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printed on June 24, 2003

Lecture - 1-1

LECTURE 1 - BACKGROUND AND PHILOSOPHY

1.1 OBJECTIVE OF THE LESSON

The objective of this lesson is to acquaint the student with thehistorical background surrounding the development of the LRFD

Specification. The material presented will provide insight into how thebasic decisions regarding the technical basis of the specification werearrived at and how the organization to actually write the specificationwas developed. The basic concepts behind the probabilistic reliability-based calibration will be introduced.

1.2 HISTORICAL DEVELOPMENT

1.2.1 Background

The apparent start of the process leading to the LRFD

Specification was the initiation of NCHRP Project 20-7/31, entitled"Development of Comprehensive Bridge Specification andCommentary", in August of 1986. In reality, the process leading to thisdecision had started almost ten years earlier.

In the late 1970's, the Ontario Ministry of Transportation andCommunication, now known as the Ministry of Transportation, decidedto develop its own bridge design specification, rather than continueutilizing the AASHTO Standard Specification for Highway Bridges. Inthe process of considering the basis for this new Specification, adecision was taken to base it on probabilistic limit states. A CodeControl Committee, chaired by Mr. Paul F. Csagoly, P. Eng., began to

develop background material on the variability of loads and thecomponents that make up resistance, including basic variabilities,such as the dispersion of the values for yield strength of metals,compressive strength of concrete and the variation of sizes in factory-made and field-made products. A major study to determine thestatistical variation in vehicle weights and configurations was alsocompleted. During the same time frame, the basic process forcalculating the statistical reliability of a bridge component, based onthe mean values of the applied loads and the parameters that wentinto the determination of resistance, and the standard deviations ofthese values was also developed. A process for determining acombination of multipliers on load and resistance to achieve a level of

reliability, to be further explained in Article 1.3, was also developed.

In 1979, the first edition of the Ontario Highway Bridge DesignCode (OHBDC) was released to the design community as NorthAmerica's first calibrated, reliability-based limit state specification.Since that time, the OHBDC has been updated in 1983 and 1993 andre-released. Very significantly, the code contained a companionvolume of commentary.

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Lecture - 1-2

As more and more U. S. engineers became familiar with theOHBDC, they recognized a certain logic in the calibrated limit statesdesign and began to question whether the AASHTO Specificationshould be based on a comparable philosophy of determining thesafety of structures. Many research projects, undertaken by theNCHRP, the National Science Foundation (NSF), and various stateswere bringing new information on bridge design faster than it could be

critically reviewed and, where appropriate, adopted into the AASHTOSpecification. It was also becoming clear that the many revisionswhich had occurred to the AASHTO Specification had resulted innumerous inconsistencies and the appearance of a patchworkdocument.

In the Spring of 1986, a group of State Bridge Engineers ortheir representatives met in Denver and drafted a letter to theSubcommittee on Bridges and Structures indicating their concern thatthe AASHTO Specification was falling behind the times. They alsoraised the concern that the Technical Committee structure, operatingunder the Subcommittee on Bridges and Structures, was not able to

keep up with emerging technologies. Presentations were made at tworegional meetings to mixed reception. Nonetheless, this group,identified below, planted the seed which led to the development of theLRFD Specification.

Name 1986 Affiliation

James E. Roberts California Department of TransportationH. Henrie Henson Colorado Department of HighwaysPaul F. Csagoly Florida Department of TransportationHo Lum Wong Michigan Department of TransportationCharles S. Gloyd Washington Department of Transportation

In July of 1986, a group of State Bridge Engineers met with thestaff of the NCHRP to consider whether a project could be developedto explore the points raised in the Denver letter. This led to NCHRPproject 20-7/31 "Development of Comprehensive Bridge Specificationsand Commentary", a pilot study conducted by Modjeski and Masters,Inc. with Dr. John M. Kulicki as Principal Investigator. The followingis a list of tasks for this project:

Task 1 - Review of the philosophy of safety and coverageprovided by other specifications.

Task 2 - Review AASHTO documents, other than the StandardSpecification, for their potential for inclusion into a standardspecification.

Task 3 - Assess the feasibility of a probability-basedspecification.

Task 4 - Prepare an outline for a revised AASHTOSpecification for Highway Bridge Design, commentary, and

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Lecture - 1-3

present a proposed organizational process for completing sucha document.

The review of other specifications and the trends in the developmentof new specifications included a review of work done in Canada(especially the Province of Ontario), Great Britain, the FederalRepublic of Germany and Japan. Personal contacts with practitioners

and researchers in other countries provided information on theemerging directions of specification development and insight into whatdesigners were doing to implement specifications, and, in some cases,what designers were choosing not to implement based on aperception of unnecessary complication. Information collected fromthese various sources indicated that most of the First World countriesappeared to be moving in the direction of a calibrated, reliability-based, limit states specification.

Task 2 can best be summarized as a search for gaps andinconsistencies in the 13th Edition of the AASHTO StandardSpecifications for Highway Bridges. "Gaps" were areas where

coverage was missing; "inconsistencies" were internal conflicts, orcontradictions of wording or philosophy. Many gaps andinconsistencies were found and they are summarized in the list shownin Table 1.2.1-1.

TABLE 1.2.1-1 GAPS AND INCONSISTENCIES

GENERAL FORMAT

Division II

Commentary

Presentation Format

ANALYSIS AND DESIGN PHILOSOPHY

Use of more Refined Design Methods forGirder Bridges

Improved Slab Design

Effective Flange Width Bridge Dynamics

Foundation Design Methods

The Current LFD Provisions in theAASHTO Specifications

Curved Girder Bridges

PERFORMANCE OF MEMBERS AND

SYSTEMS

Modern Bearing Systems

Features of Prestressed Concrete Design

Fatigue of Prestressed Girders

Shielded or Blanketed Strands

Design of Compression Members

Partial Prestressing

Prestress Losses

Local Stress Requirements

Time Dependent Concrete Properties

Foundation Design for Lateral Loads

Compression Plate Design

Anchorage Zone Stresses

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Lecture - 1-4

The Effect of Skew

ADDITIONAL LOADS

The Live Load Model

Thermal and other Environmental Loads

Ship Collision

Erection Engineering and ConstructionLoads

Combination of Load

TYPES OF CONSTRUCTION NOT

COVERED OR PARTIALLY COVERED

Segmental Concrete Bridges

Cable-Stayed Bridges

Multi-Web Box Girder Bridges

Design for Shear and Torsion in ConcreteMembers by Space Truss Analogy

United Treatment of Concrete Design

Continuity Joints for Prestressed I-BeamsMade Continuous for Live Load

Horizontal Shear Requirements andComposite Sections

Features of Steel Design

Carrying Capacity of DistinctlyUnsymmetric Plate Girders

Splices in Overdesigned Members

Net Section Requirements for Built-up

Members

K Factors for Compression Members

Friction Joints

Riveted Construction

Sealing Requirements

Deflection Criteria

Metal Deck Systems

Proprietary Wall Systems

Details which are Sensitive to Distortion-Induced Fatigue

Connection Design

The BS5400 Fatigue Detail Catalog

With respect to Task 3 and the feasibility of using probability-

based limit states design, a review of the philosophy used in a varietyof specifications resulted in three possibilities, two of which arealready included in the current specification. They are:

Allowable stress design which treats each load on thestructure as equal from the view point of statistical variability.A "common sense" approach may be taken to recognize thatsome combinations of loading are less likely to occur thanothers, e.g., a load combination involving a 160 km per hourwind, dead load, full shrinkage and temperature may bethought to be far less likely than a load combination involving

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Lecture - 1-5

the dead load and the full design live load. For example, in the13th Edition and others, the former load combination waspermitted to produce a stress equal to four-thirds of the latter.

Load Factor Design - In which a preliminary effort was madeto recognize that the live load, in particular, was more highlyvariable than the dead load. This thought is embodied in the

concept of using a different multiplier on dead and live load,e.g., a load combination involving 130% of the dead loadcombined with the 217% of the live load, and requiring that ameasure of resistance based primarily on the estimated peakresistance of a cross-section exceed the combined load.

Reliability-based design which seeks to take into accountdirectly the statistical mean resistance, the statistical meanloads, the nominal, or notional, value of resistance, thenominal or notional value of the loads and the dispersion ofresistance and loads as measured by either the standarddeviation or the coefficient of variation, i.e, the standard

deviation divided by the mean. This process can be useddirectly to compute probability of failure for a given set ofloads, statistical data and the Designer's estimate of thenominal resistance of the component being designed. Thus,it is possible to vary the nominal resistance to achieve acriteria which might be expressed in terms such as thecomponent (or system) must have a probability of failure ofless than 0.0001, or whatever variable is acceptable to society.Alternatively, the process can be used to target a quantityknown as the "reliability index" which is somewhat, but notdirectly, relatable to the probability of failure. Based on this"reliability index", it is possible to reverse engineer a

combination of load and resistance factors to achieve aspecific reliability index. This is discussed in Article 1.3.

