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ACKNOWLEDGEMENTS

I would like to give my sincere thanks to my advisor, Dr. Fred Moses, for his

continuous contributions to my research. Without his guidance, support and

encouragement, throughout the course of this research and all my M.S. and Ph. D. studies

here at the University of Pittsburgh none of this would have been possible. I am honored

and grateful to have had the opportunity to work under his supervision.

I would like to thank the members of my Dissertation committee, Dr. Christopher

J. Earls, Dr. Jeen-Shang Lin, Dr. Dipo Onipede, Jr. and Dr. Luis E. Vallejo for their time,

constructive input and contributions.

I am very thankful to Dr. Michel Ghosn of CUNY, for his contributions in the

course of this research.

I would like to thank to my wife Seda Akta, for her continuous support, patience

and love and I would like to apologize to my daughter dil Aktafor not being next to her

for about a year during the last phase of this study. I would like to thank my parents

Mefharet and Abdurrahman Akta, my mother-in-law kran Sekin, my sister Mfide

Budak, my brother Ersin Aktaand my brother-in-law Serdar Akar for their endless love

and support. I also would like to thank my grandmother mmehan encan who taught

me to go after my dreams and to my uncle Fahri encan for being my mentor.

I would like to thank all my officemates for their friendship, which I will cherish

for the rest of my life.


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I am also indebted to zmir Institute of Technology in Turkey for providing

scholarship support to my graduate study at the University of Pittsburgh.


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use of the structure. In current Load and Resistance Factor Design (LRFD) practice, the

code calibration is carried out by using predefined target (reliability) levels. Defining the

target safety level is not clear since codes involve multiple load combinations including

combination of time dependent loads, and a dilemma arises as to which target safety

index should be used in calibrating load factors for these combinations.

This study investigates use of reliability based cost optimization in calibration of

design codes. Load factors that make the total expected cost a minimum are the optimum

load factors and the corresponding safety index is the optimum safety index for each load

case. No predefined target safety levels are used herein; however, the contribution of the

current code in the calibration process is satisfied by deducing the failure cost that a

current level of safety implies in the current code. The most basic gravity load

combination is chosen to deduce the failure cost. First Order Reliability Method (FORM)

is used for reliability analysis, and the combination of time dependent loads is derived

with the Ferry-Borges Method. Since application of Ferry-Borges method requires

independency between load events, the part coming from the time independent modeling

uncertainties are separated herein from the time dependent part and calculations are

carried out by combining the FerryBorges Method with a Nested Reliability Analysis.

The approach is illustrated with application to a bridge design specification including

dead, live, wind and earthquake load combinations. Optimum total cost including failure

cost and safety indices are compared to existing code format. A recommended load factor

table is presented as a product of the calibration process.


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DESCRIPTORS

Code Calibration Code Optimization

Conditional Probability Cost Optimization

Ferry-Borges Load Model First Order Reliability Method

Load and Resistance Factor Design Nested Reliability Analysis

Rackwitz-Fiessler Algorithm Reliability Based Cost Optimization

Structural Reliability Time Dependent Reliability Analysis


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TABLE OF CONTENTS

ACKNOWLEDGEMENTS ........................................................................................... iii

ABSTRACT....................................................................................................................v

LIST OF FIGURES......................................................................................................xiv

LIST OF TABLES .......................................................................................................xvi

NOMENCLATURE....................................................................................................xxii

1.0 INTRODUCTION................................................................................................. 1

1.1 Background ................................................................................................. 1

1.2 Problem Statement ...................................................................................... 6

1.3 Objectives....................................................................................................7

1.4 Literature Review...................................................................................... 10

1.5 Methodology ............................................................................................. 15

1.6 Recommended Load Factors Table........................................................... 19

2.0 STRUCTURAL RELIABILITY THEORY AND PRACTICE.......................... 22

2.1 Structural Reliability ................................................................................. 22

2.2 Structural Codification .............................................................................. 23

2.3 Code Calibration ....................................................................................... 26


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2.4 Simulation Methods .................................................................................. 33

2.5 Load Combination..................................................................................... 34

2.5.1 Turkstras Rule.............................................................................. 35

2.5.2 Ferry-Borges Process .................................................................... 37

2.5.3 Wens Load Coincidence Method................................................. 41

2.6 Modifications to Ferry-Borges Model....................................................... 42

2.6.1 Mixed Type Distributions ............................................................. 42

2.6.2 Separation of Time Variant and Invariant Random

Variables........................................................................................ 44

2.6.3 Evaluation of Pf by Numerical Integration or Nested

Reliability Analysis ....................................................................... 46

2.6.4 Logarithmic Approximation to Yearly Safety Indices .................. 52

2.7 Sensitivity Analysis................................................................................... 54

2.8 System Reliability ..................................................................................... 55

3.0 STRUCTURAL OPTIMIZATION ..................................................................... 58

3.1 General Optimization Formulation ........................................................... 58

3.2 Solving Nonlinear Programming Problem................................................ 59

3.3 Sequential Quadratic Programming .......................................................... 60

3.4 Optimization Using MATLAB ................................................................. 61


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4.0 RELIABILITY BASED STRUCTURAL OPTIMIZATION AND

APPLICATION IN CODE CALIBRATION ..................................................... 63

4.1 Formulation of the Expected Total Cost ................................................... 65

4.1.1 Initial Cost ..................................................................................... 66

4.1.2 Failure Cost ................................................................................... 68

4.1.3 Normalization of the Terms in Total Cost Function ..................... 70

4.2 Optimization Problem ............................................................................... 71

4.3 Solution Procedure .................................................................................... 72

4.4 Numerical Example................................................................................... 75

4.4.1 Separation of Modeling Uncertainty for Live Load...................... 77

4.4.2 Separation of Modeling Uncertainty for Wind Load .................... 79

4.4.3 Statistical Data after Separation of Random Variables ................. 80

4.4.4 Load Combination Cases .............................................................. 80

4.4.5 Calibration Process........................................................................ 81

4.4.6 Recommended Load Factors ......................................................... 96

4.4.7 Discussion of Results .................................................................... 97

5.0 APPLICATION OF CODE CALIBRATION BY USING

RELIABILITY BASED COST OPTIMIZATION TO AASHTO

BRIDGE SPECIFICATIONS ............................................................................. 99


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5.1 AASHTO Bridge Specifications ............................................................... 99

5.2 Design Format and Design Checks for AASHTO LRFD....................... 102

5.3 Load and Resistance Data ....................................................................... 105

5.3.1 Resistance Data ........................................................................... 106

5.3.2 Dead Load Data........................................................................... 107

5.3.3 Live Load Model ......................................................................... 108

5.3.4 Wind Load Model ....................................................................... 113

5.3.5 Earthquake Load Model .............................................................. 117

5.4 Cost Data ................................................................................................. 123

5.5 Probabilistic Total Cost Model ............................................................... 124

5.5.1 Strength III: Combination of Dead and Wind Load.................... 129

5.5.2 Strength V: Combination of Dead, Live and Wind Load ........... 137

5.5.3 Extreme Event I: Combination of Dead, Live and

Earthquake Load ......................................................................... 146

5.6 Optimum Load Factors Table and Discussion of Results ....................... 160

6.0 SENSITIVITY ANALYSIS.............................................................................. 162

6.1 Effect of Initial Cost Slope Change, CIonKG ...................................... 162

6.2 Effect of Gravitational Load Cost Factor, KG on Deduced

Failure Cost Factor,g.............................................................................. 163


