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Computational Stochastic Mechanics – Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

RELIABILITY ASSESSMENT OF NONLINEAR MDOF SYSTEMS SUBJECT TO EVOLUTIONARY STOCHASTIC EXCITATION IOANNIS P. MITSEAS1*, IOANNIS A. KOUGIOUMTZOGLOU2, POL D. SPANOS3, M. BEER1+ 1

Institute for Risk and Uncertainty, University of Liverpool, Brodie Tower, Brownlow Street, L69 3GQ, Liverpool, United Kingdom. E-mail: *[email protected], [email protected] 2 Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, NY 10027,USA. E-mail: [email protected] 3 L.B. Ryon Chair in Engineering, Rice University MS 321, P.O. Box 1892, Houston, TX 77251, USA. E-mail: [email protected] An approximate analytical technique for determining the survival probability and first-passage probability density function (PDF) of nonlinear multi-degree-of-freedom (MDOF) structural systems subject to a non-stationary stochastic excitation vector is developed. The proposed technique can be construed as a two-stage approach. First, relying on statistical linearization and utilizing a dimension reduction approach the nonlinear n-degree-of-freedom system is cast/decoupled into (n) effective single-degree-of-freedom (SDOF) linear time-variant (LTV) systems corresponding to each and every degree of freedom of the original MDOF system. Second, an approximate technique based on stochastic averaging is employed in conjunction with the time-varying stiffness and damping elements of the effective SDOF LTV oscillator for determining the survival probability and first-passage PDF of the nonlinear MDOF system. Overall, the technique appears to be efficient and versatile since it can handle readily, at a low computational cost, a wide range of nonlinear/hysteretic behaviors as well as various stochastic excitation forms, even of the fully non-stationary in time and frequency kind. A 3-DOF structural system exhibiting hysteresis following the Bouc-Wen model is included as a numerical example. The reliability of the technique is demonstrated by pertinent Monte Carlo simulations. Keywords: first-passage problem, evolutionary stochastic processes, hysteretic system, statistical linearization, stochastic averaging, reliability assessment

1

Introduction

Structural systems are subject to stochastic excitations that exhibit strong variability in both intensity and frequency content. This fact necessitates the representation of this class of structural loads by non-stationary stochastic

1

processes. Further, structural systems under severe dynamic excitations, such as those due to earthquakes, can exhibit significant nonlinear behavior of the hysteretic kind. Thus, there is considerable interest within the engineering community for determining the response and assessing the reliability of

Reliability Assessment of Nonlinear MDOF Systems Subject to Evolutionary Stochastic Excitation I.P. Mitseas, I.A. Kougioumtzoglou, P. D. Spanos, and M. Beer

nonlinear/hysteretic systems subject to evolutionary stochastic excitations (Roberts and Spanos 2003). In engineering dynamics, the evaluation of the probability that the system response stays within prescribed limits for a specified time interval is advantageous for reliability based system design applications. The determination of the above time-variant probability, known as survival probability, has been coined in the literature as the firstpassage problem and has been a persistent challenge in the field of stochastic dynamics. In this regard, several researchers have focused on developing Monte Carlo simulation (MCS) based techniques (e.g. Schueller et al. 2004) for structural system reliability assessment. Note, however, that there are cases where the computational cost can be prohibitive; thus, rendering the development of alternative efficient approximate analytical/numerical techniques necessary (e.g. Kougioumtzoglou and Spanos 2013). In this paper, an approximate analytical technique for determining the survival probability and first-passage probability density function (PDF) of nonlinear multidegree-of-freedom (MDOF) structural systems subject to a non-stationary stochastic excitation vector is developed. Specifically, first relying on a statistical linearization based dimension reduction approach the original MDOF system is cast into effective singledegree-of-freedom (SDOF) linear time-variant (LTV) systems corresponding to every degree of freedom of the original MDOF system. Second, a stochastic averaging based approximate technique is utilized to derive the nonlinear MDOF system survival probability and first-passage PDF at a low computational cost. A Bouc-Wen 3-DOF structural system subject to an evolutionary stochastic earthquake excitation is included as a numerical example. Pertinent MCS data demonstrate the reliability of the technique.

