EVALUATION OF EXISTING BRIDGES USLNC; ADVANCED RELIABIUIY METHODS

Department of Civil Engineering And Appiied Mechanics

McGiil University

Montreal, Canada

A thesis submitted ta the Facuity of Graduate Studies and Uesearch in partiai firIfilfment of the requirements for the degree of Master of Engineering.

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ABSTRACT

A rnethodology to evaluate structurai reliability of existing bridge structures wing advanced

reliability FORM/SORM (First Order Reliability Method / Second Order Reliability Method)

methods is presented. The rnethodology is applied to evaluate the structural reliability of

bridge steel members subject to compression, tension, shear and bending. The steel truss

rnembers, floor beam and süïnger of a typical span located on the Jacques-Cartier Bridge in

Montreal are evaluated using this methodology. Data for the & d o n was obtained frorn a

comprehensive inspection and sampling program on the bridge involving tests on 74 steel

coupons. The results of the evaluation are then compared to the results of the evaluation

using the load and mistance factors method and the mean toad method outlied in the drafi

version of the CHBDC, 1998, (Canadian Highway Bridge Design Code).

In this application, the advanced reliability analysis indicates that the estimation of the

reliability of bridge members, in their current condition, using the foads and resistance

factors method specified in the code may be t w conservative. This situation was encountered

in the case of the tioor beams of the analysed span, when the failure mode considered was

shear, The advanced reliabitity anaiysis indicates also in this appIication, that the evaluation

of the reliability of bridge members, using the mean Ioad method specified in the code may

not be conservative. This situation was encountered in the case of the truss members of the

analysed span, when the filure mode considered was compression.

The evaiuation of bridge structures using advanced structural reliability methods can be used

to obtain more accurate estimates of their reliability. This information can be used to develop

better, monitoring, inspection, maintenance, and rehabilitation strategies for bridge members.

Une méthode pour i'éduation des structures &tes des ponts, utiIisant les méthodes

avancées d'analyse de fiabilité (Méthode d'anaIyse de 6abiIité de premier ardre et méthode

d'analyse de fiabilité de second ordre) est présentée. La méthode présentée est utilisée pour

l'analyse de fiabilité des membrures en acier des ponts, soumises a des efforts de

compression, de tension et de cisaillement et à des &rts engendrés par des moments de

flexion. Les membrures des fermes, la poutre transversale et le Iongeron d'une travée typique

au Pont Jacques-Cartier, à Montd , sont Mués suivant cette méthode. Les données pour

I'évaiuation ont été obtenues a partir d'un programme d'inspection et d'échantilIonnage au

pont comprenant des essais sur 74 échantillons en acier. Les résultats de l'évaluation sont

comparés a ceux de la méthode des tàcteurs de pondération des charges et de la résistance et

la méthode de charge moyenne, teks que présentées dans la version préliminaire (1998) du

code canadien des ponts-routes.

Dans cette analyse, l'estimation de la fiabilité suivant [es méthodes avancées dPanaIyse de

fiabilité indique que l'estimation de la fiabiIité des membrures des ponts, dans leur état

actuel suivant la méthode de pondération des charges et de la résistance décrite dans Ie code

des ponts, peut ètre conservatrice. Cette situation a été consratée dans le cas des poutres

tramversaies de la travée analysée, quand ie mode de rupture par cisaillement a été. considéré.

Cette d y s e indique aussi que l'estimation de la tiabiIité des membrures des ponts, en

utilisant la méthode de la charge moyenne décrite dans le code des poms peut s'avérer

parfois non c o n s e d c e . Cette situation a été constatée dans le cas des membrures des

fermes de la travée anaiysée, quand Ie mode de rupture par compression a été considéré.

L'évahation des structures des ponts en utilisant les méthodes avancées d'analyse de fiabilité

peut être utilisée pou obtenir une estimation plus précise de la fiabilité de ces structures.

Cette Uiformation peut être utiIisée pour établir de meilleures stratégies d'inspection,

d'entretien et de réfection des membrures des ponts.

1 . MTRODUCTION

1.1 SCOPE OF WORK ...................... ..., ........................................................ 1

t . 2 LITERATURE REVIEW ................... .. .................................................. - 3

.................................................................... 1 -3 MAJOR CONTRIBUTIONS -6

2 . REVEW OF ADVANCED RELIABILITY METEIODS

..... 2.1 RELïABILïïY ANALYSIS OF STRUCTURES .......................,.... ,. 9

............................... 2.2 FIRST-ORDER-SECOND-MOMENT METHODS 10

................................................................. 2.3 MEAN VALUE METHODS 11

.......... ........................ 2.4 ADVANCED RELIABLITY METHODS .. 1 4

2.5 APPROXIMATE METHODS FOR MCLLJDING NORMATION

ON DISTRIBUTIONS ............................................................................. 17

........ 2.6 COMPUTER PROGRAMS FOR THE FORM/SORM MEXHODS -18

3 . 3.1 SCOPE .................................................................................................... 19

3.3 CHBDC PROVISIONS FOR BRIDGE EVALUATION ..................... ..... 20

3.3 THE TARGET RELIABILiTY INDEX P ................................................ 20

............... .......*...................................... 3.3. I TYPE OF TRAFFIC .. 21

..................... 3 .3.2 SY STEM BEHAWOüR ................................. ...,,., -23

3 -3 -3 ELEMENT BEHAVIOUR ........................................................... 24

3.3.4 INSPECTION LEVEL ................................................................. 25

3 .3.5 iME'ORTANT STRUCTURES ................................................. -25

3.4 EVALUATION USING THE LOAD AND RESISTANCE

FACTOR METHOD ................................................................................ 26

3 .4.1 DEAD LOAD FACTORS ao ....................................................... 26

3.4.2 L M LOAD FACTORS at ........................................................ 27

3.4.3 LIVE LOAD CAPACITY FACTOR ............................................ 28

......................... 3.5 EVALUATION USING THE MEAN LOAD METHOD 29

3.6 DETERMINATION OF THE YIELD STRENGTH OF STEEL

FROM TESTS ON COUPONS ........................................................... 3 I

4 . BIUDGE EVALUATION USING FORMBORM METHODS

4.1 SCOPE .................................................................................................... 33

4.2 THE LIha STATE FUNCTION ........................................................... 34

4.3 THE LIMIT STATE FUNCTlON FOR MEMBERS UNDER

COMPRESSION AND FLEXURAL BUCKLING DUE

TO COMPRESSION ............................................................................... 37

4.3. I THE RESISTANCE VARIABLE XR ......................................... 37

4.3 -2 THE PROFESSIONAL VARIABLE Xp ....................................... 41

4.4 THE i,iMlT STATE FüNCTION FOR MEMBERS UNDER

TENSION ................................................................................................ 45

4.4.1 THE RESISTANCE VARIABLE XR .......................................... 45

4.4.2 THE PROFESSIONAL VARIABLE Xe ..................................... 46

.... 4.5 THE LIMIT S T A E FUNCTION FOR MEMBERS UNDER SHEAR 46

4.5.1 THE RESISTANCE VARIABLE XR ................. .......,. ..,.. ... -46

4.5.2 THE PROFESSIONAL VARIABLE Xp ...................................... 48

4.6 THE LIMIT STATE FUNCTION FOR LATERALLY SUPPORTED

AND LATERAUY UNSUPPORTED MEMBERS SUBJECT

TO BENDING ........................................................................................ -48

4.6.1 THE RESISTANCE VARIABLE XR .......................................... 48

...................................... 4.6.2 THE PROFESSIONAL VARIABLE X p 51

4.7 SUBROUTINES FOR THE LIMIT STATE FUNCTIONS ...................... 52

5 . EXAMPLE : EVALUATlON OF A STEEL TRUSS SPAN ON

THE JACOUESCARTlER BRIDGE

5.1 SCOPE .......................... .... ...... .... ........................... 54

5.2 DESCRIPTION OF THE JACQUES-CARTIER BRIDGE ...................... 54

................................................. 5.3 DESCRIPTION OF THE TRUSS SPAN 55

5.4 MATERIAL PROPERTES OF THE STEEL USED IN

THE CONSTRUCTION OF THE TRUSS SPAN ..............................-.-S.. 59

5.5 EVALUATION OF THE TRUSS SPAN ACCORDNG TO CHBDC ..... 65

5.6 EVALUATION OF THE TRUSS SPAN ACCORDING TO THE

FORWSORM METHOD ..... . .......+..... .........-...... ......................... ... . ... .-. .- 67

5.7 COMPARISON OF THE RESULTS ................................................. ...... 68

6. B:

6.1 SCOPE ......................................................................................... 76

6.2 MATERIAL CONDITION RATING SY STEM . . .. .. . ... . .. . ... . . .. . . . . . . . . . . . . -.. 76 6.3 RESISTANCE REDUCTION FACTOR AR .--......-.......,-.--................ ..-... 78

6.4 EXAMPLE: EVALUATION OF A DETENORATED FLOOR

BEAM IN SHEAR ON THE TRUSS SPAN OF THE JACQUES-

CARTIER BRIDGE ......... .... . . .. . .. .. . ... .... . .. .. ... ... . . . . .. . . . . . . . . . . . . . . 8 1

CONCLUSION ............................ .. .. ......... ..... .. .... ....... . ..... ... ............. ...+--..-...... 84

APPENûM A .........................................-..........-..-..---..*.-.-......................-..-..--.. 90

TABLES Al TO A6

EVALUATION OF THE TRUSS MEMBERS, THE STRINGER AND THE FLOOR BEAM USING THE LOAD AND RESISTANCE FACTORS METHOD AND THE MEAN LOAD METHOD. TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE.

APPENDIX B .. . .. . .. .. ... .. . . .. . . . . . . . . . . .-.. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

SUEROUTiNES FOR THE LIMIT STATE FUNCTIONS.

Figure 2.1

Figure 2.2

Figure 3.1

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 6.1

iiiustration of the Reliability index Concept ............................................ 12

Limit Staîe Functions in the Original and Redud Coordinate Systems ... 16

Configuration of the T d c Design Load CL1-W .................................... 22

General Layout of the Jacques-Cartier Bridge .......................................... 56

Typical Cross Section at the Truss Span of the Jacques-Cartier Bridge ..... 57

Steel Skekon of the T m Span of the Jacques-Cartier Bridge ................ 58

Numbering of the T m Members of the Tniss Span of the

Jacques-Cartier Bridge ........................................................................ 7 1

Material Condition Rating of Cornponents .............................................. 80

Table 3.1

Table 3.2

Table 3.3

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 4.7

Table 4.8

Table 5.1

Target reiiability index 8, fbr CLI-W, CLZW, CU-W,

PA, PB, and PS traffic .............................................................................. 26

Dead load factors a~ ................................................. ... ....................... 27

Live load factors a ~ , Normal TraflCic, (CL1 -W, CL2-W and CL3-W) ....... 28

Statistical parameters and distribution functions for

the dead load variables ........................................................................ 36

Statistical parameters and distribution tùnctions for

........................................................... the trafic load m-ables.. ........ ,. -36

StatisticaI parameters and distribution fiinctions for the

lateral distribution of the live load.. .......................................................... 37

Statisticai parameters and distribution functions for the

dynamic load ailowance ............ .. ......................................................... 37

Statistical parameters and diiiution fiinctions for the

material properties of roilecl and welded W shapes ................................... 39

Statisticai parameters and d ibu t ion finctions

for the geometricd properties of roileci W shaped ............................ ..,.. 41

Statisticai parameters and distribution hnctions for the

professional variable Xp of mUed W shapes under compression ............... 44

Statistical parameters and distribution functions for the

...................... professionai variable XP of mUed W shapes under bending 52

Tests r d î s on coupons of the specid carbon steeI

............................................................ from the Jacques-Cartier Bridge.. -6 1

Reiiability analysis of the auss members in compression - Tmss span

of the Jacques-Cartier bridge - Cornparison of results for the

FORMISORM method, the loads and resistance factors method,

..................................................... and the mean load method in CHBDC. 72

Table 5.3

Table 6.1

Table 6.2

Table A 1

Table A2

Table A3

Table A4

Table A5

Table A6

Reliability analysis of the tmss mernbers in tension - Truss span

of the Jacques-Cartier bridge - Comparison of results for the

FORMISORM meîhod, the loads and mistance mors method,

and ihe mean load rnethod in CHBDC ...................................................... 73

RdiabiIity analysis of the stringer in shear and bending - Truss span

of the Jacques-Cartier bridge - Cornparisan of muits f6r the

FORM/SORM method, the loads and raistance îàcton method,

and the mean lod method in CHBDC .............. ,.,, .................................... 74

Reliability analysis of the floor beam in shear a d bending - T u s span

of the JacquMartier bridge - Comparison of resuits for the

FORM/SORM method, the Ioads and resistance factors method,

and the mean load method in CHBDC ........................... ,...,.... ................... 75

Proposai values for the resistance reduction fâctor AR ............................. 81

Flour beam in shear - Truss span of the Jacques-Cartier bridge - Comparison of resu1ts for the d u a t i o n of the live Ioad capacity

factor acwrding to the material condition rating of the flmr beam,

using the loads and resistance hctors method, the mean load method

and the FORMlSORM meth od... ........ .-.. ............................................... 83

Evaluaîion of the tnrss mernbers of the tniss span of the Jaques-Cartier

........ bridge using the loads and resistance factors method in the CHBDC 91

Evaiuation of the truss members of the tniss span of the

Jacques-Cafier bridge using the mean load method in the CHBDC ......... 97

EvaIuation of the svinger of the truss span of the JacquesXartier

bridge using the loads and mistance factors method in the CHBDC ........ 99

EvaIuation of the stringer of the tm span of the Jacques-Cartier

............................... bridge using the mean Ioad method in the CHBDC 100

Evaluation of the floor beam of the tniss span of the Jacques-Cartier

...... bridge using the loads and r&stance factors method in the CHBDC IO1

Eduation ofthe ff wr beam of the truss span of the Jacqudartier

................................ bridge using the mean Ioad method in the CHBDC 102

1. INTRODUCTION

SCOPE OF WORK

The evaiuation of the structural performance of a wnstnicted hcility is

subjected to many uncertainties due to uncertainties in loads, material properties,

system responses, geometrid characteristics and anaiysis procedures.

in recognition of these uncertainties, it has become wmmon practice to

evaiuate the performance of a stnrcture in terms of its reliability or its converse, its

probability of failure.

