Seismic response analysis of nonlinear structures with uncertain parameters under stochastic ground motions

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Seismic response analysis of nonlinear structures with uncertain parameters under stochastic ground motions Jun Xua,b, De-Cheng Fengc,

T

⁎

a

Department of Structural Engineering, College of Civil Engineering, Hunan University, Changsha 410082, PR China Hunan Provincial Key Lab on Damage Diagnosis for Engineering Structures, Hunan University, Changsha 410082, PR China c Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing 210096, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Stochastic ground motions Nonlinear structures Uncertain parameters Probability density evolution method Rotational quasi-symmetric point strategy

An orthogonal expansion of stochastic ground motion model is incorporated into the probability density evolution method (PDEM). In this regard, a new methodology for seismic response analysis of nonlinear structures with uncertain parameters under stochastic ground motions can be formulated. The fundamentals and numerical algorithms of the methodology are introduced. Then, a rotational quasi-symmetrical point strategy (RQ-SPS) is developed for selecting the representative points in the random-variate space, which is of paramount importance to the tradeoff of accuracy and efficiency for the proposed methodology. An eight storey shear frame structure under seismic excitations, which exhibits strong nonlinear mechanical behaviors, is investigated. The effectiveness of the RQ-SPS is first verified, where the results by Monte Carlo simulations are also provided for comparisons. Then, case studies are implemented, in which the randomness involved in both stochastic ground motions and structural parameters are taken into account. The computational results demonstrate that the effect of randomness in structural parameters cannot be ignored compared to that in stochastic ground motions. Some features of the PDF evolutionary process of response are also discussed.

1. Introduction Performance-based seismic design or control is of paramount importance for engineers to design earthquake-resilient structures, where various parameters involved in seismic ground motions and structural properties should be treated as uncertain quantities [1,2]. On the other hand, it is almost inevitable that the engineering structures will experience nonlinearity when subjected to disastrous earthquakes [3–6]. In this regard, the randomness and nonlinearity need to be taken into account simultaneously for the performance-based seismic design [7], which provides a comprehensive understanding of structural behaviors under seismic excitations. Traditionally, the randomness is dealt with separately in stochastic dynamics of structures [8]. When the randomness in the modeling of structures is considered, the so-called stochastic finite element method (SFEM) or the random structural analysis [9,10] is established. Various approaches are available for calculating the response variability in the context of SFEM. With the development of SFEM, a variety of approaches such as the random perturbation technique [11], the path integral technique [12–15] and the orthogonal polynomial expansion method [9,16,17] have been well developed. However, great difficulties arise even for obtaining the second-order

⁎

statistics of response of strongly nonlinear structures under seismic excitations [18]. Although the Monte Carlo simulation(MCS) [19,20] or its variants [21–23] is versatile regardless of nonlinearity, the computational effort is always intractable in practical engineering. On the other hand, the method treating the randomness involved in seismic excitations is referred to as the random vibration method. Extensive developments in the random vibration method such as the pseudo-excitation method [24], the equivalent linearization method [25], the stochastic nonlinear equivalent method [26], the FPK equation method [27–29] and the Hamiltonian method [30], etc. have been well investigated. Unfortunately, the solution to multiple-degreeof-freedom (MDOF) nonlinear structures subjected to random seismic excitations is still an open challenge. To authors’ knowledge, the investigation of stochastic dynamics considering the randomness in both structural properties and external seismic excitations, which is also called the compound random vibration, is rather limited. The MCS method seems to be the only feasible method under this circumstance if the prohibitively large computational burden is ignored. However, it is widely accepted as a checking method for verification of a newly developed method. Recently, a new method named probability density evolution method (PDEM) has been well developed by Li and Chen [31,1,32,33] for nonlinear stochastic dynamic problems. This method treats the

Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (D.-C. Feng).

https://doi.org/10.1016/j.soildyn.2018.04.023 Received 19 December 2017; Received in revised form 19 March 2018; Accepted 20 April 2018 0267-7261/ © 2018 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng

where δkl denotes the Kronecker's delta, i.e.

randomness on a unified basis by invoking the random event description of the principle of preservation of probability and the embedded physical mechanism [8]. In this regard, the SFEM problems, the random vibration problems and the compound random vibration problems can be tackled in a unified framework by PDEM. Besides, the PDEM is capable of deriving the instantaneous probability density function (PDF), which is an intrinsic description of random dynamic systems; hence all the necessary information of the interested random dynamic system, e.g. statistical moments, reliability, etc. can be obtained without difficulty. It is therefore possible to conduct seismic response analysis of complex engineering structures taking into account the randomness in both ground motions and structural parameters for the performance-based design or control. The objective of the present paper is to develop a new methodology based on PDEM to carry out seismic response analysis of structures considering the randomness involved in both ground motions and structural parameters. The paper is arranged as follows. In Section 2, the stochastic ground motion model based on the orthogonal decomposition is first introduced. Section 3 devotes to providing the fundamentals of PDEM and its numerical algorithms, where the determination of representative points plays an important role in achieving the tradeoff of accuracy and efficiency. In Section 4, a new strategy is proposed for selecting the representative points in PDEM to conduct repeated deterministic dynamic response analyses. Numerical example is investigated to validate the effectiveness of the proposed methodology and case studies are also implemented in Section 4. Some features of the responses are observed and discussed. Concluding remarks are included in the final section.

