New partial interaction models for bolted-side-plate reinforced concrete beams

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NEW PARTIAL INTERACTION MODELS FOR

BOLTEDSIDE-PLATED

REINFORCED CONCRETE BEAMS

LI, LINGZHI

(李凌志)

Ph.D. THESIS

THE UNIVERSITY OF HONG KONG

2013

======================================================

New Partial Interaction Models for

Bolted-Side-Plated Reinforced Concrete Beams

by

LI, Lingzhi

(李凌志)

A thesis submitted in partial fulfillment of the requirements for

the degree of Doctor of Philosophy

at The University of Hong Kong

in August 2013

============================

Abstract of thesis entitled

“New Partial Interaction Models for

Bolted-Side-Plated Reinforced Concrete Beams”

Submitted by

LI, Lingzhi

for the degree of Doctor of Philosophy

at The University of Hong Kong

in August 2013

Existing reinforced concrete (RC) beams often need to be strengthened due to

material deterioration or a change in usage. The bolted side-plating (BSP)

technique, i.e., attaching steel plates to the side faces of RC beams using anchor

bolts, effectively enhances the bearing capacity without significant loss in

deformability thus receives wide acceptance. However, as a newly developed

technique, only limited information is available in literature, which mainly

focused on the overall load–deflection performance of lightly reinforced BSP

beams. Little studies have been conducted on the partial interaction between steel

plates and RC beams which is closely related to the performance of BSP beams.

The longitudinal and transverse slips, which control the degree of partial

interaction, have yet to be determined precisely. Accordingly, in this thesis,

extensive experimental, numerical and theoretical studies on BSP beams are

presented.

The experimental behaviour of BSP beams was investigated. For the first

time, special effort was put in precisely measuring the profiles of longitudinal and

transverse slips. In order to investigate the behaviour of BSP beams under other

load cases and beam geometries, a nonlinear finite element analysis was

conducted. The numerical method is more economical and capable of overcoming

the difficulty in measuring the transverse slips precisely. A new approach to

evaluating the transverse bolt shear force was also developed through a parametric

study.

New partial interaction models were developed by isolating and considering

the longitudinal and the transverse partial interaction separately. A longitudinal

slip model was developed based on the BSP beam section analysis, in which

different strains of steel plates and RC beams were considered but the difference

in deflection hence the difference in curvature was not taken into account.

Meanwhile, a piecewise linear model was also proposed for the transverse slip

and bolt shear transfer by introducing Winkler’s model and defining the

transverse slip as the difference in deflection. Formulas for the slips, the plate

forces, the strain and the curvature factors that indicate the degree of partial

interaction, were also deduced. Furthermore, these formulas allow us to evaluate

the effect of partial interaction in the BSP strengthening design.

A numerical program was originally developed to evaluate the performance

of BSP beams with partial interaction. The balance between strengthening effect

and strengthening efficiency was also achieved by a parametric optimization study,

which would simplify the design procedure of BSP strengthening significantly.

According to the numerical and theoretical results, a new design approach for

BSP beams, which needs only minor modification to existing design formula for

RC beams, was proposed to aid engineers in designing this type of BSP beams

and to ensure proper details for desirable performance. Compared to the

conventional design methods that assume a full interaction between steel plates

and RC beams, this new method not only retains the features such as ease of use

and fast calculation, but also yields results that are more reliable.

(478 Words)

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DECLARATION

I declare that this thesis represents my own work, except here due

acknowledgement is made, and that it has not been previously included in a thesis,

dissertation or report submitted to this University or to any other institution for a

degree, diploma or other qualification.

Signed ______________

LI LINGZHI

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ACKNOWLEDGEMENTS

First, I would like to express my deepest gratitude to both my supervisors

Prof. S.H. Lo and Dr. Ray Su for their guidance in the research work. Without

their consistent and invaluable advices, this work would have been impossible.

Their enthusiasm and strict attitude to research will influence me for my lifetime.

Special sincere thanks would also go to Dr. W.H. Siu for his selfless and

warm-hearted advice and help in the preparation of this study.

The experimental works in this thesis has benefited greatly from the technical

assistance by all technicians in the structural engineering laboratory of the

University of Hong Kong. Final Year Project students Mr Y.R. Ke and Mr K.K.

Tam are also gratefully acknowledged for their industrious technical work.

Without their assistance, the experimental testing of this study would not have

been conducted successfully.

The financial supports given by the Research Grants Council of Hong Kong

SAR (Project No. HKU7166/08E and HKU7151/10E), along with the generous

technical supports by the HILTI Corporation are gratefully acknowledged.

Finally, I am indebted to all my beloved family members for their sacrifice

and support, who nudged me when times were tough and celebrated with me

when I had done my best.

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TABLE OF CONTENTS

DECLARATION ............................................................................................................. I

ACKNOWLEDGEMENTS................................................................................................ II

TABLE OF CONTENTS ................................................................................................. III

ABBREVIATIONS AND NOTATIONS ............................................................................... IX

LIST OF FIGURES .................................................................................................... XVI

LIST OF TABLES ..................................................................................................... XXIV

CHAPTER 1

INTRODUCTION ............................................................................. 1

1.1 Overview .................................................................................................. 1

1.2 Research objectives .................................................................................. 2

1.3 Scope of thesis .......................................................................................... 4

CHAPTER 2

LITERATURE REVIEW ................................................................... 7

2.1 Overview .................................................................................................. 7

2.2 Strengthening techniques of RC beams .................................................... 7

2.2.1 Strengthened by adhesively bonded steel plates ............................. 7

2.2.2 Strengthened by adhesively bonded FRPs ...................................... 9

2.2.3 Strengthened by mechanically bolted steel plates ........................ 10

2.3 Researches related to BSP beams ........................................................... 12

2.3.1 Partial interaction between steel plates and RC beam .................. 12

2.3.2 Buckling of deep steel plates ........................................................ 13

2.3.3 Moderately reinforced BSP beams ............................................... 14

2.3.4 Other issues related to BSP beams ............................................... 15

2.4 Conclusions ............................................................................................ 16

CHAPTER 3

EXPERIMENTAL STUDY ON BSP BEAMS .................................... 17

3.1 Overview ................................................................................................ 17

3.2 Specimen preparation ............................................................................. 18

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3.2.1 Specimen details ........................................................................... 18

3.2.2 RC beam fabrication ..................................................................... 19

3.2.3 Strengthening procedure ............................................................... 19

3.3 Material properties ................................................................................. 20

3.3.1 Concrete ........................................................................................ 20

3.3.2 Reinforcing bars ............................................................................ 21

3.3.3 Steel plates .................................................................................... 21

3.3.4 Bolt connection ............................................................................. 21

3.4 Test procedure ........................................................................................ 22

3.4.1 Test set-up ..................................................................................... 22

3.4.2 Instrumentation ............................................................................. 23

3.4.3 Loading history ............................................................................. 23

3.5 Conclusions ............................................................................................ 24

CHAPTER 4

RESULT AND ANALYSIS OF EXPERIMENTAL STUDY ON

BSP BEAMS ................................................................................. 43

4.1 Overview ................................................................................................ 43

4.2 Failure mode ........................................................................................... 43

4.3 Strength, stiffness and ductility .............................................................. 46

4.3.1 Strength and stiffness .................................................................... 46

4.3.2 Ductility and toughness ................................................................ 48

4.4 Longitudinal and transverse slips ........................................................... 48

4.4.1 Longitudinal slip ........................................................................... 49

4.4.2 Transverse slip .............................................................................. 50

4.5 Strain and curvature factors .................................................................... 51

4.5.1 Strain factors ................................................................................. 51

4.5.2 Curvature factors .......................................................................... 52

4.6 Plate behaviour ....................................................................................... 52

4.7 Conclusions ............................................................................................ 53

CHAPTER 5

NUMERICAL STUDY ON BSP BEAMS .......................................... 67

5.1 Overview ................................................................................................ 67

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5.2 Numerical modelling .............................................................................. 67

5.2.1 Modelling of concrete ................................................................... 68

5.2.2 Modelling of reinforcement and steel plates ................................ 69

5.2.3 Modelling of bolt connections ...................................................... 70

5.2.4 Finite element meshes and load steps ........................................... 70

5.3 Validation of numerical model using experimental results .................... 71

5.3.1 Comparison of the load–deflection curves ................................... 71

5.3.2 Comparison of the longitudinal slip profiles ................................ 72

5.3.3 Comparison of the transverse slip profiles ................................... 72

5.4 Studies on longitudinal slip and shear transfer ....................................... 72

5.4.1 Longitudinal shear transfer ........................................................... 72

5.4.2 Influence of loading position ........................................................ 73

5.5 Studies on transverse slip and shear transfer .......................................... 74

5.5.1 Transverse shear transfer .............................................................. 74

5.5.2 A brief introduction to the parametric study ................................. 75

5.5.3 Transverse shear transfer profiles under different loading

arrangements ................................................................................ 76

5.5.4 Transverse shear transfers under different load levels and beam

geometries .................................................................................... 77

5.5.5 Half bandwidths under different load levels and beam

geometries .................................................................................... 79

5.5.6 Support–midspan ratios under different load levels and beam

geometries .................................................................................... 81

5.5.7 Evaluation of transverse shear transfer and bolt shear force in

BSP beams .................................................................................... 82

5.5.8 Worked example ........................................................................... 82

5.6 Conclusions ............................................................................................ 85

CHAPTER 6

THEORETICAL STUDY ON

LONGITUDINAL PARTIAL INTERACTION IN BSP BEAMS ......... 109

6.1 Overview .............................................................................................. 109

6.2 Basic conceptions about BSP beams .................................................... 109

6.2.1 Longitudinal and transverse slips ............................................... 109

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6.2.2 Partial interaction ........................................................................ 111

6.2.3 Strain and curvature factors ........................................................ 111

6.2.4 Axial and flexural stiffnesses ...................................................... 113

6.2.5 Plate–RC and bolt–RC stiffness ratios ....................................... 114

6.2.6 Longitudinal and transverse shear transfers ............................... 115

6.2.7 Lightly and moderately reinforced RC beams ............................ 117

6.2.8 Shallow and deep steel plates ..................................................... 118

6.3 Longitudinal slip in BSP beams ........................................................... 119

6.3.1 Longitudinal slip profile ............................................................. 119

6.3.2 Governing equation .................................................................... 120

6.4 Longitudinal slip in BSP beams under various loading conditions...... 125

6.4.1 Under four-point bending ........................................................... 125

6.4.2 Under three-point bending .......................................................... 127

6.4.3 Under a uniformly distributed load ............................................. 130

6.4.4 Under a triangularly distributed load .......................................... 131

6.4.5 Under a support moment ............................................................ 133

6.4.6 Under pure bending .................................................................... 134

6.4.6.1 Superposition of longitudinal slip .................................. 134

6.4.6.2 Longitudinal slip under pure bending by using

superposition .................................................................. 135

6.5 Verification ........................................................................................... 136

6.5.1 Verification by the experimental results ..................................... 136

6.5.2 Superposition for longitudinal slip under weak non-linearity .... 138

6.6 Conclusions .......................................................................................... 138

CHAPTER 7

THEORETICAL STUDY ON

TRANSVERSE PARTIAL INTERACTION IN BSP BEAMS ............. 155

7.1 Overview .............................................................................................. 155

7.2 Simplified piecewise linear model ....................................................... 155

7.2.1 Simplification of shear transfer profiles ..................................... 155

7.2.2 Shear transfer according to Winkler’s model ............................. 157

7.2.3 Solution based on force equilibrium and deformation

compatibility ............................................................................... 160

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7.2.4 Experimental verification ........................................................... 167

7.3 Approximate solution for strengthening design ................................... 168

7.3.1 Under four-point bending ........................................................... 169

7.3.2 Under three-point bending .......................................................... 170

7.3.3 Under a uniformly distributed load ............................................. 171

7.4 Conclusions .......................................................................................... 173

CHAPTER 8

ANALYSIS OF BSP BEAMS WITH PARTIAL INTERACTION ....... 185

8.1 Overview .............................................................................................. 185

8.2 Partial interaction in BSP beams .......................................................... 186

8.3 Program details ..................................................................................... 187

8.3.1 Material models .......................................................................... 187

8.3.2 Analysis of a BSP beam section with partial interaction ............ 188

8.3.3 Analysis of a BSP beam with partial interaction ........................ 191

8.4 Study on analysis results ...................................................................... 192

8.4.1 Verification by experimental results ........................................... 192

8.4.2 Partial interaction on strengthening effect .................................. 194

8.4.3 Recommendation on choice of strain and curvature factors ....... 195

8.5 Conclusions .......................................................................................... 196

CHAPTER 9

DESIGN OF BSP BEAMS WITH PARTIAL INTERACTION ........... 209

9.1 Overview .............................................................................................. 209

9.2 Theoretical base .................................................................................... 209

9.2.1 Material models .......................................................................... 210

9.2.2 Sectional analysis and flexural strength ..................................... 211

9.2.3 Verification by experimental results ........................................... 214

9.3 Proposed design procedure ................................................................... 214

9.3.1 Estimation of plate sizing ........................................................... 214

9.3.2 Estimation of number of bolts .................................................... 217

9.3.3 Verification of partial interaction ............................................... 218

9.3.4 General strengthening strategies and preliminary design ........... 220

9.4 Worked example ................................................................................... 222

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9.4.1 Current state of the structure needed strengthening.................... 222

9.4.2 Arrangement of steel plates ........................................................ 226

9.4.3 Arrangement of anchor bolts ...................................................... 228

9.4.4 Verification of partial interaction ............................................... 229

9.4.5 Discussion of strengthening effect and efficiency ...................... 233

9.5 Conclusions .......................................................................................... 234

CHAPTER 10

CONCLUSIONS ........................................................................... 241

10.1 Summary ............................................................................................ 241

10.2 Conclusions ........................................................................................ 243

10.3 Recommendations for future study .................................................... 245

REFERENCES ......................................................................................................... 247

PUBLICATIONS ...................................................................................................... 253

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ABBREVIATIONS AND NOTATIONS

Abbreviations

BSP Bolted side-plated, Bolted side-plating

FRP Fibre reinforced polymer

LSQ Least square fitting

LDT Linear displacement transducer

LVDT Linear variable displacement transducer

NLFEA Nonlinear finite element analysis

ODE Ordinary differential equation

RC Reinforced concrete

UDL Uniformly distributed load

Notations

(EA)c Axial stiffness of the unstrengthened RC beam

(EA)cp Axial stiffness of the BSP beam

(EA)p Axial stiffness of the steel plates

(EI)BSP Overall flexural stiffness of the BSP beam

(EI)c Flexural stiffness of the unstrengthened RC beam

(EI)cp Flexural stiffness of the BSP beam

(EI)p Flexural stiffness of the steel plates

A, Ai Parameters or undetermined constants (i = 1, 2, 3…)

AF Parameter controlled by the magnitudes of the external loads

Ac Cross-section area of the concrete

Ap Cross-section area of the steel plates

As Cross-section area of the reinforcement

Asc, Ast Cross-section areas of the compressive and the tensile reinforcement

aF Relative position of the external load F: xF/L

B Width of the RC beam section

B, Bi Parameters or undetermined constants (i = 1, 2, 3…)

C, Ci Parameters or undetermined constants (i = 1, 2, 3…)

c Depth of the neutral axis

x

D, Di Parameters or undetermined constants (i = 1, 2, 3…)

Dc Thickness of the RC beam

Dp Thickness of the steel plates

Dsl, Dsb Thicknesses of the floor slab and the secondary beam

db Nominal diameter of the anchor bolts

dc Thickness a layer of the concrete

dp Thickness a layer of the steel plates

dtc Lever arm between the tensile reinforcement and the compressive

block of the RC beam section

dδ Deflection difference between the steel plates and the RC beam

E Young’s modulus of the steel

E0 Initial modulus of the concrete

Ec Secant modulus at 0.4fc on the ascending branch of the concrete

stress–strain curve

Ecc Secant modulus at the peak compressive strength of the concrete

Ep Young’s modulus of the steel plates

Es Young’s modulus of the reinforcement

F, Fi Total external load (the ith point load, i = 1, 2, 3…)

Fb Shear force recorded in the bolt test

Fbp Peak shear force recorded in the bolt test

Ff External load at failure

Fp Peak total external load

fc Compressive strength of the concrete

fcef Effective compressive strength of the concrete

fco Cylinder Compressive strength of the concrete

fcu Cube compressive strength of the concrete

fic Stress at the inflection point in Sargin’s model

ft Tensile strength of the concrete

ftef Effective tensile strength of the concrete

fu Ultimate strength of the reinforcement

fub Ultimate tensile strength of the anchor bolt material

fup Ultimate strength of the steel plates

fy Yield strength of the reinforcement

fyp Yield strength of the steel plates

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Gf Fracture energy per unit area of a stress-free crack

g Permanent uniformly distributed load

H Hardening modulus of the steel

h Depth of the RC beam

h1, h2 Error functions used in least square fitting

hc, h0 Depths of the compressive and the tensile reinforcement

hpt, hpb Depths of the top and the bottom edge of the steel plates

Ip Second-moment of area of the steel plates

ic Effective radius of gyration of the RC beam

icp Separation between the centroids of the RC beam and the steel plate

ip Effective radius of gyration of the steel plates

Kb Shear stiffness of the anchor bolts

Kb, 0.10 Shear stiffness of the anchor bolts at Fb /Fbp = 0.10

Ke Equivalent elastic stiffness in the load–deflection curve

k Equivalent foundation modulus

km Stiffness of the connecting media

L Clear span of the RC beam

Ls Shear span of the RC beam

Lcd Band size for the fictitious compression plane model

Lph Half length of the steel plates

M Bending moment

M0, M1 Flexural strength when the strain or curvature factor equals 0 or 1

Mc Bending moment resisted by the RC beam (EI)c φc

Md Design moment caused by the external loads

MG Design moment caused by the external permanent loads

MQ Design moment caused by the external variable loads

Mp Bending moment resisted by the steel plates

MS Support bending moment

Mu Flexural strength of the BSP beam

MuRC, MuBSP Flexural strengths of the RC and the BSP beam

MuBSP, FI Flexural strength of the BSP beam under full interaction assumption

N Resultant axial force of the BSP beam

Nc Compression force of the RC beam

Np Tension force of the steel plates

xii

Nu Resultant axial force of the BSP beam corresponding to Mu

nb Number of the anchor bolts in a shear span

p Parameter including the stiffness components of the BSP beams

q Distributed external transverse load

qp Distributed external transverse load on the steel plates

qu Distributed external transverse load corresponding to Mu

Rb Shear force of an anchor bolt

Rby Yield shear force of an anchor bolt

S Shear deformation of the anchor bolts

Sb Longitudinal bolt spacing

Sby Yield shear deformation of the anchor bolts

Slc Longitudinal slip on the plate–RC interface

Str Transverse slip on the plate–RC interface

Sx Interfacial slip along the beam axis

Sy Interfacial slip along the depth of the beam

s Bond slip at the steel–concrete interface

s1, s2, s3 Bond slip parameters in the CEB-FIB Model Code 1990 (CEB 1993)

Tm Longitudinal bolt shear force

tm Longitudinal shear transfer

tp Thickness of one steel plate

Ut Modulus of toughness

V Shear force

Vc Transverse shear force of the RC beam

Vm Transverse bolt shear force

Vp Transverse shear force of the steel plates

vc Shear stress of the RC beam

vm, vm,i Shear transfer, shear transfer caused by the ith point load (i = 1, 2…)

vp Shear stress of the steel plates

w, wi Half bandwidth of the shear-transfer block (the ith block i = 1, 2…)

w’ Width of the opposite shear-transfer block

wc Crack opening in the concrete

wcr Crack opening in the concrete at complete release of stress

wcd Plastic displacement for the fictitious compression plane model

wsla Width of the ascending branch of the longitudinal slip profile

xiii

wsl Half bandwidth of the longitudinal slip profile

X, Y Axes in the horizontal and the vertical direction

x, y Axes along the beam axis and the depth of the beam

xF Position of the external load with refer to the left support

xNpm Position of the critical plate tensile force

ycc Centroidal level of the RC beam

yna Level of neutral axis in the RC beam

ypc Centroidal level of the steel plates

αv Modifier in the computation of the bolt shear strength

α Unique value for the strain and the curvature factors

αε Strain factor εp,ypc / εp,ypc

αφ Curvature factor φp / φc

β Parameter for Winkler’s model

βa Axial stiffness ratios between the steel plates and the RC beam

βm Ratio between the stiffness of the bolt connection and the flexural

stiffness of the RC beam

βp Flexural stiffness ratios between the steel plates and the RC beam

γaF, γaF-1 Parameters for the computation of the longitudinal slip

γb Partial safety factor for the bolt connection

γc Partial safety factor for the concrete material

γG Partial safety factor for the actions caused by the permanent loads

γM2 Partial safety factor for the bolt material

γQ Partial safety factor for the actions caused by the variable loads

γs Partial safety factor for the steel material

δ, δy, δp, δf Midspan deflection, midspan deflection when the yielding, the peak

load or the failure occurs

δc Deformation of the RC beam

δcX Displacement of the RC beam in the direction

δcY Displacement of the RC beam in the vertical direction

δcx Displacement of the RC beam along the beam axis

δcy Displacement of the RC beam along the depth of the beam

δcm Deflection of the RC beam under the bolt shear force

δce Deflection of the RC beam under the external loads

δp Deflection of the steel plates, or midspan deflection at the peak load

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δp0 Deflection of the steel plates at the left support

δpf Deflection of the steel plates referring to the left support

δpX Displacement of the steel plates in the horizontal direction

δpY Displacement of the steel plates in the vertical direction

δpx Displacement of the steel plates along the beam axis

δpy Displacement of the steel plates along the depth of the beam

δy Deflection

εc Strain of the concrete or the RC beam

εc0 Strain at the peak compressive stress in the concrete

εcc Maximum strain on the compression surface of the RC beam

εcd, εcu Ultimate compressive strain of the concrete

εceq

Equivalent uniaxial strain of the concrete

εcr Ultimate tensile strain of the concrete

εct Strain at the peak tensile stress of the concrete

εic Strain at the inflection point in Sargin’s model

εp Strain of the steel plates

εpt, εpb Strains at the top and the bottom edge of the steel plates

εs Strain of the reinforcement

εsc, εst Strains of the compressive and the tensile reinforcement

εy Yield strain of the reinforcement

εyp Yield strain of the steel plate

ζ Critical shear transfer ratio due to a change in the beam geometries

ζEIc Critical shear transfer ratio due to a change in (EI)c

ζEIp Critical shear transfer ratio due to a change in (EI)p

ζkm Critical shear transfer ratio due to a change in km

η Factor defining the effective strength of the concrete

θ Rotation

θc Rotation of the RC beam

θp Rotation of the steel plates

σ Stress

σ1, σ2 Principle stress

σc Stress of the concrete

σcef Effective stress of the concrete

σcn Normal stress in the crack

xv

σp Stress of the steel plates

σs Stress of the reinforcement

φ Curvature

φc Curvature of the RC beam

φc,PI Curvature of the RC beam, with the transverse partial interaction

φFI Curvature with full interaction

φPI Curvature with the longitudinal partial interaction

φp Curvature of the steel plates

φp,PI Curvature of the steel plates, with the transverse partial interaction

λ Factor for the effective depth of the concrete compression zone

ξ Relative location along the beam axis, or the error tolerance

ξF Dimensionless shear transfer ratio at the loading points

ξFp Dimensionless shear transfer ratio at the loading points (referring to

the peak load)

ξp Parameter used to compute the longitudinal slip and the strain factor

ξS Dimensionless shear transfer ratio at the supports

ξw Relative half bandwidth of the shear transfer profile

ρst Steel ratio of the tensile reinforcement

ρstb Balanced tensile steel ratio

τ Bond stress on the plate–RC interface

τf Residual bond stress

τmax Peak bond stress

Subscripts

Symbols with the following subscripts are with the general meanings:

exp, num, the Data derived from the experimental, numerical, and theoretical study

I, J, K, i, j, k Indexes of layers, iterations, increments and steps

m, n, s Numbers of layers of concrete, the steel plates, and the reinforcements

max, min Maximum and minimum of a variable value

S, LS, RS, F Values at the supports, the left and right supports, and at the loading

point

ycc, ypc Values at the centroidal level of the RC beam and the steel plates

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LIST OF FIGURES

Figure 1.1 Illustration of a typical BSP beam ...................................................... 6

Figure 1.2 Illustration of longitudinal and transverse slips .................................. 6

Figure 3.1 Cross section of specimens (a) lightly reinforced (P75B300) and

(b) moderately reinforced (the other specimens) .............................. 27

Figure 3.2 Configurations of strengthening measures (section view) for

Specimens (a) P75B300, (b) P100B300 & P100B450, (c)

P250B300 and (d) P250B300R & P250B450R ................................ 27

Figure 3.3 Configurations of strengthening measures (front view) for

Specimens (a) P75B300, (b) P100B300, (c) P100B450, (d)

P250B300, (e) P250B300R and (f) P250B450R .............................. 28

Figure 3.4 Reinforcement cages ......................................................................... 29

Figure 3.5 Details and installation of dynamic sets; (a) injection washer, (b)

installation drawing and (c) actual installation ................................. 30

Figure 3.6 Details and installation of buckling restraint devices; (a) design

diagram and (b) actual installation ................................................... 31

Figure 3.7 Measured stress–strain relationships of reinforcement (a) T10, (b)

T16 and (c) R10 ................................................................................ 33

Figure 3.8 Measured stress–strain relations of steel plates ................................ 33

Figure 3.9 Design diagram of bolt test set-up for the

“HIT-RE 500 + HAS-E” anchoring system ..................................... 34

Figure 3.10 Actual installation of bolt test set-up for the

“HIT-RE 500 + HAS-E” anchoring system ..................................... 35

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Figure 3.11 Shear force–slip curves of the “HIT-RE 500 + HAS-E”

anchoring system .............................................................................. 35

Figure 3.12 Design diagram of test setup ............................................................ 36

Figure 3.13 Experimental set-up for Specimen (a) CONTROL, (b) P75B300,

(c) P100B300, (d) P100B450, (e) P250B300R, (f) P250B450R

and (g) P250B300 ............................................................................. 39

Figure 3.14 Arrangements of (a) strain gauges and (b) LVDTs & LDTs ........... 40

Figure 3.15 Design diagram of LVDT sets for the measurement of

longitudinal and transverse slips....................................................... 41

Figure 3.16 Actual arrangement of LVDT sets for the measurement of

longitudinal and transverse slips....................................................... 42

Figure 4.1 Load–deflection curves for the reference beams .............................. 57

Figure 4.2 Load–deflection curves for the lightly reinforced beams ................. 57

Figure 4.3 Load–deflection curves for the moderately reinforced beams ......... 58

Figure 4.4 Load–deflection curves for beams with or without buckling

restraint ............................................................................................. 58

Figure 4.5 Midspan vertical slips of P75B300 at (a) the peak load and (b)

failure ................................................................................................ 59

Figure 4.6 Failure modes of (a) P100B300, (b) P100B450, (c) P250B300R

and (d) P250B450R .......................................................................... 60

Figure 4.7 Plate buckling of P250B300 ............................................................. 61

Figure 4.8 Equivalent elasto-plastic system of the load–deflection curve ......... 61

Figure 4.9 Longitudinal slip profiles along the beam axis for (a) P100B300,

(b) P100B450, (c) P250B300R and (d) P250B450R ........................ 63

xviii

Figure 4.10 Transverse slip profiles along the beam axis for (a) P100B300,

(b) P100B450, (c) P250B300R and (d) P250B450R ........................ 65

Figure 4.11 Development of (a) strain factors and (b) curvature factors ............ 66

Figure 5.1 The concrete model’s (a) biaxial failure law and (b) equivalent

uniaxial stress–strain curve ............................................................... 90

Figure 5.2 Bond–slip curve from CEB-FIB Model Code 1990 (CEB 1993) .... 91

Figure 5.3 The Bi-linear Steel Von Mises Model’s (a) biaxial failure law

and (b) stress–strain curve .............................................................. 92

Figure 5.4 Simulation of bolt connection: (a) a bolt element and (b)

load–slip curve comparison .............................................................. 93

Figure 5.5 Meshing of (a) the RC beam and (b) the steel plates for

P250B450R ....................................................................................... 94

Figure 5.6 Comparison of load–deflection curves obtained from the

experimental and numerical studies for (a) P100B300 and

P250B300R and (b) P100B450 and P250B450R ............................. 95

Figure 5.7 Comparison of longitudinal slip profiles obtained from the

experimental and numerical studies for (a) P100B300 and (b)

P100B450 ......................................................................................... 96

Figure 5.8 Comparison of transverse slip profiles obtained from the

experimental and numerical studies for (a) P100B300, (b)

P100B450, (a) P250B300R and (b) P250B450R ............................. 98

Figure 5.9 Longitudinal slip and shear transfer profiles of a BSP beam

under an asymmetrical load or two symmetrical loads .................... 99

Figure 5.10 Variation in the longitudinal shear transfer profile as the position

of imposed load ................................................................................ 99

Figure 5.11 Transverse slip and shear transfer profiles a BSP beam under an

asymmetrical load or two symmetrical loads ................................. 100

xix

Figure 5.12 Reference beam under (a) a midspan point load, (b) an

asymmetric point load, (c) two symmetric point loads, (d) a

uniformly distributed load, (e) a trapezoidal distributed load and

(f) a triangular distributed load ....................................................... 101

Figure 5.13 Variation in the transverse shear transfer profile as the location

of (a) an asymmetrical load or (b) two symmetrical loads ............. 102

Figure 5.14 Superposition of the transverse shear transfer profiles for (a) two

loads or (b) a uniformly distributed load (UDL) ............................ 103

Figure 5.15 Variation in the transverse shear transfer base on (a) the load

level and (b) the stiffnesses of RC, plates and bolt connection ...... 104

Figure 5.16 Variation in normalised transverse shear transfer profiles of a

BSP beam under three point bending based on (a) the load level,

(b) the RC stiffness, (d) the plate stiffness and (d) the bolt

stiffness ........................................................................................... 106

Figure 5.17 Variation in the half bandwidth of transverse shear transfer

profile of a BSP beam under three point bending .......................... 107

Figure 5.18 A worked example for the evaluation of transverse shear transfer

in a BSP beam................................................................................. 107

Figure 5.19 Comparison between the computed shear transfer profiles and

that derived from a numerical model .............................................. 108

Figure 6.1 Illustration of longitudinal and transverse slips .............................. 140

Figure 6.2 External and internal forces in a BSP beam ................................... 141

Figure 6.3 Definition of lightly and moderately reinforce concrete beams ..... 142

Figure 6.4 Definition of (a) shallow and (b) deep steel plates ......................... 142

Figure 6.5 Variation in the longitudinal slip as the length of steel plates (a)

wsl < Lph, (b) wsla < Lph < wsl and (c) Lph < wsla ............................... 143

xx

Figure 6.6 The profiles of shear force, bending moment and longitudinal

slip in a BSP beam under four-point bending................................. 144


slip in a BSP beam under arbitrary three-point bending ................ 144


slip in a BSP beam under a uniformly distributed load (UDL) ...... 145


slip in a BSP beam under a triangularly distributed load (TDL) .... 145


slip in a BSP beam under a support moment .................................. 146

Figure 6.11 Illustration of superposition for longitudinal slip in BSP beams;

(a) force superposition and (b) longitudinal slip superposition ...... 147

Figure 6.12 Superposition for longitudinal slip in a BSP beam under pure

bending; (a) force superposition and (b) longitudinal slip

superposition ................................................................................... 148


experimental and theoretical studies for (a) P100B300 and (b)

P100B450 ....................................................................................... 149

Figure 6.14 Comparison of longitudinal tensile force transfers obtained from

the experimental and theoretical studies for (a) P100B300 and (b)

P100B450 ....................................................................................... 150

Figure 6.15 Shear force–slip curves of the “HIT-RE 500 + HAS-E”

anchoring system ............................................................................ 151

Figure 6.16 Comparison of the maximum longitudinal slips obtained from


P100B450 ....................................................................................... 152

xxi

Figure 6.17 Comparison of the maximum plate tensile forces obtained from


P100B450 ....................................................................................... 153

Figure 6.18 Verification of superposition for longitudinal slip in BSP beams;

(a) force superposition and (b) longitudinal slip superposition ...... 154

Figure 7.1 Shear transfer profiles of a BSP beam under (a) a point load at

the midspan, (b) a point load close to the support, (c) two point

loads close to the supports and (d) two point loads close to the

midspan ........................................................................................... 176

Figure 7.2 The piecewise linear profile model for transverse slip and shear

transfer in BSP beams; (a) illustration of transverse slip and (b)

simplified profile model ................................................................. 177

Figure 7.3 Analogy of shear transfer to Winkler’s model; (a) an infinite

beam under a point load and (b) a semi-infinite beam under a

point load ........................................................................................ 178

Figure 7.4 Shear transfer in BSP beams with (a) rigid bolts or infinitely

flexible steel plates and (b) elastic bolts and rigid steel plates ....... 179

Figure 7.5 Variation of shear transfer profile (a) before and (b) after

cracking occurs ............................................................................... 180

Figure 7.6 Linear profile model for a BSP beam under four-point bending .... 181

Figure 7.7 Comparison of experimental and theoretical shear transfer

profiles at load level (a) F/Fp = 0.25, (b) F/Fp = 0.5 and (c)

F/Fp = 0.75 for P100B300 .............................................................. 182

Figure 7.8 Comparison of experimental and theoretical shear transfer

profiles at load level F/Fp = 0.5 for (a) P100B450, (b)

P250B300R and (c) P250B450R .................................................... 183

Figure 7.9 Shear transfer profile model for a BSP beam under UDL .............. 184

xxii

Figure 8.1 Stress–strain curve of concrete in compression .............................. 199

Figure 8.2 Stress–strain curve of steel reinforcement and steel plates ............ 199

Figure 8.3 Strain profiles of a BSP section with partial interaction ................ 200

Figure 8.4 Modified moment–curvature analysis of a BSP beam section

with partial interaction .................................................................... 201

Figure 8.5 Profiles of moment, longitudinal and transverse slips, strain and

curvatures in BSP beams ................................................................ 202

Figure 8.6 Modified moment–curvature analysis of a BSP beam with partial

interaction ....................................................................................... 203

Figure 8.7 Flexural strength profile of a BSP beam ........................................ 204

Figure 8.8 Flexural strength contribution ratios of the RC beam (φc (EI)c),

the steel plates (φp (EI)p) and the plate tensile force (icp Np) for (a)

P100B300 and (b) P250B300R ...................................................... 205

Figure 8.9 Moment–curvature curves of lightly reinforced (ρst = 0.59%)

BSP beams with (a) shallow and (b) deep steel plates ................... 206

Figure 8.10 Moment–curvature curves of moderately reinforced (ρst = 1.77%)

BSP beams with (a) shallow and (b) deep steel plates ................... 207

Figure 8.11 Strengthening effect and efficiency for (a) lightly and (b)

moderately reinforced BSP beams ................................................. 208

Figure 9.1 Stress–strain curve of concrete in compression condition.............. 236

Figure 9.2 Stress–strain curve of steel reinforcement and steel plates ............ 236

Figure 9.3 Shear force–slip curve of anchor bolts ........................................... 236

Figure 9.4 Sectional strain and stress profiles in a BSP beam ......................... 237

xxiii

Figure 9.5 Sectional strain and stress profiles of steel plates in a BSP beam

at the occurrence of (a) plate yielding and (b) plate

entire-sectional tension ................................................................... 237

Figure 9.6 A typical RC structural layout; (a) Plane layout and (b) Elevation

layout .............................................................................................. 238

Figure 9.7 Strengthening strategies for the RC beams of (a) Type 1 and (b)

Type 2 ............................................................................................. 239

Figure 9.8 Simplified models for (a) Beam 1 (a main girder) and (b) Beam 2

(a secondary beam) ......................................................................... 239

Figure 9.9 Strengthening details for (a) Beam 1 (a main girder) and (b)

Beam 2 (a secondary beam) ............................................................ 240

xxiv

LIST OF TABLES

Table 3.1 Beam geometries and strengthening details ....................................... 25

Table 3.2 Concrete mix proportioning ............................................................... 25

Table 3.3 Cube and cylinder compressive strengths of concrete ....................... 25

Table 3.4 Strengths and moduli of reinforcement bars ...................................... 26

Table 3.5 Strengths and moduli of steel plates .................................................. 26

Table 4.1 Concrete strengths, beam geometries and strengthening details ....... 55

Table 4.2 Load levels (F/Fp) when failure phenomena occurred ...................... 55

Table 4.3 Strengths, stiffnesses and ductility ..................................................... 56

Table 4.4 Slips on the plate–RC interface.......................................................... 56

Table 4.5 Contribution of the steel plates due to bending and tension .............. 56

Table 5.1 Comparison of experimental and numerical longitudinal slips ......... 88

Table 5.2 Comparison of experimental and numerical transverse slips ............ 88

Table 5.3 Half bandwidth and support–midspan shear transfer ratios ............... 89

Table 8.1 Comparison between experimental and analytical load capacities .. 198

Table 8.2 Enhancement of lightly and moderately reinforced BSP beams ..... 198

Table 8.3 Recommended strain and curvature factors ..................................... 198

Table 9.1 Comparison of experimental and theoretical peak loads ................. 235

Table 9.2 Summary of strengthening effect ..................................................... 235

Chapter 1 Introduction

1

CHAPTER 1

INTRODUCTION

1.1 OVERVIEW

Many old buildings all around the world need to be retrofitted or strengthened.

In the developed metropolises, a majority of reinforced concrete (RC) buildings

have served much longer than their design working life. For instance, over four

thousand private buildings have served longer than fifty years in Hong Kong. A

large proportion of these buildings are multiple-storey single-span frame

structures including the well-known five-storey building that collapsed recently

on Ma Tau Wai Road, To Kwa Wan. For these old structures, material

deterioration such as concrete carbonation or steel corrosion is a main reason of

the degradation of structure safety. On the other hand, in the developing regions

such as Mainland China, many newly built structures are also in poor condition

due to unsatisfactory quantities in design and construction. A typical example can

be referred to the notorious collapse incident of Yang Ming Tan Bridge in Harbin,

which happened just ten months after its inauguration.

In these dilapidated RC structures under the requirement of strengthening, RC

beams are the most common members needed to be retrofitted. There are several

methods available to enhance RC beams, for instance (1) shortening the length of

span by installing additional supports, (2) increasing the cross section area by

adding newly cast concrete, and (3) enhancing the cross section by attaching steel

plates or fibre reinforced polymers (FRP) to the soffit face or the side faces. The

utilisation of the first two methods is very limited because they shorten the clear

span or the clear height under the beams and require lots of labour. In contrast, the

latter method has been accepted worldwide over the past several decades for its

small space occupancy and execution convenience.

Steel plates attached to RC beams by adhesive bonding usually suffer from

serious debonding and peeling. To overcome these shortcomings, steel plates can


2

be anchored to RC beams with bolts. Although bolting steel plates to the beam

soffit can effectively increase the flexural strength and stiffness, it may lead to

over reinforcement thus decrease the ductility of the strengthened beams. There is

also a potential risk of destroying the congested tensile reinforcement near the

soffit faces in the fabrication of bolt holes. Therefore, the bolted side-plating (BSP)

technique, i.e., attaching steel plates to the beam side faces using anchor bolts, has

received extensive acceptance. RC beams strengthened by this technique, as

shown in Figure 1.1, are known as bolted side-plated (BSP) beams.

