reliabilityy analysis of prestressed concrete …

RELIABILITYY ANALYSIS OF PRESTRESSED CONCRETE BRIDGE

GIRDER BEAM

GUNA KUMAR

A dissertation submitted in partial fulfillment for the requirements for the award of

Master of Engineering (Structures)

Faculty of Civil Engineering

Universiti Teknologi Malaysia

NOVEMBER, 2005

iii

To Chechi the one and only.

iv

ACKNOWLEDGEMENT

This endeavor without a doubt has been time consuming, mentally taxing and

inevitably expensive, however, towards the end of this journey the pieces began to

fall in its place and the feeling of the saw has been sharpened after all, has sunk in.

For this my deepest gratitude goes to Assoc.Prof. Dr. Nordin Yahaya, my

dissertation supervisor, for all the guidance, knowledge and making the time cross

300 miles to extend help. My special thanks to my fellow colleague Mr. Yugabalan

for the help extended in the MATLAB script. To the special people around me (Mr.

Jim Balan) my gratitude for the patience and support which, was an impetus to

sustain me during this dissertation.

My appreciation to En. Khairul at SPS UTM City Campus for his unreserved

diligence in allowing me to use the computer lab especially at odd hours. Finally my

warm thanks to all my fellow master students for their friendship and support.

Thank you

Guna Kumar

December 2005.

v

ABSTRACT

Recent studies undertaken in the United States and Europe have shown that

there are many highway bridges rated as structurally deficient requiring

rehabilitation or replacement due to the increase traffic flow and modern

truck loads. The objective of this dissertation is to compare the reliability

based load factor for prestressed bridge girders against the deterministic load

factor given in six (6) codes namely; BS5400 Pt.2, AASHTO, Canadian

Code (OHBDC), Eurocode, Australian Code (Austroads) and JKR

specifications. ‘Y5’ prestressed bridge girder with a resistance of 4192kNm

assumed to be a constant are considered. The load parameters and the bridge

spans are treated as random variables. The statistical parameters are based on

available literature and test data. The reliability indices for load factor are

calculated by iteration utilizing Monte Carlo simulation. Based upon the

results obtained it can be concluded with confidence that in the design of

primary highway loads, the load factor given in the codes can be reduced by

approximately 10 to 13%. The results also indicate that AASHTO is the most

conservative code while BS5400 Pt.2 and Eurocode are the most permissive.

This would mean that using BS5400 Pt.2 or Eurocode will result in a cost

effective bridge girder beam.

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ABSTRAK

Objektif dissertasi ini adalah untuk memperbandingkan prinsip

keboleharapan rasuk jambatan prategangan berdasarkan kepada faktor beban

hidup keboleharapan dengan faktor beban hidup konvensional

(deterministik) yang diberikan dalam enam (6) piawai-piawai rekabentuk

iaitu; BS5400 Pt.2, AASHTO, Piawaian Kanada (OHBDC), Eurocode,

Piawaian Australia (Austroads) dan spesifikasi JKR. Rasuk jambatan

prategasan ‘Y5’ dengan nilai keupayaan masimum 4192kNm adalah

dianggap tetap ataupun sebagai satu konstan. Parameter beban dan panjang

rasuk jambatan adalah dianggap sebagai pembolehubah rawak. Parameter

statistik beban dengan panjang rasuk jambatan adalah berdasarkan kepada

kajian literatur sediada serta kajian makmal. Prinsip keboleharapan rasuk

jambatan berdasarkan kepada faktor beban hidup adalah dijalankan dengan

kaedah simulasi Monte Carlo secara berulang. Melalui kajian keboleharapan

dengan simulasi Monte Carlo, keputusan yang didapati menunjukan bahawa

faktor beban hidup yang diberikan dalam piawaian rekabentuk boleh

dikurangkan sebanyak 10% hingga 13%. Keputusan kajian ini mendapati

bahawa piawan ASSHTO merupakan satu piawan yang paling konservatif

manakala piawaian BS5400 Pt.2 dan Eurocode adalah piawaian-piawaian

yang paling permisif. Ini bermakna dengan menggunakan piawaian BS5400

Pt.2 atau Eurocode akan memberikan satu rekabentuk rasuk jambatan yang

kos effektif.