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6.3 Effect of Design Life, Ton Deduced Failure Cost Factor,g.................. 164

6.4 Effect of Real Interest Rate,jon Deduced Failure Cost Factor,

g............................................................................................................... 165

6.5 Effect of Gravitational Load Cost Factor, KG on Optimum

Load Factors............................................................................................ 166

6.6 Effect of Design Life, Ton Optimum Load Factors ............................... 167

6.7 Effect of Real Interest Rate, jon Optimized Load Factors and

Safety Index............................................................................................. 168

6.8 Effect of Failure Cost Factor,gon Optimized Load Factors and

Safety Index............................................................................................. 169

6.9 Effect of Separation of Time Variant and Invariant Parts on

Probability of Failure Evaluation............................................................ 170

6.10 Effect of Separation of Time Variant and Invariant Parts on

Optimum Load Factors............................................................................ 172

6.11 Effect of COVs on Optimum Load Factors ........................................... 172

7.0 CONCLUSIONS AND FUTURE RESEARCH

RECOMMENDATIONS .................................................................................. 176

7.1 Conclusions............................................................................................. 176

7.2 Future Research....................................................................................... 178


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APPENDIX ................................................................................................................. 180

BIBLIOGRAPHY ....................................................................................................... 191


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LIST OF FIGURES

FigureNo Page No

1 Algorithm for current calibration practice of a structural design code

(Melchers (1999)).................................................................................................. 28

2 The load and resistance factors for a predefined target safety level

(Ellingwood et al. (1980)).....................................................................................29

3 The safety index vs. cost....................................................................................33

4 Ferry-Borges Model ............................................................................................. 40

5 Mixed Type Distribution...................................................................................... 42

6 Approximation to safety index,.........................................................................53

7 Parallel and Series Systems.................................................................................. 57

8 Component Safety index, vs. sum of total cost factors, TCF.........................87

9 Component Safety Index vs. ICF, FCFand TCF.........................................88

10 Kinzua Bridge, McKean County, PA (Built in 1900)........................................ 100

11 HL-93 truck and lane loading (AASHTO LRFD(1994))................................... 112

12 Modeling uncertainties,XWfor wind load.......................................................... 115

13 Extremal Type II fit to single earthquake event ................................................. 122


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14 Effect of variable separation on probability of failure calculations .................... 171


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LIST OF TABLES

Table No: Page No

1 Load factor table for ASCE 7-95 (1996).................................................................8

2 Current Code Load Factors for AASHTO ............................................................ 19

3 Recommended Load Factors ................................................................................. 20

4 The statistical data for sample analysis ................................................................. 48

5 Comparison of number of discrete points used (range={-4 to

+4})...................................................................................................................49

6 Comparison of number of discrete points used (range={-3 to

+3})...................................................................................................................49

7 Cpu-time for NRA and Numerical Integration ..................................................... 52

8 Logarithmic Approximations to safety index, beta along the lifetime.................. 54

9 Lifetime Statistical Data (T=75 years) .................................................................. 75

10 Statistical data after time variant and invariant distinction................................... 80

11 Data for calculation of gravitational load cost factor,KG ....................................83

12 KG values...............................................................................................................83


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13 Deducing an averagegvalue using Equation (4-29.a)..........................................85

14 Deducing an averagegvalue using Equation (4-29.b) ......................................... 85

15 Optimum componentand W(g=58) for Wind Load Alone Case......................91

16 Optimum systemand W(g=3528) for Wind Alone Case..................................91

17 Current code safety levels and costs for D+W case .............................................. 93

18 Optimum component,Dand W(g=58) for Dead +Wind Load Case ...............93

19 Optimum system, Dand W(g=3528) for Dead +Wind Load Case..................94

20 Optimum componentand W(D=1.25,g=58) for Dead +Wind Load

Case.......................................................................................................................95

21 Optimum system and W (D=1.25, g=3528) for Dead +Wind Load

Case.......................................................................................................................96

22 Recommended Load Factors ................................................................................. 97

23 Resistance Statistical Parameters (Nowak 1993)................................................ 107

24 Dead Load Statistic Data (Nowak (1993)).......................................................... 108

25 The statistical data at time T for one truck loading............................................. 110

26 Live Load Model Statistical Data........................................................................ 112

27 Wind speed statistical data (Ellingwood et al.(1980)) ........................................116

28 Wind load statistical data .................................................................................... 117


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29 Distribution for a single earthquake event .......................................................... 121

30 Earthquake Load Statistical Data ........................................................................ 123

31 Calculated KG values.......................................................................................... 127

32 Deduced failure cost factor,gvalues (component safety) .................................. 128

33 Deduced failure cost factor,gvalues (system safety)......................................... 128

34 Statistical data for D+W case .............................................................................. 132

35 Current Code safety levels and total cost factors for D+W................................. 132

36 Optimized component safety index, compand load factors, D and W

for D+W forg=30. .............................................................................................. 133

37 Optimized component safety index, comp and load factors, D=1.25

and Wfor D+W forg=30.................................................................................... 134

38 Optimized system safety index, sys and load factors, D and W for

D+W forg=1727................................................................................................. 135

39 Optimized system safety index,sysand wind load factor, Wfor D+W

for D=1.25 andg=1727. ..................................................................................... 136

40 Statistical data for D+W+L case ......................................................................... 138

41 The safety index and cost values for D+W+L case using current code

load factors .......................................................................................................... 141


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42 Optimized component safety index, compand load factors, W, and L

for D+W+L (D=1.25 g=30)................................................................................ 142

43 Optimized component safety index,compand wind load factor, Wfor

D+W+L (D=1.25, L=1.35g=30). ...................................................................... 143

44 Optimized system safety index, sys and load factors, W, and L for

D+W+L (D=1.25g=1727). ................................................................................ 144

45 Optimized system safety index, sys and wind load factor, W for

D+W+L (D=1.25, L=1.35g=1727). .................................................................. 145

46 Statistical data for D+L+E case........................................................................... 147

47 Current code safety index and cost factors for D+E+L caseg=1727 .................150

48 Optimized system safety index, sys and load factors, E and L for

D+E+L (D=1.25,g=30)...................................................................................... 151

49 Optimized system safety index, sys and load factors, E and L for

D+E+L (D=1.25,g=1727).................................................................................. 153

50 Current code safety index and cost factors for D+E+L case g=1727

(with updated range)............................................................................................ 155

51 Optimized component safety index,sysand load factor, Efor D+E+L

(D=1.25, L=0.50,g=30 with updated range). ................................................... 156


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52 Optimized system safety index, sys and load factor, E for D+E+L

(D=1.25, L=0.50,g=1727 with updated range). ............................................... 158

53 Optimum Load Factors........................................................................................ 161

54 KGvalues for changing CI................................................................................ 163

55 Deducedgvalues for differentKGvalues ........................................................... 164

56 Failure cost factor,gfor different design life, T values...................................... 164

57 The effect of real interest rate, j on failure cost factor, g (D+W

combination)........................................................................................................ 165

58 Effect ofKGon optimum load factors and optimumranges for D+W

case...................................................................................................................... 166

59 Effect of T on optimum load factors and optimum ranges (D+W

combination)........................................................................................................ 167

60 Optimized safety index, and load factors, D and W for different j

values (D+W combination)................................................................................. 168

61 Optimized safety index and load factors for different g values

(D+W+L combination)........................................................................................ 170