1, k = l δkl = ⎧ ⎨ ⎩ 0, otherwise

Further, the correlated random vector Γ = [γ1 (ϖ ), γ2 (ϖ ), …, γN (ϖ )]T can be transformed to be the function of a set of standard uncorrelated random variables {ξ j (ϖ ), j = 1, 2, …N } such that N

Γ=

ρij =

ϕk (t ) ϕl (t ) dt = δkl

T

RX (τ ) ϕi (t1) ϕj (t2) dt1 dt2,

i, j = 0, 1, …, (N − 1)

(6)

N

γk (ϖ ) =

∑

λj ξ j (ϖ ) φj, k

(7)

j=1

where φj, k denotes the k-th element of Φj . Then, the stochastic process can be approximated by N

X (ϖ , t ) =

N

N

∑∑

λj ξ j (ϖ ) φj, k ϕk (t ) =

k=1 j=1

∑

λj ξ j (ϖ ) f j (t )

j=1

(8)

where N

f j (t ) =

∑ φj,k ϕk (t ) k=1

(9)

It is obvious that the functions {f j (t ), j = 1, 2, …, N } are orthogonal with each other with the time domain, i.e.

fi (t ), f j (t ) =

∫0

T

fi (t ) f j (t ) dt = δij

(10)

The series expansion in Eq. (8) is therefore referred to as the orthogonal expansion of a stochastic process. In many practical applications, the eigenvalues λj may quickly decrease to be zero with the increase of j, which means a large number of correlated random variables can be represented by a small number of uncorrelated ones. Thus, Eq. (8) can be reduced to be r

X (ϖ , t ) ≈

∑

λj ξ j (ϖ ) f j (t )

j=1

(11)

where r ⪡N . It is seen that the key issue is to determine the orthogonal base functions to efficiently implement the orthogonal expansion of a stochastic process. In this paper, the Hartley basis function is specifically adopted, i.e.

2⎞ 2kπt ⎞ ⎟ cas ⎛ ϕk (t ) = ⎜⎛ ⎝ T ⎠ ⎝ T⎠

(12)

where (1)

cas (t ) = cos (t ) + sin (t )

where the symbol ϖ represents the random nature of the corresponding quantity and {ϕk (t ), k = 1, 2, …, ∞} is a set of orthogonal base functions satisfying T

T

∫0 ∫0

where T is the time duration and RX (τ ) = RX (t1, t2) is the auto-covariance function. In this regard, the random variable γk (ϖ ) can also be written as

N

∫0

(5)

where the correlation coefficient ρij is

Consider a real-valued stochastic process, which is denoted as X (ϖ , t ) , with a zero mean and a finite second order moment. It is known that the stochastic process X (ϖ , t ) can be expanded by orthogonal expansion such that [51,52]

ϕk , ϕl =

(4)

R = [ρij ]N × N

2.1. Orthogonal expansion of stochastic processes

k=1

λj ξ j (ϖ )

where λj s and Φj s are the eigenvalues and standard eigenvectors of the correlation matrix R , i.e. RΦj = λj Φj and

Since Housner [34] first described earthquake ground motions as stochastic processes, extensive methods have been developed for rationally describing and modeling the stochastic seismic excitations. This research has spawned the development of methods rooted in two categories: the physical modeling and the mathematical expansion. In the physical modeling, the physical mechanism of an earthquake is incorporated to formulate the explicit expression of the random function, where the basic random variables are determined by the observed real data [35–39]. As for the mathematical expansion, the methods such as Karhunen-Loeve (KL) decomposition [40,41], the spectral representation method (SRM) [42–48], the stochastic harmonic function method [49,50] and the orthogonal decomposition method(ODM) [51], etc. have been well studied. As is known, a large number of random variables could be involved in the KL decomposition and the SRM, which may lead to significant numerical errors or infeasible computational efforts. In the present paper, the orthogonal decomposition method is specifically adopted to model the non-stationary seismic ground stochastic process because it is feasible to represent the stochastic process with only a few of random variables.

∑ γk (ϖ ) ϕk (t )

∑ Φj j=1

2. Stochastic ground motion model

X (ϖ , t ) =

(3)

(13)

2.2. Orthogonal expansion of non-stationary seismic ground motions Generally, the non-stationary seismic acceleration stochastic process can be expressed as [51]

(2) 150

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng

X¨ g (t ) = g (t )·X¨ (t )

parameters; and X¨ g (ξ , t ) is the aforementioned stochastic ground motion represented by orthogonal expansion. Further, if we denote Θ = [ζ , ξ ] = [Θ1, Θ2, …, Θd]T , where d is the total number of random variables involved in the structural dynamic system, Eq. (21) can be rewritten as

(14)

in which X¨ (t ) is the real zero-mean stochastic process with the power density function (PSD) S X¨ (ω) and g (t ) is the envelope function defined as 2 ⎧ (t / t1) , ⎪1, g (t ) = ⎨ exp [−c (t − t2)], ⎪ 0, ⎩

0 ≤ t < t1 t1 < t ≤ t2 t2 < t ≤ T t≥T

¨ (t ) + C (Θ) Ẏ (t ) + G (Y (t ) Ẏ (t ), Θ) = −M (Θ) IX¨ g (Θ, t ) M (Θ) Y

For a well-posed structural dynamic system, the solution to Eq. (23) is unique and continuously dependent on the random vector Θ, i.e.

