Seismic Analysis of Existing Bridges with
Detailing Deficiencies
A Dissertation Submitted in Partial Fulfilment of the Requirements
of the Master Degree in
Earthquake Engineering
By
Davide Kurmann
Supervisor(s): Prof. Dr. A Dazio, IBK, ETH Zürich
February, 2009
Istituto Universitario di Studi Superiori di Pavia
Università degli Studi di Pavia
The dissertation entitled “Seismic Analysis of Existing Bridges with Detailing Deficiencies”,
by Davide Kurmann, has been approved in partial fulfilment of the requirements of the Master
Degree in Earthquake Engineering.
Prof. Dr. A. Dazio …… …
Mr. M. Bimschas………… …
………
……
Abstract
ABSTRACT
The main objective of this work is to investigate the influence of detailing deficiencies, such as low
transverse reinforcement ratios and lap-splices in potential plastic hinge regions of existing bridge
piers with wall-type cross sections, as compared with equivalent members properly detailed to
undergo large inelastic deformations without loss of lateral load carrying capacity when subjected to
earthquake excitations.
Both, capacity and demand were investigated, by performing inelastic time-history analyses on singledegree-of-freedom systems representing isolated bridge piers and multi-degree-of-freedom systems as
models for entire bridge structures subjected to transverse earthquake excitations. Special attention
was focused on the potentially increased demand as a result of initiation of strength degradation.
Accurate modelling of the monotonic and cyclic behaviour of the considered bridge piers has been
achieved by means of numerical analyses, existing capacity prediction methods and experimental
observations. Although, previous studies and theories demonstrating strength degradation dealt almost
exclusively with splice lengths of 20dbl and were focused on square columns, experimental evidence,
based on quasi-static cyclic loading tests carried out at the ETH Zürich [Bimschas et al., 2008], was
used to characterize the hysteretic behaviour of the bridge piers considered in this study. The ETH
experiments were performed on two identical wall-type bridge piers (aspect ratio h/lw=2.2) with the
only difference being a 43dbl lap-splice of the longitudinal reinforcement in one test unit.
Results of dynamic analyses show, that differences arising from ductility-based versus energy-based
strength degradation rules do not affect maximum displacement demands, if energy dissipation is
properly accounted for. Moreover results indicate, that for members with spliced longitudinal
reinforcement at pier base and deformations above initiation of strength degradation, an increased
displacement demand arises compared with same member detailed with continuous longitudinal
reinforcement. Nevertheless, due to redundancy of entire bridge structures, modelled as multi-degreeof-freedom systems with elasto-plastic abutments, degradation effects are less important than in the
case of isolated piers (single-degree-of-freedom systems) subjected to same input motion.
A displacement factor is proposed in order to account for higher pier demands arising from strength
degradation when predicting displacement demand from simplified procedures.
Keywords: bridge assessment; pier; strength degradation; lap-splice; abutment
i
Acknowledgements
ACKNOWLEDGEMENTS
It has been a pleasure and honour for me, to take part in the master program at the ROSE School. I
really appreciate the time spent in Pavia with its unique opportunity to encounter teachers and
classmates from all over the world with such different backgrounds and cultures but a common
passion for Earthquake Engineering. In particular, I will never forget the good time spent at the CAR
College and in Pavia with Antonio and his family, Riccardo, Beatriz, Angie, Sara, Ilaria, Fabrizio,
Antonio, Carlos, Joao, Mario, Gabriele, Giovanni and Tiziana with Marianella and Cristina.
Moreover, I would like to especially thank Prof. H. Bachmann, Dr. M. Koller, Dr. D. Somaini and
Mr. B. Basler, Mr. H. Hader, Mr. Beat Weiss, for the encouragement, trust and financial support over
the past year. Attending the master program at ROSE would never have been possible for me without
the scholarships provided by the Foundation for Structural Dynamics and Earthquake Engineering and
Ernst Basler and Partners Ltd., the company I’m working for.
During the last three months of the master program I had the possibility to work on my thesis at the
Institute for Structural Engineering of the ETH Zürich under the supervision of Prof. A. Dazio with the
support of Mr. M. Bimschas. For the guidance and patient support, they gave me throughout this time
I express here my sincere gratitude.
Finally I could never describe in words, the patience, support, encouragement and love of my wife,
Evelyn. Her unlimited and contagious “joie de vivre” accompany me every day and are simply
irreplaceable. I’m so proud of you!
ii
Index
TABLE OF CONTENTS
Page
ABSTRACT ............................................................................................................................................i
ACKNOWLEDGEMENTS....................................................................................................................ii
TABLE OF CONTENTS ......................................................................................................................iii
LIST OF FIGURES ...............................................................................................................................vi
LIST OF TABLES................................................................................................................................xx
LIST OF SYMBOLS .........................................................................................................................xxiii
1 INTRODUCTION ...........................................................................................................................28
1.1 Statement of the problem .........................................................................................................28
1.2 Definition of terms and previous research ...............................................................................28
1.2.1 Strength degradation ......................................................................................................28
1.2.2 Lap-splice failure ...........................................................................................................30
1.3 Objective and scope of the work..............................................................................................31
2 OVERVIEW OF THE REPORT.....................................................................................................32
2.1 Previous research .....................................................................................................................32
2.2 Experimental data used in this work ........................................................................................33
2.3 Evaluation of monotonic and cyclic member response............................................................34
2.4 Definition of target seismicities ...............................................................................................34
2.5 Single-degree-of-freedom-analyses .........................................................................................35
2.6 Multi-degree-of-freedom-analyses...........................................................................................36
3 PREVIOUS STUDIES ON STRENGTH DEGRADATION..........................................................37
3.1 Strength degradation model according to [Priestley et al., 1996]............................................37
3.2 Experimental studies................................................................................................................38
3.2.1 Chai et al., 1991 .............................................................................................................38
3.2.2 Lynn et al., 1996 ............................................................................................................39
iii
Index
3.2.3 Melek et al., 2004...........................................................................................................41
3.2.4 Considerations on test specimen from past studies........................................................42
3.3 Comparison of experimental evidence with the method according to [Priestley et al., 1996] 43
3.3.1 Unit with circular a cross section...................................................................................43
3.3.2 Units with quadratic cross sections................................................................................44
3.3.3 Summary considerations ................................................................................................45
4 EXPERIMENTAL DATABASE ....................................................................................................46
4.1 Motivation of experimental work ............................................................................................46
4.1.1 Code provisions dependence on required splice length .................................................46
4.1.2 Experimental test units...................................................................................................46
4.1.3 Effects of cross section type on member .......................................................................47
4.2 Test setup and units..................................................................................................................47
4.3 Quasi-static cyclic experiments ...............................................................................................49
5 FUNDAMENTAL MODELLING ASSUMPTIONS .....................................................................51
5.1 Sectional and member response...............................................................................................51
5.1.1 Flexural response without lap splice..............................................................................51
5.1.2 Flexural response with lap splice ...................................................................................54
5.1.3 Shear capacity envelope.................................................................................................56
5.2 Modelling of reinforced concrete members without lap splices ..............................................58
5.2.1 Monotonic member behaviour .......................................................................................58
5.2.2 Cyclic member behaviour ..............................................................................................59
5.3 Modelling of reinforced concrete members with lap splices ...................................................62
5.3.1 Monotonic member behavior .........................................................................................62
5.3.2 Cyclic member behaviour ..............................................................................................63
5.4 Calibration of monotonic and hysteretic behaviour on experimental yielding points .............66
5.4.1 Monotonic member behaviour .......................................................................................67
5.4.2 Cyclic member behaviour ..............................................................................................68
5.5 Scaling Factors.........................................................................................................................68
6 TARGET SEISMICITY FOR THE ANALYSES...........................................................................70
6.1 Selection of target response spectra .........................................................................................70
6.1.1 Selection and scaling procedure of real ground motions ...............................................71
6.2 Artificial Ground Motion Database .........................................................................................74
6.2.1 Generation Procedure.....................................................................................................74
7 SINGLE-DEGREE-OF-FREEDOM ANALYSES .........................................................................76
7.1 Introduction..............................................................................................................................76
7.2 Evaluation of responses from NLTHA on SDOF systems ......................................................79
iv
Index
7.2.1 Influence of strength degradation modelling .................................................................79
7.2.2 Influence of reinforcement detailing..............................................................................82
7.2.3 Influence of bridge pier height.......................................................................................84
7.2.4 Influence of damping ratio.............................................................................................89
7.2.5 Influence of critical damping coefficient .......................................................................90
7.2.6 Displacement increasing factors f(δ) and f(μ) ...............................................................92
7.3 Simplified procedures for maximum displacement demand prediction...................................95
7.3.1 Brief overview and main considerations........................................................................95
7.3.2 Miranda et al., 2003 .......................................................................................................97
7.3.3 Priestley et al., 2007.....................................................................................................100
7.3.4 Iwan, 1980....................................................................................................................109
7.3.5 Guyader et al., 2004 .....................................................................................................114
7.3.6 Probabilistic evaluation of simplified displacement prediction procedures.................122
8 MULTI-DEGREE-OF-FREEDOM ANALYSES.........................................................................129
8.1 Introduction............................................................................................................................129
8.2 Parametrical study on abutment elastic stiffness ...................................................................132
8.2.1 Bridge model M1-CRA-SLM-EL................................................................................134
8.2.2 Bridge model M2-CRA-SLM-EL................................................................................135
8.2.3 Bridge model M3-CRA-SLM-EL................................................................................136
8.2.4 Bridge model M4-CRA-SLM-EL................................................................................137
8.2.5 Conclusions..................................................................................................................138
8.3 Influence of spliced reinforcement at pier base .....................................................................139
8.3.1 Bridge model M1 .........................................................................................................140
8.3.2 Bridge model M2 .........................................................................................................142
8.3.3 Bridge model M3 .........................................................................................................144
8.3.4 Bridge model M4 .........................................................................................................146
8.4 Discussion of results from multi-degree-of-freedom analyses ..............................................148
9 CONCLUSIONS AND OUTLOOK .............................................................................................153
9.1 Conclusions............................................................................................................................153
9.1.1 General considerations.................................................................................................153
9.1.2 Single-degree-of-freedom analyses..............................................................................154
9.1.3 Multi-degree-of-freedom-analyses...............................................................................155
9.2 Outlook for further studies.....................................................................................................155
10 REFERENCES ..................................................................................................................................1
v
Index
LIST OF FIGURES
Page
Figure 1.1: Classification of reinforced concrete column failure modes as ATC-6 (1981)...................29
Figure 1.2: Shear failure prevention due to shear strength degradation in lap-splice............................30
Figure 1.3: Lap-splice failure of longitudinal bars in columns [Priestley et al., 1996] .........................30
Figure 2.1: Examples of force displacement relation according to [Priestley et al., 1996] method
versus experimental results on rectangular column units performed by [Melek et al., 2004].......32
Figure 2.2: Examples of typical bridge column cross section [Bimschas et al., 2008] .........................33
Figure 2.3: Experimentally recorded hysteretic (force deformation) behaviour of test unit VK1 (left,
continuous reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008] .......33
Figure 2.4: Experimentally obtained cyclic behaviour of test units VK1 (left) and VK2 (right) versus
hysteretic loops using software code Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006] ................34
Figure 2.5: Assumed elastic response spectra of acceleration (left) and displacement (right) according
to code provisions [SIA 261, 2003] for PGA of 0.16g, 0.35g, 0.50g (agd), stiff soil conditions
(Soil B), structure importance factor III and elastic viscous damping ξ=0.02 .............................35
Figure 2.6: Idealized single-degree-of-freedom systems representing a single bridge pier with lumped
mass subjected to transverse earthquake excitation (left). Front (middle) and lateral view (right)
of a typical bridge pier and superstructure considered in this study..............................................35
Figure 2.7: Example of bridge model (top). Superstructure and piers maximum displacement demands
from single NLTHA and PGA 0.35g are represented by grey lines. Mean NLTHA displacement
pattern (black thick line) and pier displacement demands P1 to P4 (points) are evidenced for case
of continuous longitudinal reinforcement (centre) and spliced longitudinal reinforcement at pier
base (bottom). ................................................................................................................................36
Figure 3.1: Idealized moment-curvature relationship for members without or with lap-splices in
potential plastic hinge regions [Priestley et al., 1996]...................................................................37
Figure 3.2: Tension stress induced by force transfer in lap splices [Priestley et al., 1996]...................38
Figure 3.3: Test setup and reinforcement details for the columns tested by [Chai et al., 1991]............39
vi
Index
Figure 3.4: Hysteretic response of reference columns without (left) and with lap-splice (right) from
experiments performed by [Chai et al., 1991]...............................................................................39
Figure 3.5: Load assembly and specimen location [Lynn et al., 1996] .................................................40
Figure 3.6: Column test unit and cross section with different transverse detailing [Lynn et al., 1996] 40
Figure 3.7: Load-displacement relations for the test units with spliced reinforcement [Lynn et al.,
1996]..............................................................................................................................................40
Figure 3.8: Test setup, reinforcing details and unit cross sections [Melek et al., 2004].......................41
Figure 3.9: Normalized moment drift relations for all test units [Melek et al., 2004]..........................41
Figure 3.10: Example of a force-displacement rel. computed according to [Priestley et al., 1996]
compared to the experimental results of a circular column (left) and relevant testing setup (right)
[Chai et al., 1991] ..........................................................................................................................43
Figure 3.11: Examples of force-displacement relationships computed according to [Priestley et al.,
1996] compared to experimental results of rectangular columns ..................................................44
Figure 4.1: Examples of typical bridge column cross section ...............................................................47
Figure 4.2: Setup for the quasi-static test on existing Swiss bridge piers carried out at the ETH Zürich
[Bimschas et al., 2008] ..................................................................................................................47
Figure 4.3: Cross section, reinforcement detailing and elevation of test units VK1-2 [Bimschas et al.,
2008]..............................................................................................................................................48
Figure 4.4: Experimentally recorded hysteretic (force deformation) behaviour of test unit VK1 (left,
continuous reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008] .......49
Figure 4.5: Quasi-static cyclic displacement application on test unit VK1 (left, continuous
reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008] ..........................49
Figure 4.6: Quasi-static cyclic horizontal load application on test unit VK1 (left, continuous
reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008] ..........................50
Figure 5.1: Curvature ductility for members without and with lap splices in column end regions .......56
Figure 5.2: Analytical and experimental results for test unit VK1 (continuous longitudinal
reinforcement) ...............................................................................................................................58
Figure 5.3: Hysteretic rules used for the analysis: Modified Takeda with degrading stiffness (left,
IHYST=4) and Origin-Centred (right, IHYST=7) according to Ruaumoko [Carr, 2004].............59
Figure 5.4: Comparison of analytical and experimental hysteretic responses for test unit VK1...........60
Figure 5.5: Hysteretic area for damping calculation in one cycle of loading [Priestley et al., 2007]....61
Figure 5.6: Comparison of analytical and experimental cumulative dissipated energy ........................61
Figure 5.7: Analytical and experimental results for test unit VK2 (spliced longitudinal reinforcement)
.......................................................................................................................................................62
Figure 5.8: Strength degradation model implemented in Ruaumoko [Carr, 2004] ...............................64
vii
Index
Figure 5.9: Ductility based (left) and energy based (right) strength degradation in Idarc [Idarc, 2006]
in a force-displacement-relationship..............................................................................................64
Figure 5.10: Comparison of analytical and experimental hysteretic responses for test unit VK2.........65
Figure 5.11: Comparison of analytical and experimental cumulative dissipated energy ......................66
Figure 5.12: Capacity curve envelope for test specimen VK1 and VK2 based on experimental data
according to [Bimschas et al., 2008] .............................................................................................67
Figure 5.13: Experimentally obtained cyclic behaviour of test units VK1 (left) and VK2 (right) versus
hysteretic loops using software code Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006] based on
backbone assumptions presented in Table 5.11 and Figure 5.12 (left).........................................68
Figure 6.1: Assumed elastic response spectra of acceleration (left) and displacement (right) according
to code provisions [SIA 261, 2003] for PGA of 0.16g, 0.35g, 0.50g (agd), stiff soil conditions
(Soil B), structure importance factor III and elastic viscous damping ξ=0.02 .............................70
Figure 6.2: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching
procedure on displacement spectra for different period ranges.....................................................73
Figure 6.3: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching
procedure on displacement spectra for a period range of T=0.50-2.00sec. (these motions are
identified as RGM) ........................................................................................................................74
Figure 6.4: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching
procedure on displacement spectra for a period range of T=0.35-1.00sec. (these motions are
identified as RGM N) ....................................................................................................................74
Figure 7.1: Idealized single-degree-of-freedom systems representing a single bridge pier with lumped
mass subjected to transverse earthquake excitation (left) and front, lateral view of a typical bridge
pier and superstructure considered in this study (middle, right)....................................................76
Figure 7.2: Capacity curves from Table 7.1 displayed in Acceleration Displacement Response Spectra
(ADRS) versus seismic demands (left) and force-deformation relations of
bridge piers
considered in dynamic single-degree-of-freedom analyses...........................................................77
Figure 7.3: Nomenclature used in this study for dynamic analyses on single-degree-of-freedom
systems ..........................................................................................................................................78
Figure 7.4: Comparison of hysteretic loops obtained with Ruaumoko [Carr, 2004] vs. Idarc [Idarc,
2006] in dynamic analyses on single-degree-of-freedom systems for input motion AGM44 and
2% damping ratio. Backbone curve without strength degradation(left) and with strength
degradation (right) .........................................................................................................................79
viii
Index
Figure 7.5: Comparison of total dissipated hysteretic energy for analyses carried out using Ruaumoko
[Carr, 2004] vs. Idarc [Idarc, 2006], considering members with continuous and spliced
reinforcement, experimental and numerical backbone assumptions for systems with 100H100MI.......................................................................................................................................................80
Figure 7.6: Influence of software code on maximum displacement demand for member with
continuous reinforcement and influence of strength degradation modeling (energy vs. ductility
based) for members with spliced longitudinal reinforcement at pier base. Legend: input ground
motion (AGM = Artificial, RGM, Real) and backbone assumption (Exp=Experimental,
Num=Numerical)...........................................................................................................................81
Figure 7.7: Influence of strength degradation modeling (energy vs. ductility based) on lateral force
carrying capacity corresponding to maximum displacement demand for members with spliced
end reinforcement. Legend: input ground motion (AGM = Artificial, RGM, Real) and backbone
assumption (Exp=Experimental, Num=Numerical) ......................................................................82
Figure 7.8: Maximum displacement demand ratio from NLTHA as a function of drift (left) and
displacement ductility (right) based on experimental backbone assumption. Analyses assumed a
SDOF system with a pier height of h=6.6m and a lumped mass of m=550t. Aspect ratio
correspond to h/lw = 2.2 .................................................................................................................83
Figure 7.9: Maximum displacement demand ratio from NLTHA as a function of drift (left) and
displacement ductility (right) based on numerical backbone assumption. Analyses assumed a
SDOF system with a pier height of h=6.6m and a lumped mass of m=550t. Aspect ratio
correspond to h/lw = 2.2 .................................................................................................................83
Figure 7.10: Pseudo spectral acceleration vs. spectral displacement (ADRS response spectra) and
initial periods of different SDOF systems considered in this study on the influence of pier height
(left) and initial stiffness (or stiffness to yield) of the same members in a force-displacement
relation (right). A lumped mass to m=550to has been considered for all cases. ...........................84
Figure 7.11: Trend of mean value, standard deviation and single values of displacement ductility
demands obtained from NLTHA for different pier heights as presented in Figure 7.10 and Table
7.4 assuming systems without strength degradation, initial stiffness prop. elastic viscous damping
ratio of ξ=0.02 and a seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type
(AGM = Artificial, RGM = Real GM, RGM N = Real newly scaled) ..........................................86
Figure 7.12: Trend of mean value, standard deviation and single values of drift demands obtained from
NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming systems
without strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a
seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM =Artificial,
RGM = Real GM,
RGM N = Real newly scaled)......................................................................86
ix
Index
Figure 7.13: Trend of mean value, standard deviation and single values of displacement ductility
demands obtained from NLTHA for different pier heights as presented in Figure 7.10 and Table
7.4 assuming systems with strength degradation, initial stiffness prop. elastic viscous damping
ratio of ξ=0.02 and a seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type
(AGM = Artificial, RGM = Real RGM, RGM N = Real newly scaled) .......................................87
Figure 7.14: Trend of mean value, standard deviation and single values of drift demands obtained from
NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming systems
with strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a
seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM=Artificial,
RGM=Real GM, RGM N=Real newly scaled) .............................................................................87
Figure 7.15: Comparison of the trend of mean value (continuous line) and standard deviation (dotted
line) of displacement ductility demands obtained from NLTHA for different pier heights as
presented in Figure 7.10 and Table 7.4 assuming systems with and without strength degradation,
initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a seismicity of PGA 0.16g (left)
and PGA 0.35g (right). Ground motion type (AGM = Artificial, RGM = Real GM, RGM N =
Real newly scaled).........................................................................................................................88
Figure 7.16: Comparison of the trend of mean value (continuous line) and standard deviation (dotted
line) of drift demands obtained from NLTHA for different pier heights as presented in Figure
7.10 and Table 7.4 assuming systems with and without strength degradation, initial stiffness prop.
elastic viscous damping ratio of ξ=0.02 and a seismicity of PGA 0.16g (left) and PGA 0.35g
(right). Ground motion type (AGM=Artificial, RGM=Real GM, RGM N=Real newly scaled) ..88
Figure 7.17: Influence of initial stiffness proportional elastic viscous damping ratio ξel assumed in
NLTHA on maximum displacement demand for member with continuous and spliced
longitudinal reinforcement. Legend: input ground motion (AGM = Artificial, RGM=Real, RGM
N = Real newly scaled)..................................................................................................................90
Figure 7.18: Influence of damping coefficient c on maximum displacement demand for member with
continuous and spliced longitudinal reinforcement, experimental backbone assumption and a pier
height of 6.60m and a lumped mass of 550t. (AGM = Artificial Ground Motion) .......................91
Figure 7.19: Comparison of proposed prediction (orange line) using displacement increasing factor
versus effective obtained mean and standard deviation values of displacement ductility (left) and
drift ratio (right) for systems suffering strength degradation from NLTHA. Pier heights on
horizontal axes are obtained considering a reference pier of h=6.60m. Only systems presented in
Figure 7.10 and Table 7.4 have been considered here...................................................................93
Figure 7.20: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility (left) and drift ratio (right) for systems suffering strength
degradation. Displacement ductilities and drift ratios on horizontal axis are referred to systems
x
Index
with strength degradation. Only systems presented in Figure 7.10 and Table 7.4 have been
considered here. .............................................................................................................................93
Figure 7.21: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility (left) and drift ratio (right) for systems suffering strength
degradation. Displacement ductilities and drift ratios on horizontal axis are referred to systems
without strength degradation. Only systems presented in Figure 7.10 and Table 7.4 have been
considered here. .............................................................................................................................94
Figure 7.22: Proposed prediction (red line) for consideration of increased displacement demand as a
function of drift ratio for systems suffering strength degradation. Drift ratios on horizontal axis
are referred to systems with strength degradation. Only single-degree-of-freedom systems with
pier height of h=6.60m, a lumped mass of m=550to and an experimental backbone (left) and
numerical backbone (right) are presented here. Ground motion type (AGM=Artificial,
RGM=Real GM, RGM N=Real newly scaled) .............................................................................94
Figure 7.23: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility for systems suffering strength degradation. Displacement
ductilities on horizontal axis are referred to systems with strength degradation. Only singledegree-of-freedom systems with pier height of h=6.60m, a lumped mass of m=550t and an
experimental backbone (left) and numerical backbone (right) are presented here. Ground motion
type (AGM=Artificial, RGM=Real GM, RGM N=Real newly scaled) ........................................95
Figure 7.24: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Miranda et al., 2003] and sorted by
longitudinal reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g
left to right). All NLTHA assumed a SDOF system with a pier height of 6.60m and a lumped
mass of 550t...................................................................................................................................99
Figure 7.25: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Miranda et al., 2003] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering artificial GM only. All NLTHA assumed a SDOF system with a pier height of 6.60m
and a lumped mass of 550t ............................................................................................................99
Figure 7.26: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Miranda et al., 2003] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real GM only. All NLTHA assumed a SDOF system with a pier height of 6.60m and
a lumped mass of 550t .................................................................................................................100
Figure 7.27: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Miranda et al., 2003] and sorted by
xi
Index
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real, newly s. GM only. All NLTHA assumed a SDOF system with a pier height of
6.60m and a lumped mass of 550t ...............................................................................................100
Figure 7.28: Concept of effective (secant) stiffness proposed by Priestley [Priestley et al., 2007] ....101
Figure 7.29: Capacity curves from Table 7.9 displayed in Acceleration Displacement Response
Spectra (ADRS) versus seismic demands (left) and force-deformation relations of bridge piers
considered in capacity spectrum method (CSM) according to the procedure of [Priestley et al.,
2007]............................................................................................................................................102
Figure 7.30: Comparison of hysteretic damping obtained from elastic, linear time-history analysis
ELTHA based on effective period Te and maximum displacement demand from NLTHA
performed in this study versus prediction according to [Priestley et al., 2007] procedure assuming
a Thin Takeda model (TT)...........................................................................................................104
Figure 7.31: Examples of iterative procedure adopted in CSM for a real scaled ground motion (left)
and an artificial ground motion (right) in order to find the performance point (PP), defined as
intersection of capacity and demand on locus of performance points, considering effective period
Te and equivalent viscous damping ξe as proposed by [Priestley et al., 2007]. ...........................105
Figure 7.32: Examples of iterative procedure adopted in CSM for a case with multiple solutions (left)
and a case without solution (right) assuming a capacity curve with strength degradation
considering effective period Te and equivalent viscous damping ξe as proposed by [Priestley et
al., 2007]......................................................................................................................................106
Figure 7.33: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Priestley et al., 2007] and sorted by
longitudinal reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g
left to right). All NLTHA assumed a SDOF system with a h=6.60m, m=550t and neglected
strength degradation on capacity .................................................................................................107
Figure 7.34: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Priestley et al., 2007] and sorted by
longitudinal reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g
left to right). All NLTHA assumed a SDOF system with a h=6.60m, m=550t and considered
strength degradation on capacity .................................................................................................107
Figure 7.35: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Priestley et al., 2007] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering artificial GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................108
xii
Index
Figure 7.36: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Priestley et al., 2007] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected
strength degradation. ...................................................................................................................108
Figure 7.37: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Priestley et al., 2007] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real, newly s. GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................109
Figure 7.38: Early effort to define optimal equivalent linear parameters [Iwan, 1980] ......................109
Figure 7.39: Comparison of equivalent linear parameters based on the procedure proposed by [Iwan,
1980], based on optimal stiffness and damping, versus the procedure proposed [Priestley et al.,
2007], based on secant stiffness and equivalent viscous damping. Periods are intended as initial
period T=T0 .................................................................................................................................110
Figure 7.40: Comparison of predicted displacement demand according to [Iwan, 1980] (left) using
equivalent linear optimal parameters (modified CSM) versus conventional CSM according to
[Priestley et al., 2007] (right) using secant stiffness for same ground motion, elastic viscous
damping and capacity curve (RGM42, 2%, numerical backbone). .............................................111
Figure 7.41: Comparison of equivalent viscous damping obtained from elastic, linear time-history
analysis ELTHA based on effective period Teff on secant stiffness and maximum displacement
demand from NLTHA performed in this study versus prediction according to [Iwan., 1980].
