Reduction of Random Variables in Structural Reliability Analysis S. - PowerPoint PPT Presentation

Reduction of Random Variables in Structural Reliability Analysis S. A DHIKARI AND R. S. L ANGLEY Cambridge University Engineering Department Cambridge, U.K. Random Variable Reduction in Reliability Analysis – p.1/18

Outline of the Talk • Introduction • Approximate Reliability Analyses: FORM and SORM • Proposed Reduction Techniques • Numerical examples • Conclusions Random Variable Reduction in Reliability Analysis – p.2/18

Structural Reliability Analysis Finite Element models of some engineering structures Random Variable Reduction in Reliability Analysis – p.3/18

The Fundamental Problem Probability of failure: � P f = p ( y ) d y (1) G ( y ) ≤ 0 • y ∈ R n : vector describing the uncertainties in the structural parameters and applied loadings. • p ( y ) : joint probability density function of y • G ( y ) : failure surface/limit-state function/safety margin/ Random Variable Reduction in Reliability Analysis – p.4/18

Main Difficulties • n is large • p ( y ) is non-Gaussian • G ( y ) is a complicated nonliner function of y Random Variable Reduction in Reliability Analysis – p.5/18

Approximate Reliability Analy- ses First-Order Reliability Method (FORM): • Requires the random variables y to be Gaussian. • Approximates the failure surface by a hyperplane. Second-Order Reliability Method (SORM): • Requires the random variables y to be Gaussian. • Approximates the failure surface by a quadratic hypersurface. Asymptotic Reliability Analysis (ARA): • The random variables y can be non-Gaussian. • Accurate only in an asymptotic sense. Random Variable Reduction in Reliability Analysis – p.6/18

FORM • Original non-Gaussian random variables y are transformed to standardized gaussian random variables x . This transforms G ( y ) to g ( x ) . • The probability of failure is given by β = ( x ∗ T x ∗ ) 1 / 2 P f = Φ( − β ) with (2) where x ∗ , the design point is the solution of following optimization problem: � ( x T x ) 1 / 2 � g ( x ) = 0 . min subject to (3) Random Variable Reduction in Reliability Analysis – p.7/18

Gradient Projection Method • Uses the gradient of g ( x ) noting that ∇ g is independent of x for linear g ( x ) . • For nonlinear g ( x ) , the design point is obtained by an iterative method. • Reduces the number of variables to 1 in the constrained optimization problem. • Is expected to work well when the failure surface is ‘fairly’ linear. Random Variable Reduction in Reliability Analysis – p.8/18

Example 1 Linear failure surface in R 2 : g ( x ) = x 1 − 2 x 2 + 10 6 5 x * Failure domain: 4 g(x) = x 1 −2x 2 +10 < 0 β 3 x 2 2 Safe domain g(x) = x 1 −2x 2 +10 > 0 1 0 −1 −10 −8 −6 −4 −2 0 x 1 x ∗ = {− 2 , 4 } T and β = 4 . 472 . Random Variable Reduction in Reliability Analysis – p.9/18

Main Steps For k = 0 , select x ( k ) = 0 , a small value of ǫ , (say 0 . 001 ) and a large value of β ( k ) 1. (say 10 ). � ∂g ( x ) � Construct the normalized vector ∇ g ( k ) = ∂x i | x = x ( k ) , ∀ i = 1 , .., n so that 2. | ∇ g ( k ) | = 1 . Solve g ( v ∇ g ( k ) ) = 0 for v . 3. Increase the index: k = k + 1 ; denote β ( k ) = − v and x ( k ) = v ∇ g ( k ) . 4. Denote δβ = β ( k − 1) − β ( k ) . 5. (a) If δβ < 0 then the iteration is going in the wrong direction. Terminate the iteration 6. procedure and select β = β ( k ) and x ∗ = x ( k ) as the best values of these quantities. (b) If δβ < ǫ then the iterative procedure has converged. Terminate the iteration procedure and select β = β ( k ) and x ∗ = x ( k ) as the final values of these quantities. (c) If δβ > ǫ then go back to step 2. Random Variable Reduction in Reliability Analysis – p.10/18