While some specifications are being developed in terms of the"probability of failure", it was generally agreed that it would be moreappropriate to use the reliability index process to develop load andresistance factors. That way, design could proceed in a processdirectly analogous to load factor design as it appeared in the 13 th

Edition of the Standard Specifications.

In May of 1987, the findings of NCHRP Project 20-7/31 werepresented to the AASHTO Subcommittee on Bridges and Structures

outlining the information above and indicating that seven optionsappeared to be available for consideration. They were:

Option 1 - Keep the Status Quo

Option 2 - Table Consideration of LRFD for the Short-Term

Option 3 - Immediate Adoption of the OHBDC

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Lecture - 1-6

Option 4 - Replace Current Specification with LRFDImmediately

Option 5 - Replace Current LFD with LRFD in the near term

Option 6 - Develop LRFD for evaluation only

Option 7 - Develop LRFD as a Guide Specification

A recommendation was made to proceed to:

develop a probability-based limit states specification,

fill as many of the gaps and inconsistencies as possible, and

develop a commentary to the specification.

Under the direction of then Chairman Robert Cassano of California,the Subcommittee directed the NCHRP to develop a project to

complete this task. This led to NCHRP Project 12-33 which was alsoentitled "Development of Comprehensive Specification andCommentary", which was started in July of 1988, by Modjeski andMasters, Inc.

1.2.2 Organization of Project

1.2.2.1 RESEARCH TEAM

A hierarchial structure was established consisting of a PrincipalInvestigator and Co-Principal Investigator from Modjeski and Masters,Inc., a Code Coordinating Committee, as identified in Table 1.2.2.1-1,

and 15 working groups called task groups also identified in Table1.2.2.1-1. Additionally, an Editorial Committee was developed incharge of the responsibility of assembling the information and makingit editorially and technically consistent.

The original plan was to have the Code CoordinatingCommittee meet on a regular basis and to judicate the technicalcontent of the Specification. While the Code Coordinating Committeemet several times in the early part of the development of thespecification, it became apparent that to meet the schedule imposedon the project, the Editorial Committee would have to deal directly withthe Task Group Chairman.

TABLE 1.2.2.1-1 - NCHRP 12-33 PROJECT TEAM

John M. Kulicki, Principal InvestigatorDennis R. Mertz, Co-Principal Investigator

Scott A. Sabol, Program Officer (1992-1993)Ian M. Friedland, Senior Program Officer (1988-1993)

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Lecture - 1-7

C O D E C O O R D I N A T I N GCOMMITTEE

John M. Kulicki, Chairman -Modjeski and Masters, Inc.John J. Ahlskog -FHWA

Richard M. Barker -Virginia Polytechnic Institute and State UniversityRobert C. Cassano -Imbsen & Associates, Inc.Paul F. Csagoly -SRD Engineering, Inc.James M. Duncan -Virginia Polytechnic Institute and State UniversityDennis R. Mertz -University of DelawareTheodore V. Galambos -University of Minnesota

Andrzej S. Nowak -University of MichiganFrank D. Sears -Consultant

NCHRP PANEL

Veldo M. Goins, Chairman -Oklahoma DOTRoger Dorton -Buckland and Taylor Ltd

Steven J. Fenves -Carnegie-Mellon UniversityRichard S. Fountain -Parsons, Brinckerhoff, Quade and Douglas, Inc.C. Stewart Gloyd -Corridor Design Management GroupStanley Gordon -FHWAGeerhard Haaijer -AISCClellon L. Loveall -Tennessee DOT

Basile Rabbat -Portland Cement Assoc. of

Research & DevelopmentJames E. Roberts -California DOT

Arunprakash M. Shirole -New York State DOTJames T. P. Yao -Texas A & M UniversityLuis Ybanez -Texas State DOT (Retired)

EDITORIAL COMMITTEE*

John M. Kulicki, ChairmanPaul F. CsagolyDennis R. MertzFrank D. Sears

*with appreciation to Modjeskiand Masters, Inc.'s staffmembers:

Diane M. LongScott R. EshenaurChad M. ClancyRobert P. BarrettDavid M. BarrettDonald T. PriceNancy E. KauhlMalden B. WhippleWagdy G. Wassef

Raymond H. RowandCharles H. Johnson

TASK GROUPS

General Design Features

Frank D. Sears, ChairmanStanley R. Davis -Ivan M. Viest -Consultant

Loads and Load Factors

Paul F. Csagoly, ChairmanPeter G. Buckland -Buckland and Taylor Ltd.Eugene Buth -Texas A & M UniversityJames Cooper -FHWAC. Allin Cornell -Stanford University

James H. Gates -CALTRANSMichael A. Knott Greiner, Inc.Fred Moses -University of Pittsburgh

Andrzej S. NowakRobert Scanlan -Johns Hopkins University

Analysis and Evaluation

Paul F. Csagoly, ChairmanPeter BucklandIan G. Buckle -State University at BuffaloRoy A. Imbsen -Imbsen & Associates, Inc.Jay A. Puckett -University of WyomingWallace W. Sanders, Jr.

-Iowa State UniversityFrieder Seible -University of California - SanDiegoWilliam H. Walker -University of Illinois at Urbana-Champaign

Concrete Structures Steel Structures Aluminum Structures

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Lecture - 1-8

Robert C. Cassano, ChairmanJohn H. Clark -Anderson, Bjornstad, Kaneand Jacobs, Inc.Michael P. Collins -University of TorontoPaul F. Csagoly

David P. Gustafson -CRSIAntoine E. Naaman -University of MichiganPaul Zia -North Carolina StateUniversityDon W. Alden -Imbsen & Associates, Inc.

Frank D. Sears, ChairmanJohn Barsom -USS Division of USX Corp.Karl Frank -University of Texas at AustinWei Hsiong -Consultant

William McGuire -Cornell UniversityDennis R. MertzRoy L. Mion -AISC Marketing,Inc.Charles G. Schilling - ConsultantIvan M. ViestMichael A.Grubb -AISC Marketing, Inc.

Frank D. Sears, ChairmanTeoman Pekoz -Cornell University

Wood Structures

Andrzej S. Nowak, Chairman

Baidar Bakht -Ministry of Transp. of OntarioR. Michael Caldwell -TBTADonald J. Flemming -Minnesota DOTHota V. S. Gangarao -West Virginia UniversityJoseph F. Murphy -Structural ReliabilityConsultants

Michael A. Ritter -USDA Forest Products Lab

Raymond Taylor -Ministry of Transp. of OntarioThomas G. Williamson -American Institute of Timber

Construction

Deck Systems

Paul F. Csagoly, Chairman

Barrington deVere Batchelor -Queens UniversityDaniel H. Copeland -Grid Manufacturer's Assoc.Gene R. Gilmore -IKG/GreulichRichard E. Klingner -University of Texas at AustinRoman Wolchuk -Consultant

Foundations

J. Michael Duncan, Co-

ChairmanRichard M. Barker, Co-Chairman

Walls, Piers and Abutments

J. Michael Duncan, Co-ChairmanRichard M. Barker, Co-ChairmanJames Withiam

-D'Appolonia

Buried Structures

James Withiam

Bridge Railings

Ralph W. Bishop, Chairman -California DOTEugene ButhJames H. Hatton, Jr. -FHWA

Teddy J. Hirsch -Texas A & M UniversityRobert A. Pege - New Jersey DOT

J o i n t s , B e a r i n g s a n dAccessories

Charles W. Purkiss, Chairman -California DOTIan G. Buckle

Earthquake Provisions AdvisoryGroup

Ian Buckle, ChairmanRobert CassanoJames Cooper

Calibration

Andrzej S. Nowak, ChairmanC. Allin CornellDan M. Frangopol

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Lecture - 1-9

John J. Panak -Texas State Dept. ofHighways and Public TransportationDavid Pope -Wyoming DOTCharles W. Roeder

-University of WashingtonJohn F. Stanton -NYSBA

James GatesRoy ImbsenGeoffrey Martin -University of SouthernCalifornia

-University of ColoradoTheodore V. GalambosRoger Green -University of WaterlooFred MosesKamal B. Rojiani -VPISU

1.2.3 Project Schedule

The original plan called for three drafts, which were releasedand reviewed as follows:

The first draft was released in April of 1990 and was totallyuncalibrated. The primary intent was to show coverage andorganization. This draft was released to the AASHTO BridgeEngineers, the FHWA, all members of the NCHRP Panel and

Task Group Members, and several private authorities. All told,it was reviewed by about 250 engineers, because many of theDepartment of Transportations circulated it to in-houseexperts. Approximately 4,000 comments were receivedconcerning the first draft, all were read and reviewed, andmany were discussed with Task Group Chairmen or sentdirectly to them. Many of the comments were included in thesecond draft, but there was no written response to thequestions.