Effective period on horizontal axis (right) correspond to a secant stiffness................................112
Figure 7.42: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Iwan, 1980] and sorted by
longitudinal reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g
left to right). All NLTHA assumed a SDOF system with a h=6.60m, m=550t and neglected
strength degradation on capacity .................................................................................................112
Figure 7.43: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Iwan, 1980] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering artificial GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................113
Figure 7.44: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Iwan, 1980] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
xiii
Index
considering real GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected
strength degradation. ...................................................................................................................113
Figure 7.45: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Iwan, 1980] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real, newly s. GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................114
Figure 7.46: Mean value contours (left) and standard deviation contours (right) of εD error distribution
over two-dimensional parameter space for entire ensemble [Guyader et al., 2004] ...................115
Figure 7.47: Engineering acceptability range applied to distribution of εD considering an additive
relation for effective damping and a ratio for effective period [Guyader et al., 2004] ...............115
Figure 7.48: Force (f) vs. displacement (x) for bilinear (BLH), stiffness degrading (KDEG), strength
degrading (STRDG) and pushover backbone models from a THA with a sinusoidal acceleration
function (left) and schematic diagram and hysteresis loops for the pinching models (right)
[Guyader et al., 2004b]................................................................................................................117
Figure 7.49: Comparison of equivalent linear parameters based on procedure proposed by [Guyader et
al., 2004], based on optimal stiffness and damping, versus procedure proposed [Priestley et al.,
2007], based on secant stiffness and equivalent viscous damping. Periods are intended as initial
period T=T0 .................................................................................................................................118
Figure 7.50: Comparison of equivalent linear parameters based on procedure proposed by [Guyader et
al., 2004] versus previous study carried out by Iwan [Iwan, 1980], both based on optimal
stiffness and damping. Periods are intended as initial period T=T0 ............................................118
Figure 7.51: Comparison of equivalent viscous damping obtained from elastic, linear time-history
analysis ELTHA based on effective period Teff on secant stiffness and maximum displacement
demand from NLTHA performed in this study versus prediction according to [Guyader et al..,
2004]. Effective period on horizontal axis (right) corresponds to a secant stiffness. ..................119
Figure 7.52: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Guyader et al., 2004] and sorted by
longitudinal reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g
left to right). All NLTHA assumed a SDOF system with a h=6.60m, m=550t and neglected
strength degradation on capacity .................................................................................................119
Figure 7.53: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Guyader et al., 2004] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering artificial GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................120
xiv
Index
Figure 7.54: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Guyader et al., 2004] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected
strength degradation. ...................................................................................................................120
Figure 7.55: Ratio of maximum displacement demand from NLTHA divided by maximum
displacement demand according to procedure proposed by [Guyader et al., 2004] and sorted by
longitudinal reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and
considering real, newly s. GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and
neglected strength degradation. ...................................................................................................121
Figure 7.56: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and four methods, considering
members with continuous longitudinal reinforcement only (no strength degradation of backbone
curve) and seismicities of PGA 0.16g, 0.35g, 0.50g. All NLTHA assumed a SDOF with a
h=6.60m, m=550t ........................................................................................................................123
Figure 7.57: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone
curve in NLTHA) but without considering it when predicting displacement. All presented data
includes seismicities of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t
.....................................................................................................................................................124
Figure 7.58: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution the four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone
curve in NLTHA) but without considering it when predicting displacement and applying to the
displacement prediction the factor f(δ) as in section 7.2.6. All presented data includes seismicities
of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t.................................125
Figure 7.59: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution the four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone
curve in NLTHA) but without considering it when predicting displacement and applying to the
displacement prediction the factor f(μ) as in section 7.2.6. All presented data includes seismicities
of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t.................................126
Figure 7.60: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and the method [Priestley et
al., 2007], considering four different assumptions for the estimation of displacement prediction
xv
Index
and considering members with spliced longitudinal reinforcement only. All presented data
includes seismicities of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t
.....................................................................................................................................................127
Figure 7.61: Performance Point error results for stiffness degrading model (KDEG) with second slope
ratio of 0% - two far-field ground motion databases [Guyader et al., 2004] and results for this
study according to Figure 7.56 as points, considering roughly μ(mean)= 3.0 for this distribution
.....................................................................................................................................................128
Figure 8.1: Nomenclature used in this study for dynamic analyses on multi-degree-of-freedom systems
.....................................................................................................................................................129
Figure 8.2: Front and lateral view of bridge piers considered in this study. Pier height is defined as the
height starting from connection pier footing to pier up to centre of superstructure mass. ..........130
Figure 8.3: Multi-degree-of-freedom systems for dynamic analyses considering an homogeneous
repartition of mass on superstructure (top), with constant lumped masses of mi=78.6t (abutments
masses mi=38.3t), respective a concentrated tributary mass on each pier (bottom), with constant
lumped masses of mi=550t , neglecting mass on abutments. ......................................................130
Figure 8.4: Plan and longitudinal view of bridge model 1 (short M1) considered. Piers P1-P4 have a
pier height of h=6.60m to superstructure centre of mass of; all piers and abutments are assumed
to be laterally restrained for transverse excitations .....................................................................131
Figure 8.5: Plan and longitudinal view of bridge model 2 (short M2) considered. Piers P1 and P4 have
a pier height of h=6.60m while central piers have an height of h=13.20m to the superstructure
centre of mass; all piers and abutments are assumed to be laterally restrained for transverse
excitations....................................................................................................................................131
Figure 8.6: Plan and longitudinal view of bridge model 3 (short M3) considered. Piers P1 and P4 have
a pier height of h=13.20m while central piers have an height of h=6.60m to the superstructure
centre of mass; all piers and abutments are assumed to be laterally restrained for transverse
excitations....................................................................................................................................131
Figure 8.7: Plan and longitudinal view of bridge model 4 (short M4) considered. Piers P1 and P2 have
a pier height of h=13.20m while piers P3 and P4 have an height of h=6.60m to the superstructure
centre of mass; all piers and abutments are assumed to be laterally restrained for transverse
excitations....................................................................................................................................131
Figure 8.8: Initial stiffness Rayleigh damping model (ICTYPE=0) described in Ruaumoko [Carr,
2004]............................................................................................................................................132
Figure 8.9: Abutment force-displacement relationships considered in this parametrical study. Equal
elastic stiffness assumption has been modelled in both abutments (AL and AR) .......................133
xvi
Index
Figure 8.10: Bridge model M1 (top), superstructure and piers displacement demands from mean
NLTHA (bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m
(grey thick line) have been evidenced; point represents pier positions P1 to P4.........................134
Figure 8.11: Abutments (AL and AR) strength demands for bridge model M2 in order to remain
elastic (left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF
(empty points) vs. mean NLTHA on SDOF (full points and dotted lines)..................................134
Figure 8.12: Bridge model M2 (top), superstructure and piers displacement demands from mean
NLTHA (bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m
(grey thick line) have been evidenced; point represents pier positions P1 to P4.........................135
Figure 8.13: Abutments (AL and AR) strength demands for bridge model M2 in order to remain
elastic (left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF
(empty points) vs. mean NLTHA on SDOF (full points and dotted lines)..................................135
Figure 8.14: Bridge model M3 (top), superstructure and piers displacement demands from mean
NLTHA (bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m
(grey thick line) have been evidenced; point represents pier positions P1 to P4.........................136
Figure 8.15: Abutments (AL and AR) strength demands for bridge model M3 in order to remain
elastic (left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF
(empty points) vs. mean NLTHA on SDOF (full points and dotted lines)..................................136
Figure 8.16: Bridge model M4 (top), superstructure and piers displacement demands from mean
NLTHA (bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m
(grey thick line) have been evidenced; point represents pier positions P1 to P4.........................137
Figure 8.17: Abutments (AL and AR) strength demands for bridge model M4 in order to remain
elastic (left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF
(empty points) vs. mean NLTHA on SDOF (full points and dotted lines)..................................137
Figure 8.18: Maximum abutments strength demands for bridge models M1-M4 according to NLTHA
analyses presented in sections 8.2.1 to 8.2.4 for a seismicity level of PGA 0.16g (left) and PGA
0.35g (right).................................................................................................................................138
Figure 8.19: Elasto-plastic hysteresis implemented in Ruaumoko [Carr, 2004] and used in this study to
represents abutment behaviour in multi-degree-of-freedom analyses .........................................139
Figure 8.20: Bridge model M1 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and displacement demands P1 to P4 (points) have been evidenced...........................140
Figure 8.21: Bridge model M1 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
xvii
Index
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and displacement demands P1 to P4 (points) have been evidenced...........................141
Figure 8.22: Bridge model M2 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................142
Figure 8.23: Bridge model M2 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................143
Figure 8.24: Bridge model M3 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................144
Figure 8.25: Bridge model M3 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................145
Figure 8.26: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................146
Figure 8.27: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black
thick line) and pier displacement demands P1 to P4 (points) have been evidenced....................147
Figure 8.28: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for PGA 0.35g (grey lines). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced for the case of continuous
longitudinal reinforcement (centre) and with spliced longitudinal reinforcement at pier base
(bottom). ......................................................................................................................................148
Figure 8.29: Comparison of displacement demand ratio from multi-degree-of-freedom analyses on
entire bridge structure carried out in section 8.3 versus predictions developed in this study (see
section 7.2.6) assuming capacity curves from numerical analyses in NLTHA. Displacement
ductilities and drift ratios on horizontal axis are referred to systems without strength degradation
(thus Model-CRA-SLM-EP) .......................................................................................................150
xviii
Index
Figure 8.30: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution arising from differences
between prediction (red line) and effectively obtained displacement ratios presented in Figure
8.29 ..............................................................................................................................................150
Figure 8.31: Comparison of displacement demand ratio from multi-degree-of-freedom analyses on
entire bridge structure carried out in section 8.3 versus predictions developed in this study (see
section 7.2.6) assuming capacity curves from numerical analyses in NLTHA. Displacement
ductilities and drift ratios on horizontal axis are referred to systems without strength degradation
(thus Model-CRE-SLM-EP)........................................................................................................ 151
Figure 8.32: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution arising from differences
between prediction (red line) and effectively obtained displacement ratios presented in Figure
8.31 ..............................................................................................................................................151
xix
Index
LIST OF TABLES
Page
Table 3.1: Column lap-splice requirements according to the ACI Code Provisions from years 1951 to
1989 ...............................................................................................................................................42
Table 3.2: Test specimen from previous studies considered here and having a circular section...........42
Table 3.3: Test specimen from previous studies considered here and having a quadratic cross section
.......................................................................................................................................................43
Table 4.1: Characteristic data of test units.............................................................................................48
Table 5.1: Evaluation of sectional response quantities according to [Priestley et al., 2007]................53
Table 5.2: Results of bilinear idealization of member analysis with CUMBIA for test unit VK1 ........58
Table 5.3: Modeling assumptions for the cyclic inelastic analysis in Ruaumoko [Carr, 2004].............59
Table 5.4: Modelling assumptions for the cyclic inelastic analysis in Idarc [Idarc, 2006]....................60
Table 5.5: Comparison of calculated damping for single cycling of loading in analyses and experiment
.......................................................................................................................................................61
Table 5.6: Results of linear idealization of member analysis by application of [Priestley et al., 1996]
model .............................................................................................................................................62
Table 5.7: Results of linear idealization of member analysis by application of modified [Priestley et
al., 1996] model in order to fit experimental data .........................................................................63
Table 5.8: Strength degradation modelling assumption for analysis in Ruaumoko [Carr, 2004]..........64
Table 5.9: Modelling assumptions for the cyclic inelastic analysis in Idarc [Idarc, 2006]....................64
Table 5.10: Comparison of calculated damping for single cycling of loading in analyses and
experiment .....................................................................................................................................66
Table 5.11: Capacity curve assumptions considered in this study for NLTHA.....................................67
Table 5.12: Scaling factors experimental test units to real bridge piers dimensions .............................69
Table 5.13: Scaling factors for pier height variation starting from real bridge piers dimensions..........69
Table 6.1: Soil conditions for stiff soil (Soil B) according to code provisions [SIA 261, 2003]...........71
Table 6.2 Typical frequencies generated by different seismic sources..................................................72
xx
Index
Table 6.3 Post-processing procedures of real accelerograms ................................................................72
Table 6.4: Criteria for consideration of real ground motion data ..........................................................72
Table 6.5: Accepted real ground motions from past earthquakes..........................................................73
Table 6.6: Scaling factors based on Displacement Response Spectra for different seismicities in period
range between T = 0.50-2.00sec. (3th-4th columns) and T = 0.35-1.00sec. (5th-6th columns),
respectively....................................................................................................................................73
Table 6.7: Artificial ground motion generation parameters used in SIMQKE ......................................75
Table 7.1: Capacity curve assumptions considered in this study for NLTHA on single-degree-offreedom systems representing bridge piers subjected to transverse earthquake excitations..........77
Table 7.2: Modeling assumptions for reference, tested bridge pier (scaled to real dimensions) ...........77
Table 7.3: Variability of considered SDOF models for NLTHA ..........................................................78
Table 7.4: Single-degree-of-freedom systems considered in this section. Each system follow the
backbone assumptions presented in Table 7.1 for members with and without strength degradation
obtained with numerical analyses. Ductility dependent strength degradation is considered
according to Table 5.8 ...................................................................................................................85
Table 7.5: Modified damping ratios to be consistent in terms of critical damping coefficients when
performing NLHTA.......................................................................................................................91
Table 7.6: Stiffness and damping model assumptions for the considered methods...............................95
Table 7.7: Statistical data of earthquakes used in [Miranda et al., 2003] ..............................................97
Table 7.8: Site dependent coefficients for inelastic displacement ratios CR [Miranda et al., 2003]......98
Table 7.9: Capacity curve assumptions considered in this study for NLTHA.....................................101
Table 7.10: Secant stiffness correction factors λ for elastic damping [Priestley et al., 2007].............103
Table 7.11: Equivalent viscous damping coefficients for hysteretic damping component according to
the procedure proposed by [Priestley et al.,2007] .......................................................................104
Table 7.12: Coefficients for effective linear parameters according to [Guyader et al., 2004b] for farfield ground motions, different hysteretic models and second slope ratios α .............................117
Table 7.13: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distribution according to error bound defined in [Guyader et al., 2004]......123
Table 7.14: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.56 .............123
Table 7.15: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.57 .............124
Table 7.16: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.58 .............125
Table 7.17: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.59 .............126
xxi
Index
Table 7.18: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.60 .............127
Table 8.1: Superstructure properties according to a study carried out by Kuhn [Kuhn, 2008] on an
existing Swiss roadway bridge with box-girder cross section.....................................................131
Table 8.2: Elasto-plastic modelling assumptions for abutments considered in this study...................138
xxii
Index
LIST OF SYMBOLS
a
agd
Factor
Peak ground acceleration according to [SIA 261, 2003]
b
Factor
c
concrete cover of concrete; distance from extreme compression fibre to neural axis
c
Damping coefficient; Factor
ccr
d
Critical damping ratio
Factor
dbl
Longitudinal reinforcing bar diameter
ds
Reinforcing bar diameter of transverse bars
fc
Concrete compressive cylinder strength
fcc
Compressive strength of confined concrete compressive
fco
Maximum feasible compressive strength of unconfined concrete
f(δ)
Drift dependent, displacement increasing factor
f(μ)
Ductility dependent, displacement increasing factor
fl
Lateral confining stress
flx
Lateral confining stress in x-direction
fly
Lateral confining stress in y-direction
fs
Stress of longitudinal reinforcement
fy
Yield strength of longitudinal reinforcement
fyh
Yield strength of transverse reinforcement
fyt
Yield strength of transverse reinforcement
ft
Concrete tensile strength
fu
Ultimate strength of longitudinal reinforcement
g
Acceleration due to gravity
ke
Effective stiffness
kinit
h
h’
href
Initial stiffness
Bridge pier height
Rectangular section core dimension measured to inside of longitudinal reinf. cage
Pier height of reference pier
xxiii
Index
ke
Confinement effectiveness coefficient
kel
Elastic stiffness (of an abutment)
lw
Wall length
ls
Lap-splice length
m
Lumped mass
me
Effective mass
p
Perimeter of crack surfaces around bar in lap-splice failure
r
Bilinear factor for second slope stiffness
s
Longitudinal spacing of transverse reinforcement
vs
Soil shear wave velocity
A
Cross section area
ADRS
Aeff
Ag
AGM
Aloop
As
Acceleration Displacement Response Spectra
Acceleration in response spectra according to effective period
Gross section area
Artificial ground motion
Area enclosed in a force-displacement loop
Effective shear area
Asec
Acceleration in response spectra according to secant period
Asx
Total area of transverse reinforcement within spacing s, in x-direction
Asy
Total area of transverse reinforcement within spacing s, in y-direction
Asp
Bar area of spiral reinforcement
Bc
Column section width
CR
Inelastic correction factor
CSM
D
D’
Capacity spectrum method
Diameter of circular column
Core diameter of circular column
Dlin
Maximum displacement amplitude of the linear time-history
Dinel
Maximum displacement amplitude of the non-linear time-history
E
Elastic modulus of elasticity
Ec
Concrete modulus of elasticity
Es
Modulus of elasticity of reinforcement
Esec
Secant modulus of elasticity
Etot
Total energy dissipation in hysteresis loops
F
Force
Fdeg
Force corresponding to initiation of degradation in capacity curve of a member
Feff
Force at effective (secant) stiffness
Fel
Elastic stiffness (of an abutment)
Fm
Force at maximum displacement
Fres
Fy
Force corresponding residual displacement capacity
Yielding force
xxiv
Index
Fult
Force corresponding ultimate displacement capacity
G
Shear modulus
Hc
Column section depth
K0
Initial stiffness
Iv
Moment of inertia
L
Column length
Lc
Member length
Leff
Effective length of the member
Lp
Effective length of plastic hinge
Lsp
Strain penetration length
M
MADRS
Moment; Earthquake Magnitude; Modification factor
Modified Acceleration Displacement Response Spectra
Mn
Nominal flexural strength
Ms
Moment capacity of section, as limited by lap-splice strength
Ms
Moment magnitude of surface waves
Mr
Residual capacity of section, after failure by lap-splice
My
Yield moment
Mu
Ultimate moment capacity
N
NLTHA
P
Axial load
Nonlinear time-history analysis
Axial load
PGA
Peak Ground Acceleration
PGD
Peak Ground Displacement
PGV
Peak Ground Velocity
PSA
Peak pseudo spectral acceleration
R
RGM
S
SA
Sa, real
Sa, target
SD
T
Force reduction factor
Real ground motion
Soil factor
Peak spectral acceleration
Peak spectral value in real ground motion response spectra
Peak spectral value in target response spectra
Peak spectral displacement
Period
TB
Period for construction of response spectra
TC
Period for construction of response spectra
TD
Period for construction of response spectra
Te
Effective period
Teff
Effective period
Tinit
Initial vibration period
T0
Initial vibration period
Tsec
Secant vibration period
xxv
Index
V
Shear force; shear strength
Vc
Shear capacity from concrete mechanism
Vp
Shear capacity from axial load mechanism
Vs
Shear capacity from steel truss mechanism
α
Factor
αPGA
β
δ
δdeg
δmax
εc
εcc
εco
εcu
εD
εs
εsm
φ
φy
φn
φu
γ
η
λ
μ
μdeg
μΔ
μφ
ξ
ξe
ξeff
ξel
ξel,NLTHA
ξhyst
ξ0
ρcc
ρl
ρs
ρx
ρy
Scaling factor of ground motion
Factor
Drift ratio
Member drift at initiation of strength degradation
Maximum drift demand
Concrete compression strain
Strain at peak stress for confined concrete
Strain at peak stress for unconfined concrete
Ultimate compression strain in concrete
Error measure of displacement error
Strain of longitudinal reinforcement
Maximum Strain of longitudinal reinforcement
Curvature
Yield curvature
Nominal curvature
Ultimate curvature
Factor
Damping reduction factor
Secant stiffness correction factor
Ductility factor; Factor
Displacement ductility factor for initiation of strength degradation
Displacement ductility factor
Curvature ductility factor
Viscous damping ratio
Effective viscous damping ratio
Effective viscous damping ratio
Elastic viscous damping ratio
Elastic viscous damping ratio assumed in NLTHA
Hysteretic viscous damping ratio
Elastic viscous damping ratio
Volumetric ratio of transverse reinforcement
Longitudinal reinforcement ratio
Volumetric ratio of transverse reinforcement
Area ratio of transverse reinforcement in x-direction
Area ratio of transverse reinforcement in y-direction
xxvi
Index
ρt
σ
Transverse reinforcement ratio
Δ
Displacement
Δdemand
Δelastic
Δfy
Δfu
Δinelastic
Δm
Δmax
Δres
Δy
Δtop
Δult
ϑ
Standard deviation
Displacement demand
Displacement demand for an elastic analysis
Flexural yield displacement of a member
Ultimate flexural displacement of a member
Displacement demand for an inelastic analysis
Maximum displacement
Maximum displacement experienced in a NLTHA
Residual displacement capacity
Yield displacement
Displacement at top of member (i.e. bridge pier)
Ultimate displacement capacity
Angle of inclination of steel truss mechanism
xxvii
Chapter 1. Introduction
1 INTRODUCTION
1.1 Statement of the problem
In many countries the majority of bridges as built before the establishment of modern seismic
design codes. As a consequence, often these structures do not comply with current standards,
and their seismic safety is, even in countries of moderate seismicity such as Switzerland,
rather uncertain. On one hand, this may be the result of the higher seismic demand in modern
codes compared to older provisions. On the other hand, from today’s perspective, older
structures often show insufficient detailing which limits the deformation capacity
significantly. Typical examples for deficient detailing are very low transverse reinforcement
ratios or lap-splices in the potential plastic regions of the piers.
Although in past decades significant progress has been made in the development of analysis
procedures for newly designed structures, these same methods may not always be directly
applicable to existing bridges. This is mainly the case because modern design principles
preclude certain undesired failure modes, thus warranting a ductile and predictable
deformation behaviour. However, these failure modes, shear failure, lap-splice failure or
failure of the soil-foundation system for example, cannot generally be ruled out in existing
structures, making their analysis more demanding. This is particularly the case if a failure
mode does not only influence the deformation capacity but rather also the deformation
demand, as for example in the case of a lap-splice failure.
1.2
Definition of terms and previous research
1.2.1 Strength degradation
Considering a reinforced concrete member with continuous longitudinal reinforcement first,
an idealized flexural response can usually be assumed as bilinear, with an elastic and an
inelastic branch of deformation. The behaviour in the inelastic range starts at a yielding point,
defined by a yield displacement and by a yielding lateral force and terminates at failure in
correspondence with the ultimate displacement capacity. Over this range of deformations, the
28
Chapter 1. Introduction
lateral load carrying capacity of the member is assumed to be approximately constant without
degradation effects.
On the other hand, in recognising the fact that the shear strength of reinforced concrete
members is ductility dependent, different design and assessment models have been developed,
experimentally tested and calibrated with the aim to describe the real shear capacity at
different ductility levels. The first conceptual shear model can be found in the 1981 Seismic
Design Guidelines for Highway Bridges [ATC-6, 1981] of the Applied Technology Council
and more recently Kowalski and Priestley [Kowalski et al., 2000], Sezen and Moehle [Sezen
et al., 2004] proposed revised or new models that explicitly take into consideration additional
factors such as aspect ratio and axial load level.
Together with the idealized flexural response, these shear models are used to categorize three
different failure modes of columns subjected to lateral displacements:
a) Flexural failure
b) Flexural-shear failure
c) Brittle shear failure
Figure 1.1: Classification of reinforced concrete column failure modes as ATC-6 (1981)
It has been experimentally demonstrated that the flexural behaviour of members with lap
splices in plastic hinge regions suffer strength degradation starting in the inelastic branch of
deformation (or even before yielding for short lap splices). As a consequence, the bilinear
idealization of flexural strength capacity presented in Figure 1.1 must be corrected for this
detailing deficiency in order to capture the real behaviour of the member.
A conceptual strength degradation model that accounts lap-splice failure has been proposed
by Priestley, Seible and Calvi in [Priestley et al., 1996] starting from considerations on the
force-mechanism transfer in lap-splices and analyzing the interaction between longitudinal
reinforcement, concrete matrix and lateral confinement at failure state. In their model
Priestley el al. recognise the fact that a concrete matrix degrades with increasing ductility
demand, thus lap splices capacity also diminishes with increasing ductility. A lap-splice
failure is achieved when a rupture surface parallel and perpendicular to the member cross
section is formed (see also Figure 1.3).
Assuming the same shear strength envelope for members with or without lap splices, Figure
1.2 shows the capability of flexural strength degradation to avoid a shear failure reducing the
shear demand in the inelastic range of deformations for certain cases. Moreover, it has to be
noticed that strength degradation is always associated with the partial loss of lateral load
29
Chapter 1. Introduction
capacity, with an increase of member displacement demand, and with a deterioration of the
critical concrete section which can potentially lead to a deterioration of the axial load carrying
capacity mechanism. Any improvement in shear failure prevention for members with splicedend reinforcement must always be regarded critically and weighted against these negative
consequences.
a) Failure without lap-splice
b) Behaviour with lap-splice
Figure 1.2: Shear failure prevention due to shear strength degradation in lap-splice
1.2.2 Lap-splice failure
As described in [Priestley et al., 1996], “lap splice failure involves relative longitudinal
movement of the spliced bars, and requires the formation of a fracture both perpendicular and
parallel to the member surface in order to permit the bars to slide relative to the RC-member
core. Depending on the spliced bars, the concrete, and reinforcement but mostly on splice
length, the splice failure can occur before or after any yielding of the longitudinal bars,
governing the displacement capacity of the member.”
Figure 1.3: Lap-splice failure of longitudinal bars in columns [Priestley et al., 1996]
30
Chapter 1. Introduction
1.3 Objective and scope of the work
Objective of this master thesis is the proper modelling of the monotonic and hysteretic
behaviour of single-degree-of-freedom systems (SDOF) and multi-degree-of-freedom systems
(MDOF) representing on one hand single existing bridge pier and on the other hand entire
existing bridges featuring detailing deficiencies. Among the several possible deficiencies, the
work focuses on the cyclic strength degradation resulting from the failure of lap-splices in
potential plastic hinge regions.
Accurate, realistic modelling of the hysteretic behaviour of bridge piers with or without
spliced longitudinal-reinforcement at the pier base is a main goal of this study. To this
purpose, recommendations and models given in the literature but also experimental results
coming from test performed at ETH Zürich by Bimschas [Bimschas et al., 2008] on wall type
bridge piers will be used to calibrate the numerical models.
Firstly, the behaviour of single-degree-of-freedom systems representing single bridge piers
with tributary superstructure mass will be studied and compared by means of dynamic
analyses for the case of members with or without spliced longitudinal reinforcement at pier
base. This will show the effect of a cyclically degrading system in an isolated and pure form,
without any influence of other parts of the structure. In a later stage, entire bridge modelled as
multi-degree-of-freedom systems are used to study the effect of strength degradation on
integral systems, as well as the interaction between degrading and non-degrading parts of the
structure.
31
Chapter 2. Overview of the report
2 OVERVIEW OF THE REPORT
2.1 Previous research
Research on lap splice behaviour in reinforced concrete members under cyclic loading
performed in the past was mainly intended for formulation of adequate development length
for code provisions and retrofit assessment. Behavioural assessments only became major goal
in recent times [Melek et al., 2004]. Moreover, these past studies were mostly focused on
quadratic cross sections that were built to meet the requirements of the previous code
provisions of the West Coast of the United States (pre 1971, according to Table 3.1), thus
having a typical splice length of longitudinal reinforcement of roughly 20dbl (dbl longitudinal
bar diameter). These very short (compression)-lap splices did not usually meet the
requirements for tension yielding of the longitudinal bars, with consequently dramatically
reduced strength capacity even at low ductilities.
Figure 2.1: Examples of force displacement relation according to [Priestley et al., 1996] method versus
experimental results on rectangular column units performed by [Melek et al., 2004]
32
Chapter 2. Overview of the report
The influence of longer splice lengths, as it is the case of even older Swiss Code provisions,
as well as of wall-type cross section shape on the backbone of the strength degradation curve
can be quantified only very approximatively by means of existing test data, leading to the
experimental works undertaken at the ETH Zürich by Dazio and Bimschas [Bimschas et al.,
2008].
2.2 Experimental data used in this work
I the framework of an ongoing research project at the ETH Zürich, two test units were
designed to represent real existing Swiss bridge piers from the 1960s with detailing
incorporating seismic deficiencies typical for structures of that time. Both test units had the
same concrete dimensions, same steel properties and similar concrete properties, the only
difference being that one unit had a 43dbl lap-splice at the base of the pier while the other had
continuous reinforcement [Bimschas et al., 2008].
Figure 2.2: Examples of typical bridge column cross section [Bimschas et al., 2008]
Figure 2.3: Experimentally recorded hysteretic (force deformation) behaviour of test unit VK1 (left,
continuous reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008]
33
Chapter 2. Overview of the report
Despite the only difference between the two specimens being a lap-splice of the longitudinal
reinforcement in test unit VK2, the observed hysteretic behaviour of the two units strongly
differed especially in the inelastic deformation range. While VK1 exhibits hysteretic loop
with practically no loss of strength up to shear failure, VK2 suffers significant cyclic strength
degradation, but achieves longer drifts because of the "shear failure prevention provided by
the splice" (section 1.2.1), being able to sustain the full axial load up to ultimate drift capacity.
2.3 Evaluation of monotonic and cyclic member response
Based on existing theories on strength degradation, experimental evidence observed in
[Bimschas et al., 2008] and appropriate hysteresis rules implemented in the software codes
Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006], monotonic and hysteretic member behaviour
representing existing bridge piers with or without spliced longitudinal reinforcement at pier
base has been reproduced.
Figure 2.4: Experimentally obtained cyclic behaviour of test units VK1 (left) and VK2 (right) versus
hysteretic loops using software code Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006]
The calibration of the shape of numerical hysteresis loops focused particularly on the
matching of the hysteretic energy dissipated during the tests. In the numerical models,
strength degradation was reproduced by means of two different theories. 1) based on pure
ductility dependence in Ruaumoko [Carr, 2004] or 2) based on pure energy dissipation in
Idarc [Idarc, 2006]. In this way, the influence of the strength degradation modelling on the
response in dynamic analyses could be studied.
2.4 Definition of target seismicities
Appropriate hazard levels covering a wide range of seismicities, from moderate to high, have
been considered with the aim of producing inelastic member deformations of different
magnitudes in single- and multi-degree-of-freedom systems.
34
Chapter 2. Overview of the report
Based on target displacement response spectra given in the Swiss design code provisions [SIA
261, 2003], a total of twelve real and artificial ground motions were scaled or generated in
order to match the target spectra for each hazard level.
Figure 2.5: Assumed elastic response spectra of acceleration (left) and displacement (right) according to
code provisions [SIA 261, 2003] for PGA of 0.16g, 0.35g, 0.50g (agd), stiff soil conditions (Soil B), structure
importance factor III and elastic viscous damping ξ=0.02
2.5 Single-degree-of-freedom-analyses
Firstly bridge piers were studied as isolated systems, in order to properly observe the
influence of spliced longitudinal reinforcement in potential plastic hinge regions at pier base
on their response. Moreover, parametrical variation of damping ratio, pier height, backbone
curve assumptions were considered and their influence on member response (drift ratio,
ductility demand) evaluated.
