Calibration of LRFD Bridge Code

CALIBRATION OF

LRFD

BRIDGE CODE

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Andrzej S. Nowak, l Member, ASCE ABSTRACT: This paper reviews the code development procedures used for the new load and resistance factor design (LRFD) bridge code. The new code is based on a probability-based approach. Structural performance is measured in terms of the reliability (or probability of failure). Load and resistance factors are derived so that the reliability of bridges designed using the proposed provisions will be at the predefined target level. The paper describes the calibration procedure (calculation of load and resistance factors). A new live load model is prop.osed, which provides a consistent safety margin for a wide spectrum of spans. The dynamic load model takes mto account the effect of road roughness, bridge dynamics, and vehicle dynamics. Statistical models of resistance (load-carrying capacity) are summarized for noncomposite steel, composite steel, reinforced concrete, and prestressed concrete. The reliability indices for bridges designed using the proposed code are compared with the reliability indices corresponding to the current specification. The proposed code provisions allow for a consistent design with a uniform level of reliability.

INTRODUCTION

The objective of this paper is to present the procedures used in the calibration of a new load and resistance factor design (LRFD) bridge code. The allowable stress method and load factor design, specified in the current AASHTO code (Standard 1992), do not provide for a consistent and uniform safety level for various groups of bridges. One of the major goals of the new code is to provide a uniform safety reserve. The main parts of the current AASHTO (Standard 1992) specification were written about 50 yr ago. There were many changes and adjustments at different times, which resulted in gaps and inconsistencies. Therefore, the work on the LRFD code also involves rewriting the document based on the stateof-the-art knowledge about various branches of bridge engineering. This paper summarizes some of these changes related to load and resistance models. The theory of code writing has advanced in the last 20 yr. Some of the important contributions were summarized by Madsen et al. (1986), Melchers (1987), Ellingwood et al. (1980), and Nowak and Lind (1979). The major tool in the development of a new code is the reliability analysis procedure. Structural performance is measured in terms of the reliability or probability of failure. The code provisions are formulated so that structures designed using the code have a consistent and uniform safety level. The available reliability methods are reviewed in several textbooks (Thoft-Christensen and Baker 1982; Madsen et al. 1986; Melchers 1987). The methods vary with regard to accuracy, required input data, computational effort, and special features (time variance). In an LRFD code, the basic design formula is

L

'V)(

< R n

(1)

where Xi = nominal (design) load component i; 'Vi load factor i; R n = nominal (design) resistance; and = resistance factor. The objective of calibration is to determine load and resistance factors so that the safety of bridges designed according to the code will be at the preselected target level. This paper presents the calibration procedure, including load models, resistance models, reliability analysis, and the development of load and resistance factors. Bridge load and resistance models are only summarized here because they are 'Prof., Dept. of Civ. Engrg., Univ. of Michigan, Ann Arbor, MI 48109-2125. Note. Associate Editor: Dennis R. Mertz. Discussion open until January 1, 1996. To extend the closing date one month, a written request must be filed with t~e ASCE Manager of Journals. The manuscript for thiS paper was submitted for review and possible publication on January 8, 1993. This paper is part of the Journal of Structural Engineering, Vol. 121, No.8, August, 1995. ©ASCE, ISSN 0733-9445/95/0008-1245-1251/ $2.00 + $.25 per page. Paper No. 5403.

described in other papers (Nowak 1993; Nowak and Hong 1991; Hwang and Nowak 1991; Tabsh and Nowak 1991; Ting and Nowak 1991; Nowak et al. 1993). Load and resistance are treated as random variables and are described by bias factors (ratio of mean to nominal), denoted by A, and by coefficients of variation, denoted by V. CALIBRATION PROCEDURE