(15)

Y (t ) = H (Θ, t )

where c is the coefficient of attenuation, t1 and t2 are the start and end time instants of the stationary portion of the ground motion. Usually, it is quite difficult to represent the seismic acceleration process via orthogonal expansion with a few of terms. Instead, the seismic displacement process can be first captured by orthogonal expansion. The earthquake displacement PSD, denoted as SX (ω) can be obtained by

SX (ω) = ω−4S X¨ (ω)

RX (τ ) =

∫−∞

SX (ω) exp (iωτ ) dω

Y˙ (t ) = h (Θ, t )

(16)

Z (t ) = ψ [Y (t ), Y˙ (t )] = HZ (Θ , t )

(17)

It is noted that all the randomness can be depicted by Θ; the augmented system (Z (t ), Θ) is consequently a probability preserved system where the joint PDF of (Z (t ), Θ) is denoted as pZΘ (z , θ, t ) . According to the principle of preservation of probability, we have [1]

D Dt

∑

λj ξ j (ϖ ) f j (t )

(18)

j=1

r

S0

∑

λj ξ j (ϖ ) Fj (t )

+

(19)

j=1

where N −1

N −1

Fj (t ) =

∑ ηk +1 φj,k +1 ϕ¨k (t ) = − ∑ k=1

k=0

(20)

where ηk s are defined as the harmonic modulated coefficients, which are given empirically [51]. Practically, if the stochastic ground motion is assumed to be a zeromean Gaussian process, the random variables ξ = [ξ1, ξ2, …, ξr ]T are independent normal random variables. The effectiveness of this stochastic ground motion model has been verified in Ref. [51].

t=0

= Y˙ 0

(29)

D p (z , θ, t ) dzdθ ∫ Dt Ωt × ΩΘ Z Θ ∂p (z , θ, t ) ∂p (z , θ, t ) ⎤ dzdθ + Z˙ (θ, t ) Z Θ = ∫Ω × Ω ⎡ Z Θ t Θ ⎢ ⎥ t ∂ ∂z ⎣ ⎦ =0

(30)

Considering the arbitrariness of Ωt × ΩΘ , we will have a partial differential equation that governs the evolving joint PDF, i.e.

∂pZ Θ (z , θ, t ) ∂t

(21)

+ Z˙ (θ, t )

∂pZ Θ (z , θ, t ) ∂z

=0

(31)

which is called the generalized density evolution equation (GDEE) [32,33,53]. Under the initial condition

with deterministic initial conditions

= Y0, Y˙ (t )

(28)

Then, Eq. (28) can be rearranged as

Without loss of generality, to determine the dynamic response of complex structures considering the double sources of randomness, the stochastic equation of motion could be written as follows:

t=0

∂pZ Θ (Z , θ, t ) dΘ ⎤ dZdθ ∂θ dt ⎥ ⎦

Θ˙ = 0

3.1. Fundamentals of PDEM

Y (t )

(27)

For a stochastic system, Θ is time invariant, i.e.

3. Probability density evolution method and its numerical algorithms

¨ (t ) + C (ζ ) Y˙ (t ) + G (Y (t ) Y˙ (t ), ζ ) = −M (ζ) IX¨ g (ξ , t ) M (ζ ) Y

pZΘ (z , θ, t ) dzdθ = 0

∂p (Z , θ, t ) ∂p (Z , θ, t ) + Z˙ (θ, t ) Z Θ = ∫Ω × Ω ⎡ Z Θ Θ ⎢ t ∂t ∂Z ⎣ . ∂p ( Z , θ , t ) ⎤ dZdθ + Θ ZΘ ⎥ ∂θ ⎦

2

⎛ 2kπ ⎞ ηk + 1 φj, k + 1 ϕk (t ) ⎝ T ⎠

Θ

t

D p (Z , θ, t ) dZdθ ∫ Dt Ωt × ΩΘ Z Θ D = ∫Ω × Ω [pZ Θ (Z , θ, t )] dZdθ Θ Dt t ∂p (Z , θ, t ) ∂p (Z , θ, t ) dZ + ZΘ · = ∫Ω × Ω ⎡ Z Θ Θ ⎢ t ∂ ∂Z t dt ⎣

where S0 is the spectral intensity factor. According to the relationship between the displacement process and the acceleration process, the earthquake acceleration process can be approximately expressed as

X¨ (ϖ , t ) ≈

∫Ω ×Ω

(26)

where Ωt and ΩΘ are the distribution areas of Z and Θ, respectively. The total derivative in Eq. (27) can be expanded as

r

S0

(25)

where h = ∂H/ ∂t . Usually, some one-dimension physical quantity, denoted by Z (t ) , is of special interest in practical engineering, which could be expressed as

In this regard, the correlation matrix (Eq. (5)) of the earthquake displacement process can be determined. Similarly, the orthogonal expansion of the earthquake displacement process can be approximated by [51]

X (ϖ , t ) ≈

(24)

and the corresponding velocity is assumed to take the form

and the corresponding auto-covariance function is +∞

(23)

(22)

pZ Θ (z , θ, t )

where and C are n × n mass and damping matrices, respectively; Y , Y˙ ¨ are the n × 1 node displacement, velocity and acceleration vecand Y tors, respectively; G is the nonlinear restoring force vector; I is the unit vector; ζ = [ζ1, ζ2, …, ζm]T are the random variables involved in structural

t = t0

= δ (z − z 0) pΘ (θ )

(32)

where δ is the Dirac's delta function, the GDEE (Eq. (31)) can be solved and the instantaneous PDF of response can be readily obtained such that 151

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng

pZ (z , t ) =

∫Ω

Θ

pZΘ (z , θ, t ) dθ

is actually evaluated; thus numerical methods suitable for the high-dimensional integral, usually called cubature formulas, could be applicable to this problem. It has been also demonstrated in Ref. [7] that the cubature formula with negative weights may yield spurious results for PDEM and only the cubature formula with all positive weights is preferred. The quasi-symmetric point strategy (Q-SPS) can fulfill the aim [64,7,61].