62 Optimized safety index and load factors for differentgvalues (D+E+L

combination)........................................................................................................ 170

63 Statistical data for separation example................................................................ 171


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64 Comparison of separation of time variant and invariant random

variables .............................................................................................................. 172

65 Resistance,RCOV effect on optimum (D+W combination).............................. 173

66 Dead Load,DCOV effect on optimum (D+W combination)............................. 174

67 Wind Speed, VCOV effect on optimum (D+W combination) ........................... 175

68 Wind Load Modeling, XW COV effect on optimum (D+W

combination)........................................................................................................ 175


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NOMENCLATURE

Symbol Explanation

A acceleration coefficient

Cf cost of failure

CI initial cost

COV coefficient of variation

Cp pressure coefficient

Csm elastic response coefficient

CT total Cost

D dead load random variable

Dn nominal dead load

E earthquake random variable

En nominal earthquake load

Ez exposure coefficient

f number of failures in MC Simulation

FCF failure cost factor

g deduced failure cost ratio (CF/C0)


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g limit state function

G gust factor

G limit state function

H Hessian matrix

h limit state function

ICF initial cost factor

ID dynamic effect of live load on bridge

IM dynamic amplification factor

j discount rate

K marginal cost slope

L live load random variable

Ln nominal live load

Pf probability of failure

Pfi probability of failure in ithyear

pi probability of load iexceeding the limit state

Ps probability of survival

QE time dependent earthquake load effect

QL time dependent live load effect


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QW time dependent wind load effect

R resistance random variable

Rm response modification factor

Rn nominal resistance

S Load effect

S site coefficient

TCF total cost factor

Tm mthvibration mode period

V wind velocity

Vn nominal wind velocity

W weight of the structure

W wind load random variable

Wn nominal wind load

XE earthquake load modeling uncertainty

XE earthquake load modeling uncertainty

Xi random variables

xi* design values for random variables

XL live load modeling uncertainty


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XW wind load modeling uncertainty

Z(t) total load effect at time t

CI change in initial cost

L change in live load factor

nominal load to nominal dead load ratio

safety index

resistance factor

load factor

i0 reference load factors

marginal cost slope ratio

I rate of occurrence for load i

j Lagrange multipliers

mean of a random variable

rate of occurrence of events per unit time

standard deviation of a random variable


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1.0 INTRODUCTION

1.1 Background

Engineered structures have to be designed and used according to specified

requirements. The collection of these requirements is referred herein as the codes or

specifications. The first written document on building regulations is the Code of

Hammurabi and it is dated back to about 2000 B.C. (Nowak and Lind (1995)*

, Ersoy

(1991)). Although it is far from defining any technical requirements, this Code strictly

emphasizes safety by providing severe penalties such as the death penalty for any

individual responsible for a structural failure causing a fatality. This clearly indicates that

safety was an important issue even in ancient civilized societies.

Structures have been built since civilization was first established. There are many

structures, which were built centuries ago and still stand. Experience has been a key

parameter in the course of structural design development. Every failure has been a lesson

for the constructor. Deterministic experience was the most acceptable safety decision

criteria. The use of probabilistic methods in structural mechanics was only introduced

into design practice in the late 1960's. Since then it is also important to introduce past

experience by scientific approaches. The developments in structural analysis techniques,

statistical data collections, reliability theory, faster computing abilities etc. now make it

possible to generate more robust structural design codes.

*Parenthetical references refer to the bibliography.


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Pre-1970 codes were based on experience rather than on probabilistic background;

both the lack of data and the theoretical basis for reliability analysis are the main reasons

for not earlier achieving more sophisticated design codes. Applicability of reliability

theory to structural design began with researchers including Freudenthal (Freudenthal

(1956), Freudenthal et al. (1966)), Cornell (1969), Turkstra (1970), Lind (1970), Moses

(Moses and Kinser (1967), Moses and Stevenson (1970)) etc. The evolution of the design

codes comes to its current stage by use of the Reliability Theory. The loads, resistances,

mathematical models, and analysis methods are all treated as random variables. In order

to express these variables in the design process it is important to consider this

randomness. In reliability analysis, the statistical data used for describing loads and

resistances must also contain the variability due to modeling and analysis in addition to

the randomness of the physical event. The effects of the physical events are modeled

mathematically, but a mathematical model may not replicate the real effects perfectly.

Modeling is an approximation to the real behavior of a phenomenon, such as earthquake

or wind effects on structures; using such approximate models introduces uncertainty into

the system. Also, analysis techniques applied to predict behavior of a physical event or

structures do not replicate the exact behavior with the exception of few cases. Analysis

may produce conservative results relative to what is observed, or the results of the

analysis may also be unconservative compared to observations. In any case, analysis

techniques introduce uncertainties into the design picture.

It is not enough to use the probabilistic approach alone to calibrate a new code;

deciding on the safety levels that a new code should adopt is a challenging step in the


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calibration process. It is common practice to reach a decision based on experience and

intuition. Although, experience and intuition are always very important, rationalization of

the decision process where possible is worth pursuing. For example, there is an

inconsistency in current codes; different load combinations have different level of safety.

Increasing safety level would introduce higher costs; therefore, lower safety levels are

adapted to keep cost feasible, such as in case of load combinations involving wind or

earthquake loads. It is one of the objectives of this research to overcome the above-

mentioned inconsistency.

Design codes are categorized into four different levels (Melchers (1996)). A Level I

code is in a deterministic format and load factors (in US) and partial coefficients (in

Europe) are used to design the structures. Current codes such as AISC-LRFD (1986),

API-LRFD (1989), AASHTO-LRFD(1994) etc. are in the Level I format and designers

do not involve any probabilistic calculations in use of design codes in this format. Partialfactors as in Europe or load and resistance factors (LFRD) as in US are used in design

checks. The safety is defined with a safety index, , which is simply for a normal

distribution, a distance of mean reserve capacity to zero (failure) in terms of the number

of standard deviations. A typical LRFD check is as in Equation (2-5). Level II format

deals with the probabilistic nature of random variables. Distribution types with mean and

standard deviation are directly used to represent the random variables and reliability is

estimated by an approximate solution such as First Order Reliability Method (FORM)

which approximates the probability of failure using first order terms of Taylor Series

expansion of the limit state function (See Appendix). A conversion to normal distribution


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is performed at the most probable failure points during the evaluation of the probability

of failure. A Level III format involves a full probabilistic approach; probability of failure

is calculated using a convolution integral. Level IV codes involve all available tools

mentioned above as well as economical data; a minimum total cost or maximum utility is

used as the code objective. It is proposed herein to use Level IV format to calibrate a

Level I (LRFD) code. Total cost is minimized to find optimum safety level and load

factors.

Throughout the text, structural design codes will be referred to as design tools to

proportion the structural members. These codes prescribe the minimum strength

requirements to assure a reasonable safety level as defined by the code committee. In

code calibration practice it is common to use the current codes safety levels as

predefined safety levels for calibration process. Safety index, is calculated along the

design space defined by design points, which are represented by different values of

nominal environmental load to nominal dead load ratio. For example using different

values of Ln/Dn ratios can represent bridges with different span lengths. Ln/Dn ratio is

smaller for larger span bridges where dead load governs, and it is higher for shorter span

bridges. The safety indices calculated for discrete design points using the present code are

averaged and become the target safety level, tfor calibration of a new code. Herein an

alternate approach to replace the current practice is studied i.e., using reliability based

cost optimization to calibrate a new code. The sum of expected total costs (initial plus

expected failure cost) is minimized. Since it is hard to quantify the failure cost, the

simplest load combination case is used as a reference and the failure cost is deduced from


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that reference combination case. The reason for selecting the basic load combination case

is that it would be the most agreed load combination. Using the total cost model

presented herein, the failure cost that basic combination case imposes can than be the

failure cost for all load combination cases covered in the calibration. Doing so means that

even if the safety indices for different combinations are not uniform, the costs of

consequences for these combinations are identical. This helps to implement the trade-off

between the safety and cost explicitly and rationally.

