j======================================================
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
xiv
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
xvi
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
xvii
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
Figure 6.7 The profiles of shear force, bending moment and longitudinal
slip in a BSP beam under arbitrary three-point bending ................ 144
Figure 6.8 The profiles of shear force, bending moment and longitudinal
slip in a BSP beam under a uniformly distributed load (UDL) ...... 145
Figure 6.9 The profiles of shear force, bending moment and longitudinal
slip in a BSP beam under a triangularly distributed load (TDL) .... 145
Figure 6.10 The profiles of shear force, bending moment and longitudinal
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
Figure 6.13 Comparison of longitudinal slip profiles obtained from the
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
the experimental and theoretical studies for (a) P100B300 and (b)
P100B450 ....................................................................................... 152
xxi
Figure 6.17 Comparison of the maximum plate tensile forces obtained from
the experimental and theoretical studies for (a) P100B300 and (b)
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
Chapter 1 Introduction
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
Chapter 1 Introduction
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.
Chapter 1 Introduction
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.
Chapter 1 Introduction
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.
Chapter 1 Introduction
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
Chapter 2 Literature review
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.
Chapter 2 Literature review
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
Chapter 2 Literature review
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.
Chapter 2 Literature review
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.
Chapter 2 Literature review
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
Chapter 2 Literature review
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.
Chapter 2 Literature review
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.
Chapter 2 Literature review
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.
Chapter 2 Literature review
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.
Chapter 3 Experimental Study on BSP Beams
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.
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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.
Chapter 3 Experimental Study on BSP Beams
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.
Chapter 3 Experimental Study on BSP Beams
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.
Chapter 3 Experimental Study on BSP Beams
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.
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
(%) Sample 1 Sample 2 Sample 3
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
(%) Sample 1 Sample 2 Sample 3
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
(%) Sample 1 Sample 2 Sample 3
6 337.6 313.5 330.2 327.1 3.1
Thickness
(mm)
Ultimate strengths of different samples (MPa) fup
(MPa)
Standard
derivation
(%) Sample 1 Sample 2 Sample 3
6 460.4 460.0 455.8 458.7 0.6
Thickness
(mm)
Elastic moduli of different samples (GPa) Ep
(GPa)
Standard
derivation
(%) Sample 1 Sample 2 Sample 3
6 225.5 210.7 220.0 218.7 2.8
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
29
Figure 3.4 Reinforcement cages
Chapter 3 Experimental Study on BSP Beams
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)
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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)
Chapter 3 Experimental Study on BSP Beams
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)
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
37
(c) P100B300
(b) P75B300
Chapter 3 Experimental Study on BSP Beams
38
(e) P250B300R
(d) P100B450
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 3 Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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.
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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.
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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.
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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.
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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 -
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Midspan deflection (mm)
CONTROL
CONTROL*
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Midspan deflection (mm)
P100B300
CONTROL
P250B300 P250B300R
0 20 40 60 80 100 120
0
100
200
300
400
P250B300R
P250B450R
P100B300
P100B450
CONTROL
Load
(kN
)
Midspan deflection (mm)
P100B450 CONTROL
P250B300R
P250B450R
P100B300
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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)
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Distance from midspan (mm)
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Distance from midspan (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
)
Distance from midspan (mm)
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Distance from midspan (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
)
Distance from midspan (mm)
(a)
(b)
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
)
Distance from midspan (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
)
Distance from midspan (mm)
(c)
(d)
Chapter 4 Result and Analysis of Experimental Study on BSP Beams
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
Midspan deflection (mm)
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
Midspan deflection (mm)
(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).
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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.
Chapter 5 Numerical Study on BSP Beams
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:
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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:
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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.
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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.
