Sub-critical excitations of SDOF elasto-plastic systems

International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108 www.elsevier.com/locate/nlm

Sub-critical excitations of SDOF elasto-plastic systems Siu-Kui Au ∗ Department of Building and Construction, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 6 June 2006; received in revised form 10 October 2006; accepted 20 November 2006

Abstract The critical excitation of a dynamical system is defined as the excitation that drives the system from one state to another with minimum energy. It plays an important role in both deterministic and stochastic problems of vibrations. For linear-elastic systems it can be directly obtained by calculus of variation, but the approach is not applicable to general non-linear-hysteretic systems. For single-degree-of-freedom (SDOF) elasto-plastic systems, the critical excitation has been found recently using a time-domain parameterization scheme, which also suggested the existence of ‘sub-critical excitations’ stemming from the local optima of the associated optimization problem. This paper presents a study of the sub-critical excitations based on the theoretical background laid out in the previous work. The sub-critical excitations are investigated in terms of their time-domain characteristics, energy, abundance and distribution. It is found that sub-critical excitations exist in abundance and their number grows in a combinatorial manner with the target duration. When mapped on a polar plot relative to the critical excitation, their distribution exhibits structures of progressively fine scale as the target duration increases. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Critical excitation; Elasto-plasticity; Hysteretic; Sub-critical excitation

1. Introduction Single-degree-of-freedom (SDOF) elasto-plastic systems have been frequently studied in the seismic response of structures subjected to severe earthquake loads and they provide insights into multi-degree-of-freedom (MDOF) systems [1–8]. Let y(t) denote the displacement response of an elasto-plastic system. Its dynamics is governed by y(t) ¨ + 2y(t) ˙ + Fr (y) = f (t),

(1)

where  is the viscous damping ratio,  is the elastic natural frequency (in rad/s), f is the excitation and Fr is the restoring force. The restoring force is given by 2 y before first yielding at y = ±b0 and is equal to ±2 b0 during plastic loading in the positive/negative direction. In the context of this study, the critical excitation is the one that drives the response y from rest to the threshold level bF at a specified time instant tF with minimum energy. The energy ∗ Tel.: +852 2194 2769; fax: +852 2788 7612.

E-mail address: [email protected]. 0020-7462/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2006.11.005

E of an excitation f is defined by  1 tF E(f ) = f (t)2 dt. 2 0

(2)

Utilizing critical excitations for solving dynamic problems was independently advocated by Drenick [9] and Papoulis [10]. Critical excitations are relevant in both deterministic and stochastic problems of vibrations. In the deterministic context, they are related to the control law with minimum energy that drives a system from one state to another [11]. In a stochastic context, they are the most probable excitation in the first passage failure region of the standard normal load space for systems subjected to white noise excitations. They provide bounds for maximum response under constraints in the input energy. They have also been used as ‘design points’ for constructing importance sampling densities for efficient estimation of first passage failure probabilities (e.g., [12–15]). Specifically, in the standard Gaussian load space the importance sampling density can be constructed as a weighted sum of Gaussian distributions centered among the design points with a view to accounting for the probability contribution from their neighborhood. Such a class of importance sampling densities has been shown to be

1096

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

applicable in high dimensions (i.e., with a large number of random variables, as featured by first-passage problems) provided that the covariance matrices of the centered Gaussian distributions are properly chosen [16]. In addition, important insights can be gained from the critical excitations and their dependence on system parameters of interest. When the system is linear-elastic, the critical excitation can be obtained readily using the calculus of variation. Other studies have been devoted to the linear problem under the context of random vibrations with constraints in the power spectral density function and with non-stationary characteristics [17–19]. Critical excitations for general non-linear systems are much more difficult to obtain than for linear systems because the calculus of variation cannot be applied directly. Most studies focused on SDOF systems, but even in this case exact solutions have not been obtained. Iyengar [20] studied non-linear-elastic systems and provided bounds for their maximum response under deterministic and stochastic inputs. Westermo [21] considered maximizing the input energy density (i.e., work done by excitation per cycle) within the class of excitations spanned by the displacement and velocity response. Approximate periodic solutions were obtained numerically for the critical response of SDOF elasto-plastic systems. The relationship between resonance frequency and response amplitude was numerically obtained, which exhibited complex behavior such as multiple critical harmonics not observed in linear systems. Equivalence linearization [22] has also been used for approximate solution in the frequency domain [23–25]. Koo and co-workers [15] recently showed that for non-linear-elastic systems at sufficiently large first passage time the critical excitation is identical to one that generates the mirror image of the free-vibration response when the system is released from the target level. This result also holds for linear-elastic systems and is a consequence of the elastic nature of the restoring force. However, this result does not apply hysteretic systems. Au [26] recently obtained the critical excitation for SDOF elasto-plastic systems using a time-domain parameterization scheme that allows for its efficient solution. The solution exhibits characteristics that are different from their linearelastic counterparts. The critical excitation first drives the system from rest to achieve the maximum amount of elastic strain energy, which prepares the system to make plastic deformations. The target threshold level is reached by accumulating successive plastic displacements that are driven by positive pulses developed in the critical excitation. The study in [26] focused on the critical excitation whose energy is globally minimal among all possible excitations that can reach the target threshold within the designated duration. It nevertheless indicates the possibility of other local optimal solutions, referred as ‘sub-critical excitations’ in this work. Investigation of these sub-critical excitations can provide additional insights into the first-passage failure of the system. They may also be used as design points for constructing the importance sampling distribution, supplementing the critical excitations for further possible variance reduction. This is relevant because the neighborhood of the critical excitations may not account fully for the first-passage failure probability content; it is possible that