(33)

It is seen that the GDEE is just a one-dimensional partial differential equation, which holds for arbitrary degree-of-freedom (DOF) structural dynamic systems. This equation possesses some significant technical advances over the classical probability density evolution equations such as the FPK equation, the Liouville equation, etc. [54], which could be unfeasible to be solved for problems with a large number of DOFs. Besides, it is quite difficult to explicitly obtain the coefficient Z˙ (θ, t ) in GDEE; thus the analytical solutions only exist for some special cases. Since only a one-dimensional partial differential equation is involved, numerical algorithms could be of great necessity for general cases.

4.1. Quasi-symmetric point strategy As is mentioned, the Q-SPS is appropriate for the high-dimensional numerical integral such that

∫R

3.2. Numerical algorithms The numerical algorithms for obtaining the probabilistic response can be expressed as follows [1,55,56].

U0 = (βγ , 0, …, 0),

γ=

∂pZ Θ (z , θq, t ) ∂z

=0

t = t0

= δ (z − z 0) Pq

(34)

(35)

This equation can be numerically solved by the finite difference method, e.g. total-variation diminishing scheme. Step (4). Synthesize all the results obtained by Step (3) and finally give

∑ pZ Θ (z, θq, t ) q=1

(38)

d+2 2

χ=

d+2 d−2

(39)

q = 1, 2 .., 2d (βγ , 0, …, 0), uq = ⎧ k ⎨ ⎩ (βχ , βχ , …, βχ ), q = 2d + 1, 2d + 2 .., 2d + 2

Nsel

pZ (z , t ) =

U1 = (βχ , βχ , …, βχ )

and β is the permutation of ± 1. It is seen that a total of + 2d points are involved, where the set of 2d points (U1) having a product structure and the same weight and the set of 2d points (U1) locate on the coordinate axes. To reduce the number of points in (U1) without losing accuracy, the orthogonal array can be applied where a subset of points in (U1) could be adequate for numerical integration. In this regard, the subset of points in (U1) is called the quasi-symmetric points and this strategy for numerical integration is therefore named as the quasisymmetric point strategy (Q-SPS), which has 2k + 2d points, with k ≤ d . For simplicity, the exponent k as a function of dimension d is given in Table 1 for d ≤ 24 [64]. Therefore, the integration points in Q-SPS are

with the corresponding initial condition

pZ Θ (z , θq, t )

(37)

q=1

2d

Step (3). Introduce Z˙ (θq, t ) into the GDEE (Eq. (31)). By doing so, the discretized form of GDEE can be obtained such that

+ Z˙ (θq, t )

N

∑ aq J (uq)

where

Step (2). Substitute Θ = θq into Eq. (23), generate the time history of ground motion (Eq. (14)) and then solve the deterministic equation of motion via a time-domain numerical integration scheme. In this regard, the numerical solution Z˙ (θq, t ) can be obtained.

∂t

1 T e−u uJ (u) d u ≈ (2π )d /2

where Rd is the d-dimensional infinite region; u = (u1, u2, …, ud ) is the standard normal vector; J is the integrand; aq s and u q s are the constant weights and the selected discrete points, respectively. In Q-SPS, the selected discrete points can be determined by the following two fully symmetric point sets, i.e.

Step (1). Select a set of representative points in the random-variate space ΩΘ , which are denoted as θq = {θ1, q, θ2, q, …, θd, q, q = 1, 2, …, Nsel} , where Nsel is the total number of points. The corresponding assigned probabilities can be specified accordingly, which are defined as Pq s, q = 1, 2, …, Nsel .

∂pZ Θ (z , θq, t )

d

(40)

The corresponding weights can be given as [64]

(36)

4/(d + 2)2 , q = 1, 2 .., 2d aq = ⎧ 2 /[2k (d + 2) 2], q = 2d + 1, 2d + 2 .., 2d + 2k ⎨ − ( d 2) ⎩

By using the previous numerical algorithms, seismic response of complex structures with random parameters subjected to stochastic ground motions can be estimated. Actually, this is the called the “point evolution” scheme of PDEM, whereas the “ensemble evolution” scheme, which provides the solution with higher accuracy, is still being studied [57]. It is noted that the determination of representative points together with their assigned probabilities plays an important role in achieving the tradeoff of accuracy and efficiency for estimating the evolutionary PDF. Although some smart strategies have been well developed [58–60,7,61,53,62,18], the investigation on this subject is still of paramount significance. To this end, a new strategy will be developed for seismic response analysis of complex structures with uncertain parameters under stochastic ground motions based on PDEM.

(41)

2d + 2k ∑q = 1

aq = 1. Obviously, we have aq > 0 and Then, if Q-SPS is applied for PDEM, the integration points is taken as the representative points and the weights are considered as the assigned probabilities, i.e. [7] −1 ⎧θq = T (u q),q = 1, 2, …, N sel = P a ⎨ q, ⎩ q

(42)

T −1

denotes the inverse Rosenblatt/Nataf Transformation and where Nsel = 2k + 2d . This strategy for the determination of representative points and assigned probabilities has been successfully applied to PDEM to carry out stochastic response analysis of structures. Nonetheless, due to the

4. Points selection in PDEM based on rotational quasi-symmetric point strategy

Table 1 Exponent k as a function of dimension d.

