Research Papers: Design Automation

Efficient Reliability Assessment With the Conditional Probability Method

[+] Author and Article Information
Rami Mansour

Department of Solid Mechanics,
Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: ramimans@kth.se

Mårten Olsson

Department of Solid Mechanics,
Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: mart@kth.se

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 8, 2017; final manuscript received April 18, 2018; published online June 1, 2018. Assoc. Editor: Samy Missoum.

J. Mech. Des 140(8), 081402 (Jun 01, 2018) (12 pages) Paper No: MD-17-1747; doi: 10.1115/1.4040170 History: Received November 08, 2017; Revised April 18, 2018

Reliability assessment is an important procedure in engineering design in which the probability of failure or equivalently the probability of survival is computed based on appropriate design criteria and model behavior. In this paper, a new approximate and efficient reliability assessment method is proposed: the conditional probability method (CPM). Focus is set on computational efficiency and the proposed method is applied to classical load-strength structural reliability problems. The core of the approach is in the computation of the probability of failure starting from the conditional probability of failure given the load. The number of function evaluations to compute the probability of failure is a priori known to be 3n + 2 in CPM, where n is the number of stochastic design variables excluding the strength. The necessary number of function evaluations for the reliability assessment, which may correspond to expensive computations, is therefore substantially lower in CPM than in the existing structural reliability methods such as the widely used first-order reliability method (FORM).

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Fig. 1

Presentation of the main steps in the proposed CPM and the FORM

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Fig. 2

An example of a (a) highly nonlinear and (c) linear conditional probability of failure Pf|u(u) with constant gradient direction; and corresponding integrand Pf|u(u)fu(u) for the (b) nonlinear and (d) linear case

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Fig. 3

Illustration of the CPM: (a) linearize the load, Lu, at u = 0, (b) find umax,lin according to Eq. (29), (c) interpolate a polyharmonic surrogate model Lupolyh(u) of the form given in Eq. (31), and (d) compute probability of failure using Eq. (35)

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Fig. 4

Illustration of the steps used to compute the total probability of failure using CPM for the problem defined in Eq. (36)

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Fig. 5

True load and conditional probability of failure for the problem defined in Eq. (36)

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Fig. 6

(a) CDF computed using MV and AMV and (b) performance function evaluated at the nominal value of the strength

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Fig. 8

(a) Polyharmonic spline approximation of the load, (b) conditional probability of failure computed using the polyharmonic approximation of the load, (c) true load, and (d) true conditional probability for the problem defined in Eq. (44)

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Fig. 9

A shaft subject to torsion and flexure: (a) structure and (b) cross section

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Fig. 10

(a) Polyharmonic spline approximation of the load, (b) conditional probability of failure computed using the polyharmonic approximation of the load, (c) true load, and (d) true conditional probability for the problem defined in Eq. (45)

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Fig. 12

(a) Generalized reliability index computed using different mean values of the strength for bar 5 and bar 7, (b) relative error in probability of failure, and (c) number of function evaluations

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Fig. 13

A box plot of the absolute relative error of the generalized reliability index for the problems studied in this paper



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