a significant contribution comes from the neighborhoods of the sub-critical excitations, as a result of their abundance, for example. This paper presents a study of the sub-critical excitations for SDOF elasto-plastic systems. The study will focus on their time-domain characteristics, energy contents, distribution and abundance. The parameterization scheme and characteristics of the critical excitation developed in the previous work will be summarized first, providing a background for the current study. This will be followed by discussion of local optima in the governing objective function that leads logically to the existence of sub-critical excitations. The characteristics of sub-critical excitations will be studied in detail by analytical and numerical means. 2. Critical excitation The critical excitation for SDOF elasto-plastic systems with zero initial conditions has been obtained in [26]. The solution approach starts with a general consideration of the time history of hysteretic response and then eliminates sub-optimal candidates based on physical and mathematical arguments. Fig. 1 shows a schematic diagram for the response due to the critical excitation. It depicts the case where the response takes n = 3 plastic excursions to reach the target threshold level; the actual value of n is the one that minimizes the energy of the excitation while meeting the target. The critical excitation drives the response from rest to point A, at which the system touches the elastic–plastic boundary and has accumulated the maximum elastic spring energy. The excitation then by means of a positive pulse drives the response upward to make a plastic excursion of b1 beyond the yield level b0 (point B). The response unloads elastically from C to D and rises again to make another plastic excursion. The process is repeated until the final target level bF is reached at tF , at which the system may be assumed to be stationary for large tF [15,26]. The configuration of the critical response has been arrived using mathematical and physical arguments. First, it can be shown analytically that the critical response does not have negative plastic deformation, essentially because the latter opposes the positive plastic displacements and hence always leads to a sub-optimal configuration. Second, the segments A–B–C, D–E–F and G–H–I are identical due to the symmetrical dependence of the governing objective function on their associated parameters, and so are the segments C–D and F–G. The configuration shown in Fig. 1 corresponds to a ‘boundary critical’ mode [26] where the response is driven to touch the negative elastic–plastic boundary (at points A, D and G before rebound to make a plastic excursion. This configuration has shown to be valid when tF is not too small compared to the natural period, a situation of typical interest in vibration problems. The critical response, and hence the critical excitation, is completely specified by n, the ‘boundary state parameters’ {t0 , t1 , s1 , t2 , v1 , b1 } and the time history within each segment. By construction, the dynamics is governed by only one equation (linear-elastic or plastic loading) within each segment.

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

y(t) b1 v1 B

+ b0

b1 v1 E

C

I b1 v1 H

F

Fr (y)

1097

B C,E F,H I

O

O

G

y

t

D

− b0

A t0

t1

s1

t2

t1

s1

t2

t1

s1

A

D

G

Fig. 1. Schematic diagram for critical response.

The critical time history within each segment can thus be determined using the calculus of variation for a given n and a set of boundary state parameters. The critical excitation time history for the linear-elastic segments, i.e., O–A, A–B and C–D, is given by ˙ d − t), f (t) = 1 h(td − t) + 2 h(t ∗

0 t td ,

(3)

where td is the duration of the segment (td = t0 for O–A; t1 for A–B; t2 for C–D) and the time origin is defined at the initial point of the segment. The values of 1 and 2 are found from the following equation: H(td ) = L(td )x,

(4)

where  = [1 , 2 ]T , x = [y1 , v1 , y2 , v2 ]T is a vector collecting the response states at the beginning (y1 , v1 ) and at the end (y2 , v2 ) of the segment 

 h11 (td ) h12 (td ) H(td ) = , h21 (td ) h22 (td ) 

td

h11 (td ) = h22 (td ) =

0 td

h dt,

(5) td

h12 (td ) = h21 (td ) = 0

h˙ 2 dt,

−g(td ) −h(td ) ˙ d) −g(t ˙ d ) −h(t 

g(t) = e

−t

cos d t + 

 0 , 1

 1 − 2

(7) 

sin d t

(8)

is the linear-elastic free-vibration response  with unit initial displacement but zero velocity; d =  1 − 2 is the damped natural frequency, and h(t) =

e−t sin d t d

R(s1 ) = L2 (s1 )x

(11)

with x = [b0 , v1 , b1 ]T ,   r (s ) r12 (s1 ) R(s1 ) = 11 1 , r21 (s1 ) r22 (s1 )  s1  s1 2 r dt, r12 (s1 ) = r21 (s1 ) = r r˙ dt, r11 (s1 ) = 0 0 s1 r22 (s1 ) = r˙ 2 dt,

(12)

(13)

0

s 2 0 1 r dt L2 (s1 ) = 2 r(s1 )

(6)

1 0

(10)

where, again, the origin is defined at the initial point of the phase;  = [1 , 2 ]T is determined from

r(t) =

and 

0 t s1

−r(s1 ) −˙r (s1 )

 1 , 0

(14)

and

hh˙ dt,

0

L1 (td ) =

f ∗ (t) = 1 r(s1 − t) + 2 r˙ (s1 − t),



 2

For the elasto-plastic segments (i.e., B–C, E–F, H–I) the segment of critical excitation is given by

(9)

is the linear-elastic free-vibration response with zero initial displacement but unit velocity, also commonly known as the unit impulse response.

1 (1 − e−2t ). 2

(15)

The whole time history of the critical excitation is constructed by patching up the different segments. Its energy can be expressed in terms of n and the boundary state parameters t0 , t1 , s1 , t2 , v1 and b1 (Fig. 1). Determining the critical excitation necessitates optimization of these parameters. For a given n, the up-crossing constraint requires that b1 = (bF − b0 )/n. The optimal value of v1 can be obtained analytically in terms of other parameters because it is found that the energy is a quadratic function of v1 . The optimal value of the remaining parameters should minimize the following objective function: J (t0 , t1 , t2 , ) = E0 (t1 ) + nE 1 (t1 , s1 ) + (n − 1)E2 (t2 ) + [tF − t0 − n(t1 + s1 ) − (n − 1)t2 ],

(16)

where the Lagrange multiplier  incorporates the time constraint; E0 , E1 and E2 are the contributions of energy from the segments O–A, A–B–C and C–D, respectively, E0 =

b02 h22 (t0 ) , 2(t0 )

(17)

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S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

where (t0 ) = h11 (t0 )h22 (t0 ) − h12 (t0 ), ⎤ ⎤ ⎡ −b0 T −b0 1⎢ 0 ⎥ ⎢ 0 ⎥ T −1 E1 (t1 , s1 ) = ⎣ ⎦ L1 (t1 ) H(t1 ) L1 (t1 ) ⎣ ⎦ b b0 2 0 vˆ1 vˆ1    T b0 1 b0 + vˆ1 L2 (s1 )T R(s1 )−1 L2 (s1 ) vˆ1 , 2 b b1 1

(18)