Example 2 25 ( x 1 − 1) 2 − x 2 + 4 g ( x ) = − 4 5 Failure domain 4 g(x) < 0 1 2 Safe domain 3 g(x) > 0 3 4 5 x 2 2 1 0 −1 −5 −4 −3 −2 −1 0 1 2 3 4 x 1 x ∗ = {− 2 . 34 , 2 . 21 } T and β = 3 . 22 . Random Variable Reduction in Reliability Analysis – p.11/18

Example 3 25( x 1 + 1) 2 − ( x 2 − 5 / 2) 2 ( x 1 − 5) g ( x ) = − 4 − x 3 + 3 10 x ∗ = { 2 . 1286 , 1 . 2895 , 1 . 8547 } T and β = 3 . 104 . Random Variable Reduction in Reliability Analysis – p.12/18

Dominant Gradient Method • More than one random variable is kept in the constrained optimization problem. • Dominant random variables are those for which the failure surface is most sensitive. • Variables for which the failure surface is less sensitive is removed in the constrained optimization problem. • Is expected to work well for near-linear failure surface. Random Variable Reduction in Reliability Analysis – p.13/18

Relative Importance Variable Method • Based on the entries of ∇ g the random variables are grouped into ‘important’ and ‘unimportant’ random variables. • Unimportant random variables are not completely neglected but represented by a single random variable. • Is expected to work well for near-linear failure surface. Random Variable Reduction in Reliability Analysis – p.14/18

Multistoried Portal Frame Random Variables: 11� 12� 18� P� 2� Axial stiffness (EA) and the bending stiffness 17� 19� 20� P� (EI) of each member are uncorrelated Gaussian 1� 10� 14� 9� random variables (Total 2 × 20 = 40 random 15� 13� 16� variables: x ∈ R 40 ). 8� 7� 10� EI (KNm 2 ) EA (KN) 9� 11� 12� Element Standard Standard Mean Mean 6� 5� 6� Type Deviation Deviation 5� 7� 5.0 × 10 9 6.0 × 10 4 5 @� 2.0m� 8� 1 7.0% 5.0% 3.0 × 10 9 4.0 × 10 4 2 3.0% 10.0% 4� 3� 2� 1.0 × 10 9 2.0 × 10 4 3 10.0% 9.0% 1� 3� 4� Failure surface: 1� 2� g ( x ) = d max − | δh 11 ( x ) | 3.0� m� δh 11 : the horizontal displacement at node 11 Nel=20 , Nnode=12 d max = 0 . 184 × 10 − 2 m P 1 = 4 . 0 × 10 5 KN, P 2 = 5 . 0 × 10 5 KN Random Variable Reduction in Reliability Analysis – p.15/18

Multistoried Portal Frame Results (with one iteration) Method 2 Method 3 FORM MCS ‡ Method 1 ( n reduced = 1 ) n d = 5 n d = 5 n = 40 (exact) β − 3.399 3.397 3.397 3.397 P f × 10 3 0.338 0.340 0.340 0.340 0.345 ‡ with 11600 samples (considered as benchmark) Random Variable Reduction in Reliability Analysis – p.16/18

Conclusions & Future Research • Three iterative methods, namely (a) gradient projection method, (b) dominant gradient method, and (c) relative importance variable method, have been proposed to reduce the number of random variables in structural reliability problems involving a large number of random variables. • All the three methods are based on the sensitivity vector of the failure surface. • Initial numerical results show that there is a possibility to put these methods into real-life problems involving a large number of random variables. Random Variable Reduction in Reliability Analysis – p.17/18

Conclusions & Future Research • Future research will address reliability analysis of more complicated and large systems using the proposed methods. This would be achieved by using currently existing commercial Finite Element softwares. • Applicability and/or efficiency of the proposed methods to problems with highly non-linear failure surfaces, for example, those arising in structural dynamic problems will be investigated. Random Variable Reduction in Reliability Analysis – p.18/18