SEISMIC PERFORMANCE EVALUATIONOF REINFORCED CONCRETE STRUCTURES USINGMULTI-DIRECTIONAL POLYGONAL 3D LATTICE MODEL
Mauro Ricardo Simão

Supervisor: Associate Professor Tomohiro Miki

A thesis submitted to the
Faculty of the Graduate School of
Kobe University in partial fulfillment of
the requirements for the degree of
DOCTOR OF PHILOSOPHY IN ENGINEERING

Department of Civil Engineering
Kobe University
Kobe, Japan

January 2017

The thesis entitled
Seismic Performance Evaluation of Reinforced Concrete Structures
Using Multi-Directional Polygonal 3D Lattice Model
written by Mauro Ricardo Simão
has been submitted to the Department of Civil Engineeringof Kobe University

Supervisor:
Associate Professor Tomohiro Miki

Supervising Committee:
Professor Shinichi Akutagawa
Professor Hidenori Morikawa
Professor Takashi Nagao

ABSTRACT
Recently, with the advance of research and improved design methods, a big overflow ofdesign procedures of reinforced concrete (RC) structures has been proposed in thecontext of performance based design. These design procedures are in many cases basedon concrete mechanics and are applied by the nonlinear analysis. On the other hand,with recent seismic events and developments in the field of earthquake engineering, a
new shift in the seismic evaluation of RC structures has occurred and the specifications
in many countries try to keep up with the most recent developments in this field.
With the advent of three dimensional (3D) analytical methods, the seismic performancein can now consider the effects of combination of bilateral bending, shear and torsion
that actual structures are subjected to during actual earthquakes. However with all thesedevelopments little attention has been paid to geometrical particularities of RC members
such as circular cross-section geometry. On the other hand, is seismic design, while the
new keywords revolve around performance evaluation, there is lack of clarity on themeaning of performance based design in some cases, and especially when performanceis based on the numerical verification of damage conditions.
With the above in mind, the aim of this study is to generally propose the 3D latticemodel which has been enhanced from a 2D lattice model to assess the seismic
performance of RC structures. In order to do that, two conceptual categories were used:
first is the development of a 3D lattice model aimed at circular-cross section columns.
In this conceptual part, issues of performance are evaluated based on response of RC
members and analysis of criteria related to force and displacement. In order to achieve
that, a multi-directional polygonal 3D lattice model is proposed and the applicability todiscretize and correctly represent the structural response is verified by performing staticand dynamic analyses in RC columns.
In seismic events, after cracking of concrete, there is a change in the gross shear
resisting area capacity of concrete as well as degradation of stiffness. In the modelingusing the multi-directional polygonal 3D lattice model these two important issues arestudied .The applicability of the multi-directional polygonal 3D lattice model is further
extended to perform cumulative seismic damage evaluation in RC columns. In seismic
events, RC structures are subjected in many times to multiple sequences of motion;
therefore it is important to study this effect from the force and displacement point ofview, as well as understanding the effect of residual displacement in every new loadingof RC structures, and in this study that is carried out by the multi-directional polygonalI3D lattice model.
On the other hand in this study, performance evaluation of RC structures under seismic
motion is evaluated from the energy approach. A numerical verification method of
performance of RC structures in seismic analysis is proposed by performing damage
evaluation of existing RC structures from the material point of view. One of the key
points from the fracture mechanics is that the global structural behavior of a structure isaffected first and foremost by the material behavior. On the other hand in the mechanics
of solids it is said that the strain energy density of the material can inform us about thenature and behavior of the material, whether that is a brittle material or ductile. In thisstudy based on that a choice was made to perform damage evaluation from the material
point of view based on the energy dissipation criteria.
One of the key characteristics about the way the discretization of structure is performed
using the lattice model is that it allows the individual analysis of stress-strain
relationship of each element. That means that the strain energy density can be evaluatedindividually and from that the global behavior of structure understood. Based on that, adamage index can be proposed from the stress-strain relationship of each one of theseelements.
The way the proposed damage index works, is by establishing a ratio between ultimate
energy dissipation capacity of a RC member and the accumulated energy dissipationthat is captured from the response under seismic motion. In the analysis special
attention is given to concentration of damage in the RC structures. The method isverified by performing numerical verification of RC columns that have been subjected
to cyclic loading and also by performing dynamic analysis of RC column that has beentested by a shake-table test.
In order to extend the concept of energy based damage evaluation proposed in this study,
the nonlinear analysis using the 3D dynamic lattice model is performed on actual RC
structures. The target used in the analysis is actual rigid-frame RC viaducts damaged at
1995 Hyogo-ken Nanbu Earthquake. Tis RC viaduct has slight damage with diagonal
cracks and the buckling of longitudinal reinforcement. The results of dynamic analysis
are compared with actual damages of RC viaducts. The comparison reveals thereliability of the 3D dynamic lattice model at the structural system level. It is also foundthat the analysis can predict the damage conditions.
IIACKNOWLEDGEMENTSFirst and foremost I would like to express my deep gratitude to my supervisor Associate
Professor Tomohiro Miki for his superb guidance and unconditional support during the
6 years in which I have been a member of the structural concrete laboratory. For that, a
lifetime of gratefulness will never be enough to express my gratitude towards myProfessor and I am forever indebted to him.
I would like express my gratitude to Professor Shinichi Akutagawa, Professor Hidenori
Morikawa and Professor Takashi Nagao, members of the supervising committee, fortaking the time to be in the committee and contributing to the refinement of this work.
My sincere words of thanks go to Professor Satoru Oishi, Professor Hideyuki Kita,
Associate Professor Yasuko Kuwata and Associate Professor Toshimori Otazawa for theimmeasurable kindness and advice throughout all these years of graduate studies.
I would like to acknowledge Professor Óscar Mário Cavele my undergraduate professorof structural mechanics, who recently passed away, for early on strongly encouraging
me to pursue my interest in structural engineering and for patiently teaching me thefundamentals of engineering mechanics.
I would like to acknowledge the members of Professor Miki’s Laboratory and Professor
Kuwata’s Laboratory past and present for the great atmosphere and shared sense of duty
and for always helping me out with Japanese language.
I want to thank my family, especially my father Engenheiro Amândio Santos Simão forbringing engineering into our household and unconditionally supporting my vision.
I want to thank my friends in Japan and Mozambique, especially Dr. Nader Al-Juma’I
from Kwansei Gakuin University with whom I have shared this journey from day one.
I want to thank Dr. Letícia Sarmento dos Muchangos from Osaka University, my fellow
engineer, my best friend and my life partner for all the love, continuous debate of ideas,
support and the years of her life she has been giving me. Our journey has just begun.

III

TABLE OF CONTENTS
ABSTRACT ...................................................................................................................... IACKNOWLEDGEMENTS ........................................................................................... III
1. GENERAL INTRODUCTION .................................................................................... 1
1.1. BACKGROUND ......................................................................................................... 1
1.2. LITERATURE REVIEW............................................................................................... 3
1.2.1. 3D nonlinear analysis of concrete structures ................................................... 3
1.2.2. Damage Evaluation of RC Structures .............................................................. 4
1.2.3. Seismic Design Philosophies of RC Structures ............................................... 6
1.3. SCOPE AND OBJECTIVES .......................................................................................... 8
1.4. CONTENTS OF THE THESIS ....................................................................................... 9
REFERENCES OF CHAPTER 1 ................................................................................... 10
2. ANALYTICAL MODEL ............................................................................................ 13
2.1. INTRODUCTION ...................................................................................................... 13
2.2. OUTLINES OF 2D LATTICE MODEL ........................................................................ 13
2.3. MATERIAL MODELS ............................................................................................... 15
2.3.1. Compressive stress-strain relationships for concrete..................................... 15
2.3.2. Tensile stress-strain relationships of concrete ............................................... 17
2.3.3. Reinforcement model .................................................................................... 18
2.4. OUTLINES OF 3D LATTICE MODEL ........................................................................ 19
2.4.1. Configuration of elements ............................................................................. 21
2.5. NONLINEAR ANALYSIS PROCEDURE ...................................................................... 24
2.6. CYCLIC ANALYSIS OF RC COLUMNS USING 2D LATTICE MODEL.......................... 25
2.6.1. Outline of experiment and analysis ............................................................... 25
2.6.2. Lateral force-displacement characteristics of RC column ............................. 26
2.7. STATIC ANALYSIS OF RC COLUMNS SUBJECTED TO BILATERAL LOADING USING 3D
LATTICE MODEL .......................................................................................................... 28
2.7.1. Outline of experiment and analysis ............................................................... 28
2.7.2. Analytical results and discussion ................................................................... 29
REFERENCES OF CHAPTER 2 ................................................................................... 32
3. DEVELOPMENT AND APPLICATION OF MULTI-DIRECTIONAL
POLYGONAL 3D LATTICE MODEL .......................................................................... 34
3.1. INTRODUCTION ...................................................................................................... 34IV

3.2. MODELING AND GEOMETRY OF MEMBERS ............................................................ 34
3.3. CROSS-SECTIONAL AREA OF LATTICE MEMBERS .................................................. 36
3.3.1. Cross-sectional area of arch members ........................................................... 37
3.3.2. Cross-sectional area of truss members .......................................................... 39
3.4. PARAMETRIC ANALYSIS OF RELATIONSHIP BETWEEN ARCH AND TRUSS MEMBERS 40
3.4.1. Introduction ................................................................................................... 40
3.4.2. Effect of value of t ......................................................................................... 40
3.5. LINEAR-ELASTIC ANALYSIS OF CIRCULAR-CROSS SECTION RC COLUMNS........... 42
3.5.1. Introduction ................................................................................................... 42
3.5.2. Outline of analytical target ............................................................................ 42
3.5.3. Discretization of target .................................................................................. 42
3.5.4. Analytical results and discussion ................................................................... 43
3.6. STATIC ANALYSIS OF CIRCULAR RC COLUMNS SUBJECTED TO CYCLIC LOADING . 44
3.6.1. Outline of analysis ......................................................................................... 44
3.6.2. Analytical results and discussion ................................................................... 46
3.7. DISCRETIZATION METHOD ANALYSIS OF CIRCULAR RC COLUMNS ....................... 47
3.7.1. Introduction ................................................................................................... 47
3.7.2. Outline of analysis ......................................................................................... 47
3.7.3. Analytical results and discussion ................................................................... 49
3.8. CUMULATIVE SEISMIC DAMAGE ASSESSMENT IN CIRCULAR RC COLUMNS .......... 52
3.8.1. Introduction ................................................................................................... 52
3.8.2. Outline of analysis ......................................................................................... 53
3.8.3. Analytical results and discussion ................................................................... 55
REFERENCES OF CHAPTER 3 ................................................................................... 59
4. NUMERICAL VERIFICATION OF SEISMIC PERFORMANCE OF CIRCULAR
RC COLUMNS USING MULTI-DIRECTIONAL POLYGONAL 3D LATTICE
MODEL .......................................................................................................................... 60
4.1. INTRODUCTION ...................................................................................................... 60
4.2. ENERGY DISSIPATION CAPACITY IN RC COLUMNS ................................................ 61
4.2.1. Introduction ................................................................................................... 61
4.2.2. Hysteretic energy dissipation......................................................................... 61
4.2.3. Elemental energy dissipation ......................................................................... 62
4.3. DAMAGE RANGE EVALUATION .............................................................................. 63
4.3.1. Introduction ................................................................................................... 63
4.3.2. Evaluation of seismic damage using damage index ...................................... 64V

4.4. NUMERICAL VERIFICATION OF SEISMIC PERFORMANCE ........................................ 65
4.4.1. Analytical frame-work ................................................................................... 65
4.4.2. Evaluation of performance states .................................................................. 67
4.5. NUMERICAL VERIFICATION OF SEISMIC PERFORMANCE OF RC COLUMN SUBJECTEDTO CYCLIC LOADING .................................................................................................... 68
4.5.1. Outline of analysis ......................................................................................... 68
4.5.2. Analytical results and discussion ................................................................... 69
4.6. PERFORMANCE EVALUATION OF RC COLUMNS SUBJECTED TO SEISMIC MOTIONDESIGNED ACCORDING TO 2002 JRA SPECIFICATIONS ................................................. 71
4.6.1. Outlines of Analysis ...................................................................................... 71
4.6.2. Analytical results and discussion ................................................................... 72
REFERENCES OF CHAPTER 4 ................................................................................... 75
5. PERFORMANCE EVALUATION OF RC VIADUCT ............................................. 77
5.1. INTRODUCTION ...................................................................................................... 77
5.2. DAMAGE DISTRIBUTION IN RC VIADUCTS BASED ON ENERGY DISSIPATION ......... 78
5.2.1. Outline of target structure .............................................................................. 78
5.2.2. Configuration of lattice model....................................................................... 79
5.2.3. Damage distribution of Hansui R5 viaduct ................................................... 81
5.3. NUMERICAL VERIFICATION OF SEISMIC PERFORMANCE OF HANSUI R5 VIADUCT . 84
5.4. SEISMIC DESIGN AND RETROFIT CONSIDERATIONS BASED ON 3D LATTICE MODEL..................................................................................................................................... 87
REFERENCES IN CHAPTER 5.................................................................................... 88
6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ...... 89
6.1. GENERAL CONCLUSIONS ....................................................................................... 89
6.2. RECOMMENDATIONS FOR FUTURE RESEARCH ....................................................... 91
6.3. FURTHER CONTRIBUTIONS AND DISCUSSION ......................................................... 92

VI

1. GENERAL INTRODUCTION
1.1. BackgroundIn the last few decades, a considerable number of very strong earthquakes have occurredin many parts of the world. That has been the case in the United States of America with
the Loma Prieta Earthquake in 1989 and the Northridge Earthquake in 1994. In the
People`s Republic of China the 2008 Sichuan Earthquake caused great damage to social
infrastructure, in New-Zealand the Christchurch Earthquake in 2011 caused great socialand financial damage and in japan, a country very vulnerable to earthquakes, a numberof strong seismic events such as the Great Hanshin Earthquake (Kobe Earthquake) in
1995, the Great Eastern Japan Earthquake (Tohoku Earthquake) in 2011 and more
recently the Kumamoto Earthquake in 2016 exposed the vulnerabilities that social
infrastructures are subjected to in the event of strong seismic actions.
With such potential to cause financial damage and destruction of social infrastructure,
the effect earthquakes have on public infrastructures such as water and gas pipelines,
roads and viaducts should be object of close attention to the Civil Engineeringcommunity. In the event of earthquakes, reinforced concrete (RC) structures are very
vulnerable to damage in general, but that is particularly the case of RC bridges andviaducts. Because of the complexity of different earthquakes and the response thatdifferent RC structural members tend to show under seismic excitation, the study andanalysis of seismic response is inherently highly nonlinear. In recent times, design
engineers are encouraged by design codes to address these nonlinearities during thedesign process. In japan the benchmark to this shift in the design process occurred after
the Great Hanshin Earthquake in 1995. For structural design, the standard specifications
for seismic performance verification of concrete structures (JSCE 2002) have been
extensively revised. The core concept in this revision was that considerable inelastic
deformation of RC structures is possible after the longitudinal reinforcement yield. This
concept moved from the previous consideration of only the elastic behavior when the
structure is subjected to strong earthquake motion. Furthermore in these revisions, theductility design criterion is adopted, replacing the criterion for load carrying capacity todetermine the deformation capacity of RC structures. It is also noteworthy that underthese revisions, the designers are required to perform verifications using dynamic
analysis. The applicability of this requirement has been greatly enhanced by the recent
developments in the field of earthquake engineering, which allowed the use of the input
ground motion that has been recorded during actual earthquakes.
1

The upgrade in the design requirements of concrete structures has been made possible
by important contributions in analysis technology and methods. For example thestructural response in dynamic analysis for RC columns is considered analyzing theprincipal axes of structural systems, but in reality the actual ground-motion is applied inthree-dimensional (3D) way in the actual structures from different directions of its
principal axes. Under these conditions, the behavior becomes more complex that under
uniaxial motion and 3D analysis is indispensable to clarify the seismic performance ofRC columns subjected to multi-directional ground motion (Miki 2004). In the current
research for advanced analytical technology where 3D constitutive models with special
focus on cracking of concrete applied to nonlinear analysis are the target, nonlinear
finite element analysis using the solid element has been proposed (Hauke et al. 2000,
Maekawa et al. 2001, Tsuchiya et al. 2002) and nonlinear analysis using
three-dimensional lattice model (Miki and Niwa 2004) has also been proposed. The
principal merit of the mentioned analysis tools is that they can reliably capture theresponse of RC structures. However the 3D finite solid element analysis requires
powerful computational capability and the computation times is quite large due to thelarge number of degrees of freedom, which contrasts with the nonlinear analysis using
three-dimensional lattice model where due to its simple and efficient 3D numericalapproach the analysis difficulties are tackled with more ease.
Based on the outcome of recent research on design of RC structures, a new era isinaugurated which is focus on the notion of performance based design of structures.
Many propositions have been made. At the core of it are the damage evaluation models,
which generally state that performance verification of RC structures is realized though
the evaluation of seismic damage of RC structures. Early models tended to evaluate
seismic damage in terms of deformation capacity, but notably Park and Ang (1985)
conveyed that seismic damage on a structure is centered around the earthquake energy
input and the structural mechanical dissipated energy, therefore in their proposition, thePark-Ang damage model, damage is evaluated by a combination of excessive
deformation and a ratio of plastic strain energy. More and more the research on theperformance evaluation of RC structures under seismic motion recognizes theimportance of energy based criteria for analysis as well as the need to reduce thecomplexities of analysis by combining efficient but powerful analytical tools for the
analysis of the behavior of RC structures with simple but practical methods to performdamage evaluation of RC structures under seismic motion and ultimately performance
evaluation and performance based design.
21.2. Literature Review
1.2.1. 3D nonlinear analysis of concrete structures
In order to perform structural analysis of concrete structures, the target structure should
be discretized using one of many modeling tools currently developed. On the modelingprocess it is important to clarify the scale or level in which the discretization takes place.
Generally three levels are recognized in the analytical modeling that is: macro-level,meso-level and micro-level. Furthermore in recent literature, a very strong focus on 3Dmodeling has emerged. In these 3D models two major issues are at the center: the first istreating structural members in 3D space which has implications in geometry of thestructures and loading conditions and second is expressing the stress field by means offull 3D solid elements. Below a brief description of 3D nonlinear analysis research is
given.
(a) Fiber model
According to Feng et al. (2012) in fiber-based models, materials have only
uni-directional strength and stiffness whose behavior is defined in terms of its
stress-strain response. Fiber sections are assumed to remain plane throughout theanalysis. In reinforced concrete structures, the fiber section is assembled withpre-defined concrete and steel materials. The section is divided into a number ofconcrete patches where the steel fiber will be located. Strain compatibility between
reinforcement and surrounding concrete is assumed. The sectional reactions under force
and moment are in terms of axial strain at mid-section and curvature. A unique solutionof this deformation combination will be obtained based on the moment-curvature
analysis of the section. Spacone et al. (1996a, 1996b) proposed the formulation of afiber beam-column element for nonlinear static and dynamic analysis of RC frames. In
their work flexibility-based formulations that allowed a more accurate description of theforce within the element have been presented. The flexibility-based model offers a clear
and reliable procedure for their implementation in finite element analysis.
(b) Finite element analysis of concrete structures
The finite element method is a powerful computational tool, which allows complex
analyses of the nonlinear response of RC structures to be carried out in a routine fashion.
In their study Ngo and Scordelis (1967) analyzed with a finite element model simple
beams in which concrete and reinforcing steel were represented by constant strain
triangular elements, and a special bond link element was used to connect the steel to theconcrete and describe the bond-slip effect. Nilson (1972) introduced nonlinear material3properties for concrete and steel and a nonlinear bond-slip relationship into the analysis
and used an incremental load method of nonlinear analysis. Four constant strain
triangular elements were combined to form a quadrilateral element by condensing out
the central node. Cracking was accounted for by stopping the solution when an element
reached the tensile strength, and reloading incrementally after redefining a new cracked
structure. More recently Hauke and Maekawa (2000) have presented a 3D constitutive
model for nonlinear finite element analysis of RC members with special attention to
cracking. In their research, post-cracking formulations derived from uniaxial tension
tests are generalized into spatially arbitrarily inclined cracks in multiple directions. The
development of computing allowed more recently the performance of nonlinear 3D
analysis using solid elements and Noguchi et al. (2001, 2002) conducted the 3D finite
element analysis for performance verification of steel beam-RC joints. These analyses
are largely dependent of the computational capabilities due to large degree of degrees of
freedom. From that the lattice model (Niwa et al. 1995, Miki et al. 2003a, 20003b) has
been proposed, based on decreased degrees of freedom which shortens computing timeand is also more practical while remaining accurate in the prediction of RC structural
behavior considering material nonlinearity.