⎡

J2 (t0 , t2 ) = E0 (t0 ) + (n − 1)E2 (t2 ) (19)

(22)

that is a function of t0 and t2 . The sub-critical excitations correspond essentially to local minima of J2 . 3.1. Local optima of J2

with vˆ1 =

It is clear that if the time constraint is satisfied the objective function in (16) reduces to the energy of the excitation because then the Lagrange multiplier terms vanishes. In this case, the energy can be viewed for a given n as sum of two terms, one being J1 = nE 1 (t1 , s1 ) that is a function of t1 and s1 , and the other being

P41 (t1 ) − P43 (t1 ) − Q21 (t1 ) b0 P44 (t1 ) + Q22 (t1 ) Q23 (s1 ) − b1 . P44 (t1 ) + Q22 (s1 )

(20)

Pij denotes the (i, j ) entry of P = L1T H−1 L1 ; Qij denotes the (i, j ) entry of Q = L2T R−1 L2 ; and ⎡ ⎤ ⎤ −b0 −b0 T 1⎢ 0 ⎥ ⎢ 0 ⎥ T −1 E2 (t2 ) = ⎣ ⎦ L1 (t2 ) H(t2 ) L1 (t2 ) ⎣ ⎦, b0 2 b0 0 0 b02 = {h22 (t2 )[1 + g(t2 )]2 − 2h12 (t2 )[1 + g(t2 )] 2(t2 ) × g(t ˙ 2 ) + h11 (t2 )g(t ˙ 2 )2 }. (21) ⎡

It is not efficient to optimize directly the objective function in (16) numerically because it is found that multiple optima exist, essentially with respect to t0 . In view of this, an iterative scheme has been developed for optimizing the parameters t0 , t1 , s1 and t2 for a given n. The scheme starts by omitting the time constraint (with  = 0) and optimizes in individual subspaces with respect to (t1 , s1 ) followed by (t0 , t2 ), then iterates to include the time constraint through an updated value of  until convergence. Details can be found in Appendix A. Typically, the optimal values correspond to t1 +s1 ≈ T /2, t2 ≈ T /2 and t0 = tF− n(t1 + s1 ) − (n − 1)t2 ≈ tF − T (n − 1/2), where T = 2/ 1 − 2 is the natural period. These values conform to physical intuitions [26]. 3. Local optima and symmetric sub-critical excitations The critical excitation presented in the previous section is the excitation in the function space on [0, tF ] that drives the system from rest to the level bF with globally the least energy. It corresponds to the one whose value of n and the boundary state parameters t0 , t1 , s1 and t2 are global minimizer of the objective function in (16). However, the objective function has other local minima. The excitations that correspond to these local minima are referred as ‘sub-critical excitations’ in this work. The subcritical excitations can still drive the system from rest to the target within the designated duration, but their energy is greater than the global minimum value of the critical excitation.

For a given n, consider fixing the values of t1 and s1 at their global optimal values, say, t1 = t1∗ and s1 = s1∗ , and study the variation of J2 as a function of t0 and along the path where t2 = [tF − n(t1 + s1 ) − t0 ]/(n − 1), i.e.,   tF − n(t1∗ + s1∗ ) − t0 ˜ J2 (t0 ) = J2 t0 , n−1   t0 = E0 (t0 ) + (n − 1)E2 t  − , (23) n−1 where t  = [tF − n(t1∗ + s1∗ )]/(n − 1) is defined for convenience. Since t1 and s1 are fixed at their optimal values, the variation of the total energy of the excitation follows that of J˜2 . Following the path t2 = [tF − n(t1 + s1 ) − t0 ]/(n − 1) implies that the time constraint is always satisfied. As a result, the variation of J˜2 with respect to t0 reflects directly the variation of the energy of excitation along the subspace where the time (and target threshold) constraint is always satisfied. The local minima of J2 under the time constraint correspond to those unconstrained minima of J˜2 where the applicable domain of t0 should be confined to (0, tF − n(t1∗ + s1∗ )) since t2 > 0. The discussion here assumes that tF /T is not small, e.g., tF /T > 5, which is often the case in vibration problems. It is clear that the local optima of t0 must be interior to this domain because t0 = 0 or t2 = 0 are clearly not optimal. At the local minima, J˜2 (t0 ) = E0 (t0 ) − E2 (t  − t0 /(n − 1)) = 0

(24)

and so the positions of the local minima are governed by the derivatives of E0 and E2 . Appendix B presents an asymptotic analysis of E0 and E2 , which shows that for large t0 (e.g., t0 /T > 2), |E0 (t0 )|>|E2 (t  − t0 /(n − 1))| and hence J˜2 (t0 ) ∼ E2 (t  − t0 /(n − 1)).

(25)

This implies that the moderate to large local minima of J˜2 correspond essentially to those of E2 (t  − t0 /(n − 1)). We next examine the locations of the local minima of J˜2 (t0 ) based on those of E2 (t  − t0 /(n − 1)). It can be shown that E2 (t) = 23 b02

1 + cos d t + O(2 ) d t − sin d t

(26)

and so for small  the local minima of E2 can be approximately obtained based on the minima of the dominant term. Analysis

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

20

Partial energy

106

n=2

0

105

n=1 n=2

104

n=3 n=4

1099

103 102

1 5 10

-20 20

n=3

0

t1 + s1

t0

0 t1 + s1

t2

1

5

6

t0∗ = tF − n(t1∗ + s1∗ ) − (n − 1)(m + 21 )T

and

t1 + s1

t2

2 t0 (sec)

3

4

-20 1

2

20

3

4

5

6

n=2

0 4

-20 20

n=4

0 -20 20

n=2

0 -20 20

3 n=3

0 Critical excitation

-20 20

n=2 2

0 -20 20

n=1

1

0 -20 1

2

3

4

5

6

Fig. 2. Sub-critical excitations (symmetric) for tF = 6, bF = 2.

of the dominant term shows that its local minima occur at t = (m + 21 )T ,

m = 0, 1, 2, . . . , 

i.e., (27)

where T = 2/ 1 − 2 is the natural period. The large local minima of J˜2 will occur at t  − t0 /(n − 1) = (m + 1/2)T ,

t2∗ = (m + 21 )T ,

m = 0, 1, 2, . . . ,

(28)

where m proceeds as long as J˜2 is dominated by E2 at the local optimum. Note that the largest optimum of t0 is given by