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LIST OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

CONTENT vii

LIST OF TABLES xii

LIST OF FIGURES xiv

NOMENCLATURE xvii

CHAPTER 1 INTRODUCTION 1

1.1 General Introduction 1

1.2 Background Of The Problem 2

1.3 Objectives Of Study 7

1.4 Scope Of Study 8

1.5 Importance Of Study 9

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1.6 Limitation Of Study 10

1.7 Thesis Organization 10

CHAPTER 2 LITERATURE REVIEW 12

2.1 Introduction 12

2.2 Definition Of Reliability 17

2.3 Reliability Based Design Concepts 18

2.4 Limit States In Deterministic Based Design 20

2.5 Load Safety Factor In Limit States Design 21

2.6 Concept Of Reliability Evaluation 23

2.6.1 Continuous Probability Distribution 24

2.7 Reliability With a Single Loading 26

2.8 Reliability Computation Of Moments 29

2.9 Monte Carlo Simulation 31

2.9.1 Simulation Procedures 31

2.10 Reliability Index 32

2.11 Previous Researches 35

2.11.1 System Reliability Models For Bridge 36

Structures

2.11.2 Probabilistic Limit State Analysis 38

Of Framed Structures

2.11.3 Reliability Based Design Of Steel 40

Bridges Under Fatigue Loading

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2.11.4 Reliability Analysis Of Bridge Girders, 41

A Comparison Of Code Of Practice

CHAPTER 3 METHODOLOGY 43

3.1 Introduction 43

3.2 Primary Highway Bridge Loadings 44

3.3 Assumptions On Arbitrary Bridge 47

3.4 Ultimate Limit State Analysis (Deterministic) 49

3.5 Reliability Analysis Of Bridge Girders 52

3.6 Monte Carlo Simulation 53

3.6.1 Application Of Monte Carlo Simulation 53

3.7 Steps To Derive Matlab Script For Monte Carlo 58

Simulation

3.8 Summary Of Methodology 60

CHAPTER 4 RESULTS 61

4.1 Introduction 61

4.2 Load Model 63

4.3 Results 64

4.3.1 Code Of Practice – BS5400 Pt.2 64

4.3.2 Code Of Practice – AASHTO 65

4.3.3 Code Of Practice – Canadian Code 66

4.3.4 Code Of Practice – Eurocode 67

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4.3.5 Code Of Practice – Australian Code 68

4.3.6 JKR Specifications 69

CHAPTER 5 DISCUSSION ON RESULTS 71

5.1 Introduction 71

5.2 General Overview On The Results 73

5.3 Discussion On Case Study No.1 – BS5400 76

5.4 Discussion On Case Study No.2 – AASHTO 78

5.5 Discussion On Case Study No.3 – Canadian 79

Code (OHBDC)

5.6 Discussion On Case Study No.4 – Eurocode 80

5.7 Discussion On Case Study No.5 – Australian 82

Code (Austroads)

5.8 Discussion On Case Study No.6 – JKR 83

Specification

5.9 Discussion On 35m Bridge Span With Various 85

Codes


Codes


Codes


Codes

xi


Codes


Codes

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 98

6.1 Introduction 98

6.2 Recommendation For Further Work 100

REFERENCES 102

xii

LIST OF TABLES

LIST TITLE PAGE

Table 2.1 Parameters In The Functions Of R and S 17

Table 2.2 Load Factor at ULS For Various Codes 22

Table 2.3 Quantitative Values between Pf and Β 35

Table 2.4 Bias Factor λ & V For Bridge Structures 36

Table 3.1 Load Factor For Various Codes 51

Table 3.2 Statistical Parameters For MC Computations 54

(Nowak & Yamani, 1994).