1.2.2. Damage Evaluation of RC Structures
Structural damage characterization under seismic loading has extensively been studied
in recent years and many models with different concepts have been proposed. One of
the relatively simplest one was the prediction of damage in terms of ductility demands.
Although much effort has been spent on estimation of damage with ductility demand in
reinforced concrete frames, there is still a large gap between the ductility demand andactual observed damage in experiments. Eventually, researchers constructed more
realistic damage models with different parameters. In the light of these findings,
different models have been proposed by researchers using either the classical definitionof low-cycle fatigue or energy dissipation of a member during its inelastic response, or
stiffness deterioration under reversed cyclic loading.
(a) Damage models based on ductility concepts
The development of damage models starting from the ductility concept led to thedevelopment of the first damage models. Newmark and Rosenblueth (1971), proposed
the ductility factor as a mean to assess damage. The factor can be expressed either as afunction of curvature, rotation or displacement. Using a similar approach as Newmark
and Rosenblueth (1971), Lybas and Sozen (1977) came up with a similar damage
4

concept which correlates the initial elastic stiffness and maximum elastic stiffness do
produce a damage index. Using the stiffness based damage index, Banon et al. (1981)
presented a flexural damage index, computed according to the relation between theultimate bending moment as resulting from a pushover analysis, maximum bending
moment and corresponding curvatures. In this approach, cosndierations of ultimate
displacement capacity, softening due to the buckling of longitudinal reinforcement areconsidered as shown in Figure 1.1.
(b) Damage models based on energy concepts
The benchmark moment in which energy based damage models became widely
proposed is a study by Park and Ang (1985) which proposed a damage index as thelinear combination of the maximum displacement and the dissipated energy namely.
The Park and Ang index can take into account both maximum plastic displacement andplastic dissipated energy and is supported by a wide correlation with observed damage.
Tembulkar and Nau (1987) carried out an analytical study on inelastic structuresmodelled as SDOF systems with two different hysteretic models, in order to investigate
the role of models on seismic energy dissipation. They have stated that damage attainedby a reinforced concrete member under dynamic action cannot be predicted adequately
by response spectrum concepts. Therefore, a well-constructed hysteresis model should
be constructed as a tool for the damage assessment of concrete structural members.
Load

Occurrence of reinforcement buckling

P3

3

P2

Neglecting buckling

2

Yield of longitudinal
reinforcement

P1

1

Considering buckling
behavior (actual behavior)

Initiation of flexural cracking
Displacement

0
1

dy
P1

duc

dun

P2

3

2

P3

Figure 1.1 Load-displacement relationship of a RC column
5Stojadinovic and Thewalt (1996) proposed a pair of energy balanced hysteresis models
based on experiments that were conducted on knee joint sub-systems. The proposedmodels were piecewise linear and segments on envelope curves were defined by somespecial points such as first crack, yield and ultimate resistance.

1.2.3. Seismic Design Philosophies of RC Structures(a) Japan
The Great Hanshin Earthquake in 1995, marked a turning point in terms of prescribedseismic design specifications is Japan due to elevated degree of damage sustained by
RC structures. The most evident result of that was the publication of the Standard
Specifications for Concrete Structures- Seismic Performance Verification (2002) by theJapan Society of Civil engineers (JSCE). In these specifications, the concept of
performance-based design is nuclear which represented a major change from thespecifications published in 1986 which was based on the limit state design method. In
these new specifications, two levels of design earthquake ground motion are prescribed:
Level 1 corresponds to an earthquake that is likely to occur a few times within the lifeof a structure and level 2 corresponds to a very rare strong earthquake. The Seismic
performance level of a structure is verified considering three seismic performancelevels: in seismic performance level 1 function of the structure is maintained without
any repair after the earthquake, in seismic performance level 2 function of the structure
can be restored within a short period after the earthquake and in seismic performancelevel 3 there is no overall collapse of the structure, however it does not remainfunctional after the earthquake. With that seismic design should be carried out so thatthe structure satisfies seismic performance 1 against level 1 earthquake ground motion
and seismic performances 2 or 3 against level 2 earthquake ground motion.
(b) United States of America
In the United States of America a considerable number of design specifications exist
from the federal level to the state level. However with all its diversity, the AASHTO
Bridge Design Specifications (2012) and Caltrans (2015) give a very informative
outlook on the design philosophies of RC Structures in the USA. The design
philosophies of Caltrans (2013) are centered on the contributions of AASHTO Bridge
Design Specifications (2012). In the design, the Load and Resistance Factor Design(LRFD) method is prescribed. With focus on seismic design of RC structural members,
Caltrans (2013) states that columns, shafts, Pile/Shaft groups in soft or liquefiable soils,
6pier walls, and pile/pile-extensions in slab bridges designed and detailed to behave in aductile manner are designated as seismic-critical members. Seismic-critical members
may sustain damage during a seismic event without leading to structural collapse or lossof structural integrity. Other bridge members such as dropped bent cap beams, outrigger
bent cap beams, “C” bent cap beams, and abutment diaphragm walls shall be designed
and designated as seismic-critical if they will experience any seismic damage as
determined by the Project Engineer and approved during Type Selection. All othercomponents not designated as seismic-critical shall be designed to remain elastic in aseismic event.
For structural applications, seismic demand is represented using an elastic 5% damped
response spectrum. In general, the Design Spectrum (DS) is defined as the greater of: aprobabilistic spectrum based on a 5% in 50 years probability of exceedance (or 975-year
return period); a deterministic spectrum based on the largest median response resulting
from the maximum rupture of any fault in the vicinity of the bridge site and a statewide
minimum spectrum defined as the median spectrum generated by a Magnitude 6.5
earthquake on a strike-slip fault located 12 kilometers from the bridge site.
(c) Europe
In the context of the European Union (EU), member countries of the EU have umbrella
codes which are the Eurocodes. On a country local level, the specifications are subject
to adjustment; however the Eurocodes prescribes the global design philosophy in the
EU space. According to Eurocode 8 (2004) which deals with seismic design ofstructures, the design of earthquake resistant concrete structures shall provide thestructure with an adequate capacity to dissipate energy without substantial reduction of
its overall resistance against horizontal and vertical loading.
Structures in seismic regions shall be designed and constructed in such a way that thefollowing requirements are met: No collapse requirement - the structure shall be
designed and constructed to withstand the design seismic action without local or global
collapse. Damage limitation requirement - the structure shall be designed andconstructed to withstand a seismic action having a larger probability of occurrence thanthe design seismic action, without the occurrence of damage and the associatedlimitations of use. Seismic action is considered dividing the territories in seismic zones.
The earthquake motion at a given point on the surface is represented by an elastic
ground acceleration response spectrum called elastic response spectrum.
71.3. Scope and Objectives
Currently, in the field of 3D nonlinear analysis of RC structures the 3D finite solid
element analysis tends to be a dominant force. Constitutive models focused on crackingof concrete are applied to finite element analysis, with very accurate results. Althoughthese developments mean more accuracy and better performance prediction, the fact ofthe matter is that the process remains very complex and sensitive to changes in theanalytical input information, as well as the ability of the design engineer to deal withoften complex FEM packages, which means that a simpler and yet effective method to
perform nonlinear seismic performance analysis of RC structures in 3D space is
indispensable.
On the other hand the JSCE standard specifications for concrete structures state very
important principles when it comes to seismic design and can be resumed as being
important to carry seismic design not only to ensure the safety of structures throughout
the occurrence of an earthquake, but also ensure the prevention of fatal damage
affecting human life, on a social point of view and on an engineering point of view Miki(2004) refers that in order to obtain good performance of structures during earthquakes,
it is essential to analyze in detail the dynamic characteristics of the actual 3D structural
system, and clearly address the subjective nature of observations of seismic
performance.
Miki et al. (2004) developed the 3D lattice model concept for nonlinear analysis of RC
structures, based on the 2D lattice model presented by Niwa et al. (1995), in order toenhance its capabilities. This 3D lattice developed, offers reasonable prediction of theshear carrying capacity of RC structural members, and by discretizing a RC structuralmember into truss elements, internal stress flows are easy to determine.
With that, the main objective of this study is performance evaluation of existing RC
structures subjected to seismic excitation using the 3D lattice model. In a more specificway, three conceptual phases are developed: one, detailed development of a 3D latticemodel analysis for circular cross-section RC columns, two investigate and propose
appropriate quantitative damage measurement index to induced seismic damage on an
existing structures and three propose a seismic performance evaluation method for RC
structures using the 3D Lattice model.

81.4. Contents of the Thesis
The study is divided in six chapters. In chapter 1 background of the study is discussed
including the most recent advances in the nonlinear analysis of RC structures subjected
to seismic motion as well as major seismic design philosophies in Japan, United Statesof America and Europe.
Chapter two presents the analytical model used in the study. In this study the latticemodel is used. In this review the initial development of 2D lattice model is briefly
presented, followed by a more detailed focus of the development of 3D lattice model.
Here, material considerations as well as geometry properties are presented. The
constitutive models of concrete and reinforcement are explained.
Chapter three presents the full development and application of the multi-directional
polygonal 3D lattice model. In this chapter full consideration of the application of themodel is given to static analysis of RC columns subjected to cyclic loading. On theother hand analysis of methodology od discretization of circular cross section columns
based on circular to rectangular equivalence are compared to a more realistic
representation of geometry using the multi-directional polygonal 3D lattice model.
Cumulative seismic damage evaluation of RC columns is performed.
In chapter four the numerical evaluation of seismic performance of RC structures isperformed. Here the energy approach is explained and a damage index is proposed
based on the strain energy of the lattice model elements. To perform the analysis, themulti-directional polygonal 3D lattice model is used and the applicability of the method
is verified by performing analysis in RC column subjected to cyclic loading anddynamic analysis of RC column tested by shake table test,
In chapter five the damage evaluation using energy dissipation is subject to an
expansion of scope. An actual viaduct is analyzed. This is a viaduct that suffereddamage during the Hyogo-ken Nanbu earthquake. The damage is compared to the actual
registered damage in the structure.
Finally chapter six includes conclusions based on the study and the multiple aspects ofit and from that a few recommendation for a future study on the topic discussed in thisthesis.

9References of Chapter 1
AASHTO LRFD Design Specifications (2012): American Association of State Highway
and Transportation Officials Specifications for Bridge Structures, Washington, DC.
Banon, H., Biggs J.M., Irvine H.M. (1981): Seismic Damage in Reinforced Concrete
Frames, Journal of ASCE- Structural Division, vol. 107, pp. 1713-1729.
CALTRANS (2013): Seismic Design Criteria-2013, California Department oftransportation.
Eurocode 8 (2004): Design of structures for earthquake resistance – Part 1: Generalrules, seismic actions and rules for buildings, December
Feng, Y., Kowalsy, M. J. and Nau, Y.M (2012): Fiber-Based Modeling for Investigating
the Effect of Load History on the Behavior of RC Bridge Columns, 15 WCEE Proc.,
Lisboa.
Hauke, B. and Maekawa, K. (2000): Three-Dimensional Modeling of Reinforced
Concrete with Multi-Directional Cracking, Concrete Library of JSCE, No.36,
pp.181-206, December.
JSCE (2002): Standard Specifications for Concrete Structures-2002, Seismic
Performance Verification, December.
Lybas, J.M. and Sozen, M.A. (1977): Effect of Beam Strength and Stiffness on Dynamic
Behavior of Reinforced Concrete Coupled walls, Technical Report on Civil Eng.
Studies, No. 444, University of Illinois, Urbana.
Maekawa, K., Fukuura, N., and An. X. (2001): 2D and 3D Multi-directional Cracked
Concrete Model under Reversed Cyclic Stresses, Modeling of Inelastic Behavior of RC
Structures Under Seismic Loads, ASCE, pp.56-78.
Miki, T. (2004): Nonlinear Analysis of Reinforced Concrete Structures Subjected to
Seismic Loading by Using Three-dimensional Lattice Model, Doctoral thesis, Tokyo
Institute of Technology, February.

10

Miki, T. and Niwa, J. (2004): Nonlinear Analysis of RC Structural Members Using 3D
Lattice Model, Journal of Advanced Concrete Technology, JCI, Vol. 2, No.3,
pp.343-358, October.
Miki, T., Niwa, J., and Lertsamattiyakul, M. (2003a): Earthquake Response Analysis for
RC Bridge Piers Considering Reinforcement Buckling Behavior, Journal of Materials,
Concrete Structures and Pavements, JSCE, No.732/V-59, pp.225-239, May. (inJapanese)Miki, T., Niwa, J., and Lertsamattiyakul, M. (2003b): Numerical Evaluation of Seismic
Performance of Reinforced Concrete Bridge Piers Using Dynamic Lattice Model,
Concrete Library of JSCE, Vol.41, pp.49-64, June.
Newmark, N.M and Rosenblueth, E. (1971): Fundamentals of Earthquake Engineering,
Prentice-Hall, Englewood Cliffs, New Jersey.
Ngo, D. and Scordelis, A.C. (1967): Finite Element Analysis of Reinforced Concrete
Beams, Journal of ACI, Vol. 64, No. 3, pp. 152-163.
Nilson, A. H. (1972): Internal Measurement of Bond Slip, Journal of ACI, Vol. 69, No.
7, pp 439-441.
Niwa, J., Choi, I. C., and Tanabe, T. (1995): Analytical Study for Shear Resisting
Mechanism Using Lattice Model, Concrete Library of JSCE, No.26, pp.95-109,
December.
Noguchi, H., Kashiwazaki, T., Uchida, K., and Nozaki, Y. (2001): FEM Analysis for
Structural Performance Design of Concrete Structures, Modeling of Inelastic Behaviorof RC Structures under Seismic Loads, ASCE, pp.257-275.
Noguchi, H., Zhang, D, and Kashiwazaki, T. (2002): Three-dimensional Finite Element
Analysis of RC Beam-column Joints Reinforced by New Reinforcing Way, Proceedingsof the JCI, Vol.24, No.2, pp.397-402. (in Japanese)
Park, Y. J. and Ang, A.H.S (1985): Mechanistic Seismic Damage Model for Reinforced
Concrete, Journal of the Structural Division, ASCE, Vol. 111, No. 4, pp. 722-739.
11

Tembulkar, J.M. and Nau, J.M. (1987): Inelastic Modeling and Seismic Energy
Dissipation. Journal of Structural Engineering, ASCE, Vol. 113, No. 6, pp. 1373-1377.
Stojadinivic, B. and Thewalt, C.R. (1996): Energy Balanced Hysteresis Models, Report
No. UCB/EERC-96/01, Earthquake Enginerring Research Center, University ofCalifornia, Berkeley

12

2. ANALYTICAL MODEL2.1. Introduction
It is fundamental that the performance of RC structures can be simulated using
nonlinear analytical models that are both objectively simple to handle and accurate. inseismic analysis of RC structures, many nonlinearities related to geometry and thematerial require that the analysis criteria is handled with appropriate analogies, and themost essential methods in order to do this treat the structure as a 2 dimensional (2D)
element. However in a seismic event, actual ground motions imposed on RC structures
work against the actual 3 dimensional (3D) structures from different directions along theprincipal axis. With that it is important to be able to carry the nonlinear analysis of RC
structures in 2D and 3D spaces with a method that is both accurate and objective while
preserving a sensible need for calculation time and CPU processing power. To meetthese requirements, in this study the lattice model first proposed by Niwa et. al (1995)
for 2D space and enhanced by Miki and Niwa (2004) for 3D space is focused. Thismodel predicts the shear carrying capacity of RC members based on arch and truss
analogy to treat concrete as a material. In this chapter the formulations of 2D latticemodel and 3D lattice model which are described.

2.2. Outlines of 2D Lattice Model
The lattice model consists of members representing concrete and reinforcement. Figure
2.1 schematically shows the lattice model of RC column. In the lattice model, arch
action and truss action are considered as the shear resisting mechanism of a structuralmember is. For the 2D lattice model, a RC member in 3D is presented as the 2D model
based on the plane stress condition. The concrete is modeled into flexural compression
members, flexural tension members, diagonal compression members, diagonal tension
members, vertical members, or arch members. For RC column, longitudinal andtransverse reinforcement are modeled into horizontal and vertical member, respectively.
The diagonal compression members and the diagonal tension members represent thetruss action of shear resistance mechanism in the lattice model. The 2D lattice model iscategorized as a fixed angle truss model in which the diagonals are assuming as the 45°
angle of inclination trusses. By incorporating the arch member to represent the arch
action, the lattice model can be used to estimate the changing direction of internal
compressive stress flows after diagonal cracking. For the lattice model, the arch partshould be located along the compressive stress flows inside a RC member.
13

Lateral force

Lateral forcedLoading direction

0.5 d

d

Arch member

Flexural tension
memberCross section

Flexural compression
member

Diagonal tension
memberd: Effective
Depth

Diagonal compression
memberLongitudinal
reinforcement

Concrete
Steel
Node

Transverse
reinforcement(a) RC Column(b) 2D Lattice model

Figure 2.1 Outline of 2D lattice model (Miki 2004)
The arch and truss analogy allows the consideration of shear resisting mechanism. In
other words, in the discretization process the concrete is divided into truss and archmembers. When the value of t is defined as the ratio of the width of the arch part to thewidth of cross section b, the widths of the arch part and the truss part are given by bt
and b ( 1 − t ), respectively, where 0 < t < 1. The value of t is determined based on thetheorem of the minimization of the total potential energy for the 2D lattice model with
the initial elastic stiffness. Figure 2.2 shows a detailed diagram of cross section of RC
column modeled using 2D lattice model.

bbtTruss part: b( 1 t )dArch part: bt

Figure 2.2 Cross section of RC member modeledby 2D lattice model (Miki 2004)
14

In the modeling due to the existence of reinforcing bars for flexural tension members
the concrete still contributes to tensile resistance even after cracking. In that manner the
cross-sectional area of concrete flexural members, for simplicity, is determined byconsidering the bond effect between concrete and reinforcing bars, and thecross-sectional area of the flexural tension or compression member is assumed to be theproduct of the double depth of cover concrete and the width of cross section.

2.3. Material Models
2.3.1. Compressive stress-strain relationships for concrete
In this study, in order to take into account the confinement effect by transverse
reinforcement that is observed when a suitable amount of transverse reinforcement isused to confine the concrete, the stress-strain relationship proposed by Mander et al.
(1988) as expressed by Equation 2.1 and illustrated in Figure 2.3, is used as a material
model for the diagonal compression members and the arch members. Under thesecircumstances a significant increase in both compressive strength and ductility can be
expected

 c’

f cc ’xrr 1  x r

(2.1)

where

x

 c’
 cc’

(2.2)



  1.254  2.254 1  7.94 f l ’ 2 f l ’f cc’
 f c 0’

f c 0’
f c 0’


(2.3)

f l  K e  w f wy

(2.4)

'






 f cc’  1
 f c 0’  cc’  c 0’
1  5

(2.5)

Ec 0
Ec 0  Esec

(2.6)

Ec 0  5000 f c 0’

(2.7)

f cc’
 cc’

(2.8)

r

Esec 

15c’

Confined by transverse reinforcement

 fcc’

c’

c’

c’

f c’

 f c’

 fc’

c’

Uncracked concrete(=1.0)
0

Ec

o’

Ec

t

c’

0

cc’

c’

o’

Figure 2.3 Compressive model of concrete

Figure 2.4 Reduction in compressive
strength of cracked concrete

where, fc’ is the uniaxial compressive strength of the concrete, rw is the transverse
reinforcement ratio (= Aw / bws), Aw is the cross-sectional area of the transverse
reinforcement, bw is the width of web concrete of a RC member, s is the transverse
reinforcement spacing, and fwy is the yield strength of the transverse reinforcement.
Moreover Vecchio and Collins (1986) demonstrated that the compressive stress of
diagonally cracked concrete decreases as the transverse tensile strain, εt, increases, as
shown in Figure 2.4. Therefore, the value of εt, for the diagonal tension members,
which are normal to the diagonal compression members, is used to determine thecoefficient to express concrete compressive softening, η. The behavior of the cracked
concrete in compression is then characterized by Equation 2.9. For the arch member,the transverse tensile strain of the diagonal tension member near the loading point is
used.



1.0
 
0.8  0.34 t'  o 

 1.0

(2.9)

where  o’ is equal to 0.002. The tensile strain  t of the diagonal tension members,
which is the normal direction to the diagonal compression member, is used to determine
the coefficient of the compressive softening of concrete.

16

t

t

ft

ft

f t: Tensile strength of
concrete
L : Element length

ft /4
0

t

cr

t
0

Figure 2.5 Tension stiffening modelcr
0.75GF/(ftL)

5GF/(ftL)

Figure 2.6 Tension softening model (1/4 model)

Conversely for flexural compression members including the cover concrete, thequadratic stress-strain relationship illustrated in Figure 2.3.that has been proposed by
Vecchio and Collins (1986) is adopted as Equation 2.10 shows.