The development of a new code involves the following steps: 1. Selection of representative bridges: About 200 structures were selected from various geographical regions of the United States. These structures cover materials, types, and spans, which are characteristic of the region. Emphasis is placed on current and future trends; instead of on very old bridges. For each selected bridge, load effects were calculated for various components. Loadcarrying capacities were also evaluated. 2. Establishment of statistical database for load and resistance parameters: The available data on load components, including results of surveys and other measurements, were gathered. Truck survey and weigh-in-motion (WIM) data were used for modeling live load. There was little field data on the dynamic load and, therefore, a numerical procedure was developed to simulate the dynamic bridge behavior. Statistical data for resistance include material tests, component tests, and field measurements. Numerical procedures were developed to simulate the behavior of large structural components and systems. 3. Development of load and resistance models: Loads and resistance are treated as random variables. Their variation is described by cumulative distribution functions (CDF) and correlations. The CDFs for loads were derived using the available statistical database (step 2). The live load model includes the multiple presence of trucks in one lane and in adjacent lanes. Multilane reduction factors were calculated for wider bridges. The dynamic load was modeled for single trucks and two trucks, side-by-side. Resistance models were developed for girder bridges. The variation of the ultimate strength was determined by simulations. 4. Development of reliability analysis procedure: Structural performance is measured in terms of the reliability or probability of failure. Reliability is measured in terms of the reliability index 13, calculated by using an iterative procedure. The developed load and resistance models (step 3) are part of the reliability analysis procedure. JOURNAL OF STRUCTURAL ENGINEERING / AUGUST 1995/1245

J. Struct. Eng., 1995, 121(8): 1245-1251

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laJ Standard HS20 Truck

5. Selection of target reliability index: Reliability indices were calculated for a wide spectrum of bridges designed according to the current AASHTO report ("Standard" 1992). The performance of existing bridges was evaluated to determine whether their reliability level is adequate. The target reliability index 131' was selected to provide a consistent and uniform safety margin for all structures. 6. Calculation of load and resistance factors: Load factors 'Yare calculated so that the factored load has a predetermined probability of being exceeded. The relationship among nominal (design) load X m mean load m x , and factored load 'YiXn is shown in Fig. 1. The corresponding terms for resistance are shown in Fig. 2. Resistance factors, , are calculated so that the structural reliability is close to the target value 131"

142.3 kN

* I

____*

35.6kN

142.3 kN

I

t

_

(hI HS20 Lane Loading 80 kN (for moment) 115 kN (for shear)

9.3 kN/m

leI Military Loading

LOAD MODELS

Load models are an important part of the code. The major components of bridge loads are dead load, live load, dynamic load (impact), environmental loads (e.g., wind, earthquake, temperature, water pressure), and special loads (e.g., collision forces). The code specifies design loads, load combinations, and load factors. However, the calculation of load factors requires a knowledge of the statistical models, in particular the distribution of magnitude, rate of occurrence, time variation, and correlation with other load components. The development of the live load model for bridges is presented in another paper (Nowak 1993). Therefore, it is only summarized in this paper with the resulting statistical parameters. Dead load D is the weight of structural and nonstructural members and is calculated using specified densities of materials. In the reliability analysis, several categories of Dare considered, because of differences in the degree of variation. For factory-made components (structural steel, precast concrete) the bias factor A is 1.03 and the coefficient of variation V is 0.08; for cast-in-place concrete X. is 1.05 and V is 0.10; and for asphalt surface the mean thickness is equal to 75 mm (3 in.) and Vis 0.25 (Nowak 1993). In the current AASHTO code (Standard 1992), a dead load factor of 1.3 is applied to all components of dead load. The current design live load is based on a HS-20 truck, lane loading, or military loading, as shown in Fig. 3 (Standard 1992). The corresponding moments and shears are lower than the actual load effects of heavy traffic observed on the highways in the United States today. The actual moments and

fleW

o FIG. 1.

o

mx Nominal Load, Mean Load, and Factored Load

R

FIG. 2. Nominal Resistance, Mean Resistance, and Factored Resistance

10_7_kN_L~_10_7_kN

_____

_

1~·2m.1 FIG. 3. 1992)

Nominal Live Load; HS-20 Truck and Lane Load (Standard

2

Shear

o-l--.----,.-...--,-~-r__~__r-~-r-...,..._j

o

10

20

30

40

50

60

Span (m)

FIG. 4. 1992)