The practice of analysing and ver@ing the structural reliability of a structure

is accepted in seved areas such as offshore and marine engineering, nuclear

engineering and aerospace engineering and in the broader field of civil engineering.

Advanced reliability analysis procedures in combination with efficient

computational algorithm are the basic twls for the evaluation of these structures.

The present research thesis proposes the use of advanced stnictrrrai reiiabiIity

methods and computationd tools for the evaiuation of existing bridges and as an

objective basis for establishimg optimal inspection, monitoring, maintenance and

replacement stmtegies.

in the current bridge code C M S A S6-88 (1988), and in the draft version of

the CHBDC (CANADIAN HlGHWAY BRlDGE DESIGN CODE) (1998), which

will supetsede the present d e , bridge structures are evaiuated reIative to a target

reiiability index B. The targd teliability index is a hnction of the type of traffic for

which the bridge is evaiuated, the system behavior, the behavior of the efement being

evaluated, the andysis method used, and the inspection levei of the bridge. For the

given target reliability index, &ce and load fàctors are provided by the code and

used to evaiuate the bridge and its components. The load and resistance factors are

meant to address a wide spectrwn of loads and resistances and may lead to inaccurate

but usually conservative evaiuations for specific structures.

As an alternative to the load and resistance factors method, the commentary of

CAN/CSA-S6-88 (1988), descriies the Mean Load Method. This method is

scheduld to be forrnaily included in the new code according to the dmft version of

the CHBDC (1998). The advantage of this method is that it dows the evaluator to

include specific information on loads and resistance for a given structure. This is

achieved through the use of the bias coefficient 6 (ratio of the mean to nominal

values) and the coefficient of variation V (ratio of the standard deviation to the mean

value) for any variable considered in the evaiuation, when the evaiuator has

information on these statistical parameters. However, this method does not aIIow the

evaluator to include information on the statistical distribution of the variables, if this

information is available. A h , the uncertainty on a given failure mode is introduced

through a bias factor and a coefficient of variation; however, the actuai mechanid

fomuiation of the tàiiure mode is not considered.

in the present research thesis, a methodology for evaiuating existing bridge

structures using the FORM/SORM methods (FIRST ORDER RELIABILITY

METHODISECOND ORDER RELIABILITY METHOD) is introduced. The use of

these advanceci reliability methods aiiows the evaiuator to introduce information on

the distribution finction for any variable used in the evaluation and the i i i i t state

function for each failure mode considered. This additionai information can be used to

obtain more accurate evaluations of the reliability of a given structure.

in the foiiowing sections, the FORhiYSORM reliability method is briefly

described. This method is then applied to the dua t ion of a steel bridge structure

where the statistical distribution of the yield stress of the steel members is knom

Difrent mechanid failure modes, namely tension, compression, shear and bending

are cansidered in the apptication of this method. The results are then wmpared to the

results obtained with the load and resistance h o r s method and the Mean Load

Method as specified in the CHBüC.

The use of advanced structurai reEability methods in structurai analysis !us

been mainly aimed at deriving loads and resistance factors to be used in building and

stmcturai codes. Elîingwood et al. (1980) used firstsrder-second-moment reiiability

procedures to develop load factors and correspondhg load definitions for the

American National Standard A58 for different types of buildiig matends (e-g.,

structurai steel, reinforced and pre-stressed concrete, heavy timber, engineered

masonry, cold fonned steel, aluminum). The methodology aIsa included resistance

factors ($) for the various matenal groups consistent with the load factors and the

reliability [evels.

Kennedy and Gad M y (1980) used first-order-second-moment reliability

procedures to determine resistance factors for colurnns and beams made fiom rolled

W, welded W, and class H holiow structural steel sections as produced in Canada,

based on material data and geometnc properties obtained fiom Canadian mills.

Kennedy and Baker (1984) derived resistance factors for steel highway

bridges using advanced reliabiIity methods. They used Monte Car10 simulation

techniques to derive resistance fàctors for the fully plastic moment resistance, the

yield moment resistance, the inelastic buckiing moment resistance, the moment

resistance of composite sections, and column resistance for dendemess parameter

vaiues of0.8, 1.0 and 1.2 ( refer to section 4.3.1 of the present research thesis for the

definition of the slenderness pammeter A). Based on the findings of their research,

they recommended the use of a generai resistarice factor of 0.93 for aH types of

resistance and bridges studied.

Moses and Verna (1987) used advanced reliability methods to formulate a

reliabiiity-based strategy for the evaluation of existing steel girders and prestressed

concrete highway bridges in the United States. Load and resistance fkctors have been

recommended that lead to consistent and uniform reliability levels for the evaluation

of these bridges. Detailed guidelines for the evaluation of this type of bridges have

been given in a format suitable for inclusion in the American Association of Staîe

Highway and Transportation Oficiais (AASHTO) Maintenance Inspection Manual.

Tabsh and Nowak (1991) used Monte CarIo simulation techniques to establish

the reliability of highway girder bridges. Reliability indices were dculated for non-

composite and composite steel girderq reinforceci concrete T-beams and prestressed

concrete girders. The dculations were performed for girders and structurai systems

composed of 5 steel girders. The effect of correlation between the strength of girders

in the same bridge was considered. ReiiabiIity indices for the stnictual systems were

found to be higher than for the girders. Sensitivity functions were developed for the

various parameters related to the considerd girders. The results indicate the

importance of resistance parameters such as yield stress of steel or steel cross-

sectional area.

Kennedy et ai. (1992) developed load and resistance factors for the evaluation

of Canadian highway bridges using firstsrder second-moment analysis techniques.

The load and resistance factors were developed for a range of target values between

2.0 and 3.75, of the retiabili index B @fer to section 2.3 of the present research

thesis for the definition of the reliability index B). Dead load factors were estaùlished

for steel girders, concrete girders, concrete bridge decks, and wearing d a c e s - Live

load factors were aiso deveioped for four categories of traffic.

Nowak and Gmuni (1994) used a 6rst order retiability method to caiculate

load and resistance factors for the Ornario Highway Bridge Design Code (OHBDC)

1991 edition The work invoIvd the devdopment of load and resistance models and

the calculation of reliability indices. The calculations were perfomed for bridge

girders designed using OHBDC (1983). The reSuIting reliability indices were between

3 and 4 for steel girders and reinforced concrete T-beams and between 3.5 and 5 for

prestressed concrete girders. The acceptance criterion in the selection of load and

resistance factors was the closeness to the target reiiability level.

Nowak et al. (1994) used Monte Cario techniques and statisticai models to

calculate the bias factor (mean-to-nominal ratio) and coefficient of variation of the

existing moment and shear capacity of reinforced concrete T-beams and prestressed

concrete AASHTO girders. The statistical parameters can be used as a basis for the

deveiopment of design and evaluation criteria for concrete bridge components.

Nowak (1995) used a tirst order reliability method to derive load and

resistance factors to be used in the AASHTO load and resistance factor design

(LRFD) bridge code. A new load model was proposeci, which provides a consistent

safety margin for a wide spectnim of spans. The proposeci ci-ynamic load model takes

into account the effect of road roughness, bridge dynamics and vehicle dynamics.

Statisticai models of resistance (load-carrying capacity) were summarited for non

composite steel, composite steel reinforced concrete and prestressed concrete bridges

Advanced reliability rnethods have also been used with cost benefit analysis to

establish optimum repair strategies for existing bridges.

Enright and Frangopol (1998) used Monte Carlo simulation to find the

cumulative-time system fiilure reliability of reinforced concrete highway girder

bridges subject to damage h m environmemal attack, such as alkali-silica reaction,

corrosion and fieeze-thaw. A time-variant series system reliability-approach, in which

both load and resistance are thedependent, was used. An existing reinforced-

concrete-T-beam bridge Iocated near PuebIo, Colorado, was investigated. The effects

of various parameters on the time Vanant bridge reliability, such as variability in dead

and iive Ioads, iive Ioad occurrence rate, strength Ioss rate, degradation initiation

tirne, resistance correlation and wmber of girders under attack, were studied. The

results can be used to better predict the remaining life of deteriorating reinforced

concrete bridges, and to develop optimal Iifetime reliability-based maintenance

strategies for these bridges.

Stewart and Val (1999) us4 Monte Car10 simulation techniques and a

reliability-based approach for the assessrnent of aging bridges. The effect of load

history (proof loads and prior senrice loads) was used to evaluate the reliability of a

reinforced concrete bridge subject to varying degrees of deterioration. The estimates

of reliability were used in a riskast benefit analysis. It was found that proof load

testing may not be cost effective if the wsts of bridge Mure (unsuccessfùl test) and

the test itseif are considered in a pretiminary cost-benefit analysis. The information

on load history increased significantIy the reliabiIity estimates of existing bridges.

Estes and Frangopol (1999), used a 6rst order reliability method approach,

similar to the one used in the present research, to determine the reliability of different

components of a Colorado State Highway bridge. Compoaents reliabilities were then

used in a system reliability approach to optimize the lifetime repair strategy of the

bridge, The bridge was modeted in a series-parailel combination of failure modes, and

the reliability of the overd1 system was computed using tirnedependent deterioration

models and live load modets. Based on an established repair criterion, available repair

options, repair costs, and updating, the optimal repair strategy was developed.

1.3 MAJOR CONTRIBUTIONS

Most of the previous wodc done b r the evaluation of bridge structures was

aimed at establishing load and resistance &ors to be used in order to ensure tbat the

target reliability index B is achievd Therefore, the Ioad and resistance tàctors

derived had to encompass diment modes of Eiilure and diierent scenarios for the

combination of the Ioads and resistance.

In the present research thesis, as in the recent work of Estes and FrangopoI

(1999), the reliability of each component of the bridge is evaluated individually. The

mathematical formuiation of the failure modes of each component is defined in limit

state fùnctions. The statistical properties of each randorn variable used in the limit

state fimctions are estimated or determined f?om published data The FORMISORM

method is then used to estimate the current reliabiiii of the component mder

andysis and compare it to the target reliability. Reliabilities are only computed at the

component level and the model does not attempt to predict the changes in reliability

as a function of the . These features could be inchdeci in fbture development of the

model but are beyond the sape of the research thesis. Only component reiiabilities

are necessary in the context of the thesis since the major objective of the research is to

compare the results of an advanced reliability analysis with the provisions of the

current and proposeci bridge code.

The present research thesis is in line with the research needs, as outlined in the

position statements of the Working Group on Reliability Concepts, Techniques and

Implementation. These statements were prepared during the U.S. National Science

Foundation Workshop, Reliability in Bridge Engineering, held in Boulder, Colorado

in 1996, {FrangopoI, Ghosn, Hearn and Nowak (1 998)). We quote the following

fiom the above statements wnceming the research nmds in the category of

"Modehg of Unoertainties, Bridge Resistance and Loading Models":

"...techniques me needed to explain h m to mclude in situ &ta fiom the observed

behaviaur and the observed loading of an existing bridge structure in the reliability

evaluation. Also, because of the limitations of the avoilable &a, the type of the

probability distribution firnctions is ofien unknown. Resemch is then needed to

dmelop techniques to aktennine the types of probabtlity distribution firnctiom of the

r d o m m'ables t h nifuence the reiiabitity cai~~Iaîio~t~. Tlae "stanaàrdiza~~on" of

such &tu d &ta collection techniques is epcialty needed as more and more

engineers are perfonning these reliability calcfllatiom ofien trsing wry Iimited &CI

bases. '"

in the present research thesis, in situ data on the material properties of the steel used

in the bridge being evaiuated as an example has been obtained through a sampling

program An attempt has been made to include this information in the evaluation

model, This technique can be applied to any other random variable used in the

evaluation.

Also, the work of the present research thesis compares the resuits of the

evaluation of an existing bridge using the FORMfSORM method and the methods

outlined in the Canadian bridge code. This has not been done previously to the best of

our knowledge.

As the procedures described in the present research are more elaborate than

the procedures specified in the code, they provide more accurate estimates of the

reliability of each component of the bridge. The use of these procedures may be

warranted if the results can lead to economical savings in the rehabilitation of the

anal yzed component.

The engineer can apply these methods in the evaluation of bridge structures by

using one of the different commercial computer prograrns for struchuai reliabiüty

analysis, which are readily available in the market.

2. REVIEW OF ADVANCED RELIAB~X'TY METHODS

2.1 RELIABILITY ANALYSIS OF STRUCTURES

in the conceptual fiamework of struduraI reliability, the variables

characterising loads and raistances are assumed to be random variables and the

statistical information necessary to describe their probability distribution tiinctions

are assumed to be known.

A mathematicai mode[ is first derived which relates the resistance and load

variables for the limit state of interest. Suppose th this relation is given by

where Xi are resistance or load variab1es. The fiilme is assumed to o c w when g < 0.

A desired level of reliability is obtained by checkhg that the probability of

failure pf is smaller than the target probability p,

p f = !--.IfE(xt,xz3+.., xa) dxid X Z . . . ~ < PO ( 2-2 ER

in which f5 is the joint probability density fiction for X and the integration is

performed over the region 52 where { X : gO < O ).

In the initial applications of this concept to stnictueai safety probtems, the

limit state was wnsidered wîth only two variables; a resistance Rand a Ioad e£Fii Q,

dimensionally consistent with R The Mure event in this case is R - Q < O and the

probability of failure is computed as,

where FR is the cumulative probability distribution function (c-df.) of R and f;l is the

probability density fiinction for Q. UR and Q bo t . have lognormal distributions, the

probabiüty of failure can be approximated by,

when the coeficient of variation of R FR), and the coefficient of variation of Q (VQ),

are l e s than 0.30; O [ ] is the standard normal cumulative distribution function.

