RELIABILITY OF HIGHWAY GIRDER BRIDGES

By Sami W. Tabsh1 and Andrzej S. Nowak,2 Member, ASCE

ABSTRACT: Reliablity procedures are developed for girder bridges. Load models are based on the available statistical data. The derivation of load models is de­scribed in other papers. Resistance models are developed by simulations using the available test results for materials and components. The girder behavior is de­scribed by a moment curvature relationship. Bridge capacity is determined in terms of the maximum truck load before failure. The analysis is performed for a single unit truck, a semitrailer, and combinations of these trucks. For a given truck po­sition, the load is gradually increased until the deformation (maximum deflection) exceeds the critical level. The developed resistance models are used in the reli­ability analysis. Reliability indices are calculated for noncomposite and composite steel girders, reinforced concrete T-beams, and prestressed concrete girders. The calculations are performed for girders and structural systems. Reliability indices for the system are higher than for girders. The effect of correlation between girder strengths is analyzed. Girder bridges behave as parallel systems. Sensitivity func­tions are developed for various parameters related to the considered girders. The developed approach is demonstrated on typical girder bridges.

INTRODUCTION

There is a growing need for efficient methods for evaluation of bridge performance. Load and resistance parameters are random variables. There­fore, deterministic approaches usually do not reveal the actual safety reserve. On the other hand, in the past 20 years, probabilistic methods were devel­oped and are available for practical applications. The objective of this paper is to present a practical procedure to calculate the reliability for girder bridges.

The analysis utilizes the available statistical data on bridge load and re­sistance. The basic parameters describing a random variable are the mean and standard deviation. It is convenient to use nondimensional parameters, bias factor, and the coefficient of variation. Bias factor is the ratio of mean to nominal, where nominal is the design value and the coefficient of vari­ation is the ratio of standard deviation and the mean. The load model is based on truck-weight surveys. Resistance (load-carrying capacity) is con­sidered for steel girders (noncomposite and composite), reinforced concrete T-beams, and prestressed concrete girders. The moment-curvature relation­ships are developed using the incremental load approach. The interaction between the girders is considered in calculation of the system reliability.

Structural reliability is measured in terms of the reliability index. Calcu­lations are performed for a wide spectrum of girder bridges, including struc­tural members (girders) and whole systems.

LOAD MODEL

The loads on a highway bridge include dead load, D, live load, L, dy­namic load, / , environmental loads, and special loads (breaking and collision

'Struct. Engr., Gannett Fleming, Inc., Harrisburg, PA 17105; formerly, Grad. Stud. Res. Asst., Dept. of Civ. Engrg., Univ. of Michigan, Ann Arbor, MI 48109.

2Prof. of Civ. Engrg., Univ. of Michigan, Ann Arbor, MI. Note. Discussion open until January 1, 1992. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 22, 1990. This paper is part of the Journal of Structural Engineering, Vol. 117, No. 8, Au­gust, 1991. ©ASCE, ISSN0733-9445/91/0008-2372/$1.00 + $.15 per page. Paper No. 26075.

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8000

7000

E" 6000

£" ^ J, g 5000

CO

£ *~ 4000 o II E -° "J 3000

2000

1000

0 0 100 200

Span (ft) ( 1 f t „ 3 0 5 m m )

FIG. 1. Live-Load Moments versus Span Length: Mean Maximum 75-Year Mo­ment, Mean Maximum One-Month Moment, and HS-20 Moment {Standards 1989)

forces). For short and medium spans, dead load, live load, and dynamic load dominate the design. The live-load effect also depends on truck position and distribution factors.

Dead load is the weight of structural and nonstructural members. Because of differences in the statistical parameters, it is convenient to distinguish between factory-made members (steel and precast concrete), cast-in-place members (T-beams, slab), and wearing surface (asphalt). In this study, the mean-to-nominal ratio was taken to be 1.03 for factory-made members and 1.05 for cast-in-place members, with the coefficient of variation, V, equal to 0.08 and 0.10, respectively. The mean asphalt thickness was assumed equal to 3.5 in. (90 mm) and V equal to 0.15.