2

2 2.1

MDOF System Dimension Reduction Statistical linearization treatment

Consider an n-degree-of-freedom nonlinear structural system governed by the equation 𝑴𝒒 + 𝑪𝒒 + 𝑲𝒒 + 𝒈 𝒒, 𝒒 = 𝑭 𝑡

(1)

where q denotes the response acceleration vector, q is the response velocity vector, q is the response displacement vector; M, C and K denote the n × n mass, damping and stiffness matrices, respectively. g q, q is an arbitrary nonlinear n × 1 vector function of the variables q and q. F t T = (f1 t , f2 t , … , fn t ) is a n × 1 zero mean, non-stationary stochastic excitation vector process defined as F t = γa t , where γT = (γ1 , γ2 , … , γn ) is an arbitrary constant vector of weighting coefficients, and a t is a non-stationary process with an evolutionary power spectrum (EPS) Sa ω, t . In this regard, F t possesses the EPS matrix 𝑺𝑭 𝝎, 𝒕 = 𝛾1 2 𝑆𝑎 𝜔, 𝑡 0 ⋯ 0 2 0 𝛾2 𝑆𝑎 𝜔, 𝑡 ⋯ 0 ⋮ ⋱ ⋯ ⋮ 0 0 ⋯ 𝛾𝑛 2 𝑆𝑎 𝜔, 𝑡

(2)

Further, the non-stationary stochastic excitation process is regarded to be a filtered stationary stochastic process according to the concept proposed by Priestley (Priestley 1965); see also (Dalhaus 1997, Spanos and Kougioumtzoglou 2012, Kougioumtzoglou 2013). Thus, the excitation EPS matrix Eq.(2) can be given by 𝑺𝑭 𝜔, 𝑡 = 𝑨 𝜔, 𝑡 𝑺𝑭 𝜔 𝜜(𝜔, 𝑡)𝑻∗

(3)

where the superscripts (T) and (*) denote matrix transposition and complex conjugation, respectively; A ω, t is the modulating matrix which serves as a time-variant filter; and


SF ω is the power spectrum matrix corresponding to the stationary stochastic vector process F(t). Note that both separable and non-separable EPS can be defined considering Eq.(3). In this manner, realistic excitations exhibiting variability in both intensity and frequency content can be considered. Focusing next on the frequency domain, the response determination problem is defined as seeking the corresponding system response EPS matrix Sq ω, t . In the following a statistical linearization approach (Roberts and Spanos 2003) is employed for determining the response EPS matrix Sq ω, t . In this regard, a linearized version of Eq.(1) is given in the form

𝑡

𝒉 𝑡 − 𝜏 𝑨 𝜔, 𝜏 𝑒 −𝑖𝜔 (𝑡−𝜏) 𝑑𝜏

(8)

0

In Eq.(8) h t denotes the impulse response function matrix. Furthermore, the time dependent cross – variance of the response can be evaluated by the expression ∞

𝐸 𝑞𝑖 𝑞𝑗 = −∞

𝑆𝑞 𝑖 𝑞 𝑗 𝜔, 𝑡 𝑑𝜔

(9)

Next, omitting the convolution of the impulse response function matrix with the modulating matrix can lead to substantial reduction of the computational effort, especially for the case of MDOF systems (Kougioumtzoglou and Spanos 2013, Mitseas et al. 2014). In this manner, Eq.(8) takes the form

𝑴𝒒 + 𝑪 + 𝑪𝒆𝒒 𝒒 + 𝑲 + 𝑲𝒆𝒒 𝒒 = 𝑭 𝑡 (4)

𝑯𝒈𝒆𝒏 𝜔, 𝑡 = 𝑯 𝜔 𝑨 𝜔, 𝑡

Following (Roberts and Spanos 2003) and adopting the standard assumption that the response processes are Gaussian, the timedependent elements of the equivalent linear matrices Ceq and K eq are given by the expressions

where H ω is the frequency response function (FRF) matrix defined as

𝑒𝑞

𝑐𝑖,𝑗 = 𝐸

𝜕𝑔𝑖 𝜕𝑞𝑗

(5)

𝑒𝑞 𝑘𝑖,𝑗

𝑯 𝜔 = (−𝜔2 𝑴 + 𝑖𝜔 𝑪 + 𝑪𝒆𝒒 + ⋯ … + (𝑲 + 𝑲𝒆𝒒 ))−𝟏

𝜕𝑔𝑖 =𝐸 𝜕𝑞𝑗

(6)

Further, for a linear MDOF system subject to evolutionary stochastic excitation a matrix input-output spectral relationship of the form ∗

