Reliability-Based Pavement Design Model Accounting for ... · Reliability analysis techniques in a pavement design procedures must provide a rational framework for quickly addressing

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RELIABILITY-BASED PAVEMENT DESIGN MODEL ACCOUNTING FOR INHERENT VARIABILITY OF DESIGN

PARAMETERS

by

Hyung Bae Kim, Ph.D. (Corresponding Author) Chief Researcher

Highway Research Institute Korea Highway Corporation

293-1 Kumto-dong Sujong-gu, Songnam-shi,

Kyungki-do, Korea Phone: (82-2) 2230-4851

Fax: (82-2) 2230-4184 E-Mail: [email protected]

and

Neeraj Buch, Ph.D. Associate Professor

Department of Civil and Environmental Engineering Michigan State University 3553 Engineering Building

East Lansing, MI 48824-1226 Phone: (517) 432-0012 Fax: (517) 432-1827

E-Mail: [email protected]

Prepared for

82nd Transportation Research Board Annual Meeting

Washington D.C.

January 12-16, 2003

Total Number of Words : 3,786 with 6 tables and 6 figures

TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal.

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ABSTRACT

The basic premise of pavement design procedures employed in Mechanistic

Empirical (M-E) approaches are widely accepted. However, several issues need to be

resolved before implementing M-E pavement design procedure in practice. One of the

issues is that the design procedure should provide a consistent pavement performance

level considering inherent variability associated with design input parameters. For a

complete M-E pavement design procedure, the effects of the inherent variability of input

design parameters on predicted pavement performance must be addressed and quantified.

Some of the key principles for applying the reliability concept to M-E pavement

design are presented to account for the inherent variability of design parameters in this

paper. In particular, the selection of an appropriate reliability assessment technique and

careful characterization of design input variability were considered because of their

central role in calculating the reliability of pavement performance and determining the

reliability-based safety factor of the pavement design procedure.

In addition, a reliability analysis model for evaluating uncertainties in the M-E

flexible pavement design procedure and a reliability-based pavement design approach

using Load and Resistance Factor Design (LRFD) format are introduced.

Key Word : Pavement Design, Reliability-Based Design, Reliability Index, LRFD


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BACKGROUND

The basic premise of pavement design models and procedures employed in

Mechanistic Empirical (M-E) approaches are widely accepted. However, couples of

issues need to be resolved before implementation. One of the issues is that the design

procedure should provide a consistent pavement performance at a desired level of

reliability by considering various sources of uncertainties. Adequate reliability techniques

should be incorporated into the M-E pavement design procedure to allow the designer to

consider various uncertainties of pavement design and produce a consistent pavement

performance level. Presently, several M-E pavement design procedures are adopting the

reliability modules to account for the inherent variability within the design process (4, 14).

However, there are couples of drawbacks in utilizing these procedures in practice. They

provide only the reliability of design result from a set of design input parameters. It

means that the design procedure should be run a number of times to yield the pavement

thickness meeting a target reliability level that the design engineer expects. Furthermore,

in these design procedures the design engineer is required to use the Monte Carlo

Simulation method to determine reliability with a great deal of computing time.

Reliability analysis techniques in a pavement design procedures must provide a rational

framework for quickly addressing uncertainties in predicting performance of the

pavement and simply determining the optimum pavement cross-section for target

reliability. In this study, a reliability-based pavement design format assessing effects of

variability of pavement input parameters on performance and establishing appropriate

safety factors has been developed.


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RELIABILITY CONCEPTS

The pavement design reliability is defined as the probability that the pavement’s

traffic load capacity exceeds the cumulative traffic loading on the pavement or the

amount of pavement distress accumulated during desired service life does not exceed a

specified level. In terms of rut-depth, this definition can be presented as followings;

SM RD RDrut predict= −max [1]

where :

SMrut = safety margin between maximum allowable and predicted rut-depth, RDmax = maximum allowable rut depth in the design period, and RDpredict = predicted rut depth in the design.

Since there are uncertainties in the major input parameters of pavements such as moduli

of layers, thickness of layers, traffic volume, etc., it is reasonable to define each

parameter as a random variable with its mean and standard deviation or its complete

probability distribution. Once the statistical information for each random variable is

obtained, one can calculate mean and standard deviation of the pavement performance

function, which in this study, is taken as SMrut.

