Master Thesis Rocking-Final, SROUR-SAVOIA

ALMA MATER STUDIORUM – UNIVERSITA’ DI BOLOGNA
FACOLTA’ DI INGEGNERIA
Corso di Laurea Magistrale in Civil Engineering
D.I.C.A.M.
Dipartimento di Ingegneria Civile, Ambientale e dei Materiali
Tesi di Laurea in
Earthquake Engineering
ROCKING SYSTEM FOR SEISMIC PROTECTION OF
REINFORCED CONCRETE STRUCTURES
Tesi di Laurea di: Relatore:
MAHDI YOUSSEF SROUR Chiar.mo Prof. Ing. MARCO SAVOIA
Correlatori:
Dott. Ing. NICOLA BURATTI
Sessione III
Anno Accademico 2010/2011

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DEDICATION
To Mariam and Youssef

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TABLE OF CONTENTS
DEDICATION.......................................................................................................................i
TABLE OF CONTENTS ....................................................................................................iii
LIST OF SYMBOLS..........................................................................................................vii
LIST OF FIGURES.............................................................................................................xi
LIST OF TABLES .............................................................................................................xv
AKNOWLEDEMENTS ......................................................................................................xvii
ABSTRACT.......................................................................................................................xix
1. INTRODUCTION TO ROCKING SYSTEMS IN CONCRETE STRUCTURES .............1
1.1. Foreword .......................................................................................................................1
1.2. Introduction to Rocking Structures ...............................................................................1
1.3. Experimental Tests of the Behaviour of Jointed, Precast, Post-Tensioned Rocking
Structures.......................................................................................................................2
1.3.1. The U.S. and Japan PRESSS Research Program .............................................2
1.3.2. Monotonic and Cyclic Quasi Static Tests ........................................................8
1.4. Post-Tensioned, Precast Wall Systems .......................................................................11
1.5. Conclusion...................................................................................................................14
2. LITERATURE REVIEW OF ROCKING THEORY ........................................................15
2.1. Rocking Wall Base Sliding .........................................................................................15
2.2. Mechanics of a Rocking Wall .....................................................................................19
2.2.1. Equations of Motion.......................................................................................19
2.2.2. Forces in a Rocking System ...........................................................................22
2.2.2.1. Forces in the System before Impact......................................................22
2.2.2.2. Forces in the System at Impact .............................................................27
2.2.3. Energy Dissipation Capacity of a Rocking Rigid Wall..................................31
2.3. Adapting Rocking Walls to meet a Target Performance.............................................37
2.3.1. Forces in a Rocking System with Hysteretic Energy Dissipators at the Base 38
2.3.1.1. Forces in the System before Impact......................................................38
2.3.1.2. Forces in the System at Impact .............................................................39
2.3.2. Total Accelerations in the System..................................................................40

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2.3.3. Energy Dissipation Capacity of the Rocking Rigid-Wall with Hysteretic
Energy Dissipators................................................................................................41
2.4. Modeling Techniques for Post-Tensioned Rocking Systems......................................45
2.4.1. Lateral Response of Post-Tensioned Connections..........................................45
2.4.2. Section Analysis Methods for Post-Tensioned Rocking Connections............46
2.5. Design Recommendations ...........................................................................................55
2.5.1. Introduction.....................................................................................................55
2.5.2. Proposed Procedure for the Seismic Design of the Structure .........................56
2.5.2.1. Displacement Based Seismic Design of the System .............................56
2.5.2.2. Assessment of the Overall Performance................................................58
2.5.3. Modifying the Response of Rocking Walls ....................................................61
2.6. Non Linear Time History Analyses.............................................................................62
2.7. Conclusion ...................................................................................................................65
3. CONCRETE BRIDGE PIER IN SEISMIC ACTIONS......................................................67
3.1. Introduction..................................................................................................................67
3.2. Hybrid Systems............................................................................................................67
3.2.1. Introduction to Hybrid Systems ......................................................................67
3.2.2. Overview.........................................................................................................69
3.2.3. Background of Post-Tensioned Precast Bridge Pier Systems.........................71
3.3. Hybrid Details Investigated .........................................................................................72
3.4. Cyclic Modeling Techniques for Post-Tensioned Rocking Connections....................74
3.5. Conclusion ...................................................................................................................80
4. NUMERICAL MODELING OF BRIDGE PIER WITH MONOLITHIC
CONNECTION...................................................................................................................83
4.1. Introduction..................................................................................................................83
4.2. Column Description.....................................................................................................83
4.3. Moment-Curvature Analysis........................................................................................86
4.3.1. Input Properties...............................................................................................87
4.3.1.1. General Input Parameters And Section Geometry ................................87
4.3.1.2. Longitudinal Reinforcing Steel .............................................................88
4.3.1.3. Concrete.................................................................................................89
4.3.2. Results.............................................................................................................91
4.3.3. Trilinear Idealization.......................................................................................93

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4.4. Structural Analysis Model...........................................................................................94
4.4.1. Description of the Model................................................................................94
4.4.1.1. Geometry, Masses and Applied Loads..................................................94
4.4.1.2. Elastic Cross-Section Properties ...........................................................98
4.4.1.3. Hysteresis Rule .....................................................................................98
4.4.1.4. Plastic Hinge Length...........................................................................100
4.4.1.5. Structural Damping.............................................................................101
4.4.2. Monotonic Displacement-Driven Pushover Analysis ..................................101
4.4.2.1. Results.................................................................................................101
4.4.3. Cyclic Displacement-Driven Pushover Analysis .........................................103
4.4.3.1. Input Time History at Top Node.........................................................103
4.4.3.2. Results.................................................................................................103
4.4.4. Non-Linear Time History Analysis ..............................................................105
4.4.4.1. Input Ground Motion ..........................................................................105
4.4.4.2. Results.................................................................................................107
4.5. Conclusion.................................................................................................................114
5. NUMERICAL MODELING OF BRIDGE PIER WITH HYBRID CONNECTION......115
5.1. Introduction ...............................................................................................................115
5.2. The Numerical Model ...............................................................................................115
5.2.1. Hybrid Connection Properties ......................................................................115
5.2.2. Description of the Model..............................................................................116
5.2.2.1. Geometry, Masses and Applied Loads................................................116
5.2.2.2. Elastic Cross-Section Properties .........................................................118
5.2.2.3. Hysteresis Rule ...................................................................................119
5.2.2.4. Structural Damping.............................................................................120
5.2.3. Cyclic Displacement-Driven Pushover Analysis .........................................122
5.2.3.1. Results.................................................................................................122
5.2.4. Non-Linear Time History Analysis ..............................................................123
5.2.4.1. Results.................................................................................................124
5.3. Conclusion.................................................................................................................126

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6. COMPARATIVE RESULTS OF MONOLITIC AND HYBRID BRIDGE PIER
CONNECTIONS.............................................................................................................. 127
6.1. Introduction................................................................................................................127
6.2. Pushover Analysis Results.........................................................................................127
6.3. Non-Linear Time History Analysis Results...............................................................128
7. CONCLUSION AND FUTURE DEVELOPEMENT......................................................133
7.1. Summary....................................................................................................................133
7.2. Conclusions................................................................................................................134
7.3. Recommendations for Future Development..............................................................135
REFERENCES..................................................................................................................137
APPENDIXES ..................................................................................................................141
Appendix A Column Full Information..............................................................................141
Appendix B Columna Input File.......................................................................................143
Appendix C Loma Prieta Earthquake 1989 ......................................................................146
Appendix D Ruaumoko Input files...................................................................................147
Appendix E Model Calibration.........................................................................................155

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LIST OF SYMBOLS
Ad dissipation bar area
APT post tensioning strand area
B column cross section size
be confined concrete region thickness
Bf foundation dimension
ccover concrete cover
cd viscous damping coefficient
cNA neutral axis depth
de elastic displacement
di inelastic displacement
Ec concrete Young modulus
Edissipated energy dissipated in one cycle
Eelastic elastic energy at maximum response
Es steel Young modulus
fcc confined concrete strength
fck concrete cylindrical strength
Fd damping force
Fe elastic force
Fi i-floor design force
Fl maximum confining lateral stress
fl minimum confining lateral stress
Fp0 initial prestress
FPT post tensioning force
Fu ultimate lateral force
Fy yield lateral force
fyk steel yield stress
g acceleration of gravity
G soil shear modulus
Gred reduced soil shear modulus
H structure height
Heff structural effective height
Ieff effective modulus of inertia

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Igross gross modulus of inertia
k structural stiffness
keff effective stiffness
ki initial stiffness force-displacement relationship
ki’ initial stiffness moment-curvature relationship
Ks superstructure stiffness
ku unloading stiffness force-displacement relationship
ku’ unloading stiffness moment-curvature relationship
Kx foundation horizontal stiffness
ky yield stiffness force-displacement relationship
Kz foundation vertical stiffness
Kq foundation rotational stiffness
Lp plastic hinge length
lunb_d dissipation bar unbonded length
lunb_PT tendon unbonded length
Lw wall depth
m seismic mass
meff effective seismic mass
Mu design moment
My yield moment
N axial load
P gravity load
q force reduction factor
r post-yield stiffness ratio force-displacement relationship
r’ post-yield stiffness ratio moment-curvature relationship
Sa spectral acceleration
SD spectral displacement
T0 structural period at secant stiffness at yield
Teff effective period
Vb base shear
vs shear wave velocity