Based on results arising from dynamic analyses, simplified displacement increasing factors
accounting for larger displacement demands in members suffering strength degradation, when
compared with the same members with no degradation were developed.
Figure 2.6: Idealized single-degree-of-freedom systems representing a single bridge pier with lumped mass
subjected to transverse earthquake excitation (left). Front (middle) and lateral view (right) of a typical
bridge pier and superstructure considered in this study
35
Chapter 2. Overview of the report
Finally, by means of different displacement prediction methods based on initial stiffness,
secant stiffness or effective stiffness, displacement demands were estimated and compared
with results stemming from dynamic analyses for members without and with strength
degradation. In the latter case, improvements in the application of the displacement increasing
factors mentioned previously were evaluated.
2.6 Multi-degree-of-freedom-analyses
Entire bridge structures, formed by a series of bridge piers linked by a superstructure and
restrained by elasto-plastic abutments were finally modelled and analyzed. Effects of strength
degradation on multi-degree-of-freedom systems were thus studied and compared with
observations obtained from single-degree-of-freedom analyses.
Figure 2.7: Example of bridge model (top). Superstructure and piers maximum displacement demands
from single NLTHA and PGA 0.35g are represented by grey lines. Mean NLTHA displacement pattern
(black thick line) and pier displacement demands P1 to P4 (points) are evidenced for case of continuous
longitudinal reinforcement (centre) and spliced longitudinal reinforcement at pier base (bottom).
As a general consideration, due to the higher redundancy of complex bridge models versus
“isolated” bridge piers, strength degradation in multi-degree-of-freedom systems was less
important and severe than that single-degree-of-freedom analyses. In fact, loss of lateral load
carrying capacity in one pier was compensated by adjacent piers and abutments connected
through the superstructure, thus preventing a local and global structural failure.
36
Chapter 3. Previous studies on strength degradation
3 PREVIOUS STUDIES ON STRENGTH DEGRADATION
3.1 Strength degradation model according to [Priestley et al., 1996]
Using the description presented in [Priestley et al., 1996], “[…] curves 1 and 2 of Figure 3.1
describe the moment-curvature interaction of a section without spliced longitudinal
reinforcement. The flexural moment capacity at yielding is equal to the nominal moment Mn
and correspond to a curvature ductility of μφ=1; beyond this point, in members detailed with
significant lateral confinement the moment capacity increase up to the ultimate moment
capacity Mu and curvature ductility μ1 while by members with low transverse confinement the
moment capacity remains constant up to failure, at a ductility μ1.
On the other hand, curves 3 and 4 of Figure 3.1 represent the idealized sectional behaviour for
members with spliced reinforcement. In this case the flexural strength will degrade from the
initial strength Ms to a residual strength Mr as cyclic inelastic response develops. For sections
where splice failure initiates before Mn is achieved, an ultimate curvature ductility of μφ=8 for
attainment of Mr can be assumed otherwise a value of μφ=(8+μ3) is set for the damage-control
limit state. The value of μ3 corresponds to an extreme compression fibre compression strain of
εc=0.002.”
Figure 3.1: Idealized moment-curvature relationship for members without or with lap-splices in potential
plastic hinge regions [Priestley et al., 1996]
37
Chapter 3. Previous studies on strength degradation
The maximum bar force capacity Tb, that can be transferred without the assistance of special
transverse reinforcement confining the splice depends on the splice length ls, and tension
strength of the concrete ft together with the idealized perimeter of the concrete failure block p
as follows:
Tb = Ab f s = f t pl s
(2.1)
Therefore, once the transferring force has been defined for a specific detailing, the member
behaviour can be estimate (see section 5.1.2).
Figure 3.2: Tension stress induced by force transfer in lap splices [Priestley et al., 1996]
3.2 Experimental studies
Researche on lap-splice behaviour in reinforced concrete members under cyclic loading
performed in the past can be categorized into three different groups, depending on the main
goal: a) required development length, b) retrofit assessment and c) behaviour assessment.
Among the previous experimental studies, work done by Chai [Chai et al., 1991], Lynn [Lynn
et al., 1996] and Melek [Melek et al., 2004] can be considered of particular interest. In the
latter case, research was focused on full scale columns covering a wide range of parameters
such as different levels of axial load, lateral load history (cyclic and near fault) and
considering the interaction between lateral load capacity and loss of axial load-carrying
capacity.
3.2.1 Chai et al., 1991
Chai [Chai et al., 1991] investigated primarily the retrofit of circular columns by encasing
plastic hinge regions in a bonded steel jacket. As reference they tested two identical bridge
columns built according to 1960s standards in California with the following detailing
deficiencies: 20dbl splice at column base (only for one case) and low transverse reinforcement
ratio.
According to the behaviour of members with lap splices, the backbone curve of this reference
column suffered from rapid, asymptotically strength degradation after the first cycle to
displacement ductility of 1.5, thus decreasing the lateral load carrying capacity down to a
residual level dictated from pure axial load eccentricity. On the contrary, the reference
specimen with continuous longitudinal reinforcement was able to carry the high lateral load
38
Chapter 3. Previous studies on strength degradation
capacity up to failure which occurred at approximately a displacement ductility of 5.0 as can
be seen in Figure 3.4.
Figure 3.3: Test setup and reinforcement details for the columns tested by [Chai et al., 1991]
Figure 3.4: Hysteretic response of reference columns without (left) and with lap-splice (right) from
experiments performed by [Chai et al., 1991]
3.2.2 Lynn et al., 1996
Lynn [Lynn et al., 1996] conducted experiments on vulnerable building columns with typical
proportion and details based on surveys of RC buildings built before the 1970's on the West
Coast of the United States of America in order to provide new information on the lateral and
axial load carrying capacity of these structural elements. Column test units were built with a
quadratic cross section, two different configurations of longitudinal bars and three different
detailing of transverse reinforcement. Out of a total of eight test units, three were built with a
20dbl splice of longitudinal reinforcement at the column base.
39
Chapter 3. Previous studies on strength degradation
Figure 3.5: Load assembly and specimen location [Lynn et al., 1996]
Figure 3.6: Column test unit and cross section with different transverse detailing [Lynn et al., 1996]
Experimental evidence of degradation of the lateral load carrying capacity mechanism with
increasing displacement ductility demand was noticed for two of the three units with spliced
reinforcement subjected to the double bending. Nevertheless, consistently different backbones
for these specimens were obtained. Comparing the results of specimen 3SLH18 with these of
2SLH18, the only difference between them being in longitudinal reinforcement ratio (3%
against 2%), the different level of strength degradation is evident (Figure 3.7).
Figure 3.7: Load-displacement relations for the test units with spliced reinforcement [Lynn et al., 1996]
40
Chapter 3. Previous studies on strength degradation
The authors also observed that the likelihood that gravity load can be maintained beyond the
point where lateral load capacity is dramatically reduced depends upon the nature of the
lateral load failure, the column details, the level of axial load and the lateral load history. If
primarily shear failure occurs, this results in rapid loss of gravity load carrying capacity
regardless of the gravity load level (specimen 3SMD12) but as experimental evidence
supports, if failure initiates in the lap-splice, the column may simply "hinge at the splice"
temporarily relieving shear demands and leaving a column capable of sustaining gravity loads
until splice failure degenerates to shear failure.
3.2.3 Melek et al., 2004
Melek and Wallace [Melek et al., 2004] performed experiments with the aim to overcome the
lack of knowledge on the lateral load behaviour of columns with splice of the longitudinal
reinforcement, and to study the influence of important parameters such as axial load and shear
for a level as well as load history. The lap splice implemented in the test units was 20dbl for
all cases.
Figure 3.8: Test setup, reinforcing details and unit cross sections [Melek et al., 2004]
Evidence of degradation of the lateral load resisting mechanism can be seen in Figure 3.9,
where normalized base moment versus lateral drift is plotted. All specimen exhibit similar
responses with lateral strength degradation stating at drift levels of δ=1-1.5% (μ=1.5-2.5).
Figure 3.9: Normalized moment drift relations for all test units [Melek et al., 2004]
41
Chapter 3. Previous studies on strength degradation
As a consequence lateral strength degradation started either just prior to or just after yielding
of the longitudinal starter bars, thus leading to non-ductile member responses with limited
energy dissipation.
3.2.4 Considerations on test specimen from past studies
Among the test specimen presented in sections 3.2.1 to 3.2.3, nine out of ten presented a
quadratic cross section while column tested in [Chai et al., 1991] had a circular shape.
Moreover, test specimen were built according to previous code provisions in West Coast of
United Stated (pre 1971, Table 3.1), thus featuring a typical splice length of roughly 20dbl (dbl
longitudinal bar diameter). These very short (compression)-lap splices usually did not fullfil
the requirements for tension yielding of the longitudinal bars, with consequently dramatically
reduced strength capacity even at low ductilities.
Table 3.1: Column lap-splice requirements according to the ACI Code Provisions from years 1951 to 1989
fy
f'c
(ksi)
(psi)
40
≥ 3000
(20.7MPa)
(275.8MPa)
1951
1956
1963
19711
19712
19771
App(A)
40
< 3000
(275.8MPa)
(20.7MPa)
50
≥ 3000
(344.7MPa)
(20.7MPa)
50
< 3000
(344.7MPa)
(20.7MPa)
60
≥ 3000
(413.7MPa)
(20.7MPa)
60
< 3000
(413.7MPa)
(20.7MPa)
19772
19831
App(A)
19833
19891
19894
Ch.21
App(A)
20dbl
20dbl
20dbl
20dbl
30dbl
20dbl
30dbl
20dbl
39dbl
20dbl
30dbl
27dbl
27dbl
27dbl
27dbl
30dbl
27dbl
30dbl
27dbl
39dbl
27dbl
30dbl
20dbl
20dbl
20dbl
25dbl
30dbl
25dbl
30dbl
25dbl
49dbl
25dbl
38dbl
27dbl
27dbl
27dbl
33dbl
33dbl
33dbl
33dbl
33dbl
49dbl
33dbl
38dbl
24dbl
24dbl
24dbl
30dbl
30dbl
30dbl
30dbl
30dbl
59dbl
30dbl
45dbl
32dbl
32dbl
32dbl
40dbl
40dbl
40dbl
40dbl
40dbl
59dbl
40dbl
45dbl
1
Values may be reduced 17% for confinement by "minimum ties"
Appendix A of the ACI code, "Special Provisions for Seismic Design"
3
Appendix A of the ACI code, Class "C" Lap Splice calculated for f'c=3000psi"
4
Chapter 21 of the ACI code, Class "B" Lap Splice calculated for f'c=3000psi"
2
Compared with Swiss Code requirements of the same years (previous 1970), the development
length according to Table 3.1 are consistently lower. In fact, even in older Swiss provisions a
minimal development length of 40dbl had always been used in practice, because of a tension
splice requirements. Therefore, experimental evidence stemming from studies performed by
[Chai et al., 1991], [Lynn et al., 1996] and [Melek et al., 2004] as well as the model proposed
by [Priestley et al., 1996] (see Figure 3.10, Figure 3.11 as well as Annex A) can be considered
just as a starting point for the study of the behaviour of existing building and bridges
constructed in Switzerland, thus underlying the lack of research in this field.
Table 3.2: Test specimen from previous studies considered here and having a circular section
Study
Spec.
D
(m)
L
(m)
fy
(MPa)
fu
(MPa)
f’c
(MPa)
dbl
(mm)
ρl
(-)
ties
(-)
P/(f’cAg)
(-)
ls
(-)
[Chai et al., 1991]
Col. 1
0.61
3.658
315.1
497.8
38.2
19.05
2.53
S
0.177
20dbl
42
Chapter 3. Previous studies on strength degradation
Table 3.3: Test specimen from previous studies considered here and having a quadratic cross section
Study
Spec.
b..h
(m)
L
(m)
fy
(MPa)
fu
(MPa)
f’c
(MPa)
dbl
(mm)
ρl
(-)
ties
(-)
P/(f’cAg)
(-)
ls
(-)
[Lynn et al., 1996]
3SLH18
0.46
2.95
331
495
25.6
31.8
3
H
0.09
20dbl
[Lynn et al., 1996]
2SLH18
0.46
2.95
331
495
33.1
25.4
2
H
0.07
20dbl
[Lynn et al., 1996]
3SMD12
0.46
2.95
331
495
25.5
31.8
3
D
0.28
20dbl
[Melek et al., 2004]
2S10M
0.46
1.83
510
818
36
25.4
2
H
0.1
20dbl
[Melek et al., 2004]
2S20M
0.46
1.83
510
818
36
25.4
2
H
0.2
20dbl
[Melek et al., 2004]
2S30M
0.46
1.83
510
818
36
25.4
2
H
0.3
20dbl
[Melek et al., 2004]
2S20H
0.46
1.68
510
818
35
25.4
2
H
0.2
20dbl
[Melek et al., 2004]
2S20HN
0.46
1.68
510
818
35
25.4
2
H
0.2
20dbl
[Melek et al., 2004]
2S30X
0.46
1.52
510
818
35
25.4
2
H
0.3
20dbl
3.3
Comparison of experimental evidence with the method according to [Priestley et
al., 1996]
3.3.1 Unit with circular a cross section
Based on test results of an "as-built" circular column in [Chai et al., 1991] a comparison
between predicted force-displacement behaviour and real capacity curve can be seen in Figure
3.10. Priestley degradation model [Priestley et al., 1996] describes with accuracy the member
response in terms of force and deflection. In this case the splice length of 20dbl was able to
develop the full nominal moment capacity, thus ensuring a moderate inelastic deformation
without loss of lateral load capacity, before strength degradation started at displacement
ductility of approximately μ=1.5. Moreover, it can be noticed that the rate of strength
degradation is somewhat smaller compared to that observed in tests on columns with
quadratic cross section. This is a positive consequence of the distribution of the longitudinal
reinforcement over the cross section of the column (see chapter 4.1.3).
Figure 3.10: Example of a force-displacement rel. computed according to [Priestley et al., 1996] compared
to the experimental results of a circular column (left) and relevant testing setup (right) [Chai et al., 1991]
43
Chapter 3. Previous studies on strength degradation
3.3.2 Units with quadratic cross sections
The nine units with quadratic cross section tested by [Lynn et al., 1996] and [Melek et al.,
2004] considered in this study and summarized in Table 3.3 all had a sectional dimension of
457 time 457millimeters. According to the [Priestley et al., 1996] degradation model only one
specimen was able to develop full nominal moment capacity while in eight cases yielding of
longitudinal bars could not be achieved. Figure 3.11 shows for four of these nine columns
capacity curve experimentally obtained versus analytical prediction.
Also in these cases, the [Priestley et al., 1996] degradation model accurately describes the
force displacement behaviour of the specimens, particularly for those with "no yielding of
longitudinal bar" but seems to be conservative in the prediction of the lateral load carrying
capacity for the member with yielding of longitudinal bars (Figure 3.11, top left). In one case
(Figure 3.11, bottom left) simulation of a near fault directivity pulse was tested, hence a
different experimental backbone curve for positive and negative loading can be observed.
Figure 3.11: Examples of force-displacement relationships computed according to [Priestley et al., 1996]
compared to experimental results of rectangular columns
44
Chapter 3. Previous studies on strength degradation
3.3.3 Summary considerations
Because of the complexity and the number of factors governing cyclic strength degradation of
a single member, a linear idealization of an irregular backbone curve is always difficult to
achieve. Nevertheless comparison of experimental results on circular and quadratic columns
performed in previous studies with the proposed analytical procedure by Priestley [Priestley et
al., 1996] have shown, that for short lap splice lengths (20dbl), the analyses accurately
described the general member force deformation behaviour, capturing the point of initiation
and the shape of strength degradation for most of the cases (see also Annex A).
As previously stated, all the test units were built with a lap splice of 20dbl according to older
ACI Code provisions and for 90% of the cases the considered column cross section was
quadratic with dimensions 457mm squared. Influence of longer splice length and wall-type
cross section shape on initiation and backbone of strength degradation curve can not be
quantified with these data. This lead to the experimental work undertaken at ETH Zürich by
Dazio and Bimschas presented in [Bimschas et al., 2008] and considered for the next sections
of this work.
45
Chapter 4. Experimental database
4 EXPERIMENTAL DATABASE
4.1
Motivation of experimental work
4.1.1 Code provisions dependence on required splice length
As described in Chapter 3, cyclic failure of lap-splices in plastic regions has been studied
experimentally and a model to describe its influence on the cyclic strength degradation of the
member has been proposed, but previous research has been conducted with short lap-splices
of 20dbl length, which were common in the U.S. older provisions as pure compression splices.
However, in Switzerland for example, even in the past compression as well as tension lapsplices have been designed with a length of at least 40dbl which should warrant to some extent
a better performance than that of the short lap-splices mentioned above.
4.1.2 Experimental test units
Two of the most common deficiencies of older existing bridge piers are low transverse
reinforcement ratios and lap-splices at the pier base. The former may cause premature shear
failure, thus reducing the displacement capacity of the pier, while the latter can lead to a
degradation of flexural strength during inelastic cyclic loading which can cause an increase in
displacement demand. As the flexural strength defines the shear demand, there may also be an
interaction between the lap-splice behaviour and potential shear problems. In order to quantify
the effects of lap-splice on member behaviour two 1:2 scale quasi-static cyclic tests on bridge
piers with wall type cross section were carried out at ETH Zürich [Bimschas et al., 2008].
The test units were designed to represent real existing Swiss bridge piers from the 1960s with
detailing incorporating seismic deficiencies which are typical for structures of that time. All
units have the same concrete dimensions, the same steel properties and similar concrete
properties with the only difference being that one unit had a lap-splice at the base of 43dbl
while the other one had continuous reinforcement.
46
Chapter 4. Experimental database
4.1.3 Effects of cross section type on member
Previous research was mainly conducted on squared cross sections and only in the work
conducted by Chai [Chai et al., 1991] was a circular cross section studied. Squared cross
sections, contrary to circular and wall cross sections have the particularity that reinforcement
is mainly concentrated at the extremities, thus leading to consistently higher contribution of
"end reinforcement bars" on the overall flexural strength. As a consequence, loss of tension
strength at the extremities due to splice degradation for increasing sectional curvature leads to
a faster degradation of the overall member behaviour in comparison with wall and circular
sections.
Figure 4.1: Examples of typical bridge column cross section
For this reason and because wall shape cross sections are widely used in older existing Swiss
bridge piers, this has been the preferred cross section in the experiments carried out by
Bimschas [Bimschas et al., 2008].
4.2 Test setup and units
The experimental test setup consisted of a reaction wall with a horizontal actuator able to
provide lateral force on the tested specimen and two vertical actuators for axial force. In order
to provide a strong footing for the tested specimens, the footings were also bi-directionally
pre-stressed. The experiments consisted of displacement controlled, quasi-static loading
history applied to the cantilever pier with a lateral load application at 3.3m height above the
pier base.
Figure 4.2: Setup for the quasi-static test on existing Swiss bridge piers carried out at the ETH Zürich
[Bimschas et al., 2008]
47
Chapter 4. Experimental database
Table 4.1: Characteristic data of test units
Test Unit VK1
Test Unit VK2
Axial load
N (kN)
1370
1370
Concrete compressive cylinder
strength
fc (MPa)
35
39
Yield strength of longitudinal
reinforcement
fy (MPa)
515
515
Ultimate strength of
longitudinal reinforcement
fu (MPa)
630
630
Yield strength of transverse
reinforcement
fyh (MPa)
500
500
Longitudinal reinforcement
ratio
ρl (%)
0.82
0.82
Transverse reinforcement ratio
ρt (%)
0.08
0.08
Aspect ratio
H/lw
2.2
2.2
Lap-splice length
Ls (mm)
-
600 (∼43dbl)
In order to simulate bridge pier behaviour, the test units were subjected to a constant axial
load of 1370kN during the whole test, corresponding to a moderate axial load ratio of
N/(Agf'c)=0.075 respective 0.067 for VK1 and VK2.
Figure 4.3: Cross section, reinforcement detailing and elevation of test units VK1-2 [Bimschas et al., 2008]
48
Chapter 4. Experimental database
4.3 Quasi-static cyclic experiments
Load application consisted of the quasi-static loading of the tested specimen to target top
displacements. Small cycles of loading were interposed between two identical large cycles up
to a target displacement of about 60mm; beyond this only large cycles were performed up to
the failure of the tested units.
Figure 4.4: Experimentally recorded hysteretic (force deformation) behaviour of test unit VK1 (left,
continuous reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008]
Figure 4.5: Quasi-static cyclic displacement application on test unit VK1 (left, continuous reinforcement)
and VK2 (right, spliced reinforcement) [Bimschas et al., 2008]
A detailed description of the test results can be found in [Bimschas et al., 2008]; relevant here
is, that despite the only difference between the two specimens being a splice of the
longitudinal reinforcement in specimen VK2, the observed hysteretic behaviour of the two
units strongly differed in the inelastic range of deformations. While VK1 exhibits hysteretic
loop with practically, no loss of strength up to shear failure, VK2 suffers high cyclic strength
49
Chapter 4. Experimental database
degradation but achieves higher drifts because of the "shear failure prevention" effect
provided by the lap-splice (see section 1.2.1), thus being able to sustain the full axial load up
to ultimate drift capacity.
Figure 4.6: Quasi-static cyclic horizontal load application on test unit VK1 (left, continuous
reinforcement) and VK2 (right, spliced reinforcement) [Bimschas et al., 2008]
50
Chapter 5. Fundamental modelling assumptions
5 FUNDAMENTAL MODELLING ASSUMPTIONS
5.1
Sectional and member response
5.1.1 Flexural response without lap splice
Sectional response is obtained by performing a moment curvature analysis and idealized into
a bilinear curve. Subsequently, by making use of the concentrated plasticity concept, a forcedeformation relationship is obtained by integrating section curvatures over member length.
The moment curvature diagram is constructed with the aid of the computer software Cumbia
[Montejo et al., 2007] to carry out the moment curvature analysis using the properties of the
critical section. The program is written in software code Matlab and performs the sectional
analysis using the unconfined and confined concrete models proposed by Mander [Mander et
al., 1988a,b] and the reinforcing steel model proposed by King [King et al., 1986].
Alternatively the user is able to define any other model for unconfined or confined concrete
and for reinforcing steel. Default material models in Cumbia [Montejo et al., 2007] used for
the analyses are presented herein.
(a) Model for the Confined Concrete
The longitudinal compressive stress f’c is given by
fc =
f cc' xr
r −1+ xr
(5.1)
where:
x=
εc
ε cc
(5.2)
⎡
⎛ f cc'
⎞⎤
− 1 ⎟⎟ ⎥
'
⎝ f co
⎠ ⎥⎦
ε cc = ε co ⎢1 + 5 ⎜⎜
⎣⎢
(5.3)
51
Chapter 5. Fundamental modelling assumptions
Ec
E c − E sec
r =
f cc'
E sec =
⎛
ε cu = 1 .4 ⎜⎜ 0 .004 +
⎝
(5.4)
(5.5)
ε cc
1 . 4 ρ s f yh ε su ⎞
⎟⎟ , with εcu 40% larger as in [Mander et al., 1988a,b]
f cc'
⎠
(5.6)
The level of confinement is defined differently for circular and rectangular sections and
depends on the amount of transverse reinforcement in the section.
For circular sections:
⎛
7 . 94 f l '
fl'
2
f cc' = f co' ⎜ − 1 . 254 + 2 . 254 1 +
−
⎜
f co'
f co'
⎝
f l' =
ρs =
1
k e ρ s f yh
2
(5.9)
dss
⎛
s' ⎞
⎟
⎜⎜ 1 −
2 d s ⎟⎠
⎝
ke =
1 − ρ cc
ke =
(5.7)
(5.8)
4 A sp
1−
⎞
⎟
⎟
⎠
2
s'
2d s
1 − ρ cc
for circular hoops
(5.10)
for circular spirals
(5.11)
For rectangular sections:
f lx' = k e ρ x f yh
and
ρx =
⎛
⎜1 −
⎜
ke = ⎝
n
Asx
sH c
and
(w )
∑ 6B
i =1
f ly' = k e ρ y f yh
ρy =
⎞⎛
⎟⎜ 1 − s '
⎟⎜
2 Bc
c H c ⎠⎝
1 − ρ cc
' 2
i
(5.12)
A sy
(5.13)
sB c
⎞⎛
s'
⎟⎟ ⎜⎜ 1 −
2H c
⎠⎝
⎞
⎟⎟
⎠
(5.14)
To determine the confined concrete compressive strength in rectangular sections Cumbia
[Montejo et al., 2007] assumes an average effective lateral confined stress of
f l ' = 0 . 5 ( f lx' + f ly' )
(5.15)
52
Chapter 5. Fundamental modelling assumptions
(b) Model for the Unconfined Concrete
The unconfined concrete follow the same curve as the confined concrete with a lateral
confined stress of f'l = 0. The part of the falling branch for strains larger than 2εo is assumed
to be a straight line which reaches zero at εsp.
(c) King Model for the Reinforcing Steel
The stress-strain relation for the reinforcing steel used the model proposed by King [King et
al., 1986] is
f s = E sε s
εs ≤ εy
(5.16)
fs = f y
εy ≤ εs ≤ εsh
(5.17)
⎡ m (ε s − ε sh ) + 2 (ε s − ε sh )(60 − m ) ⎤
fs = f y ⎢
+
⎥ εsh ≤ εs ≤ εsm
2
2 (30 r + 1)
⎣ 60 (ε s − ε sh ) + 2
⎦
(5.18)
⎛ f su
⎜
⎜ f
y
m= ⎝
⎞
⎟ (30 r + 1)2 − 60 r − 1
⎟
⎠
15 r 2
and r = εsm - εsh
(5.19)
(d) Idealization of the moment curvature diagram
The bilinear idealization of the moment curvature diagram is estimated according to the
recommendations given by [Priestley et al., 2007] as follow:
Table 5.1: Evaluation of sectional response quantities according to [Priestley et al., 2007]
Condition (at fist
occurrence of either)
Point
First Yield
My, φy
εs = εy = fy/Es
εc = 0.002
Nominal Moment
Mn
εs = 0.015
εc = 0.004
Description
Steel strain at yielding
Strain at peak stress of unconfined concrete
Strain limit on steel deformations after yielding
(onset of 1mm crack width)
Ultimate strain of unconfined concrete (onset of
spalling)
Nominal Curvature
φn
φn = φn Mn/My
Linearly extrapolated nominal curvature from first
yielding
Ultimate Capacity
Mu, φu
εs = 0.6εsu
Effective ultimate strain of steel
Ultimate strain of confined concrete
εc = εcu
The member response is obtained using the plastic hinge method proposed by [Priestley et al.,
1996]. The plastic hinge method replaces the real curvature distribution with an equivalent
curvature distribution in order to find the displacements in the member.
L p = kL c + L sp ≥ 2 L sp
(5.20)
53
Chapter 5. Fundamental modelling assumptions
L sp = 0 . 022 f s d bl
f sp ≥ f y
⎛ f
⎞
k = 0 . 2 ⎜ su − 1 ⎟ ≤ 0 . 08
⎜ f
⎟
⎝ y
⎠
(5.21)
(5.22)
An effective length of the member is then defined as:
L eff = L + L sp
single bending
(5.23)
L eff = L + 2 L sp
double bending
(5.24)
The flexural displacement of the member at yielding considers the effective length of the
member and is calculated as:
Δ fy =
φ n L2eff
single bending
(5.25)
double bending
(5.26)
Δ fu = (φ u − φ n )L p (L + L sp − 0 . 5 L p ) + Δ fy
single bending
(5.27)
Δ fu = (φ u − φ n )L p (L + 2 (L sp − 0 . 5 L p )) + Δ fy
double bending
(5.28)
Δ fy =
3
φ n L2eff
6
The flexural displacement at ultimate capacity is given by:
5.1.2 Flexural response with lap splice
The maximum bar force Tb that can be transferred without the assistance of special transverse
reinforcement confining the splice according to the procedure of [Priestley et al., 1996] is
Tb = Ab f s = f t pl s
(5.29)
And the perimeter p of the characteristic block is
p=
p=
πD '
2n
+ 2 ( d bl + c ) ≤ 2 2 ( c + d bl )
s
+ 2 ( d bl + c ) ≤ 2 2 ( c + d bl )
2
circular cross section
(5.30)
rectangular cross section
(5.31)
where n is the number of longitudinal bars of diameter dbl evenly spaced around the core of
diameter D' with cover c and s is the average spacing between spliced pairs of bars along the
critical member face. The upper limit to Equations (5.30) and (5.31) applies when bars are
widely spaced and failure by the starter bars pulling off the core with a 45° wedge result in a
lower effective perimeter.
Taking a conservative estimate the tension strength as
54
Chapter 5. Fundamental modelling assumptions
f t = 0 . 33
f c'
( MPa )
(5.32)
the required splice length so that the longitudinal bar reach a given stress fs is
l s = 0 . 48 d bl f s /
f c'
(5.33)
Assuming fs=fy we obtain the necessary development length of the longitudinal bar in order to
reach yielding thus the considered lap splice length of each member can be checked for this
condition.