The probability of failure, Pr(f) can be determined by constructing a probability

density function (pdf) on the performance function (e.g. SMrut) and calculating the area

under the curve that is less than the value of the limit state. Alternatively, the reliability of

the performance function can be characterized by a reliability index βHL, which is the

shortest distance between the design point on the failure surface and the origin in a

standardized normal space (7). Figure 1 graphically illustrates how to obtain β from

design input parameters that are defined as random variables. For normal random


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variables, the probability of failure, Pr(f), is estimated using the approximate relationship:

dzzgf HL ∫−

∞−=−Φ=

ββ )()()Pr( [2]

where :

( )HLβ−Φ = area under the pdf of standard normal variate from -∞ to -β, and g(z) = pdf of pavement performance.

The reliability of pavement performance can now be expressed as: 1-Pr(f).

The βHL can be obtained by an iteration method suggested by Rackwitz and

Fiessler (11).

SOURCES OF UNCERTAINTIES IN THE M-E FLEXIBLE PAVEMENT

DESIGN

Uncertainties affecting pavement performance can be grouped into the following four

categories:

1. Spatial variability that includes a real difference in the basic properties of materials from one point to another in what are assumed to be homogeneous layers and a fluctuation in the material and cross-sectional properties due to construction quality.

2. Variability due to the imprecision in quantifying the parameters affecting

pavement performance (i.e. random measurement error in determining the strength of subgrade soil, and estimation of traffic volume in terms of average daily traffic and mean truck equivalency factor).

3. Model bias due to the assumption and idealization of a complex pavement analysis

model with a simple mathematical expression. 4. Statistical error due to the lack of fit of the regression equation.

The first and second sources can be combined into uncertainties of design parameters

representing the spatial variability and inconsistent estimation of the parameters. The


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third and fourth sources can be combined into systematic errors consistently deviating

from predicted actual pavement performance. Uncertainties of design parameters cause

the variation within the probability distribution of the performance function, whereas

systematic errors cause the variation in possible location of the probability distribution of

the performance function (3). This concept is graphically presented in Figure 2.

DEVELOPMENT OF A RELIABILITY MODEL FOR EVALUATING THE

VARIABILITY OF PAVEMENT PERFORMANCE

The reliability model in this paper consists of two subsystems: an analytically

derived mechanistic subsystem for predicting pavement performance and a reliability

subsystem for analyzing the limit state function. An iterative loop including these

subsystems is established as is presented in Figure 3. In the mechanistic subsystem, the

structural analysis of a pavement section is conducted to obtain the relevant pavement

structural responses due to traffic loads and the prediction of pavement distress using the

transfer function as done in general M-E flexible pavement design procedure.

Considering the variability of design input parameters, the mechanistic subsystem

produces a set of predicted pavement distresses by varying the material and cross-

sectional properties. Using these predicted distresses, the reliability subsystem then

estimates the reliability of the pavement section in terms of the reliability index and

produces revised design points that are closer to the failure surface representing the limit

state condition (SMrut = 0 for example). Based on these revised design input parameters,

the subsystems are run in regular sequence. The iterative running of subsystems continues

until the revised design points reach to failure surface and βHL converges.


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In order to effectively quantify systematic errors of the design procedure, a

professional factor concept, defined as a representative ratio of the measured to predicted

pavement performance is introduced (12). The professional factor, P, reflects

uncertainties of the assumptions and simplifications used in design models. These

uncertainties could be the result of using approximations for theoretically exact formulas.

When this suggested reliability model is applied to design the pavement with rutting

failure criterion, the limit-state function of the model incorporating the professional factor

can be expressed as follows:

predictrut RDPRDSM ⋅−= max [3]

where :

predict

measure

RD

RDP = [4]

RDmeasure = Measured Rut-Depth

RDpredict = Predicted Rut Depth by the Transfer Function

In order to better explain how to execute the pavement reliability model

suggested in this study, an illustrative example is introduced. The eight pavement

sections were designed using the AASHTO 93 protocol. The parameter inputs include the

asphalt concrete (AC) modulus (2069 – 3103MPa) and subgrade resilient modulus (12-

76MPa). The thickness of the base and subbase are fixed at 203 and 406mm respectively.