𝑺𝒒 𝜔, 𝑡 = 𝑯𝒈𝒆𝒏 𝜔, 𝑡 𝑺𝑭 𝜔 𝑯𝑻

𝒈𝒆𝒏

𝜔, 𝑡 (7)

can be derived (Kougioumtzoglou and Spanos 2009), where 𝑯𝒈𝒆𝒏 𝜔, 𝑡 =

3

(11)

Consequently, taking into account Eq.(3), Eq. (7) becomes 𝑺𝒒 𝜔, 𝑡 = 𝑯 𝜔 𝑺𝑭 𝜔, 𝑡 𝑯𝑻∗ 𝜔

and

(10)

(12)

Note that Eq.(12) constitutes a straightforward generalization of the celebrated spectral relationship based on stationarity and on the Wiener-Khinchin theorem. In this regard, the abovementioned approximation can be construed as a quasi-stationary approach which, in general, yields satisfactory accuracy in cases of relatively stiff systems (Hammond 1973, Jangid and Datta 1999). Note in passing that the spectral input-output relationship of Eq.(12) is exact for the case of stationary processes (Roberts and Spanos 2003). Further, adopting the aforementioned quasi-stationary approach, it can be readily seen that for the i-


th degree of freedom, using Eq.(2), Eq.(9) and Eq.(12) yields ∞ 2

𝐸 𝑞𝑖 𝑡

=

( 𝐻𝑖1 𝜔

2

𝛾1 + ⋯

𝜔2

−∞

−∞

(13)

and ∞

=

+ 𝐻𝑖𝑛 (𝜔)

−∞ 2 2 𝛾𝑛 )𝑆𝑎

𝜔2 ( 𝐻𝑖1 𝜔

2

𝛾1 2 + ⋯

𝜔, 𝑡 𝑑𝜔

(14)

Eqs.(13) and (14) hold true in the approximate quasi-stationary sense delineated earlier. Clearly, Eq.(12) constitutes an approximate formula for determining the MDOF system response EPS matrix at a low computational cost; thus, circumventing computationally intensive Monte Carlo simulations. 2.2 Effective SDOF linear time-variant system Adopting next the system dimension reduction approach developed in (Kougioumtzoglou and Spanos 2013), an auxiliary effective SDOF LTV system corresponding to the i-th degree of freedom can be defined as 2 𝑞𝑖 + 𝛽𝑒𝑞 ,𝑖 𝑡 𝑞𝑖 + 𝜔𝑒𝑞 ,𝑖 𝑡 𝑞𝑖 = 𝑓𝑖 𝑡

(15)

where the time-varying stiffness and damping elements of the effective LTV system can be determined by equating the variances of the response displacement and velocity expressed utilizing Eq.(15) with the corresponding ones (Eqs.(13-14)) evaluated via the statistical linearization approach delineated in the previous section; that is, E 𝑞𝑖 2 𝑡 ∞

−∞

=

1 2 2 )2 + (𝛽 2 (𝜔𝑒𝑞 𝑡 − 𝜔 𝑒𝑞 ,𝑖 𝑡 𝜔) ,𝑖

× 𝛾𝑖 2 𝑆𝑎 𝜔, 𝑡 𝑑𝜔

E 𝑞𝑖 2 𝑡 ∞

2

+ 𝐻𝑖𝑛 (𝜔) 2 𝛾𝑛 2 )𝑆𝑎 𝜔, 𝑡 𝑑𝜔

𝐸 𝑞𝑖 2 𝑡

and

(16)

4

= 1 2 2 )2 + (𝛽 2 (𝜔𝑒𝑞 𝑡 − 𝜔 𝑒𝑞 ,𝑖 𝑡 𝜔) ,𝑖

× 𝛾𝑖 2 𝑆𝑎 𝜔, 𝑡 𝑑𝜔

(17)

Clearly, Eqs.(16) and (17) in conjunction with Eqs.(13) and (14) constitute a nonlinear system of two algebraic equations for the evaluation of the LTV system time-varying stiffness and damping elements. Further, considering the case where the LTV system of Eq.(15) is a lightly damped system, and relying on a combination of deterministic and stochastic averaging (e.g. Spanos 1978), it was shown in (Spanos and Lutes 1980, Kougioumtzoglou and Spanos 2009) that the non-stationary LTV system response amplitude PDF is given by the time-dependent Rayleigh distribution of the form 𝑝 𝑎𝑖 , 𝑡 =

𝑎𝑖 𝑎𝑖 2 exp − 𝑐𝑖 (𝑡) 2𝑐𝑖 (𝑡)

(18)