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Greek symbols:
 Takeda model parameter force-displacement relationship
’ Takeda model parameter moment-curvature relationship
 Takeda model parameter force-displacement relationship
’ Takeda model parameter moment-curvature relationship
d target displacement
f displacement due to foundation rotation
res residual displacement
s structural displacement
u ultimate displacement
y inelastic displacement
cu maximum concrete compressive strain
y steel yield strain
p plastic curvature
res residual curvature
u ultimate curvature
y yield curvature
 spectrum damping dependence
 displacement ductility
 curvature ductility
 axial load ratio
soil soil Poisson modulus
 second to first order moment ratio
f foundation rotation
0 yield to gross stiffness ratio
l longitudinal steel ratio
soil soil density
 angular frequency
f foundation angular frequency
s structure angular frequency
initial stiffness elastic damping
tangent stiffness elastic damping
eq equivalent viscous damping

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f foundation equivalent viscous damping
hyst hysteretic damping
s structural equivalent viscous damping

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LIST OF FIGURES
Figure 1.1 Five storeys precast post-tensioned frame building tested at the University of
California, San Diego (Priestley et al. [1999]).....................................................................8
Figure 1.2 Rocking motion of Hybrid connection (Dissipater is anonymous in this figure)9
Figure 1.3 The states of rocking wall behaviour ..................................................................9
Figure 1.4 Post-tensioned rocking wall tested at University of Canterbury, Rahman and
Restrepo [2000]...................................................................................................................12
Figure 1.5 Detailing of a post-tensioned wall, providing load paths for the high
compression forces during rocking, Holden [2001]...........................................................13
Figure 2.1 The relationship between Base Moment and Rotation .....................................16
Figure 2.2 - Rocking wall free body diagram (Housner’s block).......................................19
Figure 2.3 Mechanical properties of a rigid rocking block, (a) Rocking period of vibration
with amplitude and (b) Moment rotation response of rocking block ..............................22
Figure 2.4 Forces and actions in a rocking rigid wall ........................................................23
Figure 2.5 The impact process............................................................................................27
Figure 2.6 Aslam et al. [1980] post-tensioned rocking block ............................................34
Figure 2.7 Moment rotation response of a post-tensioned rigid block...............................34
Figure 2.8 Makris and Zhang [1999] restrained post-tensioned rocking block..................35
Figure 2.9 Makris and Zhang [1999] moment rotation response of a restrained post-
tensioned rigid block ..........................................................................................................36
Figure 2.10 Static lateral loading of a rocking wall with dissipators, Toranzo [2002]….. 37
Figure 2.11 Rocking wall with hysteretic energy dissipators.............................................38
Figure 2.12 Areas for the calculation of equivalent viscous damping due to dissipators...42
Figure 2.13 Correction of the hysteretic loop of dissipators ..............................................44
Figure 2.14 Structural limit states of a post-tensioned rocking system..............................45
Figure 2.15 The Monolithic Beam Analogy (MBA)..........................................................48
Figure 2.16 Three regions of the revised monolithic beam analogy (rMBA), Palermo
[2004]..................................................................................................................................54
Figure 2.17 Hybrid System concept (Flag shape equivalent representation).....................63
Figure 3.1 An overview of idealized lateral response for various concrete columns.........70
Figure 3.2 HYB-1 System overview ..................................................................................73
Figure 3.3 Lumped plasticity model consisting of two rotational springs in parallel........ 75
Figure 3.4 DRAIN-2DX model for a hybrid joint, Kim [2002].........................................76

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Figure 3.5 Multi-spring element within Ruaumoko, Carr [2005].......................................77
Figure 3.6 Two spring model of a precast, post-tensioned coupled wall, Conley et al.
[1999]..................................................................................................................................79
Figure 3.7 Fibre element model of a post-tensioned beam-column joint subassembly
(modified from El-Sheikh et al. [1998]).............................................................................79
Figure 3.8 Influence of the length of the rocking fibre segment on the moment-rotation
response...............................................................................................................................80
Figure 4.1 Column Details - Elevation and Cross section view .........................................85
Figure 4.2 Flowchart of moment-curvature analysis ..........................................................87
Figure 4.3 Column cross-section in Columna.....................................................................88
Figure 4.4 Stress-Strain relationship of the longitudinal reinforcing steel .........................89
Figure 4.5 Stress-Strain relationship of concrete................................................................90
Figure 4.6 Moment-Curvature relationship and strain limit states .....................................92
Figure 4.7 Trilinear idealization of the moment-curvature relationship.............................94
Figure 4.8 Ruaumoko structural model...............................................................................95
Figure 4.9 Diagrams of the moment along the column height............................................97
Figure 4.10 Schoettler-Restrepo hysteresis rule .................................................................99
Figure 4.11 Base shear-Top displacement from the push over analysis...........................102
Figure 4.12 Time history displacement at the top node (cyclic).......................................103
Figure 4.13 Moment-Curvature behaviour at the column base (cyclic) ...........................104
Figure 4.14 Base shear-Top displacement behaviour at the column base (cyclic)...........104
Figure 4.15 Input analysis ground motion - LGPC...........................................................105
Figure 4.16 Acceleration Response Spectra of input ground motion(5% damping ratio)106
Figure 4.17 Velocity Response Spectra of input ground motion (5% damping ratio)......106
Figure 4.18 Displacement Response Spectra of input ground motion (5% damping
ratio)..................................................................................................................................107
Figure 4.19 Moment-Curvature behaviour at the column base ........................................108
Figure 4.20 Base Shear-Top displacement behaviour at the column base........................109
Figure 4.21 Deformed configurations along the column height .......................................110
Figure 4.22 Bending moment diagram along the column height......................................110
Figure 4.23 Curvature diagram along the column height .................................................111
Figure 4.24 Displacement Time History at node 2 ...........................................................111
Figure 4.27 Displacement Time History at the top node ..................................................113

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Figure 4.28 Relative Velocity Time History at the top node............................................113
Figure 4.29 Relative Acceleration Time History at the top node.....................................114
Figure 5.1 Ruaumoko structural model.............................................................................117
Figure 5.2 Hybrid lumped plasticity modeling of bridge system.....................................118
Figure 5.3 Ramberg-Osgood rule (Ruaumoko Appendices) ............................................119
Figure 5.4 Multi-linear rule .............................................................................................120
Figure 5.5 Moment-Rotation relationship of dissipating spring.......................................121
Figure 5.6 Moment-Rotation relationship of self-centering spring..................................121
Figure 5.7 Moment-Curvature behaviour at the column base (cyclic).............................122
Figure 5.8 Base shear-Top displacement behaviour at the column base (cyclic).............123
Figure 5.9 Moment-Rotation behaviour at the column base ............................................125
Figure 5.10 Base Shear-Top Displacement behaviour at the column base ......................125
Figure 5.11 Displacement time history at the top node....................................................126
Figure 6.1 Base shear-Top displacement curves for hybrid and monolithic connection
(Pushover).........................................................................................................................127
Figure 6.2 Base shear-Top displacement curves for hybrid and monolithic connection
(Time History)...................................................................................................................128
Figure 6.3 Peak response parameters of non-linear time history analysis for monolithic
and hybrid......................................................................................................................... 130
Figure 6.4 Displacement time history at the top node for hybrid and monolithic
connection.........................................................................................................................130
Figure E.1 Calibration of the experimental and analytical results ...................................155

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LIST OF TABLES
Table 4.1 Measured yield and ultimate strengths of the column reinforcing steel.............84
Table 4.2 Longitudinal reinforcing steel properties ...........................................................88
Table 4.3 Concrete properties.............................................................................................90
Table 4.4 Significant points of the modified trilinear idealization.....................................93
Table 4.5 Schoettler-Restrepo hysteresis rule parameters................................................100
Table 4.6 Moment, Curvatures, Base shear and Displacement of the significant limits
state...................................................................................................................................102
Table 4.7 Peak response parameters of non-linear time history analysis (monolithic
connection)....................................................................................................................... 107
Table 5.1 Peak response parameters of the non-linear time history analysis (hybrid
connection)....................................................................................................................... 124
Table 6.1 Peak response parameters of the non-linear time history analysis...................129

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ACKNOWLEDGEMENTS
First and foremost I offer my sincerest gratitude to my supervisor, professor Marco Savoia,
who has supported me throughout my thesis with his patience and knowledge while also
allowing me the freedom to work in my own way. I attribute the level of my Masters degree
to his encouragement and effort. Without him this thesis would not have been completed or
written. One simply could not wish for a better or friendlier supervisor.
I would like also to thank, professor Francesco Ubertini, professor Angelo Di Tommaso,
doctor Nicola Buratti, and all the other professors and doctors in the University of Bologna.
Their time, comments and encouragement during the last year in my master process has
allowed me to gain great knowledge and understanding.
I would like also to thank, my best friend Francesco Carrea who had been more than a
friend. I want to thank him for his support in everything regarding references, materials and
connections with other professors and doctors.
I acknowledge the University of Bologna, the oldest university in the world, for providing me
with such a great opportunity for a new study and research experience.
It has been a smooth journey leading to the completion of my thesis at the University of
Bologna, and this could not have been possible without the help and encouragement of a
number of people in my life. For their support, I owe great gratitude and appreciation, as I
would not have been able to make it where I am today without their support and continuous
motivation.
Last, but not least, I thank my family for unconditional support and encouragement to pursue
my interests, for sharing their experience, for listening to my complaints and frustrations, and
for believing in me. I must acknowledge the unwavering support of my family. Mom, Dad,
Mariam and Youssef: I will ever never forget your words, “Nothing is impossible if you have
the will.”
Finally, I would like to remember my father Youssef, who is watching me from up there, I say
to you dad, “I finished the first and the second step in our deal and the third is on the way; my
supreme leader.”