The live-load model was developed by Nowak and Hong (1991). It is based on the Ontario truck survey data (Agarwal and Wolkowicz 1976). Other available data were also considered in the load analysis, in particular weigh-in-motion (WIM) measurements from Wisconsin and Florida, and Michigan State Police citation files. The model includes the effect of mul­tiple presence (in lane and side by side). It was determined that the maxi­mum 75-year live load is the result of two trucks side by side, with correlated weights, each corresponding to a maximum monthly truck. The moments (per lane) due to the maximum 75-year truck, maximum monthly truck, and the American Association of State Highway and Transportation Officials (AASHTO) (Standard 1989) HS-20 truck are shown in Fig. 1 for spans up to 200 ft (60 m). For the maximum 75-year live load, the coefficient of variation is 0.14.

The mean maximum 75-year live load was derived for individual girders (Nowak and Hong 1991). The calculations were performed using a finite element procedure. AASHTO (Standard 1989) girder distribution factor is, in most cases, s/5.5, where s = girder spacing. The actual distribution is a nonlinear function of s. Therefore, the resulting' mean-to-nominal live load ratios vary depending on girder spacing. The coefficient of variation ex-

75 Year Moment

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Strain Structural Steel

Strain Prestressing Steel

Stress

Strain Reinforcing Steel

Strain Concrete

FIG. 2. Typical Stress-Strain Diagrams for Steel and Concrete

pressing the uncertainty in live-load analysis (analysis or influence factor) is 0.12.

The dynamic load model was developed by Hwang and Nowak (1991). It is based on bridge tests and numerical simulations. The dynamic load depends on three major factors: bridge dynamics, vehicle dynamics, and road roughness. In practice, it is very difficult to evaluate the contribution of these factors. It was observed that dynamic load (as a fraction of live load) is reduced for multiple trucks and for heavy trucks. In this study dynamic load is treated as an additional static equivalent added to live load. Based on the results of bridge tests and analytical simulations, it is further assumed that the mean maximum dynamic load is 0.15 of the mean maximum 75-year live load with a coefficient of variation of 0.8.

The total load, Q, is a combination of D, L, and /,

Q = D + L + I (1)

The mean of Q is a sum of means of its components and the variance is a sum of variances. Q = a sum of three components (1), therefore it is treated as a normal random variable.

RESISTANCE MODELS

Resistance, R, is considered as a product of three factors: material prop­erties (strength), fabrication (dimensions), and professional (analysis). The statistical parameters were derived on the basis of the available data. As a product, the distribution of R is treated as lognormal.

The capacity of a bridge depends on the resistance of its components and connections. The major parameters include span length, number of girders, girder spacing, and transverse stiffeners (number and position of dia­phragms). The component resistance is determined mostly by material strength and dimensions.

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TABLE 1. Statistical Parameters of Material Properties

Variable

(1)

Fy

fy f, fp. fc f'c E,

Nominal (ksi) (2)

36 40 60

270 3 5

29,000

Mean (ksi) (3)

38 45 67

281 2.76

4.03 29,000

Mean-to-nominal ratio (4)

1.055 1.125 1.115 1.04 0.92

0.805 1.0

V (5)

0.10 0.12 0.10 0.025 0.18 0.15 0.06

Note: 1 ksi = 6.895 MPa.

In this study, the behavior of girder bridges is considered at two levels. The first level involves the failure analysis of a single girder. The second level requires the analysis of the whole bridge as a structural system.