(a) Yielding of longitudinal bars achieved
If yielding of longitudinal reinforcement bars is achieved, the multi-linear idealized momentcurvature follows the same path as bilinear idealized curve up to the point of nominal capacity
Mn. Beyond this point the flexural capacity is assumed to be constant up to an extreme fibre
compression strain of concrete εc=0.002. After this point the capacity degrades linearly up to
the residual strength moment Mr (based on axial force eccentricity alone):
⎛ h '− a ⎞
M r = P⎜
⎟
⎝ 2 ⎠
rectangular section
(5.34)
⎛ D' ⎞
M r = P⎜
⎟
⎝2− x⎠
circular section
(5.35)
It has been experimentally estimated, that this residual moment capacity can be conservatively
associated with a sectional curvature of:
φ = 8φ n +
φ (ε c = 0 . 002 )
φn
(5.36)
Priestley, Seible and Calvi [Priestley et al., 1996] recommend that this sectional curvature be
considered the ultimate curvature capacity for assessment of damage-control limit state,
although experimental evidence from a limited database indicates that larger curvature
ductilities can be sustained.
The idealized moment-curvature relationship for the case where yielding of the longitudinal
reinforcement is not achieved is shown in Figure 5.1 by means of curve number four.
(b) Yielding of longitudinal bars not achieved
If yielding of longitudinal reinforcement bars is not achieved, because of short lap splice, the
nominal moment capacity will never be achieved and the maximum flexural strength of the
section can be estimated assuming a reinforcing steel stress fs. The corresponding curvature
can be calculated as
φs = φn
Ms
Mn
(5.37)
55
Chapter 5. Fundamental modelling assumptions
The flexural strength of the member will degrade after this point from Ms to a residual
moment Mr as cyclic inelastic response develops. Experimental results support a residual
section curvature, associated with Mr of:
φ = 8φ n
(5.38)
Figure 5.1: Curvature ductility for members without and with lap splices in column end regions
5.1.3 Shear capacity envelope
The shear capacity envelope for each column is estimated using the revised UCSD Model by
Kowalsky and Priestley [Kowalsky et al., 2000] and the Sezen [Sezen et al., 2004] shear
model. Both models are ductility dependent for increasing member curvature and explicitly
consider the effects of axial load on total shear strength capacity.
(a) Modified UCSD shear model
The model predicts shear strength capacity of the member as the sum of three separate
components: steel truss mechanism Vs, axial load mechanism Vp and concrete shearing
mechanism Vc.
V = Vs + V p + Vc
(5.39)
The steel truss mechanism assumes an angle of inclination of ϑ=30° from the longitudinal
axis of the member and is calculated as follow:
Vs =
π
2
D − c lb +
A sp f yh
V s = A xx f yh
s
H − c lb +
s
dh
−c
2
cot( ϑ )
dh
−c
2
cot( ϑ )
circular section
(5.40)
rectangular section
(5.41)
The concrete shear strength mechanism is ductility dependent and accounts for degradation of
concrete the matrix with increasing member curvature:
56
Chapter 5. Fundamental modelling assumptions
f c' (0 . 8 A g )
V c = αβγ
(5.42)
M
≤ 1 .5
VD
β = 0 .5 + 20 ρ l ≤ 1
(5.43)
0 . 05 ≤ γ = 0 . 37 − 0 . 04 μ Δ ≤ 0 . 29 uni-axial bending
(5.45)
0 . 05 ≤ γ = 0 . 33 − 0 . 04 μ Δ ≤ 0 . 29 bi-axial bending
(5.46)
1≤α = 3−
(5.44)
Where the yield displacement corresponds to Δfy, calculated considering only the flexural
deformation and the ratio M/VD is equivalent to the aspect ratio Lc/D. For rectangular sections
Lc/lw is assumed.
The axial load contribution describes the positive effect of compression load on shear strength
and is considered as:
D−c
2L
D−c
Vp = P
L
Vp = 0
Vp = P
P>0
single bending
(5.47)
P>0
double bending
(5.48)
P<0
(5.49)
The previous equations are used for the assessment of existing structures. In the case of a
design for a new structure a more conservative approach is used, reducing the acting axial
load by 15%, increasing the angle of inclination up to ϑ=35° and applying a global strength
reduction factor of 0.85.
(b) Sezen shear model
The model predicts the shear strength capacity of the member as the sum of two separate
components: the steel truss mechanism Vs and the concrete shearing mechanism Vc. In this
procedure the shear strength degradation is related to both the concrete and the steel truss
mechanism, with the latter one being the result of anchorage degradation and reinforcement
misalignment.
V n = k Δ (V c + V s )
Vc =
Vs =
f c'
0 .5
a
d
Ast f yh d
s
1+
(5.50)
P
0 .5
f c' A g
0 .8 A g
(5.51)
(5.52)
As noted by Miranda [Miranda et al., 2004] “[…] the formulation of concrete shear strength
corresponds to the principal tension stress of un-cracked concrete members subjected to both
normal and shear stresses and is limited by 0 .5 f c' . Moreover the aspect ratio a/d correction
factor was introduced to calibrate the equation as the result of experimental data, but the
specimen covered only a small range between 2.0 and 4.0, thus caution is needed outside this
range.”
57
Chapter 5. Fundamental modelling assumptions
5.2 Modelling of reinforced concrete members without lap splices
With the aim to perform NLTHA on single piers and on whole structures, the experimental
member behaviour (monotonic and cyclic) of test unit VK1 (continuous longitudinal
reinforcement) was estimated analytically based on assumptions of Chapter 5.1.
5.2.1 Monotonic member behaviour
Assuming material and sectional properties given in Chapter 4, member deformations were
obtained from sectional analysis and using the theory of lumped plasticity. Finally by means
of an idealization a bilinear approximation of member force deformation relation was
calculated.
Table 5.2: Results of bilinear idealization of member analysis with CUMBIA for test unit VK1
Sectional response
-1
Member response
Description of considered point
φ (m )
M (kNm)
Δ top (mm)
F (KN)
nominal yielding (φy, Mn)
0.00325
2357.9
14.52
714.51
ultimate capacity (φu, Mu)
0.03439
2230
48.97
697.0
Comparison of analytical results versus experiment can be seen in Figure 5.2. The analytical
pushover curve is very similar to the backbone curve of the experiments in terms of
displacement and lateral load capacity, but conservative regarding ultimate displacement
capacity. This latter consideration arises because Cumbia uses a conservative hinge length Lp
for walls, neglecting the wall length contribution on plastic hinge length, that for the
considered case correspond to about 50% of Lp.
Figure 5.2: Analytical and experimental results for test unit VK1 (continuous longitudinal reinforcement)
58
Chapter 5. Fundamental modelling assumptions
It is interesting to notice, that even if a shear failure was reported in [Bimschas et al., 2008],
analytical verification predicts flexural failure. Shear capacity is greater than flexural
capacity; [Kowalski et al., 2000] prediction is 10% above [Sezen et al., 2004] model.
5.2.2 Cyclic member behaviour
Starting from the analytical results of section 5.2.1, the cyclic member response of specimen
VK1 subjected to quasi-static cyclic loading was reproduced by means of appropriate
hysteretic rules in software Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006]. Consistency of
analyses with experimental results were validated in terms of lateral load capacity,
displacement capacity, total energy dissipation and cyclic dissipation.
(a) Model using software code Ruaumoko
Quasi-static cyclic analysis with identical displacement history, such as in experiments of
[Bimschas et al., 2008], was performed using two spring in series with different hysteretic
rules in order to achieve the desired loop behaviour. By setting the point of nominal yielding
as in section 5.2.1 and modelling the two independent springs as in Table 5.3 a consistent
analysis response was obtained in Ruaumoko [Carr, 2004].
Figure 5.3: Hysteretic rules used for the analysis: Modified Takeda with degrading stiffness (left,
IHYST=4) and Origin-Centred (right, IHYST=7) according to Ruaumoko [Carr, 2004]
Table 5.3: Modeling assumptions for the cyclic inelastic analysis in Ruaumoko [Carr, 2004]
Hysteretic Rule
Percent
Stiffness
K0
Force Fy
(positive/negative)
Stiffness
rK0
Unloading
Reloading
Factor α
Factor β
Modified Takeda
85%
41’828kN/m
607kN
0
0.5
0.0
Origin-Centred
15%
7’381kN/m
107kN
0
-
-
Total Response
100%
49’209kN/m
714kN
0
(b) Model using software code Idarc
As for the prior analysis, a quasi-static cyclic analysis was performed in Idarc [Idarc, 2006]
using a single hysteretic rule able to independently describe stiffness degradation, strength
deterioration (ductility of energy dependent) and slip (or pinching) behaviour. In this case the
sectional and member properties of the column were defined and by means of lumped
plasticity theory a force drift relation was automatically defined. Table 5.4 shows the relevant
modelling assumptions.
59
Chapter 5. Fundamental modelling assumptions
Table 5.4: Modelling assumptions for the cyclic inelastic analysis in Idarc [Idarc, 2006]
Hysteretic
Rule
Trilinear
Δy
Member
Length
Defined
Plastic
Hinge Lp
Post
Yield
Stiffness
HC
HBD
HBE
HS
14.5mm
3300mm
L/6=167mm
0.0
2.0
0.01
0.01
0.54
Percent
Force
Fy
Displ.
100%
714kN
(c) Comparison of results
By means of these analyses the experimental behaviour of test unit VK1 could accurately be
reproduced in terms of hysteretic loops, cumulative energy dissipation and comparative
hysteretic cyclic damping for single cycles of loading according to the procedure proposed by
Jacobsen [Jacobsen, 1960].
Figure 5.4: Comparison of analytical and experimental hysteretic responses for test unit VK1
The method proposed by Jacobsen for damping estimation compare the effective energy
dissipated in one cycle of loading with the energy arising from a substitute structure behaving
perfectly elastic up to a target displacement Δm and force Fm (see Figure 5.5) as
ξ =
Ah
2π F m Δ m
(5.53)
By considering first the experimental data, assuming analytical yield displacement and
considering the relevant cycles of loading, an elastic material damping for test unit VK1 has
been found to be ξel = 5.0%.
60
Chapter 5. Fundamental modelling assumptions
Figure 5.5: Hysteretic area for damping calculation in one cycle of loading [Priestley et al., 2007]
By subtracting the equivalent amount of energy dissipated in the experiment before yielding,
the cumulative energy dissipated in the quasi-static cyclic analyses in Ruaumoko [Carr, 2004]
and Idarc [Idarc, 2006] were compared with test results.
Figure 5.6: Comparison of analytical and experimental cumulative dissipated energy
Table 5.5: Comparison of calculated damping for single cycling of loading in analyses and experiment
Displacement ductility μΔ
ξExperimental (%)
ξRUAUMOKO (%)
ξIDARC (%)
1.09
7.44
2.02
1.44
1.44
8.33
7.22
8.01
2.17
12.8
12.2
13.4
2.89
14.6
13.6
14.2
3.63
15.6
14.9
15.7
4.35
16.3
15.5
15.8
61
Chapter 5. Fundamental modelling assumptions
5.3 Modelling of reinforced concrete members with lap splices
As for members with continuous longitudinal reinforcement, the experimental behaviour
(monotonic and cyclic) of test unit VK2, detailed with a lap splice of 43dbl at the column base
was estimated analytically based on the assumption of Chapter 5.1.
5.3.1 Monotonic member behavior
Neglecting the marginal effect of a slightly higher measured concrete compression strength in
test unit VK2, but considering that the full nominal moment capacity can develop in the
section and applying the strength degradation model of Priestley [Priestley et al., 1996] a
linear idealized member force deformation relationship was calculated. Because of the high
scatter of the original model in the prediction of strength degradation compared with
experimental results, a modification of the model in [Priestley et al., 1996] was adopted with
the aim to better fit the real observed strength degradation.
Table 5.6: Results of linear idealization of member analysis by application of [Priestley et al., 1996] model
Sectional response
-1
Member response
Description of considered point
φ (m )
M (kNm)
Δ top (mm)
F (KN)
nominal yielding (φy, Mn)
0.00325
2357.9
14.5
714.5
degrading curvature φc(εc = 0.002)
0.00641
2357.9
16.8
714.5
ultimate curvature φ = φc/φy + 8φy
0.003241
828
36.8
250.9
Because from experimental report of test unit VK2 a real member failure was not observed
even beyond the maximum test displacements of 110mm, a linear idealization of strength
decay after this point was interpolated down to zero lateral load carrying capacity.
Figure 5.7: Analytical and experimental results for test unit VK2 (spliced longitudinal reinforcement)
62
Chapter 5. Fundamental modelling assumptions
As for previous predictions of backbone curves, the lumped plasticity model described in
Chapter 5.1 was used to get member response from sectional analysis. It should be noted, that
assumptions regarding modification of the [Priestley et al., 1996] model proposed here are
purely base on experimental data on test unit VK2, thus further use on members with different
characteristics should be evaluate on a case by case basis.
Table 5.7: Results of linear idealization of member analysis by application of modified [Priestley et al.,
1996] model in order to fit experimental data
Sectional response
-1
Member response
Description of considered point
φ (m )
M (kNm)
Δ top (mm)
F (KN)
nominal yielding (φy, Mn)
0.00325
2357.9
14.5
714.5
degrading curvature φc(εc = 0.005)
0.019
2357.9
25.9
714.5
residual capacity φ = φc/φy + 20φy
0.084
828
72.9
250.9
-
0
140
0
no capacity, by interpolation
Comparison of analytical and experimental results in Figure 5.7 underlines the need for
improved strength degradation models for columns with splice length greater than 20dbl and
for wall shaped cross section. Because of the limited database and the simplicity of the
original model with respect to real behaviour, general considerations of experimental
evidence versus proposed idealization (modification) of [Priestley et al., 1996] prediction
cannot be done for splices of 43dbl. Nevertheless the model greatly underestimates the
initiation of strength degradation and overestimates the strength decay, leading to a much
lower ultimate displacement as observed in test unit VK2, underling the fact that a calibration
has been undertaken for splice length of roughly 20dbl.
5.3.2 Cyclic member behaviour
Based on results of section 5.3.1 the cyclic member response of test unit VK2 subjected to
quasi-static cyclic loading was reproduced by means of appropriate hysteretic rules in
software Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006]. Because the only difference
between the two test units is a lap-splice at column base in one specimen, identical hysteretic
rules will be assumed in these cyclic analyses with the only difference being definition and
implementation of strength degradation. As for previous test unit, consistency of analyses
with experimental results was validated in terms of lateral load capacity, displacement
capacity, total energy dissipation and cyclic dissipation.
(a) Model using software code Ruaumoko
According to the strength degradation implementation in Ruaumoko [Carr, 2004], both
hysteretic rules were degraded as a function of maximum ductility as showed in Figure 5.8.
As for previous analyses, a wall top yield displacement of Δy = 14.5mm and a lateral yielding
force equal to Fy = 714.5kN were assumed.
63
Chapter 5. Fundamental modelling assumptions
Table 5.8: Strength degradation modelling assumption for analysis in Ruaumoko [Carr, 2004]
Strength degradation ILOS
DUCT 1
DUCT 2
RDUCT
DUCT 3
3 (function of max. ductility)
1.78
5.01
0.35
9.71
Figure 5.8: Strength degradation model implemented in Ruaumoko [Carr, 2004]
(a) Model using software code Idarc
Modelling of cyclic strength degradation in Idarc [Idarc, 2006] can be achieved in three ways:
by purely ductility or energy dependence or as a combination of both, thus resulting in
different hysteretic loops as shown in the next figure.
Figure 5.9: Ductility based (left) and energy based (right) strength degradation in Idarc [Idarc, 2006] in a
force-displacement-relationship
Because of experimental evidence of strength degradation observed in two cycles to equal
displacement ductility and with increasing ductility, pure energy based strength degradation
was assumed for the analysis, thus the only parameter changing from test unit VK1 is HBE.
Table 5.9: Modelling assumptions for the cyclic inelastic analysis in Idarc [Idarc, 2006]
Δy
Member
Length
Defined
Plastic
Hinge Lp
Post
Yield
Stiffness
HC
HBD
HBE
HS
14.5mm
3300mm
L/6=167mm
0.0
2.0
0.01
0.65
0.54
Hysteretic
Rule
Percent
Force
Fy
Displ.
Trilinear
100%
714kN
64
Chapter 5. Fundamental modelling assumptions
(c) Comparison of results
Reproduction of experimental behaviour for test unit VK2 has been accurately modelled with
both software and comparison of hysteretic loops and cumulative energy dissipation concur
well with observations. Nevertheless hysteretic cyclic damping for single cycles of loading
according to the procedure proposed by Jacobsen [Jacobsen, 1960] were 10-20% longer as
reported in Table 5.10.
Figure 5.10: Comparison of analytical and experimental hysteretic responses for test unit VK2
As explained in section 5.2.2 comparisons of dissipated energy did not account for
experimental energy before yielding. In this case an elastic material damping for test unit
VK2 has been found to be ξel = 4.6%.
65
Chapter 5. Fundamental modelling assumptions
Figure 5.11: Comparison of analytical and experimental cumulative dissipated energy
Table 5.10: Comparison of calculated damping for single cycling of loading in analyses and experiment
Displacement ductility μΔ
ξExperimental (%)
ξRUAUMOKO (%)
ξIDARC (%)
1.08
8.60
1.95
1.35
1.43
7.79
7.05
8.06
2.17
13.8
12.7
14.0
2.90
13.7
14.2
15.8
3.63
15.0
15.3
16.8
4.35
15.1
16.5
17.5
5.01
13.8
16.6
17.7
5.76
14.8
17.3
18.1
6.48
14.3
17.8
17.6
7.20
14.5
18.5
17.5
5.4 Calibration of monotonic and hysteretic behaviour on experimental yielding points
Beside the calibration of monotonic and cyclic behaviour of test units VK1 and VK2 based on
numerical analyses using the theory described in section 5.1, force deformation capacity has
been reproduced using experimental data, considering the effective yielding force and
displacements for specimens VK1 and V K2 recorded during experiments carried out at the
ETH Zürich by Bimschas [Bimschas et al., 2008]. Moreover cyclic behaviour has been
66
Chapter 5. Fundamental modelling assumptions
calibrated using software codes Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006] as described
in section 5.2.2 and 5.3.2, but using experimental backbone curves.
In this way, dynamic single-degree-of freedom analyses and multi-degree-of-freedom
analyses could be performed twice, firstly considering capacity estimated on an analytical
basis (section 5.2.2 and 5.3.2) and then considering calibration on experimental data presented
in this section (5.4).
5.4.1 Monotonic member behaviour
Experimental results for specimen VK1 and VK2 shows, that although equal sectional
properties, yielding force and displacement differs for the two test units, the member with
spliced end-reinforcement is stiffer, as a consequence of the double reinforcement ratio at pier
base due to this bar detailing.
Table 5.11: Capacity curve assumptions considered in this study for NLTHA
yielding point
degradation
residual cap.
failure point
Capacity curve [kN, mm]
Fy
Δy
Fdeg
Δdeg
Fres
Δres
Fult
Δult
Continuous, Experimental data
700
9.77
-
-
-
-
700
64
Lap-splice, Experimental data
715
8.22
715
31.32
171.5
94.55
0
148
As can be seen, the level of strength capacity can easily be predicted using numerical
analyses, but yield displacement sometimes varies up to 50% when compared with
experimental observed behaviour versus application of lumped plasticity theory starting from
sectional analysis.
Figure 5.12: Capacity curve envelope for test specimen VK1 and VK2 based on experimental data
according to [Bimschas et al., 2008]
67
Chapter 5. Fundamental modelling assumptions
5.4.2 Cyclic member behaviour
Considering the capacity curves presented in Figure 5.12, the hysteretic behaviour has been
calibrated with software codes Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006] in order to
match the experimentally observed loops. The amount of energy dissipation has particularly
been optimized in order to avoid an overestimation of the hysteretic energy dissipation in
within the numerical loops compared with experiment.
Figure 5.13: Experimentally obtained cyclic behaviour of test units VK1 (left) and VK2 (right) versus
hysteretic loops using software code Ruaumoko [Carr, 2004] and Idarc [Idarc, 2006] based on backbone
assumptions presented in Table 5.11 and Figure 5.12 (left).
The calibration of the hysteretic behaviour presented in this section has been performed by
Bimschas (ETH Zürich) and was kindly shared to the author for this study.
5.5 Scaling Factors
Because tested specimens were built on a 50% scale compared to reality dimensions, scaling
factors on Table 5.12 must be applied to achieve real bridge pier dimensions and capacities.
As a consequence the yielding point of the 6.6m height real bridge pier will correspond to a
top displacement of Δy = 29.04mm and a lateral force equal to Fy = 2858kN (backbone from
numerical analyses).
68
Chapter 5. Fundamental modelling assumptions
Table 5.12: Scaling factors experimental test units to real bridge piers dimensions
Unit to scale
Scaling factors
Density
1
Pressure
1
Length
2
Area
4
Force
4
Mass
8
Volume
8
Moment
8
Starting from real pier dimensions, member length will be varied in a range between 6.6m 13.2m in order to cover different possible pier height of existing bridges, keeping sectional
properties constant. As a consequence, following scaling factors must be applied to the
different piers:
Table 5.13: Scaling factors for pier height variation starting from real bridge piers dimensions
Unit to scale
Scaling factors
Yield displacement
(L/Lreal)2
Yield force
(Lreal/L)
69
Chapter 6. Target seismicity for the analyses
6 TARGET SEISMICITY FOR THE ANALYSES
6.1 Selection of target response spectra
Considering the aim of the analyses being mostly focused on the evaluation of possible
increased displacement demand for existing bridge piers with detailing deficiencies compared
with those without, three level of seismicity ranging from moderate to very high have been
selected in order to cover a wide range of hazard levels. Elastic response spectra of
acceleration (left) and displacement (right) can be seen in Fig. 6.1. According to [SHA, 2004]
a PGA 0.16g implies a return period of 475 years or 10% probability of exceedance in 50
years, while a PGA 0.35g roughly corresponds to a return period of 2500 years or 2%
probability of exceedance in 50years for the highest hazard region of Switzerland (Wallis).
Figure 6.1: Assumed elastic response spectra of acceleration (left) and displacement (right) according to
code provisions [SIA 261, 2003] for PGA of 0.16g, 0.35g, 0.50g (agd), stiff soil conditions (Soil B), structure
importance factor III and elastic viscous damping ξ=0.02
70
Chapter 6. Target seismicity for the analyses
Elastic response spectra of acceleration according to [SIA 261, 2003] are calculated as
⎛
(2 .5η − 1)T
PSA = a gd γ f S ⎜⎜ 1 +
TB
⎝
⎞
⎟⎟
⎠
PSA = 2 . 5 a gd γ f S η
T < TB
(6.1)
T B < T < TC
(6.2)
PSA = 2 . 5 a gd γ f S η
TC
T
TC < T < T D
(6.3)
PSA = 2 . 5 a gd γ f S η
TC T D
T2
T > TD
(6.4)
Where agd is the peak ground acceleration (PGA), γf is the structure importance factor (1.4 for
this study) and η is the damping correction factor (or damping reduction factor Rξ)
implemented in code provisions [SIA 261, 2003] and [EC8, 1998] for generation of response
spectra with an elastic viscous damping differing from 5% critical damping
⎛
1
⎞
0 . 50
η = Rξ = ⎜⎜
⎟⎟
⎝ 0 . 5 + 10 ξ ⎠
≥ 0 . 55
(6.5)
Parameters S, TB, TC and TD account for site soil condition and are listed below in the case of
a stiff soil assumption.
Table 6.1: Soil conditions for stiff soil (Soil B) according to code provisions [SIA 261, 2003]
Soil
vs (m/s)
NSPT (-)
Su (kPa)
S (-)
TB (s)
TC (s)
TD (s)
B
400-800
> 50
> 250
1.20
0.15
0.50
2.00
Based on elastic response spectrum of pseudo-acceleration (PSA), displacement response
spectrum (SD) is estimated using the following period dependence as
SD =
PSA
ω2
2
T2
⎛ T ⎞
= PSA ⎜
⎟ = PSA
4π 2
⎝ 2π ⎠
(6.6)
Based on elastic displacement response spectra real ground motion are selected and scaled
while artificial ground motion are generated.
6.1.1 Selection and scaling procedure of real ground motions
Considering the fact, that due to many factors the frequency range over which recording
instruments work can differ from station to station, but a typical frequency band over which
modern strong motion instruments work is 0.01Hz to 25Hz (response characteristics), and
considering that the typical frequency band of interest in earthquake engineering is in a range
0.01Hz to 10Hz, a reasonable band pass filter between 0.01 and 25Hz is adopted for all
engineering applications. Nevertheless, for very low frequencies (up to say 0.1-0.2Hz) noise
could strongly influence the response and the use of strong motions for structures with
vibration period Te > 5s should consider appropriate post-processing analyses.
71
Chapter 6. Target seismicity for the analyses
Table 6.2 Typical frequencies generated by different seismic sources
Frequency
Type of measurement
0.00001-0.0001
Earth tides
0.0001-0.001
Earth free oscillations, earthquakes
0.001-0.01
Surface waves, earthquakes
0.01-0.1
Surface waves, P and S waves, earthquake with M>6
0.1-10
P and S waves, earthquakes with M>2
10-1000
P and S waves, earthquakes with M<2
Table 6.3 Post-processing procedures of real accelerograms
Application of
Polinomial
type
Baseline correction
Linear
Butterworth filter
4th order
Filtering
type
Filtering
configuration
Frequency
Pass range
Butterworth
Bandpass
0.01 to 25Hz
The selection and scaling procedure of real ground motions from past earthquakes consisted
first of post-processing the recorded accelerograms by performing a baseline correction, a
filtering procedure as described before and subsequently a matching procedure of the actual
displacement response spectra with target spectra over a period range that approximately
described the vibration periods of the structures considered. Because of the high variability of
real ground motions with respect to target spectra in code provisions care was taken while
choosing the shape and scaling factor of selected motions.
An important issue in the matching of real ground motion displacement response spectra with
target displacement spectra is represented by the scaling factor α, defined as the factor for
which the minimum of squared error of the area enclosed by two periods is achieved, as
T2
⎛ T2
min = ⎜ ∫ S a ,t arg et ( T )dT −α ∫ S a ,real ( T )dT
⎜T
T1
⎝ 1
⎞
⎟
⎟
⎠
2
(6.7)
Because this scaling procedure implicitly modifies the characteristics of the real ground
motion, according to [NZS 06] matching of the target response spectra should be achieved for
factors corresponding to 30% to 300% of the real ground motion.
Table 6.4: Criteria for consideration of real ground motion data
Ground motion
scaling factor
Range of periods
to be scaled
lower bound of
acceptance
upper bond of
acceptance
α
T1 - T2 = 0.5 – 2.0s
0.30
3.00
α
T1 - T2 = 0.35 – 1.0s
0.30
3.00
72
Chapter 6. Target seismicity for the analyses
Figure 6.2: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching procedure on
displacement spectra for different period ranges.
A total of four strong ground motions coming from four different earthquakes were selected
from different databases according to criteria listed in Table 6.4. For each ground motion the
SD and PSA response spectra for ξ=2% was calculated and compared with the target
spectrum of displacement.
Table 6.5: Accepted real ground motions from past earthquakes
RGM
record
Earthquake
Date
Magnitude
(Richter)
Location
Fault
Comp.
PGA
(g)
PGV
(m/s)
PGD
(m)
1
Kobe
17.01.1995
7.2
Kobe JMA
NS
0.819
0.878
0.220
2
Friuli
6.05.1976
6.5
unknown
-
0.473
0.297
0.0859
3
Kocaeli
17.08.1999
7.4
Sakaria
-
0.595
0.811
0.320
4
Loma Prieta
18.10.1989
7.1
Corralitos
-
0.810
0.602
0.197
Because of the broad period range initially considered in the scaling procedure (T=0.502.00sec.) a new scaling was performed for a shorter period range (T=0.35-1.00sec.) in a
subsequent phase. NLTHA were performed for both scaling factors and results are presented
in the next chapter in a separated form.
Table 6.6: Scaling factors based on Displacement Response Spectra for different seismicities in period
range between T = 0.50-2.00sec. (3th-4th columns) and T = 0.35-1.00sec. (5th-6th columns), respectively
RGM
record
Earthquake
Scaling Factor
Scaling Factor
Scaling Factor
Scaling Factor
αPGA 0.16g , ξ = 2- 5%
αPGA 0.35g , ξ = 2- 5%
αPGA 0.16g , ξ = 2- 5%
αPGA 0.35g , ξ = 2- 5%
1
Kobe
0.30
0.65
0.24
0.52
2
Friuli
1.32
2.88
0.87
1.90
3
Kocaeli
0.40
0.88
0.54
1.18
4
Loma Prieta
0.53
1.16
0.32
0.70
73
Chapter 6. Target seismicity for the analyses
Figure 6.3: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching procedure on
displacement spectra for a period range of T=0.50-2.00sec. (these motions are identified as RGM)
Figure 6.4: Comparison of target, elastic response spectra for a PGA 0.35g with ground motion scaled
response spectra of acceleration (left) and displacement (right) using the described matching procedure on
displacement spectra for a period range of T=0.35-1.00sec. (these motions are identified as RGM N)
6.2
Artificial Ground Motion Database
6.2.1 Generation Procedure
Artificial, statistically independent ground motions has been generated using the software
Simqke [Carr, 2004] based on a target response spectra matching procedure. The generation
of artificial strong ground motions implied a trapezoidal envelope of the accelerogram,
according to [Carr, 2004] and used following parameter:
74
Chapter 6. Target seismicity for the analyses
Table 6.7: Artificial ground motion generation parameters used in SIMQKE
Generation parameter
Definition
Value
Minimum period of simulation
Ts
0.1s
Maximum period of simulation
Tl
4s
Acceleration duration
DUR
20-30s
Acceleration rise time
TRISE
2s
Acceleration level time
TLVL
15s
Time step for accelerogram
DELT
0.02s
AGMAX
0.16-0.5g
DAMP
2%
Maximum acceleration
Damping ratio
75
Chapter 7. Single-degree-of-freedom analyses
7 SINGLE-DEGREE-OF-FREEDOM ANALYSES
7.1 Introduction
Based on the cyclic modelling assumptions presented in chapter 5, NLTHA on single-degreeof-freedom systems have been carried out assuming member capacity behaviours and lumped
mass consistent with experiments carried out by [Bimschas et al., 2008], but scaling all
involved parameters to match real dimensions (pier height h=6.60m) and studying members
with and without strength degradation (see Table 7.1). Moreover, a set of different parameters
have been varied according to Table 7.3 in order to study their influence on displacement
demand.