A second draft was released in late April of 1991 to the samegroup of people. Additionally, it was noted at several regional

and national conferences on bridge engineering that allinterested parties could obtain a copy of the draft specificationat their cost, and that they would be free to submit reviewcomments. This second draft contained a preliminary set ofload and resistance factors which changed relatively little insubsequent drafts. Approximately 6,000 comments werereceived for this draft and were processed as outlined above.

The third draft was submitted in April of 1992 and wasreviewed in the same process that was used for the seconddraft. For this draft, about 2,000 comments were received andthey were processed as described above.

After reviewing the third draft, the NCHRP Panel determinedthat the specification was approaching a draft that could be consideredfor a ballot item, but that additional work would be worthwhile andwould reduce modifications needed in the future. Accordingly, theproject was extended to include a fourth draft, whose scope includedthe following items:

Continue to review the distribution factors developed underNCHRP Project 12-23 and which were included in theproposed specification,

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Continue to refine calibration,

Consider further the need for special short-span live loads,

Further refine and verify the proposed strip width method forcalculating moments and deck slabs,

Develop an index to the specification,

Convert to the SI system of measurement,

Develop further trial designs, and

Complete the text for fourth draft.

This fourth draft was submitted in March of 1993 and wasaccepted as a ballot item at the May 1993 meeting of theSubcommittee of Bridge and Structures.

One of the most valuable features of the process of developingthis specification was two rounds of trial designs. In 1991, and againin 1992, various States and industry groups volunteered to docomparative designs using the 14th edition of the StandardSpecifications and the LRFD Specifications. Additionally, interestedindustry groups also organized their own series of trial designs andcontributed information and critiques based on that work. FourteenStates and several industry groups participated in the initial 1991designs, and 22 States and several industry groups worked on the1992 set. As would be expected, the 1992 set were more completeand included nine slab bridges, 20 concrete beam bridges, 9 steel

beam and girder bridges, 1 truss, 1 segmental concrete bridge, 2wood bridges and 5 culverts, and a series of retaining wall designs.The designs included substructure, superstructure and pile and spreadfooting foundations. Additionally, a comprehensive set of prestressedbeam bridges were evaluated by industry and contributed to theproject.

These two series of trial designs achieved several importantobjectives:

They exposed areas where further development of loadmodels and resistance formulations was necessary, and where

further calibration was advisable.

They demonstrated that the specification, though considerablylonger and more comprehensive than the StandardSpecification, was nonetheless readable and workable.

They pointed to numerous areas where improvements andclarification in the wording could be made.

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They vastly broadened the base of practicing engineers whowere becoming conversant with the LRFD Specification.

All things considered, the two trial design sessions, whichrequired supplementary meetings of the Subcommittee on Bridgesand Structures, proved to be one of the most important steps in thedevelopment and adoption of the LRFD Specification.

1.2.4 Project Objectives

There were several objectives in the development of this newspecification. They may be summarized as:

To develop a technically state-of-art specification which wouldput U. S. practice at or near the leading edge of bridge design.

To make the specification as comprehensive as possible andinclude new developments in structural forms, methods ofanalysis and models of resistance.

To the extent consistent with the thoughts above, keep thespecification readable and easy to use, bearing in mind thatthere is a broad spectrum of people and organizations involvedin bridge designs.

To keep specification-type wording and not to develop atextbook.

To encourage a multi-disciplinary approach to bridge design,particularly in the area of hydraulics and scour, foundationdesign and bridge siting.

To place increasing importance on the redundancy andductility of structures.

Many changes had to be made in the content and appearanceof the Standard Specification to achieve the objectives outlined above.Areas of major changes are identified below:

The introduction of a new safer philosophy of safety - LRFD.

The identification of four limit states to be discussed later.

The development of new load factors.

The development of new resistance factors.

The relationship of the chosen reliability level, the load andresistance factors, and load models through the process ofcalibration.

The development of improved load models necessary toachieve adequate calibration, including a new live load model.

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Revised techniques for analysis and the calculation of loaddistribution.

A combined presentation of plain, reinforced and prestressedconcrete.

The introduction of limit state-based provisions for foundation

design and soil mechanics.

Expanded coverage on hydraulics and scour.

Changes to the earthquake provisions to eliminate the seismicperformance category concept by making the method ofanalysis a function of the importance of the structure.

Inclusion of large portions of the Guide Specification forSegmental Concrete Bridge Design.

Inclusion of large portions of the FHWA Specification for ship

collision.

Expanded coverage on bridge rails based on crash testing,with the inclusion of methods of analysis for designing thecrash specimen.

The introduction of the isotropic deck design process.

The development of a parallel commentary.

It was the underlying principle of NCHRP 12-33 to make asmuch use of existing research findings as possible. The project was

not supposed to involve the development of new information, althoughsome limited work was necessary to tie information together to makea comprehensive specification.

Any effort to develop a Specification on the scale of theAASHTO LRFD Specification for Highway Bridge Design has to reacha point where upgrading the technical content for new ideas must bestopped in order to finish the test and publish the document. Thisdoes not stop the tide of new ideas. Future changes must beexpected. Some of the areas where continued development shouldbe expected and encouraged include:

Continue development of a database upon which to projectbridge loads. This is particularly true of live load for which itwas initially thought that much information would bedetermined from weight-in-motion (WIM) studies. For variousreasons, much of the WIM data was not directly usable in thedevelopment of the live load model. Coincident with NCHRP12-33, an FHWA-sponsored project involving theinstrumentation of bridges throughout the Country and themeasurements of loads in correlation of analysis. This projecthas generated large amounts of data, and continuing efforts

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should be made to extract from that data information for furtherrefinements of load models and analysis techniques.

There is a continuing need to refine and verify foundationresistance and deformation. More work needs to be done onthe large scale testing of foundations, the determination ofgroup action, and the amount of movement which is

acceptable. This latter point was a subject of considerablediscussion during the development of NCHRP 12-33 asStructural Engineers and Foundation Engineers are apt to viewthis issue differently. Clearly, a multi-disciplinary consensusis necessary.

The issue of temperature gradient was also hotly debated.There appears to be general agreement that the temperaturegradient does exist, as has been demonstrated in manystudies. The real problem appears to be related to thestructural effect actually caused by the gradient and its jointaction with other loads.

Further simplification of load distribution is warranted, as isextension to explicitly include unequal spans and splayedgirders. This could take the form of further refinement oforthotropic plate models, which were considered in thedevelopment of the LRFD specification, and suitable PC-typecomputer program to do a detailed analysis.

The joint probability of load occurrence remains an issue ofmuch interest. How much live load should be applied with anearthquake loading, should other loads be applied with thatsame combination? How much more refinement of wind

loadings can be done before site-specific studies arenecessary? How should ice, wind and other loads becombined? Should ship and vehicle collision be appliedsimultaneously with scour and earthquake and other loadings?

Continued development of reliability theory should involvemore emphasis on system rather than component reliability,the use of second order methods, improved methods ofprojecting the all important "tails" of measured data, thedevelopment of larger and more inclusive databases, and theinclusion of aging and deterioration models.

Re-evaluate countrywide loadometer survey, future projectionsof truck traffic, etc., and relate them to fatigue criteria.

Continue further refinement on temperature gradient, iceloads, stream flow forces, debris forces, including developmentof threshold design values below which there is no need toapply these forces.

Finally, it is time to start developing a "North AmericanHighway Bridge Design Specification". The common LRFD

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Lecture - 1-14

(probabilistic limit states) philosophy utilized in Canadian and U. S.Specifications should make such a joint specification, applicable toMexico if they agreed and helped to develop it, more attainable nowthan at any time in the last 15 years. Such a multi-Nationalspecification would be consistent with the Eurocode approach and theconcept of a North American Free Trade Zone.

1.3 SUMMARY OF RELIABILITY CONSIDERATION

1.3.1 Overview of a Probability-Based Specification

The investigation of probability-based limit states designstarted on a note of some skepticism regarding whether thisphilosophy was mature enough in its development to encompass thecombination of art and science involved in bridge engineering. Afterconsidering the underlying principles of service load design, loadfactor design and limit states design, it became apparent that of thesethree possibilities, probability-based limit states design was

unquestionably the most comprehensive and rational way to proceed.Finally, for clarity, it was decided that the AASHTO version of limitstates design would be termed Load and Resistance Factor Design(LRFD), like the AISC Specification.

A consideration of probability-based reliability theory can besimplified considerably by initially considering that natural phenomenacan be represented mathematically as normal random variables, asindicated by the well-known bell-shaped curve, Figure 1.3.1-1. Use ofthis assumption leads to closed form solutions for areas under partsof this curve as given in Table 1.3.1-1, which can conveniently answerthe following questions:

What percentage of the total number of values fall within agiven range Y X Z? The answer to this question is givenby the area bounded by the two values Y and Z, as shown inFigure 1.3.1-1.

What percentage of the total values are such that X Z? Thisis shown by the shaded area in Figure 1.3.1-2.

