Köylüoglu, H. Ugur

Nielsen, Søren R.K.

English

Path integration (cell-to-cell mapping) method is applied to evaluate the joint probability density function (jpdf) of the response of the structural systems, with uncertain properties, subject to white noise excitation. A general methodology to deal with uncertainties is outlined and applied to the friction controlled slip of a structure on a foundation where the friction coefficient is modelled as a random variable. Exact results derived using the total probability theorem are compared to the ones obtained via path integration

    Aalborg UniversitetPath Integration Applied to Structural Systems with Uncertain PropertiesNielsen, Søren R.K.; Köylüoglu, H. UgurPublication date:1996Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Nielsen, S. R. K., & Köylüoglu, H. U. (1996). Path Integration Applied to Structural Systems with UncertainProperties. Dept. of Building Technology and Structural Engineering. Structural Reliability Theory Vol. R9602No. 150General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: November 29, 2020INSTITU TTET FOR BYGN INGSTEKN IK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGTNEERING AALBORG UNIVERSITET • AUC • AALBORG • DANMARK STRUCTURAL RELIABILITY THEORY PAPER NO. 150 To he presented at the ASCE Joint Specialty Conference on Probabilistic Me-chanics and Structural Reliability, Worcester, USA, August Hl96 S. R . K. NIELSEN & H. U. KOYLUOGLU PATH INTEGRATION APPLIED TO STRUCTURAL SYSTEMS WITH UN-CERTAIN PROPERTIES MARCH 1996 ISSN 1395-7953 R9602 The STRUCTURAL RELIABILITY THEORY papers are issued for early dissemina-tion of research results from the Structural Reliability Group at the Department of Building Technology and Structural Engineering, University of Aalborg. These papers are generally submitted to scientific meetings, conferences or journals and should there-fore not be widely distributed. Whenever possible reference should be given to the final publications (proceedings, journals, etc.) an.d not to the Structural Reliability Theory papers. I Printed at Aalborg University I INSTITUTTET FOR BYGN INGSTEKN IK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERII'{G AALBORG UNIVERSITET • AUC • AALBORG e DANMARK STRUCTURAL RELIABILITY THEORY PAPER NO. 150 To be presented at the ASCE Joint Specialty Conference on Probabilistic Me-chanics and Structural Reliability, Worcester, USA, August 1996 S. R. K. NIELSEN & H. U. KOYLUOGLU PATH INTEGRATION APPLIED TO STRUCTURAL SYSTEMS WITH UN-CERTAIN PROPERTIES MARCH 1996 ISSN 1395-7953 R9602 PATI-1 INTEGRATION APPLIED TO STRUCTURAL SYSTEMS WITH UNCERTAIN PROPERTIES Sj~jren R. K. Nielsen1 and H. Ugur Koyliioglu2 ABSTRACT Path integration (cell-to-cell mapping) method is applied to evaluate the joint prob-ability density function (jpdf) of the response of structural systems, with uncertain properties, subject to white noise excitation. A general methodology to deal with un-certainties is outlined and applied to the friction controlled slip of a structure on a foundation where the friction coefficient is modelled as a random variable. Exact re-sults derived using the total probability theorem are compared to the ones obtained via path integration. INTRODUCTION Path integration is an approximate method to solve the Fokker-Planck equation of nonlinear dynamic systems driven by excitations with independent increments. Path integratibn methods have been applied to nonlinear dynamic systems subject to white noise by Sun and Hsu (1990), N aess and Johnsen (1991 ), Poisson driven train of impulses Koyliioglu et al. (1995.a). In all of these studies, very accurate results are reported and it is pointed out that the path integration methods can only be applied to problems of low dimensionaliy due to the large memory and cpu time requirements of the numerical calculations. In this paper, new equations are introduced to represent the random variables that model the uncertain properties. A deterministic system equation but with random initial conditions is then obtained for the integrated state vector made up of the displacement, velocity and random