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ABSTRACT
“Analysis should be as simple as possible, but no simpler”
Albert Einstein (1879-1955)
The world has seen a dramatic increase in its population in the previous decades. And
consequently many civil structures with frequent design action, including bridges, no longer
comply with the requirements they have been designed for, because of the increasing in the
current load. This raises the problem of rehabilitation or replacement of these structures. A
way for fulfilling the need to replace these structures has been identified in the use of
prefabricated systems and elements. The hybrid jointed ductile connections originally
developed for either precast concrete frames and wall systems have been shown to exhibit
superior performance complemented with a reduced level of damage and negligible residual
deformations of the structural systems. These innovative advanced systems, consisting of
relatively simple construction methods based on post-tensioning techniques, have been
recently proposed to be adopted in bridge piers and systems as a viable and highly
competitive alternative to traditional monolithic cast-in-place construction.
From the performance based design, we can realize that it is not economical or even practical,
to design structures to remain elastic after a major earthquake. Therefore, traditional seismic
design methodologies require structures to respond inelastically by detailing members to
accommodate significant plasticity. It can be appreciated that, while life-safety of the
occupants is ensured, structures conforming to this traditional design philosophy will be
subjected to excessive physical damage following an earthquake. Thus, excessive economic
loss and large social impact, due to extensive damage and operational problems in structures
are expected to be great.
Alternative solutions for precast concrete buildings based on ductile joint connections have
introduced an innovative concept in the seismic design of frame and shear wall systems. In
this contribution, the feasibility and efficiency of the application to bridge piers and systems
of hybrid solutions, where self-centering and energy dissipating properties are adequately
combined to achieve the target maximum displacement with negligible residual deformations.

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This research includes the theory behind the rocking system and its efficiency in structural
protection. In addition, two modeling of bridge piers with different connections, the first is
the traditional monolithic connection and the second is the hybrid rocking one. Where a
numerical comparison of hybrid bridge pier systems in a cantilever configuration - pier to
foundation connection- and the traditional monolithic connection is carried out through static
pushover analysis and a non-linear time history analyses considering lumped plasticity
model. Full modelings of both columns were done and the results are presented.

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CHAPTER 1 INTRODUCTION TO ROCKING SYSTEMS IN
CONCRETE STRUCTURES
1.1 Foreword
Rocking systems have self centering properties -given by post tensioning unbonded tendons-
and accommodate the seismic lateral displacement demand with a base rotation which leads
to only one concentrated opening at the foundation to column joint compared to the crack
spreading and damage typical of the plastic region of classical reinforced concrete columns.
1.2 Introduction to Rocking Structures
The rocking motion of structures first dates back as analytical study to Housner (1963) where
he tried to explain why during the Valdivia earthquake of Chile in 22/05/1960, several golf-
ball-on-a-tee types of elevated water tanks survived the shaking despite the appearance of
instability, while much more stable-appearing reinforced-concrete elevated water tanks were
severely damaged. This apparently strange behavior has been explained by studying the
dynamics of a rigid block resting upon a rigid horizontal base and excited into rocking
motion. The dynamic characteristics of these types of structures are sensibly different from
non-linear elastic structures. Housner showed the low amount of shear force and bending
moment generated during an earthquake designing slender structures to act as rigid blocks;
however it was not clear how to reduce the probability of overturning through the design
procedure. The advantage of rocking in a structure is the self-centering upon unloading and
lack of residual drifts after an earthquake.
Rocking solutions are usually applied to precast concrete systems although recent efforts
have been made to integrate the rocking self-centering concept to steel moment resisting
frames (Garlock et al. 2007).
To increase the self centering capacity, rocking systems have been coupled with unbonded
post tensioned tendons which provide restoring force with considerably less concrete tension
cracking in the system, if compared to monolithic solutions, as the concrete is not bonded to
the tendons and does not go into tension. These systems perform well under the self centering
point of view but low energy dissipation is associated to the rocking process, mainly related
to radiation damping and concrete crushing in the impact region. The low energy dissipation
could lead to greater system displacements and to a higher number of high displacement
peaks if compared to monolithic solutions (Kurama 2002). To increase the energy dissipation

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without affecting the self centering capacity, the systems can be coupled with frictional
dampers (Priestley et al. 1999), viscous dampers (Kurama 2000, Marriott et al. 2008) and
mild steel dissipation devices (Kurama 2002, Holden et al. 2003, Restrepo & Rahman. 2007,
Marriott et al. 2008). When such additional energy dissipation devices are placed, the system
is then called hybrid.
1.3 Experimental Tests of the Behaviour of Jointed, Precast, Post-Tensioned Rocking
Structures
1.3.1 The U.S. and Japan PRESSS Research Program
Cooperation between the joint U.S.-Japan research program in the early 1990’s under the title
of the PREcast-Seismic-Structural-Systems (PRESSS) program was a major force in the
development of jointed ductile precast connections, Priestley [1991]. The intent of this major
research program was to improve the inelastic response, analytical modeling, design
recommendations, and to improve the understanding of “ductile” precast buildings. The
US.PRESSS program was divided into three phases (Priestley [1991]); Phase I was
concerned with the conceptual development and evaluation of newly proposed structural
concepts, specifically concerning practicality, economy and seismic performance (ductility
and dissipation capabilities). This was followed by Phase II involving detailed experimental
studies of precast components and sub-assemblages, paralleled with analytical studies. Phase
III involved the testing of a 60% scale, multi-storey precast building having both structural
moment resisting frames and shear wall elements.
Phase II and III of the research program deals entirely with experimental confirmation and
analytical modeling. As part of the inter-program co-ordination of the US.PRESSS program,
the National Institute of Standards and Technology (NIST), carried out numerous
experimental tests related to Phase II. One of the earlier tests carried out by the PRESSS
program is discussed in Priestley and Tao [1993]. They present the experimental results of a
pre-stressed, pre-cast beam-column-joint sub-assembly with fully grouted post-tensioned
tendons carried out at NIST. While comparable ductility demands to monolithic reinforced
concrete elements could be achieved, the response was subjected to extensive stiffness
degradation, pinching and hence unreliable energy dissipation. Furthermore, as the tendons
are likely to exceed the limit of proportionality (yielding in tension), the shear transfer
mechanism at the beam interface may be lost, resulting in a loss in the gravity load carrying
capacity. Priestley and Tao [1993] then proposed the idea of partially unbonded post-
tensioning tendons, whereby the tendon would be debonded for some length either side of the

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beam-column joint. This would reduce the strains in the tendon, and provide a non-linear
elastic response with a marked increase in stability. The system however would have
relatively little energy dissipation and be the subject of large concrete compressive strains. It
is for this reason special detailing at the beam end region, consisting of spiral reinforcement,
was suggested. Based on a number of non-linear time-history analyses, and the force-
displacement relationships adopted, they stated that the difference in displacement response
between a non-linear elastic pre-stressed concrete frame with unbonded tendons, and an
equivalent reinforced concrete frame, may be less than 38%.
The natural progression within phase II of the PRESSS program was to provide a similar
system with fully debonded (unbonded) tendons. Thus, Priestley and MacRae [1996]
constructed and tested a 67% scale pre-cast, post-tensioned interior and exterior beam-
column joint subassembly with unbonded tendons. While the experimental results indicated a
stable force-displacement response up to 4% of inter-storey drift, the units were subjected to
significant stiffness degradation (initial stiffness reduced by approx 65% at a design inter-
storey drift ratio of 2.5%). It is likely this degradation resulted from crushing of the beam
cover concrete and inelastic compression stresses in the concrete at the beam end region, in
addition to cracking within the column, joint and beam elements. The beam end regions were
detailed with special spiral confinement reinforcing (2.5% by volume), while joint transverse
reinforcement was keep to a minimum as it was envisaged that joint shear would be resisted
entirely by a single diagonal strut from corner to corner of the joint due to the pre-stressed
nature of the system. Diagonal shear cracking developed within the joint, but stabilized as the
lateral load reached a maximum - this was in addition to yielding of the transverse
reinforcement. It is for these reasons that the authors suggest to adopt a more conservative
design approach for the design of the joint transverse reinforcement.
The inter-program co-ordination of the PRESSS program allowed a number of experimental
tests to run in parallel. Cheok and Stone [1994] tested a total of twenty 33% scale, precast
beam-column joint subassemblies consisting of both post-tensioned tendons (bonded,
partially debonded and fully unbonded) and mild steel reinforcement (bonded, partially
debonded or fully unbonded). This work was also reported in similar publications by Stone et
al. [1995] and Stanton et al. [1997]. Of the 20 tests conducted, 4 specimens were accepted for
a second round of proof testing and are discussed in detail. The four specimens comprised of
two mild steel reinforcement ratios and two different material types: grade 60 (fy = 414MPa)
and a ductile grade of stainless steel (fy = 304MPa). All four specimens had partially grouted
post-tensioned tendons, grouted over a length equal to 37.7% of the bay length. Furthermore,