This dissertation contains the following chapters:

Reliability Theory and it is applications are described in second chapter. All the

necessary tools to use in the approach used herein are explained.

In Chapter 3, the optimization techniques are investigated. The software

MATLAB that is also used for the optimization is also introduced in this chapter.

The Reliability Based Cost Optimization with its development in last three

decades is presented in the Chapter 4. The proposed model for code calibration is also

given in detail in this chapter.

In Chapter 5, application of the method to the AASHTO Bridge Specification is

described. The statistical data on random variables are provided in this chapter.

In Chapter 6, the sensitivity analyses are described for the proposed method is

explained. The outputs are investigated for changes in applied data, and results are

discussed.


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In last chapter discussions on the method and recommendations for future

research are presented.

In the Appendix, the basic reliability problem and the Rackwitz-Fiessler

Algorithm is presented.

1.2 Problem Statement

Structural design specifications impose minimum safety requirements; therefore the

design code sets the standard for the engineering practice. Since the early days of the

application of reliability theory to codified design, the question of how safe is safe

enough is asked both by researchers and engineers. The question is still unanswered and

both researchers and practicing engineers are still working on reaching a sound solution,

but the answer does not seem so straightforward. There is always a risk of failure with

any structure; in other words the probability of failure of a structure is always greater

than zero. So, absolute reliability can never be attained. Since the probability of failure

can be expressed quantitatively, this allows bringing the economical considerations into

the picture. Increasing the safety level is costly, so there must be a balance between the

safety and the cost.

Forssell (1924) stated that a design should be made to maximize the utility to the

owner, including the expected losses (loss of structure, tangible losses, etc.). The

expected cost of the losses can be expressed by multiplying the probability of failure,Pf

with the cost of the losses. Because the loss has not occurred yet and in order to quantify


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the probable failure cost of losses, the expected value has to be calculated. Forsells

statement has played a very important role in optimization of the structural design codes.

Consideration of the expected losses in the optimization process would allow reaching a

balance between the safety and economy. The expected losses include property damage,

function losses, and personal injuries and life losses. Intangible losses such as personal

injuries and life losses are very hard to determine. Assigning monetary values for a

human life is not easy, and creates many debates. Since intangible losses also have to be

included in order to reach a reasonable solution the structural optimization is rarely

achieved explicitly in current applications.

In order to reach more sound design codes it is important to explicitly express the

code optimization through the total cost optimization. In this study the ways to find a

reasonable approach to that problem is investigated. In some examples of current design

practice the structures are optimized on a project-by-project basis. Generally thisoptimization cannot violate the minimum requirements dictated by the code. Therefore if

a code is not optimized, such structural optimization does not give the real optimum

solution.

1.3 Objectives

In this research it is aimed to establish a procedure to calibrate structural design

codes. A balance between safety and economy is sought using Reliability Based Cost

Optimization.


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Table 1Load factor table for ASCE 7-95 (1996)

Dead Live Snow Wind EQ

1 Dead 1.60 - - - -

2 Dead+Live 1.20 1.60 - - -

3 a. Dead+Snow+Live 1.20 0.50 1.60 - -

b. Dead+Snow+Wind 1.20 - 1.60 0.80 -

4 Dead+Live+Wind 1.20 0.50 - 1.30 -

5 a. Dead+EQ+Live 1.20 0.50 - - 1.50

b. Dead+EQ+Snow 1.20 - 0.20 - 1.50

6 a. Dead-Wind 0.90 - - 1.30 -

b. Dead-EQ 0.90 - - - 1.50

Load Combination

A sample of a load factor table can be seen in Table 1. The load combinations and

corresponding load factors are given in a code along with the code specified nominal

design loads. In the calibration the load factors such as in Table 1, are calibrated with the

updated statistical data to achieve a reliability criteria.

Current applications of the code calibration provided pre-assigned target safety

indices. These safety levels are mostly deduced from the previous versions of the codes

by selecting some sample designs, designing those samples according to current code,

and finding the average safety index for those sample designs. The target safety levels are

different for loads such as gravity, wind, earthquake, etc. For example in ANSI A.58

(Ellingwood et al. (1980)) the target safety index for combinations of dead load and snow

load is 3.0. On the other hand, the target safety index drops to 2.5 and 1.75, when wind

load and earthquake loads are involved, respectively. It is important to decide on the

target safety levels; a consistency should be sought in the decision process. The common


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trend in calibration process is to reach uniform target safety levels for the combinations

(Ditlevsen (1997)).

It has also to bear in mind that different loads may affect the structure differently;

for instance, the effect of lateral load such as wind and earthquake might have a higher

relative impact on the cost than the gravitational loads. Also, the loads have different

statistical data (such as COV), which affects cost of failure differently because of the

influence on probability of failure. It might be highly costly to strengthen a structure to a

high reliability target for earthquake then to gravitational loads. In this study the target

safety levels are not predefined, instead the target safety levels are attained implicitly,

using the balance between cost and safety.

In code calibrations, for the load combinations involving time dependent load

effects, probabilistic load combination techniques such as Turkstras Rule, Ferry-Borges

Method and Wens Load Coincidence Method are used; Ferry-Borges Method is selected

to perform load combination analysis for this research. This method relies on the

independence among the repetitions of load events. This is not precise because of

modeling uncertainties involved. A better representation of time dependent load effect is

used herein by separating modeling uncertainties that are constant throughout lifetime

and time dependent events such as environmental events (wind speed, earthquake

acceleration, etc.) or vehicular weight load.

Available data on statistics of resistance and load events along with cost data such

as cost slopes for different type of loads were gathered to use in the proposed calibration

procedure.


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It is common practice to perform calibration considering only reliability of a

member rather than a system. It would also be important to investigate the system

reliability because the structures are a system of the elements and the reliability of a

structure is the reliability of a system. In this research reliability analysis are mostly

based on element safety, and the effect of system safety has not been studied extensively.

In order to generalize the results of the calibration process, sensitivity analysis are

run and the effect of the variations in the assumed or deduced parameters are

investigated.

As a product of the calibration process, load factors that give the optimal code are

illustrated.

1.4 Literature Review

Structural design codes are widely used. Although every designer should look for

an optimized design, he or she is constrained with the minimum proportioning

requirements imposed by the current code. So a real optimum design highly depends on

an optimum code.

Reliability theory and the optimization techniques have to be used together in

order to optimize a code. Loads and resistance show variability in nature. Therefore

assuming the loads and resistance deterministic would not be a realistic approach.

Reliability theory helps to capture the probabilistic nature of the loads and resistance.