The first question and its statistical ramifications are already includedin the AASHTO Standard Specifications for Highway Bridges in thefatigue design provisions of Article 10.3.1, in which the allowablefatigue stress ranges of the various categories were defined by the

"95% confidence" limits. To put this in the prospective of Figure 1.3.1-1, this is equivalent to saying that 95% of all the details tested at agiven stress range failed at a number of cycles bounded by the valuesof Y and Z. In this context, the value of Y and the value of Z are eachlocated two standard deviations on either side of the mean or averagevalue.

The question illustrated by Figure 1.3.1-2 deals very explicitlywith the problem of defining loads and resistance of members.

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Lecture - 1-15

z 0 1 2 3 4 5 6 7 8 9

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0754

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2258 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2518 0.2549

0.7 0.2580 0.2612 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852

0.8 0.2881 0.2910 0.2939 0.2967 0.2996 0.3023 0.3051 0.3078 0.3106 0.3133

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830

1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767

2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.48172.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890

2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4971

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981

2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986

3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993

3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995

3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997

3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998

3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4999

3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4993 0.4999

3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999

3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999

3.9 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000

Table 1.3.1-1 - Areas Under Portion of Normal Probability Density Curve

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Lecture - 1-16

Figure 1.3.1-1 - Normal Distribution Curve Showing DistributionBounded by the Values "Y" and "Z"

Figure 1.3.1-2 - Normal Distribution Curve Showing Portion ofDistribution Less than or Equal to "Z"

If we now accept the notion that both load and resistance arenormal random variables, we can plot the bell-shaped curvecorresponding to each of them in a combined presentation dealing

with distribution as the vertical axis against the value of load, Q, orresistance, R, as shown in Figure 1.3.1-3. The mean value of loadand the mean value of resistance is also shown, as is a second valuesomewhat offset from the mean value, which is the "nominal" value,or the number that designers calculate the load or the resistance tobe. The ratio of the mean value divided by the nominal value is calledthe "bias". The objective of a design philosophy based on reliabilitytheory, or probability theory, is to separate the distribution ofresistance from the distribution of load, such that the area of overlap,i.e., the area where load is greater than resistance, is tolerably small,say one in 10,000. The objective of a LRFD approach to a designspecification is to be able to define load factors, shown as Qn in

Figure 1.3.1-3, and resistance factors, shown as Rnin Figure 1.3.1-3,in a way that forces the relationship between the resistance and loadto be such that the area of overlap is less than or equal to the valuethat a code-writing body accepts. Note in Figure 1.3.1-3 that it is thenominal load and the nominal resistance, not the mean values, whichare factored.

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Lecture - 1-17

Figure 1.3.1-3 - Separation of Loads and Resistance

The relationship between the nominal (design) load, mean loadand factored load is shown in Figure 1.3.1-4. The shaded area inFigure 1.3.1-4 is equal to the probability of exceeding the factored loadvalue.

Figure 1.3.1-4 - Probability Density Function, fx (q), of Load

Component, Qi; Mean Load, Qm, Nominal (design) Load, Qn, andFactored Load, iQn(Nowak, 1993)

The relationship between the nominal (design) resistance,mean resistance and factored resistance is shown in Figure 1.3.1-5.The shaded area in Figure 1.3.1-5 is equal to the probability ofexceeding the factored resistance value.

Figure 1.3.1-5 - Probability Density Function, fR(X), of Resistance, R;Mean Resistance, Rm, Nominal (design) Resistance, Rn, and FactoredResistance, Rn(Nowak, 1993)

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Lecture - 1-18

A conceptual distribution of resistance minus load, combiningthe individual curves discussed above, is shown in Figure 1.3.1-6,where the area of overlap from Figure 1.3.1-3 is shown as negativevalues, i.e., those values to the left of the origin.

Figure 1.3.1-6 - Definition of Reliability Index,

It now becomes convenient to define the mean value of resistanceminus load as some number of standard deviations, , from the origin.The variable is called the "reliability index". The problem with thispresentation is that the variation of the quantity, resistance minus load,is not explicitly known. Much is already known about the variation ofloads by themselves or resistances by themselves, but the differencebetween these has not yet been quantified. However, from probabilitytheory, it is known that if load and resistance are both normal andrandom variables then the standard deviation of the difference is:

(1.3.1-1)(R Q)

2

R

2

Q

Given the standard deviation, and considering Figure 1.3.1-6, we cannow define the reliability index, , as:

(1.3.1-2)

R Q

2

R 2 Q

Comparable closed-form equations can also be established for otherdistributions of data, e.g., log-normal distribution. It is very important

to realize that a "trial and error" process is available for solving for when the variable in question does not fit one of the already existingclosed-form solutions.

In a reliability-based code in the purest sense, the designer isasked to calculate the value of provided by his design and thencompare that to a code-specified tolerable value. A designer wouldrequire much knowledge of reliability theory to apply such a purereliability-based code. Alternatively, through a process of calibrating

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Lecture - 1-19

load and resistance factors by trial designs, it is possible to developload and resistance factors so that the design process looks verymuch like the existing Load Factor Design Methodology.

The process of calibrating load and resistance factors startswith Equation 1.3.1-2 and the basic design relationship; the factoredresistance must be greater than or equal to the sum of the factored

loads:

(1.3.1-3) R Q ixi

Solving for the average value of resistance yields:

(1.3.1-4)R Q 2

R

2

Q R

1

ix

i

Using the definition of bias, indicated by the symbol , Equation 1.3.1-4, leads to the second equality in Equation 1.3.1-4. A straightforwardsolution for the resistance factor, is:

(1.3.1-5)

ix

i

Q 2

R

2

Q

Unfortunately, Equation 1.3.1-5 contains three unknowns, i.e., theresistance factor, , the reliability index, , and the load factors, .

The acceptable value of the reliability index, , must be chosenby a code-writing body. While not explicitly correct, we can conceiveof as an indicator of the fraction of times that a design criteria will bemet or exceeded during the design life, analogous to using standarddeviation as an indication of the total amount of population included ornot included by a normal distribution curve. Utilizing this analogy, a of 2.0 corresponds to approximately 97.3% of the values beingincluded under the bell-shaped curve, or 2.7 of 100 values notincluded. When is increased to 3.5 for example, now only twovalues in approximately 10,000 are not included. It is more accurateto think of as a basis for comparing the relative safety of populationsstructures.

Assuming that a code-writing body has established a value of, Equation 1.3.1-5 still indicates that both the load and resistancefactors must be found. One way to deal with this problem is to selectthe load factors which apply to all materials and structural action,assume trial resistance factors for each material and action beingconsidered and then calculate the reliability indicies. This process hasbeen used by several code-writing authorities.

The steps in the process is as follows:

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Lecture - 1-20

Factored loads can be defined as the average value of load,plus some number of standard deviation of the load, as shownas the first part of Equation 1.3.1-6 below.

(1.3.1-6)ixi xi n i xi n Vixi

Defining the "variance", Vi, as being equal to the standarddeviation divided by the average value, leads to the secondhalf of Equation 1.3.1-6. Utilizing the concept of bias one moretime, Equation 1.3.1-6 can now be condensed into Equation1.3.1-7.

(1.3.1-7)i 1 n Vi

Thus, it can be seen that load factors can be written in termsof the bias and the variance, as depicted in Figure 1.3.1-7.

This gives rise to the philosophical concept that load factorscan be defined so that all loads have the same probability ofbeing exceeded during the design life. This is not to say thatthe load factors are identical, just that the probability of theloads being exceeded is the same.

Figure 1.3.1-7 - Graphical Presentation of Equation 1.3.1-7 (Nowakand Lind, 1979)

Using Equation 1.3.1-5, for a given set of load factors and trialresistance factors, the value of the reliability index can becalculated for various types of structural members and forvarious load components, e.g., shear, moment, etc. on thevarious structural components. Computer simulations of arepresentative body of structural members can be done,yielding a large number of values for the resistance factor.

Reliability indices are then grouped by structural member andby load component to determine if they cluster around the

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Lecture - 1-21

target reliability index. If close clustering results, a suitablecombination of load and resistance factors has been obtained.

If close clustering does not result, a new trial set of load and/orresistance factors can be used and the process repeated untilthe reliability indices do cluster around the target.

The resulting load and resistance factors taken together willyield reliability indices close to the target value selected by thecode-writing body as acceptable.

The process above appears to be rather illusive. Fortunately,other jurisdictions had utilized this calibration process and found it toyield reasonable load and resistance factors, which was also the casein NCHRP 12-33. Figure 1.3.1-8 shows the dispersion of the reliabilityindices observed for bridges designed to the AASHTO StandardSpecifications within the Province of Ontario. After defining load andresistance factors through the process outlined above, analysis ofthese same set of bridges produced reliability indices clustered around

the target value of 3.5, as shown in Figure 1.3.1-9. The reason thatthe values do not plot exactly on the horizontal straightline, indicatedby a reliability index of 3.5, is that the resistance factors did not clusterat exactly the same number. Thus, when reasonable andconservative interpretations of the resistance factor values are utilized,the reliability indices will generally be above the target value and anunavoidable amount of scatter will still result. As can be seen inFigure 1.3.1-9, a handful of the comparative values were significantlybelow the target reliability index, and this indicated that additionaldesign provisions were necessary for this particular group of bridges.