1100

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

t0∗ = tF − n(t1∗ + s1∗ ) − (n − 1)T /2 and successive optima are separated by (n − 1)T . When  = 0, the value of E2 vanishes at its local minima, which can be reasoned directly from (26) or intuitively by noting that in the absence of damping the response can go by free vibration from b0 to −b0 in durations T /2, T /2+T , T /2+2T , etc. without the action of external force. In general, for small  > 0, it can be shown by directly substituting (28) into (21) that √ −(2m+1)/ 1−2 31 − e 1 E2 (T (m + 2 )) = 2 √ 2 1 + e−(2m+1)/ 1− = 2m3 2 + O(4 ), m = 0, 1, 2, . . . (29) and therefore the value of J˜2 at the local minima is dominated by E0 , i.e., J˜2 (t0∗ ) = E0 (t0∗ ) + O(2 ).

(30)

Since E0 is a non-increasing function, the global minimum of J˜2 occurs at the largest optimal value of t0 . To illustrate the results established so far, consider a SDOF elasto-plastic oscillator with natural frequency  = 2 (1 Hz), damping ratio  = 1% and yield displacement b0 = 1. The target threshold level and first passage time are assumed to be bF = 2 and tF = 6 s. For this case, the critical excitation corresponds to n = 3 (i.e., b1 = 13 ), t0 = 3.44 s, t1 = 0.39 s, s1 = 0.13 s and t2 = 0.50 s; these values are obtained by the iterative algorithm presented in Appendix A. Note that t1 + s1 = 0.52 ≈ T /2 and t2 = 0.50 ≈ T /2, as expected. The top right plot in Fig. 2 shows the variation of the ‘partial energy’ J˜2 with respect to t0 and for different values of n. Note that this plot only indicates the optimal value of t0 for a given n; the optimal value of n cannot be determined from this plot because the total energy of the excitation depends also on J1 = nE 1 which is a function of n. For n = 1, J˜2 is a non-increasing function of t0 because in this case J˜2 = E0 (t0 ). For n = 2, J˜2 has multiple minima that are separated by the natural period T. In general, for n2, the separation between successive minima is equal to (n − 1)T , as indicated by (28). The values of J˜2 for different n coincide visually at the local minima because they only differ by O(2 ) at these locations, according to (30). 3.2. Symmetric sub-critical excitations The sub-critical excitations that correspond to different optima of t0 for each n are plotted on the left column in Fig. 2. For a given n, the sub-critical excitations correspond to different duration t0 of the initial linear-elastic segment, and accordingly different duration t2 of the subsequent elastic-unloading segments. From the plot of J˜2 , the local optima are located at five distinct energy levels and they are numbered in ascending order of their magnitude. The sub-critical excitations with the same local minimum value of J˜2 (up to O(2 )) have been grouped together in the figure. Recall that J˜2 accounts for the contribution of energy from only the initial linear-elastic segment (of duration t0 ) and the subsequent elastic-unloading segments (each of duration t2 ). These segments are shaded in the

plots of the excitations. The energy of the positive pulses in the excitation are not included in J˜2 . The sub-critical excitations can be viewed as generated by shortening the initial segment (of duration t0 ) by (n − 1)T and then allocating this duration to the (n − 1) subsequent elastic-unloading segments, with the duration of each segment lengthened by T. Taking the case of n = 3 as an example, the partial energy curve (center line) has two local minima at t0 = 1.43 and 3.43 s, which are separated by two natural periods (2 s) as can be expected from (28). For t0 = 1.43 s, t2 = [tF − n(t1 + s1 ) − t0 ]/(n − 1) = [6 − (3)(0.39 + 0.13) − 1.43]/(3 − 1) = 1.51 s. The corresponding sub-critical excitation is drawn in the second plot from the top of Fig. 2, where the durations t0 , t1 and t2 have been marked for illustration. The construction of other sub-critical excitations can be followed similarly. As mentioned before, the values of J˜2 for different values of n appear to coincide at the local minima, but in fact they differ by O(2 ) at those locations. Examination of the corresponding sub-critical excitations gives an alternative account of this fact. For example, consider the sub-critical excitations at energy level 3, i.e., the fifth and sixth plots from the top in Fig. 2. The initial segments of the excitation up to t0 (i.e., just before the first positive pulse) for n = 2 and 3 are identical, because they both correspond to the excitation that drives the system from rest to y(t0 ) = −b0 with minimum energy. Since the subsequent elastic-unloading segments for n = 2 and 3 are different, their resulting values of J˜2 are not the same. Nevertheless, the difference is onlyO(2 ). Essentially, the subsequent elastic-unloading segments are responsible for providing sufficient energy to compensate for the energy dissipated through damping as the response goes from +b0 to −b0 (relative to the current neutral axis). Energy balance reveals that the energy of these segments is O(2 ). 4. Unsymmetric sub-critical excitations The sub-critical excitations shown in Fig. 2 all correspond to ‘symmetric modes’, in the sense that all the (n − 1) subsequent elastic-unloading segments are identical with the same duration t2 . The symmetry assumption is inherent in the expression for J in (16) and J2 in (22) in that the contribution of energy from the elastic-unloading segments is simply an integer multiple of E2 (t2 ). One consequence is that for a given n 3, symmetric sub-critical excitations only appear at energy levels n, n + (n−1), n + 2(n − 1), . . . (as long as optima exist). For example, for n = 3, the sub-critical excitations appear only at energy levels 3 and 5 in Fig. 2. Note that for n 2 symmetry is irrelevant because for n = 1 the elastic-unloading segments are not defined while for n = 2 they adjoin each other. Symmetry was used for arriving at the critical excitation (the global optimum) in [26] but it should be relaxed when subcritical excitations are considered. Relaxing symmetry leads to the discovery of unsymmetric sub-critical excitations that can exist at consecutive energy levels for a given n. The symmetry assumption enforces the elastic-unloading segments to have the same duration of t2 . Relaxing this assumption, the function J2 in (22) that accounts for the energy in the initial and elastic

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

unloading segments now reads J2 = E0 (t0 ) +

n−1 

1101

10

E2 (t2i ),

(31)

0

i=1

where {t2i : i = 1, . . . , n − 1} are the durations of the (n−1) elastic-unloading segments, not necessarily identical. Using the dominance of E2 (·) over E0 (·) as before, we obtain that jJ2 /jt2i ∼ E2 (t2i ) near the optima, and so minimizing the total energy J = J1 (t1 , s1 ) + J2 ({t2i : i = 1, . . . , n − 1}) with respect to t2i (i = 1, . . . , n − 1) requires that E2 (t2i ) = 0, yielding ∗ t2i = (mi + 21 )T ,

mi = 0, 1, 2, . . . ,

n−1 

∗ t2i = tF .