The BSP retrofitting technique not only supresses the pre-mature debonding

failures and the risk of destroying tensile reinforcement, but also provides space

on the soffit face to prop up the RC beams. The steel plates in the BSP beams

usually cover a large portion of the side faces, from the tensile to the compressive

region. In this way, the RC beams can be enhanced in terms of both the tensile

and the compressive reinforcement thus be significantly enhanced in flexural

strength without a visible decrease in deformability. This feature is particularly

beneficial to the moderately reinforced RC beams, since their degree of

reinforcement is already very close to the balanced degree of reinforcement.

Despite all their advantages over the RC beams retrofitted by other

retrofitting techniques, the BSP beams are also accompanied by many

shortcomings. The partial interaction caused by a combination of longitudinal and

transverse slips on the plate–RC interface (see Figure 1.2) is the main concern for

the performance of BSP beams. Unless it is restrained properly, the plate buckling

which exists in the compressive region of the deep steel plates may be detrimental

to the overall performance as well.

1.2 RESEARCH OBJECTIVES

Although the BSP technique brings great benefits, the behaviour of BSP

beams, especially the partial interaction as a result of the longitudinal and

transverse slips, is not well understood. Limited studies were found in literature,

and most of them focused on the overall load–deflection behaviour of the lightly


3

reinforced RC beams. However, for the majority of moderately reinforced RC

beams in our buildings, studies can be hardly found. Due to the lacking of reliable

analytical models for the partial interaction behaviour, the assumption of full

interaction, i.e., the strains of steel plates and RC beams are assumed the same, is

usually accepted by structural engineers in their strengthening design practice.

With the aim of achieving better comprehension of the behaviour of BSP

beams and developing reliable analytical models for the partial interaction, a

comprehensive study has been conducted. The main objectives are listed below:

(1) To carry out experiments on the performance of moderately reinforced BSP

beams, especially the influence of partial interaction caused by the interfacial

longitudinal and transverse slips.

(2) To simulate the behaviour of BSP beams with different geometries and under

various loading conditions, especially the variation in the longitudinal and

transverse slips and shear transfers.

(3) To develop analytical models for the longitudinal and transverse partial

interaction, thus provide an available approach to integrate the effect of partial

interaction in the performance evaluation of BSP beams.

(4) To propose a design approach for the retrofitting of existing RC beams using

the BSP technique, which is simple to understand and convenient to use.

In order to achieve the aforementioned objectives, an extensive study, which

consists of experimental testing, numerical simulation, theoretical analysis,

program developing, and design procedure proposing, has been conducted:

(1) A total of seven full-scale BSP beams with different steel plate depths and

various bolt spacings are tested under four-point bending. Their behaviour is

compared to the available test results of lightly reinforced BSP beams

obtained by other researchers. Special efforts are focused on the investigation

of the longitudinal and transverse slips along the beam span. The indicators

which quantify the degree of partial interaction, i.e., the strain and the

curvature factors, are also studied.


4

(2) A nonlinear finite element model is established to simulate the behaviour of

BSP beams and investigate the variation in longitudinal and transverse slips

and shear transfers. The influence of different beam geometries and load

conditions is investigated in detail. A parametric study on the behaviour of

transverse slip and shear transfer is also carried out.

(3) Analytical models for the longitudinal and transverse partial interaction are

presented respectively. The profiles of the longitudinal and the transverse slips

of BSP beams under various load cases are proposed. Formulas for the strain

and the curvature factors, which indicate the degree of longitudinal and

transverse partial interaction, are further developed.

(4) A program to evaluate the overall performance of BSP beams, which

considers the influence of both the longitudinal and the transverse partial

interaction in terms of the strain and the curvature factors, is developed. A

parametric study is also conducted to find a balance between the strengthening

effect and the strengthening efficiency.

(5) A design method for BSP beams, which needs only little modification to the

existing design formula of RC beams, is also proposed.

1.3 SCOPE OF THESIS

This thesis consists of ten chapters. The first chapter gives a brief

summarization of the research background, the objectives of this study and an

outline of the remaining chapters.

Chapter 2 expresses a brief literature review on the previous studies on

various retrofitting techniques at first. Then a detailed review on the previous

efforts devoted to BSP beams is further presented.

Chapter 3 describes the test scheme of BSP beams. The specimen geometries,

the material properties, the strengthening methods and procedures, the test setups

and instrumentation are introduced in detail.


5

Chapter 4 reports the study outcomes on the experimental results. The overall

performance of the specimens, for instance the failure modes, the load–deflection

performance, the strength, stiffness and ductility enhancements are studied. The

longitudinal and transverse slip, the strain and curvature factors, and the flexural

and tensile contribution of the bolted steel plates are also investigated.

Chapter 5 presents a numerical simulation of the behaviour of BSP beams by

a nonlinear finite element analysis (NLFEA). The details of the numerical model

are firstly reported. Parametric studies are then conducted and special focus is

placed on the behaviour of longitudinal and transverse slips and shear transfers.

Chapter 6 provides an analytical model for the longitudinal slip and shear

transfer along the beam span, based on the BSP beam section analysis. Design

formulas of the maximum longitudinal slip, plate tensile force and strain factor are

also developed for BSP beams subjected to several simple loading conditions.

Chapter 7 proposes a piecewise linear profile model for the transverse shear

transfer in BSP beams, based on the force superposition principle and the analogy

of transverse shear transfer to the foundation reaction in Winkler’s model. Design

formulas of the maximum transverse slip and the curvature factor are also

developed for BSP beams subjected to several simple load cases.

Chapter 8 develops a numerical program to evaluate the performance of BSP

beams. The partial interaction as a result of the longitudinal and the transverse

slips is taken into accounts in terms of the strain and the curvature factors. An

optimization study is also conducted and a unique value of strain and curvature

factors is also recommended for the strengthening design of BSP beams.

Chapter 9 proposes a design procedure for the strengthening of BSP beams.

The recommended value of strain and curvature factors is directly introduced to

the existing strength formula of RC beams to determine the size of steel plates.

Then the design formulas developed in Chapters 7 and 8 are used to determine the

bolt arrangement and verify the degree of partial interaction.

Chapter 10 gives the summary and conclusions of the present study, along

with recommendations for future study on the behaviour of BSP beams.


6

Figure 1.1 Illustration of a typical BSP beam

Figure 1.2 Illustration of longitudinal and transverse slips

RC beam Steel plate

Anchor bolt Column

1

1

1-1

Anchor bolt

Steel plate

RC beam

Str

Slc

Longitudinal slip: Slc

Transverse slip: Str RC beam

Steel plate

Original

position

Deformed

position

Relative slip

Plate position with slip

Plate position without slip

Chapter 2 Literature review

7

CHAPTER 2

LITERATURE REVIEW

2.1 OVERVIEW

In this chapter, a brief summary of existing researches done by other

researchers on different kinds of external strengthening techniques for RC beams

will be given firstly. Then more detailed review will be focused on the previous

efforts on the strengthening technique of BSP beams.

2.2 STRENGTHENING TECHNIQUES OF RC BEAMS

2.2.1 Strengthened by adhesively bonded steel plates

Since its first application in the 1960s (Fleming and King 1967; L’Hermite

and Bresson 1967), the strengthening technique bonding steel plates to the tension

face of existing RC beams has gained universal acceptance. This is a convenient

method of increasing flexural strength and stiffness, decreasing flexural crack

widths, with negligible changes in the member dimensions.

The majority of research focused on the flexural strengthening of RC beams

by bonding steel plates to the tension soffits. The externally bonded steel plates

act as additional longitudinal reinforcements and its flexural behaviour can be

predicted by the beam theory. However, as the plate is not enclosed by the

concrete, much research has gone into studying premature peeling failure due to

separation between the plate and the concrete. Roberts and Hajikazemi (1989a)

conducted a theoretical investigation of RC beams strengthened on the tension

faces by externally bonded steel plates. It was indicated that the shear and normal

stresses, in and adjacent to the adhesive layers, increase rapidly towards the ends

of the steel plates and depend on the shear and normal stiffness of the connection

and on the thicknesses and points of termination of the steel plates. Oehlers and


8

Moran (1990) tested 57 plated RC beams subjected to pre-cracking and

pre-cambering, to study peeling induced by increasing curvature; A method is

derived for determining the moment at which peeling starts and the moment that

causes complete separation of the plate. Oehlers (1992) studied RC beams and

slabs strengthened by gluing steel plates to their soffits and found peeling due to

shear force depends on diagonal shear crack and cannot be prevented by adding

stirrups and limiting the shear flow at the steel plate–concrete interface.

Furthermore, a strong interaction between debonding due to shear forces and

debonding due to flexural forces was also found. Based on these findings, Oehlers

proposed a design procedure to prevent debonding due to peeling and suggested

this strengthening technique is better suited for RC slabs than RC beams.

Hamoush and Ahmad (1990) conducted an analytical study on the behaviour of

damaged concrete beams strengthened by externally bonded steel plates using

linear elastic fracture mechanics and the finite element method. It was found that

the failure by debonding was dependent on the stress near the interfacial crack tip

and on the critical strain energy release rate required for crack propagation.

In addition to the steel plates bonded to the tension face, vertical steel strips

can be bonded to the beam webs to enhance the shear strength of RC beams. The

main disadvantages of this method result in the need to anchor the top and bottom

of each steel plate and peeling due to the small shear-resisting area of the

individual strips. Adhikary et al. (2000) bonded continuous horizontal steel plates

to the beam web to improve the ultimate shear strength and provide additional

stiffness against bending and contribute to flexural strength too. Sharif et al. (1995)

presented test results for shear-damaged RC beams with deficient shear strength,

strengthened by externally bonded steel plates. Different arrangements of steel

plates were used in order to eliminate shear failure and develop ductile behaviour.

The strength of all repaired beams was increased and the degraded stiffness of the

beams was restored. However, the failure was abrupt due to plate separation with

the exception of beams repaired with full encasement at the shear zone. Such

jacket-type repair enhanced the shear capacity and was so effective that flexural

failure occurred.


9

2.2.2 Strengthened by adhesively bonded FRPs

High performance fibre reinforced polymers (FRP) gained widespread use as

strengthening materials for RC structures after its first utilisation in 1990s for their

unique advantages. Compared with steel material, FRP material can offer a high

strength-to-weight ratio, an excellent resistance to electrochemical corrosion,

great conformability with enhanced surface, less increase in the size of structure

member, fast execution and lower labour costs.

Experiments and retrofitting practice proved that by adhesively bonding

carbon/glass FRP plates/sheet to the tension soffit of RC beams, the flexural

strength can be significantly increased. In addition, the cracking behaviour of the

beams was improved by delaying the formation of visible cracks and reducing

crack widths at higher load levels. Teng et al. (2002; 2003) conducted a

comprehensive review of the flexural strengthening of RC beams with FRP

materials. The general way is bonding unstressed or prestressed FRP plates or

sheets to the soffit of beams and anchor plates are used to prevent anchorage

failures at the plate ends. If the ends of the plates are properly anchored, beams

fail in flexure or shear. Otherwise, several types of debonding failure modes can

be observed: (a) those associated with high interfacial stresses near the ends of the

bonded plate and (b) those induced by a flexural or flexural-shear crack

(intermediate crack) away from the plate ends.

An et al. (An et al. 1991) studied the strengthening of beams by bonding

GFRP plates to the tension flanges. The increase in the flexural strength and

improvement in the cracking behaviour as well as decrease in the ductility of the

beams were observed. Sharif et al. (1994) tested concrete beams strengthened

using different patterns of glass FRP plates to increasing the flexure strength. It

was found that among different flexure strengthening patterns, only the I-jacket

FRP plates can develop flexural strength and provide enough ductility despite the

brittleness of FRP plates. Malek et al. (1998) presents a method for calculating

shear and normal stress concentration at the cut-off point of the plate, based on

linear elastic behaviour of the materials. Etman and Beeby (2000) conducted an

experimental investigation of the bond stress along the concrete–epoxy–plate

interface. It was found that the plate breadth to thickness ratio was a significant


10

factor, which affects the bond stress concentration at the plate end. It was also

found that the plate end cut-off may affect the bond stress concentration.

Al-Sulaimani et al. (1994) tested concrete beams strengthened using different

patterns of glass FRP plates to increasing the shear strength. It was found that the

increase in shear capacity was almost identical for both strip and wing shear

repairs and not adequate to cause beams to fail in flexure, while that by U-jacket

repair was sufficient and flexural failure occurred. Grace et al. (1999) tested 14

simply supported cracked beams strengthened with carbon/glass FRP sheets and

plates. The U-shape vertical fibres around the beam cross section were found to

not only significantly reduce deflections and increase load carrying capacity, but

also eliminate the potential rupture of the longitudinal sheets. Chen and Teng

(2003a; 2003b) also developed several design proposals to deal with the shear

failures caused by FRP rupture and debonding.

Many other researchers have engaged in the development of the strengthening

of RC beams by bonding FRP plates or sheets to the tensile face (Buyukozturk et

al. 2004; Smith and Teng 2002a; Smith and Teng 2002b; Soudki and Sherwood

2000; Zhu 2006). These efforts made this strengthening technique familiar to

everyone and accepted worldwide, but the premature debonding failures of the

FRP plates occurring at or near the plate ends have always been a serious

problem.

2.2.3 Strengthened by mechanically bolted steel plates

The strengthening technique attaching steel plates on to RC beams by

adhesive generates an even stress distribution between the interface and provides

a smooth external surface, but suffers from peeling stresses and depends on the

tensile strength of the concrete near the surface. While the technique attaching

steel plates on to RC beams by anchoring bolts overcomes the problem of peeling.

Barnes and Subedi (Subedi and Baglin 1998; Barnes et al. 2001) studied the shear

strengthening of RC beams, compared the experimental results from the two

methods of plate attachment, namely adhesive bonding and bolting. All the plated

beams show increased strength and stiffness when compared with the control.


11

Among them, all the adhesive-plated beams failed in the form of progressive or

explosive peeling caused by tensile splitting of the concrete cover beneath the

steel plates, they behaved in a similar manner to the control and showed the brittle

failure associated with the tensile splitting of concrete. On the other hand, all the

bolt-plated beams failed in shear, with a diagonal crack extending from the edge

of the loading plate to the edge of the support plate, thus exhibited a more ductile

response when a large proportion of the plate became plastic.

The steel plates can be bolted to either soffit or side faces of the RC beams.

Roberts and Haji-Kazemi (1989b) conducted an experimental study on under-

reinforced RC beams strengthened by bolting thin steel plates to the tensile face.

A significant increase in both flexural strength and stiffness was achieved and the

improved performance was quantifiable by conventional calculations. Foley and

Buckhouse (1999) presented a simple method for increasing the flexural strength

and stiffness of existing RC beams by bolting structural steel U-shape channels to

the tension face utilising expansion and epoxy-adhered threaded shafts. The

sectional size of the U-shape channels was determined based on fundamentals of

RC design and the tear-off behaviour near the channel termination prior to design

load was inhibited by anchoring bolts. However, great care should be taken to

avoid drilling into the tensile rebars in the procedure of holes preparation, and the

ductility of the strengthened RC beams was severely reduced.

A great deal of efforts have been devoted to the analytical study on the partial

interaction between the RC beams/slabs and the bolted steel plates on the bottom

surfaces. Newmark et al. (1951) presented a linear elastic partial interaction

theory on composite steel and concrete T-beams based on the assumption of the

discrete shear connectors embedded in concrete as a continuous imperfect

connection exist between the steel–concrete interface. Szabo (2006) developed an

energy method using the Euler-Lagrange equation based on variational calculation

for determining the internal axial force between the steel or timber beam and the

concrete slab. Kim and Choi (2011) proposed an approximate analysis method for

a simply supported composite beam with partial interaction. The internal axial

force was approximated by Fourier series to solve the governing differential

equation in linear elastic partial interaction theory.


12

The flexural strength and ductility capacity of RC beams can be increased

significantly by mechanically bolting steel plates to their soffit, supposing that the

shearing strength of the RC beams is sufficient. Otherwise the RC beams would

fail in shear. Many researchers (Oehlers et al. 1997; Nguyen et al. 2001; Su and

Zhu 2005) therefore proposed to attach steel plates to the beam side faces using

anchor bolts. RC beams strengthened by this technique, i.e., bolted side-plated

(BSP) beams, have proved to be significantly enhanced in terms of flexural

strength without a visible decrease in the ductility.

2.3 RESEARCHES RELATED TO BSP BEAMS

Su and Zhu (2005) conducted experimental and numerical studies on

coupling beams strengthened by steel plates mechanically bolted on the vertical

faces. It was observed that the attached plates increased the ultimate capacity,

stiffness and deformability, and slightly reduced the ductility of the coupling

beams. The results revealed that the external bolted steel plates can significantly

improve the inelastic behaviour in terms of higher energy dissipation and lower

strength degradation of the coupling beams. The results were compared with those

from a nonlinear finite element analysis (NLFEA) and showed great coincidence.

2.3.1 Partial interaction between steel plates and RC beam

Distinct from RC beams strengthened by bolting steel plates to the soffit face

in which only the partial interaction caused by longitudinal slip exists, the BSP

beams are more complicated for there is the partial interaction caused by a

combination of both longitudinal and vertical slips on the plate–RC interface.

Oehlers et al. (1997) conducted theoretical studies on the vertical partial

interaction of RC beams strengthened by steel plates bolted to its web sides and

proposed a fundamental mathematical model to establish the relationship between

the degree of vertical partial interaction and the stiffness as well as plastic

deformability of the anchoring bolts utilised. The concrete and steel materials are


13

assumed to remain elastic while the bolts are assumed to be plastic thus all the

bolts are fully loaded and there is a unique shear force distribution along the

longitudinal axis on the interface. This proposed model is easy to understand and

utilise despite the unique shear distribution on the plate–RC interface is hardly

accordant with the real stress distribution. Based on this model, Nguyen et al.

(2001) derived the relationship between the vertical and longitudinal partial

interactions, which were further developed to determine the distribution of slip

strain, slip and the neutral axis separation of the steel plates and the RC beam in

terms of degrees of vertical and longitudinal interaction. The difference between

the curvatures of the steel plates and the RC beam was neglected in the calculation

of the neutral axis separation.

Su and Zhu (2005) conducted experimental and numerical studies on BSP

coupling beams and showed that small slips on the plate–RC interface could

significantly affect the overall response of BSP beams. Siu and Su conducted

comprehensive experimental, numerical and theoretical studies on the behaviour

of BSP beams. They proposed some numerical procedures for predicting the

nonlinear load–deformation response of bolt groups (Su and Siu 2007; Siu and Su

2009) along with the longitudinal and transverse slip profiles of BSP beams under

symmetrical loading conditions such as four-point bending and uniformly

distributed load (UDL) (Siu 2009; Siu and Su 2011). Their predicted longitudinal

slips were in good agreement with the test results obtained at some discrete

locations on the beams (Siu and Su 2010), despite the complete longitudinal and

transverse slip profiles along the beam span were not measured.

2.3.2 Buckling of deep steel plates

Besides the partial interaction caused by the longitudinal and transverse slips

on the plate–RC interface, the behaviour of BSP beams is also controlled by the

buckling which might exist in the compressive region of the strengthening steel

plates, because the steel plates are only constrained at discrete point. This

detrimental effect is especially serious for the beams strengthened by very deep

steel plates, for a very large portion of the plates is in the compressive region.


14

Smith et al. (Smith and Bradford 1999a; Bradford et al. 2000) conducted a

comprehensive theoretical study on the buckling problem of BSP beams. This

problem was treated as a contact problem and simplified as a unilateral local

buckling of steel plates restrained at discrete boundary points. The steel plate was

discretised into rectangular grids and the point restraints and free edges were

simplified by certain boundary conditions. The Rayleigh-Ritz method with a

nonlinear elastic foundation that exhibits sign-dependent foundation stiffness was

employed to consider the plate buckling towards or away from the RC beam. A

so-called local buckling push test was also undertaken on bolted plates of various

configurations (Smith and Bradford 1999b; Smith et al. 2001), in which the

strengthening steel plate was divided into several portions isolated from one

another by a group of anchor bolts. Within each loading run, each portion was

subjected to a unique combination of in-plane axial, bending and shear plate

actions. The analytically proposed expression for local buckling study was

verified by the testing results and can be used in design practice as a guideline for

bolt arrangement to prevent local plate buckling.

Cheng and Su (2011) improved the shear strengthening method for coupling

beams by introducing a buckling restraint device to the steel plates bolted on the

vertical faces. The experimental study revealed that the deformation and energy

dissipation of the deep RC coupling beams retrofitted with restrained steel plates

improved while the flexural stiffness did not increase. Moreover, by using

laterally restrained steel plates, the specimens had better post-peak behaviour, a

more ductile failure mode, and better rotation deformability.

2.3.3 Moderately reinforced BSP beams

The structural behaviours of RC beams are controlled by the tensile steel

ratios and can be classified by the balanced steel ratio ρstb, at which the yielding of

the outermost tensile-reinforcement-layer and the crushing of concrete occur

simultaneously. If an RC beam is lightly reinforced with a tension steel ratio of

ρst << ρstb, it will fail in a ductile mode, and both its strength and stiffness can be

increased significantly by external reinforcement with a small sacrifice of ductility.


15

In contrast, if an RC beam is over-reinforced with ρst > ρstb, its strength and

stiffness are controlled by the compressive strength of the concrete rather than the

strength of the tensile reinforcement, and adding external tensile reinforcement

will cause the beam to fail in a brittle mode with very little ductility. It is noted

that over-reinforced RC beams are forbidden for use in structural design, and a

strengthening design for this type of beam is rarely needed. However, there are a

large number of moderately reinforced RC beams in existing buildings whose

tensile steel ratios are lower than but very close to the balanced steel ratio ρstb.

Most of the available strengthening techniques up to now have focused on the

lightly reinforced RC beams. Roberts and Hajikazemi (1989b) proposed a method

to strengthen under-reinforced RC beams with a ρstb of 1.21% by bolting steel

plates to the beam soffit, Foley et al. (1999) proposed a technique by bolting steel

channels to the tension face of the lightly reinforced RC beams with a ρst of 0.54%,

Ruiz et al. (1999) studied the size effect and bond–slip dependence of lightly

reinforced RC beams with a ρst less than 0.3%, and Siu and Su (2011) studied the

partial interaction of lightly reinforced BSP beams with a ρst of 0.85%.

Although the BSP retrofitting technique is particularly suitable for the

strengthening of these moderately reinforced RC beams, rigorous studies on the

behaviour of the moderately reinforced BSP beams is still outstanding.

2.3.4 Other issues related to BSP beams

Although both flexural strength and deformability of RC beams can be

enhanced significantly by the side-bolted steel plates, these exposed steel plates

are liable to corrosion and fire, thus the durability and the range of usage of BSP

beams are limited. Galvanization (Dreulle 1980), which has been widely used in

steel structures, could increase the resistance to corrosion of steel plates, thus

enhance the durability of BSP beams. However, fire retardant coatings (Li and

Qin 1999) should be used to increase the fire resistance of the steel plates, thus

retain the loading capacities of BSP beams at a high level even in the elevated

ambient temperature.


16

Compared with the strengthening techniques bonding steel plates or FRPs to

RC beams, which can provide a smooth external surface and a continuous shear

stress transfer on the interface of the two components, the shear force is

transferred by discrete anchor bolts in BSP beams. Therefore, considerable stress

concentration may exist in both the steel plates and the existing concrete. The

protruded anchor bolts might cause an esthetical problem as well.

2.4 CONCLUSIONS

The BSP beams are undeniably accompanied by shortcomings such as the

partial interaction caused by both the longitudinal and transverse slips, the local

plate buckling, the cost of time and labour in the fabrication of bolt holes, the

damage to the existing concrete and the aesthetic problems caused by the

protrusive anchor bolts. However, compared to the RC beams strengthened by

other conventional strengthening techniques, the BSP beams have proved to be

immune to the premature debonding failures and possess both enhanced flexural

strength and stiffness without a visible reduction in ductility. These features make

the BSP retrofitting technique especially attractive for the strengthening of the

moderately reinforced RC beams.

Comprehensive theoretical and experimental efforts have devoted to the

behaviour of BSP beams and it was illustrated that the partial interaction between

the steel plates and the RC beam, which is a result of the longitudinal and

transverse slips caused by the shear transfers, controls the performance of the BSP

beams. However, the complete longitudinal and transverse slip profiles along the

entire beam span have yet to be measured. Although limited analysis methods

have been developed, the requirement of symmetrical loading conditions in the

analysis of the longitudinal slip, along with the linear profile model in the analysis

of the transverse slip, limited the application of these theoretical approaches to

study the partial interaction of BSP beams. In addition, most of the available

strengthening techniques up to now have focused on the behaviour of the lightly

reinforced RC beams, the performance of the moderately reinforced BSP beams

have yet to be studied comprehensively.

Chapter 3 Experimental Study on BSP Beams

17

CHAPTER 3

EXPERIMENTAL STUDY ON BSP BEAMS

3.1 OVERVIEW

Although there have been experiments conducted to investigate the behaviour

of BSP beams, most of these studies focused on the overall load–deflection

performance. The partial interaction existing on the interface between the steel

plates and the RC beam, which is the result of the longitudinal and transverse slips

caused by the shear transfers, has yet to be assessed. The profiles of the

longitudinal and transverse slips along the whole beam span have yet to be

measured and their internal mechanism is still unknown. In addition, the plate

buckling, which might occur in the compressive regions of the steel plates, should

be studied and restrained by some appropriate measures.

The structural behaviours of RC beams are controlled by the cross-sectional

tensile steel ratio. However, almost all previous researches corresponding to BSP

beams focused on the RC beams that are lightly reinforced. Since the moderately

reinforced RC beams represent a major portion of the existing building stock, a

comprehensive experimental study on the behaviour of the moderately reinforced

BSP beams is of practical interest.

Aiming at a better understanding of the behaviour of moderately reinforced

BSP beams, especially the effect of the partial interaction on the plate–RC

interface, an experimental study was designed. It included seven RC beams with

different sectional properties and strengthening arrangements. Four-point bending

was employed to study the bending performance of the specimens with or without

the influence of shear. The behaviours of load–deflection, failure mode, flexural

strength, stiffness, ductility, roughness, longitudinal and transverse slips as well as

the flexural and tensile contribution of the steel plates were investigated in detail.


18

3.2 SPECIMEN PREPARATION

3.2.1 Specimen details

Seven full-scale RC beams with the same properties but different tensile steel

ratios were fabricated. The length of the beams was 4000 mm, and the cross

section was 225 mm (breadth) × 350 mm (depth). The reinforcement details of the

specimens are shown in Figure 3.1. The high-yield steel deformed bars and the

mild steel round bars are chosen for the longitudinal and transverse

reinforcements and denoted with ‘T’ and ‘R’ respectively. Compressive

reinforcement of 2T10 was used to facilitate the fabrication of the reinforcement

cages. Transverse reinforcement of R10-100 was employed to insure that no

premature shear failure would occur before the peak loads were achieved. Tensile

reinforcement of 3T16 was chosen for Specimen P75B300, and 6T16 was used

for the rest of the specimens. The corresponding tensile reinforcement ratios were

0.85% and 1.77%, respectively.

A control RC beam, namely CONTROL, without any retrofitting measures

was used as a reference to demonstrate the beam performance before

strengthening. The other specimens were strengthened with two steel plates

anchored to their side faces and were named according to the design parameters,

such as the depth of steel plate and the horizontal bolt spacing, which have

primary effects on strengthening. Table 3.1 summarises the names of the

specimens and the design parameters of the steel plate, anchor bolt and buckling

restraint arrangements for all the specimens. And Figures 3.2 and 3.3 show the

section views and the elevations of the steel plate and anchor bolt arrangements

for all the specimens. Steel plates with a thickness of 6 mm and a length of

3950 mm were used for all BSP beams. Three different plate depths, 75 mm,

100 mm and 250 mm, were chosen to yield distinct strengthening effects.

For Specimen P75B300, two steel plates with a depth of 75 mm were fixed

onto the side faces by ten bolts located at the centroidal level of the plates with a

horizontal bolt spacing of 300 mm. All bolts were assigned to the shear span, and

none were located in the pure bending zone.


19

For Specimens P100B300 and P100B450, two shallow steel plates with a

depth of 100 mm were installed by a row of anchor bolts with a uniform spacing

of 300 mm and 450 mm, respectively.

For Specimens P250B300, P250B300R and P250B450R, two deep steel

plates with a depth of 250 mm were fixed by two rows of anchor bolts with a

horizontal spacing of either 300 mm or 450 mm. To study the influence of plate

buckling, which might occur in the compressive zones of the steel plates, buckling

restraint devices were introduced to Specimens P250B300R and P250B450R but

not to Specimen P250B300.

3.2.2 RC beam fabrication

All the RC beams were cast in a same wooden formwork to insure identical

dimensions. The reinforcement cage was first fabricated and the strain gauges

were also attached at the designated position on the compressive and tensile rebars

before the reinforcement cages were placed at the required position in the

formwork, as shown in Figure 3.4. The spacing of the drilled holes for the

installation of the anchor bolts and that of the transverse stirrups were carefully

designed, the position of one or two stirrups were adjusted slightly to avoid

damage to the stirrups in the fabrication of the holes with a rotary hammer. The

reinforcement cages were also fixed firmly in the formwork to prevent dislocation

during the concrete casting. The specimens were then cast and left for at least

three weeks of curing before the strengthening procedures were undertaken.

3.2.3 Strengthening procedure

The anchor bolt installation followed the instructions in the technology

manual provided by Hilti Corporation (2011). Strengthening measures were

conducted three weeks after the RC beams were cast. Drilled holes with a

diameter of 12 mm and a depth of 105 mm were formed using a rotary hammer on

the side faces and cleaned thoroughly. HIT-RE 500 adhesive mortar was then


20

injected into the holes, and HAS-E anchor shafts with a diameter of 10 mm were

turned into the mortar until they reached the required depth of 95 mm. Then the

specimen was isolated for a minimum of 24 hours for the curing of the adhesive

mortar to achieve the designed strength.

Drilled holes with a diameter of 12 mm were also formed in the steel plates.

After the adhesive mortar in the RC beams was cured, the steel plates were fixed

to the side faces of the beam by the dynamic sets. The HIT-RE 500 adhesive

mortar was also injected into the gaps between the anchor bolt shafts and the steel

plates using dynamic sets for all specimens except Specimen P75B300 to study

the effects of slips at the shaft–plate gaps. The newly injected adhesive mortar

was also left for curing at least 24 hours before the specimen was put to test. A

dynamic set, as shown in Figure 3.5, was composed of an injection washer used to

inject adhesive mortar, a spherical washer designed to prevent the mortar from

leaking and an ordinary nut to fix the steel plates and the washers on the concrete

surface.

The buckling restraint device shown in Figure 3.6 was composed of steel

angles L63 × 5 mm, which were used to prevent the steel plates from buckling.

Steel plates with a thickness of 10 mm were installed at the top row of anchor

bolts to fix the steel angles. To avoid adding extra strength and stiffness to the

BSP beams, discrete short steel angles were employed and connected to the thick

steel plates by bolt connections with slotted holes, which allow the steel angles to

rotate and translate in the longitudinal direction. The interface between the steel

angles and the thick steel plates was carefully sanded and lubricated to reduce

friction.

3.3 MATERIAL PROPERTIES

3.3.1 Concrete

The concrete mix proportion adopted in this study is tabulated in Table 3.2.

The mix used 10 mm coarse aggregate with a water-to-cement ratio of 0.72, an

aggregate-to-cement ratio of 6.68 by weight, and a measured slump of 50 mm. For


21

each specimen, four 150 mm × 150 mm × 150 mm concrete cubes and four

Ø150 mm × 300 mm cylinders were cast, and compressive tests were performed

on the test day to obtain the compressive strengths, which are listed in Table 3.3.

3.3.2 Reinforcing bars

High-yield steel deformed bars (T) were used for compressive and tensile

reinforcement while mild steel round bars (R) were used for transverse

reinforcement. Three bar samples with a length of 500 mm were taken from each

type of reinforcement for tensile tests to obtain the yield strength and Young’s

modulus. The measured stress–strain relationships of the reinforcement bars are

illustrated in Figure 3.7. The material properties, i.e., the yield strengths and

elastic moduli, are tabulated in Table 3.4.

3.3.3 Steel plates

The side plates were made of mild steel. Three 500 mm × 50 mm strips were

used for tensile tests to determine the yield strength and Young’s modulus of the

steel plates. The measured stress–strain relationships of the steel plates are

illustrated in Figure 3.8. The material properties, i.e., the yield strengths and

elastic moduli, are tabulated in Table 3.5.

3.3.4 Bolt connection

The “HIT-RE 500 + HAS-E” chemical anchoring system (Hilti 2011), which

was provided by Hilti Corporation, was chosen as the connecting medium

between the steel plates and the RC beams. The HAS-E anchor shafts were

Grade 5.8 and covered by a galvanised surface with a thickness of at least 5 µm.

The HIT-RE 500 adhesive mortar was a two-component, ready mix epoxy resin,

and its working and curing time were 30 minutes and 12 hours, respectively, at a

base-material temperature of 20 °C.


22

To determine the shear force–slip response of the “HIT-RE 500 + HAS-E”

anchoring system, three RC blocks with the same sectional properties as the RC

beams and with a length of 200 mm were cast and fabricated. Holes were drilled,

and bolts were planted with HIT-RE adhesive mortar following the

aforementioned procedure. A specifically designed transfer plate, as shown in

Figure 3.9, was used to conduct compression shear tests on the anchor shafts. The

samples were loaded using a hydraulic jack, and a monotonic displacement

controlled load was applied to the transfer plate. The two strengthened steel plates,

which simulated the bolted-side plates, transferred the compression force to shear

forces and applied them to the two anchor bolts. A photograph of the test set-up is

also shown in Figure 3.10. The load increased at a rate of 0.01 mm/sec and

terminated when either bolt failed. Figure 3.11 shows the shear force–slip

responses. The peak bolt shear force was 53 kN, and the slip at the peak force was

4 mm. The secant modulus at 25% of the peak shear force, which could be chosen

to represent the initial elastic stiffness, was 112 kN/mm.

3.4 TEST PROCEDURE

3.4.1 Test set-up

The experiments were conducted in a test frame in the Structural Engineering

Laboratory at The University of Hong Kong. The clear span between the two

roller supports, which were bolted to the strong floor, was 3600 mm. A monotonic

load provided by a 500 kN capacity hydraulic jack was equally divided into two

concentrated forces by a steel transfer beam and applied at the two trisectional

points of the specimen under test. Hence, a pure bending zone with a length of

1200 mm was generated in the middle part of the specimen. The design diagram

and the actual setup arrangements for all the specimens are illustrated in Figures

3.12 and 3.13.


23

3.4.2 Instrumentation

The longitudinal tensile and compressive strains in the reinforcement and

steel plates were measured by strain gauges. The shear strains in the steel plates

were determined by rosette strain gauges. The arrangement of strain gauges is

shown in Figure 3.14(a).

To measure the deformation of the specimen under testing, LDTs were

employed to measure the vertical deflections at several sections along the

specimen; four LVDTs were also designed to determine the rotations at both

supports, as shown in Figure 3.14(b).

The rhombic set of LVDTs proposed by Siu (2009) was firstly employed in

the measuring of the longitudinal and transverse slips in Specimen P75B300 (see

Figure 3.13(b)), but the accuracy was unfortunately inadequate. Therefore, a new

slip measuring device was tailor-made, as shown in Figures 3.15 and 3.16. This

device was composed of aluminium angles, plates and bolting connectors. It

included two sets: Set A was embedded into the RC beam through two expansion

bolts, where one was located in the compressive region of the side face and the

other was in the beam soffit, and Set B was fixed onto and moved with the steel

plate when relative slips occurred. Three LVDTs were installed on Set A. One set

was in the transverse direction with the probe tip in contact with the lower edge of

the steel plate, and the other two were in the longitudinal direction with the probe

tips pointing at the upper and lower sides of Set B. Hence, if slips occurred, the

first LVDT measured the transverse slip, and the other two recorded the

longitudinal slips.

3.4.3 Loading history

To investigate the load–deflection behaviour, especially in the post-peak

region, a displacement controlled loading process was designed for the specimens

in this study. The loading rate was chosen to be 0.01 mm/sec up to 50% of the

theoretical peak load. Then, it was increased to 0.02 mm/sec until the post-peak

load decreased to 80% of the actual peak load, and the test was terminated.


24

3.5 CONCLUSIONS

In this chapter, the detail of the experimental study on BSP beams was

reported. The beam geometry, especially the tensile reinforcement ratio was

chosen to be on the lower side but close to the balanced steel ratio to cover the

majority of RC beams existing in the building stock. The steel plate and anchor

bolt arrangements were designed in a way that they were not only feasible for

retrofitting operation, but also within the practical range of the major parameters

of BSP beams, such as the depth of steel plates, the longitudinal spacing and the

number of rows of anchor bolts. This study focused on the partial interaction

between the steel plates and the RC beams, and due consideration was taken to

precisely quantify the longitudinal and transverse slips and the shear force–slip

relationship of the anchor bolts. Hence, a new measuring device for the calibration

of the longitudinal and transverse slips, along with a new force transfer device for

the bolt test, was designed for the purpose.


