APPLYING THE METHOD APPLYING THE METHOD OF MOMENTS TO OF MOMENTS TO DEVELOP RELIABILITY DEVELOP RELIABILITY FUNCTION FOR RIGID FUNCTION FOR RIGID PAVEMENTS PAVEMENTS Ivan Damnjanovic & Zhanmin Zhang The University of Texas at Austin
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 optimization models.
Outline Outline � Introduction � Structural Reliability Models � Method of Moments � Development of Reliability Function Using AASHTO Design Procedure � Case Study � Conclusions
Stress- -Strength Interference Method Strength Interference Method Stress STRESS PROBABILITY DENSITY STRENGTH FAILURE REGION
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 ] T = G X ( ). X [ ,..., x x 1 n The structural failure expressed as an event and the probability of failure can { } ≤ G ( ) 0 X be expressed as an n -dimensional probability integral: = ∫ = ≤ Pr[ ( ) 0] ( ) pF G X f X d X ≤ G X ( ) 0
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 β failure probability using the existing standardized functions.
Reliability Indices Reliability Indices µ G β = 2 M σ G ⎡ ⎤ ⎛ ⎞ − α β sign ( ) ln X -µ 2 X ⎢ ⎥ ' β = 3 G ⎜ + 2 M ⎟ X = A 1 2 2 σ 3 M ⎝ ⎠ ⎣ u ⎦ ln( ) A X 2 b 2 α − β + α β − 3( 1) ( 1) β = 4 G 2 M 3 G 2 M 4 M 2 α − α − α − (9 5 9)( 1) FAILURE STATE 4 G 3 G 4 G SAFE STATE β X -µ 1 X ' X = 1 1 σ X 1 G(X ,X ) = 0 1 2
Failure Probability Failure Probability With the estimated reliability indices, the failure probability can be calculated as: = Φ − β pF ( ) • • M M The cumulative failure and the reliability functions can be expressed in terms of the reliability index, as follows: ( ) = Φ − β • F t ( ) ( ) t M ( ) = − = −Φ − β • R t ( ) 1 F t ( ) 1 ( ) t M
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 : ∆ − log[ PSI /(4.5 1.5)] = + − + + log W 7.35log( D 1) 0.06 18 7 8.46 + × + 1 1.624 10 ( 1) D ⎧ ⎫ 0.75 − ( 1.132) S C D ⎨ ⎬ + − c d (4.22 0.32 ) 215.63 [ p t 0.75 0.25 ⎩ − ⎭ J D 18.42 ( E k ) ] c � The time-dependant stress: t + − (1 ) 1 r = × = × N t ( ) ESAL TGF t ( ) ESAL 0 0 r
Modeling Limit State Function (2 of 2) Modeling Limit State Function (2 of 2) Then, the limit state function: = − G D S ( , , , k E , ESAL r t , , ) log W log N t ( ) C C 0 18 and the failure domain defined as: { } ESAL r t ≤ G D S ( , , , k E , , , ) 0 C C 0 establish the time-dependent probability integral.
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 0.001 MCS 0.0009 pF(2M) pF(3M) 0.0008 Method Max Abs Error Mean Error pF(4M) 2M 0.00272331 0.015986295 0.0007 3M 0.002307601 0.009565282 Failure Probability 0.0006 4M 0.000989728 0.006322845 0.0005 0.0004 0.0003 0.0002 0.0001 0 4 4.5 5 5.5 6 6.5 7 Time
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 over Monte Carlo simulation is its ability to express the reliability in a general functional form.
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