(33)

i=1

= tF − n(t1∗

+ s1∗ ) −

0 -10 1

2

3 Time (sec)

4

5

6

Fig. 3. Sub-critical excitations (unsymmetric) at energy level 4 for tF = 6, bF = 2.

10 0 -10 10

Rearranging and using (32) yields t0∗

n=3

(32)

which is analogous to (27). When all the mi ’s are identical, the sub-critical excitation is symmetric. On the other hand, for a given n and a given local optimal solution with t1 = t1∗ and s1 = s1∗ , the failure time constraint now reads t0∗ + n(t1∗ + s1∗ ) +

-10 10

n−1   n−1 T − mi T . 2

(34)

n=3

0

i=1

The sub-critical excitations are generated from different possible values of {mi : i =1, . . . , n−1}. The case n−1 i=1 mi =0, i.e., mi =0(i=1, . . . , n−1) corresponds to the sub-critical excitation with n−1the lowest energy among these possibilities. Since the term sub-critical i=1 mi in (34) is a non-negative integer, other n−1 ∗ excitations will have in t0 one period less ( i=1 mi = 1),  two periods less ( n−1 i=1 mi = 2) and so on (as long as the local optimum exists). Their energy increases with the value of n−1 m . Following this hierarchy, the local optima of {t2i : i=1 i i = 1, . . . , n − 1} corresponding to the sub-critical excitations (including both symmetric and unsymmetric ones) of increasing energy levels can be generated by exhausting all legitimate  combinations of {mi : i=1, . . . , n−1} for n−1 m i=1 i =1, 2, . . . . n−1 For each value of i=1 mi , starting from the symmetric sub-critical excitation with mi = 0 (i = 1, . . . , n − 1), all the otherthe sub-critical solutions can be generated by taking n−1 out i=1 mi periods from t0 and then allocating it among {t2i : i = 1, . . . , n − 1}, exhausting all possible combinations. For example, Fig. 3 shows the unsymmetric sub-critical excitations developed from the critical excitation in Fig. 2 (the third plot from the bottom). The two possibilities correspond to taking out a period T from t0 and allocating it to the first and the second elastic-unloading segments, respectively. These two excitations have the same value of t0 that is one period less than the critical excitation from which they are developed. Their initial segments are identical to that of the two symmetric subcritical excitations at energy level 4 in Fig. 2 and therefore they belong to this energy level. They only differ from the symmetric sub-critical excitations at level 4 in their elastic-unloading segments, but such difference only leads to a negligible

-10 10 0 -10 10 n=4

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Fig. 4. Sub-critical excitations (unsymmetric) at energy level 5 for tF = 6, bF = 2.

energy difference of O(2 ). Fig. 4 shows the unsymmetric sub-critical excitations at energy level 5 that are developed from the symmetric sub-critical excitations for n = 3 and 4. For n = 3, the two possibilities correspond to allocating the duration 2T to the first and second elastic-unloading segments, respectively. For n = 4, the three possibilities correspond to allocating the duration T to the first, second and third elasticunloading segments, respectively. It is clear that the number of

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Partial energy

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Fig. 5. Energy plot for tF = 6, bF = 2.

5. Population of sub-critical excitations 5.1. Abundance of sub-critical excitations We next investigate the abundance of sub-critical excitations. For illustration purposes consider the sub-critical excitations for a larger target time tF = 8 s. Fig. 6 shows the partial energy J˜2 for this case. There is obviously a larger number of local optima than that in Fig. 2 because the admissible range of t0 is wider. The energy plot in Fig. 7 now consists of more subcritical states. The sub-critical excitations for n = 3, 4 and 5 are shown in Fig. 8. Their corresponding displacement responses calculated using non-linear dynamic analysis are shown in Fig. 9, which verifies that all the responses meet the target y(tF ) = bF , as expected. For each n, the sub-critical excitations have been generated by sequentially taking out one period

2

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4 t0 (sec)

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Fig. 6. Partial energy versus t0 for tF = 8, bF = 2.

55 ---1

50

Energy

possible sub-critical excitations grows in a combinatorial manner with the energy level and the number of elastic-unloading segments, suggesting that unsymmetric sub-critical excitations can be quite abundant. This will be further explored in the next section. As mentioned before, the partial energy J˜2 shown in Fig. 2 does not account for the energy contribution from the positive pulses. Fig. 5 shows the ‘total’ energy of the subcritical excitations for different n. For each value of n, the lowest energy value is marked by a hollow circle; this corresponds to the largest optimal value of t0 for the given n. The optimal value of n (corresponding to the critical excitation) is marked by a solid circle. The energy of other sub-critical configurations is marked by a dot, with the associated number giving the number of sub-critical excitations at that energy level. For example, the energy with number ‘2’ of n = 3 refers to the two sub-critical excitations shown in Fig. 3, while the level with number ‘3’ of n = 3 refers to the symmetric subcritical excitation for n = 3 at level 5 in Fig. 2 and the two unsymmetric ones in the top two plots of Fig. 4.