4
the mild steel was either fully bonded or de-bonded over a length equal to 50mm at the
connection interface. The mild steel units were found to perform well, with first rupture of
the mild steel occurring at 2.9% for one unit and 3.5% for the second unit, with the tendons
remaining elastic in both tests. The stainless-steel units did not perform well as the strain
capacity of the stainless steel was reduced because deformed ribs were machined along the
bar to help improve bond performance. In one test, rupture occurred at a lateral drift ratio of
2.0% while in the second test bond failure occurred at lateral drift ratio of 2.0% due to a
relatively optimistic anchorage length. In all cases the tendons remained elastic except for the
test unit having bond failure which was tested to 6.0% of lateral drift. Losses within the
tendon load were recorded in all four specimens as a result of strain penetration within the
grouted portion of the tendon (in addition to yielding of the tendons for one specimen),
reducing the average strain over the unbonded length. However, given the reduction in
tendon load, it was concluded that the gravity load carrying capacity (due to friction) could
still be maintained.
El-Sheikh et al. [1999] presented two analytical models to model the experimental lateral
response of a single NIST beam-column joint test, one being a fibre element model, the
second being a lumped plasticity spring model. In general, both models were able to
accurately capture the experimental response – including the initial stiffness and lateral
strength. El-Sheikh et al. [1999] divided the push-over response of a post-tensioned frame
into three (displacement) limit states. These three limit states were defined via a tri-linear
representation of the moment-rotation behaviour of the beam-column connections; 1) the
linear limit state, defining a reduction in lateral stiffness from the initial elastic stiffness of the
system; 2) the yield limit, corresponding to yielding of the unbonded post-tensioned tendons;
3) the ultimate limit state, being failure of the confined concrete due to rupture of the spiral
reinforcement (or rupture of the post-tensioned reinforcement). A design procedure is
outlined by associating the three displacement limit states with three corresponding seismic
design intensities (elastic, design and survival). The proposed design adopts an equivalent
lateral force procedure (as per BSSC [1997]) and is based on the equal displacement principle
for ductile structures in order to respect the three structural displacement limit states. The
elastic design spectrum is reduced by the response modification factor, R= 8 for “special
moment resisting frames with ductile connections”. However, the non-linear time-history
analyses revealed that the equal displacement principle violated the design requirements for
frame structures located on medium or soft soil conditions in regions of high seismicity.

5
In addition to pre-cast beam-column joints, the PRESSS program also investigated the
behaviour of pre-cast, post-tensioned walls within Phase II. Much of this work was carried
out at Lehigh University, Pennsylvania. The experimental results of Mueller [1986],
presented by Armouti [1993], were used to validate a fibre model used within the DRAIN-
2DX program reported by Kurama et al. [1998]. Mueller [1986] constructed and tested five
precast, concrete walls of 1/3 scale; one wall being representative of an unbonded, post-
tensioned precast concrete wall appropriate for an analytical comparison with a fibre model.
The precast wall comprised of 3 segments, where only the pre-stressing bars crossed each
precast wall panel. That is, the only form of inelastic deformation would come from material
nonlinearity of the concrete and post-tensioning steel in addition to friction within the PVC
grouted ducts (assumed to be unbonded). That is, the longitudinal reinforcement was
curtailed within each precast element and did not pass between the precast wall panels. The
precast wall was constructed with spiral reinforcement around the toe regions to confine the
concrete under excessive axial strains. It was for this reason that the concrete was modeled as
a multi-linear inelastic spring using the confinement model of Mander et al. [1988]. The
experimental results exhibited a significant amount of stiffness degradation and material non-
linearity (with an appreciable amount of energy dissipation). The fibre model accurately
captured the cyclic behaviour, albeit for under predicting the loading branch: this was
attributed to errors in accurately recording the experimental lateral load. Kurama et al. [1998]
concluded that while the experimental test verification was not based on unbonded tendons
(the tendons were grouted within smooth electrical conduits), the fibre model required further
verification with more experimental tests to fully confirm its accuracy.
Kurama et al. [1999] proposed a seismic design approach for precast concrete walls with
unbonded post-tensioned tendons. A performance-based design approach was developed,
incorporating structural limit states and a seismic hazard based on the equivalent lateral force
method in BSSC [1997]. In particular Kurama et al. [1999] provided a number of structural
limit states that should be satisfied considering two seismic design intensities. For a design
level event (corresponding to an immediate occupancy performance level) the yield
displacement (and lateral capacity) of the wall should not be exceeded: this corresponds to
yielding of the post-tensioned tendons. Furthermore, inter-storey displacements should be
limited to prevent damage to the non structural elements. For the survival limit state
(corresponding to the collapse prevention performance level) yielding of the tendons can be
accepted, however, the self-centering capability of the system should be preserved to some
extent. The displacement should not exceed the maximum displacement capacity of the wall

6
(corresponding to fracture of the special confinement reinforcement) and crushing of the
precast panel should be avoided (in those regions of concrete not having special confinement
reinforcement). Furthermore, at the survival limit state, shear slip of the precast wall panel
units is to be avoided and the lateral displacements of the gravity- load carrying elements
should be controlled to ensure their vertical load carrying capacity is not compromised.
Using a fibre-element model, Kurama et al. [1999] investigated the lateral load behaviour of
precast walls with unbonded post-tensioned tendons by varying a number of structural
parameters. The initial post-tensioned force, amount of prestressed reinforcement, wall
length, location of the post-tensioning tendons, unbonded length and the confinement
reinforcement ratio were investigated. Kurama et al. [1999] commented on the effect these
parameters have on three design limit states corresponding to: 1) onset of softening; 2)
yielding of the prestressed reinforcement; and 3) rupture of the confinement reinforcement.
Kurama et al. [1999] concludes by stating that the equal displacement assumption provides
some correlation to time history analysis for walls located on stiff soil sites only and noted
that further research is required to improve this estimation. The base shear was found to be
significantly influenced by higher modes for post-tensioned walls due to the elongation of the
modal periods when the structure entered the non-linear range.
Further analytical studies were also carried out by Kurama [2000] investigating the
application of unbonded post-tensioned walls utilizing linear viscous dampers connected to
adjacent column braces. A fibre model was verified against a finite element model using
nonlinear rectangular plane-stress elements in addition to gap/contact elements within the
finite element program ABAQUS. Kurama [2000] proposed a design procedure using an
Acceleration Displacement Response Spectrum (ADRS), combined with non-linear time-
history analyses subjected to increasing white noise intensity.
As part of Phase III of the PRESSS program, a 60% scale, five-storey precast test building
was designed and tested at the University of California, San Diego (UCSD). The test building
comprised of precast moment resisting frames in one direction and post-tensioned coupled
walls in the orthogonal direction, as reported in Priestley et al. [1999] and pictured in Figure
1.1. In particular, the coupled walls provided strength and dissipation through the use of U-
Shaped Flexure Plates (UFP) fixed between the two walls and activated via the relative
vertical displacement incompatibility between the two wall elements. This form of dissipation
was found to be extremely stable and efficient following component testing carried out under
the NIST program and reported in Priestley [1996]. The building was tested under a pseudo-
dynamic loading protocol, simulating the response to real earthquake excitation. Furthermore,

7
as the earthquake intensity was increased, push-over tests were carried out after each major
event to determine the structure flexibility and energy dissipation characteristics. Three
design intensity earthquake records were run through the structure with a fourth record equal
to 150% of the design intensity to represent the response under a maximum credible event.
The structure experienced only minor damage to the cover concrete at the toe of the coupled
wall, along with some cracking and deformation of the adjoining precast floor units at each
floor level. In general, a simple macro model (using lumped plasticity modeling techniques)
was able to accurately model the time history response; however, higher modes associated
with contact stiffness at the base- foundation rocking interface were sensitive to the spring
stiffness adopted in the model, Conley et al. [1999]. Furthermore, it was found that in order to
return a proper agreement between the experimental results and the analytical model, the
inclusion of an external column element was required to represent the out-of-plane stiffness
of the orthogonal frames. It was concluded that the use of UFP plates provided an outstanding
level of structural performance, while the performance of the structure further verified the
displacement-based design methodology for precast buildings.

8
(a) Post-tensioned hybrid frame (b) Post-tensioned coupled wall
Figure 1.1 Five storeys precast post-tensioned frame building tested at the University of
California, San Diego (Priestley et al. [1999])
1.3.2 Monotonic and Cyclic Quasi Static Tests
Also, monotonic and cyclic quasi static tests have been carried out to evaluate the base shear
– top displacement relationship applying a lateral load, by means of an actuator, at the point
corresponding to the resultant of a lateral force distribution according to the flexural first
mode of vibration. Among these tests Holden et al. (2003) added milled mild steel re-bars
across the base joint to provide a satisfactory level of hysteretic damping. The authors
detailed the wall reinforcement layout according to the internal force flow obtained from a
strut and tie model analysis and adopted fiber reinforced concrete for the test unit. The same
type of energy dissipation bars have been used to test hybrid walls by Restrepo & Rahman
(2007) with a wall reinforcement layout made by low amount of steel ratio in the horizontal (r
= 0.0025) and vertical (r = 0.0084) directions and stirrups confined region at the wall toes.
The results outlined the main states of behavior under lateral loads and the “flag shape”
hysteresis loop typical of these hybrid systems (Figure 1.3).