Single-degree-of-freedom dynamic analyses in this study aim to represent, in a simplified
manner, bridge pier behaviours of multi-span bridges, when they are subjected to transverse
earthquake excitations. Therefore “isolation” of the single pier starting from a multi-degreeof-freedom system is obtained by consideration of an appropriate superstructure bridge
portion to concentrate as lumped mass on top of the pier as presented in Figure 7.1 (left).
Figure 7.1: Idealized single-degree-of-freedom systems representing a single bridge pier with lumped mass
subjected to transverse earthquake excitation (left) and front, lateral view of a typical bridge pier and
superstructure considered in this study (middle, right).
76
Chapter 7. Single-degree-of-freedom analyses
Table 7.1: Capacity curve assumptions considered in this study for NLTHA on single-degree-of-freedom
systems representing bridge piers subjected to transverse earthquake excitations
yielding point
degradation
residual cap.
failure point
Fy
Δy
Fdeg
Δdeg
Fres
Δres
Fult
Δult
Continuous, Experimental data
2800
19.54
-
-
-
-
2800
295.9
Continuous, Numerical data
2858
29.04
-
-
-
-
2858
281.9
Lap-splice, Experimental data
2860
16.44
2860
62.47
686
189.1
0
295.9
Lap-splice, Numerical data
2850
29.04
2858
51.69
1000
145.5
0
281.9
Capacity curve [kN, mm]
Figure 7.2: Capacity curves from Table 7.1 displayed in Acceleration Displacement Response Spectra
(ADRS) versus seismic demands (left) and force-deformation relations of bridge piers considered in
dynamic single-degree-of-freedom analyses
Due to differences in estimation of yield displacements arising from experimental
observations versus numerical methods, three different initial vibration periods were
considered in dynamic analyses (Table 7.2).
Table 7.2: Modeling assumptions for reference, tested bridge pier (scaled to real dimensions)
Initial period
Experimental backbone
Numerical backbone
Δy (mm)
Reinforcement
Height and Mass
Tinit (s)
Δy (mm)
Fy (kN)
Continuous
6.60m, 550t
0.39-0.47
19.54
2800
29.04
2858
Lap-Splice
6.60m, 550t
0.35-0.47
16.44
2860
29.04
2858
Fy (kN)
An important issue of the study has been the choice of a conservative elastic viscous damping
ratio. In contrast to experimental evidence, where 5% damping was obtained, initial stiffness
proportional elastic viscous damping has been considered to be either ξel, NLTHA = 2% or 5%.
A damping ratio of 2% is in our opinion the minimal, realistic assumption for reinforced
concrete columns. Moreover, other parameters were varied as described in Table 7.3.
77
Chapter 7. Single-degree-of-freedom analyses
Table 7.3: Variability of considered SDOF models for NLTHA
Parameter
Considered range
Symbol
Pier height (see section 7.2.3)
6.6m to 13.2m
H
Pier mass
550t
M
Initial stiffness prop., elastic viscous damping ratio
2 to 5%
I
Reinforcement detailing
Continuous, Lap-splice
C, L
Software code
Ruaumoko, Idarc
R, I
Backbone curve assumption
Experimental, Numerical
E, A
Ground motion type
Artificial, Real (scaled)
Seismicities
0.16g, 0.35g, 0.50g
AGM, RGM
11/14, 41/44, 61/64
The explained nomenclature used for the description of each dynamic analysis carried out in
this work is listed below and provides information about the single-degree-of-freedom
system, elastic viscous damping, reinforcement detailing, software code, backbone curve,
ground motion type and the seismicity considered.
Figure 7.3: Nomenclature used in this study for dynamic analyses on single-degree-of-freedom systems
The results of each dynamic analysis performed on the SDOF system have been evaluated in
terms of maximum response in: displacement demand, ductility, drift, force at peak
displacement and total energy dissipation from cycles of inelastic deformations. The
comparison of results presented below acknowledges the influence of strength degradation
modelling (energy or ductility based), longitudinal reinforcement detailing (continuous or
spliced), elastic damping ratio assumed in NLTHA and backbone assumption (experimental
or numerical) on different responses.
Moreover, displacement demands obtained with NLTHA were predicted by means of four
different methods: [Miranda et al., 2003], [Priestley et al., 2007], [Iwan et al., 1980],
[Guyader et al., 2006] using straightforward or iterative procedures such as the capacity
spectrum method (CSM) or the modified capacity spectrum method (MADRS) performing
elastic linear time-history analyses with the chosen accelerograms to obtain damped response
spectra.
78
Chapter 7. Single-degree-of-freedom analyses
7.2
Evaluation of responses from NLTHA on SDOF systems
7.2.1 Influence of strength degradation modelling
Based on the different hysteretic models, input ground motions and parameter variations
presented in previous sections, NLTHA were carried out using two different software Idarc
[Idarc, 2006] and Ruaumoko [Carr, 2004]. The aim was to provide comparison and
consistency in the results, and to study the differences in response, mostly in terms of
maximum displacement demand and energy dissipation, depending on the degradation
modelling assumptions.
Figure 7.4: Comparison of hysteretic loops obtained with Ruaumoko [Carr, 2004] vs. Idarc [Idarc, 2006]
in dynamic analyses on single-degree-of-freedom systems for input motion AGM44 and 2% damping
ratio. Backbone curve without strength degradation(left) and with strength degradation (right)
Strength degradation in software code Idarc [Idarc, 2006] is implemented on a pure
(hysteretic) energy dissipative basis while in Ruaumoko degradation is implemented on a pure
displacement ductility basis. Due to these considerations we first compare the total energy
dissipated by Ruaumoko Etot,Ruaumoko with the total energy dissipated by Idarc Etot,Idarc for the
same NLTHA as:
E tot , Ruaumoko
E tot , Idarc
(7.1)
As it can be seen from the comparison of hysteretic loops in Figure 7.4 respective in data
presented in Figure 7.5(left PGA 0.16g, right PGA 0.35g) the mean value of the total energy
dissipation ratio lies only 6% off the optimum (defined as a ratio of 1.00) with a coefficient of
variation CV ranging between 0.089-0.14=8.9-14%. The ratio diminishes with increasing
PGA and generally a slightly higher energy is dissipated by software code Ruaumoko [Carr,
2004]. This means, that for a given NLTHA, differences in terms of total energy dissipation
depending on the software code used can be neglected for all the analyses.
79
Chapter 7. Single-degree-of-freedom analyses
Figure 7.5: Comparison of total dissipated hysteretic energy for analyses carried out using Ruaumoko
[Carr, 2004] vs. Idarc [Idarc, 2006], considering members with continuous and spliced reinforcement,
experimental and numerical backbone assumptions for systems with 100H100MI-
The optimum corresponds to a ratio of 1.00 because means that an equal amount of energy is
dissipated in the Ruaumoko and Idarc analysis for a defined SDOF model and ground motion.
This remark is also valid for the next plots, where displacement demand, maximum force or
others indicators are analized.
The influence of the strength degradation modelling on maximum displacement demand is
shown in Figure 7.6 (left PGA 0.16g, right PGA 0.35g). As it can be seen, the mean value of
the maximum displacement demand ratio defined as the maximum displacement obtained by
Ruaumoko Δmax,Ruaumoko divided by the maximum displacement obtained by Idarc Δmax,Idarc for
the same NLTHA:
Δ max, Ruaumoko
Δ max, Idarc
(7.2)
lies 1-6% off the optimum (defined as a ratio of 1.00) with a coefficient of variation CV
ranging between 0.04-0.094 = 4.0-9.4% and generally a slightly higher displacement demand
obtained by means of the software code Ruaumoko [Carr, 2004].
80
Chapter 7. Single-degree-of-freedom analyses
Figure 7.6: Influence of software code on maximum displacement demand for member with continuous
reinforcement and influence of strength degradation modeling (energy vs. ductility based) for members
with spliced longitudinal reinforcement at pier base. Legend: input ground motion (AGM = Artificial,
RGM, Real) and backbone assumption (Exp=Experimental, Num=Numerical)
As a consequence of previous comparison remarks between Ruaumoko and Idarc analyses,
responses obtained with NLTHA on SDOF systems in terms of maximum displacement
demands (or drift, or ductilities) are not affected by the art of degradation rule used (energy or
ductility based). Moreover, approximately the same variability can be noticed in terms of
lateral force at maximum displacement demand (or at effective period), comparing the
effective force at secant stiffness obtained by Ruaumoko Feff,Ruaumoko divided by equivalent
force obtained by Idarc Feff,Idarc for the same NLTHA:
Feff , Ruaumoko
Feff , Idarc
(7.3)
Figure 7.7 shows that the mean of the ratio defined in Equation 7.3 lies only 2-4% off the
optimum (defined as a ratio of 1.00) with a CV ranging between 0.029-0.16 =2.9-16% and
that increases for high seismicities.
An example of the difference in level of effective force at maximum displacement demand
can be seen in Figure 7.4: While in Ruaumoko [Carr, 2004] to a defined displacement
ductility correspond one lateral force capacity, in Idarc [Idarc, 2006] due to the energy based
degradation rule, the level of Feff is affected by the energy history of the previous peak pulse.
Typically, this means a lower force drop in the energy based hysteretic rule, as confirmed in
Figure 7.7.
81
Chapter 7. Single-degree-of-freedom analyses
Figure 7.7: Influence of strength degradation modeling (energy vs. ductility based) on lateral force
carrying capacity corresponding to maximum displacement demand for members with spliced end
reinforcement. Legend: input ground motion (AGM = Artificial, RGM, Real) and backbone assumption
(Exp=Experimental, Num=Numerical)
7.2.2 Influence of reinforcement detailing
Detailing deficiencies at the pier base, i.e. lap-splices of the longitudinal reinforcement, have
as a consequence a degradation of the plastic hinge region, and thus a degradation of the
lateral load carrying capacity mechanism with increasing displacement demand. Comparison
of models representing bridge piers with or without lap-splice when subjected to the same
earthquake ground motions was carried out in terms of displacements defining the ratio of
maximum displacement demand for a SDOF system with spliced reinforcement Δmax,L
divided by the maximum displacement demand for the same system with continuous
reinforcement Δmax,C (Equation 7.4)
Δ max, L
Δ max, C
(7.4)
Because of the different modelling of strength degradation depending on which software code
(energy or ductility based) or backbone curve (experimental or numerical) were used, four
different cases should be distinguished. Nevertheless, because it has been observed in section
7.2.1, that the influence of strength degradation modelling generally doesn’t affect
displacement demand, the results obtained from Ruaumoko [Carr, 2004] and Idarc [Idarc,
2006] are presented together in Figure 7.8 and. Figure 7.9. Drifts and ductility demands on
horizontal axes are relative to the models with spliced reinforcement (letter L).
82
Chapter 7. Single-degree-of-freedom analyses
Figure 7.8: Maximum displacement demand ratio from NLTHA as a function of drift (left) and
displacement ductility (right) based on experimental backbone assumption. Analyses assumed a SDOF
system with a pier height of h=6.6m and a lumped mass of m=550t. Aspect ratio correspond to h/lw = 2.2
Figure 7.9: Maximum displacement demand ratio from NLTHA as a function of drift (left) and
displacement ductility (right) based on numerical backbone assumption. Analyses assumed a SDOF
system with a pier height of h=6.6m and a lumped mass of m=550t. Aspect ratio correspond to h/lw = 2.2
It is interesting to notice, that considering experimental evidence described in [Bimschas et
al., 2008], thus taking into account the higher initial stiffness (stiffness to yield) for a member
with spliced reinforcement kinit,L compared with the stiffness of the same member with
continuous reinforcement kinit,C, generally results in a reduced displacement demand for the
member with spliced reinforcement for low ductilities (or drift ratios), because of the shorter
vibration period of piers detailed with lap-splices. Nevertheless, primarily due to the shortperiod for a model with a splice Tinit,L = 0.35s and the variability of ground motions, (see
Figure 7.8 and Figure 7.9) it can occur, that for drifts below initiation of strength degradation
(δ = 0.95%, experimental backbone) the model with larger initial stiffness kinit,L experiences
higher displacement demands than the model having kinit,C (see Figure 7.8).
83
Chapter 7. Single-degree-of-freedom analyses
Adopting an engineering approach, thus neglecting the higher initial stiffness for a member
with spliced reinforcement kinit,L compared with the stiffness of the same member with
continuous reinforcement kinit,C, reduces the scatter of the data below initiation of strength
degradation (δ = 0.78%, numerical backbone) and show a clearer trend of degradation effects
on displacement demand (see Figure 7.9).
Neglecting the differences between experimental and numerical backbone assumption in
initiation of strength degradation, a mean drift ratio of δdeg = 0.85% can be defined as the
point where systems with spliced longitudinal reinforcement start to degrade.
7.2.3 Influence of bridge pier height
As previously mentioned in this study, experiments on wall-type bridge piers carried out at
the ETH Zürich and presented in [Bimschas et al., 2008] were scaled (scale 1:2) in order to
represent single-degree-of-freedom-systems with a pier height of h=6.60m and a lumped mass
of m=550t.
The influence of bridge pier height on maximum displacement demand has been studied here
performing NLTHA on twenty-two different systems with eleven pier heights ranging from
h=6.60-13.2m and assuming backbone curves from numerical analyses and scaling factors as
in Table 5.13. Finally, cyclic behaviour in dynamic analyses considered both degradation and
non-degradation depending on longitudinal reinforcement detailing and were performed using
an initial stiffness proportional elastic viscous damping ratio of ξ=0.02.
Figure 7.10: Pseudo spectral acceleration vs. spectral displacement (ADRS response spectra) and initial
periods of different SDOF systems considered in this study on the influence of pier height (left) and initial
stiffness (or stiffness to yield) of the same members in a force-displacement relation (right). A lumped
mass to m=550to has been considered for all cases.
As can be seen from Figure 7.10 and Table 7.4, the initial vibration periods of the SDOF
considered in this study lie in the acceleration and velocity proportional portions of the elastic
response spectra and due to the level of chosen yielding forces, displacement ductilities
greater than one are expected for all systems even for the lower seismicity with PGA 0.16g.
84
Chapter 7. Single-degree-of-freedom analyses
To ensure consistency between the experiments performed by Bimschas [Bimschas et al.,
2008] and the sectional analyses presented in chapter 5.2 and 5.3, the lumped mass has not
been varied with increasing pier height, thus meaning that axial load ratio remains constant
for all systems.
Table 7.4: Single-degree-of-freedom systems considered in this section. Each system follow the backbone
assumptions presented in Table 7.1 for members with and without strength degradation obtained with
numerical analyses. Ductility dependent strength degradation is considered according to Table 5.8
Name
(-)
Pier height
(m)
Height ratio h/href
(-)
Pier mass
(t)
Fy
(kN)
Δy
(mm)
Tinit
(sec)
100H100M
6.60
1.00
550
2858
29.04
0.47
110H100M
7.26
1.10
550
2598
35.14
0.54
120H100M
7.92
1.20
550
2382
41.82
0.62
130H100M
8.58
1.30
550
2198
49.08
0.70
140H100M
9.24
1.40
550
2041
56.92
0.78
150H100M
9.90
1.50
550
1905
65.34
0.86
160H100M
10.56
1.60
550
1786
74.34
0.95
170H100M
11.22
1.70
550
1681
83.93
1.04
180H100M
11.88
1.80
550
1588
94.09
1.13
190H100M
12.54
1.90
550
1504
104.83
1.23
200H100M
13.20
2.00
550
1429
116.16
1.33
From the comparison of maximum displacement ductility demands for different levels of
seismicity (Figure 7.11 and Figure 7.13 for single-degree-of-freedom systems with continuous
and spliced longitudinal reinforcement respectively) the following considerations can be
made: a) the shape of the mean displacement ductility values (continuous lines) for systems
with spliced reinforcement are similar to those for systems with continuous reinforcement, b)
the shapes of mean values depends on seismicity, c) generally a decrease of displacement
ductility is observed for increased pier height, but d) a roughly constant or slightly increased
ductility demand is observed for the lowest seismicity (PGA 0.16g) up to a pier height ratio of
h/href=1.5.
As a consequence, the slope of the mean value of maximal drift ratio vs. pier height is steeper
for the lowest seismicity (Figure 7.12, left and Figure 7.14, left) when compared with the drift
ratios for a PGA 0.35g (Figure 7.12, right and Figure 7.14, right), this means that for a PGA
0.16g displacement ductility demands of a single-degree-of-freedom are roughly constant
independently of the pier height (μΔ≅1.25), while for PGA0.35g displacement ductility
demands decrease with increased pier height.
85
Chapter 7. Single-degree-of-freedom analyses
Figure 7.11: Trend of mean value, standard deviation and single values of displacement ductility demands
obtained from NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming
systems without strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a
seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM = Artificial, RGM = Real
GM, RGM N = Real newly scaled)
Figure 7.12: Trend of mean value, standard deviation and single values of drift demands obtained from
NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming systems without
strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a seismicity of
PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM =Artificial, RGM = Real GM,
RGM N = Real newly scaled)
86
Chapter 7. Single-degree-of-freedom analyses
Figure 7.13: Trend of mean value, standard deviation and single values of displacement ductility demands
obtained from NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming
systems with strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a
seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM = Artificial, RGM = Real
RGM, RGM N = Real newly scaled)
Figure 7.14: Trend of mean value, standard deviation and single values of drift demands obtained from
NLTHA for different pier heights as presented in Figure 7.10 and Table 7.4 assuming systems with
strength degradation, initial stiffness prop. elastic viscous damping ratio of ξ=0.02 and a seismicity of
PGA 0.16g (left) and PGA 0.35g (right). Ground motion type (AGM=Artificial, RGM=Real GM, RGM
N=Real newly scaled)
Comparing the trend of mean values and standard deviations for increasing pier height and
considering the lowest seismicity (PGA 0.16g) shown in Figure 7.15, left and Figure 7.16, left
it can be noticed, that strength degradation does not affect the member response because the
maximum displacement demand remains below the initiation of degradation (see Table 5.8).
On the other hand, for an higher seismicity (PGA 0.35g) shown in Figure 7.15, right and
87
Chapter 7. Single-degree-of-freedom analyses
Figure 7.16, right it can be noticed, that strength degradation increases the maximum
displacement demand (and drift ratio) for all pier heights considered (h=6.60-13.2m), but this
seems to be more important for shorter piers. Moreover, strength degradation (Figure 7.16,
right) introduces more uncertainty in the member response, reflected in a larger standard
deviation when compared with the trend obtained for the case without strength degradation.
Figure 7.15: Comparison of the trend of mean value (continuous line) and standard deviation (dotted line)
of displacement ductility demands obtained from NLTHA for different pier heights as presented in Figure
7.10 and Table 7.4 assuming systems with and without strength degradation, initial stiffness prop. elastic
viscous damping ratio of ξ=0.02 and a seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground
motion type (AGM = Artificial, RGM = Real GM, RGM N = Real newly scaled)
Figure 7.16: Comparison of the trend of mean value (continuous line) and standard deviation (dotted
line) of drift demands obtained from NLTHA for different pier heights as presented in Figure 7.10 and
Table 7.4 assuming systems with and without strength degradation, initial stiffness prop. elastic viscous
damping ratio of ξ=0.02 and a seismicity of PGA 0.16g (left) and PGA 0.35g (right). Ground motion type
(AGM=Artificial, RGM=Real GM, RGM N=Real newly scaled)
88
Chapter 7. Single-degree-of-freedom analyses
7.2.4 Influence of damping ratio
Dynamic analyses carried out in this study are based on initial stiffness proportional viscous
damping, thus maintaining constant the elastic part of system damping throughout the
analysis. As a consequence, in order to reduce the risk of a possible overestimation of the
damping in the analyses and because of the interest in studying the effects of different elastic
viscous damping ratios ξel assumptions on maximum displacement demands, all NLTHA
were performed with two elastic damping ratios, i.e. ξel=0.02 and ξel=0.05.
Because artificial and real ground motions were generated or scaled to match a determined
response spectra with ξel=0.02 and considering the fact that this spectrum was calculated
starting from a spectrum with ξel=0.05 by means of an elastic viscous damping reduction
factors Rξ implemented in the code provisions [SIA 261, 2003] and [EC8, 1998] as
⎛
⎞
1
Rξ = ⎜⎜
⎟⎟
⎝ 0 . 5 + 10 ξ ⎠
0 . 50
≥ 0 . 55
(7.5)
The responses (displacement demand) for systems with ξel=0.02 is expected to be about 20%
larger than those with ξel=0.05. The maximum displacement demand ratio is defined as the
maximum displacement obtained with 2% elastic damping Δmax,0.02 divided by the maximum
displacement obtained by the same analysis with 5% elastic damping Δmax,0.05 for the same
NLTHA, i.e.
Δ max, 0 .02
Δ max, 0 .05
(7.6)
Figure 7.17 shows that the mean value of the maximum displacement demand ratio defined in
Eq. 7.6 corresponds for systems without strength degradation to a value of 1.18-1.22 being
constant over the whole seismicity range considered (left to right PGA 0.16g, 0.35g, 0.50g)
and having a CV of about 10%. Ratios about 6-8% larger were obtained for systems suffering
strength degradation for maximal drift demands up to δmax = 2%. This underlying the fact
underline the fact, that especially for some moderate seismicities the level of elastic viscous
damping assumed in NLTHA can both a) determine if a system suffer strength degradation or
not and b) consequently increase the displacement demand gap between the two elastic
viscous damping ratios. In any case it is interesting to notice, that also below initiation of
strength degradation, especially for the case with the shorter vibration period Tinit = 0.35s
(experimental backbone, spliced reinforcement) a larger ratio can not be excluded for some
ground motions.
Generally it can be stated, that independently of any consideration of strength degradation or
not, the predicted increase in displacement demand for different damping ratios can generally
be accurately predicted by means of simplified methods present in code provisions.
89
Chapter 7. Single-degree-of-freedom analyses
Figure 7.17: Influence of initial stiffness proportional elastic viscous damping ratio ξel assumed in
NLTHA on maximum displacement demand for member with continuous and spliced longitudinal
reinforcement. Legend: input ground motion (AGM = Artificial, RGM=Real, RGM N = Real newly
scaled)
7.2.5 Influence of critical damping coefficient
The comparisons presented in section 7.2.4 studied the influence of different initial stiffness
proportional elastic viscous damping ratios ξ used in NLTHA on maximum displacement
demands assuming that the ratio of the damping coefficient c divided by the critical damping
ccr was constant for all initial stiffness, i.e.
ξ =
c
c
=
= const.
c cr
2 k init m
(7.7)
independently of the initial stiffness kinit considered in the NLTHA. Therefore, for a given
elastic viscous damping ratio ξ, different damping coefficients c were implicitly assumed.
In section 7.1 it was shows that the determination of the initial stiffness kinit yields different
results depending if the determination is based on experimental evidence or on an analytical
model. Thus, it seems reasonable to estimate and compare results of NLTHA for the same
damping coefficients c (or different damping ratios ξ) and comment on the effects of different
initial stiffness kinit on displacements demands.
Equivalent elastic damping ratios ξ to be used for NLTHA considering experimental
backbone, when compared with analyses using ξ=0.02−0.05 and numerical backbone are
listed in Table 7.5. As an example, when performing NTHA with ξ=0.02 (numerical
backbone, continuous reinforcement) this corresponds to an initial stiffness proportional
damping ratio of ξ=0.01657 (experimental backbone, continuous reinforcement) due to the
fact, that the experimental data describe a stiffer system than the numerical one.
90
Chapter 7. Single-degree-of-freedom analyses
Table 7.5: Modified damping ratios to be consistent in terms of critical damping coefficients when
performing NLHTA
Num. BB
Exp. BB
ξ (-)
Reinforcement
Height and Mass
ξ (-)
ccr (kNs/m)
Continuous
6.60m, 550t
0.02
294
0.01657
355
Continuous
6.60m, 550t
0.05
736
0.04144
888
Lap-Splice
6.60m, 550t
0.02
294
0.01504
391
Lap-Splice
6.60m, 550t
0.05
736
0.03761
978
ccr (kNs/m)
The maximum displacement demand ratio defined as the maximum displacement obtained
with 2% respective 5% elastic damping Δmax,(2-5%) (continuous or spliced reinforcement, Exp.
BB) divided by the maximum displacement obtained with the same analysis using a damping
ratio of 1.657% or 4.144% (continuous reinforcement, Exp. BB) respective 1.504% or
3.761% (spliced reinforcement, Exp. BB) Δmax,ξ(c,crit) for the same NLTHA is defined as:
Δ max, 2 − 5 %
Δ max, ξ ( c ,crit )
(7.8)
Figure 7.18 shows that the mean value of the maximum displacement demand ratio defined
by Eq. 7.8 corresponds for systems without strength degradation to a value of 0.97 being
constant over the whole seismicity range (left to right PGA 0.16g, 0.35g) and having a
covariance of about 8%. Once again, the displacement demand decrease for a damping of
1.657% to 2% or for one of 4.144% to 5% can be roughly predicted according to the ratio of
the viscous damping reduction factors Rξ for damping of 5% versus 4.144% implemented in
code provisions [SIA 261, 2003] and [EC8, 1998] and described by equation (7.7). The
former case has a ratio of 1.23/1.20=0.98 and the latter one a ratio of 1/1.05=0.95; both cases
are very close to the mean of 0.97 in Figure 7.18.
Figure 7.18: Influence of damping coefficient c on maximum displacement demand for member with
continuous and spliced longitudinal reinforcement, experimental backbone assumption and a pier height
of 6.60m and a lumped mass of 550t. (AGM = Artificial Ground Motion)
91
Chapter 7. Single-degree-of-freedom analyses
7.2.6 Displacement increasing factors f(δ) and f(μ)
Based on to the results obtained from NLTHA performed in this study and because of the
need of simplified relations to describe strength degradation effects in an earthquake
engineering performance-based design and assessment domain, two displacement increasing
factors f(δ) and f(μ) are defined here. The aim of these factors is to roughly describe the
increased displacement demand arising from a ductility or energy based cyclic strength
degradation when compared with the same single-degree-of-freedom system without loss of
lateral load carrying capacity. Thus, the general definition of these factors is somewhat
proportional to the ratio of maximum displacement demand for a system with strength
degradation Δmax,L divided by the maximum displacement demand for the same system
without strength degradation Δmax,C, i.e.:
f (δ ) or f ( μ ) ∝
Δ max, L
Δ max, C
(7.9)
As it can be seen in Figure 7.8 and Figure 7.9, a mean drift ratio of δdeg = 0.85% can be
defined as the drift value where for systems with 100H100M strength degradation initiates.
This drift ratio corresponds to a ductility demand of μΔ, LE = 3.41 (spliced reinforcement,
experimental backbone) or μΔ, LA = 1.93 (spliced reinforcement, numerical backbone), thus
well beyond yielding especially for the first case. As a consequence, the initial period (or
initial stiffness kinit,L) marginally affects the displacement demand at this point and whatever
the backbone assumption was (experimental or numerical) a linear increase of displacement
demand for single-degree-of-freedom systems with spliced reinforcement Δmax,L with respect
to the same system with continuous reinforcement Δmax,C can be predicted as a function of
lateral drift taking into account a bridge pier aspect ratio h/lw as follows
f (δ ) =
Δ max, L
=1
Δ max, C
⎡
⎢
δ max − δ deg
Δ max, L
f (δ ) =
= 1+ ⎢
Δ max, C
⎢ 2 ⋅ ⎛⎜ h ⎞⎟ − δ deg
⎣⎢ 2 . 2 ⎝ l w ⎠
⎤
⎥
⎥ ⋅ 0 . 20
⎥
⎦⎥
for δ max ≤ δ deg
(7.10)
for δ deg < δ max ≤ 2 % ⋅ ⎛⎜ h ⎞⎟ (7.11)
⎝ lw ⎠
Assuming that strength degradation occurs at approximately the same inelastic deformation
for systems with pier height other than h=6.60m (reference pier), the drift ratio for the
initiation of degradation can be estimated as follows
δ deg =
0 . 85 % ⎛ h ⎞
⋅⎜
⎟
2 .2 ⎝ l w ⎠
(7.12)
In the same way, a ductility dependent displacement factor f(μ) that describes the same
behaviour has been found to be
f (μ ) =
Δ max, L
=1
Δ max, C
for μ ≤ μ deg
(7.13)
92
Chapter 7. Single-degree-of-freedom analyses
f (μ ) =
Δ max, L
⎛ μ − μ deg
= 1 + ⎜⎜
Δ max, C
⎝ 2 . 00
⎛ 1 . 80
⎞
⎟⎟ ⋅ 0 . 20 ⋅ ⎜
⎜μ
⎠
⎝ deg
⎞
⎟
⎟
⎠
for μ deg < μ ≤ 2 ⋅ μ deg
(7.14)
Due to the relatively small number of performed NLTHA and to the increased scatter in the
data, an upper limit for the use of these approximation has been set to be a drift ratio of δmax =
2% (for h=6.60m) and a displacement ductility of μmax=2 μdeg. The parameters have been
estimated in order to fit the trend in Figure 7.19 (mean value for spliced reinforcement).