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Freudenthal's (1947) work can be assumed to be the first step to introduce the

reliability approach to structural design. His later work (Freudenthal (1956), Freudenthal

et al. (1966)) helped researchers to realize the structural reliability concept can be an

efficient way to deal with structural design and analysis. Cornell's (1969) work on the

probabilistic code approach for ACI gave a momentum to the applications of structural

reliability theory to structural design. But lack of data made it difficult to apply. Turkstra

(1970) defined the modern approach to the codification and used a Bayesian approach for

the decision analysis. Bayesian approach is very suitable for structural design because it

uses the available experience and data to improve the estimate of structural reliability in

the decision process. Bayesian approach can simply be identified as systematically

combining judgment, experience and indirect data with the observed data (Ang and Tang

(1975)). Bayesian Approach is one of the vital issues that make a continuous

improvement in structural design codes.

The idea of using optimization along with the reliability analysis has been an

important step to reach balanced designs. Forsell (1924) long before the structural

reliability was in the picture formulated the design process as a minimization of total cost

that covers construction, maintenance, and expected failure cost. The work of Moses and

his colleagues (Moses and Kinser (1967), Moses (1969), Moses and Stevenson (1970))

showed how to integrate the optimization with reliability analysis. Rosenbluth and

Mendoza (1971) formulated the optimum of structural design using minimization of the

expected total cost. Ravindra and Lind (1972) introduced the optimization of code

concept as, minimization of the sum of expected total costs of all structures designed


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using a structural code. Researchers started to accept the notion of probabilistic based

design and in 1970s the researchers were focused on creating the basis for a probabilistic

and more rationale structural design codes.

American Concrete Institute (1977) is the first organization that adopted the

probability based code concept in USA, but it is the ANSI A.58 Building Code

Requirements for Minimum Design Loads in Buildings and Other Structures (1980) that

set an example for calibrating the design codes. Ellingwood et al. (1980) predefined the

target safety levels using the current code then, and determined the load factors required

for that target safety level, and then obtained the resistance factors. The LRFD studies

involving code calibration followed these same footsteps that Ellingwood and his

colleagues developed.

The number of probability based codes following LRFD philosophy start to

increase. In 1986 American Institute for Steel Construction migrated to a LFRD based

design code. The American Petroleum Institute, API code in 1989 then followed AISC.

Next, American Association of State Highways Officials, AASHTO (1994) migrated to a

load and resistance factor based specification for design of highway bridges.

Developments in the First Order Second Moment (FOSM) reliability methods, and

collection of statistical data helped to provide more rational codes.

Although the theoretical basis of code calibration matured during the last three

decades, the selection of target safety index is still a subject of many debates. The

decision process to decide about the target safety index is controversial and still more

political than it is scientific. Another problem arises since the load combinations do not


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show a uniform safety level, which brings the question; are the structures less safe for

loads such as wind or earthquake, because it is the current practice to have lower target

safety indices for the combination of loads such as wind and earthquake. The reason

having lower target safety levels is due to the fact that there is a trade off between safety

and cost. This is an implication of putting cost and reliability issues together in the

decision process. But unfortunately, this is only done qualitatively. Intuition is the main

factor in decision process for current practice.

Ellingwood (1994) in his paper summarized the past accomplishments and the

future challenges on codification. He pointed out that target reliability selection is

difficult and has always been the soft side of the probability-based codes. According to

Ellingwood (1994) decisions based on the intuition and judgment fails when the rare

events like earthquake and winds are the basis of the design. He points, Better estimates

of limit state probability are also required in support of life cycle cost analyses.

Ditlevsen (1997) also stressed that the decision on the target reliability levels has

to be based on decision theoretical principles such as expected total cost optimization. He

points that the formal codes safety levels can be used to back calculate the cost of

failure. Ditlevsen (1997) states that Since the costs of consequences of structural

damage or collapse embrace the intangible socio-economic costs such as injury or loss of

human lives and possibly irremediable damages on the environment, there seem to be no

other way than to establish some fix-points in the most frequent existing practice and

calibrate the reliability level to these fix-points. He also proposes to migrate from

partial safety factor code to a full probabilistic code to obtain a code with uniform safety.


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Rackwitz (2000) gave the basis for the code making process. In his model he also

used the total cost minimization as the decision tool to determine the target safety for the

calibration process where the maintenance and reconstruction costs are also considered.

The renewal processes are used to represent the failure in time, which assumes

independent and identical distribution of load processes, and each reconstruction assumes

a new structure. In load combination analysis, time variant loads are represented by a

Poisson process and outcrossing rates are used to define the failure probability for these

loads.

Researchers including Kanda and Ellingwood (1991), Kanda (1996), Wen and

Kang (1997), Ang and De Leon (1997), Rackwitz (2000), proposed to establish the cost

of failure through the current practice by assigning direct monetary values for the failure.

This approach brings the problem of the assigning values for the intangible cost which is

a political issue and also, it is difficult to assign a value for a life loss or injuries. Theappraisal for human life is mostly based on the work by Viscusi (1993). In his work

Viscusi estimates the value of human life using the labor market data. The outcome of

that approach would be so sensitive to the type of structure, location, type of failure and

to injury and fatality costs, etc. Therefore reaching to a generalized solution would be

difficult and open to many debates. It is believed, it would be more rational to use the

current code as a fix point and back calculate the failure cost using the risk that society is

currently willing to accept.

It is proposed in this study to deduce an implied cost of failure through the most

basic and most agreed load combination from the previous code; such as dead plus live


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load combination in the AASHTO Bridge Specifications (Aktas et al. (2000, 2001a,

2001b)). There is a consensus on the safety level of this load combination case, so the

risk amount a society is willing to pay to avoid failure can be deduced. As mentioned

earlier considering the trade-off between safety and cost, it is not realistic to assume a

uniform safety level for all the combination cases. When it is not costly to increase

safety, the safety level may be increased as in present criteria wherein a connection has a

higher safety index than other elements (see AISC). Assuming the failure represents the

complete failure, the cost of failure should be the same for all the combinations. The

expected losses do not change from one to another load combination (such as

gravitational loads, earthquake, wind, etc.), i.e. consequences of system failure should be

same. Basically the proposed method is based on that assumption and used herein for

calibrating a reliability-based consistent structural design code.

1.5 Methodology

An optimized code means that the recommended load and resistance factors give

the optimum solution for the total of the expected designs described in a design space.

The total costs for every design has been summed and the optimization is processed for

the sum of the total costs of the designs. Each individual design may not in themselves be

the optimum design. Therefore, the total cost over the applicability of the design code in

the whole design space is optimized instead of individual designs.


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The total cost of the design consists of the initial cost, CIthat is a linear function

of the load and resistance factors, and expected failure cost, CF that is a product of the

cost of failure, Cfand probability of failure,Pf. The expected total cost function is given

in Equation (1-1).

ffIT PCCC += (1-1)

Cost of failure is the cost incurred due to a failure, such as property damage, loss

of function, and fatality. The failure can occur anytime during the lifetime of a structure;

hence the probability of failure is calculated annually. In order to make a comparison all

costs have to be on the same basis, such as present worth. Therefore, assuming a constant

cost of failure throughout the design life of the structure, the failure costs are calculated

annually and discounted to present worth value at the time of construction. In order to

simplify the problem a constant nominal real interest rate that is the rate adjusted for

inflation is introduced for the design life. The total cost now has a linear part and a non-

linear part of the probability of failure calculation through FORM iteratively. Non-linear

programming is used to find the optimum load factors.

The calculation of the annual probability of failure involves finding the probability

of failure of the structure in nyears, where nis from 1to lifetime, and the differences of

the two consecutive years gives the probability of failure of a structure specifically in that

year.