9
Figure 1.2 Rocking motion of Hybrid connection (Dissipater is anonymous in this figure)
Figure 1.3 The states of rocking wall behaviour
According to Figure 1.3 the main behavior states can be summarized in:
1. Decompression, which identifies the initiation of gap opening at the wall base to
foundation joint.
2. Softening or geometric non-linearity, which is associated to the beginning of significant
reduction of the wall lateral stiffness due to gap opening or nonlinear behavior of the toe
concrete in compression (depending on the initial level of vertical load due to gravity and
post tensioning). Marriott et al. (2008) identify this state with the neutral axis at mid depth of
the wall section.
3. Yielding of mild steel reinforcement, with consequent further decrease of the wall lateral
stiffness.
4. Yielding of post tensioning reinforcement. The post tensioning steel reaches the limit of
proportionality, the wall self centering property can be reduced.

10
5. Rupture of mild steel reinforcement, which can be avoided by controlling the steel strain
by means of an unbonded length.
6. Failure state, associated to the confined concrete crushing or to the post tensioning steel
rupture.
The principal parameters controlling these states are the position and amount of the energy
dissipation bars and the post tensioning tendons, the initial amount of vertical load, the
section geometry and the initial strain on the post tensioning tendons. The main drawbacks of
these quasi-static tests are the inability to capture the acceleration spikes in the vertical and
horizontal directions due to impact and the inability to capture the dynamic associated to the
system, in fact when rocking is triggered the system stiffness decreases and therefore the
mode of vibration and the lateral load distribution change.
To better understand the dynamic associated to rocking systems Toranzo (2002) applied the
self centering rocking wall idea to “confined” masonry constructions. A three story 40% scale
“confined” (by means of reinforced concrete beams and columns) masonry wall and slab
subsystem was tested on a shake table. The tests involved also the use of steel hysteretic
energy dissipation bars between the wall toes and the foundation element. The test results
showed vertical and horizontal acceleration spikes due to wall impact during rocking, the
latter being larger in the upper levels. These spikes lead to peak absolute accelerations and
peak inter-story shear forces higher than expected, in some cases more than doubled,
although the effects on the base shear demand was not so pronounced and therefore in
agreement with the analysis estimation.
The tests showed that the amount of the horizontal acceleration spikes are reduced with the
application of the energy dissipation devices while the vertical ones are not. The vertical
acceleration spikes could temporary reduce the shear friction capacity of the wall base and
lead to a horizontal slip of the wall.
The preliminary results published show the beneficial effects of additional dissipation devices
to the post tensioned wall in damping the response after the main peaks although the
maximum displacement associated to some ground motions could be larger in the case of
additional dissipation devices than without.

11
1.4 Post-Tensioned, Precast Wall Systems
A significant amount of experimental and analytical work on precast concrete systems with
unbonded post-tensioning has been investigated outside of the PRESSS program. In
particular, Rahman and Restrepo [2000] tested three half scale unbonded post-tensioned
precast concrete wall units at the University of Canterbury, Figure 1.4. The post-tensioned
walls were tested with and without grouted mild steel reinforcement. Unit-1 was detailed with
two unbonded post-tensioned tendons each stressed to approximately 95kN. The
experimental response was very stable, with some stiffness degradation as a result of crushing
of the cover concrete and tendon losses. The toe regions were adequately detailed for the
expected high concrete compression strains. This limited damage to crushing of the cover
concrete. Unit-2 was detailed with two unbonded post-tensioned tendons each stressed to
approximately 95kN in addition to two grouted mild steel reinforcing bars. The mild steel bar
had a machined diameter of 12mm over a length of 200mm to confine the inelastic strain to
the machined region. The experimental response was very stable with some stiffness
degradation due to damage to the cover concrete and losses within the post-tensioning
tendons. Furthermore, rupture of one dissipater occurred at 3% of drift. Unit 3 was identical
to unit 2 except for a more heavily detailed toe region, anticipating higher concrete strains
resulting from a) 16mm diameter mild steel dissipaters, b) 200kN additional post-tensioning
representing gravity loading. The experimental response was stable with almost no strength
deterioration but significant stiffness degradation. Furthermore, residual deformations were
almost completely non-existent. Again, damage was limited to loss of cover concrete and
minor flexural cracking.

12
Figure 1.4 Post-tensioned rocking wall tested at University of Canterbury, Rahman and
Restrepo [2000]
Similar work by Holden [2001], also carried out at the University of Canterbury, investigated
the cyclic response of two precast concrete wall specimens. One wall unit combined
unbonded post-tensioned tendons (using carbon fibre tendons) and grouted mild steel
reinforcement, with details being similar to the walls tested by Rahman and Restrepo [2000].
This hybrid wall was detailed with steel base plates at the toe of the wall welded to a diagonal
strut made from steel bars which met at the middle of the wall, Figure 1.5. The steel diagonal
strut was designed to resist the large compression forces as the wall rocked from toe to toe.
The second specimen was an emulation of a cast-in-place monolithic wall: this was used as a
benchmark to compare the response of the hybrid wall. The experimental response of the
hybrid specimen was found to have very little energy dissipation, significant stiffness
degradation and significant pinching. Holden [2001] concluded that the poor behaviour was

13
associated with a combined bearing/push-out failure of the internal steel dissipaters beneath
the foundation block. This prevented the mild steel from yielding in compression, limiting the
equivalent viscous damping to approximately 3.5%-8% for the 3rd and 1st
cycle respectively.
The hybrid unit sustained virtually no cosmetic damage, and residual deformations were
negligible.
Wall configuration with steel diagonal struts to resist compression
Figure 1.5 Detailing of a post-tensioned wall, providing load paths for the high compression
forces during rocking, Holden [2001]

14
1.5 Conclusion
We can summarize that, while there is a general acceptance regarding the desirable behaviour
of precast systems with unbonded post-tensioned systems, the degree of damage sustained to
the structural elements and the efficiency (and stability) of the energy dissipation lies within
the detailing and design of the critical rocking regions. Detailing of the rocking toe region,
anchorage of the dissipation (either internally or externally) and tendon details appear to vary
from test to test. A standardized method is required to maximize the efficiency of precast
systems with unbonded post-tensioned tendons.
To a greater extent, while significant analytical and experimental work has been carried out
on precast systems with unbonded post-tensioned tendons (with or without energy
dissipation), little work to confirm the dynamic response has been carried out. A significant
amount of analytical work has focused on relatively complex fibre-element models in
addition to simple macro-models. While the complex modeling techniques are very accurate,
they require a degree of competency to be used correctly. Simple macro models appear
extremely attractive as they can achieve comparable accuracy at a fraction of the
computational cost.
Moreover, while supplementary viscous dampers have been investigated within precast
systems, little work has been devoted to experimental testing and to the development of
adequate and simple design procedures. The combination of both hysteretic and viscous
dampers has great potential for the seismic protection of structures located in either near-field
or far field seismic regions. Furthermore, precast systems with unbonded post-tensioning
appear to be an attractive solution in which to implement this hysteretic-viscous combination.

15
CHAPTER 2 LITERATURE REVIEW OF ROCKING THEORY
2.1 Rocking Wall Base Sliding
In the case of rocking walls the base shear capacity is an important issue, because if it relies
on the shear friction at the wall to foundation joint, this could be not sufficient, especially
when the joint is closed. In fact when the gap opens the elongation of the post tensioning steel
will increase the vertical load and therefore the shear friction capacity. Yielding in the post
tensioning steel will reduce the prestress and therefore the shear friction capacity. The instant
when the gap closes could be critical under a base shear demand-capacity point of view: the
demand could be significant due to impact horizontal acceleration spikes while the capacity is
at minimum due to the low level of vertical load (the post tensioning force is at minimum, the
vertical force in the energy dissipation bars is acting upward and the vertical acceleration
spikes due to impact are acting upward). A base shear demand greater than the capacity leads
to a slip of the base joint with detrimental effects on the post tensioning tendons and on the
energy dissipation bars which may kink and prematurely failing in the following cycles.
Other than that, no recentering capacity is associated to horizontal slip which should therefore
be avoided. Knowing the shear friction capacity and demand allows, if necessary, to design
shear resistance passive methods like steel dowels or mechanical keys connections, although
from a constructability point of view it is better to erect the wall directly on top of the
foundation avoiding additional work related to shear key install. To evaluate if shear friction
capacity is sufficiently large to avoid sliding, Restrepo & Rahman (2007) proposed a formula
based on the effective height to wall depth ratio of the wall. The formula contains a parameter
to take into account the “sensitivity of rocking systems to feed high frequency energy caused
by impact of the wall toes” although no value of the parameter is indicated. To estimate the
maximum base-shear demand Kurama et al. (2002) adopted a formula based on the sum of
the first mode component base shear and a higher mode component which is a function of the
first and second mode effective height and mass and of the peak ground acceleration.
Although the results of time history analyses showed that the formula adopted provides a
good upper bound base shear estimate for the case study, the equation does not capture the
horizontal acceleration spikes, and so inertia forces, associated to gap closing as rocking
occurs: based on that formula, considering a rocking wall with one single rigid floor (i.e. a
single degree of freedom system), only the first mode base shear component exists with no

16
account of the base shear demand increase due to the horizontal acceleration peaks as gap
closes.
These horizontal acceleration peaks, are more likely associated to the change in lateral system
stiffness when gap closes.
To justify this statement a post tensioned hybrid wall, whose hysteretic energy dissipation
devices are yielding in compression when gap is closing, is considered. The stiffness (kb) of
the base moment - rotation (Figure 2.1) when the gap is approaching closure is given by the
contribution of the post tensioning tendons (kPT), whose behavior is considered linear elastic,
and the hysteretic energy dissipation bars (kd) which, if the gap opening is big enough, are
yielding in compression to allow gap closure.
Figure 2.1 The relationship between Base Moment and Rotation
The post tensioning tendons force associated to base rotation is:
Where:
Fp0 is the initial prestress
Lw is the wall depth
lunb_PT is the tendons unbonded length
CNA is the neutral axis
APT is the tendons area
EPT is the tendons steel elastic modulus