Other distri'butions than normal or lognormal may be specified for R and Q. In

these cases, Eq. 2.3 must ûequently be evaluated numeridy.

This procedure provides a basis for quantitatively estimating the structural

reliability of a component. It is tacitly assumed that al1 uncertainties, in the design or

the evaluation, are contained in the joint probability density fiinction & and that fx is

known However, in structural reliability analysis, these probability density fiinctions

are seldom known precisely due to a generd swcity of data. In fact, it may be

difficult in many instances to determine the probability densities for the individual

variabtes, let alone the joint density fx, In some cases, only the first and second order

moments, Le. mean and variance may be known with any confidence. Moreover, the

limit state hnction may be highly non-linear in the basic variables. Even in those

instances where statistical information may be sufficient to define the marginal

distributions of the individual variables, it usually is impractical to perforrn

numericaiIy the operations necessary to evaiuate Eq. 2.2.

The difficuities outIined above have motivated the development of first-order-

second-moment (FOSM) reliability anaiysis methods, so cded because the random

variabtes are descriïed ody by their first two moments and the Iimit state fiiaction is

l ineuid in order to compute the reliability index. While any wntinuous

matheniatical form of the limit state equation is possibie, it must be lin- relative

to some point for the purpose of performing the reliability analysis. LiiearUation of

the tàilure criterion dehed by equation 2.1 at a given point 1 4 s to the foiiowing

expression,

where s i , X*Z, ..., X',) is the tineuhing point. The reiiabiiity analysis is then

performed with respect to this Linearized version of Eq. 2.1. As might be expected,

one of the key considerations is the selection of an appropriate l h m k n g point-

2.3 MEAN VALUE METBODS

in eariier structurai reliability studies, the I i n e n g point Ki, X'z, . . ., XWn)

- - was set equd to the mean value ( XI,X2.. . . ,% ). The mean and standard deviatioa in

Z are approximated by

The accuBq of Eqs. 2.6 and 2.7 depends on the importance of higher order

terms in Eq. 2.5 and on the coefficients of variation of the randorn variables.

The reliability index or the safety index B is defmed by

which is the reciprocai of the C.O.V. in z. This is illustrated U1 Fig. 2.1 which shows

the probability densiiy fiuiction of z for the simple two-variable problem

Z=g(R,Q)=in(R/Q)=O

Note that p is the distance h m 2 or In ( EUQ ) to the ongin in units of

standard deviatioa As such, P is a meam of the probability that g ( ) will be less

than zero. The shaded area to the left of the origin is eqyd to the probability of

- - faiIure. Note that if oz or a m remains constant, a positive shift in Z or In(R/Q)

shifts the density to the right, reducing the failure probability. Thus an ïncrease in P

r = 41 Ml

Figure 2.1 - Iliustration of the Reiiabiiity Index Concept.

Using the smdl - variance approximation

As outlined in the "Guidelines for the Development of L h i t States Design"

by the Canadian Standards Association (1981), Eq. 2.9 cari be used as the basis for

the devetopment of probability-based load and raistance factors for the design and

evaluation of civil engineering structures.

In this development, no mention has been made of the probability distribution

functions of the random variables. The reliability index P depends ody on measures

of central tendency (5 ) and dispersion ( 0 2 ) of the limit aate hinction. However, it

is important to note that if the probability laws governing the variables in the Iimit

date equaîion are known, there is a relation beween B and pf . In the example just

considered, if R and Q are Iognormat and statisticaliy independent, then R- Q is

lognormal with mean in R/Q and variance ah WQ. The probability of failure is then

Comparing Eqs 2.1 1 and 2.9, the reliabflity index B is reIaîed to the

cumulative distribution fiinction of the standard normal distribution according to the

foIiowing expression,

in cases when the probability distribution fiindons cannot be determineci

exactly, B is a usehl comparative measure of reliability and can serve to evaluate the

relative safety of various design or evaluation alternatives, provided that the first and

second moment statistics are handled cunsistently. in such cases the probability of

failure computed h m Eq.2.I 1 is referred to as a " notionai " probability, indicating

that it should be interpreted, at bat, in a comparative sense as opposed to a classicai

or relative fiequency sense.

2.4 ADVANCED RELIABILITY METHODS

Mean value FOSM methods have two basic shortcomings. First, the limit state

tiinction is Iinearized at the mean values of the X-variables. When the limit state

function is nonlinear, significant emrs may be introduced at increasing distances

from the IineariWng point by neglecting higher order terms. In most structural

reliability problems, the mean point is, in fact, some distance from the faiIure region,

and thus unacceptable ermn are likely in approximating Eq. 2.1 by Eq. 2.5 when g( )

is nonlinear. Second, the mean d u e methods are not invariant to diierent

mechanically quivalent formulations of the same problem. In effect, this means that

B depends on how the Iirnit state is fomulated. The lack of invariance arises because

the linear expansion is taken about the mean d u e point. This problem may be

avoided by linearinng g ( ) at some point at the Eülure &ce. This is because g ( )

and its partial derivatives in Eq. 2.5 are independent of how the problem is f o d a t e d

onIy on the surface g ( ) = 0.

The selection procedure can be explained as follows. With the limit sate and

its variables as given in Eq. 2.1, the variables )[i are first transfonned to reduced

variables with zero mean and unit variance through

in the space of reduced coordinates ui, the Iimit state is

gi ( U I , U ~ , ..., &)=O (2.15)

with tàilure occurring when gi < O. This is illustrated in Fig. 2.2.

We now define a reliabiIity index as the shortest distance betweea the

surface g l = O and the origin. The point (U*~ ,U*~ , . . .,LI*,) on gl = O which corresponds

to this shortest distance is refmed to as the design point.

if we consider

u*; = -a$ ( 2.16 )

where cosine ai is the direction which minimises b, searching for B amounts to

searching for ai Several cornputer program dgorithms c m be used in the search for

ai . Note fiom Fig 2.2 that the procedure for searching for B or is equivaleut to

Iinearizing the limit state equatiou in reduced variables at the point (u*~,u*~,.. -,Pa),

and computing the reliability associated with the [inearited rather than origind limit

state.

Figure 2.2 - Limit State Rmction in the Original and

Rednced Coordinate Systems.

2.5 APPROXMATE METHODS FOR INCLüDiNG INFORMATION ON

The first-order-seand-moment pmeedure outlined in the prwious section

gives values of the reliability index P which may be ntated to a pbability of failure

in cases when the variables X, are normalIy distributeci and the fiinction g is linear in

. In other cases, Eqs. 2.12 and 2.13 are not exact. Many structurai problems involve

random variables, which are clearly non-normal. It seems appropriate that the

information on the distributions of the miables be incorporated in the anaiysis in a

way that does not require the multidimensional integration in Eq.2.2.There are a

number of approaches for doing this. The one used in this research thesis is the

F O U 4 (Fust Order Reliability Method). The SORM (Second Order Reliabiiii

Method) is similar and would Iead to siIIiilar results.

The basic idea in the FORM methad is to transfonn the non-normal variables

into equivalent normal variables prÎor to the search for the design point. The main

advantage of doing this is that sums and differences of independent normal variables

are aIso normal with easily calculated meam and variances. The ability to calculate

failure probabilities in accordance with Eqs. 2.12 and 2.13 is thereby retained. This

transformation may be accomplished by approximating the true distribution of

variable Xi by a n o r d distribution at the value X*i correspondhg to a point on the

failure SurFdce- The justification for this is that if the normaiisation takes place at the

point dose to that where &hue is most iikely, (Le. minimum p), the estimates of the

failure probability obtained by the approximaie procedure shodd approximate the

true ( but unknown ) Fdilure probability quite closely.

FoUowing Rackwitz and Fiessler (19?6), we determine the mean and standard

deviation of the equivaleut normai variaHe such that at the value X*i , the cumulative

probability and probability density of the actuaI and approximating normai variables

are equai. Thus,

in which:

5 and fi are respectively the original cumulative and probability density ftnctions of

X, , $ ( ) is the probability density tùnction of the standard normal variable and O[ 1 is the cumulative probability distribution function of the standard n o d variable.

Hawig deterrnined kNi and &"i of the equivalent nomial distributions, the

solution proceeds exactly as describeci in the previous section. inasmuch as the

design point X*, changes with each iteration, the parameters kNi and 67 must be

recomputed during each iteration cycle also. The iterations are repeated und the

vaiues of p on successive iterations differ oniy by some srnaII tolerance.

The SORM procedure is similar to the FORM procedure outlined above

except that, in the search of the design point, the Iinearization of the Iimit state

function is replaced with a second-degree surface.

2.6 COMPUTER PROGRAMS FOR TBE FORMISORM METHODS

Several computer programs have been deveIoped which use the

FORM/SORM procedures as outlined above, as a tool for the analysis of structural

safety. To name a few, Nessus / SwRI, PROBAN, COMREL, BRITE are such

program. in the present research thesis we wiI1 use the program SYSREL developed

by RCP GmbH, Munich, Germany.

3. BRIDGE EVALUATION IN CHBDC

3.1 SCOPE

in the present Chapter, the load and resistance adjustment factors method and

the mean load method contained in the drafi of the new Canadian code CHBDC

(Canadian Highway Bridge Design Code), (1998) for the evaluation of bridge

components at the ULS (ultimate limit state) are described.

The new code contains also provisions for the evaluation of bridge

components at the ultimate litnit state, using the load testing method. These

provisions will not be reviewed in this section.

The provisions for the evaluation of bridge components using the load and

resistance adjustment factors method in the new CHBDC are similar to the ones

contained in the bridge code CANICSA-S6-88 (1988). ïhe evaluation using the mean

Ioad rnethod was part of the comrnentary of CAN/CSA-S6-88 (1988); these

provisions are now part of the new CHBDC.

We will also show in this chapter the procedure used in the code for the

calculation of the yield strength of structurai steel used in evahation, based on the

results of tests of coupon specirnens. This procedure will be used for the caldation

of the yield strength parameters of the steel for the structure evaluated as an example

in the present research thesis.

3 3 CHBDC PROVISIONS FOR BRJDGE EVGtUATION

The steps in the evaluation of a bridge or a bridge component are as follows:

1. Definition of the geometry of the structure (clauses 14.5 and 14.6);

2. Definition of the t r a c category for which the bridge is being rated

(clause 14.8);

3. Definition of dead Ioads (clause 14.7);

4. Analysis of the bridge for live and dead Ioads (clauses 14.1 and 14. IO);

5. For each wmponent being considered, determine:

a. the desired reiiability level (clause 14.11)

b. if the load and resistance adjustment factors method is use& h d the

factors comsponding to the desired reliability Ievel (clause 14. L2),

if the mean load method is used, find the bias coefficients and

coefficients of variation for the Ioads and the resistance h m the code,

fiom reported values in technical publications, or fiom field

rneasurements (clause 14.15 and commentary).

c. detennine the live toad wpacity factor, F (clauses 14.13 and 14.14),

which is the factor by which live Ioad has to be muhiplied so that the

factored capacity of the bndge is not exceeded for the combination of

permanent and [ive Ioads under consideration, and

6. dependmg on the value of the capacity factor obtained, decide what action

is necessary. PossibIe choices inchde posting the bndge (clause 14-19,

repairing substandard elements, wnducting more detaireci andysis or tests,

or, if the rating is sarisfactory, taking no h h e r action

3.3 THE TARGET RELIABILITY INDEX 6

in the CHBDC, the target reIiability index B, which can vary between 2.0 and

3.75, is chosen by the evahator as a finction of the structud behaviour, leveI of

inspection and evduation, and haflic situation for which the evaiuation is made. The

target reliabifity indices are based on life d e t y considerations and have been

calibrated using previous design and evaluation procedures as well as economic

considerations.

The evaluator considers the following factors when choosing the target

reliability index B: 1. Type of t r a c .

2. System behaviour

3. Element behaviour

4. inspection level

5 . Importance of the structure

3.3.1 TYPE OF TRAFFIC

The foilowing types of tmtlic are considered:

1. Normal Trafic

Normal trafic is divided into three categories, CL1-W. CL2-W, and CL3-W

a. CL1-W is for vehicle trains consisting of a tractor and more than one

tsailer.

b. CIL2-W is for üucWtraiIer or tractorisemitraiIer combinations.

c. CL3-W is for single unit vehicles.

The bridge evaluation preseated in this study is based on a CL\-W vehicle.

The configuration and the Ioads for this design vehicle are shown in Fig.3.2

C U - W rndbad

-

( a m ) D 8

Figure 3.1 - Confignration of the Traffic Design Load CLI-W

2. Permit-Annual PA) Tdc

Vehicles for this type of traffic are issued permits on an annual basis or for the

duration of a specific project to carry an individual load. The vehicles with

annuai permits will be allowed to travel without restriction along with the

nonnai t r a c on any route for which the permit is valid.

3. Permit-Bulk Haul PB) Trfic

Vehicles for this type of M c are issued permits by some jurisdictions to

carry divisible bulk load under a permit program, such as grains, chemicals,

etc. These vehicles have been found to maintain good wntrol on the loads

within the specified limits of the permit program with iittie or no occurrence

of overloads.

4. Permit-Controiled (PC) T r S c

This category includes extremely heavy permit vehicles that are required to

travel with specified travel restriction under supervision to cary indivisible

Ioads on a specified route. The vehicle is generaily required to cross a bridge

alone.

5. Permit-Single tri^ PS) Traffic

This category includes vehicles tbat are issued permits for a single trip along a

specified route to cary an individual load.

3.3.2 SYSTEM BEHAVIOüR

The following categories of system behaviour are considered:

1. Cateeorv SI, where element fidure leads to total wllapse. Tbis would

inchde failure of main members with no benefit fiom continuity or

muttiple load paths, such as a simpIy çupported @der in a 2-girder

systern

2. Category 52. where element mure d l probably not i d to total

wllapse. This would inciude main bad-carrying members in a mdti-

girder system, or continuous main members in bending.