Loads including live, wind, earthquake, etc. show variation in both in time and

space. The possibility of all the load effects to affect the structure at their expected


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combinations would also be attained, because the cost of failure used would be same for

all the load combinations.

AASHTO bridge design specifications are used to illustrate the proposed

approach. The AASHTO LRFD specification consists of load combinations including

dead, live, wind and earthquake. The first load combination involves gravitational loads;

dead and live load. Live load is the effect of heavy vehicles crossing the span. The work

on the calibration of the AASHTO in 1993 was mostly based on this combination case.

The lack of the available data made it not possible to do an extensive calibration for other

load combinations. The calibrations of the AASHTO specification for extreme loads (e.g.

earthquake, wind, vessel collision, etc.) are now studied under the NCHRP-12-48 project.

The target safety index used for the dead and live load combination case was reported to

be 3.5 (Nowak (1993)). This value is used herein to back calculate the cost of failure by

assuming that the optimum solution to that specific load combination is at the safety levelof 3.5. This assumption is based on the Linds postulate that the current code is already

close to the optimum (Lind (1977)). The deduced cost of failure will hold for all the load

combination cases. The optimized load factors are found for the other load combination

cases and the load factors are developed herein.

One final step is to check for the sensitivity of the optimized load factors to the

random variable data and other assumed input data such as relative cost value and the

discount rate. The analyses are rerun for the changing values and the behavior of the

optimized load factors is investigated.


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1.6 Recommended Load Factors Table

In this study an example for the proposed calibration approach is presented. The

AASHTO Specification has current code load factors given in Table 2. As a result of the

calibration load factors tabulated in Table 3 are recommended.

The dead load factor for all the combinations studied are kept at the current code

specified value of 1.25. In the load combination case of dead and wind load effects, the

wind load factor has been increased from 1.40 to 1.60, because the safety level obtained

using the current code values gives a high expected failure probability and expected total

cost. The recommended load factors are concluded using the system safety (system safety

index is equal to component safety index plus one) and assuming wind load marginal cost

slope is three times the marginal gravitational load cost slope. The current code gives an

expected total cost of 4.4, where the recommended load factors reduces that to 4.2, which

corresponds to a 5% reduction in the sum of expected total cost for the design space.

Table 2 Current Code Load Factors for AASHTO

Load Combination D L W

D+L 1.25 1.75 --- ---

D+W 1.25 --- 1.40 ---

D+L+W 1.25 1.35 0.40 ---D+L+E 1.25 0.50 --- 1.00

Load Factors


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Table 3 Recommended Load Factors

Load Combination D L W D+L 1.25 1.75 --- ---

D+W 1.25 --- 1.60 ---

D+L+W 1.25 1.35 1.10 ---

D+L+E 1.25 0.50 --- 1.45

Load Factors

In dead, live and wind load combination case the load factors required to keep a

balance between cost and safety are higher than the current code specified values (see

Table 2). The current code load factor value of 0.40 for wind load is due to the

assumption that there would be no truck present on the bridge when the wind speed is

higher than 55 mph. Since Ferry-Borges Method does not allow the change in the limit

state function in time, this constraint is introduced by having lower values for Wn/Dn

range. Doing so decreases the domination of the wind load in the reliability analysis,

therefore the probability of wind speed star values to go beyond the 55 mph would be

relatively low. Considering system safety (component safety index plus one) it is

recommended to increase the wind load factor to 1.10 in order to have minimum

expected total cost. The sum of expected total costs using current code factors for

combination of dead, wind and live loads is about 29.2. The recommended load factors

for this case lower the sum of expected costs to less than one-half of the current code cost

to 13.0.

In the last combination case, which is the combination of the dead, live and

earthquake load, the safety index calculated is the system safety index, because the


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earthquake model used in this study already accounts for the system effects. The load

factors for dead and live loads are kept at current code values of 1.25 and 0.50,

respectively. Since the earthquakes marginal cost is the dominating factor the

optimization is run for the earthquake load factor. The load factor for the earthquake load

is found to be 1.44 and 1.45 is recommended as the calibration result. The recommended

earthquake load factor of 1.45 lowers the sum of expected total cost 30%, from the

current codes cost of 27.0 to 20.0.


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2.0 STRUCTURAL RELIABILITY THEORY AND PRACTICE

2.1 Structural Reliability

Structural reliability is based on the rational quantification of a structures chance

to satisfy the performance requirements. In the design process of structures, which is

mainly proportioning of the elements, the strength of material used and the loads they

will be subjected are not known exactly. These input show variation in space and time

and can be best represented by probabilistic and statistical approaches.

Freudenthal (1947) introduced the reliability concept to structural engineering in

the United States. The structural reliability theory had shown a slow improvement until

mid 1960s, but since then structural reliability and its applications draw considerable

amount of attention from researchers. Especially following the work for the American

National Standard on Minimum Design Loads (Ellingwood et al. 1980), it has been more

accepted by the engineering practice.

Structural reliability is basically the probability of demand not to be greater than

capacity. Demand is the load effects on the structure and capacity is the resistance or

strength of the structure. Structural reliability can be represented by either the probability

of survival, Ps or the probability of failure, Pf. The relation between these two

probabilities is

sf PP = 1 (2-1)


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In structural reliability since Ps is so close to 1 it is preferable to use the probability of

failure to not lose the significance of the calculated probability values.

As mentioned earlier both demand (load effects) and capacity (strength) show

variation in time (wind, snow, earthquake loads, deterioration of structures over time

reduces the resistance), in location (earthquake prone regions are susceptible to higher

earthquake loads), use (live loads in office and residential buildings, or traffic loads on

bridges). Unfortunately to quantify the variability is not easy and most important this

variance is not unique. The load effects and the strength are both random in nature. For

example, it is a slim chance to test two elements produced, and obtain the same resistance

from these two. Therefore, the best solution to represent variations is using the statistical

theory.

Historically, the main decision tool for designers was performance experience.

One of the main reasons that structural reliability theory and applications show a slow

improvement until 1960s is the lack of sufficient data to represent the design variables

rationally. As the data from the observations of environmental effects and from the

laboratory tests of construction materials of common practice become more abundant, it

made it possible to represent variables reasonably.

2.2 Structural Codification

Although the approach behind the current structural codes is probabilistic, in

order to ease the application and obtain uniformity among the designers, codes are


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written in deterministic formats. The procedure is called Load and Resistance Factor

Design, LRFD in US, and Partial Safety Factor Design in Europe. Following the code

specifications allows designers to attain a sufficient level of safety. In a sense, structural

design codes may serve as a legal shield to a designer.

Research on structural design codes showed a considerable increase in 1970s

after establishing First Order Second Moment Theory, FOSM, which is also called First

Order Reliability Method, FORM and the gathering of statistical data on random

variables. FOSM is based on representation of the limit state function, G (Equation (A-

6)) in a linear form by using the first term of the Taylor Series Expansion (Equation (A-

8)). The safety is represented by an index, that is simply the number of standard

deviations from origin to mean of a standard normal distributed variable. So the FOSM or

FORM name comes from using the linear (first order) approximation of limit state

function and using the first and second derivatives of a normal distribution, mean and

standard deviation, to define the probability of failure (or safety). When the second

term of the Taylor Series expansion in Equation (A-8) is used to evaluate the limit state

function, the solution becomes more complex. When the second order terms are used, the

method is called Second Order Reliability Method (SORM). In most reported cases, the

accuracy obtained using FORM is sufficient for the Code calibration issues. Therefore

FORM is used in this study instead of SORM. This conclusion is supported by

comparisons with Monte Carlo Simulation.