17
The tendons moment contribution is:
And the tendons stiffness is therefore:
The dissipation bar force associated to base rotation when the dissipation bars are yielding in
compression when the gap is closing is:
Where:
Fy is the yield force
l unb_d is the dissipation bars unbonded length
cy is the neutral axis as dissipation bars are yielding in compression
Ad is the dissipation bars area
Ed is the dissipation bars steel elastic modulus
k is the post yield dissipation bars stiffness
The dissipation bars moment contribution is:
And the dissipation bars moment contribution is:

18

The base moment rotation stiffness is obtained considering that the tendons and the energy
dissipation bars act in parallel:
The displacement stiffness associated to the contribution of a floor at a level Hi is now
considered. This stiffness is obtained considering two systems in series: the first one
(stiffness kb_∆) is the contribution of the base moment rotation relationship considering the
wall acting as a rigid body while the second one (stiffness kel) is the wall flexural stiffness
(including shear stiffness will not change the findings) considering the wall base as fixed.
Before gap closes, the first contribution is obtained from the following relations:
and therefore:
The second contribution is simply:
Therefore the lateral displacement stiffness is:
Once the gap is closed the lateral displacement stiffness is:
The stiffness increase once the gap is closed is

19
This stiffness increase when gap closes is associated to an horizontal “impact” for the system
and therefore explains the horizontal acceleration spikes when gap is closing but does not
explain why these spikes are bigger in the upper floors. This behavior is explained by the
horizontal velocity of the floor before impact. The floor velocity is associated to the
momentum (define as velocity times mass) which affects the system impulsive response.
When gap is closing (with a velocity Ɵ’), the upper floors are subjected to a tangential
velocity higher than the one associated to lower floors. This sensibly affects the impulsive
response where the spikes at the upper floors are bigger.
2.2 Mechanics of a Rocking Wall
The simplest rocking scheme is the one that assumes that the rocking wall is rigid. A rocking
rigid wall may be defined as an oscillating system, though it is different from the common
harmonic-type oscillating system, as it presents a rather bilinear stiffness. Housner (1963)
was among the first interested in the rocking system as a structural type and, certainly, the
first in publishing some of the following kinematic equations.
2.2.1 Equations of Motion
The free vibrations of a rocking wall acting as a rigid block have been studied by Housner
(1963). The rigid block is considered oscillating about the centers of rotation O and O’
(Figure 2.2)
Figure 2.2 - Rocking wall free body diagram (Housner’s block)

20
When the block is rotated from the vertical by an angle the self weight will exert a
restoring moment. Assuming that there is no horizontal sliding, the equation of motion can be
written as (Housner 1963):
Where:
IO is the polar moment of inertia or mass moment of inertia about point O
R is the distance between the center of rotation and the wall center of gravity
α is the angle between R and the vertical axis
W is the wall self weight
To simplify the equation to an ordinary differential equation, Housner proposed to substitute
the sine with its argument. A more rigorous approximation related to a small rocking
amplitude Ɵ is proposed here considering:
This leads to:
Where is the angular acceleration of the block in rad/s2
The solution of the previous equation is obtained considering the wall released from an initial
rotation and with zero initial velocity:
Where cosh (x) is the hyperbolic cosine and it is defined as
And from there:

21
Note that the above solution is evaluated for the case when the rocking wall is coming down
from = o. The preceding stage, when the wall is rotating upwards, can be derived with
negative values of time. Therefore, the total time range covered by this equation is
–T/4 ≤ t ≤ T/4. It is important to observe that the rotational acceleration is always negative.
A full cycle of rocking consists of a wall rotation around O from until reaching the
vertical position, a following rotation around O’ until reaching (neglecting impact energy
losses) and then back again to the vertical position and rotating around O. The time T to
complete a cycle is the period of free vibration and it is four times the time required to go
from to zero:
Once the wall geometry is defined, the period of rocking depends therefore on the initial gap
opening.
The response of a rigid block under free vibration is highly non-linear; the period of the
system is highly dependant on the rotation amplitude . The relationship describing the
period is shown in Figure 2.3 (a), where represents the release amplitude. As the release
amplitude approaches α, the period extends to infinity before over-turning. The period is
equal to zero when the release amplitude is zero (assuming a rigid block). By equating
moments about the rocking toe O, the lateral response, in terms of overturning moment
versus base rotation is easily computed and shown in Figure 2.3 (b). Intuitively, the negative
bilinear stiffness of the system implies a highly non-linear, unstable system.

22
Figure 2.3 Mechanical properties of a rigid rocking block, (a) Rocking period of vibration
with amplitude and (b) Moment rotation response of rocking block
2.2.2 Forces in a Rocking System
It is necessary to identify the forces in the system for design purposes. The first part deals
with the definition of the forces before impact, while the second part deals with the definition
of the forces at impact.
2.2.2.1 Forces in the System before Impact
Fig. 2.4 allows for the definition of the forces Fh and Fv in terms of the rotation . They
would be:

23
Figure 2.4 Forces and actions in a rocking rigid wall
In the definition of Fv above, the inertial forces due to the radial and tangential acceleration of
the mass are always going to be negative (see equation 2.17-b) and therefore the static
solution is going to be an upper limit for the dynamic solution. Since the equations
above relate the forces to the time through a hyperbolic function, the maximum and minimum
values for these expressions will be found in the extremes when t = 0 and t =  T/4. It can be
demonstrated that with the contribution of the inertial forces, Fv will be between the extremes
presented below:

24
The ratio MR2
/Io can at its largest value be equal to 1 (single lumped mass), the most
common values being equal or less than 0.75 (MR2
/Io = 0.75 for a rectangular wall).
Combining this ratio with practical values of α and o, one can see that the inertial forces
might account for forces in the order of 0.05W, which will be always opposing the load W.
Therefore, it is proposed to use, for practical purposes, the static solution of Fv:
The definition of the horizontal force Fh can be dealt with in a similar way. The maximum
and minimum values of Fh will be found when t = 0 and t = ± T/4:
As it can be observed, the difference between these two extreme values depends on the ratio
o / α. The difference can be significant large for values of o/α above 0.5. The upper limit of
both extreme values though is the same, and can be used for design purposes:
This is not only an upper limit for Fv but also will differ from the actual analytical solution by
less than 10% for practical values of α and o. One can then rewrite this expression in the
following way:

25
The reason for this change is that no one can define αeff in terms of the geometry of the wall
and find that it is possible to use an effective radius, Reff, with a definition very close to that
commonly used to represent a multi-storey building by a SDOF oscillator.
Where:
In summary, the reacting forces Fv and Fh can be represented by an equivalent static solution
where the static lateral load is applied at a height, Reff, as defined above:
There is also the necessity of defining a simple relationship between the two forces, Fh and
Fv. The fact that the equivalent static solution is an upper bound to the dynamic problem,
does not guarantee that one can use equations 2.23 and 2.33 to relate Fh to Fv. However, a
different approach leads to a solution equivalent to the use of the static one. Taking moments
about the centre of gravity of the rocking wall, c.g., then:

26
Again, the maximum and minimum values of these expressions will be when t = 0 and t = ±
T/4:
In the equations above, for small values of α and knowing that MR2
is of the same order of Io
and Icg, and that o/α is always in the range from 0 to 1, one can say simplify the equations
above into:
or:
From these results, the relationship between Fh and Fv, derived from the equivalent static
solution, is only a good approximation of the actual forces when the ratio o/α is small or
when the rotation is close to = 0. For these cases, therefore, one can say that:
This relationship is needed to define in a practical manner the base shear developed at impact
in the rocking wall. This is addressed in the next section.

27
2.2.2.2 Forces in the System at Impact
The forces developed at impact are expected to be the largest forces during the rocking
process. For design purposes it is important, therefore, to define a close equation for the
expected impact load at the base of the wall. The impact load will be defined using an impact
amplification factor fimp, applied to the approximate equivalent static solution for Fh and Fv
defined in equations 2.23 and 2.33. The impact problem in deformable bodies is rather
complex as it involves the analysis of travelling shock waves through the deformable body.
However, for the purposes of this analysis, simplified energy considerations will be used. All
the flexibility of the system will be constrained to the contact elements, which in the case of a
rigid wall will represent the stiffness of the foundation. In the case of a non-perfectly rigid
wall, it will be defined as two springs in series, combining the stiffness of the foundation and
the stiffness of the wall. Figure 2.5 defines the stages to be used for the evaluation of the
process. Figure 2.5 also shows the horizontal and vertical contact springs at the base of the
rocking wall, with stiffness kx and ky respectively.
Figure 2.5 The impact process
The initial conditions of the system are defined by the uplifting of one end of the wall up to a
height ui. Then, the wall is released. Three stages will be defined, on which the energy is to
be compared. The initial conditions of the system, where the wall presents it maximum uplift,
define stage 1. Stage 2 is defined immediately before impact, and stage 3 is defined at the
maximum deformation of the contact elements.