Figure 7.19: Comparison of proposed prediction (orange line) using displacement increasing factor versus
effective obtained mean and standard deviation values of displacement ductility (left) and drift ratio
(right) for systems suffering strength degradation from NLTHA. Pier heights on horizontal axes are
obtained considering a reference pier of h=6.60m. Only systems presented in Figure 7.10 and Table 7.4
have been considered here.
Figure 7.20: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility (left) and drift ratio (right) for systems suffering strength degradation.
Displacement ductilities and drift ratios on horizontal axis are referred to systems with strength
degradation. Only systems presented in Figure 7.10 and Table 7.4 have been considered here.
93
Chapter 7. Single-degree-of-freedom analyses
Figure 7.21: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility (left) and drift ratio (right) for systems suffering strength degradation.
Displacement ductilities and drift ratios on horizontal axis are referred to systems without strength
degradation. Only systems presented in Figure 7.10 and Table 7.4 have been considered here.
Even if the displacement increasing factor was calibrated mostly following the trend of Figure
7.19, this simplified approach seems to represent in a conservative manner the expected
displacement demands for systems with backbone assumption based on experimental data
(see Figure 7.22 and Figure 7.23). As noticed in section 7.2.2, the scatter in the data prior to
initiation of strength degradation depends here on the differing initial stiffness of systems
without loss of lateral load capacity, compared with those suffering it (see also [Bimschas et
al., 2008]).
Figure 7.22: Proposed prediction (red line) for consideration of increased displacement demand as a
function of drift ratio for systems suffering strength degradation. Drift ratios on horizontal axis are
referred to systems with strength degradation. Only single-degree-of-freedom systems with pier height of
h=6.60m, a lumped mass of m=550to and an experimental backbone (left) and numerical backbone (right)
are presented here. Ground motion type (AGM=Artificial, RGM=Real GM, RGM N=Real newly scaled)
94
Chapter 7. Single-degree-of-freedom analyses
Figure 7.23: Proposed prediction (red line) for consideration of increased displacement demand as a
function of displacement ductility for systems suffering strength degradation. Displacement ductilities on
horizontal axis are referred to systems with strength degradation. Only single-degree-of-freedom systems
with pier height of h=6.60m, a lumped mass of m=550t and an experimental backbone (left) and numerical
backbone (right) are presented here. Ground motion type (AGM=Artificial, RGM=Real GM, RGM
N=Real newly scaled)
7.3
Simplified procedures for maximum displacement demand prediction
7.3.1 Brief overview and main considerations
The maximum displacement demand obtained by NLTHA has been compared with different
simplified approaches proposed by [Miranda et al., 2003], [Priestley et al., 2007], [Iwan,
1980] and [Guyader et al., 2004]. The theories underlying these methods are based on
different stiffness and damping model assumptions.
Table 7.6: Stiffness and damping model assumptions for the considered methods
Method
Stiffness model
Damping model
Other considerations
[Miranda et al., 2003]
initial (elastic)
equivalent viscous
damping
Δinelastic based on correction factor
(CR) applied on Δelastic
[Priestley et al., 2007]
effective (secant)
equivalent viscous
damping
iterative procedure using capacity
spectrum method (CSM) to estimate
Δdemand
[Iwan, 1980]
effective (optimal)
equivalent viscous
damping (optimized)
iterative procedure using modified
capacity spectrum method (CSM) to
estimate Δdemand
[Guyader et al., 2004]
effective
(statistically optimal)
equivalent viscous
damping (statistically
optimized)
iterative procedure using modified
capacity spectrum method (CSM) to
estimate Δdemand
Except for the procedure proposed by Miranda [Miranda et al., 2003], that considers an
inelastic displacement correction factor CR as linearization (see section 7.3.2), the other three
methods use an approximate analysis technique by replacing the actual nonlinear system with
95
Chapter 7. Single-degree-of-freedom analyses
an equivalent linear system in order to estimate displacement demand of inelastic system. A
brief description of this technique will be presented here and can be found in [Guyader et al.,
2004].
“Based on the analysis of linear system response, different conclusions of the nonlinear
system response may be deduced. Recalling the equation of motion for the SDOF system:
where f ( x , x& ) represents a linear viscous damped system, the differential equation of motion
may be expressed as
m &x&lin + c eff x& lin + k eff x lin = − m u&&(t )
(7.15)
Where m=mass of the system, ceff=effective viscous damping coefficient and keff=effective
spring stiffness. For a given ground excitation, ü(t), the solution xlin(t) may be computed using
any convenient numerical solution procedure. For an inelastic system, the restoring force
f ( x , x& ) may take a variety of forms. The solution for an inelastic system is designated as
xinel(t).
[…] Dividing by the mass Equation (7.15) may be written as
&x&lin +
4πξ eff
x& lin
Teff
⎛ 2π
+⎜
⎜T
⎝ eff
2
⎞
⎟ x lin = − u&&( t )
⎟
⎠
(7.16)
Where the effective fraction of critical damping ξeff and the effective period are expressed as
ξ eff =
c eff
2 k eff m
Teff = 2π
m
k eff
(7.17)
(7.18)
Numerous approaches can be employed for making a comparison between displacement time
histories xinel(t) and xlin(t). Within the framework of performance-based engineering, the key
performance variable is the maximum relative displacement amplitude that a structure
experiences from the demand earthquake. The relative displacement for the inelastic and
linear SDOF systems is xinel(t) and xlin(t) respectively.
The effective linear parameters obtained based on a comparison of displacement values would
not be appropriate for use in a velocity or force-based design procedure. […] The maximum
acceleration or maximum pseudo-acceleration as compared to &x&linel (t ) would be a much better
comparison parameter for determining effective linear parameters intended for use in a forcebased analysis procedure”.
Effective linear parameters can be found in literature for different input motions based on
steady-state harmonic, stationary random response or earthquake excitations. The effective
linear parameters developed by [Priestley et al., 2007] and [Guyader et al., 2004] are based on
earthquake excitations and assume a secant stiffness as effective stiffness for the former and a
statistically optimized effective stiffness and damping for the latter. Iwan [Iwan, 1980] used
96
Chapter 7. Single-degree-of-freedom analyses
statistically random response as input motions and optimal effective parameters, thus being in
some sense the precursor of the work proposed by [Guyader et al., 2004].
7.3.2 Miranda et al., 2003
The procedure proposed by Miranda [Miranda et al., 2003] arises from studies based on
constant relative strength inelastic displacement ratios to estimate maximum lateral inelastic
displacement demands on structures from maximum lateral elastic displacement demand. A
relatively large number of recorded earthquake ground motions and three different soil
conditions were considered and the influence of initial (elastic) period of vibration, level of
lateral yielding strength, site conditions, earthquake magnitude, distance to source and strainhardening ratio were evaluated.
Table 7.7: Statistical data of earthquakes used in [Miranda et al., 2003]
Ms
Earthquakes
Accelerograms
Soil Class A
Soil Class B
Soil Class C
5.8-7.7
12
216 (72 each soil)
vs30 = 760-1525m/s
vs30 = 360-760m/s
vs30 = 160-360m/s
The inelastic displacement ratio CR, is defined as the maximum lateral inelastic displacement
demand Δinelastic, divided by the maximum lateral elastic displacement demand Δelastic, on
systems with same mass and initial stiffness when subjected to the same earthquake ground
motion.
CR =
Δ inelastic
Δ elastic
(7.19)
The inelastic displacement demand is computed in systems with constant yielding strength
relative to the strength required to maintain the system elastic, and the relative lateral strength
is characterized by the strength ratio R, which is defined as
R=
mS a
Fy
(7.20)
where m is the mass of the system, Sa correspond to the spectral acceleration ordinate and Fy
is the lateral yielding strength of the system.
Based on nonlinear regression analyses, the following simplified expression of inelastic
displacement ratios to estimate maximum inelastic displacement demand from maximum
elastic displacement demand for structures with given lateral strength ratio R has been
proposed:
CR =
Δ inelastic
=1+
Δ elastic
⎡
1
1⎤
− ⎥ (R − 1)
⎢
b
c⎦
⎣ a (T / T s )
(7.21)
where T is the period of vibration of the system, Ts is the characteristic period at the site and
a, b, c are constants that depend on site conditions.
97
Chapter 7. Single-degree-of-freedom analyses
Table 7.8: Site dependent coefficients for inelastic displacement ratios CR [Miranda et al., 2003]
Site class
a
b
c
Ts (s)
B
42
1.60
45
0.75
C
48
1.80
50
0.85
D
57
1.85
60
1.05
simplified, all classes
50
1.80
55
a-c class dependent
Site independent (simplified) coefficients a, b, c are used in the study presented here to
estimate the displacement demand for all the systems because of the relative low scatter in the
data when using these values of a, b, c instead of site class dependent parameters. Moreover,
because the artificial and real ground motions generated for this study were respectively
scaled to match a defined response spectra with a soil class B according to [SIA 261, 2003],
thus explicitly assuming a soil shear wave velocity vs = 400-800m/s, a characteristic period at
the site equal to Ts = 0.85s was assumed (see also chapter 6).
In the following figures, comparison of the maximum displacement demand ratio defined as
the maximum displacement obtained by a dynamic analysis Δmax,NLTHA divided by the
maximum displacement obtained by procedure [Miranda et al., 2003] Δmax,Miranda et al. for the
same NLTHA as
Δ max, NLTHA
Δ max, Miranda
(7.22)
el al .
is plotted.
As it can be seen in next figure, the mean value of the displacement ratio correspond to
approximately 0.90 (or 90%) with a CV ranging between 0.11-0.23=11-23% over the whole
seismicity range considered for members with continuous reinforcement. As can be expected,
considering a determined ground motion type (Figure 7.25, Figure 7.26, Figure 7.27), the CV
increases with increasing seismicity (or drift ratio) because higher inelastic displacement
demands are less reliable and dependent from initial period than those close to an elastic
response using this method of prediction.
For members with spliced longitudinal reinforcement this method doesn’t capture any
degradation because it relies on initial stiffness only. Nevertheless, as it can be seen in Figure
7.24 the mean value of the displacement ratio corresponds to about 0.94 (or 94%) for the
lowest seismicity, thus being very close to the value of the member with continuous
reinforcement. Increases of respectively 10% and 20% are reached at each higher hazard
level. Considering a determined ground motion type (Figure 7.25, Figure 7.26, Figure 7.27)
we can observe that the CV also increases with increasing seismicity implying higher
uncertainty for higher drift (and ductility) demand.
98
Chapter 7. Single-degree-of-freedom analyses
Figure 7.24: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Miranda et al., 2003] and sorted by longitudinal
reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g left to right). All NLTHA
assumed a SDOF system with a pier height of 6.60m and a lumped mass of 550t
Figure 7.25: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Miranda et al., 2003] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering artificial GM
only. All NLTHA assumed a SDOF system with a pier height of 6.60m and a lumped mass of 550t
99
Chapter 7. Single-degree-of-freedom analyses
Figure 7.26: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Miranda et al., 2003] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real GM only.
All NLTHA assumed a SDOF system with a pier height of 6.60m and a lumped mass of 550t
Figure 7.27: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Miranda et al., 2003] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real, newly s.
GM only. All NLTHA assumed a SDOF system with a pier height of 6.60m and a lumped mass of 550t
7.3.3 Priestley et al., 2007
Procedure proposed by Priestley [Priestley et al., 2007] arises from studies on effective period
Te, based on secant stiffness of structures to peak displacement demand, and on an equivalent
viscous damping ξe as a result of elastic and hysteretic damping. Calculation of the expected
100
Chapter 7. Single-degree-of-freedom analyses
displacement demand for existing structures is based on an iterative procedure called capacity
spectrum method (CSM) where a performance point has to be found. The performance point
is the intersection of the capacity spectrum and the locus of performance points (demand)
where for both sides (capacity and demand) compatibility in terms of effective period Te and
equivalent viscous damping ξe is accomplished.
Capacity is defined by means of a backbone curve that represents the envelope of the forcedeformation relation assumed for a SDOF system in a NLTHA. Each point on the capacity
curve corresponds to a unique displacement and consequently to a defined ductility μ and
effective period Te.
Figure 7.28: Concept of effective (secant) stiffness proposed by Priestley [Priestley et al., 2007]
Table 7.9: Capacity curve assumptions considered in this study for NLTHA
yielding point
degradation
residual cap.
failure point
Fy
Δy
Fdeg
Δdeg
Fres
Δres
Fult
Δult
Continuous, Experimental data
2800
19.54
-
-
-
-
2800
295.9
Continuous, Numerical data
2858
29.04
-
-
-
-
2858
281.9
Lap-splice, Experimental data
2860
16.44
2860
62.47
686
189.1
0
295.9
Lap-splice, Numerical data
2850
29.04
2858
51.69
1000
145.5
0
281.9
Capacity curve [kN, mm]
By CSM the capacity curves with strength degradation where idealized in the degrading
branch as linear up to a lateral force carrying capacity equal to zero with a line described by
points Fdeg and Fres
101
Chapter 7. Single-degree-of-freedom analyses
Figure 7.29: Capacity curves from Table 7.9 displayed in Acceleration Displacement Response Spectra
(ADRS) versus seismic demands (left) and force-deformation relations of bridge piers considered in
capacity spectrum method (CSM) according to the procedure of [Priestley et al., 2007]
Displacement ductility μ, is defined as the ratio of a given target displacement Δe, divided by
the yield displacement Δy as
μ =
Δe
Δy
(7.23)
Structural effective period Te, is defined as the period at maximum deflection Δe, thus depends
on effective stiffness ke and effective mass me of system as
T e = 2π
me
= 2π
ke
me
Fe / Δ e
(7.24)
Considering that effective stiffness can always be expressed as a function of initial stiffness
and backbone assumptions, the effective period for a structure without strength degradation
can be expresses by means of displacement ductility as
T e = 2π
me
= 2π
ke
me
= 2π
Fe / Δ e
me
= 2π
F y /( Δ y μ )
μ
me
= 2π
Fy / Δ y
μ
me
(7.25)
ky
Assuming that for systems with strength degradation, drop in lateral force capacity can be
expressed by means of reduction factor Rμ, c for a given displacement ductility μ as the ratio
of the yield force Fy divided by the effective force at Fe as
R μ ,c =
Fy
Fe
(7.26)
the effective period for a structure suffering strength degradation is
102
Chapter 7. Single-degree-of-freedom analyses
T e = 2π
me
= 2π
ke
me
= 2π
Fe / Δ e
me
= 2π
( F y / R μ ,c ) /( Δ y μ )
μ
R μ ,c
me
ky
(7.27)
As a consequence, a period shifting (lengthening) occurs for inelastic deformation at all
periods and for an equal displacement ductility a structure suffering strength degradation has
a longer inelastic vibration period than one without loss of lateral load capacity. For this
reason, because the proposed methodology is primarily intended for systems without strength
degradation, for the case of systems with spliced reinforcement the CSM was performed
twice: firstly neglecting degradation on capacity side, then considering it.
According to the procedure by [Priestley et al., 2007] there exists for each displacement
demand an equivalent viscous damping ξe equal to the damping that a structure must develop
in order to remain elastic so that the maximum displacement response Δe of the elastic system
be equal to those achieved in the NLTHA based on the same input accelerogram and
considering the effective period of structure Te obtained from NLTHA. Equivalent viscous
damping ξe is defined as the sum of an elastic viscous damping ξel (material damping) and a
hysteretic damping ξhyst coming from cycles of inelastic deformation as
ξ eq = ξ el + ξ hyst
(7.28)
The elastic damping according to [Priestley et al., 2007] is not equal to the elastic damping
assumed in NLTHA, but is corrected by a ductility dependent factor λ that accounts for the
damping proportional stiffness model considered in NLTHA (initial or tangent) and for the
hysteretic rule.
ξ el = ξ el , NLTHA μ λ
(7.29)
The hysteretic damping according to [Priestley et al., 2007] accounts for the damping
proportional to the energy dissipated in cycles of inelastic deformations and in its complete
form is ductility and period dependent (see Eq. 7.30). Moreover, hysteretic damping depends
on the hysteretic rule adopted, since some rules dissipate considerable more energy than
others for the same force-deformation envelope.
⎛
⎞
⎛
1 ⎞
1
⎟
ξ hyst = a ⎜⎜ 1 − b ⎟⎟ ⎜⎜ 1 +
μ ⎠⎝
(Te + c )d ⎟⎠
⎝
(7.30)
Table 7.10: Secant stiffness correction factors λ for elastic damping [Priestley et al., 2007]
Model
Initial stiffness
Tangent stiffness
EPP
0.127
-0.341
Bilinear (BI)
0.193
-0.808
Takeda Thin (TT)
0.340
-0.378
Takeda Fat (TF)
0.312
-0.313
Flag
0.387
-0.430
Ramberg-Osgood
-0.060
-0.617
103
Chapter 7. Single-degree-of-freedom analyses
Table 7.11: Equivalent viscous damping coefficients for hysteretic damping component according to the
procedure proposed by [Priestley et al.,2007]
Model
a
b
c
d
EPP
0.224
0.336
-0.002
0.250
Bilinear (BI)
0.262
0.655
0.813
4.890
Takeda Thin (TT)
0.215
0.642
0.824
6.444
Takeda Fat (TF)
0.305
0.492
0.790
4.464
Flag
0.251
0.148
3.015
0.511
Ramberg-Osgood
0.289
0.622
0.856
6.460
In comparison of damping prediction according to [Priestley et al., 2007] metohd, equivalent
viscous damping ξe was estimated performing a series of elastic, linear time-history analyses
based on effective period Te and maximum displacement demand Δe obtained from NLTHA
with different backbone assumptions and with or without degradation; results are presented
below and shows the high scatter in data even for low ductility demands. Nevertheless,
because displacement does not linearly depend on damping (damping is usually considered in
a squared-root as in Equation 7.5) this error in damping prediction will not be directly
reflected in displacement demand prediction.
Figure 7.30: Comparison of hysteretic damping obtained from elastic, linear time-history analysis ELTHA
based on effective period Te and maximum displacement demand from NLTHA performed in this study
versus prediction according to [Priestley et al., 2007] procedure assuming a Thin Takeda model (TT)
Demand in CSM is estimated by means of the response spectra the accelerograms considered
in the NLTHA. Loci of performance points (LPP) are estimated at small intervals of
equivalent viscous damping ξe based on the effective period Te assumed on capacity side and
performance point is found at the intersection of the capacity curve with the LPP (Figure
7.31).
104
Chapter 7. Single-degree-of-freedom analyses
Figure 7.31: Examples of iterative procedure adopted in CSM for a real scaled ground motion (left) and
an artificial ground motion (right) in order to find the performance point (PP), defined as intersection of
capacity and demand on locus of performance points, considering effective period Te and equivalent
viscous damping ξe as proposed by [Priestley et al., 2007].
Multiple solutions may occur in CSM, meaning that the locus of performance points intersects
the capacity curve more than once (Figure 7.31 and Figure 7.32, left). In these cases different
assumptions may be made as to which is the most representative solution of the CSM;
[Guyader et al., 2006] use the most conservative solution, but in our opinion, based on
analyses performed here, this assumption usually leads to a large overestimation of
displacement demand for most cases. Depending on the shape of the locus of performance
points compared to the capacity curve, multiple solutions may be categorized into those with
a) LPP tangent to the capacity curve (Figure 7.31, left) and those with the b) LPP intersecting
capacity curve at single, separated displacements (Figure 7.32, left).
The first case means, that more solutions are very close to each other and a slight difference in
prediction of effective period and equivalent viscous damping leads to a relatively high shift
in displacement demand but also implies, that for a given ground motion there is a high
probability of peak response being achieved at approximately the same time by a single peak
of acceleration. In the second case the solutions are usually divided by a relatively high gap in
effective period and equivalent viscous damping, meaning that we are dealing with two
clearly separated systems.
It is also possible, that the CSM does not have a solution meaning that the locus of
performance points will never intersect the capacity curve (Figure 7.32, right). In our analyses
this was mostly the case for large ductility demands (PGA 0.50g) and systems suffering
strength degradation. As it can be seen in Figure 7.30, the assumed equivalent viscous
damping ξe according to [Priestley et al., 2007] procedure is about constant for ductilities μ >
6, implying a constant damped response spectra for these cases. Moreover, for members
suffering strength degradation the capacity curve decays for large ductilities, thus falling
approximately parallel to the damped ground motion response spectra (Figure 7.32, right).
105
Chapter 7. Single-degree-of-freedom analyses
Figure 7.32: Examples of iterative procedure adopted in CSM for a case with multiple solutions (left) and
a case without solution (right) assuming a capacity curve with strength degradation considering effective
period Te and equivalent viscous damping ξe as proposed by [Priestley et al., 2007].
Comparisons of the maximum displacement demand ratio defined as the maximum
displacement obtained by a dynamic analysis Δmax,NLTHA divided by the maximum
displacement obtained by procedure [Priestley et al., 2007] Δmax,Priestley et al. for the same
NLTHA as
Δ max, NLTHA
Δ max, Priestley
(7.31)
el al.
are plotted in Figure 7.33 to Figure 7.36. As discussed previously the CSM was performed
twice for systems suffering strength degradation; in Figure 7.33 the results for the case
without consideration of capacity degradation are presented while in Figure 7.34 degradation
is considered for the members with spliced reinforcement. As expected, difference in
displacement ratio for members with spliced reinforcement presented in Figure 7.33 and
Figure 7.34 are observed for drift ratios δ >δdeg and are very high for a PGA 0.50g. At this
seismicity level, considering an initial stiffness proportional elastic damping ratio of
ξel,NLTHA=2% all performed CSM analyses (artificial ground motions) had no solution for both
numerical and experimental backbone assumption but leaded to a solution for ξel,NLTHA=5%.
At moderate to high seismicities (PGA 0.16g, 0.35g) a mean value of displacement ratio
corresponding to about 1.05-1.10 (or 105-110%) with a CV of 0.12-0.25 (or 12-25%) is
obtained for members with spliced reinforcement in both CSM analyses.
As for the prediction proposed by [Miranda et al., 2003] a better fit of the data was obtained
for members with continuous reinforcement, a mean value of displacement ratio
corresponding to about 1.01-1.07 (or 101-107%) and a CV of 0.11-0.21 (or 11-21%) were
obtained for the whole seismicity range considered.
Treating members with continuous reinforcement only and dividing the results depending on
ground motion type it can be observed, that the CV remains constant for artificial ground
motions and increases with increasing seismicity (or drift ratio) for real, scaled ground
106
Chapter 7. Single-degree-of-freedom analyses
motions. For real ground motion a shifting of the mean displacement ratio into the
conservative sector (ratio < 1.00) has been observed for increasing seismicity while for
artificial accelerograms a mean value around 1.00 was obtained.
Figure 7.33: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Priestley et al., 2007] and sorted by longitudinal
reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g left to right). All NLTHA
assumed a SDOF system with a h=6.60m, m=550t and neglected strength degradation on capacity
Figure 7.34: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Priestley et al., 2007] and sorted by longitudinal
reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g left to right). All NLTHA
assumed a SDOF system with a h=6.60m, m=550t and considered strength degradation on capacity
107
Chapter 7. Single-degree-of-freedom analyses
Figure 7.35: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Priestley et al., 2007] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering artificial GM
only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
Figure 7.36: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Priestley et al., 2007] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real GM only.
All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
108
Chapter 7. Single-degree-of-freedom analyses
Figure 7.37: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Priestley et al., 2007] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real, newly s.
GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
7.3.4 Iwan, 1980
The procedure proposed by Iwan [Iwan, 1980] was one the of early efforts to define optimal
linear parameters for earthquake excitation based on making an adjustment to the linear
response spectrum. As described in [Guyader et al., 2004], in the study of Iwan [Iwan, 1980]
“ductility dependent inelastic response spectra were compared with elastic response spectra,
and displacement preserving shifts of the inelastic spectra were determined which minimized
the average absolute value difference between the inelastic and equivalent linear spectra over
a range of periods.
Figure 7.38: Early effort to define optimal equivalent linear parameters [Iwan, 1980]
109
Chapter 7. Single-degree-of-freedom analyses
A family of hysteresis behaviour was considered including bilinear hysteretic as well as
pinching hysteretic models.[…] Using the stated procedure, the following relationships were
obtained for the optimal effective linear parameters:
Teff
− 1 = 0 . 121 (μ − 1)
(7.32)
ξ eff − ξ 0 = 5 . 87 (μ − 1)0 .371
(7.33)
T0
0 . 939
The proposed equations were not developed to have dependence on the second slope ratio and
in addition, the optimal effective period defined by the above relationship is significantly less
than the period associated with the secant stiffness (or, the optimal stiffness is significantly
greater than the secant stiffness). The period employed is the optimal effective linear period in
the conventional CSM (in this study defined as Teff in [Priestley et al., 2007]). It is also
observed that the damping value used in the conventional CSM approach (defined as ξeq in
[Priestley et al., 2007]) is significantly greater than the optimal damping parameter”.
Figure 7.39 shows the comparison of predicted effective periods and damping, for three
different initial periods T=0.35-1.00s It can be seen how the optimal equivalent linear
parameters proposed in [Iwan, 1980] are considerably lower than those proposed by [Priestley
et al., 2007]. Effective (equivalent) viscous damping has been estimated assuming a Thin
Takeda (TT) hysteretic rule, initial stiffness proportional damping and the relationship in
Equation (7.15) to obtain the effective period Teff .
Figure 7.39: Comparison of equivalent linear parameters based on the procedure proposed by [Iwan,
1980], based on optimal stiffness and damping, versus the procedure proposed [Priestley et al., 2007],
based on secant stiffness and equivalent viscous damping. Periods are intended as initial period T=T0
As an example, considering an elastic viscous damping ratio of ξel=0.02 (NLTHA), an initial
period of T0=0.50s and a displacement ductility of μ=4.00 we obtain ξeff=0.16, Teff=1.00s for
[Priestley et al., 2007] versus ξeff=0.11, Teff=0.67s for [Iwan, 1980], thus a difference of
roughly 45% for both parameters.
110
Chapter 7. Single-degree-of-freedom analyses
Seismic demand in the procedure of [Iwan, 1980] is represented by means of a damped
response spectrum as for a conventional CSM, but due to the different definition of effective
period (optimal stiffness instead of secant stiffness) some modification must be undertaken in
order to allow for the use of the optimal linear equations (7.22 and 7.23). Considering that the
conventional CSM, described in section 7.3.3 for the procedure of [Priestley et al., 2007],
implicitly use the relation Te=Tsec or Teff=Tsec for ADRS seismic demand, a modification of
the response spectra MADRS that allows for use of any period T different from Tsec is used in
this procedure.
A modification factor M is estimated for a single ADRS (see Figure 7.40), where each value
of PSA at every spectral displacement SD may be multiplied by the ratio of secant stiffness
proportional pseudo-acceleration Asec divided by the effective stiffness proportional pseudoacceleration Aeff as
M =
Asec ⎛ Teff
= ⎜⎜
Aeff
⎝ Tsec
⎞
⎟⎟
⎠
2
(7.34)
Figure 7.40: Comparison of predicted displacement demand according to [Iwan, 1980] (left) using
equivalent linear optimal parameters (modified CSM) versus conventional CSM according to [Priestley et
al., 2007] (right) using secant stiffness for same ground motion, elastic viscous damping and capacity
curve (RGM42, 2%, numerical backbone).
Comparing the results from a conventional CSM with a modified CSM for a single ground
motion (Figure 7.40) is interesting to notice, that not only damping and effective period are
different but also the shape of locus of performance points is affected by the modification
proposed by Iwan [Iwan, 1980]. As it can be seen in Figure 7.40 a multiple-solution arises
using the conventional CSM and the shape of locus of PP lies parallel to the capacity curve in
a wide range of displacement, but a unique, defined solution is achieved in MADRS.
111
Chapter 7. Single-degree-of-freedom analyses
Figure 7.41: Comparison of equivalent viscous damping obtained from elastic, linear time-history analysis
ELTHA based on effective period Teff on secant stiffness and maximum displacement demand from
NLTHA performed in this study versus prediction according to [Iwan., 1980]. Effective period on
horizontal axis (right) correspond to a secant stiffness.
Figure 7.42: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Iwan, 1980] and sorted by longitudinal reinforcement
detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g left to right). All NLTHA assumed a
SDOF system with a h=6.60m, m=550t and neglected strength degradation on capacity
112
Chapter 7. Single-degree-of-freedom analyses
Figure 7.43: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Iwan, 1980] and sorted by longitudinal reinforcement
detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering artificial GM only. All NLTHA
assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
Figure 7.44: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Iwan, 1980] and sorted by longitudinal reinforcement
detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real GM only. All NLTHA
assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
113
Chapter 7. Single-degree-of-freedom analyses
Figure 7.45: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Iwan, 1980] and sorted by longitudinal reinforcement
detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real, newly s. GM only. All
NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
7.3.5 Guyader et al., 2004
The procedure proposed by Guyader [Guyader et al., 2004] arises from early studies carried
out by Iwan [Iwan, 1980] and improves and extends the procedure described in section 7.3.4
for several hysteretic systems including bilinear, stiffness degrading, strength degrading and
pinching models and considers the effects of different second slope ratios (post-yield divided
by initial stiffness). The main improvement of this modified CSM compared with Iwan [Iwan,
1980] is the fact, that effective linear parameter equations have been statistically optimized in
order to minimize errors in terms of accuracy and precision, thus giving insight into the
sensitivity of the performance point prediction.