Material Properties Four basic materials are considered in the paper: structural steel, rein­

forcing steel, prestressing steel, and concrete. Typical stress-strain relation­ships are shown in Fig. 2. The statistical parameters for material and fab­rication variables were derived from the available test data (Ellingwood et al. 1980; Kennedy and Baker 1984; Siriaksorn 1980) and by analysis. The most important ones include yield stress for structural steel, Fy, yield stress for reinforcing steel, fy, ultimate prestressing stress, fps, compressive strength of concrete, f'c, and modulus of elasticity of steel, Es. The nominal (design) values, means, and coefficients of variation are shown in Table 1. Dimen­sions were also considered. It was observed that the mean-to-nominal ratio is 1.0 and the coefficient of variation is very small, 0.01-0.02 in most cases.

Girder Resistance Moment-curvature relationship is developed for bridge girders using the

strain incremental approach. The girder cross section is idealized by a set

I

[

Actual Section Idealized Section

FIG. 3. Actual and Idealized Girder Section

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[L + O

n - q

M- = Mean a = Standard Deviation

0.0000 0.0002 0.0004 0.0006 0.0008

Curvature (rad/ln) (1ln = 25mm)

FIG. 4. Moment versus Curvature for Noncomposite W36 x 210 Steel Section

4800-

(i =Mean a = Standard Deviation

0.0001 0.0002 0.0003 0.0004

Curvature (rad/ln) (1ln = 25mm)

FIG. 5. Moment versus Curvature for Composite W36 x 210 Steel Section

of uniform layers, as shown in Fig. 3. For each strain level the correspond­ing stress diagram is generated using the nonlinear stress-strain relationships for the materials (Fig. 2). The position of a neutral axis is determined from the force equilibrium equation. Strain is increased gradually by increments and the procedure is continued until an ultimate strain of 0.0035 at the top of the concrete slab is reached. Tensile strength of concrete is neglected (below neutral axis). For steel girders, compact sections are considered.

The realization of moment versus curvature depends on material properties and geometry. Moment-curvature relationships were generated for the con­sidered types of girders. Thousands of Monte Carlo simulations were per­formed using the statistical parameters for materials (Table 1) (Tabsh 1990).

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I to m

FIG.

H + a

H r H - a

H = Mean a = Standard Deviation

1 1 i 1 1 0.0000 0.0001 0.0002 0.0003 0.0004

Curvature (rad/ln) (1ln = 25mm)

6. Moment versus Curvature for Reinforced Concrete T-Beam

1 II

i

? A

I

4000

3000

2000

1000

0 Q 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 m

Curvature (rad/ln) (1ln = 25mm)

FIG. 7. Moment versus Curvature for Prestressed Concrete AASHTO Type III Girder

TABLE 2. Statistical Parameters of Girder Resistance Girder type (1)

Noncomposite steel Composite steel Reinforced concrete Prestressed concrete

Mean-to-nominal ratio (2)

1.11 1.10 1.14 1.05

V (3)

0.115 0.12 0.13 0.075

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M

Curvature

FIG. 8. Bilinear Idealization of Moment-Curvature Relationship for Composite Steel Girder

The results of these simulations are summarized in Figs. 4, 5, 6, and 7 for noncomposite steel girder, composite steel girder, reinforced concrete T-beam, and prestressed concrete girder, respectively. Three curves are shown for each girder. The middle one represents the mean, the other two are one standard deviation above and one standard deviation below the mean. The statistical parameters of resistance (moment carrying capacity) are summa­rized in Table 2. Nominal moment carrying capacities are calculated using the AASHTO specifications (Standard 1989).

System Resistance Bridge capacity is usually defined in terms of the ultimate truck weight

that can be carried by the structure. However, this definition depends on truck configuration, truck weight distribution to axles, and truck position on the bridge. Furthermore, the maximum load effect is often due to multiple trucks (e.g., two trucks side by side). For a given truck (or trucks) type and position on the bridge, the bridge capacity can be determined by gradually increasing axle loads until deformations exceed acceptable values.