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
91
Figure 5.2 Bond–slip curve from CEB-FIB Model Code 1990 (CEB 1993)
s s1 s2
s3
τf
τmax
τ
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
)
Midspan deflection (mm)
0 20 40 60 80 100
0
100
200
300
400
P250B450R, Numerical
P250B450R, Experimental
P100B450, Numerical
P100B450, Experimental
Load
(kN
)
Midspan deflection (mm)
(a)
(b)
Chapter 5 Numerical Study on BSP Beams
96
Figure 5.7 Comparison of longitudinal slip profiles obtained from the
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
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Longit
udin
al s
lip S
lc (
mm
)
Distance from midspan (mm)
-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
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Longit
udin
al s
lip S
lc (
mm
)
Distance from midspan (mm)
(a)
(b)
Chapter 5 Numerical Study on BSP Beams
97
(a)
(b)
-1800 -1200 -600 0 600 1200 1800
0.30
0.00
-0.30
F/Fp = 0.75 , Experimental
F/Fp = 0.75 , Numerical
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Tra
nsv
erse
sli
p S
tr (
mm
)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 1800
0.30
0.00
-0.30
F/Fp = 0.75 , Experimental
F/Fp = 0.75 , Numerical
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Tra
nsv
erse
sli
p S
tr (
mm
)
Distance from midspan (mm)
Chapter 5 Numerical Study on BSP Beams
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
F/Fp = 0.75 , Experimental
F/Fp = 0.75 , Numerical
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Tra
nsv
erse
sli
p S
tr (
mm
)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 1800
0.30
0.00
-0.30
F/Fp = 0.75 , Experimental
F/Fp = 0.75 , Numerical
F/Fp = 0.25 , Experimental
F/Fp = 0.25 , Numerical
Tra
nsv
erse
sli
p S
tr (
mm
)
Distance from midspan (mm)
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
Longit
udin
al s
hea
r tr
ansf
er (
kN
/m)
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
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
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
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
Distance from left support (x/L)
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)
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
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)
Distance from left support (x/L)
Shea
r tr
ansf
er v
m (
kN
/m)
vm, LS
w
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
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
Distance from left support (x/L)
Norm
aliz
ed s
hea
r tr
ansf
er
(b)
(b)
(a)
Chapter 5 Numerical Study on BSP Beams
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
Distance from left support (x/L)
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
Distance from left support (x/L)
Norm
aliz
ed s
hea
r tr
ansf
er
(c)
(d)
(c)
Chapter 5 Numerical Study on BSP Beams
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
Chapter 5 Numerical Study on BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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.
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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.
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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.
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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:
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
The solution of the ODE is given by:
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
The solution of the ODE is given by:
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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.
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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.
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
144
Figure 6.6 The profiles of shear force, bending moment and longitudinal slip in
a BSP beam under four-point bending
Figure 6.7 The profiles of shear force, bending moment and longitudinal slip in
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
145
Figure 6.8 The profiles of shear force, bending moment and longitudinal slip in
a BSP beam under a uniformly distributed load (UDL)
Figure 6.9 The profiles of shear force, bending moment and longitudinal slip in
a BSP beam under a triangularly distributed load (TDL)
L
q
V
M
Slc
L
q
V
M
Slc
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
146
Figure 6.10 The profiles of shear force, bending moment and longitudinal slip in
a BSP beam under a support moment
L
V
M
Slc
MS
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
149
Figure 6.13 Comparison of longitudinal slip profiles obtained from the
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
F/Fp = 0.25, Experimental
F/Fp = 0.25, Theoretical
Longit
udin
al s
lip S
lc (
mm
)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 1800
-1.5
0.0
1.5
F/Fp = 0.75, Experimental
F/Fp = 0.75, Theoretical
F/Fp = 0.25, Experimental
F/Fp = 0.25, Theoretical
Longit
udin
al s
lip S
lc (
mm
)
Distance from midspan (mm)
(b)
(a)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
150
Figure 6.14 Comparison of longitudinal tensile force transfers obtained from the
experimental and theoretical studies for (a) P100B300 and (b) P100B450
-1800 -1200 -600 0 600 1200 1800
0
250
500
F/Fp = 0.75, Experimental
F/Fp = 0.75, Theoretical
F/Fp = 0.25, Experimental
F/Fp = 0.25, Theoretical
Pla
te t
ensi
on f
orc
e N
p (
kN
)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 1800
0
250
500
F/Fp = 0.75, Experimental
F/Fp = 0.75, Theoretical
F/Fp = 0.25, Experimental
F/Fp = 0.25, Theoretical
Pla
te t
ensi
on f
orc
e N
p (
kN
)
Distance from midspan (mm)
(b)
(a)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
152
Figure 6.16 Comparison of the maximum longitudinal slips obtained from the
experimental and theoretical studies for (a) P100B300 and (b) P100B450
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
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
Fb /Fbp = 0.75
Fb /Fbp = 0.10
Experiment
Fb /Fbp = 0.75
Fb /Fbp = 0.10
Experiment
(b)
(a)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
153
Figure 6.17 Comparison of the maximum plate tensile forces obtained from the
experimental and theoretical studies for (a) P100B300 and (b) P100B450
0.00 0.25 0.50 0.75 1.00
0
200
400
Upper boundary prediction :
Kb=Kb, 0.10 at Fb /Fbp = 0.10
Lower boundary prediction :
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
Upper boundary prediction :
Kb=Kb, 0.10 at Fb /Fbp = 0.10
Lower boundary prediction :
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)
Chapter 6 Theoretical Study on Longitudinal Partial Interaction in BSP Beams
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
Distance from midspan (mm)
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.
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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.
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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).
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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:
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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.
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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):
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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.