 `c  . f `c 2( `c /  `0 )   `c /  `0 2

(2.10)In the case of flexural compression members, because the direction of compression
stress is assumed to correspond to the direction of the principal tensile strain, the effectof compressive softening behavior represented by the Equation 2.10 is neglected, thus
 = 1.0.

2.3.2. Tensile stress-strain relationships of concrete
For the flexural tension members prior to cracking, a linear-elastic relationship isapplied, while the tension stiffening curve proposed by Okamura and Maekawa (1991)
defined by Equation 2.11 and illustrated in Figure 2.6 is applied after cracking. The
strain at crack initiation,  cr is assumed to be 0.0001 (100 μ).

 t  f t ( cr /  t ) 0.4(2.11)

where , ft is the uniaxial tensile strength of concrete.

17s
Es /100
fy

Es

Es

s

0

Figure 2.7 Stress-strain relationship of reinforcement under cyclic loading
The diagonal tension members exhibit elastic behavior prior to cracking. However, oncea crack occurs, concrete is assumed to exhibit tension softening behavior. In this study,
softening behavior, expressed by the 1/4-model (Uchida et al. 1991) shown in Figure
2.7 and Equations 2.12 and 2.13, is applied to the diagonal tension members

 1   cr  0.75GF /( f t .L)(2.12)

 2   cr  5.0GF /( f t .L)(2.13)

where, ft is the splitting tensile strength of concrete and L is the element length. Here,the fracture energy of concrete, GF, is assumed to be standard value of 0.1 N/mm.

2.3.3. Reinforcement model
The stress-strain relationship of the reinforcement is expressed as an elasto-plastic
model under monotonic loading. As shown in Figure 2.7, the stress-strain relationshipof the reinforcement is bi-linear, having a tangential stiffness after yielding of Es /100
(where Es indicates the elastic modulus of reinforcement). The unloading and reloading
paths are also shown. After yielding, the Bauschinger effect which is when the stiffnessof the reinforcement decrease as the stress state moves from tension to compression,
while similar behavior is observed when the stress changes from compression to tension.
is incorporated into the analysis using the model proposed by Fukuura et al. (1997).

18

2.4. Outlines of 3D Lattice Model
The discretization of a RC structure using a 3D model allows the consideration ofgeometries and loading conditions that cannot be expressed by a 2D model. In this study,
the extension from the 2D lattice model to the 3D lattice model presented is based on
the concept of the conventional 2D lattice model which has been proved be previous
studies that can appropriately predict the 2D response of RC structural members
subjected to the monotonic or reversed cyclic loading (Niwa et al. 1995, Miki et al.
2003a, 2003b).
The detailed development of the 3D lattice model for RC columns is described by Mikiand Niwa (2004) and furthermore by Miki (2004). Since in the lattice model, the shear
resisting mechanism is divided into arch action and truss action, to represent the truss
action, it is assumed that 3D space is comprised of an orthogonal coordinate system thatis defined by three planes, such as x-y plane, y-z plane, and z-x plane. Two crossed truss
members are located on each truss plane so that unit element consists of 12 truss
members in six truss planes as shown in Figure 3.1 (a). In each truss plane, the in-plane
2D constitutive law of concrete, with the consideration of the softening of compressive
strength of diagonally cracked concrete depending on the transverse tensile strain(Vecchio and Collins 1986).
Assumption of plane stressb(1-t)/b(1-t)/2dnd

d: Effective depth
nd

d

d

y-z
plane

md

x-y

z-x
plane

md

b
bt

b: Width of cross section

y

y
z

x
Truss part

Truss members

Arch partx
Arch members(b) 3D lattice model(a) 2D lattice model

Figure 3.1 Discretization for 2D and 3D lattice models (Miki 2004)
19

b
btb

dtd

Truss partdArch partyzx

Figure 3.2 Partition of cross section in 3D lattice model (Miki 2004)
In order to represent the arch action in the 3D lattice model for RC column, four archmembers are arranged by connecting between the top and the bottom of the column at
the opposite corners. The resisting mechanism of a RC column subjected to one certain
load consists of two arch members crossing each other. The stiffness of these archmembers is assumed to be equivalent to one of two arch members in the 2D latticemodel. Two pairs of arch members are incorporated into the model symmetrically.
When lateral load is applied on the RC column from the diagonal direction of thesection, it is assumed that the corner-to-corner arch action (from the loading point to thebottom of a column at opposite corner) inside the RC member is idealized as acompressive strut.
The value of t is defined by the ratio of the arch part width to the cross-sectional widthof a RC member in the 2D lattice model as mentioned previously. Assuming the global
stiffness of 3D lattice model to be equivalent to that of 2D lattice model, thecross-sectional area of the arch member can be identified. The schematic representation
about the division of cross-section of the 3D lattice model is shown in Figure 3.3. Here,the ratios of the arch part width to the width, b and the depth, d in the cross section of
the column are defined by tb and td, respectively.
With the determination of value of t in the 3D lattice model, both the cross-sectional
width of the column and the cross-sectional depth of the column are evaluated inpreliminary analysis. According to the 2D lattice model, the values of tb and td in the 3D
lattice model are determined based on the theorem of minimization of the total potential
energy.

20

P
0

nd
P

d

P
0

u
Equivalent in termsof global stiffness ofstructural system

z

d

u
md

md

y

nd
P

y
z

x

x

(a) 2D lattice model(b) 3D lattice model

Figure 3.3 Arch members in 2D and 3D lattice model (Miki 2004)
2.4.1. Configuration of elements(a) Concrete Elements
Arch members
The arch members assumed in the 2D lattice model and 3D lattice model are illustrated
in Figure 3.3. The global stiffness of 2D lattice model and 3D lattice model arecompared with each other, in that way the cross-sectional areas of arch member arepresented in Equation 3.1 and Equation 3.2 as follows:1  m2  n2 Aarch3 D  2
 1 m

3/ 2

 Aarch2 d

(3.1)

1
Aarch2 D  bt b .dt d . sin 
2

(3.2)

where Aarch-3 is the cross-sectional area of arch members in the 3 lattice model, arch-2D isthe cross-sectional area of arch members in the 2D lattice model. The values of m, n aredetermined in which md and nd correspond to the width of cross section and the heightof the column in the model, respectively. Here, d is the effective depth of cross-section.
In the modeling, the height of the lattice model is not always equal to the height of thecolumn because the horizontal or vertical distance of two adjoining nodes is determinedbased on the half of the effective depth d. Hence, the height and the width of the 3D
lattice model are set on the dimensions of the structure comparatively close to actual
dimensions.

21

d

d

d

d

d

d

y
z

(a) x-y truss plane(b)y-z truss planex(c) z-x truss plane

Figure 3.4 Truss planes in 3D lattice model (Miki 2004)
Truss members
In the 3Dlattice model the horizontal or vertical distance of two adjoining nodes is equal
to 0.5d, and then the thickness of the truss members on the side view of the column isequal to 0.5d⋅sin45°. In order to represent the truss mechanism along the principal
direction in 3D space, it is assumed that the cross-sectional area of truss members inside
a RC column is a half of that at the surface of the column. The cross-sectional areas oftruss members in x-y plane, y-z plane, and z-x plane illustrated in Figure 3.4 byhatching are expressed as the following equations (Equations 3.21 to 3.23).

Atruss xy b ( 1  tb ) d sin 452
2

(3.21)h( 1  t d ) d sin 4522
a d
Atruss zx  sin 452n 2Atruss yz 

(3.22)(3.23)

where, Atsuss-xy, Atsuss-yz, and Atsuss-zx are the cross-sectional areas of truss members in x-yplane, y-z plane, and z-x plane, respectively. b, h, and a represent the cross-sectional
width, the cross-sectional height, and the shear span of the column, respectively.

22

1 :2

ns

:1

nl

L

on
irect i
d
g
n
oadi

As1

1

:
:2

1

As3
As2

As1
As1

As2
As3

As1

Lattice modeling(a) Cross section of a column(b) 3D lattice model

Figure 3.5 Arrangements of longitudinal and transverse
reinforcements with single layer (Miki 2004)(b) Reinforcement Elements
For longitudinal reinforcement and transverse reinforcement, horizontal and verticalmembers are used. Each reinforcement member is determined based on the actual crosssection and actual location of reinforcements. Figure 3.5 shows the example of thearrangements of longitudinal and transverse reinforcements at the single layer in which
there are no intermediate ties. Here, the longitudinal reinforcement is divided into eightelements in one layer as illustrated in Figure 3.5 (b). It is assumed that thecross-sectional areas of reinforcement along the side perpendicular or parallel tolaterally loading direction are divided at a ratio of 1:2:1 (Figure 3.5 (a)). The
cross-sectional areas of members of longitudinal reinforcement are expressed as thefollowing equations (Equations 3.24 to 3.26).
n n
As1   l  s  1  Al
4 4
n
As 2  l  Al
2

As 3 

(3.24)(3.25)ns Al
2(3.26)

where, As1, As2, and As3 are the cross-sectional areas at the corner of section, at the
middle of tension extreme fiber, and at the middle of the side parallel to laterallyloading direction, respectively. Similarly, Al is the cross-sectional area of longitudinal
reinforcement. Here, nl and ns are the number of longitudinal reinforcement along with23

the side perpendicular and parallel to laterally loading direction, respectively.
In the lattice model, the transverse reinforcement is arranged at intervals of 0.5d
throughout the model uniformly so that the actual transverse reinforcement ratio equals
the model transverse reinforcement ratio. The transverse reinforcement ratio, rw of an
actual RC structural member is calculated by the following equation:

rw 

Aw
bw s

(3.27)

where Aw is the cross-sectional area of a couple of transverse reinforcements, bw is thewidth of the cross section of the RC member, and s is the transverse reinforcement
spacing.

2.5. Nonlinear Analysis Procedure
The nonlinear analysis procedure is done based on computer programs that have beendeveloped to facilitate the process (Miki et al.2004). The static analysis case isperformed for a monotonic and cyclic loading, and the dynamic analysis is done
assuming that the mass equivalent to the self-weight of the structure is distributed over
all the nodal points. Moreover it is assumed that a concentrated mass having the weightof superstructure applies uniformly on the nodal points at the top of the structure.
The numerical procedure implemented in the computer program is explained below.
The equations of motion are formulated to satisfy the equilibrium condition for the
structure. Then, prior to time integration, nodal displacements of the lattice model areconverted into those in the generalized coordinates by using the mode shape vector. The
mode shape vector of vibration is obtained as the solution for the free vibration
equations neglecting the damping. The mode of vibration can be obtained by solving the
Eigen problem. In this study, the subspace iteration method is used to solve the Eigen
problem.
The stiffness matrix, K can be obtained from the tangent stiffness considering thematerial nonlinearities of concrete and reinforcement. The damping is assumed to be
proportional to a combination of the mass and the stiffness matrices, so-called theRayleigh damping. However, this damping formulation has no physical meaning and24

may lead to the damping with unexpected vibration mode shapes. From a previous
studies (Hilber et al.1977, Sing et al.1991), it was pointed out that when the factor in the
Newmark method,  was given as the value that was present, and consequently it waspossible to attain unconditionally stability and a favorable energy dissipation property if  1 / 4  0.52 . Therefore, in the analysis, it is assumed that the viscous damping isneglected (h = 0). In addition, the numerical damping of the Newmark method withfactors   0.36 and   0.70 is used as time integration (committee 311 2002). Herea time interval is set as 0.01 sec. Moreover, since the nonlinear responses appear when
RC structures are subjected to large ground motions, it is necessary to iterate thecalculation until a sufficiently converged solution is obtained. In this study, theNewton-Raphson iteration method is used to iterate until an adequately converged
solution is obtained.
In order to check the convergence, the out-of-balance force and the energy increment
are compared with each initial value during the iteration. The convergence tolerances
for the out-of-balance force and energy are set at 0.001 and 0.01, respectively(Miki.2004).

2.6. Cyclic Analysis of RC Columns Using 2D Lattice Model
In order to demonstrate the applicability of 2D lattice model , analysis of RC column instatic case under cyclic loading is performed and compared the analytical results toexperimental results on RC column experimentally loaded laterally and expected to faileither in flexure of shear. An analysis previously carried by Miki (2004) is here
presented to demonstrate the applicability of the 2D lattice model.

2.6.1. Outline of experiment and analysis
The experiment was carried out by Takemura et al. (1997) on RC bridge piers subjected
to static reversed cyclic loading are analytical target. The specimen and arrangement ofreinforcement are illustrated in Figure 4.1. The specimen is a cantilever RC bridge pierwith a cross section of 400 mm by 400 mm. The reversed cyclic loading was applied bycontrolled horizontal displacement at a point 1,245 mm above the base of the pier. The
uniaxial compressive strength of concrete was 35.7 MPa, yield strength of longitudinal
reinforcement is 363 MPa and nominal diameter SD295 D13 and yield strength oftransverse reinforcement is 368 MPa and nominal diameter SD295 D6.
25

SD295
D6

5@64
=320
D6@70

5@64
=320

SD295
D13

400

400

19@70 = 1330

Loading point

1245

2030

400

20 D13600

Cross section
Unit: mm1000

Figure 4.1 Specimen details and test setup (Takenuma et al. 1997)
In the experiment, the displacement amplitude was increased stepwise in the incrementsof nδy (n = 1, 2, 3…) at each cyclic loading step. Here, δy is the lateral displacement at
the initial yielding of longitudinal reinforcement at the bottom of the pier and is taken asδy= 6mm. During the test, a constant axial compressive load of 156.7 kN was applied at
the top of the pier; this is equal to an applied axial compressive stress of 0.98 MPa.
The specimen subjected to cyclic loading is analyzed by the 2D static lattice model. In
Figure 4.2 analytical model is presented. To simulate RC piers subjected to reverse
cyclic loading, the flexural compression members and flexural tension members are
assumed to have the same cross-sectional area. In addition, since the specimen is acantilever RC pier, two intersecting arch members connecting the loading points at the
top of the pier and the opposite pier-footing connections are provided. Here, from theresults of pre-analysis already described, the value of t is obtained as 0.20. The applied
axial compressive load at the top of the pier is uniformly distributed over the top threenodes.

2.6.2. Lateral force-displacement characteristics of RC column
The lateral force-lateral displacement relationships obtained in the experiment and static
lattice model analysis are shown in Figure 4.3. The experimental result shows that thelongitudinal reinforcement initially yields on the flexural tension side at the bottom of
the RC bridge pier.

26

Constant axial
force 157kN

Lateral force

Arch members

Diagonal concrete
memberFigure 4.2 Static lattice model
As the lateral displacement increases gradually after reversing the loading direction, thelongitudinal reinforcement behaves plastically and deforms laterally outwards in aprocess referred to as buckling. Ultimately, the lateral load-lateral displacement curve
reaches the post-peak region accompanied by the buckling of longitudinal reinforcement
and the spalling of cover concrete. In the analytical result (shown in Figure 4.3b)), thebehavior of the RC bridge pier is found to be close to the experimental result. The
comparison of two results indicates that the analytical method is applicable to the

200

200

100

100

0
-60

-30

0

30

60

-100

Lateral force (kN)

Lateral force (kN)prediction of the initial stiffness, the load carrying capacity, and the cyclic behavior of
RC bridge piers after the yielding of longitudinal reinforcement. The behavior can be
successfully predicted. However, it is also confirmed that further softening behavior in
the post-peak region is not properly predicted by the analysis.

0
-60

-30

0

30

-100
-200
Lateral displacement (mm)-200
Lateral displacement (mm)(a) Experiment(b) Analysis
Figure 4.3 Lateral force and lateral displacement relationships
2760

2.7. Static Analysis of RC Columns Subjected to Bilateral Loading Using 3D
Lattice Model
The 3D lattice model has been developed based on the 2D lattice model as previously
stated. One of the new and key features of analysis in 3D space is that the assumption ofplane stress is no longer used and therefore the bilateral loading of RC columns can beused to verify the expansion of scope of 3D lattice model from 2D case. In that sense, toshow the consequent extension from 2D analysis to 3D analysis using the lattice model,
static analysis of RC columns subjected to bilateral loading is performed. The analysis
previously carried out by Miki (2004) in his work is used.

2.7.1. Outline of experiment and analysis
A set of experiments were carried out by Kawashima et al. (1991, 1993) on RC bridgepiers subjected to static reversed cyclic loading. In these experiments in order toinvestigate the effect of biaxial loading, the test was conducted for RC bridge piers
subjected to bending from the diagonal direction of the section. The tests were for thesquare sectional piers. The dimensions and reinforcement arrangement of the specimen
are illustrated in Figure 4.4 and loading directions in Figure 4.5.
The piers were three RC bridge cantilevers with a square cross section of 500 mm ×
500 mm. All reinforcing bars had a minimum of 35 mm of concrete cover. The diameterof longitudinal reinforcing bars was 13 mm, and consequently the longitudinal
reinforcement ratio was 2.03 %.
Lateral force

Cross section
500

2500

500
Longitudinal
reinforcement: D13

752.5

Transverse
reinforcement:φ 9@250

Unit: mm
1600

Figure 4.4 Details and arrangement of specimen
28




P10 ( = 0)P33 ( = 30)P34 ( = 45)Figure 4.5 Loading condition
The transverse reinforcements were 9 mm diameter round bars with 250 mm spacing.
Hence, the transverse reinforcement ratio was 0.10 %. The longitudinal reinforcement inall piers had the nominal yield strength of 295 N/mm2, while the transverse
reinforcement had nominal yield strength of 235 N/mm2. The average compressive
strength of concrete was 31.3 N/mm2 in the column P−10 and 39.8 N/mm2 in columns
P−33 and P−34. The piers had an identical dimension and arrangement of reinforcement,
while the loading direction is different each other. In the column P−10, the lateral load
was provided along the principal axis, while in the columns P−33 and P−34, the load
was provided in the direction of a diagonal of the pier cross section. For the specimensof P−33 and P−34, the loading stub at the top portion of the pier was inclined from theprincipal axis as were 30 and 45, respectively.
In the experiment, the displacement amplitude was increased step wisely in incrementsof n⋅δy (n = 1, 2, 3…) at each loading step. Here, δy was determined as the lateraldisplacement when the measured strain of the longitudinal reinforcement at the bottomof the pier firstly reached the yield strain of 1,800μ, and was taken as δy= 13 mm in thecolumn P−10. The yield displacement was used to control the displacement in bothcolumns P−33 and P−34. The loading cycles were controlled of ten each cycle at thesame amplitude.

2.7.2. Analytical results and discussion
The experimental lateral load-lateral displacement relationships at the top of the pier foreach specimen are shown in Figure 4.6. In all specimens, similar behavior wasobserved while the development of damage was significantly different. However inanalysis For the specimen P−10, the analytical and experimental results are found toshow the good agreement with each other.

29

200

150

Experiment

150

Analysis100

P33 ( =100

P33 ( = 30)

Lateral load (kN)

Lateral load (kN)

200

50
0
-50
-100
-150

-50
-100

-200
-90

-60

-30
0
30
60
Lateral displacement (mm)

90

-90

200

-60

-30
0
30
60
Lateral displacement (mm)

90

200

150

Experiment

150

Analysis100

P33 ( =100

P33 ( = 30)

Lateral load (kN)

Lateral load (kN)

0

-150

-200

50
0
-50
-100
-150

50
0
-50
-100
-150

-200

-200
-90

-60

-30
0
30
60
Lateral displacement (mm)

90

-90

200

-60

-30
0
30
60
Lateral displacement (mm)

90

200

150

Experiment

150

Analysis100

P33 ( =100

P33 ( = 30)

Lateral load (kN)

Lateral load (kN)

50

50
0
-50
-100
-150

50
0
-50
-100
-150

-200

-200
-90

-60

-30
0
30
60
Lateral displacement (mm)

90

-90

-60

-30
0
30
60
Lateral displacement (mm)90Figure 4.6 Lateral load-lateral displacement relationships obtained by thecyclic loading test and static 3D lattice model
Matching with experimental observation, the buckling behavior of longitudinal
reinforcement is predicted at the lateral displacement of more than 55 mm. However,the divergence of analytical results from the experimental results is observed at the large
deformation range. This is because that the fracture of longitudinal reinforcing bars dueto the low-cycle fatigue is not incorporated in the analysis.
In the specimen P−33, the analytical load-displacement relationship is also found to beclose to the experimental result. It is found that the flexural ductility of these square30piers subjected to bending from the direction of section diagonal is almost similar tothat for bending from the direction of a principal axis of the section. In both theexperiment and analysis, the slight increase in the load carrying capacity is observed.
After the analytical displacement exceeded around 40 mm, the gradual decrease in thelateral force can be observed in the post-peak region of load-displacement relationship.
That is similar to the experimental observation. This point is corresponding to thecompressive softening of concrete at the base of the pier and the initiation of thebuckling of longitudinal reinforcement.
For specimen P − 34, it can be observed that the experimental and analyticalload-displacement curves and its envelope curves are very close to each other. The
analytical and experimental comparison of the load carrying capacity of RC piers
indicates that there is slight difference with the load carrying capacity if the lateral load
is applied from the direction of a principal axis of the section. The tenth cycle for thesame displacement produced nearly the same response and the degradation of stiffness
cannot be observed until the lateral displacement reached at 50 mm.