The improvements in codes are still in progress and studies in US and Europe are

still looking for improved structural design codes. In order to represent the loads and


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resistances more realistically more data is required. As the time goes by, researchers will

access more accumulated data to represent the loads better. The resistance does reflect the

reality rather well since the low coefficient of variation, COV values, especially for

factory produced members, show that controlled production helps to reduce the

deviations from the nominal properties such as strength, geometry, etc. Most likely, since

the computing power is easy to access with improving technology, even designers may be

asked by design codes to use probabilistic approaches to verify the safety of the

individual designs. At present, some national codes such as the Dutch code allow using

probabilistic computations to check the designs (Hoogenboom and Kasbergen (1998)).

Codes are used through the safety-checking format, which are the mathematical

representation of a limit state. Limit state might be an ultimate limit state (ULS) or

serviceability limit state (SLS). Ultimate limit state represents the physical limit of an

element before failure. When an element is beyond that limit it is nonfunctional or lost. Incase of serviceability it is the level of functioning that is in question; floor excessive

vibration or deflection is an example of a serviceability limit state.

The current safety format used in US is called LRFD. American Concrete

Institute (ACI), American Petroleum Institute (API), American Institute of Steel

Construction (AISC), and American Association of State Highway and Transportation

Officials (AASHTO) now provide the LRFD format for their design specification. The

safety checking equation for LRFD proposed by Galambos and Ravindra (1978) was

im

n

i

in SR =

1

(2-2)


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where is the resistance factor, Rn is the nominal resistance, iare load factors and Sim

are the mean load effects. This design check equation is applied for different types of

load cases, for different members such as beams, columns, etc. and connections. The

design check equation allow a designer to determine minimum required design resistance

among the different load combinations, such as dead and live load effects, dead and wind

load effects, etc.

2.3 Code Calibration

In a code calibration process, the steps of Ellingwood et al. (1980) are mostly

followed. These steps are composed of selecting the design space; use of existing code to

design sample cases; defining the limit states; statistical data evaluation; use of reliability

methods (FOSM theory or simulation techniques); selection of target safety levels by

using current applications (if the current code gives satisfactory designs the safety index

obtained from the sample designs are averaged and used as the target safety level for

these designs) and finally deciding about the load and resistance factors by minimizing

the deviations from the target safety level for the range of designs (Melchers (1999)). The

current code calibration process can be summarized as the algorithm shown in Figure 1,

and a calibration example of the dead, wind and live load combination using current

calibration practice is given in Figure 2.

It is important to define what type of material and structures are covered by the

code. The codes cover a wide range of design space but in order to make a code more


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efficient limiting it to a reasonable range would be appropriate. If the deviations from the

target reliability index are high even after minimization it might be an indication that the

range shall be narrower.

Design experience is a vital input of code calibration. It is important to use

experience to improve the codes, because it is the application that can check the

satisfaction of a design code. Code calibration can be viewed as an ongoing process, the

new challenges, disasters, and observations can help to improve the current code. The

information gathered need to be reflected during calibration of a new code. Bayesian

Theory, basically updating the present data by available new information is the keystone

of the calibration.


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Figure 1Algorithm for current calibration practice of a structural design code(Melchers (1999))


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Figure 2 The load and resistance factors for a predefined target safety level(Ellingwood et al. (1980))


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A design code checks the safety through the limit state equations. Failure modes,

such as flexure in a beam or plate buckling, etc. have to be defined and necessary

analytical and empirical equations are used to rationalize the safety check. A LRFD code

contains the limit state check equations, nominal loads and resistance values and load and

resistance factors.

In current calibration practice, if present applications give successful results then

the safety index obtained with the present code can be prescribed as the target safety

levels for the new code. In this study instead of obtaining just safety levels, the failure

cost is also deduced from the current code. This value is used to express total cost, which

is then minimized along the design space defined by the possible ranges of the random

variables. It is a common practice to represent the nominal loads as a ratio with respect to

the dead load to define the design space. For instance in ANSI A58 (Ellingwood et al.

(1980)) the loads such as live, wind and snow loads are represented by a factor of dead

load ranging between 0.2 and 5. Assume that the design requirement is in the form of

nWnLnDn WLDR ++ (2-3)

whereRn,Dn,Lnand Wnare the nominal values for the resistance, dead, live, and wind

loads respectively and s are the corresponding load factors. Both sides of the Equation

(2-3) are divided by nominal dead loadDnand that gives

n

n

W

n

n

L

n

n

D

n

n

D

W

D

L

D

D

D

R

+

+

(2-4)

So the design equation is normalized by the nominal dead load. Therefore, instead of

dealing with actual nominal values the ratios of resistance and loads to dead load are


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used. It is useful to use the normalized form (Equation (2-4)) to generalize the calibration

process rather than working with a single case. So Equation (2-4) can be rewritten as

nWnLnDn WLDR ++ (2-5)

where the nominal value for the dead load,Dnis equal to unity and the nominal values for

the other loads and the resistance is a factor of 1.0. The same procedure for limit state

function; for the design criteria given in Equation (2-3), limit state function, g is

''''' LWDRg = (2-6)

where prime represent the non-normalized random variables. Dividing and multiplying

random variables with their nominal values and normalizing with dead load nominal

value,Dngives

=

n

n

nn

n

nn

n

nn

n

nn D

L

L

L

D

W

W

W

D

D

D

D

D

R

R

R

D

g

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

'

' (2-7)

nnnn LLWWDDRRg = (2-8)

where R, D, W and L are the normalized random variables. Since the original random

variables are normalized by their nominal values the mean for these random variables are

equal to the bias of these random variables. Also, nominal values Rn, Wn and Ln are

normalized with respect toDnvalues so they are equal to a factor ofDn.

The normalization eases the consideration of a range for the possible nominal

values. For instance in case of highway bridge specifications the ranges of nominal loads

such asL, Wetc. to the nominal dead loads represents the different spans for the highway


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bridges; a low Ln/Dnor Wn/Dn ratio represent that the bridge has a long span and dead

load dominates the design equation (2-8).

The approach followed herein requires the optimization of the total cost, which is

given in Equation (2-9)

{ } +=spaceDesignSpaceDesign

fFIT PCCC (2-9)

where, CT, CIand CFare total, initial and failure costs andPfis the failure probability for

a single realization of the design space. One has to realize when optimization of a code is

concerned, the optimization should cover the full design space; therefore, selecting load

and resistance factor for an optimum code may not give the optimum design for every

individual design problem covered in a code. Involvement of probability of failure makes

the minimization a Reliability Based Cost Optimization; because the failure associated

cost is assessed as the multiplication of the cost of consequences, CF due to a failure

times the probability of a failure to occur. The optimum solution is as shown in Figure 3.

The optimum solution is at the point where the slopes of the initial cost and cost of failure

are equal and opposite, and the corresponding safety index is called the optimum safety

level.


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CI

CT

CF

Cost

opt.

CT (min)

Figure 3 The safety index vs. cost

2.4 Simulation Methods

Simulation is a tool for especially complex reliability problems; however, it is

mostly preferred to confirm the results obtained out of FORM or SORM solutions. But

when the limit state function is highly non-linear it may be necessary to use a simulation

technique to compute failure probabilities. In this study simulation is used only for

confirming the outcome of the conditional probability calculations performed for the

Ferry-Borges Method.