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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)
1.4 for shallow plates
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
0.7 for shallow plates
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:
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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:
0.155 for shallow plates
0.250 for deep platesw
(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)
0.45 for shallow plates
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:
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
171
3
4
,max 3
1
0
4 1
for shallow plates0.114 1 77.0
for deep plates0.092 1 77.0
m pc
m p
tr x
c
FL
EI
FL
EI
LS
L
(7.47)
,max 0
,max 2
,max 0
2.2 for shallow plates
1.0 for deep plates
tr x
tr x L
tr x
SS
S
(7.48)
The minimum curvature factor (αφ, min) occurs at the midspan (i.e., x = L/2)
and reads:
11
4
,min 12 14
0.93 0.07 625 for shallow plates
2.21 1.21 1840 for deep plates
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.
0.133 for shallow plates
0.145 for deep platesw
(7.50)
By introducing Equation (7.50) into the solution technique based on force
equilibrium and displacement compatibility, the maximum shear transfer ratios at
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
20 29300 1 for shallow plates
25 21100 1 for deep plates
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)
The minimum curvature factor (αφ, min) occurs at the midspan (i.e., x = L/2)
and reads:
4
,min 2 4
4
4
0.72 5400 1 for shallow plates
0.63 10300 1 for deep plates
m p m p
m p m p
x L
L L D
L L D
(7.57)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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.
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Distance from left support (x/L)
Shea
r tr
ansf
er (
kN
/m)
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
Simplification as a polyline
Distance from left support (x/L)
Shea
r tr
ansf
er (
kN
/m)
(b)
(a)
Opposite shear
transfer
Positive shear transfer
Negative shear
transfer
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Simplification as a polyline
Distance from left support (x/L)
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
Simplification as a polyline
Distance from left support (x/L)
Shea
r tr
ansf
er (
kN
/m)
(d)
(c)
Opposite shear
transfer
Positive shear transfer
Negative shear
transfer
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Positive shear transfer
(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’
Positive shear transfer
x
Opposite shear transfer
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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’
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
180
Figure 7.5 Variation of shear transfer profile (a) before and (b) after cracking
occurs
(a)
(b)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Distance from midspan (mm)
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
Distance from midspan (mm)
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)
Distance from midspan (mm)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Piecewise linear model
ExperimentP100B450
Shea
r tr
ansf
er (
kN
/m)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 180060
40
20
0
-20
-40
-60
Piecewise linear model
ExperimentP250B300R
Shea
r tr
ansf
er (
kN
/m)
Distance from midspan (mm)
-1800 -1200 -600 0 600 1200 180040
20
0
-20
-40
P250B450R Piecewise linear model
Experiment
Shea
r tr
ansf
er (
kN
/m)
Distance from midspan (mm)
Chapter 7 Theoretical Study on Transverse Partial Interaction in BSP Beams
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
Distance from left support (x/L)
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.
Chapter 8 Analysis of BSP Beams with Partial Interaction
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:
Chapter 8 Analysis of BSP Beams with Partial Interaction
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 .
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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)
Chapter 8 Analysis of BSP Beams with Partial Interaction
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).
Chapter 8 Analysis of BSP Beams with Partial Interaction
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.
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial 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:
Chapter 8 Analysis of BSP Beams with Partial Interaction
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.
Chapter 8 Analysis of BSP Beams with Partial Interaction
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)
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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)
Chapter 8 Analysis of BSP Beams with Partial Interaction
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 :
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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)
Chapter 8 Analysis of BSP Beams with Partial Interaction
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)
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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
Chapter 8 Analysis of BSP Beams with Partial Interaction
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.
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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:
Chapter 9 Design of BSP Beams with Partial Interaction
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:
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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
M f b c E A c h A h cc
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
M f b c E A c h A h cc
cE t c h t h c
f
fc
(9.17)
Chapter 9 Design of BSP Beams with Partial Interaction
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).
Chapter 9 Design of BSP Beams with Partial Interaction
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.
Chapter 9 Design of BSP Beams with Partial Interaction
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
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
(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
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
(9.28)
Chapter 9 Design of BSP Beams with Partial Interaction
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
0.72 5400 1 for shallow plates
0.63 10300 1 for deep plates
m p m p
m p m p
x L
L L D
L L D
(9.32)
4 4
4 4
20 29300 1 for shallow plates
25 21100 1 for deep plates
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.
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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:
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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:
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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:
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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.
Chapter 9 Design of BSP Beams with Partial Interaction
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%
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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)
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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
Chapter 9 Design of BSP Beams with Partial Interaction
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.
Chapter 10 Conclusion
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.
Chapter 10 Conclusion
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.
Chapter 10 Conclusion
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.
Chapter 10 Conclusion
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.
Chapter 10 Conclusion
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
REFERENCES
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reinforced concrete beams using steel plates bonded on beam web:
experiments and analysis.” Construction and Building Materials, 14(5), pp.
237–244.
An, W., Saadatmanesh, H. and Ehsani, M.R. (1991). “RC beams strengthened with
FRP plates. II: Analysis and parametric study.” Journal of Structural
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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.
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