25

Table 3.1 Beam geometries and strengthening details

Specimen ρst

(%)

Dp

(mm)

Sb

(mm)

Rows of

bolts

Midspan

bolts

Adhesive in

shaft–plate gaps

Buckling

restraint

CONTROL 1.77 - - - - - -

P75B300 0.85 75 300 1 None None No

P100B300 1.77 100 300 1 Yes Yes No

P100B450 1.77 100 450 1 Yes Yes No

P250B300 1.77 250 300 2 Yes Yes No

P250B300R 1.77 250 300 2 Yes Yes Yes

P250B450R 1.77 250 450 2 Yes Yes Yes

Table 3.2 Concrete mix proportioning

Water Cement w/c

Fine

aggregate

Coarse

aggregate

Maximum

aggregate size Slump

(kg/m3) (kg/m

3) (kg/m

3) (kg/m

3) (mm) (mm)

200 279 0.72 1025 838 10 50

Table 3.3 Cube and cylinder compressive strengths of concrete

Specimen Cube strengths of different samples (MPa) fcu

(MPa)

Standard

derivation

(%) Sample 1 Sample 2 Sample 3 Sample 4

CONTROL 37.3 36.0 38.5 - 37.3 7.6

P75B300 39.0 40.9 40.0 39.0 39.7 4.7

P100B300 32.9 34.7 34.1 33.8 33.9 2.2

P100B450 41.1 41.7 40.8 39.5 40.8 2.3

P250B300 37.3 40.5 34.4 31.8 36.0 10.4

P250B300R 36.5 37.5 34.1 35.2 35.8 4.1

P250B450R 37.4 37.3 38.2 37.7 37.7 1.1

Specimen Cylinder strengths of different samples (MPa) fco

(MPa)

Standard

derivation

(%) Sample 1 Sample 2 Sample 3 Sample 4

CONTROL 30.7 32.5 35.2 - 32.8 6.9

P75B300 32.4 35.5 34.6 33.2 33.9 4.2

P100B300 29.7 29.2 29.1 27.5 28.9 3.3

P100B450 31.3 34.6 33.3 33.7 33.2 4.2

P250B300 30.0 32.7 30.8 25.4 29.7 10.4

P250B300R 24.2 27.1 26.9 28.0 26.6 6.2

P250B450R 26.0 27.3 28.2 26.4 27.0 3.6


26

Table 3.4 Strengths and moduli of reinforcement bars

Specimen Yield strengths of different samples (MPa) fy

(MPa)

Standard

derivation

(%) Sample 1 Sample 2 Sample 3

T10 493.8 497.7 511.0 500.8 1.5

T16 520.8 522.1 521.7 521.6 0.1

R10 299.2 297.2 297.5 298.0 0.3

Specimen Ultimate strengths of different samples (MPa) fu

(MPa)

Standard

derivation


T10 627.8 628.3 650.8 635.6 1.7

T16 628.0 628.0 626.8 627.6 0.1

R10 375.0 372.6 373.9 373.8 0.3

Specimen Elastic moduli of different samples (GPa) Es

(MPa)

Standard

derivation


T10 198.2 217.6 219.0 211.5 4.5

T16 200.8 200.7 200.3 200.6 0.1

R10 198.0 197.0 199.1 198.0 0.4

Table 3.5 Strengths and moduli of steel plates

Thickness

(mm)

Yield strengths of different samples (MPa) fyp

(MPa)

Standard

derivation


6 337.6 313.5 330.2 327.1 3.1

Thickness

(mm)

Ultimate strengths of different samples (MPa) fup

(MPa)

Standard

derivation


6 460.4 460.0 455.8 458.7 0.6

Thickness

(mm)

Elastic moduli of different samples (GPa) Ep

(GPa)

Standard

derivation


6 225.5 210.7 220.0 218.7 2.8


27

Figure 3.1 Cross section of specimens (a) lightly reinforced (P75B300) and (b)

moderately reinforced (the other specimens). (dimensions in mm)

Figure 3.2 Configurations of strengthening measures (section view) for

Specimens (a) P75B300, (b) P100B300 & P100B450, (c) P250B300, and (d)

P250B300R & P250B450R. (dimensions in mm)

(c)

50

2

50

5

0

50

1

50

5

0

(b)

50

100

200

50

50

63

75

2

13

38 3

8

(a)

(d)

Buckling

restraint

device

225

2T10

R10-100

6T16

35

0

(b)

(a)

35

0

225

R10-100

3T16

2T10


28

Figure 3.3 Configurations of strengthening measures (front view) for

Specimens (a) P75B300, (b) P100B300, (c) P100B450, (d) P250B300, (e)

P250B300R and (f) P250B450R. (dimensions in mm)

(a)

(d)

(c)

(e)

(b)

(f)

450

300

300

300

450

300


29

Figure 3.4 Reinforcement cages


30

Figure 3.5 Details and installation of dynamic sets; (a) injection washer, (b)

installation drawing and (c) actual installation

(b)

Adhesive mortar

(except P75B300)

Nut

Spherical washer

Injection washer

Steel plate

Anchor rod Concrete

Adhesive mortar

(c)

Top view Bottom view

(a)


31

Figure 3.6 Details and installation of buckling restraint devices; (a) design

diagram and (b) actual installation

(a)

Concrete

Steel angle

Steel plate

Dynamic set

Thick steel plate

Steel angle

Concrete

Top view

Front view

(b)

Top view

Front view


32

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0

100

200

300

400

500

600

700

800

T10

Sample 1

Sample 2

Sample 3

Str

ess

(MP

a)

Strain

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

0

100

200

300

400

500

600

700

800

T16

Sample 1

Sample 2

Sample 3

Str

ess

(MP

a)

Strain

(a)

(b)


33

Figure 3.7 Measured stress–strain relationships of reinforcement (a) T10, (b)

T16 and (c) R10

Figure 3.8 Measured stress–strain relations of steel plates

0.00 0.05 0.10 0.15 0.20 0.25

0

100

200

300

400

500

Sample 1

Sample 2

Sample 3

Str

ess

(MP

a)

Strain

Steel plate

0.00 0.05 0.10 0.15 0.20 0.25

0

100

200

300

400

500

Sample 1

Sample 2

Sample 3

Str

ess

(MP

a)

Strain

R10

(c)


34

Figure 3.9 Design diagram of bolt test set-up for the “HIT-RE 500 + HAS-E”

anchoring system (dimensions in mm)

Set A

Transfer plate

LVDT

Steel plate

Steel angle

350

20

0

5

0

Set B

1

1

1-1

225

Anchor rod

Set A

Transfer plate

LVDT

Steel plate

Steel angle

Set B


35

Figure 3.10 Actual installation of bolt test set-up for the

“HIT-RE 500 + HAS-E” anchoring system

Figure 3.11 Shear force–slip curves of the “HIT-RE 500 + HAS-E” anchoring

system

0 1 2 3 4 5 6

0

10

20

30

40

50

60

Sample 1

Sample 2

Sample 3

Shea

r fo

rce

(kN

)

Slip (mm)

Initial elastic stiffness


36

Figure 3.12 Design diagram of test setup (dimensions in mm)

(a) CONTROL

500-kN hydraulic jack

Steel transfer beam

200 1200 1200 1200 200

4000


37

(c) P100B300

(b) P75B300


38

(e) P250B300R

(d) P100B450


39

Figure 3.13 Experimental set-up for Specimen (a) CONTROL, (b) P75B300, (c)

P100B300, (d) P100B450, (e) P250B300R, (f) P250B450R and (g) P250B300

(g) P250B300

(f) P250B450R


40

Figure 3.14 Arrangements of (a) strain gauges and (b) LVDTs & LDTs

(dimensions in mm)

(b)

(a)

1-1 2-2

Strain gauge LVDT for rotation

LVDT for longitudinal slip

LVDT for transverse slip

LDT for deflection

150 300 300 300 300 300 150

Strain gauge Rosette type of strain gauge

1

1

600 600 600 120

20

0

LVDT & LDT

2

2


41

Figure 3.15 Design diagram of LVDT sets for the measurement of longitudinal

and transverse slips

Top view

Set A

Set B

LVDT

Set B

Set A

LVDT

Front view Side view


42

Figure 3.16 Actual arrangement of LVDT sets for the measurement of

longitudinal and transverse slips

Side view

Upward view

Chapter 4 Result and Analysis of Experimental Study on BSP Beams

43

CHAPTER 4

RESULT AND ANALYSIS OF EXPERIMENTAL

STUDY ON BSP BEAMS

4.1 OVERVIEW

The results of the experiments on BSP beams described in Chapter 3 are

reported and analysed in this chapter. The overall behaviours of the specimens,

such as the failure mode, the enhancement of strength and stiffness, and the

variation in ductility and toughness are presented. The profiles of the longitudinal

and the transverse slips along the beam span, together with their development are

discussed in detail. The partial interaction on the plate–RC interface and the

indicators used to denote the degree of partial interaction, i.e., the strain and the

curvature factors, are also studied. The behaviour and moment contribution of the

steel plates are discussed as well.

To show the difference in responses between the lightly and the moderately

reinforced BSP beams, the results of tests on three lightly reinforced BSP beams

conducted by Siu (2009), CONTROL*, P75B300*, P150B400*, were also

extracted for comparison. A complete key parameter comparison for all the BSP

beams is listed in Table 4.1.

4.2 FAILURE MODE

The macroscopic failure modes of RC beams can be categorised as two

primary types: (1) Flexural failure preceded by the yielding of the tensile

reinforcement, which is common in under-reinforced beams; (2) Brittle failure

caused by crushing of the concrete, which occurs in over-reinforced beams. For

BSP beams, two more failure modes can be found: (3) Flexural failure proceeded


44

by the yielding of the tensile regions of the steel plates; (4) Brittle failure

attributed to the buckling of the compressive regions of the steel plates.

The microscopic phenomena that initiate the corresponding macroscopic

failure modes can be described, respectively, as follows: (1) the strain of the

outermost tensile-reinforcement-layer reaches its yield strain εst > εy ; (2) the

maximum compressive strain of the concrete exceeds its crushing strain εcc > εcu ;

(3) the maximum tensile strain at the bottom edge of the steel plates reaches its

yield strain εpb > εyp ; and (4) the maximum compressive strain at the top edge of

the steel plates decreases suddenly Δεpt < 0 .

To classify the failure modes of the specimens, the orders of occurrence of

these microscopic phenomena with respect to load levels F/Fp are computed and

tabulated in Table 4.2. The load–deflection curves at the midspan of the

specimens are also shown in Figures 4.1 ~ 4.4.

The failure of Specimen CONTROL for the moderately reinforced reference

beam was initiated by the yielding of the tensile reinforcement (at F/Fp = 0.91)

and followed closely by the crushing of the concrete (at F/Fp = 0.94). Figure 4.1

shows that the beam failed in a flexural mode, but its ductility was lower than the

lightly reinforced reference beam CONTROL* (Siu 2009) due to the use of more

tensile steel.

Figure 4.2 shows that the lightly reinforced BSP beams P75B300* and

P150B400* failed in very brittle modes compared to CONTROL* (Siu 2009).

Because there were no anchor bolts assigned to the pure bending zones of these

beams, enormous transverse slips occurred after the formation of plastic hinges, as

shown in Figure 4.5. Hence, the effective lever arms provided by the steel plates

were seriously reduced, and the load-carrying capacities and stiffnesses decreased

rapidly in the post-peak region producing the steep descending branches. In

contrast to the RC beams with steel plates on the beam soffits, for which plate-end

anchor bolts are sufficient, the BSP beams require a uniform distribution of

anchor bolts over the entire span.

Specimen P75B300 did not suffer from this detrimental effect and behaved in

a more ductile manner than its counterpart P75B300*. Its failure was caused by


45

the yielding of the tensile reinforcement (at F/Fp = 0.77) because the gaps

between the bolt shafts and the steel plates of P75B300 were not filled with

adhesive mortar. The slips between the bolt shafts and steel plates weakened the

connection stiffness and hence the strength contribution from the steel plates and

caused substantial reductions of the degree of reinforcement and the flexural

strength of the beam.

The failure of both P100B300 and P100B450 was caused by the crushing of

the concrete (at F/Fp = 0.78 and 0.80, respectively). Figure 4.3 shows that their

descending branches are shorter and steeper compared to that of Specimen

CONTROL. The reason is that the shallow steel plates attached to the tensile

region of the RC beams acted as additional tensile reinforcement, which caused

over-reinforcement and brittle failure. It is evident from Figures 4.6(a) and (b) that

a large portion of concrete was crushed when the steel plates were only slightly

deformed for both specimens. These phenomena reveal that attaching shallow

steel plates to the beam soffit or the tensile regions at the side faces of the beam is

not suitable for moderately reinforced RC beams.

In contrast, the steel plates in P250B300R and P250B450R yielded in tension

at a very early loading stage (at F/Fp = 0.44 and 0.29, respectively). Thus, the

strength contributions from the steel plates were significant, and if thicker steel

plates were used, the strengths of these specimens could increase. The yielding of

the tensile reinforcement occurred relatively late (both at F/Fp = 0.83) and was

followed by the crushing of the concrete (at F/Fp = 0.84 and 0.89, respectively),

mainly at the concrete covers, as shown in Figures 4.6(c) and (d). These two

specimens failed in flexural modes with very high strengths and deformations.

The comparison of their load–deflection curves is presented in Figure 4.3.

The performances of Specimens P250B300 and P250B300R were very

similar at the early loading stages, as shown in Figure 4.4. The steel plates of

P250B300 yielded in tension at a very early loading stage (when F/Fp = 0.26).

The crushing of the concrete (at F/Fp = 0.85) occurred prior to the yielding of the

tensile reinforcement (at F/Fp = 0.88). Subsequently, serious buckling occurred on

the compressive edges of the steel plates (see Figure 4.7) before reaching the peak

load. The compressive region of the steel plates lost its strength, and the specimen


46

behaved as an over-reinforced RC beam with shallow steel plates attached to its

tensile region. The beam then failed rapidly.

4.3 STRENGTH, STIFFNESS AND DUCTILITY

RC beams in a building are expected to have sufficient strength and stiffness

within the intended design life and deform significantly before failure under

extreme loads. To compare the strength, stiffness and ductility of the lightly (Siu

2009) and moderately reinforced BSP beams, an equivalent elasto-plastic system,

as shown in Figure 4.8, is used to represent the simplified load–deflection curves

of the specimens. The peak load Fp is chosen as the yield strength. A line starting

from the origin, crossing the point on the ascending branch at the load level of

0.75 and terminated at the peak load is defined as the elastic branch, and its slope

represents the stiffness of the beam Ke. A horizontal line with a capacity equal to

Fp is the plastic branch. The point on the descending branch, where the load is

equal to 0.8Fp, is chosen as the end of the plastic branch. Ductility can be

quantified by the modulus of toughness Ut, where Ut is the area under the entire

load–deflection curve, which represents the amount of energy absorbed before

failure.

The primary parameters (Fp, Ke, and Ut) that indicate the overall behaviours

(strength, stiffness, and ductility) of the lightly and moderately reinforced BSP

beams are tabulated and compared with the corresponding reference RC beams

(CONTROL* and CONTROL, respectively) in Table 4.3. The numbers preceding

the parentheses are the absolute values of the parameters, while those inside the

parentheses are the ratios of the parameters relative to those of the corresponding

reference beam.

4.3.1 Strength and stiffness

Table 4.3 shows that the strength and stiffness improvements (1.43 and 1.15,

respectively) of P75B300* are higher than those (1.18 and 1.04, respectively) of


47

P100B300, and the improvements (1.59 and 1.34, respectively) of P150B400* are

also higher than those (1.43 and 1.26, respectively) of P250B300R. Therefore, the

improvements in terms of the strengths and stiffnesses of all the lightly reinforced

BSP beams are higher than those of the moderately reinforced BSP beams, even

with shallower steel plates and fewer anchor bolts. This result shows that it is

more difficult to enhance RC beams with a higher degree of reinforcement.

However, for the lightly reinforced Specimen P75B300 in this study, these

improvements are much lower than those of its counterpart P75B300* due to the

delayed response of the steel plates caused by slips at the shaft–plate gaps.

Among the moderately reinforced specimens, the improvements in terms of

the strength and stiffness (1.43 and 1.26, respectively) of P250B300R with a plate

depth of 250 mm are much higher than those (1.18 and 1.04, respectively) of

P100B300 with a plate depth of 100 mm. In addition, the improvements (1.43 and

1.26, respectively) of P250B300R with a bolt spacing of 300 mm are nearly the

same as those (1.41 and 1.27, respectively) of P250B450R with a bolt spacing of

450 mm. Thus, the strength and stiffness improvements increase significantly with

the depth of the steel plates but not the bolt spacing. Furthermore, the

improvements (1.18 and 1.04, respectively) of P100B300 are even slightly lower

than those (1.22 and 1.16, respectively) of P100B450 because these two

specimens were over-reinforced by shallow steel plates. The failure was due to the

concrete crushing, and their strengths were controlled by the concrete strength.

Specimen P100B300 had the lowest concrete cube strength (see Table 4.1), which

resulted in the lowest strength and stiffness among all the moderately reinforced

specimens.

The strength improvement was increased from 1.34 for Specimen P250B300

without plate buckling restraint to 1.43 for Specimen P250B300R with buckling

restraint devices. However, the stiffness improvements of these two specimens

(1.26 and 1.27, respectively) are almost the same. Hence, the improvement due to

the use of buckling restraint devices is significant for the beam strength but not for

the stiffness. The reason is that plate buckling usually occurs just before reaching

the peak load. It does not affect the stiffness, which is mainly controlled by the

elastic behaviour at the initial loading stages.


48

4.3.2 Ductility and toughness

As mentioned earlier, the modulus of toughness (Ut) represents the amount of

energy absorbed before the failure of a beam and can be used to measure the

ductility. Table 4.3 shows that due to the use of uniformly distributed anchor bolts

along the entire steel plates, the modulus of toughness of all the moderately

reinforced BSP beams is higher than that of the lightly reinforced BSP beams. For

example, the ratio of Ut is 0.80 for Specimen P100B300, which is higher than that

of 0.64 for Specimen P75B300*, and the value 1.48 of P250B300R is much

higher than the value 0.67 of P150B400*. Due to the slips at the shaft–plate gaps

for Specimen P75B300, its modulus of toughness ratio 1.39 is much higher than

the value 0.64 of its counterpart (Specimen P75B300*).

Among the moderately reinforced BSP beams, the modulus of toughness

ratios for specimens with shallow steel plates (Specimens P100B300 and

P100B450) are reduced (0.80 and 0.89, respectively) due to the increase in the

degrees of reinforcement. Specimen P100B300 had a very low ratio due to the

low concrete strength and hence a high degree of reinforcement. On the other

hand, the ratios of Ut of the plate buckling restrained Specimens P250B300R and

P250B450R are enhanced significantly (1.48 and 1.37, respectively) because the

compressive zone of the steel plates significantly reduced the degrees of

reinforcement. However, when plate buckling was not restrained, the ratio of Ut

dropped from 1.48 for Specimen P250B300R with buckling restraints to 0.66 for

Specimen P250B300 without buckling restraints.

4.4 LONGITUDINAL AND TRANSVERSE SLIPS

The longitudinal and transverse slips are attributed to the looseness of the

axial strain or the curvature of the steel plates, therefore controlling the degree of

partial interaction and affecting the behaviour of the BSP beams significantly. The

longitudinal slip Slc is controlled by the plate–RC axial stiffness ratio

βa = (EA)p / (EA)c, the plate–RC flexural stiffness ratio βp = (EI)p / (EI)c and the

bolt–RC stiffness ratio βm = km / (EI)c, where km is the stiffness of the bolt


49

connection. The transverse slip Str is controlled by the plate–RC flexural stiffness

ratio βp and the bolt–RC stiffness ratio βm.

The longitudinal and transverse slip profiles from midspan to one of the

supports of the moderately reinforced BSP beams at four different load levels

(F/Fp = 0.25, 0.5, 0.75 and 1) are illustrated in Figures 4.9 and 4.10, respectively.

The values of those at the supports and the loading points at two load levels

(F/Fp = 0.75 and 1) are tabulated in Table 4.4. Because the longitudinal slip varies

along the section depth, the value at the centroidal level of the steel plates is

adopted as the nominal longitudinal slip.

4.4.1 Longitudinal slip

It is shown in Figure 4.9 that the longitudinal slips in all the BSP beams were

initiated at the plate-ends and spread progressively toward the midspan region.

The longitudinal slips of the specimens with shallow steel plates, Specimens

P100B300 and P100B450, decreased from the plate-ends and vanished near the

midspan. In contrast, the longitudinal slips of the specimens with deep steel plates,

Specimens P250B300R and P250B450R, were more complicated. The direction

of slips in the middle portion of beam span changed alternately because the

centroidal level of the steel plates and the neutral axis of RC beams were close to

each other, and thus small variations on the neutral axis level led to alternations of

the slip direction. The figure also shows that the incremental slip in each load step

is approximately double that in the previous step. Thus, the longitudinal slip is

proportional to the square of the load level (F/Fp)2 because the increase of the

axial stiffness ratio βa caused by the stiffness deterioration of the RC beam was

accelerated by the development of concrete cracking and crushing as the load

levels increased.

Table 4.4 shows that for specimens with the same plate depth, the plate-end

longitudinal slip (2.67 mm) of P100B450 with a bolt spacing of 450 mm is

approximately 1.7 times of that (1.50 mm) of P100B300 with a bolt spacing of

300 mm. The results demonstrate that the longitudinal slip is inversely

proportional to the bolt spacing. Specimens P250B300R and P100B300 had the


50

same bolt spacing of 300 mm, and the longitudinal slips (0.29 mm) of

P250B300R with deep steel plates was only approximately 1/5 of that (1.50 mm)

of P100B300 with shallow steel plates at F/Fp = 1. Hence, the longitudinal slip is

no longer a dominant factor for evaluating the performance of BSP beams with

deep steel plates.

4.4.2 Transverse slip

Figure 4.10 shows that the transverse slips are close to zero at the midspan,

negative near the plate-ends and positive with a maximum magnitude near the

loading points for all BSP beams. Obviously, the transverse slips are caused by

the shear force transferred from the RC beams to the steel plates, and the

directional reversal reveals the bolt force equilibrium in the transverse direction.

The high plate–RC flexural stiffness ratio βp due to the serious stiffness

deterioration at the plastic hinge regions caused the largest slip to occur near the

loading points.

Table 4.4 shows that for the BSP beams with the same size of steel plates, the

transverse slips at the loading points of P100B300 and P100B450 were 0.07 mm

and 0.12 mm, respectively, at F/Fp = 0.75 but then increased to 0.30 mm and

0.23 mm, respectively, at F/Fp = 1. The results imply that the transverse slip

increases with the number of anchor bolts when F/Fp 0.75, but beyond that point

it is controlled by the concrete strength. The enormous increase (from 0.07 mm to

0.30 mm) of P100B300 in the load interval F/Fp = 0.75~1 was caused by the

significant increase in the plate–RC flexural stiffness ratio βp due to the rapid

deterioration of the flexural stiffness of the reinforced concrete component after

reaching the peak load. The numbers in the table also show that when the BSP

beams have the same bolt spacing, the transverse slips increase significantly with

the increase in plate depth. As an illustration, the transverse slips at the loading

point (0.07 mm and 0.30 mm) of P100B300 are much lower than those (0.17 mm

and 0.46 mm) of P250B300R at load levels of both 0.75 and 1.

It can be found by comparing the longitudinal and transverse slips in

Table 4.4 that for the BSP beams with shallow steel plates, the transverse slip is


51

less than 10% of the longitudinal slip; however, for the BSP beams with deep

steel plates, the longitudinal and transverse slips are of the same order of

magnitude. Hence, the effects of transverse slips on BSP beams with deep steel

plates cannot be ignored.

4.5 STRAIN AND CURVATURE FACTORS

The strain and curvature factors can be used to quantify the longitudinal and

transverse partial interaction, i.e., the degrees of the axial strain looseness or the

curvature reduction of the steel plates due to the longitudinal or transverse slips.

The factors are controlled by the stiffness ratios (βa , βp and βm) and decrease as

the slips increase. Figure 4.11 illustrates the variations in the strain and the

curvature factors as the midspan deflection for the moderately reinforced BSP

beams, a curve of the lightly reinforced BSP beam P75B300* is also plotted for

comparison.

4.5.1 Strain factors

As shown in the figure, the strain factors of all the BSP beams with shallow

steel plates were approximately 0.7 at the beginning of the loading process and

decreased gradually to approximately 0.4 at the peak load. The strain factors of

P75B300* were the highest due to its weakest steel plates and therefore lowest

plate–RC axial stiffness ratio βa. The strain factors of P100B300 were higher than

those of P100B450 as a result of its smaller bolt spacing and hence higher

bolt–RC stiffness ratio βm. The strain factors of the two BSP beams with deep

plates, P250B300R and P250B450R, were very small because their axial stiffness

ratios βa were high and the centroidal levels of their steel plates and RC beams

were close together, resulting in negligible centroidal strains in the steel plates.


52

4.5.2 Curvature factors

The curvature factors of all the moderately reinforced BSP beams remained

unchanged at a high level of 0.8 over the whole loading process. Furthermore, the

curvature factors of the specimens with shallow plates were even higher due to the

lower flexural plate stiffness and thus lower plate–RC flexural stiffness ratio βp.

The curvature factor of P75B300* also remained at a high level at the initial

loading stages but decreased significantly after the yielding of the tensile

reinforcement as a consequence of the enormous transverse slips caused by the

lack of anchor bolts at the midspan.

4.6 PLATE BEHAVIOUR

The steel plates in a BSP beam retrofit the RC beam in two primary ways: (1)

behaving as additional tensile reinforcement to apply an eccentric compressive

force Np to the RC beam, thus providing an additional coupling moment icp Np,

where icp is the eccentricity, and (2) resisting the lateral loads due to their flexural

stiffness directly and hence providing an additional bending moment φp (EI)p. The

latter strengthening effect is unique and distinct from that of the steel plates

attached to the beam soffit.

The tensile forces and bending moments of the steel plates in the BSP beams

with shallow or deep plates at two load levels (F/Fp = 0.75 and 1) are tabulated in

Table 4.5, and their contribution to the flexural strength of the BSP beams is also

compared. The values within the parentheses are the ratio of the tensile force to

the yield strengths of the steel plates. The steel plates in the BSP beams with

shallow plates, P100B300 and P100B450, contributed almost half of their tensile

strengths. For the BSP beams with deep steel plates, P250B300R and P250B450R,

the tensile force was relatively low and was only approximately a quarter of their

tensile strengths. However, the bending moments of the steel plates in P100B300

and P100B450 were very limited and almost less than 10% of those in

P250B300R and P250B450R. The ratio of the flexural contributions of the

bending moment provided by their flexural stiffness φp (EI)p to the coupling


53

moment icp Np provided by their tensile axial force is tabulated in Table 4.5. The

bending moment φp (EI)p taken by the shallow plates was only 15% of the

coupling moment icp Np, whereas the bending moment φp (EI)p in the deep plates

was approximately 7 times of the coupling moment icp Np.

4.7 CONCLUSIONS

A comprehensive study of the results of the experiments reported in Chapter

3 was carried out. The behaviours of moderately reinforced BSP beams under

four-point bending were studied and compared with the available test results for

lightly reinforced BSP beams reported by other researchers. The main findings of

this study are summarised as follows:

(1) The experimental results reveal that unlike those of the lightly reinforced RC

beams, the strengths and stiffnesses of the moderately reinforced RC beams

are controlled by the concrete strength, thus can only be improved by adding

very deep steel plates to the side faces.

(2) Deep steel plates in BSP beams are prone to buckling on their compressive

edge. This phenomenon has serious adverse effects on strength and ductility

but not stiffness. Buckling restraints should be added to prevent the plate

from buckling and to improve the post-peak performance of the beam.

(3) In contrast to the RC beams strengthened by steel plates attached to the beam

soffit, for which plate-end anchor bolts are sufficient, BSP beams require a

uniform distribution of anchor bolts over the entire beam span; otherwise,

enormous transverse slips will occur at midspan and jeopardise the

load-carrying capacity of the beam.

(4) The gaps between the bolt shafts and the steel plates weaken the connection

between the steel plates and the RC beam, thus decrease the strength of the

BSP beam. However, the reduction in the degree of connection can also

increase the ductility to some extent.


54

(5) The strengthening effect of BSP beams is affected by the properties of the

connecting medium, which is determined by the bolt spacing and the shear

force–slip response of the anchor bolts.

(6) Longitudinal slip is initiated from the plate-ends and decreases progressively

toward the midspan. In BSP beams with deep steel plates, the longitudinal

slips at the centroidal level of the steel plates may reverse in direction.

Longitudinal slips increase with the bolt spacing and the stiffness ratios of the

steel plates to the RC beams.

(7) A transverse slip changes its direction from the plate-ends to the midspan, and

reaches its maximum magnitude at the loading points. Transverse slips

increase with the plate–RC flexural stiffness ratios and hence the plate-depth.

They also increase with the bolt spacing before reaching the load level of

0.75, above which they are controlled by the concrete strength.

(8) For BSP beams with shallow steel plates attached to the tensile region of the

side faces, longitudinal slips are the dominant factor for evaluating the

performance of the beams, and transverse slips can be neglected. However,

for BSP beams with deep steel plates, longitudinal slips are no longer a

dominant factor, and the transverse slips control the behaviour of the beams.

(9) Both the strain and curvature factors increase with the number of anchor bolts

and the reduction of the plate–RC stiffness ratio. The strain factors of BSP

beams with shallow steel plates decrease as the loading process, and those of

BSP beams with deep steel plates remain very low over the whole loading

process. The curvature factors remain at a relative high level over the entire

loading process.

(10) The steel plates in BSP beams contribute to the overall flexural strength by

both the coupling moment provided by their axial tensile force and the

bending moment provided by their flexural stiffness. Shallow plates

contribute mainly to the former, whereas deep plates contribute mainly to the

latter.


55

Table 4.1 Concrete strengths, beam geometries and strengthening details

Specimen fcu

(MPa)

fco

(MPa)

ρst

(%)

Dp

(mm)

Sb

(mm)

Rows of

bolts

Midspan

bolts

Adhesive in

shaft–plate gaps

Buckling

restraint

CONTROL* 35.2 - 0.85 - - - - - -

P75B300 39.7 33.9 0.85 75 300 1 None None No

P75B300* 35.3 - 0.85 75 300 1 None Yes No

P150B400* 34.6 - 0.85 150 400 2 None Yes No

CONTROL 37.3 32.8 1.77 - - - - - -

P100B300 33.9 28.9 1.77 100 300 1 Yes Yes No

P100B450 40.8 33.2 1.77 100 450 1 Yes Yes No

P250B300 36.0 29.7 1.77 250 300 2 Yes Yes No

P250B300R 35.8 26.6 1.77 250 300 2 Yes Yes Yes

P250B450R 37.7 27.0 1.77 250 450 2 Yes Yes Yes

Note: Specimens marked by * were extracted from the experimental study by Siu (2009).

Table 4.2 Load levels (F/Fp) when failure phenomena occurred

Specimen

(1). Reinforcement

tensile yielding

εst > εy

(2). Concrete

compressive crushing

εcc > εcu

(3). Steel plate

tensile yielding

εpb > εyp

(4). Steel plate

compressive buckling

Δεpt < 0

P75B300 0.77 0.84 0.85 -

CONTROL 0.91 0.94 - -

P100B300 0.87 0.78 0.86 -

P100B450 0.85 0.80 0.89 -

P250B300 0.88 0.85 0.26 0.96

P250B300R 0.83 0.84 0.44 -

P250B450R 0.83 0.89 0.29 -


56

Table 4.3 Strengths, stiffnesses and ductility

Specimen Strength Fp (kN) Stiffness Ke (kN/mm) Toughness Ut (kN·mm)

CONTROL* 169.0 (1.00) 9.2 (1.00) 16064 (1.00)

P75B300 222.5 (1.32) 9.4 (1.02) 22264 (1.39)

P75B300* 241.0 (1.43) 10.5 (1.15) 10299 (0.64)

P150B400* 269.2 (1.59) 12.3 (1.34) 10791 (0.67)

CONTROL 267.6 (1.00) 11.5 (1.00) 22915 (1.00)

P100B300 316.9 (1.18) 12.0 (1.04) 18344 (0.80)

P100B450 326.5 (1.22) 12.1 (1.06) 20359 (0.89)

P250B300 359.4 (1.34) 14.6 (1.27) 15021 (0.66)

P250B300R 382.0 (1.43) 14.5 (1.26) 33805 (1.48)

P250B450R 376.7 (1.41) 14.6 (1.27) 31395 (1.37)

Table 4.4 Slips on the plate–RC interface

Specimen

Longitudinal slip (mm)

at supports

Transverse slip (mm)

At supports At loading points

F/Fp = 0.75 F/Fp = 1 F/Fp = 0.75 F/Fp = 1 F/Fp = 0.75 F/Fp = 1

P100B300 0.72 1.50 -0.05 -0.09 0.07 0.30

P100B450 1.12 2.67 -0.06 -0.09 0.12 0.23

P250B300R 0.14 0.29 -0.12 -0.21 0.17 0.46

P250B450R 0.17 0.39 -0.17 -0.33 0.19 0.52

Table 4.5 Contribution of the steel plates due to bending and tension

Specimen

Tensile force Np

(kN)

Bending moment φp (EI)p

(kN·m)

Bending–coupling ratio

(φp (EI)p / icp Np)

F/Fp = 0.75 F/Fp = 1 F/Fp = 0.75 F/Fp = 1 F/Fp = 0.75 F/Fp = 1

P100B300 150 (0.38) 195 (0.50) 2.8 4.6 0.13 0.17

P100B450 144 (0.37) 189 (0.48) 2.9 5.6 0.15 0.20

P250B300R 192 (0.20) 296 (0.30) 42.2 50.8 7.16 6.74

P250B450R 113 (0.12) 196 (0.20) 45.8 54.2 13.08 6.11


57

Figure 4.1 Load–deflection curves for the reference beams

Figure 4.2 Load–deflection curves for the lightly reinforced beams

0 20 40 60 80 100 120

0

100

200

300

P150B400*

P75B300*

P75B300

CONTROL*

Load

(kN

)

Midspan deflection (mm)

P150B400*

CONTROL*

P75B300*

P75B300

0 20 40 60 80 100 120

0

100

200

300

CONTROL

CONTROL*

Load

(kN

)


CONTROL

CONTROL*


58

Figure 4.3 Load–deflection curves for the moderately reinforced beams

Figure 4.4 Load–deflection curves for beams with or without buckling restraint

0 20 40 60 80 100 120

0

100

200

300

400

P250B300R

P250B300

P100B300

CONTROL

Load

(kN

)


P100B300

CONTROL

P250B300 P250B300R

0 20 40 60 80 100 120

0

100

200

300

400

P250B300R

P250B450R

P100B300

P100B450

CONTROL

Load

(kN

)


P100B450 CONTROL

P250B300R

P250B450R

P100B300


59

Figure 4.5 Midspan vertical slips of P75B300 at (a) the peak load and (b)

failure (dimensions in mm)

20

0

(b)

15

5

(a)


60

Figure 4.6 Failure modes of (a) P100B300, (b) P100B450, (c) P250B300R and

(d) P250B450R

(a)

(c)

(b)

(d)

Steel plate

removed

Steel plate

removed


61

Figure 4.7 Plate buckling of P250B300

Figure 4.8 Equivalent elasto-plastic system of the load–deflection curve

Ke

F

Fp

Ff = 0.8Fp

0.75Fp

δy δp δf δ

Ut

Ut


62

(a)

(b)

0 600 1200 1800

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Longit

udin

al s

lip (

mm

)

Distance from midspan (mm)

0 600 1200 1800

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Longit

udin

al s

lip (

mm

)



63

Figure 4.9 Longitudinal slip profiles along the beam axis for (a) P100B300, (b)

P100B450, (c) P250B300R and (d) P250B450R

(c)

(d)

0 600 1200 1800

-0.5

-0.3

0.0

0.3

0.5

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Longit

udin

al s

lip (

mm

)


0 600 1200 1800

-0.5

-0.3

0.0

0.3

0.5

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Longit

udin

al s

lip (

mm

)



64

0 600 1200 1800

-0.4

0.0

0.4

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Tra

nsv

erse

sli

p (

mm

)


0 600 1200 1800

-0.4

0.0

0.4

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Tra

nsv

erse

sli

p (

mm

)


(a)

(b)


65

Figure 4.10 Transverse slip profiles along the beam axis for (a) P100B300, (b)

P100B450, (c) P250B300R and (d) P250B450R

0 600 1200 1800

-0.8

-0.4

0.0

0.4

0.8

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Tra

nsv

erse

sli

p (

mm

)


0 600 1200 1800

-0.8

-0.4

0.0

0.4

0.8

(F/Fp) = 1.00

(F/Fp) = 0.75

(F/Fp) = 0.50

(F/Fp) = 0.25

Tra

nsv

erse

sli

p (

mm

)


(c)

(d)


66

Figure 4.11 Development of (a) strain factors and (b) curvature factors

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

P75B300*

P100B300

P100B450

P250B300R

P250B450R

Str

ain f

acto

r


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

P75B300*

P100B300

P100B450

P250B300R

P250B450R

Curv

ature

fac

tor


(a)

(b)

Chapter 5 Numerical Study on BSP Beams

67

CHAPTER 5

NUMERICAL STUDY ON BSP BEAMS

5.1 OVERVIEW

The experimental study reported in Chapters 3 and 4 revealed the behaviour

of BSP beams with the same RC beam geometry but different plate and bolt

arrangement under four-point bending. The performance of BSP beams was

proved to be controlled by the partial interaction caused by the longitudinal and

transverse slips on the plate–RC interface.

In this chapter, a nonlinear finite element analysis (NLFEA) using the

computer software ATENA is conducted to investigate the behaviour of BSP

beams with different beam geometries and under various loading conditions.

Special emphasis is put on the investigation of the partial interaction caused by

the longitudinal and transverse slips and shear transfers. Without a doubt, the use

of the NLFEA is more cost-effective than conducting more experiments. It also

alleviates the difficulty of taking precise measurements of the transverse slip in

tests. The experimental results in Chapters 4 are employed to validate the NLFEA.

The NLFEA is then used to conduct a parametric study to evaluate the transverse

shear transfer of BSP beams. Based on which, a new design approach to

estimating the transverse shear transfer profile is developed. An example is also

presented to illustrate the effectiveness of the proposed approach in the

determination of the transverse shear transfer profile of a BSP beam under

realistic loading conditions.

5.2 NUMERICAL MODELLING

ATENA is a two-dimensional NLFEA program developed by Cervenka V.

and Cervenka J. (2012) for modelling the nonlinear behaviour of RC members

considering both material and geometric nonlinearities (Cervenka et al. 2012).


68

The main assumptions and methodologies used in the numerical model are briefly

presented below.

5.2.1 Modelling of concrete

The concrete in ATENA is idealised as a two-dimensional body with a unit

thickness. The behaviour of the concrete is simulated by the concrete constitutive

model SBETA, which considers (1) the nonlinear behaviour of concrete in

compression, including hardening and softening; (2) fracturing of concrete in

tension, based on nonlinear fracture mechanics; (3) a biaxial strength failure

criterion; (4) the reduction in compressive strength after cracking and (5) the

reduction in shear stiffness after cracking. The SBETA model is based on the

biaxial failure criterion proposed by Kupfer and Gerstle (1973) and the equivalent

uniaxial stress–strain curve proposed by Darwin and Pecknold (1977). The

effective concrete strengths were determined as functions of the current stress

states according to the Kupfer failure criterion, which considers four different

cases (see Figure 5.1(a)): (1) compression–compression, (2) tension–compression,

(3) compression–tension and (4) tension–tension:

1

2

1

2

2 1

2

1 3.650

1,ef

c cf f

(5.1)

2 111 5.3278 0

53.,

278

ef cc c

c

ff

ff

(5.2)

2

2 11 0.8 053.278

,ef ct t

cff

ff

(5.3)

2 1, 0t

e

t

ff f (5.4)

The effective principal stresses were determined from the equivalent uniaxial

strains according to the modified equivalent constitutive curve, which also

considers four states (see Figure 5.1(b)): (1) concrete in tension before cracking is


69

idealised as a linearly elastic material, (2) concrete after cracking is considered

using a fictitious crack model based on the exponential crack opening model and

fracture energy (Hordijk 1991), (3) concrete in compression before the peak stress

is described by CEB-FIP Model Code 90 (CEB 1993), and (4) concrete after the

peak stress is described by a fictitious compression plane model (van Mier 1986).

0 , 0ef eq ef

c cc tE f (5.5)

3

3 101 exp 6.93 6.93 5.14,

fcn c c ccref ef

cr cr crc t

Gw w wexp w

w wf w f

(5.6)

2

0

0

0

0

21

eq

c

cc cef ef

eq

c

c

c c

c c

E

E

Ef

E

(5.7)

0cd

cd c

cd

w

L (5.8)

To represent the material properties of the locally mixed concrete used in the

experiment, the compressive strength and elastic modulus were chosen as the

values obtained in the experiment, and the strain at peak stress and the plastic

displacement for the fictitious compression plane model were taken as the

following (Lam 2006):

0.75

00 3.46

6 mmc

c cu

d

f

w

E

(5.9)

5.2.2 Modelling of reinforcement and steel plates

A bilinear elastic material model with hardening was chosen to represent

reinforcement. The transverse reinforcement was modelled by adding a smeared

reinforcement layer to the concrete layer. The longitudinal reinforcement was

modelled in ATENA using the discrete bar element CCBarWithBond to consider


70

the bond–slip effect, according to the CEB-FIP Model Code 90 (CEB 1993), as

shown in Figure 5.2.