Bias Factors for Moments and Shears; AASHTO (Standard

shears caused by the heaviest vehicles observed in truck surveys range from 1.5 to 1.8 times the design moments and shears (calculated using the HS-20 load). In the United States an average lifetime for bridges is about 75 yr. Therefore, this time period is used as the basis for the calculation of loads. Using the available data, a statistical model was developed for the mean maximum 75-yr moments and shears by extrapolation of the truck survey data. For a single lane, the bias factors (ratios of the mean maximum 75-yr moment/shear and HS-20 moment/shear) are plotted versus the span length in Fig. 4. Various new design loads were considered to obtain a uniform bias factor. Good results were achieved by superposition of the HS-20 truck and a uniformly distributed load of 9.3 kN . m (640 lb/ft). A tandem is specified for shorter spans. The proposed live load is shown in Fig. 5. The bias factors for moments and shears calculated for various spans using the new load are presented in Fig. 6. For multiple lanes, the girder moment or shear is a result of more than one truck load. It was determined by simulations that for two-lane bridges, two side-by-side trucks govern, with fully correlated weights. The probability of having two very

1246/ JOURNAL OF STRUCTURAL ENGINEERING / AUGUST 1995

J. Struct. Eng., 1995, 121(8): 1245-1251

(a) Truck and Uniform Load

TABLE 1.

Proposed Design Multilane Factors Number of Lanes

142.3kN

.

I

142.3kN 9.3kN/m

4.27m

I

~.

4.27-9.14m

~

I

ADTT (1)

One

Two

Three

(2)

(3)

(4)

(5)

100 1,000 5,000

1.15 1.20 1.25

0.95 1.00 1.05

0.65 0.85 0.90

0.55 0.60 0.65

--

(h) Tandem and Uniform Load

Span:

I

- ---

++

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1l1.2kN

2

Ill1.2kN

--+--

9.3kN/m

Four or more

12m 18m 27m 36m 60m

IJ·2m~1 FIG. 5.

Proposed Nominal Live Load (LRFD Load)

2

--

O+---.----r--.--,....-~--r-~--l

....

o

~

2

FIG. 7.

o-l--.---r-~-.,.---,.-.---.---.-..---.----.-----l

20

30

40

50

60

Span (m)

FIG. 6. Load

Bias Factors for Moments and Shears; Proposed LRFD

heavy trucks simultaneously on the bridge is lower than having a single heavily loaded truck. It was calculated that in a side-by-side occurrence, each truck is about 85% of the mean maximum 75-yr truck, which corresponds to the rriean maximum 2-month truck. Traffic frequency is very important in the statistical analysis of the heaviest trucks. Average daily truck traffic (ADTT) varies depending on local conditions. The calculations were performed for ADTT equal to 100, 1,000 and 5,000 trucks (in one direction). The proposed multilane factors are presented in Table 1. The total moment is distributed to girders. The girder moment can be determined by using a girder distribution factor (GDF). In the present AASHTO code (Standard 1992), GDFs for moments are specified only as a function of girder spacing, s GDF

0.4 + (s/6) - (s/25)2

=

(4)

where Span = span length in ft (1 ft = 0.305 m). For comparison, GDFs calculated using (3) and (2) are shown in Fig. 7. Dynamic load is defined as the ratio of dynamic deflection and static deflection. The current AASHTO (Standard 1992) report specifies impact I as a function of span length only

+ Span)

(5)

where Span = span length in ft (1 ft = 0.305 m). The actual dynamic load depends on three major factors: road roughness, bridge dynamics (natural period of vibration), and vehicle dynamics (type and condition of suspension system). The derivation of the statistical model for the dynamic behavior of bridges is presented by Hwang and Nowak (1991). The simulations indicated that the mean dynamic load is less than 0.17 for a single truck and less than 0.12 for two trucks, for all spans. The coefficient of variation of the dynamic load is 0.80. The coefficient of variation of a joint effect of the live load and dynamic load is 0.18. The proposed new design dynamic load is 0.33, applied to the truck effect only, with no dynamic load applied to the uniformly distributed portion of the live load (Fig. 5). RESISTANCE MODELS