The moduli of the based and subbase are also fixed at 276 and 103 MPa respectively. All

sections were designed to carry the traffic volume of 5, 10, 13, and 18.5 million ESALs

with a reliability of 90% and an overall standard deviation So of 0.42 while the design


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∆PSI was fixed at 1.7. The design summary is presented in Table 1. The rut prediction

model used for this illustrative example is summarized in equation 5 (5). The model is

shown as ;

( ) ( )( )

( ) ( ) ( )

−+++−

⋅−++−=

SG

ACSGvbasev

annualAC

E

EN

KVTSDHRD

ln034.0ln258.0271.0657.0703.2

ln01.0011.0ln033.0016.0

883.0,

097.0, εε

[5]

where: RD : Average rut-depth along a specified wheel path segment (inch) SD : Pavement surface deflection (in.), KV : Kinematic viscosity (centistroke), Tannual : Annual ambient temperature (oF), HAC : Thickness of asphalt concrete (in.), N : Cumulative traffic volume (ESAL), εv,base : Vertical compressive strain at the top of base layer (10-3), εv,SG : Vertical compressive strain at the top of subgrade (10-3), EAC : Resilient modulus of asphalt concrete (psi) corrected at the reference temperature of 20oC , and ESG : Resilient modulus of subgrade (psi).

The statistical information for this rut prediction model is summarized in Table 2. A rut-

depth of 12.7mm was considered as a limit state in this study. The strains and deflections

induced in the pavement were computed using MICHPAVE (6). It is very important to

estimate the variability of material properties involved in producing design outputs.

When sufficient data from in-situ and laboratory tests are not available, a possible

approach to characterize realistic statistical properties of design input parameters is to use

estimates based on published values, which are most conveniently expressed in terms of

the coefficient of variation (COV) :

x

x

mCOV

σ==

valueavaerage

deviationstandard [6].

General ranges of COV for common design input parameters are summarized in Table 3.


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It should be noted that the values shown in Table 3 provide only a rough guide for

estimating values of COV for any given case and the engineer’s judgment must be

primarily used in determining appropriate values of COV from published sources. Table 3

also shows the values used in this study.

The reliability indices computed for the 13 pavement sections using the FORM

(Rackwitz and Fiessler’s algorithm) are summarized in Table 4. The difference in the

reliability indices between 13 sections indicates that, in general, the AASHTO design

method does not yield cross sections with uniform reliability even though same reliability

level and serviceability index are assigned to all sections. It means that the existing

AASHTO method, which is based on empirical relationship between key elements of the

pavement performance, does not properly account for inherent variability of design

parameters in terms of mechanistic failure criterion such as pavement rutting. This

motivates the development of a design approach that tries to achieve uniform reliability

for all mechanistic-empirical pavement designs with failure modes such as fatigue

cracking, rutting or low-temperature cracking.

DEVELOPMENT OF PRACTICAL RELIABILITY-BASED PAVEMENT DESIGN

FORMAT

Basic Concept in Reliability-Based Design (RBD)

The reliability associated with an appropriate design equation should equal a

target value representing a certain degree of structural safety. The reliability-based design

of the pavement should guarantee that the probability of failure of a pavement lies below

an intended target level. Employing a design criterion of the pavement performance such

as pavement rut depth, this concept can simply be expressed as follows;


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Pr (SMrut < 0) ≤ pT [7]

where pT is the acceptable target probability of failure.

Reliability-based design in form of equation 7 involves the repeated use of

reliability assessment routines, such as the FORM to evaluate the probabilities of failure

of trial designs until computed probability of failure is extremely close to pT. This

approach is not suitable for designs that are carried out on a routine basis such as

pavement thickness design. In this study, a practical reliability-based M-E flexible

pavement design procedure was established involving the use of Load and Resistance

Factor Design (LRFD) format that is a worldwide prevalent form of the limit-state design

philosophy. The basic concept of the reliability-based approach applied to mechanistic-

empirical pavement designs can be expressed as follows:

),.......,( 21 nRoverallthreshold qqqfD γ≥ [8]

where:

Dthreshold = Threshold amount of pavement distress, γoverall = Overall safety factor reflecting a specified target reliability, fR = Procedure for predicting pavement performance in terms of pavement distress, and qi = Parameters in a pavement design procedure.