In Eq.(18) 𝑎𝑖 represents the LTV oscillator response amplitude defined as 𝑎𝑖 2 𝑡 = 𝑞𝑖2 𝑡 +

𝑞𝑖 𝑡 𝜔𝑒𝑞 ,𝑖 (𝑡)

2

(19)

where ci (t) accounts for the non-stationary variance of the LTV system of Eq.(15). As it was shown in (Spanos and Lutes 1980, Kougioumtzoglou and Spanos 2009) the nonstationary LTV system response is given by 𝑐𝑖 𝑡 = −𝛽𝑒𝑞 ,𝑖 𝑡 𝑐𝑖 𝑡 +

𝜋𝑆𝑓 𝜔𝑒𝑞 ,𝑖 𝑡 , 𝑡 (20) 2 𝜔𝑒𝑞 ,𝑖 𝑡

Eq.(20) constitutes a first-order nonlinear ordinary differential equation (ODE) which can be solved utilizing standard numerical schemes such as the Runge-Kutta; see references (Spanos and Lutes 1980, Roberts and Spanos 2003, Kougioumtzoglou and


Spanos 2013) for a more detailed presentation of the topic.

𝑚

𝑃𝑖𝐵 𝑇 =

𝐵 1 − 𝐹𝑖,𝑗

(24)

𝑗 =1

3

MDOF Nonlinear System Reliability Assessment

The survival probability 𝑃𝑖𝐵 is defined as the probability that the system response amplitude 𝑎𝑖 stays below a prescribed barrier 𝐵 over the time interval [0, 𝑇], given that 𝑎𝑖 𝑡 = 0 < 𝐵. In the ensuing analysis, a recently developed approximate technique (Spanos and Kougioumtzoglou 2014) is employed for determining the survival probability and the first-passage PDF of the LTV system of Eq.(15). The first-passage PDF 𝑝𝑖𝐵 𝑇 and the survival probability 𝑃𝑖𝐵 are related according to the expression 𝑝𝑖𝐵 𝑇 = −

𝑑𝑃𝑖𝐵 𝑇 𝑑𝑇

(21)

Next, adopting the discretization scheme employed in (Spanos and Solomos 1983) yields intervals of the form

𝑇𝑒𝑞 ,𝑖 (𝑡 𝑖,𝑗 −1 ) 2

(22)

where Teq ,i represents the LTV system equivalent natural period given by 𝑇𝑒𝑞 ,𝑖 𝑡 =

2𝜋 𝜔𝑒𝑞 ,𝑖 𝑡

where is defined as the probability that the response amplitude 𝑎𝑖 will exceed the prescribed barrier B over the time interval t i,j−1 , t i,j , given that no crossings have occurred prior to time t i,j−1 . Next, invoking the Markovian property of the response amplitude 𝑎𝑖 , one gets 𝐵 𝐹𝑖,𝑗 = 𝑃𝑟𝑜𝑏 𝑎𝑖 𝑡𝑖,𝑗 ≥ 𝐵 ∩ 𝑎𝑖 𝑡𝑖,𝑗 −1 < 𝐵

𝑃𝑟𝑜𝑏 𝑎𝑖 𝑡𝑖,𝑗 −1 < 𝐵

(23)

=

𝐵 𝐻𝑖,𝑗 −1,𝑗

=

(25)

𝐵 𝐻𝑖,𝑗 −1

where ∩ denotes the intersection of sets. B Utilizing Eq.(18) Hi,j−1 can be determined as 𝐵 𝐵 𝐻𝑖,𝑗 −1 =

𝑝 𝑎𝑖,𝑗 −1 , 𝑡𝑖,𝑗 −1 𝑑𝑎𝑖,𝑗 −1 = 0

= 1 − 𝑒𝑥𝑝 −

𝑡𝑖,𝑗 −1 , 𝑡𝑖,𝑗 , 𝑗 = 1,2, … , 𝑚, 𝑡𝑖,0 = 0, 𝑡𝑖,𝑚 = 𝑇, 𝑡𝑖,𝑗 − 𝑡𝑖,𝑗 −1 =

B Fi,j

𝐵2 2𝑐𝑖 (𝑡𝑖,𝑗 −1 )

(26)

B whereas Hi,j−1,j is defined as a double integral of the form ∞ 𝐵 𝐻𝑖,𝑗 −1,𝑗 = 𝐵

𝑑𝑎𝑖,𝑗 × 𝐵

𝑝 𝑎𝑖,𝑗 −1 , 𝑡𝑖,𝑗 −1 ; 𝑎𝑖,𝑗 , 𝑡𝑖,𝑗 𝑑𝑎𝑖,𝑗 −1

(27)