Since non-normal random variables can be transformed to be standard normal ones by some transformations, e.g. Rosenblatt/Nataf transformation [63], the determination of representative points and assigned probabilities in the standard normal random-variate space is of special interest. It is seen from Eq. (33) that a high-dimensional integral 152

d

3–5

6

7

8–9

10–16

17–20

21–24

k

d

5

6

7

8

9

10

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng

Uiρ =

0⎤ ⋮⎥ 0⎥ i − th row ⋮⎥ ⎥ j − th row 0⎥ ⋮⎥ 1⎦

j=0

⎝ ⎠

ρ

(45)

ρ

j=0

⎝ ⎠

(49)

(50)

ρ

∫U (ui + i

bi ) ρl pUi (ui ) dui ,

i = 1, 2, …, d

(51)

where l = 1, 2, …Nl , Nl is the number of fractional moments. Also, the marginal fractional moments can be estimated by RQ-SPS such that

d

where RT (α ) = ∏i = 1 ∏ j = i + 1 Wij (αi, j ) is the rotational matrix and a total of d (d − 1)/2 times of plane rotations are involved. Besides, it can be proven that the rotational matrix is an orthogonal matrix such that [61]

RT = 1

∞

∑ ⎜⎛ j ⎟⎞ Uiρ −j E [(Ui − Ui) j]

μ Λil = E [Λ i l]=

Wij (αi, j ) u q = RT (α ) u q d

(48)

is a constant vector, which ensures that Λ where B = { b1 , b2 , …, bd a positive random vector and b1, b2 , …, bd are the truncated boundaries for random variables U1-Ud . Then, the exact marginal fractional moments of Λ reads

(44)

d

i=1 j=i+1

RTT RT = I ,

ρ

}T

In this regard, the rotation of the point in the space can be extended d

∞

Λ=U+B

as

∏∏

(47)

When ρ is a real number, the binomial coefficient will never become zero. It could be observed that a single fractional moment actually contains the information of a large number of central moments. In this regard, a few number of marginal fractional moments could be able to sufficiently characterize the marginal PDF and its tail distribution. Therefore, if the RQ-SPS can provide very accurate marginal fractional moments of U , it is anticipated that very good results of probabilistic response could be obtained by PDEM with RQ-SPS. Therefore, the basic idea to formulate the RQ-SPS is to minimize the relative errors of marginal fractional moments of U . To avoid the fractional operation of a negative random variable, a translation is required such that

where

∼ u q (α ) =

i = 1, 2, …, d

∑ ⎜⎛ j ⎟⎞ Uiρ −j (Ui − Ui) j

μUρi = E [Uiρ] =

(43)

0 0 … … … ⋱ ⋮ ⋮ … cos α … −sin α … ⋮ ⋱ ⋮ … sin α … cos α … ⋮ ⋮ ⋱ 0 0 … … … thcolumn j − th column

uiρ pUi (ui ) dui ,

where Ui denotes the mean value of random variable Ui . Then, Eq. (47) can be rewritten as

Before we proceed, we would like to introduce how to rotate a point in high-dimensional spaces by the Givens transform [65]. Let us start from rotating the point u q = {u1, q, u2, q, …, ud, q} in the (i, j ) plane by an angle α (in rad) counterclockwise. The point after rotation, denoted as ∼ u q , gives

⎡1 ⎢⋮ ⎢0 ⎢⋮ Wij (α ) = ⎢ ⎢0 ⎢⋮ ⎣0 i−

i

where ρ is the fractional order, which is a real number, and μiρ is the marginal fractional moment of U in the i-th dimension. Perform the Taylor series expansion over Uiρ about its mean yields

4.2. Marginal fractional moments based rotational quasi-symmetric point strategy

∼ u q = Wij (α ) u q

∫U

μUρi = E [Uiρ] =

quasi-symmetry and sparseness of Q-SPS, the information of the marginal probability density function may be not captured sufficiently [61], especially for the tail distribution, which may decrease the accuracy of PDF evaluation by PDEM. To improve the accuracy, the basic idea is to rotate the integration points in Q-SPS. Hence, the strategy, which could better reproduce the marginal probabilistic information, is of great interest for seismic response analysis of random structures under stochastic ground motions by PDEM. To this end, a new rotational quasisymmetric point strategy will be introduced.

ρ

μ͠ Λli =

(46)

Nsel

∑ aq (∼ui,q +

Nsel

bi ) ρl =

q=1

∑ aq [R (α ) uq + q=1

bi ]ρl (52)

where ui, q represents the i-th coordinate of the point u q . To optimally determine the rotational angle vector α , the objective function could be adopted as the maximum relative error of marginal fractional moments, i.e.

where RT is the determinant of RT . It should be emphasized that the associated weight still keeps the same after the rotation. Thus, the crux for obtaining the points after rotation is to specify the rotational angle vector α = {α1,2, α1,3, …, αd − 1, d} . Since the quasisymmetric points are rotated herein, the resulting points after rotation is called the rotational quasi-symmetric points and this strategy for numerical integration is named as the rotational quasi-symmetric point strategy (RQSPS). As is mentioned, the marginal probabilistic information needs to be better reproduced by using the RQ-SPS in PDEM. Thus, the rotational angle vector needs to be optimally determined, where the indices related to the marginal probabilistic information are employed to formulate the objective function [18]. Besides, such indices need to be computed efficiently even in high dimensions. In the present paper, the marginal fractional moments of input random vector is employed to formulate the objective function to find the optimal angle vector. This is because a single fractional moment actually contains the information about a large number of central moments [66]. It is known that the marginal probabilistic information could be related to the high-order central moments [18]. Thus, the marginal fractional moments could be adopted as the appropriate indices to formulate the objective function. Besides, it is quite easy to calculate the marginal fractional moments of input random vector even in high dimensions. The marginal fractional moment of random vector U in the i-th dimension can be expressed as

ρ

Q (α ) =

μ͠ Λli (α ) − μ ρΛli μ ρΛli

i = 1, 2, …, d l = 1, 2, …, Nl

∞

(53)

where . ∞ denotes the infinite norm. Therefore, the following optimization problem needs to be dealt with, i.e.

