ABSTRACT: Classical structural reliability methods remain challenges in the application of engineering structures. A reliability method by means of theory of extreme value distribution has been developed in recent years which submits to the framework of physical stochastic systems and resorts to the probability density evolution method (PDEM). In this method, a hypothesis of virtual random process on extreme values of response samples is included pushing the static extreme value into dynamic process so as to be integrated with the generalized density evolution equation (GDEE) in the scheme of PDEM. The uniqueness of formulation, however, of virtual random process cannot be secured that results in the difficulty of quantifying the numerical error of solution. A new structural reliability method is proposed in this paper through the re-construction of response random process sticking with the extreme value once it occurs, whereby the uniqueness of re-constructed process can be secured. The reliability solution can be directly derived from the probability density function of the re-constructed process at the terminal instant using the PDEM. Numerical investigations against the method of extreme-value distribution and the Monte Carlo simulation reveal the validity and applicability of the proposed extreme-response evolution method.

KEY WORDS: Dynamic reliability; hysteretic structures; probability density evolution method; extreme-value evolution; Monte Carlo simulation.

1 INTRODUCTION

Safety assessment and risk analysis of engineering structures under extreme hazard excitations have received great appealing in recent years. The argument quantifying the structural safety usually refers to structural reliability, which provides the principle of structural design for decision makers on the balance between construction investment and repairment cost due to structural damage resulting from unexpected hazard in the period of structural life. There are typically three family of structural reliability methods. The first which is usually used in the engineering application refers to Cross-Level Process Theory [1]. It cannot offer a sufficient accuracy since artificial hypothesises on the behavior of cross-level events have to be made bypassing the assessment of joint probability density function for caculation of mathmatical expectation of crossing rate. Diffusion-process theorey, underlying the second structural reliability method, possesses a rigorous theoretical principle while it cannot be employed in the case of multi-degree-of-freedom systems [2]. The third method is random simulation technique. It has a wide applicability in principle while there needs a over-head computation effort especially in case of direct Monte Carlo simulation. Although a collection of embeded techniques for number reduction of samples, e.g. variance reduction and importance sampling, has gained impressive progresses, their applicability still remains open [3].

To overcome the challenges inherent in classical structural reliability methods, a framework of physical stochastic systems, together with the probability density evolution method (PDEM), was developed in the past decade [4-5]. In this methodology, the randomness is treated on a unified basis of the principle of preservation of probability, and the systems

are decoupled by involving the embedded physical mechanism. A family of generalized density evolution equation (GDEE) was thus established, of which dimension is essentially irrelevant to the number of components and the degree of complexibility of structural systems. The GDEE, moreover, provides a new perspective for structural reliability through constructing virtual random process on extreme values of response samples merely meeting with the terminal-value condition. It is referred as the theory of extreme-value distribution (EVD) which has been demonstrated to be validated and applicability in reliability assessment of engineering structures [6]. The extension to global reliability of structural systems was carried out by introducing the principle of equivalent-extreme-value events [7].

The critical principle of EVD is the hypothesis of virtual random process on extreme values of response samples. This treatment is to push the static argument such as the extreme value into dynamic process so as to be integrated with the GDEE and be readily solved by the PDEM. It exposes evidently that the formualtion of virtual random process is not unique due to the hypothesis upon which the numerical error relies. In this paper, an attempt into reliability assessment of structural systems is performed through focusing on the extreme values of response samples and their evolution. The technique used in this method is the process sticking with the extreme value once it occurs. By doing so, the structural reliability can be directly derived from the probability density function of the re-constructed process at the terminal instant using the PDEM. It is indicated that the re-constructed process is unique where no hypothesis is introduced. The sections arranged in this paper are distributed as follows. Section 2 details the basic principle and implementation procedure of

Global reliability via probability density evolution analysis of the development process of nonlinearity of structures

Jie Li1,2, Yongbo Peng1, Jianbing Chen1,2 1State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, P.R.China

2School of Civil Engineering, Tongji University, Shanghai 200092, P.R.China email: pengyongbo@tongji.edu.cn, chenjb@tongji.edu.cn, lijie@tongji.edu.cn

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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the PDEM. The theory of extreme value distribution and its difference from the method appealed in this paper are provided in Section 3. An eight-storey shear frame with Bouc-Wen nonlinearities subjected to random seismic ground motions is investigated for illustrative purposes, which is included in Section 4. The concluding remarks are included in Section 5.