Table 4.1 Code Of Practice: BS 5400 Part 2 64

Table 4.2 Code Of Practice: AASHTO 1994 65

Table 4.3 Code Of Practice: Canadian Code (OHBDC) 66

Table 4.4 Code of Practice : Eurocode 67

Table 4.5

Code of Practice : Austroads 68

Table 4.6

JKR Specifications 69

Table 5.1 Comparison Of Data For 35m Bridge Span 85

Table 5.2 Comparison Of Data For 40m Bridge Span. 87


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Table 5.4 Comparison Of Data For 50m Bridge Span 90



Table 5.7 Conservativeness Of Codes To Bridge Span. 96

xiv

LIST OF FIGURES

LIST FIGURE TITLE PAGE

Figure 2.1 Service Life Model of RC Bridge Decks 14

Exposed To Various Risks

Figure 2.2 Basic Concept of Structural Reliability 18

Figure 2.3 System Reliability Index Vs. Girder Reliability 37

Index.

Figure 2.4 Density Function Of Load fλ After N= 105 38

Simulations.

Figure 2.5 Cumulative Density Function Of Load Fλ 39

With N=105

Figure 2.6 Fatigue Reliability Of Bridges In 50 years 40

Figure 2.7 Reliability Indexes For Various Code Of Practice 41

Figure 3.1 Location Of Distributed and Knife Edge Loads 44

To Produce Maximum Mid Span Moment.

Figure 3.2 Uniformly Distributed Loads On Notional Lanes 45

To Produce Maximum Mid Span Moment.

Figure 3.3 HA UDL & KEL With Loaded Length 46

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Figure 3.4 Footway Distributed Load 47

Figure 3.5 Typical Load Diagram For Arbitrary Bridge 48

Figure 3.6 Distribution Functions 55

Figure 3.7 Histogram Frequency Distribution For 35m Bridge 56

Span With Loads And Load Factor To AASHTO.

Figure 3.8 Probability Curve For 35m Bridge Span With 57

Loads And Load Factor To AASHTO.

Figure 4.1 Comparison Of Bridge Span To Load Factor 64

: (BS5400 Pt.2)


: AASHTO


: OHBDC


: Eurocode


: Austroads


: JKR

Figure 5.1 Comparison Of Reliability Based Live Load 73

Factor For Various Codes.

Figure 5.2 Reliability Based Reduction Factor In Live Loads 75


: BS5400 Pt.2

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: AASHTO


:OHBDC


: Eurocode


: Austroads


: JKR

Figure 5.9 Load Factor Variance (%) For 35m Bridge Span 86






NOMENCLATURES

(S) - loads

(R) - resistance

Pf - probability of failure

φ - resistance factor

γi - load factor

Rn - nominal design resistance

Sn - nominal load effects

G ( ) - limit state function

xi - random variable

μx - mean value

n - number of observation

Var (x) - variance

σx - standard deviation

COV - coefficient of variation

∞ - infinity

mx - random variable of moment

β - reliability index

Φ-1 - inverse standard normal distribution function

∂ - partial differential

fcu - concrete strength

fy - steel strength

λ - bias factor

γf3 - live load factor

Mav - average moment

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l - length

b - breath/width

d - effective depth

fcn - complementary function

ULS - ultimate limit state

SLS - serviceability limit state

f ( ) - probability density function

∑ - summation equation

E (x) - expected value of X

PDF - probability density function

CDF - cumulative density function

(N) - number of iterations

w - loads

KEL - knife edge load

M - moment

M (x) - mid span moment distribution

% - percentage

e - exponent

Z - safety margin

mR - mean of resistance

mS - mean of loads

ln - lognormal

UDL - uniformly distributed loads

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LIST OF APPENDICES

APPENDIX TTILE PAGE Appendix A Example of Monte Carlo Simulation Results

for 55m Bridge Span.

