System Reliability-Based Design Optimization of Structures Constrained by First Passage Probability

Transcription

1 System Reliability-Based Design Optimization of Structures Constrained by First Passage Probability Junho Chun* University of Illinois at Urbana-Champaign, USA Junho Song Seoul National University, Korea Glaucio H. Paulino Georgia Institute of Technology, USA June 17 th, 2015

2 Structural Engineering under Natural Hazards and Risks One of the most fundamental requirements on building structures is to withstand various uncertain loads such as earthquake ground motions, wind loads and ocean waves. 1 2 San Francisco Earthquake, 1907 Kobe Earthquake, 1995 The structural design needs to ensure safe and reliable operations over a prolonged period of time despite random excitations caused by hazardous events. Acceleration (gal) HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.14gal x - time / DT = 0.02 Random Excitations Random process Non-deterministic excitations Many possibilities of the process Photograph: Kimimasa Mayama/Reuters 2

3 Acceleration Elevation, m Motivation Reliable Structural Design under Stochastic Excitations Research aims to find the optimal structure and system under stochastic excitations Structural Design Courtesy of Skidmore, Owing and Merrill, LLP Structural elements optimization Structural system Structural performance optimization Time, s Story Displacement, mm 3

4 Outline Discrete Representation Method First Passage Probability / Structural Engineering Constrains Reliability-Based Design Optimization Formulation / Sensitivity Analysis Numerical Applications / Discussion 4

5 Discrete Representation of Stochastic Excitation The stochastic excitation is represented by a linear combination of basis functions, s(t), with standard normal independent random variables, v: n T f ( t) ( t) visi( t) ( t) s( t) v i1 Modeling Ground Excitations - Filtered Gaussian Process Stochastic ground excitations can be modeled by using a filter representing the characteristic of soil medium and Gaussian process. t f ( t) v( ) s( t ) d 0 n n v s ( t) W h ( t t ) t i i i f i i1 i1 n 2π 0 / t vi h f ( t ti ) t T s( t) v i1 Gaussian process Soil Medium (Filter) Filter parameter: ω g, ζ g Ground acceleration (Filtered Gaussian Process) Der Kiureghian, A. (2000). The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1),:

6 Discrete Representation of Responses of Linear Structures The convolution integral for determining the responses of linear systems subjected to the stationary process can be developed with the impulse response function. Dynamic Responses t u( t) f (τ) h ( t τ) dτ 0 s t n n T i i s i i a v 0 i1 i1 u( t) v s ( ) h ( t τ) dτ v a ( t) ( t) Instantaneous Failure Probability Deterministic, time-dependent - filter + structure Random, time-independent Failure event of a linear system at a certain time t i E : g( t, u ) 0 E : u t u E : a( t ) v u T f i 0 f i 0 f i 0 Failure Probability : (, 0) 0 β, 0 P E g t u t u f f i i β t, u i 0 u a 0 t i 6

7 First Passage Probability In the reliability analysis of dynamic system subjected to stochastic excitations, a significant problem is to determine the first passage probability that any one of output states of interest exceeds a certain threshold value within a given time duration T. Stress Displacement Ssiger International Plaza Courtesy of Skidmore, Owing and Merrill, LLP First passage probability is defining the problem as a series system problem such as: n P ( E ) P( u max u( t) ) P u( t ) u fp sys 0 0 t tn i 0 i1 Song, J., and A. Der Kiureghian (2006). Joint first-passage probability and reliability of systems under stochastic excitation. J. Engineering Mechanics, ASCE, 132(1): Fujimura, K. and A. Der Kiureghian (2007). Tail-Equivalent Linearization Method for Nonlinear Random Vibration. Probabilistic Engineering Mechanics, 22:

8 Reliability-Based Design Optimization under Constraints on First Passage Probability Optimization Formulation min f ( d) d obj nt s. t P E E ( t, d) : g ( t, d) 0 P, i 1,..., n fp, i sysi fi k i k fi c k1 lower upper d d d Objective function Probabilistic constraints with M( d) u( t, d) C( d) u( t, d) K( d) u( t, d) f ( t, d ) f ( t, d)= M( d) lu ( t)= M( d) lf ( t) Probabilistic Constraints in Structural Engineering Stress Maximum Displacement Inter-Story Drift Ratio g Hearst Tower (New York City) Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation. 8