variables. Assuming that the random parameters are stochastically independent of the white noise excitation, the integrated state vector becomes Marko-vian, which further simplifies the problem. Then, the Chapman-Kolmogorov equation is discretized in space and time, and the Markov process is represented by a Markov chain. The proposed method has been applied to the friction controlled slip of a struc-ture on a foundation where the friction coefficient is modelled as a random variable. The transitional probability matrix is calculated numerically using Gaussian closure applied for a small time step. 2. DEALING WITH UNCERTAIN PROPERTIES The equation of motion of a SDOF system with mass m and random stiffness and damping subject to Gaussian white noise W(t) with intensity 50 reads in state vector 1 Aalborg University, DK-9000 Aalborg, Denmark. 2 Ko<; University, TR-80860 istinye, istanbul, Turkey. 2 form as Z(t) = a(Z(t)) + b(t)W(t) Z(O) = Zo (1) The state vector Z(t), drift-vector a(Z(t)) and diffusion vector b(t) are given as [ Y(t) l [ ol [ Y(t) l [ 0 l Z(t) = ~~:~ , Zo = ~ , a(Z(t)) = _! H(~, Y, V) , b(t) = ~ (2) where H(Y, Y, X) is the nonlinear restoring force depending on the displacement Y(t), velocity Y(t) and mean-zero random variables xr = [X1 , X 2 , ···X a] specifying the stiff~ ness and damping properties. X is assumed to be stochastically independent of the white noise excitation W(t), and the jpdf fx(x) is known. In (1), the random parameters have been introduced as extra state variables V(t) by the equation V(t) = 0 , V(O) =X. Then, the equation of motion with random parameters X and deterministic initial con-ditions Y(O) = Y(O) = 0 has been transformed into an equivalent system with deter-ministic coefficients and random initial conditions. Due to the white noise excitation and 1as Zo is independent of W(t), the state vector Z(t) of dimension N = 2 + d is Markovian. Such a treatment has been investigated in Koyliioglu et al. (1995.b ). 3. PATH INTEGRATION Let qz( z, t; lzo, ti-1) signify the transitional jpdf of the Markov process from the state Zo at time ti-1 to a state z at a later time t; after the ith transition. The jpdf fz(z, i;) of Z(t;) at timet; then fulfills the Chapman-Kolmogorov equation : fz(z,t;) = r qz(z,t;izo,t;_l)Jz(zo,t;_1)dzo Jzo i = 1,2, ... to = 0 (3) The initial distribution at timet= 0 is given as fz(z, 0) = fyvv(Y, y, v, 0) = 5(y)5(y )Jx (v). Path integration demands a suitable discretization of the state space, and constant time intervals b..t = i; - t;_1 between the transitions. Then, the time and space continuous Markov vector Z(t) is carried into a Markov chain with the transition equation (4) Using a uniform mesh, the jth component Pi,j = ffl.z· fz(z, i;)dz of Pi of (4) being the probability lumped to the jth nodal point with posftion Zj, centered in a cell b..zj = flz1 · · · b..z N. flzz denotes the spacings of the mesh in the lth coordinate direction. The component Qjk = ft>z qz(z, to + fltizk, io)dz of the transition probability matrix Q signifies the probability of being in node j at the time t = t0 + b..t on condition of starting at the kth node at time t = t0 • Resuming this procedure a number n of times, the state Pn is given as Pn = Qnpo, attaining a stationary state as n --. oo. 3 4. FRICTION CONTROLLED SLIP OF A STRUCTURE ON A FOUN-DATION The equation of motion for the slip displacement S(t) of a structure on a foundation is S(t) + vg sign(S) = -G(t) (5) where vis the friction coefficient and G(t) is the horizontal surface acceleration. In what follows, G(t) is assumed to be white noise with intensity So. Then, the exact stationary conditional pdf and the variance for the slip velocity conditioned on v are . vg ( vglsl) fs(siv) = 2?rSo exp - ?rSo (6) vis a random variable with pdf fv(l/), mean E[v] and coefficient of variation Cv. The displacement S(t) is not considered in the state vector as there is no stiffness term involved in this dynamic system. Then, the state vector consists of the velocity S(t) and the dummy random process V(t) = v. This yields Z(t)T = [S(t), V(t)], thus N = 1 + 1 = 2. 