A nonlinear analysis of girder bridges was performed using the grid model and the direct stiffness method. The work is a continuation of the approach suggested by Tantawi (1986). System resistance is calculated for composite steel girders. Moment-curvature relationship for the girders is idealized by a bilinear model, as shown in Fig. 8. In Fig. 8, M* = the average of plastic moment and ultimate moment. Linear analysis is used to determine the load level corresponding to the first plastic hinge. Then, the stiffness matrix is updated and additional loading is applied until the next plastic hinge is formed. The procedure is repeated until either a large level of permanent deformation or an unstable grid is obtained.

The two most common truck configurations are used in this study: a single unit and a semitrailer (Moses and Garson 1973). The axle spacings and weight distributions of each truck type are shown in Fig. 9. When the wheel load

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20%

(a)

• U U if IT U W

40% (each) 11.1% 22.2% (each) 22.2% (each) (V o 0.3m)

(b)

FIG. 9. Truck Configurations Used in System Reliability Analysis: (a) Single-Unit Truck; (h) Semitrailer Truck

ouu -

500 -

400 -

300 -

200 -

100 -

0 -

H + a

-~ .... I ' ^~

1 1 I )x = mean

/ a = standard deviation

— L i 1 1 1 1 1 1 1 1 1 1 1 1

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Maximum Bridge Deflection (ft) (1ft = 0.3m)

FIG. 10. Truck Load versus Maximum Bridge Deflection

position does not coincide with a grid node, then a linear distribution in the longitudinal direction is assumed. However, a nonlinear load distribution is considered in the transverse direction.

The system resistance is defined as the ultimate weight of a combination of trucks. For short spans only one truck per lane is considered, a single unit, S or a semitrailer, T. For longer spans, the following combinations are considered for each lane: S, T, SS, TT, and ST. For multilane cases, various side-by-side combinations of lane loads are considered. For each truck com­bination, the ultimate load is calculated. The obtained resistances are com­pared with extreme load values calculated using the available statistical models

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(Nowak and Hong 1991), The governing combination is used in the system reliability analysis.

The system resistance is a random variable. The variation is determined by variation in moment curvature for girders, and properties of the con­necting media (slab and transverse stiffeners). The statistical parameters of the system resistance are obtained using a special sampling technique (Zhou and Nowak 1988). Load-deflection curves for a composite steel girder bridge are shown in Fig. 10. The span is 60 ft (18 m) with five W33 X 118 girders spaced at 8 ft (2.4 m). The middle curve represents the mean. The other two curves are one standard deviation above and below the mean.

RELIABILITY ANALYSIS

Reliability analysis starts with the formulation of a limit state function. The limit state function, g, is

g = R~Q ! (2)

where R = resistance and Q = load effect. The safety reserve is measured in terms of a reliability index, p, (Thoft-

Christensen and Baker 1982)

p = -$>~l(PF) (3)

where PF = probability of failure; and O - 1 = inverse of the standard normal distribution function.

The reliability index was calculated using an iterative procedure (Rackwitz and Fiessler 1978). The method is based on normal approximations of non-normal variables at the so-called design point.

Reliability of Bridge Girder This is a function of R and Q. The resistance, .R, is the moment carrying

capacity and the load, Q, is the total applied moment. Reliability index is a function of the statistical parameters of R and Q. The calculations were carried out for each case of material type, span, and girder spacing. The girders were designed according to AASHTO load factor design (Standard 1989). The factored load, Qn, is

Q„ = 1.3D + 2.17(L + I) (4)

where D, L, and / are load effects (moments) per girder calculated using the AASHTO Standard Specifications for Highway Bridges (1989). The mean load per girder is calculated using the statistical models described earlier.

Required nominal (design) resistance, R„, is

Rn = ^ (5) <t>

where 4> = resistance factor; cj> is equal to 1 for steel girders, 0.90 for rein­forced concrete (flexure), and 1 for pretensioned concrete girders. The mean resistance is calculated by multiplying the nominal by the bias factor (mean to nominal) (Table 2).