The steel plates were idealised as a plane stress layer and the steel material

was simulated using the bilinear steel Von Mises model, which considers a biaxial

failure law and a bilinear stress–strain curve, taking into account both the elastic

state and the hardening of the steel, as shown in Figure 5.3.

5.2.3 Modelling of bolt connections

The bolt connections between the steel plates and the RC beam were

simulated by discrete bolt elements to allow for interfacial slip, as shown in

Figure 5.4(a). The connection was composed of two element types that were both

simulated by the bilinear steel Von Mises model. Four internal triangular elements

were employed to simulate the behaviour of the bolt shaft, and four external

quadrilateral elements were utilised to simulate the shear force–slip relationship

obtained from the experiments in Chapters 3 and 4, as shown in Figure 5.4(b).

5.2.4 Finite element meshes and load steps

Both the concrete and smeared transverse reinforcement layers were

composed of 4-node isoparametric plane stress elements with an element size of

12.5 mm. Their meshes were identical and connected to each other at every node

so that perfect bonding could be assumed. The discrete longitudinal reinforcement

was modelled by 2-node bar elements, and the bond–slip effect was taken into

account by introducing the bond–slip relation (see Figure 5.2) into the difference

in displacement between the bar nodes and the corresponding concrete-layer

nodes. The steel plates were modelled by a layer of 4-node isoparametric plane

stress elements with an element size of 12.5 mm. The nodal coordinates of the

concrete and steel plate layers were designed so that the nodes located at the

anchor bolts were exactly coincident with the outer and central nodes of the bolt

elements. The nodes on the steel plate layer were then connected to the central


71

nodes of the bolt elements, and those on the concrete layer were connected to the

outer nodes of the bolt elements, as shown in Figure 5.4(a). The hinge and rollers

at the supports and the loading points were simulated by 4-node isoparametric

plane-stress rigid plates to prevent high stress concentration. The finite element

meshes of specimen P100B450, which will be discussed in more detail in the next

section, are shown in Figure 5.5. Only half of the meshing is illustrated, owing to

the symmetry of the geometry and loading.

Monotonic displacements were induced at the two loading points, and the

modified Newton–Raphson method was used to determine the complete

load–deflection curve, including the post-peak descending branch.

5.3 VALIDATION OF NUMERICAL MODEL USING

EXPERIMENTAL RESULTS

In this section, the results of the experimental study described in Chapter 4

are extracted to verify the finite element modelling. The simulation of the overall

load–deflection behaviour and the longitudinal and transverse slips is validated.

5.3.1 Comparison of the load–deflection curves

The overall load–deflection curves derived from the numerical and

experimental studies are compared in Figure 5.6. The numerical results generally

capture the full range behaviour of all the specimens with shallow (P100B300 and

P100B450) and deep steel plates (P250B300R and P250B450R) in the tests,

except for a slight overestimation of both the stiffness and the peak load. This

outcome may be due to the difference in concrete strengths between the RC

beams and concrete cubes and the plate buckling that occurred despite the use of

buckling restraint devices. It should be noted that there is small overestimation of

the peak load; the values are off by only 2.0%, 0.5%, 1.7%, and 3.7% in

Specimens P100B300, P100B450, P250B300R, and P250B450R, respectively.


72

5.3.2 Comparison of the longitudinal slip profiles

The experimental and numerical longitudinal slip profiles for Specimens

P100B300 and P100B450 at two load levels (F/Fp = 0.25 and 0.75) are compared

in Figure 5.7. The longitudinal slips at the plate ends are also tabulated in

Table 5.1 for all specimens at four different load levels (F/Fp = 0.25, 0.50, 0.75,

and 1.00). It can be observed that the numerical predictions agree very well with

the experimental longitudinal slips and the average numerical to experimental slip

ratios are 1.00, 0.94, 1.07, and 1.10, respectively.

5.3.3 Comparison of the transverse slip profiles

The transverse slip profiles for Specimens P100B300, P100B450,

P250B300R, and P250B450R at two load levels (F/Fp = 0.25 and 0.75) are

compared in Figure 5.8. The numerical and experimental profiles are in good

agreement. The numerical and experimental transverse slip at the loading points

are listed in Table 5.2 for all specimens at four load levels (F/Fp = 0.25, 0.50, 0.75,

and 1.00). The numerical predictions agree very well with the experimental results

for P100B300 and P100B450, and the average numerical to experimental slip

ratios at the loading point are 1.13 and 0.93, respectively. The predicted slips for

P250B300R and P250B450R are also acceptable, despite some overestimation,

with the average numerical to experimental ratios of 1.10 and 1.28, respectively.

The discrepancy may be due to the buckling that occurred in the deep steel plates,

which reduced their flexural stiffness thus the measured transverse slips.

5.4 STUDIES ON LONGITUDINAL SLIP AND SHEAR

TRANSFER

5.4.1 Longitudinal shear transfer

The longitudinal slip Slc is the result of the deformation of anchor bolts under

the longitudinal bolt shear force Tm:


73

mlc

b

TS

K (5.10)

where Kb = Rby /Sby is the bolt stiffness, which can be determined from bolt shear

tests; Rby is the yield shear force of an anchor bolt; and Sby is the corresponding

yield deformation. Assuming that the bolt behaves in elasto-plastic manner, the

bolt stiffness of the shear force–slip relation in the elastic region is denoted by Kb.

The longitudinal shear stress transfer tm is defined as the longitudinal bolt

shear force Tm divided by the bolt spacing Sb, i.e.,

mlcm c

b b

lb

m

T Kt S k S

S S (5.11)

where km = Kb /Sb is the bolt stiffness per unit length. If uniform bolt spacing is

used, km is a constant along the beam span. Theoretically, the longitudinal shear

transfer tm and the longitudinal bolt shear force Tm can be estimated once the

longitudinal slip Slc is measured.

The normalised longitudinal slip and shear transfer profiles for a BSP beam

subjected to two symmetrically arranged point loads, along with those for a BSP

beam under an asymmetrical point load, are compared in Figure 5.9. The

difference between the normalised profiles of the longitudinal slip and the shear

transfer is negligible. This behaviour is reasonable because according to the linear

elastic connectivity assumption, the relationship between the longitudinal bolt slip

and shear transfer is linear (see Equation (5.11)), and hence their normalised

profiles should coincide with each other. Therefore, it is convenient to estimate

the longitudinal bolt shear forces from the measured longitudinal slips.

5.4.2 Influence of loading position

The longitudinal shear transfer profiles caused by a single point load located

at various positions are shown in Figure 5.10. When the point load is applied at

the midspan, the longitudinal shear transfer profile is antisymmetrical with regard


74

to the midspan. As the load moving toward the left support, the magnitude of the

longitudinal shear transfer reduces due to the reduction of external bending

moment. The position where the longitudinal shear transfer is zero also moves

leftward as the point load at a lower speed, thus locates at neither the midspan nor

the loading point but another point between them. Furthermore, the magnitude of

the longitudinal shear transfer at the right hand side decreases more significantly

in order to keep the longitudinal bolt shear force in equilibrium.

5.5 STUDIES ON TRANSVERSE SLIP AND SHEAR

TRANSFER

5.5.1 Transverse shear transfer

The transverse slip Str is the result of the deformation of anchor bolts under

the transverse bolt shear force Vm:

mtr

b

VS

K (5.12)

The transverse shear stress transfer vm is defined as the transverse bolt shear

force Vm divided by the bolt spacing Sb, i.e.,

bm tr

mm tr

b b

V Kv S k S

S S (5.13)

If uniform bolt spacing is used, km is a constant along the beam. Theoretically,

the transverse shear transfer vm and the transverse bolt shear forces Vm can be

estimated once the transverse slip Str is measured. However, the experimental

results reported in Chapter 4 have shown that the transverse slip, which usually

ranges from 0.01 to 0.5 mm, is hard to measure accurately.

The normalised transverse slip and shear transfer profiles for a BSP beam

subjected to two symmetrically arranged point loads, along with those for a BSP

beam under an asymmetrical point load, are compared in Figure 5.11. It is seen


75

that the difference between the normalised profiles of the transverse slip and the

shear transfer is negligible. This can also be explained by the linear elastic

connectivity assumption as shown in Equation (5.13).

5.5.2 A brief introduction to the parametric study

Figure 5.12(a) shows the reference BSP beam used in the parametric study,

which has the same geometry as Specimen P100B300. The flexural stiffnesses of

the RC beam and the steel plates and the stiffness of the bolt connection of the

reference beam are as follows:

2' 8000 kM mc

EI (5.14)

2' 220 kM mp

EI (5.15)

2' 370 kN/mmk (5.16)

Six basic loading cases, illustrated in Figure 5.12, were considered in the

parametric study, including (a) a midspan point load, (b) an asymmetrically

arranged point load, (c) two symmetrically arranged point loads, (d) a uniformly

distributed load (UDL), (e) a trapezoidal distributed load and (f) a triangularly

distributed load. The influences of the different load levels (F/Fp), the flexural

stiffness of the RC beam (EI)c, and the plate–RC and bolt–RC stiffness ratios

(βp = (EI)p /(EI)c and βm = km /(EI)c) on the transverse shear transfer profile were

investigated. By varying the location of the applied point load, the transverse

shear transfers at specific locations, such as at the left support (vm,LS), the right

support (vm,RS) and the loading point, for concentrated load cases (vm,F), were

obtained. For the distributed load cases, vm,F is the transverse shear transfer at the

midspan. The half bandwidth of the transverse shear transfer profile w is a

distance measured from the location of vm, F to the first intersection of the

transverse shear transfer profile and the beam axis, as shown in Figures 5.13 and

5.14. The computed transverse shear transfers at specific locations, together with


76

the half bandwidth, are useful for evaluating the entire transverse shear transfer

profile for the basic loading cases. By employing the superposition principle, the

transverse shear transfer profile under any arbitrary combination of external loads

can be evaluated.

5.5.3 Transverse shear transfer profiles under different loading

arrangements

By varying the position of the point loads acting on the reference beam, the

influence of the load location on the transverse shear transfer profile was

investigated. Figure 5.13 shows the typical variations of transverse shear transfer

profiles for both the asymmetrically arranged single point load and symmetrically

arranged two-point load cases. For the single-point-load case, when the point load

was close to the left support (xF /L = 1/6), the negative transverse shear transfer at

its right side was negligible, but was concentrated at its left side with a very steep

slope (vm, LS > vm, F > vm, RS). As the load moved toward the midspan, the positive

and negative transverse shear transfers on its right side increased gradually, while

those on its left side decreased and acquired a gentler slope (the ratio

vm, LS /vm, F decreased, whereas vm, F and the ratio vm, LS /vm, F increased). As shown

in Figure 5.13(b), when two point loads were relatively far apart and close to the

supports (xF /L = 1/12), the positive transverse shear transfer from the RC beam to

the steel plates was resisted mainly by the negative transverse shear transfer at the

supports, and the transverse shear transfers at the supports were more critical than

those under the point loads (vm, LS = vm, RS > vm, F). As the two loads got closer to

each other and eventually became a single load (xF /L = 1/2), the positive

transverse shear transfer near the midspan increased and the transverse shear

transfer vm, F increased gradually. Meanwhile, the negative transverse shear

transfers at the supports and the slopes of the negative transverse shear transfers

between the loads and the supports became more and more gentle (vm, LS /vm, F =

vm, RS /vm, F decreased). In other words, the magnitude of the transverse shear

transfer (vm, F) and the transverse shear transfer ratios (vm, LS /vm, F and vm, RS /vm, F)

were highly dependent on the locations of the external loads. Furthermore, the

half bandwidth w also varied significantly as the locations of the external loads


77

changed. The transverse shear transfer ratios (vm, LS /vm, F and vm, RS /vm, F) of the

shallow plates and the dimensionless half bandwidth w/L of the deep and shallow

plates under various loading cases are presented in Table 5.3.

The transverse shear transfer profile under a point load at the left trisectional

point was added to that under a point load at the right trisectional point, and the

resultant shear transfer profile was compared with the transverse shear transfer

profile under two point loads at both the trisectional points. The comparison,

shown in Figure 5.14(a), indicates that the two profiles are very similar. When

five point loads with a uniform spacing were applied, the profile obtained from

the superposition was very close to that obtained from the NLFEA under a UDL,

as shown in Figure 5.14(b). It is evident that the transverse shear transfer profile

of complicated load arrangements can be estimated by superimposing the

transverse shear transfer profiles from the basic load cases.

5.5.4 Transverse shear transfers under different load levels and

beam geometries

The magnitudes of the external loads (F or q in Figure 5.12) were varied to

study the influence of load level F/Fp on the transverse shear transfer vm, F. For

brevity, vm, F was divided by the peak total external load Fp and the span length L

to obtain a dimensionless transverse shear transfer ratio ξFp as follows:

,

,p

Fp m F

p

m F

b

F V Lv

L F S

(5.17)

The stiffnesses of the RC beam (EI)c, the steel plates (EI)p, and the bolt

connection km were also varied to study their effects on vm, F, which can be

quantified by the transverse shear transfer factor ζ defined as follows:

,

,

' '

' '

Fp p pm F

Fp m F p pF

F SV

V S

(5.18)


78

The transverse shear transfer factors due to the changes in (EI)c, (EI)p, and km are

denoted by ζEIc, ζEIp, and ζkm, respectively. Combining the dimensionless

transverse shear transfer ratio ξFp and the transverse shear transfer factors (ζEIc,

ζEIp, and ζkm), the transverse shear transfer vm, F can be evaluated as:

,

p

m F EIc EIp km Fp

Fv

L (5.19)

The dimensionless transverse shear transfer ratios ξFp at different load levels

F/Fp are depicted in Figure 5.15(a). Under the working load condition (F/Fp <

0.75), the square root of the dimensionless transverse shear transfer ratio ξFp1/2

, in

general, increases linearly with the load level F/Fp. However, when F/Fp > 0.75,

the results from the NLFEA revealed that serious degradation of concrete occurs

and the steel plates take up more of the loading. As a result, the dimensionless

transverse shear transfer ratio increases rapidly. Because the transverse shear

transfer vm, F increases drastically when the load level approaches unity, a working

load level of F/Fp < 0.75 should be adopted for the design of BSP beams. When

F/Fp < 0.75, the dimensionless transverse shear transfer ratios for all single point

load cases and all distributed load cases are estimated as:

2

2

0.65 under a point load

0.30 under a distributed load

p

Fp

p

F F

F F

(5.20)

The variations in the transverse shear transfer factors ζEIc, ζEIp, and ζkm for the

corresponding stiffnesses (EI)c, (EI)p, and km (under a load level F/Fp < 0.75) are

plotted in Figure 5.15(b). After some trials of different curve-fitting functions, it

was found that the variation of the transverse shear transfer factors could be

approximated as follows:

1

26

1 0.13 log ' , ' 8000 kM mEIc c c cEI EI EI (5.21)

8

21 0.19 log ' , ' 220 kM mp p pEIp EI EI EI

(5.22)


79

1133

log '

21.8 1.8, ' 370 kN/m

1 0.8 '1 0.8 10 m mkm m

m mk k

kk k

(5.23)

It can be observed from the figures that as the stiffnesses ((EI)c, (EI)p, or km) is

reduced to 1% or increased by 100 times, the variation in ζEIc1/16

, ζEIp1/8

, and ζkm3

are all within the range of 0 to 2. However, the rates of change of the various

transverse shear transfer factors (ζEIc, ζEIp, and ζkm) are very different, due to the

differences in the magnitudes of the exponents (1/16, 1/8, and 3). Because the

transverse shear transfer vm, F decreases (or increases) drastically as (EI)c (or (EI)p)

increases, an excessive plate–RC stiffness ratio (βp= (EI)p /(EI)c) should be

avoided in the design of BSP beams.

5.5.5 Half bandwidths under different load levels and beam

geometries

The transverse shear transfer profiles of BSP beams subjected to a single

point load at the midspan for different F/Fp, (EI)c, βp, and βm were evaluated. The

computed transverse shear transfer profiles were normalised by vm, F so that the

normalised transverse shear transfer at the midspan was equal to one. Figure 5.16

presents the normalised transverse shear transfer profiles. As shown in Figures

5.16(a) and (b), the shapes of the normalised profiles for different F/Fp and

different (EI)c were very similar, and the half bandwidth w remained almost

unchanged. However, it is evident in Figures 5.16(c) and (d) that w increased with

increasing βp and decreasing βm. The results further revealed that w is a constant

when the plate–bolt stiffness ratio remains unchanged, i.e., (EI)p /km = βp /βm = C1,

where C1 is a constant.

In other words, the half bandwidth w is independent of the load level F/Fp and

the stiffness of the RC beam (EI)c, but is controlled by the plate–bolt stiffness

ratio βp /βm. The variation in the relative half bandwidth w/L as (βp /βm)1/4

is plotted

in Figure 5.17 and can be expressed approximately by the following linear

relation:


80

1

4

0.07 0.10p

m

w

L

(5.24)

Therefore, for a BSP beam under three-point bending, w can be obtained

using Equation (5.24). For a proper strengthening design, the number of anchor

bolts used should be proportional to the area of the steel plates so that yielding of

the steel plates happens prior to failure of the anchor bolts. Thus

1by pb yp

b

n R f A

(5.25)

where γb is a partial safety factor, nb is the number of anchor bolts in a shear span,

and fyp and Ap are the yield strength and the cross-sectional area, respectively, of

the steel plates.

As Equation (5.26) shows, the ratio of the axial plate stiffness to the bolt

connection stiffness, βa /βm , is a constant.

22

byby by

mpb yp by b b yp

ba

m

LS En R REA k E C

f S S f

(5.26)

where nb Sb = L/2 is the length of a shear span. However, the flexural stiffness

ratio βp /βm, which controls the length of the half bandwidth w, is not a constant

but rather increases with increasing plate depth Dp:

2 222

24 12

p by

m p m p ppb ym

pp

LS E CEI k i EA k D D

f

(5.27)

where ip is the radius of gyration of the steel plates. Substituting Equation (5.27)

into Equation (5.24) yields the following expression for the half bandwidth:

1 1

4 220.038 0.10p

wC D

L (5.28)


81

Equation (5.28) demonstrates that the half bandwidth w can be determined once

the strengthening layout is known. It is also evident that w varies linearly with

Dp1/2

and thus is not very sensitive to changes in the plate depth. Hence, in real

strengthening design, BSP beams can be roughly categorised into two types with

respect to the plate depth Dp: shallow plate (Dp < Dc/3) and deep plate (Dp > Dc/2)

cases. Two single values (w/L = 0.155 and 0.250, respectively) can be chosen for

them. The dimensionless half bandwidths w/L of BSP beams with shallow and

deep plates for all basic load cases are listed in Table 5.3.

5.5.6 Support–midspan ratios under different load levels and

beam geometries

Figures 5.16(a) and (b) shows that the variations in the support–midspan

transverse shear transfer ratios (vm, LS /vm, F and vm, RS /vm, F) for different load levels

(F/Fp) and different RC stiffnesses (EI)c are small. However, it is evident from

Figures 5.16(c) and (d) that the ratios vary significantly with increasing βp and

decreasing βm.

Although curve-fitting results similar to Equation (5.28) can be obtained for

the support–midspan transverse shear transfer ratios (vm, LS /vm, F or vm, RS /vm, F),

they are omitted for brevity. This approach is used because their variations in

βp /βm and Dp are similar to that of the half bandwidth (w/L). The ratios for a BSP

beam with shallow steel plates under different load cases are listed in Table 5.3.

For deep steel plates, these ratios can be slightly modified by multiplying them by

the ratio of the w values for deep and shallow plates. For instance, the ratio

vm, LS /vm, F for BSP beams with deep plates under a UDL can be computed as 2.70

= 2.43×(0.400/0.360), where 2.43 is the ratio vm, LS /vm, F for those with shallow

plates and 0.400/0.360 is the ratio of w values for deep and shallow plates.


82

5.5.7 Evaluation of transverse shear transfer and bolt shear force

in BSP beams

The procedure for evaluation of the transverse shear transfer profile and bolt

shear forces in a BSP beam is described in this section. When the geometry of a

BSP beam, its material properties, and the external loads are defined, the values of

parameters such as F, (EI)c , (EI)p, and km , as well as those of the stiffness ratios

((EI)c /(EI)c’, (EI)p /(EI)p’, and km /km’), can be determined. From the sectional

analysis and the loading arrangement, the peak load Fp and hence the load level

F/Fp can be evaluated. Using Equation (5.20), the value of the dimensionless

transverse shear transfer ratio ξFp , which is a function of F/Fp, can then be

obtained. Employing Equations (5.21) ~ (5.23), the values of the transverse shear

transfer factors ζEIc, ζEIp, and ζkm can be computed. The magnitude of the

transverse shear transfer vm, F at the loading points or the midspan of the beam can

then be determined using Equation (5.19).

From Table 5.3, the support–midspan transverse shear transfer ratios

(vm, LS / vm, F and vm, RS / vm, F) and therefore the transverse shear transfers at the

supports can be evaluated. The dimensionless half bandwidth w/L, as shown in

Table 5.3, can be used to locate the point of zero transverse shear transfer. By

combining the transverse shear transfers at specific locations using a piecewise

polyline, the entire transverse shear transfer profile can be determined.

The transverse shear transfer profile of a complicated loading arrangement

can be determined by superimposing the transverse shear transfer profiles of the

individual basic load cases. Using Equations (5.12) and (5.13), the transverse bolt

shear forces can be derived from the transverse shear transfer profile.

5.5.8 Worked example

Consider a simply supported RC beam under a point load (F1 = 250 kN) and a

UDL (q2 = 160 kN/m), as shown in Figure 5.18. The clear span is 4200 mm and

the cross section is 300 mm × 600 mm. Compression reinforcement of 3T10 and

tension reinforcement of 4T25 are employed. Two steel plates of 6 mm × 200 mm


83

are bolted to the side faces of the RC beam by a row of anchor bolts at a spacing

of 350 mm. The material properties are as follows:

30 MPa , 23 GPa

460 MPa , 211GPa

355 MPa , 210 GPa

58 kN , 0.5 mm

c c

y

yp

by

s

p

by

f E

f E

f E

R S

(5.29)

The stiffnesses of the RC beam, the steel plates and the bolt connection can

be computed based on the geometry of the beam and the material properties,

which are given by:

2

2

2

31400 kN m

168 kN m

320 kN m

c

p

m

EI

EI

k

(5.30)

Substituting the stiffnesses into Equations (5.21) ~ (5.23) yields the following

transverse shear transfer factors:

16

8

1

3

1 0.13 log 31400 80000 0.043

1 0.19 log 168 220 13.90

1.80.980

1 0.8 370 320

EIc

EIp

km

(5.31)

The ultimate bending moment, computed from a moment–curvature analysis,

is Mu = 576 kN∙m. Thus, the peak loads when only the point load (F1) or the UDL

(q2) is imposed can be obtained as follows:

,1

1

,2 2

576549 kN

1.05

576 81101 kN

8 4.2

up

up u

MF

L

MF q L L

L

(5.32)


84

Substituting the peak forces (Fp1 and Fp2) into Equation (5.20) yields the

following values:

2

2

2

,1

,

2500.65 0.087

549

160 4.20.30 0.034

1101

Fp

Fp

(5.33)

Substituting Equations (5.31) and (5.33) into Equation (5.19) yields the following

values for transverse shear transfer in the midspan:

, ,1

, ,2

5490.043 13.9 0.98 0.087 6.76 kN m

4.2

11010.043 13.9 0.98 0.034 5.21 kN m

4.2

m F

m F

v

v

(5.34)

Multiplying the midspan transverse shear transfer by the support–midspan

transverse shear transfer ratios in Table 5.3 yields the following transverse shear

transfers at the supports (x = 0 mm and 4200 mm):

, ,1

, ,1

, ,2 , ,2

1.04 6.76 7.0 kN m

0.32 6.76 2.2 kN m

2.43 5.21 12.7 kN m

m LS

m RS

m LS m RS

v

v

v v

(5.35)

By superimposing the transverse shear transfers for both load cases, the

transverse shear transfer as well as the transverse bolt shear force can be evaluated

as follows:

,

,

,

7.0 12.7 19.7kN m

2.2 12.7 14.9 kN m

6.8 5.2 12.0 kN m

m LS

m RS

m F

v

v

v

(5.36)

,

,

,

19.7 0.35 6.9 kN

14.9 0.35 5.2 kN

12.0 0.35 4.9 kN

m LS

m RS

m F

V

V

V

(5.37)


85

The maximum transverse shear transfer and bolt shear force occur at the left

support. Their magnitudes are 19.7 kN/m and 6.9 kN, respectively.

The dimensionless half bandwidths (w/L) for F1 and q2 are 0.133 and 0.360

(see Table 5.3). Because the negative transverse shear transfer near the left

support is influenced by both F1 and q2, and that near the right support is mainly

controlled by q2, the locations where transverse shear transfer is zero can be

approximately computed as follows:

0.5 0.360 0.25 0.1334200 540 mm

2

4200 0.5 0.360 3610 mm

L

R

x

x

(5.38)

The transverse shear transfer profile is obtained by connecting the transverse

shear transfers at specific locations using a piecewise polyline:

0, 540, 1050, 3610, 4200 mm

19.7, 0.0, 12.0, 0.0, 14.9 kN/mm

x

v

. (5.39)

A comparison between the computed transverse shear transfer profile and that

obtained by a NLFEA is shown in Figure 5.19. Very good agreement between the

two profiles is observed.

5.6 CONCLUSIONS

This chapter presented the results of a NLFEA of the longitudinal and

transverse slips and shear transfers in BSP beams. A comprehensive parametric

study of the transverse shear transfer profiles in BSP beams with various beam

geometries under different loading conditions was conducted. The main findings

of this study are summarised as follows:

(1) Bolt connections in BSP beams can be simulated using discrete bolt elements,

which comprise the outer quadrilateral elements simulating the bolt–slip

relationship and the inner triangular elements simulating the bolt shafts. The


86

numerical results derived from the NLFEA show promising agreement with

the experimental results in terms of both the overall load–deflection curve and

the specific longitudinal and transverse shear transfer behaviour.

(2) The profiles of longitudinal and transverse slips correlated very well to those

of the corresponding longitudinal and transverse shear transfers due to the

nearly linear bolt shear–slip properties under working loads conditions.

(3) The longitudinal and transverse shear transfer profiles are affected by the load

arrangement and support condition. The principle of superposition can be

used to estimate the bolt forces under working load conditions. The bolt

forces can be conveniently estimated by the measured bolt slips.

(4) The longitudinal shear transfer profile of a BSP beam subjected to

symmetrical loads is antisymmetrical with regard to the midspan. The

longitudinal shear transfer under an asymmetrical point load is less than that

under a point load at the midspan, and its magnitude at the farther support is

less than that at the nearer support due to longitudinal bolt force equilibrium.

(5) The positive transverse shear transfer in a BSP beam under a point load is

concentrated in the vicinity of the applied load, and the negative transverse

shear transfer is concentrated at the supports. The positive and negative

transverse shear transfers balance each other and satisfy the vertical bolt force

equilibrium requirement.

(6) The half bandwidth of the transverse shear transfer profile and the

support–midspan transverse shear transfer ratios are independent of the

magnitude of the applied load and the flexural stiffness of the RC beam. The

half bandwidth increases with increasing flexural stiffness of the plate and

decreases with increasing bolt stiffness. The half bandwidth increases linearly

with the fourth root of the plate–bolt stiffness ratio, or in other words, the

square root of the plate depth.

(7) The magnitude of the transverse shear transfer is controlled by the magnitude

of the applied load. Because the transverse shear transfer increases drastically

when the load level approaches the peak load, a working load level limit of


87

0.75 should be imposed in the design of BSP beams to avoid excessive

transverse bolt shear force demand.

(8) The transverse shear transfer demand decreases significantly as the flexural

stiffness of the RC beam increases, and increases rapidly as the flexural

stiffness of the plate increases. Therefore, the plate–RC stiffness ratio should

be limited to ensure an acceptable bolt shear force demand.

(9) The design table and formulas provided in this chapter can be used to

determine the transverse shear transfer profiles of BSP beams subjected to six

basic load cases. By superimposing the transverse shear transfer profiles from

individual basic load cases, the transverse shear transfer profile and hence the

critical transverse bolt shear force of BSP beams under more complicated

external load conditions can be evaluated.


88

Table 5.1 Comparison of experimental and numerical longitudinal slips

Specimen F/Fp Slc,exp Slc,num Slc,num/Slc,exp Average

P100B300 1.00 1.503 1.246 0.83 1.00

0.75 0.716 0.635 0.89

0.50 0.306 0.360 1.18

0.25 0.126 0.138 1.10

P100B450 1.00 2.670 2.199 0.82 0.94

0.75 1.120 0.809 0.72

0.50 0.470 0.444 0.94

0.25 0.150 0.193 1.29

P250B300R 1.00 0.290 0.203 0.70 1.07

0.75 0.140 0.198 1.41

0.50 0.090 0.113 1.26

0.25 0.040 0.036 0.90

P250B450R 1.00 0.390 0.325 0.83 1.10

0.75 0.170 0.255 1.50

0.50 0.110 0.146 1.33

0.25 0.060 0.045 0.75

Table 5.2 Comparison of experimental and numerical transverse slips

Specimen F/Fp Str,exp Str,num Str,num/Str,exp Average

P100B300 1.00 0.300 0.283 0.94 1.13

0.75 0.070 0.075 1.07

0.50 0.030 0.036 1.20

0.25 0.010 0.013 1.30

P100B450 1.00 0.230 0.285 1.24 0.93

0.75 0.120 0.089 0.74

0.50 0.040 0.033 0.83

0.25 0.010 0.009 0.90

P250B300R 1.00 0.460 0.426 0.93 1.10

0.75 0.169 0.196 1.16

0.50 0.080 0.088 1.10

0.25 0.030 0.037 1.23

P250B450R 1.00 0.520 0.585 1.13 1.28

0.75 0.190 0.239 1.26

0.50 0.090 0.115 1.28

0.25 0.030 0.044 1.47


89

Table 5.3 Half bandwidth and support–midspan shear transfer ratios

Force location Dimensionless half bandwidth

(w/L < 1/2)

Support–midspan ratios

for shallow plates

(xF /L ≤ 1/2) Shallow plates Deep plates vm, LS / vm, F vm, RS / vm, F

1/12 0.038 0.040 1.22 0.11

1/6 0.100 0.105 1.20 0.21

1/4 0.133 0.145 1.04 0.32

1/3 0.139 0.167 0.92 0.45

5/12 0.145 0.203 0.78 0.56

1/2 0.155 0.250 0.66 0.66

TDL 0.330 0.360 1.33 1.33

UDL 0.360 0.400 2.43 2.43


90

Figure 5.1 The concrete model’s (a) biaxial failure law and (b) equivalent

uniaxial stress–strain curve

(a)

fc

fc

fcef

(3) Compression–tension

(1) Compression–compression

(2) Tension–compression

(4) Tension–tension

ft

ft

σ2

σ1

(b)

εceq

εcd εc0

εct

εcr

ft ef

fcef

E0

Ecc

σcef

(3) Prior peak stress

(1) Prior cracking

(2) Post cracking

wcd

wcr

wc

(4) Post peak stress


91

Figure 5.2 Bond–slip curve from CEB-FIB Model Code 1990 (CEB 1993)

s s1 s2

s3

τf

τmax

τ


92

Figure 5.3 The Bi-linear Steel Von Mises Model’s (a) biaxial failure law and

(b) stress–strain curve

fy

-fy

-fy

fy σ1

σ2

(a)

ε

H

E

fy

-fy

σ

(b)


93

Figure 5.4 Simulation of bolt connection: (a) a bolt element and (b) load–slip

curve comparison

(a)

(b)

0 1 2 3 4 5 6

0

20

40

60

Experimental

Numerical

Bolt

shea

r fo

rce

(kN

)

Slip (mm)

Simulation of the load-slip relationship

Internal node

External node

Simulation of the bolt shaft

Bolt element

Steel plate layer

Concrete layer


94

Figure 5.5 Meshing of (a) the RC beam and (b) the steel plates for P250B450R

Co

ncr

ete

or

stee

l pla

te e

lem

ent

nod

es

wh

ich

are

co

nn

ecte

d t

o b

ond

ele

men

ts

Dis

cret

e re

info

rcem

ent

(a)

(b)

Concr

ete

layer

Ste

el p

late

lay

er

Su

pp

ort

pla

te

lay

er

Lo

adin

g p

late

lay

er


95

Figure 5.6 Comparison of load–deflection curves obtained from the

experimental and numerical studies for (a) P100B300 and P250B300R and (b)

P100B450 and P250B450R

0 20 40 60 80 100

0

100

200

300

400

P250B300R, Numerical

P250B300R, Experimental

P100B300, Numerical

P100B300, Experimental

Load

(kN

)


0 20 40 60 80 100

0

100

200

300

400

P250B450R, Numerical

P250B450R, Experimental

P100B450, Numerical

P100B450, Experimental

Load

(kN

)


(a)

(b)


96


experimental and numerical studies for (a) P100B300 and (b) P100B450

-1800 -1200 -600 0 600 1200 1800

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

F/Fp = 0.75 , Experimental

F/Fp = 0.75 , Numerical



Longit

udin

al s

lip S

lc (

mm

)


-1800 -1200 -600 0 600 1200 1800

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5





Longit

udin

al s

lip S

lc (

mm

)


(a)

(b)


97

(a)

(b)

-1800 -1200 -600 0 600 1200 1800

0.30

0.00

-0.30





Tra

nsv

erse

sli

p S

tr (

mm

)


-1800 -1200 -600 0 600 1200 1800

0.30

0.00

-0.30





Tra

nsv

erse

sli

p S

tr (

mm

)



98

Figure 5.8 Comparison of transverse slip profiles obtained from the

experimental and numerical studies for (a) P100B300, (b) P100B450, (a)

P250B300R and (b) P250B450R

(c)

(d)

-1800 -1200 -600 0 600 1200 1800

0.30

0.00

-0.30





Tra

nsv

erse

sli

p S

tr (

mm

)


-1800 -1200 -600 0 600 1200 1800

0.30

0.00

-0.30





Tra

nsv

erse

sli

p S

tr (

mm

)



99

Figure 5.9 Longitudinal slip and shear transfer profiles of a BSP beam under

an asymmetrical load or two symmetrical loads

Figure 5.10 Variation in the longitudinal shear transfer profile as the position of

imposed load

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Distance from left support (x/L)

1F , Shear transfer

1F , Longitudinal slip

2F , Shear transfer

2F , Longitudinal slip

Norm

aliz

ed l

ongit

udin

al s

hea

r tr

ansf

er t

m

Norm

aliz

ed l

ongit

udin

al s

lip S

lc

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-12

-8

-4

0

4

8

12

1F@1/6

1F@1/3

1F@1/2


Longit

udin

al s

hea

r tr

ansf

er (

kN

/m)


100

Figure 5.11 Transverse slip and shear transfer profiles a BSP beam under an

asymmetrical load or two symmetrical loads

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Norm

aliz

ed s

hea

r tr

ansf

er v

m

Norm

aliz

ed t

ransv

erse

sli

p S

tr

1F , Shear transfer

1F , Transverse slip

2F , Shear transfer

2F , Transverse slip


101

Figure 5.12 Reference beam under (a) a midspan point load, (b) an asymmetric

point load, (c) two symmetric point loads, (d) a uniformly distributed load, (e) a

trapezoidal distributed load, and (f) a triangular distributed load. (dimensions in

mm)

225

35

0

xF = L/2

300

L = 3600 mm

F

50

100

6

xF F

q, F = q L

q, F = q L / 2

(a)

(b)

(c)

(d)

q, F = q (L – xF)

xF

xF

(e)

(f)

F/2 F/2 xF

6T16

2T10


102

Figure 5.13 Variation in the transverse shear transfer profile as the location of (a)

an asymmetrical load or (b) two symmetrical loads

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4

0

-4

vm, F

vm, RS

vm, LS

F

1F , xF/L=1/6

1F , xF/L=1/3

1F , xF/L=1/2


Shea

r tr

ansf

er v

m (

kN

/m)

xF

w

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3

0

-3

vm, RS


2F , xF/L=1/12

2F , xF/L=1/3

2F , xF/L=5/12

2F , xF/L=1/2

Shea

r tr

ansf

er v

m (

kN

/m) vm, LS

vm, F

xFF/2xF F/2

w

(a)

(b)


103

Figure 5.14 Superposition of the transverse shear transfer profiles for (a) two

loads or (b) a uniformly distributed load (UDL)

(a)

(b)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

0

-1

F/2

Superposition

NLFEA


Shea

r tr

ansf

er v

m (

kN

/m)

F/2

vm, F

w

vm, LS vm, RS

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4

0

-4

-8

vm, F

vm, RS

F/5 F/5 F/5 F/5 F/5

Superposition

UDL (NLFEA)


Shea

r tr

ansf

er v

m (

kN

/m)

vm, LS

w


104

Figure 5.15 Variation in the transverse shear transfer base on (a) the load level

and (b) the stiffnesses of RC, plates and bolt connection

0.00 0.25 0.50 0.75 1.000.0

0.2

0.4

0.6

0.8

1.0

1F x/L=1/6

1F x/L=1/4

1F x/L=5/12

1F x/L=1/2

TriangDL

TrapezDL

UDL

1/2

Fp

Load level (F/Fp)

1/2

Fp = 0.65(F/Fp)

1/2

Fp = 0.30(F/Fp)

-2 -1 0 1 2

0.0

1.0

2.0

3.0

1/16

EIc =10.13log[(EI)c/(EI)c']

1/8

EIp =1+0.19log[(EI)p/(EI)p']

3

km =1.8/[1+0.810-log(km/km')

]

1/1

6

EIc

,

1

/8

EIp

,

3

km

log[(EI)c/(EI)c'] , log[(EI)p/(EI)p'] , log(km/km')

(EI)c = 8000 kNm2

(EI)p = 220 kNm2

km = 370 kN/m2

1/16

EIc

3

km

1/8

EIp

(a)

(b)


105

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

0.0

-1.0 F/Fp = 0.25

F/Fp = 0.50

F/Fp = 0.75

F/Fp = 1.00


Norm

aliz

ed s

hea

r tr

ansf

er

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

0.0

-1.0 (EI)c'/(EI)c = 0.01

(EI)c'/(EI)c = 0.1

(EI)c'/(EI)c = 1

(EI)c'/(EI)c = 10

(EI)c'/(EI)c = 100


Norm

aliz

ed s

hea

r tr

ansf

er

(b)

(b)

(a)


106

Figure 5.16 Variation in normalised transverse shear transfer profiles of a BSP

beam under three point bending based on (a) the load level, (b) the RC stiffness,

(c) the plate stiffness, and (d) the bolt stiffness

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

0.0

-1.0


Norm

aliz

ed s

hea

r tr

ansf

er

m'/m = 100

m'/m = 10

m'/m = 1

m'/m = 0.1

m'/m = 0.01

(d)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

0.0

-1.0 p'/p = 0.01

p'/p = 0.1

p'/p = 1

p'/p = 10

p'/p = 100


Norm

aliz

ed s

hea

r tr

ansf

er

(c)

(d)

(c)


107

Figure 5.17 Variation in the half bandwidth of transverse shear transfer profile

of a BSP beam under three point bending

Figure 5.18 A worked example for the evaluation of transverse shear transfer in

a BSP beam (dimensions in mm)

4200

xF = 1050

350

q2 = 160 kN/m F1 = 250 kN

300

60

0

20

0

6

4T25

3T10

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

NLFEA

w/L = 0.07 (p m)1/4

+0.10

w /

L

1

4p m


108

Figure 5.19 Comparison between the computed shear transfer profiles and that

derived from a numerical model

0 1050 2100 3150 4200

15

10

5

0

-5

-10

-15

-20

-25

NLFEA

Piecewise

Distance from left support (mm)

Shea

r tr

ansf

er v

m (

kN

/m)

Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams

109

CHAPTER 6

THEORETICAL STUDY ON LONGITUDINAL

PARTIAL INTERACTION IN BSP BEAMS

6.1 OVERVIEW

The behaviour of BSP beams is very unique and different from normal RC

beams and those retrofitted by attaching steel plates or FRPs to the beam soffit.

