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7/25/2019 MSCE Thesis Final Draft (Baylon as of 02 04 2016).pdf
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Reliability Analysis of a bridge pier using
Interval Uncertainty Analysis
A Thesis presented tothe Faculty of Civil Engineering
College of Engineering
De La Salle UniversityManila
________________________________________
In Partial Fulfillmentof the Requirements for the Degree ofMaster of Science in Civil Engineering
Major in Structural Engineering________________________________________
BAYLON, MICHAEL BAUTISTA
Thesis Adviser:
DR. LESSANDRO ESTELITO O. GARCIANO
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FEBRUARY 2016
TABLE OF CONTENTS
LIST OF FIGURES ..................................................................................................................................... 4
LIST OF TABLE .......................................................................................................................................... 6
CHAPTER 1 ................................................................................................................................................ 7
PROBLEM SETTING ............................................................................................................................. 7
1.1 Background of the Study ........................................................................................................... 7
1.2 Statement of the Problem........................................................................................................ 12
1.3 Significance of the Study ......................................................................................................... 12
1.4 Objective of the Study .............................................................................................................. 13
1.5 Scope and Limitations ............................................................................................................. 13
1.6 Assumptions in the Study ........................................................................................................ 14
CHAPTER 2 .............................................................................................................................................. 15
REVIEW OF RELATED LITERATURE............................................................................................. 15
2.1 Interval Analysis ........................................................................................................................ 15
2.2 Fragility analysis ....................................................................................................................... 21
2.3 Seismic Fragility Analysis in the Philippines......................................................................... 26
2.4 Synthesis ................................................................................................................................... 28
CHAPTER 3 .............................................................................................................................................. 30
CONCEPTUAL AND THEORETICAL FRAMEWORK................................................................... 30
3.1 Conceptual Framework............................................................................................................ 30
3.2 Theoretical Framework ............................................................................................................ 33
3.2.1 PGA Normalization ................................................................................................................. 33
3.2.2 Pushover Analysis (Nonlinear Static Analysis).................................................................. 33
3.2.3 Time History Analysis (Nonlinear Dynamic Analysis)....................................................... 34
3.2.4 Ductility Factors ...................................................................................................................... 35
3.2.5 Damage Index and Damage Rank ...................................................................................... 36
3.2.6 Interval Arithmetic Operations .............................................................................................. 36
3.2.7 Interval Uncertainty Analysis (IUA)...................................................................................... 38
3.2.8 Probability of Exceedance .................................................................................................... 40
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CHAPTER 4 .............................................................................................................................................. 42
RESEARCH METHODOLOGY .......................................................................................................... 42
4.1 Input data .................................................................................................................................... 42
4.4 Interval Uncertainty Analysis method of reliability analysis ................................................. 47
4.5 A library of MatLab functions implementing interval arithmetic operations ...................... 49
4.6 Research paradigm ................................................................................................................... 51
CHAPTER 5 .............................................................................................................................................. 58
RESULTS AND DISCUSSIONS ........................................................................................................ 58
5.1 Fragility Analysis ........................................................................................................................ 58
5.2 Probability of Occurrence ......................................................................................................... 63
5.3 Fragility Curves (Conventional) ............................................................................................... 73
5.4 Fragility Curves by Interval Uncertainty Analysis ................................................................. 76
5.4.1 Interval Uncertainty Analysis (Lower Bound)..................................................................... 76
5.4.2 Interval Uncertainty Analysis (Upper Bound)..................................................................... 80
5.5 Comparison of Fragility Curves: Conventional vs. IUA........................................................ 85
5.6 Interval Uncertainty Analysis after First Pass...................................................................... 105
CHAPTER 6 ............................................................................................................................................ 115
SUMMARY, CONLUSION, & RECOMMENDATION ................................................................... 115
6.1 Summary ................................................................................................................................... 115
6.2 Conclusion ................................................................................................................................ 115
6.3 Recommendation ..................................................................................................................... 116
BIBLIOGRAPHY ..................................................................................................................................... 118
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LIST OF FIGURES
Figure 1. Example of seismic fragility curves. ...................................................................................... 8
Figure 2. Reliability analysis results under a 4-subinterval-BPA-structure case........................... 15
Figure 3. Typical deformation behavior as seen in the finite element model................................. 16
Figure 4. A graph of displacement at the plastic hinge with interval holonomic elastoplastic
alpha-umaxand alpha-uminresponses..................................................................................................... 17
Figure 5. Single objective optimization of example wing 2 under landing loads in progress,
deterministic, probabilistic, interval analyses....................................................................................... 18
Figure 6. Probability of Failure for the Performance Function Z=R-S........................................... 19
Figure 7. Summary of Seismic Fragility Curves of some lifelines in CAMANAVA area of
Damage Rank As ..................................................................................................................................... 27
Figure 8. Summary of Seismic Fragility Curves of some lifelines in CAMANAVA area of
Damage Rank A ....................................................................................................................................... 28
Figure 9. Conceptual Framework of the Study .................................................................................... 31Figure 10. Interval Uncertainty Analysis after the First Pass........................................................... 32
Figure 11. General Elevation Plan of the Bridge Showing Pier 2..................................................... 43
Figure 12. Elevation Plan of the Pier.................................................................................................... 43
Figure 13. Detail of Bored Pile .............................................................................................................. 44
Figure 14. Section A of the Bored Pile................................................................................................. 44
Figure 15. Section C (Coping) ............................................................................................................... 45
Figure 16. Ground motion data of Bohol October 2013 earthquake normalized in 0.2g............. 47
Figure 17. Normal probability function assumption of an interval value......................................... 49
Figure 18. Research paradigm ............................................................................................................. 51
Figure 19. Pushover curve from nonlinear static analysis of SAP2000.......................................... 58
Figure 20. Pushover curve (zoomed in) to compute the energy at yield Ee................................... 59
Figure 21. Hysteresis Model of Tohoku-Kanto Fukushima-2g (1 out of 300 hysteresis model)
using SAP2000. ........................................................................................................................................ 60
Figure 22. Hysteretic energy computation using Autodesk's AutoCAD......................................... 61
Figure 23. Probability of Occurrence for Conventional Fragility Curves......................................... 65
Figure 24. Probability of Occurrence for IUA-Lower Bound Fragility Curves................................ 65
Figure 25. Probability of Occurrence for IUA-Upper Bound Fragility Curves................................ 66
Figure 26. Plot of Lognormal of PGA to the Damage Ratio of DR="D".......................................... 67
Figure 27. Plot of Lognormal of PGA to the Damage Ratio of DR="C".......................................... 68
Figure 28. Plot of Lognormal of PGA to the Damage Ratio of DR="B".......................................... 69
Figure 29. Plot of Lognormal of PGA to the Damage Ratio of DR="A".......................................... 70
Figure 30. Plot of Lognormal of PGA to the Damage Ratio of DR="As"........................................ 71
Figure 31. Conventional fragility curves for the different damage ranks........................................ 75
Figure 32. IUA (Lower Bound) fragility curves for different damage ranks.................................... 80
Figure 33. IUA (Upper Bound) fragility curves for different damage ranks.................................... 84
Figure 34. Bounded fragility curve of DR="D" c.o.v.=5% in X-direction......................................... 85
http://c/Users/michael%20baylon/Documents/MSCE%20Thesis%202016/MSCE%20Thesis%20Final%20Draft%20(Baylon%20as%20of%2002%2002%202016).docx%23_Toc442202704http://c/Users/michael%20baylon/Documents/MSCE%20Thesis%202016/MSCE%20Thesis%20Final%20Draft%20(Baylon%20as%20of%2002%2002%202016).docx%23_Toc442202704http://c/Users/michael%20baylon/Documents/MSCE%20Thesis%202016/MSCE%20Thesis%20Final%20Draft%20(Baylon%20as%20of%2002%2002%202016).docx%23_Toc4422027047/25/2019 MSCE Thesis Final Draft (Baylon as of 02 04 2016).pdf
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Figure 35. Bounded fragility curve of DR="C" c.o.v.=5% in X-direction......................................... 86
Figure 36. Bounded fragility curve of DR="B" c.o.v.=5% in X-direction.......................................... 87
Figure 37. Bounded fragility curve of DR="A" c.o.v.=5% in X-direction.......................................... 88
Figure 38. Bounded fragility curve of DR="As" c.o.v.=5% in X-direction........................................ 89
Figure 39. Conventional and IUA (Mean) fragility curves of DR="D" c.o.v.=5% in X-direction... 90
Figure 40. Conventional and IUA (Mean) fragility curves of DR="C" c.o.v.=5% in X-direction... 91
Figure 41. Conventional and IUA (Mean) fragility curves of DR="B" c.o.v.=5% in X-direction... 92
Figure 42. Conventional and IUA (Mean) fragility curves of DR="A" c.o.v.=5% in X-direction... 93
Figure 43. Conventional and IUA (Mean) fragility curves of DR="As" c.o.v.=5% in X-direction. 94
Figure 44. Bounded fragility curves for damage rank of "No Damage" of c.o.v.=5% .................. 95
Figure 45. Bounded fragility curves for damage rank of "Slight Damage" of c.o.v.=5% ............. 96
Figure 46. Bounded fragility curves for damage rank of "Moderate Damage" of c.o.v.=5% ...... 97
Figure 47. Bounded fragility curves for damage rank of "No Damage" of c.o.v.=10% ............... 98
Figure 48. Bounded fragility curves for damage rank of "Slight Damage" of c.o.v.=10% ........... 99
Figure 49. Bounded fragility curves for damage rank of "No Damage" of c.o.v.=20% ............. 100
Figure 50. Bounded fragility curves for damage rank of "Slight Damage" of c.o.v.=20%.......... 101
Figure 51. Pushover curve with lower and upper bounds................................................................ 106
Figure 52. Pushover curve (inset) with lower and upper bounds.................................................. 107
Figure 53. Interval Uncertainty Analysis seismic fragility curve of "No Damage" rank.............. 110
Figure 54. Interval Uncertainty Analysis seismic fragility curve of "Slight Damage" rank.......... 111
Figure 55. Interval Uncertainty Analysis seismic fragility curve of "Moderate Damage" rank... 112
Figure 56. Interval Uncertainty Analysis seismic fragility curve of "Extensive Damage" rank.. 113
Figure 57. Interval Uncertainty Analysis seismic fragility curve of "Complete Damage" rank.. 114
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LIST OF TABLE
Table 1. Summary of Results of Ordinary Monte Carlo Simulation................................................. 20
Table 2. Relationship between the damage index and damage rank based from HAZUS (2013).