find α min Q (α ) s. t . 0 ≤ αi, j ≤ 2π ,

∀ α i, j ∈ α

(54)

This optimization problem can be solved by using some global optimization algorithms, such as the genetic algorithm. Once the rotational angle vector α is obtained, the resulting rotational points could be obtained without difficulty. As is mentioned, the assigned probability associated with each point after the rotation is still the same with that before the rotation. Then, the representative points and their assigned probabilities are 1 ∼ ⎧θ͠ q = T (u q), q = 1, 2, …, Nsel ⎨ Pq = aq , ⎩

153

(55)

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng 0.04

PDEM(RQ−SPS) MCS

Mean(m)

0.02 0 −0.02 −0.04 −0.06 0

5

10

15

20

Time(sec) 2

x 10 PDEM(RQ−SPS) MCS

Std.D(m)

1.5 1 0.5

Fig. 1. The 8 storey shear frame structure.

0 0

5

10

15

Time(sec)

In this regard, the points θ͠ q s together with the assigned probabilities Pq s will be employed and verified first in PDEM to implement seismic response analysis of structures with uncertain parameters under stochastic seismic ground motions.

Fig. 2. Mean and standard deviation comparisons.

5. Numerical example It is mentioned that the RQ-SPS plays an important role if applied to PDEM, which affects the accuracy and efficiency. Therefore, the effectiveness of RQ-SPS needs to be first verified for an MDOF nonlinear structural dynamic problems with uncertainties. 5.1. Case 1: validation of RQ-SPS in PDEM Consider an 8-DOF planar shear frame structure with uncertain parameters subjected to deterministic ground motion for this purpose, which is shown in Fig. 1. The lateral inter-storey stiffness of each storey are assumed to be independent random variables, whose distribution properties are listed in Table 2. The deterministic masses m1 to m8 are 1.8, 1.8, 1.8, 1.8, 1.8, 1.8, 2.2 and 2.2 (× 105 kg), respectively. Rayleigh damping C = κ1 M + κ2 K is considered, where κ1 = 0.4591 and κ2 = 0.0041. First, the El Centro accelerogram (N-S component) is adopted as the deterministic excitation. Besides, the extended Bouc-wen model is employed to describe the nonlinear behavior, where the parameters in this model are αB = 0.05, AB = 1, nB = 1, qB = 0.25, pB = 1000 , d ψ = 5, λB = 0.5, ψB = 0.05, βB = 40 , γB = 35, d ν = 2000 , d η = 2000 and ζB = 0.99, where the subscript “B” denotes for Bouc-wen model. In this regard, seismic response analysis of structures with uncertain parameters under deterministic ground motion is investigated. In this case, a total of 8 independent random variables are involved and thus 144 representative points and their assigned probabilities are generated by the proposed Q-SPS. Simultaneously, Monte Carlo simulations (MCS)(20,000 runs) are carried out for comparisons. Fig. 2 shows the mean and the standard deviation of the first-storey drift by PDEM with Q-SPS and MCS, respectively. It can be observed that the results by PDEM with the proposed Q-SPS accord very well with those by MCS, indicating the efficacy of Q-SPS in PDEM for statistical moments assessment. By adopting PDEM, one can capture the PDF Table 2 Distribution information of lateral inter-storey stiffness. Stiffness 7

Mean (× 10 kN/m) Coefficient of variation Distribution

k1

k2

k3

k4

k5

k6

k7

k8

1.4 0.2 LN

1.4 0.2 LN

1.4 0.2 LN

1.4 0.2 LN

1.4 0.2 LN

1.4 0.2 LN

1.6 0.2 LN

1.6 0.2 LN

Fig. 3. PDF surfaces in typical time intervals.

Note: LN = lognormal. 154

20

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300 250

Acceleration(m/s2)

J. Xu, D.-C. Feng

PDF at 1.8 sec PDF at 3.2 sec PDF at 5.1 sec

0 −2 0

150 Acceleration(m/s2)

PDF

200

2

100 50

−0.05

−0.04 −0.03 −0.02 Inter−storey drift(m)

−0.01

(a) 1 0.9 0.8 0.7

CDF

0.6

1.8s

5.1s

6

8 10 Time(sec)

12

14

16

2

4

6

8 10 Time(sec)

12

14

16

2

4

6

8 10 Time(sec)

12

14

16

5

0

−5 0

2 1 0 −1 −2 0

Fig. 5. Typical time histories.

3.2s

medium soil (Site II) is considered and the earthquake acceleration PSD is represented by a modified Kanai-Tajimi model is adopted such that [67,68]

0.5 0.4 0.3

S X¨ (ω) =

0.2 0.1 0 −0.06

4

0 Acceleration(m/s2)

0 −0.06

2

PDEM(RQ−SPS) MCS −0.05

−0.04

−0.03

−0.02

−0.01

ωg4 + 4ςg2 ωg2 ω 2 1 ω4 · · S0 1 + (Dω)2 (ω2 + ω02)2 (ω2 − ωg2)2 + 4ςg2 ωg2 ω 2

(56)