3. Cataorv S3, where element failure leads to local M u r e only. This wouId

iaclude deck slabs, stringers and bearings in compression

The folIowing categories of element behaviour are considered:

1. Categ:oryEl

Where the element being considered is subject to sudden los of capacity with

tittIe or no warning. This rnight include failure by buckling.

2. Cateszoy E2

Where the element being considered is subject to sudden failure with Little or

no warning but wiIl retain post-failure capacity. This might inchde steel

plates in compression with post-buckling capacity.

3. Catercory E3

Where the eIement being considered is subject ta gradua1 Mure with warning

of probable failure. This might inciude steel beams in bendimg or shear or steel

in tension at p s s section

3.3.4 INSPECTION LEVEL

The following categories of inspection 1eveIs are considered:

1. Level iNSP 1

Where a component can not be inspected.

2. Level iNSP2

Where inspection is to the satisfaction of the evaluator, with the results of

each inspection recorded and available to the evaiuator.

3. Level iNSP3

Where the evaluator has camed out inspection of criticai and substandard

components and final eduation calculations account for al1 information

obtained during this inspection.

In Table 3.1, the target reliabiIity index B, for CLI-W, CL2-W, CL3-W, Pk

PB and PS traffic, is given as a hnction of the system behaviour, the element

behaviour and the inspection level, (CHBDC, (1998)).

3.3.5 WORTANT STRUCTURES

For structures, which couid affect the life safety of people under or near the

bridge, or are essentid to the local economy, or are necessary for the

rnovement of emergency vehicles, a value of B at lest 0.25 p a t e r than those

given in the code sMl be used,

Table 3.1 - Target Reliablity Index p, for CL1-W, CU-W, CW-W, PA, PB

and PS TrnZtic, (CEIBDC, 1998).

System Element 1 Inspection k t 1

Behaviour

3.4 EVALUATION USING TBE LOAD AM) RESISTANCE FACTOR

Once the evaluator detemines the target reliability index for the component

of the bridge to be evaluated, he can select the dead load factors a~ and the live load

factor a ~ . These factors are then used to calculate the live load capacity factor F,

which is an indication of the reliability of the component under evaluation, This is

done according to the following procedure:

The dead Ioad factors are detenniaed according to the target retiabili index P and the foflowing dead load categories:

1. Dl, which includes the dead load of fiictory produced components,

and cast in place concrete excluding decks.

2. D2, which includes bituminous concrete surfacing based on field

measured thickness,

3. D3, which includes bituminous wncrete surfking based on nominal

thickness or thickness specified on drawings

4. D4, for cast in pIace concrete decks, includiig voided decks and

cementitious concrete overlays, wood and non-stnictural components.

The dead load factors ab correspondmg to the different values of are given

in Tab le 3.2 (CHBC, (1 998)).

Table 3.2 - Dead load factors ab (CHBDC, 1998).

I I

Target reliability i n d u B

3.4.2 LnrE LOAD FACTORS. aL

The live Ioad Fdctor is determineci accordmg to the target reiïability index P, the type of traEc as o u t i i i above and the method used in dcuiating the

lateral disiribution of iive bads to the element considerd The type of

distribution can be one of the folIowing categories:

1. StaticaIly determinate.

2. Sophisticated, laterd disûibution is statically indeterminate and is

calculated by a sophisticated analysis rnethod.

3. Simplified, lateral distribution is calculated in accordance with the

simplified mettiods given in the code.

The [ive load facton ut, for NonnaI T&c (CLl-W, CL2-W and CL3-W),

correspondhg to the different d u e s of B are given in Table 3.3 {CHBC, (1998)}.

Table 3.3 - L N ~ load facton UL, Normal Traffic, (CU-W, GI3-W, CL3-W)

(CHBDC, 1998).

.-

3.4.3 LiVE LOAD CAPACITY FACTOR

Type of analysis

Statically determinate

Sophisticated

Simplifiecl

The Iive load capacity factor, F, at the uitimate state, is the fàctor by which the

evaluation iive load has to be multiplieci so that the factored capacity of the

bridge is not exceeded for the combination of permanent and Sie Ioads under

consideration The live load capacity factor gives therefore an indication of

2.25

1.42

1.48

1.36

2.50

1.47

1.55

1.42

2.75

1.5 1

1.62

1.49

3.25

1.63

1.75

1.62

3.00

1.57

1.69

1.55

3.50

1.69

1.83

1.70

3.75

1.75

1.90

1.78

the reliability of the bridge component under the loads considered. The live

load capacity Factor, F, is calculated as follows:

where,

D dead load by category-

L live load.

1 dynamic load ailowanceance

Rf, factored resistance of the element calculated according to the

applicable provisions in the code.

U, resistance adjustment factor in order to fine tune the resistance factors

used in the rest of the code. WhiIe approximations made to the

resistance factors in the interest of simplicity are appropriate for the

design of new bridges, in the evaluation of existing bridges their use

may lead to unnecessary postings or strengthening.

3.5 EVALUATION USING TBE MEAN LOAD METHOD

As an alternative to the load and resistance factor method, the capacity factor,

F, at the ultimate limit state, c m be caicuiated using the mean load method according

to the following equation:

where,

Notation:

D. non-factored dead load by category.

L, non-factored live load.

1, dynamic load allowance.

R, non-factored mean resistance.

6m, 6a, 6&, SL, 6 ~ , bias coefficients (ratios of mean to nominal effects) for dead

load analysis method, live load analysis rnethod, dead loads, dynamic Ioad allowance,

Iive load and resistance respectively.

Va Vd, VD, VI, VL, VL coefficients of variation for dead load andysis method,

[ive load analysis method, dead loads, dynamic load allowance, Iive load and

resistance respectively.

Bias coefficients and coefficients of variations to be used in the Mean Load Method

may be taken fkom the cornrnentary of the code, corn reported vahm in techcal

publications or fiom field measurements.

3.6 DETERMINATION OF TBE YELD STRENGTH OF STEEL FROM

TESTS ON COUPONS

When the resuits of tests on coupon specimens are available, the yield strength

(in MPa) used for evaiuation is calculateci as foiiows:

where,

- Fy, average value of the yield strength of the coupons in MPa

V, coefficient of variation of the yield men@ of the coupons.

K, is a coefficient of variation modification factor. It depends on the number

n of test and reflects the uncertainty in the standard deviation when it is

calailated fiom a smaiI sampie. Table 3.4 gives the values of K, as a hnction

of the number of tests.

The vdue of 28 MPa in the equation reflects the difference between the yield

strength measured fiom a test on a coupon and the static yield strength. Mi11

tests are performed at loading rates that are higher than the rate of loading in

structures, and the concept of the "static yield stress levelu has been used to

d e h e the yield stress under zero strain rate (GaIambos and Ravincira, 1978).

Table 3.4 - CaiKcient of variation modüïcatioo firtar Id, correspondiag

ta the numbcr of tests n.

n IG

3 I 3 -46

4

5

6

8

1 O

12

16

20

25

30 or more

2.34

1 -92

E -69

1.45

1.32

1.24

1.14

1 .O8

1.03

1 .O0

4.BRIDGE EVALUATION USING FORMBORM

METHODS

4.1 SCOPE

in Chapter 2, the theory for estimation of structura1 reliability using the

FOWSORM method was outlined. The program SYSREL was used in this study to

perfonn this type of analysis.

in order to determine the system reliability index B of a bridge component

using the FOWSORM method, the evaiuator has to define the limit state ftnctions

defining the modes of failure considered.

In this Chapter the limit state ttnctions for steel bridge components are

defined for four îàilure modes at the ultimate Iimit state:

failure under compression and flexurai buckling due to compression,

failure under tension,

failure under shear, and

failure for doubly symmetric, laterally supporteci and Iaterally unsupported

beam members under bending.

The limit state hnctions for al1 the modes of failure are consistent with the

code, the commentary of the code and the references in the code.

For each mode of failwe, the d i r e n t random variables are defined and a

distribution hncbon and parameters are specified based on the information contained

in the code and the references mentioued in the code. Some parameters are estimateci

fiom field measwements when this information is available to the evaiuator-

in generai, the b t state fünction for bridge members cari be dehed as

follows:

g(X)=xkXp-XQ (4.1

where:

XR is the variable deniing the member resistance. This variable is a firnction

of other variables definmg the geometric and material properties of the

section under consideration.

Xp is the variable defining the professionai factor, which is the ratio of the test

capacity of the rnember to the predicted resistance of the member.

XQ is the variable defining the Ioad effects on the member. This variable is a

fitnction of other variables defining the effects of different loads on the

member such as the different categories of dead loads, the live load and

dynamic load allowance.

The variation of Xp and XR is related to the type of failure mode and will be

discussed in the following sections where the tirnit state function for each particular

failure mode will be discussed.

The variation of XQ for bridge structures, is related to the type of dead and

trafEic loads and is as follows:

Dl, D2, D3, and D4 are the nominal dead load effects per category of dead

load as presented in section 3 -4.1.

Xoi, XD2, XD3 and Xw are variables reflecting the variation of the nominal

dead load effects Dl, D2, D3 and D4 respectively. in the absence of specific field

meaSuTements, the parameters for the statisucal distributions for the &ove variabtes

cm be assumed as given in the commentary of the code. W e present in Table 4.1 the

statistical parameters for these variables as given in the code and the type of

distribution hctions assumexi in the present research thesis.

L is the nominal traffic load efféct.

XL is a vatiabIe reflecting the variation in the nomina1 traffic load effect and is

dependent on the type of considered as odined in section 3.3.1 of the present

research thesis. Table 4.2 gives the statisticd parameters for this variable as

mentioned in the commentary of the code and the distribution function assumed in the

present research thesis. In the absence of field data for the type of trafic being

considered, the evaiuator can use the staîïstical parameters as shown in the table.

Xa is a variable reflecting the variation in the lateral distribution of the [ive

load and is dependent on the type of lateral distribution of live Ioad considered by the

evaluator as mentioned in section 3.4.2 of the present research thesis. Table 4.3 givs

the statistical parameters for this variable as mentioned in the commentary of the code

and its distribution bnction assumed in the present research thesis.

I is the nominal dynamic traffic load ailowance.

Xt is a variable reflecting the variation in the dynarnic traffic Ioad allowaace.

TabIe 4.4 gives the statîsticai parameters for this variable as mentioned in the

commentary of the code and its distribution fiinction assurneci in the present research

thesis. In the absence of field data for the dynamic load alIowance, the evduator c m

use the st&tical parameters as shown in the table.

Table 4.1 - Sîatistical parameters and distribution functions for the dead

1 variable/ Dud load category 1 8 ( V 1 Distribution Cuection I 1 XDI 1

1 1 1 1

Di ( S t d ) 1 1.012 1 0.03 1 Lognormai(l~012,0.03036)

I I I I

ote: 6 is the bias coefficient (ratio of mean to nominal values). V is the coefficient of

1 variation (ratio of the standard deviation to the mean value).

1 For the definition of dead load category, refa to section 3.4.1. I

Table 4.2 - Statisticai parameten and distribution functions for tbe traîfic

load variables, (CHBDC, 1998). , ,' , -- 1 ~ a r i a b k r Traffîe load type Distribution function

1 I 1 I

e: For the definition of T&c Load Type, refk to section 3.3.1.

1 Vaber for PA fiPBOc were not available at the t h e of publication of the present l

Lognortuai(i.06,0.00954)

tognod(1.0020.03908)

1 6 is the bias coefficient ( d o of mean to nominal values). V is the coefficient of I

XL(PB)

XL(PC)

1 variation (ratio of the standard dwiaîion to the mean value). I

1 .O6

1.002

Permit BuIk (PB)

Permit Controlled (PC)

0.009

0.039

Tabk 4.3 - Strtisticai pammeters and distribution functions for the faterai

distribution of the lm loid, (CHBDC, 1998).

I I I

Xa 1 Stmidly determinate 1 1.00 1 0.00 1 -

l 1 l 1

Note: 6 is the bias coefficient (ratio of mean to nominal values). V is the ~ ~ c i e n t O

]variation (ratio ofthe standard deviaîioa b the mean value). I

Table 4.4 - Statisticai paramtten and distribution function for the dynamic

load aiiowance, (CHBDC,1998)

k: 6 ir the bias coef3kient (do of mean to nominal values). V is the coetiïcient of

Variable

XI

1 variation (ratio of the standard dmation io the mean value).

4.3 THE LMiT STATE FUNCITON FOR MEMBERS UrVDER

Typt

Dynamic Load AlIowance

COMPRESSION AND FLEXURAL BUCKLING DUE TO

COMPRESSION

4.3.1 THE RESISTANCE VARIABLE XR

6

0.40

The resistance variable XR for memben under compression and flexurai

buckiing due to compressioa can be expresseci as follows:

V

1 .O0

Distribution function

Norma1(0.40,0.40)

where 3L is the slendemess parameter,

In the above equation, the foiiowing parameters are considered as constant:

K, effective length factor.

L, Iength of the member,

n = 1.34 for hot-rolled W-shapes, fabricated box-shapes and hollow structural

sections Class C.

n = 2.24 for welded H-shapes having flame-cut flange edges and hollow

structural sections Class H.

The following material ~roperties of the member under consideration are

considered as variables:

Fy = Fy, Xq , where Fy, is the nominal yield strength and is constant, X F ~ , is

a variable reflecting the staîisticd variation in the yieId strength of the

material used for the fabrication of the member under consideration.

E = E, XE , where E,, is the nominal modulus of elasticity and is constant, XE,

is a variable refiecting the statisticai variation in the modulus of elasticity of

the matenai used for the tabrication of the member under consideration.

