loads were increased gradually with an

increment of approximately 5kN until the ultimate carrying capacities of SSPC segment

specimens. Displacement transducers and Strain gauges located on the key points shown

in Fig.6.4 were used to measure strains and deflections respectively.

SSPC segment

CL

Spreader beam

Load cellP

300

1113.3

Unit：mm


(a)Test setup for positive bending

SSPC segment

CLLoad cellP

300

1116.7

Unit：mm


Spreader beam

(b)Test setup for negative bending Fig. 6.3 Test setup for SSPC segment specimens


114

S2,S3,S4,S5 (The outside of main girder)

20

20

30

40

100

S6,S7,S8,S9 (The inside of main girder)

S1


S11,S12,S13,S14(The outside of main girder)

S10

1200

The numbers S1,S2,S3 etc. denote the serial numbers of strain gauges　　:displacement transducer Unit : mm

125

Fig. 6.4 The arrangement of measuring instruments on SSPC segment specimens

6.2.3 Test Results and Discussion Table6.3 gives the values of the yield load of each structural member and the ultimate

load, and cracking load for SSPC segment specimens and steel segment specimens. The

cracking load was taken as the applied load corresponding to crack occurring. The

ultimate load of SSPC segment specimen of Case2 equals to about 1.54 times of the

ultimate load of steel segment of Case2 under positive bending. The ultimate load of

SSPC segment specimen of Case4 equals to about 3.69 times of the ultimate load of steel

segment specimen of Case3 under negative bending. It can be observed that concrete

infill obviously affects the ultimate load of SSPC segment specimen under negative

bending.


115

Table 6.3. Experimental results of SSPC segment specimens

Specimen Cracking

load (kN)

Main girder yielding load

(kN)

Skin plate yielding load

(kN)

Ultimate load (kN)

Concrete infill

Loading method

Ratio of ultimate load for SSPC to

Steel (%)

Case1 --- 88.9 134.2 144.1 No Positive

Case2 --- 97.0 136.7 210.7 Yes Negative

Case3 37.2 93.6 --- 222.5 No Positive

Case4 --- 474.5 531.1 778.1 Yes Negative

u-case3

u-case1

100 154.4PP

× =

u-case4

u-case2

100 369.3PP

× =

Note: the dimensions of steel segment specimens are same as SSPC segment specimens, including thickness of each structural member

(a)Load-deflection response

Load and midspan deflection relationships for all tested specimens can be described

by the load-midspan deflection curves shown in Fig.6.5. Case1 and Case3 are steel

segment specimens without concrete infill. It can be observed that the ultimate load and

deflection of SSPC segment are larger than those of the steel segment under positive

bending. However, the ultimate load of SSPC segment is larger than that of the steel

segment, and ultimate deflection is smaller than that of the steel segment under negative

bending.

0 3 6 9 12 150

50

100

150

200

250

8.0

938.0

3.0

Case1

Case3

Load

(kN

)


Positive bendingCase1 steel segment specimenCase3 SSPC segment specimen

0 5 10 15 20 250

100

200

300

400

500

600

700

8008.0

938.0

3.0

Case2

Case4

Load

(kN

)


Negative bendingCase2 steel segment specimenCase4 SSPC segment specimen

(a) Positive bending (b) Negative bending

Fig.6.5. Load-deflection curves of SSPC segment specimens at midspan


116

(b)Strain distribution

Fig.6.6 shows the measured strain distribution along SSPC segment height (with the

origin at the edge of the main girder to tunnel side) on the double sides of the main

girders. Such curves are displayed under different loading levels for the sections of the

100mm distance from the midspan section. The strain values for the main girders were

taken as the average strain values (from the measured strain based on the

symmetrically-installed strain gauges in the sections of the 100mm distance from the

midspan section). It can be observed that the plane sections remain plane after loading,

because the strain distribution along the height main girder is linear, if the strain values

of the origin are neglected.

-1200 0 1200 2400 3600 4800

0

20

40

60

80

100

120


2020

30

40

100


S1,S10



24.5kN 49.0kN 73.5kN 88.2kN 122.5kN

Case1

Hei

ght o

f mai

n gi

rder

(mm

)

ground side

Microstrain

-1200 0 1200 2400 3600 4800

0

20

40

60

80

100

120


2020

30

40

100


S1,S10



ground side

24.5kN 49.0kN 73.5kN 122.5kN 147.0kN

Case3 with concrete infill

Hei

ght o

f mai

n gi

rder

(mm

)

Microstrain

(a) Positive bending

-3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500120

100

80

60

40

20

0

-20


100


S1,S10



202030

40

24.5kN 49.0kN 73.5kN 88.2kN 117.6kN

Case2

Hei

ght o

f mai

n gi

rder

(mm

)

Tunnel side

Microstrain

-1500 -1000 -500 0 500 1000120

100

80

60

40

20

0


100


S1,S10



202030

40

58.8kN 96.0kN 156.8kN 196.0kN 264.6kN 294.0kN 343.0kN


Hei

ght o

f mai

n gi

rder

(mm

)

Tunnel side

Microstrain

(b) Negative bending Fig.6.6. Measured strain distribution along the height main girder


117

(a) Longitudinal cracks (b) Concrete crushing

Fig.6.7. Failure modes of SSPC segment specimens

(c)Failure modes

The failure modes of the SSPC segment are shown in Fig.6.7. Failure of the SSPC

segment specimen occurs in tensional region under positive bending due to the bending

cracks shown in Fig.6.7 (a). For failure of the SSPC segment specimen under negative

bending, compression concrete infill exhibits crushing failure when reaching its ultimate

strain as shown in Fig.6.7 (b).

6.2.4 FEM Results and Discussion The full scale SSPC segment specimens are modeled and shown in Fig6.8. The simply

support boundary conditions are applied at the support points. 8-Node 3D solid elements

with tri-linear interpolation and nonlinear interpolation are used to model steel plates and

concrete, respectively. Vertical load is distributed to all the nodes on the loading

positions. Contact analysis with a friction coefficient of 0.1, and the relative sliding velocity ( VR ) of default value 0.1 is used in FEM analysis.


118

(a) Steel segment (b) FEM model of steel segment

(c) SSPC segment (d) FEM model of SSPC segment

(e) Negative bending (f) Positive bending

Fig.6.8 Finite element model of an SSPC segment


119


Among the analysis results, load vs. midspan deflection response histories of SSPC

segment specimens (with shear ribs) are shown in Fig.6.9. Comparing the analyzed

results with the experimental results, it can be seen that the load-deflection curves from

numerical analysis are agree with the experimental load- load-deflection curves.

Load-relative slip response histories of SSPC segment specimens (with shear ribs)

are shown in Fig.6.10. It can be observed that the vertical relative slip between the steel

tube and concrete infill occurs, especially in the applying positive bending load. It

should be noted that the vertical relative slip affect the behavior of composite segment in

design.

0 3 6 9 12 150

100

200

300

400

500

600

700

800

SSPC segment

CLLoad cellP

300

1116.7

Unit：mm


Spreader beam

SSPC negative bending

Load

(kN

)


Exprimental result FEM result

0 3 6 9 12 15 180

50

100

150

200

250

SSPC segment

CL

Spreader beam

Load cellP

300

1113.3

Unit：mm


SSPC Positive bending

Load

(kN

)


Exprimental result FEM result

Fig.6.9 Load-deflection curves for tested SSPC segment specimens at midspan

0.0 0.1 0.2 0.3 0.40

50

100

150

200

250

Key location


SSPCPositive bending

Lo

ad(k

N)

Vertical relative slip(mm)

0.00 0.03 0.06 0.09 0.12 0.150

100

200

300

400

500

600

700

800

Key location


SSPCNegative bending

Lo

ad(k

N)


Fig.6.10 Load-relative slip relationship for SSPC segment specimens


120


Fig.6.11 shows the strain distributions at the measured locations. The strain

distributions of nonlinear analysis show good agreement with the experimental results in

main girders. The tendency of the strain distribution of nonlinear analysis shows close

agreement with the experimental value in the skin plates.

-2000 -1000 0 1000 2000 3000 4000 5000

0

20

40

60

80

100

120

140 Line PlotFEM result Experimental result


2020

30

40

100


S1,S10



ground side

24.5kN 49.0kN 73.5kN 122.5kN 147.0kN


Hei

ght o

f mai

n gi

rder

(mm

)

Microstrain

-1500 -1000 -500 0 500 1000140

120

100

80

60

40

20



100


S1,S10



202030

40

58.8kN 96.0kN 156.8kN 196.0kN 264.6kN 294.0kN 343.0kN


Hei

ght o

f mai

n gi

rder

(mm

)

Tunnel side

Microstrain

Fig.6.11 Strain distribution for main girder of SSPC segment specimens

(c)Failure modes

Fig.6.12 shows images of the finite element model and the experiments for SSPC

segment specimens, with those deformed geometry. It can be seen that the crack pattern

predicted using the finite element model is similar to the observed pattern.

(a) Equivalent of cracking strain in tension (b) Longitudinal cracks in tension


121

(c) Plastic strain in compression (d) Concrete crushing in compression

Fig.6.12 Concrete infill failure of SSPC segment specimens

(d)Contact status

Fig.6.13 shows the images of contact status and contact force of SSPC segment

specimens. It can be seen that the steel tube separates from the concrete infill under

positive bending. However, the steel tube contacts the concrete infill throughout the

applied loading process.

(a) Contact normal force (Concrete infill) (b) Contact normal force (Steel tube)

Positive bending


122

(c) Contact normal force (Concrete infill) (d) Contact normal force (Steel tube)

Negative bending

Fig.6.13 Contact status of SSPC segment specimens

(e)Stress Distribution


plates of SSPC segment specimens. It can be seen that the longitudinal stress

(x-direction) in the steel tube is non-uniformly distributed along the width under positive

bending, and is uniformly distributed along the width under negative bending. Therefore,

the skin plate in tension can be assumed as the member of full cross-section effective.



123

(b) Negative bending

Fig.6.14 x-directional stress in steel tube of SSPC segment specimens

Transverse joint


Ductile cast iron

Infilled concrete

Reinforcing bar

Fig.6.15 DRC segment [2]

6.3 DRC SEGMENT

6.3.1 Test Specimens DRC segments (Ductile Cast Iron and Reinforced Concrete Segment) shown in

Fig.6.15 were used in the fourth section of the Tokyo Metropolitan Area Outer

Underground Discharge Channel. The Outer Underground Discharge Channel is about

6.3 km in length for flood control, connecting Naka River, Ayase River, and Edo River,

and locates underground as shown in Fig.6.16 [2].


124

1st section 2nd section3rd section

4th section

(a) The location of the Outer Underground Discharge Channel

(b) Vertical section of geological profile at the fourth section

Fig.6.16 Arrangement of the Discharge Channel tunnel and ground condition of 4th section


125

Details of the fourth section of the Outer Underground Discharge Channel tunnel are

shown in Fig.6.17. The tunnel is 11.8m outside diameter, 10.9m inside diameter and

about 50m overburden. A segmental ring consists of 1 wedge-shaped segment (KP,

inserted from the tunnel axial direction), 2 adjacent segments (BP) and 6 standard

segments (A1P~A6P). These DRC segments are 40.0° in central angle, 1200mm in

width and 450mm in thickness.

Segment arrangement S=1/50(To face side) Face side

Shaft side

Lift positionGrouting position

A6

B7

K8

B9

A1

A2

A3A4

A5

Fig.6.17 Segmental ring arrangement


126

Details of the standard segments (A1P~A6P) are shown in Fig.6.18. The standard

segments (A1P~A6P) consist of 11mm thickness skin plate, 20mm thickness main

girders(including outside main girders and inside main girders), and 20mm thickness

joint plates of ductile cast iron, and reinforced concrete infilling(reinforcing bars

including D29 main reinforcements and D13 structural reinforcements).

C

C

A

AFace side

Shaft side

SM490

A1~A6 segments S=1/50

B

B

face side shaft sideView C-C S=1/10

face side shaft sideView A-A S=1/10

face side shaft sideView A-A S=1/10

Fig.6.18 Configuration and reinforcing bar layout of A1P~A6P segments


127

In general, the lining member of tunnel is subjected to combined bending, axial and

shear forces. Therefore, it is important to study the mechanical behavior of segments

(called as tunnel lining) under confined bending and axial forces. However, it is very

difficult that the axial force is applied on the segment with curvature in the laboratory

experiments. If the ratio of the thickness of segment to the radius of the tunnel is smaller

than 1/20, the segment without curvature can be used to study the mechanical behavior

of the segment under combined bending and axial forces. In this study, the DRC segment

specimens without curvature of the same dimensions as the actual used DRC segments

in the fourth section of the Outer Underground Discharge Channel tunnel were design

and tested. Details of the standard segment specimens (AP) are shown in Fig.6.19 and

the dimensions of DRC segments are summarized in Table6.4. Mechanical properties of

ductile cast iron, reinforcing bar, and concrete are shown in Table6.5.

400

100

1200

100

350

465

125100 125

365

150 125125100 100

400400

D29 (SD295)150

20

10

100

125

125

100

150

400

100

125

125

100

150

1200

540 475 475 410 540475475410

465

3800

540 475 410475 410 475 475 540

A A

View A-A

B

B

View B-B

Main reinforcements(D29) structural reinforcements(D13)

Inside main girder

Unit: mm

Fig.6.19. Configuration and reinforcing bar layout of DRC segment specimens


128

Table 6.4. Details of DRC segment specimens

Dimension Thickness Main girders

Skin plate

Main girder

Joint plate outside inside Specimen Width

(mm) LengthL(mm)

HeightH(mm)

ts(mm) tm(mm) tj(mm) Height (mm)

DRC(Positive bending ) 1200 3800 465 11.0 20.0 20.0 450 350

DRC(Negative bending ) 1200 3800 465 11.0 20.0 20.0 450 350

Note: Positive bending is the load applied on the outside (ground side) of DRC; Negative bending is the load applied on the inside (tunnel side) of DRC.

Table 6.5. Mechanical material properties for DRC segment specimens

Member Material Yield strength

(N/mm2)

Tensile strength /Compressive

strength(N/mm2)

Young modulus

(N/mm2)

Ductile cast iron FCD500-7 340 516 1.7×105

Concrete Design strength

30 N/mm2 -- 49.3 3.3×104

Main reinforcement

(D29) SD295 349 536 2.1×105

Structural reinforcement

(D13) SD295 393 531 2.1×105

6.3.2 Test Setup The DRC segment specimens were tested under constant axial load and increasing

bending load. Fig.6.20 gives a schematic view the test setup. The axial tensional/

compressive loads were applied and maintained constant by the 4 hydraulic actuators.

The bending load was applied by vertical loading in the 500mm range from the left/right

of the midspan of the DRC segment specimen via a hydraulic actuator. The axial loading

history was generally based on the spring-beam model analysis for the segment design

of the fourth section of the Tokyo Metropolitan Area Outer Underground Discharge

Channel.


129

4300

Note: Test setup is the joint bending testing, segment testing is the half width of the above test setup

1650

8270

500

1700

1900

950

2400

3800

Abutment

Specimen

Loading position

Cable used to apply axial force

Pushing actuator applying axial force

Frame used to apply axial force

Actuator used to apply axial force

Actuator used to apply axial force Pushing actuator applying axial force Cable used to apply axial force

Abutment

Loading position

Fig.6.20 Test setup for DRC segment specimens [2](Unit:mm)

(Cable)8270

1200

6050

1700 1700

500 500

3800

950 650

950 650

Vertical loading actuator

Frame used to apply axial force

Actuator used to apply axial force

Abutment

Pushing actuator applying axial force

Specimen

Cable used to apply axial force Load P

Axial force N N Abutment

Axial force


130

The testing steps of DRC segment specimens under combined positive bending/

negative bending and axial load are shown in Fig.6.21, and consist of the 4 test steps.

The first three steps are the applied load reached the yield carrying capacity under

compressive axial load, no axial load, and tensional axial load respectively. The last step

is the applied load reached the DRC segment specimens failure under no axial load.

Displacement transducers and Strain gauges located on the key points shown in Fig.6.22

and Fig.6.23 were used to measure strains and deflections, respectively.

Case1No inner water pressure

Case2Inner water pressure

N=0kN(positive)N=0kN(negative)


Design load(Case2)P=199kN(positive)P=223kN(negative)

Allowable carrying capacityP=1050kN(positive)P=1215kN(negative)


N=-4767kN(positive)N=-5283kN(negative)

Full plastic LoadP=3645kN(positive)P=4145kN(negative)

Yield LoadP=1757kN(positive)P=2839kN(negative)

Allowable carrying capacityP=1050kN(positive)P=1215kN(negative)

allowable carrying capacityP=997kN(positive)P=720kN(negative)

Actual allowable carrying capacityP=1499kN(positive)P=2442kN(negative)

Design load(Case1)P=672kN(positive)P=471kN(negative)

Ver

tical

load

ing(

bend

ing

load

)

Axial load

Fig.6.21 Testing steps

The numbers 1,2,3 etc. denote the serial numbers of displacement transducer　　:displacement transducer Unit : mm

1

3

26 4

200 3800

8

7

500 9

5

200450500450

1200

1011

12

14

13

15

600

Fig. 6.22 The arrangement of displacement transducers on DRC segment specimens


131

475 475 475 475450 400 100 500 450

200 200

600

600

125

125

100

150

150

100

125

125

Mian measuring section

Abutment Abutment

Loading position Loading position

WFLA-6-11-5LT

FLA-5-11-5LT

WFLA-6-11-5LT

FLA-5-11-5LT

WFLA-6-11-5LT

FLA-5-11-5LTFLA-5-11-5LT

FLA-5-11-5LT WFLA-6-11-5LT with waterproof

Strain fixed only the outsidefor main girder

(a) The arrangement of strain gauges on skin plate

125

125

100

150

125

125

100

150

1200

475 475 475 475450 400100

500 450

200 200


Abutment Abutment


465

100

100

100

200 200Abutment Abutment10

010

010

010

0200

200

Longtitudinal direction(WFLA-6-11-5LT ;FLA-5-11-5LT )

Transversal direction(WFLA-6-11-5LT ;FLA-5-11-5LT )

Outside main girder

Inside main girder

(b) The arrangement of strain gauges on main girders


132

475 475 475 475450 400100

500 450

200 200

600

600

125

125

100

150

150

100

125

125


Abutment Abutment


PML-60-5L (Mortar strain gauage)

PL-90-11 for concrete

465

100

100

100

100

(c) The arrangement of strain gauges on concrete infill

125

125

100

150

125

125

100

150

1200

3800

475 475 475 475450 400100

500 450

200 200


Abutment Abutment


465

Main reinforcement (WFLA-6-11-5LT with waterproof )Structural reinforcement (WFLA-6-11-5LT with waterproof )(bottom side)Structural reinforcement (WFLA-6-11-5LT with waterproof )(Vertical)

(d) The arrangement of strain gauges on reinforcing bars

Fig. 6.23 The arrangement of strain gauges on DRC segment specimens


133

6.3.3 Test Results and Discussion (a)Load-deflection response

Load and midspan deflection relationships for the tested specimens can be described

by the load-midspan deflection curves shown in Fig.6.24. It can be observed that the

deflections of DRC segment specimens under only bending load are different from those

of DRC segment specimens under combined bending and compressive/tensional axial

loads. However, this difference of the deflection is very small under combined negative

bending and compressive/tensional axial loads.