The γoverall to obtain a target reliability index, βtarget, can be determined as follows;

),......,(

),,,.........,(

121

112211

nnR

nnnnRoverall qqqqf

qqqqf

−

−−=γγφφγ [9]

where φι or γj is a partial safety factor of each variable for reduction or amplification of its

amount. For a specified βtarget, φι and γj can be computed through


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+==

i

iii

i X

NXettX

NX

X

ii n

m

n

X σβαφ arg

**

[10]

+==

j

jjj

j X

NXettX

NX

X

jj n

m

n

X σβαγ arg

**

[11]

where:

NX i

m = Equivalent normal mean values of design variable Xi, NX i

σ = Equivalent normal standard deviation of design variable Xi,

iXn = Nominal values of design variable Xi, and *

iXα = Direction cosine associated with the failure point of design variable Xi.

In order to have a constant of γoverall for all design situations, φι and γj must depend on the

particular variability of all basic variables in the design model. If a constant set of φι, γj,

and γoverall are prescribed, the associated reliability indices will deviate from a target

reliability index for certain design situations. However, it is possible to select a value of

γoverall that minimizes these deviations when considered over all likely combinations of

design features presented in the practice.

Illustrative Example of RBD

A RBD format employing a rut prediction model, which was introduced in the

previous section, as the major limit-state function can be written as follows:

[ ]),,,,( NEEEHfPRD SGBaseACACRoverallthreshold ⋅⋅≥ γ [12] where :

),,,,,(

),,,,,(

NEEEEHfP

NEEEEHfP

SGSBBaseACACR

NSGESBEBaseEACEACHRPoverall

SGSBBaseACAC

⋅

⋅=

γφφφφφγγ [13]

),,,,,( NEEEETf

RD

RD

RDP

SGSBBaseACACR

measured

predicted

measured == [14]


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The partial safety factors for all variables can be computed using equation 10 through 11.

The nominal values of design variables were assumed the mean values of them. In order

to determine rational constant values of overall safety factors corresponding to target

reliability indices, a factorial experiment matrix was established as summarized in Table

5. Each cell represents a specific design feature. The factorial matrix provides a simple

but effective way to relate design features to site conditions. Three major design variables

of traffic volume, AC thickness, and resilient modulus of subgrade that were examined

factors mostly affecting reliability for predicting pavement rut depth were selected and

included in the matrix (8). High, moderate, and low values for each variable were

determined based on the findings reported in NCHRP 1-32 project and Michigan DOT

pavement design practice (15). In Table 5, eighteen cells shaded in the matrix were

selected regarding most likely combinations of design features presented in the practice.

The reliability indices, failure points of design variables, and direction cosines associated

with the design points in 18 shaded cells were determined by the FORM with 12.7 mm of

rut depth as the limit state. Then, partial safety factors and overall safety factors

corresponding to specified target reliability indices were calculated. In the next step, the

constant set of partial and overall safety factors for various target reliabilities of 75 to

99% were determined and summarized in Table 6.

A flowchart showing a M-E flexible pavement design procedure based on

equation 12 is illustrated in Figure 4. In this design procedure, the cross-section of a

pavement is optimally determined by following steps :

1. Input design parameters including expected traffic volume during pavement service life, a desired target reliability level, a threshold rut-depth as failure criterion (RDthreshold), ambient annual temperature around the site, and effective resilient modulus of the subgrade soil of the site.


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2. Set up trial pavement cross-section and material properties. 3. Determine a certain degree of uncertainty accompanied with the design procedure in

terms of an overall safety factor ( overallγ ) of the design model: From Table 6, select a value in accordance with the desired reliability level set up in step 1.

4. Compute the surface deflection and compressive vertical strains at the top of the base layer and subgrade using the pavement analysis computer program.

5. Compute the predictive pavement rut-depth (RDpredicted). 6. Modify the pavement cross-section if the difference between RDthreshold and

overallγ *P*RDpredicted is higher than a specified tolerance level and repeat step 3

through 5 until the difference will be less than the tolerance level.

In order to compare design results using the AASHTO and RBD procedures

suggested in this study, the thirteen pavement sections shown in Table 1 were re-designed

by the RBD procedure where the AC thickness was changed so that the revised section

would accommodate the design traffic and satisfy the threshold rut-depth at a given target

reliability. The target reliability of 90% (βtarget = 1.28) was assigned to both design

procedures. Figure 5 shows the difference of design results obtained by the AASHTO and

RBD methods in terms of AC thickness. The reliability indices for the pavement sections

determined by both methods were computed using the FORM in which equation 3 is

employed as the limit state function. Figure 6 presents the comparison of reliability

indices indicating that the RBD procedure does successfully yield cross-sections whose

reliability indices are close to the target reliability index while the AASHTO method does

not generally produce designs of uniform reliability for actual mechanistic failure

criterion.