Reliability indices are calculated for girders in typical slab on girders bridges. Concrete slab thickness is 8 in. (200 mm), with 3-in. (75-mm) asphalt sur-

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X •a c

£ 3 i3 (9

0)

Reinforced Concrete Prestressed Concrete

Composite Steel Noncomposite Steel

10 20 30 —r~ 40 50 60

Span (m)

FIG. 11. Reliability Indexes versus Span for Bridge Girders

face. There are five girders spaced at 8 ft (2.4 m). The results are plotted versus span in Fig. 11, for noncomposite steel and composite steel, rein­forced concrete, and prestressed concrete. The reliability varies with the span length and material type.

The load and/or resistance parameters may be underestimated or over­estimated because of insufficient data, future changes, or human errors. There is a need to evaluate the importance of these parameters. Therefore, sensi­tivity functions are developed relating the reliability with the nominal values of load components (D, L, and/) and resistance parameters, such as material strength (Fy, fy, fps, and f'c) and section geometry (effective depth, d, slab effective width, b, slab thickness, ts, plastic section modulus, Z, reinforcing steel area, As, and prestressing steel area, Aps).

First, the calculations are performed using the nominal values of param­eters according to AASHTO (Standard 1989). Then, for selected parameters (one at a time) it is assumed that the nominal value is either higher or lower than that specified by the code (whichever reduces reliability).

Reliability indices are calculated for various degrees of departure from the specified value. Bridges with span of 100 ft (30 m) are considered. The results are shown in Fig. 12 for noncomposite steel, Fig. 13 for composite steel, Fig. 14 for reinforced concrete, and Fig. 15 for prestressed concrete girders. Horizontal axis is the percentage departure from the specified nom­inal value.

The results indicate that the reliablity of steel girders depends mostly on the plastic section modulus and yield stress. For reinforced and prestressed concrete girders the most important parameters are the reinforcing or pre-

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"8 bo to

f x u

•a "2

g . s

c S H ^ o .a o O

•a « 0*

"T 10

FIG.

0 10 20 30 40 50

Percent Change from Nominal

12. Sensitivity Functions for Noncomposite Steel Girder

Percent Change from Nominal

FIG. 13. Sensitivity Functions for Composite Steel Girder

stressing steel area and effective depth. On the other hand, dynamic load, slab dimensions (effective width and thickness), or concrete strength do not affect the structural reliability considerably.

Reliability of Bridge System This depends on the strength of components (girders, slab, and transverse

stiffeners). From the system reliability theory point of view, a girder bridge is a mixed parallel and series system. Usually, several girders must reach their ultimate load before the structure goes down. An example of load-deflection curves for a 60-ft- (18-m-) span composite steel girder bridge is shown in Fig. 16. The calculations are performed for two truck configura-

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-»q

O

is

2 5

§

•a

ei -0.2

0 10 20 30 40 Percent Change from Nominal

FIG. 14. Sensitivity Functions for Reinforced Concrete T-Beam

0 10 20 30 40

Percent Change from Nominal

FIG. 15. Sensitivity Functions for Prestressed Concrete Girder

tions, a single three-axle truck and a five-axle semitrailer, as shown in Fig. 9.

The first plastic hinge is formed at the total truck weight of 350 kips (1,550 kN) for a single truck and 500 kips (2,220 kN) for a semitrailer (truck is placed in one lane only). The ultimate loads are 500 kips (1,550 kN) for a single truck and 750 kips (3,340 kN) for a semitrailer. The difference between the ultimate load for the first girder and ultimate load for the bridge can be considered as a measure of redundancy..

Reliability indices can be calculated using the system resistance and load models. For each truck configuration (one or two trucks per lane, one or

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•a

•4

1 1

800

600

400

200

# ™ ^

Single Truck

Semi-trailer

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Maximum Bridge Deflection (ft) (1ft s 0.3m)

FIG. 16. Load Deflection Curves for Composite Steel Girder Bridge with Span of 60 ft (18 m); Due to Single Truck and Semitrailer

IX

l ft.