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500

4000

Tensional axial load(N=345kN)for considering inner water pressure

No axial load(N=0kN)Allowable stress state

No axial load(N=0kN)Ultimate limit state

Compressive axial load(N=-4767kN)No inner water pressure

Ver

tical

load

(kN

)


First step (Vertical load +Compressive axial load) Second step (Vertical load to allowable carrying capacity)Thrid step (Vertical load +Tensional axial load) Fourth step (Vertical load to ultimate carrying capacity)


0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500

4000

4500


No axial load(N=0kN)Allowable stress state



Ver

tical

load

(kN

)


First step (Vertical load +Compressive axial load) Second step (Vertical load to allowable carrying capacity)Thrid step (Vertical load +Tensional axial load) Fourth step (Vertical load to ultimate carrying capacity)


Fig.6.24. Load-deflection curves of DRC segment specimens at midspan


134


Fig.6.25 shows the measured strain distributions on each composite structural

member of DRC segment specimens. Such curves are displayed under different loading

levels for the measured sections. It is observed from Fig.6.25 (a) that the strain values in

the skin plates increase and decrease repeatedly, and the strain distribution shows non-

uniform distribution along the width. The strain response is attributed to the shear lag

effect. Some values of the strains in the skin plates are larger than the values of the

strains in the edges of the main girders. The confinement effect of the main girders on

the deflection of the skin plates is greater. The strain values for the main girders were

taken as the average strain values (from the measured strain based on the

symmetrically-installed strain gauges in the sections of the 100mm distance from the

midspan section). It can be observed from Fig.6.25 (b) that the plane sections remain

plane after loading, because the strain distribution along the height main girder is linear.

The strain values for the main girders were taken as the average strain values (from the

measured strain based on the symmetrically-installed strain gauges in the sections of the

100mm distance from the midspan section). The strain distribution of main

reinforcements show markedly change under positive bending and negative bending

shown in Fig.6.25 (c), it can be seen that main reinforcements resist the large tensional

force under combined positive bending and axial loads, because the cracks are

progressive and through growth.

-1000 -750 -500 1500 2000-600

-400

-200

0

200

400

600Positive bending

475 475 450 400 100

600

600

1251

2510

0150

1501

0012

512

5

1900

Skin platePloting position

Microstrain

Segm

ent w

idth

(mm

)

N=-4757kN, allowable carrying capacity N=0, allowable carrying capacity N=345kN, allowable carrying capacity N=0, yield load

-500 -250 0 250 500 750 1000 1250 1500 1750 2000-240

-180

-120

-60

0

60

120

180

240Positive bending

Concrete side(tunnel inside)

100

100

100

100

200

465

475 475 450 400100

Outside main girder

Skin plate side(ground side)



Ploting position

Microstrain

Segm

ent t

hick

ness

(mm

)


(a) Skin plate (b) Outside main girders


135

-2000 -1600 -1200 -800 -400 0 400 800 1200 1600 2000-600

-400

-200

0

200

400

600

Positive bending

125

1251

0015

012

512

510

015

0

1200

1900

475 475 450 400 100

Main reinforcement

Ploting position

Microstrain

Segm

ent w

idth

(mm

)

N=-

4757

kN, a

llow

able

car

ryin

g ca

paci

ty N

=0,

a

llow

able

car

ryin

g ca

paci

ty N

=345

kN,

allo

wab

le c

arry

ing

capa

city

N=0

,

yie

ld lo

ad-200 -150 -100 -50 0 50 100 150

-600

-400

-200

0

200

400

600

465

100

100

225

365

225

365

1200

475 475 450 400 100CL

Positive bendingCentral rib

Ploting position

Microstrain

Segm

ent w

idth

(mm

)


(c) Main reinforcements (d) Central rib Positive bending

-3500-3000-2500-2000-1500-1000 -500 0 500 1000 1500 2000-600

-400

-200

0

200

400

600Negative bending

475 475 450 400 100

600

600

1251

2510

0150

1501

0012

512

5

1900


Microstrain

Segm

ent w

idth

(mm

)


-1500 -1000 -500 0 500 1000 1500 2000-240

-180

-120

-60

0

60

120

180

240

Negative bendingSkin plate side(ground side)


100

1001

0010

020046

5

475 475 450 400 100

Outside main girder




Ploting position

Microstrain

Segm

ent t

hick

ness

(mm

)

(a) Skin plate (b) Outside main girders

-600 -400 -200 0 200 400 600 800 1000-600

-400

-200

0

200

400

600


Negative bending

1251

2510

0 150

125

125

100

150

1200

1900

475 475 450 400 100

Main reinforcement

Ploting position

Microstrain

Segm

ent w

idth

(mm

)

-150 -100 -50 0 50 100 150 200-600

-400

-200

0

200

400

600


465

100

100

225

365

225

365

1200

475 475 450 400 100CL

Negative bendingCentral rib

Ploting position

Microstrain

Segm

ent w

idth

(mm

)

(c) Main reinforcements (d) Central rib

Negative bending

Fig.6.25. Measured strain distribution


136

Fig.6.25 (d) shows the measured strain distribution along transversal direction on the

central rib. It is observed that the strain values are very small, and maximum deflection

on transversal direction is about 0.14mm. Therefore, this deflection on transversal

direction can be neglected, because of considering the constrained effect by the adjacent

segmental ring on longitudinal direction in actual tunnel.

(b)Relative slip

Fig.6.26 shows the relative slip of the interface between outside main girders and

concrete infill. The relative slips were measured by the cantilever displacement

transducers. It is observed that the relative slips occur on the interface between outside

main girders and concrete infill of transversal and longitudinal directions. Therefore, the

-0.1 0.0 0.1 0.2 0.3 0.4 0.50

500

1000

1500

2000

2500

3000

3500

4000

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C an tile v e r d isp lacem e n t tran sd ucers

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)


Full loading steps under positive bending

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

0

500

1000

1500

2000

2500

3000

3500

4000

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C an tile v e r d isp la ce m e n t tran sd u ce rs

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)

Horizontal relative slip(mm)

Positive bending

-0.1 0.0 0.1 0.2 0.3 0.4 0.50

500

1000

1500

2000

2500

3000

3500

4000

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C an tile v e r d isp lacem e n t tran sd u ce rs

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)


Full loading steps under positive bending

-0.2 -0.1 0.0 0.1 0.2

0

500

1000

1500

2000

2500

3000

3500

4000

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C a n tile v e r d isp la ce m e n t tra n sd u ce rs

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)


Positive bending



137

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

600

1200

1800

2400

3000

3600

4200

4800

600

4 7 5 4 7 5 4 5 0 4@ 1 0 0 10 0

465

C an tile v e r d isp lacem e n t tran sd u cers

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)


Full loading steps under negative bending

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

0

600

1200

1800

2400

3000

3600

4200

4800

Ver

tical

load

(kN

)60

0

4 75 47 5 4 5 0 4 @ 1 00 1 0 0

465

C an tilev e r d isp lacem en t tran sdu cers

L 4 L 2a L 1 a

Measured point



-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.10

600

1200

1800

2400

3000

3600

4200

4800

Ver

tical

load

(kN

)60

0

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C an tile v e r d isp lacem e n t tran sd u ce rs

L 4 L 2 a L 1 a

Measured point



-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

0

600

1200

1800

2400

3000

3600

4200

4800V

ertic

al lo

ad(k

N)

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C a n tile v e r d isp la ce m e n t tra n sd u ce rs

L 4 L 2 a L 1 a

Measured point




Fig.6.26. Load-relative slip curves of DRC segment specimens

effect of relative slip on the deflection and load carrying capacity of composite structure

should be considered.

(c)Failure modes

The failure modes of the DRC segment specimens are shown in Fig.6.27. Failure of

the DRC segment specimen occurs in tensional region under positive bending due to the

bending cracks. Failure of the DRC segment specimen under negative bending was not

observed, because the testing condition was limited. The crack pattern seen on the

tensional region is progressive and through growth of the small spacing transversal

cracks under positive bending; and on compressive region is progressive and

non-uniform of the spalling cracks


138

Width of the cracks(Unit: mm)： ①0.08，②0.15，③0.10，④0.08，⑤0.10，⑥0.06，⑦0.06，⑧less than0.04，⑨less than 0.04，⑩0.04

①② ③④⑤⑥

⑦⑧ ⑨

(a) The cracks distribution of third loading step

(b) The cracks distribution at failure

Fig.6.27. Failure modes under combined positive bending and axial loads [2]

6.3.4 FEM Results and Discussion Only one quarter of each the DRC segment specimens are modeled taking advantage

of symmetry in two mutually perpendicular vertical planes shown in Fig6.28. Symmetric

boundary conditions are applied at the two vertical planes of symmetry. 8-Node 3D solid

elements with tri-linear interpolation and nonlinear interpolation are used to model steel

plates and concrete, respectively. 8-Node 3D solid element with bi-linear interpolation is

used to model reinforcing bars. The vertical and horizontal constraints are applied at the

bottom of the support along its centre line and mid-span section respectively.

Longitudinal load is distributed to all the nodes on top of the loading plate, with nodes at

symmetry plane getting half of the load at rest of the nodes. Axial load is given as a

uniform pressure load at the DRC segment end. It is applied in first and third loading


139

steps, which is the case in the experiment too. For instance, axial load is given as zero;

the vertical load is increased until to actual allowable carrying capacity and ultimate

carrying capacity in second and fourth loading steps respectively. Contact analysis with a friction coefficient of 0.1 and the relative sliding velocity ( VR ) of default value 0.1 is

used in FEM analysis

(a) FEM model of DRC segment

(b) Negative bending (c) Positive bending

Fig.6.28 Finite element model of a DRC segment


Among the analysis results, load vs. midspan deflection response histories of DRC

segment specimens (with shear ribs, reinforcements) are shown in Fig.6.29.

Comparing the analyzed results with the experimental results, it can be seen that the

load-deflection curves from numerical analysis are agree with the experimental load-

load-deflection curves.


140

0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000

4500

5000




Ver

tical

load

(kN

)


Exprimental result(Full steps)FEM result(Compressive axialload + positive bending load)FEM result(Tensional axial load + positive bending load) FEM result(No axial load, the applied load to failure)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500




Ver

tical

load

(kN

)


Exprimental result(Full steps)FEM result(Compressive axialload + positive bending load)FEM result(Tensional axial load + positive bending load) FEM result(No axial load, the applied load to failure)


Fig.6.29 Load-deflection curves for tested DRC segment specimens at midspan


141

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

500

1000

1500

2000

2500

3000

3500

4000

4500

5000 Line+Plot PlotFEM result Experimental result

600

4 7 5 4 7 5 4 5 0 4 @ 1 0 0 1 0 0

465

C an tile v er d isp lacem e n t tran sd u cers

L 4 L 2 a L 1 a

Measured point

Ver

tical

load

(kN

)


Full loading steps under positive bending Four loading step under positive bending

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.20

500

1000

1500

2000

2500

3000

3500

4000

4500

5000 Line+Plot PlotFEM result Experimental result

Full loading steps under positive bending Four loading step under positive bending

600

47 5 47 5 450 4@ 100 100

465

C antilever d isp lacem en t transd ucers

L 4 L 2a L 1a

Measured point

Ver

tical

load

(kN

)

Horizontal relative slip(mm) Fig.6.30 Load-relative slip relationship for DRC segment specimens

Load-relative slip response histories of DRC segment specimens (with shear ribs,

reinforcements) are shown in Fig.6.30. It can be observed that the vertical relative slip

between the steel tube and concrete infill occurs, especially in the applying positive

bending load. It should be noted that the vertical relative slip affect the behavior of

composite segment in design.


142

(b)Strain distribution Fig.6.31 shows the strain distributions at the measured locations. The strain

distributions of nonlinear analysis show good agreement with the experimental results in

main girders. The tendency of the strain distribution of nonlinear analysis shows close

agreement with the experimental value in the skin plates. In addition, the strain

distribution along the skin plate width for DRC segment specimens is uniform. It can be

concluded that the evaluation of shear studs is necessary to design composite segments.

-1000 -800 -600 -400 0 200 400 600 800

-600

-300

0

300

600Positive bending


475 475 450 400 100

600

600

1251

2510

0150

1501

0012

512

5

1900


Microstrain

Segm

ent w

idth

(mm

)


-3500-3000-2500-2000-1500-1000 -500 0 500 1000 1500 2000-600

-400

-200

0

200

400

600


Negative bending

475 475 450 400 100

600

600

1251

2510

0150

1501

0012

512

5

1900


Microstrain

Segm

ent w

idth

(mm

)


(a) Strain distribution for skin plate

-1200-900 -600 -300 0 300 600 900 12001500180021002400-300

-240

-180

-120

-60

0

60

120

180

240 Line Plot Positive bendingFEM result Experimental result



100

100

100

100

200

465

475 475 450 400100

Outside main girder




Ploting position

Microstrain

Segm

ent t

hick

ness

(mm

)

-1500 -1000 -500 0 500 1000 1500

-240

-180

-120

-60

0

60

120

180

240


Negative bendingSkin plate side(ground side)


100

1001

0010

020046

5

475 475 450 400 100

Outside main girder




Ploting position

Microstrain

Segm

ent t

hick

ness

(mm

)

(b) Strain distribution for main girder


143

-800 -400 400 800 1200 1600-600

-400

-200

0

200

400


N=-

4757

kN, a

llow

able

car

ryin

g ca

paci

ty N

=0,

a

llow

able

car

ryin

g ca

paci

ty N

=345

kN,

allo

wab

le c

arry

ing

capa

city

N=0

,

yie

ld lo

ad

Positive bending12

512

5100

150

125

125

100

150

1200

1900

475 475 450 400 100

Main reinforcementPloting position

Microstrain

Segm

ent w

idth

(mm

)

- 6 0 0 -4 0 0 -2 0 0 0 2 0 0 4 0 0-6 0 0

-4 0 0

-2 0 0

0

2 0 0

4 0 0

6 0 0

L in e P lo tF E M r e s u l t E x p e r im

N = -5 2 8 3 k N , a N = 0 , a N = 5 2 1 k N , a N = 0 , y

N e g a t iv e b e n d in g

1 9 0 0

4 7 5 4 7 5 4 5

M a in r e in fo rc e m e n tP lo

M ic ro s tr a in

Segm

ent w

idth

(mm

)

(c) Strain distribution for main reinforcements

Fig.6.31 Strain distribution for the members of DRC segment specimens

(c)Failure modes

Fig. 6.32 shows images of the finite element model and the experiments for DRC

segment specimens, with those deformed geometry. It can be seen that the crack pattern

predicted using the finite element model is similar to the observed pattern.

(a) Equivalent of cracking strain in tension


144

Width of the cracks(Unit: mm)： ①0.08，②0.15，③0.10，④0.08，⑤0.10，⑥0.06，⑦0.06，⑧less than0.04，⑨less than 0.04，⑩0.04

①② ③④⑤⑥

⑦⑧ ⑨

(b) Longitudinal cracks in tension

(c) Plastic strain in compression

Fig.6.32 Concrete infill failure of DRC segment specimens


145

(d)Contact status

Fig.6.33 shows the images of contact status and contact force of DRC segment

specimens. It can be seen that the steel tube separates from the concrete infill under

positive bending. However, the steel tube contacts the concrete infill throughout the

applied loading process.

(a) Contact status (Positive bending)

(b) Contact status (Negative bending)

Fig.6.33 Contact status of DRC segment specimens


146

(e)Stress Distribution


plates of DRC segment specimens. It can be seen that the longitudinal stresses

(x-direction) in the skin plate are uniformly distributed along the width under positive

bending and negative bending. Therefore, the composite degree is defined by shear

connector in design.



Fig.6.34 x-directional stress in skin plate of DRC segment specimens


147

6.4 RESULTS AND DISCUSSION OF THE MECHANICAL MODEL

The applicability of the proposed mechanical model based on Closed-composite

segment is verified by comparing the experimental results of SSPC segment and DRC

segment. The following contents should be replaced, when the proposed mechanical

model is used in the structural analysis of others type segment.

(1) 0 Positive bending0 negative bending

bslip

t

γγ

γ=⎧

= ⎨ =⎩, and s-t 0k = / s-b 0k = , because of one skin

plate.

(2) No relative slip occurs between concrete and reinforcement. (3) The post-buckling stress udf ′ is defined by the parameters of the used materials.

(4) Stud shear connector is replaced by rib shear connector. The shear strength of a

rib shear connector is calculated by Eq.(2.26).