.................................................................................................................................................................... 36
Table 3. Pertinent ground motion (East-West) data of the Bohol, Philippines earthquake......... 46Table 4. Pertinent ground motion (Up and Down) data of the Bohol, Philippines earthquake.... 46
Table 5. Summary of calculations of ductility factors, damage indices, and damage ranks using
MS Excel Spreadsheet. For Tohoku-Kanto Eq. (Fukushima).......................................................... 62
Table 6. Summary of Damage Ratio for the Conventional fragility curves.................................... 63
Table 7. Summary of Damage Ratio for the IUA fragility curves (Lower Bound).......................... 63
Table 8. Summary of Damage Ratio for the IUA fragility curves (Upper Bound).......................... 64
Table 9. Tabulation of ln(PGA) with number of occurrences per damage rank............................ 72
Table 10. Tabulation of the product of number of occurrences to ln(PGA)................................... 72
Table 11. Tabulation of the square of the difference of the mean value of ln(PGA) to a PGA
value. .......................................................................................................................................................... 73Table 12. Tabulation of the values for the (X - Mean ) / Standard Deviation................................. 74
Table 13. Tabulation of the values for the Probability of Exceedance per PGA/g values........... 74
Table 14. Tabulation of ln(PGA) with number of occurrences per damage rank.......................... 76
Table 15. Tabulation of the product of number of occurrences to ln(PGA)................................... 77
Table 16. Tabulation of the square of the difference of the mean value of ln(PGA) to a PGA
value. .......................................................................................................................................................... 78
Table 17. Tabulation of the values for the ( X - Mean ) / Standard Deviation.............................. 79
Table 18. Tabulation of the values for the Probability of Exceedance per PGA/g values........... 79
Table 19. Tabulation of ln(PGA) with number of occurrences per damage rank.......................... 81
Table 20. Tabulation of the product of number of occurrences to ln(PGA)................................... 81Table 21. Tabulation of the square of the difference of the mean value of ln(PGA) to a PGA
value. .......................................................................................................................................................... 82
Table 22. Tabulation of the values for the ( X - Mean ) / Standard Deviation................................ 83
Table 23. Tabulation of the values for the Probability of Exceedance per PGA/g values........... 83
Table 24. Checklist of bounded fragility curves that subscribe to the "norm"................................ 94
Table 25. Tabulation of bounded fragility curves for DR="D","C", "B" for c.o.v.=5%.................. 101
Table 26. Tabulation of bounded fragility curves for DR="D","C" for c.o.v.=10%....................... 102
Table 27. Tabulation of bounded fragility curves for DR="D","C" for c.o.v.=20%....................... 102
Table 28. Summary of calculated c.o.v. based from the bounded fragility curves which follow
the set norm. ........................................................................................................................................... 103Table 29. Parameters from Nonlinear Static Analysis to be used in computing ductility factors
using Interval Analysis. .......................................................................................................................... 107
Table 30. Parameters from Nonlinear Dynamic Analysis to be used in computing ductility
factors using Interval Analysis. ............................................................................................................. 108
Table 31. Ductility factors computed from Octave script. ............................................................... 108
Table 32. Damage indices and damage rank based from HAZUS............................................... 109
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CHAPTER 1
PROBLEM SETTING
1.1 Background of the Study
Nowadays, an increasing social awareness was brought by nations media
practitioners on vulnerability of structures in the occurrence of every major earthquake.
As a consequence of this increased awareness, research has been the forefront to ease
the quantification of the potential social and economic losses of communities across the
globe. Bridge fragility curves have grown from this surge in research as they are essential
component to the risk assessment methodology.
The risk assessment of lifeline presented in Figure 1 shows that one of the key
links in the assessment methodology is to estimate the damage to the lifeline
components. This is done by estimating the performance of the various highway bridges
in the network as a function of a ground motion intensity parameter. This bridge
performance is commonly represented in either a damage probability matrix or a fragility
function (Nielson, 2005).
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A fragility function is a conditional probability that gives the likelihood that a
structure will meet or exceed a specified level of damage for a given ground motion
intensity measure (Ramanathan, 2012).
Figure 1. Example of seismic fragility curves.
Source: (Nielson, 2005)
In dealing with real world problems, uncertainties are unavoidable. As Civil
Engineers, it is important to recognize and quantify the presence of all major sources of
uncertainty in the analysis and design of structures. The sources of uncertainty may be
classified into two broad types: (1) aleatory those that are associated with natural
randomness; and (2) epistemic those that are associated with inaccuracies in our
prediction and estimation of reality (Ang & Tang, 2007). Uncertainties can arise from
(Modares, Taha, & Mohammadi, 2014):
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1) modeling due to oversimplification of the system and the boundary
conditions,
2) manufacturing errors in structural member dimensions,
3) load variations compiled from historical records or through instrumentations in
the field or laboratory,
4) interaction of structure leads to complexities,
5) variations in material properties, and
6) lack of sufficient test data.
Different methods of structural reliability were already in use. From the most
sophisticated method of inclusion of general method called evidence-based-theory
reliability analysis (Jiang, Zhang, Han, & Liu, 2013), a proposed interval uncertain multi-
objective optimization method for structures with uncertain-but-bounded parameters (Li,
Luo, Rong, & Zhang, 2013), interval elastoplastic analysis of structures (Yang,
Tangaramvong, Gao, & Tin-Loi, 2015), application of interval-based optimization of
aircraft wings under landing loads (Majumder & Rao, 2008), and to the relatively simple
method using interval uncertainty analysis in the reliability assessment of structures,
specifically that of probability of failure of plane truss (Modares, Taha, & Mohammadi,
2014).
In applications, interval analysis provides rigorous enclosures of solutions to model
equations. In this way, one can at least surely know what a mathematical model tells him,
and, from that, one might determine whether it adequately represents reality. Without
rigorous bounds on computational errors, a comparison of numerical results with physical
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measurements does not tell how realistic a mathematical model is (Moore, Baker
Kearfott, & Cloud, 2009).