The statistical parameters for the variables XFy and XE are repxted in the

Iiterature and have been used to derive the resistance factors used in the American

and Canadian structural steel codes {Galambos and Ravindra (1978), Kennedy and

Gad Aly (1 980) and Kennedy and Baker (1984)).

The statistical parameters for the variables X F ~ and XE for roiled and welded

W shape sections are listed in Table 4.5 {Kennedy and Gad Aly (1980)).

Note that the evaiuator can use other sources of data or he can perfonn his

own tests in order to derive the distribution parameters for these variables. In the

present research thesis, we followed this latter approach and derived the parameten

of the statisticai distributions of the yield stress ( X F ~ ).

Table 4.5 - Statisticai parameters and distribution functions for the material

properties of rollcd and welded W shapes, (Kennedy and Gad Aly,

Materid Property

Y ield strength of cross

section of rolled W

shapes, Fy.

Y ield strength of cross

section of welded W

shapes, Fy.

Moduius of Elasticity, E. I I I I p: 6 is the bias coefficient (ratio of mean to nominal values). V is the coefficient of

1 variation (ratio of the standard deviation to the mean vaiue). 1

Distribution function S V

The following geometricaI properties of the section under consideration are

considered as variables:

A = A, XA , where A, is the nominal cross-sectional a m , XA is a variable

refiecting the variation in the area of the section.

r = rn J? , where rn is the nominal radius of gyration, X, is a variable reflecting

the variation in the radius of gyration of the section of the rnember, along the

weakest axis.

Table 4.6 gives the statistical parameters for the variables defining the

geomeuical properties of roiied W-shapes, {Kennedy and Gad Aly (1980)).However,

the evahator can use data fiom field tests in order to define the statistical parameters

of the above variables.

Table 4.6 - Statisticd psramctcrs and distribution functions for the

geometrical propeFtiu of rollcd W s b a p (Kennedy and Gad Aly,

Variable Gcomctric Propcrty 6 V Distribution function l 1 1 I 1 & 1 Web thickness, w [ 1.017 1 0.0384 1 tognotmaI(t.Oi7.0.0391)

Xt, 1 Plastic section modulus, 2x1 0.990 1 0.0380 1 rognod(0.!W0,000376) I I I I

XS. 1 Uastic section moduius, Sx 1 0.990 1 0.0210 ( Lognoima1(0.990,0.0208)

Warping torsional bguormai(O.99û,0.0891)

x l ~

Xc

xr

i I

Note: 6 is the bias coefficient (ratio of mean to nomina[ vatues). V is the coefficient of

variation (ratio of the standard deviation to the mean value). 1

Moment of inertik ly

Radius of gyration, r

S t. Venant to rsional

constant J

4.3.2 THE PROFESSIONAL VARTABLE XE

In the case of members under compression, the statistical parameters of the

professional variable Xp are given in Table 4.7 for mlled W shapes {Kennedy and

Gad Aly ( 1980)).

1.000

1 .O00

0.960

Ke~edy and Baker (1984) have also used the statistïcal parameters of these

variables to recommend a value for the tesistance factor to be used in the Ontario

Highway Bridge Design Code (OHBDC), {MTO, 2 99 1 ) .

0.0580

0.0230

O. 1 O00

bgnorma1(1.000,0.0580)

~ognormai(~.ûûû,~.OUO)

~0gnod(0.%0.0.0960)

The work of Kennedy and Gad M y (1980) is based on the design equations

for members under compression as given in the Canadian structurai steel design

standard CANKSA S16.1-1974. The formulation of the design curve for members

under compression in the draft of CHBDC and in the new S 16.1 standard is different

Eom the one used in the standard CAN/CSA S 16.1-1974. However, since the end

results, when applying the old and new standards, are practically very close, we

consider that the results of the work of Kennedy and Gad Aly can be used to define

the professional factor in the present research thesis.

In the present research thesis, the formulation of the resistance variable for

members under compression as defined in Eqs. 4.1 and 4.2, is based on the design

equation given in the cirat? of the CRBDC.

The statistical parameters (6p, bias coeEcient, and Vp coefficient of variation)

for the overall professionai factor for members under compression is expresseci in

terms of two professional factors whose statistical parameters are (6pr, Vpi) and (&,

Vp2), { K e ~ e d y and Gad Aly C 1980)). The first two parameters, (&, Vp,) express

the variation between the experimental results for the ultimate strength of columns

under compression and results of the ultimate strength theory based on the analysis of

1 12 column curves Sy Bjorhovde (I972),

Cr is the maximum compressive force of a member, Cy is the axid

compressive load at yieId stress.

The second set of parameters, (6~2, Vn) expresses the variation between the

results of the ultimate strength theory and the resuits based on the design curves in

CSA standard S16.1-1974.

The relationship between the different parameters (6p, Vp), (6~1, Vpl) and (an,

Vpz) can be expressed as follows:

The values (6p, Vp), @pl, Vpl) and (6p2, VpZ) as a hnction of the siendemess

parameter are given in table 4.7.

Tabk 4.7 - Statistical parameters and distribution functioas for the

professional variable XP of mUtd W shapes under compression,

(Ktnneây and Gad Aly, 1980).

A, slenderness parneter.

6pi=(C4Cy experiment)/ (C&ultimate men@ theory)

&=(WCy ultimate strength theory)/ (C&predicted by the code)

6p=8flx&

ote: 6 is the bias coefficient (ratio of mean to nominal values). V is the coefficient of

variation (ratio of the standard deviation to the mean due) .

5

0.0

0.2

VPI

0,050

0.050

an

1 .O3

1 .O3

6PZ

1 .O00

0.986

7

VPZ

0.000

6 p

1.030

0.014

Vp

0.050

1 .O16

Disîribution function

~ognorma1(1.~30,0.0~0)

0.052 ~ o g n ~ n a a l ( l . o ~ p . o ~ ~

4.4 TEE LIMIT STATE FUNLITON FOR MEMBERS UNDEI TENSION

4.4.1 THE RESISTANCE VARIABLE XR

The resistance variable XR for members under tension can be expressed as

follows:

w here,

A = A,, XA , where k, is the nominal gros cross-sectional area, XA is a

variable reflecting the variation in the gros area of the section.

Fy = Fyn X F ~ , where Fy, is the nominal yield suength and is constant, XF~, is

a variable reflecting the statistical variation in the yield strength of the

material used for the fabrication of the member under consideration.

The statistical parameters for the variable XA for roUed W-swes are listed in

Table 4.6 {Kennedy and Gad Aly (1980)). However, the evaluator can use other

parameters if the information is available fiom field tests.

Table 4.5 gives the statistical parameters for the variable X F ~ {Kennedy and

Gad AIy (1980)).

We have presented above the formulation of the resistance variable to be used

in the limit state function of the members under tension based on the gross cross-

sectional area and the yield strength of the section, which is the dominant case for the

limit state most of the time. However, in some cases, the limit state can depend on the

net area of the section or the reduced net area of the section and the ultimate tende

strength of the material; for these cases, the formulation of the resistance variable

shouid be expressed accordingly.

4.4.2 THE PROFESSIONAL VARIABLE Xe

in the case of tension, the professionai factor is set equal to 1.0, since no data

wuld be found on the variable Xp in the case of tension.

4.5 TEE WMIT STATE mTNCTION FOR MEMBERS UNDER SHEAR

4.5.1 THE RESISTANCE VARIABLE XR

In the case of shear, the resistance variable XR, in Newtons, can be expressed

as follows:

where,

5.34 k v = 4 + , when aih < 1

tn the above equations,

a, is the spacing of transverse stBeners in mm and is considered as constant.

The following material Dropeq of the member under consideration is

considered as variable:

Fy = Fyn XFy , where Fy, is the nominal yield strength in MPa and is

constant, XFy, is a variable refledng the statistical variation in the yield

.strength of the material used for the fabrication of the member under

consideration.

Table 4.5 gives the statistid parameters for the variable X F ~ {Kennedy and

Gad Aly (1980)).

The following pmetrical ~to~erties of the member under consideration are

considered as variables:

h = h, &, where h, is the norniaal depth in mm of web between flanges, is

a variable reflecting the variation in the depth of web between the flanges of

the section

w = w, X , where w, is the nominal web thickness in mm, Xw, is a variable

reflecting the variation in the web thickness of the section-

Table 4.6 gives the statisticai parameters for the variables & and X, for rolled

W-shapes {Kennedy and Gad AIy (1980))

3.5.2 THE PROFESSIONAL VARIABLE Xp

in the case of shear, the professional factor is set equal to 1.0, since no data

could be found on the variable Xp, for the case of shear.

4.6 TEE LIMIT STATE FUNCTION FOR LATERALLY SUPPORTED

AND LATERALLY ZTNSUPPORTED MEMBERS SUBJECT TO

BENDING

4.6. t THE RESISTANCE VARIABLE XR

In the following, we will consider the case of doubly symmetrk sections.

Considering,

Mp = Zx Fy

My = S, Fy

L

We cm express the resistance variable XR as follows:

For Class 1 and 2 Sections

When continuous lateral support is provided to the compression flange and the

member is subject to bending about its major axis,

When the section is subject to bending about its major axis and laterally

unbraced for a length L,

0.28 Mp a) X ~ = l . l S M p [ l - ] s Mp, for M,, > 0.67 Mp ( 4.9 )

For Class 3 Sections

When continuous lateral support is provided to the compression flange and the

member is wbject to bending about its major ais,

When the section is subject to bending about its major axis and Iaterally

unbraced for a length L,

b) XR=&, for M,, 1 0.67 My ( 4.13 )

In the above equations, the fouowing parameters are constant:

49

L, the unbraced length,

oz the coefficient to account for increased moment resistance of a Iatedly

unsupported beam segment when subjecî to a moment gradient.

0 2 = 1, when the bending moment at any point within the unbraced length is

larger than the larger end moment or when there is no effective lateral support

for the compression flange at one of the ends of the unsupported length.

0 2 = 1.75 + 1.05 K + 0.3 K~ , where K is the ratio of the smaller factored

moment to the Iarger factored moment at opposite ends of the unbraced

length, positive for double curvature and negative for single amahire.

The following material ~ro~erties of the member are considered as variables:

Fy = Fyn X F ~ , where Fy, is the nominal yieId strength and is constant, X F ~ is

a variable reflecting the statistical variation in the yield strength of the

material used for the fabrication of the member under consideration.

E = E, XE , where E, is the nominal modulus of elasticity and is constant, XE,

is a variable reflecting the statistical variation in the modulus of elasticity of

the material used for the Eibrication of the member under considecation.

G = G, &, where G, is the nominal shear modulus, &, a variable reflecting

the statisticai variation in the shear modulus of the materid used for the

fabrication of the member under consideration.

Table 4.5 gives the statisticai parameters for the variables X F ~ and XE

( K e ~ e d y and Gad Aly (1980)).

The statistical parameters for the variable XG are considered to be the same as

the statistical parameters for the variable XE.

The following geometric ~ro~erties of the member are considered as

variables:

2, = 2, Xzx, where Z, is the nominal plastic section moddus, Xa, is a

variable reflecting the variation in the plastic section modulus of the section

S, = S, Xsx, where Sm is the nominal elastic section modulus, Xs, is a

variable reflecting the variation in the elastic section modulus of the section.

1, = 1, XI,, where 1, is the nominal moment of inertia wiîh respect to the y

ais, XI,, a variable reflecting the variation in the moment of inertia of the

section with respect to the y axis.

J = Jn XI, where Jn is the nominai St-Venant torsional constant, XJ, is a

variable reflecting the variation of St-Venant toniond constant of the section.

C, = C, Xcw, where C, is the nominal warping tonional constant, L, is a

variabie reflecting the variation in the warping torsional constant of the

section.

Table 4.6 gives the statistical parameters for the variables &, GXsx, Xcy, XJ,

and X., for roIled W-shapes {Kennedy and Gad Aly (1980)).

4.6.2 THE PROFESSIONAL VARIABLE Xp

In the case of members under bending, the values defining the statistical

parameters of the professionai variable Xp to be used in the limit state ftnction

(Eq.4.I) are given in Table 4.8 for roiied W shapes {Kennedy and Gad AIy (1980)).

The variation of the profescional variable under bending depends on the type

At the plastic moment,

At the yield moment.

By inelastic buckling.

By elastic buckling.

Table 4.8 - Statisticai paramtttrs and distribution functions for the

professioui variable Xp of miid W shapcs under bending,

Kennedy and Gad Aly, 1980).

l Mode of failure 1 6 1 V 1 Distribution f i~nctbi

bote: 6 is the bias coefficient (ratio of mean to nominal values). V is the coeficient of

Me, plastic moment.

My, yield moment.

Inelastic buckli~g.

Elastic buckling.

1 variation (ratio of the standard deviation to the mean due).

4.7 SUBROUTINES FOR TEE LIMIT STATE FUNCTIONS

1.10

1 .O7

1-05

I .O3

The four limit state fictions descn'bed above have ben programmed in

FORTRAN language. The programming has been done so that the subroutines

describing the limit state tiinctions ca. be used with the system reliabiiii anaiysis

program SYSREL. These subroutines can be slightiy modifieci to suit any other

0. t 10

0.060

0.078

0.093

hgno~(1.10,0.121)

hgaormai(1.07,0.0o4]

tagnormai(1.05,0.082)

Lognormai(i.OS,O.O%)

system reliability program, provided that the program is written in FORTRAN

language.

The listing for the four foiiowing subroutines describing the iimit state

fiinctions are included in Appendix B:

VCOM-FOR, subroutine for the Limit state tiinction for members subject to

compression and flexural bucklmg due to compression

VTEN.FOq subroutine for the limit state ftnction for rnembers subject to

tension.

VSHEAKFOEt, subroutine for the limit state ftnction for members subject to

shear.

VBEND.FOR, subroutine for the tirnit state fùnction for doubly syrnmetric,

laterally supported and laterally unsupporteci beam members under bending.