1

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---4 ---6

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30 1

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n Fig. 7. Energy plot for tF = 8, bF = 2.

from the duration of the initial segment and allocating it to the elastic-unloading segments in a combinatorial manner. Note that sub-critical excitations of the same n and t0 have the same energy (up to O(2 )). To analyze the abundance of sub-critical excitations in a systematic manner, we group them in a two-dimensional array according to their value of n and their ‘relative energy level’, as shown in Fig. 8. Specifically, for each n the lowest energy level is denoted by ‘0’. The next higher level whose sub-critical excitations are generated by taking out one period from t0 and allocating to elastic-unloading segments is denoted by ‘1’; and so on for higher energy levels. Grouping the sub-critical excitations according to their number of elastic-unloading segments n and their relative energy level r, the (n, r)-energy group consists of the sub-critical excitations that are possible combinations of allocating r natural periods among (n − 1) segments, each with an integer

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Fig. 8. Sub-critical excitations for tF = 8, bF = 2.

multiple of natural period. The number of natural periods allocated to each segment may be denoted using a (n − 1)-dimensional vector, say,

number is equal to

c = [m1 , m2 , . . . , mn−1 ],

where r! = 1 × 2 × · · · × r is the factorial of r. Note that N (2, r) = 1 (for all r 1) and N (n, 1) = n − 1 (for all n2), which can be easily verified from (36) and agree with intuitions. The following recursive relationship can be readily established by breaking the original combinatorial problem into (r + 1) sub-problems of allocating i (=0, 1, . . . , r) periods to the first elastic-unloading segment and (r − i) periods to the remaining (n − 2) elastic-unloading segments:

(35)

where mi is the number of natural periods allocated to the ith elastic-unloading n−1 segment. It is clear that mi 0 are integers and i=1 mi = r. For example, for n = 4, r = 2 (lowest group in the middle column of Fig. 8), the six sub-critical excitations shown correspond to c = [0 0 2], [0 1 1], [0 2 0], [1 0 1], [1 1 0] and [2 0 0]. The number N (n, r) (n 2, r 1) of sub-critical excitations within the (n, r)-energy group is simply equal to the number of combinations of allocating r natural periods among (n − 1) elastic-unloading segments, each with an integer multiple of natural periods. This is equivalent to the number of combinations of choosing r balls, one at a time and with replacement, from (n − 1) distinct balls, where ordering is immaterial. From standard combinatorial theory [27], this

N (n, r)=n+r−2 Cr =

N (n, r) =

r 

(n + r − 2)! , (n − 2)!r!

N (n − 1, r − i).

(36)

(37)

i=0

It should be noted that by definition N (n, r) includes the number of possible symmetric and unsymmetric sub-critical excitations within the (n, r)-energy group. Eq. (36) shows that N (n, r) grows in a combinatorial manner with the number of

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45 Energy

segments n and hence with the target failure time tF . On the other hand, recall that for each n the symmetric sub-critical excitations occur only at relative energy levels r = 0, (n−1), 2(n − 1), . . ., and so their number only grows linearly with n. Thus, the abundant population of sub-critical excitations is dominated by unsymmetric ones. Figs. 10 and 11 show the energy plot of the sub-critical excitations for tF = 10 and 20 s, which illustrates the growth of the population of the sub-critical excitations as the failure time increases.

---1 ---1 ---1 ---1

40 35

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---1 0

5.2. Distribution of sub-critical excitations 30

As a means of visualizing the distribution of the sub-critical excitations (which are points in the infinite-dimensional function space), we take the critical excitation as a reference and consider the location of the sub-critical excitations relative to it. Specifically, consider the hyper-angle between a sub-critical excitation fs and the critical excitation f ∗ , defined by (38)

---6 ---3

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n Fig. 10. Energy plot for tF = 10, bF = 2.

where f ∗ , fs  , = cos−1 ∗ f fs

---2 0

f ∗ , fs  =



tF 0

f (t)fs (t) dt

(39)

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

---78 ---66 ---55 ---45 ---286 ---36 ---28 ---220 ---21 ---15 ---10 ---165 ---715 ---6 ---3 ---120 ---84 ---495 ---1287 ---56 ---330 ---35 ---20 ---210 ---792 ---1716 ---10 ---4 ---126 ---1716 ---70 ---462 ---924 ---1287 ---35 ---252 ---15 ---126 ---462 ---792 ---715 ---5 ---286 ---56 ---210 ---330 ---495 ---78 ---21 ---84 ---13 ---220 ---6 ---66 ---28 ---120 ---165 ---7 ---36 ---45 ---55 ---12 ---8 ---9 ---10 ---11

Energy

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10

1105

Fig. 11 (tF = 20 s), whose numbers are 135 and 16,398, respectively. It is seen that the sub-critical excitations do not cluster around the critical excitations but rather among themselves. They do not populate as a set of points on a hyper-plane nor a hyper-sphere. Blow-up views of increasing scales show that the sub-critical excitations of each energy group (n, r) cluster among themselves, exhibiting structures of progressively smaller scales. For example, Fig. 13 shows views of the polar plot for tF = 20 s with progressive magnification of the bracketed regions. It is seen that details of finer scales are revealed upon magnifications. The fine scales stem from the variety of sub-critical excitations within an energy group, whose number increases in a combinatorial fashion with the number of elastic-unloading segments (n − 1). Of course, for the present case a single point will eventually appear upon a sufficiently large but finite number of magnifications because the number of sub-critical excitations is still finite. Nevertheless, it suggests that the polar image of the sub-critical excitations may exhibit a fractal structure that has an arbitrary fine scale as n → ∞ [28]. This shall be left for future research.

6. Conclusions

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Fig. 12. Polar plot of sub-critical excitations for tF =10 s (circle) and tF =20 s (dot). The radius and angle of each point represent the L2 -norm (see (40)) of the corresponding sub-critical excitation and the hyper-angle (see (38)) it makes with the critical excitation.

is the inner product between f ∗ and fs ; and  tF 1/2 fs = fs (t)2 dt

(40)

0

is the L2 -norm of fs . We map the sub-critical excitations on a polar plot using their norm as a radius and as their angle. By this construction, the critical excitation is always mapped as a point on the horizontal axis in the polar plot; a hyper-plane and a hyper-sphere in the Euclidean space will be mapped as a vertical line and a circle in the polar plot, respectively. This mapping gives a perspective of visualization akin to that of the failure region in the neighborhood of a design point in structural reliability analysis. Fig. 12 shows the polar plot of the sub-critical excitations for tF = 10 s (circle) and 20 s (dot). The sub-critical excitations mapped correspond to those listed in Fig. 10 (tF = 10 s) and