In his dissertation, Nielson (2005) tackled the uncertainty in the seismic demand
function which was placed on highway bridges. He added that it is not always easy to
separate out the different sources of uncertainty in a given problem. Usually, the
randomness of seismic ground motion can be considered to be aleatoric in nature. But
when one deals with synthetic ground motions, the uncertainty innate into these ground
motion cannot be only be attributed to seismological mechanisms but also path and site
characteristics. More knowledge that pertains to attenuation relationships and effects of
soil can reduce some of this uncertainty, thereby alluding to its epistemic nature. He also
added that there is uncertainty associated with each median which was defined. This
uncertainty is given in the form of a lognormal standard deviation or dispersion. His study
claimed that when enough data is unavailable for the assessment of the dispersion for
each limit state, it is still beneficial to account for some degrees of uncertainties. The
assignment of this uncertainty can be assumed in a subjective manner, that is, by
estimating the coefficient of variation or c.o.v. In his doctoral research study,
Ramanathan (2012) considered uncertainty by citing the works of Melchers (1999) and
Ellingwood and Wen (2005), which stated that treatment of uncertainty in seismic
reliability and performance assessment has been a subject of research for many years.
In his part, Ramanathan used normal distribution in his model of reinforced concrete
materials, with the median and coefficient of variation used statistical parameters.
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The use of Latin Hypercube sampling (LHS) is a stratified sampling procedure that
provides an efficient way of sampling variables from their distribution while limiting the
required sample size. This procedure was considered to different bridge realizations
statistically different and yet nominally identicalfor the probabilistic treatment of fragility
curves. LHS was implemented to reduce the computational effort. An algorithm
suggested by Iman and Conover (1982) was implemented in the frames of the proposed
methodology for the treatment of uncertainty in capacity and demand (Stefanidou &
Kappos, 2013).
Traditional reliability analysis requires probability distributions of all the uncertain
parameters. These include the computations of reliability index using First-Order Second-
Moment (FOSM), First-Order Reliability Method (FORM), Second-Order Reliability
Method (SORM), Hasofer-Lind, Rackwitz-Fiessler Procedure, Monte Carlo Simulation
(MCS). All except except MCS has closed form solution; whilst MCS, gives sub-optimal
solution, depending on the observance of uncertainty in the estimate of the probability as
it decreases as the total number of simulations, N, increases (Nowak & Collins, 2013).
However, in many practical applications, the variation bounds can be only determined for
the parameters with limited information (Han, Jiang, Liu, Liu, & Long, 2014).
From the above mentioned methods of reliability analysis, there is a research gap
in bounded fragility curves. In the light of introducing a method to develop these curves
for the unknown-but-bounded uncertainty, seismic fragility curve will now have
probability of exceedance values with an overlay of quantifying how low and how high a
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decision-maker will adopt in his design, re-design, or retrofit of built structures. The
researcher adopts interval analysis to produce this interval. Thus, an interval uncertainty
analysis form of seismic fragility curves is being developed in this present study.
1.2 Statement of the Problem
There is a need to study the application of a technique that can respond to the
uncertainties during the evaluation of damage index formula used (Park & Ang, 1985).
Interval uncertainty analysis is chosen to embed in this reliability analysis, since the
outcome of this proposed method is an interval value of probability of exceedance for a
given intensity measurement, i.e., peak ground acceleration (PGA); thus, seismic fragility
curves by interval uncertainty analysis (IUA).
1.3 Significance of the Study
This study is significant due to the following reasons:
A. Epistemic uncertainty can be quantified through interval analysis; thus, its
embedment to the probability of exceedance.
B. Re-assessing built structures using the proposed method of reliability analysis
particularly that of seismic fragility curves.
C. An alternative reliability analysis aside from the existing ones, where computer
simulation is possible through fast computing requirements.
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1.4 Objective of the Study
This research aims to study the reliability of piers of a reinforced concrete deck
girder bridges using interval uncertainty analysis applied to fragility curves.
Specifically, this research aims:
A. To quantify the uncertainty in the construction of seismic fragility curves by
using interval analysis.
B. To develop a methodology in assessing the performance of bridge piers using
interval uncertainty analysis under shear mode of failure
C. To compare the difference between a conventional fragility curve analysis and
an interval uncertainty analysis when applied to a reinforced concrete deck
girder
1.5 Scope and Limitations
The study is focused only in the reliability analysis using the shear mode of failure
of bridge pier.
Moreover, the study limits to the fragility analysis and the non-linear static
(Pushover Analysis) and non-linear dynamic (Time History Analysis) analyses in
constructing fragility curves by interval uncertainty analysis.
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A number of strong significant ground motion data is limited to four (4) sets which
will be obtained from PHIVOLCS, PEER, and K-net.com, namely, Mindoro December
1999 earthquake, Bohol October 2013 earthquake, The Great Hanshin Kobe earthquake
in 1995, and the Tohoku-Kanto March 2011 earthquake. Peak ground accelerations
(PGAs) are divided into ten discrete values, from 0.2g to 2.0g.
1.6 Assumptions in the Study
This study assumes that the nonlinear static analysis (pushover analysis) and
nonlinear dynamic analysis (time history analysis) are the suitable model for the shear
mode of failure. Coefficient of variation used are 1%, 5%, 10%, and 20% for the
computation of damage indices in the interval uncertainty analysis.
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CHAPTER 2
REVIEW OF RELATED LITERATURE
2.1 Interval Analysis
A novel evidence-theory-based reliability analysis method for structures with
epistemic uncertainty is one those methods that deals with a specific theory based on a
more general than other uncertainty modeling techniques. This was equivalent to the
classical probability theory, possibility theory, p-box approach, fuzzy sets and convex
models. In a further perspective, these basic axioms of evidence theory allowed one to
combine alleatory and epistemic uncertainty in a straightforward manner without any prior
knowledge (Jiang, Zhang, Han, & Liu, 2013). Fig. 2 is one example of the outputs of the
reliability analysis under the case of a 4-subinterval-BPA-structure, as used in the
applications of their developed general method using evidence-theory-based.
Figure 2. Reliability analysis results under a 4-subinterval-BPA-structure case.
Source: (Jiang, Zhang, Han, & Liu, 2013)
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A proposed systematic design optimization method for structures with uncertain-
but-bounded parameters is another method. Using the interval method, Li et. al used to
describe the uncertainty and the Kriging model was applied in order to generate for the
approximation model. The interval number programming method was utilized to
transform each uncertain optimization problem with a single objective function into a
deterministic multi-objective optimization problem. Typical numerical examples applied
to engineering demonstrated that the proposed method can effectively search the Pareto
frontier using estimated approximation models. Moreover, this method has the ability to
retain an unchanged approximation space. Refer to Fig. 3, the crashworthiness vehicle
design was used as the engineering application problem, using a closed-hat beam with
interval uncertain parameters (Li, Luo, Rong, & Zhang, 2013).
Figure 3. Typical deformation behavior as seen in the finite element model.
Source: (Li, Luo, Rong, & Zhang, 2013)
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As of this writing, a work that considers the effects of uncertainties in a classical
elastoplastic analysis was started. The uncertainties were related to the applied forces
and the plastic material capacities, both of which were taken to lie within deterministic but
bounded intervals. This proposed novel interval approach obtained full spectra of the
extreme responses under interval force and yield capacity data. The efficiency and
robustness of this method were illustrated through a number of practically-motivated
engineering structure examples. The results have been partially validated by Monte Carlo
simulations and interval limit analyses. With these examples (see Fig. 4), it was
highlighted by this study the importance of assessing the influence of uncertain applied
forces and yield limits for practical application, with special attention to higher load levels,
that is, a sufficient number of plastic hinges had been developed (Yang, Tangaramvong,
Gao, & Tin-Loi, 2015).