28
Only potential and strain energy is present in stage 1. The potential energy is defined by the
initial uplifting of one end and the depression due to the static deformation of the vertical
springs, δst = W/ky (see Fig 2.5). The strain energy is given by the deformation of the springs
due to the initial value of the forces Fh and Fv. The static approximate solution (equations
2.23 and 2.33) is used to define the forces Fh and Fv.
In stage 2, most of the potential energy has been transformed in kinetic energy. The uplifting
ui has been reduced to zero and only the depression due to the deformation of the vertical
springs defines the potential energy of the system. The strain energy will be taken as the same
as it was in stage 1, as it was found in the previous section that the reaction forces remain
fairly constant for practical values. In the definition of the kinetic energy, the angular velocity
will be set in terms of the tangential velocity v: .
Finally, stage 3 defines the maximum deformation in the springs that are impacted upon by
the rocking wall. It will be assumed that at this stage, the springs at the other end have been
unloaded and, therefore, all the strain energy is due to the deformation in the impact springs.
The potential energy is also defined only by the depression δimp. It also will be assumed that
impact only affects the vertical velocity of the c.g. of the masses, which is zero when the

29
impact springs reach the maximum deformation. The horizontal and the angular velocity of
the system will be regarded as the same as they were immediately before impact occurred.
That was observed in preliminary numerical models. Note that the horizontal velocity vx is
related to the total velocity by vx = v cosα.
The evaluation of the different expressions for the energy in the system at any stage leads to
some simplifications, as some of the energies are found to be much smaller than the total
energy in the system. It can be observed that the initial strain energy Es1 is very small
compared to the initial potential energy Ep1. After operating one can find that the ratio Es1/Ep1
is:
In the numerator, ky and kx are of the same order in a rocking wall, which means that after its
ratio is multiplied by tan2
α, the value in the numerator will be close to 1. In the denominator,
the values of ui that one may have at the design stage are much greater than the initial static
deformation δst and, therefore, one expects to get a large number in the denominator. One can
find that when practical values are placed in the above equation, the ratio Es1/Ep1 is less than
0.02. For practical purposes, therefore, Es1 (and consequently Es2) can be neglected. As the
design value ui is much larger than δst one also can ignore this last term in the definition of
the potential energy of the system. Neglecting the contribution of δst is conservative as δst is
always going to reduce the total height that defines the initial potential energy of the system.

30
With the simplifications, the total energy at every stage would be:
Comparing the energy of stages 1 and 2 one can obtain the velocity immediately before
impact occurs:
And equating the total energy in stage 2 to the total energy in stage 3 one gets the following
second order equation for δimp:
The solution of this second-degree equation is:
Finally, the factor of amplification due to impact will be:

31
The amplified forces Fv imp and Fh imp can now be calculated with:
Even a greater simplification may be achieved if the angle αeff is small enough to make
(ky/kx)tan2
α eff << 1. If this is the case the impact factor fimp may be taken as:
Notice that, in the previous and following analysis, the excitation at the base was not taken
into account. This means that, in the event of an earthquake, the work done by the base shear
at the foundation is not taken into account. This affects the balance of energy as developed
above. It is expected that, as rocking will significantly uncouple the first mode of vibration
from the oscillation at the base, the equations above will still be valid.
2.2.3 Energy Dissipation Capacity of a Rocking Rigid Wall
If there is no energy lost during rocking, the system would oscillate indefinitely at the period
defined by the release amplitude. In reality, some energy is lost during rocking defined as
contact damping. Housner expresses this energy loss as the ratio of the kinetic energy before
and after impact and assumes the impact is purely inelastic, i.e. no elastic rebound force
occurs (no bouncing).
The coefficient of restitution e is defined by Eq.(2.62)
Where
= angular velocity immediately before impact
= angular velocity immediately after impact

32
Housner went on to show that by equating moment equilibrium immediately before and after
impact, the change in kinetic energy is related entirely to the geometry of the block
A relationship is derived relating the amplitude after n successive rocking impacts when
released from an initial amplitude .
Where
n = number of impacts from release
Φ = dimensionless amplitude, Φ= /α, Φ0 defines the dimensionless release
amplitude and Φn is the dimensionless amplitude after n impacts.
In a later publication, Priestley et al. [1978] related this reduction in kinetic energy to
equivalent viscous damping derived considering free-vibration of rocking blocks.
Eq.(2.65) can be approximated with the following empirical equation
Tso and Wong [1989a] note that the highly non-linear nature of rocking blocks gives rise to
complicated dynamic characteristics during forced or earthquake excitation. Makris and
Konstantinidis [2001] and Makris and Konstantinidis [2003] dispute the use of Eq.(2.66) for
the design of rocking blocks. They argue that the dynamic characteristics of a rocking system
are not compatible with a response spectrum derived from either a single-degree-of-freedom
inelastic bilinear oscillator or an elastic damped oscillator. The conclusion that the rocking
response is quite different to that of a traditional elastic or ductile structural response has
been supported by Aslam et al. [1980] and was further confirmed after experimental tests
were found difficult to repeat. Makris supported his argument with numerical analyses
indicating that at a 1.6% change in the excitation amplitude resulted in a 125% change in
displacement (rotation) response, whereas similar increases to the response of a bilinear

33
oscillator would be minimal. Following these studies by Makris and Konstantinidis [2001]
the use of a rocking spectrum was suggested as a design tool for rocking blocks.
Experimental studies have shown that the reduction in kinetic energy predicted by Eq.(2.62)
is almost always greater than that observed during testing. Housner’s theorem states that the
impact must be purely inelastic; however, free-vibration studies have shown that some energy
is returned to the rocking system via elastic bouncing during impact, Evison [1977]. On the
contrary, other experimental studies noted other sources of energy dissipation, such as
friction within the system, resulting in greater energy loss than that predicted by Eq.(2.62),
Tso and Wong [1989b].
Aslam et al. [1980] provided a practical extension of the pure rocking block by locating
prestressed tendons within the centre of the rocking block to increase the stability of the
system (Figure 2.6).

34
Figure 2.6 Aslam et al. [1980] post-tensioned rocking block
The addition of the post-tensioned tendons dramatically improves the overturning response of
the block. Figure 2.7 compares the lateral response of a post-tensioned block with a pure
rocking block. In Figure 2.7 the distance from the rocking toe to each post-tensioned tendon
group is defined as di, the total (initial) prestressing force is denoted as TPT,0, while the
stiffness of each tendon group is defined as K.
Figure 2.7 Moment rotation response of a post-tensioned rigid block

35
The addition of prestressed tendons has two key advantages; first, the bilinear stiffness is
significantly increased: if the tendon stiffness K is large enough the bilinear stiffness will be
positive. Second, toppling of the block is prevented. While the system is still non-linear, the
dynamic characteristics are now more consistent with traditional ductile systems. In the case
of a prestressed rocking block, the rocking spectrum is no longer appropriate and the use a
traditional response spectrum can be adopted for design.
Another extension to the free-standing block was carried out by Makris and Zhang [1999] by
adding ductile elements at the edge of the rocking section (Figure 2.8). The addition of the
ductile elements adds strength and energy dissipation to the section, increasing the overall
stability. Makris and Zhang [1999] found that the ductile restrainers provided only a marginal
improvement to the response under a sine-pulse acceleration time history (representing a
simplified near-fault ground motion). Under these ground motion events the level of
mechanical damping has little relevance, rather the strength and stiffness of the loading
envelope is of greater importance. The lateral response of Figure 2.8 is graphically illustrated
in Figure 2.9.
Figure 2.8 Makris and Zhang [1999] restrained post-tensioned rocking block

36
Figure 2.9 Makris and Zhang [1999] moment rotation response of a restrained post-tensioned
rigid block
Specific Conclusions to the Rocking Response of Rigid Blocks:
The peculiarities of free-standing blocks are of little relevance to post-tensioned systems
whose dynamic characteristics are more akin to that of traditional ductile systems. Post-
tensioned systems have a larger bilinear stiffness: this increases the stability of the system by
mitigating/preventing toppling. Hence, this suggests that a conventional response spectrum,
as opposed to a rocking spectrum, is more appropriate.
The coefficient of restitution (defining the energy lost during impact) is a useful concept
when modeling rocking blocks based on the fundamentals of block mechanics, i.e. by
numerically solving the equation of motion for an inverted pendulum. A more practical
alternative for design and modeling is to equate this energy loss to equivalent viscous
damping (EVD). While it is argued that the coefficient of restitution cannot be related to
EVD for free-standing rocking blocks, such a relationship may be valid for post-tensioned
systems whose dynamic characteristics are similar to traditional ductile systems.