Therefore, this chapter introduces the theoretical basis and special terminologies

corresponding to BSP beams in detail.

The formulations of the longitudinal slip, the longitudinal bolt shear force,

and the strain factor that indicates the degree of longitudinal partial interaction are

deduced based on the cross sectional analysis of BSP beams. Then formulas are

developed for BSP beams under various loading conditions, which can be used in

the design practice. The outcomes of the experimental and the numerical studies

reported in Chapters 3 ~ 5 are also extracted to verify the analytical model.

6.2 BASIC CONCEPTIONS ABOUT BSP BEAMS

6.2.1 Longitudinal and transverse slips

Unlike RC beams strengthened with steel plates on the beam soffit, in which

only longitudinal slip exists, both longitudinal and transverse slips coexist on the

plate–RC interface of BSP beams, as illustrated in Figure 6.1. By introducing a

coordinate system with the origin at the left support and the x axis along the beam

axis, which is parallel to the global horizontal and vertical XY coordinate system,

the location of an arbitrary point on the RC beam or the steel plates can be

expressed as a vector or a pair of coordinates as is shown in Figure 6.1(a).

, ,x y X Y OA (6.1)


110

As the BSP beam deforms under external loads as shown in Figure 6.1(b), the

point on the RC beam, which coincides with the point A on the aforementioned

reference coordinates, moves to a new position A1, and its displacement can be

expressed as:

, cx cy 1AA (6.2)

On the other hand, the point on the steel plates, which also coincides with the

point A on the reference coordinates, moves to a different new position A2 due to

the deformability of the bolt connection. Its displacement can be written as:

2, px py AA (6.3)

The difference between these two displacements is the relative slip happening on

the plate–RC interface and can be written as a vector:

,px cx py cy 1 2 2 1S A A AA AA (6.4)

The slip vector S can be divided in x and y directions and expressed as a resultant

combination of the longitudinal and transverse slips as shown in Figure 6.1(c):

,x yS SS (6.5)

px cxxS (6.6)

py cyyS (6.7)

Under the hypothesis of small deformation, the discrepancy between the

deformed xy coordinates and the reference XY coordinates is negligible. Therefore

the longitudinal and transverse slips (Sx and Sy) can be approximated as the

differences between the horizontal and vertical deformations of the steel plates

and the RC beam as:

pX cXxS (6.8)


111

pY cYyS (6.9)

In most cases, the longitudinal and transverse slips vary with the measured

point’s location on the beam. And even on the same cross section, they usually

change very much along the depth of the beam. For the convenience of discussion,

the longitudinal and transverse slips of the points on the centroidal level of the

steel plates (y = ypc), can be chosen as the nominal slips:

, , , , ,pc pc pc pc pcpx y cx y pX ylc x cX yyS S (6.10)

, , , , ,pc pc pc pc pcpy y cy y pY ytr y cY yyS S (6.11)

6.2.2 Partial interaction

Because of the combination of longitudinal and transverse slips, there is large

delay and release of strain and stress in the steel plates compared to its RC beam

counterpart. This phenomenon is named as partial interaction.

The longitudinal slip causes the delay in the longitudinal deformation of the

steel plates and hence reduces their axial strain. Meanwhile, the transverse slip

leads to the reduction in the vertical deflection of the steel plates thus decreases

their curvature.

6.2.3 Strain and curvature factors

The degree of partial interaction, which is caused by the longitudinal and

transverse slips, controls the performance of the BSP beams. Its effect can be

quantified by two indicators, the strain factor and the curvature factor (Siu 2009).

As illustrated in Figure 6.2, the strain factor αε is defined as the longitudinal strain

ratio between the steel plates and the RC beam at the centroidal level of the steel

plates, and is used to denote the axial strain looseness of the steel plates due to the

longitudinal slip; the curvature factor αφ is defined as the curvature ratio between


112

the steel plates and the RC beam, and is used to denote the curvature reduction of

the steel plates due to the transverse slip.

,

,

pc

pc

p y

c y

(6.12)

p

c

(6.13)

Where the plate-centroidal plate and RC strains equals the derivatives of the

plate and RC longitudinal displacements at the plate centroidal level and read:

,

, ,

d'

d

pc

pc pc

px y

p y px yx

(6.14)

,

, ,

d'

d

pc

pc pc

cx y

c y cx yx

(6.15)

Corresponding to the plane strain assumption, the RC strain at the plate-centroidal

level can be expressed by that at the RC-centroidal level as:

, ,p cccc y c y cp ci (6.16)

Where icp is the separation between the RC and plate centroidal axes:

cp pc ccyi y (6.17)

And the plate and RC curvatures (φp and φc) are:

p

p

p

M

EI (6.18)

c

c

c

M

EI (6.19)


113

6.2.4 Axial and flexural stiffnesses

The capacity of a member resisting extension can be measured by the axial

stiffness, i.e., the product of its cross sectional area and the material’s elastic

modulus (EA). And the capacity of a member resisting bending can be measured

by the flexural stiffness, in other words the product of its cross sectional moment

of inertia and the material’s elastic modulus (EI). For a BSP beam, it is composed

of two components connected by a series of anchor bolts, i.e., the RC beam and

the steel plates. Their axial and flexural stiffnesses can be written as:

d d

c s

c sc

A A

E E A E AA (6.20)

d

p

pp

A

A EE A (6.21)

2 2d d

c s

c sc

A A

E y A EE y AI (6.22)

2d

p

pp

A

E yE AI (6.23)

By defining the effective radii of gyration of the RC beam and the steel plates

referring to their centroidal axes ic and ip, the flexural stiffness can be expressed in

terms of the axial stiffness as:

2,c

c cc c

c

EIi EI EA i

EA (6.24)

2,

p

p pp p

p

EIi EI EA i

EA (6.25)

In most cases, the RC beam of a BSP beam performs as a compressive

bending member while the steel plates behave as tensile bending members as

shown in Figure 6.2. The bending resistant capacity of the BSP beam can be


114

measured to some extent by the sum of the flexural stiffnesses of the two

components as following:

cp c p

EI EI EI (6.26)

However, its extension resistant capacity cannot be measured by the sum of

the axial stiffnesses of the two components, since the RC beam is under

compression while the steel plates are subjected to tension. The resultant axial

stiffness can be expressed as the harmonic mean of the axial stiffnesses as:

1

1 1c p

cp

c p

c p

EA EAEA

EA EA

EA EA

(6.27)

In addition to the flexural stiffnesses provided by those of these two

components, the coupling behaviour offered by the RC compression and the plate

tension also contribute to the bending bearing capacity of the BSP beam. So the

overall flexural stiffness should be as:

2

cp pSP cp cBEI EI EA i (6.28)

6.2.5 Plate–RC and bolt–RC stiffness ratios

The behaviour of longitudinal and transverse slips of BSP beams is highly

controlled by the relative stiffness ratios among the three components, the RC

beam, the steel plates, and the anchor bolt connection. For convenience of

discussion, three parameters, i.e., the ratios between the axial and the flexural

stiffnesses of the steel plates and the RC beam, and that between the stiffness of

the anchor bolt connection and the flexural stiffness of the RC beam can be

defined as follows:

p

a

c

EA

EA (6.29)


115

p

p

c

EI

EI (6.30)

mm

cI

k

E (6.31)

Where km depends on both the stiffness of the anchor bolts Kb and the bolt spacing

Sb as:

b

m

b

kK

S (6.32)

6.2.6 Longitudinal and transverse shear transfers

In order to conduct the cross-sectional analysis of a BSP beam, an elemental

segment with a length of dx is illustrated in Figure 6.2. The BSP beam is

subjected to an external load q, which causes an internal bending moment M and

an internal shear force V in the BSP beam section.

Shear forces (Tm, i and Vm, i) are transferred from the RC beam to the steel

plates in both longitudinal and transverse directions as shown in Figures 6.2(b)

and (c). The internal moment Mp, shear force Vp and tensile force Np arise in the

steel plates. The internal moment Mc, shear force Vc and compressive force Nc that

is the opposite force of the plate tension Np also arise in the RC beam. The two

components work together to resist the bending moment, and the total resisting

moment M is a result of those provided by the flexural stiffnesses of the RC beam

and the steel plate (Mc and Mp) and the coupling effect offered by the plate tension

and the RC compression (Tm·icp), which are given by:

c p m cpM iM TM (6.33)

d

c

c

A

c ccM y A EI (6.34)


116

d

p

p pp

A

pM y A EI (6.35)

where Tm is the longitudinal bolt shear force, and it is equal to the plate tension Nc

and the RC compression Np due to the pure bending condition:

,m m i c pT T N N (6.36)

,dcc

c

c c y cc

A

N A EA (6.37)

,dpc

p

p pp p

A

yN A EA (6.38)

The longitudinal bolt shear force Tm is the sum of the discrete bolt shear

forces Tm, i , Tm, i+1 , … and Tm, i+n . It can be divided by the bolt spacing Sb, thus

simplified as a continuous shear stress tm, which is termed as the longitudinal

shear transfer as:

d

d

m b lcm m lc

b

Tx S

x S

St

Kk (6.39)

, dm m i mT T t x x (6.40)

where Kb = Rby /Sby is the bolt stiffness, which can be determined from bolt shear

tests; Rby is the yield shear force of the bolt; and Sby is the corresponding yield

deformation. Assuming that the bolt behaves in elasto-plastic manner, the bolt

stiffness of the shear force–deformation relation in the elastic region is denoted by

Kb. km = Kb /Sb is the bolt stiffness per unit length. If uniform bolt spacing is used,

km is a constant along the beam thus:

dm m lcT k S x (6.41)

The transverse bolt shear force Vm can also be simplified as a continuous

shear stress vm, which is termed as the transverse shear transfer as:


117

d

d

m b trm

b

trm

Vx S

x S

Sv

Kk (6.42)

, dm m i mV V v x x (6.43)

dtm rmV k S x (6.44)

Both the strain and curvature of the steel plates are smaller than those of the

RC beam, due to the partial interaction caused by the longitudinal and transverse

shear transfers (εp, ypc < εc, ypc and φp < φc).

The hypothesis of Bernoulli beam is applied to the RC beam, the steel plates,

and the BSP beam, and the following basic derivatives are available:

d

'd

y

yx

(6.45)

d

'dx

(6.46)

d

'd

MV M

x (6.47)

d

'd

Vq V

x (6.48)

6.2.7 Lightly and moderately reinforced RC beams

The structural behaviours of RC beams are controlled by tensile steel ratio ρst,

and can be classified into two categories as under- and over-reinforced beams by

the balanced steel ratio ρstb, at which the yielding of the outermost tensile

reinforcement layer and the crushing of concrete occur simultaneously. As the

over-reinforced beams are seldom used in structural design, they are not

considered in this study. As an under-reinforced beam performs differently for


118

different tensile steel ratios, the following definition of lightly and moderately

reinforced beams is adopted in the subsequent discussion as shown in Figure 6.3.

A lightly reinforced beam, whose reinforced degree ρst /ρstb is less than 1/3,

fails in a ductile mode. Its flexural strength is less than 40% of that of the

balanced-reinforced beam, thus it can be enhanced significantly by adding

external reinforcement with a small sacrifice in ductility.

In contrast, a moderately reinforced beam, whose reinforced degree ρst /ρstb is

greater than 2/3, fails in a brittle mode. Its flexural strength is already more than

80% of that of the balanced-reinforced beam, thus adding external tensile

reinforcement cannot increase its flexural strength significantly but cause a very

brittle failure with little ductility.

6.2.8 Shallow and deep steel plates

The steel plates in a BSP beam retrofit the RC beam by both their flexural

stiffness φp (EI)p and the additional eccentric-compression-force effect icp Np . The

proportion of these two effects can be identified by the modulus ratio Ip: Apicp2,

which is the ratio between the second moments of area of the steel plates with

regards to the plate-centroid and the RC-centroid as shown in Figure 6.4:

2

23

232 1

212 3 3 3 3

2 1 12

12 2 2 4 3

1: ,

12:

: ,2

p pc c cp

c

p p cp

p pc c cp

c

t D D Dt

DA

t

D

D

ID D

iD

tD

(6.49)

For the shallow steel plates whose depth Dp /Dc < 1/3, the modulus ratio

Ip: Apicp2 is less than 1/12; thus the error caused by neglecting the flexural stiffness

(EI)p and treating them as additional tensile rebars might be acceptable. However,

for the deep steel plates whose depth Dp /Dc > 1/2, the modulus ratio Ip: Apicp2 is

great than 1/3; thus their flexural stiffness can no longer be neglected.


119

6.3 LONGITUDINAL SLIP IN BSP BEAMS

6.3.1 Longitudinal slip profile

Longitudinal slip is a result of the longitudinal shear transfer tm between the

steel plates and the RC beams. Behaving as additional tensile reinforcements, the

steel plates resist considerable tensile forces in the region where significant

bending moment exists. This tensile force needs to be transferred back to the RC

beams through the anchor bolts.

The longitudinal slip on the plate–RC interface is quite similar to the bond

slip on the rebar–concrete interface. For an infinite BSP beam as shown in

Figure 6.5(a), the longitudinal slip Slc and hence the longitudinal shear transfer tm

concentrate in a finite region near the maximum bending region. The longitudinal

slip is zero at the point where the plate tensile force Np reaches its maximum

Np, max, and increases on both sides as the increase of the longitudinal shear

transfer tm. Beyond a certain distance wsla, both Slc and tm cease ascending and

reach a maximum, then begin to descend and diminish to zero at a distance wsl.

If the half-length of the steel plates Lph is less than the length required for the

longitudinal shear transfer in an infinite BSP beam, i.e., Lph < wsl, the longitudinal

slip and tensile force transfer will take place along the whole steel plates. Their

magnitudes are magnified to offer the same amount of tension resistance within a

shorter distance as shown in Figure 6.5(b). Furthermore, if the half-length of the

steel plates is further reduced such that Lph < wsla, the longitudinal slip and tensile

force transfer reach a maximum at the plate ends. Their magnitudes are further

magnified as shown in Figure 6.5(c).

For typical retrofitting practices, the span of the RC beams to be strengthened

and the length of the steel plates utilized are relatively short. Therefore, the

longitudinal slip profiles are limited to the situation as shown in Figure 6.5(c),

thus certain simplifications described hereafter can be adopted.


120

6.3.2 Governing equation

According to Equation (6.10), the longitudinal slip at the centroidal level of

the steel plates (y = ypc) can be chosen as the nominal longitudinal slip:

, ,pc pcpc y xl x c yS (6.50)

By differentiating Equation (6.50) with respect to x and substituting Equations

(6.14), (6.15), (6.16), (6.37), (6.38), (6.36) and (6.27) into it, we have:

, ,

, ,

, ,

d dd

d d d

1 1

1

pc

pc

cc

pc

pc

pc

px y cx y

p y c y

p y c y cp c

cp c

m cp c

m c

lc

p c

p c

p

p

p

c

c

c

S

x x x

i

N Ni

EA EA

T iEA EA

T iEA

(6.51)

Substituting Equations (6.34) and (6.35) into Equation (6.33) gives the total

resistant moment M as:

ppc cp mcEI EIM i T (6.52)

According to the results obtained from the experimental and the numerical

studies reported in Chapters 4 and 5, it is evident that the magnitude of the

transverse slip is less than 1/10 of that of the longitudinal slip. So it is acceptable

to neglect the effect of the transverse slip in the formulation of the longitudinal

slip. Under this hypothesis, the vertical deflections of the RC beam and the steel

plates are identical along the entire beam span. Therefore the curvatures of the

two components are the same.

c p (6.53)


121

By substituting Equation (6.53) into Equation (6.51) we have:

d 1

d

lc

cp

m cp

ST i

x EA (6.54)

Substituting Equations (6.53) and (6.26) into Equation (6.52) yields:

cp mc

cp mc

p

p

EI EI i T

EI i

M

T

(6.55)

cp m

cpI

M i T

E

(6.56)

By substituting Equation (6.56) into Equation (6.54), and further substituting

Equation (6.28) into it we have:

2

2

d 1

d

1

cp m

m cp

cp

cp cp

m

cp cp

cpcp cp

lc

cp

cp

cp

B

cp

m

cp cp

cp

m

cp cp

SP

cp

i TST i

x EA EI

i iT

EA EI EI

EI EA i iT

EA EI EI

M

EI iT

EA EI I

M

M

ME

(6.57)

Differentiating Equation (6.57) with respect to x and substituting Equations (6.39)

and (6.47) we have:

2

2

d d d

d d d

cpm

cp cp

cp

m

cp cp

lc BSP

cp

BSP

cp

MEI iS T

x EA EI x EI x

EI it

EA EIV

EI

(6.58)

Differentiating Equation (6.41) twice gives:


122

2

2

d d1

d d

lc m

m

S T

x k x (6.59)

Substituting Equation (6.59) into Equation (6.57) yields the governing equation of

Tm as:

2

2

d0

d

BSP

cp

m m cpmm

cp cp

k EI k iTT

x EA EI IM

E

(6.60)

Substituting Equation (6.39) into Equation (6.58) gives the governing equation of

Slc as:

2

2

d

d0lc BSP

cp

m cp

lc

cp cp

k EI iSS

x EA EI EIV

(6.61)

Equations (6.60) and (6.61) give two ordinary differential equations (ODE) of

the second order for the longitudinal slip Slc and the longitudinal bolt shear force

Tm . Similar formulations were developed by Newmark et al. (1951) for a

composite beam that was composed of an RC slab and a steel beam. By

introducing a parameter as following:

2 BS

p

m

p

P

c c

k EI

EAp

EI

(6.62)

The governing equations can be simplified and read:

2

2

2

d0

d

m cpm

m

cp

xx

k iTp T

x EIM x

(6.63)

2

2

2d0

d cplc

lc

cp

iS xp S x V x

EIx (6.64)


123

After a series of transformations and substitutions of Equations (6.30) and (6.31),

the parameter p2 can also be expressed in the form of the plate–RC and bolt–RC

stiffness ratios as:

2

2

2

2 22

1 1

1

1 1

p c

p c

p cpcm

cp

m

c p

cpm c c

c p c

pp

ik

EA EA EI EI

EI EI ik

EI EA EA EI

p

i ii

EI

(6.65)

From Equation (6.56), we have:

cp

m

cp

M EIT

i

(6.66)

Differentiating Equation (6.66) twice and introducing Equations (6.47) and (6.48):

2 2 2

2 2 2

2

2

d d d

d d d

d

d

1

1

m

cpcp

cpcp

T MEI

x i x x

q EIi x

(6.67)

Substituting Equations (6.66) and (6.67) into Equation (6.63) yields:

2 2

2

2

d 10

dcp BSP

pp q

xM

x EI EIx x x

(6.68)

Substituting Equation (6.45) into Equation (6.46) yields:

2

2

d''

d

y

yx

(6.69)

Then replace the curvature φ in Equation (6.68) with δy’’, we obtain:


124

2

4 2 2

2 2

d d 10

d d

y y

cp BSP

pp q

x xM

x x EI Ex

Ix

(6.70)

In conclusion, the equations of the longitudinal slip, shear transfer, and bolt

shear force, along with those of the curvature and the vertical deflection, are all

obtained by introducing the following two hypotheses: (1) The influence of

transverse slip is neglected, thus the vertical deflections and curvatures of the steel

plates and the RC beams are synchronized along the beam span; (2) The shear

force–slip performance of the anchor bolts follows a linear relation.

Both the governing equations for the longitudinal slip Slc and the longitudinal

bolt shear force Tm are second order ODEs. The general solutions of

Equations (6.64) and (6.63) are as follows:

1 2 3e epx px

lcS x C C C (6.71)

1 2 3e epx p

m

xx D DT D (6.72)

Thus the profiles of both the longitudinal slip Slc and the longitudinal bolt shear

force Tm can be easily obtained by the combination of the general solutions and

appropriate boundary conditions.

By substituting Equations (6.51) and (6.41) into Equation (6.12), the profile

of the strain factor αε , which indicates the degree of the longitudinal partial

interaction, can be obtained as:

, ,

,,

1 1

d d d1 1d d d d

pc pc

pc

pc

l

p y p y

c yp y

m lc

c p plc lc

m

S EA EAS Sx x T x k S x

(6.73)

Hence, substituting Equation (6.71) into Equation (6.73) gives the expression of

the strain factor profile.


125

6.4 LONGITUDINAL SLIP IN BSP BEAMS UNDER

VARIOUS LOADING CONDITIONS

6.4.1 Under four-point bending

For a simply supported BSP beam subjected to two point forces imposed at

the two trisectional points as shown in Figure 6.6, only the left half of the beam

needs to be considered owing to the symmetry of the beam geometry and loading

arrangement. The distribution of transverse shear force V and bending moment M

can be represented as:

, 0 3

0 , 3 2

F x L

LV x

x L

(6.74)

, 0 3

3, 3 2M x

x

F x x L

F L L L

(6.75)

Both V and M are piecewise linear functions, thus substituting Equations (6.71)

and (6.74) into Equation (6.64) yields the governing equations expressed by

piecewise functions as:

2

,1 ,1

2

, ,2 2

'' 0 0 31

2

,

' 0 , 3'

cp

lc lc

p c

lc lc

ix

FS x p S x

EI

S x p S x x

L

L L

(6.76)

The general solution of the above SODEs can be written as:

,1 1 1 2

,2 2 2

e e 0 3

e

,1

e 3 2,

cppx px

lc

p c

px px

lc

iS x A B x L

S

F

p

x xA B

EI

L L

(6.77)

According to Equation (6.51), we have:

, ,'pc pcp y c ylcS x x x (6.78)


126

Because the bending moment at the supports is zero, therefore both strains in the

RC beam and the steel plates equal to zero. The longitudinal slip (Slc) should be

zero at midspan due to the symmetry. Furthermore, the longitudinal slip and its

first derivative should satisfy the continuity conditions at the loading point. In

conclusion, the boundary and continuity conditions can be stated as:

,1

, 2

,1 ,

,1

2

2,

' 0 0

2 0

3 3

' 3 ' 3

lc

lc

lc lc

lc lc

S

S L

S L S L

S L S L

(6.79)

Substituting Equation (6.77) into Equation (6.79) gives the longitudinal slip

profile as:

2

cosh1 , 3

3 2

where

02cosh 3 1

sinh 3sinh ,2

cosh 2

1

:

F

lc

F

cp

F

p c

pxA x

pL

pLA p xL x

pL

L

S

F iA

p

L

E

x

I

L

(6.80)

The maximum longitudinal slip occurs at the plate ends (i.e., x = 0):

,max

11

2cosh 3 1lc FSpL

A

(6.81)

The longitudinal tensile force in the steel plates reaches its maximum at the

midspan (i.e., x = L/2), and the magnitude reads:

,max

2sinh 6

3 2cosh 3 1Fp m

pLLN A

p pLk

(6.82)

Substituting Equation (6.80) into Equation (6.73) gives the strain factor

profile as:


127

1

2

1

2

1 31

h

1 , 3 22cos

, 02cosh 3

h 13 1

6 sinh 6 cosh 2

1

sin

p m

p m

L

px

L LL

EA kx

x pL

p px

EA kx

pL

pp pL p L x

(6.83)

With reference to Equation (6.80), parameter AF is controlled by the magnitude of

the external load F. However, parameter AF no longer appears in Equation (6.83),

indicating that the strain factor is independent of the magnitude of the external

loads. The minimum strain factor occurs at the loading point (i.e., x = L/3) as:

1

,min

2

12cosh 13 1

3 sinh 3

p m

L pL

EA

p pL p

k

(6.84)

The minimum strain factor can also be approximated by the value at the midspan

(i.e., x = L/2), where Np, max occurs, we have:

,mi

1

,

n

max

sinh 31

cosh 2

p

F

p

pLEApA

pLN

(6.85)

6.4.2 Under three-point bending

For a simply supported BSP beam under a point force F applied at a location

with a distance of

, ( 0.5)F F Fx a L a (6.86)

away from the left support as shown in Figure 6.7. The distribution of shear force

V and bending moment M can be expressed as:


128

1 , 0

,F F

FFa F x x

a LV x

xF x

(6.87)

1 , 0

,F

F F

F

M xL

a Fx x x

a F x x Lx

(6.88)

Both V and M are piecewise linear functions, thus substituting Equations (6.86)

and (6.87) into Equation (6.64) yields the governing equations as well as the

boundary and continuity conditions:

2

,1 ,1

2,2

2

,

10'' 0 ,

1

'' 0 ,1

F cp

lc lc F

p

F cp

lc lc F

p

c

c

a FS x p S x

EI

a FS x p S x x

E

ix x

I

ix L

(6.89)

,1

,2

,1 ,

,1 ,

2

2

' 0 0

' 0

' '

lc

lc

lc F lc F

lc F lc F

S

S L

S x S x

S x S x

(6.90)

Similarly, solving the ODE problem gives the longitudinal slip profile as:

1

2

1

cosh

cosh

where:

1

, 01

.

1

sinh;

sinh

sinh.

sinh

F

F

F

F

F a FF

lc

F a F F

cp

c

F

F

p

a

F

a

x xa

a x L

F iA

p EI

p L

pL

p L

pL

A pxS x

A p x L x

a

a

(6.91)

The maximum longitudinal slip occurs at the plate end closer to the imposed load

(i.e., x = 0), hence:


129

1,max1

Fa Flc FaS A (6.92)

The plate tensile force in the steel plates reaches its maximum at the location

where there is no longitudinal slip on the plate–RC interface (i.e., Slc (xNpm) = 0).

Its magnitude is:

,max

21 1

arcc

1sinh

osh n 1

1

l

F

F FF

Npm Npma Fm F

F FF

a aa

p

Npm

p x xN ak

a a

Ap

aL

px L

p

(6.93)

Comparison of Equations (6.86) and (6.93) shows that the location of the

maximum plate tension Np, max, i.e., the location of zero longitudinal slip, does not

coincide with the position of the maximum bending moment. This phenomenon is

very different from the common conception and should be born in mind when

conducting related practical design.

The maximum value can also be approximated by the value at xNpm and the

general expression for the strain factor is:

12

1

1 , 01 1 csch

F

p mF

a

p EA kx x

a px pxx

(6.94)

When aF = 0.5, the BSP beam is subjected to three-point bending at the

midspan point, and Equation (6.91) is simplified to:

cosh2

cosh 2

11 , 0

2lc FA x

pxS x L

p L

(6.95)

The maximum longitudinal slip, tensile plate tensile force and the minimum

strain factor are simplified to:

,max

1 sec 2

2

hlc FS

p LA

(6.96)


130

,max

sech coth

4 2mp F

pL pLLN A

pk

(6.97)

,mi

1

,

n

1 cos

21

h

sinh

p

F

p max

E pLpA

N pL

A

(6.98)

6.4.3 Under a uniformly distributed load

Let’s consider a simply supported BSP beam under a uniformly distributed

load (UDL) q applied along the whole span as shown in Figure 6.8. The

distribution of shear force V and bending moment M can be expressed as:

2

LV x q x

(6.99)

2

L x xM x q

(6.100)

Substituting Equation (6.99) into Equation (6.64) yields the governing equation as

well as the boundary and continuity conditions:

2 2'' 0

1

cp

lc lc

p c

Lx q

S x p S xEI

i

(6.101)

' 0 0

02

lc

lc

S

LS

(6.102)

The solution of the ODE is given by:


131

2

sinh 2 22, 0

2 cosh 2

where:1

2lc F

cp

F

p c

L xL xA x

p pL

q iA

L

E

S x

p I

p

(6.103)

The maximum longitudinal slip occurs at the plate ends (i.e., x = 0), the

maximum plate tensile force and the minimum strain factor occur at the midspan

(i.e., x = L/2). Their magnitudes are as follows:

,max

tanh 2

2lc F

pLLS A

p

(6.104)

2

2,max

sech 12

8m Fp

pLLN A

pk

(6.105)

1

,

,min1 1 sech 2

p

F

p max

EAA pL

N

(6.106)

6.4.4 Under a triangularly distributed load

Again, let’s consider a simply supported BSP beam subject to a triangularly

distributed load along the whole span, in which the load at the left support is zero

and that at the right support is q as shown in Figure 6.9. The distribution of shear

force V can be expressed as:

2 23

6

L xV x q

L

(6.107)

Substituting Equation (6.107) into Equation (6.64) yields the governing equation

as well as the boundary and continuity conditions:


132

2 2

2

3

6'' 01

cp

lc lc

p c

L xq

LS x p S xE

i

I

(6.108)

' 0 0

' 0

lc

lc

S

S L

(6.109)


2

2

2

cosh11

sinh6 2

where:1

lc F

c

F

p c

p

S xp

pxL xA

p LL p

q iA

EI

L

p

(6.110)

The maximum longitudinal slip occurs at the right plate end (i.e., x = L), the

maximum plate tensile force and the minimum strain factor attain at the location

(xNpm) where there is no longitudinal slip on the plate–RC interface, which can be

easily obtain by solving the equation Slc (xNpm) = 0, and their magnitudes read:

,max 2

1 1coth

3lc F

LS A L

p pp

L

(6.111)

2 2

,max 2

sinh1

6 sinh

Npm Npm Npm Npm

Fp m

x L x px xN A

L p Lk

pL

(6.112)

1

,

,min

sinh1

sinh

Npmp Npm

F

p max

pxxEAA

N L pL

(6.113)


133

6.4.5 Under a support moment

Let’s consider a simply supported BSP beam under a moment MS at the left

support as shown in Figure 6.10. The distribution of shear force V and bending

moment M can be expressed as:

SMV x

L (6.114)

1S

xM x M

L

(6.115)

Substituting Equation (6.115) into Equation (6.63) yields the governing equation

as well as the boundary and continuity conditions:

2

1

'' 01

m cp

l lc

S

c

p c

xk M

LS x p S x

E

i

I

(6.116)

' 0 0

' 0

lc

lc

S

S L

(6.117)


2

1sinh cosh tanh 2

where:1

lc F

m cp

F

p c

S

AS x

M

L x px px pLp

k iA

p E LI

(6.118)

The maximum longitudinal slip occurs at the left plate end (i.e., x = 0), the

maximum plate tensile force and the minimum strain factor attain at the location

(xNpm) where there is no longitudinal slip on the plate–RC interface, which can be

easily obtain by solving the equation Slc (xNpm) = 0, and their magnitudes read:

,max

1tanh

2lc F

pLS A L

p

(6.119)


134

2,max

2 cosh tanh 2 sinh 1

2

Npm Npm Np Npm

m

m

Fp

L x x px pL pk

xA

pN

(6.120)

,min

1

,

1 1 cosh tanh sinh2p

Npm NpmF

p max

EApx pxpA

NL

(6.121)

6.4.6 Under pure bending

6.4.6.1 Superposition of longitudinal slip

Because linear material properties are assumed in the formulation of the

longitudinal slip in BSP beams, the superposition principle should be able to

utilize in the analysis. The basic conception of the superposition of longitudinal

slip is illustrated in Figure 6.11. Substituting the independent variable x with L−x

in Equation (6.110) gives the longitudinal slip profile of a BSP beam under a

triangularly distributed load along the whole span, in which the load at the left

support is q and that at the right support is zero, as follows:

2

2

2

cosh11

sinh6 2

where:1

lc F

c

p C

p

F

L xL L x pS x

pA

p LL p

q iA

p EI

L

(6.122)

Adding up Equation (6.110) and the negative of Equation (6.122), in other

words superimposing the longitudinal slip profiles under these two loading

conditions (see Figure 6.11) gives the resultant superimposed profile as:

2

cosh cosh1

sinh2

where:1

2lc F

cp

F

p C

L xpxLA

p L

q iA

S

p I

pxx

p

E

(6.123)


135

After several steps of transformation as shown in Equation (6.124), it is

evident that this resultant superimposed profile is equal to that under a UDL as

shown in Equation (6.103):

2

cosh cosh cosh sinh sinh1

2sinh 2 cosh 22

cosh cosh sinh sinh2sinh 2 11

2sinh 2 cosh 2

2

2

2

cosh cosh1

sinh

2

2

F

F

lc F

F

Lpx pL px pxLA

p L L

Lpx px px

px

p p

ppx LLA

p L Lp

LA

p

L xpxL

L

pxS x

pA

p

22sinh 2 cosh 2sinh 2 cosh 2 sinh1

2sinh 2 cosh 22

sinh 2 cosh cosh 2 sinh1

co

2

2

sh 22

sinh 2 21

cosh 22

2

F

F

L L Lpx px

p L L

L Lpx pxLA

p L

p p px

p p

p px

p

L xLA

p

pp L

x

(6.124)

Therefore, the longitudinal slip profile of a BSP beam under a UDL can be

derived from the superposition of those under two triangularly distributed loads

(see Figure 6.11).

6.4.6.2 Longitudinal slip under pure bending by using superposition

In the case of a simply supported BSP beam under pure bending, as the

previous case, the superposition can be used to obtain the longitudinal slip profile.

Substituting the independent variable x with L−x in Equation (6.118) gives the

longitudinal slip profile of a BSP beam under a moment MS at the right support

and reads:

2

1sinh cosh tanh 2

where:1

S

lc F

m cp

F

p c

A x pLp pL x LS x

M

xp

k iA

p LEI

(6.125)


136

Adding up Equation (6.118) and the negative of Equation (6.125), in other

words superimposing the longitudinal slip profiles under these two loading

conditions gives the resultant superimposed profile as shown in Figure 6.12:

2

22 sinh cosh tanh

2

where:1

lc F

m cp

F

p

S

c

pLA L x px px

p

k iA

LEI

x

M

p

S

(6.126)

The maximum longitudinal slip occurs at the two plate ends (i.e., x = 0 and L),

the maximum plate tensile force and the minimum strain factor achieve at the

midspan (i.e., x = L/2) and read:

,max

2tanh

2lc F

pLS A L

p

(6.127)

2

2,max

sech 2 1

8Fp m

pLLN A

pk

(6.128)

1

,

,min1 1 sech 2

p

F

p max

EAA pL

N

(6.129)

6.5 VERIFICATION

6.5.1 Verification by the experimental results

The experimental and theoretical profiles of the longitudinal slip Slc and the

plate tensile force Np of Specimens P100B300 and P100B450 at two load levels

(F/Fp = 0.25 and 0.75) are shown in Figures 6.13 and 6.14 respectively. The

figures indicate that the experimental and theoretical profiles are in good

agreement for both the longitudinal slip and the plate tensile force of BSP beams,

despite some minor discrepancies at several discrete points such as at the plate

ends for the longitudinal slip and at the midspan for the plate tensile force.


137

In order to study the variation of the maximum longitudinal slip Slc, max with

the external load, the secant moduli Kb, 0.10 = 231 kN/mm, Kb, 0.30 = 104 kN/mm,

and Kb, 0.75 = 37 kN/mm at the shear force level Fb /Fbp = 0.10, 0.30, and 0.75,

respectively, were chosen in the shear force–slip response curves of the

“HIT-RE 500 + HAS-E” anchoring system (see Figure 6.15) for the subsequent

analysis of the maximum longitudinal slip and the maximum plate tensile force in

the specimens. The comparison between the experimental and theoretical

maximum longitudinal slips (Slc, max) at various load levels is illustrated in

Figure 6.16. The figures show that the theoretically predicted Slc, max is

proportional to the load level F/Fp. It can be seen that the predicted Slc, max reduces

significantly as the increase in the stiffness of bolt connection km (i.e., the secant

modulus of anchor bolts Kb); for instance, the upper-boundary prediction (by

using Kb, 0.75 = 37 kN/mm) is about 4 times of the lower-boundary prediction (by

using Kb, 0.10 = 231 kN/mm). When compared to the linear variation of the

theoretical prediction, the ascending rate of the experimental Slc, max increases as

the increasing F/Fp. This is because Slc, max occurs at the plate ends, and hence it is

mainly controlled by the plate-end anchor bolts whose behaviours at high load

levels are highly nonlinear. In short, an upper and a lower boundary solution are

needed for the estimation of the maximum longitudinal slip Slc, max in practical

design. When the load level is low (F/Fp ≤ 0.50), the lower-boundary prediction

using a nearly elastic bolt modulus Kb, 0.10 gives an accurate prediction. On the

other hand, when the load level is relatively high (F/Fp ≥ 0.75), the

upper-boundary prediction using a lower bolt modulus Kb, 0.75 should be chosen to

yield a conservative prediction.

The comparison between the experimental and theoretical maximum plate

tensile forces (Np, max) at various load levels is illustrated in Figure 6.17. Similar to

the previous discussion, the predicted Np, max also increases proportionally to the

load level F/Fp. However, its variation is bounded by a smaller range of the bolt

modulus Kb. The upper-boundary prediction (by using Kb, 0.10 = 231 kN/mm) is

nearly less than 1.2 times of the lower-boundary prediction (by using

Kb, 0.30 = 104 kN/mm). Moreover, the experimental Np, max also increases nearly

proportional to F/Fp, despite a slight reduction in the ascending rate. This is

because Np, max yields at the midspan, and hence it mainly depends on the shear


138

resistance of the mid-region anchor bolts that deform only slightly and remain

almost linear elastic during the whole loading process. In general, the

upper-boundary prediction using a nearly elastic bolt modulus Kb, 0.10 yields a

satisfactory conservative prediction for the whole loading process.

6.5.2 Superposition for longitudinal slip under weak non-linearity

The outcomes of the numerical study as described in Chapter 5 are employed

to check if the superposition principle is still valid for the real BSP beams in their

early stage of loading where weak material non-linearity exists. As shown in

Figure 6.18, the longitudinal slip profile of a BSP beam under a point load at the

left trisectional point is added to that under a point load at the right trisectional

point, and the resultant superimposed profile is compared with the longitudinal

slip profile under two point loads of the same magnitudes at both the trisectional

points. This figure indicates that the two profiles coincide very well. Therefore,

the superposition principle is proved to be applicable to the analysis of the

longitudinal slip in BSP beams at their early loading stage.

6.6 CONCLUSIONS

In this chapter, a new analytical model for the longitudinal partial interaction

was proposed. The longitudinal slip and shear force transfer in BSP beams were

deduced based on the BSP beam section analysis. The formulation considered

force equilibrium, deformation compatibility, and continuity requirements. Linear

elastic material properties and simply supported boundary conditions were

assumed for simplicity in the analysis. The results of the experimental study

reported in Chapters 3 and 4 were introduced to verify the theory for a loading

case of four-point bending. Then the theoretical analysis was extended to solve

other practical loading cases. The main outcomes of this study are as follows:

(1) In BSP beams, the steel plates act as additional reinforcement and develop

tensile force through the interfacial shear transfer of bolt connection. For an


139

infinite BSP beam, the longitudinal slip is zero at the location of the maximum

plate tensile force, and increases on both sides then begins to reduce and

becomes negligible after a certain distance. In practice, the span of BSP beams

is short and the maximum longitudinal slip usually occurs at the plate ends.