Displacement equilibrium can be written as follows: Substituting 0bγ = , 0tγ = into Eqs.(5.24) and (5.25) gives:

1 1t t Vγ α γ η′′ = − + (6.1)

2 2b b Vγ α γ η′′ = − + (6.2)

For the load cases shown in Fig.5.4, solving Eqs. (6.1) and (6.2), and using the boundary conditions that

0 00t bx x

γ γ= == = ,

/ 2 / 20t bx L x L

γ γ= =

′ ′= = and / 2V P= gives

the relative slip solution

( )( )

1

2 1

L x L x

L

P e e e

e

λ λ λ λ

λ

ηγ

− − −

−

+ − −=

+ (6.3)

Correspondingly, the relative slip strain solution is

( )( )2 1

x x L

s L

P e e

e

λ λ λ

λ

ηλε

− −

−

−=

+ (6.4)

where 1 2 or λ α α= , 1 2 or η η η= , t1

1

hEI

η = , b2

2

hEI

η =


148

1 t t m mt m mb c cuEI E I E I E I E I= + + + , 2 b b m mt m mb c cuEI E I E I E I E I= + + + , 2

t1

t 1 t t m mt c cu

1 1tK hs EI E A E A E A

α⎛ ⎞

= + +⎜ ⎟+⎝ ⎠,

2b

2b 2 b b m mb c cu

1 1bK hs EI E A E A E A

α⎛ ⎞

= + +⎜ ⎟⎜ ⎟+⎝ ⎠.

The total deflection calculated from elastic deformation and slip-induced deflection

is

( )( )

3

1

1δ

48 4 2 1

L

L

ePL LPEI h h e

λ

λη

λ

−⎛ ⎞−⎜ ⎟= + +⎜ ⎟+⎝ ⎠

(6.5)

( )( )2

2δ 2 312 2 2 4 2 1

b L b

L

P L L L b e eb b b L b PEI h h e

λ λ λ

λη

λ

−⎛ ⎞− −⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎜ ⎟⎝ ⎠ ⎝ ⎠ +⎣ ⎦ ⎝ ⎠ (6.6)

( )

( )4 2 / 2

3 2

5 2 1δ384 8 1

L L

L

qL L e eqEI h h e

λ λ

λη

λ⎛ ⎞− −

= + +⎜ ⎟⎜ ⎟+⎝ ⎠ (6.7)

In the performance-based analysis of composite segments under axial load and

bending, moment-curvature responses are obtained by the analysis procedure described

in Chapter 5.

Comparison of the predicted results derived from the analysis procedure and the experimental results of SSPC segment and DRC segment is shown in Table 6.6. ytM is

measured yield bending capacity. It shows that the analysis results are much closer to the

measured values with a mean value of 1.028.

Table 6.6. Comparison of elastic bending capacities of composite segment specimens

Specimens ytM (kNm)

M (kNm)

MΔ (kNm)

pM (kNm)

p yt/M M

SSPC(Positive) 23.7 25.9 1.1 24.8 1.04

SSPC(Negative) 90.8 94.3 1.5 92.8 1.02

DRC(Positive) 1059.3 1147.1 59.4 1087.7 1.03

DRC(Negative) 1704.6 1809.6 69.1 1740.4 1.02

Mean value 1.028



149

Table 6.7 shows the comparison between the experimental results and calculation values from the present study, where utM is measured ultimate bending capacity. It

shows that the simplified calculation for the ultimate bending moment is applicable in

practical design of others type segment.

Table 6.7. Comparison of ultimate bending capacities of composite segment specimens

Specimens s-tk s-bk utM (kNm)

uM (kNm) ut u/M M

SSPC(Positive) 0.92 0 46.2 44.9 0.97

SSPC(Negative) 0 0.91 162.1 155.8 0.96

DRC(Positive) 0.89 0 2186.8 2242.3 1.025

DRC(Negative) 0 0.86 2693.6(*) 3086.9 ---

Note: * means the bending moment is not ultimate bending moment due to the limitation of the test setup.

The moment-curvature response is obtained by subdividing the cross-section into a

large number of horizontal layers. The ultimate strain is obtained by the compressive

tests of the unconfined concrete. For each step or increment of strain, the depth to the

neutral axis is determined by the strain distribution in main girders，when the sum of all

the forces acting on the section becomes zero (i.e., equilibrium of forces is satisfied);

then the moment of all the forces (acting on the section layers) about the neutral axis is

calculated and the curvature is determined by dividing the concrete strain by the depth of

the neutral axis. The entire curve is plotted by repeating the above procedure until the

ultimate strain in the confined concrete was reached. The last point in the

moment-curvature curve was the moment (flexural) capacity of the section. The

moment-curvature curves obtained using mechanical analysis of the proposed model are

compared with the curves obtained by the experiments as shown in Fig.6.35. It can be

observed that the curvatures predicted by the proposed model are closer to the

experimental curvatures.


150

0.00 0.02 0.04 0.06 0.08 0.10 0.120

10

20

30

40

50

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)

SSPC segment (Positive)


0.00 0.02 0.04 0.06 0.08 0.100

20

40

60

80

100

120

140

160

180

SSPC segment (Negative)

Experimental result Proposed modelBe

ndin

g m

omen

t(kN

m)

Curvature(1/m)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

500

1000

1500

2000

2500

Fourth step (No axial load and ultimate carrying capacty )

DRC segment (Positive)

Experimental result Proposed modelBe

ndin

g m

omen

t(kN

m)

Curvature(1/m) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

400

800

1200

1600

2000

2400

2800

3200

Bend

ing

mom

ent(k

Nm

)

Curvature(1/m)

Fourth step (No axial load and ultimate carrying capacty )

DRC segment (Negative)


Fig.6.35 Bending moment-curvature curves of composite segments

Fig.6.36 shows the load-midspan deflection curves of the experiments and mechnical

analysis of the proposed model. It can be observed that the load-midspan deflection

response predicted using the proposed model is similar to the corresponding

experimental plots. Therefore, the formula and the mechanical model based on

Closed-composite segment have reasonable accuracy, when which are used in the

structural analysis of others type composite segment.


151

0 3 6 9 12 15 180

50

100

150

200

250

SSPC Positive bending

Load

(kN

)


Exprimental result Proposed model

0 3 6 9 12

0

100

200

300

400

500

600

700

800

Exprimental result Proposed model

SSPC negative bending

Load

(kN

)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000

DRC segment (Positive bending)Ver

tical

load

(kN

)


Exprimental result(Full steps) Proposed model

0 5 10 15 20 25 30

0500

1000150020002500300035004000450050005500

DRC segment (Negative bending)

Exprimental result(Full steps) Proposed model

Ver

tical

load

(kN

)

Deflection at midspan(mm) Fig.6.36 Load-deflection curves of composite segments at midspan

6.5. SUMMARY

From experimental results of composite segments (including SSPC segment, and

DRC segment), it can be seen that the failure modes of SSPC segment and DRC segment

behave differently from that of the Closed-composite segment, where the cracks are

progressive and through growth under positive bending. The relative slip occurs at

interface between concrete infill and the steel tube for SSPC segment and DRC segment.

The proposed finite element model using MSC.Marc shown in Chapter 4 is able to

simulate the mechanical behavior of composite segments (including SSPC segment, and

DRC segment).

Comparisons have been made between experimental results of SSPC segment and


152

DRC segment and the mechanical predictions of behavior using the proposed method.

Good agreement has been found and this indicates that the method is suitable for general

analysis of others type composite segments.

6.6. REFERENCES

1) Japan Steel Segment Association (JSSA), 1995. The report of development of Steel

Segment with Pre-filled Concrete(SSPC) . (In Japanese)

2) Masami, Shirato, et al.2003. Development of new composite segment and application

to the tunneling project. Journal of JSCE, No.728, 157-174.(In Japanese)


153

Chapter 7. Design of Composite Segment

7.1 INTRODUCTION

Following the planning works for the tunnel, the lining of a shield tunnel (hereinafter

called as segment rings) is designed according to the following sequence, as a rule [1]:

1) Adherence to specification, and code or standard.

The tunnel to be constructed should be designed according to the appropriate

specification standard, and code or standards, which are determined by the persons in

charge of the project or decided by discussion between these persons and the designers.

2) Decision on inner dimension of tunnel.

The inner diameter of the tunnel to be designed should be decided in consideration of

the space that is demanded by the functions of the tunnel. This space is determined by:

a) The construction gauge and car gauge, in the case of railway tunnels;

b) The traffic volume and number of lanes, in the case of road tunnels;

c) The discharge, in the case of water tunnels and sewer tunnels;

d) The kind of facilities and their dimensions, in the case of common ducts.

3) Determination of load condition.

The loads acting on the lining include earth pressure and water pressure, dead load,

reaction, surcharge and thrust force of shield jacks, etc. The designer should select the

cases critical to the design lining.

4) Determination of lining conditions.

The designer should decide on the lining conditions, such as dimension of the lining

(thickness), strength of material, arrangement of reinforcement, etc.

5) Computation of member forces. The designer should compute member forces such as

bending moment, axial force, and shear force of the lining, by using appropriate models

and design methods.

6) Safety check.

The designer should check the safety of the lining against the computed member

forces.

7) Review.

Design of Composite Segment

154

If the designed lining is not safe against design loads, the designer should change the

lining conditions and design lining. If the designed lining safe but not economical, the

designer should change the lining conditions and redesign the lining.

8. Approval of the design.

After the designer judges that the designed lining is safe, economical, and optimally

designed, a document of design should be approved by the persons in charge of the

project. In Fig.7.1, these steps are shown on a flow chart for designing tunnel linings.

No

No

Yes

Yes

Execution of construction works

Approval

Safty and ecnomical

Check of safety of lining

Computation of member forces

Model to compute member forces

Assumption of lining condition (Thickness, etc.)

Load condition

Inner diameterSpecification/code/standard to be used

Function/Capacityto be given to tunnel

Alignment plane/Profile cross section

Survey/Geology

Planning of tunnel project

Fig.7.1 Flow chart of shield tunnel lining design


155

It is well known that the segments production cost accounts for a large part of the

total shield tunnel construction cost. One of the effective methods is to design the

segments more efficiently for reducing the shield tunnel construction cost. The common

design method for shield segments is to determine firstly the load acting on the tunnel

lining, then determine the material and the cross sectional dimensions of the segments by

structural calculations. Therefore, it is important to evaluate the load accurately. In

structural calculation, the stress in each part of a segment is usually calculated with a

design model, by taking account of the structural property of segmental lining and

interaction between the ground and lining. According to the calculation result of member

forces, the safety of the most critical sections must be checked using the limit state

design method or the allowable stress design method. The safety factors should be based

on the ground loading and defined in accordance with the structural requirements and

codes (e.g. National standard specification for design and construction of concrete

structures). Therefore, construction procedure and performance of underground

structures should be linked with the factors of safety.

The following sections will describe the calculation of the loads, structural model,

and the material properties of the limit state design method or the allowable stress design

method. Finally, a design example of shield tunnel composite lining will be expressed.

7.2 CALCULATION OF LOADS

Overburden earth pressure or loosening earth pressure calculated by Terzaghi’s

formula is very useful and has generally been adopted as the vertical load acting on the

tunnel lining for the segment design on the basis of previous field measurement data.

Therefore, in this study, the loads are calculate using the Terzaghi’s formula and the

proposed method by guideline of design segment of JSCE[2].

To take the relationship of the outer diameter of a tunnel and overburden, soil

condition, and ground water level effects into account, the design earth pressure and

water pressure are calculated by guideline of design segment of JSCE[3] shown in

Fig.7.2.


156

*8 shown in Table.6.245±*7 range of the tunnel spring

*6 shown in Table.6.1

minp*5 is the minimum vertical load in design; ie, 196kN/m2 of JR code

cN*4 is assumed as 8 for JSCE codesN*3 is assumed as 10

H′*2 Determine the range of by the soil condition between tunnel spring and tunnel crown

is the minimum overburden for considering the effect of arching;ie, 1~2D of JSCE code; 1.0D of double track and1.5D of single track of Metro code; 2.0D of JSWC

H′*1

Calculation of water pressure

Calculation of earth pressure

based on Terzaghi's loosing earthpressure by rock condition

based on Terzaghi's loosing earthpressure by rock condition

*6 *51 min(> )ep D pα γ′=

*6 *51 min(> )ep D pα γ ′=No

Soil-water separated is safty than soil-water integrated ?

Rock with cracks ?Mixed sandy stratum ?

1

*5min

full overburden pressure(> )

epp

=

No

No

Yes

Yes

Yes

A

Organizing soil condition for calculating the ground reaction and lateral earth pressure (determine k,λ)*7

Calculation of q

e1,q

e2

1 1 1 2

1 1 1

2 2 2

v e w v

h e w

h e w

p p p pq q qq q q

= + == += +

Calculation of kδ

ENDqw2

qe2

qe1 q

w1

0RcR

iR

H

Hw

Ground reaction

Wat

er p

ress

ure

Earth

pre

ssur

e

Earth

pre

ssur

e

Wat

er p

ress

ure

90o

g

pg

Ground reaction by self weight

pe2Earth pressure

pw2Water pressure

Ground reaction

pe1Earth pressure

pw1Water pressure

Surcharge p0

1 2 1e w wp p p gπ′ = − −

Vertical pressure at tunnel crowncosidering ground reaction at crowninduced the bottom water pressure

2 1 1w w wp p p gπ≤ + +

Check the bottom water pressure*8

1*5

min

full overburdenpressure(> )

epp

=1*5

min

Terzaghi's loosing

earth pressure(> )ep

p

=

A

1 1 2, ,w w wp q qCalculation of

Design ground water tablebased on the others conditions

NoYes*4

cN N≥

*3sN N≥

NoYes

C

B


Calculation of water pressure

NoYesSandy soil ?


Calculation of water pressure C

BA

NoYesOrganizing soil condition forcalculating the vertical loads(By c,φ,N)*2

Yes

No

Sandy soil ?

Soil-water separated Soil-water integrated*1H H′>Overburden

Start

Fig.7.2 Flow chart of calculation of the loads [3]


157

Table 7.1. Earth pressure acting on the lining by Terzaghi[4]

Rock condition Rock load factor H(m) Remarks

(1) Hard and intact 0 Simple support may be required

(2) Hard, stratified or schistose

0~0.25B

(3) Massive, moderately jointed

0~0.5B

Simple support may be required; nonuniformly varying load

(4) Moderately blocky, seamy and jointed

0~0.25B;0.35(B+Ht) lateral earth pressure is equal to zero

(5) Very blocky and seamy, shattered

(0.35~1.10)(B+Ht) lateral earth pressure is very little or equal to zero;

(6) Completely crushed but chemically intact

1.10(B+Ht) Inverts essential, arched roof essential

(7) Squeezing rock at moderate depth

(1.10~1.0)(B+Ht)

(8) Squeezing rock at great depth

(2.10~4.5)(B+Ht)

Inverts essential in excavation, arched roof essential

(9) Swelling rock Up to 80m irrespective of (B+Ht)Inverts essential in excavation, arched roof essential

Note: 1) The table shows the rock load factor for overburden up to 1.5(B+Ht) B is width or span of opening; Ht is height of opening

2) The table shows tunnel is located below groundwater table. If tunnel is located above groundwater table, the value of (6) reduces 50%.

Table 7.2. Coefficient (λ ) of lateral earth pressure and coefficient ( k ) of ground reaction [5]

k (MN/m3) Type of soil λ

During grouted After grouted

N value

guideline

Very dense sandy soil 0.35-0.45 35.0-47.0 55.0-90.0 30N ≤

Dense sandy soil 0.45-0.50 21.5-35.0 28.0-55.0 15 30N≤ <

Medium, loose sandy soil 0.50-0.60 21.5 28.0 15N <

Hard clayey soil 0.40-0.50 31.5 41.0 8 25N≤ ≤

Medium and stiff clayey soil 0.50-0.60 13.0-31.5 15.0-46.0 4 8N≤ <

Soft clayey soil 0.60-0.70 3.5-7.0 3.8-7.5 2 4N≤ <

Very soft clayey soil 0.70-0.80 3.5 3.8 2N <


158

0p

γ′

cφ

cφ

γ

vσ

1vσ

wH

H WL

12B

45 /2φ°+

2φ

1φ1c

2c

1γ

245 /2φ°+

2γ

H2

0p

vσ

1vσH1

12B

(a) Single soil strata with ground water (b) Alternation of strata

Fig.7.3 Calculation model of loosing earth pressure

7.2.1 Earth Pressure and Water Pressure Earth pressure and water pressure are calculate by flow chart of calculation of the

loads (Fig.7.2) shown in Table 7.3 and Table 7.4.

(1) Loosing earth pressure by Terzaghi’s formula

Loosing earth pressure for Single soil strata with ground water shown in Fig.7.3(a) is

calculated as

( ) ( )0 1 0 1/ tan / tan1 10

0

/1

tanK H B K H B

v

B c Be p e

Kφ φγ

σφ

− −−= − + (7.1)

1 0/ 4 / 2cot

2B R π φ+⎛ ⎞= ⎜ ⎟

⎝ ⎠ (7.2)

where, vσ is loosing earth pressure; 0K is Ratio between lateral earth pressure and

vertical earth pressure(equal to 1);φ is angle of internal friction of soil; 0p is surcharge;

c is cohesion of soil; γ is unit weight of soil; and 0R is outer radius of the lining.


159

(2) Loosing earth pressure of below ground water table (a) wH H<

( ) ( )0 1 0 1( ) / tan ( ) / tan1 11 0

0

/1

tanw wK H H B K H H B

v

B c Be p e

Kφ φγ

σφ

− − − −−= − + (7.3)

( ) ( )

( ) ( )

( ) ( )

( ) ( )

0 1 0 1

0 1 0 1 0 1

0 1

0 1 0 1

/ tan / tan1 11

0

( ) / tan ( ) / tan / tan1 10

0

/ tan1 1

0

/ tan / tan1 10

0

/1

tan

/1

tan

/1

tan/

1tan

w w

w w w

w

w

K H B K H Bv v

K H H B K H H B K H B

K H B

K H B K H B

B c Be e

K

B c Be p e e

K

B c Be

KB c B

e p eK

φ φ

φ φ φ

φ

φ φ

γσ σ

φ

γφ

γφ

γφ

− −

− − − − −

−

− −

′ −= − +

⎧ ⎫−= − +⎨ ⎬⎩ ⎭

′ −+ −

′ −= − +

( ) ( )0 1 0 1( ) / tan / tan1 1

0

/1

tanw wK H H B K H BB c B

e eK

φ φγφ

− − −−+ −

(7.4)

(b) wH H≥

( ) ( )0 1 0 1/ tan / tan1 10

0

/1

tanK H B K H B

v

B c Be p e

Kφ φγ

σφ

− −′ −= − + (7.5)

where, 1 0/ 4 / 2cot

2B R π φ+⎛ ⎞= ⎜ ⎟

⎝ ⎠.