Figure 4. A graph of displacement at the plastic hinge with interval holonomicelastoplastic alpha-umaxand alpha-uminresponses
Source: (Yang, Tangaramvong, Gao, & Tin-Loi, 2015)
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Reliability of structures is not only applied to vertical and horizontal structures but
also to aircraft structural components such as wings under landing loads. In order to
optimize the aircraft wing structures subjected to landing loads, an interval-based
automated optimization method was used. The interaction between landing gear and
flexible airplane structure was considered as a coupled system. Here, uncertainties were
considered in terms of the system parameters and described as interval numbers. In the
aspects of computing, the optimization procedure was illustrated using two distinct
applications, i.e., symmetric double-wedge airfoil, and supersonic airplane wing. This
interval analysis-based multicriteria optimum design of airplane wing structures under
landing loads was demonstrated. The comparison of results indicated that, for
comparable data and for the same value of the permissible landing stress, the minimum
value of any specific objective function obtained by interval analysis was in good
agreement with the ones obtained by deterministic and probabilistic analyses (Majumder
& Rao, 2008). Fig. 5 shows the progress of individual objective functions with the number
of iterations for all the three types of analyses.
Figure 5. Single objective optimization of example wing 2 under landing loads inprogress, deterministic, probabilistic, interval analyses.
Source: (Majumder & Rao, 2008)
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Regarded as significant in the literature of structural reliability, the Taylor-series
inspired technique has been used frequently. One of which is the First Order Reliability
Method (FORM). But with the harmonious mix with the interval analysis method, a newly-
formed Interval First Order Reliability Method (IFORM) was introduced by Modares et. al.
This method was an enhanced FORM after considering the uncertainties using interval
analysis. In this method, the intervals selected for using the method can be based on a
pre-determined multiple of standard deviation of the performance function. Again, this
developed method was compared to the Monte Carlo Simulations. The results from the
traditional first-order analysis were within the lower and upper bound results from the
interval method, as seen in Fig. 6. It was found from this method that for a small amount
of uncertainty in the resistance and applied load, the traditional first-order approximation
method results in probability values that correspond to bounds equal to once standard
deviation of performance function (Modares, Taha, & Mohammadi, 2014).
Figure 6. Probability of Failure for the Performance Function Z=R-S.Source: (Modares, Taha, & Mohammadi, 2014)
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The author attempted to come up with an assessment of a transportation lifeline
pier using ordinary Monte Carlo Simulation under a large magnitude earthquake.
According to their paper, the Philippines capital has its mass transit, the Light Rail Transit
System (LRT), constructed in the 1980s as part of the governments modernization efforts
in the field of transportation. Over the past thirty years, the LRT has withstood a number
of natural hazards including a strong earthquake in July of 1990. Due to this event, the
Philippine government initiated the earthquake reconstruction project and made
recommendations to retrofit important bridges. The paper investigated the reliability index
of the columns of the LRT under a Level 1 (El Centro) earthquake and Level 2 (Tohoku-
Kanto) earthquake using ordinary Monte Carlo Simulation. Based from the maiden
structural plans of LRT, the slenderness ratio of columns based from the ACI 318 was
observed and checked for buckling failure. Referring to Table 1, a significant +12%
difference between the reliability indices of unconfined (3.47) and confined (3.89)
reinforced concrete column was computed in the said simulation (Baylon, Garciano, &
Koike, 2012).
Table 1. Summary of Results of Ordinary Monte Carlo SimulationSource: (Baylon, Garciano, & Koike, 2012)Based from NSCP 2001
Tohoku-Kanto Earthquake ( M9.0) El Centro Earthquake of 1940
Unconfined Confined % diff Unconfined Confined % diff
Pf 0.00146 0.00012 92% 0.00029 0.00013 55%
2.976037 3.672701 23% 3.440799 3.652203 6%
Based from NSCP 2010
Tohoku-Kanto Earthquake ( M9.0) El Centro Earthquake of 1940
Unconfined Confined % diff Unconfined Confined % diff
Pf 0.00109 0.00039 64% 0.00023 0.00002 91%
3.064547 3.359796 10% 3.503029 4.10748 17%
% diff Pf 28% 36%
% diff 13% 11%
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2.2 Fragility analysis
Civil engineering facilities such as bridges, buildings, power plants, dams
and offshore platforms are all intended to contribute to the benefit and quality of life. It is
important that the benefit of the facility can be identified considering all phases of the life
of the facility. These facilities are established, operated, maintained and decommissioned
in such a way that was optimized or enhance the possible benefits to society and
individuals of the society (Sorensen, 2004). It is advisable to have clear expectations
about those portions of the structure that are expected to undergo inelastic deformations
and to use the analyses to (1) confirm the locations of inelastic deformations and (2)
characterize the deformation demands of yielding elements and force demands in non-
yielding elements. In this regard, capacity designs concepts are encouraged to help
ensure reliable performance (Deierlein, Reinhorn, & Willford, 2010).
Bridges are potentially one of the most seismically vulnerable structures in
the highway system. While performing a risk analysis of a highway system, it is imperative
to identify seismic vulnerability of bridges associated various states of damage
(Shinozuka, Feng, Kim, Uzawa, & Ueda, 2003). In the probabilistic seismic risk
assessment of highway transportation networks, fragility curves are used to represent the
vulnerability of a bridge. Because these networks have hundreds or thousands of
bridges, it is not possible to study each bridge individually. Instead, bridges with similar
properties are grouped together and represented by the same fragility curve (Gomez,
Torbol, & Feng, 2013).
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Seismic fragility is the probability that a geotechnical, structural, and/or non-
structural system violates at least a limit state when subjected to a seismic event
of specified intensity. Current methods for fragility analysis use peak ground acceleration
(PGA), pseudo spectral acceleration (PSa), velocity (PSv), or spectral displacement (Sd)
to characterize seismic intensity (Kafali & Grigoriu, 2004). These fragility curves indicate
the evolving potential for component and system damage under seismic loading
considering time-dependent corrosion-induced deterioration. The results indicate that
while corrosion may actually decrease the seismic vulnerability of some components,
most critical components suffer an increase in vulnerability (Ghosh & Padgett, 2010).
A method was used to track changes of the structural parameters of a bridge
throughout its service life. Based on vibration data the fragility curves are updated
reflecting a change in structural parameters. Fragility curves based on vibration data,
whenever these are available, represent the vulnerability of a bridge with greater accuracy
than fragility curves based only on the geometry and material properties (Gomez, Torbol,
& Feng, 2013). Most of the studies states that fragility curves can be empirically and
analytically generated.
The empirical fragility curves are usually developed based on the damage reports
from past earthquakes (Jernigan & Hwang, 2002). While the analytical fragility curves
are developed through seismic response data from the analysis of bridges. The fragility
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analysis generally includes three major parts: (a) the simulation of ground motions,
(b) the simulation of bridges to account for uncertainty in bridge properties, and (c) the
generation of fragility curves from the seismic response data of the bridges.
Nonlinear analysis requires thinking about inelastic behavior and limit states that depend
on deformations as well as forces. They also require definition of component models that
capture the force-deformation response of components and systems based on expected
strength and stiffness properties and large deformations (Deierlein, Reinhorn, & Willford,
2010). The seismic response data can be obtained from nonlinear time history
analysis, elastic spectral analysis, or nonlinear static analysis (Choi, DesRoches, &
Nielson, 2004).
The nonlinear static analysis is normally used for determining the capacities
beyond the elastic limit. One type of nonlinear static analysis is the pushover analysis
wherein which it is mainly used for estimating the strength and drift capacity of a structure
when subjected to selected earthquake. It incorporates the nonlinear-deformation
characteristics of individual components and subjects the elements of a structure
to monotonically increasing lateral load within a height-wise distribution until a
predetermined displacement is attained (Requiso, Balili, & Garciano, 2013). The static
pushover analysis has no rigorous theoretical foundation. It is based on the assumption
that the response of the structure can be related to the response of an equivalent single
degree-of-freedom (SDOF) system. This implies that the response is controlled by a
single mode, and that the shape of this mode remains constants throughout the time
history response. Clearly, both assumptions are incorrect, but pilot studies carried out by
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several investigators have indicated that these assumptions lead to rather good
predictions of the maximum seismic response of multi degree-of-freedom (MDOF)
structures, provided their response is dominated by a single mode (Krawinkler &
Seneviratna, 1998). In the pushover analysis, it is assumed that the target displacement
for the MDOF structure can be estimated as the displacement demand for the
corresponding equivalent SDOF system (Krawinkler & Seneviratna, 1998).