0

Considering the slowly time-varying behavior of the response amplitude 𝑎𝑖 , it is assumed that it is constant over the time interval t i,j−1 , t i,j . In this regard, the survival probability PiB also has a constant value over the same time interval. Obviously, the survival probability is given by

5

Assuming next that βeq ,i t i,j = βeq ,i t i,j−1 and ωeq ,i t i,j = ωeq ,i t i,j−1 over the time interval t i,j−1 , t i,j , it has been shown in (Spanos and Solomos 1983, Spanos and Kougioumtzoglou 2014) that the joint response amplitude PDF p αi,j−1 , t i,j−1 ; αi,j , t i,j is given in the form


𝐿𝑖,𝑛 𝑁 2 𝑛 =1 2𝑛

𝑝 𝛼𝑖,𝑗 −1 , 𝑡𝑖,𝑗 −1 ; 𝛼𝑖,𝑗 , 𝑡𝑖,𝑗 = 𝛼𝑖,𝑗 −1 𝛼𝑖,𝑗 × 2 𝑐𝑖 𝑡𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗 1 − 𝑟𝑖,𝑗 2 2 𝛼𝑖,𝑗 𝑐 𝑡𝑖,𝑗 −1 + 𝛼𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗 𝑒𝑥𝑝 − 2 2𝑐𝑖 𝑡𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗 1 − 𝑟𝑖,𝑗

with ×

𝐿𝑖,𝑛 =

2 1 − 𝑟𝑖,𝑗

𝑐𝑖 𝑡𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗

𝑛+1

4𝑛 𝑐𝑖 𝑡𝑖,𝑗 −1

𝛼𝑖,𝑗 𝛼𝑖,𝑗 −1 𝑟𝑖,𝑗

𝐼0

(33)

(28)

𝛤𝑖 1 + 𝑛,

𝑛+2

×

2 2𝑐𝑖 𝑡𝑖,𝑗 −1 1 − 𝑟𝑖,𝑗

− 𝛤𝑖 1 + 𝑛, 0

where 2 𝑟𝑖,𝑗 =

2 𝑐𝑖 𝑡𝑖,𝑗 1 − 𝑟𝑖,𝑗 𝐵2

𝑐𝑖 𝑡𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗

1 − 𝛽𝑒𝑞 ,𝑖 𝑡𝑖,𝑗 −1 (𝑡𝑖,𝑗 −1 − 𝑡𝑖,𝑗 )

.

(29)

Next, I0 (. ) denotes the modified Bessel function of the first kind of order zero. I0 (. ) can be further expanded in the form 𝑥2 𝑥4 𝑥6 𝐼0 𝑥 = 1 + 2 + 2 2 + 2 2 2 + ⋯ 2 2 4 2 4 6

(30)

Considering Eqs.(27), (28) and (30) and manipulating yields N 𝐵 𝐻𝑖,𝑗 −1,𝑗

= Ai,0 +

Ai,n

(31)

n=1

+

𝐵2 𝑐𝑖 𝑡𝑖,𝑗

− 𝛤𝑖 1 + 𝑛,

𝐴𝑖,0 = 𝑒𝑥𝑝 1 − 𝑒𝑥𝑝

ii. × ×

2 2𝑐𝑖 𝑡𝑖,𝑗 −1 1 − 𝑟𝑖,𝑗

iii.

2 1 − 𝑟𝑖,𝑗

(32) iv.

and 𝐴𝑖,𝑛

2𝑛 𝑟𝑖,𝑗

= 𝑐𝑖 𝑡𝑖,𝑗 −1 𝑐𝑖 𝑡𝑖,𝑗

𝑛+1

2𝑛+1

1−

v.

×

2 𝑟𝑖,𝑗

6

2𝑐𝑖 𝑡𝑖,𝑗

−𝑛

𝛤𝑖 1 + 𝑛 𝐵2 1 − 𝑟𝑗2

𝐵2𝑛 (34)

Concisely, the developed technique comprises of the following steps: i.