(3) Loosing earth pressure for alternation of strata

Loosing earth pressure of alternation of strata shown in Fig.7.3(b) is calculated as

( ) ( )0 1 1 1 0 1 1 1/ tan / tan1 1 1 11 0

0 1

/1

tanK H B K H B

v

B c Be p e

Kφ φγ

σφ

− −−= − + (7.6)

( ) ( )

( ) ( )

( ) ( ) ( )

0 2 1 2 0 2 1 2

0 2 1 2

0 1 1 1 0 2 1 2

0 1 1 2 2 1

/ tan / tan1 2 2 11 1

0 2

/ tan1 2 2 1

0 2

/ tan / tan1 1 1 1

0 1( tan tan ) /

0

/1

tan/

1tan

/1 1

tan

K H B K H Bv v

K H B

K H B K H B

K H H B

B c Be e

KB c B

eK

B c Be e

K

p e

φ φ

φ

φ φ

φ φ

γσ σ

φγ

φγ

φ

− −

−

− −

− +

−= − +

−= −

−+ − −

+

(7.7)


160

Table 7.3. Examples of notation used in the guidelines (Soil condition) [3]

1H H≥

1H H<

Sandy stratum is main stratum Clayey stratum is main stratum

a) Loosing sandy soil

N<N s

b) Medium stiff sandy soil

HH

w

H H H

H H H H

Hw

Hw

Hw

a) Soft and medium clayey soil

N<N c

b) Hard clayey soil

p0 p0 p0 p0

p0 p0 p0 p0

p0 p0 p0 p0

p0 p0 p0 p0

p0 p0 p0 p0

WLWL

WL WL

WL WLWL

WLWLWL

WL WL WL WL

Loosing sandy soil Medium stiff sandy soil

Soft and medium clayey soil

Hard clayey soil

Single soilstratum

Alternation of soilstrata

H H

H

HH H

HHHHH H

Hw

Hw

HwHw

Hw

Hw

Hw

Hw

Hw

Hw

Hw H

w

І

II

(1)

H H ′<

(1)

(2)

(3)

(2)

H H ′≥

sN N≥ cN N≥

H H ′≥

Note: 1) H ′ is the minimum overburden for considering reduced earth pressure;

2) 10sN = or 8sN = of design guideline of JSCE;

3) sλ is coefficient of lateral earth pressure of sandy soil; sk is coefficient of subgrade reaction of

sandy soil; cλ is coefficient of lateral earth pressure of clayey soil; ck is coefficient of subgrade

reaction of clayey soil;

4) If H H ′< , the earth pressure is calculate by the full overburden.


161

Table 7.4. Examples of notation used in the guidelines [3]

I Single soil stratum

(1) H H ′< (2) H H ′≥

a) Loosing sandy soil

sN N<

1 0( )e w wp H H H pγ γ ′= − + +

1 1( / 2)e s eq p tλ γ ′= +

{ }2 1 ( / 2 2 )e s e cq p t Rλ γ ′= + +

1w w wp Hγ=

1 ( / 2)w w wq H tγ= +

2 ( / 2 2 )w w w cq H t Rγ= + +

subgrade reaction sk δ=

In according with

A-a)- I-(1)

A. Sandy stratum is main stratum

b) Medium stiff sandy soil

sN N≥

In according with A-a)- I-(1)

From Terzaghi’s formula ( ) ( )

( ) ( )

0 1

0 1 0 1

0 1

/ tan*1 1 11

0

( ) / tan / tan1 1

0/ tan

0

/1

tan/

1tan

w

w w

K H Be

K H H B K H B

K H B

B c Bp e

KB c B

e eK

p e

φ

φ φ

φ

γφ

γφ

−

− − −

−

′ −= −

−+ −

+

1 0/ 4 / 2cot

2B R π φ+⎛ ⎞= ⎜ ⎟

⎝ ⎠


cN N<

1 0ep H pγ= +

1 1( / 2)e c eq p tλ γ ′= +

{ }2 1 ( / 2 2 )e c e cq p t Rλ γ ′= + +

subgrade reaction ck δ=

In according with

B-a)- I-(1) B. Clayey stratum is main stratum b) Hard clayey

soil cN N≥

n according with B-a)- I-(1)

1ep Hαγ=

1 1( / 2)e c eq p tλ γ ′= +

{ }2 1 ( / 2 2 )e c e cq p t Rλ γ ′= + +

subgrade reaction ck δ=

α is reduced factor shown in Table.7.1

Note: *1) if 2 1 1w e wp p p gπ> + + , calculation of earth pressure at crown should be considered the effect

of the subgrade reaction induced the bottom water pressure; Existing ground water and

alternation of soil strata, loosing earth pressure is given by Eqs.(7.3)-(7.7).

Continue


162

Continue Table 7.4.

II Alternation of soil strata

(1) (2) (3)

a)Loosing sandy

soil

sN N<

In according with

A-a)- I-(1)

Coefficient of lateral earth pressure of sandy soil and coefficient of subgrade reaction of sandy soil are equal to cλ and

ck , respectively; others calculation in according with A-a)- I-(1)

In according with

A-a)- I-(1)

A. S

andy

st

ratu

m is

mai

n st

ratu

m

b) Medium stiff

sandy soil

sN N≥

1ep is equal to the loosing earth pressure of the above side of clayey stratum adding earth pressure of the overburden of clayey stratum


ck , respectively ; others calculation in according with A-b)- I-(2)

In according with

A-b)- I-(2)


cN N<

In according with

B-a)- I-(1)

1 0ep H pγ= +

*21 1( / 2)e s e w wq p H tλ γ λ′= − +

1*22 ( / 2 2 )

e w we s

c

p Hq

t Rγ

λγ

− +⎧ ⎫= ⎨ ⎬′ +⎩ ⎭

*21 ( / 2)w w wq H tγ= +

*22 ( / 2 2 )w w w cq H t Rγ= + +


In according with

B-a)- I-(1)

B. C

laye

y st

ratu

m is

mai

n st

ratu

m

b) Hard clayey Soil*3

cN N≥

In according with

B-b)- I-(1)


ck , others calculation in according with B-b)- II-(1)

1ep Hαγ=

*21 1( / 2)e s e w wq p H tλ γ λ′= − +

1*22 ( / 2 2 )

e w we s

c

p Hq

t Rγ

λγ

− +⎧ ⎫= ⎨ ⎬′ +⎩ ⎭

*21 ( / 2)w w wq H tγ= +

*22 ( / 2 2 )w w w cq H t Rγ= + +


Note: *2) Determining water pressure should be careful in alternation of soil strata. In case, soil-water integrated should be considered for the safety;

*3) if considering water pressure is safety, full water pressure is adopted by soil-water separated.


163

7.2.2 Self Weight Self weight is the vertical load acting along the centroid of the cross section of tunnel

and is calculated as

2 c

WgRπ

= (7.8)

where W is weight of lining per meter in longitudinal direction; cR is radius of controid

of the lining.

7.2.3 Surcharge The surcharge increases earth pressure acting on the lining. The following act on the

lining as the surcharge:

1) Road traffic load,

2) Railway traffic load, and

3) Weight of buildings.

7.3 STRUCTURAL CALCULATION

The structural models for the segmental lining are shown in Fig.7.4. The segments

are assembled in a staggered pattern to compensate for the decrease in the bending

rigidity of the circumferential joint. The multiple hinge ring model can not be applied.

Uniform rigidity ring Multi-hinge ring Beam-spring Shell-spring

Hinge

cL cL

Rotation spring

Shear spring at radial direction

Axial spring at circumferential direction

Longitudinal spring

Shear spring at radial direction

Axial spring at circumferential direction

Rotation spring

Shear spring

Rotation spring

Rotation spring

Longitudinal spring Shear spring


Longitudinal joint

Fig.7.4 Structural models of the segmental lining


164

As present, the uniform rigidity ring model has been applied. The circumferential

joint is assumed to have the same rigidity as that of the segment, the moment for the

design of joint is overestimated, and that occurring at the segment is not calculated

correctly. Therefore, it is difficult to design the bending moment at the joint area.

An average uniform rigidity ring model was proposed in order to make up for the

disadvantage of the uniform rigidity ring model. In this model, the ratio between the rigidity of a ring with joints and that without any joints is assumed to be η , where η is

the effective ratio of the bending rigidity. In poor ground condition, the effect of η on

member forces is unobvious, but in good ground condition, this effect is very obvious. In addition, a bending moment whose magnitude is additional rate ζ times that of the

segment is considered to be distributed to the adjacent segment in the joint area (Fig.7.5).

At this point, the values of the bending moments used for the segment and the joint are assumed to be M(1+ )ζ and M(1- )ζ , respectively. Even in this model, the effective

ratio of the bending rigidity (η ) is described as being universally determined by the

profile of the joint and the shape and the size of the segment. However, if all joint

surfaces are compressed, the rigidity of the circumferential joint with the same rigidity of the segment varies with the stress conditions. Therefore, η changes depending on

the load. Although the distribution of bending moment near the circumferential joints is considered using the additional rate (ζ ), the bending moment obviously varies with the

bending rigidity of the circumferential joint. The actual structural property is not

considered shown in Fig.7.5, where the moment between the two rings is simply

distributed along the joint area. It is impossible to calculate actual distribution of the

bending moment by using this model.

Fig.7.5 Concept of the additional rate of bending rigidity [2]

M: Bending moment calculated by uniform ring with EIη

M0: Design bending moment for main cross section; ( )0 2M M+M 1 Mζ= = +

M1: Design bending moment for circumferential section; ( )1 2M M-M 1 Mζ= = −

M2: Bending moment transferred to adjacent rings due to staggered arrangement

Width

2 / 2M

2 / 2M

1M2 / 2M

2 / 2M


165

For the above reason, the beam-spring model and the shell-spring were proposed

and applied in design of segmental lining nowadays. In the beam spring model,

segments are modeled using beams. The circumferential joints are modeled using the

rotation of springs and their rigidities are expressed by the constants of the springs

concerning the bending moments. The longitudinal joints are modeled using shear

springs to simulate the relative displacement occurring between the two adjacent rings in

the longitudinal direction. However, the bending moment distribution along the width of

a ring is non-uniform due to the split effect of the longitudinal joints. Therefore, the

shell-spring model is applied, in which tunnel lining is modeled using shell, and the

joints are modeled using springs.

7.3.1 Elastic Equation Method The tunnel is assumed to be made of a rigid material and the soil reaction is

determined to be independent of the tunnel deformation caused by active loads. This

model is called the ‘conventional model’/ the ‘modified conventional model’ shown in

Fig.7.2, and the calculations of member force based on elastic equation method is given

by Table 7.5

Table 7.5. Equations of member forces for conventional model/ modified conventional model[3]

Load Bending moment Axial force Shear force

Uniform load in vertical direction

1 1( )e wp p+

21 1

1 (1 2S)( )4 e w cM p p R= − +

1 1( ) S2e w cN p p R= + 1 1( ) SCe w cQ p p R= − +

Uniform load in lateral direction

1 1( )e wq q+ 2

1 11 (1 2C2)( )4 e w cM q q R= − +

1 1( ) C2e w cN q q R= + 1 1( ) SCe w cQ q q R= +

Triangularly varying load in lateral direction

2 2 1 1( )e w e wq q q q+ − − 22 2 1 1

1 (6 3C 12C2 4C3)48

( )e w e w c

M

q q q q R

= − − +

+ − −

2 2 1 1

1 (C 8C2 4C3)16

( )e w e w c

N

q q q q R

= + −

+ − − 2 2 1 1

1 (S 8SC 4SC2)16

( )e w e w c

Q

q q q q R

= + −

+ − −

Subgrade reaction in lateral direction

( )kδ 0

4πθ≤ <

2(0.2346 0.3536C) cM k Rδ= −

04πθ≤ <

0.3536C cN k Rδ=

04πθ≤ <

0.3536S cQ k Rδ=


166

4 2π πθ≤ ≤

2

( 0.3487 0.5S2+0.2357C3) c

Mk Rδ

= − +

4 2π πθ≤ ≤

( 0.7071C C2+0.7071S2C) c

Nk Rδ

= − +

4 2π πθ≤ ≤

(SC 0.7071C2S) cQ k Rδ= −

Dead load ( )g 0

2πθ≤ ≤

23 5( S C)8 6 cM gRπ θ= − −

2π θ π≤ ≤

2( )S

85 C S26 2

cM gR

π π θ

π

⎧ ⎫− + − −⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪−⎪ ⎪⎩ ⎭

02πθ≤ ≤

1( S C)6 cN gRθ= −

2π θ π≤ ≤

( S+ S+ S21 C)6 c

N

gR

π θ π= −

−

02πθ≤ ≤

1( C S)6 cQ gRθ= − −

2π θ π≤ ≤

( ) 1C SC C6 cQ gRπ θ π⎧ ⎫= − − −⎨ ⎬

⎩ ⎭

Lateral displacement at spring ( )δ

No considering soil reaction due to dead weight of tunnel lining

{ }( )

41 1 1 1 2 2

4

2( ) ( ) ( )

24 0.0454e w e w e w c

c

p p q q q q R

EI kRδ

η

+ − + − +=

+

Considering soil reaction due to dead weight of tunnel lining

{ }( )

41 1 1 1 2 2

4

2( ) ( ) ( )

24 0.0454e w e w e w c

c

p p q q q q g R

EI kR

πδ

η

+ − + − + +=

+

Where EI is bending rigidity of segment per unit width

Note: 1) θ is angle from crown; S=sinθ , 2S2=sin θ , 3S3=sin θ ; C=cosθ , 2C2=cos θ , 3C3=cos θ ; 2) The values of effective ratio of the bending rigidity of η and additional rate of ζ are not clear,

and the design examples are shown in Table 7.6.

Table 7.6. Effective ratio of the bending rigidity of η and additional rate of ζ [3]

Segment type Joint type η ζ

Plate type 0.1~0.3 0.5~0.7 RC segment(Flat type) Pin

(Box type) (0.3~0.5) (0.3~0.6)

RC segment(Box type) Bolt --- ---

Steel and ductile cast iron segment Bolt 0.5~0.7 0.1~0.3

Note: 1) For conventional model, 1η = , 0ζ =


167

7.3.2 Calculation of Shell-Spring Model

The constants of the springs of the joints (circumferential joint and longitudinal

joint) and soil reaction should be determined, when Beam-Spring Model and

Shell-Spring Model are used to calculate the shield tunnel structure. In general, these

constants are derived from the experimental results, theoretical calculation, and FEM. In

this study, the calculation method of the constants of these springs proposed by Koizumi,

Murakami, and Kimura[6-9] are described in the following sections, which is applied in

the segment design in Japan, and Moleman software package(Structural analysis

program for underground structure).

(a) Spring constant of circumferential joint

Springs of circumferential joint consist of axial spring with rigidity of cak , shear

spring of rigidity of csk , and rotational spring with rigidity of ckθ . The calculation

model of circumferential joint is shown in Figs.7.6 and 7.7.

The rigidity of axial spring cak is given by

22 (Tightening)

2 2

(Separated)2

(In compression)

b pupl

b puca b pu

b pu

k kk

k kk k k

k k

⎧+⎪ +⎪⎪= ⎨

⎪ +⎪⎪∞⎩

(7.9)

where bk is the axial rigidity of bolt; puk is the axial rigidity of joint plate in

compression; plk is the axial rigidity of joint plate considering the initial tightening bolt

induced compressive strain released; and the subscript and superscript of cak , c

sk , and ckθ express the directions and joint location, respectively.

2t

T/2 T/2

T/2 T/2

kpu

kpu

kpL kb

kpu

kb

kpu

(a) Tightening (b) Separated (a) Tightening (b) Separated Fig.7.6 Stress condition of joint member Fig.7.7 Modeled joint member by springs


168

lH

le

ln l1

AeAb1

Nut

Bolt

Fig.7.8 Bolt and nut

11

1

(Single bolt)

(Double bolts)

e

en e

bb

e

n e

EAAl l lAk

EAl l l

⎧⎪

+ +⎪= ⎨⎪⎪

+ +⎩

(7.10)

where E is elastic modulus of a bolt; 0.6e Hl l= ; and 1l , nl , Hl , eA , and 1bA is shown in

Fig.7.8.

puk and plk are calculated as for single bolt

Single bolt0.5

0 Double bolts

u

wpu

EAt tk

⎧⎪ += ⎨⎪⎩

(7.11)

Single bolt

2 Double bolts

l

pl

EAtkEAt

⎧⎪⎪= ⎨⎪⎪⎩

(7.12)

where 2 2( )u u aA r rπ= − ; 2 2( )l l aA r rπ= − ; 2 2( )e aA r rπ= − ; t is the thickness of a joint

plate; wt is the thickness of a washer; /12u wr r t= + ; / 3l wr r t= + ; / 6e wr r t= + ; ar is the

radius of a bolt hole; and wr is the radius of a washer.

Shear spring of rigidity of csk can be assumed as infinite, if the relative slip induced

the shear force is not occurring, when friction force from initial tightening of a bolt is

very large.


169

The rotational spring with rigidity of ckθ is given by

sep

sep

max sep

max sep

(Tightening)

(Separated)

c

M

kM Mθ

θ

θ θ

⎧⎪⎪= ⎨ −⎪⎪ −⎩

(7.13)

sep 0

(2 )2( / 3)

(2 )b pu pl b pu

bopl b pu

k k k k kM d y N

k k k+ +

= −+

(7.14)

sepsep

0

2d yδ

θ =−

(7.15)

where, d is distance between the central axial of a circumferential bolt and the upper extreme of a joint plate; 0y is the neutral axis depth; and boN is the initial tightening

force of a bolt.

The semi-experiential value of ckθ is given by Koizumi based experimental study

[7].

