Reliability and Uncertainty in Structural Dynamics S. A DHIKARI In - PowerPoint PPT Presentation

Reliability and Uncertainty in Structural Dynamics S. A DHIKARI In collaboration with Prof. R S. Langley Cambridge University Engineering Department Cambridge, U.K. Reliability and Uncertainty in Structural Dynamics – p.1/23

Outline of the Talk Introduction: Research Interests Structural Reliability Analysis Reliability Analysis for Dynamics Conclusions Reliability and Uncertainty in Structural Dynamics – p.2/23

Research Areas 1. Identification of damping in vibrating structures 2. Deterministic and stochastic structural dynamics 3. Sensitivity analysis of damped structures 4. Statistical Energy Analysis (SEA) 5. Structural reliability analysis Reliability and Uncertainty in Structural Dynamics – p.3/23

Structural Reliability Analysis Reliability and Uncertainty in Structural Dynamics – p.4/23

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/ Reliability and Uncertainty in Structural Dynamics – p.5/23

Main Difficulties n is large p ( y ) is non-Gaussian P f is usually very small (in the order of 10 − 4 or smaller) G ( y ) is a complicated nonlinear function of y Reliability and Uncertainty in Structural Dynamics – p.6/23

Approximate Reliability Analyses 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. Reliability and Uncertainty in Structural Dynamics – p.7/23

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 � ( x T x ) 1 / 2 � min subject to g ( x ) = 0 . (3) Reliability and Uncertainty in Structural Dynamics – p.8/23

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. Reliability and Uncertainty in Structural Dynamics – p.9/23

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 . Reliability and Uncertainty in Structural Dynamics – p.10/23

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 . Reliability and Uncertainty in Structural Dynamics – p.11/23

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 . Reliability and Uncertainty in Structural Dynamics – p.12/23

Multistoried Portal Frame Random Variables: Axial stiffness (EA) and the bending stiffness (EI) of P 1 = 4 . 0 e 5 KN, P 2 = 5 . 0 e 5 KN each member are uncorrelated Gaussian random Nel=20, Nnode=12 11� variables (Total 2 × 20 = 40 random variables: 12� 18� P� 2� x ∈ R 40 ). 17� 19� 20� P� 1� 10� EI (KNm 2 ) EA (KN) 14� 9� 15� Element 13� Standard Standard 16� Mean Deviation Mean Deviation 8� Type 7� 10� 9� 11� 5.0 × 10 9 6.0 × 10 4 1 7.0% 5.0% 12� 6� 3.0 × 10 9 4.0 × 10 4 2 3.0% 10.0% 5� 6� 5� 7� 1.0 × 10 9 2.0 × 10 4 3 10.0% 9.0% 8� 5 @� 2.0m� 4� Failure surface: g ( x ) = d max − | δh 11 ( x ) | , 3� 2� 1� 3� 4� δh 11 : horizontal displacement at node 11, 1� 2� d max = 0 . 184 × 10 − 2 m Reliability and Uncertainty in Structural Dynamics – p.13/23 3.0m�

Multistoried Portal Frame Results (with one iteration) Approximation FORM MCS ‡ ( n reduced = 1 ) n = 40 (exact) 3.399 3.397 β − P f × 10 3 0.338 0.340 0.345 ‡ with 11600 samples (considered as benchmark) Reliability and Uncertainty in Structural Dynamics – p.14/23

Dynamic Reliability Problem The central issues: The failure surface is discontinuous (hence not differentiable) and multiple-connected FORM and SORM, in its classical form, is not applicable Reliability and Uncertainty in Structural Dynamics – p.15/23

A 2 DOF System 1� 2� m� m� 1� 2� k� 2� k� 1� k� 3� k 1 = ¯ k 1 (1 + x 1 / 3) , k 2 = ¯ k 2 (1 + x 2 / 3) , ω 1 = 32 . 22 and ω 2 = 35 . 52 Reliability and Uncertainty in Structural Dynamics – p.16/23

Transfer Function 10 2 H 11 ( ω )/y s 10 1 10 0 30 32 34 36 38 40 42 ω (rad/s) Reliability and Uncertainty in Structural Dynamics – p.17/23

Static Failure Surface g ( x 1 , x 2 ) = H 11 ( ω ) / ¯ y s − α max = 0 , ω = 0 2 1 0 x1 −1 −2 −3 −3 −2 −1 0 1 2 x2 α max : maximum allowable amplification=6 Reliability and Uncertainty in Structural Dynamics – p.18/23

Static Reliability Integral Reliability and Uncertainty in Structural Dynamics – p.19/23

Dynamic Failure Surface g ( x 1 , x 2 ) = H 11 ( ω ) / ¯ y s − α max = 0 , ω = 33 . 26 rad/s 2 1 0 x 1 −1 −2 −3 −3 −2 −1 0 1 2 x 2 α max : maximum allowable amplification=6 Reliability and Uncertainty in Structural Dynamics – p.20/23

Dynamic Reliability Integral Reliability and Uncertainty in Structural Dynamics – p.21/23

Conclusions & Future Research A gradient projection method based on the sensitivity vector of the failure surface is developed to reduce the number of random variables in structural reliability problems involving a large number of random variables. Current methods work well when the failure surface is close to linear (static problems). For dynamic problems the failure surface becomes highly non-linear, discontinuous and multiple-connected. Reliability and Uncertainty in Structural Dynamics – p.22/23

Conclusions & Future Research Further research is needed to develop non-classical methods for solving dynamic reliability problems. Reliability and Uncertainty in Structural Dynamics – p.23/23