(2) The ODEs for the longitudinal slip of BSP beams under various loading cases,

such as four-point and three-point bending, uniformly and triangularly

distributed load, support moment and pure bending, were established based on

the governing equation. Then the profile of longitudinal slip was obtained

according to appropriate boundary and loading conditions. The formulas for

the maximum longitudinal slip, the maximum plate tensile force, and the

minimum strain factor were obtained as well.

(3) Comparison between the theoretical and the experimental profiles of the

longitudinal slip and plate tensile force of two BSP beams under four-point

bending were conducted, and good agreements were observed.

(4) The maximum longitudinal slip Slc, max of BSP beams occurs at the plate ends.

Its magnitude depends on the load level and the bolt modulus used in the

calculation. When the load level is low (F/Fp ≤ 0.50), the lower-boundary

prediction using a nearly elastic bolt modulus Kb, 0.10 gives an accurate

prediction. On the other hand, when the load level is high (F/Fp ≥ 0.75), the

upper-boundary prediction using a lower bolt modulus Kb, 0.75 should be

chosen to yield a conservative prediction.

(5) The plate tensile force of BSP beams reaches its maximum Np, max near the

midspan and increases almost proportionally to the load level F/Fp. In general,

the upper-boundary prediction using a nearly elastic bolt modulus Kb, 0.10 can

yield a conservative prediction of Np, max during the whole loading process.

(6) The superposition principle is applicable to the analysis of longitudinal slip

and shear transfer of BSP beams in their early stage of loading where weak

material non-linearity exists.


140

Figure 6.1 Illustration of longitudinal and transverse slips

Str

Slc

S

A

A

x

y

A1

A2

A1

A2

S

x

y

A

(a)

(b)

(c)

O

O

O

Y

X

Y

X

Y

X

RC beam

Steel plate

RC beam

Steel plate

Original

position

Deformed

position


141

Figure 6.2 External and internal forces in a BSP beam

(a)

(b)

(c)

ycc ypc

ycc

ypc

M+dM

V+dV

M

V

q

φc

φp

εp, ypc

εc, ypc

εc, ycc

φc

εc, ycc

Mc

Nc

Mc+dMc

Nc+dNc

Vc+dVc

Mc

Vc

Nc

q

Vm, i+2 Vm, i+1

Vm, i

Tm, i+2 Tm, i+1 Tm, i

Tm, i+2 Tm, i+1 Tm, i

Vm, i+2 Vm, i+1 Vm, i

φp

εp, ypc

Mp

Np

Mp+dMp

Np+dNp

Vp+dVp

Mp

Vp

Np

dx

dx

dx

vc

vp

ypc

icp


142

Figure 6.3 Definition of lightly and moderately reinforce concrete beams

Figure 6.4 Definition of (a) shallow and (b) deep steel plates

Dc /2 icp

(a) (b)

Dp

icp

Dc /3

Dp

Np

φp(EI)p Np

φp(EI)p

The centroid of RC beam

The centroid of steel plate

Balanced-reinforecd

Moderately reinforecd

Lightly reinforecd

0.00 0.01 0.02 0.03 0.04 0.050.0

0.2

0.4

0.6

0.8

1.0

1.2

st /stb = 1

st /stb = 2/3

st /stb = 1/3

Norm

aliz

ed m

om

ent

M/M

b,m

ax

Curvature (rad/m)


143

Figure 6.5 Variation in the longitudinal slip as the length of steel plates (a)

wsl < Lph, (b) wsla < Lph < wsl and (c) Lph < wsla

wsl

wsla

Lph

Longitudinal

slip profile

Steel plate RC beam

wsl

Lph

wsla

wsl

wsla

Lph

(a)

(b)

(c)


144

Figure 6.6 The profiles of shear force, bending moment and longitudinal slip in

a BSP beam under four-point bending


a BSP beam under arbitrary three-point bending

F

L

xF = aF L aF ≤ 0.5

(1−aF) L

V

M

Slc

xF = aF L (1−aF) L

xNpm L−xNpm

F F

L

L/3 L/3 L/3

M

V

Slc

x


145


a BSP beam under a uniformly distributed load (UDL)


a BSP beam under a triangularly distributed load (TDL)

L

q

V

M

Slc

L

q

V

M

Slc


146


a BSP beam under a support moment

L

V

M

Slc

MS


147

Figure 6.11 Illustration of superposition for longitudinal slip in BSP beams; (a)

force superposition and (b) longitudinal slip superposition

(b)

(a)

q

q

q

Slc = Slc,1 + Slc,2

Slc,2

Slc,1


148

Figure 6.12 Superposition for longitudinal slip in a BSP beam under pure

bending; (a) force superposition and (b) longitudinal slip superposition

(b)

(a)

MS

MS

MS

MS

Slc = Slc,1 + Slc,2

Slc,2

Slc,1


149


experimental and theoretical studies for (a) P100B300 and (b) P100B450

-1800 -1200 -600 0 600 1200 1800

-1.5

0.0

1.5

F/Fp = 0.75, Experimental

F/Fp = 0.75, Theoretical



Longit

udin

al s

lip S

lc (

mm

)


-1800 -1200 -600 0 600 1200 1800

-1.5

0.0

1.5





Longit

udin

al s

lip S

lc (

mm

)


(b)

(a)


150

Figure 6.14 Comparison of longitudinal tensile force transfers obtained from the


-1800 -1200 -600 0 600 1200 1800

0

250

500





Pla

te t

ensi

on f

orc

e N

p (

kN

)


-1800 -1200 -600 0 600 1200 1800

0

250

500





Pla

te t

ensi

on f

orc

e N

p (

kN

)


(b)

(a)


151

Figure 6.15 Shear force–slip curves of the “HIT-RE 500 + HAS-E” anchoring

system

0 1 2 3 4 5 6

0

20

40

60

Sample 1

Sample 2

Sample 3

Mean value

Kb, 0.10 at Fb/Fbp = 0.10

Kb, 0.30 at Fb/Fbp = 0.30

Kb, 0.75 at Fb/Fbp = 0.75

Bolt

shea

r fo

rce

Fb (

kN

)

Slip (mm)

Kb, 0.30

Kb, 0.10

Kb, 0.75


152

Figure 6.16 Comparison of the maximum longitudinal slips obtained from the


0.00 0.25 0.50 0.75 1.00

0.0

1.0

2.0

Upper boundary prediction :

Kb=Kb, 0.75 at Fb /Fbp = 0.75

Lower boundary prediction :

Kb=Kb, 0.10 at Fb /Fbp = 0.10

Experimental

Cri

tica

l lo

ngit

udin

al s

lip S

lc,

max

(m

m)

Load level F/Fp

0.00 0.25 0.50 0.75 1.00

0.0

1.0

2.0


Kb=Kb, 0.75 at Fb /Fbp = 0.75


Kb=Kb, 0.10 at Fb /Fbp = 0.10

Experimental

Cri

tica

l lo

ngit

udin

al s

lip S

lc,

max

(m

m)

Load level F/Fp

Fb /Fbp = 0.75

Fb /Fbp = 0.10

Experiment

Fb /Fbp = 0.75

Fb /Fbp = 0.10

Experiment

(b)

(a)


153

Figure 6.17 Comparison of the maximum plate tensile forces obtained from the


0.00 0.25 0.50 0.75 1.00

0

200

400


Kb=Kb, 0.10 at Fb /Fbp = 0.10


Kb=Kb, 0.75 at Fb /Fbp = 0.30

Experimental

Cri

tica

l pla

te t

ensi

le f

orc

e T

pm

(kN

)

Load level F/Fp

0.00 0.25 0.50 0.75 1.00

0

200

400


Kb=Kb, 0.10 at Fb /Fbp = 0.10


Kb=Kb, 0.75 at Fb /Fbp = 0.30

Experimental

Cri

tica

l pla

te t

ensi

le f

orc

e T

pm

(kN

)

Load level F/Fp

Fb /Fbp = 0.10 Fb /Fbp = 0.30

Fb /Fbp = 0.10

Fb /Fbp = 0.30

(b)

(a)


154

Figure 6.18 Verification of superposition for longitudinal slip in BSP beams; (a)

force superposition and (b) longitudinal slip superposition

-1800 -1200 -600 0 600 1200 1800

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3


1F + 1F superposed

2F concurrently

Longit

udin

al s

lip (

mm

)

F F

F

F

(a)

(b)

F F

Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams

155

CHAPTER 7

THEORETICAL STUDY ON TRANSVERSE

PARTIAL INTERACTION IN BSP BEAMS

7.1 OVERVIEW

Due to the complicated nature of the transverse partial interaction of the

transverse slips and shear transfer in BSP beams, it is almost impossible to obtain

a closed-form analytical solution. In this chapter, a simplified piecewise linear

analytical model is proposed for the transverse shear transfer in BSP beams, based

on a set of shear transfer profiles obtained from the nonlinear finite element

analysis (NLFEA) as described in Chapter 5. Winkler's model and the force

superposition principle are employed to evaluate the shape of the proposed

piecewise linear model. The magnitude of the piecewise linear shear transfer

profile is determined by considering the force equilibrium and displacement

compatibility conditions. The results of the experimental study as shown in

Chapters 3 and 4 are used to verify the analytical model.

For the convenience of strengthening design, the outcomes of the numerical

study reported in Chapter 5 are also introduced to the analytical model to achieve

simple formulas for the maximum transverse slips and the minimum curvature

factor, which is the indicator of the degree of the transverse partial interaction.

7.2 SIMPLIFIED PIECEWISE LINEAR MODEL

7.2.1 Simplification of shear transfer profiles

Figure 7.1 presents the shear transfer profiles of a simply supported BSP

beam under different loading arrangements, extracted from the previous numerical

non-linear finite element analysis (NLFEA) reported in Chapter 5. Figure 7.1(a)

shows a load case in which a BSP beam is subjected to a point load at midspan.


156

Both the positive shear transfer arising from the applied point load and the

negative shear transfer caused by the support reactions are found to be localised in

a small region; there is no interaction between the positive and negative shear

transfer profiles. The magnitude of the opposing shear transfer caused by the

applied load is relatively small. Figure 7.1(b) shows a load case in which the point

load is closer to the right support. The positive and negative shear transfer profiles

overlap with each other. The opposing shear transfers caused by the applied load

and the reactions in the overlapping regions cancel each other out. The negative

shear transfer at the support closer to the applied load increase, while that at the

other end decreases to achieve force equilibrium. Figure 7.1(c) presents a load

case in which two widely separated point loads are imposed on the BSP beam

simultaneously. There is no interaction between the two shear transfer regions.

Figure 7.1(d) shows the last load case, in which the two point loads are close to

each other. The two shear transfer profiles are found to overlap and interact with

each other. The positive shear transfer in the overlap region accumulates due to

the force superposition effects and the opposing shear transfers outside this region

increase to maintain vertical force equilibrium.

It is worth noting that each of these profile curves can be simplified as a

piecewise linear polyline. Therefore, a simplified piecewise linear model may be

developed for determining the shear transfer profile in BSP beams. The basic

assumptions of the proposed model are as follows:

(1) The shear force–slip relationship of bolt connections is linearly elastic.

(2) The small deformation flexural theory, i.e., the Bernoulli hypothesis, is

adopted for both the RC beam and the steel plates.

(3) The parabolic positive shear transfer distribution is simplified as a triangular

profile composed of piecewise straight lines, as shown in Figures 7.1(a) ~ (c).

(4) When adjacent loads are close to each other, the increase in shear transfer in

the overlap region is neglected, as shown in Figure 7.1(d).

(5) The negative shear transfer distribution near the support is also simplified as a

linear profile.


157

Based on the above assumptions, the proposed piecewise linear shear transfer

model for a simply supported BSP beam under arbitrarily point loads is illustrated

in Figure 7.2. It can be observed that each applied point load (Fi) acting on the

beam span induces an isosceles-triangle-shaped stress block for the positive shear

transfer, with a maximum magnitude of vm, i and a width of 2wi. The support

reactions induce right-triangle-shaped stress blocks for negative shear transfers,

with a peak value of vm, LS for the left support and vm, RS for the right support.

7.2.2 Shear transfer according to Winkler’s model

According to Winkler’s model (Kerr 1964), if a point load is imposed on an

infinite elastic beam resting on an elastic half-space, the reaction acting on the

elastic beam is concentrated in the vicinity of the applied load. The maximum

reaction force occurs directly under the applied load and diminishes in both

directions. Alternatively, if a point force acts at the end of a semi-infinite beam on

an elastic half-space, the reaction reaches a maximum at the loading end and

approaches zero at the other end.

In this study, the shear transfer between an RC beam and the steel plates in a

BSP beam (see Figure 7.3) is simulated using Winkler’s model. The RC beam

behaves as an elastic beam supported by an elastic medium, which is formed by

the bolt connections and the bolted steel plates. For an infinite BSP beam

subjected to a point load at the midspan, the positive shear transfer is concentrated

in the vicinity of the applied load, with a maximum at the loading point. However,

unlike an elastic half-space, which can support all the applied loads, the bolted

steel plates transfer the shear forces away from the loading region and back to the

RC beam to achieve equilibrium of the vertical forces. This action of bolted steel

plates is referred to herein as “opposite shear transfer.” According to

Saint-Venant’s principle, the opposite shear transfer is relatively large near the

positive shear transfer region and diminishes to zero at some distance away from

the positive shear transfer region (see Figure 7.3(a)).

Because anchor bolts have finite shear stiffness, when the vertical shear force

is transferred through the bolt connections, shear deformations may occur. In this


158

study, shear transfer is denoted by vm. The transverse slip Str is defined as the

deflection difference between the steel plates and the RC beam. The intensity of

shear transfer in the BSP beams is assumed to be proportional to the transverse

slip (vm = km Str). For a coordinate system whose origin is at the point load F and

whose x axis is coincident with the beam axis, the equation that governs the

transverse slip profile Str (x) of the BSP beam can be expressed as follows:

4

4·

d

dtr

trc

xx

x

SEI k S (7.1)

01 p

cm

Ck EI

e

(7.2)

where k is the equivalent modulus of the supporting medium, the formula for

which contains an undetermined constant C0.

When the steel plates are very flexible, e.g., βp = 0, or the stiffness of the

connection medium is very high, e.g., βm = ∞, there will be no transverse slip

between the steel plates and the RC beam. The equivalent modulus k will be very

high, as shown in Figure 7.4(a). Alternatively, when the flexural stiffness of the

steel plates is very high, i.e., βp = ∞, the equivalent modulus becomes a function

of the connection stiffness only (or k = km), as shown in Figure 7.4(b). When the

connection stiffness is zero, i.e., βm = 0, there will be no connectivity between

these two components, thus k = 0. The extreme conditions of connectivity can be

summarised as follows:

0,0

,

,

,0

k

kk

k

k

m

mp

m

p

(7.3)

If a parameter β, as shown in Equation (7.4), is introduced, the general

solution of the governing equation can be written in terms of several

undetermined integration constants, as shown in Equation (7.5).


159

0

1 1

4 41

4 14 p

m

c

CeEI

k

(7.4)

1 2 3 4cossin si con sx x

trS C Ce x C x e x C xx (7.5)

Because the steel plates are not connected to any other supports, the positive and

the opposite shear transfers should attain self-equilibrium. Furthermore, the shear

transfer at infinity should be equal to zero. Given these two conditions, some of

the undetermined integration constants can be computed directly:

0

1

2

3 4

00 0

d 0

tr

tr

CSC

S xxC C

(7.6)

Substituting these integration constants back into Equation (7.5), the transverse

slip profile Str (x) and the shear transfer profile vm (x) can be written as follows:

4

4

sin

sin

cos

cos

t

x

m

x

r x C

v

S e x x

k x xx eC

(7.7)

where both profiles are a combination of cosine and sine functions with periods of

2π/β. The remaining undetermined integration constant C4 is governed by the

support conditions.

Similarly, the shear transfer of a semi-infinite BSP beam subjected to a point

load at its end can also be solved using Winkler’s model. For brevity, the

formulation is omitted from this paper. The typical shear transfer profiles

computed from Equation (7.7) for an infinite and a semi-infinite BSP beams are

illustrated in Figures 7.3(a) and (b), respectively.

The half bandwidth of the positive shear transfer is denoted as w, while the

width of the opposite shear transfer is expressed as w’. The magnitudes of w and

w’ can be derived as follows:


160

0

0

1

4

1

4

1

1'

44 4

4

p

p

m

m

C

C

w

ew

e

(7.8)

It is obvious that neither w nor w’ is independent of the intensity of the point load

F but rather increase as the plate–RC stiffness ratio βp increases and the bolt–RC

stiffness ratio βm decreases. The widths defined in Equation (7.8) are very useful

in determining the shape of the piecewise linear shear transfer profile.

Figure 7.3 also indicates that both the positive and the negative shear transfer

profiles can be represented by a polyline. The opposite shear transfer is ignored

due to its small intensity.

It should be noted that after cracking occurs in the concrete, the flexural

stiffness of the RC beam reduces. The region of the RC beam between the two

point loads might deform slightly upward (see Figure 7.5) due to the vertical

reaction forces exerted by the steel plates on the concrete beam. Hence, the

positive shear transfer between the two loads might reduce.

7.2.3 Solution based on force equilibrium and deformation

compatibility

To illustrate the implementation of the piecewise linear model, the shear

transfer profile of a simply supported BSP beam under four-point bending (see

Figure 7.6(a)) is expressed as a piecewise linear function controlled by several

undetermined constants. The force equilibrium and deformation compatibility

requirements are then used to determine these undetermined constants.

If the x axis is defined along the undeformed beam axis and originating from

the left support, the shear transfer profile vm, which is controlled by three

undetermined constants, ξS, ξF, and ξw, can be expressed as a piecewise linear

function that is symmetrical with respect to the midspan.


161

, ,

1

1 3 3

1 3 11

3

1 3 11

3

10

2

where

1

, ,: ,

S

F

m

F

F

w

w

w

w

w

m F m S

w S

v vx

Fv

L

F L F L

w

L L

(7.9)

As Figure 7.6(b) shows, w is the half bandwidth of the positive shear transfer, vm, F

and vm, S are the shear transfers at the loading point and the support. Because the

transverse load on the steel plates qp is equivalent to the shear transfer vm as

shown in Figure 7.6(d), the vertical force equilibrium of the steel plates gives:

1 1

0 0d d 0p mq v (7.10)

By substituting Equation (7.9) into Equation (7.10) and further solving it, one of

the unknown constants, for instance ξS, can be expressed in terms of the others:

2

1 3

wS

w

F

(7.11)

Substituting Equation (7.11) back into Equation (7.9) yields the following:

2

1 3 36 1

31 3

1

3 3

1

3 3

10

1 3

3

2

3

1 3

w w

w

w

w

m p F

w

w

w

w

Fv q

L

(7.12)


162

Using the Bernoulli hypothesis for both the RC beam and the steel plates, the

relationship between the vertical deflection and the transverse shear force can be

expressed as follows:

4

4

d

dEI q

(7.13)

Hence, the relative deformation of a steel plate with respect to its left end, i.e., the

free shape of the steel plates under the shear transfer vm, can be expressed as

follows:

3

2

32 2

2

3

2 2

2 2

1

324

13 4 3 36

3

136 1 3 9 4 15 12 4 1 3

3

19 4 15 9 4 17 12 1 6 41 15 18

3

19 4 21 9 4 21 4 45 72 162

2

p

F w

w w w

w w w w

w w w w w w w

w w w w w

pf

FL

EI

(7.14)

Equation (7.14) can be expressed in matrix form as follows:

2

2

3

2 3

3

2

4

1,

3241

0 0 12 9 108 0 0 0 0

36 108 36 135 108 4 36 108 108where:

36 135 36 153 108 4 39 108 108

36 189 36 189 0 4 45 72 162

1 3

pf F

T

w w

w w w

w w w w

p

FL

EI

pf

pf

p

f

f

pA W Ω

A

W

Ω1 3

1 3

1 2

w

w

(7.15)


163

Assuming that δp0 is the deflection of the steel plates at the left end caused by the

transverse slip, the vertical deflection of the steel plates δp can be written as (see

Figure 7.6(f)):

0p p pf (7.16)

Because the RC beam is of a different flexural stiffness (EI)c = (EI)p / βp but

subjected to the same opposite shear transfer as the steel plates (see Figure 7.6(c)),

its deformation under the shear transfer should be multiplied by a factor −βp as:

cm p pf (7.17)

The deflection of the RC beam under the applied four-point bending is as follows:

3

2

2

3

3

127 18

3

127 18

1 3

116227 27 1

3

127 27 1

2

w

ce

cw

FL

EI

(7.18)

Equation (7.18) can also be expressed in the matrix form (see Figure 7.6(c)) as

follows:

3

231,

162

1

27 0 18 0

27 0 18 0where:

0 27 27 1

0 27 27 1

ce

c

FL

EI

ce

ce

A Ω

A

(7.19)


164

According to the superposition principle, the deflection of the RC beam is

controlled by the sum of that under the applied four-point bending and that under

the opposite transverse shear transfer (see Figures 7.6(c) & (e)). Thus,

c ce cm c pfpe (7.20)

The deflection difference between the steel plates and the RC beam is as

following (see Figure 7.6(g)):

0 1+p c p p pf ced (7.21)

If the deflection difference is defined as the transverse slip, using the linear shear

force–slip relationship, the shear transfer becomes the following (see Figures

7.6(h) & (i)):

0' 1+m m m p cp pf ev k d k (7.22)

Using the shear force equilibrium condition of the steel plates (i.e., Equation

(7.10)), the deflection of the steel plates at the left end (δp0) can be expressed as:

13

2 3 4 5

0

11152 15 648 540 648

486 5832w w w w w

c

p

Fp

FL

EI

(7.23)

Equation (7.23) may be expressed in matrix form as follows:

13

2 3 4 5

0

111

486 5832

where: 52 15 648 540 648

p

p F

T

w w w w w

c

FL

EI

p0 p0

p0

p0

A W

A

W

(7.24)

Substituting Equations (7.15), (7.19) and (7.24) into Equation (7.22) yields the

following expression for shear transfer vm’(ξ):


165

2

1

3

2

4 1 1' 1

5832 3241

1 11

162 486

1

m Fm p

Fv L

L

p0 p0 pf

ce

pfA W A W

A D

(7.25)

Equation (7.25) shows that the resultant shear transfer vm’ is directly proportional

to the bolt–RC stiffness ratio βm and linearly proportional to the applied load F

and the undetermined constant ξF . It should be noted that vm’ is a polynomial

function of order 5 with respect to ξw and of order 3 with respect to ξ.

According to the deformation compatibility requirement, the resultant shear

transfer (vm’ in Equation (7.25)) derived from the deflection difference should be

equal to the assumed shear transfer (vm in Equation (7.12)) along the whole beam

span. It should be noted that vm is a piecewise linear function, while vm’ is a cubic

polynomial function. Although their forms are different, both of them are

controlled by the undetermined constants ξw and ξF. If deformation compatibility

must be satisfied along the whole beam span, these undetermined constants can be

determined by least-squares fitting (LSF):

1

1

2

2 20

, , '

, , , d

w F m m

w F w F

h v v

h h

(7.26)

2

2

0

0

w

F

h

h

(7.27)

For computational efficiency, deformation compatibility is enforced at several

crucial points. In this way, the bolt forces obtained will be of acceptable accuracy

for general engineering applications. For brevity, only this method is discussed

herein. The plate end (i.e., ξ = 0) and the loading point (i.e., ξ = 1/3) are selected

to be the maximum points for a beam under four-point bending.


166

' 0 0m mv v (7.28)

1 1

'3 3

m mv v

(7.29)

Substituting ξ = 0 into Equations (7.12) and (7.25), then equating them (i.e.,

Equation (7.28)), the following polynomial equation of order 1 with respect to ξF

and order 6 with respect to ξw is obtained.

6

5

4

1

3

2

4 1

0 1944

0 972

0 2484 31 132 0

0 693 1 1

0 141

34992 52

w

w

ww

w

w

T

T

p F

m wL

(7.30)

The above equation indicates that ξF can be expressed explicitly in terms of ξw as:

3 4 5 12 6 4

132 1 3

1 1 52 141 693 2484 972 1944 34992

w

w w w w w w mwp

FL

(7.31)

By substituting ξ = 1/3 and Equation (7.31) into Equations (7.12) and (7.25), and

then into Equation (7.29), the following 6th-order polynomial equation with

respect to ξw is obtained.

6

5

4

1

3

4 1

2

0 9720

0 16524

0 89641 132 0

0 1881

0 111

110808 4

w

T

m

w

w

p

w

w

w

F

L

L

(7.32)

Equation (7.32) does not have an explicit solution. However, when the

plate–RC and bolt–RC stiffness ratios (βp and βm), the clear span (L) and the

applied load (F) are known, this polynomial equation can be easily solved by


167

available numerical methods, such as the bisection method, Newton’s method or

the secant method. Noted that for a feasible solution, 0 ≤ ξw ≤ 1/3, the solution

should also satisfy the following condition because the total shear transfer

reaction should not be great than the applied load:

1 3

1 3d

w

wmv F

(7.33)

By discarding all the unreasonable solutions, a unique solution for ξw can be

determined. The maximum shear transfer at the loading point (vm, F = ξF F/L) can

be obtained by substituting the solution ξw into Equation (7.31). Furthermore,

inserting ξw and ξF into Equations (7.11) and (7.12), the maximum shear transfer

at the support (vm, S = ξS F/L) and the profile function vm(ξ) along the entire beam

span can be computed. Once vm is known, by dividing it by the bolt connection

stiffness km , the transverse slip along the whole beam span can be determined

from the following expression:

m

tr

m

vS

k

(7.34)

The curvature factor, which is the indicator of the degree of the transverse

partial interaction and equal to the ratio between the curvatures of the steel plates

and the RC beam (αφ = φp /φc), can also be obtained by its definition.

7.2.4 Experimental verification

Comparisons between the experimental and theoretical shear transfer profiles

for P100B300 at three load levels (F/Fp = 0.25, 0.5, and 0.75) are shown in

Figure 7.7. It is evident that the piecewise linear model predicts the shear transfer

profiles very well for the whole loading process, even though the concrete

material response becomes highly nonlinear in the later loading stages.

Comparisons between the experimental and theoretical shear transfer profiles

for the other specimens (P100B450, P250B300R and P250B450R) at the load


168

level F/Fp = 0.5 are shown in Figure 7.8. These figures indicate that the piecewise

linear model is generally capable of predicting the behaviour of shear transfer in

BSP beams of different beam geometries within an acceptable degree of error.

According to Equation (7.8), the width of the shear transfer block (2w) is

independent of the applied load, although a constant term involving the applied

load F can be found in the approximate solution (see Equation (7.32)). The

resultant shear transfer profiles derived from Equation (7.32) also indicate that the

variation in w is very small for all three load levels (see Figure 7.7). Furthermore,

according to Equation (7.8), the ratio between the shear transfer half bandwidths

of the specimens with a bolt spacing of 300 mm and those with a bolt spacing of

450 mm is as follows:

1 14 4, B450B300

, B300B450

300 0.90

450

m

m

kw

kw

(7.35)

The experimental results indicate that the corresponding ratios obtained from

P100B300 and P100B450 and those from P250B300R and P250B450R are 0.91

and 0.93, respectively. The theoretical ratio agrees very well with the

experimental ratios.

7.3 APPROXIMATE SOLUTION FOR

STRENGTHENING DESIGN

It is obvious that Equation (7.32) it is not convenient for the strengthening

design of BSP beams due to its complicated form without an explicit solution. On

the other hand, the numerical study in Chapter 5 showed that the half bandwidth

of shear transfer profile w is independent of the load level F/Fp and the stiffness of

RC beam (EI)c, but varies as the square root of the depth of steel plates Dp1/2

. So

in the real strengthening design, the BSP beams can be roughly categorized into

two different types corresponding to the plate depth Dp. Two single values of w

can be chosen for them respectively, and the subsequent discrepancy is acceptable.

Therefore, approximate solutions are presented in this section by introducing the


169

proposed relative half bandwidth of shear transfer ξw to the solution technique

based on force equilibrium and displacement compatibility for BSP beams under

different loading cases.

7.3.1 Under four-point bending

Substituting the following numerical results of ξw in Chapter 5 into Equations

(7.31) and (7.11) yields the maximum shear transfer ratios ξF and ξS as following:

0.139 for shallow plates

0.167 for deep platesw

(7.36)

11 1

1

4

14 1

0.046 1 63.2 for shallow plates

0.050 1 89.1 for deep plates

p m

F

p m

L

L

(7.37)


2.0 for deep platesS

F

F

(7.38)

Further substituting Equations (7.36) ~ (7.38) into Equations (7.12) and (7.34),

the maximum transverse slips (Str, max) at the supports (i.e., x = 0) and the loading

points (i.e., x = L/3) can be computed as follows:

3

4

,max 3

1

0

4 1

for shallow plates0.032 1 44.4

for deep plates0.025 1 44.4

m pc

m p

tr x

c

FL

EI

FL

EI

LS

L

(7.39)

,max 0

,max 3

,max 0


0.5 for deep plates

tr x

tr x L

tr x

SS

S

(7.40)

The minimum curvature factor (αφ, min) occurs at the midspan (i.e., x = L/2)

and its magnitude is:


170

11

4

,min 112 4

1.8 0.8 2500 for shallow plates

3.6 2.7 6500 for deep plates

p p m

p p m

x L

L

L

(7.41)

7.3.2 Under three-point bending

Similarly, for a BSP beam under three-point bending, by introducing the

corresponding numerical results of ξw in Chapter 5, the maximum shear transfer

ratios at the midspan (ξF at x = L/2) and the supports (ξS at x = 0) can be obtained

as follows:



(7.42)

11 1

1

4

14 1

0.051 1 34.6 for shallow plates

0.092 1 76.8 for deep plates

p m

F

p m

L

L

(7.43)


1.00 for deep platesS

F

F

(7.44)

Then the transverse slip (Str) can be obtained as following:

2

1

4

2

1

4

2.9, 0.3

0.114 1 77.0

0.0

5

2.9. 0.

23 1 15

5.60

for shallow platesc

c

m p

m p

tr

FL L xx L

EI

FL L xx L

EI

L

L

S x

(7.45)

3

4 1

4.

0.092 1 7

0

7.0for deep plates

mc p

tr

FL L x

LEIS x

(7.46)

The maximum transverse slips (Str, max) at the supports (i.e., x = 0) and the

midspan (i.e., x = L/2) can be computed as:


171

3

4

,max 3

1

0

4 1



m pc

m p

tr x

c

FL

EI

FL

EI

LS

L

(7.47)

,max 0

,max 2

,max 0


1.0 for deep plates

tr x

tr x L

tr x

SS

S

(7.48)


and reads:

11

4

,min 12 14



p p m

p

L

p m

x

L

L

(7.49)

7.3.3 Under a uniformly distributed load

It is evident from the numerical study in Chapter 5 that the shear transfer

profile of a BSP beam under a uniformly distributed load (UDL) can be simulated

by several uniformly spaced point loads. For brevity, instead of simulating the

UDL by too many point loads, the shear transfer profile under UDL is

approximated by that under three point loads with some modification (see

Figure 7.9). The shear transfer between adjacent point loads is simulated by

connecting the maximum shear transfers at the loading points using piecewise

lines. The value of ξw derived from the numerical study in Chapter 5 is employed

for the two point loads at the quartering points (ξw) as following.



(7.50)

By introducing Equation (7.50) into the solution technique based on force

equilibrium and displacement compatibility, the maximum shear transfer ratios at


172

the quartering points (ξF,1), the midspan (ξF,2) and the supports (ξS) can be

obtained as follows:

4 4

,14 4

10 2600 1 for shallow plates

7 5600 1 for deep plates

m p p m p

m p

Sp

F

Dm pp p

L L C

L L C

(7.51)

4 4

,24 4

1.6 254900 1 for shallow plates

3.6 72300 1 for deep plates

m p p m Sp

F

p

m p p m p Dp

L L C

L L C

(7.52)

4 4

4 4



m p p m p

m p p

Sp

S

Dpm p

L L C

L L C

(7.53)

22 48

2 48 2

1 28200 1 5500

1 16900 1 7200

for shallow plates

for deep plates

m p p m p p

m p p m p p

L L

L L

C

(7.54)

The maximum transverse slips (Str, max) at the supports (i.e., x = 0) and the

midspan (i.e., x = L/2) can be computed as follows:

0

4

4,maxS

tr xm c

qL

L EIS

(7.55)

2

4

4,max 2

Ftr x L

m m c

SqL

L EI

(7.56)


and reads:

4

,min 2 4

4

4



m p m p

m p m p

x L

L L D

L L D

(7.57)


173

2

8 2

8

4

4

1 0.65

20100 0.9 5000

1 0.65

14600 0.8 5500

for shallow plates

for deep plates

m p p

p m p p

m p p

p m p p

L

L

L

L

D

(7.58)

7.4 CONCLUSIONS

Since a closed-form analytical solution for the transverse partial interaction of

BSP beams is difficult to obtain, this chapter proposed a simplified piecewise

linear analytical model for the transverse shear transfer of BSP beams. The shape

of the piecewise linear shear transfer profile is derived from Winkler’s model and

the force superposition principle. The magnitude of the piecewise linear profile is

obtained by considering force equilibrium and deformation compatibility.

Available experimental results were used to verify the accuracy of the proposed

model. Approximate formulas convenient for strengthening design are also

proposed. Based on the results of this study, the following conclusions are drawn:

(1) The magnitude of the shear transfer is found to be controlled by the magnitude

of the applied load. However, the widths of the positive and opposite shear

transfer blocks are controlled by the stiffnesses of the RC beam, the steel

plates and the bolt connection and not by the applied load.

(2) After concrete cracking, the RC beam deforms upward slightly due to the

degradation of its flexural stiffness. This reduction in flexural stiffness due to

cracking causes a decrease in the positive shear transfer.

(3) The proposed piecewise linear shear transfer model has been proved to be

capable of predicting shear transfer behaviour during the entire loading

process for BSP beams under four-point bending loads.

(4) The experimental results support the theoretical conclusion that although the

width of the shear transfer block is independent of the applied load, it


174

increases with increasing plate–RC stiffness ratio and decreasing bolt–RC

stiffness ratio.

(5) The proposed piecewise shear transfer model and the approximate formulas

for the maximum transverse slip and the minimum curvature factor, which can

be used for the strengthening design of BSP beams, is of great practical

significance.


175

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.0

-0.6


Shea

r tr

ansf

er (

kN

/m)

[email protected]

Simplification as a polyline

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.6

0.0

-0.6

[email protected]



Shea

r tr

ansf

er (

kN

/m)

(b)

(a)

Opposite shear

transfer

Positive shear transfer

Negative shear

transfer


176

Figure 7.1 Shear transfer profiles of a BSP beam under (a) a point load at the

midspan, (b) a point load close to the support, (c) two point loads close to the

supports, and (d) two point loads close to the midspan

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2

0

-2

[email protected] , 0.83



Shea

r tr

ansf

er (

kN

/m)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2

0

-2

[email protected] , 0.58



Shea

r tr

ansf

er (

kN

/m)

(d)

(c)

Opposite shear

transfer


Negative shear

transfer


177

Figure 7.2 The piecewise linear profile model for transverse slip and shear

transfer in BSP beams; (a) illustration of transverse slip and (b) simplified profile

model

F1 F2 F3

w1 w1 w2 w2

w3 w3 F1 F2

F3

(b)

vm,LS vm,RS

vm,1 vm,2 vm,3

(a)

Negative shear transfer block

Positive shear transfer block


178

Figure 7.3 Analogy of shear transfer to Winkler’s model; (a) an infinite beam

under a point load and (b) a semi-infinite beam under a point load


(b)

(a)

w w’ RC beam

Steel plate

Equivalent spring

of bolt connection

Opposite shear transfer

RC beam

Steel plate

Equivalent spring

of bolt connection

w w’


x

Opposite shear transfer


179

Figure 7.4 Shear transfer in BSP beams with (a) rigid bolts or infinitely

flexible steel plates and (b) elastic bolts and rigid steel plates

(b)

(a)

RC beam

Steel plate

Equivalent rigid

bolt connection

Positive shear transfer from RC

Opposite shear

transfer back to RC

RC beam

Rigid steel plate

Equivalent spring

of bolt connection

w w’


180

Figure 7.5 Variation of shear transfer profile (a) before and (b) after cracking

occurs

(a)

(b)


181

Figure 7.6 Linear profile model for a BSP beam under four-point bending: (a)

loading condition, (b) piecewise linear model, transverse loads of (c) RC beam

and (d) steel plates, vertical deflections of (e) RC beam and (f) steel plates, (g)

difference in deflection, (h) transverse slip, and (i) transverse shear transfer

w w w w

F F vm,s

L/3 L/3 L/3

L

vm,s

vm,F vm,F

y

x

(d)

(a)

(b)

(c)

(f) (e)

(g)

(h)

(i)

−vm F F

F F

Four-point bending

vm or qp

δp0

δp = δp0 + δpf δc = δce + δcm

dδ = δp − δc

Str = dδ

vm’ = km Str

F F


182

Figure 7.7 Comparison of experimental and theoretical shear transfer profiles

at load level (a) F/Fp = 0.25, (b) F/Fp = 0.5, and (c) F/Fp = 0.75 for P100B300

(b)

(a)

(c)

-1800 -1200 -600 0 600 1200 180040

20

0

-20

-40


Piecewise linear model

Experiment

Shea

r tr

ansf

er (

kN

/m)

F/Fp = 0.25

-1800 -1200 -600 0 600 1200 180040

20

0

-20

-40


F/Fp = 0.50 Piecewise linear model

Experiment

Shea

r tr

ansf

er (

kN

/m)

-1800 -1200 -600 0 600 1200 180040

20

0

-20

-40

F/Fp = 0.75 Piecewise linear model

Experiment

Shea

r tr

ansf

er (

kN

/m)



183

Figure 7.8 Comparison of experimental and theoretical shear transfer profiles

at load level F/Fp = 0.5 for (a) P100B450, (b) P250B300R, and (c) P250B450R

(b)

(a)

(c)

-1800 -1200 -600 0 600 1200 180040

20

0

-20

-40


ExperimentP100B450

Shea

r tr

ansf

er (

kN

/m)


-1800 -1200 -600 0 600 1200 180060

40

20

0

-20

-40

-60


ExperimentP250B300R

Shea

r tr

ansf

er (

kN

/m)


-1800 -1200 -600 0 600 1200 180040

20

0

-20

-40

P250B450R Piecewise linear model

Experiment

Shea

r tr

ansf

er (

kN

/m)



184

Figure 7.9 Shear transfer profile model for a BSP beam under UDL

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4

0

-4

-8

UDL

Piecewise linear patter


Shea

r tr

ansf

er (

kN

/m)

Chapter 8 Analysis of BSP Beams with Partial Interaction

185

CHAPTER 8

ANALYSIS OF BSP BEAMS WITH PARTIAL

INTERACTION

8.1 OVERVIEW

From the results obtained in the previous chapters, it can be concluded that

the performance of BSP beams is controlled by the plate–RC interfacial partial

interaction, which is a result of both the longitudinal and transverse slips caused

by the shear transfers. Due to this partial interaction, there would be a large loss in

the additional enhancement provided by the side-bolted steel plates. Hence a

simplified analysis based on the assumption of full interaction would result in an

overestimation in the flexural strength and stiffness along with an underestimation

in the deformability of the BSP beams.