31

References of Chapter 2
Fukuura, N. and Maekawa, K. (1997): Computational model of reinforcing bar underreversed cyclic loading for RC nonlinear analysis, Journal of Materials, Concrete
Structures and Pavements, JSCE, Vol. 564, No.35, pp.291-295.
Kawashima, K. (1991): Effect of Oblique Excitation on Dynamic Strength and Ductilityof Reinforced Concrete Bridge Piers, Civil Engineering Journal, Public Works
Research Center (PWRC), No.33, Vol.8, pp.27-33. (in Japanese)
Kawashima, K., Hasegawa, K., Nagashima, H., Koyama, H., and Yoshida, T.
(1993):Seismic Design Method of Reinforced Concrete Bridge Piers Based on Dynamic
Strength and Ductility, Journal of Research PWRI, No.190, pp.51-57, May. (inJapanese)
Mander, J. B., Priestley, M. J. and Park, R. (1988): Theoretical stress-strain model for
confined concrete, Journal of Structural Engineering, ASCE, Vol. 114, No(8), pp.
1804-1826.
Miki, T. (2004): Nonlinear Analysis of Reinforced Concrete Structures Subjected to
Seismic Loading by Using Three-dimensional Lattice Model, Doctoral thesis, Tokyo
Institute of Technology, February.
Miki, T. and Niwa, J. (2004): Nonlinear Analysis of RC Structural Members Using 3D
Lattice Model, Journal of Advanced Concrete Technology, JCI, Vol. 2, No.3,
pp.343-358, October.
Miki, T., Niwa, J., and Lertsamattiyakul, M. (2003a): Earthquake Response Analysis for
RC Bridge Piers Considering Reinforcement Buckling Behavior, Journal of Materials,
Concrete Structures and Pavements, JSCE, No.732/V-59, pp.225-239, May. (inJapanese)Miki, T., Niwa, J., and Lertsamattiyakul, M. (2003b): Numerical Evaluation of Seismic
Performance of Reinforced Concrete Bridge Piers Using Dynamic Lattice Model,
Concrete Library of JSCE, Vol.41, pp.49-64, June.

32

Niwa, J., Choi, I. C., and Tanabe, T. (1995): Analytical Study for Shear Resisting
Mechanism Using Lattice Model, Concrete Library of JSCE, No.26, pp.95-109,
December.
Takemura, K. and Kawashima, K. (1997): Effect of Loading Hysteresis on Ductility
Capacity of Reinforced Concrete Bridge Piers, Journal of Structural Engineering, JSCE,
Vol.43A, pp.849-858, March. (in Japanese)
Vecchio, F. J. and Collins, M. P. (1986): The Modified Compression Field Theory for
Reinforced Concrete Elements Subjected to Shear, ACI Journal, Vol.83, No.2,
pp.219-231, March/April.

33

3. DEVELOPMENT AND APPLICATION OF MULTI-DIRECTIONAL
POLYGONAL 3D LATTICE MODEL3.1. Introduction
Circular cross-sectional RC columns are favored for bridge piers, because of relative
simplicity of construction as well as omnidirectional strength characteristics under windand seismic loads (Ang et al. 1989). Under those conditions, some very important issues
arise: the first one is related to the ability of the column shear resistance considering thatthe design procedures and formulas that are adopted are based on experimental work
performed essentially of rectangular shaped cross-sections. Therefore it is important toconsider the particular case of circular cross-sectional geometry in RC columns.
Based on previous studies (Miki and Niwa 2004), the 3D lattice model has shown
capabilities of prediction of shear behavior of RC structural members with relative
simplicity of analytical procedure. Because of that, an analysis mechanism has beendeveloped for circular cross-section RC columns using the 3D lattice model based on
the concept of a more realistic multi-directional polygonal discretization. The 3D latticemodel for a circular column is named multi-directional polygonal 3D lattice model(Simão and Miki 2015).

3.2. Modeling and Geometry of Members
In the multi-directional polygonal 3D lattice model, the discretization of the circularcolumn from solid concrete to the 3D lattice model is performed so that the actual
cross-sectional diameter, D of the analytical model corresponds to that of the original
target which means that target diameter defines mesh size and model height, or
geometry of the model.
The arch and truss discretization is shown in Figure 3.1. The arch action is defined byfour arch members, as shown in Figure 3.2, connected from the loading point toopposite bottom of the column which is representative of internal stress flow. This islargely a simplification from the general 3D lattice model case where it is assumed thatthe compressive portion in arch action is sufficiently represented in the pair of archmembers. Along the principal axis, after the loading is reversed, the direction of thecompressive stress changes to the orthogonal direction, this is a fundamental for the
representation of symmetrical pairs of arch members. The representation of the truss
action, schematically represented in Figure 3.3, is performed on a 3D space.
34

Loading
A
Truss
A`

mD

Arch

Loading

D

Arch
Part
mD

Truss
Part
D
Dt

A-A`

D×(1-t)Fig.3.1 3D lattice model for a circular column

In the 3D space, the representation of the truss elements is divided in three parts asshown in Figure 3.3, which are inner diagonal members (IDM) in Figure 3.3 (a),surface diagonal members (SDM) in Figure 3.3 (b) and diagonal members in transverse
direction (DMT) in Figure 3.3 (c), respectively. The modeling of diagonal members is
preceded by the distribution of sixteen peripheral nodes for every horizontal layer. The
distance between two successive layers is equal to half of the diameter i.e. 0.5D, thein-plane nodal positions are set according to their polar coordinates defined by x = r
cosφ and y = r sinφ in an orthogonal system, where r is the radius and φ is the internalangle defined by a triangle formed by joining two successive node to each other and thecenter node. In the case of inner diagonal members, each node at the surface is directly
LoadArch members

Truss members

Vertical and horizontal
members

Figure 3.2 Schematic Diagram of multi-directional polygonal 3D lattice model
35

Inner diagonal members (IDM)

Surface diagonal members (SDM)(a)

Diagonal members in transverse
direction (DMT)(b)
Figure 3.3 Representation of the truss elements
connected to the lower or upper node in the central group on nodes.

(c)

For the surface diagonal members the nodes are connected successively along thesurface area and in the case of diagonal members in transverse direction as shown in
Figure 3.3 (b), sixteen peripheral nodes are connected to a center node for every layerof nodes as shown in Figure 3.3 (c), and every node is connected to the second
following node. In the lattice model, the height of the analytical model does not always
correspond to the height of the target, but rather the closest dimension. This is due to thefact that distance between two nodes in vertical direction is fixed as half of the diameter.
Longitudinal reinforcement is represented as vertical reinforcement member along thesixteen nodes defining the geometry of the model. Regarding transverse reinforcement,
it is represented in the form of horizontal reinforcement members uniformly distributed
at intervals of 0.5D throughout the model as the intervals of arrangement are not taken
into account. In the model it considered that concrete and steel are perfectly bonded andno bond degradation is observed for simplicity in analysis.

3.3. Cross-Sectional Area of Lattice Members
The process of determination of cross-sectional area of arch and diagonal members in
the multi-directional 3D lattice model is performed considering the particularities of thecross section of the analytical target. As previously stated, the diameter of the model isinvariant for the definition of geometry of the analytical model. Previous researches(Pique and Burgos 2008, Merta and Kolbitsch 2006) refer to the fact that under severe
seismic loading inelastic response and cracking of concrete is observed. Under thoseconditions, the shear resisting area of concrete and the structural stiffness conditions
36

will be affected. In order to take those conditions into account in analysis the notion ofanalytical diameter is introduced in order to facilitate the calculation of thecross-sectional area of lattice members considering the effects of such conditions andgenerally improve the performance of the analytical model. With that, analytical
diameter is the defined as the diameter obtained from analytical conditions to calculate
the cross-sectional area of arch and truss members.
With that, the transformation from solid concrete to truss and arch members analogy
tries to offer a more realistic discretization of RC columns. In the 3D lattice model,material nonlinearity is already considered; however in this study reduction factors areused to better consider material nonlinearity, especially reduction of cross-sectional areaof trusses to account the effect of cracking. Based on that, in this research two different
approaches are studied: the first one is the shear resisting capacity of concrete as
discussed by Merta and Kolbitsch (2006). In their work they derived the effective shear
area of circular cross-section columns purely analytically. The ratio of the effective
shear area to gross section area was expressed as a function of the neutral axis depth fordifferent values of the concrete cover. For a typical value of neutral axis depth, it was
shown that the effective shear area ranges between 0.6 and 0.8 times the sections grossarea depending on the depth of the concrete cover.
A second approach used to consider the change is mechanical properties after crackingof concrete is the degradation of stiffness EI. Pique and Burgos (2008) extensively
studied the effective rigidity of reinforced concrete elements in seismic analysis andobserved in their work that stiffness degradation can be observed along the element
length after cracking of concrete. However it is not practical to evaluate the stiffness
degradation in various cross-sectional portions individually, because of that reasonableaverage reduction factor values must be adopted. Reduction factors of moment of inertia
to consider the averaged effect of stiffness degradation ranging from 0.4 to 0.7 times
gross moment of inertia are considered in the pre-analysis in this study based on thework presented by the previous study by Pique and Burgos (2008).

3.3.1. Cross-sectional area of arch members
For the determination of the stiffness matrix in the arch members, it is assumed that asingle arch member is representative of the stress flow for analysis purposes. The
equivalence of global stiffness of structural systems in 2D and 3D is assumed. The
37

geometrical properties of the circular cross section column are considered the same in
the transverse plane directions.
The structural stiffness matrices for the 2D and 3D lattice model, K2D and K3D, areexpressed using the direct stiffness method as follows:1 m 0  1  m
m2 0  m  m2

0 00
K2D =  2 D 
2
1 m 
1m
 sym.
m2



0

0
0

0
0

0

1 m
1 1  m
2m
m  m  m2


1 1  m2

K3D = 3 D 2 
2m 
1
m
 sym.
m2



(3.1)

1 

 m
1 

1 
m

1 

(3.2)

In this case, 2D 

Ec Aarch2 Dl arch2 D

 3D 

Ec Aarch3 Dl arch3 D

Ec Aarch2 D(3.3a)

D 2 ana (1  m 2 )Ec Aarch3 D(3.3b)

D 2 ana (2  m 2 )where, Aarch-2D and Aarch-3D are representatives of the cross-sectional areas of archmembers for both the 2D and 3D lattice models. larch-2D and larch-3D are the lengths ofarch members in the 2D and 3D lattice models. Ec is the Young’s modulus of concrete.
m is set so that that mDana corresponds to the height model, furthermore D is thediameter of cross section of the column.
Regarding the cross-sectional area of an arch member of concrete in the 3D latticemodel, the equivalence in global stiffness is used for the computation of the relationshipbetween the arch members in 2D and in 3D assuming a single representation of thestress flow in the arch part and expressed by Equation 3.4.
38

3

 2  m2  2
  Aarch2 D
Aarch3 D  2 
 1 m (3.4)

Aarch2 D  D 2 ana t  sin (3.5)

where, in the 2D case it is assumed that the length of the arch band corresponds to thefull length of diameter for simplicity, and  is the inclination of the arch member in the
2D lattice model.

3.3.2. Cross-sectional area of truss members
The determination of the truss action in the multi-directional polygonal 3D lattice model
is performed based on the omnidirectional nature of the circular cross section. Therefore,
the cross sectional area of diagonal truss members will be determined according to theformulas shown in Equations 3.6, 3.7 and 3.8, respectively. Because of thehigh-contribution to stiffness that the IDM have in response, a stiffness adjusting factor
β empirically determined in pre-analysis between 0.5 and 0.9 is necessary in thecalculation of the cross-sectional area of the IDM.

D ana( 1  t ) Dana
sin 452
2(3.6)

Dana ( 1  t ) Dana
sin 452
2(3.7)

a Danasin 45
2m 2

(3.8)

AIDM  

ASDM 

ADMT 

where AIDM, ASDM and ADMT, represent the cross-sectional areas of the inner diagonal
members, surface diagonal members and diagonal members in transverse direction in
the 3D space. Here a represents the shear span, Dana is the analytical diameter, m is set
so that that mDana corresponds to the height model.

393.4. Parametric Analysis of Relationship between Arch and Truss Members
3.4.1. Introduction
In the modeling of RC structures using the multi-directional polygonal 3D lattice model,
a certain level of uncertainty can be observed due to the interaction between the archand truss analogy and the use of reduction factors as well an multiple other analogies. In
order to extend the arch and truss analogy to a 3D case using the multi-directional
polygonal 3D lattice model, it is important to understand first how much the response of
the analytical target is affected by using different relationships between the arch andtruss members through the value of t. From that a clear tendency in behavior can be
grasped and the use of reduction factors can be done much more effectively.

3.4.2. Effect of value of t
In order to investigate the effect of value of t fundamental 2D lattice model analysis ofrectangular cross-section RC column is presented. The arch and truss analogy used in
the multi-directional polygonal 3D lattice model is largely an extension of the approach
proposed in previous studies (Miki and Niwa 2004), therefore using a 2D approachallows a simple and clear validation of the assumption. The analysis presented has been
conducted by Miki (2004) and is here used. The analytical target is shown in Figure 3.4.
The parameters of the column are as follows: fc’ = 27.0 N/mm2, fy = 380 N/mm2, Es = 72
kN/mm2, fwy = 360 N/mm2, and Esw =175 kN/mm2. The longitudinal and transverse
reinforcement ratios are set as 2.38 % and 0.056 %, respectively. The Constantcompressive axial force of 79.7 kN is applied at the top of the column.

35 10@38 35

D6@250

1500

D13
D6
D10

Cross section

400

2090

Loading direction
300
4@58
35
35

Lateral load

450

190

Constant
axial load

Unit: mm
1500

Figure 3.4 Experimental setup (Miki 2004)
40

0
-50

-100
-150
-100

Exp.
Ana.
-50

0

50
0
-50

-100

-150
100
-100

50

Lateral displacement (mm)

-150
-100

Exp.
Ana.
-50

0

50

100
50
0
-50

-100

Exp.
Ana.
-50

0

0
-50

Exp.
Ana.
-50

0

50

Lateral force (kN)

50

100

Exp.
Ana.
-50

0

50

100

t = 0.6

50
0
-50

-100

-150
100
-100

50

Exp.
Ana.
-50

0

50

100

Lateral displacement (mm)150

t = 0.7

Lateral displacement (mm)100

Lateral displacement (mm)

150

-150
-100

-50

150

Lateral displacement (mm)

-100

0

Lateral displacement (mm)t = 0.5

-150
100
-100

50

50

-150
100
-100

Lateral force (kN)-50

Lateral force (kN)

Lateral force (kN)

0

-100

Lateral force (kN)0

t = 0.3-100

150

t = 0.4

50

100

-50

100

Lateral displacement (mm)

150
100

Exp.
Ana.

Lateral force (kN)

50

100

150

t = 0.2150

t = 0.8

50
0
-50

-100

-150
100
-100

Exp.
Ana.
-50

0

50

Lateral force (kN)

100

150

t = 0.1

Lateral force (kN)

Lateral force (kN)

150

100

t = 0.9

50
0
-50

-100

-150
100
-100

Lateral displacement (mm)

Exp.
Ana.
-50

0

50

100

Lateral displacement (mm)Figure 3.5 Effect of value of t
The analytical results are shown in Figure 3.5. In his calculation, Miki (2004)
determined the value of t based on the theorem of minimization of total potential energy
as 0.2. However by extending the limits from 0.1 to 0.9 it is visible that the smaller
values show a higher energy dissipation capacity up to the large deformation range.
With the increase of t the maximum force is slightly increased. Based on that, it ispossible to assume that in the multi-directional 3D lattice model, any changes in therelationship between arch and truss part will likely affect the energy dissipation capacity
based on visible changes in the maximum displacement while changes in the maximum
force are marginal in their contribution. In the modeling of RC structures using the
proposed analytical model it becomes then necessary to pay attention at all steps to theenergy dissipation capacity. It remains to be studied to real arch effect in circular RC
columns, however this analysis gives a valuable benchmark to be used in circularcolumns in conjugation with reduction factors.
41

3.5. Linear-Elastic Analysis of Circular-Cross Section RC Columns
3.5.1. Introduction
In order to verify the applicability of the two analytical cases previously discussed, one
related to the hypothesis of reduction of shear area of concrete and a second one related
to degradation of stiffness EI, linear-elastic analysis will be conducted on a circularcross-section column and the analytical results compared to a linear-elastic curve
response determined by the elasticity theory. Here the analysis is centered in stressconditions that will not produce any yielding in the target column. The analytical
relationship that has been applied the determination of the load-displacement
relationship in elastic range is presented in Equation 3.9.

d

Pl 3
3EI

(3.9)

where d is the displacement, P represents the load, l represents the length of the column,
E is the Young’s modulus and I the moment of Inertia.

3.5.2. Outline of analytical target
The analytical target is a circular cross sectional column tested by E-Defense. The targetof this study is the C1-5 column (Kawashima et al.2010). The detailed description of thetarget of analysis, including arrangement of reinforcement is shown in Figure 3.4. The
specimen C1-5 is a cantilever circular column with diameter 2000 mm. The heights ofthe column and footing correspond to 7500 mm and 1800 mm respectively. The
longitudinal and transverse reinforcement have a nominal strength of 345 MPa and thedesign concrete strength of 27 MPa. Sixty four deformed 35 mm diameter longitudinalbars are presented in two layers, while deformed 22 mm circular ties are set at 150 mm
and 300 mm intervals in the outer and inner longitudinal bars.

3.5.3. Discretization of target
In order to analyze the target presented in Figure 3.6, two analytical cases are set for
static analysis. For analytical case one (AC-1) and analytical case two (AC-2) the
multi-directional polygonal 3D lattice model is used. In the modeling for AC-1 andAC-2 geometrical properties defined by the mesh size and height of the models are
precisely the same.
42

1500

D35@36bars

6000
9300

56@150=8400
27@300=8100

D35@36bars
150
100

D22@150mm
2000

1800

Y: Up/Down
Z: Transverse

X: Longitudinal

7000
Unit:mmFigure 3.6 C1-5 specimen details (Kawashima et al.2010)
The geometry of AC-1 and AC-2 is shown in Figure 3.7. The dimension of the truss
members and arch members differ between AC-1 and AC-2. In the AC-1 Dana isobtained from effective shear area of concrete determined to be a value corresponding to
0.7 of the gross area of concrete. In the AC-2, Dana is derived from the flexural stiffness
EI, and corresponds to 0.5 of the gross moment of Inertia. These reduction factor values
were obtained in pre-analysis by calibrating the analysis each time until satisfactoryload-displacement curve was obtained. However the benchmarks used are those referredby the literature (Pique and Burgos 2008, Merta and Kolbitsch 2006). The analytical
diameters obtained for analysis are 1673.32 mm for AC-1 and 1829.38 mm for AC-2.

3.5.4. Analytical results and discussion
The analytical results of the load-displacement for the target column C1-5 are presented
and subjected to discussion for the proposed analytical cases. The results for analyticalcase AC-1 Figure 3.7 (a) and analytical case AC-2 are presented in Figure 3.7 (b). The
analytical results of AC-1 shows acceptable degree of agreement between the analysis
and the linear-elastic theory plot regarding elastic stiffness. The proposed cases capture
the elastic stiffness in elastic phase both acceptably. AC-1 produced a slightly bigger
analytical diameter then AC-2. One of the key issues during the development process ofthe multi-directional polygonal 3D lattice model was a tendency to overestimate plastic
stiffness. These two approaches for all their merits also address this issue.
43

3000

Linear-elastic

3000

2000

2000

Linear-elastic

0
-40

-20

0

20

-1000
-2000

AC-1

40

Load (KN)

Load (KN)

1000

1000
0

-40

Analysis

-20

-1000

0

-2000

AC-2

-3000
Displacement (mm)2040Analysis-3000
Displacement (mm)(a)(b)
Figure 3.7 Load-displacement of AC-1 and AC-2

With the above results it is confirmed the validity of the effective shear hypothesis andthe effective stiffness hypothesis based on comparison with the elastic stiffness obtained
from the elastic theory. Since the results are quite similar, one of the conclusions thatcan be drawn is that both approaches can equally be applied to discretize a circularcross-section column using the multi-directional polygonal 3D lattice model.