The pushover is expected to provide information on many response characteristics
that cannot be obtained from an elastic static and dynamic analysis. The following are the
examples of such response characteristics:
a. The realistic force demands on potentially brittle elements (axial, shear,
moment).
b. Estimates of the deformation demands for elements that have to deform
inelastically in order to dissipate the energy imparted to the structure by ground
motions.c. Consequences of the strength deterioration of individual elements on the
behavior of the structural system.
d. Identification of the critical regions in which the deformation demands are
expected to be high and that to become the focus of thorough detailing,
e. Identification of the strength discontinuities in plan or elevation that will lead to
changes in the dynamic characteristics in the elastic range.
f. Verification of the completeness and adequacy of load path, considering all the
elements of the structural system, all the connections, the stiff non-
structural elements of significant strength, and the foundation system.
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In contrast to the nonlinear static procedure, the nonlinear dynamic procedure,
when properly implemented, provides a more accurate calculation of the structural
response to strong ground shaking. Since the nonlinear dynamic analysis model
incorporates inelastic member behavior under cyclic earthquake ground motions, the
nonlinear dynamic procedure explicitly simulates hysteretic energy dissipation in the
nonlinear range (Nielson, 2005). In nonlinear dynamic analysis, the detailed structural
model subjected to a ground-motion record produces estimates of component
deformations for each degree of freedom in the model and the modal responses
are combined using schemes such as the square-root-sum-of-squares (Requiso D.
A., 2013).
Time-history analysis is a step-by-step integration analysis performed in the time
domain. It is the most rational method for earthquake response analysis because it can
account for all sources of nonlinearity. Although this method is computationally intensive,
it is the preferred method for investigating the response of bridges subjected to seismic
excitation. Similarly to the pushover analysis, SAP2000 can also directly perform the time
history analysis to obtain the nonlinear behavior, which includes generation of the
hysteresis models (Requiso D. A., 2013).
In establishing the seismic fragility curves, there is no universally applicable best
method for calculating fragility curves. Different methods may be preferred depending on
the circumstances (Requiso, Balili, & Garciano, 2013). The information that would be
derived from the fragility curve can be used by design engineers, researchers, reliability
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experts, insurance experts and administrators of critical systems to analyze evaluate
and improve the seismic performance of both structural and non-structural systems
(Requiso D. A., 2013). In principle, the development of bridge fragility curves will
require synergistic use of the following methods: (1) professional judgment, (2) quasi-
static and design code consistent analysis, (3) utilization of damage data associated
with past earthquakes, and (4) numerical simulation of bridge seismic response
based on structural dynamics (Shinozuka, Feng, Kim, Uzawa, & Ueda, 2003).
2.3 Seismic Fragility Analysis in the Philippines
In the local setting, seismic assessment of bridge piers and a fish port was recently
implemented by students in one of the universities in the countrys capital. Their study
was based from works of Karim-Yamazaki, Shinozuka et. al, and Ang-Park type of fragility
curves with emphasis on the nonlinear static and nonlinear dynamic analyses. Figures 7
and 8 summarize this seismic assessment depending in the damage rank classification
based from HAZUS (Baylon M. B., 2015).
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Figure 7. Summary of Seismic Fragility Curves of some lifelines in CAMANAVA area ofDamage Rank As
Source: (Baylon M. B., 2015)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
PROBABILITYO
FEXCEEDANCE(SHEAR)
PEAK GROUND ACCELERATION (in g)
FRAGILITY CURVES
DAMAGE RANK ="As"
Navotas Fishport
Lambingan
Tullahan(Mal-Val)
Bangkulasi
Tullahan (Ugong)
LRT1 South
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Figure 8. Summary of Seismic Fragility Curves of some lifelines in CAMANAVA area ofDamage Rank A
Source: (Baylon M. B., 2015)
2.4 Synthesis
Based from the previous studies reviewed, theres no definite approach of interval
uncertainty analysis to the construction of seismic fragility curves. Albeit, the interval
analysis has been in the research arena for quite a long time, reliability analysis of
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
PROBABILITYOFE
XCEEDANCE(SHEAR)
PEAK GROUND ACCELERATION (in g)
FRAGILITY CURVES
DAMAGE RANK = "A"
Navotas Fishport
Lambingan
Tullahan(Mal-Val)
Bangkulasi
Tullahan (Ugong)
LRT1 South
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structures using this method was already applied to sophisticated analysis, such as finite
element approach, but not yet with the fragility analysis. Some of the reliability analyses
presented in this chapter has been reviewed and motivated the present study to pursue
with one thing in mind: assessing the performance of bridge pier using fragility analysis.
The present study attempts to apply the interval analysis with the incorporation of
uncertainty in the results of nonlinear structural analyses, namely, the pushover curve
and hysteresis area; thus, an interval uncertainty analysis fragility curve or IUA-FC.
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CHAPTER 3
CONCEPTUAL AND THEORETICAL FRAMEWORK
3.1 Conceptual Framework
Figure 9 illustrates the conceptual framework of the present study. Using the
structural model and normalized ground motion data, nonlinear static and dynamic
analyses were used in the preliminary steps for the conventional seismic fragility
development. The current study used shear as the mode of failure. The output of these
two nonlinear analyses were the parameters as inputs to the damage index formula.
Using interval uncertainty analysis, these parameters were calculated to obtain the lower
bound, upper bound, and mean value of the damage indices for every ground motion
datas peak ground accelerations (PGA) from 0.2g to 2.0g . These damage ranks were
counted as frequencies to compute the probabilities of occurrence for various peak
ground acceleration (PGA) values. For every damage level, probabilities of occurrence
were used to compute the mean and standard deviations to be used in the lognormal
equation for the fragility analysis. Plotting the cumulative lognormal probability versus the
peak ground acceleration for every damage levels creates the seismic fragility curves.
Comparing these fragility curves to the conventional seismic fragility curves creates
another plot of the difference of IUAs mean probabilities to that of conventional
probabilities versus PGA.
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Figure 9. Conceptual Framework of the Study
Nonlinear Static
Analysis
(Pushover
Analysis)
Nonlinear
Dynamic Analysis
(Time History
Analysis)
IUA Seismic
Fragility Curves
Structural Model
Ground motion data
Mode of Failure
SHEAR
Parameters for
Damage Index
Interval
Uncertainty
Analysis (IUA)
DAMAGE
INDICES
Checking for Optimum Coefficient of Variation to be used for
the 2ndPass of IUA
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Nonlinear Static
Analysis
(Pushover
Analysis)
Nonlinear
Dynamic
Analysis (Time
History Analysis)
IUA Seismic
Fragility Curves
Structural Model
Ground motion data
Mode of Failure
SHEAR
Parameters for
Damage Index
DAMAGE
INDICES
Interval Uncertainty
Analysis (IUA)
Figure 10. Interval Uncertainty Analysis after the First Pass.
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3.2 Theoretical Framework
3.2.1 PGA Normalization
Using structural model of the building and normalized ground motion data as an
input subjected to two nonlinear methods namely nonlinear static analysis and nonlinear
dynamic analysis to develop seismic fragility curves. Peak Ground Acceleration (PGA)
normalization is done by generating the original data to create another data for different
PGAs. This is basically the same graph but different extent depending on the PGA. In the
present study, the PGA normalization ranges from 0.2 g to 2.0 g.
3.2.2 Pushover Analysis (Nonlinear Static Analysis)
Nonlinear static analysis, also called as pushover analysis, is used to investigate
the force-deformation behavior of a structure for a specified distribution of forces, typically
lateral forces (Chopra, 2012). In this study, the pushover analysis is applied to produce
pushover curve showing the relationship between the force and the displacement that
would be used for further analysis.
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3.2.3 Time History Analysis (Nonlinear Dynamic Analysis)
In the present study, the researcher is adopting the concept of time history analysis
considering the Bangkulasi Bridge as a single-degree-of-freedom (SDOF) system which
is subjected to normalized ground motion having different excitations.
The formula adopted from Karim and Yamazaki (2001) shown in equation 1 where
the ground motion data is multiplied by the ratio of the normalized and original peak
ground acceleration defines the relationship of various earthquakes while maintaining its
time history pattern.
SOURCENEW uAu
0 (1)
Where:
SOURCENEW uAu
0 = the normalized ground motion data.
SOURCEu = the source ground motion data.
0A =a coefficient factor to normalize the source of ground motion =
source
normalized
PGA
PGA
Using software, the acceleration time histories obtained was used as an input
producing another relationship between the force and displacement called the hysteresis
model (bilinear model).