i. Typical Load Diagram 107

ii. MATLAB Script 108

iii. Monte Carlo Simulation Result BS5400 Pt.2 110

iv. Monte Carlo Simulation Result AASHTO 112

v. Monte Carlo Simulation Result OHBDC 114

vi Monte Carlo Simulation Result Eurocode 116

vii. Monte Carlo Simulation Result Austroads 118

viii. Monte Carlo Simulation Result JKR Specification 120

CHAPTER 1

INTRODUCTION

1.1 General Introduction

Highway bridges and associated structures in some ways influence the visual

quality of our surroundings with their sophisticated appearance and they are in fact

highly functional artifacts with long service life which, is regularly used and seen by

the masses. Highway bridges are critical link and form a considerable investment in

infrastructure that should be kept safe and serviceable. The design aspects of

highway bridges on the other hand are influenced by the application of loads and

these form the fundamental data in designing the bridge. The basic philosophy of the

application of loads is that the worst case scenario of the loads is taken as the basis

of the bridge design.

The last ten (10) years has seen the rapid development of reliability based

assessment methods to help engineers tackle the analysis, quantification, monitoring

and assessment of structural risks, undertake sensitivity analysis of inherent

uncertainties and make the appropriate decisions about the performance of a

structure.

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The structure maybe at the design stage, under construction or in actual use.

Highway bridge damage or failure especially those involving the loss of life, are

very rare and usually have causes outside the realm of typical design specifications.

The primary emphasis in bride design is the application on live loading effects and

in this instance the reliability with respects to live loadings due to HA and HB loads

(Frangopol, 1999). Highway bridges assessed with the reliability methods have been

found to be structurally deficient and required replacement to allow them to carry

modern truck traffic and the increase in annual traffic (Ibid). The cost to the relevant

infrastructure agencies to rehabilitate these bridges is enormous. In short, reliability

methods used in highway bridge design can be an effective method in producing a

high degree of performance structure which is cost effective. This forms the

quintessential requirement of awareness in using the reliability methodology to

highway bridge design.

1.2 Background of The Problem

Recent studies undertaken in the United States and Europe have shown that

there are many highway bridges rated as structurally deficient requiring

rehabilitation or replacement to allow them to carry increased traffic flow and

modern trucks loads (Nowak, 2000). The deficiency of these highway bridges are

primarily from the live loading effect and not from the threats like corrosion,

collision, wind and scouring. There is now an underlying realization that the

analytical techniques developed for the bridge design are in many cases unable to

accurately model the structural behavior of highway bridges.

3

This has resulted in underestimating the actual load capacity of bridge girders in the

design assessments. In the design of highway bridges the designer must have an

understanding of the types and magnitude of the loads that are expected to act on the

bridge during its lifetime. It has been expounded earlier that the basis of bridge

design relies on the worst case loading scenario or in other words, load combinations

giving the worst bending moments and stresses. The HA and HB loads used in

combinations are derived from the British Standard 5400 (Part 2) or from local

specification such as JKR which uses LTAL and SV loadings in lieu of HA and HB

loads although both uses the same methodology to derive the ultimate and

serviceability limits. There are five (5) load combinations and combinations one (1)

to three (3) are called the principle combinations while combinations four (4) and

five (5) are called the secondary combination. These formed the basic data to assess

the bridge comparing it to the ultimate or the serviceability limit state. This method

of analysis can be described as deterministic, resulting in bridges that are an

underestimation of the actual load carrying capacity. The fact that the loading

combinations are random variables and thus absolute safety or zero probability of

failure cannot be achieved. Vehicle comes in various shapes and sizes, traffic

passing over a bridge fluctuates with time and at any given time it is impossible to

quantify the number of vehicles and its specifications passing over a bridge. This

makes the load a random variable.

The loading on a highway bridge is further compounded by the issue of

heavy vehicles being modified to carry heavier loads either legally or illegally (lorry

hantu, overloading of lorries etc.).

4

The HB or SV loads are thus constantly changing compared to what is given in the

codes and specifications. This uncertainty in the context of codes and specification is

taken care by the “safety factor” however; the accuracy of using this safety factor is

subjective and cannot be taken as the absolute although engineers design with the

later in mind.