5. APPROXIMATING TRANSITIONAL PROBABILITY MATRIX During a transition from the state Zk = (Sk, vk) at the time t 0 , the system is assumed to behave locally Gaussian assuming 6.t to be small. The conditional mean value function JL(i + 6.t) and conditional covariance matrix K.(t0 + 6.t) are determined from the differential equations, Koyliioglu et al. (1995.a). jL(t) = E [a(Z(t))] JL(O) = Zk (7) K.(t) = E [! a(Z(t))] K.(t) + K.(t)E [%z a(Z(t))] T + b(t)bT(t) "'(0) = 0 (8) (7) and (8) provide the solutions E[vlzk] = Vk and "'sv = "'vv = 0, so qz(s, v, to + 6.tlzk,to) = 8(v- vk)cp((s- /Js)/~)/~, i.e. diffusion only takes place in the S direction. f..Ls and "'ss are obtained by numerical integration of (9) and (10) over the interval [0, 6.t]. d ( -f..L. (t) ) -d f..Ls(t) = -vkg 1- 2ci>( 5 ) t J "'ss(t) (9) (10) 4 6. NUMERICAL EXAMPLE In the numerical example, the conditional variance of the slip velocity conditioned on the mean friction coefficient a-1(E[v]) is set to 1, i.e., ~~~fg2 = 1. v is assumed to be uniformly distributed in the interval [E[v](1 - J3)c11 , E[v](1 + J3cv )] where E[v] = 0.5 and c11 denotes the coefficient of variation, which will be varied. Path integration results are obtained for the unconditional pdf Js( s) of S using a mesh with 40 x 40 cells, extending ±4a-s in the S direction and covering the v direction. The transition time interval is selected as flt = 0.01 1~'"(~]9 . Numerical results shown with full line are compared in Figures 1 and 2 with the exacts ones shown with dashed signature obtained by evaluating the compound probabilities analytically as given in eq. (11 ). As seen, a-~ in (11) is very different than (6). These are equal if Vc--> 0. fs(s) = 1 fs(siv)fv(v)dv 0 - 1 - 2 .-. •Cl) ·Cii" -3 I -- 4 Q 0 ..J -5 -6 - 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 . s Figure 1: Js(s) , Cv = 0.1. 5 0 -1 -2 .-. •CO .u; -3 I --4 Q 0 _, -5 -6 -7 -5 -4 -3 -2 -1 0 1 2 3 4 5 • s Figure 2: Js(s) , C11 = 0.3 . 7. REFERENCES 1. Koyliioglu, H.U., Nielsen, S.R.K. and (;akmak, A.S. (1995.a) Fast Cell-to-Cell Mapping (Path Integration) with Probability Tails for the Stochastic Response of Nonlinear White Noise and Poisson Driven Systems. Structural Safety, 17, pp. 151-165. 2. Koyliioglu, H.U., Nielsen, S.R.K. and <;akmak, A.S. (1995.b) Solution of Random Structural System subject to Nonstationary Excitation: Transforming the Equa-tion with Random Coefficients to one with Deterministic Coefficients and Random Initial Conditions . Soil Dynamics and Earthquake Engineering, 14, pp. 219-228. 3. Naess, A. and Johnsen, J.M. (1991), Response Statistics of Nonlinear Dynamic Systems by Path Integration. Proc. IUTAM Symposium on Nonlinear Stochastic Mechanics (Bellomo and Casciati, eds.), Torino, Italy, July, 1991, Springer-Verlag. 4. Sun, J.Q. and Hsu, C.S. (1990) The Generalized Cell Mapping Method in Nonlin-ear Random Vibration based upon Short- Time Gaussian Approximations, J. Appl. Mech., 57, 1018-1025. STRUCTURAL RELIABILITY THEORY SERIES PAPER NO. 125: H. L Ha.nsen, P. H. Kirkega.a~1 & S. R. K. Nielsen: Modelling of Deteriorating RC-Structures under Stochast?:c Dynamic Loading by Neural Networks. ISSN 0902-7513 R9409. PAPER NO. 126: H. U. Koyliioglu, S. R. K. Nielsen & A. ~ . Qa.kma.k: Reliability Approximations for MDOF Structures with Random Properties s'l.(,bject to Random Dy-namic E:Ecitation in Modal Subspaccs. ISSN 0902-7513 R9440. PAPER.NO. 127: H. U. Koyli:ioglu , S. R. K. Nieisen and A. S. Qakrnak: A Fa· ster Sim.ulation Method for:' the Stochastic Response of Hysteretic Structures subject to Earthquakes. ISSN ®902-7513 R9523. PAPER NO . 128: H. U. Koylii0glu, S. R. K. Nielsen, A. S. Qakmak & P. H. Kirkegaard: Prediction of Global and Localized Damage and Future Reliability for RC Structures subject to Earthquakes . ISSN 0901-7513 R9426. PAPER NO. 129·: C. Pedersen & P. Th0ft-Christensen: Interactive Structural Optimi-zation with Qua.s i-Newt'on Algorithms. lSSN 0902-7513 R9436. PAPER NO. 130: I. Enevoldsen & J. D. S(llrensen: Decomposition Tech'TI!iques and Effective Algorithms in Reliability-Based Optimization. ISSN 0902-7513 R9412. PA.PER NO. 131: H. U. Koyliiog1u, S. R. K. Nielsen & A. S. Qakma~k: Appr:'oximate F'orwarll Difference Equations for the Lower Order Non-Stationary Statistics of Gepme-trically N on-Li'TI!eo,r Systems subject to Random Excitation. ISSN 0902-7513 R9422. PAPER NO. 132: I. B. Kwon: Decision Theory applied tJo Structu.ml Engineering Problems. Ph. D.