34 ft

right lane left lane A

0 © © 0 © •4-^ t^ 4 @ 8 ft. = 32 ft. M ^ - # 1 ft.

(1ft = o.3m)

FIG. 17. Cross Section of Considered Girder Bridge

two loaded lanes were considered) the ultimate strength is calculated using the load incremental method. The mean maximum load is calculated ac­cordingly, due to one or multiple trucks.

Bridge girders share the live load. At the higher load levels several girders may reach their ultimate load, and there is a nonlinear redistribution of ex­ternal forces. An important consideration is the effect of correlation between girder strengths on the system reliability. Correlation is measured in terms of the coefficient of correlation, p. There is practically no data available to estimate the correlation. Therefore, two extreme cases are considered. The calculations are performed for full correlation (p = 1) and no correlation (p = 0).

The system reliability analysis is demonstrated on a 60-ft (18-m) com­posite steel girder bridge with 5 W33 X 135 girders spaced at 8 ft (2.4 m), as shown in Fig. 17. The reliability index of each girder is 3.80. Various

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a a

u OS

I >-.

CO

o 00 03

9-j

8 -

7 -

6 -

5 -

4 -

3 -

2 -

1 -

0 -

J3 p=0

^ ^ / * P = 1

^ ^ ' ^ s '

y S ^ S ^

/ T y / ^

/yr ^\S • p = Coefficient of Correlation.

,—__ , , , , , ,. .,,

Girder Reliability Index

FIG. 18. Girder versus System Reliability Indices for Full Correlation and No Correlation

x o •o c

CO

to T3

a

u -

8 -

6 -

4 -

2 -

0 -

AB

e

, .__

«f G

(1ft 3 0.3m)

' I " I ' 1

f " •

H Span = 40 ft. « Span = 60 ft.

a Span = 80 ft.

O Span = 100 ft

• 1 •» 1 '

0 1 2 3 4 5 6 7

Girder Reliability Index

FIG. 19. Girder versus System Reliability Indices for Spans 40, 60, 80, and 100 ft (12, 18, 24, and 30 m)

combinations of trucks (Fig. 9) are considered. It turned out that two side-by-side trucks govern. The system reliability indices are calculated for cor­related and uncorrelated girder strengths, and the results are 4.95 and 6.20, respectively.

The increase of system reliability over that of girder reliability reflects the level of redundancy girder bridges have. The difference between the cor­related and uncorrelated cases indicates that girder bridges behave like par­allel systems.

Reliability indices are also calculated for other steel sections. The span,

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slab, and girder spacing remain the same. The comparison of system and girder reliabilities for correlated and uncorrelated cases is presented in Fig. 18. The relationship is almost linear in the practical range of values.

A practically linear relationship between the system and girder reliability is observed for other spans too. Composite steel girder bridges are consid­ered with spans 40, 60, 80, and 100 ft (12, 18, 24, and 30 m). For uncor­related girders spaced at 8 ft (2.4 m) the results are plotted in Fig. 19.

CONCLUSIONS

Reliability is a convenient meausre of the structural performance. There is a need for an efficient methodology to be used in the development of bridge evaluation and design criteria.

Reliability indices are calculated for girders and bridge systems. Load and resistance models are based on the available data. The analysis is performed for composite and noncomposite steel girders, reinforced concrete T-beams, and prestressed concrete girders. The results indicate that the reliability of bridges designed according to AASHTO Specifications {Standard 1989) vary depending on span and type of material. For noncomposite steel the reli­ability indices are 3 -3 .5 , for composite steel they are 2 .5-3.5 and for rein­forced concrete and prestressed concrete 3 .5-4.

The effect of correlation between the strength of girders in the same bridge is also considered. The reliability indices are calculated for full-correlation and no-correlation cases. The calculated system reliability is higher than for girders. Therefore, the considered girder bridges can be considered as par­allel systems.