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Lecture - 1-22

Figure 1.3.1-8 - Range of Reliability Indices Obtained Using AASHTOStandard Specifications (Nowak and Lind, 1979)

Figure 1.3.1-9 - Range of Reliability Indices Obtained with Reliability-Based OHBDC (Nowak and Lind, 1979)

The chief advantages for a probability-based LRFDspecification are as follows:

A more uniform level of safety throughout the system willresult.

That measure of safety will be a function of the variability ofloads and resistance.

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Lecture - 1-24

1.4 OVERVIEW OF THE CALIBRATION PROCESS

1.4.1 Outline of the Calibration Process

The following steps are the major phases of the calibration ofthe load and resistance factors for the LRFD Specification:

Develop a database of sample current bridges

Extract load effects by percentage of total load

Develop a simulation bridge set for calculation purposes

Estimate the reliability indices implicit in current designs

Revise loads-per-component to be consistent with the LRFDSpecification

Assume load factors

Vary resistance factors until suitable reliability indices result asdescribed in Article 1.3.1

The outline above assumes that suitable load factors are assumed.If the process of varying the resistance factors and calculating thereliability indices does not converge to a suitable narrowly grouped setof reliability indices, then the load factor assumptions must be revised.In fact, several sets of proposed load factors were investigated todetermine their effect on the clustering of reliability indices.

1.4.2 Development of a Sample Bridge Database

Approximately 200 representative bridges were selected fromvarious regions of the United States by requesting sample bridgeplans from various states. The selection was based on structural-typematerial and geographic location to represent a full-range of materialsand design practices as they vary around the Country. Anticipatedfuture trends were also considered by questionnaires sent to selectedstates. One-hundred and seven sets of plans were received fromwhich the 200 representative bridges were selected. Obviously, someplan sets contained more than one bridge or the bridge containedseveral separable units. The list of structures provided by the StateDepartments of Transportation is given in Table 1.4.2-1.

For bridges selected from within this database, moments andshears were calculated for the dead load components, the live loadand the dynamic load allowance. Nominal or design values werecalculated using the 1989 edition of the AASHTO StandardSpecifications. The statistically projected live load and the notionalvalues of live load force effects were calculated. Resistance wascalculated in terms of moment and shear capacity. For each structure,both the nominal design resistance, indicated by the cross-sectionshown on the plans, and the minimum actual required resistance,according to the 1989 AASHTO, were developed. Generallyspeaking, the nominal resistance is larger than the minimum value,

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Lecture - 1-25

because standard available sizes of plates and rebar and othercomponents that comprise resistance do not exactly meet thetheoretical requirement. Additionally, a phenomenon known as the"designers bias" is implicit in actual designs. This factor results fromthe tendency of designers to "bump up" a given component to the nextavailable commercial size.

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Lecture - 1-26

Table 1.4.2-1 - Selected Bridges

STRUCTURAL TYPE REQUESTED SPAN (m) PROVIDED SPAN (m)/STATE

Steel, Simple Span

Rolled Beams, Non-composite 12 to 25 14.6 - PA18.0 - MI25.3 - PA

Rolled Beams, Composite 15 to 25 14.6 - PA14.9 - PA15.2 - PA15.5 - PA20.4 - PA23.2 - PA24.4 - PA26.2 - PA

Plate Girder, Non-composite 30 to 45 23.8 - PA30.5 - PA

Plate Girder, Composite 30 to 55 31.4 - MI33.2 - PA37.2 - MI

Box Girder 30 to 55 None

Through-truss 90 to 120 91.4 - PA92.4 - PA94.8 - PA

121.0 - PA

Deck Truss 60 to 120 61.0 - NY76.2 - NY91.4 - NY121.9 - NY

Pony Truss 45 30.5 - OK31.4 - PA91.4 - PA

Arch 90 to 150 109.7 - NY132.9 - NY

192.0 - NY222.5 - NY

Tied Arch 90 to 180 163.1 - NY

Steel, Continuous Span

Rolled Beams 15-20-15 to 25-30-25 22.6-18.3 - PA25.9-24.4-25.9 - MI

23.2-29.3-24.4-18.3 - PA

Plate Girder 30-36-30 57.9-54.9 - MI36.6-45.7-36.6 - MI61.0-61.0-61.0 - KY

91.4-91.4-91.4-91.4 - KY59.4-59.4-59.4-59.4 - KY

61.0-61.0-61.0-61.0-61.0-61.0 - KY

Box Girder 30-36-30 to 90-120-90 31.4-31.4-31.4 - MD37.5-37.5-37.5 - MD43.3-45.7-31.4 - MD37.2-49.4-37.2 - IL

35.4-42.1-42.1-42.1-35.4 - IL45.7-50.9-53.3-53.3-50.9-45.7 - IL

Through-truss 120 None

Deck Truss 120 None

Tied Arch 90-150 None

Reinforced Concrete, Simple Span

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Lecture - 1-27

Slab 6 to 12 9.1 - OK

T-beam 12 to 24 12.2 - IL12.2 - OK13.1 - IL

15.2-15.2 - OK18.3 - IL

Arch-barrel 12 None

Arch-rib 18 None

Reinforced Concrete, Continuous Span

Slab, Two-span 9-912-12

None

Slab, Three-span 8-8-8 None

Solid Frame 12 12.2 - CA

T-beam, Frame 16 None

T-beam, Two-span 15-1521-21

18.9-18.9 - CO21.6-21.6 - CO

T-beam, Three-span 12-15-12 to 15-21-15 11.6-15.2-11.6 - TN12.2-15.5-12.2 - TN12.7-15.5-12.2 - TN14.0-17.1-11.9 - TN14.3-19.8-14.3 - TN16.2-22.3-16.2 - TN15.2-21.6-12.8 - TN

Arch None

Box, Three-span 18-24-18 to 23-27-23 21.0-36.3-29.3 - MD

Prestressed Concrete, Simple Span

Slab 9 to 12 None

Voided Slab 9 to 15 None

Double T 12 to 18 11.9 - CO

Closed Box CIP 38 None

AASHTO beam 15 to 30 23.2 - MI23.2 - CO31.1 - TX31.1 - PA32.0 - PA31.4 - MI33.5 - CO36.0 - TX36.0 - CO39.6 - TX42.1 - CO

Bulb 18 to 36 None

Box Girder 24 to 36 22.6 - PA22.6 - PA25.0 - CA29.0 - CA31.1 - CA31.7 - CA35.4 - CA36.0 - CA36.0 - CA38.1 - CA38.1 - CA

Prestressed Concrete, Continuous Span

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Lecture - 1-28

Slab 10-10 to 12-15-12 None

Voided Slab 15-21-15 to 32-32 None

AASHTO Beam 25 to 33 None

Post-tensioned AASHTO Beam 30-30 None

Bulb None

Box 19.8-19.8 - CA26.5-25.9 - CA28.3-26.2 - CA31.4-31.1 - CA32.6-31.1 - CA33.5-48.8 - CA36.0-30.8 - CA61.0-61.0 - CA

18.3-24.4-18.3 - CA21.0-25.0-18.0 - CA22.9-27.4-22.9 - CA21.0-28.0-21.0 - CA23.2-27.4-23.2 - CA21.6-25.9-21.6 - CA20.1-25.9-15.8 - CA

Wood

Saw Beam 5.5 - MN

Glulam Beam - Nailed 14.9-15.2-14.9 - MN

Glulam Beam - Dowelled None

Glulam Beam - Composite None

Truss 15.2-30.5-30.5-14.9 - MN

Arch None

Deck - Nailed 9.8-9.8-9.8 - MN

Deck - Composite None

Deck - Prestressed Transversely 13.4 - MN

Deck - Prestressed Longitudinally

1.4.3 Extraction of Load Effects

For each of the bridges in the database, the load indicated bythe contract drawings was subdivided by the following characteristiccomponents:

The dead load due to the weight of factory-made components

The dead weight of cast-in-place components

The dead weight due to asphaltic wearing surfaces whereapplicable

The dead weight due to miscellaneous items

The live load due to the HS20 loading The dynamic load allowance or impact prescribed in the 1989

AASHTO Specification

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Lecture - 1-29

Full tabulations for all these loads for the full-set of bridges in thedatabase are presented in Nowak, 1993.

In summary, the combination of the tandem with the uniformload and the HS20 with the uniform load, were shown to be anadequate basis for a notional design load in the LRFD Specification.

1.4.4 Development of the Simulated Bridge Set

Based on the relative amounts of the loads identified in thepreceding article for each of the combination of span and spacing andtype of construction indicated by the database, a simulated set of 175bridges was developed which was comprised of:

Twenty-five non-composite steel girder bridge simulations forbending moment with spans of 9, 18, 27, 36 and 60 m, and foreach of those spans, spacings of 1.2, 1.8, 2.4, 3.0 and 3.6 m.