37
2.3 Adapting Rocking Walls to meet a Target Performance
Research conducted on RC rocking walls has shown that the presence of hysteretic energy
dissipators may improve the seismic response of a rocking system. Rahman and Restrepo,
(2000) and Holden et al. (2002) have used pieces of mild steel connecting the base of the
rocking wall with the foundation expecting them to yield axially during the uplift of the wall.
The cyclic static tests conducted by these researchers confirmed the contribution of the
dissipators towards creating flag-shaped hysteresis-loops in their force-displacement
response. Rahman and Restrepo, (2000) reported that the observed hysteresis-loops
represented up to 14% of equivalent viscous damping. The effect of the yielding pieces of
steel at the base of the wall is shown schematically in Figure 2.10.
Figure 2.10 Static lateral loading of a rocking wall with dissipators, Toranzo [2002]

38
Rocking walls diminish the impact actions and provide a controlled source of dissipation of
energy without damaging the rest of the structure. By having a controlled source of energy
dissipation one can use some of the design methodologies suitable for a proper performance-
based design.
2.3.1 Forces in a Rocking System with Hysteretic Energy Dissipators at the Base
2.3.1.1 Forces in the System before Impact
Again, the static solution provides an upper bound to the definition of the forces Fh and Fv.
For this case, they will be defined as:
Where Fy is the yielding force of the dissipators.
Figure 2.11 Rocking wall with hysteretic energy dissipators

39
2.3.1.2 Forces in the System at Impact
The same approach that was used for rocking walls without dissipators in section 2.2.2 will
be used here. Apart from the energy at the three defined stages, however, one must account
for the work done by the hysteretic energy dissipators as they yield all the way until the
impact process finishes. The depression due to the initial deformation of the springs, δst, will
not be accounted for in this case. However, one does have to account for the elastic energy
stored in the dissipators. It will be assumed that in all three stages the steel dissipators are
yielding. The elastic energy stored in the dissipators will be the same in any case and,
therefore, will be cancelled out when comparing the cases.
Where kd is the stiffness of the dissipators and 2Fy/kd is the elastic deformation that the
dissipators undergo before yielding in the opposite direction. Note that for Eq. (2.69) to be
valid, ui/2 > 2Fy/kd.
Comparing stages 1 and 2, and accounting for the work done by the dissipators, W1-2, one can
define the velocity immediately before impact:
Comparing stages 2 and 3, and accounting for the negative work done by the dissipators, one
gets a second order equation that leads to the following definition of δimp:

40
In the equation above, ky and kx are usually of a similar order, therefore, if α is rather small (α
< 10o
) then tan2
α will be a very small number and the expression (ky/kx tan2
α + 1) can be
replaced by 1. In that case, δimp can be calculated with:
However, this is not the maximum deformation that the springs at the base can sustain.
Towards the end of the impact process, the forces in the dissipators can change direction due
to the uplifting of the other end of the rocking wall. In that case, equilibrium conditions
would require an increase in the reaction at the base of the wall of a magnitude equal to the
absolute change in the actions in the dissipators. The maximum change one could expect
would be 4Fy (from -2Fy to +2Fy). If the dissipators are very stiff, this change could occur
very quickly, and the increase in the reaction at the impacting corner may be fully developed
by the time the impact deformation reaches its peak. The maximum expected impact
deformation would occur in that case and it would be:
The actual impact deformation would be between these two extremes and would depend on
the flexibility of the dissipator. A soft dissipator will lead to impact actions close to δimp,
while a stiff dissipator would lead to impact deformations close to δ*
imp. Conservatively, until
experimental information is found, the design impact amplification factor can be defined as:
2.3.2 Total Accelerations in the System
Total accelerations are an important parameter within a performance-base scheme, and the
following analysis is intended to predict them. As the rocking system uncouples, at some
extent, the oscillation of the wall from the shaking at the base, one can attempt to predict the
total accelerations that might occur in the system during an earthquake from those expected
in the simple rocking model. The acceleration can be obtained from Eq. 2.17-b or derived
from the expected inertial forces in the system. The second option is more appealing as
simple close equations have been derived to define the expected forces in the centre of

41
gravity of the system (they are the same as the reactions at the base). Following this
approach, the expected total horizontal acceleration, ahi, at any height of the wall, hi, can be
derived from the next equation. Noise must be expected from the higher modes of free
vibration in the structure, as they are not uncoupled at all.
The impact amplification factor, fimp, has to be used to define the peak accelerations when
impact occurs. The same approach might be used to define vertical accelerations, although in
this case there is no uncoupling from the vertical base shaking.
2.3.3 Energy Dissipation Capacity of the Rocking Rigid-Wall with Hysteretic Energy
Dissipators
In large levels of seismic demand, the energy dissipators alone provide most of the hysteretic
damping in the system. Figure 2.12 represents the cyclic lateral loading of a rocking wall
with perfectly rigid-plastic dissipators. The enclosed areas can be used to calculate the
equivalent viscous damping (EVD) of the system using the following equation (Kramer,
1996):
Where Ahl is the area enclosed by the hysteretic loops and Aext is the total rectangular area
defined by the coordinates of the maximum force-deformation point. Figure 2.12 also allows
determining the magnitude of the areas in terms of other already known parameters.

42
Figure 2.12 Areas for the calculation of equivalent viscous damping due to dissipators
After operating, Eq. (2.77) becomes:
In most cases, the deformation of the rocking wall before rocking is much smaller than the
total deformation of the system, Δ. If that is expected, Eq. (2.78) may be reduced to:
For large lateral displacements, this ideal equivalent viscous damping could reach values of
the order of 15% to 25%. This value, however, has been defined assuming a perfect rigid-
plastic dissipator. Since it is not possible to find such a dissipator in a real structure,
correction factors need to be defined to account mainly for the smaller hysteretic loops that
one should expect when using a real imperfect dissipator.

43
Since the wall is behaving elastically, the hysteretic loops depicted in Figure 2.12 have the
same area as the total area of the hysteretic loops produced by the dissipators, ΣA1i (Figure
2.13). The correction of the dissipated energy, therefore, can be made in the areas depicted in
Figure 2.10. Three factors were used to that end. Factor C1 accounts for the non-perfect
rigidity of the dissipators; C2 accounts for the curved shape of the hysteretic loops; and factor
C3 accounts for corrections that the experimental evidence would require one to do. With
these corrections, the design hysteretic equivalent viscous damping is:
The definition of the three factors takes into account possible different locations of the
dissipators in the base of the wall (Figure 2.11). The initial stiffness kd and the yielding load
Fy, are recommended to be constant to maintain symmetry in the dynamic response of the
rocking wall and, therefore, will be treated as such in the following analysis. Factor C1 can be
defined as the sum of the ratios between the areas A2i and A1i (Figure 2.13):
Where Δdi is the vertical deformation and stiffness of each dissipator. For certain
combinations of Fy, Δd and kd, the equation above might produce negative values, this only
means that the dissipator is still within the elastic region and is not dissipating any energy. In
that case the ratio should be taken as zero. C1 can also be defined in terms of the lateral
displacement of the structure, Δ. Assuming that the deformation of the wall is small
compared to the total lateral displacement after rocking has occurred one can define C1 as:
The definition of C2 requires the testing of the dissipators as the shape of the loops depends
on properties of the material and the type and dimensions of the dissipator. Values between
0.80 and 0.90 were found in experimental work.

44
Figure 2.13 Correction of the hysteretic loop of dissipators
The hysteretic damping due to the dissipators is not the only source of energy dissipation in
the structure. Equivalent viscous damping (EVD) due to impact and other mechanisms
intrinsic to the structure need to be added to get the total EVD of the system. The EVD
intrinsic to the structure, ξo, has been found to be very small when a structure is behaving
elastically, usually below 2% (Early, 1989). The EVD due to impact, , has been defined in
a previous section in this chapter. Mander and Cheng (1997) have pointed out that the
definition of the EVD due to impact, following Houssner’s approach is not compatible with
the EVDs calculated following an energy approach, and therefore they cannot be added up.
Mander and Cheng, however, found that the results are close enough when the lateral
displacement is less than 0.25 the width of the rocking wall.
When dissipators are used, the contribution of the impact to the total energy dissipation
capacity of the system is small. In the design stage, acknowledging that one is getting at least
10% of equivalent viscous from the hysteretic dissipators, one could either define the EVD
from the impact process simply as 2% or, conservatively, neglect it. This is reinforced by the
experimental evidence exposed in section 2.2.3 that showed the unreliability of the impact
energy dissipation mechanism.
As the EVDs described above represent parallel energy dissipation mechanisms, and
acknowledging the limitation pointed out by Mander and Cheng in the definition of the EVD
due to impact, the expressions can be added up to produce the total EVD of the system. The
total equivalent viscous damping of the system is therefore:

45
2.4 Modeling Techniques for Post-Tensioned Rocking Systems
2.4.1 Lateral Response of Post-Tensioned Connections
The lateral response of a generic post-tensioned system is discussed below and illustrated in
Figure 2.14. The lateral response is characterized by a number of discrete points.
Figure 2.14 Structural limit states of a post-tensioned rocking system
Decompression is the deformation state where the strain at the outer most fibre approaches
zero and uplift (rotation) of the base is initiated. The neutral axis depth is located at the edge
of the section (c = Lw) and is characterized by a sudden reduction in stiffness when compared
to the initial (gross) section stiffness.
Geometric non-linearity occurs when the neutral axis of the section approaches the mid
height of the section (c = Lw/2) and defines a further reduction in stiffness.
Yielding of the mild steel reinforcement can occur either before or after the geometric non-
linearity point depending on the section dimensions and location of the steel reinforcement
within the section. While some stiffness is lost the system still retains significant post-yield
stiffness due to the elongation of the prestressed reinforcement along the rocking interface.
Yielding of the prestressed reinforcement will result in a total loss in stiffness. Prestressing
tendons are inherently brittle with minimal strain ductility. The recentring capacity of the
section can be jeopardized if the prestressed reinforcement yields: this reduces the effective
tendon prestressing force. Some permanent displacements may be tolerated for very rare
earthquake events. A reduction in the prestress force can be detrimental in some cases, such

Master Thesis Rocking-Final, SROUR-SAVOIA

Master Thesis Rocking-Final, SROUR-SAVOIA

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