The flexural strength of an RC beam is conventionally regarded as a sectional

property, thus can be obtained by a moment–curvature analysis. However, in a

BSP beam, the flexural strength is affected by the partial interaction, which varies

along the beam span and cannot be incorporated into the section analysis directly.

Therefore, although the section properties retain unchanged along the beam axis,

the flexural strengths at different locations distinguish from one another.

In this chapter, an analysis method, which incorporates both the conventional

moment–curvature analysis and the longitudinal and transverse partial interaction,

is proposed. The BSP beam is decomposed into a series of end-to-end segments

along the beam axis. The longitudinal and transverse slip profiles derived from the

previous Chapters 6 and 7 are used to decide the degrees of partial interaction for

every segment, and the modified moment–curvature analyses are conducted

segment by segment. The overall load–deflection behaviour of the BSP beam is

yielded by integration of the behaviours of all the discrete segments.


186

8.2 PARTIAL INTERACTION IN BSP BEAMS

In a BSP beam, the RC beam and the steel plates work together to resist the

bending moment M by their flexural strengths φc (EI)c, φp (EI)p, and the coupling

action icpTm between these two components as:

p pc cp mcEI EI iM T (8.1)

Due to the shear deformation of anchor bolts subjected to the bolt shear forces

Tm and Vm, the relative slips Slc and Str occur on the plate–RC interface in both

longitudinal and transverse directions. Because of these slips, both strain and

curvature of the steel plates are smaller than those of the RC beam (εp, ypc < εc, ypc

and φp < φc). Hence the strain and the curvature factors αε and αφ, which are used

to indicate the degrees of partial interaction, are less than unity as:

,

,

1pc

pc

p y

c y

(8.2)

1p

c

(8.3)

According to the studies in Chapters 6 and 7, the moment–curvature (M – φ)

relations under full interaction (FI), partial interaction (PI) caused by longitudinal

slip Slc or transverse slip Str can be obtained as follows:

2

,,

+ Full interaction

Longitudinal partial interaction

Transverse partial interaction

cpc cp

PI m cpc

c

FIp

p

pc PIpPI

EI EI EA i

M EI EI T i

EI EI

(8.4)

Evidently, the magnitudes of the curvatures in Equation (8.4) are different. The

curvature of the RC beam under partial interaction is smaller than that under full

interaction. The curvature of the steel plates under partial interaction is greater

than that under full interaction. Thus the following inequality exists:


187

, , ,p PI FI c PI PI (8.5)

However, the partial interaction on the plate–RC interface of a BSP beam is a

result of both the longitudinal and transverse slips. So the moment–curvature

relation is even complicated than those of the aforementioned three cases. An

analytical solution, which combines the partial interactions in both directions, is

usually of great difficulty. Therefore a program, which considers the influence of

the overall longitudinal and transverse partial interaction profiles to every single

cross section by discretizing the beam into a series of end-to-end beam segments,

is a feasible way to analyse the behaviour of the BSP beams.

8.3 PROGRAM DETAILS

8.3.1 Material models

A constitutive relation proposed by Attard at el. (Attard and Setunge 1996;

Attard and Stewart 1998), which was based on the Sargin’s model (Sargin 1971),

is adopted as shown in Figure 8.1.

2

0 0

0

2

0

0

0

0

0

1 2 1

c c

cc

cc c

c

c c

A B

fA B

(8.6)

where σc is the stress at strain εc , fc is the peak stress at εc0. Both the ascending

and descending branches are governed by the same formula with different values

of constants A0 and B0. All the variables needed are determined from the uniaxial

compressive cylinder strength fc . To represent the material properties of the

locally mixed concrete, the modification to the parameters Ec and εc0 advised by

Lam (2006) were adopted .


188

0

0

00

2

0

0

c cc c

c

ic cicc c

c ic c ic

E

fA

f

f f

(8.7)

2

00

0

0 11

0.55

0

c c

c c

A

B

(8.8)

0.52

0.750.75

0

1.253.46 3.46

4370c c

ccu

cc

c

E f

ff

E E

(8.9)

0

1.41 0.17ln

2.50 0.30ln

ic c c

ic c c

f f f

f

(8.10)

For simplification, the tensile strength of concrete is ignored and the subsequent

error is estimated to be less than 0.2%.

Both the reinforcement and steel plates are considered as elasto-plastic

materials as shown in Figure 8.2.

where: s s s y

s

y s

y s

y

yff

EE

(8.11)

where :p p yp

p

p

yp y

p

p

p yp

p

y

E

ff E

(8.12)

8.3.2 Analysis of a BSP beam section with partial interaction

Although the centroidal level of steel plates ypc translates due to the transverse

slip Str, the experimental study reported in Chapters 3 and 4 indicated that its


189

magnitude is small compared to the depth of the beam (0.46 mm / 350 mm =

0.13%). A layered model is employed as illustrated in Figure 8.3. The concrete

and the steel plates are divided into m and n layers; the reinforcement is also

divided into s layers according to the actual arrangement of the rebars. The strains

of concrete εc,i and those of the reinforcement εs,k can be expressed as follows:

, , 1, ...,cc ii c nay y i m (8.13)

, , 1, ...,ss kk c nay y k s (8.14)

The concrete strain at the plate-centroidal level is:

, pcc y c pc nay y (8.15)

And the strain of steel plates can be expressed by that at the plate-centroidal level

as follows:

, ,, 1, ...,pc p p j pp j p y cy y j n (8.16)

Substituting Equations (8.2), (8.3) and (8.15) into Equation (8.16) gives:

, , 1, ...,pc na p j pp cj c cy y y y j n (8.17)

The internal axial force N can be obtained by introducing the material models

(see Equations(8.6) ~ (8.12)), and the pure bending condition of the BSP beam

section should be satisfied such that:

, , ,

1 1 1

,2 0m n s

c i p pc c j sp p s sk k

i j k

N Bd t d A

(8.18)

Where B is the width of the RC beam section. The internal bending moment M

can also be obtained as follows:

, , ,, , ,,

1 1 1

2c c i c p p

m n s

c p j si p p j s s k

i k

k s k

j

M Bd y t d y y A

(8.19)


190

And the bending moment components carried by the RC beam, the steel plates

and the coupling action are as follows:

, ,

1 1

,

1

,

, , ,

,

1

2 2

2

2 2

c c c i c c s k c s kc

p p p

m s

c i s s k

i k

n

j pc

p p pc

p p p jpj

n

cp m cp p p p jc

j

EI Bd y D y D A

EI t d y y

i T i N t d y D

(8.20)

It should be noted that the sum of the three components in Equation (8.20) is

equal to the internal bending moment M in Equation (8.19).

It is evident from Equations (8.13), (8.17) and (8.14) that if the strain and the

curvature factors (αε and αφ) are known, the strains of concrete, steel plates, and

reinforcement (εc,i, εp,k, and εs,j) depend on the curvature and the neutral axis level

of the RC beam only (φc and yna). Therefore under a given bending moment MI,

the following modified moment–curvature analysis process (see Figure 8.4) can

be conducted to obtain the curvature of a BSP section φc,I with partial interaction

αε,I and αφ,I :

(1) An initial trial of curvature φc,I is chosen, and then the neutral axis level of the

RC beam yna,I is the only unknown quantity.

(2) An initial trial of the neutral axis level of the RC beam yna,I is also chosen. So

the strains of concrete, steel plates, and reinforcement (εc,I, εp,I, and εs,I) can be

expressed in terms of φc,I and yna,I as shown in Equations (8.13), (8.17), and

(8.14).

(3) The internal axial force NI is computed using Equation (8.18). If |NI | > ξ

(where ξ is the selected tolerance), the pure bending condition (NI = 0) is not

satisfied, the value of yna,I has to be modified, and the iteration starting from

step (2) is repeated.

(4) If |NI | < ξ, the pure bending condition (NI = 0) is satisfied and yna,I is the

required neutral axis level, the internal bending moment MI’ is computed

using Equation (8.19).


191

(5) If |MI’ – MI | < ξ, φc,I is the required curvature under the designated bending

moment MI. Otherwise if |MI’ – MI | > ξ, φc,I is not the required curvature,

which is required to be modified and the iteration starting from step (1) to step

(4) needs to be repeated until |MI’ – MI | < ξ.

(6) However, if we fail to achieve |MI’ – MI | < ξ even when MI’ reaches the

maximum moment MI’max, i.e., MI’max < MI, this means that MI exceeds the

bearing capacity and the section fails.

By incorporating the strain and the curvature factors, only minor modification

is needed for the conventional moment–curvature analysis to study a BSP section

with partial interaction.

8.3.3 Analysis of a BSP beam with partial interaction

In order to obtain the overall performance of a BSP beam, the magnitude of

external load F is increased step by step (FJ +1= FJ + ΔF). At each load step FJ, the

BSP beam is decomposed into K segments. The bending moment, the strain and

the curvature factors at each segment are computed as shown in Figure 8.5; by

using the aforementioned modified moment–curvature analysis with partial

interaction (see Figure 8.4), each segment is checked until failure is located. The

detailed flowchart is depicted in Figure 8.6 and the steps are as follows:

(1) The geometry of the beam, the boundary conditions such as supports and

loading positions, the material properties of concrete, reinforcement, steel

plate and anchor bolts are defined.

(2) The first load step is imposed on the BSP beam (i.e., FJ = 0 = ΔF). The profiles

of bending moment MJ and shear force VJ are computed.

(3) The profiles of longitudinal and transverse slips (Slc and Str), along with those

of strain and curvature factors (αε and αφ) are calculated. Since both αε and αφ

are not dependent on the magnitude of FJ , there is no need to recalculate them

in the subsequent load steps.


192

(4) The BSP beam is decomposed into K segments, the external bending moment,

the strain and the curvature factors at each segment (MJ, I , αε, I and αφ,I , …

where I = 1, 2, …, K) are computed.

(5) At each segment, the curvature φc,I under a specific moment MJ,I is obtained

by the modified moment–curvature analysis with partial interaction (αε,I and

αφ,I) until the curvatures of all the segments along the beam are achieved

(i.e., I = K), then increase the load (i.e., FJ +1= FJ + ΔF) and go back to step (2)

to the next load step.

(6) If convergence is failed to achieve at any segment I, this means MJ,I exceeds

the flexural strength of this segment (i.e., Mu,I < MJ,I ), and the maximum

moment of the segment I can be chosen as Mu,I ≈ MJ −1,I and the BSP beam

reaches its load capacity (Fp ≈ FJ −1).

The accuracy of the load capacity prediction Fp is controlled by the step

increment ΔF in the last step (6), which can be refined by dividing it into several

smaller load steps.

8.4 STUDY ON ANALYSIS RESULTS

8.4.1 Verification by experimental results

The results of the experimental study reported in the Chapters 3 and 4 are

extracted to verify the validation of the computer program developed.

The load capacities of the BSP specimens are computed using the program

(Fp,the = 2Mu / Ls) and compared with the experimental results (Fp,exp) as shown in

Table 8.1. It can be seen that the load capacity of the unstrengthened beam

CONTROL is predicted accurately with an error of only 0.3%. When full

interaction is assumed on the plate–RC interface (αε = αφ = 1), the load capacities

of the BSP beams with both shallow and deep steel plates are overestimated by a

maximum of 8.0% and 5.5% on the average. When the partial interaction is taken

into account (αε < 1 and αφ < 1), the accuracy of the prediction is improved


193

significantly. The load capacities are overestimated by a maximum of 3.6% and

only 2.2% on average.

The flexural strength profiles Mu of the specimens are computed with

consideration of both longitudinal and transverse partial interaction along the

entire beam span. The flexural strength profile of the Specimen P100B300 is

plotted in Figure 8.7. The flexural strength based on full interaction assumption is

also plotted for comparison. It is observed that if partial interaction is taken into

account, the analytical flexural strength varies along the beam span even under the

linear material assumption. Compared with the profiles of strain and curvature

factors (αε and αφ) in Figure 8.5, the loading point section is the critical section

corresponding to the minimum αε and αφ . In fact, the failure of all the BSP

specimens, either those with shallow (P100B300 and P100B450) or deep steel

plates (P250B300R and P250B450R), occurred near the loading point (see

Chapter 4 for details). Thus the prediction of the program shows good agreement

with the experimental results. It is also seen from Figures 8.5 and 8.7 that the

section at the middle of shear span is at the maximum flexural strength

corresponding to the maximum αφ and αε .

As shown in Equation (8.1), the flexural strength of a BSP beam is the sum of

the plate and RC flexural strengths φc (EI)c, φp (EI)p, and the coupling action icpTm.

The contributions of these three components of the specimens are also computed

using the program and plotted in Figure 8.8 for the Specimens P100B300 and

P250B300R. The moment–curvature curves are also plotted for the same loading

process. It can be seen that for the BSP beam with shallow plates (P100B300), the

RC beam carries more than 85% of the total bending moment. The coupling

action accounts for 13%, and the flexural strength of the steel plates is almost

negligible (2%). The ratio between the coupling action and the plate flexural

strength is icp Np : φp (EI)p ≈ 6.5 : 1. On the other hand, for the BSP beam with deep

steel plates (P250B300R), the RC beam carries less moment (74% < 85%) and the

deep plates take up more moment (23% > 15%), of which the majority is due to

their flexural strength (icp Np : φp (EI)p ≈ 1 : 8). According to the experimental

results, the bending moment taken by the shallow plates was only 15% of the

coupling action (icp Np : φp (EI)p ≈ 7 : 1), whereas that for the deep plates was


194

approximately 7 times of the coupling action (icp Np : φp (EI)p ≈ 1 : 7). It is seen that

the experimental and analytical icp Np : φp (EI)p ratios agree well with each other

for the performance of deep and shallow steel plates in the BSP beams.

8.4.2 Partial interaction on strengthening effect

In order to compare the effect of longitudinal and transverse partial

interaction on the sectional behaviour of lightly and moderately reinforced BSP

beams (for instance ρst /ρstb = 0.23 and 0.69) strengthened by shallow and deep

steel plates (for instance Dp / Dc = 0.29 and 0.71), a parametric study is conducted

using the computer program. A control RC beam without any strengthening

measures (αε = αφ = 0) is employed for reference, and its flexural strength is

denoted by M0. At first, the curvature factor is fixed to zero (αφ = 0), and no

transverse interaction occurs; The strain factor is increased progressively from

αε = 0.1 to 0.9, in other words the longitudinal interaction is increased from almost

no interaction to nearly full interaction. By doing so, the variation of the flexural

behaviour of a BSP section with respect to the longitudinal partial interaction is

investigated. Similarly, the strain factor is fixed to zero (αε = 0) and the curvature

factor is increased progressively (αφ = 0.1 ~ 0.9) to study the transverse partial

interaction. The resultant moment–curvature curves are compared in Figures 8.9

and 8.10, and the relative enhancement in flexural strength with regard to the

control beam (M’ – M0)/M0 are compared in Table 8.2.

As shown by Figure 8.9 and Table 8.2, when the shallow plates (Dp/Dc= 0.29)

are employed, the behaviour of the lightly reinforced BSP beams is mainly

controlled by the strain factor αε . The flexural strength is enhanced by 66% when

αε = 0.1. As αε increases from 0.1 to 0.5, the strength enhancement increases

significantly (by 66% ~ 115%). However, as αε further increases to 0.9, the

enhancement is negligible (by 115% ~ 117%). It is also noted that as the flexural

strength increases, the deformability reduces considerably. On the other hand,

neither the flexural strength ((M’ – M0)/M0 = 4 ~ 14%) nor the shape of the M – φ

curves change much as the curvature factor varies (αφ = 0.1 ~ 0.9). However,

when the deep plates (Dp/Dc = 0.71) are utilized (see Figure 8.9(b)), the flexural


195

strength increases tremendously with the increase of both the strain and the

curvature factors, and the enhancements are 47 ~ 80% and 78 ~ 96% respectively.

It is also noted that the enhancements as αφ increases from 0.1 to 0.5 are

significant (74% – 47% = 27% and 92% – 78% = 14%), but those between 0.5

and 0.9 are relatively small (80% – 74% = 6% and 96% – 92% = 4%).

Furthermore, the deformability does not reduce with the increase of the curvature

factor. In other words, shallow steel plates attached to the side faces of a lightly

reinforced beam increase the flexural strength but reduce the ductility.

As shown by Figure 8.10 and Table 8.2, unlike their lightly reinforced

counterparts, the moderately reinforced RC beams can hardly be enhanced by

shallow steel plates (enhanced by 16% and 5% at αε = 0.9 and αφ = 0.9). Even

when deep plates are utilized, a significant enhancement can only be achieved

when a large curvature factor is employed (by 6% at αε = 0.9, and by 39% at

αφ = 0.9 respectively). Furthermore, the same conclusion can be drawn that an

excessive degree of partial interaction is not essential in the strengthening of

moderately reinforced BSP beams. For instance, when αε or αφ increases from 0.5

to 0.9, the enhancements are not significant (from 14%, 4%, 5%, and 31%

increases to 16%, 5%, 6%, and 39%, respectively).

8.4.3 Recommendation on choice of strain and curvature factors

The increase in the degree of partial interaction certainly enhances flexural

strength, along with greater connection stiffness achieved by a larger number of

anchor bolts. However it is evident from Table 8.2, Figures 8.9 and 8.10 that the

enhancement effects are different for lightly and moderately reinforced RC beams

as well as BSP beams with shallow and deep steel plates. Nevertheless, the

enhancement is not significant for strain or curvature factor greater than 0.5. This

means that as the number of anchor bolts reaches a given value, the significance

of each newly added anchor bolt decreases rapidly. In order to find a balance

between the strengthening effect (a satisfactory strength enhancement) and the

strengthening efficiency (an economic number of anchor bolts), a parametric

study is conducted. The strength enhancement under an ideally full interaction


196

(M1 – M0, where M1 is obtained at αε or αφ = 1) is set as the target strength

enhancement, and the enhancement M’ – M0 under different αε or αφ are compared

with M1 – M0. The variation of the relative strengthening effect (i.e., the

normalized strength enhancement (M’ – M0) / (M1 – M0)) with respect to the strain

and the curvature factors (αε and αφ) are shown in Figure 8.11. The numbers of

anchor bolts nb needed to achieve the specific αε and αφ are also normalized and

plotted for comparison.

Figure 8.11(a) shows the variation of the relative strengthening effect and the

corresponding number of anchor bolts for a lightly reinforced BSP beam with

shallow plates, and Figure 8.11(b) shows that for a moderately reinforced BSP

beam with deep plates. The increase of strengthening effect with the degree of

partial interaction in Figure 8.11(b) is slower than that in Figure 8.11(a), thus it is

less efficient to strengthen a moderately reinforced RC beam. In both cases, a

minimum relative strengthening effect of 0.90 can be attained if αε and αφ are

chosen to be 0.6. Furthermore, as αε and αφ increase from 0 to 0.6, the required

number of anchor bolts increases almost linearly, beyond that it begins to increase

rapidly. Hence, this point strikes a balance between the strengthening effect and

efficiency in the use of anchor bolts. For simplicity, a common value of 0.6 for

both the strain and the curvature factors is recommended as shown in Table 8.3. It

is noted that although the shallow plates can be used for lightly reinforced RC

beams, they are not suggested for moderately reinforced RC beams. On the other

hand, the deep steel plates are in general more reliable for strengthening both

lightly and moderately reinforced RC beams.

8.5 CONCLUSIONS

In this chapter, a theoretical study and a computer simulation are presented

for the analysis of BSP beams taking into account the partial interaction in terms

of the strain and the curvature factors. The following findings are highlighted

based on the analytical results:


197

(1) A lightly reinforced RC beam with a degree of reinforcement less than 1/3 can

be strengthened by adding external reinforcement with an acceptable reduction

in ductility. However, a moderately reinforced RC beam with a degree of

reinforcement greater than 2/3 can only be strengthened effectively by

attaching deep steel plates to the side faces of the beam.

(2) The flexural strengths of BSP beams would be overestimated by a

conventional moment–curvature analysis based on the assumption of full

interaction. More accurate results can be obtained by taking the partial

interaction on the plate–RC interface into account.

(3) If the partial interaction is considered, the flexural strength varies along the

beam span even under a linear material assumption. In the case of four-point

bending, the loading points are the critical sections with the minimum strain

and curvature factors.

(4) The shallow steel plates can be used to strengthen lightly reinforced RC beams

effectively. In contrast, the deep steel plates can be used for both lightly and

moderately reinforced RC beams. The strengthening effect is controlled by the

compatibility of both the strain and the curvature for the lightly reinforced

BSP beams, and mainly by only the compatibility of the curvature for the

moderately reinforced BSP beams.

(5) As the strain or the curvature factor increasing from 0.1 to 0.5, the

strengthening effect increases significantly. However, further increase of these

factors does not result in considerable increase in strength enhancement.

Therefore an excessive connection between the steel plates and the RC beam

is neither economic nor necessary.

(6) A strain or curvature factor of 0.6 can attain a relative enhancement of 0.9

with a reasonable number of anchor bolts, which is a balance between the

strengthening effect and efficiency. Consequently, a common value of 0.6 is

recommended for both the strain and the curvature factors in the strengthening

design of BSP beams.


198

Table 8.1 Comparison between experimental and analytical load capacities

Specimen Fp,exp (kN)

Fp,the (kN) (Fp,the - Fp,exp) / Fp,exp

Full

interaction

Partial

interaction

Full

interaction

Partial

interaction

CONTROL 268 269 0.3%

P100B300 317 329 323 3.8% 1.9%

P100B450 327 353 338 8.0% 3.6%

P250B300R 382 398 376 4.2% −1.7%

P250B450R 377 400 383 6.1% 1.6%

Mean absolute error 5.5% 2.2%

Table 8.2 Enhancement of lightly and moderately reinforced BSP beams

ρst /ρstb αε (αφ)

Dp /D = 0.29 < 1/3

(Shallow plates)

Dp /D = 0.71 > 1/2

(Deep plates)

αε > 0, αφ = 0 αε = 0, αφ > 0 αε > 0, αφ = 0 αε = 0, αφ > 0

0.23 < 1/3

(Lightly

reinforced)

0.9 117% 14% 80% 96%

0.5 115% 13% 74% 92%

0.1 66% 4% 47% 78%

0.69 > 2/3

(Moderately

reinforced)

0.9 16% 5% 6% 39%

0.5 14% 4% 5% 31%

0.1 6% 0% 2% 8%

Table 8.3 Recommended strain and curvature factors

Beam type Strengthened by

Shallow plates (Dp /D < 1/3) Deep plates (Dp /D > 1/2)

Lightly reinforced αε = 0.6

αε = 0.6

(ρst /ρstb < 1/3) αφ = 0.6

Moderately reinforced Not recommended αφ = 0.6

(ρst /ρstb > 2/3)


199

Figure 8.1 Stress–strain curve of concrete in compression

Figure 8.2 Stress–strain curve of steel reinforcement and steel plates

εy , εyp

Es , Ep

σs , σp

fy , fyp

O

εs , εp

εc0

Ec σc

fc

εc O


200

Figure 8.3 Strain profiles of a BSP section with partial interaction: (a) Section

and (b) Strain profile

yc,i

ypc

dc εc,i

φp

As,k

Ac,i

yp,j

dp Ap,j

φc

εp,j

εc,ypc

εp,ypc

φp < φc

εp,ypc < εc,ypc

(a) (b)


201

Figure 8.4 Modified moment–curvature analysis of a BSP beam section with

partial interaction

Moment, strain and curvature

factors (MI, αε,I & αφ,I)

Initial trial of curvature of the

RC beam (φc,I)

Net axial force (NI) is obtained

Iteration for

new trial yna,I

Initial trial of neutral-axis level

of the RC beam (yna,I)

No

Moment (MI’) is obtained

Iteration for

new trial φc,I

No

Yes

Yes

No

MI’max < MI ? |MI’ − MI| < ξ ?

|NI| < ξ ?

Yes

End

Section I fails End

φc,I yielded

Solve φc,I :

Solve yna,I :


202

Figure 8.5 Profiles of moment, longitudinal and transverse slips, strain and

curvatures in BSP beams

αφ

Slc

M

StrI

SlcI

MI

1 2 3 … … I K

P P

Str

αε

αφI

αεI


203

Figure 8.6 Modified moment–curvature analysis of a BSP beam with partial

interaction

M – φ analysis

under MJ,I, αε,I & αφ,I

(see Figure 8.4)

External moment and shear

(MJ, I & VJ, I)

Strain and curvature factors

(αε,I & αφ,I), in 1st load step

Decompose beam along axis:

I = 1,2,…K

FJ =0 = ΔF

I = I + 1

No

Yes End

Fp ≈ FJ −1

FJ +1 = FJ + ΔF

I = K ?

For each segment I :

For each loading step FJ :

Materials, geometries, support

Segment I fails

φc,I yielded


204

Figure 8.7 Flexural strength profile of a BSP beam

-1800 -1200 -600 0 600 1200 1800190

192

194

196

198

200

Mu , Full iteraction

Mu , Partial iteraction

Fle

xura

l st

rength

Mu (

kN

.m)

Distance from mid-span (mm)


205

Figure 8.8 Flexural strength contribution ratios of the RC beam (φc (EI)c), the

steel plates (φp (EI)p) and the plate tensile force (icp Np) for (a) P100B300 and (b)

P250B300R

0.00 0.01 0.02 0.03 0.04 0.050.0

0.2

0.4

0.6

0.8

1.0

M

c(EI)c

/ M

p(EI)p

/ M

icpNp

/ M

Fle

xura

l co

ntr

ibuti

on r

atio

Curvature (rad/m)

0

50

100

150

200

Mo

men

t M

(k

N.m

)

0.00 0.01 0.02 0.03 0.04 0.050.0

0.2

0.4

0.6

0.8

1.0

M

c(EI)c

/ M

p(EI)p

/ M

icpNp

/ M

Fle

xura

l co

ntr

ibuti

on r

atio

Curvature (rad/m)

0

50

100

150

200

250 M

om

ent

M (

kN

.m)

(a)

(b)


206

Figure 8.9 Moment–curvature curves of lightly reinforced (ρst = 0.59%) BSP

beams with (a) shallow and (b) deep steel plates

0.0 0.1 0.2 0.3

0

30

60

90

120

150

= 0.9, = 0.0

= 0.5, = 0.0

= 0.1, = 0.0

= 0.0, = 0.9

= 0.0, = 0.5

= 0.0, = 0.0

Mom

ent

M (

kNm

)

Curvature (rad/m)

0.0 0.1 0.2 0.3

0

30

60

90

120

150

= 0.9, = 0.0

= 0.5, = 0.0

= 0.1, = 0.0

= 0.0, = 0.9

= 0.0, = 0.5

= 0.0, = 0.0

Mom

ent

M (

kNm

)

Curvature (rad/m)

(a)

(b)

Dp /Dc = 0.29

Dp /Dc = 0.71


207

Figure 8.10 Moment–curvature curves of moderately reinforced (ρst = 1.77%)

BSP beams with (a) shallow and (b) deep steel plates

0.00 0.05 0.100

30

60

90

120

150

180

210

240

= 0.9, = 0.0

= 0.5, = 0.0

= 0.1, = 0.0

= 0.0, = 0.9

= 0.0, = 0.0

Mom

ent

M (

kNm

)

Curvature (rad/m)

0.00 0.05 0.100

30

60

90

120

150

180

210

240

= 0.9, = 0.0

= 0.0, = 0.9

= 0.0, = 0.5

= 0.0, = 0.1

= 0.0, = 0.0

Mom

ent

M (

kNm

)

Curvature (rad/m)

(a)

(b)

Dp /Dc = 0.29

Dp /Dc = 0.71


208

Figure 8.11 Strengthening effect and efficiency for (a) lightly and (b)

moderately reinforced BSP beams

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

M' due to

nb due to

Norm

aliz

ed s

tren

gth

enhan

cem

ent

(M'-

M0)

/ (M

1-M

0)

Strain factor

Norm

aliz

ed b

olt

num

ber

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

M' due to

nb due to

Norm

aliz

ed s

tren

gth

enhan

cem

ent

(M'-

M0)

/ (M

1-M

0)

Curvature factor

Norm

aliz

ed b

olt

num

ber

(a)

(b)

Chapter 9 Design of BSP Beams with Partial Interaction

209

CHAPTER 9

DESIGN OF BSP BEAMS WITH PARTIAL

INTERACTION

9.1 OVERVIEW

According to the detailed investigation reported in the previous chapters, the

behaviour of BSP beams was found to be very different from that of RC beams

strengthened by attaching steel plates or FRPs to the beam soffit. Hence, the

normal analysis and design methods for normal steel–RC composite beams would

not be applicable to the BSP beams.

In light of this situation, recommended design procedure is proposed in this

chapter. The formula used to compute the flexural strength of normal RC beams is

modified, by introducing the recommended strain and curvature factors proposed

in Chapter 8 to take the partial interactions in both longitudinal and transverse

directions into account. The dimension of steel plates is computed by this formula

and further used to determine the bolt arrangement. Then the maximum plate–RC

slips and the minimum strain and curvature factors are checked by employing the

formulas developed in Chapters 6 and 7. A worked example, which includes the

strengthening design of both a lightly and a moderately reinforced beams

subjected to different loading arrangements, is also presented for reference.

9.2 THEORETICAL BASE

In the computation of the ultimate moment resistance of a BSP beam section,

the following assumptions are employed:

(1) The bond–slip effect of both tensile and compressive reinforcement is ignored,

i.e., the strain in the rebars is the same as that in the surrounding concrete.


210

(2) The effects of both longitudinal and transverse slips between the bolted steel

plates and the RC beam are considered.

(3) The cross-sections of both the steel plates and the RC beams remain plane

respectively after deformation.

(4) The tensile strength of concrete is ignored; the compressive stress of concrete,

the tensile and compressive stresses in reinforcing steel and plate steel are

derived from the design stress–strain relations given in the Eurocodes (BSEN

1992 2004).

(5) The shear strength of anchor bolts is computed according to the Eurocodes

(BSEN 1993 2005).

9.2.1 Material models

The stress–strain relation for the design of concrete material in the Eurocodes

(BSEN 1992 2004) is adopted as shown in Figure 9.1:

0

2

0

0

1 1 0cc c c

c

c c c cu

c

f

f

(9.1)

Where σc is the stress at strain εc , εc0 is the strain at the maximum strength fc , εcu

is the ultimate strain.

Both the reinforcement and steel plates are considered as elasto-plastic

materials (BSEN 1992 2004) as shown in Figure 9.2.

, where: s s s y

s

y s y

s y y

Ef

fE

(9.2)

, wher e :p p yp

p

yp p y

p

p yp yp

p

EE f

f

(9.3)


211

Since the shear failure of anchor bolts is a brittle failure, the elastic shear

force–slip relation is simplified for anchor bolts as shown Figure 9.3 and the

maximum slip in BSP beams should be always less than Sby .

2

, where

4

:

b by by

b b

b v

b

uby

yb

R K S S

K R S

S dR f

(9.4)

Where fub and db are the ultimate tensile strength and the nominal diameter of

anchor bolt, αv is a modifier and a value of 0.5 or 0.6 is conventionally chosen

(BSEN 1993 2005).

9.2.2 Sectional analysis and flexural strength

In order to obtain the flexural strength of a BSP beam, the cross-sectional

strain and stress profiles at the ultimate limit state are illustrated in Figure 9.4. The

concrete strain at the compressive surface reaches the ultimate strain εcu, therefore

the curvature of the RC beam can be expressed by the depth of neutral axis c as:

cuc

c

(9.5)

The strains of the compressive and tensile reinforcement can be written by their

depths (hc and h0) as follows:

sc c cc h (9.6)

0st c h c (9.7)

From the discussion in Chapter 8, strengthening effect of 90% is guaranteed when

the strain or the curvature factor is chosen to be no less than 0.6. Therefore, for

brevity a unique value (α = 0.6) is chosen for both αε and αφ , thus the strains of

the steel plates at their top and bottom edges can be written as following:


212

0.6ptpt c c ptc h c h (9.8)

0.6pb c cpb pbh c h c (9.9)

For a satisfactory strengthening design, the outmost layer of tensile

reinforcement should yields before concrete crushing occurs, thus its tensile stress

is the yield strength fy at the ultimate limit state. By substituting the strains in

Equations (9.6) ~ (9.9) into the material constitutive relations, the internal

sectional axial force Nu and bending moment Mu can be obtained. Furthermore,

the pure bending condition should be satisfied as:

2 2

0p

cuu c s sc c y st

cu cup pt p pbp

N f b c E A c h Ac

E t c h t cc

f

E hc

(9.10)

Where λ is a factor defining the effective depth of the concrete compression zone

and η is a factor defining the effective strength as shown in Figure 9.4(c), and a

value of 0.8 and 1.0 is conventionally adopted for λ and η respectively if the

concrete grade is lower than C50 (BSEN 1992 2004).

It can be found that c is the only unknown in Equation (9.10) and it is

convenient to solve this quadratic equation to yield the neutral axis depth c as

following:

2

2 2

4

2

where: 2

c

s sc cu y st p cu

s sc cu c p cu

p pb pt

p pb pt

B AC Bc

A

A f b

B E A A E t h h

C E A h h h

f

E t

(9.11)

Then then ultimate moment resistance Mu can be expressed as following:


213

3

2

3

2

012

2 2

3 3

cuu c s sc c y st

cu cup pp pt p pb

M f b c E A c h A h cc

E t c h E t

f

h cc c

(9.12)

However, the neutral axis depth c solved from Equation (9.10) must be

substituted into Equations (9.5) ~ (9.9) to check if the strains of the reinforcement

and the steel plates (εsc , εpt, and εpb) surpass their corresponding yield strain (εy

and εyp) or change their directions as following:

(1) If the yielding of the compressive reinforcement happens (εsc > εy), the second

terms in Equations (9.10) and (9.12) should be replaced by

and respectively.y sc y sc cA A c hf f

(2) If the yielding of the top edge of steel plates happens (εpt > εyp), the

corresponding triangular stress block in Figure 9.4(c) should be replaced by an

echelon stress block as shown in Figure 9.5(a) and the corresponding fourth

terms in Equations (9.10) and (9.12) should be replaced respectively by

2

2 12 and respectively.

3yp p pt yp p pt

yp yp

cu cu

c ct c h t cf f h

(3) If the yielding of the bottom edge of steel plates happens (εpb > εyp), the

corresponding triangular stress block in Figure 9.4(c) should be replaced by an

echelon stress block as shown in Figure 9.5(a) and the corresponding fifth

terms in Equations (9.10) and (9.12) should be replaced by

2

2 12 and respectively.

3yp p pb yp p p

yp y

c cu

b

p

u

c ct h c t h cf f

(4) If the strain of the top edge of steel plates is negative (εpt < 0), this means the

steel plates is in tension for entire section as shown in Figure 9.5(b). Actually,

this phenomenon implies the steel plates are shallow ones and attached to the

tensile region of the RC beam. In this case, no modification is needed for

Equations (9.10) and (9.12). Of course, when shallow steel plates are

employed in a strengthening design, and the occurrence of entire-section

tension of steel plates can be pre-assured, the signs of the third terms in these


214

two equations can be reversed so that only positive strains and plate depths are

yielded.

Finally, it is also worth noting that the plate thickness (tp) in all the equations

is the thickness of a single plate and the thickness of both plates is 2tp .

9.2.3 Verification by experimental results

For a BSP beam subjected to four-point bending, the peak load can be

expressed conveniently as Fp = 2Mu / Ls (where Ls is the shear span). The results

extracted from the experimental study reported in Chapters 3 and 4 were

employed to verify the aforementioned sectional analysis method as shown in

Table 9.1. It is evident that the proposed sectional analysis method can predict the

peak load of the specimens with a satisfactory mean discrepancy of about 5.2%.

9.3 PROPOSED DESIGN PROCEDURE

9.3.1 Estimation of plate sizing

In the BSP strengthening design for a specific RC beam, the geometry of the

RC beam and the loading arrangement are known, thus its flexural and shear

capacity (Mu and Vu) and the required bending moment and shear force (Md and Vd)

are known. Furthermore, the available depth and position of the side-bolted steel

plates can be determined as:

max 0 , , , ...pt sl sbh D D (9.13)

min , ...pb ch D (9.14)

Which means that the top-edge depth hpt of the steel plates should be greater than

the depths of existing secondary beams and slab (Dsb and Dsl), and the

bottom-edge depth hpb should be less than the depth of the RC beam Dc. If any


215

other restraints such as ceiling installation fitments and beam-crossing pipes exist,

the depth of the steel plates may be more limited.

Once the depth of the steel plates is chosen, the thickness of the steel plates

(2tp) can be determined according to the chosen steel plates material property (fyp

and Ep) and the required bending moment (Mu ≥ Md) by using the aforementioned

Equations (9.10) and (9.12) with safety factors as following:

(1) In case no yielding occurs at the compressive reinforcement and both edges of

steel plates (i.e., εsc < εy , εpt < εyp and εpb < εyp), which barely exists in the

real strengthening practice, the thickness of one steel plate tp can be

determined:

22

0

3 3

1 11

2

1 2

3

cuu c s sc c y st

c

cup

s

p pt pb

s


E t c h h cc

f

(9.15)

2

2 2

4

2

1

where: 1

2

1

p pb pt

s

p

c

c

s sc cu y st p cu

s sc cu c p pb pt

s

cu

B AC Bc

A

A f b

B E A A E t h h

C E A h E t h h

f

(9.16)

(2) In case yielding occurs at the compressive reinforcement and the bottom edge

of steel plates but not at the top edge of steel plates (i.e., εsc > εy , εpt < εyp and

εpb > εyp), which is the most common situation in the real strengthening design

of BSP beams, the thickness of one steel plate tp can be determined as:

22

0

2

23

1 11

2

1 2 1

3 3

cuu c s

s

py

p pt

s cu

sc c y st

c

cup py p pb


cE t c h t h c

f

fc

(9.17)


216

2

2

4

2

1 12

where: 1

2

1

c p cu py p

c cu

y sc st p py c

py

p

s

pb p ptu

p cu

s

p pt

s

B AC Bc

A

A f b E t t

B A A t h E h

C

f

E

f

f

t h

(9.18)

(3) In case yielding occurs at the compressive reinforcement and both the edges of

steel plates (i.e., εsc > εy , εpt > εyp and εpb > εyp), which also barely exists in

the real strengthening design of BSP beams, the thickness of one steel plate tp

can be determined as:

22

0

2

2 2

1 11

2

1 2

3

cuc s sc c y st

c

py p p

u

s

py

t b

s cu

p

M f b c E A c h f

f

A h cc

ct c h h c

(9.19)

1 1where: 4

12

s

pb

c py p

c

py p y s s

s

tt cp

f

f

Bc

A

A f b t

B t h h A Af

(9.20)

In Equations (9.15) ~ (9.20), the coefficients 𝛾c and 𝛾s are the partial factors

for concrete and steel materials, the left-hand-side quantity Mu ≥ Md, where Md is

the design value of the applied bending moment, which takes account of the

combination of partial-factored actions as follows:

...d GG Q QM M M (9.21)

Where MG , MQ , 𝛾G and 𝛾Q are the bending moment actions caused by permanent

and variable loads and their corresponding partial factors (BSEN 1992 2004).


217

In most cases, only Equations (9.17) and (9.18) in case 2 are needed for the

strengthening design of BSP beams. However, it is always recommended to check

the yield states of the compressive reinforcement and the top edge of steel plates.

If yielding occurs, the corresponding equations in cases 1 and 3 can be used. After

the minimum plate-thickness is obtained, the size of the steel plates (2tp and Dp)

can be chosen from the practical and available inventory.