* c c cRk kEIθ θ= (7.16)

where, EI is the bending rigidity of a segment; cR is radius of controid of the lining; *ckθ is a dimensionless parameter, * 210ckθ > (uniform rigidity ring), * 0.1 ~ 10ckθ = (ring

with joints), * 310ckθ−< (multiple hinge ring).

(b) Spring constant of longitudinal joint

Springs of longitudinal joint consist of axial spring with rigidity of Lak , shear spring

of rigidity of Lsk , and rotational spring with rigidity of Lkθ .

The rigidity of axial spring Lak in longitudinal direction is given by

{ }96 (1 ) (In tension)

192 (1 )

(In compression)

ca l

cLa ca

Dk hk l D ek

αα

⎧ +⎪ + += ⎨⎪ ∞⎩

(7.17)

where 3

12(1 )eb tD Eν

=−

; cl is the length of a main girder (joint plate); lh is distance

between the central axial of a longitudinal bolt and the lower extreme of a main girder;

e is eccentricity between center of working force and the lower extreme of a main


170

girder; ν is Poisson ratio of a main girder; the effective width of eb , and the

dimensional parameter α are calculated as

( )2 / 6e wb r t= + (7.18)

3 3

3 3

3

3

( ) (for steel segmental ring)64

(for RC segmental ring)64

c u l

u l

c

l

l h hh h

lh

α

⎧ +⎪⎪= ⎨⎪⎪⎩

(7.19)

where uh is distance from the central axial of a longitudinal bolt and the upper extreme

of a main girder.

Shear spring of rigidity of Lsk can be assumed as infinite, if the relative slip

between the two adjacent rings induced the shear force is not occurring, when friction

force from initial tightening of a bolt is very large.

The rotational spring with rigidity of Lkθ is given by

( ) ( )

2

0 0

(for steel segmental ring)2

/ 3(for RC segmental ring)

2

La

LLa

d k

kd y d y kθ

⎧⎪⎪= ⎨

− −⎪⎪⎩

(7.13)

where, d is distance between the central axial of a longitudinal bolt and the upper extreme of a main girder; and 0y is the neutral axis depth.

(c) Spring constant of soil reaction

Spring constant of soil reaction shown in Table 7.7 is given by Kimura.

Table 7.7. Spring constant of soil reaction [9]

Passive soil reaction spring 0

3(1 )(5 6 )

grc

g g

EK

R ν ν=

+ −

Active soil reaction spring , / 2.0rt rc rcK K K= Spring constant of soil

reaction

Tangential soil reaction spring , / 3.0t rc rcK K K=

gE is modulus of deformation of soil; 0R is outer radius of the lining; gν is Poisson ratio of

soil.


171

7.4 CHECK OF SAFETY OF SEGMENTAL LINING

According to the calculation result of member forces, the safety of the most critical

sections must be checked using the limit state design method or the allowable stress

design method. These are as follows:

1) Section with the maximum positive moment.

2) Section with the maximum negative moment.

3) Section with the maximum axial force.

The safety of the lining against the thrust force of the shield jacks should be checked.

7.4.1 Allowable Stress Design Method The allowable stresses in lining materials are described as the following sections [2].

(a) Concrete for segment

Allowable stresses of concrete for segment are shown in Table 7.8.

Table 7.8. Allowable stresses of concrete for segment (N/mm2) [2]

Characteristic compressive strength ckσ 42 45 48 51 54 57 60

Allowable compressive stress for bending moment caσ 16 17 18 19 20 21 22Allowable standard shear stress Shear stress by bending1) aτ 0.73 0.74 0.76 0.78 0.79 0.81 0.82

Allowable bond stress(deformed bar) 0τ 2.0 2.1 2.1 2.2 2.2 2.3 2.3

Overall loading baσ 15 16 17 18 18 20 21Allowable carrying stress

Local loading2) baσ 1/ 2.8ba ck aA Aσ σ≤ ⋅ and ba ckσ σ≤

Note: 1) aτ is calculated by the effective height for a segment of d=20cm and tensional reinforcement ratio=1%, therefore, aτ must be adjusted as follows:

a) Adjustment for different effective height and tensional reinforcement ratio The adjusting method can be applied by multiplying a coefficient ofα , α is calculated by the

following formula: 43 20 /wp dα = , where wp is tensional reinforcement ratio(%);d is effective height(cm)

When 3.3%wp ≤ , d 20cm≥ , and d 20cm< , d is equal to 20cm. b) Addition of allowable shear stress

When the combined bending moment and axial compressive force apply on a segment, allowable shear stress can be calculated by multiplying a coefficient of nβ , nβ is calculated by the following formula:

01 / 2n dM Mβ = + ≤ ; where dM is design bending moment; and 0M is bending moment neglecting tensional concrete.

2) A is the influenced area of the applying load; and A0 is area of the applying load. 3) / 2.8 1ca ckσ σ= + (N/mm2) for bending moment; / 2.8ca ckσ σ= (N/mm2) for overall loading.[10]


172

(b) Cast-in-place reinforced concrete

Allowable stresses of cast-in-place reinforced concrete are shown in Table 7.9.

Table 7.9. Allowable stresses of cast-in-place reinforced concrete (N/mm2) [2]

Characteristic compressive strength ckσ 18 21 24 27 30

Allowable compressive stress for bending moment3) caσ 7 8 18 19 20 Allowable standard shear stress Shear stress by bending1) aτ 0.55 0.58 0.60 0.63 0.65

Allowable bond stress(deformed bar) 0τ 1.4 1.5 1.6 1.7 1.8

Overall loading baσ 6 7 8 9 10 Allowable carrying stress

Local loading2) baσ 1/ 3ba ck aA Aσ σ≤ ⋅ and ba ckσ σ≤

Note: 1) aτ is calculated by the effective height for a segment of d=20cm and tensional reinforcement ratio=1%, the adjustment of aτ is the same with the above segment concrete;

2) A and A0 are the same with the above segment concrete; 3) / 3 1ca ckσ σ= + (N/mm2) for bending moment; / 3ca ckσ σ= (N/mm2) for overall loading.[10]

(c) Cast-in-place plain concrete

Allowable stresses of cast-in-place plain concrete are shown in Table 7.10.

Table 7.10. Allowable stresses of cast-in-place plain concrete (N/mm2) [2]

Characteristic compressive strength ckσ 18 21 24 27 30

Allowable compressive stress for bending moment3) caσ 5.5 6.3 7.0 7.8 8.5

Allowable tensional stress3) taσ 0.72 0.80 0.87 0.95 1.00

Allowable bond stress(deformed bar) 0τ 1.4 1.5 1.6 1.7 1.8

Overall loading baσ 6 7 8 9 10 Allowable carrying stress

Local loading2) baσ 1/ 3ba ck aA Aσ σ≤ ⋅ and ba ckσ σ≤

Note: 2) A and A0 are the same with the above segment concrete; 3) / 4 1ca ckσ σ= + (N/mm2) for bending moment; / 3ca ckσ σ= (N/mm2) for overall loading;

( )2 / 30.42 / 4ta ckσ σ= (N/mm2) [10]

(d) Reinforcement

Allowable stresses of reinforcement are shown in Table 7.11.

Table 7.11. Allowable stresses of reinforcement (N/mm2) [2]

Category of reinforcement SR235 SR295 SR295 A,B SD345 SD390

Allowable stress 140 180 180 200 220

For increasing durability of tunnel, the value of allowable stress is lower that characteristic allowable stress.


173

(e) Steel material and welded part

Allowable stresses of steel and welds are shown in Table 7.12-7.14. Values of no

buckling and buckling are shown in Table 7.12 and Table 7.13, respectively. Allowable

stresses for local buckling of steel segment are shown in Table 7.14.

Table 7.12. Allowable stresses of steel material and welds (N/mm2) [2] Type of steel

Type of stress

SS400 SM400 SMA400STK400

SM 490 STK490

SM 490 Y

Axial stress Allowable tensional stress staσ Bending stress

Axial stress Allowable compressive stress Bending stress

160 215 240

Allowable shear stress 90 125 140

Steel material structure

Allowable carrying stress Between steel material 220 300 336 Allowable tensional stress Allowable compressive stress

160 215 340 Groove welding

Allowable shear stress 90 125 140 Allowable tensional stress, compressive stress in bead 160 215 240

Factory welding

Fillet welding Allowable tensional stress, compressive stress and shear stress about throat depth

90 125 140

Welded part

Site welding The design allowable stress is equal to 90% the above value.

Table 7.13. Allowable buckling stresses (N/mm2) [2]

SS400,SM400 SMA400, STK400

SM 490 STK490

SM 490 Y

Axial stress

0 9l r< ≤ : staσ9 130l r< ≤ :

0.91( 9)sca l rσ − −

0 8l r< ≤ : staσ8 115l r< ≤ :

1.42( 8)sca l rσ − −

0 8l r< ≤ : staσ 8 105l r< ≤ :

1.68( 8)sca l rσ − −

i)

Compressive Stress, overall area Bending

stress

(1) For bending around strong axis, equivalent slenderness ratio is calculated by the following formula: ( )el r F l b= ⋅ , where, 12 2 /F β α= + for I-shaped cross section, and for box-shaped cross section:

0β β< : 0F = ; 0 1β β≤ < : 0 01.05( ) /(1 ) 3 1F b lβ β β α= − − + ;

1 2β≤ < : 0.74 (3 )( 1)F b lα β β= + + ; 2β ≥ : 1.28 (3 )F b lα β= + ; 0 (14 12 ) /(5 21 )β α α= + +

For U-shaped cross section, 1.1 12 2 /F β α= + (2) For bending around weak axis: scaσ

ii)

Note: 1) in i) l is length of buckling member; r is radius of rotation of the whole cross sectional area 2) in ii) l is distance between points fixed on the flange; b is width of the flange for I-shaped cross section, and distance between web plates for box-shaped and U-shaped cross section; α is ratio of the thickness of a flange to the thickness of a web plate; and β is ratio of the height of a web plate to the thickness of a flange.


174

Table 7.14. Allowable stresses for local buckling of steel segment (N/mm2) [2]

Undisturbed from local buckling Under the influence local buckling Type of steel

Ratio of width and thickness

Allowable stress (N/mm2)



SS400, SM400

13.1r r

ht f K

≤⋅ ⋅

160 13.1 16r r

ht f K

≤ ≤⋅ ⋅

2

27200 r rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

SM490 A 11.2r r

ht f K

≤⋅ ⋅

215 11.2 16r r

ht f K

≤ ≤⋅ ⋅

2

27200 r rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

20.65 0.13 1.0f ϕ ϕ= + + ; 1 2

1

σ σϕσ−

= ( 1σ and 2σ are the minimum edge stress and the maximum edge

stress applied to the main girder ); 2

2.33 1.0( )r

r

Kl h

= +

Where, rl is length of the buckling of main girder(mm); h is height of main girder(mm); rt is thickness of

main girder(mm); and rK is buckling coefficient.

(f) Spheroidal graphite cast iron

Allowable stresses of spheroidal graphite cast iron are shown in Table 7.15-7.17.

Values of no buckling and buckling are shown in Table 7.15 and Table 7.16, respectively.

Allowable stresses for local buckling of ductile cast iron segment are shown in Table

7.17.

Table 7.15. Allowable stresses of spheroidal graphite cast iron (N/mm2) [2] Type of cast iron Type of stress FCD 450-10 FCD 500-7

Allowable tensional stress for bending moment 170 190

Allowable compressive stress for bending moment 200 220 Allowable shear stress 110 130

Table 7.16. Allowable stresses for buckling of spheroidal graphite cast iron (N/mm2) [2] Type of cast iron Type of stress FCD 450-10 FCD 500-7

Allowable axial stress 0 7l r< ≤ : 200 7 105l r< ≤ : 200 1.42( 7)l r− −

0 7l r< ≤ : 220 7 100l r< ≤ : 220 1.63( 7)l r− −

Note: l is length of buckling member; r is radius of rotation of the whole cross sectional area


175

Table 7.17. Allowable stresses for local buckling of ductile cast iron segment (N/mm2) [2]

Undisturbed from local buckling Under the influence local buckling

Type of steel Ratio of width and thickness




FCD 450-10 15.2r

ht f K

≤⋅ ⋅

200 15.2 21.7r

ht f K

≤ ≤⋅ ⋅

2

46500 rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

FCD 500-7 14.3r

ht f K

≤⋅ ⋅

220 14.3 20.5r

ht f K

≤ ≤⋅ ⋅

2

46500 rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

20.65 0.13 1.0f ϕ ϕ= + + ; 1 2

1

σ σϕσ−



2 2 4 2

4 40 15 203

K α να π π π

= + + − ( 2.26α ≤ ), min 2.37K = ( 2.26α = );

Where, rl is length of the buckling of main girder (mm); h is height of main girder (mm); rt is thickness of

main girder (mm); and K is buckling coefficient; rl hα = ; andν is Poisson’s ratio of cast iron.

(g) Steel casting for welded structure

Allowable stresses of steel casting for welded structure are shown in Table 7.18.

Table 7.18. Allowable stresses of steel casting for welded structure (N/mm2) [2] Type of steel Type of stress SCW 480

Allowable tensional stress for bending moment 180

Allowable compressive stress for bending moment 180

Allowable shear stress 100

(h) Bolt

Allowable stresses of bolt are shown in Table 7.19.

Table 7.19. Allowable stresses of bolt (N/mm2) [2] Type of bolt Type of stress 4·6 6·8 8·8 10·9

Allowable tensional stress 120 210 290 380

Allowable shear stress 90 150 200 270


176

7.4.2 Limit State Design Method

(1) Design Values of Material

Characteristic values of the strength in lining materials are described as the following

sections [2].

(a) Concrete for segment

Characteristic values of the strength of concrete for segment are shown in Table 7.20.

Table 7.20. Characteristic values of the strength of concrete for segment (N/mm2) [2]

Characteristic compressive strength ckf ′ 42 45 48 51 54 57 60

Tensional strength tkf 2.7 2.9 3.0 3.1 3.2 3.4 3.5

Bending crack strength1) bckf 2.7 2.8 2.9 3.0 3.1 3.2 3.3

Bond strength (deformed bar) bokf 3.3 3.5 3.6 3.8 4.0 4.1 4.2

Carrying strength (Overall loading) akf ′ 42 45 48 51 54 57 60

Carrying strength (Local loading2)) akf ′ ak ckf fη′ ′= , 2aA Aη = <

Note: 1) bokf is calculated by the following formula based on the assumptions that the thickness of segment is 250mm and nominal largest size of coarse aggregate is 20mm.

0 1bok b b tkf k k f= , 011

0.85 4.5( / )bch

kh l

= ++

, 10.55 0.4/1000bk

h= ≥ , 21000 /ch F c tkl G E f=

Where, 0bk is a factor of concrete softening; 1bk is a factor of crack strength decreasing from drying and heat of hydration etc. h is thickness of member(mm)>200; chl is characteristic length(mm);

FG is fracture energy(N/mm); cE is elastic modulus of concrete(kN/mm2); and tkf is characteristic tensional strength of concrete(kN/mm2).

FG can be calculated by the following formula for plain concrete 3 3max1/100F ckG d f ′=

Where, maxd is nominal largest size of coarse aggregate; and ckf ′ is characteristic compressive strength of concrete(kN/mm2).

2) A is the influenced area of the applying load; and A0 is area of the applying load shown in the following Figure.

3) 2 / 30.23( )tk ckf f ′= (N/mm2); and 2 / 30.28( ) 4.2bok ckf f ′= ≤ (N/mm2)[10].

2aA rπ=

21( )A r rπ= +

1 2aA b b= ⋅

1 1 2 2( 2 ) ( 2 )A b c b c= + ⋅ +

c2 c2b2

c1b1

c1

2rr1


177

(b) Cast-in-place reinforced concrete

Characteristic values of the strength of cast-in-place reinforced concrete are shown

in Table 7.21.

Table 7.21. Characteristic values of the strength of cast-in-place reinforced concrete (N/mm2) [2]

Characteristic compressive strength ckf ′ 18 21 24 27 30

Tensional strength tkf 1.5 1.7 1.9 2.0 2.2

Bending crack strength1) bckf 1.6 1.8 1.9 2.0 2.2

Bond strength (deformed bar) bokf 1.9 2.1 2.3 2.5 2.7

Carrying strength (Overall loading) akf ′ 18 21 24 27 30

Carrying strength (Local loading2)) akf ′ ak ckf fη′ ′= , 2aA Aη = <

Note: 1) bokf is calculated by the following formula based on the assumptions that the thickness of segment is

250mm and nominal largest size of coarse aggregate is 20mm. The calculation of bokf is the same

with the above segment concrete;

2) A and A0 are the same with the above segment concrete;

3) 2 / 30.23( )tk ckf f ′= (N/mm2); and 2 / 30.28( ) 4.2bok ckf f ′= ≤ (N/mm2)[10].

(c) Reinforcement

Characteristic values of the strength of reinforcement are shown in Table 7.22.

Table 7.22. Characteristic values of the strength of reinforcement (N/mm2) [2]

Category of reinforcement SR235 SR295 SR295 A,B SD345 SD390

Tensional yield strength ykf 235 295 295 345 390

Compressive yield strength ykf ′ 235 295 295 345 390

Shear yield strength vykf 1) 135 170 170 195 225

Note: 1) 3yk

vyk

ff =

(d) Steel material and welded part

Characteristic values of the strength of steel and welds are shown in Table 7.23-7.25.

Values of no buckling and buckling are shown in Table 7.23 and Table 7.24, respectively.

Characteristic values of the strength for local buckling of steel segment are shown in

Table 7.25.


178

Table 7.23. Characteristic values of the strength of steel material and welds (N/mm2) [2] Type of steel

Type of stress

SS400 SM400 SMA400STK400

SM 490

STK490

SM 490Y SMA490 SM 520

SM570SMA570

Tensional yield strength ykf 235 315 355 450

Compressive yield strength ykf ′ 235 315 355 450

Shear yield strength vykf

(overall area) 135 180 205 260 Steel material structure

Carrying strength akf ′ (between

steel material) 350 470 530 675

Tensional yield strength ykf

Compressive yield strength ykf ′235 315 355 450

Groove welding

Shear yield strength vykf

(overall area) 135 180 205 260

Tensional and compressive yield strength in bead 235 315 355 450

Factory welding

Fillet welding Tensional , compressive and

shear yield strength about throat depth

135 180 205 260

Welded part

Site welding The design allowable stress is equal to 90% the above value.