=

sid

(2)

where s = girder spacing in ft (1 ft = 0.305 m); and d = 5.5 for steel girders and prestressed concrete girders, and 6.0 for reinforced-concrete T-beams. GDF is applied to half of the lane moment. For shear, the GDF in (2) is specified, except for the axle of the HS-20 truck placed directly over the support. New GDFs were derived by Zokaie et al. (1991), which relate GDF to girder spacing and span length. For the moment

and for shear

=

I = 50/(125

GDF

4

GDFs Specified by AASHTO and Proposed LRFD Code

GDF

10

3

Girder Spacing (m)

Shear

o

AASlITO (1992)

0.15 + (s/3)OIi(s/Span)02

(3)

The capacity of a bridge depends on the resistance of its components and connections. The component resistance R is mostly determined by materials strength and dimensions. R is treated as a random variable. The causes for uncertainty can be put into the following categories: 1. Material: strength of material, modulus of elasticity, cracking stress, and chemical composition. 2. Fabrication: geometry, dimensions, and section modulus. 3. Analysis: approximate method of analysis, and idealized stress and strain distribution model. JOURNAL OF STRUCTURAL ENGINEERING / AUGUST 1995/1247

J. Struct. Eng., 1995, 121(8): 1245-1251

The resulting variation of resistance has been modeled by tests, observations of existing structures, and by numerical simulations. In this study, R is considered a product of the nominal resistance R and three parameters: strength of material M, fabrication (dimensions) factor F, and analysis (professional) factor P, as was suggested by Ravindra and Galambos (1978) II

(6)

R = R"MFP

The mean value of R is a product of the mean values of M, F, and P, and the coefficient of variation V R is

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(7)

where V M, Vh and VI' = coefficients of variation of M, F, and P, respectively. For this calibration, the resistance parameters are based on the available material and component tests. Flexural capacity is established by simulation of the moment-curvature relationship, as described by Tabsh and Nowak (1991). The shear capacity of concrete components is calculated using the modified compression field theory (Collins and Mitchell 1991). The statistical parameters were developed by Nowak et al. (1993). The models of resistance are considered for noncomposite steel girders, composite steel girders, reinforced-concrete Tbeams, and prestressed-concrete AASHTO-type girders. Nominal (design) value of resistance is calculated using (1). The bias factors and coefficients of variation for the considered materials and limit states are summarized in Table 2.

where -1 = inverse standard normal distribution function. Examples of {3s and corresponding PfoS are shown in Table 3. The available procedures to calculate {3 vary with regard to accuracy, required input data, and computing effort. In this study, the reliability analysis is performed using an iterative method based on normal approximations to non normal distributions at the so-called design point [Thoft-Christensen and Baker (1992), Melchers (1987)]. The design point is the point of the maximum probability (maximum joint probability density of Rand Q) on the failure boundary (limit state function). The mathematical representation of the failure boundary is the limit state function equal to zero, g = R - Q = O. The design point, denoted by (R*, Q*), is located on the failure boundary, so R* = Q*. Let F R be the cumulative distribution function (CDF) and fR the probability density function (PDF) for R. Similarly, F(} and fQ are the CDF and PDF for Q. An initial value of R* (design point) is guessed first. Next, F R is approximated by a normal distribution F~, such that (II, 12)

The standard deviation and the mean of R' are rr~ =

The available reliability methods are presented in several publications [e.g., Thoft-Christensen and Baker (1982), Madsen et al. (1986)]. In this study the reliability analysis is performed using an iterative procedure. Let R represent the resistance (e.g., moment carrying capacity) and Q represent the load effect (e.g., total moment applied to the considered beam). Then, the corresponding limit state function g can be written as

(13)

R* - rr;, I[FR(R*)]

( 14)

m~ =

where 4>11 = PDF of the standard normal random variable; and = CDF of the standard normal random variable. Similarly, F Q is approximated by a normal distribution F Q, such that F~(Q*)

RELIABILITY ANALYSIS

4>,,{-I[FR(R*)]}/fR(R*)

= FQ(Q*);

f~(Q*)

= fQ(Q*)