3.6. Static Analysis of Circular RC Columns Subjected to Cyclic Loading
3.6.1. Outline of analysis
In order to further confirm the performance of the multi-directional 3D lattice model,
the model was verified by the simulation of RC column that has been subjected to cyclicloading experimentally. The target is a circular cross section column tested named C-29
(Hoshikuma et al. 2013). In previous sub-chapter a discussion on the hypothesis ofeffective shear area and effective flexural area was introduced. One of the conclusions
at that level was that these two approaches could equally be applied in themulti-directional polygonal 3D lattice model. Based on that, a more detailed look will
be presented here on the sensitivity analysis of reduction factor based on degradation of
stiffness. In other words in this study, the stiffness degradation is analyzed considering
the applicability of stiffness reduction factors to represent the effect that a cracked
section has in the actual design and response of RC columns. To perform the analysis,
the focus will be on the degradation of flexural stiffness. In a more specific way,parametric analysis was conducted in order to verify the effective applicability ofmoment of inertia (second moment area) reduction factors. The assessment of stiffness44

degradation induced by seismic damage is verified by the simulation of RC column thathas been subjected to monotonic loading experimentally.
Column C-29 details are shown in Figure 3.8. The cross section is circular withdiameter of 600 mm and column height of 3410 mm, respectively. The compressive
strength and Young’s modulus of concrete are 31.8 N/mm2 and 28000 N/mm2,
respectively. The longitudinal reinforcement of D10 has yield strength of 397 N/mm2
and the transverse reinforcement D6 has yield strength of 397 N/mm2, respectively
The benchmark inertia reduction factors suggested by previous studies range from 0.4 to
0.8 times gross moment of inertia (Pique and Burgos 2008, Paulay and Pristley 1992)
and were considered in pre-analysis. However, during the pre-analysis, the decision wasmade to select three analytical cases, each corresponding to a specific reduction factor
of inertia that presented the most effective demonstration of the effect of inertia
reduction factors. In that way analytical case one denoted as C-29(1) corresponds to aneffective inertia of concrete of 0.8 times the gross moment of inertia of concrete,
analytical case two denoted as C-29(2) corresponds to the effective moment of inertia of
0.6 times the gross moment of inertia, and analytical case three denoted as C-29(3)corresponds to an effective inertia of 0.4 times the gross moment of inertia. The
obtained effective moments of inertias were the used to calculate analytical diameterused to discretize the RC column in the multi-directional 3D lattice model. In all
analytical cases the value of t is set as 0.4.

800

6@85=510

145

3410

2610

35@75=2700

D10

40D6@9=360

40
600

700

Unit:mm

1400

Figure 3.8 C-29 specimen details (Hoshikuma et al. 2013)
45

300
200

C-29 (1)C-29 (2)

200

150
100

Load (KN)

Load (KN)

100
0
-150

-100

-50

0

50

100

150

-100

50

0
-150

-100

-50

50

100

150

-100

Experimental

-200

0
-50

-150

Analysis

Experimental

Analysis
-200

-300

Displacement (mm)Displacement (mm)
200
150
100

Load (KN)

50
0

-150

-100

-50

0

50

100

150

-50
-100
Experimental-150
analysis-200
Displacement (mm)Figure 3.9 C-29 load-displacement relationships

3.6.2. Analytical results and discussion
The analytical results presented in Figure 3.9 show that there is a considerable
difference in the load-displacement relationships for the analytical cases. Based on
hysteretic agreement, C-29(3) presents more agreement between the analysis and theexperiment, while C-29(1) presents less agreeability regarding the predicted thehysteresis of the target.
Looking at the results, it is possible to infer that that in the case of AC1 and AC2, thetendency is to underestimate the displacement and overestimate the lateral load. The
case of AC3 shows that actually a more significant reduction in effective inertia is
necessary in analysis, when compared to the literature, to more accurately represent the
response. One of the reasons behind it is related to how reinforcement is treated. In thediscussion about reduction factors proposed, Pique and Burgos (2008) mention that intheir analysis there is no special consideration on the effects of reinforcement ratio,
meaning that reinforcement is treated as a typical reinforcement ratio; i.e. it is not given
special attention to the ratio between confined concrete and the amount of reinforcement.
However, the pre-analysis of the target column C-29 in this study proved that thedominance of concrete part of the column cannot be neglected, and one of the effects isthe need to further decrease the reduction factors in order to account for this effect.
46

In terms of hysteretic energy dissipation capacity, it is valuable to focus first on cases
C-29(2) and C-29(3) because they produce close values of energy dissipation, andcompare them to the energy dissipation produced by the experimental results. The
experimental hysteretic energy dissipation corresponds to 27.5 KN-m while AC2corresponds to 30.2 KN-m and C-29(3) corresponds to 30.5 KN-m. However, looking at
the load-displacement relationships presented in Figure 3.9, it is clear that althoughC-29(2) and C-29(3) present very similar hysteretic energy dissipation, the distributionof energy based on hysteretic response of C-29(3) is far more agreeable with theexperiment than C-29(2). In other words the hysteric response similarity between
C-29(3) and the experiment suggests a much more similar mechanism of energy
dissipation than that of C-29(2) and the experiment. On the other hand the tendency of
C-29(1) is to overestimate energy dissipation capacity. Here the hysteretic energy
dissipation of AC1 corresponds to 47.9 KN-m.

3.7. Discretization Method Analysis of Circular RC Columns
3.7.1. Introduction
The highly nonlinear behavior that reinforced concrete (RC) structures exhibit in theoccurrence of an earthquake is at the top of priorities in the analytical development of
numerical techniques to study seismic behavior. Many techniques are available toperform seismic analysis of RC columns. Columns are especially vulnerable against
lateral loads induced by seismic action result in many times in shear failure as adominant mode of failure. In seismic design of RC columns, the geometry of the section
has a strong influence on the shear capacity of the member. However the majority of thecodes simply assume that the shear capacity of a circular cross section equals thecapacity of an equivalent rectangular section (Merta 2004). With that in mind, it isimportant to evaluate the merits of treating a circular column just as an equivalentcross-sectional area rectangular column against a more realistic discretization of circularcross-section.

3.7.2. Outline of analysis
In order to analyze the applicability of circular-rectangular equivalence against a more
direct discretization, the analytical target is a circular cross section column named C1-5
tested using a shake-table by E-Defense (Kawashima et al. 2013) which has beenpreviously introduced in Sub-chapter 3.4.3 and specimen details in Figure 3.6.
47

600

Acceleration (gal)

400
200
0

0

5

10

15

20

-200
-400
-600
-800

Time (sec)(a) Longitudinal direction

600

Acceleration (gal)

400
200
0
0

5

10

15

20

-200
-400
-600
-800

Time (sec)(b) Transverse direction

Figure 3.10 100% E-Takatori ground motion (Kawashima et al. 2010)

Column C1-5 was designed according to the 2002 JRA Design Specifications ofHighway Bridges (JRA 2002), based on the design response spectrum (Kawashima et al.
The specimen is a cantilever circular column with diameter 2000 mm. The heights ofthe column and footing correspond to 7500 mm and 1800 mm respectively. Column
C1-5 was excited using a near-field ground motion which was recorded at the JR
Takatori Station during the 1995 Kobe earthquake. Taking into account thesoil/structure interaction, a ground motion with 80% of the original intensity of JR
Takatori was imposed as command to the table in the experiment. In the experimental
program, C1-5 has been excited under different conditions, however in this study the
analysis will focus on C1-5(1) which corresponds to the first excitation. The groundmotion corresponding to 100% E-Takatori in longitudinal and transverse directions isshown in Figure 3.10.
In order to analyze column C1-5 two analytical situations have been set. First is theconversion of circular cross sectional shape into an equivalent rectangular crosssectional shape, based on the fact that most design methods for shear and flexure of RC
members are mainly based on the rectangular cross-sectional shape for the analysis.
48

2000

7975.90

8000

1000

886.231772.45

Unit:mm

Unit:mm(b) Analytical models AM-2 andAM-3(a) Analytical model AM-1

Figure 3.11 Analytical models

Upon this, the 3D lattice model as developed in previous studies (Miki and Niwa 2004)is used in an analytical model further addressed as AM-1 shown in Figure 3.11 (a).
The second analysis corresponds to the application of the multi-directional polygonal
3D lattice model, newly developed for circular cross section columns. Here, the shear
capacity of reinforced concrete is considered by assuming that of the actual resistingarea of concrete is reduced in seismic response. It is in that way that an analytical model
addressed as the AM-2 is developed, where an effective area of concrete corresponding
to 0.7 of the gross area of concrete is assumed to determine the cross-sectional area ofthe lattice members.
The reduction in flexural stiffness is considered to perform analysis. Assuming that
there is stiffness degradation in seismic response, an analytical model AM-3 isdeveloped; where in the case effective area moment of inertia corresponding to 0.7 of
the gross moment area of inertia is assumed for the analytical diameter used todetermine the cross-sectional area of lattice model elements. It should be noted that thegeneral geometry of AM-2 and AM-3 is the same as detailed in Figure 3.11 (b).

3.7.3. Analytical results and discussion
The analytical results using analytical models AM-1, AM-2 and AM-3 are shown in
Figure 3.12 and Figure 3.13 respectively, and compared to the experimental results.

49

3000.0
3000.0AM-1

AM-22000.02000.01000.0

Load (KN)

Load (KN)

1000.00.0-100.0

-50.0

0.0

50.0

0.0

-100.0

100.0

-50.0

0.0

50.0

100.0

-1000.0-1000.0-2000.0-2000.0

Experimental

Experimental
AnalysisAnalysis-3000.0-3000.0

Displacement (mm)Displacement (mm)3000.02000.0AM-3Load (KN)

1000.00.0-100.0

-50.0

0.0

50.0

100.0

-1000.0-2000.0

Experimental
Analysis-3000.0
Displacement (mm)(a) Longitudinal direction
2000.02000.0AM-1

AM-21500.01000.01000.0

Load (KN)

-40.0

0.0
-20.0
0.0
-500.0

20.0

40.0

60.0

80.0

Load (KN)

500.0

500.0
-60.0

1500.00.0-100.0-50.0-500.0

0.0

50.0

100.0

-1000.0-1000.0-1500.0-1500.0

Experimental

Experimental-2000.0-2000.0

AnalysisAnalysis-2500.0-2500.0

Displacement (mm)Displacement (mm)2000.01500.0AM-31000.0

Load (KN)

500.0-150.00.0-100.0-50.0-500.0

0.0

50.0

100.0

-1000.0-1500.0
Experimental-2000.0

Analysis-2500.0
Displacement (mm)(b) Transverse direction
Figure 3.12 Load-displacement relationships computed using analytical models
The analytical response using AM-1 shows reasonable agreement with the experiment
before cracking occurs; the initial stiffness is very consistent in both directions, however,
it is in the longitudinal direction that the after cracking behavior differ visible, but whilein the transverse direction the hysteretic response though is agreeable underestimates
50

80.0

Displacement in longitudinal direction (mm)Displacement in longitudinal direction (mm)

80.0
60.0

AM-1

40.0
20.0

0.0
-100.0

-50.0

0.0

50.0

100.0

-20.0
-40.0
Experimental

60.0

AM-2

40.0
20.0
0.0

-100.0

-50.0

50.0

100.0

-40.0
-60.0

Analysis

-60.0

0.0
-20.0

Displacement in tranverse direction (mm)-80.0

Experimental-100.0Analysis

Displacement in tranverse direction (mm)Displacement in longitudinal direction (mm)

80.0
60.0
40.0

AM-3

20.0

0.0
-100.0

-50.0

-20.0

0.0

50.0

100.0

-40.0

-60.0
-80.0
-100.0Experimental-120.0Analysis

Displacement in tranverse direction (mm)Figure 3.13 Displacement relationships computed using analytical models
maximum force as well as displacement. Thus supporting that, the assumption ofequivalence of responses between a rectangular cross sectional member and circularcross section member has large limitation in inelastic phase.
The analysis performed using the multi- directional polygonal lattice model, in bothproposals presented in this paper show that largely better hysteretic agreement isobtained. Looking at the response using AM-2 when consideration for reduction ofshear area of concrete is used, it is visible that the prediction of maximum displacement
and force is reasonably performed, especially in the longitudinal direction. However inthe transverse direction the hysteretic response overestimates the deformation capacity.
On the other hand, AM-3 shows the most acceptable prediction of response whencompared to the other models presented. This suggests that consideration for anequivalent flexural stiffness in the analysis proved more accurate than for shear area ofconcrete. It is nonetheless important to mention that using the multi-directional
polygonal lattice models the initial stiffness is highly estimated.
The reason behind that is the elevated number of elements necessary to describe theelement, when compared to the rectangular lattice model. The consequence of this is51

that at the beginning, regardless of the approach chosen, that is AM-2 and AM-3, themodel consistently has large initial stiffness. This shortcoming should be object ofattention in further research on the topic. A detailed look at Fig.10 allows thecomparison of displacement capacity prediction in longitudinal and diagonal directions.
Comparing the analytical models to the experiment, AM-2 and AM-3 present more
acceptable agreement with the experiment in comparison to AM-1; this is a fundamental
outcome that comes from the more realistic discretization of target that AM-2 andAM-3 propose.

3.8. Cumulative Seismic Damage Assessment in Circular RC Columns
3.8.1. Introduction
Experimental and field evidence indicates that the strength, stiffness and ultimate
deformation capacity of reinforced concrete (RC) elements and structures deteriorate
during excursions into the plastic range (Teran-Gilmore and Jirsa 2005).
Furthermore, a number of seismic events such as the Loma Prieta earthquake in 1989,
and more recently Kumamoto earthquake in 2016 are evidence that RC structures andspecially columns are in a relatively short time-period subjected to strong plastic
deformation due to the sequence of earthquakes` foreshock, main shock and after-shockin one hand, and on the other hand the great Tohoku earthquake in 2011, the strongest
ever recorded in Japan proved that RC structures can be hit by a very powerfulearthquake followed by numerous strong after-shocks.
It is fundamental that in seismic damage and seismic performance analyses of RC
structures, the cumulative effect that an earthquake`s main shock and after-shocks haveon the behaviour of RC members, especially columns, is considered, in order toimprove the understanding of the real structural behaviour under seismic loading,especially the cumulative effect on the stiffness degradation, energy dissipation as wellas parameters related to strength and displacement on RC members.
With the above in mind, this study is focused on the applicability of themulti-directional polygonal 3D lattice model to perform cumulative seismic damage
evaluation in circular cross section RC Columns (Simão and Miki 2016).

52

1000

800

800

600

600

400

400

200
0

-200

0

5

10

15

20

-400

Acceleration (gal)Acceleration (gal)

1000

200
0

-200

0

10

20

30

40

-400
-600

-600

-800

-800

C1-5(1)-1-1000

C1-5(1)-2-1000

Time (sec)Time (sec)

1000
800

Acceleration (gal)

600
400
200

0
-200

0

10

20

30

40

50

60

-400
-600
-800

-1000

Time (sec)C1-5(2)

Figure 3.14 Sequential loading in longitudinal direction for column C1-5

3.8.2. Outline of analysis
The analytical target is a circular cross section column named C1-5 tested using a
shake-table by the E-Defense presented in Sub-chapter 3.4.3 and specimen details in
Figure 3.6. In the experiment Vertical load is in the form of tributary mass to thecolumn by two decks including four weights of 307 ton and 215 ton in the longitudinal
and transverse directions, respectively. The geometry of the model corresponds to thesame geometry presented in Figure 3.11(b). For reduction factors, Pique and Burgos(2008) suggest that for typical columns in order to balance precision and simplicity
based on studies previously presented, reduction factor of 0.7 to the gross flexural
stiffness is applied in this analysis. The analytical diameter corresponds to 1825.85 mm.
According to Kawashima et al. (2010) column C1-5 was excited in longitudinal,transverse and vertical direction. However this study will neglect vertical direction to
focus on the principal directions. The convention and loading sequences used in theexperiment are also applied in the analysis.
With that C1-5(1)-1 corresponds to one 100% E-Takatori to a time of 20 seconds ofloading, C1-5(1)-2 to two 100% E-Takatori corresponding to 40 seconds of aggregate

53

3000

2000
1500

2000

1000

Load (KN)

Load (KN)

1000
0
-100

-50

0

50

100

500
0

-80

-60

-40

-20

-1000

-500

0

20

40

60

80

-1000

Analysis C1-5(1)-1-2000

Experimental

-3000

-1500

Analysis C1-5 (1)-1-2000

Experimental-2500

Displacement (mm)Displacement (mm)

3000

4000
3000

2000

2000

0
-150

-100

-50

0

50

100

150

-1000
-2000

Load (KN)

Load (KN)

1000
1000
0
-150

-100

-50

0

50

100

150

-1000

C1-(5)-2-2000

Experimental

-3000

-3000
Displacement (mm)C1-(5)-2
Experimental-4000
Displacement (mm)

3000

4000
3000

2000

2000
1000

0
-300

-200

-100

0

100

200

300

-1000
-2000

Load (KN)

Load (KN)

1000

0
-300

-200

-100

100

200

300

-2000

C1-5(2)C1-5(2)-3000

Experimental

(a)

0
-1000

Experimental

-3000

-4000

Displacement (mm)Displacement (mm)Longitudinal direction(b)Transverse direction

Figure 3.15 Load-displacement relationships for C1-5 analytical cases

loading, and after 21% increase in the tributal mass at the top of the pier, from 307 ton
to 372 ton, C1-5(2) was excited to three 100% E-Takatori corresponding to 60 secondsof aggregate loading. In the experiment, column C1-5 was subjected to further loading.
However, for the scope of this study, the focus will be on the three initial loading
sequences as explained, because it was concluded during the pre-analysis, that on
further loadings, the nonlinear behavior of the column became much more difficult to
describe. The loading sequences for longitudinal direction for C1-5(1)-1, C1-5(1)-2 andC1-5(2) are shown in Figure 3.14, respectively.

54

3.8.3. Analytical results and discussion
The analytical results for load-displacement relationships are shown in Figure 3.15 andthe results for time-history are shown in Figure 3.16, for both longitudinal andtransverse direction. Looking at Fig. 6, generally the comparison of hysteresis response
between the analytical cases and the experiment shows the highly nonlinear behaviour
that accumulated damage causes in the response of the RC column, and this complexity
is response calls for a very detailed analysis of the response. It is noticeable that caseC1-5(1)-1 shows the best agreement in hysteresis between analysis and the experimentfor both directions, when compared to cases C1-5(1)-2 and C1-5(2). Overall thestrength capacity prediction is considered to be acceptable in both directions for thethree analytical cases.
On the other hand, with increase in the number of loading sequences, going to cases
C1-5(1)-2 and C1-5(2), there is a visible tendency to underestimate the maximum
displacement capacity due to overestimation of stiffness in response in the longitudinal
direction, especially in the quadrant corresponding to positive displacement and positive
load. This situation is directly related to the high initial stiffness that is observed in theanalytical response of all analytical cases, and according to results previously reported
by Simão and Miki (2015), by modelling a circular column using the multi directional
polygonal 3D lattice model, elastic stiffness is by tendency overestimated due to theincreased number of diagonal members used to discretize a circular column into
polygonal 3D lattice model, which increases the size of the stiffness matrix.
In the analytical results, it is visible that after cracking, for the longitudinal direction,
the response continues to underestimate the stiffness degradation, thus the response fails
to reach the maximum displacement capacity, remaining stiffer than the experiment.
This is a point that should be addressed in further stages of this study. However in thetransverse direction the response better captured the strength and stiffness degradation
going from case C1-5(1)-1 to C1-5(1)-2 and C1-5(2).
The interpretation of analytical results can be further performed by evaluating thehysteretic energy dissipation capacity of all cases. Hereafter, the hysteretic energy
dissipation corresponds to the enclosed area of the load-displacement relationships,
calculated for analytical response and experimental cases. The energy dissipation is
calculated at the end of each analytical cycle, corresponding to total loading time. The
analytical calculated energy dissipation for C1-5(1)-1 at 20 seconds of total loading
55

80

60

60

40
20
0
-20

0

5

10

15

20

Displacement (mm)

100

80

Displacement (mm)

100

40
20

0
0

-20

5

10

15

20

-40

-40

Analysis C1-5(1)-1-60

Analysis C1-5(1)-1-60

Experimental-80

Experimental

-100

-80

time (sec)time (sec)

150
150

100

50
0
0

5

10

15

20

25

30

35

40

-50

Displacement (mm)Displacement (mm)

100

50

0
0

5

10

15

20

25

30

35

40

-50

-100
C1-5(1)-2

Experimental-150

Time (sec)C1-5(1)-2

Experimental-100

Time (sec)

300
250

150
100

150
100

50
0
-50

0

10

20

30

40

50

60

Displacement (mm)Displacement (mm)

200

50
0
0

10

20

30

40

50

60

-50
-100

-100
-150

Time (sec)C1-5(2)C1-5(2)-150

Time (sec)Experimental(a)Longitudinal direction(b)