CONCLUSIONS AND RECOMMENDATIONS

A reliability analysis model for quantifying uncertainties in common M-E

flexible pavement design procedures and identifying the design parameters that the most

significantly affect the variability of pavement performance was presented. A M-E

flexible pavement design procedure using the RBD format was illustrated in an effort to


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design a pavement cross-section satisfying performance threshold with a target reliability

during its intended design life.

Based on the findings presented here, the following conclusions and

recommendations are made;

1. The performance reliability in a given pavement section can be reasonably expressed

as the reliability index, βHL, the invariant minimum distance between the origin and

the failure surface.

2. Uncertainties affecting pavement performance consist of two portions: uncertainties of

design parameters and systematic errors. Uncertainties of design parameters reflect the

spatial variation and random measurement error of the pavement material properties

and systematic errors are associated with the model bias in predicting pavement

performance.

3. The effects of systematic errors can be quantified by employing the professional factor

that is defined as a representative ratio of the measured to predicted pavement

performance.

4. Based on the proposed reliability analysis procedure, partial and overall safety factors

accounting for inherent variability of pavement design parameters can be developed

and classified to different functional road classes such as interstate, principal arterials,

and residential streets for each performance measure.

5. The AASHTO design method, which is being used in practice, does not properly

account for inherent variability of design parameters in terms of mechanistic failure

criterion.

6. The suggested RBD format employing the actual mechanistic failure criterion is


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capable of yielding the design results with the uniform reliability.

7. The suggested RBD format, appropriately calibrated using considerably more data

than used in this illustrative example, could be implemented in future pavement design

guides.

8. It is recommended that the target reliabilities that can be varied with traffic levels and

pavement serviceabilities should be calibrated to inherent past practices associated

with pavement design.

Acknowledgement

The Authors would like to thank the Advanced Highway Research Center in Hanyang

University and the Pavement Research Center of Excellence in Michigan State University

for providing financial support for this study.


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REFERENCES

1. 1993 AASHTO Guide for Design of Pavement Structures, American Association of

State Highway and Transportation Officials, Washington, D.C., 1993.

2. Ang, A. and Tang, W. Probability Concept in Engineering Planning and Design

Volume- I and II, John Wiley and Sons, Inc., 1975 and 1984.

3. Christian, J.T., Ladd, C.C., and Baecher, G.B. Reliability Applied to Slope Stability

Analysis, Journal of Geotechnical Engineering (ASCE), Vol. 120, Dec. 1994,

pp.2180-2207.

4. Hallin, J.P, Darter, M.I., and Witzack, M.W., Development of the 2002 Guide for the

Design of New and Rehabilitated Pavement Structures, presented in the Workshop for

2002 Guide for Mechanistic Pavement Design: Issues in Development and

Implementation, 80th Annual Meeting of TRB, Washington, D.C., 2001.

5. Harichandran, R. S., Buch, N., and Baladi, G. Y. Flexible Pavement Design in

Michigan : Transition from Empirical to Mechanistic Methods, Presented in 80th

Annual Meeting of Transportation Research Board, Washington, D.C., 2001.

6. Harichandran, R.S., Yeh, M.S., and Baladi, G.Y., MICHPAVE: a Nonlinear Finite

Element Program for Analysis of Flexible Pavements, Transportation Research

Record 1286, TRB, National Research Council, Washington, D.C., 1990, pp. 123-132.

7. Hasofer, A.M., and Lind, N.C., Exact and Invariant Second-Moment Code Format,

Journal of the Engineering Mechanics, (ASCE), Vol. 100, 1974, pp 111-121.

8. Kim, H.B., Framework for Incorporating Rutting Prediction Model in the Reliability-

Based Design of Flexible Pavements, Ph.D. Dissertation, Michigan State University,

1999.


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9. Noureldin, S. A., Sharaf, E., Arafah, A., and Al-Sugair, F., Estimation of Standard

Deviation of Predicted Performance of Flexible Pavements using AASHTO Model,

Transportation Research Record 1449, Transportation Research Board, National

Research Council, Washington, D.C., 1994, pp. 46-56.

10. Proposed AASHTO Guide for Design of Pavement Structures, NCHRP Project 20-

7/24, Vol. 2, American Association of State Highway and Transportation Officials,

Washington, D.C., 1985.