Representative composite steel girder bridges for bending

moments having the same parameters as those identifiedabove.

Representative reinforced concrete T-beam bridges forbending moments having spans of 9, 18, 27 and 39, withspacings of 1.2, 1.8, 2.4 and 3.6 m in each span group.

Representative prestressed concrete bridges for momentshaving the same span and spacing parameters as those usedfor the steel bridges.

Representative steel girder bridges for shear having the same

span and spacing parameters as those identified for bendingmoment.

Representative reinforcing concrete T-beams for shear havingthe same span and spacing parameters indicated previouslyfor bending moment.

Representative prestressed concrete girder bridges for shearhaving the same span and spacing parameters as previouslyindicated for prestressed beams.

Full tabulations of these bridges and their representative amounts of

the various loads are presented in Nowak, 1993.

1.4.5 Calculated Reliability Indices and Selection of Target Value

The reliability indices were calculated for each simulated andeach actual bridge for both shear and moment. The range of reliabilityindices which resulted from this phase of the calibration process ispresented in Figure 1.4.5-1. It can be seen that a wide range ofvalues were obtained using the current specifications, but this was

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Lecture - 1-30

anticipated based on previous calibration work done for the OntarioHighway Bridge Design Code (OHBDC).

The most important parameters which determine the reliabilityindex for beam and girder bridges are the girder spacing and the spanlength. In general, reliability indices are higher for larger girderspacing, due to the conservatism of the S/N-type distribution factors

utilized for conventional beam and girder bridges in the 1989 AASHTOSpecification.

Figure 1.4.5-1 - Reliability Indices Inherent in the 1989 AASHTOStandard Specification

These calculated reliability indices, as well as past calibrationof other specifications, serve as a basis for the selection of the target

reliability index, T. A target reliability index of 3.5 was selected for theOHBDC and is under consideration for other reliability-basedspecifications. A consideration of the data shown in Figure 1.4.5-1indicates that a of 3.5 is indicative of past practice. Hence, thisvalue was selected as a target for the new calibration.

1.4.6 Load and Resistance Factors

1.4.6.1 LOAD FACTORS

The parameters of bridge load components are summarizedin Table 1.4.6.1-1. These data and Equation 1.3.1-7 enable an initial

set of trial load factors to be calculated in a rational way.

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Lecture - 1-31

Table 1.4.6.1-1 - Parameters of Bridge Load Components

LOAD COMPONENTBIAS

FACTOR

COEFFICIENTOF

VARIATION

Dead Load, D1 1.03 0.08

Dead Load, D2 1.05 0.10

Dead Load, D3 1.00 0.25

Live Load (with impact) 1.10-1.20 0.18

Various sets of load factors, corresponding to different valuesof n, are presented in Table 1.4.6.1-2. The relationship is also shownin Figure 1.4.6.1-2.

Table 1.4.6.1-2 - Considered Sets of Load Factors

LOADCOMPONENT n = 1.5 n = 2.0 n = 2.5

Dead Load, D1 1.15 1.20 1.24

Dead Load, D2 1.20 1.25 1.30

Dead Load, D3 1.375 1.50 1.65

Live Load (withimpact)

1.40-1.50 1.50-1.60 1.60-1.70

Figure 1.4.6.1-2 - Load Factors vs. n (Nowak, 1993)

Recommended values of load factors correspond to n = 2. Forsimplicity of the designer, one factor is specified for D1and D2, =

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Lecture - 1-32

1.25. For D3, weight of asphalt, = 1.50. For live load and impact, thevalue of load factor corresponding to n = 2 is = 1.60. However, amore conservative value of = 1.75 is utilized in the LRFD code.

1.4.6.2 RESISTANCE FACTORS

The acceptance criterion in the selection of resistance factors

is how close the calculated reliability indices are to the target value ofthe reliability index, T. Various sets of resistance factors, , areconsidered. Resistance factors used in the code are rounded off tothe nearest 0.05. For each value of , the minimum requiredresistance, RLRFD, is determined from the following equation:

(1.4.6.2-1)RLRFD

1.25D 1.50 DA

1.75 (L I)

where D is dead load, except of DA, which is the weight of the asphalt

surface. The load factors are equal to the recommended values fromSection 1.4.6.1.

The calculations were performed using the load componentsfor each of the 175 simulated bridges. For a given resistance factor,material, span and girder spacing, a value of RLRFDis calculated usingEquation 1.4.6.2-1. Then, for each value of RLRFDand correspondingloads, the reliability index is computed. For easier comparison with1989 AASHTO, a resistance ratio, r, is defined as

(1.4.6.2-2)rR

LRFD

RHS20

Resistance ratio is an indication of the actual change of the coderequirements. Value of r > 1 corresponds to LRFD code being moreconservative than 1989 AASHTO, and r < 1 corresponds to LRFDbeing less conservative than 1989 AASHTO.

Values of r and were calculated for live load factors, = 1.75.For comparison, the results are also shown for live load factor, =1.60. The calculations are performed for the resistance factors, ,listed in Table 1.4.6.2-1.

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Lecture - 1-33

Table 1.4.6.2-1 - Considered Resistance Factors

MATERIAL LIMITSTATE

RESISTANCEFACTORS,

LOWER UPPER

Non-Composite Steel Moment 0.95 1.00

Shear 0.95 1.00

Composite SteelMoment 0.95 1.00

Shear 0.95 1.00

Reinforced ConcreteMoment 0.85 0.90

Shear 0.90 0.90

Prestressed ConcreteMoment 0.95 1.00

Shear 0.90 0.95

1.4.6.3 RECOMMENDED LOAD AND RESISTANCE FACTORS

The recommended load factors are listed in Table 1.4.6.3-1and recommended resistance factors are given in Table 1.4.6.3-2.

Table 1.4.6.3-1 - Recommended Load Factors

LOAD COMPONENT LOAD FACTOR,

Dead Load (except asphalt overlay) 1.25Dead Load (asphalt overlay andutilities)

1.50

Live Load (including impact) 1.75

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Lecture - 1-34

Table 1.4.6.3-2 - Recommended Resistance Factors

MATERIALLIMIT

STATERESISTANCEFACTOR,

Non-Composite Steel

Moment 1.00

Shear 1.00

Composite SteelMoment 1.00

Shear 1.00

Reinforced ConcreteMoment 0.90

Shear 0.90

Prestressed ConcreteMoment 1.00

Shear 0.90

Reliability indices were recalculated for each of the 175simulated cases and each of the actual bridges from which thesimulated bridges were produced. The range of values obtained usingthe new load and resistance factors is indicated in Figure 1.4.6.3-1.

Comparing the values of reliability indices obtained with the1989 AASHTO Specification and the LRFD Specification indicates thata considerable improvement in the clustering of reliability index valueshas been obtained. This is a direct result of the integration of the loadfactor, resistance factor, accurate load models and suitable resistancemodels. NCHRP 12-33 was not charged to make a wholesale re-adjustment of the inherent safety in the highway system. Selection ofthe target reliability index of 3.5 is consistent with that view. However,a fully consistent philosophy has been established for thespecification, as indicated by the tightly clustered reliability index valueshown in Figure 1.4.6.3-1.

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Lecture - 1-35

Figure 1.4.6.3-1 - Reliability Indices Inherent in LRFD Specification

At any future time, AASHTO may decide that more or less safety(reliability) is desired. Should such a decision be made, a consistentmeans is now available to adjust load factors and resistance factorsto achieve any increment in reliability.

One of the early concerns about the development of aprobability-based LRFD Specification was that it would be used as abasis for reducing the strength of bridges. A comparison was madeof the apparent resistance demands required by the 1989 AASHTOSpecification and the LRFD Specification. For purposes ofcomparison, the "demand" is taken as a sum of the factored loads

divided by a resistance factor. Such a comparison is shown in Figure1.4.6.3-2 for the simulated bridges used in the calibration process,based on the second draft. Some small variations are possible due torefinements made as the Specification developed. This figureindicates that generally slightly more structure will be required basedon the factored loads and the resistance factors alone. A totalcomparison would have to also include any advances in more realisticanalysis methods and resistance formulations which are included inthe LRFD Specification. Some of these features have been, and moremay be, adopted into the Standard Specifications as time goes on.

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Lecture - 1-36

Figure 1.4.6.3-2 - Structural Demand - LRFD vs. Standard

Specification

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Lecture - 1-37

REFERENCES

American Association of State Highway and Transportation Officials,"Standard Specifications for Highway Bridges", Thirteenth Edition,1983, as amended by the Interim Specifications - Bridges, 1984, 1985,Washington, D. C.