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3.2.4 Ductility Factors
From the output of the nonlinear static and nonlinear dynamic analyses which are
the push-over curve and hysteresis model respectively, the ductility factors were obtained
using the following equations, also adopted from Karim and Yamazaki (2001).
y
dynamic
d
max (2)
y
static
u
max (3)
e
h
h
E
E (4)
Where:
= displacement ductility
= ultimate ductility
= hysteretic energy ductility
= displacement at maximum reaction at the push over curve (static)
= maximum displacement at the hysteresis model (dynamic)
= yield displacement from the push-over curve (static)
= hysteretic energy, i.e., area under the hysteresis model
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= yield energy, i.e., area under the push-over curve (static) but until yield point
only
3.2.5 Damage Index and Damage Rank
Once ductility factors are obtained, damage indices for the conventional seismic
fragility curves can be determined using equation 5, taking which is the cyclic loading
factor as 0.15 according to Jiang, et. al (2012), for bridges.
u
hd
DI
(5)
After computing the damage indices, damage rank for each damage index
was determined using Table 2.
Table 2. Relationship between the damage index and damage rank based from HAZUS (2013).
Damage Index () Damage Rank (DR) Definition
0.00
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by upper and lower bound of an interval in both axis. Given two interval numbers X~
and
Y~
whose lower and upper bound has this symbol:
XXX ~
; YYY ~
(6)
With this definition, the following interval arithmetic operationsequations adopted
from Moore et al. (2009) are computed with sets;
Addition
= [ ] 7
Subtraction
= [ ] 8
Multiplication
= min max 9
Where:
= { } (10)
Division
It can be accomplished via multiplication by the reciprocal of the second operand.
= (1 ) (11)
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Where:
1 = [1 1 ] (12)
Assume that 0
3.2.7 Interval Uncertainty Analysis (IUA)
Applying the interval uncertainty analysis to the preceding topics discussed
generates an interval of damage indices. According to Modares et al. (2014) interval
analysis is widely applied in engineering especially for civil engineering. In designing,
some values are assumed knowing that it is unknown but bounded. The concept of the
interval uncertainty analysis is to obtain a range of values incorporating the uncertainties.
Using interval analysis, the ductility factors and the damage indices equations are
now bounded as shown in the following equations:
y
dynamic
d
~
~
~ max (13)
y
static
u
~
~
~ max (14)
e
h
h
E
E~
~
~ (15)
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Where:
= interval of displacement ductility
= interval of ultimate ductility
= interval of hysteretic energy ductility
= interval of displacement at maximum reaction at the push over curve
(static)
= interval of maximum displacement at the hysteresis model
(dynamic)
= interval of yield displacement from the push-over curve (static)
=interval of hysteretic energy, i.e., area under the hysteresis model
= interval of yield energy, i.e., area under the push-over curve (static) but until
yield point only
After the interval of ductility factors are obtained, damage indices for the IUA
seismic fragility curves can be determined using equation 5 and taking as 0.10 for
vertical structures and 0.15 for bridges.
=
+
(16)
Where:
;
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The output of these equations can now be used to determine the range of damage
indices for every ground motion data from 0.2g to 2.0g peak ground acceleration (PGA).
Having the damage indices interval obtained, damage rank per PGA for upper and
lower bound can now be obtained using the Table 2.
3.2.8 Probability of Exceedance
Once the parameters have been obtained, the cumulative probability of occurrence
() of the damage, equal or higher than the damage rank, is computed using equation
15.
XP
r
ln (17)
Where:
rP =Cumulative Probability of Exceedance
= Cumulative Normal Distribution Function
X = Peak Ground Acceleration
= Mean
Standard Deviation
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Plotting the values of cumulative lognormal probability against the peak ground
acceleration creates seismic fragility curve. After the conventional and IUA seismic
fragility curves are developed, evaluation of probability of exceedance difference would
take place.
The statistical formulas used in deriving the mean and standard deviation were
based from an ungrouped data premise.
N
i
i
N
i
ii
f
xf
1
1
ln
(18)
1
ln
1
2
N
x
N
i
i
(19)
Where:
f = frequency of damage rank per PGA.
x =the PGA in cm/s2.
=the mean of the natural logarithm of PGA, in cm/s2.
=the standard deviation of the PGA, in cm/s2.
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CHAPTER 4
RESEARCH METHODOLOGY
4.1 Input data
Structural Plans
The structural plans of Bangkulasi Bridge were obtained from Bureau of Designs
in Department of Public Works and Highways-National Capital Region.
Pier 2 of the Bangkulasi Bridge as it can be seen in its general elevation plan of
Figure 11was assumed as the more probable to fail when subjected to earthquake
load. The elevation plan of the pier is shown in Figure 12. Referring to Figures 13 to 15,
Pier 2 is composed of 4-1200mm piles as seen in its detailed bored pile elevation, the
section of a typical pier, and that of the section of coping.
Based from these sectional properties of the structural plans, dimensions were
incorporated in both the resistance and load effect models.
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Figure 11. General Elevation Plan of the Bridge Showing Pier 2
Figure 12. Elevation Plan of the Pier
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Figure 13. Detail of Bored Pile
Figure 14. Section A of the Bored Pile
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Figure 15. Section C (Coping)
Ground Motion Data
The secured ground motion data of significant earthquakes from Philippine
Institute of Volcanology and Seismology (PHIVOLCS), Pacific Earthquake Engineering
Research (PEE), and Kik-Net were used as loads for the structure. Tables 2 and 3 are
pertinent data of the ground motion used for the simulation. These ground motion data
were normalized in such a way that a total of ten (10) were created from the base data.
This is based from the previous study used in developing fragility curves by Requiso et.
al.
The ground motion data that was used in this study are summarized as follows:
1. Tohoku-Kanto-FKS March 11, 2011 Magnitude 9.0
2. Tohoku-Kanto-AIC March 11, 2011 Magnitude 9.0
3. Tohoku-Kanto-HYG March 11, 2011 Magnitude 9.0
4. Tohoku-SIT March 11, 2011 Magnitude 9.0
5. Bohol October 15, 2013 Magnitude 7.2
6. Mindoro (Cainta, Rizal station) November 15, 1994 Magnitude7.1
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7. Mindoro (Station Quezon City) November 15, 1994 Magnitude 7.1
8. Mindoro (Station Marikina City) November 15, 1994 Magnitude 7.1
9. Kobe Shin-Osaka January 16, 1995 Magnitude 6.9
10. Kobe Takarazuka January 16, 1995 Magnitude 6.9
11. Kobe Takatori January 16, 1995 Magnitude 6.9
12. Kobe Nishi-Akashi January 16, 1995 Magnitude 6.9
13. Kobe Kakogawa January 16, 1995 Magnitude 6.9
14. Kobe KJM January 16, 1995 Magnitude 6.9
15. Kobe HIK January 16, 1995 Magnitude 6.9
Table 3. Pertinent ground motion (East-West) data of the Bohol, Philippines earthquakeFILE: 201310150012.TBPS.HNE.cor
DATE: 10/15/2013 0:12STATION: TBPSStation LATITUDE: 9.691Station LONGITUDE: 123.862SR: 100 HzEarthquake Latitude: 9.8Earthquake Longitude: 123.8
Earthquake Depth: 10.0 kmEarthquake Magnitude: 7.2Earthquake MECH: 1 [0 = null,1 = reverse,2 = strike-slip, 3 = normal ]RHYP: 17.1 kmPGA: 2.137010 m/s/s (0.217907 g)PGV: 0.668706 m/sInstrument corrected time histories filtered using a 4th orderButterworth bandpass between 0.4-50.0 Hz
Record Processed: 2014-04-03 13:50
Table 4. Pertinent ground motion (Up and Down) data of the Bohol, Philippinesearthquake
FILE: 201310150012.TBPS.HNZ.cor
DATE: 10/15/2013 0:12
STATION: TBPSStation LATITUDE: 9.691Station LONGITUDE: 123.862SR: 100 HzEarthquake Latitude: 9.8Earthquake Longitude: 123.8Earthquake Depth: 10.0 kmEarthquake Magnitude: 7.2Earthquake MECH: 1 [0 = null,1 = reverse,2 = strike-slip, 3 = normal ]RHYP: 17.1 kmPGA: 1.364399 m/s/s (0.139125 g)PGV: 0.370004 m/s
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Instrument corrected time histories filtered using a 4th orderButterworth bandpass between 0.4-50.0 Hz
Record Processed: 2014-04-03 13:50
This ground motion data can be plotted in MS Excel as shown in Figure 16.