A study of the sub-critical excitations for SDOF elastoplastic systems has been presented in this paper. The candidates of critical excitation are completely defined by their number of plastic excursions and a set of boundary state parameters. The sub-critical excitations stem from the local optima of the objective function in the space of the boundary state parameters. For a given number of plastic-excursions, they are generated by shortening the initial linear-elastic segment of the critical excitation in integer multiples of natural periods and allocating to the elastic-unloading segments in a combinatorial manner. They exist in abundance and are dominated by unsymmetric ones, where the durations of different elastic-unloading segments can be possibly different. The sub-critical excitations may be grouped according to their number of elastic-unloading segments and relative energy level; the latter is equal to the number of periods shortened in the initial linear-elastic segment relative to the lowest energy excitation in the group. When the sub-critical excitations are mapped on a polar plot relative to the critical excitation, they cluster among themselves and exhibit a fine scale structure as the target duration increases, possibly admitting a fractal structure in the limit. A detailed study of the latter is left for future research. The abundance of sub-critical excitations and their timedomain characteristics indicate that there is a large variety of ways for generating large elasto-plastic response with locally minimal energy. These ways differ essentially in their temporal sequence of accumulating plastic excursions to reach the target thresholds and this complexity is not observed in linearelastic systems. One direction of future research is applying the critical and sub-critical excitations for solving stochastic dynamics problems of elasto-plastic systems, e.g., as design points for importance sampling for estimating first passage probabilities [29].

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a

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Fig. 13. Polar plot of sub-critical excitations for tF = 20 s with progressive magnifications: (a) ×1 (original); (b) ×10; and (c) ×100. The radius and angle of each point represent the L2 -norm (see (40)) of the corresponding sub-critical excitation and the hyper-angle (see (38)) it makes with the critical excitation.

Acknowledgment

terms of t0 :

The work described in this paper was fully supported by a grant from the City University of Hong Kong (Project No. 7200053). The financial support is gratefully acknowledged.

t2 =

Appendix A. Iterative scheme for optimization This appendix presents an iterative algorithm for finding for a given n the optimal values of t1 , s1 and t2 that minimize the objective function in (16). For details the reader is referred to [26]. First, we start with  = 0 and obtain tˆ1 and sˆ1 by numerically minimizing J˜1 (t1 , s1 ) = E1 (t1 , s1 ) − (t1 + s1 ),

(41)

where a robust initial guess for tˆ1 and sˆ1 is found to be half of the elastic natural period. Given such tˆ1 and sˆ1 , the remaining parameters t0 and t2 are to satisfy the time constraint t0 + (n − 1)t2 + n(tˆ1 + sˆ1 ) = tF , which allows us to express t2 in

tF − n(tˆ1 + sˆ1 ) − t0 . n−1

(42)

Substituting this expression into (16), the optimal value of t0 can be found by minimizing   tF − n(tˆ1 + sˆ1 ) − t0 . (43) J˜0 (t0 ) = E0 (t0 ) + (n − 1)E2 n−1 on 0 < t0 tF − n(tˆ1 + sˆ1 ), where the last term in (16) vanishes because the time constraint has already been enforced. As multiple minima exist for t0 , the global minimum should be found by calculating the value of J˜0 for a grid of values of t0 on [0, tF − n(tˆ1 + sˆ1 )] and then selecting the minimizing value. This strategy is computationally inexpensive because analytical expression for J˜0 is available and only a one-dimensional grid is involved. The value of  should be updated by =

b02 jE0 (tˆ0 ) ˙ tˆ0 )]2 . (44) =− [h22 (tˆ0 )h(tˆ0 ) − h12 (tˆ0 )h( jt0 2(tˆ0 )2

With the updated value of , the procedure is iterated until tˆ0 , tˆ1 , sˆ1 and tˆ2 have all converged. Experience shows that only a

S. Au / International Journal of Non-Linear Mechanics 41 (2006) 1095 – 1108

small number of iterations (e.g., <5) are required to achieve a relative tolerance of 1%. This iterative procedure allows the determination of optimal boundary state parameters and hence the energy of the critical excitation for a given value of n. The energy of the critical excitation should then be calculated for different values of n. The optimal value of n is obtained as the one that minimizes the energy. For implementation, the starting value of n may be set at half of the integral number of natural periods within the target duration tF . Appendix B. Asymptotic behavior of J˜2 In this appendix we establish that the derivative of J˜2 in (23) is dominated by that of E2 for moderate to large values of t0 . It is assumed that 0  < 1 and the target failure time tF is not small (e.g., tF /T > 5). We first obtain some asymptotic relationships that will be used later. B.1. Dominance of h11 (t) and h22 (t) over h12 (t) Let h11 (t), h12 (t) and h22 (t) be defined in (6): 

t

h11 (t) = h22 (t) =

0 t

 h(s)2 ds,

t

h12 (t) = 0

˙ h(s) ds.

(45)

0

We first show that for large t,  h12 (t) h12 (t) O(t −1 ), , = O(e−t ), h11 (t) h22 (t)

 = 0,  > 0,

(46)

and therefore h12 (t) is dominated by h11 (t) and h22 (t) for large t. To prove (46), note that when  = 0, h(t) = sin t/ is oscillatory and does not decay with t (see (9)). By direct integration, it can be readily obtained that h11 (t) ∼ t/22 and h22 (t) ∼ t/2 for large t. Since h12 (t) = h(t)2 /2 remains bounded, the first claim of (46) follows. When  > 0, h(t) is exponentially decaying with exp(−t) and so h12 (t)=h(t)2 /2=O(exp(−2t)). On the other hand, it can be obtained by direct integration that h11 (t) → 1/43 = O(1) and h22 (t) → 1/4 = O(1) as t → ∞, and therefore the second claim of (46) follows. B.2. Asymptotic behavior of E0 (t) and E2 (t) Using the result in (46), the following asymptotic results for large t can be derived: (t) = h11 (t)h22 (t) − h12 (t)2  2 t /42 ,  = 0, ∼h11 (t)h22 (t) ∼ 1/162 4 ,  > 0, E0 (t) ∼

b02 ∼ 2h11 (t)



b02 2 /t,  = 0, 23 b02 ,  > 0,

  b02 (1 + g(t))2 g(t) ˙ 2 + E2 (t) ∼ 2 h11 (t) h22 (t)  2 2 2 4b0  cos (t/2)/t,  = 0, ∼ 23 b02 [1 + 2g(t)],  > 0.