(15. 16)

The standard deviation and mean of Q' are rr~ =

m~

4>,,{-I[FQ(Q*)]}/fQ(Q*)

= Q* -

rr~-I[FQ(Q*)]

(17) (18)

The reliability index is

13 =

(m~

- m~)/(rr;l

+

rrJ)1I2

(19)

(8)

Next, a new design point can be calculated from the following equations:

If g > 0 the structure is safe, otherwise it fails. The probability of failure P r is equal to

(20)

g=R-Q

PF

=

Prob(R -

(21 )

Q < 0) = Prob(g < 0)

(9)

It is convenient to measure structural safety in terms of a reliability index {3, defined as a function of P r (10) TABLE 2.

Statistical Parameters of Resistance

Type of structure (1 ) Noncomposite steel girder Moment Shear Composite steel girders Moment Shear Reinforced-concrete T-beams Moment Shear Prestressed-concrete girders Moment Shear

Bias factor

Coefficient of variation

(2)

(3)

1.11 1.14

0.115 0.12

1.11 1.14

0.12 0.12

1.14 1.165

0.13 0.16

1.05 1.165

0.Q75 0.16

Then, the second iteration begins; the approximating normal distributions are found for F R and F Q at the new design point. The reliability index is calculated using (19), and the next design point is found from (20) and (21). Calculations are continued until R* and Q* do not change in consecutive iterations. Resistance is a product of parameters M, F, and P; therefore, it is assumed that the cumulative distribution function of R is lognormal. The CDF of the load is treated as a normal distribution function because Q is a sum of the components TABLE 3.

Reliability Index and Probability of Failure

Reliability index (1 )

o 1

1248/ JOURNAL OF STRUCTURAL ENGINEERING / AUGUST 1995

J. Struct. Eng., 1995, 121(8): 1245-1251

Probability of failure (2)

0.5 0.159

2

(1.0228

3 4 5 6

0.00135 (J. 0000317 0.000000287 0.000000000987

5-ro--------------,

5 ....-------------,

s = girder spacing

s -3.6m s =3.0m -

s=2.4m

~

8=I.am

j

f .l!

s • girder spacing

s.3.6m

4

s .3.0m 3

2

-----

s = 2.4 m

~

s-1.8m

s. 1.2 m

0.1-------------. .. o 50

s

= 1.2 m

0.1----~---------'

o

50

Span (JIll Span (m)

FIG. 8.

13s for Steel Girders-Moments; AASHTO (Standard 1992)

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5....---------------,

FIG. 12. 13s for Reinforced-Concrete T-Beams-Shears; AASHTO (Standard 1992)

s = girder spacing

s·3.6m

5.,.------------,

s • girder spadng

s .3.0m

s.3.6m

- . - - s=2.4m 2

-

s -3.0m

s-1.8m --

s=2.4m

--0--

sal.am

s - 1.2 m O.L..-~-~----------'

o

s - 1.2 m

50 Span (JIll

O.L..-~------------'

o

FIG. 9. 13S for Reinforced-Concrete T-Beams-Moments; AASHTO (Standard 1992)

5....--------------,

50 Span (m)

FIG. 13. 13s for Prestressed-Concrete Girders-Shears; AASHTO (Standard 1992)

s = girder spacing

s=3.6m 2

s=3.0m .-.-- s = 2.4 m

-

.....

s=1.8m s. 1.2 m

------

O.!--~--------~---J

o

50

o

Span (JIl)

1.5

RG. 10. 13s for Prestressed-Concrete Girders-Moments; AASHTO (Standard 1992)

5....-----------------,

s • girder spadng

2.5

Load Factors versus k

---+---------------

s =3.0m

~---_et'---~---~-----------

s" 2.4 m 1.8m

----.------'"0'-.-

s" 1.2 m 0.1----~------~-..J

o

2.0

5~--------------'

s -3.6m

Se

L Uncludlrlg dynamic loadl

It

FIG. 14.

-a--

o (factory-made component') o (cast-ln-pJace concrete) o 'asphalll

1

50

q,-O.95."I"1.70 41=0.95."1"1.60