Figure 16. Ground motion data of Bohol October 2013 earthquakenormalized in 0.2g.
4.4 Interval Uncertainty Analysis method of reliability analysis
The MatLab script named IUA_DI.m (Algorithm 1 in the Appendices) was then
developed for computing the damage indices using Interval Uncertainty Analysis (IUA).
Equations to calculate in the section of chapter 3 were used and implemented.
To calculate an interval, i.e., lower bound and upper values, one has to consider
the function:
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]cov)1(cov)1([][~ xx
xxx (20)
where:
x~ = a 2x1 vector, known as an interval value.
xx, = the lower bound value and the upper bound value, respectively.
x = the mean value
cov = coefficient of variation value
xx =the formula for the lower bound
xx = the formula for the upper bound
It is assumed that the probability distribution for each interval to be normal
distribution. According to Modares, Taha, & Mohammadi (2014), a unity multiplier for the
standard deviation is recommended for both the lower and upper bound of the interval.
Figure 17 shows the illustration of this assumption.
Where X is the interval, is the lower bound, and is the upper bound
To get the lower and upper bound, a coefficient of variation (c.o.v.) is assumed to
1%, 5%, 10%, and 20%, to set the value of standard deviation, , given the mean value,
.
. . .= (21)
= ... (22)
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= 1 . . . 1 . . . (23)
Input intervals in Equations 13 to 15 were used to implement Interval Uncertainty
Analysis (IUA). To come up with the values of the damage index intervals, the parameters
which were obtained from the nonlinear static and dynamic analyses maximum and
yield displacements, hysteretic energy, and elastic energy were used to apply
applicable interval arithmetic operations.
Figure 17. Normal probability function assumption of an interval value.
4.5 A library of MatLab functions implementing interval arithmetic operations
In the implementation of interval uncertainty analysis, it was convenient to develop
separate customized functions that implements interval arithmetic operations. Algorithm
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2 is a collection of MatLab functions created for this purpose. These functions are based
from the equations in section 3.2.2, that is, basic discussion of interval arithmetic
operations.
The following steps were implemented to Algorithm 2.
Step 1: Impose a value for the coefficient of variation (COV). In this study a value of 5%
was used.
Step 2: The COV value will be used in computing the standard deviation by multiplying
the mean value by the COV.
Step 3: The mean value and standard deviation value will be used in calculating the lower
and upper bounds of any interval value.
Step 4: Using the different fundamental arithmetic operations of intervals, one can
compute for the sum, difference, product, and quotient of an interval arithmetic operation.
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4.6 Research paradigm
The research paradigm is shown in Figure 18.
Figure 18. Research paradigm
Once all necessary section properties have been defined the actual simulation can
now be performed. To do so, the lateral ground motion of Types 1 & 2 earthquake would
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be considered and a nonlinear static procedure would be used to account for the shear
failure namely the pushover analysis. Using the software (SAP 2000), the implementation
of the pushover procedures as prescribed in ATC-40 and FEMA-273 and the time history
analysis would be faster, reliable and easier since it was already integrated into the
software. All the results then are obtained and be ready for the seismic fragility curve
development. Using MatLab and MS Excel Spreadsheet, damage indices were
determined using interval arithmetic operations. These damage indices in interval form
were separated into lower bound, upper bound, and mean for the processing of the
corresponding fragility curve and combination thereof. From these processed values,
probability of occurrence values were calculated thru spreadsheet solution. The statistical
parameters to obtain the probability of exceedance values, i.e., mean and standard
deviation, were computed using conventional statistical formulas used for ungrouped
data. For given PGA, probability of exceedance values were calculated using the
lognormal transformation as proposed by Shinozuka (2000).
4.6.1 SAP2000
The following procedures, a step-by-step in conducting the push-over analysis and
time history analysis in SAP 2000. (Requiso, 2013)
1. Create a model without the pushover analysis
2. Define the properties and acceptance criteria for the pushover hinges.
3. Establish the pushover hinges on the model by selecting one or more frame
member/s and assign its hinge properties and location.
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4. Define the load cases of the pushover analysis. These load cases are
combination of dead load and the pushover load case itself.
5. Run the basic static analysis, if desired, dynamic analysis. Then run the static
nonlinear pushover analysis
6. SAP2000 can demonstrate the pushover curve by simply clicking the display
menu and selecting the show pushover curve function at the topmost toolbar of
the program. This would show you a table which gives the coordinates of each
step of the curve. It also allows the user to print the pushover curve or convert it
to an excel file to analyze the results
Once done with the steps pushover analysis will now be obtained, the results
would be used in the next procedure which is the nonlinear dynamic analysis (Time history
analysis). The following step-by-step procedure of Karim and Yamazaki (2001) in
performing the nonlinear dynamic analysis (Time History Analysis) is used.
1. Select the ground motion data records.
2. Normalize the PGA of the strong motion data records to be used. For this step,
you can normalize the peak ground acceleration by depending on the highest
value among the records. After picking the highest value in the records, you can
now use equation 1 which is mentioned in chapter 3 of the study. The result
would now be considered as normalized record to be used.
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3. Create a computer model in SAP 2000, you can use the exact computer model
from the pushover model but make sure to remove all the pushover data and
load cases.
4. Obtain the structures stiffness by nonlinear static analysis.
5. Plot the stiffness of the structure in order to obtain its yield point and maximum
displacement.
6. Perform the nonlinear dynamic response analysis (time-history analysis), using
the selecting strong motion records. In order to get the hysteresis area, you need
to plot the coordinates of the hysteresis area to AutoCAD.
7. Compute the ductility factors of the structure by performing the ductility equations
2, 3 and 4.
8. Obtain the damage indices of the structure in each excitation level using eq.5.
Use table 3.1 to calibrate the index of the damage done.
9. Obtain the total occurrence for each damage rank and get the damage ratio.
10. Construct the fragility curve by using the computed damage ratio and the ground
motion indices for each rank.
The damage ratio is defined as the number of occurrence of each damage rank
(no, slight, moderate, extensive, and complete) divided by the total number of records.
Once obtained, the damage ratio is plotted with the in (PGA) on a lognormal probability
paper to obtain the necessary parameters (mean and standard deviation).
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Once the mean and standard deviation have been obtained, the cumulative
probability (PR) of occurrence of the damage equal or higher than the damage rank can
be computed using equation 6. Then by simply plotting acquired cumulative probability
with the peak ground acceleration (PGA normalized to different excitation), the fragility
curve can now be obtained.
4.6.2 Conventional Seismic Fragility Curves
Conventional seismic fragility curves are created by plotting the cumulative
lognormal probability against peak ground acceleration. The steps to develop
conventional seismic fragility curves are as follows:
1. Obtain ductility factors. After obtaining the results from the two nonlinear analyses,
the ductility factors namely displacement ductility, ultimate ductility and hysteric
energy ductility factors are computed using the equations 2,3 and 4, respectively
which was adopted from Karim and Yamazaki (2001)
2. Obtain damage indices. Using equation 5 taking as 0.15, damage indices for
every PGA are calculated considering the values obtained from the previous step.
3. Determine the damage rank. After damage indices are obtained, damage rank can
be determined referring to table 2.
4. Obtain the probability of occurrence. Determine the probability of occurrence by
dividing the number of occurrence to the total number of occurrence.
5. Obtain the cumulative lognormal probability. Using equation 16, the probability of
exceedance is calculated for every PGA.
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6. Plot the conventional seismic fragility curves. From the statement above, the
fragility curves is developed by plotting the lognormal probability versus the peak
ground acceleration.