2 2𝑐𝑖 𝑡𝑖,𝑗 1 − 𝑟𝑖,𝑗 2 −𝐵

2 1 − 𝑟𝑖,𝑗

𝛤𝑖 1 + 𝑛

In Eq.(34) Γi z represents the Gamma ∞ function defined as Γi z = 0 t z−1 e−t dt; and Γi [γ, z] represents the incomplete Gamma ∞ function defined as Γi [γ, z] = z t γ−1 e−t dt.

where −𝐵2

𝑛

2 1 − 𝑟𝑖,𝑗

− 𝑐𝑖 𝑡𝑖,𝑗

×

Determination of the MDOF system response variances (Eqs.(13-14)). Determination of the time-varying elements 𝛽𝑒𝑞 ,𝑖 𝑡 and 𝜔𝑒𝑞 ,𝑖 𝑡 by solving the 2x2 system of algebraic equations (Eqs.(16-17)). Determination of 𝑐𝑖 𝑡 via numerically integrating the ODE Eq.(20). Determination of the equivalent natural period 𝑇𝑒𝑞 ,𝑖 𝑡 (Eq.(23)) and discretization of the time domain via Eq.(22). Determination of the parameters 𝐵 𝐵 𝐻𝑖,𝑗 −1 and 𝐻𝑖,𝑗 −1,𝑗 via Eqs.(26) and (27).


vi.

4

Determination of the survival probability 𝑃𝑖𝐵 𝑇 via Eq.(24) and of the corresponding first-passage PDF 𝑝𝑖𝐵 𝑇 via Eq.(21). Numerical Application

In this section, a nonlinear three-degree-offreedom structural system following the BoucWen hysteretic model (Wen 1980, Ikhouane and Rodellar 2007) subject to evolutionary stochastic earthquake excitation is considered to demonstrate the reliability of the technique. The survival probabilities and the firstpassage PDFs obtained via the developed approximate technique are compared with survival probability and first-passage PDF estimates obtained via pertinent Monte Carlo simulations (10,000 realizations). A standard fourth-order Runge-Kutta numerical integration scheme is employed for solving the nonlinear system differential equation of motion (Eq.(1)), whereas the barrier level B is expressed as a fraction λ of the maximum value of the non-stationary response displacement standard deviation of the first floor, i.e. B = λmax⁡ ( c1 (t)). Further, considering relative displacements, the 3-DOF nonlinear structural system is governed by the equation 𝑴𝒒 + 𝑪𝒒 + 𝑲𝒒 + 𝒈 𝒒, 𝒒 = 𝑭 𝑡

(35)

where 𝐪T = q1 q2 q3 z1 z2 z3

(36)

and 𝐌=

𝐌𝟏𝟏 𝐌𝟐𝟏

𝐌𝟏𝟐 𝐌𝟐𝟐

(37)

with

𝐌𝟏𝟏

m1 m = 2 m3

0 m2 m3

0 0 m3

and 0 𝐌𝟏𝟐 = 𝐌𝟐𝟏 = 𝐌𝟐𝟐 = 0 0

0 0 0

0 0 0

(39)

In Eq.(38) m1 = 2.0615 × 105 , m2 = 5 2.0559 × 10 and m3 = 2.0261 × 105 . Further, 𝐊=

𝐊 𝟏𝟏 𝐊 𝟐𝟏

𝐊 𝟏𝟐 𝐊 𝟐𝟐

(40)

with 𝐊 𝟏𝟏 =

αk1 0 0

−αk 2 αk 2 0

𝐊 𝟏𝟐 (1 − α)k1 0 = 0

0 −αk 3 , αk 3

−(1 − α)k 2 (1 − α)k 2 0

(41)

0 −(1 − α)k 3 (42) (1 − α)k 3

and 0 𝐊 𝟐𝟏 = 𝐊 𝟐𝟐 = 0 0

0 0 0

0 0 0

(43)

In Eqs.(41) and (42) k1 = 3.9668 × 108 , k 2 = 3.5007 × 108 and k 3 = 2.6927 × 108 whereas α stands for the rigidity ratio which can be viewed as a form of post-yield to preyield stiffness ratio. Further, the 6-by-6 damping matrix of the structural system C is assumed to be partly proportional to the stiffness matrix; that is, 𝐂=

𝐂𝟏𝟏 𝐂𝟐𝟏

𝐂𝟏𝟐 𝐂𝟐𝟐

with 𝐂𝟏𝟏 = c𝐊 𝟏𝟏 and

7

(38)

(44)


0 𝐂𝟏𝟐 = 𝐂𝟐𝟏 = 0 0 1 C22 = 0 0

0 1 0

0 0 0

0 0, 0

and (45) 𝐊 𝐞𝐪 =

0 0 1

(46)

where c is taken equal to 2 × 10−3 . For the specific example γ = 𝑚𝑖 , and the loading vector becomes 𝐅(t)T = f t f t f t 0 0 0