11. Rackwitz, R., and Fiessler, B., Structural Reliability under Combined Random Load

Sequences, Computers & Structures, Vol. 9, 1978, pp. 489-494.

12. Ravindra, M.K., and Galambos, T., Load and Resistance Factor Design for Steel,

Journal of Structural Division (ASCE), Vol. 104, Sept. 1978, pp.1337-1353.

13. Timm, D., Birgisson, B., and Newcomb, D., Variability of Mechanistic-Empirical

Flexible Pavement Design Parameters, Proceedings of the Fifth International

Conference on the Bearing Capacity of Roads and Airfields, Vol. 1, Norway, 1998,

pp.629-638.

14. Timm, D.H., Newcomb, D.E., Birgisson, B., and Galambos, T.V., Incorporation of

Reliability into the Minnesota Mechanistic-Empirical Pavement Design Method, Final

Report Prepared to Minnesota Department of Transportation, Minnesota Univ.,

Department of Civil Engineering, Minneapolis, July, 1999.

15. Von Quintus, H., Killingsworth, B.M., Darter, M.I., Owusu-Antwi, E., and Jiang, J.,

Catalog of Recommended Pavement Design Features, Final Report, NCHRP 1-32,

Transportation Research Board, National Research Council, Washington, D.C., 1997.


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List of Tables and Figures Table 1 Summary of Pavement Cross-Section and Statistical Information

Table 2 Statistical Information of the Rut Prediction Model

Table 3 Summary of Variability of Design Input Parameters

Table 4 Summary of the Computations of Reliability Indices

Table 5 Factorial Experiment Matrix with Major Design Variables

Table 6 Summary of Partial and Overall Safety Factors with Various Target Reliabilities

Figure 1 Graphical Illustration of Reliability Index

Figure 2 Integrated Presentation of Types of Uncertainties Associated with M-E Flexible

Pavement Design

Figure 3 Flow Diagram for Pavement Reliability Analysis

Figure 4 Flowchart for M-E Flexible Pavement Design Procedure using RBD Format

Figure 5 AC Thickness Determined by RBD and AASHTO (1993) Methods

Figure 6 Comparison of Reliabilities for Pavement Sections Designed by RBD and

AASHTO (1993) Methods


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Table 1 Summary of Pavement Cross-Section and Statistical Information

Section No.

1 2 3 4 5 6 7 8 9 10 11 12 13

AC Thickness

(mm) 241 140 102 152 127 178 229 203 73 69 102 152 127

AC Modulus (MPa)

2758 2758 2758 2069 3103 2758 2069 3103 2758 3103 3103 3103 3103

Subgrade Modulus (MPa)

21 52 76 52 52 34 34 34 55 55 55 34 55

Traffic (KESAL)

18,522 18,522 18,522 18,522 18,522 18,522 18,522 18,522 5,000 5,000 10,000 10,000 13,000

* The thickness of the base and subbase are fixed at 203 and 406mm. **Moduli of the base and subbase are fixed at 276 and 103 MPa


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Table 2 Statistical Information of the Rut Prediction Model DEPENDENT VARIABLE IS RD

SOURCE SUM-OF-SQUARES

DF MEAN-SQUARE

F P-value

REGRESSION 2.336 9 0.260 43.333 2.044E-18

RESIDUAL 0.188 42 0.004

TOTAL 2.583 51

CORRECTED 0.477 50

RAW R-SQUARED (1-RESIDUAL/TOTAL) = 0.905

CONFIDENCE INTERVAL (95%) PARAMETER ESTIMATE A.S.E.

LOWER BOUND UPPER BOUND

a1 -0.016 0.036 -0.089 0.058

a2 0.033 0.094 -0.157 0.223

a3 0.011 0.023 -0.037 0.058

a4* -0.010

a5 -2.703 0.181 -3.708 -1.698

a6 0.657 3.300 -6.002 7.317

a7 0.097 0.670 -1.255 1.448

a8 0.271 0.912 -1.569 2.111

a9 0.883 1.601 -2.348 4.114

a10 0.258 0.587 -0.926 1.443

a11 0.034 0.114 -0.196 0.264

*A4 is assumed to be a constant value before running statistical analysis due to the difficulty of convergence in the regression model


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Table 3 Summary of Variability of Design Input Parameters