Table 7.24. Characteristic values of steel buckling (N/mm2) [2]

SS400,SM400 SMA400, STK400

SM 490 STK490

SM 490Y SMA490SM 520

SM570 SMA570

Axial strength

0 9l r< ≤ : ykf ′ 9 130l r< ≤ :

1.33( 9)ykf l r′ − −

0 8l r< ≤ : ykf ′ 8 115l r< ≤ :

2.06( 8)ykf l r′ − −

0 8l r< ≤ : ykf ′ 8 105l r< ≤ :

2.46( 8)ykf l r′ − −

0 7l r< ≤ : ykf ′ 7 95l r< ≤ :

3.51( 7)ykf l r′ − −

i)

Com

pres

sive

yie

ld S

treng

th O

vera

ll ar

ea

Bending strength

(1) For bending around strong axis, equivalent slenderness ratio is calculated by the following formula: ( )el r F l b= ⋅ , where, 12 2 /F β α= + for I-shaped cross section, and for box-shaped cross section:

0β β< : 0F = ; 0 1β β≤ < : 0 01.05( ) /(1 ) 3 1F b lβ β β α= − − + ;

1 2β≤ < : 0.74 (3 )( 1)F b lα β β= + + ; 2β ≥ : 1.28 (3 )F b lα β= + ; 0 (14 12 ) /(5 21 )β α α= + +

For U-shaped cross section, 1.1 12 2 /F β α= +

(2) For bending around weak axis: ykf ′

ii)

Note: 1) in i) l is length of buckling member; r is radius of rotation of the whole cross sectional area 2) in ii) l is distance between points fixed on the flange; b is width of the flange for I-shaped cross section, and distance between web plates for box-shaped and U-shaped cross section; α is ratio of the thickness of a flange to the thickness of a web plate; and β is ratio of the height of a web plate to the thickness of a flange.


179

Table 7.25. Characteristic values for local buckling of steel segment (N/mm2) [2]

Undisturbed from local buckling Under the influence local buckling Type of steel


Strength (N/mm2)


Strength (N/mm2)

SS400, SM400

13.1r r

ht f K

≤⋅ ⋅

235 13.1 16r r

ht f K

≤ ≤⋅ ⋅

2

40800 r rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

SM490 11.2r r

ht f K

≤⋅ ⋅

315 11.2 16r r

ht f K

≤ ≤⋅ ⋅

2

40800 r rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

20.65 0.13 1.0f ϕ ϕ= + + ; 1 2

1

σ σϕσ−



2.33 1.0( )r

r

Kl h

= +

Where, rl is length of the buckling of main girder(mm); h is height of main girder(mm); rt is thickness of main girder(mm); and rK is buckling coefficient.

(e) Spheroidal graphite cast iron

Characteristic values of the strength of spheroidal graphite cast iron are shown in

Table 7.26-7.28. Values of no buckling and buckling are shown in Table 7.26 and Table

7.27, respectively. Characteristic values of the strength for local buckling of ductile cast

iron segment are shown in Table 7.28.

Table 7.26. Characteristic values of the strength of spheroidal graphite cast iron (N/mm2) [2] Type of cast iron Type of stress FCD 450-10 FCD 500-7

Tensional yield strength ykf 280 320

Compressive yield strength ykf ′ 320 360

Shear yield strength vykf ( /(1 )ykf ν= + ) 220 250

Table 7.27. Characteristic values of the strength for buckling (N/mm2) [2] Type of cast iron Type of stress FCD 450-10 FCD 500-7

Axial compressive yield strength Overall area

0 7l r< ≤ : ykf ′

7 105l r< ≤ : 2.34( 7)ykf l r′ − −

0 7l r< ≤ : ykf ′

7 100l r< ≤ : 2.79( 7)ykf l r′ − −


180

Table 7.28. Characteristic values for local buckling of ductile cast iron segment (N/mm2) [2]

Undisturbed from local buckling Under the influence local buckling

Type of steel Ratio of width and thickness

Strength (N/mm2)


Strength (N/mm2)

FCD 450-10 15.2r

ht f K

≤⋅ ⋅

320 15.2 21.7r

ht f K

≤ ≤⋅ ⋅

2

75400 rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

FCD 500-7 14.3r

ht f K

≤⋅ ⋅

360 14.3 20.5r

ht f K

≤ ≤⋅ ⋅

2

75400 rt f Kh

⋅ ⋅⎛ ⎞⎜ ⎟⎝ ⎠

20.65 0.13 1.0f ϕ ϕ= + + ; 1 2

1

σ σϕσ−



2 2 4 2

4 40 15 203

K α να π π π

= + + − ( 2.26α ≤ ), min 2.37K = ( 2.26α = );

Where, rl is length of the buckling of main girder (mm); h is height of main girder (mm); rt is thickness of

main girder (mm); and K is buckling coefficient; rl hα = ; andν is Poisson’s ratio of cast iron (=0.27).

(f) Steel casting for welded structure

Characteristic values of the strength of steel casting for welded structure are shown

in Table 7.29.

Table 7.29. Characteristic values of steel casting for welded structure (N/mm2) [2] Type of steel Type of stress SCW 480

Tensional yield strength ykf 275

Compressive yield strength ykf ′ 275

Shear yield strength vykf 155

(g) Bolt

Characteristic values of the strength of bolt are shown in Table 7.30.

Table 7.30. Characteristic values of the strength of bolt (N/mm2) [2] Type of bolt Type of stress 4·6 6·8 8·8 10·9

Tensional yield strength ykf 240 480 660 940

Shear yield strength vykf 135 275 380 540


181

(2) Stress-Strain Curve of Material

Stress-strain curves of material must be determined in limit state. Fig.7.9 shows the

stress-strain curves of concrete, steel, reinforcement, spheroidal graphite cast iron, and

bolt, which are used in limit state for considering failure of the cross-section in

“Standard specifications for tunneling-2006, Shield tunnels” in Japan [2].

1 cdk f ′

0

: Compressive strength of concrete(N/mm2)

0.002 cuε ′

ckf ′

1 1 0.003 0.85ckk f ′= − ≤

0.0025 0.0035cuε′≤ ≤

15530000

ckcu

fε′−′ =cσ

1 20.002 0.002

c cc ckk f ε εσ ⎛ ⎞′= −⎜ ⎟

⎝ ⎠

0

ydfydf

s ydfσ =

s sEσ ε=sσ

: Tensional yield strength (N/mm2)

ε (a) Concrete (b) Steel, reinforcement, spheroidal graphite cast iron, bolt

Fig.7.9 Stress-strain curve

Linear stress-strain curves of material are used in the serviceability limit state. The

elastic moduli of materials are shown in Table 7.31 and Table 7.32.

Table 7.31. Elastic modulus of concrete (segment) [2]

Characteristic compressive strength ckf ′ (N/mm2) 42 45 48 51 54 57 60

Elastic modulus cE (kN/mm2) 31.4 32.0 32.6 33.2 33.8 34.4 25.0

Table 7.32. Elastic modulus of steel, reinforcement, and spheroidal graphite cast iron [2]

Type of material Elastic modulus(kN/mm2)

Steel and steel casting sE 210

Spheroidal graphite cast iron dE 170

Poisson’s ratio of concert, steel, steel casting, and spheroidal graphite cast iron is

shown in Table 7.33

Table 7.33. Poisson’s ratio [2] Material Poisson’s ratio

Elastic range 0.17 Concrete Cracks occurred 0

Steel and steel casting 0.30 Spheroidal graphite cast iron 0.27


182

(3) Safety Factor

It should address safety factors in the limit state design method. These are as

follows: 1) Material factor mγ

2) Member factor bγ

3) Load factor fγ

4) Structural analysis factor aγ

5) Structure factor iγ

(a) Material factor

Material factor is shown in Table 7.34.

Table 7.34. Nominal standard for material factor [2]

Material factor mγ

Concrete Steel material Limit state

Segment Cast-in place

Reinforcement bar Main girder

Longitudinal ribSkin plate

Spheroidal graphite cast iron

Bolt

Ultimate limit state 1.2 1.3 1.00 1.05 1.00 1.10 1.05Serviceability limit state 1.0 1.0 1.00 1.00 1.00 1.00 1.00

(b) Member factor

Member factor is shown in Table 7.35 and Table 7.36.

Table 7.35. Nominal standard for member factor (Concrete segment) [2]

Member factor bγ

Main part Limit state

Bending Compression Shear


Longitudinal joint

Metallic fitting for hanging

Ultimate limit state 1.10 1.30 1.301)

1.102) 1.10 1.15 1.30

Serviceability limit state 1.0 1.0 1.00 1.00 1.00 1.00

Note: 1) The calculation of shear capacity obtained by the strength of concrete.

2) The calculation of shear capacity obtained by the strength of steel.


183

Table 7.36. Nominal standard for member factor (Steel and cast iron segment) [2]

Member factor bγ

Main part Limit state

Bending Compression Shear


Longitudinal joint

Metallic fitting for hanging

Ultimate limit state 1.10 1.30 1.15 1.10 1.15 1.30

Serviceability limit state 1.0 1.0 1.00 1.00 1.00 1.00

(c) Load factor

Load factor is shown in Table 7.37.

Table 7.37. Nominal standard for load factor fγ [2]

Earth pressure

Limit state Loosening pressure

Full overburden

pressure

Coefficient of lateral

earth pressure

Water pressure

Coefficient of subgrade reaction

Dead weight Surcharge others

Ultimate limit state 1.1-1.31) 1.05 0.8-1.0 0.9-1.0 0.9-1.0 1.0-1.1 1.0-1.3 1.0-1.3

Serviceability limit state 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Note: 1) fγ can be assumed as 1.0 in the lower limit of the vertical earth pressure.

(d) Structural analysis factor

Structural analysis has an uncertainty in the analyzed results. Therefore, it is

necessary to determine structural analysis factor for designing the tunnel lining. Structural analysis factor aγ is shown in Table 7.38.

Table 7.38. Nominal standard for structural analysis factor [2]

structural analysis factor aγ

Ultimate limit state 1.0-1.1

Serviceability limit state 1.0


184

(e) Structure factor

It is normal design practice to ensure the safety of the structure, especially for important structures. Therefore, structure factor sγ shown in Table 7.39 is used in

designing the tunnel lining.

Table 7.39. Nominal standard of structure factor [2]

structural analysis factor sγ

Ultimate limit state 1.0-1.3 Serviceability limit state 1.0

7.5 DESIGN EXAMPLE

The design example is the fourth section of the Tokyo Metropolitan Area Outer

Underground Discharge Channel. The Closed-composite segment and RC segment will

be designed for evaluating the practicability in the following design examples.

7.5.1 Dimensions of Segment

Type of segment: Composite segment/RC segment, flat type

Diameter of segmental lining: D0=11800mm

Radius of centroid of segmental lining: Rc=5550 mm

Width of segment: B=2000mm

Thickness of segment: t=350mm (This value is assumed for calculating loads acting

on the segmental lining)

7.5.2 Ground Conditions

Overburden: H=51.35m

Groundwater table: G.L.-6.33m Hw=51.35-6.33=45.02 m Surcharge: P0=3.1×10×1.0=31kN/m2 (Height of flood=3.1m; fγ =1.0)

Soil condition: Alternation of soil strata (Clay, Sandy, and Gravel)

Unit weight of soil, submerged unit weight of soil, and angle of internal friction of


185

soil, and N value are shown in Fig.7.10.

N=6.0C=25 kN/m3

C=6.3 kN/m3

N=2.9

N=6.0

N=22.9C=120 kN/m3

C=0 kN/m3

N=39C=0 kN/m3

φ=0 degree

φ=40 degree

φ=0 degree

φ=0 degree

38.0kN/mγ ′ =

318.0kN/mγ =

39.6kN/mγ′=

319.6kN/mγ =

39.6kN/mγ′=

319.6kN/mγ =

35.3kN/mγ ′ =

315.3kN/mγ =

37.4kN/mγ′ =

317.4kN/mγ =

6.55

m25

.1m

9.1m

3.7m

11.8

m51

.35m

-44.75m

-6.33m Ground water Table

+6.60m

+9.70m Flood water Table

Clay in alluvial (Ac1)

Clay in alluvial (Ac2)

Sand in alluvial (Ns1)

Gravel in alluvial (Ng1)

Clay in diluvial (Dc2)

Clay in diluvial (Dc3)

Sand in diluvial (Ds3)

φ=0 degree

Fig.7.10 Ground condition

7.5.3 Load Conditions

The forces applying on tunnel lining are calculated based on combined load case

shown in Table 7.40.

Table 7.40. Combined load case Earth pressure Groundwater table Inner water pressure Load

case Tunnel

condition Large small High Low Normal Abnormal

Self weight

Subgrade reaction

1 No inner water pressure × × × ×

2 No inner water pressure × × × ×

3 Normal inner water pressure × × × × ×

4 Normal inner water pressure × × × × ×

5 Abnormal inner water pressure × × × × ×

6 Abnormal inner water pressure × × × × ×


186

The vertical earth pressure at the tunnel crown ( 1ep ) is calculated by Terzahgi’s

formula based on flow chart of calculation of the loads shown in Fig.7.2. The tunnel is

located in alternation of soil strata, and main stratum is hard clayey of soil stratum.

Therefore, earth pressure calculate is calculated using Soil-Water integrated method.

Vertical pressure at tunnel crown: Earth pressure: 2

1 468.2kN / me f vp γ σ= =

21 1 468.2kN / mv ep p= =

Lateral pressure at tunnel crown: Earth pressure: ( ) 2

1 0.5 211.9kN / me f c v iq tγ λ σ γ= + =

21 1 211.9kN / mh eq q= =

Lateral pressure at tunnel bottom:

Earth pressure: ( ){ } 22 0 0.5 265.4kN / me f c v iq D tγ λ σ γ= + − =

22 2 265.4kN / mh eq q= =

7.5.4 Calculation of Member Forces

The member forces are calculated by Shell-Spring model shown in Fig.7.11. The

segments are assembled in a staggered pattern.

Anchor joint (AS) joint and axial slide (Fig.7.12) were used as circumferential joint,

and longitudinal joint respectively, in Tokyo Metropolitan Area Outer Underground

Discharge Channel. These constants of springs were obtained by the experiments, and

described as follows:

Constants of spring of circumferential joint:

Axial spring: 65.0 10 kN / mc

ak = ×

Shear spring: 52.2 10 kN / mc

sk = ×

Rotational spring: 5

5

1.77 10 kNm/rad (Positive bending)1.67 10 kNm/rad (Negative bending)

ckθ⎧ ×

= ⎨×⎩

Constants of spring of longitudinal joint:

Axial spring: 64.8 10 kN / mL

ak = ×

Shear spring: 52.1 10 kN / mL

sk = × Rotational spring: 42.2 10 kN / mLkθ = ×


187

Constants of spring soil reaction are shown in Table 7.41.

Table 7.41. Spring constant of soil reaction

Modulus of deformation; Constants of Spring of soil reaction Remark

Modulus of deformation of soil gE (kN/m2) 48.0 10× 4g mE E=

Passive soil reaction spring rcK 42.0 10× 0

3(1 )(5 6 )

grc

g g

EK

R ν ν=

+ −

Active soil reaction spring rtK 41.0 10× / 2.0rt rcK K=

Constants of Spring of soil reaction

(kN/m2) Tangential soil reaction spring 36.7 10× / 3.0t rcK K=

4 22.0 10 kN/mmE = × ; 0 5.9mR = ; 0.45gν =

Table 7.42 shows the result of calculation of member forces of the segmental lining

(The segments are assembled in a staggered pattern). The maximum positive moment

occurs at the tunnel crown (Section B) and the maximum negative moment occurs at the

spring (Section B) which is located at 277.5 degrees from the tunnel crown (Clockwise).