2 AXIOMS OF PROBABILITY DENSITY EVOLUTION METHOD

Without loss of generality, consider the equation of motion of an n-degree-of-freedom structural system subjected to random seismic excitations as follows

( , , ) ( , )gt x t+ = -MX f X X MI Q (1)

where T1 2= ( , , , )nX X XX denotes n by 1 displacement

vector; overdots denotes differentiation with regard to t; M denotes n by n mass matrix; f denotes internal forces of the structures including the damping and restoring forces; I denotes a n-order column vector with all the components being 1; ( , )gx tQ denotes the earthquake ground motion;

Q denotes random vector characterizing the randomness involved in the ground motion, which was detailed in previous sections. The joint PDF of Q is identified as ( )pQ q .

It is evident that any arbitrary response of interest ( )tX

depends upon Q and could be written in the form of a function as

= ( , )tX H Q

(2)

Using the principle of preservation of probability, one can deduce the generalized density evolution equation (GDEE)

governing the joint PDF of ( , )X Q [4,8]

( , , ) ( , , )( , ) 0X Xp x t p x tX t

t x

¶ ¶+ =

¶ ¶Q Qq qq (3)

where ( , , )Xp x tQ q indicates joint PDF of ( , )X Q ;

( , )X t q indicates the velocity of X under the

condition{ = }Q q .

The initial condition for Eq. (3) reads

0 0( , , ) | ( ) ( )X tp x t x x pd= = -Q Qq q (4)

where ( )d ⋅ denotes the Dirac’s delta function; 0t is the initial

time; 0x is the deterministic initial value.

The joint PDF ( , , )Xp x tQ q could be given by solving

the initial-value partial differential Eqs. (3) and (4) using the

numerical procedures. The instantaneous PDF of ( )X t could

be obtained by

( , ) ( , , )dX Xp x t p x tW

= òQ

Q q q (5)

where WQ indicates the distribution domain of Q .

To get instantaneous PDFs and other probabilistic indices of responses, a collection of equations consisting of the motion equation Eq. (1), the generalized density evolution equation Eq. (3) and the integration equation Eq. (5) are to be solved. The numerical implementation involves the following steps [8]:

Step 1: Select representative point set

1, 2, ,{ = ( , , , ) 1,2, , }

res q q q s q resq nq q q = qs in the

domainWQ , where resn is the total number of the selected

points. Step 2: For the prescribed{ }q=Q q , solve Eq. (1) with a

deterministic time integration method to evaluate the value

of q( , )mX tq , where =mt m t , = 0,1,2,m ; t denotes

the time step used in the time integration method.

Step 3: Introduce q( , )mX tq into the generalized density

evolution equation Eq. (3), and solve Eq. (3) under the initial condition Eq. (4) using a finite difference method whereby the

component of numerical solution ( , , )Xp x tQ q , denoted by

( , , )X j q kp x tQ q is obtained, here 0= +jx x j x ,

= 0, 1, 2,j ; x denotes the space step; ˆ=kt k t ,

= 0,1,2,k ; ˆt denotes the time step used in the finite

difference method. Step 4: Repeat steps 2 and 3 running

over 1,2, , resq n= , and then take numerical integration in

Eq. (5) to calculate the numerical solution ( , )Xp x t .

3 RELIABILITY EVALUATION USING EXTREME VALUE DISTRIBUTION AND EXTREME VALUE EVOLUTION

The dynamic reliability of the structure defined in Eq. (1), using the first passage failure criterion, could be denoted by

Pr{ ( ) , [0, ]}sR X t t T= ÎW Î (6)

where Pr{}⋅ indicates the probability of the random event;

sW denotes the safe domain; ( )X t denotes physical quantities

of the structure against which the reliability is defined. The expression of Eq. (6) in fact equals to

Pr{ ( , ) }sR W T= ÎWQ (7)

where ( , )W TQ denotes extreme value of ( )X t over time

interval [0, ]T corresponding to the failure criterion, i.e.

