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Research Papers: Design Automation

# Time-Dependent Reliability Analysis Using the Total Probability Theorem

[+] Author and Article Information
Zissimos P. Mourelatos

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309
e-mail: mourelat@oakland.edu

Monica Majcher

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309
e-mail: mtmajch2@oakland.edu

Vijitashwa Pandey

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309
e-mail: pandey2@oakland.edu

Igor Baseski

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309
e-mail: ibaseski@oakland.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 11, 2014; final manuscript received November 28, 2014; published online January 9, 2015. Assoc. Editor: Xiaoping Du.

J. Mech. Des 137(3), 031405 (Mar 01, 2015) (8 pages) Paper No: MD-14-1417; doi: 10.1115/1.4029326 History: Received July 11, 2014; Revised November 28, 2014; Online January 09, 2015

## Abstract

A new reliability analysis method is proposed for time-dependent problems with explicit in time limit-state functions of input random variables and input random processes using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using time-dependent conditional probabilities which are computed accurately and efficiently in the standard normal space using the first-order reliability method (FORM) and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation (MCS) or adaptive importance sampling are used based on a Kriging metamodel of the conditional probabilities. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

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## Figures

Fig. 1

Schematic of a composite limit state

Fig. 3

Cross section of turbine blade at the root [11]

Fig. 2

Illustration of probability of failure calculation

Fig. 5

Realizations of velocity v(t)

Fig. 6

Reliability index with random variables at mean values

Fig. 4

Fig. 8

Composite limit state for t = 8.2 months and Δt = 0.05 months

Fig. 9

Sensitivity of conditional probability of failure P(F/W) at t = 8.2 with h1, h2, ɛa, and l1

Fig. 10

Conditional probability of failure at all 50 training points

Fig. 11

Comparison of estimated Pf(0,T) with MCS as a function of Δt

Fig. 7

Composite limit state for t = 8.2 months and Δt = 0.2 months with random variables at mean values

## Errata

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