Fig.7.11 Structural model

(a) Anchor joint

(b) Axial slide (AS) joint Fig.7.12 Schematic of joint [11]


188

Table 7.42. Member forces of segmental lining (A-Ring)

θ (deg)

Bending moment M

(kN m/Ring)⋅

Axial force N

(kN/Ring)

Shear forceQ

(kN/Ring)θ

(deg) Bending moment

M (kN m/Ring)⋅

Axial force N

(kN/Ring)

Shear forceQ

(kN/Ring)

0.0 606.1 3485.9 -90.0 101.3 -450.7 4714.8 -24.1

3.8 635.5 3487.9 -28.9 105.0 -439.8 4677.3 -70.6

7.5 628.5 3487.7 32.7 108.8 -396.2 4639.1 -116.5

11.3 610.4 3512.9 62.5 112.5 -349.9 4590.3 -122.8

15.0 580.2 3560.0 139.7 116.3 -301.5 4527.5 -135.2

18.8 502.6 3608.1 212.4 120.0 -245.6 4466.5 -152.3

22.5 416.3 3671.7 232.3 123.8 -184.0 4397.4 -159.6

26.3 323.4 3751.8 252.7 127.5 -130.1 4334.3 -141.4

30.0 221.3 3832.9 269.5 131.3 -75.0 4267.2 -147.5

33.8 115.5 3918.9 274.3 135.0 -16.3 4200.8 -157.5

37.5 16.7 4004.0 244.0 138.8 46.5 4134.1 -157.2

41.3 -72.7 4095.1 223.4 142.5 104.9 4069.6 -150.9

45.0 -155.6 4179.5 204.6 146.3 162.8 4008.7 -149.4

48.8 -230.5 4259.8 164.4 150.0 220.1 3954.6 -132.1

52.5 -282.4 4342.3 124.3 153.8 264.7 3903.7 -110.5

56.3 -326.4 4413.8 106.3 157.5 305.3 3857.7 -102.2

60.0 -364.3 4472.8 65.8 161.3 343.4 3820.0 -100.1

63.8 -377.0 4532.8 26.8 165.0 382.5 3784.3 -83.3

67.5 -384.9 4582.5 18.0 168.8 407.6 3753.3 -65.2

71.3 -390.8 4618.9 15.0 172.5 450.9 3729.1 -108.2

75.0 -396.4 4657.3 2.4 176.3 491.2 3703.2 -96.0

78.8 -392.6 4689.9 -9.8 180.0 525.0 3691.8 -77.3

82.5 -410.3 4708.2 47.1 183.8 550.9 3690.4 -27.7

86.3 -429.0 4732.0 47.8 187.5 546.4 3688.8 22.5

90.0 -447.2 4740.4 45.0 191.3 533.6 3707.4 45.5

93.8 -463.7 4738.1 14.3 195.0 511.4 3744.0 110.6

97.5 -458.3 4735.5 -17.0 198.8 448.2 3781.0 172.8


189

Table 7.42. (Continue)

θ (deg)

Bending moment

M (kN m/Ring)⋅

Axial force N

(kN/Ring)

Shear forceQ

(kN/Ring)θ

(deg)

Bending moment

M (kN m/Ring)⋅

Axial force N

(kN/Ring)

Shear forceQ

(kN/Ring)

202.5 378.0 3830.3 189.0 303.8 -311.0 4451.5 -156.9

206.3 302.4 3893.9 206.0 307.5 -246.1 4372.9 -168.3

210.0 219.1 3957.8 221.6 311.3 -190.5 4298.9 -147.7

213.8 131.4 4027.1 227.3 315.0 -132.2 4220.7 -157.7

217.5 50.9 4093.5 199.3 318.8 -68.9 4141.1 -172.9

221.3 -22.3 4166.3 184.2 322.5 1.1 4060.1 -179.8

225.0 -91.2 4234.6 171.7 326.3 69.8 3981.2 -179.3

228.8 -154.7 4300.6 142.0 330.0 139.3 3904.4 -181.1

232.5 -200.7 4368.8 112.0 333.8 209.4 3834.3 -167.2

236.3 -241.0 4429.4 100.1 337.5 268.3 3768.4 -147.3

240.0 -277.8 4481.2 72.3 341.3 322.9 3707.5 -138.0

243.8 -296.7 4534.4 45.7 345.0 374.7 3656.5 -134.8

247.5 -313.0 4580.1 42.4 348.8 426.9 3608.6 -113.6

251.3 -329.3 4615.7 45.3 352.5 231.1 3568.1 -45.9

255.0 -347.8 4653.6 38.5 356.3 516.9 7071.4 -134.4

258.8 -359.0 4685.9 28.8 360.0 606.1 3485.9 -90.0

262.5 -390.7 4707.3 83.7

266.3 -423.5 4733.4 84.6

270.0 -456.0 4744.2 81.8

273.8 -486.7 4743.6 48.0

277.5 -493.1 4742.9 12.5

281.3 -496.3 4722.7 2.4

285.0 -495.0 4683.2 -51.4

288.8 -456.7 4642.8 -105.3

292.5 -413.8 4589.9 -116.2

296.3 -367.1 4519.9 -133.2

300.0 378.0 3830.3 189.0


190

Figs 7.13 and 7.14 show the distributions of bending moment, axial force, and shear

force in A-Ring and B-Ring, respectively.

639.9-472.3

3698.34968.9

Note: θ is angle from crown(Clockwise)

Axial force distribution of A Ring

ABA

Minimum (θ=0)

Bending moment(kNm/Ring)Axial force(kN)Maximum (θ=90)

Note: θ is angle from crown(Clockwise)Shear force distribution of A Ring

ABA

-251.7205.0

Minimum (θ=322.5)

Shear force(kN)Maximum (θ=37.5)

(a) Axial force distribution of A-Ring (b) Shear force distribution of A-Ring


4958.53701.0

Axial force(kN)

Bending moment distribution of A Ring

ABA

-508.3654.4

Minimum (θ=82.5)

Bending moment(kNm/Ring)

Maximum (θ=356.3)

-1000 -500 0 500 1000

320

325

330

335

340

Staggered pattern

Bending moment distrubution Average value

B AA

Segment width(mm)

Bend

ing

mom

ent (

kNm

)

(c) Moment distribution of A-Ring (d) Moment distribution along width of A-Ring

Fig.7.13 Distributions of member forces of A-Ring in circumferential direction


191

3698.34968.8


Axial force distribution of B Ring

ABA

639.9-472.3

Minimum (θ=0)

Bending moment(kNm/Ring)Axial force(kN)Maximum (θ=270)

-181.1274.3

A

Note: θ is angle from crown(Clockwise)Shear force distribution of B Ring

BA

Minimum (θ=330)

Shear force(kN)Maximum (θ=33.8)

(a) Axial force distribution of B-Ring (b) Shear force distribution of B-Ring


4963.73684.4

Axial force(kN)

Bending moment distribution of B Ring

ABA

-524.0666.8

Minimum (θ=277.5)

Bending moment(kNm/Ring)

Maximum (θ=3.7)

-1000 -500 0 500 1000

325

330

335

340

345

Staggered pattern

Bending moment distrubution Average value

B AA

Segment width(mm)

Ben

ding

mom

ent (

kNm

)

(c) Moment distribution of B-Ring (d) Moment distribution along width of B-Ring

Fig.7.14 Distributions of member forces of B-Ring in circumferential direction


192

(c) Contour of bending moment in longitudinal direction

(a) Contour of shear force

Figs 7.15 shows Contour of member forces of segmental lining assembled in a

staggered pattern. It can be seen that splice effect of longitudinal joints occurs.

7.5.5 Designing Segmental Lining and Checking Safety

According to the above calculation result of member forces, the safety of the most critical sections is checked using the limit state design method. The structure factor sγ

(b) Contour of bending moment in circumferential direction

(d) Contour of axial force in circumferential direction

Fig.7.15 Contour of member forces of segmental lining assembled in a staggered pattern


193

of 1.3 is used in designing segmental lining, because the tunnel is a permanent structure.

In the design of RC segment, the design axial capacity and the design bending

capacity of the member cross-sections subjected to axial load and bending moment can

be calculated using Eqs.(7.14) and (7.15). Bending moment-axial force curve of

cross-section can be determined based ultimate limit state shown in Fig. 7.16.

Stress-strain relationship of concrete and steel shown in Fig.7.9 is adopted.

( )h/2

ud s-h/2N ( )bd / + / c m s my y T Tσ γ γ′ ′= +∫ (7.14)

( ) ( ){ }h/2

ud s-h/2 M ( )b d / + h / 2 t h / 2 t / c m s my y y T Tσ γ γ′ ′= − − −∫ (7.15)

where s s sT Aσ ′= ; s s sT Aσ′ ′ ′= ; b is width of RC segment; h is height of RC segment;

mγ is material factor; t′ and t are outer and inner concrete cover thickness, respectively.

0lε′<

0lε′ =u cuε ε′ ′=(c)Ultimate limit state(3)

(b)Ultimate limit state(2)

u cuε ε′ ′=

u l cuε ε ε′ ′ ′= =

cuε′

cuε′

lε′

uε′

M

N

N

1 cdk f ′

1 cdk f ′

A's

As

As

A's

h/2 x

h/2 x

h/2

x

(a) Strain (b) Stress (c) Force



Section

Section

Section

(a)Ultimate limit state(1)

MN

1 cdk f ′

A's

As

Fig.7.16 Transition of ultimate limit states for RC segment


194

2sA 6193.6 mm=

2sA 6193.6 mm′ =

Strength of concrete: 48N/mm2

350

2000

7565

D22 16@120

Unit: mm

D22 (SD345)

D22 (SD345)

D22 16@120

Fig.7.17 Section of RC segment and arrangement of main reinforcements

0 200 400 600 800 1000 12000

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000 Maximum positive moment Maximum negative moment

Axi

al fo

rce

(kN

)

Bending moment (kNm)

Fig.7.18 Axial force-moment interaction diagram of RC segment

Fig.7.17 shows the design cross-section of RC segment. Axial force-moment

interaction diagram of cross-section is described by Fig.7.18. Therefore, as a rule, the

safety for combined axial load and bending moment is examined by confirming that the

point (Md, Nd) is located inside of the (Mud, N'ud) curve. It can be seen that both Section

A and Section B are safe.


195

In the design of composite segment, the following design rules must be ensured.

(a) Width to Thickness Ratio

Elastic buckling stress for a rectangular plate subjected to an axial compression can

be calculated using Eq. (7.16), where ν is a Poisson’s ratio (0.3 for steel), and k is a

bucking coefficient that is determined from boundary conditions and aspect ratios of the

plate. When the plate is sufficiently long, the coefficient k depends only on the boundary

conditions. The values of k are given as 4.0, 5.42, and 6.97 for both edges simply

supported, one edge simply supported with the other fixed, and both edges fixed, respectively [12]. In order to prevent elastic buckling, the buckling stress, crF , should be

greater than or equal to the yield stress, syf , which results in the limit of width to

thickness ratio for simple support conditions and for fixed support condition:

( ) ( )

2

22

12 1s

crEF k

s tπν

=−

(7.16)

s is spacing of shear studs; Es is the elastic modulus of steel plate; and t is taken as

the minimum thickness of skin plates.

It can be seen from Fig.4.17 of FEM analysis in Chapter 4 that stress of the buckled

top skin plate still increases as the applied load increases, and is greater than the

post-buckling stress calculated by Eq. (7.17). Considering the design tendency to safety

side, according to Guideline of buckling design of JSCE code, the post-buckling stress udf ′ is calculated using Eq. (7.17):

( )/ud s syf t s E f′ = (7.17)

(b) Confined Concrete

Montoya et al.[13] proposed a concrete confinement model for steel tube confined concrete is adopted in limit state design. The stress cσ is related to the strain cε using

the following formula:

2

1.0

ccc

c c

cc cc

f

A B Cf f

σε ε

=⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(7.18)


196

(c) Shear Stud and Steel Plate

Shear stud can be bonded to the steel element using an automatic welding procedure. The most important dimension is the diameter of the shank shd which varies from

about 13 mm to about 22 mm. The diameter of the head is usually about 1.5d shd , and

the height of the connector is usually greater than 4 shd , in order to ensure that the

connector does not pull out of the concrete element. The weld collar has a diameter of about 1.3 shd , and a height of about 0.3 shd . The minimum spacing of connectors in a

composite segment is controlled by this concrete bearing zone because it is necessary to

ensure that these zones do not overlap, otherwise the dowel strengths of the connectors

will be reduced. This is achieved in design through detailing rules which require that the longitudinal spacing of stud shear connectors is greater than about 5 shd , and that the

lateral spacing is greater than about 4 shd . The distance from the edge of the stud to the

edge of the steel plate is greater than about shd [14].

The design axial capacity and the design bending capacity of the member

cross-sections subjected to axial load and bending moment can be calculated using the

state of stress and strain distribution shown in Figs.7.19. Mud, N'ud can be calculated by

the proposed method described in Chapter 5.

The number of shear studs can be calculated using Eqs.(7.19) and (7.20). Axial force

and shear force of members can be determined based ultimate limit state shown in Figs.

7.20, and 7.21.

tttb

tc

tm

,t1sf

,b1sfAs

c

Main reinforcement

s,mε

b

φ

cc st( )ε ε+

Equivalent Stressdistribution

Concrete

Mai

n gi

rder

Mai

n gi

rder 3d

tf ′tf ′

1 cfα ′

1 1dβcf ′

ccf2d

1d

dccf

cf ′

N.A

ctε

ccε

cε ′

Actual Stress distribution

Steel tubeConcrete

Strain distribution

Skin plate

Concrete infill

Skin plate

Composite section

Fig.7.19 Transition of ultimate limit states for Closed-composite segment


197

Section Force

Tsu

Tc

Tsl

Fig.7.20 Resultant axial force in each element

B8 A1 A2 A3 A4 A5 B6 K7K7

(a) Segmental ring arrangement

0 5 10 15 20 25 30 35 40

-200

-150

-100

-50

0

50

100

150

200

250

300

A4=280.1kNA3=670.2kNA2=259.7kNShear force of one segmentA1=920.9kN

A4

A3

A2

A1

Shea

r for

ce (k

N/m

)

Arclength from tunnel crown (m)

(b) Shear force distribution of B-Ring

Fig.7.21 Distribution of connectors in segment with distributed loads

The number of longitudinal shear studs in a segment:

( ){ }su cmin , /SL u mN T T Q γ= (7.19)

where uQ is shear capacity of a shear stud accounting for the effects of shear stud

spacing and plate thickness; mγ is safety factor of shear stud material.


198

The shear studs are once again distributed according to the areas of the shear force

shown in Fig.7.21. It is also standard convention to distribute the shear studs uniformly

within each of the zones A1-K7.

The number of circumferential shear studs in a segment:

( )maxwhen ii SL u

SCL i m L

AL N QNS L Sγ

⎧ ⎫ ⎛ ⎞= =⎨ ⎬ ⎜ ⎟⎝ ⎠⎩ ⎭

(7.20)

where iA is area of shear force; iL is length of area iA ; LS is spacing of longitudinal shear

studs.

Fig.7.22 shows the design cross-section of Closed-composite segment based on the

previous rules.

230

2000

Unit: mm

Thickness(mm)

Skin plate Main girder Joint plate

6.0 12.0 12.0

Details of shear stud

Characteristic strength of concrete: 48N/mm2Yield strength of steel: 235N/mm2 (SM400)Tensional strength of shear stud: 530N/mm2

10.5

33

100

22

120 Number of shear studs: 17@120mm

Number of top shear studs is equal to number of top shear studs in the design of composite segment for taking in accout the tunnel lining resisting positive and negative moment

Note:

(a) Longitudinal direction


199

230

4550

Unit: mm87.5 Number of lateral shear studs: 36@125mm125

(b) Circumferential direction

Fig.7.22 Section of Closed-composite segment and arrangement of shear studs

Axial force-moment interaction diagram of cross-section is described by Fig.7.23.

Therefore, as a rule, the safety for combined axial load and bending moment is examined

by confirming that the point (Md, Nd) is located inside of the (Mud, N'ud) curve. It can be

seen that both Section A and Section B are safe.

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000 Maximum positive moment Maximum negative moment

Axi

al fo

rce

(kN

)

Bending moment (kNm) Fig.7.23 Axial force-moment interaction diagram of Closed-composite segment

7.6. SUMMARY

Based on the proposed mechanical model in Chapter 5, the tunnel lining of the fourth

section of the Tokyo Metropolitan Area Outer Underground Discharge Channel are

designed as RC segmental lining and Closed-composite segmental lining using limit

state design method, respectively.

Design of C

omposite Segm

ent

200

Closed-com

posite segment

RC

segment

Cutoff

performance of

circumferential

joint

Excellent cutoff performance of circum

ferential joint because

of high

manufacturing

accuracy (e.g.

Tolerance is equal to 1.2mm

in segment dim

ension, and 10m

m in assem

bly dimension )

Cutoff perform

ance of circumferential joint is less than C

C

segment, because m

anufacturing accuracy (Tolerance) of R

C segm

ent is larger. (e.g. Tolerance is equal to 1.0mm

in segm

ent dimension, and 20m

m in assem

bly dimension )

FunctionalityD

urability (Protection m

ethod for corrosion)

Exposed steel material of inside of segm

ent is covered by anticorrosion m

aterial, durability can be obtained.

Protection method for corrosion can be neglected, if the

thickness

of concrete

cover is

ensured.( H

owever,

maintenance

and repair

are necessary,

because cracks

occurred in main body of segm

ent )

Assem

bling of segm

ents

Assem

bling of

segments

and fastening

are sim

ultaneously com

pleted, m

anual handling

is not

necessary. Therefore,

excellent construction

and reduced construction period can be obtained.

It is necessary to fasten circumferential joints using m

anual handling or partially m

echanical machine. O

ne pass type is w

idely used in longitudinal joints.

Construction

Safety of dam

age in assem

bling

Not be dam

aged easily for handling and jack thrust in assem

bling stage because of concrete infill covered by the steel tube.

The corner edges of RC

segment are easily dam

aged because of the segm

ent weight and w

eak of tensile strength. C

areful attention must be paid w

hen the concrete form is

removed and R

C segm

ents are transported and assembled.

Effect on the environm

ent M

ucking

Com

pared to RC

segment, the low

er thickness of segm

ent can be obtained. Therefore, the reduced muck

is advantage

in the

reduction of

the effect

on environm

ent.

Com

pared to composite segm

ent, the lower thickness of

segment can be not obtained. Therefore, a lot of m

uck is disadvantage in the reduction of the effect on environm

ent. The tunnel diam

eter more increases, and the effect on

environment is large, because the over excavation of from

200m

m to 300m

m is necessary.

Maintenance

The completion m

aintenance is slightness inspection and repair (e.g. Steel tube painting) because of few

dam

age of segment in construction and com

pletion periods. The deterioration of facilities in tunnel is sm

aller for

leakage, because

of excellent

cutoff perform

ance. It is advantage in maintenance.

Com

pared to composite segm

ent, protection method for

damage is difficult in construction and com

pletion periods. The dam

age occurred in use period, and large scale repair and

reinforced construction

are necessary.

The cost

of m

aintenance increases, because the amount of leakage is

large.

Table 7.43. Com

parison of structural type

200


The M

echanical Behavior and D

esign of Com

posite Segment for Shield Tunnel

201

Economy

Num

ber of segment ring( Length of the tunnel: 6.3 km

): 6300m

2.0m=3150R

ing÷

C

ost of segment:

1210003150=381150000 $

×

Segment

117250 $ Seal m

aterial2500 $

Caulking

1250 $Total

121000 $/Ring

⎧⎫

⎪⎪

⎪⎪

⎨⎬

⎪⎪

⎪⎪

⎩⎭

The increased cost of mucking:

0 $ The

reduced construction

cost for

the reduction

of construction period: 3 m

onths 300000 $ =900000 $

×

Investment effects for the reduction of construction period:

3 months

23000000 $ =69000000 $×

Total cost: 3861150000-900000-69000000 = 311250000 $

Num

ber of segment ring( Length of the tunnel: 6.3 km

): 6300m

2.0m=3150R

ing÷

C

ost of segment:

1027503150=323662500 $

×

Segment

99000 $Seal m

aterial2500 $

Caulking

1250 $W

aterproofing6700 $

Total102750 $/R

ing

⎧⎫

⎪⎪

⎪⎪

⎪⎪

⎨⎬

⎪⎪

⎪⎪

⎪⎪

⎩⎭

The increased cost of mucking:

6615000 $ The

reduced construction

cost for

the reduction

of construction period: 0 $ Investm

ent effects for the reduction of construction period: 0 $

Total cost: 323662500+6615000 = 330277500 $

Evaluation

The C

losed-composite

segments

have advantages

in cutoff perform

ance, construction, structural safety, and the effects on environm

ent. The

completion

maintenance

is excellent

(slightness inspection and repair), because of few

damage of segm

ent in construction and com

pletion periods.