where the values of parameters in Eq. (56) and the envelope function (Eq. (15)) are listed in Table 3. The number of truncated term is r = 10 and therefore only 10 independent standard normal random variables are considered in this case. By using the advocated RQ-SPS, a total of 276 representative points can be generated. Then, the corresponding time histories of the acceleration processes can be obtained accordingly. Fig. 5 pictures three typical time histories of the acceleration processes. It could be observed that not only the amplitudes but also the frequencies exhibit obvious non-stationarity, which reveals that the stochastic excitation is a non-stationary process. By using the PDEM with RQ-SPS, one can obtain the probabilistic response of the first inter-storey drift for Case 2. Fig. 6 shows the PDF surface and contour during the time interval [4.5,11]s. Besides, the PDF curves at three typical time instants are shown in Fig. 7. From these figures, it is seen that the PDF evolution process of the inter-storey drift is obvious a non-stationary process. Unlike the PDFs in Case 1 (Fig. 4a), the PDFs are symmetrical about zero, which manifests that the mean of the inter-storey drift is zero when deterministic structure is driven by stochastic seismic excitations. Clearly, the response of random structures under deterministic excitation is not a zero-mean process(see Fig. 2). Further, if the shear force of the first storey is concerned, its PDF evolution process could be also captured by PDEM with RQ-SPS, which is shown in Figs. 8 and 9. It can be seen that the PDF evolution process of shear force process is still symmetrical and non-stationary. An interesting phenomenon is also observed herein that, the PDF of shear force could be bi-modal. As is known, some regular uni-modal

0

Inter−storey drift(m)

(b) Fig. 4. PDFs and CDFs at three typical time instants.

evolution process of the first inter-storey drift. Fig. 3 shows the PDF surfaces in the time intervals [3,5]s and [10.5,13.5]s, respectively, which indicate that the evolution process of a random structure under deterministic seismic excitation is quite complicated and irregular. Fig. 4a shows the PDF curves at three typical time instants, say 1.8 s, 3.2 s and 5.1 s. The corresponding cumulative density function (CDF) curves can be also obtained in Fig. 4b, where the results by MCS are also plotted. Remarkably, it is seen that the CDF curves by PDEM with the proposed RQ-SPS and MCS have very good agreements with each other, which further demonstrates the effectiveness of the proposed RQ-SPS in PDEM for evaluating the probabilistic response. 5.2. Case 2: deterministic structure under stochastic seismic excitations In this case, all the structural parameters are assumed to be deterministic, where the coefficients of variation in Table 2 are all 0. The aforementioned stochastic ground motion model is adopted as the stochastic seismic excitation. In this regard, only the randomness in the excitation is considered. In the stochastic ground motion model, the Table 3 Parameters in the PSD. Item

S0 (cm2/s3)

ωg (rad/s)

ςg

ω0 (rad/s)

D (s)

T(s)

t1 (s)

t2 (s)

c

Value

107.50

15.71

0.72

1.83

1/(28π )

17

0.8

7.0

0.35

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(a) PDF surface

(a) PDF surface

0.05

60

Shear force (N)

Inter−storey drift (m)

5

−6

x 10

40

0

20 −0.05

x 10 4

5

3 0

2 1

−5 5

6

7

8 Time (sec)

9

10

11

5

6

(b) PDF contour

7

8 Time (sec)

9

10

11

(b) PDF contour

Fig. 6. PDF evolution process of inter-storey drift (Case 2).

Fig. 8. PDF evolution process of shear force (Case 2). 80 70 60

PDF at 4.1 sec PDF at 6.2 sec PDF at 8.5 sec

−6

5 4

50 40

PDF

PDF

x 10

30 20

PDF at 4.1 sec PDF at 6.2 sec PDF at 8.5 sec

3 2

10 0 −0.05

1 0 Inter−storey drift (m)

0.05

0 −1

Fig. 7. PDF curves of inter-storey drift at three typical time instants (Case 2).

distribution models, e.g. normal distribution, lognormal distribution, Weibull distribution etc. could be widely found in literatures to represent the PDF of stochastic response for reliability analysis. However, as is revealed in Fig. 9, spurious results could be achieved if these unimodal distributions are employed to characterize the PDF of shear force.

−0.5

0 Shear force (N)

0.5

1 6

x 10

Fig. 9. PDF curves of shear force at three typical time instants (Case 2).

Fig. 11. Since the PDF evolution process is still symmetrical, it could be concluded that the stochastic seismic excitation contributes more to the probabilistic response than the randomness in structural parameters. Comparing Fig. 11 with the counterpart in Fig. 6b, we can see that the peak value of PDF evolution process becomes larger, where the peak value in Figs. 11 and 6b are around 125 and 80, respectively. These phenomena indicate the distribution range of the inter-storey drift in Case 3 is narrowed due to the existence of randomness in structural parameters. To make this point much clearer, Fig. 12 shows the PDF comparison of the inter-storey drift at the time instant 8.5 s. The most obvious feature one can identify is that the distribution ranges and the peak values of the PDFs of Case 3 and Case 2 are different, which demonstrates the randomness in structural parameters cannot be ignored. Pictured in Fig. 13 are the comparisons of mean and standard deviation of the first inter-storey shear force in Case 3 and Case 2.

5.3. Case 3: random structure under stochastic seismic excitations In this case, the randomness in structural parameters (Table 2) and stochastic seismic excitations are simultaneously taken into account. In this regard, a total of 18 independent random variables are considered and thus 548 representative points are generated by using the RQ-SPS. In Fig. 10 shown are the comparisons of the statistical moments of the first inter-storey drift in Cases 2 and 3. Obviously, the PDF evolution process in Case 3 is still a zero mean process, whereas the standard deviation is smaller than that of Case 2. The PDF contour is pictured in 156

Soil Dynamics and Earthquake Engineering 111 (2018) 149–159

J. Xu, D.-C. Feng 4

0.01

Case 3 Case 2

0.005

2

x 10

Case 3 Case 2

Mean (N)

Mean (m)

1 0 −0.005

0 −1

−0.01 0

2

0.04

6

8 10 Time (sec)

12

14

16

−2 0

Case 3 Case 2

0.03

0 0

2

4

6

8 10 Time (sec)

12

14

6

8 10 Time (sec)

12

14

16

4

6

8 10 Time (sec)

12

14

16

Case 3 Case 2

0.05

2 1

16

0 0

Fig. 10. Mean and standard deviation of inter-storey drift comparisons.