4.6.3 Interval Uncertainty Analysis
The process for the interval uncertainty analysis is quite the same as the
conventional procedure but the parameters obtained are bounded. The procedure for
generating IUA seismic fragility curves are the following:
1. From the results of the two nonlinear analyses; maximum displacement for static
static, maximum displacement for dynamic dynamic, yield
displacement for static (), hysteretic energy (Eh), and yield energy Ee, the
interval showing the lower bound and upper bound of the parameters are
computed using the following equations taking COV or coefficient of variation as
5%.
static = 1COV 1COV (24)
dynamic = 1 C O V 1COV (25)
= 1COV 1 C O V (26)
Eh = Eh1 C O V Eh1 C O V (27)
Ee = Ee1 C O V Ee1 C O V (28)
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2. Obtain ductility factors. For the IUA process, the ductility factors are also obtained
with boundaries using equations 13, 14 and 15. The ductility factors for the
boundaries (lower and upper) are computed separately.
3. Obtain damage indices. Using equation 16 taking as 0.15, damage indices for
every PGA are calculated considering the values obtained from the previous step.
Damage indices are computed differently for the lower bound and upper bound.
4. Determine the damage rank. After damage indices are obtained, damage rank for
each boundary can be determined referring to Table 2.
5. Obtain the probability of occurrence. Determine the probability of occurrence by
dividing the number of occurrence to the total number of occurrence.
6. Obtain the cumulative lognormal probability. Using equation 17, the probability of
exceedance is calculated for every PGA and for each boundary. Using equations
18 and 19, the statistical parameters of the lognormal probability is computed.
7. Plot the IUA seismic fragility curves. Plot different fragility curve for the lower bound
and another for the upper bound.
8. Compare conventional fragility curves to IUA fragility curves. Determine the mean
of the lower and upper bound fragility curve to come up with a new IUA seismic
fragility curve which will be compared to the conventional seismic fragility curve.
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CHAPTER 5
RESULTS AND DISCUSSIONS
5.1 Fragility Analysis
The results of the Nonlinear Static Analysis or Pushover analysis is a pushover
curve as seen in Figure 19. It can be seen from this pushover curve the yield
displacement of 0.016479 m and a maximum displacement of 0.138127 m.
Figure 19. Pushover curve from nonlinear static analysis of SAP2000.
0.016479, 10652.435
0.138127, 11717.678
0
2000
4000
6000
8000
10000
12000
14000
0 0.2 0.4 0.6 0.8 1 1.2
Force(kN)
displacement (m)
Pushover Curve
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The pushover curve in Figure 19 is zoomed-in, thus, Figure 20 can be seen
showing the area bounded by the origin to the yield point. This area is called the energy
yield or Ee. The computed value of the energy at yield point is 87.77073818 kJ.
Figure 20. Pushover curve (zoomed in) to compute the energy at yield Ee.
After the researcher ran the model in the pushover analysis, he used the time-
history analysis with the result of hysteresis model as shown in Figure 21. This hysteresis
figure was then plot to Autodesks AutoCAD to compute for the area of the hysteresis.
The coordinates of the hysteresis were derived from the SAP2000 results. Thus,
computing the area known in this literature as Hysteretic energy, Eh. This can be seen
in Figure 22. The maximum displacement in dynamic analysis for a PGA of 0.2g in the
0.016479, 10652.435
0
2000
4000
6000
8000
10000
12000
14000
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
0.02
Force(k
N)
displacement (m)
Pushover Curve (zoomed in)
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Tohoku-Kanto (Fukushima station) is found to be 0.03564 m and hysteretic energy of
3.45667199 kJ.
Figure 21. Hysteresis Model of Tohoku-Kanto Fukushima-2g (1 out of 300 hysteresismodel) using SAP2000.
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Figure 22. Hysteretic energy computation using Autodesk's AutoCAD.
Abovementioned result values are then collected as follows derived from the
nonlinear analyses of SAP2000:
y
0.016479 m; static
max 0.138127 m;
eE 87.77073818 kJ
dynamic
max 0.03564 m;
hE 3.45667199 kJ
From these values, the following ductility factors can be solved.
2.162753016479.0
03564.0max
y
dynamic
d
8.382001016479.0
138127.0max
y
static
u
0.03938377073818.87
45667199.3
e
h
h
E
E
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The damage index can now be calculated with the cyclic loading be 0.15 for
bridges.
0.2587288.382001
0.0393830.152.162753
u
hd
DI
Based from the damage rank of Table 2, the corresponding damage rank for a
damage index of 0.258728 falls in the closed interval [0.14, 0.40], that is, D which is
equivalent to Slight Damage definition. Using MS Excel spreadsheet, these sample
computations were done as shown in Table 5.
Table 5. Summary of calculations of ductility factors, damage indices, and damageranks using MS Excel Spreadsheet. For Tohoku-Kanto Eq. (Fukushima)
PGA/gNLSA NLDA DUCTILITY FACTORS
DAMAGE
INDEX
DAMAGE
RANK
max y Ee max y Eh d u h (DI) (DR)
0.2 0.138 0.016 87.771 0.036 0.016 3.457 2.163 8.382 0.039 0.259 C
0.4 0.138 0.016 87.771 0.071 0.016 13.827 4.326 8.382 0.158 0.519 B
0.6 0.138 0.016 87.771 0.107 0.016 31.110 6.488 8.382 0.354 0.780 A
0.8 0.138 0.016 87.771 0.143 0.016 55.307 8.651 8.382 0.630 1.043 As
1.0 0.138 0.016 87.771 0.178 0.016 86.417 10.814 8.382 0.985 1.308 As
1.2 0.138 0.016 87.771 0.214 0.016 124.440 12.977 8.382 1.418 1.574 As
1.4 0.138 0.016 87.771 0.249 0.016 169.377 15.139 8.382 1.930 1.841 As
1.6 0.138 0.016 87.771 0.285 0.016 221.227 17.302 8.382 2.521 2.109 As
1.8 0.138 0.016 87.771 0.321 0.016 279.990 19.465 8.382 3.190 2.379 As
2.0 0.138 0.016 87.771 0.356 0.016 345.667 21.628 8.382 3.938 2.651 As
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5.2 Probability of Occurrence
After calculating the conventional and IUA damage indices using the Algorithm 1
and MS Excel spreadsheets, Tables 5, 6, and 7 summarize the respective damage ratios.
For this tabulation, the initial value of coefficient of variation (c.o.v.) of 5% was used.
Table 6. Summary of Damage Ratio for the Conventional fragility curves
PGA
DAMAGE RANK
D C B A As
0.2 g 0.4 0.046875 0 0 0
0.4 g 0.2333333 0.125 0 0 0
0.6 g 0.2333333 0.109375 0.0909091 0 00.8 g 0.1333333 0.0625 0.5454545 0.047619 0
1.0 g 0 0.109375 0.3636364 0.1904762 0
1.2 g 0 0.109375 0 0.3333333 0.041667
1.4 g 0 0.109375 0 0.2857143 0.083333
1.6 g 0 0.109375 0 0.0952381 0.25
1.8 g 0 0.109375 0 0.047619 0.291667
2.0 g 0 0.109375 0 0 0.333333
Table 7. Summary of Damage Ratio for the IUA fragility curves (Lower Bound)
PGA
DAMAGE RATIO
D C B A As
0.2 g 0.4285714 0.0461538 0 0 0
0.4 g 0.25 0.1230769 0 0 0
0.6 g 0.25 0.0923077 0.1818182 0 0
0.8 g 0.0714286 0.0923077 0.5454545 0.05 0
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1.0 g 0 0.1076923 0.2727273 0.25 0
1.2 g 0 0.1076923 0 0.35 0.038462
1.4 g 0 0.1076923 0 0.3 0.076923
1.6 g 0 0.1076923 0 0.05 0.269231
1.8 g 0 0.1076923 0 0 0.307692
2.0 g 0 0.1076923 0 0 0.307692
Table 8. Summary of Damage Ratio for the IUA fragility curves (Upper Bound)
PGA
DAMAGE RATIO
D C B A As
0.2 g 0.5 0.1481481 0 0 0
0.4 g 0.5 0.1296296 0.0416667 0 0
0.6 g 0 0.1296296 0.25 0.1538462 00.8 g 0 0.1296296 0 0.5384615 0.022222
1.0 g 0 0.1296296 0 0.3076923 0.088889
1.2 g 0 0.1296296 0 0 0.177778
1.4 g 0 0.1296296 0 0 0.177778
1.6 g 0 0.0740741 0.125 0 0.177778
1.8 g 0 0 0.2916667 0 0.177778
2.0 g 0 0 0.2916667 0 0.177778