Experimental

Transverse direction

Figure 3.16 Time-history relationships for C1-5 analytical cases

time correspond to174 KNm for longitudinal direction and 181 KNm in transverse
direction. The calculated values of experimental energy dissipation for C1-5(1)-1
correspond to 180 KNm for longitudinal direction and 140 KNm for transverse
direction. In this case, comparing the values of energy dissipation for analysis andexperiment it is clear that the analysis acceptably predicts the level of energy dissipatedby the column, although it is slightly overestimated in transverse direction.
Looking at the cases of C1-5(1)-2 at 40 seconds of loading time and C1-5(2) at 60
seconds of loading time, in transverse direction the values or energy dissipation ofanalysis and experiment are somehow close to each other. In case C1-5(1)-2,
analytical energy dissipation corresponds to 275 KNm while experimental corresponds56

to 318 KNm. In case C1-5(2), analytical energy dissipation corresponds to 376 KNm
and experimental energy dissipation corresponds to 409 KNm.
On the other hand, for longitudinal direction, although some portions of the hysteretic
response are acceptably described, the failure to accurately grasp the maximum
displacement in the analysis for C1-5(1)-2 and C1-5(2), produces an importantunderestimation in the analytical energy dissipation in this direction. With that, in thecase of C1-5(1)-2, analytical energy dissipation corresponds to 219 KNm while
experimental corresponds to 281 KNm. For C1-5(2), analytical energy dissipation
corresponds to 292 KNm and experimental energy dissipation corresponds to 497 KNm.
The interpretation of analytical results can be further performed by evaluating thehysteretic energy dissipation capacity of all cases. Hereafter, the hysteretic energy
dissipation corresponds to the enclosed area of the load-displacement relationships,
calculated for analytical response and experimental cases. The energy dissipation is
calculated at the end of each analytical cycle, corresponding to total loading time. The
analytical calculated energy dissipation for C1-5(1)-1 at 20 seconds of total loading
time correspond to174 KNm for longitudinal direction and 181 KNm in transverse
direction. The calculated values of experimental energy dissipation for C1-5(1)-1
correspond to 180 KNm for longitudinal direction and 140 KNm for transverse
direction. In this case, comparing the values of energy dissipation for analysis andexperiment it is clear that the analysis acceptably predicts the level of energy dissipatedby the column, although it is slightly overestimated in transverse direction.
Looking at the cases of C1-5(1)-2 at 40 seconds of loading time and C1-5(2) at 60
seconds of loading time, in transverse direction the values or energy dissipation ofanalysis and experiment are somehow close to each other. In case C1-5(1)-2,
analytical energy dissipation corresponds to 275 KNm while experimental corresponds
to 318 KNm. In case C1-5(2), analytical energy dissipation corresponds to 376 KNm
and experimental energy dissipation corresponds to 409 KNm.
On the other hand, for longitudinal direction, although some portions of the hysteretic
response are acceptably described, the failure to accurately grasp the maximum
displacement in the analysis for C1-5(1)-2 and C1-5(2), produces an importantunderestimation in the analytical energy dissipation in this direction. With that, in thecase of C1-5(1)-2, analytical energy dissipation corresponds to 219 KNm while57

experimental corresponds to 281 KNm. For C1-5(2), analytical energy dissipation
corresponds to 292 KNm and experimental energy dissipation corresponds to 497 KNm.
Looking at Fig. 7, the time-histories are presented for the analysis. In case C1-5(1)-1 theresults correspond to a one 100% E-Takatori corresponding to 20 seconds, while inC1-5(1)-2 are two 100% E-Takatori corresponding to 40 seconds and in C1-5(2) are
three 100% E-Takatori corresponding to 60 seconds. In the case of C1-5(1)-2 andC1-5(2) the last 20 seconds are compared to the experiment, which performed
accumulation of damage by loading separately each case in sequences of 20 seconds(Kawashima et al. 2010), and since in the analysis the loading sequences are continuous,
the last 20 seconds correspond to the displacement-time relationship that is co-related todamage accumulation or cumulative damage.
In these results, the longitudinal direction shows acceptable capability of predicting theresidual displacement from all three cases. Case C1-5(1)-1 in longitudinal direction
shows the best agreement between analysis and experiment among all the cases. The
effect of accumulation of damage is acceptably described for longitudinal direction, in
the transition from first to second loading in case C1-5(1)-2 and from second to third
loading in case C1-5(2), as shown by the analytical and experimental displacements in
those portions. The transverse direction however came short in accurately describing
this transition.

58

References of Chapter 3
Ang, B. G., Priestley, M. J. N. and Paulay, T. (1989): Seismic Shear Strength of
Circular RC Columns, ACI structural journal, Vol. 86, No.1, pp. 45-59,
Hoshikuma, K., Sakai, J., Komori, N. and Sakayanagi, N. (2013): Evaluation of Limit
States of Reinforced Concrete Columns for Seismic Design of Highway Bridges,
Technical Note of PWRI, No. 4262,
Merta, I. (2004): Shear Strength Model of Reinforced Concrete Circular Cross-Section
Members, Proc. of 5th International Conference on Fracture Mechanics of Concrete
and Concrete Structures, pp. 1297-1302,
Merta, I. and Kolbitsch, A.: Shear Area of RC Circular Cross-Section Members, proc.
of 31st Conference on our World in Concrete and Structures, 2006.
Miki, T. and Niwa, J. (2004): Nonlinear Analysis of RC Structural Members using 3D
Lattice Model, Journal of advanced concrete technology, Vol.2, No.3, pp. 343-358,
Paulay, T, Priestley M. J. N. (1992): Seismic Design of Reinforced Concrete andMasonry Buildings, Wiley Interscience, 1992
Pique, J. R. and Burgos, M. (2008): Effective Rigidity of Reinforced Concrete Elements
in Seismic Analysis and Design, Proceeding of 14th World Conference on Earthquake
Engineering, Beijing, ChinaSimão, M.R., and Miki, T. (2015): Dynamic Analysis for RC Columns with Circular
Cross Section Using Multi-Directional Polygonal 3DLattice Model, Proceeding of JCI,
No.2 , 757-762
Simão, M.R., and Miki, T. (2016): Cumulative Seismic Damage Assessment Circular
RC Columns Using Multi-Directional Polygonal 3DLattice Model, Proceeding of fib,No.2 , 757-762
Teran-Gilmore, A. and Jirsa, O.J. (2005): A Damage Model for Practical Seismic
Design that Accounts for Low Cycle fatigue. Earthquake spectra, Vol. 21, No.3

59

4. NUMERICAL VERIFICATION OF SEISMIC PERFORMANCE OF
CIRCULAR RC COLUMNS USING MULTI-DIRECTIONAL POLYGONAL 3D
LATTICE MODEL4.1. Introduction
Reinforced concrete (RC) structures reluctantly exhibit severe damage subjected to astrong earthquake motion. In Japan in the context of seismic performance of RC
structures, the assessment of serviceability and reparability is a primary task after a huge
earthquake. In most post-earthquake structural damage evaluation, the analysis isqualitative. However, it is important to establish a quantitative correlation between theobserved damage state and an applicable damage index, so as to make thepost-earthquake evaluation of damage level in structures more reliable and lesstime-consuming.
The damage index normalizes the damage evaluation criteria and the threshold values
influence the decision related to repair and retrofitting, and they should be connected tothe damage states, with clear definition of the damage level and failure mechanism. Thisis specially the case on the context of performance-based design, where performance
objectives are set so that the structural capacity for each performance level is related toa state of damage and is quantified using one or more engineering limit states.
With a new focus in seismic design on the performance of reinforced concrete bridges,commonly current seismic design standards for reinforced concrete bridges do notprovide adequate performance design requirements. Although previous research on the
response of reinforced concrete bridge columns is extensive, these studies are notadequate to develop all aspects of performance-oriented design. Development of
performance-based design methods requires further experimental and analyticalinvestigations to evaluate intermediate damage levels and to develop analytical models
and appropriate design methodologies (Lehman and Mohele 2000).
Based on the above considerations, the multi-direction polygonal 3D lattice model is
proposed to perform the verification of performance level of RC columns from thematerial point of view based on sustained seismic induced damage (Simão and Miki2015).

60

4.2. Energy Dissipation Capacity in RC Columns
4.2.1. Introduction
In the field of engineering mechanics the evaluation of behavior of structures has
traditionally focused on strength and deformation capacity. In those cases to a certain
level of applied force, a certain level of displacement is expected and from that the
behavior of the structure understood from pure force-deformation criteria. It is
nonetheless true that in modern design of structures, the structural members tend to becomposite structures. That means that more than one material is used in the same
member and thus the global structural response is very much dependent on material
behavior. At the material level a division between brittle materials, quasi-brittle material
and ductile materials can be found. That is specially the case in RC structures; whereconcrete is a quasi-brittle material and steel reinforcement a ductile material. At thislevel the behavior of materials is best understood by looking and the stress and strain
conditions, and the criteria that differentiates the material type is the energy dissipation
criteria. In this sub-chapter energy capacity of materials will be studied in the context ofseismic damage evaluation (Popov 1990).
4.2.2. Hysteretic energy dissipation
Tembulkar & Nau (1987) stated that damage attained by a RC member under dynamic
loading can be assessed by a well-constructed hysteretic model. Furthermore, withwell-defined parameters, energy based hysteresis models may successfully represent
seismic response considering the deteriorating behavior of RC members.
On the other hand, Bousias et al. (1995) reported that the strong coupling between the
two transverse plane directions of columns produces an apparent reduction of strengthand stiffness in each of the two transverse plane directions when considered separately,
but also an increase in hysteretic energy.
With that, the cumulative hysteretic energy dissipation is a very important notion in theanalysis of cyclic behavior of RC structures, because it relates to the global damage
potential and energy capacity. The hysteretic energy is calculated considering the areaof each loading cycle in the X (longitudinal) and Y (transverse) direction for the lateralforce and displacement relationships, and the total energy dissipation is calculated as thesum of these two parts. Equations 4.1, 4.2 and 4.3 express the analytical relationships.
Ed  x   Fx d x

(4.1)

61

Ed  y   Fy d y(4.2)

Ed total   Fx d x   Fy d y(4.3)

where Ed-x is hysteretic energy dissipation in X direction; Ed-y is hysteretic energy
dissipation in Y direction; Ed-total is total hysteretic energy dissipation; Fx is force in X
direction and Fy is force in Y direction

4.2.3. Elemental energy dissipation
The 3D lattice model offers some unique characteristics in terms of analytical capacityof the composing analytical elements. Because of the truss and arch analogy, theanalytical response is obtained individually for every component of the system. The
lattice model comprises several elements, and it is assumed that an average stress andstrain relationship governs each one of them. In other words, the stress and strain
relationships define the strain energy for each element in the lattice model, that is
concrete and reinforcement elements.
This definition can be very useful for the calculation of strain energy after the target ofanalysis has reached the yielding point, and especially because of highly nonlinear
behavior of concrete after cracking. That way, the strain energy density for one element
is defined as presented in Equation 4.4, and the accumulated strain energy density of amaterial is shown in Equation 4.5.

Eei 

1
 i (t )   i (t  1) i (t )   i (t  1)
2

(4.4)

Es mat  t 1 Ee (t )
n

(4.5)

where Eei is the elemental strain energy density, εi (t) is strain in element i at time step t,σi(t) is the strain in element i at time step t, Es-mat is the accumulated material strainenergy.

62

By taking into account the energy dissipation in individual elements, the distribution ofenergy dissipation in a RC column can be evaluated by the lattice model. Based on thisassumption the energy dissipated in each element can be calculated from the product of
the strain energy dissipated in each element, where the strain energy is the area enclosedby the stress-strain relationship for the unloading and reloading curves, and theelemental volume. RC member accumulated energy dissipation will be the sum of all
the elemental energy dissipation histories as shown in Equation 4.5 below.
Ed mat  Es mat  Ve

(4.5)

where Ed-mat is the accumulated energy dissipation of the material in the member Es-mat
is the accumulated strain energy density for all element and Ve is the Volume ofelements which for simplicity is assumed to be constant

4.3. Damage Range Evaluation
4.3.1. Introduction
The evaluation of seismic damage of concrete structures is very important in order totake countermeasures, such as repair and strengthening the deteriorated structures after
severe earthquakes. In technical literature, a large number of damage indexes have been
proposed. Some of them are based on cyclic fatigue concepts (Krawinkler and Zohrei
1983) and others make use of structural mode (Dipasquale and Cakmak 1990,).
Furthermore, a group of damage indexes include ductility ratio or plastic deformation(Powell and Allahabadi 1988; Cosenza et al.1988) whilst some concern hystereticenergy absorption (Dipasquale and Cakmak 1989; Fajfar 1992). Other damage indexes
are a integration of different parameters. For example, Park's and Ang's model (Park andAng 1985], and Reinhorn's and Valles's model (Reinhorn and Valles 1995) consist ofboth deformation and energy terms.
From the material point of view however it is very useful to derive a damage index fromthe energy criteria point of view, because in that way a useful understanding of theactual damage range capacity of the material can be attained, based on the strain energy
density analysis, issues of resilience and toughness can be studied and thus providing abetter understanding of the damage distribution in material that is connected tostructural performance. However, global damage condition of a structure can only beassessed using energy methods if there is knowledge about the total energy capacity.
63Based on the discretization method of the lattice model it is possible to determine froman elemental level the expected energy capacity of single elements and compare them tothe response obtained from the analytical results

4.3.2. Evaluation of seismic damage using damage index
In order to realize the performance evaluation of RC columns under seismic loading, it
is very important to quantitatively verify the damage states. In this study, this isperformed by looking at the damage level, based on the measured damage range
measures at the end of the relevant loading sequence. To do that, damage range
evaluation is proposed using a damage index defined from the material point of view
based on energy criteria for concrete and steel reinforcement. In order to use the energy
dissipated as an indicator of seismic damage in concrete structures, the total energy
dissipated must somehow be normalized so as to compare the results of different size
specimens (Inoue 1994).
In this study damage index is defined from the energy dissipation point of view. In thatway, damage index in material is defined as the ratio between the calculated materials
accumulated energy dissipation and the ultimate material energy dissipation. The
ultimate material energy dissipation is obtained from the constitutive models for
ultimate condition of materials and for concrete for concrete in compression is used themodel proposed by Mander et al.(1988), concrete in tension proposed by Uchida et al.
(1991) and reinforcement proposed by Fukuura and Maekawa (1997). In the case of
concrete the ultimate strain in compression is assumed to be 0.0035 when concrete
crushes and the ultimate compressive stress corresponds to 0.002 of compressive strain.
The tensile strain is calculated using the 1/4 model (Uchida et al. 1991). The damage
index in material is presented as:

DI mat Ed matEd ultimate(4.6)

here DImat is the damage index of the material, Ed-mat is the accumulated energy
dissipation of the material in the member and Ed-ultimatet is the ultimate energy dissipationof the material in the member.

64

4.4. Numerical Verification of Seismic Performance
4.4.1. Analytical frame-work
In the context of performance-based seismic design most codes define a single level ofseismic hazard and a single level of performance that is generally understood to belife-safety i.e. performance objectives other than the life-safety are not evaluatedexplicitly. As seismic performance objectives can be defined based on expected
performance levels, according to Li et al. (2013), structural damage conditions can beassessed from 0-1. Here damage condition 0 means no damage at all, while damage
condition 1.0 means total failure of structure. This convention is quite simple and usefulto evaluate the damage potential in structures.
In this study, this convention will be applied at the material level, where to the same
extent damage level 0 corresponds to sound material and damage level 1 corresponds tototal failure of material. In this study a damage potential separation point named damage
limitation point (DLP) is proposed. The DLP corresponds to the point in the damage
scale where damage potential changes from moderate to high. The DLP corresponds toa theoretical threshold in the damage scale where damage potential in the material
becomes critical under the given loading conditions and is defined for concrete andreinforcement as follows:
To understand this concept it is useful to fall back to classic engineering mechanics ofsolids and explained by Popov (1990). According to the later resilience corresponds tothe ability of material to absorb energy without suffering plastic strain. On a
stress-strain diagram of material the area of the elastic region represents the density ofstrain energy that can be absorbed without any permanent damage to the material or theso called modulus of resilience UR. On the other hand, toughness corresponds to theability of the material to absorb energy prior to fracture, and in the stress-strain diagramof the material it is represented by the full area of the enclosure prior to fracture or theso called modulus of toughness UT. With that two important threshold values arepresented: resilience and toughness.
In order to use this analogy and extend these concepts to numerical evaluation of RC
structures a few assumptions are important: the first one is that the constitutive models
of material used in this study are to be best extent possible representatives of the actual
behavior of material is seismic response, and in the analysis process a few
simplifications are necessary to ensure practicability of the method.

65

t
ft

ft/4
t

0

cr

2

1

(a) Concrete in tension
c’

Concrete that has been confined
Fracture of the
rebar

fcc’
fco’

Unrestrained concrete

Ec
0

c ’
co’

cc’

cu’

(b) Concrete in compression
Figure 4.1 Damage limitation points in concrete
Based on the explained, DLP will be the ratio between modulus of resilience andmodulus of toughness.
DLPmat 

U R matU T mat(4.7)

where DLPmat is the damage limitation point of material, UR-mat is the modulus ofresilience of material and UT-mat is the modulus of toughness of material. In concrete themodulus of toughness corresponds to the sum of total enclosed area of stress-strain
diagram in tension side and in compression side or in other words the ultimate strainenergy density in concrete.
66

Figure 4.2 Damage limitation points in steel reinforcement
On the other hand, modulus of resilience in tension side is the sum between the tensile
strain energy density at cracking of concrete represented by the hatched area in Figure
4.1(a) and the compressive strain energy density at maximum stress which isrepresented by the hatched area in Figure 4.1(b).
The same approach is used for reinforcement. In that case the modulus of toughness
corresponds to the sum of total enclosed area of stress-strain diagram until fracture of
reinforcement or the total strain energy density of reinforcement. On the other hand themodulus of resilience corresponds to the an closure represented in hatch in Figure 4.2,
which is the yielding strain energy density of reinforcement.
In the numerical evaluation of seismic performance the damage index of material iscompared to DLP. The detailed explanation of the method is shown in the next
sub-chapter.

4.4.2. Evaluation of performance states
In order to establish a co-relation between the analogies describe in sub-chapter 4.4.1,
and real structural damage condition and from that derive structural performance of RC
member, it is important to look at the JSCE specifications (2002) and experimental
evidence to co-relate damage index and observed structural performance.

67

With that, based on the seismic performance criteria indicated by JSCE (2002) three
seismic performance levels of structures are defined. First, the seismic performance thatis assumed to correspond to small residual deformation on a structure and in principle
yield load of the member is not reached. In the seismic performance II the load carrying
capacity does not deteriorate after an earthquake and finally in the seismic performance
III it is required that the whole structural system does not collapse.
On the other hand, Kawashima et al. (2010) clearly described the seismic performanceof RC columns from the material point of view based on experimental evidence.
According to their experimental observations, in this study, the seismic performance ofRC columns will be evaluated based on damage index and correlated to structural
damage as discussed in the previous research (JSCE specifications 2002; Kawashima etal. 2010)
With that, in order to evaluate the performance, damage range between 0 to DLP
corresponds to a seismic performance category I (SP-I), and damage range between
DLP to 1.0 corresponds to a seismic performance category II (SP-II), respectively. As
for the SP-I of concrete, the maximum structural damage corresponds to cracking of
cover concrete. For reinforcement the maximum structural damage corresponds toyielding and buckling of reinforcement.
On the other, hand in the SP-II for concrete the maximum structural damage condition isspalling of cover concrete, crushing of concrete and fracture of confined concrete, forreinforcement the maximum structural damage is fracture of reinforcement.

4.5. Numerical Verification of Seismic Performance of RC column subjected to
Cyclic Loading
4.5.1. Outline of analysis
The experimental case (Hoshikuma et al. 2013) described in sub-chapter 3.5.1 will be
used to perform the numerical verification of seismic performance of RC column thathas been subjected to cyclic loading. The target is a circular cross section column named
C-29. The cross section of this column is circular with diameter of 600 mm and column
height of 3410 mm. The compressive strength and the Young’s modulus of concrete are
31.8 N/mm2 and 28000 N/mm2, respectively.
68

200
150
100

Load (KN)

50
0

-150

-100

-50

0

50

100

150

-50
-100
Experimental-150
analysis-200
Displacement (mm)Figure 4.3 C-29(3) load-displacement
relationships
The longitudinal reinforcement of D10 and transverse reinforcement D6 have yieldstrength of 397 N/mm2 and the transverse reinforcement D6 has yield strength of 397
N/mm2, respectively. In order to conduct the analysis, three scenarios related to different
inertia reduction factors were studied, corresponding to C-29(1), C-29(3) and C-29(3).
In this cub-chapter the focus will be case C-29(3) which offered the best agreement
between analysis and experiment in load-displacement curve. Detailed explanation of
all the cases can be found in chapter 3.5.1. The load-displacement curve of C-29(3) isshown in Figure 4.3.