100

5

Shear force (N)

60 40 20

−0.05 6

7

8 Time (sec)

9

10

−6

x 10

x 10

80 0

2

Fig. 13. Mean and standard deviation of shear force comparisons.

120

5

4

x 10

3

0.02 0.01

Inter−storey drift (m)

2 5

4

Std.D (N)

Std.D (m)

4

11

4

5

3 0 2 −5

1

Fig. 11. PDF contour of inter-storey drift (Case 3). 5 120

Case 3 Case 2

100

6

7

8 Time (sec)

9

10

11

Fig. 14. PDF contour of shear force (Case 3).

PDF

−6

80

1.6

60

1.4

x 10

Case 3 Case 2

1.2 40

PDF

1

20 0 −0.04

0.8 0.6

−0.02 0 0.02 Inter−storey drift (m)

0.04

0.4 0.2

Fig. 12. PDF comparison of inter-storey drift at the time instant 8.5 s.

0 −1.5

Likewise, the shear force is still a zero-mean stochastic process in Case 3, however, the standard deviation could be enlarged when the randomness in structural parameters is considered. In this regard, the distribution range of the PDF contour of the shear force is also enlarged compared to that of Case 2, which is shown in Fig. 14. This feature could be noticed more clearly in Fig. 15 where the PDFs at the time instant 6.1 s in Case 3 and Case 2 are simultaneously pictured. In addition, with consideration of the randomness of structural parameters, the bi-modal PDF of shear force may change to be a multi-modal one. These properties of shear force show that the probabilistic information of the internal force is quite different from that of the inter-storey drift. As a matter of fact, they are in contrast with each other. Remarks: It is noted that in the proposed methodology, the

−1

−0.5 0 0.5 Shear force (N)

1

1.5 6

x 10

Fig. 15. PDF comparison of shear force at the time instant 6.1 s.

randomness inherent in both seismic ground motions and structural parameters is considered and tackled for stochastic seismic response analysis of nonlinear structures, which could be a very difficult task in the framework of classical random vibration theory. Besides, although the proposed RQ-SPS has been combined with the maximum entropy method in Ref. [66] for structural reliability analysis, its applicability for MDOF nonlinear structural dynamic problems has not been addressed. In this regard, the effectiveness of RQ-SPS has been verified as a proper strategy for selecting the representative points in PDEM. Then, 157

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stochastic seismic response analysis of nonlinear structures can be readily performed based on PDEM with RQ-SPS. Third, it is always considered that the randomness in structural parameters could be ignorable compared to that in stochastic dynamic excitations. In this study, the contributions of different sources of randomness to the seismic dynamic response are studied. It is revealed that both the randomness in seismic excitations and structural parameters should be simultaneously taken into account to achieve a comprehensive understanding of structural behaviors for performance-based design or control.

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6. Concluding remarks This paper has presented a new methodology for seismic response analysis of nonlinear structures with uncertain parameters subjected to stochastic ground motion. Actually, this methodology has been established based on the probability density evolution method(PDEM), in which an orthogonal expansion model of stochastic ground motion has been employed and a new rotational quasi-symmetric point strategy has been proposed for the selection of representative points in the randomvariate space. The numerical algorithms have been also outlined. A planar frame structure under seismic excitations, which exhibits strong nonlinearity, has been investigated in detail. Three cases including: Case 1: only considering the randomness in structural parameters, Case 2: only considering the randomness in stochastic ground motions and Case 3: considering the randomness in both structural parameters and stochastic ground motions, have been implemented. The following conclusions could be reached. (1). The proposed RQ-SPS is effective in PDEM for nonlinear structural dynamic problems with uncertainties. Both the statistical moments and the probabilistic information can be efficiently and accurately captured by PDEM with RQ-SPS. (2). When only the randomness in structural parameters is considered, the PDF evolution of response is a non-zero mean and unsymmetrical process. On the contrary, the PDF evolution process of response is a zero mean and symmetrical one once the stochastic ground motions are applied. (3). It seems that the randomness in stochastic seismic excitations plays much more important role than the randomness in structural parameters. Nevertheless, the randomness in structural parameters cannot be ignored, which may result in significant change on the evolutionary PDF. (4). When Case 3 is compared with Case 2, it is observed that the randomness in structural parameters may narrow the distribution range of inter-storey drift, whereas the distribution range of shear force may exhibit the opposite characteristic. The applications of the proposed methodology to practical largescale engineering structures will be further investigated. Acknowledgments The research reported in this paper is supported by the National Natural Science Foundation of China (Grant No. 51608186), the Fundamental Research Funds for the Central Universities (No. 531107040890) and the National Natural Science Foundation of Hunan Province (No. 2017JJ3016). The support is gratefully appreciated. The reviewers are highly appreciated for their constructive comments to improve the original manuscript. References [1] Li J, Chen JB. Stochastic dynamics of structures. Singapore: Wiley; 2009. [2] Xu J, Kong F. An adaptive cubature formula for efficient reliability assessment of nonlinear structural dynamic systems. Mech Syst Signal Process 2018;104(104):449–64. [3] Feng D, Kolay C, Ricles JM, Li J. Collapse simulation of reinforced concrete frame structures. Struct Des Tall Spec Build 2016;25(12):578–601.

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