5. EXAiMPtE: EVALUATION OF A STEEL TRüSS

SPAN LOCATED ON THE JACOUES-CARTIER

BRIDGE

In the present Chapter we evaluate a steel tms span located on the Jacques-

Cartier bridge using the resistance and load faetor method and the mean load method

given in the drafk of the CHBDC code. We dso evaiuaîe âhis structure using the

FORMiSORM structurai reliability mettiod presented in the present research thesis

and compare the results of this evaluation to the m i t s of the evduation using the

methods outiined in the code.

Before doing these evaluations, we will present the results of the evaiuation of

the yield strength of the steel used in the consmaion of the tniss span and the

parameters for the statisticd distribution for the yietd strength based on tests of steel

coupon specimens taken h m the bridge.

5.2 DESCRIPTiON OF TEE JACOüESCARTIER BRIDGE

A generai Iayout of the Jacquedartier bridge is given in Fig. 5.1.

The Jacques-Cartier bridge is a steel bridge connecting the City of Montreal

with the south shore City of Longueuil across the St-Lawrence river.

This bridge is composed of 39 tmss spans ranging in Iength from

approximately 60 feet to 245 feet, and a main span of appmxhateIy 1925 fw-

The main span is composeci of two anchor sections of 420 feet each, two

cantilever anns of 354 feet each and a suspension span of 3 7 î fe approximately.

We propose in this study to evaluate a t y p i d tnrss span, which is 98 feet in

length. There are 9 spans of this type as shown in Fig.5.1.

5.3 DESCRIPTION OF THE TRUSS SPAN

The mss span, having 98 feet in length, is composed of two main tmsses, une

on the upstream side of the bridge and one on the downstream side. There are 5 floor

beams perpendicdw to the traffic direction, ~ ~ e c t e d to the nodes of the bottom

chords of the trusses. The distance between two adjacent floor beams is

approximately 24.5 feet. The floor beams support the roadway stringers, which are

approximately 4 feet apart and are paraiIel to M c direction The stringers suppon

the roadway which is composeci of an 8 inch concrete slab and a 2 inch asphalt

overlay. There is a concrete sidewalk and a steel guard-rai[ on each side of the

roadway. Figure 5.2 shows a typicai cross section of the tniss span. Figure 5.3 is an

illustration of the steel skekton of the span showing the tniss rnernbers and the floor

bearns.

Figure 5.1 - General Layout of the Jacques-Cartier Bridge.

STRINGER (typ.) CONCRET€ SIDEWALK \ WlTH C O N C R E T Y L A B ASP LT OVER Y \ ,

Figure 5.2 - Typical Cross Section of the Truss Span of the Jacques-Cartier Bridge.

Figure 5.3 - Steel Skeleton of the Truss Span of the Jacques-Cartier Bridge.

CONSTRUCLlON OF TEE TRUSS SPAN

Accordhg to the book by LX. Wilson (I930), descniing the comct ion of

the Jacques-Cartier Bridge in Montreai, two types of steel have been used in the

çonstniction of the bridge, "Speciai Carbon Steel" and "Silicon Steel".

In generai, the Silicon steel bas been used for the fàbrication of the truss

rnembers of the main span and for the 245 feet spans of the bridge. Ai1 other steel

elements of the bridge have been fabricated using Special Carbon Steel. The

construction shop drawings mention the type of steel used for the hbrication of each

member.

Therefore, ail the members of the tniss van, having 98 feet in length, to be

evaluated in the present research thesis are made of Special Carbon Steel.

Wilson (1930) mentions the following values for the ultimate tende strength

and the yield strength for the Specd Carbon Steel used:

0 Ultimate tende strength: 414 MPa - 483 MPa.

Minimum yield strength: 248 MPa.

In order to verie the actuai tende strength of the steel, 74 coupon specimens

were taken fiom the bridge, h m members identified on the construction shop

drawings as members fabticated with Speciai Catbon Steel. Speciai Gare was taken in

order not to damage the members where the steel coupons were taken. The coupons

were tested according to Standard CANKSA-G4020M. The results of these tests are

shown in Table 5.1.

The compilation of the tesuIts of these tests gïves the following:

Average value of the yield straigth of the coupons, = 268 MPa.

a Coefficient of Vanation of the yield strength, V=0.0634.

The cainilation of the nominal yield strength is done using Eq. 3.3, (CHBDC,

1998). The constant K was set at 1.0 since more than 30 coupons were tested. The

nominal yield strength calculated according to the above procedure is:

Fp = 221 MPa

It is interesting to note that ifEq.3.3 is applied without subtracting the value

of 28 MPa from the average value of the yield strength of the coupons, the value of

F, would be 247 MPa , which is very close to the reported value in the book by

Wilson (1 930). The value of 28 MPa reflects the difference between the yield strength

observed during a coupon test and the static yield strength.

In order to be consistent with the assumptions in the CHBDC, we have to

consider the static yield strength. For every coupon, the static yield strength is

caiculated by subtracting 28 MPa fiom the reported value of the yield measured

during testing. The static yield strengths for the coupons are shown in Table 5.1.

The average value of the static yield strength for aii coupons is 240 MPa, and

the coefficient of variation is 0.0708.

We will therefore consider the foiiowing for the yield strength used in the

evahation of the truss span:

Nominal YieId strength, Fp = 221 MPa

The variable X F ~ (to be used in the formulation of the limit state

functions as shown in Chapter 4), which rdects the statisticai

variation in the yield strength has the foiiowing parameters:

0 Bias coefficient S = average/nominal= 2401221 = 1.086

Co&cient of vanation V = 0.0708

(Note that the value of S and V repoaed by Kennedy and Gad Aly,

(1980), and shown in tabIe 4.5 for roiied and weided W shapes, are

1 -07 and 0.065 respectively).

The variable XF, has a Lognormal (1 .O86,O.O?69) distribution.

Table 5.1 - TESTS RESULTS ON COUPONS OF THE SPECIAL CARBON STEEL

COUPON

FROM THE JACQUES-CARTIER BRIDGE. I I

I Tests Results I I I I

Yield Ultimate strength tensile obsewed

strength durlng test Elongation

Static Yleld

strength MPa), set

note below

Table 5.1 Sheel 1 of 4

Table 5.1 -TESTS RESULTS ON COUPONS OF THE SPECIAL CARBON STEEL

COUPON

FROM THE JACQUES-CARTIER BRIDGE.

I Tests Results

Yield Ultlmate strength tenslle observed

strength during test Elangation

Static Yleld

strength MPa), ses

note below -

235 264

Table 5.1 heel2 of 4

COUPON

Table 5.1 - TESTS RESULTS ON COUPONS OF THE SPECIAL CARBON STEEL FROM THE JACQUESLCARTIER BRIDGE.

1

3 I I

T - - T - - T - - r - - T - - T - - r - - r - - r - - r - r - r - I - 1 - -

1

- , - 1

- - - i

ests Results

Yleld Statlc Yleld

strength (MPa), sw

note

Table 5.1 hm! 3 of 4

Table 5.1 - TESTS RESULTS ON COUPONS OF THE SPECIAL CARSON STEEL FROM THE JACQUES-CARTIER BRIDGE.

COUPON

Ultimate tenslle

rtrengîh Zeport date (MPa)

Yleld strength otmerved durlng tesi

( M W

Statlc Yleld

strengih MPa), see

note bdow

243 229 243 246 231 234 24% 228 243 236 235 23 1 240 . .

Standard deviation(74 coupons) = 25 i 7 17 Coef. Var.(74 coupons) = 0.0559 0.0634 0.0708

ote: tatlc Yield strength (MPa) = Yield strength obsewed during test (MPa) - 28 (MPa)

1 Table 5.1

Sheet 4 of 4

5.5 EVALUAïïON OF TEE TRUSS $PAN ACCORDING TO CHBDC

The folIowing members of the tniss span were evaluated accordiig to the

CHBDC (1998):

The tniss members in tension and compression.

A typicai stringer in shear and bending.

A typical floor beam in shear and bending.

The numbenng of the truss members is shown in Fig. 5.4.

The computer program for structurai analysis S A F i was used to derive the

internal forces in the above mentioned members of the bridge. This program has the

advantage of generating internai forces for movable truck and lane loading as

specified in the CHBDC (1998). The movable truck and lane loads for CLI-625 were

usai to calculate the internai forces in the members.

The following categories of dead loads were considered in the evaluation:

a Category 1, dead load of factory produced steel components.

Category 3, dead load of bituminous concrete surfacing based on nominal

thickness.

a Category 4, dead load of cast-in-place concrete deck.

The internal forces generated in the members for each category of dead loads

were calcuiated separately.

The provisions of the code (CHBDC, 1998) for the modification factor for

multiple Iane loading (clause 14.8.4), and the dynamic Ioad aiiowance (clause

14.8.1.6) were applied.

The results of the eduation according to the resistance and load hctors

method and to the mean load method are included in Tables Al to A6 of

MPENDIX A

The target reliabiiii index B for each member of the bridge beimg evaiuated

was set according to the type of tratliic, the system behaviour, the &ment betiaviour

and the inspection IeveI considered for the evaiuated member. This information is

included in the appropriate table in APPEM)iX A

Since the Jacques-Cartier bridge is essential to the local economy of the

Montreal area, a value of 0.25 was added to the value of B given in the code.

in the load and resistance hcton method, the dead Ioad &ors were selected

according to the target reliabiiii index B and the category of dead load. The live load

factor was detennined according to the target reliability index B and the type of lateral

distribution of live load assumed. AI1 this information is included in the tables.

The live load capacity factors F, in the case of the load and resistance factors

method were calculateci according to Eq. 3.1.

in the case of the mean load method, the live load capacity factors F were

caIcuLated according to Eq. 3.2.

The values given in the commentary of the code for the bias coefficients 6-

BAL. &3,6r, &, 6 ~ , and the coefficients of variation Va Vd, VD, VI, VL, VR, were

used in applying the above equatioa Al1 this information is included in the tables of

APPENDIX A

in the case of the mean Ioad method, the actual reliability index of the

rnember considered ($ at F=l), was caicuiated by setting Eq.3.2 equal to 1 and

solving this equation to obtain the value of B. This d u e represents the d u e of the

relîabiiiity index of the member under the loads considered, Al1 the results for the

actual reliability index of the rnembers are presented in the Tables of APPENDK A

5.6 EVALUATION OF TEE TRUSS SPAN ACCORDING TO TEE

F0RMtSOR.M METHOD

The methodotogy presented in Chapter 4 was used in the evaiuation of the

tmss span according to the FORMISORM method.

The system reliability program SYSREL, as developed by RCP GmbY

Munich, Germany, was used.

The lirnit state fiinctions for each mode of failure were written in FORTRAN

and were compiled with the above program.

The results of the evaluation using the F O W S O R M method and the

comparison of these results with the resdts of the evaluation using îhe load and

resistance factors method and the mean load method are presented in the follouhg

Tables:

Table 5.2 for the truss members in compression.

r Table 5.3 for the truss members in tension.

Table 5.4 for the stringer in shear and bending.

Table 5.5 for the floor beam in shear and bending.

The same nominal values for the geometric properties, the material properties

arid the Ioads, used in the previous evaluation procedures were used in the

FORM/SORM evaluation procedure.

The vaIues of the statisticai parameters and the distniution fùnctions for the

Merent variables are listed in Cbapter 4, except for the steel yield strensth, wbich

are listed in section 5.4.

The program SYSREL calculates the actual reliability index under the

considered loads for each member evaluated using the FORMISORM method. In

order to calculate the live load capacity factor according to the FORMISORM

method, the foilowing iterative process was foilowed:

The live load was varied until the target reliability index as called for Ui

the code was reached according to the SYSREL program Al1 the dead

loads were kept the same in this process.

0 The live load capacity factor is therefore the ratio of the live Load at the

target reliability index fl to the Iive load for which the member is being

evaluated.

The values of the a d reliability index and the values of the Iive Ioad

factors, caicuiated according to the FORMtSORM method, are reported in Tables 5.2,

5.3, 5.4 and 5.5.

5.7 COMPARISON OF THE RESULTS

In cornparhg the results shown in Tables 5.2 to 5.5, we c m condude the

foIlowing :

0 For the truss memben in com~ression:

The live load capacity factors calculatecl acwrding to the

FORMISORM method caa vary between 97% and 11 1% of the live

load capacity m o r s calcuIated accordiig to the load and resistance

factors method.

The Ioad and resistance &ors method is conservative in this case.

This is expected since the load factors encompass rnany combinations

of dead loads, tive loads and resistance values.

The Iive load capacity factors calculated according to the

FORMlSORM methad can vary between 88% and IWh of the live

load capacity hctors caicuiated accordkg to the mean load method.

The mean Load method is not conservative at al1 in this case.

For the truss mtmbers in tension:

The live Ioad capacity factors dculated according to the

FORM/SORM method are Iarger (up to 5%) than the live load

capacity factors caldateci acwrding tc the load and resistance factors

method.(Note: the values for member No.30 are not wnsidered since

there is practidly no Ioad in this member)

Here again, the load and mistance fhctor method is consemative.

The live load capacity factors calculated according to the

FORMISORM method are aimost equal to the values of the live load

capacity factors calculated according to the mean load method.(Here

again member No.30 is not considered)

For the strinptr in sbcar:

The values of the live load factor calculated according to the

FORMf SORM method are higher than the values calculated according

to the load and &stance factors method (13%) and the values

calculated according to the mean load method (8.5%).

For the floor beam in shear:

The values of the [ive Ioad hctor calculated acwrding to the

FORMJSORM method are higher than the values calculated acwrding

to the load and resistance Mors method (31%) and the values

caicuIated according to the mean Ioad method (30%).

In this case, both the load and resistance factors method and the mean

load rnethod are very conservative.

For the strincer in btnd in~

0 The values of the iive load factor calculated according to the

FORMlSORM method are qua1 to the d u e s calcuIated according to

the load and resistance factors method.