British Standards Institute, "British Standard 5400 - Parts 1 through10", London (1978-1984)

Dorton, R. A., "Implementing the New Ontario Bridge Code",International Conference on Short- and Medium-Span Bridges,Toronto (1982)

Gellert, W., Kustner, H., Hellwich, M., and Kastner, H. - Editors, TheVNR Concise Encyclopedia of Mathematics, Van Nostrand Reinhold(1977) pp. 591-600

Harr, M. E., Reliability-Based Design in Civil Engineering, McGraw-Hill

Book Company (1987) pp. 3-4

International Standards Organization, ISO 2394, "General Principlesfor the Verification of the Safety of Structures"

Japan Road Association "Specifications for Highway Bridges, Part I:Common Specifications and Part III: Concrete Bridges" (March 1984)

Nowak, A. S. and Lind, N. C., "Practical Bridge Code Calibration",ASCE, Journal of the Structural Division, Vol. 105, No. ST12(December 1979) pp. 2497-2510

Nowak, A. S., "Calibration of LRFD Bridge Design Code", Departmentof Civil and Environmental Engineering Report UMCE 92-25,University of Michigan, 1993

Ontario Ministry of Transportation and Communications, "OntarioHighway Bridge Design Code", Toronto, Ontario, Canada (1983)

Vincent, G. S., "Load Factor Design of Steel Highway Bridges", AISIBulletin 15 (March 1969)

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Lecture - 2-1

LECTURE 2 - GENERAL FEATURES OF DESIGNS

2.1 OBJECTIVE OF THE LESSON

The objective of this lesson is to familiarize the student with thecontents of the general features portion of the LRFD Specification.

This section of the Specification sets forth a series of safety,geometric, hydraulic and other requirements that need to beconsidered when establishing the site for a bridge.

Further, it defines the objectives of good design in very generalterms. The intent of this section was more than simply a matter ofestablishing certain "laundry lists" to be checked off during design.Rather, it was a reaction to the compartmentization of the designprocess which sometimes separates the Bridge Engineer, theHydraulics Engineer, the Geotechnical Engineer, the TransportationPlanner and the Maintenance Engineer from each other, rather thanallowing them to work collegially to establish the best overall solution

to a transportation system need. It was hoped that reminding theBridge Engineer of these collegial requirements in the bridgespecification would enable the Bridge Engineer to be injected earlierinto the overall consideration of planning route design. If allcomponents of the design team work together, then not only will theimportant hydraulic, geometric and other requirements be met, butthey can be met also accounting for and accommodating, wherepossible, simplified bridge geometrics.

2.2 LOCATION FEATURES

This section of the Specification provides general guidance onbridge siting and introduces the general requirements for crossingflood plains and other waterways. Bridges and bridge approaches onflood plains are required to be compatible with the goals andobjectives of flood plain management as practiced by the variousstates. Basically, it is a question of identifying and mitigating thehydraulic consequences of having substructures units and roadwayembankments in areas apt to be flooded by various design floods.Similarly, the effect of engineered construction on the long-termaggradation or degradation of channel boundaries and depths must beconsidered, both on a local, i.e., local scour, level, but on a watercourse level as well.

The safety considerations related to the bridge as anobstruction to land and marine traffic must also be considered.Various guidelines in the AASHTO Roadside Design Guide areavailable to assist the Bridge Designer in determining whensubstructure units must be protected by guardrail or other barriers.Provisions for confining or redirecting errant vehicles on the roadway,supported by the bridge, are provided in Section 13 of theSpecifications which deals with barriers and railings.

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Lecture - 2-2

Wherever possible and practical, bridges crossing navigablewaterways should be designed with piers outside of the waterway.Where the economic consequences of doing this are severe, thenbridges must be protected against vessel collision by suitable fenders,dikes or dolphins. Guidance on the design of dolphin and fendersystems, as well as ship collision forces, are given in Section 3 of theSpecification. Further guidance can be obtained from several

AASHTO publications, including highway drainage guidelines,hydraulic analysis for the location of design of bridges and theAASHTO Guide Specification for vessel passage.

Vertical and horizontal clearances for highway bridges, andclearances for railroad overpasses are identified in the Specification.Generally speaking, these clearances will be consistent with theAASHTO policy on geometric design of highway and streets and theAREA Manual for Railway Engineering, Chapters, 7, 8, 9, 15 and 18.Various numerical data are presented in this portion of theSpecification, which may be subject to jurisdictional differences, butthe following general principles should apply:

Reductions and vertical clearances, due to settlement,overlaying of roadways being crossed or other factors shouldbe included in establishing the overall clearance requirement.

Somewhat higher clearances should be established for signsattached to structures and for the vertical clearance tooverhead structural members, such as truss portals.

No object on or under a bridge other than barriers should belocated closer than 1200 mm to the edge of the designatedlane, and the inside face of the barrier should not be closer

than 600 mm to either the face of the object or the edge of thedesignated traffic light.

2.3 FOUNDATION INVESTIGATIONS

Foundation investigations are a necessary part of the siting ofany structure. Article 10.4 of the Specification provides a list ofpossible requirements of a subsurface investigation program.

2.4 DESIGN OBJECTIVES

This group of articles in the Specification identifies a series ofobjectives for all bridge design. They are: safety, serviceability,constructibility, economy and bridge aesthetics.

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Lecture - 2-3

2.4.1 Safety

The Specification clearly establishes that public safety is theprimary responsibility of the Design Engineer. All other aspects ofdesign, including serviceability, maintainability, economics andaesthetics are secondary to the requirement for safety. This does notmean that other objectives are not important, but safety is paramount.

All comprehensive design specifications are written to establishan acceptable level of safety, as explained in Lecture 1. There aremany methods of attempting to quantify safety and the methodinherent in the LRFD Specification is probability-based reliabilityanalysis. The design vehicle for treating safety issues in the LRFDSpecification is the establishment of "limit states" to define groups ofevents or circumstances which could cause a structure to beunserviceable for its original intent.

2.4.1.1 LIMIT STATES

The LRFD Specification is written in a probability-based limitstate format requiring examination of some, or all, of the four limitstates defined below for each design component of a bridge.

Service Limit States

Service limit states are restrictions on stress, deformation andcrack width under regular service conditions. They are intended toallow the bridge to perform acceptably for its service life.

Fatigue and Fracture Limit States

Fatigue and fracture limit states are restrictions on stress rangeunder regular service conditions reflecting the number of expectedstress range excursions. They are intended to limit crack growthunder repetitive loads to prevent fracture during the design life of thebridge.

Strength Limit States

Strength limit states are intended to ensure that strength andstability, both local and global, are provided to resist the statisticallysignificant load combinations that a bridge will experience in its designlife. Extensive distress and structural damage may occur under

strength limit states, but overall structural integrity is expected to bemaintained.

Extreme Event Limit States

Extreme event limit states are intended to ensure the structuralsurvival of a bridge during a major earthquake, or when collided by avessel, vehicle or ice flow, or where the foundation is subject to thescour which would accompany a flood of extreme recurrence, usuallyconsidered to be 500 years. They are considered to be unique

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Lecture - 2-4

occurrences whose return period is significantly greater than thedesign life of the bridge.

Limit States Design Equation

As the basis of LRFD methodology, each component andconnection are to satisfy Equation 2.4.1.1-1 for each limit state. All

limit states are of equal importance.

iiQi Rn= Rr (2.4.1.1-1)

where:

i = DRI: = DRI 0.95 for loads for which amaximum value of i is appropriate, and,

for loads for which a minimumi1

IDR

1.0

value of iis appropriatei = Load Factor: A statistically based multiplier on forceeffects

= Resistance Factor: A statistically based multiplierapplied to nominal resistance

i = Load Modifier D = A factor relating to ductilityR = A factor relating to redundancyI = A factor relating to operational importanceQi = Nominal Force Effect: A deformation, stress or stress

resultantRn = Nominal Resistance: Based on the dimensions as

shown on the plans and on permissible stresses,deformations or specified strength of materials

Rr = Factored Resistance: Rn

Ductility, redundancy and operational importance aresignificant aspects affecting the margin of safety of bridges. While thefirst two directly relate to the physical strength, the last concerns theconsequences of the bridge being out of service. The grouping ofthese aspects is, therefore, arbitrary, however, it constitutes a firsteffort of codification. In the absence of more precise information, eacheffect, except that for fatigue and fracture, is estimated as 5%,accumulated geometrically, a clearly subjective approach. With time,improved quantification of ductility, redundancy and operationalimportance, and their interaction, may be attained.

Ductility

The response of structural components or connections beyondthe elastic limit can be characterized by either brittle or ductilebehavior. Brittle behavior is undesirable because it implies the suddenloss of load carrying capacity immediately when the elastic limit isexceeded. Ductile behavior is characterized by significant inelastic

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Lecture - 2-5

deformations before any loss of load carrying capacity occurs. Ductilebehavior provides warning of structural failure by large inelasticdeformations. Under cyclic loading, large reversed cycles of inelasticdeformation dissipate energy and have a beneficial effect on structureresponse.