RC

segments have disadvantages in structural safety and

maintenance, because dam

age of segment w

ill occur in construction and com

pletion periods.


201


202

Comparisons have been made between the cross-section of RC segment and the

cross-section of Closed-composite segment shown in Table 7.43. The reduction of

segment thickness is obtained using Closed-composite segment and this indicates that

Closed-composite segment is suitable for decreasing muck, construction period and the

outside diameter of the shield machine. In general, the costs of construction, risk, and

maintenance will be decreased

7.7. REFERENCES

1) International tunnelling association, 2000. Guidelines for the design of shield tunnel

lining. Tunnelling and Underground Space Technology, Vol. 15, No. 3, pp. 303-331.

2) Japanese Society of Civil Engineers (JSCE), 2007. Standard specifications for

tunneling-2006, Shield tunnels.

3) Japanese Society of Civil Engineers (JSCE), 1994. Standard specifications for design

segment (in Japanese).

4) Terzaghi, K., 1946. Rock Defects and Load on Tunnel Supports. In: Proctor, R.V.,

White, T.C. (Eds.), Introduction to Rock Tunnelling with Steel Support. Commercial

Shearing and Stamping Co., Youngstava, OH, USA.

5) Railway Technical Research Institute (RTRI), 1997. Design Standard for Railway

Structures (Shield-Driven Tunnel), Maruzen, pp. 47-61. (In Japanese)

6) Murakami, Koizumi, et al, 1978. Study on rational design method of the

circumferential joint of a segment. Waseda Univ. report, No.82. (In Japanese)

7) Koizumi, 1979. Study on segment design. Ph.D. thesis, Waseda Univ., Tokyo. (In

Japanese)

8) Koizumi, Murakami, Nishino,1988. Study on the analytical model for shield tunnel in

longitudinal direction. Journal of JSCE, No.394, 1-10. (In Japanese)

9) Kimura, Koizumi, 1999. A design method of shield tunnel lining taking in account of

the interaction between the lining and the ground. Journal of JSCE, No.624, 123-134.

(In Japanese)

10) Japanese Society of Civil Engineers (JSCE), 2003. Standard specifications for

concrete structures-2002, Structural performance verification.


203

11) Masami, Shirato, et al.2003. Development of new composite segment and

application to the tunneling project. Journal of JSCE, No.728, 157-174.(In Japanese)

12) Galambos TV, 1998. Guide to stability design criteria for metal structures.5th ed.

New York: Wiley.

13) Montoya, E.,2003. Behavior and analysis of confined concrete. Ph.D. thesis, Univ. of

Toronto, Toronto.

14) Deric J Oehlers, Mark A Bradford, 1995. Composite steel and concrete structural

members. UK: Pergamon, Elsevier Science.


205

Chapter 8. Conclusions

The mechanical behavior of composite segment with different composited designs is

investigated in this paper. The local buckling, slip and effective strength formulas are

incorporated into the nonlinear analysis procedures to account for local buckling and slip

effects on the strength and ductility performance of composite segments under combined

loads. From the results described in previous chapters, several conclusions may be

summarized as follows.

(a) The accuracy of proposed FEM models

A comparison between the analyzed and experimental results indicates that the

proposed finite element model using MSC.Marc can simulate the mechanical behavior

of composite segments (including Closed-composite segment, SSPC segment, and DRC

segment).

Contact analysis was found to be able to simulate the slip at the interface between

steel tube and concrete infill. For the composite segments with no axial load, frictionless

condition could simulate the load-deflection curves from the experimental results. For

axially loaded the composite segments, a friction coefficient of 0.1 produced convincing

results.

Discrete shear studs modeled using 3D beam elements is able to simulate the

mechanical behavior of shear connectors.

The effective width of the skin plate in compression should be adopted in analysis

and design of composite segment, according to different degree of shear connection. The

skin plate in tension can be assumed as the member of full effective cross-section.

(b) The accuracy of proposed mechanical model

A nonlinear fiber element analysis method is proposed for the inelastic analysis and

design of concrete infill steel tubular composite segments with local buckling and slips.

Sectional geometry, residual stresses and strain hardening of steel tubes and confined

concrete models were considered in the proposed mechanical model. The local buckling,

slip and effective strength formulas were incorporated into the nonlinear analysis

Conclusion

206

procedures to account for local buckling and slip effects on the strength and ductility

performance of composite segments under combined loads.

The formulation of differential equations is suitable for the solution of partially

interactive steel tube composite segment elements. Closed form solutions for single,

double and uniform load arrangements on simple span segments have been presented. A

step-wise linearization method has been described which allows the non-linear affects of

concrete cracking and shear stud stiffness to be incorporated.

The analysis involves non-dimensional constants which affect the steel plate

interaction. The effects are either direct when the constant relates to the plate for which

it is formulated or crossed when it affects the opposite plate for which it is formulated.

More attention should be paid on the shear capacity of shear studs in composite

segments. In current codes, only some coefficients are suggested for considering the

influences of the concrete strength, steel strength, and shear stud dimension. It was

found that the shear stud spacing and the thickness of the welded steel plate also have the effects on the shear capacity, so the coefficients η and Vβ are recommended.

Comparisons are made between experimental results and the mechanical predictions

of behavior using the proposed method. Good agreement is found and this indicates that

the proposed method is suitable for general analysis of tunnel lining of composite

segment.

(c) Design tunnel lining of composite segment

A comparison is made between the cross-section of RC segment and the cross-section

of Closed-composite segment. The reduction of segment thickness is obtained using

Closed-composite segment and this indicates that Closed-composite segment is suitable

for decreasing muck, construction period and the outside diameter of the shield machine.

In general, the costs of construction, risk, and maintenance will be decreased

The Mechanical Behavior and Design of Composite

Segment for Shield Tunnel

Abstract in Japanese

合成セグメントの力学的挙動および設計法

に関する研究

（概要）張穏軍

大都市部の地下には，地下鉄をはじめとして，上下水道，電気用とう道，通信用とう

道，ガス管路などのインフラ構造物が数多く構築されている．これらの施設は主な公共

用地である道路下に構築されることが多い．東京都内の国道下だけに限っても，道路1km

あたり約33kmの管路が埋設されている．このほかに都道や区道の下にもトンネルや管路

類が輻輳し，現在では都市部の地下は非常に混雑した状態となっている．

これまでの都市部の地下は，建設が容易な浅いところから順に利用されてきており，

新たに建設される施設は既存の施設より深い道路下に設置せざるを得ないため，その構

築深度は年々深くなりつつある．また，最近では，深い地下を利用した地下河川や地下

貯留管などの整備も数多く行われ，さらにこれに加えて，道路用の大断面のトンネルも

造られるようになってきている．

地下利用においては，大深度化だけではなく，トンネルの大断面化，従来の円形断面

に代わって矩形断面や楕円形断面，多心円形断面などの異形断面化が進んできている．

このため，トンネルには大きな土水圧が作用したり，大きな断面力が発生するなど，シ

ールドトンネルの覆工に用いられるセグメントに付与すべき性能も多様化してきてい

る．

円形断面をもつ大断面のトンネルでは，掘削土量は掘削外径の二乗に比例して増える

ことから，建設発生土の処理コストが増加する．また，シールド機の仮組立て，分割，

運搬，現場での再組立てなどのシールド機の製作コストも増加する．一方で，地下河川

や地下貯留管などを除けば，一般にトンネル断面の中央部のみを有効に利用することが

多く，その上部と下部は余剰な断面となることが多い．トンネルが大断面化するほどこ

れらの影響が顕著になってくる．このため，円形以外の断面を採用して，掘削土量を減

少させる一方で，断面を有効に使用しようとするケースも増えてきている．円形以外の

断面のトンネルを構築する場合には，覆工に発生する断面力や変形が大きくなることが

多く，鉄筋コンクリート製のセグメント（以下，RC セグメントと呼ぶ）を採用した場

合にはセグメントの厚さが厚くならざるを得ない．セグメントが厚くなると，その分重

量が重くなり，セグメント工場における製作性やハンドリング，工場から現場までの運

搬，工事ヤードから坑内への搬入，切羽での組立てなどに大きな労力と神経を使うこと

になり，また，セグメントの損傷なども起きやすくなる．

一方，一次覆工に鋼製セグメントを採用した場合には，トンネルの掘削深度が大きく

なれば大きな土水圧やジャッキ推力，裏込め注入圧などの荷重が作用することになる．

これに必要な耐荷力や剛性を確保するためには相当に厚い鋼板を使わざるをえず，溶接

などの製作上の問題を含めて経済性に疑問が生じる．ダクタイルセグメントを採用した

場合には溶接の問題はないが，鋼製セグメントと同様に製作コストに問題が生じる．こ

れらのことから，鋼のもつ力学的な優位性やコンクリートのもつ経済性に加えて，鋼殻

によるコンクリートの変形拘束効果を期待した合成セグメントが開発されている．合成

セグメントには，RC セグメントの鉄筋を鋼板やラチスガーダー，形鋼などに置き換え

てその外側をコンクリートで覆うもの，鋼製セグメントの内径側に鉄筋を配置してその

内部にコンクリートを中詰めしたもの，4 面または 6 面を鋼殻の内部にコンクリートを

中詰めしたものなど，いくつかのタイプがある．最近では鋼製セグメントや鋼殻の内部

にコンクリートを中詰めしたものが多く使われている．しかし，これらの合成セグメン

トの現行の設計では，主げたを無視して外側および内側のスキンプレートを鉄筋として

評価したり，主げたは考慮してもスキンプレートには有効幅を考えたりすることが多い．

また，その内部に打設されたコンクリートは考慮されずに，二次覆工の代わり程度の役

割として扱われる場合がほとんどである．すなわち，設計上の取り扱いは合成セグメン

トごとに異なるのが実状である．これらの合成セグメントのうちで，スタッドを溶植し

たスキンプレート，主げたプレート，継手プレートからなる六面体の鋼殻の内部に高流

動性のコンクリートを中詰めした合成セグメント（以下，密閉式合成セグメントと呼ぶ）

は，その外側の 6 面が鋼板で囲われているため，内部のコンクリートの挙動が直接観測

できないこと，内部のコンクリートは鋼板やその内側に溶植されたスタッドジベルによ

る変形の拘束効果を受けると想像されることなどから，その耐荷機構は力学的にもっと

も複雑なものであると考えられる．

本研究は，密閉式合成セグメントの耐荷機構を明らかにして，その合理的な設計手法

を提案するとともに，それが類似の合成セグメントにも適用できることを示したもので

ある．

本研究では，まず，密閉式合成セグメントの模型供試体を作製して載荷試験を行い，

その結果に考察を加え，有限要素法による非線形解析や鉄筋コンクリート理論（以下，

RC 理論と略称する）に準拠した解析を行った．それらの結果から，密閉式合成セグメ

ントの耐荷機構を明らかにし，その降伏曲げ耐力，終局耐力，変形の具体的な算定手法

を提案した．つぎに，鋼製セグメントにコンクリートを中詰めしただけのセグメントと

ダクタイルセグメントの内径側に鉄筋を配置してコンクリートを中詰めしたセグメン

トの実験結果を用いて，提案した算定手法がそのほかの類似の合成セグメントにも適用

可能かどうかを検証した．最後に，密閉式合成セグメントの優位性を検討する目的で，

提案した算定手法により，首都圏外郭放水路第４工区のセグメントを対象として，限界

状態設計法にもとづいた試設計を行った．結果として，密閉式合成セグメントは，RC

セグメントと比べて，セグメントの薄肉化が可能であり，総合的な工費を縮減できるこ

とがわかった．

本論文は 8 章より構成されおり，その概要は以下のとおりである．

第 1 章は，研究の背景，目的および論文の構成について記述した章である．

第 2 章では，合成セグメントに用いられる材料の材料特性に関する既往の研究をレビ

ューし，精度が高い材料モデルを検討している．

第 3 章は，密閉式合成セグメント模型の載荷試験にもとづいて，その挙動を評価した

章である．ここでは，まず，模型供試体の寸法，使用した材料，ずれ止めの配置，測定

項目および載荷方法などについて詳述した．つぎに，密閉式合成セグメント模型の載荷

試験を行い，その結果から，耐荷性能を損なう要因として，鋼とコンクリートとの境界

面に生じるずれ，スキンプレートの局部座屈，ずれ止め位置におけるコンクリートの局

所破壊などが考えられ，これらが密閉式合成セグメント模型の曲げ耐力に大きな影響を

与えることが明らかになった．すなわち，密閉式合成セグメント模型は，隣接する 2 本

のスタッドジベル間の圧縮側スキンプレートに局部座屈が生じた後も圧縮側スキンプ

レートは圧縮耐力を保持し，圧縮ひずみが終局ひずみに達したときに，中詰めコンクリ

ートが圧壊することにより破壊に至る．

第 4 章は 3 次元材料非線形 FEM を用いて，密閉式合成セグメント模型の耐荷機構を

検討した章である．3 次元材料非線形 FEM による解析結果は，載荷試験におけるスタッ

ドジベルの効果をよく表現できた．また，曲げ耐力および破壊モードの解析結果は実験

結果のそれらと一致し，両者の変形挙動もほぼ表現できた．

第 5 章は，載荷試験結果と FEM による解析結果をもとに，曲げと軸力との組合せ荷

重を受ける密閉式合成セグメント模型の降伏曲げ耐力，終局曲げ耐力および変形の算定

手法を提案した章である．この章では，密閉式合成セグメント既往の実験結果を用いて，

提案した算定手法の妥当性も検証した．結果として，提案した算定手法は密閉式合成セ

グメントの降伏曲げ耐力，終局曲げ耐力および変形を精度よく算定できることを示した．

第 6 章は，その他の合成セグメントへの適用性を検討したものである．この章では，

コンクリート中詰め鋼製セグメントおよびコンクリート中詰めダクタイルセグメント

についての FEM 解析を行い，その結果とそれらの既往の実験結果とを比較検討すると

ともに，提案した算定手法および FEM モデルの適用性も検証した．

第 7 章は提案した算定手法を用いて合成セグメントを試設計したものである．

試設計の対象は首都圏外郭放水路第４工区であり，密閉式合成セグメントを限界状態

設計法にもとづいて設計した．試設計の結果をみると，密閉式合成セグメントは，RC

セグメントと比べて，セグメントの薄肉化が図れ，セグメントの組立時間の短縮などを

総合的に考えると，工費の縮減が期待できることがわかった．

第 8 章は結論を述べた章であり，第 2 章から第 7 章で得られた主要な研究成果を要約

して述べている．

List of Papers

早稲田大学博士（工学）学位申請研究業績書

氏名張穏軍印

（2009 年 10 月現在）

種類別題名、発表・発行掲載誌名、発表・発行年月、連名者

（申請者含む）

1．論文

論文

○ 論文

○ 論文

○ 論文

○ 論文

○ 論文

A study of the localized bearing capacity of reinforced concrete K-segment，Tunnelling and Underground Space Technology, ITA, Vol.22,No.4, pp.467-473, 2007,7. Wenjun Zhang, Atsushi Koizumi.

密閉式合成セグメントの鋼コンクリート界面の相対ずれに関する研究，

現代トンネル技術, 第4回日中シールド技術交流会,広州,中国，pp.40-45, 2007,9．張穏軍，小泉淳．

Flexural rigidity of closed composite segment, Proceedings of the International Symposium on Tunnelling for Urban Development, ITA, Pattaya, Thailand, pp.205-212, 2007,12. Wenjun Zhang, Atsushi Koizumi.

Stiffness and Deflection of Composite Segment for Shield Tunnel, Tunnelling Technology & The Environment, TAC, Canada, pp.95-105, 2008,10. Wenjun Zhang, Atsushi Koizumi.

合成セグメントの曲げ挙動に関する研究,土木学会論文集 F, Vol.65,No2, pp.246-263, 2009.張穏軍，小泉淳．

Design of composite segment for underground discharge channel, Proceedings of the fifth China-Japan conference on shield tunnelling, Chengdu, China, pp.10-19, 2009,9. Wenjun Zhang, Atsushi Koizumi.

2．報告

報告

ジャッキ推力による鋼製セグメントの座屈挙動に関する研究，土木学会

トンネル工学研究発表会論文報告集，第 16 巻，pp.319~324,2006，鈴木哲

太郎，張穏軍，小泉淳．

3．著書

訳書

訳書

シールド工法の調査・設計から施工まで(中国語版)，中国建築工業出版

社，2008，ISBN 978-7-112-09246-8, 監訳：張穏軍．

シールドトンネルの耐震検討(中国語版)，中国建築工業出版社，2009,

ISBN 978-7-112-10949-4, 監訳：張穏軍．

4．講演

○ 講演講演講演講演

○ 講演

合成セグメントの設計，JR 東日本発表会，2007 年 3 月，張穏軍.

覆工版の設計，北京交通大学，2006 年 7 月，小泉淳, 張穏軍.

シールド工法，瀋陽地下鉄会社，2006 年 7 月，小泉淳, 張穏軍.

シールド工法とセグメントの設計，大連理工大学，2006 年 7 月，小泉淳,

張穏軍.

合成セグメントの力学的挙動および設計法に関する研究，みずほ情報総

研株式会社，2009 年 9 月，張穏軍.

5．その他

鋼構造研

究助成費

合成セグメントの力学的なメカニズムに関する研究，社団法人日本鉄鋼

連盟(JISF)，2008, 300 万円，張穏軍.

shield tunnel

Documents