In order to compare the maximum displacement amplitude of the nonlinear time history
xinel(t), designated as Dinel and the maximum displacement amplitude of the linear time history
xlin(t), designated as Dlin Guyader defines an error measure capable of distinguishing between
a conservative displacement prediction and a non-conservative displacement prediction as
D − D inel
(7.35)
ε D = lin
D inel
With this definition, a negative value of εD corresponds to a non-conservative displacement
prediction while a positive value reflects a conservative displacement prediction. As observed
in [Guyader et al., 2004],”[…] εD might be considered to have a positive bias as it ranges
from -1 to ∞”. However, for the range of systems and excitations considered in this study
([Guyader et al., 2004]), the slight positive bias in statistical distribution of εD is
inconsequential.
114
Chapter 7. Single-degree-of-freedom analyses
Figure 7.46: Mean value contours (left) and standard deviation contours (right) of εD error distribution
over two-dimensional parameter space for entire ensemble [Guyader et al., 2004]
It is interesting to notice, that there is a nearly diagonal (linear) contour of zero error in Figure
7.46 (left) thus meaning, that for certain combinations, overprediction of effective period and
damping or underprediction of both parameters leads to consistent maximum displacement
response.
Assuming a normal distribution of results and using mean and standard deviation Guyader
completely described the error distribution and defined an engineering acceptability range of
error εD ranging from -10% to 20% in order to estimate the effective linear parameters to be
used as
ℑ EAR ≡ 1 − P ( − 0 . 10 < ε D < 0 . 20 ) = minimum
(7.36)
Figure 7.47: Engineering acceptability range applied to distribution of εD considering an additive relation
for effective damping and a ratio for effective period [Guyader et al., 2004]
The general form of statistically optimized linear parameter equations for effective damping
and effective period developed by [Guyader et al., 2004] are
115
Chapter 7. Single-degree-of-freedom analyses
ξ eff − ξ 0 = A (μ − 1)2 + B (μ − 1)3
for
μ < 4 .0
(7.37)
ξ eff − ξ 0 = C + D (μ − 1)
for
4 .0 ≤ μ ≤ 6 .5
(7.38)
− 1 = G (μ − 1 ) + H (μ − 1 )
for
μ < 4 .0
(7.39)
− 1 = I + J (μ − 1)
for
4 .0 ≤ μ ≤ 6 .5
(7.40)
Teff
T0
Teff
T0
2
3
As noticed in [Guyader et al., 2004] “for large values of ductility, a semiempirical approach
has been adopted to extend the effective linear parameter equations. Data points were
calculated at ductilities of 8.0 and 10.0. The equations are based on the asymptotic behaviour
of the secant period and equivalent viscous damping”. The following equations are proposed
ξ eff
F (μ − 1) − 1 ⎛ Teff
⎜
− ξ0 = E
[F (μ − 1)]2 ⎜⎝ T0
⎞
⎟⎟
⎠
2
⎞
⎛
(μ − 1)
− 1 = K ⎜⎜
− 1 ⎟⎟
T0
⎠
⎝ 1 + L [(μ − 1) − 1]
Teff
for
μ > 6 .5
(7.41)
for
μ > 6 .5
(7.42)
Because this study is mainly focused on the comparison of behaviour of members without
strength degradation with those suffering strength degradation, the parameters for estimation
of effective damping and period have been considered for a stiffness degrading model
(KDEG) without post-yield stiffness (α=0%) and for ground motions in the far-field as in
Table 7.12. Observing the hysteresis loops proposed in Figure 7.48 the energy dissipation for
stiffness degrading models seems to be more appropriate when compared with the hysteresis
loops assumed in NLTHA and presented in Figure 5.4 and Figure 5.10 due to the shape and
amount of area included in the loops.
116
Chapter 7. Single-degree-of-freedom analyses
Figure 7.48: Force (f) vs. displacement (x) for bilinear (BLH), stiffness degrading (KDEG), strength
degrading (STRDG) and pushover backbone models from a THA with a sinusoidal acceleration function
(left) and schematic diagram and hysteresis loops for the pinching models (right) [Guyader et al., 2004b]
Table 7.12: Coefficients for effective linear parameters according to [Guyader et al., 2004b] for far-field
ground motions, different hysteretic models and second slope ratios α
Model
TRange
α
BLH
Tshort
BLH
Tshort
BLH
Tshort
BLH
Tshort
BLH
Tshort
BLH
A
B
C
0%
3.1922
−0.6598
10.5687
2%
3.3338
−0.6405
9.3792
5%
4.1504
−0.8260
10%
5.0731
−1.0826
20%
4.64
Tshort
60%
STRDG
Tshort
D
E
F
G
H
I
J
K
L
0.1156
19.13
0.73
0.1108
−0.0167
0.2794
0.0892
0.57
0
1.1101
18.85
0.42
0.1034
−0.0142
0.2107
0.1125
0.665
0.02
10.1243
1.6428
22.35
0.4
0.1145
−0.0178
0.1777
0.124
0.768
0.05
11.6899
1.5791
24.38
0.36
0.1262
−0.0224
0.1713
0.1194
0.87
0.1
−0.9900
11.75
1.13
25.25
0.37
0.0952
−0.0149
0.1748
0.093
0.98
0.2
2.377
−0.6125
4.8036
0.0169
13.42
0.35
0.0433
−0.0091
0.0677
0.0257
0.96
0.6
−5%
5.6014
−1.2944
13.6407
0.608
22.012
0.9
0.195
−0.0379
0.1843
0.1825
0.71
−0.05
−0.03
STRDG
Tshort
−3%
5.2749
−1.1635
13.9824
0.6924
23.7334
0.9
0.1801
−0.0331
0.2128
0.1716
0.76
KDEG
Tshort
0%
5.1261
−1.1090
12.1052
1.3622
20.66
0.62
0.1725
−0.0317
0.1673
0.1767
0.85
0
KDEG
Tshort
2%
5.3031
−1.1722
11.2724
1.6023
19.79
0.51
0.1756
−0.0335
0.1637
0.1708
0.88
0.02
KDEG
Tshort
5%
5.642
−1.2962
10.182
1.8661
19.51
0.38
0.1809
−0.0366
0.1472
0.164
0.92
0.05
KDEG
Tshort
10%
5.3056
−1.2203
8.8425
1.9861
21.14
0.37
0.1652
−0.0338
0.1419
0.144
0.97
0.1
KDEG
Tshort
20%
4.5877
−1.0250
9.6022
1.3379
23.38
0.34
0.1343
−0.0267
0.224
0.0839
1.01
0.2
KDEG
Tshort
60%
2.414
−0.6128
2.9263
0.7511
20.2018
0.295
0.0639
−0.0157
0.0715
0.0262
0.976
0.6
PIN1
Tshort
0%
3.2857
−0.6870
5.4427
1.8604
12.4
0.49
0.2222
−0.0445
0.1628
0.212
0.987
0
PIN1
Tshort
2%
3.4226
−0.7156
5.6695
1.9379
13.3673
0.42
0.2057
−0.0412
0.1507
0.1963
0.988
0.02
PIN1
Tshort
5%
3.3888
−0.7083
5.6711
1.9015
13.5806
0.375
0.2034
−0.0417
0.1367
0.1898
1.05
0.05
PIN1
Tshort
10%
3.3443
−0.7438
5.5659
1.4835
13.4024
0.37
0.199
−0.0430
0.1581
0.1575
1.08
0.1
PIN1
Tshort
20%
2.7945
−0.6374
6.9221
0.3397
11.6507
0.4
0.1682
−0.0363
0.2829
0.0839
1.06
0.02
PIN1
Tshort
60%
0.8507
−0.2201
1.7454
−0.0106
4.4852
0.4
0.071
−0.0168
0.1278
0.0189
1.06
0.6
PIN2
Tshort
0%
5.0641
−1.1737
9.4127
1.4917
16.2
0.5
0.2109
−0.0435
0.186
0.1785
0.866
0
PIN2
Tshort
2%
5.2207
−1.2100
9.7038
1.5378
17.3304
0.45
0.1962
−0.0405
0.173
0.166
0.872
0.02
PIN2
Tshort
5%
4.9926
−1.1225
9.3702
1.7518
18.1582
0.395
0.182
−0.0365
0.1704
0.1604
0.94
0.05
PIN2
Tshort
10%
4.7203
−1.0514
10.0604
1.3451
18.6225
0.39
0.168
−0.0338
0.1923
0.1361
0.99
0.1
PIN2
Tshort
20%
3.6915
−0.7815
10.9408
0.394
17.8952
0.4
0.1301
−0.0242
0.2756
0.0806
1.02
0.2
PIN2
Tshort
60%
1.5464
−0.3964
3.4977
−0.0937
8.355
0.45
0.058
−0.0129
0.1213
0.0175
1.02
0.6
PB
Tshort
NA
5.6683
−1.4363
12.8666
−0.2112
11.705
0.89
0.1691
−0.0344
0.1115
0.1609
0.738
0
117
Chapter 7. Single-degree-of-freedom analyses
Figure 7.49: Comparison of equivalent linear parameters based on procedure proposed by [Guyader et
al., 2004], based on optimal stiffness and damping, versus procedure proposed [Priestley et al., 2007],
based on secant stiffness and equivalent viscous damping. Periods are intended as initial period T=T0
Figure 7.50: Comparison of equivalent linear parameters based on procedure proposed by [Guyader et
al., 2004] versus previous study carried out by Iwan [Iwan, 1980], both based on optimal stiffness and
damping. Periods are intended as initial period T=T0
The comparison in Figure 7.49 of equivalent linear effective parameter obtained by Guyader
[Guyader et al., 2004] with those proposed by [Priestley et al., 2007] shows that the amount
of effective damping is actually lower in [Priestley et al., 2007] for a ductility range up 3 and
greater for higher displacement ductilities while the effective period, based on secant stiffness
is the optimal effective period for all ductilities. According to Figure 7.46 this qualitatively
means, that for ductilities up to 3, overestimations of both parameters in [Priestley et al.,
2007] may lead to consistent maximum displacement responses, but for higher ductilities,
because of overprediction of effective period combined with underprediction of effective
damping, the risk of substantial errors in predictions [Priestley et al., 2007] is higher.
118
Chapter 7. Single-degree-of-freedom analyses
In fact for this ductility range we are moving roughly “perpendicular” to the line of zero error
in Figure 7.46 (left).
Figure 7.51: Comparison of equivalent viscous damping obtained from elastic, linear time-history analysis
ELTHA based on effective period Teff on secant stiffness and maximum displacement demand from
NLTHA performed in this study versus prediction according to [Guyader et al.., 2004]. Effective period on
horizontal axis (right) corresponds to a secant stiffness.
Figure 7.52: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Guyader et al., 2004] and sorted by longitudinal
reinforcement detailing, ground motion type, seismicity (PGA 0.16g, 0.35g, 0.50g left to right). All NLTHA
assumed a SDOF system with a h=6.60m, m=550t and neglected strength degradation on capacity
119
Chapter 7. Single-degree-of-freedom analyses
Figure 7.53: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Guyader et al., 2004] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering artificial GM
only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
Figure 7.54: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Guyader et al., 2004] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real GM only.
All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
120
Chapter 7. Single-degree-of-freedom analyses
Figure 7.55: Ratio of maximum displacement demand from NLTHA divided by maximum displacement
demand according to procedure proposed by [Guyader et al., 2004] and sorted by longitudinal
reinforcement detailing, seismicity (PGA 0.16g, 0.35g, 0.50g left to right) and considering real, newly s.
GM only. All NLTHA assumed a SDOF with a h=6.60m, m=550t and neglected strength degradation.
121
Chapter 7. Single-degree-of-freedom analyses
7.3.6 Probabilistic evaluation of simplified displacement prediction procedures
Comparison of the proposed simplified procedures versus results from NLTHA in terms of
precision and accuracy is obtained considering a probabilistic approach. As a first step,
definition of a displacement response error εD as in [Guyader et al., 2004] is presented. “The
displacement response error is defined as the ratio of the difference between the linear
displacement at the effective period and damping and the inelastic displacement to the
inelastic displacement expressed as
ε D (α , μ , HYST ) =
[
]
D lin Teff (T0 , α , μ ), ξ eff (ξ 0 , α , μ ) − D inel (T0 , ξ 0α , μ , HYST
D inel (T0 , ξ 0α , μ , HYST
)
)
(7.43)
The maximum displacement of nonlinear system, Dinel, is a function of the initial period, T0,
linear viscous damping, ξ0, second slope ratio, α, response ductility, μ, and hysteretic model,
denoted HYST. The linear system response, Dlin, is a function of the two linear system
parameters: effective period and effective damping. In the analysis, Dinel, is a function of
ductility. Therefore, scaling ground motions will have no effect on the results obtained.”
In a more general form, the displacement response error can be defined as the ratio of the
difference between the displacement from a simplified procedure described in previous
sections and the inelastic displacement form NLTHA to the inelastic displacement obtained
by NLTHA expressed as
εD =
D lin ,Prediction − D nltha
D nltha
(7.44)
Motivation and advantages to describe a displacement difference between simplified
prediction and dynamic analysis in this way has been discussed in section 7.3.5. As stated in
[Guyader et al., 2004], it is generally assumed, that distributions of errors are normal in shape.
Therefore, for a given simplified prediction, knowing the mean m and standard deviation σ of
the distribution, the normal density function can be described as
f (x , μ , σ ) =
⎛
⎜−
1
e⎝
2π σ
(x−m ) ⎞
⎟
2σ 2 ⎠
(7.45)
thus, the probability density function (PDF) is the integral from -∞ to ∞ of the considered
distribution is:
CDF = f ( x , μ , σ ) =
−∞
∫
−∞
⎛
⎜−
1
e⎝
2π σ
(x−m ) ⎞
⎟
2σ 2 ⎠
dx
(7.46)
According to [Guyader et al., 2004], the probability that the displacement prediction error for
the procedures proposed by [Miranda et al., 2003], [Priestley et al., 2007], [Iwan, 1980] and
[Guyader et al., 2004] lies in ranges listed in Table 7.14 and folllowing. They have been
estimated considering two separate groups (or distributions) a) continuous reinforcement and
b) spliced longitudinal reinforcement at the pier base. For members with spliced
reinforcement two different approaches have been considered in [Priestley et al., 2007],
[Iwan, 1980] and [Guyader et al., 2004] when computing the performance point; first
neglecting strength degradation on capacity curve, than considering it.
122
Chapter 7. Single-degree-of-freedom analyses
Moreover, in the case of strength degradation, distributions that accounted for the effects of
displacement increasing factors defined in 7.2.6 have been estimated and evaluated.
Table 7.13: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distribution according to error bound defined in [Guyader et al., 2004]
Method
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
From an engineering point of view, a desirable range of acceptability (EAR) of displacement
error εD can be assumed between values of -0.10 < εD < 0.20 meaning that a conservative
displacement prediction up to 20% and a non-conservative underprediction up to -10% lies
between these boundaries.
As can be seen in Figure 7.56 (members with continuous reinforcement), the procedure
proposed by [Priestley et al., 2007] has the highest probability (53%) to be within the EAR
compared with the other procedures. Due to its generally conservative prediction, the
procedure of [Miranda et al., 2003] has the highest probability (83%) to be within the second
range of error (-20% to 40%), with [Priestley et al., 2007] prediction being slightly lower.
Table 7.14: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.56
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
[Miranda et al., 2003]
0.12
0.22
0.48
0.83
0.07
0.10
[Priestley et al., 2007]
-0.01
0.20
0.53
0.81
0.17
0.02
[Iwan, 1980]
-0.15
0.15
0.36
0.63
0.37
0.00
[Guyader et al., 2004]
-0.09
0.15
0.50
0.77
0.23
0.00
Method
Figure 7.56: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and four methods, considering
members with continuous longitudinal reinforcement only (no strength degradation of backbone curve)
and seismicities of PGA 0.16g, 0.35g, 0.50g. All NLTHA assumed a SDOF with a h=6.60m, m=550t
123
Chapter 7. Single-degree-of-freedom analyses
Nevertheless, dispersion of the procedures proposed by [Iwan, 1980] and [Guyader et al.,
2004] are considerably lower than in [Miranda et al., 2003], [Priestley et al., 2007] in Figure
7.56, but looking at the results for members with spliced reinforcement in Figure 7.57 this
difference becomes smaller. As expected, the probability to be in EAR decreases for all
methods and shapes “shifted” to the non-conservative range, because displacement prediction
Dlin,prediction is now estimated neglecting strength degradation. However strength degradation is
considered in displacement from NLTHA Dnltha have considered it. The method of [Priestley
et al., 2007] has the highest probability to be within the EAR (49%) and [Miranda et al.,
2003] those in the second range of error (-20% to 40%), with 76%.
In this case predictions proposed by [Iwan, 1980] and [Guyader et al., 2004] have
considerably lower probabilities to lie within EAR with 28% and 40% when compared with
other the two methods.
Table 7.15: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.57
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
[Miranda et al., 2003]
0.04
0.25
0.45
0.76
0.17
0.07
[Priestley et al., 2007]
-0.07
0.19
0.49
0.75
0.25
0.01
[Iwan, 1980]
-0.20
0.18
0.28
0.50
0.50
0.00
[Guyader et al., 2004]
-0.13
0.19
0.40
0.64
0.36
0.00
Method
Figure 7.57: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone curve in
NLTHA) but without considering it when predicting displacement. All presented data includes seismicities
of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t
124
Chapter 7. Single-degree-of-freedom analyses
With the aim of better predict displacement demands of members suffering strength
degradation based on simplified approaches, thus neglecting strength degradation on capacity
side when performing CSM or modified CSM but considering it with a displacement
increasing factor as found in 7.2.6, comparison of displacement errors based on this approach
are presented in Figure 7.58 and have bee estimated as
εD
eff
=
f (δ ) ⋅ D lin ,Prediction − D nltha
D nltha
(7.47)
Compared with results presented in Figure 7.57, a roughly equal or reduced scatter in the data
and a shift of all distributions in the conservative range can be observed in Figure 7.58. As a
consequence the probability that the margin of error lies within EAR increases for three out of
four predictions underlying the positive effects of a displacement factor f(δ) on correction of
error.
Table 7.16: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.58
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
[Miranda et al., 2003]
0.08
0.26
0.43
0.75
0.14
0.11
[Priestley et al., 2007]
-0.03
0.21
0.49
0.77
0.21
0.02
[Iwan, 1980]
-0.17
0.17
0.33
0.57
0.43
0.00
[Guyader et al., 2004]
-0.11
0.17
0.44
0.70
0.30
0.00
Method
Figure 7.58: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution the four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone curve in
NLTHA) but without considering it when predicting displacement and applying to the displacement
prediction the factor f(δ) as in section 7.2.6. All presented data includes seismicities of PGA 0.16g, 0.35g,
0.50g and SDOF systems with a h=6.60m, m=550t
125
Chapter 7. Single-degree-of-freedom analyses
Roughly the same considerations can be applied to the case where a ductility dependent
displacement increasing factor f(μ) instead of drift dependent factor f(δ) is used.
Table 7.17: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.59
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
[Miranda et al., 2003]
0.08
0.27
0.42
0.73
0.15
0.12
[Priestley et al., 2007]
-0.03
0.22
0.48
0.75
0.22
0.03
[Iwan, 1980]
-0.17
0.17
0.33
0.57
0.43
0.00
[Guyader et al., 2004]
-0.11
0.18
0.44
0.69
0.31
0.00
Method
Figure 7.59: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution the four methods, considering
members with spliced longitudinal reinforcement only (with strength degradation of backbone curve in
NLTHA) but without considering it when predicting displacement and applying to the displacement
prediction the factor f(μ) as in section 7.2.6. All presented data includes seismicities of PGA 0.16g, 0.35g,
0.50g and SDOF systems with a h=6.60m, m=550t
By the procedure [Priestley et al., 2007] a conventional CSM has been utilized for the
estimation of the performance point. Strength degradation on capacity for members with
spliced reinforcement was firstly neglected (Figure 7.31) and then considered (Figure 7.32)
leading to the distributions presented in Figure 7.57 (and Figure 7.60, curve 1) and Figure
7.60 (curve 2) respectively. Moreover, based on displacement increasing factors, two more
distributions have been obtained in Figure 7.58 (and Figure 7.60, curve 3) and Figure 7.59
(and Figure 7.60, curve 4) leading to a total of four possibilities to predict displacement
demand for members with spliced reinforcement using [Priestley et al., 2007].
Comparison of these four different way underline the fact, that neglecting strength
degradation in dynamic analyses and when performing CSM, but considering it by means of a
displacement increasing factor f(μ) or f(δ) leads to better results. As in the case of
consideration of strength degradation on capacity when carrying out a conventional CSM,
126
Chapter 7. Single-degree-of-freedom analyses
where a higher scatter in the data has been found (Figure 7.60, curve 2). Moreover, for this
case the probability of multiple solutions or no solutions in CSM is higher, as discussed in
7.3.3.
Table 7.18: Mean m, standard σ deviation of displacement error εD and probabilities for some ranges
assuming a normal distributed distribution according to data presented in Figure 7.60
m
σ
Probability
Probability
Probability
Probability
(-10%<PP error<20%)
(-20%<PP error<40%)
(PP error<-20%)
(PP error>40%)
[Priestley et al., 2007] (1)
-0.07
0.19
0.49
0.75
0.25
0.01
[Priestley et al., 2007] (2)
-0.03
0.24
0.45
0.72
0.24
0.04
[Priestley et al., 2007] (3)
-0.03
0.21
0.49
0.77
0.21
0.02
[Priestley et al., 2007] (4)
-0.03
0.22
0.48
0.75
0.22
0.03
Method
Figure 7.60: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution and the method [Priestley et al.,
2007], considering four different assumptions for the estimation of displacement prediction and
considering members with spliced longitudinal reinforcement only. All presented data includes
seismicities of PGA 0.16g, 0.35g, 0.50g and SDOF systems with a h=6.60m, m=550t
In his study [Guyader et al., 2004] proposes probability ranges of performance point error
(displacement error) as a function of ductility for the cases listed in Table 7.12. When
considering Figure 7.61, representing the case utilized here (KDEG, α=0, far-field ground
motions) we can observe for example, how the probability of the displacement error lying
within the EAR (-10% to 20%) rapidly decrease for μ<3 and remains roughly constant to a
value of 35-40% for higher ductilities.
As a validation of results founded in [Guyader et al., 2004], considering results from
members with continuous reinforcement only (distribution of Table 7.14), we compare the
probabilities obtained in this study using the method [Guyader et al., 2004] assuming a mean
displacement ductility of μ=3 (red points in Figure 7.61) with the dotted lines in the next
figure. Even though this study considered a relatively small database of input ground motions,
127
Chapter 7. Single-degree-of-freedom analyses
approximately the same values as proposed by [Guyader et al., 2004] were found here. In fact,
even if in [Guyader et al., 2004] optimization of effective parameters based on probabilistic
approach was performed, a slight bias in the data was found in this study, thus meaning that
the mean values of the predicted displacement errors for method [Guyader et al., 2004] were
found to be on the non-conservative side (negative error, see Figure 7.56 to Figure 7.59).
Figure 7.61: Performance Point error results for stiffness degrading model (KDEG) with second slope
ratio of 0% - two far-field ground motion databases [Guyader et al., 2004] and results for this study
according to Figure 7.56 as points, considering roughly μ(mean)= 3.0 for this distribution
128
Chapter 8. Multi-degree-of-freedom analyses
8 MULTI-DEGREE-OF-FREEDOM ANALYSES
8.1 Introduction
Based on modelling assumptions and on analyses and observations on single-degree-offreedom systems from the previous chapter, the effects of spliced longitudinal reinforcement
detailing in potential plastic hinge regions at pier ends have been implemented and studied in
a series of multi-degree-of-freedom systems representing both regular and irregular five span
bridge configurations. Due to the potentially high influence of abutment modelling and
restraining on single piers and overall bridge responses, firstly a parametrical study of elastic
abutment restraining has been performed and in a second step, due to the level of strength
demand observed in the abutments an appropriate elasto-plastic abutment model has been
used for further studies.
Backbone curve assumptions considered both experimental observation and results from
numerical analyses as it has been the case for single-degree-of-freedom systems. A
nomenclature has been defined for dynamic analyses of multi-degree-of-freedom systems as
follows
Figure 8.1: Nomenclature used in this study for dynamic analyses on multi-degree-of-freedom systems
129
Chapter 8. Multi-degree-of-freedom analyses
The majority of multi-degree-of-freedom analyses considered a series of lumped masses
distributed homogeneously along the superstructure, modelled as an elastic beam. I however,
to allow a more straightforward check of the results obtained during single-degree-of-freedom
analyses, a series of dynamic analyses considered lumped masses at each pier top only. In all
cases, the masses were supposed to be lumped at an height corresponding to the
superstructure centre of mass, thus pier height was defined as for single-degree of-freedom
analyses according to Figure 8.2.
Figure 8.2: Front and lateral view of bridge piers considered in this study. Pier height is defined as the
height starting from connection pier footing to pier up to centre of superstructure mass.
Figure 8.3: Multi-degree-of-freedom systems for dynamic analyses considering an homogeneous
repartition of mass on superstructure (top), with constant lumped masses of mi=78.6t (abutments masses
mi=38.3t), respective a concentrated tributary mass on each pier (bottom), with constant lumped masses of
mi=550t , neglecting mass on abutments.
As for the majority of dynamic analyses on bridges, the superstructure has been assumed to
remain elastic during dynamic excitations. Moreover, superstructure properties has been set
equal to a previous study carried out by Kuhn [Kuhn, 2008] at the ETH Zürich on an existing
Swiss roadway bridge with box-girder cross section (Table 8.1). Starting from a
superstructure cross-sectional area A=5.13m2 and a reinforced concrete specific weight of
γconc=25kN/m3 this corresponds to a superstructure unitary mass of msuperstructure,G=12.8t/m’.
Bridge spans have been estimated in order to obtain a constant pier tributary mass of m=550t
due to compatibility reasons with experiments carried out by Bimschas [Bimschas et al.,
2008]. Therefore, choosing a bridge central span of L=36m it follows that a total mass per
length of msuperstructure,tot=550t/36m=15.3t/m’; subtracting msuperstructure,G a reasonable residual
mass of msuperstructure,res=2.5t/m can be attributed to non-structural elements.
130
Chapter 8. Multi-degree-of-freedom analyses
Table 8.1: Superstructure properties according to a study carried out by Kuhn [Kuhn, 2008] on an
existing Swiss roadway bridge with box-girder cross section
Elastic (Young’s)
Modulus E
Shear
Modulus G
Cross-sectional
Area A
Effective-Shear
Area As
Moment of inertia
of section in vertical axis Iv
30’000MPa
11’600MPa
5.13m2
1.74m2
41.24m4
Figure 8.4: Plan and longitudinal view of bridge model 1 (short M1) considered. Piers P1-P4 have a pier
height of h=6.60m to superstructure centre of mass of; all piers and abutments are assumed to be laterally
restrained for transverse excitations
Figure 8.5: Plan and longitudinal view of bridge model 2 (short M2) considered. Piers P1 and P4 have a
pier height of h=6.60m while central piers have an height of h=13.20m to the superstructure centre of
mass; all piers and abutments are assumed to be laterally restrained for transverse excitations
Figure 8.6: Plan and longitudinal view of bridge model 3 (short M3) considered. Piers P1 and P4 have a
pier height of h=13.20m while central piers have an height of h=6.60m to the superstructure centre of
mass; all piers and abutments are assumed to be laterally restrained for transverse excitations
Figure 8.7: Plan and longitudinal view of bridge model 4 (short M4) considered. Piers P1 and P2 have a
pier height of h=13.20m while piers P3 and P4 have an height of h=6.60m to the superstructure centre of
mass; all piers and abutments are assumed to be laterally restrained for transverse excitations
131
Chapter 8. Multi-degree-of-freedom analyses
Damping in dynamic analyses has been considered in form of initial stiffness proportional
Rayleigh damping, meaning that the damping matrix is based on the elastic stiffness of the
structure at the beginning of the time-history and remains constant throughout the timehistory analysis. As noted in [Carr, 2004] “The tangent, secant and elastic damping matrices
are identical. This means, that as the structure softens, by yielding etc., the effective damping
increases because the Rayleigh coefficients α and β were computed for the initial natural
frequencies of free-vibration and some of the frequencies have now decreased. […] the
Rayleigh damping model shows that the level of damping in the higher model of free
vibration can be very large.”
Coefficients α and β were computed for each multi-degree-of-freedom model considering an
elastic damping ratio ξ=0.02 for the first and fourth natural frequencies, thus implicitly
keeping a roughly constant damping ratio between vibration periods T1 and T4.
