APPLYING THE METHOD APPLYING THE METHOD OF MOMENTS TO OF MOMENTS - - PowerPoint PPT Presentation

Ivan Damnjanovic & Zhanmin Zhang The University of Texas at Austin

APPLYING THE METHOD APPLYING THE METHOD OF MOMENTS TO OF MOMENTS TO DEVELOP RELIABILITY DEVELOP RELIABILITY FUNCTION FOR RIGID FUNCTION FOR RIGID PAVEMENTS PAVEMENTS

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

Background Background

Performance of a pavement can never be predicted with an absolute certainty. Failing to recognize such a fact can often lead to improper design and management decisions.

Sources of Uncertainties Sources of Uncertainties

Aleatory uncertainty, or irreducible

uncertainty - due to an inherent irregularity in the properties and behavior.

Epistemic uncertainty, or the uncertainty -

due to the lack of knowledge about the system’s behavior.

Uncertainty in the occurrence of both

acknowledged and unacknowledged errors.

Modeling Performance Uncertainty

Time-series models Markov process Generalized-Markov process Discrete choice models Reliability models Lifetime testing models Structural reliability models

Motivation Motivation

The motivation of this paper is to investigate

the applicability of the analytical method, the Method of Moments, to estimate the failure probability and to model pavement reliability function.

The advantage of such an approach is the

ability to use the reliability function in pavement economic analyses and

• ptimization models.

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

Stress Stress-

• Strength Interference Method

Strength Interference Method

STRESS FAILURE REGION PROBABILITY DENSITY STRENGTH

Structural Reliability Models (1 of 2) Structural Reliability Models (1 of 2)

The fundamental considerations in the structural reliability theory are:

Mathematical formulation of the limit

state function.

Characterization of the basic random

variables.

Evaluation of the multidimensional

probability integral.

Structural Reliability Models (2 of 2) Structural Reliability Models (2 of 2)

A structural reliability model is formulated in terms of n basic random variables and a limit state function The structural failure expressed as an event and the probability of failure can be expressed as an n-dimensional probability integral:

1

[ ,..., ]T

n

x x = X

( ). G X

{ }

( ) G ≤ X

( ) 0

Pr[ ( ) 0] ( )

G X

pF G f d

≤

= ≤ = ∫ X X X

Evaluation of Probability Integrals Evaluation of Probability Integrals

Monte Carlo Simulation (MCS) First-order Reliability Method (FORM) Second-order Reliability Method (SORM) First-order Third Moment Method (FOTM) Method of Moments (MM)

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

Method of Moments Method of Moments

The method of moments is based on two sequential steps.

Estimate moments of the limit state function

using the point estimates obtained in the standard normal space.

Calculate the reliability index and the

failure probability using the existing standardized functions. β

Reliability Indices Reliability Indices

2 G M G

µ β σ =

3 2 3

( ) ln 1 ln( )

G M M b

sign A u A α β β ⎡ ⎤ ⎛ ⎞ − = + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦

2 4 2 3 2 4 2 4 3 4

3( 1) ( 1) (9 5 9)( 1)

G M G M M G G G

α β α β β α α α − + − = − − −

1 1

1 X ' 1 X

X -µ X = σ

FAILURE STATE SAFE STATE

2 2

2 X ' 2 X

X -µ X = σ

1 2

G(X ,X ) = 0

β

Failure Probability Failure Probability

With the estimated reliability indices, the failure probability can be calculated as: The cumulative failure and the reliability functions can be expressed in terms of the reliability index, as follows:

( )

M M

pF β

• = Φ −

( )

( ) ( )

M

F t t β• = Φ −

( )

( ) 1 ( ) 1 ( )

M

R t F t t β• = − = −Φ −

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

AASHTO Design Procedure AASHTO Design Procedure

One of the most widely used methods for

designing pavement structures is the AASHTO design method.

The design procedure is based on the

results from the accelerated pavement testing experiment, known as the AASHO Road Test.

Modeling Limit State Function (1 of 2) Modeling Limit State Function (1 of 2)

The strength of a pavement : The time-dependant stress:

(1 ) 1 ( ) ( )

t

r N t ESAL TGF t ESAL r + − = × = ×

18 7 8.46 0.75 0.75 0.25

log[ /(4.5 1.5)] log 7.35log( 1) 0.06 1 1.624 10 ( 1) ( 1.132) (4.22 0.32 ) 215.63 [ 18.42 ( ) ]

c d t c

PSI W D D S C D p J D E k ∆ − = + − + + + × + ⎧ ⎫ − + − ⎨ ⎬ − ⎩ ⎭

Modeling Limit State Function (2 of 2) Modeling Limit State Function (2 of 2)

Then, the limit state function: and the failure domain defined as: establish the time-dependent probability integral.

{ }

( , , , , , , )

C C

G D S k E ESAL r t ≤

18

( , , , , , , ) log log ( )

C C

G D S k E ESAL r t W N t = −

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

Comparison of the Reliability Estimates Comparison of the Reliability Estimates

In order to test the applicability of Method of

Moments, a numerical study was conducted.

The failure probabilities were estimated

using Method of Moments and Monte Carlo Simulation (1 million samples).

The Results of the Comparison The Results of the Comparison

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 4 4.5 5 5.5 6 6.5 7 Time Failure Probability MCS pF(2M) pF(3M) pF(4M)

Method Max Abs Error Mean Error 2M 0.00272331 0.015986295 3M 0.002307601 0.009565282 4M 0.000989728 0.006322845

Outline Outline

Introduction Structural Reliability Models Method of Moments Development of Reliability Function

Using AASHTO Design Procedure

Case Study Conclusions

Conclusions Conclusions

The most accurate predictions of failure

probability are obtained using 4M reliability index.

Method of Moments is a robust approach

for estimating reliability functions since it can accommodate different types of failure mechanisms.

The key advantage of Method of Moments

• ver Monte Carlo simulation is its ability to

express the reliability in a general functional form.

Thanks & Questions? Thanks & Questions?