-Thesis. ISSN 0902-7513 R.9421. PAPER 133: N. U. Koyliioglu, S. R. K. Nieisen & A. ~. Qakmak: Stochastic Dyna-mics of Nonlinear Strnctv1res with Random Properties subject to Ra'TI!dom Stationary Excitation. ISSN 0902-7513 R9520. PAPER NO. 134: H. U. Koyliiogll!l., S. R. K Nielsen & A. S. Qakmak: Solution of Random Struct·ural System s11.bject to N on-Strdionary Excitation: Transforming the Equation with Randem Coefficients to One with Deterministic Coefficients and Random Initial Co'TI!ditio.ns . ISSN 0902-7513 R9429. PAPER NO. 135: S. Engelund, J. D. S!llreRsen & ·s. Krenk: Estimation of the Time to Initiati·on of Corrrosion in Existing Uner:'a,cked Concrete Structures. ISSN 0902-7513 R9438. PAPER. NO. 136:. H. U. Koyliioglu, S. R. K. Nielsen & A. S. Qakmak: Solution Method8 for Structures with Random Properties subject to Random Excitation. ISSN 0902-7513 R9444. PAPER NO. 137: J. D. S!llrensen, M. H. Faber & I. B. Kroon: Optimal Reliability-Based Planning of Experiments for POD Curves. ISSN 0902-7513 R9455. PAPER NO . 138: ~.R.K. Nielsen & P.S. Skjrerbrek, H.U. Koyliioglu & A.~ . Qakmak: Prediction of Global FJama.ge and Reliability based upon Sequential Identification and Updating .of RC Structures subject to Ea'flthquake8. ISSN 0904-751g R.9505. STRUCTURAL RELIABILITY THEORY SERIES PAPER NO. 139: R. Iwankiewicz, S. R. K. Nielsen & P. S. Skjrerbrek: Sensitivity of R eliability Estimates in Partially Damaged RC Structures subject to Earthquakes, using Reduced Hysteretic Models . ISSN 0902-7513 R9507. PAPER NO 141: H. U. Koyluoglu, S. R. K. Nielsen & A. ~- Qakmak: Uncertain Buckling Load and Reliability of Columns with Uncertain Properties. ISSN 0902-7513 R9524. PAPER NO. 142: S. R. K. Nielsen & R. Iwankiewicz: Response of Non-Linear Systems to Renewal Impulses by Path Integmtion. ISSN 0902-7513 R9512. PAPER NO. 145: H. U. Koyluoglu, S. R. K. Nielsen, Jamison Abbott and A. ~­Qakmak: Local and Modal Damage Indicators for Reinforced Concrete Shear Frames subject to Earthquakes. ISSN 0902-7513 R9521 PAPER NO. 146: P. H. Kirkega.ard, S. R. K. Nielsen, iJt. C. Micaletti and A. ~- Qakmak: Identification of a Maximum Softening Damage Indicator of RC-Structures using Time-Frequency Techniques. ISSN 0902-7513 R9522. PAPER NO . 147: R. C. Mica.letti, A. ~- Qakmak, S. R. K. Nielsen & P. H. Kirkegaa.rd: Construction of Time-Dependent Spectra using Wavelet Analysis for Determination of Global Damag e. ISSN 0902-7513 R9517. J . PAPER NO . 148: H. U. Koyluoglu, S. R. K. Nielsen & A. ~- Qakmak: , Hysteretic MDOF Model to Quantify Damage for TC Shea,r Frames .mbject to Ea.rthquakes. lSSN 1395-7953 R9601. PAPER NO . 149: P. S. Skjrerbrek, S. R. K. Nielsen & A. ~- Qakmak: Damage Lo6:a-tion of Severely Damaged RC-Structures based on Measured Eigenperiods from a Single Response. ISSN 0902-7513 R9518. PAPER NO. 150: S. R. K. Nielsen & H. U. Koyluoglu: Path Integration applied to Structural Systems with Uncertain Properties. ISSN 1395-7953 R9602. PAPER NO. 151: H. U. Koyluoglu & S. R. K. Nielsen: System Dynamics and Modified Cv.mwlant Neglect Closure Schemes. ISSN 1395-7953 R9603. PAPER NO. 154: J. D. S\?lrensen, M. H. Faber, I. B. Kroon: Optimal Reliability-Based Planning of Experiments for POD Curves. ISSN 1395-7953 R9542. PAPER NO. 155: J. D. S\?lrensen, S. Engelund: Stochastic Finite Elements in Reliability-Bas ed Structural Optimization. ISSN 1395-7953 R9543. PAPER NO. 161: S. Engelund, J. D. S0rensen: Stochastic Models for Chloride-initiated Corrosion in Reinforced Concrete. ISSN 1395-7953 R9608. 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Path Integration Applied to Structural Systems with Uncertain Properties

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Path Integration Applied to Structural Systems with Uncertain Properties

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