(47)

Further, 𝐠 𝐪, 𝐪

𝐓

= ( 0 0 0 −g1 q1 , z1 … −g 2 q2 , z2 − g 3 q3 , z3 )

(48)

In the Bouc-Wen model the additional state zi is associated with the displacement qi via the equation z i = g i q i , zi

(49)

𝐊 𝐞𝐪𝟏𝟏 𝐊 𝐞𝐪𝟐𝟏

𝐊 𝐞𝐪𝟏𝟐 , 𝐊 𝐞𝐪𝟐𝟐

(54)

0 𝐊 𝐞𝐪𝟏𝟏 = 𝐊 𝐞𝐪𝟏𝟐 = 𝐊 𝐞𝐪𝟐𝟏 = 0 0 k eq 1 0 0

𝐊 𝐞𝐪𝟐𝟐 =

0 k eq 2 0

0 0 0

0 0 0

0 0

(55)

(56)

k eq 3

Further, the elements ceq i and k eq i are given by the expressions

ceq i =

2 E(qi zi ) γ + β E(zi 2 ) − A π 2 E(qi )

(57)

2 E(qi zi ) γ E(qi 2 ) + β π E(zi 2 )

(58)

and k eq i =

where g i q i , zi = −γ q i zi zi … + Αq i

n−1

− βqi zi

n

… (50)

The parameters γ, β, Α and n are capable of representing a wide range of hysteresis loops (Ikhouane and Rodellar 2007). In the study herein the following values are considered: a = 0.15, β=γ=0,5, n=1 and A=1. The equivalent linear matrices take the form (e.g. Roberts and Spanos 2003) 𝐂𝐞𝐪 =

𝐂𝐞𝐪𝟏𝟏 𝐂𝐞𝐪𝟐𝟏

𝐂𝐞𝐪𝟏𝟐 , 𝐂𝐞𝐪𝟐𝟐

0 𝐂𝐞𝐪𝟏𝟏 = 𝐂𝐞𝐪𝟏𝟐 = 𝐂𝐞𝐪𝟐𝟐 = 0 0 𝐂𝐞𝐪𝟐𝟏

ceq 1 = 0 0

0 ceq 2 0

0 0 0 0 0

0 0 0

respectively. Furthermore, the excitation EPS Sα ω, t takes the form Sα ω, t = g t

2

SCP ω

(59)

where SCP ω represents the widely used in earthquake engineering Clough-Penzien power spectrum (Clough and Penzien 1993) and g t denotes a time-modulating envelope function given by

(51)

g t = k e−at − e−bt

(52)

where a=0.1 and b=0.3; and k is a normalization constant so that g t max = 1. The Clough-Penzien spectrum is given by

(53)

ceq 3

8

SCP ω =

(60)


S0

(ω/ωf )4 1 − (ω/ωf )2 2 + 4ξf 2 (ω/ωf )2 ωg 4 + 4(ξg )2 ωg 2 ω2

(ωg 2 − ω2 )2 + 4ξg 2 ωg 2 ω2

× (61)

where S0 is the amplitude of the bedrock excitation spectrum, modeled as a white noise process. The parameters values used are S0 = 20m2 /sec 3 , ξg = 0.7, ωg = 2rad/sec, ξf = 0.6 and ωf = 12.5rad/sec (see Fig.1). The total duration of the excitation is 20 sec.

Figure 1.Nonstationary separable excitation power spectrum Sα ω, t

Figure 3: Time-varying equivalent linear damping βeq ,i t coefficients.

Figure 4: Time-varying survival probability of the system’s first degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

Figure 2: Time-varying equivalent linear stiffness ωeq .i t coefficients. Figure 5: First-passage PDF of the system’s first degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

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Figure 6: Time-varying survival probability of the system’s second degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

Figure 7: First-passage PDF of the system’s second degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

Figure 8: Time-varying survival probability of the system’s third degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

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Figure 9: First-passage PDF of the system’s third degree-of-freedom for various values of the parameter λ; comparisons with MCS (10,000 realizations).