The values of the live load îàctor caiculated according to the

FORMISORM method are higher than the values caiculated according

to the mean Ioad method (5%)

@!g

w The values of the live load factor caldateci accordhg to the

FOWSORM method are airnost equal to the values caIculated

aecording to the load and resistance factors method and the mean load

method.

Tabk 6.2 - REUABIUM ANALYSIS OF THE TRUSS MEMBERS IN COMPRESSION - TRUSS SPAN OF JACQUES-CARTIER BRIDOE - COMPARlSON OF RESULTS FOR THE FORMISORM METHOD, THE LOADS AND RESISTANCE FACTORS METHOD, AND THE MEAN

No& Cornergence to a .oluUon for p could not be obtained wiai.the program SYSREL. *nism h practkally no kad on rwtnber No.12

Tabb 6.3 - REUABIUTY ANALYSIS OF THE TRUSS MEMBERS IN TENSION - TRUSS SPAN OF JACQUES-CARTIER BRIDQE - COMPARISON OF RESULTS FOR THE FORMlSORM METHOD, THE LOADS AND RESISTANCE FACTORS METHOD, AND THE MEAN

F3E2

1 .O21 0.989 0.982 1,002 0.982

1.019 1.021 0.995 0.928

FSFl

. 1 .M7 1.019 1.015 1.031 1.028 1.M 1.M8 1 .O36 0.978

Muril relkbilHy

Inka using F ~ R ~ W ) ( I Y

method

5.993 7.390 7.872 6.131 7.076 6.059 5.903 6.703 16.030

LOAD F1 U v e W

factor 8ccordlng to

ioadand resldrnce

fmors M h o d ln CHBOC

1.67 1.92 2.05 1.73 2.40 1.74 1.70 2.1 3

161.63

M m r

10 21 32 4 26

27 26 2s *30

U w loid (in KN) al

CHBOC Iirfpî

ralkblUty i ~ r uaing -

mahod

1512.0 893.0

913.0 151 1.0 1394.0 2354.0 2336.0 1342.0 3071.0

Dud loldRlva

lod

1.63 1.04 1.00 1.62 1.65

1.75 1.75 1.70 1.25

F3 Llve lord factor according to FORW - mm

1 .748 1.956 2.081 1.784 2.166 1.820 1.781 2.207

158.066

METHOD IN F i Uveloiid

lador wcordlng to Mean Lord Method in c n m

-

1 .71 2 1.978 2.1 19 1.780 2.51 1 1.787 1.714 2.21 8

170.326

Tirgel mikbiliîy

in de^ accordlng 1oCHBOC

3 . ~ ~ ~

3.25 3.25 3-25 3-25 3.25

3.25

3.25 3-00

-ad loiid, unlrctorad,

In UN

1409.2 476.5 437.1 1375.3 931.6 2257.9 2292.1 1034.0 24.3

CHBDC. Acîual

retkbiltty Inder using the Mein

Lord i ah al ln CHsOc

6.044 7.431 7.974 6.293 8.497 6.239 6.095

7.61 3 44.104

Liveload, unladord

(wlthno knp.cth In

UN

885.0 456.6 438.8 847.1 565.2 1293.1 1311.6 608.1 19.4

Table 5.4 - RELlABlLlïY ANALYSIS OF THE STUINGER IN SHEAR AND BENDING - TRUSS SPAN OF JACQUES-CARTIER BRIDGE - COMPARISON OF RESULTS FOR THE FORWSORM METHOD. THE LOAOS AND RESISTANCE FACTORS METHOD, AND THE MEAN

- StRlNOEf IN SHEAR

STRINQEF IN

BENDINO

bondlng shsar, KN. for bendlnp 1

factors mothod ln CHBDC A

AETHOD H i F2 Uvo loiid

triaor iccording la Mean Load Mslhod in

CHBDC

indm uslh th. Mean

Loid Melhod In CH6Dc

KN for shiir, KN.m f i

bendlna) al

rslïablllty Indes uslna

F3 Livi load hcta aocordln[ ta POAUl

8aRM rnethod

Table 5.4 Sheet 1 of 1

Table 5.5 - RELlABlLlN ANALYSIS OF THE FLOOR BEAM IN SHEAR AND BENDlNG - TRUSS SPAN OF JACQUES- CARTIER BRIDGE - COMPARISON OF RESULTS FOR THE FORMISORM METHOD, THE LOADS AND RESISTANCE

Member

, FLOOR

BEAM IN SHEAR

FLOOR BEAM IN BENDlNG

Dead load, rnfactored, In KN for shear,

KN.m for bendlng 1

FACTORS METHOD, AND Tt Live load, 1 Dead 1 Target IF1 Llve loac unfactored

(wlth no Impact), ln

KN for shear,

KN.m for

loadl Llve loa

rellability index

accordlng I o CHBDC

factor accordlng t(

load and resistance

factors method ln

bendlng CHBDC

factor accordlng I o Mean

Load Method in CHBDC

rellablllr lndex

uslng th1 Mean Load

Method l CHBDC

4OD IN CHBDC. Actual 1 Live load (ln

rellability lndox uslnu

:ORW so method i

KN for shear, KN.m for

bendlng) at CHBDC Mtgd

rellablllty Index using

load facto

@O FORMI

F ORMISORM method

5.816

Table 6.5 Sheet 1 d 1

6. BRIDGE INSPECTION AND EVALUATION

According to CHBDC (1998) requirements (clause 14.3.31, the d u a t o r has

to take into consideration the defms and deterioration Ievel in the member being

evaluated. Men, the evaluator is Fdced with the diicdty of determinhg the extent

of the deterioration and its consequenus on the evduation.

In the present Chapter, a methodology is pranted wtiere the evaluator can

use the maîerial condition rating of the members to assess the effects of the

deterioration on the evaiuation.

The Material Condition Rating Systems included in the inspection m a l s of

the Ministère de Transports du Québec WQ), (199 1) and the Ontario Ministry of

Transport (MTO), (1991), are presented. A method is proposeci, which relates this

system to the evaiuation procedures using the Ioad and resistance factors method, the

mean load method and the FORWSORM methods.

This method is then applied ta the evaluatian of the floor beams of the miss

van of the Jacques-Cartier bridge for different IeveIs of deterioration. The resuits

illustrate how each method caa influence the evahation of the floor beams and

consequently the decisions concerning their rehbilitation.

The material condition rating system is a numericd system where a number

h m 1 to 6 is assigned to each component of the structure based upon observed

material defects and the resuIting effect on the abiity of the componern to perform its

fiinction in the structure. In addition, the number O is assigneci tu a component when it

does not exist in the particular structure under inspection; and the number 9 is

assigned when a component is not visible or accessibie at the time of inspection.

Al1 components of a stniaure are ciassiied as primary, secondary or

awùliary. The classification is given in the MTQ and MT0 inspection m u a i s and is

generaily done dong traditional classifidon of components based on structural

behaviour.

Primary components include embankments supporting foundations, piers,

abutment walls, pin and hanger bearings, beams, girders, stringers, floor beams,

tnisses, arches, load bearing diaphragms, wnnection of primary components, decks

and structural steel watings on pcimuy components.

Secondary components include embankments not supporting foundations,

ballast walls, wing walls, retaining wails, bearing seats, other bearings, joints, non-

load bearing diaphragms, bracing, wnnection of secondary components, curbs,

sidewalks, approach slabs, bankr wails, railings and structural steel coatings on

secondary components.

Auxiliary components include siope protections, signs and utilities.

In the case of the t w span of the Jacques Cartier bridge, the truss members,

the floor beams and the stringers are aü ciassified as primary components.

The materiai condition d n g system h r the components of a structure

represents the condition of the componeut b d upon observed defects. For example

cornmonly occurrîng defects in steel wmponents are the foiiowing:

Corrosion

Permanent deformation

Cracking

Loosewmections.

These defects are described and categorised as to their severity in the MTQ

and MT0 inspection manuals. General guideiines based upon the severity and extent

of observed defects are given in Fig 6.1 {IWQ, (1991)). The inspection manual of

the MT0 has similar provisions {MTO, (1991)).

The variable AR, is defined as a percentage denoting the reduction of the

resistance of the component due to deterioration

Where the deterioration in bridge components is mainly due to corrosion, it is

proposed that the vdue of AR be estimated as a fùnction of the material condition

ratings of the components.

For each material condition rating number, it is proposed to consider a value

of AR equal to the average value of the correspondhg upper and lower limit of the

percentage loss of cross section area, as given in Fig. 6.1, in the case of veq severe

materiai defect.

Based on this assumption and Fig.6.1 we propose to use values of AR as given

in Table 6.1.

The evaiuator wi however conduct his own measurements in order to

determine the percentage loss of the cross section am of the component and use it as

a value for AR.

The proposa1 to use the resistance reduction factor in the d u a t i o n of the

resistance of bridge cornponents is consistent with the code. The commentary of the

code recommends the use of a reduced resistance adjustmeut factor in order to

account for the deterioration and the reduction in the value of the nominal resistance.

The live load capacity factor is thea,

a usine the lord and raïstance factors method:

In the above equation, AR is the resistance reduction factor, al1 the other tenns

are as defined in Eq. 3.1.

or usiw the mern load mtîhod:

In the above equation, AR is the resistance reduction hctor, al1 the other terms

are as defined in Eq. 3.2.

The Iimit state îùnction, in the case ofthe FORWSORM method, is then:

In the above equation, AR is the resistance reduction hcto~; al1 the other terms

are as defined in Eq. 4.1.

Primary Element Secondary Element Auxiliary EIement

R

% Loss of Cornpanent Cross Sedon, Surface A m or Lmgth Affxted

Figare 6.1 - Material Condition Raîing of Components {(MTQ (1991))

Table 6.1 - Proposeci values for tbt resistincc reduction factor AR

Mattriai condition 1 AR

6.4 EXAMPLE: EVALUATION OF A DETEWRATED FLûOR BEAM

Rathg

IN SEEAR ON THE TRUSS SPAN OF TEE JACOUESICARTIER

Primary Element I~econday Element 1 Auriliary Eltment

We have evaluated the floor beam in shear on the tmss span of the laques-

Cartier bridge, for the case where the inspection of this element reveais the existence

of detenorateci sections at the locations where the maximum shear can occur. This

situation is oflen encountered in bridges where de-king salts are used on the madway.

The evaluation was done considering diierent materiai condition ratings and

using the methodology presented in the previous sections. The three evaiuation

methods presented in the present research thesis were used. The results of this

exercise are summarised in Table 6.2.

The results shown in Table 6.2 suggest the following:

0 When the results of the Ioad and resistance @ors method or when the

resuits of the mean load method are considered, an immediate decision

should be taken to rehabilitate the beam when the material condition rating

is 3 or les.

0 When the results of the FORMISORM method are considececi, a decision

to delay the rehabilitation of the floor beam can still be considered men if

the material rating of the beam is 3 or 2.

Considering the large expendinire involved in the rehabilitation of the floor

beams of the Jacques Cartier bridge, the decision to delay the rehabilitation of the

floor beams can have a signiticanr economical impact. For example, in lieu of

replacing the floor beams or strengthening the corrodeci web parts of the floor beams

in order to increase the resistance in shear, a decision can be taken to onIy apply a

new protective coating on these parts in order to stop the progress of corrosion.

The above exercise illustrates how the type of anaiysis wed in the evduation

c m have an infiuence on the decision making process for the rehabilitation of

deteriorated bridge components.

Note that a more sophisticated limit state bction can be used with the FORM

procedure that includes the rate of corrosion of steel to determine inspection

schedules of critical components as weii as a tentative schedule for the replacement or

rehabilitation of critical components.

Table 6.2 - FLOOR BEAM IN SHEAR - TRUSS SPAN OF JACQUES-CARTIER BRIDGE - COMPARISON OF RESULTS FOR THE EVALUATION OF THE LlVE LOAD CAPACITY FACTOR

ACCORDING TO THE MATERIAL CONDITION RATING OF THE FLOOR BEAM, USlNG THE LOADS AND RESISTANCE FACTORS METHOD, THE MEAN LOAD METHOD AND THE FORMISORM

Target factor AR reliability F!

MET).

Loads and resistance

factors method

Live load capacity factor F I

Actual Live load Actual Live load reliabilty capacity reliabilty capacity

index Ifactor FZ 1 index Ifartor F1

Mean load method 1

Mcr Jc,xls ,OB-08-26

FORMlSORM Method I

Table 6.1 Shed 1 of 1

7. CONCLUSION

The FOWSORM method cari be used as a computational twl to evaluate

components of existing bridge structures. It can be adapted in order to follow the

same philosophy in the evaluation of bridge components accordhg to the CHBDC,

and therefore to seek the same level of safety sought for in the code. The

FORMISORM method wi dso be adapted to seek different levels of d e t y and

different variations in the uncertainty of the loadq the method of analysis and the

resistance than the ones assurnecl in the code.

When campard ta the load and &stance factors method and the mean load

method contained in the code, the FORMlSORM method has the following

advantages:

a The formulation of the mechanical kilure mode of the evaiuated component

can be introduced in the evaiuation.

The statistical distribution fiindons of al1 the variables can be introduced in

the evaluation.

Information gathered fiom fieid data can be introduced in the iÎmit state

function used in the evaluation using the FORM/SORM methods. In the present

research thesis, information on the material properties of the steel used in the

fabrication of the bridge members was introduced in the evaIuation The same

procedure cm be used to introduce any oiher information availabie to the evduator

(information on dead loads, traffic loads, dynamic load allowance, material propdes

and gametrical properties ofcorisüuction materials, etc.)

In the FORMfSORM method, each eduatiou of a bridge component cm be

treated as a unique problem; therefore greater accuracy cih be achieved in the

evaluation Although the application of the FO W S O R M method is more eiaborate

than the methods outlined in the code, its apphation may be warrant4 if the

evaiuation results can Iead to economical savings and can help to develop better

monitoring, inspection, maintenance and rehabiIitation strategies for bridge members.