9.3.2 Estimation of number of bolts

Since the type (fyp and Ep) and size (hpb, hpt, and 2tp) of the steel plates are

eventually chosen, the number of anchor bolts is estimated as:

2

1

1

14

yp pb pt p

sbb

by

M

f h h t

n

R

(9.22)

where Rby is the yield shear force of an anchor bolt (see Equation (9.4)), γM 2 is the

partial safety factor for bolts and a value of 1.25 is conventionally chosen (BSEN

1993 2005). γb is a coefficient taking account of the varying distribution of bolt

shear force along the beam span, and a value from 1.5 to 2.0 can be chosen, since

the shear transfer profiles are between triangle and parabola as shown in Chapters

6 and 7. The leading constant 1/4 at the left hand side is attributed to two steel

plates and two shear spans for each plate. This means in order to guarantee the

failure occurring in form of the flexural plate yielding but not the brittle bolt

shearing, the shear capacity of bolt connection should be greater than the axial

strength of steel plates.

When the minimum number of anchor bolts is determined, the actual

plate–bolt layout can be adjusted corresponding to practical plate size and the

minimum bolt spacing (BSEN 1993 2005). Then the preliminary strengthening

scheme can be determined. Of course, the partial interaction of BSP beams is

highly dependent on not only the beam geometries but also the load arrangement,

thus should be verified according to each specific case.


218

9.3.3 Verification of partial interaction

The partial interaction, which is caused by the longitudinal and transverse

slips between the steel plates and the RC beams, should be checked in terms of the

maximum longitudinal and transverse slips (Slc, max and Str, max), and the minimum

strain and curvature factors (αε, min and αφ, min) as following:

2 2

,ma ,x maxtl rc bySS S (9.23)

,min ,minmin , 0.6 (9.24)

For a BSP beam under four-point bending, the maximum longitudinal and

transverse slips occur at the plate end, the minimum strain factor occurs at the

loading point, and the minimum curvature factor occurs at the midspan. Their

magnitudes are given by:

20,max

11

2cosh 3 1

cp

x

c p

lcSpLI

i

E

F

p EI

(9.25)

3

4

,max 3

1

0

4 1



m pc

m p

tr x

c

FL

EI

FL

EI

LS

L

(9.26)

1

,min 32

1cosh 1 1

3 si 3n

3

h

p m

x LL pL

pL p

E k

p

A

(9.27)

11

4

,min 112 4



p p m

p p m

x L

L

L

(9.28)


219

For a BSP beam under uniformly distributed load (UDL), the maximum

longitudinal and transverse slips occur at the plate end, the minimum strain and

curvature factors occur at the midspan. Their magnitudes are given by:

,ma 30x tanh2 2

l

cp

x

c p

c

pL pLS

EI

q

EI

i

p

(9.29)

0

4

4,maxS

tr xm c

qL

L EIS

(9.30)

1

2

,min 22

1

1 sec

1

h2

8

p m

x L

L

ppL

EA k

(9.31)

4

,min 2 4

4

4



m p m p

m p m p

x L

L L D

L L D

(9.32)

4 4

4 4



m p p m p

m p p m p

S

L L C

L L C

(9.33)

22 48

2 48 2

1 28200 1 5500

1 16900 1 7200

for shallow plates

for deep plates

m p p m p p

m p p m p p

L L

L L

C

(9.34)

2

8 2

8

4

4

1 0.65

20100 0.9 5000

1 0.65

14600 0.8 5500

for shallow plates

for deep plates

m p p

p m p p

m p p

p m p p

L

L

L

L

D

(9.35)

The formulas of Slc, max , Str, max , αε, min , and αφ, min for a BSP beam subjected

to other loading cases are given in Chapters 6 and 7, thus are not listed herein.


220

9.3.4 General strengthening strategies and preliminary design

The flexural strength of an RC beam can be simply expressed as M = fy Ast dtc,

where dtc is the lever arm controlled by the depth of the beam h. Therefore, the

flexural strength of an RC beam can be augmented by increasing either the

strength of tensile reinforcement fy Ast or the depth of the beam h. In the structural

design, which measure should be taken is highly controlled by the position of each

RC beam in the whole structure. Figure 9.6 shows different types of RC beams in

a typical plane and elevation layout of RC buildings. Figure 9.7 presents the

available BSP strengthening schemes for the RC beams of different types.

As shown in Figure 9.6, the beams Type 1 are usually main girders with a

very large clear span. Large clear heights are usually not required because the

space under these beams is usually occupied by infilled walls or furniture.

However, the external loads imposed on these beams are usually of great

magnitude, including those transferred from floor slabs, secondary beams, and

infilled walls. Furthermore, since the ductility of main girders and the principle

“strong-column-weak-beam” are required by design codes, a steel ratio less than

2/3 of the balance steel ratio (ρst < 2/3ρstb) is usually preferable. Therefore, the

beams Type 1 are usually designed with a large beam depth h, but lightly

reinforced to achieve both a great flexural strength and ductility. Although the

depth of the beam is large, there is limited area on the side faces to be used for the

side-bolted steel plates due to the connection with secondary beams. When

bearing capacity greater than the original design is required, the strengthening

technique of BSP beams with shallow steel plates is especially suitable for the

beams Type 1 as shown in Figure 9.7(a). The shallow steel plates serve as

additional external tensile reinforcement and increase the degree of reinforcement

thus enhance the flexural strength.

How much the beams Type 1 can be strengthened is governed by the

difference between the current steel ratio and the preferred steel ratio (2/3ρstb − ρst).

The available area on the side faces is also a parameter controlling the

strengthening effect. This is because although thicker steel plates can always be

chosen to achieve a greater reinforcement, the degree of partial interaction is

limited by the available number of anchor bolts, which is highly governed by the


221

available side-face area as the minimum bolt spacing is strictly regulated in the

design codes.

The main failure mode of the beams Type 1 strengthened with shallow steel

plates is the yielding of the tensile reinforcement and the bolted plates. In order to

prevent the compressive concrete from crushing, the degree of reinforcement, i.e.,

the sectional area of the steel plates, should be limited strictly. To prevent the

undesirable shear failure happening at the anchor bolts, enough bolts should be

provided as well.

As shown in Figure 9.6, the beams Type 2 are usually secondary beams or

main girders with a shorter beam span and subjected to lower external loads. For

these beams, large clear heights below the beams are usually required for the

installation of equipment, pipelines and ceilings. Therefore, the beams Type 2 are

usually designed to be shallow beams with a small beam depth h, but moderately

reinforced with large tensile reinforcement Ast. For the beams Type 2, the deep

steel plates increase both their tensile and compressive reinforcement thus

enhance the flexural strength without a visible reduction in ductility, as shown see

Figure 9.7(b).

Since the deep steel plates increase both the tensile and compressive

reinforcement, the tensile steel ratio ρst is no longer an obstacle to the

strengthening effect of the beams Type 2. The available side-face area becomes

the key parameter, for it controls both the available plate depth and the maximum

number of anchor bolts.

The buckling in the compressive region of the deep steel plates is the greatest

potential risk for the strengthened beams Type 2. It should be suppressed by

taking appropriate measures, such as using more anchor bolts, installing or

welding steel angles to the compressive edge of the steel plates.

As reported in Chapter 8, the variation in strength enhancement is not

significant when the strain and the curvature factors (αε and αφ) are greater than

0.5, and a strengthening effect of 0.90 can be guaranteed if αε and αφ are chosen to

be 0.6. Therefore, in the preliminary design, the sectional area of the steel plates

can be multiplied by a factor of 0.90 and treated as the additional tensile and


222

compressive reinforcement of the RC beam sections. Then the approximate

formula M = fy Ast dtc can be used to estimate the flexural strength of the BSP

beam section. Since the shear failure of anchor bolts should not occur prior the

yielding of steel plates, the number of anchor bolts can be roughly determined by

equating the total shear strength of all anchor bolts with the entire-sectional yield

strength of the steel plates (see Equation (9.22)).

9.4 WORKED EXAMPLE

9.4.1 Current state of the structure needed strengthening

For brevity and illustration, let us assume that Figure 9.6(a) shows the plane

layout of a prefabricated RC structure factory building and all beams are simply

supported. The originally designed lived load was 5 kN/m2 but now needs to

increase to 12 kN/m2 for a change in usage. Therefore proper retrofitting measures

should be imposed to the structure. For illustration, only the strengthening designs

of a main girder and a secondary beam labelled as Beam 1 and Beam 2 in

Figure 9.6(a) are discussed herein. The simplified models are shown in Figure 9.8

and the section details are shown in Figure 9.9.

The originally designed loads before a change in usage are as following:

Floor:

Dead load (g): 25 mm cement floor finishing 0.025×21 = 0.5 kN/m2

100 mm RC slab 0.100×25 = 2.5 kN/m2

20 mm cement ceiling finishing 0.020×17 = 0.4 kN/m2

3.4 kN/m2

Live load (q): 5.0 kN/m2

Total 1.35g + 1.5q = 12.5 kN/m2

Beam 2 (secondary beam):

Self-weight: 200×400 mm 1.35×0.20×0.40×25 = 2.7 kN/m

From slab: 2.4 m span 12.5×2.4 = 30.1 kN/m

Total q2 = 32.8 kN/m


223

Beam 1 (main girder):

Self-weight: 350×700 mm q1 = 1.35×0.35×0.70×25 = 8.3 kN/m

From Beam 2 6.0 m span F1 = 32.8×6.0 = 197 kN

Therefore, the originally designed moments on Beams 1 and 2 are as:

2

d,1

18.3 7.2 197 2.4 525.7 kN m

8M (9.36)

2

d,2

132.8 6.0 147.5 kN m

8M (9.37)

The designed material properties are as follows:

30 MPa , 460 MPa , 200 GPack y sf f E (9.38)

The section properties of Beams 1 and 2 are as follows respectively:

2

,1

2

,1

,

0,1

0,1

1

700 37 667 mm

350 667 232050 mm

20 3 942 mm4

25 5 2453 mm4

24531.06 %

232050

sc

st

st

h

A

A

A

(9.39)

2

,2

2

,2

,2

0,2

0,2

400 37 367 mm

200 367 72600 mm

20 2 632 mm4

20 4 1256 mm4

12561.73 %

72600

sc

st

st

h

A

A

A

(9.40)

The originally flexural strengths of Beams 1 and 2 can be computed as follows:


224

uRC,1 1 1 ,1 ,1

,1 ,1

1

1

,1 1 ,1

1

1 10

1.5 460 2453 942108 mm

1.15 1.0 30 350 0.8

0.0035108 35 0.0023 0.002

108

c y sc y st

c

c y st sc

c

cuc

s

c

s

ys

N f b c f A A

f A A

f b

c

f

c

hc

(9.41)

2

uRC,1 1 1 ,1 1 ,1 ,1 0,1 1

2

1 11

2

1 0.81.0 30 350 0.8 108 1

1.5 2

1460 942 108 35 2453 667 108

1.15

615.2 kN m

c y sc c

s

y st

c

M f b c f A c h A h cf

(9.42)

2 2 ,2 2 ,2 ,2

2

2

2 2

2

2

,2 2 ,2

2

uRC,2

1 10

3200 62832 15394000 0

62832 62832 4 3200 1539400080 mm

2 3200

0.003580 35 0.00196 0.002

108

cuc s sc c y st

c

cusc c

s

y

N f b c E A c h f Ac

c c

c

c hc

(9.43)

22

2 2 ,2 2 ,2 ,2 0,2 2

2

2

uRC,2

2

1 11

2

1 0.81.0 30 200 0.8 80 1

1.5 2

1 0.00352E5 624 80 35 460 1256 367 80

1.15 80

166.1 kN m

cuc s sc c y st

c s

M f b c E A c h hf A cc

(9.44)

Therefore the originally designed structure is safe before a change in usage, for

the bearing moments are less than the flexural strengths as following:

d,1 uRC,1525.7 kN m < 615.2 kN mM M (9.45)


225

uRC,2d,2 157.5 kN m < 166.1 kN mM M (9.46)

However, as the lived load increasing to 12 kN/m2 due to a change in usage,

the actual loads are as following:

Floor:

Dead load (g): slab, floor and ceiling finishing 3.4 kN/m2

Live load (q): 12.0 kN/m2

Total 1.35g + 1.5q = 23.7 kN/m2

Beam 2 (secondary beam):

Self-weight: 200×400 mm 1.35×0.20×0.40×25 = 2.7 kN/m

From slab: 2.4 m span 23.7×2.4 = 57.0 kN/m

Total q2 = 59.7 kN/m

Beam 1 (main girder):

Self-weight: 350×700 mm q1 = 1.35×0.35×0.70×25 = 8.3 kN/m

From Beam 2 6.0 m span F1 = 59.7×6.0 = 358 kN

Therefore, the design moments on Beams 1 and 2 increase significantly as:

2

d,1

1' 8.3 7.2 358 2.4 912.8 kN m

8M (9.47)

2

d,2

1' 59.7 6.0 268.5 kN m

8M (9.48)

Thus, the original RC Beams 1 and 2 are no longer safe after a change in usage,

because the design moments are much greater than the flexural strengths as

following:

d,1 uRC,1' 912.8 kN m >> 615.2 kN mM M (9.49)

d,2 uRC,2' 268.5 kN m >> 166.1 kN mM M (9.50)


226

9.4.2 Arrangement of steel plates

As the top part on the side faces of Beam 1 is occupied by the secondary

beams, and only the bottom part is available for the installation of steel plates, a

shallow plate-depth of dp,1 = 250 mm is chosen. On the other hand, considering

the moderate steel-ratio of Beam 2, a largest possible plate-depth of dp,2 = 300 mm

is chosen. A trial plate-thickness of tp = 6 mm is also chosen for both the beams

and can be adjusted accordingly in case insufficient flexural strength is proven.

By implementation of Equations (9.17) and (9.18), the designed flexural

strengths can be computed for the Beams 1 and 2 respectively as follows:

1

1

1

2

1

E4

40

11.0 30 350 0.8

1.5

1 0.00176 210E3 0.6 0.0035 355 2

1.15 0.6 0.0035

1.420

1942 2454

1.15

1

0

5.973E6

5.3

2 6 355 450 210E3 0.6 0.0035 7001.15

1210E3 0.6 0.0035 450

1.80E8

15

A

B

C

c

24 1.420

289 mm2

5.973E6 E4 5.380E8 5.973E6

E1.420 4

(9.51)

uBSP,1

22

3

2

21 0.81.0 30 350 0.8 289 1

1.5 2

1200E3 942 289 35

1.15 289

1400 2454 667 289

1.15

1 2 0.0035210E3 6 289 450

1.15 3 289

1 1 289 0.00176 700 289

1.15 3 0.6 0.

0.0035

0.6

3550035

M

1039.7 kN m

(9.52)


227

0.0031 0.002

0.0011 0.0017

0.0029 0.0017

pt py

pb

sc s

y

c

p

Equations (9.17), (9.18) are suitable. (9.53)

2

2

2

2

2

E4

40

11.0 30 200 0.8

1.5

1 0.00176 210E3 0.6 0.0035 355 2

1.15 0.6 0.0035

1.180

1628 1256

1.15

1

0

2.481E6

.6

2 6 355 100 210E3 0.6 0.0035 4001.15

1210E3 0.6 0. 510035 100 2

1 1E8

. 5

A

B

C

c

24 1.180 2

199 mm2

2.481E6 E4 .651E8 2.481E6

E1.180 4

(9.54)

uBSP,2

22

3

2

21 0.81.0 30 200 0.8 199 1

1.5 2

1200E3 628 199 35

1.15 199

1400 1256 367 199

1.15

1 2 0.0035210E3 6 199 100

1.15 3 199

1 1 199 0.00176 400 199

1.15 3 0.6 0.

0.0035

0.6

3550035

M

277.7 kN m

(9.55)

0.0029 0.002

0.0010 0.0017

0.0023 0.0017

pt py

pb py

sc sc

Equations (9.17), (9.18) are suitable. (9.56)

Thus, the BSP Beams 1 and 2 is safe after a change in usage, for the bearing

moments are less than the flexural strengths as following:


228

uBSP,1d,1' 912.8 kN m < 1039.7 kN mM M (9.57)

d,2 uBSP,2' 268.5 kN m < 277.8 kN mM M (9.58)

Furthermore, it is evident from Equation (9.53) that the top edge of the shallow

steel plates are inverse to our pre-set sign convention, which means the entire

sections of the shallow steel plates are subjected to tension force.

9.4.3 Arrangement of anchor bolts

The anchor bolts of Grade 5.8 (fub = 500 MPa, Sby = 1.5 mm) with a diameter

of 12 mm can be chosen for this strengthening design. The yield shear force of an

anchor bolt is:

2

by 0.5 500 28.3 k12

N4

R

(9.59)

Substituting Equation (9.59) and the geometry and material properties of the

steel plates in to Equation (9.22) gives the estimated number of anchor bolts

respectively as:

,1

1355 700 450 6

1 1.152.0 41 pcs14

28.3E31.25

bn

(9.60)

,2

1355 400 100 6

1 1.152.0 50 pcs14

28.3E31.25

bn

(9.61)

Because the depth of steel plates for Beam 1 is 250 mm, 2 rows of anchor

bolts can be used, and the corresponding computed bolt spacing is


229

,1

7200 2176 mm

41 2bS (9.62)

Of course, the computed bolt spacing is an approximate estimation, and 2 rows of

bolts with a bolt spacing of Sb, 1 = 150 mm is actually chosen for fabrication

convenience, thus the total number of bolts for Beam 1 is

,1

72002 2 1 196 pcs

150bn

(9.63)

Because the depth of steel plates for Beam 2 is 300 mm, 3 rows of anchor

bolts can be used, and the corresponding computed bolt spacing is

,2

7200 2216mm

50 3bS (9.64)

For fabrication convenience, 3 rows of bolts with a bolt spacing of Sb, 2 = 150 mm

is actually chosen, thus the total number of bolts for Beam 2 is

,2

72002 3 1 246 pcs

150bn

(9.65)

The bolt spacing in the vertical direction can be arranged corresponding to the

steel structure design codes, and the final strengthening layouts are shown in

Figure 9.9.

9.4.4 Verification of partial interaction

The stiffnesses of the RC beams, the steel plates, and the bolt connections,

along with the corresponding stiffness ratios, are computed according to their

geometry and material properties for Beams 1 and 2, respectively:


230

2

2

,1 ,1

,1 ,1

,1 ,1

6.30E8 N , 3.28E12 N mm

1.66E9 N , 9.04E13 N mm

6.30E8 3.28E120.380 , 0.036

1.66E9 9.04E13

p p

c c

a p

EA EI

EA EI

(9.66)

2

2

,2 ,2

,2 ,2

,2 ,2

7.56E8 N , 5.67E12 N mm

6.91E9 N , 1.25E13 N mm

7.56E8 5.67E121.09 , 0.454

6.91E9 1.25E13

p p

c c

a p

EA EI

EA EI

(9.67)

28.3E3

18900 N mm1.5

bK (9.68)

2

1

,1

,

4

189002 251 N mm

150

2522.78E-12 mm

9.04E13

m

m

k

(9.69)

2

2

,2

,

4

189003 378 N mm

150

3783.02E-11mm

1.25E13

m

m

k

(9.70)

The parameters p and ξp, which are used for the computation of the

longitudinal slips and strain factors, can be computed for Beams 1 and 2,

respectively, as follows:

,1

,1

,1

234 mm

72 mm

225 mm

c

p

cp

i

i

i

(9.71)

,2

,2

,2

135 mm

87 mm

50 mm

c

p

cp

i

i

i

(9.72)


231

2 2

2

1

72 2252.78E-12 234 8.28E-4

0.036 1 0.036p

(9.73)

2 2

2

2 87 503.02E-11 135 1.05E-3

0.454 1 0.454p

(9.74)

,1

8.28E-4 7200ξ 1.9

3 39p

pL (9.75)

,2

1.05E 6000ξ 3.15

2

-3

2p

pL (9.76)

The peak loads for Beams 1 and 2 can be derived from the ultimate flexural

strengthen Mu as follows:

,1

1039.7433.1kN

7.2 3pF (9.77)

,2 2

277.861.7 kN/m

6.0 8pq (9.78)

Then the maximum longitudinal slips Slc, max at the peak loads can be obtained

for Beams 1 and 2, respectively:

2,ma 1 0x,

18.28E-4 9.04E1

433.1E3 225 11

2cosh 1.3 3.28E1 92 91.47 mmlc x

S

(9.79)

2,max,2 0 1.05E-3 1.2

61.7E3 503.15 tanh 3.15

5.675E13 E120.31mmlc x

S

(9.80)

The maximum transverse slips (Str, max) at the peak loads can also be obtained

for Beams 1 and 2, respectively:

3

4 1,max,1 0

433.1E3

9.04E13 0.032 2.78E-12 7200 1 0.036 44

0

.

72

4

00.26 mmt xrS

(9.81)


232

4

,max, 402

61.7E3 60

1.25E13

00 6.901.11mm

6000 2.78E 12t xrS

(9.82)

4 4

,2

2.78E-12 0.454 29300 0.454 2.78E-12 0.4520 6000 6000

3.62 9

4 1

Eξ

6.90

S

(9.83)

8

2

4

226000

600

2.78E-12 0.454 1

16900 0.454 2.78E-12 0.454 1 72000

3.62E

0.454

9

C

(9.84)

Therefore, the resultant slips can be verified as following:

2 2 2 2

,max,1 ,max10

1.47 1.0.26 mm 1.5 49 mmlc tr byx

S S S

(9.85)

2 2 2 2

,max,2 ,max,20

0.31 1.1.11 mm 1.5 15 mmlc tr byx

S S S

(9.86)

The minimum strain and curvature factors (α𝜀, min and αφ, min) can also be

obtained for Beams 1 and 2, respectively, as follows:

1

2

min,1

6.30E8 2521

7200 cosh 1 0.61 0.611.99

3 8.28E-4 sinh 1.99 8.28E-4

(9.87)

1

2

2min,2

7.56E8 3781

6000 1 0.67 0.6

8 1 sech 3.15 1.05E-3

(9.88)

1

4min,1

2500 0.0361.8 0.8 0.036 0.55 0.6

7200 2.78E-12

(9.89)

4 4

min,2

0.63 6000 2.78E-12 10300 0.454 6000 2.78E-12 0.454 1

2.75E9

0.56 0.6

(9.90)


233

8

4

2

2 2.78E-12 0.454 1

14600 0.454 2.78E-12 0.45

6000

60 4 0.8 5500 0.45400

2.75E9

D

(9.91)

It is evident from Equations (9.85) ~ (9.90) that the maximum resultant slips

and the minimum strain and curvature factors can satisfy the requirements, despite

the minimum curvature factors are slightly less than the required limit. This

strengthening arrangement will be still accepted, thanks to the conservation in the

flexural strengths (see Equations (9.57) and (9.58)) and the insensitive variation of

the flexural strength as the strain and curvature factors when α𝜀, min and αφ, min are

greater than 0.5 (see Chapter 5 for details). Of course, further computation shows

the actual flexural strengths of Beams 1 and 2 are still conservative, even when

the smaller curvature factors are obtained. For brevity the computation is omitted

and the results are as follows:

uBSP,1d,1' 912.8 kN m < ' 1037.1 kN mM M (9.92)

d,2 uBSP,2' 268.5 kN m < ' 275.3 kN mM M (9.93)

9.4.5 Discussion of strengthening effect and efficiency

Let Md and Md’ be the design moments before and after a change in usage,

MuRC and MuBSP be the flexural strengths before and after strengthening, the

corresponding factors of safety and enhancement percentages are tabulated in

Table 9.2. The flexural strengths under a full interaction assumption (MuBSP, FI),

together with the corresponding factors of safety and enhancement percentages,

are also computed for comparison.

The original flexural strengths of Beams 1 and 2 are much smaller than the

required design moments for a change in usage (the factors of safety are 0.67 and

0.62 respectively), thus both beams need to be strengthened (the required

enhancements are 48% and 62% respectively). After appropriate strengthening are

employed (see Figure 9.9), the actual enhancements are greater than the


234

requirements (69% > 48% and 66% > 62%, respectively), thus the structure is safe

(the factors of safety are 1.14 and 1.03, respectively).

It is also evident that the utmost enhancements when full interaction

assumption is employed are just slightly greater than the actual enhancements

(71% > 69% and 73% > 66% for Beams 1 and 2, respectively). The strength

losses due to partial interaction are negligible (only 1% and 4%, respectively), and

the relative strengthening effect (see Chapter 8 for details) is greater than 90%

thus very satisfactory (97% and 90%, respectively). Therefore the stiffness of the

bolt connection is sufficient, and it is neither necessary nor economical to arrange

too many anchor bolts for the strengthening of these two BSP beams.

9.5 CONCLUSIONS

In this study, a design procedure is proposed for the strengthening of RC

beams using the BSP technique. The following findings can be concluded based

on the results of the analysis:

(1) By employing the recommended strain and curvature factors, only minor

modification is needed for the conventional flexural strength formula of RC

beams to cover the computation of the flexural strengths of BSP beams.

(2) The recommended strain and curvature factors facilitate the strengthening

design considerably, by dividing the design procedure into two parts: (a) the

evaluation of plate size using the modified flexural strength formula and (b)

the evaluation of number of bolts by the plate size, which is followed by the

verification of the degree of partial interaction using the simplified formulas

proposed in Chapters 6 and 7.

(3) The worked example shows the effectiveness and efficiency of the proposed

design procedure in the strengthening design of RC beams using the BSP

technique.


235

Table 9.1 Comparison of experimental and theoretical peak loads

Specimen Fp,exp Fp,the (Fp,the − Fp,exp) / Fp,exp

CONTROL 268 278 3.7%

P100B300 317 335 5.6%

P100B450 327 364 11.3%

P250B300R 382 369 -3.4%

P250B450R 377 375 -0.3%

Mean absolute error : 5.2%

Table 9.2 Summary of strengthening effect

Description Expression Member type

Beam 1 Beam 2

Design moment before a change in usage Md 526 148

Design moment after a change in usage Md’ 913 269

Flexural strength before strengthening MuRC 615 166

Flexural strength after strengthening MuBSP’ 1037 275

Flexural strength under full interaction MuBSP, FI 1050 288

Factor of safety before a change in usage MuRC / Md 1.17 1.13

Factor of safety after a change in usage MuRC / Md’ 0.67 0.62

Factor of safety after strengthening MuBSP’ / Md’ 1.14 1.03

Required enhancement (Md’ / MuRC) − 1 48% 62%

Actual enhancement (MuBSP’ / MuRC) − 1 69% 66%

Utmost enhancement under full interaction (MuBSP, FI / MuRC) − 1 71% 73%

Strength loss due to partial interaction 1 − (MuBSP’ / MuBSP, FI) 1% 4%

Relative strengthening effect (MuBSP − MuRC) / (MuBSP, FI − MuRC) 97% 90%


236

Figure 9.1 Stress–strain curve of concrete in compression condition

Figure 9.2 Stress–strain curve of steel reinforcement and steel plates

Figure 9.3 Shear force–slip curve of anchor bolts

εc0

σc

fc

εcu

εc O

εy , εyp

σs , σp

fy , fyp

O

εs , εp

Sby

Rb

Rby

O

S


237

Figure 9.4 Sectional strain and stress profiles in a BSP beam: (a) Section, (b)

Strain profile, and (c) Stress profile

Figure 9.5 Sectional strain and stress profiles of steel plates in a BSP beam at

the occurrence of (a) plate yielding and (b) plate entire-sectional tension

c εpt > εyp fyp

fyp

c

εpb > εyp

(a)

(b)

εpt < 0

εpb > εyp fyp

Strain profile

Strain profile

Stress profile

Stress profile

c

φc 0.6φc

εpb εst

εpt

εsc

εcu Es εsc Asc

ηfc

Ep εpt

Ep εpb

fy Ast

λc

h h0

hc

hpt

hpb

(a)

(b)

(c)


238

Figure 9.6 A typical RC structural layout; (a) Plan layout and (b) Elevation

layout

Equipment pipelines

Infill wall Furniture

Equipment pipelines

Type 1 Type 2 Type 1

Ty

pe

2

Ty

pe

2

Ty

pe

2

Type 2 Type 1 Type 1 Type 2

(a)

(b)

Beam 1

Bea

m 2


239

Figure 9.7 Strengthening strategies for the RC beams of (a) Type 1 and (b)

Type 2

Figure 9.8 Simplified models for (a) Beam 1 (a main girder) and (b) Beam 2 (a

secondary beam) (dimensions in mm)

(b)

(a)

Dsl

Ast

Dsb

hpt ≥

Dsl,

Dsb

hpb ≥

Dc

Dc

Dp

(b)

(a)

6000

2400

q2

F1 F1

2400 2400

7200

q1


240

Figure 9.9 Strengthening details for (a) Beam 1 (a main girder) and (b) Beam 2

(a secondary beam) (dimensions in mm)

(b)

(a)

350

200

40

0

40

0

70

0

10

0

10

0

70

11

0

70

60

9

0

60

25

0

30

0

Sb = 150 Sb = 150

90

3T10

5T25

2T10

4T20

Chapter 10 Conclusion

241

CHAPTER 10

CONCLUSIONS

10.1 SUMMARY

The BSP retrofitting technique not only combines the great performances of

steel in tension and concrete in compression, but also provides a lot of unique

advantages over the other strengthening methods. For instance, it prevents the

premature peeling failures, enhances load capacities without a significant

reduction in deformability, avoids the potential risk of destroying the tensile

reinforcement in the preparation of bolt holes, and provides space on the soffit

faces to prop below the beam during construction process.

However, as a newly arising technique, corresponding studies are lacking in

literature. A comprehensive study on the partial interaction caused by both

longitudinal and transverse slips at the plate–RC interface has yet to be carried out.

Furthermore, most of existing studies focused on the strengthening techniques of

lightly reinforced RC beams, while the retrofitting of moderately reinforced RC

beams have not attracted enough attention of previous researchers.

Aiming at developing reliable analytical models for the longitudinal and

transverse partial interaction, thus providing a more accurate approach to evaluate

the performance of BSP beams, comprehensive studies were conducted by the

author, as reported in the previous chapters of this thesis.

Several moderately reinforced BSP beams with different plate depths and bolt

spacings were tested under four-point bending. Their behaviour was investigated

and compared to the available test results for lightly reinforced BSP beams. Both

the overall load–deflection performance and the parameters controlling the degree

of partial interaction between steel plates and RC beams, which are essential to

the overall performance of BSP beams, were investigated in detail.


242

A nonlinear finite element model was formulated to investigate both the

overall load–deflection performance and the specific behaviours such as the

longitudinal and transverse slips and shear transfers in BSP beams. Compared to

experimental studies, the numerical method provides a more economical approach

to the analysis of BSP beams with different beam geometries and loading

conditions, and overcomes the difficulty of measuring the transverse slip precisely

in tests.

Since the major factor controlling the performance of BSP beams is the

degree of partial interaction due to the longitudinal and transverse slips on the

plate–RC interface, the main effort of this study was the development of

analytical models for both the longitudinal and the transverse slips and shear

transfers. An analytical approach for the longitudinal slip and shear transfer was

developed based on the BSP section analysis which takes account of the different

stress and strain profiles of steel plates and RC beams. A piecewise linear model

was also developed for the transverse slip and shear transfer based on the

outcomes of the numerical study, Winkler’s model and the force superposition

principle. These new analytical models allow us to evaluate the degree of partial

interaction between steel plates and RC beams in terms of the strain and the

curvature factors.

Based on the proposed analytical models, a computer program was developed

to evaluate the performance of BSP beams with partial interaction. A parametric

optimization study was also conducted. As a result, the balance between the

strength enhancement and the economical number of bolts, in terms of a unified

value for both the strain and the curvature factors, was established.

A design procedure has been proposed at the end of this thesis. The modified

flexural strength formula, which includes the influence of partial interaction, is

used to determine the plate dimension. The simplified formulas for the maximum

bolt slips and the minimum strain and curvature factors are used to determine the

bolt arrangement and verify the degree of partial interaction. This new design

approach is not only easy to use but also more accurate than the conventional

design methods using the assumption of full interaction.


243

10.2 CONCLUSIONS

In this section, the main findings achieved through the experimental,

numerical and theoretical studies in the previous chapters are summarised and

concluded. These conclusions provide a comprehensive view of the behaviour of

BSP beams, which can provide a valuable source of information to structural

engineers in their strengthening design of BSP beams.

The strength and stiffness of lightly reinforced RC beams with a degree of

reinforcement less than 1/3 can be strengthened by attaching shallow steel plates

to the tensile region of the side faces with a small sacrifice of ductility. On the

other hand, moderately reinforced RC beams with a degree of reinforcement

greater than 2/3 can be retrofitted effectively only by adding deep steel plates,

which cover both the tensile and compressive regions of the side faces.

The steel plates in BSP beams contribute to the overall flexural strength in

forms of both the coupling moment provided by their axial tensile forces and the

bending moment provided by their flexural stiffness. The shallow steel plates in

the lightly reinforced BSP beams, which serve as additional tensile reinforcement,

contribute mainly in the former form. On the other hand, the deep steel plates in

the moderately reinforced BSP beams, which serve as both additional tensile and

compressive reinforcement, contribute mainly in the latter form.

The strengthening effect of BSP beams is governed by the degree of partial

interaction on the plate–RC interface, which is the result of both longitudinal and

transverse slips and can be quantified by the strain and the curvature factors. For

the lightly reinforced BSP beams with shallow steel plates, the longitudinal slip is

the dominant factor in evaluating the overall performance, and the transverse slip

can be neglected; the strengthening effect is controlled by both the strain and the

curvature factors. However, for the moderately reinforced BSP beams with deep

steel plates, the longitudinal slip is no longer the dominant factor, and the

transverse slip also controls the overall performance; the strengthening effect is

controlled mainly by the curvature factor.


244

The magnitudes of the longitudinal and transverse slips increase as increasing

load level, plate–RC stiffness ratio, and bolt spacing. The longitudinal slip reaches

a maximum value at the plate ends and vanishes near the midspan where the plate

tensile force reaches the maximum value. The transverse slip concentrates and

reaches a maximum at the point of applied load or at the supports. The shape of

transverse slip profile is independent of the magnitude of applied load and the

flexural stiffness of the RC beam, but is controlled by the plate-bolt stiffness ratio.

Since the magnitudes and locations of maximum bolt slips are highly controlled

by the location of external loads, uniform bolt spacing is recommended in the

strengthening design of BSP beams. The strain factor reaches a minimum value at

the positions where the external loads located or the longitudinal slip is zero. The

curvature factor reaches a minimum at the midspan in most cases.

For the lightly reinforced BSP beams with shallow steel plates, the ultimate

flexural strength enhancement is governed by the difference between the current

steel ratio and the balanced steel ratio. The available area on the side faces is also

a controlling parameter because the degree of partial interaction, in other words

the available number of anchor bolts, is limited by the available side-face area.

However, for the moderately reinforced BSP beams with deep steel plates, the

tensile steel ratio is no longer an obstacle to the ultimate strength enhancement,

since the deep steel plates increase both tensile and compressive reinforcement.

The available side-face area becomes the key parameter, for it controls both the

available plate depth and the maximum number of anchor bolts.

The load capacity of BSP beams would be overestimated if the assumption of

full interaction is employed in the calculation. Results that are more accurate can

be obtained by taking the partial interaction on the plate–RC interface into

account. A strain or curvature factor of 0.6 can attain an optimal enhancement

with a reasonable number of anchor bolts, and an excessive connection is neither

economic nor necessary. The recommended strain and curvature factors facilitate

the strengthening design considerably by dividing the design procedure into two

parts: (a) the evaluation of the plate size by the modified flexural strength formula

and (b) the evaluation of the number of anchor bolts by the plate size and the

verification of the degree of partial interaction.


245

For the lightly reinforced BSP beams with shallow steel plates, the main

failure mode is the yielding of the tensile reinforcement and the bolted shallow

plates, which is ductile thus is the preferable failure pattern. In order to prevent

the compressive concrete from crushing, the size of the bolted steel plates should

be limited. However, for the moderately reinforced BSP beams with deep steel

plates, the plate buckling is the greatest potential risk thus should be suppressed

by appropriate buckling restraint measures. The brittle shear failure of the anchor

bolts is undesirable for BSP beams of both types, thus should be avoided by

arranging enough anchor bolts.

10.3 RECOMMENDATIONS FOR FUTURE STUDY

The formulation of the analytical models for longitudinal and transverse

partial interaction is based on the simply supported beams. Certain modifications

should be made and more experimental studies ought to be conducted to cover the

continuous RC beams, which represent a major portion in our building stock.

Proper connections can also be employed between the plate-ends and the

supporting columns if main girders are strengthened using the BSP technique, or

between the plate-ends and the supporting main girders when secondary beams

are strengthened. In this way, the steel plates can provide additional compressive

reinforcement to the sections near the supports, thus improve the flexural strength

and ductility significantly.

In the experimental study, moderately reinforce RC beams with a tensile steel

ratio of 1.77% were investigated. Further experimental studies on RC beams with

higher degree of reinforcement should be carried out, for a steel ratio up to 2.50%

is also widely used in the industry.

Buckling in the compressive region is a main concern for the deep steel plates

in BSP beams. Although it was constrained by the buckling restraint measures in

the experimental study, the effect was barely satisfactory. Thick steel angles or

channels are suggested to be welded directly to the compression region of steel

plates in the strengthening practice.


246

For brevity, friction between the steel plates and the RC beams was ignored

in this study, further theoretical and experimental studies might be necessary for

further clarification and justification.

Resistance to corrosion and fire is an important issue for steel plates and

anchor bolts, to which due attention and consideration should be paid. Stainless

steel and galvanized steel can be used to enhance the durability of BSP beams,

and certain fire retardant coating can be used for steel to help retain the loading

capacity of BSP beams under elevated ambient temperature.

247

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253

PUBLICATIONS

==

International refereed journal publications:

Li, L.Z., Lo, S.H. and Su, R.K.L. (2012). “Experimental study of moderately

reinforced concrete beams strengthened with bolted-side steel plates.”

Advances in Structural Engineering, 16(3), pp. 499-516.

Su, R.K.L., Li, L.Z. and Lo, S.H. (2013). “Shear Transfer in Bolted Side-Plated

Reinforced Concrete Beam.” Engineering Structures, (in press).

Lo, S.H., Li, L.Z. and Su, R.K.L. (2013). “Optimization of partial interaction in

bolted side-plated reinforced concrete beams.” Computers and Structures,

(revision submitted).

Su, R.K.L., Li, L.Z. and Lo, S.H. (2013). “Longitudinal partial interaction in bolted

side-plated reinforced concrete beams.” Advances in Structural Engineering,

(revision submitted).

Su, R.K.L., Li, L.Z. and Lo, S.H. (2013). “A piecewise linear shear transfer model

for bolted side-plated RC beams.” Engineering Structures, (under review).

Lam, W.Y., Li, L.Z., Su, R.K.L., Pam, H.J. (2013). “Behaviour of plate anchorage

in plate-reinforced composite coupling beams.” Structural Engineering and

Mechanics, (under review).

Conference publications:

Li, L.Z., Lo, S.H. and Su, R.K.L. (2013). “Study of moderately reinforced concrete

beams strengthened by bolted-side steel plates.” Design Fabrication and

Economy of Metal Structures, Miskolc, Hungary.

Documents

New partial interaction models for bolted-side-plate reinforced concrete beams