4.5.2. Analytical results and discussion
The seismic performance verification of column C-29(3) is presented in Table 1. The
results show that energy accumulated dissipation of concrete is 35.4 kN-m and ultimate
energy dissipation of concrete is 151.1 kN-m, while in reinforcement the accumulated
energy dissipation is 202 kN-m and ultimate energy dissipation is 6072 kN-m. Thismeans that by large in the given conditions, the damage potential is bigger in concrete
than in steel as and also the idea that energy the energy dissipation potential of steel
reinforcement is by large most significant than that of concrete in damage resistancecapacity due to higher ductility of steel reinforcement.
Looking at the seismic performance of materials, it can be assessed that the damage
index of concrete corresponds to 0.59 and is placed between the DLP and 1, and thedamage index of steel reinforcement is 0.00008 and is placed between 0 and DLP.
69

Table 1: seismic performance verification of column C-29(3)Material

Concrete

Steel

Energy dissipation

Accumulated

35.4

202

KN-m

Ultimate

151.1

3036

DLP

0.40

0.004

Damage index

0.59

0.00008

Seismic performance

SP-II

SP-I

With that, the concrete performance is that of SP-II and the performance ofreinforcement is that of SP-I. In terms of structural seismic performance it is important
to correlate the observed structural damage reported by the experiment. Based on thedefinition provided, analytical seismic performance criteria concrete has been affectedby cracking followed by spalling of cover concrete. Reinforcement on the other hand,
has been affected by yielding of reinforcement at the plastic hinge followed by localized
buckling.
Sensitivity analysis of concrete suggests that because the damage index is not far fromthe value of DLP perhaps the sustained damage is not critical. On the other hand,
reinforcement is less affected by the seismic action based on the calculated damage
index, which is confirmed by visual inspection of the experimental target (Hoshikuma etal. 2013).
The report on the experimental results, clearly show the progress of damage in thecolumn C-29. The mechanism of failure is flexural, and flexural cracking developed
along the column, with visible concentration on the plastic hinge. The observed damage
condition of concrete in experiment corresponds to cracking and some spalling, which ispredicted by the analytical model. In reinforcement only yielding is observed in theexperiment, which corresponds to SP-I in the analytical performance verification
method proposed.

70

1000

2000

A`

8000

A
Local damage
analysis target

Unit:mmFigure 4.4 C1-5 Analytical model
4.6. Performance Evaluation of RC Columns Subjected to Seismic Motion
Designed according to 2002 JRA Specifications
4.6.1. Outlines of Analysis
In order to perform the analysis the analytical target is a circular cross section column
named C1-5 tested using a shake-table by E-Defense. The complete detailing of column
C1-5 is shown in sub-chapter 3.4.2. The specimen is a cantilever circular column withdiameter 2000 mm. The heights of the column and footing correspond to 7500 mm and1800 mm respectively. The analytical model is shown in shown in Figure 4.4. Column
C1-5 was excited using a near-field ground motion which was recorded at the JR
Takatori Station during the 1995 Kobe earthquake. Taking into account the soil/structureinteraction, a ground motion with 80% of the original intensity of JR Takatori name
E-Takatori was imposed as command to the table in the experiment and corresponds to100% E-Takatori. In this study the analysis will focus on C1-5(1) corresponding to
100% E-Takatori and C1-5(3) corresponding to 125% E-Takatori and 21% top mass
increase, respectively. The input accelerations are shown in Figure 4.5 for longitudinal
direction.
Simão & Miki (2014) reported that on a cantilever bridge column, about 50% ofdamage is concentrated below the column mid-height. Based on that, it is reasonable toassume that the most significant concentration of damage at the bottom is very
representative of damage condition. In order to analyze Localized damage evaluation,
numerical verification of damage concentration is performed in case C1-5(1). The
height at the bottom of the column portion where it is assumed that the plastic hinge
71

600

Acceleration (gal)

400
200
0

0

5

10

15

20

-200
-400
-600
-800

Time (sec)(a) 100% E-Takatori
1200
1000

Acceleration (gal)

800

600
400
200
0
-200 0

5

10

15

20

-400
-600

-800
-1000

Time (sec)(b) 125% E-Takatori
Figure 4.5 E-Takatori acceleration - longitudinal direction (Kawashima et al. 2010)
will develop and thus numerical verification of damage concentration is performed is1000 mm corresponding to the first layer of lattice model elements as shown in Figure
4.4.
4.6.2. Analytical results and discussion
The results of analysis are presented for C1-5(1) and C1-5(3) regarding theload-displacement relationships in Figure 4.6. The calculated analytical andexperimental hysteretic energy dissipations are presented in Table 2. In Table 3 ispresented the analysis on levels of performance for C1-5(1) for concrete and steel
reinforcement.
A detailed look at Figure 4.6 shows that for both C1-5(1) and C1-5(3) the
multi-directional polygonal 3D lattice model acceptably shows the hysteretic behavior,comparing the analytical response and the experiment in longitudinal and transverse
directions. In general, there is an over-estimation in initial stiffness, and this is due tothe fact that by discretizing a circular column using the multi-directional polygonal 3D
lattice model, more elements are needed to accurately describe the detailing of thecolumn as in opposition to a 3D lattice model applied to a rectangular column (Miki and
72

4000
3000

C1-5(1)3000

C1-5(3)
Load (KN)

2000

Load (KN)

1000
0
-100

-50

0

50

100

2000
1000

-500

-300

-1000

0
-100
-1000

100

300

500

-2000
Analysis C1-5(1)-2000Experimental-3000
Displacement (mm)-3000

Analysis C1-5 (3)-4000

Experimental

Displacement (mm)(a) Longitudinal direction
2500

2000

2000

1500

C1-5(3)

1000

1500
1000

500

Load (KN)

Load (KN)

C1-5(1)

0
-80

-60

-40

-20

-500

0

20

40

60

80

-1000
-1500
-2000

500
0
-300

-200

-100

-500

0

100

200

300

-1000
Analysis C1-5 (1)-1500

Experimental

-2000

-2500

-2500

Displacement (mm)Displacement (mm)Analysis C1-5 (3)Experimental(b) Transverse direction
Figure 4.6 Load-displacement relationship computed using analytical modelNiwa 2004) or representing a circular cross-section column using a circular-rectangularcross-sectional area equivalence. On the other hand the maximum displacement
capacity and the strength have acceptable agreement between analysis and experiment.
The hysteretic energy dissipation that has been calculated and shown in Table 2 foranalysis and experiment shows that by looking at the ratios between analysis andexperiment, in the case of C1-5(1) corresponds to 1.24 and for C1-5(3) corresponds to
0.87. These values suggest that generally the method grasp the behavior of the column
acceptably and furthermore the impact of the slight overestimation if initial stiffness is
not very big on energy dissipation capacity.
In Table 3, the results for performance level of column are shown. Looking at theanalytical results, in concrete the response strain energy corresponds to 3.60 N/mm2
while the total strain energy corresponds to 8.91 N/mm2. Furthermore accumulated
response energy dissipation corresponds to 12.05 KN-m while total energy dissipation
corresponds to 41.87 KN-m. Furthermore accumulated response energy dissipation
corresponds to 12.05 KN-m while total energy dissipation corresponds to 41.87 KN-m.
For steel reinforcement on the other hand, response strain energy is equal to 12.13
N/mm2 while the total strain energy is equal to 316.02 N/mm2.
73

Table 2: Hysteretic energy dissipation of column C1-5
Hysteretic Energy dissipation (KN-m)

C1-5(1)C1-5(3)

Analysis (1)

353

2466

Experimental (2)

283

2808

(1)/(2)

1.24

0.87

Table 3: Concentrated seismic performance evaluation of C1-5(1)Material

Concrete

Steel

Accumulated

3.60

12.13

N/mm

Ultimate

8.91

316.02

Energy dissipation

Accumulated

12.05

20.75

KN-m

Ultimate

41.87

201.9

Damage limitation point (DLP)

0.39

0.43

Damage index

0.33

0.04

Performance Level

SP-I

SP-I

Strain energy
2

On the accumulated material energy dissipation, response energy dissipation is equal to20.75 KN-m while total accumulated energy dissipation is equal to 201.9 KN-m. These
results show that by a large margin steel reinforcement is more ductile than concrete in
one hand, and also based on energy dissipation levels, in concrete the C1-5(1) caseproduced almost a quarter of total energy dissipation capacity, while in reinforcement it
had a small significance. The objective of assessing seismic performance levels inmaterial in this study is realized comparing the damage index of material to the damage
scale previously explained. In that manner, for concrete the damage index calculated
corresponds to 0.33, which falls between 0 and DLP that corresponds to 0.39. This
corresponds to performance level SP-I. On the other hand, in the case of steel
reinforcement, the calculated damage index is 0.04 and that corresponds to performancelevel SP-I. For both materials based on the performance levels, the damage range
corresponding is moderate. However a more detailed look at both damage indexes
suggests that concrete by far as been more exposed to greater damage such as
occurrence of some level of cracking because it`s damage index of 0.33 is close to thethreshold DLP value of 0.39. This is further confirmed by the report on the experimental
program (Kawashima et al. 2010) which states that after the first shaking corresponding
to C1-5(1) concrete shows some visible damage cracks.

74

References of Chapter 4
Cosenza, E, Manfredi, G, Ramasco, R. (1988): The Use of Damage Functionals in
Earthquake Engineering: A Comparison Between Different Methods, Journal of
Earthquake Engineering and Structural Dynamics, Vol.22, No.10, pp. 855-868
Dipasquale, E., Cakmak, A. S. (1989): On the Relation between Local and Global
Damage Indices, Technical Report No. NCEER- 89/34, National Centre for Earthquake
Engineering Research, State University of New York at Buffalo, NY
Dipasquale, E, Cakmak, A. S. (1990): Seismic Damage Assessment Using Linear
Models, Journal of Soil Dynamics and Earthquake Engineering, Vol.9, No.4,
pp.194-215
Fajfar, P. (1992): Equivalent Ductility Factors Taking into Account Low Cycle Fatigue,
Journal of Soil Dynamics and Earthquake Engineering, Vol.21, No.10), pp.837-848.
Fukuura, N. and Maekawa, K. (1997): Computational Model of Reinforcing Bar Under
Reversed Cyclic Loading for RC Nonlinear Analysis. Journal of Materials, Concrete
Structures and Pavements, JSCE,Vol. 564, No.35, pp.291-295.
Hoshikuma, K., Sakai, J., Komori, N. and Sakayanagi, N., (2013): Evaluation of Limit
States of Reinforced Concrete Columns for seismic Design of Highway Bridges,
Technical Note of PWRI, No. 4262. (in Japanese)
Inoue, S. 1994. Ductility and energy dissipation of concrete beam members and their
damage evaluation based on hysteretic energy dissipation, Kyoto: Kyoto University
Press
JSCE (2002): Standard specifications
Performance Verification, December.

forconcrete structures-2002, Seismic

Kawashima, K., Sasaki, T., Ukon, H., Kajiwara, K., Unjoh, S., Sakai, J., Kosa, k.,
Takahashi, Y., Yabe, M., and Matsuzaki, H. (20010): Evaluation of Seismic Performanceof a Circular Reinforced Concrete Bridge Column Designed in Accordance With theCurrent Design Code Based on E-Defense Excitation, Journal of JSCE, vol.66, no.2, pp.
324-343.
75

Krawinkler, H, Zohrei, M. (1983): Cumulative Damage in Steel Structures Subjected to
Earthquake Ground Motions, Journal of Computers and Structures, Vol.16, No.1-4,
pp.531-541
Li, D. Peng, Y. Ning, M. and Long, G., (2013): A New Strategy to Assess the Seismic
Energy Dissipation Safety of Reinforced Concrete Bridge Piers, International Journalof Energy Engineering, Vol.3, pp.55-61.
Mander, J. B., Priestley, M. J. and Park, R. (1988): Theoretical Stress-Strain Model for
Confined Concrete, Journal of Structural Engineering, ASCE, Vol. 114, No.8, pp.
1804-1826.
Park, Y. J., Ang, A. H. S. (1985): Mechanistic Seismic Damage Model for Reinforced
Concrete, Journal of Structural Division (ASCE), Vol.111, No.4, pp.722-739
Popov, E. P. (1990): Engineering Mechanics of Solids, Prentice –Hall Inc. Vol.2.
Powell, G. H., Allahabadi, R. (1988): Seismic Damage Prediction by Deterministic
Methods: Concepts and Procedures, Journal of Earthquake Engineering and Structural
Dynamics, Vol.16, No.5, pp.719-734
Reinhorn, A. M., and Valles, R. E. (1995): Damage Evaluation in Inelastic Response of
Structures: A Deterministic Approach, Report No. NCEER-95, National Centre for
Earthquake Engineering Research, State University of New York at Buffalo,Simão, M.R. and Miki, T. (2015): Damage Evaluation of RC Columns Subjected to
Seismic Loading by Energy Dissipation Using 3D Lattice Model. Proceeding of 2015
fib Conference: Leipzig.
Simão, M.R. and Miki, T. (2014): Seismic Damage Range Evaluation of RC Columns
with Circular Cross Section by Energy Dissipation Using 3D Lattice Model. Proc. 2014
JSCE International Summer symposium: Osaka.
Uchida, U., Rokugo, K., and Koyanagi, W. (1991): Determination of Tension Softening
Diagrams of Concrete by Means of Bending Tests, Journal of Materials, Concrete
Structures and Pavements, JSCE, No.426, Vol.14: 203-212.
76

5. PERFORMANCE EVALUATION OF RC VIADUCT
5.1. Introduction
Viaducts as civil structures are in the epicenter of societies. Good and sound viaducts
provide the infrastructure from which socio-economic development can be put to place.
This is the case especially in Japan, because the country has a very extensive network ofelevated viaducts, mainly in urban areas. Whenever an earthquake takes place in Japan,
viaducts are very much exposed to seismic induce damage. This was the case with the
Hyogo-ken Nanbu earthquake, which cause the collapse and ultimately failure of many
reinforced concert structures, but especially viaducts.
The evaluation of damage remains a highly difficult task, because so many factors
influence the level of damage and response of an RC (reinforced concrete) structure,even within the same structure with similar arrangements of reinforcement and concrete
member’s cross sections. Factors such as soil conditions, applied ground motions andother properties of the earthquakes such distance from the fault line and epicenter,material and structural specifications such as strength and stiffness of material anddeterioration of concrete and reinforcement based on the age of materials, ensure that
this task will remains ever complex. But still, even with the complex nature of theanalysis, it is ever important to study the behavior of RC viaducts under seismic
loadings, if not for their interesting and challenging analysis process, definitely for theirsocial and economic importance.
In this study, the dynamic lattice model was introduced and used for the evaluation of
structural response under cyclic loading. Furthermore, from the dynamic analysis,
energy dissipation was proposed as means to evaluate damage range under seismic
loading from a RC column, either for the whole target or more specifically the mostdamaged zone, according to a specific case assessment. On that note, using the conceptof energy dissipation presented ion chapter four, this chapter will perform analysis ofdamage in a RC rigid-frame viaduct that suffered damage during the Hyogo-ken Nanbu
earthquake. In this analysis the buckling behavior of reinforcement at the end ofcolumns is taken to account, but the bond-slip behavior of reinforcing bars at the joint
portion as well as soil and structure interaction are not considered.

77

5.2. Damage Distribution in RC Viaducts Based on Energy Dissipation
5.2.1. Outline of target structure
The seismic performance evaluation is performed for Hansui R5 which is a rigid-frame
railroad viaduct. The target is a beam-slab type rigid-frame RC viaduct with three-span.
The viaduct was designed according to Structural Design Standards of Japan National
Railways enacted in 1970. Here, the viaducts were designed using the seismic intensity
method with a design horizontal seismic intensity of 0.2 (Miki. 2004). In theobservation of actual structures, the degree of damage was determined according tomost heavily damaged members in the viaduct. Figure 5.1 shows the dimensions andreinforcement arrangement of Hansui R5. In Figure 5.2, the ground-motion acceleration
for the viaduct in transverse and longitudinal direction is shown.
In the viaduct, the cross section of a column was a square of 900 mm. All reinforcing
bars in the columns had a minimum concrete cover of 60 mm. The beams had
rectangular cross section with 700 mm width and 1,000 mm depth for the upper portionin transverse direction, while with 700 mm width and 1,100 mm depth for otherportions. Heights of columns were 5,000 mm and 4,000 mm in lower and upper portions,
respectively. The compressive strength of concrete was 29.1 MPa, while the tensile
strength was 1.27 MPa to a Young`s modulus of 18.4 GPa. The longitudinal
reinforcement had yield strength of 322 MPa, ultimate strength of 521 MPa and Young’s
modulus of 203 GPa. While the transverse reinforcement had yield strength of 263 MPa,
ultimate strength of 380 MPa and Young`s modulus of 183 GPa.
6500

60

60

900

900

7@111
= 780

6@130
= 780

60

60

60

4000

00
249

60

BB



7@111
= 780

900
D32

(b) AA section in
the column

0
710

60

0
710

AA

4500

60

0
710

FF
DD

900

11000

EE
CC

7@111
= 780

5000

EE

D32

(a) RC viaduct

Figure 5.1 Dimensions and arrangements of reinforcement in Hansui R5 Viaduct(Miki. 2004)78(c) BB section in
the column

600

Acceleration (gal)

400
200
0
0

2

4

6

8

10

8

10

-200
-400

-600

Time (sec)(a) Longitudinal (NS)800

Acceleration (gal)

600
400
200
0
0

2

4

6

-200
-400
-600

Time (sec)(b) Transverse (EW)

Figure 5.2 Hansui R5 Input ground motion

5.2.2. Configuration of lattice model
The 3D dynamic lattice model and the boundary conditions used in the analysis areillustrated in Figure 5.3. The three-span viaduct is treated as a unit of the analyticalmodel. As seen in the figure, the model consists of beams and columns, while the slab is
not included. In the analysis, it is assumed that the masses corresponding to theself-weight of viaducts are uniformly distributed over all nodal points, using the
lumped-mass idealization. It is also assumed that there is a concentrated mass, which isequal to the weight of the superstructure and the slab, acting on the top nodes ofcolumns and beams. It is considered that the joints of each column and beam are rigidly
connected between each member. According to the flexural deformation of columns andbeams, it is appropriate to provide arch members in two layers along the longitudinal
direction of the member axis.

79

P1

P2
P3

P4
P5

P6
P7

P8

Unit: mmFigure 5.3 Lattice model of Hansui R5 Viaduct (Miki.2004)

Hence, the arch members in both columns and beams are modeled into eight concrete
members. Similarly, at the column-beam joint portion, four arch members are modeled
as. In order to determine the cross-sectional area of arch members in the 3D latticemodel, the values of t b and td are determined based on the theorem of minimization of
the total potential energy. The calculations for the values of t b and t d are conducted oneach structural member individually.
Viaducts are founded on the sufficiently stiff ground; it is assumed that the input groundmotion is directly applied to the bottom of each lower column. The 3D lattice model
treats the RC viaduct that is disregarded the foundation, and consequently theinteraction between the structure and soil is not considered in the analysis. For theboundary between neighboring viaducts in the longitudinal direction, the horizontal
direction is assumed to be a free condition.
In the dynamic analysis it is assumed that the viscous damping is neglected (h=0) andthe numerical damping of the Newmark method with factors β=0.36 and γ= 0.70 is used
as time integration. Here a time interval of 0.01 sec. is set, the Newton-Raphson
iteration method is used for the calculations until an adequately converged solution is
obtained, since the nonlinear responses appear in RC structures when they are subjectedto large ground motions.
In order to check the convergence, the out-of-balance force and energy increment are
80

compared with initial values during the iteration. The convergence tolerances of theout-of-balance force and energy are set to be 0.001 and 0.01, respectively (Miki 2004).

5.2.3. Damage distribution of Hansui R5 viaduct
The damage distribution evaluation is performed according to the principle of energy
dissipation previously presented. The damage will be focused on the columns of Hansui
RC. In order to evaluate the damage, the structure, the elemental stress and strain of the
8 columns that are part of the structure will be used for the calculation of strain energy.
This same strain energy will be multiplied by each individual volume, related to eachspecific elemental stress and strain and thus energy dissipation computed. The analysis
will evaluate the amount of damage in terms of energy dissipation for the whole column
height, followed by a specific focus on the bottom layer, in which case corresponds tothe height of 2100 mm corresponding to six lattice layers. Furthermore, for column P1,
the joint between the column and the beam will be object of calculation. The damage
range of the pier of the Hansui R5 has been calculated, based on their energy dissipation.
The assumption that more energy dissipated means potentially a bigger range anddegree of damage allows a clear and practical assessment of damage sustained because
of seismic excitation. Figure 5.4 shows the damage of eight piers that are part of thetarget.