Figure 8.8: Initial stiffness Rayleigh damping model (ICTYPE=0) described in Ruaumoko [Carr, 2004]
8.2 Parametrical study on abutment elastic stiffness
The influence of abutment elastic stiffness assumption on single elements and overall bridge
transverse response in terms of forces and deformations has long been recognised. Moreover,
in existing bridges a sort of detailing deficiency can arise from the fact, that “shear keys” or
“abutment restraining systems” strength have not been estimated from a capacity design
methodology due to the force-based philosophy of the older code provisions leading to
undesired abutment mode of failure during earthquakes.
For these reasons, a parametric study on the effects of these important members for the four
bridge configurations (Figure 8.4 to Figure 8.7) has firstly been performed with the aim to
define abutments demand levels and further adjustments of abutment modelling for further
studies.
For this purpose, modelling assumptions defined with the following nomenclature (M1 to
M4)-CRA-SLM-EL were considered. Abutment elastic stiffness spaced between kel=1MN/m
to 100MN/m covering a wide range of bridge applications, from very stiff to soft abutment
restraining level. Furthermore a set of 28 accelerograms from either real or artificial ground
132
Chapter 8. Multi-degree-of-freedom analyses
motions, scaled into three seismicity levels (PGA 0.16g, 0.35g, 0.50g), presented in chapter 6
and considered in single-degree-of-freedom analyses (chapter 7) were used for the scope of
this study. For each hazard level, the mean NLTHA was used to define the demands for
abutments, piers and superstructure.
Figure 8.9: Abutment force-displacement relationships considered in this parametrical study. Equal
elastic stiffness assumption has been modelled in both abutments (AL and AR)
133
Chapter 8. Multi-degree-of-freedom analyses
8.2.1
Bridge model M1-CRA-SLM-EL
Figure 8.10: Bridge model M1 (top), superstructure and piers displacement demands from mean NLTHA
(bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m (grey thick line)
have been evidenced; point represents pier positions P1 to P4.
Figure 8.11: Abutments (AL and AR) strength demands for bridge model M2 in order to remain elastic
(left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF (empty points) vs.
mean NLTHA on SDOF (full points and dotted lines)
134
Chapter 8. Multi-degree-of-freedom analyses
8.2.2
Bridge model M2-CRA-SLM-EL
Figure 8.12: Bridge model M2 (top), superstructure and piers displacement demands from mean NLTHA
(bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m (grey thick line)
have been evidenced; point represents pier positions P1 to P4.
Figure 8.13: Abutments (AL and AR) strength demands for bridge model M2 in order to remain elastic
(left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF (empty points) vs.
mean NLTHA on SDOF (full points and dotted lines)
135
Chapter 8. Multi-degree-of-freedom analyses
8.2.3
Bridge model M3-CRA-SLM-EL
Figure 8.14: Bridge model M3 (top), superstructure and piers displacement demands from mean NLTHA
(bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m (grey thick line)
have been evidenced; point represents pier positions P1 to P4.
Figure 8.15: Abutments (AL and AR) strength demands for bridge model M3 in order to remain elastic
(left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF (empty points) vs.
mean NLTHA on SDOF (full points and dotted lines)
136
Chapter 8. Multi-degree-of-freedom analyses
8.2.4
Bridge model M4-CRA-SLM-EL
Figure 8.16: Bridge model M4 (top), superstructure and piers displacement demands from mean NLTHA
(bottom). Displacement patterns for k,el=1MN/m (black thick line) and k,el=100MN/m (grey thick line)
have been evidenced; point represents pier positions P1 to P4.
Figure 8.17: Abutments (AL and AR) strength demands for bridge model M4 in order to remain elastic
(left) and pier displacement ductility demands (P1 to P4) from mean NLTHA on MDOF (empty points) vs.
mean NLTHA on SDOF (full points and dotted lines)
137
Chapter 8. Multi-degree-of-freedom analyses
8.2.5 Conclusions
The elastic restraining level of abutments (elastic stiffness kel) directly influences vibration
periods and mode shapes of any multi-degree-of-freedom system. Stiffer abutments lead to
shorter vibration periods when compared with the same bridge configurations with lower
values of kel and usually restrain more superstructure at the abutments. This reduces pier
displacement demands on external piers, but increases the demand for the central spans (as
showed in sections 8.2.1 to 8.2.4). On the other hand, stiff abutments capture more forces than
softer abutments thus increasing the strength demands in order to remain elastic (see Figure
8.18). It is usually very difficult to predict the real behaviour of structure at abutments, due to
the influence of many factors, but results presented in sections 8.2.1 to 8.2.4 can be
considered as a possible upper and lower limit of solutions.
If we consider that existing bridges have usually been restrained at abutments by means of
very stiff shear keys, lets say with elastic stiffness in the order of kel=50-100MN/m and low
level of strength being in a range of 2-5% of vertical load, it appears clear from Figure 8.18,
that insufficient strengthening is provided in these members for a fully elastic assumption. In
fact, 2-5% of vertical load corresponds in this study for bridge models M1-M4 to a lateral
abutment force capacity of Fel=440-1100kN, while lateral force demands on abutments, even
for a PGA 0.16g lies above these values (Figure 8.18, right).
Figure 8.18: Maximum abutments strength demands for bridge models M1-M4 according to NLTHA
analyses presented in sections 8.2.1 to 8.2.4 for a seismicity level of PGA 0.16g (left) and PGA 0.35g (right)
As a consequence, for further analyses on multi-degree-of-freedom systems presented in the
following sections, an elasto-plastic modelling of abutments according to Table 8.2 will be
considered.
Table 8.2: Elasto-plastic modelling assumptions for abutments considered in this study
Elastic
stiffness k0
100MN/m
Yielding
Force Fy
Yield
Displacement Δy
Post-Yield
Stiffness ratio r
Hysteretic rule
in Ruaumoko [Carr, 2004]
500kN
5.00mm
0.00
IHYST = 1
138
Chapter 8. Multi-degree-of-freedom analyses
Figure 8.19: Elasto-plastic hysteresis implemented in Ruaumoko [Carr, 2004] and used in this study to
represents abutment behaviour in multi-degree-of-freedom analyses
Pier displacement ductility demands have also been particularly observed in this parametrical
study on elastic abutments. As showed in Figure 8.11(right), Figure 8.13 (right), Figure 8.15
(right) and Figure 8.17 (right) maximum pier ductility demands recorded in dynamic analyses
on multi-degree-of-freedom systems representing entire bridge structures under transverse
earthquake excitations are usually higher compared with single-degree-of-freedom systems
having short piers (h=6.60m), but are lower for taller piers (h=13.20m). Particularly critical
seems to be the case of a short, stiff pier (h=6.60m) close to a taller piers (h=13.20m) and
both piers being placed in central spans, thus without positive restraining effects of the
abutments (see section 8.2.3). Another critical configuration is represented by the irregular
bridge model in section 8.2.4, that increases the influence of higher modes on the overall
bridge response and particularly on stiff pier P3.
Reasons for the higher pier displacement ductility demands observed here could be the
consequence of many factors such as distribution of lumped masses on the superstructure,
stiffness assumption of superstructure and abutments, bridge configurations that in general
can be resumed in the influence of higher modes present in each multi-degree-of-freedom
system compared with a pure single-degree-of-freedom system.
Although maximum pier displacement ductility demands (without considering strength
degradation) observed in section 8.2 are sometimes higher compared with single-degree-offreedom systems, this does not necessary means, that strength degradation effects will be
amplified when analyzing entire bridge systems. In fact, due to system redundancy present in
multi-degree-of-freedom systems, when a single bridge pier suffers strength degradation,
superstructure, adjacent bridge piers and abutments are expected to control the systems for the
successive excitation steps, thus preventing a possible member and systems collapse.
8.3 Influence of spliced reinforcement at pier base
As for single-degree-of-freedom analyses, the influence of spliced longitudinal reinforcement
at pier base has been studied on multi-degree-of-freedom systems representing typical bridge
configurations, as showed in (Figure 8.4 to Figure 8.7). Both backbone assumptions, based on
experimental data and numerical analyses, were considered in a separate form here. For each
bridge model, results in terms of superstructure and pier maximum displacement demands
with or without strength degradation (continuous vs. spliced reinforcement) were compared
with each others and with single-degree-of-freedom-analyses. A set of 28 accelerograms from
either real or artificial ground motions, scaled into three seismicity levels (PGA 0.16g, 0.35g,
0.50g), presented in chapter 6 were used for the scope of this study.
139
Chapter 8. Multi-degree-of-freedom analyses
8.3.1 Bridge model M1
a) Pier capacity models considering data from numerical analyses
Figure 8.20: Bridge model M1 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and displacement demands P1 to P4 (points) have been evidenced
140
Chapter 8. Multi-degree-of-freedom analyses
b) Pier capacity models considering data from experimental observations
Figure 8.21: Bridge model M1 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and displacement demands P1 to P4 (points) have been evidenced
141
Chapter 8. Multi-degree-of-freedom analyses
8.3.2 Bridge model M2
a) Pier capacity models considering data from numerical analyses
Figure 8.22: Bridge model M2 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
142
Chapter 8. Multi-degree-of-freedom analyses
b) Pier capacity models considering data from experimental observations
Figure 8.23: Bridge model M2 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
143
Chapter 8. Multi-degree-of-freedom analyses
8.3.3 Bridge model M3
a) Pier capacity models considering data from numerical analyses
Figure 8.24: Bridge model M3 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
144
Chapter 8. Multi-degree-of-freedom analyses
b) Pier capacity models considering data from experimental observations
Figure 8.25: Bridge model M3 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
145
Chapter 8. Multi-degree-of-freedom analyses
8.3.4 Bridge model M4
a) Pier capacity models considering data from numerical analyses
Figure 8.26: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
146
Chapter 8. Multi-degree-of-freedom analyses
b) Pier capacity models considering data from experimental observations
Figure 8.27: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for members with continuous longitudinal reinforcement (centre) and with spliced
longitudinal reinforcement at pier base (bottom). Mean NLTHA displacement pattern (black thick line)
and pier displacement demands P1 to P4 (points) have been evidenced
147
Chapter 8. Multi-degree-of-freedom analyses
8.4 Discussion of results from multi-degree-of-freedom analyses
Multi-degree-of-freedom analyses carried out in section 8.3 have demonstrated, that influence
of strength degradation as a consequence of spliced longitudinal reinforcement in the
potentially plastic hinge region of bridge piers is less critical than in the case of single-degreeof-freedom analyses carried out in chapter 7, this because of the redundancy present in whole
bridge structures compared with isolated members (bridge piers). This means, that while a
single pier starts suffering strength degradation in a bridge system adjacent piers may help it
via superstructure to retain the “softened” member, so that the global bridge behaviour
remains similar to the case without strength degradation. As a consequence of strength
degradation, the other piers and abutments may have to sustain higher demands in subsequent
cycles of earthquake shaking, leading to higher peak displacement demands than for the
continuous case, even if these piers do not experience strength degradation.
As an example we consider the mean displacement demand values from NLTHA for bridge
model M4-CRA-SLM-EP versus M4-LRA-SLM-EP (Figure 8.28) and a seismicity level of
PGA 0.35g (see chapter 6). Piers P1-P2 have a yield displacement of Δy=116.16mm, while
piers P3-P4 have a yield displacement of Δy=29.04mm. Initiation of strength degradation
starts for all piers at a displacement ductility of 1.78 (according to Table 5.8). Based on mean
displacement patterns for model M4-CRA-SLM-EP in Figure 8.28 it is expected, that only
piers P3-P4 will increase displacement demand due to strength degradation in model M4LRA-SLM-EP, but due to the reasons discussed previously a slightly higher displacement
demand is effectively obtained for pier P1-P2, even if these values are well below initiation of
strength degradation (in fact P1 and P2 remains elastic).
Figure 8.28: Bridge model M4 (top), superstructure and piers maximum displacement demands from
single NLTHA for PGA 0.35g (grey lines). Mean NLTHA displacement pattern (black thick line) and pier
displacement demands P1 to P4 (points) have been evidenced for the case of continuous longitudinal
reinforcement (centre) and with spliced longitudinal reinforcement at pier base (bottom).
148
Chapter 8. Multi-degree-of-freedom analyses
With the aim to provide consistency with observations based on Figure 8.28, comparisons of
single pier displacement values from dynamic analyses on multi-degree-of-freedom systems
will be presented in this section.
A displacement demand ratio, analogue as in section 7.2.2 , is defined as the maximum
displacement demand for a member (bridge pier P1-P4) with strength degradation Δmax,L
divided by the maximum displacement demand of the same member without strength
degradation Δmax,C, i.e.:
Δ max, L
(8.1)
Δ max, C
Ratios have firstly been calculated for each pier P1-P4 in each bridge model M1-M4 and for
every NLTHA carried out in section 8.3, then compared with prediction factors developed in
section 7.2.6. Factor f(μ) and f(δ) in section 7.2.6 which aims to account for higher pier
displacement demands due to strength degradation as a consequence of member deterioration
in potential plastic hinge regions compared with member without strength degradation.
As can be seen in Figure 8.29 and Figure 8.31, predictions (red lines) are mostly conservative
when compared with effective obtained ratios of MDOF analyses. Considerably higher scatter
in the data has been observed for the case dynamic analyses performed with backbone curves
from experimental behaviours, because of the difference in yield displacement recorded
during the tests carried out by Bimschas [Bimschas et al., 2008] while for pier capacity curves
from numerical analyses this difference was neglected.
Moreover, from previous example on Figure 8.28, in multi-degree-of-freedom analyses,
single piers that experience ductilities or drifts well below initiation of degradation based on
models with continuous reinforcement (CRA or CRE) can increase displacement demand for
the cases with consideration of strength degradation (LRA or LRE) because of system
redundancy. A generally trend can not be predicted here, as can be seen from Figure 8.29 and
Figure 8.31, but ratios Δmax,L / Δmax,C > 1.10 were seldom observed in this study.
In order to measure the effectiveness of the displacement factors f(μ) and f(δ) with results
arising from NLTHA performed on multi-degree-of-freedom systems in section 8.3, an error
has been defined as
εD =
f ( μ , δ ) ⋅ Δ max, C −
Δ max, L
Δ max, L
Δ max, C
(8.2)
Δ max, C
Analogously to the error distributions developed in section 7.3.6 a normal distribution of error
data εD has been assumed here. Results are plotted in Figure 8.30 and Figure 8.32 and
underline the increased difficulty to capture variability of data with simplified predictions for
the case of pier capacity curves based on experimental behaviour (Figure 8.32) as compared
with those from numerical analyses (Figure 8.30).
149
Chapter 8. Multi-degree-of-freedom analyses
Figure 8.29: Comparison of displacement demand ratio from multi-degree-of-freedom analyses on entire
bridge structure carried out in section 8.3 versus predictions developed in this study (see section 7.2.6)
assuming capacity curves from numerical analyses in NLTHA. Displacement ductilities and drift ratios on
horizontal axis are referred to systems without strength degradation (thus Model-CRA-SLM-EP)
Figure 8.30: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution arising from differences between
prediction (red line) and effectively obtained displacement ratios presented in Figure 8.29
Considering as an acceptable engineering range (EAR), displacements errors in predictions
lying between -10% (underestimation) and +20% (overestimation), it can be seen from Figure
8.30, that applying the displacement factors f(μ) and f(δ) to the Eq.8.2 a probability of 90% is
achieved.
It means, that by performing dynamic analyses on bridge models calibrated without
considering strength degradation, maximum pier displacement demands obtained can easily
be multiplied with factors f(μ) and f(δ) to obtain demands for the case of spliced
reinforcement. The resulting, increased bridge pier displacements lie with a probability of
150
Chapter 8. Multi-degree-of-freedom analyses
roughly 90% within the EAR range. In this way, it is not necessary to perform dynamic
analyses with implementation of strength degradation. Strength degradation can be accounted
after dynamic analyses have been performed by means of factors f(μ) and f(δ).
Figure 8.31: Comparison of displacement demand ratio from multi-degree-of-freedom analyses on entire
bridge structure carried out in section 8.3 versus predictions developed in this study (see section 7.2.6)
assuming capacity curves from numerical analyses in NLTHA. Displacement ductilities and drift ratios on
horizontal axis are referred to systems without strength degradation (thus Model-CRE-SLM-EP)
Figure 8.32: Comparison of probability density function (PDF, left) and cumulative probability density
function (CDF, right) of displacement error for normal distribution arising from differences between
prediction (red line) and effectively obtained displacement ratios presented in Figure 8.31
If we effectively perform dynamic analyses considering capacity behaviour observed in
experiments carried out by [Bimschas et al., 2008] instead of numerical analyses, prediction
using displacement increasing factors f(μ) and f(δ) are less reliable. The probability, that
displacement ratios from simplified methods lie within EAR drops to 58%. This is mostly due
151
Chapter 8. Multi-degree-of-freedom analyses
to the fact, that members with spliced reinforcement did not have same initial stiffness as
those with continuous reinforcement according to [Bimschas et al., 2008].
Nevertheless, as can be seen in Figure 8.31, simplified factors are usually conservative.
Increase in displacement demands for members suffering strength degradation is partially
compensated by stiffer initial behaviour, leading to values generally lower than 1.00 for
ductilities smaller that μdeg.
152
Chapter 9. Conclusions and outlook for further studies
9 CONCLUSIONS AND OUTLOOK
9.1
Conclusions
9.1.1 General considerations
Consequences of spliced longitudinal reinforcement in potential plastic hinge regions of wall
type bridge columns on monotonic and hysteretic behaviour have been presented in this study
and compared with the behaviour arising from identical members without this detailing
deficiency. This was done by performing inelastic analyses on both single-degree-of-freedom
and multi-degree-of-freedom systems.
Calibration of capacity curves for single elements (bridge columns) was based on both
numerical analyses carried out by means of methods developed from previous studies and
experimental observations steaming from tests performed at the ETH Zürich by Bimschas and
Dazio [Bimschas et al., 2008] on two wall type bridge columns with an aspect ratio of
h/lw=2.2. However, due to the fact that previous studies and methods dealing with strength
degradation considered almost exclusively with splice lengths of 20dbl and were focused on
square columns, while experiments at the ETH Zürich aimed to represent typical existing
Swiss bridge piers with splices of 43dbl, calibration adopted in this study has mostly been
derived empirically from [Bimschas et al., 2008]. Cyclic strength degradation for dynamic
analyses has independently been reproduced in two different ways, on one hand considering a
ductility based hysteretic rule implemented in software code Ruaumoko [Carr, 2004] and on
the other hand adopting a pure energy dissipative hysteretic rule implemented in software
code Idarc [Idarc, 2006].
Within the framework of single and multi-degree-of-freedom systems, dynamic analyses
attempted to represent three different hazard levels, from moderate to very high. Therefore,
artificial and real ground motions have been consistently generated or scaled for this purpose,
matching the target displacement spectra at each hazard level.
153
Chapter 9. Conclusions and outlook for further studies
9.1.2 Single-degree-of-freedom analyses
As a first step, inelastic dynamic analyses on single-degree-of-freedom systems representing
typical bridge piers was performed assuming continuous longitudinal reinforcement at pier
base and then compared with the same members having spliced longitudinal reinforcement in
potential plastic hinge regions. A variety of pier configurations with different pier heights and
damping ratios were considered. In this way, drop of lateral force carrying capacity in
members with spliced reinforcement due to plastic hinge region deterioration was studied
using stand alone system, without the interaction of any other external factors. Results have
shown, that maximum displacement demand increase for members suffering strength
degradation compared to “non-degrading” members in the range of 0-25%, depending on
displacement ductilities (or drift) level achieved and backbone curve assumptions (based on
experimental observations or numerical analyses). A larger scatter in the data has been
observed for the case where the backbone curve of the piers war obtained directly from the
experimental results presented in [Bimschas et al., 2008] compared to the case where the
same backbone curve was computed analytically by means of the plastic hinge method. This
difference is due to the stiffer behaviour of the pier observed during the tests. The behaviour
was stiffer because the amount of longitudinal reinforcement in the 43dbl lap-splice region is
double compared to the case without lap-splice. Moreover, it has been found that the art of
strength degradation modelling (ductility or energy based) hardly affected the maximum
displacement demand, if both model were calibrated to dissipate roughly an equal amount of
energy during inelastic deformation cycles. Nevertheless, this did not always corresponded to
the same drop in lateral force level. In fact, using an energy based strength degradation model,
the energy dissipation history influenced the drop in lateral force capacity and generally
resulted in lower strength degradation compared with a ductility based rule (in average 515%).
The results of inelastic dynamic analyses on single-degree-of-freedom systems have been
compared in terms of maximum displacement demand to results obtained by means of four
different simplified prediction methods. The four methods differ significantly regarding the
assumption of the stiffness of the considered SDOF systems, i.e. initial, secant or effective
stiffness assumptions were made. Generally, the results steaming from the simplified methods
agreed well with those of the inelastic dynamic analyses, however, the method proposed by
[Priestley et al., 2007] was the more accurate one. Nevertheless, when comparing
displacement predictions from these methodologies versus displacement demands from
dynamic analyses on systems suffering strength degradation, less reliable results and large
scatter in the data are obtained.
Displacement increasing factors f(μ) (ductility dependent) and f(δ) (drift dependent) have
finally been developed with the aim to consider strength degradation in a simplified manner,
by multiplication of maximum displacement demands obtained from dynamic analyses on
members without strength degradation. The effectiveness of these factors has been tested
within the framework of displacement predictions arising from simplified procedures.
154
Chapter 9. Conclusions and outlook for further studies
9.1.3 Multi-degree-of-freedom-analyses
Based on the observations on single piers, dynamic analyses on multi-degree-of-freedom
systems representing four entire bridge structures subjected to transverse earthquake
excitations have been carried out in order to study the interaction of strength degradation in
single members (piers) combined with such external factors as adjacent piers with equal or
different characteristics, superstructure stiffness, abutments restraining level and distributed
mass. Due to the large abutment strength demand encountered during a parametric study on
the abutment elastic stiffness, elasto-plastic abutment assumptions have been adopted when
studying strength degradation.
Results from multi-degree-of-freedom analyses have shown, that due to the higher
redundancy of these systems compared with analyses on isolated piers, strength degradation
effects are less important when considering entire bridge models. This means, that the
probability of a structural collapse as a consequence of a single pier suffering degradation is
reduced by activation of “displacement reserves” of adjacent piers or abutments and
superstructure. Nevertheless, due to the distributed mass in the superstructure and the
restraining effects of the abutment, maximum pier displacements within the framework of
bridge systems may still be larger than the demand encountered in single-degree-of-freedom
analyses.
It has also been shown, that starting from the maximum pier displacement demands of single
piers arising from dynamic analyses on entire bridge structures without strength degradation
and applying the displacement factors f( or f() obtained in the framework of SDOF system
analyses, conservative demand predictions are usually obtained for softened piers, when
compared with dynamic analyses on bridges with consideration of strength degradation.
Moreover it can be stated, that for the majority of the bridge systems encountered, strength
degradation did not have an effect on displacement demand for the lowest seismicity
considered (PGA 0.16g), due to the low level of ductility demands encountered.
As a consequence, even for the highest Swiss hazards level, with a return period of 475 years
(or 10% of exceedance in 50 years), strength degradation did not occur in the majority of the
bridge configurations having the characteristics considered, this based on degradations curves
experimentally obtained in [Bimschas et al., 2008] and reproduced in this study.
9.2 Outlook for further studies
Strength degradation effects within the framework of earthquake engineering performancebased design and retrofit of structures has mostly been neglected in analyses and little
research has been carried out on this topic up to now. Experiments were usually based on a
lap-splice length of 20dbl, and focused on square cross sections, neglecting a wide range of
possibilities that can be encountered in existing structures with circular or rectangular cross
section shapes or longer splices.
A drop in lateral force carrying capacity may or may not be combined with partial loss of
axial load carrying capacity for column members. In any case, experimental evidence shows,
that higher deterioration of the concrete matrix in plastic hinge regions compared with a
member having continuous reinforcement is expected. A consequence of this, is a higher
155
Chapter 9. Conclusions and outlook for further studies
variability in structural response leading to a greater risk of structural collapse in the presence
of this detailing deficiency.
Therefore, in my opinion, further studies should be focussed on both numerical analyses and
experimental studies.
In numerical analyses, demand should be investigated by performing parametric studies.
Initial stiffness, level of yielding force, point of initiation and shape of degradation should be
varied, preferably using a ductility dependent hysteretic rule, in order to relate increases in
displacement demand for members suffering strength and backbone assumptions. As a result,
revised displacement increasing factors f(μ) or f(δ) should be proposed for a wider range of
backbone curves, covering a wider range of engineering applications and implying different
splice length, section shapes and axial load ratios.
In addition experimental tests should be carried out on columns having “tension-splices”, i.e.
as considered in [Bimschas et al., 2008], in order to provide additional information on the
capacity behaviour of these members. In my opinion, short piers having circular cross section
shapes could be considered as a priority. Partly because of the large number of circular piers
in existing bridges and partly because these members generally degrade more rapidly than the
wall-type piers tested by [Bimschas et al., 2008] with consequent premature collapse.
In any case, when performing experimental tests, axial load carrying capacity should be
related to a drop in lateral force carrying capacity in order to provide guidelines for collapse
prevention of structures where members suffering strength degradation are evident. P-Δ
effects may be discussed in some special cases.
Regarding the other structural members governing the response when analyzing entire bridge
structures, abutments are of prime importance and should be modelled accurately on a case by
case basis, in order to achieve realistic structural response entire system.
156
References
10
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Annex A. Experimental capacity curves for members with spliced end-reinforcement versus prediction
Annex A. Experimental capacity curves for members with spliced
longitudinal reinforcement vs. [Priestley et al., 1996] prediction
Chai et al., 1991
Figure A 1: Force displacement capacity curve from test performed by [Chai et al., 1991 ] on circular
columns versus [Priestley et al., 1996] prediction method. The shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
Lynn et al., 1996
Figure A 2: Force displacement capacity curve from test performed by [Lynn et al., 1996 ] on rectangular
columns versus [Priestley et al., 1996] prediction method. Shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
A1.1
Annex A. Experimental capacity curves for members with spliced end-reinforcement versus prediction
Figure A 3: Force displacement capacity curve from test performed by [Lynn et al., 1996 ] on rectangular
columns versus [Priestley et al., 1996] prediction method. Shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
Melek et al., 2004
Figure A 4: Force displacement capacity curve from test performed by [Melek et al., 2004 ] on rectangular
columns versus [Priestley et al., 1996] prediction method. Shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
A1.2
Annex A. Experimental capacity curves for members with spliced end-reinforcement versus prediction
Figure A 5: Force displacement capacity curve from test performed by [Melek et al., 2004 ] on rectangular
columns versus [Priestley et al., 1996] prediction method. Shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
Figure A 6: Force displacement capacity curve from test performed by [Melek et al., 2004 ] on rectangular
columns versus [Priestley et al., 1996] prediction method. Shear capacity envelope has been estimated
according to [Kowalsky et al., 2000]
A1.3
Annex B. Ground Motions
Annex B. Ground Motions
Accelerogram AGM11
Figure B 1:Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.1
Annex B. Ground Motions
Accelerogram AGM12
Figure B 2: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.2
Annex B. Ground Motions
Accelerogram AGM13
Figure B 3: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.3
Annex B. Ground Motions
Accelerogram AGM14
Figure B 4: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.4
Annex B. Ground Motions
Accelerogram RGM11
Figure B 5: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.5
Annex B. Ground Motions
Accelerogram RGM12
Figure B 6: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.6
Annex B. Ground Motions
Accelerogram RGM13
Figure B 7: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.7
Annex B. Ground Motions
Accelerogram RGM14
Figure B 8: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.8
Annex B. Ground Motions
Accelerogram RGM11N
Figure B 9: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.9
Annex B. Ground Motions
Accelerogram RGM12N
Figure B 10: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.10
Annex B. Ground Motions
Accelerogram RGM13N
Figure B 11: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.11
Annex B. Ground Motions
Accelerogram RGM14N
Figure B 12: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.12
Annex B. Ground Motions
Accelerogram AGM41
Figure B 13: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.13
Annex B. Ground Motions
Accelerogram AGM42
Figure B 14: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.14
Annex B. Ground Motions
Accelerogram AGM43
Figure B 15: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.15
Annex B. Ground Motions
Accelerogram AGM44
Figure B 16: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.16
Annex B. Ground Motions
Accelerogram RGM41
Figure B 17: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.17
Annex B. Ground Motions
Accelerogram RGM42
Figure B 18: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.18
Annex B. Ground Motions
Accelerogram RGM43
Figure B 19: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.19
Annex B. Ground Motions
Accelerogram RGM44
Figure B 20: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.20
Annex B. Ground Motions
Accelerogram RGM41N
Figure B 21: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.21
Annex B. Ground Motions
Accelerogram RGM42N
Figure B 22: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.22
Annex B. Ground Motions
Accelerogram RGM43N
Figure B 23: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.23
Annex B. Ground Motions
Accelerogram RGM44N
Figure B 24: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.24
Annex B. Ground Motions
Accelerogram AGM61
Figure B 25: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.25
Annex B. Ground Motions
Accelerogram AGM62
Figure B 26: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.26
Annex B. Ground Motions
Accelerogram AGM63
Figure B 27: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.27
Annex B. Ground Motions
Accelerogram AGM64
Figure B 28: Ground motion (top), elastic response spectra of acceleration and displacement computed for
different damping ratios (middle) and constant-ductility inelastic response spectra of acceleration and
displacement generated with for a viscous damping of ξ = 2% (bottom)
B1.28