Previous Investigation This Study

Property Range of COV

(%) Type of

Distribution Reference COV

Type of Distribution

3 – 12 Normal (13) AC Thickness

(TAC) 3 – 12 Normal (9)

10 Normal

10 – 40 Lognormal (13) AC Modulus

(EAC) 10 –20 Normal (9)

25 Lognormal

10 – 30 Normal (9) Base Modulus

(EBase) 5 – 60 Lognormal (14)

15 Lognormal

10 – 30 Normal (9) Subbase Modulus (ESubbase) 5 – 60 Lognormal (14)

15 Lognormal

10 – 30 Normal (13) Subgrade Modulus

(ESG) 20 – 45 Lognormal (9) 35 Lognormal

42 Lognormal (10) 42 Traffic

(N) -

Extreme Value Type I

(14) -

Extreme Value Type I

Professional Factor

(P) 20 Normal (8) 20 Normal


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Table 4 Summary of the Computations of Reliability Indices

Section No. Reliability Index

1 2.151

2 0.574

3 0.007

4 0.718

5 0.379

6 1.149

7 2.037

8 1.694

9 0.295

10 0.323

11 0.318

12 1.102

13 0.615


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Table 5 Factorial Experiment Matrix with Major Design Variables

Traffic Volume (ESAL) 1,000,000 5,000,000 15,000,000

Subgrade Modulus (MPa) 28 55 97 28 55 97 28 55 97

AC Thickness (mm) 76

127

178

254


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Table 6 Summary of Partial and Overall Safety Factors with Various Target Reliabilities

Target Reliability Level (%) 75 80 90 95 99

βtarget (Normal Variate)

0.68 0.84 1.28 1.65 2.33

Overallγ 1.155 1.191 1.295 1.385 1.557

Pγ 1.120 1.148 1.225 1.293 1.410

Nγ 1.070 1.087 1.133 1.170 1.241

ACEφ 0.986 0.982 0.973 0.966 0.951

BaseEφ 0.996 0.996 0.993 0.991 0.988

SubbaseEφ 0.997 0.997 0.995 0.994 0.991

SGEφ 0.969 0.962 0.942 0.927 0.894

ACTφ 0.976 0.971 0.956 0.946 0.919


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Figure 1 Graphical Illustration of Reliability Index

Safe Region

Failure Region

t Hyperplane

Failure Surface g(z1,z2)=0

Design Point

βΗL

z2

z1


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Figure 2 Integrated Presentation of Types of Uncertainties Associated with M-E Flexible Pavement Design

Prediction Range Associated with Parameter Uncertainties

Prediction Range Associated with Parameter Uncertainties

Prediction Range Associated with Systematic Uncertainties

( )[ ]xmgE( )[ ]xmgE − ( )[ ]xmgE +

Possible Prediction Range


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Figure 3 Flow Diagram for Pavement Reliability Analysis

Material andCross-sectionalProperties

Traffic andEnvironmentalFactor

PavementStructureModel

StructuralResponses

PavementPerformance

Model

PredictedPavementDistress

Failure Criteria(= Threshold Amountof Distress)

Limit StateFunction, g(x)

ReliabilityIndex (βHL) atg(xi)

Revised InputPoints (xi+1)

βHL atg(xi+n) = 0

Initial Input Points (x1)

Mechanistic Subsystem Reliability Subsystem


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Figure 4 An Iterative Loop for M-E Flexible Pavement Design Procedure using RBD Format

• Characterization of Surface, Base, and Subbase MaterialsProperties

• Characterization of Subgrade Soils

� Cross-Sectional Properties

Structural Analysis of Pavement Section

Primary Response of Pavement (stress, strain, and deflection)

Calculation of Rut Depth (RDpredicted) with Predictive Model

• Traffic Information • Environmental Condition

Final Design

Yes Change Cross-Sectional Properties

No

RDthreshold –γoverall*P*RDpredicted ≤ Tolerance Level

• Threshold Rut Depth (RDthreshold)

• Professional Factor (P) • Overall Safety Factor

(γoverall) based on βtarget


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Figure 5 AC Thickness Determined by RBD and AASHTO (1993) Methods


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Figure 6 Comparison of Reliabilities for Pavement Sections Designed by RBD and AASHTO (1993) Methods


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Reliability-Based Pavement Design Model Accounting for ... · Reliability analysis techniques in a pavement design procedures must provide a rational framework for quickly addressing