Energy Dissipation (KN-m)

1400
1200

1000
800

Total Energy Dissipation

600
Bottom Layers Energy
Dissipation

400
200
0
P1

P2

P3

P4 P5
TargetP6

P7

P8

Figure 5.4 Damage distribution evaluation of Hansui R5 piers
81The damaged has been calculated to the full height of the piers, as well as bottom layers
corresponding to a height of 2100 mm. from the analysis pier P5 in the presents thebiggest damage range equivalent to a energy dissipation of 1203 kN-m, while pier P2
presents the least damage range which is equivalent to 925 kN-m. In general, the region
composed by piers P3, P4, P5 and P6 has the biggest damage range, and in the structure
this region corresponds to the transition zone from one set of piers to the following,with no stiffness beam connecting them, as visible in Figure 5.3
The calculations of damage range at the bottom of the columns suggest that about 40%
to 60% of the total damage was there located. This proved totally independent from theactual total energy dissipation. In other word piers P6 and P8 had about 60% of damage
concentrated at the bottom, the maximum percentage, but their total energy dissipation
does not correspond to the maximum values if the entire targets are object ofcomparison, thus proving the complex and nonlinear nature of this estimations.
Figure 5.5 shows the location of the joint within pier P1. The energy dissipation at thejoint has been calculated, in order to understand its contribution to the general state ofdamage range of P1. The results are presented in Figure 5.6. The total energy
dissipation at the joint is 147 KN-m which corresponds to 13% of the total energy
dissipation of the pier. The energy was calculated considering the four arch membersthat are part of the joint lattice model structure. The value of energy dissipation at thejoint and thus damage range is small when compared to the contribution of the bottom
layer.

Figure 5.5 Damage range evaluation of P1 joint82

Energy Dissipation (kN--m)

1000
800

600

Energy Dissipation(kN-m)

400

200
0
P1

Joint

TargetFigure 5.6 P1 and P1 joint comparative Energy Dissipation

In other words the level of damage range at the top of the column is much less than thelevel of damage at the bottom, and the joint, although it represents a portion of
discontinuity in the columns, that is the behavior is affected by the connection to thebeam, showed reduced damage range level. This is most likely related to the stiffness
the connection beam introduced to the system at that level.
Figure 5.7 shows the actual damage conditions of the Hansui R5 viaduct after theearthquake.

Figure 5.7 Damaged conditions of Hansui R5 (Miki.2004)
83

It is possible to visualize at first glance that the biggest range of damage is at the bottomof the piers. The damage range analysis based on energy dissipation correctly predictedthat this would be the actual damage situation. Looking closer at the joint of the piers,
the level of damage is not as elevated as it at the bottom, and this is related to thedecreased level of energy dissipation at this point that was calculated using the 3D
lattice model. Although very complex, the level of response and damage was acceptably
predicted by the 3D Dynamic Lattice Model, especially the damage range evaluationconsidering the nonlinearities, by using the stress-strain relationships for the calculationof strain energy, further applied for the estimation of energy dissipation in order topredict damage range.
By comparing between the actual damage condition and the analytical response of RC
viaducts, the validity of the 3D dynamic lattice model has been confirmed at thestructural system level. The analytical response predicted by the 3D dynamic latticemodel is used to evaluate the seismic performance of RC structures.
The maximum displacement during the earthquake and the residual displacement after
the earthquake are useful to evaluate the possibility of the restoration or rehabilitation
after the earthquake occurs. In order to evaluate the seismic performance of a RC
structure, the predicted response, such as the maximum and residual displacements, ca
be compared with the limiting values determined from the importance of target
structures.

5.3. Numerical Verification of Seismic Performance of Hansui R5 Viaduct
In order to perform the numerical verification of seismic performance of Hansui viaduct,
the analytical target has been subjected to the motion presented in Figure 5.2. In orderto perform the simulation, the footing is not modeled and at the bottom of each lower
column a fixed support is provided. In this analysis a personal computer Pentium 4 with
1.7 GHz was used and the total computation time is about 3 hours, where the total time
and time intervals were set to be 10 sec and 0.01 sec, respectively (Miki 2004).
The numerical verification method fully detailed in Chapter 4, was applied to thecolumns of Hansui viaduct and the analytical results are described in Table 4. The time
history is presented in Figure 5.8; in the figure it is visible that the calculation fails at 4
seconds of acceleration. Therefore the numerical varication has limited time scope.
84

Longitudinal
Displacement (mm)Displacement (mm)

150
100
50
0
-50
-100
150
100
50
0
-50
-100
-150

Transverse

0

2

4

6

8

150
100
50
0
-50
-100
150
100
50
0
-50
-100
-150

10

Longitudinal

Transverse

0Time (sec)

6

8

10

150

Displacement in longitudinal
direction (mm)Displacement in longitudinal
direction (mm)4Time (sec)

150
100
50
0
-50
-100

2

Direction dropped after
failure in the analysis-150-150 -100 -50

0

50

100

100
50
0
-50
-100
-150
-150 -100 -50

150Displacement in transverse direction (mm)

0

50

100

150

Displacement in transverse direction (mm)Figure 5.8 Analytical responses for Hansui R5 Viaduct
The analytical results shown in Figure 5.8 represent the response obtained at different
target points marked in black dot in the figures From the response it is possible toobserve that the longitudinal direction is dominant with in both cases for the target point
and in the case represented in the right in the figure, where the target point is an innerbeam, the maximum observable displacement is about 100mm compared to the casepresented to the left where it is a bit over 50mm. This might be to the stiffening effectobserved in the outer layer members, which present stiffer response when subjected toseismic motion.
On the other hand, one of the key objectives of this work is to make use of energy
dissipation criteria to evaluate the performance of RC structures under seismic motion.
With that, the analytical results offer a great chance to understand the performance ofthe structure from the material point of view up to the moment the calculation isinterrupted. The resumed calculations regarding the seismic performance evaluation of
the Hansui viaduct from the material point of view are shown in Table 4.
85

Table 4: Seismic performance evaluation of Hansui viaduct
Damage Limitation Point

Damage Index (DI)Seismic Performance

ColumnConcrete

Steel

Concrete

Steel

Concrete

Steel

P1

0.4

0.00380.05

0.00072

SP-I

SP-I

P2

0.4

0.00380.06

0.00070

SP-I

SP-I

P3

0.4

0.00380.06

0.00078

SP-I

SP-I

P4

0.4

0.00380.07

0.00087

SP-I

SP-I

P5

0.4

0.00380.08

0.00090

SP-I

SP-I

P6

0.4

0.00380.07

0.00089

SP-I

SP-I

P7

0.4

0.00380.06

0.00079

SP-I

SP-I

P8

0.4

0.00380.07

0.00087

SP-I

SP-I

The columns of Hansui viaduct present the same material properties and distribution ofreinforcement and because of that looking at the table the damage limitation point is thesame for all the targets. On the other hand the damage index is calculated for each target
and compared to the damage limitation point to understand the performance of thematerials at 4 seconds of shacking. The overall results suggest that both materials
remain structurally sound after the first 4 seconds. For concrete in all 8 peers, theseismic performance level is SP-I, which corresponds to the first level. At the structural
damage level in concrete in all 8 peers, cracking of concrete as well as some level ofspalling can be expected in visual inspection. On the other hand for the case of steel
reinforcement in all 8 peers, the seismic performance category is SP-I. At the material
level the maximum observable damage condition expected in yielding of reinforcement.
The analytical method proposed can generally predict the damage states from anumerical point of view. However it is necessary to conduct some sensitive analysis on
the values presented by the methods in order to be better informed about the realdamage condition of structures. With that, comparing the values of the damage indexes
calculated for concrete and for steel with the damage limitation point of each material it
is possible to understand that numerically they are considerably apart. In other words,
the damage indexes are considerably smaller than the damage limitation points. In thedefinition of damage limitation point, it has been said that this corresponds to thethreshold value where the damage potential of the material changes from moderate toconsiderable, and based on this definition at 4 seconds of shacking, it is possible toconclude that both material still present a considerable sound state.
865.4. Seismic Design and Retrofit Considerations based on 3D Lattice Model
The standard goal of every analytical model in seismic analysis is to contribute to abetter understanding of the complex factors that affect the behavior of RC structures inseismic events. Miki (2004) states in his doctoral thesis that one of the key concepts the
updated JSCE standard specification establish is the concept that inelastic deformation
can considerably be observed in RC structures after the longitudinal reinforcement
yields, in opposition to the idea that only plastic behavior of structures subjected tostrong motion is considered. On the other hand Priestley et al. (1996) stated that the useof elastic design promotes a false sense of the response levels to be expected underseismic attack and typically will result in severely underestimated displacements while
encouraging designers to ignore aspects of ductility and rational hierarchy of strengths.
In current design practice a large consideration should be given do ductility aspects.
Ductility being defined as: the ability of the structural member to displace inelasticallythrough several cycles of response without significant degradation of strength or
stiffness. This is specially the case for bridge columns, where sudden loss of strength or
stiffness can have catastrophic results and ultimately lead to structural failure andcollapse. In the discussion pertaining ductility, strength, stiffness, energy dissipation anddeformation are at the center of discussion. Although mathematically ductility is definedas the ration between deformation at a certain level of response to the level of
deformation at yielding, understanding strength, stiffness, energy dissipation anddeformation allows a much more informed decision on design aspects of RC bridge
elements, especially columns
The contribution of the 3D lattice model for the design and retrofit of RC structures canbe seen from a multitude of approaches. Recent development in earthquake engineering
allow the improved estimation of earthquake input ground motions and thus allowing
the dynamic analysis using input ground motion that has been recorded during actual
earthquakes to be used in the performance evaluation of RC structures. With adiscretization method that is simplified, the 3D lattice model allows the typical use of
dynamic analysis using actual ground motion to relatively easily estimate the seismic
performance of existing RC structures while retaining reasonable computational time.
On the other hand, aspects of design can be seen through the lenses of Priestley et al.
(1996). According to their discussion, in design several different measures of strengthmust be considered. They discuss required strength, nominal strength, expected strength,87

dependable strength, extreme strength, ideal strength and according to their discussion,in design, the presented strength characteristics follow a Gaussian or bell shaped
distribution, on a relationship between strength and frequency. That evaluation shows
how important it is to correctly grasp the response strength of a structure under seismic
loading. Throughout the analysis the 3D lattice model has shown consistency incorrectly predicting the strength to a certain load. The evidence of that is the extensive
comparison between analytical results and experimental information to give proof or
reliability. In that sense it can be useful to use to 3D lattice model to evaluate thestrength of RC structural members.
The contribution of the 3D lattice model can be extended to aspects of retrofit as well. It
is commonplace that still existing RC bridges have been designed without clear
understanding of performance states or performance demand in actual seismic events.
The result of that is that many RC bridges have been designed to substandard codes andso in seismic events, their performance is not to be taken for granted and it is necessary
to go back and verify the expected levels of performance under seismic loading so thatappropriate retrofitting action can be taken.
With a better understanding of site seismicity, which might be the single most important
aspect in retrofit consideration, the modeling and simulation of existing RC bridges caneasily be performed under dynamic analysis using the 3D lattice model. A simple
example of the applicability of the 3D lattice model is in the retrofitting of RC Columns.
In many cases they are retrofitted using steel jackets, which act as an extremely efficienttransverse reinforcement. In that manner it is possible to simulate different transverse
reinforcement ratios using the 3D lattice model to make a more informed decision on
the steel jacketing to be used.

References in Chapter 5
Miki, T.(2004): Nonlinear Analysis of Reinforced Concrete Structures Subjected to
Seismic Loads by Using Three-Dimensional Lattice model, Tokyo Institute ofTechnology, Doctoral Thesis, March
Priestley, M.J.N., Seibe, F. and Calvi, G.M.(1996): Seismic Design and Retrofit ofBridges, Willey and sons, March88

6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH
6.1. General Conclusions
This study presented the performance evaluation of reinforced concrete (RC) structuressubjected to seismic loading using 3D lattice model. The concept of 3D lattice model
defines the shear resisting mechanism in terms of arch and truss actions in RC structuralmember. The truss action is comprised of an orthogonal coordinate system defined bythree planes, as for the arch part, the internal stress flow is idealized as the compressive
strut. The fundamental 3D lattice model has been developed for a quadrangular crosssection target structure. In this study an expansion of scope is presented, with thedevelopment of 3D lattice model that targets circular cross section columns.
Furthermore the damage analysis is performed and taken further by proposing a seismic
performance evaluation method. With that a few important conclusions may be taken.
The applicability of the 3D lattice model to perform static analysis on reinforced
concrete columns with circular cross section is verified through the application of the
3D lattice model previously developed, to a circular cross section column, considering
equivalence in cross-sectional area in geometry to a rectangular cross section. A secondand new approach is proposed, the multi-directional polygonal 3D lattice model, withmore realistic geometrical discretization.
Looking at the analytical results, the circular to rectangular equivalence in crosssectional area shows acceptable performance in response prediction based on initialstiffness, however it`s applicability has a larger scope in 2D analysis, in 3D analysis amore realistic discretization is obtained using the multi-directional polygonal 3D latticemodel. The performance of the multi-directional polygonal 3D lattice model isconsidered acceptable both in terms of general response prediction of the envelope
curve comparing initially to linear-elastic curve obtained from analysis of large-scale
bridge pier followed by analysis performed on a column subjected to monotonic loading
test. In both cases good agreement was observed in initial stiffness of the columns whenreduction factors of cross-section area of concrete part of RC column are used. However,comparing analysis to the experimental results it was observed that the initial responsestrength is slightly overestimated.
The applicability of the 3D lattice model to perform dynamic analysis for reinforced
concrete columns with circular cross section is proposed. The analytical results show89that before cracking, the geometrical equivalence between circular and square crosssectional areas produces softer and more acceptable results, especially with relation toinitial stiffness. On the other hand, in the inelastic range, the multi-directional polygonal
3D lattice model shows reasonable accuracy in hysteresis analysis, either considering
the reduction of shear resisting area of concrete or reduction in the flexural stiffness that
occurs in seismic response of RC members.
In this research the multi-directional polygonal 3D lattice model is proposed in order toperform cumulative seismic damage evaluation of circular cross section RC columns
under multiple loading sequences. The analytical results suggest that after the first shakethe model works in an acceptable way to predict the transition from one loading
sequence to the following, as well as residual displacement in longitudinal direction,and the energy dissipation capacity for one, two or three of loading sequences intransverse direction.
In the analysis, the hysteretic response should be conjugated with the energy dissipation
capacity and the residual displacement in order to have a more complete evaluation of
the response. This need is proved by the increased complexity in the nonlinear behavior
that is introduced by shaking the analytical target to more than one excitation.
The applicability of this model to perform cumulative damage analysis is verified undera limited scope. Based on that, for further stages of this study, the research should focus
on reducing the intrinsic initial high initial stiffness observed in elastic response, as it
influenced the stiffness degradation in inelastic phase in some cases. Furthermore,
underestimation of response displacement in higher order shaking sequences should
also be closely looked at.
The applicability of the 3D lattice model to perform damage evaluation of reinforced
concrete columns with circular cross section is proposed using the multi-directional
polygonal 3D lattice model. The analytical results confirm the applicability of the model,
especially the hysteretic response after cracking of concrete. Damage evaluation has
been proposed from the material point of view considering the accumulated energy
dissipation of concrete and reinforcement, which are dependent on strain energy.
Analytical results show that by large scale, steel reinforcement is the most dominantmaterial in seismic resistance capacity.

90The analysis of damage considering the critical damage condition point (CDP), allowed
the reasonable prediction of real structure damage condition. At this point in thisresearch, suffices to assume that damage range potential between 0 and CDP ismoderate, while damage range from CDP to 1 is consistent with more severe damage
potential, especially for concrete which presents less ductility capacity when compared
to steel reinforcement.
In this study, numerical verification of seismic for circular RC columns using
multi-directional polygonal 3D lattice model is proposed. From the analysis, it isverified that based on static analysis, it is possible to acceptably reduce the high elastic
stiffness observed in the modeling previously proposed based on the dynamic analysis.
This suggests that a more fundamental treatment using static analysis should be given to
the analytical target when the main focus of analysis is the elastic range.
The damage index and seismic performance categories, allows the numerical evaluationof performance of material based on the seismic performance criteria broadly describedby JSCE and fundamental experimental work performed by Kawashima et al. (2010)
which are the basis considered in this study to co-relate damage index and structural
performance evaluation from the material point of view.
Furthermore a full framed rail road viaduct was subject to damage analysis using energy
dissipation developed from 3D dynamic analysis. The target structure was Hansui R5
viaduct. The results from the analysis have good level of agreement with the actual
damage structure when compared. From the analysis the bottom layer was the mostdamaged position of the piers, very much visible in the post-damage figure presented.
Furthermore, the joint of pier P1 was subjected to analysis, and based on its energy
dissipation which was about 13% of the total for the pier, it was concluded that itpresented a very small level of damage when compared to the remaining parts of thestructure, especially the bottom.

6.2. Recommendations for Future Research
While conducting this research the complex nature of the issues studied was always
present and throughout the study, decision on simplification of procedure andassumptions had to be made. With that certain aspects remain to be further researched toachieve better results.
91

In the modeling of RC columns using the multi-directional polygonal 3D lattice model
there should be a profound experimental study to clarify the arch action in circularcolumns. In this study although an analytical approach was presented using the direct
stiffness method, a more fundamental experimental and analytical research on the arch
action in circular columns is needed to reduce the degree of uncertainty.
In the discretization of circular columns using the multi-directional polygonal 3D latticemodel for the truss part, a separation between Inner Diagonal Members (IDM), Surface
Diagonal Members (SDM) and Diagonal Members in Transverse direction (DMT) has
been used. This was the geometry that was found to best work at this stage and stableenough to conduct the analysis. However other geometries should be further
investigated and in the current research elemental cross section of diagonal members in
the multi-directional polygonal 3D lattice model has been proposed using simplified
formulas, but the calculation of their cross sectional area can be better achieved using
more sophisticated formulation.
Still regarding the modeling using the multi-directional polygonal 3D lattice model, inthe case of dynamic analysis, high plastic stiffness is observed in the response. In thisstudy the applicability of the dynamic analysis was focused on the inelastic range andthus the initial stiffness of the model overestimation was not profoundly investigated.
On further research some investigation on how to reduce the elastic stiffness should be
performed, especially if the focus is the elastic design of RC members.
One of the key features of the 3D lattice model is the reduced need for computationaltime due to the reduced number of degrees of freedom. So with that in analysis,
especially the geometrical modeling it is important to keep in mind that choices have to
be made so that the computational requirements of the analysis will not grow at a very
fast pace, and it is also recommendable that computers with strong memory are used foranalysis, especially for the analysis of viaducts.

6.3. Further Contributions and Discussion
Based on the discussions and recommendations from the thesis defense sessions, anumber of points for further contribution and discussion in the refinement of scope of
this research have been presented. From a general point of view, it is important toclearly understand the scopes of the analytical contributions of the study based on a92

laboratory controlled environment to a more practical design environment. In this study,
in order to confirm the applicability of the procedures proposed, the analyticalinformation is compared to experimental information. However, in an actual RC
structural member, the degree of complexity in response is potentially larger than theobserved in laboratory environment, therefore it is important to effectively conduct
sensitivity analysis in the interpretation of the analytical and experimental in order toextend the applicability to real RC structures.
Furthermore from a general point of view, there is a big trend in the use of Finite
Elements Methods (FEM) as well as others such as the fiber model, already described inthis thesis and the Distinctive Elements Methods (DEM). All these analytical models
have strong points and weak points, but one of the successes that the Lattice Modelachieves when compared to them is the ability to give good prediction of flexural andshear behavior and obtain the appropriate failure mode, which can be verified
experimentally while allowing the discretization of RC structural members with areduced number of degrees of freedom. The consequences of that are that in one hand,the calculation time is greatly reduced. For example for a typical column in this study,
the calculation time took more than only a couple of minutes. On the other hand theunderstanding of the failure mode appropriately allows a much more efficient
application of the Lattice Model in the damage evaluation of RC structures, which isbased on the understanding of failure modes and grasping real damage progress in
structure.
From a specific point of view, another important contribution has been offered. In thecomparison between analytical results and the experimental results it is important tokeep in the interpretation the effect of pull-out of longitudinal rebar in the footing. The
question is that there is a size-dependency in the sense that in smaller column samples,the effect is much more severe than in larger samples. Therefore on a case-by-case basis,
it is important to analyze carefully this effect, especially in the case where there isunderestimation of the effect such as in smaller column samples.
The lattice model showed good ability to discretize circular columns. Based on that, afurther extension of scope would be to study its applicability to study RC piles. Basedon the geometry alone, the multi-directional polygonal 3D lattice model can discretize
RC piles, however, further research is need in the constitutive modeling as well as soil
and structure interaction models.
93