If, by means of confinement or other measures, a structural

component or connection made of brittle materials can sustaininelastic deformations without significant loss of load carrying capacity,this component can be considered ductile. Such ductile performanceshall be verified by experimental testing.

In order to achieve adequate inelastic behavior, the systemshould have a sufficient number of ductile members and either:

joints and connections which are also ductile and can provideenergy dissipation without loss of capacity; or,

joints and connections which have sufficient excess strength

so as to assure that the inelastic response occurs at thelocations designed to provide ductile, energy absorbingresponse.

Behavior which is ductile in a static context, but which is notductile during dynamic response, should also be avoided. Examplesof this behavior are shear and bond failures in concrete members, andloss of composite action in flexural members.

Past experience generally indicates that typical bridgecomponents designed in accordance with these specifications exhibitadequate ductility. Connection and joints require special attention to

detailing and the provision of adequate load paths.

For unusual and important structures in high-seismic zones,the Owner may specify a minimum ductility factor as a demonstrationthat ductile failure modes will be obtained. The factor may be definedas:

(2.4.1.1-2)u

y

where:

u- Deformation at ultimatey- Deformation at the elastic limit

The ductility capacity of structural components or connectionsmay either be established by full or large scale experimental testing,or with analytical models which are based on realistic materialbehavior. The ductility capacity for a structural system may be

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Lecture - 2-6

determined by integrating local deformations over the entire structuralsystem.

The special requirements for energy dissipating devices areimposed because of the rigorous demands placed on thesecomponents.

Given proper controls on the innate ductility of basic materials,proper proportioning and detailing of a structural system are the keysconsideration in ensuring the development of significant, visible,inelastic deformations, prior to failure, at the strength and extremeevent limit states.

It may be assumed that the requirements for ductility aresatisfied in a concrete structure in which the resistance of aconnection is not less than 1.3 times the maximum force imposed onthe connection by the inelastic and ductile actions of the adjacentcomponents.

For the fatigue and fracture limit state for fracture-criticalmembers and for the strength limit state for all members:

D 1.05 for non-ductile components and connections,

= 1.00 for conventional designs and details complyingwith these specifications

0.95 for components and connections for whichadditional ductility-enhancing measures have beenspecified beyond those required by theseSpecifications

For all other limit states:

D = 1.00

Redundancy

Multiple load path structures should be used, unless there arecompelling reasons to the contrary.

For the strength limit state:

R 1.05 for non-redundant members

= 1.00 for conventional levels of redundancy

0.95 for exceptional levels of redundancy

For all other limit states:

R = 1.00

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Lecture - 2-7

Operational Importance

The concept of operational importance is applied to thestrength and extreme-event limit states. The Owner may declare abridge or any structural component and connection, thereof, to be ofoperational importance. If a bridge is deemed of operationalimportance, Iis taken as 1.05. Otherwise, Iis taken as 1.0 for

typical bridges and may be reduced to 0.95 for relatively lessimportant bridges.

Such classification should be based on social/survival and/orsecurity/defense requirements.

2.4.1.2 LOAD FACTORS AND LOAD COMBINATIONS

The following permanent and transient loads and forces shallbe considered:

Permanent Loads

DD = DowndragDC = Dead load of structural components

attachmentsDW = Dead load of wearing surfaces and utilitiesEF = Dead load of earth fillEH = Horizontal earth pressureES = Earth surcharge loadEV = Vertical earth pressure

Transient LoadsBR = Vehicular braking forceCE = Vehicular centrifugal force

CR = CreepCT = Vehicular collision forceCV = Vessel collision forceEQ = EarthquakeFR = FrictionIC = Ice loadIM = Vehicular dynamic load allowanceLL = Vehicular live loadLS = Live load surchargePL = Pedestrian live loadSE = SettlementSH = Shrinkage

TG = Temperature gradientTU = Uniform temperatureWA = Water load and stream pressureWL = Wind on live loadWS = Wind load on structure

The load factors for various loads, comprising a design loadcombination, are indicated in Table 2.4.1.2-1. The larger of the twovalues for load factors shown for TU, TG, CR, SH and SE are to beused when calculating deformations; the smaller value shall be used

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Lecture - 2-8

when calculating all other force effects. For each load combination,every load that is indicated, including all significant effects due todistortion, should be multiplied by the appropriate load factor. Allrelevant subsets of the load combinations in Table 2.4.1.2-1 should beinvestigated.

The factors should be selected to produce the total extreme

factored force effect. For each load combination, both positive andnegative extremes should be investigated. In load combinationswhere one force effect decreases the effect of another, the minimumvalue should be applied to the load reducing the force effect.

For permanent force effects, the load factor which producesthe more critical combination shall be selected from Table 2.4.1.2-2.In the application of permanent loads, force effects for each of thespecified six load types should be computed separately. Assumingvariation of one type of load by span, length or component within abridge is not necessary. For each force effect, both extremecombinations may need to be investigated by applying either the high

or the low load factor as appropriate. The algebraic sums of theseproducts are the total force effects for which the bridge and itscomponents should be designed. This article reinforces the traditionalmethod of selecting load combinations to obtain realistic extremeeffects, and is intended to clarify the issue of the variability ofpermanent loads and their effects.

When the permanent load increases the stability or load-carrying capacity of a component or bridge, the minimum value of theload factor for that permanent load shall also be investigated. Uplift,which is treated as a separate load case in past editions of theAASHTO Standard Specification for Highway Bridges, becomes a

Strength I load combination. For example, when the dead loadreaction is positive and live load can cause a negative reaction, theload combination would be 0.9DC + 0.65DW + 1.75(LL+IM). If bothreactions were negative, the load combination would be 1.25DC +1.50DW + 1.75(LL+IM).

The various load combinations shown in Table 2.4.1.2-1 aredescribed below.

STRENGTH I - Basic load combination relating to the normalvehicular use of the bridge without wind.

STRENGTH II - Load combination relating to the use of thebridge by permit vehicles without wind. If apermit vehicle is traveling unescorted, or ifcontrol is not provided by the escorts, the otherlanes may be assumed to be occupied by thevehicular live load herein specified. Forbridges longer than the permit vehicle, additionof the lane load, preceding and following thepermit load in its lane, should be considered.

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Lecture - 2-9

STRENGTH III - Load combination relating to the bridgeexposed to maximum wind velocity whichprevents the presence of significant live loadon the bridge.

STRENGTH IV - Load combination relating to very high deadload to live load force effect ratios. This

calibration process had been carried out for alarge number of bridges with spans notexceeding 60 m. Spot checks had also beenmade on a few bridges up to 180 m spans. Forthe primary components of large bridges, theratio of dead and live load force effects israther high, and could result in a set ofresistance factors different from those foundacceptable for small- and medium-spanbridges. It is believed to be more practical toinvestigate one more load case, rather thanrequiring the use of two sets of resistance

factors with the load factors provided inStrength Load Combination I, depending onother permanent loads present. This LoadCombination IV is expected to govern when thedead load to live load force effect ratio exceedsabout 7.0.

STRENGTH V - Load combination relating to normal vehicularuse of the bridge with wind of 90 km per hourvelocity.

EXTREME EVENT - Load combination relating to ice load, collision

by vessels and vehicles, and to certainhydraulic events and earthquakes whoserecurrence interval exceeds the design life.The joint probability of these events isextremely low, and, therefore, they arespecified to be applied separately. Underthese extreme conditions, the structure isexpected to undergo considerable inelasticdeformation by which locked-in force effectsdue to TU, TG, CR, SH and SE will be relieved.The 0.50 live load factor signifies a lowprobability of the presence of maximum

vehicular live load at the time when extremeevents may occur.

SERVICE I - Load combination relating to the normaloperational use of the bridge with 90 km perhour wind. All loads are taken at their nominalvalues and extreme load conditions areexcluded.

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Lecture - 2-10

SERVICE II - Load combination whose objective is to preventyielding of steel structures due to vehicular liveload, approximately halfway between that usedfor Service I and Strength I limit state, for whichcase the effect of wind is of no significance.This load combination corresponds to theoverload provision for steel structures in past

editions of the AASHTO Standard Specificationfor the Design of Highway Bridges.

SERVICE III - Load combination relating only to prestressedconcrete structures with the primary objectiveof crack control. The addition of this loadcombination followed a series of trial designsdone by 14 states and several industry groupsduring 1991 and early 1992. Trial designs forprestressed concrete elements indicatedsignificantly more prestressing would beneeded to support the loads specified in the

proposed Specifications. There is nonationwide physical evidence that thesevehicles used to develop the notional live loadshave caused detrimental cracking in existingprestressed concrete components. Thestatistical significance of the 0.80 factor on liveload is that the event is expected to occurabout once a year for bridges with two designlanes, less often for bridges with more than twodesign lanes, and about once a day for thebridges with a single design lane.

FATIGUE - Fatigue and fracture load combination relatingto gravitational vehicular live load and dynamicresponse, consequently BR and PL need notbe considered. The load factor reflect

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