In Figs. (2) and (3) the equivalent timevarying stiffness ωeq .i t and damping 𝛽𝑒𝑞 ,i 𝑡 elements corresponding to each DOF are plotted. Note that the hysteretic/degrading behavior of the structural system is captured by the decreasing with time trend of the stiffness element, as well as the increasing with time trend of the damping element. Further, in Figs. (4) and (5) the survival probabilities PiB T as well as the corresponding first-passage PDFs pBi T for the first DOF of the hysteretic MDOF structural system are plotted for various barrier levels; comparisons with MCS utilizing a spectral representation approach (Shinozuka and Deodatis 1991) (10,000 realizations) demonstrate a satisfactory degree of agreement. Furthermore, in Figs. (6), (7), (8) and (9) the survival probabilities PiB T as well as the associated first-passage PDFs pBi T corresponding to the second and third DOF are plotted for various barrier levels. Comparisons with MCS demonstrate a satisfactory degree of accuracy in these cases as well. 5

Conclusions

An approximate analytical technique for determining the survival probability and firstpassage PDF of nonlinear/hysteretic MDOF structural systems subject to evolutionary


stochastic excitation has been developed. Specifically, based on an efficient dimension reduction approach and relying on the concepts of stochastic averaging/linearization reliability statistics of the nonlinear MDOF system have been determined at a low computational cost. A significant advantage of the technique relates to the fact that it can readily treat a broad range of nonlinear/hysteretic behaviors as well as stochastic excitations with arbitrary EPS forms. Acknowledgments The first author acknowledges the financial support of the State Scholarships Foundation in Greece (IKY).

References Journal Papers Dahlhaus R., 1997. Fitting time series models to non-stationary processes, Ann Stat; 25: 1–37. Hammond J.K., 1973. Evolutionary spectra in random vibration, J Roy Stat Soc; 35: 167–188. Jangid R.S., Datta T.K, 1999. Evaluation of the methods for response analysis under nonstationary excitation, Shock and Vibration; 6: 285–297. Kougioumtzoglou I. A., 2013. Stochastic joint time-frequency response analysis of nonlinear structural systems, Journal of Sound and Vibration, vol. 332: 7153-7173. Kougioumtzoglou I. A., Spanos P. D., 2009. An approximate approach for nonlinear system response determination under evolutionary stochastic excitation, Current Science, Indian Academy of Sciences, vol. 97: 1203-121. Kougioumtzoglou I. A., Spanos P. D., 2012. An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators, Probabilistic Engineering Mechanics, vol. 28: 125-131. Kougioumtzoglou I. A., Spanos P. D., 2013. Nonlinear MDOF system stochastic response determination via a dimension reduction approach, Computers and Structures; 126: 135148.

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Priestley M.B, 1965. Evolutionary spectra and nonstationary processes. J. Roy Stat Soc; 27: 204– 237. Schueller G. I., Pradlwarter H. J., Koutsourelakis P. S., 2004. A critical appraisal of reliability estimation procedures for high dimensions, Probabilistic Engineering Mechanics, vol. 19: 463-474. Shinozuka M., Deodatis G., 1991. Simulation of stochastic processes by spectral representation, Applied Mechanics Reviews, vol. 44, no. 4, pp. 191-204. Spanos P.D, 1978. Non-stationary random vibration of a linear structure, International Journal of Solids and Structures, vol 14: 861867. Spanos P. D, Kougioumtzoglou I. A., 2014. Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation, Journal of Applied Mechanics, 81(5), 051016. Spanos P. D, Kougioumtzoglou I. A., 2012. Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination, Probabilistic Engineering Mechanics; 27: 57-68. Spanos P. D., Lutes L.D, 1980. Probability of response to evolutionary process, J. Eng Mech. Div. Am. Soc. Civil Eng., 106, 213-224. Spanos P. D., Solomos G. P., 1983. Markov approximation to transient vibration, Journal of Engineering Mechanics, vol. 109: 1134-1150. Wen Y. K., 1980. Equivalent linearization for hysteretic systems under random excitation, Journal of Applied Mechanics, vol. 47: 150154. Books Clough R. W., Penzien J., 1993. Dynamics of structures, McGraw-Hill. Ikhouane F., Rodellar J., 2007. Systems with hysteresis: analysis, identification and control using the Bouc-Wen model, John Wiley and Sons. Roberts J. B., Spanos P. D., 2003. Random Vibration and Statistical Linearization. New York: Dover Publications.


Edited Volumes Mitseas I.P., Kougioumtzoglou I.A., Beer M., Optimal design of nonlinear structures under evolutionary stochastic earthquake excitations, Proceedings of the Intnl Conf. on Engrg. and

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Appl. Science Optimiz., OPT-i 2014, Kos, Greece, 2014.