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The simulation methods are based on realizing random numbers for random

variables by using the random number generators and testing those with limit state

function, G such as Equation (A-6). When Gis calculated repeatedly from enough points

the probability distribution of the G can be obtained. It is important to keep generated

numbers independent. The probability of failure is defined as the P(G


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consider all these loads at their lifetime maximum. Since the simultaneous occurrence of

the time variant loads at their maximum is rare, the total maximum effect of the

combined loads is not the sum of the maximum load effects of these time variant loads.

For example, there is a very low chance to have a strong earthquake along with extreme

high wind speeds. In early design codes load combinations were taken care of by

applying a reduction factor to the sum of individual design load effects (e.g. in ACI

codes 1970 edition a factor of 70% for combination of live load and wind (or

earthquake) loads), which was based on experience and intuition Wen(1977). Three

methods are currently used in load combination problems, Turkstras Rule, Ferry-Borges

Model and Wens Load Coincidence Method.

2.5.1 Turkstras Rule

Turkstras Rule is a deterministic approach to solve a Load Combination problem,

and has been used extensively in code developments because of its simplicity in

application. Since the probability of occurrence of time variant loads at their maximum

simultaneously is so small Turkstra (1972) proposed to approximate the combined effect

by assuming one of the loads is at its maximum and the rest are at their arbitrary point in

time values. A set of sub-combinations is created and the maximum of this set is the

maximum effect of the combined loads. Assuming the total effectZ(t)at time tis

( ) ( ) ( ) ( )tXtXtXtZ nL++= 21 (2-11)

then the maximum value of Z can be found by


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++

++

++

=

max21

max21

2max1

maxmax

n

n

n

XXX

XXX

XXX

Z

L

M

L

L

(2-12)

where Xi represents the instantaneous, and Xmaxi represent the maximum values of load

variable i. Consider the combination of the dead load, live load and wind load; dead load

does not show variation throughout the design life of a structure so assuming time

independent dead load is reasonable. On the other hand live load and wind load shows

variations during the life of a structure. So the combined effect of the above-mentioned

loads would be the maximum of dead load effect combined with the maximum of

combination of live load at its maximum with point in time value of wind load or wind

load at its maximum with point in time value of live load. The combination of these loads

can be represented as follows;

+

++=

++=

max

maxmaxmax

)()()(

WL

WLDZ

tWtLDtZ

apt

apt (2-13)

whereD,Land Wrepresent dead, live and wind loads, respectively and subscript apt is

the arbitrary point in time value for the corresponding load.

Turkstras rule has been widely used in code calibration processes; in most of

todays codes this rule has been the load combination technique (ANSI-A58, AISC,

AASHTO, etc.). Turkstras rule has an important drawback that avoids using this rule in

this research. Turkstras rule does not allow introducing the modeling uncertainties into

the picture because it absolutely depends on the independence of the loads repetitions, but


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the modeling makes the load effects correlated. Although Turkstras rule is easy to

apply, it is difficult to estimate the accuracy of the method. It may be conservative or

unconservative depending on the number of the repetitions of each individual load case

and relative uncertainty of the magnitude of each load.

2.5.2 Ferry-Borges Process

Ferry-Borges Method models the load process as rectangular pulses that change

after prescribed, equal and deterministic intervals (Turkstra and Madsen (1980)). It is

assumed that load intensities are independent and constant during the interval. The

lifetime of the structure is an integer multiple of the load intervals, and also the individual

loads intervals are integer factors of each other as shown in Figure 4.

In Figure 4 the t1, t2, t3are the single event duration times and Tis the lifetime of

the structure. The combination ofX1(t),X2(t)andX3(t)is given as in Equation (2-11). The

maximum effect ofZ(t)can be written as;

( ) ( ) ( ) ( ){ }

++= tXtXtXtZ

ttTT321

32

maxmaxmaxmax (2-14)

The cumulative distribution CDF of the maximum effect of the combination of

the loads can be calculated using a convolution integral (Melchers (1999)). For example

the maximum CDFfor ( ) ( ){ }

+ tXtX

t32

3

max can be calculated by

( ) ( ) ( )[ ] dxFxfxF tt

X

x

X3

2

32max =

(2-15)


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In the above equation the probability of the maximum of X2(t)+X3(t)in t2to be

less than x is calculated. ( )[ ]3

2

3

tt

X xF gives the probability of maximum in t2/t3

repetitions of the load event is less than (x-), and )(2

xfX is the probability density

function ofX2, and the multiplication is integrated from tox.

The calculation of Equation (2-15) becomes time consuming as the number of

variables increases. Instead of calculating the convolution integral by numerical

integration, FORM theory can be used to find the result. The calculation of probability of

failure,Pf, algorithm developed by Rackwitz and Fiessler (1978) based on FORM will be

used. The effects of the load repetitions are taken into account during the conversion of

non-normal variates to normal variates. For instance the CDFofX3in time span t2is

( ) ( )[ ] 32

32

*3)at t(

*3

tt

X xFxCDF = (2-16)

where *ix represent the design point or most probable point which can be defined as the

point at the limit state function with the highest probability density (Melchers (1999)). At

time t2the combined effect ofX2andX3has a CDFof

( ) ( ) ( )[ ] 32

322

*32

*)at t(

*3

*2

tt

XX xFxFxxCDF +=+ (2-17)

and at t1the CDFof the combined effect ofX2andX3is

( ) ( ) ( )[ ]2

1

3

2

321

*3

*2)at t(

*3

*2

tt

tt

XX xFxFxxCDF

+=+ (2-18)

the combined effect ofX1,X2, andX3at t1is then


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( ) ( ) ( ) ( )[ ]

++=++2

1

3

2

3211

*3

*2

*1)at t(

*3

*2

*1

tt

tt

XXX xFxFxFxxxCDF (2-19)

finally the combined effect ofX1,X2, andX3in lifetime T has a CDFof

( ) ( ) ( ) ( )[ ]n

tt

tt

XXX xFxFxFxxxCDF

++=++2

1

3

2

321

*3

*2

*1)Tat(

*3

*2

*1 (2-20)

The equivalent normal mean and standard deviation is found according to the CDF

values calculated by Equations 2-16 to 2-20.

The Ferry-Borges Method is still a simplification because the time durations for

every load pulse is assumed to be constant and without fluctuations during the duration.

But it is more accurate compared to Turkstras Rule, because in Turkstras Rule time

duration and rate of occurrence are not considered at all. Ferry-Borges Method is suitable

to handle the separated time dependent and independent parts of the load effects.

Therefore it is preferred to use this method in implementation of the proposed cost model

in this study.


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t3 t3 t3 t3t3 t3 t3 t3 t3t3 t3 t3

t2 t2 t2 t2t2 t2

t1 t1 t1

X3(t)

X2(t)

X1(t)

t

t

t

t2 / t3=2

t1 / t2=2 T / t1=n

Figure 4 Ferry-Borges Model


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2.5.3 Wens Load Coincidence Method

This model is proposed by Wen (1977), to calculate the probability of failure

when combination of two or more time dependent loads are concerned. The probability of

failure can be calculated by using (Wen (1981))

( )

++

= = =

TppTPn

i

n

i

n

j

ijijii

1 1 1

expexp1 L (2-21)

where P is the probability of failure for a duration of time T, n is the number of loads

involved in combination, i is the rate of occurrence for load i, pi is the probability of

load i exceeding the limit state when considered alone and ij and pij are the rate of

occurrence

h dissertation enginaktas

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