Without generalising the tindings of the evaluation of the truss span of the

Jacques-Cartier bridge, we can say the fOUowing conceruhg these results:

O The difference in the results of the evaluation using the FORMJSORM

method and the methods specified in the code depends on the mode of failure

considered.

The results using the load and resistance factors methoci are conservative in al1

cases. This is to be expected since the Ioad hctors detennined by the code

writers had to encompass many combinations of dead loads, Live loads and

resistance values. The results of the evaluation using the loads and resistance

factors are very conservative in the case of the floor bearns, when the shear

failure mode is considered.

O In the case of members in compression, there is a substantiai difference

between the resuits of the load and resistance fàcton method and the results of

the mean load method, aithough the same variation in the variables was used

in both cases. Such large difference in the r a i t s when applying the two

methods outlined in the code is not acceptable and can lead to confision in the

evaiuation.

O When compared to the r d t s of the FORMJSORM method, the results of the

evaiuation using the mean load method specified in the code are not

conservative in the case of tniss members subject to compression but are

conservative in the case of flwr beams subject to shear.

A provision should be included in the code to allow the experienced evaluator

to use advanced reliability methods der than the ones contained in the code, in the

evaiuation of bridge components, provided that the same Ievel of safety sought for in

the code is achieved.

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APPENDIX A

TABLES A.l TO A.6

EVALUATION OF THE TRUSS MEMBERS.

THE STRINGER AND THE FLOOR BEAM USING THE

LOAD AND RESISTANCE FACTORS METHOD AND

THE MEAN LOAD METEIOD.

TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE

TABLE At - EVALUATION OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUES-CARTIER BRIOGE USlNG THE LOAO AND

RESISTANCE FACTORS METHOD IN THE CHBDC - --

I~ype of Trafic = N m a l E (moduius of elasücity)= 200000 MPa

Truck Load = CL1625 4 (resistance factor ) = 6.95 (tension) Lateral Ditiibuîion Category for Live Load = U (resistance Statically Determinaie adjustment factor) = 1.00 (km. 6 comp.)

n(ro8ed W shapes).; 1-34 (for calculaüon of Cr)

TABLE A1 - EVALUATION OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE USlNG THE LOAD AND

RESISTANCE FACTORS-METHOD IN THE CHBDC (contlnueâ)

I Cr= factorcd compressive resistana.

(U = &stance adjustment factor

TABLE A.1- EVALUAT\ON OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUESICARTiER BRIDGE USlNG THE LOAD AND

RESISTANCE FACTORS METHOD IN THE CHBDC (continued)

I D l = unfactored dead load of sted (cabgoty 1)

M = unfacfOced dead load of cest in piaœ concrete ( 4)

I 03 =unfactored dead bad of bitu-us conarta (caegory 3)

D-total =total unfactored dead foad

Tabk Al Shcct3of6

TABLE AA - EVALUAT~ON OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUESICARflER BRIME USlNG THE LOAD AND

RESISTANCE FACTORS METHOD IN THE CHBDC (continued)

1 = target reiiabiïï index

. (ten.) = hre load with impad, member in tension

. (comp.) = h e load with impact, member in compression

Tabie A 1 Sheet 4 of 6

TABLE Al - EVALUATlON OF THE TRUSS MEMBERS OF T HE TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE USlNG THE LOAD AND

RESISTANCE FACTORS METHOD IN THE CHBDC (continued) .

p = target reliabiiii index

q = üve load factor

Iq, = dead load factor for load Category 1 (&el)

1% = dead load factor for load Category 4 (conaete deck)

TABLE A1 - EVALUAIYON OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUESICARTIER BRIDGE USlNG THE LOAD AND

RESISTANCE FACTORS METHOD IN THE CHBDC (continued)

'f = factored t e d e force in mernber

X = factored compressive force in mernber

Table Af S M 6 of 6

TABLE A.2 - EVACUATION OF THE TRUSS MEMBERS OF THE TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE USlNO THE MEAN LOAD METHOD LN THE CHBDC

R = non faclwed mean reslsmnce I = dynamk load allowance 04 = non factonad dead load for concrele = bias ~ ~ ~ ~ d s n t s (meenlnomlnal) for Ot, 04 deck (calegory 4) and 03 respecüvoly

L = non factared live load Dl = non faclored dead loed for eleel 03 = non factored dsad load for VDa,Vm,Vm = cosfliclent of varlath for D1,M and 03

(-teQorY 1) bltumlnous concroie (category 3) respecUwtly

Table A.2 Sheet 1 of 2

TABLE A.2 - EVALUATION OF THE TRUSS MEMBERS OF THE TRUSS SPAN ON THE JACQUES-CARTIER BRIDGE USlNG THE MEAN LOAD METHOD IN THE CHBDC (CONTINUED)

R = non factomd mean reil~tanw AL = live l a d analysb maiod 6&,i3,4 = b l a ~ coenrclento ( m a n I mlnal ) p = larget nliabilily Index for R,L,AL and I rerpecUvely

I L = non factwed llve load I = dynamlc load albance V,,V,,V,.V, coefîicient of varfaIlon for R,L,AL end I rerpecüvely

Table A,2 Sheat 2 of 2

TABLE A3 - OlALUATlON OF THE STRINGER OF THE TRUSS SPAN OF THE JACQUES- C-ER BRlûGE USlNG THE LOAD AND ïHE RESISTANCE FACTORS METHOD IN ïHE

CHBDC 1 SHEN? IN STRiNGER

I Type of Anaiysis = Staticaity detamiinab

Important struchne J ( S L V ~ loniaiai captais) = 1.49B-06 mm4 G(waphig--)

B (target rebbitity index) = 275 = 6.96E+11 mm6

TABLE A.4 - EVALUATION OF THE STRINGER OF THE TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE USlNG THE MEAN LOAD METHOD IN THE CHBDC

R = non faclored mean reilslence I = dynamlc loed allowence M - non factwed dead ioad for k,b,b = blai coemdsnb (meanlnominal) for concrets dedi (category 4) Dl, M and D3 respedvely

I L - non factomd llve load D l - non factored dead load for steel 03 - non factored dead load for Voi,Voc,Voi - cocimdsnt oi variation for Dl ,M and (-mgory 1) bltuminous c o m t e (calegory 3) Ds mipscüveiy I

R = non factored mean reilslence AL = live ioad analysls methoci 6&,6,6, = blas coefficients (mean I p = tami rellablfity index nomlnal) for R,L.AL and 1 rerpectively

I L = non factored llve load I = dynamlc load allowance VR.VLiVAL.Vl = coemClent of variation F = I b load capsclly factor for R,L.AL and 1 reipecthrely

STRINOER I mENmNa I A m ARI 177 iRI n mI 6 611 r 6 . A eoJ I*O1I 0.031 t .oJ o.1.J 1 . 4 0 . J

603

1.44

Member r

VOS J

0.53 -

vol

0.03

~ D Z

1.07

STRINQER b s n w

v m

0.14

501

1 .O1

STRINQER l- Table A.4

Shed 1 of 1

D4

24.66

D l

3.56

0 3

3.25

I

0.30

L

76.77

Member

STRINQCR (SHmR)

R

827.33

TABLE AS - EVALUAllON OF THE FLOOR BEAM OF THE TRUSS SPAN OF THE JACQUES- CARTlER BRIDGE USING THE LOAD AND THE RESISTANCE FACTORS METHOD IN THE

CHBDC 1 SHEAR IN FLOOR BEAM 1

Type of Analysis = Staîicaily deteminate Bernent Behavkur = E3 h (dearmdweb) = 1067 mm ~ B a h a v b u r = S 2 w (wer~~dmsss) = 11 mm

lnspedkn Level = INSP2 a(opecngdsmt+ f2TE93mm Imwrtant stnic!ure FY [ybld-mal= 221 MPa

I , - ~.

p (target reliability index) = 3 Fs(mJieorm)= 12728MPa 1

L Type of Anafysis = Staticalîy deteminate

aement Behm*our = €3 z (phairc seetbn maduira) = 1.81~+07 mm3

I Inspection Level = INSP2 ~ ( m b i a a r b m m a n t o t ~ ) = 1.62EM8 mm4 I Important structure J (s~vanra amant) = 223EW mm4

p (target reliabüii index) = 3 C,(warpriiombm~carstant)= 1.19l313 mm6

FY (yield-)= 221 MPa

TABLE A.6 - EVALUATION OF THE FLOOR BEAM OF THE TRUSS SPAN OF THE JACQUES-CARTIER BRIDGE USlNG THE MEAN LOAD METHOD IN THE CHBDC

= non factored m a n rnrilnlencs I = dynamk: load a l l 0 ~ m ~ e DI non lactored dead l a d for lbi,&,& blar coe111ciinb (mnlnomlnel) for concrete de& (calepory 4) Dl, M and 03 respscilvnly

I L non lacloreci livn ioad D l .i non factorird dead load for s lwl 0 3 = non factored dead load for Voi,Vm,Vw - confiideml of varletlon lu 01,04 and WWPY 1) bltumlnws C O ~ ' 0 t e (calsgory 3) D3 rnipectively 1

R - non facioreid m a n reslsbms AL live load analysis method &,q,6,q blas coefkhnls (mean 1 p - iarget rnfbbllity index nomlnai) for R.L.AL and I mspectlvely

I L non laclored Ilve load I dynamlc laad allowance V,,VL,VAL.V, = cwffldent of varialion for R.L.AL and I respecUvely

F live load capacity Isclor

Table A.6 Sheet 1 of 1

P

3.00

3.00

VL

0.07

0.07

6~

1.36

1.36

f

1.17

1.40

VR

0.17

0.10

Msmber

B U M (SHEAR)

BEAM (BENDINO)

p at F-1

3.492

4.852

AL

1.00

1.00

6~

1.17

1.13

VAL

0.00

0,OO

4

0.40

0.40

VI

1.00

1.00

SUBROUTINES FOR THE LIMIT STATE FUNCTIONS

BI-SUBSOPTXNE VCOM.E'ûR - LIMïT S T . FCNCTION FOR

kun-OVECI9I *cn-cL RfmRI(

Pm

SrmRMRIWL STAnr 1USIARr.m.m. l m 1

C I/O : IL0 1 1 1

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C Cor Ch. sm.rcb(mS1 for Che jolnC-bmtJ pOlntIS1 OC Cbm QC-SWa

C of zbi rya ra . (NIN-01.

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c w u l lmd *truccl+. bmu-point* n y bm altmrmd hmra. lNUï.nm.01

C Ic .sTAïîT 1s MC pmvadmd by th* uamr ch* SYSRa objuc- l l t~ruima

C contaln JII mqu~lly W QIIy mûul* IWTüiW L Pm).

C

C hiO u s a s u m woslümrmd ln *STPRTw:

C

C I I MAnT(W0 on luput 1s priamc by S B R U CO zero t s r l q i n ln

C U-*cm i f *s'mnS. l a u l lmd ac Cbm bmqlnolnq OC ch* amarcb Cor

C Chi ]OloC bata-pOliiC Of CUt-Smt lm.IWT; Ch* Cüt-Set w . I W

C m~tchia th* Iinm-nrnb.1 In IOR+ Ja deCinul on inprit t o âïSûEL.

C In mir car* ch- lnput virlablm IlWî la s i r - O . C r i , Meer h.vinq Couad C h Jolnc kCJ-point Cor CüC-SmC 00.1Cür

c SYSRLI trii= to ri& chi m . ~ - p ~ l n u OC the lnref i r r C O K U C C ~ ~ ~ U

c oc U* SU. NC-S~C I C chm wntml -n i t ch UEI ma imc -1 .nd

C .mr- 1. cJl1.d m c l l l wtch th* S a n valw oc 1 m 1 w r t a C NW-11 1.nm.O) rhirw 11 l a Cbe mPb.r of tnm io.EuVa COKUtralnt

C for rUch chm a0 ul lmd i n ~ c t l w bmta-plnt 1s amarcmd.

C QSTMtTINüI 1s prmamt Co 13. joint hca-polnc OC Cut-SmC na.1Cm

C tn t iua a s m . C

C W0t.s:

C - c - u w r r y ui. pr*siccinqi OC maRt hy snïm. 4- mf f i c imt .

C m l y L i convmrqmCi problmma o ~ ~ p r th. uamr u y up.rtimt wlLh

C r tar t loq ~ l u t t e n r q L W 0 14 PûWa hy &fiIIlng J CUat~Zmd

c *-P. h :c c m cemaci conaidmrabïi nork to &fini A s w C mlu t i sn for cbm rmarch for t b i lofnt+t~ point of mach Cut-SmC

C r Cirat t ry ta U> qiR stmm OLfamC to tbi nlp4CtW4 Si& Co a l1

C O-apte v a r l l b l ~ a rml*rtInq CO n a t a c m vart.blu

C (e-q. Dit-KI--1.1 and a p a l r t w offrmt Co O - V J r i a l u aiaarc+&

c ro ï o d n q qtmctc~ma m.q. mi-11-1.1. r u a ~ J S to b. p u f a r i i d

C oniy tn oii *PM ind I~rmspxl+. or th. riCELnq of ICü? IL*.

C Car a l 1 arc-Smtsl. In m a t UIU tbis mirm.dy u i l l

C c a n v c ~ c u n $ l d i E l b l y .

c - ni+ rcrcclrq miricion ( jo int bmtr-pincl set hy in u a i

c m . a i . o (LA. &ma si4cdun4 cor JCI ~iuccim aci-pint. Uct-11

C nmuld bm aitrird by th* uau Unly in mry r u i CMU u IL

C LS uaLtkmly t ü ~ t yau un LLad J k t f i r a t u t i p p salitlnu. If

C the Lnictira ~ c d - p l i i t Is vmy c1we CO Cbm ja ln t hmm-point

c a i c w i o d y LM- i lmt bi œnwrqano prabima JM yag m y

82-SüBRûUTINE -.FOR - LIMIT STATE ETMCTION FOR

MEMBeRS IN TENSION