(49)

B.3. Asymptotic behavior of E0 (t) and E2 (t) By direct differentiation and using the dominance relationship in (46), the followings can be concluded: b02 h(t)2



E0 (t)

∼−

O(t −2 ), O(e−2t ),

 = 0,  > 0,

(50)

E2 (t)

 b02 2 h(t) O(t −1 ), ∼− = O(e−t ), h11 (t)

 = 0,  > 0.

(51)

2h11 (t)

2

∼

Numerical experiments show that these asymptotic relationships are quite good for moderate to large values of t/T , e.g., t/T > 2. B.4. Asymptotic behavior of J2 (t0 )

˙ ds = 1 h(t)2 , h(s)h(s) 2

2

1107

(47)

(48)

To examine the asymptotic behavior for J2 (t0 ) in (24), note that the applicable domain of t0 is 0 < t0 < tF − n(t1∗ + s1∗ ). As t0 increases, the argument t0 of E0 in (24) increases while the argument t2 = t  − t0 /(n − 1) of E2 decreases. We consider the following two cases separately: (1) t0 large and t2 small and (2) t0 large and t2 large. For the first case, E0 (t0 ) = O(t0−2 ) and E2 (t2 )=O(1), implying |E0 (t0 )|>|E2 (t  −t0 /(n− 1))| and hence J˜2 (t0 ) ∼ E2 (t  − t0 /(n − 1)). For the second case, E0 (t0 ) = O(t0−2 ) and E2 (t2 ) = O(t2−1 ), implying again |E0 (t)|>|E2 (t  −t0 /(n−1))| and J˜2 (t0 ) ∼ E2 (t  −t0 /(n−1)). In conclusion, (25) holds as long as t0 is large (e.g., t0 /T > 2). References [1] G.Q. Cai, Y.K. Lin, Reliability of nonlinear structural frame under seismic excitation, J. Eng. Mech. ASCE 124 (8) (1998) 852–856. [2] A.K. Chopra, C. Chintanapakdee, Comparing response of SDF systems to near-fault and far-fault earthquake motions in the context of spectral regions, Earthquake Eng. Struct. Dyn. 30 (2001) 1769–1789. [3] N.J. Tarp-Johansen, O. Ditlevsen, Time between plastic displacements of elasto-plastic oscillators subjected to Gaussian white noise, Probab. Eng. Mech. 16 (2001) 373–380. [4] D. Vian, M. Bruneau, Tests to structural collapse of single degree of freedom frames subjected to earthquake excitations, J. Eng. Mech. ASCE 129 (12) (2003) 1676–1685. [5] J.W. Lindt, G. Goh, Effect of earthquake duration on structural reliability, Eng. Struct. 26 (2004) 1585–1597. [6] J. Hancock, J.J. Bommer, The effective number of cycles of earthquake ground motion, Earthquake Eng. Struct. Dyn. 34 (2005) 637–667. [7] J.F. Wilson, E.G. Callis, The dynamics of loosely jointed structures, Int. J. Non-linear Mech. 39 (2004) 503–514. [8] C.T. Huang, S.Y. Kuo, Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects, Int. J. Non-linear Mech. 41 (2006) 530–542. [9] R.F. Drenick, Model-free design of aseismic structures, J. Eng. Mech. ASCE 96 (1970) 483–493.

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[10] A. Papoulis, Maximum response with input energy constraints and the matched filter principle, IEEE Trans. Circuit Theory 17 (2) (1970) 175–182. [11] K. Zhou, J. Doyle, K. Glover, Robust and Optimal Control, PrenticeHall, Englewood Cliffs, NJ, 1996. [12] A. Der Kiureghian, The geometry of random vibrations and solutions by FORM and SORM, Probab. Eng. Mech. 15 (1) (2000) 81–90. [13] S.K. Au, J.L. Beck, First excursion probability for linear systems by very efficient importance sampling, Probab. Eng. Mech. 16 (3) (2001) 193–207. [14] K.V. Yuen, L.S. Katafygiotis, An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability, Probab. Eng. Mech. 20 (2005) 109–114. [15] H. Koo, A. Der Kiureghian, K. Fujimura, Design-point excitation for non-linear random vibrations, Probab. Eng. Mech. 20 (2) (2005) 136–147. [16] S.K. Au, J.L. Beck, Importance sampling in high dimensions, Struct. Safety 25 (2) (2003) 139–163. [17] R.N. Iyengar, C.S. Manohar, Nonstationary random critical seismic excitations, J. Eng. Mech. ASCE 113 (4) (1987) 529–541. [18] R. Srinivasan, R. Corotis, B. Ellingwood, Generation of critical stochastic earthquakes, Earthquake Eng. Struct. Dyn. 21 (1992) 275–288.

[19] I. Takewaki, A new method for nonstationary random critical excitations, Earthquake Eng. Struct. Dyn. 30 (4) (2001) 519–535. [20] R.N. Iyengar, Worst inputs and a bound on the highest peak statistics of a class of non-linear systems, J. Sound Vib. 25 (1) (1972) 29–37. [21] B.D. Westermo, The critical excitation and response of simple dynamic systems, J. Sound Vib. 100 (2) (1985) 233–242. [22] J.B. Roberts, P.D. Spanos, Random Vibration and Statistical Linearization, Wiley, New York, USA, 1990. [23] R.F. Drenick, The critical excitation of nonlinear systems, Int. J. Appl. Mech. 44 (2) (1977) 333–336. [24] A.J. Philippacopoulos, P.C. Wang, Seismic inputs for nonlinear structures, J. Eng. Mech. ASCE 10 (1984) 828–836. [25] I. Takewaki, Critical excitation for elastic-plastic structure via statistical equivalent linearization, Probab. Eng. Mech. 17 (1) (2002) 73–84. [26] S.K. Au, Critical excitation of SDOF elasto-plastic systems, J. Sound Vib. 296 (4–5) (2006) 714–733. [27] R. Merris, Combinatorics, PWS Publication, Boston, 1996. [28] E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, UK, 1993. [29] G.I. Schueller, Developments in stochastic structural mechanics, Arch. Appl. Mech. 75 (2006) 755–773.