In many cases, the bridges will exhibit no outward signs of distress.

Although this does not, in itself, imply that failure may not be imminent; it is likely

that some form of damage or significant deformation will precede collapse in cases

of ductile failures of concrete bridge girders. This brings into question the

appropriateness of using elastic analysis for the determination of ultimate strength

for many types of bridges; and in particular short span concrete bridge girders which

have been found to be deficient in flexure (Thoft-Christensen, 1999).

Clearly there is a need to review and refine the existing methods and to

develop improved techniques which can more realistically model the ultimate load

capacity of bridges. Current codes of practice are written with the implicit

assumption that the design and assessment of bridges will usually be undertaken

using linear elastic analysis techniques. Elastic theory is well established and is

supported by many computer software packages, and has been found most

satisfactory for the design of bridges. As a lower bound method the engineer can be

confident that the analysis method should be conservative and hence safe.

5

The questions that are glaringly unanswered are what does “failure” actually mean

in an elastic analysis and what are the consequences of such failure in terms of both

risks to life and economic terms? The conventional approach to the assessment of

concrete bridge deck is to initially perform a simple elastic beam analysis using a

representative strip of the bridge girder.

If this “quick” check shows that the structure to be inadequate, a more

detailed linear elastic analysis allowing for transverse distribution of loads would

probably be performed using either a grillage or finite-element analysis. These

results are then examined to identify individual locations at which the maximum

calculated moments or shears exceed the estimated ultimate capacity of the section.

The decision to strengthen or replace a structure is commonly made on the basis of

these results. Transverse steel bars, shear bars are added to the bridge girder to

supplement this deficiency.

In reality concrete structures will crack under heavy loads resulting in a

change in stiffness of the bridge girder. Even when the ultimate moment capacity of

a section of the bridge girder is exceeded, loads will be redistributed elsewhere in

the deck slab provided that the deck possesses sufficient ductility and it does not fail

prematurely in shear. As a result, linear elastic analysis will not accurately model the

distribution of stresses or the actual behavior in the post-elastic range where non-

linear effects dominate. Elastic methods can be very conservative since failure of

one element in the structure is typically used to define failure of the structure as a

whole. In the cases of flexural failure, the consequences are likely to be small and

only may affect the serviceability of the structure.

6

If one accepts the that serviceability criteria does not govern and collapse is a

criterion on which to base the assessment, such conservativeness is not warranted for

concrete bridge girder for which ductile flexural failure is the critical mechanism of

failure. Once an individual section has reached ultimate or yield capacity, the failure

must develop into full collapse mechanism before the structure will actually fall

down (Melchers, 1987). Elastic models are still relied upon as the primary analyses

tool for assessing concrete bridge girder and full scale loads tests conducted shows

that concrete bridges are often able to carry loads well in excess of the “theoretical”

capacity calculated using this technique (Ibid).

It is thus important to investigate the options available to an engineer if some

form of alternative assessment can be carried out in tandem with his elastic models.

Reliability modeling and assessment is an appropriate solution to the problem of

quantifying the variables (loading in this case) and providing an accurate range of

limit states based on primary highway loadings. The probability of failure of bridge

decks can be predicted with high accuracy using the reliability based assessment.

The research problems can be summarized as follows;

i) The primary highway loadings (HA loads) given in the relevant codes

are random variables and thus cannot be quantified in a deterministic

manner. What is the best approach in modeling this variable?

ii) How do we evaluate the load models that has been developed in (i)

above?

7

iii) How do we measure the probability of failure of the bridge girder

arising from primary highway loadings (HA loads)?

iv) Are there any difference between the various codified approach to

bridge girder analysis (ULS and SLS methods) and the reliability

based approach to analyze a bridge girder? If there are, what are the

major differences?

v) How the implementation of reliability based modeling and evaluation

of bridge girder based on primary highway loading can enhance the

overall safety level of a bridge girder?

vi) Are there any imminent cost saving arising from using reliability

based assessment in bridge girder analysis?

