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

Reliability and Uncertainty in Structural Dynamics

• S. ADHIKARI

In collaboration with Prof. R S. Langley Cambridge University Engineering Department Cambridge, U.K.

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Outline of the Talk

Introduction: Research Interests Structural Reliability Analysis Reliability Analysis for Dynamics Conclusions

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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

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Structural Reliability Analysis

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The Fundamental Problem

Probability of failure: Pf =

• G(y)≤0

p(y)dy

(1) y ∈ Rn: 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/

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Main Difficulties

n is large p(y) is non-Gaussian Pf is usually very small (in the order of 10−4

• r smaller)

G(y) is a complicated nonlinear function of y

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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.

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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 Pf = Φ(−β) with β = (x∗T

x∗)1/2 (2)

where x∗, the design point is the solution of min

• (xT x)1/2

subject to g(x) = 0.

(3)

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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

• btained 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.

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Example 1

Linear failure surface in R2: g(x) = x1 − 2x2 + 10

−10 −8 −6 −4 −2 −1 1 2 3 4 5 6

x1 x2 Failure domain: g(x) = x1−2x2+10 < 0 Safe domain g(x) = x1−2x2+10 > 0

β x*

x∗ = {−2, 4}T and β = 4.472.

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Example 2

g(x) = − 4

25 (x1 − 1)2 − x2 + 4

−5 −4 −3 −2 −1 1 2 3 4 −1 1 2 3 4 5

x1 x2 Failure domain g(x) < 0 Safe domain g(x) > 0 1 2 3 4 5

x∗ = {−2.34, 2.21}T and β = 3.22.

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Example 3

g(x) = − 4 25 (x1 + 1)2 − (x2 − 5/2)2(x1 − 5) 10 − x3 + 3

x∗ = {2.1286, 1.2895, 1.8547}T and β = 3.104.

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Multistoried Portal Frame

P1 = 4.0e5 KN, P2 = 5.0e5 KN Nel=20, Nnode=12

5 @ 2.0m 3.0m

2 1 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

P

2

P

1

Random Variables: Axial stiffness (EA) and the bending stiffness (EI) of each member are uncorrelated Gaussian random variables (Total 2 × 20 = 40 random variables:

x ∈ R40).

EA (KN) EI (KNm2) Element Standard Standard Type Mean Deviation Mean Deviation 1 5.0×109 7.0% 6.0×104 5.0% 2 3.0×109 3.0% 4.0×104 10.0% 3 1.0×109 10.0% 2.0×104 9.0% Failure surface: g(x) = dmax − |δh11(x)|, δh11: horizontal displacement at node 11, dmax = 0.184 × 10−2m

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Multistoried Portal Frame

Results (with one iteration)

Approximation FORM MCS‡ (nreduced = 1) n = 40 (exact) β 3.399 3.397 − Pf × 103 0.338 0.340 0.345

‡with 11600 samples (considered as benchmark)

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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

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A 2 DOF System

m

1

m

2

k1 k2 k3 1 2

k1 = ¯ k1(1 + x1/3), k2 = ¯ k2(1 + x2/3), ω1 = 32.22 and ω2 = 35.52

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Transfer Function

30 32 34 36 38 40 42 100 101 102

ω (rad/s)

H11(ω)/ys

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Static Failure Surface

g(x1, x2) = H11(ω)/¯ ys − αmax = 0, ω = 0

−3 −2 −1 1 2 −3 −2 −1 1 2

x1 x2 αmax : maximum allowable amplification=6

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Static Reliability Integral

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Dynamic Failure Surface

g(x1, x2) = H11(ω)/¯ ys − αmax = 0, ω = 33.26 rad/s

−3 −2 −1 1 2 −3 −2 −1 1 2

x1 x2 αmax : maximum allowable amplification=6

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Dynamic Reliability Integral

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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.

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Conclusions & Future Research

Further research is needed to develop non-classical methods for solving dynamic reliability problems.

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