9 Probabilistic Constraints in Structural Engineering - Detail Stress y x ue2,y 2 u l e2 ue2,x Maximum Displacement E ( t, d) : g ( t, d) 0 f k k ( t, d) E t u H tip k f ( k, d) : o 0 ue1,y ne θe A e, L e, D e u l e1 1 ue1,x g cosθe ue 1 g ue1, x g ue2, x ne sinθ ue, e1, g u e2 e u u e2 e1, y u u e2, y D D ( t, d) n ( u ( t, d) u ( t, d)) B u e g g e e e e2 e1 e e Le Le De l l ( u e2( t, d) u e1( t, d)) L e E ( t, d) : g ( t, d) 0 E ( t, d) : ( t, d) 0 fe k e k fe k oe e k Inter-Story Drift Ratio E ( t, d) : g ( t, d) 0 fi k i k ( t, d) ( t, d) E t u i k i1 k f (, ) : 0 i k d oi Hi Engineering constraints can be expressed in terms of the discrete representation form as: D ( t, d) ( a( t, d) v a( t, d) v) e T T e e2 e1 Le t d P 1 T T β, Le oe Le oe D a( t, d) a( t, d) D b ( t, d) e k fe e k e1 k e2 (, d) :Stress(, d) β, d P E t t t fe fe k k e k e e k Pfp _ e nt e, β R 9

10 Adjoint Method Verification Sensitivity Analysis of Probabilistic Constraint (Stress, Time duration = 8 secs, σ oe =35MPa) P ( E ) β, R fp sys n d i d i P f 1 ( E ) A sys 1 Finite difference method Adjoint method P f 3 ( E ) A sys 2 P f 4 ( E ) A sys 4 10

11 Numerical Application 1 2D Bracing System Optimization Image courtesy of SOM min f ( d) d obj nt s. t P E E ( t, d) : g ( t, d) 0 P, i 1,..., n fp, i sysi fi k i k fi c k1 0.02m d 1m 2 2 with M( d) u( t, d) C( d) u( t, d) K( d) u( t, d) M( d) lf ( t) Volume Stress / Maximum Drift Ratio / Inter-Story Drift Ratio Kanai-Tajimi Filter 2 (2 1) f f 2 2 f f f 2 f f f f f f f f 1 f h ( t) exp( t) sin( 1 t) exp( t)2 cos( 1 t) 11

12 ωu ζ ofoe f Numerical Application 1 2D Bracing System Opt. (Stress Constraint) Optimized Structures Initial area = 0.5m 2 Convergence History P f = P f = P f = Φ o ω f ζ f t (sec) t (sec) Init. Bars (m 2 ) Threshold 0.2 5π σ oe = 35 MPa 20,000 MPa E 12

13 Numerical Application 1 2D Bracing System Opt. (Stress Constraint) Dynamic Behavior Dynamic Response Comparison Initial System Optimized System 13

14 Numerical Application 1 2D Bracing System Opt. (Inter-Story Drift constraints) Optimized Structures Initial area = 0.3m 2 Convergence History P f = P f = P f = Φ o ω f ζ f t (sec) t (sec) Init. Bars (m 2 ) Threshold 1.0 5π u oδ = 1/50 20,000 MPa E 14

15 Numerical Application 1 2D Bracing System Opt. (Inter-Story Drift constraints) Optimized Structures Dynamic Response Comparison (P f = ) P f = P f = P f =

16 Concluding Remarks New framework integrating random vibration theories into structural optimization was developed. First passage probability was incorporated into structural optimization. SCM enables for an efficient and accurate computation of the failure probability of a large-size system reliability problem. Efficient method of sensitivity calculation was derived. Developed framework identified optimal bracing systems that can resist future realization of stochastic processes with a desired level of reliability. Junho Chun jchun8@illinois.edu 16

17 Acknowledgement National Science Foundation (NSF) - CMMI Thank you for your attention Junho Chun jchun8@illinois.edu

18 BACK-UP SLIDES

19 Numerical Application 1 2D Bracing System Opt. (Drift constraints) 19