[0, ]( , ) ext { ( , )}

t TW T X t

Î=Q Q (8)

If the symmetric double boundary criterion is used in Eq. (6), Eq. (7) could be re-written as

Pr{ ( ) , [0, ]}BR X t x t T= £ Î (9)

in which Bx denotes the symmetric boundary, then Eq. (8)

becomes

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[0, ]( , ) max{ ( , )}

t TW T X t

Î=Q Q (10)

It is seen that the extreme value over time period is a static argument, which denotes the maximum of absolute value of response sample, as shown in Figure 1.

Figure 1. Schematic diagram of extreme values of response samples.

To obtain the probability density function (PDF) of extreme values, the theory of extreme value distribution is usually used. The critical principle of EVD is the hypothesis of virtual random process on extreme values of response samples. This treatment is to push the static extreme value into dynamic process so as to be integrated with the GDEE and be readily solved by the PDEM. The theory of virtual stochastic process is employed as the following formulation [6]:

( ) ( ( , ), ) ( , )Z W T Tt j t f= =Q Q (11)

which satisfies the conditions

0( ) 0Z

tt

== ,

( ) ( ( , ), ) ( , ) ( , )c

c cZ W T W Tt t

t j t f t=

= = =Q Q Q (12a,b)

where ct denotes a prescribed value. One might realize that

Eq. (11) is in a form similar to Eq. (3) and therefore the joint PDF of the arguments ( , )Z Q , denoted by ( , , )Zp z tQ q , is

governed as the counterpart of Eq. (3) by

( , , ) ( , , )( , ) 0Z Zp z p z

z

t tf t

t¶ ¶

+ =¶ ¶

Q Qq qq (13)

where ( , ) ( , )f t f t t= ¶ ¶ q q .

The initial condition in accordance with Eq. (12a) reads

0( , , ) | ( ) ( )Zp z z ptt d= =Q Qq q (14)

One can then obtain the PDF of ( )Z t :

( , ) ( , , )dZ Zp z p zt tW

= òQ

Q q q (15)

Eqs. (13) ~ (15), as counterparts of Eqs. (3) ~ (5), could be numerically implemented by the procedures outlined in the previous section. Noting Eq. (12b) one could get the PDF ofW :

( ) ( , )c

W Zp w p z wt t

t=

= = (16)

The reliability denoted by Eq. (9) could then be evaluated through transformation of Eq. (10)

Pr{ ( , ) } ( )ds

s WR W p w wtW

= Î W = òQ (17)

where W denotes a functional variable of basic random vectorQ in the distribution domain sW .

One can see that the dynamic reliability evaluation becomes a problem of one-dimensional integration on the extreme value distribution. This avoids the main difficulties encountered in traditional dynamic reliability theories.

In principle the form of ( )j ⋅ and ( )f ⋅ could be arbitrary

once Eq. (12) is satisfied. This provides a flexibility of selecting appropriate formulation to achieve good numerical properties. We investigate a couple of function forms of the virtual stochastic process in accordance with the boundary conditions denoted by Eq. (12), e.g. ( ) ( , ) cZ W Tt t t= Q ,

( ) ( , )sin( )cZ W Tt wt t= Q with 0.5 ,2.5 , ,(2 0.5)nw p p p= + . It

is noted that the kurtosis of extreme value distribution gives rise to burrs in case that different forms of ( ( , ), )W Tj tQ are

employed, while the numerical error has little influence upon the reliability estimates. In this paper, the virtual stochastic process in Eq. (11) takes the form:

( ) ( ( , ), ) ( , )sin( )c

Z W T W Twt

t j tt

= =Q Q (18)

in which 2.5w p= . The prescribed value 1ct = .

Figure 2. Schematic diagram of response samples of re-constructed random process.