Sensitivity analysis is performed for various design parameters related to bridge girders. The results indicate the importance of resistance parameters such as yield stress of steel or steel area.

The developed reliability procedure is demonstrated on typical composite steel girder bridges. The relationship between the reliability index for a girder-and-bridge system is derived, and it is slightly nonlinear.

ACKNOWLEDGMENT

The presented research was partially supported by the National Science Foundation, under Grant No. MSME-8715496, with program director Ken­neth Chong, which is gratefully acknowledged.

APPENDIX I. REFERENCES

Agarwal, A. C , and Wolkowicz, M. (1976). Interim report on 1975 commercial vehicle survey. Res. and Development Div., Ministry of Transp. and Communi­cations, Downsview, Ontario.

Ellingwood, B., Galambos, T. V., MacGregor, J. G., and Cornell, C. A. (1980). "Development of a probability based load criterion for American National Standard A58." Report 577, Nat. Bureau of Standards, Washington, D.C.

Hwang, E.-S., and Nowak, A. S. (1991). "Simulation of dynamic load for bridges." J. Struct. Engrg., ASCE, 117(5), 1413-1434.

Kennedy, L. D. J., and Baker, K. A. (1984). "Resistance factors for steel highway bridges." Can. J. Civ. Engrg., 11(2), 324-334.

Moses, P., and Garson, R. (1973). "Probability theory for highway bridge fatigue

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stresses." Report SMSMD No. 50, Div. of Solid Mech., Structures and Mech. Design, Case Western Reserve Univ., Cleveland, Ohio.

Nowak, A. S., and Hong, Y.-K. (1991). "Bridge live load models." J. Struct. Engrg., ASCE, 117(9).

Rackwitz, R., and Fiessler, B. (1978). "Structural reliability under combined random load sequences." Comput. Struct., 9(5), 489-494.

Siriaksorn, A. (1980). "Serviceability and reliability analysis of partially prestressed concrete beams," thesis presented to the University of Illinois, at Chiago, Illinois, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Standard specification for highway bridges. (1989). American Association of State Highway and Transportation Officials, Washington, D.C.

Tabsh, S. W. (1990). "Reliability-based sensitivity analysis of girder bridges," thesis presented to the University of Michigan, at Ann Arbor, Mich., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Tantawi, H. M. (1986). "Ultimate strength of highway girder bridges," thesis pre­sented to the University of Michigan, at Ann Arbor, Mich., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Thoft-Christensen, P., and Baker, M. J. (1982). Structural reliability theory and its applications. Springer-Verlag, New York, N.Y.

Zhou, J-H., and Nowak, A. S. (1988). "Integration formulas to evaluate functions of random variables." Struct. Saf, 5(4), 267-284.

APPENDIX II. NOTATION

The following symbols are used in this paper:

Aps

As

b D d

Es

Fy

f'c Jps

fy g I L

M Mn

M* PF

Q R S

SS ST

s T

TT

*s

= = = = = = = =

= = = = = = = = = = = = = = = = = =

area of prestressing steel; area of rebars; slab effective width; dead load; effective depth; modulus of elasticity for steel; yield stress of structural steel; compressive strength of concrete; ultimate stress of prestressing tendons; yield stress of rebars; limit-state function; impact, or dynamic load; live load; moment; nominal (design) moment carrying capacity; average of plastic moment and ultimate moment; probability of failure; total load; resistance (load carrying capacity); single unit truck configuration; two side-by-side single-unit trucks; single-unit truck and semitrailer (configuration); girder spacing; semitrailer truck configuration; two side-by-side semitrailers; slab thickness;

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V ~ coefficient of variation; Z = plastic section modulus; P = reliability index; p = coefficient of correlation between girder strengths;

< J > - 1 = standard normal distribution function; and <}> = resistance factor.

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