1.3 Objectives of Study

The objectives of this study are to;

i) Conduct an investigation approach to formulate the various load

factor models from various codes which can be used to analyze

maximum mid span moment effects on a standard prestressed bridge

girder beam.

ii) Assess the reliability of the standard bridge girder based upon the

load models and load factors given by various codes and

specifications namely JKR, BS 5400 Part 2, AASHTO, Eurocode,

Austroads (Australian Highway Code) and Canadian Code

(OHBDC).

8

1.4 Scope of Study

The scope of study covered literature review from various sources and codes

of practice namely BS 5400 Part 2, JKR specifications, AASHTO, Eurocode,

Austroads (Australian Highway Code) and Canadian Codes specification on primary

loadings (HA loads) and their respective load factors. This study carried out a

statistical approach to model primary loadings (HA loadings) to highway bridge

girders and evaluate the result with the conventional elastic analysis approach to

bridge girder designs. The data for the conventional and reliability based method of

analysis are based on an arbitrary data as follows;

1) Bridge Span = 42 meters.

2) Carriageway Width = 12 meters

3) Footway width at both shoulders = 2 meters.

4) Bridge Girder = Standard Prestressed Girder Beam.

Computer software namely MATLAB Ver. 6.5.1 was utilized to run statistical

process (probabilities) approach to model the maximum moments at mid span

arising from primary highway loadings (HA & KEL loads). The results from

utilizing various load factors from the codes will then be compared to the

conventional elastic model analysis of the bridge girder. This would result in some

difference which then can be used to evaluate which codes gives rise to a

conservative design and which codes give rise to a permissible design.

9

The reliability based assessment of bridge girder data would be crossed referenced

with the most permissive code to check if there are can be any significant reduction

in member size and subsequent cost savings.

Finally the scope of study demonstrated that by using the reliability based

assessment of highway bridge decks, can supplement if not enhance the final

analysis for any bridge girder.

1.5 Importance of Study

This study basically showed on how to model primary highway loads (HA

loadings) to bridge girder using reliability approach (probability approach) and how

to evaluate the models based on the load factors given in various codes mentioned

above. The data from reliability analysis could perhaps be used to develop reliability

based design codes or it can be used to supplement the load factors used in the

various codes.

The result data from the reliability analysis can be used by design engineers to

enhance their decision making in analyzing highway bridge girders. This would in

some parameters benefit engineers to improve their accuracy and reduce some

uncertainties in the design of highway bridges. Lastly on the whole the design would

be envisaged to be safe for its lifetime and cost effective.

10

1.6 Limitations of Study

The limitation of this study is that other loads such as transient highway

loadings such as wind loads, collision loads, dynamic effects, scouring, braking and

centrifugal loads, shrinkage and thermal effects are not considered in this study. This

is primarily due to the fact that modeling these loads would be too large an endeavor

to undertake. Therefore only primary highway loading namely HA, KEL and

Footway Loading are taken into account to demonstrate highway bridge load

modeling.

1.7 Thesis Organization

This thesis is organized into six (6) chapters. Chapter one (1) will consist of

the problem background, research objectives, scope and importance of the study and

the limitations. Chapter two (2) covers literature review which, discusses the topic of

reliability assessment methods, primary bridge loadings , elastic analysis of bridge

deck using ULS (Ultimate Limit State) and SLS (Serviceability Limit State) and the

current practice of reliability analysis on bridge decks. Chapter three (3) presents the

methodology in basic theory of reliability engineering on modeling of primary loads

(HA loads) for an arbitrary bridge. Chapter four (4) would present the comparison

data from reliability assessment derived from the load factor from various codes on

bridge girders. The data would be tabulated and the major differences identified. The

differences in result would then be related to the design aspects to ascertain which

codes are permissible and which are deemed conservative.

11

Chapter five (5) would consist of the discussion of the results. Chapter six (6) will

consist of the conclusions and appropriate recommendations.

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