Since the formualtion of virtual random process is not unique, the numerical error of this method relies upon the trival result. We note that the reason introducing virtual random process is the extreme values always occurring at

t

t

t

0

0

0

,X XPDF

,X X

,X X ( )X T

T

T

T

X

X

Bx

Bx-

t

t

t

0

0

0

EVD

X

X

X

PDF

( )W T

Bx

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different instants for different response samples except their static behavior of extreme values. Referring to Eq. (3), its critical item is the velocity component of physical quantity of interest ( , )X tq , which indicates that if the velocity

component is zero the sample process has no contribution to the fluctuation of random process We could image there exists an accurate PDF of exreme values at the terminal instant T using Eq. (3) by means of the re-construction of the random process sticking with the extreme value once it occurs; see Figure 2. The re-constructed random process is denoted by

( , )X t q .

The accurate PDF of exreme values at the terminal instant T is thus governed by the generalized density evolution equation:

( , , ) ( , , )( , ) 0X X

p x t p x tX t

t x

¶ ¶+ =

¶ ¶Q Qq q

q

(19)

4 NUMERICAL EXAMPLE

A randomly base-excited eight-storey shear frame with Bouc-Wen hysteretic behavior is taken as the numerical example. The structural parameters are follows: m1= m2= 1.0×105 kg, m3= m4= 0.9×105 kg, m5= m6= 0.9×105 kg, m7= m8= 0.8×105 kg; k1= k2= 3.6×101 kN/mm, k3= k4= 3.2×101 kN/mm, k5= k6= 3.2×101 kN/mm, k7= k8= 2.8×101 kN/mm. Rayleigh’s damping a b= +C M K is employed, where a = 0.01, b = 0.005, resulting in a damping ratio of 1.05% for the first vibrational mode. The natural frequencies of the unyielded building are respectively 3.64, 10.40, 16.46, 22.45, 27.91, 31.89, 34.68, and 36.81 rad/sec. The parameters included in the Bouc-Wen model [9] take value 0.01a = , 1.0A = ,

140.0b = , 20.0g = , 1.0n = , 0.002vd = , 0.001hd = ,

0.2y = , 0.005yd = , 0.1l = , 0.95sz = , 0.25q = ,

and 2000.0p = , respectively. The typical hysteretic curve

of structural component subjected to non-stationary harmonic waves is shown in Figure 3, which indicates a strong nonlinear state inherent in the structural system.

-300 -200 -100 0 100 200 300-600

-400

-200

0

200

400

600

800

Displacement [mm]

Res

torin

g F

orce

[kN

]

Figure 3. Typical restoring force v.s. inter-story drift.

A Clough-Penzien filtered process with non-stationary modulation modelled by stochastic harmonic function [10] is used as the seismic excitation:

4 2 2 2 4CP

02 2 2 2 2 2 2 2 2 2 2 2

4( )

( ) 4 ( ) 4g g g

g

g g g f f f

S Sw z w w w

ww w z w w w w z w w

+=

- + - + (20)

where ,z wg g denote the equivalent damping ratio and

fundamental frequency of the first-layer soil of local site,

which take 0.85 and 20 rad/sec, respectively; ,f fz w denote

the equivalent damping ratio and fundamental frequency of the second-layer soil of local site, which take 0.85 and 2 rad/sec, respectively.

In this case, just 20 independent random variables are included in the stochastic harmonic function of seismic excitations where 10 variables are used to represent the phases, and 10 variables are used to represent the frequencies. 1000 representative processes are generated. For the purpose of validating PDEM, 9999 times of Monte Carlo simulation is carried out. A simple time integration scheme with just the current state is employed in the deterministic non-linear dynamic analysis of the structure. It is proved to be efficient in this numerical example for obtaining a logical physical mechanism of Bouc-Wen components. Besides, the hysteretic displacement z is solved by the fourth-order Runge-Kutta scheme.

Using the procedure detailed in the above sections, we gain the re-constructed response process of the 1000 samples. Figure 4 shows the re-constructed and original time history of a sample of displacement response .

0 2 4 6 8 10 12 14 16 18 20-100

-80

-60

-40

-20

0

20

40

60

Time [sec]

Dis

plac

emen

t [m

m]

OriginalRe-constructed

Figure 4. Re-constructed (red dot) and original (blue solid) time history of a sample of displacement random process.

It is indicated that the re-constructed random process eccentially excludes the contribution to the fluctuation of probability density function of the process thereafter the extreme value occurs. This treatment results the benefit that the reliability of structural systems can be directly derived from the probability density function of re-contructed response process at the terminal instant; see Figure 2.

The PDF contour of re-constructed and original displacement random process at interval [5 20] sec are shown in Figures 5 and 6, respectively. There exposes a significant difference from the two PDF contours: the re-constructed one gives rise to be symmetric concentration of proability especially in the later interval of period where most sample processes locate at their extreme values; while the original one gives rise to be almost normal distribution with zero-mean. It is also indicated that the original one includes more abundant probaility information than the re-constructed one since the sample process thereafter its extreme value occurs has no contribution to the fluctuation of probability density function of random process, which is exposed as the parallel lines of Figure 5. A clearer performance are shown in Figures 7 and 8, respectively.

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Figure 5. PDF contour of re-constructed displacement random

process at interval [5 20] sec.

Figure 6. PDF contour of original displacement random process at interval [5 20] sec.

-400 -300 -200 -100 0 100 200 300 4000

0.002

0.004

0.006

0.008

0.01

0.012

Displacement [mm]

PD

F

PDF at 20 sec

Figure 7. PDF of re-constructed displacement random process at terminal instant.

-400 -300 -200 -100 0 100 200 300 4000

0.005

0.01

0.015

0.02

0.025

0.03

Displacement [mm]

PD

F

PDF at 20 sec

Figure 8. PDF of original displacement random process at terminal instant.

The reliability results of the nonlienar structural system using the extreme-reponse evolution method (ERE) proposed in ths paper, the theory of extreme-value distribution (EVD), and the Monte Carlo simulations (MCS) in case of three different threshold values xB= 200 mm, xB= 300 mm and xB= 350 mm are shown in Table 2, respectively. It is seen that the proposed method has an evident error aganst the standard

Monte Carlo simulations in case of low thresholds, and it gains an accurate result in case of high thresholds. While the results of EVD meet well with those of the MCS. It is explained that the error in case of low thresholds exclusively derives from the numerical inaccuracy of PDF of re-constructed random process.

Table 2. Reliability of nonlinear system using three schemes.

Threshold [mm]

ERE EVD MCS

200 0.8545 0.9168 0.9108

300 0.9654 0.9768 0.9770 350 0.9953 0.9957 0.9960

The PDF of virtual random process of extreme

displacement involved in the EVD is shown in Figure 9. One might see that the PDF of the EVD exhibits a smoothing curve while that of the ERE exposes to be of fluctuation in the neighborhood of thresholds. The reliability assessment, nevertheless, using the extreme-reponse evolution method proposed in this paper could gain a sound result, and the higher of the threshold, the lower of the computational effort for deriving a smoothing curve of PDF.

0 50 100 150 200 250 300 350 4000

0.004

0.008

0.012

0.016

0.02

Displacement [mm]

PD

F

Figure 9. PDF of virtual random process of extreme displacement.

5 CONCLUDING REMARKS

Structural reliability is a critical argument quantifying the safety of engineering structures. A new structural reliability method is proposed in this paper through the re-construction of response random process sticking with the extreme value once it occurs. The structural reliability can be directly derived from the probability density function of the re-constructed process at the terminal instant using the PDEM. Numerical investigations against the method of extreme-value distribution and and the Monte Carlo simulation reveal the validality and applicability of the proposed extreme-response evolution method. Its benefit from the method of extreme-value distribution is that re-constructed process is unique where no hypothesis is introduced.

ACKNOWLEDGMENTS

The study is financially supported partly by the National Natural Science Foundation of China (NSFC Grant Nos. 11172210, 51108344), the Shuguang Program of Shanghai (Grant No.11SG21), the Exploratory Program of State Key Laboratory of Disaster Reduction in Civil Engineering at Tongji University (Grant No. SLDRCE11-B-04) and the Fundamental Research Funds for the Central Universities.

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[7] J. Li, J.B. Chen and W.L. Fan, The equivalent extreme-value event and evaluation of the structural system reliability, Structural Safety, 2007, 29(2): 112-131.

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